A system of N non-interacting classical spins, where each spin can take values σ = - 1, 0, 1 , is placed in a magnetic field h. The single spin Hamiltonian is given by $$H = -\mu_B h \sigma + \Delta (1 - \sigma^2)$$​​
where Delta are positive constants with appropriate dimensions. If M is the magnetization, the zero-field magnetic susceptibility per spin $$\frac{1}{N} \frac{\partial M}{\partial h} \bigg|_{h \to 0}$$​.at a temperature T = 1//βk_B is given by

Correct option is B

Solution:

​σ = - 1, 0, 1

​$$H = -\mu_B h \sigma + \Delta (1 - \sigma^2) \quad .\\\ \\ \quad \text{Draw energy levels using } \sigma = -1, 0, 1 \text{ and } H = -\mu_B h \sigma + \Delta (1 - \sigma^2)\\\ \\Z = e^{\beta \mu_B h} + e^{-\beta \mu_B h} + e^{-\beta \Delta}\\F = -k_B T \ln \left( e^{\beta \mu_B h} + e^{-\beta \mu_B h} + e^{-\beta \Delta} \right)\\dF = -S dT - P dV - \mu dh\\\ \\\mu = \left( \frac{\partial F}{\partial h} \right)_{T, V} = k_B T \frac{e^{\beta \mu_B h} - e^{-\beta \mu_B h}}{e^{\beta \mu_B h} + e^{-\beta \mu_B h} + e^{-\beta \Delta}} \cdot \beta \mu_B\\\ \\ = k_B T \beta \mu_B \frac{e^{\beta \mu_B h} - e^{-\beta \mu_B h}}{e^{\beta \mu_B h} + e^{-\beta \mu_B h} + e^{-\beta \Delta}}\\\ \\\text{In the limit where h is small}\\\ \\M = \frac{\left(1 + \beta \mu_B h - 1 + \beta \mu_B h \right) \mu_B}{2 + e^{-\beta \Delta}}\\\ \\ = \frac{\beta \mu_B^2}{2 + e^{-\beta \Delta}}$$​​