Temporary HW: Difference between revisions
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<math> G(r) = \langle s_k s_{k+r} \rangle - \langle s_k \rangle \langle s_{k+r} \rangle </math> | <math> G(r) = \langle s_k s_{k+r} \rangle - \langle s_k \rangle \langle s_{k+r} \rangle </math> | ||
where <math> r </math> is the separation between the two spins in units of the lattice constant. The statistical average <math> \langle \ldots \rangle </math> is over all microstates. Because all lattice sites are equivalent, <math> G(r) </math> is independent of the choice of specific site <math> k </math> and depends only on the separation r (for a given temperature T and external field H). Since the average value of spin <math> \langle s_k \rangle </math> at site <math> k </math> is independent of the choice of that site (for periodic boundary conditions) and equals <math> m=M/N </math>, where <math> M <math> is magnetization, the correlation function can also be written as: | where <math> r </math> is the separation between the two spins in units of the lattice constant. The statistical average <math> \langle \ldots \rangle </math> is over all microstates. Because all lattice sites are equivalent, <math> G(r) </math> is independent of the choice of specific site <math> k </math> and depends only on the separation r (for a given temperature T and external field H). Since the average value of spin <math> \langle s_k \rangle </math> at site <math> k </math> is independent of the choice of that site (for periodic boundary conditions) and equals <math> m=M/N </math>, where <math> M </math> is magnetization, the correlation function can also be written as: | ||
<math> G(r) = \langle s_k s_{k+r} \rangle - m^2 </math>. | <math> G(r) = \langle s_k s_{k+r} \rangle - m^2 </math>. |
Revision as of 14:25, 7 April 2011
Problem 1: Three Ising spins
Assume three spins are arranged in an equilateral triangle with each spin interacting with its two neighbors. The energy expression for the Ising model in an external magnetic field is give by:
(a) Find canonical partition function for this model.
(b) Find average spin .
(c) Find internal energy .
Problem 2: Spin-spin correlation function in the Ising model
We can gain further insight into the properties of the one-dimensional Ising model of ferromagnetism
by calculating the spin-spin correlation function
where is the separation between the two spins in units of the lattice constant. The statistical average is over all microstates. Because all lattice sites are equivalent, is independent of the choice of specific site and depends only on the separation r (for a given temperature T and external field H). Since the average value of spin at site is independent of the choice of that site (for periodic boundary conditions) and equals , where is magnetization, the correlation function can also be written as:
.
The spin-spin correlation function is a measure of the degree to which a spin at one site is correlated with a spin at another site. If the spins are not correlated, then trivially . At high temperatures the interaction between spins is unimportant, and hence the spins are randomly oriented in the absence of an external magnetic field. Thus in the limit , we expect that Failed to parse (syntax error): {\displaystyle G(r) → 0 for any r. For fixed T and H, we expect that, if spin k is up, then the two adjacent spins will have a greater probability of being up than down. For spins further away from spin k, we expect that the probability that spin k + r is up or correlated will decrease. Hence, we expect that G(r) → 0 as r → ∞. The magnetic susceptibility is proportional to <math> G(r=0) } , spins with free boundary conditions in equilibrium with a heat bath at temperature and in zero magnetic field. Enumerate all microstates and calculate and for k = 1, the first spin on the left.