利用倒易点阵计算晶面间距的注记  

Notes on the calculation of interplanar spacing using reciprocal lattice

作　　者：杨军[1,2] YANG Jun(State Key Laboratory of Mesoscience and Engineering Institute of Process Engineering,Chinese Academy of Sciences,Beijing 100190,China;School of Materials Science and Optoelectronics Engineering,University of Chinese Academy of Sciences,Beijing 100049,China)

机构地区：[1]中国科学院过程工程研究所介科学与工程全国重点实验室,北京100190 [2]中国科学院大学材料科学与光电技术学院,北京100049

出　　处：《大学物理》2025年第1期23-26,37,共5页College Physics

基　　金：支持国科大加快世界一流大学和一流学科建设。

摘　　要：利用倒易点阵性质可以推导出计算晶面间距的通用公式,并可在计算一些具有特殊点阵参数晶体的晶面间距时进行简化.但这一公式并不能直接用来计算复杂晶胞的晶面间距.使用倒易点阵性质导出的通用公式计算复杂晶胞的晶面间距时,要么考虑该晶面位向上符合出现倒易点约束条件的代表晶面,要么找到复杂晶胞对应的原胞,使用其原胞的三条晶棱构建新基矢并求得倒易点阵的基矢,进而利用同名倒易基矢模长的倒数获取原复杂晶胞的晶面间距.但在新基矢体系中,原晶胞标记晶面的指数也常常相应改变,而这一点在学生计算晶面间距时常常被忽略而得不到正确结果.本文先对复杂晶胞晶面出现倒易点的约束条件进行简单描述,然后结合一个体心立方晶胞的具体示例给出利用代表晶面或构建新基矢计算晶面间距的过程,以期通过具体计算加深学生对这些知识点的理解并在实践环节中采用.Using the properties of reciprocal lattice,a general formula for calculating the crystal interplanar spacing is derived,which can be simplified when calculating the interplanar spacing of some crystals with special lattice constants.However,this formula cannot be directly used to calculate the interplanar spacing of complex unit cells.When the general formula derived from the reciprocal lattice properties is used to calculate the interplanar spacing of complex unit cells,either a representative crystal plane that meets the constraint conditions for the appearance of reciprocal points in that plane orientation should be considered,or the primitive cell corresponding to the complex unit cell should be found.Then,new basis vectors are constructed using the three edges of the primitive cell,and the basis vectors of the reciprocal lattice are obtained.Furthermore,the interplanar spacing of the original complex unit cell can be obtained by taking the reciprocal of the modulus length of the reciprocal basis vector with the same indices.Nevertheless,in the new basis vector system,the indices of the crystal planes marked in the original unit cell often change accordingly,which is often overlooked by students when calculating interplanar spacing,leading to incorrect results.In this paper,we first briefly describe the constraints for the appearance of reciprocal points on crystal planes of complex unit cell.Then,combining a specific example of a body-centered cubic unit cell,we present the process of calculating interplanar spacing using representative crystal planes or constructing new basis vectors,aiming to deepen students'understanding of these knowledge points through specific calculations and adopt them in practical applications.

关 键 词：倒易点阵 晶面间距 晶胞 体心立方晶胞 

分 类 号：O711.4[理学—晶体学]