In the case of symmetric wave functions the occupation numbers of quantum states are not limited and can take on any values. The series
(2.27)
converges only if
. This condition is valid for any
including zero. Thus the chemical potential must be
negative while for fermions it can take both positive and negative values.
The series (2.27) represents a geometric progression and its value can easily be obtained:
![$\displaystyle \Omega_{k}=k_{B}T\ln\biggl(1-\exp\biggl(\frac{\mu-\epsilon_{k}}{k_{B}T}\biggr)\biggr).$](img222.png) |
(2.33) |
Using (2.29) one obtains from (2.33) for the average number of bosons:
![$\displaystyle \langle n_{k} \rangle = \frac{1}{\exp\bigl(\frac{\epsilon_{k}-\mu}{k_{B}T}\bigr)-1}.$](img223.png) |
(2.34) |
Considering the equilibrium phonon gas the number of phonons in the phase space element
can now be written as2.9:
![$\displaystyle dN_{ph}=\frac{1}{\exp\bigl(\frac{\hbar\omega(\vec{k})}{k_{B}T}\bigr)-1}\frac{d^{3}kd^{3}r}{(2\pi)^{3}},$](img226.png) |
(2.35) |
where
is the phonon frequency.
S. Smirnov: