On the evaluation of phonon widths and shifts

Abstract

I n this paper a method is described for the numerical evaluation of phonon widths and shifts due to three-phonon processes in an anharmonic crystal. With this method thed-function and principal value restrictions on the frequencies of the phonons participating in the three-phonon processes do not have to be taken into account explicitly. The method is based on a dispersion relation satisfied by the width and shift, and on their expansion in Legendre functions of the first and second kinds, respectively. It is illustrated by a numerical example. Es wird eine Methode beschrieben zur numerischen Berechnung vonPhononenbreiten und Phononenverschiebungen bei Drei-Phononenprozessen in einem anharmonischen Kristall. Bei dieser Methode ist eine explizite Berucksichtigung der &Funktion und der Hauptwertbedingungen fur die Frequenzen der an diesen Prozessen beteiligten Phononen nicht notwendig. Die Methode beruht auf einer Dispersionsrelation fur Breite und Verschiebung sowie auf deren Entwicklung in Legendrefunktionen erster bzw. zweiter Art. Das geschilderte Verfahren wird durch ein numerisches Beispiel illustriert.

Phonon anharmonicity of monoclinic zirconia and yttrium-stabilized zirconia

“…37,38 A quasiharmonic analysis cannot, of course, account for linewidth broadening of phonons at elevated temperatures. Anharmonic phonon perturbation theory shows that the leading term in phonon lifetime broadening originates from the cubic anharmonicity 39,40 . The cubic anharmonicity may be especially large in monoclinic zirconia because its atoms do not have inversion symmetry, so it is possible for some phonons to have an odd-order terms in their potential energy.…”

Many-Particle Theory of Interacting Phonons in a Crystal

“…11. First-order basic diagrams for the calculation of <Usj^>rj, Using (27), (33), and the rules described in the previous chapter, we find from these diagrams the following correction to (47):…”

“…For a detailed proof compare the lecture notes of Van Hove. (3,4) The new propagator (27) now explicitly contains a correction for the presence of thermal phonons in the system. By application of the above theorem, the total contribution to the trace in (26) from all processes characterized by diagram a in Fig.…”

“…Again such an internal instantaneous phonon line can be broken and run around the cylinder (diagrams 1 and m). Just as in the foregoing, we can fully account for these turns by adding to each loop in a basic diagram a factor which derives from the right-hand side of (27) by letting Aj3 go to zero. We find the contribution to Zo of all the processes characterized by the basic diagram k, thus including the related diagrams: There are three such terms, namely, A2 A3 A3 , A3A*A*, and A3A*A*, and similarly three terms for the left vertex.…”

“…It was demonstrated in the foregoing (see the theorem on page 13) that in this case we can restrict ourselves to just so-called basic diagrams (diagrams without external lines), provided we substitute for each internal line in a basic diagram the propagator (27) and for each loop the factor (33). With this limitation (together with the limitation to connected diagrams) only a finite number of diagrams are required for a calculation of FA to a given order in powers of u/d.…”