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Linear chain with m th nearest neighbor interactions
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[QUOTE="wildroseopaka, post: 4536972, member: 491011"] [b]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. [h2]Homework Equations[/h2]: Included in part 1 [h2]The Attempt at a Solution[/h2] 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. [/b] [/QUOTE]
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Linear chain with m th nearest neighbor interactions
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