Magnetic Anisotropy. Magnetoelastic Effects. Magnetic Domain Walls and Domains. Magnetization Process. Soft Magnetic Materials. Amorphous Materials: Magnetism and Disorder. Hard Magnetic Materials. Magnetic Annealing and Directional Order. Electronic Transport in Magnetic Materials. Please select Ok if you would like to proceed with this request anyway.
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The specific requirements or preferences of your reviewing publisher, classroom teacher, institution or organization should be applied. The E-mail Address es field is required. It is hoped that this text will contribute in some measure to the growing appreciation of the field of magnetic materials as a paradigm of the new scientific order in which the stubborn disciplinary barriers of physics, chemistry, mathematics, and metallurgy are not only breached but also reformed as bridges for scientific understanding and technological development.
I would like to begin by thanking by thesis advisor, Helmut Juretschke, who introduced me to the exciting field of magnetism and magnetic materials.
Particular gratitude goes to my students, postdocs and other colleagues over the years at MIT who have helped me learn various aspects of this fascinating field. I would especially like to thank Nick Grant, who saw a need for returning magnetism to the Department of Materials Science and Engineering, and mentored me in how to make it work here.
I have been privileged to teach much of the material in this book to graduate students at MIT since Also, with generous support from the National Magnet Lab through Jack Crow, Reza Abbascian hosted my sabbatical at the University of Florida in , where much of this material was given as a graduate course.
Hans-Joachim Giintherodt graciously hosted my visit to the University of Basel, where, with the support of the Swiss National Science Foundation, much of the material was presented to the diploma and graduate students.
I am truly grateful for these valuable experiences. Thanks go sincerely to my editors at Wiley, Greg Franklin, John Falcone, and Rosalyn Farkas who had faith in the project as well as patient hope for its delivery before the new millennium. The anonymous, Wiley-selected reviewers of the early drafts were extremely helpful in their specific comments and broader recommendations.
Particular and heartfelt thanks go to those friends and colleagues who selflessly read, corrected, and commented on drafts of various chapters. P deeply appreciate the time they took to improve the treatment of various topics and search out errors. While much of the material presented here has appeared elsewhere, the selection, arrangement, and presentation of the material is, for better or worse, my own. I apologize in advance to those whose important contributions have been overlooked here because of my own unfamiliarity with them or my inability fully to appreciate their significance.
H am extremely grateful to Mrs. Lee Ward, who worked tirelessly on the drawings and patiently accepted my suggestions and reversals of opinion on many of the figures. Special thanks go to Mr. Robin Lippincott, who worked with the manuscript as it evolved, through several generations of hardware and software, from class notes to its present form.
The formidable task of keeping track of the references, figures, tables, and permissions as material was rearranged, was accomplished only by his tireless work. Robin dealt calmly with my unending revisions and changes of notation. He also applied his own professional writing skills to editing my drafts.
Although his literary style soars above the dry technical genre, he was still able to detect and expunge many of my awkward and incorrect constructions. Thank you so much, Robin. Finally, I would like to thank my wife, Carol, and children, Kevin, Meghan, and Kara, who shared my time with this book for too many nights, weekends, and years. Their love and encouragement made it easier to persevere when the task grew bigger than H ever would have imagined. O'Handley is a Senior Research Scientist in the Department of Materials Science and Engineering at the Massachusetts Institute of Technology, where he has taught several undergraduate and graduate courses and conducted research on magnetic materials, superconductors, and a variety of rapidly solidified materials.
Prior to M. He earned his Ph. He received his B. O'Handley has authored more than technical and scientific publications, including several book chapters and review articles. He holds more than ten U. He has been active in the annual Conference on Magnetism and Magnetic Materials since serving as Publications Cochairman from to It reviews many elementary concepts Maxwell's equations in differential and integral form, units, concepts of magnetic fields and magnetic moments, types of magnetism, and generic applications that may be familiar to some readers.
The shao shih is a ladle-shaped magnet that balances and pivots on a brass plate. The handle of the ladle is the north-seeking pole of the compass. The English word magnet came from Magnesia, the name of a region of the ancient Middle East, in what is now Turkey, where magnetic ores were found. But the development of a magnetic abacus, that is, a computer with binary magnetic information storage, took thousands more years to achieve. The late Dr.
An Wang pioneered the use of magnetic core memories in his early Wang computers. These magnetic memories see Section AWer a current pulse of critical magnitude related to the coercive field of the core at a given address, the core remained magnetized a remanent magnetization in a given direction persists in zero field. The core could be read or overwritten by later pulses. Today, information technologies ranging from personal computers to mainframes use magnetic materials to store information on tapes, floppy diskettes, and hard disks.
The dollar value of magnetic components coming out of Silicon Valley is greater than that of the semiconductor components made there. Our seemingly insatiable appetite for more computer memory will probably be met by a variety of magnetic recording technologies based on nanocrystalline thinfilm media and magnetooptic materials. Personal computers and many of our consumer and industrial electronics components are now powered largely by lightweight switch-mode power supplies using new magnetic materials technology that was unavailable 20 years ago.
Magnetic materials touch many other aspects of our lives. Each automobile contains dozens of motors, actuators, sensors, inductors, and other electromagnetic and magnetomechanical components using hard permanent as well as soft magnetic materials.
Finally, magnetic materials are the backbone of expanding businesses: electronic article surveillance, asset protection, and access control. Tiny strips or films of specially processed magnetic materials store one or more bits of infomation about an item or about the owner of an identification badge. Access to secure areas or inappropriate removal of merchandise or property can be monitored and controlled.
The purpose of this text is to introduce readers to the basic concepts needed to understand magnetism, magnetic materials, and their applications and bring readers to a point from which they can appreciate the technical literature.
The text moves from an exposition of the principles of magnetism to a consideration of some of the major classes and applications of magnetic materials. The introductory chapters consider magnetism from three perspectives: 1. What requirements do Maxwell's equations place on magnetism?
And does classical electron theory indicate? Why is quantum mechanics needed to understand magnetism? The major energies controlling magnetic processes, domain wall formation, and technical magnetism are considered next. These principles are then applied to specific magnetic materials used in various devices. Several chapters are dedicated to the properties of specific magnetic materials in three classes: soft, nanocrystalline, and hard magnetic materials. A number of more advanced topics are included to meet the needs of specific audiences.
Throughout the text the treatment is intended to be empirical, moving from observation to understanding. An effort is made to build new concepts on familiar ones. It is worth reviewing what may be three fairly familiar observations related to magnetism. Their quantitative understanding will lead to a discussion of Maxwell's equations. Here the thumb indicates the direction of the positive current I and the fingers indicate the direction of the magnetic field lines B.
It is important to know the magnitude and direction of this field as well as its dependence on current and on distance from the wire. The topology of the field has the cylindrical symmetry of a torus, and its sense is again given by a right-hand rule. With the fingers this time indicating the direction of the current, the thumb gives the direction of the magnetic field inside the solenoid. The field outside follows from the symmetry of a torus.
Because solenoids are often used to provide fields for testing magnetic materials, it is important to be able to calculate the strength of the field along the axis of the solenoid. Figure 1. It is believed that the magnetic field of the earth results from a current in its molten iron core. What would the direction of such a current have to be for the black end of a compass needle like that in Figure 1.
The voltage results from a change in magnetic flux inside the coil. This induced voltage is a result of Lena' law --if there is a change in flux in a coil, a voltage is induced in the coil with a sense that would produce a current whose magnetic field opposes the initial change.
Note that the voltage in Figure 1. The sense of the voltage is also given by a right-hand rule with assignments similar to those in Figure 1. Figanre 1. Observations similar to 1 and 2 above were first reported by Hans Christian Oersted, and in the early nineteenth century Andrt Ampere was able to describe them mathematically. Observation 3 was first recorded by Michael Faraday, who used it to write the mathematical form of the law of magnetic induction.
In order to be able to calculate the magnitude and direction of the fields described here qualitatively, the set of magnetic and electric fields must be defined and the equations that relate them to each other and to charge and current distributions, must be understood.
However, because the literature abounds with data in cgs and other units, cgs equations and quantities will often be given along with mks.
An Appendix to this chapter summarizes the equations and conversion factors for the important magnetic quantities in mks and cgs units. It is known that an electric field can cause positive and negative charges in a material to be displaced relative to each other creating an electric dipole moment p coulomb-meter.
The parameters pr and X, are different ways of describing the response of a material to magnetic fields. In general, the permeability and susceptibility are tensors because they relate two vector quantities that need not be parallel. Thus 14 is the cause and M as the material effect.
B is a field that includes both the external field poM, due to macroscopic currents, and the material response, p,M, due to microscopic currents.
Ferromagnets represent a low reluctance path for magnetic field lines; hence they draw in the flux of a nearby field and add to it by their magnetization Fig. An explanation will be sought for the strength of the atomic magnetic dipole moment strength after a review of Maxwell's equations and their consequences.
However, inside a material there is an additional contribution to B from the sample magnetization M. H erg ]. The general form of these equations is such that a characteristic of a given field, such as its divergence or its rotational quality, is equated to a source term, namely, charge or current density or to a time change in a complementary field. Comparing Eqs. The B field curls around J in a right-hand sense Figs. The negative sign in the Maxwell-Faraday equation is a manifestation of kenz' law; a changing B field induces a back electromotive force EMF opposing the current change that gave rise to the B field change.
Alternatively, a changing B field induces an electric field whose current generates a magnetic field that opposes the change in the first B field. There will often be interest in the use of Eqs. In these cases, the integral forms of Maxwell's equations are more useful. This can be seen more clearly on integrating this differential equation over a volume containing the charge density.
Integrating both divergence Eqs. Magnetic poles always come in pairs, usually designated north and south, called dipoles. The latter two, originating with Faraday and AmpGre, respectively, are at the foundation of our traditional understanding and application of magnetism.
Equations 1. Amp2re -the normal component of a current density J passing through an area A gives rise to a B field circulating around that area. But allowance should be made for the existence of currents, J 0 that is why this case is sometimes called magnetoquasistatics; charges can still move, but the fields are independent of time. In this case, Maxwell's differential equations become SI: v. Complete magnetostatics apply in the limit that there is no current flowing.
This case is treated in Chapter 2. These f o m s of Maxwell's equations form the basis of electrostatics and magnetostatics. The most fundamental applications of these equations to magnetism are now reviewed. Symmetry suggests that B circulates around the wire, so it can be assumed that B is circular and has a constant value at a distance R from the wire Fig.
From Figure 1. The area integral of the RNS gives p o l. Construction for calculation of strength of B field is shown on right. The same result as Eq. It is done most conveniently in cylindrical coordinates. The magnetic field shape about the current loop is toroidal as can be deduced from the right-hand rule for a current-carrying wire Fig. This equation will be derived later.
At large distances, the field of a current loop is the same as that of a small bar magnet, a magnetic dipole Fig. A solenoid is now formed consisting of IV of these loops, each carrying a current I. The field inside the solenoid can be calculated simply by integrating the Maxwell-Ampere equation over an area normal to several adjacent turns Fig. Inside the solenoid the B field is strong because the field lines are compressed; outside the solenoid the B field is weak because the field lines are spread out.
The return field just outside the middle of an infinite solenoid is zero; along the outside branch of the rectangular path in Fig. Right, shape of dipole field about a current loop above and about a permanent magnet below.
XX xxxxxx Bin Figure 1. The fact that the magnetic moments of materials arise from microscopic, atomic-scale, currents is now justified.
What is the atomic magnetic moment p,? Ht will be shown that atomic magnetic moments come from microscopic current loops. Consider a line of N circular atomic current loops with a common axis Fig. This represents a number of atoms with their atomic orbitals aligned along a given direction in a material. The solenoid equation, Eq. Thus, Eq. This is a crude plausibility argument for a very important relation.
Thus, atomic magnetism has its origin in microscopic currents and the atomic magnetic moments can be calculated if the currents and the areas they enclose are known. Another microscopic current, in addition to electron orbital motion, also important to magnetism, is one due to the intrinsic angular momentum or spin of electrons.
Spin will be discussed in Chapter 3. For now, note that I A can be replaced by p, in Eq. The p, notation was used till now in order to avoid confusion with the permeability and also to draw attention to the analogy between electric, p,, and magnetic, p,, dipole moments.
But atomic magnetic moments are not necessarily aligned in all materials. The way the local atomic moments couple to each other, parallel, antiparallel, or not at all, provides the first way of classifying magnetic materials.
The individual atomic moments ,urnmay be randomly oriented if they do not interact with each other. Such uncoupled magnetic moments may be aligned partially depending on thermally induced agitation in an applied magnetic field H.
This weak field-induced magnetization behavior defines a paramagnet. This defines an ordered magnetic material, examples of which are ferromagnets, antiferromagnets, and ferrimagnets. The dimensionless in SI. This value of magnetization corresponds to a flux density poM of order T, which is much less than the 1 T that our hydrogenic model suggests for fully aligned moments.
What is keeping the moments in a paramagnet from aligning in an external field? Perhaps it is thermal energy, kB T The degree to which a paramagnetic moment will respond to a field can be appreciated by considering the potential energy U of the moment pm in an applied field B : [In mks units the energy is written as -p; B because the factor p, needed to Figure 1.
Inset shows a schematic of the distribution of local moments in the paramagnetic case. So it can be seen why paramagnetic susceptibilities are so small.
The field has only a weak linear effect in aligning the moments because thermal energy is large relative to the magnetic energy. Proper derivations of susceptibilities will be given in Chapters 3 and 4. The spontaneous, long-range magnetization of a ferromagnet is observed to vanish above an ordering temperature called the Curie temperature T, Fig. T, is the Curie temperature. Where does the energy for magnetic alignment in a ferromagnet come from?
Is not the moment-disordering effect of thermal energy just as strong in a ferromagnetic solid; could there be something other than an external field contributing to the tendency to align the local moments? The molecular field will be examined in Chapters 4 and 5. But if ferromagnets have such strong magnetizations, why do two pieces of iron not attract each other the way they are attracted to a permanent magnet? Mow is iron magnetized or demagnetized? Weiss came up with an hypothesis for that, too.
He postulated the existence of magnetic domains, regions ranging in size upwards from approximately 0. Domains are separated from each other by domain walls, surfaces over which the orientation of p, changes relatively abruptly within about IOOnm.
The domain walls will be the subject of Chapter 8 and domains will be examined more closely in Chapter 9. The magnetizations in different domains have different directions so that over the whole sample their vector sum may vanish. The image was taken with a scanning electron microscope fitted with a special detector to reveal magnetization direction; see Chapter Hn this figure the directions of the crystal are parallel to the figure edges.
It is not a coincidence that M inside the domains is parallel to these crystallographic directions. Crystallographic axes lie in the image plane along the horizontal and vertical directions.
Left panel shows magnetic contrast when the instrument is sensitive to the horizontal component of magnetization; dark is magnetized to the left, light to the right.
In the right panel, the contrast is sensitive to vertical component of magnetization; dark is magnetized down, light is magnetized up.
Courtesy of R. Celotta et al. Note also that domains often form so as to create closed flux loops. This minimizes magnetostatic energy Chapter 2. In an ideal magnet, does it matter much where the domain wall lies?
If the potential energy of a domain wall were independent of its position, it would only take a relatively weak field to move the wall, much as it is relatively easy to move a ripple in a carpet. In some soft magnetic materials, domain walls can be moved with fields of order 0. However, defects such as grain boundaries and precipitates cause the wall energy to depend on position, so in most materials, higher fields are required to move domain walls Chapter 9.
It is now possible to understand, at least qualitatively, the B-H loops of ferromagnets and connect them with domain wall motion and other magnetization processes. But before describing the B-H loop of a ferromagnet, the classification of strongly magnetic materials should be completed. Such materials are called antiferrornagnets. Chromium has the BCC structure with the body-center atoms having one direction of spin parallel to 1 and typically of order lop4. At higher fields B increases sharply and the permeability increases to its maximum value p,,,.
When most domain wall motion has been completed there often remains domains with nonzero components of magnetization at right angles to the applied field direction. The magnetization in these domains must be rotated into the field direction to minimize the potential energy - M - B. This process generally costs more energy than wall motion because it involves rotating the magnetization away from an "easy" direction [which may be fixed in the sample by sample shape Chapter 2 , crystallography Chapter 6 , stress Chapter 7 , or atomic pair ordering Chapter 14 ].
On decreasing the magnitude of the applied field, the magnetization rotates back toward its "easy" directions, generally without hysteresis i.
As the applied field decreases further, domain walls begin moving back across the sample. Because energy is lost when a domain wall jumps abruptly from one local energy minimum to the next Barkhausen jumps , wall motion is an irreversible, lossy process.
The B-H loop opens up, that is, it shows hysteresis, when lossy magnetization processes are involved. The induction and magnetization remaining in the sample when the applied field is zero are called the residual induction B, and remanence M,, respectively.
The reverse field needed to restore B to zero is called the coercivity or coercive field H,. It is a good measure of the ease or difficulty of magnetizing a material. The field needed to restore M to zero is called the intrinsic coercivity, ,H,. The distinction between H, and ,H, is of importance only in permanent magnets because in a soft magnetic material H , -- t- Figure 2.
Upper right, free poles at surfaces and field lines that they give rise to. Lower left, geometry for calculation of field components due to surface charge. Lower right, variation of the magnitude of the internal field with position inside a thicker sample.
In cgs units the right-hand side expressions in Eqs. For a thicker sample, the field from each surface drops off with distance from the surface, so the internal field varies approximately as sketched in the lower right of Figure 2. The boundary conditions on B and H are derived from Maxwell's equations. Figure 2. Equation 2. In the limit that the height h of the pillbox shrinlcs to zero, the only contributions to the surface integral come from the normal component of B passing through the top and bottom surfaces: Mere 8, is the angle between the B field in each medium and the surface normal a,not the pillbox normal.
Also from Eq. Near the center of a magnetically charged surface, the perpendicular components of the field due to the surface charge are equal and opposite Eq.
For another boundary condition on the H field, the integral form of V x H and Stokes theorem are used to give [cf. An edge view of a right-handed, closed line element with long segments on opposite sides of the interface.
Note the difference between n, the interface normal, and n', the normal to the area enclosed by the path of integration. The normal n' to the area enclosed by this path is determined by a right-hand rule; n' is tangent to the interface. Again, as the dimension of the rectangular path normal to the material interface shrinks to zero, Eq.
We can write Eq. We summarize these boundary conditions on B and H. These boundary conditions will help us to understand the phenomena in Figures 2.
The constructions in Figure 2. Expressing the RHS of Eq. Choose the y direction normal to the interface so that the limits of y integration go to zero. In this case the integral in Eq. Thus, K is the average current along the surface per unit length transverse to the current path.
It is concentrated right at the surface. At the left is sketched the magnetized bar and its surface poles or the equivalent surface current around the bar , which are the sources for the H and B fields, respectively. Only one quarter of the bar is sketched at right because of the symmetry of the situation. It is the output of micromagnetic calculations of the field distribution inside and around a uniformly magnetized bar.
Only the upper right quadrant of the bar, of finite extent out of the paper, is shown. Note that the surface charges are sources for the H field inside and outside the sample. An equivalent current through the surface windings is the source of the B field inside and outside the sample. Note also that the boundary conditions on B normal component continuous across surface and H tangential component continuous across surface are properly satisfied in these calculated fields. M is proportional to B - H, which in this case is uniform inside the bar.
The boundary conditions on B and H are properly satisfied. The H field inside the magnetized sample in Figure 2. In a sample that is not so constrained, this internal field would tend to demagnetize the sample. We now consider such demagnetizing fields separately. At the end of a sample that is assumed to be magnetized perpendicular to the ends without application of an external field, Eq. It is now shown how this explains the introductory examples. What happens in Figures 2. This surface pole field or dipole field is the demagnetizing field H, mentioned earlier.
The internal field is sometimes written Comparison with Eq. We have so far assumed M is normal to the surface. For an arbitrarily shaped sample, the demagnetizing field for a given direction of M relative to the sample axes may be approximated as The constant of proportionality N is called the demagnetization factor and, in general, it is a tensor function of sample shape.
For ellipsoids N is a diagonal tensor and can be calculated because for those shapes, the internal field turns out to be uniform. It tells us the component of flux density normal to the surface for a given shape and direction of magnetization.
It expresses the effect of the unit vector IZ in Eq. Thus, for soft magnetic materials, where a large magnetization results from a relatively weak external field, the internal field can be much less than the applied field even if the shape factor N is very much less than unity.
However, for permanentmagnet materials, where very large external fields are required to achieve appreciable magnetization, shape effects become important only for much smaller-aspect-ratio, larger N, samples Fig. We can quantitatively interpret Figure 2. The coercivity is defined as the external field needed to reduce B, to zero.
From Eq. If H , f 0, the remanent flux density is given by Eq. It should be clear from Figure 2. Hence, as was stated above, N should be a tensor function of position and orientation in a given sample. However, N is a double-valued or triple-valued diagonal tensor for magnetization along the principal axes of an ellipsoid.
For a truly ellipsoidal specimen in a uniform field oriented along one of the major axes, the magnetization is uniform throughout the sample. This is not true for samples of other shapes where the magnetic response is greatly reduced by dipole fields near corners of rectangular specimens. The dashed lines rotated into the vertical axis in each case relate one loop to the other. Clearly, it costs less field energy to magnetize ferromagnetic materials along their long directions, preferably in closed circuits, because then there are no surface poles to cause opposing fields.
The exact demagnetization factors for various ellipsoids have been calculated for magnetization along the three axes, for arbitrary aspect ratios Osborn These equations may be used as approximations for samples of similar geometries. The curves for nonellipsoidal bodies depend on the permeability of the material.
There is an approximation for a two-dimensional problem where the field is essentially uniform in the third direction. The demagnetization factors are taken from the demagnetization fields calculated from Eqs. For example, in Figure 2. As a cautionary note, it is emphasized that N is not a constant inside any magnetized sample that is not an ellipsoid; N is merely approximated. Consider a thin film of dimensions w :h :t 25 : 5 : 1 Fig. The two-dimensional approximation, Eq. Use of Eq. The large disparity in these various approximations is due to the square shape and small aspect ratio of this thin film.
The preference for the magnetization to lie in a particular direction in a polycrystalline sample is given by the shape anisotropy.
See Figure 2. The Zeeman energy density due to the magnetization orientation in the applied field is Figure 2. The magnetization takes on an orientation that minimizes the total energy density u. The minimum energy is found at a value of 8 for which dulde, the negative of the torque on M , is zero and u has positive curvature. The torque is zero when the energy is an extremum: Divide both sides of Eq.
Below saturation Eq. It cannot be met for " N M ,. This is clear by considering Figure 2. For H c H,,,,,,,io,, the magnetization increases linearly with H. M Figure 2. Application of a field H transverse to the EA results in rotation of the domain magnetizations but no wall motion.
Note that the demagnetizing factor can be determined from the field at the knee of the M-H curve if Ms is known and if there is no anisotropy other than shape. Alternatively, Ms can be determined from the saturation field if N is known. This field is about 0. If domain walls parallel to the easy direction are present, this simple result is unchanged because the hard-axis field causes no wall motion and rotates the domain magnetization by equal amounts Fig.
Even for application of a field along an axis of relatively easy magnetization Fig. The wall motion that results from application of the field will be justified in Chapter The internal field seen by the material is the applied field plus the demagnetizing field [Eq. Note that as the shape of each domain changes, its demagnetizing factor changes Fig. The M-H curve is no longer linear in H but is sublinear Fig. Thus the loop is sheared over by the sample's shape as indicated in Figure 2.
Note that it is not possible from this measurement alone to distinguish shape anisotropy from some other source of anisotropy. Example 2. In the case of thin ribbons of amorphous magnetic material 25 pm thick and mm wide , changes in length from cm dramatically alter the M-H curves Fig. We can calculate the demagnetization effects on the M-Hex, loop by expressing the measured magnetization in terms of the external and demagnetizing fields 2.
See Problem 2. The effective susceptibility, which measures the initial slope of M-He,,, is sharply reduced in shorter samples even though N is of order The data for the longest sample in Figure 2. Approximating the ribbon as an extremely prolate ellipsoid, Eq. Cutting the ribbon length from 10 to 5 cm increases N derived from the data by a factor of 3.
It costs energy to place dipoles adjacent to each other if they have the same orientation as in Figure 2. This energy is stored in the fields about the dipole configuration. The energy cost is less when you assemble dipoles head to tail as in Figure 2.
Two end-to-end dipoles have a negative energy that gets more negative the closer they are. Let us make these concepts quantitative. The same result can be extended to a rigid assembly of dipoles. The factor of enters because dipole 4 Figure Assembly of dipoles in high-energy configuration a and low-energy configuration b. These relations are very useful.
It is appropriate to examine more carefully the process of magnetizing a sample. Consider the magnetization curve in Figure 2. For the process of magnetizing the sample from the demagnetized state to any point MI,B, , three energy densities can be defined: the potential energy of the magnetized sample in B,, --M,B,, as well as the energies A, and A,.
What is the physical significance of these two contributions to the energy density? A , is the work done by thefield to bring the sample to the state of magnetization M,. Consider a magnetic sample of cross section A and volume A1 inside a solenoid of N turns over the length 1 of the sample. This dzfferentinl energy is the area inside of the rectangular elements between the M - B curve and the M axis in Figure 2.
Its integral is A, as defined in Eq. The term A, is proportional to the energy given up by the magnetized material as it is drawn into a field. Thus, A, represents work done on the sample and A, describes work done by the sample.
Note that A, and A, are functions of the path by which the sample is magnetized. The internal energy of the sample is its potential energy, -p,M,H,,plus the work done to magnetize it: If a sample is already magnetized in the absence of a field i.
The internal energy of a magnetic sample is decreased in the presence of a field; a magnetic sample can do worlr when exposed to a magnetic field. In Chapter 6 the values of these integrals A, and A, will be shown to depend on the direction in which a crystal is magnetized. Their variation with direction defines the anisotropy energies. In situations for which? This definition of the scalar magnetic potential is used by Jackson Here the practice more common in the engineering literature is followed.
M defines a volume magnetic charge density p,, again in analogy with electrostatics. Solutions for 4 , may be obtained from the differential Eq. M , surface charges, A4. Brown has shown that Eq. The 2D equations used for surface poles [Eqs. It was shown in Section 2. The volume term in Eq.