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

In summary, the conversation discusses finding the partition function and expected value of the dipole moment in a cubic lattice. The dipole moment is represented as a unit vector pointing to one of the 8 corners of the system. The mean-field Hamiltonian is given by the product of the dipole moment and the average dipole moment. The speaker is unsure about the direction of the average dipole moment and wonders if they should assume a random direction and proceed with the summation over the 8 possible directions.
  • #1
George444fg
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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
 
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  • #2


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.
 

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

What is the partition function of BaTiO3?

The partition function of BaTiO3 is a mathematical concept used in statistical mechanics to describe the distribution of energy among the different possible states of a system. It is denoted by the symbol Z and is related to the thermodynamic properties of the material.

How is the partition function of BaTiO3 calculated?

The partition function of BaTiO3 can be calculated using the Boltzmann distribution, which takes into account the energy levels and degeneracy of the system. It is also dependent on the temperature and other external factors such as pressure and magnetic field.

What is the significance of the partition function of BaTiO3 in materials science?

The partition function of BaTiO3 is important in materials science as it provides information about the thermodynamic properties of the material. It can be used to calculate quantities such as heat capacity, entropy, and free energy, which are crucial in understanding the behavior of the material under different conditions.

How does the partition function of BaTiO3 change with temperature?

The partition function of BaTiO3 is directly proportional to the temperature. As the temperature increases, the number of accessible energy states also increases, leading to a larger partition function. This relationship is important in predicting the behavior of materials at different temperatures.

What factors can affect the partition function of BaTiO3?

The partition function of BaTiO3 can be influenced by various factors such as temperature, pressure, magnetic field, and the crystal structure of the material. Additionally, any changes in the energy levels or degeneracy of the system can also impact the partition function.

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