1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Susceptibility of a simple metal (Problem 31.6 in Ashcroft's

  1. May 14, 2017 #1


    User Avatar
    Gold Member

    1. The problem statement, all variables and given/known data
    The susceptibility of a simple metal has a contribution ##\chi_{c.c}## from the conduction electrons and a contribution ##\chi_{ion}## from the diamagnetic response of the closed-shell core electrons.
    Taking the conduction electron susceptibility to be given by the free electron values of the Pauli paramagnetic and Landau diamagnetic susceptibilities, show that:

    $$ \frac{\chi_{ion}}{\chi_{c.c}} = -\frac{1}{3} \frac{Z_c}{Z_v}\langle (k_F r)^2 \rangle $$

    where ##Z_v## is the valence, ##Z_c## is the number of core electrons, and ##<r^2>## is the mean square ionic radius defined in (31.26).

    2. Relevant equations
    $$(31.26) <r^2> = \frac{1}{Z_i} \sum <0|r_i^2 |0>$$

    $$\chi^{molar} = -Z_i (e^2/(\hbar c))^2 \frac{N_A a_0^3}{6}\langle (r/a_0)^2 \rangle$$

    $$\chi_{pauli} = \bigg(\frac{\alpha}{2\pi}\bigg)^2 (a_0k_F)$$

    $$\chi_{Landau} = -1/3 \chi_{Pauli}$$

    3. The attempt at a solution
    I thought that ##\chi_{molar}=\chi_{ion}## and that ##\chi_{c.c} = \chi_{Landau}+\chi_{Pauli} = 2/3 \chi_{Pauli}##.

    But when I divide between the two susceptibilities I don't get the right factors, has someone already done this exercise from Ashcroft and Mermin?

    I tried searching google for a solution but to a veil.
  2. jcsd
  3. May 14, 2017 #2

    Charles Link

    User Avatar
    Homework Helper
    Gold Member

    I was able to find something in a google search. For the closed shell electrons, this form of diamagnetism is described by the Langevin theory, as opposed to free electron diamagnetism, which is the Landau theory.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted

Similar Discussions: Susceptibility of a simple metal (Problem 31.6 in Ashcroft's