Ferromagnetism

The spins of a regular Ising lattice interact by the energy

(i)

where B is an external field, μ is the magnetic moment, and the prime indicates that the summation is only over the nearest neighbors. Each spin σ_i has z nearest neighbors. The spins are restricted to equal ±1. The coupling constant J is positive. Following Weiss, represent the effect on σ_i of the spin–spin interaction in (i) by the mean field set up by the neighboring spins σ_j. Calculate the linear spin susceptibility χ(τ) using this mean field approximation. Your expression should diverge at some temperature τ = τ_c. What is the physical significance of this divergence? What is happening to the spin lattice at τ = τ_c?

SOLUTION

Using the mean field approximation, we may write the magnetization M of the lattice as

(1)

where n is the density of the spins and B_eff is the sum of the imposed field and the field at spin σ_i produced by the neighboring spins:

(2)

where λ is a constant. We may rewrite (1) as

(3)

The susceptibility χ is given by

(4)

For B and M small we may rearrange (4), yielding

(5)

where τ_c ≡ λnμ². The divergence of χ at τ = τ_c indicates the onset of ferromagnetism. The spins will align spontaneously in the absence of an applied magnetic field at this temperature.

مواضيع ذات صلة

مصطلحات الفيزياء النووية

إزاحة حمراء REDSHIFT

المركبات النانوية (Nanocomposites)

مولد فان دي جراف الالكتروستاتي

التطويرات التقنية الحديثة في المعجلات الالكتروستاتية

معجلات الالكترونات الالكتروستاتية

معجلات الفان دي جراف الترادفية

الحد من زيادة الطاقة في معجل فان دي جراف

أنبوبة التعجيل قي معجلات الجسيمات من نوع فان دي جراف

ارتكاز معجل فان دي جراف

مبدأ تشغيل معجل الفان دي جراف

القضايا الأخلاقية المتعلقة بتكنولوجيا النانو