JOURNAL OF APPLIED PHYSICS
VOLUME 84, NUMBER 7
1 OCTOBER 1998
Influence of the real shape of a sample on the pinning
induced magnetostriction
A. Nabiałeka) and H. Szymczak
Institute of Physics, Polish Academy of Science, Al. Lotników, 32/46, 02-668 Warsaw, Poland
V. A. Sirenko
Institute for Low Temperature Physics and Engineering, 47 Lenin Avenue, 310164 Kharkov, Ukraine
A. I. D’yachenko
Donetsk Physico-Technical Institute, Ukrainian Academy of Sciences, ul. R. Luxemburg, 72, 340114
Donetsk, Ukraine
~Received 3 November 1997; accepted for publication 3 July 1998!
The pinning induced magnetostriction in an isotropic superconductor was calculated for different
sample shapes. We analyzed some special shapes for which the solution can be found analytically.
The magnetostriction of a finite slab was considered. In order to determine the influence of
demagnetization effects, the pinning induced magnetostriction of an infinitely long and thin strip
was calculated. A simple, approximate formula can be used in particular cases to analyze the
magnetostriction induced by pinning forces. In this formula the magnetostriction of a sample is
connected directly with its magnetization. We also present some experimental results on high
temperature superconductors which are analyzed in frames of the developed theory. © 1998
American Institute of Physics. @S0021-8979~98!04319-9#
cylinder.3,4 In this special case the magnetostriction can be
described by the formula4
I. INTRODUCTION
Conventional superconductors are characterized by a
relatively low magnetostriction, which is usually described
in frames of a thermodynamic model.1 In this model changes
of the sample dimensions were calculated by differentiation
of an appropriate thermodynamic potential. The pinning induced mechanism of the magnetostriction was first proposed
by Ikuta et al.2 in order to explain giant magnetostriction in a
Bi2Sr2CaCu2O8 single crystal. A force, opposite to one
which acts on flux lines, acts on the sample and causes the
changes of the sample dimension ~the pinning induced magnetostriction!. Ikuta et al.2 developed a one-dimensional
model of the magnetostriction induced by the pinning forces
for the infinite slab of thickness 2d. They neglected the demagnetization effects. When the external magnetic field is
applied parallel to the slab, the changes of the thickness of
the sample can be described by the formula2
S D
Dd
d
52
parallel
1
* d @ B 2 2B 2 ~ x !# dx,
2E m 0 d 0 0 e
12 n
DR
5
R
E m 0R 2
R
0
r @ B 2 ~ r ! 2B 2e # dr,
~2!
where R is the radius of the sample and n is the Poisson’s
ratio.
If the magnetic field profile in the sample is known, one
can calculate magnetostriction hysteresis loops similarly to
the magnetization ones. Ikuta et al. have performed such calculations for three most physically acceptable critical state
models.5
In the case of infinite slab2 the stress is uniaxial. In the
case of a cylinder4 the state of stress must be analyzed in two
dimensions. However, because of the cylindrical symmetry
of the sample and critical currents, we do not expect any
changes in the shape of the sample but only the changes of
the cylinder radius. In the case of a sample of an arbitrary
shape we can also expect the changes of the shape of the
sample. The complication of the problem increases because
the stress caused by the pinning forces is not homogeneous.
In this work we are going to show that for different
shapes of the sample we can use the same approximate formulas on the magnetostriction induced by the pinning forces.
These formulas connect the pinning induced magnetostriction of the sample directly with its magnetization. In Sec. II
the evidence will be given that the influence of currents
flowing in ends of a finite slab on the magnetostriction is of
the same order of magnitude as the influence of the currents
flowing parallel to the surface of the slab. Hence, the state of
stress of a finite sample in the plane perpendicular to an
external magnetic field must always be analyzed in two dimensions. In order to study the influence of the demagneti-
~1!
where E is Young’s modulus, d is half of the thickness of the
sample, m 0 is the magnetic permeability of vacuum, B e is the
external magnetic field value, and B(x) is the magnetic field
profile inside the sample. Index ‘‘parallel’’ denotes that the
changes of the thickness of the sample are caused by the
currents flowing parallel to its surface.
Recently, the problem of stress and of the pinning induced magnetostriction was also solved in the case of an
isotropic superconductor in a form of infinitely long
a!
Electronic mail: nabia@ifpan.edu.pl
0021-8979/98/84(7)/3770/6/$15.00
E
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© 1998 American Institute of Physics
Nabiałek et al.
J. Appl. Phys., Vol. 84, No. 7, 1 October 1998
3771
zation effects, we will consider ~in Sec. III! a model of a
critical state of an infinitely long and thin strip with magnetic
field applied perpendicularly to the surface of the strip; it
means the sample with an extremely large demagnetization
factor. The currents’ and magnetic field’s distribution in this
model was recently found analytically by Brandt et al.6 We
will consider only the model with the isotropic critical current density in the plane perpendicular to external magnetic
field. This approximation seems to be reasonable for high
temperature superconductors ~HTSs! also, when we measure
a transverse magnetostriction ~the changes of the dimension
of the sample are measured perpendicularly to the external
magnetic field! and magnetic field is oriented parallel to the
c axis ~it means that the shielding currents flow in the ab
plane! of the layered HTS. In Sec. IV, some experimental
results on HTSs will be presented. These results confirm the
main conclusions following theoretical considerations.
II. INFLUENCE OF CURRENTS FLOWING IN THE ENDS
OF A FINITE SLAB
Following Refs. 2 and 5 we will consider only the pinning induced part of magnetostriction. We will neglect the
reversible part of magnetostriction connected with the Meissner shielding currents on the surface of the sample. In HTSs
the values of the first critical field are low and a reversible
part of the magnetostriction is negligible. The reversible part
of the magnetostriction can be easily calculated. This was
done by Brändli.1 In his work1 an analysis of experimental
data of the reversible part of the magnetostriction for conventional superconductors was also performed. In some superconductors we can observe large magnetostriction caused
by the presence of magnetic rare-earth ions.7 Since the
mechanism of this part of the magnetostriction is completely
different, we will not discuss it in the present work.
At this point, we would like to make a short comparison
between the model of the magnetostriction proposed by Ikuta
et al.2 and that developed by Brändli.1 The magnetic induction profile inside a sample determines the pinning induced
magnetostriction. Instead of talking about pinning forces as
the cause of the magnetostriction in the superconductor ~like
in the work of Ref. 2!, we can talk about the pressure of the
magnetic field ~similarly like in the work of Ref. 1!. Such
pressure acts on every sample with nonzero magnetization in
an external magnetic field. In his work1 Brändli calculated
the magnetostriction of superconductors as a consequence of
magnetic field pressure, which acts on the surface of the
sample. In his experiments on conventional superconductors
the irreversible part of magnetostriction was small in comparison to the reversible one. What makes type-II superconductors different from other magnetics is the fact that magnetic induction inside the superconductor changes according
to the critical state condition. In this case it is not enough to
consider the magnetic field pressure on the sample surface
~like in the work of Ref. 1! but we have also to remember
about magnetic forces, which act on the whole volume of the
sample. These forces are equivalent to the pinning forces.
In this section we will consider a slab, in which the cross
section in the plane perpendicular to an external magnetic
field is presented in Fig. 1~a!. At first we assume the length
FIG. 1. The cross section of a finite slab in the plane perpendicular to an
external magnetic field. ~a! The sample is divided into a central part in form
of a parallelepiped and two ends in the form of halves of a cylinder; the flow
of screening currents is also shown. ~b! The influence of the ends of the
finite slab is equivalent two opposite forces, which act on the central part of
the finite slab. ~c! The integration of the y component of the pinning forces
over the end of a finite slab.
of the sample in the direction perpendicular to an external
magnetic field to be large in comparison to the thickness of
the sample—2d. The length of the sample parallel to an
external magnetic field is infinite. It means that similarly as
in the works of Ref. 2–5 we can neglect the demagnetization
effects. We can divide this sample into three parts @as is
shown in Fig. 2~a!#: the central part, whose cross section is in
form of a rectangle with dimensions 2d3L and two ends of
the sample in form of two halves of the cylinder. In the
central part of the sample, currents are flowing only parallel
to the surface of the slab. These currents induce in the central
part of the sample the uniaxial stress. This state of stress we
can consider in frames of the model proposed by Ikuta et al.2
We have assumed the ends of the slab to be relatively small
in comparison to the central part. Hence, we will study only
the influence of the ends on the central part of the slab and
we will neglect the deformation of the ends. Bearing in mind
the above discussion, one can consider the influence of the
ends as two opposite forces F y and -F y . These forces are
applied to the sample @as is shown in Fig. 1~b!# and cause an
uniaxial and homogeneous stress with s y 5F y /2dh, where
h→` is the length of the slab parallel to an external magnetic field. In this case the changes of the length L of the
sample, caused by the forces 6F y , will be described by a
simple formula:
S D
DL
L
5
Fy
sy
.
E
~3!
Index ‘‘F y ’’ denotes that the changes of the slab length are
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Nabiałek et al.
J. Appl. Phys., Vol. 84, No. 7, 1 October 1998
currents flowing parallel to its surface.2,5 In this case the
relative changes of the thickness of the finite slab are given
by the formula (Dd/d) total5(Dd/d) parallel2 n (DL/L) F y .
The relative changes of the length of the sample are given
by the approximate formula (DL/L) total'(DL/L) F y
2 n (Dd/d) parallel . Using these formulas as well as Eqs. ~1!
and ~5! we obtain finally
S D
Dd
d
FIG. 2. The irreversible part of the transverse ~a,c,e! and longitudinal ~b,d,f!
magnetostriction of ceramics YBa2Cu3O72d measured at 4.2 ~a,b!, 20 ~c,d!,
and 40 K ~e,f!. Dotted lines show the fitting curves.
caused by forces 6F y . In order to calculate the F y force, we
integrate the y component of the pinning forces over the ends
of the sample @as it is shown in Fig. 1~c!#.
F y 5h
E
p
0
sin w d w
E
d
0
J c ~ x ! B ~ x ! xdx,
~4!
where * p0 sin w dw52 and J c (x) is the critical current distribution given by the critical state condition
J c ~ x ! 52
1 dB ~ x !
.
m 0 dx
On the basis of Eqs. ~3! and ~4! we finally obtain
S D
DL
L
52
Fy
1
2E m 0 d
E
d
0
@ B 2e 2B 2 ~ x !# dx.
~5!
The relative changes of the length of the finite slab caused by
its ends ~5! are exactly the same as the relative changes of
the thickness of the slab caused by the currents flowing parallel to its surface ~1!. According to the Poisson’s rule, in the
case of uniaxial and homogeneous stress, the changes of the
length (DL/L) F y cause the changes of the thickness
2d @ (Dd/d) F y 52 n (DL/L) F y # . The currents flowing parallel to the surface of the slab will also ~according to the Poisson’s rule! cause the changes of its length (DL/L) parallel .
Unfortunately, the stress induced by these currents is not
homogeneous
and
the
formula
@ (DL/L) parallel5
2 n (Dd/d) parallel# can be used only as an approximation. We
will further consider the state of stress in the fininte slab as a
superposition of the uniaxial homogeneous stress caused by
the ends of the slab and the state of stress induced by the
52
total
12 n
2E m 0 d 0
E
d
0
@ B 2e 2B 2 ~ x !# dx'
S D
DL
L
.
total
~6!
The only difference between the formula on the magnetostriction of an infinite slab ~1! and a finite slab ~6! is a coefficient of (12 n ). However, even for a very long ~but finite!
slab the state of stress is two dimensional and we cannot
neglect the influence of the ends of the sample on the total
magnetostriction.
When an external magnetic field is swept between two
extreme values 2B m and B m a hysteresis loop of the magnetostriction is observed. Applying an original Bean model
~it means, assuming that J c is independent on a magnetic
field!, one can obtain the formula on the width of the magnetostriction hysteresis loop. In the range 2B m 12B p ,B e
,B m 22B p ~where B p is the field of full penetration; B p
5 m 0 J c a; a5d for a slab or a5R for a cylinder! it was
found:4,5
D
S D
Dd
Dd Dd 12 n
5
J B d
5 2
d
d↓ d↑
E c e
~7!
in the case of finite slab and
D
S D
S D
DR
DR DR 2 12 n
2
5
5
J cB eR
R
R↓ R↑ 3 E
~8!
in the case of a cylinder. The arrows ↓↑ denote a decreasing
and an increasing magnetic field, respectively. Bearing in
mind the formulas on the widths of the magnetization hysteresis loops DM 5M ↓ 2M ↑ 5J c d in the case of slab and
DM 5(2/3)J c R in the case of a cylinder and Eqs. ~7! and
~8!, it is easy to find a formula, which is valid both for a
cylinder and for a finite slab. This formula connects directly
the width of the transverse magnetostriction hysteresis loop
and the magnetization hysteresis loop:
D
S D
Dd
d
5
transverse
S D
12 n
B e DM .
E
~9!
In some experiments, we measure also the longitudinal magnetostriction. It means, the changes of the sample dimensions
are measured parallel to the direction of an external magnetic
field. According to our approximation, the state of stress of a
finite slab in the plane perpendicular to an external magnetic
field can be treated as symmetric similarly as in the case of a
cylinder, because (Dd/d) total'(DL/L) total @see formula ~6!#.
In the case of homogeneous and symmetric stress in the
plane perpendicular to an external magnetic field the longitudinal magnetostriction is connected with the transverse one
by the formula
Nabiałek et al.
J. Appl. Phys., Vol. 84, No. 7, 1 October 1998
D
S D
Dh
h
FS D
522 n D
longitudinal
Da
a
transverse
G
.
~10!
However, as the stress induced by the pinning forces is in
general not homogeneous, this formula should be treated
only approximately.
According to our assumptions the formula ~9! is valid
only for two extreme cases ~a cylinder and a very long slab!.
Until now in our considerations we have neglected the deformation of the ends of a finite slab assuming the length L
of the slab to be large in comparison to its diameter 2d.
However, we have assumed the ends of the slab in form of
two halves of a cylinder. It seems reasonable to treat approximately the deformation of such two halves of the cylinder ~connected by a central part in the form of a finite slab!
as a deformation of a cylinder. In this case formula ~9! can
also be used ~approximately! even in the case d'L.
The generalization of the formula ~9! on the sample of
an arbitrary shape can be done only approximately and for
some special shapes of the sample. The exact solution of the
problem of the state of stress of the sample with an arbitrary
shape is not easy. In most cases numerical calculations are
necessary. However, in our opinion, formula ~9! can be used
as a relatively good approximation for a wide class of
samples studied in experiments.
The currents’ distribution in samples with extremely
large demagnetization factors ~it means in strips and disks!
has been studied by many authors.6,8 We have chosen the
model proposed by Brandt et al.,6 because in this model the
currents’ and magnetic field’s distributions in the superconductor can be expressed by relatively simple analytical formulas. In this model we consider a strip with the width 2a
~along the y axis! and the thickness d ~along the x axis!
assuming the sample to be infinitely long ~along z axis!. We
assume the current to be constant over the whole thickness of
the strip and J c to be independent on a magnetic field, similarly as in the Bean model. An external magnetic field is
applied along the x axis perpendicularly to the surface of a
strip. For simplicity we will consider only the zero field cooling (ZFC) magnetostriction curve (the magnetostriction is
measured after cooling the sample in zero external magnetic
field).
In order to simplify the calculation, let us introduce
some new relative variables and parameters: h 5y/a; h
5B e /B * where B * 5 m 0 J c d, k51/cosh(ph), c5 A12k 2
5tanh(ph). According to Ref. 6 the current distribution in
the sample is given by the following formulas:
J z~ h ! 5
2J c
arctan
p
J z ~ h ! 5J c for
III. MAGNETOSTRICTION OF AN INFINITELY LONG
AND THIN STRIP
In this section, we will try to answer the question—how
do the demagnetization effects influence the pinning induced
magnetostriction in the superconductor. The simplest way to
consider demagnetization effects in magnetic materials is the
introduction of demagnetization factor. Such approximation
in the case of the pinning induced magnetostriction was also
suggested by Ikuta et al.2 The simple introduction of a demagnetization factor in case of superconductors is not fully
appropriate. This is mainly because the magnetization of the
superconductor is caused by macroscopic screening currents.
The self component of the magnetic field could change radically the distribution of currents in the sample.
Below, we will consider a sample with extremely large
demagnetization factor. We will study an infinitely long and
thin strip with an external magnetic field applied perpendicular to the surface of the strip. The aim of our considerations
is to show an example of analytical solution of the influence
of demagnetization effects on the magnetostriction induced
by the pinning forces. However, if one analyzes the magnetostriction of a real superconducting sample, whose shape is
similar to a thin strip, some additional factors should be
taken into account.
In our consideration we have neglected the effect of
buckling. This effect plays an important role in the case of
thin strip, when the forces act parallel to its surface. The
most common samples, whose shapes are similar to the thin
strips, are thin films. Thin films, however, are deposited on a
substrate. In this case the interaction between the film and
the substrate plays an important role. We do not consider this
interaction in our model.
3773
S
ch
Ak
2
2h2
D
for
0, h ,k,
~11!
k, h ,1.
The distribution of the x component of a magnetic field in the
sample is given by
H x ~ h ! 50 for
H x~ h ! 5
0, h ,k,
J cd
arctanh
p
SA D
h 2 2k 2
ch
~12!
for
k, h ,1.
The y component of the pinning force per unit volume is
f y ( h )5J z ( h )B x ( h )5 m 0 J z ( h )H x ( h ). The y component of
an internal local stress one can obtain by integrating s y ( h )
5a * f y ( h )d h 1C. The constant C was obtained from the
boundary condition s y (1)50. Finally, the following stress
distribution is found:
s y~ h !5
m 0 J 2c ad
ln~ k ! for
p
0, h ,k,
~13!
m 0 J 2c ad
s y~ h !5
@ I ~ k, h ! 1ln~ k !# for
p
where
I ~ k, h ! 5 h arctanh
SA D
k, h ,1,
S
D
Ah 2 2k 2
h 2 2k 2
2arctanh
.
ch
c
In order to obtain the relative changes of the width of the
strip we should integrate « y 5 s y /E over the whole width of
the strip, which means
Da 1
5
a
E
Es
a
0
y~ h !d h .
~14!
From Eqs. ~13! and ~14! it was found:
S D
1
Da
Be
52
J c aB e tanh p
.
a
2E
B*
~15!
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Nabiałek et al.
J. Appl. Phys., Vol. 84, No. 7, 1 October 1998
On the other hand, the magnetization in Brandt et al.6 model
is given ~for ZFC curve! by the simple formula:
S D
1
Be
M 52 J c a tanh p
.
2
B*
~16!
Hence, by comparison of Eqs. ~15! and ~16! the magnetostriction of an infinitely long and thin strip could be connected with its magnetization by a very simple formula:
Da 1
5 M Be .
a
E
~17!
According to formula ~17! the magnetostriction of a strip is
proportional to the product of its magnetization and an external magnetic field. Hence we expect the influence of demagnetization effects on the pinning induced magnetostriction in the superconductor to be the same ~at least in the case
of the strip! as on its magnetization. For an external magnetic
field B e !B * 5 m 0 J c d ~where d is the thickness of the
strip! tanh(pBe /B*)'pBe /B* and Da/a'2(1/2E)(B 2e / m 0 )
3( p a/d). In this range the pinning induced magnetostriction of a strip is higher than the magnetostriction of an infinite slab5 by a factor of ( p a/d). When an external magnetic
field B e @B * , Da/a'2(1/2E)J c aB e . In this case we have
a linear dependence of the magnetostriction with the same
coefficient as for an infinite slab.5
Similarly as in the case of an infinite slab,2,5 we have
assumed our strip to be infinitely long. However, as we have
proved in Sec. II, even for a very long but finite slab we
expect the influence of the currents flowing in its ends on the
magnetostriction to be the same order of magnitude as the
influence of the currents flowing parallel to the surface of the
slab. It seems to be reasonable to expect a similar effect in
the case of the strip with finite length also. Hence we presume that in the Eqs. ~15! and ~17! a coefficient of (12 n )
should appear in the case of a strip with finite length. In this
case the formula ~9!, which correlates the widths of the magnetostriction and magnetization hysteresis loops in case of a
cylinder and a finite slab, should also be valid in the case of
a strip with finite length.
At this point one can see an additional convenience of
application of the approximate formula ~9! in the analysis of
the experimental results of the pinning induced magnetostriction, as in this formula the influence of demagnetization effects is also included.
IV. COMPARISON WITH THE EXPERIMENTS ON HTSs
In this section we are going to discuss the applications of
the isotropic theory based on the pinning induced mechanism
of the magnetostriction in the analysis of the experimental
results on HTSs. We have investigated the magnetostriction
in HTSs by the strain gauge technique. The measurements
were performed below T c in an external magnetic field up to
12 T. We have measured both transverse and longitudinal
magnetostriction. HTSs are highly anisotropic and the isotropic theory in general is not applicable ~except for the case of
B e parallel to the c axis!. However, in this section we are
going to show that the formulas ~9! and ~10! which were
derived for some special shapes of the isotropic supercon-
FIG. 3. The magnetostriction of La1.865Sr0.135CuO4 single crystal measured
at 4.2 K. The sample dimensions were 532.530.6 mm3. The c axis was
parallel to the 2.5 mm edge. In ~a!–~f! is shown the orientation of the sample
with respect to an external magnetic field and the strain gauge.
ductor can also be useful in analysis of the HTS for different
sample orientations with respect to an external magnetic
field.
Figure 2 presents the transverse and longitudinal magnetostriction measurements on YBa2Cu3O72d ceramics at different temperatures. The dotted lines show fitting curves.
These curves were obtained using formulas ~9! and ~10!,
magnetization data ~the measurements of the magnetization
were performed exactly on the same sample as the magnetostriction!, and elastic constants of YBa2Cu3O72d crystals
taken from other experiments. One can see in the range of
strong magnetic fields or temperatures above 4.2 K that the
agreement between the theory and the experiment ~both
qualitative and quantitative! is very good. The discrepancy at
4.2 K in the range of relatively small magnetic field we assume to be caused by the weak links of ceramic HTS. Using
the elastic constants characteristic for HTS ceramic and formulas ~9! and ~10! we are able to explain this discrepancy as
the influence of the intergrain critical currents in ceramic
superconductor. Similar results and good agreement between
the theory and the experiment are also found in the ceramic
Hg0.8Pb0.2Ba2Ca2Cu3O8 superconductor.
Figure 3 presents the magnetostriction measurements on
La1.865Sr0.135CuO4 single crystal at 4.2 K. The sample dimension were 532.530.6 mm3 and the c axis of this crystal was
oriented parallel to the 2.5 mm edge. The strain gauge was
fixed in the 2.535 mm2 plane of the sample and the changes
of the sample dimensions were measured in the direction
parallel to the 2.5 or 5 mm edge. The magnetic field was
parallel to one of the edges of the sample. The sample ori-
Nabiałek et al.
J. Appl. Phys., Vol. 84, No. 7, 1 October 1998
entation with respect to external magnetic field as well as
with respect to the strain gauge for each measurement is
shown in Fig. 3. In the case of this sample we expect the
influence of the anisotropy connected with the layered structure of the superconductor as well as the anisotropy connected with the real shape of the sample on the magnetostriction. The results in Fig. 3 are grouped in pairs ~a,b!, ~c,d! and
~e,f!. For each pair the orientation of the sample with respect
to an external magnetic field is the same. Hence, for each
pair we expect also the magnetization of the sample to be the
same. The only difference in each pair is the direction in
which the changes of the sample dimensions were measured.
According to formulas ~9! and ~10! we expect the irreversible part of the magnetostriction to be proportional to the
irreversible part of the magnetization. One can easily find in
Fig. 3 that the shapes of the magnetostriction hysteresis
loops in each pair are almost the same. These results suggest
that the idea of correlation of the irreversible part of both
transverse and longitudinal magnetostriction directly with
the magnetization is good, even in the case of highly anisotropic HTS. The sign of the longitudinal magnetostriction
~a,f! is always opposite to the transverse one. This is in
agreement with formula ~10!. This also concurs with the results presented by other authors.2,9 By comparison of the
pairs ~c,d! and ~e,f!, one can see the influence of demagnetization effects on the magnetostriction in HTSs. This influence is pronounced in the range when the direction of the
magnetic field sweep is changed ~it means in the range near
12 T!. In the case when the demagnetization factor is large
~c,d!, the magnetostriction changes its sign in the range
lower than about 1 T ~between 12 and 11 T!. In the case ~e,f!,
where the demagnetization factor is relatively low, the magnetostriction changes its sign in the range of about 5 T ~between 12 and 7 T!. Similar influence of the demagnetization
factor is usually observed in the case of magnetization hysteresis loops.
We have performed the measurements on different
HTSs. Similarly as in the case of other works2,9 the qualitative agreement between the experimental results and the calculations performed in frames of the magnetostriction induced by the pinning forces is relatively good. However in
most cases, if we measure the magnetostriction on large
single crystals or melt-textured samples, the absolute values
of the experimentally obtained magnetostriction are higher
than those obtained from theoretical calculations. Similar results were also found by other authors.2,9 We found that this
discrepancy cannot be explained by demagnetization effects.
The cause for this discrepancy is not fully clarified yet. In
3775
our opinion, it is connected with determination of elastic
constants in large HTS samples. This problem, as well as
more detailed analysis of experimental data, will be published elsewhere.
V. CONCLUSIONS
We have compared an isotropic model of the magnetostriction of a finite slab with the model of the magnetostriction of a cylinder. The obtained results show that in both
cases we expect the irreversible part of the magnetostriction
to be connected with the irreversible part of the magnetization by the same simple formula. On the basis of calculations
for an infinitely long and thin strip we can conclude that the
influence of demagnetization effects on the pinning induced
magnetostriction is similar as on the magnetization of the
investigated samples. The comparison of theory with experiment shows that this theory can, to a certain extent, be useful
in analysis of the magnetostriction of highly anisotropic
HTS.
ACKNOWLEDGMENT
This work was partly supported by the Polish Government Agency KBN under Contract No. 2P03B06111.
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