In this appendix we derive the expression for Landau diamagnetism of free electrons. Our approach will be to derive the result first for 2- from which we will easily be able to obtain the 3- result by integrating over .

Firstly, we remind ourselves of some statistical mechanics. It is convenient to
calculate the *grand partition function*, where

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which implicitly determines . is related to the free energy by

Note that is considered as a function of and so differentiating the right hand side of Eq. (A4) with respect to (keeping constant) gives

This will be useful later. Note that a lot of confusion in statistical mechanics comes from a lack of understanding of what is being kept constant in partial derivatives. Once the correct value of has been determined for a given , we determine the magnetization from

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so we have to compute the term in the free energy.

A useful feature of the grand canonical ensemble is that, for
non-interacting particles, the grand potential factorizes into a product of
grand potentials for each single particle state, and so the grand potential is
a *sum* of grand potentials for single particle states. Hence, for free
electrons,

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Consider first . We are interested in the low- limit where, as
discussed in class, the difference between and its limit,
, is negligible.
As discussed above is a constant
.
and so, for small and

where the last equality is from Eq. (14). For , where is negligible, is just with , and so

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Now we add a field, which changes in two ways. Firstly it changes the density of states, as we have discussed in detail in the main part of the text. Secondly it changes the chemical potential, so . However, we shall see that the change in due to the change in with does not affect the free energy , so we just focus here on the change in due to the modification of the density of states.

In the presence of a field ,
the energy levels take the discrete values
and so

where and . This approximation is good provided the function varies smoothly over a single interval of width . For the present problem, is replaced by , and the integrand varies rapidly on a scale of . Hence use of Eq. (A15) will be valid for . In this situation, many Landau levels are partially occupied and the oscillations found in the main part of the text are washed out. In our case the integral goes to infinity but both the function and its derivative vanish in this limit, and so, evaluating the derivative of the integrand at , we get, to order ,

where corresponds to the integral in Eq. (A15) and to the sum, and, from now on, we work per unit area. The last term in Eq. (A16) is the change in from , the change in the chemical potential due to the field, which is also of order .

Substituting Eq. (A16) into in Eq. (A4) gives

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The magnetization is given by
which
yields
with
, the diamagnetic
susceptibility of free electrons per unit area in two dimensions, given by

To get the corresponding result in 3- we need to add the motion in the
-direction, specified by , and integrate from to .
This is easy because Eq. (A18) is a constant independent of the
(2-) density of electrons (which would vary with ) and hence
the diamagnetic susceptibility of free electrons in 3- per unit *volume*
is given by

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