



Introduction to FORC Diagrams
First Order Reversal Curve (FORC) diagrams provide a detailed
characterization of hysteresis behavior, and can be a powerful
tool in the study of magnetic materials. We have developed an
efficient method for the measurement of high resolution FORC
diagrams, and have measured the FORC diagrams of a wide range of
samples, including thin films, rare earth magnets, geological
samples, and frozen ferrofluids.
Definition of a FORC Diagram
The acquisition of a FORC diagram involves the measurement of First
Order Reversal Curves (FORCs). As shown in Figure 1, the
measurement of a FORC begins by saturating a sample in a large
positive applied field. The field is decreased to a reversal (or
return or recoil) field H_r, and the FORC is the magnetization curve
that results when the applied field is increased back up to saturation.
The magnetization at the applied field H_a on the FORC with reversal
field H_r by M(H_r,H_a).
Figure 1: Definition of a First Order Reversal Curve (FORC).
In our method, we typically measure 100 FORCs, and obtain a FORC data
set such as that shown in Figure 2, in which the sample was a
volcanic rock from Nevada containing weakly interacting single domain
particles.
Figure 2: 100 measured FORCs for a sample of volcanic rock.
It is useful to transform a set of FORC data into a FORC diagram
contour plot. The FORC distribution is defined as the negative of the
mixed second derivative of the magentization with respect to H_r and H_a.
In a FORC diagram, we plot the FORC distribution after a change of
coordinates from {H_r,H_a} to {H_c = (H_a  H_r)/2,
H_b = (H_a + H_r)/2}. The FORC diagram in Figure 3
was calculated from the data set in Figure 2. In the calculation
of the FORC distribution from data, a certain amount numerical
smoothing is unavoidable; the degree of numerical smoothing is
denoted by SF. Further details on the measurement of a FORC diagram
have been described elsewhere.
Figure 3: FORC diagram (SF=2) calculated from the data in Figure 2.
The white region indicates large positive values of the FORC distribution
defined above. This sample demonstrates the highly elongated
contours characteristic of a noninteracting single domain particle system.
Relationship to the Preisach Model
The original motivation for our measurements was to obtain
Preisach distributions. However, as explained next, further
consideration has led us to use the terms FORC distribution
and FORC diagram.
In our measurements we frequently encounter distributions with
dramatic asymmetries with respect to the central horizontal axis.
We also frequently encounter regions where the distribution has
negative values. Such negative values and this asymmetry are not
allowed features of a Preisach distribution. However, they are
important aspects of our experimental data. We could mathematically
manipulate the data in an effort to erase these apparent ``anomalies''.
But instead, we have decided to study the measured distributions in
their raw form, independent of the Preisach model or any other model.
In order to emphasize that these distributions are independent of
the Preisach model and not subject to the requirements that constrain
a Preisach distribution, we refer to them as FORC distributions.
We have found that FORC distributions and diagrams can be useful tools
in the characterization of magnetic materials.
Experimental Details
In our measurement of a FORC data set, we choose evenly spaced reversal fields,
where the field spacing is denoted by FSpc. The data points on each individual FORC
are taken with the same field spacing. The same measurement time, denoted by, t_m,
is spent at every data point. A typically value for t_m would be 0.3 seconds. To
calculate the FORC distribution at a specific data point, a local polynomial fit of
the magnetization M(H_r,H_a) is done around that point, and the mixed second
derivative with respect to H_r and H_a can be easily obtained from one of
the parameters of this fit.
Most of our measurements have been done on AFGMs (Alternating field gradient
magnetometer) made by Princeton Instruments Corporation. The Paleomagnetics group
at the University of Utrecht, together with Chris Pike, have modified the software
system for the AFGM in order to tailor it to the requirements of a FORC diagram
measurement. This has made the measurement procedure much more time efficient than
it would otherwise be. Presently, a FORC diagram measurement on a typical sample of
magnetic recording media takes roughly an hour. Samples with weaker signal can take
longer.
At UC Davis, we have written a software program in Mathematica which can
directly take the output file of the Utrecht software and conveniently output a FORC
diagram.
The local polynomial fit described above is done over a grid of
(SF+1)*(SF+1) data points. For a wellbehaved sample, SF
is set to 2. For samples
with poor signal to noise ratio, SF is increased to 3 and sometimes 4. The larger
SF is, the more the resulting diagram is smoothed. As a results, some finer
features may be lost when SF is increased.
In general, the signal to noise
requirements of a FORC diagram measurement are much greater than the requirements
for a major hysteresis loop, due to the second derivative one has to perform.
In the case
of geological samples, where signals are generally weak, sample preparation is
critical. We have developed a simple mold which enables us to get the maximum
amount of material on the AFGM probe tip, hence affording the maximum possible
signal. Finally, the results can also be improved by increasing the
measurement time t_m.
Selected FORC Diagram Results
We have performed FORC diagram measurements on nearly 100 samples spanning a
wide range of systems. You can find some selected samples
HERE.
