Question

The Ising approximation - what exactly is it?

Answer 1

I am slightly confused about the nature of the Ising model to study ferromagnetism. Consider the Heisenberg Hamiltonian with Zeeman term:

H=−12∑i≠jJijSi⋅Sj+gμBB∑iSziH=−12∑i≠jJijSi⋅Sj+gμBB∑iSiz

This can also be written as:

H=−12∑i≠jJijSzi⋅Szj+gμBB∑iSzi−12∑i≠jJijS−iS+jH=−12∑i≠jJijSiz⋅Sjz+gμBB∑iSiz−12∑i≠jJijSi−Sj+

Ashcroft and Mermin define the Ising model to be simply the exclusion of the last term in the above expansion. Whilst other sources (e.g. Simon, 2013) say that it is saying the atoms are in spin states +S+S or −S−S. These do not appear equivalent - not even in mean field approximation where (correct me if I am wrong) the last term vanishes anyway due to rotational symmetry. Thus which is the standard definition and are they linked in any way

hope this helps....

Answer 2

sting theory should be done s+1,s+2................
so derived from this the formula s has been taken.

H=−12∑i≠jJijSi⋅Sj+gμBB∑iSziH=−12∑i≠jJijSi⋅Sj+gμBB∑iSiz



H=−12∑i≠jJijSzi⋅Szj+gμBB∑iSzi−12∑i≠jJijS−iS+jH=−12∑i≠jJijSiz⋅Sjz+gμBB∑iSiz−12∑i≠jJijSi−Sj+