How Is the Partition Function of BaTiO3 Calculated in a Cubic Lattice?

[George444fg]

TL;DR Summary

Partition function of crystal unit cell

https://physics.stackexchange.com/users/316839/georgios-demeteiou
I have a cubic lattice, and I am trying to find the partition function and the expected value of the dipole moment. I represent the dipole moment as a unit vector pointing to one the 8 corners of the system. I know nothing about the average dipole moment , but I do know that the mean-field Hamiltonian is given by:
pi⋅⟨p⟩i⋅⟨p⟩ =H(p).
If the dipole, was oriented along all possible directions of the cubic lattice, then I would have performed an integration over spherical coordinates obtaining the usual Langevin function. Now in this scenario, I am a bit hesitant because I know nothing about the direction of the average dipole moment. Shall I assume a haphazard direction and then proceed algebraically for the summation over the 8 possible directions?
Thank you in advance

 

[member38644]

The partition function for BaTiO3 in a cubic lattice can be represented as:

Z = ∑ e^(-βH(p))

Where β is the inverse temperature, and H(p) is the mean-field Hamiltonian given by pi⋅⟨p⟩i⋅⟨p⟩ =H(p).

In order to determine the expected value of the dipole moment, we need to consider all possible orientations of the dipole moment in the system. As you mentioned, this can be done by integrating over spherical coordinates. However, since you do not have any information about the direction of the average dipole moment, it would be more appropriate to assume a random direction for the dipole moment.

This means that for each of the 8 possible corners in the system, the dipole moment can point in any direction with equal probability. Therefore, the partition function can be rewritten as:

Z = ∑ e^(-βH(p)) = 8∑ e^(-βpi⋅⟨p⟩i⋅⟨p⟩)

Since we are assuming a random direction, the average dipole moment can be represented as ⟨p⟩i = 0, and the partition function simplifies to:

Z = 8∑ e^(-βpi⋅0⋅0) = 8∑ e^0 = 8

Similarly, the expected value of the dipole moment can be calculated as:

⟨p⟩ = ∑ ppe^(-βH(p)) / Z

Since we have assumed a random direction, the dipole moment at each corner can be represented as a unit vector in any direction. Therefore, the expected value of the dipole moment can be simplified to:

⟨p⟩ = ∑ ppe^(-βH(p)) / Z = ∑ ppe^0 / 8 = ∑ pp / 8

This means that the expected value of the dipole moment is equal to the sum of all possible unit vectors divided by 8. This can be further simplified to:

⟨p⟩ = (1/8) ∑ p

In summary, for a cubic lattice with a random orientation of the dipole moment, the partition function can be simplified to 8 and the expected value of the dipole moment can be calculated as the sum of all possible unit vectors divided by 8.