倒易点阵

对于以${\displaystyle {\boldsymbol {a}}}$为基矢的一维晶格，其倒格子的基矢为

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

对于以${\displaystyle ({\boldsymbol {a_{1}}},{\boldsymbol {a_{2}}})}$为基矢的二维晶格，定义其二维平面法线向量为${\displaystyle {\boldsymbol {n}}}$，其倒格子的基矢为

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

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

對三維晶格而言，我們定義素晶胞的基矢 ${\displaystyle ({\boldsymbol {a_{1}}},{\boldsymbol {a_{2}}},{\boldsymbol {a_{3}}})}$，可以用下列公式決定倒晶格的晶胞基矢${\displaystyle ({\boldsymbol {b_{1}}},{\boldsymbol {b_{2}}},{\boldsymbol {b_{3}}})}$

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

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

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

倒晶格与正晶格的基矢满足以下关系

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

定义三维中的倒晶格向量G

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

其中(h,k,l)为密勒指数，向量G的模长与正晶格的晶面间距有以下关系

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

向量G和正晶格向量R有以下关系

${\displaystyle \mathbf {R} =c_{1}{\boldsymbol {a_{1}}}+c_{2}{\boldsymbol {a_{2}}}+c_{3}{\boldsymbol {a_{3}}}}$

${\displaystyle \mathrm {e} ^{\mathrm {i} \mathbf {G\cdot R} }=1}$

三维倒晶格中的晶胞体积Ω_G和正晶格的晶胞体积Ω有以下关系

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

簡單立方晶體的素格子基矢可以寫成

${\displaystyle {\boldsymbol {a_{1}}}=a{\hat {x}}}$

${\displaystyle {\boldsymbol {a_{2}}}=a{\hat {y}}}$

${\displaystyle {\boldsymbol {a_{3}}}=a{\hat {z}}}$

體積為

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

可推得倒晶格的素格子基矢

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

${\displaystyle {\boldsymbol {b_{2}}}={2\pi \over a}{\hat {y}}}$

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

所以簡單立方晶體的倒晶格同樣為簡單立方晶體，但是晶格常數為 ${\displaystyle 2\pi \over a}$。

面心立方晶體的素格子基矢可以寫成下列三項

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

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

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

體積為

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

可推得倒晶格之素格子基矢

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

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

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

面心立方晶體的倒晶格為體心立方晶體。

體心立方晶體的素格子基矢可以寫成下列三項

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

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

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

體積為

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

可推得倒晶格之素格子基矢

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

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

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

可得知體心立方晶體之倒晶格為面心立方晶體。

在布拉菲晶格中,三軸互為九十度的${\displaystyle ({\boldsymbol {a_{1}}},{\boldsymbol {a_{2}}},{\boldsymbol {a_{3}}})}$ (立方, 正方, 斜方)的晶體結構,是很容易被證明其倒晶格空間之三軸${\displaystyle ({\boldsymbol {b_{1}}},{\boldsymbol {b_{2}}},{\boldsymbol {b_{3}}})}$與其真實晶格之三軸有垂直的關係.