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

© International Union of Crystallography 2006
International Tables for Crystallography (2006). Vol. B. ch. 4.1, p. 402   | 1 | 2 |

Section 4.1.2.3. Einstein and Debye models

B. T. M. Willisa*

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

4.1.2.3. Einstein and Debye models

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In the Einstein model it is assumed that each atom vibrates in its private potential well, entirely unaffected by the motion of its neighbours. There is no correlation between the motion of different atoms, whereas correlated motion – in the form of collective modes propagating throughout the crystal – is a central feature in explaining the characteristics of the TDS. Nevertheless, the Einstein model is occasionally used to represent modes belonging to flat optic branches of the dispersion relations, with the frequency written symbolically as [\omega({\bf q})=\omega_E] (constant).

In the Debye model the optic branches are ignored. The dispersion relations for the remaining three acoustic branches are assumed to be the same and represented by[\omega({\bf q})={\bf v}_sq,\eqno(4.1.2.9)] where [{\bf v}_s] is a mean sound velocity. The Brillouin zone is replaced by a sphere with radius [q_D] chosen to ensure the correct number of modes. The linear relationship (4.1.2.9[link]) holds right up to the boundary of the spherical zone. In an improved version of the Debye model, (4.1.2.9[link]) is replaced by the expression[\omega({\bf q})={\bf v}_s(2q_D/\pi)\sin(\pi{}q/2q_D),\eqno(4.1.2.10)] which is the same as (4.1.2.9[link]) at q = 0 but gives a sinusoidal dispersion relation with zero slope at the zone boundary.








































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