This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Physics 240B, Spring 2008 Homework 1 Solutions Jesse Noffsinger February 14, 2008 Solutions are in general not the original work of the author. Problem 1) Chapter 6 of Kittel is very helpful for this problem. Portions not worked out explicitly here appear in the recommended texts. a) The total number of electrons up to wavevector k is: N e = 2 k 3 (2 /L ) 3 = V 3 2 k 3 (1) The density of states, D ( k ) is: D ( k ) = dN dk = V 2 k 2 (2) while: = h 2 k 2 2 m (3) Yielding D ( ) = dN d = V 2 2 parenleftbigg 2 m h 2 parenrightbigg 3 / 2 1 / 2 (4) b) The total number of electrons is: N = V 3 2 parenleftbigg 2 m F h 2 parenrightbigg 3 / 2 1 / 2 F = 3 N F bracketleftBigg V 2 parenleftbigg 2 m h 2 parenrightbigg 3 / 2 bracketrightBigg 1 (5) Entering this expression for 1 / 2 F into Eqn (4) gives: D ( F ) = 3 N 2 F (6) 1 c) C v = U T (7) The total energy of the sytem being U = integraldisplay f ( ,T ) D ( ) d (8) which depends on temperature only through the FermiDirac distribution. The Sommerfield expansion [Eqn 2.70 in Ashcroft and Mermin] can be applied, or a direct following of Kittels derivation gives: C v = 2 3 k 2 b TD ( F ) (9) d) If we associate an energy H with spin up/down electrons, we can find the paramagnetic susceptibility of the free electron gas, defined as = M H (10) The total magnetic moment per volume is M = ( n + n ), arising from the difference in the number of spin up and spin down electrons. M = 1 2 integraldisplay [ f ( H ) f ( + H )] D ( ) d (11) The 1/2 accounts for the spin in the standard density of states formula. Taking H to be small with respect to the electronic energies [Ziman pg.332], and the energy derivative of the FermiDirac distribution to be a delta function peaked at F : M = 2 HD ( F ) (12) or = 2 D (...
View
Full
Document
This note was uploaded on 08/01/2008 for the course PHYSICS 240B taught by Professor Cohen during the Spring '08 term at University of California, Berkeley.
 Spring '08
 Cohen
 Work

Click to edit the document details