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

Fig. 7.10 Variation with u of the anisotropy energy E and the torque L (¼2dE/du) for a uniaxial crystal. u is the angle between Ms and the easy axis.

are energy minima, and u ¼ 908, which is a direction of difficult magnetization, is a position of instability. The slope of the torque curve at L ¼ 0 is negative for the positions of stability (u ¼ 0 and 1808) and positive for the unstable position (u ¼ 908). At a stable position, a clockwise (positive) rotation of the sample produces a negative (counterclockwise) torque, and vice versa. At an unstable position, a clockwise rotation of the sample produces a positive (clockwise) torque. The value of K1 can found simply from the maximum amplitude of the torque curve (¼+K1), or from the values of the slope at the zero crossings (¼+2K1), or by fitting the entire curve to Equation 7.6 with the value of K1 as a fitting parameter.

The preceding analysis is valid only if the field is strong enough so that the magnetization Ms is aligned with the field H for all values of u. This condition is often not met, and we have instead the situation shown in Fig. 7.11. Here c is the angle from the c direction to the applied field, which is known from the measurement; u is the angle from the c direction to the magnetization Ms, and w (¼c 2 u) is the angle from the field H to the magnetization Ms. Neither u nor f is known directly, but the angular position of Ms is determined by the balance between two torques, LK ¼ 2K1 sin 2u and LH ¼ MsH sin w. Here LK is the anisotropy torque, acting to rotate the magnetization toward the easy direction, and LH is the torque exerted by the field, acting to rotate the magnetization toward the field. Since these torques are balanced, we have K1 sin 2u ¼ MsH sin w. We also know that the torque exerted on the sample by the anisotropy must be balanced by the measured torque Lmeas, so that

So if we measure L, and we know Ms and H, we can find sin f and therefore f from sin f ¼ (Lmeas/MsH ). And knowing f, we can correct the measured value of c for each measured

Fig. 7.15 Measured torque of a f011g disk of an Fe þ 3.85 wt% Si alloy. The fitted curve is drawn for K1 ¼ 2.87 105 erg/cm3 and K2 ¼ 1 105 erg/cm3. [R. M. Bozorth, Ferromagnetism, reprinted by IEEE Press (1993).]

prepare in single-crystal form than pure iron, for reasons given in Section 13.4, while they remain like iron in having k100l easy directions. Similar Fe–Si alloys are widely used in the magnetic cores of electrical machines.

Distortion of the torque curve occurs when the field is not strong enough to align the magnetization exactly in the field direction, just as in the case of uniaxial anisotropy discussed above, and can be corrected in the same way, if the magnetization and field are known. The value of sin w is found from Lmeas ¼ MsH sin f, and the value of f is used to correct the measured value of c to give u.

If a disk is cut parallel to f111g and Ms makes an angle u with a k110l direction, the crystal anisotropy energy may be written

The calculated torque curve is a simple sine curve, repeating itself every 608, with an amplitude of +K2/18 and zero-crossing slope +K2/3. In principle, a disk cut parallel to f111g is a better specimen for the determination of K2 than one cut parallel to f110g, because the torque on a f111g specimen is determined only by K2, whereas the torque on a f110g specimen is a function of both K1 and K2. In practice, however, f111g disks often yield imperfect

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sin 6u curves, because slight misorientation of the specimen gives a relatively large contribution from K1. In general, accurate K2 values are not easy to obtain from torque measurements, and the values reported in the literature tend to be inconsistent.

Fourier analysis of experimental torque curves offers a means of sorting out the various contributions to the observed torque. The torque is expressed as a Fourier series:

Fourier analysis of the experimental curve then yields the values of the various coefficients An and Bn, which in turn describe the kinds of anisotropy present. For example, suppose a torque curve is obtained from a disk cut parallel to f100g in a cubic crystal. If conditions are perfect, Fourier analysis of the experimental curve would show that all Fourier coefficients are zero except B4, because Equation 7.10 shows that the torque varies simply as sin 4u. But suppose that the specimen or torque magnetometer, or both, is slightly misaligned and that this misalignment introduces a spurious uniaxial component into the torque curve. This will be reflected in a nonzero value for B2, because Equation 7.6 shows that uniaxial anisotropy causes a sin 2u variation of torque. The nonzero value of B2, in this particular example, discloses the misalignment, while the value of B4 yields the quantity desired, namely,

K1 ¼ 22B4.

Slight experimental imperfections do not distort a torque curve so much that its basic character is unrecognizable. Thus, in the example just described, the experimental curve would be somewhat distorted but still recognized as basically similar to the curve of Fig. 7.13, which describes pure biaxial anisotropy. The function of Fourier analysis is then to separate out the spurious torques and leave only the torque due to the crystal itself. But specimens are also encountered in which two, or even three, sources of anisotropy are simultaneously present, and with more or less the same strength. Fourier analysis of the torque curve then becomes not merely a means of refining slightly imperfect experimental data, but a necessary method for disentangling the various components of the anisotropy. Before computers became ubiquitous, Fourier analysis of experimental torque data was a fairly tedious computational task. It is now quick and easy.

7.5.2Torque Magnetometers

The instrument for making torque measurements is called a torque magnetometer. Depending on the material to be measured, the size of the sample, and the temperature, the maximum torque may vary over many orders of magnitude, so that no single design is universally useful. Commercial instruments are available, but many torque magnetometers are specially built for particular uses.

Early designs made use of a torsion fiber, or a spiral clock spring, to measure the torque, as suggested by Fig. 7.16. In this case, the deflections (the twist in the torsion fiber) are large enough to be read by eye, but there is no simple way to record the data automatically. A further difficulty is that over substantial portions of the torque curve, centered around the unstable hard axes, there may be no stable angular position of the sample. The condition for stability is that the net torque on the sample (the sum of the anisotropy torque and the torsion fiber torque) is zero and has negative slope. As noted above, negative slope means that a positive (clockwise) rotation of the sample produces a negative (counterclockwise) torque, and vice versa. Therefore the condition for stability is that the stiffness of the

Fig. 7.16 Basic mechanical torque magnetometer (schematic).

torsion fiber, given by the (negative) slope of the torque vs angle curve, must be greater than the positive slope of the torque curve in the hardest direction. In making this comparison, the absolute value of the torque must be used; that is, the torque per unit volume times the volume of the sample.

Again using the uniaxial case as the simplest example, the absolute value of the slope in the hard direction is 2K1v, where v is the volume of the sample. The torque acting on the sample due to the torsion fiber is 2kb, where k is the torsion constant (dyne-cm or N-m per radian) and b is the twist in the fiber. The slope of this torque (vs angle) is simply 2k. The zero net torque condition gives kb ¼ K1v, and the negative slope condition gives 2K1v 2 k , 0, or k . 2K1v. Solving for b gives b , 12 (radian), or less than 308. Thus the suspension fiber must be made stiff enough so that the maximum twist angle is less than 308 to permit stable readings through a full 3608 rotation of the sample. This analysis does not take into account the distortion of the torque curve when the field is not strong enough to align the magnetization parallel to the field, as discussed above. The distortion increases the measured slope in the hard direction, and thus increases the required stiffness of the torsion fiber and decreases the maximum twist in the fiber. Exactly similar reasoning applies to the torque curves for the various cubic anisotropy cases.

The conclusion is that, in order to insure stable readings, the simple torsion fiber version of the torque magnetometer must be made so stiff that its maximum deflection is limited to about +208 in the most favorable case, which limits the precision of the torque readings. This limitation can be overcome to some extent by using a softer torsion fiber and limiting the angular motion of the sample to a small range, say +18, with mechanical stops, and measuring the torque required to jump the sample position from one limit to the other over the range of unstable positions. However, this virtually eliminates any possibility of automatic recording of the torque data.

of a D’Arsonval analog meter; if the coil carries a current, it experiences a torque proportional to the current. A sensing circuit, usually based on a light beam, mirror, and photocell or photodiode as in the figure, provides a feedback signal that drives a current through the coil to balance the anisotropy torque. If the system response is fast enough (which is not difficult), the sample can be held at any angle to the field. The value of the current through the coil is proportional to the torque on the sample. This design can be made very sensitive for the measurement of small anisotropies in small samples. Generally some low-friction method of keeping the sample centered in the air gap of the magnet is required; for example, a jewel bearing, an airbearing, or a second torsion fiber.

In any torque magnetometer, either the electromagnet may be rotated around the sample, or the magnetometer and sample may be rotated in a stationary magnet gap. Neither is easy, but either is possible.

The ideal specimen shape is that of an ellipsoid of revolution (planetary or oblate spheroid), which is relatively easy to saturate. However, an ellipsoidal specimen is difficult to make and also somewhat difficult to mount securely in the sample holder, and most investigators settle for a disk with a diameter/thickness ratio of 10 or more. Thin film samples have inherently a very large diameter/thickness ratio, and make very satisfactory samples if the sensitivity of the instrument is high enough. Simple theory predicts that the measured value of the anisotropy should remain constant with increasing field once the field is large enough to saturate the sample in any direction, but in practice, at least with samples other than thin films, the measured anisotropy increases with increasing measuring field. This is presumably partly due to lack of saturation in small volumes of the sample where the local demagnetizing fields are large, and partly due to the real increase in saturation magnetization with field as the field overcomes thermal vibration (the paraprocess). Whether the measured values should be extrapolated to infinite field or back to zero field, and how the extrapolation should be done, are matters that remain unresolved. They rarely have a major impact on the measured values of the anisotropy, but they do limit the accuracy of experimental values.

Torque measurements in a superconducting magnet are difficult, because access to the sample is normally only possible in a direction parallel to the field. Split-coil superconducting magnets can be built to provide access perpendicular to the field direction, but they must be designed to withstand the large attractive force between the two coils, and the perpendicular access path must penetrate the thermal insulation around the superconducting windings.

7.5.3Calibration

The torsion constant of a fiber (dyne-cm or N-m per radian) is given by

where r is the fiber radius, l is the length, and G is the shear modulus, all in consistent units. However, this equation does not give values accurate enough for most measurements,

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mainly because uncertainty in the value of the radius r of a small fiber is magnified in the r4 term.

The value of the torsion constant can be determined quite accurately by using it as a torsion pendulum. The wire is suspended from a fixed support and its lower end is attached to the center of a heavy disk of radius R and mass M. By rotating the disk through a small angle and then releasing it, the system will go into torsional oscillation. The period of oscillation T (sec) is measured by counting complete cycles of oscillation in a measured time interval, and the torsion constant is obtained from

where I is the moment of inertia, equal to MR2/2 for a disk. Therefore

which requires no knowledge of the fiber dimensions or material. If k is large (a stiff suspension), the period of oscillation T may be small, and some kind of electronic recording system will be needed.

Torque magnetometers that do not rely on the properties of a torsion fiber must be calibrated directly. In principle, any torque measuring system can be calibrated with a string wrapped around the shaft and a set of appropriate weights and pulleys. In practice, this does not work well for a sensitive instrument, and a sample of known anisotropy is needed instead. The usual choice is to rely on the shape anisotropy of a thin straight wire of a material of known saturation magnetization, commonly nickel. The anisotropy energy (per unit volume) is uniaxial, and is given by

where DN is the difference in demagnetizing factor parallel and perpendicular to the wire axis. For a long thin wire, N is 2p (cgs) or 12 (SI) when the magnetization is perpendicular to the wire axis, and effectively zero when the magnetization is parallel to the axis. The sample behaves as described previously for a uniaxial material, and the torque (per unit volume of wire) is

The field must be high enough to saturate the wire in the perpendicular direction [H . 2pMs (cgs) or .1/2Ms (SI)]. The sample volume can be obtained by direct measurement, or (preferably) from the sample mass and density.