Revision of the statistical mechanics of phonons to include phonon line widths Page: 3 of 4
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Use of the two former functions leads to summations similar to that 1n (3)
which are Invalid. However, use of the phonon free energy per mode
■tt u)/2 + k T ln[ 1 - exp(- 3R <u)] In (1) Instead of the B-E function leads to
a result that Is well behaved at all temperatures. The purpose of_this work
1s to derive the expression for the average free energy per mode f for a
crystal having large phonon Unewldths and to test the properties of the
thermodynamic functions derivable from f.
The procedure 1s to Insert f in (1) and to assume m is complex. The line
Integration 1s accomplished by contour integration over the w-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 pos1 ’'ve
semicircle. Poles occur at w = + 2mri (kT/h); n=l ,2.....but the residues
are zero for all n. The clockwise cortou" for the lower half plane consists
of a line e below the real axis (with small semicircle at origin) plus the
large negative semicircle. Poles occur at <u = - 2mM (kT/h); n = 1,2,...,
but, again, all of the residues are zero. The only residues of Importance
are at the poles id = 13+ 1T (upper half plane) and oj» 13- 1T (lower half
plane). Results for the two half planes are averaged as discussed by Morse
and Feshback , a procedure which leads to,
7 ■ h io/2 + (kT/2) In [1 - 2 exp(- 7) cos y + exp(- 2 7)] , (4)
where 7 ■ h ui/kT and y ■ h r/kT. This same procedure was used 1n deriving
n 1 n (3) from (1).
3. Thermodynamic Applications
The specific heat per mode 1s obtained by the formula c ■ - T 327/3T2.
However, the exact form of c depends on the assumptions made as to the
temperature dependence of the anharmonlc shift A(qj) and half width T(qj).
Maradudln and Fein  derived expressions for A and r and found approxima-
tions suitable for the very low temperature range and for the high tempera-
ture range with T > Debye G. At low temperatures, A and r are nearly Indepen-
dent of T while at high temperatures they vary linearly with T. They also
calculated formulas for A and T for high temperatures based on an anharmonlc
potential function (Morse potential) which, 1n turn, was based on the heat
of sublimation of lead. Using their data, we thus have a basis for testing
the possible validity of (4), at least for the case of crystalline lead.
Thus, for the high temperature approximation, we may write,
A ■ d T; 7 ■ h wQ/kT + h d/k -1 xQ + 6 ;
I’ ■ *j T; y ’ h g/k , (5)
where w Is the harmonic (or quasiharmonic) elgentrequency. Using (5) 1n (4),
we obtain the mode specific heat,
cy ■ k xQ2 o^("(e2^ + 1) cos y ■ 2ex]/(e2x - 2e^cos y + 1)^ (6)
Since r Is small In the high temperature approximation, we may replace cos y
by (1 - Y7/2). The shift A 1s also small and we may replace exp(x) by
exp(x )(1 + <S + 62/2). Thus, (6) can be rewritten 1n the form,
cv/k ■ x2 ex/(ex - l)2 - A x2 ex(ex + l)/(ex - 1)^
■i (l/2)(i'r - yJ) xL cx(e2x + 4 ex + l)/(eX - 1)^ , (7)
1n which x now denotes x0 - h <o0/kT. The leading term 1n (7) 1s the ordi-
nary harmonic specific. When Integrated over the spectrum, this leads io
the Dulong-Petlt value of 3R per mole for T > Debye 0, Upon expanding
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Overton, W.C. Jr. Revision of the statistical mechanics of phonons to include phonon line widths, article, January 1, 1983; New Mexico. (digital.library.unt.edu/ark:/67531/metadc1093195/m1/3/: accessed February 15, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.