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°.