Find Density of States with Heisenberg Model

[CNX]

Homework Statement

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
$S^z$ = project of total spin S=0,1.

Homework Equations

$$Z=\sum_{S m_s} <S m_s| \exp(-\beta H) |S m_s>$$

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?

 

[mark726]

Your solution looks close, but there are a few things that could be improved. Here are some suggestions:

1. It would be helpful to define all of your variables (e.g. \beta, \gamma, h) at the beginning of your solution so the reader knows what they represent.

2. In your first step, you use the expression for Z from the homework equations, but you don't actually use the Hamiltonian given in the problem. Instead, you use a modified Hamiltonian with the constants \alpha and \gamma. It would be more accurate to use the original Hamiltonian and then manipulate it to get to your final expression for Z.

3. In the second step, you have an extra bracket around the exponential expression. It should just be \exp(\beta \alpha h m_s^i - \beta \gamma).

4. In the third step, you have an extra 2 in front of the first exponential term. This should just be \exp(-\beta \gamma) without the 2.

5. In your final expression for Z, you should have a factor of z (the number of nearest neighbors) instead of N (the total number of spins). This is because each spin only interacts with z other spins, not all N spins.

With these changes, your final expression for Z should be:

Z = \left (2\exp[-\beta \gamma] + \exp [\beta(h\alpha-\gamma)] + \exp[-\beta(h\alpha+\gamma)]\right )^z

I hope this helps!