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

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measuring the magnetization curve and the hysteresis loop of a strongly magnetic substance, and finally, in the last section of this chapter, the methods of measuring the susceptibility of a weakly magnetic substance. Methods of measuring more specialized magnetic properties (e.g. anisotropy, magnetostriction, and core losses) will be dealt with at the appropriate place in later chapters.

The student who wishes to gain a good understanding of magnetic materials cannot afford to slight the contents of this chapter, even if she is not particularly interested in measurements, because some quite basic magnetic phenomena are first introduced here. For example, the demagnetizing fields discussed in Section 2.9 have an importance not restricted to measurements; these fields can affect the magnetic state and magnetic behavior of many specimens.

2.2FIELD PRODUCTION BY SOLENOIDS

Solenoids are useful for measurements on specimens of almost any shape, but are particularly suited to rods and wires. They can be designed to produce fields ranging up to more than 200 kilo-oersteds (kOe) or 16 MA/m or 20 tesla (T), although simple solenoids are limited to fields below about 1 kOe or 0.1 T. [Although the SI unit of field is the A/m,

2.2.1Normal Solenoids

These are usually made with insulated copper wire, wound on a tube of any electrically insulating material, such as polyvinyl chloride (PVC) pipe. For the dimensions shown in Fig. 2.1, the field H at a point P on the axis, distant x from the center, is given by

Fig. 2.1 Single-layer solenoid and its field distribution. The axial field at point P is expressed as the fraction of the field at the center of an infinitely long solenoid.

where n is the number of turns and i is the current in amperes. With C1 ¼ 1 and L measured in meters, the field is in A/m; with C1 ¼ 4p/10 and L measured in centimeters, the field is in oersteds. The terms inside the brackets are dimensionless when all quantities are measured in the same units. Note that the coil diameter is measured to the center of the winding; that is, the value of D is the outer diameter of the tube plus twice the radius of the wire.

At the center of the solenoid (x ¼ 0), this reduces to

and, when L2 D2, to

In any solenoid, the field is highest at the center and decreases towards the ends. The field at the end of a long solenoid is just one-half of the field at the center. But the field over the middle half is quite uniform, as shown by Fig. 2.1, and the values in Table 2.1, which are derived from Equation 2.1. In this table, Hinf is the field, given by Equation 2.3, at the center of an infinitely long solenoid. When the L/D ratio is 20, for example, the field over the middle half is uniform to within 0.15% and is only about 0.13% less than that produced by an infinitely long solenoid. The field variation in the radial direction is generally negligible.

To achieve a higher field, it is preferable to increase n/L by winding the wire in two or more layers rather than to increase the current. Although H is proportional to i, the heat developed in the winding is proportional to i2R, where R is its resistance. Thus doubling the number of layers and keeping the current constant will double H, R, and the amount of heat; whereas doubling the current will double H, but quadruple the heat. Solenoids consisting of relatively few layers can be treated mathematically as a series of concentric solenoids of increasing diameter whose fields add together. Very thick solenoids (many layers) are the subject of an extensive literature, which is well summarized by Montgomery [E. Bruce Montgomery, Solenoid Magnet Design, Wiley-Interscience (1969)].

Cooling of the winding becomes necessary for continuous fields larger than about 1 kOe or 0.1 T. This can be accomplished in a variety of ways, such as by blowing air over the solenoid with a fan, by immersing the solenoid in a cooling liquid, usually water but possibly liquid nitrogen, by winding the wire on a water-cooled tube; or by forming the winding from copper tubing so that it can carry both electric current and cooling water.

TABLE 2.1 Field Uniformity in Solenoids

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Solenoids are usually made from high-purity copper wire insulated with a thin, flexible enamel insulating coating. Such wire is made and sold as magnet wire; it is available in a wide range of diameters, and with various types and thicknesses of insulating coatings. The thinnest coating is generally adequate for solenoid construction, but a high-temperature insulation may be desirable. The enamel insulations are organic materials, and are limited to working temperatures of 2408C or lower. For higher temperatures, braided fiberglass tubing may be used, although it is bulky compared to enamel coatings. Also available, but expensive, is nickel-plated copper wire with a bonded surface layer of very small insulating ceramic particles. Its operating temperature is limited by oxidation of the wire rather than failure of the insulating layer.

Solenoid design is a matter of balancing several conflicting requirements, and the following points should be kept in mind:

1.D is determined by the working space required within the solenoid.

2.The ratio L/D is fixed by the distance over which field uniformity is required. Because the specimen to be tested must normally be subjected to a reasonably uniform field, this means that the maximum specimen length effectively determines the ratio L/D, with the coil length L 1.5–2 times the length of the sample. Specimen length is in turn governed by the factors discussed in Section 2.6.

3.For a given L, the field is proportional to the number of ampere-turns ni, and the power required (which is equal to the rate of heat generated) is proportional to i2R.

4.For a given current, the voltage required from the power source is proportional to R, which in turn is proportional to n. For the most effective use of a given power supply, the solenoid resistance R should be chosen so that at maximum field the power supply operates at both its maximum current and maximum voltage ratings.

Helmholtz coils will produce an almost uniform field over a much larger volume than a solenoid. Two identical thin parallel coils, ideally with square cross-section, are placed at a distance apart equal to their common radius r (Fig. 2.2). The field parallel to the axis of the coils at a point P on the axis a distance z from the midpoint is given by

The distance between the two coils is L, so for the Helmholtz configuration L ¼ r. Here C1 is defined as in Equation 2.1, dimensions are in centimeters for cgs and meters for SI, and n is the number of turns in each coil. At the center of the coil system (z ¼ 0)

Figure 2.3 shows the field from each of the individual coils (normalized to unity), and the total field from both. G. G. Scott [J. Appl. Phys., 28 (1957) pp. 270–272] gives equations for both components of the field at points off the axis. For the same power consumption,

Fig. 2.2 Helmholtz coils. The spacing between coils L is equal to the coil radius r.

Helmholtz coils produce a field which is only a few percent of that produced by a solenoid of length r. They are thus confined to low-field applications.

By increasing the coil spacing slightly, the length of the uniform field region can be increased at the cost of a slight dip in field at the center. Figure 2.4 compares the computed fields of a “pure” Helmholtz coil with one where the coil spacing has been increased by 10%. The arrows show the distance over which the field is within 0.7% of the maximum field. Increasing the coil spacing by 10% lowers the maximum field by about 5%, but increases the length of the uniform field by almost 50%. Equation 2.4 is written in a form than permits the effect of changing the coil spacing L to be easily calculated.

Fig. 2.3 Field distribution in Helmholtz coils. Position is specified in units of the coil radius r. The light lines show the field (normalized to unity) from the individual coils (gray bars), located at +0.5 r, and the heavy line is the total field.

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Fig. 2.4 Field distribution in a “pure” Helmholtz coil (upper curve) compared with field distribution in a Helmholtz coil pair whose coil spacing has been increased by 10%. The dimension lines show the distance over which the field is uniform within 0.7% of the maximum field.

The power supply to a solenoid or Helmholtz coil pair must normally provide variable direct current. Variable dc power supplies, dry cells, lead-acid storage batteries, or alternating current rectifiers (with suitable filtering) can all be used. If the field needs to be reversed during the measurement, so as to record a hysteresis loop, either a reversing switch is needed or a bipolar power supply can be used. The bipolar supply permits a smooth and unbroken reversal of the field, which is often a significant advantage. Solenoids may be driven with alternating current (ac) when required. The inductance of the solenoid must be considered, which may require balancing capacitors and can lead to high voltages appearing across the terminals of the solenoid.

2.2.2High Field Solenoids

To produce very high fields with normal solenoids requires very large power input, and two major design problems must be addressed. The first is the removal of large amounts of heat. (Note that maintaining a steady magnetic field by means of an electric current is a process of exactly zero efficiency. All the input power goes into heat.) The second is providing sufficient mechanical strength to resist the large forces acting on current carriers in the presence of large fields.

Beginning about 1936, Francis Bitter [Rev. Sci. Instrum., 10 (1939) pp. 373–381] began the development of high-field solenoids of a new type. The coil of a Bitter magnet is sketched in Fig. 2.5, and a photograph of two partly assembled coils in shown in Fig. 2.6. (Any device which produces a field is commonly referred to as a “magnet,” whether or not it contains iron.) The winding is composed, not of wire, but of thin disks of copper or a copper alloy. These disks, usually 1 ft (30 cm) or more in diameter and about 0.04 in (1 mm) thick, have a central hole and a narrow radial slot and are insulated from each other by similarly cut sheets of thin insulating material. Each copper disk is rotated about 208 with respect to its neighbor, so that the region of overlap provides a conducting path for the current to flow from one disk to the next. The current path through the entire stack of disks is therefore helical, and the stack of disks acts as a solenoid. The disks are clamped tightly together and enclosed

in a case (not shown in the sketch). Cooling water is pumped axially through the magnet, through a large number of small holes or slots cut in each disk. At the National High Magnetic Field Laboratory (NHMFL) in Florida, dc fields of 45 T (450 kOe) can be

magnets of various sizes, maximum field, and field uniformity. The photographs in Fig. 2.7 show top and side views of a Bitter magnet ready for use.

Many aspects of the design of high-field Bitter magnets are treated in the book by Montgomery, previously cited. Bitter magnets require large motor-generator sets, consisting of an ac motor driving a dc generator, or else high power rectifier systems, to provide the necessary large variable dc currents. Such installations, together with the necessary cooling water supply and pumps, are expensive and only a few exist in the world.

Pulsed fields offer a less costly approach to the problem of measurements in high fields. If the measurement can be made quickly, by means of high-speed data collection, then only a transient field is needed. This is most commonly done by slowly charging a bank of capacitors and then abruptly discharging them through a solenoid. A large pulse of current, lasting a fraction of a second, is produced, and the problem of heat removal is thus greatly minimized or even eliminated, depending on the magnitude of the field required. Pulsed fields up to 200 kOe (20 T) or more can be attained in special watercooled or cryogenically-cooled solenoids, and pulsed fields of moderate strength (10–30 kOe, 1–3 T) in conventional, wire-wound, uncooled solenoids are easily obtained. Even higher fields and longer pulse durations of 0.5 s or more are reached by extracting the stored rotational energy of a large dc generator in a relatively long current pulse. The design of pulsed field solenoids must also take into account the large forces acting on the current carriers.

For even higher fields, flux compression devices can be used. A large field is created inside a heavy copper tube, which is then compressed radially inward, usually by explosive charges. A large induced current flows around the tube, which effectively keeps all the flux lines within the tube. As the tube area shrinks, the flux density, or field, increases. In this case, the sample is destroyed as the experiment is carried out.

2.2.3Superconducting Solenoids

The phenomenon of superconductivity provides a radically different approach to the highfield problem. When a normal metal is cooled near 0K, its electrical resistivity decreases to a low but nonzero value rr, called the residual resistivity (Fig. 2.8a). However, the resistivity of some metals and alloys decreases abruptly to zero at a critical temperature Tc. These materials are called superconductors; lead (Tc ¼ 7.2K) and tin (Tc ¼ 3.7K) are examples. (The magnetic properties of superconductors are also important; see Chapter 16.) If a current is once started in a circuit formed of a superconductor maintained below Tc, it will persist indefinitely without any power input or heat generation, because the resistance is zero. The attractive possibility at once presents itself of producing very large magnetic fields by making a solenoid of, for example, lead wire and operating it below Tc by immersing the windings in liquid helium (4.2K). However, soon after the discovery of superconductivity in 1911, it was found that an applied magnetic field decreased Tc and a field of a few hundred oersteds or several tens of milli-tesla destroyed the superconductivity completely (Fig. 2.8b). Thus, when the field produced by the solenoid itself exceeds a critical value

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Fig. 2.8 (a) Variation of electrical resistivity with temperature for a normal metal and a superconductor. (b) Dependence of critical temperature on magnetic field.

Hcr, the normal resistivity of the wire returns, along with the attendant problems of heating and power consumption.

The solution to this problem was not found until 1961 when Kunzler et al. [J. E. Kuntzler, E. Buehler, F. S. L. Hsu, and J. H. Wernick, Phys. Rev. Lett., 6 (1961) pp. 89–91] discovered that the niobium-tin intermetallic compound Nb3Sn remains superconducting even at a field of 88 kOe or 8.8 T or 7 MA m. It was later found that the critical field of this alloy at 4.2K, the temperature of liquid helium, is 220 kOe or 17.6 MA/m. Nb3Sn is very brittle, and various metallurgical problems had to be solved before it was successfully made in the form of a composite tape suitable for a solenoid winding. It was later found that Nb–Zr and Nb–Ti alloys, which are reasonably ductile, are superconducting up to fields of about 80 kOe or 6.4 MA/m at 4.2K. Superconducting solenoids of all three of these materials have been constructed.

Once current is flowing in a superconducting solenoid, no power input is required for the solenoid itself. However, a superconducting short-circuit link must be provided while the field is constant, and opened when the field needs to be altered. For this reason, superconducting solenoids in which the field needs to be swept, or frequently changed, are commonly operated with external power supplies and with nonsuperconducting leads carrying the current to the magnet.

A superconducting solenoid must be maintained at or near the temperature of liquid helium, which means that liquid helium must be purchased or a helium liquifier must be operated. The sample environment is therefore naturally at liquid helium temperature, but can be maintained at temperatures up to room temperature or even higher with appropriate equipment. Temperatures significantly above room temperature are hard to obtain. Superconducting solenoids are the common choice when fields above about 20 kOe or 2 T are required, up to a maximum of about 200 kOe or 20 T (unless the sample needs to be at high temperature). Superconducting materials with much higher critical temperatures have been discovered, but have not yet (2007) been made into successful high-field magnets.

The great interest in high magnetic fields extends beyond studies of their effects on the magnetic properties of materials. They are needed for a wide range of experiments in solidstate physics and biology, and for magnetic resonance imaging (MRI) systems.