The properties of metals at low temperatures

The properties of metals at low temperatures

5 THE PROPERTIES OF METALS AT LOW TEMPERATURES H. M. Rosenberg IN an earlier article in this series MAcDonALD (1) gave a review of some aspects of th...

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5 THE PROPERTIES OF METALS AT LOW TEMPERATURES

H. M. Rosenberg IN an earlier article in this series MAcDonALD (1) gave a review of some aspects of the low temperature behaviour of metals. In particular, he dealt with the specific heat and the electrical conductivity of metals and alloys. The purpose of this article is to bring these two subjects up to date and also to describe some other properties of metals which have been the subject of low temperature research. These are the thermal conductivity, on which a very large amount of work has been done, and the mechanical properties of metals, in which systematic investigations have been started only in the last few years. Whilst the domain of low temperatures has no defined upper limit, we shall, in general, be dealing with the range of temperatures below t h a t of liquid oxygen, 90°K. THE

THERMAL

CONDUCTIVITY OF M E T A L S A N D ALLOYS

There are two processes by which heat can be transported through a metal. The first, and usually the more important, method is conduction by the electrons. The second process is t h a t of conduction by the lattice vibrations (or to use the term suggested by Peierls, "phonons") but this need only be considered in alloys and in metals and semi-conductors which have very few electrons. The total thermal conductivity of the metal will be the sum of the two. A full treatment of the theory of these two processes will not be attempted here, but an outline will be given in order t h a t the underlying ideas might be appreciated. For a more complete discussion the reader is ~eferred to the review article by KLEIVIENS. (2) Both conduction processes are limited by various scattering mechanisms without which infinite conductivity would result, and the theoretical problem at the outset is to decide what types of scattering are possible, which ones are the most effective (i.e. which ones give an appreciable thermal resistance) and finally to calculate t h a t thermal resistance. The problem is usually simplified by assuming t h a t one scattering process is not influenced by the others t h a t are present at the same time. This is not always rigorously true but, in general, the errors involved are small. This simplification {which in the case of electrical conductivity is called Matthiessen's rule) means t h a t the 339

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thermal resistivity due to each scattering process can be calculated separately and the sum of these resistivities will give the total thermal resistivity for that conduction mechanism. In our case we have two conduction mechanisms, that by the electrons and that b y the phonons, and these will have thermal resistivities We and Wg respectively. These thermal resistivities will each be the sum of a number of resistivities, e.g. W , = W a + W B + . . . due to the scattering mechanisms A, B,.... The total heat transport of the metal, however, will be the sum of the heat transported by the electrons and the phonons and hence the total thermal conductivity, K, is the sum of the conductivities of the electrons and the phonons, i.e. K = K 6 + Kg

(1)

where K, = 1 / W 6 and K~ = 1 / W a. We shall now consider the terms W, and Wg in more detail. The Electronic Thermal Resistivity

There are two main mechanisms b y which the electrons m a y be scattered and hence give rise to a thermal resistance. These are the scattering of the electrons b y the thermal vibrations of the lattice, giving a resistivity Wi, and the scattering of the electrons b y impurities, giving a resistivity We. These two scattering mechanisms are, of course, the ones which a/so have to be considered in the case of electrical resistance. There is another possible extra resistivity mechanism, the scattering of electrons b y one another, b u t this has not as yet been detected. Let us first consider the term We, the resistance due to impurities. This can be simplified b y using the formula from kinetic theory whereby the thermal conductivity, K, can be calculated from the expression K = Aclv (2) where A is a constant usually taken to be ~, c is the specific heat per unit volume, 1 the mean free path and v the velocity of the heat carriers. In the case where the electrons are being scattered b y impurities (these can be either physical or chemical impurities) the mean free path will not be temperature dependent, v can be assumed to be constant since the electron distribution is degenerate, and hence K will be proportional to the specific heat of the electrons, i.e. K will be proportional to the temperature, and therefore We will be of the form tilT. It is this scattering of electrons b y impurities which gives rise to the constant residual electrical resistance, Re, at low temperatures (1) and this is connected with We b y the Wiedemann-Franz law Ro

=

WeT 34O

L

(3)

THE

PROPERTIES

OF M E T A L S

AT L O W T E M P E R A T U R E S

where L is the Lorenz constant and is equal to 2.45 x 10 -s if Ro is expressed in ohm-cm and Wo in w a t t -1 cm deg. Hence = Ro/L

(4)

The second term Wt, due to the scattering of the electrons b y the lattice vibrations, cannot be dealt with so simply. Qualitatively one can see that as the temperature is reduced the amplitude of the lattice vibrations gets less and hence their scattering effect becomes smaller. This will give, therefore, a thermal resistance which decreases as the temperature is decreased. A proper solution, however, can only be achieved b y solving the Boltzmann transport equation and this leads to a complicated integral equation which has only been solved for the case of quasi-free electrons. This has been done b y several authors to various orders of approximation and the reader is referred to WrLSON'S TM book for these solutions. They are also reviewed b y K~EMENS [~ and b y 0 L S E N a n d ROSENBERG. {4) The result of these calculations is that, at low temperatures (T < {9/10), the thermal resistance Wi is of the form =T 2, the numerical value of ~ being given b y GN~I3 :~ =

~

K ~ 02

(5)

where _hr is the number of electrons per atom, K ~ is the limiting thermal conductivity at high temperatures, 0 is the I)ebye temperature and G is a numerical factor which has a value of about 70 depending on what approximation is used. It will be noted that = can only be calculated with the aid of other experimental data, in particular K ~. This is because it is not possible to calculate the true interaction function of the electrons with the lattice vibrations. If, however, N is taken as unity and experimental values of K ~ and 0 are used, the calculated value of ~ appears to be about four times greater than that obtained from actual measurements of the thermal conductivity at low temperatures (see HULM~5)). I t has been suggested b y BLXCKM~N~) that one reason for the discrepancy might be that since the Bloch theory only assumes scattering of electrons b y the longitudinal lattice vibrations a special value of 0 should be used, say OL, which only takes into account such vibrations. The specific heat 0, of course, takes account of lattice vibrations of all polarizations. The value of such a 0 L is about 1.5 times higher than the specific heat 0 and this would help to remove the discrepancy. There is, however, a more serious di~cnlty. When the theoretical solution for temperatures higher than 0/10 is examined, it is found that W i should pass through a maximum at about 0/5, i.e. that there should be a minimum in the thermal conductivity. Such a minimum has been 341

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looked for very carefully in several experiments (see for example B E R ~ N and M~cDONXLD (7) and ~OSEI~BERG (8)) and has not been found. This problem has been considered by ZI~AN (9) who showed that whilst the theoretical treatment outlined above was correct as far as it went, there was one electron-phonon scattering process which had been neglected. This was t h e mechanism first suggested b y PmERLS (I°) and called b y him an "umklapp-process." In this form of interaction momentum is not conserved among the interaction components themselves, b u t some is transferred to the lattice as a whole. Ziman has developed the theory, taking into account umklapp-processes and the fact that only longitudinal phonons should be considered. He shows that the maximum in Wi is then almost entirely removed and that the numerical values obtained are in quite good agreement with the measurements on sodium of BER~_~ and M_~cDO~xLD. (7) Unfortunately, his treatment does not give an explicit expression for the calculation of W o•

The general form of the electronic thermal resistance at temperatures below 0/10 is therefore W. = W, + Wo = ~ T ~ + fl/T

(6)

It will be noted that a, the lattice scattering coefficient, should be a constant for a particular metal, b u t that fl, the impurity coefficient, will depend on the particular sample of the metal which is being measured. Therefore from a fundamental point of view, a is the more important term. Providing that fl is fairly small it can be seen that there will be a minimum in the total electronic thermal resistance, W,, at low temperatures, i.e. there will be a maximum in the thermal conductivity. The smaller the value of fl for a particular metal (i.e. the purer the sample) the higher will be this maximum and the lower the temperature at which it will occur. Fig. 1 shows typical curves for the thermal conductivity of high purity metal. Whilst one would expect that Eqn. (6) would hold most satisfactorily for a metal such as sodium, it is found that the thermal resistivity of all pure metals is of this form. The simplest w a y of checking this is to plot W,T against T a, when a straight line should be obtained with a slope a. Deviations from such a straight line have been obtained (see below, p. 350) but these have been explained in such a way as not to throw Eqn. (6) into disrepute.

The Phonon Thermal Resistivity There is a basic difference between phonon scattering and electron scattering. This is that at the temperatures which we are considering, the electron energy distribution is degenerate and hence to a first approximation can be considered as independent of the temperature. 342

THE

PI~OPERTIES

OF METALS

A T LO%¥ T E M P E R A T U R E S

The vibrational spectrum of the lattice, however, changes continuously with the temperature and hence over a given temperature range we can consider the lattice and its defects to be "scanned" b y a band of changing wavelength. Depending on the wavelength used some types of defects will be detected and others will not---hence a whole range of scattering processes is possible as the temperature is changed. The most important scattering mechanisms for phonons are, in the

t/+ 1

0

30

40

:

50

70

80

90

100

"rernperatur~ °K Fig. 1. T h e t h e r m a l c o n d u c t i v i t y , K , of t w o s p e c i m e n s of l i t h i u m ~16> s h o w i n g t h e g e n e r a / b e h a v i o u r of m e t a l s of h i g h p u r i t y

order of their effectiveness as the temperature is increased, the scattering of the waves at the specimen boundaries which occurs at the lowest temperatures, the scattering b y electrons, b y dislocations, b y point defects, and lastly, the scattering of the waves b y interaction with one another. A full discussion of these mechanisms would be out of place in this article as several of them are only of importance in the conductivity of non-metals and for details the reader is referred to the articles by B ~ , (n) Pm~RL~ (1°) and KLEM~S. (~) In metals at low temperatures the main contribution to Wg is that arising from the scattering b y electrons although at higher temperatures the resistivity due to defects and impurities must also be taken into account. The resistance arising from the scattering of the phonons b y the electrons, Wge, has been considered b y M~Er~so~, (1~> KL~+ME~S(2) and Znvu~. (la) These authors show that Wg, is of the form ? T -~. This can be seen from the kinetic theory, Eqn. (2). The mean free path, l, of the phonons will be inversely proportional to the number of electrons 343

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which are able to scatter them. The only electrons which can do this will be those whose energy lies within ~T of the Fermi surface Eo and the proportion of these electrons to the total number present will be of the order kT/Eo. Hence the number of electrons available for scattering will be proportional to T and therefore l will vary as T -i. Assuming that the phonon velocity, v, is constant, and that the specific heat is proportional to T a gives K proportional to T 2, i.e. Wg, = yT -2. The calculation of the value of ? b y the authors mentioned in the previous paragraph is uncertain, since assumptions must be made about the form of the interaction constant between the electrons and the lattice waves--the same difficulty as arises in the case of the electronic thermal conductivity. MAKINSO~'S calculation (12~ overcomes this by using the limiting high temperature value of the electronic thermal conductivity, K~, in his expression for ~,. This reduces to = 0.204N202/K~

(7)

K~EMENS (2~ has argued that since it is known that the Bloch theory (on which Ma]dnson's work is based) does not take into account effects such as umklapp-processes which probably influence the value of K®, it might be safer to modify Eqn. (7) so as to use the value of the low temperature electronic thermal resistance, Wi, instead of K w. His value of r is given b y ~, ~ 3"2 × 10-3o¢0~N4/a (8) where ~ is defined b y W~ ---- ~T 2 as in Eqn. (5). Z n v ~ ' s (13~contribution deals mainly with a non-degenerate electron distribution such as one meets in the case of semi-conductors with a very small concentration of electrons. In the limit, however, his expressions reduce to the ones given above. The scattering of phonons b y impurities and lattice defects can be treated b y considering the analogous example of the scattering of light waves b y small particles (Rayleigh scattering). In this case the mean free path of the waves is proportional to the fourth power of the wavelength. Since for lattice waves (as in the case of Wien's displacement law for radiation) the dominant lattice wavelength is inversely proportional to the temperature, this gives l proportional to T -4. I f we assume, as before, that we are in the region where c is proportional to T 3, then substituting in Eqn. (2) gives K proportional to T-l; i.e. the thermal resistance of the phonons due to point defects and individual impurity atoms, WD, should be of the form eT. At higher temperatures this flattens off because the specific heat tends to become constant and also because, as the phonon wavelength becomes shorter, Rayleigh scattering no longer holds, the scattering tending to become 344

THE PROPERTIES OF METALS AT LOW TEMPERATURES independent of the wavelength. At 1OUTtemperatures, however, the total lattice thermal resistance Wg is of the form Wo = Woe + WD = 7'/T 2 + eT.

(9)

There will also be other contributions to Wg from dislocations and grain boundaries, b u t for the time being we shall ignore these. I t is found b y experiment that the thermal resistance of a high purity metal (with a purity say better than 99.9 per cent) can be fitted very well to the form of the expression for We [Eqn. (6)]. This indicates that practically all the heat is transported b y the electrons and only a very small amount b y the lattice. The main reason for this is that the term Wo,, due to the scattering of the phonons b y the electrons, is generally quite large, and this reduces the total phonon conductivity until it is very much smaller than the electronic contribution. There are, however, two main ways in which the phonon conductivity can become appreciable. The first arises when the number of electrons per atom is very small as, for example, in bismuth, antimony, germanium and silicon. There are then very few electrons to carry the heat and hence K, is very much reduced. At the same time there are fewer electrons to scatter the lattice vibrations, Woe is therefore smaller and so the lattice conductivity can rise to quite large values (in the case of germanium and silicon it can be as high as 10 w a t t units or more). The second w a y in which the lattice conductivity can be easily detected occurs when impurity atoms are added to the metal as in an alloy. This has the effect of tending to reduce only K~ at low temperatures because, as has already been noted, W D is proportional to T for impurity scattering and hence Kg will be unaffected in the liquid helium range. Thus Ke can be reduced until it is of the same order of magnitude as Kg. I t should be noted that this method of detecting Kg differs from the first in that the electronic conductivity is reduced and the total conductivity is usually quite small (a few tenths of a w a t t unit or sometimes very much less). In the first case, however, w h e n / ¥ has a low value, the lattice conductivity is actually enhanced and the total conductivity can be quite high. Once experiments have been made the problem arises as to how one should separate the measured conductivity into its various components. It is obvious that some assumptions must be made as to what scattering mechanisms need or need not be considered. If they were all taken into account in the analysis the problem would be far too complicated. For a pure metal one generally assumes that all the heat transport is b y the electrons and hence that the total thermal resistance, W, is equal to W~ as given b y Eqn. (6). This means that, as has already been explained, if we plot WT against T s, a straight line should 23

345

PROGRESS

IN M E T A L P H Y S I C S

be obtained. If one has a sample in which it is suspected that there is an appreciable proportion of lattice conductivity present, then one generally assumes that at the lowest temperatures the dominant term in K s is that due to the phonons being scattered by the electrons (i.e. 1/Wg,), and the dominant term in K~ is that due to the electrons being scattered b y the impurities (i.e. 1/Wo). By referring to Eqns. (6) and (9) the form of the total thermal conductivity, K, is in these circumstances

K ~- K~ --k Ks ---- Tiff -t- T~IF

(10)

I f this is so, a plot of K I T against T will give a straight line with a slope of 1/F and an intercept on the K / T axis of I/ft. A check on this can be obtained b y measuring the residual electrical resistance, Ro, of the specimen. As has already been shown, fl = Ro/L. This relation usually holds exceedingly well. Due to the fact that other scattering mechanisms come into play at higher temperatures the straight line of the K I T against T plot does not usually extend very far, but it is nevertheless very useful in giving an idea of the magnitude of the lattice conductivity and also of the value of the parameter F.

Recent Experimental Work In the past few years the thermal conductivity of a very wide range of pure metals has been measured and in many cases the values of ~ and fl have been calculated. For details the reader is referred to the review papers b y KLE~E~S (~) and b y POWELL and BLANPIED(14) ( t h ~ has most of the data in the form of graphs, and includes information on alloys and non-metals). The experiments show that all pure metals do have thermal conductivity curves similar to those shown in Fig. 1. In many cases the conductivity at the maximum can be very high, 50 to 100 watt units, or even more, particularly when the metal can be obtained in a very high state of purity and is carefully annealed or made into a single crystal (for comparison it should be noted that the conductivity of copper at room temperature is about 4 watt units). The alkali metals are, of course, the ones to which one would expect the theory to apply most successfully. Unfortunately, they are rather difficult to handle and prepare as thermal and electrical conductivity specimens. The earlier work of BERMX~ and MAcDO~XLD (7~ on sodium has been continued b y MACDONALD et al. (zS) on all the alkali metals and b y ROSENBERG(z6) on lithium. These all confirm that whilst the form of the conductivity is that which one would expect from the theory as outlined above, the numerical value of the thermal resistance I'V~is about four times smaller than the theoretical value. As has been mentioned, ZL~L~N'S¢9) modification of the theory does now give good 346

THE

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OF

METALS

AT

LOW

TEMPERATURES

agreement in the case of sodium but it has not yet been applied to the other alkali metals. One rather general result t h a t has been shown is obtained from Eqn. (5) for m. I f the form of this expression is correct, even though the values of G and ~Y are uncertain, one might expect t h a t metals with similar electron configurations would have the same values for G and N and t h a t the value of ~K~0 2 would be the same for those metals. ROSENBERGcs) has analysed his results and has shown t h a t for metals of a given chemical group the values for ~K~0 ~ are the same. A similar type of analysis has since been made by KLEMENS(2) and MACDONALD et al. (15) show t h a t it holds for their results on the alkali metals. One interesting problem t h a t must be decided is whether the electrons do only interact with the longitudinal lattice vibrations (as the Bloeh theory suggests) or whether they are: in fact, able to interact with phonons of all polarizations. OLSEI~and I~OSENBERG[4) suggested t h a t if only longitudinal phonons were involved then there might be an appreciable lattice conductivity due to the transverse phonons and this had not been observed. Zn~L~N19) has pointed out, however, t h a t there would probably be sufficient interaction between the longitudinal and transverse phonons themselves to reduce this conductivity. A very useful attack on the problem, however, has been made by KEMP et al. Clv~ KLEMENS(ls) had shown t h a t the low temperature ratio of the lattice to the electronic thermal resistivity of a metal, W g d W ~, depends very much on which type of coupling scheme between electrons and phonons is assumed. I f there is interaction with the longitudinal lattice waves only, then W g J W i will be t w e n t y times larger t h a n ff there was interaction with all lattice waves. The problem was to measure this ratio for a pure metal. I t was not possible to do this directly because, as has been indicated previously, the lattice conduction of a pure metal is very small. The metal which was chosen was silver and the problem was solved (1~) by measuring the conductivities of various silver-cadmium and silver-palladium alloys of known concentration. These alloys, which were in effect impure silver, had an appreciable lattice conduction and this could be extracted from the total conductivity as has been shown earlier in this article. The values of I V~ so found were plotted against the percentage concentration of impurity and the curves were extrapolated back to zero impurity. Since the curves were quite smooth this seems quite a justifiable procedure. The results, which are reproduced in Fig. 2, show t h a t the value of W~ which was obtained was t h a t to be expected if the electrons were able to interact with phonons of all polarizations (the theoretical calculation gives the point M on the ordinate). I f interaction had been with the longitudinal phonons only, then an intersection with the 347

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PHYSICS

ordinate at the point B should have been obtained. The authors conclude t h a t in the case of silver one must assume t h a t electrons can interact with phonons of all polarizations. Another interesting point which arises from this work is the effect of crystal perfection on the lattice conductivity. Some of the specimens were measured in the "as drawn" condition as well as after careful annealing and it was found that, as was to be expected, the lattice conductivity was increased by the annealing treatment. The extent of the improvement, however, was surprising when compared with the We T s (emdeg) W -1) 140xl0 z

120 !00

B - " ~80 60

\

\

40

o\ %

I . . . . 100

20 I

5O

....

. . . .

0

!

50

. . . .

!

100

% Pd Fig. 2. T h e coefficient of t h e l a t t i c e c o n d u c t i v i t y W~ T s for silver p l o t t e d as a f u n c t i o n of c a d m i u m a n d p a l l a d i u m i m p u r i t y . (17) T h e p o i n t B i n d i c a t e s w h e r e t h e c u r v e s h o u l d c u t t h e axis if o n l y t h e longituc]inA.l v i b r a t i o n s i n t e r a c t e d w i t h t h e electrons, w h e r e a s t h e a c t u a l p o i n t of i n t e r s e c t i o n implies i n t e r a c t i o n b y all l a t t i c e v i b r a t i o n s

theoretical treatment given by KLEME~S(19) who showed t h a t the additional phonon thermal resistance due to dislocations should be proportional to T -2. The resistance of the strained specimens was indeed of this form, but in order to account for the magnitude of this resistance in terms of dislocations it was necessary to postulate t h a t there were of the order of 1013 dislocations per cm2. * I t might be thought t h a t it would be easier to see the effect of dislocations on the lattice conductivity ff one used a non-metal. In such a case, however, the presence of the dislocations often introduces extra electron energy levels (see R E A D (21)) and this can lead to complicated behaviour in the thermal conductivity. Experiments by GEB.~LLEet al.(22) on germanium, which has had dislocations introduced by bending, show this. • Klemens (private communication) has recently suggested t h a t this figure should be reduced b y a factor of 15.

348

THE

PROPERTIES

OF

METALS

AT

LOW

TE]~iPERATURES

The fact that dislocations can be detected by heat conductivity measurements has been utilizedby LOMER and ROSENBERG. (189) They have strained crystals of alpha brass and have shown that the build-up of dislocations with increasing amounts of cold work can be easily investigated. It would seem that this might be a useful method for determining the dislocation density of a cold-worked metal. Now that the general form of the thermal conductivity of pure metals has been established, experiments have tended to be designed

119o \

o--

~lso

\

J,7o

O-O6 g

. I

3 -

0

I

I

50

I

I

i

t

I

100

2

Cd5

o,2 1

0

tOO0

2OOO

3OOO

T3

Fig. 3. A p l o t of W T a g a i n s t T a for a c a d m i u m single c r y s t a l , s h o w i n g t h e d i f f e r e n t slopes o b t a i n e d a t h i g h a n d low t e m p e r a t u r e s d u e t o t h e v a r i a t i o n of e c=S)

to investigate particular scattering processes such as have been described in the previous paragraphs. Other work, however, has been started in order to see whether there is at low temperatures any deviation from the strict form of We [Eqn. (6)]. In general this requires quite careful measurements (to better than 1 per cent) which is difficult, though not impossible, to achieve. There is one case, however, where a deviation is quite easy to detect. This is in the case of cadmium. It will be recalled that to check Eqn. (6), W T is plotted against T a and a straight line should be obtained. When this is done for a cadmium single crystal a straight line is indeed obtained up to about 4°K, b u t at this temperature the curve starts to bend upwards and at a higher temperature it becomes a straight line again ~4th a slope about three times that of the original line (see Fig. 3). This was rather difficult to explain as the detailed theory shows that whereas at higher temperatures the slope of the W T curve might be expected to decrease it seemed unlikely that it should increase. I t was suggested (RosENBERG (2S)) 349

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IN M E T A L P H Y S I C S

t h a t the increase in slope was probably connected with the very marked drop in the value of the Debye 0 (by ~-, 30 per cent) which had been found for cadmium from specific heat measurements (SMITH(24~). Since the slope of the W T curve [i.e. ~ in Eqn. (5)] depends on 1/04 (K~ depends on 02) even a small change in 0 might affect ~. One must, of course, be careful in this connexion not to take the value of 0 too literally when one calculates ~. 0 is only inserted into the equation as a convenient way of averaging over the phonon spectrum. I t is not introduced because of the specific heat as such. Since the averaging process is different for specific heat and conductivity a slight change in 0 should not be taken very seriously in its effect on ~. In the case of cadmium, however, 0 changes by such a large amount t h a t it is not surprising t h a t some change in ~ is detected. The temperature at which the change of slope occurs corresponds quite well with t h a t at which 0 starts to drop in value. I f there were a strict correspondence between ~ and 0, however, one would have expected ~ to have increased five times instead of by a factor three. To check t h a t this change in 0 was indeed the cause of the increase in slope of the WT curve, measurements of the conductivity of a zinc crystal were also taken C~a~ since Smith's work showed t h a t zinc also exhibited a sharp drop in 0. The increase in ~ was shown by zinc as it was for cadmium, although once again the increase in ~ was not exactly in accord with Eqn. (5). Another type of thermal conductivity research which has proved popular is to see whether anomalous behaviour in the electrical resistance is reflected in the thermal resistance. In general, as might be expected, similar changes are found. Such experiments have centred around magneto-resistance effects, anisotropic effects and the minimum in the electrical resistance which is found in some metals at low temperatures (see below, p. 379). Several workers have investigated this minimum with particular reference to magnesium, which was known (M~cDoN~D and ~¢~ENDELSSOHN (25)) to exhibit a marked minimum in its electrical resistivity. The thermal conductivity at first sight shows no evidence of any anomaly, but if WT is plotted against T a it is then found t h a t there is a minimum in t h a t curve which does occur at about the same temperature as does the minimum in the electrical resistivity. There is a difference in the details from the various workers (see Fig. 4); KEMP et al. (~6) and I:~OSENBERG(27) find t h a t the value of W T dips below the continuation of the linear part of the curve, but the former find t h a t at the lowest temperatures it then rises to a value greater than it would have done if there had been no anomaly, whereas the latter found t h a t at temperatures below the minimum, W T increased until it fell on the extrapolation of the linear part of the curve again. SPOHR and WEBBER(2s) found t h a t W T increased in value at low temperatures, but t h a t there was no minimum in the curve. KLEMENS(~) 35O

THE

PROPERTIES

OF M E T A L S

AT L O W T E M P E R A T U R E S

has pointed out t h a t the shape of all these curves could be partially explained ff one assumed t h a t some lattice thermal conductivity was present, b u t since the minima in the first two experiments cited do coincide with the temperature of the electrical resistance minimum, it seems certain the effect has something to do with this anomaly as well.

The Effect of a Magnetic Field on the Thermal Conductivity I t has been found t h a t the thermal resistivity of m a n y metals increases on the application of a magnetic field in a similar manner to the way

5oC

..S.W

"~ 3 " 0

0"5

0

2

4

(10 "K)

6

8 (20 °K)

10

T-~x 10-3 (OK)

Fig. 4. P l o t s of W T a g a i n s t T 3 for m a g n e s i u m b F v a r i o u s w o r k e r s , s h o w i n g t h e a n o m a l y w h i c h is a s s o c i a t e d w i t h t h e electrical r e s i s t a n c e m i n i m u m . SW--SPOHR a n d WEBBER, (28) K S W - - K E ] ~ P et al., ~s) R~I:~OSENBERG (27)

in which the electrical resistivity is also increased. I t is rather unfort u n a t e t h a t this phenomenon cannot be explained at all on the free electron model of a metal since such a model gives rise to zero magnetoresistance. This means t h a t a more complicated model must be adopted and this has been done by SONDHEIMER and WILSO~ (~9~ and b y KOHLER. (3°1 fl~ll these workers derive the same results although their methods are different. The model which t h e y t reat is one consisting of two overlapping bands (i.e. the s and the d bands) and t hey assume t h a t there is no interaction between one band and the other. The general form of the results depends on whether the number of carriers in one band is equal or unequal to the num ber of carriers in the 351

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~ETAL

PHYSICS

other band. I f A W is the change in thermal resistivity in a field H compared with the value W(O) in zero field at the same temperature, then, if the number of carriers in each band is unequal, AW/W(O) ---- AHU/(1 + B / H 2)

(11)

whereas if the number of carriers is equal then AW/W(O) = CH 2

(12)

where A, B and C are functions of the temperature, W(0), and the number of carriers in each band. It will be noted that whilst both equations show that for small fields AW is proportional to H e, Eqn. (11) leads to a saturation effect in high fields whereas Eqn. (12) does not. SO~DHmZ~.~ (Physical Society Conference on Crystal Dynamics, 1957) has recently shown, however, that at high fields both Eqns. (11) and (12) should tend to a saturation value equal to the lattice thermal resistance. Experimental results have not so far shown any saturation effect (except for bismuth and antimony which will be discussed below) and that whereas in small fields A W is approximately proportional to H e this usually changes to a linear relation in higher fields. In view of the rather artificial nature of the model it is not surprising that the theory does not agree very precisely with the experimental observations. Early work on the magneto-resistance effect was done b y G~O~.mE~ and ADE~ST~DT (al) who measured the effect of a 12 kG field on the thermal and electrical resistances of copper, tungsten and beryllium single crystals and of platinum and silver polycrystals at liquid hydrogen temperature. They found that it was only with the single crystals that A W was appreciable and that the relative increases in the thermal resistance were not as large as the increases in the electrical resistance, i.e. that the Wiedemann-Franz law did not hold. This under certain circumstances can be shown to follow from SONDHV.~.~ and WmsoN's (2~) theory. They (al) showed that AW was proportional to H 2 for small fields b u t that the relationship tended to become linear. The largest change in W which they noted was for beryllium whose thermal resistance increased sixty times in a 12 kG field. DE HAAS and DE NOBEL(3~' aa) have measured the magnetoresistance of a tungsten single crystal in fields up to 36.4 kG between 14 and 20°K. At 15°K they found that the thermal resistance increased 222 times in a field of 36.4 kG. At liquid helium temperatures the effect of a longitudinal field on pure tin has been determined b y I-IULM(5) up to 1"5 kG, and M~.~D~LSSOEN and ROSEZ~BERG(a4~ carried out a general survey and found appreciable effects in polycrystalline indium and thallium and in single crystals of cadmium, zinc, tin, lead and gallium. Some of their measurements were made in fields up to 18-5 kG but no sign of saturation of 352

THE

PROPERTIES

OF

METALS

AT

LO~ r TEMPERATURES

AW was detected. In general t h e y found t h a t AW/W(O), in a given field, increased at lower temperatures, and t h a t a transverse field had much more effect t h a n a longitudinal one. The effect of the field can be very marked. For a cadmium single crystal at 2°K the thermal resistance increased over 1000 times in a field of 18.5 kG. ALERS(~) has recently taken measurements on zinc crystals in fields up to 60 kG and he too detects no saturation effect. Two further interesting points arise from his work. He shows t h a t the value of AW/W(0) can depend very critically on the orientation of the magnetic field with respect to the crystal axis--a change of about 5 ° can change AW/W(O) by a i Electricol

Ther'mal

6

resls~flce

resistance ] F

'E 5

1

t'4

ZiJinc o~.

~" 2 1 0

,

0.5

i

1-0

I

1.5

2.0 0

0"5

1.0

1.

I ~ / gous~ I x l f f 4

Fig. 5. The oscillatory n a t u r e of t h e t h e r m a l a n d electrical m a g n e t o resistance of zinc in high m a g n e t i c fields, H . ~aS) T h e e x t r a a n o m a l o u s resistance is p l o t t e d against 1/H

factor of three at 60 k G - - a n d this effect was also shown by Grfineisen and his co-workers. Alers also showed t h a t if the field was parallel to the hexagonal axis of the crystal an oscillatory variation was impressed on the more or less linear curve of AW/W(0) against H. I t was found t h a t these oscillations were periodic in 1]H (see Fig. 5). These variations are characteristic of the de H a a s - v a n Alphen effect (i.e. the periodic variation of the magnetic susceptibility with the field, which is found in some metals at low temperatures) and seem to be a reflection of this effect in the behaviour of the thermal resistivity. Alers also found a similar effect in the electrical resistivity. STEELE and BABISKI~(aSI have found the same type of oscillatory behaviour in the electrical and thermal resistivities of bismuth. Whilst the details of the magneto-resistive effect are difficult to account for properly, KOHLER(3°) has derived a general equation which fits the results quite well. This is

A W~W(O) -~- J[H/W(0) TL] 353

(13)

PROGRESS

IN M E T A L

PHYSICS

where J is some function which m a y or m a y not be of the form of Eqns. (II) or (12). Eqn. (13), which is an extension of Kohler's rule for electrical resistivity, shows that we should expect a larger magneto-

resistive effect if we are at a lower temperature, or if W(O) is smaller, i.e. if we have a purer specimen. This is borne out b y experiment; e.g. Hulm showed that whilst he got a magneto-resistive effect for pure tin, his specimens with small amounts of impurities showed no effect. Most of the other workers in the field have also shown this. In particular if the specimen is a single crystal it usually shows a much greater value of AW/W(O). If one assumes that L is constant then one would expect from Eqn. (13) that a plot of AW/W(O) against H/W(O)T would be a single curve for points taken on the same specimen at different temperatures. Both KOHLER(a°) and MENDELSSOHN and ROSENBERG(34) have shown that this is approximately true, although there is a slight spreading of the points at the highest fields. There is one important aspect of the magneto-resistive effect that has yet to be mentioned. This arises when one considers metals which have an appreciable lattice thermal conductivity Kg. Even though this conductivity .is itself usually limited b y electron scattering, So~I)H~.n~ER and W ~ s o ~ 129) have shown that Kg will not be affected b y the application of a magnetic field. Thus in such a metal (e.g. bismuth or antimony) only the electronic part of the thermal conductivity will be reduced b y the field and hence b y quite a simple analysis (for details see the review articles already quoted (~, 4)), one can obtain the value of the lattice thermal conductivity of the metal. A considerable amount of work on these lines has been done on bismuth b y DE I-L~s et al. ( ~ (liquid nitrogen and hydrogen temperatures), b y GRi~EISEN et al. (38~ (liquid air temperatures) and by S~J~YT (39) (liquid helium to nitrogen temperatures). The lattice conductivities of antimony and beryllium have been estimated in a similar manner. (3L 40) I t has already been remarked that the magnitude of the magnetoresistive effect can depend very considerably on the orientation of the field to the crystal axis, and many workers have investigated this b y rotating the magnetic field with respect to the specimen axis. In general very complicated variations are obtained. These are due to the anisotropy and fine structure of the Fermi surface and the nature of the overlap of the bands, and in principle it should be possible to use this data in order to obtain such detailed information on the nature of the electron distribution. Up to now, however, no theory has been devised in which such measurements can be used.

The Thermal Conductivity of Superconductors Soon after the phenomenon of superconductivity had been established experiments were made to see whether, at the temperature at which 354

TH~

PROPeRTIeS

OF ~ T A L S

AT L O W

TEMPERATURES

the electrical resistance dropped to zero, there was any corresponding change in the thermal conductivity. Early experiments by O N ~ E S and HOLST ca~ published in 1914 showed that at the superconducting transition temperature, To, no discontinuous change in the thermal conductivity accompanied the disappearance of the electrical resistivity. I t was not until the 1930s, however, t h a t further investigations were made, mainly by I)E HAAS and his co-workers ~42,4a) and also by MEND~.LSSOF_W and PONTIUS.~44~ They showed t h a t when a metal became superconducting the thermal conductivity had a lower value t h a n when a magnetic field was applied to bring it back to the "normal," non-superconducting state. For a detailed report of this work and the much greater volume which has been done since 1946, the reader is referred to the review by MENDELSSOHN.(45} S i n c e , however, this t)Te of research shows yet another facet of thermal conductivity, and particularly since it has recently shown itself to be of considerable practical importance in cryogenic technique, the salient features will be described. Whilst the phenomenon of superconductivity has yet to be fully explained* and, as yet, no fully satisfactory theory is available, m a n y of the experimental observations can be explained in terms of the "two fluid" model of GORTER and CASnUIR.~4e~ According to this theory the electrons form two groups, or fluids, the "normal" electrons, which occupy higher energy levels, and the "superconducting" electrons, which exist in a lower set of levels. Below the superconducting transition temperature a certain fraction, x, of the electrons remain in the normal state, whilst (1 -- x) go into the superconducting state. As the temperature is reduced, more electrons become superconducting, and at the absolute zero x----0. The properties of the superconducting electrons are such t h a t t h e y are not scattered by the lattice waves or the impurities. Below the transition temperature, even though ( 1 - - x) might be quite small, there will be a thread or a fine network of superconducting electrons which will "short circuit" the resistance due to the normal ones, and hence the metal will have zero resistivity. The experiments of DAUNT and MENDELSSOHN(47) led to the conclusion t h a t the specific heat of these superconducting electrons is zero and hence, from Eqn. (2), t h e y cannot contribute to the heat transport. Hence, as the temperature is reduced below To, fewer and fewer electrons can carry the heat, and so the thermal conductivity becomes much less t h a n it is when the metal is in the normal state. In pure metals this is indeed the general observation, but the behaviour can be shown to fall into two groups. When the transition temperature is on the high temperature side of the thermal conductivity maximum (i.e. * For a detailed description of superconductivity the reader is referred to Shoenberg's

Superconductivity (Cambridge University Press, 1952).

355

PROGRESS

METAL PHYSICS

IN

where the electrons are being scattered by the lattice vibrations) the thermal conductivity curve for the superconducting state meets the curve for the normal state at quite a large angle, as in Fig. 6. Such behaviour is typical of metals with a small Debye 0 and a fairly high transition temperature, e.g. lead and mercury. If on the other hand To is on the low temperature side of the thermal conductivity maximum (where impurity scattering is dominant) then the superconducting curve gradually falls below the curve for the normal state as is shown 30

k&

4

E

2

~zo Pbl

I

I ,,|

I

2

4

!

I

6

I

I

8

•.~ I0

E

0

I0

! 20 30 ~-9--7- 40 temperature (°K)

Fig: 6. T h e t h e r m a l c o n d u c t i v i t y of l e a d i n t h e n o r m a l a n d s u p e r c o n d u c t i n g s t a t e s s h o w i n g t h e s h a r p b r e a k a w a y of t h e s u p e r c o n d u c t i n g c u r v e w h e n t h e t r a n s i t i o n t e m p e r a t u r e , T o lies o n t h e h i g h t e m p e r a t u r e side of t h e m a x i m u m c o n d u c t i v i t y . T h e i n s e t s h o w s t h e s u p e r c o n d u c t i n g c u r v e o n a larger scale (8~

in Fig. 7. Most superconductors fall into this latter category. By making certain assumptions regarding the mean free path of the normal electrons when the metal is in the superconducting state, HEIS~.I~BERG(4s~ and KOPPE (49~ derived a function which describes the thermal conductivity, when limited by impurities, which fits the experimental data fairly well down to about 0.4T~. Up to the present, however, there is no really satisfactory explanation for the behaviour of the conductivity when, as in Fig. 6, it is limited by lattice scattering. So far in this discussion we have assumed that the conductivity in the superconducting state is entirely electronic, and at temperatures just below To this is probably true. If, however, as the electrons fall into the superconducting state they cannot be scattered by the phonons, then, conversely, it seems reasonable to assume that the phonons themselves will not be scattered by these electrons (HuLM(5~). As the 356

THE

PROPERTIES

OF M E T A L S

AT L O W T E M P E R A T U R E S

temperature is reduced, more and more electrons leave the normal state and become superconducting [according to the Gorter-Casimir theory the fraction of normal electrons is (T/T~) 4] and hence the lattice waves are no longer scattered so much. This means that the lattice conductivity will start to increase appreciably below, say, about 0-3T~, and at lower temperatures one might expect the conductivity to be almost entirely due to the phonons. To check this, experiments must be done below I°K. This was, until recently, an

"•

3

4

5

6

.=. 1

1

10

I,

~ ?0

1

, I

30

I

.I

40

temperature (°K) Fig. 7. The t h e r m a l c o n d u c t i v i t y of t i n i n t h e n o r m a l a n d s u p e r c o n d u c t i n g s t a t e s s h o w i n g t h e g r a d u a l s h i f t of t h e s u p e r c o n d u c t i n g c u r v e f r o m t h e n o r m a l one w h e n t h e t r a n s i t i o n t e m p e r a t u r e , To, is o n t h e low t e m p e r a t u r e side of t h e c o n d u c t i v i t y m a x i m u m . T h e i n s e t s h o w s t h e s u p e r c o n d u c t i n g c u r v e o n a larger scale ~sJ

unknown region for thermal conductivity measurements, and new thermometric techniques had to be devised before satisfactory results were obtained. The earliest experiments were made b y HEF,R and DAUNT~5°~ and later b y GOOD~L~, ~sl~ b u t more precise work has since been published b y OLSE~, RE~TO~~ and M_E~DELSSOH~~5~, 53, ~ and b y LAREDOJ~5} These workers show that at the lowest temperatures the thermal conductivity in the superconducting state is proportional to T ~. This is the same temperature dependence as is obtained for crystals of non-metals (e.g. diamond and AltOs) at low temperatures, and this lends weight to the lattice conduction hypothesis. For non-metals it can be shown that the T 3 dependence for the thermal conductivity is due to the fact that the phonon mean free path is so long that it is being limited b y the size of the specimen, i.e. 1 in Eqn. (2) should equal the diameter of the specimen. B y inserting measured values of the conductivity in Eqn. (2) 1 m a y be calculated, and it does indeed agree quite well with the specimen diameter (see BER~NCn~). In the case of superconductors, however, the agreement between l and the 357

PROGRESS

IN

.WIETAL P H Y S I C S

diameter of the specimen is not entirely satisfactory. LAREDO'S results on tin(aS) show that I is less than three-quarters of the specimen diameter and MENDELSSOHN and RENTON(53) show t h a t for their tin specimen, l was about one-third of the diameter. At the moment it is not possible to say whether these discrepancies are due to crystal imperfections or whether there is also some other type of scattering process present. There seems to be no doubt, however, t h a t the heat transport in this region is by the phonons, even though the details are not yet properly understood. When a superconductor is placed in a gradually increasing magnetic field, the field suddenly starts to penetrate the specimen (at a strength dependent on the shape of the specimen and the direction of the field), until at a field strength H c all the metal has become normal (H c depends on the metal and the temperature). The electrical resistance returns and the electronic properties are those of an ordinary metal. The range between where the field first starts to penetrate and H e is called the transition region. I f now the field is once more reduced to zero, the specimen does not always return to its original superconducting state, although its electrical resistance again drops to zero. Magnetic susceptibility measurements, however, quite often show t h a t some parts of the metal are still normal in zero field and t h a t there remains a so-called frozen-in flux. I t will be seen at once t h a t this type of behaviour is very suitable for investigation by thermal conductivity measurements since it should be possible to get an idea of the amount of metal which, trapped by this frozen-in flux, remains normal. Accordingly m a n y experiments have been made on these lines. The thermal conductivity has been measured as the field has been increased up to H c and then decreased to zero once more. In addition, the transition region itself has been investigated to determine how the thermal conductivity changes as the field penetrates the specimen. The general experimental procedure has been to take thermal conductivity measurements as the field was increased in steps from zero to H e and then reduced back to zero again. Usually two sets of measurements were taken, in fields which were transverse and longitudinal to the specimen axis. It is unfortunate t h a t such experiments were not so easy to interpret as might have at first been supposed. In general the conductivity as the field was increased was not exactly the same as when it was decreased, and so a form of hysteresis curve was obtained. At higher temperatures the results were, perhaps, what might have been expected. At the point where the magnetic field started to penetrate the specimen (in a transverse field with a cylindrical specimen this should occur at ½He) the thermal conductivity started to increase as parts of the specimen became normal, and with increasing field it gradually rose until at Hc the normal conductivity 358

THE

PROPERTIES

OF METALS

AT LOW

TEMPERATURES

was a t t a i n e d . As the field was r e d u c e d this curve was m o r e or less retraced, a l t h o u g h t h e r e m i g h t be some hysteresis (Fig. 8). A t low t e m p e r a t u r e s , p a r t i c u l a r l y w i t h s u p e r c o n d u c t i n g alloys, m u c h m o r e complicated b e h a v i o u r was observed. As t h e transverse field first p e n e t r a t e d the specimen t h e c o n d u c t i v i t y dropped, passed t h r o u g h a m i n i m u m and t h e n rose to its n o r m a l value a t H c. On reducing t h e field this b e h a v i o u r was n o t r e p e a t e d - - n o m i n i m u m was observed and

!

~ 1.2 5"2g °K longTt,udTnal

I

fi

q 0"8

u

1'2

eo

5 4 0 oK

transvePse/

/

0"8 0

51 )0 gauss

1000

Fig. 8. The change in thermal conductivity of a lead-bismuth alloy at 5.4°K when superconductivity is destroyed by a magnetic field, c56) Full line--field increasing; broken line--field decreasing the c o n d u c t i v i t y in zero field was usually lower t h a n it h a d been in the original s u p e r c o n d u c t i n g state. Such b e h a v i o u r was first shown in n i o b i u m a n d in l e a d - b i s m u t h alloys b y MENDELSSOHN a n d 0LSEN cse) (Fig. 9) b u t it has since been f o u n d in p u r e metals, e.g. tin, lead a n d indium.(5~, 5sl T h e e x p l a n a t i o n of t h e m i n i m u m in increasing field is still n o t generally a g r e e d - - o n e possible reason is t h a t it is due to lattice c o n d u c t i o n which is r e d u c e d because the lattice waves are being s c a t t e r e d a t the boundaries of the n o r m a l / s u p e r c o n d u c t i n g phases ( S L A D E K (59), R E N T O I ~ (54), L A R E D O a n d P I P P A R D ( e ° ) ) . O t h e r writers (HELM (sl), DETWmER a n d FAIRBAI~'K(57), WEBBER a n d SPOHR (s8)) suggest t h a t in certain cases it is due to the scattering of electrons at the phase b o u n d a r y . 359

PROGRESS I ~ IVIETAL PHYSICS T h e fact t h a t the c o n d u c t i v i t y does n o t r e t u r n to its original value after the completion of a magnetic cycle has in some cases been t a k e n as evidence o f considerable frozen-in flux in the specimen. S ~ F M A N , (eg) however, in work on t i n - i n d i u m single crystals, has t a k e n b o t h magnetic a n d t h e r m a l c o n d u c t i v i t y m e a s u r e m e n t s on the same specimens and he showed t h a t some specimens which exhibited no frozen-in flux

Eo 0 . 8

//

2"g2"K

1

Iongitud{nol

to

/J A'

0"4~" . . . . . . . . .

|

E 0.8 oI ee

~" . . . . "~'-'~'"

//

2.89°K transverse

-

//

L "0

0'4

~---P ......... /ff¢'~ /

t

~--~-'o~-~'>° "~ 0

500

1000

gauss

Fig. 9. The change in thermal conductivity of a lead-bismuth alloy at 2-89°K when superconductivity is destroyed by a magnetic field, showing the m~nlmum which appears as the transverse field first penetrates the specimen. (Se) Full line--field increasing; broken line--field decreasing f r o m their m a g n e t i c b e h a v i o u r still showed a considerable frozen-in flux effect in their t h e r m a l c o n d u c t i v i t y . I t seems as ff the t h e r m a l c o n d u c t i v i t y m i g h t be a v e r y m u c h more sensitive measure o f the s t a t e of t h e specimen t h a n h a d h i t h e r t o been supposed. A n o t h e r t y p e of e x p e r i m e n t on t h e t h e r m a l c o n d u c t i v i t y o f superc o n d u c t o r s has been m a d e on various series of s u p e r c o n d u c t i n g alloys, in particular, l e a d - b i s m u t h , i n d i u m - t h a l l i u m a n d t i n - i n d i u m . T h e general results will n o t be p r e s e n t e d here as t h e y are v e r y complicated due to t h e fact t h a t usually b o t h lattice a n d electronic conductivities are involved. Since the b e h a v i o u r of each is different a n d varies with 360

THE

PROPERTIES

OF

METALS

AT

LOW

TEMPERATURES

the temperature and the impurity content, the interpretation of the results leads to considerable uncertainties. The reader is referred to KLEMENS (2) for a general review. The Thermal Switch

I t has probably struck the reader t h a t the above brief review of the thermal conductivity of superconductors has been a description of experiments which have been extremely academic. As so often happens in scientific research, however, the most academic experiment suddenly becomes of practical importance. This has happened most decidedly in the thermal conductivity of superconductors. One of the problems which have to be tackled in any cryogenic experiment is t h a t of thermal contact. I f a specimen has to be cooled it must be connected with a heat sink, e.g. a bath of liquid helium or a paramagnetic salt below I°K. At helium temperatures this is fairly simple to achieve as the contact can usually be a rod of pure metal such as copper. I t is sometimes desirable, however, to break the contact so t h a t the specimen should be isolated and, ff the contact has to be made again at a later stage of the experiment, this is difl~cnlt to achieve. I t is possible to use helium exchange gas to provide contact but below I°K the vapour pressure is very low and so the heat is not conducted away very quickly. The gas is also difficult to pump away when isolation is required. Mechanical contacts can be used, but, as the work of B E R ~ N ~eS~has shown, these usually have a high thermal resistance and moreover this resistance tends to increase if the pressure on the contacts is released and then reapplied. I t has been suggested independently by several authors ~ ) t h a t the difference in the thermal conductivities of a metal in the normal and superconducting states could be utilized in a thermal switch which entailed no moving parts and would operate instantaneously when a field greater than H c was applied to or removed from the metal. The lower the operating temperature below the transition temperature, the larger is the ratio between the two conductivities, and it is therefore advisable to use a metal which has a high transition temperature. Accordingly, lead (To ~ 7.2°K) is the most convenient to choose and since it can be obtained in a very pure form the conductivity in the normal state (when the switch will be "made") will be lfigh. Such a switch will be most effective in the demagnetization region below l ° K since here, as has already been discussed in this article, the superconducting thermal conductivity seems to be mainly due to the lattice vibrations and is very low. Hence the heat flow when the switch is "open" will be small. As the ratio of the two conductivities is proportional at least to T 2 / T 2 this ratio can be of the order of several hundreds or thousands to one, depending on the operating temperature below 1°K. ~4

361

PROGRESS

IN METAL PHYSICS

Such a switching device has been used successfully in two main types of experiment. DA~B¥ et al. (651 in their work on two stage demagnetization used a lead contact between some paramagnetic salt which was demagnetized to 0.25°K and a second salt which was cooled via this contact by the first salt. The second salt could be demagnetized further after it had been isolated by making the lead superconducting. In this way very low temperatures could be reached by the second salt using relatively low magnetic fields. By demagnetizing from a field of 9000 G a temperature of approximately 10-a°K could be reached. The switch was so efficient t h a t the heat influx to the salt was only 1 erg/min and the temperature was kept below 10-2°K for 40 min. The superconducting switch has also been used by H~,ER et al. (66) in their so-called magnetic refrigerator. Two lead strips are used to connect a working substance with a paramagnetic salt, and the salt with a helium bath boiling at about I°K. B y alternately connecting the demagnetized paramagnetic salt to the working substance and then magnetizing it again when it is in contact with the helium bath (to take away the heat of magnetization) and then reconnecting it to the working substance, one can go round a refrigeration cycle. By operating this cycle once every two minutes it is possible to keep the working substance at a temperature of 0-3°K for an indefinite period, with a temperature variation less t h a n O.01°. This concludes the review of the thermal conductivity of metals at low temperatures. I t will be seen t h a t besides the technological data which has been collected, the subject embraces a very wide range of problems and is a most useful tool in m a n y investigations. The Mechanical Properties of ~letals at Low Temperatures

Whilst demonstration experiments with liquid air show t h a t the mechanical properties of m a n y everyday substances deteriorate at low temperatures, this is by no means the case with most metals. The early experiments of BoAs and SCHMID (67) o n cadmium single crystals showed t h a t the critical shear stress increased very considerably as the temperature was reduced to 20°K and further work (ss) indicated a similar effect for zinc and magnesium. At about the same time DE H ~ s and HADFIELD(~9) measured the yield strength and tensile strength of m a n y polycrystalline metals and alloys and they showed t h a t in most cases the strength at 20°K was much higher than it was at room temperature. Until recently, however, very few such measurements have been taken, but experiments in the last few years have shown t h a t it is a very interesting and fruitful field of investigation. Besides research on the strength of metals, we shah also describe recent work on the elastic moduli, creep, fatigue and internal friction 362

THE

PROPERTIES

OF METALS

AT

LOW

TEMPERATURES

at low temperatures. Many of these topics have been dealt with in detail in earlier articles in this series and we shall therefore not go into theoretical details except in so far as t h e y are necessary for an understanding of the low t e m p e r a t u r e results. In general we shall be concerned with work of a more fundamental nat ure which has been carried out at 90°K or below. A comprehensive compilation of engineering d ata on metals below room t em pe r at ure is given by TEED. ~7°~

The Strength of Metals at Low Temperatures As has been mentioned above, the first experiments on the tensile strength at 20°K were made by DE 14AAS and I~IADFIELD.{~9} T hey made measurements on iron of 99.85 per cent purity, four carbon steels, t h i r t y alloy steels, four non-ferrous alloys, and commercial high p u r i t y copper and nickel. T h e y found t h a t the tensile strength in all cases increased between room and liquid air temperature, but t h a t for iron and most of the steels there was no further increase in strength at 20°K because brittle fracture occurred. O~ly in the case of steels with high nickel content was this brittle fracture inhibited. For copper, nickel and the non-ferrous alloys, however, the strength increased down to 20°K. F o r example, the tensile strength of copper at 20°K was about twice its room t em pe rat ure value and its ductility was also improved at low temperatures. Since the war much more work has been started on these lines. Whilst a great deal of this has been concerned with tests on technological alloys, measurements on pure metals have also been reported. ELDIN and COLLr~S(71~ have designed a machine which can apply loads of up to 30 tons at liquid helium temperatures. T h e y describe measurements on 1020 steel at temperatures down to 12°K. I n confirmation of the work of DE HAAS and HADFIELD c69~ t h e y also found a t e m p e r a t u r e (61.5°K) below which brittle fracture took place, although t h e y found t h a t as the t e m p e r a t u r e was decreased below this to 12°K th e brittle fracture strength did not remain constant but increased slightly. Uzl~K (~2~ describes an apparatus in which loads of 8 tons can be applied and this has been used at room temperature, 78°K and 20°K to obtain the stress-strain curves of several steels, bronze and duralumin. He also found t h a t brittle fracture occurred below 78°K and he concluded t h a t the lower the critical shear strength (or yield point) at room temperature, the greater was the percentage change in shear strength at low temperatures and the more was the likelihood of brittle fracture occurring. H e also found t h a t a nickelchromium steel showed ductility at 20°K thus being in agreement with the observations of DE H~_~s and HADFIELD~eg) t h a t nickel inhibits brittle fracture. WESSEL ~S~ has recently described an apparatus in which loads of up 363

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to 10 tons can be applied at temperatures down to 4-2°K. This has been used for investigating the brittle fracture and the ductile-brittle transition which occurs in some metals. In a preliminary paper (~4~ he reports an investigation of the stress-strain curves down to 4.2°K of a number of metals from the various crystallographic types. He shows that for the face-centred cubic metal, nickel, and for the hexagonal close packed, zirconium, the yield strength was not very dependent on temperature whereas the ultimate tensile strength improved considerably as the temperature was decreased. For body-centred cubic metals (beta brass and two alloy steels), on the other hand, the yield strength was very temperature dependent whereas the ultimate tensile strength 0"8 .,0"~ •~ 0.4 o

~ o.2 W

0

1~

2~

3~

~°K

4~

5~

Fig. 10. T h e v a r i a t i o n of p r e - y i e l d p l a s t i c s t r a i n w i t h t e m p e r a t u r e for m o l y b d e n u m , s h o w i n g t h e m a x i m u m w h i c h is o b t a i n e d a t t h e b r i t t l e d u c t i l e t r a n s i t i o n t e m p e r a t u r e c75;

was not. I t was this very rapid increase in the yield stress of the bodycentred cubic metals at low temperatures which gave rise to the brittle fracture which was observed with most of them at these temperatures. In later work WESSET.(~6) has concentrated on using his technique as a means of investigating the brittle fracture in body-centred cubic metals and the transition temperature range in which the fracture changes from the ductile to the brittle type. Together with BECHTOLDc76) work has been done on molybdenum, tantalum, niobium and ship plate steel. In all these metals the yield strength increases very much between room temperature and 4.2°K (by a factor of three or four) and it is in the region where the yield stress is increasing rapidly that the ductile-brittle transition occurs. At higher temperatures in the transition range there is usually considerable plastic flow after the yield point followed b y a brittle fracture, b u t at lower temperatures a brittle fracture occurs very soon after yielding has taken place with very little or no reduction in cross-sectional area of the specimen. The experiments show that the plastic strain which precedes the yield point reaches a maximum value at the ductile--brittle transition temperature (see Fig. 10) and Wessel uses this observation as strong evidence in favour of his theory of brittle fracture.* He suggests that * Note added in proof. The reader is also referred to another theory of brittle fracture recently proposed b y COTT~EV.r..~19O;

364

THE

PROPERTIES

OF

METALS

AT

LOW

TEMPEI~ATURES

the pre-yield plastic strain is due to the movement of dislocations which are then pried up at grain boundaries or other barriers. Due to the very high stresses produced in the region of these pile-ups, new dislocation sources are eventually activated which break through the grain boundary barriers and produce the yield point phenomenon. At lower temperatures, due to there being less thermal activation, the yield stress increases (i.e. the stress at a prie-up must be increased) before any break-through can occur, and hence the pre-yield plastic strain will increase. Eventually, however, the strains required will be so large that the highly localized stresses which are produced in the dislocation pile-ups will cause cleavage in unfavourably oriented regions and hence micro-cracks will develop. In favourably oriented regions plastic flow will still occur, b u t due to the spreading of the micro-cracks the amount of pre-yield extension will be reduced. At still lower temperatures, as the yield stress increases further, many micro-cracks will form after a very sma]J amount of plastic strain, and brittle fracture results after a very limited extension. Thus this theory of brittle fracture does explain the observations on the pre-yield plastic strain. Wessel points out that whilst it does not seem possible to avoid brittle fracture in some metals, it might be possible to lower the brittle transition temperature ff the yield stress could be reduced. This could be achieved b y reducing the possibility of the formation of large dislocation prie-ups either b y having a very small grain size (or perhaps b y producing a dispersed second phase within the grain) or b y having a very much purer metal. Measurements on the shear strength of metals between 4.2 and 300°K have been reported b y SIMON et al. ~7~ The prime object of their experiments was to determine the temperature variation of the coefficient of friction, b u t in some subsidiary work the shear strength was measured b y finding the force necessary to shear the metal with a punch and die. They point out that due to the uncertainty in the actual distribution of stress, this method might not give accurate values b u t that there seemed to be no reason to doubt the relative changes which were measured. Measurements were made on Fe, Ni, Cu, Pb, Zn and Sn. For Sn and Zn there was little change in shear strength between 4.2 and 300°K, but for Cu, Ni and Fe its value had approximately doubled b y 4-2°K and for Pb it had increased b y over four times. A very detailed study of the tensile properties of single crystals of high-purity iron has been made by ALLE~" et al. ~78) Stress-strain curves were taken from 20 to 373°K and they cover the range from cleavage (brittle) up to fully ductile fractures. Due to the fact that single crystals were used, the type of fracture obtained at low temperatures was very dependent on the crystallographic orientation of the 365

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specimen. In agreement with Wessel they found the very rapid rise in yield strength at low temperatures, which one expects in a bodycentred cubic metal, and they also observed that, below 90°K, the cleavage strength was not constant, but that it rose to a higher value at 20°K. This is a similar observation to t h a t of ELDIN and COLLr~S(u) on the fracture strength of steel to which we have already referred. Measurements of the ultimate tensile strength of a number of polycrystalline pure metals between 4 and 300°K have been made by

I

Do

4 .~

AI ~

Ag\

J Zn 0

50

Cd 100

150

2 O0

250

I

300

Temperature7 °K F i g . 11. T h e v a r i a t i o n w i t h t e m p e r a t u r e of t h e u l t i m a t e t e n s i l e s t r e n g t h of s o m e m e t a l s . N o t e t h a t e v e n b e t w e e n 20 a n d 4 - 2 ° K t h e r e is still a s l i g h t i n c r e a s e . ~79) T h e c u r v e for zinc is d i f f e r e n t f r o m t h e o t h e r s a s it h a s a b r i t t l e t r a n s i t i o n a t low t e m p e r a t u r e s

McC~_MMO~ and ROSENBERO.(79) Some representative curves are shown in Fig. 11 and it can be seen t h a t in most cases the increase in strength at low temperatures is very considerable. In aluminium, for example, the tensile strength increases about four times between room temperature and 4-2°K, and most of the other metals double their strength. I t is a pity t h a t at the moment a quantitative theory is not available to account for these observations and the others of a similar nature which have been described. There seems to be no doubt t h a t the increased strength of metals at low temperatures is due to the fact t h a t the external stress must be increased to move the dislocations in order to compensate for the lack of thermal activation. Such ideas can be developed in order to derive the temperature variation of the critical shear stress (SEEGER(S°~). The final strength of the metal, however, is much more dependent on its work-hardening characteristics, and this makes a full solution of the problem far less tractable. 366

THE

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METALS

AT

LO~ r TEMPERATURES

I t has been possible in this section to mention only t h a t work which has covered the range doom to 20°K or below, and we have not been able to describe the ver y extensive experiments which have been made in the last few years at temperatures down to t h a t of liquid nitrogen (78°K) as these would merit a separate article. For a report on some of this work the reader is referred to the review article by M A D D I N and CHEN~sl~ in an earlier volume of this series. Whilst n o t actually concerned with the measurement of strength, mention should be made of the work of BLEWITT et al. ~s2~ who have tak en stress-strain curves at 4°K on several single crystals of copper. This work is of particular interest as the curves instead of being of the usual parabolic shape have a ver y extended linear work-hardening range. The authors suggest t h a t such a linear work-hardening occurs before the dislocation pile-ups break through their barriers (e.g. mosaic boundaries) and at low temperatures, due to lack of thermal activation, this break-through will only occur at relatively high stresses. A detailed t h e o r y on these lines has been recently published b y ~RIEDEL. {83) I~ONTA~'A and his co-workers c1~, 105, lo6~ have measured the tensile strength, elastic moduli and hardness of m a n y metals used in aircraft construction and their work is referred to in more detail below in the section on fatigue. Serrated S t r e s s - S t r a i n C u r v e s

Many observers have reported t h a t the stress-strain curves of several metals at 4.2 or 20°K exhibit serrations, instead of giving the smooth curve usually obtained at higher temperatures. UZHIK(72) shows such a curve for an austenitic steel at 20°K and WESSEL (73, v4, 75, 7e~ obtained similar curves for m a n y of his specimens at 4-2°K. These curves are v e r y similar to those obtained for cadmium single crystals (see for example THOM:PSO~" and MZLLXRDm6)) in which twinning occurred during the extension, and BLEWITT et al. (se) have ascribed the term " a p p a r e n t twinning" to serrated curves which t h e y obtained for copper single crystals pulled at 4.2°K. T h e y have since shown (nT~ by x - r a y methods t h a t this is, in fact, true mechanical twinning. I t seems unlikely, however, t h a t this would be the mechanism in every case when one considers the wide range of metals in which it has been observed. This t y p e of behaviour does occur sometimes at higher temperatures and here it can be explained on the basis of dislocations breaking away from an impurity atmosphere which then diffuses so as to re-anchor the dislocations. This does not seem to be a possible mechanism for the observations at 4.2°K, where the probability of diffusion must be v er y small indeed. Wessel suggests t h a t some more general dislocation mechanism might be invoked. U nde r the high stresses necessary for 367

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deformation at low temperatures a "burst" of dislocations might be produced in which there would be considerable plastic flow. These would then interact with one another and hence restrict the flow until another burst was produced in another region of the specimen.

More light has been shed on the problem recently b y BASINSKI{ns~ who has shown that for pure aluminium the serrated curves can only be obtained at 4.2°K b u t that they can be produced at higher temperatures with alumim'um alloys. These have a much lower thermal conductivity than pure aluminium at higher temperatures and this suggested that the effect is connected with the rise in temperature of parts of the specimen so that unstable flow can then occur. This was checked b y using a specimen with a niobium wire through the middle; this wire is superconducting at 4.2°K and becomes normal at 9-2°K. Basinski shows that when the specimen was pulled at 4.2°K the niobium wire did become normal indicating that a very appreciable temperature rise occurred in the specimen, and he noted that this coincided with the drop in the stress-strain curve. It does seem, therefore, as if the effect might sometimes be due to the unstable flow which he suggests. The Elastic Moduli at Low Temperatures Whilst it is possible to determine the various elastic moduli by tensile, torsion and compressibility measurements, low temperature determinations of these quantities have tended to rely on ultrasonic techniques. In some experiments the resonant frequency of the specimen has been determined, whilst in others the transit time of very high frequency pulses (of about 10 Mc/s) has been measured. From the technological point of view the variation of the elastic constants is very small. There is a slight increase in their value as the temperature is decreased but between room temperature and 4.2°K this increase is usually less than 10 per cent. Fundamentally, however, the low temperature values are important, since these are the values which should be used in checking the theory of cohesion and specific heats. It should be noted that from the third law of thermodynamics the slope of the curve of elastic constant against temperature should tend to zero as the absolute zero of temperature is approached. Ultrasonic methods have been used in measurements on aluminium single crystals b y SUTTON(84) and on copper single crystals b y OVERTON and GAFFNEY.(Ss) A typical curve from the latter work is given in Fig. 12 in which the variation of the constant c~ with temperature from 4 to 300°K is shown. The decrease in slope towards zero at the lowest temperatures will be noted. Above about 0/3 the curve is linear. Sutton's work on aluminium has been done between 63 and 773°K and this gives very similar results to that on copper. The only other measurements on these lines have been on crystals 368

THE PROPERTIES

OF M E T A L S

AT L O W

TEMPERATURES

of germanium and silicon. FINE ~Se~determined the resonant frequency of his germanium crystals between 1.7°K and 275°C whereas McSKn~IN ~S~ measured the velocity of ultrasonic waves (of from 10 to 30 Mc/s) in his specimens of germanium and silicon. Considering the difference in their methods and in their specimens, there is very good agreement in the results of these two workers. The general form of the results is similar to those for A1 and Cu. I t should be remarked t h a t ultrasonic experiments give the adiabatic elastic constants and t h a t the isothermal constants must be calculated from these. Since the correction t er m is a function of the ratio of the specific heat at constant pressure to t h a t at constant volume, there is ,Elastic constants of copper C44 vs temperature

I

8.4

I

% ,~ s.o

!

I I

--ox

7.6

I

!

I

I 0

!

!

C44

L

30

60

go

120

150

180 210

240 270

300

r,°K F i g . 12. T h e v a r i a t i o n of t h e e l a s t i c coefficient c44 of c o p p e r w i t h t e m p e r a t u r e ~8~)

very little difference at low temperatures between the adiabatic and the isothermal values. OVERTO:Nand GAFF~EY (85) have made the necessary calculation for copper and t h e y show t h a t even at 100°K the d ~ e r e n e e is less t h a n 0.5 per cent although by room t em perat ure a correction of ab o u t 3 per cent is needed. Low temp er a t ur e compressibility measurements have been made by SWENSON(s8) who measured the volume change of all the alkali metals for pressures up to 10,000 arm at 4.2 and 77°K. As one would have expected from the other work which has been described in this section, there was v e r y little change in the compressibility below 77°K. The values of AV/V (where AV is the change in volume) are shown in Fig. 13. I t will be noted t h a t as one goes from lithium to caesium the metals become ve r y much softer. The dilatation observed is quite considerable. F o r caesium at 4.2°K there is a change in volume of ab o u t 25 per cent when a pressure of 10,000 arm is applied. The results have been compared with calculations by BARDEEN(89) which are 369

PROGRESS

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PHYSICS

based on the Wigner-Seitz theory of cohesion but it is shown that, except for sodium, discrepancies exist which are outside the experimental error and are probably due to the over-simplified nature of the model. Much better agreement (except for lithium) is achieved if the lattice constants (from density determinations) are compared with those calculated by BROOKS.(9°) One interesting observation in these experiments was the discovery of a possible transformation in rubidium at 77°K. In one run there was a sudden change in volume as the pressure 0.30

Cornpr~sslons of the aika|i metals at 77°K and 4.2°K 0,25 /s

0-20

/

"~ 0"15

,/

y

7 '~'Rb

.

,9'

I t

~'~

0.10

//

/

~-

4.f

0.05

x~

4ooo Pr.essu r'e~, atrrl.

)0

10000

Fig. 13. The change in volume, AV/Vo, with pressure, P, of the alkali metals at 4.2 and 77°K(aS) was being applied, and this change persisted even when the sample was warmed to room temperature under zero pressure. The density was about 10 per cent higher than for the normal metal, and instead of melting at 38°C, the specimen did not melt even at 100°C. I t was unfortunate t h a t further experiments failed to reproduce this effect, but the author states t h a t it would be very difficult to ascribe it to any error in observation. I t is possible therefore t h a t it might be due to a crystal transformation. Since some of the other alkali metals are known to undergo low temperature transformations, a transition in rubidium would not be surprising. One interesting change in elasticity occurs when a metal becomes superconducting. Thermodynamic reasoning shows t h a t when a metal becomes superconducting there should be a very small change in the 37O

THE

PROPERTIES

OF METALS

AT LOW

TEMPERATURES

elastic moduli (see SHOENBERG(91)). At one time it was thought t h a t this would be much too small to measure, but very delicate static torsional experiments have been made by 0LSEN (92) in order to detect this change in polycrystalline tin. He found t h a t on destroying superconductibility ~ i t h a magnetic field the relative change in the shear modulus AG]G was of the form 3-5 × 10-6[1 -- (TITs)4], where T~ is the superconducting transition temperature (3.7°K). Such an expression is in good agreement with theoretical calculations by PIPPXRD. (93) Similar work using an ultrasonic method has been reported by LA~DAUER(94) who used tin single crystals. Whilst he also found t h a t the effect was larger at lower temperatures, the magnitude of his changes was very much greater t h a n those found by Olsen. At the moment there seems to be no explanation for the discrepancy between these two sets of work.

The Creep of Metals at Liquid Helium Temperatures Experiments over most of the normal experimental temperature range show t h a t creep is very temperature dependent. The extension per unit time under a given load increases very markedly as the temperature is raised. All theories of creep have had to take account of this fact, and in the general explanation it is assumed t h a t under the applied load, dislocations could not break through the barriers which were obstructing their movement and t h a t thermal activation was necessary to provide the extra energy for the barriers to be overcome. Independent of the details of the model used, the extension, e, is always given by an expression of the form e ~ kTf (stress, time), i.e. the extension should be proportional to the absolute temperature. Most of the experimental evidence supports this result; for details the reader is referred to the review article by SULLY:cnS) In 1930, MEISS~ER et al. (95) made experiments to see ff t h e y could detect creep in cadmium crystals at liquid helium temperatures, and t h e y showed t h a t quite measurable extensions (of the order of a few tenths of a per cent) were recorded in the first few minutes of a test. They made experiments at 4.2 and at 1.2°K and the curves which t h e y published indicate t h a t the creep rate was very similar at both temperatures. This, of course, contradicts the prediction of the theor)T which has just been outlined. Experiments to check this early work have been made recently by GLE~ (961 and he too measured the creep on cadmium crystals in the liquid helium range. This work confirmed the fact t h a t creep does occur at these low temperatures and t h a t it is not very temperature dependent. Glen shows t h a t the amount of creep observed is very much more than would be expected from the theory, although some measurements which he took at 77 and 90°K are in agreement with it. The experiments show t h a t at low temperatures 371

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thermal energy cannot be the mechanism which governs creep. He therefore suggests t h a t an activation process for dislocation movement might be due to a quantum mechanical tunnel effect whereby there is always a finite possibility of a dislocation overcoming an energy barrier. MorT (9~) has calculated what form of creep would be expected if such a tunnel effect was the activation mechanism, and with m a n y simplifying assumptions he shows t h a t at temperatures around 1°K the amount of creep due to thermal activation and to the tunnel effect would be about the same. I t does seem, therefore, t h a t in this region, temperature-independent creep due to the tunnel effect might be possible, although the case is by no means proved conclusively. Experiments on Internal Friction

Work on internal friction usually involves setting a specimen into vibration at its resonant frequency and then measuring the decay of the amplitude of these oscillations. The fact t h a t the amplitude decreases with time even when the specimen is suspended in a vacuum shows t h a t there must be an anelastic interaction in the crystal lattice. This can arise by the interaction of the crystal lattice with impurities, point defects or dislocation networks. At room temperature and above, the study of the variation of the internal friction has been a very useful means of following structural changes, e.g. of precipitation and ordering, and it can be a very sensitive method of detecting various dislocation mechanisms. Quite often the main difficulty is to decide precisely what mechanism is being detected. A full review of the subject is given by NowIcK (gs) in an earlier volume in this series. Low temperature experiments have been concerned with the effect of cold work on the internal friction. The initial measurements were made by BORDONI(99~ between 4 and 300°K who observed t h a t in a pure annealed metal the internal friction decreased as the temperature was reduced. If, however, the metal was strained before testing, the internal friction rose on lowering the temperature, passed through a maximum and then started to decrease again. This behaviour was shown very markedly by copper and the experiments on this metal have been repeated in more detail between 20 and 300°K by NIBLETT and "vVI~KS.(1°°~ In their experiments the specimen was driven electrostatically and was made to oscillate in a transverse mode at 1 kc/s (Bordoni used longitudinal vibrations at 10-40 kc/s). They observed two peaks in the internal friction, one at 75°K corresponding to t h a t found by Bordoni and a smaller one at 32°K (Fig. 14). The amount of strain (provided it was greater than about 2 per cent) had no effect on the height or on the position of the absorption peaks. I t seems most likely t h a t these maxima are due to some type of relaxation 372

THE

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OF METALS

AT

LOW

TEMPERATURES

process in which a relaxation time, t, exists of the form t = t o exp (U]RT) where U is an activation energy and R is the gas constant. The value of U for this process is very small--about 0.1 eV. This means t h a t the energy absorption cannot be due to the actual almihflation of any form of defect, because ff this were so, these defects would anneal out at room temperature. I n actual fact it was found t h a t the copper had to be heated to about 350°C before the low temperature absorption peaks were removed. This indicated t h a t they were due to some dislocation mechanism. MASON'101) has suggested t h a t this mechanism xl~ 4

20

A!

15

t

IC

.=.=)

100

0

2(

Ternperature,°K Fig. 14. T h e i n t e r n a l f r i c t i o n of c o l d - w o r k e d c o p p e r a t low t e m p e r a t u r e s . A - - s t r a i n e d 8.4 p e r c e n t ; B - - a n n e a l e d 1 h r a t 250°C; C - - a n n e a l e d a f u r t h e r h o u r a t 350°C c1°°)

might be the vibration of segments of dislocation lines which are pinned at either end by impurity atoms or by other dislocations. Niblett and Wilks point out t h a t this does not seem very likely since to obtain a well-defined activation energy one requires a constant length of segment which can vibrate. Since the position of the peaks is not dependent on impurity or on the exact amount of cold work (both of which would affect the lengths of the pinned segments) it does not appear likely t h a t Mason's theory is correct. A more likely explanation has been given by SEEGER.~1°21 This also involves the oscillation of segments of dislocation line, but in this theory it is shown t h a t there is a unique length of line along the close packed direction which ~ill bulge out alternately to the atomic planes on either side of its original position. This unique length is determined solely by considerations of minimum line energy and it is not dependent on any mechanical pinning at the ends of the segment. I t is thus more in accord with the experimental results. Quantitatively the theoretical value of the activation energy is in reasonable agreement with t h a t found by experiment, although for complete agreement a value of the Peierls force must be used which is an order of magnitude higher t h a n t h a t found by conventional tension experiments. Seeger points out, 373

PROGRESS IN METAL PHYSICS however, t h a t tension experiments might give a value of the Peierls force which is too low because a large fraction of the dislocation lines will not lie along close packed directions and hence the effective Peierls force will be reduced. NIBLETT(1°3) mentions that it is possible t h a t internal friction experiments might, in actual fact, be the best method of determining the correct value of the Peierls force and this m a y well be so, provided t h a t one or two other points in the theory (such as the value for the lille energy of a dislocation) can be cleared up. The fact t h a t two internal friction peaks are observed can be explained by noting t h a t in a face-centred cubic lattice there are two principal directions in the close packed plane in which vibrations can occur and these would be expected to have different activation energies.

Fatigue at Low Temperatures Whilst the years since the Second World War have seen a very large increase in the amount of work on fatigue, it is surprising t h a t few tests have been extended to very low temperatures. When one considers the ever-increasing problems of aircraft operating at great heights at high speeds it is not surprising t h a t the first reports of low temperature fatigue experiments concerned themselves with aircraft alloys. These experiments were made by F o n t a n a and his co-workers and the first paper by Z~vmRow and FONTANAa~) dealt with measurements on magnesium and aluminium alloys and on various stainless steels. I t was shown t h a t in all cases the fatigue lifetime at liquid nitrogen temperature under a given alternating stress was about ten times greater t h a n the lifetime at room temperature. Tests at -- 78°C gave intermediate curves. Hardness and tensile tests were also made on the specimens and it was noted t h a t there was a strong correlation between the tensile and the fatigue behaviour. For metals in which a decrease in temperature produced a large change in tensile strength there was a correspondingly large change in fatigue strength, whereas those specimens whose tensile properties were not very temperature dependent did not show very much change in the fatigue strength. Later papers (1°5, 10e) gave similar data on titanium and titanium alloys. I t was also shown t h a t the effect of temperature on notched specimens was very much less than on unnotched ones. In a paper describing a new type of helium liquefier in which a brass bellows was used to produce a change in volume at low temperatures, L o n g and Sn~IoN(1°7) noted t h a t the lifetime of these bellows at low temperatures was very much greater than it was at room temperature. This observation started a more fundamental investigation on the low temperature fatigue of brass and later of copper. This work, by McCAM~O~ and ROSENBERG,(108)has been concerned with the measurement of fatigue behaviour down to 4.2°K. They have confirmed the 374

THE

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AT

LOW

TEMPERATURES

work of F o n t a n a and his group and have shown t h a t the fatigue characteristics of copper continue to improve at 20°K and t h a t there is a further improvement between 20 and 4.2°K (Fig. 15). They have also measured the ultimate tensile strength at these temperatures and have shown t h a t the increases in the ultimate tensile strength are in almost exactly the same ratio as are the increases in the fatigue strengths. Later work has extended the measurements to the other face-centred cubic metals, silver, gold, aluminium and nickel, and to hexagonal cadmium and magnesium, where similar results were found. IO X

% t~

"--~..

20 °K~-r",

n

Cycles to fo~lure

Fig. 15. The fatigue of copper at room temperature, 90, 20 and 4.2oK(los) The case of aluminium was particularly interesting because its tensile strength increases over four times between room temperature and 4.2°K. This large increase was reflected in the fatigue curves and gave rise to very large changes in lifetime when the temperature was altered. Between room temperature and 4-2°K the lifetime under a given stress increased by a factor of about 107. Anomalous behaviour was shown by zinc and iron which have a transition to brittle fracture at low temperatures. Below this temperature the fatigue strength decreased and it was, in fact, very difficult to fatigue these metals at all in this range. These low temperature experiments throw useful light on the fundamental mechanism of fatigue, in t h a t t h e y show t h a t thermal diffusion is not necessary in order to initiate fatigue cracks. I f it were, then there should be virtually no difference between the fatigue curves taken at 20 and at 4.2°K, whereas there is a considerable change in this temperature region. This does not mean t h a t at higher temperatures diffusion need not be taken into account, but it is clearly not fundamental to the problem, l~rom electron microscope studies of copper specimens fatigued at 4.2°K, COTTRELL and HULL(l°s) have proposed a purely geometrical mechanism involving the interaction of intersecting slip 375

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planes, to produce surface micro-cracks. Once the fatigue crack has formed, its growth will depend on whether the stress concentration at its tip is such t h a t the metal can work-harden until the fracture strength is exceeded in t h a t region (see HEAD(n°~). I f it can, then the crack will spread. On this basis one can see why there is such a strong correlation between the fatigue and tensile strengths although no detailed explanation has yet been given. The fact t h a t fatigue behaviour does vary so much with temperature is very useful in studying the fundamental aspects of the problem because we now have a method of altering the fatigue characteristics of a metal without recourse to alloying, special heat treatments or prestraining, all of which introduce unknown factors into the structure of the metal. Another aspect of the work has been concerned with the measurement of the amount of work hardening which has been introduced into the metal when it is fatigued. I t was found t h a t when a metal was fatigued at a certain stress for a given time at room and at lower temperatures, then stress-strain curves taken afterwards on the same specimens at room temperature showed t h a t the lower the fatiguing temperature, the softer was the specimen. In order to determine whether a longer period of fatiguing at the low temperatures would have increased the hardness, measurements were taken to see how the hardness increased as a function of the number of fatigue cycles. I t was found that, independent of the temperature, the hardness rose to a final constant value after the first few thousand cycles. Thus the softness at low temperatures could not be explained on this basis. Experiments were also made after fatiguing at low temperatures with a stress which had been increased to t h a t at which the lifetime would have been the same as t h a t of the specimens fatigued at room temperature. These showed t h a t the specimens which had been fatigued at the low temperature were now harder t h a n those fatigued at room temperature. There does not seem to be, therefore, a n y simple correlation between work hardening and fatigue as a function of temperature. Nevertheless, the cold work introduced during fatigue is of interest because there is a growing body of evidence to show t h a t this cold work is of a different nature from t h a t which is introduced during unidirectional extension. M c C ~ o ~ and ROSE,BERG
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expected to occur in fatigue since no large dislocation movements occur. This is particularly the case in their experiments because the final fatigue stress was reached only after a very gradual increase in load. Corroboration for this type of effect is given by McCxMMON and ROSENBERG(lla) who showed that the fatigue lifetime at 20 and 4.2°K depended very much on whether the full load was applied to the specimen immediately, or whether the loading was brought up from zero to its final value in about l0 sec (at 225 c/s). In the first case one might expect hardening b y dislocation interaction since quite large movements would occur if the full load was suddenly applied to a soft annealed metal; in the second ease the point defect hardening, as postulated b y Broom and Ham, might become the dominant mechanism. These two types of hardening might lead to the different fatigue properties which were observed. Broom and his co-workers have also shown that certain processes which are activated b y fatigue can be quenched if the fatiguing is done at 90°K. For example, they show that if a metal is strained b y a certain stress and is then fatigued at a lower stress level, considerable softening of the metal occurs. Such a process, which presumably involves the movement of point defects, is almost inhibited at 90°K and verylittle softening is observed. In some work on aluminium alloys ~n4~ it was shown that the poor fatigue behaviour of D.T.D.683 is probably due to over-ageing during the test. This was demonstrated b y the fact that at room temperature the fully hardened and the initially over-aged specimens both gave very similar fatigue curves, indicating that the fully hardened material had become over-aged due to the fatiguing. Confirmation was given b y tests at 90°K. At this temperature the over-ageing during fatigue was quenched and hence the fatigue characteristics of the fully hardened metal were now much better than those of the over-aged sample. ELECTRICAL COI~DUCTMTY AT L o w TEMPERATURES An earlier article in this series Ill has given an outline of the theory of electrical conductivity and has described much of the experimental work up to about 1951. Since then several hundred papers have been published on the subject and it will be possible to mention only a few selected topics in this article. A more detailed treatment of the theory and of some of the experimental results has been given b y MACDONALD. (Ll9) The general behaviour of the electrical resistivity at low temperatures is similar to that already described for the thermal resistivity. There are two resistive mechanisms; one is due to the scattering of the electrons b y the lattice vibrations, the "ideal" resistivity, and it therefore decreases at low temperatures, usually being proportional to T 5 for

~5

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T ~ 0. The other is the scattering b y static imperfections and impurity atoms, and this gives a constant term which at very low temperatures is the only one of any significance. Thus the resistance at low temperatures tends to this constant value--usually called the residual resist a n c e - w h o s e magnitude depends on the particular sample of the metal which is being measured. The ideal resistivity, on the other hand, should be the same for all samples of the same metal. Most experimental values of the ideal resistivity are compared with the theoretical expression of Griineisen (see ref. 119) which is based on the free-electron model. Whilst the temperature dependence predicted b y this theory does tend to T 5 at low temperatures, there are serious discrepancies when it is applied in detail. Firstly, the theory uses a characteristic temperature, OR . This is assumed to be constant, b u t if experimental values of the resistivity are fitted to the Griineisen expression, the value of OR varies very considerably in many cases. Even for the alkali metals, which one might have hoped to be good approximations to the idealized model, OR changes quite markedly except for sodium which does have a constant value. That the value of OR is not constant is readily understandable, since it is well known that a crystal lattice does not vibrate strictly with a Debye spectrum and that the specific heat (see below, p. 385) also cannot be represented b y a single value of 0. The effect of such deviations from the Debye spectrum on the resistivity has been calculated b y CORNISH and M_~cDO~ALD~1~°~ and they show that a much better agreement is obtained for lithium if a modified spectrum is used. As in the case of thermal conductivity there is considerable confusion as to whether OR should correspond to the specific heat 0, or whether it should be 0L representing only the longitudinal lattice vibrations. M~cDo~qALD~ngl points out, however, that the differences in the behaviour of OR might not be due entirely to the lattice spectrum but to variation in the actual electron behaviour in different metals. To support this he cites the case of sodium and potassium. The specific heat 0 of both these metals exhibits the same type of temperature dependence, but whereas OR has a constant value for sodium, it varies considerably with temperature for potassium. I f the anomalies in 0R were due only to the lattice spectrum then one would have expected a similar variation for both metals. In spite of the uncertainty in the value of OR it is nevertheless possible to choose some kind of average value so that the electrical resistivity theory can be tested quantitatively. It is then found that, for the alkali metals in particular, the theoretical value of the resistivity at T ~ 0/2 is about twice that which is observed. At low temperatures comparison is best achieved b y "anchoring" the experimental to the theoretical curve at high temperatures. I f this is done, considerable deviations between the curves occur at low temperatures. The theory 378

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can be modified by assuming t h a t the interaction between two electrons is not determiued by the expression -- e2/r but t h a t a screened field of the form -- e2/r exp (-- br) is used. HOVSTON(1~1) shows t h a t such a screened potential gives results in much better agreement with experiment, although he still has to use some arbitrarily chosen parameters. ZIMAN(9) has also considered the problem using BAI~DEEN'S(122) work as a starting point. This takes into account scattering by um]dapp-processes. He shows t h a t in the case of sodium quite good agreement can be achieved, particularly if the scattering formula of Bardeen is modified. On the experimental side the electrical conductivities of all the alkali metals have been measured by MACI)ON~D et al. (15~and they show that, with the exception of sodium, the ideal resistivity is proportional to T 5 at low temperatures. For sodium, however, an unexpected result was shown, because whilst there was a proportionality to T 5 between 8 and 15°K, the temperature dependence was raised to T s below about 9°K (see also Woo])s(128~). The reason for the T s term is not yet clear, although the authors suggest t h a t it might be due to the fact t h a t at very low temperatures it might not be possible for the lattice vibrations to remain in equilibrium during the passage of a current. They also show t h a t the use of a screening parameter improves the agreement between theory and experiment. For copper, silver and gold, I~EME~S (l~a) suggests t h a t the theory would be in better accord with experiment if it were assumed t h a t the Fermi surface touched the zone boundary in some places. PIPP~R])'S (ls7) work on the anomalous skin effect of copper shows t h a t this is probably true. When an electron drifted to such a part of the surface, it would be scattered to the other side of the zone and this would introduce an extra resistance.

The Resistance M i n i m u m The previous review article described an anomaly which is sometimes observed in electrical resistivity at low temperatures, namely a minimum in the resistance of some metals. In the past few years further work has been reported although we still have no satisfactory explanation of the existence of the minimum. Most of the work on the resistance minimum has been made on copper, silver and gold, or their dilute alloys. A minimum can be produced by a very small amount of added impurity; for example PEA~SO~ (12s) has shown t h a t copper containing 0.005 atomic per cent iron produces a very marked minimum. GERR1TSEN and LINDE (126) have extended their earlier measurements on silver-manganese alloys to copper-manganese, copper-chromium, gold-manganese and goldchromium alloys. The copper and gold alloys show the same very 379

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complicated behaviour exhibited previously by their silver alloys; in that they have a resistance minimum followed at a lower temperature b y a resistance maximum. KORRINGA and GERI~ITSENC122~have attempted to account for these effects by assuming a resonant type of scattering near the Fermi surface, but it is not clear whether this would apply in all cases, in which a minimum has been observed. BlmWlTT et al. ~1~ reported that for pure copper single crystals a minimum was only produced if the specimen was strained and then recrystallized, and they suggested that the presence of grain boundaries was necessary for the production of a minimum. SCHMITT and FISKE, ~2~ however, showed that the size of the minimum in their copper specimens was unaffected b y alterations in the grain structure. The results of Blewitt et al. were supported b y a theory of grain boundary scattering b y KOEHLER, (1S0) but COHEN and BARRETT (1S1) showed that this implied a considerable pressure on the grain boundary. They tried to observe this b y seeing if they could detect the bulging of a grain boundary in an aluminium wire when it was recrystallized b y passing a high electric current through it. No bulging was detected and they concluded that grain boundary electron interaction must be much smaller than that suggested b y Koehler. PEXRSON~132~suggested that the minimum observed b y Blewitt et al. might be due to the segregation of impurities along the grain boundaries, and he also pointed out how careful one must be in the preparation of specimens in order not to introduce impurities into solid solution which might affect, or produce, a minimum. The detailed experiments by M~tcDONALD and PEarSON (~3s) on copper and its dilute alloys show that the minimum is very much enhanced if the specimen is prepared under reducing conditions, probably due to iron impurity being brought into solid solution, since this, as has been mentioned above, has a very large effect. They also show that very marked anomalies occur in the thermo-electric power for those specimens which exhibit minima. Whilst most workers have considered the minimum as an extra scattering mechanism which occurs at low temperatures, it could also be considered as a decrease in resistance below the ordinary residual resistance. Some support for this latter view comes from experiments b y ROSENBERG (27) o n the minimum observed in magnesium, and PEARSON~32~ also suggests this possibility. Recent work by OwEN et al. clss~ on the electron and nuclear spin resonance and the magnetic susceptibility of copper-manganese alloys also shows unexpected results. In particular, the shift due to the copper in the paramagnetic resonance peak is only about 0.02 of the theoretical value, and it is suggested that this is because there is only very weak coupling between the 3d ~¢[n electrons and the 48 conduction electrons of the Cu. They also found that for more than 380

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1 per cent ~ the alloys became anti-ferromagnetic. Specific heat measurements by Smith (unpublished) on similar alloys show t h a t there is a peak in the liquid helium range for those containing less than 1 per cent Mn. I t would seem likely t h a t these observations should also be connected with whatever mechanism produces the resistance minimum, but it is not yet clear how all these phenomena should be linked together. An interesting point arises as to whether at very low temperatures the resistance continues to rise, or whether it flattens off to a constant value. I t appears t h a t either can happen. In the case of dilute copper alloys WlrrrE (184) has shown t h a t the resistance has become constant by about 2°K whereas MV,~DOZA and T H o ~ s (ls5) (for copper, silver and gold) and CROFT et al. c13e~ (for gold) showed t h a t the resistivity was still increasing in the demagnetization region. The Resistance due to Cold W o r k and Irradiation

In the last few years m a n y measurements have been made on the change of electrical resistivity when a metal is deformed or irradiated. Whilst such a change is not necessarily a low temperature effect, it will be appreciated t h a t since the defects which are introduced affect only the temperature-independent residual resistance, a much larger percentage change in resistance is obtained ff the measurements are taken at a low temperature t h a n ff t h e y are made at room temperature. In addition, however, there is always the possibility t h a t some of the defects which are introduced might anneal out at room temperature, and hence the initial deformation must be carried out at a low temperature if the effect of all the damage is to be observed. I f after deformation at a low temperature the temperature of the specimen is raised in steps to higher temperatures, then any change in the initial resistance will indicate t h a t some of the damage has annealed out. The study of such recovery curves, coupled with the resistance change produced in the original deformation, should give information which ought to assist us in determining the nature of the original damage. In some cases the recovery curve exhibits steps where no annealing occurs in between two temperature regions of resistance recovery, and in these cases certain temperature regions can be recognized in which one presumes t h a t well-defined, unique recovery processes occur. From a detailed study of such recovery curves one can calculate the activation energy of the defect which has annealed and hence one can make assumptions as to what the defect was. Provided t h a t the curves can be interpreted unambiguously, it can be seen t h a t such resistance recovery experiments could provide very useful information. It is unfortunate t h a t at the moment there are m a n y conflicting ideas as to the various recovery mechanisms and a coherent picture cannot yet 381

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be given. A detailed review of earlier experiments is given by BROOM.{137) The outstanding thing about all the experiments, however, is the very great amount of damage which can anneal out below room temperature. I f copper, say, is strained at 20°K, then the resistivity will be increased by about 15 per cent for a strain of 10 per cent. I f the specimen is allowed to warm up to room temperature and is then cooled down again and the resistivity is measured again at 20°K, it will be found t h a t about half of the original increase has recovered, although there is no recovery in the cold work introduced into the specimen. I t seems certain t h a t the resistivity recovery must be due to the migration of point defects. Such experiments were first made at 78°K on copper, silver and gold, by MOLE~XAARand AARTS.~13sJ More detailed work on the recovery curves by DRUYVESTEYX and MAN~TVELD(139, 140) showed the existence of two regions of recovery, one between -- 130 and -- 60°C and another between -- 30 and + 50°C. The activation energy in these two regions was about 0.2 and 0.88 eV respectively. Later work has been reported by EGGL~STO~~m~ at 4°K and by vA~ BUEREI~ and JONGENBURGER (142, 143,144) at 20°K. The last named authors have suggested t h a t the extra resistivity due to dislocations can be separated from t h a t due to point defects by measuring the magneto-resistance of their strained specimens and then plotting the results on a Kohler diagram. Point defects, being isotropic, will introduce no additional magneto-resistance above t h a t of the unstrained metal, whereas if dislocations are present the curve on the Kohier diagram will be displaced. This displacement can only be removed if the specimens are heated above 200°C and hence it seems likely t h a t it is caused by the dislocations. VA~ BUEREI~c144~ has analysed the results and comes to the conclusion t h a t about half of the extra resistivity originally introduced by cold work is due to dislocations and the other half to point defects. The recovery of the electrical resistivity introduced by low temperature irradiation has yielded very interesting results. COOPER et al. ~145~ irradiated copper, silver and gold at 10°K with 12 MeV deuterons. On warming up t h e y found t h a t about 40 to 50 per cent of the copper resistance recovered at about 40°K and the activation energy of the process was of the order of 0.1 eV. Similar results were found with silver. A recovery process at such a low temperature is very surprising and there has been much speculation as to the mechanism involved. At one time it was thought to be the motion and annihilation of an interstitial atom, but recent work by BLEWITT et aZ.{188} on the release of stored energy in the temperature range of the resistivity recovery gives a value which is so small that the interstitial mechanism appears highly unlikely. 382

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L O M E R and COTTRELL (147) have shown that the m e a n number of jumps a defect has to m a k e before it is annihilated, calculated from experimental data, is about l0 T for the 0.2 eV annealing stage (which is rather high if the concentration of defects is about I0-4), but only about I0 for the higher temperature, 0.88 eV stage. This figure is m u c h too small and they suggest that all the defects might be of the same type, but that whilst some are annihilated in the first stage, others might remain trapped at, for example, impurity atoms. Energy would then be required to release the defects for these traps before they could migrate. They show that such a mechanism yields reasonable results. They suggest that the high figure of 107 for the 0.2 eV stage (where it was assumed that vacancies and interstitials annihilate one another) might be overcome by assuming that the interstitial cannot m o v e freely in three dimensions but that it is constrained to m o v e as a "crowdion" (PANETH C14s~)in one line. If this is so then the rather high number of jumps before annihilation can be understood. For a further discussion on the mechanism of the various recovery steps the reader should refer to SEEGER c14e~and B R O O M ~IS7~in addition to the authors already mentioned. M u c h more work must be done, however, before we have a clear picture of all the processes involved.

Superconductivity Besides the work referred to in other sections of this paper, perhaps some of the most interesting superconductivity experiments have been those in which superconducting behaviour has been altered or has been produced by mechanical means. Experiments by BUCKEL and H~SCH c149) have been made on the resistance of thin films of metal which have been condensed on to a support at liquid helium temperatures. Under these conditions of preparation the atoms are in a considerable state of strain and disarray, and it has been found t h a t the superconducting transition temperatures of these films are very different from those of the bulk metal. In some cases the transition temperature is raised and in others it is lowered. One of the most interesting results, however, is t h a t bismuth, which is not a superconductor, becomes superconducting when prepared as a film in this way, its transition temperature being 6°K. W h a t is just as interesting is t h a t warming the bismuth film up to about 130K destroys its superconducting properties. Here we have more evidence of the fact pointed out in the previous section t h a t even at these very low temperatures migration in the crystal lattice is possible. More recent work by BARTH(15°1 has shown t h a t the superconducting properties can be maintained even after annealing at 200°K if small amounts of other metals are deposited with the bismuth. Experiments on the superconducting transition at very high 383

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pressures (104 and above) b y JonEs and his co-workers (151,ls2~ and by HATTOl~(153, 154) have also shown that considerable shifts in the transition temperature can occur, and, once more, under these conditions bismuth becomes superconducting. (151, 152) Both these and the experiments of Buckel and Hilsch emphasize the fact that superconductivity is not a property dependent on the atom itself, but that the crystal structure plays a very large part in determining whether or not a metal is a superconductor. One of the more complicated aspects of superconductivity has been that associated with the intermediate state which arises when a transverse magnetic field is gradually applied to a superconductor. At some stage the field penetrates the specimen and parts of it become normal. This is called the intermediate state. As the field is increased further the normal regions extend until at the critical field the whole of the metal is in the normal state. The structure of the intermediate state is of great interest since there have been many theoretical estimates of the shape and thickness of the normal and superconducting regions. These are very dependent on the surface energy of the superconducting/normal boundary. Some experiments on the structure of the intermediate state were made some time ago by MESHKOVSKY and SH~L~lXOV (15s) who used a bismuth micro-probe to detect changes in the field along the specimen, b u t recently a new technique has been developed (18e) which enables the different regions to be seen very clearly. The principle is very similar to the magnetic powder method for the investigation of ferro-magnetic domains. Niobium powder (a superconductor), which has both a high transition field and temperature, is sprinkled on to the specimen surface to be investigated. A magnetic field is applied to produce the intermediate state in the specimen, and this field penetrates the specimen through the normal regions. Since the niobium is still superconducting and hence strongly diamagnetic, it is forced away from the regions where the field can penetrate and hence it only remains over the parts of the surface which are still superconducting. Thus the structure of the intermediate state is shown. Besides giving values for the surface energy, these experiments have also confirmed the theoretical prediction that the lamluae produced in a long rod are transverse to the rod when a transverse field is applied to the specimen. There is still no satisfactory theory to account for superconductivity. Both FROHLICH(156) and BARDEEN(157) have developed models but the most promising is that of BARDEEN eta/. (xTs) in which a specialized electron-phonon interaction gives rise to a state which has zero electrical resistance and which also exhibits the magnetic properties of superconductors (Meissner effect). For this state to be realized a certain condition has to be fulfilled which serves to separate superconductors 384

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from other metals. Calculations, on the basis of this theory, of the specific heat and of the penetration depth of a magnetic field agree very well with experimental results. A detailed account of the experimental basis of superconductivity is given by SERn~~lss) and the earlier theoretical work is dealt with by BARDEEN(ls9) to which papers the reader is referred for further details. THE SPECIFIC HEAT OF ~4~ETALS AT LOW TEMPERATURES

The Lattice Specific Heat BLACrZ~__~'S theoretical treatment (le°) of crystal vibrations shows t h a t in a discrete lattice the frequency distribution is dependent on the square of the frequency, n, for small values of n, but t h a t the dependence becomes greater t h a n n z for higher values of n. He shows t h a t this leads to a specific heat, c, which is proportional to T a in the n ~ region (i.e. the same as the Debye theory) but t h a t c increases more rapidly t h a n the Debye theory predicts as higher values of n are excited. I f the specific heat data are represented by the variation of the Debye 0 with temperature, this means t h a t 0 should be constant at very low temperatures (say below 0/100) b u t as the temperature is increased, 0 should decrease, pass through a minimum value and then gradually increase to a constant value at high temperatures. Such a prediction has been confirmed by recent experiments and these are illustrated in Fig. 16 by the very marked changes in 0 which are shown for zinc and cadmium. These "experiments and others by SMITH and WOLCOTT(lel, 18~)show t h a t this pattern for the variation of 0 exists for nearly all metals. They show t h a t the discrepancies between the theoretical calculations of 0 using the elastic constants and the experimental value are due in m a n y cases to the fact t h a t the experiments have not been taken to low enough temperatures and t h a t a "constant" value of 0 has been used which is really at about the mioimum in the 0 curve. The constant value of 0 from their own results does fit the theoretical calculations quite well, but this emphasizes the fact which Blackman first pointed out, t h a t one must go to about 0/100 before one can be sure of being in the true Debye T 3 region. I t should be noted t h a t experiments such as these call for very accurate measurements in the liquid helium region, because in order to calculate the lattice specific heat, the electronic contribution must first be subtracted and this can be a large fraction of the total value. One must also work with very pure metals, as m a n y anomalies which were noted in early work can now be ascribed to impure samples. I t is difficult, however, to see how impurities can account for the small humps in each of the specific heat curves of l~a, K, Rb and Cs which DAUPHI-NEE eta/. (163, 164) have found in measurements between 20 and 330°K. These are slight increases 385

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above the main specific heat curve which occur at about 200°K for sodium and at about 150°K for the other three metals. The reason for their presence is not clear, but even here the authors do not entirely discount the possibility of impurities being the cause, even though most of their specimens (particularly the sodium) were of high purity. Measurements on the specific heat of sodium at lower temperatures (1.4-20°K) by PARKII~SON and QUARRINGTON(z65) and of lead by HOROWITZ et al. (lse) show a similar variation of 0 with temperature as has been described for zinc and cadmium. Many other workers have

3oo 35c

250

Zn

e 200 150 ~

v

J j

L

, . : . - ~ c--.~-E--

f

10C

500

25

50

75

100

125

T, °K Fig. 16. The variation of 0 with temperature for zinc and cadmium from low temperature specific heat measurements. Note the very sharp drop in ~ at low temperatures (zSz~ made measurements down to I ° K so t h a t more reliable values of 0 are now available (CoR~-K et al. (le~) for Cu, Ag and Au; ESTERMANI~ et al. ~1e8) for Mg, Ti, Zr and Cr; RAYNE and KEMP ~1e9) for Cr and Ni; HILL and SMITH~1~°)for Be). I n some cases, however, 0 had not become constant even b y about I°K (SMITH and WoLco~r (le~ Hg; CLEMENT and QUINNELLa~l~ In). For the heavier alkali metals one would not expect to get a reliable low temperature value of 0 until about 0-5°K or below. The Electronic Specific Heat As the earlier article m in this series showed, there is a considerable contribution to the low temperature specific heat by the valency electrons of the metal. This, in agreement with the Sommerfeld free electron theory, gives a term of the form ? T . The value o f ? is dependent on the density of electrons g(E) at the Fermi surface and it can be written in the form ? = l~.~k~g(Eo) (14) 386

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where ~ is Boltzmann's constant and Eo is the energy at the Fermi surface. The value of g(Eo) from experimental data can be compared with that to be expected from the free electron model and from more complicated overlapping band models. Recent experiments have confirmed the earlier observations that for the transition metals the value of g(Eo) is much higher than that to be expected from the free electron theory, and that it can be accounted for by assuming an overlap of the bands which can give rise to a very high density of states ( M o r t and JONES~I~). Careful work has shown that the electronic specific heat is proportional to T even at the lowest temperatures. RAYNE c1~3~has taken measurements from I°K down to 0.1°K for Cu, Ag, Pt, Pd, W and Mo and finds a linear relation in this region. The only exception which he found was for sodium which gave an anomalous peak at 0.87°K. This peak was reproducible from one run to the next and is difficult to explain, except in terms of some impurity or crystal transition. This is a pity, because results on the electronic specific heat of sodium would be those which should agree best with the free electron theory. Measurements b y P~RKr~soN and QUARRINGTON~1651 above 1°K give no anomaly l~ut their value of ~ is about 60 per cent above that given by the Sommerfeld theory. BUCKINGHAM and SCHAFROTH(174} have suggested that an extra contribution to the theoretical value might arise from the interaction of the electrons with the lattice vibrations, but it is not yet clear whether this would account for the high experimental value. Recent work (Smith and Theodossiou, private communication) shows that the other alkali metals also exhibit values of ~ which are too high, and this gives another warning against treating them with too much confidence as free electron metals. Such a state of affairs has already been mentioned in the sections on conductivity. For the divalent hexagonal metals SMITH and WOLCOTT{161,162}show that the values of ~ for Be, Mg, Zn and Cd can be accounted for by the baud structure to be expected from an overlapping s and d band. For small axial ratios the density of states changes rapidly. This accounts for the small value of ~ for Be and the much larger ~ for Mg although the axial ratio is only slightly larger. For high axial ratios g(Eo) does not change very much, and this explains the similarity in the values of ~ for Zn and Cd. By measuring the specific heat of a series of palladium-silver alloys (up to about 50 per cent Ag) in which the Fermi level is gradually changed, HOA_~E et al. ~175~ have been able to plot the shape of the curve relating g(E) and E. There is some uncertainty in their results as the addition of the silver might affect the shape of the band. Nevertheless such work shows the important use to which specific heat measurements can be put. 387

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The Specific Heat of Superconductors Calorimetric measurements are yet another tool in the investigation of the properties of superconductors. From a consideration of the entropy change when a superconductor is made to enter the normal state b y the application of the critical magnetic field H, it is quite simple to show that when a metal becomes superconducting at its transition temperature in zero field, T0, there should be a discontinuous rise in the electronic specific heat (see, for example, SHOENBERG(91)). This change is related quite simply to the slope of the critical field curve at To. This result was first shown b y KEESOM and KOK (176) for tin, and later work has confirmed it for most of the other superconductors. The change in specific heat, Ac, is related quite simply to the slope of the critical field at Tc by the formula 4~ \ d T / T = r ~

(15)

where V is the specific volume. This expression has been found to hold very well in most cases. Below T¢ in zero magnetic field the superconducting electronic specific heat has been found to vary as T a. To calculate this from the experimental data it is necessary, of course, to subtract the lattice specific heat. This can be deduced from the normal state measurements. This T a term was very reassuring since ff this is assumed then, again from thermodynamic arguments, one can show that the curve of the critical field against temperature should be parabolic. This had in fact been found experimentally to be the case for most superconductors. The T 3 specific heat can be derived on fairly reasonable assumptions from the GORT~R and CASIMm(46~two-fluid model of superconductivity. This theory, briefly described in the earlier section on thermal conduetivity, assumes there to be two interpenetrating electron fluids, the normal electrons and the superconducting electrons. The fraction of superconducting electrons increases as the temperature is reduced. On this artificial--but surprisingly successful--model a cubic specific heat law can be deduced. There has been very much calorimetric work on superconductors to see whether the T a law is strictly obeyed, and it appears that it is a good approximation provided the readings are not taken to too low a temperature. CLEMENT and QUINNELL(177) have shown that Eqn. (15) holds very well for lead, and HOROwr,'z et al. (16e~have shown that a T 3 law holds for lead from 1 to 4°K. CLEMENTand QUI~NELL (1:1~ have also shown a T 3 relation for indium. Nevertheless there is a considerable amount of evidence that the T a law is not quite good enough at the lower temperatures. CORAX et al. (1~9) show that the electronic specific heat of vanadium can be well represented b y an exponential 388

THE

PROPERTIES

OF METALS

AT LOW

TEMPERATURES

curve. Exponential behaviour is also shown by tin below 0.7T0 ( C o B ~ and SATTERTI-FWAITE(180)) and GOODMANCxsl~ in measurements between 0.25 and 1.2°K has found a similar relation for aluminium. Such a law of the form exp ( -- b/kY) might be expected if there was an energy gap of magnitude b which defined the spacing of a condensed zone of electrons. Such a gap is predicted by the recent theory of B~DEEI~" ¢t al. ~17s~ and, as has been noted earlier, this theory agrees well with the results of specific heat experiments.* I t should be mentioned t h a t the deviations from the Y3 law which have been described have only been detected by means of calorimetric work of very high accuracy. In no small part this has been due to the use of semi-conducting thermometers to record the temperature of the specimen. In particular, the use of an ordinary carbon composition radio resistor (of a few tens of ohms room temperature resistance) has been very successful ( C L E M E N T and QUINNELL(177)). In the liquid helium range these have a very temperature dependent resistivity and good reproducibility can be obtained from one run to the next. In addition t h e y are unaffected by magnetic fields and so are particularly useful for work on superconductors. In some cases ( G O O D ~ clslJ) a layer of carbon paint has been used as a thermometer and ESTER~NN et al. u~s~ used germanium alloy thermometers. These latter thermometers have been further developed by GEBALLE et al. as2~ As the reader can see from the above, a considerable amount of information is still required about the specific heat of superconductors. In particular, measurements at very low temperatures below 1°K are important and are very difficult to obtain accurately. We have already mentioned the work of GOODM~NuS1) and RA~-NE~73) below 1°K and we should also refer to the work of S~MOILOV[183) on cadmium down to 0.3°K. These workers have all used fairly conventional techniques. Recently, however, another method has been suggested which might be of great use below 1°K. This is not a static method but involves the use of a sinusoidal power input to one end of the specimen, which is in the form of a rod. Two resistance thermometers of short time constant are spaced along the rod, the other end of which is thermally anchored. Temperature waves pass down the rod and the phase difference between the two thermometers is measured electronically. From this the thermal diffusivity (K/c d) can be calculated (K is the thermal conductivity, c the specific heat and d the density). A constant current through the heater enables K to be determined and hence c can be calculated. The advantage of the method is t h a t the specimen can remain in contact with the heat sink all the time, i.e. * Note added in proof. The existence of a n energy gap is shown m o s t convincingly by GLOVEI~ a n d TINKHAMi191) in their e x p e r i m e n t s on the absorption of micro-waves by superconductors.

389

P R O G R E S S I ~ METAL P H Y S I C S t h e p a r a m a g n e t i c s a l t for w o r k b e l o w I ° K , w h e r e a s i n t h e o r d i n a r y m e t h o d t h e s p e c i m e n m u s t b e i s o l a t e d a f t e r i t h a s b e e n cooled, or i f t h i s is n o t d o n e t h e n t h e specific h e a t o f t h e s a l t m u s t b e s u b t r a c t e d f r o m t h e f i n a l r e s u l t a n d t h i s l e a d s t o i n a c c u r a c i e s . Also t h e s p e c i m e n n e e d n o t be v e r y large in c o n t r a s t to the q u a n t i t i e s u s u a l l y required. Such a m e t h o d of m e a s u r e m e n t has been i n v e s t i g a t e d in the range a b o v e I ° K b y H o w L r ~ et al. (1~) a n d t h e y h a v e s h o w n t h a t i t is c a p a b l e o f y i e l d i n g a c c u r a t e r e s u l t s for c o p p e r a n d a l u m i n i u m i n g o o d a g r e e m e n t with other workers.* REFEREI~'CES ~1~ i~I~.cDoNAJ~U,D. K. C. ; Progress in Metal Physics, Vol. 3, p. 42, Pergamon Press, London, 1952. (2~ KLE~ENS, P. G. ; HandSuch der Physik, Vol. 14, 1956. ~s~ WILSON, A. H. ; Theory of Metals, Cambridge University Press, 2nd edition, 1953. (4) OI~EN, J. L. and ROSENBER(~, H. 1VI.; Advanc. Phys. 2 (1953) 28. ~5~ H u I ~ , J. K.; Proe. Roy. Soc. ~204 (1950) 98. c6~ B ~ C K ~ , 1~I.; Proc. Phys. Soc. Zond. A64 (1951) 681. [vJ BEP,M)_N, R. and MAcDoNALD, D. K. C.; Proc. Roy. Soc. A209 (1951) 368. (s~ ROSENBERG, H. M.; Phil. Trans. A247 (1955) 441. (9~ ZI~rA~, $. M.; Proc. Roy. Soc. A226 (1954) 436. ~1o~ P s ~ . ~ , s , R. E. ; Quantum Theory of Solids, Clarendon Press, Oxford, 1955. (11~ B E R ~ N , R.; Advanc. Phys. 2 (1953) 103. (12~ I~I~NSON, R. E. B.; Proc. CamS. Phil. Soc. 34 (1938) 474. ~1,~ Z~M.tLN, J. M.; Phil. Mag. 1 (1956) 191. ~4~ POWELL, R. L. and BL~NPIED, W. A.; N.B.S. circular 556 (1954). (la~ I~IAcDoNALD, D. K. C., WHITE, @. K. and WOODS, R . B . ; Proc. Roy. Soc. A235 (1956) 358. ~16) ROSENBERG, H. M.; Phil. Mag. 1 (1956) 738. ~7) ~ , W. R. G., KLE~NS, P. G., SREEDHAR, A. K. and WroTE, G. K.; Proc. Roy. Soe. _4233 (1956) 480. cla~ Kr.~.~r~,Ns, P. G.; Aust. J. Phys. 7 (1954) 57. (~) ; Proc. Phys. Soc. Zond. A68 (1955) 1113. ~0~ MOT'r, ~ . F.; Phil. Mag. 43 (1952) 1151. ( ~ RE.~D, W . T. ; Phil. Mag. 45 (1954) 775. ~ GEB~r.~.~.~T. H., PEarSON, @. L. and ROSENBERG, H. 1~. ; to be published. {as} ROSENBERG, IT. ~ . ; Phil. Mag. 2 (1957) 541. (~a~ See references 161, 162. ( ~ MAcDoNALD, D. K. C. a n d MENDELSSOHn, K.; Proc. Roy. Soc..~202 (1950) 523. ( ~ KE~P, W. R. G., SREEDHAR, A. K. and WHITE, G. K. ; Proc. Phys. Soc. Lond. A66 (1953) 1077. {~7} ROSENBERO, H . . ~ . ; Phil. Mag. 45 (1954) 73. [~s~ SPOHR, D. A. and WEBBER, R. T.; Conf. Phys. des basses Temps., p. 453, Paris, 1955. Phys. Rev. 95 (1954) 602. ( ~ SONDHEI~ER, E. H. and WILSON, A. H.; Proc. Roy. Soc. A190 (1947) 435. (a0~ KOHLER, 1)I.; Ann. Phys., Zpz. 5 (1949) 181; 6 (1949) 18. * Note added in proof. The temperature wave method has now been used successfully down to 0.15°K in experiments on tin by Z~V~R~S~rII.[~) 390

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L O ~ ~" T E M P E R A T U R E S

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~73~ WESSEL, E. T.; Bull. Amer. Soc. Test. Mat. No. 211 (1956) 40. (74) ; Trans. Amer. Soc. Metals 49 (1957) 149. (75) ; TO b e p u b l i s h e d ( A I M E ) . (76) BECHTOLD, 3. H . a ~ d WESSEL, E . T. ; Proc. O.N.R. Symp. on Molybdenum, D e t r o i t (Sept. 1956). (~7~ S ~ o N , L., MCMA~ON, H . O. a n d BOWEN, R. 3. ; J. Appl. Phys. 22 (1951) 177. {78| AT.T,I~.N, H. P., HOPKINS, B. E. a n d ~cLENNAN, 3. E. ; Proc. Roy. Soc. A 2 3 4 (1956) 221. (7~ McCAM~ON, R . D. a n d ROSENBERG, H. M. ; Proc. Roy. Soc. A242 (1957) 203. (s0) SEEGER, A.; Phil. ~]lag. 45 (1954) 771. (Sl; lYIADDIN, R. a n d CHEN, H. K. ; Progress in Metal Physics, Vol. 5, p. 53, P e r g a m o n Press, L o n d o n , 1955. (s2~ BL~wITT, T. H., C O L T ~ N , R. R. a n d REDMAN, 5. K. ; Conf. on Defects in Cryst. Solids, p. 369, P h y s i c a l Society, L o n d o n , 1955. (sa~ ~ D E L , 3.; Phil. ~]lag. 46 (1955) 1169. ~s4) SU~'rON, P. M.; Phys. Rev. 91 (1953) 816. (ss) OV:ERTON, W. L. a n d GAFFNEY, 3.; Phys. Rev. 98 (1955) 969. (s6; F ~ E , M. E . ; J. Appl. Phys. 26 (1955) 862. (s,~ i V I c S ~ N , H. 3.; J. Appl. Phys. 24 (1953) 988. (ss~ SWENSON, C. A . ; Phys. R~v. 99 (1955) 423. csa~ B A R D ~ N , J . ; J. Chem. Phys. 6 (1938) 364, 372. (~0~ BROOKS, H . ; Phys. Rev. 91 (1953) 1027. (91~ SHOENBEP~, D. ; Superconductivity, C a m b r i d g e U n i v e r s i t y Press, 1952. cgz) OLSEN, 3. L.; Nature, Zond. 175 (1955) 37. (~a) PIPPARD, A. B.'; Phil. itlag. 46 (1955) 1115. (94~ L A N D A ~ , 3. K. ; Phys. R,ev. 96 (1954) 296. (gs~ MEISSNER, W., POL~NYI, M. a n d Sc~r~r~, E . ; Z. Phys. 66 (1930) 477. (~a* GLEN, 3. W . ; Phil. Mag. 1 (1956) 400. ( ~ MOTT, N. F. ; Phil. Mag. 1 (1956) 568. ~s~ HOWICK, A. S.; Progress in Metal Physics, Vol. 4, p. 1, P e r g a m o n Press, L o n d o n , 1953. (~*~ BOP,DONI, P. G. ; J. Aeoust. Soc. Amer. 26 (1954) 495. (~0o~ HIBLETT, D. H . a n d WILES, 3.; Phil. Mag. 2 (1957) 1427. ~0~ MASON, W . P. ; Bell Syst. Tech. J. 34: (1955) 903. (lOa~ SEEOER, A . ; Phil. ~ a g . 1 (1956) 651. (~0s~ I~BLETT, D. H . ; D. Phil. Thesis, O x f o r d (1956). (~0a) ZAMBROW, 3. L. a n d F O N T ~ A , IV(. G.; Trans. Amer. Soe. Metals 41 (1949) 480. (lOS) SPI~TNA.~, 3. W . , FONTANA, M. (~r. a n d BROOKS, H . E. ; Trans. Amer. Soe. M e t a l s 43 (1951) 547. (~0s~ BISHOP, S. l~I., SPRETNAK, 5. W . a n d FONTANA, i~. ~ . ; Trans. Amer. Soc. Metals 45 (1953) 993. (~0~ LONG, H . HI. a n d S ~ o N , F. E . ; Appl. Sci. Res., Hague 4 (1954) 237. (~os) McCA~r~oN, R. D. a n d ROSENBER~, H. ~I. ; Conf. Phys. des basses Temps., p. 482, P a r i s , 1955. [10~ CO~RELL, A. H. a n d HULL, D . ; Proe. Roy. Soe. A242 (1957) 211. (~o~ HEAD, A. K.; Phil. Mag. 44 (1953) 925. ( n ~ ~ ¢ I c C ~ o N , R. D. a n d ROSENBE~O, H. i~I.; Phil. Mag. 1 (1956) 964. (~z~ BROO~, T. a n d H ~ , R. K . ; Prov. Roy. Soc. A242 (1957) 166. (ua~ M c C ~ i o N , R. D. ~ n d ROSENRER(~, H . M. ; Proe. Roy. Soe..~242 (1957) 203. (n4) BROO~, T., i~IoIzNEUX, J . H . a n d Wm'rrAKER, V. H.; J. Inst. Met. 84 (1956) 357.

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AT

LOV~ ~ TEMPERATURES

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IN METAL

PHYSICS

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