I get that the spins can interact not just the nearest neighbour in the general sense. But why in Ising's paper when he solves the linear chain model, he consider spins can only interact with nearest neighbour spins? In his paper, he got mention the reasons, but I don't quite understand as I don't understand German. Why did he say that the forces exert by the spins fade with distance?
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2 Answers
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You can think of each little spin as being a small little magnetic dipole. The magnetic field of a dipole reads: $$ {\bf B} = \frac{\mu_0}{4\pi}(\frac{3{\bf r}({\bf{m}}\cdot{\bf{r}})}{r^5}-\frac{{\bf m}}{r^3}) $$ The first term vanishes in the plane perpendicular to the dipole. You can see that the resulting field drops off in strength cubically with distance. If the spins are arranged at equal intervals, then the interaction energy coming from next to nearest neighbors will be 1/8 of that coming from nearest neighbors. Since this is an order of magnitude smaller, it is reasonable to neglect everything but nearest neighbor interactions.
Also, it was not Ising who decided that he should work on this particular model. The model was given to him by his Advisor, Wilhelm Lenz.
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While the original publication is in German, a translation has been made available. The key statements for your questions are:
- [Weiss explanation] proposes electrical dipole effects for the effects of the individual elements (= elementary magnets). But then very considerable electrical field strengths would result through the summation of very slowly decreasing dipole fields which would be destroyed by the conducting power of the material.
- Therefore, we propose, in contrast to Weiss, that the forces which the elements exert upon each other quickly fade with distance so that, in a first approach, only neighboring atoms influence each other.
- We want to apply these propositions to a model as simple as possible.
