# Mechanism before Ising model

1. Mar 4, 2008

### jostpuur

The Ising model starts with an assumption that the nearby spins reach lower potential by pointing at the same direction. However, two magnets would reach lower potential by pointing at different directions. What is the mechanism, that in the first place makes spins attempt to align themselves in the same directions, in some materials?

2. Mar 27, 2008

### rayohauno

I´m not an expertise on the field, but there are ferro interactions (spins tend to align), and anti ferro interactions.
ferromagnetism its a quantum mechanical effect. classically it can´t be explained. I quote to wiki:

According to classical electromagnetism, two nearby magnetic dipoles will tend to align in opposite directions (which would create an antiferromagnetic material). In a ferromagnet, however, they tend to align in the same direction because of the Pauli principle: two electrons with the same spin cannot also have the same "position", which effectively reduces the energy of their electrostatic interaction compared to electrons with opposite spin. (Mathematically, this is expressed more precisely in terms of the spin-statistics theorem: because electrons are fermions with half-integer spin, their wave functions are antisymmetric under interchange of particle positions. This can be seen in, for example, the Hartree-Fock approximation to lead to a reduction in the electrostatic potential energy.) This difference in energy is called the exchange energy.

look at wiki:
http://en.wikipedia.org/wiki/Ferromagnetism

best regards
rayo

Last edited: Mar 27, 2008
3. Mar 28, 2008

### jostpuur

In fact I found this wiki page some time after starting this thread. This is the part that didn't make sense:

Wait a minute! Two electrons will repel each other anyway. If they have the same spins, it becomes even more difficult to push them close, because exclusion principle produces some effective force. So effectively, having the same spins, should make the electrostatic interaction stronger? The Wikipedia seems to be explaining this precisely the opposite way

Last edited: Mar 28, 2008
4. Mar 28, 2008

### rayohauno

2 electrons that have (nearby) the ¨same position¨ (i put it under quotes, because it is not the classical position, in fact it is about the same quantum orbital state) will have very high energy (because electrons tend to repel each other because they have the same electric charge). so diferent positions will lower the energy, and this (is forced to) happen when they have the same spin (because the pauli exclusion principle), so, one of the electrons change its orbit to a higher energy one (for itself), but overall system energy system fall dawn.
but anyway this is a kind of classical interpretation, the better one is the quantum:

http://en.wikipedia.org/wiki/Exchange_energy

so essentially u must consider 4 states and its energy levels (here for simplicity I consider only 2 electrons (the las two outer free electrons of some kind of atoms), but the analogous calculations can be dozen for more complicated atomic (molecular) configurations):

1. same orbit, different spins (may be anti ferromagnetism)
2. different orbit, different spins (this is not ferromagnetism)
3. different orbit, same spins (this is this case, ferromagnetism)
4. same orbit, same spins (forbidden by the exclusion principle)

why system goes to state 3 and not to state 2 ?? it seems that 1 has lower energy than 2. so i believe that
state 2 is unstable, it falls (falls because energy falls dawn) to state 1. because 1 has lower energy than 2.
so when spins are in opposite directions, lower state energy its 1. when spins are in same direction, lower
state energy its 3.

why some systems goes to 3, and others to 1 ?? that is... when 3 has lower energy than 1 or the converse ??
I think that u must do specific calculation about the kind of atom (material) u have in consideration. u must carry out the hartree-fock aproximations. so u will have the exchange energy (and the orbital_energy see below). so in one side (3) u have the energy cost because one electron change its orbit to a more energetical one (lets call this, orbital_energy). in the other side (1) both electrons remains in the same orbit, but u have to pay the cost of the exchange energy because both electrons have the same spins.

so if orbital_energy > exchange energy then state 1 has the lower energy and then it is the state.
so if orbital_energy < exchange energy then state 3 has the lower energy and then it is the state.

here orbital_energy is the energy cost to pay because u have 2 electrons in different orbits (here this energy cost is positive because instead of have 2 electrons in the lowest orbital energy level available u have one of them in one level above).

here exchange energy is the energy cost to pay because u have 2 electrons in same spin orientation (this
amount of energy depends on the orbital state as is shown in wiki, hartree-fock aproximation).

best regards
rayo

Last edited: Mar 28, 2008
5. Mar 28, 2008

### kdv

You should not think of the exclusion principle as "adding" to th erepulsion.

Think of it this way. First, ignore electrostatic repulsion. because of the exclusion pricniple (because of the exchange force if you want), electrons with the same spin will be found further away from each other than electrons in antisymmetric spin states, right? Now consider the electrostatic repulsion. Since the symmetric spin state electrons are on average further away from each other, their electrostatic energy will be smaller

6. Mar 28, 2008

### jostpuur

I see. But now it goes like this: First decide what effect we want. Then choose which way to think about the exclusion principle.

I believe that's the qualitative part of the explanation. How quantitative does it get? Is it understood why some materials are ferromagnetic and others are not?

7. Mar 28, 2008

### jostpuur

Another question. Is it clear what particles cause the magnetization? Electrons for sure? The atom nucleus have spins too. And about electrons... are they all mobile, or are some of the electrons on bound states in some atoms? This is relevant, because we are dealing with the distances of the magnetization causing particles.

8. Mar 28, 2008

### rayohauno

of course, l have been writed a lot... u just do the calculations, ti depends on the material, and then u have the values of orbital cost energies and exchange energies

best regards
rayo

Last edited: Mar 28, 2008