The thermal conductivity of niobium—zirconium alloys at low temperatures

The thermal conductivity of niobium—zirconium alloys at low temperatures

The thermal conductivities of three samples of heat-treated niobium-zirconium alloy have been measured at low temperatures, and analysed into componen...

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The thermal conductivities of three samples of heat-treated niobium-zirconium alloy have been measured at low temperatures, and analysed into components due to phonon and electron carrier~ The phonon carrier component is found to be dominant at most temperatures, particularly below the superconducting critical temperature. The observed variations in the magnitude of the conductivity are correlated with the metallurgical structures

The thermal conductivity of niobium--zirconium alloys at low temperatures N. Morton, B. W. James, G. H. Wostenholm, R. J. Sanderson, and M. A. Black

The low temperature thermal conductivities of a wide range of pure metal and alloy superconductors are discussed in an early paper by Hulm, 1 who has conveniently summarized the systematics of likely contributions to the conductivity both above and below the superconducting critical temperature Tc. The thermal conductivity of normal metals at low temperatures is reviewed by Klemens 2 and by Mendelssohn and Rosenberg. 3 In general the thermal conductivity of a metal K is a sum of two contributions, Ke and Kg, due to electron and phonon carriers respectively, thus K = Ke + Kg

(1)

Various scattering mechanisms contribute to the two corresponding thermal resistivities We = Ke'l and Wg = Kg"1. The normal state thermal resistivity for electron carriers is given by the expression Wen = (LooT)'I + D T 2

(2)

where the two terms represent elastic scattering due to lattice defects and inelastic scattering by phonons respectively. L is the Lorentz number (which takes the value 2.445 x 10-8 V 2 m "2 if the mean free paths for electrons between scatters by lattice defects are identical during thermal and electrical conduction) and o 0 is the residual electrcal conductivity. D is a constant given approximately by the expression 70 N 2/3 D=

(3)

Keoo0D 2

where N is the effective number of electrons per atom, Keoo is the electronic component of the thermal conductivity at high temperature, limited by phonon scattering, and 0 D is the Debye temperature. For a wide range of metals it is found that 70 N2/3 ~ 15. In the limit of high temperatures ( T > 0D) the electrons are scattered elastically by phonons and defects, and the Wiedemann-Franz law holds to a good approximation, thus Keoo= L

ooo

T

(4)

where o,,~ is the high temperature electrical conductivity. In the superconducting state the electronic contribution to the conductivity Kes falls relative to Ken due to the reduced number of carriers. Values of Ken below Tc are obtained in practice either by appropriate extrapolation from thermal conductivity values above Tc, or by direct measurement with the application of a magnetic field of sufficient strength to induce the phase transition. The thermal resistivity is given by the expression Wes = (L o 0 T f) "1 + D T 2 g-1

(5)

where f and g are unique functions of the reduced temperature T Tc'l. Bardeen, Rickayzen, and Tewordt 4 have derived analytic expressions for b o t h f a n d g, and a more reliable numerical calculation o f g has been made subsequently by Tewordt. 5 The analytic function l a n d the numerical function g are in agreement with experimental data, at least for weak-coupling superconductors. In general several different phonon scattering mechanisms contribute to W.g. Two mechanisms operate at low temperatures, namely electron scattering and defect scattering. In the normal state the appropriate expression for the thermal resistivity is (6)

Wgn = A T -2 + Z, Cm T m

m

where A and Cm are constants which characterize the electron and defect scattering mechanisms respectively. The electron-phonon scattering constants D and A appearing in (5) and (6) are related by the expression A = 3.2 0D 4 N 4D D x 10-3

(7)

The temperature variations of the defect scattering resistivity contributions are determined by the relative sizes of the defects and the typical phonon wavelength~ at low temperature, which is inversely proportional to temperature and typically • 10-8 m. Scattering by small defects, such as isotopes or impurity atoms, follows the Rayleigh scattering law, that is, the scattered intensity is proportional to ~-4. The mean freepath between scatters l is correspondingly proportional to-~4, thus I ~ T-4

The authors are with the Department of Pure and Applied Physics,

University of Salford, Salford, Lancashire, UK. Received 22 June 1973.

CRYOGENICS

. NOVEMBER

1973

(8)

A well-known general relationship exists between 1, the appropriate thermal resistivity W, the heat capacity S per

665

unit volume, and the carrier velocity v, thus I4/-1 =

1 --S

3

vl

(9)

Combining (8) and (9), assuming the Debye formula for S and v to be constant yields W ~ T. The magnitude of the appropriate constant of proportionality C 1 is proportional to the square of the difference in masses for isotope defects, and is correspondingly small in this case. C 1 for the case of chemical impurities is less easy to predict, but may be small for an impurity with mass close to that of the host atom. For large defects and for crystal boundaries the scattering is independent of wavelength, and correspondingly W cc/-3. Dislocations produce a resistivity W = / - 2 . In the superconducting state the electron scattering coefficient A is reduced, due to the smaller number of normal state electrons. As C m will be identical in both physical states the appropriate expression for the thermal resistivity in the superconducting state is given by Wgs = A T "2 h "1 + ~, C m T m m

(10)

Bardeen, Rickayzen, and Tewordt 4 have derived an expression for the factor h as a function of reduced temperature, which is, however, only in qualitative agreement with experimental data. Klemens and Tewordt 6 have recalculated h for samples with varying purities, obtaining better agreement with experiment. Strictly (2), (5), (6) and (10), are only accurate when one term predominates, due to possible interference between scattering mechanisms, and experimentally other temperature dependences may be observed. Reports of low temperature thermal conductivity measurements on samples of type II (alloy or compound) superconductors with technological potential have been infrequent to date. For example, measurements have been described by Cody and Cohen 7 for Nb3Sn, by Radosevich and Williams 8 for NbC0.96 and by Dubeck and Setty 9 for 67% Nb 33% Ti in the limited range 2 to 4 K. Two extremes of behaviour are observed in the wider range measurements, depending whether the major contribution is due to electrons (Nb3Sn) or phonons (NbC0.96). The data for Nb 3 Sn show that the conductivity falls sharply below the critical temperature, due simply to the reduction in the number of active electron carriers. Below 6.5 K the phonon conductivity is enhanced due to the weakened electron scattering, and is in turn limited by crystal boundary scattering at very low temperatures. In contrast the data for NbCo.96 show that the conductivity rises sharply as the temperature is reduced below Tc, due to the reduced scattering of phonon carriers by electrons, and it only falls to zero in the usual manner due to crystal boundary scattering at very low temperatures. The thermal conductivity in the normal state exhibits a peak at approximately 50 K for NbCo.96, but is almost constant for Nb3Sn. Dubeck and Setty state that their data imply a phonon conduction mechanism predominates in niobium-titanium alloy at low temperatures. We report here measurements of the thermal conductivities of three samples of nominal 75% Nb 25% Zr alloy, which have been quenched from 1 500°C and annealed at 800°C for periods up to 5 hours to enhance their suitability for technological applications. This work was partially motivated by the desire to obtain thermal conductivity data for use in investigations of the thermo-magnetic instabilities, 666

known as flux jumps, observed in many type II superconductors. The heat treatment, metallurgical examination and structures, and critical current carrying capacity Jc of these alloys have been described in detail by Waldron, 10;,11 who quenched many samples from 1 500°C and subsequently annealed them within a range of temperatures up to 900°C for periods of up to 20 hours. According to the phase diagram of Rogers and Atkins 12 a miscibility gap exists at temperatures below 970°C, and correspondi.ngly the annealed alloys segregate into zirconium rich and niobium rich regions. For quenched alloys annealed at 800°C Waldron found that Jc increases for periods of anneal up to approximately 2 hours, then reduces to a minimum after approximately 5 hours, and increases again slightly after 20 hours. The upper critical field H, of the material at 4.2 K reduces to approximately 5 T afte~ annealing for 1 hour, and, by comparison with results obtained for samples annealed at 900°C, presumably reduces rather more and remains substantially constant for longer annealing periods. The x-ray and chemical analyses, and optical and electron microscopy studies showed that a zirconium rich phase,/3Zr, precipitates within grains and particularly at grain boundaries, although it is only readily observed after annealing for five hours at 800°C. The matrix is a niobium rich phase,/3Nb, as shown clearly by the observed reduction in magnitude o f H c .13,14 Annealing 2 for 20 hours produces a cellular structure in the grain boundary precipitates. According to the phase diagram of Rogers and Atkin the 13Zrprecipitates and the flNb matrix will contain 34 and 83 atomic percent of niobium respectively, and the final proportion of/3Zr will be 20% by weight. The large Jc values observed for samples annealed for one to two hours were not explained by these observations, but were very reasonably attributed by Waldron to the existence of a very finely divided/3Zr precipitate within the matrix, with dimensions below the limits of resolution available. The peak in Jc was attributed to the growth to and beyond an optimum size of/3Zr particles. According to the magnetic energy flux pinning theory of Anderson and Kim 15 the optimum size would be of the order of the coherence length for the material, that is, ,x, 10-8 m. Alternatively, if the flux is pinned largely by the strain field surrounding a defect, the optimum defect size could be less than the coherence length. The alternative suggestion made by Waldron for the existence of an optimum inter-particle spacing appears less likely, especially in the light of the recent analysis of flux pinning mechanisms given by Kramer. 16 The cellular structure observed at the grain boundaries may be responsible for the small increase observed in Jc for well annealed samples. The/3Zr precipitate is in fact a less efficient flux pinner than some other types of defect, possibly due to the fact that it is itself a superconductor with a higher critical field than the matrix, 13,14 although a rather lower critical temperature.17,18 It is not clear, therefore, whether the precipitates represent attractive or repulsive potentials for the flux lattice. The dynamic pinning model of Yamafuji and Irie 19 allow~ an understanding of the pinning mechanism in the latter case. It is important to note that any effect of dislocations on the Jc values of these alloys was largely eliminated by the high temperature anneal.

Experimental details Three strip samples of nominal 75% Nb 25% Zr alloy, with dimensions approximately 20 x 6.4 x 0.28 mm were selected from a series investigated by Waldron. The strips chosen had CRYOGENICS

. NOVEMBER

1973

been spark cut from a sheet rolled from batch 1 (B1) material, 10,11 parallel to the rolling direction. Subsequently the strips had been heated in a dynamic vacuum for one hour at 1 500°C and quenched in silicone oil, followed by annealing in vacuo in sealed quartz ampoules for 10 minutes, one, and five hours, respectively at 800°C. The samples had been inserted and removed from the furnace at this temperature. The samples showed some surface contamination, but this was not removed to avoid altering the microstructure, and the samples were simply wiped clean with acetone before mounting in the cryostat. A conventional Searle's arrangement was used in the thermal conductivity measurements. The samples were mounted in a cryostat comprising an outer chamber immersed in liquid helium, with an inner chamber, which acted as a heat shield, containing the samples. The shield temperature was controlled by current fed to a heating coil from a regulating circuit, which sensed the resistance of a carbon resistance thermometer attached to the shield. The necessary small heat leak for temperature control, between the shield and the helium bath, was maintained by a low pressure of helium in the space between the outer chamber and the shield. Four copper clamps were provided to attach one end of a sample to a copper post attached to the heat shield, to connect the electrical heater at the other end, and to attach the two thermometers at intermediate points. The facewidth of the thermometer clamps was 1 mm, and they were spaced 9 mm apart between centres. The thermometers and heaters were nominal 47 ~2 Allen-Bradley 1/10 W carbon resistors, smeared with vacuum grease and pushed into holes drilled in the copper clamps. All electrical connexions from the shield to the samples were of long lengths of 36 swg constantan wire. Sufficient space was available to mount two samples within the heat field in a back-to-back configuration. A temperature difference of 0.4 to 1 K was maintained between the thermometers by the heaters, and the resistances of the thermometer and the heater powers were measured potentiometrically. The carbon thermometers were calibrated after each run, at approximately degree intervals, against a germanium resistance thermometer pre-calibrated at tempexatures up to 20 K by The National Physical Laboratory, Teddington (NPL), with an estimated precision of 0.02 K. This calibration was in the form of an eighth order polynominal of the form 8 T"l=

~ai(logeR)i

The germanium thermometer was mounted in the copper post, and a low pressure of helium exchange gas was introduced into the heat shield to equalize temperatures. The resistance temperature data for the carbon thermometers were fitted to the interpolation formula for the AllenBradley resistors suggested by Clement and Quinnell 20 (12)

Very similar, systematic discrepancies of up to 0.1 K were obtained for all the carbon thermometers used, between the temperatures calculated from (11) and (12). Since the germanium thermometer temperatures departed rather abruptly from the Clement and Quinnell formula (12) temperatures

CRYOGENICS. NOVEMBER 1973

The critical temperatures were determined electrically in a separate experiment. The samples were mounted within the heat shield, filled with a low pressure of helium exchange gas. Current and voltage leads were clamped to the specimens, and the voltages generated across the samples by a small, direct current were amplified and displayed on the y-axis of an x - y recorder. The voltage developed across the germanium resistance thermometer, connected in an independent circuit, was amplified and fed to the x-axis. A potentiometer was used to back-off the bulk of the voltage before amplifica{ion in this case. The temperature of the samples was varied slowly by adjusting the current fed to the heating coil around the heat shield. As a precaution against thermal lag effects the phase transitions were observed during both increasing and decreasing temperature excursions. The normal state electrical conductivities were determined at 17.25 K and at room temperature (measured after the liquid helium had evaporated) by direct potentiometric measurements of the voltages developed by known currents. In these measurements the length dimension was taken to be the distance between the inner faces of the voltage sensing clamps.

Analysis of results (11)

i=1

loge R + a (loge R) -1 = b + c T "1

at certain points, which is physically improbable, it seemed that the estimate of precision of the NPL interpolation polynomial (11) of 0.02 K was slightly too low. A spurious oscillation of the high order polynomial was evidently present. With this in mind, smoothed correction graphs were plotted of the discrepancies between the two sets of temperatures for use in processing the thermal conductivity data. The random deviation of calibration data from the smooth curves was generally less than + 0.02 K, with the exception of the region noted above, where systematic deviations up to -+0.03 K occurred. The errors in the thermal conductivity data, due to temperature instabilities and the smoothing procedure errors, were investigated in simulated conduction experiments made with zero heater power. The calculated temperature difference under these circumstances did not exceed 0.03 K, and for typical temperature differences of 0.7 K, the corresponding error in the conductivity data was 4.5%. The precision was further limited by uncertainties in the thermometer separations and other sample dimensions, and by the heat leaks along connecting wires that would occur during data recording. The mean uncertainty of the thermal conductivity data was finally estimated to be approximately -+7%.

The measured thermal conductivities of the three samples are shown as functions of temperature in Figs 1 and 2 a, b, c and the critical temperature and resistivity data are summarized in Table 1. The thermal conductivity graphs clearly exhibit a behaviour which is intermediate between the two extremes observed previously for NbC0.96 and Nb3Sn, hence both phonon and electron carriers are likely to be important in the present case. Accordingly the thermal conductivity will be represented by an expression which combines equations 1-10, thus K=Ke+Kg = [ ( L o 0 f T ) ' I + D T 2 g ' I ] "1

+(A T-2 h"1 + ~ CmTm)-I m

(13)

where f, g and h reduce to unity for temperatures above

667

X

3.0

m i

2.C

X

.

'

/x'~f..~ * /

1.0

X

~

I

o

t

5

t

IO

15

r,K Fig.1 Thermal conductivity versus temperature for 75% Nb 25% Zr alloys annealed at 800°C for 10 minutes (x), one hour (+), and five hours (o) denoted a, b, and c respectively

a

3

x

x x x

b

'T'I~ ~ "0

3

re

2

+

++

= ~

t

C

o

o

0 I

Of'-

O

I~ , 5

0

0

o

o

o

.~-I 15

Thermal

conductivity

versus temperature

for 75% Nb

25% Zr alloys annealed at 800°C for (a) 10 minutes, (b) one hour,

and (c) five hours respectively. The full and broken curves show the proposed variation of the phonon and electron carrier contributions respectively Table 1

Annealing period

T c, K

o t ~Ol'm "1 x 106

o*, Q-1 m-1 x 106

10 min

10.89

2.55

1.66

1h

10.89

3.67

2.10

5 h

10.85

3.57

1.98

t

A t 17.25 K

* A t 293 K

668

where P0 and p are the resistivities at 17.25 K and room temperature respectively. The ratio of D to D' was close to the value 30 found for Nb3Sn by Cohen, Cody, and Goldstein. 21 As the effect of D on the thermal conductivity is limited below the critical temperature, the values obtained for h will not be seriously in error.

T2(K-Ken) " I = A + ~

T, K F i g . 2 a , b, c

(14)

First estimates for the values of A and Cm were obtained by assuming D to be zero, hence Ken ,x, L o 0 T, and writing (13) for the normal state in the form

0 m

I IO

Since (L o 0 T)"1 >>D T 2 in the temperature range employed it proved impossible to obtain accurate values for D directly from the thermal conductivity data. Nominal values for D were estimated from (7) using the A values determined, and assuming a value for OD ,x, 220 K (obtained by linear interpolation between values quoted by Morin and Maita 18) and 70 N 2 / 3 . 1 5 . These values were compared with values, D', calculated from (3), withthe same assumptions for the magnitudes of 0 D and N, and extrapolating the measured phonon induced electrical resistivity linearly to high temperatures and using the Wiedemann-Franz law. (4), to estimate Keoo, thus Keoo= L (293 - 17.25) (p - p0) "1

+

+

0

Tc. An attempt has been made to fit the data to this expres~on. This analysis largely ignores the separation into twc distinct phases, since the measured critical temperatures are those of the/3Nb matrix, although the measured electrical conductivities will correspond to appropriately averaged values. The analysis must therefore be regarded as tentative. The functions l a n d g in the superconducting state have been assumed to be the successful expressions developed by Bardeen, Rickayzen and Tewordt,4 and by Tewordt 5 respectively, whereas the function h has been estimated from the data. In brief, the coefficients A, Cm, and D were estimated from normal state data, Kes was then calculated directly, and KIZs values were obtained by subtraction from the measured data. The function h was estimated from the K~ values using (10) and the values of A and Cm. Since h w~ll presumably be approximately a unique function of the reduced temperature, the satisfactory achievement of this condition should confirm the internal consistency of the analysis.

m

Cm T m +2

(15)

The simplest form for the right-hand side of(15) is obtained by assuming that effective defects are relatively small compared with the phonon wavelen~thX, when m = 1. Graphs of T 2 (K - L o 0 T)"1 against T? were indeed approximately straight lines, with intercepts A and slopes C 1 . Values of D were then calculated using (7), as described above, followed by the calculation of more accurate values for Ken using (2). New graphs of T 2 (K - Ken)"1 against T 3 allowed second estimates for A (and hence D) and C 1 to be obtained. The procedure was repeated as necessary until stationary values were obtained. The values for h were then calculated, as described above, with satisfactory agreement apparent between values for the three samples. The separate contributions of Ke and K~ to the thermal conductivity are shown in Fig.2a, b, c and the thermal resistivity contributions are shown in Fig.3a, b, c. The numerical values estimated for A, C1, D, and D' are summarized in Table 2, and the values for the function h in the superconducting state are plotted as a function of reduced temperature in Fig.4.

CRYOGENICS.

NOVEMBER

1973

Table 2

Annealing period

~-l

m deg3

~l

D' w.'l m m deg"1 x 10 -4 deg-1 x 10 -5 D/D'

C1 W"1 m

10 min

68

'~ 0

2.0

0.97

1h

94

'v0.014

2.8

0.94

29.8

5h

120

0.014

3.5

1.03

34

1.38

0.46

30

Nb3Sn* *

20.6

From reference 21

I

a

\

\

\

2

I O 4 o~ u

i

x~ E

3



2

I

I .m

scattering at low temperatures was mvestigated using (9), assuming a crystal size • 40/~,10,11 and published values for the atomic weights and the heat capacity,18 and estimated density and sound velocity. This contribution was shown to be negligible within the range of temperatures employed. Conclusions

The form of the thermal conductivity graphs in Fig. 1 suggest that both electron and phonon carders contribute substantially to the thermal conductivity of niobium-zirconium alloys. However, the phonon contribution is generally dominant, and the electronic contribution becomes completely negligible within the liquid helium range of temperatures, in agreement with the results obtained for a niobiumtitaniiJm alloy by Dubeck and Setty. The observed differences in the thermal conductivities are therefore due chiefly to variation in the magnitude of the parameter A, with secondary variations due to the variation of the magnitudes of the parameters C 1, o 0, and D. The magnitude of the parameter A, which characterizes the degree of scattering of phonon carriers by electrons, increases with annealing time, although a less marked change is observed after the first hour. Combining (3) and (7) yields an expression for A, thus

~-m

b x\

A = 0.224N 2 0D 2 Keoo"1

I O 4

C

I \

3

\ %

2

I O

5

IO

15

r,K Fig.3a, b, c Thermal resistivity contributions versus temperature for 75% Nb 25% Zr alloys annealed at 800°C for (a) 10 minutes, (b) one hours, and (c) five hours respectively. The full and broken curves show the proposed variation of contributions to the phonon and electron carrier resistivities respectively. The upper pairs of curves in each figure show resistivity due to phonon scattering by electrons, and electron scattering by lattice defects respectively. The lower pairs of curves show resistivity due to phonon scattering by lattice defects, and electron scattering by phonons respectively. The curve showing scattering of phonons by lattice defects is omitted from (a)

(16)

The increase in the magnitude of A must be attributed directly to the formation of the BNb matrix; however, the value of 0 D will only increase slightly,18 and values for Keoocalculated from (14) are approximately constant. The latter result is found quite generally in alloy systems,2 and is perhaps further confirmed by the constancy of the electronphonon coupling parameters calculated for a wide range of niobium alloys by Morin and Maita, 18 employing the wellknown BCS expression for the magnitude of Tc. Hence, according to (16), the observed increase in magnitude of A is possibly due to a modest increase in the value of N ( b y approximately 30%). The corresponding magnitude of D would be more constant according to (3).

X +

4 X 4"

~0 ÷

Since it was observed that the magnitude of A increased substantially with increasing annealing time a second analysis was made. Here A was held constant (equal to the smallest value obtained previously) and a possible defect scattering contribution to Wg of the form C.2 T "2 postulated to explain the reduced ~onductivity. It was felt that such a contribution might conceivably arise from the presence of suitably sized defects, although the effect of dislocations, which produce a resistivity of this type, may of course be ignored in these samples. The values of h were calculated as before, but were found in general to be negative so this approach was rejected. The further possibility of a contribution to Wg of the form C. 3 T -3, due to crystal boundary

CRYOGENICS . NOVEMBER

1973

X ÷

2

0

X +

0 x

0

+

I

x

÷

I

O

Fig.4

0.2

I

I

I

0,4

0.6

O.8

~.

O

I.O

Variation of the factor h as a function of temperature

669

The magnitude of 6"1, which characterizes the scattering of phonons by point defects, increases from approximately zero, becoming constant after annealing for one hour. A small value for C 1 is probably to be expected in a quenched niobium-zirconium alloy, due to the almost identical masses of the two species. The marked increase of C 1 with annealing must be attributed to the formation of a very finely divided/3Zr precipitate, with a particle size smaller than the typical phonon wavelength, that is, less than 10 -8 m. Since in principle up to 20% by weight (or volume) of the material may be contained in these precipitates, it is easy to demonstrate that the mean inter-particle spacing will be little larger than the actual size. The mean free path for phonons between scatters will be correspondingly small. In fact, a thermal resistivity corresponding to this scattering length calculated from (9), employing published values of the atomic weights and heat capacity, 18 and estimated density and velocity of sound, is in rough agreement with the observed defect induced resistivity. The small size and spacing of the ~zt precipitates is evidently responsible for their effectiveness as a magnetic flux pinner, with corresponding high values of the critical current. Since the particles are probably of dimensions rather smaller than the coherence length (~, 10 -8 m), it is possible that their effectiveness as flux pinners is enhanced by the larger surrounding strain field regions. The observed increase in the low temperature electrical conductivity after annealing for one hour reflects the appearance of the higher conductivity/~N-b matrix. The magnitude of the residual resistivity Po for the alloy annealed

670

for 10 minutes is in fair agreement with published values 13 for 75% Nb 25% Zr, and the decrease on annealing is compatible with the assumed decrease to 17% Zr in the matrix as PO is proportional to the zirconium content. The values of the function h shown in Fig.4 fall below the theoretical values,4,6 but are self-consistent for the three samples.

References 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

Hulm, J. K. ProcRoySocA204(1951)98 Klemens,P. (3. Solid State Physics 7 (1958) 1 Mendelssohn,K., Rosenbetg, H. M. Solid State Physics 12 (1961) 223 Bardeen,J., Riekayzen, G., Tewordt, L. Phys Rev 113 (1959) 982 Tewotdt, L. Phys Rev 129 (1963) 657 Klemens,P. G., Tewotdt, L. RevModPhys36(1964) 118 Cody, G. D., Cohen, R. W. RevModPhys36(1964) 121 Radosevich,L. G., Williams, W. S. Phys Rev 188 (1969) 770 Dubeck~L., Setty, K. S. L. Phys Lett 27A (1968) 334 Waldxon,G. W. J. JLess-Common Metals 17 (1969) 167 Waldton,G. W. ]. JMaterSci4(1969) 290 Rogers,B. A., Atkins, D. F. TransAIME 203 (1955) 1034 Bedincoutt, T. G., Hake, R. R. PhysRev131(1963) 140 Coffey, H. T., Hulm, J. K., Reynolds, W. T., Fox, D. K., Span, R. E. JAppIPhys 36 (1965) 128 Anderson,P. W., Kim, Y. B. Rev Mod Phys 36 (1964) 39 Ktam¢~,E. J. JApplPhys 44 (1973) 1360 Hulm,J. K., Blaugh~, R. D. PhysRev123(1961) 1569 Motin,F. J.,Maita, J. P. PhysRev 129(1963) 1115 Yamafuji,K., Ifie, F. Phys Lett 25A (1967) 387 Clement,J. R., Quinnell, E. H. Rev SciInstr 23 (1952) 213

Cohen,R. W., Cody, G. D., Goldstein, Y. Tech Rept AFMLTR-65-169 (1965) 162

CRYOGENICS. NOVEMBER 1973