1 The model
Equilibrium condition in the Stoner-Wohlfarth model In its simplest form, theStoner-Wohlfarth model [1] deals with the calculus of the equilibrium state of the total magnetic moment of a single-domain ferromagnetic particle with uniaxial anisotropy. The model assumes that all the magnetic moments in the particle are always parallel. The magnetisation processes of the “Stoner-Wohlfarth particle” are done through coherent rotation of all these moments. At any time the directions of the magnetic moments within the particle are parallel. The magneto-crystalline free energy density is, for a single-domain uniaxial crystal, given by:
(1) where 
is the angle between the easy axis (OZ axis) of the particle and the saturation polarisation moment of theparticle (see Figure 1).
The magnetostatic free energy density maybe written as:
(2)
where the vectors 
(saturation polarisation of the particle) and 
(applied field vector) were expressed in spherical co-ordinates: 
and 
, respectively.
Introducing the anisotropy field, defined as:
(3) the total free energy density for one single-domain particle may be written as:
(4)
From the minimum condition for the j co-ordinate,that is:
(5)
Fig. 2
one obtains:
(6) with the solution: 
. So, at the equilibrium, the polarisation vector, , will be in the same plane as the applied field. Without restricting the generality of the discussion we shall take 
as the plan where all the vectors are. Taking that into account, the total freeenergy becomes:
(7) and making the following notations:
, 
(8) one may transform the expression of W, as:
(9)
The stable equilibrium conditions are givenby:
(e) 
(10)
(s) 
(11)
and after simple calculations one obtainsthe system in the form:
(e) 
(12)
(s) 
(13)
The system of equations which results byreplacing in(s) the sign “>” with “=” has as solution the parametricequations 
and
of the geometrical locus of the critical fields which separates in the 
plane the region where the free energy function has two minima as a function of from theregion where this function has only one minimum.
They may be written as:
(14) and if one eliminates the parameter one obtains:
(15) which is an astroid in the plane (see Figure 3). Inside the astroid W has two minima as a function of 
, while outside there is only one. To prove that, we may write the (e) and (s) equations
(16)
which represents two lines in the plane. These lines have properties which shall be used in the calculus of the equilibrium state:
1. The line (e) is tangent to the astroidin the point determined by the angle 
(17) which is the slope of the (e) line.
2. The line (s) is perpendicular on the (e) line due to the fact that their slope product is –1. The (s) line divides the plane into two half-planes. In one the equilibrium is stable while in the other is unstable. We noted the line region within the stable half-plane isnoted with (es).
3. Analysing the (es) lines we obtain the configuration presented in the Figure 5. If the point of the vector is within the hashed region there will be with certainty an equilibrium solution between 0° and 90°. The region for 
is presented in the Figure 6.
In conclusion, when the field is applied and its point is inside the astroid, there are two equilibrium solutions (inside the astroid we have two hashed regions corresponding to differente quilibrium orientations, see Figure 7); outside the astroid it is possible only one equilibrium solution.
In many real cases, one may consider more then one anisotropy source.
Let’s see if we can still use the geometrical method we developed previously in the calculus of the equilibrium state. For example, let’s consider that there is another anisotropy source of magneto-elastic type. The magneto-elastic free energy density is given by:
(18)
where 
is the saturation magnetostriction coefficient and 
is the mechanical stress vector which makes the angle 
with the magneto-crystallineanisotropy axis.
The total free energy density is in thiscase:
(19)
If we are making the notations:
(20) the expression for W becomes:
(21) or
(22), where
(23) and may be transformed as:
(24)
If one chooses 
so
(25) one obtains:
(26) and with
(27) after simple algebra,
(28) where 
is an equivalent total anisotropy field and is the angle between the easy axis of the particle and its magneto-crystalline easy axis.
The total free energy density may bewritten in the terms of equivalent anisotropy, as:
(29)
and the stable equilibrium condition is given by:
(30)
(31)
which may be written as:
(32)
(33)
Replacing the sign “>” with “=” in (s) we are obtaining a system of equations whose solution is the critical curve with the same physical signification the astroid in the absence of the magneto-elastic term. The parametric equations of this critical curve are:
(34)
To provide an easier explanation we transform these equations in a system 
rotated with respect to the system withthe angle :
(35) which may be transformed in the simple form:
(36) which represents in the system the equations of an astroid.
| Activity: | |
| Run the following Maple program in order to get familiar with the calculus of the critical curves. The program contains animated representation of the (e) and (s) curves as a function of the angle. |
|
########################
# Stoner-Wohlfarth model######################## > restart: with(plots): > w:=(h, theta, theta0)->-cos(theta)^2-2*h*cos(theta-theta0): > w_d1:=(h, theta, theta0)->diff(w(h, theta, theta0), theta): > w_zx:=(hz, hx, theta)->-cos(theta)^2-2*hx*sin(theta)-2*hz*cos(theta): > w_zx_d1:=(hz, hx, theta)->diff(w_zx(hz, hx, theta), theta): > w_zx_d2:=(hz, hx, theta)-> diff(w_zx_d1(hz, hx, theta), theta): > hx_c:=sin(theta)^3: hz_c := -cos(theta)^3: > astro_plot:=plot([hz_c, hx_c, theta=0..2*Pi], color=black, thickness=3): > hx_e := (hz, theta)-> hz*tan(theta) + sin(theta): > hx_s := (hz, theta)->-hz/tan(theta) -cos(2*theta)/sin(theta): > ech_anim:=animate(hx_e(hz, theta), hz=-10..10, theta=0..2*Pi, view=[-1..1, -1..1], color=blue, numpoints=200, frames=100): > stb_anim:=animate(hx_s(hz, theta), hz=-10..10, theta=0..2*Pi, view=[-1..1, -1..1], color=red, numpoints=200, frames=100):
> display([astro_plot, ech_anim, stb_anim]); > theta_s := (1/2)*arctan(hs*sin(2*psi)/(1+hs*cos(2*psi))): > hk_s := sqrt(1+hs^2+2*hs*cos(2*psi)): > hz_c := -hk_s*((1/2)*sin(2*(theta-theta_s))*sin(theta)+cos(2*(theta-theta_s))*cos(theta)): > hx_c := hk_s*((1/2)*sin(2*(theta-theta_s))*cos(theta)- cos(2*(theta-theta_s))*sin(theta)): > psi:=Pi/8; > animate([hz_c, hx_c, theta=0..2*Pi], hs=0..2, color=black): psi := 1/8 Pi; > hs:=2: astro_plot_s := plot([hz_c, hx_c, theta=0..2*Pi], color=black, thickness=3 ): > hx_e_s := (hz, theta) -> hz*tan(theta) + hk_s*sin(2*(theta-theta_s))/2/cos(theta): > hx_s_s := (hz, theta) -> -hz/tan(theta) - hk_s*cos(2*(theta-theta_s))/sin(theta): > ech_anim_s := animate(hx_e_s(hz, theta), hz=-10..10, theta=0..2*Pi, view=[-hk_s..hk_s, -hk_s..hk_s], color=black, numpoints=200, frames=100): > stb_anim_s := animate(hx_s_s(hz, theta), hz=-10..10, theta=0..2*Pi, view=[-hk_s..hk_s, -hk_s..hk_s], color=blue, numpoints=200, frames=100): > display([astro_plot_s, ech_anim_s, stb_anim_s]); |
2 Magnetisation processes in the Stoner-Wohlfarth model
The most common hysteresis loop which maybe calculated represents the magnetic moment vector projection on the fielddirection as a function of the applied field: 
(see Figure 9). The next Maple program implements the algorithm for the calculus of hysteresis loops using the Stoner-Wohlfarth model. It may be used to calculate the loops for 
between 0° and 90°.
|
####################################### > restart; with(plots): with(plottools): > w := -Ps/2*(Hk*cos(theta)^2+2*H*cos(theta-psi)); > dw := diff(w, theta); > equ := dw=0; > g := (sin(psi)^(2/3) + cos(psi)^(2/3))^(-3/2); ##### > psi := 35.0*Pi/180.0; H := 0.1; Ps := 1.0; Hk := 1.0; > Hc := evalf(Hk * g); theta := 'theta'; > theta_s := evalf(Pi - arctan( tan(psi)^(1/3) )); > equ; > H := 'H'; np := 100; H_max := 1.0; > h_p := array(1..np); > h_n := array(1..np); > m_p := array(1..np); > m_n := array(1..np); > h_p[1] := 0.0; m_p[1] := cos(psi); > for i from 2 to np do > h_n[1] := 0.0; m_n[1] := cos(Pi-psi); > for i from 2 to np-1 do > h_n[np] := Hc; m_n[np] := cos(theta_s-psi); > pl_pp := array(1..np); > pl_pn := array(1..np); > pl_np := array(1..np); > pl_nn := array(1..np); > for i from 1 to np do > display([seq(pl_pp[i], i=1..np) seq(pl_np[i], i=1..np), seq(pl_pn[i], i=1..np), seq(pl_nn[i], i=1..np)]); #####  |
2 The "cross-over" of the hysteresis branches in the Stoner-Wohlfarth model
One problem of the coherent rotation Stoner-Wohlfarth model is that it is not giving the correct dependence of the critical field on the angle between the easy axis and the applied field direction. In order to solve this discrepancy, a modification of the switching criterion is frequently used: the rotation of the magnetisation is governed by the full Stoner-Wohlfarth astroid and the switching occurs according to a truncated astroid [2], [3]. Usually the astroid is truncated in the region of small angle between the easy axis and the applied field direction Fig.1.
Fig. 1
Another problem is known as the "cross-over" [4], [5], that is, the cross of the hysteresis branches, which is observed at the hysteresis loops calculated for angles near 90° between the easy axis and the applied field direction Figure 2.
Fig. 2 The Stoner-Wohlfarth hysteresis loops for two angles between the easy axis and the applied field direction: 
On the 85° loop one observes the "cross-over" at 
field.
As we proven before, for an orientation of the applied field, 
, one obtains two solutions (corresponding to the two branches of the hysteresis loop) if the applied field is less than the critical field, 
, which is given by:
(1)
where
(2)
For 
close enough to 90° one observes two intersections (A and B in Fig. 2) between the hysteresis branches, which will be referred to as the "cross-over". From the equilibrium condition one may obtain the following expressions for the ascending and the descending branches:
(3)
where 
is the magnetic moment projection on the applied field direction normalised to the saturation magnetic moment, 
and 
are the values of the applied field corresponding to the same value of the magnetic moment projection, m.
Fig. 3 The hysteresis loop given by (6) for 
=65°. The ascending and the descending branches (solid line) are limited by the critical point (B); the “cross-over” point (A) is not reached.
Fig. 4 The hysteresis loop given by (6) for 
=85°; the “cross-over” point (A) is reached.
The "cross-over" condition is:
(4)
Introducing (3) in (4) one obtains: 
(5)
Solving the equation 
one gets the minimum value of 
at which the "cross-over" is observed:
(6)
One observes that:
· the "cross-over" occurs at the same value of the magnetic moment projection;
· the angle between the applied field and the polarisation vector is 45° in both equilibrium positions at the "cross-over";
· the "cross-over" occurs only for 
.
In order to eliminate in the Stoner-Wohlfarth model the branches “cross-over” one may use for the corrected critical field the expression:
(7)
Fig.5 The "cross-over" points for three values of the angle between the easy axis and the applied field direction: 80°, 85° and 90°.
Fig.6 The critical field given by (10) with solid line. With dashed line is represented Hcl field given by (8), for 
(region EB) and Hc given by (3) for 
(region BC)
In Fig.6 one presents with solid line the critical field given by (2) and with dashed line the field Hcl, given by (7), for angles between the easy axis and the applied field direction smaller than 
- region EB and the critical field Hc, given by (1), for angles bigger than
.
The "cross-over" in the "truncated astroid" model
As we previously shown one problem related to the use of the classical coherent rotation model is the critical field dependence on the applied field orientation. The customary solution to that problem is the use of the "truncated astroid" for the critical field and the full astroid for the calculus of the equilibrium orientations. In the Interacting Particle System (IPS) [6], [7] model we used for the critical field the expression:
(8)
with
(9)
where 
has been used as fitting parameter.
In order to correctly determine the magnetic moment in this case, one has to change the total free energy density expression by adding a correction term dWT:
(10)
One may easily observe that this term is equivalent to a supplementary field parallel to the easy axis direction. The new formula for the total free energy allows obtaining a model in which the critical field is given by (8) and, in the same time, the magnetic moment is calculated in a consistent way to this hypothesis; the use in the calculus of the magnetic moment of the whole astroid is no more needed. We will refer to this model as the "truncated astroid" model. The parametric equations of the truncated astroid are:
(11)
Using (11) one may find the angle between the magnetic moment and the easy axis at the critical field. This angle and the magnetic moment are given by:
(12)
To determine the two "cross-over" points one uses an analogous method as in the case of the classical coherent rotation model. From the equilibrium equation one obtains the following analytical expressions for the ascending and the descending branches:
(13)
where 
. From the condition 
one determines the normalised magnetisation values in the two "cross-over" points:
(14)
The intersection field Hcl is found using these values in the expressions of the ascending and the descending branch:
(15)
where 
.
The minimum angle 
at which the "cross-over" occurs, which will be referred to as 
, is determined as the solution of the equation:
(16)
One may observe that the two values of the magnetic moment corresponding to the same field 
characterise the stable and the unstable equilibrium. These values are equals only in the critical point 
which are a maximum point of the 
function; this is why the closure field 
it is smaller then the critical one. In this case, 
and the angle 
is an extreme point of this function.
To increase the accuracy and the time-efficiency of the numerical calculus it is allowed to use the differential:
(17)
This function has the same root as 
but the large slope near 
provide a good precision in the numerical algorithm which determines
. In order to eliminate the non-physical hysteresis loops for angles between the easy axis and the applied field direction superior to 
one may use the following expression for the critical field:
(18)
Fig.7 The functions 
and 
have the same root 
; In this point 
present a large slope which allows an increasing of the accuracy and the time efficiency of the 
calculus.
Fig.8 With solid line is represented 
which is an interpolation of the equation 
numerical solution versus the 
parameter.
In the Fig.9 one presents the graphs of the critical field calculated with (18) with solid line; with dashed line is the critical field given by (8) for angles superior to 
. One observes that the "cross-over" problem is not solved in the "truncated astroid" model.
Fig.9 The critical field given by (21) with solid line. With dashed line is Hc given by (8) for 
for three values of 
parameter.
Fig.10 The configuration of the magnetic moment of the particle in the "cross-over" point.
A physical motivation for the "cross-over"
In order to give a physical explanation to the "cross-over" problem, one may start from the observation that this phenomenon appears at angles between the easy axis and the applied field direction near 90° at sufficiently high fields. In the classical coherent rotation model the anisotropy free energy density formula used in (1) is only the first term of a series expansion:
(19)
The first order approximation is satisfactory for small angles between the polarisation vector and the anisotropy axis; this condition is not fulfilled near the "cross-over" point. To analyse the effect of the supplementary terms in the series expansion (19) on the "cross-over" condition let us observe more carefully the configuration of the magnetic moments in this point.
In Fig.10 one observes that the angles between the magnetic moment of the particle and the easy axis are different for the two equilibrium orientations while the angles between the magnetic moment and the applied field direction are equal to 45°. One may conclude that, in this case, there are two different angles between the easy axis and the magnetic moment of the particle for which the moment of the anisotropy couple has the same absolute value and different signs. It is easy to show that this is a necessary and sufficient condition for the "cross-over" to occur; this fact is closely related to the shape of the anisotropy free energy density function of
. This condition could be satisfied only if this function has an inflection point for
.
In Fig.11 one presents the effect of taking into account an anisotropy free energy density formula, as in (19), with seven terms (K0=0,K1,...,K6, with Kn/K1=2-(n-1), n=1..6). The "cross-over" point is shifted to higher magnetic fields and the change in the magnetic moment projection at the switching field is approximately two times smaller than in the classical case.
On the free energy density graph, presented in Fig.12 one observes that these curves have inflection points for any number of terms in the series expansion (19); this is motivated by the fact that the functions derivatives are zero both at 0° and 90°.
To see the effect of an anisotropy term without inflection point in the domain
, we simulated the hysteresis loops for an anisotropy free energy density term:
(20)
in Fig.13 one observes that the "cross-over" disappears as expected 
.
Fig.11 The hysteresis loops for 
in the coherent rotation model: (1) with the classical Stoner-Wohlfarth model; (2) with the anisotropy free energy density given by (19), with seven terms (K0=0,K1,...,K6, with Kn/K1=2-(n-1), n=1..6).
Fig.12 The graphs of the anisotropy free energy density as a function of 
in four cases; (1) the classical Stoner-Wohlfath case; (2) the anisotropy free energy given by (19) with three terms (K0=0,K1,K2, with Kn/K1=2-(n-1), n=1,2).; (3) the anisotropy free energy given by (19) with seven terms (K0=0,K1,...,K6, with Kn/K1=2-(n-1), n=1..6); (4) the anisotropy free energy given by (20).
Fig.13 The hysteresis loops in the coherent rotation model for 
: (1) with the classical Stoner-Wohlfarth model; (2) with the anisotropy free energy density given by (20).
Fig.14 The hysteresis loops when the anisotropy free energy density is given by (20) for three values of 
: 5°,45°,85°.
3. 2D & 3D coherent magnetization rotation model
We shall add to the previous presentation a vector generalization of the Stoner-Wohlfarth model, developed by Thiaville [8].
To start, one presents the 2D generalization. The same principles will apply to the 3D generalization. The total free energy density will be written in the general form:
(1)
where 
is the anisotropy energy density, which depends in the 2D case on the angle between the easy axis and the polarization vector. This expression could be written as:
(2)
if one makes the notations:
(3)
(4)
(5)
(6)
(7)
where 
is a constant (in the uniaxial case the anisotropy constant), and 
is a unit vector on polarization vector direction.
The equilibrium and stability conditions are obtained deriving (2); one obtains:
(e) (8)
and
(s) 
(9)
where
(10)
and is a unit vector, perpendicular to 
.
If one consider the inverse problem (for a given 
find the locus of the applied field vector tip), (e) and (s) represent the equations of two straight lines:
(e) is a line perpendicular to the 
vector (parallel to the 
vector)
(s) is a line perpendicular to the 
vector (parallel to the 
vector).
So, the (e) and (s) lines are reciprocally perpendicular, (e) being parallel to the equilibrium orientation of the polarization vector. The system defining the critical curve 
is given by:
(11)
and has the solution:
. (12)
Example: For the uniaxial case:
and
. (13)
which represents the equation of an astroid.
If one makes the derivative of the 
in order to find the direction of the tangent to the critical curve, one finds:
(14)
that is, the tangent is parallel to the equilibrium orientation of the polarization vector, as the (e) line. We can introduce an orientation of the critical curve: the positive sense of the critical curve may be given by
.
| Problem: | |
| Prove that the orientation is reversing in the cusp points of the critical curve. | |
Note
One observes that all the properties we derived for the uniaxial case in the previous sections applies to the general 2D case.
Constant energy curves
The locus of the applied field vector tip for one given value of the energy, w, in the equilibrium state is the solution of the system:
(15)
that is,
. (16)
|
#######################################
|
The tangent to the constant energy curve is given by:
(17)
that is, it is perpendicular to the equilibrium orientation,
(see Figure 16).
Defining by 
the arc-length on the constant energy curve we have:
(18)
Fig. 16
where 
is the curvature radius of the constant energy curve.
In order to find the position of the curvature centre, noted with
, we can use the relation (see Figure 16):
. (19)
Introducing in (19) the expressions (16) and (18) one obtains:
(20)
that is, the curvature centre is on the critical curve. From the mathematical point of view one may say that the critical curve is the common evolute of all constant energy curves.
Geometric construction for energy calculations
Thiaville [9] also proved that it is possible to use a simple geometric construction to estimate the free energy density when a field is applied to the ferromagnetic particle. The normalized total free energy density may be expressed as:
(21)
and if one uses (11), one obtains:
(22)
Taking into account (14), the arc-length on the critical curve is given by:
(23)
which is equivalent to:
. (24)
In conformity with (24) the difference between the values of 
in two equilibrium states on the critical curve characterized by q0 and q is twice the arc-length measured on the critical curve. This property could be used in (22) for expressing the sum between the first and the second term. The final term in (22) is minus twice the length of the segment on the equilibrium line (tangent to the critical curve) linking the point on the critical curve with the tip of the applied field vector (see Figure). In fact, in this way one can calculate the difference of energy for two states characterized by two values of the applied field. For example, when the field 
is applied as presented in the Figure, the equilibrium (stable) orientation of the polarization vector is the orientation of the vector 
. Using the consequences of the previous geometrical discussion we can say that the difference of energy between the “C” state and the “A” state (see Figure, the “A” state is characterized by an applied field vector whose tip is on the point “A”) is twice the sum of the lengths of the arc AB and the segment BC, that is:
(25)
Fig. 17
Note:
The points characterized by 
(A, see Figure 18) and 
(B) have the same energy; the points A and B could be connected by two quarters of the astroid one positive (AC) and one negative (CB).
Fig. 18
Fig.19
When the tip of the applied field vector is inside the astroid there are three tangents to the critical curve passing through it: two for stable equilibrium states and one for an unstable equilibrium state (see Figure 19).
3D coherent magnetization rotation model
The method applied in the previous section could be used for developing a similar set of geometrical rules for the 3D coherent magnetization model [10].
The most general form of the free energy density (normalized to 2K this time in order to obtain a simpler form) is:
(25)
where:
(26)
is the unit vector of the polarization vector.
To obtain the stable equilibrium conditions we need the following expressions:
(27)
(28)
(29) (30)
(31)
In the calculus we used:
(32)
(33)
(34)
(35)
(36)
(37)
where 
is a set of three unit vectors which are an orthogonal basis.
From the equilibrium conditions:
(38)
(39)
one can find a general expression for the 
vector. From (38) and (39), which represent the equations of two planes, we can calculate the components on 
and 
directions:
(40)
(41)
The intersection of these planes is the straight line oriented on the 
direction. The general solution of the system (38)-(39) is:
(42)
If (
,
) are the spherical angles defining the orientation of the polarization vector at the equilibrium the free energy density could be developed in the Taylor series near this equilibrium orientation as follows:
(43)
where
, (44)
and
. (45)
Taking into account the equilibrium conditions (38) and (39) the stability condition,
(46)
may be written as:
(47)
or, in a simpler form;
(48)
where
(49)
In order to have (48) for any value of x, the determinant should be negative and the coefficient of 
should be positive:
(50)
Introducing the equilibrium condition given by (42) in the first equation in (50) we are founding two solutions:
(51)
From the other equation in (50) we can say that the stable equilibrium part of the line defined by (42) is given by the condition:
. (52)
Introducing 
in (42) one finds the equations of two surfaces, 
.
|
###############################
> l_minus := (1/2)*(-c1-sqrt(c2^2+4*c3^2)):
# Case 2: Cubic anisotropy
|
4. Transverse susceptibility of the Stoner-Wohlfarth particle
The well-known experimental method of transverse susceptibility (TS) is a method for direct determination of the magnetic anisotropy in particulate magnetic systems. This is due to the fact that, as predicted by Stoner-Wohlfarth model, for non-interacting uniaxial single-domain fine particle systems, the field dependence of transverse susceptibility presents characteristic peaks, located at the anisotropy and switching fields. The conditions, as uniaxial anisotropy, single-domain non-interacting particles, in which the anisotropy can be determined accurately using this method might appear to be very restrictive. In fact, the same conditions that apply to other popular methods for determining the anisotropy in the case of particulate systems, as single detection point techniques or rotation hysteresis method, are satisfied in many practical cases. For TS experiments an important step in this direction was done by advancing the classical model for TS due to Aharoni [11], by taking into account the influence of the higher order terms of the uniaxial anisotropy [12], [13], [14], [15] fact that is important for many usual magnetic uniaxial materials as cobalt.
In this section we present the methods used in the TS evaluation for both uniaxial and cubic single-domain particles.
The experimental method of transverse susceptibility (TS) is used for direct measurement of the anisotropy in ferromagnetic systems due to the fact that usually these systems show sharp peaks (see also Fig. 1_a) located at the anisotropy field which makes possible a precise detection and the calculation of these important physical parameters as it was proven for uniaxial single particles within the coherent rotation Stoner-Wohlfarth theory in [16]. The series expansion for the magneto-crystalline uniaxial anisotropy is given by:
(1)
where 
is the angle between the easy axis and the magnetic moment of the particle.
However, this is accurate only when the angle between the easy axis and the magnetic moment is sufficiently small. In the TS experiment, it was shown that the particles with the easy axis oriented near 90° to the DC field direction are responsible for the peaks located at
, where 
is the anisotropy field; the shape of the TS curve is significantly influenced by the particles with the easy axis oriented near 90°. For these particles, when the field is near
, the angle between their easy axis and the magnetic moment is close to 90º. Therefore, neglecting the higher order terms in the TS calculation is a major source of errors. For materials with high values of 
this error is more significant.
In the TS experiment one applies to the sample a system of two magnetic fields (see Fig.1), one DC field that is considered on the Oz direction, and a small amplitude AC field, on the Ox direction in Fig.1. The material easy axis (EA) orientation is given by the spherical angles 
and, if one can define a single orientation of the sample magnetization, as in the coherent rotation model, the orientation of the magnetic moment is given by the radial unit vector in the spherical coordinates system
. The angle between the total applied field and the Oz axis at a certain moment is qh. In the TS experiment, one measures the limit:
(2)
that will be referred to as the transverse susceptibility in the experimental setup described in Fig.1.
Fig. 1_a Typical TS signal with anisotropy peaks and coercivity peak. The DC field is decreasing in this experiment
Aharoni [17, using the Stoner-Wohlfarth model has calculated the components of the reversible susceptibility tensor and the TS susceptibility:
(3’)
Applying a similar methodology, we have found the expression for the TS when the second term in the series expansion of the anisotropy free energy is also taken into account [18], [19]:
(3)
where 
is the reduced DC field, 
, 
, 
and M is the saturation magnetization of the particle.
Fig.1 Transverse susceptibility experiment for an uniaxial single-domain ferromagnetic particle
Critical curve approach for the uniaxial case
The TS evaluation is simpler for 
due to the fact that the free energy has only two minima which can be selected with the well-known SW astroid critical curve. When
, the free energy landscape is more complicated. The number of minima is higher and the selection of the stable state in the TS measurement is more complex and the use of the critical curves formalism in this case is a helpful tool. Using the same strategy as Thiaville [20], [21] one express the normalized free energy density as:
(4)
where 
is the normalized anisotropy free energy, 
, H is the applied field, and the orientation of the magnetization is given by the unit vector 
. For a system with uniaxial anisotropy the easy axis, the applied field direction and the magnetic moment direction are in the same plane. So, one can choose the Oz axis direction on the easy axis and one can solve a 2D problem instead of the initial 3D problem. In spherical coordinates, the anisotropy free energy density can be expressed as a function of the 
angle only, and
. The equilibrium and stability conditions are given by:
(5)
where we have used the relations: 
. (6)
Replacing the inequality sign with equal in the stability condition, one obtains the condition for the critical field that separates the stable from the unstable regime. The critical field vector 
is defined by both equilibrium and critical stability conditions:
(7)
In fact, the first equation gives the 
component of the critical field vector and the second the radial component. The equilibrium equation in (5) represents in the plane a line perpendicular to the vector
, that is parallel to
, and the critical stability equation is a line perpendicular to, which is parallel to
. The tip of the critical field vector is at the intersection of these two lines (see Fig. 2). Taking this into account, the critical field vector can be written as:
(8)
where 
are the unit vectors in the Cartesian coordinates associated to the same 2D region.
The tip of the critical field vector, 
, gives the critical curve for a certain expression of the anisotropy free energy density which is in agreement with the assumption that allowed the 2D calculus. The derivative of the critical field vector with respect to the angle q gives the direction of the tangent direction to the critical curve. Using (4) in the calculus of this derivative one obtains the expression:
(9)
We observe that, for a given value of the angle q, the tangent to the critical curve is parallel to the direction of the equilibrium orientation of the magnetization given by 
.
Fig.2 Critical curve, critical field vector and the equilibrium (e) line and critical stability line (s)
The critical curve has a cusp when the absolute value of the critical field vector has a extremum value. So, using (7), the zeros of the equation:
(10)
gives the q angles for which a cusp appear on the critical curve. Since the derivative of the critical field vector with respect to the 
angle in cusps is zero (see (7)), and the derivatives signs are opposite before and after the cusp, one may also say that the critical curve sense, given by the orientation of the derivative of the critical field vector, is also changing in the cusps. To see how these rules are working, one can check them on the simplest case, the particle with uniaxial anisotropy when taking into account only the first term in the series expansion of the anisotropy free energy density, that is, the case
. In this case, the anisotropy free energy is given by:
(11)
and 
. (12)
Using (10) in (6) and (7), one obtains the expression of the critical field vector:
(13)
and for the critical field vector derivative,
. (14)
The expression of the critical field vector, (11), is the parametric formulation of the well-known astroid curve and the derivative (12) is showing that on the curve there are four cusps, for
, as presented on Fig.3.
When 
is not zero, the anisotropy free energy density is given by:
(15)
the derivatives are:
(16)
the critical field vector is:
(17)
and the derivative of the critical field vector is:
. (18)
From (16) one can see that supplementary cusps are obtained in comparison with the ones observed in the 
case, only if the equation
(19)
has a solution, that is, if the square of the sinus from 
is in the interval [0,1], which is equivalent with:
(20)
If 
the critical curve has no supplementary cusp. It can be shown that in the case of 
values bigger than (1/4) supplementary pairs of cusps are appearing in 
. For values smaller than –(1/6) such pair of cusps are appearing in 
. A more complete image of these regions can be observed in the following table [22]:
Fig. 3 Critical curve for the uniaxial anisotropy, 
.
| Region | Examples | ||
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| “region 2” |  |
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| “region 3” |  |
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To calculate the TS using the critical curves calculated previously essentially implies to find the stable equilibrium state of the magnetic moment at a certain moment from these curves. In fact, the critical curve can provide an interval of values for the 
angle in which there is only one solution.
This is quite simple when 
and the critical curve has only four cusps, but it is not so obvious in the cases with eight cusps. In the first case inside the critical curve, in each point, there are possible two stable equilibrium states and one of instable equilibrium. In an equivalent way one can say that there are two energy minima separated by a minimum. In the exterior of the critical curve, in the same case, only one minimum is possible. The eight-cusp systems have inside the critical curve regions with more than two minima which give the difficulty in choosing the right one, followed by the magnetic moment in its dynamic.
Fig.4 The stable equilibrium orientation of the magnetization vector (
)
In Fig. 4 one presents the stable equilibrium states for a uniaxial anisotropy single-domain particle with
. The external field (which in the TS experiment is the DC field) is applied on the direction (a’b’c’d’e’’f’’) decreasing from (a’) to (f’’). One observes that the tangents from the critical curves corresponding to stable equilibrium orientations of the magnetization vector are starting for the fields (a’,b’,c’,d’) from the region (abcd) on the critical curve. When the applied field passes in the region (d’,e’’,f’’) the stable equilibrium states are associated to tangents started from the (ef) region.
Fig.5 (up) The critical curve for k2=1.0; (down) Detail of the region marked on the left figure.
In Fig. 5 one presents the critical curve for k2=1.0. It can be easily observed that the cusp characterized by q=0 (point b on the critical curve shown in Fig. 4) is in this case transformed in three cusps. The critical curve follows the path (B1, B2, B3, B4, B5) where B2, B3 and B4 are the cusp points. The cusp B3 is now the one for q=0. As shown by Thiaville[23] it can be found a rule that can be applied even in these cases to find the stable equilibrium orientation for the system.
A systematic analysis has shown that the critical curve approach can be applied for the 2D case quite efficiently. We have compared the results obtained with this method with the micromagnetic method, which is presented below, and a good agreement was found in each case. However, even in the case of uniaxial anisotropy there are cases in which the bi-dimensional image is not sufficient.
For example, uniaxial systems
, for different values of the
, parameter corresponds to systems with the easy axes forming a cone (an easy cone). In these cases one observe in certain conditions jumps from one cusp point to another. This is due to the fact that the magnetic moment can move freely around the easy cone surface. Especially due to these cases, the critical curve approach has a limited value in the calculation of the TS curves.
For cubic anisotropy, the 2D critical curve can not be used anymore because the applied field direction, the easy axis direction and the magnetization direction aren’t in the same plane. The cubic anisotropy case is essentially a 3D problem that needs a full 3D approach.
The complexity of the critical approach [24], [25] is an argument in the favor of the micromagnetic method that will be presented in the next section.
Micromagnetic algorithm
The micromagnetic model we have used is based on the Landau-Lifshitz-Gilbert equation [26]. The dynamics of the magnetization vector 
of each particle in the applied field 
is described by the Landau-Lifshitz-Gilbert (LLG) equation:
(21)
where 
is assumed to be invariable, a is the phenomenological damping constant assumed to be positive and g is the gyromagnetic factor. With the following notations:
(22)
and using spherical co-ordinates, equation (19) can be written as:
(23)
with
(24)
and
(25)
named equivalent anisotropy fields, where wa is the anisotropy free energy density.
Uniaxial anisotropy
For uniaxial anisotropy, if the easy axis orientation is given by the unit vector
, and the orientation of the magnetization vector is on the direction of the 
unit vector (in spherical coordinates), the anisotropy free energy can be expresses as:
(26)
and the equivalent anisotropy fields are given by:
. (27)
Cubic anisotropy
For cubic anisotropy, the anisotropy free energy density can be expresses as a function of the relative orientation of the [100], [010] and [001] axes with respect to the magnetization vector. If one uses the Euler angles to define the orientation of these axes
, one obtain,
(28)
where 
are the unit vectors of the Cartesian coordinates of the laboratory system. The free energy density in these conditions is given by:
(29)
Using (27) in (23) one obtains the equivalent anisotropy fields.
Fig. 6 The free energy surfaces for uniaxial (left) and cubic (right) anisotropies.
The TS process was simulated by a sequence of fields, identical to those applied in the experiment. At each step, the LLG equation is integrated until the motion of the magnetic moments can be neglected.
Simulated TS curves
In Figs.7-11 one show results obtained with the LLG algorithm for single-domain and for systems of particles. In Figs. 7-10 the TS curves are for uniaxial single-domain particles. Due to the possibility to use the critical curve approach, the uniaxial case is a very important tool for testing the micromagnetic model. However, our analyses have shown that in certain cases the sensitivity of the micromagnetic algorithm to factors like the AC field amplitude, is much higher that usual. For an assembly of non-interacting single domain particles the TS response is given by the integral of the transverse susceptibility of each particle over the easy axis distribution. Fig. 11 displays the results obtained for a randomly oriented system for different values of the second order anisotropy parameter, 
. The effect of on the TS curve of the ensemble can be observed especially on the peaks position. The other parameters, like the orientation distribution is influencing the shape of the TS curve.
Figs. 7 TS signal when qa=89°, ja=0° for a uniaxial single-domain particle.
Figs. 8 TS signal when qa=45°, ja=0°for a uniaxial single-domain particle.
Figs. 9 TS signal when qa=90°, ja=90° for a uniaxial single-domain particle.
Figs. 10 TS signal when qa=45°, ja=90° for a uniaxial single-domain particle.
Fig.11 The TS signal for a system of non-interacting uniaxial particles randomly oriented
TS for uniaxial vs. cubic anisotropy
We have systematically simulated the TS for a cubic single-particle with 
J/m3, 
(values characteristic for Ni) to compare the results with the ones given by the uniaxial anisotropy systems which can be calculated with the critical curve approach or with the micromagnetic one. The first case analyzed was when the easy axis [111] is oriented on the Oz direction, that is, 
. The DC field is applied in the yOz plane, making the angle qf with the axis [111] (see. Fig.12). The results are compared with the TS curve for an uniaxial with 
. The field is normalized to the anisotropy field, 
for the uniaxial case. A supplementary normalization factor of (7/9) is used for the single-domain with cubic anisotropy. This is motivated as follows: the free energy density expression when the orientation of the [111] easy axis is on the Oz direction (and the stable equilibrium orientation of the magnetization is on the 
semi-plane) is given by:
Fig.12 Cubic (a) and uniaxial (b) single domain particles. The surfaces are the representation of the free magneto-crystalline energy density for the two cases
(30)
with 
and 
. The coefficient of the 
is 
. With the simplified expression of the free energy one can use the critical curve approach, as described for the uniaxial case. The critical curve is presented in the Fig.13. The detail presented in the right figure show that the critical curve touches the axis in 0.777.
We should mention that the configuration presented by the critical curve, especially for very small angles with the easy axis [111], explains the problems for any micromagnetic calculus to determine precisely the critical field in this case. Another observation that can be made is that the minimum of the free energy in the easy axis direction has a different symmetry in the cubic case as in the uniaxial one as it can be obser
References
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1 The Model
The term of hysteresis [1-3] from the etymological point of view is related to the lag that can be observed between the input, in magnetism, the applied magnetic field and the output, the magnetic moment of the sample. The most common lag-effect is observed when one applies a field that depends on time as sinusoidal function and the magnetic moment follows the same time dependence of the field with a certain phase-lag. When the applied field is zero, the magnetic moment has a remanence, and we have to apply a (coercive) field in the opposite direction to obtain the demagnetized state. This hysteresis type is dependent on the frequency and in many cases when the frequency is very small, the hysteresis disappears as well. This is a rate-dependent hysteresis. However, for many ferromagnetic materials, one observes even for measurements made at very low frequencies that the system still has a hysteresis loop which does not change much if the field rate in the measurement is modified in quite a wide range of values. This is a rate-independent hysteresis and will be discussed on details. We have to mention that this is however an approximation and that in many cases one can still evidence deviations from the rate-independent hysteresis behaviour. Magnetic relaxation which can include phenomena in a wide range of time scale is responsible in fact for these deviations.
As a first conclusion, one can introduce with a certain approximation the notion of rate-independent hysteresis, which will exclude both the phase-lag and magnetic relaxation. We can link this type of hysteresis with the existence in the system of entities with metastable states. To make our analysis simpler, one can concentrate our discussion on a system of ferromagnetic single-domain particles (a typical magnetic recording medium known as a particulate medium). Each isolated particle has its individual hysteresis loop that depends on the particle’s shape, anisotropy, volume, etc. This is also a rate-independent type hysteresis if the particles are not too small in which case they become superparamagnetic.
So, in a classical particulate magnetic medium each particle has its rate independent hysteresis loop. It is characterized by a free energy function that has for a definite domain of the applied field (between the negative and positive switching fields) two minima separated by a maximum. When the applied field is outside this domain the free energy has only one minimum and as a consequence the moment has only one equilibrium position. When the applied field is between the negative and positive switching fields the particle has two equilibrium positions. However, for a given value of the applied field, the particle will stay in one minimum and will not change that position (the relaxation is considered at the timescale of the experiment to be negligible). Only the applied field can produce an irreversible change between the two equilibrium states and this change is obtained when this field has a value equal with one of the two characteristic switching fields. As long as the particle maintains an equilibrium position and no switches occur, the changes in the moment position are reversible.
The behaviour of an ensemble of particles will also display hysteresis that will be controlled not only by the hysteretic properties of each particle seen as isolated but also by the interactions between the magnetic moments of the particles. We shall see that there are qualitative differences between the hysteresis of one particle and the hysteresis of a system.
Using the same example of a particulate medium one can explain in a simple manner how it was developed the well-known Preisach [4] hysteresis model, which will be referred to as the Classical Preisach Model (CPM).
For simplicity, let's suppose that all the particles have their anisotropy axes parallel to each other and with the applied field direction. In this case, if the field value is between the two switching fields, two equilibrium positions are possible; they are characterized by the angle between the anisotropy axis and the magnetic moment vector of the particle. In the case previously described, the equilibrium positions are characterized by the angles 0° and 180°. The hysteresis loop, in the absence of interactions, is symmetric with respect to the origin of the (H, m) co-ordinate system and has a rectangular shape. The negative and positive switching fields have the same absolute values.
For the beginning we consider that the interaction field is positive and parallel to the applied field direction. The effect of this interaction field is a displacement of the hysteresis loop to the left (towards smaller values of the applied field); the positive switch is assisted by the positive interaction field. In a similar way one may say that a negative interaction field will assist the negative switch, so a displacement of the hysteresis loop to the right (to higher fields) should be observed. In conclusion, the effect of the interaction field is in any case an asymmetry of the hysteresis loop of the constituent particles. The positive and negative switching fields will be unequal in absolute value (see Fig. 1). Even if this is a rather strong simplification for the effect of the interaction field on the particle’s hysteresis loops, we shall see that this will allow us to understand the main problems related to the effect of interactions in particulate systems.
In the Preisach model the switching fields of the particles are used as co-ordinates in a plane, named the Preisach plane (see Fig. 2), that is, the magnetic moment of each particle is associated to a point with the switching fields as co-ordinates (in the Preisach plane Hα is the positive switching field and Hβ is the negative switching field). Associating all the magnetic moments from a sample to this plane one obtains a distribution called the Preisach distribution which may be seen, taking into account the big number of particles in the system, as a continuous function of the switching fields of the particles (see Fig 2).
In any real system the interaction fields are not identical for all the particles. The particle anisotropy, and in consequence, the switching fields absolute values, in the absence of interactions, are not identical for all the particles, as well. For any dispersion, some of the particles are sufficiently far from the neighbours, so their hysteresis loops are symmetrical. The magnetic moments of these particles will be associated to the second bisector of the Preisach plane (Hα = -Hβ). The particles with the same coercivity, Hcc, but with different interaction fields will be associated to a line parallel to the first bisector:
Hc = (Hα - Hβ) / 2 = const. = Hcc → Hβ = Hα - 2Hcc (1)
and those with the same interaction field, Hii, are associated to a line parallel to the second bisector:
Hi = -(Hα + Hβ) / 2 = const. = Hii → Hβ = - Hα - 2Hii (2)
Usually, the positive and negative interaction fields, when the sample is demagnetized, are equally probable, so the Preisach distribution will be symmetrical with respect to the second bisector which corresponds to the zero interaction field. This property is confirmed by the symmetry of the major hysteresis loop of the sample with respect to the origin of the (H, m) system. Most of the real systems Preisach distributions have a maximum value corresponding to the most probable value of the anisotropy field of the particles from the system which is noted with Hc0 in Fig. 3.
To be able to calculate the magnetic moment of the sample in the CPM one has to know its Preisach distribution. The determination of the Preisach distribution of a sample, also called identification, requires a number of experimental data. However, we wish to present first how one calculate a magnetization curve if we already know the distribution. This will help us to understand also how an identification process can be designed.
Fig.1 - A representative rectangular hysteresis loop associated to the point of co-ordinates (Hα , Hβ) in the Preisach plane. The coercive field of particle is Hc = (Hα - Hβ) / 2 and the characteristic interaction field is Hi = -Hs = - (Hα + Hβ) / 2 (Hs is the shift of the rectangular hysteresis loop in the H direction). We noted the magnetic moment of the particle with m0 and the saturation magnetic moment with m0s
Fig.2 - The Preisach plane. The saturation magnetic moment m0 of the particle whose hysteresis loop is represented below is associated to the point M. The T0 triangle is containing all the particles in the system. The Everett integral in the point M is calculated as the integral of the Preisach distribution over the T triangular region (hashed region). The P0Q0 line is the first bisector of the Preisach plane.
Fig.3 - The Preisach plane and a continuous Preisach distribution
Fig.4 - The Preisach plane. One applies the field H to the system. The limit triangle is partitioned into three zones. The "non-local memory" appears due to the existence of the region III, marked with "memory"
Let's suppose that we know the Preisach distribution of the system and let's observe what is happening in the Preisach plane when a (positive) field H is applied. As we may see on Fig. 4, two lines, L1 and L2, could be plotted in the Preisach plane; they are obtained from the condition that either the positive switching field, Hα, is equal to the applied field or the inferior switching field, Hβ, equals the same value. One mention that all the particle magnetic moments are associated to a triangular zone, which will be referred to as the limit triangle, bounded by three lines: the first bisector, L1s corresponding to a field sufficient to saturate the sample in the positive direction (Hα,max) and L2s for a field sufficient to saturate the sample in the negative direction (Hβ,min). Due to the symmetry of the major hysteresis loop, these positive and negative saturation fields are equal in absolute value. Outside this triangular limit there is no magnetic moment associated to the Preisach plane since i) the left-up zone of the Preisach plane is characterized by hysteresis loops with the superior switching field less than the inferior switching field, which is non-physical, and ii) the absolute value of all the switching fields in the system is less than the saturation field.
As one sees on Fig. 4, the L1 and L2 lines are generating from their intersection with the limit triangle three zones, marked with Roman numbers. The particles associated to the zone “I” have positive switching fields less than the applied field, so it will be all on the positive branch of their hysteresis loops, we shall say that they are on the "+" position. The particles associated to the zone “II” have their negative switching field greater than the applied field, so they will be on the negative branch, named "-" position. For the particles from the zone “III” the applied field is between the inferior and superior switching fields so their state could not be changed by the applied field. The existence of this zone makes the memory effect possible in ferromagnetic particulate medium. This memory is qualitatively different from the memory exhibited by one isolated particle which is also characterized by a hysteresis loop. The single-domain particle has a local memory while the system has a non-local memory. That is, if the actual value of the magnetic moment projection in one single particle case is completely determined by the values of the field and magnetic moment at an infinitely near previous moment, the actual value of the magnetic moment projection in the particle system case is also determined by the history of applied fields. However, when the field value is changing it is not necessary to see what is happening with all the three zones discussed previously. It is easy to observe that if the field value is increasing only the particles which are entering in the “I” zone change their state to "+" state. The others remain unchanged. In the same way, for the decreasing fields only the particles entering in the “II” zone are changing their state. So, a simple rule may be formulated (see Fig. 5):
- for the increasing fields the L1 line is going to the right (higher fields) and all the moments at the left of this line are switched to the “+” position and
- for the decreasing fields, the line L2 is going down (lower fields) and all the moments over that line are switched to the “-” position.
After a sequence of fields a staircase line, L, separates the “+” and “-” zones in the limit triangle (see Fig. 6).
Fig.5 - Left: increasing field applied to the sample; Right: decreasing field applied to the sample.
Fig.6 - The Preisach plane after a field sequence
The memory of the system is "concentrated", in fact, in this staircase line. One observes that the co-ordinates of the segments which are forming the L staircase line are given by the local extreme values of the applied fields (local minima and maxima). Finally, the total magnetic moment of the system is the difference between the magnetic moment on the “+” and “-” zones, that is:
(3)
One also observes that the calculus of these two integrals over the S+ and S- regions can be decomposed in a sum of integrals over triangular zones such us the one represented in Fig. 2. These integrals are named Everett integrals. The Everett integral for the triangle T(Hα1 , Hβ1) is given by:
(4)
which may be written in simplified manner for any point in the Preisach plane as:
(5)
If one performs a normalization at the saturation value of the magnetic moment which is
ms = E(Hm , -Hm) (6)
where Hm = Hα,max = -Hβ,min is the maximum absolute value of the critical field of the particles in the system and one uses the notation
(7)
then the normalized Everett integral will be:
(8)
[1] G. Bertotti, Hysteresis in magnetism: for physicists, materials scientists, and engineers. San Diego: Academic Press, 1998.
[2] I. D. Mayergoyz, Mathematical models of hysteresis and their applications, 1st ed. Amsterdam; Boston: Elsevier, 2003.
[3] E. Della Torre, Magnetic hysteresis, New York: IEEE Press, 1999.
[4] F. Preisach, "Über die magnetische Nachwirkung," Z.Phys., vol. 94, pp. 277-302, 1935