PHS2061 - Quantum & Thermal Physics - Physics Assignment Help

Assignment Task

Question 1
Paramagnet
A paramagnet is a collection of spins which exhibits an internal magnetic field due to spin alignment, in the same direction as an external field. The spins could be protons (hydrogen nuclei), unpaired electrons in liquids or solids, or composite spins such as entire nuclei or monatomic gases.
The induced internal field is known as the magnetization of the paramagnet. A simple model of a paramagnet is N spins, which are either spin-up (aligned with the external field) or spin-down (anti-aligned). One possible microstate for an N = 30 paramagnet is:
↑↓↑↑↓↑↑↑↑↑↓↑↑↑↑↑↑↑↓↓↑↑↑↑↑↑↓↑↑↑ (1)
Let the number of spins up be N↑ and the number of spins down be N↓, so that N = N↑ + N↓.
The energy of a spin due to the Zeeman interaction depends on the external magnetic field strength B. Spins ‘up’ (parallel to external field) have energy −µB and spins down (anti-parallel) have energy µB, where µ is the magnetic moment of the individual spins (a constant).
(a) Give an expression for the internal energy U of a paramagnet.
(b) What variable or variables fully describes the macrostate of a paramagnet?
(c) Use Stirling’s approximation to show that the entropy of a paramagnet is approximated by

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