Cullity B.D. Introduction to Magnetic Materials. Second Edition (2008)

114 DIAMAGNETISM AND PARAMAGNETISM

3.3The susceptibility of FeCl2 obeys the Curie–Weiss law over the temperature range 90 K to room temperature, with u ¼ 48K. Its molecular susceptibility at room temperature is 1.475 1022 emu/Oe/(g mol).

a.What is the effective magnetic moment in Bohr magnetons?

b.What are the spin-only values of J and mH (max)?

c.At an applied field of 8000 Oe, what is the value of the molecular field at 08C and at 1008C?

3.4Show that the Brillouin function B(J, a0) reduces to Equation 3.35 for J ¼ 1, to Equation 3.40 for J ¼ 12, and to Equation 3.45 when a0 is small.

3.5Plot the relation between M/M0 and H, according to quantum theory, for a material with g ¼ 2 and J ¼ 12 for fields up to 10 T and temperatures of 208C and 2K.

a.What is the effective moment?

b.What is the atomic susceptibility at 208C?

c.What percentage of the saturation magnetization is attained at a field of 10 T at 208C?

d.What field is needed to produce 85% of saturation at 2K?

3.6Repeat the plots of Problem , according to classical theory.

a.What is the atomic susceptibility at 208C?

b.What percentage of the saturation magnetization is attained at a field of 10 T at 208C?

c.How can the susceptibilities be the same (for classical and quantum theories) but the saturation percentages be different?

3.7The susceptibility of a-Mn is

The susceptibility of MnCl2 obeys the Curie–Weiss law, with u ¼ 3.0K and an effective moment of 5.7 mB per molecule. Calculate the susceptibility per atom of Mn in a-Mn as a percentage of the susceptibility per atom of Mn in MnCl2.

116 FERROMAGNETISM

Fig. 4.1 Magnetization curves of iron, cobalt, and nickel at room temperature (H-axis schematic). The SI values for saturation magnetization in A/m are 103 times the cgs values in emu/cm3.

Above its Curie temperature Tc a ferromagnet becomes paramagnetic, and its susceptibility then follows the Curie–Weiss law, with a value of u approximately equal to Tc. The value of u is therefore large and positive (over 1000K for iron), and so is the molecular field coefficient. This fact led Weiss to make the bold and brilliant assumption that a molecular field acts in a ferromagnetic substance below its Curie temperature as well as above, and that this field is so strong that it can magnetize the substance to saturation even in the absence of an applied field. The substance is then self-saturating, or “spontaneously magnetized.” Before we consider how this can come about, we must note at once that the theory is, at this stage, incomplete. For if iron, for example, is self-saturating, how can we explain the fact that it is quite easy to obtain a piece of iron in the unmagnetized condition?

Weiss answered this objection by making a second assumption: a ferromagnet in the demagnetized state is divided into a number of small regions called domains. Each domain is spontaneously magnetized to the saturation value Ms, but the directions of magnetization of the various domains are such that the specimen as a whole has no net magnetization. The process of magnetization is then one of converting the specimen from a multi-domain state into one in which it is a single domain magnetized in the same direction as the applied field. This process is illustrated schematically in Fig. 4.2. The dashed line in Fig. 4.2a encloses a portion of a crystal in which there are parts of two domains; the boundary separating them is called a domain wall. The two domains are spontaneously magnetized in opposite directions, so that the net magnetization of this part of the crystal is zero. In Fig. 4.2b a field H has been applied, causing the upper domain to grow at the expense of the lower one by downward motion of the domain wall, until in Fig. 4.2c the wall has moved right out of the region considered. Finally, at still higher applied fields, the magnetization rotates into parallelism with the applied field and the material is saturated, as in Fig. 4.2d. During this entire process there has been no change in the magnitude of the magnetization of any region, only in the direction of magnetization.

The Weiss theory therefore contains two essential postulates: (1) spontaneous magnetization; and (2) division into domains. Later developments have shown that both of these

Fig. 4.4 Effect of temperature on the value of spontaneous magnetization. Curve 1 is the Langevin function.

Therefore the slope of the straight line representing the molecular field is, at any temperature,

But the slope of the line determines the point of intersection P with the Langevin curve and hence the value of Ms/M0. Therefore Ms/M0 is determined solely by the ratio T/Tc. This means that all ferromagnetic materials, which naturally have different values of M0 and Tc, have the same value of Ms/M0 for any particular value of T/Tc. This is sometimes called the law of corresponding states.

This statement of the law is very nearly, but not exactly, correct. In arriving at the Langevin law in Equation 3.13, we considered the number n of atoms per unit volume and set nm ¼ M0. But n changes with temperature because of thermal expansion. Therefore, values of M=M0 at different temperatures are not strictly comparable, because they refer to different numbers of atoms. When dealing with magnetization as a function of temperature, a more natural quantity to use is the specific magnetization s, which is the magnetic moment per unit mass, because then thermal expansion does not affect the result.

If ng is the number of atoms per gram and m¯ the average component of magnetic moment in the direction of the field, then we can write Equation 3.13 as

If we then define, for a ferromagnetic material, ss and s0 as the saturation magnetizations at TK and 0K, respectively, an exact statement of the law of corresponding states is that all

Fig. 4.6 Curie points (Tc), recrystallization temperatures [R], and phase changes in Fe, Co, and Ni. BCC ¼ body-centered cubic; FCC ¼ face-centered cubic; HCP ¼ hexagonal close packed. Ni is FCC at all temperatures.

which heavily cold-worked specimens will recrystallize; thus iron and cobalt can be recrystallized while still ferromagnetic, but nickel cannot. The three curves of Fig. 4.5 have similar shapes and, when the data are replotted in the form of ss/s0 vs T/Tc as in Fig. 4.7, the points conform rather closely to a single curve. Thus Weiss prediction of a law of corresponding states is verified. However, the shape of the curve of ss/s0 vs T/Tc predicted by the Weiss–Langevin theory does not agree with experiment. We can see this by finding graphically the points of intersection of the curve of Equation 4.7 and the lines of Equation 4.12 for various values of T/Tc. The result is shown by the curve labeled “classical, J ¼ 1” in Fig. 4.7. This disagreement is not surprising, inasmuch as we have already seen in Chapter 3 that the classical Langevin theory, which was the only theory available for Weiss to test in this manner, does not conform to experiment.

The Weiss theory may be modernized by supposing that the molecular field acts on a substance having a relative magnetization determined by a quantum-mechanical Brillouin function B(J, a0), as discussed in Chapter 3. In terms of specific magnetization, we have

where a0 ¼ mHH/kt from Equation 3.40. The straight line representing the molecular field is given by

which can be solved for ss/s0 as a function of T/Tc. The theoretical curves for J ¼ 12 and J ¼ 1 are plotted in Fig. 4.7. Either one is in fairly good agreement with experiment, with the curve for J ¼ 12 perhaps slightly better.

If J equals 12, the magnetic moment is due entirely to spin, the g factor is 2, and there is no orbital contribution. That this condition is closely approximated by ferromagnetic substances is also suggested by experimental values of g. Table 4.1 lists observed g factors for Fe, Co, Ni, and several ferromagnetic alloys, and they are all seen to be close to 2. We may therefore conclude that ferromagnetism in transition metals is due essentially to electron spin, with little or no contribution from the orbital motion of the electrons. At 0K the spins on all the atoms in any one domain are parallel and, let us say, “up.” At some higher temperature, a certain fraction of the total, determined by the value of the Brillouin function at that temperature, flips over into the “down” position; the value of that fraction determines the value of ss.

Up to this point we have put the applied field equal to zero and considered only the effect of the molecular field. If a field H is applied, the total field acting on the substance is (H þ Hm), where by H we mean the applied field corrected for any demagnetizing effects.

University Press (1997), p. 69.