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

Fig. 9.44 Method for establishing a point on the anhysteretic magnetization curve. Dashed line shows the full curve.

amplitude large enough to saturate the specimen. The amplitude of the alternating field is then slowly reduced to zero while the constant field remains applied. The magnetization M1, or induction B1, resulting from this treatment is then measured. Figure 9.44 shows how the final state is arrived at. This process is repeated for a series of values of H1, and the resulting anhysteretic curve, shown by the dashed line, is then plotted.

An alternate method is to produce a series of symmetrical minor hysteresis loops, and then draw a line joining the tips of these loops. Although the two methods are not equivalent, they lead to very similar results except near the origin.

The anhysteretic curve has no points of inflection, lies above the normal magnetization curve, passes approximately through the midpoints of horizontal chords of the major loop, and is independent of the previous magnetic history of the specimen. It has an important bearing on the behavior of magnetic tapes for sound recording (Section 15.3).

9.16EFFECT OF PLASTIC DEFORMATION (COLD WORK)

Plastic deformation makes magnetization more difficult. The M, H (or B, H ) curve is lower than that for the fully annealed condition. The hysteresis loop rotates clockwise, becomes wider (larger coercivity), and has a bigger area (larger hysteresis loss) for the same maximum value of M (or B). The mechanical strength and hardness also increase. Cold-working is generally possible only for metals and alloys; ceramics are brittle materials and will break rather than deform under stress.

These several changes are caused by the increased numbers of dislocations and other lattice defects. In polycrystalline materials, severe cold work multiplies the dislocation density by a factor of about 104. The resulting microstress impedes both domain wall motion and domain rotation, increasing the magnetic hardness. It also impedes the motion of the dislocations themselves and the generation of new dislocations, thus increasing the mechanical hardness.

In the low-field region the effect of cold work is illustrated by Fig. 9.45, which applies to the hydrogen-annealed ingot iron mentioned earlier. The hysteresis curves shown were calculated by Equation 9.52 from values of mi and n derived from the m, H curves of Fig. 9.40.

330 DOMAINS AND THE MAGNETIZATION PROCESS

Fig. 9.45 Hysteresis loops for polycrystalline iron, calculated from measured Rayleigh constants.

Note that the coercive field Hc is the same for both the annealed and prestrained specimens, which is unusual. This agrees with the fact that mi/n was found to be independent of strain in this material. The Rayleigh equations predict that Hc, for a given value of Hm, depends only on the ratio of mi to n and not on the individual values of these constants.

It is instructive to consider the changes in magnetic and other properties caused by still larger amounts of cold work, and how these properties change during subsequent annealing. Figure 9.46a shows how the hardness of commercially pure nickel increases with tensile deformation, up to 35% elongation (26% reduction in area). The X-ray line width b increases similarly, because of increased microstress and grain fragmentation. Conversely, the maximum permeability mm (see Fig. 1.14) precipitously decreases with even a small amount of cold work.

Specimens elongated 35% were then annealed for 1 h at various temperatures up to 8008C, with the results shown in Fig. 9.46b. Three phenomena occur, depending on the temperature.

1.Recovery, up to about 6008C. The main process taking place here is dislocation rearrangement, leading to partial relief of residual stress (macro and micro). As a result, at the higher temperatures in this range, b begins to decrease and mm to increase. However, the dislocation density remains high and so does the hardness.

2.Recrystallization, in the range 600–7008C. The cold-worked structure is replaced by an entirely new, almost stress-free, grain structure, and many properties change abruptly. The hardness and dislocation density drop to low values, and b continues to decrease. There is a moderate increase in mm.

3.Grain growth, above about 7008C. The average grain size increases as larger grains grow at the expense of smaller ones, causing a further small decrease in hardness due to the reduction in the amount of grain boundary material per unit volume. Residual stress, as evidenced by b, decreases to its minimum level. The increase in grain size, permitting more extensive motion of domain walls, and the decrease in stress, result

Fig. 9.46 Changes in physical properties of nickel during tensile deformation and subsequent annealing. RF ¼ hardness on Rockwell F scale; b ¼ width (degrees) of the (420) X-ray line, using Cu Ka1 radiation; mm ¼ maximum permeability. [H. Chou and B. D. Cullity, unpublished.]

in an abrupt increase in mm, and magnetic softness is restored. The fact that the hardness, for example, of the material annealed at 8008C is less than that of the unworked material, shown at the extreme left of Fig. 9.46a, shows that this material had been incompletely annealed before the cold-working experiments were begun.

It is clear from these measurements that a cold-worked material can be restored to a condition of maximum magnetic softness only by annealing above the recrystallization temperature. This temperature depends on the material, and, for a given material, it is lower the greater the purity and the greater the amount of previous cold work. Figure 4.6 shows the approximate recrystallization temperatures of iron, cobalt, and nickel.

Plastic deformation, in addition to lowering the permeability, can also produce magnetic anisotropy, particularly in certain alloys. This effect will be described in Sections 10.5 and 10.6

The results in Fig. 9.46 and similar studies are evidence of the close connection between magnetic and mechanical hardness, in that both increase when a metal is cold worked. And when carbon is added to iron to form steel, both the magnetic and mechanical hardness increase with the carbon content. In general, anything that increases mechanical hardness in metals and alloys also increases magnetic hardness. While this is a fairly good rule, it does have important exceptions.

1.Any element that goes into solid solution in a metal will increase the mechanical hardness, because the solute atoms interfere with dislocation motion. However, the effect on magnetic behavior is not predictable. The addition of silicon to iron, for example,

PROBLEMS 333

demagnetizing factor, show that

where the expressions for S and N are better approximations for larger values of k. The total energy is

E t ¼ Sg þ 2NMs2V:

(If the ellipsoid were isolated in a nonmagnetic medium, its magnetostatic energy would be NM2s V/2, but here it is embedded in a medium of magnetization, Ms, which doubles the surface pole strength and quadruples the energy.) Show that the value of k that minimizes the total energy is given by

Verify the values given below for iron (g ¼ 1.5 erg/cm2, Ms ¼ 1714 emu/cm3). Here E0 is the magnetostatic energy of the isolated cube, taken as equal to the magnetostatic energy of a sphere of the same volume.

These values show that for particles a few tenths of a mm in size, the formation of spike domains reduces the total energy only slightly. However, for particles in the size range of a few mm, the formation of spike domains is strongly favored.

9.3Consider magnetization by rotation in a single domain particle with uniaxial anisotropy, as illustrated by the hysteresis loops of Fig. 9.36.

a.Compute and plot the loop when the angle a between the easy axis and the field is 208.

b.Show that the reduced intrinsic coercivity (the field h that makes m zero) is equal to hc for a less than 458 and equal to 2(sin 2a)/2 for a greater than 458.

9.4Find g, d, and L for barium ferrite, with K ¼ 33 105 erg/cm3, Tc ¼ 4508C, and Ms ¼ 380 emu/cm3. Find d from Equation 9.11 with a set equal to 30 nm. This calculation is approximate, because Equation 9.11 was derived for a simple cubic structure, and because the appropriate value of a is uncertain.

336 INDUCED MAGNETIC ANISOTROPY

10.2MAGNETIC ANNEALING (SUBSTITUTIONAL SOLID SOLUTIONS)

When certain alloys are heat treated in a magnetic field and then cooled to room temperature, they develop a permanent uniaxial anisotropy with the easy axis parallel to the direction of the field during heat treatment. They are then magnetically softer along this axis than they were before treatment. The heat treatment may consist only of cooling through a certain temperature range in a field, rather than prolonged annealing; the cooling range or annealing temperature must be below the Curie point of the material and yet high enough, usually above 4008C, so that substantial atomic diffusion can occur. Unidirectional and alternating fields are equally effective, since the field determines an axis, rather than a direction, of easy magnetization. The field must be large enough to saturate the specimen during the magnetic anneal if the resulting anisotropy is to develop to its maximum extent. Usually a field of a few oersteds or a few hundred A/m is sufficient, since the material is magnetically soft to begin with, and its permeability at the magneticannealing temperature is higher than at room temperature. The term “magnetic annealing” is applied both to the treatment itself and to the phenomenon which occurs during the treatment; i.e., an alloy is often said to magnetically anneal if it develops a magnetic anisotropy during such an anneal.

single crystal, with k100l sides, of an alloy of Fe þ 6.5%Si; heat treatment in a field increased its maximum permeability from 50 103 to 3.8 106.

However, most of the research on magnetic annealing has been devoted to the binary and ternary alloys of Fe, Co, and Ni. Compositions which respond well to magnetic annealing are 65–85% Ni in Fe, 30–85% Ni in Co, 45–60% Co in Fe, and ternary alloys containing 20–60% Ni, 15–35% Fe, balance Co. Magnetic annealing has been studied most often in binary Fe–Ni alloys, for which the equilibrium phase diagram is shown in Fig. 10.1. Both the a (body-centered cubic) and the g (face-centered cubic) phases are ferromagnetic. There is a large thermal hysteresis in the a ! g and g ! a transformations because of low diffusion rates below about 5008C, and the equilibrium state shown in Fig. 10.1 is very difficult to achieve. For example, the g ! a transformation on cooling is so sluggish that it is easy to obtain 100% g at room temperature in alloys containing more than about 35% Ni, by air cooling g from an elevated temperature.

Typical of the magnetic-annealing results obtained on Fe–Ni alloys are those shown for 65 Permalloy in Fig. 10.2. Comparison of the hysteresis loop of Fig. 10.2c with Fig. 10.2a or b shows the dramatic effect of field annealing: the sides of the loop become essentially vertical, as expected for a material with a single easy axis. Conversely, if the loop is measured parallel to the hard axis, i.e., at right angles to the annealing field, the shearedover, almost linear loop shown in Fig. 10.2d is obtained, where the change in the H scale should be noted. Alloys which show the magnetic-annealing effect commonly have the peculiar, “constricted” loop shown in Fig. 10.2b when they are slowly cooled in the absence of a field; this kind of loop will be discussed later.

A note on experimental techniques: magnetic annealing of sheet samples is usually done with a coil, or system of coils, located outside the annealing furnace. The field produced by the heating elements of the furnace may be significant, and must then be taken into account

Fig. 10.1 Equilibrium phase diagram of Fe–Ni alloys. [E. J. Swartzendruber, V. P. Itkin, and C. B. Alcock, ASM Handbook, Volume 3 (1992).]

in finding the field acting on the sample. The annealing field may be applied in different directions either by providing two sets of coils, or by rotating the sample in the furnace. If the specimen is in the form of a disk, the annealing field can be applied in any direction in the plane of the disk. Specimens the form of rods are difficult to magnetize in a direction perpendicular to the rod axis, because of the large demagnetizing field. Instead, a current is passed along the rod axis during the anneal, producing a circular field around the axis (Section 1.6). This field can easily be made strong enough to saturate the specimen circumferentially, except for a small volume near the axis. If a magnetic measurement is subsequently made parallel to the rod axis in the usual way, the measurement direction is at right angles to the annealing field. A ring sample, or a picture-frame sample, can be magnetized circumferentially by running a large current through a single conductor that passes through the ring. It is much easier to provide electrical insulation at high temperatures for a single conductor than for a multi-turn winding on a toroidal sample. It is generally difficult to anneal a ring sample in a field oriented perpendicular to the circumference, because of the large demagnetizing field.

Magnetic annealing evidently creates a preferred domain orientation in the demagnetized state, with the direction of magnetization in each domain tending to lie parallel to the axis

338 INDUCED MAGNETIC ANISOTROPY

Fig. 10.2 Hysteresis loops of a 65 Ni–35 Fe alloy after various heat treatments: (a) annealed at 10008C and cooled quickly, (b) annealed at 4258C or cooled slowly from 10008C, (c) annealed at 10008C and cooled in a longitudinal field, (d) same as (c) but with a transverse field. [R. M. Bozorth, Ferromagnetism, reprinted by IEEE Press (1993).]

of the annealing field. Subsequent magnetization in this direction can then take place preferentially by 1808 wall motion. As a result, magnetostriction and magnetoresistance measured in the annealing direction are reduced to very small values. Figure 10.3 illustrates this point for a Ni–Fe alloy of slightly different composition. The central curve is for a specimen cooled slowly from 10008C in the absence of a field; slow cooling from 6008C in a longitudinal field (lower curve) greatly decreases the longitudinal magnetostriction and the same treatment in a transverse field increases it. Furthermore, when the data for the upper curve are replotted in the form of l vs (B H)2 ¼ (4pM)2 (cgs) or l vs (B m0H)2 ¼ (m0M)2 (SI), the result is a straight line in agreement with Equation 8.31. The magnetization process for this specimen (annealing field transverse to axis) is therefore one of pure rotation of the domain vectors through 908 from their initial positions transverse to the axis.

The anisotropy created by magnetic annealing is due to directional order in the solid solution, an idea originated by S. Chikazumi [J. Phys. Soc. Japan, 5 (1950) p. 327].

Fig. 10.3 Magnetostriction of a 68 Ni–32 Fe alloy after various treatments. [H. J. Williams et al. Phys. Rev., 59 (1941) p. 1005.]

The theory of this effect was described independently by several authors [L. Ne´el, J. de Physique et Radium, 15 (1954) p. 225; S. Taniguchi and M. Yamamoto, Sci. Rep. Tohoku Univ., A6 (1954) p. 330; S. Taniguchi, Sci. Rep. Tohoku Univ., A7 (1955) p. 269; S. Chikazumi and T. Oomura, J. Phys. Soc. Japan, 10 (1955) p. 842]. By directional order is meant a preferred orientation of the axes of like-atom pairs. An example, for atoms arranged on a two-dimensional square lattice, is shown in Fig. 10.4. The arrangement in Fig. 10.4a approximates a random solid solution of a 50–50 “alloy” of A and B; the atom positions were determined by drawing black and white balls from a box. If the solution were ideally random, there would be 56 black-white (AB) nearest-neighbor pairs, 28 AA pairs, and 28 BB pairs; the arrangement in Fig. 10.4a comes close to this, since it has 54 AB, 28 AA, and 30 BB pairs. Perfect ordering is shown in Fig. 10.4b; all nearest-neighbor pairs are AB, and no AA or BB pairs exist. The arrangement in Fig. 10.4c has directional order in the vertical direction. It has the same 28 AA pairs as Fig. 10.4a, but in Fig. 10.4c 20 of these have vertical axes and only eight have horizontal axes. The same kind of preferred orientation is shown by BB pairs; 19 are vertical and nine horizontal.

Note that directional order can be achieved in a solid solution which is perfectly random in the usual crystallographic sense. Thus, in terms of nearest neighbors only, a 50–50 solution is random if the neighbors of any given A atom, for example, are on the average half A and half B. This solution can deviate from randomness in either of two ways: (1) short-range order, in which more than half of the neighbors of an A atom would be B atoms; and (2) clustering, in which more than half of the neighbors would be A atoms. The solution of Fig. 10.4c shows neither a tendency to short-range order (a preponderance of AB pairs) nor to clustering (a preponderance of AA and BB pairs); it has 56 unlike-atom pairs and 56 like-atom pairs. The like-atom pairs are, however, preferentially oriented. The word “order” in the term “directional order” should therefore not be misconstrued. Whether short-range order or clustering exists depends on the number and kind of atoms surrounding a given atom but not on their relative positions.

How does directional order create magnetic anisotropy? The basic hypothesis of the theory is that there is a magnetic interaction between the axis of like-atom pairs and