text: # 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
  
  ### 简单立方晶体
  
  ${\displaystyle {\boldsymbol {a_{1}}}=a{\hat {x}}}$
  ${\displaystyle {\boldsymbol {a_{2}}}=a{\hat {y}}}$
  ${\displaystyle {\boldsymbol {a_{3}}}=a{\hat {z}}}$
  
  ${\displaystyle {\boldsymbol {a_{1}}}\cdot {\boldsymbol {a_{2}}}\times {\boldsymbol {a_{3}}}=a^{3}}$
  
  ${\displaystyle {\boldsymbol {b_{1}}}={2\pi \over a}{\hat {x}}}$
  ${\displaystyle {\boldsymbol {b_{2}}}={2\pi \over a}{\hat {y}}}$
  ${\displaystyle {\boldsymbol {b_{3}}}={2\pi \over a}{\hat {z}}}$
  
  ### 面心立方晶体(FCC)
  
  ${\displaystyle {\boldsymbol {a_{1}}}={a \over 2}\left({\hat {y}}+{\hat {z}}\right)}$
  ${\displaystyle {\boldsymbol {a_{2}}}={a \over 2}\left({\hat {z}}+{\hat {x}}\right)}$
  ${\displaystyle {\boldsymbol {a_{3}}}={a \over 2}\left({\hat {x}}+{\hat {y}}\right)}$
  
  ${\displaystyle {\boldsymbol {a_{1}}}\cdot {\boldsymbol {a_{2}}}\times {\boldsymbol {a_{3}}}={a^{3} \over 4}}$
  
  ${\displaystyle {\boldsymbol {b_{1}}}={2\pi \over a}\left(-{\hat {x}}+{\hat {y}}+{\hat {z}}\right)}$
  ${\displaystyle {\boldsymbol {b_{2}}}={2\pi \over a}\left(+{\hat {x}}-{\hat {y}}+{\hat {z}}\right)}$
  ${\displaystyle {\boldsymbol {b_{3}}}={2\pi \over a}\left(+{\hat {x}}+{\hat {y}}-{\hat {z}}\right)}$
  
  ### 体心立方晶体(BCC)
  
  ${\displaystyle {\boldsymbol {a_{1}}}={a \over 2}\left(-{\hat {x}}+{\hat {y}}+{\hat {z}}\right)}$
  ${\displaystyle {\boldsymbol {a_{2}}}={a \over 2}\left(+{\hat {x}}-{\hat {y}}+{\hat {z}}\right)}$
  ${\displaystyle {\boldsymbol {a_{3}}}={a \over 2}\left(+{\hat {x}}+{\hat {y}}-{\hat {z}}\right)}$
  
  ${\displaystyle {\boldsymbol {a_{1}}}\cdot {\boldsymbol {a_{2}}}\times {\boldsymbol {a_{3}}}={a^{3} \over 2}}$
  
  ${\displaystyle {\boldsymbol {b_{1}}}={2\pi \over a}\left({\hat {y}}+{\hat {z}}\right)}$
  ${\displaystyle {\boldsymbol {b_{2}}}={2\pi \over a}\left({\hat {z}}+{\hat {x}}\right)}$
  ${\displaystyle {\boldsymbol {b_{3}}}={2\pi \over a}\left({\hat {x}}+{\hat {y}}\right)}$
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