In a hydrogen atom the electron and proton spins
can combine in four ways:
the SINGLET state
has an energy
J
below the three TRIPLET states
,
and
which are degenerate in zero magnetic field.
The energies of the singlet and triplet
states
are not shifted much by a weak magnetic field
(where
is the magnitude of the [negative] magnetic moment
of the triplet state),
but the
states split by
1,±1
= ±mB.
ANSWER:
This is a simple Boltzmann distribution:
probability of singlet state
where
is the
partition function. For B=0,
, giving
.
At
J,
and
,
giving
or
= fraction of atoms in the singlet state.
ANSWER:
or
.
ANSWER:
As
,
and
the exponentials all approach unity; the two in
the numerator cancel, giving
.
As
,
and
. However, since
,
even faster, so
.
Of course, in between
must have
nonzero values and there must be a peak in the magnitude
somewhere. This could be found by differentiating
with respect to
and setting the derivative
equal to zero, but this was not requested.
An example is shown below for the case of
.
ANSWER:
This is easy if we apply the principles of diffusive
equilibrium to a given single-particle state
of energy
:
it may be populated by either one particle (N=1)
or none (N=0); all other values of N are explicitly
forbidden by the Pauli principle. We use
,
where is the Gibbs sum. In this case,
.
Since
this gives
.
ANSWER:
The graph below shows f as a function of
(i.e.
in units of
) for
and for
.
For z=1
(
or x=0)
we always get
and for
(same as
or
) we always get
.
For
(same as
or
)
we get
if
and
if
. That case
(
) also gives
f = 0.119 at
(z=2), as shown.
The main thing to realize here is that
becomes more and more like a step function
as
gets larger and larger (i.e. at low temperature).
ANSWER: At
the distribution becomes a simple step function
at
:
all states below the Fermi energy
are filled (N=1) and all states above
are empty (N=0).
ANSWER:
Of the many ways to approach this part, the easiest is to just
think about it! Since the chemical potential is the incremental
change in free energy as the last particle is added
(),
and the free energy
at
,
is just the energy level at which the last particle
goes in. For spin-1/2 particles, two can go into each state
(
and
) and the states
are evenly spaced,
,
so the last particle goes in at
and that is the chemical potential at
or FERMI
ENERGY:
.
Note: it has been brought to my attention that some people had seen a derivation of the Fermi-Dirac distribution function before, while others had not; this was correlated with which Engineering specialization people had been following. Since this question is easy to answer if you grasp the basic approach and almost impossible to solve by plugging in standard formulas, it was an excellent test of your ability to synthesize ideas (the whole point of a Physics education) but an unfair one because of the importance of ``prior knowledge.'' I will try to take this into account when making up final marks, by noting which programs people are in and which courses you have taken previously. I may need to get that information from you on the Web.