# Heisenberg Model

by CNX
Tags: heisenberg, model
 P: 28 1. The problem statement, all variables and given/known data Find density of states $$H = \frac{-JzM}{g\mu_B} \sum_i S_i^z + \frac{JzNM^2}{2g^2\mu_b^2} = -\alpha \sum_i S_i^z + \gamma[/itex] z = # nearest neighbors J = exchange M = magnetization $S^z$ = project of total spin S=0,1. 2. Relevant equations [tex]Z=\sum_{S m_s}$$ 3. The attempt at a solution {let h = h-bar} $$Z = \sum_{S m_s}$$ $$= \Pi_i \sum_{S m_s} \left [ \exp(\beta\alpha h m_s^i - \beta\gamma) \right]}$$ for S = 0, $m_s = 0$; for S = 1, $m_s = -1,0,1$ $$= \Pi_i \left (2\exp[-\beta \gamma] + \exp [\beta(h\alpha-\gamma)] + \exp[-\beta(h\alpha+\gamma)]\right )$$ $$= \left (2\exp[-\beta \gamma] + \exp [\beta(h\alpha-\gamma)] + \exp[-\beta(h\alpha+\gamma)]\right )^N$$ So, $$Z = \left (2\exp[-\beta \gamma] + 2\exp [-\beta\gamma]\cosh(h\beta\alpha)]\right )^N$$ right or close?

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