Thermal expansion at low temperatures of anisotropic metals: Indium

Thermal expansion at low temperatures of anisotropic metals: Indium

In this paper the authors describe their work on two 99.99 per cent pure samples of indium at temperatures down to 1"5° K. The expansion characteristi...

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In this paper the authors describe their work on two 99.99 per cent pure samples of indium at temperatures down to 1"5° K. The expansion characteristics have been determined to this temperature with three points in mind: first the effect of anisotropy ; second the electronic contribution and its correlation with normal-to-superconducting length change; third the thermodynamic consistency of the expansion data with other measurements

THERMAL EXPANSION AT LOW TEMPERATURES OF ANISOTROPIC METALS: INDIUM J. G. COLLINS, J. A. COWAN, and G. K. WHITEr

1 N D 1 u M has a crystal structure which may be considered as tetragonally-distorted face-centred cubic. It is stretched in the direction of the c-axis so that c/a = 1.076 at room temperature. Pearson~ quotes widely differing values of the linear expansion coefficients 0t~ and 0t± measured parallel and perpendicular to the tetrad axis, respectively. The more recent measurements are in slightly better agreement. For example, macroscopic measurements by Vernon and Weintroub, 2 and X-ray determinations by Graham et al. 3 and by Smith and Schneider 4 give values of ct± between 49 ~ind 56 x 10-6 degK -1 and of 0tH between - 4 and - 1 4 x 10-6 degK -~ at room temperature. The X-ray data extend down to 90 ° K, where 0t± and ct:i each have comparable positive values, around 25 x 10-6 degK-L Madaiah and Graham s have also obtained some macroscopic values from single crystals at low temperatui-es; their values seem more scattered but agree qualitatively with the X-ray data. Swenson 6 measured a polycrystalline rod down to about 25 ° K and obtained values of 0t (average) which agree quite well with those obtained from the X-ray determinations (see Figure 1). At low temperatures (below 20 ° K) the only data previously available were the differences in length between normal and superconducting states of indium measured by Olsen and Rohrer. ~-9 We have now determined the expansion characteristics of indium down to 1.5 ° K with three principal problems in mind: primarily, to see how the anisotropy affects the lattice contribution to the expansion and to compare it t Division of Physics, C.S.I.R.O., Sydney, Australia. J.A.C. is on sabbatical leave from University of Waterloo, Ontario, Canada. Received 17 March 1967. CRYOGENICS " A U G U S T 1967

with other anisotropic metals such as cadmium, zinc, magnesium and tin; ~o,n secondly, to determine the electronic contribution and its correlation with the normal-to-superconducting length change; and thirdly, to study the thermodynamic consistency of the expansion data with critical field and specific heat measurements for superconducting indium.

Experimental Details The expansion was measured in a differential cell,

relative to copper, using the three-terminal capacitance method. 12,t3 The only recent alterations to the cell were in the thermometers: the platinum thermometer used for measuring temperatures above 10° K was a Meyers-type (no. 459) calibrated at the National Bureau of Standards on the N.B.S. 1955 Scale; the Honeywell germanium thermometer used below 11 ° K has been calibrated recently in the National Standards Laboratory against a helium gas thermometer. TM The magnetic field of ~ 103 Oe needed for some of the measurements below 3.4 ° K, the transition temperature of indium, was provided by a superconducting solenoid (niobium-25 per cent zirconium) wound on to the outer brass vacuum jacket. The two specimens were single crystals of 99.99 per cent purity supplied by Metals Research Ltd., Cambridge, in the form of cylinders of ¼ in. diameter and just over 2 in. length; the cylinder axes were within 2 degrees of (1) the tetrad axis and (2) a plane normal to the tetrad axis, respectively. They were each reduced to the required length of 5.08 cm with flat, parallel end-faces in the following manner, avoiding damage as far as possible. 2111

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Figure 1. Principal linear coefficients of thermal expansion of indium, parallel (11) and perpendicular (J_) to the tetrad axis and the average value ~ = (2c~j_ + ~zli)/3

The crystals were supported in a 2 in. diameter brass cylinder with bees-wax. The ends were turned on a lathe using a very sharp tool and taking cuts of less than I mil. The surface was bathed in cutting oil to prevent welding of the turnings. The ends were then lightly ground with 8 p carborundum on a flat surface plate, cleaned and lightly etched with a solution of hydrochloric acid, picric T A B L E 1. V A L U E S OF THE LINEAR EXPANSION COEFFICIENT (a × 105 degK-~), V O L U M E COEFFICIENT (p x 106 degK-~), and GRONEISEN P A R A M E T E R S T(°K)

a_L

atl

284 85 75 65 57.5 28 26 25 " 24 22 20 18 16 15 14 12 10 8 7 6 5 4 3.5 3 2.5 2

51.4 26.15 24.2 22.1 20.25 6.49 5.37 4.82 4.27 3.26 2.36 1,63 1.09 0.925 0.815 0.77 0.94 1.125 1.18 1.15 0-995 0.70 0.54 0.383 0.24 0,127

- 7.7 19.55 20.45 21 "35 22.0 22.55 21.84 21.41 20.93 19.71 18-09 16.00 13.41 11.98 10.47 7,30 4.01 0.77 -- 0.325 -- 0.99 -- 1.26 -- 1.05 --0.835 -- 0.60 -- 0"385 --0.20

P 95.1 71.85 68.8 65.5 62.5 35.53 32.58 31-05 29.48 26.23 22.80 19.26 15.59 13.83 12.10 8.84 5.89 3.02 2.035 1.31 0.73 0.35 0.245 0.16 0.095 0,054

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2.43 2.29 2.25 2.25 2.26

2.34 2.30 2.27 2.27 2.26

2.21

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2.53

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2.56

2.38 2.30 2.27 2.26 2.26 2.22 2.24 2.24 2.27 2,30 2.32 2.34 2.33 2.33 2.32 2.32 2.35 2.2o 2.2 2.25 2.25 2.3 2.4 2.4 2.4 2.4

2.3 2.2

2.5 2.2

2.3 2.35 2.4 2.6 2,7 2.7 2.7

2.0 1 "65 1.2 1.15 1.15 1.05 1.1

acid, and alcohol. Finally they were wiped quickly with strong nitric acid. The parallelism of the ends was checked with an autocollimator to ensure they were within 0.5 min of arc. After removal from the brass support a copper wire was cemented into a fine hole at one end. This later served to hold the cylindrical sample firmly in position in the expansion cell (see reference 13). Below 30 ° K, linear thermal expansions were measured for temperature intervals varying from about 1 degK (above 20 ° K) to about 0.2 degK (below 4 ° K). The inaccuracy is governed by the uncertainty in the ,cell calibration (about 1 per cent) or by the lack of sensitivity of the capacity measurement, whichever is the greater. The limitation of sensitivity corresponds to a possible error ~ 10-8 degK -1 in the expansion coefficient. Also measurements of expansion were made between 55 and 90 ° K and between 0 ° C and room temperature; in these regions the probable error in the linear expansion coefficients is around 0.I × 10-6 degK -I, which comes from uncertainty in the cell calibration. Results

The experimental values for the principal linear coefficients are shown in Table 1 and illustrated in Figures 1 and 2. The first figure includes some previous values obtained by Vernon and Weintroub 2 above room temperature using an interferometer, by Graham et al. 3 above 90 ° K using X-rays, and by Smith and Schneider 4 above 100 ° K using X-rays. Points attributed to Smith and Schneider are each an average of the values reported for their two purest specimens. Also, data for a polyCRYOGENICS

" AUGUST

1967

crystalline sample measured by Swenson 6 are c o m p a r e d with some average values calculated from our single crystal data =a,, = ill3 = (2ct j_ + ¢ 11)/3F r o m the figures it is obvious that the behaviour pattern of the principal expansion coefficients is more complex for indium than for most other anisotropic (or isotropic) systems. Parallel to the tetrad axis, ~ reaches a negative value of - 1.3 x 10 -6 at 5 ° K and then rises to a m a x i m u m positive value of approximately 24 x I0 -~ at 35-40 ° K, after which it decreases and is again negative at r o o m temperature. On the other hand ct± is always positive but has an unusual m a x i m u m and m i n i m u m near 7 and 13 ° K, respectively. The volume coefficient fl, calculated from fl = 2¢±+ct11, shows no unusual features in its behaviour and agrees quite well with values measured by Swenson ~ on a polycrystalline sample with little or no preferred orientation. In Table 1 are also included values for the principal Griineisen parameters defined by ~s ~' -

and fl = 2= ± + ~lt, can be represented by such graphs, and they give

A V / V = (0.39 + 0.1)T ~ + (0.12 + 0.01)T 4 x 10 -s, and fl = (0.66 + 0 . 2 ) T + (0.52 + 0.02)T a x 10 -s degK -1, respectively, for T < 3.5 ° K. C o m p a r i s o n with the corresponding expression for heat capacity between 1-5 and 4 ° K leads to ye ( e l e c t r o n i c ) = 2"9 + 0.8 and yz (lattice) = 2.3 +_ 0.1. There is a rather large .r

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Inj

/

fl VKs Cp - (2=j_ + oql)VKs/Cp

~,± = yt = ~,, = [(c. + c~2)a±+ cI3~I]V/Co ~' II = 9'3 = [2c~3~j_ + c33~ Ii] V/Cp where V is the atomic volume, Ks is the adiabatic bulk modulus, Cp is the heat capacity at constant pressure, and c~j are adiabatic values of the elastic stiffnesses. In calculating the tabulated values, we have used V = 15-71 cm 3 at 284 ° K, V = 15-43 at 75 ° K, and V = 15-39 below 30 ° K. Elastic constants are from C h a n d r a s e k h a r and Rayne 16 from which we deduced the following values of Ks: 4-23 x 10 jl dyn/cm 2 at 284 ° K, 4-58 x 1011 (75 ° K), and 4.61 x 1011 ( T < 30 ° K). Specific heat data are taken from O ' N e a l and Phillips: t7 T < 4 ° K ; Clement and Quinnell:

TM

4 < T < 20 ° K ;

Bryant and K e e s o m 9 9 T < 4 ° K; Clusius and Schachinger: ~° T > 20 ° K. At the lowest temperatures (T~Oo) it is c o m m o n l y assumed that the expansion coefficient like the specific heat can be represented by a polynomial of the form a T + bT 3+ cT 5 + . . . in which the first term can be identified with the contribution of the conduction electrons (ct~) and higher order terms with the lattice waves (¢z). Generally the T 5 term is sufficiently small that at temperatures T < 0/30 we can construct linear plots for the expansion coefficient ~ and the expansivity AI/I using the relations

or~T= a + bT z

and

AI/IT 2= ½a + ¼bT 2

However, in anisotropic systems such as indium, zinc, and c a d m i u m (cf. reference 10), higher order terms are too large to allow this simple analytical representation of the linear measurements. But the volume data, viz.

A V / V = (2All + AIH)/I CRYOGENICS • AUGUST 1967

Temperolure -

20

(*K)

Figure 2. Linear coefficients of thermal expansion of indium at low temperatures

uncertainty in the electronic Griineisen parameter )'e because the expansion due to the electrons is much smaller than that due to the lattice at temperatures above about 2 ° K (0/50). Figure 3 shows the differences in length and volume between the s-state and the n-state. At a given temperature the difference is small (AI[I,,., 5 x 10 -8) c o m p a r e d with the thermal expansion or contraction between, say, 2 and 4 ° K (Al/l,,, 150 x 10-8). This is because the expansion due to the electron gas is much less than the expansion due to the lattice. F r o m the slopes o f these curves at T = Te we find the discontinuities in the expansion coefficients to be (Ctn - as)It --- (13"3 + 0"5) x 10 -8 degK -~ (an - ~s)j_ --- - (2"8 + 0"2) x 10 -8 degK -1 fin - fls ~ (7"7 + 0"6) x 10 -8 degK -1 Our data for the length differences disagree markedly with those of Rohrer 8 (see also Andres, Olsen, and Rohrer 2t for a graph of (Is - In)/ln), whereas our values for A V/V do agree quite well with his values for polycrystalline indium. This point is discussed further in the final section.

Thermodynamic Inferences Because the superconducting transition is a reversible phase transition, there exist a n u m b e r o f t h e r m o d y n a m i c relationships between the changes which occur in the macroscopic parameters of state at the phase b o u n d a r y 221

(see Shoenberg, 2: ch. 3). In 1962 Mapother 23 showed that the best available specific heat data were consistent with magnetic measurements of the t e m p e ~ t u r e dependence of the critical field He to a high degree of accuracy. We are now able to extend the comparison to include our new data on volume changes at the transition. In an anisotropic superconductor the discontinuities in length and volume at the transition are related to the stress dependence of the critical-field by 22

We estimate C , - Cs to be 9.37 + 0.4 mJ/mol, degK from the measurements of O'Neal and Phillips, whence

dTe/dP = - 43"0 + 4 mdegK/kbar These two values for dTe/dP agree well with each other and also with the direct pressure measurement made by Jennings and Swenson 26 who obtained a value

bile 4rt /In --is~ bpi -- He k Is ,/l +½HeZI and

bPbtle_ He4rr( ~ - - ' - - sVs) + ½HeZ Here pi, it and Zl are a uniaxial stress, the length of the sample, and the linear compressibility, respectively, in some arbitrary direction in the crystal, and P and Z represent the hydrostatic pressure and the bulk compressibility, respectively. The second term on the righthand side of the above equations arises from magnetostriction in the superconducting state; it is greater for the perpendicular (.L) orientation and reaches about 5 per cent at 1-5° K. We have calculated the dependence of the critical field on uniaxial and on hydrostatic stress using our length and volume measurements in conjunction with the measurements by Finnemore and Mapother 24 of the critical field as a function of temperature. When the calculated values of ~He/~pi are plotted as functions of T 2 and extrapolated to 0 ° K and to Te (Figure 4),t we obtain

(~He/bptl)~,=o = - 7"3 + 0"30e/kbar, (bHe/bpll)re = - 10"4 + 0" 3 0 e / k b a r (bHe/bp~)r--o = 1"9 + 0"10e/kbar, (bHe/bp~)re = 1"9 + 0 " 2 0 e / k b a r (bHe/bP)r=o = - 3"9 + 0 - 1 0 e / k b a r , (~He/bP)re = - 6"6 + 0 " 3 0 e / k b a r The subscripts refer to directions parallel ( l ! ) a n d perpendicular (_L) to the tetragonal axis. There are a number of published values for bHe/bP at 0 ° K and at Te from both static pressure measurements and volume changes. These are shown by arrows in Figure 4, and were taken from Olsen and Rohrer. 2s The extrapolated values at Te of the stress derivatives o f He can be used to calculate the stress dependence of the critical temperature by using the equation

~Tc/bpi = - (bHc/bpl)/( bnc/b T)r=re We estimate bHe/bTto be - 154 + 7 0 e / d e g K at Te from Finnemore and Mapother's data, and this yields

bTe/bpH = - 65"3 + 3, bTe/bp± = 11"9 + 1, dTe/dP = - 42"8 + 3 m d e g K / k b a r A value for dTe/dP can be obtained independently of the magnetic measurements by using the Ehrenfest relation connecting the discontinuities in expansion coefficient and specific heat at the critical temperature 2z

dTe/dP = VTe(J3n- #s)l(C. - c~) "i" The extrapolation allows for a slight non-linearity in the data. This is not shown in Figure 4 where the straight lines may indicate slightly different values at (7 K and at To.

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Figure 3. Differences in length (Is - - /n)/I and v o l u m e (Vs - Va)/Vn f o r indium in the s u p e r c o n d u c t i n g (s - ) and normal ( n - ) state. Data marked - - - are f r o m RohrerS. o

- 4 3 - 0 + 0-1 mdegK/kbar. From the pressure dependences of Te and Ho given above, we may calculate also the corresponding volume derivatives dinTe dlnV

-

5-7

and

dinHo - 6.4 dlnV

(cf. RohrerS). The free-energy difference between the 's' and 'n' states as T approaches zero is 2~ ½rr 2 =

v(H~ - Ho~)/8.

where F is the electronic specific heat coefficient in the normal state. It follows that

bH 2

dH 2

b-if- - d P -

4FZT, + ~Y=-

l)T2

We have used this equation to estimate 7e from our extrapolated values of bHe/bP and the critical field curve as T--~ 0 ° K. The result is Ye = 3 " 2 + 0 " 4 in satisfactory agreement with the direct calculation from the volume expansion curve. These results are, of course, not entirely independent because bHeb/P was calculated from our measurements of (Vn - Vs). A simple parabolic curve is often used for the critical field and it leads to the result (see, for example, reference 8) CRYOGENICS

• AUGUST

1967

7e = I + 2 K r \

(din Te dP

dlnHo~ dP ] = 2"3+0"7

The parabolic law is not a good approximation for indium and we therefore give little weight to this value for )'e.

Discussion The volume expansion coefficient of indium depends on temperature in much the same way as does the heat capacity, and the Griineisen parameter y remains between 2.2 and 2.4 from room temperature, where 7"/0,,. 3, down to liquid helium temperatures where T/O,,, 0.03. This is consistent with the behaviour of the lattice contribution observed in most other metals, whether cubic or anisotropic. ~s Although the electronic contribution to the expansion is relatively small, even at 3 or 4 ° K, we have determined an approximate value for 7e of about 3.0. This is considerably greater than the free-electron value of 2/3, although comparable with the values of around 2 observed for lead, aluminium, and some transition elements) 5 The behaviour of the principal coeffÉcients of linear expansion of indium differs noticeably from that of other anisotropic metals such as cadmium, tin, and zinc (Figure 5). For the latter we may may conclude with Griineisen and Goens 2s that the anisotropy of expansion is an obvious corollary of the elastic anisotropy. At the lowest temperatures lattice vibrations are first excited thermally in directions of greater elastic softness, i.e. in the loosely packed c-direction in zinc and cadmium. Anharmonicity of the vibrations causes an initial expansion in such a direction together with a cross-contraction or negative expansion in the plane normal to this. At higher temperatures vibrations are also excited in this normal direction and expansion takes place which may eventually lead to a decrease in the expansion coefficient in the original direction. In the equations t5 ~_

=

~X:1 =

[(s~, + s,9~'1 + s~,~,lCv/V, [S33?;11 -t- 2 S l 3 ) '

x]Cv/V,

we can attribute the negative values for ~± at low temperature and the decrease in c% at higher temperature, which are observed in cadmium and zinc, to negative values of the cross-compliance s~3. Indium is distorted from a close-packed structure by an 8 per cent elongation in the direction of the c or tetrad axis. This direction may be variously considered as the softer, because S33/Slt~2, o r the stiffer, because X±/XII" 3. This ambiguity is reflected by the linear expansion coefficients. At liquid helium temperatures ~± increases rapidly and is accompanied by a negative expansion (0%) parallel to the tetrad axis. Then at 5 or 6 ° K ~ 11 begins to increase in the positive direction and =, in turn begins to decrease. This continues until the temperature is nearly 20 ° K when ¢± once more begins to increase while = It goes through a maximum at 30 ° K and then decreases. Obviously these meanderings are related via the elastic compliances. Because Sl3 happens to be nearly equal in magnitude to sn + s~, and to ½s33, while having the opposite sign, there is a delicate balance in the above equations, and the sign and magnitude of the CRYOGENICS

• AUGUST

1967

resultant expansions are quite sensitive to small changes with temperature in the Griineisen parameters and in the compliances. This interrelation of the elastic and expansion properties of indium still leaves the microscopic origin a mystery. Ultrasonic velocity measurements by Winder

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Figure 4. Values for the pressure dependence of the critical field calculated from length differences ( I n x and In11)and volume differences in Figure 3. The arrows on the ordinate indicate values obtained by previous authors from effect of hydrostatic pressure (compilation of Olsen and Rohrer) =~

and Smith 29 indicate that there are low-lying frequency branches in the lattice spectrum of indium which may well be responsible for the peculiar expansion behaviour. But until there are more details of the frequency spectrum and its dependence on strain the exact expansion mechanism is uncertain. The length differences in the two principal directions between normal and superconducting states differ considerably from those observed by Rohrer. s The present values for the expansion coefficients at 85 and --

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T/eo~ Figure 5. Variation of linear expansion coefficients with reduced temperature (T18o) for c a d m i u m (0o = £10 ° K). tin (80 = 201 ° K) and indium (0o = 110° K) 121

284 ° K lie close enough to the best X-ray values for us to believe that our cylindrical samples were representative of correctly aligned single crystals, although there was undoubtedly local damage and misoeientation within a millimetre or so of each end. Cody 3° has measured ~He/~pI for two highly oriented, polycrystalline samples of indium and has deduced values at Te of bHe/bp it ~ - 9 0 e / k b a r ,

bHe/bp± ~ 20e/kbar,

both of which are in good agreement with our own results. REFERENCES

1. PEARSON, W. B. Handbook of Lattice Spacings of Metals (Pergamon, London, 1958) 2. VER_~ON,E. V., and WEnwrRooa, S. Proc. phys. Soc. Lond. 1366,

887 (1953) 3. GRAHAM,J., MOORE,A., and RAYNOR,G. V. J. Inst. Metals 84, 86 (1955) 4. SMITH,J. F., and SCHNEIDER,V. L. J. less-common Metals 7, 17 (1964) 5. MADAIAH,N., and GRAHAM,G. M., Canad. J. Phys. 42, 221

6. 7. 8. 9. 10.

(1964) SWENSON,C. A. Phys. Rev. 100, 1607 (1955) OLSEN,J. L., and ROARER,H. Helo. phys. acta 30, 49 (1957) R o ~ , H. Phil. Mag. 4, 1207 (1959) ROHRER,H. Heir. phys. acta 33, 675 (1960) MCCAMMON,R. D., and WHITE,G. K. Phil. Mag. 11, 1125 (1965)

11. WrirrE, G. K. Physics Lett. 8, 294 (1964) 12. WHITE,G. K. Cryogenics 1, 151 (1961) 13. CARR,R. H., MCCAMMON,R. D., and WHITE, G. K. Proc. roy. Soc. A2,80, 72 (1964) 14. ROGERS,J. S., SWENSON,C. A., and TAINSH,R. J. Unpublished

work 15. COLLINS,J. G., and WHITE, G. K. Progress in Low Temperature Physics 4, 450 (North-Holland, Amsterdam, 1964) 16. CHANORASEKHAR,B. S., and RAYNE,J. A. Phys. Rev. 124, 1011 (1961) 17. O'NEAL, H. R., and PHILLIPS, N. E. Phys. Rev. 137, A748 (1965) 18. CLEMENT,J. R., and QUINNELL,E. H. Phys. Rev. 92, 258 (1953) 19. BRYANT,C. A., and KEESOM,P. H. Phys. Rev. 123, 491 (1961) 20. CLusIus, K., and SCHACHINGER,L. Z. angew. Phys. 4, 442 (1952) 21. ANDRES, K., OLSEN, J. L., and ROHRER, H. I . B . M . J . Res. Devel. 6, 84 (1962) 22. SHOENBERG, D. Saperconductivity (Cambridge University Press, 1952) 23. ]~APOTHER, D. E. Phys. Rev. 126, 2021 (1962) 24. FINNEMORE,D. K., and MAPOTHER,D. E. Phys. Rev. 140, A507 (1965) 25. OLSEN,J. L., and ROHRER,H. Helvet.phys. acta 33, 872 (1960) 26. JENNINGS,D., and SWENSON,C. A. Phys. Rev. i12, 31 (1958) 27. SWENSON,C. A., I.B.M. J Res. Devel. 6, 82 (1962) 28. GRONEISEN,E., and GOENS, E. Z. Phys. 29, 141 (1924) 29. WINDER,D. R., and SMI'rH, C. S. J. Phys. Chem. Solids 4, 128

(1958) 30. COPY,G. D. Phys. Rev. 111, 1078 (1958); Dr Cody (private communication) has pointed out an error in the values quoted for ~He/~p,, and ~He/~p± in the text and in Table I of his paper. The correct values are those given above.

SECOND INTERNATIONAL CRYOGENIC ENGINEERING CONFERENCE: EUROPE (I.C.E.C.2) Plans for the continuation of the series of cryogenic engineering conferences started in Japan in April 1967 have now reached an advanced stage. The next conference will be held in Brighton, England, on 7-10 May 1968 under the title given above. The primary aim of the Conference is to provide exchange of information on problems of cryogenic engineering, including the scientific aspects associated with the field. Emphasis will be laid in the main branches of the subject, such as: refrigeration technology, various applications of superconductivity, storage, insulation, properties of materials, and thermometry. However, papers of sufficient merit dealing with any other aspect of cryogenics will be accepted. Contributed papers should be of about 1 200 words, a time of 15 minutes per paper (including discussion) being allocated. Abstracts should not exceed 150 words and should provide a clear indication of the content of the paper. Final date for the receipt of Abstracts is 15 November 1967. Accepted papers must be received by the Conference Secretary by 1 March 1968. Application forms for registration can be obtained from the Conference Secretary. The Conference fee, including preprints and Conference Proceedings, is £14. Details of the social programme and a ladies' programme will be announced at a later date. The scientific programme of the Conference has been arranged by an International Committee consisting of: 224

L. Weil, University of Grenoble, France (Chairman); K. Mendelssohn, University of Oxford, U.K. (ViceChairman); A. A. Smailes, CRYOGENICS(Conference Secretary); L. Bewilogua, Technical University of Dresden, Germany; J. B. Gardner, British Oxygen Company Ltd, London, U.K.; E. F. Hammei, Los Alamos Scientific Laboratory, New Mexico, U.S.A. ; W. Klose, D.P.G., Erlangen, Germany; J. W. L. K6hler, Philips Research Laboratories, Eindhoven, Netherlands; L. Lefevre, Soci6t6 l'Air Liquide, Sassenage, France; J. L. Olsen, Eidg. Technische Hochschule, ZiJrich, Switzerland; K. Oshima, University of Tokyo, Japan; R. S. Safrata, Nuclear Research Institute, R.e~, Czechoslovakia. An invitation to provide a member of this Committee has also been sent to the Council of Ministers of the U.S.S.R. Support for the aims of the Conference has already been expressed by a number of national and international bodies concerned with the field of low temperature engineering and research. Organization of the Conference has been undertaken by the publishers of CRYOGENICS. All correspondence, including Abstracts and applications for registration, should be addressed to: The Conference Secretary, I.C.E.C.2, 32 High Street, Guildford, Surrey, England. CRYOGENICS

• AUGUST

1967