Page 180 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Systems and Ensembles
and is called the grand canonical distribution. For a system which may
occupy different energy states E the total energy of a specific imple-
k
mentation is given by E = Σ n E , where the number of particles is
i k k k
N = Σ n fixed. Here n is the number of subsystems or particles with
i k k k
energy E . To calculate the normalization for (5.18) we have to sum over
k
all possible numbers of particles N′
(
Q = ∑ ZN′)exp – ( αN′) (5.19)
N′
Grand Q is called the grand partition function. Z N′( ) is the canonical ensem-
Partition ble partition function. As was observed for the partition function , we
Z
Function
shall see that Q is a key quantity for a system. Let us analyze Q in more
detail. It consists of a product of two functions ZN′( ) and exp – ( αN′) ,
where the first term increases with increasing N′ , and the second term
decreases with increasing N′ . As a product they result in a sharp maxi-
mum at the equilibrium particle number . For the case of large enough
N
N and a sufficiently sharp peak with width ∆N , we may write that
Q = ZN()exp – ( αN)∆N . This implies that we replace the sum of prod-
ucts in (5.19) by the argument function’s value at the equilibrium particle
number N times the width of the argument function. Hence we obtain
–
lnQ = lnZN() αN + ln∆N . For ∆N « N , the last term in may be
dropped. This gives

lnQ = lnZN() αN (5.20)
–
and for the grand partition function we obtain that

] )exp
Q = ∑ exp – ( βΣ n E[ i i i k – ( αN ) (5.21)
k
k
In (5.21) we see that the sum over different implementations of the prod-
] )exp
uct exp – ( βΣ n E[ i i i k – ( αN ) is nothing else than the product of
k
sums over each energy level E i , and hence we may write that

Semiconductors for Micro and Nanosystem Technology 177