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

Fig. 12.9 Ideal switching astroid.

turned to magnetic data storage materials, and to heads for writing and reading digital information. Since about 2000, various new forms of magnetic random access memory (MRAM) have been proposed and developed (see Chapter 15), and the switching behavior of magnetic films is again a subject of research. In the new devices, often a magnetic film is biased to be magnetized in a specific direction by an exchange interaction with an adjacent antiferromagnetic film. As a result the measured switching asteroid does not have its origin at zero field.

12.5TIME EFFECTS

We turn now to a group of phenomena loosely known as “time effects,” which manifest themselves on a time scale ranging from seconds to days. The most important of these effects, which have been given a bewildering variety of names, can be classified as follows:

1.Time decrease (or time decay) of permeability or disaccommodation.

2.Magnetic after-effect or magnetic viscosity. Two different kinds have been observed:

(a)Diffusion after-effect or reversible after-effect or Richter after-effect.

(b)Thermal fluctuation after-effect or irreversible after-effect or Jordan after-effect.

This discussion excludes time-dependent magnetic changes caused by metallurgical phase changes. The magnetic properties of some alloys will change with time if the alloy is in an unstable state, such that a second phase is being precipitated; an example is the aging of steel, in which carbon is withdrawn from solution in the iron to precipitate as cementite Fe3 C. Or a magnetic phase may precipitate from a nonmagnetic alloy, causing pronounced changes in the magnetic properties; an example is the Cu–Co alloy described

422 MAGNETIZATION DYNAMICS

in Section 11.7. Usually such changes are so slow as to be unmeasurable at room temperature, but they may become significant at temperatures of a few hundred degrees Celsius. These effects are not included in this section, which is restricted to time effects occurring in pure substances or single-phase solid solutions.

12.5.1Time Decrease of Permeability

This effect has been most often studied in iron containing interstitial carbon and/or nitrogen in solution. The effect is dependent on these interstitials and does not occur in pure iron. If a specimen of iron containing some carbon in solution is demagnetized with a gradually decreasing alternating field, its initial permeability mi decreases with the time elapsed since demagnetization. The measurement must be made at field levels so low that no long-range domain wall motion occurs. This process is shown schematically in Fig. 12.10a and by actual measurements in Fig. 12.10b. This effect is substantial, in that the drop in permeability can amount to more than 50% of the initial value.

This decrease in permeability is due to the preferential distribution of carbon atoms with respect to the local direction of the magnetization, specifically, in a plane at right angles to the magnetization in iron, as described in Section 10.3. Once this distribution is set up, the domain walls tend to become stabilized. The kind of potential wells caused by the carbon atoms, and the forces required to free domain walls from these wells, have been sketched in Fig. 10.7. During demagnetization, domain walls are in continual motion over large distances, and in any local region there is no preferred direction of magnetization for any length of time and therefore no preferred distribution of carbon atoms; x, y, and z sites are equally populated. When demagnetization ceases, at time t ¼ 0, the permeability is high, because the isotropic carbon distribution does not stabilize domain walls. But the carbon atoms immediately begin to distribute themselves preferentially in each domain, i.e., the material self-magnetically anneals, and potential wells begin to build up with time at each domain wall, somewhat as sketched in Fig. 12.11. The walls become increasingly difficult to move, so the permeability mi decreases.

Fig. 12.10 Time decrease of permeability. (a) Schematic. (b) Measurements on iron containing carbon. [J. L. Snoek, Physica, 5 (1938) p. 663.]

effect occurs when H is suddenly reduced to zero from some higher value; B then decreases slowly to its final value.

It is important to ensure that eddy currents do not interfere with the effect to be measured. Even in materials which do not show an after-effect, B is delayed by eddy currents in reaching its final value, as described in Section 12.2 and as illustrated in an exaggerated way in Fig. 12.12. A rather unrealistic break in the B, t curve is there indicated. Actually the two portions of the curve run smoothly together, so that it may be difficult to establish an origin for the measurement of the after-effect. The eddy-current delay should be made as small as possible, by reducing the specimen thickness.

In iron containing carbon, permeability decay and the diffusion after-effect have the same basic cause: the tendency of carbon atoms to take up preferred positions and thus stabilize domain walls. But the two effects are not the same. To measure permeability decay, we measure a time-dependent change in the ease of magnetization with a small temporarily applied ac field; wall motion is very restricted. To measure the after-effect, we measure a time-dependent change in the magnetization itself under a constant applied field; wall motion proceeds as far as it can. The after-effect is the more complex of the two, because the domain walls are in continuing motion until the effect ceases. When the fast change Bi is over, carbon atoms immediately start to arrange themselves so as to stabilize walls at their new positions. The stabilization is at first weak, and the stillapplied field is able to push the walls on at a fairly high velocity. But the slight slowing down allows the carbon atoms more time for stabilization at the later position of the wall, the effect becomes cumulative, and wall motion finally ceases. Despite the very rapid diffusion rate of carbon at room temperature, wall motion can persist for 10 to 15 sec, simply because of the difficulty of stabilizing a moving wall.

In the simplest case the rate of change of Ba is governed by a single time constant t:

and a plot of log Ba against t will be a straight line. This behavior has been found at very low applied fields, but generally more complex results are obtained, suggesting the presence of a range of relaxation times. S. Chikazumi [Physics of Ferromagnetism, Oxford University Press (1997) Chapter 20] deals with the mathematics of this situation. The problem is that fitting experimental results to a mathematical model with several adjustable parameters provides little physical insight.

The diffusion after-effect is strongly dependent on temperature. The relaxation time or times are proportional to eQ/kT, where Q is the activation energy for diffusion of the interstitials.

The quasi-static experiment of Fig. 12.13 is one way of demonstrating the effect of dissolved carbon, say, on wall motion. Another quite different way is to apply a small alternating field and vary the frequency over a wide range. If the frequency is so low that the carbon atoms always have time to reach their equilibrium positions with respect to the local magnetization, B and H will be in phase and there will be no power loss due to the diffusion after-effect. At the other extreme, if the frequency is so high that the carbon atoms never have time to reach equilibrium positions; again B and H will be in phase. At intermediate frequencies, the carbon atoms are mobile enough to make B lag behind H by a phase angle f. There is then an energy loss per cycle proportional to tan f, which is called the loss factor. If the after-effect is characterized by a single relaxation time t and if we put Bma/Bi ¼ R, then

426 MAGNETIZATION DYNAMICS

where v is the angular frequency, equal to 2p times the frequency in Hz. This function has a maximum value of

R p

2 1 þ R

when

p

v ¼ 1 þ R : t

Because R is of the order of 0.1–0.4, the frequency that maximizes tan f is not much larger than 1/t. Y. Tomono [J. Phys. Soc. Japan, 7 (1952) p. 174] measured the total losses of iron specimens under alternating magnetization at various frequencies and temperatures. By subtracting the losses due to hysteresis and to eddy currents, he obtained the loss factor due to the after-effect. From the observed variation of tan f with v, the values of t and R can be found from the relations given above. At constant frequency, tan f also goes through a maximum as a function of temperature; at 200 Hz, for example, this maximum occurred at about 1008C, while at 4000 Hz it occurred at 1508C. This behavior is due to the temperature dependence of the relaxation time, given by

where Q is the activation energy of the process that controls the rate of the after-effect. At low temperatures, t is so large that the after-effect essentially does not exist (carbon atoms cannot reach equilibrium positions); B and H are in phase. At high temperatures, the after-effect is so fast that B and H are again in phase. At intermediate temperatures, losses will occur, and tan f will have its maximum value when

The temperature at which the maximum in tan f occurs should increase with increasing frequency v, in agreement with experiment. From values of t determined at different temperatures, the value of Q was found to be 0.99 eV/atom or 23 kcal/mol, which is in reasonable agreement with the value of Q for diffusion of carbon in alpha iron (20.1 kcal/mol).

This result gives a satisfying explanation of the observed single relaxation time and establishes carbon diffusion as the rate-controlling process of the after-effect in the reversible range of magnetization. In the irreversible range other processes must play an additional role.

12.5.3Thermal Fluctuation After-Effect

This effect differs from the diffusion after-effect in three respects: (1) it does not require the diffusion of anything and therefore can occur in pure metals or other pure substances; (2) it is much less sensitive to temperature; and (3) it causes a loss factor tan f in alternating fields which does not depend on frequency. The magnitude of the fluctuation after-effect

is larger in magnetically hard than in magnetically soft materials, and has mostly been measured in permanent magnet materials.

The original paper in this field is by R. Street and J.C. Woolley [Proc. Phys. Soc., A62 (1949) p. 562]. They studied an alnico permanent-magnet alloy. The after-effect was measured on the descending part of the hysteresis loop. The abrupt change in field DH was such that the final value of H was near the coercive field, and this DH produced an instantaneous magnetization change DMi of about 200 emu/cm3 o kA/m. This was followed by the after-effect change DM shown in Fig. 12.14, where the curves have been shifted vertically and drawn from arbitrary origins for clarity. Near room temperature, the after-effect change in 20 min is about 35 emu/cm3 or kA/m, about 17% of the instantaneous change. The results of Fig. 12.14, including the temperature dependence, conform to the relation

where a and b are constants. Street and Woolley ascribed the after-effect to fluctuations of thermal energy sufficient to cause irreversible rotations in small volumes of the material. This view is consistent with the current opinion that alnico contains single-domain regions. The probability that the energy of a particular volume will fluctuate by an amount Q above the average is proportional to e2Q/kT. One would therefore expect the after-effect DM to vary in the same way, whereas it is found to vary linearly with T, and would expect the rate of the after-effect d(DM )/dt to vary as e2t/t, whereas actually it varies as 1/t (see Equations 11.24 and 11.27). However, Street and Woolley showed that Equation 12.21 could be accounted for by assuming that the activation energy Q was not

Fig. 12.14 Fluctuation after-effect in alnico [R. Street and J. C. Woolley, Proc. Phys. Soc., A62 (1949) p. 562].

resistance is one, and there may be some friction at the point of support of the spring. But when the greatest care is taken to reduce these sources to a minimum, the oscillations still die out eventually. Evidently, there are some frictional processes within the material of the spring itself. This “internal friction,” or damping, occurs in any vibrating material.

Internal friction may be measured and specified in various ways. The most fundamental is the fractional energy loss per cycle, also called the specific damping capacity:

For the weight on the spring, the value of W is the elastic strain energy stored in the spring at either extreme extension of the cycle. A value of DW/W ¼ 0.1 is considered large, and values as low as 1025 have been measured.

Another way to measure damping is to measure how rapidly the oscillations of the system die out. If An is the amplitude of any one oscillation and Anþ1 that of the next, then the damping is characterized by the logarithmic decrement d, defined by

If d is small, d ¼ DA/A, where DA is the difference between successive amplitudes and A is their mean. If DW/W is 0.1, the amplitude will decrease to one-tenth of its original value in about 46 cycles; if DW/W is 1025, about 460,000 cycles will be required. The vibrational energy W is always proportional to A2, no matter how complex the stress system. For example, suppose a rod is stretched by a stress s to a strain, or amplitude, 1. Then the stored energy ¼ work done ¼ 12s1 ¼ 12(1E)(1) ¼ 12E12 where E is the elastic modulus. Therefore, if c is a constant,

The spring-weight system described above is a freely vibrating system. Alternatively, we may choose to drive the system at constant amplitude, with the input power balancing the losses. The strain will then lag behind the applied stress by a phase angle f, and tan f, which equals f when f is small, is a measure of the damping. This is entirely analogous to the loss resulting from B lagging behind H in alternating magnetization, as described in the previous section. In an electrical circuit, “vibration” consists in the oscillation of charge back and forth, and a quantity Q21 describes the damping of these oscillations. These various measures of damping are related as follows, when the damping is small:

430 MAGNETIZATION DYNAMICS

When an alternating stress is applied to a solid, the various imperfections within the solid move in particular ways, each according to its own nature and the kind of stress imposed. Dislocations move back and forth in slip planes in response to a shear stress parallel to those planes. Solute atoms jump from crystal axes that are shortened by the applied stress to nearby positions on elongated axes, where they have more room, and back again when the stress reverses. If the solid is polycrystalline, a kind of rubbing action occurs at the grain boundaries. In addition, if the material is magnetic, domain walls will move when the material is stressed. The work done in moving these imperfections during a quarter cycle of stress is not returned to the system during the next quarter cycle, as is the work of purely elastic deformation. The work done in moving these imperfections back and forth during a cycle constitutes the internal friction loss DW. It is converted into heat.

The relative magnitude of any one of these damping mechanisms (dislocations, solute atoms, grain boundaries, domain walls) depends markedly on the frequency and amplitude of the vibration imposed on the solid, and may depend strongly on temperature. One mechanism can make a far greater contribution to the total damping in a certain frequency or amplitude range than some other mechanism, and yet be practically negligible in some other range.

Internal friction has been studied at frequencies ranging from about one Hz to a few MHz. The nature of the apparatus required differs from one frequency range to another and, for the lowest range, usually takes the simple form of a torsion pendulum. The specimen is a wire suspended vertically from a fixed support. The lower end of the wire is attached to a fairly heavy weight, either a horizontal rod called an inertia bar or a disk as discussed in Section 7.5.3. The weight is turned a few degrees, so as to twist the wire, and then released. The resulting torsional oscillations of the wire are indicated by a light beam reflected from a mirror, fixed to the lower end of the wire, onto a graduated scale or a photodetector.

Some adjustment of the torsional frequency is possible by altering the moment of inertia of the suspended weight or the diameter of the wire, but the range of adjustment is not very large. If the specimen is made thick enough to be called a rod, rather than a wire, and other parts of the apparatus are made more robust, frequencies of some tens of Hz can be attained. Higher frequencies generally require different experimental methods.

Domain wall damping, often called magneto-mechanical damping, is easily separated from other effects. The specimen is enclosed in a solenoid capable of producing a saturating magnetic field. Damping is then measured with the specimen first demagnetized, then saturated to remove domain walls. The difference between the two measurements gives the magnetic contribution. Figure 12.15 vividly demonstrates the large difference in damping between the two states.

At extremely low amplitudes magnetic damping is small and independent of amplitude, as shown by the data of Fig. 12.16 for longitudinal vibration of a rod. In these measurements any one region of the rod is alternately in states of axial tension and compression. This is accomplished by cementing a quartz “driver” crystal to the end of the rod and applying an alternating voltage to opposite faces of the quartz. Because quartz is piezoelectric, this voltage causes the quartz to vibrate, and the vibrations are transmitted to the specimen rod. Another quartz crystal cemented to the rod senses the vibrations and generates a voltage between opposite faces proportional to the vibrational amplitude. The specimen will vibrate substantially (resonate) only at its natural frequency, which is determined by its dimensions, or at integral multiples (harmonics) of this frequency. The vibration frequency is determined by the frequency of the applied voltage. At each selected