442 lines
413 KiB
Plaintext
442 lines
413 KiB
Plaintext
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{
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"cells": [
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{
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"cell_type": "code",
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"execution_count": 1,
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"id": "eaed683c-c6f1-45e4-aaee-ae4e57209f5f",
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"metadata": {},
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"outputs": [],
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"source": [
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"import scipy as scp\n",
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"from scipy.spatial import Voronoi, voronoi_plot_2d\n",
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"import numpy as np\n",
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"import matplotlib.pyplot as plt\n",
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"from mpl_toolkits.mplot3d import Axes3D\n",
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"from mpl_toolkits.mplot3d.art3d import Poly3DCollection\n",
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"%matplotlib widget"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"id": "efd84ecd-9fb3-4d2f-9f7a-47cd1ff09eea",
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"metadata": {},
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"outputs": [],
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"source": [
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"class Lattice:\n",
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" def __init__(self, *vecs):\n",
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" # if the vecs were put in an iterable\n",
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" if len(vecs) == 1:\n",
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" vecs = vecs[0]\n",
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" if len(vecs) == 3:\n",
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" pass\n",
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" elif len(vecs) == 2:\n",
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" pass\n",
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" else: raise ValueError(\"Vecs must contain either 2 or 3 vectors\")\n",
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" self.dim = len(vecs)\n",
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" self.vecs = list(vecs)\n",
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" for i, v in enumerate(self.vecs):\n",
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" if type(v) != np.ndarray:\n",
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" self.vecs[i] = np.array(v)\n",
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" if self.vecs[i].shape != (self.dim,):\n",
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" raise ValueError(f\"Got {self.dim} vectors, therefore all vectors must be {self.dim} dimensional but vector {i+1} has shape {self.vecs[i].shape}\")\n",
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" self.vecs = np.array(self.vecs)\n",
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" self.vec_lengths = np.array([np.linalg.norm(v) for v in self.vecs])\n",
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" self.center = np.zeros(self.dim)\n",
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"\n",
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" def get_point(self, *ns):\n",
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" if len(ns) != len(self.vecs): raise ValueError(f\"Requires one index for each lattice vector {len(self.vecs)}, but got only {ns}\")\n",
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" point = self.center.copy()\n",
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" for i, n in enumerate(ns):\n",
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" point += n * self.vecs[i]\n",
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" return point\n",
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"\n",
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" \n",
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" def get_points_around_center(self, n):\n",
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" points = []\n",
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" import itertools\n",
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" ns = [i for i in range(-n, n+1)]\n",
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" for n in itertools.product(*[ns for _ in range(self.dim)]):\n",
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" # print(n)\n",
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" points.append(self.get_point(*n))\n",
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" return points"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 3,
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"id": "515b9133-233b-4a6a-8b2f-a2fca10fea3d",
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"metadata": {},
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"outputs": [],
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"source": [
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"def rot_mat_2D(rad):\n",
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" return np.array([[np.cos(rad), -np.sin(rad)], [np.sin(rad), np.cos(rad)]])\n",
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"\n",
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"\n",
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"def get_reciprocal_lattice(lattice: Lattice):\n",
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" if lattice.dim == 2:\n",
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" rot_90_deg = rot_mat_2D(np.pi / 2)\n",
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" a1, a2 = lattice.vecs\n",
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" b1 = 2 * np.pi * rot_90_deg @ a2 / (np.dot(a1, rot_90_deg @ a2))\n",
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" b2 = 2 * np.pi * rot_90_deg @ a1 / (np.dot(a2, rot_90_deg @ a1))\n",
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" return Lattice(b1, b2)\n",
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" elif lattice.dim == 3:\n",
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" a1, a2, a3 = lattice.vecs\n",
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" V = np.dot(a1, np.cross(a2, a3))\n",
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" b1 = 2 * np.pi/V * np.cross(a2, a3)\n",
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" b2 = 2 * np.pi/V * np.cross(a3, a1)\n",
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" b3 = 2 * np.pi/V * np.cross(a1, a2)\n",
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" return Lattice(b1, b2, b3)\n",
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" else: raise NotImplementedError(f\"Dim must be 2 or 3, but is {lattice.dim}\")"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 4,
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"id": "2855d08e-70d8-4ef7-ba19-2c06316caf15",
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"metadata": {},
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"outputs": [],
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"source": [
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"\n",
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"def get_elementary_cell(lattice: Lattice):\n",
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" points = lattice.get_points_around_center(1)\n",
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" return points\n",
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"\n",
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"def get_orthogonal_2D(vec):\n",
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" return np.array((vec[1], -vec[0]))\n",
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"\n",
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"def get_unit_cell_vertices(lattice: Lattice, voronoi: Voronoi):\n",
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" \"\"\"regard only voronoi vertices which are closest to the center <=> their norm is <= 0.5*(norm of the unit vectors added together\n",
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" \"\"\"\n",
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" lattice_vec_norm = np.sqrt(np.sum(lattice.vec_lengths**2))\n",
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" return voronoi.vertices[np.linalg.norm(voronoi.vertices, axis=1) <= 0.5 * lattice_vec_norm]\n",
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" \n",
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"def plot_unit_cell(lattice, fig_ax=None, vec_label=\"a\", subplot_kw={}):\n",
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" # get voronoi of the points around the center\n",
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" points = get_elementary_cell(lattice)\n",
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" voronoi = Voronoi(points)\n",
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"\n",
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" if fig_ax:\n",
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" fig, ax = fig_ax\n",
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" else:\n",
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" if lattice.dim == 3:\n",
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" fig = plt.figure()\n",
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" ax = fig.add_subplot(1,1,1, projection=\"3d\") \n",
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" else:\n",
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" fig, ax = plt.subplots(**subplot_kw) \n",
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"\n",
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" # unit cell vertices\n",
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" cell_points = get_unit_cell_vertices(lattice, voronoi)\n",
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" # sort by polar angle for the fill function\n",
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" cell_points = list(cell_points)\n",
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" # print(cell_points)\n",
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" # print([i for i in map(lambda p: np.arctan2(p[1],p[0]), cell_points)])\n",
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" if lattice.dim == 2:\n",
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" cell_points.sort(key=lambda p: np.arctan2(p[1],p[0]))\n",
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" x_cell, y_cell = zip(*cell_points)\n",
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" ax.fill(x_cell, y_cell, color=\"#4444\")\n",
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" ax.scatter(x_cell, y_cell, color=\"orange\")\n",
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" \n",
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" # lattice points\n",
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" x_lat, y_lat = zip(*lattice.get_points_around_center(3))\n",
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" ax.scatter(x_lat, y_lat, color=\"blue\")\n",
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" \n",
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" # lattice vectors\n",
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" arrowprops = dict(arrowstyle=\"-|>,head_width=0.4,head_length=0.8\", color=\"black\", shrinkA=0,shrinkB=0)\n",
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" for i, vec in enumerate(lattice.vecs):\n",
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" ax.annotate(f\"\", xy=lattice.vecs[i], xytext=lattice.center, arrowprops=arrowprops)\n",
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" if vec_label is not None:\n",
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" # add name of vector at a perpendicular offset starting at half length\n",
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" ax.annotate(r\"$\\vec{\"+f\"{vec_label}}}_{i+1}$\", xy=0.7*lattice.vecs[i], xytext=0.7*lattice.vecs[i] + 0.06*get_orthogonal_2D(lattice.vecs[i]))\n",
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" elif lattice.dim == 3:\n",
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" ridges = voronoi.ridge_vertices\n",
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" for ridge in ridges:\n",
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" verts = voronoi.vertices[ridge]\n",
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" # print(verts, type(verts), verts.shape, verts.ndim)\n",
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" ax.add_collection3d(Poly3DCollection([voronoi.vertices[ridge]], edgecolor=\"black\"))\n",
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" ax.set_xlim(-2, 2)\n",
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" ax.set_ylim(-2, 2)\n",
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" ax.set_zlim(-2, 2)\n",
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" else: raise NotImplementedError(f\"Dim must be 2 or 3, but is {lattice.dim}\")\n",
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"\n",
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" # limit to 2*lattice vectors\n",
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" def calc_lim(axis):\n",
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" lim = 2.05 * np.max(np.abs(lattice.vecs[axis,:]))\n",
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" return -lim, lim\n",
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" ax.set_xlim(*calc_lim(0))\n",
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" ax.set_ylim(*calc_lim(1))\n",
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" if lattice.dim == 3: ax.set_zlim(*calc_lim(2))\n",
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" return fig"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 5,
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"id": "c2faef81-5f2a-4950-b986-e891ddfa7da8",
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"metadata": {},
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"outputs": [],
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"source": [
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"sphere_point = lambda rad: np.array([np.cos(rad), np.sin(rad)])\n",
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"\n",
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"square_lattice = Lattice([1, 0], [0, 1])\n",
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"tilted_lattice = Lattice([1, 0.5], [0, 1])\n",
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"honeycomb_lattice = Lattice(sphere_point(0), sphere_point(np.pi * 2/3))"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 6,
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"id": "a6b6ccf4-34c7-419d-8bcc-f3ce2870ba25",
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"metadata": {},
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"outputs": [],
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"source": [
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"# fig, axs = plt.subplots(3, figsize=(4, 12))\n",
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"# plot_unit_cell(square_lattice, fig_ax=(fig, axs[0]))\n",
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"# plot_unit_cell(tilted_lattice, fig_ax=(fig, axs[1]))\n",
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"# plot_unit_cell(honeycomb_lattice, fig_ax=(fig,axs[2]));"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 7,
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"id": "5314dd3a-17a7-402d-b3ee-ddbf69a6c4ed",
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"metadata": {},
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"outputs": [],
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"source": [
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"def plot_lattice(lattice: Lattice):\n",
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" reci = get_reciprocal_lattice(lattice)\n",
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" print(reci.vecs)\n",
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" if lattice.dim == 3:\n",
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" fig = plt.figure()\n",
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" axs = [fig.add_subplot(1,2,i, projection=\"3d\") for i in [1,2]]\n",
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" else:\n",
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" fig, axs = plt.subplots(1, 2)\n",
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" plot_unit_cell(lattice, fig_ax=(fig, axs[0]))\n",
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" plot_unit_cell(reci, fig_ax=(fig, axs[1]), vec_label=\"b\")"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 8,
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"id": "39a49d00-fc93-4f2a-8645-d5e440e7a092",
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"[[ 6.28318531e+00 -3.84734139e-16]\n",
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" [ 3.84734139e-16 6.28318531e+00]]\n",
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"[[ 6.28318531e+00 -3.84734139e-16]\n",
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" [-3.14159265e+00 6.28318531e+00]]\n",
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"[[6.28318531e+00 3.62759873e+00]\n",
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" [4.44252717e-16 7.25519746e+00]]\n"
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]
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},
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{
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"data": {
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"application/vnd.jupyter.widget-view+json": {
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"model_id": "d387139d3da141268f8c9c7345f26575",
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"version_major": 2,
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"version_minor": 0
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},
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" </div>\n",
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" "
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||
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],
|
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|
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"text/plain": [
|
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|
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"Canvas(toolbar=Toolbar(toolitems=[('Home', 'Reset original view', 'home', 'home'), ('Back', 'Back to previous …"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
"metadata": {},
|
||
|
|
"output_type": "display_data"
|
||
|
|
}
|
||
|
|
],
|
||
|
|
"source": [
|
||
|
|
"plot_lattice(square_lattice)\n",
|
||
|
|
"plot_lattice(tilted_lattice)\n",
|
||
|
|
"plot_lattice(honeycomb_lattice)"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"execution_count": 11,
|
||
|
|
"id": "ef0a5d74-739f-4688-ad7f-fed91c39dcaa",
|
||
|
|
"metadata": {},
|
||
|
|
"outputs": [],
|
||
|
|
"source": [
|
||
|
|
"simple_cubic = Lattice([1,0,0], [0,1,0], [0,0,1])\n",
|
||
|
|
"graphite = Lattice([0.5,-0.5 * 3**(0.5),0], [0.5,0.5 * 3**(0.5),0], [0,0,1])"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"execution_count": 12,
|
||
|
|
"id": "364f3ddb-21d0-4586-8b3b-acad8ba20d9b",
|
||
|
|
"metadata": {},
|
||
|
|
"outputs": [
|
||
|
|
{
|
||
|
|
"name": "stdout",
|
||
|
|
"output_type": "stream",
|
||
|
|
"text": [
|
||
|
|
"[[6.28318531 0. 0. ]\n",
|
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|
|
" [0. 6.28318531 0. ]\n",
|
||
|
|
" [0. 0. 6.28318531]]\n",
|
||
|
|
"[[ 6.28318531 -3.62759873 0. ]\n",
|
||
|
|
" [ 6.28318531 3.62759873 -0. ]\n",
|
||
|
|
" [-0. 0. 6.28318531]]\n"
|
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]
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},
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{
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"data": {
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"application/vnd.jupyter.widget-view+json": {
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"model_id": "3eb3e14cd16741d085826e020df23362",
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"version_major": 2,
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"version_minor": 0
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},
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{
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" Figure\n",
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" </div>\n",
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||
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" "
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],
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"text/plain": [
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||
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"Canvas(toolbar=Toolbar(toolitems=[('Home', 'Reset original view', 'home', 'home'), ('Back', 'Back to previous …"
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||
|
|
]
|
||
|
|
},
|
||
|
|
"metadata": {},
|
||
|
|
"output_type": "display_data"
|
||
|
|
}
|
||
|
|
],
|
||
|
|
"source": [
|
||
|
|
"plot_lattice(simple_cubic);\n",
|
||
|
|
"plot_lattice(graphite);"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"execution_count": null,
|
||
|
|
"id": "c6512444-6c86-4e31-bc6a-2c8b1e00c28d",
|
||
|
|
"metadata": {},
|
||
|
|
"outputs": [],
|
||
|
|
"source": []
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"execution_count": null,
|
||
|
|
"id": "2dce1256-7afa-4e86-8aff-5db7b13b4d4d",
|
||
|
|
"metadata": {},
|
||
|
|
"outputs": [],
|
||
|
|
"source": []
|
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|
|
}
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||
|
|
],
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||
|
|
"metadata": {
|
||
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|
"kernelspec": {
|
||
|
|
"display_name": "conda",
|
||
|
|
"language": "python",
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||
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"name": "conda"
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},
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"language_info": {
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"codemirror_mode": {
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|
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"name": "ipython",
|
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
|
||
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.12.3"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 5
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}
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