Derive the magnetization and susceptibility for the free spin (J=S=1/2). You may use the relation:
To solve for the magnetization and susceptibility of a free spin system (with \( J = S = 1/2 \)), we begin by using the thermodynamic relation and the given equation for the hyperbolic cotangent function. 1. Magnetization: The magnetization \( M \) can be expressed in terms of the partition function and the external magnetic field \( H \). For a spin system, the average magnetization \( \langle M \rangle ______ ____ _______ ______ _________ __________ _____.
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