We propose an exact expression to describe the hysteresis loops of an ensemble of Stoner–Wohlfarth particles undergoing an alternating quasi-static magnetic field. A statistical approach, which treats the quantities characterizing each particle as random variables, is adopted to get the orientation distribution of the local polarizations with respect to the applied field direction and the constitutive equation of the whole particle assembly. The hysteresis loop area gives the energy loss figure, but we have also obtained a straightforward integral expression for this quantity. The analytical relationships for the symmetric loops and the losses are successfully tested against numerical results, and the mathematical method adopted also displayed the ability to reproduce the “elemental loop” associated with any given particle of the system. While having a fundamental character, the proposed approach bears applicative interest, representing a versatile tool as the core of codes that simulate the behavior of devices employing magnetic components.

Magnetic materials exhibit a broad variety of compositions and microstructures and a wide spectrum of responses to a varying magnetizing field H.1–3 

Although such behavior is the result of three-dimensional (3D) magnetization process, a two-dimensional theoretical approach is often regarded as appropriate, in addition to being perfectly adequate for a large class of materials, such as steel sheets, amorphous and nanocrystalline alloys, and thin films. It is also possible, in many cases, to consider a given material as an ensemble of mesoscopic regions (“particles”), either interacting or independent, each characterized by the local polarization Js and a magnetic anisotropy {uniaxial [see Fig. 1(a)], biaxial, cubic, etc.}, and even single-domain, if sufficiently small.

FIG. 1.

Panel (a) shows, for a particle of the system, the relative orientation of the local anisotropy K and the local polarization Js with respect to the reference direction determined by the applied field H. In the (H, H) plane, the corresponding astroid representation, with vertices ±HK, is displayed in (b), with uH being the associated Gibbs free energy. (c) Branches of a general hysteresis loop, drawn starting from the demagnetized state (DEM). (d) Correspondence between the Q quadrants and the values of the γ angle.

FIG. 1.

Panel (a) shows, for a particle of the system, the relative orientation of the local anisotropy K and the local polarization Js with respect to the reference direction determined by the applied field H. In the (H, H) plane, the corresponding astroid representation, with vertices ±HK, is displayed in (b), with uH being the associated Gibbs free energy. (c) Branches of a general hysteresis loop, drawn starting from the demagnetized state (DEM). (d) Correspondence between the Q quadrants and the values of the γ angle.

Close modal

The Stoner–Wohlfarth (SW) theory,4–8 in its native formulation (which considers single-domain particles as independent, endowed with uniaxial anisotropy, and at T = 0 K), fits the simplifications mentioned above and is, thus, widely exploited in modeling. The main fields where the SW approach (often with some modifications) plays an important role are (1) magnetic nanoparticles,9–12 notably for magnetic hyperthermia;13,14 (2) ferromagnetic thin films15–17 and magnetic random access memories;18–20 (3) amorphous and nanocrystalline materials;21–25 (4) soft magnetic composites;26,27 (5) permanent magnets;28 and (6) study of the effect of spin-polarized currents.29,30

In the SW model, the response to an applied field H of a particle characterized by the local uniaxial anisotropy K is described by a vector hysteresis transducer providing the constitutive law for the reversible coherent rotations and the irreversible jumps of the local polarization Js. In other words, one gets the equilibrium orientation γ of each Js as a function of the anisotropy constant K, its orientation φ, and H [see Fig. 1(a)]. With J being the polarization of the whole particle assembly, the J(H) constitutive equation of the system comes out after integration over the probability density functions (pdfs) of K and φ. The design of magnetic components needs to implement J(H) in numeric modeling codes: a step with high computational cost, scaling up with the device size. Indeed, an explicit γ = γH(K, φ) solution for the Js orientation is not available so far, being the equilibrium angle obtained either following numeric procedures25,31 or using analytic approximations.

In the first case, the problem is commonly faced by adopting the astroid representation for the SW transducer [see Fig. 1(b)] and exploiting the so-called “tangent method,”7 as well as searching for the minimum of the Gibbs free energy of each particle utilizing, e.g., the Newton–Raphson method. These are time-consuming approaches, not efficient when implemented in codes used to design magnetic devices.

Researchers following the second way tried, for example, to recover the SW loop branches with ad hoc analytic functions,32 or proposed an H = H(J) “inverse” constitutive law,12 or limited the analytic solution to the low-field regime.33 Alternatively, an SW-like hysteron with different kinds of approximations for the anisotropy energy term,34,35 or with an equivalent field playing the role of the anisotropy,36 has been also adopted.

This work is not focused on the search of the γ = γH(K, φ) single particle solution [see Fig. 1(a)], afterward integrated over the whole system, but it deals at the start from the entire particle assembly by handling each quantity as a random variable (rv) with associated pdf. Statistical methods37,38 are then utilized to retrieve pH(γ): the equilibrium angle pdf, ending with an exact expression for the hysteresis loops (and consequently for the energy losses) of the whole system. The single particle behavior comes out as a particular case. Apart from its importance from the basic viewpoint, this result turns out to be useful for applications (chiefly in the magnetic loss prediction39), shortening the cumbersome and slow procedure needed for the design of devices.

In this paper, after a review of the SW model features essential for our purposes (Sec. II A), and some hints about the numerical approach and the adopted statistics (Sec. II B), the core of the work is discussed in Sec. III. Here, the statistical approach is outlined, and the pdfs controlling the equilibrium orientation of the local polarization vs a quasi-static H are worked out, separately accounting for the irreversible (Sec. III A) and reversible (Sec. III B) phenomena. In both these subsections, the pdf evolution is described starting from the demagnetized state and following the system magnetization along the branches of the hysteresis loop driven by an alternating field [Fig. 1(c)], separately for the four Q quadrants shown in Fig. 1(d). Eventually, the obtained constitutive equation (and thus the hysteresis loop and loss) is successfully tested against the numerical procedure, both for the entire assembly (Sec. III B 5) and for the case of a single particle (Sec. III B 6). An approximate analytic expression for the energy loss, not involving the knowledge of the hysteresis loop, is worked out and effectively checked (Sec. IV), and finally, in Sec. V, some possible advancements of the proposed approach are listed.

The SW approach considers the physical system as an ensemble of non-interacting single-domain “particles” at T = 0 K, each characterized by a local anisotropy axis with modulus K (and anisotropy field HK = 2 K/Js, where Js = |Js|), forming an angle φ with a reference direction defined by the applied field H [Fig. 1(a)]. After introducing the Gibbs free energy density of the particle, together with its first and second derivative (where H = |H|),
uH(K,φ;γ)=Ksin2(φγ)HJscosγ,
(1a)
uHuHγ=Ksin(2φ2γ)+HJssinγ,
(1b)
uHuHγ=2Kcos(2φ2γ)+HJscosγ,
(1c)
the γ equilibrium orientations of the local polarization Js (corresponding to the minima of uH) are derived.4–7 This can be done starting from the conditions uH=uH=0 (identifying the uH horizontal inflection point generated by the merging of one minimum and one maximum), which allows one to draw, in the (HH cos φ, HH sin φ) Cartesian reference frame, a closed curve (astroid) whose contour (with vertices ±HK) represents the border separating the (H, H) plane region corresponding to two minima of uH (inside) from the one where one minimum only appears [Fig. 1(b)]. On passing to the polar coordinates (φ, H), after defining the quantity
A(φ)=(sin2/3φ+cos2/3φ)3/2,
(2)
the field threshold corresponding to the astroid is written as
Hc,K(φ)=HK/A(φ).
(3)
If H is instead given, in the (φ, K) plane, where each point represents a particle of the system, one finds the corresponding always positive anisotropy threshold,
Kc,H(φ)=12|H|JsA(φ),
(4)
with one (two) minimum (minima) when K is lower (higher) than Kc,H.

The introduction reports the strategies followed in the literature to find the γ equilibrium angle for each particle. In the following, the statistics of the system is described by the ψ(K, φ) joint probability density function, with K > 0 and φ limited to the [0;π/2] range for symmetry reasons.

The numerical approach adopted in this work to test the analytic form of J(H), worked out in the following, finds the local Js equilibrium orientations via an iterative procedure searching for the uH absolute minimum and follows its time evolution by tracking the memory of the irreversible rotations (IRs), for each particle. On passing to an assembly of SW particles, one gets the behavior of the whole system after integration over the ψ(K, φ) pdf. When we compare the analytic expressions for the hysteresis loop and loss to the numerical simulations, φ and K are assumed to be statistically independent variables. Consequently, we can write ψ(K, φ) = f(K)g(φ), and in particular, we have chosen for the tests, with ⟨K⟩ the average anisotropy value,
f(K)=2K2Kexp2KK,
(5a)
g(φ)=2/π.
(5b)

In this work, we have abandoned the problem of finding the exact analytical solution for the local Js equilibrium orientation γ of a single SW particle in favor of a statistical approach that considers the whole assembly of SW particles from the beginning. Accordingly, the quantities K, φ, and γ (−π to +π) are treated as random variables, along with their ψ(K, φ) and pH(b,Q)(γ) pdfs (the first normalized to 1 for each φ angle). In the latter, “b” indicates the loop branch considered [see Fig. 1(c)], “Q” is the quadrant to which γ pertains [see Fig. 1(d)], and H plays the role of a parameter. All along the paper, the evolution of the γ values, which are bound to increase from −π to π [see Fig. 1(d)], dictates the sequence adopted to discuss the Q quadrants (IIIIVIII).

As a response to an alternating field, with peak values ±Hp, the corresponding hysteresis loop is obtained as follows: The magnetization process is assumed to start from the demagnetized state (DEM), where H = J = 0 [Fig. 1(c)]—a situation that, for symmetry reasons, can be obtained by formulating the pdf for the Js equilibrium orientation still in terms of K and φ instead of γ, the local polarization lying along the easy axis direction in each particle,
pDEM(Q=I)(K,φ)=pDEM(Q=III)(K,φ)=12ψ(K,φ),
(6a)
pDEM(Q=II)(K,φ)=pDEM(Q=IV)(K,φ)=0.
(6b)
[An example, for a given φ = φ1 and a postulated ψ(K, φ1) vs K behavior, is reported in Fig. 3(a).] The J vs H evolution is then followed and investigated in sequence [see Fig. 1(c)] along the First Magnetization Curve (FMC), the RECoil curve (REC), and the Negative Descending Branch (NDB) of the hysteresis loop, both in Secs. III A and III B. Note that the ascending branch (↑) of the major loop of vertex (Hp, Jp) can be obtained from the descending one (↓), build-up of the REC and NDB curves, being J(H) = −J(−H) for symmetry reasons.

Along each loop branch, the applied field triggers the irreversible rotations (IRs) and drives any local Js of the whole particle ensemble to its equilibrium angle (reversible rotations) at the same time. Despite the ensuing complex interplay between reversible and irreversible processes, the technique developed here treats them separately, and the γ pdf is consequently worked out in two steps: before finding pH,IRR(b,Q)(K,φ), i.e., the pdf determined by the IRs only, which bring the local Js to their absolute minimum (Sec. III A), and subsequently accounting for the coherent reversible rotations (by means of the Js relaxation to the equilibrium orientation), ending with pH(b,Q)(γ) (Sec. III B). This approach is sketched in Fig. 2.

FIG. 2.

Evolution of the probability density function, from the K and φ statistics to the γ statistics. The two steps describing the irreversible and reversible processes in sequence are put in evidence. The superscripts (b, Q) of the pdfs are understood for the sake of simplicity. Note the dependence on (K, φ) of pDEM and pH,IRR, and on γ of the final pH pdf.

FIG. 2.

Evolution of the probability density function, from the K and φ statistics to the γ statistics. The two steps describing the irreversible and reversible processes in sequence are put in evidence. The superscripts (b, Q) of the pdfs are understood for the sake of simplicity. Note the dependence on (K, φ) of pDEM and pH,IRR, and on γ of the final pH pdf.

Close modal

Eventually, after the integration of pH(b,Q)(γ)cos γ over the [−π; +π] γ domain, one gets the constitutive law of the system [Sec. III B 5, Eq. (28)].

At this stage, we assume H to drive the irreversible processes only, again leaving the local Js to point along one of the two sides of the φ easy axis directions of the particle to which it pertains. For this reason, the magnetization distribution among the quadrants, rearranged by the occurrence of the IRs, is again described by a pdf stated in terms of K and φ instead of γ: pH,IRR(b,Q)(K,φ), for the three loop branches of Fig. 1(c), as listed in Sections III A 1 -III A 4, and illustrated in Fig. 3 for a given φ = φ1.

FIG. 3.

For a given φ = φ1, and with a postulated ψ(K, φ1), the evolution of the pdfs (see the left part of Fig. 2) is described for the four Q quadrants [see Fig. 1(d)]. As H drives the system along the branch sequence of Fig. 1(c), the pDEM(Q) [Eq. (6)] describing the demagnetized state (DEM) transforms into the pH,IRR(FMC,Q), pH,IRR(REC,Q), and pH,IRR(NDB,Q) as a response to the irreversible rotations (IRs) displayed by the gray arrows.

FIG. 3.

For a given φ = φ1, and with a postulated ψ(K, φ1), the evolution of the pdfs (see the left part of Fig. 2) is described for the four Q quadrants [see Fig. 1(d)]. As H drives the system along the branch sequence of Fig. 1(c), the pDEM(Q) [Eq. (6)] describing the demagnetized state (DEM) transforms into the pH,IRR(FMC,Q), pH,IRR(REC,Q), and pH,IRR(NDB,Q) as a response to the irreversible rotations (IRs) displayed by the gray arrows.

Close modal

1. First magnetization curve (FMC)

For H increasing from DEM [Fig. 1(c)], Eqs. (1b) and (1c) state that the Js in quadrant III smoothly (i.e., without IRs) crosses the γ = −π/2 threshold and move to quadrant IV when φ > π/4 and H > HIII→IV = (K/Js)sin 2φ [a value always lower or equal to Hc,K(φ): the threshold for IRs]. As stated above, we are not interested at this stage in the equilibrium position of Js, so we account for these Js reversible rotations by assigning the same pdfs to these quadrants: pH,IRR(FMC,IV)=pH,IRR(FMC,III). Note that this “doubling” of the pH,IRR(FMC,Q) does not affect the normalization because each Js does not pertain to Q = III and Q = IV for the same field, the Step 2 (Sec. III B) taking charge of finding the actual Js orientation vs H. As H increases from zero [where Kc,0 = 0, see Eq. (4)] to the peak value Hp, the Js in quadrant III or IV switch to quadrant I if K < Kc,H(φ), and the corresponding pH,IRR(FMC,Q)(K,φ) is reported in Fig. 3(b). Note that pH,IRR(FMC,II) = 0 because the system geometry [Fig. 1(a)] does not allow irreversible rotations to the second quadrant, a fact in agreement with the detailed discussion reported in the  Appendix, which shows that no energy minima of uH [Eq. (1a)] can appear in Q = II.

2. Recoil curve (REC)

When the field is reduced from Hp to zero, the polarization decreases from its peak value Jp = J(Hp) to the remanence Jr. Along this branch, no IRs occur, and thus, pH,IRR(REC,Q)(K,φ) is frozen to the value reached along FMC at H = Hp: pHp,IRR(FMC,Q)(K,φ) [see Fig. 3(c)]. Here, again, we have pH,IRR(REC,IV) = pH,IRR(REC,III) because the “doubling” of the pdfs, pointed out in FMC persists, but, as above, it does not affect the normalization.

3. Negative descending branch (NDB)

Eventually, the system is brought from the remanence to the negative tip of the hysteresis loop, lowering H from 0 to −Hp, with the magnetization evolving as described by pH,IRR(NDB,Q)(K,φ) [Fig. 3(d)]. For what concern the IRs and the smooth Js transitions discussed in Sec. III A 1, a “symmetrical” reasoning applies here, with the quadrants I, II, and IV correspondingly playing the role of Q = III, IV, and I in FMC, and the γ = π/2 crossing occurring at the threshold HIII = −HIII→IV [≤Hc,K(φ)], when φ > π/4.

4. Thresholds of the pH,IRR(b,Q)(K,φ) functions

The quantities Kc,H(φ1) and Kc,Hp(φ1) of Fig. 3, signaling a variation of the pH,IRR(b,Q) values for a given φ = φ1, become the Kc,H(φ) and Kc,Hp(φ) threshold functions when considering the entire 0 ≤ φπ/2 range. As H, starting from the demagnetized state (DEM), oscillates between the ±Hp peak values of the hysteresis loop, the Kc,H(φ) threshold follows it, swinging between Kc,0 = 0 and Kc,Hp(φ), with the pH,IRR(b,Q)(K,φ) function that accordingly changes. An example for Q = I across NDB is depicted in the (φ, K) plane of Fig. 4.

FIG. 4.

An example of the “irreversible” pH,IRR(NDB,I)(K,φ1) pdf (on the right) selected by the Kc,H(φ) and Kc,Hp(φ) thresholds (on the left). See Fig. 3 for other cases.

FIG. 4.

An example of the “irreversible” pH,IRR(NDB,I)(K,φ1) pdf (on the right) selected by the Kc,H(φ) and Kc,Hp(φ) thresholds (on the left). See Fig. 3 for other cases.

Close modal

Given H and γ, and considering the scheme of Fig. 2, for the three branches of the hysteresis loop indicated in Fig. 1(c), the pH(b,Q)(γ) pdf, which accounts for irreversible and reversible processes, is obtained through standard statistical methods.37,38

One starts by computing the cumulative distribution function PH(b,Q)(γ), i.e., the probability for a general γ′ to satisfy the request γlowγγ, where the lower limit is −π, −π/2, 0, and π/2 when γ pertains to Q = III, IV, I, and II, respectively [Fig. 1(d)]. Having available the pH,IRR(b,Q)(K,φ) pdfs, this approach can be stated in terms of K and φ, i.e., identifying in the (K, φ) plane (limited to the 0 ≤ φπ/2 range for symmetry reasons), a DH(γ) domain that satisfies the demand above (γlowγγ)—a result achievable if a relationship connecting the rv γ′ to the rv’s K and φ: γ ′ = γH(K, φ) exists. In the case of an SW particle, for which it is not possible to get such an explicit γH function, the connection between the rv’s can be stated in an implicit way from the request that uH(K, φγ′) [Eq. (1a)] shows a minimum, i.e., from a couple of conditions,
uH=0,
(7a)
uH>0.
(7b)
The first one, solved with respect to K [see Eq. (1b)], provides a relationship valid for H and φ values making it positive,
KH*(φ;γ)=HJssinγsin(2φ2γ).
(8)
The second one, with the constraint K = KH*, becomes [see Eq. (1c)]
uH*(φ;γ)=HJssinγ2tan(2φ2γ)+1tanγ>0.
(9)
By setting Eq. (9) to zero, one gets the φ values where it changes sign in the 0 ≤ φπ/2 interval,
φ0(Q)(γ)=γ12arctan(2tanγ)+n(Q)π2,
(10)
where n(Q) = 2, 1, 0, and −1 when Q = III, IV, I, and II, respectively. Furthermore, in the 0 ≤ φπ/2 range, both the KH* and uH* functions have a vertical asymptote at
φ(Q)(γ)=γ+n(Q)π2.
(11)
To delimit the DH(γ) domain (magenta regions in Fig. 5), one has to find before, for each γ′, the Δφ*(γ) interval, i.e., the φ values range making uH(K, φγ′) minimum, which is defined by the overlap region where both KH*(φ;γ) and uH*(φ;γ) are positive. This process is described in the  Appendix, for the three loop branches [Fig. 1(c)] and the four quadrants [Fig. 1(d)].
FIG. 5.

The magenta regions represent the DH(γ) integration domains of the PH(b,Q) cumulative distribution functions [Eq. (13a)], for the three “b” branches (FMC, REC, and NDB) reported in Fig. 1(c), and the four Q quadrants. Note that when Q = II in FMC and REC, and Q = IV in NDB, no DH domain appears, according to the fact that, in these conditions, the local Js cannot find any stable equilibrium orientation. In Q = IV of FMC and REC, and in Q = II of NDB, the top-left DH border is correspondingly given by the function KH*(φ;π/2) = |H|Js/sin(2φ) [Eq. (8) with γ′ = ∓π/2], with asymptotes φ = π/2 in both cases. Observe the thin white curves KH*(φ;γ), whose evolution from γlow to γ (thick black curves) devises the DH(γ) regions.

FIG. 5.

The magenta regions represent the DH(γ) integration domains of the PH(b,Q) cumulative distribution functions [Eq. (13a)], for the three “b” branches (FMC, REC, and NDB) reported in Fig. 1(c), and the four Q quadrants. Note that when Q = II in FMC and REC, and Q = IV in NDB, no DH domain appears, according to the fact that, in these conditions, the local Js cannot find any stable equilibrium orientation. In Q = IV of FMC and REC, and in Q = II of NDB, the top-left DH border is correspondingly given by the function KH*(φ;π/2) = |H|Js/sin(2φ) [Eq. (8) with γ′ = ∓π/2], with asymptotes φ = π/2 in both cases. Observe the thin white curves KH*(φ;γ), whose evolution from γlow to γ (thick black curves) devises the DH(γ) regions.

Close modal
Once this is done, the DH(γ) domain emerges as the (φ, K) plane region swept by all the KH*(φ;γ) functions [each of them limited to its Δφ*(γ) range] for increasing γ′ between γlow to γ (white curves in Fig. 5). This procedure is summarized by the D̂-operator,
D̂H(γ)=KH*φ;γγlowγ,
(12)
where γlow is the lower limit of the quadrant considered, as detailed in the next paragraphs.
The knowledge of DH(γ) allows one to write the cumulative distribution function, given H and γ, and its derivative—the probability density function,
PH(b,Q)(γ)=DH(γ)pH,IRR(b,Q)(K,φ)dKdφ,
(13a)
pH(b,Q)(γ)=dPH(b,Q)(γ)/dγ,
(13b)
with the integrand reported in Fig. 3 (red curves). When performing the integration vs K in Eq. (13a), one must remember that the thresholds Kc,H(φ) and Kc,Hp(φ) [see Eq. (4)] indicate, for each φ, the K values where the pH,IRR(b,Q)(K,φ) pdf changes, as discussed in Sec. III A 4. As an example, Fig. 4 displays the case Q = I across NDB.

1. First magnetization curve (FMC)

According to Sec. III A 4 and Fig. 3(b), along this branch, the Kc,Hp(φ) curves do not play any role (and thus are not depicted in Fig. 5), whereas the Kc,H(φ) ones not only signal the K thresholds where the pH,IRR(FMC,Q) pdfs [retrieved from Fig. 3(b)] change but also represent a portion of the DH integration domains (magenta regions in Fig. 5, for Q = III and Q = IV) border.

a) πγπ/2(Q=III)̄

The DH(γ) domain is obtained employing the D̂-operator [Eq. (12)] with γlow = −π. Being φ(III)(γ) = γ + π [Eq. (11) with n(Q) = 2], Eq. (13a) becomes
PH(FMC,III)(γ)=0φ0(III)(γ)Kc,H(φ)pH,IRR(FMC,III)(K,φ)dKdφ+φ0(III)(γ)γ+πKH*(φ;γ)pH,IRR(FMC,III)(K,φ)dKdφ.
(14)
Observing [Eq. (8)] that KH*(γ+π;γ) (where any pdf vanishes), and considering Eq. (A2), Eq. (13b) becomes
pH(FMC,III)(γ)=γ+πφ0(III)(γ)IH(FMC,III)(φ;γ)dφ,
(15)
where φ0(III) is supplied by Eq. (10) with γ′ = γ and n(Q) = 2, and the integrand is defined as follows:
IH(FMC,III)(φ;γ)pH,IRR(FMC,III)KH*(φ;γ),φKH*(φ;γ)γ,
(16)
with
KH*(φ;γ)γ=HJscosγsin(2φ2γ)+2sinγcos(2φ2γ)sin2(2φ2γ)
(17)

b) π/2γ0(Q=IV)

The DH(γ) domain is obtained by utilizing the D̂-operator [Eq. (12)] with γlow = −π/2. The top-left border of DH(γ) is given [Eq. (8)] by the curve KH*(φ;π/2)=HJs/sin2φ, with asymptote φ = π/2—a value not accounted for by Eq. (11). The cumulative and density functions [the second one remembering again Eq. (A2)] become
PH(FMC,IV)(γ)=π/4φ0(IV)(γ)Kc,H(φ)KH*(φ;π/2)pH,IRR(FMC,IV)(K,φ)dKdφ+φ0(IV)(γ)π/2KH*(φ;γ)KH*(φ;π/2)pH,IRR(FMC,IV)(K,φ)dKdφ,
(18)
pH(FMC,IV)(γ)=π/2φ0(IV)(γ)IH(FMC,IV)(φ;γ)dφ,
(19)
with φ0(III) given by Eq. (10) with γ′ = γ and n(Q) = 1, and the integrand defined similarly as in Q = III [Eqs. (16) and (17)]

c) 0γπ/2(Q=I)

The DH(γ) domain is obtained using the D̂-operator [Eq. (12)] with γ′ increasing from γlow = 0. The cumulative function [being φ(γ) = γ from Eq. (11) with n(Q) = 0], and the pdf associated [observing that KH*(γ;γ), from Eq. (8)] become
PH(FMC,I)(γ)=0γ0pH,IRR(FMC,I)(K,φ)dKdφ+γπ/20KH*(φ;γ)pH,IRR(FMC,I)(K,φ)dKdφ,
(20)
pH(FMC,I)(γ)=γπ/2IH(FMC,I)(φ;γ)dφ,
(21)
again with the integrand defined similarly as in Q = III [Eqs. (16) and (17)].

d) π/2γπ(Q=II)

As discussed in the  Appendix, the absence of a stable equilibrium orientation for the local Js is reflected in the lack of the integration domain DH, and agrees with the fact that, in this region, we have pH,IRR(FMC,II) = 0 (Sec. III A and Fig. 2). Therefore,
pH(FMC,II)(γ)=0.
(22)

2. Recoil curve (REC)

The very same path driving to the FMC pdfs can be adopted here, the Kc,H threshold decreasing with H, from Kc,Hp to zero. However, as shown in Fig. 3(c), due to the absence of IRs along this branch, the “irreversible” pdfs are frozen to the last value reached across FMC, i.e., when H = Hp,
pH,IRR(REC,Q)=pHp,IRR,(FMC,Q)
(23)
for all Q quadrants. Accordingly, the integrands [Eq. (16)] turn into
IH(REC,Q)(φ;γ)=pHp,IRR(FMC,Q)KH*(φ;γ),φKH*(φ;γ)γ,
(24)
with the derivative given by Eq. (17).

3. Negative descending branch (NDB)

Along this branch, a very similar and, to some extent, specular scenario (specular to the FMC and REC cases) appears, with the DH shapes in Q = III, IV, I, and II recovering the ones found along FMC (or REC) in Q = I, II, III, and IV, respectively, but with different φ integration limits (see Fig. 5). Now, the top-left border of DH in Q = II is given [Eq. (8)] by the curve KH*(φ;π/2)=HJs/sin2φ, with asymptote φ = π/2—a value not accounted for by Eq. (11). Thus, a procedure like the one described above supplies the γ pdfs for the four quadrants,
pH(NDB,III)(γ)=γ+ππ/2IH(NDB,III)(φ;γ)dφ,
(25a)
pH(NDB,IV)(γ)=0,
(25b)
pH(NDB,I)(γ)=γφ0(I)(γ)IH(NDB,I)(φ;γ)dφ,
(25c)
pH(NDB,II)(γ)=π/2φ0(II)(γ)IH(NDB,II)(φ;γ)dφ,
(25d)
with the IH(NDB,Q) defined similarly to IH(FMC,Q), and the pH,IRR(NDB,Q)(K,φ) values changing over both the thresholds Kc,H(φ) and Kc,Hp(φ), as shown in Fig. 3(d).

4. Compact expression for the pH(b,Q)(γ) pdfs

It is possible to write the pdfs for the local Js making an angle γ with H in a very compact integral form, depending on the “b” branch (FMC, REC, or NDB) and the Q (=III, IV, I, and II) quadrant occupied by the equilibrium angle γ,
pH(b,Q)(γ)=φLOW(b,Q)φHIGH(b,Q)IH(b,Q)(φ;γ)dφ,
(26)
with
IH(b,Q)(φ;γ)pH,IRR(b,Q)KH*(φ;γ),φKH*(φ;γ)γ.
(27)
The pH,IRR(b,Q) are retrieved from Fig. 3 [remember Eq. (23) as well], the derivative from Eq. (17), and the integration limits summarized in the scheme of Fig. 6. In it, to make Eq. (26) formally valid for any “b” and “Q,” the integration limits of Q = II in FMC and REC and of Q = IV in NDB are “artificially” put equal to zero, thus obtaining pH(FMC,II) = pH(REC,II) = pH(NDB,IV) = 0.
FIG. 6.

Limits of the integral appearing in Eq. (26), for the FMC, REC, and NDB branches of the hysteresis loop [see Fig. 1(c)], and the Q quadrants. The threshold φ0(Q)(γ) is given by Eq. (10), with γ′ = γ.

FIG. 6.

Limits of the integral appearing in Eq. (26), for the FMC, REC, and NDB branches of the hysteresis loop [see Fig. 1(c)], and the Q quadrants. The threshold φ0(Q)(γ) is given by Eq. (10), with γ′ = γ.

Close modal

5. Constitutive law and hysteresis loop

Considering a particle assembly magnetized by an alternating quasi-static field, from the constitutive equation, calculated over the whole γ domain,
J(b)(H)=JsππpH(b,Q)(γ)cosγdγ,
(28)
with pH(b,Q) given by Eq. (26), the hysteresis loop is drawn, and the corresponding energy loss W is given by its area. Moreover, as described below (Sec. IV), it is also possible to work out a semi-empirical but very accurate integral expression for W not entailing the knowledge of the hysteresis loop.

Figure 7 shows, for the whole −πγπ range, an example of pH(b,Q)(γ) vs H evolution, calculated with Eq. (26), and the corresponding hysteresis loop [Eq. (28)] with Js = 1.61 T. We have assumed ψ(K, φ) =f(K)g(φ), with the marginal pdfs supplied by Eqs. (5a) and (5b) and K = 3000 J/m3.

FIG. 7.

When a quasi-static alternating field is applied, the evolution of pH(γ) [Eqs. (26) and (27)] is displayed for the FMC, REC, and NDB branches [Eq. (28)] of the hysteresis loop drawn on the top, where the fields in correspondence of which the pH(γ) are calculated are marked with open dots of the same color. The dashed black lines represent the demagnetized state (H = J = 0) realized when Eq. (6) is satisfied. On the other extreme, pH(γ) becomes the Dirac delta function δD(γ) or δD(γ + π) when H → ±, respectively. Observe the symmetry of the pH(γ) corresponding to the ±Hp peak fields.

FIG. 7.

When a quasi-static alternating field is applied, the evolution of pH(γ) [Eqs. (26) and (27)] is displayed for the FMC, REC, and NDB branches [Eq. (28)] of the hysteresis loop drawn on the top, where the fields in correspondence of which the pH(γ) are calculated are marked with open dots of the same color. The dashed black lines represent the demagnetized state (H = J = 0) realized when Eq. (6) is satisfied. On the other extreme, pH(γ) becomes the Dirac delta function δD(γ) or δD(γ + π) when H → ±, respectively. Observe the symmetry of the pH(γ) corresponding to the ±Hp peak fields.

Close modal

The effectiveness of the statistical method is apparent in Fig. 8, where the quasi-static hysteresis loops, computed at different peak polarization Jp using Eq. (28) and the corresponding loss figures (inset), perfectly recover the ones obtained via the numerical procedure presented in Sec. II B [again it is assumed ψ(K, φ) = f(K)g(φ), with the marginal pdfs furnished by Eqs. (5a) and (5b)].

FIG. 8.

Alternating quasi-static magnetizing field. The excellent agreement between the First Magnetization Curve (open dots and dashed lines) and the hysteresis loops numerically found, and the ones obtained via the statistical method, are displayed (HK = 2K/Js; Js = |Js|). The inset shows the behavior of losses, with the red curve representing both the values given by the loop areas and the empirical ones of Eq. (33), this last plotted in correspondence with the Jp(Hp) peak polarizations of Eq. (34). One cannot appreciate significant differences between the two loss figures.

FIG. 8.

Alternating quasi-static magnetizing field. The excellent agreement between the First Magnetization Curve (open dots and dashed lines) and the hysteresis loops numerically found, and the ones obtained via the statistical method, are displayed (HK = 2K/Js; Js = |Js|). The inset shows the behavior of losses, with the red curve representing both the values given by the loop areas and the empirical ones of Eq. (33), this last plotted in correspondence with the Jp(Hp) peak polarizations of Eq. (34). One cannot appreciate significant differences between the two loss figures.

Close modal

6. Elemental hysteresis loops

The statistical approach developed was dictated by the impossibility to solve Eq. (8) with respect to the γ equilibrium angle for the local polarization Js [see Fig. 1(a)]—a goal that would directly supply the analytic expression for the elemental loops associated with a single particle. This difficulty has been circumvented by the mathematical technique outlined above, which supplies, in a sense, a “statistical solution” of Eq. (8), working out pH(γ), the probability to find Js laid along the γ direction. In this framework, the elemental hysteresis loops corresponding to a particle characterized by φ = φ1 and K = K1 come out defining the K and φ joint statistics as follows:
ψ(K,φ)=δD(KK1)δD(φφ1),
(29)
where δD is the Dirac delta function.
As usual, the starting point is the demagnetized state (DEM), identified by Eq. (6), which can now be rewritten in terms of γ, for any K1,
pDEM(Q=I)(γ)=12δD[γφ1],
(30a)
pDEM(Q=III)(γ)=12δD[γ(φ1π)],
(30b)
pDEM(Q=II)(γ)=pDEM(Q=IV)(γ)=0.
(30c)
For increasing HHc,K1(φ1) [Eq. (3)], all the local Js reversibly rotate toward γ = 0 [in particular, the Js belonging to the third quadrant when H = 0 remains confined in it if φπ/4 or, when φ > π/4, as long as HHIIIIV (see Sec. III A 1)], and no loop appears. When the alternating H overcomes the inversion field Hp = Hc,K1(φ1), the Js particles in Q = III or Q = IV jump to the first quadrant in the 0 ≤ γφ1 range [with pH(γ) → δD(γ) when H → +]. A similar and, to some extent, specular behavior is found following the recoil curve and the negative descending branch.

The ability of the analytic procedure to reproduce the elemental loops obtained when H oscillates with peak values |Hp| = Hc,K(φ1) [Eq. (3)] and J = Js cos γ measured along the H direction is apparent in Fig. 9, from the comparison with the cycles numerically computed.

FIG. 9.

Alternating quasi-static magnetizing field. For different φ1 angles between the local easy axis K and the H applied field direction (HK1 = 2K1/Js; Js = |Js|), it is apparent the capacity of the statistical approach to perfectly cover the elemental hysteresis loops obtained through the numerical procedure outlined in Sec. II B.

FIG. 9.

Alternating quasi-static magnetizing field. For different φ1 angles between the local easy axis K and the H applied field direction (HK1 = 2K1/Js; Js = |Js|), it is apparent the capacity of the statistical approach to perfectly cover the elemental hysteresis loops obtained through the numerical procedure outlined in Sec. II B.

Close modal
Let us consider the elemental hysteresis loops introduced above (Sec. III B 6), with local coercive fields ±Hc,K(φ) [Eq. (3)], peak polarization Jp(φ), remanence Jr(φ), and loss W. Now, we take as peak value the polarization reached immediately after the irreversible switch occurring at Hc,K under increasing field: Jp(φ) ≔ J[Hc,K+] (a larger H moves J along a reversible branch, non-modifying the loop area). An example obtained for φ = 35° (Js = 1 T, K = 1 J/m3) is given by the green solid line in Fig. 10(a), where instead the orange dashed rectangle corresponds to the case where irreversible rotations (180° switches between φ and φπ of the local Js) only are present. The area (giving the energy loss in J/m3) of the latter, with peak irreversible polarization JIRR,pJr = Js cos φ, is supplied by the following relationship:
WIRR(K,φ)=2Hc,K(φ)2Jr(φ)=8KcosφA(φ),
(31)
which is obtained from Eq. (3), and with A(φ) given by Eq. (2).
FIG. 10.

(a) The continuous green line describing the elemental hysteresis loop transforms into the dashed orange rectangle when only irreversible rotations are accounted for (Js = 1 T, K = 1 J/m3, φ = 35°, Hc,K(φ) = 1.02 A/m, Jc = 0.243 T, Jr = 0.819 T, Jp = 0.92 T). The difference between the loop areas is put in evidence by the gray pseudo-triangles. (b) Behavior vs φ of the loss W, and its irreversible component WIRR (both independent of Js), of the corresponding elemental loops. For φ = 0, the absence of irreversible processes entails W = WIRR = 8K [Eq. (31)]. (c) Interpolation of the K independent quantity W/WIRR vs φ, obtained (see inset) through the least squares linear fit of such a ratio vs log10[1/(12πφ)].

FIG. 10.

(a) The continuous green line describing the elemental hysteresis loop transforms into the dashed orange rectangle when only irreversible rotations are accounted for (Js = 1 T, K = 1 J/m3, φ = 35°, Hc,K(φ) = 1.02 A/m, Jc = 0.243 T, Jr = 0.819 T, Jp = 0.92 T). The difference between the loop areas is put in evidence by the gray pseudo-triangles. (b) Behavior vs φ of the loss W, and its irreversible component WIRR (both independent of Js), of the corresponding elemental loops. For φ = 0, the absence of irreversible processes entails W = WIRR = 8K [Eq. (31)]. (c) Interpolation of the K independent quantity W/WIRR vs φ, obtained (see inset) through the least squares linear fit of such a ratio vs log10[1/(12πφ)].

Close modal
For any φ, the difference between W and WIRR is concentrated in the gray pseudo-triangles with vertices ±Jr, ±Jc, and ±Jp [Fig. 10(a)]: all linear quantities dependent on Js and independent of K [Jr = Js cos φ, and for Jc and Jp, see Eqs. (8.34), (8.38), and (8.41) of Ref. 5]. Being Hc,K [Eq. (3)] inversely proportional to Js, as this last varies the loops aspect ratio modifies, but W and WIRR [Eq. (31)] do not. Moreover, Hc,K is the same for both loops, and one gets WIRR(aK, φ) = aWIRR(K, φ) and W(aK, φ) = aW(K, φ). For the arguments above, the ratio between the two losses [these last plotted in Fig. 10(b)] turns out to be φ dependent only, as reported in Fig. 10(c), where it is interpolated by the following general empirical law:
rW(φ)W(K,φ)WIRR(K,φ)=1+0.289log1012πφ,
(32)
with the coefficient 0.289 found via a least square linear fit of W/WIRR vs log10[1/(12πφ)] [Fig. 10(c) inset].
Eventually, for H = Hp, the energy loss of the whole particle ensemble is obtained after integration over the anisotropy axes values and orientations [from Eqs. (31) and (32), and with Kc,Hp given by Eq. (4)],
W(Hp)=0π/20Kc,Hp(φ)ψ(K,φ)W(K,φ)dKdφ=80π/2rW(φ)cosφA(φ)0Kc,Hp(φ)ψ(K,φ)KdKdφ.
(33)
One can also formulate this expression in terms of the peak induction corresponding to Hp, calculated on the First Magnetization Curve with Eq. (28),
JpJ(FMC)(Hp)=JsππpHp(FMC,Q)(γ)cosγdγ.
(34)
The energy loss values supplied by Eq. (33) are almost identical to the ones given by the hysteresis loops areas, as shown in the inset of Fig. 8.

The developed mathematical tool analytically predicts the response of an assembly of non-interacting SW particles driven by a quasi-static alternating magnetic field. The exact integral relationships for the hysteresis loops and losses allow one to bypass the computationally slow numeric procedure necessary to implement the SW model in software used to predict the behavior of devices containing magnetic components.

The versatility of this approach lies in its skills listed below.

  1. It works with any anisotropy distribution, permitting one to easily predict the role played by the sample texture (e.g., the effect of a macroscopic easy axis induced either by annealing or by applied stress).

  2. The same analytic-statistical procedure, although more complicated, can be exploited with any field history (e.g., rotating fields,31,34 loops with bias, and asymmetrical minor loops).

  3. Marginal modifications of the approach allow it to account for the presence of domain walls inside the particles, following the idea suggested in Ref. 22.

  4. The whole mathematical structure is prone to a generalization for systems of particles endowed with biaxial anisotropy.

  5. Under whatever field history and accounting for interactions, in Ref. 40, a mathematical technique capable to track the irreversible switches in a particle assembly described by the SW model has been worked out (so preserving the non-local memory of the system), together with a graphic representation that turns out to be an extension of the one operating in the Preisach model. One can then envisage a coupling between such a tool and the one developed here, with the aim to assess a more general analytical approach to the particle systems description.

This research work was carried out in the framework of the Grant No. 19ENG06 HEFMAG project, which was funded by the EMPIR program, and co-financed by the Participating States and the European Union’s Horizon 2020 research and innovation program.

The author has no conflicts to disclose.

C. Appino: Conceptualization (lead); Data curation (lead); Formal analysis (lead); Investigation (lead); Methodology (lead); Software (lead); Supervision (lead); Validation (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (lead).

The data that support the findings of this study are available within the article.

For a general γ′ equilibrium orientation of Js, with γlowγγ (where γ depends on the Q quadrant considered, and γlow is its lower limit), the Δφ*(γ) range, corresponding to a minimum of the particle Gibbs energy uH(K, φγ′) [Eq. (1a)], is determined by the overlap between two φ regions: the first one, where KH*(φ;γ) [Eq. (8)] is positive, and the second one satisfying Eq. (9), uH*(φ;γ)> 0, as illustrated in Fig. 11 (remember that φ is limited between 0 and π/2 for symmetry reasons).

FIG. 11.

The Δφ*(γ) intervals, where the Gibbs free energy uH(K, φγ′) of a “particle” displays a minimum, are shown for each branch and quadrant (note that no Δφ* exists for Q = II in FMC or REC, and for Q = IV in NDB). The values of φ0(Q)(γ) and of the asymptotes φ(Q)(γ) of KH*(φ;γ) and uH*(φ;γ) are given by Eqs. (10) and (11), respectively. In the case Q = I for FMC and REC, the picture displays the situation occurring when γπ/4, corresponding to the of KH* and uH* crossing. A similar situation is found in the case Q = III for NDB, when γ ≤ −3π/4.

FIG. 11.

The Δφ*(γ) intervals, where the Gibbs free energy uH(K, φγ′) of a “particle” displays a minimum, are shown for each branch and quadrant (note that no Δφ* exists for Q = II in FMC or REC, and for Q = IV in NDB). The values of φ0(Q)(γ) and of the asymptotes φ(Q)(γ) of KH*(φ;γ) and uH*(φ;γ) are given by Eqs. (10) and (11), respectively. In the case Q = I for FMC and REC, the picture displays the situation occurring when γπ/4, corresponding to the of KH* and uH* crossing. A similar situation is found in the case Q = III for NDB, when γ ≤ −3π/4.

Close modal

The Δφ*(γ) interval is found before for the field values 0 ≤ HHp, i.e., when the First Magnetization Curve (FMC) and the RECoil curve (REC) are covered (upper strip of Fig. 11). In this case, for Q = III and Q = IV, the two regions partially overlap, and the Δφ*(γ) limits are [φ0(III); φ(III) = γ′ + π] and [φ0(IV); π/2], respectively [see Eqs. (10) and (11)]. In Q = I, the entire region where KH*> 0 falls in the range where uH*> 0; φ0(I), which drops outside, does not play any role, and the Δφ*(γ) limits are [φ(I); π/2]. Eventually, in Q = II, the two regions do not overlap, signaling the fact that, in this quadrant, no stable equilibrium position for Js exists. This fact agrees with the result pH,IRR(FMC,II) = 0, which is reported at the end of Sec. III A, and is derived simply by observing the system geometry [Fig. 1(a)].

When H < 0 (lower strip of Fig. 11), i.e., along the Negative Descending Branch (NDB), a very similar and, to some extent, specular scenario appears—the situations in Q = III, IV, I, and II now recovering the ones found along FMC and REC in Q = I, II, III, and IV, respectively.

Eventually, let us contemplate the case when φ0(Q) constitutes the lower limit of the Δφ*(γ) interval (Q = III and Q = IV for FMC and REC, and Q = I and Q = II for NDB). In this circumstance, inserting the φ0(Q) value [Eq. (10)] into Eq. (8), one gets (considering that K > 0)
KH*φ0(Q)(γ);γ=JsHsinγsin[arctan(2tanγ)],
(A1)
a quantity that, together with function (10), constitutes a couple of parametric equations (with γ′ parameter) identifying, in the (φ, K) plane, a curve that coincides with Kc,H, being this last obtained under the conditions uH = uH = 0, as well. Consequently, φ0(Q) represents the abscissa of the tangent point between KH* and Kc,H, and one gets
KH*φ0(Q)(γ);γ=Kc,Hφ0(Q)(γ),
(A2)
an essential result for the calculation leading to pH(γ), in Sec. III B 1.
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