一、定义
正格子基矢:。
倒格子基矢:。
需要满足关系:。
1. 三维的情况
倒格子基矢可以写为以下形式:
2. 二维的情况
倒格子基矢可以写为以下形式:
其中,表示的是垂直于二维平面的单位矢量[1]。
1. 一维的情况
直接写出倒格子基矢表达式:
此外,也可以写为以下形式:
其中,表示的是垂直于
的两个相互垂直单位矢量。
二、数值计算
以方格子的基矢为例,数值计算代码如下:
"""
This code is supported by the website: https://www.guanjihuan.com
The newest version of this code is on the web page: https://www.guanjihuan.com/archives/15978
"""
import numpy as np
from math import *
def main():
a1 = [0, 1]
a2 = [1, 0]
b1, b2 = calculate_two_dimensional_reciprocal_lattice_vectors(a1, a2)
print(b1, b2)
def calculate_one_dimensional_reciprocal_lattice_vector(a1):
b1 = 2*pi/a1
return b1
def calculate_two_dimensional_reciprocal_lattice_vectors(a1, a2):
a1 = np.array(a1)
a2 = np.array(a2)
a1 = np.append(a1, 0)
a2 = np.append(a2, 0)
a3 = np.array([0, 0, 1])
b1 = 2*pi*np.cross(a2, a3)/np.dot(a1, np.cross(a2, a3))
b2 = 2*pi*np.cross(a3, a1)/np.dot(a1, np.cross(a2, a3))
b1 = np.delete(b1, 2)
b2 = np.delete(b2, 2)
return b1, b2
def calculate_three_dimensional_reciprocal_lattice_vectors(a1, a2, a3):
a1 = np.array(a1)
a2 = np.array(a2)
a3 = np.array(a3)
b1 = 2*pi*np.cross(a2, a3)/np.dot(a1, np.cross(a2, a3))
b2 = 2*pi*np.cross(a3, a1)/np.dot(a1, np.cross(a2, a3))
b3 = 2*pi*np.cross(a1, a2)/np.dot(a1, np.cross(a2, a3))
return b1, b2, b3
if __name__ == '__main__':
main()
运算结果:
[-0. 6.28318531] [ 6.28318531 -0. ]
三、符号计算
以石墨烯格子的基矢为例,符号计算代码如下(用到Python符号运算库SymPy):
"""
This code is supported by the website: https://www.guanjihuan.com
The newest version of this code is on the web page: https://www.guanjihuan.com/archives/15978
"""
import sympy
def main():
print('bases in the real space')
a = sympy.symbols('a')
a1 = sympy.Matrix(1, 2, [3*a/2, sympy.sqrt(3)*a/2])
a2 = sympy.Matrix(1, 2, [3*a/2, -sympy.sqrt(3)*a/2])
print('a1:')
sympy.pprint(a1)
print('a2:')
sympy.pprint(a2)
print('\nbases in the reciprocal space')
b1, b2 = calculate_two_dimensional_reciprocal_lattice_vectors_with_sympy(a1, a2)
print('b1:')
sympy.pprint(b1)
print('b2:')
sympy.pprint(b2)
def calculate_one_dimensional_reciprocal_lattice_vector_with_sympy(a1):
b1 = 2*sympy.pi/a1
return b1
def calculate_two_dimensional_reciprocal_lattice_vectors_with_sympy(a1, a2):
a1 = sympy.Matrix(1, 3, [a1[0], a1[1], 0])
a2 = sympy.Matrix(1, 3, [a2[0], a2[1], 0])
a3 = sympy.Matrix(1, 3, [0, 0, 1])
cross_a2_a3 = a2.cross(a3)
cross_a3_a1 = a3.cross(a1)
b1 = 2*sympy.pi*cross_a2_a3/a1.dot(cross_a2_a3)
b2 = 2*sympy.pi*cross_a3_a1/a1.dot(cross_a2_a3)
b1 = sympy.Matrix(1, 2, [b1[0], b1[1]])
b2 = sympy.Matrix(1, 2, [b2[0], b2[1]])
return b1, b2
def calculate_three_dimensional_reciprocal_lattice_vectors_with_sympy(a1, a2, a3):
cross_a2_a3 = a2.cross(a3)
cross_a3_a1 = a3.cross(a1)
cross_a1_a2 = a1.cross(a2)
b1 = 2*sympy.pi*cross_a2_a3/a1.dot(cross_a2_a3)
b2 = 2*sympy.pi*cross_a3_a1/a1.dot(cross_a2_a3)
b3 = 2*sympy.pi*cross_a1_a2/a1.dot(cross_a2_a3)
return b1, b2, b3
if __name__ == '__main__':
main()
运算结果:
bases in the real space
a1:
⎡3⋅a √3⋅a⎤
⎢─── ────⎥
⎣ 2 2 ⎦
a2:
⎡3⋅a -√3⋅a ⎤
⎢─── ──────⎥
⎣ 2 2 ⎦
bases in the reciprocal space
b1:
⎡2⋅π 2⋅√3⋅π⎤
⎢─── ──────⎥
⎣3⋅a 3⋅a ⎦
b2:
⎡2⋅π -2⋅√3⋅π ⎤
⎢─── ────────⎥
⎣3⋅a 3⋅a ⎦
四、使用GUAN开源项目软件包
目前以上六个计算倒格子基矢的函数已加入开源项目Guan:https://py.guanjihuan.com,使用命令”pip install --upgrade guan“安装到最新版,安装后可直接调用。代码如下:
import guan
import sympy
a1 = [0, 1]
a2 = [1, 0]
b1, b2 = guan.calculate_two_dimensional_reciprocal_lattice_vectors(a1, a2)
print(b1, b2, '\n')
print('bases in the real space')
a = sympy.symbols('a')
a1 = sympy.Matrix(1, 2, [3*a/2, sympy.sqrt(3)*a/2])
a2 = sympy.Matrix(1, 2, [3*a/2, -sympy.sqrt(3)*a/2])
print('a1:')
sympy.pprint(a1)
print('a2:')
sympy.pprint(a2)
print('\nbases in the reciprocal space')
b1, b2 = guan.calculate_two_dimensional_reciprocal_lattice_vectors_with_sympy(a1, a2)
print('b1:')
sympy.pprint(b1)
print('b2:')
sympy.pprint(b2)
参考资料:
[1] 黄昆《固体物理学》
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谢谢老哥!非常有用!