Heisenberg Model


by CNX
Tags: heisenberg, model
CNX
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#1
Nov20-10, 07:36 PM
P: 28
1. The problem statement, all variables and given/known data


Find density of states

[tex]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
[itex]S^z[/itex] = 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>[/tex]


3. The attempt at a solution

{let h = h-bar}

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

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

for S = 0, [itex]m_s = 0[/itex]; for S = 1, [itex]m_s = -1,0,1[/itex]

[tex]= \Pi_i \left (2\exp[-\beta \gamma] + \exp [\beta(h\alpha-\gamma)] + \exp[-\beta(h\alpha+\gamma)]\right )[/tex]

[tex] = \left (2\exp[-\beta \gamma] + \exp [\beta(h\alpha-\gamma)] + \exp[-\beta(h\alpha+\gamma)]\right )^N[/tex]
So,

[tex]Z = \left (2\exp[-\beta \gamma] + 2\exp [-\beta\gamma]\cosh(h\beta\alpha)]\right )^N[/tex]

right or close?
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