Heat flow measurements of superfluid 4He in wide tube

Research and technical notes

Heat flow measurements s u p e r f l u i d 4He in w i d e tube

Pressure gauge N ~=~ UCN valve d r i v e ~

of

3He p u m p ~ g H e

Y. Fujii, K. Sakai* and M. Ogura* Faculty of Science, Okayama University of Science, 1-1 Ridai-cho, Okayama, 700 Japan " Bubble Chamber Physics Laboratory, Tohoku University, Aramaki, Aoba-ku, Sendal, 980 Japan

UCN tube

pump

..2K pot iHHH IK pot.~ ~ / IIIII [II sup°r,eak ttE I

Irql UCN g u i d e ~

Received 8 May 1992: revised 7 January 1993 Figure I

The effective thermal conductivity of superfluid 4He was measured in the temperature range between 0.48 and 0.67 K using a tube 8 cm in diameter, which is sufficiently wide to neglect the influence of phonon scattering on the wall. The data are compared with the calculations of Landau et al. and Marls and the experimental results of Greywall. The obtained results are useful for analysis of thermal resistance in the ultracold neutron production equipment at KEK.

Keywords: superfluid helium; thermal conductivity; ultra-low temperatures

A 3He refrigerator has been constructed to refrigerate superfluid 4He (He II) for ultra-cold neutron (UCN) production experiments at the Booster Synchrotron Facilities of the National Laboratory for High Energy Physics (KEK). The He II container (UCN tube) is a long horizontal tube, 8 cm in diameter. To increase the U C N density by the super-thermal method 1, He II has to be cooled below 0.8 K to suppress up-scattering by phonons and rotons. This means that the energy of the phonons emitted by the interaction of cold neutrons with He II has to be efficiently carried away via the He II. Although there are some experimental data z-4 relating to heat flow in He II below 1 K, no data are available for calculation of the effective thermal conductivity x in a U C N tube whose diameter is more than one order of magnitude larger than the phonon mean free path 2 at 0.5 K. Thus we have carried out steady state heat flow measurements in a U C N tube under saturated vapour pressure in the temperature range 0.48-0.67 K.

Experimental details A schematic view of the 3He refrigerator for U C N production is shown in Figure 1. The U C N tube was made of stainless steel (SUS 316L), with an 8 cm outer diameter, 0.04 cm thick and 300 cm long. The thermal conductivity measurement was carried out using this tube filled with He II which was transferred from a 1 K pot through a superleak packed with fine A120 3 powder to eliminate 3He. The ratio of 3He to 4He in the U C N tube was measured to be less than 1 x 10 -9 (reference 5). The tube was thermally connected to a 3He pot via an 0011-2275/93/101010-03

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Cryogenics 1993 Vol 33, No 10

Schematic view of 3He refrigerator for UCN production

oxygen-free high conductivity ( O F H C ) copper wall (see Figure 2). The surface area of copper on the He II side was 0.65 × 103 cm 2 and that on the liquid 3He side was increased to 4.2 x 10 3 cm 2 by copper fins. The He II temperature was measured using two calibrated Ge thermometers (from Scientific Instruments, Inc.), Ge 1 and Ge 2. The arrangement of the thermometers and manganin wire heaters in the U C N tube is shown in Figure 2. The effective thermal conductivity was measured using the conventional, technique. A known quantity of heat per unit time Q was forced to flow through the He II by dissipating power in heater 1, and the temperature difference AT between Ge 1 and Ge 2 was measured at the steady state. The effective thermal conductivity x is expressed as

x = 0 x Ax/(A x AT)

(1)

where A is the cross-sectional area of He II and Ax is the distance between the two thermometers. (Hereafter the effective thermal conductivity is abbreviated to thermal conductivity). The cross-section A was 49.3 cm 2 and Ax was 251.5 cm*. The electric resistance of the Ge thermometer was measured with an a.c. resistance bridge (RV-Elektroniikka, AVS-45), applying a constant excitation voltage of 1 x 10 -4 V. The dissipation power at the thermometer was of the order of 10-11 W at the lowest temperature. No self-heating effect was observed. The lowest temperature of the He II without heat input was 0.45 K. In the present case, the coefficient of heat transfer from the He II-Cu boundary to the liquid 3He was found to be small compared to x for He II. That is, by application of heat at heater 1 the resulting temperature difference between the He II and the liquid 3He became much larger than the temperature difference in the He II. Owing to this situation, the t¢ measurement was carried out with a low level of heat flux, which resulted in AT always being smaller than 1 mK. To measure the small temperature difference, two Ge thermometers were calibrated reciprocally at several temperatures by using heater 2 instead of heater 1. The experimental results are shown in Figure 3 as a function of temperature on a log-log scale. The data * As shown in Figure 2, Ge 2 was positioned at the centre of the 3He pot, which was the heat reservoir. Thus, when considering the heat flow, the effective value of Ax was taken to be 243.5 cm in the calculation of x

Research and technical notes UCN tube (SUS, o . d . = , t = 0 . 0 4 ) Ge thermometer 2

Ge thermometer 1

3He pot ] (OFHC c o p p e r ) ~ ,

S u p p o r t pipe

Heater/1 / / ( S U S ,

t =0.05)

// [

o.d. = 0.3,

~,(o/

Heater 2 \ ~ .

i', ~-34 , "9 ~"-0,6

HeII

.

._=

~..=. I 8 ~--"-~13 ; 13.4-- - ~ ~ ,

--238.1 300

Unit: cm

Figure 2

Arrangement of Ge thermometers and manganin wire heaters for steady state heat f l o w measurement in He II

I

104

I

I

I

thermal resistance in He II is regarded as negligibly small compared with the thermal resistance between He II and liquid 3He. We now discuss the x values on the basis of the theoretical and experimental results. At very low temperatures where 2 is larger than the tube diameter d, the heat flow is restricted by the scattering of long wavelength phonons from the wall of the tube. With increasing temperature, 2 is limited by phonon-phonon collisions, and the phonons can be treated as a gas with a definite viscosity. Therefore, the thermal conductivity of He II depends on the temperature and the diameter of the tube. In the regime 2 ~ d, K is described as followsa

I

- 32p,(e,)2

1 + 8s

(2)

/, //

103

/.J/

10 2

II

i

/ /

i

/

Greywall d = 0 . 8 cm

T 101

,/'

,,/"

,/ I0 0

~,°

.I

°f. j°"

/

o

j.° , /"" 10 -1

10 -2 0.3

Greywall d ~ 0 . 1 4 cm

,1

I

I 0.5

I

I

I

I

T[K)

Figure 3 Temperature dependence of thermal conductivity in He II. + , Present experimental result for tube of d = 8 cm at saturated vapour pressure; . . . . , Greywall's experimental results 4 for tubes of d = 0.14 cm and 0.8 cm at P = 2 bar. , Calculated curves for tube of d = 8 cm from p h o n o n mean free paths given by Landau et al., Maris (and Benin) and Greywall, respectively

fluctuate considerably because of the difficulty in measuring the small temperature difference. The accuracy of K was about + 20~. The experimental results of Greywall using a tube o f d = 0.14 cm and 0.8 em at P = 2 bar are also shown in Figure 3 for comparison. As will be discussed below, the thermal conductivity of He II in a tube of d = 8 cm was calculated using the mean free path evaluated from his results.

Discussion When there is a heat input at the opposite end of the UCN tube from the 3He pot (see Figure 1), we can calculate the resultant temperature increase of He II at that end using the present data. For a heat input of 3 W, which is the estimated heat input from cold neutrons in the UCN production experiment at KEK, the calculated temperature increase is 0.027 K at 0.6 K. Therefore the

where S is the entropy per unit volume, Pn is the normal fluid density, (cg) is the average group velocity of the phonons and s is the slip coefficient at the boundary. Landau et al. 6 calculated the phonon mean free path assuming the phonon dispersion relation was normal and that the viscosity mean free path was dominated by a four-phonon process. Meanwhile, Maris 7 calculated 2 assuming an abnormal phonon dispersion relation and the dominance of a three-phonon process, in order to explain Whitworth's result 3. Benin 8 proceeded along the same lines as Maris by using a variational calculation of the eigen-values of the completely continuous part of the collision integral operator in the Boltzmann phonon equation. Their results showed that 2 increased more slowly with decreasing temperature than the T - 9 temperature dependence proposed by Landau et al. Recently, Greywalla measured the thermal conductivity of He II between 0.1 and 0.7 K under pressures up to 25 bar using five tubes with diameters of 0.14 1.4 cm. Some of these values are shown in Figure 3. He evaluated the mean free path from his results using Equation (2). These three sets of 2 values (from Landau et al,, Maris and Benin, and Greywall) indicate that the present tube diameter of 8 cm is more than one order of magnitude larger than the phonon mean free path above 0.48 K. We then calculated the thermal conductivities for the present tube diameter using Equation (2) with the three sets of 2 values. We employed the He II entropy calculated from the specific heat data of Greywall9 and the value of s = 0.592 given by Benin and Maris 1°. These results are also shown in Figure 3. The temperature dependence of the mean values of the present experimental data is consistent with the curve calculated using the Maris (and Benin) theory, although our absolute values were 20~o smaller than the theory. Nowadays, it is known that the phonon dispersion of He II is anomalous to the experimental results for specific heat 9 and neutron scattering 11.12, which means that a three-phonon scattering process can take place 13, Thus the present data suggest that the dominant scattering mechanism in viscous flow is the three-phonon process. We also have to refer to a particular feature of the present experimental set-up (Figure 2), that is, that heat flowed horizontally in the wide tube and that heater 1 was located lower than the centre line of the He II column. This arrangement could potentially cause convection flow. However, in the present case, the Rayleigh number is several orders of magnitude smaller than the

Cryogenics 1993 Vol 33, No 10

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Research and technical notes critical value for the onset of convection flow owing to the small expansion coefficient, the large thermal conductivity and the small temperature gradient. Thus the influence of convection flow on the result will be ignored.

Summary The thermal conductivity x of He II in a very wide tube was measured between 0.48 and 0.67 K. The data are available for estimation of the temperature increase of He II caused by heat input in the U C N production experiment. The x result was compared with curves calculated using 2 values proposed by Landau et al., Maris (and Benin) and Greywall. The temperature dependence of the mean values of the present x data is consistent with the temperature dependence deduced from the theory by Marls (and Benin), but the absolute value was ~ 20% smaller than the theoretical value. To investigate phonon interaction in He II, more accurate measurement is desirable in a wide tube with various levels of heat flux.

Acknowledgements The authors would like to thank Professor H. Yoshiki of

K E K for giving them the chance to conduct these experiments using the 3He refrigerator constructed for the U C N experiment and for his helpful advice. They are also grateful to Professor H. J. Marls of Brown University for his kind comments and to Professor T. Shigi of Okayama University of Science, Professor K. Yamada of Nagoya University and Professor M. Tsubota of Tohoku University for useful discussions.

References 1 Golub,R. and Pendlebury,J.M. Phys Lett (1977) A62 337 2 Fairbank, H.A. and Willm, J. Proc Roy Soc London (1955) Ser A231 545 3 Whitworth,R.W. Proc R Soc London (1958) Ser A246 390 4 Greywall,D.S. Phys Rev B (1981) 23 2152 5 Yosldki,H., Morimoto,K., Kndo,N., Kiryu,Y. et aL Bull 35th Mto Jpn Phys Soc (1980) 27a 1 6 Landau, L.D. and Khalatnikov,LM. Zh Eksp Teor Fiz (1949) 19 637 and 709 7 Maris, H.J. Phys Rev A (1973) 8 1980 8 Benin,D. Phys Rev B (1975) 11 145 9 Greywall,D.S. Phys Rev B (1978) 18 2127 10 Benin,D. and Marls, H.J. Phys Rev B (1978) 18 3112 11 Woods,A.D.B. and Cowley,R.A. Phys Rev Lett (1970) 24 646 12 Cowley,R.A. and Woods,A.D.B. Can J Phys (1971) 49 177 13 Maris,H.J. Rev Mod Phys (1977)49 341

Novel method of preparation of nuclear orientation thermometers by melting

Nuclear orientation thermometers utilize the temperature dependence of the anisotropic angular distribution of gamma rays emitted by an assembly of nuclei orientated in a ferromagnetic matrix by means of hyperfine interaction. In the case of 54Mn(Ni) and 57Co(Fe) thermometers, which are often used for the measurement of very low temperatures in nuclear orientation experiments, the angular distribution can be written 1' 2 as

J. Dupbk, J. KoniEek* and A. Machovht

I4(0, T) = (1 -- 3e)B2(T)A2U2Q2P2(cos O)

Institute of Scientific Instruments, Academy of Sciences of the Czech Republic, Krblovopolskb 147, 612 64 Brno, Czech Republic * Department of Nuclear Sciences and Physical Engineering, Czech Technical University, B~ehov~5 7, 115 19 Prague 1, Czech Republic t" Institute of Thermomechanics, Academy of Sciences of the Czech Republic, Dolej~,kova 5, 182 00 Prague 8, Czech Republic Received 12 November 1992: revised 15 April 1993

A simple and fast melting procedure for the preparation of nuclear orientation thermometers is proposed. S4Mn(Ni) and s7Co(Fe) thermometers prepared by the new melting technique and the usual diffusion technique are compared in the very low temperature region.

Keywords: thermometers; nuclear orientation thermometers; melting technology 0011-2275/93/101012-03 © 1993 Butterworth-HeinemannLtd 1012

Cryogenics 1993 Vol 33, No 10

+ (1 - IOe)Bg(T)A4U4Q4P4(cos O) where: ~ = ( M s - M ) / M s denotes the degree of magnetic saturation; M and M s represent the actual and the saturated magnetization, respectively, of the matrix orientated in the direction of the external magnetic field; T is the temperature; Bk(T) are the orientation parameters which also depend on the hyperfine splitting; A s are angular distribution parameters; Uk are disorientation coefficients; Qk are the solid angle correction factors which account for the finite detector angular resolution; Pk are the normalized Legendre polynomials; and ® is the angle between the external magnetic field and the measured gamma ray emission. Since the parameters of hyperfine interaction and the coefficients A k and Uk are well known 1 for the 54Mn and 57C0 nuclei, the temperature can be evaluated by comparison of the experimental normalized intensities W=xp(o, T) and the theoretical values W(®, T). The parameters e and Bk(T ) are sensitive to sample preparation technology, which can influence the evaluation of the temperature. In this paper, thermometers prepared by the usual diffusion technology ~'3'4 and those prepared by the melting technique are compared.