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

at 208C. Therefore, writing M0 (molecular weight) instead of A (atomic weight) in Equation 3.17 because the constituents of oxygen are molecules, we have

m ¼ 3M0kTx 1=2 N

¼ (3)(32 g=mole)(1:38 10 16 erg=K)(293K)(1:08 10 4 emu=g Oe) 1=2(cgs) 6:02 1023 molecules=mole

¼ (3)(0:032 kg=mole)(1:38 10 23 J=K)(293 K)(1:36 10 6(J=T)=(kg Am 1) 1=2(SI) (6:02 1023 molecules=mole)(4p 10 7 T=Am 1)

¼2:64 10 20 erg=Oe per molecule (cgs)

¼2:64 10 23 Am2=molecule or J=T per molecule (SI)

and dividing by the value of the Bohr magneton mB.

This value of m is typical. Even in heavy atoms or molecules containing many electrons, each with orbital and spin moments, most of the moments cancel out and leave a net magnetic moment of only a few Bohr magnetons.

We can now calculate a and justify our assumption that it is small. Typically, H is about 10,000 Oe or 1T or 800 kA/m in susceptibility measurements. Therefore, at room temperature,

a ¼ mH ¼ (2:64 10 20 erg=Oe)(104Oe) kT

¼ 0:0065,

which is a value small enough so that the Langevin function L(a) can be replaced by a/3. The effect of even very strong fields in aligning the atomic moments of a paramagnetic is very feeble compared to the disordering effect of thermal energy at room temperature. For example, there are N/32 oxygen molecules per gram, each with a moment of

of saturated iron. But the magnetization acquired in a field as strong as 100,000 Oe or 10 T or 8 MA/m at room temperature is only s ¼ xH ¼ (1.08 1024)(105) ¼ 10.8 emu/g (cgs) ¼ (1.36 1026)(8 106) ¼ 10.9 Am2/kg (SI), about 2% of the saturation value.

There is nothing in our previous discussion of the diamagnetic effect to indicate that it was restricted to atoms with no net magnetic moment. In fact, it is not; the diamagnetic effect occurs in all atoms, whether or not they have a net moment. A calculation of the susceptibility of a paramagnetic should therefore be corrected by subtracting the diamagnetic contribution from the value given by Equation 3.17. This correction is usually small (of the

Fig. 3.4 Variation of mass susceptibility with absolute temperature for paraand diamagnetics.

Therefore, u (¼ rCg) is a measure of the strength of the interaction because it is proportional to the molecular field constant g. For substances that obey Curie’s law, u ¼ g ¼ 0.

Figure 3.4 shows how x varies with T for paraand diamagnetics. If we plot 1/x versus T for a paramagnetic, a straight line will result; this line will pass through the origin (Curie behavior) or intercept the temperature axis at T ¼ u (Curie–Weiss behavior). Data for two paramagnetics which obey the Curie–Weiss law are plotted in this way in Fig. 3.5, and we note that both positive and negative values of u are observed, positive for MnCl2 and negative for FeSO4. Many paramagnetics obey the Curie–Weiss law with small values of u, of the order of 10 K or less. A positive value of u, as illustrated in Fig. 3.5, indicates that the molecular field is aiding the applied field and therefore tending to make the elementary magnetic moments parallel to one another and to the applied field. Other things being equal, the susceptibility is then larger than it would be if the molecular field were absent. If u is negative, the molecular field opposes the applied field and tends to decrease the susceptibility.

It is important to note that the molecular field is in no sense a real field; it is rather a force, which tends to align or disalign the atomic or molecular moments. The strength of this force depends on the amount of alignment already attained, because the molecular field is proportional to the magnetization. Further discussion of the molecular field will be deferred to the next chapter.

Early in this section it was stated that the effect of an applied field on the atomic or molecular “magnets” was to turn them toward the direction of the field. This statement requires qualification, because the effect of the field is not just a simple rotation, like that of a compass needle exposed to a field not along its axis. Instead, there is a precession of the atomic moments about the applied field, because each atom possesses a certain

98 DIAMAGNETISM AND PARAMAGNETISM

Fig. 3.5 Reciprocal mass susceptibilities of one diamagnetic and two paramagnetic compounds. Note change of vertical scale at the origin.

amount of angular momentum as well as a magnetic moment. This behavior is analogous to that of a spinning top. If the top in Fig. 3.6a is not spinning, it will simply fall over because of the torque exerted by the force of gravity F about its point of support A. But if the top is spinning about its axis, it has a certain angular momentum about that axis; the resultant of the gravitational torque and the angular momentum is a precession of the axis of spin about the vertical, with no change in the angle of inclination u. In an atom, each electron has

Fig. 3.6 Precession of (a) a spinning top in a gravitational field, and (b) a magnetic atom in a magnetic field.

100 DIAMAGNETISM AND PARAMAGNETISM

between the two, first for orbital and then for spin moments. The orbital magnetic moment for an electron in the first Bohr orbit is, from Equation 3.4,

The angular momentum due to spin is sh=2p where s is a quantum number equal to 12. Therefore, from Equation 3.5,

Therefore the ratio of magnetic moment to angular momentum for spin is twice as great as it is for orbital motion. The last two equations can be combined into one general relation between magnetic moment m and angular momentum p by introducing a quantity g:

where g ¼ 1 for orbital motion and g ¼ 2 for spin. The factor g is called (for historical reasons) the spectroscopic splitting factor, or g factor.

In an atom composed of many electrons the angular momenta of the variously oriented orbits combine vectorially to give the resultant orbital angular momentum of the atom, which is characterized by the quantum number L. Similarly, the individual electron spin momenta combine to give the resultant spin momentum, described by the quantum number S. Finally, the orbital and spin momenta of the atom combine to give the total angular momentum of the atom, described by the quantum number J. Then the net magnetic moment of the atom, usually called the effective moment meff , is given in terms of g and J, as we might expect by analogy with Equation 3.25. The relation is

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corresponding resolved moments mH are then mB and 2mB, parallel and antiparallel to the field, as illustrated in Fig. 3.7b.

2.J ¼ 1. Here there are an infinite number of J values, corresponding to an infinite number of moment orientations. This is equivalent to the classical distribution of Fig. 3.7a.

To compute mH or meff we must know g, as well as J, for the atom in question. The g factor is given by the Lande´ equation:

If there is no net orbital contribution to the moment, L ¼ 0 and J ¼ S. Then Equation 3.29 gives g ¼ 2 whatever the value of J. On the other hand, if the spins cancel out, then S ¼ 0, J ¼ L, and g ¼ 1. The g factors of most atoms lie between 1 and 2, but values outside this range are sometimes encountered.

At this point the calculation of the net magnetic moment of an atom would seem straightforward, simply by a combination of Equations 3.29 and 3.28. However, the values of J, L, and S are known only for isolated atoms; it is, in general, impossible to calculate m for the atoms of a solid, unless certain simplifying assumptions are made. One such assumption, valid for many substances, is that there is no orbital contribution to the moment, so that J ¼ S. The orbital moment is, in such cases, said to be quenched. This condition results from the action on the atom or ion considered of the electric field, called the crystalline or crystal field, produced by the surrounding atoms or ions in the solid. This field has the symmetry of the crystal involved. Thus the electron orbits in a particular isolated atom might be circular, but when that atom forms part of a cubic crystal, the orbits might become elongated along three mutually perpendicular axes because of the electric fields created by the adjoining atoms located on these axes. In any case, the orbits are in a sense bound, or “coupled,” rather strongly to the crystal lattice. The spins, on the other hand, are only loosely coupled to the orbits. Thus, when a magnetic field is applied along some arbitrary direction in the crystal, the strong orbit–lattice coupling often prevents the orbits, and their associated orbital magnetic moments, from turning toward the field direction, whereas the spins are free to turn because of the relatively weak spin–orbit coupling. The result is that only the spins contribute to the magnetization process and the resultant magnetic moment of the specimen; the orbital moments act as though they were not there. Quenching may be complete or partial.

Fortunately, it is possible to measure g for the atoms of a solid, and such measurements tell us what fraction of the total moment, which is also measurable, is contributed by spin and what fraction by orbital motion. Experimental g factors will be given later.

3.7.1Gyromagnetic Effect

Two entirely different kinds of experiments are available for the determination of g. The first involves the gyromagnetic effect, which depends on the fact that magnetic moments and angular momenta are coupled together; whatever is done to change the direction of one will change the direction of the other. From the magnitude of the observed effect a quantity g0, called the magnetomechanical factor or g0 factor, can be calculated. The g

factor can then be found from the relation

If the magnetic moment is due entirely to spin, then g ¼ g0 ¼ 2. If there is a small orbital contribution, g is somewhat larger than 2 and g0 somewhat smaller.

Two methods of measuring the gyromagnetic effect, and thus the value of g0, have been successful:

1.Einstein–de Haas Method. A rod of the material to be investigated is suspended vertically by a fine wire and surrounded by a magnetizing solenoid. If a field is suddenly applied along the axis of the rod, the atomic moments will turn toward the axis. But this will also turn the angular momentum vectors toward the axis. Since angular momentum cannot be created except by external torques, this increase in the axial component of momentum of the atoms must be balanced by an increase, in the oppo-

site direction, of the momentum of the bar as a whole. The result is a rotation of the bar through a very small angle, from which the value of g0 can be computed. The

experiment is extremely difficult with a ferromagnetic rod, and even more so with a paramagnetic, because the size of the observed effect depends mainly on the magnetization that can be produced in the specimen.

2.Barnett Method. The specimen, again in the form of a rod, is very rapidly rotated about its axis. The angular momentum vectors therefore turn slightly toward the

axis of rotation and cause the magnetic moments to do the same. The rod therefore acquires a very slight magnetization along its axis, from which g0 can be calculated.

The two methods may be described as “rotation by magnetization” and “magnetization by rotation.” Detailed accounts of both are given by L. F. Bates in Modern Magnetism, Cambridge University Press (1961). These are difficult experiments, and the results of various authors do not always agree. The available values were almost all published before 1950.

3.7.2Magnetic Resonance

The second kind of experiment is magnetic resonance, which measures g directly. The specimen is placed in the strong field Hz of an electromagnet, acting along the z-axis. It is also subjected to a weak field Hx acting at right angles, along the x-axis; Hx is a highfrequency alternating field generally in the microwave region near 10 GHz. The atomic moments precess around Hz at a rate dependent on g and Hz. Energy is absorbed by the specimen from the alternating field Hx, and, if the intensity of Hz or the frequency n of Hx is slowly varied, a point will be found at which the energy absorption rises to a sharp maximum. In this resonant state the frequency n equals the frequency of precession, and both are proportional to the product gHz, from which g may be calculated.

Assuming that g and J are known for the atoms involved, we can proceed to calculate the total magnetization of a specimen as a function of the field and temperature. The procedure is the same as that followed in deriving the classical (Langevin) law, except that:

1.The quantized component of magnetic moment in the field direction mH(¼ gMJ mB) replaces the classical term m cos u.