BGU Physics Department

Experimental Superconductivity Laboratory,

Prof. Grzegorz Jung - group leader,
Dr. Valery Dikovsky;
Ilan Barboy, M.Sc. student.

Basic Research: Quantum Coherence, Vortex Matter, High-Tc Superconductivity, Metastable Superconductivity, Random Processes in Superconductors Applied Research: High and Low Current Applications of Superconductors.

Keywords: Josephson and Quasi-Josephson effects, Vortex Dynamics, Flux and Voltage Noise, Random Telegraph Noise, Flux Metastability.

Applied Research:
Superconducting Electronics, Thin Superconducting Films, Quasi Josephson Vortex Devices, Fault Current Limiter, AC losses, Superconducting Particle Detectors.

General description of the field and its place in the overall physical picture: Along with the laser and superfluidity, superconductivity is the only quantum phenomenon observable on the macroscopic scale, and as such, Superconductivity is regarded one of the most important subjects of the solid state physics in the 20-th century. Discovered in 1911 went through several periods of astonishing developments starting from early spectacular experimental discoveries, through phenomenological models to microscopic quantum theory and development of practical applications. Recent discoveries of high-Tc superconductivity (1986) that still awaits for the theoretical explanation renewed the general interest in the field in all its aspects.

Scientific background: High critical temperature superconductors (HTSC) are characterized by strong anisotropy of their superconducting parameters, very short coherence length, and low pinning energies. This features lead to random movements of vortices at low magnetic fields, strong random voltage and flux fluctuations, and pronounced manifestations of surface barriers.Departure of the vortex system from equilibrium caused by a change of external parameters may bring about smooth relaxation to the equilibrium conditions or may result in pronounced magnetic metastabilities. We study the dynamics of vortex nucleation and penetration into a superconducting strip, including the instabilities in penetrating flux front in increasing magnetic field and/or transport current, as well as an opposite process of flux expulsion at decreasing field. Metastability of a diluted Abrikosov lattice and activation energies of vortex processes are investigated by means of statistical analysis of random telegraph voltages resulting from intrinsic flux fluctuations and direct measurements of the flux noise. Geometrical metastabilities in superconducting strip are being employed for developing radiation detectors and spectroscopy.

Periodic potential structures enforce coherence in current driven vortex motion in superconducting specimens thus resulting in the appearance of quasi-Josepson effects similar to the well known phenomena occurring in classical Josephson weak links. We investigate the dynamics of coherent vortex motion by means of studying dc current voltage characteristics and interactions between vortices and microwave fields.
The applied research project based on collaboration between academic institutions of Israel and Germany and German industrial enterprises is aimed at development of a superconducting fault current limiter for electrical networks, a device that is likely to become the first practical large-scale application of high-temperature superconductors. The concept of a superconducting fault current limiter is very simple, nevertheless the apparently straightforward path to build a practical device is hampered by severe problems associated with thermo-mechanical stability of an active superconducting element. We have originally elaborated the concept of an open core inductive device and designed and tested several small-scale models of the fault current limiter. Recently we have developed the concept of flux creep based current limiter that overcomes the above mentioned obstacle. In connection with the limiter project we are pursuing extensive investigations of ac losses and flux motion associated dissipation in superconducting cylinder. Significance and possible applications (technological, interdisciplinary, basic, etc.): High-Tc superconducting system containing a system of Abrikosov vortices can be regarded as a laboratory model for a general class of nonlinear systems in which one can easily change external parameters controlling the dynamics of the system such as magnetic field, current flow, or temperature. The experimental investigations of vortex matter dynamics, by means of measuring electric, magnetic, and noise properties enable one to get a deeper insight into the performance of such systems. Noise measurments performed with the superconducting systems enable one to eliminate background noise due to dissipation of electric charge carriers and obtain a clear picture of the intrinsic nature of stochastic processes associated with magnetic flux motion.

Technological applications of this research are concentrated around the concept of a superconducting fault current limiter for power distribution networks, superconducting particle detectors based on geometrical flux metastability in superconducting strips, and cryoelectronic devices based using quasi-Josephson effects associated with enforced coherent vortex motion in artifical periodic potential structures.

 Achievements in a general physical language understandable for physicists working in far fields: Pronounced random telegraph fluctuations (RTN) of the voltage and magnetic flux in current biased superconducting strips have been detected. These fluctuations have been modeled theoretically and associated with various forms of vortex matter dynamics depending on the type and the structure of the superconducting material. Original, new and efficient  techniques for statistical analysis of (RTN) signals and underlying background fluctuations have been developed and successfully applied.

New technique enabling reversible enforcment of arbitrary pinning structures in superconducting films has been developed and employed to enforce the coherence in current driven vortex flow in large high-Tc superconducting thin strips. New phenomena of quasi-Josephson self-induced resonant current steps in the current-voltage characteristics of thin film strips has been observed and explained by the theoretical model.

Several small and medium scale models of a superconducting fault current limiter (FCL) have been constructed and tested in the laboratory conditions. In 1996 one of our models reached the best power limiting parameters in the World for the inductively coupled device. Recently a feasibility of an inductive (FCL) model operating in the flux creep mode has been practically demonstrated. This model eliminates the problem of the overheated domains in the active superconducting ring that constitutes one of the most severe obstacles hampering practical development of the full scale superconducting FCL.

External Collaborations

a Israeli Collaborations:

- Groupe de Physique des Solides, Universite Paris 7, Prof. Georges, Waysand: Metastable Superconductivity, superconducting particledetector, low noise underground physics.
- Ecole Politechnique, Dr. Marcin Konczaykowski: Vortex Dynamics. CEA, DRECAM/SPEC
- Orme des Merisiers, Orsay, Dr. Miguel Occio: Random processes insuperconductors and CMR manganites.
- Instytut Fizyki PAN Warszawa, Prof. S.J. Lewandowski: Superconducting thin films, Josephson effects, microwave cryoelectronics
- Istituto di Cibernetica, Arco Felice Napoli, Prof. Antonio Barone, Dr. Carlo Camerlingo: High-T$_c$ thin films, vortex dynamics
- University of Salerno, Prof. Bonaventura Savo: Random telegraph noise in superconductors.
- The Institute for Technical Physics (ITP), Research Centre Karlsruhe (KfK), Prof. Peter Komarek, Superconducting fault current limiter;
- Aventis AG (Hoechst) Frankfurt A/M. - Industrial partner cooperating in the fault current limiter development in the framework of a GIF grant
- Badenwerk AG Karlsruhe, Industrial partner cooperating in the fault current limiter development.
- University of Rochester and Laser Energetic Laboratory, Prof. Roman Sobolewski: Laser induced artificial pinning structures and laser film patterning, femtosecond scale sampling measurements of voltages due to nucleation and motion of flux vortices.
- Argonne National Laboratory, Dr. Valery Vinokur: Modeling of channeled flow of vortices in superconducting thin films.


Cryogenic Equipment
Modified Variox cryostat, Oxford Instruments; Differential temperature cryostat with fixed 4.2 K and variable (2.17 to 300 K) temperature zones.
Closed cycle refrigerator Janis CCS 200 for 8-300 K range.
Variable temperature liquid nitrogen cryostat with the microwave insert
dc SQUID designed to operate inside the Variox + control electronics.
Temperature controllers; Lake Shore 330 and Oxford Instruments ITC 503
Helium 4 bath cryostat for insertion into an electromagnet.
Liquid nitrogen bath cryostats
Liquid cryogen storage dewars.
Self pressurized nitrogen gas and liquid dispenser.

Signal analysis and data acquisition equipment:
Two channel Spectrum and Cross-correlation analyzer SR 780 with extended record memory.
FFT Network Analyzer SR 770 Stanford Research, (0.476 mHz - 100 kHz.
RF Spectrum Analyzer HP8568B (100 Hz - 2.5 GHz.
Microwave Spectrum Analyzer HP8592L, up to 26 GHz
Noise source HP 346A (10MHz-18 GHz.
X-band Microwave Waveguide Setup - in acquisition
Digital oscilloscope Tektronics TDS 500 (4-channel, 500 MHz
Digital Lock-in Amplifier SR850 Stanford Research.
Two-phase Lock-in Amplifier SR 570 Stanford Research.
Programmable Current Sources, Keithley 2243, Keithley 236, Yokagushi 120.
Programmable Band pass filter/amplifier SR650 Stanford Research.
Programmable Low Noise Amplifiers, SR560 Stanford Research
Programmable Arbitrary Function Synthesizer, Keithley 3940
Nanovoltmeters Keithley 182, DMM Keithley 196, DMM Keithley 2001.
Low noise preamplifiers: low frequency dc to 100 kHz (SR 550, SR 552, SR 554), high frequency up to 35 MHz Hypres HA-103-35, and UHF 0.1-1500 MHz HP 8447D.
Several home made low noise instrumental amplifiers dc - 100 kHz, noise level 1-2 nV/Hz1/2
Several desktop computers with GPIB interface, optoelectronics GPIB bus isolators/extenders, Keithley Metrabyte Viewdac and National Instruments Labview data acquisition packages.

High Current Superconducting Systems:
 System for measurement of I-V characteristics and critical currents up to 250 A (pulse method(
 System for measurement of I-V characteristics and critical currents up to up to 10,000 A (inductive method),
 Experimental set-up for testing current limiting devices and switches with ac and dc currents up to 10 A, and single phase voltages up to 380 V.
 Several small and middle scale prototypes of Superconducting Fault Current Limiter

Grants executed in the last 5 years

 1995-96 French Israeli AFIRST found,
 "Dynamics Manifestations of Magnetic Metastabilities in High-Tc Superconductors".
 1995-97 Israeli Ministry of Science and the Arts,
 "Dynamic of Vortices in High-Tc Superconducting Films with Periodic Magnetic Inhomogenities"
 1996-98 GIF, German-Israeli Foundation for Scientific Research and Development, "Superconducting Fault Current Limiter"
 1997-98 National Research Council of Italy,
Research and Training Programme for Third Mediterranean Countries, "Superconductor Based Devices".
 1998 Arc-en-Ciel French-Israeli Exchange Program,
 "Dynamics of Magnetic Flux Metastabilities in Superconductors".
 1998-2000 Israeli Academy of Sciences,
 "Quasi-Josephson Effects in Coherent Motion of Abrikosov Vortices in Periodic Magnetic Structures"
 1998-2000 Israeli Ministry of Science and Polish Komitet Badan Naukowych, Bilateral Polish-Israeli Exchange program
"Josephson Effects due to Enforced Vortex Coherence in High-T$_c$ Superconductors”
 1999-2001 Italian-Israeli Cooperation Program
"  Nonlinear Dynamics of Josephson Networks"
 2001-2003 Arc-en-Ciel French-Israeli Exchange Grant
" Vortex Matter in Superconductors.”
 2001-2004 Binational Science Foundation Grant (BSF),
"  Dynamics of Channelled Vortex Motion.”
 2001-2004 The Israel Science Foundation,
 “Phase Separation and Percolation Effects in CMR Manganites”

Perspectives and plans for the next 3-5 years:
 Continuation of the current research lines:
We plan to develop in cooperation with Weizmann Institute a new technique for noise measurements which will be based on measuring the cross-correlation between different sensors. This, if successful will enable us to push the sensitivity of noise measurements orders of magnitude further with respect to the current possibilities.
Geometrical and flux front instabilities will be investigated in the conditions of extremely low noise background in the underground facility in Rustrell, France. We are actively participating in the creation of this facility and establishment of the research plans.
Coherent vortex motion and quasi-Josephson research will be developed towards detection of the microwave radiation generated by the moving vortex system. Relevant theoretical picture will be developed. In collaboration with Ecole Polytechnique the transport and noise measurements will be correlated with the magneto-optic observations.
Fault current limiter: We plan to establish procedures for modeling a full scale device performance using the medium scale device by means of physical modeling procedures. We have already developed the formal background for this modeling and plan to verify it experimentally in the near future.

New research project:

We plan to apply the methodology developed for noise investigations in superconductors to studies of the properties of other classes of materials, like e.g., giant magnetoresistance materials.
We plan to start investigations of the vortex matter dynamics in very high magnetic fields in cooperation with Tel Aviv University.

Recent publications, last 5 years

 M. Bonaldi, G. Jung, B. Savo, A. Vecchione, and S. Vitale
Inrinsic High-T$_c$ Josephson Junctions in Random Telegraph Noise
Fluctuators Phys. Rev. B 53, 90-93 (1996)

M. Bonaldi, G. Jung, B. Savo, A. Vecchione, and S. Vitale,
Intrinsic Josephson junctions and Random Telegraph Voltage Noise in
Granular HTSC Thin Films.
In Applied Superconductivity , ed. D. Dew Huges,
Institute of Physics Publishing, Bristol, 1007--1010 (1996)

 G. Jung, B. Savo, A. Vecchione,
Amplitude Modulated Random Telegraph Voltage Noise.
in Applied Superconductivity , ed. D. Dew Huges,
Institute of Physics Publishing, Bristol, 1003--1006 (1996)

V. Meerovich, V. Sokolovsky, G. Jung, S. Goren,
High-Tc Superconducting Fault Current Limiter.
In Applied Superconductivity , ed. D. Dew Huges,
Institute of Physics Publishing, Bristol, 603--606 (1996).

 V.D. Ashkenazy, G. Jung, B.Ya. Shapiro,
Nuclear Spin Relaxation Rate due to Random Motion of Vortex Bundles.
Phys. Rev. B 53 , 2686--2690 (1996).

 V.D. Ashkenazy, M. Bonaldi, G. Jung, B.Ya. Shapiro, and S. Vitale
 Non--monotonic High Frequency Flux--Flow Noise Spectra in
 High-Tc Superconductors.
 Solid State Communications , 98 , 517--521 (1996).

 G. Jung and B. Savo
 Elementary and Macroscopic Two Level Fluctuations in High--Tc
 J. Appl. Phys. 80 , 2939--2948 (1996).

 V.D. Ashkenazy, G. Jung, B.Ya. Shapiro,
 Cross-Correlation of Thermal Flux Noise in Layered Superconductors.
 Phys. Rev. B  54 , 9428--9435 (1996).

 V.D. Ashkenazy, C. Coccorese, G. Jung, B. Savo, and B.Ya. Shapiro,
 Random Telegraph Voltages in High-Tc Thin Film at Zero Magnetic Field.
 Czechoslovak J. Phys. 46 Supl., 1363--1364 (1996).

 C. Coccorese, G. Jung, B. Savo,
 Voltage Fluctuations in Bi2Sr2CaCu208+x Films at Zero Field.
 Czechoslovak J. Phys. 46 Supl., 1365--1366 (199.(6

 V.D. Ashkenazy, G. Jung, B.Ya. Shapiro,
 Cross--Correlation of Magnetic Noise of Layered Superconductors.
 Czechoslovak J. Phys. 46 Supl., 1741--1742 (1996).

 V. Meerovich, M. Sinder, V. Sokolovsky, S. Goren, G. Jung, G. E. Shter,
 and G.S. Grader,
 Penetration Dynamics of a Magnetic Field Pulse into High-Tc Superconductors
 Supercond. Sci. Technol. 9 , 1024--1047 (1996).

 G. Jung and B. Savo
 Elementary and Macroscopic Two Level Fluctuators in High—Tc Superconductors.
 Molecular Phys. Repts. , 15 , (1996).

 G. Jung, Y. Yuzhelevski, B. Savo, C. Coccorese, V. D. Ashkenazy, B.Ya. Shapiro,
 Random Telegraph Voltages in High--Tc Superconducting Films.
 in Unsolved Problems of Noise ed. Ch. R. Doering, L. Kiss, and M. F. Shlesinger, World Scientific, 285--290 (1997)

 M. Bonaldi, G. Jung, A. Vecchione, and S. Vitale,
 Hot Gas Temperature Controller for a Cryostat Insert Having High Stability.
 Rev. Sci. Instr. , 68 , 2071--2075 (1997)

 G. Jung, B. Savo, and Y. Yuzhelevski
 Quiet and Noisy Metastable Voltage States in High--Tc Superconductors.
 in Noise in Physical Systems and 1/f Fluctuations ,
 Eds. C. Claeys and E. Simon, World Scientific, 325--328 (1997)

 V. Meerovich, V. Sokolovsky, S. Goren, G. Jung, G.E. Shter,
 and G.S. Grader,
 A Hybrid Magnetic Shield Employing Ferromagnetic Iron and HTSC Ring.
 in "Applied Superconductivity 1997", ed. H. Rogalla and D.H. Blank,
  IoP Conference Series, 158 , 1651--1654 (1997)

 V. Meerovich, V. Sokolovsky, S. Goren, G. Jung, I. Vajda, A. Szalay, N. Gobi
 Application of Inductive HTSC Current Limiters in Distribution Networks.
 in "Applied Superconductivity 1997", ed. H. Rogalla and D.H. Blank,
  IoP Conference Series, 158 , 1227--1230 (1997)

 C. Coccorese, G. Jung, B. Savo, and Y. Yuzhelevski
 Voltage Fluctuations Below the Critical Current in a Superconducting BSCCO Thin Film.
 in "Applied Superconductivity 1997", ed. H. Rogalla and D.H. Blank,
  IoP Conference Series, 158 , 17--20 (1997)

 V. Meerovich, V. Sokolovsky, J. Bock, S. Gauss, S. Goren, and G. Jung,
 Performance of BSCCO Cylinders in a Prototype of Inductive Fault Current Limiter.
 in "Applied Superconductivity 1997", ed. H. Rogalla and D. H. Blank,
  IoP Conference Series, 158 , 1179--1182 (1997)

Y. Yuzhelevski, C. Camerlingo, M. Ginovker, M. Guilloux-Viry, R. Monaco, A. Perrin, B.Ya. Shapiro, and G. Jung,
 Coherent Motion of Vortices in Periodic Magnetic Structures.
 in "Applied Superconductivity 1997", ed. H. Rogalla and D. H. Blank,
  IoP Conference Series, 158 , 81--84 (1997)

 G. Jung and B. Savo
 Josephson Mechanism in Random Telegraph Voltage Noise in High--Tc Superconductors.
 Applied Superconductivity , 6 , 391-397 (1998)

 V. Sokolovsky, V. Meerovich, S. Goren, and G. Jung,
 Analytical Approach to AC Loss Calculation in High-Tc Superconductors.
 Physica C 306 , 154-162 ((1998

 V. Sokolovsky, V. Meerovich, S. Goren, J. Bock, S. Gauss, and G. Jung,
 AC Losses in BSCCO Cylinders Operating in Inductive Current Limiter.
 IEEE Trans. Appl. Supercond. , 9 , 1361-1364 (1999).

 C.Camerlingo, C.Nappi, M.Russo, and G.Jung,
 In-Plane Properties of (103)/(013) Oriented YBCO Films.
 Int. J. Mod. Phys. B 13 , 1091-96 (1999)

 F. Bass, G. Jung, B.Ya. Shapiro, I. Shapiro,
 Pinning Modulation of Flux-Antiflux Dynamics.
 Physica C 314 , 254-262 (1999)

 Y. Yuzhelevski and G. Jung
Artificial Reversible and Programmable Magnetic Pinning for High-Tc Superconducting Thin Films.
 Physica C 314 , 163-171 (1999)

 V. Sokolovsky, V. Meerovich, S. Goren, and G. Jung,
 AC Losses in BSCCO Hollow Cylinders with Induced Current.
 Physica C 319 , 238-248 (1999)

 Y. Yuzhelevski, G. Jung, C. Camerlingo, M. Russo, M. Ghinovker, B.Ya. Shapiro,
 Current Driven Vortex Dynamics in a Periodic Potential.
 Phys. Rev. B 60 , 9726-9733, (1999)

 V. Meerovich, V. Sokolovsky, J. Bock, S. Gauss, S. Goren, and G. Jung,
 Performance of Inductive Fault Current Limiter Employing BSCCO Superconducting Cylinders.
 IEEE Trans. Appl. Supercond. , 9 , 4666-4676 (1999)

 B.Ya. Shapiro, M. Ghinovker, I. Shapiro, and G. Jung,
 Nucleation of Superconductivity in an Overcooled Normal Domain.
 Physica B 284-288 , 765-766 (2000).

 G. Jung, V. Jeudy, D. Limagne, G. Waysand, T.A. Girard, M.J. Gomes, and B.Ya. Shapiro,
  Real Time Dynamics of Flux Expulsion from Type-I Superconducting Strips.
  Inst. Phys. Conf. Ser. 162 , 955-958 (2000).

 V. Sokolovsky, V. Meerovich, S. Goren, and G. Jung,
  Quenching in BSCCO Hollow Cylinders Employed in an Inductive Fault Current Limiter.
  Inst. Phys. Conf. Ser. 162 , 747-750 (2000(

 Y. Yuzhelevski, M. Yuzhelevski, and G. Jung
  Random Telegraph Noise Analysis in Time Domain
  Rev. Sci. Instr. , 71 1681-1688 (2000).

 G. Jung, S.J. Lewandowski, B.Ya. Shapiro, and Y. Yuzhelevski
 Quasi-Josephson Effects in Coherent Vortex Motion in a Periodic Potential.
 Physica C  332 , 51-57 (2000).

 C. Camerlingo, C. Nappi, M. Russo, and G. Jung,
 Angle Dependence of Vortex Pinning in (103)/(013) Oriented YBCO Films
 Physica C 341 , 1349-1350 (2000)

 S.J. Lewandowski, G. Jung, V.D. Okunev, Z.A. Samoilenko, A. Abal'oshev ,P. Gier{\l owski, and A. Klimov,
 Laser Irradiation Effects in Crystalline and Amorphous YBaCuO Thin Films.
 Proc. SPIE , 4086 , 518-523 (2000).

 Y. Paltiel, E. Zeldov, Y. Myasoedov, M.L. Rappaport, G. Jung, S. Bhattacharya, M.J. Higgins, Z. L. Xiao, E.Y. Andrei, P.L. Gammel, and D.J. Bishop
 Instabilities and Disorder-driven First-order Transition of the Vortex Lattice.
  Phys. Rev. Lett. , 85 , 3712-3715 (2000).

 G. Jung, B. Savo and Y. Yuzhelevski,
 Quiet and Noisy Metastable Voltage States in High-Tc Superconductors.
 Phys. Rev. B  62 , 6674-6680 (2000).

 V. Sokolovsky, V. Meerovich, M. Gladstein, Y. Yuzhelevski, G. Jung, and S. Goren,
 An Experimental Study of Heating Due to the ac Loss in a BSCCO Cylinder.
 Physica C  336 , 102-106 (2000).

 G. Jung, M. Ocio, Y. Paltiel, H. Shtrikman, E. Zeldov
 Magnetic Noise Measurements Using Cross-Correlated Hall Sensor Arrays
 Appl. Phys. Lett. 78 , 359-361 (2001).

 V. Markovich, E. Rosenberg, Y. Yuzhelevski, G. Jung, G. Gorodetsky, D. A. Shulyatev, Ya. M. Mukovskii
 Observation of Correlation Between Electroresistance and Magnetoresistance
 in La0.82 Ca0.18MnO3 Single Crystal.
 Appl. Phys. Lett. 78 , 3499-3501 (2001)

 Y. Yuzhelevski, V. Dikovsky, V. Markovich, G. Gorodetsky, G. Jung,  D. A. Shulyatev, Ya. M. Mukovskii
  Current Induced Telegraph Noise in CMR Manganites
  Fluctuation and Noise Letters 1 , L105-L109 (2001)

 Ghassan Yassin, Stafford Withington, Cavendish, Grzegorz Jung, Valery Dikovsky, Ilan Barboy, M. Kambara, D. A. Cardwell, Stafford Withington,
 Investigation of Microwave Propagation in High-Temperature Superconducting Waveguides,
 IEEE Microwave and Guided Wave Letters , 11 , 413-415 (2001).

 Y. Yuzhelevski, V. Markovich, V. Dikovsky, G. Gorodetsky, G. Jung, D. A. Shulyatev,
 and Ya. M. Mukovskii
 Current-Induced Metastable Resistive States and Memory Effects in Low-Doped Manganites.
 Phys. Rev. B 64 , 224428 (2001).

 Y. Paltiel, G. Jung, Y. Myasoedov, M. L. Rappaport, E. Zeldov, S. Bhattacharya, M. J. Higgins,
  Dynamic Creation and Annihilation of Metastable Vortex Phase as a Source of Excess
  Noise. Europhys. Lett. , in print.

 G. Jung, V. Jeudy, D. Limagne, G. Waysand, T.A. Girard and M.R. Gomes
 Expulsion of Magnetic Flux from a Type I Superconducting Strip
 Physica C , in print.

 G. Yassin, G. Jung, I. Barboy, V. Dikovsky, M. Kambara, D.A. Cardwell, S. Withington, and G. Jung.
 Microwave Transmission Through High-Temperature Superconducting Waveguides.
 Physica C , in print.

 M.I. Tsindlekht, I. Felner, G. Jung,
 Giant Nonlinear Response of Nb Single Crystal in Slowly Varying Magnetic Field
 Physica C , in print.