Solving a problem in statistical mechanics, I obtained the following expression:
where $k_B$ is the Boltzmann constant, $\theta_D$ is the debye temperature and N the number of particles.
I wish to evaluate at High and low temperature limits, but I am running intro trouble. For $T>>\theta_D$, the upper limit of the integral becomes 0, and thus the first term is 0. Then, the second term is divided by T, but T -> infinity, so it is 0.
How can this be? What am I doing wrong here?
In the second case, for $T<<\theta_D$, the upper limit of the first integral goes to infinity, and thus the contribution of the first term is of the order $T^3$. However, the second therm is divided by T, which is almost 0 and an exponential that is infinite. What can I do?
I'm sure I am doing something very wrong, but I cannot figure out what