Why Does My Spin Chain Energy Calculation Differ from the Textbook?

[jorgen]

Hi all,

I am supposed to calculate the energy of chain of spins where the magnetic field H = 0. For the first chain the spins are all aligned in the same direction - up - hence the energy

U = -NJ

where N is the total number of spins. Next, the half the chain is spin up and the other half is spin down with periodic boundary conditions first and last interact.

UpUpUp...UpDownDown...Down

here I get the total energy U = -(N-2)*J but the book says -(N-4)*J which I really don't understand. Any comments appreciated - thanks in advance

Best Jorgen

 

[alphysicist]

Hi jorgen,

Hi all,

I am supposed to calculate the energy of chain of spins where the magnetic field H = 0. For the first chain the spins are all aligned in the same direction - up - hence the energy

U = -NJ

where N is the total number of spins. Next, the half the chain is spin up and the other half is spin down with periodic boundary conditions first and last interact.

UpUpUp...UpDownDown...Down

here I get the total energy U = -(N-2)*J but the book says -(N-4)*J which I really don't understand. Any comments appreciated - thanks in advance

Best Jorgen

I hope I am understanding the situation you describe, but the book answer makes sense to me. Each location of where there is an up-down neighbor represents an energy change (relative to the up-up or down-down case) of +2J (where J is the interaction energy magnitude between spin neighbors), because it goes from -J to +J. With the periodic boundary conditions, there are two of these locations.

 

[baba89]

Hello Jorgen,

Thank you for sharing your question with us. It seems like you have a good understanding of the energy of a chain of spins with a magnetic field of H = 0. When all the spins are aligned in the same direction, the energy is simply the product of the number of spins and the interaction energy, as you correctly stated.

In the case where half of the chain is spin up and the other half is spin down with periodic boundary conditions, the total energy should indeed be -(N-2)*J. This is because the first and last spins, which are interacting, contribute an energy of -2J, leaving the remaining N-2 spins to contribute an energy of -J each.

It is possible that the book you are referring to made a mistake in their calculation. I would recommend double checking your work and also consulting other sources to confirm the correct energy expression.

I hope this helps and good luck with your calculations! Keep up the good work in your research.

Best,