倒易点阵

倒易点阵（英語：reciprocal lattice），又称倒（易）晶格、倒（易）格子，是物理学中描述空间波函数的傅立叶变换后的周期性的一种方法。相对于正晶格所描述的实空间周期性，倒晶格描述的是动量空间，亦可认为是k空间的周期性。根据位置和动量所满足的庞特里亚金对偶性，布拉菲晶格的倒晶格仍然是一种布拉菲晶格，而倒晶格的倒晶格就会变回原始晶格（正晶格）。 对于以 a {\displaystyle {\boldsymbol {a}}} 为基矢的一维晶格，其倒格子的基矢为 b = 2 π a a 2 {\displaystyle {\boldsymbol {b}}=2\pi {\frac {\boldsymbol {a}}{a^{2}}}} 对于以 ( a 1 , a 2 ) {\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 n {\displaystyle {\boldsymbol {n}}} ，其倒格子的基矢为 b 1 = 2 π a 2 × n a 1 ⋅ ( a 2 × n ) {\displaystyle {\boldsymbol {b_{1}}}=2\pi {\frac {{\boldsymbol {a_{2}}}\times {\boldsymbol {n}}}{{\boldsymbol {a_{1}}}\cdot ({\boldsymbol {a_{2}}}\times {\boldsymbol {n}})}}} b 2 = 2 π n × a 1 a 2 ⋅ ( n × a 1 ) {\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 ( a 1 , a 2 , a 3 ) {\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 ( b 1 , b 2 , b 3 ) {\displaystyle ({\boldsymbol {b_{1}}},{\boldsymbol {b_{2}}},{\boldsymbol {b_{3}}})} b 1 = 2 π a 2 × a 3 a 1 ⋅ ( a 2 × a 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}}})}}} b 2 = 2 π a 3 × a 1 a 2 ⋅ ( a 3 × a 1 ) {\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}}})}}} b 3 = 2 π a 1 × a 2 a 3 ⋅ ( a 1 × a 2 ) {\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 a i ⋅ b j = 2 π δ i j = { 2 π , i   =   j 0 , i   ≠   j {\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 G = h b 1 + k b 2 + l b 3 {\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 | G h k l | = 2 π d h k l {\displaystyle \mathbf {|G_{hkl}|} ={\frac {2\pi }{d_{hkl}}}} 向量G和正晶格向量R有以下关系 R = c 1 a 1 + c 2 a 2 + c 3 a 3 {\displaystyle \mathbf {R} =c_{1}{\boldsymbol {a_{1}}}+c_{2}{\boldsymbol {a_{2}}}+c_{3}{\boldsymbol {a_{3}}}} e i G ⋅ R = 1 {\displaystyle \mathrm {e} ^{\mathrm {i} \mathbf {G\cdot R} }=1} 三维倒晶格中的晶胞体积ΩG和正晶格的晶胞体积Ω有以下关系 Ω G = ( 2 π ) 3 Ω {\displaystyle \Omega _{G}={\frac {(2\pi )^{3}}{\Omega }}} 在此以一维晶格为例。在一个以 a {\displaystyle {\boldsymbol {a}}} 为基矢的一维晶格中，其波函数应该为布洛赫波 ψ k ( x ) = e i k ⋅ x u k ( x ) {\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 G = n b ,   n = 0 , 1 , 2 , ⋯ {\displaystyle {\boldsymbol {G}}=n{\boldsymbol {b}},\ n=0,1,2,\cdots } b = 2 π a a 2 {\displaystyle {\boldsymbol {b}}=2\pi {\frac {\boldsymbol {a}}{a^{2}}}} G ⋅ a = 2 π n {\displaystyle {\boldsymbol {G}}\cdot {\boldsymbol {a}}=2\pi n} 以及一个函数 u k + G ( x ) = e − i G ⋅ x u k ( x ) u k + G ( x + a ) = e − i G ⋅ x e − i G ⋅ a u k ( x + a ) = e − i G ⋅ x u k ( x + a ) {\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 u k ( x ) {\displaystyle u_{\boldsymbol {k}}({\boldsymbol {x}})} 是一个布洛赫波包，满足 u k ( x + a ) = u k ( x ) {\displaystyle u_{\boldsymbol {k}}({\boldsymbol {x+a}})=u_{\boldsymbol {k}}({\boldsymbol {x}})} 所以 u k + G ( x + a ) = u k + G ( x ) {\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 ψ k ( x ) = e i k ⋅ x u k ( x ) = e i ( k + G ) ⋅ x e − i G ⋅ x u k ( x ) = e i ( k + G ) ⋅ x u k + G ( x ) = ψ k + G ( 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}}} 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 b i {\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 2 d sin ⁡ θ = n λ 2 × 2 π λ sin ⁡ θ = 2 π d n {\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 k {\displaystyle {\boldsymbol {k}}} ，则上述公式可变换为 | k | = 2 π λ | G h k l | = 2 π d h k l 2 | k | sin ⁡ θ = | G | {\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 k i {\displaystyle {\boldsymbol {k_{i}}}} ，反射波波矢为 k o {\displaystyle {\boldsymbol {k_{o}}}} ，可以得到 Δ k = k o − k i = G {\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連續的。 簡單立方晶體的素格子基矢可以寫成 a 1 = a x ^ {\displaystyle {\boldsymbol {a_{1}}}=a{\hat {x}}} a 2 = a y ^ {\displaystyle {\boldsymbol {a_{2}}}=a{\hat {y}}} a 3 = a z ^ {\displaystyle {\boldsymbol {a_{3}}}=a{\hat {z}}} 體積為 a 1 ⋅ a 2 × a 3 = a 3 {\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 b 1 = 2 π a x ^ {\displaystyle {\boldsymbol {b_{1}}}={2\pi \over a}{\hat {x}}} b 2 = 2 π a y ^ {\displaystyle {\boldsymbol {b_{2}}}={2\pi \over a}{\hat {y}}} b 3 = 2 π a z ^ {\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 2 π a {\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 a 1 = a 2 ( y ^ + z ^ ) {\displaystyle {\boldsymbol {a_{1}}}={a \over 2}\left({\hat {y}}+{\hat {z}}\right)} a 2 = a 2 ( z ^ + x ^ ) {\displaystyle {\boldsymbol {a_{2}}}={a \over 2}\left({\hat {z}}+{\hat {x}}\right)} a 3 = a 2 ( x ^ + y ^ ) {\displaystyle {\boldsymbol {a_{3}}}={a \over 2}\left({\hat {x}}+{\hat {y}}\right)} 體積為 a 1 ⋅ a 2 × a 3 = a 3 4 {\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 b 1 = 2 π a ( − x ^ + y ^ + z ^ ) {\displaystyle {\boldsymbol {b_{1}}}={2\pi \over a}\left(-{\hat {x}}+{\hat {y}}+{\hat {z}}\right)} b 2 = 2 π a ( + x ^ − y ^ + z ^ ) {\displaystyle {\boldsymbol {b_{2}}}={2\pi \over a}\left(+{\hat {x}}-{\hat {y}}+{\hat {z}}\right)} b 3 = 2 π a ( + x ^ + y ^ − z ^ ) {\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 a 1 = a 2 ( − x ^ + y ^ + z ^ ) {\displaystyle {\boldsymbol {a_{1}}}={a \over 2}\left(-{\hat {x}}+{\hat {y}}+{\hat {z}}\right)} a 2 = a 2 ( + x ^ − y ^ + z ^ ) {\displaystyle {\boldsymbol {a_{2}}}={a \over 2}\left(+{\hat {x}}-{\hat {y}}+{\hat {z}}\right)} a 3 = a 2 ( + x ^ + y ^ − z ^ ) {\displaystyle {\boldsymbol {a_{3}}}={a \over 2}\left(+{\hat {x}}+{\hat {y}}-{\hat {z}}\right)} 體積為 a 1 ⋅ a 2 × a 3 = a 3 2 {\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 b 1 = 2 π a ( y ^ + z ^ ) {\displaystyle {\boldsymbol {b_{1}}}={2\pi \over a}\left({\hat {y}}+{\hat {z}}\right)} b 2 = 2 π a ( z ^ + x ^ ) {\displaystyle {\boldsymbol {b_{2}}}={2\pi \over a}\left({\hat {z}}+{\hat {x}}\right)} b 3 = 2 π a ( x ^ + y ^ ) {\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 ( a 1 , a 2 , a 3 ) {\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 ( b 1 , b 2 , b 3 ) {\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. 晶体学 对偶空间 电子衍射 埃瓦尔德球（英語：Ewald's sphere） 密勒指数（英語：Miller index） 布里渊区 维基共享资源中相关的多媒体资源：倒易点阵 （英文）http://newton.umsl.edu/run//nano/known.html （页面存档备份，存于互联网档案馆） - Jmol-based electron diffraction simulator lets you explore the intersection between reciprocal lattice and Ewald sphere during tilt. （英文）DoITPoMS Teaching and Learning Package on Reciprocal Space and the Reciprocal Lattice（页面存档备份，存于互联网档案馆）