Investigation of magnetic flux gradients in hard superconductors
Analytical calculations of the transverse magnetostriction in a thin hard superconductor are presented in relation to the distributions of currents and fields within a superconducting specimen in a varied magnetic field. The approach is successfully tested on high-temperature superconductors. The fl...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2001
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| Cite this: | Investigation of magnetic flux gradients in hard superconductors / V.V. Bruk, V.V. Eremenko, N.I. Makedonskaya, Yu.A. Shabakayeva, V.A. Sirenko // Физика низких температур. — 2001. — Т. 27, № 4. — С. 419-424. — Бібліогр.: 13 назв. — англ. |
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| author | Bruk, V.V. Eremenko, V.V. Makedonskaya, N.I. Shabakayeva, Yu.A. Sirenko, V.A. |
| author_facet | Bruk, V.V. Eremenko, V.V. Makedonskaya, N.I. Shabakayeva, Yu.A. Sirenko, V.A. |
| citation_txt | Investigation of magnetic flux gradients in hard superconductors / V.V. Bruk, V.V. Eremenko, N.I. Makedonskaya, Yu.A. Shabakayeva, V.A. Sirenko // Физика низких температур. — 2001. — Т. 27, № 4. — С. 419-424. — Бібліогр.: 13 назв. — англ. |
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| description | Analytical calculations of the transverse magnetostriction in a thin hard superconductor are presented in relation to the distributions of currents and fields within a superconducting specimen in a varied magnetic field. The approach is successfully tested on high-temperature superconductors. The flux distribution derived from magnetostriction measurements is in satisfactory agreement with that obtained from computer processing of magnetooptical images. The magnetic flux distribution below the irreversibility line of hard superconductors is derived from both original magnetization and magnetostriction measurements and image processing. Perfect consistency of the results is obtained for a family of high-temperature superconductors (La₂₋xSrxCuO₄, Bi₂Sr₂CaCu₂Ox, YBa₂Cu₃O₇₋d).
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Fizika Nizkikh Temperatur, 2001, v. 27, No. 4, p. 419–424 Br uk V. V., E re men ko V. V., Maked onskaya N. I. , Sha baka yeva Yu. A ., an d Sire nko V. A .Investig ation of m ag netic flux g rad ie nts in har d super con duct orsBr uk V. V., E re men ko V. V., Maked onskaya N. I. , Sha baka yeva Yu. A ., an d Sire nko V. A .Investig ation of m ag netic flux g rad ie nts in har d super con duct ors
Investigation of magnetic flux gradients in hard
superconductors
V. V. Bruk, V. V. Eremenko, N. I. Makedonskaya, Yu. A. Shabakayeva,
and V. A. Sirenko
B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine
47 Lenin Ave., Kharkov 61164, Ukraine
E-mail: shabakayeva@ilt.kharkov.ua
Received October 26, 2000
Analytical calculations of the transverse magnetostriction in a thin hard superconductor are pre-
sented in relation to the distributions of currents and fields within a superconducting specimen in a
varied magnetic field. The approach is successfully tested on high-temperature superconductors. The flux
distribution derived from magnetostriction measurements is in satisfactory agreement with that obtained
from computer processing of magnetooptical images. The magnetic flux distribution below the irre-
versibility line of hard superconductors is derived from both original magnetization and magnetostriction
measurements and image processing. Perfect consistency of the results is obtained for a family of
high-temperature superconductors (La2−xSrxCuO4 , Bi2Sr2CaCu2Ox , YBa2Cu3O7−δ).
PACS: 75.30.Kz
Introduction
Measurements of the irreversible «giant» magne-
tostriction in HTSC [1–4] appears to be an effective
tool for examination of the pinning-related phe-
nomena in hard superconductors. The 1D case was
considered in detail for a thin long strip in a
longitudinal magnetic field [5], and good agreement
was achieved with the conventional models [6-9] of
the magnetic flux distributions in type-II supercon-
ductors. In Ref. 10 another typical experimental
situation is analyzed which corresponds to an exter-
nal magnetic field normal to the flat face of a thin
superconductor. Such a situation is frequently real-
ized during magnetization and magnetic flux dis-
tribution measurements the thin films and single
crystals of HTSC. It is not straightforwardly de-
scribed by the classical models and their extensions,
and it requires taking the demagnetizing effects into
account. The problems of the current and flux
distributions in these geometries are widely dis-
cussed in the literature (see, e.g., Ref. 11 and the
references therein). In this work we present analy-
tical estimations of the relevant magnetostrictions
and show that their measurements can elucidate
some aspects of the problem.
Calculations
It follows from Refs. 1–4 that the irreversible
magnetostriction of HTSC with high critical cur-
rent densities jc is caused by the pressure of the
magnetic field, which smears the magnetic flux
within a specimen with strong pinning. The rele-
vant reduced strain of a superconductor in a mag-
netic field is related to the magnetic flux distribu-
tion inside the sample by the elastic equations
taking into account the balance of forces in a system
«flux line lattice imperfection», provided that the
fully penetrated state has been established [1]:
∆L
L
= −
1
L
∫
−L/2
L/2
Be
2 − B2(x)
µ0E
dx , (1)
where L is the sample size, Be = µ0He is the mag-
netic induction corresponding to the external mag-
netic field He , B(x) is the local flux density in the
sample, E is the Young’s modulus along the length
change and µ0 is the permeability of vacuum. We
consider a thin flat sample of thickness d and width
2a under the action of a varied field perpendicular
to the face of the sample [10] (Fig. 1). If the
specimen is assumed to be infinite in one direction z
© V. V. Bruk, V. V. Eremenko, N. I. Makedonskaya, Yu. A. Shabakayeva, and V. A. Sirenko, 2001
and Be to be uniform along the x axis, the flux
gradients are in the y direction. In this case the
longitudinal component of Be merely corresponds to
the shielding currents, while the transverse compo-
nent is responsible for the flux profiles, and so it
alone determines the magnetostrictive strain, which
is consequently directed along y. B(y) is determined
by the Maxwell equations.
The elastic stresses in the sample are calculated
following [10]. Accordingly, the transverse magne-
tostriction (1) reads:
∆L
L
=
1
E
∫
0
1
σ(η) dη , (2)
which is equal to [10]
∆L
L
=
1
E
µ0 jcda
π
π
2
(cm
− c1)hm
+
πc1
2
h −
− cm
Artanh
√k1
2 − km
2
k1
+ c1
2 Artanh
√k1
2 − km
2
cmk1
+
+
1
E
µ0 jcda
π
F(km , k1) , (3)
where σ(η) is elastic stresses in the sample;
η ≡
y
a
∈ [− 1, 1] ; cm
≡ √1 − km
2 = tanh (πhm
) ;
c1 ≡ √1 − k1
2 = tanh (πh1) ; hm
≡
Bm
µ0 jcd
;
h ≡
Be
µ0 jcd
; h1 ≡
Bm
− Be
2µ0 jcd
;
k1 ≡ cosh−1 (πh1) ; km
≡ cosh−1 m(πhm
) ;
F(km , k1) =
=
2
π
k1M(km , k1 , x = k1) − ∫
k
m
k
1
M(km
, k1 , x) dx
;
Bm corresponds to the maximum field in a magneti-
zation cycle.
The calculations of ∆L/L using flux gradients
determined from magnetization measurements are
presented in Fig. 1. It was anticipated that estimate
of the flux gradients from magnetooptical image
processing would also be used in these calculations.
This is illustrated below on the YBa2Cu3O7−δ
samples.
The process of determination of magnetic flux
distribution from magnetooptical image consists of
two stages using the magnetooptical method and
the method of computer processing.
The principle of visualization of magnetic flux
structure in high-Tc superconductors, based on the
Faraday effect, is shown in Fig. 2,a. The supercon-
ducting sample was covered with a garnet film
doped with bismith. The external magnetic field
was applied parallel to the c axis of the sample. The
magnetic flux profile was determined by observing
changes of the domain structure. The domain struc-
ture (in a garnet film placed above the supercon-
ducting sample) was observed as a function of
magnetic field intensity and may be regarded as
magnetic flux sensor.
The visual pattern of magnetic flux trapping in
the YBa2Cu3O7−δ sample when the external mag-
netic field is decreased from some maximum value
to zero is shown in Fig. 2,b. The magnetooptical
image obtained is subjected to preliminary com-
puter processing.
Applying computer methods for image processing
allowed us to examine the trapped magnetic flux
distribution both on a qualitive level and to perform
quantitative estimations of the local magnetic in-
duction (Be) distribution over the sample surface.
When a Bi-containing iron garnet film was used as
Fig. 1. Simulated magnetostriction loops of the
La1.85Sr0.15CuO4 in a transverse magnetic field for different
hm : 2.5 (1); 5 (2); 12.5 (3); 25 (4).
V. V. Bruk, V. V. Eremenko, N. I. Makedonskaya, Yu. A. Shabakayeva, and V. A. Sirenko
420 Fizika Nizkikh Temperatur, 2001, v. 27, No. 4
indicator, the preliminary stage of computer pro-
cessing included low-frequency spatial image filter-
ing in order to eliminate the disturbances induced
by the labyrinth magnetic structure in the regions
of Be below the saturation field. For this reason the
initial image was scanned by a sliding window of
square shape with a size a few times larger than the
period of the labyrinth structure. For each selected
window a two-dimensional Fourier transform was
performed, the high-frequency harmonics associated
with the labyrinth structure were filtered out, and
inverse Fourier transform was performed. When a
film with in-plane anisotropy was used as a indica-
tor, there was no need of the preliminary image
processing.
In order to obtain a quantitative estimate of the
local magnetic induction after the low-frequency
image filtering a segmentation of the latter was
performed by separation of the regions with the
comparatively uniform values of beam light inten-
sity. This segmentation was based on analysis of a
histogram of the numerical equivalents of the image
brightness. In the interactive mode the brightness
values were shown on the histogram, which corre-
sponded to the boundaries between the separated
grades of the image elements. To visualize the
segmented image, all elements within the separated
grade were coded by a definite color.
The transition to quantitative estimates was
based on the following assumptions:
a) the angle between the transmission planes of
the polarizer and analyzer did not differ much from
the normal one;
b) trapped magnetic flux rotates the plane of
polarization by small angle.
The first assumption means that the intensity of
the registered light beam is linear in the second
power of the sine of the rotation angle ϕ of the
plane of polarization: I = I0 sin
2 ϕ. The second as-
sumption allows the sine of the small angle to be
replaced by the angle itself expressed in radians:
I ≈ I0ϕ2. Taking into account that ϕ ∼ B, one ob-
tains I ∼ α2B2, where α is a linear factor deter-
mined by the sample surface, B is the integral value
of the magnetic induction. A pattern of the obtained
distribution of the local magnetic induction is
shown in Fig. 3.
In addition the computer processing allows one
to analyze the surface distribution of the local
values of the Be and Be
2 gradients, the latter being
directly connected with the pinning force.
In order to study the spatial distribution of the
Be
2 gradient the image was spatially differentiated
using the Sobel operator [12]. For this reason the
projections of the image brightness gradients were
calculated (in arbitrary integer units) along the
axes x and y by the method of image convolution
with masks of the Sobel operator:
Hx
=
−1, 0, 1
−2, 0, 2
−1, 0, 1
; Hy =
1, 2, 1
0, 0, 0
−1, −2, −1
,
∂I
∂x(y)
(jx , jy) =
= ∑
i
x
=−1
1
∑
i
y
=−1
1
I(jx + ix , jy + iy)Hx(y)(ix + 2, iy + 2) ,
(4)
Fig. 2. Magnetooptical scheme: a) 1 — HTSC, 2 — indicator;
−k, +k — transmitted and reflected light, respectively, Be —
external magnetic field; b) visualization of trapping of magnetic
flux in the sample of YBa2Cu3O7−δ (T = 14 K).
a
b
Investigation of magnetic flux gradients in hard superconductors
Fizika Nizkikh Temperatur, 2001, v. 27, No. 4 421
where jx , jy are the numbers of the row and co-
lumn, respectively, with the intersection corre-
sponding to the image element for which the gra-
dient value is estimated.
Further, the modulus of the gradient is calcu-
lated according to the equation:
∂Be
2
∂ r
=
α
12b
∂I
∂x
2
+
∂I
∂y
2
1/2
, (5)
where b is the size of a single pixel of the image.
The appearance of the factor α/12b may be
explained by the following equation:
∂B
∂x
=
α
3
I13 − I31
2 √2 b
1
√2
+
I23 − I12
2b
+
I33 − I11
2 √2b
1
√2
=
=
α
12b
[2(I23 − I12) + (I13 − I31) + (I33 − I11)] , (6)
where Ii,j is the intensity for the image element
with the row number i and column number j in the
selected window (i, j = 1–3). A similar formula
may be written for ∂B/∂y also. Then
∂B
∂ r
=
∂B
∂x
2
+
∂B
∂y
2
1/2
.
In Fig. 4 a typical distribution of pinning force
is presented. It is seen from the figure that the
maximum values of |∂Be
2/∂r| are observed within the
boundary of the trapped magnetic flux region.
Estimaties of the |∂Be/∂r| values were made in
order to study the dependence of this characteristic
on Be and hence to estimate magnetostriction with
respect to (3).
The calculations were performed along different
lines within the image, on going from the boun-
daries of the trapped magnetic flux region to the
region of maximum values of I using the following
equations:
∂Be
∂r
=
1
2Be
∂Be
2
∂r
, (7)
Be
= α √I . (8)
For qualitative analysis of the obtained depend-
ence the dispersion diagrams were constructed in
the plane Be − |∂Be/∂r|. A typical scatter diagram is
presented in Fig. 5. As is seen from the figure the
gradient initially increases with Be and then de-
creases.
Fig. 3. Distribution of local magnetic induction, mT (filtered
pattern).
Fig. 4. Distribution of pinning force Fp , N/mm2 for a frag-
ment outlined in Fig. 3.
V. V. Bruk, V. V. Eremenko, N. I. Makedonskaya, Yu. A. Shabakayeva, and V. A. Sirenko
422 Fizika Nizkikh Temperatur, 2001, v. 27, No. 4
Experimental data
The magnetostriction and magnetization meas-
urements were performed on a single crystal of
YBa2Cu3O7−δ . The superconducting transition
temperature was determined from the temperature
dependence of zero-field-cooled magnetization and
equalled TSN = 92 K. The magnetostriction was
measured by strain gauges along different crystal-
lographic directions with an external magnetic field
perpendicular to the face of the sample. The magne-
tization of the sample was measured in a vibrational
magnetometer. The experimental geometry used al-
lowed us to control the surface flux distributions
with magnetooptical sensors by utilizing demagne-
tizing effects.
Figure 6 shows the magnetostriction of YBa2Cu3O7−δ
at 14 K. A more detailed analysis of the magneto-
striction gives good agreement between the theory
and experiment. We have obtained the width of the
magnetostriction hysteresis loop at 14 K in a mag-
netic field of 12 T about 2⋅10−5 both theoretically,
using formula (3), and experimentally.
The Young’s moduli of YBa2Cu3O7−δ along the
direction of the magnetostriction measurements
were calculated from the following equation:
1
E
=
c22c33 − c23
2
c11c22c33 + 2c12c13c23 − c11c23
2 − c22c13
2 − c33c12
2 +
+
ny
2[c33(c11 − c22) + c13
2 − c23
2 ] + nz
2[c22(c11 − c33) + c12
2 − c23
2 ]
c11c22c33 + 2c12c13c23 − c11c23
2 − c22c13
2 − c33c12
2 +
+ nx
2ny
2
1
c66
+
2(c13c23 − c12c33) − (c22c33 − c23
2 ) − (c11c33 − c13
2 )
c11c22c33 + 2c12c13c23 − c11c23
2 − c22c13
2 − c33c12
2
+
+ nx
2nz
2
1
c44
+
2(c12c23 − c22c13) − (c11c22 − c12
2 ) − (c22c33 − c23
2 )
c11c22c33 + 2c12c13c23 − c11c23
2 − c22c13
2 − c33c12
2
+
Fig. 6. Experimental and simulated magnetostriction loops for a
YBa2Cu3O7−δ single crystal.
Fig. 5. Typical scattering diagram Be − |∂Be/∂r|.
Investigation of magnetic flux gradients in hard superconductors
Fizika Nizkikh Temperatur, 2001, v. 27, No. 4 423
+ ny
2nz
2
1
c55
+
2(c12c13 − c11c23) − (c11c33 − c13
2 ) − (c11c22 − c12
2 )
c11c22c33 + 2c12c13c23 − c11c23
2 − c22c13
2 − c33c12
2
(9)
where the elastic moduli cik are derived from the
acoustic measurements [13], and n is the unit vector
along a direction of magnetostriction measurement
in the coordinates related to the principle lattice
axes.
Conclusion
A model description of magnetostriction is pro-
posed for a thin superconductor in a magnetic field
perpendicular to a flat face of the sample. The
simulated magnetostriction loop calculated using
flux gradients derived from magnetooptical image
processing is in a semi-quantitative agreement with
the transverse magnetostriction measurements on
the single crystal YBa2Cu3O7−δ .
The authors acknowledge Dr. S. B. Feodosyev
for his assistance in computations. We are grateful
to Prof. H. Szymczak for stimulating discussions.
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V. V. Bruk, V. V. Eremenko, N. I. Makedonskaya, Yu. A. Shabakayeva, and V. A. Sirenko
424 Fizika Nizkikh Temperatur, 2001, v. 27, No. 4
|
| id | nasplib_isofts_kiev_ua-123456789-130021 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0132-6414 |
| language | English |
| last_indexed | 2025-12-07T17:58:53Z |
| publishDate | 2001 |
| publisher | Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| record_format | dspace |
| spelling | Bruk, V.V. Eremenko, V.V. Makedonskaya, N.I. Shabakayeva, Yu.A. Sirenko, V.A. 2018-02-04T16:50:06Z 2018-02-04T16:50:06Z 2001 Investigation of magnetic flux gradients in hard superconductors / V.V. Bruk, V.V. Eremenko, N.I. Makedonskaya, Yu.A. Shabakayeva, V.A. Sirenko // Физика низких температур. — 2001. — Т. 27, № 4. — С. 419-424. — Бібліогр.: 13 назв. — англ. 0132-6414 PACS: 75.30.Kz https://nasplib.isofts.kiev.ua/handle/123456789/130021 Analytical calculations of the transverse magnetostriction in a thin hard superconductor are presented in relation to the distributions of currents and fields within a superconducting specimen in a varied magnetic field. The approach is successfully tested on high-temperature superconductors. The flux distribution derived from magnetostriction measurements is in satisfactory agreement with that obtained from computer processing of magnetooptical images. The magnetic flux distribution below the irreversibility line of hard superconductors is derived from both original magnetization and magnetostriction measurements and image processing. Perfect consistency of the results is obtained for a family of high-temperature superconductors (La₂₋xSrxCuO₄, Bi₂Sr₂CaCu₂Ox, YBa₂Cu₃O₇₋d). The authors acknowledge Dr. S. B. Feodosyev
 for his assistance in computations. We are grateful
 to Prof. H. Szymczak for stimulating discussions. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур Низкотемпеpатуpная магнитостpикция магнетиков и свеpхпpоводников Investigation of magnetic flux gradients in hard superconductors Article published earlier |
| spellingShingle | Investigation of magnetic flux gradients in hard superconductors Bruk, V.V. Eremenko, V.V. Makedonskaya, N.I. Shabakayeva, Yu.A. Sirenko, V.A. Низкотемпеpатуpная магнитостpикция магнетиков и свеpхпpоводников |
| title | Investigation of magnetic flux gradients in hard superconductors |
| title_full | Investigation of magnetic flux gradients in hard superconductors |
| title_fullStr | Investigation of magnetic flux gradients in hard superconductors |
| title_full_unstemmed | Investigation of magnetic flux gradients in hard superconductors |
| title_short | Investigation of magnetic flux gradients in hard superconductors |
| title_sort | investigation of magnetic flux gradients in hard superconductors |
| topic | Низкотемпеpатуpная магнитостpикция магнетиков и свеpхпpоводников |
| topic_facet | Низкотемпеpатуpная магнитостpикция магнетиков и свеpхпpоводников |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/130021 |
| work_keys_str_mv | AT brukvv investigationofmagneticfluxgradientsinhardsuperconductors AT eremenkovv investigationofmagneticfluxgradientsinhardsuperconductors AT makedonskayani investigationofmagneticfluxgradientsinhardsuperconductors AT shabakayevayua investigationofmagneticfluxgradientsinhardsuperconductors AT sirenkova investigationofmagneticfluxgradientsinhardsuperconductors |