The chemical and physical properties of an element depend on its atomic

number, _i. e._, on the positive charge carried by the nucleus of an

atom of that element. And this positive charge is, as we have seen, a

_resultant charge_. It is a result of the combination of a number of

unit positive charges with a smaller number of unit negative charges.

We can obviously reach the same resultant figure in as many ways as

we please. If the resultant charge on the nucleus is to be 5, for

instance, then we could take the combinations +6 and −1, or +7 and −2,

or +8 and −3, and so on. Each of these arrangements would give atoms

having identical chemical and physical properties. But their atomic

weights would be different. The atomic weights depend, not on the

resultant positive charge, but on the actual number of positive charges

present in the nucleus, including those that are compensated for by

negative charges as well as those that are not. In the above case, for

instance, our atoms would have atomic weights 6, 7, 8, and so on. And

this is the only difference they would have. By no other chemical or

physical properties could they be distinguished one from another.

It is therefore a very interesting fact, and one fitting in beautifully

with our theory, that many elements have been shown to consist of a

mixture of atoms having different atomic weights, but identical in

every other respect. The element chlorine, for example, has the atomic

weight 35·46. This number is about as far removed as it could be from

being a whole number, and is therefore specially fatal to the theory

that all atomic weights are simple multiples of the same unit. But

it has been shown that chlorine is really a mixture of two groups of

atoms, the atomic weight of the atoms of one group being 35 and the

other 37. These groups are mixed together in about the proportion of 3

to 1, and the ordinary measured atomic weight of 35·46 is really the

average weight of the mixture. Neon, again, whose atomic weight is

ordinarily given as 20·2, is found to consist of two groups of atoms

with atomic weights 20 and 22. Much more complicated groupings have

been discovered. Thus krypton, whose atomic weight is put as 82·92,

is made up of groups of atoms having the weights 78, 80, 82, 83, 84,

86. Such elements are called _Isotopes_, the name indicating that the

groups of atoms belonging to these elements occupy the _same place_ in

the periodic table. It will be noticed that the atomic weights of these

groups of atoms are all whole numbers. This is on the basis of oxygen

taken as 16. On this basis hydrogen is not exactly unity, but is 1·008.

It appears probable, then, that all atoms have atomic weights which are

nearly, but not quite, whole multiples of hydrogen.

The existence of isotopes definitely destroys the great importance

that chemists had always assigned to atomic weights. We have atoms

of different atomic weights but of the same properties. Further, as

we saw in studying the disintegration of radium into lead, we have

elements of the same atomic weight but with wholly different chemical

and physical properties. The atomic weight of an element, therefore, by

no means suffices to determine its chemical and physical properties.

We see once more that the really important quantity to be known about

an element is its _atomic number_, _i. e._, its position in the

periodic table. It is worth noting that the existence of isotopes was

not suspected until comparatively recent times, although very delicate

determinations of atomic weights have been practised for decades.

The most refined measurements customary in such determinations never

varied from sample to sample of the same element. Exactly the same

mixture of atoms constitutes chlorine, for instance, wherever the

chlorine is obtained, and always has done so ever since men began to

study chlorine. The groups of atoms which make up chlorine or any other

isotope must have been thoroughly and universally mixed long ages

ago—in all probability before the formation of the earth’s crust, when

such universal and complete diffusion would have been possible.

§ 5. _Relativity and the Atom_

It is necessary, now, to say a little about the Restricted Principle

of Relativity, since certain points about the modern theory of atomic

structure cannot be understood without it. But it is not necessary to

explain the principle itself. It is only necessary to describe the

relation between energy and mass that the theory shows to exist. In

pre-Relativity mechanics, it was always assumed that the mass of a

body was completely independent of its velocity. There was no reason

to suppose otherwise. Whether a body was moving fast or slow, or

whether it was at rest, its mass, when measured, was always found to

be the same. But the theory of relativity asserts that the mass of a

body does vary with its velocity. As the body moves faster its mass

increases. The mass increases in such a way that, at the speed of

light, it becomes infinite. This can only mean that the velocity of

light is a natural limit, that no material body could possibly exceed

this speed. If it be true that the mass of a body increases with its

velocity, it might be thought that experiment would long ago have led

us to suspect that fact. But the law according to which the increase

occurs is such that the increase is not measurable except at very

great speeds. Now we are not familiar with bodies moving at very great

speeds. We know of velocities such as 100 miles per hour and even,

in astronomy, of velocities which reach a few miles per second. But

velocities which are a considerable fraction of the velocity of light,

viz., 186,000 miles per second, are practically unknown. It is the α-

and the β-particles which furnish us with examples of bodies moving at

speeds comparable with that of light. And experiments on these bodies

show that their mass does increase with their velocity, and precisely

in the way predicted by Einstein’s theory. So that the ratio _e/m_, the

ratio of charge to mass of an electron, varies with the velocity of the

electron. As the velocity increases _m_ increases and therefore _e/m_

grows smaller. The value of _e/m_ usually given, viz., 1·77 × 10^{7}

(electromagnetic) units, is the value for low velocities, when m may be

taken as the mass of the electron at rest. We will denote this value

of _m_ as _m__{o}; _m__{o} is the mass of the stationary electron. At

half the velocity of light, the mass of the electron is 1·15 _m__{o},

_i. e._, it is about one-seventh greater. If the electron is moving at

nine-tenths the velocity of light its mass is 2·3 _m__{o}, or nearly

two and a half times greater. At ninety-nine-hundredths of the velocity

of light the mass of the electron is seven times its value at rest, and

at the velocity of light itself, as we have said, its mass is infinite.

This result of relativity theory must obviously be borne in mind in

any attempts to ascertain in detail what goes on inside an atom. If

the mass of the rotating electrons is a quantity which enters into our

calculations, then obviously we must remember that the mass varies

with the velocity of rotation we ascribe to the electrons. Another

important aspect of this theory is that it shows we must ascribe _mass_

to _energy_. This, again, is a very novel conception. We are used to

thinking of energy as something to which the property of possessing

mass cannot be ascribed. The two things seem to have nothing to do with

one another. But it can be shown that energy certainly does possess

inertia, and the property of possessing inertia is what we really mean

by mass. The mass of a body is, indeed, only another way of measuring

the total amount of energy it contains. Every piece of matter possesses

a vast store of internal energy. If the piece of matter begins to

move its energy is increased in virtue of its motion. Its mass also

is increased. But the increase in mass due to increase in energy is

usually extremely small. In any chemical combination which is attended

by the development of heat, for instance, there is a certain loss of

mass due to the energy radiated away during the process of combination.

The resultant mass of the compound is less than the sum of the original

masses of its constituents. But the loss which occurs in this way

during any chemical process is exceedingly small, and the old law of

the invariability of mass is, in all such cases, quite good enough.

But there appears to be a beautiful and highly interesting exception.

We have seen that there is reason to suppose that the helium nucleus,

which is shot out of radioactive bodies as an α-particle, is a very

stable structure. It is composed, as we have said, of four hydrogen

nuclei and two electrons. The great stability of this structure

suggests that its formation was attended by a great expenditure

of energy, so that an enormous amount of energy would have to be

communicated to it to break it up. Now the atomic weight of helium is

4, and the atomic weight of hydrogen is not 1, but 1·008. Four times

the mass of the hydrogen atom would give an atomic weight of 4·032. The

suggestion is that the difference between this value and the actual

measured value of 4, represents the _mass_ of the _energy_ lost in the

process of combining the four hydrogen nuclei into the helium nucleus.

This gives a measure, also, of the amount of energy that would be

required to split up the helium nucleus into its original components.

The amount of energy represented by this figure is really enormous.

It is sixty-three million times greater than the energy expended in

ordinary chemical processes, and this figure is a measure of how much

more stable the helium nucleus is than an ordinary chemical compound.

A chemical compound can often be dissociated merely by raising its

temperature a few degrees, but even the enormous energy possessed by

the fastest α-particles is only about a third of that required to

dissociate the helium nucleus.

CHAPTER V: _Quantum Theory_

Chapter V

_Quantum Theory_

§ 1. _The Stability of the Atom_

We have seen that the theory we have been describing, called the

nuclear theory of the atom, gives a very satisfactory account of a

large number of phenomena. The observed scattering of α-particles

on passing through thin sheets of metal, the existence of isotopes,

the changes which occur in radioactive phenomena, all receive very

convincing explanations. There can be no doubt that the nuclear theory

of the atom is essentially true, that the atomic models we have

imagined correspond closely to actual atoms. But there is a fatal

objection to this theory of the atom, as we have presented it hitherto.

Such an atom could not continue to exist!

According to the classical theory of electrodynamics every change of

motion on the part of an electrically charged body is attended with

a radiation of energy. In wireless telegraphy, for instance, it is

the rapid oscillations of the electrons in the sending apparatus which

produce the electromagnetic waves. Each time an electron suffers a

change in the direction or speed of its motion, or in both, it sends

out an electromagnetic wave. Such a process cannot be kept up without a

continual supply of energy. In the atomic model, as we have presented

it, the outer electrons, which we imagine to be continually circulating

round the nucleus, would be continually sending out energy. For a

circular motion is a perpetually changing motion, and every change of

motion on the part of a charged body is accompanied by the emission

of energy. For an electron not to radiate energy, according to the

classical theory, it must either be at rest, or be moving uniformly in

a straight line. It is obvious that the outer electrons of our atom

cannot be imagined as at rest. They would be attracted by the nucleus

and simply fall into it, just as the planets would fall into the sun if

they were robbed of their orbital motion. In order to counterbalance

the attraction of the nucleus the outer electrons must have a circular

or elliptical or some such motion. And any such motion would be

attended by a radiation of energy. As a result of this radiation of

energy it can be shown that the orbit of the rotating electron would

grow smaller and its velocity of rotation greater. This process would

continue until finally the electron fell into the nucleus. That is to

say, the atom, as we have depicted it, is, on the classical theory of

electrodynamics, essentially unstable. The whole material world, as we

know it, ought to have vanished long ago. Further, the spectrum of any

element contains perfectly sharp lines which are situated in perfectly

definite parts of the spectrum. The radiations from the atoms of a

given element are perfectly definite; they do not assume all values.

But if the outer electrons, from which these radiations proceed, are

continually changing their orbital distances and velocities, then there

ought to be a continuous succession of lines in the spectrum of that

element, instead of the perfectly distinct and permanent arrangement

which exists in fact.

So we see that, when we come to investigate the mathematical theory

of our atomic model it turns out to be highly unsatisfactory. Are we,

therefore, to abandon our model completely? Before we answer this

question we will look at one or two other phenomena where similar

extraordinary difficulties have been found. We will consider, in the

first place, the phenomena of heat radiation, since it is here that

the insufficiency of the old theory of electromagnetic radiation

was first demonstrated. Let us consider the heat rays radiated by

what is called a “black body.” A black body is defined as one which

absorbs the whole of the radiant energy that it receives. There is no

substance which exactly satisfies this condition, but it is possible

to produce the equivalent of it by artificial means. It was shown by

the German physicist Kirchhoff that a space enclosed by an opaque

envelope, and maintained at a uniform temperature, is filled with a

radiation identical with that which would be emitted by a black body

at the same temperature. If, therefore, a small hole be made in the

opaque envelope, the rays which escape through it will be the same as

those that would be produced by a black body. Several physicists have

studied these rays, and they have reached extraordinarily interesting

results. The way in which the total amount of energy radiated is

distributed amongst the different rays has been the chief object of

their researches. The rays which come from the enclosure are of very

different wave-lengths; they vary between wide limits. Corresponding to

each wave-length is a certain fraction of the total energy radiated,

and this fraction depends upon the length of the actual wave concerned

in a rather complicated way. It is found, as the result of actual

measurements, that the longest waves have very little energy. As the

wave-lengths decrease the energy increases until a certain wave-length

is reached where the energy has a maximum value. As we go on past this

point to shorter and shorter wave-lengths the energy decreases, until

for very short wave-lengths it is practically zero. Now this result is

in the most flagrant contradiction with the theoretical calculations.

According to the mathematical theory, the energy contained in the

very short wave-lengths should be very great. As the wave-lengths get

shorter and shorter, tending towards zero, the energy contained in

them should, according to the calculations, tend towards infinity.

Observation shows that it tends towards zero. The contradiction is as

striking as it could be.

We can see how extraordinary this observed result is if we consider

an analogous case. Suppose that we have a number of corks floating

on the surface of a bowl of water. Now suppose that, by some means,

we agitate these corks, causing them to oscillate up and down in the

water, and then leave them to themselves. We know that the oscillations

will, after a time, die down. The whole mass, water and corks, will

once again become quiescent. The difference is that the water will be

slightly warmer. The energy which was contained in the oscillating

corks is ultimately transferred to the molecules of the water and

appears as heat energy. Now this result is quite in accord with the

calculations. But if the result were to be analogous to the radiation

result mentioned above, the corks would have to go on oscillating

for ever with undiminished vigour. We should all agree that such a

phenomenon was highly mysterious. The results obtained in the radiation

experiments are no less mysterious.

Let us turn to yet another phenomenon which is entirely contradictory

of our expectations. It is found that light of high frequency, _i.

e._, of short wave-length, when allowed to fall on a metal, liberates

electrons from the metal. The old scientific question of “How much?”

immediately, of course, comes to the fore. We want to know the

number of the electrons liberated and their velocities. And we want

to know how these two quantities depend on the light which is used.

We find, as the result of careful experiment, that the _number_ of

electrons liberated depends on the _intensity_ of the light, but that

the _velocity_ of the electrons depends on the _frequency_ of the

light. This result is very surprising. We should have expected that

the more intense the beam of light the higher the velocity of the

liberated electrons. But only the number of electrons is influenced by

the intensity. A very weak beam of high frequency light will cause

electrons to be shot out of the metal with high velocity. We get a

firmer grasp of the paradoxical nature of this result if we first

create _X_-rays by bombarding an anti-cathode with electrons, and then

use the _X_-rays to liberate electrons from a metal. _X_-rays, as we

have said, may be regarded as extremely high frequency light-waves.

Now let us suppose that we produce some electrons in a cathode tube,

and cause them to bombard the anti-cathode, so producing _X_-rays. The

electrons will have a certain velocity, depending upon the voltage

applied to the tube, and they will generate _X_-rays having a certain

frequency. The higher the velocity of the electrons the higher the

frequency of the resulting _X_-rays. These _X_-rays are now allowed

to fall on a sheet of metal. Immediately electrons are liberated from

the metal, and the astonishing discovery is made that the electrons

so produced have the same velocity as the electrons which generated

the _X_-rays. We may illustrate this result by an analogy used by Sir

William Bragg. Imagine that we drop a plank, from the height of a

hundred feet, into the ocean. The impact produces waves in the ocean

which spread out in circles around the point of impact. As the waves

spread out they naturally get feebler and feebler, since the same total

amount of energy is distributed over a longer and longer circumference.

After travelling, say, two miles, let us suppose that the outermost

wave reaches a ship. We are to imagine that, immediately the wave

reaches the ship, it causes a plank to be shot up out of the ship to

a height of one hundred feet. This case seems precisely analogous to

the liberation of electrons by _X_-rays. The _X_-rays have spread out

in ever increasing spheres from the point of impact, and yet, wherever

they touch a metal, they liberate electrons having precisely the energy

of the electrons which generated the _X_-rays.

The key to these extraordinary results is to be found in Planck’s

Quantum Theory. It was at the end of the year 1900 that Max Planck

published his theory that energy is not emitted in a continuous

fashion, but only in little finite packets, as it were. An oscillating

atom, for instance, is to be conceived as sending out little doses of

energy, one after the other. It does not emit energy continually. And

Planck asserted that the size of these little packets depended on the

frequency of the oscillation, being greater the greater the frequency.

Such an hypothesis is very strange, and is in entire contradiction to

the classical dynamical theory on which the whole science of physics

had been built. Yet, strange as the theory was, the results it was

invented to explain certainly existed, and it could be shown that the

old dynamics not only had not explained them, but could not possibly

explain them. It was clear that any satisfactory explanation would have

to be something quite revolutionary in character. And Planck’s theory

did, as a matter of fact, explain the observed radiation phenomena

extremely well. Planck calculated, on his theory, how the energy of

radiation should be distributed amongst the different wave-lengths,

and his calculations precisely agreed with the experimental results.

It is possible that, even so, this revolutionary theory would not have

obtained general acceptance. But Einstein applied the theory to the

phenomena attending the liberation of electrons from metals under

the influence of light, and his calculations, also, were shown to be

in precise agreement with the evidence. The quantum theory, then,

although strange and, in many respects, little understood, has become

one of the great arms of modern physical research. It is still attended

with very grave difficulties. The phenomena of electron emission from

metals, for instance, certainly suggests that light energy exists in

small bundles, dotted about round the surface of the sphere which was

regarded as forming the old “wave-front.” Each bundle, we may suppose,

contains sufficient energy to liberate an electron with the velocity

of the electron which gave rise to the bundle. On the other hand,

certain well-known phenomena in light, particularly the phenomenon of

“interference,” seem utterly irreconcilable with this assumption; they

are perfectly well explained on the old wave theory of light, but they

seem quite inexplicable on the new quantum theory of light. It depends

on which phenomenon we want to explain which theory we employ. Neither

of them seem in the least adequate to explain all the known phenomena,

and they also seem quite irreconcilable with one another. The physicist

must keep both and yet they cannot live together. A compromise has been

tried. Sommerfeld and Debye, for instance, have endeavoured to work

out a theory whereby the energy brought by the light waves has been

regarded as continuous, but as being able, in some way, to accumulate

until the amount contained in a quantum “bundle” is reached. Having

accumulated to this amount, the energy is then supposed to work

suddenly and to shoot out the electron with the requisite velocity.

But the period required for this accumulation can be calculated, and

it is found that, to explain the effects produced by _X_-rays, an

accumulation period amounting to some years is required. So that the

emission of electrons under the influence of _X_-rays should not take

place until some years had elapsed. It is found, however, that the

emission takes place immediately the _X_-rays are applied, and ceases

instantly when they are discontinued. The contradiction is complete.

But the quantum theory, however puzzling it may be in certain aspects,

has shown itself competent to deal with very baffling phenomena.

It was natural, therefore, faced by the great puzzle presented by

the stability of the atom, to surmise that here, also, the quantum

theory would prove competent to overcome the difficulties. In its

original form, the theory could not be applied to the atom. It was

first necessary to extend it. This was first done, and the theory

successfully applied, by a brilliant young Danish physicist, Niels Bohr.

§ 2. _Bohr’s Atom_

Before we go on to describe Bohr’s conception of the atom we must

make a few remarks about spectra, since the explanation of spectrum

lines is one of the most important duties that an atomic model has to

fulfil. The whole science of spectrum analysis began with Fraunhofer’s

discovery that light from the sun, if spread out in a coloured band

by a prism, contained, besides its different colours, a large number

of fine dark lines crossing the band at right angles to its length.

Kirchhoff found that the light from incandescent gases, when treated

in the same way, also gave lines, although in this case the lines were

bright lines. But he further found that a gas will absorb the same

lines that it emits, so that if light be passed through a gas, dark

lines will occur at the same positions as the bright lines occur when

the gas is incandescent. Each chemical element was found to have its

own appropriate series of lines, and these lines serve, with remarkable

delicacy and exactitude, as a means of recognising the presence of

these elements. The lines in the sun’s spectrum, for instance, can

be disentangled into the groups belonging to each separate chemical

element in the sun. A similar analysis, performed on the light from

various stars, enables us to say what chemical elements are present in

those stars.

Every incandescent substance sends out light of several different

wave-lengths. These different rays are, in the ordinary way, jumbled

together, but, on being passed through a prism, they are separated

out in an orderly manner. The spectrum ranges from the red to the

violet. The waves giving red light are the longest waves and those

giving violet the shortest. Waves longer than red waves, the so-called

infra-red waves, do not affect the retina of our eye as light at

all, and the same remark applies to the waves shorter than violet

waves, the so-called ultra-violet waves. But such waves, although

they do not affect our eyes, can be made to affect certain chemical

preparations; with ultra-violet waves, for instance, photographs may

be taken of invisible objects, a fact perfectly well known to certain

“spirit” photographers. Now each line on a spectrum corresponds to a

definite wave-length. Light which is all of one wave-length is called

monochromatic light; each line on a spectrum corresponds to a certain

wave-length of monochromatic light. Corresponding to each line in the

spectrum of a given element the wave-length can be measured, and the

interesting question arises as to whether there is any relation between

the lengths of the different waves emitted by that element. We shall

see that there are such relations, and that Bohr’s theory of the atom

takes us some way towards explaining them.

In the first place, we have to assume, in applying quantum theory to

the atom, that an electron describes a circle or an ellipse round

the nucleus without radiating any energy. This assumption is in flat

disagreement with the classical theory of electrodynamics, but it is

in agreement with the quantum theory. Another assumption we must make,

and which is not in agreement with the old theory, is that the electron

can only move in certain orbits. If the orbit be a circular one, for

instance, then an electron can only circulate round the nucleus at

certain definite distances from it. It could not describe a circle

whose radius was intermediate between two of these distances. Whatever

one of the possible circles the electron is on, it will continue to

traverse that circle indefinitely unless some external force acts on

it. If an external force does act on it, then the electron passes

directly to another of the possible circles. During this transition

from one possible circle to another the electron radiates energy, and

this energy is monochromatic, that is, it is energy of a perfectly

definite wave-length. And the amount of energy so emitted is a quantum

of energy. The quantum of energy belonging to a certain frequency

depends upon that frequency. Its amount is, in fact, equal to the

frequency multiplied by a certain extremely small figure called

Planck’s constant. Thus the quantum, or the atom of energy, is not

an invariable thing. Like the atoms of matter, energy atoms are of

different sizes. The higher the frequency the greater the atom of

energy. As we have said, a monochromatic radiation is emitted by the

electron in passing from one possible orbit to another. This radiation

has, of course, a definite wave-length and therefore a definite

frequency corresponding to it. This frequency, multiplied by the

quantity called Planck’s constant, is equal to the total energy emitted

by the electron in passing from one orbit to the other.

We have said that certain relations have been found to exist between

the lines in the spectrum of a given element. It was in 1885 that

Balmer discovered that the lines in the spectrum of hydrogen could

be represented by a certain very simple formula. The frequencies

corresponding to a certain prominent group of lines in the hydrogen

spectrum may be represented by multiplying a certain constant figure

by the quantity (1/4 − 1/_n_^{2}) where _n_ takes on the values 3, 4,

5, 6, 7. These are the five strongest hydrogen lines, and for them

the quantity in the brackets becomes (1/4 − 1/9), (1/4 − 1/16), (1/4

− 1/25), (1/4 − 1/36), (1/4 − 1/49). Each of these values is to be

multiplied by a certain figure, the same in each case, and the results

will be the frequencies corresponding to each of these five lines

respectively. Another series of lines in the hydrogen spectrum is

obtained by using, instead of the general quantity in brackets given

above, the quantity 1/9 − 1/_n_^{2}, where _n_ takes on the values 4, 5,

6, etc. Still another series can be obtained from the general formula

1/1 − 1/_n_^{2}, where _n_ has the values 2, 3, 4, etc. It is easy to see

that the most general formula, including all these cases, is 1/_m_^{2} −

1/_n_^{2}. In the first formula we gave, for instance, _m_ = 2. In the

second _m_ = 3, and in the third _m_ = 1. Formulæ which are a trifle

more complicated were discovered later, and were found to represent

still other series of lines. And these formulæ were applied to other

elements besides hydrogen.

On Bohr’s theory, when an electron passes from one orbit to another,

it emits a certain quantity of energy, and the energy so radiated has

a certain frequency. If, therefore, we subtract the energy possessed

by the electron in its second orbit from the energy it possessed in

its first orbit, we have the total quantity of energy emitted by it

in passing from one to the other. The frequency, therefore, could be

calculated from the subtraction of these two quantities. Now it is

suggestive that Balmer’s formula for the frequency, given above, is

expressed by the subtraction of two quantities. Bohr showed that this

was no accident, and that the two quantities in Balmer’s formula do

indeed correspond to the energies before and after the transition of

the electron from one orbit to the other. In fact, Bohr was able, on

his theory, to deduce Balmer’s formula. It no longer appeared as a mere

empirical rule, but as a theoretical consequence of the structure of

the atom. This result was a most striking success for Bohr’s theory to

achieve. He also deduced, from his theory, the value of the constant

figure which is used to multiply the different quantities in brackets

given above; his calculated figure and the empirically ascertained

figure were in precise agreement. The values of the different possible

radii on which the electron in a hydrogen atom can move were also

deduced by Bohr. The electron is most stable when on its first orbit,

the orbit nearest the nucleus. This is the normal condition for a

hydrogen atom. The actual diameter of a hydrogen atom in this condition

can be calculated on Bohr’s theory, and the value so obtained is found

to be in agreement with the value obtained by quite other methods.

The fact that the spectrum of hydrogen possesses a number of lines,

therefore, shows us that, in the immense number of atoms present in any

specimen of hydrogen, there are always many whose electrons are passing

from one orbit to another. In one atom an electron will be passing from

the second to the first orbit, or from the third to the second, or

from the fourth to the third, and so on. Such transitions must always

be going on, for it is only in virtue of them that the hydrogen atoms

radiate any energy at all. The state to which all these changes tend

is the most stable state, when the electron is on its first orbit. We

may say, then, as Bohr puts it, that the spectrum of hydrogen shows

us the formation of the hydrogen atom, since the transition to the

successively decreasing orbits may be regarded as stages in the process

by which the hydrogen atom reaches its normal condition.

§ 3. _The Fine Structure of Hydrogen Lines_

We may summarise the theory of the hydrogen atom we have given hitherto

by saying that the hydrogen atom consists of a positive nucleus

carrying one unit of charge, and that a single electron is describing

an elliptical orbit about it. We can imagine a number of ellipses,

of different sizes, enclosing the nucleus. Each of these ellipses

will have a common focus, and it is at this focus that the nucleus is

situated. The single electron can move on any one of these ellipses,

but only on these; it cannot describe an intermediate orbit. Under

the influence of an external force the electron may jump from one

of these ellipses to another. During this jump it radiates energy in

the form of monochromatic waves. As long as it remains on any one of

these ellipses it is not radiating energy. These elliptical motions

are called _stationary states_. The electron only radiates energy,

then, in passing from one stationary state to another. This simple

theory suffices to explain the positions of the lines in the hydrogen

spectrum. We reach a formula which is exactly like Balmer’s formula

showing the distribution of these lines.

When we come to look more closely into the matter, however, we find

that there is a factor we have neglected in our calculations. We have

said that the electron describes an ellipse about the nucleus. We

suppose that it describes this ellipse in obedience to the ordinary

laws which regulate the motion of a single planet about the sun. It is

a peculiarity of such motion that the speed is not uniform. A planet,

in its elliptical orbit about the sun, is sometimes moving faster

and sometimes slower, depending upon which part of the ellipse it is

describing. At those parts of the ellipse which are nearest the sun

the planet is moving fastest. At the parts most remote from the sun it

is moving most slowly. As precisely the same laws apply to our electron

moving round the nucleus, we have to take into account the fact that

the speed of the electron in its orbit is continually changing. This

is where the theory of relativity comes in. We have seen that it is a

consequence of that theory that the mass of an electron varies with

its velocity, becoming greater the greater the velocity. Our electron,

therefore, is not only moving with a varying speed; it is also moving

with a varying mass. What influence will this variation of mass have on

the motion?

This problem was solved by Sommerfeld. The result is that the electron

continues to move in an almost elliptical orbit, but this orbit itself

is slowly and uniformly rotating. The actual motion of the electron

in space is a combination of these two motions. The effect of this on

the spectrum of hydrogen will be that corresponding to each hydrogen

line there will be two or three lines extremely close together. Each

hydrogen line will really consist of more than one line. These lines

will be so close together that it would be almost impossible to see

them separately. Nevertheless, measurements have been made, and these

measurements are in agreement with Sommerfeld’s theory. The fine

structure, as it is called, of the hydrogen lines, is due to the

variations in mass of the electron in describing its orbit about the

hydrogen nucleus. The complete explanation of the hydrogen spectrum

requires both quantum theory and relativity theory; conversely, the

striking agreement between calculation and observation in the hydrogen

spectrum greatly supports both these theories.

So far we have considered the theory, in detail, only in its

application to the hydrogen atom. The hydrogen atom is the simplest

atom, and we should expect the theory to be most adequate in dealing

with this case. But although the sheer complexity of the heavier

atoms has hitherto prevented so complete a description of them being

formulated, the general theory of their structure, as we shall proceed

to show, has much that is of interest and value to tell us.

CHAPTER VI: _The Grouping of Atoms_

Chapter VI

_The Grouping of Atoms_

§ 1. _The Outer Electrons_

Any theory of the atom which is to secure our assent must, in broad

lines if not in detail, account for the remarkable periodicity in

the properties of the chemical elements. We have already shown that

this periodicity, the recurrence of similar physical and chemical

properties, led Mendeléev to construct the Periodic Table. In each of

the columns of the diagram in Chapter IV the elements run through a

cycle of chemical properties which is approximately repeated in the

next column. This repetition of properties is not perfectly regular.

The columns are of unequal length; some contain 8, some 18, and some

32 elements. But the periodicity, although not perfectly simple, is

quite unmistakable, and is one of the most important and outstanding

properties of the chemical elements. Instead of arranging the atoms by

their general chemical properties we may arrange them according to some

specific property. We may, for instance, arrange them in accordance

with what is called their “atomic volumes.” If we do this we again get

a periodic relation. Still other properties, such as “compressibility,”

“expansion coefficient,” etc., show the same intriguing peculiarity.

All these properties seem to be connected with the actual amount of

space taken up by the atom. They are not properties of the nucleus;

they are dependent upon the outer electrons. And also, if we study

the visible spectra of the various elements, we find the same curious

recurrence. All those elements which are called Alkalis, for example,

have spectra which seem to be constructed on the same ground plan.

The different alkalis differ enormously from one another in the

complication of their structure; their atomic numbers are 3, 11, 19,

37, 55, so that we pass from a system containing three circulating

electrons to one containing fifty-five. And yet their spectra are

fundamentally the same. Here, also, we are concerned with the outermost

electrons. The nuclei of these atoms are not concerned in their

visible spectra. The nuclei of the various atoms, arranged according

to their atomic numbers, simply show a straightforward advance from

complexity to complexity. There is no kind of repetition or recurrence

in the properties which depend on the nucleus. But the way in which

the outer electrons are arranged does show a recurrence, and all those

physical and chemical properties which show a recurrence may be assumed

to depend on the outer electrons. Thus we may say that _the chemical

properties of an atom depend not on its nucleus, but on its outer

electrons_.

The visible spectrum, as we have said, depends upon the arrangement

of the outer electrons. But _X_-rays also possess a spectrum. The

_X_-rays emitted from any source are not all of the same wave-length

and these waves, by a method of which we shall learn more later, can

be arranged in order like those of visible light. Now the _X_-ray

spectra of the elements do not manifest a recurrence. They advance, in

a straightforward way, with the atomic number. They depend upon the

inner part of the atom and not upon the outer electrons. And it is

because they originate in the neighbourhood of the nucleus, where the

atomic forces are most intense, that the _X_-rays possess their great

penetrative power. Periodicity is not an inner, but only an outer,

property of the atom.

The great dominating factor which governs the properties of an atom

is the charge on its nucleus. We see this very clearly in the case of

isotopes. Two isotopic varieties of an element cannot be distinguished

from one another by their chemical properties. The outer electrons are

arranged in the same way in the two varieties of atoms, and it is this

arrangement which determines the chemical properties. Their visible

spectra are also the same, and so are the spectra in the ultra-violet

region. This fact furnishes an even more exact proof that their outer

electrons are arranged in the same way than does the identity of their

chemical properties. Two isotopic elements also have the same _X_-ray

spectrum; therefore the inner structure of the atoms, also, is the

same in the two cases. The whole structure of the atom is evidently

dependent on the charge carried by the nucleus; where this charge is

the same the atomic structure is the same.

The general question of how the electrons in a heavy atom are to

be supposed to be arranged is one of great difficulty, and no

perfectly precise answer can yet be given. There are certain general

considerations, however, which enable us to give a partial answer

to the question. In the periodic table each column ends with what

is called an “inert gas.” These elements are so called because they

possess great stability; they are not in the least eager to enter into

combination with other elements. Let us consider the element argon,

for instance. It is an inert gas, and occurs at the end of the second

group of eight in the periodic table. Its atomic number is 18. It

therefore contains 18 electrons rotating round the nucleus. Owing to

the marked stability of argon we must suppose that these 18 electrons

are arranged in some peculiarly stable configuration. The natural ideal

of every atom would be to reach so stable a condition. It is a state

to which every atom aspires. The atom of chlorine, which just precedes

argon in the table, and therefore possesses only 17 electrons, shows a

marked disposition to capture one additional electron. It is striving

towards the perfect state of possessing 18 electrons—not because it

requires the extra electron to become electrically neutral, of course,

but because the extra electron gives it greater mechanical stability.

On the other hand, the element potassium, which immediately follows

argon in the table, shows a marked tendency to get rid of one of its

19 electrons—again in order to attain the perfect state of possessing

18 electrons. If we go back to sulphur, which has 16 electrons, or

forward to calcium, which has 20 electrons, the same tendency manifests

itself. Sulphur has a tendency to capture two electrons and calcium has

a tendency to lose two electrons. We can understand, therefore, why the

Germans call the inert gases, like argon, the “noble” gases. It is not

only that they are sublimely inactive, but their condition is that to

which all the others aspire.

The great stability of the inert gases, and the fact that they occur

at the end of each period in the periodic table, so that, immediately

after each inert gas, the whole cycle of chemical properties begins

again, show us that they are, as it were, the natural terminations

of the building schemes which led up to them. After each inert gas a

fresh building scheme has to be adopted for the next group of atoms. We

have seen that each step along the periodic table means the addition

of a fresh electron. We can imagine the outer electrons of an atom

to be arranged in a ring, or on the surface of a sphere, or in what

configuration we like. When a fresh electron has to be added to produce

the atom one step on in the periodic table, we may imagine that, in

general, this new electron joins the rest. It takes its place in the

ring or on the sphere or whatever it may be. But there will come a

moment when the addition of a fresh electron will spoil the stability

of the whole structure. There will be no place for it in the ring,

and it will have to start a new ring, by itself, outside the existing

one. When yet another electron joins up, it will help the first one in

establishing the new ring. Presently the new ring will itself have all

the members it can stably hold, and further electrons will have to

build up yet another ring. The process will not be quite so simple as

this, for as outer rings continue to be built they will so influence

the inner rings that these will be able to take more members than

they could originally accommodate. But, in the broadest outlines, the

process is as we have described. Now the inert gases form such points

of departure. By the time an inert gas is reached the system which led

up to it has done all it could; it has fulfilled itself in producing an

inert gas. If atom building is to continue, it must be on a different

system, although the system of the new atom will, of course, resemble

the system of the atom to which it is connected by a line in the

periodic table. In just the same way, all the different inert gases

have systems which resemble one another.

The inert gas which precedes argon is neon, an element possessing ten

electrons. There is reason to suppose that it contains 2 inner and 8

outer electrons. The inert gas preceding neon is helium, the first of

the inert gases, and helium has 2 electrons. With the obvious exception

of helium it is supposed that the chemical similarity of all the inert

gases is due to their possessing 8 outer electrons, however many groups

of inner electrons they may have. The following table, showing the

number and arrangement of the electrons in successive groups, going

outwards from the nucleus, has been proposed by *Bohr*.

Helium 2

Neon 2, 8

Argon 2, 8, 8

Krypton 2, 8, 18, 8

Xenon 2, 8, 18, 18, 8

Radium Emanation 2, 8, 18, 32, 18, 8

How are we to consider these different groups of electrons to be

arranged? No precise answer can yet be given to this question, but the

whole trend of the most modern speculations is to emphasise the fact

that it is not sufficient to regard the different groups as lying in

plane rings. It is necessary to investigate the _spatial_ configuration

of the electronic orbits. It is very probable that different electrons

move in orbits which are inclined at various angles to one another.

Even in the solar system, the planets do not all rotate in precisely

the same plane. In the electronic orbits, we must imagine these

differences to be much greater. It has even been suggested that the

number 8, which occurs with such frequency in electronic groups, may

indicate that these groups of eight are arranged like a cube, one

electron being at each corner. The idea has something to recommend it,

although it appears that such an arrangement cannot be explained by the

known forces within the atom. But it serves as an indication of the

direction in which a solution is being sought. We shall now deal with

this question in more detail.

§ 2. _Hydrogen and Helium_

Hydrogen and helium are the two members of the first group in the

periodic table, and we shall now proceed to examine their atomic

structure. We have already described the structure of the hydrogen atom

in some detail. We know that it consists of a single electron rotating

about a nucleus. The electron can circulate on a number of different

orbits, but its most stable state is obtained when it circulates on

the first orbit, the one nearest the nucleus.

When we come to the helium atom the question is much more complicated.

A helium atom which had but one electron would be essentially similar

to a hydrogen atom, with the difference that the nucleus would carry

two positive charges instead of one. The real problem of the helium

atom is to determine the way in which the second electron enters into

its constitution. A spectrum of helium consists of two complete series

of lines, and for this reason helium was supposed to consist of two

different gases, called “orthohelium” and “parhelium.” But it is now

known that these two series of lines arise from the fact that the

second electron can enter into the constitution of the helium atom

in two different ways. The two electrons may be describing orbits of

the same kind, but inclined at an angle to one another, or they may

be describing orbits of different kinds, one outside the other. The

first case gives the most stable state for the atom, and in reaching it

the atom emits the spectrum which used to be referred to “parhelium.”

The second case is less stable and the process of reaching it gives

the “orthohelium” spectrum. This state was produced experimentally

by bombarding helium atoms with electrons. Such a bombardment could

produce what was called a “metastable” condition of the helium atom,

and it was found that the atom could not return to its normal condition

merely by radiating energy. The bombardment had caused the second

electron to move in an orbit outside that of the first electron—an

orbit of a different kind. And, having once done this, the second

electron could not make a jump back to its original orbit. Before it

could return to normal the “metastable” atom had to interact with atoms

of other elements—it had to go through a sort of chemical reaction.

In its normal state, then, the helium atom may be said to consist of

two electrons moving round the nucleus in similar circles, these two

circles being inclined at an angle of 120° to one another. And owing

to the interaction between the two electrons the planes of these two

circles are slowly moving. So that already, and when we are dealing

with an atom containing only two electrons, we are in the presence

of very considerable complications. The detailed working out of the

constitution of more complicated atoms would obviously be a task of

immense difficulty. Bohr has been able, however, to say something about

the broad lines of their structure.

§ 3. _Lithium—Neon_

We now come to the second group of the periodic table, a group

possessing eight members. We begin with lithium. Its atomic number is

3, and therefore an atom of lithium consists of 3 electrons revolving

about a nucleus. We shall assume that two of these electrons move in

orbits similar to those characteristic of the normal helium atom, that

is, in orbits which are not in the same plane but which are otherwise

similar. This assumption is very natural, for the normal structure of a

helium atom is a very stable condition. It is distinctly more stable,

for instance, than the structure of the hydrogen atom. Helium is the

first of the inert gases. We assume, then, that two of the electrons

in a lithium atom move as do the two electrons of the helium atom. How

are we to suppose the third electron to move? The spectrum of lithium

shows us that the third electron moves in orbits which are altogether

outside the region containing the first two electrons. The spectrum

also shows that the third electron sometimes moves in orbits which,

although they lie outside the region of the first two electrons for

the greater part of their length, yet, at their nearest point to the

nucleus, approach it as closely as do the first two electrons. These

are the orbits characteristic of the lithium atom in its normal state.

The firmness with which the outer electron is held in these orbits is

only about one-third of that with which the electron in a hydrogen

atom is held, and only about one-fifth of that with which the helium

electrons are held. The chemical properties of these three elements,

therefore, depending, as they do, on the outer electrons, should be

very different, as, in fact, they are.

We may assume that, in any atom, the third electron moves in the same

kind of orbit as does the third electron of the lithium atom. This

orbit is, as we have said, very excentric. It may be regarded as

markedly elliptical. That part of it nearest the nucleus is within the

region in which the two inner electrons move. The rest of it extends

far beyond this region. We may imagine that the fourth, fifth, and

sixth electrons move in similar orbits. There is reason to suppose

that the four electrons, from the third to the sixth inclusive, which

move in these excentric orbits, are so distributed as to form an

exceptionally symmetrical configuration. Each of these outer electrons

penetrates to the region occupied by the inner electrons, but not at

the same moment. Bohr supposes that the outer electrons reach their

nearest point to the nucleus separately at equal intervals of time.

This structure carries us as far as carbon, which has six electrons.

Lithium has one outer electron, beryllium two, boron three and carbon

four. Each of these outer electrons moves in very excentric orbits

which enclose and partly penetrate the approximately circular orbits

within which the two inner electrons move. This method of building

reaches completion in the carbon atom. If yet another electron were

added to the four outer electrons of carbon the symmetry of the

arrangement would be destroyed. There is, as it were, no room for

five such orbits. Also, the fact that the elements in the second half

of this group in the periodic table have very different properties

from those in the first half suggests that a new system of building

comes into existence directly we proceed beyond carbon. Bohr supposes

that in nitrogen, an element possessing seven electrons, the seventh

electron moves in a large and approximately circular orbit. It lies

completely outside the two inner electrons, although the outermost

parts of the excentric orbits on which the other four outer electrons

lie extend beyond it. The eighth, ninth and tenth electrons also move

in large circular orbits of this kind. The great stability of the last

element reached in this way, the inactive gas neon, suggests that the

final arrangement possesses great symmetry. We must suppose that, with

this element, the four large circular orbits are not only symmetrical

amongst themselves, but also in relation to the four elliptical orbits.

§ 4. _Sodium—Argon_

We have, so far, considered two types of orbits, the approximately

circular, and the markedly elliptical. For the first two electrons we

assume circular motion. For the next four we assume elliptical motion,

and for the next four we again assume circular motion. In this way we

have got as far as neon, and we now begin another group of the periodic

table. Acting on the same general principles, we shall assume that the

eleventh electron inaugurates a new era of elliptical orbits. These

orbits are very elliptical. For the most part they are well outside the

orbits of the first ten electrons, but for part of their course they,

like the first group of elliptical orbits, penetrate even closer to

the nucleus than do the two innermost electrons. The existence of such

markedly elliptical orbits, passing, during part of their course, so

close to the nucleus, greatly helps the stability of the atom. In the

distribution of the twelfth, thirteenth, and fourteenth electrons we

meet conditions similar to those we encountered when considering the

fourth, fifth, and sixth electrons. There seems to be an exception in

the case of aluminium, the element whose atom contains 13 electrons.

In this case the thirteenth electron seems to move in a less markedly

elliptical orbit. But Bohr does not regard this behaviour as typical

for the thirteenth electron in all atoms; it is peculiar to aluminium,

where the thirteenth electron is also the last electron. By the time

we get to silicon, containing 14 electrons, we find the thirteenth

electron, like the eleventh, twelfth, and fourteenth, moving in the

markedly elliptical type of orbit inaugurated by the eleventh electron.

As in the preceding cases, we suppose this type of construction to be

completed when we have four electrons describing these new orbits.

The fifteenth electron introduces a new system, just as the seventh

electron did. But whereas the seventh electron introduced an almost

circular type of orbit, the fifteenth electron continues with the

excentric type of orbit, although the excentricity is not so marked

as in the case of the orbits we have just left. These new orbits

are excentric enough to penetrate closer to the nucleus than do the

circular orbits inaugurated by the seventh electron, but they do not

reach the region of the two innermost electrons. These orbits will

accommodate four members, so that this type of construction will carry

us up to the atom possessing 18 electrons, _i. e._, up to the element

argon. Thus we again reach an inert gas, and, allowing for the greater

complexity, we see that its symmetrical properties closely correspond

to those of the preceding inert gas, neon.

§ 5. _The Remaining Elements_

The development we have been describing hitherto is straightforward

in the sense that fresh groups of electrons have been regarded as

possessing fresh types of orbits which are, as it were, independent of

those previously existing. We have not considered the later electrons

as causing any development of the inner groups of electrons. That

there must be some interaction is obvious, but we have not found

it necessary to assume that the interaction is sufficient to cause

any fundamental modification of the orbits already established.

When we come to the fourth period of the periodic table, however,

matters are different. As we see from the diagram in Chapter IV, the

fourth period contains 18 elements. At the beginning of this period

the atom continues to develop in a way analogous to that we have

already studied. The first two elements of this period are, as shown

by the connecting lines, analogous to the first two elements of the

third period. But then occurs a group of eight elements which do not

correspond, in our diagram, to anything in the third period. And

after this group occur two elements which are again analogous to the

first two elements of the third period. Why is it that we have this

interregnum, as it were, lasting over eight elements? Bohr’s answer is

that we are here concerned with the development of one of the inner

groups of electrons. The normal system of atom building, as we have

sketched it, cannot now proceed. The later electrons captured will

now be concerned in the internal rearrangement, and only when this is

completed will the normal process be able to proceed.

This theory gives an interesting explanation of the facts that many

elements of the fourth period differ markedly from the elements of

the preceding periods in their magnetic properties and also in the

characteristic colours of their compounds. That highly magnetic

substance, iron, for instance, occurs in the fourth period. To

understand the explanation, offered by the theory, of the magnetic

properties of these elements, we must revert to the familiar fact that

an electric current is always attended by magnetic force. Electric

currents are constituted by the movements of electrons, and the

moving electrons within an atom will give rise to their appropriate

magnetic forces. Now we may imagine that, in any thoroughly symmetrical

arrangement of the electrons within an atom, these magnetic forces

form a closed system within the atom, so that no resultant external

effects are manifested. With any markedly unsymmetrical arrangement of

the electrons, however, we may expect appreciable external magnetic

effects to manifest themselves. Bohr supposes, therefore, that the

process of reorganisation within the atom which characterises a group

of elements in the fourth period, is attended by the lack of symmetry

which would result in magnetic forces being exhibited. Where the

symmetry is at last restored the magnetic effects cease. In Bohr’s

words: “On the whole a consideration of the magnetic properties of the

elements within the fourth period gives us a vivid impression of how a

wound in the otherwise symmetrical inner structure is first developed

and then healed as we pass from element to element.”

The characteristic colours to which we have alluded also find an

explanation on this theory. These colours are due, of course, to the

absorption of light, and they are thus evidence that energy changes

are going on comparable with those giving visible spectra. This is in

contrast to the elements of the earlier periods, where the electrons

are more firmly held and where the less rigid conditions, due to the

development of an internal group of orbits, do not occur.

--------+-------+-----------------------------------------------------------

| | NUMBER OF ELECTRONS IN DIFFERENT TYPES OF ORBITS.

ELEMENT.|ATOMIC |-+-+-+-+-+-+-+-+-+--+--+--+--+--+--+--+--+--+--+--+--+--+--

|NUMBER.|1|2|3|4|5|6|7|8|9|10|11|12|13|14|15|16|17|18|19|20|21|22|23

--------+-------+-+-+-+-+-+-+-+-+-+--+--+--+--+--+--+--+--+--+--+--+--+--+--

Helium | 2 |2| | | | | | | | | | | | | | | | | | | | | |

Neon | 10 |2|4|4| | | | | | | | | | | | | | | | | | | |

Argon | 18 |2|4|4|4|4|-| | | | | | | | | | | | | | | | |

Krypton | 36 |2|4|4|6|6|6|4|4|-| -| | | | | | | | | | | | |

Xenon | 54 |2|4|4|6|6|6|6|6|6| -| 4| 4|--|--|--| | | | | | | |

Niton | 86 |2|4|4|6|6|6|8|8|8| 8| 6| 6| 6|--|--| 4| 4|--|--|--|--| |

? | 118 |2|4|4|6|6|6|8|8|8| 8| 8| 8| 8| 8|--| 6| 6| |--|--|--| 4| 4

--------+-------+-+-+-+-+-+-+-+-+-+--+--+--+--+--+--+--+--+--+--+--+--+--+--

Table showing distribution of Electrons in the Inert Gases, including a

hypothetical element of atomic number 118.

The building up of the rest of the elements, up to and including the

seventh period, may be supposed to take place on the broad lines we

have now laid down. The process is a double one. New groups of outer

orbits will be formed, and also there will be a development of groups

of inner orbits. The whole process is very complex, and no attempt has

yet been made to examine it in detail.

The seventh period ends abruptly with uranium, whose atomic number is

92. The last elements in this period are all radioactive, and, as we

have said before, it seems probable that an element of higher atomic

number than 92 would be too unstable to exist. Nevertheless, on the

principles we have followed hitherto we can construct theoretically,

and in its main lines, the structure of an atom having a higher atomic

number than 92. The last inert gas known to us, niton, has an atomic

number 86. The next inert gas, if it existed, would have an atomic

number 118. Bohr gives a table, which we reproduce, showing in some

detail the number of electrons and the characters of their orbits for

the six inert gases. He includes, as a seventh, the imaginary gas

having an atomic number 118 and shows its hypothetical construction.

CHAPTER VII: _The Inner Regions_

Chapter VII

_The Inner Regions_

§ 1. _X-Ray Spectra_

In speaking of _X_-rays we have referred to their wave-lengths and

to their spectra, but we have not yet given any indication as to

how these wave-lengths are measured. The most satisfactory method

of determining the wave-lengths of ordinary light is by means of a

“diffraction grating.” This apparatus consists, essentially, of a sheet

of glass on which a large number of very fine lines have been ruled

very close together. The lines should be parallel and equidistant.

Now the distance between two adjacent lines should be of the same

order of magnitude as the wave-lengths to be measured. The lengths

of visible light waves are comprised between 4 × 10^{-5} cm. and 7 ×

10^{-5} cm., _i. e._, they lie between 4 and 7 hundred-thousandths of a

centimetre. _X_-rays, as we have said, have wave-lengths about 10,000

times smaller than this. It is difficult enough to rule lines close

enough together for the distances to be comparable with the lengths of

light-waves; it is utterly impossible to rule them ten thousand times

closer still. The distance between adjacent lines would have to be of

the order of 10^{-8} cm., _i. e._, of the same order of magnitude as

the molecular distances in a solid body. Manifestly such an apparatus

is impossible to construct. But it so happens that nature has provided

such an apparatus.

Certain mineralogists and mathematicians were long ago concerned to

elucidate the regular shape and structure of crystals in terms of

regular arrangements of their molecules or atoms. These molecules or

atoms were supposed to be arranged in definite patterns, so that a

crystal consisted of layers, arranged one behind the other, containing

these regular assemblages. The distance between the molecules or atoms,

so arranged, would be of the order of 10^{-8} cm. A crystal of salt,

for instance, would have as the distance between its molecules 5·6 ×

10^{-8} cm. The brilliant idea occurred to a German scientist named

Laue that such an arrangement really constituted a sort of diffraction

grating and one, moreover, of just the right dimensions to serve for

the measurement of _X_-ray wave-lengths. The realisation of this idea

was highly successful, and the employment of crystals has not only

served to measure _X_-ray spectra, but has also taught us a great deal

about the structure of the crystals themselves. It is now possible,

employing this method, to obtain photographs of _X_-ray spectra.

As a result of these researches we now know that the wave-lengths of

_X_-rays vary within fairly wide limits, according to the conditions of

their emission. The longest waves are about 12 × 10^{-8} cm. in length,

while the shortest are about 3 × 10^{-8} cm. The shorter the wave the

greater its penetrative power or “hardness.” Now we have already said

that each of the chemical elements, on being bombarded by cathode

rays, emits a group of _X_-rays which is characteristic of it. The

hardness of these _X_-rays varies with the substance that emits them,

and in such a way that the greater the atomic number of the substance

the harder are the emitted rays. We are concerned here with a wholly

atomic phenomenon, for if a substance be chosen as the anti-cathode

which is a compound of two or more elements, it is found that the

resultant _X_-ray emission, when the anti-cathode is bombarded, is

really a combination of the _X_-ray groups which would be emitted

separately by the elements that have gone to make up the compound.

These important facts were discovered by Barkla, who also discovered

that there were two series of _X_-rays in the characteristic _X_-ray

emission from an element. He called these two series the _K_-group and

the _L_-group. He observed that the lighter elements (up to silver)

gave the _K_-group of _X_-rays, and that heavy metals (such as gold and

platinum) gave the _L_-group. Of these two groups, the _K_-group is

the more penetrating. The harder or more penetrating the _X_-rays, the

greater the impact of the cathode rays necessary to produce them, and

Barkla saw that the _K_-group, in the case of the heavy metals, would

be so hard that the experimental methods known to him would not suffice

to produce them. Similarly, the _L_-group for the lighter elements

would be too little penetrating, too soft, to be observable by the then

known means. Barkla had determined the hardness of the rays he obtained

by measuring their absorption by thin sheets of aluminium. And he had

established a relation, as we have said, between hardness and atomic

weight.

These results were made much more precise when the analysis of _X_-ray

spectra by crystals replaced the absorption method of measurement, and

when the wave-lengths so determined were related, not to the atomic

weight, but to the atomic number. Besides the _K_- and _L_-groups, a

third group, called the _M_-group, has been discovered. The _M_-group

of rays is still softer than the _L_-group. The _K_-group, so far as

our means of observation carry us, begins with sodium, whose atomic

number is 11. With this light element the _K_-group, the hardest of

the three groups, is distinctly weak. As the atomic number advances

the _K_-groups emitted by the corresponding elements grow harder and

harder, reaching their extreme degree of hardness with wolfram, whose

atomic number is 74. For one and the same element, emitting both the

_K_-group and the _L_-group, the _L_-group is much the softer. The

_L_-group has been observed with copper, whose atomic number is 29,

and here it is even weaker than the _K_-group of sodium. From copper

onwards the _L_-group gets harder and harder, and it has been observed

right up to the last of the elements, uranium. The still weaker

_M_-group has only been observed so far with the heaviest elements, and

even then special precautions have to be taken to observe it at all.

These three groups of rays together make up the _X_-ray spectrum.

§ 2. _The K-Group_

Moseley, probably the most gifted of the young English men of science

killed in the war, was the first to make a considerable advance on

Barkla’s work. His first photographs (1913) were devoted to the

_K_-group, and extended from calcium, with atomic number 20, to copper,

with atomic number 29. These elements were used, successively, to form

the anti-cathode of a cathode tube, and were therefore bombarded

directly by electrons. To obtain the _X_-ray spectra he used, of

course, the method of crystal analysis, but not in its most modern

form. He established the following results.

As the atomic number increases the corresponding lines in the spectrum

move regularly in the direction of smaller wave-lengths, that is, the

hardness of the lines increases with the atomic number. This result, in

a less definite form, was, as we have seen, already reached by Barkla.

Each element gives two _K_-lines. The stronger, more obvious, line

corresponds to the longer wave-length. This line in the _K_-spectrum of

an element is called the _K_α-line. The weaker line is the harder line,

_i. e._, it corresponds to the shorter wave-length. This line is called

the _K_β-line.

The _X_-ray spectrum of an element is purely a property of the atoms

of that element. Brass, for instance, which is an alloy of copper and

zinc, gives four _K_-lines, of which two are the _K_-lines of copper,

while the other pair are the _K_-lines of zinc. Thus the _K_-spectrum

of a complex substance is obtained by merely adding together the

_K_-spectra of its elementary constituents.

The fourth result is of particular interest. It will be remembered

that, for a few places in the periodic table, we inverted the order

of the elements as given by their atomic weights. There are two or

three places where a heavier element is put before a lighter one. The

whole complex of the chemical and physical properties of such pairs of

elements is allowed to determine their position in the periodic table,

even when this is not in agreement with the atomic weight. Nickel and

cobalt form such a pair. Cobalt is heavier than nickel, with an atomic

weight of 58·97 as against an atomic weight of 58·68. Nevertheless,

cobalt is written before nickel. This order, justified by general

considerations, was completely confirmed by the _X_-ray spectra of

these elements. The _K_-group for nickel has harder lines than the

_K_-group for cobalt, and the increase in hardness corresponds to an

advance of one step in the periodic table. Here we have a clear proof

that the _X_-ray spectra follow the order of the atomic numbers, not

the order of the atomic weights. To settle this point was the original

object of Moseley’s research.

The fifth result which emerges from these researches is also of great

interest. We have spoken of gaps in the periodic table and we have

left spaces for elements which have not yet been discovered, but

to which we have ascribed appropriate atomic numbers. In Moseley’s

original research there was a gap between calcium and titanium. This

gap was immediately revealed by the _X_-ray spectra. The advance in

hardness from one element to another is quite uniform, and in passing

from calcium to titanium a sudden jump was found, corresponding to

the omission of one element. This missing element is known. It is the

rare substance named Scandium, with atomic number 21. Its absence from

Moseley’s series was at once revealed by the _X_-ray spectra. The

regularity of the growth in hardness of the _X_-ray spectra enables

us, without ambiguity, to say precisely how many elements (up to

uranium) are yet undiscovered, and exactly whereabouts they occur in

the periodic table. Thus, corresponding to atomic number 43, there is

a missing element. It has received the name Ekamanganese. Other gaps

in the system occur at atomic numbers 61, 75, 85, and 87. The study

of the _X_-ray spectra of the elements, therefore, enables us to say

definitely that five elements are missing.

Moseley’s results have been followed up, and his experiments repeated

with better apparatus. The main discoveries that have been made by

these later researches on the _K_-group are that there is a third line

belonging to the group, and that the _K_α-line really consists of two

lines very close together—what is called a doublet. The third line

of the _K_-group is even weaker and harder than the _K_β-line. It is

called the _K_γ-line. The same law holds for this third line as for

the other two. Like them, it increases in hardness for elements of

increasing atomic number.

The beautiful simplicity and precision of the results make this

research on the _K_-group one of the most interesting in all the modern

work on the atom.

Of the _L_- and _M_-groups we need only say at present that they

contain a large number of lines of which many are doublets. The general

law of their variation in hardness with the atomic number is the same

as for the _K_-group.

§ 3. _The Electrons near the Nucleus_

We shall now proceed to show how these experimental facts are explained

by the theory of atomic structure that we have outlined. In doing so we

shall present the problem in a rather simplified form, but one which

serves, in its main lines, as the basis for the detailed examination

which Bohr, and one or two others, are attempting. We recall again

the fact that the atom is regarded as a kind of planetary system, of

which the nucleus is the central body and the electrons the revolving

planets. We have already discussed the way in which we may suppose

these electrons to be arranged. They exist in groups; each member of

any one group moves in the type of orbit characteristic of that group.

We shall find that it simplifies our ideas and does not essentially

disturb the main lines of the theory if we imagine these groups of

electrons to be situated on circles all centring about the nucleus.

The circles get larger and larger, of course, as we proceed outwards

from the nucleus. The circle closest to the nucleus we shall call the

_K_-circle and the others, as we go outwards from the nucleus, the

_L_-, _M_-, _N_-, etc., circles.

Let us now consider how we may suppose a radiation belonging to the

_K_-group to be caused. We may suppose the first step to consist in the

removal of an electron from the _K_-ring to the periphery of the atom,

or else outside the atom altogether. If this removal be effected by a

cathode-stream bombardment, we may imagine that it is the result of

the direct impact of one of the bombarding electrons on the electron

of the _K_-ring. A certain minimum amount of energy is necessary for

this impact to be powerful enough to remove the electron. The electrons

of the _K_-ring, the innermost ring, are powerfully attracted by the

nucleus, and the bombarding electron must be moving sufficiently fast

for its impact to overcome this attraction. There is therefore a

certain minimum velocity below which the cathode-stream bombardment

cannot detach an electron from the _K_-ring. The higher the atomic

number the greater the charge on the nucleus, and the more firmly,

therefore, the electrons in the _K_-ring are held. For elements of high

atomic numbers, therefore, only the most intense bombardment would

suffice to detach an electron from the _K_-ring.

When the electron is detached, the _K_-ring is left incomplete, and

an electron from another ring will rush to take the vacant place.

Now we must remember that each ring corresponds to a different level

of energy, in accordance with Bohr’s quantum theory of the atom. In

passing from one ring to another an electron passes from one energy

level to another. A certain amount of energy is liberated by the

process, and this energy manifests itself as a radiation. It will

be what we have called a “monochromatic” radiation, that is, it

will be of one definite wave-length. It will furnish a line in the

_K_-spectrum. Now it may happen that the electron which rushes to

take the vacant place comes from the ring next to the _K_-ring, or

from the ring next but one, or from the ring next but two, and so

on. It is not at all likely to come from a very far-off ring, so we

may say that it will come from the _L_-ring, or the _M_-ring, or the

_N_-ring. But the farther off the ring from which it comes the greater

is the energy liberated, and the higher the frequency of the resultant

radiation or, what comes to the same thing, the greater the hardness

of the resultant radiation. So that an electron which falls from the

_N_-ring to the _K_-ring will give a harder radiation, _i. e._, one of

smaller wave-length, than an electron which falls from an _M_-ring to

a _K_-ring, and harder still, of course, than an electron which falls

from an _L_-ring to a _K_-ring. At the same time, it is more _likely_

that the missing _K_-ring electron will be replaced from the ring next

to it, the _L_-ring, than from the other more distant rings. And, as

between the _M_-and the _N_-rings, an electron is more likely to come

from the _M_-ring than from the _N_-ring. So that we should expect the

least hard line in the _K_-spectrum, the one due to the passage of an

electron from the _L_-ring to the _K_-ring, to be also the strongest

line, since the proportion of atoms where this particular change

is occurring is the largest proportion. And, by the same reasoning,

we should expect the hardest line, the one due to the passage of an

electron from the _N_-ring to the _K_-ring, to be also the weakest

line. The other line, the one due to the passage from the _M_-ring to

the _K_-ring, would be intermediate, of course, both in strength and

hardness. Thus our theory explains the observed fact that the hardest

line is the weakest and that the softest line is the strongest, while

the other line is, of course, intermediate in both respects.

A similar explanation holds good for the _L_- and _M_-groups of the

_X_-ray spectrum. The bombardment will sometimes detach an electron

from the _L_-ring. It is to be noticed that the energy necessary to

do this is less than in the case of the _K_-ring, and that for two

reasons. In the first place, the electrons in the _L_-ring are farther

removed from the nucleus than the electrons in the _K_-ring, and in the

second place they are subject to a certain repulsive effect from the

electrons of the _K_-ring. All electrons repel one another. An electron

belonging to any ring is repelled by the other members of that ring,

as well as by the members of other rings. As we get farther away from

the nucleus this effect becomes more marked, and it acts, to a first

approximation, as if the charge on the nucleus had been reduced, and

therefore exerted a less firm binding effect on the electron. It

requires less energy, therefore, in the case of any given element,

to detach an electron from the _L_-ring than from the _K_-ring. The

electron having been detached the vacant place may be occupied by an

electron from any of the farther outlying rings. And, here again, an

electron is more likely to come from the next farther ring than from a

more distant ring. At the same time, the passage of the electron from

the nearer ring will radiate less energy than the passage from a more

distant ring. So that in this case also the weaker line should be the

harder. This agrees with the experimental results.

The same general remarks apply to the detachment and replacement of an

electron from the _M_-ring, with the difference that the detachment is

still easier than in the case of the _L_-ring. The _M_-ring is farther

from the nucleus and also the repulsive effect of the inner electrons

is more noticeable.

§ 4. _Doublets_

The simplicity of the _K_-spectrum is due to the fact that we are here

concerned only with the innermost electrons of the atom and, in this

region, the charge on the nucleus exerts a very firm control. The

stability of the electrons close to the nucleus is very considerable.

As we get farther away from the nucleus, however, the conditions

become more complicated and the resulting spectra, indicating what

changes are going on, also become more complicated. When we reach the

outer electrons, those concerned in producing the visible spectrum,

the changes going on are of the greatest complexity. This growth

in complexity is apparent directly we pass from the _K_-spectrum

to the _L_-spectrum. The _L_-spectrum contains more lines than the

_K_-spectrum, and their explanation is less simple. An interesting

feature of the _L_-spectrum is the large number of doublets it

contains. It is found that close pairs of lines may be distinguished in

the _L_-spectrum and that the distance between these pairs is constant.

It is this fact, that the doublets are of the same size, as it were,

that calls for explanation. The explanation offered by the theory is

that the _L_-ring is not really a single ring, but consists of two or

more rings, each ring corresponding to a slightly different energy

level. Let us suppose that the _L_-ring really consists of two rings.

Then an electron from the _M_-ring may fall on to one or the other of

these two rings. We shall denote these two _L_-rings by _L__{1} and

_L__{2}.

If an electron falls from the _M_-ring on to the _L__{1}-ring, it will

radiate an amount of energy slightly different from that radiated

by an electron falling from the _M_-ring on to the _L__{2}-ring.

The wave-lengths corresponding to these radiations will therefore

be slightly different and the corresponding lines in the spectrum,

indicating these wave-lengths, will therefore consist of a pair of

lines close together—a doublet. Now let us consider what happens when

the electrons fall from the _N_-ring. The electron which falls from

the _N_-ring, whether it falls on the _L__{1}- or the _L__{2}-ring,

will radiate more energy than the electron from the _M_-ring. But the

_difference_ in the energy radiated, depending on whether it falls

on to the _L__{1}- or the _L__{2}-ring, is obviously the same as the

corresponding difference in the case of an electron falling from the

_M_-ring. It is the difference in the two paths which is concerned,

and this difference is simply the distance between the _L__{1}- and

the _L__{2}-rings. In the case of electrons falling from the _N_-ring,

therefore, although the actual amounts of energy radiated are greater,

and therefore the corresponding wavelengths are shorter, yet the

resulting pair of lines are at the same distance apart as in the case

of the _M_-ring. Similar reasoning applies to any pair of electrons

which start from the same outer ring and fall, one on the _L__{1}-ring,

and the other on the _L__{2}-ring. In each case, a doublet is produced,

and all these doublets have their component lines separated by the same

interval.

This theory gives a satisfactory explanation of the observed equality

of the _L_-doublets, but we can go on to deduce a result from it

which can be used as a test. We have seen that the _K_ α-line in

the _K_-spectrum is produced by the passage of an electron from the

_L_-ring to the _K_-ring. But there are two _L_-rings. The passage of

an electron from each of these to the _K_-ring should produce a line in

the _K_-spectrum. These two lines should be very close together; they

should form a doublet. Now we have seen that the _K_ α-line actually is

a doublet. But we can go further. The _difference_ in path, according

to whether the electron falls on the _K_-ring from the _L__{1}- or

the _L__{2}-ring, is the same as the difference in the paths of two

electrons, coming from the same outer ring, but falling one on the

_L__{1}-ring and the other on the _L__{2}-ring. So that the interval

between the components of the _K_-doublet should be the same as the

interval between the components of the _L_-doublets. This result is

confirmed by actual measurement. The “interval” or “distance” between

the components of a doublet is really a difference, of course, in the

hardness of the corresponding waves.

We have said sufficient to make the main lines of the theory clear. We

need only add that the further explanation of the _L_-spectrum and also

of the _M_-spectrum requires us to assume that the _M_-ring also is not

a single ring, but consists of two or more.

INDEX

A

α-RAYS, 65

Aluminium, 154

Anti-cathode, 73

Argon, 141

Atom, 33

Atomic numbers, 82

Avogadro, 34

B

ß-rays, 65

Balmer, 127

Barkla, 168

Black body, 114

Bohr, 123

Bragg, 118

Brownian movement, 45

C

Cathode rays, 55

Compounds, formation of, 29

Crystal analysis, 166

D

Dalton, 23

Debye, 122

Diffusion, 40

E

Einstein, 46, 105, 120

Electron, mass of, 60

—— radius of, 62

Element, 28

F

Fraunhofer, 123

G

γ-rays, 65, 74

Gases, Kinetic Theory of, 42

Geological strata, age of, 70

Gold atom, 88

Gramme-atom, 37

H

Helium atom, 86, 147

—— nucleus, 87, 108

Hertz, 54

Hydrogen atom, 85

I

Inert gases, 142

Ions, 56

—— charge on, 58

Isotopes, 102

K

Kirchhoff, 114, 123

L

Laue, 166

Light, velocity of, 54

Lithium atom, 90, 149

M

Magnetic properties of the elements, 157

Maxwell, 42, 53

Mendeléev, 77

Molecule, 33

Monochromatic light, 125

Moseley, 170

N

Neon, 144

Newlands, 77

Nitrogen, 152

P

Periodic system, 77

Perrin, 46

Planck, 119, 127

Prout, 25

R

Radium, 63

—— disintegration of, 95

Relativity and the atom, 104, 133

Rutherford, 93

S

Sommerfeld, 122, 134

Stokes, 57

T

Temperature, absolute zero of, 42

U

Uranium, 79

W

Waves, longitudinal and transverse, 124, 165

X

_X_-rays, 70, 167, 168