Critical-state
and cylinder
magnetization
geometries
of type-11 superconductors
in rectangular
slab
T. H. Johansen and H. Bratsberg
Department of Physics, University of Oslo, P.0. Box 1048, Blindern, 0316 Oslo 3, Norway
(Received24 October 1994; acceptedfor publication 20 December 1994)
A scheme is described for analytical calculation of critical-state magnetization 11/1of
superconductorsin the geometry of long rectangularslabs and cylindrical specimensin a parallel
magnetic field. The simplicity of the general scheme is demonstratedby deriving compact
expressionsfor the ascending and descending field branches of M in the exponential model
Jc=jcO exp(-BIB,) and in the Rim, Hempstead,and Strnad model [Phys. Rev. 129, 528 (1963)],
Jc=jcol( 1 + BIBo). The analysesfocus on the vertical width AM of large field magnetization
hysteresis loops. While Bean’s result [Phys. Rev. Lett. 8, 250 (1962)-j,J,mAM, today is used
extensively to infer the critical current, it is well known that the method lacks consistencywhen a
field dependenceis seenin AM. For the two models it is shown explicitly that in the expansionof
the functional relation AM( J,), Bean’sresult correspondsto the lowest-orderterm. Also to the next
order in the functional expansionwe find a unifying form of expressingthe model behaviors.This
term containsthe secondderivative of J:(B) with a prefactor that dependson the samplegeometry.
A model-independentproof for the two first terms in the expansionof AM(J,) is also given, which
allows the significanceof size and shape,i.e., thicknessand aspectratio, to be discussedon a general
basis. New methods to extract J, from AM data, one of them without having to invoke specific
critical-state models, are indicated. 0 1995 American Institute of Physics.
I. INTRODUCTION
Whenever flux pinning centers are present in a type-II
superconductorthe distribution of penetrated flux will be
nonuniform and dependon the past magnetic history of the
specimen. To describe this behavior, Bean’ introduced in
1962 the critical-state model, which in recent years also has
been applied extensively to studies of high-temperaturesuperconductors.In this model one assumesthat the local current density, which by Ampere’s law is associatedwith the
nonuniform flux distribution, has a certain magnitude J,.
This critical current, which originally was taken as independent of the flux density, has in later refinementsbeen given
various dependenceson the local induction magnitudeB. Up
to now the most frequently used B-dependentcritical-state
functions are due to Kim, Hempstead,and Strnad,2
J,=
JcO
1 +BIBo’
and Fietz et a1.,3
J,= j,ge-B’Bn.
Here jc, and B, are positive parameters.See,e.g., Ref. 4 for
a review of J,(B) functions introducedin the literature. Even
more model functions are discussedin Refs. 5-7.
Measurementsof the irreversible behavior in magnetization has become a standardmethod to determineJ, and its
dependenceon B in a superconductingsample.On the other
hand, one has today no meansto derive J,(B) directly from
a given set of data for the vertical width AA4 of the magne
tization hysteresisloop. This is due to the fact that A.M is not
a function of J, , but rather a functional. For a given magnetized state the magnetization dependson an integral of an
J. Appl. Phys. 77 (8), 15 April 1995
a priori unknown function J,(B). Thus, in order to analyze
data one relies on the applicability of a chosenmodel function which also should be integrable.
Most commonly one uses the simple result of Bean’s
model, namely, that J, is proportional to AM, or
J,===AM/d,
(3)
where d is a length characteristic of the sample size and
geometry.A measuredfield dependenceof AM is then used
together with Eq. (3) to get the B dependenceof J,. This
method is not self-consistent,and errors are inevitable. The
inherent problem has been discussedby several authors,4.8
and it has been shown that Eq. (3) gives misleading results
for small fields if the sampleactually behavesin accord with
Eqs. (1) and (2).
A second method is to choose a particular function
J,(B), from which one computesAM and fits model parameters in a consistent manner.6V7
For this far more attractive
approachto be versatile, one needsthe solution of different
models in various geometriesat hand.
Based on the Kim and co-workers model, Chen and
Goldfarb4 derived analytically the magnetizationcurves of
an infinitely long sample of any rectangular cross section
placed in a parallel field. They also showed that the results
are applicable to other simple shapessuch as the circular
cylinder. In their treatment of A&f versus applied field H
they found that for this model Eq. (3) is modified according
to
AM(H)
7=J.‘H)[
l+s( ~j*++
where 2w is the samplethickness,and s is a numberdepending on the geometry.Chen and Goldfarb state that s = l/4 for
0021-8979/95/77(8)/3945/8/$6.00
D 1995 American Institute of Physics
3945
/
/
ILH
In this work we consideronly systems with isotropic
propertiesin the xy plane.As arguedin Ref. 4, a valid assumptionfor this caseis that the fiux penetratesequallyfrom
all sides.This implies that the current flows in a patternof
concentricrectangularloops in the xy plane, where each
loop is equidistantfrom the externalboundary.By definition,
the magnetizationis evaluatedby integratingthe magnetic
momentof theseloops divided by the total samplevolume.
This is expressedby
W
-X
a!
M=--T
/
an infinite slab and ~=I/20 for a cylinder, and conclude
thereforethat the error in using the simpler Eq. (3) is much
smallerfor the cylindrical case.
Applying the same schemeof calculation,Chen, Sanchez,and M u iiozglater derivedexpressionsfor the magnetization using the exponentialm o d e l. That article, however,
containsno analytical discussionof AM. The long expressionscontainingnumerousnesteddefinitionsof terms make
suchfurther analysisdifficult.
In the following sectionwe describea different scheme
of calculatingthe critical-statemagnetizationfor the general
slab and cylinder geometries.The schemereadily leadsto
compactformulas for the ascendingand descendingfield
branchesof the hysteresisloop. This is demonstrated
in Sets.
III and IV, where the two m o d e ls, Eqs. (1) and (2), are
treated. In the discussionof the results we focus on the
analysisof AM, which is the central quantity for comparisonswith experiment.For the Kim andco-workersm o d e lwe
confnm Eq. (4), althoughwith a different value for s in the
cylinder geometry.W e also derive a similar relation for the
exponentialm o d e l. A m a in result of this article is to show
that the secondterm in the expansioncan be expressedin a
unified way, i.e., written on a m o d e l-independent
form. In
Sec.V this result is generalized,and we derive the two first
terms in the expansionof the functional AM(J,), valid for
any critical-statefunction J,(B) proposedin the literature.
I
!
i
I
II. MAGNETIZATION
Consideran infinitely long superconductor
of rectangular crosssection.Let 2w and2w/a! denotethe short andlong
side of the rectangle,respectively.The parameterQ , which is
the aspectratio of the cross section, can have any value
within the range O=%Y=G~,
where rw=Ocorrespondsto the
infinite slab geometry.W e use a coordinatesystem aligned
with the sampleas shown in F ig. 1. The appliedmagnetic
field H is directedalong the z axis, and we defineB, from
H=(B,/&.
3946
J. Appl. Phys., Vol. 77, No. 8, 15 April 1995
(5)
4w- I 0 Ab)j,bW.
Herej,(x) denotesthe currentdensitycrossingthe x axis at
a given positive x, and A(x) is the area enclosedby the
currentloop,
Ajx)=4x
FIG. 1. Cross section of an infinitely long type-II superconductor placed in
a parallel magnetic field. In the critical state the supercurrent flows in concentric rectangular loops, each loop equidistant from the external boundary.
w
i-,+x
(-
i
.
Note that in Eq. (5) the sign of j, with respectto the y axis
correctly gives the sign of M .
W e want to determineM as function of appliedfield
whencycled througha m a jor hysteresisloop. As usualwhen
discussingcritical-statem o d e ls,we neglectthe lower critical
field and surface effects. F igure 2 illustrates then the sequenceof different situationsthat occur during sucha cycle.
Except for an interval immediatelyafter the field sweepdirection is reversed,which we o m it in the presentdiscussion,
the flux density profile will always be such that x(B) is a
single-valuedfunction whereverjY#O; thus, B can be used
insteadof x as integrationvariable.From Ampere’slaw,
j,dx=
- dBlpo ,
(7)
andEqs. (5) and (6), the magnetizationcanthereforebe written as
X[l+a(X-l)]dB.
p&f=- I BCl
BtIl
Here X=x/w, and B, is the induction at the m idpoint.
The profile function X(B) is obtainedby integrationof
Eq. (7), which gives
(9)
The current densityj,(B) has the m a g n itudeof the actual
critical-statefunction J,(B), and the sign is the oppositeof
that of the slopedBldx.
One seesthat in order to calculateM analyticallyin the
presentgeometrya requirementon the critical-statefunction
is that its inverseis twice integrable.The secondintegration
involvesthe squareof the first obtainedfunction, which representsthe inductionprofile. A m a jority of the critical-state
functions proposedin the literature satisfiesthis condition,
and in fact leadsto integralsthat are easily solved.
The aboveschemeof calculationis differentfrom that of
Chen et aL4,’ for the same geometry.The connectionbetweenthemis seenby integratingEq. (8) by parts.This gives
the alternativeformulas
T. H. Johansen and H. Bratsberg
as a dimensionlessparameter.Moreover, throughoutthis article, wheneverinduction symbols are typed in lower caseit
indicates that the quantity is normalized by Bo; we define
ba= B,IBo, etc.
Ill. EXPONENTIAL
A. Increasing
MODEL
field
Assumethat initially the superconductorhas beencooled
below T, in zero field. As the field is steadily increasedthe
penetratedflux will, in the part of the sample used in the
calculations,have a density profile with dBldx>O [see Fig.
2(a)]. Thus, jY= -J, where in this model J,(e) is given by
Eq. (2).
From EZq.(9) we obtain the profile function
Increasing
x+;
Ba
B”eB’~~odB’/Bo = 1 - $a - ,&y/k.
s6
03)
ln the first stageof magnetizationthe flux entersthe sample
only in an outer layer of thickness S. This penetrationdepth
is given by
b)
s=l-X(b=O)=(eb+l)lk.
(141
The initial stagelasts until b, reachesa value b, corresponding to complete penetration,or 6= 1. It follows that
Decreasing
BS
b,=ln(l+k).
(15)
1. O<b,sb6,
The virgin branch of the magnetizationcurve is found
from Eq. (8) setting the lower limit of integration equal to
zero. With X given by Eq. (13) one obtains
1+a
-[I-(1--b,)ebr]+$$l-4e”a
k
WOM
FIG. 2. Schematic picture of the local flux density distribution in the sample
along the positive x axis (see top) at various stages of a magnetization
hysteresis loop.
-=-b,+
Bo
+(3-2b,)e”ba].
(16)
By expandingthe expressionin powers of 6, one finds
poM=
-B,+
1
I
B(X)(2aX+
1 -a)dX,
0
(10)
POM
w--o-=.-b
+ ‘+a b;+...
a 2k
Bo
,
(17)
which shows that the initial slope of ,@4 vs B, equals - 1.
1 -a)dX.
ill)
One can benefit substantially by properly choosing the
schemeof integration accordingto the aim of the analysis.
As pointed out in Refs. 4 and 9 also geometriesother
than rectangularcan be directly included in such analyses.In
particular, the case a= 1, in this article, applies to both
squareand circular cross sections,with the cylinder diameter
being equal to 2w.
In the following two sectionswe apply Eqs. (8) and (9)
to derive M(B,) for the two important critical-state models,
Eqs. (1) and (2). In both caseswe define
2. b,ab,,
When the applied field exceedsthe full penetrationvalue
the midplane induction b,n, becomesnonzeroand dependent
upon b,. We find b, as the b satisfying X=0 in Eq. (13),
i.e.,
b,=ln(eba--k).
To obtain M the integrationbecomesthe sameas that in Sec.
III A 1 and with the nonzerob, we obtain
a
k = ,uunjcow/Bo,
J. Appt. Phys., Vol. 77, No. 8, 15 April 1995
(12)
(18)
o!
- 1- 2- + k eba.
T. H. Johansen and H. Bratsberg
3947
B. Decreasing field
The situations under discussion are illustrated in Fig.
2(b). When the appliedfield has beenloweredfrom its maximum valueby a sufficient amountthe induction gradienthas
changedsign throughoutthe sample.In these situationswe
havej,=+J,.
From Eq. (9) one sees immediately that changing the
sign of j, leadsto a new profile function given by Eq. (13)
with the substitutionk-+ -k, i.e.,
x= 1 -(eb-ebqk.
(20)
Again, settingX=0 we find now
I
b,=ln(eba+k).
(21)
Since k-+-k in both X and b, , the same substitutionapplies for the entire expressionfor M in Eq. (8). Thus, from
Eq. (19) we immediately obtain
(22)
3
I
-2
I
-1
I
I
ia
I
I
I
1
I
2
I
I
3
FIG. 3. Magnetization curves for the exponential model. The branches for
ascending (lower) and descending (upper) field are calculated for (~=l, 0.5,
and 0 (infinite slab). The virgin curves merge with the major loops at the
point of full penetration, b,=ln(l+k)=ln(2).
POM 1-ff
-=
--k--[1-(1-b,)cbm]-s[l-4ebm+(3
Bo
-2b,)e2bm]-
1+a
- k
[l-(l+b,)evba]
2. -b,,G b,GO
In this interval the induction becomesnegative in an
outer regionX>Xa [see Fig. 2(b), curve (IV)]. Since J, depends only upon the absolute value of the induction two
separatefunctions describethe profile. Note first that X0 satisfies
X0=1-6(jb,])=1+(1-e-b~)lk.
(23)
In the inner region, 0 =GXGXa, integrationof Eq. (7) yields
1
x(i) = __
-=-(ebm-eb)lk.
(24)
POW
The midplaneinduction is found using that X(‘)( b = 0) =X0,
which gives
b,=ln(2+k-e-‘a).
(25)
In the outer region, XaGXGl , Eq. (9) can againbe applied.
Since B<O we set j,,(B) =J,( - B) and obtain
(26)
The magnetizationintegral, Eq. (8),:now becomes
I
poM=-
’ X(‘)[ 1 + a(X(‘)-- l)]dB*,
I
0
and we find
3948
J. Appl. Phys., Vol. 77, No. 8, 15 April 1995
(27)
(28)
where b, is given by Eq. (25).
3. b&-b,
It is evident that in this range of b,, illustrated in Fig.
2(b) curve (V), the physical situation is the same as for
baa b, , except now the direction of the supercurrentis reversed.Indeed, the entire magnetizationloop possessesthe
symmetry M(-b,)=-M(b,).
In the specialcasesof cr=O (infinite slab) and CY=1 (cylinder) all our expressionsfor M coincide with the results
given in Ref. 8.
Fire 3 shows the ascending and descendingfield
branchesof the magnetizationloops obtainedfor the aspect
ratios a=O, l/2, and 1. The plot covers the range
-4b,~b,~4b,,
and we have set k=l. ‘By this choice one
can seethat our curvesagreewith Fig. 3(b) in Ref. 9 (Ref. 9
usesthe full penetrationfield and not B, as normalizingparameter).
C. Width of hysteresis
*Cl
X(O’[ 1
-t a(X@)- l)]dB,
-$[1-4e-b~+(3+2b,)e-2b~]-b,,
loop
In Fig. 4(a) we show how the vertical width, p&MIB,,
betweenthe branchesin Fig. 3 is reducedas the appliedfield
increases.When plotting the samedata with a logarithmicy
axis, as in Fig. 4(b), it becomesevident that the curvesform
a set of nearly perfect straight lines over the entire range
b,>b,.
This suggests that in this region AM to an
T. H. Johansen and H. Bratsberg
which indeedcan be expressedas
AM=(
1-~)wJ,(B,)[
l+&E(ke-ba)2+***
1
1’
(3 1)
It is clear that as 6, exceedsb, even the lowest-ordercorrectionterm very rapidly becomesnegligible.In fact, already
at b,= b,=ln(l fk), whereone can show that the total correction is maximum,the term has the value
m"
0.4
2
4
k*
1 5-30~
10 3-a (l+k)”
9
-~
0.2
0
2
I
3
ba
(a)
W ith k= 1 this amountsto l/24 for (Y=O,and even smaller
for a=1 where it becomesl/40. Note that since‘k2/( 1
+ k)*<l
the correction to the leading behavior
AM(B,)~J,(B,)
can never be large at b,=bp, and it always decaysexponentiallyfast as b, increases.
Becauseof the rapid convergencetoward a straight-line
behaviorit shouldin practicebe possibleto judge critically
from a log plot of experimentalAM datawhetherthe exponential m o d e lis a properchoice.Providedthat the measurements include a range abovethe full penetrationfield, and
that this m o d e l is a reasonableone, the high field part of a
log plot is fitted well by a straight line. The parametersjCo
and B. can then be found from the line’s slopeand intersection with the vertical axis accordingto
ln(AM)=ln[(l-cu/3)wj,a]-B,IBs.
(32)
If one finds a significantsystematicdeviationfrom such
behavior,other critical-statem o d e lsshouldbe considered.
IV. KIM AND CO-WORKERS MODEL
-0
(b)
2
1
3
A. Increasing field
In this m o d e l J,(B) is given by Eq. (1). The profile
function, the analogof Eq. (13), now becomes
ba
FIG. 4. (a) Magnitude of the hysteresis in magnetization for the exponential
model. The graphs represent the vertical separation of the magnetization
branches in Fig. 3. (b) Logarithmic plot of the curves in (a). The straight
lines appearing as distinct near b,=b,
are the high-field asymptotes. Note
the rapid convergence.
extremelygood approximationis proportionalto the exponential function J,(B,), with a proportionality factor depending on cr. To verify this analytically we use the
expressionsfor M found in Sets. III A 2 and III B 1, which
gives
X= 1 -[(b,+
l)*-(b+
1j2]/2k.
The full penetrationfield is found by setting X( b = 0) =O,
which gives
b,= Ji?k-
1.
(34)
1. Ocb,cb,,
By integrationof Eq. (8) one gets for the virgin magnetization
POM
-=-bo+F(
Bo
1+cY
lf
; b, ) b,2
(35)
By expandingin the quantity ke -ba we find
(ke-ba)3+*+*
J. Appl. Phys., Vol. 77, No. 8, 15 April 1995
(33)
, (30)
One sees directly that the initial slope of m
equals- 1.
T. H. Johansen and H. Bratsberg
vs B,
3949
2. b,ab6,
The Eq. (8) now gives
0.4
1+CY
POM
I-a
-=--b,-l+~(,b0+1)3Bo
3k
ib,+ 1J3
0.2
2ff
-IskZ~tb,+1)5--ib,+1)sl,
im
a?
where the midplaneinduction is
b,= &T,+ 1)2-2k-
0.0
2
1.
(3jj
I"
_
-0.2
B. Decreasing field
1. baa0
As in the treatmentof the exponentialmodel we can for
this region make use of the substitutionk~-- k, and from
the resultsof Sec. IV A 2, we immediatelyget
1+cY
POM
~=-h,-l-~(b~flj3+3k(b,+lj~
Bo
-04
I
-3
-2
-1
I
4
IO
I
1
2
I
I
3
1-a
2a
“-y-y-p [(b,+- ljSU(b,f
1F1,
(38)
FIG. 5. Magnetization curves for the Kim and co-workers model. As in Fig.
3, the branches are calculated for LY=1, 05, and 0 (infinite slab). Here the
virgin curves merge with the major loops at the point, b, = i/i?%
-1=\/5I.
where
b,= \I(&+ 1)2+2k- 1.
(39)
2. -bps b,<O
Following the procedureof Sec. III B 2 we find that this
time the midplaneinduction dependson 6, accordingto
b,= J2+2k--(b,-
1)2- 1.
The magnetizationbecomes
I-a
POM
-zz
6k
Bo
(3+2b,)b;-!-
1fCY
- 6k
60kS
cy (8b;+25bm+20)b;
(3-2b,)bz-
&
valid for b,&b, .
Expandingthis time in the quantity l/(1 + b,) we tind a
high degreeof term cancellation,and obtain
5-3ff
20
(8bz-25b,+20)b;
k3
(l+bJ5
+-.. ’
i43)
-b,.
(41)
The remainingpart of the magnetizationcurve againfollows from the condition M( - b,) = - M(b,). For both models we have verified analytically that the expressionscombine to give a continuousascendingand descendingbranch
in the magnetizationcurve.
C. Width of hysteresis
loop
The magnetizationfor ascendingand descendingapplied
field is shownin Fig. 5 for cu=O,0.5 and 1, againusing k=l.
The correspondinghysteresiswidth AM is plotted in Fig.
6(a) as a function of applied field. Also for the Kim and
co-workersmodel it is possibleto produceessentiallylinear
graphsabove the full penetrationfield; this is now obtained
by plotting l/AM as shown in Fig. 6(b).
To derive the analog of the expansionEq. (31) we subtract Eqs. (36) and (38) using the respectiveexpressionsfor
6,. The result is
3950
J. Appl. Phys., Vol. 77, No. 8, 15 April 1995
It follows that AM can be written as
AM--(~-T)wJ,(B,)~
l+$~(l~~,)~+...).
(44)
This result can be comparedwith the previously published
Eq. (4). One seesthat both the first and secondterm in the
expansionshave the samefield dependence,and we identify
3 5-3cY
s=syq-.
For the infinite slab case (cx=O)we get s-1/4 in agreement
with Chen and Goldfarb.” For cylinder geometry (a=l),
however, the above relation gives s=3/20, three times the
value given in Ref. 4. Thus, the benefit of using cylindrical
samplesin a Bean model type of analysisis not as dramatic
as previously thought.
Like in the exponentialmodel, we find that even the
lowest-ordercorrectionis very small. At b,= 6, it equals
T. H. Johansen and H. Bratsberg
PIG. 7. Two flux density distribution consistent with a given applied field.
BI(X) is the induction profile when B, is reached from above (descending
field), while Bt(X) is the profile when coming from the lower side (ascending field).
0.0 1
I
1
0
I
1
I
I
2
I
AM&)=(
1-+,(B,)(
3
d2
XdB” J;(&)+***
ba
(a)
l+;it,Ew2,;
a
1
(45)
>
valid for B,>B,.
We now show that this unified form in fact holds for any
critical-state modelsfor the presentgeometry.Let BT(X) and
B I (X) denotethe internal induction profile in the ascending
and descendingfield situationsillustrated in Fig. 7. The differencein magnetizationbetweenthe two statesequals
h
4
m”
,u~AM=
‘[LQ(X) --BT(X)](2cuX+ 1 - a)dX,
J0
(461
where we here have used the alternative magnetizationformula, Eq. (IO). Expanding the profiles in a Taylor series
about X= 1 gives
I
I
II
0
1
1
I
2
I
I
3
PIG. 6. (a) Magnitude of the hysteresis in magnetization for the Kim and
co-workers model. The graphs represent the vertical separation of the curves
in Fig. 5. (b) The field dependence of I/A&f. Plotted this way the high-field
asymptotes appear as straight lines near b,=b, also for this model. Like in
Fig. 4(b) the convergence is strong.
3 5-3a
20 3-cu
+$?;‘(l)(x-l)3+...
,
(47)
bs
(4
k2
(1-t2k)Z’
which amounts from l/36 (a=O) to l/60 (a=l) for k=l.
Again the rapid convergenceof the expansionjustifies the
use of a simple graphical test of applicability, similar to the
one suggestedfor the exponential model. This time l/AM
should be plotted against the applied field, and again if the
high-field behavior appearslinear, the model parametersj,
and B, are determinedby the fit of a straight line.
V. DISCUSSION
By inspectionwe seethat in both Eqs. (31) and (44) the
expansionfor AA4 can be written in terms of the critical-state
function J,(B) in the following way:
J. Appl. Phys., Vol. 77, No. 8, 15 April 1995
and a similar seriesfor BL(X). From Ampere’s law one has
B;(l) = powJ,(l) and B;(l) = -,~~wJ,(l), where in
this context J,(l) meansJ,(X= 1). It is readily seenthat the
secondderivatives of BT and BL becomeequal, also in sign,
and the terms cancel in Eq. (46). The third-order terms again
have oppositesign. Thus, one obtains
B~(X)-B~(X)=-2~~w~,(.l)(x-1)-~~w~~(1>
X(X-
1)3/3f***
.
(48)
Equation (46) then integratesto give
A,=(
1-;)w,(l,(
l+$-,~~+.+
(49)
This expansiongeneralizesa similar result obtainedby Fietz
and Webb.” It relates the magnetizationhysteresiswidth to
the local current distribution near the surfaceof the sample.
The secondterm in this expansiondependson the curvature
of the distribution profile. Note now that Ampere’s law
yields
T. H. Johansen and H. Bratsberg
3951
JkYw=(Pow)2J,
d
-&
Applied to the point where X=1, B=B,
tains
one thereforeob-
M=(l-a)M,fcU&,,
Togetherwith Eq. (49) this provesthe generalvalidity of Eq.
(45). Since the Taylor expansionis a good approximation
only as long as the induction profile is an analyticalfunction,
the above derivation is restricted to B,>B, due to the
nonanalyticityin the B dependenceof J, .
A nice feature of Eq. (45) is that the sample’sgeometry
and magneticproperties,representedby the function J,(B),
enter as separatefactors. This allows us to discussthe significanceof shapeand size on a generalbasis.
First, we note that the leading term representsthe Bean
model result derived in Ref. 4. The secondterm, which obviously vanishesin the Bean model, has an a-dependentcoefficient which monotoneouslyincreasesfrom 1 to 513as LY
is varied from 1 to 0. Thus, the geometricalaspectratio of
the sampledeterminesonly weakly the contribution of the
secondterm. Nevertheless,the deviation from the leading
order behavior will always be minimum for a sample of
squareor circular cross section. For the two models treated
in the previous sections,the graphs in Figs. 4(b) and 6(b)
illustrate this point.
It is clear that the relative weight of the secondterm in
Eq. (45) can be altered much more efficiently through the
quadraticfactor w2. Physically,this is due to the fact that the
secondterm representsdeviationsfrom a linear flux distribution. Such nonlinearitiessoon becomeless pronouncedthe
thinner the sample.
We saw in the previous sectionsthat in both modelsthe
size of the hysteresisloops is largestfor the infinite slab case,
a=O. Although this observationis consistentwith Eq. (45), a
generalargumentis most easily formulatedfrom Eq. (8). If
M, denotesthe magnetizationof an infinite slab of a given
thickness,the magnetizationof an equally thick finite slab
can be written
Ba
p&f = poM, + CY X( 1 -X)dB.
I Bl?t
(52)
Along the ascendingfield branch one has B,< B, , and the
integralis positive. For the descendingbranchB,> B, , and
the integral becomesnegative.Thus, when a is increased
from zero both branchesof M are shifted away from M,
toward smaller magnitudes,i.e., the loop shrinks monotonically with larger a’.
3952
J. Appl. Phys., Vol. 77, No. 8, 15 April 1995
Another point can be seen directly from Eq. (8). Together with the fact that a=1 includes the cylinder case
where diameter equals slab thickness,4it follows that the
magnetizationof a rectangularslab can be expressedas
(53)
where M,, is the magnetizationof the cylinder. Thus, the
generalslab has a magnetizationgiven by a weightedsum of
the results for two simpler geometricalcases.
Returningto the result, Eq. (45), we seethat the relation
between.J,(B,) and AM(B,) also can be written as follows:
1 5-3a
l-40
3-a
J,(B,) =
w2&
(54)
AM@,)
%w(l-a/3)
1 I-3a/5
1-z(1-cr13)3
i-4
(55)
This describeshow to go beyond the Bean model approximation when extractingJ, from AM datawithout invoking a
specific model. The very simple critical-state functions
J,(B) that one uses today are, with few exceptions,motivated on a phenomenologicalbasis.For future developments
of microscopically derived models, the possibility shown
here, to extract on a more generalbasis the function J,(B)
from experimentaldata, seemsto be important.
ACKNOWLEDGMENTS
The authors wish to thank Professor Y. Galperin for
stimulating discussions.The financial supportfrom the Research Council of Norway and the Nansen Foundationis
gratefully acknowledged.
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