VARIFORC User Manual: Chapter 8 (FORC Tutorial) Manual Chapter08 Tutorial

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VARIFORC User Manual: 8. FORC tutorial

8.1

VARIFORC User Manual
Chapter 8:

FORC tutorial

1,1

M0

H
(M1 − M0 )′

3,1
1,1

M1

2,1

H

1,1

(M2 − M1 )′

2,1

2,2

M2

(M3 − M2 )′
3,2
2,2
0

3,3

M3

H

2,1
2,2
0

3,2

µ0 Hb

cr 3,1

3,3

µ0 Hc

VARIFORC User Manual: 8. FORC tutorial

8.2

© 2014 by Ramon Egli and Michael Winklhofer. Provided for non-commercial research
and educa onal use only.

VARIFORC User Manual: 8. FORC tutorial

8.3

This chapter is based on contribu ons by Ramon Egli and Michael Winklhofer to the Interna onal Workshop on Paleomagne sm and Rock Magne sm (Kazan Ins tute of Geology and
Petroleum Technology, Russia, October 7-12, 2013), summarized in the following conference
proceeding ar cle:

Recent developments on processing and interpreta on aspects of
ﬁrst-order reversal curves (FORC)
Ramon Egli
Central Ins tute for Meteorology and Geodynamics, Hohe Warte 38, 1190 Vienna, Austria
(r.egli@zamg.ac.at)

Michael Winklhofer
Department for Earth and Environmental Sciences, Munich University, Theresienstrasse 41,
80333 Munich, Germany (michael@geophysik.uni-muenchen.de)

Abstract  Several recent developments in paleo- and environmental magne sm have been based on
measurement of ﬁrst-order reversal curves (FORC). Most notable examples are related to the detecon of fossil magnetosomes produced by magnetotac c bacteria and to absolute paleointensity es mates for temperature-sensi ve samples, such as meteorites. Future developments in these scien ﬁc
disciplines rely on improved characteriza on of natural magne c mineral assemblages. Promising
results have been obtained in several cases with the parallel development of FORC processing protocols on one hand, and models for idealized magne c systems on the others. Un l now, FORC diagrams
have been used mainly as a qualita ve tool for the iden ﬁca on of magne c domain state ﬁngerprints,
with missing quan ta ve links to other magne c parameters. This ar cle bridges FORC measurements
and conven onal hysteresis parameters on the basis of three types of FORC-related magne za ons
and corresponding coercivity distribu ons. One of them is the well-known satura on remanence, with
corresponding coercivity distribu on derived from backﬁeld demagne za on data in zero-ﬁeld FORC
measurements. The other two magne za on types are related to irreversible processes occurring
along hysteresis branches and to the inversion symmetry of magne c states in isolated par cles, respec vely. All together, these magne za ons provide precise informa on about magne za on processes in single-domain, pseudo-single-domain, and mul domain par cles. Unlike hysteresis parameters
used in the so-called Day diagram, these magne za ons are unaﬀected by reversible processes (e.g.
superparamagne sm), and therefore well suited for reliable characteriza on of remanent magne zaon carriers. The so ware package VARIFORC has been developed with the purpose of performing
detailed FORC analyses and calculate the three types of coercivity distribu ons described above. Key
examples of such analyses are presented in this ar cle, and are available for download along with the
VARIFORC package.

VARIFORC User Manual: 8. FORC tutorial

8.4

VARIFORC User Manual: 8. FORC tutorial

8.5

8.1 Introduc on
Several measurement protocols have been developed over the last 50 years for understanding complex magne za on processes related to technological applica ons [Chikazumi,
1997; Coey, 2009], the origin and stability of rock magne za ons [Dunlop and Özdemir, 1997;
Tauxe, 2010], and environment-sensi ve magne c minerals in sediments [Evans and Heller,
2003; Liu et al., 2012]. First-order reversal curves (FORC) provide one of the most advanced
protocols for probing hysteresis processes and represent them in a two-dimensional parameter space (i.e. coercivity ﬁeld Hc and bias ﬁeld Hb ). The interpreta on of hysteresis has evolved from mathema cal formalisms based on the superposi on of elemental source contribuons, called hysterons [Preisach, 1935; Mayergoyz, 1986; Hejda and Zelinka, 1990; Fabian and
Dobeneck, 1997], toward physical models of speciﬁc magne c systems, such as non-interacng [Newell, 2005; Egli et al., 2010] and interac ng [Woodward and Della Torre, 1960; Basso
and Berto , 1994; Pike et al., 1999; Muxworthy and Williams, 2005; Egli, 2006a] single-domain (SD) par cles, pseudo-single-domain (PSD) par cles [Muxworthy and Dunlop, 2002; Carvallo et al., 2003; Winklhofer et al., 2008], mul domain (MD) crystals [Pike et al., 2001b;
Church et al., 2011], and spin glasses [Katzgraber et al., 2002]. These models provide prototype signatures for speciﬁc magne za on processes (e.g. switching, vortex nuclea on, domain wall pinning), which can be recognized in FORC diagrams of geologic samples [Roberts
et al., 2000, 2006]. Some of these signatures occur within a limited subset of FORC space, as
for instance along Hc » 0 (viscosity and MD processes) or along Hb » 0 (weakly interac ng
SD par cles). Therefore, it is possible to iden fy the corresponding sources in FORC diagrams
of samples containing complex magne c mineral mixtures [e.g. Roberts et al., 2012], and, in
some cases, to es mate the abundance of magne c par cles associated with these processes
[Roberts et al., 2011; Yamazaki and Ikehara, 2012; Egli, 2013; Ludwig et al., 2013]. Up to the
few examples men oned above, FORC diagrams of geologic materials are mostly interpreted
in a qualita ve manner. Furthermore, only loose connec ons have been established with more common magne c parameters, such as isothermal and anhystere c remanent magne zaons and domain state-sensi ve ra os, although some of these parameters can be directly
derived from FORC subsets [e.g. Fabian and Dobeneck, 1997; Winklhofer and Zimanyi, 2006;
Egli et al., 2010].
Quan ta ve interpreta on of FORC measurements is based on the calcula on of magne c
parameters associated with speciﬁc magne za on processes. Some of these processes produce FORC signatures that are representable only in terms of non-regular func ons, whose
appearance in the FORC diagram depends strongly on data processing. A meanwhile wellknown example of non-regular FORC signatures is represented by the so-called central ridge
produced by non-interac ng SD par cles [Egli et al., 2010; Egli, 2013]. Magne c viscosity is
another example associated with a ver cal ridge near Hc = 0 [Pike et al., 2001a]. On the other
hand, most magne c processes in weakly magne c natural samples produce con nuous FORC

VARIFORC User Manual: 8. FORC tutorial

8.6

contribu ons with very low amplitudes, which are below the signiﬁcance threshold a ainable
with conven onal FORC processing [Egli, 2013]. Since the introduc on of FORC measurements
to rock magne sm [Pike et al., 1999; Roberts et al., 2000], some studies have been dedicated
to selected aspects of FORC processing, such as computa onal op miza on [Heslop and Muxworthy, 2005], locally weighted regression [Harrison and Feinberg, 2008], error calcula on
[Heslop and Roberts, 2012], and variable polynomial regression smoothing [Egli, 2013]. These
improvements have been merged into a single FORC processing procedure called VARIFORC
(VARIable FORC smoothing) [Egli, 2013]. The principal advantage of VARIFORC consists in the
possibility of processing FORC data containing high-amplitude, non-regular FORC signatures
as well as low-amplitude, con nuous backgrounds, using a local compromise between high
resolu on and noise suppression requirements. First applica ons of this technique enabled
full characteriza on of SD signatures in pelagic carbonates [Ludwig et al., 2013].
Meanwhile, VARIFORC has been complemented with rou nes for the automa c separaon of diﬀerent FORC contribu ons, and the calcula on of corresponding magne za ons and
coercivity distribu ons. The full VARIFORC package, including a detailed user manual, is available at h p://www.conrad-observatory.at/cmsjoomla/en/download. VARIFORC runs on Wolfram Mathema caTM and Wofram PlayerProTM (see Chapter 2). Applica on examples of quanta ve FORC analysis performed with VARIFORC are discussed in this paper.

VARIFORC User Manual: 8. FORC tutorial

8.7

8.2 A brief introduc on to FORC diagrams
8.2.1 Reversible and irreversible hysteresis processes
Ferrimagne c materials are characterized by complex magne c proper es that depend on
their past magne c and thermal history. Memory of previously applied ﬁelds gives raise to the
well-known phenomenon of magne c hysteresis. The discovery magne c hysteresis is credited to Sir Alfred James Ewing (1855-1955), who measured the ﬁrst hysteresis loop (Fig. 8.1)
on a piano wire [Ewing, 1885]. While the main characteris cs of a hysteresis loop are summarized by four magne c parameters yielding the well-known Day diagram [Day et al., 1977;
Dunlop, 2002a,b], much more detailed informa on on magne za on processes can be obtained by accessing the inner area of hysteresis loops. This is possible by in-ﬁeld measuring protocols involving a sequence of ﬁeld sweep reversals. The oldest example of such sequences is
the alterna ng-ﬁeld (AF) demagne za on [Chikazumi, 1997], in which the ﬁeld sweep is reversed at increasingly small ﬁeld amplitudes, un l a demagne zed, so-called anhystere c state
H = M = 0 is reached (Fig. 8.1).

Fig. 8.1: Original ﬁgure from Ewing [1885] showing the hysteresis measurement of a piano wire (see
the IRM Quarterly Vol. 22 for a story about Sir Alfred Ewing’s ﬁrst hysteresis measurements). The
measurement shown here represents an AF demagne za on curve, as a possible method for accessing
the inner area of hysteresis loops.

VARIFORC User Manual: 8. FORC tutorial

8.8

Other measuring protocols for accessing the inner area of a hysteresis loop are possible,
and the FORC protocol described by Pike et al. [1999] is just one of them. All protocols start
from a well-deﬁned magne c state obtained by satura ng the sample in a large ﬁeld. The ﬁrst
magne za on curve obtained by sweeping the magne c ﬁeld from posi ve or nega ve satura on coincides with one of the two major hysteresis loop branches M (H) . Hysteresis branches are also known as a zero-order curves, because they originate directly from a saturated
state. If the ﬁeld sweep producing a zero-order curve is reversed at a reversal ﬁeld Hr , before
satura on is reached, a new magne za on curve M(Hr , H) originates from the major hysteresis loop (Fig. 8.2a). This curve represents a ﬁrst-order magne za on, also known as ﬁrstorder reversal curve in case of FORC measurements. A set of ﬁrst-order curves branching from
the major hysteresis loop at diﬀerent reversal ﬁelds covers the en re area enclosed by the
loop, accessing a much larger number of magne za on states that cannot be obtained with
simple hysteresis measurements. If the ﬁeld sweep is reversed again while a ﬁrst-order curve
is measured, a second-order curve is obtained, and so on. Within this context, AF demagne za on is a sequence of nested magne za on curves with increasing order.
When describing magne za on curves, an important dis nc on is made between magne za on changes due to reversible and irreversible processes. The two types of processes
occurring along any magne za on curve are dis nguished by comparing a small por on
MA  MB of the curve between close ﬁelds HA and HB with the magne za on MA* obtained
by sweeping the ﬁeld from HB back to HA (Fig. 8.2a). Hysteresis, known in this context as
magne c memory, ensures that MB  MA* does not follow the same path as MA  MB , in
which case MA* ¹ MA [Mayergoyz, 1986]. The diﬀerence MA* - MA is the irreversible magne za on change occurring when sweeping the ﬁeld from HA to HB , while MB - MA* is the reversible change. The sum of the two contribu ons gives MB - MA , as expected.
8.2.2 Preisach diagrams

Because n-th order magne za on curves depend on n + 1 parameters (i.e. n reversal
ﬁelds and one measuring ﬁeld), interpreta on of ﬁrst- and higher order curves requires a parameter space model. The best known bivariate hysteresis model has been implemented by
Preisach [1935] for the characteriza on of transformer steel. The Preisach model assumes that
magne za on curves are the result of magne c switching in elemental rectangular hysteresis
loops (so-called hysterons). Hysterons are characterized by two switching ﬁelds HA £ HB where the magne za on jumps discon nuously from the lower to the upper branch and viceversa (Fig. 8.2b). Each hysteron is thus represented by a point in (HA , HB ) -space, and macroscopic magne c volumes or magne c par cle assemblages are described by a bivariate sta s cal distribu on P(HA , HB ) of hysteron switching ﬁelds, known as the Preisach distribu on.

VARIFORC User Manual: 8. FORC tutorial
(a)

(b)

8.9

HA
Hb

ΔMrev

B

A*
A

ΔMirr
nd

2
1st

A
B

3rd

A*

HB

Hsat

−Hsat
Hc
Fig. 8.2: Preisach theory in a nutshell. (a) The major hysteresis loop (black lines with large arrows) is
composed of two zero-order magne za on curves star ng from posi ve and nega ve satura on,
respec vely. First-order magne za on curves originate from the major hysteresis if the ﬁeld sweep is
reversed (black curve labeled with 1st). Higher-order magne za on curves (curves labeled with 2nd and
3rd) are obtained a er successive ﬁeld sweep reversals. Any point inside the major hysteresis loop can
be accessed by ﬁrst-order magne za on curves (dashed black line). For any of these points (e.g. point
A at the end of the dashed line), magne za on changes can be decomposed into a reversible ( ΔMrev )
and an irreversible ( ΔMirr ) component by sweeping the ﬁeld a li le further to point B and then back
to the original ﬁeld, ending with point A* , which, because of ΔMirr , does not coincide with A. The
ini al parts of ﬁrst-order curves origina ng from the upper hysteresis branch (blue segments) deﬁne
the irreversible component (red bars) of magne za on changes along this branch. (b) The Preisach
diagram is a representa on of hysteresis processes as the sum of elemental contribu ons from rectangular hysteresis loops (hysterons, sketched in red) with switching ﬁelds HA and HB . Because HB ³ HA
by deﬁni on, hysteron coordinates (HA , HB ) plot below the HA = HB diagonal, over a triangular area
(colored) limited by the satura on ﬁeld Hsat above which magne c hysteresis is fully reversible.
Further dis nc ons can be made between (1) closed hysterons with HA = HB , (2) hysterons with only
one possible state in zero ﬁeld (posi ve or nega ve satura on, blue areas), and (3) hysterons with two
possible states in zero ﬁeld (so-called magne c remanence carriers, green square). The Preisach space
can also be expressed in transformed coordinates represen ng coercivity (i.e. hysteron opening
Hc = (HB - HA )/2 ) and the bias ﬁeld (i.e. hysteron horizontal shi s Hb = (HB + HA )/2 ). Hysteron
examples (red) are given for selected points of the Preisach space, which can be understood as samples
of the Preisach distribu on. Contour lines over the region occupied by remanence-carrying hysterons
(green) represent a Preisach distribu on obtained for interac ng SD par cles by Dunlop et al. [1990].
In the Preisach-Néel model, Hc - and Hb -coordinates coincide with coercivi es and interac on ﬁelds
of real SD par cles, respec vely.

VARIFORC User Manual: 8. FORC tutorial

8.10

Hysterons are merely a mathema cal construct and do generally not correspond to discrete par cles or sample volumes. Nevertheless, the bivariate Preisach distribu on provides
intrinsically more informa on than any one-dimensional magne za on curve. The simplest
physical interpreta on of a Preisach distribu on has been proposed by Néel [1958] with what
is known as the Preisach-Néel model of single-domain (SD) par cles. This model relies on the
resemblance between hysteresis loops of individual SD par cles with uniaxial anisotropy [Stoner and Wohlfarth, 1948] on one hand, and symmetric Preisach hysterons (i.e. HA = -HB ) on
the other hand. Both are characterized by only two magne za on states (one for each hysteresis branch) with discon nuous transi ons at HA = -Hc and HB = +Hc . The Preisach distribu on of isolated SD par cles is thus concentrated along the HA = -HB diagonal of the Preisach space and coincides with the well-known coercivity or switching ﬁeld distribu on.
In interac ng SD par cle assemblages, magne c switching of individual par cles occur in
a total ﬁeld given by the sum of the applied ﬁeld and an internal, so-called interac on ﬁeld
Hb , which is the sum of dipole ﬁelds produced by the magne c moments of the other parcles. Whenever Hb ¹ 0 , elemental hysteresis loops are shi ed horizontally, so that magne c
switching occurs at HA = Hb - Hc and HB = Hb + Hc . Because the interac on ﬁeld is a local
variable determined by the spa al arrangement and magne za on of neighbor par cles, the
Preisach distribu on of interac ng SD par cles can be represented as the product of a coercivity distribu on f (Hc ) and an interac on ﬁeld distribu on g(Hb ) :
P = f (Hc ) g(Hb )

(8.1)

with Hc = (HB - HA )/2 and Hb = (HB + HA )/2 (Fig. 8.2b). More generally, Hc and Hb are
known as the coercivity ﬁeld and the bias ﬁeld of hysterons. The appealing simplicity of the
Preisach-Néel model has promoted the use of the transformed coordinates (Hc , Hb ) (whereby
Hb is also called Hu or Hi ), instead of the original Preisach ﬁelds HA and HB .
The Preisach space spanned by hysteresis processes that are saturated in ﬁelds |H|< Hsat
is a triangular region delimited by the diagonal line HB ³ HA (by deﬁni on of hysteron switching ﬁelds), and by HA >-Hsat , HB < +Hsat , respec vely (Fig. 8.2b). This space can be further
subdivided into a square region with HA < 0 and HB > 0 where hysterons can have two magne za on states in zero ﬁeld, and the remaining space where hysterons are nega vely or posively saturated when no external ﬁelds are applied. The square region is par cularly relevant
to paleo- and rock magne sm, because remanent magne za ons originate only from hysterons located within it. In par cular, the satura on remanent magne za on Mrs corresponds
to the integral of the Preisach func on over this region, i.e.

Mrs = ò

0

HA =-Hsat

+Hsat

òH =0

P(HA , HB )dHA dHB

(8.2)

B

On the other hand, the satura on magne za on Ms corresponds to the integral of the
Preisach func on over the en re domain deﬁned by HB ³ HA .

VARIFORC User Manual: 8. FORC tutorial

8.11

8.2.3 The FORC distribu on

Several measurement protocols have been developed in order to obtain experimental
Preisach func on es mates. What is nowadays known as the FORC protocol has been ﬁrst
described by Hejda and Zelinka [1990]. With this protocol, ﬁrst-order magne za on curves
M(Hr , H) , measured upon posi ve sweeps of H (i.e. H increases) from reversal ﬁelds Hr , deﬁne the so-called FORC func on
ρ(Hr , H) = -

1 2 M
2 Hr H

(8.3)

[Pike, 2003]. This func on coincides with the Preisach distribu on in case of measurements
performed on samples that are correctly described by the Preisach model. Because real samples rarely sa sfy this condi on, empirical distribu ons such as eq. (3) do generally not coincide with the Preisach distribu on up to few excep ons [e.g. Carvallo et al., 2005]. For example, the Preisach distribu on is a strictly posi ve probability func on, while FORC diagrams
can have nega ve amplitudes [Newell, 2005]. Several modiﬁca ons of the original Preisach
model have been developed in order to account for such diﬀerences. So-called moving Preisach models [Vajda and Della Torre, 1991] take the eﬀect of macroscopic magne za on states
on the intrinsic hysteron proper es into account, and are used for instance to describe magne za on-dependent interac on ﬁelds. Magne c viscosity, on the other hand, is accounted
by Preisach models with stochas c inputs simula ng thermal ﬂuctua ons of switching ﬁelds
[Mitchler et al., 1996; Borcia et al., 2002].
Modiﬁca ons of the Preisach formalism are not suﬃcient to explain all aspects of FORC
func ons, especially in case of non-SD magne c systems. Therefore, physical FORC models
have been developed in order to properly interpret magne c processes in isolated [Newell,
2005] and interac ng [Muxworthy and Williams, 2005; Egli, 2006] SD par cles, nuclea on of
magne c vor ces in PSD par cles [Carvallo et al., 2003; Winklhofer et al., 2008], domain wall
displacement in MD crystals [Pike et al., 2001b; Church et al., 2011], and magne c viscosity
[Pike et al., 2001a]. Magne c models of idealized systems yield characteris c signatures of the
FORC func on that can be used as ﬁngerprints for the iden ﬁca on of magne c minerals in
geologic samples [Roberts et al., 2000]. In some cases, these signatures are precisely determined to the point that quan ta ve analysis is possible [Winklhofer and Zimanyi, 2006; Egli et
al., 2010; Ludwig et al., 2013].
The remaining part of this sec on is dedicated to the implementa on of a general FORC
model that will be used to interpret the proper es of SD, PSD, and MD samples presented in
this ar cle. For this purpose, a rela vely simple magne c system with few magne za on states is considered. This system corresponds to the micromagne c hysteresis simula on of a
cluster of seven strongly interac ng SD par cles (Fig. 8.3). The upper branch of the major hysteresis loop contains three magne za on jumps produced by abrupt transi ons between four

VARIFORC User Manual: 8. FORC tutorial

8.12

magne c states with magne za ons M0 , M1 , M2 , and M3 . These states represent con nuous segments of the upper hysteresis branch.

(b)
H

1,1

M0
30

Hr,1

1,1

2,1

2,2

M2
3,3

−Hr,3

H

20

cr 3,1

2,2

10

Hr,1 3,2

(M2 − M1 )′
0

2,1

(M3 − M2 )′

0

Hr,2

H

Hr,2

3,2

0

Hr,3

0

−1

+50 mT

µ0 Hb [mT]

−50

30

2,1

1,1

2,2
0

M3

10

M1

(M1 − M0 )′

20

3,1 cr

20

(a)

−3

0

−2

0

−1

0

3,3

Hr,3
0

20

40

µ0 Hc [mT]
Fig. 8.3: FORC model of a linear chain of 7 SD magne te par cles with elonga on e = 1.3 and long
axes perpendicular to the chain axis. This model represents the simula on of a collapsed magnetosome
chain according to Fig. 9a in Shcherbakov et al. [1997]. In this example, the chain axis forms an angle
of 75° with the applied ﬁeld direc on. Magne za on jumps along the upper branch of the major
hysteresis loop are indicated by dashed lines. Cursive number pairs are used to count discon nui es
of ﬁrst-order curves M1 , M2 , and M3 (blue lines). For example, (2,2) is the second jump (counted from
the right) occurring along M2 . Any measurable FORC coincide with M0 , M1 , M2 , or M3 . The amplitude
of the last magne za on jump on M3 (magenta) deﬁnes the contribu on Mcr of the central ridge to
the FORC diagram shown in (b). (b) FORC diagram calculated from (a), consis ng of three diagonal
ridges deﬁned by ﬁrst deriva ves (Mi - Mi-1 )¢ of diﬀerences between M0 , M1 , M2 , and M3 . The
ridges width is exaggerated in order to show the color coding for posi ve (orange to magenta) and
nega ve (blue) contribu ons. Cursive number pairs indicate peaks of the FORC func on produced by
magne za on jumps with same labels as in (a). One of the peaks, labeled with CR, contributes to the
central ridge and is generated by the last magne za on jump (i.e. 3,1) of the last FORC (i.e. M3 ). All
FORC contribu ons are enclosed in a triangular region deﬁned by ver ces with coordinates (0, Hsat )
and (Hsat ,0) , where Hsat » 40 mT is the ﬁeld above which hysteresis becomes fully reversible.

VARIFORC User Manual: 8. FORC tutorial

8.13

If the ﬁeld sweep is reversed within the posi vely saturated state M0 , the resul ng ﬁrstorder magne za on curves will always coincide with M0 . Because these curves are iden cal,
M/ Hr = 0 and no contribu on to the FORC func on is obtained. If the reversal ﬁeld is decreased below the ﬁrst magne za on jump at Hr,1 , ﬁrst-order curves will start from M1 instead of M0 , and con nue along M1 un l a magne za on jump (labeled with ‘1,1’ in Fig. 8.3a)
will bring the magne za on M1 back to posi ve satura on (i.e. M0 ). The ﬁnite diﬀerence
between the last ﬁrst-order curve coinciding with M0 and the ﬁrst one coinciding with M1
creates a contribu on

1

(M1 - M0 )
ρ = δ(Hr - Hr,1 )
2
H

(8.4)

to the FORC distribu on, where δ(Hr - Hr,1 ) is the Dirac impulse func on accoun ng for the
magne za on jump at Hr,1 . Because δ(Hr - Hr,1 ) is zero everywhere, except for Hr - Hr,1 =
0, eq. (8.4) produces a diagonal ridge in FORC space (Fig. 8.3b). Using the coordinate transforma ons Hc = (H - Hr )/2 and Hb = (H + Hr )/2 , the ridge loca on is given by a line with equaon Hb = Hr,1 + Hc . FORC contribu ons along this line are propor onal to the deriva ve of
M1 - M0 and are of two fundamental types. The ﬁrst type occurs at points where M0 and M1
are con nuous, and is propor onal to diﬀerences between their slopes. Such FORC contribuons are magne cally reversible, because a small change of the applied ﬁeld H does not nucleate magne c state transi ons. On the other hand, magne cally irreversible contribu ons
occur at magne za on jumps occurring along M1 (e.g. jump ‘1,1’ in Fig. 8.3b). In this case, the
deriva ve of M1 - M0 is a Dirac impulse with amplitude ΔM1,1 , contribu ng with a point peak

1
ρ = ΔM1,1 δ(Hr - Hr,1 ) δ(H - H1,1 )
2

(8.5)

to the FORC distribu on. Equa ons (8.4-5) can be generalized to any pair of ﬁrst-order curves,
giving raise to as many diagonal ridges in FORC space, as discrete magne za on jumps are
encountered along the upper hysteresis branch. The FORC func on is thus fully described by
the sum of all diagonal ridges, i.e.
ρ=

1 n

(Mi - Mi-1 )
δ(Hr - Hr,i )
å
2 i=1
H

(8.6)

An important characteris cs of this FORC model is that both reversible and irreversible contribu ons can have posi ve and nega ve amplitudes, depending on the slopes of ﬁrst-order
curves, and on whether a magne za on jump occurs along Mi or Mi-1 .
The FORC func on of a simple system with few magne za on states, such as in the example of Fig. 8.3, is given by a certain number of inﬁnite, isolated peaks corresponding to discrete
transi ons between magne c states. Each peak is preceded by a sort of diagonal “shadow”
produced by the pronounced curvature of magne za on curves in proximity of magne c state

VARIFORC User Manual: 8. FORC tutorial

8.14

transi ons. Peak posi ons deﬁne so-called switching or nuclea on ﬁelds in which magne c
state transi ons occur. Small modiﬁca ons of the magne c system, as for instance the introduc on of an addi onal par cle in the SD cluster model of Fig. 8.3, modify cri cal ﬁelds and
eventually produce addi onal magne za on states with corresponding transi ons. Therefore,
samples containing large numbers of heterogeneous magne c par cles generate a dense
“cloud” of peaks merging into a con nuous FORC distribu on. Because individual peaks can
be posi ve or nega ve, some regions of the FORC diagram might be characterized by nega ve
amplitudes. In general, all FORC contribu ons are contained within a triangular region deﬁned
by ver ces with coordinates (0, Hsat ) and (Hsat ,0) .
An important characteris c of the general FORC model described above is related to the
inversion symmetry of magne c states. This symmetry ensures that the last magne za on
jump along the upper branch (i.e. the transi on from M2 to M3 in Fig. 8.3a) is always accompanied by an iden cal jump along the following ﬁrst-order curve, which coincides with
the lower hysteresis branch (i.e. jump ‘3,1’ in Fig. 8.3a). This jump produces an inﬁnite peak
on the last diagonal ridge of the FORC diagram (Fig. 8.3b), which is located exactly at Hb = 0 .
This is because the last diagonal ridge starts at a certain nega ve reversal ﬁeld Hr,last and ends
with a jump occurring at H = -Hr,last , so that Hb = Hr,last - Hr,last = 0 . While other peaks can
occur everywhere in FORC space, the peak associated with Hr,last is always placed exactly at
Hb = 0 .
A sample containing many isolated (i.e. non-interac ng) par cles with few magne c states
will produce a corresponding number of FORC peaks along Hb = 0 , while other peaks contribute to a distributed background. The superposi on of all peaks with Hb = 0 appears as an
inﬁnitely sharp, so-called central ridge [Egli et al., 2010]. Its existence has been ﬁrst predicted
for non-interac ng uniaxial SD par cles [Newell, 2005], which represent the simplest possible
case of par cles with two magne c states, and observed for a magnetofossil-bearing lake sediment [Egli et al., 2010]. Because of the theore cally inﬁnite sharpness of the central ridge,
high-resolu on FORC measurements and proper processing are necessary for its iden ﬁcaon. Since its ﬁrst observa on, the central ridge has been found to be a widespread signature
of freshwater and marine sediments containing magnetofossils [Roberts et al., 2012]. Two
condi ons must be met for the existence of a central ridge: ﬁrst, magne c par cles should not
interact with each other, since the presence of an interac on ﬁeld destroys the inversion symmetry of single par cle hysteresis loops by shi ing them horizontally. Second, individual par cles should have only few magne za on states, so that the lower hysteresis branch merges
directly with the upper branch, without joining any other ﬁrst-order curve. For example, MD
par cles with many domain wall pinning sites produce a large number of individual FORC
peaks, none of which must forcedly occur at Hb = 0 (Fig. 8.4).

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8.15

In any case, the central ridge is not an exclusive feature of SD par cles, as it can occur in
ensembles of non-interac ng par cles with few magne za on states (e.g. PSD). Some examples will be provided with the discussion of PSD magne za on processes in sec on 8.4.2.

(b)
(a)
ΔMn

0

Mbf

M+
Mn−1

+50

µ0 Hb [mT]

norm. magnetization

+1

0

−1

−100

0

µ0 H [mT]

+100

−50

0

+50

µ0 Hc [mT]
Fig. 8.4: (a) Model hysteresis loop (black) and FORCs (gray) generated by three MD par cles with
demagne zing factors of 0.1, 0.2, and 0.3, respec vely, calculated according to Pike et al. [2001b]. Only
FORCs necessary for measurement of the backﬁeld demagne za on curve Mbf (blue dots) are shown
for clarity. The last FORC Mn-1 that does not coincide with the lower hysteresis branch is show in
purple. It merges with the lower hysteresis branch before the last magne za on jump ΔMn has occurred, so that no central ridge contribu ons are produced. (b) FORC diagram corresponding to the MD
hysteresis model shown in (a). Gray diagonal lines are individual FORC trajectories along which irreversible magne za on processes are recorded as posi ve (orange dots) and nega ve (blue dots) peaks.
The dashed line is a quadra c ﬁt to the dots showing clustering around the crest of a ‘crescent-shaped’
distribu on as discussed in Church et al. [2011].

VARIFORC User Manual: 8. FORC tutorial

8.16

8.3 Coercivity distribu ons derived from FORC measurements
FORC measurements subsets deﬁne three types of coercivity distribu ons that provide a
bridge with conven onal parameters used in rock magne sm since several decades. These
coercivity distribu ons originate from three par cular FORC segments (Fig. 8.5): (1) the ini al
part M(Hr , H  Hr ) of each curve and its departure from the upper hysteresis branch, (2) the
remanent magne za on M(Hr ,0) of each curve, and (3) the point H = -Hr of each curve
where the applied ﬁeld equals the reversal ﬁeld amplitude. These regions deﬁne magne zaon curves that will be discussed in the following.

+1

norm. magnetization M/Ms

ΔMbf
Mrs
−Hr

H=0
ΔMbf

Hr = Hc

0

Hr = Hcr
ΔMirr

−1
−50

0

+50

ﬁeld µ0 H [mT]
Fig. 8.5: Relevant magne za on processes captured by FORCs. ΔMirr (red bar) is the irreversible magne za on change along the upper hysteresis branch, deﬁned by the ini al diﬀerence between FORCs
origina ng at consecu ve reversal ﬁelds Hr . In this example, Hr -values have been chosen to coincide
with the coercivity Hc and the coercivity of remanence Hcr for didac c purposes. The diﬀerence
between the same two FORCs in zero ﬁeld ( H = 0 , blue bar) deﬁnes a contribu on ΔMbf to the backﬁeld demagne za on curve. Finally, the abrupt slope change of FORCs at H = Hr deﬁnes the contribuon ΔMcr (green) to the central ridge. The FORC star ng at Hr = 0 , where M = Mrs , is called saturaon ini al curve [Fabian, 2003]. In ensembles of non-interac ng SD par cles, this curve coincides with
the upper hysteresis branch because nega ve ﬁelds are required to switch them from posi ve saturaon.

VARIFORC User Manual: 8. FORC tutorial

8.17

8.3.1 Backﬁeld coercivity distribu on

Backﬁeld or DC demagne za on of a posi vely saturated sample is obtained by measuring
its remanent magne za on a er applica on of increasingly large nega ve ﬁelds [Wohlfarth,
1958]. The applied nega ve ﬁelds are equivalent to reversal ﬁelds Hr of the FORC protocol
(Fig. 8.6), so that the backﬁeld demagne za on curve is given by FORC remanent magneza ons M(Hr ,0) . The corresponding backﬁeld coercivity distribu on is deﬁned as the ﬁrst
deriva ve of M(Hr ,0) , i.e.

fbf (x) = -

1 dM(-x ,0)
2
dx

(8.7)

The backﬁeld coercivity distribu on can be determined very precisely with the same polynomial regression method used to calculate FORC diagrams. The factor ½ in eq. (8.7) ensures
that the integral of fbf over all ﬁelds yields the satura on remanence Mrs of the sample.
Moreover, fbf is deﬁned only for posi ve arguments, which correspond to nega ve reversal
ﬁelds, because the remanent magne za on of curves star ng at Hr > 0 cannot be measured.
Within the Preisach model, the argument of fbf coincides with the coercivity ﬁeld Hc of hysterons, and fbf (Hc )dHc coincides with the Mrs -contribu on of all hysterons with coercivi es
comprised between Hc and Hc + dHc .

+2

Hr,n

0

Hr,n

0
–1

–60

–2

–40

–20

M (Hr , 0)

M , μAm2

+1

0

Hr , mT
–100

0

+100

H , mT
Fig. 8.6: Construc on of a backﬁeld demagne za on curve (right) from FORC measurements (le ) of
a magnetofossil-bearing pelagic carbonate. FORC por ons that are actually swept during backﬁeld
measurements are shown in blue, and some zero-ﬁeld measurements are highlighted with blue circles.
Remanent magne za on measurements Mr = M(Hr ,0) on FORCs beginning at -Hr deﬁne the backﬁeld curve coordinates (Hr , Mr ) .

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8.18

8.3.2 Reversal coercivity distribu on

Ini al FORC slopes can be used to calculate irreversible magne za on changes
ΔMirr = M(Hr + δH , Hr + δH) - M(Hr , Hr + δH)

(8.8)

along the upper hysteresis branch (Fig. 8.7), where δH is the (constant) ﬁeld increment used
for the measurements. The sum of all ΔMirr ’s obtained from consecu ve FORCs star ng at
reversal ﬁelds Hr £ x deﬁnes a magne za on curve Mirr (x) with the following meaning: if
reversible magne za on processes are removed from the upper hysteresis branch, the resul ng ‘irreversible hysteresis’ branch would coincide with Mirr up to a constant (Fig. 8.7b).
The so-called reversal coercivity distribu on is deﬁned by analogy with backﬁeld demagne za on as

firr (x) = -

1 dMirr (-x)
1 M(Hr , H)
=.
2
dx
2 Hr
Hr = H =-x

(8.9)

The factor ½ in eq. (8.9) has been introduced to ensure that the total integral Mirs of firr is a
magne za on with the following proper es: 0 < Mirs £ Ms for any sample, and Mirs = Ms in
absence of reversible processes. Unlike Mrs , the parameter Mirs includes all irreversible processes occurring along the major hysteresis loop and shall therefore be called irreversible satura on magne za on.
A fundamental property of Mirs is that it coincides with the total integral of the FORC
func on, because, using eq. (8.6):

òòH , H
r

ρ(Hr , H)dHr dH =

1 n
1 n
(
)
M
M
=
å i-1 i H=Hr,i 2 å ΔMi = Mirs
2 i=1
i=1

(8.10)

This important result implies that, while reversible magne za on processes can contribute
locally to the FORC func on, these local contribu ons cancel each other out upon integra on
in FORC space.
The deﬁni ons of fbf and firr are analogous, since they both rely on diﬀerences between
consecu ve FORCs evaluated at H = 0 and H = Hr , respec vely, and are both related to the
FORC star ng at Hr = -x (Fig. 8.5). An important diﬀerence, on the other hand, is that the
argument of firr can be posi ve as well as nega ve, unlike other coercivity distribu ons.
Posi ve arguments of firr correspond to measurements of the upper hysteresis branch in nega ve ﬁelds and vice versa. Similarly, posi ve arguments of fbf correspond to nega ve ﬁelds
used for DC demagne za on. Furthermore, firr (0) = fbf (0) by the deﬁni ons given with eq.
(8.7) and eq. (8.9).

VARIFORC User Manual: 8. FORC tutorial
(a)

8.19

+1

M

M+
0
ΔMirr
M−

−1
−50

0

+50

µ0 H [mT]

(b)

M + Ms

M+(H) + Ms
1

Mirr(H)

Mirr(H)

2Mirs

ΔMirr
0
−50

0

µ0 H [mT]

+50

Fig. 8.7: Construc on of the magneza on curve corresponding to irreversible processes along the upper
hysteresis branch. (a) Irreversible
magne za on changes ΔMirr (red)
deﬁned by the ini al parts of all
FORCs (blue). (b) Upper hysteresis
branch M+ (black curve) a er adding the satura on magne za on
Ms , and irreversible magne za on
curve Mirr (H) reconstructed by integra ng all magne za on steps
ΔMirr shown in (a). The satura on
value Mirr (H  ¥) is the total irreversible magne za on 2Mirs of the
hysteresis branch.

8.3.3 Central ridge coercivity distribu on

The central ridge coercivity distribu on is best explained by considering an isolated magne c par cle with any domain state. A ﬁrst-order curve star ng from the upper hysteresis
branch just a er a magne za on jump has occurred at Hr < 0 will contribute to the central
ridge if another magne za on jump is encountered at H = -Hr while sweeping the ﬁeld back
towards posi ve values, before merging with the previous FORC. Usually, magne za on
jumps can occur at any ﬁeld and there is no par cular reason for having one exactly at H =
-Hr . In this case FORC contribu ons accumulate at Hb = 0 as over any other place in FORC
space, genera ng a con nuous FORC distribu on. An excep on is provided by the FORC origina ng from the upper hysteresis branch just a er the last magne za on jump. This curve
coincides by deﬁni on with the lower hysteresis branch. Because of inversion symmetry, the
lower branch will contain a symmetric magne za on jump at H = -Hr . If the lower branch

VARIFORC User Manual: 8. FORC tutorial

8.20

merges with the previous FORC curve before H = -Hr is reached, as it is commonly the case
for MD hysteresis (Fig. 8.4a), the jump at H = -Hr will not contribute to the FORC diagram,
because the two curves are iden cal and M/ Hr = 0 . Otherwise, there will be a contribu on
to the central ridge in form of a peak with FORC coordinates (Hc , Hb ) = (-Hr ,0) (Fig. 8.3).
Because these contribu ons are placed exactly at Hb = 0 , they produce a ridge of the form

1
ρcr (Hc , Hb ) = fcr (Hc ) δ(Hb )
2

(8.11)

where fcr is the so-called central ridge coercivity distribu on [Egli et al., 2010]. In FORC diagrams obtained from real measurements, the inﬁnite sharpness of ρcr is regularized by replacing the Dirac impulse with an appropriate func on of Hb that takes the smoothing eﬀects
of measurements performed with ﬁnite ﬁeld increments into account. A rigorous treatment
of such eﬀects is given in Egli [2013]. The central ridge coercivity distribu on is obtained from
real measurements in two steps: ﬁrst, the central ridge contribu on ρcr to the FORC diagram
is isolated from the con nuous background produced by other processes, then ρcr is integrated over Hb so that
fcr (Hc ) = ò

+¥

-¥

ρcr (Hc , Hb )dHb .

(8.12)

While the amplitude of ρcr depends on the resolu on of FORC measurements and on FORC
processing, fcr is independent of any measuring and processing parameter and reﬂects intrinsic magne c proper es. The complex opera on of isola ng the central ridge and calcula ng
fcr is performed automa cally by VARIFORC with few controlling parameters described in this
user manual (Chapter 6).
The total magne za on Mcr associated with the central ridge is obtained by integra ng
fcr over Hc and represents the total contribu on of last magne za on jumps in isolated
magne c par cles. Accordingly, Mcr /Mirs is the ra o between the last magne za on jump
ΔMn of a hysteresis branch and the sum of all magne za on jumps over the same branch. In
case of non-interac ng, uniaxial SD par cles, ΔMn = ΔM1 is the only magne za on jump of
single par cle hysteresis, so that Mcr /Mirs = 1 . As soon as addi onal magne c states begin to
exist in small PSD par cles, the rela ve amplitude of ΔMn decreases with respect to the sum
Mirs of all magne za on jumps, and Mcr /Mirs < 1 , un l Mcr = 0 is reached in MD-like systems.

VARIFORC User Manual: 8. FORC tutorial

8.21

8.4 Examples
The physical meaning of FORC diagrams and derived coercivity distribu ons is best illustrated with topic examples related to SD, PSD, and MD magne c par cle assemblages. The
hysteresis proper es of samples discussed in this sec on are summarized by the Day diagram
of Fig. 8.8.

1
L uncultured cocci
[Lin and Pan, 2009]

P uncultured MB
(mainly cocci with double chains)
[Pan et al., 2005]

cultured MB
L P

0.5

CBD
extr. 20%

10%

20%

Mrs/Ms

30%

30%

AV-109

40%
50%

volc. ash

50%

60%

MS disp.

60%

SP saturation envelope

40%

40%

0.2

AMB-1 intact and collapsed
[Li et al., 2012]

SP+SD
mixing curves

50%
15 nm

70%
70%

SP+PSD
mixing curves

0.1
80%

EF-3
MD-20

interacting uniaxial
SD particles
[Muxworthy et al., 2003]

0.05
1

2

SD+MD mixing curves
5

80%

magnetofossil-rich sediments:

10 nm

Roberts et al., 2011
Abrajevitch and Kodama, 2011
Gehring et al., 2011
this study

10

20

Hcr/Hc
Fig. 8.8: Day diagram summarizing the hysteresis proper es of samples discussed in this paper (red
circles for SD samples, red triangles for PSD samples, red squares for MD samples), compared with
proper es of magnetofossil-bearing sediments (colored dots). The Day diagram with mixing curves
between domain states (gray) is drawn from Dunlop [2002b]. Cultured magnetotac c bacteria (‘cultured MB’) plot exactly on the expected spot for non-interac ng uniaxial SD par cles. The eﬀect of
magnetosta c interac ons on such par cles is shown with models from Muxworthy et al. [2003] and
with disrupted magnetosome chains (green circle, from Li et al., 2012]). In general, interac ng SD parcles follow the SD+MD mixing curve. Magnetofossil-bearing sediments follow a diﬀerent trend with
end-members deﬁned by CBD-extractable magne c minerals on one hand (red circle labeled as ‘CBD
extr.’, from Ludwig et al. [2013]) and the central region of the diagram on the other hand, possibly
represented by a clay mineral dispersion of SDS-treated Magnetospirillum cells (red circle labeled as
‘MS disp.’). Iron nanodots with single-vortex states (red triangle labeled as AV-109, from Winklhofer et
al. [2008]) do not plot on the expected trend line for PSD par cles.

VARIFORC User Manual: 8. FORC tutorial

8.22

8.4.1 SD magne c assemblages

The ﬁrst example is a conceptual model of a sample containing a small number of noninterac ng, uniaxial SD par cles with rectangular (Fig. 8.9a) and curved (Fig. 8.9c) single-parcle hysteresis loops. Reversible processes (i.e. magne c moment rota on in the applied ﬁeld)
are absent in the ﬁrst case. The SD par cles have two stable magne za on states in ﬁelds
| H |< Hs , where Hs is a par cle-speciﬁc switching ﬁeld. Transi ons from one magne za on
state to the other in individual par cles once their speciﬁc Hs -values have been reached is
seen in Fig. 8.9 as a series of magne za on jumps. These jumps represent irreversible magneza on processes, while reversible magne c moment rota ons (Fig. 8.9c) occur along con nuous segments of the magne za on curves.
(c)

(a)

+1
−Hr

ΔMcr

norm. magnetization

norm. magnetization

+0.5

M+(H )
0
ΔMirr
Hr
ΔMbf
−0.5

−Hr
ΔMcr
M+(H )

0
ΔMirr
Hr

ΔMbf

−1
−0.1

0

−0.1

+0.1

0

µ0 H [T]

+0.1

µ0 H [T]

(b)

(d)
f bf = f irr = fcr

f bf

f irr = fcr
−0.05

0

+0.05

µ0 H [T]

+0.1

−0.05

0

+0.05

+0.1

µ0 H [T]

Fig. 8.9: Modeled FORC proper es of few uniaxial, non-interac ng SD par cles. Switching of individual
par cles appears as magne za on jumps. (a) Preisach-Néel model with rectangular single par cle hysteresis loops (inset). This case is characterized by ΔMirr = ΔMbf = ΔMcr , so that the coercivity distribuons in (b) are iden cal. ΔMirr and ΔMcr are magne za on jumps produced by the same par cle in
Hr and -Hr , respec vely. (c) Model with Stoner-Wohlfarth single par cle hysteresis loops (inset).
Magne za on jumps occur at same ﬁelds as in (a), but their size is smaller, because magne za on
changes are caused in part by magne c moment rota ons over the con nuous segments. Because
magne za on jump sizes of single par cle hysteresis loops are smaller than satura on remanent magne za ons, ΔMirr = ΔMcr < ΔMbf , and the backﬁeld coercivity distribu on is larger than the other two
coercivity distribu ons, as shown in (d).

VARIFORC User Manual: 8. FORC tutorial

8.23

Each magne za on jump along the upper hysteresis branch is the star ng point of a FORC
that does not coincide with the previous one, while all FORCs star ng from the same con nuous hysteresis segment are iden cal.
Non-interac ng, uniaxial SD par cles have rela vely simple FORC proper es. First, no switching occurs when the ﬁeld is reduced from posi ve satura on to zero. Therefore, all FORCs
M(Hr ³ 0, H) star ng at posi ve reversal ﬁelds are iden cal to the upper branch M+ of the
major hysteresis loop and their shape is en rely determined by reversible magne c moment
rota on. Departure from M+ of the FORC M(0, H) origina ng at Hr = 0 (called satura on
ini al curve Msi ), can be used as a measure of how much real hysteresis loops diﬀer from the
ideal non-interac ng SD case characterized by Msi = M+ [Fabian, 2003]. As soon as nega ve
ﬁelds are reached along M+ (Hr ) , all par cles with Hs >-Hr are switched: accordingly, FORCs
star ng at Hr < 0 are produced by a mixture of switched and unswitched par cles. While the
applied ﬁeld is increased from Hr < 0 to H = 0 , magne c moments rotate reversibly without
further switching. Moreover, the remanent magne za on Mbf = M(Hr ,0) obtained at H = 0
reﬂects the same conﬁgura on of switched par cles created at the beginning of the corresponding FORC.
In both examples of Fig. 8.9, the last magne za on jump of each FORC contributes to the
central ridge and has the same amplitude as the magne za on jump on M+ (H) from which
the FORC is branching, because both jumps are produced by the two switching ﬁelds Hs of
same par cles. Therefore, the coercivity distribu ons associated with M+ (Hr ) and with the
central ridge are iden cal, i.e. fcr (x) = firr (x) over x ³ 0 and firr (x < 0) = 0 (Fig. 8.9b,d). The
backﬁeld coercivity distribu on, on the other hand, is based on magne za on diﬀerences
measured in zero ﬁeld instead of the switching ﬁelds, and is therefore dis nct from the other
two coercivity distribu ons in case of SD par cles with curved elemental hysteresis loops, such
as Stoner-Wohlfarth par cles (Fig. 8.9c,d). In case of randomly oriented Stoner-Wohlfarth parcles, the mean size of magne za on jumps in single-par cle hysteresis is S = Mirs /Mrs =
Mcr /Mrs = 0.5436 [Egli et al., 2010], and Mrs /Ms = 0.5 . Single-par cle hysteresis loops become much closer to rectangular loops as soon as thermal ac va ons are taken into considera on, because switching occurs in smaller ﬁelds where reversible magne c moment rota on
is less pronounced. FORC measurements yield S » 0.8-0.9 for SD par cles in a pelagic carbonate [Ludwig et al., 2013].
The FORC proper es discussed above are important for the iden ﬁca on of SD par cles
in geologic samples, notably magnetofossils in freshwater and marine sediment, but also welldispersed SD par cles in rocks. In par cular, the occurrence of sedimentary SD par cles in
isolated form or as linear chains produced by magnetotac c bacteria is the ma er of an ongoing debate. For example, the unusually strong SD signature of sediments from the Paleocene-Eocene thermal maximum (PETM) has been a ributed to magnetofossils produced by
magnetotac c bacteria thriving in a par cularly favorable environment [Kopp et al., 2007;

VARIFORC User Manual: 8. FORC tutorial

8.24

Lippert and Zachos, 2007], as well as, at least in part, to isolated SD par cles produced by a
cometary impact [Wang et al., 2013]. In the following, some examples of FORC and coercivity
distribu on signatures of sedimentary SD par cles are discussed.
The ﬁrst example is based on high-resolu on FORC measurements by Wang et al. [2013]
of a pure culture of the magnetotac c bacterium MV-1, which produces single chains of prisma c 3553 nm magne te crystals [Sparks et al., 1990]. The original measurements have
been reprocessed with VARIFORC and results are shown in Fig. 8.10. Isolated magnetosome
chains behave as a whole like SD par cles with uniaxial anisotropy, because the magne c
moments of individual crystal are switched in unison due to strong magnetosta c coupling
[Jacobs and Bean, 1955; Egli et al., 2010]. Magnetosta c interac ons between chains, on the
other hand, are minimized by the good separa on naturally provided by the much larger cell
volume.
Because of intrinsic magnetosome elonga on and well-constrained dimensions, MV-1 cultures provide a close analogue to random dispersions of nearly iden cal, uniaxial SD par cles.
The resul ng coercivity distribu ons are rela vely narrow with virtually no contribu ons at
Hc = 0 (Fig. 8.10f), as expected for SD par cles with minimum uniaxial anisotropy provided
by crystal elonga on and chain geometry. Hysteresis parameters ( Mrs /Ms = 0.496 , Hcr /Hc =
1.19, Fig. 8.8) prac cally coincide with those of randomly oriented Stoner-Wohlfarth par cles.
Lack of strong magnetosta c interac ons is conﬁrmed by the negligible intrinsic ver cal extension of the central ridge, as predicted by theore cal calcula ons [Newell, 2005].

Fig. 8.10 (front page): Cultures of the magnetotac c bacterium MV-1 represent one of the best
material realiza ons of non-interac ng SD par cle assemblages with minimum uniaxial magne c anisotropy. These bacteria contain a single chain of SD magne te crystals that switch in unison, behaving
eﬀec vely as an equivalent SD par cle with elonga on along the chain axis. (a) Set of FORC measurements where every 4th curve is plo ed for clarity. (b) Same as (a), a er subtrac ng the lower hysteresis branch from each curve. Every 2nd curve is shown for clarity. The bell-shaped envelope of all
curves is the diﬀerence between upper and lower hysteresis branches, i.e. the even component
Mrh = (M+ - M- )/2 of the hysteresis loop mul plied by a factor 2 [Fabian and Dobeneck, 1997]. (c)
FORC diagram calculated with VARIFORC from the measurements shown in (a). (d) Same as (c), a er
subtrac on of the central ridge. Most contribu ons in this diagram are due to reversible magne za on
processes (i.e. in-ﬁeld magne c moment rota ons). (e) Central ridge isolated from (c) and plo ed with
a 2 ver cal exaggera on. Zero-coercivity contribu ons are completely absent, as expected for a
system of par cles with intrinsic shape anisotropy along chain axes. The central ridge’s ver cal extension slightly exceeds the minimum extension expected from data processing of an ideal ridge, revealing
residual magnetosta c interac ons between magnetosome chains. The associated interac on ﬁeld
amplitudes are <0.5 mT.

VARIFORC User Manual: 8. FORC tutorial
(a)

(b)
+2

2

∆M , μAm2

M , μAm2

8.25

0

1

0

–2
0

–200

+200

–200

0

H , mT

+200

H , mT

(c)

(d)
+40

+40

mAm2
T2

mAm2
T2

0

Hb , mT

4

2

–40

1.0

Hb , mT

0

0.5
–40
0.0

0
–80

–80
0

40

80

0

40

Hc , mT

(f)

mAm2
T2

40

4

2×

0
2

–5
0

40

Hc , mT

80
0

dM/dH , μAm2/T

Hb , mT

(e)

+5

80

Hc , mT

fbf

μAm2:
Mrs = 1.01
Mirs = 0.889
Mcr = 0.568
20

0
–40

fcr
firr
0

40

80

−Hr or Hc , mT

Fig. 8.10 (con nued): (f) Coercivity distribu ons derived from FORC measurements and corresponding
magne za ons calculated by integra on of the distribu ons over all ﬁelds. The condi on Mirs = Mcr
expected for these par cles is not exactly met, because of residual FORC contribu ons not corresponding to non-interac ng, uniaxial SD par cles. On the other hand, firr (x < 0) = 0 , as expected from
posi vely saturated SD par cles that cannot be switched in posi ve ﬁelds. High-resolu on FORC measurements have been kindly provided by Wang et al. [2013].

VARIFORC User Manual: 8. FORC tutorial

8.26

Ideally, the three types of coercivity distribu ons shown in Fig. 8.10f should be characterized by firr º fcr £ fbk and firr (x) = 0 for nega ve arguments, so that Mcr /Mirs = 1 . The measured ra o Mcr /Mirs = 0.64 reﬂects residual FORC contribu ons of unspeciﬁed nature clearly
visible over Hb > 0 , where ρ = 0 is expected from non-interac ng SD par cles [Newell, 2005].
These contribu ons are probably associated with a small frac on of collapsed magnetosome
chains (Fig. 8.10d).
The second example is also based on a magnetotac c bacteria sample, but its magne c
proper es are less straigh orward. The sample is a synthe c sediment analogue obtained by
dispersing cultured cells of the magnetotac c bacterium Magnetospirillum magnetotac cum
MS-1 in a clay slurry (kaolinite) while dissolving the cell material with addi on of 2% sodium
dodecyl sulfate (SDS) during con nuous s rring. The purpose of this experiment was to check
the stability of magnetosome chains in sediment once the cell material is dissolved. Analogous
experiments performed directly in aqueous solu on yielded strongly interac ng magnetosome clusters [Kobayashi et al., 2006]. FORC analysis of this sample (Fig. 8.11) poses a formidable problem in terms of data processing, because of the simultaneous presence of (1) a
sharp superparamagne c (SP) overprint, and (2) a double discon nuity at Hr = H = 0 , due to
the overlap of a central ridge and a ver cal ridge in the FORC diagram.

Fig. 8.11 (front page): FORC measurements of a specially prepared sample containing equidimensional
magne te magnetosomes. The sample was obtained by dispersing a Magnetospirillum culture in clay
(kaolinite) with subsequent 2% SDS addi on under con nuous s rring. Dissolu on of cell material by
SDS is expected to produce clay-magnetosomes aggregates of some form. (a) Set of FORC measurements where every 12th curve is plo ed for clarity. The insert shows a zoom around the origin, where
a sigmoidal SP contribu on is recognizable. The SP signature saturates in <2 mT, and, although not
contribu ng to the FORC diagram, it poses a processing problem, because polynomial regression provides a correct ﬁt only if unsuitably small smoothing factors are chosen (SF = 2 in this case). (b) Same
as (a), a er subtrac ng the lower hysteresis branch from each curve. Every 3rd curve is shown for clarity. The exponen al-like envelope of all curves is the diﬀerence between the upper and lower hysteresis branches, and the cusp at H = 0 denotes a system with zero-coercivity contribu ons. The SP
contribu on shown in the inset of (a) is naturally eliminated from measurement diﬀerences, which
therefore no longer pose FORC processing problems. (c) FORC diagram calculated with VARIFORC from
the measurement diﬀerences shown in (b). The only signiﬁcant contribu ons are the central ridge,
indica ve of non-interac ng SD par cles, and a ver cal ridge at Hc = 0 , which is produced by magne c
viscosity. The absence of other signiﬁcant FORC contribu ons, and in par cular the typical signature
for reversible magne c moment rota on, indicate that single-par cle hysteresis loops are prac cally
rectangular. (d) Same as (c), a er subtrac on of the central ridge. Residual contribu ons around the
former central ridge loca on reveal addi onal magne za on processes, which, given the SD nature of
the sample, must arise from magnetosta c interac ons. (e) Central ridge isolated from (c) and plo ed
with a 3 ver cal exaggera on. The central ridge peak at Hc = 0 denotes a system containing SD parcles with vanishing coercivity.

VARIFORC User Manual: 8. FORC tutorial
(a)

8.27

(b)

0

∆M , μAm2

M , μAm2

+0.4

0.2

−4

0.1

+4

–0.4

0
–50

0

+50

–50

0

H , mT

+50

H , mT

(c)

(d)
mAm2
T2

+20

mAm2
T2

+20

0.6

3

0

0

0.4

Hb , mT

Hb , mT

2

0.2

1

–20

–20

0

0
–40

–40
0

20

40

0

20

Hc , mT

mAm2
T2

Hb , mT

+5

3×

2

0
1
0

20

Hc , mT

40

0

(f)
10

dM/dH , μAm2/T

(e)

–5

40

Hc , mT

μAm2:
Mrs = 80
Mirs = 120
Mcr = 74

fbf
5

fcr
0

–20

0

firr
20

40

−Hr or Hc , mT

Fig. 8.11 (con nued): (f) Coercivity distribu ons derived from FORC measurements and corresponding
magne za ons calculated by integra on of the distribu ons over all ﬁelds. The condi on Mirs » Mrs »
Mcr met by this sample is typical for non-interac ng SD par cles with squared hysteresis loops and
represents a physical realiza on of a Preisach-Néel system. Residual firr -contribu ons over nega ve
arguments are caused by non-zero FORC amplitudes over Hb > 0 in (d).

VARIFORC User Manual: 8. FORC tutorial

8.28

Because the sigmoidal SP overprint extends only over few measurement points, satura ng
in <2 mT (Fig. 8.11a), it cannot be adequately ﬁ ed by polynomial regression with smoothing
factors required for adequate measurement noise suppression [Egli, 2013]. The SP overprint
is eliminated by subtrac ng the lower branch of the major hysteresis loop from all curves, in
which case no par cular features are seen at H = 0 (Fig. 8.11b). This opera on does not aﬀect
FORC calcula ons, because the Hr -deriva ve of any magne za on curve added or subtracted
to all measurements is zero. For this reason, subtrac on of the lower hysteresis branch is an
op on provided by VARIFORC for processing quasi-discon nuous measurements. Moreover,
FORC measurement diﬀerences reveal details that are o en completely hidden in hysteresis
loops with Mrs /Ms  0 and/or large paramagne c contribu ons.
The hysteresis loop of this sample is clearly constricted at H = 0 , in what is o en called a
‘wasp-waisted’ shape [Tauxe et al., 2006]. The interpreta on of corresponding Day diagram
parameters ( Mrs /Ms = 0.177 , Hcr /Hc = 5.12 , Fig. 8.8) is ambiguous, because it involves mixtures of SD, PSD, and SP par cles. On the other hand, the FORC diagram (Fig. 8.11c), contains
two precisely interpretable signatures, namely a central ridge, as expected for non-interac ng
SD par cles, and a ver cal ridge due to magne c viscosity. Addi onal FORC contribu ons outside of the two ridges are very weak (Fig. 8.11d). Coercivity distribu ons (Fig. 8.11f) are characterized by exponen al-like func ons peaking at Hc = 0 . Because this is also true for fcr , many
par cles must have vanishingly small switching ﬁelds. Such features can be explained by a
combina on of thermal ac va on eﬀects and the absence of chain-derived uniaxial anisotropy, as expected for equidimensional MS-1 magnetosomes if their original arrangement is
destroyed. On the other hand, the presence of magnetosome clusters similar to those obtained from cell disrup on in aqueous solu ons [Kobayashi et al., 2006] can be excluded, because of the absence of magnetosta c interac on signatures otherwise reported with FORC
diagrams of extracted magnetosomes [e.g. Chen et al., 2007; Wang et al., 2013]. The apparent
contradic on between lack of uniaxial chain anisotropy and magnetosta c interac on signatures can be reconciled by assuming that magnetosomes have been individually dispersed in
the clay matrix.
The three coercivity distribu ons derived from FORC measurements are almost iden cal;
approaching the limit case fbk = fcr = firr predicted for non-interac ng SD par cles with rectangular single-par cle hysteresis loops. Rectangular loops can be explained by the strong
switching ﬁeld reduc on in thermally ac vated SD par cles close to the SD/SP threshold. This
example demonstrates the level of detailed informa on that is provided by high-resolu on
FORC measurements. Results shown in Fig. 8.10 and Fig. 8.11 can be considered representa ve for well dispersed SD par cles with and without minimum uniaxial shape anisotropies.
The eﬀect of shape anisotropy is much less evident with samples of interac ng SD par cles,
because local interac on ﬁelds act as an addi onal magne c anisotropy source.

VARIFORC User Manual: 8. FORC tutorial

8.29

The third SD example is based on high-resolu on FORC measurements of a magnetofossilbearing pelagic carbonate from the Equatorial Paciﬁc [Ludwig et al., 2013]. Typical sediment
magne za ons of the order of few mAm2/kg, as for this sample, yield FORC measurements
with important noise contribu ons that need to be adequately suppressed in order to obtain
useful FORC diagrams. FORC processing becomes cri cal in such cases, as shown in Fig. 8.12.
Conven onal data processing based on constant smoothing factors yields signiﬁcant values of
the FORC distribu on only over a limited region around the central ridge (Fig. 8.12a), unless
the high resolu on required in proximity of Hb = 0 and Hc = 0 is sacriﬁced. The VARIFORC
variable smoothing algorithm, on the other hand, ﬁnds a locally op mized compromise between resolu on preserva on and noise suppression. With this approach, signiﬁcant domains
of the FORC distribu on are drama cally expanded (Fig. 8.12b), revealing a broad, con nuous
background around the central ridge, as well as nega ve FORC amplitudes characteris c for
SD par cles.

(a)

(b)

SF = 4

variable smoothing

Am2
T2kg

0

Am2
T2kg

–50

Hb , mT

Hb , mT

4

8

0

8

4
–50

0
0

0

50

Hc , mT

100

0

50

100

Hc , mT

Fig. 8.12: Example showing the importance of proper FORC processing for extrac ng detailed informa on from weak natural samples. The two FORC diagrams have been obtained from the same set of
high-resolu on measurements (ﬁeld step size: 0.5 mT) of a pelagic carbonate from the Equatorial Paciﬁc [Ludwig et al., 2013]. The red contour(s) enclose signiﬁcant regions of the FORC diagram, i.e. regions
where the FORC func on is not zero at a 95% conﬁdence level according to the error calcula on method implemented by Heslop and Roberts [2012]. (a) Conven onal FORC processing with a constant
smoothing factor SF = 4. The central ridge is the only signiﬁcant FORC feature that can be resolved.
Larger smoothing factors would extend the signiﬁcant region at the cost of blurring the central ridge
to the point where it can no longer be iden ﬁed as such (see Fig. 1 in Egli [2013]). (b) VARIFORC processing obtained with a variable smoothing factor op mized for the best compromise between noise
suppression and detail preserva on. Low-amplitude features, such as nega ve contribu ons, are now
signiﬁcant over large por ons of the whole FORC space.

VARIFORC User Manual: 8. FORC tutorial

8.30

The last example of this sec on (Fig. 8.13) is based on a special technique used to isolate
the contribu on of secondary SD magne te par cles from the same pelagic carbonate sample
of Fig. 8.12. For this purpose, iden cal FORC measurements has been performed before and
a er trea ng homogenized sediment material with a citrate-bicarbonate-dithionite (CBD) solu on for selec ve magnetofossil dissolu on [Ludwig et al., 2013]. Large magne te crystals,
as well as SD par cles embedded in a silicate matrix, are not aﬀected by this treatment. Therefore, diﬀerences shown in Fig. 8.13 between the two sets of measurements represent the
intrinsic magne c signature of CBD-extractable par cles. Hysteresis proper es ( Mrs /Ms =
0.44, Hcr /Hc = 1.34 , Fig. 8.8) are close to the limit case of randomly oriented, non-interac ng
SD par cles with uniaxial anisotropy, despite evident magnetosta c interac on signatures
deducible from posi ve FORC contribu ons over the upper quadrant (Fig. 8.13d). Interpretaon of interac on signatures in terms of collapsed magnetosome chains or authigenic SD magne te clusters requires further inves ga on [Ludwig et al., 2013]. Coercivity distribu ons (Fig.
8.13f) display minor contribu ons near Hc = 0 , and their overall shape is be er associable
with intact magnetotac c bacteria cultures (Fig. 8.10) than dispersed magnetosomes in clay
(Fig. 8.11). Coercivity distribu ons of magnetofossil-bearing sediment are wider than those of
individual bacterial strains, because of the natural diversity of magnetosome and chain morphologies. On the other hand, no systema c diﬀerences are observed between FORC-related
magne za on ra os (Table 8.1), as long as chain integrity is not evidently compromised. In
par cular, FORC proper es of PETM sediment appear to be compa ble with those of similar
magnetofossil-bearing samples, rather than dispersions of equidimensional SD par cles.

Fig. 8.13 (front page): FORC analysis of a pelagic carbonate sample from the Equatorial Paciﬁc, obtained from diﬀerences between iden cal measurements of the same material before and a er selecve SD magne te dissolu on [Ludwig et al., 2013]. This approach, combined with the fact that the
main magne za on carriers are magnetofossils, ensures that results shown here represent the uncontaminated signature of secondary SD minerals. (a) Set of FORC measurements where every 8th curve is
plo ed for clarity. (b) Same as (a), a er subtrac ng the lower hysteresis branch from each curve. Every
4th curve is shown for clarity. The bell-shaped envelope of all curves is the diﬀerence between the
upper and lower hysteresis branches. Its shape is intermediate between the examples shown in Fig.
8.10-11, albeit closer to Fig. 8.10b. (c) FORC diagram calculated with VARIFORC from the measurements shown in (a). The central ridge is overlaid to addi onal low-amplitude contribu ons (<10% of
the central ridge peak), which, because of their extension over the FORC space, represent as much as
50% of the total magne za on Mirs ‘seen’ by the measurements. (d) Same as (c), a er subtrac on
of the central ridge. The lower quadrant partly coincides with the signature of reversible magne c
moment rota ons as predicted by Newell [2005]. Because non-SD contribu ons are excluded by the
special prepara on procedure, posi ve FORC amplitudes over Hb > 0 must represent the signature of
magnetosta c interac ons between SD par cles. (e) Central ridge isolated from (c) and plo ed with a
3 ver cal exaggera on.

VARIFORC User Manual: 8. FORC tutorial
(a)

(b)
6

∆M , mAm2/kg

+5

M , mAm2/kg

8.31

0

4

2

–5
0
–100

0

+100

–100

0

H , mT

+100

H , mT

(c)

(d)
Am2
T2kg

Am2
T2kg

8

4

0

0.4

Hb , mT

Hb , mT

0

0

–50

–50

0

0

50

–0.4

0

100

50

Hc , mT

Hc , mT

(f)

Am2
T2kg

Hb , mT

8

3×

0

4

–5
0

50

Hc , mT

100

0

dM/dH , mAm2/(T kg)

(e)

+5

100

mAm2/kg:
Mrs = 3.25
Mirs = 3.18
Mcr = 2.12

40

fcr

20

fbf

firr
0

–50

0

50

100

150

−Hr or Hc , mT

Fig. 8.13 (con nued): (f) Coercivity distribu ons derived from FORC measurements and corresponding
magne za ons calculated by integra on of the distribu ons over all ﬁelds. Magne za on ra os (e.g.
Mirs / Mrs , Mcr / Mirs , Table 8.1) are similar to those of the MV-1 example in Fig. 8.10 and representa ve for magnetofossil-bearing sediment.

VARIFORC User Manual: 8. FORC tutorial

8.32

Table 8.1: Hysteresis parameters Hcr /Hc and Mrs /Ms , and ra os between FORC-derived magne zaons Mrs , Mirs , and Mcr , for samples described in this ar cle.
Material

Hcr /Hc

Mrs /Ms

Mirs /Mrs

Mcr /Mrs

Mcr /Mirs

Strictly SD examples
MS-1 dispersion in clay
MS-1
AMB-1 a
MV-1 b
CBD-extractable in pelagic carbonate c

5.12
1.233
1.267
1.190
1.340

0.177
0.494
0.500
0.496
0.442

1.397
0.928
0.893
0.879
0.815

0.885
0.510
0.698
0.561
0.651

0.633
0.550
0.782
0.638
0.667

Magnetofossil-rich sediments
Pelagic carbonate c
PETM b
Soppensee d

1.690
1.677
1.503

0.399
0.418
0.411

1.011
0.953
1.066

0.569
0.550
0.387

0.563
0.576
0.364

PSD examples
AV-109 e
EF-3
Volcanic ash b

2.578
4.489
2.421

0.267
0.069
0.219

1.856
2.500
1.976

0.598
0.097
0.024

0.322
0.039
0.012

MD par cles
MD20

3.147

0.075

2.873

0

0

a

FORC data kindly provided by Li et al. [2012].
FORC data kindly provided by Wang et al. [2013].
c
FORC data from Ludwig et al. [2013].
d
FORC data from Kind et al. [2011].
e
FORC data from Winklhofer et al. [2008].
b

8.4.2 PSD magne c assemblages

The next two FORC examples are based on PSD par cle assemblages, star ng with the
simplest case of an array of iden cal, weakly interac ng Fe nanopar cles with grain sizes slightly larger than the upper SD limit [Winklhofer et al., 2008]. These par cles can have two pairs
of an parallel magne c states: so-called ‘ﬂower’ states with nearly homogenous magne zaon and SD-like magne c moments (SD+ and SD), and single vortex states with nearly zero
magne c moments (SV+ and SV). Hysteresis proper es are shaped by the transi on sequences SD+  SV+  SD and SD  SV  SD+ between posi ve and nega ve satura on. Similar
transi ons in magne te cubes have been modeled micromagne cally [Newell and Merrill,
2000], yielding the single-par cle hysteresis loops shown in Fig. 8.14.

VARIFORC User Manual: 8. FORC tutorial
(a)

8.33

(b)
Hb

M
SD+
2,1

M0
ΔMbf

ΔMirr
Hr2

1,1

SV+

(M

0

−

M

(1,1) − M0

)′
0

1

M2 − (1,1)
(2,1) − M1

Hr1

Hc

H
Hr1

SV− 2,2

M1

2

(M

ΔMirr
M2

M

1

ΔMbf

(2,2) − M1

)′

SD−

−

Hr2

(c)

(d)
Hb

M

(1,1) − M0

SD+
M0

1,1

ΔMirr

Hr1

(2,1) − M1

0

M1

SV−

H

M

)′

Hc

M2 − (1,1)

1

Hr1
(2,2) − M1

2,2

2

ΔMbf

−

1

2,1

SV+
Hr2

(M

)′
M0

(M

−

ΔMirr
M2

SD−

Hr2

Fig. 8.14: Two examples of single par cle hysteresis loops (le plots) and corresponding FORC diagrams (right plots), generated by micromagne c simula ons of 0.1 µm (a) and 0.11 µm (c) magne te
cubes by Newell and Merrill [2000]. In both cases, the par cles have two SD-like (SD) and two vortexlike (SV) magne za on states. SD-like states in (c) exist only in suﬃciently large applied ﬁelds and
cannot contribute to remanent magne za ons. Transi ons between magne c states occur at magne za on jumps (dashed lines and red lines), deﬁning three groups of iden cal FORCs M0 , M1 and M2 .
Magne za on jumps relevant for FORC calcula ons are labeled by number pairs like in Fig. 8.3. Corresponding posi ve and nega ve peaks of the FORC func on (b,d) are shown with ‘+’ and ‘’ symbols,
respec vely. Gray diagonal lines with arrows are the only FORC trajectories producing non-zero contribu ons. Only peaks located to the right of the dashed lines contribute to the backﬁeld demagne za on
curve and thus to Mrs , determining large diﬀerences in magne c remanence proper es of otherwise
similar FORC diagrams.

VARIFORC User Manual: 8. FORC tutorial

8.34

The complex FORC signature of Fe nanopar cles (Fig. 8.15) can explained by a combina on
of the two micromagne c models in Fig. 8.14, with individual peaks corresponding to magne c
transi ons between SD and SV states. The SV  SD+ transi on along the lower hysteresis
branch produces a central ridge peaking at Hc » 0.15 T . Addi onal pairs of posi ve FORC
peaks at Hc » 0.06 T and nega ve peaks just above and below the central ridge are produced
by the remaining transi ons, while nega ve FORC amplitudes peaking at Hb » -0.15 T can
be explained by reversible magne za on changes of the SV+-state in proximity of its denuclea on ﬁeld. All relevant FORC contribu ons occur at or in proximity of SV nuclea on in
±0.01 T and SV denuclea on in ±0.15 T (Fig. 8.15c), producing a constricted hysteresis loop
(Fig. 8.15a) and bimodal coercivity distribu ons (Fig. 8.15f).
Unlike the case of isolated SD par cles, magne c state transi ons from posi ve satura on
(i.e. SD+) occur already in posi ve ﬁelds. These transi ons (e.g. SV nuclea on) are not captured
by remanent demagne za on measurements, therefore contribu ng to firr (x < 0) , but not to
fbk . In the example of Fig. 8.15, SV denuclea on is the only process captured by the central
ridge, so that fcr is characterized by a single peak at Hc » 0.15 T , instead of two peaks, as for
the other two coercivity distribu ons.

Fig. 8.15 (front page): FORC analysis of Fe nanodots [Winklhofer et al., 2008]. The ar ﬁcial sample (AV109) is a two-dimensional, quasi-hexagonal array of polycrystalline Fe nanodots with a diameter of
6713 nm and 20 nm thickness. The nanodots center-to-center spacing is 2 dot diameters [Dumas et
al., 2007]. FORC measurements have been performed in the array plane. (a) Set of FORC measurements where every 2nd curve is plo ed for clarity. (b) Same as (a), a er subtrac ng the lower hysteresis
branch from each curve. Every 2nd curve is shown for clarity. Hysteresis loop constric on at H = 0 and
the double peak of the curve envelope in (b) are produced by a bimodal distribu on of nuclea on
ﬁelds. As evident in (b), some FORCs cross each other, as well as the lower hysteresis branch. This
means that regions outside the major hysteresis loop can in principle be accessed by FORC measurements (e.g. Fig. 8.14a), albeit rarely seen with natural samples and impossible with non-interac ng SD
par cles. (c) FORC diagram calculated with VARIFORC from the measurements shown in (a), featuring
localized peaks typical for magne c transi ons between four magne c states: two SD-like states with
large magne c moments, and two states corresponding to a single magne c vortex with small net
magne c moment. Because magne c par cles in this sample are prac cally iden cal, magne c transi ons occur collec vely, appearing as dis nct FORC func on peaks. In case of less homogenous samples, FORC peaks would merge into a con nuous background with triangular contour lines, as commonly seen with natural PSD assemblages. The dashed lines mark the rectangular domain of FORC
amplitudes associated with remanent magne za ons. Accordingly, only about half of the two peaks
at Hc » 0.06 T contribute to Mrs . (d) Same as (c), a er subtrac on of the central ridge. The two
nega ve peaks around the central ridge in (c) now appear as a single contribu on produced by vortex
denuclea on. (e) Central ridge isolated from (c) and plo ed with a 2 ver cal exaggera on, featuring
a single peak at 0.15 T.

VARIFORC User Manual: 8. FORC tutorial
(a)

8.35

(b)
60

∆M , nAm2

M , μAm2

+0.1

0

40

20

0

–0.1
0

–0.5

+0.5

–0.5

0

H, T

+0.5

H, T

(c)

(d)
μAm2
T2

+0.1

μAm2
T2

+0.1
8

0

Hb , T

Hb , T

4

4

0

2

0
–0.1

–0.1

0

–2

–0.2

–0.2
0

0.1

0.2

0

0.1

Hc , T

0.2

Hc , T

(e)

(f)
μAm2:
Mrs = 28.8
Mirs = 53.5
Mcr = 17.5

Hb , mT

+20

2×

0

8

4

–20
0

0.1

0.2

Hc , T

0

dM/dH , μAm2/T

0.4

0.2

0
–0.2

firr

fbf
fcr
0

0.2

−Hr or Hc , mT

Fig. 8.15 (con nued): (f) Coercivity distribu ons derived from FORC measurements and corresponding
magne za ons calculated by integra on of the distribu ons over all ﬁelds. The bimodal character of
firr and fbf arises from the existence of two diﬀerent ﬁelds for nuclea on (±0.01 T) and denuclea on
(±0.15 T) of vortex states, producing the constricted hysteresis loop seen in (a). Only vortex denucleaon is captured by the central ridge, so that fcr consists of a single peak.

VARIFORC User Manual: 8. FORC tutorial

8.36

Hysteresis parameters of small PSD crystals are very sensi ve to vortex nuclea on ﬁelds:
if nuclea on from posi ve satura on occurs in nega ve ﬁelds, SD-like values of Mrs /Ms are
obtained from SD states that are stable in zero ﬁelds. If, on the contrary, vortex states can
nucleate from SD+ in posi ve ﬁelds, Mrs /Ms drops well below 0.5, because of their small net
magne c moment. On the other hand, PSD par cles are always characterized by Mcr < Mbk
< Mirs , unlike the case of non-interac ng SD par cles, where Mbk ³ Mirs .
The FORC diagram in Fig. 8.15c does not resemble the typical signature of PSD magne te
in synthe c and geologic samples, which consists of a unimodal distribu on peaking near
Hc = Hb = 0 , with triangular contour lines having their maximum ver cal extension at Hc = 0
[Roberts et al., 2000; Muxworthy and Dunlop, 2002]. This signature is explainable on the basis
of the abovemen oned PSD processes by taking the following factors into considera on: ﬁrst,
magne c states of small PSD par cles are very sensi ve to grain sizes and shapes, so that
magne za on jumps of single-par cle hysteresis generate dis nct FORC peaks only in case of
excep onally homogeneous samples, such as the Fe nanopar cles discussed above. Second,
the number of possible magne c states grows rapidly with increasing grain size, along with
the number of FORC peaks, which eventually merge into a unimodal func on. As discussed in
sec on 3, the FORC func on occupies a triangular area of the FORC space limited by ver ces
with coordinates (0, Hsat ) and (Hsat ,0) , where Hsat is the ﬁeld in which the two branches of
the hysteresis loop merge. This limit is consequently imposed to the shape of contour lines.
Finally, the central ridge is broadened by magnetosta c interac ons, which are probably not
negligible in most synthe c magne te powders.
Some natural materials, such as olivine-hosted Fe-Ni par cles in chondri c meteorites
[Lappe et al., 2011], contain weakly interac ng PSD par cles with suﬃciently homogeneous
proper es for producing FORC diagrams with dis nguishable contribu ons from vortex nuclea on and denuclea on. In such cases, high-resolu on measurements are essen al for capturing details of PSD magne za on processes. Faint evidence of such processes persists in
FORC diagrams of many natural rocks (Fig. 8.16).

Fig. 8.16 (front page): Two FORC examples based on natural samples containing PSD par cle assemblages, i.e. (a-c) basalt (sample EF-3), and (d-f) volcanic ash [Ludwig et al., 2013]. (a,d) FORC measurements, with results a er lower hysteresis branch subtrac on shown in the insets. (b,e) FORC diagrams
calculated with VARIFORC from the measurements shown in (a,d). Some features typical for singlevortex PSD par cles (Fig. 8.15) can be recognized, namely the existence of a central ridge, albeit much
weaker, and the inﬂuence of localized nega ve FORC amplitudes (labeled with a minus sign) on the
overall shape of contour lines. Furthermore, two nearly symmetric posi ve peaks above and below the
central ridge are dis nguishable in (b). Unlike the example of Fig. 8.15, the existence of such contribu ons cannot be directly inferred from the hysteresis loop. The ver cal ridge along Hc = 0 in (e) is a
signature of magne c viscosity.

VARIFORC User Manual: 8. FORC tutorial

8.37

(d)

(a)

M , mAm2

+0.1
0
0.02

–0.1

M , mAm2/kg

+50

–0.1

0

20

–50

0
–0.1

0

0

+0.1

–0.1

0

H, T

–0.1

+0.1

H, T

(e)

(b)

+50

mAm2
T2

+40

Am2
T2kg

0

10

20

Hb , T

Hb , T

20

0
10

0
–40

–50
0

0

40

0

80

50

μAm2:
Mrs = 13.5
Mirs = 33.7
Mcr = 1.3

0.4

firr
fbf

0.2

fcr
0

–50

0

50

−Hr or Hc , mT

100

dM/dH , mAm2/(T kg)

(f)

(c)
dM/dH , mAm2/T

100

Hc , T

Hc , T

mAm2/kg:
Mrs = 12.7
Mirs = 25.1
Mcr = 0.3

0.4

fbf

firr

0.2

fcr
0

–50

0

50

100

150

−Hr or Hc , mT

Fig. 8.16 (con nued): (c,f) Coercivity distribu ons derived from the FORC measurements shown in
(b,e). Residual bimodality is s ll recognizable for firr and fbf in (c). The central ridge distribu on fcr is
much smaller than the other two, but signiﬁcantly 0. Unlike the case of SD par cles, the contribu on
of firr over nega ve arguments is not negligible, as expected from large posi ve FORC amplitudes over
the upper quadrant.

VARIFORC User Manual: 8. FORC tutorial

8.38

Because of its sharpness, the central ridge remains dis nguishable even with contribu ons
as low as few % of Mirs . Localized nega ve amplitudes over the lower quadrant, on the other
hand, produce characteris c contour line indenta ons when overlaid to a posi ve background. Resolu on of such signatures in PSD samples can beneﬁt paleomagne c applica ons,
where FORC analysis has been proposed as a selec on [Wehland et al., 2005; Carvallo et al.,
2006; Acton et al., 2007] and modeling [Muxworthy et al., 2011a,b] tool for paleointensity
determina ons. For example, Mcr /Mirs is a measure for the rela ve magne za on of vortex
states, which probably represent the preferen al PSD contribu on to natural remanent magne za ons.
8.4.3 MD par cles

Domain wall displacement models [Pike et al., 2001b; Church et al., 2011] explain the ideal
hysteresis proper es and FORC signature of MD par cles (Fig. 8.4). These proper es are met
by annealed magne te crystals [Pike et al., 2001b], while FORC diagrams of unannealed parcles (Fig. 8.17) can be explained by the superposi on of MD and PSD signatures. As far as
coercivity distribu ons are concerned, ideal MD proper es are characterized by hysteresis
branches and corresponding irreversible contribu ons being are quasi-symmetric about H =
0, which means that firr (x) » firr (-x) . The typical firr -shape of MD par cles resembles a
Laplace (double exponen al) distribu on (Fig. 8.17c). Because the magne za on of ideal
crystals with weak domain wall pinning is almost completely reversible, irreversible processes,
which occur in form of so-called Barkhausen magne za on jumps, represent only a small fracon of the satura on magne za on, i.e. Mirs /Ms  1 . For example, Mirs /Ms = 0.22 for the
20-25 µm magne te crystals of Fig. 8.17. Only a small frac on of all Barkhausen jumps yield a
remanent magne za on, so that Mrs /Mirs  1 ( Mrs /Mirs = 0.35 for the example of Fig. 8.17).
Since fbk (0) = firr (0) , the backﬁeld coercivity distribu on is a func on that decays much more
rapidly than firr to zero for posi ve arguments. Finally, fcr = 0 because of the absence of a
central ridge.
In summary, ideal MD crystals without strong domain wall pinning are characterized by
Mcr = 0 and Mrs  Mirs  Ms . For comparison, PSD and SD par cles yield Mrs < Mirs and
Mrs ³ Mirs , respec vely. Therefore, the ra o Mrs /Mirs can be considered as a sort of domain
state indicator analogous to Mrs /Ms , with the important advantage that Mrs /Mirs is insensi ve
to reversible magne za on processes (e.g. SP contribu ons).

VARIFORC User Manual: 8. FORC tutorial
(b)

(a)
+0.2

M , mAm2

8.39

+50

mAm2
T2

0
0.04

–0.2

60

0

–0.1

0

–0.1

+0.1

(c)
dM/dH , mAm2/T

0

Hb , T

H, T

40

μAm2:
Mrs = 28.4
Mirs = 81.7
Mcr ≈ 0

1.0

20
–50

firr

0.5

0
–60

0

fbf
fcr
–40

–20

0

20

−Hr or Hc , mT

40

60

0

50

Hc , T

Fig. 8.17: FORC analysis of synthe c MD magne te par cles with 20-25 µm mean grain size. (a) Set of
FORC measurements where every 8th curve is plo ed for clarity. The inset shows the same set of curves
(every 4th is plo ed) a er subtrac on of the lower hysteresis branch. Measurement details are recognizable only in this plot, because of the very small hysteresis loop opening. (b) FORC diagram obtained
with VARIFORC from the FORC measurements shown in (a). The MD nature of this sample is determined by the large ver cal spread of the FORC func on in proximity of Hc = 0 . Addi onal PSD signatures
are recognizable over Hb < 0 and along a sort of blurred central ridge. These features are typical of
unannealed MD crystals. (c) Coercivity distribu ons derived from the FORC measurements shown in
(c). firr is a quasi-even func on with similar contribu ons over nega ve and posi ve ﬁeld, as expected
for MD par cles. The central ridge does not exist, so that fcr = 0 .

VARIFORC User Manual: 8. FORC tutorial

8.40

8.5 Conclusions
FORC diagrams are two-dimensional parameter representa ons of hysteresis processes
providing a lot more details than the conven onal bulk hysteresis parameters Mrs /Ms and
Hcr /Hc . The simplest interpreta on of FORC diagrams is based on Preisach theories, which,
however, rarely describes real magne za on processes. Dedicated models have been developed for explaining the FORC proper es of ideal SD, PSD, and MD par cle assemblages. Such
models describe ‘end-member’ FORC signatures that can be used for qualita ve interpreta on
of FORC diagrams in terms of domain states. This has been the ﬁrst rock magne c applica on
of FORC measurements. Meanwhile, con nuous modeling improvements and the use of highresolu on measurements resulted in ﬁrst quan ta ve FORC analyses of samples containing
non-interac ng SD par cles.
An important forward step in FORC analysis consists in overcoming the quan ta ve gap
with conven onal magne c parameters. As shown in this ar cle, FORC measurements deﬁne
three diﬀerent types of irreversible magne za ons and corresponding coercivity distribuons: the ﬁrst type is represented by the well-known satura on remanence Mrs and associated coercivity distribu on derived from backﬁeld demagne za on. The second magne za on type deﬁned by FORC measurements is the irreversible satura on magne za on Mirs ,
which is the sum of all irreversible magne za on changes occurring along the upper or lower
branch of the hysteresis loop. Mirs is also the total integral of the FORC func on. The coercivity
distribu on associated with Mirs represents irreversible processes occurring on the upper
hysteresis branch at a given ﬁeld. The third, so-called central ridge magne za on Mcr is generated by the last magne za on jump occurring in single par cle hysteresis loops of isolated
SD and small PSD magne c par cles. The central ridge coercivity distribu on is derived from
the corresponding signature of the FORC func on along Hb = 0 .
Unlike Mrs /Ms and Hcr /Hc , these magne za ons are unaﬀected by reversible magne za ons and provide more robust domain state informa on than the Day diagram. For example, the Day diagram characteris cs of clay-dispersed, SDS-treated magnetosome chains (Fig.
8.8) suggest a mixture of SD, SP, and PSD par cles. In reality, as seen with FORC diagram measurements, the whole remanent magne za on of this sample is produced by non-interac ng
SD par cles. This sample represents a possible end-member of a trend deﬁned by magnetofossil-bearing sediments on the Day diagram. The magne c proper es of these sediments are
therefore not necessarily interpretable as mixtures of magnetofossils on one hand, and SP and
PSD par cles on the other. Another example is represented by the Day diagram proper es of
small PSD par cles (Fig. 8.15), which plots on the same trend deﬁned by sediments, while its
purely PSD nature is clearly recognizable on the basis of Mirs /Mrs and Mcr /Mirs .
All FORC processing aspects described in this paper, including the calcula on of FORCrelated magne za ons and coercivity distribu ons, have been implemented in the VARIFORC
so ware package, so that they can become a rou ne magne c analysis tool.

VARIFORC User Manual: 8. FORC tutorial

8.41

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