The dynamical matrix

Eckold, G.

doi:10.1107/97809553602060000638

International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

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International Tables for Crystallography (2006). Vol. D. ch. 2.1, pp. 267-268

Section 2.1.2.3. The dynamical matrix

G. Eckold^a ^*

^a Institut für Physikalische Chemie, Universität Göttingen, Tammannstrasse 6, D-37077 Göttingen, Germany
Correspondence e-mail: geckold@gwdg.de

2.1.2.3. The dynamical matrix

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If the ansatz (2.1.2.10a) is inserted into the equation of motion (2.1.2.9), the following eigenvalue equation is obtained: $[\eqalignno{&\omega _{{\bf q},j}^2 {\bf e}_\kappa ({\bf q},{j})\cr &\quad= \sum\limits_{\kappa 'l'}\sqrt {{1 \over {m_\kappa m_{\kappa '}}}}{\bf V}(\kappa l,\kappa 'l')\exp[i{\bf q}({\bf r}_{l'} - {\bf r}_l)]{\bf e}_{\kappa '}({\bf q},{j})& \cr &\quad= \sum\limits_\kappa \sqrt {{1 \over {m_\kappa m_{\kappa '}}}}\left [\sum\limits_{l'}{\bf V}(\kappa l,\kappa 'l')\exp[i{\bf q}({\bf r}_{l'} - {\bf r}_l)]\right]{\bf e}_{\kappa '}({\bf q},{ j)}. &\cr &&(2.1.2.15)} ]$ The summation over all primitive cells on the right-hand side of equation (2.1.2.15) yields the Fourier-transformed force-constant matrix $[{\bf F}_{\kappa \kappa '}({\bf q}) = \textstyle\sum\limits_{l'}{\bf V}(\kappa l,\kappa 'l') \exp[i{\bf q}({\bf r}_{l'} - {\bf r}_{l})], \eqno (2.1.2.16) ]$ which is independent of l for infinite crystals. $[{\bf F}_{\kappa \kappa '}({\bf q}) ]$ contains all interactions of type $[\kappa]$ atoms with type $[\kappa']$ atoms. Using this notation, equation (2.1.2.15) reduces to $[\omega _{{\bf q},j}^2 {\bf e}_\kappa ({\bf q},{j)} = \sum\limits_{\kappa '}{\sqrt{{1 \over {m_\kappa m_{\kappa '}}}}} \,{\bf F}_{\kappa \kappa '}({\bf q}) \,{\bf e}_{\kappa '}({\bf q},{ j)}. \eqno (2.1.2.17) ]$ If for a given vibration characterized by $[({\bf q},j) ]$ we combine the three-dimensional polarization vectors $[{\bf e}_{\kappa}({\bf q},j) ]$ of all atoms within a primitive cell to a 3N-dimensional polarization vector $[{\bf e}({\bf q},j)]$ , $[{\bf e}({\bf q},j) = {\pmatrix{ {{\bf e}_1 ({\bf q},j)} \cr \vdots \cr {{\bf e}_N ({\bf q},j)} \cr }} = {\pmatrix{ {e_1^x ({\bf q},j)} \cr {e_1^y ({\bf q},j)} \cr {e_1^z ({\bf q},j)} \cr \vdots \cr {e_N^x ({\bf q},j)} \cr {e_N^y ({\bf q},j)} \cr {e_N^z ({\bf q},j)} \cr }} \eqno (2.1.2.18) ]$ and simultaneously the $[3 \times 3]$ matrices $[{\bf F}_{\kappa \kappa '}({\bf q}) ]$ to a $[3N\times 3N]$ matrix F(q) $[{\bf F}({\bf q}) ={\pmatrix{ {F_{11}^{xx}}& {F_{11}^{xy}}& {F_{11}^{xz}}& {}& {}& {} & {F_{1\,N}^{xx}}& {F_{1\,N}^{xy}}& {F_{1\,N}^{xz}} \cr {F_{11}^{yx}}& {F_{11}^{yy}}& {F_{11}^{yz}}&{} & \,s &{} & {F_{1\,N}^{yx}}& {F_{1\,N}^{yy}}& {F_{1\,N}^{yz}} \cr {F_{11}^{zx}}& {F_{11}^{zy}}& {F_{11}^{zz}}&{} &{} & {}& {F_{1\,N}^{zx}}& {F_{1\,N}^{zy}}& {F_{1\,N}^{zz}} \cr {}& {}& {}& {F_{\kappa \kappa '}^{xx}}& {F_{\kappa \kappa '}^{xy}}& {F_{\kappa \kappa '}^{xz}}& {}& {}& {} \cr {}& \vdots & {} & {F_{\kappa \kappa '}^{yx}}& {F_{\kappa \kappa '}^{yy}}& {F_{\kappa \kappa '}^{yz}}& {} & \vdots & {} \cr {}& {}& {}& {F_{\kappa \kappa '}^{zx}}& {F_{\kappa \kappa '}^{zy}}& {F_{\kappa \kappa '}^{zz}}& {}& {}& {} \cr {F_{N1}^{xx}}& {F_{N1}^{xy}}& {F_{N1}^{xz}}& {}& {}& {} & {F_{NN}^{xx}}& {F_{NN}^{xy}}& {F_{NN}^{xz}} \cr {F_{N1}^{yx}}& {F_{N1}^{yy}}& {F_{N1}^{yz}}& {}& \,s & {} & {F_{NN}^{yx}}& {F_{NN}^{yy}}& {F_{NN}^{yz}} \cr {F_{N1}^{zx}}& {F_{N1}^{zy}}& {F_{N1}^{zz}}& {}& {}& {}& {F_{NN}^{zx}}& {F_{NN}^{zy}}& {F_{NN}^{zz}} \cr }}, \eqno (2.1.2.19) ]$ equation (2.1.2.17) can be written in matrix notation and takes the simple form $[\omega _{{\bf q},j}^2 \, {\bf e}({\bf q},j) = [{\bf M}\, {\bf F}({\bf q})\, {\bf M} ] \,{\bf e} ({\bf q}, j) = {\bf D}({\bf q})\, {\bf e}({\bf q},j), \eqno (2.1.2.20) ]$ where the diagonal matrix $[{\bf M}= \pmatrix{ {1 \over {\sqrt {m_1 }}} &0 &0 &{} &{} &{} &{} \cr 0 &{{1 \over {\sqrt {m_1 }}}} & 0 &\,s &{} &{} &{} \cr 0 &0 &{{1 \over {\sqrt {m_1 }}}} &{} &{} &{} &{} \cr {} &\vdots &{} &{} &{} &\vdots &{} \cr {} &{} &{} &{} &{{1 \over {\sqrt {m_N }}}} &0 &0 \cr {} &{} &{} & \,s & 0 &{{1 \over {\sqrt {m_N }}}}& 0 \cr {} &{} &{} &{} & 0 &0 &{{1 \over {\sqrt {m_N }}}} \cr } \eqno (2.1.2.21) ]$ contains the masses of all atoms. The $[3N\times 3N]$ matrix $[{\bf D}({\bf q}) = {\bf M}\, {\bf F}({\bf q}) \,{\bf M} \eqno (2.1.2.22) ]$ is called the dynamical matrix. It contains all the information about the dynamical behaviour of the crystal and can be calculated on the basis of specific models for interatomic interactions. In analogy to the $[3\times 3]$ matrices $[{\bf F}_{\kappa \kappa '}({\bf q}) ]$ , we introduce the submatrices of the dynamical matrix: $[{\bf D}_{\kappa \kappa '}({\bf q}) = {1 \over {\sqrt {m_\kappa m_{\kappa '}}}}\, {\bf F}_{\kappa \kappa '}({\bf q}). \eqno (2.1.2.22a) ]$ Owing to the symmetry of the force-constant matrix , $[V_{\alpha \beta}(\kappa l,\kappa 'l') = V_{\beta\alpha}(\kappa 'l',\kappa l), \eqno (2.1.2.23) ]$ the dynamical matrix is Hermitian:¹ $[{\bf D}^{T}({\bf q}) = {\bf D}^ * ({\bf q}) = {\bf D}(- {\bf q}) \eqno (2.1.2.24) ]$ or more specifically $[D_{\kappa \kappa '}^{\alpha \beta }({\bf q}) = D_{\kappa '\kappa }^{\beta \alpha ^*} ({\bf q}) = D_{\kappa '\kappa }^{\beta \alpha }(- {\bf q}). \eqno (2.1.2.24a) ]$ Obviously, the squares of the vibrational frequency $[\omega_{{\bf q}, j} ]$ and the polarization vectors $[{\bf e}({\bf q},j)]$ are eigenvalues and corresponding eigenvectors of the dynamical matrix. As a direct consequence of equation (2.1.2.20), the eigenvalues $[\omega _{{\bf q},j}^2]$ are real quantities and the following relations hold: $[\eqalignno{\omega _{ {\bf q},j}^2 &= \omega _{- {\bf q},j}^2, &(2.1.2.25)\cr {\bf e}^ * ({\bf q},j) &= {\bf e}(- {\bf q},j). &(2.1.2.26)}%fd2.1.2.26 ]$ Moreover, the eigenvectors are mutually orthogonal and can be chosen to be normalized.