Page 183 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Statistics
5.2.2 Bose-Einstein Statistics
α
If the total particle number is tuned by a parameter , then we have to
N
use the grand partition function to derive the average number of particles
per energy-level. For the case of Bose-Einstein (BE) statistics, arbitrary
numbers n are allowed. This yields for (5.22)
k
∞
1
Q = ∏ ∑ exp ( – ( βE + α)n ) = ∏ ------------------------------- (5.25)
i
i
βE i )
α +
( –
–
i i 1 e
n i
Thus the logarithm of (5.25) is
(
lnQ = – ∑ ln 1 e ( – α + βE i ) ) (5.26)
–
i
and inserting this into (5.20), we obtain
(
–
lnZ = αN – ∑ ln 1 e ( – α + βE i ) ) (5.27)
i
The partition function already includes the constraint of the total number
of particles and thus may be used to calculate the average number of par-
ticles n i for a specific energy level E i

∂
∂
1 lnZ 1 βe ( – α + βE i ) ∂ lnZ α 1
n = – ---------------- = – --- – ------------------------------- + --------------------- = ---------------------- (5.28)
i β ∂ E β ( – α + βE i ) ∂ α ∂ E α + βE i
–
i 1 e i e – 1
α
The derivative lnZ ∂⁄∂ α comes from the fact that may depend on the
position of the energy levels. The fact that, for a given , the parameter
N
α maximizes the grand partition function, results in lnZ ∂⁄∂ α = . 0
Equation (5.28) is called the Bose-Einstein distribution.
Chemical The important relation
Potential
∂ lnZ µ
α = ------------- = – ------ (5.29)
∂ N kT

180 Semiconductors for Micro and Nanosystem Technology