Specific heat of a particle on the cone

This work investigates how a conical singularity can affect the specific heat of systems. A free nonrelativistic particle confined to the lateral surface of a cone—conical box—is taken as a toy model. Its specific heat is determined as a function of the deficit angle and the temperature. For a vanishing deficit angle, the specific heat is that of a particle in a flat disk where a characteristic temperature separates quantum and classical behaviors, as usual. By increasing the deficit angle the characteristic temperature also increases, and eventually another characteristic temperature (which does not depend on the deficit angle) arises. When the cone gets sufficiently sharp, at low and intermediate temperatures the azimuthal degree of freedom is suppressed. At low temperatures the specific heat varies discontinuously with the deficit angle. Connections between certain theorems regarding common zeros of the Bessel functions and this discontinuity are reported.

Figure 1

Numerical plots of $C∕\kappa$ versus $T∕\Theta$. The plots from the left correspond to $\alpha =1$, $0.3$, $0.2$, $0.15$, $0.1$ and $\alpha \to 0$, respectively. The plots for $\alpha =1$ to $\alpha =0.15$ display maxima corresponding to specific heat of about $1.119\kappa$, $1.125\kappa$, $1.068\kappa$, and $1.042\kappa$ at temperatures of about $0.52\Theta$, $0.72\Theta$, $1.59\Theta$, and $3.49\Theta$, respectively. At $T\simeq 0.94\Theta$, the plot for $\alpha \to 0$ displays a maximum corresponding to $C\simeq 0.535\kappa$.