Page 178 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Systems and Ensembles
A
system . This average energy is the first moment of the probability dis-
tribution E = Σ E P( ) . The factor β , which we can also write as
i i i
β = ( k T) – 1 , is the inverse of the thermal energy. Here is the temper-
T
B
ature and k is the Boltzmann constant. Note that the sum in the denom-
B
inator of (5.11) accounts for the normalization of P . Equation (5.11) is
i
called the canonical probability distribution.
Average If we write Z = k ∑ exp – ( βE ) for the denominator of (5.11), then the
k
Energy average energy is given by
∂
1 Z ∂ lnZ
E = – --------- = – ----------- (5.12)
Z β ∂ β
∂
Z
Canonical We call the canonical partition function. Suppose that we know the
Partition total number of particles for a specific implementation to be the sum
k
Function
N = Σ n , where n i is the number of particles with energy E i such that
i i
k
E = Σ n E i holds. Then we may write the partition function as
k
i i
Z = ∑ exp – ( βΣ n E[ i i i k i i N k (5.13)
] ), with the constraint Σ n =
k
k
Average The bracket Σ n E[ ] denotes one specific implementation of E .
i i i k k
Number of This gives us direct access to the average number of particles n j in a spe-
Particles
j
cific state . For a canonical distribution this is given by
∑ n exp – ( βΣ n E ] )
[
j
i i
i k
1 ∂
∂
Z
k
n = --------------------------------------------------------- = – --------------- = – 1 lnZ (5.14)
----------------
∂
∂
j βZ E β E
∑ exp – ( βΣ n E ] ) j j
[
i k
i i
k
Sum Over The canonical partition function is sometimes also called the sum over
Z
States states. We have already seen that Z is a central term in statistical
mechanics, from which many other system properties may be derived.
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Let us consider the energies of all realizations that the system may
k
assume for a small energy interval EE +,[ δE] . In this case the probabil-
A
ity of finding with the energy is given by the sum over all imple-
E
mentations k
Semiconductors for Micro and Nanosystem Technology 175