PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Condensed Matter, Materials & Soft Matter · Origins of quantum theory
Thermal Physics, Ch Photons and phonons: blackbody radiation and the heat capacity of solids
Problem
A solid niobium block is held at 15.0K in a cryostat. Machined inside it is an evacuated cavity of volume 2.0cm3 whose walls are at the block's temperature, connected to the outside by an aperture of area 0.50mm2 that opens into a space cold enough that nothing comes back in. Niobium has a Debye temperature θD=275K; ignore the contribution of the conduction electrons to its heat capacity.
((a)) Using the standing-wave state count in a box, find how many electromagnetic modes the cavity holds with frequency below 1.00THz, counting both polarisations.
((b)) Find the mean number of photons ⟨n⟩ in one such mode at 1.00THz, and state in one sentence why the chemical potential to be used for the photons is zero.
((c)) Using the effusion argument (a flux 41cu for quanta of speed c and energy density u), find the radiant power leaving through the aperture.
((d)) Find the molar heat capacity of the niobium at 15.0K from the Debye low-temperature law CV=(12π4/5)R(T/θD)3, and check that T≪θD. Use kB=1.381×10−23JK−1, h=6.626×10−34Js, c=2.998×108ms−1, R=8.314JK−1mol−1, σSB=5.67×10−8Wm−2K−4.