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

64 EXPERIMENTAL METHODS

Fig. 2.34 Measured and corrected magnetization curves of cold-worked iron, from the data of Table 2.2.

applies. The low-field permeability is unknown, but we may make the reasonable assumption that it lies somewhere between 102 and 1. From Fig. 2.28 we read Nd ¼ 0:002 (SI) or Nd ¼ (0:002)(4p) ¼ 0:025 (cgs). Using Equation 2.37, this leads to the demagnetizing fields Hd listed in the table (which is in cgs units), and we see that they form a very substantial fraction of the applied fields. The flux density is plotted in Fig. 2.34 as a function of both applied and true fields. It is clear that the apparent permeability, given by B/Ha, is much less than the true permeability, or B/Htr. If the two conditions noted above are met, it may be shown that

This suggests an experimental method to determine a value for the demagnetizing factor.

If a material has a high permeability mtrue, say 5000 or higher, then 1=mtrue is negligible and Equation 2.38 gives:

A measurement of B vs Ha on a sample of this material will give an initial straight line with a slope

The reciprocal of the numerical value of this slope gives the experimental value of Nd. So a sample of high-permeability material, such as Ni–Fe permalloy or high-purity annealed iron or nickel, made in the same size and shape as the sample(s) to be measured, can be

used to establish an experimental demagnetizing factor for all high-permeability samples of the same size and shape, measured in the same apparatus.

If M is measured directly instead of B, then we can write

The value of x at low field can be measured on a ring sample where N ¼ 0. For soft magnetic materials, 1=x may be negligible. Then the reciprocal of the slope of the measured M vs Ha curve gives an experimental value for N. This is the method used to determine the experimental points in Fig. 2.31.

When the field applied to a specimen on open circuit is reduced to zero, the induction remaining is always less than in a ring specimen, because of the demagnetizing field. In Fig. 2.33b the induction in a ring specimen would be Br, because Hd ¼ 0 and Ha ¼ H. But in an open-circuit specimen, the remanent induction is given by the intersection of the demagnetizing line OC or OC0 with the second quadrant of the hysteresis loop. If the specimen is long and thin, this line (OC) will be steep and the residual induction B1 will not differ much from that of the ring. If the specimen is short and thick, the line (OC0) will be so nearly flat that the residual induction B2 will be very small.

The demagnetizing effect can assume huge proportions in short specimens of magnetically soft materials. For example, suppose a sample of iron can be brought to its saturation value of Ms ¼ 1700 emu/cm3 (1.7 106 A/m) by a field of 10 Oe (800 A/m) when it is in the form of a ring. If it is in the form of a sphere, Hd at saturation will be (4p=3)(1700) ¼ 7120 Oe ([1/3][1.7 106] ¼ 570 kA/m), and the applied field necessary to saturate it will be 7120 þ 10 ¼ 7130 Oe or 570 þ 0.8 ¼ 571 kA/m. The M, Ha (or B, Ha) curve will be a straight line almost to saturation, with a slope determined by the value of Nd, and the details of the true curve, such as the initial permeability, will be unobservable.

In open-circuit measurements on very soft magnetic materials, the Earth’s magnetic field of 0.5 Oe or 40 A/m may be significant. This influence can be minimized by orienting the long axis of the sample perpendicular to the Earth’s field, as determined by a magnetic compass. It may be important to consider that the Earth’s field has a component normal to the Earth’s surface, except near the equator.

We have seen three possible methods of calculating or eliminating the demagnetizing field correction:

1.Make the specimen in the form of an ellipsoid. Then Hd can be exactly calculated, but at the cost of laborious specimen preparation.

2.Use a rod or strip or thin film specimen of very large length–area ratio. The demagnetizing factor is then so small that any error in it has little effect on the computed value of the true field.

3.Apply a correction using a calculated or tabulated or measured demagnetizing factor. The uncertainties in this approach have been noted above.

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Fig. 2.37 Vibrating sample magnetometer or VSM (schematic).

mechanical vibrator. The oscillating magnetic field of the moving sample induces an alternating emf in the detection coils, whose magnitude is proportional to the magnetic moment of the sample. The (small) alternating emf is amplified, usually with a lock-in amplifier which is sensitive only to signals at the vibration frequency. The lock-in amplifier must be provided with a reference signal at the frequency of vibration, which can come from an optical, magnetic, or capacitive sensor coupled to the driving system. The detection-coil arrangement shown in Fig. 2.37 is only one of several possible ones described by Foner, all of which involve balanced pairs of coils that cancel signals due to variation in the applied field. The coil arrangement of Fig. 2.37 is very commonly used. The apparatus is calibrated with a specimen of known magnetic moment, which must be of the same size and shape as the sample to be measured, and should also be of similar permeability.

The driving system may be mechanical, through a cam or crank and a small synchronous motor, or in a recent commercial instrument, with a linear motor. In this case the vibration frequency is generally below 40 Hz, and the vibration amplitude is a few millimeters. The

amplitude is fixed by the geometry of the mechanical system or by the drive signal to the linear motor. Alternatively, the driving system may be based on a loudspeaker, as in Fig. 2.37, in which case the frequency is generally about 100 Hz, and the amplitude is near 0.1 mm. In this case, the amplitude may vary, depending on the mass of the sample and its interaction with the magnetic field, so there must be some provision for monitoring and/or controlling the amplitude as well as the frequency of vibration. One method is to attach a small permanent magnet to the sample rod, outside the varying field region, with a second set of sensing coils. Then the signal from these coils can be used in a feedback loop to maintain a constant amplitude of vibration. Alternatively, a portion of the signal from the permanent magnet can be balanced against the signal from the unknown sample, making the method a null method.

Fig. 2.38 VSM working in a superconducting solenoid. The oscillating sample motion is provided by a linear motor. Courtesy of Quantum Design, Inc.

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Care is necessary to minimize vibration of the sensing coils in the field, and to keep the measuring field from influencing other parts of the system. Note that the VSM measures the magnetic moment m of the sample, and therefore the magnetization M, whereas the fluxmeter method ordinarily measures the flux density B.

The VSM is very versatile and sensitive. It may be used for both weakly and strongly magnetic substances, and standard versions can detect a magnetic moment of about 1025 emu ¼ erg/Oe or 1028 A m2. This corresponds to the saturation magnetization of about 0.04 mg of iron, which suggests the attention to cleanliness that is necessary when measuring small or weakly magnetic samples. The VSM must be used with care in the determination of the magnetization curves or hysteresis loops of magnetically soft materials. The specimen has to be short, and the demagnetizing field may then be such a large fraction of the applied field that the true field is uncertain. However, if the sample is or can be made very thin, as is often the case in the computer and electronics industries, the demagnetizing correction can be small or negligible. The demagnetizing correction is also generally unimportant in the determination of saturation magnetization Ms.

The VSM can be adapted for measurements at high and low temperatures, since only the sample and vibrating rod must be heated or cooled. (The organic insulation on magnet wire does not suffer at low temperatures, but has limited high temperature endurance.) VSMs of normal design are limited to small samples, generally less than 1 g.

Figure 2.10 shows a commercial VSM mounted to operate in an electromagnet, and Fig. 2.38 shows a VSM driven by a linear motor, working in a superconducting solenoid magnet. The arrangement in Fig. 2.38 can operate as a VSM or as an extraction magnetometer.

2.11.3 Alternating (Field) Gradient Magnetometer—AFGM or AGM (also called Vibrating Reed Magnetometer)

Sensitivities even higher than the VSM can be achieved by mounting the sample at the end of a fiber, and subjecting it to a fixed dc field plus an alternating field gradient, produced by an appropriate coil pair, as indicated in Fig. 2.39. The field gradient produces an alternating force on the sample, which causes it to oscillate and flexes the fiber. If the frequency of vibration is tuned to a resonant frequency of the system, the vibration amplitude increases by a factor equal to the quality factor Q of the vibrating system, which can be of the order of 100. The first system of this kind used an optical microscope to observe the vibration amplitude, but later versions [P. J. Flanders, J. Appl. Phys., 63 (1988) pp. 3940–1345] use a piezoelectric crystal to generate a voltage proportional to the vibrational amplitude, which in turn is proportional to the sample moment. The limiting sensitivity of the commercial version is about 1026 emu or 1029 Am2. At this level of sensitivity, sample cleanliness and mounting methods become critical. The AGM, also called the resonant reed or vibrating reed magnetometer, is more limited than the VSM in the maximum mass of the sample that can be measured, and tuning the vibration frequency to resonance complicates the measurement. The necessary presence of a field gradient means the sample is never in a completely uniform field, which is sometimes a limitation.

2.11.4Image Effect

Open-circuit measurements made with an electromagnet may show errors due to the image effect. Figure 2.40a shows the exterior field of a short magnetized specimen. If this

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Fig. 2.41 Magnetization curve showing image effect.

specimen is now placed in the gap of an unmagnetized electromagnet, the lines of force from the specimen swing around to the positions indicated in Fig. 2.40b. These new positions are just those which would result if the lines of force were connected to the “magnetic images” of the specimen in the pole pieces, as shown by the dashed lines. An analogous effect occurs in electrostatics, when the lines of force from a point charge shift their position when a conducting plate is placed nearby. The magnetic moment of the magnetic image exactly equals that of the specimen when the permeability m of the pole caps is infinite, decreases as m decreases, and vanishes when the pole caps become magnetically saturated [m (cgs) or mr (SI) ¼ 1]. The change in field distribution caused by this image effect generally increases the flux linkage with the search coil and therefore increases the fluxmeter or VSM signal. This increased signal disappears as the material of the pole caps approaches saturation, which results in an apparent magnetization curve like that in Fig. 2.41. The image effect appears as a drop in signal at high fields, although physically it results from an increase in signal at low fields. An analogous effect occurs in a superconducting magnet, as magnetic flux can increasingly penetrate the coil windings at high field levels.

The image effect can be minimized by keeping the sample and sensing coils as far as possible from the pole faces of the electromagnet, or by increasing the working gap of the magnet. However, these are not usually practical methods, because they reduce either the working space in the magnet or the maximum field that can be applied to the sample. The best solution in principle is to measure a sample of known constant magnetization, and of the same size and shape as the sample to be measured, over the full range of fields available in the electromagnet, and use these data to establish a field-dependent correction factor. This requires a sample whose magnetic moment remains constant over a wide range of applied fields. Such a sample can be achieved in the form of an artificial specimen, consisting of a small coil of the same shape and size as the real specimen, in which a constant current is maintained in order to produce a constant magnetic moment of known magnitude. However, the total magnetic moment of such a coil will be much less than the moment of a strongly magnetic sample of the same size. It is also difficult to make a coil in any shape other than a cylinder. Another possibility is to use a sample of a modern permanent magnet material, whose remanent moment is unaffected by a wide range of applied fields, or a soft magnetic material that saturates in low fields and keeps a constant saturated magnetization in high fields. In both these cases it is hard to be sure that the sample magnetization is exactly constant over a wide range of fields, but the correction may be adequate for many purposes.

Precise measurements by any of the methods noted above require that the position of the specimen be accurately reproducible, both with respect to the coils and, because of the

image effect, with respect to the pole pieces. In addition, the standard specimen used to calibrate the apparatus must have the same size and shape as the unknown, and also have similar susceptibility or permeability.

2.11.5SQUID Magnetometer

The SQUID device described previously, when used as a magnetometer, acts as a very high-sensitivity fluxmeter, in which the integration is performed by counting voltage steps. It is of such high sensitivity that in a working instrument the magnetic field is held exactly constant by a superconducting shield, and the sample is moved slowly through a superconducting pickup coil coupled to the SQUID while flux quanta are counted. Since the SQUID is a superconducting device, it is usually incorporated in a system including a superconducting magnet. Measurements over a range of fields and temperatures are time consuming, and the systems are normally operated unattended, under computer control. The sensitivity of a SQUID magnetometer is generally about 1027 emu or 10210 Am2, an order of magnitude more sensitive than the AGM.

2.11.6Standard Samples

The U.S. National Institute of Standards and Technology (NIST) includes in its list of Standard Reference Materials three different magnetic moment standards: one of yttrium iron garnet (YIG) and two of nickel, and a magnetic susceptibility standard of platinum. (www.nist.gov/srm).

2.11.7Background Fields

Any kind of work involving weak fields, such as the measurement of low-field properties of materials, the measurement of magnetic fields produced by small current flows in the human body, or the calibration of magnetometers for use in satellite systems, is likely to be influenced by variations in the Earth’s magnetic field. In addition to short-term and long-term variations in the Earth’s field itself, which are small but not zero, substantial fields can be generated locally by electric power lines, large motors, transformers, and the like. One solution is to build a laboratory in a remote location, avoiding the use of any iron or steel construction materials (such as nails!). A more practical solution is to control the field in an existing laboratory. This control can be achieved by building a magnetically shielded room and/or by generating compensating fields. The latter are produced by an arrangement of three large Helmholtz-coil pairs on mutually perpendicular axes. Three magnetometers (or a single three-axis magnetometer) at the center sense the fields there and through appropriate circuits control the currents in the Helmholtz coils to control the three field components. In this manner the field in the central region can be made to have any required (low) value, including zero, regardless of field changes outside the system.

2.12MAGNETIC CIRCUITS AND PERMEAMETERS

A permeameter is a device for testing straight bars or rods in a closed magnetic circuit. To understand this method requires a general knowledge of the magnetic circuit, which we will now consider. The concept of a magnetic circuit is a very useful one, not merely with