Statistical Mechanics - Physics at Oregon State University
Statistical Mechanics - Physics at Oregon State University
Statistical Mechanics - Physics at Oregon State University
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82 CHAPTER 4. STATISTICS OF INDEPENDENT PARTICLES.<br />
we need absolute convergence, th<strong>at</strong> is we need to be able to find a minimum value<br />
of N for which the partial sum is a good approxim<strong>at</strong>ion for all temper<strong>at</strong>ures.<br />
Hence we need for every ɛ > 0 we a value Nɛ such th<strong>at</strong><br />
N > Nɛ ⇒ |SN(T ) − S(T )| < ɛ (4.58)<br />
In th<strong>at</strong> case we can interchange summ<strong>at</strong>ion and limits, and can integr<strong>at</strong>e the<br />
sum of the series by integr<strong>at</strong>ing each term and sum the resulting integrals.<br />
In our case, for large values of the quantum numbers the energy is very large<br />
and the distribution function can be approxim<strong>at</strong>ed by<br />
lim<br />
ɛ→∞ fF D(ɛ) ≈ e µ−ɛ<br />
kB T (4.59)<br />
Hence the terms in the series for N decay very rapidly. If we consider a temper<strong>at</strong>ure<br />
interval [0, Tmax] we always have th<strong>at</strong> e −ɛ −ɛ<br />
kB T k e B Tmax . Therefore we<br />
can always use the value <strong>at</strong> Tmax to condition the convergence of the series.<br />
Hence we find th<strong>at</strong> the series converges uniformly on the interval [0, Tmax], with<br />
endpoints included. We may interchange the limit T → 0 and the sum and get<br />
N = <br />
Θ(ɛF − ɛo) (4.60)<br />
orb<br />
If we use free particle energies, which are always positive, we find immedi<strong>at</strong>ely<br />
th<strong>at</strong> ɛF > 0 because N > 0. Note th<strong>at</strong> the only problem is when we take<br />
T → ∞. Th<strong>at</strong> limit needs to be analyzed separ<strong>at</strong>ely, since we now have an<br />
infinite number of infinitesimally small terms to sum. We will see l<strong>at</strong>er how<br />
to deal with th<strong>at</strong> limit. We already know the answer, though, from a previous<br />
section. At large temper<strong>at</strong>ures we can replace the sum by an integral, if the<br />
orbital energies are free particle energies. This is always true for large quantum<br />
numbers, due to Bohr’s correspondence principle.<br />
Grand partition function.<br />
The grand partition function for a Fermi gas involves a sum over st<strong>at</strong>es.<br />
If we enumer<strong>at</strong>e the orbitals for the independent particles, a st<strong>at</strong>e of the total<br />
system can be specified by the occup<strong>at</strong>ion of each orbital, or by a set of numbers<br />
{n1, n2, . . .}. Here ni denotes the number of particles in orbital i. Since we are<br />
dealing with fermions ni is zero or one only. Examples of st<strong>at</strong>es are {0, 0, 0, . . .}<br />
for a st<strong>at</strong>e with no particles or {1, 1, 0, . . .} for a st<strong>at</strong>e with one particle in orbit<br />
1 and in orbit 2.<br />
The grand partition function is<br />
Z(T, µ, V ) = <br />
{n1,n2,...}<br />
The total number of particles is easy to find:<br />
e 1<br />
k B T (µN({n1,n2,...})−E({n1,n2,...}))<br />
(4.61)