Basic Permanent Magnetism:

a magnet's intrinsic property.

by Dr. Peter Campbell

In Section 1A we showed that a permanent magnet's magnetization M will provide a magnetizing force H which establishes a flux density B, these three being directly related. When acting on a microscopic magnetic dipole moment µ_m, this flux causes work to be done to rotate µ_m into alignment with B. Combining the equations that describe these two facts, we find that this work E may be considered to comprise two components:

The first term represents the work done by the magnetizing force (applied field) upon an individual moment µ_m, while the second term represents the work done by the net distribution of moments (i.e. the magnetization) upon an individual moment µ_m.

Magnetocrystalline Anisotropy

A study of this energy within a magnetic material will indicate whether it is a permanent magnet, and will yield a magnet's most fundamental property, its intrinsic coercivity. We will illustrate this in a simplified way, but the explanation will provide an understanding of how permanent magnets work.

Firstly, let's just look at the second term, in which it is instructive to replace the cosine function with its expansion, in order to separate the constant from the varying quantities:

The varying quantity is only the second term of this equation, and this is really all that concerns us when studying the energy of the magnet's crystallographic structure. Because of this, the second term is called the magnetocrystalline anisotropy energy E_k, the minimization of which expresses any preferred direction for an individual moment µ_m within a crystal structure having a net magnetization M:

Conversely, the maximization of this E_k function indicates when the orientation of an individual moment µ_m is unstable within a crystal structure. This function is maximized at 180^o, being the unstable condition when µ_m is directly opposed to the net M (the stable state is when µ_m is aligned with M, at 0^o/360^o).

Real crystal structures are much more complicated than this, but we can at least take our simplified analysis one step further, and consider iron, which is one of the most common constituent elements in the alloys from which permanent magnets are made. Iron has a crystal structure which places individual moments µ_m in a "body-centered-cubic" lattice. In this case, the stable states for an individual moment µ_m are whenever it is aligned with any of the planes which occur every 90^o, so the unstable condition now first occurs at 45^o. To apply to iron, the last equation should therefore be modified to force the maximum to first occur at 45^o rather than 180^o, which is achieved by setting:

While you don't need to worry about this, it is common to define a crystallographic constant for the material K₁ = 8µ_oµ_mM, so that:

The benefit of doing this is that it moves us from consideration of microscopic quantities such as µ_m towards just dealing with the material's macroscopic properties. We can now look at the first term of the energy E expression, which represented the work done E_f by a field H upon an individual moment µ_m, and we can apply this also to the macroscopic property M. Remember that we defined M in Section 1A as the summation of µ_m over an elemental volume (if you don't remember, click on the button below to see that definition). If all the moments are aligned within this volume, M will assume its saturation value of M_sat, and the field H will try to rotate M_sat (i.e. all of the µ_m) away from one of the preferred crystal planes and into alignment with H.

This situation is illustrated by this diagram, for which the first term is modified to yield a macroscopic version of E_f:

Now, the total energy E within a permanent magnet is the sum of the two terms: its magnetocrystalline anisotropy energy E_k plus its field energy E_f:

In any practical application, a permanent magnet works best if its properties are optimized to provide magnetic field in a "preferred" direction, which means that M_sat should be aligned along a "preferred" axis. Now, the material is of little use as a permanent magnet if M_sat can be easily disrupted by an externally applied field, particularly one on the "preferred" axis which directly opposes M_sat. So, we can use the energy equation to investigate the ability of M_sat to withstand a reverse applied H, and use this information to characterize the quality of the material as a permanent magnet. This is done by setting H at 180^o to M_sat , differentiating E to find the minimum total energy, then differentiating a second time and setting the result =0 to find the reversal of the rate of change of energy:

This procedure yields the applied field -H which is just strong enough to flip over M_sat into the reverse direction, i.e. the amount of reverse applied field that the magnet can withstand. Since the angle of M_sat is 0^o, this equation yields a unique value for this critical field which is known as the intrinsic coercivity H_ci of the material:

Notice that H_ci is a most fundamental characteristic property of a permanent magnet, being dependent only upon the material's crystallographic constant and saturation magnetization. Its derivation was based upon the material's magnetocrystalline anisotropy, which occurs in all types of permanent magnet, exclusively in many.

We can now illustrate this behavior graphically. The permanent magnet maintains a magnetization of +M_sat until it experiences a reverse applied field of magnitude-H_ci, at which point the magnetization flips over to -M_sat. It then requires a forward applied field +H_ci to flip the magnetization back to +M_sat again. This is the intrinsic magnetization characteristic of the material. Actually, for the practical application of a permanent magnet, we are only interested in the upper half of this diagram. The first quadrant (top right) represents the region in which the magnet is initially magnetized, and the second quadrant (top left) represents the region in which a magnet is doing work against a reverse applied field, but hopefully not exceeding-H_ci. Therefore, only the second quadrant is needed to design a magnet to work in an application, and this is known as the intrinsic demagnetization curve.

In the next section, we relate this ideal intrinsic demagnetization curve to practical curves that are measured on actual magnet materials, which can be used for permanent magnet design.