## Heisenberg Model

1. The problem statement, all variables and given/known data

Find density of states

$$H = \frac{-JzM}{g\mu_B} \sum_i S_i^z + \frac{JzNM^2}{2g^2\mu_b^2} = -\alpha \sum_i S_i^z + \gamma[/itex] z = # nearest neighbors J = exchange M = magnetization $S^z$ = project of total spin S=0,1. 2. Relevant equations [tex]Z=\sum_{S m_s} <S m_s| \exp(-\beta H) |S m_s>$$

3. The attempt at a solution

{let h = h-bar}

$$Z = \sum_{S m_s} <S m_s| \exp \left[-\beta(-\alpha \sum_i S_i^z + \gamma) \right] |S m_s>$$

$$= \Pi_i \sum_{S m_s} \left [ \exp(\beta\alpha h m_s^i - \beta\gamma) \right]}$$

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?
 PhysOrg.com science news on PhysOrg.com >> King Richard III found in 'untidy lozenge-shaped grave'>> Google Drive sports new view and scan enhancements>> Researcher admits mistakes in stem cell study