A ferromagnet is a material (like iron) that magnetizes spontaneously, even in the absence of an externally applied magnetic field. This happens because each elementary dipole has a strong tendency to align parallel to its neighbors. At the magnetization of a ferromagnet has the maximum possible value, with all dipoles perfectly lined up; if there are atoms, the total magnetization is typically, where 碌a is the Bohr magneton. At somewhat higher temperatures, the excitations take the form of spin waves, which can be visualized classically as shown in Figure . Like sound waves, spin waves are quantized: Each wave mode can have only integer multiples of a basic energy unit. In analogy with phonons, we think of the energy units as particles, called magnons. Each magnon reduces the total spin of the system by one unit of and therefore reduces the magnetization by . However, whereas the frequency of a sound wave is inversely proportional to its wavelength, the frequency of a spin-wave is proportional to the square of .. (in the limit of long wavelengths). Therefore, sinceand .. for any "particle," the energy of a magnon is proportional
In the ground state of a ferromagnet, all the elementary dipoles point in the same direction. The lowest-energy excitations above the ground state are spin waves, in which the dipoles precess in a conical motion. A long-wavelength spin wave carries very little energy because the difference in direction between neighboring dipoles is very small.
to the square of its momentum. In analogy with the energy-momentum relation for an ordinary nonrelativistic particle, we can write *, where* is a constant related to the spin-spin interaction energy and the atomic spacing. For iron, m* turns out to equal , abouttimes the mass of an electron. Another difference between magnons and phonons is that each magnon ( or spin-wave mode) has only one possible polarization.
(a) Show that at low temperatures, the number of magnons per unit volume in a three-dimensional ferromagnet is given by
Evaluate the integral numerically.
(b) Use the result of part () to find an expression for the fractional reduction in magnetization, Write your answer in the form , and estimate the constantfor iron.
(c) Calculate the heat capacity due to magnetic excitations in a ferromagnet at low temperature. You should find , where differs from To only by a numerical constant. Estimatefor iron, and compare the magnon and phonon contributions to the heat capacity. (The Debye temperature of iron is .)
(d) Consider a two-dimensional array of magnetic dipoles at low temperature. Assume that each elementary dipole can still point in any (threedimensional) direction, so spin waves are still possible. Show that the integral for the total number of magnons diverge in this case. (This result is an indication that there can be no spontaneous magnetization in such a two-dimensional system. However, in Section we will consider a different two-dimensional model in which magnetization does occur.)
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