The resistance of liquid metalsrspa.royalsocietypublishing.org/content/royprsa/146/857/465.full.pdf · We therefore suggest the hypothesis')* that in normal metals the change of resistance

465

The Resistance of Liquid Metals

By N. F. Mott, the H. H. Wills Physical Laboratory, University of Bristol

(Communicated by R. H. Fowler, F.R.S.—Received March 28, 1934)

The electrical resistance of most normal metals in the liquid state just tbove the melting point is about twice as great as that of the solid metal just >elow the melting point. Certain abnormal metals, however, such as bismuth, gallium and antimony, which are rather poor conductors in the solid state, ncrease their conductivity on melting. The purpose of this paper is to discuss his change of resistance from the point of view of the modern theory of dectronic conduction which is based on the wave mechanics, and to obtain a ormula for the change of resistance which is in quantitative agreement with experiment for normal metals. No quantitative theory can as yet be given for the abnormal metals, but we shall show that their behaviour is explicable in i qualitative way.

In a solid the atoms vibrate about mean positions which are fixed in the solids in a liquid at temperatures near the melting point, it is now generally recognized that the atoms vibrate about mean positions which, though not fixed, move slowly compared with the velocity, of order of magnitude y / (&T/M), with which the atoms vibrate. The most direct evidence for this is afforded by the specific heats of monatomic metals, which have, within the limits of experimental error (~ 7%), the same values (in the neighbourhood of 3R) for a given metal m the solid and liquid states near the melting point.* Further evidence is given by the rates> of diffusion of gold in mercury*)- (0*72 sq cm/day) or of thorium B (Pb) in non-radioactive liquid leadj (2*2 sq cm/day). If one compares these numbers with the formula for the diffusion coefficient in gases,

D = \lc ,

where l is the mean free path, and c the mean molecular velocity, one finds, on setting c equal to a quantity of the order of magnitude of y'(&T/M), that l must be taken to be about one-hundredth of the interatomic distance.

* Euken, c H andb. exp. P hysik ,’ vol. 6/1, p. 333 (1929). t Cf. L andolt-B ornstein’s 4 T ables,’ 5 th ed., vol. I, p. 249. t Groli and von Hevesy, 4 Ann. P hysik .’ vol. 63, p. 85 (1920).

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466 N. F. Mott

A theory of the viscosity of liquids has recently been given by Andrade * based on the same hypothesis.

In discussing the phenomenon of melting, therefore, and the accompanying change of resistance, we shall treat the atoms of the liquid as though they were vibrating about mean positions which remain fixed. We shall also suppose tha t each atom vibrates with a definite frequency vL, which is the same for each atom, although it is probable th a t in a real liquid the frequencies will not all be exactly the same. We shall denote by vs the frequency with which the atoms vibrate in the solid, so tha t

©s —

is the characteristic temperature of the solid, if Einstein’s rather than Debye’s formula for the specific heats is used.

We shall denote by U the work required to remove an atom at rest in the solid to a position of equilibrium in the liquid. If the temperature of melting Tm is such th a t &TM ^ Avs, then for both solid and liquid the mean energy of vibration is 3&TM per atom, so th a t U is equal to the latent heat of fusion, expressed in ergs per atom.

At the temperature TM a t which the liquid and solid are in equilibrium at zero pressure, the free energy F is a minimum. Thus, if one atom is transferred from the liquid to the solid, the change in the free energy must vanish. It is a w^ell-known result of statistical mechanics that, for a body consisting of N atoms,

F = N (—JcT log f + E),

where f is the partition function of each atom, and E the energy of an atom at rest in its position of equilibrium. Thus, if Ns atoms are in the solid phase and Nl in the liquid, the free energy is

Ns ( - JcT log /s) + ( - JcT log /L + U),

where / s, / L are the partition functions of atoms in the solid and liquid respectively. At the melting point, T = TM, this must be a minimum for any change in Ns and NL, subject to the condition tha t Ns + NL must remain constant. We obtain

JcT m log / s = &Tm log / l U,whence

f h e -V/ kT^L _ i

/ s

* 4 P h il. M ag.,’ vol. 17, p. 698 (1934).



The Resistance of Liquid Metals 407

nee, for a simple harmonic oscillator at temperatures such that &T &v, / proportional to (&T/v)3, we have*

_ e -iU/ArTM> n )Vs ’

r _ e-40(L/TM)jvs

t L is the latent heat in kilojulesf per gram atom and TM the melting point in legrees K.

According to the theory of metallic conduction developed^ by Bloch and others, electrons can pass freely through a perfect crystal lattice, which con- equently has infinite conductivity. Resistance is due to irregularities in the attice, which may be due to the heat motion of the atoms, or to the presence )f impurities or foreign atoms in solid solution. The theory shows that the resistance due to heat motion is proportional to the square of the distance through which the atoms vibrate ; at temperature T, this distance x is givenby

bfx2~ k T (//M = 4tcV )

fx being the restoring force ; hence,§ if a is the conductivity,

l/a oc x 2 oc T/Mv2

a*2T/M02, (2)where M is the mass of an atom.

The conductivity also depends on the extent to which the electrons may be considered “ free,” i.e., to the ease with which they move from atom to atom under the influence of an electric field. Bethe|| gives the following formula for the conductivity:

= 2n0 M /K dE\* 1 Jc@2 mCT 7T2 m\C dKJ hKa0 T ' U

* Cf. Ratnoweky, ‘ Verh. deuts. phys. Ges,,’ vol. 16, p. 1033 (1914), who obtains th e same formula. Ratnow sky calculates vL/vs for various m etals ; his experim ental values for L, however, are no t the same as those used here.

1* A kilojoule is 1010 ergs.% 1 Z. Physik,’ vol. 59, p. 208 (1930).§ A formula of th is type was proposed by W ien (‘ S itzBer. Preuss. Akad. W iss,’ p. 184

(1913)) before the application of quantum m echanics; Gruneisen, ‘ Verli. deuts. phys. Ges., voJ. 15, p. 194 (1913); Bridgem an (‘ Phys. R ev.,’ vol. 9, p. 269 (1917), and Beckm an ( Phys. Z., vol. 16, p. 59 (1917)) have shown th a t the change of conductiv ity under pressure da/dp can be explained for m any (but no t all) m etals, by tak ing for d S /d p a value deduced from the compressibility and therm al expansion coefficient.

|| ‘ Handb. Physik,’ vol. 24/11, p. 507 (1933).



468 N. F. Mott

Here n0 denotes the number of free electrons per atom ; M, m are the masses of a vibrating atom and of an electron respectively, and a0 is the radius of the first Bohr orbit. K denotes the wave number of an electron at the top of the Fermi distribution, and E the kinetic energy of such an electron. G is a constant depending on the interaction between the metallic ion and a free electron and is a property of the ion rather than of the crystal structure.

We must now ask which of the properties of (3) may be expected to alter when the metal melts. K depends only on the specific volume, and so will not alter appreciably. dEjdK, on the other hand, depends on the structure, and will alter considerably on melting, if the arrangement of the atoms relative to each other alters. However, for good conductors, and especially for monovalent conductors, E and K are likely to have values approximating to the values for free electrons, namely

E = \m v2, K = 2nmvjh ;

so th a t E = h2K.2/8rc2m, and

K = 2E — twice the maximum energy of the Fermi distribution. dK

Since the energy of the Fermi distribution depends only on the volume, we should not expect any great change on melting.

For strongly diamagnetic metals, such as bismuth, it is well known* that the diamagnetism and low conductivity are related to an abnormally small value of dE/dK .. Since the diamagnetism disappears on melting, we should expect an increase of dEjdK ..

We therefore suggest the hypothesis')* th a t in normal metals the change of resistance on melting is due mainly to the change in v, or 0 , discussed above. We should therefore expect th a t

T k — / _ 6 - f u / * t m _ e — 80L/t m^°s Vvs /

where L is the latent heat, in kilojoules per gram atom. Table I shows the extent to which this hypothesis is in agreement with experiment. The values

* P eierls, ‘ Z. P h y s ik ,’ vol. 80, p. 763 (1933).t S im on, c Z. P h y s ik ,’ vol. 27, p. 157 (1924), h as suggested th a t th e change o f resistance

is p ro p o rtio n a l to vL/vs, a n d h as rem a rk e d th a t th e observed change o f resistance is of th e sam e o rder o f m ag n itu d e as th e change in vL/vg deduced by R a tn o w sk y (loc. cit.).




f Tm and L are taken from the International Critical Tables,! the values of g/<rL from Gruneisen.J

'able j — CTl) cts are the conductivities L is the latent heat in kilojoules point.

Element L Tm (degrees K)

of the liquid and solid respectively ; per gram atom, and TM the melting

e (**) (\ glJ calc. \yLyobs.

A ................. 3-5 459 510 1-84 1-68fa ................... 2-65 370-5 200 1-77 1-45

................... 2-38 335-3 126 1-75 1-55tb ............... 2 18 311-5 85 1-76 1-61s................... 2-i 299 68 1-75 1-66u .................... 11-5 1356 310 1-97 2-07Ig ................... 11 1233-5 215 2-0 1-9Ln ................... 13*3 1336 175 2-22 2-28a. ..................... 8-0 933 400 2-0 1-64

................... 6 -2 593-9 168 2-3 2-0Pb ................... 4-70 600-5 90 1-87 2-07in ................... (7)* 504-8 — (3)* 2-11̂ ................... 6 - l5 580-5 96 2-3 2-0

Jn ................... 7 1 692-4 235 2-3 2-09

*g ................... 2-33 234 97 2-23 3 2 -4 -9&i....................... 10-9 544 — 5-0 0-43}a ................... 5-56 302-7 — 4-5 0-58ib ................... 19-5 903-5 — 5-6 0-67Te ................... 11*2 1808 — 1-65 —ffi ................... 18 17 1725 — 2-34 —

n ............... 22 2028 —* L not known accurately.

2-40

Table I shows also the Debye temperature 0 of some of the metals. For hi, Na, K and A1 the assumption upon which the calculation is based, namely I'm ^ ©> is hardly justified. I t would therefore be more correct to use, insteadof (1)

e^,,s/fcTM — 1 eju/*T M( 1. 1)

This formula gives rather smaller values of vs/vL than those shown in TableI. Assuming as before that <ts/<tl = (vs/vL)2, one obtains the followingvalues :—

Li Na K A1as/CTL calc, from (1) .......... 1-84 1-77 1-75 2-0

calc, from (1.1) ___ 1-50 1*58 1-67 1-8

os/crL observed .................(1-57)1-68 1-45 1-55 1-64

t Approximately the same figures are given by Euken, H andb. exp. P hysik ,’ vol. 8/1, p. 592 (1929).

t ‘ Handb. exp. Physik,’ vol. 13, pp. 28, 29 (1928) ; cf. Landolt-Bornstein, vol. 2, p. 1052.



470 N. F. Mott

The agreement with experiment is improved. On the other hand, a small correction must be applied in the other direction to take account of the fact tha t the law, cr cc 0 2/T, depends also on the assumption that T ^> ©. However, according to the formulae of Bloch, which are in agreement with experiment* as regards the temperature variation of resistance of copper and gold in the region T — 0 , this correctionf is only about 1% when T ~ 20 and 5% when T cr 0 . For lithium this correction is appreciable, and the corrected value is shown in brackets above.

For the monovalent metals, considering the considerable experimental uncertainties in the measurement of the latent heats and of crs/crL, the agreement is good, and is fair for some of the other metals. For bismuth, gallium and antimony, rrs < crL, and we must conclude that the effective number of free electrons, (dF/dK.)2, increases by a factor of about 10 when the metal melts.

Of the metals for which Gs/ctl has been measured, mercury is the only one in which the increase of resistance on melting is considerably greater than that predicted by the theory, so that we must conclude tha t dF/dK is smaller in the liquid than in the solid state. This view is supported by the fact that the admixture of a small quantity of Au, Cd, Sn, Pb or Bi increases the conductivity, whereas in most other liquid metals, as in solids, the resistance is raised by the presence of foreign atoms.

The agreement obtained between theory and experiment for the normal metals is rather surprising, since it shows tha t the extra resistance in the liquid is due to the greater amplitude of the atomic oscillations, and not, to any large extent, to the irregularity of the arrangement of the atoms, as contrasted with their regular arrangement in the crystalline solid. From this we must, I think, conclude that, in a region large compared with the electron’s wave-length (i.e., the interatomic distance) the atoms of the liquid are arranged in a regular way, as in a crystal; and also that these small crystals are not separated by sharp surfaces, as are the crystallites of a polycrystalline metal, but merge gently into one another. Any gradual change of crystalline phase would not have the effect of scattering the electron waves as they travel through the liquid. This view is strengthened by the fact recorded by Nor- buryj tha t the addition of about 1% of Al, Ni, Ag, Sn or other metals to molten copper resulted in a rise of resistance, independent of temperature, of about

* Griineisen, 6 A nn. P h y s ik ,’ vol. 16, p. 530 (1933).t cf. B ethe , loc. cit. p. 532.t ‘ P roc. F a rad a y Soc.,’ vol. 16, p. 581, fig. 6 (1920).




the same amount as for solid copper. Thus it appears that foreign atoms have the same scattering power in the solid and liquid states, showing that the disorder in a liquid is not so great that it cannot be increased by the pressure of foreign atoms.

The change of electrical resistivity with temperature has been measured for very few liquid metals. For the resistivity, p = 1/a, just above the melting point, Northrup* finds for liquid Cu, Ag, Au, and Bridgeman* for Li, Na, K,the following values :—

Li Na K Cu Ag Au

103 l ^ P / exP.......... 1,45 3-2 3-6 0-38 0-71 0-46p dT\calc...... 2-2 2-7 3-0 0-74 0-81 0-75

On the theory given above,

and hence, as for a solidp = const Tj

1 dp _ 1 p dT T (4)

The values calculated from (4) at the melting point are shown above. The theoretical formula gives a t any rate the order of magnitude of the observed effect.

On the other hand, the temperature coefficient of liquid mercury is equal to about one quarter of the theoretical value (4), and various investigators* have found that the resistance of liquid zinc is almost independent of temperature. One can only interpret these facts by assuming tha t the effective number of free electrons increases with T, which is, a priori, quite probable for a divalent metal.

[Note added in proof, May 15, 1934—While this paper was in press, a paper by Schubin (‘ Phys. Z. Sowjet Union,’ vol. 5, p. 83, 1934) has appeared, dealing with the same subject. Schubin considers that the extra resistance in liquids is due to a process in which the electrons are scattered without loss of energy, the configuration of the ions changing at the same time from one state of relative equilibrium to another. The probability of such a process is shown to be independent of temperature.

If Schubin’s explanation of the resistance is correct, the temperature coefficient should be less than that given by (4), as is in fact the case for a great many metals. On the other hand, it seems to the author that, except for the

* References in 4 H andb. Exp. P h y sik ,’ vol. 13, p. 28 (1928).



472 The Resistance of Liquid Metals

monovalent metals silver and the alkalis, it is rather dangerous to assume that the free electron number in a liquid is independent of temperature, since, according to the theory of Bloch, when an allowed zone of energies is nearly full, the number of electrons which are free to move depends rather sensitively on the structure. According to the experimental values given above, however, it seems th a t it is just for the monovalent metals th a t (4) is approximately valid.

Schubin also considers, in agreement with the present author, tha t the type of disorder which exists in a liquid is unlikely to increase the resistance appreciably.]

Summary

I t is assumed th a t the atoms in a liquid metal vibrate about slowly varying mean positions with a frequency vL ; the ratio of vL to the atomic frequency of the solid is calculated from the observed latent heat and melting point. It is shown th a t the change of resistance on melting can be accounted for by the change in atomic frequency, for normal metals ; and the bearing of this fact, on theories of liquid structure is discussed.



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The resistance of liquid metalsrspa.royalsocietypublishing.org/content/royprsa/146/857/465.full.pdf · We therefore suggest the hypothesis')* that in normal metals the change of resistance