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At low temperatures, the internal energy of a system of free electrons may be written as an expansion
Obtain the value of the constant A, and indicate the order of magnitude of the terms that have been discarded.
Physics
This is again a straightforward (although somewhat la- borious) application of the method of Sommerfeld to obtain as- ymptotic low-temperature results.
In the grand-canonical ensemble, the number of particles and the internal energy of the gas of free fermions are give by
and
where
At low temperatures, ƒ (ε) is almost a step function, so the derivative dƒ /dε has a pronounced peak at ε= µ, which provides the basis for the development of a low-temperature expansion. Let us write
At low temperatures, βµ ~ βεT → ∞, so that we may discard exponential corrections, of order
exp (-βεT ), and write
which is the main idea of Sommerfeld´s scheme. Except for these exponential corrections, we have the series expansion
where
and so on (as you can check in the integral table of Gradshteyn and Ryzhik, for example). We then have the expansion
from which we write
We left for the reader the task of obtaining the numerical coef- ficients (M2 and M4). It is then easy to write an expansion of the internal energy in terms of (even) powers of (T/TF )
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