(*Science Progress*, * v. 27*, 1933, pp. 634-649)

LOSCHMIDT'S NUMBER

By S. E. VIRGO, M.Sc.

Physics Department, University, Sheffield

LOSCHMIDT's number, N, is defined as the number of atoms in a gram-atom or the number of molecules in a gram-molecule.

This number is frequently referred to as "Avogadro's Number," the term "Loschmidt's Number" being then reserved for the number of molecules in a cubic centimetre of a gas under standard conditions. Unfortunately, these designations are often interchanged. Avogadro's important hypothesis on the identity of the numbers of molecules in equal volumes of different gases at the same pressure and temperature was formulated in 1811, and is appropriately associated with his name; but Avogadro made no quantitative estimate of either of the above-mentioned constants. The first actual estimate of the number of molecules in one cubic centimetre of a gas under standard conditions was made in 1865 by Loschmidt, and from this the number of molecules (atoms) in a gram molecule (atom) was later evaluated. From the quantitative view-point it thus seems preferable to speak of "Loschmidt's number per gram-molecule (atom)," and of "Loschmidt's number per cubic centimetre," as is almost invariably done in the German scientific literature. This terminology avoids ambiguity, and has been adopted here.

More than eighty different experimental determinations of this number have been made [1], and as it is a basic atomic constant its most probable value is of great importance in atomic physics. It is, therefore, the purpose of this article to outline the main methods by which Loschmidt's number has been evaluated, and to give some indication of current opinions of its most probable value.

Although as early as the seventeenth century Boyle had attempted to estimate the size of atoms, it was not until 1865 that the first successful attempt was made, by the Viennese physicist Loschmidt, to calculate the number which bears his name. This number is, by virtue of its definition, the same for atoms and molecules of all kinds. Though molecules may vary in size, shape and mass, the number of molecules in a gram-molecule is a universal constant for all solids, liquids and gases, elements and compounds.

Loschmidt's method was based on the kinetic theory of gases, which had been developed with great success largely by the efforts of his contemporaries, Maxwell and Clausius.

In the kinetic theory, the molecules of a gas are supposed to be hard elastic spheres moving rapidly and unceasingly in all directions. When two molecules collide, a redistribution of energy takes place, and the molecules rebound in different directions with new velocities. The movements of individual particles are governed solely by chance, and from these assumptions Maxwell was able to deduce that the viscosity of a gas is given by the relation :

where ρ is the density of the gas,

υ is the average velocity of the molecules,

and l is the mean free path, or average distance between two impacts.

He was also able to show that if each molecule is a sphere of diameter σ , then :

where N is Loschmidt's number, and V is the gram-molecular volume.

Since η and ρ are known with
accuracy, these two equations can be solved for Nσ ^{2},
but to find either N or σ , one more equation is still
necessary. To obtain this, Loschmidt assumed that if a gas were condensed, its molecules
would be very closely packed so that volumes of the molecules in a gram-molecule could be
regarded as approximately equal to the volume of the liquid formed on condensation. This
gave the third equation for finding N. The result is really a lower limit for the value of
N, because it is very dot doubtful whether the molecules of a liquid at ordinary
temperatures can truly be regarded as small spheres in actual contact; it is more probable
that they possess some slight freedom of movement , which would make them seem to be
slightly larger in volume. The most reliable data give, for mercury, N>44 x 10^{22}.

Van der Waals' equation leads to a more exact relation between N and σ , since the constant b in that equation is of the form:

From this Perrin has found for mercury N = 62 x 10^{22}, while
Ghose has obtained the same number for helium.

The hypotheses on which the kinetic theory rests render these calculations liable to considerable error, although the molecules of monatomic gases are most likely to approximate to perfect spheres, and thus give the most reliable results [2].

Brownian Movement

More direct methods of finding N have been developed from the Brownian movement in liquids. The random movements of tiny particles suspended in a liquid strongly resemble the supposed movements of gas molecules in the kinetic theory. In fact the motion is due to impacts between the visible particles and the invisible molecules, and for feeble concentrations the pressure p exerted on the walls of the vessel by the particles obeys the gas law, if it is written in the form

where R is the gas constant per gram-molecule,

c is the concentration in grams per c.c.

and m is the mass of a particle.

It follows, therefore, that the particles of a colloid in dilute suspension should exhibit in equilibrium a similar statistical distribution to the molecules of a perfect gas in equilibrium, under its own weight. In other words, if n and n are the numbers of particles present at any instant in equal areas of two horizontal layers in the liquid, one at a height h above the other, then:

(4)

provided that n and n_{o} are very large numbers.

If measurements are made of the displacements which a number of uniform particles undergo in any given direction in a second, they will have widely differing values, since they are subject to the laws of chance. But, as Einstein showed in 1905, the mean square displacement, , will have a very definite value connected with the viscosity of the liquid η , and the radius of the particles a, by the relation:

(5)

He also showed that if D be the coefficient of diffusion,

(6)

Each of these equations has been used to determine Loschmidt's number, In 1908, Perrin
commenced a series of exhaustive researches with very uniform gamboge emulsions, obtained
by fractional centrifuging. In succeeding years he varied the conditions of the experiment
within wide limits, and in 1911 he published the results of counting over 13,000 grains at
four different levels in a very uniform suspension. From equation (4) he obtained N = 68.3
x 10^{22}. By measuring the displacements of 1,500 gamboge particles under a
microscope, equation (5), N = 68.5 x 10^{22}, in good agreement with his other
result; but Brillouin, working under his direction, measured the diffusion coefficient of
the granules, and his results give only N = 44 x 10^{22} [3].

A number of other workers have performed similar researches. The most reliable
experiments on vertical distribution were performed by Westgren in 1914 [4] on the
vertical distribution of extremely fine gold sols; he obtained N = (68.5 ± 0.2) x 10^{22}. From Einstein's displacement formula
Svedberg, also working with gold sols, obtained N == 60.8 x 10^{22} [5], while in
1923 Shaxby found N = 60.8 x 10^{22} from uniform cultures of staphylococci [6].

Three or four workers have restricted themselves to examining the movements of one
particle only. When numerous particles are observed, no matter how nearly uniform the
emulsion, the radius a is necessarily a mean value but when a single particle is studied,
the radius a is a definite quantity. Nordlund devised a method of recording on a
photographic plate travelling in a horizontal direction, the position, at regular
intervals of time, of a minute mercury particle falling in water. The photographs, one of
which is shown in Fig. I, show beautifully the influence of both gravity and the Brownian
movement ; and by comparing these two effects, Nordlund obtained = 59.1 x 10^{22}.

The main diffusion results are: Svedberg, 58 x 10^{22}; Westgren, 65.5 x 10^{22},
(both observers used gold sols; and Shaxby, 59 x 10^{22} from the diffusion of
cocci.

Constantin has shown that more concentrated emulsions obey a distribution law based on
van der Waals' equation, and thus obtained N = 60 x 10^{22}

The Brownian movement has also been studied in gases. Einstein's displacement formula
has been applied to the movements of charged mercury globules falling in air by Fletcher
[7], who obtained N = (60.3 ± 0.3) x 10^{22}; while E.
Schmid [8] found from similar experiments on selenium particles N = 59.3 x 10^{22}.

An ingenious variation has been recently devised by Kappler. When a thin quartz fibre
with a mirror attached to its lower end is suspended in air at ordinary pressures, no
marked movement of the mirror is noticeable. When, however, the pressure is reduced so
that the effect of each individual air molecule impinging on the system plays its part,
the mirror exhibits frequent irregular oscillations, which can be detected by means of
light reflected from the mirror, and recorded on a photographic plate moving parallel to
the axis of the wire. A typical plate is shown in Fig. 2. From the Brownian movement of
quartz fibres, Kappler obtained N = 60.5 x 10^{22 }[27].

Although, with the exception of Perrin's work, the Brownian movement results are extremely consistent, the chief criticism seems to be that, since they all depend on measuring a statistical mean, they are all open to a probable error of , where n is the number of observations taken. This means that in order to obtain a result with a probable error within ± 1 percent, at least 20,000 observations must be made; or to obtain a probable error of 0.1 percent, no less than 2 million readings are needed! This is quite irrespective of any other incidental errors.

Fluctuations

Loschmidt's number has also been obtained by measuring other fluctuating quantities
[9]. Density fluctuations, necessarily a molecular phenomenon, gave N = 6o x 10^{22}
(Constantin); critical opalescence yielded the value 75 x 10^{22} (Keesom and
Onnes), and the critical miscibility of two liquids gave N = 77 x 10^{22} (Filrth)
and N = 62-65 x 10^{22}, (Zernike). These results are liable to large errors, and
from our point of view can only be regarded as confirming the order of magnitude of N.

Linked with fluctuation phenomena is the explanation of the colour of the sky, why Rayleigh attributes to the scattering of the incident light by the individual air molecules in the atmosphere.

If E_{o} is the intensity of the incident light of wavelength λ and E is the intensity of the beam scattered in a
direction at right angles to the primary beam, then

(7)

where h = 22.42 x 10^{3} x (8)

In these equations, s is the total length of the path traversed by the beam between the two points where the intensity is measured, and μ is the refractive index of the medium.

If the intensity is measured at regular intervals during the day, the corresponding
values of s may be regarded as proportional to sec A, where A is the zenith distance of
the sun. The experiment of course requires a pure atmosphere free from dust and water
vapour. The ratio E/E_{o} is generally obtained by means of a spectrophotometer.
Dember [10] at Teneriffe found N = 64 x 10^{22}, and Pacini [11] obtained N = 62 x
10^{22}. Cabanes [12] studied the scattering of light in specially purified air
and argon; with the former he obtained 55 x 10^{22}, and with the latter, 69 x 10^{22}.
These experiments involve measuring photometrically a ratio of the order of 10 million:1,
a matter of no small difficulty ; they can hardly be regarded as leading to precise values
of N.

Fowle [13] has modified the method so that, instead of measuring E/E_{o}
directly, he correlates it with the quantity of water vapour present in the atmosphere,
and then measures the atmospheric transmissibility. This process, though indirect, is very
ingenious, and the result, which was obtained from readings taken over three years, was
given as N = (6o.6 ± .4) x 10^{22}. It is probably the most accurate value of
Loschmidt's number at present obtained by means of Rayleigh's law.

Radioactivity

More direct methods have arisen from a study of radioactivity. It is well known that α particles are doubly ionised helium atoms, and that only one α particle is emitted from any one atom in the transformation of a radioactive substance. Radioactive decay is therefore a process in which the behaviour of single atoms may be studied, and the observer who counts α particles emitted during the decay of a radioactive substance is actually counting individual atoms.

All the radioactive methods of finding Loschmidt's number depend on a value of Z, the number of α particles emitted per second by 1 gram of radium. Z was first measured in 1908 by Rutherford and Geiger, who allowed α particles from a standardised radium-C preparation to pass down a tube about 450 cm. long, through a mica window, and excentrically into a cylindrical vessel with a central insulated wire electrode. This wire was connected to a quadrant electrometer, while the outer case was raised to a high negative potential. When the gas pressure in the counter was reduced to 2 or 3 cm. of mercury, the influence of each α particle entering it was magnified by ionisation by collision with the molecules of the residual gas.

Thus the entry of each α particle into the counter produced
a ballistic deflection of the electrometer. By counting the deflections produced by a
radioactive preparation of known strength, Rutherford and Geiger found Z = 3. 4 x 10^{10}. In 1918, Hess and Lawson
[14], using a greatly improved counter, obtained Z = 3. 72 x 10^{10}. More recently, in 1929,
Ward, Wynn-Williams and Cave [15] designed a modification of the apparatus in which the
ions produced by the α particles were collected on a disc
connected to the grid of the first valve of a five-valve circuit. The impulses could then
be detected by means of a loudspeaker or recorded on a photographic film and counted at
leisure. From a count of 92,000 particles they concluded that Z = 3. 66 x 10^{10}.

Rutherford and Geiger also devised a means of measuring the charge transported by the α-particles. If the charge carried per second by the α-particles. If the charge emitted from 1 gram of radium is Q, since each α -particle is a doubly ionised helium atom:

where e is the electronic charge.

The most recent and probably the most reliable value of Z by this method was published
by Braddick and Cave in 1928 [16]. By assuming Millikan's value of e, they obtained Z = 3. 69 x 10^{10}.

The mean of the results of Hess and Lawson, Ward, Williams and Cave, and Braddick and
Cave is Z = 3. 69 v 10^{10}, and
although for several years physicists seemed generally inclined to favour a lower value,
3.4 x 10^{10}, it now seems probable that the higher value cannot be far from the
truth.

To obtain N, however, more information is necessary. N could be found if we knew the volume V of helium in c.cs. at N.T.P. produced by 1 gram of radium per second, for then:

(9)

Now the determination of V is a matter of extreme difficulty. The volume of helium
given off by 1 gram of radium in equilibrium with its decay products is about 150 mm^{3}
per year, but as it is not usually possible to work with more than a small fraction of a
gram of radium, the volume actually collected in an experiment is considerably less. A
greater source of trouble is the part played by occlusion by the walls of the tube. Dewar
[17] observed the growth of helium from two radium preparations, one experiment continuing
for six weeks, and the other for two months. The method involved observing the increase in
the gas pressure due to the helium generated in an apparatus of known volume, and applying
Boyle's law. His mean value was V = 0.48 mm^{3} so that N = 6o x 10^{22}.
Boltwood and Rutherford [18] performed similar experiments; their mean result gives V =
0.43 mm^{3}, from which N = 66 x 10^{22}. These numbers can hardly be
regarded as accurate to more than the first significant figure.

A second radioactive method of finding Loschmidt's number arises from combining Z with λ , the decay constant of radium. λ has
been found by Ellen Gleditsch from a careful study of the decay of four ionium
preparations as 4.14 x 10^{-4} year^{-1}. This gives N = 63.5 x 10^{22}.

The third radioactive method of finding N depends on the volume V, of 1 curie of radon
at N.T.P. If λ _{r}, is the decay constant of
radon, then:

(10)

is known with a fair degree of accuracy as 0.192 day^{-1} but the determination
of V, is again not an easy matter. The most reliable method was designed by Wertenstein,
who measured the change in pressure in the apparatus due to purified radon from a
standardised radium preparation by means of a decrement gauge. His result, V_{r}
= 6.39 x 10^{-4} c.c., leads to a value of N = 61.6 x 10^{22}, which is
probably as reliable as any hitherto obtained by a radioactive method. But neither Z nor
V, is known so precisely that the resulting value of Loschmidt's number can be regarded as
having a probable error of much less than 1 or 2 per cent.

More Accurate Methods of Finding Loschtmidt's Number

It is interesting at this stage to review our position. Loschrnidt's number has been
determined by studying widely different phenomena, and there is remarkable agreement
between the results. Indeed all the methods indicate a number in the region of N = 61-62 x
10^{22}. But none of the methods we have yet discussed can be regarded as a
precision method, mainly because in each case the argument turns at some stage on
estimating a statistical average, which to be accurately calculated needs a very large
number of observations. The most direct of these methods are undoubtedly based on the
Brownian movement, and a mean of the results of Nordlund, Svedberg, Shaxby, Westgren,
Schmid, and Fletcher (giving equal weight to each observer), leads to N = 60.0 x 10^{22},
which ought to be very near the true value of Loschmidt's number.

It seems probable that the most exact value of Loschmidt's number is to be obtained by some method which does not depend on the measurement of a fluctuating quantity, the individual values of which are distributed purely by chance. Four methods which satisfy this condition have been devised.

The oldest is based on the relation of Faraday's constant to electronic charge. When in 1833 Faraday formulated his famous laws of electrolysis, he gave to the world the first indication of the atomic nature of electricity; for by his laws a gram-atom of every substance is always associated with the same quantity of electricity (or at least, a small integral multiple of the quantity associated with each monovalent element), whence each atom must always bear the same elementary charge e, or some simple multiple of it. In other words :

Ne = F

where F is Faraday's constant, or 9649.1 ± 0.1 international em.u. The history of the
various methods of finding e, from Townsend's pioneer work (e = 3 x 10^{-10}
es.u.: N = 96 x 10^{22}), through Thomson's modification and Wilson's
improvements, up to the classical work of Millikan, is too well known to be repeated here.

In 1917 Millikan gave the final results of his experiments as e = 4.774 x 10^{-10}
es.u. and N = (60-62 ± 0.06) x 10^{22}. His experiments show remarkable
consistency; for several years his work was examined for flaws by some of the keenest
brains in Europe, and it has withstood every adverse criticism. Every link seemed
substantiated by experimental evidence, and for over ten years Millikan's 1917 value of N
was accepted by physicists as the best obtained at that time. In 1929 however, Birge [20]
showed that a small correction was necessary, as Faraday's constant is given in
international coulombs (based on the definition of an ampere as that quantity of
electricity which, under standard conditions, will deposit 0.00111827 grams of silver per
second), and Millikan's p.d. was read on a voltmeter calibrated in international volts.
Now although the international units were originally intended to be exactly equal to the
corresponding absolute units, and indeed seemed so when they were defined, more recent
work wnhas shown that there are, in some cases, slight discrepancies; for instance:

1 international coulomb = (0.99995 ± 0.00005) absolute coulombs and 1 international volt = (1.00046 ± 0.00005) absolute volts.

These differences are slight enough, but they have changed Millikan's value to N =
(6o.64 ± 0.06) x 10^{22}.

Birge recalculated e by using Millikan's readings and weighting them differently. The result is practically identical with Millikan's, and undoubtedly Millikan's value must be considered among the most reliable determinations of Loschmidt's number.

It is not generally known that other experiments of this type have been performed [21].
In 1914 Lee found N = 60.8 x 10^{22} using Millikan’s apparatus wheras in
Switzerland in 1913, Schidlof and Mile. Murzynowska, working with a smaller condenser and
a p.d. of only 100 volts (compared with Millikan's 3,000 volts) found e = 4.738 x 10^{-10}
and N = 61.1 x 10^{22}. Schidlof and Karpowicz obtained in 1915 e = 4.82 x 10^{-10}
and N = 6o.o x 10^{22}, while in the same year Targonski obtained e = 4.68 x 10^{-10},
giving N = 61.9 x 10^{22}. Though not so exact as Millikan's, these results are in
good agreement with his and strongly enhance his value. Indeed a mean (unweighted) of all
balanced droplet results gives 6o.85 x 10^{22} which is within one-third per cent.
of Millikan's estimate.

The second method, which is independent of fluctuation phenomena, was devised by duNoò y [22], who made a careful study of the surface tension of dilute solutions of sodium oleate. His results are shown graphically in Fig. 3.

The dotted curve was obtained by measuring the surface tension directly after preparing the solutions. The continuous curve was obtained from observations taken after the solutions had been allowed to stand for two hours. This curve exhibits three well-defined minima, which can only indicate unique arrangements of the molecules of the dissolved substance. It is well known that the dissolved molecules in very dilute solutions tend to form a monomolecular layer covering the whole surface of the liquid. DuNoò y supposes that if the concentration were so adjusted that in a given sample all the dissolved molecules were adsorbed in the surface layer, then the surface energy would be a minimum, and would give rise to a corresponding minimum in the surface tension.

There are three possible methods of packing the molecules in the surface layer, the thickness of the layer corresponding to the depth, and height of the molecule respectively. Strictly speaking the terms "length," " breadth," and "height" can hardly be applied to molecules, which are probably very irregular in shape; but here we are considering not the actual molecles, but the space they occupy when symmetrically packed with others.)

Let ρ = density of the dissolved species

A = area of the adsorption surface

V = volume of the solution under examination

And *l _{I}* = thickness of the adsorbed layer corresponding

Then the total volume of adsorbed substance = A*l _{1}* =

(11)

Similarly for the other two minima:

(12)

Hence the number of molecules per c.c. =

and if M = molecular weight of the adsorbed substance,

(13)

For sodium oleate, duNoò y found N =
(60.04 ± 0.09) x 10^{22}. The principle of the method
is extremely simple. The chief criticism seems to be that there is at present no evidence
to justify the assumption that all the dissolved molecules are adsorbed in the surface. If
this is not true, the calculated value of N is probably lower than the true value. At
present du Noò y work is unique, but the
simplicity of the method makes it probable that, in the future, it will lead to very
accurate values of N.

The most recent experimental values of Loschmidt's number rest on the discovery by Doan and Compton that X-ray wavelengths may be determined directly by means of ruled diffraction gratings. Before this discovery, all X-ray spectroscopy was based on the reflexion of X-rays at crystal surfaces, first observed by Sir W. H. and W. L. Bragg. They showed that reflexion takes place when the experimental arrangements satisfy the relation:

nλ = 2d sin θ

where λ is the wavelength of the incident beam,

θ the angle of is the glancing angle (the complement of incidence in optics),

and d is the lattice constant of the crystal.

A more exact form of the relation takes into account the refraction of the X-rays at the crystal surface. This is

(14)

where μ is the index of refraction.

Equation (14) connects λ and d. To determine either quantity one more equation is necessary. This involves Loschinidt's number, for in rock salt of molecular weight M and

density ρ :

Actually, for accurate work, rock salt has been superseded by calcite, and the corresponding equation is

(15)

where K is the number of molecules in an elementary unit of the crystal and β is a quantity which depends on the slope of the crystal faces.

Until recently all X-ray wavelength measurements were based on equations (14) and (15) and an arbitrary value of N; Millikan's was generally selected. But if an X-ray wavelength could be found by an independent experiment, Loschmidt's number could then be calculated.

Doan and Compton's discovery that X-rays can be diffracted by ruled gratings allows us to do this. The principle is the same as in the diffraction of visible radiation, although of course a photographic method has to be used. The angles to be measured are very small (only a few minutes of arc), so that the problem is now one of measuring small angles with a high degree of accuracy.

Two lines of attack have been devised. it is possible either to work in air with short
wavelengths of 1 or 2A, or in *vacuo* with longer wavelengths of 10-20A. The former
method gives smaller angular separation of the diffracted beam, but has the advantage of
easier adjustment. Both methods have been used, the former by Doan and Compton (N = 60.60
x 10^{22}, Wadlund (N == 60.6o x 10^{22}), and Bearden [23}, whose work
was verv thorough and painstaking. His result, (60.19 ± 0.03) x 10^{22}, is
rather lower than Millikan's, and it seems difficult to attribute the discrepancy entirely
to experimental errors.

Long-wave experiments have been performed by Böcklin
(60.34 x 10^{22}), Howe (6o.19 x 10^{22}), and Cork (60.06 x 10^{22}),
though it is doubtful whether any of these can be regarded as having the same accuracy as
Bearden's work.

Relations between e, N, and h

In 1929, W. N. Bond [24] devised an ingenious method of calculating e (and hence N) from the various methods of finding Plsnck’s constant h. There is no experiment by which Plank’s constant can be found without assuming an arbritary value of e, but in all the various experiments from which Planck's constant has been found, the final calculation is based on an equation of the form :

*h = Ae ^{n}* (16)

where A is an experimentally determined quantity, and n is an index which may be either
1, 4/3, or 5/3. Any two sets of experimental data involving different values of n could be
solved for h and e (N being obtained through Faraday's constant). This is essentially what
Planck did in 1902 when he found N by combining his radiation formula with Stefan's law.
Bond has generalised the principle
, and by solving 36 sensibly independent sets of
experimental results gathered from widely different branches of physics, he has obtained
the value, N == (60.54 ± .03) x 10^{22}. Birge, while accepting the principle has
questioned the choice of data; he has made a very thorough recalculation with carefully
chosen sets of results, and has obtained N = (60.62 ± .03) x 10^{22} [25].

Conclusion

In reviewing the methods of finding Loschmidt's number, we cannot fail to be impressed by their great variety and the remarkable agreement among their results. But to decide on the most probable value is a matter of considerable difficulty, for while there can be little doubt about the first two figures, only three methods can reasonably be expected to give the third with reasonable precision. These are the balanced drop method, the X-ray method, and Bond's method.

The discrepancy between Bond's and Birge's results shows how important is the selection of data for this calculation. Birge's figure is, however, in very close agreement with Millikan's value, although the X-ray value is slightly lower.

The X-ray value has been the subject of much discussion. At one time it was suggested
that possibly Millikan's result was too high, owing to some unsuspected source of error.
But the close agreement with Birge's calculation makes that now almost impossible. It was
also suggested that Bragg's theory of crystal structure perhaps only represented a first
approximation to the true form, and that really some kind of secondary structure was
superimposed on it. But Bearden has recently redetermined N by a dispersion method based
on an elaborate expression for the refractive index of quartz in terms of the wavelength
of the incident X-radiation and *e/m* (1.761 x 10^{7} em.u). The result is
almost identical with Millikan's value, and appears to indicate that the optical theory of
gratings is inexact when applied to X-rays. The question is, however, by no means settled.

On the other hand, no flaws have been detected either in the theory or the execution of Millikan's experiment, and the close agreement of his value with that calculated by Birge is strongly in favour of the most probable value of Loschmidt's number being at the present time :

N = (60.62 ± .03) x 10^{22}.

In conclusion, I wish to express my sincere gratitude to Prof. S. R. Milner, F.R.S., and Dr. R. W. Lawson for their guidance and encouragement while the material for this article was being collected.

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