- #1
Cryg9
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Homework Statement
A 1-D lattice consists of a linear array of N particles (N>>1) interacting via spring-like nearest neighbor forces. The normal node frequencies are given by
[tex]\omega_n=\omega_0\sqrt{\,\,2-2\cos\left(2\pi n/N\right)}[/tex]
where [itex]\omega_0[/itex] is a constant and n an integer ranging from -N/2 and +N/2. The system is in thermal equilibrium at temperature T. Let cv be the constant length specific heat.
a) Compute cv for the T -> infinity.
b) For T -> 0, [itex]c_v\rightarrow A \omega^{-\alpha}T^{\gamma}[/itex] where A is a constant you need not compute. Compute the exponents [itex]\alpha\text{ and }\gamma[/itex].
Homework Equations
[tex]U=(1/2+n)\hbar\omega\\\\c_v=\left.\frac{d U}{d T}\right|_V[/tex]
The Attempt at a Solution
a) Using the energy of a harmonic oscillator listed above, we have
[tex]U=N(1/2+n)\hbar\omega_0\sqrt{\,\,2-2\cos\left(2\pi n/N\right)}[/tex]
which does not have T in it explicitly. It seems n should depend on T, that in order to cause it to go to the next excited mode, one would have to add enough energy to increment n but I am not sure how to express that.
b) For T->0, the lowest energy state is when n=0 -> wn=0, no motion/energy. That does not really tell me anything useful though. I feel its the same question, how does the energy (or n) depend on T?