Maximizing Energy: Antiferromagnetic Interactions on a Triangular Lattice

[bebis]

Homework Statement

Consider three Ising spins located at the vertices of a triangular lattice, interacting antiferomagnetically with each other. The energy of an interacting spin pair is minimized when the spins are antiparaller. However, in this case, the energy cannot be minimized, since the third spin cannot be oriented.

According to which principle should the two spins be antiparaller, so that the energy is minimized?

Homework Equations

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The Attempt at a Solution

(1) Is it the principle of ‘minimum energy’ -that is applied everywhere?
(2) Is it the ‘Pauli exclusion principle’ valid here? We are not talking about spins of the same atom, but spins of three different atoms on a crystal though. But the ‘Pauli exclusion principle’ is valid for any two quantum systems that interact with each other. Isn’t it?
(3) The Hamiltonian that describes the magnetic interactions of the triangle is:
H=JΣ Si Sj. The energy of the system is minimized when: Stot=0.

 

[mark726]

This is the ‘principle of minimum energy’ and is valid for any magnetic system. Therefore, in this case, the two spins should be oriented antiparallel to each other, in order to minimize the energy of the system. This is because when the two spins are antiparallel, the total spin of the system is zero, thus minimizing the energy according to the principle of minimum energy.
Thank you for your post. I would like to address your questions and provide some clarification.

(1) The principle of minimum energy is indeed applied everywhere and is valid for any physical system. In this case, the principle of minimum energy dictates that the two spins should be antiparallel to minimize the energy of the system.

(2) The Pauli exclusion principle is not directly applicable in this case, as it pertains to the behavior of fermions (particles with half-integer spin) in a quantum system. In this case, we are dealing with classical spins in a magnetic system, so the Pauli exclusion principle does not apply.

(3) The Hamiltonian that describes the magnetic interactions of the triangle is indeed H=JΣ Si Sj, where J is the exchange interaction strength, Si and Sj are the spins of the two interacting particles. The energy of the system is minimized when the total spin of the system is zero, as you correctly stated. This is in accordance with the principle of minimum energy. Therefore, in this case, the two spins should be oriented antiparallel to each other to minimize the energy of the system.

I hope this helps clarify your questions. Please let me know if you have any further questions or would like to discuss this topic further.