Eigenvalues and phonon dispersion, acoustic modes

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. 268-270

Section 2.1.2.4. Eigenvalues and phonon dispersion, acoustic modes

G. Eckold^a ^*

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

2.1.2.4. Eigenvalues and phonon dispersion, acoustic modes

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The wavevector dependence of the vibrational frequencies is called phonon dispersion. For each wavevector q there are 3N fundamental frequencies yielding 3N phonon branches when $[\omega_{{\bf q}, j}]$ is plotted versus q. In most cases, the phonon dispersion is displayed for wavevectors along high-symmetry directions. These dispersion curves are, however, only special projections of the dispersion hypersurface in the four-dimensional q–ω space. As a simple example, the phonon dispersion of b.c.c. hafnium is displayed in Fig. 2.1.2.3. The wavevectors are restricted to the first Brillouin zone (see Section 2.1.3.1) and the phonon dispersion for different directions of the wavevector are combined in one single diagram making use of the fact that different high-symmetry directions meet at the Brillouin-zone boundary. Note that in Fig. 2.1.2.3, the moduli of the wavevectors are scaled by the Brillouin-zone boundary values and represented by the reduced coordinates ξ. Owing to the simple b.c.c. structure of hafnium with one atom per primitive cell, there are only three phonon branches. Moreover, for all wavevectors along the directions [00ξ] and [ξξξ], two exhibit the same frequencies – they are said to be degenerate. Hence in the corresponding parts of Fig. 2.1.2.3 only two branches can be distinguished.

Figure 2.1.2.3 | top | pdf |

Phonon dispersion of b.c.c. hafnium for wavevectors along the main symmetry directions of the cubic structure. The symbols represent experimental data obtained by inelastic neutron scattering and the full lines are the results of the model. From Trampenau et al. (1991). Copyright (1991) by the American Physical Society.

Whereas in this simple example the different branches can be separated quite easily, this is no longer true for more complicated crystal structures. For illustration, the phonon dispersion of the high-T_c superconductor Nd₂CuO₄ is shown in Fig. 2.1.2.4 for the main symmetry directions of the tetragonal structure (space group [I4/mmm] , seven atoms per primitive cell). Note that in many publications on lattice dynamics the frequency $[\nu =\omega/2\pi]$ is used rather than the angular frequency ω.

Figure 2.1.2.4 | top | pdf |

Phonon dispersion of Nd₂CuO₄ along the main symmetry directions of the tetragonal structure. The symbols represent experimental data obtained by inelastic neutron scattering and the full lines are drawn to guide the eye. Reprinted from Pintschovius et al. (1991), copyright (1991), with permission from Elsevier.

The 21 phonon branches of Nd₂CuO₄ with their more complicated dispersion reflect the details of the interatomic interactions between all atoms of the structure. The phonon frequencies ν cover a range from 0 to 18 THz. In crystals with strongly bonded molecular groups, like SiO₄ tetrahedra in quartz or SO₄ tetrahedra in sulfates, for example, the highest frequencies are found near 35 THz and correspond to bond-stretching vibrations. Soft materials like organic molecular crystals, on the other hand, exhibit a large number of phonon branches within a rather small frequency range which cannot easily be separated. Deuterated naphthalene (C₁₀D₈) is a well investigated example. The low-frequency part of its phonon dispersion is shown in Fig. 2.1.2.5.

Figure 2.1.2.5 | top | pdf |

Low-frequency part of the phonon dispersion of deuterated naphthalene at 6 K. The symbols represent experimental data obtained by inelastic neutron scattering and the full lines are drawn to guide the eye. Reproduced with permission from Natkaniec et al. (1980). Copyright (1980) IOP Publishing Limited.

Whereas neutron inelastic scattering is the most powerful method for the determination of phonons at arbitrary wavevectors, long wavelength $[({\bf q}\rightarrow {\bf 0})]$ phonons may also be detected by optical spectroscopy. The determination of phonon frequencies alone is, however, not sufficient for a concise determination of dispersion branches. Rather, individual phonons have to be assigned uniquely to one of the 3N branches, and this may prove a rather hard task if N is large. Here, symmetry considerations of eigenvectors are of special importance since phonons belonging to the same branch must exhibit the same symmetry properties. Moreover, inspection of Figs. 2.1.2.3 to 2.1.2.5 shows that some of the branches cross each other and others do not. It is a general statement that crossing is only allowed for branches with different symmetries – a property which yields a classification scheme for the different phonon branches. The symmetry of fundamental vibrations of a lattice will be discussed in some detail in Section 2.1.3.

In the limit of long wavelengths, there are always three particular modes with identical polarization vectors for all atoms, which will be considered in the following. At exactly $[{\bf q}={\bf 0}]$ (the Γ point) or infinite wavelength, the eigenvalue equation (2.1.2.15) reduces to $[\omega _{{\bf 0},j}^2 \, {\bf e}_\kappa ({\bf 0},j) = \sum\limits_{\kappa 'l'}{{1 \over {\sqrt {m_\kappa m_{\kappa '}}}}\, {\bf V}(\kappa l,}\kappa 'l')\, {\bf e}_{\kappa '}({\bf 0},j). \eqno (2.1.2.27) ]$ One immediately recognizes that there are special solutions with $[{1 \over {\sqrt {m_\kappa }}}\, {\bf e}_\kappa ({\bf 0},j) = {\bf u}_{o} \hbox{ for all }{\kappa}, \eqno (2.1.2.28) ]$ i.e. the (mass-weighted) eigenvectors of all atoms are identical. There are three orthogonal eigenvectors of this kind and the displacement pattern of such phonons corresponds to rigid translations of the whole lattice along the three orthogonal coordinates in direct space. These motions do not affect any interatomic interaction. Hence, there is no change in potential energy and the condition of translational invariance (cf. Section 2.1.2.2) guarantees that the frequencies of these modes are zero: $[\omega _{{\bf 0},j}^2 = \textstyle\sum\limits_{\kappa 'l'}{{\bf V}(\kappa l,}\kappa 'l') = 0 \hbox{ for } j=1,2,3. \eqno (2.1.2.29) ]$ The phonon branches that lead to zero frequency at the Γ point ( $[{\bf q}={\bf 0}]$ ) are called acoustic, whereas all other branches are called optic. The dispersion of acoustic branches in the vicinity of the Γ point can be investigated by expanding the phase factor in equation (2.1.2.15) in powers of q. Using (2.1.2.28) one obtains $[\eqalignno{&m_\kappa\, \omega _{{\bf q}\to {\bf 0},j}^2 \, {\bf u}_o &\cr&= \textstyle\sum\limits_{\kappa 'l'}{{\bf V}(\kappa l,\kappa 'l')} \, \{{1 + i{\bf q}\, ({\bf r}_{l^\prime } - {\bf r}_l) - {\textstyle{1 \over 2}} [{{\bf q}\, ({\bf r}_{l^\prime } - {\bf r}_l)} ]^2 + \ldots } \} \, {\bf u}_o.&\cr&&(2.1.2.30)} ]$ Neglecting higher-order terms, summing up both sides of equation (2.1.2.30) over κ and multiplying by $[{\bf u}_{o}]$ yields $[\eqalignno{& M\, \omega _{{\bf q}\to {\bf 0},j}^2 \, {\bf u}^{2}_o &\cr&\quad= \textstyle\sum\limits_\kappa \textstyle\sum\limits_{\kappa 'l'}\textstyle\sum\limits_{\alpha \beta }u^\alpha _o \, {\bf V}_{\alpha \beta }(\kappa l,\kappa 'l')\, u^\beta _o &\cr&\quad\quad + i\textstyle\sum\limits_\gamma q_\gamma \textstyle\sum\limits_\kappa \textstyle\sum\limits_{\kappa 'l'}\textstyle\sum\limits_{\alpha \beta }u^\alpha _o \, V_{\alpha \beta }(\kappa l,\kappa 'l')\, (r_{l^\prime }^{o^\gamma} - r_l^{o^\gamma})\, u^\beta _o \cr &\quad\quad - {\textstyle{1 \over 2}}\textstyle\sum\limits_\delta \textstyle\sum\limits_\gamma q_\gamma \, q_\delta \textstyle\sum\limits_\kappa \sum\limits_{\kappa 'l'}\textstyle\sum\limits_{\alpha \beta }u^\alpha _o\, V_{\alpha \beta }(\kappa l,\kappa 'l')&\cr&\quad\quad\quad\times (r_{l^\prime }^{o^\gamma} - r_l^{o^\gamma})\, (r_{l^\prime }^{o^\delta} - r_l^{o^\delta})\, u^\beta _o, &\cr&& (2.1.2.31)} ]$ M being the total mass of all atoms within the primitive cell ( $[M = \textstyle\sum_{\kappa = 1}^N {m_\kappa }]$ ). The first term on the right-hand side is zero according to equation (2.1.2.29). The second term vanishes due to the symmetry property of the force-constant matrices , equation (2.1.2.23). Hence (2.1.2.31) is simplified to $[\eqalignno{M\left({{{\omega _{{\bf q}\to {\bf 0},j}^{}}\over |{\bf q}|}}\right)^2 &= - {\textstyle{1 \over 2}}\sum\limits_\delta {\sum\limits_\gamma {{{q_\gamma \, q_\delta }\over {{\bf q}^2 }}\sum\limits_\kappa {\sum\limits_{\kappa 'l'}{\sum\limits_{\alpha \beta }{V_{\alpha \beta }(\kappa l,\kappa 'l')}}}} }&\cr&\quad\times (r_{l^\prime }^\gamma - r_l^\gamma)\, (r_{l^\prime }^\delta - r_l^\delta)\, {{u^\alpha _o \, u^\beta _o }\over {{\bf u}_{o}^2}}. &(2.1.2.32)} ]$ The right-hand side no longer depends on the moduli of the wavevector and displacement but only on their orientation with respect to the crystal lattice. Consequently, acoustic dispersion curves always leave the Γ point as a straight line with a constant slope ( $[\omega/|{\bf q}|]$ ).

The displacement pattern of these long-wavelength modes corresponds to a continuous deformation of a rigid body. Hence, acoustic phonons near the Γ point can be regarded as sound waves and the slope of the dispersion curve is given by the corresponding sound velocity, $[v_s = {\omega / |{\bf q}|}.\eqno (2.1.2.33)]$ Sound velocities, on the other hand, can be calculated from macroscopic elastic constants using the theory of macroscopic elasticity (cf. Chapter 1.3 ). Thus we are able to correlate macroscopic and microscopic dynamic properties of crystals. Using the generalized Hooke's law, the equation of motion for the dynamic deformation of a macroscopic body may be written as $[\rho{{\partial ^2 u_j (t)}\over {\partial t^2 }}= \sum\limits_{k = 1}^3 {\sum\limits_{l = 1}^3 {\sum\limits_{m = 1}^3 {c_{jklm}}}}{{\partial ^2 u_l }\over {\partial r^k \partial r^m }}, \eqno (2.1.2.34) ]$ $[{\rho}]$ being the macroscopic density, $[u_{i}]$ the ith Cartesian component of the deformation and ( $[c_{jklm}]$ ) the symmetric tensor of elastic stiffnesses, which is discussed in detail in Chapter 1.3 . The solution of this differential equation using plane waves, $[{\bf u}= {\bf u}_o \, \exp[i({\bf qr}- \omega t)], \eqno (2.1.2.35) ]$ leads to the following relation: $[\rho \left({{\omega \over |{\bf q}|}}\right)^2 u^j_o = \sum\limits_{k = 1}^3 {\sum\limits_{l = 1}^3 {\sum\limits_{m = 1}^3 {c_{jklm}{{q_k \, q_m }\over {{\bf q}^2 }} u^l_o }}}. \eqno (2.1.2.36) ]$ If we define the components of the propagation tensor by $[\Gamma _{jl} = \sum\limits_{k = 1}^3 {\sum\limits_{m = 1}^3 {c_{jklm}{{q_k \, q_m }\over {{\bf q}^2 }}}}, \eqno (2.1.2.37) ]$ equation (2.1.2.36) may be written as the eigenvector equation $[\rho \, v_s^2 \, {\bf u}_o = \boldGamma \, {\bf u}_o.\eqno (2.1.2.38) ]$ For a given propagation direction as defined by the Cartesian components of q, the eigenvectors of the corresponding propagation tensor yield the polarization of three mutually orthogonal deformation waves. Its eigenvalues are related to the respective sound velocities $[v_{s}=\omega/|{\bf q}|]$ . If the tensor of elastic stiffnesses is known, the elements of $[\boldGamma]$ and, hence, the velocity of elastic (sound) waves can be calculated for arbitrary propagation directions (see Section 1.3.4 ). These data, in turn, allow the prediction of the slopes of acoustic phonon dispersion curves near $[{\bf q}={\bf 0}]$ .