A general probabilistic model of electrochemical nucleation

0013-4686,83/02023747 0 3913. Pergamon

A GENERAL PROBABILISTIC MODEL ELECTROCHEMICAL NUCLEATION

SO,.W,O Press ‘ki.

OF

S. FLETCHER CSIRO,

Instilute

of Energy and Earth Resources, Division of Mmeral Chemistry, P.O. Box 124, Port Melbourne, Victoria 3207. Australia and T. LWIN

CSIRO.

Division

of Mathematics

and Statistics,

(Recricx~~ 25 February

E-0. Box 310, South 1982;

in rvcisedftirm

Melbourne,

3 Auyusr

Victoria 3205, Australia

1982)

Abstract-A general probabilistic mode1 is developed for the random time at which crystals appear in electrochemical nucleation. Using the model, formulae are found for the distribution functions, the density functions, and the moments of crystal radii during nucleation and growth. Similar formulae are found for crystal volumes and for electrical currents. The subsampling ofcrystals using small elements of electrode surface is also considered. General formulae are obtained for the moments of electrical currents observed on microelectrodes, and a special case is considered, namely, nucleation consisting of a Poisson point process along the time axis. In the final section of the paper, a model of electrochemical nuclealion is developed which takes into account the fact that active sites on electrode surfaces may have a disrribution of activities towards electrochemical nucleation LIST

OF

1. INTRODUCTION

SYMBOLS

activation rate of sites (number cm-‘5-l) densityofactivityofsites(number~’~m~s) mean value of c( in the density b(a) (number cm 2 s- ’ ) time (s) time of nucleation (s) number of nuclei activated up to time I (number cm - “) total number of sites which are potential nuclei (number cm-2) appearance rate of nuclei at time I (number cm-‘s-l) probability density function of times at which nuclei appear (s- ‘) pth power (dimensionless) growth rate of crystals at time I (cm s- ‘) radius of single cryslal at Lime r, which appwred at the earlier time r (cm) radius of a single crystal at time I, which appeared at t = 0 (cm) distribution func1ion of R,(r, t) (dimensionless) density function of R,(r, I) (cm- ‘) volume of a single crystal of radius R,(T, I) (cm3) electrical current caused by a single crystal of radius R,(r. t) (cm” s- ‘) number of crystals in a subsample at time I (dimensionless) total volume of crystals in a subsample km? total electrical current caused by crystals in a subsample (cm3 s- ‘) a constant activation rate of sites (number cm-Lsm’ ) specified quantities of various parameters (mdiczted m text). 237

Electrochemical nucleation is probabilistic in space [ 1, 21, timeC3.41, and energy[5]. Consequently, tests of electrochemical nucleation theory must be carefully devised either to take these factors into account or to minimize them by ensuring that measured quantities take their probabilistically expected (mean) values. However, since there is only a small literature on the theory of probabilistic effects in electrochemical nucleation, most workers have confined their investigations to cases where probabilistic effects were assumed to be unimportant. This is unfortunate because, as has been pointed out elsewhere[6], many new areas would become accessible to experimental investigation if the appropriate probabilistic theory could be developed. The purpose of the present work is to develop such a theory for the special case of probabilistic effects in time. Before embarking on our probabilistic treatment we first point out a difficulty with the existing theory “in the mean” which needs to be clarified before progress can be made. In order to understand this difficulty, note the conditions under which experimental data are usually acquired. Firstly, probabilistic effects in space are minimized by using large electrodes and then applying nearest-neighbour theory to deal with the moments of spatial coverage[7]. Secondly, probabilistic effects in time are minimized by observing nucleation over long time scales so that nucleation can be considered “smooth”. Thirdly, it is implicitly asiumed that heterogeneous nucleation occurs at active sites on the electrode surface, and that each of these sites has precisely the same activation energy for nucleation. Taken together, the above three conditions

238

S. FLETCHER AND T. LWIN

constitute the theoretical basis of most research on electrochemical nucleation[9]. When the mathematical counterparts of the above three conditions are considered, one is led to the standard “text-book” formulae for nucleation and growth “in the mean”[9]. However, although the first two conditions can easily be shown to minimize probabilistic effects in space and time, the third condition is really an assumption that ought to be justified for each individual experiment. In practice, of course, this is never done. The assumption that there is a unique activation energy for nucleation is thus a weakness of many present-day electrochemical nucleation studies. This becomes particularly clear when it is remembered that nucleation rates depend non-linearly on activation energiesCl0, 111, so that only a small spread in activation energies at different active sites would be sufficient to create a rather broad distribution of nucleation rates. It follows that the existing theory “in the mean” ought to be modified to take the possible distribution of nucleation rates into account, and in this paper we do this for the case where the distribution is independent of time. Having touched on the question of the distribution of activation energies at dinerent active sites, we now turn our attention to probabilistic effects in space and time. As we have shown previously[12]. a completely general theory incorporating both effects would be well nigh impossible to obtain. This followed from the fact that the derivation of the expectation value of the surface coverage for only two circular crystals of the same radius, nucleated at the same instant and distributed uniformly and independently on a circular electrode, required considerable ingenuity and a large number of error checks[l2J. For n 3 3 nuclei, the sheer tedium of calculating the full statistics of coverage was too daunting to contemplate; and if the nuclei were to appear probabilistically in time too then it can be appreciated that a completety general theory does not seem feasible. However, all is not lost because considerable progress is possible if probabilistic effects in time are considered separately from probabilistic effects in space. Although somewhat artificial from a mathematical point of view, this separation can be justified experimentally because conditions can often be Located where one or the other probabilistic effect dominates. When probabilistic effects in space were considered [I23 we found, not surprisingly, that crystal/crystal collisions and crystal/electrode-boundary collisions were the most important factors. On the other hand, in order to now develop a general theory of probabilistic effects in time, we are forced (by the arguments of the preceding paragraph) to consider that individual crystals behave independently, ie, crystal/crystal collisions and crystal/electrode-boundary collisions are assumed not to occur. Of course, this assumption is only justified for that element of the experimental time-scale for which intercrystal collisions are statistically improbable, and therefore we stress that the theory developed in the present work cannot be applied to experiments in which collisions occur. Experimentally, the nucleation and growth rates of crystals are determined by the applied electrode poten-

tial, which in turn is usually generated as a function of time according to some predetermined programme. Often, for practical reasons, the most useful potential/time waveform turns out to be a piecewise function, though usually not exceeding two pieces. This occurs, for example, with the double potential step technique or the triangular potential scan technique, both of which are widely used in electrochemical nucleation studies[9]. This being the case, we define the nucleation rate in two pieces throughout the present work. The two-piece formulation will also be of use later on, when we report the results of a computer study ofcommonly used potentialMime perturbations. Some of the statistical operations used in the present work are standard ones which can be found in textbooks on probability theory, such as that by Feller[l3]. For this reason, and for the sake wf brevity, the detailed derivations of certain formulae are omitted from the text. In the usual theory of electrochemical nucleation, it is assumed that the number of nuclei appearing on active sites is given by the first-order expression[9]

where N(t) is the observed number of nuclei (as a function of time, (t), N, is the total number of active sites, and a(t) is an activation rate of sites which depends on the applied electrode potential. Equation (1.1) is based on the foIlowing assumptions: (i) Some mechanism is operatingon the sites, such that the experimentally observed activation rate is arbitrarily close to the probabilistically expected activation rate. (ii) Only one type of active site is responsible for the appearance of nuclei, this being characterized by a

unique CL(~). The main purpose of this paper is to develop a stochastic theory for the random time of nucleation using the piecewise function

a(t) = and Eq. (1.1). For

a,(z) (I) i OL*

such a case (1.1) becomes

-j;_ that

(1.2)

(~1

N(r) =

Note

forO
(1 S)

a,Ody)l

(I.51

A general probabilistic model of electrochemical nucleation

239

which is consistent with the basic relation (1.1) Eventually, in Section 9 of the paper, we drop assumption (ii) andconsider the effect ofa distribution of active sites contributing towards nucleation.

termines their radius. Let the growth rate of the radius of a single crystal be

2. A GENERAL PROBABILITY MODEL FOR THE RANDOM TIME OF NUCLEATION

and let R,(r, t) be the radius of a single random crystal whose nucleation occurred at time r and which was observed at time t. Then

Consider have

the basic hypothesis

for 0 -z t < T, for T < f,

k, (4

k(r) =

b(t)

(3.1)

given by (1.1). We for t i T, k,(ti)du

for 0 < 1 < Co,

(2.1)

which is customarily called the “nucleation rate” of the crystals. This rate (2.1) is considered in the present work as the appearance rate of random points along the time axis. Assuming that the theoretical time-span of the whole appearance mechanism is (0, n;), we may define a probability density function of the time, z, when a nucleus p(r)

was formed. =

I

(2.2)

N(T)

and

N, = &),,=

k,@)du

for 2 c T,

kz(my

for z > T,

A (z) dz

z)

r

=

u;l(Y)

otherwise,

0

= [z: V,(z) = Yl,

(3.4) (3.5) (3.6)

From these relations it follows that

=.

(2.3)

In an experiment, a crystal observed at some time t would have a characteristic time of nucleation, which we denote by z. Evidently z < 1. Thus a more natural probability model of T is the conditional density gioen the event 0 < T < t for 0 < T < I.

p(rlO
(3.3)

Z

u,(T,

U;~(T,y)=~z:U2(T,2)=y].

T

where

=

(3.2)

c

for 0 i r < 00,

0

>T.

We introduce the following notations:

This is[4,6]

$L)

fort

(i) The distribution for t & T, H,(x)

= Pr[R,(r,

=

1

function of R,(r, t) is I) i x]

_%w;'Cw+-l$ N, (0 for 0 < x < U,(t);

(3.7)

for t > T, (2.4)

H,(x)

= Pr[R,(r,

t) 6 XI

If we further assume that the activation rate is given by the double step function (1.2), we have for 0 < t < T,

1[

=

A(t) =

N(VNN,W -. N(t) 1

forT

A,(r)/N(t)

o

for 0 < 5 < t otherwise.

(2.6)

For ! > T,

u;‘[u(r)--r]

(3.8)

h,(x) =

A I CUT’ (~11

N,W,(z)

I

.r=

for 0 < x i U,(t); u,(t)-x

(3.9)

for I > T,

p(rlO < z < t) =

A,(r)/N(r)

for 0 < T
A2(7)/N(t)

for T < 7,

(

(2.7)

(3.10)

h,(x) = OF A SINGLE

CRYSTAL

Suppose that crystal nuclei grow in the shape of hemisnheres and that some erowth mechanism de.

A,CG’(T,41 N(t)k,G) 1z= U*(T.O-X

for 0 -= x < U,(T, t)

where N(t) is as given by (1.3) 3. RADIUS

=

(ii) The density function of R,(r, f) is for t < T,

For the case of (2.5), the density (2.4) becomes: for I < T,

p(rlO
I

I

for iJ,(T, t) < x -e U(t).

A~CU~‘(Z)] {

N@)k, (2)

1 I = t/t*,-.%

i \ for Cr,(T, t) 4 x s u(t).

240

S. FLETCHER AND T. LWIN

(iii)

Expressions for the pth R,(r, Z) are obtained as: for I 6 T,

order

moments

of

ECRJL 1)l” =

4.

VOLUME

OF A SINGLE

);I$. 1. Subsamplmg

CRYSTAL

a set of crystals. It is assumed that the and that edge elects are negligible.

crystals are independent

Since the crystals are of hemispherical shape, the volume of a random crystal with radius R ,(r. I) is given

by V,(r, r) = +~[R,(T> r)]”

(4.1)

and the following results can be established: (i) The distribution function of Y,(r, t) is

Pr [ V&T, t) s x] = [H,bd],= (ii) The density

%W) = Ih(v

(4.2)

1

= (f$,“’

@,#i3

,3x)*,3

;

(4.3)

of V&T, I) is

(iii) The pth moment

= (fx)PE[R,(z,

E[ P’,(r, I)]’

5.

(+p;

of V,(r, r) is

function

t)]+.

(4.4)

CURRENT CAUSED BY A SINGLE RANDOM CRYSTAL

The current

caused

RS(q t) is given

by

P&.

t) = ;

by a random

V&T, t) -

crystal

with radius

27&(r) [R&t, I)]‘.

(5.1)

In the figure, the rectangle represents the whole space of crystals at time t. In a thought experiment circular discs all of the same size are tossed on independently and at random and the crystals trapped inside each circle are studied. This can be regarded as mathematically equivalent to using a microelectrode, or viewing a random element of surface through a microscope. The crystals are assumed to be sufficiently small for edge effects to be neglected. The number M(t) of crystals trapped inside each circle is a random variable. Thus, the total volume V*(r), of the subsample ofcrystals can be defined as the variables V,(t,, t), of the M(r) random sum which have the same distribution as V&2, t), V,(r, r). M(t) V’(f) = (6.1) v,(Tj, t). c j-l Thus E[ v*(t)]

= E[M(Q]

E[ V&z, t,l,

var [V*(r)]

= E[M(t)]

var [Y,(r,

+ var [M(r)] One can deduce VL(r, I): (i) The distribution Pr [ v:(t, (ii) The density

the

following

function

results for the current of V;(r, t) is

t) 6 ~1 = CH, (Y)] y = (*)l:z; function

(iii) The pth order

caused

by

P*(t)

=

v;t'Fj>

c

crystals

(6.4)

f)Y

j=l

E[ P(c)]

t)]?

(5.4)

var [ V’*(t)]

= E[M(z)] = E[M

OF CRYSTALS

Suppose that at time ta number ofcrystals is observed in a subsample. For an illustration, refer to Fig. I.

(6.5)

E[ T/;(r, r,],

(r)] var [ Vi(r,

+ var [ M(f)]E” 6. SUBSAMPLING

all the

can be defined as

where V$(r,. t), v;(rZ, t), . . are independently and identically distributed as P,(r, t), this latter being the currenl due to a single random crystal. We then have

of V;(T, t) is

E[ V&r, r)]” = [27rk(~)]~E[R~(r,

the total current

RZ [ V,(r, [)I. (6.3)

.44ct1

(5.2)

of V;(r, t) is

moment

Similarly,

in the subsample

(6.2) t)]

t)]

[ Y:(T, t)].(6.6)

The main assumption made in deriving the formulae (6.5) and (6.6) is that the crystals are independent of each other and of the boundaries of the sample space. As we. indicated ..__ ~._ ~~ in the ~~ introduction.

anv< relaxation

of this

241

A general probabilistic model of electrochemical nucleation assumption would lead to formidable difficulties, since we would then have to quantify the statistics of spatial coverage of the sample space using arguments in geometrical probability. (For a discussion of this problem, see [7] and ES].) 7. THE TOTAL SIZE OF AN ENSEMBLE OF CRYSTALS OBSERVED AT A GIVEN TIME At the given time t of observation, the total number of nuclei formed is

and from (2.2) the density of nucleation time r is p(r) = A exp (-AL)

A(r) dr. (7.1) s0 uolume of all the crystals observed at

Thus the total time t is

ie,

k(r) = k, for all t(0, co).

(8.7)

Then U(t) = k,t N(t)=

No[l-exp(-At)]

for 0 < t < 00,

(8.8)

for 0 <

(8.9)

and from (8.1) and (8.2):

=

H,(x)

= l-

f -c

m,

forO
1 N(t)

(8.6)

for 0 i f < 00.

Suppose also that the growth rate is constant,

* 1

1 -exp(-Ait)

(8.10)

for 0 < x < k,r. (8.11) (7.2) which is the formula of Kolmogoroff [4]. Similarly the total electrirnl current caused by all the crystals at time t is Y(C) = N(t)E[V;,r,t)]

= Zn*(11/;

[J:

k(u)du]

Next, using the above expressions in the formulae of Section 5, we get the density of current V’,(r, t) as Rexp(-_Rt)exp

w [ b,(x) = 2~)k~~2[l-exp(-&)]x”2

Ix”*

1

(8.12)

for 0 < x < 2nk~t’. x A(r)dr. 8. SPEClAL

(7.3)

CASES

8.1 Single step nucleation rate function The results for the single step nucleation rate function are easily deduced from our general model by letting A,(t) = 0 for all t and letting T ---f cc. This leads to the distribution function for R,(r, t) as

The above result is now revealed to be equivalent to Bquation (50) of Bindra et a!.[41 since .,/(2x)kz” of our paper is equivalent to C of their paper. However, note that we obtain the proper density function for the current due to a single crystal directly, without renormalization as in Bindra et al.141. This simpbfication is due to the fact that we started with the conditional approach of (2.4) explained in Section 2. The other results of Bindra et aL[4] can also be deduced from our general formulae in a similar manner. 9. NUCLEATION

and the density function of R,(T, I) as

Further, E[R,(r,

t)]” = j;

[j-:

k(u) dt$$dr.

(8.3)

Results for the volume and current of a random crystal follow from those of Sections 4 and 5 by using (8.1) and (8.2) in the general formulae given there. 8.2 Nucleation

governed

by ~1 poisson

process

Suppose that in (1.1) we insert the special case that the activation rate is constant, ie, a(y) = 2 for all y E (0, a)).

(8.4)

Then the nucleation rate is [from (2.1)] A(r) = N,Iexp(-It)

m a/i-R

for 0 < t -c 00 (XS)

ON ACTIVE

SITES

As we indicated in the Introduction, (1.1) applies only when one type of active site is present on an electrode surface and, furthermore, this one type of site is associated with a unique x(t). This is quite unlikely to be found in many experiments so (1.1) isstrictly limited in its practical applications. It is worthwhile, therefore, to generalize (1.1) to the case where a distribution of activities towards nucleation exists on an electrode surface. Consider an electrode having a large surface area, such that a set of sites exists having an essentially continuous distribution of “activities” towards nucleation. Let the activation rate of sites be a. Then define $(a).da to be the fraction of active sites having activation rates in the range a to (a + da), so that $(a) is an “activity density function” with the property : +(a). da = 1. (9.1) I On this model the electrode surface is no longer characterized by a unique a, but rather by a set of a.

242

S.

FLETCHER

This set is determined by the numbers and values of activities of sites towards nucleation. Thus a plot of $(a) us a would produce a “spectrum” of activities such as that illustrated in Fig. 2. The peaks labelled (I) and (II) in Fig. 2 might represent, for example, two different classes of active site, each class character&d by a local maximum in CL.Such local maxima in 4(a) could arise because of the presence of two fundamentally different types of defects present on an electrode surface, or they might arise because of two crystallographically separate orientations exposed on the surface of a polycrystalline material. Whatever their origin, we expect that peaks such as I and II will have a finite width, and hence a range of activities towards nucleation will exist, because of local variations in surface structure. As a matter of fact, even on a “perfect” surface containing only one type of highly reproducible site, a peak in b(a) must still have a finite width for purely quantum-mechanical reasons. Thus, any proper model for N(l) must neressari!y take into account a distribution of a, rather than a singlevalued CL

a -s

Fig. 2. An example of an activity density function 4 (a) of the activation rate GI.Roman numerals refer to local maxima.

T. L WIN

AND

and the total A(t)=z;[N(1)]

rale

of appearance =

of nuclei

YNg$(a)aexp(--olt).da. I 0

is

(9.5)

Equation (9.4) may be compared with Eq. { 1.1). It can be seen that in the distribution model N(r) is no longer simply related to N,; when there is a distribution of activities it is necessary to evaluate the weighted sutn given by the right-hand side of (9.4) in order to obtain A(t). Equations (9.4) and (9.5) could then be used in formulae (g.lk(8.3) to develop the complete theory for the distribution model. To get some idea of how the distribution model behaves, let us suppose that the distribution of ~$(a) in (9.4) is Gaussian, and compare this with the “singlevalued a” case given by (1.1). Figure 3 illustrates the density 4(a) in the two cases; we let the mean value of a be identical for both curves. In the N(f) behaviour, N(C) now rises faster for (9.4) than for (1.1) because of the upper, leading edge of the density &(a) in the Gaussian case. However, at later times, N(t) approaches N, more slowly for (9.4) than for (1.1X because of the trailing edge of the density 4(a). This result is sketched in Fig. 4. The fact that N(t) approaches N, only slowly when the distribution of a is broad accounts for the common experimental observation that the “upper limit” of N(t) is found to be potential-dependent; potential-indepenwhereas N, is, by definition, dent[5,9]. This mistake arises because, experimentally, insufficient time is allowed to elapse for N, to be properly measured.

-

0

-8

As in the simpler, “single-valued a” model, let N, be the total number of active sites on the electrode surface. Note that N, is, by our definition, independent of electrode potential since its value depends only on the structure of the electrode surface. This fact distinguishes our work from that of Markov and Kaschiev[S], who assumed that N, was dependent on potential. For a given type of site, having some specified activity, there will be a corresponding a. The total number of this one type of site will be NO 4 (a).da. The number of nuclei which appear on this one type of site is given by N,(t)

=N,~$(a).da[l-exp(-at)]

and the rate at which

these nuclei appear

$[N.(r)] = N,4(a).

da[cr exp (-

a

Fig. 3. Two possible forms of the activity density function. (A) Gaussian density (B) Unique-valued a.

(9.2) is K!r)].

(9.3)

F

Using the above two equations, we can sum over the whole range of a. Thus the total number of nuclei activated up to time t, due to all rates a, is rL.(r)=~,(ijda=~~N,,lp(=)[l-exp(-ar)Jda

Fig. 4. Number (9.4)

of nuclei as a function of time, for the densities indicated in Fig. 3.

A general probabilistic

model of electrochemical

A mathematical proof that N(t) approaches a limit not exceeding N, for a general nun-singular density 4(a) can easily be obtained. Let a, be the mean value of a. Then, from (9.4), N(t)

=iV,

l-

w $(a) 50

cNo[l [

exp

-exp(-mot)]

( - at) da

1 (9.6)

by Jensen’s inequality[ 131 since the exponential function is convex. Hence N(t) approaches N, more slowly as r --* cc for a general density +(a) than it would in the case of a singular density of +(a) which has only a unique value, a,.

243

nucleation

tained for the appearance rate of crystals. This formulation provided a reason why the “upper limit” of the number of nucleation sites observed experimentally appears potential-dependent. Acknowledgements-The authors would like to thank the Austrahan National Energy Research, Development and Demonstration Council, and the International Lead/Zinc Research Organization Project ZE-295 for financial support. Conversations with Dr D. Gates and Dr R. Tweedie, of CSIRO Division of Mathematics and Statistics, arc also gratefully acknowledged.

REFERENCES IO.

CONCLUSION

We have formulated a stochastic theory of electrochemical nucleation, for the case where crystals are independent of each other and of electrode edge effects. The theory is based on the definition of a probability distribution for the time at which nucleation occurs (z), this being related to the nucleation rate function. A conditional density of r given 0 i T < t (2.4) was defined where L was the time of observation. Using this conditional density, we obtained general formulae for the size distribution of crystals (as a function of time) in terms of any specified forms of the nucleation and growth rate functions. We also obtained formulae for the moments. We then proceeded to investigate subsampling processes, such as viewing random elements of electrode surface through a microscope, or using microelectrodes for recording data. Once again, general formulae were obtained. As an illustration of these formulae, we showed that the results of Bindra et a!.[43 were special cases of our more general model. In the final section of the paper, we considered the case of a distribution of active sites being responsible for electrochemical nucleation. A formula was ob-

1. E. B. Budevski, m Progress in Surjiiace and Membrane Scirnce (Edited by D. A. Cadenhead and J. F. Danielli)

Vol. II, pp. 71-116. Academic Press, New York 2. S. K. Rangarajan, J. elecfroana/. Chem. 46, I19 3. S. Toschev, A. Milchev and S. Sloyanov, J. Crysr. 13, 123 (1972). 4. P. Bindra, M. Fleischmann, J. W. Oldfield Singleton, Faraday Discussions Qf the Chemical

(1976). (1973). Growth

and D.

Society 56 180 (1974). 5. I. Markov and D. KaFhchiev, J. Cryst. Grnwrh 16 170

(1972). 6. M. Fleischmann, Sur+re Sri. 101, 7. A. Smith and S. 8. S. Fletcher and

M. Labram, C. Gabrielli

and A. Satar,

583 (1980).

Fletcher, Can. J. Chem. 57 1304 (1979). A. Smith, Electrochim. Acta 25 1019

(1980).

and H. R. Thirsk. in Electroanolytical by A. J. Bard) Vol. V, pp, 67-147. Academic Press, New York (1971). 10. D. Walton. J. rhem. Phys. 37 2182 (1962). II. A. Milchev. S. Stovanov and R. Kaischev.Thin SolidFilms 9. J. A. Harrison Chemistry

(Edited

22, 255 (1474).

’

12. A. Smith and S. Fletcher, Electrochim. Acra 25.889 (I 980). and i& 13. W. Feller, in An Introduction to ProbabilityTheory kpplku~ions, 2nd edn, Vol. II. John Wiley, New York (1968). Math. 14. A. N. Kolmogoroff, Bull. Acad. Ser. U.R.S.S./Sci. Nul.

3, 355 (1937).