Page 179 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Statistics
∑
PE() =
k
k P = cΩ E()exp – ( βE) (5.15)
,
k
Continuous with such that E ∈ [ EE + δE] . Equation (5.15) contains only the
k
Probability energy E for the same reasons that are valid for the microcanonical
ensemble: the energies cannot be distinguished by measurement. Since
Ω E() is the number of implementations for the heat reservoir R with
E
R
energy and is, due to the large size of , a continuous function, the
P
probability becomes a continuous function of energy.
5.1.3 Grand Canonical Ensemble
For a grand canonical ensemble the system A with a given number of
particles N k is allowed to exchange particles with its reservoir R (see
Figure 5.1). The total number of particles is conserved so that
N tot = N + N i . Following the same arguments as for the canonical
R
ensemble, the number of particles N is included in the argument list of
Ω . The concept of a reservoir now also implies that, in addition to
E « E R , we have that N tot » N i . We expand Ω into a Taylor series
i
around E = 0 and N = 0 which yields
i
i
,
(
lnΩ E – E N – N )
tot i tot i
∂ lnΩ ∂ lnΩ (5.16)
(
= lnΩ E , N )– ------------ E – ------------ N …
tot tot ∂ E i ∂ N i
R E i = 0 R N i = 0
An additional parameter arises of course from the derivative with respect
to the particle number
∂ lnΩ
------------ = α (5.17)
∂ N
R 0
α
This parameter as we shall see later will be interpreted as the chemical
potential. The probability to have a specific implementation then assumes
the form

P ∼ exp – ( βE – αN ) (5.18)
i
i
i
176 Semiconductors for Micro and Nanosystem Technology