Thermal conductivity of some non-superconducting alloys at low temperatures

Thermal conductivity of some non-superconducting alloys at low temperatures

Chari, M. S. R. De Nobel, J. 1959 Physica 25 84-96 THERMAL CONDUCTIVITY OF NON-SUPERCONDUCTING ALLOYS TEMPERATURES SOME AT LOW by M.. S. R. CHARI ...

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

Physica 25 84-96

THERMAL CONDUCTIVITY OF NON-SUPERCONDUCTING ALLOYS TEMPERATURES

SOME AT LOW

by M.. S. R. CHARI *) and J. DE NOBEL Suppl. No. 114b to the Communications from the Kamerlingh Onnes Laboratorium, Leiden, Nederland

Synopsis In this paper, the results Of investigations on the t h e r m a l conductivity of dilute alloys of manganese and of indium in silver (also in strong magnetic fields up to 25 kOe), and of some steels at low temperatures are compared with theoretical predictions. In this way, values for several coefficients in t h e terms representing different scattering processes are obtained.

I. Introduction. In the discussion of the thermal conductivity of alloys, we are concerned with two conduction mechanisms: lattice thermal conduction (which is the means of heat transport in dielectric solids) and electronic thermal conduction (which is the major means of heat transport in pure metals). In disordered alloys, in particular, since a perfectly ordered alloy has the properties of an ideal metal, especially at rather low temperatures, where the lattice conductivity 2g has a maximum and the electronic conductivity 2e is small compared to that in pure metals, these two components could even be of the same order of magnitude. It is therefore convenient to discuss separately the thermal conduction in dielectric solids and in pure metals before taking up the case of alloys. 2. Thermal conduction in non-metallic substances. In order to explain the results of E u c k e n ' s 1) thermal conductivity measurements on dielectric crystals and amorphous solids, D e b y e 2) considered a crystal as a continuum, wherein heat transport occurs through the medium of travelling elastic waves If these waves were purely harmonic, there would be no coupling between the waves and the conductivity would be infinite, there being no attenuation by mutual scattering. D e b y e explained the required coupling between the waves in terms of the anharmonicity which is assumed to account for thermal expansion. P e i e r l s a) treated the crystal as a lattice of atoms rather than as a *) Now in the Low Temperature Division, National Physical Laboratory of India, New Delhi. --

84

--

THERMAL CONDUCTIVITY OF SOME NON-SUPERCONDUCTING ALLOYS

8.5

continuum. In this theory, the coupling between the normal modes, which is essential for thermal equihbrium and for a finite thermal conductivity is ascribed to the anharmonicities arising from the third and higher order terms in the potential energy of the displaced atoms. The normal modes are quantised and the quanta of vibrational energy referred to as phonons, in analogy with photons of radiation theory. P e i e r l s has shown that for a discrete lattice, there could be threephonon interaction processes which do not conserve the wave-vector (the so-called "umklapp" processes). These would give rise to the umklapp resistivity wu (sometimes called the "intrinsic" resistivity) whic~h at temperatures of about OD and above, would be proportional to T, whilst at low temperatures, wu oc T-2 e-0Dm'. (l) The thermal resistivity due to scattering of phonons a t t h e grain boundaries or the external boundaries of the specimen, has been worked out by C a s i m i r 4) and can be written in the form wB

=

BIT s

(2)

where the suffix B on the left-hand side and the coefficient B on the fighthand side indicate that the phonons are scattered at the boundaries. For scattering of the phonons by point defects like impurity atoms or vacancies and by single dislocations, K l e m e n s 5) finds a scattering probability proportional to the fourth and the first power respectively of the frequency, so that the heat resistivity arising on these accounts are given by wD = D T Wd = d / T ~ .

(3) (3')

The suffixes D and d on the left-hand sides and the coefficients D and d on the right-hand sides indicate that the phonons are scattered by point defects and by dislocations respectively. Thus, for the over-all lattice thermal resistivity of a dielectric solid, we can write 1/)l s = wg = w u q- WB q- WD q- wa

(4)

to a satisfactory approximation. Actually, if r be the effective relaxation time and r~ the relaxation time for an individual scattering process indicated by the index e, then 1/~" = Z,, 1/'e,

this additivity relation being applicable to each individual frequency. It can be shown ( K l e m e n s 5)) that under conditions when two of these scattering terms are of comparable magnitudes, the total wg is larger than given by the simple additivity relation (4).

86

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

W i l s o n s) showed that the electronic thermal resistivity of metals (We) can be ex-" pressed as the sum of two components (in analogy with Matthiessen's rule for electrical resistivity) caused by the scattering of the conduction electrons respectively by the impurities (or static imperfections such as strains, displaced atoms etc.) and by phonons. Thus 3. Electronic thermal conduction he "in metals and alloys.

We = w0 + wi

(s)

where Wo is the impurity or "residual" thermal resistivity and is connected with the "residual" electrical resistivity p0 by the relation po/woT = Lo = vr2k2/3e 2.

(6)

L0 is the ordinary Lorenz parameter valid for high temperatures. S o n d h e i m e r 7) has shown that the simple summation implied in equation (5), though not strictly correct, involves an error of the order of only 1 ~ . At low temperatures (below about 0/10) the term wi due to scattering by lattice vibrations, can be written in the form wi = ~ T 2

(7)

a = (A/2oo 0 2) Na'.

(8)

where a can be expressed as

Here, Na is the number of conduction electrons per atom, 2** the limiting thermal conductivity at high temperatures, A is a dimensionless constant and O the Debye temperature. 4. Lattice thermal conduction in metals and alloys. The presence of conduction electrons in metallic substances, while constituting a medium for heat conduction, also forms an extra scattering mechanism for the m o m e n t u m transfer of the phonons. This reduces the lattice thermal conductivity. In the case of metals or alloys, which have a small 2e, the )tg m a y be comparable to ,~e. It was first pointed out by K ~ S n i n g s b e r g e r 8) that the heat transport by electrons and by the lattice could be taken as independent processes wherefore one could express the total thermal conductivity 2 as

2 = he + 2g.

(9)

For obtaining 2g, one can extend the theory of thermal conduction in dielectric solids to the present case, considering the phonon-electron interaction as an additional scattering mechanism, giving rise to an extra resistivity wE. M a k i n s o n 9) has shown that where electron interactions play a dominant role in limiting the phonon free paths, the lattice thermal resistivity wE,, at temperatures T ~ O, could be written as wE, = E ' / T 2

(10)

THERMAL CONDUCTIVITY O F SOME N O N - S U P E R C O N D U C T I N G ALLOYS

87

E ' being the coefficient of electron-scattering of the phonons. We might add here that single dislocations also lead to a thermal resistivity with a temperature dependence similar to w~, is). Taking the two resistivities for the scattering of the phonons by the dislocations and by the electrons, both inversely proportional to T 2 together to one term w~. = E / T 2, the over-all lattice thermal resistivity wg of metals and alloys can thus be written as wg = WB + w~. + WD + W~.

(1 1)

5. Klemens's work. It was assumed by M a k i n s o n 9) that phonons of all modes of polarisation interact equally with the conduction electrons. Since the actual value of the phonon-electron interaction constant C is not known, M a k i n s o n expressed 2E, for a free-electron gas in terms of the ideal electronic thermal conductivity at high temperatures 21coo~. Thus, 2E, =

27 X 7.18 (_T_T~ z 4.93 T221co.) 4~z2Na2 \ ¢ Y / 2i(oo)- OZNa------~

(12)

where Na is the number of conduction electrons per atom and 2ic~ refers not to the measured electronic thermal conductivity but to the "ideal" value obtained after taking the residual thermal resistivity into account. Since the original Bloch theory does not take umklapp processes nor the dispersion of phonons (that is, the dispersion of the velocity of sound) into account, and these do influence the 2ic~, K l e m e n s 1°) has shown that it would be more appropriate to compare 2w with the ideal electronic thermal conductivity at low temperatures, 21. In this manner, one would be comparing two quantities which are governed by the same mechanisms, thus eliminating the effect of any variation of the interaction constant C with the phonon frequency. Assuming the Makinson coupling scheme (that phonons of all polarisations interact equally with the conduction electrons), he obtains for a spherical Fermi surface at temperatures T ~ O, ,~E' = l/WE, = 3132i ( T /OD)4Na -413.

(13)

In equation (13), it is appropriate to use OD, the Debye temperature obtained from low temperature specific heat measurements, because it is the average over all polarisations. On the other hand, if one assumes the Bloch coupling scheme (namely that only longitudinal waves can directly interact with the electrons), the equation takes the form

2~., = 1/wE" = 10521 (T/OL)4Na -.13.

(14)

Here we use the value of O appropriate for longitudinal lattice waves, namely OL ( B l a c k m a n l l ) ) . Estimation of Or. in relation to OD is not easy for real metals. Monovalent metals like copper and sodium are elastically

88

M. S. R. C H A R I A N D

J. D E N O B E L

very anisotropic, so that averaging has to be done over the. different directions relative to the crystal axes. Such a calculation has been made by B l a c k m a n and we usually take OT, as N 1.5 OI). The numerical constant 105 appears in equation (14) in place of 313 since C a = C2L/3. Further if w~, in equations (13, 14) refers to an alloy, 21 and 21c~ also refer to the same alloy and not to the pure solvent metal.

6. Choice o//ormula/or 2E'. We have seen that equations (13) and (14) are obtained for the Makinson and Bloch coupling schemes of the phononelectron interactions. If one assumes Na ---- 1 for pure silver and calculates ~t~,,(= 1~wE, = T2/E ') from these equations, it is found that the Makinson scheme gives a value about 15 to 20 times that given by the Bloch scheme. It appears therefore that the lattice thermal conductivity (when only phonon-electron scattering processes exist) depends sensitively upon whether the electrons interact with phonons of all polarisations (Makinson scheme) or only with longitudinal phonons (Bloch scheme). Direct determination of 2~, of a pure metal being not possible, the following method is employed for testing which of these schemes holds better. Considering a series of alloys having one and the same solvent metal, the experimental values of lattice thermal resistivity in the temperature region where it shows a T -2 temperature dependence, give E for each of the alloys. We plot E against the solute concentration ( K l e m e n s l Z ) , K e m p , K l e m e n s , S r e e d h a r and Whitel3)). The value (E0) of E for the pure solvent metal is obtained by extrapolation to zero solute concentration. Since we know the expected values of E0 l~aklns°n and E0 moth for Na = 1, this constitutes an immediate check as to which of these coupling schemes describes the situation b e t t e r . . 7. The anomaly in the thermal conductivity. The existence of an anomaly in the thermal conductivity of an Ag-0.55 at.% Mn alloy rod was reported by us to the Int. conference on low temperature physics, Paris, 195517). Since an anomaly in the residual electrical resistance and the electrical magneto-resistance of Ag-Mn alloys had been observed by G e r r i t s e n and L i n d e 14) and discussed theoretically by K o r r i n g a and G e r r i t s e n l 5 ) , the first impression was that the present anomaly was the thermal counterpart of the anomaly in the residual electrical resistance, and that it should be attributed to the presence of the transitional metal ion in the alloy (see, for instance, G e r r i t s e n 16)). When later, the same anomalous behaviour was noticed in the dilute Ag-In alloy and also with the steels, it became evident that the present anomaly had probably an entirely different basis. Our preliminary communication 17) contained a fallacy in that the fig. 1 therein showed a flattening of the hump of the 2/T versus T curve on the application of increasingly strong magnetic fields. That was based on in-

THERMAL CONDUCTIVITY OF SOME NON-SUPERCONDUCTING ALLOYS

89

sufficient data. Fig. I of the present paper gives the results of a few geries of measurements in magnetic fields, obtained b y us later. It can be seen that the shape of the ~ versus T curve persists without any significant change, even in the strongest fields used, and should therefore be associated with lattice thermal conduction.

j

1

T

2

f

3

4

°K

Fig. 1. Silver-base alloys: ~ in watt/cm-deg versus temperature.

o

•H:O

¢ A



H = 19kOe

v!u=12 []

• H=

kOe

25.5kOe.

T h e s e t o f - f o u r c u r v e s a t t h e t o p are for A g - 0 . 1 4 a t . % Mn. T h e s e t of t h r e e c u r v e s a t t h e m i d d l e are for A g - 0 . 3 2 at.~/oMn. T h e s e t of four c u r v e s a t t h e b o t t o m a r e for A g - 0 . 5 5 at. % M n . D a t a for A g - 0 . 2 4 at. ~/o I n are n o t p l o t t e d since t h e effect of m a g n e t i c field is n o t m e a s u r a b l e .

Measurements on dielectric solids b y De H a a s and B i e r m a s z l S ) and B e r m a n 19, among others, do not show this effect, whereas the measurements on quartz glass b y W i l k i n s o n and W i l k s 2°) and b y B e r m a n 21) show something akin to the present anomaly in having a narrow region of temperature-independent 4. In pure metals, lattice thermal conduction is masked b y the electronic thermal conduction and plays such an insignificant role that we do not expect this feature in lattice thermal conduction to be

90

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

noticeable. In very impure metals and in alloys containing a large amount of solute metal, impurity scattering of the phonons begins to take effect at temperatures probably as low as liquid helium temperatures so that the anomaly would not be well-marked. It looks as if "suitably dilute" alloys would be the material wherein one should look for this effect 2s). 8. 2/T versus T curves. In fig. 2 we plotted ~/T versus T for the silverbase alloys measured b y us. For purposes of comparison, a few of the measured points of W hi t e and W o o d s 22) on Cu-0.056 at.% Fe (as read off the graph in fig. 5 of their paper) are also plotted. It was evident at the 0.12

W

v ~v O~2

oo!

I

J

.J

006

004

Fig. 2. Silver-base alloys: ~/T in watt/cm-deg 2 versus temperature. (D Ag-0.S5 at.% Mn A Ag--0.32 at.% Mn The curve marked . . . . . . . . . . .

[] Ag-0.24 at.% I n V Ag--0.14 at.% Mn

is for Cu-0.056 at.% Fe ( W h i t e and W o o d s ) .

very outset that the anomaly in the thermal conductivity at the liquid helium temperatures would make it rather difficult to analyse our measurements. The theoretical prediction of M a k i n s o n 9) and the experimental results - of H u l m 2s) and E s t e r m a n n and Z i m m e r m a n n 24) on coppernickel alloys, of B e r m a n 9.5)011industrial alloys, of W hi t e and W 0 0 d s 22) 2s) on very dilute copper-iron alloys, of S 1a d e k 27) 011 indium-thallium alloys, and of K e m p , K l e m e n s , S r e e d h a r and W h i t e zs) on silver-palladium

THERMAL CONDUCTIVITY OF SOME NON-SUPERCONDUCTING ALLOYS

91

and silver-cadmium alloys - have shown that at sufficiently low temperatures, lattice thermal conduction is limited mainly b y the phonons being scattered b y the conduction electrons (-- the E/T 2 term in the equation for wg). We have therefore as a first approximation, ignored the anomaly in the liquid helium region. In uther words, we considered it rather as a spread of the points, and boldly drew a straight line through the point marked Lo/po (= 1/woT) on the y-axis. The correspondence with W h i t e and W o o d s' curve for Cu-0.056 at.% Fe lends support to this step. The gradient of this line equals 1/E, whence the phonon-electron scattering coefficient E for each of the alloys is obtained. Boundary scattering of the phonons is also probably present but we neglected it, considering for example, that B e rm a n 25) has shown this to be at most about ½% of the total thermal resistivity, for a grain size of the order 0.02 mm ; and our alloy specimens have certainly much larger grain sizes. The values of E thus obtained for the silver-base alloys are respectively 570, 400, 540 and 400 cm deg3]watt, in the order of decreasing solute concentration. Since we do not expect the additional of such small percentages of solute atoms to significantly affect E, we take 400 as a reasonable value for E of pure silver. The comparatively higher value of E in the case of Ag-0.55 at.% Mn and Ag-0.24 at.% In could be probably attributed to the presence of dislocations. The value of E = 400 is in agreement with the estimate of E = 430 for pure silver (assuming the Makinson scheme - see K e m p , K l e m e n s , S r e e d h a r and W h i t e l 3 ) ) whereas it should be 8300 on the Bloch coupling scheme. Thus our measurements seem to show 2s) that in the case of silver, the conduction electrons interact more or less equally strongly with longitudinal and transverse phonons. From a discussion of the various conduction properties of monovalent metals K l e m e n s l°) was led to expect this behaviour, and this has already been confirmed for silver b y K e m p et a/13) and for copper b y K l e m e n s 12) and b y W h i t e and Woods22)

26).

Using the value of ~ = 5 × 10-5 ( R o s e n b e r g 2 9 ) ) and of E = 4 0 0 , we can estimate Na for pure silver thus: ~g = T2/400; ~ti = 1/c~T2 and O = 215°K. But we have also 2¢/~i : Therefore Na ~

313(T/O)4Na -4/3

1.1.

Referring again to the ~/T versus T curves, we find that they deviate from rectilinearity (as does also the curve of W h i t e and W o o d s ) above about 6-7°K. The lowering of the curves below the straight lines indicates the setting-in of phonon-impurity scattering and this is in agreement with the requirements of Makinson's theory. It appears, a priori, from the relative

92

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

depressions of the curves below rectilinearity, that the first small additions of impurity (or solute) atoms are much more effective in scattering phonons than are further additions. Though we do not have measurements at temperatures intermediate to liquid hydrogen and liquid helium temperatures, the trend of the ~/T versus T curves for the silver-base alloys in these two regions points to the existence of a maximum in the intermediate temperature region. A possible explanation is as follows: At these temperatures, where mutual scattering of phonons can be neglected, we have, ~, ~- I/wg = I/(B/T a + E / T 2 + DT) or

$g/T ~ 1/(BIT 2 + E / T + DT2).

For ~g/T to be maximum, we should have the minimum value for (BIT 2 + + E / T + D T 2) which would be the case if 2DT 4 = E T + 2B.

Since the boundary scattering coefficient B is hkely to amount to only a few per cent, we can write the condition in the form 2DT3max = E.

(15)

To test this conclusion, we took the measured points at liquid hydrogen temperatures and tried to split the lattice thermal resistivity into that due to scattering by electrons and that due to scattering by impurities and defects. Now, wg = E / T 2 + DT, or wgT 2 = E + D T 3. Plotting wgT 2 against T3 should give a straight line from which E and D could be found out. Actually the plotted points are rather scattered: the spread being ± 5% for Ag-0.55 at.% Mn and Ag-0.32 at.% Mn, and as much as 20% and 35% respectively for Ag-0.24 at.% In and Ag-0.14 at.% Mn. Since we cannot expect two resistance mechanisms with such diversity in temperature-dependence (especially when they are of comparable magnitudes) to be strictly additive, and since we wanted only a rough idea of the relative magnitudes of the two resistance components, we made estimates of the values of E and D from straight lines drawn symmetrically through the rather scattered points. Table I gives the values of E and D obtained in this manner compared with the value of E obtained from the hehum temperature data. The values for E in the last column were obtained thus: in double-logarithmic plots of wg versus T, there are short temperature regions just above the liquid helium remperafures, where the extrapolated curves indicate a proportionality of wg with T -2 suggesting that phononelectron scattering is dominant in these short intervals. From the value of 2 at these temperatures, a rough estimate of E could be made.

T H E R M A L C O N D U C T I V I T Y OF SOME N O N - S U P E R C O N D U C T I N G ALLOYS

TABLE I

93

* E in em degS/watt

Specimen

Ag-0.55 Ag-0.32 Ag-0.24 Ag-0.14

at.% at.% at.% at.%

Mn Mn In Mn

D in cm/watt

From wgT a vs T s curves at liquid hydrogen temperatures

From ~./T vs T curves at liquid helium temperatures

0.082 0.066 0.092 0.147

750 460 560 400

570 400 540 400

From w g v s T curves covering liquid hydrogen and helium temperatures 720 400 425 • 510

TABLE II Tmax in °K

Specimen

cale. Ag-0.55 Ag-0.32 Ag-0.24 Ag-0.14

at.% at.% at.% at.%

Mn Mn In Mn

15.5 15.5 14.5 ll.l

[

(~.g/T)ms.x in W/era deg 2

obs.

calc.

13.0 12.5 12.5 11.5

0.015 0.022 0.017 0.018

1

obs. 0.015 0.024 0.0176 0:022

Using the values of E and D, we employed equation (15) for estimating the temperature Tmax at which 2g/T would attain its maximum value; and also the corresponding (maximum) value of 2g/T. Table II gives the values thus obtained for Tmax and for (2g/T)max against their experimental values. The agreement is reasonably satisfactory.

9. The 2/T versus T curves/or the steels. Fig. 3 gives the variation of 2/T (expressed in mW/cm deg 2) with T, for the steel specimens studied by us. For purposes of comparison, we have also plotted the measurements on steels by Berman21), E s t e r m a n n and Z i m m e r m a n n 2 4 ) , K a r w e i l and SchtiferS0)) and W i l k i n s o n and Wilks2°). B e r m a n has not given the actual values of 2 in his paper; we have therefore read off the values from the plotted points in the National. Bureau of Standards Circular No. 556, fig. 23. The rest of the above-mentioned investigators have given the values of 2 at specified temperatures, which we have directly plotted. The measurements of W i l k i n s o n and W i l k s 20) do not extend to liquid helium temperatures.Those of K a r w e i l and Sch~iferS0) and of E s t e r m a n n and Z i m m e r mann24) are rather scattered, so that a straight line had to be drawn for the entire region plotted. The measurements of B e r m a n ~1) are very satisfactory; they show that the curve at liquid hehum temperatures is a straight line (indicating that the scattering of phonons by electrons is the dominant mechanism limiting lattice thermal conductivity) and the curve bends away from rectilinearity above about 6°K, indicating the onset of phonon- impurity scattering and a consequent reduction in lattice thermal

94

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

conduction. This behaviour shown l~y B e r m a n ' s curve is completely in accordance with theory and lends support to our method of analysis of the 2/T versus T curves already described. /

tm.~ z

/



v

v ~

I 1

0

T

5

10

15

20

OK

Fig. 3. The steels: ~ / T in m W / c m - d e g 2 versus temperature. [] 1287 D W 3703 <~ 1287 I (9 1798 H A 3754 K a r w e i l and S c h ~ f e r {D W i l k i n s o n and W i l k s Berman ~k E s t e r m a n n and Z i m m e r m a n n H u l m (Cu 80 Ni 20).

The phonon-impurity scattering in the steels can be seen to be effective even at such low temperatures as 6°K. This a n d the anomaly obtaining a.t liquid helium temperatures proper could possibly account for the observed fact that the proportionality "of wg with T -2 (already referred to) was noticeable only in a very restricted temperature interval. The procedure for analysis of the 2/T versus T curves is the same as in the case of the silverbase alloys. The values of 2o/T and Lo/po fit nicely here and the anomaly

THERMAL CONDUCTIVITY OF SOME NON-SUPERCONDUCTING

ALLOYS

95

is not very marked in the case of the steels No. 1287 I, 1798 H and 3754. In fact, the two measured points of E s t e r m a n n and Z i m m e r m a n n in the liquid helium region suggest that det.ailed measurements would probably have given a slight anomaly as in the case of our stainless steel No. 3754. The points of B e r m a n 2I) and those of H u l m 2s) on CusoNi20, both plotted irL fig. 3, are also suggestive. 15000

f

W

IO00C

5000

"~~

O

e

10

20 at~

30

Fig. 4. The steels: phonon-electron scattering coefficients E in cm.-degS/watt versus percentage of foreign metal content "d'. The values of the phonon-electron scattering coefficient E for the steels obtained from the 2 / T versus T curves are 9620, 7000, 14300, 15900 and 15400, in the order of increasi~lgforeign metal content. Fig. 4 gives a plot of E in cm degS/watt against the percentage "c" of the foreign metal content Extrapolation of the curve to c = 0 gives E = 11000 for pure iron. We can evaluate Na for pure iron, assuming the Makinson coupling scheme and taking = 1.8 × 10-4 from M e n d e l s s o h n and R o s e n b e r g Sl). We also take OD = 460°K for pure iron from K e e s o m and K u r r e l m e y e r S 2 ) . We have then, 2g = T2/11000 and 21 = 1/~T 2, so that 2,/2, = 313 (T/O) 4Na-'/" We thus obtain Na ---~ 0.5. A c k n o w l e d g e m e n t s . The work described in this paper and in the two earlier papers b y us 2s) was done under the over-all supervision of Prof. C. J. G o r t e r , while also Dr. G. J. V a n d e n B e r g took a stimulating interest in the investigations and played a great part in preparing the final version of their publications. It formed part of the research programme of the "Stichting voor Fundamenteel Onderzoek der Materie (F.O.M.)" and was made possible b y 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.)".

96

THERMAL C O N D U C T I V I T Y OF SOME N O N - S U P E R C O N D U C T I N G ALLOYS

One of us (M.S.R.C.) is grateful for the Netherlands Government Fellowship which brought him to the Kamerlingh Onnes Laboratorium, Leiden, and to the Council of Scientific and Industrial Research, Government of India, for partial financial support. From February 1955 up to the end of October 1956, he received a research stipend from the Netherlands "Stichting voor Fundamenteel Onderzoek der Materie (F.O.M.)". Received 4-I 1-58

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