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wildroseopaka
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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.
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.
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.
Homework Equations
: Included in part 1The 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.