On anharmonicity in zinc and cadmium

Rossmanith, E.

doi:10.1107/s0108768184002056

Cited by 5 publications

(3 citation statements)

References 9 publications

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“…e.g. Merisalo & Larsen, 1977, 1979Field, 1983;Rossmanith, 1984;Kumpat & Rossmanith, 1990) and/or a modification of the continuous and uniform phonon density of states assumed in the Debye theory (cf. e.g.…”

Section: Introductionmentioning

confidence: 99%

Thermodynamics of lattice vibrations in non-cubic crystals: the zinc structure revisited

Tomaschitz¹

2021

Acta Cryst Sect A

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A phenomenological model of anisotropic lattice vibrations is proposed, using a temperature-dependent spectral cutoff and varying Debye temperatures for the vibrational normal components. The internal lattice energy, entropy and Debye–Waller B factors of non-cubic elemental crystals are derived. The formalism developed is non-perturbative, based on temperature-dependent linear dispersion relations for the normal modes. The Debye temperatures of the vibrational normal components differ in anisotropic crystals; their temperature dependence and the varying spectral cutoff can be inferred from the experimental lattice heat capacity and B factors by least-squares regression. The zero-point internal energy of the phonons is related to the low-temperature limits of the mean-squared vibrational amplitudes of the lattice measured by X-ray and γ-ray diffraction. A specific example is discussed, the thermodynamic variables of the hexagonal zinc lattice, including the temperature evolution of the B factors of zinc. In this case, the lattice vibrations are partitioned into axial and basal normal components, which admit largely differing B factors and Debye temperatures. The second-order B factors defining the non-Gaussian contribution to the Debye–Waller damping factors of zinc are obtained as well. Anharmonicity of the oscillator potential and deviations from the uniform phonon frequency distribution of the Debye theory are modeled effectively by the temperature dependence of the spectral cutoff and Debye temperatures.

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Section: Introductionmentioning

confidence: 99%

Thermodynamics of lattice vibrations in non-cubic crystals: the zinc structure revisited

Tomaschitz¹

2021

Acta Cryst Sect A

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“…'s), i.e. the isolated-atom-potential expansions of the Boltzmann function (Willis, 1969;Willis & Pryor, 1975;Tanaka & Marumo, 1983;Scheringer, 1985a), the Fourierinvariant expansions of the Boltzmannn function (Merisalo & Larsen, 1977;Kurki-Suonio, Merisalo & Peltonen, 1979;Rossmanith, 1984) and the Gram-Charlier series (International Tables for X-ray Crystallography, 1974, p. 316;Zucker & Schulz, 1982;Kuhs, 1983) do not exclude that they may become negative for certain values of the parameters. By definition, a t Deceased 17 February 1987.…”

Section: Introductionmentioning

confidence: 99%

Negative volumes and central moments of anharmonic probability density functions

Scheringer¹

1988

Acta Cryst Sect A

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The suitability of different functions used for anharmonic temperature factors in describing anharmonicity in structures is studied from the point of view of whether the respective probability density functions (p.d.f.'s) (isolated-atom-potential expansions, Gram-Charlier series, p.d.f, of the cumulant expansion of the temperature factor) are non-negative for the final values of the parameters. Certain central moments of the p.d.f, should also not become negative. Analytical formulae have been derived for the central moments of several (different) p.d.f.'s. 20 p.d.f.'s of atoms in published structures have been examined, their negative volumes determined by numerical integration and their central moments calculated. For Al(4) in the structure of VAl~o.42 at 293 K (strong anharmonicity) the negative volume was -0.238 and several moments were negative. Here the temperature factor and p.d.f. used are not acceptable. For the remaining structures (weak anharmonicity) the negative volumes found were <1-0.031 and the respective central moments were positive. Temperature AbstractEven when direct methods fail to give a clear solution, E maps from some of the phase sets obtained contain correctly oriented fragments. It is shown that such

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“…A collection of these parameters, a description of the models used and a detailed discussion of the reliability of the temperature parameters are given by Rossmanith (1984). Parameters which are used in this Merisalo & Larsen (1979), model 3 (10-,9 j ,~-2) (10-,9 j t~-2) (10-,9 j A-s) (10-19 j/~k-4) (10-,9 j/~-4) (10-,9 j/~-4) Merisalo & Larsen (1979) are tabulated.…”

Section: Comparison Of T(q)num and T(q)serie S For CD And Znmentioning

confidence: 99%