(Solved): 1) Consider a system of N localized weakly-interacting spin- 1/2 particles of magnetic momen ...

1) Consider a system of $$N$$ localized weakly-interacting spin- $$?1/2$$ particles of magnetic moment $$?$$ in an external magnetic field B. The quantum energy levels for a single particle are $$E=\pm ?B$$. The total energy of the system is $$E=?\left(N_{?}?N_{?}\right)?B$$, and the total number of configurations with a given energy is $$?(m)=\frac{N!}{[(N+m)/2]![(N?m)/2]!},$$ where $$m=N_{?}?N_{?}$$, as shown in the lecture. a) Using the definition of temperature $$\frac{1}{k_{B}T}=\frac{?\ln ?}{?E}$$ and Stirling's approximation for the factorial, obtain a relation between the energy of the system and the temperature, valid for arbitrary temperatures. b) Under what circumstances is $$T$$ negative? c) What is the total magnetic moment $$M$$ of the system as a function of $$B$$ and $$T$$ ?