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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).

[FORC definition]
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.

[FORC loops]
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.

[FORC diagram]
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 non-interacting 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 well-behaved 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.