Page 177 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Statistics
Taking into account that (5.8) is valid for spin up and down particles it
must be multiplied by a factor 2.
5.1.2 Canonical Ensemble

A
Let us now consider a small system which stays in contact with a very
R
large heat reservoir (see Figure 5.1b). We only allow to exchange
A
heat with R (think of a tin-can filled with liquid that is immersed in a
lake). In the following we assume that A « R . The total energy of the
A
system, given by E = E + E , is conserved. If can be found in a
tot A R
well defined state A with energy E , then the energy of the heat reser-
i i
voir is given by E = E – E . The probability P to find exactly an
R tot i i
implementation A , is proportional to the number of possible implemen-
i
tations Ω E( ) for the heat reservoir , each of which has energy E
R
R R
⋅
(
P = c Ω E – E ) (5.9)
i tot i
where is the normalization factor. The prerequisite A « R immediately
c
implies that E « E . We next expand the logarithm of Ω by a Taylor
i R
series about E = 0 , (or about E ) to obtain a good approximation
i R
∂ lnΩ
(
(
lnΩ E – E ) = lnΩ E ) – ------------ E + … (5.10)
tot i tot ∂ i
E
R E i = 0
Now imagine that, for one specific E i , the reservoir only has to fulfil the
condition that E = E tot – E i . Due to its huge size compared to , the
A
R
number of implementations of R is obviously very large and also
increases strongly with increasing energy. Inserting (5.10) in (5.9) yields
⋅
P = c exp – βE i and performing the normalization we obtain
i
exp – ( βE )
i
P = ----------------------------------- (5.11)
i
∑ exp – ( βE )
k
k
Temperature where ∂ [ lnΩ ∂⁄ E ] = β is independent of the energy E . The
R E i = 0 i
β
parameter specifies the average energy per degree of freedom of the
174 Semiconductors for Micro and Nanosystem Technology