Thermal conductivity of some steels at low temperatures

Chaff, M. S . R . De Nobel, J. 1959

Physica 25 73-83

THERMAL CONDUCTIVITY OF SOME STEELS AT LOW TEMPERATURES by M. S. R. CHARI *) and J. DE NOBEL Communication No. 313c from the Kamerlingh Onaes Laboratorium, Leiden, Nederland

Synopsis The measurements made in the Kamerlingh Onnes Laboratory on the t h e r m a l conductivity of steels at low temperatures were extended to liquid helium temperatures and the technique was improved. Five specimen of different composition and heat t r e a t m e n t have been investigated. The lattice conductivity is approximately proportional to T x.9". With increasing "nickel content" from 2% to 27~/o (Ni + Cr) it decreases b y about 60, 58 and 3 7 ~ at the temperatures 70, 20 and 4°K respectively. The electronic thermal conductivity decreases b y about 80% for the mentioned temperatures.

Studies on the mechanical properties of steels at low temperatures were pioneered by H a d fi e I d 1) in collaboration with D e w a r, K a m e r l i n g h O n n e s and De H a a s . A large number of steels was also placed by H a d f i e l d at the disposal of the Kamerlingh Onnes Laboratorium, Leiden. Low temperature thermal conductivity measurements down to liquid hydrogen temperatures, on a good number of these, have already been made by one of us (D e N o b e 19.)) using gas thermometry and lead resistance thermometers. The present investigation is an extension of these to liquid helium temperatures and an incidental repetition of the earlier measurements, using a rather improved technique. Such studies on steels and other low-conductivity alloys supply information which could be of use in t h e design of cryogenic equipment (vide W i l k i n s o n and WilksS), W e x l e r a) B e r m a n 5)). They also furnish data for testing theoretical predictions on the temperature variation of the electronic and lattice components of thermal conductivity in various types of solids. Earlier studies in this direction were made by J a e g e r and D i e s s e 1h o r s t e) on carbon and silicon steels, and by G r i i n e i s e n 7) and L e e s s) on carbon steels. There has been more recent work on carbon steels by D e N o b e 1 9.) and by P o w e r s , Z i e g l e r and J o h n s t o n 0). Corrosion-resisting steels were investigated by K a r w e i l and Sch/iferl0), Z l u n i t z i n and S a v e l j e v l l ) , 1. Introduction.

*) Now in the Low Temperature Division, National Physical Laboratory of India, New Delhi, India. --

Physica 25

73

--

74

M. S. R. C H A R I A N D J. D E N O B E L

S a v e l j evil), W i l k i n s o n and W i l k s s), S c h m e i s s n e r and M e i s s n e r l 3 ) , B e r m a n s ) , De N o b e l ' ) , P o w e r s , Z i e g l e r a n d J o h n s t o n g ) , E s t e r m a n n and Z i m m e r m a n n 14) and T y l e r and W i l s o n (quoted in N.B.S. circular, No. 556). Preliminary results on a stainless steel (No. 3754) were reported by De N o b e l and CharilS). Table I gives the compositional and other details of the various steels discussed in this paper. TABLE I Com ~ositional and other details for the steels measured

Steeltype

Diameter of rod

1287D

5½ mm

3703 1287 1 1798H 3754

7~ him 5½ mm 7½ mm 7~t mm

Heat-

treatment Heated to 800°C and cooled in furnace Ibidem Ibidem Ibiden H e a t e d to 1150°C and quenched in water

Brinell hardness

Percentual composition C

153

[ Si I Mn0.14 0.21 0.72

157 277 179 172

0.11 0.16 0.34 0.18 0.22 0.93 0.43 1.09 0.12 0.43 0.24

Cr,

I Ni I P

I s

1.92

18.80

5,10 0.041 II.39 19.64 8.10

0.04

2. Description o/the apparatus. The apparatus was similar to that used by De H a a s and B i e r m a s z 18) and by D e Nobel2). Two carbon-film resistors as described in an earlier paper 21) took the place of the lead-wire thermometers of the former investigations. 3. Results. (See CharilT)). For temperatures below about O/10, M a k i n s o n ' s 18) theoretical expression for electronic thermal resistivity can be written, to a good approximation, in the form We = Wl + wo =

o~T2 + f l / T

where the first term on the right hand side is the "ideal" thermal resistivity (caused by the scattering of the conduction electrons by the thermal vibrations of the ionic lattice) and the second is tile "impurity" or "residual" thermal resistivity (caused by the scattering of the electrons by the impurity atoms and small-scale lattice defects), c¢is known to be reasonably constant for a given metal and fl = po/Lo, in the usual notation. Rather than obtain a theoretical value for ~, we used the experimental value of 18 × 10-5 cm/watt-deg for pure iron, obtained by M e n d e l s s o h n and R o s e n b e r g l 9 ) . From our experimental values of the residual electrical resistances of the steels, it was found that the &T2(-- wi) term' amounted to less t h a n 1% of fl/T (-----po/LoT-= w0) near about 25°K, but rose to about 16% of w0 at about 70°K, for the 1287D steel. At lower temperatures, it rapidly became negligible compared to w0. For tile No. 3754 stainl_ss steel, Wl was ab3ut

THERMAL

CONDUCTIVITY

75

OF SOME STEELS

18% "of w0 at 70°K, falling to less than 1% at about 40°K and becoming negligibly small at lower temperatures. TABLE II T h e r m a l c o n d u c t i v i t y 2 (in m W / c m - d e g ) a g a i n s t T (in °K) for the steels No. 1287D

No. 3703

z 4.116 3.595 3.055 2.568 2.116 1.860 1.614 19.98 z 19.393 18.573 17.528 16.175 15.168 87.70 80.36 75.34 71.49

12.76 10.27 8.90 7.43 6.35 5.41 4.53 77.9 74.5 71.5 63.8 60.3 56.1 286 289 263 232

4.044 3.626 3.070 2.616 2.062 1.663 20.600 19.148 18.15 0 18.13 9 16.165 15.04 z

No. 1287I

I

No. 1798H

r 10.26 8.43 6.89 5.80 4.81 3.81 63.8 58.6 52.5 48.1 45.9 42.2

4.080 3.648 3.503 3.046 2.590 2.209 2.045 1.690 19.830 18.346 16.12 o 15.084

4.72 4.06 3.85 2.36 2.79 2.36 2.22 1.85 26.7 25.1 21.8 20.9

3.888 3.446 2.952 2.519 2.126 1.750

3.17 2.68 2.29 1.93 1.60 1.32

3.937 2.516 1.989 1.718

3.34 1.91 1.58 1.30

19.688

80.05

79.80 73.76

227 211 195 199 175

76.13 73.62 70.39 68.04 65.93

102.3 96.9 93.9 91.8 86.1

4.004 3.702 3.350 2.930 2.535

3.11 2.64 2.41 2.17 1.82

3.884 3.434 3.026 2.500 1.843

2.80 2.35 1.99 1.75 1.25

16.84 x 15.190

20.6 19.1 17.2 15.9

19.835 17.100 16.294 15.169

20. I 16.8 16.0 14.6

76.34 73.52 71.12

77.0 74.9 73.5

71.18 68.32

73.7 74.6

19.820 18.685 17.080 16.410 15.314

20.8 19.2 16.4 16.3 15.9

92.8 *) 76.3 *)

81.4 * 71.4 *)

18.31 ~

87.64 82.92

No. 3754

I

D a t a a t l i q u i d o x y g e n t e m p e r a t u r e s for the s t a i n l e s s steel No. 3754 ( m a r k e d *), are from D e N o b e l 2).

Table II gives the thermal conductivity 2 of the steels against temperature. Fig. 1, depicts the same graphically, the curves marked ;re being the function ]re = I/we -= [[3/T -~- aT2] -1 --~ [po/LoT + 18 × 10-5 T~] -1.

The values used for p0 for the steel rods No. 1287D, 3703, 1287 I, 1798 H and 3754 were respectively 9.3, 12.9, 27.7, 41.5 and 47.8 pQ-cm. 2 of the steel reported by K a r w e i l and Sch/iferl0), containing 0.5-0.7% Mn, 0.4% C, 0.3% Si, 0.3% P and 0.03% S, is close to that for our 1287D steel at liquid helium temperatures. At 3°K they found 2 = 7.5 mW/cm-deg whereas we obtained 8.7 mW/cm-deg. The values of ;t given by W i l k i n s o n and W i l k s 8) for a stainless steel specimen between 10 and 20°K are about 3/4 of the values obtained by us for the 3754 stainless steel at the same temperatures. For the AeJu 2 steel (containing 16.05% Cr, 9.89% Ni, 0.66% Mn,

76

M. S. R. CHARI AND J. DE NOBEL

0.88% Si and 0.26%C), Z l u n i t z i n and S a v e l j e v 11) reported A - 18.9 mW/cm-deg at 18°K which corresponds closely with the value 18.8 mW/cmdeg at 18°K, obtained b y us for the 1798 H nickel-steel, whereas our stainless steel specimen No. 3754 gave 17.8 mW/cm-deg at 18°K. 05

(32

"~

,1

"

I'

c/

• ~f" /" j ' ~ / , " ¢ - - I ,"

t~; •

./~,

oo,

ooo5

I" // / °ooz

""./J.~-/~

"'

s'/ 1

T -- 2

5

10

20

50

IO0"K

Fig. 1. The thermal conductivity Aand its electronic part he of steels, both in watt/cm--deg, versus temperature in °K. (D 1287D (curve A), A 3703 (curve B), 12871 (curve C), ~7 1798 H (curve D), [] 3754 (curve E). A, B, C, D and/~ are the ~ curves, and A', B', C', D" and E' are the Aecurves (the data for the No. 3754 stainless steel in the liquid oxygen region are taken from De NobelS). • Our results for the 1798 H and 3754 steels correspond to those of S c h m e i s s n e r and M e i s s n e r l 3 ) on Chroman B2Mo (an alloy containing b y weight, 61.4% Ni, 18.5% Cr, 14.5% Fe, 3% Mn, 2% Mo and 0.6% Si). These authors give 2 at 3.9°K as 2.6 mW/cm-deg, and their electrical resistivity at liquid helium temperatures is more than twice that of our steels, No. 1798 H and 3754. The type 303 stainless steel (18% Cr, 9% Ni, 0.15% C) studied b y E s t e r m a n n and Z i m m e r m a n n 14) has at liquid helium temperatures, a thermal resistivity 1.5 times as large as these, b u t at liquid hydrogen temperatures, its thermal resistivity is similar to our No. 3754 steel. The values of the over-all thermal conductivity, as also of the lattice

THERMAL CONDUCTIVITY OF SOME STEELS

77

component reported by B e r m a n 5) for a stainless steel specimen ( 18.t~~ Cr, 7.9°/~ Ni, I ~ Ti, 0.7% Si, 0.1~/o C) are close to the corresponding values obtained by us for the Nos. 1798 H and 3754 steels. The lattice thermal conductivity. Various scattering mechanisms contribute to the lattice thermal resistivity and function as thermal resistances in series. Amongst these, the dominating one would be that with the highest contribution to the total resistivity or; in other words, the one that tends to limit the conductivity to the lowest value. The wg versus T curves for the steels measured b y us show that no single simple power law.is obeyed throughout the temperature regions. There are however narrow temperature regions between about 4 and 6°K, where the lattice thermal resistivity seems closely proportional to T-2. From these, one could make a rough estimate of the corresponding scattering coefficient. We can express wg as w~ = E / T 2

when the dominant scattering mechanism is afforded by the conduction electrons and by the dislocations. It appears therefore that the scattering coefficient derived in this manner is E. The estimated values of E are 9850, 7000, 15000, 16200 and 16200 cm-deg3/watt respectively for the five steel specimens, in the order of increasing foreign metal content. We wish to make it clear here that our measurements do not cover the intervening region between liquid helium and liquid hydrogen temperatures. By interpolation between the measured points, the values of 2 at intermediate temperatures are obtained (see fig. 1), from which wg is derived in the usual manner. The values thus obtained for E give us only a rough idea of the magnitude. One could express the lattice thermal conductivity of the steels between 6 and 25°K, by a relation of the form 2g = constant × T n. In order of increasing foreign metal content, these relations were, 2g=5.5 2g = 5.6 ~tg = 2.6 2s = 2.5 2g = 2.2

× × × × x

10- 4 T 1.z6 10-4 T 1-24 10-4 T 1.21 10- 4 T 1.21 10-4 T 1"26

where 2g is in watt/cm-deg. The power of T being less than 2 in these equations, suggests that in addition to the scattering by electrons, the phonons are scattered also by impurities, rather than by the mutual scattering b y phonons, since the last-mentioned varies too rapidly with T to give a resultant 2g of the type actually observed. One of us (De Nobel2)) concluded from his results at liquid hydrogen

78

M. S. R. CHARI AND J. DE NOBEL

temperatures, that the over-all thermal conductivity for the steels is proportional to T n where n lies between 1.07 and 1.47. Our reslflts for ;t give for n, the values 1.06, 1.04, 1.05, 1.09 and 1.11 respectively for the steels 1287 D, 3703, 1287 I, 1798 H and 3754. The Wiedemann-Franz-Lorenz parameter. Using the values of I read off the t versus T graphs, and the electrical resistivity p from the p versus T graphs, the Wiedemann-Franz-Lorenz parameter L ( = p / w T = 2p/T) is calculated at different temperatures and plotted in fig. 2. S

t Z-I~20

40

60

80 °K

Fig. 2. T h e W i e d e m a n n - F r a n z - L o r e n z p a r a m e t e r L for t h e steels in (volt/deg) 2 v e r s u s t e m p e r a t u r e . A 1287 D,

(Z) 3703,

~7 1287 I,

<~ 1798 H,

[] 3754.

The value of L for the 1287 D steel at the lowest temperatures closely correspond to those reported by K a r w e i l and Sch/iferl0) for his steel specimen containing less than 1% impurity. At 3, 10 and 20°K, they obtain L - - 2.5, 3.3 and 5.0 respectively while we find for the 1287 D steel, the values 2.6, 3.25 and 3.6 at the same temperatures. The values obtained for the No. 3754 stainless steel are very similar to those obtained by E s t e r m a n n and Z i m m e r m a n n 14) for their stainless steel type 303. Both have a broad m a x i m u m with a nearly constant value of L between 20 and 70°K, with just this difference that whereas L ranges between 4.8 and 5.1 in our No. 3754 stainless steel specimen (in that temperature interval), it is between 5.4 and 5.9 in their specimen. Variation o~ ~ with/oreign metal content. Fig. 3 shows the variation of the thermal conductivity with the percentage of foreign metal content. In the stainless steel No. 3754, since there are 18.80 percent atoms of chromium and 8.10 percent of nickel, we considered it roughly as containing 27 percent atoms of foreign metal. In the 1287 D steel, there are 0.72 atoms of manganese to 1.92 of nickel and we consider it as having 2.6 percent atoms of foreign metal. For the remaining three steels, the manganese content .is ignored. In the graphs of fig. 3, and elsewhere in this paper, the portions of the curves leading to 27 °/o foreign metal content are shown dashed, indicating that we just added up the chromium and nickel contents. With this reservation, we shall treat all the five steels as "nickel-steels".

THERMAL CONDUCTIVITY OF SOME STEELS

79

Three curves are shown in fig. 3., representative of the three low ter'nperature regions, liquid nitrogen, hydrogen and helium. It is seen from these curves that the thermal conductivity falls with increasing impurity metal content, this fall being steeper, the higher the temperature. With further increase of impurity metal content, the thermal conductivity would probably rise again, for it was shown by one of us (De Nobel2)) from extensive measurements on nickel-steels containing from 0 to 99.4 percent of nickel, that for liquid air and liquid hydrogen temperatures, the thermal conductivit y diminishes with increasing nickel content, reaching a minimum at about 26% Ni, and rising thereafter.

Q20

0]~

010

\

0

¢ • I0

20

at'/. 3 0

Fig. 3. The thermal conductivity ;t for the steels in w a t t / c m - d e g v e r s u s percentage of foreign metal content " c " at different temperatures. A 70°K,

(D 20°K,

~7 4°K.

The fall in thermal conductivity of the "nickel-steels" studied by us could be attributed mainly to the increase in nickel-content, because, for one thing, they all had a similar heat treatment. Of course, the difference in amounts of the other constituents could also have some influence. S a v e l j e v 1~), for instance, found from his measurements on chrome-nickel steels (containing up to 1% Cr and 5% Ni) at liquid hydrogen, liquid air and at room temperatures, that the thermal conductivity decreased with increase of carbon content. The measurements by one of us (De NobelZ)) on the manganese steel No. 1010 (containing 12.69% Mn, 1.27~/o C and 0.12% Si) and No. 1379E (containing 12.95% Mn, 0.09% C, 0.12% Si, 0.103% S and 0.05% P) also agreed with this conclusion. But in our case, the nickel content was in an overwhelmingly large amount in comparison with the other impurities, so that we are justified in attributing the observed change in ~t, to the "nickel-content". Even though the fall in 2 with increase in "nickel-content" is more marked in the 70°K curve, the percentuM decrease in thermal conductivity,

80

M. S. R. CHARI AND J. DE NOBEL

in passing from 2% Ni to 27% Ni-Cr, is nearly the same whichever part of the low temperature region we might consider. The percentages referred to are actually 72, 74 and 75 respectively at 70, 20 and 4°K. The lattice thermal conductivity ;tg plotted against the impurity metal content is given in fig. 4 and presents the same general characteristics as fig. 3. Both the sets of curves show a linear fall in conductivity on increasing the "nickel-content" from 2 to 12% and then they flatten out somewhat, exhibiting a minimum at about the same nickel concentration. The percentage fall in ~lg, in passing from 2 to 27~/o foreign metal content amounts to 60, 58 and 37 at the temperatures 70, 20 and 4°K respectively, while the corresponding fall in ;re is about 80% whichever part of the low temperature region be considered. Thus the addition of impurity seems to have affected te more than ;tg. This is not surprising because at these temperatures, he is limited considerably by impurity scattering, whereas in the case of Ag, there oioo

0075

0.050 _ ~

°°i% ~ , o "

~0'

,o=,

Fig. 4. T h e lattice thermal conductivity Ig in watt/cm-deg versus percentage of foreign metal content "d' at different temperatures.

A 70°K.

(9 20°K,

~

4°K.

are also other phonon-scattering mechanisms (besides impurity scattering) which are not much affected by the addition of impurity. Further the effect of impurity on ~tg is noticeable more at liquid hydrogen temparatures and above rather than at liquid helium temperatures. This is also understandable because at liquid hydrogen temperatures and above, impurity scattering is very effective in limiting lattice thermal conduction. E s t e r m a n n and Z i m m e r m a n n 1~) conclude from a comparison of their measurements on a cupro-nickel alloy Cu9o Nilo, with those by H u 1m 2o) on Cus0Ni2o, and those by W i l k i n s o n and W i l k s 8) on CuToNiso, that the lattice thermal conductivitytg, when limited mainly by phonon-electron scattering, varied in a roughly inverse proportion to the nickel-content This is also roughly true in the case of the steels measured by us, but only in the concentration range 2-12% Ni.

THERMAL CONDUCTIVITY OF SOME STEELS

81

4. The grain sizes. Below are shown microphotographs of the etch patterns, two for each steel under the magnification 100 × and 750 X respectively *). Figs. 5a and b for steel 1287 I show a non-homogeneous martensitic steel with 96 grains/inchk Etchings were done electrolytically. According to figs. 5c and d the steel 1798 H is austenitic and very homogeneous with 12-24

¢1~"

-,. "

•

. ~ . . .

:2-.

. ,,":t ,,

i!"

•

Fig. 5. M i c r o p h o t o g r a p h s of e t c h p a t t e r n s o f t h e s t e e l s a t t w o a. no. 1 2 8 7 I (100 x ) b. no. 1 2 8 7 I (750 x ) c. no. d. no. 1 7 9 8 H (750 x ) e. no. 1 2 8 7 D (100 x ) 1. no. g. no. 3703 (100 x ) h. no. 3703 (750 i. no. 3754 (100 X) j. no. 3754 (750

magnifications.

1 7 9 8 H (100 x ) 1 2 8 7 D (750 x ) x) X)

*) Messrs. J. L o o s and J. J o n g e n e e 1 from t he Koninklij ke Nederlandsche Grofsmederij, Leiden, were kind enough to make these m i c r o p h o t o g r a p h s and to discuss t h e m w i t h us. P h y s i c a 25

82

M. S. R. C H A R I A N D J. D E N O B E L

grains/inch 2 (No. 5). Figs. 5e and / however show for steel 1287 D, m a n y inclusions; strongly lamellar, with 48-96 grains/inch 2. Figs. 5g and h indicate that steel 3703 contains much cementite at the borders with m a n y inclusions and more than 96 grains/inch 2. Figs. 5i and j show that steel 3754 is rather homogeneous with few inclusions, with 12-24 grains/inch 2. Etchings were done by means of aqua regia and glycerine. All these figures ycerd made after the research on heat conduction had been finished. It would have b e e n better to start the investigations on steels (in general on metals or alloys) by investigating grain size, homogeneity of tile samples and tb-finish the low temperature research with the same metallurgical invesfigatibns in order to be sure that no transitions have occurred. T h g electrical resistivities of the steels showed irregularities. A further investigation has to be started in order to determine the source of this anomalous behaviour. A c l ~ n o w l e d g e m e n t s . The work described in this paper forms part of the thesis submitted by one of us (M.S.R.C.) for the degree of Doctor in Mathematics and Physics of the Leiden University, in October 1956. It is part of the research programme of the "Stichting voor Fundamenteel Onderzoek der Materie (F.O.M.)" and was made possible by a financial support from the "Nederlandse Organisatie voor Zuiver Wetenschappelijk Onderzoek (Z.W.O.)" and from the "Nederlandse Centrale Organisatie voor Toegepast Natuurwetenschappelijk Onderzoek (T.N.O.)". During the earlier part of the measurements, we had the co-operation of Mr..L. K a p e l , while in the later part Mrs J. P o l l - - K e y s e r gave valuabel assistance. We had also the benefit of valuable discussions with Mr. A. R. D e V r o o m e n , nat. phil. drs. The glassblowing work was done for us by Mr. A. R. B. G e r r i t s e , while Messrs. D. De J o n g and J. V a n W e e s e l gave useful technical assistance. Cryogenic technician Mr. A. O u w e r k e r k supplied the liquid helium. Received 4- l 1-58

REFERENCES I) D e w a r , J. and H a d f i e l d , R. /~., Proc: roy Soc. A 74 (1904) 326 H a d f i e l d , R. A., J. Iron Steel Inst. I (1905) 147. H a d f i e l d , R. A., W o l t j e r , H. R. a n d K a m e r l i n g h O n n e s , H., Proc. roy Soc. A 99 (1921} 174; Commun. Kamerlingh Onnes Lab., Leiden, No. 155; De H a a s , W. J. and H a d f i e l d , R. A., Phil. Trans. (A) 232.(1933) 297. 2) De N o b e l , J., Commun. No. 285b; Phy~ica 17 (1951) 551 ; Thesis Leiden, 1954. 3) W i l k i n s o n , K. R. and W i l k s , J., J. sei. Instrum. ° 6 (1949) 19. 4) W e x l e r , A., J. appl. Phys. 2 2 (1951) 1463. 5) B e r m a n , R., Phil. Mag. 42 (1951) 642. 6) J a e g e r ~ W. and D i e s s e l h o r s t , H., Wiss. Abh. phys. techn. Reichsanstalt 3 (1900) 269.

T H E R M A L CONDUCTIVITY OF SOME STEELS

83

7) G r f i n e i s e n , E., Ann. Phys. (Leipzig) 3 (1900) 43. 8) L e e s , Ch., Phil. Trans. roy. Soc. (London) A 2@8 (1908) 381. 9) P o w e r s , R. B., Z i e g l e r , J. and J o h n s t o n , H. L., T. R. 264-7, Cryogenic Lab., Ohio State University ( 195 I). I0) K a r w e i l , J. and S e h ~ f e r , K., Ann. Phys., Leipzig (5) 36 (1939) 567. II) Z l u n i t z i n , S. and S a v e l j e v , I., J. techn. Phys. U.S.S.R. Leningrad 9 (1939) 805. 12) S a v e l j e v , I., J. Phys. U.S.S.R. Moscow 4 (1941) 383. 13) S c h m e i s s n e r , F. and M e i s s n e r , H., Z. angew. Phys. 2 (1950) 423. 14) E s t e r m a n n , I. and Z i m m e r m a n n , J. E., J. appl. Phys. 23 (1952) 578. 15) De N o b e l , J. and C h a r i , M. S. R., Suppl. au Bull. Inst. internat. Froid, Anuexe 1955 - 3, 467. 16) De H a a s , W. J. and B i e r m a s z , Th., Commun. No. 236e; Physica 2 (1935) 673. 17) C h a r i , M. S. R., Thesis Leiden, 1956. 18) M a k i n s o n , R. E. B., Proc. Cambridge phil. Soc. 34 (1938) 474. 19) M e n d e l s s o h n , K. and R o s e n b e r g , H. M., Proe. phys. Soc. {London) A :3@9 (1952) 388. 20) H u l m , J. K., Proc. phys. Soc. (London) A 64 {1951) 207. 21) C h a r i , M. S. R. and De N o b e l , J., Commun. No. 313b; Physica 25 (1959) 60.