Magneto-optical imaging of transport current densities in superconductors Page: 4 of 7
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too small by at least a factor of 2, the error becoming arbitrarily large as the sample
thickness shrinks relative to its width.
The errors associated with using the local form of Ampere's law with incomplete
information are avoided in principle by using the Biot-Savart law. Here the field outside the
sample is properly treated as a non-local integral over all the currents, and the shape of the
sample is explicitly taken into account. While exact expressions can be easily written, it is
difficult to invert the non-local integral to recover J from B, and the inversion is, in general,
not unique [see, for example, 10]. However, for the case of a long wire of rectangular cross-
section, the expansion of the exact solution given by Eq. (3) handles the non-locality and
provides a linear relation between the current density and the field gradient for a specific
geometry. We take this solution as the basis for our inversion, and use Eq. (3) ignoring the
geometrical correction w/nt. This approach makes two approximations: that the filaments
have a small aspect ratio w/t and that their current density is uniform. The first is a
controlled approximation, since we know the size of the correction if we know w and t. For
our imaging geometry, t corresponds to the long and w to the short transverse dimension of
the filament, so the aspect ratio correction is small.. The approximation of uniform current
density can be analyzed as follows. The contribution of a given current element to the field
at a distant point falls off as 1/r2. Thus the field at a given point is quite sensitive to the
current density nearby, ahd much 'less sensitive to currents flowing farther away. Given the
large attenuation of 1/r2, even substantial changes in J at distant points have little effect on
the value of the field. The use of Eq. (3) to measure local variations in the current is
strongly confirmed by the results reported below: if the local field depended sensitively on
distant currents, strong local variations in current density would be unresolvable by this
technique. The main effect of using Eq. (3) for local measurements is a smearing of the
derived current density. The measured J is an average over a finite region effectively
defined by the 1 /r2 attenuation of the Biot-Savart law, and the actual variation in J will be
stronger than the measured one.
RESULTS
The multifilament wire imaged in these experiments is a 37-filament BSCCO 2223 powder-
in-tube composite [11,12] fabricated by Intermagnetics General Corporation. It had a
critical current density of approximately 1.7 x104 A/cm2 in self-field at 77 K, corresponding
to a current of 31 A. -A.1.5 cm long section of this wire was polished to about half its original
width, giving final transverse dimensions of 1.37 x 0.20 mm. Magneto-optical images were
taken of the polished surface (defined by the current direction and the short transverse
dimension of the wire) while an applied current up to 21 A was passed through the wire at
zero applied field. To avoid heating, the current was applied in pulses of 0.2 - 0.7 sec, while
the images were recorded with a charge coupled device camera and the voltage. developed
across the sample was recorded on a storage oscilloscope. Four full filaments were visible
on the imaged surface of the wire, as shown in the scanning electron microscope (SEM)
image in Fig. la. Light gray represents the Ag sheath, dark areas correspond to the
superconductor. Also visible in the SEM image are narrow regions of superconductor
(labeled "T") which correspond to the tops of filaments buried deeper in the wire, as has
been confirmed by additional polishing.
A magnetic field map of the surface produced by magneto-optical imaging is shown in Fig.
lb. The gray scale in the image has been compressed by subtracting a linear background to
show all the qualitative features. There is a sequence of light and dark bands running
roughly along the current direction, indicating substantial field gradients in the horizontal
direction. These gradients are the signature of the current distribution in the filaments,
expected on the basis of Eq. (3). The field gradients are shown quantitatively in Fig. 2a,
which plots the variation of BZ along the line indicated at the bottom of Fig. 1. In the region
outside the superconducting filaments, the field decays as expected far from a finite current
distribution. If the current were carried uniformly in the Ag sheath, the field would show a
constant gradient confined to the region of current flow, as is seen for the data at 100 K
where the BSCCO filaments are normal. At lower temperature, where the BSCCO becomes
superconducting, sharp gradients appear over the filaments themselves indicating that
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Crabtree, G. W.; Welp, U.; Gunter, D. O.; Zhong, W.; Balachandran, U.; Haldar, P. et al. Magneto-optical imaging of transport current densities in superconductors, report, December 31, 1995; Illinois. (https://digital.library.unt.edu/ark:/67531/metadc665016/m1/4/: accessed April 24, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.