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Topics in electromagnetic fluctuations at low temperatures and in superconductivity Fink, Hermann Josef 1959

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TOPICS IN ELECTROMAGNETIC FLUCTUATIONS AT LOW TEMPERATURES AND IN SUPERCONDUCTIVITY by HERMANN JOSEF FINK B.A.Sc., University of British Columbia, 1955 M.A.Sc, University of British Columbia, 1956 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the department of PHYSICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October, 1959 ABSTRACTS i i I . CURRENT FLUCTUATIONS IN A SUPERCONDUCTING CIRCUIT CARRYING A CIRCULATING CURRENT Persistent currents i n superconducting lead are free from fluctuations -9 to l e s s than 1.1 x 10 of f u l l shot noise at approximately 2.4 Mc/s. Superconducting currents are also unaffected by the surface condition of the metal to the same l i m i t as stated above. I I . A NEW ABSOLUTE NOISE THERMOMETER AT LOW TEMPERATURES I f three r e s i s t o r s , which are kept at di f f e r e n t temperatures, are arranged i n form of a ^ network and i f two of the thermal noise voltages appearing across the if network are mu l t i p l i e d together and averaged with respect to time, then under certain conditions the correla t i o n between those voltages can be made zero. This condition i s used to calculate the temperature of one noise source provided a l l the resistance Values and the other temperatures are known. A noise thermometer of t h i s kind was constructed which i s capable of measuring temperatures below approximately 140°K. The b o i l i n g points of l i q u i d oxygen and l i q u i d nitrogen were determined absolutely within 0.2 percent using the ice-point as reference. Between 1.3°K and 4.2°K the thermometer had to be calibrated due to errors a r i s i n g i n the equipment and the measured temperatures were then accurate within - 1 percent. 111. QUASI-PERSISTENT CURRENTS IN RINGS COMPOSED OF SUPERCONDUCTING AND NON-SUPERCONDUCTING REGIONS i i i ABSTRACT A number of rings composed of a superconductor (Pb, In) apart from a small insert of normal metal (Cu) perpendicular to the current flow have been investigated between 1.30°K and 4.33°K for Pb-Cu and between 1.30°K and 3.20°K for In-Cu. It was found that for samples with good electrical contact the decay of the magnetic field due to the current is exponential and that the effective resistance increased compared with the bulk resistance of Cu by approximately 2.1 for the Pb-Cu rings and by 18.5 for the Ih-Cu rings. Two different thicknesses of the Cu inserts (6.0125 cm and 0.0053 cm) were used and it was found that the resistivity of the thin Cu insert increased with respect to the thick foi l by lt>% for the Pb-Cu system and by 3d% for the In-Cu system. Part of this relative increase can be explained as a size effect due to electron scattering in the Cu insert. The effective resistance of the Pb-Cu rings shows a max:!mum at approximately 3.4°K. The resistance of the Ih-Cu samples decreases by about 10$ between 3.2°K and 1.3°K. The resistivity of the Cu foil when measured separately was constant for the above temperature range. For samples with "poor" electrical contact (probably due to some copper oxide on the insert) two definite relaxation times were observed. For these samples the effective resistance was current and temperature dependent and it was decreasing for decreasing currents and decreasing temperatures. This can be explained in terms of a rectification effect of the two oxide layers on the insert. The decay of the magnetic field of the ring is consistent with the decay of a current in an L-R circuit. iv IV. THE DESTRUCTION OF SUPERCONDUCTIVITY IN TANTALUM WIRES BY A CURRENT ABSTRACT .The transition from the superconducting to the normal state of various pre-stretched tantalum wires carrying current was investigated. When the resistance of the wire jumps discontinuously from the superconducting to the normal or intermediate state as a current is passed through i t , then this current is defined as the critical current,Ic. For temperatures T < (T c -5 millidegrees K) the resistance of the wire jumps directly from zero resistance to its normal value at the critical current, such that the total cross section of the wire goes effectively into the normal state. Between (T c -5 millidegrees K) and Tc the resistance of the wire jumps at I c to any fraction of the normal resistance between approximately zero and one. For constant temperatures the resistance-current plots show a large hysteresis effect... The transition temperature, Tc, of the various samples is strongly dependent upon their normal resistivity at helium temperatures. If the wires with a small constant current ( 4 . 2 ma) flowing in them are .cooled from above the transition temperature, the resistance decreases above T and approaches zero at approximately Tc where T c is defined by the extrapolation of the T^ -T curve to IQ M 0 . If the wires are heated from below T c the same resistance-temperature curves are reproduced. Faculty of Graduate Studies P R O G R A M M E O F T H E FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY of HERMANN J. FINK B. A. Sc., University of British Columbia, 1955 M. A. Sc., University of British Columbia, 1956 IN ROOM 301 PHYSICS BUILDING TUESDAY, SEPTEMBER 1st, 1959 AT 2:00 P. M. COMMITTEE IN CHARGE DEAN F. H. SOWARD: Chairman J. B.BROWN F.A. KAEMPFFER R. E . BURGESS E . V. BOHN J. M. DANIELS V. GRIFFITTHS K. C. MANN B. A. DUNELL External Examiner: D. K. C. MACDONALD National Research Council, Ottawa TOPICS IN ELECTROMAGNETIC FLUCTUATIONS A T LOW TEMPERATURES'AND IN SUPERCONDUCTIVITY A B S T R A C T Persistent currents in superconducting lead are free from fluctua-Q tions to less than about 10 of full shot noise at approximately 2. 4 Mc/s . Superconducting currents are also unaffected by the surface condition of the metal to the same limit as stated above. If three resistors, which are kept at different temperatures, are ar-ranged in form of a IT network then the covariance of two of the thermal noise voltages appearing across the network can be made zero by adjust-ing one of the resistors. This condition is used to calculate the tempera-ture of one noise source provided all the resistance values and the other temperatures are known. A noise thermometer of this kind was construct-ed which is capable of measuring temperatures below approximately 140°K. The boiling points of liquid oxygen and liquid nitrogen were determined absolutely within 0. 2 percent. Between 1. 3°K and 4. 2°K the thermometer had to be calibrated due to errors arising in the equip-ment and the measured temperatures were then accurate within +_ 1 per-cent. h A number of rings composed'of a superconductor (Pb, In) apart from a small insert of normal metal (Cu) perpendicular to the current flow have been investigated'between' 1. 30°K and'4'. 33°k'for Pb-Cu and between 1. 3 K and 3. 2°K for In-Cu. It was found that for samples with good electrical contact the decay of the magnetic field due to the current is exponential and that the' effective resistance in-creased compared'with the bulk resistance of Cu by approximately 2.1 for the Pb-Cu rings and by 18. 5 for the In-Cu rings. The effective resistance of the Pb-Gu rings-shows a maximum at approximately 3. 4°K. The resistance of the In-Cu samples decreases by about 10% between 3. 2 K and 1. 3QK. The resistivity of the Cu foil when'measured sep-arately was constan^ for the above temperature range. For samples with "poor" electrical contact (probably due to some copper oxide on the insert) two definite relaxation times were observed. This can be explained in terms of a rectification effect of the two oxide layers on the insert. The transition from the superconducting to the normal state of various pre-stretched tantalum wires carrying current was in-vestigated. When the resistance of the wire jumps discontinuously from the superconducting to the normal or intermediate state as a current is passed through it, then this current is defined as the criti-cal current I c . For temperatures T •< ( T c - 5 millidegrees K) the resistance of the wire jumps directly from zero resistance to its nor-mal value at the critical current, such that the total cross section of the "wire goes effectively into the normal state. . P U B L I C A T I O N S Quasi-persistent Currents in Rings Composed of Superconducting and Non-superconducting Regions. H .J . Fink, Can. J. Phys, 37, 474-484(1959) The Destruction of Superconductivity in Tantalum Wires by a Current H . J . Fink, Can. J. Phys. 37, 485-495 (1959) A New Absolute Noise Thermometer at Low Temperatures. H;'J. Fink, submitted for publication to the Can. J. Phys. G R A D U A T E S T U D I E S Field of Study: Physics Nuclear Physics J .B. Warren Quantum Mechanics G . M . Volkoff Noise in Physical Systems R. E. Burgess Electron Dynamics R. E. Burgess Advanced Electronics R. E. Burgess Statistical Theory of Matter W. Opechowski Other Studies: Diffusions in Metals . V . Griffiths Phase Transformations in Metals . W . M . Armstrong Metal Physics V . Griffiths and ]. A . H . Lund In presenting t h i s thesis i n p a r t i a l fulfilment of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It i s understood that copying or publication of this thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of P h y s i c s  The University of B r i t i s h Columbia, Vancouver 8, Canada. Date Sept.-!, 1959 v PREFACE This thesis consists of four independent papers on noise at low temperatures and superconductivity. Chapter I is a complete representation of an experiment, entitled "CURRENT FLUCTUATIONS IN A SUPERCONDUCTING CIR-CUIT CARRYING A CIRCUUTING CURRENT", performed during the winter 1 9 5 6 / 5 7 and summer 1 9 5 7 . Chapter II is also a complete paper entitled "A NEW ABSOLUTE NOISE THERMOMETER AT LOW TEMPERATURES", which was performed during the academic year 1 9 5 8 / 5 9 and summer 1 9 5 9 . Chapter HI and IV give a very brief outline of the problems and the results of two experiments entitled "QUASI-PERSISTENT CURRENTS IN RINGS COMPOSED OF SUPERCONDUCTING AND NON-SUPERCONDUCTING REGIONS" and "THE DESTRUCTION OF SUPERCONDUCTIVITY IN TANTALUM WIRES BY A CURRENT", which were performed during the winter 1 9 5 7 / 5 8 and summer 1 9 5 8 . Appendices A and B give detailed accounts of these experi-ments in form of two reprints from the Canadian Journal of Physics, Volume 3 7 ( 1 9 5 9 ) , pp. 4 7 4 - 4 8 4 and pp. 4 8 5 - 4 9 5 respectively. Diagrams and equations are numbered consecutively in each chapter and numbering starts again at the beginning of each chapter. A list of references for each chapter is given at the end of that chapter. An appendix counts as a chapter for these purposes. Acknowledgment is most gratefully given to Prof. R.E. Burgess and Dr. J.B. Brown for their supervision, their helpful advice, and their con-structive criticism of this work and for suggesting the experiment which is described in chapter I. I also acknowledge gratefully: Dr. G.M. Shrum, for extending to me the facilities for this research; Dr. J.B. Brown, for making the resources of his National Research Council grant freely available for this work; vi Prof. R.E. Burgess, for lending me electronic equipment; Dr. J.B. Garrison, Chicago University, for his suggestion of correlating voltages from noise sources at different temperatures (Chapter II); (verbal communication by Prof. A.W. Lawson to Prof. R.E. Burgess.) Prof. A. Van der Ziel for his suggestion of the tube for the firs-t stage of the noise thermometer to Dr. J.B. Brown. Dr. J.M. Daniels, for stimulating the research in noise thermometry; Dr. V. Griffiths, Dept. of Metallurgy, for supplying two of the Ta samples (Chapter IV); Mr. A.W. Greenius, B.C. Research Council for performing the microhardness test (Chapter IV); Mr. H.R.M. Zerbst, for the construction of the cryostat, for his help in performing some of the experiments, for preparing some of the drawings of this thesis, and the production of liquid helium; Mr. J . Lees, for his glass blowing; Mr. E. Price, for his helpful advice regarding problems related to instrumentation; Mr. R. Weissbach, for the production of liquid helium; The research students of the low temperature, nuclear resonance, and solid state group for their assistance in performing some of the experiments; and all others who, in various ways, have contributed to the success of these experiments. I am also indebted to the National Research Council, Canada, for two studentships ( 1 9 5 7 / 5 8 , 1 9 5 8 / 5 9 ) and three summer supplements, to the B.C. Telephone Co. for a scholarship ( 1 9 5 6 / 5 7 ) , and to the National Research Council, Canada, and the Research Corporation, New York, for their most necessary financial assistance. H . J o F . Vancouver, B.C. August 1 9 5 9 LIST OF CONTENTS CHAPTER I CURRENT FLUCTUATIONS IN A SUPERCONDUCTING CIRCUIT CARRYING A CEtCULATING CURRENT 1. Introduction 2. Experimental Procedures and Results 3* Conclusions References CHAPTER II \ A NEW ABSOLUTE NOISE THERMOMETER AT LOW TEMPERATURES 1. Introduction 2. The Thermometer and Experimental Procedures 3. Errors and Limitations of the Thermometer 4. Experimental Results and Discussion 5. Suggestions for Improvements of the Noise Thermometer References CHAPTER i n QUASI-PERSISTENT CURRENTS IN RINGS COMPOSED OF SUPER-CONDUCTING AND NON-SUPERCONDUCTING REGIONS General Outline CHAPTER 17 THE DESTRUCTION OF SUPERCONDUCTIVITY IN TANTALUM WIRES BY A CURRENT General Outline v i i i APPENDIX A Canadian Journal of Physics v Volume 37} Page QUASI-PERSISTENT CURRENTS IN RINGS COMPOSED OF SUPERCONDUCTING AND NON-SUPERCONDUCTING REGIONS (Reprint from the Canadian Journal of Physics) ZL (1959) pp. 474-484) 474 Abstract 474 1 . Introduction 474 1 1 . Experimental Procedures and Results 475 111. Uncertainties of the Measurements 481 IV. Discussion 481 References 484 APPENDIX B THE DESTRUCTION OF SUPERCONDUCTIVITY IN TANTALUM WIRES BY A CURRENT (Reprint from the Canadian Journal of Physics ZL> (1959), pp. 485-495) 485 Abstract 485 1. Introduction 485 2. Experimental Procedures and Results 487 3 . Uncertainties 491 4 . Discussion 492 References 495 ix To follow page LIST OF FIGURES CHAPTER I Fig. 1 (a) The circuit diagram of the input circuit and noise-voltage sources (b) Equivalent circuit diagram of the input at resonance 3 Fig. 2 Detailed block diagram 7 CHAPTER II Fig. I (a) The TT network which correlates noise between RQ and R2 via Rj (b) The block diagram and the equivalent noise current sources of the U network 12 Fig. 2 The block diagram of the noise thermometer 15 Fig. 3 (a) The experimental results at helium temperatures (T Q is the noise temperature calculated from equation (5) and T the temperature determined from the 1958 helium vapor pressure 3cale.) (b) The deviation € of TQ/T from the equation A/T + 1 with A s 0.385°K (O represent measurements with an integration time of 90 seconds and + with 67*5 seconds.) 22 Fig. 4. The circuit diagram of the preamplifier 23 Fig. 5 A detailed block diagram of the noise thermometer 23 Canadian Journal of Physics (vol. 37) APPENDIX A page Fig. 1. (a) Cu samples for resistance measurement (b) Potentiometer circuit for resistance measurement of Cu samples 475 Fig. 2 (a) The magnetic field distribution at a point P (b) The "flip-over" of a small compass needle for sample la at 4.22°K 477 Fig. 3 The deflection of the galvanometer due to the decaying current in the ring 479 Fig. 4 The ratio of the time constants as a function of temperature , 480 Fig. 5 The deflection of the galvanometer as a function of time for two "faulty" Pb-Cu samples 480 APPENDIX B t Fig. 1 Critical current vs. temperature difference T-Tc 488 Fig. 2 RcAn. versus temperature difference T-Tc 488 Fig. 3 The resistance of samples 1 and 3 as a function of current at constant temperature 489 Fig. 4 R/%va. temperature difference T-Tc for I = 4.20 ma 491 Fig. 5 Ro/Bn v s- aA 493 TABLES 1 Experimental Re suit s. Measured noise temperatures, T0, temperatures derived from the vapor pressure, T, and their ratios for the boiling points of oxygen and nitrogen.at barometric pressures and for helium temperatures. 1 CHAPTER 1 CURRENT FLUCTUATIONS IN A SUPERCONDUCTING CIRCUIT CARRYING A CIRCULATING CURRENT INTRODUCTION It follows from Nyquist's (1929) law that noise at radio-frequencies due to thermal fluctuations of a superconductor is zero below the trans-ition temperature, Tc, because for frequencies f<kTc/h the real part of an impedance of a superconductor approaches zero. However, i f a constant current is flowing in a superconductor then it is not obvious, a priori,, that no current fluctuations will be present. Ginsburg (1952) points out that the theoretical basis of fluctuations in superconductors is very obscure. Above T c he believes that fluctuations of the concentration of the superconducting electrons "foreshadow" the onset of superconductivity. If this assumption is correct, then one would expect that below T c fluctuations of the concentration of the normal electrons should "foreshadow" the onset of the normal phase. However, there is no experimental evidence which would support either of these suggestions. The theory of superconductivity by Bardeen, .Cooper and Schrieffer (BCS) (1957) (see also Valatin, 1958, and Bogoljubov, 1958) leaves room for current fluctuations when a persistent current is flowing in a ring. The momentum vector C| is equal to k^  + k£ of a virtual pair of electrons with opposite spin when a net current is flowing. For each value of cj thereis.a metastable state with a minimum in free energy and a unique current density. Scattering of individual electrons will not change the value of cj common to virtual states and so can only produce fluctuations about the current determined by q . Because scattering is possible according 2 to BCS's theory, current fluctuation may be also expected. It is generally accepted that shot noise in metallic conductors must. be very small because the number of free electrons in a metal is fixed and spontaneous deviations from the Fermi-Dirac distribution cannot persist longer than a few collisions which is of the order of lO" 1^ to lO" 1^ seconds. This was verified by the experiments by Bittel and Scheidhauer (1956) who found no noise in addition to thermal noise between 45 cps and 11.5 Kc/s when a current was passed through a metallic conductor. To avoid warming up the conductor the current densities in the normal metal had to be kept small compared with current densities which can be achieved in supercon-ductors. In this experiment the average current density was approximately 1.6 x 10^ amp/cm^ . Because there is no unique way of estimating the magnitude of the fluctuations of a superconducting current in a ring and because Knol and Volger's (1953) experiment on superconducting NbN was not very sensitive i t was decided to perform a similar experiment with improved technique. As pointed out above, there are arguments for and against current fluctuations in superconductors at radio-frequencies and therefore the final answer can be given by an experiment only and then only up to a certain limit determined by the accuracy of the experiment. 2. EXPERIMENTAL PROCEDURES AND RESULTS In p r i n c i p l e the experimental method t o detect f l u c t u a t i o n s i n a p e r s i s t e n t current i s e s s e n t i a l l y t h a t of K n o l and Volger (1953) but w i t h improvements of technique. Consider F i g u r e 1(a): A c o a x i a l s o l o n o i d a l c o i l , L 2 , was coupled t o a l e a d r i n g w i t h inductance L j . Because the r i n g was i n the superconducting s t a t e = 0 . The c i r c u i t o f F i g u r e 1(a) i s connected t o a s e n s i t i v e and s t a b l e a m p l i f i e r whose output was r e c t i f i e d and f e d i n t o a cathode f o l l o w e r which was d r i v i n g a pen r e c o r d e r . Two separate experiments were performed; at 1.3°K. In the f i r s t one w i t h the l e a d r i n g disconnected (M = 0 ) , the c i r c u i t was tuned t o i t s resonance frequency determined by the c o n d i t i o n <*J 1 L2C2* 1, such t h a t the noise output due t o thermal f l u c t u a t i o n s o f the input c i r c u i t was a maximum. From t h i s the resonance f r e q u e n c y ^ w a s determined. The second experiment was performed when L i was coupled t o L 2 and the resonance frequency uj 0 t the resonance r e s i s t a n c e , R r e s, and the Q 0 at u> 0 was measured. Because f o r t h i s c o n d i t i o n ' 0 0 ^ 2 ^ 2 ^ ~ ^ ) = 1, c o u l d be c a l c u l a t e d w i t h the above w i . Then the noise measurements were performed. A maximum p e r s i s t e n t c u r r e n t was induced i n the r i n g by s w i t c h i n g on and o f f a magnetic f i e l d l a r g e r than h a l f the c r i t i c a l f i e l d p e r p e n d i c u l a r t o the p l a n t of the r i n g . Obser-v a t i o n s were made of any d i f f e r e n c e between the output n o i s e i n t e n s i t i e s when there was no current f l o w i n g and when a p e r s i s t e n t c u r r e n t was f l o w i n g . The f l u c t u a t i o n s of the magnetic f i e l d are the b a s i s of observations of the radio-frequency f l u c t u a t i o n s of the c u r r e n t . The s e n s i t i v i t y of such an experiment depends on reducing the r e c e i v e r n o i s e t o a minimum. I t a l s o depends upon reducing the thermal noise of the i n p u t c i r c u i t and upon c l o s e c o u p l i n g between L^ and L 2 ; however, the l a t t e r are dependent upon each other and an optimum adjustment has t o be e 2 = I 2 / I N 2 •in ( L O L , ) F 2 2 e l d M o j L , ) 2 e* = 4 k I ( R £ i i ^ i ) d< or 4 k T H R n o :(a) A/W R RES e, = ( 4 k T R R n d f f 2 e 2 = ( 4 k c x T R R R E S d f ) ^ 2 6 3 = (pfejO-n y (b) Figure 1 (a) The ci r c u i t diagram of the input circuit and noise-voltage sources. (b) Equivalent cir c u i t diagram of the input at resonance, (to follow page 3 ) 4 found for good sensitivity. A carefully designed preamplifier with a 6AC7 pentode input stage was situated at the top of the Dewar flask and its output was >fed into a receiver. This tube kept the amplifier noise to a small value and the pentode connection assured a very «man input admittance to the amplifier by reducing the feed-back effect due to the grid-to-plate capacity. In order to interpret the magnitude of the receiver noise power out-put, the system was subjected to an overall calibration procedure in which the noise power from a saturated diode was injected at the input of the amplifier and its functional dependence on the diode current was determined. The relation was found to be linear for mr^n signals. Because the mean squared noise current of the saturated diode is pro-portional to its anode current the system acted as a quadratic device. This procedure established the equivalent input noise resistance of the receiver as corresponding to $00 ohms at 2>90°K. The resonance impedance of the tuned input circuit, which was immersed in the helium bath, was measured with a R.F. bridge at precisely the frequency,co0, at whidh noise measurements were made and the Q of the input circuit,Q 0 , was determined by measuring the frequencies of the ha l f power points. A further important parameter was the effective noise temperature of the input circuit which was determined by observing the variation of the thermal noise of the input circuit, which was kept at helium temperatures, when shunted with various known resistors at 290°K. The noise temperature of the circuit is defined as ^ ^ i T ^ R ^ and was calculated from: = ^ ^ r g 5  where R^ es i s t n e resonance resistance of the input circuit, Rg the damping resistance, © 0 the deflection of the pen recorder when the input to the preamplifier is short circuited, 63 the deflection of the pen recorder when the resonance circuit alone is connected to the preamplifier, and whan the resonance circuit is dampened by R^ . The noise temp-erature of the input circuit was found to be about 10°K which represents a marked improvement over the circuit noise in Knol and Vblger's experi-ment, whose noise temperature was approximately 400°K. Very stringent shielding procedures were required in view of the low level of noise being measured. As a precaution the intermediate frequency output of the receiver was continually monitored by an oscilliscope. In order to increase the accuracy of the measurement the DC output of the detector stage of the receiver was fed through a cathode-follower ampli-fier to a pen recorder so permitting registration and averaging of the rectified noise output over long periods. In the superconducting state the lead ring (of cross sectional radius 2.1mm and average radius 1.3 cm) at 1.3°K could be made to carry approxi-mately 620 amperes corresponding to a critical field of 760 oersted on the surface of the ring. The magnitude of the current which was flowing was checked by comparing the horizontal component of the magnetic field due to the current with the horizontal component of the magnetic field of the earth at some point. The distance of this point relative to the ring was known, and from that the current in the ring oould be calculated. It was found that the additional noise was undetectable when this persistent current was flowing. This result places an upper limit on the possible noise associated with the superconducting current and this may be expressed in terms of reduced shot noise: i 2 =2F2eIB, where i 2 is the mean squared noise current, F 2 the noise reduction factor, I the per-sistent current, B the bandwidth, e the electronic charge. By straight-forward network theory one obtains from Fig. 1(h) an expression for F 2: when the circuit is tuned to the resonance frequency U)Q.Fig. 1(b) shows a reduced equivalent circuit diagram for the above case. There $ is Boltzmann's constant, y3 is a direct observable quantity defined by: /3 m mean squared noise voltage due to I  ' mean squared noise voltage due to 21R-^ ^ and RnTR and k 2 = M2/^ L 2j With 3.4 x 1CT8 henries, <* « 0.035, ft • 0.01, k 2 - 0.665, oJo - 2ir(2.4) x 106 rad/sec, Q0 - 36, Rn - 500 ohms, Rres - 14.3 Kaand I « 820 amps, the noise reduction factor cannot exceed .1.1 x 10"^  in the light of the present experiment. Knol and Volger (1953) do not state the factor F 2 of their experiment, and the data are unfor-tunately insufficient to calculate F 2. If one assumes /3 = 0.01, and Rjj « 500 ohms then their experiment gives at 6 Mc/s a noise reduction factor which Is 82 times larger than the above. Various experiments were performed in addition with similar lead rings having various surface conditions. These were: smoothly machined, rough cast, etched with nitric acid and etched with 30# H^^Ojg acetic acid. 7 No current fluctuations were detected in these rings to the same limit as stated earlier. Figure 2 shows a detailed block diagram of the equipment. | 0 0 2 « F 3 0 0 V D.C.OVWW 27 k a M / W v 1 27Qkn== NOISE LIMITER ON OFF TO VOLUME OF AUDIO AM PL. TO C R . O . FOR MONITORING Figure 2 Detailed, block diagram (to follow page 7) RECORDER TO C.W. OSC. 8 3. CONCLUSIONS At 2.4 Mc/s persistent currents in superconducting lead are free from fluctuationsto less than 1.1 x 10"^  of full shot noise. A superconducting current is also unaffected by scattering processes determined by the surface of the metal to the same limit as stated above. A superconducting current with a limited amount of fluctuations implies a constraint on the occupancies of the ground state and the superconducting state by the electrons. The current _I is: where N is the number of superconducting electrons, v^ the velocity of the JL U i electron, e the electronic charge, Big the mass of the superconducting electron and I the length of the circuit. Because in the present experiment no fluctuations were detected Ps is a constant for a superconductor to the limit as stated above. This experiment was capable of detecting fluctuations in Pi only, not in N or v^ individually. However, it is very unlikely that there are any fluctuations in N and v^ because this would mean, for example, i f N decreases the velocity of the remaining supercon-ducting electrons must increase to conserve the momentum of the ring. Because the accuracy of the above experiment is directly proportional to the frequency, one might be able to detect noise a very high frequencies especially at frequencies At these frequencies the supercon-ducting electrons have enough energy to jump the energy gap (BCS and Biondi, Forrester, Garfunkel, and Scatterthwalte, 1958) and resistance appears. 9 REFERENCES BARDEEN, J., COOPER, L.N. and SCHRIEFFER, J.R. 1957 Phys. Rev. 108, 1175. BIONDI,, M.A., FORRESTER, A.T., GARFUNKEL, M.P., and SCATTERTHWAITE, C.B. 1958. Rev. Mod. Phys. 3J>, 1109. BrrTEL, H. and SCHEIDHAUER, K., 1956. Z. Angew. Phys. 8, 417. B0GOLJUB0V, N.N. 1958. II Nuovo Cimento 2> 794. GINSBURG, W.L. 1952, Uspechi. Fiz. Nauk 4j>, 348. See also 1953/54 Fortsch. Phys. 1, 51. KNOL, K.S. and VOLGER, J. 1953, Physica 12, 46. NIQUIST, H. 1928. Phys. Rev. ^ 2, 110. VALATIN, J.G. 1958. 31 Nuovo Cimento I, 843. CHAPTER LT A NEW ABSOLUTE NOISE THERMOMETER AT LOW TEMPERATURES 1. INTRODUCTION This paper deals with the contraction of a thermometer which makes use of the thermal fluctuations of voltage across an impedance to measure absolutely temperatures below approximately 140°k. Preliminary investigations were carried out for such a device to measure temperatures in the liquid helium region accurately.; : / According to Nyquist»s (1928) law the mean square voltage fluc-tuations arising from the thermal agitation of the electrons across an impedance, Z is given by: (1).; ~ 4 K T Re[Z] p(*i"0<tf where k is Boltzmann*s constant, T the absolute temperature, Re[zQ is the real part of the complex impedance Z, p(f ,T) the Planck factor and df the frequency interval in which the measurements are performed. The above formula can be derived from the equipartition law and the second law of thermo-dynamics and the available noise power is a universal function of the frequency and the absolute temperature, (see also Van der Ziel 1954). Equation (1) has also been proved for models which describe the random motion of the electrons in a conductor (Bernamont 1937, Bakker and Heller 1939, Spenke 1939). Nyquist's theorem can also be proved for the one-dimensional form of black-body radiation (Burgess 1941) which is received by an antenna kept in a sphere at uniform temperature. The 11 thermodynamic method has the merit that it is independent of the mechanism causing the noise. A number of papers have been published (Lawson and Long, 1946, Brown and MacDonald 1946, Gerjuoy and Forrester 1947, Cook, Greenspan and Wussler 1948) which suggest the possibility of using thermal fluctuations of voltages across an Impedance to measure low temperatures but they do not propose any practical scheme for a thermometer of this kind and no serious attempt has been reported which indicates that such an experiment has been performed at low temperature. 1 In 1946 Dicke et al. reported a radiometer which measures thermal radiation at- microwave frequencies. This method is essentially a commutation comparison technique which compares the unknown noise to that of a standard source. The radiometer has been used for observations of microwave radiation from the sun and the moon and for the measurement of atmospheric absorption at several microwave frequencies. Garrison and Lawson (1946) developed an absolute noise thermometer of the Dicke type to measure high temperatures. A chopper at the input of the amplifier is used to connect alternately the thermometer resistor and a resistor at ambient temperature (standard noise source). The principal limitation of such a switching device for comparison of noise voltages is the variation in contact potential of the chopper. Also the ultimate sensitivity of such a thermometer depends upon the signal-noise to amplifier-noise ratio. Aumont and Romand (1954) attempted an improvement of Garrison's and Lawson1s thermometer, but the final results have not yet been reported. The National Physical Laboratory (1957) reports also an improved noise thermometer for high temperatures (-^  HOO°C) based on the switching technique which is capable of comparing noise voltages to 0.05$. Cade (1958) uses an electronic switch instead of a chopper. To avoid any switching device at the inputs of the amplifier and to make noise measurement virtually independent of the amplifier noise, one can arrange 3 resistors, which are kept at different temperatures, in form of a TT network; and i f now two of the thermal noise voltages appearing across the network are multiplied together and averaged with respect to time, then under certain conditions the correlation between those voltages can be made zero. From this condition one can calculate the temperature of one noise source, provided a l l the resistance values and other temperatures are known.* Consider the network of Figure 1. By straight-forward network analysis one obtains for v 0 and v 2: (2a) «he*» Z T = Z ^ Z ^ Z 0 , Z Q = R . / ( ) * j c o R o t o ) , e tc . If one multiplies v 0 and v x and takes the time average over the product, then: (3) R e t a ^ ) Jfe *This idea was proposed by Dr. J.B. Garrison to Prof. A.W. Lawson of Chicago University (verbal communication by Prof. R.E. Burgess). c, Figure 1 (a) The K network which correlates noise between RQ and R2 via Rj. (b) The block diagram and the equivalent noise current sources of the 7f network. (to follow page 12). 13 where \\>o\~ 4WT0c.f/Ro (the Planck factor is assumed to be unity), similarly \^\ and . The time average of the products \ a \ , i , i * , and are zero because the resistors are independent noise sources. From equation (3) one sees immediately that i f either ^o^o = R i C i ^ R a Q or (OJRC)Z<£* I the product ^  (^ "o ^ *) can have a positive or a negative sign provided T~, > (T0+T2). For either of the above conditions the value of R^ required to make^(^r0 ^^j=0 can be calculated from equation (3): (ft) T,-T 0-T 2 In this experiment R 0 and R2 were both kept in the helium bath so that T 0 = T2. R2 and RQ were matched to better than i percent, and T^ was in an isothermal bath at room temperature. If T^  and the resistances are measured, T 0 can be calculated from (5) l > — ^ £ 1 . 2. THE THERMOMETER AND EXPERIMENTAL PROCEDURES The first requirement for an absolute noise thermometer of the kind described above is to find some resistors which are stable at liquid helium temperatures, whose values are preferably reproducible for several experiments, which produce no noise in addition to thermal noise, and whose resistive component is the same as the DC resistance (within a specified accuracy) over the frequency interval in which the measurements are performed. Many resistors have been tried at helium temperatures. The ones found most suitable are those manufactured by the Daven Co., series 850. They are hermetically sealed precision metal film type resistors composed of an alloy of pure, noble metals. They are stable over a period of at least seven hours to better than 1 part in and they are reproducible to that accuracy for several experiments. A 20 Kfl. resistor has a resistive component of 20 K a l i percent at 3 Mc/s. The resistance values used in equation (5) should be those appropriate to the frequency range in which the noise measurements are performed. Because no sufficiently accurate audio frequency bridge was available the metal film deposit resistors were measured at DC and at 3 Mc/sec. Since the 3 Mc/sec values differed from the DC values by less than 1 per-cent, i t seems reasonable to conclude that the deviation of the resistance in the audio frequency range from the DC value was less than 0.1 percent for the above resistors. A 20 Kn. Davohm resistor has a resistance of approximately 17.9 K-O-at liquid helium temperatures (1.3°K to 4i3?K) and the resistance value over this range varies less than 0.05 per cent. R^  was a precision wire wound resistance box and C]_ a variable con-denser, both kept at room temperature. C 0 and C 2 were the parasitic capacitance between the wires and the shielding, and the input to the ampli-fiers (including effects due to Miller capacitances) and they were equal within 3 percent. Unfortunately the parasitic capacity to ground was very large (^ 220/yjf), and about 3/5 of this was due to the shielding of Ci and R]_ which was reflected into the input of each amplifier. Fig. 2 shows the block diagram of the thermometer. The shielding requirements of the input circuit and the preamplifiers were very stringent and great care was required in avoiding ground loops and to eliminate mag-netic pick-up in the If network which in effect acts like a loop. The first tube of the preamplifier was a 6922 (American equivalent to the Philips E88CC) double triode connected in cascode followed by two RC coupled stages (7025 double triode). The cascode and the first RC coupled stage were constructed of wire wound resistors, and their filament currents were supplied by batteries. The lower and upper half power points of the ampli-fiers were approximately 3 and 7 Kc/s respectively. Both input voltages to the multiplier were constantly monitored by two oscilloscopes and two R.M.S. voltmeters to check the randomness of the noise spectrum and the gain of the amplifiers. The multiplier (a commercial type) is followed by an amplifier and a RC integrator of variable integration time (22.5 to 90 seconds). The output of the integrator was fed into a recorder via a cathode-follower to permit better registration of the time average of the signal. R0 and R2 were carefully cooled to liquid helium temperatures and then their DC value was measured with a Wheatstone bridge (R^  and C-^  were dis^-connected). RQ_ was set to approximately the value at which balance ) was expected and C^  adjusted such that R3C1- 2' This was accomplished by connecting and Z 2 in series and applying Z i A M P L I F I E R =3= A M P L I F I E R M U L T I P L I E R R E C O R D E R I N T E G R A T O R Figure 2 The block diagram of the noise thermometer According to the manufacturer's specifications the multiplier has a bandwidth from DC t° 15 Kc/s and the instantaneous error is less than 1% for DC multiplying a sine wave at maximum input (+ 50 VDC). The input voltages used in this experiment were approximately 8 to 14 volts RMS. 25 volts on either input gives unit gain from the other input to the output, (to follow page 15). 16 pulses across Z^  and Z2. was then adjusted until the shape of the pulses across the impedance + Z 2 and Z 2 were identical. This procedure adjusted the time constants t| and ^ t o approximately 10 percent. Similarly t0 and were adjusted and the average setting of was used when R^  was varied to achieve balance. If the change in Rj was large ~C\ had to be rebalanced and R^  reset. The temperature (room temperature) of R^  was read on an ordinary mercury thermometer placed on the outside of the resistance box and the value of -R^  was read from the dial setting of the resistance box. The temperature of the helium bath was then calculated from equation (5). The temperature of the helium bath was sometimes kept constant to better than 1 millidegree K by a temperature regulator (400 cps) similar to that of Boyle and Brown (1954). A stirrer was sometimes used to equalize the temperature, and the vapor pressure of the helium was measured on a mercury manometer with a cathetometer. A German silver tubing (8 mm diameter) extending into the liquid surface was connected to the manometer. The temperature determined from equation (5) was then compared with the "1958 ^ He scale of temperaturesw (Van Dijk and Durieux 1958, Brickwedde 1958). °3. ERRORS AND LIMITATIONS OF THE THERMOMETER Errors which can be represented by noise-current sources in shunt  with the If network These errors can be divided essentially into two groups: (a) errors due to the grid currents, (b) errors due to the finite input admittance. (a) The grid current is made up of 3 parts} electrons arriving at the grid (I^), electrons emitted from the grid by photoelectric emissions (I 2) and positive ions arriving at the grid ( I 3 ) . All three currents are independent of each other. The grid of the triode acts like the anode of a diode; for like the anode in the exponential part of its characteristic, for I 2 and I 3 like the anode of a saturated diode. Therefore the shot noise due to the grid current is: (6) - 2e(X,HrXz-* X3)<K. • The net grid current is I g = 1^  - I 2 - I3 and thus %\ ^2eX^.o\f. (b) The real part of the dynamic input admittance which is a function of frequency consists of 3 components; the ohmic loss in the input circuit, the cold loss of the first stage (leakage around the bulb of the tube, losses in the socket, etc.) and part of the load of the first stage which is reflected into the input due to the grid-to-plate capacity. In this experiment a cascode input was used which has the advantage that the grounded grid stage reduces the capacitive feed' back from output to input without introducing partition noise. How-ever, due to the finite feedback shot noise of the grounded grid stage contributes also to this error. The errors in (a) and (b) can be represented by noise-current sources in shunt with the Tf network. 1. The noise temperature which must be ascribed to this input conductance cannot be determined by calculation because its components and their noisiness are not readily estimated. If one assumes that the real part of the input impedance to the amplifiers is Rg and its effective temperature is T g « c* T^ , where 1 is a constant and T^ room temperature, and i f one also assumes that both amplifiers have the same input characteristics and that Rg^ > UQ and R>0— R2, then the error the absolute temperature due to shunt current sources is: ( 7 ) 6=4f = B ^ T [ e C i l - I ^ l 3 ) + 2 k ^ . where A is a constant for one particular thermometer. Errors due to current flow in the resistors Nyquist's law is based upon the assumption that the circuit is a passive network. This requires that no currents are flowing through the resistors R0, Ri and R2. To minimize the thermo-electric effects dissimilar materials in the network between the amplifiers and the resistors were avoided and voltages due to this effect were measured to be smaller than 3yuV at the in-puts of the amplifiers when Rc and R2 were at helium temperatures. The grid current Ii - (I2 + I3) was approximately 2;6 x 10~^  amperes. Because thin metal layer resistors (RQ and R2) consist = of a large number of very small con-ducting particles in loose contact, contact noise may be generated i f a current is passed through the resistors. Christenson and Pearson (1936) did not find any contact noise in thin solid carbon filaments when large currents were passed through the specimen. Also Bittel and Scheidhauer (1956) found no noise in addition to the thermal noise when a current was passed through metallic conductors between 45 cps and 11.5 Kc/s. Therefore thin solid metal layer resistors should be free of any noise in. excess of thermal noise and Nyquist's law should hold accurately for the above small currents. Son-linearities and amplifier noise Non-linearities in the amplifiers, the multiplier and the integrator are another source of error. Due to non-linearities the recorder deflection will be increased by an increment proportional to ^% ) . Because for s OO as well as for balance (seeequation 5) the correlation coefficient should be zero for no distortion of the signal, the variance should be the same for both cases. Because both amplifiers were built on different chassis and shielded from each other, the coupling capacity between the amplifiers must have been very small. The zero for balance of the recorder was then determined 18 T ' by grounding the inputs to both amplifiers. Because /Vo 1 and ^ a are functions of the equivalent noise resistance of the amplifiers, R^ , i t is desirable to make % as small as possible. for each amplifier was approximately 770 ohms at 300°K over a band width between 3 to 7 Kc/s. This was derived by measuring the recorder deflection (or the squared RMS Voltage at the inputs to the multiplier) for 'Vo 1 and w£ for various input resistances to the amplifiers at room temperature. For a band width of 3 to 12 Kc/s the equivalent noise resistance of the amplifiers was 650 ohms. If one assumes that the flicker noise is proportional to l/"f , then the equivalent noise resistance of the amplifiers at high frequencies is approximately 340 ohms. Therefore, flicker noise was the main con-tribution to Rjj between 3 and 7 Kc/s. Erroredue to mismatch of the time constants in the If network Equations (4) and (5.) were derived under the condition that (OJX)1'4&\ or that a l l the V* are equal. If this does not hold, then equation (4) for T c - T 2 is modified and one gets: (4a) To ( f to- t -FU) = ^ 1 + - ^ | [ T ' "T°^(V -k,)]  T\ - 2 T 0 1 | -v ( co r , ) 2 If T x » T c and T, ^  TD% (X + l j f ^ ^a i n e At helium temperatures i t is then sufficient to make "t0Tz^=- T,1 The deviation of f ( W j T ) from unity will increase with increasing frequency. If both f o and ' t 2 6 X 0 larger than "t, , then the error at the upper half-power frequency is less than 1%. The average error is smaller, because for lower frequencies the error decreases and t\ 20j. was always adjusted between T 0 and T 2 . For a systematic error in adjusting ~F i the fractional error in the noise temperature is almost a constant in the liquid helium range. * AC resistance of thermometer elements The deviation of the resistance of RQ, and R2 in the audio fre-quency range from their DC value was estimated to be approximately 0.1 percent (see above). Response of the integrator In the case of a narrow square noise band of width B and uniform spectral intensity a RC integrator of integration time will measure with a relative error of a single measurement, /3 ,(Burgess 1951): -v2 where A is the deflection of the recorder due to the DC component of the signal and (A-A) 2 the mean square deviation of the recorder due to the - 3 signal. For B = 4 Kc/e andy# ^ 5 x 1 0 C°-S>;) x should be at least 5 seconds. Integration times from 22 .5 to 90 seconds were used. The band-widths of both amplifiers were approximately equal. Lead Corrections If one assumes that the temperature of the leads going to RQ and r 2 are at room temperature (worst possible case), then the lead resistance should be less than 0 . 3 ohms for errors smaller than 0 . 5 percent, a require-ment not difficult to satisfy. 21 Pick-up To avoid errors due to 60 cps pick-up the lover half power points of the amplifiers were designed at approximately 3 Kc/s. Because no shielded room was available experiments could be performed only at night with fluorescent light, thyratron rectifiers, DC motors, etc. turned off. Although up. The voltages were constantly monitored oscillographically at the inputs of the multiplier, to check the randomness of the noise. 4. EXPERIMENTAL RESULTS AND DISCUSSION Table 1 shows the boiling points of liquid oxygen and liquid nitrogen measured at barometric pressure with an integration time of 90 seconds. They were found to be within 0.2 percent of the temperatures determined from the vapour pressure. The results of the noise-temperature measure-ments at helium temperatures are also shown in Table 1 and in Figure 3. Because, as pointed out in the introduction, the noise power of the real part of an impedance is a universal function of frequency and temperature, any systematic deviation of the noise-temperature can be only due to experimental error of the equipment. The plot in FigureB^a can be fitted v . best by an equation in the form: where A » 0.385°K and b • 1 for this thermometer. The term A/T can be explained due to errors which can be represented by noise-current sources in shunt with the T f network. Equation (7) 'shows that this error must be the amplifiers were protected against shock, audio noise was easily picked (9) proportional to l/T and the constant If errors due to Rs are neglected, then in T A B L E . 'I EXPERIMENTAL RESULTS Measured noise temperatures, T 0 , temperatures derived from the vapour pressure, T, and their ratios for the boiling points of oxygen and nitrogen at barometric pressures' and for helium temperatures. R 0 + R2 -n. R l -n- T l ^ - T 0 ^ T °* ToA 9,028.0 7020 * 10 296.6 90.26 ± 0.06 90.23 1.000 8,959.8 4780 * 10 299.3 77.25 * 0.08 77.33 0.999 35,740 563 298.5 4,559 4.226 1.079 n 440 a 3,586 3,201 1.120 n 360 tt 2,947 2,533 1.163 n 305 it 2,505 2,118 1.183 n 215 ti 1.774 .1.368 1.297 n 570 298.0 4.606 S 4.231 1.089 tt 452 ii 3,676 3.310 1.111 II 378 it 3,086 2,693 1.146 n 337 it 2,758 2,372 1.163 II 270 n 2,218 1,828 1.213 35,755 214 ± 1 tt 1.763 1,358 1.298 1.3 T T o T M,2 4 (a) T 0 _ 0 . 3 8 5 + , 1 1 — i 1 — 2 3 T ( ° K ) 1 4 £ o 0.01- 4- ( b ) 4 0 o 1 + ° 1 -— 1 '2 i T ( ° K ) r o I 4 o 0.01-T 0 . 3 8 5 , T + Figure 3 (a) The experimental results at helium temperatures.(T Q i s the noise temperature calculated from equation (5) and T the temperature determined from the 1958 helium vapor pressure scale.) (b) The deviation £ of To/T from the equation A/T+l with A = 0.385°K ( O represent measurements wiih an integration time of 90 seconds and + with 67.5 seconds.) (to follow page 22 ). the above experiment I i + I 2 + I 3 = 3 . 7 x 10 amperes. The measured g r i d current 1^ - ( I 2 + I 3 ) was approximately 2.6 x 10~^ amperes, which shows that shot noise due t o the g r i d current i s the main l i m i t a t i o n of the thermometer. At 15 Kc/s R g was measured to be la rge r than 10^ ohms, and because the noise temperature T g = o( T^ of Rg i s unknown the contr ibut ion due t o the second term i s uncer ta in . I f T 2 and T3 are neglected compared with I i then one can conclude t h a t ' R g / * . ^ 47 x 10° ohms. At helium temperatures equation (4b). a p p l i e s , and over t h i s temperature range a systematic e r ro r i n adjust ing "xx g ives a f r a c t i o n a l e r ro r i n the noise temperature which i s e s s e n t i a l l y a constant. This means that b i n the equation (9) i s not u n i t y . I f ^V*i/t,2= constant <l f o r a l l the measured points between 1.3°K and 4.2°K, the curve i n F i g . 3a w i l l s h i f t down. In general i t w i l l be necessary t o measure two known temperatures to c a l i b r a t e a thermometer of the above k i n d . These measurements w i l l de te r -mine A and b and when T 0 i s measured the absolute temperature, T, can be c a l c u l a t e d . However, i f one i s ce r ta in that no systematic e r ro r i s made \ 2 i n balancing the T $ , or i f (<-oX ) can be neglected wi th respect t o un i t y then b » 1 , and only one known temperature i s necessary t o ca l ib ra te the thermometer. Figure 3b shows the dev iat ion £ of TQ/T from the equation (A/T) + 1 wi th A - 0.365°K p lo t ted as a funct ion of T. The ca l ib ra ted thermometer measures temperatures accurately w i th in - 1% between 1.3°K and 4.2°K. Figure 4 shows the c i r c u i t diagram of the preampl i f ier and Figure 5 a de ta i led block diagram of the noise thermometer. This experiment makes use of the co r re la t ion of voltages from three independent noise sources at d i f f e r e n t temperatures t o determine the temperature of one (or two) noise sources. This method has the advantage that i t e l iminates any switching d e v i c e ; at the input of the ampl i f ie r . . . Figure 4 The circuit diagram of the preamplifier (to follow page 23). 50K 7M V W i MULTIPLIER MU/DIV AMLIFICATION FROM R THE SAME AS ABOVE 2 « V V V 5uf INTEGRATOR CATHODE FOLLOWER RECORDER K2-P,K2-VV AND MU/ DIV MANUFACTURED BY G. A.PHILBRICKJNC. Figure g A detailed block diagram of the noise thermometer (to follow page 23). The requirements of this method are that for good absolute accuracy of the thermometer the amplifiers, the multiplier, and the integrator must be linear and that Ti>(T 0 + T 2). At present the main limitation of the accuracy at low temperatures is shot noise generated at the grids of the first stages of the amplifiers. In principle, this method can also be used to measure high temperatures. % could be a fixed resistor at the unknown temperature and Ro and R2 could be kept at room temperature and one or preferably both of them be variable. '.. At high temperatures errors due to shot noise can be neglected. When Zx is made infinite and R0 and R2 are replaced by two antennas which are located apart from each other, then one has in principle a radio interferometer of the kind developed by Brown and Twiss (1954). In this experiment i t was demonstrated that it is possible and feasible to measure low temperatures absolutely by making use of the thermal fluctuations of voltages across an impedance. Work will continue at this university to improve the noise thermometer, and to derive an absolute temperature scale in the liquid helium region. 5. SUGGESTIONS FOR IMPROVEMENT OF THE NOISE THERMOMETER To detect reliably any discrepanciesyof the various helium vapor pressure scales i t is desirable to improve the absolute accuracy of the above noise thermometer by a factor of 300, but a factor of 30 will be sufficient i f one is willing to make a calibration of the equipment} only two temperatures have to be known accurately, say 1.3°K and 4.2°K. (a) Select a tube with a low "noise grid current" 1^ + I2 +13 -11 -12 (electrometer tube?) of the order of 10 to 10 amperes. It will be necessary to build a high gain cathode follower or a cascode input as the 25 first stage in order to guarantee that the grid-to-plate capacity is reduced and that the input impedance is very large. It is difficult to estimate the effective noise temperature of the input conductance and therefore it is advisable to make the input conductance as small as possible in order to eliminate errors due to this temperature. The tube should also have a small equivalent noise resistance (< ' ^ X L ) , with low flicker noise above 3 Ke/s and low microphony. If the proper tube is found, one can hope to reduce the present error by a factor of 100 to 300. (b) It should be possible to decrease the parasitic input capacity from - the present 220yuywf to less than 100 yuyuf by changing the mechanical out-lay of the input wiring and the shielding of R]_ and C]_. This would decrease 2 ( (x>X ) by a factor of approximately 5 and make the accuracy of the measurements less dependent upon the time constants. (c) R 0 and R2 should not be decreased, because the sensitivity of balancing R^ would suffer. For very accurate measurements i t might be necessary to increase RQ and R2 from 20 KSX. to 25 or 30 Kn, (at ambient temperature). (d) It might be necessary to build a shielded room and to operate all of the equipment from batteries in order to eliminate electrical disturbances which propagate along the power lines and radiate from them. Because the if network is a loop, it favours magnetic pick-up. (e) When an accuracy of 0.1 percent at 1.3° K is required, one must also find a reliable method of measuring the vapor-pressure-temperature of the helium bath to that accuracy in order to make a significant comparison be-tween the noise temperature and the vapor-pressure-temperature. REFERENCES AUMONT, R. and ROMAND, J. 1954 Jour. Phys. Rad. 15^  585. BAKKER, C.J. and HELLER, G. 1939, Physica 6, 262. BERNAMONT, J. 1937. Ann. Phys. 2, 71. > BITTEL, H. and SCHEIDHAUER, K. 1956. Z Angew. Phys*. S, 417. BOYLE, W.S. and BROWN, J.B. 1954. Rev. Sc. Inst. 25., 359. BRIGKWEDDE, F.G. 1959, Kamarling Onnes Conference, Leiden, June 23-28, p.128. BROWN, J.B. and MacDONALD, D.K.C. 1946. Phys. Rev. 70, 976. BROWN, R.H. and TWISS, R.Q. 1954. Phil. Mag. 4J>, 663 . BURGESS, R. 1941. Proc. Phys. Soc. j&» 293 (London). _ 1951. Phil. Mag. 42, 475 CADE, CM. 1958. J. Royal Aeron. Soc. 62, 805.' CHRISTENSON, C.J. and PEARSON, G.L. 1936. Bell System Techn. J. 15., 197. COOK, R.K., GREENSPAN, M. and WUSSLER, P.G. 1948. Phys. Rev. 7jt, 1714. DICKE, R.H. 1946. Rev. Sci. Inst. 12, 375. DICKE, R.E, and BERDIGER, R. 1946. Astrophys. J. 103. 375. DICKE, R.H. BERINGER, R., KYHL, R.L. and VANE, A.B. 1946 Phys. Rev. 22, 340. VAN DLJK, H. and DURIEUX, M. 1958. Physica 2^ ,920. GARRISON, J.B. and LAWSON, A.W. 1949. Rev. Sci. Inst. 20, 785 . GERJUOY, E. and FORRESTER, A.T., 1947. Phys. Rev. 71, 375 . * UWSON, A.W. and LONG, E.A. 1946. Phys. Rev. 70, 220 . _ _ Phys. Rev. JO, 977. NATIONAL PHYSICAL LABORATORY, 1957. Annual Report, London, Her Majesty's Stationery Office, p. 84. NYQUIST, H. 1928. Phys. Rev. ^ 2, 110. SPENKE, E. 1939. Wiss. Veroffentl. Siemens^ Werke 18, 55. VAN DER ZJEL, A. 1954. Noise (Prentice-Hall, Inc.) p. 9. CHAPTER HI QUASI-PERSISTENT CURRENTS IN RINGS COMPOSED OF SUPERCONDUCTING AND NON-SUPERCONDUCTING REGIONS GENERAL OUTLINE When a current is induced in a ring which is composed almost entirely of a metal in the super-conducting state and a normal metal perpendicular to the current flow, then one would expect the current to decay exponentially with a time constant L/R, where L is the inductance of the ring and R the resistance of the ring due to the normally conducting insert. Because under steady-state conditions the boundaries between the normal and superconducting metal must be equipotential surfaces, the quasi-persistent current must also be evenly distributed over the cross section of the normal metal. When the bulk resistance of the normal insert is compared with the resistance calculated from the measured time constant L/R, one is able to get information with regard to electron scattering at the normal and superconducting boundaries and in the normal metal. In this experiment Pb and In were used as superconductors and Cu as a normal metal. It .was'' found that most samples showed an exponential decay of the magnetic field due to the decaying current, except two which showed two definite relaxation times probably due to rectification effects of the two oxide layers on the insert. For samples with good electrical contact the resistance of the Cu-insert with respect to the bulk copper increased by a factor of 2.1 for the Pb-Cu rings and by a factor or 18.5 for the In-Cu rings. This increase in resistivity is probably due to some barrier effect, but i t is not clear why there is such a large discrepancy. When the thickness of the Insert is varied, the thinner inserts show a relative larger increase in resistivity. The measured resistance of the Pb-Cu rings shows a maximum at 3.4°K and the resistance of the In-Cu samples decreased by about 10$ between 3.2°K and 1.3°K. The maximum initial field on the surface of the ring (when the superconducting metal is in the superconducting state) is equal to the critical field of a completely superconducting ring and the ring behaves externally like an L/R circuit. Therefore, one may conclude by analogy with an L/R circuit that a current is actually flowing in the ring and that for R approaching zero the permanent magnetic field of such a ring is due to a current. Appendix A gives a detailed account of these experiments. CHAPTER 17 THE DESTRUCTION OF SUPERCONDUCTIVITY IN TANTALUM WIRES BY A CURRENT J GENERAL OUTLINE Whan a current is passed through a wire in the superconducting state, then a resistance appears when the magnetic field at the surface of the wire reaches the critical field, Hc. A number of experiments have been performed by various investigators on In and Sn and the experimental results show that they are not consistent with London's theory. Kuper's theory takes into account electron scattering at the boundaries of the disc-like intermediate state and he finds fair agreement with the experi-ments for &/t < 40, where a is the radius of the cylindrical wire and £tbs mean free path of the electrons, i The present experiments are designed to test Kuper's theory for &/£ 40. Ta wires of various degree of impurity were used and i t was found that the experimental results do not satisfy Kuper's equation. For temperatures, T, at least 5 aillidegrees below the transition temperature, Tc, the resistance of the wires jumped directly from zero to its normal value, and between Tc and (Tc - 5 m deg K) the resistance jumped to any fraction of the normal resistance between approximately zero and unity when the critical current was passed through the wires. It was also found that the transition temperature of Ta is strongly dependent upon its normal resistivity at helium temperatures, and at constant temperature the resistance - current plots show a large hysteresis effect. If the wires with a small constant current (4.2 ma) flowing in them are cooled from above T c or heated from below Tc, the resistance of the wires decreases above T c and approaches approximately zero at Tc, where T c is defined by the extrapolation of the IQ - T curve for I C • o. The transition from the normal to the superconducting state occurs over a smaller temperature interval for the purer sample, and the breadth of the transition is probably due to impurities and strains, but there is no satisfactory explanation why the resistance of the wires begins to decrease above the transition temperatures. Appendix B gives a detailed account of these experiments. APPENDIX A QUASI-PERSISTENT CURRENTS IN RINGS COMPOSED OF SUPERCONDUCTING AND NON-SUPERCONDUCTING REGIONS Reprint from the Canadian Journal of Physics, Volume 37 , April 1959, pp. 474-484. QUASI-PERSISTENT CURRENTS IN RINGS COMPOSED OF SUPERCONDUCTING AND NON-SUPERCONDUCTING REGIONS i H. J. FINK QUASI-PERSISTENT CURRENTS IN RINGS COMPOSED OF SUPERCONDUCTING AND NON-SUPERCONDUCTING REGIONS1 H. J. FINK ABSTRACT A number of rings composed of a superconductor (Pb, In) apart from a small insert of normal metal (Cu) perpendicular to the current flow have been investi-gated between 1.30° K and 4.33° K for Pb-Cu and between 1.30° K and 3.20° K for In-Cu. It was found that for samples with good electrical contact the decay of the magnetic field due to the current is exponential and that the effective resistance increased compared with the bulk resistance of Cu by approxi-mately 2.1 for the Pb-Cu rings and by 18.5 for the In-Cu rings. Two different thicknesses of the Cu inserts (0.0125 cm and 0.0053 cm) were used and it was found that the resistivity of the thin Cu insert increased with respect to the thick foil by 16% for the Pb-Cu system and by 36% for the In-Cu system. Part of this relative increase can be. explained as a size effect due to electron scattering in the Cu insert. The effective resistance of the Pb-Cu rings shows a maximum at approximately 3.4° K. The resistance of the In-Cu samples decreases by about 10% between 3.2° K and 1.3° K. The resistivity of the Cu foil when measured separately was constant for the above temperature range. For samples with "poor" electrical contact (probably due to some copper oxide on the insert) two definite relaxation times were observed. For these samples the effective resistance was current and temperature dependent and it was decreasing for decreasing currents and decreasing temperatures. This can be explained in terms of a rectification effect of the two oxide layers on the insert. The decay of the magnetic field of the ring is consistent with the decay of a current in an L-R circuit. I. INTRODUCTION When a current is flowing from a superconductor into a normal conductor, then under steady-state conditions the boundary surface will be an equi-potential surface because the electric field in the superconducting metal is zero. From Maxwell's second equation it follows that for steady-state condi-tions curl E = — B = 0 and therefore the current will be homogeneously distributed over the contact surface as it is over any cross section of the normal conductor (von Laue 1949; London 1950). Because the supercon-ducting current is a surface current and the current in the normal conductor is evenly distributed over the cross section the path of the current must be bent in the superconductor at the normal-superconducting boundary such that the above boundary conditions are satisfied. In addition one could imagine that some of the electrons moving from the superconducting into the normal metal (or vice versa) are scattered at the boundary. It was decided to investigate a closed system (circular ring) composed almost entirely of a superconductor apart from a small segment of a normal metal. When a current is induced in the ring and the normal metal is perpendicular to the current flow then one would expect the current to decay exponentially with a time constant L/R, where L is the inductance of the ring determined by its geo-metrical dimensions, and R is the resistance of the ring due to the normally 'Manuscript received December 24, 1958. Contribution from the Department of Physics, University of British Columbia, Vancouver, B.C. C a n . J . P h y s . V o l . 3 7 (1959) 474 FINK: QUASI-PERSISTENT CURRENTS 475 conducting insert. When the decay of the magnetic field due to the decay of the current in the ring is measured one is able to determine the time constant of the ring and from this one can derive the effective resistance. When the current decays in the ring curl E is no longer zero as in the steady-state case. It seems worth while to perform experiments to measure the effective resistance of the rings for different metals to see if there is any large deviation from the bulk resistance of the foil. These results should give some information about scattering at the normal and superconducting boundaries. On removal of an external magnetic field a quasi-persistent current is induced in the ring and its initial value should be proportional to the magnitude of the applied mag-netic field as long as the applied magnetic field is smaller than or approximately equal to half the critical field of the superconductor. If a magnetic field perpen-dicular to the plane of the ring which is larger than the critical field is removed from the ring, the superconducting material will be in the normal state as long as the magnetic field on the surface of the metal due to the current is larger than the critical field. When the current drops to its critical value the metal should go over into the superconducting state. Therefore it seems also worth while to investigate by direct magnetic measurements if the maximum effective initial current is determined by the critical field of the superconductor. II. E X P E R I M E N T A L P R O C E D U R E S A N D R E S U L T S The resistivity of the Cu foils was measured by a conventional current-voltage method. Two different thicknesses of the C u insert were used. Figure 1(a) shows the arrangement of the samples. A and B are the current leads and C and D are the connections to measure the potential drop across /; I was (a) (b) FIG. 1. (a) Cu sample for resistance measurement. {b) Potentiometer circuit for resistance measurement of Cu samples. approximately 10 to 11 cm, w ~ 1/2 cm, and c < 0.1 cm. The thickness of the Cu foils was 0.0125 and 0.0053 cm. Figure lib) shows the circuit diagram of the potentiometer circuit for measuring the resistance of the above samples. E L is a bank of storage batteries of 110 volts, E 2 an ordinary battery, A are ammeters, R a standard resistor (0.1 ohm), and G a galvanometer which was critically damped. The sensitivity of G was 3 . 8 8 X 1 0 - 4 (ia/mm on a scale 4 meters from G . The internal resistance of G was 22.2 ohms and the critical external damping resistance was 45 ohms. The current ii was varied between 100 ma and 500 ma. When the voltage drop across C D is the same as the voltage drop across the standard resistor R then G gives no deflection. When-ever the temperature was changed one had to wait approximately 1/2 hour 476 C A N A D I A N J O U R N A L O F P H Y S I C S . V O L . 3 7 , 1959 until the fluctuations of G died away but small fluctuations remained such that the readings could be taken only within ± 2 mm. The only limitation on the accuracy is the sensitivity of G , the fluctuations, and the accuracy of the ammeters. The resistance of the samples could then be measured within 1%. For the C u foil of thickness 0.0125 cm the resistivity was 6.62 X 10~9 ohm-cm and for the 0.0053-cm foil, 8.06 X 1 0 - 9 ohm-cm. A l l readings were constant between 1.3° K and 4.3° K . From experimental results of the anomalous skin effect by Chambers (1950) the mean free path of the electrons may be calculated from pi = 6.5 X 1 0 - 1 2 ohm-cm 2; I is then of the order of 1 0 - 3 cm. The mean free path of the electrons is of the same order of magnitude as the thickness of the foils. If one applies a correction for the size effect of " th ick" films (MacDonald 1956) the bulk resistance of the 0.0125-cm foil is approxi-mately 3% smaller and that of the 0.0053-cm foil approximately 6%. The copper foils were carefully covered with 99.98% pure lead foil or 99.95% pure indium. Heat was then applied simultaneously on both sides of the foil, the temperature being just a few degrees above the melting point of lead or indium. Some samples were prepared in air and some while blowing helium gas across the foil. There were no differences in the final result within the accuracy of the experiment. The surface of the Cu foil had to be free from oxide, otherwise In and Pb did not stick on the copper and especially the P b - C u samples were difficult to prepare. A number of lead and indium rings were cast around copper inserts in an aluminum mold. Samples Ic, 2b, 2c, and ib (see Table I) were prepared 1 day before the experiments were performed and the rest of the samples were cast approximately 3 months before measure-ments were taken. A l l the rings were of the same dimensions. The mean radius, a, of the rings was 1.305 cm and the radius of the circular cross section, b, was 0.205 cm. Therefore for an entirely superficial current the inductance of the ring was 31.7 X 1 0 - 9 henries (Shoenberg 1952). The actual field measurements were performed by comparing the horizontal components of the magnetic field due to the decaying current with the hori-zontal component of the magnetic field of the earth. The instant at which these components are equal and opposite could be observed by the "flip-over" of a compass needle. B y observing the times of flip-overs at different distances from the ring the decay of the current could be determined. Figure 2(a) shows the magnetic field components due to the current in the ring. When the compass needle flipped over, the horizontal component of the earth's magnetic field Hu balances the horizontal component of the magnetic field due to the current. Therefore (Smythe 1950) at balance: where lo is the current at t = 0, r the time constant of the ring, a the radius of the ring, p, z, and Hp the quantities as defined in Fig . 2(a), and K and E are complete elliptic integrals of the first and second kind and they are func-tions of k2, where k1 is defined by: 0) p[(a + PY + ^r2l(a-PY+z2 a2 + p2+z2 E-K (la) k2 = ±ap (a + oY+z2 FINK: QUASI-PERSISTENT CURRENTS 477 T A B L E I Superconducting materials, thickness of the Cu inserts, time constants, effective resistivity of the Cu inserts, and relations between the resistivities (po is the resistivity of the bulk Cu) Thickness Range of tempera-Supercon- of Cu tures Sample ducting insert, investi- PXIO 9 No. material X10 3 cm gated, ° K sec ohm-cm p/po P12-fc/p5-3 Pln/pPb l a Pb 12.5 4 22 21 2 1 35 16 Pb 12.5 4 23 23 .6 1 30 1.C Pb 12.5 4 21 26 .3 1 32 Id Pb 12.5 4 1 22 38 .2 Average I OO 27 .4 12.2 1 .90 2a Pb 5.3 4 23 33 .4 1 32 26 Pb 5.3 4 23 42 .4 1 31 2c Pb 5.3 4 23 52 .4 1 31 2d Pb 5.3 4 22 55 .9 1 35 Average 46 .0 17.2 2.26 0 710 3a In 12.5 3 20 3. 36 1 34 36 In 12.5 2 32 3. 82 1 33 Average 3. 59 93.4 14.5 7.65 4a In 5.3 2 31 4. 59 I 38 46 In 5.3 2 35 4. 59 1 30 Average 4. 59 172 22.6 0 543 10.0 FIG. 2. (a) The magnetic field distribution at a point P. Hp and Hz are due to the current in the ring and Ht, is the horizontal component of the magnetic field of the earth. (6) The "flip-over" of a small compass needle for sample l a at 4.22° K ; z is the distance as shown in Fig. 2(a). 478 C A N A D I A N J O U R N A L O F P H Y S I C S . V O L . 3 7 , 1 9 5 9 Figure 2(6) shows a plot when the compass needle "flipped over", where the distance z is plotted as a function of time. This method is not very accurate to determine T; its main purpose was to check if the magnetic field at t = 0 for a ring with a non-superconducting insert is the same as compared with a ring without an insert. The curve in Fig . 2(b) was extrapolated to t = 0. Then a separate experiment was performed with a completely superconducting ring of the same dimensions and the same superconducting material. In both cases the external applied field which was removed was larger than Hc/2. The current was therefore the maximum possible persistent current for the com-pletely superconducting ring. For the ring with the normal insert the current at / = 0 (due to the extrapolation) may be compared with the above current. For an applied field larger than HJ2 the initial current will be larger than Ic, but this current will decay very rapidly because the material of the ring is in the normal state. When the current reaches the critical current (within a fraction of a second) the ring (except the insert) does become superconducting and the effective initial current as determined from Fig. 2(6) is just this current and it is the same as the maximum persistent current within the accuracy of the experiment. In the further discussion of this paper the maxi-mum persistent current is always considered the maximum initial current. Figure 2(6) shows the experimental points of sample la for currents circulating in clockwise and counterclockwise direction in the ring. The plot in F ig . 2(6) was calibrated by the magnetic field measurements of the completely super-conducting sample. The ratio Io/H^ was calculated for the above sample from eq. (1) and (la) for z — 1A cm, t = 0, and p = 3.00 cm. Wi th the above ratio the solid line in Fig. 2(6) was plotted for T = 21.5 seconds. T o determine the time constants of the rings accurately (better than 2%) the rings were coupled to a coaxial coil which was in series with a sensitive galvanometer and an external critical damping resistance. The undamped period of the galvanometer was 4.7 seconds. When the external magnetic field was removed a switch in the galvanometer circuit was closed and the rate of decay of the magnetic field due to the quasi-persistent current was observed. For observation times larger than 5 to 6 seconds after removal of the external field the galvanometer follows accurately (within 1%) the induced current in the coaxial coil due to the decaying magnetic field of the ring. The rate of decay of the magnetic field for all the samples in Table I was of the. form I = I0e-'/T. Figure 3a shows the galvanometer deflection of sample 4a plotted as a function of time for various initial currents at 1.38° K and 2.31° K . For Io < I c in Fig. 3a J 0 was the same for both temperatures. Figure 36 shows a similar plot for sample la" for initial currents I 0 = Ic at various temperatures. From the slope of the galvanometer versus time plot the time constants of the various samples were determined and the average tabulated in Table I. The tabulated values of the time constants are average values over the temperature range stated in Table I for various initial currents. Jo was varied for some samples from Ic to approximately IJ1. No variation of r for different current was found within the accuracy of the experiment. Figure 4(a) shows F I N K : Q U A S I - P E R S I S T E N T C U R R E N T S 479 0 5 10 15 20 25 TIME [sec] 4 b lo = Ic il.33'K / /1.87 K "•SS^/ ,2J62-K 4.22 °K / 3.25 *K 0 10 20 30 40 50 60 70 80 90 . TIME [sec] FIG. 3. The deflection of the galvanometer due to the decaying current in the ring for (a) sample 4o and (b) for sample Id. T plotted as a function of temperature for samples l a and Id and Fig. 4(b) shows the same for sample 3a. The time constants of the Pb-Cu system showed a minimum of approximately 3% at 3.4° K as compared with the time constant at 4.22° K, and for the In-Cu samples the time constant was increasing between 3.2° K and 1.3° K by approximately 10%. However, the effective resistance of the rings was very much different for the two systems with the same thickness of the Cu insert. The effective resistance in the In-Cu systems was approximately nine times larger than in the Pb-Cu rings, and in the average the absolute value of the resistivity of the Cu insert increased by a factor of approximately 2.1 in the Pb-Cu ring as compared with 18.5 in the In-Cu ring. The ratio of the bulk resistivity of the "thick" to the "thin" 480 C A N A D I A N J O U R N A L O F P H Y S I C S . V O L . 3 7 , 1 9 5 9 foil is 0.847. When this value is compared with the measured ratios due to the decay of the current (Table I) one notices that for the Pb-Cu system the above ratio decreased by approximately 16% and for the In-Cu rings by 36%. r (a) t (b) P b - C u 1.05 NA In — Cu 1.00 ° - #lo, T42I = 2l.5sec 0.95 a — *id, T4.zz-3a5sec r2.iB= 3.27 sec 2 3 4 °K 2 3 *K FIG. 4. The ratio of the time constants as a function of temperature (a) samples la .and Id (Pb-Cu), (b) sample 3a (In-Cu). However, due to the difficulties in preparing the Pb-Cu rings the measured time constants for these systems strayed (Table I). This is probably due to different quality in electrical contact between the lead and the copper. The surface at the contact between Pb and Cu was visually inspected with a magnification of 10 and it was found that the copper foil for samples la, lb, and 2a was slightly bent, the Pb at the surface on the contact surface showed small holes, and the contact was not as clean as for samples Id, 2c, and 2d. Samples lc and 2b were of intermediate quality between the above two groups as far as visual inspection is concerned. Therefore the rings with the longer 0 , 5 10 15 20 25 30 35 TIME [sec] FIG. 5. The deflection of the galvanometer as a function of time for two "faulty" Pb-Cu samples, No. le with a = 0.0125 cm and No. 2e with a = 0.0053 cm Cu insert. The number of turns of the pick-up coil coaxial to the ring was different for the two samples. F I N K : Q U A S I - P E R S I S T E N T C U R R E N T S 481 time constants appear to be more reliable samples. Comparing the rings with the longest time constants only, one finds that the resistance of the Cu insert in the P b - C u rings increases by a factor of approximately 1.6 as compared with the bulk Cu and in the I n - C u rings it increases by 18.1. The ratio of the resistivity of the " thick" to the " th in" foil decreases for the P b - C u rings by 26% and for the In -Cu rings by 40%. The Pb of sample le (12.5 X 1 0 ~ 3 cm Cu insert) and of sample 2e (5.3 X 1 0 ~ 3 cm C u insert) had a "poor" electrical contact with the Cu foil (they are not included in Table I). Figure 5 shows the plot of the galvanometer deflection versus time for these samples at 1.33° K and 4.23° K for initial currents Io = Ic. There are two distinct time constants. The same curves were repro-duced when the current direction in the rings was reversed. After the experi-ment of sample 2e, the Pb ring could be pulled apart from one side of the C u insert which indicated a poor mechanical contact. No such curves were obtained for I n -Cu samples. III. UNCERTAINTIES OF T H E MEASUREMENTS The accuracy of the resistance measurement of the Cu samples by the current-voltage method was ~ 1 % ; the accuracy of the time constant measure-ment for various initial current (7 0 = IK to ID/7) was better than 2% for constant temperature. For constant temperature the dissipation of energy in the Cu insert due to the current was varied approximately by a factor of 50. This did not change the effective resistance within the above accuracy of the measurement, and therefore errors due to the heating of the Cu foil may be neglected. A t 1.3° K the critical current of the In samples was 3.3 times smaller than that of Pb and at higher temperatures even smaller. Therefore the increased resistance in the C u foil in the In rings compared with the Pb rings was not due to overheating of the insert. No interdiffusion data for Pb and Cu , and Cu and In could be found in the literature. Probably the diffusion rates are very small and have not been measured. The mean free path of the electrons in the bulk Cu is of the same order of magnitude as the thickness of the inserts. The main uncertainty is in the contact between the superconducting material and the Cu insert. A good contact was fairly easy to achieve for the I n - C u samples and difficult for the P b - C u rings: therefore the large stray in r for the latter. However, the author believes that the samples with the larger time constants had a good contact over the whole cross section of the ring. The thickness of both Cu foils varied by ± 3 % and their impurity content was unknown. For the determination of the effective resistivity of the rings L was calculated for an entirely superficial current. This is not exactly correct because the current in the insert is not superficial. The error in L, however, can be neglected compared with the above uncertainties. IV. DISCUSSION (a) When one performs experiments with multiply connected bodies such as rings the measurements which are taken are actually magnetic and from 482 C A N A D I A N J O U R N A L O F P H Y S I C S . V O L . 3 7 , 195!) these one deduces the electrical properties of the sample. B y measuring the ratio of the magnetic moment to the angular momentum when a magnetic field was applied to a superconducting sphere, Kiko in and Goobar (1938, 1940) and Pry, Lathrop, and Houston (1952) showed that the Lande g-factor of a superconductor is approximately unity, which indicates that the dia-magnetism of a superconductor is caused by ordinary electron flow and not in any way by the electron spin. The present experiments confirm the electrical properties of a superconductor. B y decreasing the thickness of a small non-superconducting insert the time constant of an exponential decay of a magnetic field increased correspondingly. For zero thickness of the insert the magnetic field will be equal to the initial magnetic field with finite insert. Because the maximum initial magnetic field on the surface of the ring is equal to the critical field of a completely superconducting ring and the ring behaves externally like an L-R circuit, one may conclude by analogy with an L-R circuit that a current is flowing in the ring and that for R approaching zero the permanent magnetic field of a ring is due to a persistent current; its maximum value is determined by the critical magnetic field of the superconductor. (b) The effective resistance of the thin copper foil increased with respect to the thick foil by approximately 16% for the P b - C u system and by 36% for the I n - C u system. The absolute value of the resistivity increased by a factor of 2.1 for the P b - C u rings and by 18.5 for the In -Cu samples. Because the "thick" and the " th in" copper foils are only 12.6 and 6.2 times larger than the mean free path of the electrons in the bulk material one would expect to get scattering at the boundary surfaces between the normal and superconducting material. If one assumes that such scattering surfaces exist, and if the electrons in the normal metal are scattered at the normal-super-conducting boundaries only, and if one also assumes that the increase in resistivity of the insert is the same as that of a " thick" film, then one can estimate p/p0 from (Fuchs 1938; Wilson 1954): (2) pJ~i-'3Jl P 81 where l0 is the mean free path in the bulk and t is the thickness of the insert. For U = 1.01 X 1 0 " 3 cm and 0.855 X 1 0 " 3 cm for the bulk material of the "thick" and " th in" foils, respectively, one obtains p/p 0 = 1.033 and 1.069. This checks barely with the P b - C u samples and'not at all with the I n - C u rings (see Table I). Experimentally the ratios are always larger than the calculated values. However, p/p 0 increases for the thinner foil as expected from eq. (2). Equation (2) takes into consideration only the scattering inside the C u insert. It neglects scattering at the boundary surface when the electrons are in the superconducting material. The increase in p/pa is experimentally very much larger for the I n - C u rings as compared with the P b - C u samples and this is probably due to some barrier effect but it is not clear why there is such a large discrepancy. The P b - C u samples showed a small but definite increase in resistivity at 3.4° K . Gerritson and Linde (1951) found for silver and gold manganese alloys F I N K : Q U A S I - P E R S I S T E N T C U R R E N T S 483 that the resistance after passing through a minimum passes through a subse-quent maximum at some lower temperature and falls once more with reduction of temperature. Mendoza and Thomas (1951)- observed only a resistance minimum in copper alloys. The resistance of the above Cu foil was constant between 1.3° K and 4.3° K when measured separately. However, in the P b - C u rings a resistance maximum appeared. Similarly for the I n - C u samples the effective resistance was temperature dependent and decreased with decreasing temperatures (Fig. 4). No interdiffusion data of the above metals could be found in the literature and any effect due to interdiffusion has been neglected in the above considerations. Because electron-scattering processes at low temperatures are not very well understood the present results are of considerable interest. (c) When a current is induced in a ring composed of a superconductor apart from a small insert of normal metal perpendicular to the current flow, then the maximum initial current which can be induced in the ring is the same as the maximum persistent current in an entirely superconducting ring of the same dimensions and the same superconducting material provided the bulk of the ring is in the superconducting state. (d) If the electrical contact on both sides of the insert are of good quality, then the decay of the magnetic field due to the current in the ring is exponential. (e) However, if the electrical contacts are not of good quality on both sides of the normally conducting insert, the rate of decay of the magnetic field is initially larger and then decreases (Fig. 5). F*or one sample the initial time constant is approximately the same for 4.23° K and 1.33° K , the second time constant increases for decreasing temperatures. For these samples the initial current was the critical current. The effective resistance of the above rings was decreasing for decreasing currents and the decrease was larger for lower-temperatures. The only explanation of the two time constants could be due to the rectifying effect of small patches of copper oxide on both sides of the insert (they were not visible to the naked eye) which are likely to occur. Groetzinger, Schneider, and Schwend (1956) showed that rectification between a superconductor and a semiconductor does exist at low temperatures. Bedard and Meissner (1956) measured the contact resistance between normal and superconducting metals when they were separated by their natural oxide layers. They found that two of their samples showed rectification at low currents. A n oxide layer shows barrier rectification only if it is of the order of 100 A or thicker. Very thin oxide layers show rectification because of the tunnel effect of the electrons. If one assumes that the copper insert and the oxide layers on both sides of the foil can be represented by two rectifiers back to back* connected by the resistance of the insert, then the combined voltage-current characteristic of this assembly is able to explain the decrease in resistance with decreasing current and also the fact that when the current is reversed the same galvanometer deflections are obtained. The decrease of the resistance with decreasing temperatures is consistent with the above picture *I am grateful to Professor R. E. Burgess for suggesting this explanation. 484 C A N A D I A N J O U R N A L O F P H Y S I C S . V O L . 3 7 , 1959 and it also explains the increase in resistance with increasing currents in Bedard and Meissner's experiments. ACKNOWLEDGMENTS The author wishes to acknowledge many helpful discussions with Professor R. E. Burgess and Professor J. B. Brown. Thanks are due to Mr. H. H. M . Zerbst for his help in performing the experiments and preparing the drawings. The author also wishes to thank the National Research Council, Canada, for a studentship and for the support received. REFERENCES BEDARD, F. and MEISSNER, H. 1956. Phys. Rev. 101, 20. CHAMBERS, R. G. 1950. Nature 165, 239. See also Handbuch der Phvsik, XIV, 188 (1956). FUCHS, K. 1938. Proc. Cambridge Phil. Soc. 34, 100. GERRITSEN, A. N. and LINDE, J. O. 1951. Physica, 17, 573, 584. 1952. Physica, 18, 877. GROETZINGER, G., SCHNEIDER, J., and SCHWEND, P. 1950. Bull. Am. Phys. Soc. Ser. II, 1, 58. KIKOIN, I. K. and GOOBAR, S. V. 1936. Compt. rend. acad. sci. U.R.S.S. 19, 249. 1940. J. Phys. U.S.S.R. 3, 333. VON LAUE, M. 1949. Theorie der Superleitung, Vol. 2 (Springer-Verlag), Chap. 8. LONDON, F. 1950. Superfluids, Vol. I (John Wiley & Sons, Inc.), p. 37. MACDONALD, D. K. C. 1956. Handbuch der Physik, XIV, 185. MENDOZA, E. and THOMAS, J. G. 1951. Phil. Mag. 42, 291. PRY, R. M., LATHROP, A. L., and HOUSTON, V. 1952. Phys. Rev. 86, 905. SHOENBERG, D. 1952. Superconductivity (Cambridge University Press), p. 30. SMYTHE, W. R. 1950. Static and dynamic electricity (McGraw-Hill Book Co.), p. 271. WILSON, A. H. 1954. The theory of metals, 2nd ed. (Cambridge University Press), p. 242. APPENDIX B THE DESTRUCTION OF SUPERCONDUCTIVITY IN TANTALUM WIRES BY A CURRENT Reprint from the Canadian Journal of Physios, Volume 37, April. 1959, p p . 485-495. THE DESTRUCTION OF SUPERCONDUCTIVITY IN TANTALUM WIRES BY A CURRENT' H . J . FINK A B S T R A C T The transition from the superconducting to the normal state of various pre-stretched tantalum wires carrying current was investigated. When the resistance of the wire jumps discontiiuiously from the superconducting to the normal or intermediate state as a current is passed through it, then this current is denned as the critical current J c . For temperatures T < ( r o — 5 millidegrees K) the resistance of the wire jumps directly from zero resistance to its normal value at the critical current, such that the total cross section of the wire goes effectively into the normal state. Between (Tc — S millidegrees K) and Tc the resistance of the wire jumps at I0 to any fraction of the normal resistance between approximately zero and one. For constant temperatures the resistance-current plots show a large hysteresis effect. The transition temperature, Tc, of the various samples is strongly dependent upon their normal resistivity at helium temperatures. If the wires with a small constant current (4.2 ma) flowing in them are cooled from above the transition temperature, the resistance decreases above Ta and ap-proaches zero at approximately Te, where Tc is defined by the extrapolation of the Ic-T curve to Ia = 0. If the wires are heated from below Tc the same resistance-temperature curves are reproduced. 1. I N T R O D U C T I O N When a current is passed through a superconducting wire, then a resistance appears when the magnetic field at the surface of the wire reaches the critical field, Hc. This is Silsbee's hypothesis (Silsbee 1916) and it has been experi-mentally verified by a number of authors (Scott 1948; Rinderer 1956). London (1937) showed that for J > J c a core in the wire must exist which can be neither normal nor superconducting but must be in the intermediate state. Assuming (a) the intermediate state exists in a core where the magnetic field at any point is equal to Hc,-(b) the current density is continuous at the boundary surface of the core (which is in the intermediate state) and the normal conducting shell, and (c) the electric field is constant over the total cross section of the wire, one is able to find an expression for the increase of resistance R with current J : (1) R/Rn = (1/2)[1 + V{ 1 -(/c//) 2!J for / > J c , where Rn is the resistance of the wire in the normal state. When the current approaches the current J c as defined above, the resistance of the wire should jump abruptly to 1/2 Rn and for very large currents approach the value Rn asymptotically. The above assumptions do not specify anything about the detailed structure of the intermediate state, but London (1950) assumes that the core consists of superconducting and normal disks perpendicular to the current flow in the wire. 'Manuscript received November 17, 1958. Contribution from the Department of Physics, University of British Columbia, Vancouver, B.C. C a n . J . P h y s . V o l . 37 (1959) 485 486 C A N A D I A N J O U R N A L O F P H Y S I C S . V O L . 3 7 , 1 9 5 9 Kuper (1952) employs London's suggestion of a disk-like intermediate state core. He derives an expression for R/Rn, assuming that the thickness of the normal disks is small compared with the mean free path of the electrons in the normal state. The electrons are scattered on the interfaces between the normal and superconducting disks and this increases the resistance R. The initial jump of R/R„ for I = Ic is no longer 1/2 but larger. He also assumes that distortions of the electron path do not change the resistance (only the Gibb's function), that the surface energy does not change the Gibb's function very much, and that the singularities of London's model on the axis of the wire may be neg-lected. Wi th these assumptions and some approximations Kuper obtains the following expression: where y = a/l, a is the radius of the cylindrical wire, / is the mean free path of the electrons. Therefore at / = Ic the resistance ratio Re/R„ is a function of 3'. For y = 0, Rc/Rn = 1 and for y = 00 I RQ/Rn = 1/2. Hence metals in which the mean free path of' the electrons is very small compared with the radius of the wire should satisfy eq. (1). In the present experiments y is large com-pared with previous experiments. Gorter's model (Gorter 1957) is a dynamic model which employs the phenomenon of supercooling of the normal phase. In the intermediate state the boundaries between the superconducting and normal phase are assumed to be parallel to the current. These boundaries would move continuously in the direction perpendicular to the (constant) current and the local magnetic field and the voltage drop observed in the direction of the current flow would be due to induction. If one assumes that the normal phase persists when the magnetic field falls below the critical value Hc and that an intermediate state mixture is set up everywhere II < IIC if the field has fallen to a value qTIIc in the normal phase (qr is a parameter smaller than 1), then the critical resistance is approximately 0.7 of the normal resistance (for 0.75 < q, < 1.0). The super-, cooling introduces periodicity into the dynamic model and Gorter and Potters (1958) have made some numerical calculations of the velocity of the phase boundary, the upper limit of the amplitudes of the voltage variations, and their frequency. In the above model the heat of transformation and the surface energy between the normal and superconducting phase have been neglected. The first conclusive investigations concerning the' transition from the superconducting to the normal state were performed by Schubnikow and Alekseyevsky (1936) and Alekseyevsky (1938). They observed that Rc/Rn ~ 0.8 at 1.95° K for a polycrystalline tin wire of radius 0.0056 cm and a monocrystalline wire of radius 0.0080 cm. A t approximately 2IC the resistance of the wire approaches Rn in contradiction with eq. (1) and (2). They esti-mated the temperature rise due to current heating inside the wire to approxi-mately 10-" ° K . Scott (1948) performed some experiments on polycrystalline indium wires of radius 0.005, 0.014, and 0.0175 cm between 3.34° K and 3.38° K . He (2) R/Rn = (1/2) for I > /, 1 F I N K : D E S T R U C T I O N O F S U P E R C O N D U C T I V I T Y 487 estimates that the temperature rise due to the heating effect of the current has no influence upon Rc/Rn, and he finds that R^/Rn is 0.85, 0.79, and 0.77 for the wire of radius 0.005, 0.014, and 0.0175 cm respectively in the above temperature range. He also verifies Silsbee's hypothesis. Scott's results do not satisfy eq. (1) but they fit eq. (2) with I = 2.4 X 1 0 " 3 cm. Rinderer (1956) investigated tin wires of various impurity content of radius 0.0375 cm between Tv = 3.72° K and 3.27° K . He estimates the maximum tem-perature rise for his purest sample to approximately 3 X 10~4 ° K . He confirms Silsbee's hypothesis. He finds that his purest sample satisfies approximately eq. (1); however, Rc/Rn is temperature dependent as well as dependent upon the impurity content of his samples. His samples satisfy eq. (2) only quali-tatively. Rc/Ra increases with increasing y by varying the impurity content in contradiction with Kuper's theory. For currents smaller than 4 amperes the transition becomes continuous. He states that the impurity content of his samples is responsible for the continuous transition. 2. EXPERIMENTAL PROCEDURE AND RESULTS The present experiments were designed to test London's and Kuper's equation for large y. Various experiments were performed on tantalum wires of radius 0.0127 cm and 0.00635 cm. The wires were purchased from the Fansteel Metallurgical Corporation, North Chicago. According to speci-fication they were stretched 15, 21, 22% and annealed. The impurity content was not known. Therefore it was decided to perform some metallurgical tests i i i order to find the relative strain and impurity content of the above three different samples. Three pieces of each wire were mounted in bakelite at approximately 160° C, polished, and etched with hydrofluoric acid. The shape of the grains was studied under the microscope and they were found to be equiaxed for all three samples. This implies that if there were any strains in the metal, those strains must be very small. Then another four pieces of wire were mounted in bakelite mounts and metallographically polished. These four pieces of each wire were tested in triplicate microhardness measurements using a 136° square pyramid diamond penetrator and a 200-g load. The diamond pyramid hardness (D.P.H.) is defined as the load per unit area of surface contact in kilograms per square millimeter. Table I shows that for sample 3 the D . P . H . is the smallest and that the D . P . H . increases as the normal resistivity at helium temperatures. Assuming that the strain in the wires may be neglected, sample 3 appears to be the purest because generally TABLE I Length, radius, stretch, resistance, and diamond pyramid hardness of samples Sample Length CD, cm , Radius, cm Stretch, % R300, ohms P o a t 4 . 4 ° K , ohm-cm D.P.H. 1 13.6 12.7X10"3 22 0.412 0.112 1.72X10-6 192 la 2.37 12.7X10"3 22 0.0722 0.119 1.89XH)-6 192 2 1.61 6.35X10-3 15 0.222 0.175 3.05X10-6 204 3 1.38 12.7X10-3 21 0.039 0.0703 l.OOXlO" 6 167 488 C A N A D I A N J O U R N A L O F P H Y S I C S . V O L . 3 7 , 1959 the hardness increases with increasing impurity content. The resistivity of the tantalum is fairly large at 300° K (see Table I) as well as the normal resistivity at helium temperatures compared with tin and indium. This has the advantage that one can measure the resistance of the samples fairly accurately for small currents (near the transition temperature) with conven-tional methods. For the measurement of the resistance of the above samples a standard current-voltage method was used. The current and voltage leads were copper and were spot-welded to the tantalum wire. Figure 1 shows the plot of the critical current versus the temperature difference (T—Tc). The values of Tc are compiled in Table II. The transition temperatures increase strongly with decreasing Rn/RZoo- For sample 2, two T A B L E II Transition temperature, slope at Tc, external applied field, and noise of samples Sample T0° K Slope at T = Tc, oersted/0 K -^ cxtornul Maximum noise ampli-tude per unit length in C D , / i v / c m 1 4.2328 Earth mag. ~ ± 1 0 field U 4.221 6 -276 0 ~ ± 2 0 2 4.109 s Earth mag. ~ ± 3 0 field 2 4.1097 -320 0 ~ ± 1 0 0 3 4.323o -334 0 ~ ± 2 0 separate experiments were performed: one with the magnetic field of the earth compensated, the other without compensation. Between these experi-ments the sample was kept at room temperature for about a week. The -30 -20 -10 0 (T-Tc) [mdegK] 5c -30 -20 -10 0 ( T - T ) [m deg K] FIG. 1. Critical current vs. temperature difference T—Tc. FIG. 2. Rc/Rn versus temperature difference T—Tc. F I N K : D E S T R U C T I O N O F S U P E R C O N D U C T I V I T Y 489 difference in Tc is remarkably small. For samples l c , 2, and 3 the magnetic field of the earth was compensated to less than 1% of its value with two Helmholtz coils 34 and 24 cm in diameter. Sample 2 and sample 1 were measured in the magnetic field of the earth. The samples were suspended vertically. The vertical component of the magnetic field of the earth was approximately .55 oersted and the horizontal component .20 oersted. The measurements for the uncompensated magnetic field of the earth are rather a complex case of superposition of a longitudinal and transverse field, and the field due to the current which is of the same order of magnitude as the magnetic field of the earth for small current values. Figure 2 shows a plot of Rc/Rn versus temperature for the above samples. The transition temperatures determined from the Ic versus temperature plot are not of basic importance for the sample with uncompensated magnetic field of the earth. The value Rc/Rn approaches unity for temperatures 4 to 6 millidegrees below the transition temperature. For the samples with uncom-pensated magnetic field of the earth the transition bandwidth was about 15 to 20 millidegrees. For these samples the transition was sharp even for values of Rc/Rn < 1/2 and no tails were observed. For increasing currents R/Rn approached unity. Figure 3a shows the resistance plotted versus the current for four different critical currents of sample 1. For example when RQ/Rn was 0.80, R I Rn approached unity at I/IB = 1.3, and when Rc/Rn was 0.15, R/Rn approached unity at I/Ic = 2.7. For resistances smaller than i?„ large voltage fluctuations were observed across the samples. Because the time constant of the galvanometer was 2 seconds and the galvanometer was under 1 [mA] i [mA] FIG . 3 . (a) T h e r e s i s t a n c e o f s a m p l e 1 as a f u n c t i o n o f c u r r e n t a t c o n s t a n t t e m p e r a t u r e s w h e n t h e m a g n e t i c f ie ld o f t h e e a r t h w a s n o t c o m p e n s a t e d . (6) T h e r e s i s t a n c e o f s a m p l e 3 as a f u n c t i o n o f c u r r e n t a t T = 4 . 3 0 7 ° K w h e n t h e m a g n e t i c field o f the e a r t h w a s c o m p e n s a t e d . 490 C A N A D I A N J O U R N A L O F P H Y S I C S . V O L . 3 7 , 1959 damped, the frequencies of the fluctuations which could be observed were close to 1 c.p.s. The largest amplitudes of fluctuations were not observed at the transition but approximately at J / J c = 1.1. Table II also shows the maximum noise voltages as observed by the galvanometer deflections. These values are only approximate. For the case of the compensated earth's magnetic field the fluctuations were narrowed to a region of approximately 4 to 6 millidegrees. The fluctuations were mostly of random nature as observed on the galvano-meter, but sometimes over short periods of time the galvanometer deflections were "kicks" in one direction only. The transition points (Fig. 2) of the Rc/Rn versus T—Te plot forRc/Rn < 1 (especially for the samples with compensated magnetic fields of the earth) are therefore accurate to 10 to 20% only. For small currents (< 50 ma) the voltage fluctuations were very strong as soon as the critical current was approached. Therefore for those currents it is not clear if this initial deflection of the galvanometer at / = Ic was due to the fluctuations. However, for the samples with uncompensated earth's magnetic field the initial deflection of the galvanometer at I = Ic was due to a dis-continuity in R/Rn and it is believed that the same is true when the magnetic field of the earth is compensated. W i t h the magnetic field of the earth com-pensated only the resistance at I/Ie was measured. The current was then increased to approximately 3 / c , the normal resistance of the wire was checked, and then the current was decreased until / = It, where / , is the current when the metal goes from the normal into the superconducting state. Figure 3& shows the resistance of sample 3 plotted as a function of current at T — 4.307° K . When the current was increased above the critical current and then de-creased, all the samples showed a strong hysteresis effect as observed by other investigators (Scott 1948; Rinderer 1956). For sample 3, Ii/Ic decreased linearly from approximately 0.8 at 7 C = 100 ma to 0.5 at J c = 400 ma. For sample la It/IQ was 0.55 at J c = 100 ma and 0.45 at Ic = 400 ma. For one temperature I, is not always exactly reproducible and the It/Ie values stray approximately 10%. Table I I shows also the slope of the Hc-T curves at T = Te based on the assumption that Silsbee's hypothesis holds. The transition from the superconducting to the normal state with increasing temperature and constant current was also studied. The current was 4.2 ma for samples l a , 2, and 3 for which the magnetic field of the earth was compen-sated. The smaller i? n / i? 3 0 o the sharper is the transition (Fig. 4). Sample 3 was warmed up as well as cooled at a rate of 3 millidegrees per hour, samples l a and 2 at a rate of 5 to 8 millidegrees per hour. The same curves were pro-duced for cooling and heating for the above temperature drift rates, which shows that the samples were at the same temperature as the helium bath. The transition width for sample 3 was less than 2 millidegrees for a current of 4.2 ma. Sample 3 is believed to be purer than sample l a and l a purer than 2. When a superconducting wire is cooled from above the transition tempera-ture with a constant current flowing in it, then according to eq. (1) at tem-perature Te the intermediate core should appear. A further reduction of the temperature will decrease the resistance. The core will grow until it occupies the whole wire. Then the resistance will drop suddenly and go into the super-F I N K : D E S T R U C T I O N O F S U P E R C O N D U C T I V I T Y 491 conducting state. However, this picture is not consistent with the present experiment. The extrapolation of the Ie-T curves toward the temperature axis defines Te. The origin of Fig . 4 is taken at Tc. This Tc is at R / R n — 0 and not at R / R n = 1 as implied by eq. (1). The breadth of the transition is R Rn 1.0 0.75 • f*\a 0.50 o HEATING o COOLING 0.25 i = 4.2 [mA] 0 5 10 15 20 (T-Tc) [m<teg K] FIG/4. R/Rn vs. temperature difference T—Tc for / = 4.20 ma. probably clue to impurities and strains (de Haas and Voogd 1931), but there is no satisfactory explanation why the resistance of the wires begins to decrease above the transition temperatures. It was fairly easy to keep the temperature constant with two needle valves in series. Above 4° K a temperature change of 1 millidegree corresponds approximately to a change in helium vapor pressure of 1 mm of mercury. N o other temperature-regulating devices were used. The temperature was deter-mined from the vapor pressure of helium read on a mercury manometer with a cathetometer and the 1948 temperature scale was used. 3. UNCERTAINTIES The temperature was kept constant for most measurements between 1 and 2 millidegrees, except for the measurements of Fig. 4 where the resistance readings correspond to temperature readings to better than half a millidegree. The over-all accuracy of the resistance measurement was 2 % or better. The values of the diamond pyramid hardness test strayed by approximately ± 3 % . The error in temperature due to the static pressure of the liquid helium was smaller than 1 0 - 3 ° K and was neglected. As soon as a resistance appears the wire will be heated due to the dissipation of electric energy. The increase in temperature of the wire AT" can be separated into two terms A T = A7 \ - t -A7 \ , where A7"i is the increase in temperature at the axis of the wire relative to the surface and it will depend upon the thermal conductivity K of the wire. A T o is the temperature difference between the 492 C A N A D I A N J O U R N A L O F P H Y S I C S . V O L . 3 7 . 1959 helium bath and the surface of the wire and it will depend upon the thermal contact resistance a. where Q is the dissipated energy in watts/cm length of the wire, and a is the radius of the wire. The first term does not contain any correction for an inter-mediate state core. For the smallest currents which make Rc/Rn — 1> Q varies from approximately 2 X 1 0 - 5 to 1 0 X 1 0 - 5 watts/cm length of the wires. From Hulm's (1950) measurements on 0.1% impure tantalum at 4.1° K the thermal conductivity i f = 48.2 X 10~3 watts/cm deg K . The measurements of Mendels-sohn and Olsen (1950) and Mendelssohn and Rosenberg (1952) give con-siderably larger values for 99.95% and 99.98% pure tantalum respectively. Wi th the above smaller value for K the temperature rise A T i of the above samples varies from 6 . 6 X 1 0 ~ 5 ° K for sample 3 to 3 . 4 X 1 0 ~ 4 ° K for sample 2. Only the second term of eq. (3) is of importance and there is no value of the thermal contact resistance for tantalum and helium I in the literature. If one takes the value of a for tin and helium I at 4.1° K a s ~ 1.5 watts/cm 2 deg K (Rinderer 1956), then A T 2 is of the order of 1 0 - 4 ° K for samples l a and 3, and 1 . 7 X 1 0 ~ 3 ° K for sample 2 at the above currents. Sample 2 would then have an error of larger than 2 millidegrees for current values larger than 65 ma. For smaller current values Rc/Rn < 1 and in this range the increase in tem-perature can be neglected compared with the accuracy of the temperature measurements. If one compares samples 2 and 3, one finds that Rc/Rn approaches unity when the dissipation of heat due to the current of sample 2 is five times larger than that of sample 3. This shows that the increase in Rc/Rn above 1/2 according to eq. (1) is not due to overheating of the samples. (a) For temperatures smaller than 5 millidegrees below the transition tem-perature Rc/Rn is essentially unity for the samples used above. The normal resistivity for these samples at helium temperatures was relatively large and from this one may conclude that the mean free path of the electrons at this temperature is small. Because no data for the mean free path of tantalum at helium temperatures can be found in the literature the mean free path is estimated from the Fermi energy at absolute zero. Because (3) 4. DISCUSSION and P = — 2 7 ne I one gets where n is the number of free electrons per unit volume, p the bulk resistivity of the metal, and I the mean free path of the electrons. Because tantalum has five valence electrons per atom, pi ~ 3 X 10~12 ohm-cm 2. For the above samples F I N K : D E S T R U C T I O N O F S U P E R C O N D U C T I V I T Y 493 the mean free path is then of the order of 1 0 - 6 cm. The ratio a/l is plotted versus Rc/Rn in Fig . 5. However, some authors assume that only one electron per atom contributes to the conductivity of multivalent metals. Therefore for n = 1 the values of y for T a in Fig . 5 have to be divided by 2.9. Figure 5 shows that neither eq. (1) nor eq. (2) is satisfied. Kuper's equation requires a modi-fication for large ratios of a/l. R n 1.0 0.9 0 .8 0.7 0.6 0 .5 1.25 5 2 0 8 0 320 1280 5120 FIG. 5. R0/Rn vs. a/l. + results by Scott on indium (estimated by Kuper), |—| results by Alekseyevsky on tin, O 0 results by Rinderer on tin, • results on tantalum. (6) The ratio of RJRn close to the transition temperature is believed to have any value from zero to one. However, over this narrow temperature range (5 millidegrees) no conclusive results can be derived because of large low-frequency voltage fluctuations which made accurate resistance measure-ments impossible. These fluctuations are probably due to the instability of the intermediate state over part of the cross section of the wire. The observed voltage fluctuations are not that predicted by Gorter and Potters (1958). According to their model the lowest frequencies one should expect for the above samples are of the order of 106 to 107 c.p.s. for H/He = 1.1. Voltage fluctuations in the intermediate state of T a were previously observed by Webber (1947) and Kaplan and Daunt (1955) and studied by Baird (1958,1959). Near the transition temperature the resistance of the samples is varying strongly with temperature and one could imagine that local temperature variations and minor fluctuations in the supply current give rise to magnified voltage fluctuations across the wires. For current values corresponding to RJRn — 1, the voltage fluctuations will not be magnified, because local temperature variations in the samples will not change the resistance of the metal and hence no magnification will occur. Although the magnetic field of the earth was compensated to better than 1%, the stray magnetic field due to the current in the current leads of the samples made probably the above com-pensation not better than 2%. Because the transition bandwidth in Fig . 2 494 C A N A D I A N J O U R N A L O F P H Y S I C S . V O L . 3 7 , 1959 decreases for decreasing magnetic field it is likely that Fig. 2 approaches a step function for no stray magnetic fields at all. (c) For constant temperature a marked hysteresis loop was observed when the current was increased to and decreased from above the critical current. When the resistance of the wire jumps from the normal into the superconduct-ing state the currents were only reproducible within 10%. However, for increasing / c 's the area underneath the hysteresis loop increased also. For equal 7c's the purer sample (sample 3) showed generally a smaller area under-neath the loop compared with sample la. If one makes supercooling alone responsible for the hysteresis effect, then it is not obvious why the purer sample has a smaller area. One would expect that impurities favor the growth of nuclei of the stable phase. Probably the magnetic field of the current and inhomogeneities in the metal are partly responsible for the hysteresis effect. (d) The transition temperatures decreased markedly with increasing resis-tivity of the samples at helium temperatures; the difference between samples 3 and la is more than 200 millidegrees K . According to the theory of super-conductivity by Bardeen, Cooper, and Schrieffer (1957) the transition tem-perature, 7"c, is given by: kl\ = l.Uifiu) exp(- l / iV(0)F) where (fiu) is the average phonon energy, Ar(0) is the density of Bloch states of one spin per unit energy at the Fermi surface, and V is the average of the sum of the attractive phonon interaction and the repulsive Coulomb inter-action. The criterion of the above theory is that the attractive part of V dominates the repulsive. Because tantalum has only one stable isotope, the difference in Tc of the various samples cannot be due to the isotope effect. Ar(0) as well as V could be affected by impurities, and strains in the metal would change the vibrational spectrum of the lattice and hence (fiw). Pines (1958) estimates A7(0) V = 0.296 for Ta under the assumption that fio> = &0D/2, where dD is the Debye temperature. However, the present state of the theory does not allow any quantitative calculations of the changes of the above three quantities with impurities and strain and, therefore, one cannot estimate which quantity is dominant in this experiment. (e) When a prestretched tantalum wire is cooled from above the transition temperature with a small constant current (4.2 ma) flowing in it, then the resistance decreases above the transition temperature and approaches zero at T ~ Tc, where T c is defined by the extrapolation of the Te-T curve to J c = 0. When a prestretched tantalum wire is heated from below the transi-tion temperature with a small constant current flowing in it, then at approxi-mately the transition temperature a small resistance appears and the resistance approaches its normal value well above the transition temperature. ACKNOWLEDGMENTS The author is indebted to Professor R. E. Burgess and Dr. J. B. Brown for their advice and encouragement in these experiments. Special thanks are due to Dr. V. Griffiths, who supplied two of the Ta samples; to Mr. A. W. Greenius F I N K : D E S T R U C T I O N O F S U P E R C O N D U C T I V I T Y 495 from the British Columbia Research Council, who performed the micro-hardness tests; and to M r . H . H . M . Zerbst, who prepared the drawings. The author wishes to thank the National Research Council, Canada, for a student-ship and for the support of the project. REFERENCES ALEKSEYEVSKY, N. E. 1938. J. Exptl. Theoret. Phys. U.S.S.R. 8, 342. BAIRD, D . C. 1958. Can. Met. Phys. Conf., Queen's Univ., Kingston, 3-5 Sept. 1959. Can. J. Phys. 37, 129. BARDEEN, J., COOPER, L. N., and SCHRIEFFER, J. R. 1957. Phys. Rev. 108, 1175. GORTER, C. J. 1957. Physica, 23, 45. GORTER, C. J. and POTTERS, M. L. 1958. Physica, 24, 169. DE HAAS, W. J . and VOOGD, J. 1931. Commun. Phys. Lab. Univ. Leiden, 214c. HULM, J. K. 1950. Proc. Roy. Soc. (London), A, 204, 98. KAPLAN, B. and DAUNT, J. G . 1955. Phys. Rev. 89, 907. KUPER, C. G . 1952. Phil. Mag. 43, 1264. _ . LONDON, F. 1937. Une conception nouvelle de la supraconductibilite (Herrmann & Cie, Paris). 1950. Superfluids, Vol. I (John Wiley & Sons, Inc.), p. 120. MENDELSSOHN, K. and OLSEN, J. L. 1950. Proc. Phys. Soc. (London), A, 63, 2. MENDELSSOHN, K. and ROSENBERG, H. M. 1952. Proc. Phvs. Soc. (London), A, 65, 388. PINES, D . 1958. Phys. Rev. 109, 280. RINDERER, L. 1956. Helv. Phys. Acta, 29, 339. SCHUBNIKOW, L. W. and ALEKSEYEVSKY, N. E. 1936. Nature, 138, 804. SCOTT, R. B. 1948. J. Research Natl. Bur. Standards, 41, 581. SILSBEE, F , B. 1916. J. Wash. Acad. Sci. 6, 597. WEBBER, R. T. 1947. Phys. Rev. 72, 1241. 

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