Spatial characteristics of superconductors and a 3D Hall microscope for their analysis

Based on Hall probes, a scanning microscope with a survey area of 5 $\times$ 5 mm has been developed and used for high-spatial-resolution and sensitivity measurements of the topology of magnetic field $B(X, Y, Z)$ on the surface of and around the YBa$_{2}$Cu$_{3}$O$_{7-x}$ и Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8+x}$ samples without resorting to electrodynamics expressions. The differential technique with the fixation of the beginning and end of the probe displacement ensures the scanning accuracy of about 10 and 1 $\mu$m for scanning along the $X$, $Y$, and $Z$ axes, respectively. The maximal displacement along the $Z$ axis is 25 mm. For eliminating measuring errors, normalized distributions $B(Z)/B^{max}$(0) are extrapolated to the ordinate axis shifted along the $Z$ axis by the width of the gap between the sample surface and the Hall probe.