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7 Commits
2025-02-23
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main
| Author | SHA1 | Date | |
|---|---|---|---|
| 847ad7e93b | |||
| ac4ff1d16b | |||
| eade0f6f2e | |||
| b5450708c1 | |||
| a2e6abc4b1 | |||
| e60fa83593 | |||
| d9da44ac74 |
39
scripts/ase_spacegroup.py
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39
scripts/ase_spacegroup.py
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@ -0,0 +1,39 @@
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import ase.io as io
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from ase.build import cut
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from ase.spacegroup import crystal
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a = 9.04
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skutterudite = crystal(('Co', 'Sb'),
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basis=[(0.25, 0.25, 0.25), (0.0, 0.335, 0.158)],
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spacegroup=204,
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cellpar=[a, a, a, 90, 90, 90])
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# Create a new atoms instance with Co at origo including all atoms on the
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# surface of the unit cell
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cosb3 = cut(skutterudite, origo=(0.25, 0.25, 0.25), extend=1.01)
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# Define the atomic bonds to show
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bondatoms = []
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symbols = cosb3.get_chemical_symbols()
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for i in range(len(cosb3)):
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for j in range(i):
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if (symbols[i] == symbols[j] == 'Co' and
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cosb3.get_distance(i, j) < 4.53):
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bondatoms.append((i, j))
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elif (symbols[i] == symbols[j] == 'Sb' and
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cosb3.get_distance(i, j) < 2.99):
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bondatoms.append((i, j))
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# Create nice-looking image using povray
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renderer = io.write('spacegroup-cosb3.pov', cosb3,
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rotation='90y',
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radii=0.4,
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povray_settings=dict(transparent=False,
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camera_type='perspective',
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canvas_width=320,
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bondlinewidth=0.07,
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bondatoms=bondatoms))
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renderer.render()
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50
scripts/bz.py
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50
scripts/bz.py
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@ -0,0 +1,50 @@
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# creates: bztable.rst
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# creates: 00.CUB.svg 01.FCC.svg 02.BCC.svg 03.TET.svg 04.BCT1.svg
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# creates: 05.BCT2.svg 06.ORC.svg 07.ORCF1.svg 08.ORCF2.svg 09.ORCF3.svg
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# creates: 10.ORCI.svg 11.ORCC.svg 12.HEX.svg 13.RHL1.svg 14.RHL2.svg
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# creates: 15.MCL.svg 16.MCLC1.svg 17.MCLC3.svg 18.MCLC5.svg 19.TRI1a.svg
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# creates: 20.TRI1b.svg 21.TRI2a.svg 22.TRI2b.svg
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# creates: 23.OBL.svg 24.RECT.svg 25.CRECT.svg 26.HEX2D.svg 27.SQR.svg
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# taken from https://wiki.fysik.dtu.dk/ase/gallery/gallery.html
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from formulary import *
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from ase.lattice import all_variants
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from ase.data import colors
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header = """\
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Brillouin zone data
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-------------------
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.. list-table::
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:widths: 10 15 45
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"""
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entry = """\
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* - {name} ({longname})
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- {bandpath}
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- .. image:: {fname}
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:width: 40 %
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"""
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with open('bztable.rst', 'w') as fd:
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print(header, file=fd)
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for i, lat in enumerate(all_variants()):
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id = f'{i:02d}.{lat.variant}'
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imagefname = f'out/{id}.svg'
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txt = entry.format(name=lat.variant,
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longname=lat.longname,
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bandpath=lat.bandpath().path,
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fname=imagefname)
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print(txt, file=fd)
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ax = lat.plot_bz()
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fig = ax.get_figure()
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fig.savefig(imagefname, bbox_inches='tight')
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fig.clear()
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@ -18,7 +18,7 @@ def fbutler_volmer(ac, z, eta, T):
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return fbutler_volmer_anode(ac, z, eta, T) + fbutler_volmer_cathode(ac, z, eta, T)
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def butler_volmer():
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fig, ax = plt.subplots(figsize=size_half_third)
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fig, ax = plt.subplots(figsize=size_formula_fill_default)
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ax.set_xlabel("$\\eta$ [V]")
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ax.set_ylabel("$j/j_0$")
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etas = np.linspace(-0.1, 0.1, 400)
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@ -62,7 +62,7 @@ def tafel():
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iright = i0 * np.abs(ftafel_cathode(ac, z, etas, T))
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ileft = i0 * ftafel_anode(ac, z, etas, T)
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fig, ax = plt.subplots(figsize=size_half_third)
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fig, ax = plt.subplots(figsize=size_formula_normal_default)
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ax.set_xlabel("$\\eta$ [V]")
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ax.set_ylabel("$\\log_{10}\\left(\\frac{|j|}{j_0}\\right)$")
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# ax.set_ylabel("$\\log_{10}\\left(|j|/j_0\\right)$")
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@ -91,7 +91,7 @@ def fZ_ind(L, omega):
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def nyquist():
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fig, ax = plt.subplots(figsize=(full/2, full/3))
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fig, ax = plt.subplots(figsize=size_formula_fill_default)
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split_z = lambda Z: (Z.real, -Z.imag)
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ax.grid()
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ax.set_xlabel("$\\text{Re}(Z)$ [\\si{\\ohm}]")
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@ -146,7 +146,7 @@ def fZ_tlm(Rel, Rion, Rct, Cct, ws, N):
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return Z
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def nyquist_tlm():
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fig, ax = plt.subplots(figsize=(full/2, full/4))
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fig, ax = plt.subplots(figsize=(width_formula, width_formula*0.5))
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split_z = lambda Z: (Z.real, -Z.imag)
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ax.grid()
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ax.set_xlabel("$\\text{Re}(Z)$ [\\si{\\ohm}]")
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@ -168,7 +168,7 @@ def fkohlrausch(L0, K, c):
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return L0 - K*np.sqrt(c)
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def kohlrausch():
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fig, ax = plt.subplots(figsize=(full/4, full/4))
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fig, ax = plt.subplots(figsize=size_formula_small_quadratic)
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ax.grid()
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ax.set_xlabel("$c_\\text{salt}$")
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ax.set_ylabel("$\\Lambda_\\text{M}$")
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51
scripts/cm_crystal_structures.py
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51
scripts/cm_crystal_structures.py
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@ -0,0 +1,51 @@
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from formulary import *
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from util.aseutil import set_atom_color, get_pov_settings
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"""
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Create crystal structures using ase and render them with povray
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Rotation angle:
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To get the rotation angle, open the structure in the ase.visualize.view
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and use "View->Rotation" to get the desired angles
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"""
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set_atom_color("Na", COLORSCHEME["fg-red"])
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set_atom_color("Cl", COLORSCHEME["fg-blue"])
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set_atom_color("Zn", COLORSCHEME["fg-blue"])
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set_atom_color("S", COLORSCHEME["fg-yellow"])
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from ase.lattice import compounds
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from ase.build import cut, bulk
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from ase import Atom, Atoms
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def zincblende():
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zns = compounds.Zincblende(("Zn", "S"), latticeconstant=5.0, size=(1,1,1))
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zns_cell = cut(zns, b=(0,0,1), origo=(0,0,0), extend=1.1)
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return zns_cell
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# NaCl cut
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def nacl():
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nacl = compounds.NaCl(("Na", "Cl"), latticeconstant=5.0, size=(1,1,1))
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nacl_cell = cut(nacl, b=(0,0,1), origo=(0,0,0), extend=1.1)
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return nacl_cell
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def wurtzite():
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compounds.L1_2
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wurtzite = bulk('SZn', 'wurtzite', a=3.129, c=5.017)
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wurtzite_cell = cut(wurtzite,
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a=[1, 0, 0],
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b=[-1, -1, 0],
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c=[0, 0, 1], extend=1.1)
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return wurtzite_cell
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if __name__ == "__main__":
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export_atoms(nacl(), "cm_crystal_NaCl", size_formula_half_quadratic)
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export_atoms(wurtzite(), "cm_crystal_wurtzite", size_formula_half_quadratic, rotation="70x,20y,174z")
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export_atoms(zincblende(), "cm_crystal_zincblende", size_formula_half_quadratic, rotation="-155x,70y,24z")
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w = wurtzite()
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from ase.visualize import view
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view(w)
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@ -1,5 +1,9 @@
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#!/usr/bin env python3
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from formulary import *
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from scipy.constants import Boltzmann as kB, hbar
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hbar = 1
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kB = 1
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def fone_atom_basis(q, a, M, C1, C2):
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return np.sqrt(4*C1/M * (np.sin(q*a/2)**2 + C2/C1 * np.sin(q*a)**2))
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@ -11,7 +15,7 @@ def one_atom_basis():
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M = 1.
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qs = np.linspace(-2*np.pi/a, 2*np.pi/a, 300)
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omega = fone_atom_basis(qs, a, M, C1, C2)
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fig, ax = plt.subplots(figsize=size_half_third)
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fig, ax = plt.subplots(figsize=size_formula_normal_default)
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ax.set_xlabel(r"$q$")
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ax.set_xticks([i * np.pi/a for i in range(-2, 3)])
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ax.set_xticklabels([f"${i}\\pi/a$" if i != 0 else "0" for i in range(-2, 3)])
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@ -41,7 +45,7 @@ def two_atom_basis():
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qs = np.linspace(-2*np.pi/a, 2*np.pi/a, 300)
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omega_a = ftwo_atom_basis_acoustic(qs, a, M1, M2, C)
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omega_o = ftwo_atom_basis_optical(qs, a, M1, M2, C)
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fig, ax = plt.subplots(figsize=size_half_third)
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fig, ax = plt.subplots(figsize=size_formula_normal_default)
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ax.plot(qs, omega_a, label="acoustic")
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ax.plot(qs, omega_o, label="optical")
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ax.text(0, 0.8, "1. BZ", ha='center')
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@ -57,8 +61,56 @@ def two_atom_basis():
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ax.set_xticklabels([f"${i}\\pi/a$" if i != 0 else "0" for i in range(-2, 3)])
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ax.legend()
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ax.grid()
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return fig
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def fcv_einstein(T, N, omegaE):
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ThetaT = hbar * omegaE / (kB * T)
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return 3 * N * kB * ThetaT**2 * np.exp(ThetaT) / (np.exp(ThetaT) - 1)**2
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def fcv_debye_integral(x):
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print(np.exp(x), (np.exp(x) - 1)**2)
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return x**4 * np.exp(x) / ((np.exp(x) - 1)**2)
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def heat_capacity_einstein_debye():
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Ts = np.linspace(0, 10, 500)
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omegaD = 1e1
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omegaE = 1
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# N = 10**23
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N = 1
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cvs_einstein = fcv_einstein(Ts, N, omegaE)
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cvs_debye = np.zeros(Ts.shape, dtype=float)
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integral = np.zeros(Ts.shape, dtype=float)
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# cvs_debye = [0.0 for _ in range(Ts.shape[0])] # np.zeros(Ts.shape, dtype=float)
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# integral = [0.0 for _ in range(Ts.shape[0])] # np.zeros(Ts.shape, dtype=float)
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dT = Ts[1] - Ts[0]
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dThetaT = kB*dT/(hbar*omegaD)
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for i, T in enumerate(Ts):
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if i == 0: continue
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ThetaT = kB*T/(hbar*omegaD)
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dIntegral = fcv_debye_integral(ThetaT) * dThetaT
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integral[i] = dIntegral
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# print(integral)
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integral[i] += integral[i-1]
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C_debye = 9 * N * kB * ThetaT**3 * integral[i]
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cvs_debye[i] = C_debye
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print(i, T, ThetaT, dIntegral, C_debye, integral[i])
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fig, ax = plt.subplots(1, 1, figsize=size_formula_normal_default)
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ax.set_xlabel("$T$")
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ax.set_ylabel("$c_V$")
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ax.plot(Ts, cvs_einstein, label="Einstein")
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ax.plot(Ts, cvs_debye, label="Debye")
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ax.plot(Ts, integral, label="integral")
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ax.hlines([3*N*kB], xmin=0, xmax=Ts[-1], colors=COLORSCHEME["fg1"], linestyles="dashed")
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# print(cvs_debye)
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ax.legend()
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return fig
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if __name__ == '__main__':
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export(one_atom_basis(), "cm_phonon_dispersion_one_atom_basis")
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export(two_atom_basis(), "cm_phonon_dispersion_two_atom_basis")
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export(one_atom_basis(), "cm_vib_dispersion_one_atom_basis")
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export(two_atom_basis(), "cm_vib_dispersion_two_atom_basis")
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export(heat_capacity_einstein_debye(), "cm_vib_heat_capacity_einstein_debye")
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print(kB)
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118
scripts/cm_superconductivity.py
Normal file
118
scripts/cm_superconductivity.py
Normal file
@ -0,0 +1,118 @@
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#!/usr/bin env python3
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from formulary import *
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# Define the functions
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def psi_squared(x, xi):
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return np.tanh(x/(np.sqrt(2)*xi))**2
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def B_z(x, B0, lam):
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return B0 * np.exp(-x/lam)
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def n_s_boundary():
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xs = np.linspace(0, 6, 400)
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xn = np.linspace(-1, 0, 10)
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B0 = 1.0
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fig, ax = plt.subplots(figsize=size_formula_fill_default)
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ax.axvline(x=0, color='gray', linestyle='--', linewidth=0.8)
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ax.axhline(y=1, color='gray', linestyle='--', linewidth=0.8)
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ax.axhline(y=0, color='gray', linestyle='--', linewidth=0.8)
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ax.fill_between(xn, -2, 2 , color=COLORSCHEME["bg-yellow"], alpha=0.5)
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ax.fill_between(xs, -2, 2 , color=COLORSCHEME["bg-blue"], alpha=0.5)
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ax.text(-0.5, 0.9, 'N', color=COLORSCHEME["fg-yellow"], fontsize=14, ha="center", va="center")
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ax.text(3, 0.9, 'S', color=COLORSCHEME["fg-blue"], fontsize=14, ha="center", va="center")
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ax.set_xlabel("$x$")
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ax.set_ylabel(r"$|\Psi|^2$, $B_z(x)/B_\text{ext}$")
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ax.set_ylim(-0.1, 1.1)
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ax.set_xlim(-1, 6)
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ax.grid()
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lines = []
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for i, (xi, lam, color) in enumerate([(0.5, 2, "blue"), (2, 0.5, "red")]):
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psi = psi_squared(xs, xi)
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B = B_z(xs, B0, lam)
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line, = ax.plot(xs, psi, color=color, linestyle="solid", label=f"$\\xi_\\text{{GL}}={xi}$, $\\lambda_\\text{{GL}}={lam}$")
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lines.append(line)
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ax.plot(xs, B, color=color, linestyle="dashed")
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if i == 1:
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ylam = 1/np.exp(1)
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ax.plot([0, lam], [ylam, ylam], linestyle="dashed", color=COLORSCHEME["fg2"])
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ax.text(lam/2, ylam, r'$\lambda_\text{GL}$', color=color, ha="center", va="bottom")
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yxi = psi_squared(xi, xi)
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ax.plot([0, xi], [yxi, yxi], linestyle="dotted", color=COLORSCHEME["fg2"])
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ax.text(xi/2, yxi, r'$\xi_\text{GL}$', color=color, ha="center", va="bottom")
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lines.append(mpl.lines.Line2D([], [], color="black", label=r"$\lvert\Psi\rvert^2$"))
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lines.append(mpl.lines.Line2D([], [], color="black", linestyle="dashed", label=r"$B_z(x)/B_\text{ext}$"))
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ax.legend(loc='center right', handles=lines)
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return fig
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from mpl_toolkits.mplot3d import Axes3D
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from scipy.interpolate import griddata
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def critical_type2():
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Jc0 = 100
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Bc2_0 = 30
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Tc = 90
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T = np.linspace(0, Tc, 100)
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Jc_T = Jc0 * (1 - (T / Tc)**2)
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Bc2_T = Bc2_0 * (1 - (T / Tc)**2)
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B = np.linspace(0, Bc2_0, 100)
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Jc_B = Jc0 * (1 - B / Bc2_0)
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fig = plt.figure(figsize=size_formula_normal_default)
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ax = fig.add_subplot(111, projection='3d')
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ax.plot(T, np.zeros_like(Jc_T), Jc_T, label='$J_c(T)$', color='r')
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ax.plot(T, Bc2_T, np.zeros_like(Bc2_T), label='$B_{c2}(T)$', color='g')
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ax.plot(np.zeros_like(Jc_B), B, Jc_B, label='$J_c(B)$', color='b')
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ax.set_xlim(0, Tc)
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ax.set_ylim(0, Bc2_0)
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ax.set_zlim(0, Jc0)
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# surface
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# T_grid, B_grid = np.meshgrid(T, B)
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# Jc_grid = Jc0 * (1 - (T_grid / Tc)**2) * (1 - B_grid / Bc2_0)
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# surf = ax.plot_surface(T_grid, B_grid, Jc_grid, color='cyan', alpha=0.5)
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ax.set_xlabel('$T$')
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ax.set_ylabel('$B_{c2}$')
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ax.set_zlabel('$J_c$')
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# ax.legend()
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ax.grid(True)
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ax.view_init(elev=30., azim=45)
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ax.set_box_aspect(None, zoom=0.85)
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return fig
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def heat_capacity():
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fig, ax = plt.subplots(1, 1, figsize=size_formula_small_quadratic)
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T_max = 1.7
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||||
Cn_max = 3
|
||||
f_Cn = lambda T: T * Cn_max/T_max
|
||||
Delta_C = f_Cn(1.0) * 1.43 # BCS prediction
|
||||
CsTc = f_Cn(1.0) * (1+1.43) # BCS prediction
|
||||
# exp decay from there
|
||||
f_Cs = lambda T: np.exp(-1 / T + 1) * CsTc
|
||||
|
||||
Tns = np.linspace(0.0, T_max, 100)
|
||||
Tss = np.linspace(0.0, 1.0, 100)
|
||||
Cns = f_Cn(Tns)
|
||||
Css = f_Cs(Tss)
|
||||
ax.plot(Tns, Cns, label=r"$c_\text{n}$")
|
||||
ax.plot(Tss, Css, label=r"$c_\text{s}$")
|
||||
ax.vlines([1.0], ymin=f_Cn(1.0), ymax=(CsTc), color=COLORSCHEME["fg1"], linestyles="dashed")
|
||||
ax.text(1.05, CsTc - Delta_C/2, "$\\Delta c$", color=COLORSCHEME["fg1"])
|
||||
ax.set_xlabel(r"$T/T_\text{c}$")
|
||||
ax.set_ylabel(r"$c$ [a.u.]")
|
||||
ax.legend()
|
||||
return fig
|
||||
|
||||
|
||||
|
||||
if __name__ == "__main__":
|
||||
export(n_s_boundary(), "cm_super_n_s_boundary")
|
||||
export(critical_type2(), "cm_super_critical_type2")
|
||||
export(heat_capacity(), "cm_super_heat_capacity")
|
||||
@ -4,7 +4,7 @@ import itertools
|
||||
|
||||
|
||||
def get_fig():
|
||||
fig, ax = plt.subplots(figsize=size_half_half)
|
||||
fig, ax = plt.subplots(figsize=size_bigformula_half_quadratic)
|
||||
ax.grid()
|
||||
ax.set_xlabel(f"$x$")
|
||||
ax.set_ylabel("PDF")
|
||||
|
||||
@ -31,16 +31,33 @@ COLORSCHEME = cs.gruvbox_light()
|
||||
# cs.p_tum["fg0"] = cs.p_tum["alt-blue"]
|
||||
# COLORSCHEME = cs.tum()
|
||||
# COLORSCHEME = cs.legacy()
|
||||
# COLORSCHEME = cs.stupid()
|
||||
|
||||
tex_aux_path = "../.aux/"
|
||||
tex_src_path = "../src/"
|
||||
img_out_dir = os.path.join(tex_src_path, "img")
|
||||
img_out_dir = os.path.abspath(os.path.join(tex_src_path, "img"))
|
||||
filetype = ".pdf"
|
||||
skipasserts = False
|
||||
|
||||
full = 8
|
||||
size_half_half = (full/2, full/2)
|
||||
size_third_half = (full/3, full/2)
|
||||
size_half_third = (full/2, full/3)
|
||||
def pt_2_inch(pt):
|
||||
return 0.0138888889 * pt
|
||||
def cm_2_inch(cm):
|
||||
return 0.3937007874 * cm
|
||||
|
||||
# A4 - margins
|
||||
width_line = cm_2_inch(21.0 - 2 * 2.0)
|
||||
# width of a formula box, the prefactor has to match \eqwidth
|
||||
width_formula = 0.69 * width_line
|
||||
# arbitrary choice
|
||||
height_default = width_line * 2 / 5
|
||||
|
||||
size_bigformula_fill_default = (width_line, height_default)
|
||||
size_bigformula_half_quadratic = (width_line*0.5, width_line*0.5)
|
||||
size_bigformula_small_quadratic = (width_line*0.33, width_line*0.33)
|
||||
size_formula_fill_default = (width_formula, height_default)
|
||||
size_formula_normal_default = (width_formula*0.8, height_default*0.8)
|
||||
size_formula_half_quadratic = (width_formula*0.5, width_formula*0.5)
|
||||
size_formula_small_quadratic = (width_formula*0.4, width_formula*0.4)
|
||||
|
||||
def assert_directory():
|
||||
if not skipasserts:
|
||||
@ -59,6 +76,34 @@ def export(fig, name, tight_layout=True):
|
||||
fig.tight_layout()
|
||||
fig.savefig(filename, bbox_inches="tight", pad_inches=0.0)
|
||||
|
||||
|
||||
def export_atoms(atoms, name, size, rotation="-30y,20x", get_bonds=True):
|
||||
"""Export a render of ase atoms object"""
|
||||
assert_directory()
|
||||
wd = os.getcwd()
|
||||
from util.aseutil import get_bondatoms, get_pov_settings
|
||||
from ase import io
|
||||
|
||||
tmp_dir = os.path.join(os.path.abspath(tex_aux_path), "scripts_aux")
|
||||
os.makedirs(tmp_dir, exist_ok=True)
|
||||
os.chdir(tmp_dir)
|
||||
|
||||
out_filename = f"{name}.png"
|
||||
|
||||
bondatoms = None
|
||||
if get_bonds:
|
||||
bondatoms = get_bondatoms(atoms)
|
||||
renderer = io.write(f'{name}.pov', atoms,
|
||||
rotation=rotation,# text string with rotation (default='' )
|
||||
radii=0.4, # float, or a list with one float per atom
|
||||
show_unit_cell=2, # 0, 1, or 2 to not show, show, and show all of cell
|
||||
colors=None, # List: one (r, g, b, t) tuple per atom
|
||||
povray_settings=get_pov_settings(size, COLORSCHEME, bondatoms),
|
||||
)
|
||||
renderer.render()
|
||||
os.chdir(wd)
|
||||
os.rename(os.path.join(tmp_dir, out_filename), os.path.join(img_out_dir, out_filename))
|
||||
|
||||
@np.vectorize
|
||||
def smooth_step(x: float, left_edge: float, right_edge: float):
|
||||
x = (x - left_edge) / (right_edge - left_edge)
|
||||
|
||||
@ -39,7 +39,7 @@ def transmon_cpb(wavefunction=True):
|
||||
ngs = np.linspace(-2, 2, 200)
|
||||
nrows = 4 if wavefunction else 1
|
||||
|
||||
fig, axs = plt.subplots(nrows, 3, squeeze=False, figsize=(full,full/3))
|
||||
fig, axs = plt.subplots(nrows, 3, squeeze=False, figsize=(width_line,height_default))
|
||||
axs = axs.T
|
||||
qubit.ng = 0
|
||||
qubit.EJ = 0.1 * EC
|
||||
@ -68,7 +68,7 @@ def transmon_cpb(wavefunction=True):
|
||||
|
||||
|
||||
def flux_onium():
|
||||
fig, axs = plt.subplots(1, 3, squeeze=True, figsize=(full,full/3))
|
||||
fig, axs = plt.subplots(1, 3, squeeze=True, figsize=(width_line,height_default))
|
||||
fluxs = np.linspace(0.4, 0.6, 50)
|
||||
EJ = 35.0
|
||||
alpha = 0.3
|
||||
@ -97,7 +97,7 @@ def flux_onium():
|
||||
# axs[0].set_xlim(0.4, 0.6)
|
||||
|
||||
fluxs = np.linspace(-1.1, 1.1, 101)
|
||||
fluxonium = scq.Fluxonium(EJ=9, EC=3, EL=0.5, flux=1, cutoff=100)
|
||||
fluxonium = scq.Fluxonium(EJ=9, EC=3, EL=0.5, flux=1, cutoff=30)
|
||||
fluxonium.plot_evals_vs_paramvals("flux", fluxs, evals_count=5, subtract_ground=True, fig_ax=(fig, axs[2]))
|
||||
axs[2].set_title("Fluxonium")
|
||||
return fig
|
||||
|
||||
@ -3,11 +3,19 @@ Put all scripts that generate plots or tex files here.
|
||||
You can run all files at once using `make scripts`
|
||||
|
||||
## Plots
|
||||
### `matplotlib`
|
||||
For plots with `matplotlib`:
|
||||
1. import `formulary.py`
|
||||
2. use one of the preset figsizes
|
||||
3. save the image using the `export` function in the `if __name__ == '__main__'` part
|
||||
|
||||
### `ase` - Atomic Simulation Environment
|
||||
For plots with `ase`:
|
||||
1. import `formulary.py` and `util.aseutil`
|
||||
2. Use `util.aseutil.set_atom_color` to change the color of all used atoms to one in the colorscheme
|
||||
3. export the render using the `export_atoms` function in the `if __name__ == '__main__'` part.
|
||||
Pass one of the preset figsizes as size.
|
||||
|
||||
## Colorscheme
|
||||
To ensure a uniform look of the tex source and the python plots,
|
||||
the tex and matplotlib colorschemes are both handled in `formulary.py`.
|
||||
|
||||
@ -3,4 +3,4 @@ scipy
|
||||
matplotlib
|
||||
scqubits
|
||||
qutip
|
||||
|
||||
ase
|
||||
|
||||
@ -5,7 +5,7 @@ def flennard_jones(r, epsilon, sigma):
|
||||
return 4 * epsilon * ((sigma/r)**12 - (sigma/r)**6)
|
||||
|
||||
def lennard_jones():
|
||||
fig, ax = plt.subplots(figsize=size_half_half)
|
||||
fig, ax = plt.subplots(figsize=size_formula_normal_default)
|
||||
ax.grid()
|
||||
ax.set_xlabel(r"$r$")
|
||||
ax.set_ylabel(r"$V(r)$")
|
||||
@ -29,7 +29,7 @@ def ffermi_dirac(x):
|
||||
|
||||
|
||||
def id_qgas():
|
||||
fig, ax = plt.subplots(figsize=size_half_half)
|
||||
fig, ax = plt.subplots(figsize=size_formula_normal_default)
|
||||
ax.grid()
|
||||
ax.set_xlabel(r"$\beta(\epsilon-\mu)$")
|
||||
ax.set_ylabel(r"$\langle n(\epsilon)\rangle$")
|
||||
@ -51,7 +51,7 @@ def fstep(x):
|
||||
return 1 if x >= 0 else 0
|
||||
|
||||
def fermi_occupation():
|
||||
fig, ax = plt.subplots(figsize=size_half_third)
|
||||
fig, ax = plt.subplots(figsize=size_formula_normal_default)
|
||||
# ax.grid()
|
||||
# ax.set_xlabel(r"$\beta(\epsilon-\mu)$")
|
||||
ax.set_xticks([0])
|
||||
@ -68,7 +68,7 @@ def fermi_occupation():
|
||||
return fig
|
||||
|
||||
def fermi_heat_capacity():
|
||||
fig, ax = plt.subplots(figsize=size_half_third)
|
||||
fig, ax = plt.subplots(figsize=size_formula_normal_default)
|
||||
# ax.grid()
|
||||
# ax.set_xlabel(r"$\beta(\epsilon-\mu)$")
|
||||
x = np.linspace(0, 4, 100)
|
||||
|
||||
47
scripts/util/aseutil.py
Normal file
47
scripts/util/aseutil.py
Normal file
@ -0,0 +1,47 @@
|
||||
from util.colorschemes import hex_to_rgb_float
|
||||
|
||||
def set_atom_color(symbol, hexcolor):
|
||||
from ase.data import atomic_numbers
|
||||
from ase.data.colors import jmol_colors, cpk_colors
|
||||
float_color = hex_to_rgb_float(hexcolor)
|
||||
n = atomic_numbers[symbol]
|
||||
jmol_colors[n] = float_color
|
||||
cpk_colors[n] = float_color
|
||||
|
||||
|
||||
from scipy.spatial.distance import pdist, squareform
|
||||
import numpy as np
|
||||
def get_bondatoms(atoms):
|
||||
site_positions = [site.position for site in atoms]
|
||||
pair_distances = squareform(pdist(np.stack(site_positions)))
|
||||
vs = pair_distances
|
||||
bondatoms = []
|
||||
for i in range(vs.shape[0]):
|
||||
for j in range(i):
|
||||
if vs[i, j] < 3: # up to 3 angstrom distance show a bond TODO
|
||||
bondatoms.append((i, j))
|
||||
return bondatoms
|
||||
# returns to many
|
||||
# from ase.io.pov import get_bondpairs
|
||||
# bondatoms=get_bondpairs(lat, 5)
|
||||
|
||||
|
||||
TARGET_DPI = 300
|
||||
# doc: https://github.com/WMD-group/ASE-Tutorials/blob/master/povray-tools/ase_povray.py
|
||||
def get_pov_settings(size, COLORSCHEME, bondatoms=None):
|
||||
white = hex_to_rgb_float(COLORSCHEME["bg0"])
|
||||
other = hex_to_rgb_float(COLORSCHEME["fg-yellow"])
|
||||
pixels = TARGET_DPI * size[0]
|
||||
pov_settings=dict(
|
||||
transparent=True,
|
||||
display=False,
|
||||
# camera_type='orthographic',
|
||||
camera_type='perspective',
|
||||
canvas_width=pixels,
|
||||
# point_lights : [], #[(18,20,40), 'White'],[(60,20,40),'White'], # [[loc1, color1], [loc2, color2],...]
|
||||
point_lights=[[(18,20,40), white],[(60,20,40),other]], # [[loc1, color1], [loc2, color2],...]
|
||||
background=(0, 0, 0, 1.,),
|
||||
bondlinewidth=0.07,
|
||||
bondatoms=bondatoms
|
||||
)
|
||||
return pov_settings
|
||||
@ -9,6 +9,26 @@ from math import floor
|
||||
|
||||
colors = ["red", "orange", "yellow", "green", "aqua", "blue", "purple", "gray"]
|
||||
|
||||
def duplicate_letters(color: str):
|
||||
return ''.join([c+c for c in color])
|
||||
|
||||
def hex_to_rgb_int(color: str) -> list[int]:
|
||||
color = color.strip("#")
|
||||
ctuple = []
|
||||
# turn RGBA to RRGGBBAA
|
||||
if len(color) == 3 or len(color) == 4:
|
||||
color = duplicate_letters(color)
|
||||
for i in range(len(color)//2):
|
||||
ctuple.append(int(color[i*2:i*2+2], 16))
|
||||
return ctuple
|
||||
|
||||
def hex_to_rgb_float(color: str) -> list[float]:
|
||||
clist = hex_to_rgb_int(color)
|
||||
fclist = [float(c) / 255 for c in clist]
|
||||
return fclist
|
||||
|
||||
|
||||
|
||||
def brightness(color:str, percent:float):
|
||||
if color.startswith("#"):
|
||||
color = color.strip("#")
|
||||
|
||||
@ -5,12 +5,13 @@ $out_dir = '../out';
|
||||
# Set lualatex as the default engine
|
||||
$pdf_mode = 1; # Enable PDF generation mode
|
||||
# $pdflatex = 'lualatex --interaction=nonstopmode --shell-escape'
|
||||
$lualatex = 'lualatex %O --interaction=nonstopmode --shell-escape %S'
|
||||
$lualatex = 'lualatex %O --interaction=nonstopmode --shell-escape %S';
|
||||
|
||||
# Additional options for compilation
|
||||
# '-verbose',
|
||||
# '-file-line-error',
|
||||
|
||||
ensure_path('TEXINPUTS', './pkg');
|
||||
# Quickfix-like filtering (warnings to ignore)
|
||||
# @warnings_to_filter = (
|
||||
# qr/Underfull \\hbox/,
|
||||
|
||||
@ -3,7 +3,7 @@
|
||||
\ger{Elektrochemie}
|
||||
]{el}
|
||||
\begin{formula}{chemical_potential}
|
||||
\desc{Chemical potential}{of species $i$\\Energy involved when the particle number changes}{\QtyRef{gibbs_free_energy}, \QtyRef{amount}}
|
||||
\desc{Chemical potential}{of species $i$\\Energy involved when the particle number changes}{\QtyRef{free_enthalpy}, \QtyRef{amount}}
|
||||
\desc[german]{Chemisches Potential}{der Spezies $i$\\Involvierte Energie, wenn sich die Teilchenzahl ändert}{}
|
||||
\quantity{\mu}{\joule\per\mol;\joule}{is}
|
||||
\eq{
|
||||
@ -24,7 +24,7 @@
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{activity}
|
||||
\desc{Activity}{relative activity}{\QtyRef{chemical_potential}, \QtyRef{standard_chemical_potential}, \ConstRef{universal_gas}, \QtyRef{temperature}}
|
||||
\desc{Activity}{relative activity}{\QtyRef{chemical_potential}, \fRef{::standard_chemical_potential}, \ConstRef{universal_gas}, \QtyRef{temperature}}
|
||||
\desc[german]{Aktivität}{Relative Aktivität}{}
|
||||
\quantity{a}{}{s}
|
||||
\eq{a_i = \Exp{\frac{\mu_i-\mu_i^\theta}{RT}}}
|
||||
@ -140,20 +140,20 @@
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{standard_cell_potential}
|
||||
\desc{Standard cell potential}{}{$\Delta_\txR G^\theta$ standard \qtyRef{gibbs_free_energy} of reaction, $n$ number of electrons, \ConstRef{faraday}}
|
||||
\desc[german]{Standard Zellpotential}{}{$\Delta_\txR G^\theta$ Standard \qtyRef{gibbs_free_energy} der Reaktion, $n$ Anzahl der Elektronen, \ConstRef{faraday}}
|
||||
\desc{Standard cell potential}{}{$\Delta_\txR G^\theta$ standard \qtyRef{free_enthalpy} of reaction, $n$ number of electrons, \ConstRef{faraday}}
|
||||
\desc[german]{Standard Zellpotential}{}{$\Delta_\txR G^\theta$ Standard \qtyRef{free_enthalpy} der Reaktion, $n$ Anzahl der Elektronen, \ConstRef{faraday}}
|
||||
\eq{E^\theta_\text{rev} = \frac{-\Delta_\txR G^\theta}{nF}}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\begin{formula}{nernst_equation}
|
||||
\desc{Nernst equation}{Electrode potential for a half-cell reaction}{\QtyRef{electrode_potential}, $E^\theta$ \secEqRef{standard_cell_potential}, \ConstRef{universal_gas}, \ConstRef{temperature}, $z$ \qtyRef{charge_number}, \ConstRef{faraday}, \QtyRef{activity}, \QtyRef{stoichiometric_coefficient}}
|
||||
\desc{Nernst equation}{Electrode potential for a half-cell reaction}{\QtyRef{electrode_potential}, $E^\theta$ \fRef{::standard_cell_potential}, \ConstRef{universal_gas}, \QtyRef{temperature}, $z$ \qtyRef{charge_number}, \ConstRef{faraday}, \QtyRef{activity}, \QtyRef{stoichiometric_coefficient}}
|
||||
\desc[german]{Nernst-Gleichung}{Elektrodenpotential für eine Halbzellenreaktion}{}
|
||||
\eq{E = E^\theta + \frac{RT}{zF} \Ln{\frac{ \left(\prod_{i}(a_i)^{\abs{\nu_i}}\right)_\text{oxidized}}{\left(\prod_{i}(a_i)^{\abs{\nu_i}}\right)_\text{reduced}}}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{cell_efficiency}
|
||||
\desc{Thermodynamic cell efficiency}{}{$P$ \fqEqRef{ed:el:power}}
|
||||
\desc{Thermodynamic cell efficiency}{}{$P$ \fRef{ed:el:power}}
|
||||
\desc[german]{Thermodynamische Zelleffizienz}{}{}
|
||||
\eq{
|
||||
\eta_\text{cell} &= \frac{P_\text{obtained}}{P_\text{maximum}} = \frac{E_\text{cell}}{E_\text{cell,rev}} & & \text{\gt{galvanic}} \\
|
||||
@ -172,7 +172,7 @@
|
||||
\ger[of_i]{des Ions $i$}
|
||||
|
||||
\begin{formula}{diffusion}
|
||||
\desc{Diffusion}{caused by concentration gradients}{$z_i$ \qtyRef{charge_number} \gt{of_i}, \ConstRef{faraday}, \QtyRef{diffusion_constant} \gt{of_i}, \QtyRef{concentration} \gt{of_i}}
|
||||
\desc{Diffusion}{caused by concentration gradients}{$z_i$ \qtyRef{charge_number} \gt{of_i}, \ConstRef{faraday}, \QtyRef{diffusion_coefficient} \gt{of_i}, \QtyRef{concentration} \gt{of_i}}
|
||||
\desc[german]{Diffusion}{durch Konzentrationsgradienten}{}
|
||||
\eq{ i_\text{diff} = \sum_i -z_i F D_i \left(\odv{c_i}{x}\right) }
|
||||
\end{formula}
|
||||
@ -312,8 +312,8 @@
|
||||
\ger{Massentransport}
|
||||
]{mass}
|
||||
\begin{formula}{concentration_overpotential}
|
||||
\desc{Concentration overpotential}{Due to concentration gradient near the electrode, the ions need to \hyperref[f:ch:el:ion_cond:diffusion]{diffuse} to the electrode before reacting}{\ConstRef{universal_gas}, \QtyRef{temperature}, $\c_{0/\txS}$ ion concentration in the electrolyte / at the double layer, $z$ \qtyRef{charge_number}, \ConstRef{faraday}}
|
||||
\desc[german]{Konzentrationsüberspannung}{Durch einen Konzentrationsgradienten an der Elektrode müssen Ionen erst zur Elektrode \hyperref[f:ch:el:ion_cond:diffusion]{diffundieren}, bevor sie reagieren können}{}
|
||||
\desc{Concentration overpotential}{Due to concentration gradient near the electrode, the ions need to \fRef[diffuse]{ch:el:ion_cond:diffusion} to the electrode before reacting}{\ConstRef{universal_gas}, \QtyRef{temperature}, $\c_{0/\txS}$ ion concentration in the electrolyte / at the double layer, $z$ \qtyRef{charge_number}, \ConstRef{faraday}}
|
||||
\desc[german]{Konzentrationsüberspannung}{Durch einen Konzentrationsgradienten an der Elektrode müssen Ionen erst zur Elektrode \fRef[diffundieren]{ch:el:ion_cond:diffusion}, bevor sie reagieren können}{}
|
||||
\eq{
|
||||
\eta_\text{conc,anodic} &= -\frac{RT}{\alpha \,zF} \ln \left(\frac{c_\text{red}^0}{c_\text{red}^\txS}\right) \\
|
||||
\eta_\text{conc,cathodic} &= -\frac{RT}{(1-\alpha) zF} \ln \left(\frac{c_\text{ox}^0}{c_\text{ox}^\txS}\right)
|
||||
@ -321,7 +321,7 @@
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{diffusion_overpotential}
|
||||
\desc{Diffusion overpotential}{Due to mass transport limitations}{$j_\infty$ \secEqRef{limiting_current}, $j_\text{meas}$ measured \qtyRef{current_density}, \ConstRef{universal_gas}, \QtyRef{temperature}, $n$ \qtyRef{charge_number}, \ConstRef{faraday}}
|
||||
\desc{Diffusion overpotential}{Due to mass transport limitations}{$j_\infty$ \fRef{::limiting_current}, $j_\text{meas}$ measured \qtyRef{current_density}, \ConstRef{universal_gas}, \QtyRef{temperature}, $n$ \qtyRef{charge_number}, \ConstRef{faraday}}
|
||||
\desc[german]{Diffusionsüberspannung}{Durch Limit des Massentransports}{}
|
||||
% \eq{\eta_\text{diff} = \frac{RT}{nF} \ln \left( \frac{\cfrac{c^\txs_\text{ox}}{c^0_\text{ox}}}{\cfrac{c^\txs_\text{red}}{c^0_\text{red}}} \right)}
|
||||
\eq{\eta_\text{diff} = \frac{RT}{nF} \Ln{\frac{j_\infty}{j_\infty - j_\text{meas}}}}
|
||||
@ -424,7 +424,7 @@
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{limiting_current}
|
||||
\desc{(Limiting) current density}{}{$n$ \QtyRef{charge_number}, \ConstRef{faraday}, $c^0$ \GT{c_bulk}, $D$ \qtyRef{diffusion_coefficient}, $\delta_\text{diff}$ \secEqRef{diffusion_layer_thickness}}
|
||||
\desc{(Limiting) current density}{}{$n$ \QtyRef{charge_number}, \ConstRef{faraday}, $c^0$ \GT{c_bulk}, $D$ \qtyRef{diffusion_coefficient}, $\delta_\text{diff}$ \fRef{::diffusion_layer_thickness}}
|
||||
% \desc[german]{Limitierender Strom}{}{}
|
||||
\eq{
|
||||
\abs{j} &= nFD \frac{c^0-c^\txS}{\delta_\text{diff}}
|
||||
@ -434,14 +434,14 @@
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{relation?}
|
||||
\desc{Current - concentration relation}{}{$c^0$ \GT{c_bulk}, $c^\txS$ \GT{c_surface}, $j$ \secEqRef{limiting_current}}
|
||||
\desc{Current - concentration relation}{}{$c^0$ \GT{c_bulk}, $c^\txS$ \GT{c_surface}, $j$ \fRef{::limiting_current}}
|
||||
\desc[german]{Strom - Konzentrationsbeziehung}{}{}
|
||||
\eq{\frac{j}{j_\infty} = 1 - \frac{c^\txS}{c^0}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{kinetic_current}
|
||||
\desc{Kinetic current density}{}{$j_\text{meas}$ measured \qtyRef{current_density}, $j_\infty$ \secEqRef{limiting_current}}
|
||||
\desc[german]{Kinetische Stromdichte}{}{$j_\text{meas}$ gemessene \qtyRef{current_density}, $j_\infty$ \secEqRef{limiting_current}}
|
||||
\desc{Kinetic current density}{}{$j_\text{meas}$ measured \qtyRef{current_density}, $j_\infty$ \fRef{::limiting_current}}
|
||||
\desc[german]{Kinetische Stromdichte}{}{$j_\text{meas}$ gemessene \qtyRef{current_density}, $j_\infty$ \fRef{::limiting_current}}
|
||||
\eq{j_\text{kin} = \frac{j_\text{meas} j_\infty}{j_\infty - j_\text{meas}}}
|
||||
\end{formula}
|
||||
|
||||
@ -452,10 +452,10 @@
|
||||
|
||||
\begin{formula}{butler_volmer}
|
||||
\desc{Butler-Volmer equation}{Reaction kinetics near the equilibrium potentential}
|
||||
{$j$ \qtyRef{current_density}, $j_0$ exchange current density, $\eta$ \fqEqRef{ch:el:kin:overpotential}, \QtyRef{temperature}, $z$ \qtyRef{charge_number}, \ConstRef{faraday}, \ConstRef{universal_gas}, $\alpha_{\txC/\txA}$ cathodic/anodic charge transfer coefficient, $\text{rf}$ \secEqRef{roughness_factor}}
|
||||
{$j$ \qtyRef{current_density}, $j_0$ exchange current density, $\eta$ \fRef{ch:el:kin:overpotential}, \QtyRef{temperature}, $z$ \qtyRef{charge_number}, \ConstRef{faraday}, \ConstRef{universal_gas}, $\alpha_{\txC/\txA}$ cathodic/anodic charge transfer coefficient, $\text{rf}$ \fRef{::roughness_factor}}
|
||||
%Current through an electrode iof a unimolecular redox reaction with both anodic and cathodic reaction occuring on the same electrode
|
||||
\desc[german]{Butler-Volmer-Gleichung}{Reaktionskinetik in der Nähe des Gleichgewichtspotentials}
|
||||
{$j$ \qtyRef{current_density}, $j_0$ Austauschstromdichte, $\eta$ \fqEqRef{ch:el:kin:overpotential}, \QtyRef{temperature}, $z$ \qtyRef{charge_number}, \ConstRef{faraday}, \ConstRef{universal_gas}, $\alpha_{\txC/\txA}$ Ladungstransferkoeffizient an der Kathode/Anode, $\text{rf}$ \secEqRef{roughness_factor}}
|
||||
{$j$ \qtyRef{current_density}, $j_0$ Austauschstromdichte, $\eta$ \fRef{ch:el:kin:overpotential}, \QtyRef{temperature}, $z$ \qtyRef{charge_number}, \ConstRef{faraday}, \ConstRef{universal_gas}, $\alpha_{\txC/\txA}$ Ladungstransferkoeffizient an der Kathode/Anode, $\text{rf}$ \fRef{::roughness_factor}}
|
||||
\begin{gather}
|
||||
j = j_0 \,\rfactor\, \left[ \Exp{\frac{(1-a_\txC) z F \eta}{RT}} - \Exp{-\frac{\alpha_\txC z F \eta}{RT}}\right]
|
||||
\intertext{\GT{with}}
|
||||
@ -512,7 +512,7 @@
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{rhe}
|
||||
\desc{Reversible hydrogen electrode (RHE)}{RHE Potential does not change with the pH value}{$E^0\equiv \SI{0}{\volt}$, \QtyRef{activity}, \QtyRef{pressure}, \GT{see} \fqEqRef{ch:el:cell:nernst_equation}}
|
||||
\desc{Reversible hydrogen electrode (RHE)}{RHE Potential does not change with the pH value}{$E^0\equiv \SI{0}{\volt}$, \QtyRef{activity}, \QtyRef{pressure}, \GT{see} \fRef{ch:el:cell:nernst_equation}}
|
||||
\desc[german]{Reversible Wasserstoffelektrode (RHE)}{Potential ändert sich nicht mit dem pH-Wert}{}
|
||||
\eq{
|
||||
E_\text{RHE} &= E^0 + \frac{RT}{F} \Ln{\frac{a_{\ce{H^+}}}{p_{\ce{H2}}}}
|
||||
@ -604,14 +604,18 @@
|
||||
\begin{hiddenformula}{scan_rate}
|
||||
\desc{Scan rate}{}{}
|
||||
\desc[german]{Scanrate}{}{}
|
||||
\quantity{v}{\volt\per\s}{s}
|
||||
\hiddenQuantity{v}{\volt\per\s}{s}
|
||||
\end{hiddenformula}
|
||||
|
||||
|
||||
\begin{formula}{upd}
|
||||
\desc{Underpotential deposition (UPD)}{}{}
|
||||
\desc[german]{}{}{}
|
||||
\ttxt{Reversible deposition of metal onto a foreign metal electrode at potentials positive of the Nernst potential \TODO{clarify}}
|
||||
% \desc[german]{}{}{}
|
||||
\ttxt{\eng{
|
||||
Reversible deposition of metal onto a foreign metal electrode at potentials positive of the Nernst potential.
|
||||
}\ger{
|
||||
Reversible Ablagerung von Metall auf eine Elektrode aus einem anderen Metall bei positiveren Potentialen als das Nernst-Potential.
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
\Subsubsection[
|
||||
@ -632,13 +636,13 @@
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{diffusion_layer_thickness}
|
||||
\desc{Diffusion layer thickness}{\TODO{Where does 1.61 come from}}{$D$ \qtyRef{diffusion_coefficient}, $\nu$ \qtyRef{kinematic_viscosity}, \QtyRef{angular_frequency}}
|
||||
\desc{Diffusion layer thickness}{}{$D$ \qtyRef{diffusion_coefficient}, $\nu$ \qtyRef{kinematic_viscosity}, \QtyRef{angular_frequency}}
|
||||
\desc[german]{Diffusionsschichtdicke}{}{}
|
||||
\eq{\delta_\text{diff}= 1.61 D{^\frac{1}{3}} \nu^{\frac{1}{6}} \omega^{-\frac{1}{2}}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{limiting_current}
|
||||
\desc{Limiting current density}{for a \abbrRef{rde}}{$n$ \QtyRef{charge_number}, \ConstRef{faraday}, $c^0$ \GT{c_bulk}, $D$ \qtyRef{diffusion_coefficient}, $\delta_\text{diff}$ \secEqRef{diffusion_layer_thickness}, $\nu$ \qtyRef{kinematic_viscosity}, \QtyRef{angular_frequency}}
|
||||
\desc{Limiting current density}{for a \abbrRef{rde}}{$n$ \QtyRef{charge_number}, \ConstRef{faraday}, $c^0$ \GT{c_bulk}, $D$ \qtyRef{diffusion_coefficient}, $\delta_\text{diff}$ \fRef{::diffusion_layer_thickness}, $\nu$ \qtyRef{kinematic_viscosity}, \QtyRef{angular_frequency}}
|
||||
% \desc[german]{Limitierender Strom}{}{}
|
||||
\eq{j_\infty = nFD \frac{c^0}{\delta_\text{diff}} = \frac{1}{1.61} nFD^{\frac{2}{3}} v^{\frac{-1}{6}} c^0 \sqrt{\omega}}
|
||||
\end{formula}
|
||||
|
||||
@ -6,21 +6,24 @@
|
||||
\eng{Drude model}
|
||||
\ger{Drude-Modell}
|
||||
]{drude}
|
||||
\begin{ttext}
|
||||
\eng{Classical model describing the transport properties of electrons in materials (metals):
|
||||
\begin{formula}{description}
|
||||
\desc{Description}{}{}
|
||||
\desc[german]{Beschreibung}{}{}
|
||||
\ttxt{\eng{
|
||||
Classical model describing the transport properties of electrons in materials (metals):
|
||||
The material is assumed to be an ion lattice and with freely moving electrons (electron gas). The electrons are
|
||||
accelerated by an electric field and decelerated through collisions with the lattice ions.
|
||||
The model disregards the Fermi-Dirac partition of the conducting electrons.
|
||||
}
|
||||
\ger{Ein klassisches Model zur Beschreibung der Transporteigenschaften von Elektronen in (v.a.) Metallen:
|
||||
}\ger{
|
||||
Ein klassisches Model zur Beschreibung der Transporteigenschaften von Elektronen in (v.a.) Metallen:
|
||||
Der Festkörper wird als Ionenkristall mit frei beweglichen Elektronen (Elektronengas).
|
||||
Die Elektronen werden durch ein Elektrisches Feld $E$ beschleunigt und durch Stöße mit den Gitterionen gebremst.
|
||||
Das Modell vernachlässigt die Fermi-Dirac Verteilung der Leitungselektronen.
|
||||
}
|
||||
\end{ttext}
|
||||
\begin{formula}{motion}
|
||||
\desc{Equation of motion}{}{$v$ electron speed, $\vec{v}_\text{D}$ drift velocity, $\tau$ mean free time between collisions}
|
||||
\desc[german]{Bewegungsgleichung}{}{$v$ Elektronengeschwindigkeit, $\vec{v}_\text{D}$ Driftgeschwindigkeit, $\tau$ Stoßzeit}
|
||||
}}
|
||||
\end{formula}
|
||||
\begin{formula}{eom}
|
||||
\desc{Equation of motion}{}{$v$ electron speed, $\vec{v}_\text{D}$ drift velocity, \QtyRef{scattering_time}}
|
||||
\desc[german]{Bewegungsgleichung}{}{$v$ Elektronengeschwindigkeit, $\vec{v}_\text{D}$ Driftgeschwindigkeit, \QtyRef{scattering_time}}
|
||||
\eq{\masse \odv{\vec{v}}{t} + \frac{\masse}{\tau} \vec{v}_\text{D} = -e \vec{\E}}
|
||||
\end{formula}
|
||||
\begin{formula}{scattering_time}
|
||||
@ -28,46 +31,51 @@
|
||||
\desc[german]{Streuzeit}{}{}
|
||||
\quantity{\tau}{\s}{s}
|
||||
\ttxt{
|
||||
\eng{$\tau$\\ the average time between scattering events weighted by the characteristic momentum change cause by the scattering process.}
|
||||
\eng{The average time between scattering events weighted by the characteristic momentum change cause by the scattering process.}
|
||||
}
|
||||
\end{formula}
|
||||
\begin{formula}{current_density}
|
||||
\desc{Current density}{Ohm's law}{$n$ charge particle density}
|
||||
\desc[german]{Stromdichte}{Ohmsches Gesetz}{$n$ Ladungsträgerdichte}
|
||||
\desc{Current density}{Ohm's law}{\QtyRef{charge_carrier_density}, \ConstRef{charge}, \QtyRef{drift_velocity}, \QtyRef{mobility}, \QtyRef{electric_field}}
|
||||
\desc[german]{Stromdichte}{Ohmsches Gesetz}{}
|
||||
\quantity{\vec{j}}{\ampere\per\m^2}{v}
|
||||
\eq{\vec{j} = -ne\vec{v}_\text{D} = ne\mu \vec{\E}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{conductivity}
|
||||
\desc{Drude-conductivity}{}{}
|
||||
\desc[german]{Drude-Leitfähigkeit}{}{}
|
||||
\eq{\sigma = \frac{\vec{j}}{\vec{\E}} = \frac{e^2 \tau n}{\masse} = n e \mu}
|
||||
\desc{Electrical conductivity}{Both from Drude model and Sommerfeld model}{\QtyRef{current_density}, \QtyRef{electric_field}, \QtyRef{charge_carrier_density}, \ConstRef{charge}, \QtyRef{scattering_time}, \ConstRef{electron_mass}, \QtyRef{mobility}}
|
||||
\desc[german]{Elektrische Leitfähigkeit}{Aus dem Drude-Modell und dem Sommerfeld-Modell}{}
|
||||
\quantity{\sigma}{\siemens\per\m=\per\ohm\m=\ampere^2\s^3\per\kg\m^3}{t}
|
||||
\eq{\sigma = \frac{\vec{j}}{\vec{\E}} = \frac{n e^2 \tau}{\masse} = n e \mu}
|
||||
\end{formula}
|
||||
|
||||
\Subsection[
|
||||
\eng{Sommerfeld model}
|
||||
\ger{Sommerfeld-Modell}
|
||||
]{sommerfeld}
|
||||
\begin{ttext}
|
||||
\eng{Assumes a gas of free fermions underlying the pauli-exclusion principle. Only electrons in an energy range of $\kB T$ around the Fermi energy $\EFermi$ participate in scattering processes.}
|
||||
\ger{Annahme eines freien Fermionengases, welches dem Pauli-Prinzip unterliegt. Nur Elektronen in einem Energiebereich von $\kB T$ um die Fermi Energe $\EFermi$ nehmen an Streuprozessen teil.}
|
||||
\end{ttext}
|
||||
\begin{formula}{description}
|
||||
\desc{Description}{}{}
|
||||
\desc[german]{Beschreibung}{}{}
|
||||
\ttxt{
|
||||
\eng{Assumes a gas of free fermions underlying the pauli-exclusion principle. Only electrons in an energy range of $\kB T$ around the Fermi energy $\EFermi$ participate in scattering processes. The \qtyRef{conductivity} is the same as in \fRef{::::drude}}
|
||||
\ger{Annahme eines freien Fermionengases, welches dem Pauli-Prinzip unterliegt. Nur Elektronen in einem Energiebereich von $\kB T$ um die Fermi Energe $\EFermi$ nehmen an Streuprozessen teil. Die \qtyRef{conductivity} ist die selbe wie im \fRef{::::drude}}
|
||||
}
|
||||
\end{formula}
|
||||
\begin{formula}{current_density}
|
||||
\desc{Electrical current density}{}{}
|
||||
\desc[german]{Elektrische Stromdichte}{}{}
|
||||
\eq{\vec{j} = -en\braket{v} = -e n \frac{\hbar}{\masse}\braket{\vec{k}} = -e \frac{1}{V} \sum_{\vec{k},\sigma} \frac{\hbar \vec{k}}{\masse}}
|
||||
\end{formula}
|
||||
\TODO{The formula for the conductivity is the same as in the drude model?}
|
||||
|
||||
\Subsection[
|
||||
\eng{Boltzmann-transport}
|
||||
\ger{Boltzmann-Transport}
|
||||
]{boltzmann}
|
||||
\begin{ttext}
|
||||
\eng{Semiclassical description using a probability distribution (\fqEqRef{stat:todo:fermi_dirac}) to describe the particles.}
|
||||
\ger{Semiklassische Beschreibung, benutzt eine Wahrscheinlichkeitsverteilung (\fqEqRef{stat:todo:fermi_dirac}).}
|
||||
\eng{Semiclassical description using a probability distribution (\fRef{stat:todo:fermi_dirac}) to describe the particles.}
|
||||
\ger{Semiklassische Beschreibung, benutzt eine Wahrscheinlichkeitsverteilung (\fRef{stat:todo:fermi_dirac}).}
|
||||
\end{ttext}
|
||||
\begin{formula}{boltzmann_transport}
|
||||
\desc{Boltzmann Transport equation}{for charge transport}{$f$ \ref{stat:todo:fermi-dirac}}
|
||||
\desc{Boltzmann Transport equation}{for charge transport}{$f$ \fRef{stat:todo:fermi-dirac}}
|
||||
\desc[german]{Boltzmann-Transportgleichung}{für Ladungstransport}{}
|
||||
\eq{
|
||||
\pdv{f(\vec{r},\vec{k},t)}{t} = -\vec{v} \cdot \Grad_{\vec{r}} f - \frac{e}{\hbar}(\vec{\mathcal{E}} + \vec{v} \times \vec{B}) \cdot \Grad_{\vec{k}} f + \left(\pdv{f(\vec{r},\vec{k},t)}{t}\right)_{\text{\GT{scatter}}}
|
||||
@ -79,8 +87,8 @@
|
||||
\ger{misc}
|
||||
]{misc}
|
||||
\begin{formula}{tsu_esaki}
|
||||
\desc{Tsu-Esaki tunneling current}{Describes the current $I_{\txL \leftrightarrow \txR}$ through a barrier}{$\mu_i$ \qtyRef{chemical_pot} at left/right side, $U_i$ voltage on left/right side. Electrons occupy region between $U_i$ and $\mu_i$}
|
||||
\desc[german]{Tsu-Esaki Tunnelstrom}{Beschreibt den Strom $I_{\txL \leftrightarrow \txR}$ durch eine Barriere }{$\mu_i$ \qtyRef{chemical_pot} links/rechts, $U_i$ Spannung links/rechts. Elektronen besetzen Bereich zwischen $U_i$ und $\mu_i$}
|
||||
\desc{Tsu-Esaki tunneling current}{Describes the current $I_{\txL \leftrightarrow \txR}$ through a barrier}{$\mu_i$ \qtyRef{chemical_potential} at left/right side, $U_i$ voltage on left/right side. Electrons occupy region between $U_i$ and $\mu_i$}
|
||||
\desc[german]{Tsu-Esaki Tunnelstrom}{Beschreibt den Strom $I_{\txL \leftrightarrow \txR}$ durch eine Barriere }{$\mu_i$ \qtyRef{chemical_potential} links/rechts, $U_i$ Spannung links/rechts. Elektronen besetzen Bereich zwischen $U_i$ und $\mu_i$}
|
||||
\eq{
|
||||
I_\text{T} = \frac{2e}{h} \int_{U_\txL}^\infty \left(f(E, \mu_\txL) -f(E, \mu_\txR)\right) T(E) \d E
|
||||
}
|
||||
|
||||
@ -2,17 +2,17 @@
|
||||
\eng{Condensed matter physics}
|
||||
\ger{Festkörperphysik}
|
||||
]{cm}
|
||||
\TODO{Bonds, hybridized orbitals}
|
||||
\TODO{Lattice vibrations, van hove singularities, debye frequency}
|
||||
\TODO{van hove singularities, debye frequency}
|
||||
|
||||
\begin{formula}{dos}
|
||||
\desc{Density of states (DOS)}{}{\QtyRef{volume}, $N$ number of energy levels, \QtyRef{energy}}
|
||||
\desc[german]{Zustandsdichte (DOS)}{}{\QtyRef{volume}, $N$ Anzahl der Energieniveaus, \QtyRef{energy}}
|
||||
\quantity{D}{\per\m^3}{s}
|
||||
\eq{D(E) = \frac{1}{V}\sum_{i=1}^{N} \delta(E-E(\vec{k_i}))}
|
||||
\end{formula}
|
||||
\begin{formula}{dos_parabolic}
|
||||
\desc{Density of states for parabolic dispersion}{Applies to \fqSecRef{cm:egas}}{}
|
||||
\desc[german]{Zustandsdichte für parabolische Dispersion}{Bei \fqSecRef{cm:egas}}{}
|
||||
\desc{Density of states for parabolic dispersion}{Applies to \fRef{cm:egas}}{}
|
||||
\desc[german]{Zustandsdichte für parabolische Dispersion}{Bei \fRef{cm:egas}}{}
|
||||
\eq{
|
||||
D_1(E) &= \frac{1}{2\sqrt{c_k(E-E_0)}} && (\text{1D}) \\
|
||||
D_2(E) &= \frac{\pi}{2c_k} && (\text{2D}) \\
|
||||
@ -20,53 +20,3 @@
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\Section[
|
||||
\eng{Lattice vibrations}
|
||||
\ger{Gitterschwingungen}
|
||||
]{vib}
|
||||
|
||||
\begin{formula}{dispersion_1atom_basis}
|
||||
\desc{Phonon dispersion of a lattice with a one-atom basis}{same as the dispersion of a linear chain}{$C_n$ force constants between layer $s$ and $s+n$, $M$ \qtyRef{mass} of the reference atom, $a$ \qtyRef{lattice_constant}, $q$ phonon \qtyRef{wavevector}, $u$ Ansatz for the atom displacement}
|
||||
\desc[german]{Phonondispersion eines Gitters mit zweiatomiger Basis}{gleich der Dispersion einer linearen Kette}{$C_n$ Kraftkonstanten zwischen Ebene $s$ und $s+n$, $M$ \qtyRef{mass} des Referenzatoms, $a$ \qtyRef{lattice_constant}, $q$ Phonon \qtyRef{wavevector}, $u$ Ansatz für die Atomauslenkung}
|
||||
\begin{gather}
|
||||
\omega^2 = \frac{4C_1}{M}\left[\sin^2 \left(\frac{qa}{2}\right) + \frac{C2}{C1} \sin^2(qa)\right] \\
|
||||
\intertext{\GT{with}}
|
||||
u_{s+n} = U\e^{-i \left[\omega t - q(s+n)a \right]}
|
||||
\end{gather}
|
||||
\fig{img/cm_phonon_dispersion_one_atom_basis.pdf}
|
||||
\end{formula}
|
||||
\TODO{Plots}
|
||||
\begin{formula}{dispersion_2atom_basis}
|
||||
\desc{Phonon dispersion of a lattice with a two-atom basis}{}{$C$ force constant between layers, $M_i$ \qtyRef{mass} of the basis atoms, $a$ \qtyRef{lattice_constant}, $q$ phonon \qtyRef{wavevector}, $u, v$ Ansatz for the displacement of basis atom 1 and 2, respectively}
|
||||
\desc[german]{Phonondispersion eines Gitters mit einatomiger Basis}{}{$C$ Kraftkonstanten zwischen Ebene $s$ und $s+n$, $M_i$ \qtyRef{mass} der Basisatome, $a$ \qtyRef{lattice_constant}, $q$ Phonon \qtyRef{wavevector}, $u, v$ jeweils Ansatz für die Atomauslenkung des Basisatoms 1 und 2}
|
||||
\begin{gather}
|
||||
\omega^2_{\txa,\txo} = C \left(\frac{1}{M_1}+\frac{1}{M_2}\right) \mp C \sqrt{\left(\frac{1}{M_1}+\frac{1}{M_2}\right)^2 - \frac{4}{M_1M_2} \sin^2 \left(\frac{qa}{2}\right)}
|
||||
\intertext{\GT{with}}
|
||||
u_{s} = U\e^{-i \left(\omega t - qsa \right)}, \quad
|
||||
v_{s} = V\e^{-i \left(\omega t - qsa \right)}
|
||||
\end{gather}
|
||||
\fig{img/cm_phonon_dispersion_two_atom_basis.pdf}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{branches}
|
||||
\desc{Vibration branches}{}{}
|
||||
\desc[german]{Vibrationsmoden}{}{}
|
||||
\ttxt{\eng{
|
||||
\textbf{Acoustic}: 3 modes (1 longitudinal, 2 transversal), the two basis atoms oscillate in phase.
|
||||
\\\textbf{Optical}: 3 modes, the two basis atoms oscillate in opposition. A dipole moment is created that can couple to photons.
|
||||
}\ger{
|
||||
\textbf{Akustisch}: 3 Moden (1 longitudinal, 2 transversal), die zwei Basisatome schwingen in Phase.
|
||||
\\ \textbf{Optisch}: 3 Moden, die zwei Basisatome schwingen gegenphasig. Das dadurch entstehende Dipolmoment erlaubt die Wechselwirkung mit Photonen.
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
\Subsection[
|
||||
\eng{Debye model}
|
||||
\ger{Debye-Modell}
|
||||
]{debye}
|
||||
\begin{ttext}
|
||||
\eng{Atoms behave like coupled \hyperref[sec:qm:hosc]{quantum harmonic oscillators}. The finite sample size leads to periodic boundary conditio. The finite sample size leads to periodic boundary conditions for the vibrations.}
|
||||
\ger{Atome verhalten sich wie gekoppelte \hyperref[sec:qm:hosc]{quantenmechanische harmonische Oszillatoren}. Die endliche Ausdehnung des Körpers führt zu periodischen Randbedingungen. }
|
||||
\end{ttext}
|
||||
|
||||
|
||||
|
||||
@ -17,7 +17,7 @@
|
||||
\eng[bravais_lattices]{Bravais lattices}
|
||||
\ger[bravais_lattices]{Bravais Gitter}
|
||||
|
||||
\newcommand\bvimg[1]{\begin{center}\includegraphics[width=0.1\textwidth]{img/bravais/#1.pdf}\end{center}}
|
||||
\newcommand\bvimg[1]{\begin{center}\includegraphics[width=0.1\textwidth]{img_static/bravais/#1.pdf}\end{center}}
|
||||
\renewcommand\tabularxcolumn[1]{m{#1}}
|
||||
\newcolumntype{Z}{>{\centering\let\newline\\\arraybackslash\hspace{0pt}}X}
|
||||
|
||||
@ -71,7 +71,12 @@
|
||||
\eq{\vec{R} = n_1 \vec{a_1} + n_2 \vec{a_2} + n_3 \vec{a_3}}
|
||||
\end{formula}
|
||||
|
||||
\TODO{primitive unit cell: contains one lattice point}\\
|
||||
\begin{formula}{primitive_unit_cell}
|
||||
\desc{Primitve unit cell}{}{}
|
||||
\desc[german]{Primitive Einheitszelle}{}{}
|
||||
\ttxt{\eng{Unit cell containing exactly one lattice point}\ger{Einheitszelle die genau einen Gitterpunkt enthält}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{miller}
|
||||
\desc{Miller index}{}{Miller family: planes that are equivalent due to crystal symmetry}
|
||||
\desc[german]{Millersche Indizes}{}{}
|
||||
@ -116,8 +121,8 @@
|
||||
\desc{Matthiessen's rule}{Approximation, only holds if the processes are independent of each other}{\QtyRef{mobility}, \QtyRef{scattering_time}}
|
||||
\desc[german]{Matthiessensche Regel}{Näherung, nur gültig wenn die einzelnen Streuprozesse von einander unabhängig sind}{}
|
||||
\eq{
|
||||
\frac{1}{\mu} &= \sum_{i = \textrm{\GT{\fqname}}} \frac{1}{\mu_i} \\
|
||||
\frac{1}{\tau} &= \sum_{i = \textrm{\GT{\fqname}}} \frac{1}{\tau_i}
|
||||
\frac{1}{\mu} &= \sum_{i = \textrm{\GT{:::scatter}}} \frac{1}{\mu_i} \\
|
||||
\frac{1}{\tau} &= \sum_{i = \textrm{\GT{:::scatter}}} \frac{1}{\tau_i}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
@ -135,8 +140,8 @@
|
||||
}
|
||||
\end{formula}
|
||||
\begin{formula}{bcc}
|
||||
\desc{Body centered cubic (BCC)}{Reciprocal: \fqEqRef{cm:bravais:fcc}}{\QtyRef{lattice_constant}}
|
||||
\desc[german]{Kubisch raumzentriert (BCC)}{Reziprok: \fqEqRef{cm:bravais:fcc}}{}
|
||||
\desc{Body centered cubic (BCC)}{Reciprocal: \fRef{::fcc}}{\QtyRef{lattice_constant}}
|
||||
\desc[german]{Kubisch raumzentriert (BCC)}{Reziprok: \fRef{::fcc}}{}
|
||||
\eq{
|
||||
\vec{a}_{1}=\frac{a}{2} \begin{pmatrix} -1\\1\\1 \end{pmatrix},\,
|
||||
\vec{a}_{2}=\frac{a}{2} \begin{pmatrix} 1\\-1\\1 \end{pmatrix},\,
|
||||
@ -145,8 +150,8 @@
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{fcc}
|
||||
\desc{Face centered cubic (FCC)}{Reciprocal: \fqEqRef{cm:bravais:bcc}}{\QtyRef{lattice_constant}}
|
||||
\desc[german]{Kubisch flächenzentriert (FCC)}{Reziprok: \fqEqRef{cm:bravais:bcc}}{}
|
||||
\desc{Face centered cubic (FCC)}{Reciprocal: \fRef{::bcc}}{\QtyRef{lattice_constant}}
|
||||
\desc[german]{Kubisch flächenzentriert (FCC)}{Reziprok: \fRef{::bcc}}{}
|
||||
\eq{
|
||||
\vec{a}_{1}=\frac{a}{2} \begin{pmatrix} 0\\1\\1 \end{pmatrix},\,
|
||||
\vec{a}_{2}=\frac{a}{2} \begin{pmatrix} 1\\0\\1 \end{pmatrix},\,
|
||||
@ -158,25 +163,41 @@
|
||||
\desc{Diamond lattice}{}{}
|
||||
\desc[german]{Diamantstruktur}{}{}
|
||||
\ttxt{
|
||||
\eng{\fqEqRef{cm:bravais:fcc} with basis $\begin{pmatrix} 0 & 0 & 0 \end{pmatrix}$ and $\begin{pmatrix} \frac{1}{4} & \frac{1}{4} & \frac{1}{4} \end{pmatrix}$}
|
||||
\ger{\fqEqRef{cm:bravais:fcc} mit Basis $\begin{pmatrix} 0 & 0 & 0 \end{pmatrix}$ und $\begin{pmatrix} \frac{1}{4} & \frac{1}{4} & \frac{1}{4} \end{pmatrix}$}
|
||||
\eng{\fRef{:::fcc} with basis $\begin{pmatrix} 0 & 0 & 0 \end{pmatrix}$ and $\begin{pmatrix} \frac{1}{4} & \frac{1}{4} & \frac{1}{4} \end{pmatrix}$}
|
||||
\ger{\fRef{:::fcc} mit Basis $\begin{pmatrix} 0 & 0 & 0 \end{pmatrix}$ und $\begin{pmatrix} \frac{1}{4} & \frac{1}{4} & \frac{1}{4} \end{pmatrix}$}
|
||||
}
|
||||
\end{formula}
|
||||
\begin{formula}{zincblende}
|
||||
\desc{Zincblende lattice}{}{}
|
||||
\desc[german]{Zinkblende-Struktur}{}{}
|
||||
\ttxt{
|
||||
\includegraphics[width=0.5\textwidth]{img/cm_zincblende.png}
|
||||
\eng{Like \fqEqRef{cm:bravais:diamond} but with different species on each basis}
|
||||
\ger{Wie \fqEqRef{cm:bravais:diamond} aber mit unterschiedlichen Spezies auf den Basen}
|
||||
\fsplit{
|
||||
\centering
|
||||
\includegraphics[width=0.9\textwidth]{img/cm_crystal_zincblende.png}
|
||||
}{
|
||||
\ttxt{
|
||||
\eng{Like \fRef{:::diamond} but with different species on each basis}
|
||||
\ger{Wie \fRef{:::diamond} aber mit unterschiedlichen Spezies auf den Basen}
|
||||
}
|
||||
}
|
||||
\end{formula}
|
||||
\begin{formula}{rocksalt}
|
||||
\desc{Rocksalt structure}{\elRef{Na}\elRef{Cl}}{}
|
||||
\desc[german]{Kochsalz-Struktur}{}{}
|
||||
\fsplit{
|
||||
\centering
|
||||
\includegraphics[width=0.9\textwidth]{img/cm_crystal_NaCl.png}
|
||||
}{
|
||||
|
||||
}
|
||||
\end{formula}
|
||||
\begin{formula}{wurtzite}
|
||||
\desc{Wurtzite structure}{hP4}{}
|
||||
\desc[german]{Wurtzite-Struktur}{hP4}{}
|
||||
\ttxt{
|
||||
\includegraphics[width=0.5\textwidth]{img/cm_wurtzite.png}
|
||||
Placeholder
|
||||
\fsplit{
|
||||
\centering
|
||||
\includegraphics[width=0.9\textwidth]{img/cm_crystal_wurtzite.png}
|
||||
}{
|
||||
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
|
||||
@ -1,198 +0,0 @@
|
||||
\def\L{\text{L}}
|
||||
\def\gl{\text{GL}}
|
||||
\def\GL{Ginzburg-Landau }
|
||||
\def\Tcrit{T_\text{c}}
|
||||
\def\Bcrit{B_\text{c}}
|
||||
\def\ssc{\text{s}}
|
||||
\def\ssn{\text{n}}
|
||||
|
||||
\Section[
|
||||
\eng{Superconductivity}
|
||||
\ger{Supraleitung}
|
||||
]{sc}
|
||||
\begin{ttext}
|
||||
\eng{
|
||||
Materials for which the electric resistance jumps to 0 under a critical temperature $\Tcrit$.
|
||||
Below $\Tcrit$ they have perfect conductivity and perfect diamagnetism, up until a critical magnetic field $\Bcrit$.
|
||||
\\\textbf{Type I}: Has a single critical magnetic field at which the superconuctor becomes a normal conductor.
|
||||
\\\textbf{Type II}: Has two critical
|
||||
}
|
||||
\ger{
|
||||
Materialien, bei denen der elektrische Widerstand beim unterschreiten einer kritischen Temperatur $\Tcrit$ auf 0 springt.
|
||||
Sie verhalten sich dann wie ideale Leiter und ideale Diamagnete, bis zu einem kritischen Feld $\Bcrit$.
|
||||
|
||||
}
|
||||
\end{ttext}
|
||||
|
||||
\begin{formula}{perfect_conductor}
|
||||
\desc{Perfect conductor}{}{}
|
||||
\desc[german]{Ideale Leiter}{}{}
|
||||
\ttxt{
|
||||
\eng{
|
||||
In contrast to a superconductor, perfect conductors become diamagnetic only when the external magnetic field is turned on \textbf{after} the material was cooled below the critical temperature.
|
||||
(\fqEqRef{ed:fields:mag:induction:lenz})
|
||||
}
|
||||
\ger{
|
||||
Im Gegensatz zu einem Supraleiter werden ideale Leiter nur dann diamagnetisch, wenn das externe magnetische Feld \textbf{nach} dem Abkühlen unter die kritische Temperatur eingeschaltet wird.
|
||||
(\fqEqRef{ed:fields:mag:induction:lenz})
|
||||
}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{meissner_effect}
|
||||
\desc{Meißner-Ochsenfeld effect}{Perfect diamagnetism}{}
|
||||
\desc[german]{Meißner-Ochsenfeld Effekt}{Idealer Diamagnetismus}{}
|
||||
\ttxt{
|
||||
\eng{External magnetic field decays exponetially inside the superconductor below a critical temperature and a critical magnetic field.}
|
||||
\ger{Externes Magnetfeld fällt im Supraleiter exponentiell unterhalb einer kritischen Temperatur und unterhalb einer kritischen Feldstärke ab.}
|
||||
}
|
||||
\end{formula}
|
||||
\Subsection[
|
||||
\eng{London equations}
|
||||
\ger{London-Gleichungen}
|
||||
]{london}
|
||||
\begin{ttext}
|
||||
\eng{
|
||||
Quantitative description of the \fqEqRef{cm:sc:meissner_effect}.
|
||||
}
|
||||
\ger{
|
||||
Quantitative Beschreibung des \fqEqRef{cm:sc:meissner_effect}s.
|
||||
}
|
||||
|
||||
\end{ttext}
|
||||
% \begin{formula}{coefficient}
|
||||
% \desc{London-coefficient}{}{}
|
||||
% \desc[german]{London-Koeffizient}{}{}
|
||||
% \eq{\Lambda = \frac{m_\ssc}{n_\ssc q_\ssc^2}}
|
||||
% \end{formula}
|
||||
\begin{formula}{first}
|
||||
% \vec{j} = \frac{nq\hbar}{m}\Grad S - \frac{nq^2}{m}\vec{A}
|
||||
\desc{First London Equation}{}{$\vec{j}$ current density, $n_\ssc$, $m_\ssc$, $q_\ssc$ density, mass and charge of superconduticng particles}
|
||||
\desc[german]{Erste London-Gleichun-}{}{$\vec{j}$ Stromdichte, $n_\ssc$, $m_\ssc$, $q_\ssc$ Dichte, Masse und Ladung der supraleitenden Teilchen}
|
||||
\eq{
|
||||
\pdv{\vec{j}_{\ssc}}{t} = \frac{n_\ssc q_\ssc^2}{m_\ssc}\vec{E} {\color{gray}- \Order{\vec{j}_\ssc^2}}
|
||||
% \\{\color{gray} = \frac{q}{m}\Grad \left(\frac{1}{2} \TODO{FActor} \vec{j}^2\right)}
|
||||
}
|
||||
\end{formula}
|
||||
\begin{formula}{second}
|
||||
\desc{Second London Equation}{Describes the \fqEqRef{cm:sc:meissner_effect}}{$\vec{j}$ current density, $n_\ssc$, $m_\ssc$, $q_\ssc$ density, mass and charge of superconduticng particles}
|
||||
\desc[german]{Zweite London-Gleichung}{Beschreibt den \fqEqRef{cm:sc:meissner_effect}}{$\vec{j}$ Stromdichte, $n_\ssc$, $m_\ssc$, $q_\ssc$ Dichte, Masse und Ladung der supraleitenden Teilchen}
|
||||
\eq{
|
||||
\Rot \vec{j_\ssc} = -\frac{n_\ssc q_\ssc^2}{m_\ssc} \vec{B}
|
||||
}
|
||||
\end{formula}
|
||||
\begin{formula}{penetration_depth}
|
||||
\desc{London penetration depth}{}{}
|
||||
\desc[german]{London Eindringtiefe}{}{}
|
||||
\eq{\lambda_\L = \sqrt{\frac{m_\ssc}{\mu_0 n_\ssc q_\ssc^2}}}
|
||||
\end{formula}
|
||||
|
||||
|
||||
|
||||
\Subsection[
|
||||
\eng{\GL Theory (GLAG)}
|
||||
\ger{\GL Theorie (GLAG)}
|
||||
]{gl}
|
||||
\begin{ttext}
|
||||
\eng{
|
||||
\TODO{TODO}
|
||||
}
|
||||
|
||||
\end{ttext}
|
||||
\begin{formula}{coherence_length}
|
||||
\desc{\GL Coherence Length}{}{}
|
||||
\desc[german]{\GL Kohärenzlänge}{}{}
|
||||
\eq{
|
||||
\xi_\gl &= \frac{\hbar}{\sqrt{2m \abs{\alpha}}} \\
|
||||
\xi_\gl(T) &= \xi_\gl(0) \frac{1}{\sqrt{1-\frac{T}{\Tcrit}}}
|
||||
}
|
||||
\end{formula}
|
||||
\begin{formula}{penetration_depth}
|
||||
\desc{\GL Penetration Depth / Field screening length}{}{}
|
||||
\desc[german]{\GL Eindringtiefe}{}{}
|
||||
\eq{
|
||||
\lambda_\gl &= \sqrt{\frac{m_\ssc\beta}{\mu_0 \abs{\alpha} q_s^2}} \\
|
||||
\lambda_\gl(T) &= \lambda_\gl(0) \frac{1}{\sqrt{1-\frac{T}{\Tcrit}}}
|
||||
}
|
||||
\end{formula}
|
||||
\begin{formula}{first}
|
||||
\desc{First Ginzburg-Landau Equation}{}{$\xi_\gl$ \fqEqRef{cm:sc:gl:coherence_length}, $\lambda_\gl$ \fqEqRef{cm:sc:gl:penetration_depth}}
|
||||
\desc[german]{Erste Ginzburg-Landau Gleichung}{}{}
|
||||
\eq{\alpha\Psi + \beta\abs{\Psi}^2 \Psi + \frac{1}{2m} (-i\hbar \Grad + 2e\vec{A})^2\Psi = 0}
|
||||
\end{formula}
|
||||
\begin{formula}{second}
|
||||
\desc{Second Ginzburg-Landau Equation}{}{}
|
||||
\desc[german]{Zweite Ginzburg-Landau Gleichung}{}{}
|
||||
\eq{\vec{j_\ssc} = \frac{ie\hbar}{m}(\Psi^*\Grad\Psi - \Psi\Grad\Psi^*) - \frac{4e^2}{m}\abs{\Psi}^2 \vec{A}}
|
||||
\end{formula}
|
||||
|
||||
\TODO{proximity effect}
|
||||
|
||||
\Subsection[
|
||||
\eng{Microscopic theory}
|
||||
\ger{Mikroskopische Theorie}
|
||||
]{micro}
|
||||
|
||||
\begin{formula}{isotop_effect}
|
||||
\desc{Isotope effect}{Superconducting behaviour depends on atomic mass and thereby of the lattice \Rightarrow Microscopic origin}{$\Tcrit$ critial temperature, $M$ isotope mass, $\omega_\text{ph}$}
|
||||
\desc[german]{Isotopeneffekt}{Supraleitung hängt von der Atommasse und daher von den Gittereigenschaften ab \Rightarrow Mikroskopischer Ursprung}{$\Tcrit$ kritische Temperatur, $M$ Isotopen-Masse, $\omega_\text{ph}$}
|
||||
\eq{
|
||||
\Tcrit \propto \frac{1}{\sqrt{M}} \\
|
||||
\omega_\text{ph} \propto \frac{1}{\sqrt{M}} \Rightarrow \Tcrit \propto \omega_\text{ph}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{cooper_pairs}
|
||||
\desc{Cooper pairs}{}{}
|
||||
\desc[german]{Cooper-Paars}{}{}
|
||||
\ttxt{
|
||||
\eng{Conduction electrons reduce their energy through an attractive interaction: One electron passing by atoms attracts the these, which creats a positive charge region behind the electron, which in turn attracts another electron. }
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\Subsubsection[
|
||||
\eng{BCS-Theory}
|
||||
\ger{BCS-Theorie}
|
||||
]{bcs}
|
||||
\begin{ttext}
|
||||
\eng{
|
||||
Electron pairs form bosonic quasi-particles called Cooper pairs which can condensate into the ground state.
|
||||
The wave function spans the whole material, which makes it conduct without resistance.
|
||||
The exchange bosons between the electrons are phonons.
|
||||
}
|
||||
\ger{
|
||||
Elektronenpaar bilden bosonische Quasipartikel (Cooper Paare) welche in den Grundzustand kondensieren können.
|
||||
Die Wellenfunktion übersoannt den gesamten Festkörper, was einen widerstandslosen Ladungstransport garantiert.
|
||||
Die Austauschbosononen zwischen den Elektronen sind Bosonen.
|
||||
}
|
||||
\end{ttext}
|
||||
\def\BCS{{\text{BCS}}}
|
||||
\begin{formula}{hamiltonian}
|
||||
\desc{BCS Hamiltonian}{for $N$ interacting electrons}{
|
||||
$c_{\veck\sigma}$ creation/annihilation operators create/destroy at $\veck$ with spin $\sigma$ \\
|
||||
First term: non-interacting free electron gas\\
|
||||
Second term: interaction energy
|
||||
}
|
||||
\desc[german]{BCS Hamiltonian}{}{}
|
||||
\eq{
|
||||
\hat{H}_\BCS =
|
||||
\sum_{\sigma} \sum_\veck \epsilon_\veck \hat{c}_{\veck\sigma}^\dagger \hat{c}_{\veck\sigma}
|
||||
+ \sum_{\veck,\veck^\prime} V_{\veck,\veck^\prime}
|
||||
\hat{c}_{\veck\uparrow}^\dagger \hat{c}_{-\veck\downarrow}^\dagger
|
||||
\hat{c}_{-\veck^\prime\downarrow} \hat{c}_{\veck^\prime,\uparrow}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{bogoliubov-valatin}
|
||||
\desc{Bogoliubov-Valatin transformation}{Diagonalization of the \fqEqRef{cm:sc:micro:bcs:hamiltonian} to derive excitation energies}{}
|
||||
\desc[german]{Bogoliubov-Valatin transformation}{}{}
|
||||
\eq{
|
||||
\hat{H}_\BCS - N\mu = \sum_\veck \big[\xi_\veck - E_\veck + \Delta_\veck g_\veck^*\big] + \sum_\veck \big[E_\veck a_\veck^\dagger a_\veck + E_\veck \beta_{-\veck}^\dagger \beta_{-\veck}\big]
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{gap_equation}
|
||||
\desc{BCS-gap equation}{}{}
|
||||
\desc[german]{}{}{}
|
||||
\eq{\Delta_\veck^* = -\sum_\veck^+\prime V_{\veck,\veck^\prime} \frac{\Delta_{\veck^\prime}}{2E_\veck} \tanh \left(\frac{E_{\veck^\prime}}{2\kB T}\right)}
|
||||
\end{formula}
|
||||
@ -84,6 +84,15 @@
|
||||
\ger{\GT{misc}}
|
||||
]{misc}
|
||||
|
||||
\begin{formula}{vdw_material}
|
||||
\desc{Van-der-Waals material}{2D materials}{}
|
||||
\desc[german]{Van-der-Waals Material}{2D Materialien}{}
|
||||
\ttxt{\eng{
|
||||
Materials consiting of multiple 2D-layers held together by Van-der-Waals forces.
|
||||
}\ger{
|
||||
Aus mehreren 2D-Schichten bestehende Materialien, die durch Van-der-Waals Kräfte zusammengehalten werden.
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\begin{formula}{work_function}
|
||||
|
||||
@ -1,9 +1,9 @@
|
||||
\Section[
|
||||
\eng{Semiconductors}
|
||||
\ger{Halbleiter}
|
||||
]{semic}
|
||||
]{sc}
|
||||
\begin{formula}{types}
|
||||
\desc{Intrinsic/extrinsic}{}{$n,p$ \fqEqRef{cm:semic:charge_density_eq}}
|
||||
\desc{Intrinsic/extrinsic}{}{$n,p$ \fRef{cm:sc:charge_density_eq}}
|
||||
\desc[german]{Intrinsisch/Extrinsisch}{}{}
|
||||
\ttxt{
|
||||
\eng{
|
||||
@ -89,7 +89,7 @@
|
||||
\end{formula}
|
||||
|
||||
\begin{bigformula}{schottky_barrier}
|
||||
\desc{Schottky barrier}{Rectifying \fqEqRef{cm:sc:junctions:metal-sc}}{}
|
||||
\desc{Schottky barrier}{Rectifying \fRef{cm:sc:junctions:metal-sc}}{}
|
||||
% \desc[german]{}{}{}
|
||||
\centering
|
||||
\resizebox{0.49\textwidth}{!}{\input{img/cm/sc_junction_metal_n_sc_separate.tex}}
|
||||
@ -145,7 +145,7 @@
|
||||
\eng[free_X]{for free Excitons}
|
||||
\ger[free_X]{für freie Exzitons}
|
||||
\begin{formula}{rydberg}
|
||||
\desc{Exciton Rydberg energy}{\gt{free_X}}{$R_\txH$ \fqEqRef{qm:h:rydberg_energy}}
|
||||
\desc{Exciton Rydberg energy}{\gt{free_X}}{$R_\txH$ \fRef{qm:h:rydberg_energy}}
|
||||
\desc[german]{}{}{}
|
||||
\eq{
|
||||
E(n) = - \left(\frac{\mu}{m_0\epsilon_r^2}\right) R_\txH \frac{1}{n^2}
|
||||
|
||||
546
src/cm/superconductivity.tex
Normal file
546
src/cm/superconductivity.tex
Normal file
@ -0,0 +1,546 @@
|
||||
\def\txL{\text{L}}
|
||||
\def\gl{\text{GL}}
|
||||
\def\GL{Ginzburg-Landau }
|
||||
\def\Tcrit{T_\text{c}}
|
||||
\def\Bcth{B_\text{c,th}}
|
||||
|
||||
\Section[
|
||||
\eng{Superconductivity}
|
||||
\ger{Supraleitung}
|
||||
]{super}
|
||||
\begin{ttext}
|
||||
\eng{
|
||||
Materials for which the electric resistance jumps to 0 under a critical temperature $\Tcrit$.
|
||||
Below $\Tcrit$ they have perfect conductivity and perfect diamagnetism, up until a critical magnetic field $\Bcth$.
|
||||
}
|
||||
\ger{
|
||||
Materialien, bei denen der elektrische Widerstand beim unterschreiten einer kritischen Temperatur $\Tcrit$ auf 0 springt.
|
||||
Sie verhalten sich dann wie ideale Leiter und ideale Diamagnete, bis zu einem kritischen Feld $\Bcth$.
|
||||
|
||||
}
|
||||
\end{ttext}
|
||||
|
||||
\begin{formula}{type1}
|
||||
\desc{Type-I superconductor}{}{}
|
||||
\desc[german]{Typ-I Supraleiter}{}{}
|
||||
\ttxt{\eng{
|
||||
Has a single critical magnetic field, $\Bcth$.
|
||||
\\$B < \Bcth$: \fRef{:::meissner_effect}
|
||||
\\$B > \Bcth$: Normal conductor
|
||||
\\ Very small usable current density because current only flows within the \fRef{cm:super:london:penetration_depth} of the surface.
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{type2}
|
||||
\desc{Type-II superconductor}{}{}
|
||||
\desc[german]{Typ-II Supraleiter}{}{}
|
||||
\ttxt{\eng{
|
||||
Has a two critical magnetic fields.
|
||||
\\$B < B_\text{c1}$: \fRef{:::meissner_effect}
|
||||
\\$B_\text{c1} < B < B_\text{c2}$: \fRef{:::shubnikov_phase}
|
||||
\\$B > B_\text{c2}$: Normal conductor
|
||||
\\ In \fRef{:::shubnikov_phase} larger usable current density because current flows within the \fRef{cm:super:london:penetration_depth} of the surface and the penetrating flux lines.
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{perfect_conductor}
|
||||
\desc{Perfect conductor}{}{}
|
||||
\desc[german]{Ideale Leiter}{}{}
|
||||
\ttxt{
|
||||
\eng{
|
||||
In contrast to a superconductor, perfect conductors become diamagnetic only when the external magnetic field is turned on \textbf{after} the material was cooled below the critical temperature.
|
||||
(\fRef{ed:em:induction:lenz})
|
||||
}
|
||||
\ger{
|
||||
Im Gegensatz zu einem Supraleiter werden ideale Leiter nur dann diamagnetisch, wenn das externe magnetische Feld \textbf{nach} dem Abkühlen unter die kritische Temperatur eingeschaltet wird.
|
||||
(\fRef{ed:em:induction:lenz})
|
||||
}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{meissner_effect}
|
||||
\desc{Meißner-Ochsenfeld effect}{Perfect diamagnetism}{$\chi=-1$ \qtyRef{magnetic_susceptibility}}
|
||||
\desc[german]{Meißner-Ochsenfeld Effekt}{Perfekter Diamagnetismus}{}
|
||||
\ttxt{
|
||||
\eng{External magnetic field decays exponetially inside the superconductor below a critical temperature and a critical magnetic field, path-independant.}
|
||||
\ger{Externes Magnetfeld fällt im Supraleiter exponentiell unterhalb einer kritischen Temperatur und unterhalb einer kritischen Feldstärke ab, wegunabhängig.}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{bcth}
|
||||
\desc{Thermodynamic cricitial field}{for \fRef[type I]{::type1} and \fRef[type II]{::type2}}{}
|
||||
\desc[german]{Thermodynamisches kritische Feldstärke}{für \fRef[type I]{::type1} und \Ref[type II]{::type2}}{}
|
||||
\eq{g_\txs - g_\txn = - \frac{\Bcth^2(T)}{2\mu_0}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{shubnikov_phase}
|
||||
\desc{Shubnikov phase}{in \fRef{::type2}}{}
|
||||
\desc[german]{Shubnikov-Phase}{}{}
|
||||
\ttxt{\eng{
|
||||
Mixed phase in which some magnetic flux penetrates the superconductor.
|
||||
}\ger{
|
||||
Gemischte Phase in der der Supraleiter teilweise von magnetischem Fluss durchdrungen werden kann.
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{condensation_energy}
|
||||
\desc{Condensation energy}{}{\QtyRef{free_enthalpy}, \ConstRef{magnetic_vacuum_permeability}}
|
||||
\desc[german]{Kondensationsenergie}{}{}
|
||||
\eq{
|
||||
\d G &= -S \d T + V \d p - V \vecM \cdot \d\vecB \\
|
||||
G_\text{con} &= G_\txn(B=0,T) - G_\txs(B=0,T) = \frac{V \Bcth^2(T)}{2\mu_0}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\Subsection[
|
||||
\eng{London Theory}
|
||||
\ger{London-Theorie}
|
||||
]{london}
|
||||
\begin{formula}{description}
|
||||
\desc{Description}{}{}
|
||||
\desc[german]{Beschreibung}{}{}
|
||||
\ttxt{\eng{
|
||||
\begin{itemize}
|
||||
\item Phenomenological theory
|
||||
\item Quantitative description of the \fRef{cm:super:meissner_effect}.
|
||||
\item Assumies uniform charge density $n(\vecr,t) = n(t)$ (London-approximation).
|
||||
\item Does not work near $T_\txc$
|
||||
\end{itemize}
|
||||
}\ger{
|
||||
\begin{itemize}
|
||||
\item Phänomenologische Theorie
|
||||
\item Quantitative Beschreibung des \fRef{cm:super:meissner_effect}s.
|
||||
\item Annahme: uniforme Ladungsdichte $n(\vecr,t) = n(t)$ (London-Näherung)
|
||||
\item Funktioniert nicht nahe $T_\txc$
|
||||
\end{itemize}
|
||||
}}
|
||||
\end{formula}
|
||||
% \begin{formula}{coefficient}
|
||||
% \desc{London-coefficient}{}{}
|
||||
% \desc[german]{London-Koeffizient}{}{}
|
||||
% \eq{\txLambda = \frac{m_\txs}{n_\txs q_\txs^2}}
|
||||
% \end{formula}
|
||||
\Eng[of_sc_particle]{of the superconducting particle}
|
||||
\Ger[of_sc_particle]{der Supraleitenden Teilchen}
|
||||
\begin{formula}{first}
|
||||
% \vec{j} = \frac{nq\hbar}{m}\Grad S - \frac{nq^2}{m}\vec{A}
|
||||
\desc{First London Equation}{}{$\vec{j}$ \qtyRef{current_density}, $m_\txs$/$n_\txs$/$q_\txs$ \qtyRef{mass}/\qtyRef{charge_carrier_density}/\qtyRef{charge} \GT{of_sc_particle}, \QtyRef{electric_field}}
|
||||
\desc[german]{Erste London-Gleichun-}{}{}
|
||||
\eq{
|
||||
\pdv{\vec{j}_{\txs}}{t} = \frac{n_\txs q_\txs^2}{m_\txs}\vec{\E} {\color{gray}- \Order{\vec{j}_\txs^2}}
|
||||
% \\{\color{gray} = \frac{q}{m}\Grad \left(\frac{1}{2} \TODO{FActor} \vec{j}^2\right)}
|
||||
}
|
||||
\end{formula}
|
||||
\begin{formula}{second}
|
||||
\desc{Second London Equation}{Describes the \fRef{cm:super:meissner_effect}}{$\vec{j}$ \qtyRef{current_density}, $m_\txs$/$n_\txs$/$q_\txs$ \qtyRef{mass}/\qtyRef{charge_carrier_density}/\qtyRef{charge} \GT{of_sc_particle}, \QtyRef{magnetic_flux_density}}
|
||||
\desc[german]{Zweite London-Gleichung}{Beschreibt den \fRef{cm:super:meissner_effect}}{}
|
||||
\eq{
|
||||
\Rot \vec{j_\txs} = -\frac{n_\txs q_\txs^2}{m_\txs} \vec{B}
|
||||
}
|
||||
\end{formula}
|
||||
\begin{formula}{penetration_depth}
|
||||
\desc{London penetration depth}{Depth at which $B$ is $1/\e$ times the value of $B_\text{ext}$}{$m_\txs$/$n_\txs$/$q_\txs$ \qtyRef{mass}/\qtyRef{charge_carrier_density}/\qtyRef{charge} \GT{of_sc_particle}}
|
||||
\desc[german]{London Eindringtiefe}{Tiefe bei der $B$ das $1/\e$-fache von $B_\text{ext}$ ist}{}
|
||||
\eq{\lambda_\txL = \sqrt{\frac{m_\txs}{\mu_0 n_\txs q_\txs^2}}}
|
||||
\end{formula}
|
||||
\begin{formula}{penetration_depth_temp}
|
||||
\desc{Temperature dependence of \fRef{::penetration_depth}}{}{}
|
||||
\desc[german]{Temperaturabhängigkeit der \fRef{::penetration_depth}}{}{}
|
||||
\eq{\lambda_\txL(T) = \lambda_\txL(0) \frac{1}{\sqrt{1- \left(\frac{T}{T_\txc}\right)^4}}}
|
||||
\end{formula}
|
||||
|
||||
\Subsubsection[
|
||||
\eng{Macroscopic wavefunction}
|
||||
\ger{Makroskopische Wellenfunktion}
|
||||
]{macro}
|
||||
\begin{formula}{ansatz}
|
||||
\desc{Ansatz}{}{}
|
||||
\desc[german]{Ansatz}{}{}
|
||||
\ttxt{\eng{Alternative derivation of London equations by assuming a macroscopic wavefunction which is uniform in space}\ger{Alternative Herleitung der London-Gleichungen durch Annahme einer makroskopischen Wellenfunktion, welche nicht Ortsabhängig ist}}
|
||||
\eq{\Psi(\vecr,t) = \Psi_0(\vecr,t) \e^{\theta(\vecr,t)}}
|
||||
\end{formula}
|
||||
\begin{formula}{energy-phase_relation}
|
||||
\desc{Energy-phase relation}{}{$\theta$ \qtyRef{phase}, $m_\txs$/$n_\txs$/$q_\txs$ \qtyRef{mass}/\qtyRef{charge_carrier_density}/\qtyRef{charge} \GT{of_sc_particle}, \QtyRef{current_density}, $\phi_\text{el}$ \qtyRef{electric_scalar_potential}, \QtyRef{chemical_potential}}
|
||||
\desc[german]{Energie-Phase Beziehung}{}{}
|
||||
\eq{\hbar \pdv{\theta(\vecr,t)}{t} = - \left(\frac{m_\txs}{n_\txs^2 q_\txs^2} \vecj_\txs^2(\vecr,t) + q_\txs\phi_\text{el}(\vecr,t) + \mu(\vecr,t)\right)}
|
||||
\end{formula}
|
||||
\begin{formula}{current-phase_relation}
|
||||
\desc{Current-phase relation}{}{$\theta$ \qtyRef{phase}, $m_\txs$/$n_\txs$/$q_\txs$ \qtyRef{mass}/\qtyRef{charge_carrier_density}/\qtyRef{charge} \GT{of_sc_particle}, \QtyRef{current_density}, \QtyRef{magnetic_vector_potential}}
|
||||
\desc[german]{Strom-Phase Beziehung}{}{}
|
||||
\eq{\vecj_\txs(\vecr,t) = \frac{q_\txs^2 n_\txs(\vecr,t)}{m_\txs} \left(\frac{\hbar}{q_\txs} \Grad\theta(\vecr,t) - \vecA(\vecr,t)\right) }
|
||||
\end{formula}
|
||||
|
||||
|
||||
\Subsubsection[
|
||||
\eng{Josephson Effect}
|
||||
\ger{Josephson Effekt}
|
||||
]{josephson}
|
||||
\begin{formula}{1st_relation}
|
||||
\desc{1. Josephson relation}{Dissipationless supercurrent accros junction at zero applied voltage}{$\vecj_\text{C}=\frac{2e}{\hbar}E_\text{J}$ critical current, $\phi$ phase difference accross junction}
|
||||
\desc[german]{1. Josephson Gleichung}{Dissipationsloser Suprastrom durch die Kreuzung ohne angelegte Spannung}{$\vecj_\text{C}=\frac{2e}{\hbar}E_\text{J}$ kritischer Strom, $\phi$ Phasendifferenz zwischen den Supraleitern}
|
||||
\eq{\vecj_\txs(\vecr,t) = \vecj_\text{C}(\vecr,t) \sin\phi(\vecr,t)}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{2nd_relation}
|
||||
\desc{2. Josephson relation}{Superconducting phase change is proportional to applied voltage}{$\phi$ phase differnce accross junction, \ConstRef{flux_quantum}, \QtyRef{voltage}}
|
||||
\desc[german]{2. Josephson Gleichung}{Supraleitende Phasendifferenz is proportional zur angelegten Spannung}{$\phi$ Phasendifferenz, \ConstRef{flux_quantum}, \QtyRef{voltage}}
|
||||
\eq{\odv{\phi(t)}{t} = \frac{2\pi}{\Phi_0} U(t)}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{coupling_energy}
|
||||
\desc{Josephson coupling energy}{}{$A$ junction \qtyRef{area}, \ConstRef{flux_quantum}, $\vecj_\txc$ \fRef[critical current density]{::1st_relation}, $\phi$ phase differnce accross junction}
|
||||
\desc[german]{Josephson}{}{$A$ junction \qtyRef{area}, \ConstRef{flux_quantum}, $\vecj_\txc$ \fRef[kritische Stromdichte]{::1st_relation}, $\phi$ Phasendifferenz zwischen den Supraleitern}
|
||||
\eq{\frac{E_\txJ}{A} = \frac{\Phi_0 \vecj_\txc}{2\pi}(1-\cos\phi)}
|
||||
\end{formula}
|
||||
|
||||
|
||||
|
||||
|
||||
\Subsection[
|
||||
\eng{\GL Theory (GLAG)}
|
||||
\ger{\GL Theorie (GLAG)}
|
||||
]{gl}
|
||||
\begin{formula}{description}
|
||||
\desc{Description}{}{}
|
||||
\desc[german]{Beschreibung}{}{}
|
||||
\ttxt{\eng{
|
||||
\begin{itemize}
|
||||
\item Phenomenological theory
|
||||
\item Improvement on the Landau-Theory of 2nd order phase transitions
|
||||
% which introduces an order parameter that is $0$ in the normal state and rises to saturation in the superconducting state.
|
||||
\item Additional complex, position-dependent order parameter is introduced $\Psi(\vecr)$
|
||||
\item Only valid close to $T_\txc$.
|
||||
\item Does not have time dependancy
|
||||
\end{itemize}
|
||||
}\ger{
|
||||
\begin{itemize}
|
||||
\item Phänomenologische Theorie
|
||||
\item Weiterentwicklung der Landau-Theorie für Phasenübergänge zweiter Ordnung,
|
||||
% in der ein Ordnungsparameter in the normalen Phase 0 ist und ein der supraleitenden Phase bis zur Sättigung ansteigt.
|
||||
\item Zusätzlicher, komplexer, ortsabhängiger Ordnungsparameter $\Psi(\vecr)$
|
||||
\item Nur nahe $T_\txc$ gültig.
|
||||
\item Beschreibt keine Zeitabhängigkeit
|
||||
\end{itemize}
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{expansion}
|
||||
\desc{Expansion}{Expansion of free enthalpy of superconducting state}{
|
||||
$g_{\txs/\txn}$ specific \qtyRef{free_enthalpy} of superconducting/normal state,
|
||||
$\Psi(\vecr) = \abs{\Psi_0(\vecr)} \e^{\I\theta(\vecr)}$ order parameter,
|
||||
$n(\vecr) = \abs{\Psi}^2$ Cooper-Pair density,
|
||||
\QtyRef{magnetic_flux_density},
|
||||
\QtyRef{magnetic_vector_potential},
|
||||
$\alpha(T) = -\bar{\alpha} \left(1-\frac{T}{T_\txc}\right)^2$,
|
||||
% $\alpha > 0$ for $T > T_\txc$ and $\alpha < 0$ for $T< T_\txc$,
|
||||
$\beta = \const > 0$
|
||||
}
|
||||
% \desc[german]{}{}{}
|
||||
\begin{multline}
|
||||
g_\txs = g_\txn + \alpha \abs{\Psi}^2 + \frac{1}{2}\beta \abs{\Psi}^4 +
|
||||
\\ \frac{1}{2\mu_0}(\vecB_\text{ext} -\vecB_\text{inside})^2 + \frac{1}{2m_\txs} \abs{ \left(-\I\hbar\Grad - q_\txs \vecA\right)\Psi}^2 + \dots
|
||||
\end{multline}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{first}
|
||||
\desc{First Ginzburg-Landau Equation}{Obtained by minimizing $g_\txs$ with respect to $\delta\Psi$ in \fRef{::expansion}}{
|
||||
$\xi_\gl$ \fRef{cm:super:gl:coherence_length},
|
||||
$\lambda_\gl$ \fRef{cm:super:gl:penetration_depth}
|
||||
}
|
||||
\desc[german]{Erste Ginzburg-Landau Gleichung}{}{}
|
||||
\eq{\alpha\Psi + \beta\abs{\Psi}^2 \Psi + \frac{1}{2m} (-i\hbar \Grad + 2e\vec{A})^2\Psi = 0}
|
||||
\end{formula}
|
||||
\begin{formula}{second}
|
||||
\desc{Second Ginzburg-Landau Equation}{Obtained by minimizing $g_\txs$ with respect to $\delta\vec{A}$ in \fRef{::expansion}}{}
|
||||
\desc[german]{Zweite Ginzburg-Landau Gleichung}{}{}
|
||||
\eq{\vec{j_\txs} = \frac{ie\hbar}{m}(\Psi^*\Grad\Psi - \Psi\Grad\Psi^*) - \frac{4e^2}{m}\abs{\Psi}^2 \vec{A}}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\begin{formula}{coherence_length}
|
||||
\desc{\GL Coherence Length}{Depth in the superconductor where $\abs{\Psi}$ goes from 0 to 1}{}
|
||||
\desc[german]{\GL Kohärenzlänge}{Tiefe im Supraleiter, bei der $\abs{\Psi}$ von 0 auf 1 steigt}{}
|
||||
\eq{
|
||||
\xi_\gl &= \frac{\hbar}{\sqrt{2m \abs{\alpha}}} \\
|
||||
\xi_\gl(T) &= \xi_\gl(0) \frac{1}{\sqrt{1-\frac{T}{\Tcrit}}}
|
||||
}
|
||||
\end{formula}
|
||||
\begin{formula}{penetration_depth}
|
||||
\desc{\GL Penetration Depth}{Field screening length\\Depth in the supercondcutor where $B_\text{ext}$ decays}{}
|
||||
\desc[german]{\GL Eindringtiefe}{Tiefe im Supraleiter, bei der $B_\text{ext}$ abfällt}{}
|
||||
\eq{
|
||||
\lambda_\gl &= \sqrt{\frac{m_\txs\beta}{\mu_0 \abs{\alpha} q_s^2}} \\
|
||||
\lambda_\gl(T) &= \lambda_\gl(0) \frac{1}{\sqrt{1-\frac{T}{\Tcrit}}}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{boundary_energy}
|
||||
\desc{Boundary energy}{Negative for \fRef{:::type2}, positive for \fRef{:::type1}}{$\Delta E_\text{B}$ energy gained by expelling the external magnetic field, $\Delta E_\text{cond}$ \fRef{:::condensation_energy}}
|
||||
\desc[german]{Grenzflächenenergie}{Negativ für \fRef{:::type2}, positiv für \fRef{:::type1}}{}
|
||||
\eq{\Delta E_\text{boundary} = \Delta E_\text{con} - \Delta E_\txB = (\xi_\gl - \lambda_\gl) \frac{B_\text{c,th}^2}{2\mu_0}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{parameter}
|
||||
\desc{Ginzburg-Landau parameter}{}{}
|
||||
\desc[german]{Ginzburg-Landau Parameter}{}{}
|
||||
\eq{\kappa \equiv \frac{\lambda_\gl}{\xi_\gl}}
|
||||
\eq{
|
||||
\kappa \le \frac{1}{\sqrt{2}} &\quad\Rightarrow\quad\text{\fRef{cm:super:type1}} \\
|
||||
\kappa \ge \frac{1}{\sqrt{2}} &\quad\Rightarrow\quad\text{\fRef{cm:super:type2}}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{ns_boundary}
|
||||
\desc{Normal-superconductor boundary}{}{}
|
||||
\desc[german]{Normal-Supraleiter Grenzfläche}{}{}
|
||||
\eq{
|
||||
\abs{\Psi(x)}^2 &= \frac{n_\txs(x)}{n_\txs(\infty)} = \tanh^2 \left(\frac{x}{\sqrt{2}\xi_\gl}\right) \\
|
||||
B_z(x) &= B_z(0) \Exp{-\frac{x}{\lambda_\gl}}
|
||||
}
|
||||
\fig{img/cm_super_n_s_boundary.pdf}
|
||||
% \TODO{plot, slide 106}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{bcth}
|
||||
\desc{Thermodynamic critical field}{}{}
|
||||
\desc[german]{Thermodynamisches kritisches Feld}{}{}
|
||||
\eq{\Bcth = \frac{\Phi_0}{2\pi \sqrt{2} \xi_\gl \lambda_\gl}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{bc1}
|
||||
\desc{Lower critical magnetic field}{Above $B_\text{c1}$, flux starts to penetrate the superconducting phase}{\ConstRef{flux_quantum}, $\lambda_\gl$ \fRef{::penetration_depth} $\kappa$ \fRef{::parameter}}
|
||||
\desc[german]{Unteres kritisches Magnetfeld}{Über $B_\text{c1}$ dringt erstmals Fluss in die supraleitende Phase ein}{}
|
||||
\eq{B_\text{c1} = \frac{\Phi_0}{4\pi\lambda_\gl^2}(\ln\kappa+0.08) = \frac{1}{\sqrt{2}\kappa}(\ln\kappa + 0.08) \Bcth}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{bc2}
|
||||
\desc{Upper critical magnetic field}{Above $B_\text{c2}$, superconducting phase is is destroyed}{\ConstRef{flux_quantum}, $\xi_\gl$ \fRef{::coherence_length}}
|
||||
\desc[german]{Oberes kritisches Magnetfeld}{Über $B_\text{c2}$ ist die supraleitende Phase zerstört}{}
|
||||
\eq{B_\text{c2} = \frac{\Phi_0}{2\pi\xi_\gl^2}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{proximity_effect}
|
||||
\desc{Proximity-Effect}{}{}
|
||||
% \desc[german]{}{}{}
|
||||
\ttxt{\eng{
|
||||
Superconductor wavefunction extends into the normal conductor or isolator
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
\Subsection[
|
||||
\eng{Microscopic theory}
|
||||
\ger{Mikroskopische Theorie}
|
||||
]{micro}
|
||||
\begin{formula}{isotop_effect}
|
||||
\desc{Isotope effect}{Superconducting behaviour depends on atomic mass and thereby on the lattice \Rightarrow Microscopic origin}{$\Tcrit$ critial temperature, $M$ isotope mass, $\omega_\text{ph}$}
|
||||
\desc[german]{Isotopeneffekt}{Supraleitung hängt von der Atommasse und daher von den Gittereigenschaften ab \Rightarrow Mikroskopischer Ursprung}{$\Tcrit$ kritische Temperatur, $M$ Isotopen-Masse, $\omega_\text{ph}$}
|
||||
\eq{
|
||||
\Tcrit &\propto \frac{1}{\sqrt{M}} \\
|
||||
\omega_\text{ph} &\propto \frac{1}{\sqrt{M}} \Rightarrow \Tcrit \propto \omega_\text{ph}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{cooper_pairs}
|
||||
\desc{Cooper pairs}{}{}
|
||||
\desc[german]{Cooper-Paars}{}{}
|
||||
\ttxt{
|
||||
\eng{Conduction electrons reduce their energy through an attractive interaction: One electron passing by atoms attracts the these, which creats a positive charge region behind the electron, which in turn attracts another electron. }
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\Subsubsection[
|
||||
\eng{BCS-Theory}
|
||||
\ger{BCS-Theorie}
|
||||
]{bcs}
|
||||
\begin{formula}{description}
|
||||
\desc{Description}{}{}
|
||||
\desc[german]{Beschreibung}{}{}
|
||||
\ttxt{\eng{
|
||||
\begin{itemize}
|
||||
\item Electron pairs form bosonic quasi-particles called Cooper pairs which can condensate into the ground state
|
||||
\item The wave function spans the whole material, which makes it conduct without resistance
|
||||
\item The exchange bosons between the electrons are phonons
|
||||
\end{itemize}
|
||||
}\ger{
|
||||
\begin{itemize}
|
||||
\item Elektronenpaar bilden bosonische Quasipartikel (Cooper Paare) welche in den Grundzustand kondensieren können.
|
||||
\item Die Wellenfunktion übersoannt den gesamten Festkörper, was einen widerstandslosen Ladungstransport garantiert
|
||||
\item Die Austauschbosononen zwischen den Elektronen sind Bosonen
|
||||
\end{itemize}
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
\def\BCS{{\text{BCS}}}
|
||||
\begin{formula}{hamiltonian}
|
||||
\desc{BCS Hamiltonian}{for $N$ interacting electrons}{
|
||||
$c_{\veck\sigma}$ creation/annihilation operators create/destroy at $\veck$ with spin $\sigma$ \\
|
||||
First term: non-interacting free electron gas\\
|
||||
Second term: interaction energy
|
||||
}
|
||||
\desc[german]{BCS Hamiltonian}{}{}
|
||||
\eq{
|
||||
\hat{H}_\BCS =
|
||||
\sum_{\sigma} \sum_\veck \epsilon_\veck \hat{c}_{\veck\sigma}^\dagger \hat{c}_{\veck\sigma}
|
||||
+ \sum_{\veck,\veck^\prime} V_{\veck,\veck^\prime}
|
||||
\hat{c}_{\veck\uparrow}^\dagger \hat{c}_{-\veck\downarrow}^\dagger
|
||||
\hat{c}_{-\veck^\prime\downarrow} \hat{c}_{\veck^\prime,\uparrow}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{ansatz}
|
||||
\desc{BCS ground state wave function Ansatz}{\fRef{comp:est:mean_field} approach\\Coherent fermionic state}{}
|
||||
\desc[german]{BCS Grundzustandswellenfunktion-Ansatz}{\fRef{comp:est:mean_field} Ansatz\\Kohärenter, fermionischer Zustand}{}
|
||||
\eq{\Ket{\Psi_\BCS} = \prod_{\veck=\veck_1,\dots,\veck_M} \left(u_\veck + v_\veck \hat{c}_{\veck\uparrow}^\dagger \hat{c}_{-\veck\downarrow}^\dagger\right) \ket{0} }
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{coherence_factors}
|
||||
\desc{BCS coherence factors}{}{$\abs{u_\veck}^2$/$\abs{v_\veck}^2$ probability that pair state is $(\veck\uparrow,\,-\veck\downarrow)$ is empty/occupied, $\abs{u_\veck}^2+\abs{v_\veck}^2 = 1$}
|
||||
\desc[german]{BCS Kohärenzfaktoren}{}{$\abs{u_\veck}^2$/$\abs{v_\veck}^2$ Wahrscheinlichkeit, dass Paarzustand $(\veck\uparrow,\,-\veck\downarrow)$ leer/besetzt ist, $\abs{u_\veck}^2+\abs{v_\veck}^2 = 1$}
|
||||
\eq{
|
||||
u_\veck &= \frac{1}{\sqrt{1+\abs{\alpha_\veck}^2}} \\
|
||||
v_\veck &= \frac{\alpha_\veck}{\sqrt{1+\abs{\alpha_\veck}^2}}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{potential}
|
||||
\desc{BCS potential approximation}{}{}
|
||||
\desc[german]{BCS Potentialnäherung}{}{}
|
||||
\eq{
|
||||
V_{\veck,\veck^\prime} =
|
||||
\left\{ \begin{array}{rc}
|
||||
-V_0 & k^\prime > k_\txF,\, k<k_\txF + \Delta k\\
|
||||
0 & \tGT{else}
|
||||
\end{array}\right.
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{gap_at_t0}
|
||||
\desc{BCS Gap at $T=0$}{}{\QtyRef{debye_frequency}, $V_0$ \fRef{::potential}, $D$ \qtyRef{dos}, \TODO{gamma}}
|
||||
\desc[german]{BCS Lücke bei $T=0$}{}{}
|
||||
\eq{
|
||||
\Delta(T=0) &= \frac{\hbar\omega_\txD}{\Sinh{\frac{2}{V_0\.D(E_\txF)}}} \approx 2\hbar \omega_\txD\\
|
||||
\frac{\Delta(T=0)}{\kB T_\txc} &= \frac{\pi}{\e^\gamma} = 1.764
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\begin{formula}{cooper_pair_binding_energy}
|
||||
\desc{Binding energy of Cooper pairs}{}{$E_\txF$ \absRef{fermi_energy}, \QtyRef{debye_frequency}, $V_0$ retarded potential, $D$ \qtyRef{dos}}
|
||||
\desc[german]{Bindungsenergie von Cooper-Paaren}{}{}
|
||||
\eq{E \approx 2E_\txF - 2\hbar\omega_\txD \Exp{-\frac{4}{V_0 D(E_\txF)}}}
|
||||
\end{formula}
|
||||
|
||||
\Subsubsection[
|
||||
\eng{Excitations and finite temperatures}
|
||||
\ger{Anregungen und endliche Temperatur}
|
||||
]{excite}
|
||||
\begin{formula}{description}
|
||||
\desc{Description}{}{}
|
||||
\desc[german]{Beschreibung}{}{}
|
||||
\ttxt{\eng{
|
||||
The ground state consists of \fRef{cm:super:micro:cooper_pairs} and the excited state of Bogoliubov quasi-particles (electron-hole pairs).
|
||||
The states are separated by an energy gap $\Delta$.
|
||||
}\ger{
|
||||
Den Grundzustand bilden \fRef{cm:super:micro:cooper_pairs} und den angeregten Zustands Bogoloiubons (Elektron-Loch Quasipartikel).
|
||||
Die Zustände sind durch eine Energielücke $\Delta$ getrennt.
|
||||
}}
|
||||
\end{formula}
|
||||
\begin{formula}{bogoliubov-valatin}
|
||||
\desc{Bogoliubov-Valatin transformation}{Diagonalization of the \fRef{cm:super:micro:bcs:hamiltonian} to derive excitation energies}{
|
||||
$\xi_\veck = \epsilon_\veck-\mu$ Energy relative to the \qtyRef{chemical_potential},
|
||||
\\ $E_\veck$ \fRef{::excitation_energy},
|
||||
\\ $\Delta$ Gap
|
||||
\\ $g_\veck$ \fRef{::pairing_amplitude},
|
||||
\\ $\alpha / \beta$ create and destroy symmetric/antisymmetric Bogoliubov quasiparticles
|
||||
}
|
||||
\desc[german]{Bogoliubov-Valatin transformation}{}{}
|
||||
\eq{
|
||||
\hat{H}_\BCS - N\mu = \sum_\veck \big[\xi_\veck - E_\veck + \Delta_\veck g_\veck^*\big] + \sum_\veck \big[E_\veck \alpha_\veck^\dagger \alpha_\veck + E_\veck \beta_{-\veck}^\dagger \beta_{-\veck}\big]
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{pairing_amplitude}
|
||||
\desc{Pairing amplitude}{}{}
|
||||
\desc[german]{Paarungsamplitude}{}{}
|
||||
\eq{g_\veck \equiv \Braket{\hat{c}_{-\veck\downarrow} \hat{c}_{\veck\uparrow}}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{excitation_energy}
|
||||
\desc{Excitation energy}{}{}
|
||||
\desc[german]{Anregungsenergie}{}{}
|
||||
\eq{E_\veck = \pm \sqrt{\xi^2_\veck + \abs{\Delta_\veck}^2}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{coherence_factors_energy}
|
||||
\desc{Energy dependance of the \fRef{:::bcs:coherence_factors}}{}{$E_\veck$ \fRef{::pairing_amplitude}, \GT{see} \fRef{:::bcs:coherence_factors}}
|
||||
\desc[german]{Energieabhängigkeit der \fRef{:::bcs:coherence_factors}}{}{}
|
||||
\eq{
|
||||
\abs{u_\veck}^2 &= \frac{1}{2} \left(1+\frac{\xi_\veck}{E_\veck}\right) \\
|
||||
\abs{v_\veck}^2 &= \frac{1}{2} \left(1-\frac{\xi_\veck}{E_\veck}\right) \\
|
||||
u_\veck^* v_\veck &= \frac{\Delta_\veck}{2E_\veck}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\begin{formula}{gap_equation}
|
||||
\desc{Self-consistend gap equation}{}{}
|
||||
\desc[german]{Selbstkonsitente Energielückengleichung}{}{}
|
||||
\eq{\Delta_\veck^* = -\sum_{\veck^\prime} V_{\veck,\veck^\prime} \frac{\Delta_{\veck^\prime}}{2E_\veck} \tanh \left(\frac{E_{\veck^\prime}}{2\kB T}\right)}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\begin{formula}{gap_t}
|
||||
\desc{Temperature dependence of the BCS gap}{}{}
|
||||
\desc[german]{Temperaturabhängigkeit der BCS-Lücke}{}{}
|
||||
\eq{\frac{\Delta(T)}{\Delta(T=0)} \approx 1.74 \sqrt{1-\frac{T}{T_\txC}}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{dos}
|
||||
\desc{Quasiparticle density of states}{}{}
|
||||
\desc[german]{Quasiteilchen Zustandsdichte}{}{}
|
||||
\eq{D_\txs(E_\veck) = D_\txn(\xi_\veck) \pdv{\xi_\veck}{E_\veck} = \left\{
|
||||
\begin{array}{ll}
|
||||
D_\txn(E_\txF) \frac{E_\veck}{\sqrt{E^2_\veck -\Delta^2}} & E_\veck > \Delta \\
|
||||
& E_\veck < \Delta
|
||||
\end{array}
|
||||
\right.}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{Bcth_temp}
|
||||
\desc{Temperature dependance of the crictial magnetic field}{Jump at $T_\txc$, then exponential decay}{}
|
||||
\desc[german]{Temperaturabhängigkeit des kritischen Magnetfelds}{Sprung bei $T_\txc$, denn exponentieller Abfall}{}
|
||||
\eq{ \Bcth(T) = \Bcth(0) \left[1- \left(\frac{T}{T_\txc}\right)^2 \right] }
|
||||
% \TODO{empirical relation, relate to BCS}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{heat_capacity}
|
||||
\desc{Heat capacity in superconductors}{}{}
|
||||
\desc[german]{Wärmekapazität in Supraleitern}{}{}
|
||||
\fsplit{
|
||||
\fig{img/cm_super_heat_capacity.pdf}
|
||||
}{
|
||||
\eq{c_\txs \propto T^{-\frac{3}{2}} \e^{\frac{\Delta(0)}{\kB T}}}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\Subsubsection[
|
||||
\eng{Flux pinning}
|
||||
\ger{Haftung von Flusslinien}
|
||||
]{pinning}
|
||||
\begin{formula}{description}
|
||||
\desc{Description}{}{}
|
||||
\desc[german]{Beschreibung}{}{}
|
||||
\ttxt{\eng{
|
||||
If a current flows in a \fRef{cm:super:type2}s in the \fRef{cm:super:shubnikov_phase} perpendicular to the penetrating flux lines,
|
||||
the lines experience a Lorentz force. This leads to ohmic behaviour of the superconductor.
|
||||
The flux lines can be pinned to defects, in which the superconducting order parameter is reduced.
|
||||
To move the flux line out of the defect, work would have to be spent overcoming the \fRef{cm:super:micro:pinning:potential}.
|
||||
This restores the superconductivity.
|
||||
}\ger{
|
||||
Wenn ein Strom in einem \fRef{cm:super:type2}s in der \fRef{cm:super:shubnikov_phase} senkrecht zu den eindringenden Flusslinien fließt, erfahren die Linien eine Lorentzkraft.
|
||||
Dies führt zu einem ohmschen Verhalten des Supraleiters.
|
||||
Die Flusslinien können an Defekten festgehalten werden, in denen der supraleitende Ordnungsparameter reduziert ist.
|
||||
Um die Flusslinie aus dem Defekt zu bewegen, müsste Arbeit aufgewendet werden, um das \fRef{cm:super:micro:pinning:potential} zu überwinden.
|
||||
Dies stellt die Supraleitfähigkeit wieder her.
|
||||
}}
|
||||
\end{formula}
|
||||
@ -21,12 +21,12 @@
|
||||
\desc[german]{Raman-Spektroskopie}{}{}
|
||||
\begin{minipagetable}{raman}
|
||||
\tentry{application}{
|
||||
\eng{Vibrational modes, Crystal structure, Doping, Band Gaps, Layer thickness in \fqEqRef{cm:misc:vdw_material}}
|
||||
\ger{Vibrationsmoden, Kristallstruktur, Dotierung, Bandlücke, Schichtdicke im \fqEqRef{cm:misc:vdw_material}}
|
||||
\eng{Vibrational modes, Crystal structure, Doping, Band Gaps, Layer thickness in \fRef{cm:misc:vdw_material}}
|
||||
\ger{Vibrationsmoden, Kristallstruktur, Dotierung, Bandlücke, Schichtdicke im \fRef{cm:misc:vdw_material}}
|
||||
}
|
||||
\tentry{how}{
|
||||
\eng{Monochromatic light (\fqEqRef{Laser}) shines on sample, inelastic scattering because of rotation-, vibration-, phonon and spinflip-processes, plot spectrum as shift of the laser light (in \si{\per\cm})}
|
||||
\ger{Monochromatisches Licht (\fqEqRef{Laser}) bestrahlt Probe, inelastische Streuung durch Rotations-, Schwingungs-, Phonon und Spin-Flip-Prozesse, plotte Spektrum als Verschiebung gegen das Laser Licht (in \si{\per\cm}) }
|
||||
\eng{Monochromatic light (\fRef{Laser}) shines on sample, inelastic scattering because of rotation-, vibration-, phonon and spinflip-processes, plot spectrum as shift of the laser light (in \si{\per\cm})}
|
||||
\ger{Monochromatisches Licht (\fRef{Laser}) bestrahlt Probe, inelastische Streuung durch Rotations-, Schwingungs-, Phonon und Spin-Flip-Prozesse, plotte Spektrum als Verschiebung gegen das Laser Licht (in \si{\per\cm}) }
|
||||
}
|
||||
\end{minipagetable}
|
||||
\begin{minipage}{0.45\textwidth}
|
||||
@ -44,18 +44,18 @@
|
||||
\desc[german]{Photolumeszenz-Spektroskopie}{}{}
|
||||
\begin{minipagetable}{pl}
|
||||
\tentry{application}{
|
||||
\eng{Crystal structure, Doping, Band Gaps, Layer thickness in \fqEqRef{cm:misc:vdw_material}}
|
||||
\ger{Kristallstruktur, Dotierung, Bandlücke, Schichtdicke im \fqEqRef{cm:misc:vdw_material}}
|
||||
\eng{Crystal structure, Doping, Band Gaps, Layer thickness in \fRef{cm:misc:vdw_material}}
|
||||
\ger{Kristallstruktur, Dotierung, Bandlücke, Schichtdicke im \fRef{cm:misc:vdw_material}}
|
||||
}
|
||||
\tentry{how}{
|
||||
\eng{Monochromatic light (\fqEqRef{Laser}) shines on sample, electrons are excited, relax to the conduction band minimum and finally accross the band gap under photon emission}
|
||||
\ger{Monochromatisches Licht (\fqEqRef{Laser}) bestrahlt Probe, Elektronen werden angeregt und relaxieren in das Leitungsband-Minimum und schließlich über die Bandlücke unter Photonemission}
|
||||
\eng{Monochromatic light (\fRef{Laser}) shines on sample, electrons are excited, relax to the conduction band minimum and finally accross the band gap under photon emission}
|
||||
\ger{Monochromatisches Licht (\fRef{Laser}) bestrahlt Probe, Elektronen werden angeregt und relaxieren in das Leitungsband-Minimum und schließlich über die Bandlücke unter Photonemission}
|
||||
}
|
||||
\end{minipagetable}
|
||||
\begin{minipage}{0.45\textwidth}
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
% \includegraphics[width=0.8\textwidth]{img/cm_amf.pdf}
|
||||
% \includegraphics[width=0.8\textwidth]{img_static/cm_amf.pdf}
|
||||
% \caption{\cite{Bian2021}}
|
||||
\end{figure}
|
||||
\end{minipage}
|
||||
@ -97,7 +97,7 @@
|
||||
\begin{minipage}{0.45\textwidth}
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\includegraphics[width=0.8\textwidth]{img/cm_amf.pdf}
|
||||
\includegraphics[width=0.8\textwidth]{img_static/cm_amf.pdf}
|
||||
\caption{\cite{Bian2021}}
|
||||
\end{figure}
|
||||
\end{minipage}
|
||||
@ -122,7 +122,7 @@
|
||||
\begin{minipage}{0.45\textwidth}
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\includegraphics[width=0.8\textwidth]{img/cm_stm.pdf}
|
||||
\includegraphics[width=0.8\textwidth]{img_static/cm_stm.pdf}
|
||||
\caption{\cite{Bian2021}}
|
||||
\end{figure}
|
||||
\end{minipage}
|
||||
@ -168,7 +168,7 @@
|
||||
\end{minipagetable}
|
||||
\begin{minipage}{0.45\textwidth}
|
||||
\centering
|
||||
\includegraphics[width=\textwidth]{img/cm_cvd_english.pdf}
|
||||
\includegraphics[width=\textwidth]{img_static/cm_cvd_english.pdf}
|
||||
\end{minipage}
|
||||
\end{bigformula}
|
||||
|
||||
|
||||
102
src/cm/vib.tex
Normal file
102
src/cm/vib.tex
Normal file
@ -0,0 +1,102 @@
|
||||
\Section[
|
||||
\eng{Lattice vibrations}
|
||||
\ger{Gitterschwingungen}
|
||||
]{vib}
|
||||
|
||||
\begin{formula}{dispersion_1atom_basis}
|
||||
\desc{Phonon dispersion of a lattice with a one-atom basis}{same as the dispersion of a linear chain}{$C_n$ force constants between layer $s$ and $s+n$, $M$ \qtyRef{mass} of the reference atom, $a$ \qtyRef{lattice_constant}, $q$ phonon \qtyRef{wavevector}, $u$ Ansatz for the atom displacement}
|
||||
\desc[german]{Phonondispersion eines Gitters mit zweiatomiger Basis}{gleich der Dispersion einer linearen Kette}{$C_n$ Kraftkonstanten zwischen Ebene $s$ und $s+n$, $M$ \qtyRef{mass} des Referenzatoms, $a$ \qtyRef{lattice_constant}, $q$ Phonon \qtyRef{wavevector}, $u$ Ansatz für die Atomauslenkung}
|
||||
\begin{gather}
|
||||
\omega^2 = \frac{4C_1}{M}\left[\sin^2 \left(\frac{qa}{2}\right) + \frac{C2}{C1} \sin^2(qa)\right] \\
|
||||
\intertext{\GT{with}}
|
||||
u_{s+n} = U\e^{-i \left[\omega t - q(s+n)a \right]}
|
||||
\end{gather}
|
||||
\newFormulaEntry
|
||||
\fig{img/cm_vib_dispersion_one_atom_basis.pdf}
|
||||
\end{formula}
|
||||
\begin{formula}{dispersion_2atom_basis}
|
||||
\desc{Phonon dispersion of a lattice with a two-atom basis}{}{$C$ force constant between layers, $M_i$ \qtyRef{mass} of the basis atoms, $a$ \qtyRef{lattice_constant}, $q$ phonon \qtyRef{wavevector}, $u, v$ Ansatz for the displacement of basis atom 1 and 2, respectively}
|
||||
\desc[german]{Phonondispersion eines Gitters mit einatomiger Basis}{}{$C$ Kraftkonstanten zwischen Ebene $s$ und $s+n$, $M_i$ \qtyRef{mass} der Basisatome, $a$ \qtyRef{lattice_constant}, $q$ Phonon \qtyRef{wavevector}, $u, v$ jeweils Ansatz für die Atomauslenkung des Basisatoms 1 und 2}
|
||||
\begin{gather}
|
||||
\omega^2_{\txa,\txo} = C \left(\frac{1}{M_1}+\frac{1}{M_2}\right) \mp C \sqrt{\left(\frac{1}{M_1}+\frac{1}{M_2}\right)^2 - \frac{4}{M_1M_2} \sin^2 \left(\frac{qa}{2}\right)}
|
||||
\intertext{\GT{with}}
|
||||
u_{s} = U\e^{-i \left(\omega t - qsa \right)}, \quad
|
||||
v_{s} = V\e^{-i \left(\omega t - qsa \right)}
|
||||
\end{gather}
|
||||
\newFormulaEntry
|
||||
\fig{img/cm_vib_dispersion_two_atom_basis.pdf}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{branches}
|
||||
\desc{Vibration branches}{}{}
|
||||
\desc[german]{Vibrationsmoden}{}{}
|
||||
\ttxt{\eng{
|
||||
\textbf{Acoustic}: 3 modes (1 longitudinal, 2 transversal), the two basis atoms oscillate in phase.
|
||||
\\\textbf{Optical}: 3 modes, the two basis atoms oscillate in opposition. A dipole moment is created that can couple to photons.
|
||||
}\ger{
|
||||
\textbf{Akustisch}: 3 Moden (1 longitudinal, 2 transversal), die zwei Basisatome schwingen in Phase.
|
||||
\\ \textbf{Optisch}: 3 Moden, die zwei Basisatome schwingen gegenphasig. Das dadurch entstehende Dipolmoment erlaubt die Wechselwirkung mit Photonen.
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
\Subsection[
|
||||
\eng{Einstein model}
|
||||
\ger{Einstein-Modell}
|
||||
]{einstein}
|
||||
\begin{formula}{description}
|
||||
\desc{Description}{}{}
|
||||
\desc[german]{Beschreibung}{}{}
|
||||
\ttxt{\eng{
|
||||
All lattice vibrations have the \fRef[same frequency]{:::frequency}.
|
||||
Underestimates the \fRef{:::heat_capacity} for low temperatures.
|
||||
}\ger{
|
||||
Alle Gittereigenschwingungen haben die \fRef[selbe Frequenz]{:::frequency}
|
||||
Sagt zu kleine \fRef[Wärmekapazitäten]{:::heat_capacity} für tiefe Temperaturen voraus.
|
||||
}}
|
||||
\end{formula}
|
||||
\begin{formula}{frequency}
|
||||
\desc{Einstein frequency}{}{}
|
||||
\desc[german]{Einstein-Frequenz}{}{}
|
||||
\eq{\omega_\txE}
|
||||
\end{formula}
|
||||
\begin{formula}{heat_capacity}
|
||||
\desc{\qtyRef{heat_capacity}}{according to the Einstein model}{}
|
||||
\desc[german]{}{nach dem Einstein-Modell}{}
|
||||
\eq{C_V^\txE = 3N\kB \left( \frac{\hbar\omega_\txE}{\kB T}\right)^2 \frac{\e^{\frac{\hbar\omega_\txE}{\kB T}}}{ \left(\e^{\frac{\hbar\omega_\txE}{\kB T}} - 1\right)^2}}
|
||||
\end{formula}
|
||||
|
||||
\Subsection[
|
||||
\eng{Debye model}
|
||||
\ger{Debye-Modell}
|
||||
]{debye}
|
||||
\begin{formula}{description}
|
||||
\desc{Description}{}{}
|
||||
\desc[german]{Beschreibung}{}{}
|
||||
\ttxt{\eng{
|
||||
Atoms behave like coupled \fRef[quantum harmonic oscillators]{sec:qm:hosc}. The finite sample size leads to periodic boundary conditio. The finite sample size leads to periodic boundary conditions for the vibrations.
|
||||
}\ger{
|
||||
Atome verhalten sich wie gekoppelte \fRef[quantenmechanische harmonische Oszillatoren]{sec:qm:hosc}. Die endliche Ausdehnung des Körpers führt zu periodischen Randbedingungen.
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{phonon_dos}
|
||||
\desc{Phonon density of states}{}{\QtyRef{volume}, $v$ \qtyRef{speed_of_sound} of the phonon mode, $\omega$ phonon frequency}
|
||||
\desc[german]{Phononenzustandsdichte}{}{\QtyRef{volume}, $v$ \qtyRef{speed_of_sound} des Dispersionszweigs, $\omega$ Phononfrequenz}
|
||||
\eq{D(\omega) \d \omega = \frac{V}{2\pi^2} \frac{\omega^2}{v^3} \d\omega}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{debye_frequency}
|
||||
\desc{Debye frequency}{Maximum phonon frequency}{$v$ \qtyRef{speed_of_sound}, $N/V$ atom density}
|
||||
\desc[german]{Debye-Frequenz}{Maximale Phononenfrequenz}{$v$ \qtyRef{speed_of_sound}, $N/V$ Atomdichte}
|
||||
\eq{\omega_\txD = v \left(6\pi^2 \frac{N}{V}\right)^{1/3}}
|
||||
\hiddenQuantity{\omega_\txD}{\per\s}{s}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{heat_capacity}
|
||||
\desc{\qtyRef{heat_capacity}}{according to the Debye model}{$N$ number of atoms, \ConstRef{boltzmann}, \QtyRef{debye_frequency}}
|
||||
\desc[german]{}{nach dem Debye-Modell}{$N$ Anzahl der Atome, \ConstRef{boltzmann}, \QtyRef{debye_frequency}}
|
||||
\eq{C_V^\txD = 9N\kB \left(\frac{\kB T}{\hbar \omega_\txD}\right)^3 \int_0^{\frac{\hbar\omega_\txD}{\kB T}} \d x \frac{x^4 \e^x}{(\e^x-1)^2} }
|
||||
\end{formula}
|
||||
|
||||
|
||||
|
||||
@ -3,12 +3,12 @@
|
||||
% \ger{}
|
||||
]{ad}
|
||||
\begin{formula}{hamiltonian}
|
||||
\desc{Electron Hamiltonian}{}{$\hat{T}$ \fqEqRef{comp:est:kinetic_energy}, $\hat{V}$ \fqEqRef{comp:est:potential_energy}, $\txe$ \GT{electrons}, $\txn$ \GT{nucleons}}
|
||||
\desc{Electron Hamiltonian}{}{$\hat{T}$ \fRef{comp:est:kinetic_energy}, $\hat{V}$ \fRef{comp:est:potential_energy}, $\txe$ \GT{electrons}, $\txn$ \GT{nucleons}}
|
||||
\desc[german]{Hamiltonian der Elektronen}{}{}
|
||||
\eq{\hat{H}_\txe = \hat{T}_\txe + V_{\txe \leftrightarrow \txe} + V_{\txn \leftrightarrow \txe}}
|
||||
\end{formula}
|
||||
\begin{formula}{ansatz}
|
||||
\desc{Wave function ansatz}{}{$\psi_\text{en}^n$ eigenstate $n$ of \fqEqRef{comp:est:hamiltonian}, $\psi_\txe^i$ eigenstate $i$ of \fqEqRef{comp:ad:bo:hamiltonian}, $\vecr,\vecR$ electron/nucleus positions, $\sigma$ electron spin, $c^{ni}$ coefficients}
|
||||
\desc{Wave function ansatz}{}{$\psi_\text{en}^n$ eigenstate $n$ of \fRef{comp:est:hamiltonian}, $\psi_\txe^i$ eigenstate $i$ of \fRef{comp:ad:hamiltonian}, $\vecr,\vecR$ electron/nucleus positions, $\sigma$ electron spin, $c^{ni}$ coefficients}
|
||||
\desc[german]{Wellenfunktion Ansatz}{}{}
|
||||
\eq{\psi_\text{en}^n\big(\{\vecr,\sigma\},\{\vecR\}\big) = \sum_i c^{ni}\big(\{\vecR\}\big)\, \psi_\txe^i\big(\{\vecr,\sigma\},\{\vecR\}\big)}
|
||||
\end{formula}
|
||||
@ -34,30 +34,30 @@
|
||||
\ger{Born-Oppenheimer Näherung}
|
||||
]{bo}
|
||||
\begin{formula}{adiabatic_approx}
|
||||
\desc{Adiabatic approximation}{Electronic configuration remains the same when atoms move (\absRef{adiabatic_theorem})}{$\Lambda_{ij}$ \fqEqRef{comp:ad:coupling_operator}}
|
||||
\desc{Adiabatic approximation}{Electronic configuration remains the same when atoms move (\absRef{adiabatic_theorem})}{$\Lambda_{ij}$ \fRef{comp:ad:coupling_operator}}
|
||||
\desc[german]{Adiabatische Näherung}{Elektronenkonfiguration bleibt gleich bei Bewegung der Atome gleichl (\absRef{adiabatic_theorem})}{}
|
||||
\eq{\Lambda_{ij} = 0 \quad \text{\GT{for} } i\neq j}
|
||||
\end{formula}
|
||||
\begin{formula}{approx}
|
||||
\desc{Born-Oppenheimer approximation}{Electrons are not influenced by the movement of the atoms}{\GT{see} \fqEqRef{comp:ad:equation}, $V_{\txn \leftrightarrow \txn} = \const$ absorbed into $E_\txe^j$}
|
||||
\desc{Born-Oppenheimer approximation}{Electrons are not influenced by the movement of the atoms}{\GT{see} \fRef{comp:ad:equation}, $V_{\txn \leftrightarrow \txn} = \const$ absorbed into $E_\txe^j$}
|
||||
\desc[german]{Born-Oppenheimer Näherung}{Elektronen werden nicht durch die Bewegung der Atome beeinflusst}{}
|
||||
\begin{gather}
|
||||
\Lambda_{ij} = 0
|
||||
\shortintertext{\fqEqRef{comp:ad:bo:equation} \Rightarrow}
|
||||
% \shortintertext{\fRef{comp:ad:bo:equation} \Rightarrow}
|
||||
\left[E_e^i\big(\{\vecR\}\big) + \hat{T}_\txn - E^n\right]c^{ni}\big(\{\vecR\}\big) = 0
|
||||
\end{gather}
|
||||
\end{formula}
|
||||
\begin{formula}{surface}
|
||||
\desc{Born-Oppenheimer surface}{Potential energy surface (PES)\\ The nuclei follow Newtons equations of motion on the BO surface if the system is in the electronic ground state}{$E_\txe^0, \psi_\txe^0$ lowest eigenvalue/eigenstate of \fqEqRef{comp:ad:bo:hamiltonian}}
|
||||
\desc[german]{Born-Oppenheimer Potentialhyperfläche}{Die Nukleonen Newtons klassichen Bewegungsgleichungen auf der BO Hyperfläche wenn das System im elektronischen Grundzustand ist}{$E_\txe^0, \psi_\txe^0$ niedrigster Eigenwert/Eigenzustand vom \fqEqRef{comp:ad:bo:hamiltonian}}
|
||||
\desc{Born-Oppenheimer surface}{Potential energy surface (PES)\\ The nuclei follow Newtons equations of motion on the BO surface if the system is in the electronic ground state}{$E_\txe^0, \psi_\txe^0$ lowest eigenvalue/eigenstate of \fRef{comp:ad:hamiltonian}}
|
||||
\desc[german]{Born-Oppenheimer Potentialhyperfläche}{Die Nukleonen Newtons klassichen Bewegungsgleichungen auf der BO Hyperfläche wenn das System im elektronischen Grundzustand ist}{$E_\txe^0, \psi_\txe^0$ niedrigster Eigenwert/Eigenzustand vom \fRef{comp:ad:hamiltonian}}
|
||||
\begin{gather}
|
||||
V_\text{BO}\big(\{\vecR\}\big) = E_\txe^0\big(\{\vecR\}\big) \\
|
||||
M_I \ddot{\vecR}_I(t) = - \Grad_{\vecR_I} V_\text{BO}\big(\{\vecR(t)\}\big)
|
||||
\end{gather}
|
||||
\end{formula}
|
||||
\begin{formula}{ansatz}
|
||||
\desc{Ansatz for \secEqRef{approx}}{Product of single electronic and single nuclear state}{}
|
||||
\desc[german]{Ansatz für \secEqRef{approx}}{Produkt aus einem einzelnen elektronischen Zustand und einem Nukleus-Zustand}{}
|
||||
\desc{Ansatz for \fRef{::approx}}{Product of single electronic and single nuclear state}{}
|
||||
\desc[german]{Ansatz für \fRef{::approx}}{Produkt aus einem einzelnen elektronischen Zustand und einem Nukleus-Zustand}{}
|
||||
\eq{
|
||||
\psi_\text{BO} = c^{n0} \big(\{\vecR\}\big) \,\psi_\txe^0 \big(\{\vecr,\sigma\},\{\vecR\}\big)
|
||||
}
|
||||
@ -88,10 +88,14 @@
|
||||
\begin{formula}{forces}
|
||||
\desc{Forces}{}{}
|
||||
\desc[german]{Kräfte}{}{}
|
||||
\eq{\vec{F}_I = -\Grad_{\vecR_I} E \explOverEq{\fqEqRef{qm:se:hellmann_feynmann}} -\Braket{\psi(\vecR_I) | \left(\Grad_{\vecR_I} \hat{H}(\vecR_I)\right) | \psi(\vecR) }}
|
||||
\eq{
|
||||
\vec{F}_I = -\Grad_{\vecR_I} E
|
||||
\explOverEq{\fRef{qm:se:hellmann_feynmann}}
|
||||
-\Braket{\psi(\vecR_I) | \left(\Grad_{\vecR_I} \hat{H}(\vecR_I)\right) | \psi(\vecR)}
|
||||
}
|
||||
\end{formula}
|
||||
\begin{formula}{ionic_cycle}
|
||||
\desc{Ionic cycle}{\fqEqRef{comp:est:dft:ks:scf} for geometry optimization}{}
|
||||
\desc{Ionic cycle}{\fRef{comp:est:dft:ks:scf} for geometry optimization}{}
|
||||
\desc[german]{}{}{}
|
||||
\ttxt{
|
||||
\eng{
|
||||
@ -99,11 +103,11 @@
|
||||
\item Initial guess for $n(\vecr)$
|
||||
\begin{enumerate}
|
||||
\item Calculate effective potential $V_\text{eff}$
|
||||
\item Solve \fqEqRef{comp:est:dft:ks:equation}
|
||||
\item Solve \fRef{comp:est:dft:ks:equation}
|
||||
\item Calculate density $n(\vecr)$
|
||||
\item Repeat b-d until self consistent
|
||||
\end{enumerate}
|
||||
\item Calculate \secEqRef{forces}
|
||||
\item Calculate \fRef{:::forces}
|
||||
\item If $F\neq0$, get new geometry by interpolating $R$ and restart
|
||||
\end{enumerate}
|
||||
}
|
||||
@ -146,8 +150,8 @@
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{harmonic_approx}
|
||||
\desc{Harmonic approximation}{Hessian matrix, 2nd order Taylor expansion of the \fqEqRef{comp:ad:bo:surface} around every nucleus position $\vecR_I^0$}{$\Phi_{IJ}^{\mu\nu}$ \secEqRef{force_constant_matrix}, $s$ displacement}
|
||||
\desc[german]{Harmonische Näherung}{Hesse matrix, Taylor Entwicklung der \fqEqRef{comp:ad:bo:surface} in zweiter Oddnung um Atomposition $\vecR_I^0$}{}
|
||||
\desc{Harmonic approximation}{Hessian matrix, 2nd order Taylor expansion of the \fRef{comp:ad:bo:surface} around every nucleus position $\vecR_I^0$}{$\Phi_{IJ}^{\mu\nu}$ \fRef{::force_constant_matrix}, $s$ displacement}
|
||||
\desc[german]{Harmonische Näherung}{Hesse matrix, Taylor Entwicklung der \fRef{comp:ad:bo:surface} in zweiter Oddnung um Atomposition $\vecR_I^0$}{}
|
||||
\eq{ V^\text{BO}(\{\vecR_I\}) \approx V^\text{BO}(\{\vecR_I^0\}) + \frac{1}{2} \sum_{I,J}^N \sum_{\mu,\nu}^3 s_I^\mu s_J^\nu \Phi_{IJ}^{\mu\nu} }
|
||||
\end{formula}
|
||||
|
||||
@ -166,13 +170,13 @@
|
||||
\eq{\Phi_{IJ}^{\mu\nu} \approx \frac{\vecF_I^\mu(\vecR_1^0, \dots, \vecR_J^0+\Delta s_J^\nu,\dots, \vecR_N^0)}{\Delta s_J^\nu}}
|
||||
\end{formula}
|
||||
\begin{formula}{dynamical_matrix}
|
||||
\desc{Dynamical matrix}{Mass reduced \absRef[fourier transform]{fourier_transform} of the \fqEqRef{comp:ad:latvib:force_constant_matrix}}{$\vec{L}$ vector from origin to unit cell $n$, $\alpha/\beta$ atom index in th unit cell, $\vecq$ \qtyRef{wave_vector}, $\Phi$ \fqEqRef{comp:ad:latvib:force_constant_matrix}, $M$ \qtyRef{mass}}
|
||||
\desc{Dynamical matrix}{Mass reduced \absRef[fourier transform]{fourier_transform} of the \fRef{comp:ad:latvib:force_constant_matrix}}{$\vec{L}$ vector from origin to unit cell $n$, $\alpha/\beta$ atom index in th unit cell, $\vecq$ \qtyRef{wavevector}, $\Phi$ \fRef{comp:ad:latvib:force_constant_matrix}, $M$ \qtyRef{mass}}
|
||||
% \desc[german]{}{}{}
|
||||
\eq{D_{\alpha\beta}^{\mu\nu} = \frac{1}{\sqrt{M_\alpha M_\beta}} \sum_{n^\prime} \Phi_{\alpha\beta}^{\mu\nu}(n-n^\prime) \e^{\I \vec{q}(\vec{L}_n - \vec{L}_{n^\prime})}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{eigenvalue_equation}
|
||||
\desc{Eigenvalue equation}{For a periodic crystal, reduces number of equations from $3N_p\times N$ to $3N_p$. Eigenvalues represent phonon band structure.}{$N_p$ number of atoms per unit cell, $\vecc$ displacement amplitudes, $\vecq$ \qtyRef{wave_vector}, $\mat{D}$ \secEqRef{dynamical_matrix}}
|
||||
\desc{Eigenvalue equation}{For a periodic crystal, reduces number of equations from $3N_p\times N$ to $3N_p$. Eigenvalues represent phonon band structure.}{$N_p$ number of atoms per unit cell, $\vecc$ displacement amplitudes, $\vecq$ \qtyRef{wave_vector}, $\mat{D}$ \fRef{::dynamical_matrix}}
|
||||
\desc[german]{Eigenwertgleichung}{}{}
|
||||
\eq{\omega^2 \vecc(\vecq) = \mat{D}(\vecq) \vecc(\vecq) }
|
||||
\end{formula}
|
||||
@ -186,7 +190,7 @@
|
||||
\desc{Quasi-harmonic approximation}{}{}
|
||||
\desc[german]{}{}{}
|
||||
\ttxt{\eng{
|
||||
Include thermal expansion by assuming \fqEqRef{comp:ad:bo:surface} is volume dependant.
|
||||
Include thermal expansion by assuming \fRef{comp:ad:bo:surface} is volume dependant.
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
@ -194,7 +198,7 @@
|
||||
\desc{Pertubative approaches}{}{}
|
||||
% \desc[german]{Störungs}{}{}
|
||||
\ttxt{\eng{
|
||||
Expand \fqEqRef{comp:ad:latvib:force_constant_matrix} to third order.
|
||||
Expand \fRef{comp:ad:latvib:force_constant_matrix} to third order.
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
@ -210,14 +214,13 @@
|
||||
\desc[german]{Beschreibung}{}{}
|
||||
\ttxt{\eng{
|
||||
\begin{itemize}
|
||||
\item Exact (within previous approximations) approach to treat anharmonic effects in materials.
|
||||
\item Computes time-dependant observables.
|
||||
\item Assumes fully classical nuclei.
|
||||
\item Assumes fully classical nuclei
|
||||
\item Macroscropical observables from statistical ensembles
|
||||
\item System evolves in time (ehrenfest). Number of points to consider does NOT scale with system size.
|
||||
\item Exact because time dependance is studied explicitly, not via harmonic approx.
|
||||
\item Number of points to consider does NOT scale with system size
|
||||
\item System evolves in time (\absRef{ehrenfest_theorem})
|
||||
\item Computes time-dependant observables
|
||||
\item Does not use \fRef{comp:ad:latvib:harmonic_approx} \Rightarrow Anharmonic effects included
|
||||
\end{itemize}
|
||||
\TODO{cleanup}
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
@ -244,7 +247,7 @@
|
||||
\ttxt{\eng{
|
||||
\begin{enumerate}
|
||||
\item Calculate electronic ground state of current nucleui configuration $\{\vecR(t)\}$ with \abbrRef{ksdft}
|
||||
\item \hyperref[f:comp:ad:opt:forces]{Calculate forces} from the \fqEqRef{comp:ad:bo:surface}
|
||||
\item \fRef[Calculate forces]{comp:ad:opt:forces} from the \fRef{comp:ad:bo:surface}
|
||||
\item Update positions and velocities
|
||||
\end{enumerate}
|
||||
\begin{itemize}
|
||||
@ -375,7 +378,7 @@
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{vdos} \abbrLabel{VDOS}
|
||||
\desc{Vibrational density of states (VDOS)}{}{$S_{v_i}$ velocity \secEqRef{spectral_density} of particle $I$}
|
||||
\desc{Vibrational density of states (VDOS)}{}{$S_{v_i}$ velocity \fRef{::spectral_density} of particle $I$}
|
||||
\desc[german]{Vibrationszustandsdicht (VDOS)}{}{}
|
||||
\eq{g(\omega) \sim \sum_{I=1}^N M_I S_{v_I}(\omega)}
|
||||
\end{formula}
|
||||
|
||||
@ -14,7 +14,7 @@
|
||||
\eq{\hat{V}_{i \leftrightarrow j} &= -\sum_{k,l} \frac{Z_i Z_j e^2}{\abs{\vecr_k - \vecr_l}}}
|
||||
\end{formula}
|
||||
\begin{formula}{hamiltonian}
|
||||
\desc{Electronic structure Hamiltonian}{}{$\hat{T}$ \fqEqRef{comp:est:kinetic_energy}, $\hat{V}$ \fqEqRef{comp:est:potential_energy}, $\txe$ \GT{electrons}, $\txn$ \GT{nucleons}}
|
||||
\desc{Electronic structure Hamiltonian}{}{$\hat{T}$ \fRef{comp:est:kinetic_energy}, $\hat{V}$ \fRef{comp:est:potential_energy}, $\txe$ \GT{electrons}, $\txn$ \GT{nucleons}}
|
||||
\eq{\hat{H} &= \hat{T}_\txe + \hat{T}_\txn + V_{\txe \leftrightarrow \txe} + V_{\txn \leftrightarrow \txe} + V_{\txn \leftrightarrow \txn}}
|
||||
\end{formula}
|
||||
\begin{formula}{mean_field}
|
||||
@ -64,8 +64,8 @@
|
||||
\begin{ttext}
|
||||
\eng{
|
||||
\begin{itemize}
|
||||
\item Assumes wave functions are \fqEqRef{qm:other:slater_det} \Rightarrow Approximation
|
||||
\item \fqEqRef{comp:est:mean_field} theory obeying the Pauli principle
|
||||
\item Assumes wave functions are \fRef{qm:other:slater_det} \Rightarrow Approximation
|
||||
\item \fRef{comp:est:mean_field} theory obeying the Pauli principle
|
||||
\item Self-interaction free: Self interaction is cancelled out by the Fock-term
|
||||
\end{itemize}
|
||||
}
|
||||
@ -76,14 +76,14 @@
|
||||
$\varphi_\xi$ single particle wavefunction of $\xi$th orbital,
|
||||
$\hat{T}$ kinetic electron energy,
|
||||
$\hat{V}_{\text{en}}$ electron-nucleus attraction,
|
||||
$\hat{V}_{\text{HF}}$ \fqEqRef{comp:dft:hf:potential},
|
||||
$h\hat{V}_{\text{HF}}$ \fRef{comp:est:dft:hf:potential},
|
||||
$x = \vecr,\sigma$ position and spin
|
||||
}
|
||||
\desc[german]{Hartree-Fock Gleichung}{}{
|
||||
$\varphi_\xi$ ein-Teilchen Wellenfunktion des $\xi$-ten Orbitals,
|
||||
$\hat{T}$ kinetische Energie der Elektronen,
|
||||
$\hat{V}_{\text{en}}$ Electron-Kern Anziehung,
|
||||
$\hat{V}_{\text{HF}}$ \fqEqRef{comp:dft:hf:potential},
|
||||
$\hat{V}_{\text{HF}}$ \fRef{comp:est:dft:hf:potential},
|
||||
$x = \vecr,\sigma$ Position and Spin
|
||||
}
|
||||
\eq{
|
||||
@ -158,7 +158,7 @@
|
||||
\eq{n(\vecr) = \sum_{i=1}^N \abs{\phi_i(\vecr)}^2}
|
||||
\end{formula}
|
||||
\begin{formula}{functional}
|
||||
\desc{Kohn-Sham functional}{}{$T_\text{KS}$ kinetic enery, $V_\text{ext}$ external potential, $E_\txH$ \hyperref[f:comp:est:dft:hf:potential]{Hartree term}, $E_\text{XC}$ \fqEqRef{comp:est:dft:xc:xc}}
|
||||
\desc{Kohn-Sham functional}{}{$T_\text{KS}$ kinetic enery, $V_\text{ext}$ external potential, $E_\txH$ \fRef[Hartree term]{comp:est:dft:hf:potential}, $E_\text{XC}$ \fRef{comp:est:dft:xc:xc}}
|
||||
\desc[german]{Kohn-Sham Funktional}{}{}
|
||||
\eq{E_\text{KS}[n(\vecr)] = T_\text{KS}[n(\vecr)] + V_\text{ext}[n(\vecr)] + E_\text{H}[n(\vecr)] + E_\text{XC}[n(\vecr)] }
|
||||
\end{formula}
|
||||
@ -186,7 +186,7 @@
|
||||
\begin{enumerate}
|
||||
\item Initial guess for $n(\vecr)$
|
||||
\item Calculate effective potential $V_\text{eff}$
|
||||
\item Solve \fqEqRef{comp:est:dft:ks:equation}
|
||||
\item Solve \fRef{comp:est:dft:ks:equation}
|
||||
\item Calculate density $n(\vecr)$
|
||||
\item Repeat 2-4 until self consistent
|
||||
\end{enumerate}
|
||||
@ -212,27 +212,25 @@
|
||||
}}
|
||||
\end{formula}
|
||||
\begin{formula}{lda}
|
||||
\desc{Local density approximation (LDA)}{Simplest DFT functionals}{$\epsilon_\txX$ calculated exchange energy from \hyperref[f:comp:qmb:models:heg]{HEG model}, $\epsilon_\txC$ correlation energy calculated with \fqSecRef{comp:qmb:methods:qmonte-carlo}}
|
||||
\desc{Local density approximation (LDA)}{Simplest DFT functionals}{$\epsilon_\txX$ calculated exchange energy from \fRef[HEG model]{comp:qmb:models:heg}, $\epsilon_\txC$ correlation energy calculated with \fRef{comp:qmb:methods:qmonte-carlo}}
|
||||
\desc[german]{}{}{}
|
||||
\abbrLabel{LDA}
|
||||
\eq{E_\text{XC}^\text{LDA}[n(\vecr)] = \int \d^3r\,n(r) \Big[\epsilon_\txX[n(\vecr)] + \epsilon_\txC[n(\vecr)]\Big]}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{gga}
|
||||
\desc{Generalized gradient approximation (GGA)}{}{$\epsilon_\txX$ calculated exchange energy from \hyperref[f:comp:qmb:models:heg]{HEG model}, $F_\text{XC}$ function containing exchange-correlation energy dependency on $n$ and $\Grad n$}
|
||||
\desc{Generalized gradient approximation (GGA)}{}{$\epsilon_\txX$ calculated exchange energy from \fRef[HEG model]{comp:qmb:models:heg}, $F_\text{XC}$ function containing exchange-correlation energy dependency on $n$ and $\Grad n$}
|
||||
\desc[german]{}{}{}
|
||||
\abbrLabel{GGA}
|
||||
\eq{E_\text{XC}^\text{GGA}[n(\vecr)] = \int \d^3r\,n(r) \epsilon_\txX[n(\vecr)]\,F_\text{XC}[n(\vecr), \Grad n(\vecr)]}
|
||||
\end{formula}
|
||||
|
||||
\TODO{PBE}
|
||||
|
||||
\begin{formula}{hybrid}
|
||||
\desc{Hybrid functionals}{}{}
|
||||
\desc[german]{Hybride Funktionale}{}{$\alpha$ mixing paramter, $E_\txX$ exchange energy, $E_\txC$ correlation energy}
|
||||
\eq{\alpha E_\txX^\text{HF} + (1-\alpha) E_\txX^\text{GGA} + E_\txC^\text{GGA}}
|
||||
\ttxt{\eng{
|
||||
Include \hyperref[f:comp:dft:hf:potential]{Fock term} (exact exchange) in other functional, like \abbrRef{gga}. Computationally expensive
|
||||
Include \fRef[Fock term]{comp:est:dft:hf:potential} (exact exchange) in other functional, like \abbrRef{gga}. Computationally expensive
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
@ -246,7 +244,7 @@
|
||||
\end{gather}
|
||||
\separateEntries
|
||||
\ttxt{\eng{
|
||||
Use \abbrRef{gga} and \hyperref[comp:est:dft:hf:potential]{Fock} exchange for short ranges (SR) and only \abbrRef{GGA} for long ranges (LR).
|
||||
Use \abbrRef{gga} and \fRef[Fock]{comp:est:dft:hf:potential} exchange for short ranges (SR) and only \abbrRef{GGA} for long ranges (LR).
|
||||
\abbrRef{GGA} correlation is always used. Useful when dielectric screening reduces long range interactions, saves computational cost.
|
||||
}}
|
||||
\end{formula}
|
||||
@ -255,7 +253,7 @@
|
||||
\desc{Comparison of DFT functionals}{}{}
|
||||
\desc[german]{Vergleich von DFT Funktionalen}{}{}
|
||||
% \begin{tabular}{l|c}
|
||||
% \hyperref[f:comp:est:dft:hf:potential]{Hartree-Fock} & only exchange, no correlation \Rightarrow upper bound of GS energy \\
|
||||
% \fRef[Hartree-Fock]{comp:est:dft:hf:potential} & only exchange, no correlation \Rightarrow upper bound of GS energy \\
|
||||
% \abbrRef{lda} & understimates e repulsion \Rightarrow Overbinding \\
|
||||
% \abbrRef{gga} & underestimate band gap \\
|
||||
% hybrid & underestimate band gap
|
||||
@ -374,7 +372,7 @@
|
||||
\ger{Basis-Sets}
|
||||
]{basis}
|
||||
\begin{formula}{plane_wave}
|
||||
\desc{Plane wave basis}{Plane wave ansatz in \fqEqRef{comp:est:dft:ks:equation}\\Good for periodic structures, allows computation parallelization over a sample points in the \abbrRef{brillouin_zone}}{}
|
||||
\desc{Plane wave basis}{Plane wave ansatz in \fRef{comp:est:dft:ks:equation}\\Good for periodic structures, allows computation parallelization over a sample points in the \abbrRef{brillouin_zone}}{}
|
||||
\desc[german]{Ebene Wellen als Basis}{}{}
|
||||
\eq{\sum_{\vecG^\prime} \left[\frac{\hbar^2 \abs{\vecG+\veck}^2}{2m} \delta_{\vecG,\vecG^\prime} + V_\text{eff}(\vecG-\vecG^\prime)\right] c_{i,\veck,\vecG^\prime} = \epsilon_{i,\veck} c_{i,\veck,\vecG}}
|
||||
\end{formula}
|
||||
|
||||
@ -19,22 +19,22 @@
|
||||
\begin{formula}{accuracy}
|
||||
\desc{Accuracy}{}{}
|
||||
\desc[german]{Genauigkeit}{}{}
|
||||
\eq{a = \frac{\tgt{cp}}{\tgt{fp} + \tgt{cp}}}
|
||||
\eq{a = \frac{\tGT{::cp}}{\tGT{::fp} + \tGT{::cp}}}
|
||||
\end{formula}
|
||||
\eng{n_desc}{Number of data points}
|
||||
\ger{n_desc}{Anzahl der Datenpunkte}
|
||||
\begin{formula}{mean_abs_error}
|
||||
\desc{Mean absolute error (MAE)}{}{$y$ \gt{y}, $\hat{y}$ \gt{yhat}, $n$ \gt{n_desc}}
|
||||
\desc{Mean absolute error (MAE)}{}{$y$ \GT{::y}, $\hat{y}$ \GT{::yhat}, $n$ \GT{::n_desc}}
|
||||
\desc[german]{Mittlerer absoluter Fehler (MAE)}{}{}
|
||||
\eq{\text{MAE} = \frac{1}{n} \sum_{i=1}^n \abs{y_i - \hat{y}_i}}
|
||||
\end{formula}
|
||||
\begin{formula}{mean_square_error}
|
||||
\desc{Mean squared error (MSE)}{}{$y$ \gt{y}, $\hat{y}$ \gt{yhat}, $n$ \gt{n_desc}}
|
||||
\desc{Mean squared error (MSE)}{}{$y$ \GT{::y}, $\hat{y}$ \GT{::yhat}, $n$ \GT{::n_desc}}
|
||||
\desc[german]{Methode der kleinsten Quadrate (MSE)}{Quadratwurzel des mittleren quadratischen Fehlers (SME)}{}
|
||||
\eq{\text{MSE} = \frac{1}{n} \sum_{i=1}^n \left(y_i - \hat{y}_i\right)^2}
|
||||
\end{formula}
|
||||
\begin{formula}{root_mean_square_error}
|
||||
\desc{Root mean squared error (RMSE)}{}{$y$ \gt{y}, $\hat{y}$ \gt{yhat}, $n$ \gt{n_desc}}
|
||||
\desc{Root mean squared error (RMSE)}{}{$y$ \GT{::y}, $\hat{y}$ \GT{::yhat}, $n$ \GT{::n_desc}}
|
||||
\desc[german]{Standardfehler der Regression}{Quadratwurzel des mittleren quadratischen Fehlers (RSME)}{}
|
||||
\eq{\text{RMSE} = \sqrt{\frac{1}{n} \sum_{i=1}^n \left(y_i - \hat{y}_i\right)^2}}
|
||||
\end{formula}
|
||||
@ -48,8 +48,8 @@
|
||||
\ger{Lineare Regression}
|
||||
]{linear}
|
||||
\begin{formula}{eq}
|
||||
\desc{Linear regression}{Fits the data under the assumption of \hyperref[f:math:pt:distributions:cont:normal]{normally distributed errors}}{$\mat{x}\in\R^{N\times M}$ input data, $\mat{y}\in\R^{N\times L}$ output data, $\mat{b}$ bias, $\vec{\beta}$ weights, $N$ samples, $M$ features, $L$ output variables}
|
||||
\desc[german]{Lineare Regression}{Fitted Daten unter der Annahme \hyperref[f:math:pt:distributions:cont:normal]{normalverteilter Fehler}}{}
|
||||
\desc{Linear regression}{Fits the data under the assumption of \fRef[normally distributed errors]{math:pt:distributions:cont:normal}}{$\mat{x}\in\R^{N\times M}$ input data, $\mat{y}\in\R^{N\times L}$ output data, $\mat{b}$ bias, $\vec{\beta}$ weights, $N$ samples, $M$ features, $L$ output variables}
|
||||
\desc[german]{Lineare Regression}{Fitted Daten unter der Annahme \fRef[normalverteilter Fehler]{math:pt:distributions:cont:normal}}{}
|
||||
\eq{\mat{y} = \mat{\epsilon} + \mat{x} \cdot \vec{\beta}}
|
||||
\end{formula}
|
||||
\begin{formula}{design_matrix}
|
||||
@ -60,13 +60,13 @@
|
||||
}
|
||||
\end{formula}
|
||||
\begin{formula}{scalar_bias}
|
||||
\desc{Linear regression with scalar bias}{Using the design matrix, the scalar is absorbed into the weight vector}{$\mat{y}$ output data, $\mat{X}$ \fqEqRef{comp:ml:reg:design_matrix}, $\vec{\beta}$ weights}
|
||||
\desc{Linear regression with scalar bias}{Using the design matrix, the scalar is absorbed into the weight vector}{$\mat{y}$ output data, $\mat{X}$ \fRef{::design_matrix}, $\vec{\beta}$ weights}
|
||||
\desc[german]{Lineare Regression mit skalarem Bias}{Durch die Designmatrix wird der Bias in den Gewichtsvektor absorbiert}{}
|
||||
\eq{\mat{y} = \mat{X} \cdot \vec{\beta}}
|
||||
\end{formula}
|
||||
\begin{formula}{normal_equation}
|
||||
\desc{Normal equation}{Solves \fqEqRef{comp:ml:reg:linear:scalar_bias} with \fqEqRef{comp:ml:performance:mse}}{$\mat{y}$ output data, $\mat{X}$ \fqEqRef{comp:ml:reg:linear:design_matrix}, $\vec{\beta}$ weights}
|
||||
\desc[german]{Normalengleichung}{Löst \fqEqRef{comp:ml:reg:linear:scalar_bias} mit \fqEqRef{comp:ml:performance:mse}}{}
|
||||
\desc{Normal equation}{Solves \fRef{comp:ml:reg:linear:scalar_bias} with \fRef{comp:ml:performance:mean_square_error}}{$\mat{y}$ output data, $\mat{X}$ \fRef{::design_matrix}, $\vec{\beta}$ weights}
|
||||
\desc[german]{Normalengleichung}{Löst \fRef{comp:ml:reg:linear:scalar_bias} mit \fRef{comp:ml:performance:mean_square_error}}{}
|
||||
\eq{\vec{\beta} = \left(\mat{X}^\T \mat{X}\right)^{-1} \mat{X}^T \mat{y}}
|
||||
\end{formula}
|
||||
|
||||
@ -112,7 +112,7 @@
|
||||
\desc{Bayesian linear regression}{}{}
|
||||
\desc[german]{Bayes'sche lineare Regression}{}{}
|
||||
\ttxt{\eng{
|
||||
Assume a \fqEqRef{math:pt:bayesian:prior} distribution over the weights.
|
||||
Assume a \fRef{math:pt:bayesian:prior} distribution over the weights.
|
||||
Offers uncertainties in addition to the predictions.
|
||||
}}
|
||||
\end{formula}
|
||||
@ -123,19 +123,18 @@
|
||||
\ttxt{\eng{
|
||||
Applies a L2 norm penalty on the weights.
|
||||
This ensures unimportant features are less regarded and do not encode noise.
|
||||
\\Corresponds to assuming a \fqEqRef{math:pt:bayesian:prior} \absRef{multivariate_normal_distribution} with $\vec{\mu} = 0$ and independent components ($\mat{\Sigma}$) for the weights.
|
||||
\\Corresponds to assuming a \fRef{math:pt:bayesian:prior} \absRef{multivariate_normal_distribution} with $\vec{\mu} = 0$ and independent components ($\mat{\Sigma}$) for the weights.
|
||||
}\ger{
|
||||
Reduziert Gewichte mit der L2-Norm.
|
||||
Dadurch werden unwichtige Features nicht berücksichtigt (kleines Gewicht) und enkodieren nicht Noise.
|
||||
\\Entspricht der Annahme einer \absRef[Normalverteilung]{multivariate_normal_distribution} mit $\vec{\mu}=0$ und unanhängingen Komponenten ($\mat{Sigma}$ diagonaol) der die Gewichte als \fqEqRef{math:pt:bayesian:prior}.
|
||||
\\Entspricht der Annahme einer \absRef[Normalverteilung]{multivariate_normal_distribution} mit $\vec{\mu}=0$ und unanhängingen Komponenten ($\mat{Sigma}$ diagonaol) der die Gewichte als \fRef{math:pt:bayesian:prior}.
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{ridge_weights}
|
||||
\desc{Optimal weights}{for ridge regression}{$\lambda = \frac{\sigma^2}{\xi^2}$ shrinkage parameter, $\xi$ \absRef{variance} of the gaussian \fqEqRef{math:pt:bayesian:prior}, $\sigma$ \absRef{variance} of the gaussian likelihood of the data}
|
||||
\desc{Optimal weights}{for ridge regression}{$\lambda = \frac{\sigma^2}{\xi^2}$ shrinkage parameter, $\xi$ \absRef{variance} of the gaussian \fRef{math:pt:bayesian:prior}, $\sigma$ \absRef{variance} of the gaussian likelihood of the data}
|
||||
\desc[german]{Optimale Gewichte}{für Ridge Regression}{}
|
||||
\eq{\vec{\beta} = \left(\mat{X}^\T \mat{X} + \lambda \mathcal{1} \right)^{-1} \mat{X}^\T \vecy}
|
||||
\TODO{Does this only work for gaussian data?}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{lasso}
|
||||
@ -143,11 +142,11 @@
|
||||
\desc[german]{Lasso Regression}{}{}
|
||||
\ttxt{\eng{
|
||||
Applies a L1 norm penalty on the weights, which means features can be disregarded entirely.
|
||||
\\Corresponds to assuming a \absRef{laplace_distribution} for the weights as \fqEqRef{math:pt:bayesian:prior}.
|
||||
\\Corresponds to assuming a \absRef{laplace_distribution} for the weights as \fRef{math:pt:bayesian:prior}.
|
||||
}\ger{
|
||||
Reduziert Gewichte mit der L1-Norm.
|
||||
Unwichtige Features werden reduziert und können auch ganz vernachlässigt werden und enkodieren nicht Noise.
|
||||
\\Entspricht der Annahme einer \absRef[Laplace-Verteilung]{laplace_distribution} der die Gewichte als \fqEqRef{math:pt:bayesian:prior}.
|
||||
\\Entspricht der Annahme einer \absRef[Laplace-Verteilung]{laplace_distribution} der die Gewichte als \fRef{math:pt:bayesian:prior}.
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
@ -158,7 +157,7 @@
|
||||
% \desc[german]{}{}{}
|
||||
\ttxt{\eng{
|
||||
Gaussian process: A distribtuion over functions that produce jointly gaussian distribution.
|
||||
Multivariate normal distribution like \secEqRef{bayesian}, except that $\vec{\mu}$ and $\mat{\Sigma}$ are functions.
|
||||
Multivariate normal distribution like \fRef{:::linear_regression}, except that $\vec{\mu}$ and $\mat{\Sigma}$ are functions.
|
||||
GPR: non-parametric Bayesion regressor, does not assume fixed functional form for the underlying data, instead, the data determines the functional shape,
|
||||
with predictions governed by the covariance structure defined by the kernel (often \abbrRef{radial_basis_function}).
|
||||
|
||||
@ -168,7 +167,23 @@
|
||||
\end{formula}
|
||||
|
||||
|
||||
\TODO{soap}
|
||||
\begin{formula}{soap}
|
||||
\desc{Smooth overlap of atomic atomic positions (SOAP)}{}{}
|
||||
% \desc[german]{}{}{}
|
||||
\ttxt{\eng{
|
||||
Goal: symmetric invariance, smoothness, completeness (completeness not achieved)
|
||||
\\Gaussian smeared density expanded in \abbrRef{radial_basis_function} and spherical harmonics.
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{gaussian_approximation_potential}
|
||||
\desc{Gaussian approximation potential}{Bond-order potential}{$V_\text{rep/attr}$ repulsive / attractive potential}
|
||||
% \desc[german]{}{}{}
|
||||
\ttxt{\eng{
|
||||
Models atomic interactions via a \textit{bond-order} term $b$.
|
||||
}}
|
||||
\eq{V_\text{BondOrder}(\vecR_M, \vecR_N) = V_\text{rep}(\vecR_M, \vecR_N) + b_{MNK} V_\text{attr}(\vecR_M, \vecR_N)}
|
||||
\end{formula}
|
||||
|
||||
\Subsection[
|
||||
\eng{Gradient descent}
|
||||
|
||||
@ -53,6 +53,15 @@
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{flux_quantum}
|
||||
\desc{Flux quantum}{}{}
|
||||
\desc[german]{Flussquantum}{}{}
|
||||
\constant{\Phi_0}{def}{
|
||||
\val{2.067 833 848 \xE{-15}}{\weber=\volt\s=\kg\m^2\per\s^2\ampere}
|
||||
}
|
||||
\eq{\Phi_0 = \frac{h}{2e}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{atomic_mass_unit}
|
||||
\desc{Atomic mass unit}{}{}
|
||||
\desc[german]{Atomare Massneinheit}{}{}
|
||||
|
||||
@ -34,6 +34,7 @@
|
||||
\eq{
|
||||
\epsilon(\omega)_\txr = \frac{\epsilon(\omega)}{\epsilon_0}
|
||||
}
|
||||
\hiddenQuantity{\epsilon_\txr}{}{s}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{vacuum_permittivity}
|
||||
@ -45,7 +46,7 @@
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{electric_susceptibility}
|
||||
\desc{Electric susceptibility}{Describes how polarized a dielectric material becomes when an electric field is applied}{$\epsilon_\txr$ \fqEqRef{ed:el:relative_permittivity}}
|
||||
\desc{Electric susceptibility}{Describes how polarized a dielectric material becomes when an electric field is applied}{$\epsilon_\txr$ \fRef{ed:el:relative_permittivity}}
|
||||
\desc[german]{Elektrische Suszeptibilität}{Beschreibt wie stark ein dielektrisches Material polarisiert wird, wenn ein elektrisches Feld angelegt wird}{}
|
||||
\quantity{\chi_\txe}{}{s}
|
||||
\eq{
|
||||
|
||||
@ -2,7 +2,7 @@
|
||||
\eng{Electromagnetism}
|
||||
\ger{Elektromagnetismus}
|
||||
]{em}
|
||||
\begin{formula}{speed_of_light}
|
||||
\begin{formula}{vacuum_speed_of_light}
|
||||
\desc{Speed of light}{in the vacuum}{}
|
||||
\desc[german]{Lightgeschwindigkeit}{in the vacuum}{}
|
||||
\constant{c}{exp}{
|
||||
@ -10,7 +10,7 @@
|
||||
}
|
||||
\end{formula}
|
||||
\begin{formula}{vacuum_relations}
|
||||
\desc{Vacuum permittivity - permeability relation}{\TODO{Does this have a name?}}{\ConstRef{vacuum_permittivity}, \ConstRef{magnetic_vacuum_permeability}, \ConstRef{speed_of_light}}
|
||||
\desc{Vacuum permittivity - permeability relation}{\TODO{Does this have a name?}}{\ConstRef{vacuum_permittivity}, \ConstRef{magnetic_vacuum_permeability}, \ConstRef{vacuum_speed_of_light}}
|
||||
\desc[german]{Vakuum Permittivität - Permeabilität Beziehung}{}{}
|
||||
\eq{
|
||||
\epsilon_0 \mu_0 = \frac{1}{c^2}
|
||||
@ -25,8 +25,9 @@
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{poynting}
|
||||
\desc{Poynting vector}{Directional energy flux or power flow of an electromagnetic field [$\si{\W\per\m^2}$]}{}
|
||||
\desc{Poynting vector}{Directional energy flux or power flow of an electromagnetic field}{}
|
||||
\desc[german]{Poynting-Vektor}{Gerichteter Energiefluss oder Leistungsfluss eines elektromgnetischen Feldes [$\si{\W\per\m^2}$]}{}
|
||||
\quantity{\vecS}{\W\per\m^2}{v}
|
||||
\eq{\vec{S} = \vec{E} \times \vec{H}}
|
||||
\end{formula}
|
||||
|
||||
@ -37,8 +38,8 @@
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{hamiltonian}
|
||||
\desc{Hamiltonian of a particle in an electromagnetic field}{In the \fqEqRef{ed:em:gauge:coulomb}}{\QtyRef{mass}, $\hat{p}$ \fqEqRef{qm:se:momentum_operator}, \QtyRef{charge}, \QtyRef{magnetic_vector_potential}, \ConstRef{speed_of_light}}
|
||||
\desc[german]{Hamiltonian eines Teilchens im elektromagnetischen Feld}{In der \fqEqRef{ed:em:gauge:coulomb}}{}
|
||||
\desc{Hamiltonian of a particle in an electromagnetic field}{In the \fRef{ed:em:maxwell:gauge:coulomb}}{\QtyRef{mass}, $\hat{p}$ \fRef{qm:se:momentum_operator}, \QtyRef{charge}, \QtyRef{magnetic_vector_potential}, \ConstRef{vacuum_speed_of_light}}
|
||||
\desc[german]{Hamiltonian eines Teilchens im elektromagnetischen Feld}{In der \fRef{ed:em:maxwell:gauge:coulomb}}{}
|
||||
\eq{
|
||||
\hat{H} = \frac{1}{2m} \left[\hat{p} \ \frac{e \vec{A}}{c}\right]^2
|
||||
}
|
||||
@ -48,7 +49,7 @@
|
||||
\Subsection[
|
||||
\eng{Maxwell-Equations}
|
||||
\ger{Maxwell-Gleichungen}
|
||||
]{Maxwell}
|
||||
]{maxwell}
|
||||
\begin{formula}{vacuum}
|
||||
\desc{Vacuum}{microscopic formulation}{}
|
||||
\desc[german]{Vakuum}{Mikroskopische Formulierung}{}
|
||||
|
||||
@ -11,8 +11,8 @@
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{magnetic_flux_density}
|
||||
\desc{Magnetic flux density}{Defined by \fqEqRef{ed:mag:lorentz}}{$\vec{H}$ \qtyRef{magnetic_field_intensity}, $\vec{M}$ \qtyRef{magnetization}, \ConstRef{magnetic_vacuum_permeability}}
|
||||
\desc[german]{Magnetische Flussdichte}{Definiert über \fqEqRef{ed:mag:lorentz}}{}
|
||||
\desc{Magnetic flux density}{Defined by \fRef{ed:mag:lorentz}}{$\vec{H}$ \qtyRef{magnetic_field_intensity}, $\vec{M}$ \qtyRef{magnetization}, \ConstRef{magnetic_vacuum_permeability}}
|
||||
\desc[german]{Magnetische Flussdichte}{Definiert über \fRef{ed:mag:lorentz}}{}
|
||||
\quantity{\vec{B}}{\tesla=\volt\s\per\m^2=\newton\per\ampere\m=\kg\per\ampere\s^2}{}
|
||||
\eq{\vec{B} = \mu_0 (\vec{H}+\vec{M})}
|
||||
\end{formula}
|
||||
@ -60,6 +60,7 @@
|
||||
\eq{
|
||||
\mu_\txr = \frac{\mu}{\mu_0}
|
||||
}
|
||||
\hiddenQuantity{\mu_\txr}{ }{}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{gauss_law}
|
||||
@ -88,9 +89,10 @@
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{magnetic_susceptibility}
|
||||
\desc{Susceptibility}{}{$\mu_\txr$ \fqEqRef{ed:mag:relative_permeability}}
|
||||
\desc{Susceptibility}{}{$\mu_\txr$ \fRef{ed:mag:relative_permeability}}
|
||||
\desc[german]{Suszeptibilität}{}{}
|
||||
\eq{\chi_\txm = \pdv{M}{B} = \mu_\txr - 1}
|
||||
\hiddenQuantity{\chi}{}{}
|
||||
\end{formula}
|
||||
|
||||
|
||||
@ -101,19 +103,19 @@
|
||||
\ger{Magnetische Materialien}
|
||||
]{materials}
|
||||
\begin{formula}{paramagnetism}
|
||||
\desc{Paramagnetism}{Magnetic field strengthend in the material}{$\mu$ \fqEqRef{ed:mag:magnetic_permeability}, $\chi_\txm$ \fqEqRef{ed:mag:magnetic_susceptibility}}
|
||||
\desc{Paramagnetism}{Magnetic field strengthend in the material}{$\mu$ \fRef{ed:mag:magnetic_permeability}, $\chi_\txm$ \fRef{ed:mag:magnetic_susceptibility}}
|
||||
\desc[german]{Paramagnetismus}{Magnetisches Feld wird im Material verstärkt}{}
|
||||
\eq{\mu_\txr &> 1 \\ \chi_\txm &> 0}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{diamagnetism}
|
||||
\desc{Diamagnetism}{Magnetic field expelled from material}{$\mu$ \fqEqRef{ed:mag:magnetic_permeability}, $\chi_\txm$ \fqEqRef{ed:mag:magnetic_susceptibility}}
|
||||
\desc{Diamagnetism}{Magnetic field expelled from material}{$\mu$ \fRef{ed:mag:magnetic_permeability}, $\chi_\txm$ \fRef{ed:mag:magnetic_susceptibility}}
|
||||
\desc[german]{Diamagnetismus}{Magnetisches Feld wird aus dem Material gedrängt}{}
|
||||
\eq{0 < \mu_\txr < 1 \\ -1 < \chi_\txm < 0}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{ferromagnetism}
|
||||
\desc{Ferromagnetism}{Magnetic moments align to external magnetic field and stay aligned when the field is turned off (Remanescence)}{$\mu$ \fqEqRef{ed:mag:magnetic_permeability}, $\chi_\txm$ \fqEqRef{ed:mag:magnetic_susceptibility}}
|
||||
\desc{Ferromagnetism}{Magnetic moments align to external magnetic field and stay aligned when the field is turned off (Remanescence)}{$\mu$ \fRef{ed:mag:magnetic_permeability}, $\chi_\txm$ \fRef{ed:mag:magnetic_susceptibility}}
|
||||
\desc[german]{Ferromagnetismus}{Magnetische Momente werden am äußeren Feld ausgerichtet und behalten diese ausrichtung auch wenn das Feld abgeschaltet wird (Remanenz)}{}
|
||||
\eq{
|
||||
\mu_\txr \gg 1
|
||||
|
||||
@ -40,9 +40,30 @@
|
||||
|
||||
|
||||
\Subsection[
|
||||
\eng{Integer quantum hall effect}
|
||||
\ger{Ganzahliger Quantenhalleffekt}
|
||||
\eng{Quantum hall effects}
|
||||
\ger{Quantenhalleffekte}
|
||||
]{quantum}
|
||||
\begin{formula}{types}
|
||||
\desc{Types of quantum hall effects}{}{}
|
||||
\desc[german]{Arten von Quantenhalleffekten}{}{}
|
||||
\ttxt{\eng{
|
||||
\begin{itemize}
|
||||
\item \textbf{Integer} (QHE): filling factor $\nu$ is an integer
|
||||
\item \textbf{Fractional} (FQHE): filling factor $\nu$ is a fraction
|
||||
\item \textbf{Spin} (QSHE): spin currents instead of charge currents
|
||||
\item \textbf{Anomalous} (QAHE): symmetry breaking by internal effects instead of external magnetic fields
|
||||
\end{itemize}
|
||||
}\ger{
|
||||
\begin{itemize}
|
||||
\item \textbf{Integer} (QHE): Füllfaktor $\nu$ ist ganzzahlig
|
||||
\item \textbf{Fractional} (FQHE): Füllfaktor $\nu$ ist ein Bruch
|
||||
\item \textbf{Spin} (QSHE): Spin Ströme anstatt Ladungsströme
|
||||
\item \textbf{Anomalous} (QAHE): Symmetriebruch durch interne Effekte anstatt druch ein externes Magnetfeld
|
||||
\end{itemize}
|
||||
}}
|
||||
\end{formula}
|
||||
|
||||
|
||||
|
||||
\begin{formula}{conductivity}
|
||||
\desc{Conductivity tensor}{}{}
|
||||
@ -77,28 +98,6 @@
|
||||
\eq{\nu = \frac{1}{3},\frac{2}{5},\frac{3}{7},\frac{2}{3}...}
|
||||
\end{formula}
|
||||
|
||||
\begin{ttext}
|
||||
\eng{
|
||||
\begin{itemize}
|
||||
\item \textbf{Integer} (QHE): filling factor $\nu$ is an integer
|
||||
\item \textbf{Fractional} (FQHE): filling factor $\nu$ is a fraction
|
||||
\item \textbf{Spin} (QSHE): spin currents instead of charge currents
|
||||
\item \textbf{Anomalous} (QAHE): symmetry breaking by internal effects instead of external magnetic fields
|
||||
\end{itemize}
|
||||
}
|
||||
\ger{
|
||||
\begin{itemize}
|
||||
\item \textbf{Integer} (QHE): Füllfaktor $\nu$ ist ganzzahlig
|
||||
\item \textbf{Fractional} (FQHE): Füllfaktor $\nu$ ist ein Bruch
|
||||
\item \textbf{Spin} (QSHE): Spin Ströme anstatt Ladungsströme
|
||||
\item \textbf{Anomalous} (QAHE): Symmetriebruch durch interne Effekte anstatt druch ein externes Magnetfeld
|
||||
\end{itemize}
|
||||
}
|
||||
\end{ttext}
|
||||
|
||||
|
||||
\TODO{sort}
|
||||
|
||||
|
||||
\Section[
|
||||
\eng{Dipole-stuff}
|
||||
|
||||
@ -79,7 +79,7 @@
|
||||
|
||||
|
||||
\begin{formula}{intensity}
|
||||
\desc{Electromagnetic radiation intensity}{Surface power density}{$S$ \fqEqRef{ed:poynting}}
|
||||
\desc{Electromagnetic radiation intensity}{Surface power density}{$S$ \fRef{ed:poynting}}
|
||||
\desc[german]{Elektromagnetische Strahlungsintensität}{Flächenleistungsdichte}{}
|
||||
\quantity{I}{\watt\per\m^2=\k\per\s^3}{s}
|
||||
\eq{I = \abs{\braket{S}_t}}
|
||||
|
||||
BIN
src/img/cm_crystal_NaCl.png
Normal file
BIN
src/img/cm_crystal_NaCl.png
Normal file
Binary file not shown.
|
After Width: | Height: | Size: 178 KiB |
BIN
src/img/cm_crystal_wurtzite.png
Normal file
BIN
src/img/cm_crystal_wurtzite.png
Normal file
Binary file not shown.
|
After Width: | Height: | Size: 126 KiB |
BIN
src/img/cm_crystal_zincblende.png
Normal file
BIN
src/img/cm_crystal_zincblende.png
Normal file
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
Before Width: | Height: | Size: 432 KiB After Width: | Height: | Size: 432 KiB |
@ -1,7 +1,7 @@
|
||||
IFS=$'\n'
|
||||
for d in $(find . -type d); do
|
||||
mkdir -p "../img/$d"
|
||||
mkdir -p "../img_static/$d"
|
||||
done
|
||||
for file in $(find . -type f -name '*.svg'); do
|
||||
inkscape -o "../img/${file%.*}.pdf" "$file"
|
||||
inkscape -o "../img_static/${file%.*}.pdf" "$file"
|
||||
done
|
||||
|
Before Width: | Height: | Size: 8.6 KiB After Width: | Height: | Size: 8.6 KiB |
38
src/main.tex
38
src/main.tex
@ -3,7 +3,7 @@
|
||||
\documentclass[11pt, a4paper]{article}
|
||||
% SET LANGUAGE HERE
|
||||
\usepackage[english]{babel}
|
||||
\usepackage[left=2cm,right=2cm,top=2cm,bottom=2cm]{geometry}
|
||||
\usepackage[left=1.6cm,right=1.6cm,top=2cm,bottom=2cm]{geometry}
|
||||
% ENVIRONMENTS etc
|
||||
\usepackage{adjustbox}
|
||||
\usepackage{colortbl} % color table
|
||||
@ -20,7 +20,7 @@
|
||||
% FORMATING
|
||||
\usepackage{float} % float barrier
|
||||
\usepackage{subcaption} % subfigures
|
||||
\usepackage[hidelinks]{hyperref} % hyperrefs for \fqEqRef, \qtyRef, etc
|
||||
\usepackage[hidelinks]{hyperref} % hyperrefs for \fRef, \qtyRef, etc
|
||||
\usepackage[shortlabels]{enumitem} % easily change enum symbols to i), a. etc
|
||||
\setlist{noitemsep} % no vertical space between items
|
||||
\setlist[1]{labelindent=\parindent} % < Usually a good idea
|
||||
@ -79,38 +79,31 @@
|
||||
% \def\lambda{\temoji{sheep}}
|
||||
% \def\psi{\temoji{pickup-truck}}
|
||||
% \def\pi{\temoji{birthday-cake}}
|
||||
% \def\Pi{\temoji{hospital}}
|
||||
% \def\rho{\temoji{rhino}}
|
||||
% % \def\Pi{\temoji{hospital}}
|
||||
% % \def\rho{\temoji{rhino}}
|
||||
% \def\nu{\temoji{unicorn}}
|
||||
% \def\mu{\temoji{mouse}}
|
||||
|
||||
\newcommand{\TODO}[1]{{\color{fg-red}TODO:#1}}
|
||||
\newcommand{\ts}{\textsuperscript}
|
||||
|
||||
% Create a text file with relevant labels for vim-completion
|
||||
\newwrite\labelsFile
|
||||
\immediate\openout\labelsFile=\jobname.labels.txt
|
||||
\newcommand\storeLabel[1]{
|
||||
\immediate\write\labelsFile{#1}%
|
||||
}
|
||||
\AtEndDocument{\immediate\closeout\labelsFile}
|
||||
|
||||
\input{circuit.tex}
|
||||
\input{util/macros.tex}
|
||||
\input{util/math-macros.tex}
|
||||
\input{util/environments.tex} % requires util/translation.tex to be loaded first
|
||||
\usepackage{pkg/mqlua}
|
||||
\usepackage{pkg/mqfqname}
|
||||
\usepackage{mqlua}
|
||||
\usepackage{mqfqname}
|
||||
\usepackage{mqref}
|
||||
% TRANSLATION
|
||||
% \usepackage{translations}
|
||||
\usepackage{pkg/mqtranslation}
|
||||
\usepackage{mqtranslation}
|
||||
\input{util/colorscheme.tex}
|
||||
\input{util/colors.tex} % after colorscheme
|
||||
|
||||
\usepackage{pkg/mqconstant}
|
||||
\usepackage{pkg/mqquantity}
|
||||
\usepackage{pkg/mqformula}
|
||||
\usepackage{pkg/mqperiodictable}
|
||||
\usepackage{mqconstant}
|
||||
\usepackage{mqquantity}
|
||||
\usepackage{mqformula}
|
||||
\usepackage{mqperiodictable}
|
||||
|
||||
|
||||
\title{Formelsammlung}
|
||||
@ -127,7 +120,7 @@
|
||||
|
||||
\input{util/translations.tex}
|
||||
|
||||
% \InputOnly{comp}
|
||||
% \InputOnly{cm}
|
||||
|
||||
\Input{math/math}
|
||||
\Input{math/linalg}
|
||||
@ -152,7 +145,8 @@
|
||||
\Input{cm/crystal}
|
||||
\Input{cm/egas}
|
||||
\Input{cm/charge_transport}
|
||||
\Input{cm/low_temp}
|
||||
\Input{cm/vib}
|
||||
\Input{cm/superconductivity}
|
||||
\Input{cm/semiconductors}
|
||||
\Input{cm/misc}
|
||||
\Input{cm/techniques}
|
||||
@ -196,7 +190,7 @@
|
||||
]{elements}
|
||||
\printAllElements
|
||||
\newpage
|
||||
% \Input{test}
|
||||
\Input{test}
|
||||
|
||||
% \bibliographystyle{plain}
|
||||
% \bibliography{ref}
|
||||
|
||||
@ -51,7 +51,6 @@
|
||||
b_k &= \I(c_k - c_{-k}) \quad\text{\GT{for}}\,k\ge1
|
||||
}
|
||||
\end{formula}
|
||||
\TODO{cleanup}
|
||||
|
||||
|
||||
\Subsubsection[
|
||||
@ -170,7 +169,7 @@
|
||||
x^{\log(y)} &= y^{\log(x)}
|
||||
}
|
||||
\end{formula}
|
||||
\begin{formula}{intergral}
|
||||
\begin{formula}{integral}
|
||||
\desc{Integral of natural logarithm}{}{}
|
||||
\desc[german]{Integral des natürluchen Logarithmus}{}{}
|
||||
\eq{
|
||||
@ -253,7 +252,7 @@
|
||||
\ger{Liste nützlicher Integrale}
|
||||
]{list}
|
||||
% Put links to other integrals here
|
||||
\fqEqRef{cal:log:integral}
|
||||
\fRef{math:cal:log:integral}
|
||||
|
||||
\begin{formula}{arcfunctions}
|
||||
\desc{Arcsine, arccosine, arctangent}{}{}
|
||||
|
||||
@ -13,8 +13,8 @@
|
||||
|
||||
\begin{formula}{variance}
|
||||
\absLabel
|
||||
\desc{Variance}{Square of the \fqEqRef{math:pt:std-deviation}}{}
|
||||
\desc[german]{Varianz}{Quadrat der\fqEqRef{math:pt:std-deviation}}{}
|
||||
\desc{Variance}{Square of the \fRef{math:pt:std-deviation}}{}
|
||||
\desc[german]{Varianz}{Quadrat der\fRef{math:pt:std-deviation}}{}
|
||||
\eq{\sigma^2 = (\Delta \hat{x})^2 = \Braket{\hat{x}^2} - \braket{\hat{x}}^2 = \braket{(x - \braket{x})^2}}
|
||||
\end{formula}
|
||||
|
||||
@ -108,16 +108,16 @@
|
||||
|
||||
\begin{bigformula}{multivariate_normal}
|
||||
\absLabel[multivariate_normal_distribution]
|
||||
\desc{Multivariate normal distribution}{Multivariate Gaussian distribution}{$\vec{mu}$ \absRef{mean}, $\mat{\Sigma}$ \absRef{covariance}}
|
||||
\desc{Multivariate normal distribution}{Multivariate Gaussian distribution}{$\vec{\mu}$ \absRef{mean}, $\mat{\Sigma}$ \absRef{covariance}}
|
||||
\desc[german]{Mehrdimensionale Normalverteilung}{Multivariate Normalverteilung}{}
|
||||
\TODO{k-variate normal plot}
|
||||
\begin{distribution}
|
||||
\disteq{parameters}{\vec{\mu} \in \R^k,+\quad \mat{\Sigma} \in \R^{k\times k}}
|
||||
\disteq{support}{\vec{x} \in \vec{\mu} + \text{span}(\mat{\Sigma})}
|
||||
\disteq{pdf}{\mathcal{N}(\vec{mu}, \mat{\Sigma}) = \frac{1}{(2\pi)^{k/2}} \frac{1}{\sqrt{\det{\Sigma}}} \Exp{-\frac{1}{2} \left(\vecx-\vec{\mu}\right)^\T \mat{\Sigma}^{-1} \left(\vecx-\vec{\mu}\right)}}
|
||||
\disteq{pdf}{\mathcal{N}(\vec{\mu}, \mat{\Sigma}) = \frac{1}{(2\pi)^{k/2}} \frac{1}{\sqrt{\det{\Sigma}}} \Exp{-\frac{1}{2} \left(\vecx-\vec{\mu}\right)^\T \mat{\Sigma}^{-1} \left(\vecx-\vec{\mu}\right)}}
|
||||
\disteq{mean}{\vec{\mu}}
|
||||
\disteq{variance}{\mat{\Sigma}}
|
||||
\end{distribution}
|
||||
\TODO{k-variate normal plot}
|
||||
\end{bigformula}
|
||||
|
||||
\begin{formula}{laplace}
|
||||
@ -172,7 +172,7 @@
|
||||
|
||||
\begin{bigformula}{gamma}
|
||||
\absLabel[gamma_distribution]
|
||||
\desc{Gamma Distribution}{with $\lambda$ parameter}{$\Gamma$ \fqEqRef{math:cal:integral:list:gamma}, $\gamma$ \fqEqRef{math:cal:integral:list:lower_incomplete_gamma_function}}
|
||||
\desc{Gamma Distribution}{with $\lambda$ parameter}{$\Gamma$ \fRef{math:cal:integral:list:gamma_function}, $\gamma$ \fRef{math:cal:integral:list:lower_incomplete_gamma_function}}
|
||||
\desc[german]{Gamma Verteilung}{mit $\lambda$ Parameter}{}
|
||||
\begin{minipage}{\distleftwidth}
|
||||
\begin{figure}[H]
|
||||
@ -191,7 +191,8 @@
|
||||
\end{bigformula}
|
||||
|
||||
\begin{bigformula}{beta}
|
||||
\desc{Beta Distribution}{}{$\txB$ \fqEqRef{math:cal:integral:list:beta_function} / \fqEqRef{math:cal:integral:list:incomplete_beta_function}}
|
||||
\absLabel[beta_distribution]
|
||||
\desc{Beta Distribution}{}{$\txB$ \fRef{math:cal:integral:list:beta_function} / \fRef{math:cal:integral:list:incomplete_beta_function}}
|
||||
\desc[german]{Beta Verteilung}{}{}
|
||||
\begin{minipage}{\distleftwidth}
|
||||
\begin{figure}[H]
|
||||
@ -216,11 +217,12 @@
|
||||
\ger{Diskrete Wahrscheinlichkeitsverteilungen}
|
||||
]{discrete}
|
||||
\begin{bigformula}{binomial}
|
||||
\absLabel[binomial_distribution]
|
||||
\desc{Binomial distribution}{}{}
|
||||
\desc[german]{Binomialverteilung}{}{}
|
||||
\begin{ttext}
|
||||
\eng{For the number of trials going to infinity ($n\to\infty$), the binomial distribution converges to the \hyperref[sec:pb:distributions:poisson]{poisson distribution}}
|
||||
\ger{Geht die Zahl der Versuche gegen unendlich ($n\to\infty$), konvergiert die Binomualverteilung gegen die \hyperref[sec:pb:distributions:poisson]{Poissonverteilung}}
|
||||
\eng{For the number of trials going to infinity ($n\to\infty$), the binomial distribution converges to the \absRef[poisson distribution]{poisson_distribution}}
|
||||
\ger{Geht die Zahl der Versuche gegen unendlich ($n\to\infty$), konvergiert die Binomualverteilung gegen die \absRef[Poissonverteilung]{poisson_distribution}}
|
||||
\end{ttext}\\
|
||||
\begin{minipage}{\distleftwidth}
|
||||
\begin{figure}[H]
|
||||
@ -240,6 +242,7 @@
|
||||
\end{bigformula}
|
||||
|
||||
\begin{bigformula}{poisson}
|
||||
\absLabel[poisson_distribution]
|
||||
\desc{Poisson distribution}{}{}
|
||||
\desc[german]{Poissonverteilung}{}{}
|
||||
\begin{minipage}{\distleftwidth}
|
||||
@ -296,8 +299,8 @@
|
||||
\ger{Fehlerfortpflanzung}
|
||||
]{error}
|
||||
\begin{formula}{generalised}
|
||||
\desc{Generalized error propagation}{}{$V$ \fqEqRef{math:pt:covariance} matrix, $J$ \fqEqRef{math:cal:jacobi-matrix}}
|
||||
\desc[german]{Generalisiertes Fehlerfortpflanzungsgesetz}{$V$ \fqEqRef{math:pt:covariance} Matrix, $J$ \fqEqRef{cal:jacobi-matrix}}{}
|
||||
\desc{Generalized error propagation}{}{$V$ \fRef{math:pt:covariance} matrix, $J$ \fRef{math:cal:jacobi-matrix}}
|
||||
\desc[german]{Generalisiertes Fehlerfortpflanzungsgesetz}{$V$ \fRef{math:pt:covariance} Matrix, $J$ \fRef{cal:jacobi-matrix}}{}
|
||||
\eq{V_y = J(x) \cdot V_x \cdot J^{\T} (x)}
|
||||
\end{formula}
|
||||
|
||||
@ -308,19 +311,19 @@
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{weight}
|
||||
\desc{Weight}{Variance is a possible choice for a weight}{$\sigma$ \fqEqRef{math:pt:variance}}
|
||||
\desc{Weight}{Variance is a possible choice for a weight}{$\sigma$ \fRef{math:pt:variance}}
|
||||
\desc[german]{Gewicht}{Varianz ist eine mögliche Wahl für ein Gewicht}{}
|
||||
\eq{w_i = \frac{1}{\sigma_i^2}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{weighted-mean}
|
||||
\desc{Weighted mean}{}{$w_i$ \fqEqRef{math:pt:error:weight}}
|
||||
\desc{Weighted mean}{}{$w_i$ \fRef{math:pt:error:weight}}
|
||||
\desc[german]{Gewichteter Mittelwert}{}{}
|
||||
\eq{\overline{x} = \frac{\sum_{i} (x_i w_i)}{\sum_i w_i}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{weighted-mean-error}
|
||||
\desc{Variance of weighted mean}{}{$w_i$ \fqEqRef{math:pt:error:weight}}
|
||||
\desc{Variance of weighted mean}{}{$w_i$ \fRef{math:pt:error:weight}}
|
||||
\desc[german]{Varianz des gewichteten Mittelwertes}{}{}
|
||||
\eq{\sigma^2_{\overline{x}} = \frac{1}{\sum_i w_i}}
|
||||
\end{formula}
|
||||
@ -330,18 +333,18 @@
|
||||
\ger{Maximum likelihood Methode}
|
||||
]{mle}
|
||||
\begin{formula}{likelihood}
|
||||
\desc{Likelihood function}{Likelihood of observing $x$ when parameter is $\theta$\\in general not normalized!}{$\rho$ \fqEqRef{math:pt:pdf} $x\mapsto \rho(x|\theta)$ depending on parameter $\theta$, $\Theta$ parameter space}
|
||||
\desc[german]{Likelihood Funktion}{"Plausibilität" $x$ zu messen, wenn der Parameter $\theta$ ist\\nicht normalisiert!}{$\rho$ \fqEqRef{math:pt:pdf} $x\mapsto \rho(x|\theta)$ hängt ab von Parameter $\theta$, $\Theta$ Parameterraum}
|
||||
\desc{Likelihood function}{Likelihood of observing $x$ when parameter is $\theta$\\in general not normalized!}{$\rho$ \fRef{math:pt:pdf} $x\mapsto \rho(x|\theta)$ depending on parameter $\theta$, $\Theta$ parameter space}
|
||||
\desc[german]{Likelihood Funktion}{"Plausibilität" $x$ zu messen, wenn der Parameter $\theta$ ist\\nicht normalisiert!}{$\rho$ \fRef{math:pt:pdf} $x\mapsto \rho(x|\theta)$ hängt ab von Parameter $\theta$, $\Theta$ Parameterraum}
|
||||
\eq{L:\Theta \rightarrow [0,1], \quad \theta \mapsto \rho(x|\theta)}
|
||||
\end{formula}
|
||||
\begin{formula}{likelihood_independant}
|
||||
\desc{Likelihood function}{for independent and identically distributed random variables}{$x_i$ $n$ random variables, $\rho$ \fqEqRef{math:pt:pdf} $x\mapsto f(x|\theta)$ depending on parameter $\theta$}
|
||||
\desc[german]{Likelihood function}{für unabhängig und identisch verteilte Zufallsvariablen}{$x_i$ $n$ Zufallsvariablen$\rho$ \fqEqRef{math:pt:pdf} $x\mapsto f(x|\theta)$ hängt ab von Parameter $\theta$}
|
||||
\desc{Likelihood function}{for independent and identically distributed random variables}{$x_i$ $n$ random variables, $\rho$ \fRef{math:pt:pdf} $x\mapsto f(x|\theta)$ depending on parameter $\theta$}
|
||||
\desc[german]{Likelihood function}{für unabhängig und identisch verteilte Zufallsvariablen}{$x_i$ $n$ Zufallsvariablen$\rho$ \fRef{math:pt:pdf} $x\mapsto f(x|\theta)$ hängt ab von Parameter $\theta$}
|
||||
\eq{L(\theta) = \prod_{i=1}^n f(x_i;\theta)}
|
||||
\end{formula}
|
||||
\begin{formula}{maximum_likelihood_estimate}
|
||||
\desc{Maximum likelihood estimate (MLE)}{Paramater for which outcome is most likely}{$L$ \fqEqRef{pt:mle:likelihood}, $\theta$ parameter of a \fqEqRef{math:pt:pdf}}
|
||||
\desc[german]{Maximum likelihood-Schätzung (MLE)}{Paramater, für den das Ergebnis am Wahrscheinlichsten ist}{$L$ \fqEqRef{math:pt:mle:likelihood}, $\theta$ Parameter einer \fqEqRef{math:pt:pdf}}
|
||||
\desc{Maximum likelihood estimate (MLE)}{Paramater for which outcome is most likely}{$L$ \fRef{math:pt:mle:likelihood}, $\theta$ parameter of a \fRef{math:pt:pdf}}
|
||||
\desc[german]{Maximum likelihood-Schätzung (MLE)}{Paramater, für den das Ergebnis am Wahrscheinlichsten ist}{$L$ \fRef{math:pt:mle:likelihood}, $\theta$ Parameter einer \fRef{math:pt:pdf}}
|
||||
\eq{\theta_\text{ML} &= \argmax_\theta L(\theta)\\ &= \argmax_\theta \log \big(L(\theta)\big)}
|
||||
\end{formula}
|
||||
|
||||
@ -356,13 +359,13 @@
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{evidence}
|
||||
\desc{Evidence}{}{$p(\mathcal{D}|\theta)$ \fqEqRef{math:pt:mle:likelihood}, $p(\theta)$ \fqEqRef{math:pt:bayesian:prior}, $\mathcal{D}$ data set}
|
||||
\desc{Evidence}{}{$p(\mathcal{D}|\theta)$ \fRef{math:pt:mle:likelihood}, $p(\theta)$ \fRef{math:pt:bayesian:prior}, $\mathcal{D}$ data set}
|
||||
% \desc[german]{}{}{}
|
||||
\eq{p(\mathcal{D}) = \int\d\theta \,p(\mathcal{D}|\theta)\,p(\theta)}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{theorem}
|
||||
\desc{Bayes' theorem}{}{$p(\theta|\mathcal{D})$ posterior distribution, $p(\mathcal{D}|\theta)$ \fqEqRef{math:pt:mle:likelihood}, $p(\theta)$ \fqEqRef{math:pt:bayesian:prior}, $p(\mathcal{D})$ \fqEqRef{math:pt:bayesian:evidence}, $\mathcal{D}$ data set}
|
||||
\desc{Bayes' theorem}{}{$p(\theta|\mathcal{D})$ posterior distribution, $p(\mathcal{D}|\theta)$ \fRef{math:pt:mle:likelihood}, $p(\theta)$ \fRef{math:pt:bayesian:prior}, $p(\mathcal{D})$ \fRef{math:pt:bayesian:evidence}, $\mathcal{D}$ data set}
|
||||
\desc[german]{Satz von Bayes}{}{}
|
||||
\eq{p(\theta|\mathcal{D}) = \frac{p(\mathcal{D}|\theta)\,p(\theta)}{p(\mathcal{D})}}
|
||||
\end{formula}
|
||||
|
||||
@ -2,38 +2,56 @@
|
||||
\RequirePackage{mqlua}
|
||||
\RequirePackage{etoolbox}
|
||||
|
||||
\directLuaAux{
|
||||
if constants == nil then
|
||||
constants = {}
|
||||
\begin{luacode}
|
||||
constants = {}
|
||||
function constantAdd(key, symbol, exp_or_def, fqname)
|
||||
constants[key] = {
|
||||
["symbol"] = symbol,
|
||||
["units"] = units,
|
||||
["exp_or_def"] = exp_or_def,
|
||||
["values"] = {} -- array of {value, unit}
|
||||
}
|
||||
if fqname == "" then
|
||||
constants[key]["fqname"] = fqnameGet()
|
||||
else
|
||||
constants[key]["fqname"] = fqname
|
||||
end
|
||||
end
|
||||
}
|
||||
function constantAddValue(key, value, unit)
|
||||
table.insert(constants[key]["values"], { value = value, unit = unit })
|
||||
end
|
||||
function constantGetSymbol(key)
|
||||
local const = constants[key]
|
||||
if const == nil then return "???" end
|
||||
local symbol = const["symbol"]
|
||||
if symbol == nil then return "???" end
|
||||
return symbol
|
||||
end
|
||||
function constantGetFqname(key)
|
||||
local const = constants[key]
|
||||
if const == nil then return "const:"..key end
|
||||
local fqname_ = const["fqname"]
|
||||
if fqname_ == nil then return "const:"..key end
|
||||
return fqname_
|
||||
end
|
||||
\end{luacode}
|
||||
|
||||
% [1]: label to point to
|
||||
% 2: key
|
||||
% 3: symbol
|
||||
% 4: either exp or def; experimentally or defined constant
|
||||
\newcommand{\constant@new}[4][\relax]{
|
||||
\directLuaAux{
|
||||
constants["#2"] = {}
|
||||
constants["#2"]["symbol"] = [[\detokenize{#3}]]
|
||||
constants["#2"]["exp_or_def"] = [[\detokenize{#4}]]
|
||||
constants["#2"]["values"] = {} %-- array of {value, unit}
|
||||
}
|
||||
\ifstrempty{#1}{}{
|
||||
\directLuaAuxExpand{
|
||||
constants["#2"]["linkto"] = [[#1]] %-- fqname required for getting the translation key
|
||||
}
|
||||
}
|
||||
\newcommand{\constant@new}[4][]{%
|
||||
\directLuaAuxExpand{constantAdd(\luastring{#2}, \luastringN{#3}, \luastringN{#4}, \luastring{#1})}%
|
||||
}
|
||||
|
||||
% 1: key
|
||||
% 2: value
|
||||
% 3: units
|
||||
\newcommand{\constant@addValue}[3]{
|
||||
\directlua{
|
||||
table.insert(constants["#1"]["values"], { value = [[\detokenize{#2}]], unit = [[\detokenize{#3}]] })
|
||||
}
|
||||
\newcommand{\constant@addValue}[3]{%
|
||||
\directlua{constantAddValue(\luastring{#1}, \luastringN{#2}, \luastringN{#3})}%
|
||||
}
|
||||
% 1: key
|
||||
\newcommand{\constant@getSymbol}[1]{\luavar{constantGetSymbol(\luastring{#1})}}
|
||||
|
||||
% 1: key
|
||||
\newcommand\constant@print[1]{
|
||||
@ -50,13 +68,5 @@
|
||||
%--tex.sprint("VALUE ", i, v)
|
||||
end
|
||||
}
|
||||
% label it only once
|
||||
\directlua{
|
||||
if constants["#1"]["labeled"] == nil then
|
||||
constants["#1"]["labeled"] = true
|
||||
tex.print("\\label{const:#1}")
|
||||
end
|
||||
}
|
||||
\endgroup
|
||||
}
|
||||
\newcounter{constant}
|
||||
|
||||
@ -1,5 +1,8 @@
|
||||
\ProvidesPackage{mqformula}
|
||||
|
||||
\def\descwidth{0.3\textwidth}
|
||||
\def\eqwidth{0.65\textwidth}
|
||||
|
||||
\RequirePackage{mqfqname}
|
||||
\RequirePackage{mqconstant}
|
||||
\RequirePackage{mqquantity}
|
||||
@ -13,14 +16,11 @@
|
||||
% [1]: minipage width
|
||||
% 2: fqname of name
|
||||
% 3: fqname of a translation that holds the explanation
|
||||
\newcommand{\NameWithDescription}[3][\descwidth]{
|
||||
\newcommand{\NameWithDescription}[3][\descwidth]{%
|
||||
\begin{minipage}{#1}
|
||||
\IfTranslationExists{#2}{
|
||||
\raggedright
|
||||
\GT{#2}
|
||||
}{\detokenize{#2}}
|
||||
\IfTranslationExists{#3}{
|
||||
\\ {\color{fg1} \GT{#3}}
|
||||
\raggedright\GT{#2}%
|
||||
\IfTranslationExists{#3}{%
|
||||
\\ {\color{fg1} \GT{#3}}%
|
||||
}{}
|
||||
\end{minipage}
|
||||
}
|
||||
@ -36,14 +36,13 @@
|
||||
\begin{minipage}{#1}
|
||||
}{
|
||||
\IfTranslationExists{\ContentFqName}{%
|
||||
\smartnewline
|
||||
\noindent
|
||||
\begingroup
|
||||
\color{fg1}
|
||||
\GT{\ContentFqName}
|
||||
% \edef\temp{\GT{#1_defs}}
|
||||
% \expandafter\StrSubstitute\expandafter{\temp}{:}{\\}
|
||||
\endgroup
|
||||
\smartnewline%
|
||||
\noindent%
|
||||
\begingroup%
|
||||
\color{fg1}%
|
||||
\raggedright%
|
||||
\GT{\ContentFqName}%
|
||||
\endgroup%
|
||||
}{}
|
||||
\end{minipage}
|
||||
\end{lrbox}
|
||||
@ -54,18 +53,16 @@
|
||||
% Class defining commands shared by all formula environments
|
||||
% 1: key
|
||||
\newenvironment{formulainternal}[1]{
|
||||
% TODO refactor, using fqname@enter and leave
|
||||
% TODO There is no real need to differentiate between fqnames and sections,
|
||||
% TODO thus change the meaning of f: from formula to fqname and change sec to f
|
||||
\mqfqname@enter{#1}
|
||||
% [1]: language
|
||||
% 2: name
|
||||
% 3: description
|
||||
% 4: definitions/links
|
||||
\newcommand{\desc}[4][english]{
|
||||
% language, name, description, definitions
|
||||
\ifblank{##2}{}{\dt[#1]{##1}{##2}}
|
||||
\ifblank{##3}{}{\dt[#1_desc]{##1}{##3}}
|
||||
\ifblank{##4}{}{\dt[#1_defs]{##1}{##4}}
|
||||
\ifblank{##2}{}{\dt{##1}{##2}}
|
||||
\ifblank{##3}{}{\dt[desc]{##1}{##3}}
|
||||
\ifblank{##4}{}{\dt[defs]{##1}{##4}}
|
||||
}
|
||||
\directlua{n_formulaEntries = 0}
|
||||
|
||||
@ -73,12 +70,12 @@
|
||||
% [1]: label to use
|
||||
% 2: Abbreviation to use for references
|
||||
\newcommand{\abbrLabel}[2][#1]{
|
||||
\abbrLink[f:\fqname:#1]{##1}{##2}
|
||||
\abbrLink[\fqname]{##1}{##2}
|
||||
}
|
||||
% makes this formula referencable with \absRef{<name>}
|
||||
% [1]: label to use
|
||||
\newcommand{\absLabel}[1][#1]{
|
||||
\absLink[\fqname:#1]{f:\fqname:#1}{##1}
|
||||
\absLink[\fqname]{\fqname}{##1}
|
||||
}
|
||||
|
||||
\newcommand{\newFormulaEntry}{
|
||||
@ -98,18 +95,6 @@
|
||||
##1
|
||||
\end{align}
|
||||
}
|
||||
% 1: equation for alignat environment
|
||||
\newcommand{\eqAlignedAt}[2]{
|
||||
\newFormulaEntry
|
||||
\begin{flalign}%
|
||||
\TODO{\text{remove macro}}
|
||||
% dont place label when one is provided
|
||||
% \IfSubStringInString{label}\unexpanded{#3}{}{
|
||||
% \label{eq:#1}
|
||||
% }
|
||||
##1%
|
||||
\end{flalign}
|
||||
}
|
||||
% 1: equation for flalign environment
|
||||
\newcommand{\eqFLAlign}[2]{
|
||||
\newFormulaEntry
|
||||
@ -127,7 +112,7 @@
|
||||
\includegraphics{##1}
|
||||
}
|
||||
% 1: content for the ttext environment
|
||||
\newcommand{\ttxt}[2][#1:desc]{
|
||||
\newcommand{\ttxt}[2][text]{
|
||||
\newFormulaEntry
|
||||
\begin{ttext}[##1]
|
||||
##2
|
||||
@ -141,6 +126,9 @@
|
||||
\newFormulaEntry
|
||||
\quantity@print{#1}
|
||||
}
|
||||
\newcommand{\hiddenQuantity}[3]{%
|
||||
\quantity@new[\fqname]{#1}{##1}{##2}{##3}
|
||||
}
|
||||
|
||||
% must be used only in third argument of "constant" command
|
||||
% 1: value
|
||||
@ -159,20 +147,34 @@
|
||||
\newFormulaEntry
|
||||
\constant@print{#1}
|
||||
}
|
||||
}{}
|
||||
|
||||
\newcommand{\fsplit}[3][0.5]{
|
||||
\begingroup
|
||||
\renewcommand{\newFormulaEntry}{}
|
||||
\begin{minipage}{##1\linewidth}
|
||||
##2
|
||||
\end{minipage}
|
||||
\begin{minipage}{\luavar{0.99-##1}\linewidth}
|
||||
##3
|
||||
\end{minipage}
|
||||
\endgroup
|
||||
\newFormulaEntry
|
||||
}
|
||||
}{
|
||||
\mqfqname@leave
|
||||
}
|
||||
|
||||
\newenvironment{formula}[1]{
|
||||
\begin{formulainternal}{#1}
|
||||
|
||||
\begingroup
|
||||
\label{f:\fqname:#1}
|
||||
\storeLabel{\fqname:#1} % write label witout type prefix to aux file
|
||||
\mqfqname@label
|
||||
\par\noindent\ignorespaces
|
||||
% \textcolor{gray}{\hrule}
|
||||
% \vspace{0.5\baselineskip}
|
||||
\NameWithDescription[\descwidth]{\fqname:#1}{\fqname:#1_desc}
|
||||
\NameWithDescription[\descwidth]{\fqname}{\fqname:desc}
|
||||
\hfill
|
||||
\begin{ContentBoxWithExplanation}{\fqname:#1_defs}
|
||||
\begin{ContentBoxWithExplanation}{\fqname:defs}
|
||||
}{
|
||||
\end{ContentBoxWithExplanation}
|
||||
\endgroup
|
||||
@ -188,33 +190,28 @@
|
||||
\newenvironment{bigformula}[1]{
|
||||
\begin{formulainternal}{#1}
|
||||
|
||||
\edef\tmpFormulaName{#1}
|
||||
\par\noindent
|
||||
\begin{minipage}{\textwidth} % using a minipage to now allow line breaks within the bigformula
|
||||
\label{f:\fqname:#1}
|
||||
\storeLabel{\fqname:#1} % write label witout type prefix to aux file
|
||||
\mqfqname@label
|
||||
\par\noindent\ignorespaces
|
||||
% \textcolor{gray}{\hrule}
|
||||
% \vspace{0.5\baselineskip}
|
||||
\textbf{
|
||||
\IfTranslationExists{\fqname:#1}{%
|
||||
\raggedright
|
||||
\GT{\fqname:#1}
|
||||
}{\detokenize{#1}}
|
||||
\raggedright
|
||||
\GT{\fqname}
|
||||
}
|
||||
\IfTranslationExists{\fqname:#1_desc}{
|
||||
: {\color{fg1} \GT{\fqname:#1_desc}}
|
||||
\IfTranslationExists{\fqname:desc}{
|
||||
: {\color{fg1} \GT{\fqname:desc}}
|
||||
}{}
|
||||
\hfill
|
||||
\par
|
||||
}{
|
||||
\edef\tmpContentDefs{\fqname:\tmpFormulaName_defs}
|
||||
\IfTranslationExists{\tmpContentDefs}{%
|
||||
\IfTranslationExists{\fqname:defs}{%
|
||||
\smartnewline
|
||||
\noindent
|
||||
\begingroup
|
||||
\color{fg1}
|
||||
\GT{\tmpContentDefs}
|
||||
\GT{\fqname:defs}
|
||||
% \edef\temp{\GT{#1_defs}}
|
||||
% \expandafter\StrSubstitute\expandafter{\temp}{:}{\\}
|
||||
\endgroup
|
||||
@ -230,7 +227,6 @@
|
||||
\newenvironment{hiddenformula}[1]{
|
||||
\begin{formulainternal}{#1}
|
||||
\renewcommand{\eq}[1]{}
|
||||
\renewcommand{\eqAlignedAt}[2]{}
|
||||
\renewcommand{\eqFLAlign}[2]{}
|
||||
\renewcommand{\fig}[2][1.0]{}
|
||||
\renewcommand{\ttxt}[2][#1:desc]{}
|
||||
|
||||
@ -1,53 +1,102 @@
|
||||
\ProvidesPackage{mqfqname}
|
||||
\edef\fqname{NULL}
|
||||
\RequirePackage{mqlua}
|
||||
\RequirePackage{etoolbox}
|
||||
|
||||
|
||||
\directlua{
|
||||
\begin{luacode}
|
||||
sections = sections or {}
|
||||
|
||||
function fqnameEnter(name)
|
||||
table.insert(sections, name)
|
||||
% table.sort(sections)
|
||||
table.insert(sections, name)
|
||||
-- table.sort(sections)
|
||||
end
|
||||
|
||||
function fqnameLeave()
|
||||
if table.getn(sections) > 0 then
|
||||
table.remove(sections)
|
||||
end
|
||||
if table.getn(sections) > 0 then
|
||||
table.remove(sections)
|
||||
end
|
||||
end
|
||||
|
||||
function fqnameGet()
|
||||
return table.concat(sections, ":")
|
||||
return table.concat(sections, ":")
|
||||
end
|
||||
|
||||
function fqnameLeaveOnlyFirstN(n)
|
||||
if n >= 0 then
|
||||
while table.getn(sections) > n do
|
||||
table.remove(sections)
|
||||
if n >= 0 then
|
||||
while table.getn(sections) > n do
|
||||
table.remove(sections)
|
||||
end
|
||||
end
|
||||
end
|
||||
end
|
||||
}
|
||||
\end{luacode}
|
||||
|
||||
\begin{luacode}
|
||||
function fqnameGetDepth()
|
||||
return table.getn(sections)
|
||||
end
|
||||
|
||||
function fqnameGetN(N)
|
||||
if N == nil or table.getn(sections) < N then
|
||||
luatexbase.module_warning('fqnameGetN', 'N = ' .. N .. ' is larger then the table length')
|
||||
return "?!?"
|
||||
end
|
||||
s = sections[1]
|
||||
for i = 2, N do
|
||||
s = s .. ":" .. sections[i]
|
||||
end
|
||||
return s
|
||||
end
|
||||
\end{luacode}
|
||||
|
||||
|
||||
% Allow using :<key>, ::<key> and so on
|
||||
% where : points to current fqname, :: to the upper one and so on
|
||||
\begin{luacode*}
|
||||
function translateRelativeFqname(target)
|
||||
local relN = 0
|
||||
|
||||
local relTarget = ""
|
||||
warning('translateRelativeFqname', '(target=' .. target .. ') ');
|
||||
for i = 1, #target do
|
||||
local c = target:sub(i,i)
|
||||
if c == ":" then
|
||||
relN = relN + 1
|
||||
else
|
||||
relTarget = target:sub(i,#target)
|
||||
break
|
||||
end
|
||||
end
|
||||
if relN == 0 then
|
||||
return target
|
||||
end
|
||||
|
||||
local N = fqnameGetDepth()
|
||||
local newtarget = fqnameGetN(N - relN + 1) .. ":" .. relTarget
|
||||
warning('translateRelativeFqname', '(relN=' .. relN .. ') ' .. newtarget);
|
||||
return newtarget
|
||||
end
|
||||
\end{luacode*}
|
||||
|
||||
\newcommand{\mqfqname@update}{%
|
||||
\edef\fqname{\luavar{fqnameGet()}}
|
||||
\edef\fqname{\luavar{fqnameGet()}} %
|
||||
}
|
||||
\newcommand{\mqfqname@enter}[1]{%
|
||||
\directlua{fqnameEnter("\luaescapestring{#1}")}%
|
||||
\mqfqname@update
|
||||
\directlua{fqnameEnter("\luaescapestring{#1}")}%
|
||||
\mqfqname@update
|
||||
}
|
||||
\newcommand{\mqfqname@leave}{%
|
||||
\directlua{fqnameLeave()}%
|
||||
\mqfqname@update
|
||||
\directlua{fqnameLeave()}%
|
||||
\mqfqname@update
|
||||
}
|
||||
|
||||
\newcommand{\mqfqname@leaveOnlyFirstN}[1]{%
|
||||
\directlua{fqnameLeaveOnlyFirstN(#1)}%
|
||||
\directlua{fqnameLeaveOnlyFirstN(#1)}%
|
||||
}
|
||||
|
||||
% SECTIONING
|
||||
% start <section>, get heading from translation, set label
|
||||
% secFqname is the fully qualified name of sections: the keys of all previous sections joined with a ':'
|
||||
% fqname is the fully qualified name of all sections and formulas, the keys of all previous sections joined with a ':'
|
||||
% fqname is secFqname:<key> where <key> is the key/id of some environment, like formula
|
||||
% [1]: code to run after setting \fqname, but before the \part, \section etc
|
||||
% 2: key
|
||||
@ -55,164 +104,35 @@
|
||||
\newpage
|
||||
\mqfqname@leaveOnlyFirstN{0}
|
||||
\mqfqname@enter{#2}
|
||||
\edef\secFqname{\fqname}
|
||||
#1
|
||||
% this is necessary so that \part/\section... takes the fully expanded string. Otherwise the pdf toc will have just the fqname
|
||||
\edef\fqnameText{\GT{\fqname}}
|
||||
\part{\fqnameText}
|
||||
\label{sec:\fqname}
|
||||
\mqfqname@label
|
||||
}
|
||||
\newcommand{\Section}[2][]{
|
||||
\mqfqname@leaveOnlyFirstN{1}
|
||||
\mqfqname@enter{#2}
|
||||
\edef\secFqname{\fqname}
|
||||
#1
|
||||
\edef\fqnameText{\GT{\fqname}}
|
||||
\section{\fqnameText}
|
||||
\label{sec:\fqname}
|
||||
\mqfqname@label
|
||||
}
|
||||
\newcommand{\Subsection}[2][]{
|
||||
\mqfqname@leaveOnlyFirstN{2}
|
||||
\mqfqname@enter{#2}
|
||||
\edef\secFqname{\fqname}
|
||||
#1
|
||||
\edef\fqnameText{\GT{\fqname}}
|
||||
\subsection{\fqnameText}
|
||||
\label{sec:\fqname}
|
||||
\mqfqname@label
|
||||
}
|
||||
\newcommand{\Subsubsection}[2][]{
|
||||
\mqfqname@leaveOnlyFirstN{3}
|
||||
\mqfqname@enter{#2}
|
||||
\edef\secFqname{\fqname}
|
||||
#1
|
||||
\edef\fqnameText{\GT{\fqname}}
|
||||
\subsubsection{\fqnameText}
|
||||
\label{sec:\fqname}
|
||||
\mqfqname@label
|
||||
}
|
||||
\edef\fqname{NULL}
|
||||
|
||||
\newcommand\printFqName{\expandafter\detokenize\expandafter{\fqname}}
|
||||
|
||||
|
||||
\newcommand\luaDoubleFieldValue[3]{%
|
||||
\directlua{
|
||||
if #1 \string~= nil and #1[#2] \string~= nil and #1[#2][#3] \string~= nil then
|
||||
tex.sprint(#1[#2][#3])
|
||||
return
|
||||
end
|
||||
luatexbase.module_warning('luaDoubleFieldValue', 'Invalid indices to `#1`: `#2` and `#3`');
|
||||
tex.sprint("???")
|
||||
}%
|
||||
}
|
||||
% REFERENCES
|
||||
% All xyzRef commands link to the key using the translated name
|
||||
% Uppercase (XyzRef) commands have different link texts, but the same link target
|
||||
% 1: key/fully qualified name (without qty/eq/sec/const/el... prefix)
|
||||
|
||||
% Equations/Formulas
|
||||
% \newrobustcmd{\fqEqRef}[1]{%
|
||||
\newrobustcmd{\fqEqRef}[1]{%
|
||||
% \edef\fqeqrefname{\GT{#1}}
|
||||
% \hyperref[eq:#1]{\fqeqrefname}
|
||||
\hyperref[f:#1]{\GT{#1}}%
|
||||
}
|
||||
% Formula in the current section
|
||||
\newrobustcmd{\secEqRef}[1]{%
|
||||
% \edef\fqeqrefname{\GT{#1}}
|
||||
% \hyperref[eq:#1]{\fqeqrefname}
|
||||
\hyperref[f:\secFqname:#1]{\GT{\secFqname:#1}}%
|
||||
}
|
||||
|
||||
% Section
|
||||
% <name>
|
||||
\newrobustcmd{\fqSecRef}[1]{%
|
||||
\hyperref[sec:#1]{\GT{#1}}%
|
||||
}
|
||||
% Quantities
|
||||
% <symbol>
|
||||
\newrobustcmd{\qtyRef}[1]{%
|
||||
\edef\tempname{\luaDoubleFieldValue{quantities}{"#1"}{"linkto"}}%
|
||||
\hyperref[qty:#1]{\GT{\tempname:#1}}%
|
||||
}
|
||||
% <symbol> <name>
|
||||
\newrobustcmd{\QtyRef}[1]{%
|
||||
$\luaDoubleFieldValue{quantities}{"#1"}{"symbol"}$ \qtyRef{#1}%
|
||||
}
|
||||
% Constants
|
||||
% <name>
|
||||
\newrobustcmd{\constRef}[1]{%
|
||||
\edef\tempname{\luaDoubleFieldValue{constants}{"#1"}{"linkto"}}%
|
||||
\hyperref[const:#1]{\GT{\tempname:#1}}%
|
||||
}
|
||||
% <symbol> <name>
|
||||
\newrobustcmd{\ConstRef}[1]{%
|
||||
$\luaDoubleFieldValue{constants}{"#1"}{"symbol"}$ \constRef{#1}%
|
||||
}
|
||||
% Element from periodic table
|
||||
% <symbol>
|
||||
\newrobustcmd{\elRef}[1]{%
|
||||
\hyperref[el:#1]{{\color{fg0}#1}}%
|
||||
}
|
||||
% <name>
|
||||
\newrobustcmd{\ElRef}[1]{%
|
||||
\hyperref[el:#1]{\GT{el:#1}}%
|
||||
}
|
||||
|
||||
|
||||
|
||||
% "LABELS"
|
||||
% These currently do not place a label,
|
||||
% instead they provide an alternative way to reference an existing label
|
||||
\directLuaAux{
|
||||
absLabels = absLabels or {}
|
||||
abbrLabels = abbrLabel or {}
|
||||
}
|
||||
% [1]: translation key, if different from target
|
||||
% 2: target (fqname to point to)
|
||||
% 3: key
|
||||
\newcommand{\absLink}[3][\relax]{
|
||||
\directLuaAuxExpand{
|
||||
absLabels["#3"] = {}
|
||||
absLabels["#3"]["fqname"] = [[#2]]
|
||||
absLabels["#3"]["translation"] = [[#1]] or [[#2]]
|
||||
% if [[#1]] == "" then
|
||||
% absLabels["#3"]["translation"] = [[#2]]
|
||||
% else
|
||||
% absLabels["#3"]["translation"] = [[#1]]
|
||||
% end
|
||||
}
|
||||
}
|
||||
% [1]: target (fqname to point to)
|
||||
% 2: key
|
||||
% 3: label (abbreviation)
|
||||
\newcommand{\abbrLink}[3][sec:\fqname]{
|
||||
\directLuaAuxExpand{
|
||||
abbrLabels["#2"] = {}
|
||||
abbrLabels["#2"]["abbr"] = [[#3]]
|
||||
abbrLabels["#2"]["fqname"] = [[#1]]
|
||||
}
|
||||
}
|
||||
% [1]: text
|
||||
% 2: key
|
||||
\newcommand{\absRef}[2][]{%
|
||||
\directlua{
|
||||
if absLabels["#2"] == nil then
|
||||
tex.sprint(string.sanitize(\luastring{#2}) .. "???")
|
||||
else
|
||||
if \luastring{#1} == "" then %-- if [#1] is not given, use translation of key as text, else us given text
|
||||
tex.sprint("\\hyperref[" .. absLabels["#2"]["fqname"] .. "]{\\GT{" .. absLabels["#2"]["translation"] .. "}}")
|
||||
else
|
||||
tex.sprint("\\hyperref[" .. absLabels["#2"]["fqname"] .. "]{\luaescapestring{#1}}")
|
||||
end
|
||||
end
|
||||
}
|
||||
}
|
||||
\newrobustcmd{\abbrRef}[1]{%
|
||||
\directlua{
|
||||
if abbrLabels["#1"] == nil then
|
||||
tex.sprint(string.sanitize(\luastring{#1}) .. "???")
|
||||
else
|
||||
tex.sprint("\\hyperref[" .. abbrLabels["#1"]["fqname"] .. "]{" .. abbrLabels["#1"]["abbr"] .. "}")
|
||||
end
|
||||
}
|
||||
}
|
||||
|
||||
@ -6,13 +6,17 @@
|
||||
\newcommand\luavar[1]{\directlua{tex.sprint(#1)}}
|
||||
|
||||
\begin{luacode*}
|
||||
function warning(message)
|
||||
function warning(fname, message)
|
||||
-- Get the current file name and line number
|
||||
-- local info = debug.getinfo(2, "Sl")
|
||||
-- local file_name = info.source
|
||||
-- local line_number = info.currentline
|
||||
-- tex.error(string.format("Warning %s at %s:%d", message, file_name, line_number))
|
||||
texio.write("\nWARNING: " .. message .. "\n")
|
||||
if message == nil then
|
||||
texio.write("\nWARNING: " .. fname .. "\n")
|
||||
else
|
||||
texio.write("\nWARNING: in " .. fname .. ":" .. message .. "\n")
|
||||
end
|
||||
end
|
||||
|
||||
OUTDIR = os.getenv("TEXMF_OUTPUT_DIRECTORY") or "."
|
||||
@ -22,17 +26,15 @@ function fileExists(file)
|
||||
if f then f:close() end
|
||||
return f ~= nil
|
||||
end
|
||||
|
||||
|
||||
warning("TEST")
|
||||
\end{luacode*}
|
||||
|
||||
% units: siunitx units arguments, possibly chained by '='
|
||||
% returns: 1\si{unit1} = 1\si{unit2} = ...
|
||||
\directlua{
|
||||
\begin{luacode*}
|
||||
function split_and_print_units(units)
|
||||
if units == nil then
|
||||
tex.print("1")
|
||||
tex.sprint("1")
|
||||
return
|
||||
end
|
||||
|
||||
@ -47,21 +49,22 @@ function split_and_print_units(units)
|
||||
end
|
||||
tex.print(result)
|
||||
end
|
||||
}
|
||||
\end{luacode*}
|
||||
|
||||
% STRING UTILITY
|
||||
\luadirect{
|
||||
\begin{luacode*}
|
||||
function string.startswith(s, start)
|
||||
return string.sub(s,1,string.len(start)) == start
|
||||
end
|
||||
|
||||
function string.sanitize(s)
|
||||
% -- Use gsub to replace the specified characters with an empty string
|
||||
-- Use gsub to replace the specified characters with an empty string
|
||||
local result = s:gsub("[_^&]", " ")
|
||||
return result
|
||||
end
|
||||
}
|
||||
\end{luacode*}
|
||||
% Write directlua command to aux and run it as well
|
||||
% THESE CAN ONLY BE RUN BETWEEN \begin{document} and \end{document}
|
||||
% This one expands the argument in the aux file:
|
||||
\newcommand\directLuaAuxExpand[1]{
|
||||
\immediate\write\luaAuxFile{\noexpand\directlua{#1}}
|
||||
@ -74,15 +77,17 @@ end
|
||||
}
|
||||
|
||||
% read
|
||||
\IfFileExists{\jobname.lua.aux}{%
|
||||
\input{\jobname.lua.aux}%
|
||||
}{%
|
||||
% \@latex@warning@no@line{"Lua aux not loaded!"}
|
||||
\AtBeginDocument{
|
||||
\IfFileExists{\jobname.lua.aux}{%
|
||||
\input{\jobname.lua.aux}%
|
||||
}{%
|
||||
% \@latex@warning@no@line{"Lua aux not loaded!"}
|
||||
}
|
||||
% write
|
||||
\newwrite\luaAuxFile
|
||||
\immediate\openout\luaAuxFile=\jobname.lua.aux
|
||||
\immediate\write\luaAuxFile{\noexpand\def\noexpand\luaAuxLoaded{True}}%
|
||||
}
|
||||
\def\luaAuxLoaded{False}
|
||||
|
||||
% write
|
||||
\newwrite\luaAuxFile
|
||||
\immediate\openout\luaAuxFile=\jobname.lua.aux
|
||||
\immediate\write\luaAuxFile{\noexpand\def\noexpand\luaAuxLoaded{True}}%
|
||||
\AtEndDocument{\immediate\closeout\luaAuxFile}
|
||||
|
||||
@ -5,13 +5,28 @@
|
||||
% Print as list or as periodic table
|
||||
% The data is taken from https://pse-info.de/de/data as json and parsed by the scripts/periodic_table.py
|
||||
|
||||
% INFO
|
||||
\directLuaAux{
|
||||
if elements == nil then
|
||||
elements = {} %-- Symbol: {symbol, atomic_number, properties, ... }
|
||||
elementsOrder = {} %-- Number: Symbol
|
||||
\begin{luacode}
|
||||
elements = {}
|
||||
elementsOrder = {}
|
||||
|
||||
function elementAdd(symbol, nr, period, column)
|
||||
--elementsOrder[nr] = symbol
|
||||
table.insert(elementsOrder, symbol)
|
||||
elements[symbol] = {
|
||||
symbol = symbol,
|
||||
atomic_number = nr,
|
||||
period = period,
|
||||
column = column,
|
||||
properties = {}
|
||||
}
|
||||
end
|
||||
|
||||
function elementAddProperty(symbol, key, value)
|
||||
if elements[symbol] and elements[symbol].properties then
|
||||
elements[symbol].properties[key] = value
|
||||
end
|
||||
}
|
||||
end
|
||||
\end{luacode}
|
||||
|
||||
% 1: symbol
|
||||
% 2: nr
|
||||
@ -23,30 +38,22 @@
|
||||
% 3: description
|
||||
% 4: definitions/links
|
||||
\newcommand{\desc}[4][english]{
|
||||
% language, name, description, definitions
|
||||
\ifblank{##2}{}{\DT[el:#1]{##1}{##2}}
|
||||
\ifblank{##3}{}{\DT[el:#1_desc]{##1}{##3}}
|
||||
\ifblank{##4}{}{\DT[el:#1_defs]{##1}{##4}}
|
||||
}
|
||||
\directLuaAux{
|
||||
elementsOrder[#2] = "#1";
|
||||
elements["#1"] = {};
|
||||
elements["#1"]["symbol"] = [[\detokenize{#1}]];
|
||||
elements["#1"]["atomic_number"] = [[\detokenize{#2}]];
|
||||
elements["#1"]["period"] = [[\detokenize{#3}]];
|
||||
elements["#1"]["column"] = [[\detokenize{#4}]];
|
||||
elements["#1"]["properties"] = {};
|
||||
\directLuaAuxExpand{
|
||||
elementAdd(\luastring{#1}, \luastring{#2}, \luastring{#3}, \luastring{#4})
|
||||
}
|
||||
% 1: key
|
||||
% 2: value
|
||||
\newcommand{\property}[2]{
|
||||
\directlua{ %-- writing to aux is only needed for references for now
|
||||
elements["#1"]["properties"]["##1"] = "\luaescapestring{\detokenize{##2}}" %-- cant use [[ ]] because electron_config ends with ]
|
||||
\directlua{
|
||||
elementAddProperty(\luastring{#1}, \luastringN{##1}, \luastringN{##2})
|
||||
}
|
||||
}
|
||||
\edef\lastElementName{#1}
|
||||
}{
|
||||
% \expandafter\printElement{\lastElementName}
|
||||
\ignorespacesafterend
|
||||
}
|
||||
|
||||
@ -56,9 +63,7 @@
|
||||
\par\noindent\ignorespaces
|
||||
\vspace{0.5\baselineskip}
|
||||
\begingroup
|
||||
% label it only once
|
||||
% \detokenize{\label{el:#1}}
|
||||
\directlua{
|
||||
\directlua{
|
||||
if elements["#1"]["labeled"] == nil then
|
||||
elements["#1"]["labeled"] = true
|
||||
tex.print("\\phantomsection\\label{el:#1}")
|
||||
@ -70,12 +75,8 @@
|
||||
\directlua{
|
||||
tex.sprint("Symbol: \\ce{"..elements["#1"]["symbol"].."}")
|
||||
tex.sprint("\\\\Number: "..elements["#1"]["atomic_number"])
|
||||
}
|
||||
\directlua{
|
||||
%--tex.sprint("Hier steht Luatext" .. ":", #elementVals)
|
||||
for key, value in pairs(elements["#1"]["properties"]) do
|
||||
tex.sprint("\\\\\\hspace*{1cm}{\\GT{", key, "}: ", value, "}")
|
||||
%--tex.sprint("VALUE ", i, v)
|
||||
tex.sprint("\\\\\\hspace*{1cm}{\\GT{"..key.."}: "..value.."}")
|
||||
end
|
||||
}
|
||||
\end{ContentBoxWithExplanation}
|
||||
@ -84,6 +85,7 @@
|
||||
\vspace{0.5\baselineskip}
|
||||
\ignorespacesafterend
|
||||
}
|
||||
|
||||
\newcommand{\printAllElements}{
|
||||
\directlua{
|
||||
%-- tex.sprint("\\printElement{"..val.."}")
|
||||
|
||||
@ -1,41 +1,59 @@
|
||||
\ProvidesPackage{mqquantity}
|
||||
\RequirePackage{mqlua}
|
||||
\RequirePackage{mqfqname}
|
||||
\RequirePackage{etoolbox}
|
||||
|
||||
\directLuaAux{
|
||||
quantities = quantities or {}
|
||||
}
|
||||
% TODO: MAYBE:
|
||||
% store the fqname where the quantity is defined
|
||||
% In qtyRef then use the stored label to reference it, instead of linking to qty:<name>
|
||||
% Use the mqlua hyperref function
|
||||
|
||||
% [1]: label to point to
|
||||
\begin{luacode}
|
||||
quantities = {}
|
||||
function quantityAdd(key, symbol, units, comment, fqname)
|
||||
quantities[key] = {
|
||||
["symbol"] = symbol,
|
||||
["units"] = units,
|
||||
["comment"] = comment
|
||||
}
|
||||
if fqname == "" then
|
||||
quantities[key]["fqname"] = fqnameGet()
|
||||
else
|
||||
quantities[key]["fqname"] = fqname
|
||||
end
|
||||
end
|
||||
function quantityGetSymbol(key)
|
||||
local qty = quantities[key]
|
||||
if qty == nil then return "???" end
|
||||
local symbol = qty["symbol"]
|
||||
if symbol == nil then return "???" end
|
||||
return symbol
|
||||
end
|
||||
function quantityGetFqname(key)
|
||||
local qty = quantities[key]
|
||||
if qty == nil then return "qty:"..key end
|
||||
local fqname_ = qty["fqname"]
|
||||
if fqname_ == nil then return "qty:"..key end
|
||||
return fqname_
|
||||
end
|
||||
\end{luacode}
|
||||
|
||||
% [1]: label to point to, if not given use current fqname
|
||||
% 2: key - must expand to a valid lua string!
|
||||
% 3: symbol
|
||||
% 4: units
|
||||
% 5: comment key to translation
|
||||
\newcommand{\quantity@new}[5][\relax]{%
|
||||
\directLuaAux{
|
||||
quantities["#2"] = {}
|
||||
quantities["#2"]["symbol"] = [[\detokenize{#3}]]
|
||||
quantities["#2"]["units"] = [[\detokenize{#4}]]
|
||||
quantities["#2"]["comment"] = [[\detokenize{#5}]]
|
||||
}
|
||||
\ifstrempty{#1}{}{
|
||||
\directLuaAuxExpand{
|
||||
quantities["#2"]["linkto"] = [[#1]] %-- fqname required for getting the translation key
|
||||
}
|
||||
}
|
||||
\newcommand{\quantity@new}[5][]{%
|
||||
\directLuaAuxExpand{quantityAdd(\luastring{#2}, \luastringN{#3}, \luastringN{#4}, \luastringN{#5}, \luastring{#1})}
|
||||
}
|
||||
|
||||
% 1: key
|
||||
\newcommand{\quantity@getSymbol}[1]{\luavar{quantityGetSymbol(\luastring{#1})}}
|
||||
|
||||
% 1: key
|
||||
\newcommand\quantity@print[1]{
|
||||
\begingroup % for label
|
||||
Symbol: $\luavar{quantities["#1"]["symbol"]}$
|
||||
Symbol: $\luavar{quantityGetSymbol(\luastring{#1})}$
|
||||
\hfill Unit: $\directlua{split_and_print_units(quantities["#1"]["units"])}$ %
|
||||
% label it only once
|
||||
\directlua{
|
||||
if quantities["#1"]["labeled"] == nil then
|
||||
quantities["#1"]["labeled"] = true
|
||||
tex.print("\\label{qty:#1}")
|
||||
end
|
||||
}%
|
||||
\endgroup%
|
||||
}
|
||||
|
||||
271
src/pkg/mqref.sty
Normal file
271
src/pkg/mqref.sty
Normal file
@ -0,0 +1,271 @@
|
||||
\ProvidesPackage{mqref}
|
||||
\RequirePackage{mqlua}
|
||||
\RequirePackage{mqfqname}
|
||||
\RequirePackage{mqquantity}
|
||||
|
||||
\newcommand\luaDoubleFieldValue[3]{%
|
||||
\directlua{
|
||||
if #1 \string~= nil and #1[#2] \string~= nil and #1[#2][#3] \string~= nil then
|
||||
tex.sprint(#1[#2][#3])
|
||||
return
|
||||
end
|
||||
luatexbase.module_warning('luaDoubleFieldValue', 'Invalid indices to `#1`: `#2` and `#3`');
|
||||
tex.sprint("???")
|
||||
}%
|
||||
}
|
||||
|
||||
% LABELS
|
||||
\begin{luacode}
|
||||
-- Contains <label>: <true> for defined labels
|
||||
-- This could later be extended to contain a list of all fqnames that
|
||||
-- reference the label to make a network of references or sth like that
|
||||
labels = labels or {}
|
||||
-- Table of all labels that dont exist but were referenced
|
||||
-- <label>: <fqname where it was referenced>
|
||||
missingLabels = {}
|
||||
-- aux file with labels for completion in vim
|
||||
labelsFilepath = OUTDIR .. "/labels.txt" or "/tmp/labels.txt"
|
||||
labelsLuaFilepath = OUTDIR .. "/labels.lua.txt" or "/tmp/labels.lua.txt"
|
||||
-- aux file for debugging
|
||||
missingLabelsFilepath = OUTDIR .. "/missing-labels.txt" or "/tmp/missing-labels.txt"
|
||||
function labelExists(label)
|
||||
if labels[label] == nil then return false else return true end
|
||||
end
|
||||
function labelSet(label)
|
||||
labels[label] = true
|
||||
end
|
||||
if fileExists(labelsLuaFilepath) then
|
||||
labels = dofile(labelsLuaFilepath) or {}
|
||||
end
|
||||
\end{luacode}
|
||||
|
||||
\begin{luacode*}
|
||||
function serializeKeyValues(tbl)
|
||||
local result = {}
|
||||
-- sort by keys making a new table with keys as values and sorting that
|
||||
for k, v in pairs(tbl) do
|
||||
table.insert(result, k)
|
||||
end
|
||||
table.sort(result)
|
||||
s = ""
|
||||
for i, k in ipairs(result) do
|
||||
s = s .. k .. "\tin\t" .. tbl[k] .. "\n"
|
||||
end
|
||||
return s
|
||||
end
|
||||
|
||||
function dumpTableKeyValues(tableobj, filepath)
|
||||
table.sort(tableobj)
|
||||
local file = io.open(filepath, "w")
|
||||
file:write(serializeKeyValues(tableobj))
|
||||
file:close()
|
||||
end
|
||||
|
||||
function serializeKeys(tbl)
|
||||
local result = {}
|
||||
-- sort by keys making a new table with keys as values and sorting that
|
||||
for k, v in pairs(tbl) do
|
||||
table.insert(result, k)
|
||||
end
|
||||
table.sort(result)
|
||||
return table.concat(result, "\n")
|
||||
end
|
||||
|
||||
function dumpTableKeys(tableobj, filepath)
|
||||
table.sort(tableobj)
|
||||
local file = io.open(filepath, "w")
|
||||
file:write(serializeKeys(tableobj))
|
||||
file:close()
|
||||
end
|
||||
\end{luacode*}
|
||||
|
||||
\AtEndDocument{\directlua{dumpTableKeys(labels, labelsFilepath)}}
|
||||
\AtEndDocument{\directlua{dumpTable(labels, labelsLuaFilepath)}}
|
||||
\AtEndDocument{\directlua{dumpTableKeyValues(missingLabels, missingLabelsFilepath)}}
|
||||
|
||||
% Set a label and write the label to the aux file
|
||||
% [1]
|
||||
\newcommand\mqfqname@label[1][\fqname]{
|
||||
\label{#1}
|
||||
\directlua{labelSet(\luastring{#1})}
|
||||
}
|
||||
|
||||
% REFERENCES
|
||||
% All xyzRef commands link to the key using the translated name
|
||||
% Uppercase (XyzRef) commands have different link texts, but the same link target
|
||||
% 1: key/fully qualified name (without qty/eq/sec/const/el... prefix)
|
||||
|
||||
\begin{luacode*}
|
||||
function hyperref(target, text)
|
||||
local s = ""
|
||||
if labelExists(target) then
|
||||
s = "\\hyperref[" .. target .. "]"
|
||||
else -- mark as missing and referenced in current section
|
||||
missingLabels[target] = fqnameGet()
|
||||
end
|
||||
if text == nil or text == "" then
|
||||
tex.sprint(s .. "{" .. tlGetFallbackCurrent(target) .. "}")
|
||||
else
|
||||
tex.sprint(s .. "{" .. text .. "}")
|
||||
end
|
||||
end
|
||||
\end{luacode*}
|
||||
|
||||
|
||||
% Equations/Formulas
|
||||
% \newrobustcmd{\fqEqRef}[1]{%
|
||||
\newrobustcmd{\fAbsRef}[2][]{%
|
||||
\directlua{hyperref(\luastring{#2}, \luastring{#1})}%
|
||||
}
|
||||
|
||||
\newcommand{\fRef}[2][]{
|
||||
\directlua{hyperref(translateRelativeFqname(\luastring{#2}), \luastring{#1})}%
|
||||
}
|
||||
% [1]: link text
|
||||
% 2: number of steps to take up
|
||||
% 3: link target relative to the previous fqname section
|
||||
\newcommand{\mqfqname@fRelRef}[3][1]{
|
||||
\directlua{
|
||||
local N = fqnameGetDepth()
|
||||
luatexbase.module_warning('fRelRef', '(N=' .. N .. ') #2');
|
||||
if N > #2 then
|
||||
local upfqname = fqnameGetN(N-#2)
|
||||
hyperref(upfqname .. \luastring{:#3}, \luastring{#1})
|
||||
else
|
||||
luatexbase.module_warning('fUpRef', 'fqname depth (N=' .. N .. ') too low for fUpRef if #1');
|
||||
end
|
||||
}
|
||||
}
|
||||
\newcommand{\fThisRef}[2][]{\mqfqname@fRelRef[#1]{0}{#2}}
|
||||
\newcommand{\fUpRef}[2][]{\mqfqname@fRelRef[#1]{1}{#2}}
|
||||
\newcommand{\fUppRef}[2][]{\mqfqname@fRelRef[#1]{2}{#2}}
|
||||
|
||||
% Quantities
|
||||
% <symbol>
|
||||
\newrobustcmd{\qtyRef}[2][]{%
|
||||
% \edef\tempname{\luaDoubleFieldValue{quantities}{"#1"}{"fqname"}}%
|
||||
% \hyperref[qty:#1]{\GT{\tempname}}%
|
||||
\directlua{hyperref(quantityGetFqname(\luastring{#2}), \luastring{#1})}%
|
||||
}
|
||||
% <symbol> <name>
|
||||
\newrobustcmd{\QtyRef}[2][]{%
|
||||
$\quantity@getSymbol{#2}$ \qtyRef{#2}{}%
|
||||
}
|
||||
% Constants
|
||||
% <name>
|
||||
\newrobustcmd{\constRef}[2][]{%
|
||||
% \edef\tempname{\luaDoubleFieldValue{constants}{"#1"}{"linkto"}}%
|
||||
% \hyperref[const:#1]{\GT{\tempname}}%
|
||||
\directlua{hyperref(constantGetFqname(\luastring{#2}), \luastring{#1})}%
|
||||
}
|
||||
% <symbol> <name>
|
||||
\newrobustcmd{\ConstRef}[2][]{%
|
||||
$\constant@getSymbol{#2}$ \constRef{#2}%
|
||||
}
|
||||
% Element from periodic table
|
||||
% <symbol>
|
||||
\newrobustcmd{\elRef}[1]{%
|
||||
\hyperref[el:#1]{{\color{fg0}#1}}%
|
||||
}
|
||||
% <name>
|
||||
\newrobustcmd{\ElRef}[1]{%
|
||||
\hyperref[el:#1]{\GT{el:#1}}%
|
||||
}
|
||||
|
||||
|
||||
|
||||
% "LABELS"
|
||||
% These currently do not place a label,
|
||||
% instead they provide an alternative way to reference an existing label
|
||||
\begin{luacode}
|
||||
absLabels = absLabels or {}
|
||||
abbrLabels = abbrLabels or {}
|
||||
|
||||
function absLabelAdd(key, target, translationKey)
|
||||
absLabels[key] = {
|
||||
fqname = (target == "") and fqnameGet() or target,
|
||||
translation = translationKey or ""
|
||||
}
|
||||
end
|
||||
|
||||
function absLabelGetTarget(key)
|
||||
if absLabels[key] then
|
||||
return absLabels[key].fqname or "abs:" .. key
|
||||
else
|
||||
return "abs:" .. key
|
||||
end
|
||||
end
|
||||
|
||||
function absLabelGetTranslationKey(key)
|
||||
if absLabels[key] then
|
||||
return absLabels[key].translation or ""
|
||||
else
|
||||
return ""
|
||||
end
|
||||
end
|
||||
|
||||
|
||||
function abbrLabelAdd(key, target, label)
|
||||
abbrLabels[key] = {
|
||||
abbr = label,
|
||||
fqname = (target == "") and fqnameGet() or target
|
||||
}
|
||||
end
|
||||
|
||||
function abbrLabelGetTarget(key)
|
||||
if abbrLabels[key] then
|
||||
return abbrLabels[key].fqname or "abbr:" .. key
|
||||
else
|
||||
return "abbr:" .. key
|
||||
end
|
||||
end
|
||||
|
||||
function abbrLabelGetAbbr(key)
|
||||
if abbrLabels[key] then
|
||||
return abbrLabels[key].abbr or ""
|
||||
else
|
||||
return ""
|
||||
end
|
||||
end
|
||||
\end{luacode}
|
||||
|
||||
% [1]: translation key, if different from target
|
||||
% 2: target (fqname to point to), if left empty will use current fqname
|
||||
% 3: key
|
||||
\newcommand{\absLink}[3][]{
|
||||
\directLuaAuxExpand{
|
||||
absLabelAdd(\luastring{#3}, \luastring{#2}, \luastring{#1})
|
||||
}
|
||||
}
|
||||
|
||||
% [1]: target (fqname to point to)
|
||||
% 2: key
|
||||
% 3: label (abbreviation)
|
||||
\newcommand{\abbrLink}[3][]{
|
||||
\directLuaAuxExpand{
|
||||
abbrLabelAdd(\luastring{#2}, \luastring{#1}, \luastring{#3})
|
||||
}
|
||||
}
|
||||
|
||||
|
||||
|
||||
% [1]: text
|
||||
% 2: key
|
||||
\newcommand{\absRef}[2][]{%
|
||||
\directlua{
|
||||
local text = (\luastring{#1} == "") and absLabelGetTranslationKey(\luastring{#2}) or \luastring{#1}
|
||||
if text \string~= "" then
|
||||
text = tlGetFallbackCurrent(text)
|
||||
end
|
||||
hyperref(absLabelGetTarget(\luastring{#2}, text))
|
||||
}%
|
||||
}
|
||||
\newrobustcmd{\abbrRef}[1]{%
|
||||
\directlua{hyperref(abbrLabelGetTarget(\luastring{#1}), abbrLabelGetAbbr(\luastring{#1}))}
|
||||
% if abbrLabels["#1"] == nil then
|
||||
% tex.sprint(string.sanitize(\luastring{#1}) .. "???")
|
||||
% else
|
||||
% tex.sprint("\\hyperref[" .. abbrLabels["#1"]["fqname"] .. "]{" .. abbrLabels["#1"]["abbr"] .. "}")
|
||||
% end
|
||||
% }
|
||||
}
|
||||
@ -8,8 +8,9 @@
|
||||
|
||||
\begin{luacode}
|
||||
translations = translations or {}
|
||||
-- string to append to missing translations
|
||||
-- string to append to missing translations, for debugging
|
||||
-- unknownTranslation = "???"
|
||||
-- unknownTranslation = "!UT!"
|
||||
unknownTranslation = ""
|
||||
-- language that is set in usepackage[<lang>]{babel}
|
||||
language = "\languagename"
|
||||
@ -85,9 +86,9 @@
|
||||
end
|
||||
|
||||
|
||||
function dumpTranslations()
|
||||
local file = io.open(translationsFilepath, "w")
|
||||
file:write("return " .. serialize(translations) .. "\n")
|
||||
function dumpTable(tableobj, filepath)
|
||||
local file = io.open(filepath, "w")
|
||||
file:write("return " .. serialize(tableobj) .. "\n")
|
||||
file:close()
|
||||
end
|
||||
|
||||
@ -97,7 +98,7 @@
|
||||
\end{luacode*}
|
||||
|
||||
|
||||
\AtEndDocument{\directlua{dumpTranslations()}}
|
||||
\AtEndDocument{\directlua{dumpTable(translations, translationsFilepath)}}
|
||||
|
||||
%
|
||||
% TRANSLATION COMMANDS
|
||||
@ -117,8 +118,8 @@
|
||||
% shortcuts for translations
|
||||
% 1: key
|
||||
\newcommand{\gt}[1]{\luavar{tlGetFallbackCurrent(\luastring{\fqname:#1})}}
|
||||
\newrobustcmd{\robustGT}[1]{\luavar{tlGetFallbackCurrent(\luastring{#1})}}
|
||||
\newcommand{\GT}[1]{\luavar{tlGetFallbackCurrent(\luastring{#1})}}
|
||||
\newrobustcmd{\robustGT}[1]{\luavar{tlGetFallbackCurrent(translateRelativeFqname(\luastring{#1}))}}
|
||||
\newcommand{\GT}[1]{\luavar{tlGetFallbackCurrent(translateRelativeFqname(\luastring{#1}))}}
|
||||
|
||||
% text variants for use in math mode
|
||||
\newcommand{\tgt}[1]{\text{\gt{#1}}}
|
||||
|
||||
@ -4,7 +4,7 @@
|
||||
\Section[
|
||||
\eng{Hydrogen Atom}
|
||||
\ger{Wasserstoffatom}
|
||||
]{h}
|
||||
]{h}
|
||||
|
||||
\begin{formula}{reduced_mass}
|
||||
\desc{Reduced mass}{}{}
|
||||
@ -28,7 +28,7 @@
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{wave_function}
|
||||
\desc{Wave function}{}{$R_{nl}(r)$ \fqEqRef{qm:h:radial}, $Y_{lm}$ \fqEqRef{qm:spherical_harmonics}}
|
||||
\desc{Wave function}{}{$R_{nl}(r)$ \fRef{qm:h:radial}, $Y_{lm}$ \fRef{qm:spherical_harmonics}}
|
||||
\desc[german]{Wellenfunktion}{}{}
|
||||
\eq{\psi_{nlm}(r, \theta, \phi) = R_{nl}(r)Y_{lm}(\theta,\phi)}
|
||||
\end{formula}
|
||||
@ -50,7 +50,7 @@
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{rydberg_constant_heavy}
|
||||
\desc{Rydberg constant}{for heavy atoms}{\ConstRef{electron_mass}, \ConstRef{elementary_charge}, \QtyRef{vacuum_permittivity}, \ConstRef{planck}, \ConstRef{vacuum_speed_of_light}}
|
||||
\desc{Rydberg constant}{for heavy atoms}{\ConstRef{electron_mass}, \ConstRef{charge}, \ConstRef{vacuum_permittivity}, \ConstRef{planck}, \ConstRef{vacuum_speed_of_light}}
|
||||
\desc[german]{Rydberg-Konstante}{für schwere Atome}{}
|
||||
\constant{R_\infty}{exp}{
|
||||
\val{10973731.568157(12)}{\per\m}
|
||||
@ -61,7 +61,7 @@
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{rydberg_constant_corrected}
|
||||
\desc{Rydberg constant}{corrected for nucleus mass $M$}{\QtyRef{rydberg_constant_heavy}, $\mu = \left(\frac{1}{m_\txe} + \frac{1}{M}\right)^{-1}$ \GT{reduced_mass}, \ConstRef{electron_mass}}
|
||||
\desc{Rydberg constant}{corrected for nucleus mass $M$}{\ConstRef{rydberg_constant_heavy}, $\mu = \left(\frac{1}{m_\txe} + \frac{1}{M}\right)^{-1}$ \GT{reduced_mass}, \ConstRef{electron_mass}}
|
||||
\desc[german]{Rydberg Konstante}{korrigiert für Kernmasse $M$}{}
|
||||
\eq{R_\txM = \frac{\mu}{m_\txe} R_\infty}
|
||||
\end{formula}
|
||||
@ -85,15 +85,15 @@
|
||||
\Subsection[
|
||||
\eng{Corrections}
|
||||
\ger{Korrekturen}
|
||||
]{corrections}
|
||||
]{corrections}
|
||||
|
||||
\Subsubsection[
|
||||
\eng{Darwin term}
|
||||
\ger{Darwin-Term}
|
||||
]{darwin}
|
||||
]{darwin}
|
||||
\begin{ttext}[desc]
|
||||
\eng{Relativisitc correction: Because of the electrons zitterbewegung, it is not entirely localised. \TODO{fact check}}
|
||||
\ger{Relativistische Korrektur: Elektronen führen eine Zitterbewegung aus und sind nicht vollständig lokalisiert.}
|
||||
\eng{Relativisitc correction: Accounts for interaction with nucleus (non-zero wavefunction at nucleaus position)}
|
||||
\ger{Relativistische Korrektur: Berücksichtigt die Interatkion mit dem Kern (endliche Wellenfunktion bei der Kernposition)}
|
||||
\end{ttext}
|
||||
\begin{formula}{energy_shift}
|
||||
\desc{Energy shift}{}{}
|
||||
@ -110,7 +110,7 @@
|
||||
\Subsubsection[
|
||||
\eng{Spin-orbit coupling (LS-coupling)}
|
||||
\ger{Spin-Bahn-Kopplung (LS-Kopplung)}
|
||||
]{ls_coupling}
|
||||
]{ls_coupling}
|
||||
\begin{ttext}[desc]
|
||||
\eng{The interaction of the electron spin with the electrostatic field of the nuclei lead to energy shifts.}
|
||||
\ger{The Wechselwirkung zwischen dem Elektronenspin und dem elektrostatischen Feld des Kerns führt zu Energieverschiebungen.}
|
||||
@ -131,10 +131,10 @@
|
||||
\Subsubsection[
|
||||
\eng{Fine-structure}
|
||||
\ger{Feinstruktur}
|
||||
]{fine_structure}
|
||||
]{fine_structure}
|
||||
\begin{ttext}[desc]
|
||||
\eng{The fine-structure combines relativistic corrections \ref{sec:qm:h:corrections:darwin} and the spin-orbit coupling \ref{sec:qm:h:corrections:ls_coupling}.}
|
||||
\ger{Die Feinstruktur vereint relativistische Korrekturen \ref{sec:qm:h:corrections:darwin} und die Spin-Orbit-Kupplung \ref{sec:qm:h:corrections:ls_coupling}.}
|
||||
\eng{The fine-structure combines \fRef[relativistic corrections]{qm:h:corrections:darwin} and \fRef{qm:h:corrections:ls_coupling}.}
|
||||
\ger{Die Feinstruktur vereint \fRef[relativistische Korrekturen]{qm:h:corrections:darwin} und \fRef{qm:h:corrections:ls_coupling}.}
|
||||
\end{ttext}
|
||||
\begin{formula}{energy_shift}
|
||||
\desc{Energy shift}{}{}
|
||||
@ -146,7 +146,7 @@
|
||||
\Subsubsection[
|
||||
\eng{Lamb-shift}
|
||||
\ger{Lamb-Shift}
|
||||
]{lamb_shift}
|
||||
]{lamb_shift}
|
||||
\begin{ttext}[desc]
|
||||
\eng{The interaction of the electron with virtual photons emitted/absorbed by the nucleus leads to a (very small) shift in the energy level.}
|
||||
\ger{The Wechselwirkung zwischen dem Elektron und vom Kern absorbierten/emittierten virtuellen Photonen führt zu einer (sehr kleinen) Energieverschiebung.}
|
||||
@ -188,8 +188,8 @@
|
||||
\eq{f &= j \pm i \\ m_f &= -f,-f+1,\dots,f-1,f}
|
||||
\end{formula}
|
||||
\begin{formula}{constant}
|
||||
\desc{Hyperfine structure constant}{}{$B_\textrm{HFS}$ hyperfine field, $\mu_\textrm{K}$ nuclear magneton, $g_i$ nuclear g-factor \ref{qm:h:lande}}
|
||||
\desc[german]{Hyperfeinstrukturkonstante}{}{$B_\textrm{HFS}$ Hyperfeinfeld, $\mu_\textrm{K}$ Kernmagneton, $g_i$ Kern-g-Faktor \ref{qm:h:lande}}
|
||||
\desc{Hyperfine structure constant}{}{$B_\textrm{HFS}$ hyperfine field, $\mu_\textrm{K}$ nuclear magneton, $g_i$ nuclear g-factor \fRef{qm:h:lande}}
|
||||
\desc[german]{Hyperfeinstrukturkonstante}{}{$B_\textrm{HFS}$ Hyperfeinfeld, $\mu_\textrm{K}$ Kernmagneton, $g_i$ Kern-g-Faktor \fRef{qm:h:lande}}
|
||||
\eq{A = \frac{g_i \mu_\textrm{K} B_\textrm{HFS}}{\sqrt{j(j+1)}}}
|
||||
\end{formula}
|
||||
\begin{formula}{energy_shift}
|
||||
|
||||
@ -145,7 +145,6 @@
|
||||
\desc[german]{Kommutatorrelationen}{}{}
|
||||
\eq{[A, BC] = [A, B]C - B[A,C]}
|
||||
\end{formula}
|
||||
\TODO{add some more?}
|
||||
|
||||
\begin{formula}{function}
|
||||
\desc{Commutator involving a function}{}{given $[A,[A,B]] = 0$}
|
||||
@ -288,8 +287,9 @@
|
||||
\Subsubsection[
|
||||
\eng{Ehrenfest theorem}
|
||||
\ger{Ehrenfest-Theorem}
|
||||
]{ehrenfest_theorem}
|
||||
\GT{see_also} \ref{sec:qm:basics:schroedinger_equation:correspondence_principle}
|
||||
]{ehrenfest_theorem}
|
||||
\absLink{}{ehrenfest_theorem}
|
||||
\GT{see_also} \fRef{qm:se:time:ehrenfest_theorem:correspondence_principle}
|
||||
\begin{formula}{ehrenfest_theorem}
|
||||
\desc{Ehrenfest theorem}{applies to both pictures}{}
|
||||
\desc[german]{Ehrenfest-Theorem}{gilt für beide Bilder}{}
|
||||
@ -386,8 +386,6 @@
|
||||
\eq{E_n = \hbar\omega \Big(\frac{1}{2} + n\Big)}
|
||||
\end{formula}
|
||||
|
||||
\GT{see_also} \ref{sec:qm:hosc:c_a_ops}
|
||||
|
||||
\Subsection[
|
||||
\ger{Erzeugungs und Vernichtungsoperatoren / Leiteroperatoren}
|
||||
\eng{Creation and Annihilation operators / Ladder operators}
|
||||
@ -486,11 +484,10 @@
|
||||
\ger{Aharanov-Bohm Effekt}
|
||||
]{aharanov_bohm}
|
||||
\begin{formula}{phase}
|
||||
\desc{Acquired phase}{Electron along a closed loop aquires a phase proportional to the enclosed magnetic flux}{}
|
||||
\desc{Acquired phase}{Electron along a closed loop aquires a phase proportional to the enclosed magnetic flux}{\QtyRef{magnetic_vector_potential}, \QtyRef{magnetic_flux}}
|
||||
\desc[german]{Erhaltene Phase}{Elektron entlang eines geschlossenes Phase erhält eine Phase die proportional zum eingeschlossenen magnetischem Fluss ist}{}
|
||||
\eq{\delta = \frac{2 e}{\hbar} \oint \vec{A}\cdot \d\vec{s} = \frac{2 e}{\hbar} \Phi}
|
||||
\end{formula}
|
||||
\TODO{replace with loop intergral symbol and add more info}
|
||||
\Section[
|
||||
\eng{Periodic potentials}
|
||||
\ger{Periodische Potentiale}
|
||||
@ -526,8 +523,8 @@
|
||||
\ger{Symmetrien}
|
||||
]{symmetry}
|
||||
\begin{ttext}[desc]
|
||||
\eng{Most symmetry operators are unitary \ref{sec:linalg:unitary} because the norm of a state must be invariant under transformations of space, time and spin.}
|
||||
\ger{Die meisten Symmetrieoperatoren sind unitär \ref{sec:linalg:unitary}, da die Norm eines Zustands invariant unter Raum-, Zeit- und Spin-Transformationen sein muss.}
|
||||
\eng{Most symmetry operators are \fRef[unitary]{math:linalg:matrix:unitary} because the norm of a state must be invariant under transformations of space, time and spin.}
|
||||
\ger{Die meisten Symmetrieoperatoren sind \fRef[unitär]{math:linalg:matrix:unitary}, da die Norm eines Zustands invariant unter Raum-, Zeit- und Spin-Transformationen sein muss.}
|
||||
\end{ttext}
|
||||
\begin{formula}{invariance}
|
||||
\desc{Invariance}{$\hat{H}$ is invariant under a symmetrie described by $\hat{U}$ if this holds}{}
|
||||
@ -562,7 +559,7 @@
|
||||
\eq{H &= \underbrace{\hbar\omega_c \hat{a}^\dagger \hat{a}}_\text{\GT{field}}
|
||||
+ \underbrace{\hbar\omega_\text{a} \frac{\hat{\sigma}_z}{2}}_\text{\GT{atom}}
|
||||
+ \underbrace{\frac{\hbar\Omega}{2} \hat{E} \hat{S}}_\text{int} \\
|
||||
\shortintertext{\GT{after} \hyperref[eq:qm:other:RWA]{RWA}:} \\
|
||||
\shortintertext{\GT{after} \fRef[RWA]{qm:other:RWA}:} \\
|
||||
&= \hbar\omega_c \hat{a}^\dagger \hat{a}
|
||||
+ \hbar\omega_\text{a} \hat{\sigma}^\dagger \hat{\sigma}
|
||||
+ \frac{\hbar\Omega}{2} (\hat{a}\hat{\sigma^\dagger} + \hat{a}^\dagger \hat{\sigma})
|
||||
@ -572,7 +569,7 @@
|
||||
\Section[
|
||||
\eng{Other}
|
||||
\ger{Sonstiges}
|
||||
]{other}
|
||||
]{other}
|
||||
\begin{formula}{RWA}
|
||||
\desc{Rotating Wave Approximation (RWS)}{Rapidly oscilating terms are neglected}{$\omega_\text{L}$ light frequency, $\omega_0$ transition frequency}
|
||||
\desc[german]{Rotating Wave Approximation / Drehwellennäherung (RWS)}{Schnell oscillierende Terme werden vernachlässigt}{$\omega_\text{L}$ Frequenz des Lichtes, $\omega_0$ Übergangsfrequenz}
|
||||
@ -588,6 +585,7 @@
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\begin{formula}{slater_det}
|
||||
\desc{Slater determinant}{Construction of a fermionic (antisymmetric) many-particle wave function from single-particle wave functions}{}
|
||||
\desc[german]{Slater Determinante}{Konstruktion einer fermionischen (antisymmetrischen) Vielteilchen Wellenfunktion aus ein-Teilchen Wellenfunktionen}{}
|
||||
|
||||
@ -98,10 +98,16 @@
|
||||
\desc[german]{Volumen}{$d$ dimensionales Volumen}{}
|
||||
\quantity{V}{\m^d}{}
|
||||
\end{formula}
|
||||
\begin{formula}{heat_capacity}
|
||||
\desc{Heat capacity}{}{}
|
||||
\desc[german]{Wärmekapazität}{}{}
|
||||
\quantity{C}{\joule\per\kelvin}{}
|
||||
\begin{formula}{heat}
|
||||
\desc{Heat}{}{}
|
||||
\desc[german]{Wärme}{}{}
|
||||
\quantity{Q}{\joule}{}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{density}
|
||||
\desc{Density}{}{}
|
||||
\desc[german]{Dichte}{}{}
|
||||
\quantity{\rho}{\kg\per\m^3}{s}
|
||||
\end{formula}
|
||||
|
||||
\Subsection[
|
||||
@ -125,6 +131,12 @@
|
||||
\quantity{\rho}{\coulomb\per\m^3}{s}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{charge_carrier_density}
|
||||
\desc{Charge carrier density}{Number of charge carriers per volume}{}
|
||||
\desc[german]{Ladungsträgerdichte}{Anzahl der Ladungsträger pro Volumen}{}
|
||||
\quantity{n}{\per\m^3}{s}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{frequency}
|
||||
\desc{Frequency}{}{}
|
||||
\desc[german]{Frequenz}{}{}
|
||||
@ -136,6 +148,12 @@
|
||||
\quantity{\omega}{\radian\per\s}{s}
|
||||
\eq{\omega = \frac{2\pi/T}{2\pi f}}
|
||||
\end{formula}
|
||||
\begin{formula}{angular_velocity}
|
||||
\desc{Angular velocity}{}{\QtyRef{time_period}, \QtyRef{frequency}}
|
||||
\desc[german]{Kreisgeschwindigkeit}{}{}
|
||||
\quantity{\vec{\omega}}{\radian\per\s}{v}
|
||||
\eq{\vec{\omega} = \frac{\vecr \times \vecv}{r^2}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{time_period}
|
||||
\desc{Time period}{}{\QtyRef{frequency}}
|
||||
@ -144,10 +162,39 @@
|
||||
\eq{T = \frac{1}{f}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{conductivity}
|
||||
\desc{Conductivity}{}{}
|
||||
\desc[german]{Leitfähigkeit}{}{}
|
||||
\quantity{\sigma}{\per\ohm\m}{}
|
||||
\begin{formula}{wavelength}
|
||||
\desc{Wavelength}{}{}
|
||||
\desc[german]{Wellenlänge}{}{}
|
||||
\quantity{\lambda}{\per\m}{s}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{angular_wavenumber}
|
||||
\desc{Wavenumber}{Angular wavenumber}{\QtyRef{wavelength}}
|
||||
\desc[german]{Wellenzahl}{}{}
|
||||
\eq{k = \frac{2\pi}{\lambda}}
|
||||
\quantity{k}{\radian\per\m}{s}
|
||||
\end{formula}
|
||||
\begin{formula}{wavevector}
|
||||
\desc{Wavevector}{Vector perpendicular to the wavefront}{}
|
||||
\desc[german]{Wellenvektor}{Vektor senkrecht zur Wellenfront}{}
|
||||
\eq{\abs{k} = \frac{2\pi}{\lambda}}
|
||||
\quantity{\vec{k}}{1\per\m}{v}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{impedance}
|
||||
\desc{Impedance}{}{}
|
||||
\desc[german]{Impedanz}{}{}
|
||||
\quantity{Z}{\ohm}{s}
|
||||
\end{formula}
|
||||
\begin{formula}{resistance}
|
||||
\desc{Resistance}{}{}
|
||||
\desc[german]{Widerstand}{}{}
|
||||
\quantity{R}{\ohm}{s}
|
||||
\end{formula}
|
||||
\begin{formula}{inductance}
|
||||
\desc{Inductance}{}{}
|
||||
\desc[german]{Induktivität}{}{}
|
||||
\quantity{L}{\henry=\kg\m^2\per\s^2\ampere^2=\weber\per\ampere=\volt\s\per\ampere=\ohm\s}{s}
|
||||
\end{formula}
|
||||
|
||||
\Subsection[
|
||||
|
||||
@ -73,7 +73,7 @@
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{2nd_josephson_relation}
|
||||
\desc{2. Josephson relation}{superconducting phase change is proportional to applied voltage}{$\varphi_0=\frac{\hbar}{2e}$ reduced flux quantum}
|
||||
\desc{2. Josephson relation}{Superconducting phase change is proportional to applied voltage}{$\varphi_0=\frac{\hbar}{2e}$ reduced flux quantum}
|
||||
\desc[german]{2. Josephson Gleichung}{Supraleitende Phasendifferenz is proportional zur angelegten Spannung}{$\varphi_0=\frac{\hbar}{2e}$ reduziertes Flussquantum}
|
||||
\eq{\odv{\hat{\delta}}{t}=\frac{1}{i\hbar}[\hat{H},\hat{\delta}] = -\frac{2eU}{i\hbar}[\hat{n},\hat{\delta}] = \frac{1}{\varphi_0} U}
|
||||
\end{formula}
|
||||
@ -108,29 +108,24 @@
|
||||
\end{formula}
|
||||
|
||||
\Subsection[
|
||||
\eng{Josephson Qubit??}
|
||||
\ger{TODO}
|
||||
]{josephson_qubit}
|
||||
\begin{tikzpicture}
|
||||
\draw (0,0) to[capacitor] (0,2);
|
||||
\draw (0,0) to (2,0);
|
||||
\draw (0,2) to (2,2);
|
||||
\draw (2,0) to[josephson] (2,2);
|
||||
\eng{Josephson junction based qubits}
|
||||
\ger{Qubits mit Josephson-Junctions}
|
||||
]{josephson_qubit}
|
||||
|
||||
\draw[->] (3,1) -- (4,1);
|
||||
\draw (5,0) to[josephsoncap=$C_\text{J}$] (5,2);
|
||||
\end{tikzpicture}
|
||||
\TODO{Include schaltplan}
|
||||
|
||||
\begin{tikzpicture}
|
||||
\draw (0,0) to[sV=$V_\text{g}$] (0,2);
|
||||
\draw (0,2) to[capacitor=$C_\text{g}$] (2,2);
|
||||
\draw (2,2) to (4,2);
|
||||
\draw (2,0) to[josephsoncap=$C_\text{J}$] (2,2);
|
||||
\draw (4,0) to[capacitor=$C_C$] (4,2);
|
||||
\draw (0,0) to (2,0);
|
||||
\draw (2,0) to (4,0);
|
||||
\end{tikzpicture}
|
||||
\begin{formula}{circuit}
|
||||
\desc{General circuit}{}{}
|
||||
\desc[german]{Allgemeiner Schaltplan}{}{}
|
||||
\centering
|
||||
\begin{tikzpicture}
|
||||
\draw (0,0) to[sV=$V_\text{g}$] (0,2);
|
||||
\draw (0,2) to[capacitor=$C_\text{g}$] (2,2);
|
||||
\draw (2,2) to (4,2);
|
||||
\draw (2,0) to[josephsoncap=$C_\text{J}$] (2,2);
|
||||
\draw (4,0) to[capacitor=$C_C$] (4,2);
|
||||
\draw (0,0) to (2,0);
|
||||
\draw (2,0) to (4,0);
|
||||
\end{tikzpicture}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{charging_energy}
|
||||
\desc{Charging energy / electrostatic energy}{}{}
|
||||
@ -140,10 +135,9 @@
|
||||
|
||||
\begin{formula}{josephson_energy}
|
||||
\desc{Josephson energy}{}{}
|
||||
\desc[german]{Josephson-Energie?}{}{}
|
||||
\desc[german]{Josephson-Energie}{}{}
|
||||
\eq{E_\text{J} = \frac{I_0 \phi_0}{2\pi}}
|
||||
\end{formula}
|
||||
\TODO{Was ist I0}
|
||||
|
||||
\begin{formula}{inductive_energy}
|
||||
\desc{Inductive energy}{}{}
|
||||
@ -164,262 +158,265 @@
|
||||
\end{formula}
|
||||
|
||||
|
||||
\begin{minipage}{0.8\textwidth}
|
||||
\begingroup
|
||||
\setlength{\tabcolsep}{0.9em} % horizontal
|
||||
\renewcommand{\arraystretch}{2} % vertical
|
||||
\begin{tabular}{ p{0.5cm} |p{0.8cm}||p{2.2cm}|p{1.9cm}|p{1.9cm}|p{1.8cm}|}
|
||||
\multicolumn{1}{c}{}& \multicolumn{1}{c}{} &\multicolumn{4}{c}{$E_L/(E_J-E_L)$} \\
|
||||
\cline{3-6}
|
||||
\multicolumn{1}{c}{} & & $0$ & $\ll$ 1 & $\sim 1$ & $\gg 1$\\
|
||||
\hhline{~|=====|}
|
||||
\multirow{4}{*}{$\frac{E_J}{E_C}$} & $\ll 1$ & cooper-pair box & & & \\
|
||||
\cline{2-6}
|
||||
& $\sim 1$ & quantronium & fluxonium & &\\
|
||||
\cline{2-6}
|
||||
& $\gg 1$ &transmon & & & flux qubit\\
|
||||
\cline{2-6}
|
||||
& $\ggg 1$ & & & phase qubit & \\
|
||||
\cline{2-6}
|
||||
\end{tabular}
|
||||
\endgroup
|
||||
\end{minipage}
|
||||
\begin{minipage}{0.2\textwidth}
|
||||
\begin{tikzpicture}[scale=2]
|
||||
\draw[-latex,line width=2pt] (0,1)--++(0,1) node[midway,above,sloped] () {charge noise};
|
||||
\draw[-latex,line width=2pt] (0,1)--++(0,1) node[midway,below,sloped] () {sensitivity};
|
||||
\draw[-latex,line width=2pt] (0,0)--++(1,1) node[midway,above,sloped] () {flux noise};
|
||||
\draw[-latex,line width=2pt] (0,0)--++(1,1) node[midway,below,sloped] () {sensitivity};
|
||||
\draw[-latex,line width=2pt] (0,0)--++(1,-1) node[midway,above,sloped] () {critical current};
|
||||
\draw[-latex,line width=2pt] (0,0)--++(1,-1) node[midway,below,sloped] () {noise sensitivity};
|
||||
\end{tikzpicture}
|
||||
\end{minipage}
|
||||
\begin{bigformula}{comparison}
|
||||
\desc{Comparison of superconducting qubits}{}{$E_C$ \fRef{::charging_energy}, $E_L$ \fRef{::inductive_energy}, $E_{\txJ}$ \fRef{::josephson_energy}}
|
||||
\desc[german]{Vergleich supraleitender Qubits}{}{}
|
||||
\begin{minipage}{0.8\textwidth}
|
||||
\begingroup
|
||||
\setlength{\tabcolsep}{0.9em} % horizontal
|
||||
\renewcommand{\arraystretch}{2} % vertical
|
||||
\begin{tabular}{ p{0.5cm} |p{0.8cm}||p{2.2cm}|p{1.9cm}|p{1.9cm}|p{1.8cm}|}
|
||||
\multicolumn{1}{c}{}& \multicolumn{1}{c}{} &\multicolumn{4}{c}{$E_L/(E_J-E_L)$} \\
|
||||
\cline{3-6}
|
||||
\multicolumn{1}{c}{} & & $0$ & $\ll$ 1 & $\sim 1$ & $\gg 1$\\
|
||||
\hhline{~|=====|}
|
||||
\multirow{4}{*}{$\frac{E_J}{E_C}$} & $\ll 1$ & cooper-pair box & & & \\
|
||||
\cline{2-6}
|
||||
& $\sim 1$ & quantronium & fluxonium & &\\
|
||||
\cline{2-6}
|
||||
& $\gg 1$ &transmon & & & flux qubit\\
|
||||
\cline{2-6}
|
||||
& $\ggg 1$ & & & phase qubit & \\
|
||||
\cline{2-6}
|
||||
\end{tabular}
|
||||
\endgroup
|
||||
\end{minipage}
|
||||
\begin{minipage}{0.19\textwidth}
|
||||
\begin{tikzpicture}[scale=2]
|
||||
\draw[-latex,line width=2pt] (0,1)--++(0,1) node[midway,above,sloped] () {charge noise};
|
||||
\draw[-latex,line width=2pt] (0,1)--++(0,1) node[midway,below,sloped] () {sensitivity};
|
||||
\draw[-latex,line width=2pt] (0,0)--++(1,1) node[midway,above,sloped] () {flux noise};
|
||||
\draw[-latex,line width=2pt] (0,0)--++(1,1) node[midway,below,sloped] () {sensitivity};
|
||||
\draw[-latex,line width=2pt] (0,0)--++(1,-1) node[midway,above,sloped] () {critical current};
|
||||
\draw[-latex,line width=2pt] (0,0)--++(1,-1) node[midway,below,sloped] () {noise sensitivity};
|
||||
\end{tikzpicture}
|
||||
\end{minipage}
|
||||
\end{bigformula}
|
||||
|
||||
|
||||
\Subsection[
|
||||
\eng{Cooper Pair Box (CPB) qubit}
|
||||
\ger{Cooper Paar Box (QPB) Qubit}
|
||||
]{cpb}
|
||||
\begin{ttext}
|
||||
\eng{
|
||||
= voltage bias junction\\= charge qubit?
|
||||
}
|
||||
\ger{}
|
||||
\end{ttext}
|
||||
\begin{formula}{circuit}
|
||||
\desc{Cooper Pair Box / Charge qubit}{
|
||||
\begin{itemize}
|
||||
\gooditem large anharmonicity
|
||||
\baditem sensitive to charge noise
|
||||
\end{itemize}
|
||||
}{}
|
||||
\desc[german]{Cooper Pair Box / Charge Qubit}{
|
||||
\begin{itemize}
|
||||
\gooditem Große Anharmonizität
|
||||
\baditem Sensibel für charge noise
|
||||
\end{itemize}
|
||||
}{}
|
||||
\centering
|
||||
\begin{tikzpicture}
|
||||
\draw (0,0) to[sV=$V_\text{g}$] (0,2);
|
||||
% \draw (0,0) to (2,0);
|
||||
\draw (0,2) to[capacitor=$C_\text{g}$] (2,2);
|
||||
\draw (2,0) to[josephsoncap=$C_\text{J}$] (2,2);
|
||||
\draw (0,0) to (2,0);
|
||||
\end{tikzpicture}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\begin{formula}{hamiltonian}
|
||||
\desc{Hamiltonian}{}{}
|
||||
\desc[german]{Hamiltonian}{}{}
|
||||
\eq{\hat{H} &= 4 E_C(\hat{n} - n_\text{g})^2 - E_\text{J} \cos\hat{\phi} \\
|
||||
&=\sum_n \left[4 E_C (n-n_\text{g})^2 \ket{n}\bra{n} - \frac{E_\text{J}}{2}\ket{n}\bra{n+1}+\ket{n+1}\bra{n}\right] }
|
||||
\end{formula}
|
||||
|
||||
\Subsection[
|
||||
\eng{Transmon qubit}
|
||||
\ger{Transmon Qubit}
|
||||
]{transmon}
|
||||
\begin{formula}{circuit}
|
||||
\desc{Transmon qubit}{
|
||||
Josephson junction with a shunt \textbf{capacitance}.
|
||||
\begin{itemize}
|
||||
\gooditem charge noise insensitive
|
||||
\baditem small anharmonicity
|
||||
\end{itemize}
|
||||
}{}
|
||||
\desc[german]{Transmon Qubit}{
|
||||
Josephson-Kontakt mit einem parallelen \textbf{kapzitiven Element}.
|
||||
\begin{itemize}
|
||||
\gooditem Charge noise resilient
|
||||
\baditem Geringe Anharmonizität $\alpha$
|
||||
\end{itemize}
|
||||
}{}
|
||||
\centering
|
||||
\begin{tikzpicture}
|
||||
% \draw (0,0) to[sV=$V_\text{g}$] ++(0,3)
|
||||
% to[capacitor=$C_\text{g}$] ++(2,0)
|
||||
\draw (0,0) to ++(2,0) to ++(0,-0.5) to[josephsoncap=$C_\text{J}$] ++(-0,-2) to ++(0,-0.5) to ++(-2,0)
|
||||
to[capacitor=$C_C$] ++(0,3);
|
||||
\end{tikzpicture}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{hamiltonian}
|
||||
\desc{Hamiltonian}{}{}
|
||||
\desc[german]{Hamiltonian}{}{}
|
||||
\eq{\hat{H} &= 4 E_C\hat{n}^2 - E_\text{J} \cos\hat{\phi}}
|
||||
\end{formula}
|
||||
\eng{Charge based qubits}
|
||||
\ger{Ladungsbasierte Qubits}
|
||||
]{charge}
|
||||
\begin{bigformula}{comparison}
|
||||
\desc{Comparison of charge qubit states}{}{}
|
||||
\desc[german]{Vergleich der Zustände von Ladungsbasierten Qubits}{}{}
|
||||
\fig{img/qubit_transmon.pdf}
|
||||
\end{bigformula}
|
||||
|
||||
\Subsubsection[
|
||||
\eng{Tunable Transmon qubit}
|
||||
\ger{Tunable Transmon Qubit}
|
||||
]{tunable}
|
||||
\eng{Cooper Pair Box (CPB) qubit}
|
||||
\ger{Cooper Paar Box (QPB) Qubit}
|
||||
]{cpb}
|
||||
\begin{ttext}
|
||||
\eng{
|
||||
= voltage bias junction\\= charge qubit?
|
||||
}
|
||||
\ger{}
|
||||
\end{ttext}
|
||||
\begin{formula}{circuit}
|
||||
\desc{Frequency tunable transmon}{By using a \fqSecRef{qc:scq:elements:squid} instead of a \fqSecRef{qc:scq:elements:josephson_junction}, the qubit is frequency tunable through an external field}{}
|
||||
\desc[german]{}{Durch Nutzung eines \fqSecRef{qc:scq:elements:squid} anstatt eines \fqSecRef{qc:scq:elements:josephson_junction}s, ist die Frequenz des Qubits durch ein externes Magnetfeld einstellbar}{}
|
||||
\desc{Cooper Pair Box / Charge qubit}{
|
||||
\begin{itemize}
|
||||
\gooditem large anharmonicity
|
||||
\baditem sensitive to charge noise
|
||||
\end{itemize}
|
||||
}{}
|
||||
\desc[german]{Cooper Pair Box / Charge Qubit}{
|
||||
\begin{itemize}
|
||||
\gooditem Große Anharmonizität
|
||||
\baditem Sensibel für charge noise
|
||||
\end{itemize}
|
||||
}{}
|
||||
\centering
|
||||
\begin{tikzpicture}
|
||||
\draw (0,0) to[sV=$V_\text{g}$] (0,2);
|
||||
% \draw (0,0) to (2,0);
|
||||
\draw (0,2) to[capacitor=$C_\text{g}$] (2,2);
|
||||
\draw (2,0) to[josephsoncap=$C_\text{J}$] (2,2);
|
||||
\draw (0,0) to (2,0);
|
||||
\end{tikzpicture}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\begin{formula}{hamiltonian}
|
||||
\desc{Hamiltonian}{}{}
|
||||
\desc[german]{Hamiltonian}{}{}
|
||||
\eq{\hat{H} &= 4 E_C(\hat{n} - n_\text{g})^2 - E_\text{J} \cos\hat{\phi} \\
|
||||
&=\sum_n \left[4 E_C (n-n_\text{g})^2 \ket{n}\bra{n} - \frac{E_\text{J}}{2}\ket{n}\bra{n+1}+\ket{n+1}\bra{n}\right] }
|
||||
\end{formula}
|
||||
|
||||
\Subsubsection[
|
||||
\eng{Transmon qubit}
|
||||
\ger{Transmon Qubit}
|
||||
]{transmon}
|
||||
\begin{formula}{circuit}
|
||||
\desc{Transmon qubit}{
|
||||
Josephson junction with a shunt \textbf{capacitance}.
|
||||
\begin{itemize}
|
||||
\gooditem charge noise insensitive
|
||||
\baditem small anharmonicity
|
||||
\end{itemize}
|
||||
}{}
|
||||
\desc[german]{Transmon Qubit}{
|
||||
Josephson-Kontakt mit einem parallelen \textbf{kapzitiven Element}.
|
||||
\begin{itemize}
|
||||
\gooditem Charge noise resilient
|
||||
\baditem Geringe Anharmonizität $\alpha$
|
||||
\end{itemize}
|
||||
}{}
|
||||
\centering
|
||||
\begin{tikzpicture}
|
||||
% \draw (0,0) to[sV=$V_\text{g}$] ++(0,3)
|
||||
% to[capacitor=$C_\text{g}$] ++(2,0)
|
||||
\draw (0,0) to ++(-2,0)
|
||||
to ++(3,0) to ++(0,-0.5) \squidloop{loop}{SQUID} to ++(0,-0.5) to ++(-3,0)
|
||||
to[capacitor=$C_C$] ++(0,3);
|
||||
\draw (0,0) to ++(2,0) to ++(0,-0.5) to[josephsoncap=$C_\text{J}$] ++(-0,-2) to ++(0,-0.5) to ++(-2,0)
|
||||
to[capacitor=$C_C$] ++(0,3);
|
||||
\end{tikzpicture}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{energy}
|
||||
\desc{Josephson energy}{}{$d=(E_\text{J1}-E_\text{J2})/(E_\text{J1}+E_\text{J2})$ asymmetry}
|
||||
\desc[german]{Josephson Energie}{}{$d=(E_\text{J1}-E_\text{J2})/(E_\text{J1}+E_\text{J2})$ Asymmetrie}
|
||||
\eq{E_\text{J,eff}(\Phi_\text{ext}) = (E_\text{J1}+E_\text{J2}) \sqrt{\cos^2\left(\pi\frac{\Phi_\text{ext}}{\Phi_0}\right) + d^2 \sin \left(\pi\frac{\Phi_\text{ext}}{\Phi_0}\right)}}
|
||||
\end{formula}
|
||||
\begin{formula}{hamiltonian}
|
||||
\desc{Hamiltonian}{}{}
|
||||
\desc[german]{Hamiltonian}{}{}
|
||||
\eq{\hat{H} = 4E_C \hat{n}^2 - \frac{1}{2} E_\text{J,eff}(\Phi_\text{ext}) \sum_{n}\left[\ket{n}\bra{n+1} + \ket{n+1}\bra{n}\right]}
|
||||
\eq{\hat{H} &= 4 E_C\hat{n}^2 - E_\text{J} \cos\hat{\phi}}
|
||||
\end{formula}
|
||||
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\includegraphics[width=0.8\textwidth]{img/qubit_transmon.pdf}
|
||||
\caption{Transmon and so TODO}
|
||||
\label{fig:img-qubit_transmon-pdf}
|
||||
\end{figure}
|
||||
\Subsubsection[
|
||||
\eng{Tunable Transmon qubit}
|
||||
\ger{Tunable Transmon Qubit}
|
||||
]{tunable}
|
||||
\begin{formula}{circuit}
|
||||
\desc{Frequency tunable transmon}{By using a \fRef{qc:scq:elements:squid} instead of a \fRef{qc:scq:elements:josephson_junction}, the qubit is frequency tunable through an external field}{}
|
||||
\desc[german]{}{Durch Nutzung eines \fRef{qc:scq:elements:squid} anstatt eines \fRef{qc:scq:elements:josephson_junction}s, ist die Frequenz des Qubits durch ein externes Magnetfeld einstellbar}{}
|
||||
\centering
|
||||
\begin{tikzpicture}
|
||||
% \draw (0,0) to[sV=$V_\text{g}$] ++(0,3)
|
||||
% to[capacitor=$C_\text{g}$] ++(2,0)
|
||||
\draw (0,0) to ++(-2,0)
|
||||
to ++(3,0) to ++(0,-0.5) \squidloop{loop}{SQUID} to ++(0,-0.5) to ++(-3,0)
|
||||
to[capacitor=$C_C$] ++(0,3);
|
||||
\end{tikzpicture}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\Subsection[
|
||||
\eng{Phase qubit}
|
||||
\ger{Phase Qubit}
|
||||
]{phase}
|
||||
\begin{formula}{circuit}
|
||||
\desc{Phase qubit}{}{}
|
||||
\desc[german]{Phase Qubit}{}{}
|
||||
\centering
|
||||
\begin{tikzpicture}
|
||||
% \draw (0,0) to[sV=$V_\text{g}$] ++(0,3)
|
||||
% to ++(2,0) coordinate(top1)
|
||||
% to ++(2,0) coordinate(top2)
|
||||
% to ++(2,0) coordinate(top3);
|
||||
% \draw (0,0)
|
||||
% to ++(2,0) coordinate(bot1)
|
||||
% to ++(2,0) coordinate(bot2)
|
||||
% to ++(2,0) coordinate(bot3);
|
||||
\draw[color=gray] (0,0) to[capacitor=$C_C$] (0,-2);
|
||||
% \draw (top1) to ++(0,-0.5) to[josephsoncap=$C_\text{J}$] ++(-0,-2) to (bot2);
|
||||
\draw(0,0) to ++(2,0) to[josephsoncap=$C_\text{J}$] ++(0,-2) to ++(-2,0);
|
||||
\draw (2,0) to ++(2,0) to[cute inductor=$E_L$] ++(0,-2) to ++(-2,0);
|
||||
\node at (3,-1.5) {$\Phi_\text{ext}$};
|
||||
\end{tikzpicture}
|
||||
\\\TODO{Ist beim Fluxonium noch die Voltage source dran?}
|
||||
\end{formula}
|
||||
\begin{formula}{hamiltonian}
|
||||
\desc{Hamiltonian}{}{$\delta = \frac{\phi}{\phi_0}$}
|
||||
\desc[german]{Hamiltonian}{}{}
|
||||
\eq{\hat{H} = E_C \hat{n}^2 - E_J \cos \hat{\delta} + E_L(\hat{\delta} - \delta_s)^2}
|
||||
\end{formula}
|
||||
|
||||
\Eng[TESTT]{This is only a test}
|
||||
\Ger[TESTT]{}
|
||||
\GT{TESTT}
|
||||
\begin{formula}{energy}
|
||||
\desc{Josephson energy}{}{$d=(E_\text{J1}-E_\text{J2})/(E_\text{J1}+E_\text{J2})$ asymmetry}
|
||||
\desc[german]{Josephson Energie}{}{$d=(E_\text{J1}-E_\text{J2})/(E_\text{J1}+E_\text{J2})$ Asymmetrie}
|
||||
\eq{E_\text{J,eff}(\Phi_\text{ext}) = (E_\text{J1}+E_\text{J2}) \sqrt{\cos^2\left(\pi\frac{\Phi_\text{ext}}{\Phi_0}\right) + d^2 \sin \left(\pi\frac{\Phi_\text{ext}}{\Phi_0}\right)}}
|
||||
\end{formula}
|
||||
\begin{formula}{hamiltonian}
|
||||
\desc{Hamiltonian}{}{}
|
||||
\desc[german]{Hamiltonian}{}{}
|
||||
\eq{\hat{H} = 4E_C \hat{n}^2 - \frac{1}{2} E_\text{J,eff}(\Phi_\text{ext}) \sum_{n}\left[\ket{n}\bra{n+1} + \ket{n+1}\bra{n}\right]}
|
||||
\end{formula}
|
||||
|
||||
|
||||
|
||||
\Subsection[
|
||||
\eng{Flux qubit}
|
||||
\ger{Flux Qubit}
|
||||
]{flux}
|
||||
\TODO{TODO}
|
||||
\begin{formula}{circuit}
|
||||
\desc{Flux qubit / Persistent current qubit}{}{}
|
||||
\desc[german]{Flux Qubit / Persistent current qubit}{}{}
|
||||
\centering
|
||||
\begin{tikzpicture}
|
||||
\draw (0,0) to[josephsoncap=$\alpha E_\text{J}$, scale=0.8, transform shape] (0,-3);
|
||||
\draw (0,0) to ++(3,0)
|
||||
to[josephsoncap=$E_\text{J}$] ++(0,-1.5)
|
||||
to[josephsoncap=$E_\text{J}$] ++(0,-1.5)
|
||||
to ++(-3,0);
|
||||
\node at (1.5,-1.5) {$\Phi_\text{ext}$};
|
||||
\end{tikzpicture}
|
||||
% \begin{tikzpicture}
|
||||
% \draw (0,0) to[sV=$V_\text{g}$] ++(0,3)
|
||||
% to ++(2,0) coordinate(top1)
|
||||
% to ++(2,0) coordinate(top2)
|
||||
% to ++(2,0) coordinate(top3);
|
||||
% \draw (0,0)
|
||||
% to ++(2,0) coordinate(bot1)
|
||||
% to ++(2,0) coordinate(bot2)
|
||||
% to ++(2,0) coordinate(bot3);
|
||||
% \draw[color=gray] (top1) to[capacitor=$C_C$] (bot1);
|
||||
% % \draw (top1) to ++(0,-0.5) to[josephsoncap=$C_\text{J}$] ++(-0,-2) to (bot2);
|
||||
% \draw[scale=0.8, transform shape] (top2) to[josephsoncap=$\alpha E_\text{J}$] (bot2);
|
||||
% \draw (top3)
|
||||
% to[josephsoncap=$E_\text{J}$] ++(0,-1.5)
|
||||
% to[josephsoncap=$E_\text{J}$] (bot3);
|
||||
% \node at (5,0.5) {$\Phi_\text{ext}$};
|
||||
% \end{tikzpicture}
|
||||
\end{formula}
|
||||
\Subsection[
|
||||
\eng{Inductive qubits}
|
||||
\ger{Induktive Qubits}
|
||||
]{inductive}
|
||||
\begin{bigformula}{comparison}
|
||||
\desc{Comparison of other qubit states}{}{}
|
||||
\desc[german]{Vergleich der Zustände von anderen Qubits}{}{}
|
||||
\fig{img/qubit_flux_onium.pdf}
|
||||
\end{bigformula}
|
||||
\Subsubsection[
|
||||
\eng{Phase qubit}
|
||||
\ger{Phase Qubit}
|
||||
]{phase}
|
||||
\begin{formula}{circuit}
|
||||
\desc{Phase qubit}{}{}
|
||||
\desc[german]{Phase Qubit}{}{}
|
||||
\centering
|
||||
\begin{tikzpicture}
|
||||
% \draw (0,0) to[sV=$V_\text{g}$] ++(0,3)
|
||||
% to ++(2,0) coordinate(top1)
|
||||
% to ++(2,0) coordinate(top2)
|
||||
% to ++(2,0) coordinate(top3);
|
||||
% \draw (0,0)
|
||||
% to ++(2,0) coordinate(bot1)
|
||||
% to ++(2,0) coordinate(bot2)
|
||||
% to ++(2,0) coordinate(bot3);
|
||||
\draw[color=gray] (0,0) to[capacitor=$C_C$] (0,-2);
|
||||
% \draw (top1) to ++(0,-0.5) to[josephsoncap=$C_\text{J}$] ++(-0,-2) to (bot2);
|
||||
\draw(0,0) to ++(2,0) to[josephsoncap=$C_\text{J}$] ++(0,-2) to ++(-2,0);
|
||||
\draw (2,0) to ++(2,0) to[cute inductor=$E_L$] ++(0,-2) to ++(-2,0);
|
||||
\node at (3,-1.5) {$\Phi_\text{ext}$};
|
||||
\end{tikzpicture}
|
||||
\end{formula}
|
||||
\begin{formula}{hamiltonian}
|
||||
\desc{Hamiltonian}{}{$\delta = \frac{\phi}{\phi_0}$}
|
||||
\desc[german]{Hamiltonian}{}{}
|
||||
\eq{\hat{H} = E_C \hat{n}^2 - E_J \cos \hat{\delta} + E_L(\hat{\delta} - \delta_s)^2}
|
||||
\end{formula}
|
||||
|
||||
\Subsubsection[
|
||||
\eng{Flux qubit}
|
||||
\ger{Flux Qubit}
|
||||
]{flux}
|
||||
\begin{formula}{circuit}
|
||||
\desc{Flux qubit / Persistent current qubit}{}{}
|
||||
\desc[german]{Flux Qubit / Persistent current qubit}{}{}
|
||||
\centering
|
||||
\begin{tikzpicture}
|
||||
\draw (0,0) to[josephsoncap=$\alpha E_\text{J}$, scale=0.8, transform shape] (0,-3/0.8);
|
||||
\draw (0,0) to ++(3,0)
|
||||
to[josephsoncap=$E_\text{J}$] ++(0,-1.5)
|
||||
to[josephsoncap=$E_\text{J}$] ++(0,-1.5)
|
||||
to ++(-3,0);
|
||||
\node at (1.5,-1.5) {$\Phi_\text{ext}$};
|
||||
\end{tikzpicture}
|
||||
% \begin{tikzpicture}
|
||||
% \draw (0,0) to[sV=$V_\text{g}$] ++(0,3)
|
||||
% to ++(2,0) coordinate(top1)
|
||||
% to ++(2,0) coordinate(top2)
|
||||
% to ++(2,0) coordinate(top3);
|
||||
% \draw (0,0)
|
||||
% to ++(2,0) coordinate(bot1)
|
||||
% to ++(2,0) coordinate(bot2)
|
||||
% to ++(2,0) coordinate(bot3);
|
||||
% \draw[color=gray] (top1) to[capacitor=$C_C$] (bot1);
|
||||
% % \draw (top1) to ++(0,-0.5) to[josephsoncap=$C_\text{J}$] ++(-0,-2) to (bot2);
|
||||
% \draw[scale=0.8, transform shape] (top2) to[josephsoncap=$\alpha E_\text{J}$] (bot2);
|
||||
% \draw (top3)
|
||||
% to[josephsoncap=$E_\text{J}$] ++(0,-1.5)
|
||||
% to[josephsoncap=$E_\text{J}$] (bot3);
|
||||
% \node at (5,0.5) {$\Phi_\text{ext}$};
|
||||
% \end{tikzpicture}
|
||||
\end{formula}
|
||||
|
||||
|
||||
\Subsection[
|
||||
\eng{Fluxonium qubit}
|
||||
\ger{Fluxonium Qubit}
|
||||
]{fluxonium}
|
||||
\begin{formula}{circuit}
|
||||
\desc{Fluxonium qubit}{
|
||||
Josephson junction with a shunt \textbf{inductance}. Instead of having to tunnel, cooper pairs can move to the island via the inductance.
|
||||
The inductance consists of many parallel Josephson Junctions to avoid parasitic capacitances.
|
||||
}{}
|
||||
\desc[german]{Fluxonium Qubit}{
|
||||
Josephson-Kontakt mit einem parallelen \textbf{induktiven Element}.
|
||||
Anstatt zu tunneln, können die Cooper-Paare über das induktive Element auf die Insel gelangen.
|
||||
Das induktive Element besteht aus sehr vielen parallelen Josephson-Kontakten um parisitische Kapazitäten zu vermeiden.
|
||||
}{}
|
||||
\centering
|
||||
\begin{tikzpicture}
|
||||
% \draw (0,0) to[sV=$V_\text{g}$] ++(0,3)
|
||||
% to ++(2,0) coordinate(top1);
|
||||
\draw[color=gray] (0,0) to ++(-2,0) to[capacitor=$C_C$] ++(0,-3) to ++(2,0);
|
||||
\draw (0,0) to[josephsoncap=$C_\text{J}$] ++(-0,-3);
|
||||
\draw (0,0) to ++(2,0) to[cute inductor=$E_L$] ++(0,-3) to ++(-2,0);
|
||||
\node at (1,-0.5) {$\Phi_\text{ext}$};
|
||||
\end{tikzpicture}
|
||||
\\\TODO{Ist beim Fluxonium noch die Voltage source dran?}
|
||||
\end{formula}
|
||||
\Subsubsection[
|
||||
\eng{Fluxonium qubit}
|
||||
\ger{Fluxonium Qubit}
|
||||
]{fluxonium}
|
||||
\begin{formula}{circuit}
|
||||
\desc{Fluxonium qubit}{
|
||||
Josephson junction with a shunt \textbf{inductance}. Instead of having to tunnel, cooper pairs can move to the island via the inductance.
|
||||
The inductance consists of many parallel Josephson Junctions to avoid parasitic capacitances.
|
||||
}{}
|
||||
\desc[german]{Fluxonium Qubit}{
|
||||
Josephson-Kontakt mit einem parallelen \textbf{induktiven Element}.
|
||||
Anstatt zu tunneln, können die Cooper-Paare über das induktive Element auf die Insel gelangen.
|
||||
Das induktive Element besteht aus sehr vielen parallelen Josephson-Kontakten um parisitische Kapazitäten zu vermeiden.
|
||||
}{}
|
||||
\centering
|
||||
\begin{tikzpicture}
|
||||
% \draw (0,0) to[sV=$V_\text{g}$] ++(0,3)
|
||||
% to ++(2,0) coordinate(top1);
|
||||
\draw[color=gray] (0,0) to ++(-2,0) to[capacitor=$C_C$] ++(0,-3) to ++(2,0);
|
||||
\draw (0,0) to[josephsoncap=$C_\text{J}$] ++(-0,-3);
|
||||
\draw (0,0) to ++(2,0) to[cute inductor=$E_L$] ++(0,-3) to ++(-2,0);
|
||||
\node at (1,-0.5) {$\Phi_\text{ext}$};
|
||||
\end{tikzpicture}
|
||||
\\\TODO{Ist beim Fluxonium noch die Voltage source dran?}
|
||||
\end{formula}
|
||||
|
||||
\def\temp{$E_\text{C} = \frac{(2e)^2}{2C}, E_\text{L} = \frac{\varphi_0^2}{2L}, \delta_\text{s} = \frac{\varphi_\text{s}}{\varphi_0}$}
|
||||
\begin{formula}{hamiltonian}
|
||||
\desc{Hamiltonian}{}{\temp}
|
||||
\desc[german]{Hamiltonian}{}{\temp}
|
||||
\eq{\hat{H} = 4E_\text{C} \hat{n}^2 - E_\text{J} \cos \hat{\delta} + E_\text{L}(\hat{\delta} - \delta_\text{s})^2}
|
||||
\end{formula}
|
||||
\def\temp{$E_\text{C} = \frac{(2e)^2}{2C}, E_\text{L} = \frac{\varphi_0^2}{2L}, \delta_\text{s} = \frac{\varphi_\text{s}}{\varphi_0}$}
|
||||
\begin{formula}{hamiltonian}
|
||||
\desc{Hamiltonian}{}{\temp}
|
||||
\desc[german]{Hamiltonian}{}{\temp}
|
||||
\eq{\hat{H} = 4E_\text{C} \hat{n}^2 - E_\text{J} \cos \hat{\delta} + E_\text{L}(\hat{\delta} - \delta_\text{s})^2}
|
||||
\end{formula}
|
||||
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\includegraphics[width=\textwidth]{img/qubit_flux_onium.pdf}
|
||||
\caption{img/}
|
||||
\label{fig:img-}
|
||||
\end{figure}
|
||||
|
||||
|
||||
\Section[
|
||||
@ -432,7 +429,6 @@
|
||||
\desc[german]{Ressonanzfrequenz}{}{}
|
||||
\eq{\omega_{21} = \frac{E_2 - E_1}{\hbar}}
|
||||
\end{formula}
|
||||
\TODO{sollte das nicht 10 sein?}
|
||||
|
||||
\begin{formula}{rabi_oscillation}
|
||||
\desc{Rabi oscillations}{}{$\omega_{21}$ resonance frequency of the energy transition, $\Omega$ Rabi frequency}
|
||||
@ -440,15 +436,14 @@
|
||||
\eq{\Omega_ \text{\TODO{TODO}}}
|
||||
\end{formula}
|
||||
|
||||
\Subsection[
|
||||
\eng{Ramsey interferometry}
|
||||
\ger{Ramsey Interferometrie}
|
||||
]{ramsey}
|
||||
|
||||
\begin{formula}{ramsey}
|
||||
\desc{Ramsey interferometry}{}{}
|
||||
\desc[german]{Ramsey Interferometrie}{}{}
|
||||
\begin{ttext}
|
||||
\eng{$\ket{0} \xrightarrow{\frac{\pi}{2}\,\text{pulse}}$ precession in $xy$ plane for time $\tau$ $\xrightarrow{\frac{\pi}{2}\,\text{pulse}}$ measurement}
|
||||
\ger{q}
|
||||
\end{ttext}
|
||||
\end{formula}
|
||||
|
||||
\Section[
|
||||
\eng{Noise and decoherence}
|
||||
|
||||
18
src/readme.md
Normal file
18
src/readme.md
Normal file
@ -0,0 +1,18 @@
|
||||
# Formulary tex source
|
||||
|
||||
## Special directories
|
||||
- `pkg`: My custom Latex packages
|
||||
- `img`: Images generated by `../scripts/`
|
||||
- `img_static`: Downloaded or other images not generated by me
|
||||
- `img_static_svgs`: Downloaded or other images not generated by me that need to be converted to pdf
|
||||
|
||||
|
||||
## Subject directories
|
||||
|
||||
- `bib`: bibliography files
|
||||
- `ch`: chemistry
|
||||
- `cm`: condensed matter
|
||||
- `comp`: computational
|
||||
- `ed`: electrodynamics
|
||||
- `math`: mathematics
|
||||
- `qm`: quantum mechanics
|
||||
@ -219,26 +219,27 @@
|
||||
\eng{Material properties}
|
||||
\ger{Materialeigenschaften}
|
||||
]{props}
|
||||
\begin{formula}{heat_cap}
|
||||
\desc{Heat capacity}{}{$Q$ heat}
|
||||
\desc[german]{Wärmekapazität}{}{$Q$ Wärme}
|
||||
\begin{formula}{heat_capacity}
|
||||
\desc{Heat capacity}{}{\QtyRef{heat}}
|
||||
\desc[german]{Wärmekapazität}{}{}
|
||||
\quantity{c}{\joule\per\kelvin}{}
|
||||
\eq{c = \frac{Q}{\Delta T}}
|
||||
\end{formula}
|
||||
\begin{formula}{heat_cap_V}
|
||||
\desc{Isochoric heat capacity}{}{$U$ internal energy}
|
||||
\desc[german]{Isochore Wärmekapazität}{}{$U$ innere Energie}
|
||||
\begin{formula}{heat_capacity_V}
|
||||
\desc{Isochoric heat capacity}{}{\QtyRef{heat}, \QtyRef{internal_energy} \QtyRef{temperature}, \QtyRef{volume}}
|
||||
\desc[german]{Isochore Wärmekapazität}{}{}
|
||||
\eq{c_v = \pdv{Q}{T}_V = \pdv{U}{T}_V}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{heat_cap_p}
|
||||
\desc{Isobaric heat capacity}{}{$H$ enthalpy}
|
||||
\desc[german]{Isobare Wärmekapazität}{}{$H$ Enthalpie}
|
||||
\eq{c_p = \pdv{Q}{T}_P = \pdv{H}{T}_P}
|
||||
\begin{formula}{heat_capacity_p}
|
||||
\desc{Isobaric heat capacity}{}{\QtyRef{heat}, \QtyRef{enthalpy} \QtyRef{temperature}, \QtyRef{pressure}}
|
||||
\desc[german]{Isobare Wärmekapazität}{}{}
|
||||
\eq{c_p = \pdv{Q}{T}_p = \pdv{H}{T}_p}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{bulk_modules}
|
||||
\desc{Bulk modules}{}{$p$ pressure, $V$ initial volume}
|
||||
\desc[german]{Kompressionsmodul}{}{$p$ Druck, $V$ Anfangsvolumen}
|
||||
\desc{Bulk modules}{}{\QtyRef{pressure}, $V$ initial \qtyRef{volume}}
|
||||
\desc[german]{Kompressionsmodul}{}{\QtyRef{pressure}, $V$ Anfangsvolumen}
|
||||
\eq{K = -V \odv{p}{V} }
|
||||
\end{formula}
|
||||
|
||||
@ -432,33 +433,38 @@
|
||||
\desc{Internal energy}{}{}
|
||||
\desc[german]{Innere Energie}{}{}
|
||||
\eq{\d U(S,V,N) = T\d S -p\d V + \mu\d N}
|
||||
\hiddenQuantity{U}{\joule}{s}
|
||||
\end{formula}
|
||||
\begin{formula}{free_energy}
|
||||
\desc{Free energy / Helmholtz energy }{}{}
|
||||
\desc[german]{Freie Energie / Helmholtz Energie}{}{}
|
||||
\desc{Free energy}{Helmholtz energy}{}
|
||||
\desc[german]{Freie Energie}{Helmholtz Energie}{}
|
||||
\eq{\d F(T,V,N) = -S\d T -p\d V + \mu\d N}
|
||||
\hiddenQuantity{F}{\joule}{s}
|
||||
\end{formula}
|
||||
\begin{formula}{enthalpy}
|
||||
\desc{Enthalpy}{}{}
|
||||
\desc[german]{Enthalpie}{}{}
|
||||
\eq{\d H(S,p,N) = T\d S +V\d p + \mu\d N}
|
||||
\hiddenQuantity{H}{\joule}{s}
|
||||
\end{formula}
|
||||
\begin{formula}{gibbs_energy}
|
||||
\desc{Free enthalpy / Gibbs energy}{}{}
|
||||
\desc[german]{Freie Entahlpie / Gibbs-Energie}{}{}
|
||||
\begin{formula}{free_enthalpy}
|
||||
\desc{Free enthalpy}{Gibbs energy}{}
|
||||
\desc[german]{Freie Entahlpie}{Gibbs-Energie}{}
|
||||
\eq{\d G(T,p,N) = -S\d T + V\d p + \mu\d N}
|
||||
\hiddenQuantity{G}{\joule}{s}
|
||||
\end{formula}
|
||||
\begin{formula}{grand_canon_pot}
|
||||
\desc{Grand canonical potential}{}{}
|
||||
\desc[german]{Großkanonisches Potential}{}{}
|
||||
\eq{\d \Phi(T,V,\mu) = -S\d T -p\d V - N\d\mu}
|
||||
\hiddenQuantity{\Phi}{\joule}{s}
|
||||
\end{formula}
|
||||
|
||||
\TODO{Maxwell Relationen, TD Quadrat}
|
||||
\TODO{Maxwell Relationen}
|
||||
\begin{formula}{td-square}
|
||||
\desc{Thermodynamic squre}{}{}
|
||||
\desc{Thermodynamic square}{}{}
|
||||
\desc[german]{Themodynamisches Quadrat}{Guggenheim Quadrat}{}
|
||||
\begin{minipage}{0.3\textwidth}
|
||||
\fsplit[0.3]{
|
||||
\begin{tikzpicture}
|
||||
\draw[thick] (0,0) grid (3,3);
|
||||
\node at (0.5, 2.5) {$-S$};
|
||||
@ -470,11 +476,12 @@
|
||||
\node at (1.5, 0.5) {\color{blue}$G$};
|
||||
\node at (2.5, 0.5) {$T$};
|
||||
\end{tikzpicture}
|
||||
\end{minipage}
|
||||
\begin{ttext}
|
||||
\eng{The corners opposite from the potential are the coefficients and each coefficients differential is opposite to it.}
|
||||
\ger{Die Ecken gegenüber des Potentials sind die Koeffizienten, das Differential eines Koeffizienten ist in der Ecke gegenüber.}
|
||||
\end{ttext}
|
||||
}{
|
||||
\begin{ttext}
|
||||
\eng{The corners opposite from the potential are the coefficients and each coefficients differential is opposite to it.}
|
||||
\ger{Die Ecken gegenüber des Potentials sind die Koeffizienten, das Differential eines Koeffizienten ist in der Ecke gegenüber.}
|
||||
\end{ttext}
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\Section[
|
||||
@ -675,14 +682,14 @@
|
||||
\eq{g_s = 2s+1}
|
||||
\end{formula}
|
||||
\begin{formula}{dos}
|
||||
\desc{Density of states}{}{$g_s$ \fqEqRef{td:id_qgas:spin_degeneracy_factor}}
|
||||
\desc[german]{Zustandsdichte}{}{$g_s$ \fqEqRef{td:id_qgas:spin_degeneracy_factor}}
|
||||
\desc{Density of states}{}{$g_s$ \fRef{td:id_qgas:spin_degeneracy_factor}}
|
||||
\desc[german]{Zustandsdichte}{}{$g_s$ \fRef{td:id_qgas:spin_degeneracy_factor}}
|
||||
\eq{g(\epsilon) = g_s \frac{V}{4\pi} \left(\frac{2m}{\hbar^2}\right)^\frac{3}{2} \sqrt{\epsilon}}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{occupation_number_per_e}
|
||||
\desc{Occupation number per energy}{}{\fqEqRef{td:id_qgas:dos}, \bosfer}
|
||||
\desc[german]{Besetzungszahl pro Energie}{}{\fqEqRef{td:id_qgas:dos}, \bosfer}
|
||||
\desc{Occupation number per energy}{}{\fRef{td:id_qgas:dos}, \bosfer}
|
||||
\desc[german]{Besetzungszahl pro Energie}{}{\fRef{td:id_qgas:dos}, \bosfer}
|
||||
\eq{n(\epsilon)\, \d\epsilon &= \frac{g(\epsilon)}{\e^{\beta(\epsilon - \mu)} \mp 1}\,\d\epsilon}
|
||||
\end{formula}
|
||||
|
||||
@ -797,8 +804,8 @@
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{specific_density}
|
||||
\desc{Specific density}{}{$f$ \fqEqRef{td:id_qgas:generalized_zeta}, $g$ degeneracy factor, $z$ \fqEqRef{td:id_qgas:fugacity}}
|
||||
\desc[german]{Spezifische Dichte}{}{$f$ \fqEqRef{td:id_qgas:generalized_zeta}, $g$ Entartungsfaktor, $z$ \fqEqRef{td:id_qgas:fugacity}}
|
||||
\desc{Specific density}{}{$f$ \fRef{td:id_qgas:generalized_zeta}, $g$ degeneracy factor, $z$ \fRef{td:id_qgas:fugacity}}
|
||||
\desc[german]{Spezifische Dichte}{}{$f$ \fRef{td:id_qgas:generalized_zeta}, $g$ Entartungsfaktor, $z$ \fRef{td:id_qgas:fugacity}}
|
||||
\eq{v = \frac{N}{V} = \frac{g}{\lambda^3}f_{3/2}(z)}
|
||||
\end{formula}
|
||||
|
||||
@ -825,9 +832,9 @@
|
||||
}
|
||||
\end{formula}
|
||||
|
||||
\begin{formula}{heat_cap}
|
||||
\desc{Heat capacity}{\gt{low_temps}}{differs from \fqEqRef{td:TODO:petit_dulong}}
|
||||
\desc[german]{Wärmecapacity}{\gt{low_temps}}{weicht ab vom \fqEqRef{td:TODO:petit_dulong}}
|
||||
\begin{formula}{heat_capacity}
|
||||
\desc{Heat capacity}{\gt{low_temps}}{differs from \fRef{td:TODO:petit_dulong}}
|
||||
\desc[german]{Wärmecapacity}{\gt{low_temps}}{weicht ab vom \fRef{td:TODO:petit_dulong}}
|
||||
\fig{img/td_fermi_heat_capacity.pdf}
|
||||
\eq{C_V = \pdv{E}{T}_V = N\kB \frac{\pi}{2} \left(\frac{T}{T_\text{F}}\right)}
|
||||
\end{formula}
|
||||
|
||||
@ -1,5 +1,8 @@
|
||||
\part{Testing}
|
||||
|
||||
Textwidth: \the\textwidth
|
||||
\\Linewidth: \the\linewidth
|
||||
|
||||
% \directlua{tex.sprint("Compiled in directory: \\detokenize{" .. lfs.currentdir() .. "}")} \\
|
||||
% \directlua{tex.sprint("Jobname: " .. tex.jobname)} \\
|
||||
% \directlua{tex.sprint("Output directory \\detokenize{" .. os.getenv("TEXMF_OUTPUT_DIRECTORY") .. "}")} \\
|
||||
@ -52,7 +55,7 @@ GT: {\textbackslash}ttest:name = \GT{\ttest:name}\\
|
||||
|
||||
\paragraph{Testing hyperrefs}
|
||||
\noindent{This text is labeled with "test" \label{test}}\\
|
||||
\hyperref[test]{This should refer to the line above}\\
|
||||
\fRef[This should refer to the line above]{test}\\
|
||||
Link to quantity which is defined after the reference: \qtyRef{test}\\
|
||||
\DT[eq:test]{english}{If you read this, then the translation for eq:test was expandend!}
|
||||
Link to defined quantity: \qtyRef{mass}
|
||||
|
||||
@ -30,8 +30,8 @@
|
||||
|
||||
\hypersetup{
|
||||
colorlinks=true,
|
||||
linkcolor=fg-purple,
|
||||
linkcolor=fg-blue,
|
||||
citecolor=fg-green,
|
||||
filecolor=fg-blue,
|
||||
filecolor=fg-purple,
|
||||
urlcolor=fg-orange
|
||||
}
|
||||
|
||||
@ -1,5 +1,3 @@
|
||||
\def\descwidth{0.3\textwidth}
|
||||
\def\eqwidth{0.6\textwidth}
|
||||
|
||||
\newcommand\separateEntries{
|
||||
\vspace{0.5\baselineskip}
|
||||
@ -43,12 +41,12 @@
|
||||
% add links to some names
|
||||
\directlua{
|
||||
local cases = {
|
||||
pdf = "f:math:pt:pdf",
|
||||
pmf = "f:math:pt:pmf",
|
||||
cdf = "f:math:pt:cdf",
|
||||
mean = "f:math:pt:mean",
|
||||
variance = "f:math:pt:variance",
|
||||
median = "f:math:pt:median",
|
||||
pdf = "math:pt:pdf",
|
||||
pmf = "math:pt:pmf",
|
||||
cdf = "math:pt:cdf",
|
||||
mean = "math:pt:mean",
|
||||
variance = "math:pt:variance",
|
||||
median = "math:pt:median",
|
||||
}
|
||||
if cases["\luaescapestring{##1}"] \string~= nil then
|
||||
tex.sprint("\\hyperref["..cases["\luaescapestring{##1}"].."]{\\GT{##1}}")
|
||||
|
||||
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Reference in New Issue
Block a user