# 倒易点阵

## 数学描述

### 一维晶格

${\displaystyle {\boldsymbol {b}}=2\pi {\frac {\boldsymbol {a}}{a^{2}}}}$

### 二维晶格

${\displaystyle {\boldsymbol {b_{1}}}=2\pi {\frac {{\boldsymbol {a_{2}}}\times {\boldsymbol {n}}}{{\boldsymbol {a_{1}}}\cdot ({\boldsymbol {a_{2}}}\times {\boldsymbol {n}})}}}$
${\displaystyle {\boldsymbol {b_{2}}}=2\pi {\frac {{\boldsymbol {n}}\times {\boldsymbol {a_{1}}}}{{\boldsymbol {a_{2}}}\cdot ({\boldsymbol {n}}\times {\boldsymbol {a_{1}}})}}}$

### 三维晶格

${\displaystyle {\boldsymbol {b_{1}}}=2\pi {\frac {{\boldsymbol {a_{2}}}\times {\boldsymbol {a_{3}}}}{{\boldsymbol {a_{1}}}\cdot ({\boldsymbol {a_{2}}}\times {\boldsymbol {a_{3}}})}}}$
${\displaystyle {\boldsymbol {b_{2}}}=2\pi {\frac {{\boldsymbol {a_{3}}}\times {\boldsymbol {a_{1}}}}{{\boldsymbol {a_{2}}}\cdot ({\boldsymbol {a_{3}}}\times {\boldsymbol {a_{1}}})}}}$
${\displaystyle {\boldsymbol {b_{3}}}=2\pi {\frac {{\boldsymbol {a_{1}}}\times {\boldsymbol {a_{2}}}}{{\boldsymbol {a_{3}}}\cdot ({\boldsymbol {a_{1}}}\times {\boldsymbol {a_{2}}})}}}$

### 倒晶格与正晶格的关系

${\displaystyle {\boldsymbol {a_{i}}}\cdot {\boldsymbol {b_{j}}}=2\pi \delta _{ij}={\begin{cases}2\pi ,&i\ =\ j\\0,&i\ \neq \ j\end{cases}}}$

${\displaystyle \mathbf {G} =h{\boldsymbol {b_{1}}}+k{\boldsymbol {b_{2}}}+l{\boldsymbol {b_{3}}}}$

${\displaystyle \mathbf {|G_{hkl}|} ={\frac {2\pi }{d_{hkl}}}}$

${\displaystyle \mathbf {R} =c_{1}{\boldsymbol {a_{1}}}+c_{2}{\boldsymbol {a_{2}}}+c_{3}{\boldsymbol {a_{3}}}}$
${\displaystyle \mathrm {e} ^{\mathrm {i} \mathbf {G\cdot R} }=1}$

${\displaystyle \Omega _{G}={\frac {(2\pi )^{3}}{\Omega }}}$

## 倒晶格的物理意义

${\displaystyle \psi _{\boldsymbol {k}}({\boldsymbol {x}})=\mathrm {e} ^{\mathrm {i} {\boldsymbol {k}}\cdot {\boldsymbol {x}}}u_{\boldsymbol {k}}({\boldsymbol {x}})}$

${\displaystyle {\boldsymbol {G}}=n{\boldsymbol {b}},\ n=0,1,2,\cdots }$
${\displaystyle {\boldsymbol {b}}=2\pi {\frac {\boldsymbol {a}}{a^{2}}}}$
${\displaystyle {\boldsymbol {G}}\cdot {\boldsymbol {a}}=2\pi n}$

{\displaystyle {\begin{alignedat}{2}u_{\boldsymbol {k+G}}({\boldsymbol {x}})&=\mathrm {e} ^{-\mathrm {i} {\boldsymbol {G}}\cdot {\boldsymbol {x}}}u_{\boldsymbol {k}}({\boldsymbol {x}})\\u_{\boldsymbol {k+G}}({\boldsymbol {x+a}})&=\mathrm {e} ^{-\mathrm {i} {\boldsymbol {G}}\cdot {\boldsymbol {x}}}\mathrm {e} ^{-\mathrm {i} {\boldsymbol {G}}\cdot {\boldsymbol {a}}}u_{\boldsymbol {k}}({\boldsymbol {x+a}})\\&=\mathrm {e} ^{-\mathrm {i} {\boldsymbol {G}}\cdot {\boldsymbol {x}}}u_{\boldsymbol {k}}({\boldsymbol {x+a}})\\\end{alignedat}}}

${\displaystyle u_{\boldsymbol {k}}({\boldsymbol {x+a}})=u_{\boldsymbol {k}}({\boldsymbol {x}})}$

${\displaystyle u_{\boldsymbol {k+G}}({\boldsymbol {x+a}})=u_{\boldsymbol {k+G}}({\boldsymbol {x}})}$

{\displaystyle {\begin{aligned}\psi _{\boldsymbol {k}}({\boldsymbol {x}})&=\mathrm {e} ^{\mathrm {i} {\boldsymbol {k}}\cdot {\boldsymbol {x}}}u_{\boldsymbol {k}}({\boldsymbol {x}})\\&=\mathrm {e} ^{\mathrm {i} ({\boldsymbol {k+G}})\cdot {\boldsymbol {x}}}\mathrm {e} ^{-\mathrm {i} {\boldsymbol {G}}\cdot {\boldsymbol {x}}}u_{\boldsymbol {k}}({\boldsymbol {x}})\\&=\mathrm {e} ^{\mathrm {i} ({\boldsymbol {k+G}})\cdot {\boldsymbol {x}}}u_{\boldsymbol {k+G}}({\boldsymbol {x}})\\&=\psi _{\boldsymbol {k+G}}({\boldsymbol {x}})\end{aligned}}}

## 倒晶格与晶体衍射

{\displaystyle {\begin{alignedat}{2}2d\sin \theta =n\lambda \\2\times {\frac {2\pi }{\lambda }}\sin \theta ={\frac {2\pi }{d_{n}}}\\\end{alignedat}}}

${\displaystyle {\begin{array}{lcl}|{\boldsymbol {k}}|={\cfrac {2\pi }{\lambda }}\\\mathbf {|G_{hkl}|} ={\cfrac {2\pi }{d_{hkl}}}\\2|{\boldsymbol {k}}|\sin \theta =|\mathbf {G} |\\\end{array}}}$

${\displaystyle {\boldsymbol {\Delta k}}={\boldsymbol {k_{o}}}-{\boldsymbol {k_{i}}}=\mathbf {G} }$

## 常见布拉菲晶格的倒晶格

### 簡單立方晶體

${\displaystyle {\boldsymbol {a_{1}}}=a{\hat {x}}}$
${\displaystyle {\boldsymbol {a_{2}}}=a{\hat {y}}}$
${\displaystyle {\boldsymbol {a_{3}}}=a{\hat {z}}}$

${\displaystyle {\boldsymbol {a_{1}}}\cdot {\boldsymbol {a_{2}}}\times {\boldsymbol {a_{3}}}=a^{3}}$

${\displaystyle {\boldsymbol {b_{1}}}={2\pi \over a}{\hat {x}}}$
${\displaystyle {\boldsymbol {b_{2}}}={2\pi \over a}{\hat {y}}}$
${\displaystyle {\boldsymbol {b_{3}}}={2\pi \over a}{\hat {z}}}$

### 面心立方晶體(FCC)

${\displaystyle {\boldsymbol {a_{1}}}={a \over 2}\left({\hat {y}}+{\hat {z}}\right)}$
${\displaystyle {\boldsymbol {a_{2}}}={a \over 2}\left({\hat {z}}+{\hat {x}}\right)}$
${\displaystyle {\boldsymbol {a_{3}}}={a \over 2}\left({\hat {x}}+{\hat {y}}\right)}$

${\displaystyle {\boldsymbol {a_{1}}}\cdot {\boldsymbol {a_{2}}}\times {\boldsymbol {a_{3}}}={a^{3} \over 4}}$

${\displaystyle {\boldsymbol {b_{1}}}={2\pi \over a}\left(-{\hat {x}}+{\hat {y}}+{\hat {z}}\right)}$
${\displaystyle {\boldsymbol {b_{2}}}={2\pi \over a}\left(+{\hat {x}}-{\hat {y}}+{\hat {z}}\right)}$
${\displaystyle {\boldsymbol {b_{3}}}={2\pi \over a}\left(+{\hat {x}}+{\hat {y}}-{\hat {z}}\right)}$

### 體心立方晶體(BCC)

${\displaystyle {\boldsymbol {a_{1}}}={a \over 2}\left(-{\hat {x}}+{\hat {y}}+{\hat {z}}\right)}$
${\displaystyle {\boldsymbol {a_{2}}}={a \over 2}\left(+{\hat {x}}-{\hat {y}}+{\hat {z}}\right)}$
${\displaystyle {\boldsymbol {a_{3}}}={a \over 2}\left(+{\hat {x}}+{\hat {y}}-{\hat {z}}\right)}$

${\displaystyle {\boldsymbol {a_{1}}}\cdot {\boldsymbol {a_{2}}}\times {\boldsymbol {a_{3}}}={a^{3} \over 2}}$

${\displaystyle {\boldsymbol {b_{1}}}={2\pi \over a}\left({\hat {y}}+{\hat {z}}\right)}$
${\displaystyle {\boldsymbol {b_{2}}}={2\pi \over a}\left({\hat {z}}+{\hat {x}}\right)}$
${\displaystyle {\boldsymbol {b_{3}}}={2\pi \over a}\left({\hat {x}}+{\hat {y}}\right)}$