In two dimensional system the number of allowed values of space inside a volume is . The volume in space is the area of circle with radius , therefore the number of allowed values of within the area of circle is .
It is important to note that each allowed -value leads to two one-electron levels, one for each spin value. Therefore in order to accommodate electrons, one must have
The electron density is given as , compare this with equation ,
Therefore, the relation between and in two dimensions is .
One can relate with as:
Substitute (obtained in part (a)),
Therefore, the relation between and is .
In two dimensions, the density of particles in range to is . Therefore, one can write the electron density as:
Change the integral above to an integral over energy, to have
Compare the above integral with the usual form ,
Therefore, one can conclude that is independent of , and is equal to for whereas is zero for .
The electron density at temperature is given as:
The value of being independent of gives , and all the terms in Sommerfeld expansion vanishes, as can be seen from the equation above.
The terms in the above equation represent the value of for ground state, and can be equated for the same as:
Substitute for .
Therefore, when is independent of , the relation holds at any temperature .
The electron density is given as:
Substitute for and for ,
Substitute for , such that for or for , and the limits of integration changes to to 0 as:
Multiply both sides by and rearrange,
Substitute for ,
Take antilog on both sides,
Take log on both sides,
Therefore, the expression has been derived.
The Taylor expansion of the term in expression gives,
Therefore, differs from by an amount . The term is exponentially small and can be safely ignored.
It is important to note that does not have an analytic expansion about . This accounts for the fact that the Sommerfeld expansion does not work in two dimensions.