By H. Bilz, D. Strauch, R. K. Wehner (auth.), Professor Dr. Ludwig Genzel (eds.)
The dynamical houses of solids have lately attracted renewed curiosity in reference to the expanding figuring out of section transitions and re lated phenomena. specifically, tender modes or, extra mostly, phonon 'anom alies' appear to play a major position in structural and digital section tran sitions, corresponding to ferroelectric or superconducting transitions. the knowledge of the mechanisms liable for the prevalence of surprisingly low frequencies in phonon spectra calls for an in depth research of the microscopic forces governing the lattice vibrations. Of specific value is the impression of the electron lattice interplay within the adiabatic approximation which in lots of circumstances is the foundation of peculiarities within the phonon self-energy. during this paintings the vibrational spectra of natural non-metals and of these con taining element defects are investigated. ' In those fabrics the interrelation be tween the pseudo-harmonic forces (determining the phonon dispersion re lations) and the non-linear anharmonic and electron-phonon forces (as they act in infrared and Raman spectra) is most blatant and will be quantitatively analysed by way of applicable types. the most job is to reach at a bodily right therapy of digital levels of freedom, as for instance in an digital 'shell' version, which results in the outline of phonon spectra when it comes to long-range polarizabilities and short-range deformabilities. The pur pose of our evaluation is to stimulate extra investigations which, we are hoping, will lead to particular kin among the parameters of the semi-microscopic types and the matrix components from the digital band structure.
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D) Symmetry properties of phonons. The analysis of dynamical properties of crystals such as infrared and Raman spectra employs group-theoretic methods in order to obtain selection rules, etc. A comprehensive discussion has been given recently by BIRMAN in Vol. XXV/2b of this Encyclopedia (1974a). Useful short accounts may be found in a recent book by LAX (1974) and in a condensed article by BIRMAN (1974b). There is no need for a repetition of these elaborate treatments. Some useful tables of symmetries and selection rules in one-, twoand three-phonon processes in crystals with high symmetry are presented in Sect.
5 b) Here, Vo is the volume of the elementary Wigner-Seitz cell. The factor e2/2vo is removed from A and B for convenient comparison with the Coulomb forces. 5 Ionic crystals Sect. 4 The dimensionless quantities A and - B have values between 1 and 20 for the alkali halides while the corresponding values of the Coulomb force constants are A C = -8 and B C =4. The Coulomb force matrix is usually calculated by a specific summation technique originally due to EWALD (1917a, b) and later on improved by BERTAUT (1952) and others and extended to more general cases (DE WETTE, 1959).
LAX (1965a). 0:) Born-Huang conditions (BORN and HUANG, 1954). These conditions are symmetry relations between the different components of the symmetrized fourthrank elastic tensor (Born-Huang tensor) 39 Ionic crystals Sect. 4 ~ _1 Clm,ns=Z(Cln,ms+ CIs,mn) = ens, 1m (no initial stress). 41) They were obtained by Born and Huang from the assumption that macroscopic and microscopic theories should lead to the same result. We note that, for central forces, the tensor C becomes totally symmetric and fulfils, in crystals where every particle has inversion symmetry, the Cauchy relations.