Phonon dispersion relations

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. 405-406 | 1 | 2 |

Section 4.1.5. Phonon dispersion relations

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

4.1.5. Phonon dispersion relations

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Both X-rays and neutrons are used for determining crystal structures, but the X-ray method plays the dominant role. The reverse is true for the measurement of phonon dispersion relations: the experimental determination of $[\omega({\bf q})]$ versus q was first undertaken with X-rays, but the method has been superseded by the technique of coherent inelastic neutron scattering (or neutron spectroscopy). For phonon wavevectors lying anywhere within the first Brillouin zone, it is necessary to employ radiation of wavelength comparable with interatomic distances and of energy comparable with lattice vibrational energies. X-rays satisfy the first of these conditions, but not the second, whereas the opposite holds for infrared radiation. Thermal neutrons satisfy both conditions simultaneously.

4.1.5.1. Measurement with X-rays

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Frequencies can be derived indirectly with X-rays from the intensity of the thermal diffuse scattering. For a monatomic crystal with one atom per primitive cell, there are no optic modes and the one-phonon TDS intensity, equation (4.1.3.6), reduces to $[{{{\rm d}\sigma}\over{{\rm d}\Omega}}=Q^2f^2e^{-2W}\sum_{j=1}^{3}{{E_j({\bf q})}\over{\omega^2({\bf q})}}\cos^2\alpha_j({\bf q})\delta({\bf Q}\pm{\bf q}-{\bf Q}_{\bf h}),\eqno(4.1.5.1)]$ where $[\alpha_j({\bf q})]$ is the angle between Q and the direction of polarization of the mode (j q). There are three acoustic modes associated with each wavevector q, but along certain directions of Q it is possible to isolate the intensities contributed by the individual modes by choosing $[\alpha_j({\bf q})]$ to be close to 0 or 90°. Equation (4.1.5.1) can then be employed to derive the frequency $[\omega_j({\bf q})]$ for just one mode. The measured intensity must be corrected for multi-phonon and Compton scattering , both of which can exceed the intensity of the one-phonon scattering. The correction for two-phonon scattering involves an integration over the entire Brillouin zone, and this in turn requires an approximate knowledge of the dispersion relations. The correction for Compton scattering can be made by repeating the measurements at low temperature.

The X-ray method is hardly feasible for systems with several atoms in the primitive cell. It comes into its own for those few materials which cannot be examined by neutrons. These include boron, cadmium and samarium with high absorption cross sections for thermal neutrons, and vanadium with a very small coherent (and a large incoherent) cross section for the scattering of neutrons. An important feature of TDS measurements with X-rays is in providing an independent check on interatomic or intermolecular force constants derived from measurements with inelastic neutron scattering. The force model is used to generate phonon frequencies and eigenvectors, which are then employed to compute the one-phonon and multi-phonon contributions to the X-ray TDS. Any discrepancy between calculated and observed X-ray intensities might be ascribed to such features as ionic deformation (Buyers et al., 1968) or anharmonicity (Schuster & Weymouth, 1971).

4.1.5.2. Measurement with neutrons

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The inelastic scattering of neutrons by phonons gives rise to changes of energy which are readily measured and converted to frequencies $[\omega_j({\bf q})]$ using equation (4.1.4.1). The corresponding wavevector q is derived from the momentum conservation relation (4.1.4.4). Nearly all phonon dispersion relations determined to date have been obtained in this way. Well over 200 materials have been examined, including half the chemical elements, a large number of alloys and diatomic compounds, and rather fewer molecular crystals (Dolling, 1974; Bilz & Kress, 1979). Phonon dispersion curves have been determined in crystals with up to ten atoms in the primitive cell, for example, tetracyanoethylene (Chaplot et al., 1983).

The principal instrument for determining phonon dispersion relations with neutrons is the triple-axis spectrometer , first designed and built by Brockhouse (Brockhouse & Stewart, 1958). The modern instrument is unchanged apart from running continuously under computer control. A beam of thermal neutrons falls on a single-crystal monochromator, which Bragg reflects a single wavelength on to the sample in a known orientation. The magnitude of the scattered wavevector, and hence the change of energy on scattering by the sample, is found by measuring the Bragg angle at which the neutrons are reflected by the crystal analyser. The direction of k is defined by a collimator between the sample and analyser.

In the `constant Q' mode of operating the triple-axis spectrometer, the phonon wavevector is kept fixed while the energy transfer $[\hbar\omega]$ is varied. This allows the frequency spectrum to be determined for all phonons sharing the same q; the spectrum will contain up to 3n frequencies, corresponding to the 3n branches of the dispersion relations.

In an inelastic neutron scattering experiment, where the TDS intensity is of the order of one-thousandth of the Bragg intensity, it is necessary to use a large sample with a volume of 1 cm³, or more. The sample should have a high cross section for coherent scattering as compared with the cross sections for incoherent scattering and for true absorption. Crystals containing hydrogen should be deuterated.

Dolling (1974) has given a comprehensive review of the measurement of phonon dispersion relations by neutron spectroscopy.

4.1.5.3. Interpretation of dispersion relations

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The usual procedure for analysing dispersion relations is to set up the Born–von Kármán formalism with interatomic force constants $[\Phi]$ . The calculated frequencies $[\omega_j({\bf q})]$ are then derived from the eigenvalues of the dynamical matrix D (Section 4.1.2.1) and the force constants fitted, by least squares, to the observed frequencies. Several sets of force constants may describe the frequencies equally well, and to decide which set is preferable it is necessary to compare eigenvectors as well as eigenvalues (Cochran, 1971).

The main interest in the curves is in testing different models of interatomic potentials, whose derivatives are related to the measured force constants. For the solid inert gases the curves are reproduced reasonably well using a two-parameter Lennard–Jones 6–12 potential, although calculated frequencies are systematically higher than the experimental points near the Brillouin-zone boundary (Fujii et al., 1974). To reproduce the dispersion relations in metals it is necessary to use a large number of interatomic force constants, extending to at least fifth neighbours. The number of independent constants is then too large for a meaningful analysis with the Born–von Kármán theory, but in the pseudo-potential approximation (Harrison, 1966) only two parameters are required to give good agreement between calculated and observed frequencies of simple metals such as aluminium. In the rigid-ion model for ionic crystals, the ions are treated as point charges centred on the nuclei and polarization of the outermost electrons is ignored. This is unsatisfactory at high frequencies. In the shell model, polarization is accounted for by representing the ion as a rigid core connected by a flexible spring to a polarizable shell of outermost electrons. There are many variants of this model – extended shell, overlap shell, deformation dipole, breathing shell … (Bilz & Kress, 1979). For molecular crystals the contributions to the force constants from the intermolecular forces can be derived from the non-bonded atomic pair potential of, say, the 6-exponential type: $[\varphi(r)=-{{A_{ij}}\over{r^6}}+B_{ij}\exp(-C_{ij}r).]$ Here, i, j label atoms in different molecules. The values of the parameters A, B, C depend on the pair of atomic species i, j only. For hydrocarbons they have been tabulated for different atom pairs by Kitaigorodskii (1966) and Williams (1967). The 6-exponential potential is applicable to molecular crystals that are stabilized mainly by London–van der Waals interactions; it is likely to fail when hydrogen bonds are present.

References

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Buyers, W. J. L., Pirie, J. D. & Smith, T. (1968). X-ray scattering from deformable ions. Phys. Rev. 165, 999–1005.Google Scholar

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Cochran, W. (1971). The relation between phonon frequencies and interatomic force constants. Acta Cryst. A27, 556–559.Google Scholar

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International Tables for Crystallography (2006). Vol. B. ch. 4.1, pp. 405-406