1028 THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1951 



where the stored energy for deformation {x/xo) is 



J ^00 (x/xoY V (A7.13) 



and the average value for a transition from n = to n = 1 is hv for the 

 total energy and (i)hv for potential. This leads to 



|C/op|2 = e>Lhv/2cooV (A7.14) 



As a first approximation we may take the stiffness between the planes of 

 atoms separated by xq as the same as the macroscopic value. This leads to 



Coo = C(( (A7.15) 



Furthermore, jequal relative displacements of neighbors are produced by 

 equal values of A and x/xq. Hence approximately equal changes in energy 

 may occur so that we may assume that 



S2„ = Si„. (A7.16) 



Under these conditions 



I Uo^ p = (hv/kT) I f/A p. (A7.17) 



Since the energy of the optical modes is a maximum for Py = 0, it will 

 change only a small fraction for values of Py comparable to Pi. (See Fig. 3.) 

 Consequently, conservation of energy leads to transitions between Pi and 

 a sphere with P2 — [2m(8i — hv)Y'^. The probability is equal to each point 

 on the sphere and the transition probability is readily found" to be 



lAop = {V I C/op |WM^)z;2 (A7.18) 



where Vi = P2/W is the speed after collision. If we assume relationship 

 (A7.17) between matrix elements and introduce C as defined in (A4.3), then 



1/rop = {hv/m V2/I ^ V4p (A7.19) 



where 4p is a sort of mean free path for optical scattering. 



The dependence of V2 upon the velocity before collision vi is obtained as 

 follows: 



82 = Si - hv (A7.20) 



V2 = [2(Si - hv)/mYi'' 



) , (A7.21) 



» See F. Seitz, "Modern Theory of Solids," McGraw-Hill Book Co., 1940, p. 122. 

 * See E and 11 in S of A. 2, p. 493, for a similar treatment. 



