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

Rods and disks are never uniformly magnetized except when completely saturated. The demagnetizing field varies from one point to another in the specimen and so has no single value. Two specific effective demagnetizing factors may be defined and used, depending on the way the magnetization is measured. The ballistic or fluxmetric demagnetizing factor Nf is the ratio of the average demagnetizing field to the average magnetization, both taken at the midplane of the sample. It is the appropriate factor to use when the magnetization is measured with a small coil wound around the sample at its midpoint, using a ballistic galvanometer (now obsolete) or a fluxmeter. The fluxmetric demagnetizing factor is useful primarily for rod-shaped samples. The magnetometric demagnetizing factor Nm is the ratio of the average demagnetizing field to the average magnetization of the entire sample. It is the appropriate factor to use when the total magnetic moment of the sample is determined using a vibrating-sample, an alternating-gradient, or a SQUID magnetometer (these instruments are described later). Note, however, that strictly speaking these devices measure the total sample moment only when the sample is small enough (relative to the pickup coil dimensions) to act as a point dipole. The samples used in these instruments are commonly disks magnetized along a diameter, although they may also be rods or rectangular prisms.

Values of the demagnetizing factor depend primarily on the geometry of the sample, but also on the permeability or susceptibility of the material. Bozorth [R. M. Bozorth, Ferromagnetism, Van Nostrand (1951); reprinted IEEE Press (1993)] gives a table and graphs of demagnetizing factors for prolate and oblate (planetary) spheroids, and also of fluxmetric demagnetizing factors for cylindrical samples with various values of permeability. Bozorth’s curves have been widely reprinted and used. They are shown here as Fig. 2.28. The values for cylinders are based on a selection of early theoretical and experimental results, and should not be regarded with reverence. Note particularly that the demagnetizing factors for cylindrical (nonellipsoidal) samples given by Bozorth are fluxmetric values (although Bozorth does not use this terminology) and are only appropriate for measurements made with a short, centrally-positioned pickup coil around a cylindrical sample.

The values in Bozorth’s graph for disk samples magnetized along a diameter are calculated for planetary (oblate) ellipsoids, and so do not distinguish between fluxmetric and magnetometric values. It should also be noted that Bozorth plots and tabulates values of N=4p (cgs), not N (cgs), presumably so that the values can be multiplied by B to give demagnetizing fields Hd. This is strictly incorrect, but useful for soft magnetic materials where H B and so B 4pM. Since

Bozorth’s values are numerically correct in SI.

Better values for the demagnetizing factors of rods and disks (and other shapes, such as rectangular prisms) can be determined by experiment, or by calculation. The calculations generally assume a material of constant susceptibility x, which is in fact the differential susceptibility dM/dH measured at a point on the magnetization curve. Three specific values of x are of special significance: x ¼ 21, corresponding to a superconductor in the fullyshielded state; x 0, corresponding to a weakly magnetic material such as a paraor diamagnet, or to a fully-saturated ferroor ferrimagnet; and x ¼ 1, corresponding to very soft magnetic material. The condition x ¼ 21 requires that B ¼ 0 everywhere in the samples. The condition x ¼ 0 requires that the magnetization M be constant throughout the sample, with Hd variable. Note that x ¼ dM/dH ¼ 0 does not require M ¼ 0. The condition x ¼ 1 requires that the demagnetizing field be constant throughout the samples, exactly

56 EXPERIMENTAL METHODS

Fig. 2.28 Demagnetizing factors for various samples. [R. M. Bozorth, Ferromagnmetism, Van Nostrand (1952); reprinted IEEE (1993) pp. 846–847]. Values plotted are Ncgs/4p, which are numerically equal to NSI.

equal and opposite to the applied field, with magnetization M varying from point to point. Demagnetizing factors can be calculated for other values of x, both positive and negative, but the assumption of constant and uniform x makes them of limited usefulness.

The values for x ¼ 1 should apply for soft magnetic materials far from saturation, and values for x ¼ 0 to materials at or approaching magnetic saturation. In practice, demagnetizing field corrections are most important at low fields, where values of permeability and remanence are determined. Demagnetizing corrections are relatively unimportant (although not small) as the sample approaches saturation. Values of the coercive field are generally not much affected by demagnetizing effects, since they are determined when the magnetization

Fig. 2.28 Continued.

is at or near zero. Permanent magnet materials, in which the values of susceptibility are low and uncertain, are normally measured in closed magnetic circuits where the demagnetizing fields are kept small.

A paper by D.-X. Chen, J. A. Brug, and R. B. Goldfarb [IEEE Trans. Mag., 37 (1991) p. 3601] reviews the history of demagnetizing factor calculations and derives new values of Nf and Nm for rod samples. A later paper [D.-X. Chen, E. Pardo, and A. Sanchez, J. Magn. Mag. Matls., 306 (2006) p. 135] gives improved values for rod samples, and adds some calculated values of Nm for disk samples. Similar results for rectangular prisms are given by the same authors [IEEE Trans. Mag., 41 (2005) p. 2077]. All three of these papers include results for a range of values of susceptibility as well as for sample shape.

58 EXPERIMENTAL METHODS

Fig. 2.29 Calculated SI magnetometric demagnetizing factors for rod samples magnetized parallel to the rod axis. Central dashed line is for a prolate ellipsoid. Dotted curves are for x ¼ 1; solid curves for x ¼ 0. Upper dotted and solid curves are magnetometric factors Nm; lower curves are fluxmetric factors Nf. Data in Figs. 2.29 and 2.30 from D.-X. Chen, E. Pardo, and A. Sanchez, J. Magn. Mag. Mater., 306 (2006) p. 125.

The results are extensive and detailed, and not easy to summarize. Figure 2.29 shows calculated values of Nf and Nm for rod samples. The central dashed line is for prolate ellipsoids, where Nf and Nm are the same. The dotted lines are calculated values of Nf and Nm for x ¼ 1, i.e., for very soft magnetic materials. At large values of m (long, thin rods) Nm is slightly above the ellipsoid line, and Nf is slightly below. Note that values of m less than about 10 are largely of mathematical interest, since the measurement requires a central coil whose length is small compared to the sample length. The upper solid line is Nm for x ¼ 0, and the lower solid line is Nf for x ¼ 0. For samples of low susceptibility, or for samples approaching magnetic saturation, the demagnetizing factors can differ from those of the ellipsoid (of the same m value) by a factor approaching 10 when m ¼ 100. For samples of high susceptibility, in low fields, the demagnetizing factor for an ellipsoid of the same m value is generally a reasonable approximation, considering the various uncertanties involved.

Figure 2.30 gives some results for the magnetometric demagnetizing factor Nm for disk samples magnetized along a diameter. Fluxmetric demagnetizing factors Nf are of little interest for disk samples. The dashed curve is for oblate (planetary) ellipsoids; this is the same curve given by Bozorth. The dotted curve is for x ¼ 1 (high permeability) and the solid curve is for x ¼ 0 (uniform magnetization). In the m range of practical interest, values of Nm are always higher than for the ellipsoid of the same m value, and the difference between the x ¼ 1 and the x ¼ 0 values is much less than for rod samples.

There are some relevant experimental measurements. Figure 2.31 shows data points from vibrating-sample (VSM) measurements on a series of permalloy disks, together with the calculated curves for x ¼ 1 and x ¼ 0 from Fig. 2.30. The experimental points generally

60 EXPERIMENTAL METHODS

Clearly in experimental work it is advantageous to make the value of m large, to minimize the demagnetizing correction. Ideally, the worst-case value of Hd should be comparable to the uncertainty in the measurement of the applied field; then uncertainty in the value of N becomes unimportant.

Permanent magnet samples are usually made in the form of short cylinders or rectangular blocks, and they need to be measured in high fields, so the usual practice is to make the sample part of a closed magnetic circuit. This largely eliminates the demagnetizing effect. See the next section.

A common mathematical procedure to calculate the demagnetizing field is to make use of the magnetic pole density on the sample surface, given by rs ¼ M cos u , where M is the magnetization of the sample and u is the angle between M and the normal to the surface. Note that M cos u is the component of the magnetization normal to the surface inside the body, and that M is zero outside. Therefore, the pole density produced at a surface equals the discontinuity in the normal component of M at that surface. If n^ is a unit vector normal to the surface, then

Note that this agrees with one of the definitions of M as the pole strength per unit area of cross section. The polarity of the surface is positive, or north, if the normal component of M decreases as a surface is crossed in the direction of M. Free poles can also be produced at the interface between two bodies magnetized by different amounts and/or in different directions. If M1 and M2 are the magnetizations of the two bodies, then the discontinuity in the normal component is

This is an important principle, which we shall need later.

We also note that, at the interface between two bodies or between a body and the surrounding air, certain rules govern the directions of H and B at the interface:

1.The tangential components of H on each side of the interface must be equal.

2.The normal components of B on each side of the interface must be equal.

These conditions govern the angles at which the B and H lines meet the air—body interfaces depicted in Fig. 2.24, for example.

Free poles may exist not only at the surface of a body, but also in the interior. For example, on a gross scale, if a bar has a winding like that shown in Fig. 2.32a, south poles will be produced at each end and a north pole in the center, for a current i in the direction indicated. On a somewhat finer scale, free poles exist inside a cylindrical bar magnet, as very approximately indicated in Fig. 2.32b. The condition for the existence of interior poles is nonuniform magnetization. An ellipsoidal body can be uniformly magnetized, and it has free poles only on the surface, unless it contains domains. A body of any other shape, such as a cylindrical bar, cannot be uniformly magnetized except at saturation, because the demagnetizing field is not uniform, and so the body always has interior as well as surface poles. Nonuniformity of magnetization means that there is a net outward flux of M from a small volume element, i.e., the divergence of M is greater than zero. But if there is a net outward flux of M, there must be free poles in the volume element to supply this flux. Such a volume element is delineated by dashed lines in Fig. 2.32b, in

Fig. 2.32 Internal poles in a magnetized body.

which lines of M have also been drawn, going from south to north poles. (For clarity, the lines of M connecting surface poles on the ends have been omitted.) If rv is the volume pole density (pole strength per unit volume), then

On the axis of a bar magnet, M decreases in magnitude from the center toward each end, as indicated qualitatively by the density of flux lines in Fig. 2.32b. Suppose the axis of the magnet is the x-axis, and we assume for simplicity that M is uniform over any cross section. Then only the term @Mx=@x need be considered. Between the center of the magnet and the north end, @Mx=@x becomes increasingly negative, which means that rv is positive and that it increases in magnitude toward the end, as depicted in Fig. 2.32b. Although the interior pole distributions in Fig. 2.32a and b differ in scale, both are rather macroscopic; we shall see in Chapter 9 that interior poles can also be distributed on a microscopic scale.

The general derivations of Equations. 2.31 and 2.32 may be found in any intermediatelevel text on electricity and magnetism.

In summary:

1.Lines of B are always continuous, never terminating.

2.a. If due to currents, lines of H are continuous.

b.If due to poles, lines of H begin on north poles and end on south poles.

3.At an interface,

a.the normal component of B is continuous,

b.the tangential component of H is continuous, and

c.the discontinuity in the normal component of M equals the surface pole density rs at that interface.

4.The negative divergence of M at a point inside a body equals the volume pole density at that point.

5.The magnetization of an ellipsoidal body is uniform, and free poles reside only on the surface, unless the body contains domains. See Section 9.5.

simultaneously moving each of the experimental values of M parallel to the H-axis, keeping the distance between the line OD and the value of M fixed. This is sometimes called the shearing correction.

When the measurement gives values of flux density B rather then M, the correction becomes more complicated. M and Hd are evaluated as follows:

B ¼ Htr þ 4pM ¼ Ha NdM þ 4pM (cgs)

Then

Two simplifications are often possible. First, if Nd is reasonably small compared to 4p (cgs) or 1 (SI), the denominator in Eqs. 2.35 and 2.36 may be replaced by 4p or 1. Remembering that demagnetizing factors are not exact or well-defined except for ellipsoids, we may say that if Nd is less than about 2% of its maximum value (4p or 1), it may be neglected here. This would mean m greater than 10 for a prolate ellipsoid (cigar) or greater than 30 for an oblate (planetary) ellipsoid (disk). This does not mean that the demagnetizing field is negligible, just that Nd may be neglected in this denominator. Second, in many cases of measurement on soft magnetic materials, Ha is small compared to B (cgs), or compared to B/m0 (SI). If both these conditions hold, Equation 2.33 reduces to

How this works is illustrated by the experimental data in Table 2.2, obtained from a rod of commercially pure iron in the cold-worked condition. The rod was 240 mm long and 6.9 mm in diameter and hence had a length/diameter ratio of 35. The measurements were made with a search coil at the center of the rod, so the fluxmetric demagnetizing factor

TABLE 2.2 Magnetization of Iron Rod