9780030839931, Chapter 2, Problem 1P

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Chapter Problem 1P Problem The Free and Independent Electron Gas in Two Dimensions the relation between and in two dimensions? (b) What the relation between and two dimesions? (c) Prove that two dimesions the free electron density of levels constant independent for and What the constant? (d) Show that because every term the sommerfeld expansions for n vanishes except the Deduce that any (e) Deduce from when then (f) Estimate from amount by which differs Comment on the numerical significance this 'failure' of the sommerfeld and the mathematica reason Step step solution Step (a) two dimensional system the number of allowed values of space inside volume The volume space the area circle with radius therefore the number of allowed values of within the area of circle is note that each allowed leads two one one for each Therefore in order accommodate N one must have N 2(allowed values within area circle) The electron density given as N =nV compare this with equation Therefore the relation between and in two dimensions Step (b) relate with Substitute (obtained the relation between and Step of 5 (c) In two dimensions the density particles in range to is one can write the electron density as Change integral above an integral over to have Compare the above integra with the usual form =0 one conclude that equal to for whereas for Step (d) The electron density given as The value being independent gives and all the terms in Sommerfeld expansion vanishes as can be seen from the equation above The terms above equation represent the value for ground state, and can same as a g(c) Substitute (0) when the relation holds any temperature Step (e) The electron density given as Substitute for for Substitute such that dy or for de and the limits integration changes Multiply both sides by Substitute Take antilog on both the expression has been (f) The Taylor expansion of the term in expression Therefore, differs from The term exponentially small and can safely ignored important to note that does not have an analytic expansion about T This accounts for the fact that the Sommerfeld expansion does not work two dimensions Anonymous why analytic expansion?