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.