Word Count : 427

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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 \( \langle M \rangle \) is given by: \[ \langle M \rangle = \frac{1}{Z} \sum_{s} M_s e^{-\beta E_s} _________ ________ _______ _____ _____ ________ _____ ___ ___.
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