New trends in physics teaching, v.4; The ... - unesdoc - Unesco

Use of the Einstein solid
Introductory statistical physics
The value of the Einstein solid is that its weakly coupled oscillators have equally spaced energy
levels (non-degenerate) so that the distribution of energy amongst oscillators follows the Boltzmann
factor exp(-E/kT). In general, a system with unequally spaced energy levels wil have
quantum states populated according to exp(-E/kr), but where there are many levels within
a narrow range of energy dE, the number of particles having energy in that range wil be larger
on that account (figure 6).
Energy states
Energy E
0
U
m
'4-
0
L
a,
D
E
a,
ol C
m
L
C -
z' -
Energy states -
Gas-like svstern
Energy E
Figure 6. States and energy distributions.
Any classical system wil thus have an energy distribution with two terms in it
N(dE) = f(E) exp(-E/kr) dE
The first term f(E) results from the number of states in intervals dE at various energies E, and
comes from quantum mechanics. It has nothing essential to do with statistical mechanics. The
second term is the statistical-mechanical one, and is the one which appears alone and unencumbered
in the distribution for the Einstein solid. A minor point is that the sites of oscillators
are distinguishable, even though the oscillators would not be, so that Maxwell-Boltzmann statistics
is sufficient. The next section describes Gurney's other choice.
Unit changes
Gurney adopts the line of argument which maximizes W, but introduces a neat way of doing it.
For the Einstein solid
w =N! /no!n1 !n2!. ..
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