Relaxation methods applied to engineering problems 17

R eferences

Boussinesq, J. 1904 J. Math, pures appl. (5), 10, 11—16, 363—394.Casagrande, A. 1937 J • New Engl. Wat. Wks Ass., 51, 131—172.Christopherson, D. G. and Southwell, R. V. 1938 Proc. Roy. Soc. A, 168, 317-350. Davison, B. 1936a Phil. Mag. (7), 21, 881—903.Davison, B. 19366 Phil.Mag. (7), 21, 904-922.Davison, B. and Rosenhead, L. 1940 Proc. Roy. Soc. A, 175, 346—365. Forchheimer, P. 1886 Z. Arch. Ing. Ver. Hannover, 32, 539—564.Gandy, R. W. G. and Southwell, R. V. 1940 Phil. Trans. A, 238, 453—475. Richardson, L. F. 1908 Proc. Roy. Dublin Soc. 11 (N.S.), 295—316.Terzaghi, K. 1933 ler Congr. Grands Barrages (Stockholm).Terzaghi, K. 1934a Die Bautechnik, no. 29.Terzaghi, K. 19346 Der Bauingenieur, no. 29/30.

On the specific heat of the sodium chloride crystalBy E. W. K ellermann University of Edinburgh

(Communicated by M. Born, F.R.S.— Received 21 June 1940)

Based upon previous calculations of about 280 proper frequencies of the sodium chloride lattice, these frequencies have been used to determine the specific heat of NaCl according to Born’s theory. It is shown that this number of frequencies is sufficiently large to make the determination of the frequency distribution and, therefore, of the specific heat possible. Agree­ment with experiment is obtained. It is shown in agreement with Blackman’s investigations that the deviations from Debye’s theory as found experiment­ally are due to Debye’s assumption of a v2 law for the frequency distribution instead of the distribution following from the atomic theory.

Introduction

The desirability of comparing Born’s theory of'the specific heat of crystals (1923) with experimental data and with the results of Debye’s theory (1912) has often been emphasized. Blackman’s investigations (1935 a, b, 1937) have already clearly illustrated that Born’s theory is fitted to explain the behaviour of the specific heat at low temperatures; however, the difficulties in calculating the frequency spectrum of a crystal, under assumptions on the forces which approximate the forces in a real crystal more closely than Blackman’s models, have so far been the main obstacle in carrying out an actual comparison. [Other calculations for a crystal model have been made by P. Ch. Fine (1939).]

Vol. 178. A.

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18 E. W. Kellermann

In a recent paper (1940) (in the following quoted as (K. 1940)), I have calculated about 280 proper frequencies of the sodium chloride crystal according to Born’s theory and obtained the frequency distribution curve for this lattice.

I t will be shown here that already with this comparatively small number of frequencies and in spite of other minor approximations, a calculation of the specific heat from this distribution curve gives results in satisfactory agreement with experiment and can account for the variation of Debye’s critical temperature 6 which according to Debye’s theory should be constant.

1. Determination of the frequency spectrum andSPECIFIC HEAT

I shall calculate the specific heat of sodium chloride using the results of my previous paper.

Let N(v)be a function describing the distribution (density) of proper frequencies vin a crystal, then the specific heat of 1 mole of the crystal is given by

Cv = k ^ N ( v ) E ( ^ d v . ( 1.0 )

Here E(x) is the Einstein function

- p S * - <«>

h and k are Planck’s and Boltzmann’s constants respectively, and T is the absolute temperature.

The function N(v) is normalized so that its integral taken over the whole of the spectrum gives the total number of frequencies in one mole of the crystal. I f the cell of the lattice contains as many atoms (or ions) as the chemical molecule, say s, and there are N cells per mole (N is Avogadro’s number), then there are 3 sNproper vibrations per gramme molecule, and the normalizing integral for the function N(v) is

f * 00

I N{v)dv = 3 sN(8 = 2, for NaCl). (1.2)

In general, the frequencies of a crystal can only be obtained by numerical methods, since the derivation of an analytical expression for the function N(v) is very involved. Even to obtain a good numerical approximation is difficult since the calculation of the frequencies is already tedious, and the

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determination of an unlimited number of frequencies is not feasible. At first sight, this seems to be a disadvantage of a calculation of the specific heat, based upon a relatively meagre number of frequencies. Considering the practically infinite number of waves exp r)} (cf. K. 1940) withwave vector k which represents the normal modes of vibration of a crystal, one ought to calculate a very high number of frequencies and count the number of frequencies in equal volume elements of the phase (k —) space, in order to get a really good approximation to the distribution function N{v).

I t will, however, be seen in the following th a t the present calculation is sufficient to give a relatively close approximation.

In the previous paper (K. 1940) I have calculated about 280 frequencies corresponding to 48 points of the phase space, defined in suitable units by

0 ^ ks ky kx 10, [k = (kx, k )\

hx+'ky + kg ^ 15,

kx, ky, kz, all odd or all even.

Since from reasons of symmetry, each vector k in this interval determines a further number of frequencies in the whole of the phase space (the average weight factor is about 24), the number of frequencies calculated determines about 6700 frequencies in the whole of the phase space. As already mentioned there, the distribution curve plotted previously (K. 1940, p. 547) contained a slight error in the counting of the frequencies (more precisely, in the weight factor of several frequencies). I have, therefore, plotted the corrected distri­bution curve here (figure 1) in otherwise the same way as in the former paper.

The features of the curve are essentially the same as those of the older graphs. The main deviations are a slight shift of some of the maxima, as given below in detail, and a change of the normalization constant 6N x 0*32 x 10~13 sec., instead of 6 Nx 0*33 x 10_13sec. For small frequen­cies the distribution function is proportional to v2 (N(v) = Kv2), and the constant K is here found to be

K = M x 10-38 sec.3

There might be another small maximum at the end of the spectrum. But the number of frequencies calculated is not large enough to enable us to draw definite conclusions.

In the main one finds again (cf. K. 1940) three definite maxima (figure 1). The frequencies belonging to longitudinal optical waves (l.o.), to longitudinal

On the specific heat of the sodium chloride crystal 19

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20 E. W. Kellermann

acoustic (l.a.), transverse optical (t.o.), and transverse acoustic waves (t.a.) have been collected separately (figure 2) as before (K. 1940, p. 547). I t is seen that the main maximum at the frequency of the residual rays in

1-8 2-4 flfo 3-6 5-4 60

0) = 2 TTV10 - 13

Figure 1. Frequency distribution of the sodium chloride crystal. vR, frequency of residual rays.

2-4 [34) 3 6

(O — 2 7TVX 10~13

F igure 2. Frequency distribution, analysed. vR, frequency of residual rays.

figure 1 is due to the two maxima (t.o.) at w = 2-85 x lO^sec.-1 and (l.a.) at o) = 3-Ox 1013sec.-1 respectively, as compared with 2-85xlO 13 and 2*9 x 1013sec._1 in the older paper. Two more maxima, (t.a.) and (l.o.), are

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found, one at o) = 1*9 x 1013 sec.-1 and the other a t 4*1 x 1013 sec.-1 (previously: 1*8 x 1013 and 4*2 x 1013). For our purposes the maximum at a) = 1-9 x 1013sec.-1 is the most interesting, since it shows clearly the deviation of the spectrum of a real crystal from a Debye distribution which is a v2 curve going up to a maximum near the residual rays frequency where it is cut off.

I t is encouraging to find good agreement with Blackman’s calculation (1937). Blackman has calculated the frequencies corresponding to 30,000 points of the phase space for a simplified model of a crystal. I have not calculated so many frequencies, but Blackman’s arguments, which indicate tha t it is not necessary to calculate too large a number of frequencies, also hold in my case, quite apart from the agreement of this calculation with experiment.

In addition to Blackman's arguments (1937, p. 423), one has only to consider the graphs of the frequencies as functions of the phase vector k (K. 1940, p. 543). I t is seen tha t all these curves are smooth. Now the density N(v)is proportional to the volume in phase space between two surfaces of constant frequency and, therefore, will also be a smooth curve.

The density curve as calculated by Blackman is very similar to the curve plotted in this paper. Both graphs show the same characteristic property, namely, the four maxima. In neither case can the possibility of the presence of further maxima be ruled out, but again in neither case is the accuracy of the calculation high enough to make a definite statement possible. However, the contribution of such maxima to the specific heat can safely be neglected.

I have calculated the specific heat partly numerically and partly graphi­cally. The Einstein function has been tabulated (Nernst 1918; Landolt- Bornstein 1927), and the values of the distribution function N(v) have been taken from figure 1. In this way the function N(v) E{hvjkT) has been plotted and integrated by means of a planimeter for various values of T. I have calculated Gv for six temperature values with 10° interval in tha t region where a deviation from Debye’s theory is observed. The results are given in table 1, the units being cal./°K.

On the specific heat of the sodium chloride crystal 21

Table 1

T20 015730 0-57740 1-1950 1-8460 2-45

100 4-12

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22 E. W. Kellermann

2. Comparison with experiment

In order to compare the calculated values of the specific heat with experimental results, and, a t the same time, to illustrate the deviations from Debye’s theory, I have plotted a 0 curve (figure 3), i.e. instead of plotting the Gv values directly against the temperature, I have determined a t various temperatures the characteristic temperature 0, belonging to the Cv values according to Debye’s theory. The 0 value a t zero temperature can be calculated without the knowledge of the frequency spectrum (Born and Karman 1912).

One finds 9 = (2-0)

where Q is the atomic volume and v is the mean velocity of propagation of the waves in the crystal. At low temperatures only long waves need be considered, which means that one can use the elastic data for the calculation of the mean velocity v. An approximation method for the calculation of v has been given by Hopf and Lechner (1914; cf. also K. Fuchs 1936). Using the elastic constants as calculated in the preceding paper (K. 1940) one finds for NaCl

0 = 313°Kat zero temperature.

Results of experiments on the temperature dependence of 0 for NaCl have been published by Clusius, Goldman, and Perlick (cf. Keesom 1934). While a t high temperature 0 has a constant value (0 = 281°K), theseauthors have found a minimum of 0(0 = 275° K at = 40°) and a rise again for still lower temperatures (0 = 310° K at T = 10° K). Various suggestions have been put forward to explain this deviation from Debye’s theory. The results of this calculation, however, furnish a conclusive proof th a t Blackman’s explanation has been correct. The failure of Debye’s theory to account for the temperature dependence of 0 is due to the assumption of a law for the frequency distribution instead of a spectrum of the kind as plotted in figure 1, which conforms to the actual properties of a crystal.

The curve of the calculated 0 values (figure 3) coincides with the curve obtained from the measurements by Clusius and collaborators between T = 10° and T = 40° K. I have been unable to find experimental data, measured between T = 40° and T = 100° K, i.e. for the right-hand side of the minimum, but it is seen tha t 0 approaches the constant value 281° K which is found at higher (room) temperatures.

This agreement with experiment is rather remarkable. I t not only suggests that the number of frequencies calculated is large enough for the

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determination of the spectrum, but it also justifies some other approxima­tions in the preceding calculations (K. 1940), for instance the neglecting of polarizability and forces, other than Coulomb forces, between next nearest neighbours. The only constant entering the calculations is the compressi­bility k in the constant A of the repulsive potential (K. 1940, p. 531). Actually the experimental value of k, as measured a t room temperature, has been used. But even if this value is corrected, assuming a temperature coefficient of k of about 10-15 cm.2/dyne x degree for dic/dT, the final results do not change appreciably, as can be seen by recalculating O (2.0) for = 0 with the corrected value of k . The effect of this, as of the other approxima­tions, is within the limits of error of the calculation.

On the specific heat of the sodium chloride crystal 23

T{° K)

Figure 3. Temperature dependence of The smooth line is drawn from experiments. O, calculated values.

The result of this calculation makes it desirable to calculate the frequency spectrum of KC1 or other crystals for which more experimental data are available. I f the coefficients due to the electric forces, as tabulated (K. 1940), are used in the equation of motion and the same method for the actual calculation of the frequencies is applied as used there, this would not be too arduous.

I wish to thank Professor M. Born for the interest he has taken in this paper. I am also grateful to Dr K. Fuchs for many helpful discussions.

R eferences

Blackman, M. 1935 a Proc. Roy. Soc. A, 148, 365. Blackman, M. 19356 Proc. Roy. Soc. A, 149, 117. Blackman, M. 1937 Proc. Roy. Soc. A. 159, 416.Born, M. 1923 Atomtheoriedes Leipzig. Bom, M. and Karman, Tin 1912 Phys. Z. 13. 297.

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24 E. W. Kellermann

Debye, P. 19x2 Ann. Phys., Lpz., 39, 789Fine, P. Ch. 1939 Phys. Rev. 56, 355.Fuchs, K. 1936 Proc. Roy. Soc. A, 153, 622.Hopf, L. and Lechner, G. 1914 Verh. dtsch. Phys. Oes. 16, 643.Keesom, W. H. 1934 Phys. Z. 35, 939.Kellermann, E. W. 1940 Phil. Trans. A, 238, 513.Landolfc-Bornstein 1927 Physihalisch-Chemische 1. Erg. Bd. Berlin.Nemst, W. 1918 Die theoretischen und Grundlagen des neuen

Wdrmesatzes. Halle.

The kinetics of the polymerization of isoprene on sodium surfaces

B y J . L. B olland

British Rubber Producers’ Research Association,

{Communicated by Eric K. Rideal, F.R.S.— Received 4 November 1940— Revised 19 December 1940)

The kinetics of the polymeri zation of isoprene liquid and vapour occurring on sodium surfaces have been investigated at 60 and 25° C. In the case of the liquid phase polymerization diffusion of monomer to the catalytic surface is shown to be the rate-controlling factor, except when the sodium surface is in the form of a sphere of sufficiently small dimensions.

The kinetics of the polymerization in presence of toluene are consistent wuth the view that polymeric chains are initiated by formation of free radicles on the sodium surface, the subsequent propagation occurring while the polymer is still attached to the sodium: termination occurs exclusively by interaction with toluene. In absence of toluene an alternative termina­tion reaction, leading to cross linking and requiring 4-6 kcal. more activation energy than the process involving toluene, comes into play.

I ntroduction

In recent years an increasing amount of study has been devoted toward gaining a better understanding of the fundamental nature of polymerization reactions. Perhaps the most successful method of approach so far employed has been the systematic study of the kinetics of a wide variety of polymeriza­tions, thermal, catalytic and photochemical: As yet, however, little atten­tion has been directed toward one very important type of polymerization, namely, the polymerization of conjugated compounds in presence of alkali metals. The kinetic method has therefore been applied to the polymerization of isoprene on sodium as a representative example of this class of reactions.

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