[Paper Review] FORC+: A method for separating reversible from irreversible behavior using first order reversal curves

P. B. Visscher|arXiv (Cornell University)|Oct 28, 2016

Magnetic Properties and Applications3 citations

TL;DR

This paper introduces FORC+, a novel method that cleanly separates reversible and irreversible magnetic behaviors in first-order reversal curve (FORC) analysis by deriving a distinct 'saturation field distribution' for reversible components, while preserving the standard irreversible FORC distribution. The approach uses discrete derivatives of magnetization data to isolate reversible contributions, enabling unambiguous identification of both hard- and easy-axis Stoner-Wohlfarth particles in mixed systems.

ABSTRACT

First Order Reversal Curves (FORCs) have been used for a number of years for the extraction of information from magnetization measurements. The results are most unambiguous for irreversible processes -- for a collection of Preisach hysterons, one gets a "FORC distribution" $ρ(H_{down},H_{up})$, the number of hysterons with given downward \& upward reversal fields. There have been many proposals for dealing with reversible behavior, usually involving inserting it somehow into the irreversible FORC distribution. Here we will try to do the opposite, to separate them into another function which we will call the (reversible) "saturation field distribution", which is identically zero for a completely irreversible system of hysterons, while the irreversible FORC distribution is identically zero for a reversible system. Thus in a system with both purely reversible and purely irreversible components, such as single-domain Stoner-Wohlfarth particles with hard or easy axis along the field, this approach cleanly separates them. For more complicated systems, as with conventional FORC distributions, it at least provides a "signature" making it possible to identify microscopic models that might give a particular pair of irreversible and reversible distributions.

Motivation & Objective

• To address the long-standing challenge of distinguishing reversible from irreversible magnetic contributions in FORC analysis.
• To develop a method that cleanly separates reversible and irreversible components without contaminating the standard FORC distribution.
• To provide a physically meaningful distribution for reversible behavior, specifically saturation fields of hard-axis particles.
• To enable accurate identification of microscopic models underlying experimental FORC data by revealing distinct signatures for reversible and irreversible components.
• To overcome limitations of conventional FORC visualization, which can obscure or mix reversible and irreversible effects due to interpolation artifacts.

Proposed method

• Derives a discrete analog of the standard FORC equation using finite field spacing δ, enabling clearer visualization and derivation.
• Introduces a new quantity, $ M_H(H_R, H) $, defined as the finite difference of $ M(H_R, H) $, which vanishes at $ H = H_R $ for irreversible systems.
• Proposes a second derivative of $ M_H $, expressed in Equation (7), to extract the reversible saturation field distribution $ \rho^{rev}(H) $, which is zero for irreversible systems.
• Uses the discrete plaquette method to compute $ \rho^{irr}(H_R, H) $ as the mixed partial derivative $ -\frac{1}{2} \frac{\partial^2 M}{\partial H_R \partial H} $, ensuring clean separation from reversible contributions.
• Employs direct OpenGL-based visualization to avoid interpolation artifacts, preserving the discrete nature of the data and preventing artificial mixing of reversible and irreversible signals.
• Applies smoothing techniques (e.g., polynomial fitting) to noisy data before derivative extraction to enhance signal clarity without distorting the physical interpretation.

Experimental results

Research questions

• RQ1 Can reversible magnetic behavior be isolated from irreversible behavior in FORC analysis without contaminating the standard irreversible FORC distribution?
• RQ2 What mathematical formulation allows the reversible component to be represented as a distinct distribution, specifically the saturation field distribution?
• RQ3 How can the separation of reversible and irreversible components be achieved using only the measured magnetization data from FORC experiments?
• RQ4 To what extent does the proposed method improve the accuracy of microscopic model identification compared to conventional FORC analysis?
• RQ5 Can the method prevent visualization artifacts caused by interpolation in commercial software, especially near boundaries and at $ H_R = H $?

Key findings

• The irreversible FORC distribution $ \rho^{irr}(H_R, H) $ vanishes identically in a purely reversible system, confirming its exclusive sensitivity to irreversible processes.
• The reversible saturation field distribution $ \rho^{rev}(H) $, defined in Equation (7), vanishes identically in a purely irreversible system, confirming its exclusive sensitivity to reversible behavior.
• For a system of single-domain Stoner-Wohlfarth particles with both easy and hard axes, $ \rho^{irr} $ fully describes the irreversible (easy-axis) particles, while $ \rho^{rev} $ fully describes the distribution of both $ H_{s-} $ and $ H_{s+} $ saturation fields.
• The method successfully isolates the reversible contribution via $ M_H(H_R, H) $, which is constant in $ H_R $ for hard-axis particles and thus yields a non-zero second derivative only at saturation fields.
• Direct OpenGL-based visualization prevents interpolation artifacts that commonly obscure or distort the true discrete nature of the FORC data, especially near $ H_R = H $ and at curve endpoints.
• The approach enables unambiguous identification of microscopic models by providing distinct, non-overlapping signatures for reversible and irreversible components in the data.