Debye Approximation of Heat Capacity in 1D

  • #1
jkthejetplane
29
4
Homework Statement:
Using the Debye approximation, illustrate how the phonon heat capacity changes with
respect to temperature in 1D. Discuss your results in the low and high temperature limit
respectively.
Relevant Equations:
equations of Cv from book in 3d listed below
So really i am just unsure how to answer the last part of the question. I am unsure how to apply the low and high temperature limits the way i have done it. Do i set upper/lower limits on the integral and solve? If so i am not sure what to put
1607762298291.png


Here is what he book has for 3d
1607762546553.png
 

Answers and Replies

  • #2
pai535
13
5
After you change your variable, the upper limit of integral will be ##\hbar\omega_D/k_B T##. Then put ##T\to 0## so ##\hbar\omega_D/k_B T \to \infty## and integrate.

For high-temperature behaviour, ##T \to \infty##, ##\hbar\omega/k_B T \to 0##. In this case maybe you don't change your variable and keep the original ##\hbar\omega/k_B T##, then ##\exp(\hbar\omega/k_B T)\to 1##. Then integrate. (in denominator of integral maybe can perform a Taylor expansion)
 

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