MMSE Journal Vol.3 2016

Mechanics, Materials Science & Engineering, March 2016 – ISSN 2412-5954

Let us remember that we have, for the equilibrium distribution, nqop  nqo' p ' , because nqop depends on

 qp   q' p' .

energy only. For the conservation of energy, in the elastic process, The linearized Boltzmann equation becomes:

 v p (q)  T

 nqop



T



q p 

(n

qp



 nqop )  (nqp  nqop ) Z qqpp .

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In the equation, (nq p / T ) is given only by the contribution of the equilibrium distribution nqop . To have the Boltzmann linearized equation in the general case, let us define q p as:

nqp  nqop  qp

nqop

.

(qp )

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Let us remember that ( x  qp k BT ; kB is the Boltzmann constant): 1   1  1 nqp nqp  1 .  x  k BT x  e  1 k BT





Then, in the case of elastic scattering, the linearized equation is [7]:

 k B T v p (q)  T

nqop T





qp 

k B T v p (q)  T Qqqpp 



o o qp nqp (nqp

n qop T 





q p 



 1)  qpnqop (nqop  1) Z qqpp



q p 



 qp Q qqpp 

.

(11)

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nqop (nqop  1) Z qqpp 

In the case of three-phonon scattering, the linearized equation is [7]:

k B T v p (q)  T

nqop T



 

q p  q p 





Q qqpp,q p  qp   qp   qp 





1    Q qqpp ,qp  qp   qp   qp .

2 qp  qp 

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