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Vol.46 No.5
JournalJournal of the Magnetics Society of Japan
Electronic Journal URL: https://www.jstage.jst.go.jp/browse/msjmag

2022

Hard and Soft Magnetic Materials
Interpretation of Kronmüller Formula using Ginzburg-Landau Theory
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Journal of the Magnetics Society of Japan
Vol. 46, No. 5
Electronic Journal URL: https://www.jstage.jst.go.jp/browse/msjmag

CONTENTS
Hard and Soft Magnetic Materials
Interpretation of Kronmüller Formula using Ginzburg-Landau Theory…………………………………………………………………………………… C. Mitsumata and M. Kotsugi

90

Biomagnetism / Medical Applications
Magnetocardiography Inverse Estimation from a Realistic Ventricular Model Using Spatial Filter
Methods: A Simulation Study…………………………………………………………………………… W. Sun, M. Iwai, and K. Kobayashi

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J. Magn. Soc. Jpn., 46, 90-96 (2022)<Paper>Interpretation of Kronmüller formula using Ginzburg-Landau theory
C. Mitsumata and M. Kotsugi

Tokyo University of Science, 6-3-1, Niijuku, Katsushika, Tokyo, Japan 125-8585

The Kronmüller formula is used to qualitatively explain the coercivity of permanent magnets. Such simple expressions are usefulfor understanding the elements that inﬂuence the coercivity. This study attempts to derive Kronmüller formula within the frameworkof the Ginzburg-Landau theory. The results revealed that the coercivity of the magnetic material was deﬁned by the slope of the freeenergy landscape. The reduction parameter α for the magnetic anisotropy ﬁeld Hk in the Kronmüller formula depends on the functional
form representing the free energy landscape that changes with the magnetization process of the material, and it is shown that α < 1under the condition of absolute zero temperature T = 0. Furthermore, it was revealed that α depends on the entropy of the system andthat it may show a larger reduction rate at a ﬁnite temperature T  0. Thus, it is important to acquire the free energy landscape andclarify the order parameter dependence of entropy in the magnetization process of the magnetic material.

Key words: free energy, entropy, energy landscape, coercivity, magnetization process, permanent magnet, hard magnet
1

2

Introduction

The Kronmüller formula has been used to qualitatively explain

the coercivity of permanent magnets. This simple expression isuseful for understanding the elements that inﬂuence coercivity.
The Kronmüller formula is shown below.
Hc = αHk − Nd Ms(1)

where Hc , Hk , Nd , and Ms are the coercivity, magnetic anisotropy
ﬁeld, demagnetizing coeﬃcient, and saturation magnetization,
respectively. This suggests that the coercivity can be explainedby the magnetic anisotropy ﬁeld and demagnetizing ﬁeld. In particular, for the inﬂuence of the magnetic anisotropy ﬁeld, the exis-

tence of the reduction coeﬃcient (0 < α < 1) shows the diﬃcultyof the coercivity analysis.
The parameter α in Eq. (1) has been discussed in Kronmüller’s
theoretical model analysis1, 2) and in an experimental study by Hirosawa et al.3) . However, the determination mechanism remains

unclear, and further analysis is necessary to elucidate the coercivity mechanism.
Here, the determination principle of α in Eq. (1) is discussed
using the Ginzburg-Landau theory4) . A coercivity analysis ofpermanent magnets using the Ginzburg-Landau theory has previously been attempted . However, a uniﬁed interpretation bycomparison with the Kromüller formula has not yet been obtained. In recent years, the analysis of the energy landscape of
permanent magnets8) has been progressing, and expectations arerising for attempts to discuss coercivity from the perspective of
free energy9, 10) . Therefore, in this study, the role of the energy
5–7)

landscape in the coercivity mechanism is clariﬁed, and the eﬀectof the coeﬃcient in the expression of the Kronmüller formula isconsidered. In addition, the temperature dependence of the reduction parameter, α, is discussed.
Corresponding author: C. Mitsumata (mitsumata@rs.tus.ac.jp)

90

Derivation of Kronmüller formula

2.1 Simultaneous magnetization rotation model
In the Ginzburg-Landau theory, the free energy of the system isa function of the order parameter and the ordered state where theminimum condition of energies is determined to realize the stablestate of the system. In magnetic materials, the order parameter is
described by the magnetization state of the system, and the freeenergy density f is a function of the magnetization M.

Here, the order parameter represents an array in which themagnetization is M + > 0 or M − < 0 in the local region. In
the whole system, M = M + + M − describes the magnetization
value and the array state of the system at the same time. Magnetization as an order parameter is diﬀerent from the order of

atomic arrangement. Even if the magnetization M is the same, themagnetization arrangement in the local region may be diﬀerent,
which corresponds to the magnetic domain structure. However,the single variable Landau theory in magnetic materials is a kindof mean ﬁeld theory, and it is considered that the magnetization
process of the system follows diﬀerent energy landscapes whenthe magnetization M is equal and the magnetic domain structure
is diﬀerent. Information on diﬀerent magnetic domain structures

is an index that describes the diversity of the magnetization arrangement, and is reﬂected not in the diﬀerence in the order parameter but in the entropy corresponding to the certain order variable.
In addition, there is a deﬁnition that the order parameter in Landau theory expresses the anisotropy of the system. Therefore,
when a directed line segment such as a magnetization vector isused as a variable of the scalar quantity, the system is describedby the projection component in a certain direction and the entropy
depending on such direction. For example, if the saturation stateof the magnetization inside the material is Mz = ±Ms , all themagnetizations in the local region are aligned in one direction.
There is only one magnetization array that realizes those in each

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If the absolute value of the external magnetic ﬁeld increases, the2minimum state of energy existing at M/Ms = 1 disappears, andthe magnetization state is inverted to M/Ms = −1 along the slopeof the energy landscape. To summarize the above discussion, for

1.5• / Ku1(A)magnetization reversal from positive to negative, the absence ofa stable energy state in the region of M > 0 indicates the mag-

0.5
(B)

0

netization reversal condition, and the energy landscape becomesε′ = ε − Happ M which is the addition of Zeeman energy in Eq.

-0.5(2). That is,

-1
-1

-0.5

0
M / Ms

0.5

1

2Ku M
∂ε′
=−
+ Nd M − Happ ≥ 0∂M
Ms2

Fig. 1 Methodology for magnetization reversal on energy landscape within Ginzburg-Landau theory. (A) energy landscape of

uni-axial magneitc anisotropy without an applied ﬁeld, (B) underinﬂuence of an applied ﬁeld.

case of positive and negative, and the entropy of the correspond-

ing system is S = 0. In the case of a simultaneous rotation model,
arbitrary M x and My are accompanied in the magnetization reversal process from Mz = +M s to Mz = −M s . All magnetizations
are in the same direction and have a high degree of regularity, butdeviations from a particular direction indicate a decrease in system anisotropy. Thus, using the polar coordinate representation,∫
the entropy increases to S = log(1 + sin θ cos ϕdϕ). Therefore,

in this study, it is possible to describe the free energy by using thescalar quantity magnetization M as an order parameter.
Because the Kronmüller formula does not explicitly include avariable of temperature T , we ﬁrst consider that the free energy
density consists of only the term of magnetic energy density ε as
the condition of T = 0.

(

)
M2
1
ε(M) = Ku 1 − 2 + Nd M 22
MsThis is consistent with the monotonically increasing function of
the energy landscape in the M > 0 region. In addition, because
the result of diﬀerentiating the energy density ε′ with the magne-

tization M indicates an eﬀective magnetic ﬁeld, Eq. (3) compares

the magnitudes of magnetic ﬁelds. Therefore, the condition thatthe external magnetic ﬁeld Happ < 0 required for magnetization

reversal is satisﬁed is calculated as follows by the transformationof Eq. (3).

)
(
2Ku M
+
N
M
≥ Happ
−
d
Ms2
min.

Ku is the uniaxial magnetic anisotropy constant, Nd is the demagnetizing ﬁeld coeﬃcient, M is the magnetization, and Ms isthe saturation magnetization. Equation (2) presents an expressionthat considers only the magnetic anisotropy energy and demagnetizing ﬁeld energy.
Figure 1 shows the concept of magnetization reversal within
the framework of the Ginzburg-Landau theory. Here, we consider

simultaneous rotation as the magnetization process. Equation (2)forms the energy barrier required for magnetization reversal, andcurve (A) shows the energy landscape without an applied magnetic ﬁeld. The magnetization from the positive state (M/Ms = 1)
is reversed to the negative state (M/Ms = −1) by overcoming theenergy barrier, as indicated by the arrow. This is because thelowest energy state is M/Ms = ±1. When an external magnetic
ﬁeld (Happ < 0) is applied to the system, the energy landscape,as shown in curve (B), changes owing to the inﬂuence of Zeeman energy. Thus, the energy barrier to be overcome from thestate of M/Ms = 1 becomes smaller, as indicated by the arrow.

(4)

When the maximum Happ < 0 that satisﬁes the condition of Eq.

(4) is applied to the system, the stable point on the energy landscape in the region of M > 0 disappears, and magnetization reversal occurs, that is, Eq. (4) describes the coercivity of magnetic

materials. Because the permanent magnet satisﬁes the relationship 2Ku /Ms2 > Nd , the minimum value on the left-hand side ofEq. (4) is obtained when M = Ms . Corresponding conditional
equation is as follows:−Hc = −Hk + Nd Ms
(2)

(3)

(5)

Here, the magnetization process of the system is assumed as simultaneous rotation: therefore, the Kronmüller formula for α = 1
is obtained.
The presence of the reduction parameter 0 < α < 1 in Eq. (1)indicates a decrease in the coercivity. Therefore, in this analysis, there are two possibilities: the energy barrier is low, or themagnetic order has already climbed halfway through the energy
barrier. The change in the magnetic anisotropy constant Ku is
reﬂected in the change in Hk ; therefore, it is irrelevant for the de-termination of α. The magnetostatic energy, as the second termin Eq. (2), can also be included in Eq. (4) as a coeﬃcient for M 2 .

Among the above possibilities, it is speculated that the state of thesystem rising halfway through the energy barrier inﬂuences α. It
should be noted that the change in the arrival point on the energy
landscape means that the order parameter M also needs to change
at the same time. Considering that irrationality occurs in the si-

multaneous rotation described by Eq. (2), and the possibility thatthe stable state of the system can be described by the order parameter M  Ms , it is necessary to have a discussion that reﬂects theinﬂuence of the multi-domain structure (non-uniformity of magnetization distribution). Even with an actual magnet material,

Journal of the Magnetics Society of Japan Vol.46, No.5, 2022

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INDEX

the squareness σ = Mr /Ms (Mr is the remanent magnetization)2

shows that σ < 1, and the multi-domain structure exists stably
under the condition of no external magnetic ﬁeld (Happ = 0).

2.2 Multi-domain model

1.5

For non-uniform distribution of magnetization, the magnetization as an order parameter takes the value of M < Ms . With this• / Ku1

ments for the composition of the energy landscape are as follows.

-0.5• The energy landscape under no-magnetic-ﬁeld conditions
is symmetric with respect to M = 0.

-1

condition of order parameter M = 0, and the demagnetization state is metastable.

• The magnetic domain structure of the residual magnetization state, Mr = σMs (0 < σ < 1), forms a stable point.
Rewriting these required conditions,ε(−M) = ε(M)

(6)

)
( 2 )
∂ε
∂ε
= 0,
≤0
∂M M=0∂M 2 M∼0

(7)

)
( 2 )
∂ε
∂ε
= 0,
≥0
∂M M=Mr∂M 2 M∼Mr

(8)

(
(

However, it is assumed that there is no sudden change in the magnetic domain structure near stable and metastable points. Basedon the conditions in Eq. (7) and Eq. (8), where the double derivative of the energy landscape function ε(M) changes sign in therange of 0 < M < Mr , ε(M) is a function with at least one inﬂec-

tion point in the range of 0 < M < Mr . The order parameter thatgives this inﬂection point is deﬁned as Mc . Simultaneously, con-sidering the sign condition of the double derivative of ε(M), thederivative ∂ε/∂M has the minimum value that is (∂ε/∂M) M=Mc ,while satisfying the condition of (∂ε/∂M < 0) in the range of

0 < M < Mr . This is because at M = 0, starting from ∂ε/∂M = 0,
the derivative becomes negative as M increases. When it reaches

Mr beyond the inﬂection point, it returns to ∂ε/∂M = 0. Therefore, there is always a negative minimum value in this section.
Consequently, it is possible to calculate the coercivity of the en-ergy landscape expressing the multi-domain structure using thesame procedure as in the case of simultaneous rotation.
)
(
∂ε
(9)
−Hc =
∂M M=Mc

However, Eq. (9) is a conditional expression in the range 0 <Mc < Ms , and in −Ms < Mc < 0, the sign on the left-hand side ofthe equation is inverted.
Innumerable functions satisfy the conditions in Eq. (6) to Eq.

(8). Among them, the function that is diﬀerentially continuous
in the range −Ms ≤ M ≤ Ms and can be described by the lowest
order of M is a quartic function. When the order of M increases,

92

(A)

0

in mind, we will reconsider the energy landscape. The require-• The saddle point of the energy barrier is formed under the

(B)

0.5

-1

Fig. 2-0.5

0M / Ms0.5

1Energy landscape comparison. (A) a quartic function

representing the non-uniform magnetization distribution, (B) aquadratic function representing the simultaneous magnetization
rotation.
the rate of change of the derivative ∂ε/∂M increases, and thus
the coercivity Hc of Eq. (9) tends to increase*1 . Therefore, theframework of the Ginzburg-Landau theory using the quartic function of M can be calculated to guarantee the minimum value ofthe coercivity. Moreover, the condition of diﬀerential continuity
is a requirement of the conservation law of magnetic energy.

The energy landscape of the quartic function given by above
conditions is shown below.
)
(
Nd
2Ku
Ku
M 2 + Ku
(10)ε(M) = 4 4 M 4 − 2 2 −
2
σ Ms
σ Ms

As mentioned above, the order parameter represents the mag-netization array and the projected component of the magnetization vector. Therefore, it is possible to consider the magnetic
anisotropy energy due to the rotation of the magnetization in themagnetization process. Strictly speaking, when magnetization
rotation occurs in the local region, the height of the energy bar-rier against magnetization reversal is considered to be smaller
than Ku due to the inﬂuence of the magnetic domain structure.
However, here it is approximated that the saddle point of the energy landscape is determined by Ku for direct comparison withthe simultaneous rotation. That is, Eq. (10) assumed the maxi-

mum value of the possible energy barrier. Also, as shown in Fig.
2, the constant term Ku shifts the reference, so that it can be easily compared with the energy barrier of the simultaneous rotation

model. Because the expression of the coercivity is given by thederivative of the energy landscape, there is no inﬂuence of theconstant term on the analysis. For comparison, the energy landscapes in Eq. (10) and Eq. (2) are shown in Fig.2. (A) shows
the case of the quartic equation and (B) represents the case of aquadratic equation. Here, for simplicity, the demagnetizing ﬁeld
coeﬃcient Nd = 1/3 is used assuming a sphere as the material
shape, where the demagnetizing ﬁeld distribution inside the ma*1 Assume that the height of the energy barrier at M = 0 is invariant in thecases of quadratic and quartic functions.

Journal of the Magnetics Society of Japan Vol.46, No.5, 2022

INDEX

terial becomes uniform. In addition, function (A) is displayed

under the condition that the squareness is σ = 0.9. The magne-

tization states that satisfy the respective minimum energy values
are M = ±Mr = ±σMs in case (A) and M = ±Ms in case (B).
To determine the coercivity from the energy landscape in Eq.

(10). The inﬂection point is,σ2 Ms2 Nd σ4 Ms4
−
Mc2 =
3
12Ku

redeﬁne the free energy landscape, including the entropy eﬀect.
f (M) = ε(M) −kB T S (M)V(15)

where kB is Bolzumann’s constant, T is the temperature, S isthe entropy of the system, and V is the volume of the system.

Because S counts the degrees of freedom of the magnetization(11)

Substituting the magnetization showing the inﬂection point into

array, it is expressed as a function of the order parameter M only.
In complex systems, such as magnet materials, it is diﬃcult toprecisely calculate S . Therefore, to understanding the coercivity

the slope of the energy landscape.
)
)]
[
(
(
4Ku 2
2Ku
Nd
∂ε
Mc
= 4 4 Mc − 2 2 2 −∂M M=Mc
2
σ Ms
σ Ms
√
(
)
4
2σ
2Hk − σ2 Nd Ms
Nd Ms (12)− Hk +
−Hc =
6Hk
3σ
3

mechanism qualitatively, a simple function that allows the func-

the coeﬃcients of Hk and Ms are numerically approximated byparameters that imitate NdFeB magnets. Here, it is assumed thatσ = 0.9. That is,in permanent magnets. Considering that the positive and negativeof the magnetization in the minute region represents the state ofBecause it is diﬃcult to understand the square root coeﬃcient,

Hc = 0.84Hk − 0.34Nd Ms(13)

From the Ginzburg-Landau theory, by evaluating the energy landscape, it becames clear that a reduction factor (α < 1) that in-evitably becomes Hc < Hk can be obtained owing to the nonuniformity of magnetization.
The eﬀect of the squareness σ is even more interesting. Comparing the coeﬃcient terms proportional to Hk in Eq. (1) and Eq.
(12).
√
4
α=
3σ

2Hk − σ2 Nd Ms
6Hk

(14)

As shown, the reduction factor α is a function of σ. There arevarious possible reasons for σ  1, but the dispersion of the mag-netic anisotropy in the material is a factor that has a signiﬁcanteﬀect. Because uniaxial magnetic anisotropy is assumed here, it
is not possible to discuss the case in which the dispersion of mag-netic anisotropy is large. However, in the range where the averagemagnetic anisotropy K u can be approximated as K u ∼ Ku , α tends
to decrease as σ increases 11) . That is, a seemingly contradictory
result is derived in that the coercivity decreases as the degree oforientation of the crystal axis increases, which deﬁnes the average magnetic anisotropy. This result emphasizes the importanceof discussing energy landscapes in the coercivity mechanism, as

the energy landscape for magnetization reversal changes owing
to the inﬂuence of the magnetization process.
3 Eﬀect of entropy
In the previous section, we discussed the energy landscapewhen T = 0, and the eﬀect of entropy was not considered. In

Ginzburg-Landau theory, the stable state of the system with respect to the order parameter can be discussed using the free energy density f with the addition of the entropy term. Here, wetion f to be calculated is introduced below.

In the discussion of magnetization reversal, the magnetization

state is separated into regions with M > 0 or M < 0, and thisregion distribution forms a magnetic domain structure. Underconditions of strong uniaxial anisotropy, this assumption is considered a good approximation of the magnetic domain structure

the magnetization array, the probability state density of the mag-netization P(M) can be expressed by the binomial distributionfunction.

P(M) = nCk pk (1 − p)n−k(16)where the coeﬃcient nCk is a combination that selects k elementsfrom an array of n, and the probability p that determines the pos-

itive or negative magnetization is p = 1/2 *2 . This is consistent
with the method used to count the number of states in the Isingspin.

As the number of minute regions representing the magnetic domain structure is innumerable, the binomial distribution function
expressing the stochastic density of states can be approximatedby a Gaussian function representing the normal distribution.
P(M) =√()
M2
ϕ
exp −ϕ 2 + ϕ
π
Ms

(17)

Here, the average value of the magnetized states, M, is M = 0,
and parameter ϕ in the probability state density function with M
is a valid constant in the temperature range 0 ≤ T < T c (T c is theCurie temperature). Meanwhile, P(M) becomes a delta function
δ(M) at T ≥ T c . In Eq. (17), the constant term in the expo-

nential function guarantees that the number of states is only 1,
where an ideal saturation state is M = Ms or M = −Ms . In addition, ϕ is a constant that changes depending on the magnetization

process, and as ϕ increases, the rate of change in the stochastic density of states with respect to the magnetization change of
Ms → 0 increases. Qualitatively, ϕ tends to be small with respect to the change in magnetization due to domain wall motion,*2 Strictly speaking, the probability of magnetization reversal depends on themagnetic domain structure, that is p = 1/2 ± δ. Here, it is assumed that the
average of the reversal probabilities in all cases becomes p = 1/2, and thatthe ratio of exchange energy in the height of the energy barrier is small,thus it is set as δ ≪ 1.

Journal of the Magnetics Society of Japan Vol.46, No.5, 2022

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the residual magnetization Mr has shifted toward M = 0. It canalso be observed that the slope of the energy landscape indicated2(E)

free energy / Ku

1.5

(F)

1

by the arrow is also reduced in (F) compared to that in (E). Thisdecrease in inclination represents a decrease in coercivity owingMr

to the entropy eﬀect.
The constants that appear in Eq. (21) in terms of coercivity√
shows a functional form of Hc ∼ (1 − γT )3 where γ is a con-

(S)

0.5
0

stant coeﬃcient. The coercivity, as indicated by Eq. (21) matcheswith that in Eq. (12) when T = 0 and is approximately Hc ∼ αHk .
Therefore, for the sake of simplicity, Nd = 0 was set, and the temperature dependence of Hc was estimated using the parameter α

-0.5
-1
-1

-0.5

0
M / Ms0.5

1Fig. 3 Energy landscape consisting of internal energy and entropy densities. (E) shows the internal energy density given byeq.(10), (S) shows the entropy density given by eq.(18), and (F)represents the free energy density given by (F)=(E)-(S).

and it tends to increase as the subdivision of magnetic domains
becomes small in the multi-domain structure. Consequently, theentropy that can be obtained is
S = ln(P(M)) = −ϕ

M2
+ ϕ′
Ms2

(18)

√
where ϕ′ = ϕ + ln( ϕ/π). It was found that the entropy of strong

anisotropy system is a quadratic function of the order parameter
M. The speciﬁc free energy density is expressed by Eq. (10),with a multi-domain structure. That is,)(
kB T ϕ Nd
kB T ϕ′2Ku
Ku
M 2 + Ku −−f (M) = 4 4 M 4 − 2 2 −
2
2
V
σ Ms
σ Ms
V Ms
(19)
As the free energy landscape has been determined, the coerciv-

ity is calculated from the slope of such a function. Now, the orderparameter that gives the inﬂection point is
Mc2 =

)
(
σ4 Ms4 2Ku
kB T ϕ Nd−−
6Ku σ2 Ms2 V Ms2
2(20)

Substituting this value for the slope of the free energy landscape
gives,))]([(
k B T ϕ Nd
∂f
2Ku
2Ku
= 2Mc 4 4 Mc2 − 2 2 −−∂M M=Mc
2
σ Ms
σ Ms
V Ms2
√
)3
(
4σ2 M 2
kB T ϕ Nd
2Ku−Hc = − √ s
−
−
(21)
2
σ2 Ms2 V Ms2
3 6Ku
The temperature dependence of the coercivity Hc was obtained asshown. The entropy eﬀect is schematically shown in Fig.3. (E)is the magnetic energy density, and is depicted as an energy landscape that provides a stable state of magnetization with a squareness of σ = 0.9. (S) is the functional form of Eq. (18) in termof entropy density. (F) is the energy landscape obtained by subtracting (S) from (E) with the free energy density and using theformula of F = E−T S , the ﬁgure is displayed with T = 1. Owing
to the eﬀect of entropy, the energy stabilization point that causes

94

as a function of temperature*3 .
Let us now discuss the Landau theory of ferromagnetic andparamagnetic secondary phase transitions for a localized mag-netic moment, m, which is slightly diﬀerent from the discussionof multi-domain structures. As for the relationship between theorder parameter m and the energy landscape, the minimum condition of the free energy corresponds to the stable state of theaverage magnetic moment, as in the discussion so far. Therefore,

the Landau-style phase transition theory deﬁnes the internal energy as E = am4 + bm2 and uses the coeﬃcients a and b as afunction of temperature. In this case, the order parameter was
normalized to 0 ≤ m ≤ 1. Here, the free energy F is deﬁned as
F = E − T S and the stable state of the average magnetic moment
is analyzed. For example, a ferromagnetic state without ﬂuctuation of magnetization is realized at m = 1 at T = 0, owing to thesettings of a = 1 and b = −2 on the free energy functions. Using
Eq. (18) as the entropy term
F = m4 − (2 − ϕT )m2(22)

it is possible to investigate the eﬀect of temperature to ﬁnd theminimum conditions in such free energy. Strictly speaking, theentropy of Eq. (18) considers only the positive and negative valueof the magnetization M, so it is an expression corresponding to
the Ising model. This is diﬀerent from the expression of the
Heisenberg model that uses the classical localized magnetic mo-ment. However, the main purpose of this study is to analyze amodel in which the magnetic domain structure is composed of alarger magnetization array under strong anisotropy. In the analysis of permanent magnets, the directions of magnetic moment
are limited by strong magnetic anisotropy as in Ising model, andthus, we decided to carry out this simple analysis to conﬁrm theentropy eﬀect phenomenologically. The average value of the totalmagnetic moment is approximately m = 0.8 when T = T c /2, according to the estimation using the Langevin function. Therefore,by estimating ϕT assuming that (∂F/∂m) = 0 is realized under
the condition of m = 0.8, it can be obtained as ϕT c /2 = 0.72.
*3 The coercivity dependence of T 3/2 is due to the quartic function represent-

ing the free energy landscape. Therefore, when the order of the functionchanges, the exponent of the power function with respect to the temperature T also changes.

Journal of the Magnetics Society of Japan Vol.46, No.5, 2022

INDEX√(1 − γT )3 , it is consid-

ered that the coeﬃcients ϕT and γT are equivalent in the natural√unit system because both F and (1 − γT )3 are a normalized.
Therefore, considering ϕT c /2 = γT c /2, the coeﬃcient became√
α = 0.593/(3 3σ) at T = T c /2. In general, the ﬂuctuation of theclassical spin by the Langevin function tends to be overestimated1

entropy is a major cause of the decrease in parameter α in the

Mr

(A)

Mi

(B)

-Hc

0

in low temperature range. Considering this fact, the decrease incoercivity at a ﬁnite temperature is large, and it is considered that

-0.5
-1

Kronmüller formula. As previously mentioned, the coeﬃcient

-0.5

0

0.5

H / Hkϕ of the entropy term depends on the magnetization process. It

should be noted that experimental measurements are required toHi

0.5
M / MsFrom the result of comparing F withdetermine the exact temperature dependence of α. This is be-

Fig. 4

if ϕ ≪ 1. In this sense, the reduction factor α in the Kromüller

ergy, (B) a minor loop due to energy excitation. A hatched area
between (A) and (B) corresponding to exitation energy by an ap-

cause the eﬀect of temperature T on the coercivity is suppressed

formula is governed by the change in entropy S with respect tothe change in the magnetization state M. If this change in en-

tropy can be suppressed, coercivity may increase even at higher
temperature. Furthermore, it is presumed that the temperaturedependence of the coercivity is diﬀerent even for a similar composition of permanent magnet materials, which can be explainedby the diﬀerence in the magnetic entropy due to the inﬂuence ofthe crystal texture.
4

Analysis of free energy landscape including temperatureand entropy eﬀects

4.1 Measurement of free energy landscape
To analyze the coercivity, it is necessary to ﬁnd the free energy landscape as a function of the order parameter M. To an-alyze the coercivity, as shown by Eq. (3), the inﬂuence of theexternal magnetic ﬁeld is compared with the slope of landscape.

Therefore, it is necessary for the order parameter in the functionof the free energy landscape not to be inﬂuenced by the external ﬁeld. The procedure for this measurement is illustrated in

Fig.4. The magnetization curve of (A) shows part of a major loop,
where measurements begin with an external magnetic ﬁeld +Hs
that saturates the magnetization and passes through the remanentstate Mr (H = 0) and the coercivity state −Hc . Finally, magnetization reversal occurs. In contrast, the magnetization curve of(B) is a minor loop under the external magnetic ﬁeld condition
Hi → H = 0. To obtain (B), the measurement is started from
+Hs , passes through H = 0, and then the magnetic ﬁeld sweep isreversed at Hi that satisﬁes the condition 0 > Hi > −Hc . Here,the magnetization state obtained from the external magnetic ﬁeld

condition of Hi is deﬁned by the intersection of the minor loop(B) and H = 0, and is deﬁned as Mi . This magnetic measurement
procedure is similar to the FORC (First-Order Reversal Cureve)
analysis. Focusing on the condition of H = 0 in Fig.4 regarding
the magnetization state of magnetic materials, the order parameter was relaxed from Mr to Mi . This relaxation process is due tothe energy increase corresponding to the area ∆i surrounded by

Energy landscape measurement from magnetization

curves. (A) a major loop forming the reference of magnetic en-plied ﬁeld.

the area Mr − Hi − Mi (gray hatch) between the major loop (A)and minor loop (B). In Fig. 2 shown as an example of energy
landscape, the decrease in magnetization in the region of M > 0
corresponds to the process of increasing energy due to work fromoutside of the system. The Helmhortz-type free energy used in

Landau theory describes an open system. That is, it can be seenthat the magnetization state is destabilized by the excitation en-ergy from the outside in the process of magnetization reversal,
and the magnetization is relaxed. Because the decrease in magnetization corresponds to the decrease in the order parameter, it isexpressed that the magnetization of the system is relaxed. Therefore, based on the Mr state, the free energy density corresponding
to the order parameter Mi isf (Mi ) = f (Mr ) + ∆i

(23)

In Eq. (23), f (Mi ) is uniquely determined by ∆i ; the free energy
landscape f (Mi ) can be obtained, where a series of magnetization

Mr > · · · > Mi−1 > Mi > Mi+1 > · · · > 0 are measured under theconditions of the external magnetic ﬁelds 0 > · · · > Hi−1 > Hi >Hi+1 > · · · > −Hc .
4.2 Separation of entropy terms
The free energy density determined by the measurements

shown in Fig.4 should be in the form of Eq. (15). Here, toenable a detailed discussion of the energy landscape analysis,

we attempt to separate the entropy term from the measurement
results. Two curves f j (M) and f j+1 (M) representing the freeenergy landscape are obtained by measuring Fig.4 under two
diﬀerent temperature conditions, :0 < T j < T j+1 < T c . As thefunction ε(M) in Eq. (15) is independent of the conditions T j
and T j+1 ,S (M)f j (M) − f j+1 (M)
=−
kB δT
V

(24)

The entropy density of the system can be determined as above,
where δT = T j − T j+1 . Moreover, because the limit of δT → 0

Journal of the Magnetics Society of Japan Vol.46, No.5, 2022

95

INDEX

matches the deﬁnition formula of entropy density, it can be mea-

sured without change in the global magnetic domain structure
due to the thermal ﬂuctuation of magnetization, if the diﬀerence
between T j and T j+1 is reduced. Using the free energy landscapef determined using Eq. (23) and the S obtained by Eq. (24), thefunction form is obtained inf j (M) = ε(M) −kB T j S (M)V(25)

The internal energy density of the system ε can also be described
as above. The identiﬁcation of ε is important for understanding

the coercivity mechanism, and it should also be considered thatthe barrier height of the energy landscape may not necessarily
match Ku depending on the magnetization process. To improvethe measurement accuracy of the energy landscape, we increased
the measurement point T j to ensure statistical reliability.
5

SummaryThis study attempted to derive the Kronmüller formula byconducting an analysis within the framework of the GinzburgLandau theory. The results showed that the coercivity of themagnetic material was deﬁned by the slope of the free energy landscape. The reduction parameter α for the magnetic
anisotropy ﬁeld Hk in the Kronmüller formula depends on thefunctional form representing the free energy landscape thatchanges with the magnetization process of the material, and it
is shown that α < 1 even under the condition of absolute zerotemperature T = 0. Furthermore, α depends on the entropy ofthe system and it may show a larger reduction rate at a ﬁnite
temperature T  0. Thus, it is important to acquire the freeenergy landscape and clarify the order parameter dependence ofentropy in the magnetization process of the magnetic material.
This study does not explicitly include the interactions owing

to the spatial distribution of magnetization. For example, to in-

96

corporate the eﬀect of the domain wall as internal energy, it isnecessary to introduce the term of A(▽M)2 . Here, A is the ex-change stiﬀness constant and ▽M is the spatial derivative of themagnetization. However, to incorporate speciﬁcally the domain
wall shape into the framework of the Ginzburg-Landau theory, it
is necessary to introduce a functional representation or quantiﬁcation of the magnetic domain structure. This theory needs tobe extended to capture the eﬀects of spatial modulation associated with the magnetization distribution 9, 10) , including a moreaccurate representation of the magnetic dipole interactions.

Acknowledgments This work is supported in part by the Japan

Society for the Promotion of Science, Grants-in-Aid for Scientiﬁc Research ⟨KAKENHI⟩ Aʢ21H04656ʣ
References
1) H. Kronmüller: Phys. Stat. Sol. (b) 144 , 385 (1987).
2) H. Kronmüller and K.-D. Durst: J. Magn. Magn. Mater. 74, 291(1988)

3) S. Hirosawa, K. Tokuhara, S. Mino, T. Matsui, K. Morii and Y.
Nakayama: MRS Sym. Proc. 232, 275 (1991).
4) L. D. Landau and E. M. Lifshitz: ”Statistical Physics” (Pergamon,
Oxford,1980)
5) K. Iwano, C. Mitsumata, and K. Ono: J. Appl. Phys. 115, 17D134(2014).
6) K. Iwano, C. Mitsumata, and K. Ono: J. Appl. Phys. 117, 17A704(2015).
7) K. Iwano, C. Mitsumata and K. Ono: J. Phys. Soc. Jpn. 85, 074711(2016).
8) Y. Toga, S. Miyashita, A. Sakuma and T. Miyake: npj Computational Materials, 6, 67 (2020).
9) A. L. Foggiatto, S. Kunii, C. Mitsumata, and M. Kotsugi: To be
submitted10) K. Masuzawa, S. Kunii, A. L. Foggiatto, C. Mitsumata, and M.
Kotsugi: T. Magn. Soc. Jpn. (Special Issues) 6, 1 (2022)
11) Y. Matsuura; 40th. Ann. meeting of Magn. Soc. Jpn, 7aE-6 (2016)
Recieved Apr. 28, 2022 ; Revised Jun. 25, 2022; Accepted Jul. 19, 2022

Journal of the Magnetics Society of Japan Vol.46, No.5, 2022

INDEX
J. Magn. Soc. Jpn., 46, 97-101 (2022)<Paper>Magnetocardiography Inverse Estimation from a Realistic Ventricular
Model Using Spatial Filter Methods: A Simulation Study
Wenxu Sun, Morio Iwai*, Koichiro Kobayashi*

Faculty of Engineering, Kindai Univ., 1-Umenobe, Takaya, Higashi-Hiroshima, Hiroshima 739-2116, Japan* Faculty of Science and Engineering, Iwate Univ., 4-3-5 Ueda, Morioka, Iwate 020-8551, Japan

We used a lead field normalization and measurement covariance matrix R into a weighted minimum-variancespatial filter (WMV). An inverse estimation, with an extended source, arranged at the surface of a realistic ventricularmodel was carried by WMV, weight-normalized minimum-norm (WMN), and minimum-variance spatial filter (MV)with and without noise. The performances of these spatial filters were evaluated using the estimation error and ratiosof all positions with an estimation error below 2 cm. Moreover, the proper regularization parameter was determinedfrom the estimation error. The results of the statistical analysis, while handling varied source positions, show that
WMV has the best performance for magnetocardiography (MCG) extended source inverse estimation because it leads
to less estimation error and is capable of stable inverse estimation, even at high noise levels. In other words, thecombined lead field normalization, with the measurement covariance matrix R used in WMV, is a good choice for our
MCG system.
Key words: Magnetocardiography, inverse estimation, source localization, spatial filter
1. Introduction
Despite being fatal, frequent premature ventricular
contraction (PVC) is treated with catheter ablation to
improve the patient’s quality of life. During ablation, 3D
electro-anatomical mapping is used to precisely locate
the origin 1); however, some non-invasive techniques are
required to locate the origin. Relatively uniform magnetic
permeability enables magnetocardiography (MCG) toyield higher spatial resolution of inverse estimation of
the origin compared with electrocardiography 2). To solve
the ill-posed bioelectromagnetic inverse problem, spatial
filtering techniques are widely used.
One prominent class of techniques, known as nonadaptive spatial filters, including the original minimumnorm method (MN), weight-normalized minimum-norm
method
(WMN)
method,andlow-resolution
electromagnetic tomography (LORETA) 3)-5). MN tends toconcentrate the solution near the surface, regardless ofthe true depth of the source. This is because thesensitivity of the sensor decays rapidly with depth. The
tendency to concentrate the solution at the surface can beovercome by introducing proper depth weighting [4].
Lead field normalization is effective in compensating forthe varied sensitivities of the sensors to the current
sources at different depths 6)-7). Another popular class isthe adaptive spatial filter, which introduces thecovariance matrix of the measurements. A well-known
example of this class is the minimum-variance spatial
filter (MV) 8)-9). Sekihara et al. reported that MV exhibits
bias in the reconstructed location of a single source;                     Corresponding author: Wenxu Sun(e-mail: wxusun@gmail.com).

however, this bias can be eliminated using thenormalized lead field 10). In this study, lead fieldnormalization is used in MV, which is called the weighted
minimum-variance spatial filter (WMV).
Equivalent current dipole and distributed source
models are usually used for MEG/EEG source
localization 11)-13). An extended source may be suitable forthe prior estimation of PVC origin in MCG studies,according to the clinical reports of catheter ablation.
Moreover, a proper solution space needs to be determined.
In our previous study 14)-15), an unconstrained solution
space, with thousands of voxels that covered the humanheart, was used for MCG source localization, which
resulted in the appearance of an unexpected solution. In
this study, a realistic ventricular model was used as theallowable solution space. An extended source wasarranged at the surface of the ventricular model that
represented the endocardium and epicardium. Despite
the origin of PVC in the myocardium, the excitement can
propagate into the endocardium or epicardium rapidly.
Hence, the source was constrained at the surface of theventricle.
In this study, we aim to evaluate the accuracies of theinverse estimation using the WMN, MV, and WMV
methods based on an extended source and a realistic
ventricular model.
2. Methods
2.1 Source Model and MCG System
A realistic ventricular model, reported in our previous
study 16), was generated from computed tomography
images. A total of 530 nodes and 1064 faces weregenerated at the surface of the ventricular model using
the iso2mesh toolbox 17). Magnetic fields were simulated
to arise from an extended (about 1 cm) current sourcethat contains a center and 4–7 neighbor nodes. Because

Journal of the Magnetics Society of Japan Vol.46, No.5, 2022

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the excitement propagates along the surface and acrossthe ventricular myocardial, two dipoles in the tangential
and normal directions are arranged in each neighbornode. The MCG planar sensor array contains 64 channelsof the first-order gradiometer with a baseline of 5 cm. The
sensors are arranged in an 8 × 8 matrix with 25 mm pitch
and a 175 × 175 mm measurement area (Fig. 1).

̂𝑱𝑱 = 𝑲𝑲T (𝑲𝑲𝑲𝑲T + 𝜆𝜆2 𝑰𝑰)−𝟏𝟏 𝑩𝑩,

2(4)

where T denotes the matrix transpose, 𝜆𝜆 is theregularization parameter, and I is the identity matrix.
The MV spatial filter is a representative adaptive
spatial filter that introduces the measurement
covariance matrix R. The estimated current sources canbe written as
𝐓𝐓 −𝟏𝟏

𝑲𝑲 𝑹𝑹
𝑱𝑱̂ = 𝑲𝑲𝐓𝐓 𝑹𝑹−𝟏𝟏𝑲𝑲 𝑩𝑩.

(5)

̂𝑱𝑱 = 𝑲𝑲
̃ T (𝑲𝑲
̃ 𝑲𝑲
̃ T + 𝜆𝜆2 𝑰𝑰)−𝟏𝟏 𝑩𝑩,(6)

For bio-magnetic inverse estimation, lead fieldnormalization is usually used to compensate for thevaried sensitivity and modify the corresponding columnsof K, to make the sensitivity the same for all locations r.
Hence, the WMN spatial filter can be expressed as

̃ is the normalized lead field matrix 10).
where 𝑲𝑲
The WMV spatial filter method and can expressed as̃ T −𝟏𝟏

Fig. 1. MCG system and source model.
Because the MCG inverse problem is ill-posed, weaknoise signals can cause large changes in the solution. In
this study, Gaussian white noise was added to thesimulated magnetic field data, and the inverse solutions
were computed using the above three spatial filters. The
signal-to-noise ratio (SNR) ranges from 0 to 40 in 10 dB
increments. The SNR is defined as

SNR = 10 log

var(𝑩𝑩exact )
𝜎𝜎 2

,

 (1)

where Bexact is the variance of the simulated noise-free
measurements, and 𝜎𝜎2 is the variance of the added
Gaussian white noise.

𝑲𝑲 𝑹𝑹
𝑱𝑱̂ = 𝑲𝑲̃T 𝑹𝑹−𝟏𝟏𝑲𝑲̃ 𝑩𝑩.

2.3 Statistical Analysis
The source positions were varied within the surface ofthe ventricular model, and the MCG simulation andinverse estimation were repeated with different added
noise levels. The estimation error was the distance
between the position of the center node of the extended
source and the estimated result (Fig. 2). In this study, anestimation error of less than 2 cm is considered aneffective prediction. Hence, the ratios of all positions withan estimation error below 2 cm were used to evaluate theaccuracy of inverse estimation using the three types ofspatial filters.

2.2 Spatial Filters
The basic function of bio-magnetic forward problem isgiven by𝑩𝑩 = 𝑲𝑲𝑲𝑲 + 𝒏𝒏,(2)𝑱𝑱̂ = 𝑻𝑻𝑻𝑻.

(3)

where B is the vector of measured signals; K is the sensor
lead field matrix and can be obtained from the Biot–
Savart Law; J is the original current sources within theheart, and n is the additive noise. The estimated current
sources Ĵ can be estimated by a simple linear operator, as
follows.

Here, T is the spatial filter matrix (i.e., inverse operator).
For the well-known MN method, the estimated current
sources can be written as

98

(7)

Fig. 2. Definition of estimation error.

Journal of the Magnetics Society of Japan Vol.46, No.5, 2022

INDEX

3. Results
Fig. 3 shows the noise free estimation results of thetrue position at node index 6 (Fig. 3 (a)) and 437 (Fig. 3(b)), respectively. The blue point indicates the true
position of the center node of the extended source. The
yellow point indicates the position of the estimatedmaximum current strength. The three types of spatial
filters can estimate the source position with highaccuracy for node index 437, which is near the sensors.
However, for a relatively deep node, such as node 6,
which is far from the sensors, the WMN tends toconcentrate the solution in a shallow position, leading toa relatively large spatial extension. In contrast, MV andWMV spatial filters can both estimate the source position
with high accuracy and spatial resolution.
2
Generally, the regularization parameter 𝜆𝜆 is
introduced to solve the ill-posed problem. The proper2
value of 𝜆𝜆 is usually determined by the L-curve method
18). In this study, we evaluated the mean estimation error
through statistical analysis at varied regularization

parameters to determine the proper value. In addition,2
the regularization parameter 𝜆𝜆 can be evaluated by the
product of constant α and the sum of all elements inmatrix KKT. Fig. 4 shows that the estimation error
depended on constant α. The proper constants α of WMN
are 1.0 × 10-9 without noise, and 1.0 × 10-5, 1.0 × 10-4, 1.0
× 10-3, and 1.0 × 10-2 for noise levels of 40, 30, 20, and 10
dB. However, a relatively large constant (α = 1) would
lead to a low spatial resolution, despite showing higher
accuracy than α = 0.1. α seems to be stable below 0.001
and 0.01 for MV and WMV, respectively. Hence, in thisstudy, inverse estimation was performed with varied α
values at different noise levels for WMN. Moreover, α =0.001 was used for MV and WMV for the entire
experiment.
In the simulation experiment, the source positions
were varied within the surface of the ventricular model,
and the inverse estimation was repeated 530 times at
each noise level. The results of statistical analysis areshown in Fig. 5. Fig. 5 (a) and (b) indicate the meanvalues of the estimation error and ratios of all positions
with an estimation error below 2 cm at different levels,

Fig. 3. Estimation results of the true source located at (a) node 6 and (b) node 437 without noise.

Journal of the Magnetics Society of Japan Vol.46, No.5, 2022

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INDEX

respectively. For a relatively low noise level (SNR = 40dB), MV and WMV have 83% (MV), and 88% of all
positions with an estimation error below 2 cm.
Additionally, the mean values of the estimation error of
MV and WMV are 1.0 and 0.8 cm, respectively. However,
the ratio of all positions with an estimation error below 2
cm was only 67% and the mean value of the estimation
error was 2.1 cm for the WMN spatial filter. Conversely,

Fig. 4. Estimation error with different constant α.

under the condition of high noise level (SNR = 10~20 dB),
the accuracies of MV decreased rapidly, but WMVFig. 5. Statistical analysis results.

Fig. 6. Distribution of estimation error without noise.

100

Journal of the Magnetics Society of Japan Vol.46, No.5, 2022

INDEX

appeared stable. The mean value of the estimation error
was 1.1 cm, and more than 80% of all positions had anestimation error below 2 cm for WMV at 10 dB. Thus, MV
demonstrated stable accuracy at 20~40 dB. Furthermore,
the WMV method achieved the best performance in thisstudy.
4. Discussion
In this study, we evaluated the proper regularization
parameters by assessing the estimation error withdifferent constants α for the WMN, MV, and WMVspatial methods. Then, the mean value of the estimation
error and the ratios of all positions with an estimation
error below 2 cm were used to evaluate the performanceof the three types of spatial filters. MV and WMV showedthat they could estimate the source position with higher
accuracy than WMN. This is probably a result of using
the measurement covariance matrix R in the inverse
estimation procedure. Because the noise level is differentin every sensor, the introduction of the measurement
covariance matrix R into the inverse estimation can alsoenhance the stability of accuracy at different noise levels.
Fig. 6 shows the distribution of the estimation error for
WMN, MV, and WMV without noise. WMN cannot
estimate the source located far from the sensors. In
contrast, MV shows some areas near the sensors with low
accuracy. However, lead field normalization has solved
these problems in WMV. In other words, combining leadfield normalization with the measurement covariancematrix R, which is used in WMV, is advantageous for our
MCG system.
5. Conclusion
In this study, inverse source estimation from asimulated MCG with an extended source is conducted ina realistic ventricular model using WMN, MV, and WMVspatial filters. The performances of the three spatial
filters are evaluated with and without noise. The resultsof the statistical analysis handling varied source
positions show that WMV has the best performance for
MCG source inverse estimation. In the future, theeffectiveness of WMV method will be investigated byusing the clinical MCG data obtained from PVC subject
who was ablated successfully.
Acknowledgements This work was supported by Japan
Society for the Promotion of Science KAKENHI Grant
Number JP19K23616.
References
1) Morady F, et al.: Circulation, 82, 2093 (1990).
2) Kirsten Tolstrup, et al.: Cardiology, 106, 270 (2006).
3) Leahy RM, et al.: Electroencephalogr. clin. Neurophysiol, 107,
159 (1998).
4) Sunao Iwaki and Shoogo Ueno: J. Appl. Phys., 83, 6441 (1998).
5) Pascual-Marqui, R. D: Int. J. bioelectromagn., 1, 75 (1999).
6) Grech, R, et al.: J. NeuroEng. Rehabil., 5, 25 (2008).
7) Tamesh Halder, et al.: eNeuro., 6, 0170 (2019).
8) B. D. Van Veen, et al.: IEEE Trans. Biomed. Eng., 44, 867(1997).
9) Fernández-Corazza M, et al.: Int. J. Numer. Method Biomed.
Eng., 31, 2 (2015).
10) Kensuke Sekihara, et al.: NeuroImage, 25, 1056 (2005).
11) Rupali P. Dhond, et al.: J. Altern. Complem. Med., 14, 679(2008).
1300, 121 (2007).
12) Keith S. Cover, et al.: Int. Cong. Series,113) Kiebel SJ, et al.: Neuroimage, 39, 728 (2008).
14) W. Sun and K. Kobayashi: IEEE Trans. Mag., 53, 1 (2017).
15) W. Sun and K. Kobayashi: J. Magn. Soc. Jpn., 41, 75 (2017).
16) Takashi Ijiri, et al.: PLoS ONE, 7, e36706 (2012).
17) Qianqian Fang and David Boas: Proc. ISBI, 1142 (2009).
18) Per Christian Hansen and Dianne Prost O’Leary; SIAM J.
Sci. Comput., 14, 1487 (1993).
Received May 16, 2022; Revised Jun 09, 2022; Accepted Jul. 01,
2022

Journal of the Magnetics Society of Japan Vol.46, No.5, 2022

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INDEX

Editorial Committee Members・Paper Committee Members
T. Kato and S. Yabukami (Chairperson), K. Koike, K. Kobayashi and Pham NamHai (Secretary)
T. Hasegawa
K. Hioki
S, Inui
K. ItoK. KamataH. Kikuchi
S. Kokado
Y. Kota
T. Kouda
A. Kuwahata
S. Muroga
Y. Nakamura
H. Nakayama
T. Narita
K. Nishijima
D. Oyama
T. SatoT. Suetsuna
T. Takura
K. ThamN. Wakiya
T. Yamamoto
K. Yamazaki

Y. Kamihara
K. Masuda
T. Nozaki
T. Tanaka

N. Adachi
T. GotoH. KuraE. Shikoh
M. Toko

M. GotoT. Kojima
M. Sekino
K. Tajima
S. Yoshimura

H. AokiS. Honda
T. MakiK. Suzuki
S. YakataK. Bessho
S. Isogami
M. NaoeI. TagawaS. Yamada

M. DoiM. Iwai
M. OhtakeY. Takamura
A. YaoT. DoiY. Kanai
S. Seino
M. Takezawa
M. Yoshida

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Journal of the Magnetics Society of Japan
Vol. 46 No. 5 (通巻第 323号) 2022年9月1日発行

Vol. 46 No. 5 Published Sep. 1, 2022
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