Physics of Magnetism

CHAPTER 11. MAGNETIC ANISOTROPY 101

obtains

In this case, In the face diagonal direction, [110], one obtains

Substitution of these values into Eq. (11.9) leads to In the same way,

one finds for the [111] direction

After substitution into Eq. (11.9), one finds Combining these

results leads to

These determinations of anisotropy constants have the advantage that possible errors due
to strains are avoided, at least if these are isotropic and contribute equally to the energy
of magnetization in all directions. It is also important that the energies are determined
from curves between the remanence and the corresponding saturation value, rather than
from initial magnetization curves because various domain processes not connected with
crystalline anisotropy may contribute to the energy derived from the latter.
Other methods for determining the anisotropy constants make use of a torque mag­
netometer, by means of which it is possible to measure the torque, required to keep a
crystal with its axes inclined at various known angles with respect to an applied magnetic
field. In the ideal case, the measurements should be made with the sample cut in the shape
of an oblate ellipsoid but a thin disc is usually satisfactory, provided a field well in excess
of can be applied. The disc is rotated around an axis perpendicular to both its plane
and the applied field. It is most important that the sample have a circular shape and that
it be mounted symmetrically about its center, because otherwise spurious torques will be
introduced. It is difficult to interpret the results if the applied field does not saturate the
sample (see the example given below). For this reason, the torque magnetometer is not
frequently used for investigating permanent-magnet materials based on rare-earth elements
that have very large anisotropies.
In cubic materials, the torque curves are expected to depend on the crystal plane of the
sample. For a flat sample cut with its surface perpendicular to the [001] direction, one has
for instance

After substitution of these values into Eq. (11.9), one finds