Contains a report on a research project.Lincoln Laboratory (Purchase Order DDL B-00368)United States ArmyUnited States NavyUnited States Air Force (Contract AF19(604)-7400

Andrews, J. M., Jr.

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RADIO PHYSICSI. MICROWAVE SPECTROSCOPYProf. M. W. P. Strandberg R. Huibonhoa H. PauwelsProf. R. L. Kyhl J. G. Ingersoll Mahin RahmaniDr. B. D. N. Rao P. F. Kellen W. J. SchwabeJ. M. Andrews, Jr. J. D. Kierstead J. R. ShaneJ. C. Burgiel S. H. Lerman N. TepleyY. H. Chu J. W. Mayo C. F. TomesA. ULTRASONIC ATTENUATION IN SUPERCONDUCTORSThe work described here was initiated several months ago in the very high frequencyrange. Equipment was designed and constructed for the generation and reception of ultra-sonic pulses at 165 mc. The transmitter consists of the high-frequency and blocking-oscillator portions of the circuit used by Chick, Anderson, and Truell. The transmittingand receiving transducers are identical helical resonators2 exciting x-cut quartz rodsthat are inserted in an rf electric field produced in a gap of nearly identical geometryto that of the re-entrant cavity used in the microwave phonon experiments. 3 These hel-ical resonators are shown in Fig. I-la. The two quartz rods (0. 118 inch in diameter,0. 504 inch long) with a metal specimen (usually approximately 0. 1 inch thick) sandwichedbetween their ends are inserted through a small hole in the end of each helical resonatorinto the electric-field gap. The resonators are then clamped in this coaxial configuration,mounted on stainless-steel coaxial lines, and immersed in a standard double-dewarhelium cryostat. Signal power from the receiving transducer is mixed with local oscil-lator power in a strip-line hybrid coupler; the dual output is then passed through a bal-anced detector to a 30-mc, 120-db, if strip.This apparatus has been used to study ultrasonic attenuation in specimens of indiumand mercury, both in the superconducting and normal states. The metals were obtainedfrom standard chemical reagent stock (In, 99. 99 per cent pure; Hg, 99. 999 per centpure). No attempt was made to obtain single crystals with known orientation. In thecase of indium, the quartz transducer rods were clamped in a vee-block and the indiumwas allowed to solidify in a Teflon mold.4 The mercury specimen was contained in asmall, cylindrical aluminum capsule; the quartz transducer rods had been fitted intoaluminum bushings that were inserted into the ends of the capsule and cemented intoplace. (See Fig. I-lb.)1. ExperimentIn a typical experiment the entire transducer assembly containing the metal speci-men is cooled to some fixed temperature well below the superconducting transitionThis work was supported in part by Purchase Order DDL B-00368 with Lincoln Lab-oratory, a center for research operated by Massachusetts Institute of Technology withthe joint support of the U. S. Army, Navy, and Air Force under Air Force ContractAF19(604)-7400.QPR No. 691I i I I I I 1 111111111111111 l IIIIIIIII IIIIII -(b)Fig. I-1. (a) Helical resonators used for the transmitting andreceiving transducers.(b) Mercury capsule and the indium mold. The quartztransducer rods are shown protruding from each.QPR No. 69(a)(b)Fig. 1-2. (a) Acoustic pulses transmitted through superconductingmercury. al is rf leakage; a 2 , an acoustic pulse thathas passed through both quartz rods and the mercuryspecimen; a 3 , superposition of two acoustic pulses,one having made a double reflection in the first rod,and the other, a double reflection in the second rod.(b) Acoustic pulses transmitted through normal mercury.This is exactly the same situation as in (a), exceptthat a magnetic field H >> Hc(T) has been switched on.QPR No. 69(I. MICROWAVE SPECTROSCOPY)temperature. The transmitter initiates, in the transmitting transducer, a pulseof rf power of approximately 25 watts, peak, and 1 psec in duration. A smallfraction of this power (=10 - 3 ) is converted into a pulse of acoustic energy oflongitudinal polarization by the piezoelectric effect. This pulse passes along thefirst quartz rod, through the specimen, and into the second quartz rod. At thereceiving transducer the same fraction of the acoustic power is reconverted intoan rf signal, and the rest is reflected. Figure I-2 shows the resulting oscillo-scope traces. The first pulse is rf leakage. The second is an acoustic pulsethat has passed through both quartz rods and the metal specimen. The thirdpulse is actually a superposition of two acoustic pulses: one has made a doublereflection in the first rod; the other, a double reflection in the second rod.The superposition is caused by the fact that the two quartz transducer rods areequal in length (within ±.001 in.).An electromagnet surrounds the portion of the cryostat containing the metalspecimen, which is capable of providing a variable magnetic field normal tothe ultrasonic wave vector and the cylindrical axis of the metal specimen. Ifa magnetic field greater than the critical field of the superconductor is switchedon, the metal returns to the normal state and the attenuation increases abruptly.Figure I-2b shows the same pulse-echo pattern as that shown in Fig. I-2a, exceptthat the magnetic field H >> Hc(T) has been turned on. This change in the ultra-sonic attenuation is due entirely to the effect of the conduction electrons in themetal.The relative attenuation was measured as a function of applied magnetic fieldfor both metals, and is plotted in Fig. 1-3. Observe that it is a continuousfunction, a result that is implicit in the aluminum data of David, Van der Laan,and Poulis. 5 Data on the total change in attenuation between the normal andthe superconducting states should probably be taken from the first pulse only.The reason for this is that the acoustic energy that arrives at the receivingtransducer at the time indicated by the second pulse, since it is a superposi-tion of two acoustic pulses that may well have travelled over slightly differentpaths, has an unknown phase factor determined by the details of the paths.A specimen of mercury, approximately 0. 25-in. thick, was prepared in orderto obtain reasonably reliable velocity data. The velocity was measured at roomtemperature at 165 mc and found to be 1.47 ± .03 km/sec, which is in agree-ment with the measurements of Ringo, Fitzgerald, and Hurdle. 6 A specimenof this length, however, is more vulnerable to thermal contraction effects. Thusfar, it has not been possible to maintain satisfactory acoustic contact betweensuch a long specimen and the quartz transducer rods as the temperature islowered.QPR No. 69INDIUM T= 1.750K1st PULSE A H, =209 GAUSSI He = 120 GAUSSI100 200 300H (GAUSS)(a)14121086422MERCURY T= 1.90 K2nd PULSEI He (T) = 325 GAUSS100 200 300H (GAUSS)(b)Fig. 1-3. (a) Relative attenuation of a 165-mc ultrasonic pulse inindium as a function of magnetic field.(b) Relative attenuation of 165-mc ultrasonic pulses inmercury as a function of magnetic field.QPR No. 69 5(I. MICROWAVE SPECTROSCOPY)2. DiscussionAn important mechanism for ultrasonic attenuation in metals at sufficiently low tem-peratures that the electronic mean-free path becomes of the same order of magnitudeas the acoustic wavelength (usually below 100K) is the electron-phonon scattering. Thus,if we have P acoustic power incident on a metal specimen of length L, the power Poleaving the specimen will beP = Po exp[-(aelec+ao)L]. (1)Here, aelec is that portion of the attenuation coefficient caused by scattering from theconduction electrons; a is that portion of the attenuation coefficient caused by scat-tering from thermal phonons, lattice defects, grain boundaries, and so forth. The coef-ficient a is temperature-independent in the range of temperatures which we are con-sidering, and therefore will not be discussed further. Using the free-electron model,Pippard has calculated expressions for the electronic contribution to the attenuationcoefficient for arbitrary mean-free path. This approach has been well justified by theexperimental results of Morse at frequencies between 9 mc and 56 mc. Pippard'sresult for longitudinal polarization isS 1 1 (ql)2 tan- (ql) (2)a a' -Kn, (2)n -ql-tan (ql)wherea the electronic contribution to the attenuation coefficient for the metal in thennormal stateq the acoustic wave vector1 the electronic mean-free path.The coefficient a', which is the limiting form of an for long electronic mean-freepath, ql >> 1, is given byTrNmv wat' (3)6powhereN the number of free electrons per unit volumem the electronic massv velocity of the free electrons at the Fermi surfacew the ultrasonic frequencypo the density of the metal specimenu the longitudinal acoustic velocity in the metal specimen.Reasonable estimates based upon the free-electron model, under the assumption ofQPR No. 69(I. MICROWAVE SPECTROSCOPY)one free electron per atom, yield the following values for a' at 165 mc:-1Indium a' = 4.2 cm-1-1Mercury a' = 8. 6 cm1The Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity has been applied9to the calculation of the ultrasonic attenuation in metals in the superconducting state.Their result isaas = 2f[E(T)], (4)nwherea = the electronic contribution to the attenuation coefficient for the metal in thessuperconducting statef[E] = the Fermi function of the superconducting energy gapE(T) - the temperature -dependent superconducting energy gap.Let us represent the change, in decibels, of the ultrasonic attenuation coefficientbetween the normal and the superconducting states at any given temperature by D(T).ThenD(T)a (5)n r4(T)_10 logl0(e) L tanh 2kTwhereL a the length of the specimen in cmk Boltzmann's constantT the temperature in "K.By using the theoretical curve for E(T) given in BCS, 9 it is possible to determine anfrom our experiments on indium and mercury.Specimen Purity D(T) T L (an )exp(per cent) (db) ( 0K) (cm) (cm - 1)In 99.99 3.7 (±.5) 1.75 .172 5.0Hg 99. 999 5 (±1.0) 1.9 .252 4.6These an, determined experimentally, are of the same order of magnitude as the limitingvalues a' calculated, by using Eq. 3, for the metals on the basis of the free-electronmodel. We conclude that for 165 mc the purities of the metals are sufficient to provideelectronic mean-free paths at least of the same order of magnitude as the acoustic wave-length. That is, ql > 1.QPR No. 69(I. MICROWAVE SPECTROSCOPY)As we have already pointed out, the relative attenuation, plotted as a function of mag-netic field, does not yield a discontinuity at Hc , but rather begins a slow rise at He , wellbelow Hc . This might be due to the presence of metallic impurities. Superconductingalloys are sometimes characterized by such a slow transition as a function of magneticfield, but this generally occurs at fields greater than He for the corresponding pureelements. A more cogent explanation is that the metal enters the intermediate state atH e , and then proceeds to the normal state in a continuous manner, the transition reachingcompletion at Hc(T). This intermediate state is a property of macroscopic supercon-ductors, and depends upon the geometry of the specimen. The same behavior has beenobserved in the resistivity of macroscopic superconductors as a function of magnetic10field. The temperature dependence of H is given bycHc(T) = H (0) [1 )]. (6)This formula is an approximation, but it is sufficiently accurate for our use. Here,H (0) is the critical magnetic field at absolute zero, and T is the critical temperatureat zero magnetic field. Both are properties of the particular superconductor underinvestigation. We now evaluate Hc(T) for each specimen using their critical constantsand the temperature, which was determined from the vapor pressure of the liquid helium.Specimen H (0) T T H (T) H nc c c e exp(gauss) (0 K) (0 K) (gauss) (gauss)In 283 3.41 1.75 209 120 .43Hg 411 4. 15 1.9 325 237 .27He was determined from the graphs of the experimental data (Fig. I-3).The macroscopic theory of superconductivity predicts that the metal will enter theintermediate state at a field H that is such thateHe = - n. (7)cThis formula holds for macroscopic bodies whose dimensions are large com-pared with a characteristic length A = 10- 5 cm. The product 4wrn is the demag-netizing coefficient of the specimen which takes into account the effect of itsmagnetization on the external magnetic field while in the superconducting state.Obviously, n depends upon the shape of the specimen. We list a few theoreticaln values for simple cases:QPR No. 69(I. MICROWAVE SPECTROSCOPY)nSphere 1/3Long cylinder in transverse field 1/2Long cylinder in parallel field 0Large disc with its plane normal to field 1The shapes of our specimens were not controlled accurately, but can be describedapproximately. The indium was formed into a disc, 0. 30 cm in diameter and 0. 172 cmthick. At one edge of the disc, however, there was a sprue almost as large as the disc.The applied magnetic field was oriented parallel to the plane of the disc. The mercurywas contained in a cylinder, 0. 635 cm in diameter and 0. 635 cm long. The quartz rodspenetrated 0. 192 cm into the specimen at each end. The applied magnetic field wasperpendicular to the axis of the cylinder. The experimental values for n show, indeed,that the shape of the specimen is important in determining the onset of the intermediatestate.The author wishes to express his appreciation to E. C. Ingraham for the constructionof the helical resonators, the mercury capsule, and portions of the helium cryostat.J. M. Andrews, Jr.References1. B. Chick, G. Anderson, and R. Truell, J. Acout. Soc. Am. 32, 186 (1960).2. W. W. Macalpine and R. O. Schildknecht, Proc. IRE 47, 2099 (1959).3. P. H. Carr and M. W. P. Strandberg, J. Phys. Chem. Solids 23, 923 (1962).4. The author is indebted to Norman Tepley for the fabrication of the indium speci-men.5. R. David, H. R. Van der Laan, and N. J. Poulis, Physica 28, 330 (1962).6. G. R. Ringo, J. W. Fitzgerald, and B. G. Hurdle, Phys. Rev. 72, 87 (1947).7. A. B. Pippard, Phil. Mag. 46, 1104 (1955).8. R. W. Morse, Progress in Cryogenics, Vol. 1, edited by K. Mendelssohn(Heywood and Company, Ltd., London, 1959), p. 221.9. J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. Rev. 108, 1175 (1957).10. W. J. de Haas, J. Voogd, and J. M. Jonker, Physica 1, 281 (1934).QPR No. 69

Microwave Spectroscopy

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Microwave Spectroscopy

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