PHYSICAL REVIEW B
VOLUME 58, NUMBER 10
1 SEPTEMBER 1998-II
Superconductor disks and cylinders in an axial magnetic field: II.
Nonlinear and linear ac susceptibilities
Ernst Helmut Brandt
Max-Planck-Institut für Metallforschung, D-70506 Stuttgart, Germany
~Received 14 November 1997!
The ac susceptibility x 5 x 8 2i x 9 of superconductor cylinders of finite length in a magnetic field applied
along the cylinder axis is calculated using the method developed in the preceding paper, part I. This method
does not require any approximation of the infinitely extended magnetic field outside the cylinder or disk but
directly computes the current density J inside the superconductor. The material is characterized by a general
current-voltage law E(J), e.g., E(J)5E c @ J/J c (B) # n ( B ) , where E is the electric field, B5 m 0 H the magnetic
induction, E c a prefactor, J c the critical current density, and n>1 the creep exponent. For n.1, the nonlinear
ac susceptibility is calculated from the hysteresis loops of the magnetic moment of the cylinder, which is
obtained by time integration of the equation for J(r,t). For n@1 these results go over into the Bean critical
state model. For n51, and for any linear complex resistivity r ac( v )5E/J, the linear ac susceptibility is
calculated from an eigenvalue problem which depends on the aspect ratio b/a of the cylinder or disk. In the
limits b/a!1 and b/a@1, the known results for thin disks in a perpendicular field and long cylinders in a
parallel field are reproduced. For thin disks in a perpendicular field, at large frequencies x ( v ) crosses over to
the behavior of slabs in parallel geometry since the magnetic field lines are expelled and have to flow around
the disk. The results presented may be used to obtain the nonlinear or linear resistivity from contact-free
magnetic measurements on superconductors of realistic shape. @S0163-1829~98!05534-9#
I. INTRODUCTION
In the preceding paper part I ~Ref. 1!, the current density
J, magnetic field B, and magnetic moment m(H a ) of superconductor cylinders of finite length or disks of arbitrary
thickness in an axial magnetic field B a (t)5 m 0 H a (t) ~along
y) are calculated by time integration of the equation of motion for the circulating current density J(r,t). This method,
which does not require any approximation or cutoff of the
magnetic field in the infinite space outside the cylinder, generalizes a similar calculation for strips, bars, or slabs in a
perpendicular field.2 The material is characterized by an arbitrary ~in general nonlinear! current–voltage law E(J),
where E is the electric field, which inside the superconductor
is generated by moving Abrikosov vortices. A further assumption, which cannot be relaxed so far, is B5 m 0 H, which
means that the lower critical field B c1 of the superconductor
and its reversible magnetization B2 m 0 H are assumed to be
zero. This approximation is allowed when B is large enough
everywhere in the specimen.
In the present paper the amplitude-dependent ac susceptibility of finite cylinders is calculated from magnetization
loops which are computed by the method of part I.1 In the
case of linear resistivity E/J5 r ac( v ), the linear ac susceptibility is calculated more elegantly from an eigenvalue problem which depends only on the geometry and therefore
yields a general expression into which any linear complex
resistivity r ac( v ) may be inserted. Before these two types of
susceptibilities are considered in Secs. III and IV, first some
exact analytical expressions for both nonlinear and linear ac
susceptibilities will be compiled in Sec. II for rings, hollow
cylinders, infinite slabs, infinite cylinders, and spheres. The
particular case of Ohmic conductors, or superconductors
0163-1829/98/58~10!/6523~11!/$15.00
PRB 58
with thermally assisted flux flow ~TAFF!, is considered in
Sec. V. Finally, some concluding remarks are made in Sec.
IV.
II. ANALYTICAL ac SUSCEPTIBILITIES
In this section I compile some analytical expressions for
nonlinear and linear ac susceptibilities of superconductors
and general linear conductors in various geometries. Within
the Bean critical state model,3 the full hysteresis loop of the
magnetic moment m in a cycled applied field with amplitude
H 0 is completely determined by the virgin magnetization
curve m(H a ). One has the general prescription4–6
m ↓ ~ H a ,H 0 ! 5m ~ H 0 ! 22m
S
m ↑ ~ H a ,H 0 ! 52m ~ H 0 ! 12m
D
H 0 2H a
,
2
S
D
H 0 1H a
,
2
~1!
were m ↓ and m ↑ are the branches in decreasing and increasing H a . This construction is independent of the type of time
dependence of H a (t), provided H a increases and decreases
monotonically between the values 6H 0 . For sinusoidal
H a (t)5H 0 sinvt, one may define the nonlinear complex ac
susceptibilities x m 5 x m8 2i x m9 , m 51,2,3, . . . , 7
x m~ H 0 , v ! 5
i
pH0
E
2p
0
m ~ t ! e 2i m v t d ~ v t ! .
~2!
Usually, the x m are normalized such that for H 0 →0 or v
→` the ideally diamagnetic susceptibility x (0,v ) results;
this normalization is achieved by dividing all x m , Eq. ~2!, by
the
magnitude
of
the
initial
slope
u m 8 (0) u
6523
© 1998 The American Physical Society
6524
ERNST HELMUT BRANDT
PRB 58
5limH a →0 u ] m(H a )/ ] H a u . In this paper only the fundamental susceptibility ( m 51) will be considered, denoting x 1 by
x.
Note that in the critical state model this x depends only
on the amplitude H 0 but not on the frequency v . For thin
circular Bean disks x (H 0 ) was calculated from the magnetization curves of Refs. 6 and 8 in Refs. 6 and 9.
For a thin-walled hollow cylinder and for a narrow planar
ring the hysteresis loop is a parallelogram.4,10,11 This parallelogram applies to planar rings with arbitrary shape, made
from flat or round wires, containing no, one, or several weak
links, with the weakest one determining the critical current
I c . This is so since at small H a the screening is perfect and
no magnetic flux penetrates the ring or tube. But when H a
exceeds some critical field H p , the screening current in the
ring exceeds the critical value I c and magnetic flux can leak
into the ring such that the circulating current u I u 5I c and the
magnetic moment u m u 5m sat remain constant. The full parallelogram then follows from Eq. ~1! and the complex ac susceptibility of the ring normalized to the initial value
x (0)521 is4
becomes also complex @the measured moment is Re$ m(t) % ]
and the linear ac susceptibility is then defined as x 5 m 21,
x 8 ~ h ! 521,
l ac5 ~ r ac /i v m 0 ! 1/2
x 9 ~ h ! 50,
1 1
1
x 8 ~ h ! 52 2 arcsins1 s A12s 2 ,
2 p
p
h>1,
1
pH0
E
2p
0
m ~ t ! e 2i v t d ~ v t ! .
~4!
The linear susceptibility x ac( v ) or conductivity s ac( v )
51/r ac( v ) is obtained for various geometries by replacing in
the analytical expressions for x ac the Ohmic resistivity r by
the complex r ac( v ). For example, by solving the diffusion
equation for the magnetic field, with the diffusion constant
D5 r / m 0 replaced by the complex D5 r ac / m 0 , one obtains
for slabs of width 2a in a parallel ac field H a (t)
5H 0 exp(ivt) the permeability m 5 x 115 ^ H ac(x,t) & x /
H a (t) and the susceptibility x 5 m 21,21,22
x slab~ v ! 5
tanhu
21,
u
u5
a
,
l ac
~5!
where
~6!
is the complex penetration depth or r ac( v )5i v m 0 l 2ac . For a
long cylinder with radius a in an axial field one finds23 for
x 5 m 215 ^ H ac(r,t) & r /H a (t)21,
h<1,
4 h21 12s 2
x 9~ h ! 5
5
,
p h2
p
x~ v !5
~3!
where h5H 0 /H p and s52/h21. Note that the dissipation
~proportional to x 9 ) sets in only at amplitudes exceeding the
penetration field H p . The polar plot of Eq. ~3!, x 9 versus x 8
with h as the parameter, is symmetric; i.e., x 9 ( x 8 ) yields the
same curve as x 9 (212 x 8 ).
The ac susceptibilities obtained from the critical state
model are quasistatic; i.e., they depend only on the amplitude H 0 but not on the frequency. This statement is true even
when the critical current density J c 5J c (B) depends on the
local induction and thus Eq. ~1! does not apply, e.g., for the
magnetization loops computed in Refs. 12,4 and 13 for disks
with J c (B)5J c (0)/(11 u B u /B 1 ) ~Kim model!.
In contrast to these nonlinear quasistatic ac susceptibilities x (H 0 ), the linear ac susceptibilities of materials with
linear resistivity, r ac( v ), depend only on the ac frequency
but not on the ac amplitude H 0 . Examples for this are the
Ohmic conductors with real and constant resistivity r ac( v )
5 r , and superconductors in the Meissner state with London’s magnetic penetration depth l, which formally are described by a purely imaginary ~nondissipative! resistivity
r ac5i v m 0 l 2 . More complicated complex linear resistivities
r ac5 r 8 1i r 9 occur in superconductors with thermally activated depinning above the irreversibility line ~or depinning
line! in the B-T plane; see, e.g., the models of Refs. 14–20
and the experimentally determined r ac in Refs. 19 and 20. In
complex notation one usually writes H a (t)5Re$ H 0 e i v t %
where Re$ ••• % denotes the real part and H 0 is a complex
amplitude, e.g., H 0 52i u H 0 u if H a (t)5H 0 sinvt is chosen to
obtain Eq. ~2!. The magnetic moment m(t) then formally
x cyl~ v ! 5
2I 1 ~ u !
21,
uI 0 ~ u !
u5
a
,
l ac
~7!
with the same l ac , Eq. ~6!; I 1 (u) and I 0 (u) are modified
Bessel functions.
Remarkably, formula ~7! applies also to cylinders in
transverse field and thus, due to the linearity, to any inclined
field H a (t) forming an angle u with the cylinder axis. The
axial and circulating current components in the tilted infinite cylinder exhibit the same radial dependence
f (r)5(H a /l ac)I 1 (r/l ac)/I 0 (a/l ac):
J i 52coswsin u f(r),
Jw5cosu f(r), where w is the azimuthal angle. The magnetic
moment of cylinders in a transverse H a is twice as large as in
a parallel field; this may be ascribed to the demagnetization
factor N51/2 or to the contribution of the U turn of the
currents at the ends of the cylinder. Thus u mu contains a
factor (cos2u14sin2u)1/2, which, however, drops out in the
normalized susceptibility ~7!.
For the sphere one may use London’s solution24 for the
magnetic moment of a sphere with radius a in the Meissner
state,
S
m522 p H 0 a 3 12
D
a 3l 2
3l
coth 1 2 .
a
l
a
~8!
Defining x 5m(l)/ u m(l→0) u and replacing the real l by
the complex l ac , Eq. ~6!, one obtains the susceptibility of a
sphere with arbitrary complex resistivity r ac( v ) and radius
a,
x sphere~ v ! 5
3cothu 3
2 2 21,
u
u
u5
a
.
l ac
~9!
PRB 58
SUPERCONDUCTOR DISKS AND . . . . II. . . .
For the susceptibility of thin disks or strips in perpendicular
field, analytical expressions are not available. However, one
may express the linear x ac( v ) for any geometry by an infinite sum of the form
x ~ v ! 52w
L n b 2n
(n w1L n
Y(
n
L n b 2n .
~10!
Here the complex variable w depends on r ac and v and on
the geometry, and the dipole moments b n and eigenvalues
L n follow from an eigenvalue problem which is different for
each geometry; see the examples for thin strips25,26 and
disks20,25 and thick strips and bars in a perpendicular field,27
for rectangular bars in a parallel field,28 and for finite cylinders in an axial field, below in Sec. IV. The sum in the
denominator of Eq. ~10! provides the normalization x ( v
→`)521 ~ideal diamagnetic screening!.
For practical purposes, a finite number of terms n
51, . . . ,N in the sum ~10! is sufficient; cf. Sec. IV. When
the ~real and positive! numbers L n and b n are known for a
given geometry, then x ( v ) may be calculated from the sum
~10! for any complex resistivity r ac( v ). By inverting this
relationship between the two complex functions x ( v ) and
r ac( v ) numerically, the complex resistivity r ac( v ) may be
obtained from measured ac susceptibilities as done by Kötzler et al.20
III. NONLINEAR ac SUSCEPTIBILITIES
We model the superconductor as a nonlinear conductor
with general current-voltage dependence E5E(J)J/J, e.g.,
E(J)5E c (J/J c ) n with arbitrary creep exponent n>1. In addition; B5 m 0 H is assumed, see the discussion in Sec. V of
part I.1 The nonlinear resistivity for n@1 is caused by thermally activated depinning with an activation energy U(J)
5U 0 ln(Jc /J) yielding a creep exponent n5U 0 /kT; see Sec.
III D of part I. In axial symmetric geometry, i.e., for cylindrical specimens in an axial magnetic field along ŷ, the currents, electric field, and vector potential are directed along ŵ ,
where
w 5atan(z/x),
namely,
J5J(r,y,t) ŵ , E
5E(r,y,t) ŵ , and A5A(r,y,t) ŵ . In particular, the applied
field is now Ba5B a (t)ŷ5¹X(A a ŵ ) with A a 52(r/2)B a .
The components of the magnetic field are B r 5 ] A/ ] y and
B y 52(1/r) ] (rA)/ ] r. During our computation of J and
m, B r and B y do not have to be calculated except when one
assumes a B-dependent current-voltage law, e.g., of the form
E(J,B)5E c @ J/J c (B) # n(B) . In general, E(J,B r ,B y ) may depend on both components of B, e.g., when J c (B r ,B y ) is
anisotropic like in c-axis-oriented cylinders of uniaxial highT c superconductors. Our method applies also to these anisotropic materials.
The equation of motion for the current density J(r,y,t) in
cylinders or disks is given in part I.1 On a two-dimensional
~2D! grid of N points ri 5(r i ,y i )(i51, . . . ,N) spanning the
quarter cross section 0<r<a, 0<y<b ~or half cross section 0<r<a, 2b<y<b if the specimen has no symmetry
plane at y50) this equation of motion reads
F
G
1
m 0 J̇ i ~ t ! 5 ( R i j E ~ J j ! 2 r j Ḃ a .
2
j
~11!
6525
Here the vector J i (t)5J(ri ,t) and the applied field B a (t)
depend on the time t, E(J) is a given function, e.g., E
5J n in reduced units, and R i j 5(Q i j w j ) 21 is a reciprocal
matrix into which enters the matrix Q i j 5Q(ri ,r j ) defined in
part I and the weights w i of the grid points ~e.g., w i
5ab/N for equidistant grids!. Throughout this paper, for
increased accuracy I use nonequidistant grids which are
denser ~and have thus smaller weights! near the cylinder surface.
Equation ~11! is easily time integrated by starting at time
t50 with B a 50 and J i 50 and then increasing the applied
field B a (t) to obtain the solution J i (t). From the current
density J i (t) the magnetic moment m(t) of the cylinder ~directed along y like B a ) is obtained as
m ~ t ! 52 p
E
a
0
dr r 2
E
b
0
N
dy J ~ r,y,t ! 52 p
( r 2i J i~ t ! w i .
i51
~12!
In this section B a (t)5 m 0 H 0 sinvt is chosen; the nonlinear ac
susceptibility then follows from Eq. ~2!.
The resulting magnetization loops m„B a (t)… become stationary shortly after 1/4 cycle or even earlier; i.e., the transition from the virgin curve to the final hysteresis loop occurs rapidly ~see Figs. 14–17 in part I!. This is a
consequence of the nonlinear E(J) relation; in the Ohmic
case with E}J the transient time is somewhat larger and the
stationary loop is reached exponentially in time. For exponents n>2 it is thus sufficient to calculate x from the half
loops in the time interval 0.55p < v t<1.55p .
Without restriction of generality we may choose the circular frequency v 51 corresponding to v 5E c /( m 0 J c a 2 )
when reduced units a5J c 5E c 5 m 0 51 are used. The scaling law of part I, Sec. III E, states that when a material law
E}J n is assumed, then the susceptibility x (H 0 , v ) depends
only on combinations of the form H 0 / v 1/(n21) or v /H n21
or
0
on any function of these ratios. Therefore, the x (H 0 , v ) for
different frequencies v are obtained by rescaling the amplitude axis. For example, one has x (H 0 ,10v )
5 x (101/(n21) H 0 , v ). This scaling to a good approximation
applies also to other E(J) laws if these are sufficiently nonlinear and if the effective exponent n is defined as n
5 ] (lnE)/](lnJ) taken at J5J c where J c is the typical current
density of the experiment. The nonlinear susceptibility thus
depends only on one variable combining amplitude and frequency, further on the effective exponent n, and on the geometry expressed, e.g., by the aspect ratio b/a of the cylinder.
Figures 1–6 show the nonlinear susceptibilities x (H 0 , v )
at v 5E c /( m 0 J c a 2 ) for various geometries, for creep exponents n53, 5, 11, 51, and for constant J c and B-dependent
J c (B), using various types of plots. The real and imaginary
parts of x 5 x 8 2i x 9 of cylinders with half length to radius
ratio a/b53, 0.3, and 0.03 in an axial magnetic field are
plotted in Fig. 1 versus log(H 0 /H p ), in Fig. 3 versus
H 0 /H p , and in Fig. 5, x 9 is plotted versus 2 x 8 ~polar plot!.
Here H 0 is the amplitude of the applied ac field H a (t)
5H 0 sinvt and H p is the field where full penetration occurs
in the Bean model. For bars in a perpendicular field and
cylinders in an axial field, both with rectangular cross sections 2a32b, one has2,29
ERNST HELMUT BRANDT
6526
FIG. 1. Nonlinear magnetic susceptibility x (H 0 , v )5 x 8 2i x 9
of cylinders with aspect ratios a/b53, 0.3, and 0.03 in an axial ac
magnetic field with frequency v 5E c /( m 0 J c a 2 ) and amplitude H 0
referred to the field of full penetration H p , Eq. ~14! (H p /J c a
50.9824, 0.5757, 0.1260 for b/a53, 0.3, 0.03!. Depicted is x 8
~monotonic curves! and x 9 ~curves with maximum! on a semilogarithmic plot for creep exponents n53 ~solid lines!, n55 ~long
dashed lines!, n511 ~medium dashed lines!, and n551 ~short
dashed lines! for constant J c . The curves n551 are close to the
Bean limit.
F
S DG
F S DG
2
b
a
b 2a
arctan 1ln 11 2
H p 5J c
p b
a
b
~ bar! ,
~13!
~ cylinder! .
FIG. 2. Similar as in Fig. 1 but for a cylinder with b/a51 and
J c 5const ~top, H p /J c a50.8814), J c (B)5J c (0)/(112B/ m 0 H p )
~middle, Kim model!, and for a bar with b/a51 and J c 5const
~bottom, H p /J c a50.7206). Four creep exponents n53, 5, 11, and
51 are shown, as in Fig. 1.
M ~ H 0 ! 5 a H 0 1 b H 20 1 g H 30 1•••,
~14!
This gives H p /J c a50.9824 (0.8814, 0.5757, 0.1260) for
cylinders with b/a53 (1, 0.3, 0.03), and H p /J c a
50.7206 for a bar with b/a51. In Figs. 2, 4, and 6 similar
x 8 and x 9 are plotted for a cylinder with b/a51 and J c
5const or J c (B)5J c (0)/(112B/ m 0 H p ) ~Kim model! with
B5 u B(x,y) u , and for a bar with b/a51. Note that the depicted curves look qualitatively similar for all six cases. The
slight oscillation of some of the curves with large b/a is an
artifact due to a too small number N x of grid planes along x
~or cylinders along r); e.g., in these figures nonequidistant
grids were used with N x 59, N y 520 for b/a53 and N x
530, N y 53 for b/a50.03.
In the Bean limit n→`, the general behavior of x 8 and
x 9 at small and large amplitudes H 0 is explicitly known.
Expanding the virgin magnetization curve at small H 0 ,
~15!
one obtains, from Eqs. ~1! and ~2! x 5 x 8 2i x 9 ,
15
g H 20 1•••,
16
~16!
4
2
b H 0 1 g H 20 1•••.
3p
p
~17!
x 8~ H 0 ! 5 a 1 b H 01
x 9~ H 0 ! 5
2 1/2
a
a
H p 5J c bln 1 11 2
b
b
PRB 58
In particular, using the virgin curves of part I, Sec. II, one
obtains the following relations. For long cylinders and slabs
in a parallel field the coefficients in Eq. ~15! are a ,0, b
.0, and g <0; thus when x is normalized to x (H 0 50)
521, both 12 u x 8 u and x 9 start linearly with a term
} b H 0 / u a u . This result applies also to our numerically obtained x (H 0 ) of cylinders of finite lenght. Thus, in the polar
plots of Figs. 5 and 6, the curves x 9 (2 x 8 ) start linearly in
the lower right corner. Explicitly one has, up to terms linear
in h 0 5H 0 /H p in longitudinal geometry,
1
2
x ~ H 0 ! 5211 h 0 2i
h 1•••
2
3p 0
x ~ H 0 ! 5211h 0 2i
~ slab! ,
~18!
4
h 1••• ~ cylinder! .
3p 0
~19!
SUPERCONDUCTOR DISKS AND . . . . II. . . .
PRB 58
FIG. 3. As in Fig. 1 but with a linear abscissa.
6527
FIG. 4. As in Fig. 2 but with a linear abscissa.
For thin strips and disks in a perpendicular field one has a
,0, b 50, and g .0; i.e., there is no quadratic term in the
virgin magnetization curve @cf. Eqs. ~11! and ~14! in part I#.
Explicitly, for the thin strip one has m(H a )}tanh(h)5h
2h3/31O(h 5 ) with h5H a /H s and H s 52J c b/ p ; thus b
50 and g 52 a /(3H 2s ). For the thin disk the expansion of
the bracket in Eq. ~11! of part I yields 2h2h 3 1O(h 5 ) with
h5H a /H d and H d 5J c b; thus b 50 and g 52 a /(2H 2d ).
With these g / a values the small-amplitude expansion of x
yields
x ~ H 0 ! 5211
S
2i H 20
5
2
1O ~ H 40 ! ~ strip! ,
16 3 p H 2s
S
15 i H 20
2
1O ~ H 40 ! ~ disk! .
32 p H 2d
x ~ H 0 ! 5211
D
D
~20!
~21!
The slope of x 9 (2 x 8 ) in the lower right corner of Figs. 5
and 6 in the Bean limit should thus take the universal values
x9
11 x 8
U H
5
x 9 →0
4/3p 50.42
for b Þ0,
32/15p 50.68
for b 50,
~22!
where the case b 50 corresponds to thin films in a perpendicular magnetic field and b Þ0 to all other geometries.
These slopes are approximately reached in the depicted case
n551, but for stronger creep (n511, 5, 3) the slopes are
larger. Note also that our assumption B5 m 0 H requires that
FIG. 5. The data of Fig. 1 plotted as x 9 vs 2 x 8 . The open
circles in the lower plot show the fit, Eq. ~24!.
ERNST HELMUT BRANDT
6528
PRB 58
directly solve the original equation for the current density in
a bar @Eq. ~6! or ~44! in Ref. 2# or cylinder @Eq. ~24! in Ref.
1#. Inserting the periodic time dependences H a (t)
5H 0 e i v t , J(r,t)5J(r)e i v t , and E(r,t)5E(r)e i v t , where
now formally H 0 , J(r), and E(r)5 r ac( v )E(r) are complex amplitudes @as usual, the physical quantities are obtained by taking the real part of H a (t), J(r,t), and E(r,t)],
one obtains, for the cylinder in an axial ac magnetic field the
integral equation for J(r),
r ac~ v !
J ~ r! 5
ivm0
E
r
Q cyl~ r,r8! J ~ r8! d 2 r 8 1 H 0 .
2
S
~25!
Here r5(r,y), r85(r 8 ,y 8 ), and S5ab is the integration
area ~a quarter of the clinder cross section 2a32b). The
kernel Q cyl(r,r8) is defined by Eq. ~21! of part I. Equation
~25! is related to the eigenvalue problem
E 8E
8
4p
f n ~ r! 52L n
S
a
b
dr
0
0
dy 8 Q cyl~ r,r8! f n ~ r8! ,
~26!
with positive eigenvalues L n and real eigenfunctions
f n (r,y), n 51,2, . . . ,`. For cylindrical symmetry we may
normalize the f n (r) as25
E E
8
4p
S
FIG. 6. The data of Fig. 2 plotted as x 9 vs 2 x 8 .
the amplitudes be not too small, H 0 @H en , where H en is the
field above which the penetrating vortices can overcome the
geometric edge barrier.31,32
At large amplitudes H 0 @H p the magnetization loop in the
Bean limit may be approximated by a parallelogram for all
geometries. Therefore, x (H 0 ) is given by expressions of the
type of Eq. ~3!; see also Ref. 10. In the limit H 0 @H p one has
then x 8 (H 0 )}H 23/2
and x 9 (H 0 )}H 21
The curves
0
0 .
x 9 (2 x 8 ) in Figs. 5 and 6 in the lower left corners thus
behave as
x 9 5const3 u x 8 u 2/3.
~23!
For all computed cylinders in the Bean limit we find for the
constant in Eq. ~23! the value 0.80. A very good fit for all
values of 0<2 x 8 <1 for the cylinder with b/a50.03 in the
Bean limit is
x 9 '0.80u x 8 u 2/3~ 12 u x 8 u ! 1.19,
~24!
depicted as dots in the lower plot of Fig. 5. Similar fits are
possible for the other cylinders and the bar, but for strong
creep (n<11) and nonconstant J c (B) the limit ~23! does not
apply.
IV. LINEAR ac SUSCEPTIBILITIES
If the resistivity r 5E/J of the material is linear, the magnetic response is also linear and may be expressed in the
most general form by a linear complex susceptibility x ( v )
which depends on the linear complex ac resistivity r ac( v ).
In this case it is not necessary to time integrate the equation
of motion ~11! containing the inverse kernel R i j . One may
a
b
dr r
0
0
dy f m ~ r! f n ~ r! 5 d m n .
~27!
~A different normalization introducing the symmetric kernel
Q(r,r8) Ar/r 8 is used in Ref. 30.! The arbitrary area S 8 in
Eqs. ~26! and ~27! was introduced to make the L n and f n
dimensionless. Natural choices of S 8 are7,27 S 8 5S54ab in
the limit b@a ~long cylinder! and S 8 516a 2 / p in the limit
b!a ~thin disk!; see below.
Expanding J(r) into a series of the eigenfunctions,
(n a n f n~ r! ,
J ~ r! 5
~28!
we obtain from Eqs. ~25!–~28! the magnetic moment
m(t)5m( v )e i v t , Eq. ~12!,
m ~ v ! 52 p
E
a
dr r 2
0
E
b
0
dy J ~ r,y ! 5
S8
2
(n a n b n ,
~29!
where we have defined the ‘‘oscillator strengths’’
b n5
4p
S8
E
S
r 2 f n ~ r,y ! d 2 r.
~30!
Multiplying Eq. ~25! by f m (r) and integrating r over S on
both sides using formulas ~26! and ~27!, one obtains the expansion coefficients
a n 5H 0
2 p wL n
bn
S 8 w1L n
~31!
and the magnetic moment ~29! becomes
m ~ v ! 5H 0 p w
L n b 2n
(n w1L n .
~32!
PRB 58
SUPERCONDUCTOR DISKS AND . . . . II. . . .
The results ~31! and ~32! depend on the complex variable
w5
i v m 0S 8
S8
5
5i v t ~ v ! ,
4 pr ac~ v ! 4 p l 2ac
~33!
which combines the complex resistivity r ac( v ) with the frequency v and with the arbitrary area S 8 appearing also in the
definitions ~26!, ~27!, and ~30!. This w may also be written in
terms of the complex penetration depth l ac5(i v m 0 / r ac) 1/2;
cf. the variable u5a/l ac in Eqs. ~59!, ~7!, and ~9!.
With formula ~33! we may express the linear ac susceptibility x ( v )52m( v )/m( v →`)5 m ( v )21 of cylinders in
axial field in the form of Eq. ~10!,
x ~ v ! 52
wc
(n w1Ln n ,
c n 5L n b 2n
m~ v !5 (
n
Y(
n
c nL n
,
w1L n
L n b 2n .
~34!
~35!
Here w is the variable ~33!, L n are the eigenvalues of Eq.
~26!, and b n are the integrals ~30! over the eigenfunctions of
Eq. ~26!. The normalization in Eq. ~35! guarantees that for
v →` one has x →2 ( n c n 521 and m →0, irrespective of
the values of L n and b n . With the L n and b n computed from
Eqs. ~26!, ~27!, and ~30!, the limit m( v →`) exactly coincides with the ideal diamagnetic moment of cylinders
m 8 (0), Eq. ~42! of part I. Sum rules for the b 2n , b 2n L n , and
b 2n /L n for thin strips and disks are given in Ref. 25.
Examples for the normalized amplitudes c n and eigenvalues ~or pole positions! L n are given in Table I for cylinders
with aspect ratios b/a50.15, 0.3, 1, and 3. For this Table I
chose S 8 54ab; thus one has t ( v )5 m 0 ab/ @ pr ac( v ) # in
Eq. ~33!. The grids ri 5(r i ,y i ) were chosen to be nonequidistant ~denser near the cylinder surface! as decribed below
Eq. ~29! of part I. For transparency a constant number of grid
points was used, N5N x 3N y 536 with N x 3N y 51233, 9
34, 636, and 439. The obtained L n and c n were sorted
for increasing L n ; at large n these positive numbers are very
sensitive to the choice of the grid, but the resulting x ( v ) is
less sensitive. The x ( v ) calculated from Table I are valid
from v 50 up to at least v t 5200. For higher frequencies
more grid points are required. At extreme aspect ratios b/a
!1 and b/a@1 the N3N matrix Q i j 5Q cyl(ri ,r j ) ~cf. part
I! of the eigenvalue problem may become singular, or some
of the eigenvalues artificially may turn out to be negative,
depending on the choice of the grid.
The finite number N of grid points results in the same
finite number of eigenvalues, n 51,2, . . . ,N. At small b/a a
nearly equidistant series of L n results, which is typical for
thin strips and disks in a perpendicular field; see Eqs. ~48!
and ~49! below. The rapid increase of the L n as the index n
approaches N is an artifact caused by the finite N. With
increasing aspect ratio b/a, L n and c n arrange in groups
which for b/a→` degenerate and coincide with the result
for the long cylinder; cf. Eq. ~47! below.
V. OHMIC AND TAFF RESISTIVITY
In the Ohmic case or for thermally assisted flux flow
~TAFF! ~Ref. 22! in superconductors, the resistivity r ac5 r
6529
TABLE I. Eigenvalues L n ~sized! and amplitudes c n entering
the general linear ac susceptibility x ( v ), Eq. ~34!, of cylinders with
radius a and length 2b in an axial magnetic field. In all three examples nonequidistant grids with N5N x 3N y 536 points were
used, with N x 3N y 51233, 934, 636, and 439 for b/a
50.15, 0.3, 1, and 3. For this table S 8 54ab was used, which
means t ( v )5 m 0 ab/ @ pr ac( v ) # in Eq. ~33!.
b/a
n
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
0.15
Ln
1.125
3.092
5.900
9.621
14.25
19.62
25.19
25.85
29.11
30.66
33.52
35.39
38.75
44.43
50.27
56.57
57.29
60.20
67.40
91.61
129.2
141.6
170.6
171.1
193.6
213.1
229.9
242.1
251.1
269.6
293.0
314.4
322.9
824.8
1114.0
2483.0
0.3
3
10 c n
610.3
125.6
54.63
30.98
20.16
14.67
31.03
0.168
6.139
0.224
5.880
7.806
2.176
2.054
1.658
2.473
10.15
0.017
2.413
1.188
24.75
10.22
0.443
0.006
2.257
0.223
0.771
0.481
0.079
0.000
0.000
0.000
2.335
16.36
0.965
11.33
Ln
1.371
4.391
9.261
14.03
16.16
18.71
24.80
25.05
32.63
33.48
39.55
41.88
48.22
51.02
57.45
57.86
64.38
72.33
80.59
84.38
86.91
94.39
108.3
143.6
213.5
272.8
318.6
355.7
375.6
441.7
470.1
512.5
569.6
659.1
840.1
897.2
1
3
10 c n
590.2
129.1
59.10
32.31
25.20
9.800
23.23
0.044
2.847
2.780
14.81
6.768
0.206
9.942
1.860
0.903
1.557
1.287
1.495
23.46
0.002
0.442
1.607
1.618
18.31
0.203
2.432
1.198
0.411
0.070
23.89
0.000
0.000
4.704
4.733
3.531
Ln
2.561
7.710
11.17
17.59
18.66
27.56
29.66
34.43
35.31
46.28
47.22
50.32
55.43
63.20
65.29
65.67
67.29
74.23
78.85
80.83
87.21
97.66
101.8
112.3
133.8
300.5
321.4
407.0
409.3
467.4
471.0
555.9
562.8
674.7
707.3
794.4
3
10 c n
Ln
575.8 6.127
53.58 9.108
116.2 13.91
17.83 20.44
9.554 28.19
53.78 33.44
2.400 36.57
7.138 37.04
2.601 42.72
4.477 49.99
4.409 52.73
9.523 57.75
0.222 66.18
0.032 71.81
40.67 77.27
5.003 80.52
0.089 83.57
0.003 85.14
0.002 90.22
1.564 94.89
1.963 104.6
0.781 107.4
1.166 140.1
0.090 160.6
1.630 190.5
52.28 234.6
2.736 266.6
5.490 292.1
0.009 316.0
12.17 340.1
0.755 348.6
11.35 396.1
0.004 552.6
0.000 843.4
5.585 895.7
3.173 1280.0
3
103 c n
590.9
50.49
17.64
8.587
6.187
128.4
5.350
2.437
4.352
2.179
3.018
1.385
1.254
11.35
6.078
1.519
1.334
0.000
0.142
0.040
4.555
0.444
1.943
0.214
63.31
13.07
29.39
12.37
15.08
0.735
5.622
1.890
6.473
0.652
1.585
0.003
51/s is real and independent of v . Therefore, the time constant t 5 m 0 S 8 /(4 pr ) in Eq. ~33! is also real and constant
and the variable w5i v t is purely imaginary. With
the choice S 8 54ab one then has t 5 m 0 ab s / p and
w5i v m 0 ab s / p .
For Ohmic conductivity the time t has the following
physical meaning. When the applied field is suddenly
switched on or off at time t50, the current density J(r,t) at
t>0 may be expressed as a linear superposition of relaxing
eigenmodes,
J n ~ r,t ! 5 f n ~ r! e 2t/ t n .
~36!
6530
ERNST HELMUT BRANDT
FIG. 7. The real part ~top! and imaginary part ~bottom! of the ac
permeability m 5 x 215 m 8 2i m 9 for Ohmic bars with rectangular
cross section 2a32b in a perpendicular magnetic ac field for aspect
ratios b/a50 ~thin strip!, b/a50.01, 0.03, 0.1, 0.3, 1, 2, 3, 5, and
10 as computed from the sum ~34! ~solid lines!. The dashed curves
give the analytic expressions ~5! for long slabs with b/a55 and 10,
with the variable u5a/l ac5(i v t p a/b) 1/2 inserted. For b/a510
the numerical and analytical curves are very close. To facilitate
comparison, some m 8 curves are repeated as dotted lines in the
lower plot. The time scale is t 5 m 0 ab/( pr ).
Here f n (r) are the eigenfunctions of Eq. ~26! ~or of another
equation for different geometry! and the t n 5 t /L n are relaxation times depending on the eigenvalues L n . In particular,
for thin strips ~disks! one has ~choosing S 8 54ab) 25 L 1
50.638 523 (0.876 827) and L n 'L 1 1 n 21 for all n
51,2, . . . ,`. In general, the relaxation times are given by2
t n 5 m 0 S 8 /(4 pr L n ). For long slabs (b@a) in a parallel
field, the eigenvalues of Eq. ~26! are 4 p L n /S 8 5 p 2 ( n
2 21 ) 2 /a 2 . For S 8 54ab this gives L n 5( n 2 21 ) p b/a, but
the different choice S 8 516a 2 / p in this geometry yields
more natural ~size-independent! values L n 5(2 n 21) 2
51, 9, 25, . . . .
Figures 7–10 show the linear ac susceptibility x 5 x 8
2ix95m21 or permeability m 5 m 8 2i m 9 5 x 11 of Ohmic
bars and cylinders of various aspect ratios, calculated from
Eq. ~34! as a function of the frequency v /2p . In these plots
the time unit is t 5 m 0 ab/( pr ) where r is the Ohmic resistivity. This t means that S 8 54ab was chosen. Thus, for b
!a the plotted x ( v t ) is independent of b/a ~except at very
large frequencies; see below!, but for b@a the x ( v t ) curves
depend on b such that x ( v t a/b) becomes a universal function. This invariance may be seen from the expressions ~5!
and ~7! for x ( v ) of long slabs and cylinders in a parallel
field, which depend on the variable u5a/l ac
5(a 2 i v m 0 / r ) 1/2. In our time unit t this becomes u
5(i v t p a/b) 1/2; thus x for b@a becomes a universal function of the variable u or of the combination v t a/b. In this
longitudinal limit a more natural choice of the time unit
PRB 58
FIG. 8. As in Fig. 7, but for Ohmic cylinders in axial magnetic
ac field and for b/a50 ~thin disk!, b/a50.01, 0.03, 0.1, 0.3, 1, 2,
and 3 as computed from the sum ~34! ~solid lines!, and for b53, 5,
and 10 from the analytical expression ~7! for longitudinal cylinders
~dashed lines!. The numerical and analytical curves for b/a53 are
very close.
would thus be t long5(4a 2 m 0 / pr ) corresponding to a choice
S 8 516a 2 / p in the general expression t 5 m 0 S 8 /(4 pr ). If
desired, a function t̃ (a,b) may be fitted such that for all
ratios a/b the Ohmic x 9 ( v t̃ ) exhibit their maximum at the
same position. Alternatively, t̃ (a,b) may be chosen such
that the initial slope of x 9 ( v t̃ ) becomes unity for all a/b;
FIG. 9. As in Fig. 7 but with a linear abscissa.
PRB 58
SUPERCONDUCTOR DISKS AND . . . . II. . . .
6531
~thin disk!, b/a50.01, 0.03, 0.1, 0.3, 1, 2, and 3 as computed from the sum ~34!, and for b53, 5, and 10 from the
analytical expression ~7! for longitudinal cylinders. Again
the numerical and analytical curves for b/a53 are very
close. Note also that bars and cylinders behave very similarly, but for cylinders the longitudinal limit is reached at
lower values of a/b.
The most interesting feature in these double-logarithmic
plots is that for sufficiently thin specimens with increasing
frequency m ( v ) crosses over from the behavior in perpendicular geometry ~nonlocal flux diffusion!,25
m 8 5 x 8 11}1/v ,
m 9 5 x 9 }ln~ const3 v t ! / v , ~37!
to the parallel geometry ~local flux diffusion!,
m 8 ' m 9 }1/Av .
FIG. 10. As in Fig. 8 but with a linear abscissa.
see Eqs. ~39! and ~42! below. With both choices the resulting
curves x 8 ( v t̃ ) and x 9 ( v t̃ ) for all b/a will nearly coincide;
see Fig. 11 below.
Figure 7 shows the real and imaginary parts of the ac
permeability m 5 m 8 2i m 9 5 x 21 for Ohmic bars with rectangular cross section 2a32b in a perpendicular magnetic ac
field for aspect ratios b/a50 ~thin strip!, b/a
50.01, 0.03, 0.1, 0.3, 1, 2, 3, 5, and 10, as computed
from the sum ~34!. Also shown in this plot are the analytic
expressions ~5! for long slabs with b/a55 and 10, with the
variable u5a/l ac5(i v t p a/b) 1/2 inserted. Note that the numerical and analytical m ( v t ) for b/a510 nearly coincide.
The corresponding ac permeabilities for Ohmic cylinders
in axial magnetic ac field are shown in Fig. 8 for b/a50
This crossover is clearly seen with the curves b/a50.01,
0.03, and 0.1 in Figs. 7 and 8. One furthermore realizes that
for all aspect ratios b/a, except for the limit b50, the real
and imaginary parts at large frequencies coincide, m 8 ( v )
5 m 9 ( v ). The physical reason for this finding is that above
the frequency where the skin depth d 5 A2l ac
5(2 r /i v m 0 ) 1/2 coincides with the half thickness b, the
Ohmic conductor nearly behaves like an ideal diamagnet ~or
superconductor in the Meissner state!, screening almost all
magnetic flux from the interior of the conductor. The magnetic field lines thus cannot penetrate into the bulk but have
to flow around the bar or cylinder such that everywhere B is
parallel to the specimen surface. Therefore, at high frequencies even thin conductors ~whether Ohmic or non-Ohmic! in
a perpendicular ac field behave as if the magnetic field were
applied parallel.
Figures 9 and 10 show the same data as Figs. 7 and 8 but
with a linear ordinate. This presentation clearly shows that
with increasing specimen thickness the height of the maximum of x 9 ( v ) ~i.e., of the energy dissipation! goes through
a minimum at b/a'1.
In Fig. 11 the susceptibility of Ohmic cylinders with various lengths is plotted such that the initial slope of x 9 ( v ) is
normalized to unity. This is achieved by introducing an appropriate time unit t̃ (a,b). The plotted x 8 ( v t̃ ) and x 9 ( v t̃ )
then look qualitatively similar for all aspect ratios a/b. An
analytical expression for the time t̃ (a,b) may be found in
the following way. From the definition x ( v )5m( v )/m( v
→`)5 x 8 2i x 9 and the small-frequency behavior x
52i v t̃ 2 b v 2 1•••, where b is a positive constant, one
sees that this time constant t̃ 5 ]x 9 / ]v u v 50 may be written
as
t̃ ~ a,b ! 5
FIG. 11. The linear ac susceptibilities of cylinders in axial field
from Figs. 8 and 10, plotted vs v t̃ (a,b) such that the initial slope
d x 9 /d( v t̃ )
is
unity
for
all
plotted
values
b/a
50, 0.01, 0.03, 0.1, 0.3, 1, 2, 3, and `. The time unit t̃ (a,b)
is given by Eq. ~42!.
~38!
2m ~ v →0 !
.
i v m ~ v →` !
~39!
The magnetic moment m( v ) of an Ohmic cylinder in the
limit of low frequencies is obtained noting that in this limit
the magnetic field penetrates completely; thus the electric
field is
r
r
E ~ r,t ! 5 Ḃ a ~ t ! 5 i v m 0 H a ~ t !
2
2
~40!
6532
ERNST HELMUT BRANDT
and the current density is J(r,t)5E(r,t)/ r . With the definition ~12! the magnetic moment of the cylinder at low frequencies then becomes
p ba 4
m ~ v →0 ! 5
i v m 0H a~ t ! .
4r
~41!
The magnetic moment at high frequencies is that of an ideal
diamagnet since in this limit the magnetic field is completely
screened from the interior of the specimen by the skin effect.
One
has
m( v →`)5m 8 (0)H a (t)
where
m 8 (0)
'2(2 p ba 2 18a 3 /3) is the initial slope of the virgin magnetization curve given by Eq. ~43! of part I. With the abovedefined t 5 m 0 ab/( pr ) one may thus write for the initial
slope of x 9 ( v ) of cylinders
~ 3 p /32! t
2
t̃ 5
1
113 p b/4a1 tanh@ 1.27~ b/a ! ln~ 11a/b !#
2
. ~42!
This t̃ (a,b) and the corresponding area S 8 54 pr t̃ / m 0 have
the limits t̃ / t 53 p 2 /32'1, S 8 53 p 2 ab/8 for b/a!1 and
t̃ / t 5 p a/(8b), S 8 5 p a 2 /2 for b/a@1.
From Fig. 11 one sees that for all ratios b/a the dissipative part x 9 ( v ) has its maximum close to v 51/t̃ . More
precisely, for b/a50, 0.01, 0.03, 0.1, 0.3, 1, 2, 3, and
` one has x 9max5 x 9 ( v max)50.4406, 0.4256, 0.4045,
0.3803, 0.3521, 0.3482, 0.3591, 0.3694,
and
0.3774
occurring at v maxt̃ 50.948, 0.918, 0.877, 0.832, 0.779,
0.779, 0.792, 0.805, and 0.794.
VI. CONCLUDING REMARKS
For long slabs and cylinders (b@a) in a parallel field the
eigenvalues L n and eigenfunctions f n (r) may be derived directly from a differential equation with boundary conditions
for the magnetic induction B5Bŷ. Namely, from the Maxwell equations J5¹3H ~neglecting the displacement current!, Ḃ52¹3E, ¹•B50, and the material equations E
5 r J and B5 m 0 H one obtains the diffusion equation
Ḃ~ r,t ! 5
r 2
¹ B„r,t).
m0
~43!
Inserting here the periodic time dependence e i v t one obtains
the eigenvalue equation (i v m 0 / r )B„r…5¹ 2 B„r…. For the
slab geometry with boundary conditions B 8 (0)5B(a)50
for B(x) along y, one finds that the solution of this linear
diffusion problem is a linear superposition of eigenfunctions
g n (x) of the eigenvalue problem
g n ~ x ! 9 52k 2n g n ~ x ! ,
g 8n ~ 0 ! 5g n ~ a ! 50.
~44!
This is solved by g n (x)5cosknx with k n 5( n 2 21 ) p /a, n
51,2,3, . . . . The current density along z now is J(x)
5B 8 (x)/ m 0 , and the eigenvalues of the differential equation
~44! for B(x) and integral equation for J(x) @Eq. ~25! with
the kernel Q cyl replaced by the kernel Q sym of Ref. 2# obey
the identity2 4 p L n /S 8 5k 2n .
PRB 58
Similarly, for long cylinders in a parallel field Eq. ~43!
leads to a diffusion equation for B(r,t) in cylindrical coordinates. The corresponding eigenvalue problem is
g n~ r ! 91
g 8n ~ r !
r
g 8n ~ 0 ! 5g n ~ a ! 50. ~45!
52k 2n g n ~ r ! ,
This is solved by g n (r)5J 0 (k n r) where J 0 ( r ) is a Bessel
function and the k n 5 r 0 n /a are related to the zeros of
J 0 ( r ), r 0 n 52.405, 5.520, 8.634, . . . for n 51,2,3, . . . .
One has approximately33 r 0 n ' p ( n 2 41 )1(1/8p )/( n 2 41 ).
Thus, the eigenvalues L n 5k 2n S 8 /4p for the long slab and
cylinder are
S D
L n5 n 2
1
2
S D
L n' n 2
1
4
2
2
pS8
4a 2
pS8
4a 2
~ slab! ,
~46!
~ cylinder! .
~47!
The eigenfunctions for the current densities J(x)
5B 8 (x)/ m 0 and J(r)5B 8 (r)/ m 0 of the long slab and cylinder are f n (x)5sinknx and f n (r)5J 1 (k n r) where J 1 ( r )
52J 80 ( r ) is another Bessel function. From these solutions
f n and L n one may, after some lengthy calculation, reproduce the closed-form expressions ~5! and ~7! for the linear ac
susceptibility x ( v ) by evaluating the sum ~34! analytically.
For thin strips and disks (b!a) in a perpendicular field
the eigenvalues L n of Eq. ~26! are approximately linear
functions of the index n , 25
L n ' ~ n 2110.6385!
S8
4ab
~ strip! ,
~48!
L n ' ~ n 2110.8768!
S8
4ab
~ disk! .
~49!
The formulas ~46!–~49! show that S 8 516a 2 / p and S 8
54ab are natural choices for S 8 in the parallel and perpendicular geometries, respectively, since they yield L n values
that are independent of the specimen width 2a and length 2b
and start with lowest eigenvalues which are of order unity,
namely, L 1 51 (2.344'9/4, 0.6385, 0.8768) for the slab
~cylinder, strip, disk!. The eigenfunctions f n (x) and f n (r) for
the thin strip and disk look similar as in the parallel geometry; see Fig. 2 in Ref. 20.
For more complicated geometries like cylinders of finite
length, closed-form expressions for the linear susceptibility
x ( v ) are not available, and all the more not for the nonlinear
susceptibilities. One thus has to use sums of the form ~34!
with numerically obtained numbers c n and L n ; see Table I.
The explicit expressions ~5!, ~7!, ~9!, ~10!, and ~34! for
x @ v , r ac( v ) # are very useful since they apply to arbitrary
linear complex resistivity, including the Ohmic and TAFF
~Ref. 22! cases r ac( v )5 r 51/s and the Meissner state described by the London equation, in which formally r ac( v )
5i v m 0 l 2 where l is the London penetration depth. By in-
PRB 58
SUPERCONDUCTOR DISKS AND . . . . II. . . .
verting such a relationship between x and r ac numerically,
one may extract the linear complex resistivity from measurements of the ac susceptibility as was done in Ref. 20. In the
nonlinear case one has to fit or compare the measured x to
the computed x (H 0 ) or x (H 0 , v ) as was done, e.g., in Refs.
6–13,34, and 35.
E. H. Brandt, preceding paper @Phys. Rev. B 58, 6506 ~1998!
~referred to as part I!#.
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ACKNOWLEDGMENTS
Stimulating discussions with P. Esquinazi, P. Ziemann, J.
Gilchrist, A. Gurevich, L. Prigozhin, J. Kötzler, J.R. Clem,
and M.V. Indenbom are gratefully acknowledged. This work
was supported by the German-Israeli Foundation for Research and Development, Grant No. I-300-101.07/93.
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