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!

Linear chain with m th nearest neighbor interactions

  1. Oct 13, 2013 #1
    1. Ashcroft and Mermin 22.1
    Reexamine the theory of the linear chain without making the assumption that only nearest neighbors interact, using the harmonic potential energy of the form:
    U^harm=∑_n▒∑_(m>0)▒1/2 K_m [u(na)-u([n+m]a) ]^(1/2)
    Show that the dispersion relation must be generalized to
    ω=2(∑_(m>0)▒K_m ((〖sin〗^2 (1/2 mka)))/M )^(1/2)
    Show that, provided the sum converges, the long wavelength limit of the dispersion relation must be generalized to:
    ω=a(∑_(m>0)▒〖m^2 K_m/M)^(1/2) |k| 〗
    Show that if Km = 1/mp (1<p<3), so that the sum does not converge, then in the long wavelength limit
    ω∝ k^((p-1)/2)

    Hint: it is no longer permissible to use the small-k expansion of the sine in equation a, but one can replace the sum by an integral in the limit of small k.


    2. Relevant equations: Included in part 1



    3. The attempt at a solution
    I have no problem getting parts a and b, but part c is eluding me. I first replaced the summation with an integral and got:

    ω=2(∫_0^k▒〖m^(-p) (〖sin〗^2 (1/2 mka))/M〗 〖dm)〗^(1/2)
    I then expanded the sin^2
    ω=2(∫_0^k▒〖m^(-p) (1-cos(1/2 mka))/M〗 〖dm)〗^(1/2)
    I attempted an integration by parts, but quickly realized that I would end up in a never ending cycle. I then read a tip online (not a solution, but a hint) that one should try expanding cos x. But I keep ending up with

    ω∝∫_0^k▒〖m^(-p) (1-(1-m^2 k^2+m^4 k^4-m^6 k^6+m^8 k^8-…)〗 〖dm)〗^(1/2)An
    Which, when I integrate, doesn’t seem to give me the right answer. I feel I’m missing something very basic, but I’ve been out of school for several years, and I’ve been banging my head against a wall on this for days now. Any help is appreciated.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Oct 13, 2013 #2

    TSny

    User Avatar
    Homework Helper
    Gold Member

    Try a substitution where you let ##u## = the argument of the sine function. Convert the integral over ##m## to a numerical factor times an integral with respect to ##u##. All of the ##k## dependence will be in the numerical factor. So you will not need to worry about doing the integral over ##u## if you just want to find how ##\omega## depends on ##k##.
     
  4. Oct 14, 2013 #3
    Thanks, I think that did it.
     
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: Linear chain with m th nearest neighbor interactions
Loading...