倒易点阵

一维晶格

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

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

二维晶格

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

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

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

三维晶格

对三维晶格而言，我们定义素晶胞的基矢 ${\displaystyle ({\boldsymbol {a_{1}}},{\boldsymbol {a_{2}}},{\boldsymbol {a_{3}}})}$ ，可以用下列公式决定倒晶格的晶胞基矢${\displaystyle ({\boldsymbol {b_{1}}},{\boldsymbol {b_{2}}},{\boldsymbol {b_{3}}})}$ 

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

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

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

倒晶格与正晶格的关系

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

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

在此以一维晶格为例。在一个以${\displaystyle {\boldsymbol {a}}}$ 为基矢的一维晶格中，其波函数应该为布洛赫波

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

定义其倒晶格向量

${\displaystyle {\boldsymbol {G}}=n{\boldsymbol {b}},\ n=0,1,2,\cdots }$ 

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

${\displaystyle {\boldsymbol {G}}\cdot {\boldsymbol {a}}=2\pi n}$ 

以及一个函数

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

由于${\displaystyle u_{\boldsymbol {k}}({\boldsymbol {x}})}$ 是一个布洛赫波包，满足

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

所以

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

也是一个布洛赫波包。则波函数有以下性质

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

可见，倒晶格向量G描述了波函数在以k为基矢的动量空间（k空间）内的周期性。其向量单位，即倒晶格的基矢${\displaystyle {\boldsymbol {b_{i}}}}$ 是描述k空间中平移对称性的基矢。其最小可重复单位，即倒晶格的晶胞，称为第一布里渊区。由于波矢k和动量与波函数对应的能量密切相关，在能带理论中也用来解释能带的周期性。

晶体衍射满足布拉格定律

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

定义入射波波矢为${\displaystyle {\boldsymbol {k}}}$ ，则上述公式可变换为

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

因此满足布拉格定律的晶体衍射反映的不是正晶格，而是倒晶格。

进一步将以上公式转化为向量形式，定义入射波波矢为${\displaystyle {\boldsymbol {k_{i}}}}$ ，反射波波矢为${\displaystyle {\boldsymbol {k_{o}}}}$ ，可以得到

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

这个形式也和劳厄方程式相符。

晶体衍射的想法也可以用来解释能带结构中，为什么能量的分布是不连续的。

简单立方晶体

简单立方晶体的素格子基矢可以写成

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

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

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

体积为

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

可推得倒晶格的素格子基矢

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

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

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

所以简单立方晶体的倒晶格同样为简单立方晶体，但是晶格常数为 ${\displaystyle 2\pi \over a}$ 。

面心立方晶体(FCC)

面心立方晶体的素格子基矢可以写成下列三项

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

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

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

体积为

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

可推得倒晶格之素格子基矢

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

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

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

面心立方晶体的倒晶格为体心立方晶体。

体心立方晶体(BCC)

体心立方晶体的素格子基矢可以写成下列三项

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

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

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

体积为

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

可推得倒晶格之素格子基矢

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

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

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

可得知体心立方晶体之倒晶格为面心立方晶体。

在布拉菲晶格中,三轴互为九十度的${\displaystyle ({\boldsymbol {a_{1}}},{\boldsymbol {a_{2}}},{\boldsymbol {a_{3}}})}$  (立方, 正方, 斜方)的晶体结构,是很容易被证明其倒晶格空间之三轴${\displaystyle ({\boldsymbol {b_{1}}},{\boldsymbol {b_{2}}},{\boldsymbol {b_{3}}})}$ 与其真实晶格之三轴有垂直的关系.

• 晶体学
• 对偶空间
• 电子衍射
• 埃瓦尔德球（英语：Ewald's sphere）
• 密勒指数（英语：Miller index）
• 布里渊区

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