REVISION OF THE STATISTICAL MECHANICS OF PHONONS TO INCLUDE PHONON LINE WIDTHS Los A[ A MINTER, \\T allrms

Zubarev[1] in 1960 obtained the “smeared” Bose-Einstein (B-E) function in order to take into account the fact that the eigenenergy associated with a fixed phonon wave vector q and fixed polarization index j is not precisely defined but, instead, is smeared by phonon-phonon and phonon-electron interactions. The ratio Γ(qj)/ω(qj) is often quite small, i.e., of the order of 0.01 or less, where Γ is the phonon linewidth and ħ ω is the eigenenergy. However, in strongly anharmonic crystals Γ /ω may be as large as 0.3 at certain points of the Brillouin zone. In such dramatic cases, one would suspect that such phonon linewidths would have some observable effect on the thermodynamic properties. Zubarev represented the effect of “smearing” on the statistical properties by the infinite integral[1], 
 
$$ \overline n = \int_{ - \infty }^\infty d \omega \;{\left[ {\exp (\frac{{h\omega }}{{kT}}) - 1} \right]^{ - 1}}\;L(\omega ;\overline \omega ,\Gamma )\;, $$ 
 
(1) 
 
 
 
$$ L(\omega ;\overline \omega ,\Gamma )\; = \;(\Gamma /\pi ){\left[ {{{(\omega - \overline \omega )}^2} + {\Gamma ^2}} \right]^{ - 1}}, $$ 
 
(2) 
 
in which we have deleted the indices (q,j) for convenience. The term in square brackets in (1) is the usual B-E function, while in L, the usual Lorentzian function, \( \overline \omega \) is the average or center frequency of the distribution.


Introduction
Zubarev [l] in 1960 obtained the "smeared" Bose-Einstein (B-E) function in order to take into acccunt the fact that the eiqenenergy associated with a fixed phonon wave vector q and fixed polarization index j is not precisely defined but instead,is smeared by phonon-phonon acd phonon-electron interactions. The ratio r(qj)/u(qj) is often quite small, i.e., of the order of 0,01 or less, where 1'is the phonon linewidth andti w is the eigenenergy, However, in strongly enharmonic crystals I'/umay be as large as 0.3 at certain points of the Brillouin zone. In such drnmatic cases one would suspect that such phonon linewidths would have some observable effect un the thermodynamic properties. Zubarev represented the effect of "smearing" on the statistical properties by the infinite integral[l], in which we have deleted the Indices (q,j) for convenience. The term in square brackets in~1) is the usual B-E function, while in L, the usual Lorentzian function, u is the average or center frequency of the distribution.
Equation (1) is not usable as it stands. IJ"wever,we have found a slngie formula which is the exact equivalent of (I) hy the use of contour methods. We obtain for the average B-E function, Y 2:-Ĩ 8 (e cosy . 1)/(e 2e cosy+l) where~"l~+lfi,y~(IllI'.and1{=l/kT, The summation part of (3) is due to poles on the~maginary axis Oi the m-plane. When we use (3) to derive the specific heat and entropy, we find that the entropy slope :IS/OT~II' as T * O K, due to the summation part of (3). The flr~t part of (3) Is well behaved, Accordingly, the use of (1) is proven to be invalid,

2, A New Approach
Since we have shown that (3), which is equivalent to (1), is invalid, we hnve tried using the statistical mechanical expressions for the phonon partltlon function, entropy, and free eneray, in the place of the II-Efunction in (

Use oftt:e two forme~func~ions leads"to summations similar to that in (3) which are invalid. However, use of the phonon free energy per mode +lu/2+k
Tln[l -exp(. @I u)] II, (1) instead Gf the B-E function leads to a result that is well behaved at all temperatures. The purpose of_this work is to derive the expression for the average free energy per mode f for a crystal having large phonon linewldths And to test the properties of the t.herm~dynamicfunctions derivable from f. The procedure is to insert f in (1) and to assume w is complex. The line integration is accomplished by contour integration over the u-plane. The counterclockwise contour for the upper half plane consists of a line E above the real axis (with small semicircle at the origin) plus a large posi ive semicircle. Poles occur at w u + 2nmi (kT/h); n=l ,2,..., but the res',dues are zero for all n. The clockwise coctou~for the lower half pl~ne consists of a line E below the real axis (with small semicircle at origin) plus the large negative semicircle. Poles occur at~= -2nmi (kT/h); n = 1 ,2,..., but, again, all of the residues are zero. The only residues of importance . are here~s h ti/kTand y s h r/kT, This same procedure was used in deriving n in (3) from (l).

Thermodynamic Applications
The specific heat per mode Is obtained by the formula c s -T a2~/aT2, However, the exact form of c depends on the dssumption~made as to the temperature dependence of th~enharmonic shift A(qj) and half width r(qj). Maradudin and Fein [3] derived expressions for A and I'~nd found approximations suitable for the very low temperature range and for tliehigh temperature range with T > Debye 0, At low temperatures, A and 1'are nearly independent of Twhile at high temperatures they vary linearly with T. They also calculated formulas for A and r for high temperatures based on an enharmonic potential function (tiorsepotential) which, in turn, was based on the heat of sublimation of lead. Using their data, we thus.have a btisisfor testing the possible validity of (4), at least for the case of crystalline lead. Thus, for the hiqh temperature approxlmatlon, we may write, A=dT;~mhuo/kT+hd/k-xo+ 6; l'=gT; yahg/k, where w is the harmonic (or quasiharmonic) eigent'requency.Using (5) in (4), we obta?n the mode specific heat,