Equations of motion

Willis, B. T. M.

doi:10.1107/97809553602060000563

International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

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International Tables for Crystallography (2006). Vol. B. ch. 4.1, pp. 401-402 | 1 | 2 |

Section 4.1.2.1. Equations of motion

B. T. M. Willis^a ^*

^aChemical Crystallography Laboratory, University of Oxford, 9 Parks Road, Oxford OX1 3PD, England
Correspondence e-mail: bertram.willis@chemcryst.ox.ac.uk

4.1.2.1. Equations of motion

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As a result of thermal fluctuations, the atoms vibrate about their equilibrium positions, so that the actual position of the $[\kappa]$ th atom in the lth primitive cell is given by $[{\bf R}(\kappa{}l)={\bf r}(\kappa{}l)+{\bf u}(\kappa{}l)]$ with r representing the equilibrium position and u the thermal displacement. (In lattice-dynamical theory it is advantageous to deal with the primitive cell, as it possesses the fewest degrees of freedom.) The kinetic energy of the vibrating crystal is $[(1/2)\textstyle \sum \limits_{\kappa{}l\alpha{}}m(\kappa)\dot u^2_{\alpha}(\kappa{}l),]$ where $[m(\kappa)]$ is the mass of atom $[\kappa]$ and the index $[\alpha]$ ( $[\alpha]$ = 1, 2, 3) refers to the Cartesian components of the displacement. (The dot denotes the time derivative.)

If the adiabatic approximation is invoked, the potential energy V of the crystal can be expressed as a function of the instantaneous atomic positions. Expanding V in powers of $[{\bf u}(\kappa{}l)]$ , using the three-dimensional form of Taylor's series, we have $[V=V^{(0)}+V^{(1)}+V^{(2)}+V^{(3)}+\ldots,]$ where $[V^{(0)}]$ is the static (equilibrium) potential and $[V^{(1)}]$ , $[V^{(2)}]$ are given by $[\eqalign{V^{(1)} &=\sum_{\kappa{}l\alpha}{{\partial{}V}\over{\partial{}u_{\alpha}(\kappa{}l)}}\bigg|_0u_{\alpha}(\kappa{}l)\cr V^{(2)} &={1 \over 2}\sum_{\kappa{}l\alpha}\sum_{\kappa^{\prime}l^{\prime}\alpha^{\prime}}{{\partial{}^2V}\over{\partial{}u_{\alpha}(\kappa{}l)\partial{}u_{\alpha^{\prime}}(\kappa^{\prime}l^{\prime})}}\bigg|_0u_{\alpha}(\kappa{}l)u_{\alpha^{\prime}}(\kappa^{\prime}l^{\prime}).\cr}]$ The subscript zero indicates that the derivatives are to be evaluated at the equilibrium configuration. In the harmonic approximation, $[V^{(3)}]$ and all higher terms in the expansion are neglected.

At equilibrium the forces on an atom must vanish, so that $[V^{(1)}=0.]$ Ignoring the static potential $[V^{(0)}]$ , the quadratic term $[V^{(2)}]$ only remains and the Hamiltonian for the crystal (the sum of the kinetic and potential energies) is then $[\eqalignno{H &={1 \over 2}\sum_{\kappa{}l\alpha{}}m(\kappa)\dot u^2_{\alpha}(\kappa{}l) &\cr &\quad+{1 \over 2}\sum_{\kappa{}l\alpha{}}\sum_{\kappa^{\prime}l^{\prime}\alpha^{\prime}}\Phi_{\alpha\alpha^{\prime}}\pmatrix{\kappa & \kappa^{\prime} \cr l & l^{\prime}\cr}u_{\alpha}(\kappa{}l)u_{\alpha^{\prime}}(\kappa^{\prime}{}l^{\prime}),&\cr & &(4.1.2.1)\cr}]$ where $[\Phi_{\alpha\alpha^{\prime}}]$ is an element of the 3 × 3 `atomic force-constant matrix' and is defined (for distinct atoms $[\kappa{}l]$ , $[\kappa^{\prime}l^{\prime}]$ ) by $[\Phi_{\alpha\alpha^{\prime}}\pmatrix{\kappa & \kappa^{\prime} \cr l & l^{\prime}\cr}={{\partial{}^2V}\over{\partial{}u_{\alpha}(\kappa{}l)\partial{}u_{\alpha^{\prime}}(\kappa^{\prime}l^{\prime})}}\bigg|_0.]$ It is the negative of the force in the $[\alpha]$ direction imposed on the atom $[(\kappa{}l)]$ when atom $[(\kappa^{\prime}l^{\prime})]$ is displaced unit distance along $[\alpha^{\prime}]$ with all the remaining atoms fixed at their equilibrium sites. $[\Phi_{\alpha\alpha^{\prime}}]$ is defined differently for the self-term with $[\kappa=\kappa^{\prime}]$ and $[l=l^{\prime}]$ : $[\Phi_{\alpha\alpha^{\prime}}\pmatrix{\kappa & \kappa \cr l & l\cr}=-\sum_{\kappa{}l}\sum_{\scriptstyle \kappa^{\prime}l^{\prime} \atop \kern-1.5em\scriptstyle\kappa{}l \ne \kappa^{\prime}l^{\prime}}\Phi_{\alpha\alpha^{\prime}}\pmatrix{\kappa & \kappa^{\prime} \cr l & l^{\prime}\cr}.]$ Thus the self-matrix describes the force on $[(\kappa{}l)]$ when the atom itself is displaced with all the remaining atoms kept stationary.

There are restrictions on the number of distinct force constants $[\Phi_{\alpha\alpha^{\prime}}]$ : these are imposed by symmetry and by the requirement that the potential energy is invariant under infinitesimal translations and rotations of the rigid crystal. Such constraints are discussed in the book by Venkataraman et al. (1975).

Applying Hamilton's equations of motion to equation (4.1.2.1) now gives $[m(\kappa)\ddot u_{\alpha}(\kappa{}l) = -\sum_{\kappa^{\prime}l^{\prime}\alpha^{\prime}}\Phi_{\alpha\alpha^{\prime}}\pmatrix{\kappa & \kappa^{\prime} \cr l & l^{\prime}\cr}u_{\alpha^{\prime}}(\kappa^{\prime}l^{\prime}). \eqno(4.1.2.2)]$ These represent 3nN coupled differential equations, where n is the number of atoms per primitive cell $[(\kappa = 1,\ldots{},n)]$ and N is the number of cells per crystal $[(l = 1,\ldots{},N)]$ .

By applying the periodic boundary conditions, the solutions of equation (4.1.2.2) can be expressed as running, or travelling, plane waves extending throughout the entire crystal. The number of independent waves (or normal modes) is 3nN. Effectively, we have transferred to a new coordinate system: instead of specifying the motion of the individual atoms, we describe the thermal motion in terms of normal modes, each of which contributes to the displacement of each atom. The general solution for the $[\alpha]$ component of the displacement of $[(\kappa{}l)]$ is then given by the superposition of the displacements from all modes: $[\eqalignno{u_{\alpha}(\kappa{}l)&=[m(\kappa)]^{-1/2}\textstyle \sum \limits_{j{\bf q}}|A_j({\bf q})|e_{\alpha}(\kappa|j{\bf q})&\cr&\quad\times\exp\{i[{\bf q}\cdot{\bf r}(\kappa{}l)-\omega_j({\bf q})t]\}.&(4.1.2.3)\cr}]$ Here q is the wavevector of a mode (specifying both its wavelength and direction of propagation in the crystal) and $[\omega({\bf q})]$ its frequency. There are N distinct wavevectors, occupying a uniformly distributed mesh of N points in the Brillouin zone (reciprocal cell); each wavevector is shared by 3n modes which possess, in general, different frequencies and polarization properties. Thus an individual mode is conveniently labelled (j q), where j is an index ( $[j=1,\ldots,3n]$ ) indicating the branch . The scalar quantity $[|A_j({\bf q})|]$ in equation (4.1.2.3) is the amplitude of excitation of (j q) and $[e_{\alpha}(\kappa|j{\bf q})]$ is the element of the eigenvector $[{\bf e}(j{\bf q})]$ referring to the displacement in the $[\alpha]$ direction of the atom $[\kappa]$ . The eigenvector itself, with dimensions n × 1, determines the pattern of atomic displacements in the mode (j q) and its magnitude is fixed by the orthonormality and closure conditions $[\textstyle \sum \limits_{\alpha\kappa}e^{\ast}_{\alpha}(\kappa|j{\bf q})e_{\alpha}(\kappa|j^{\prime}{\bf q}) = \delta_{jj^{\prime}}]$ and $[\textstyle \sum \limits_{j}e_{\alpha}(\kappa|j{\bf q})e^{\ast}_{\alpha^{\prime}}(\kappa^{\prime}|j{\bf q}) = \delta_{\alpha\alpha^{\prime}}\delta_{\kappa\kappa^{\prime}}]$ with $[^\ast]$ indicating complex conjugate and $[\delta]$ the Kronecker delta.

The pre-exponential, or amplitude, terms in (4.1.2.3) are independent of the cell number. This follows from Bloch's (1928) theorem which states that, for corresponding atoms in different cells, the motions are identical as regards their amplitude and direction and differ only in phase. The theorem introduces an enormous simplification as it allows us to restrict attention to the 3n equations of motion of the n atoms in just one cell, rather than the 3nN equations of motion for all the atoms in the crystal.

Substitution of (4.1.2.3) into (4.1.2.2) gives the equations of motion in the form $[\omega_j^2({\bf q})e_{\alpha}(\kappa|j{\bf q})=\textstyle \sum \limits_{\alpha^{\prime}\kappa^{\prime}}D_{\alpha\alpha^{\prime}}(\kappa\kappa^{\prime}|{\bf q})e_{\alpha^{\prime}}(\kappa^{\prime}|j{\bf q}),\eqno(4.1.2.4)]$ in which $[D_{\alpha\alpha^{\prime}}]$ is an element of the dynamical matrix D(q). $[D_{\alpha\alpha^{\prime}}]$ is defined by $[\eqalignno{D_{\alpha\alpha^{\prime}}(\kappa\kappa^{\prime}|{\bf q}) &= [m(\kappa)m(\kappa^{\prime})]^{-1/2}\exp\{i{\bf q}[r(\kappa^{\prime})-r(\kappa)]\}&\cr&\quad\times\sum\Phi_{\alpha\alpha^\prime}\pmatrix{\kappa & \kappa^{\prime} \cr 0 & L\cr}\exp[i{\bf q}\cdot{\bf r}(L)],&(4.1.2.5)\cr}]$ where $[{\bf r}(\kappa)]$ is the position of atom $[\kappa]$ with respect to the cell origin, L is $[l^\prime-l]$ and $[{\bf r}(L)]$ is the separation between cells l and $[l^\prime]$ . The element $[D_{\alpha\alpha^\prime}]$ is obtained by writing down the $[\alpha\alpha^\prime]$ component of the force constant between atoms $[\kappa, \kappa^\prime]$ which are L cells apart and multiplying by the phase factor $[\exp[i{\bf q}\cdot{\bf r}(L)]]$ ; this term is then summed over those values of L covering the range of interaction of $[\kappa]$ and $[\kappa^\prime]$ .

The dynamical matrix is Hermitian and has dimensions 3n × 3n. Its eigenvalues are the squared frequencies $[\omega_j^2({\bf q})]$ of the normal modes and its eigenvectors $[{\bf e}(j{\bf q})]$ determine the corresponding pattern of atomic displacements. The frequencies of the modes in three of the branches, j, go to zero as q approaches zero: these are the acoustic modes. The remaining [3n-3] branches contain the optic modes. There are N distinct q vectors, and so, in all, there are 3N acoustic modes and [(3n-3)N] optic modes. Thus copper has acoustic modes but no optic modes, silicon and rock salt have an equal number of both, and lysozyme possesses predominantly optic modes.

References

Bloch, F. (1928). Über die Quantenmechanik der Elektronen in Kristallgittern. Z. Phys. 52, 555–600.Google Scholar

Venkataraman, G., Feldkamp, L. A. & Sahni, V. C. (1975). Dynamics of perfect crystals. Cambridge Mass.: MIT Press.Google Scholar

International Tables for Crystallography (2006). Vol. B. ch. 4.1, pp. 401-402