The Mathematical Theory of Electricity and Magnetism (5th ed, 1927) — part 38 of 39

We notice first that equations (768) are identical with the six equations (767) which form a system symmetrical with respect to the four coordinates x, y, z, t. This of course ensures that the equations satisfy the relativity condition. The first three of this system of six equations contain only the three coordinates x, y, z; t is entirely absent from this set of three. From the set of three equations from which r was absent we obtained equation (769). From the corresponding three sets of equations from which x, y and z in turn are absent, it is of course possible to obtain three other equations of similar type. These are found to be

1 da

dZ

dY\

Cdt

dy

dz

ldb

dX

dZ\

"dxi

Cdt''

dz

ldc

dY

dX

Cdt"

dx

dy)

.(771),

which are simply Maxwell's equations of current induction (cf. § 574).

On differentiating the three equations on the right of the system (768) with respect to x, y, z and adding, we find

dx dY d_z_ /W w 3^\ n/a/ d_G dm m

lte+dy + dz~ [dx* + dy* + dz' J C dt\dx+ dy + dz ) "'( '" Precisely as in § 640, it can be shewn that F, G, H and M* are not uniquely determined when the electric and magnetic forces are given. We again suppose them to conform to the additional equation (643), namely

dF d_G dH_ 18t 7

dx+ dy+ dz~ C dt ( }'

which was imposed on them before. This equation satisfies the relativity condition, being symmetrical in the four coordinates x, y, z,t\ in terms of the notation used in equation (764) it has the form

dF dG dH dK =

dx dy dz dr

and so expresses that the vector (F, G, H, K) is of zero divergence. If we now introduce p, defined by

dX dY dZ . ._.,

dx-+dy^+Yz=^ <774)'

equation (772) assumes the form

82^ d2^T d2S¥ U2f . /lyl_ex

w+-df + ~^~c:^ = ~ p ( }'

704, 705] Generalised Relativity 627

of which the solution, obtained as in § 645, is

^=mW\dxdydz (776)^

Since our equations have all satisfied the relativity condition, these formulae must remain true for any rotation of the axes in the x, y, z, t space. Writing

4 dx* df "" dz- C2 dt '

we notice that V42 in terms of the coordinates x, y, zy t assumes the form

4 ~ dx* + dy" + dz* + dr* ' which is obviously invariant for any rotation of the axes.

Let us now take equation (775) which, in x, y, z, r coordinates, has the form

V42^ = -4ttp (777),

and transform to a new set of orthogonal axes, obtained by rotating the old

axes in the x, y, z, r space. As has just been seen the operator V42 will be

the same in the new axes as in the old. M* is the component along the axis

of t of the vector

-iF, -iO, -%H, -iK,

while p is the corresponding component of the vector

dx dy dz PTt> PTt' Pd~r'P or of the vector

. u . v . w -1Pq> ~~1Pq> ~lP Jj> P'

where V, v, w are the components of velocity of p. Thus after transfor- mation, equation (777) becomes

V,2 (- ikF - il2 G - il3H + l^Y) = - 4tt (- ip ^ k - ip ^l2- ip ^k + p) ,

where l1} l2, l3, I4, are the direction cosines of the old axis of r.

Since this equation must be true for all values of l1} l2) l3 and l4, we deduce at once

V42JP=-47rpg (778),

and two similar equations. Returning to the coordinates x, y, z, t, these

assume the form

dF dF d2F 1 d2F . u ,,___.

dx* + df + ^VW= P~G ( l

and similar equations in G and H. These together with equation (775) constitute the equations of propagation of the four potentials F, G, H, AF.

40—2

628 The Theory of Relativity [ch. xx

By direct solution of equation (779) we obtain the value of F in the form

F = jjj[Pu]dx^dydz (780X

with similar equations for G and 17.

Using equations (779) and (768) we find

1 dnV (&F c*F &F\ ~~GdxJt\d^ + dy2 + dz* J '

and, by use of equation (773), this

~ dx [dx + dy dz) \da** dy* + dz* J

d(d_G_dF\ d_(dF_dE\ dy\bx dy) dz\dz dx J

= £-f (781),

dy dz

giving Maxwell's equations of magnetic force.

Differentiating this and the two similar equations with respect to x, y, z and adding, we obtain, by the use of equation (774),

d , . d , . d , . dp _

which is the equation of continuity (cf. § 622), shewing that electric charges have a permanent existence.

706. Einstein's gravitational theory was accepted as soon as the phenomena it predicted in opposition to the classical Newtonian theory were actually observed. It is impossible for Weyl's electromagnetic theory to establish itself in a similar manner since the phenomena it predicts are precisely identical with those of the classical theory of Maxwell. As a direct observational test is impossible, Weyl's theory can only be judged by its inherent plausibility. It may be said to be the only theory at present in the field, Maxwell's mechanism of stresses and strains in an ether having, for all practical purposes, received its deathblow by the establishment of the restricted relativity theory. In its favour may be said that it gives a consistent account of electromagnetic phenomena on lines which, in view of the convincing experi- mental confirmation obtained for the parallel theory of gravitation, must be admitted to be in accordance with the general workings of nature. The principal objection which can be brought against it has already been men- tioned (§ 704).

CHAPTER XXI

THE ELECTRICAL STRUCTURE OF MATTER

707. By the end of the nineteenth century, it was generally believed that all physical phenomena, with the possible exception of gravitation, were of electric origin. Associated with this was the belief that matter was a purely electrical structure. Positive and negative charges, arranged in combination in different ways, were supposed to give rise to all the various kinds of matter in the universe, changes in the positions and arrangements of these charges being regarded as the origin of all the phenomena of physics and chemistry.

Various conjectures were made as to the actual arrangement of the positive and negative charges in matter, but positive knowledge was only obtained when the new experimental methods made available by the discovery of radioactive substances were brought into action.

708. The special properties of radioactive substances originate from their spontaneously and continuously emitting rays of various kinds. If a beam of the emitted rays is allowed to traverse a strong magnetic field, it is found to be split up into three distinct beams, two of which are deflected in opposite directions, while the third passes straight on. The three types of rays in these three beams are known as a rays,' /? rays and 7 rays respectively. We have seen (§ 631) that a charged electric particle traversing a magnetic field will describe a circle of radius vmC/eH. The curvature of the paths of the a rays is found to be the same as if the rays were positively charged particles, that of the paths of the /3 rays is curved as though they were negatively charged particles, while the absence of curvature of the paths of the 7 rays suggests that they are not charged particles at all. The charges can be measured by shooting the rays into an electrometer. It is, found that the a rays are rapidly moving particles each with a positive charge equal to twice the charge on an electron, and with a mass almost exactly equal to that of the helium atom. The /3 rays prove simply to be negative electrons moving with velocities comparable with that of light. The 7 rays are found to be radiation of the same general nature as light or X-rays, but of exceedingly short wave-length.

If a thin piece of metal foil is placed in the path of a beam of a particles, the majority of the particles pass through without their paths shewing any appreciable deflection, but a small fraction of the total number are substantially deflected. From experiments under varied conditions, the deflections are found to be such as would be expected if only isolated small areas of the foil had the power of appreciably deflecting the particles, and the number of such areas is found to be equal to the number of atoms in the foil. Moreover the deflections observed are precisely those which would be expected if each of

630 The Electrical Structure of Matter [ch. xxi

these areas had at its centre a fixed particle which repelled the a particle according to the law of the inverse square of the distance.

Investigations of this type led Sir E. Rutherford to put forward in 1911 a theory of the structure of the atom, generally known as the nuclear theory, which has stood the test of time and has now won universal acceptance. According to this theory, an atom consists of a positively charged central nucleus surrounded by a number of negative electrons, the charge on the central nucleus being such that the total charge of the atom is zero.

The simplest atom is the hydrogen atom, consisting of only one electron and the positive nucleus. If — e is the charge of an electron, that of the positive nucleus of the hydrogen atom is of course + e. Next in order comes the helium atom consisting of two electrons and a positive nucleus of charge

• 2e. This positive nucleus is found to be exactly identical with the a particles of radium radiations.

With insignificant exceptions chemical elements are known having re- spectively 1, 2, 3, ... electrons, and consequently nuclei of charges +e, +2e, 4- Se, ..., up to 60 electrons and a nuclear charge + 60e (Neodymium). After this gaps appear in the sequence, which appears to end altogether at Uranium with 92 electrons and a charge + 92e. The number which fixes the position of an element in this sequence is called its "atomic number"; for the elements of low atomic number, the atomic number is approximately equal to half the atomic weight. The first few elements with their atomic numbers are as follows :

Atomic Number

1

2

Element Hydrogen Helium

Atomic Weight

1-008

4

Isotopes

3

Lithium

6-94

6,7

4 5

Beryllium Boron

9-02 10-82

10, 11

6

Carbon

12

7

Nitrogen

1401

8 9

Oxygen Fluorine

16 19

10

Neon

20-2

20, 22

11

Sodium

23

12

Magnesium

24-32

24, 25, 26

13

Aluminium

26-96

14

Silicon

28-06

28, 29, 30

15 16 17

Phosphorus

Sulphur

Chlorine

31-02 32-06 35-46

35, 37

18 19

Argon Potassium

39-88 39-10

36, 40 39,41

20

Calcium

40-07

40, 44

709. As we have seen the mass of the negative electron is 9"00 x 10-28 grammes. The mass of the hydrogen atom is about 1845 times this, so that

708-710] The Electrical Structure of Matter 631

the nucleus has about 1844 times the mass of the electron. In the helium atom the mass of the nucleus is about 7320 times the mass of a single electron and so outweighs the two attendant electrons in the ratio of about 3660 to one. As the ratio of atomic weight to number of electrons is about the same in all elements except hydrogen, it follows that in all elements other than hydrogen only about one part in 3660 of the total mass resides outside the central nucleus. For most purposes we may think of the centre of gravity of an atom as coinciding with its nucleus.

Assuming the mass of the negative electron to be wholly electromagnetic, we have seen that its radius must be of the order of 2 x 10~13 cms. If the mass of the nucleus also is wholly electromagnetic, its radius must be much smaller than that of the negative electron; that of the nucleus of the hydrogen atom, for instance, would be about 10-16 cms. There is direct evidence that the nucleus is exceedingly small, experiments on the scattering of a rays having shewn that a particles can pass within 2 x 10-13 cms. of the centre of an atomic nucleus, and yet be deflected in accordance with the ordinary law of the inverse square.

These figures shew that both nuclei and electrons are very small in com- parison with atoms. The hydrogen atom whose radius is approximately 0*53 x 10-8 cms. is made up of only two constituent parts, each of radius 2 x 10-13 cms. or less. Since a positive and a negative charge cannot stand in statical equilibrium at a distance apart equal to several thousands of times the radius of either, we must suppose that the two charges maintain their distance as a consequence of orbital motion. The negative electron does not fall onto the positive electron for the same reason for which the earth does not fall onto the sun. In many respects an atom may be compared to a solar system, the heavy positive nucleus at the centre of the atom representing the sun and the electrons representing planets, the law of force between the nucleus and the electrons being the same as that between sun and planets, namely a force of attraction varying as the inverse square of the distance.

710. According to the analysis of § 650, a negative electron describing an orbit about a nucleus must radiate energy. If E, e are the charges on the nucleus and electron respectively, and m the mass of the electron, the ac- celeration of the electron towards the nucleus is Ee/mr2, while the acceleration of the nucleus may be neglected in comparison, on account of its much greater mass. From formula (663), the rate of emission of radiation per unit time is

^[{Xeur + (tevy + ^ewy} = w^? (782).

For the hydrogen atom, consisting only of one electron and a nucleus of equal charge, we may put

E= - e= 4-774 x 10"10 el. stat. units, r = 0"53 x 10~8 cms.,

632 The Electrical Structure of Matter [oh. xxi

from which the rate of radiation is found to be 0*46 ergs per second. As a result of this loss of energy, the radius of the orbit ought to decrease. When the radius is r, the energy of the orbit is readily fouud to be — e2J2r, so that the rate of decrease of energy is

e2 dr ~2? di'

Putting this equal to 0"46 ergs a second, we find that — dr/dt must be equal to about 112 cms. a second. Since the radius of the orbit initially is only 0'53 x 10-8 cms., the distance between the two constituents of the hydrogen atom ought to vanish altogether in a fraction of a millionth of a second.

Even in the light of common sense such a conclusion is preposterous ; it is more so in the light of exact knowledge. So far as we know all hydrogen atoms, no matter how or where selected, are identical structures, all giving the same spectrum and all having precisely the same radius except for a reservation which will shortly be explained. There is not the slightest indication of any secular change in their properties, and a change of the rapidity of that just calculated is utterly out of the question.

The conclusion to which we are driven is not merely that a normal hydrogen atom of the type we have been considering does not radiate as rapidly as is predicted by equation (782), but that it does not radiate at all. In some way the whole theory which has led to the conclusion that an accelerated electron must radiate energy is in need of amendment.

711. This discovery, if it stood alone, would be extremely disconcerting. In actual fact it does not stand alone ; it is only one of a long series of discoveries, each of which has indicated, with very little room for doubt, that the classical mechanics of Newton and the classical electrodynamics of Maxwell both fail when applied to atomic phenomena. Since the beginning of the present century the need has been recognised for a wholly new system of dynamics, such as shall be applicable to physical phenomena on atomic and sub-atomic scales, and shall merge into the Newtonian and Maxwellian dynamics in the case of larger scale phenomena. In spite of much labour, this system of dynamics has not yet been found in its entirety. Fragments are known with fair certainty, although it is of course impossible to feel absolute confidence in any part of the system until the whole has been pieced together and seen to form a consistent structure. Fortunately the parts which are known with the nearest approximation to certainty are precisely those which are necessary in the discussion of the subject of the present chapter, the electrical structure of matter.

712. In discussing the motion of a dynamical system as predicted by the classical mechanics, the usual procedure is to start from the general equations of motion, which are differential equations of the second degree, and attempt

710-712] The Electrical Structure of Matter 633

in the first place to discover one or more first integrals of these equations. In many problems, for instance, the equations of energy and of linear and angular momentum figure as first integrals of the equations of motion. Each time a first integral is derived from the equations of motion a constant of integration is introduced, different values of this constant representing different states of the dynamical system. Under the classical mechanics these constants of integration could usually have any value we chose to assign to them, or, if this was not possible, there was at least a finite continuous range of values open to each constant of integration.

The distinguishing feature of the new dynamics is that there are no longer continuous ranges of values open to the constants of integration, but only certain definite discrete values. Generally speaking, the values available for each constant of integration are an infinite set associated with the natural integers 1, 2, 3, ... as, for instance, the set of values obtained by taking integral multiples of a given constant. Thus the constants of integration shew a sort of atomicity.

A parallel can be found in the atomicity of electricity. In the earlier chapters of this book, we treated electric charges as being capable of con- tinuous variation; for instance, in calculating the energy of a condenser in § 97 we treated the electric charge e as changing continuously and integrated with respect to de. In actual fact we know that in charging a condenser, the charge must move by whole electrons at a time. The procedure of treating the charge as capable of continuous variation was, nevertheless, legitimate so long as we were dealing with charges of billions of electrons; our step de could be quite insignificant in comparison with the total value of e, although representing perhaps a million electrons. The same procedure would, how- ever, lead to disastrous errors if it were followed in problems of atomic physics. An atom normally is an electrically neutral structure, the total charge of the positive nucleus and the negative electrons being zero. It is possible to charge it positively by withdrawing one, two, three or more electrons. Thus the atom can have a positive charge E, but E is restricted to having the discrete values e, 2e, Se, ... etc.; it may not be regarded as capable of con- tinuous variation.

In precisely the same way, although for reasons not clearly understood, the constants of integration in the new dynamics are limited to definite discrete values which may perhaps, in a similar manner, be integral multiples of a fundamental constant. It may for instance be possible for a constant of integration to have any one of the values 0, c, 2c, 3c, ... but no more possible for it to have the values he or |c than it is possible for an atom to carry the charges \e or |e.

There is a further difference between the new dynamics and the old. Under the old dynamics a constant of integration was a true constant, and retained its value absolutely unchanged until the conditions of the problem

634 The Electrical Structure of Matter [ch. xxi

altered. The new dynamics has no knowledge of such absolute constancy. If a constant of integration can have any one of the values c, 2c, 3c, ... and has one of these values, say 2c, at a given instant, there is a possibility of its value taking a jump from the value 2c to some other value. It is usual to think of these jumps as occurring absolutely spontaneously, although this conception is probably only a cover for our ignorance of some underlying mechanism.

The new mechanics was originally developed by Planck and others from a study of the phenomena of black-body radiation. In 1913 Prof. N. Bohr applied the new system to the problem of atomic motions and was led to a theory of the nature of these motions which gained immediate acceptance and which has stood the test of time. This theory we shall now explain.

Bohr's Theory.

713. Let us consider the simplest case of a single electron of charge e describing an orbit about a nucleus of charge E, which, on account of its much greater mass, is treated as a fixed centre of force. In ordinary polar coordinates the equations of motion of the electron are

eE m{r-r82) = -~ (783),

™jt(r20) = 0 (784).

Equation (784) at once yields the integral

mr20 -=cons (785),

but in accordance with the principles just explained, we may not suppose all values to be permissible for the constant on the right. We shall suppose that it is restricted to being an integral multiple of a fundamental constant which, to agree with an established notation, we shall denote by hjlir. Thus equation (785) must be written in the form

mr20=Th/2>ir (786),

where r is an integer. Using this value for 6 to eliminate the angle 0 from equation (783), we obtain

1 frhy eE ,HoH.

r3m \27r/ r which, on integration with respect to r, yields the integral

•2 , 1 fTh

r2m \zir

eE -— = cons (788).

Utilising equation (786), this assumes the form

eE £??i(r2 + r202) = cons (789),

712-715] Bohr's Theory 635

which is at once seen to be the integral of energy, but again the constant on the right must be restricted to certain definite values, just as was the case with the right-hand member of equation (785).

714. Before proceeding to the comparatively complicated general case, let us consider the simple problem of circular orbits. Assuming that circular orbits are possible, these will be obtained by putting r = 0 in equation (787) giving

T2/l2

r=i^s <70O>-

As has been seen, the hydrogen atom consists of a single electron describing an orbit about a nucleus of charge e, while the helium atom consists of two electrons describing orbits about a nucleus of charge 2e. Thus on putting E=e the foregoing analysis is applicable to a normal hydrogen atom, while on putting E = 2e it becomes applicable to a helium atom from which one electron has been removed — i.e., to a positively charged helium atom of charge e.

Thus the circular orbits in both these atoms are obtained by giving various integral values to t in equation (790). The radii of the various circular orbits which are possible for either atom are seen to be proportional to the squares of the natural numbers, and so to 1, 4, 9, 16, 25, ..., while the radii possible for the positively charged helium atom are exactly half those which are possible for the hydrogen atom.

715. Mention has already been made of the possibility of what appear to be spontaneous jumps taking place in the values of the constants of integra- tion, and therefore also in the value of t in equation (790). We proceed to discuss these changes.

Corresponding to the values of r and 6 determined by equations (786) and (790), the negative energy W of a circular orbit is found to be given by

W = -lmr6 + — = ^h% (791).

The values of W form a discrete series, proportional to the inverse squares of the natural numbers. If a spontaneous change occurs in t it can only be from a higher value of t to a lower value, since any change in the reverse direction would lessen the value of W and so increase the energy of the system. If Ta is greater than t2, both Tj and t2 being integral numbers, a spontaneous jump from t = tx to t = t2 results in the system losing energy of amount

27r2e2#2m, / 1 1 \ fYrnex,

.(792).

According to Bohr's theory this lost energy leaves the system in the form of radiation; indeed the theory supposes that the atoms we have been

636 The Electrical Structure of Matter [ch. xxi

considering do not emit radiation at all except on the occasion of jumps of the kind we have been considering.

A change in the value of t, then, results in a change in the amount of radiant energy in the space surrounding the atom. Now Planck, from his study of black-body radiation to which we have already referred, had concluded that the radiant energy of an enclosure, if it changed at all, must change by jumps. The radiation in any space or enclosure can be analysed by Fourier's theorem into trains of waves of different frequencies, and the energy of the radiation can be regarded as the sum of the energies of these trains of waves. The total energy of radiation may accordingly be thought of as the sum of the contributions from disturbances of different frequencies. Planck's con- clusion was that the energy of radiation of each frequency must be an integral multiple of a certain unit, this unit being equal to a fundamental constant h multiplied by the frequency v of the radiation in question. Planck called this unit a "quantum." Thus the quantum of energy of frequency v was equal to hv, and the total energy of frequency v, being an integral number of quanta, was restricted to being of amount rhv where t was an integral number. It followed that any change in the field of radiation must consist of a jump in the energy of the radiation of a definite frequency and must be equal in amount to an integral number of quanta of this radiation.

In view of these results of Planck, it was natural for Bohr to suppose that when energy of the amount specified by formula (792) was set free into space, it formed one quantum of energy. This supposition by itself suffices to de- termine the frequency of the radiated energy. For hv, the quantum of energy, must be equal to expression (792) in order to satisfy the principle of the conservation of energy, and this gives the relation

'-*&-£) (793>'

where

^=™> (794).

Thus Bohr's theory restricts the radiation emitted from a hydrogen atom, or from a positively charged helium atom, to one of the frequencies specified by formula (793) where t2 and rx are positive integers. This gives a series of detached frequencies, or what the spectroscopist calls a "line-spectrum," whereas it is easily seen that the classical system of electrodynamics would have predicted a continuous range of frequencies or a "continuous spectrum."

716. The spectrum of hydrogen, as observed in the light from an ordinary vacuum tube, consists of a series of lines, the strongest of which (Ha) lies at the red end of the spectrum while the remainder (Hp, Hy, Hs, ...) spread out, at ever diminishing distances, towards the violet. As far back as 1885 Balmer had found that the frequencies of the different members of this series could

715-717] Bohr's Theory 637

be represented, with very great precision (about one part in 200,000), by giving to n the values 3, 4, 5, ... in the formula

— xik-xl (795)>

,22

with iV= 32902 x 1015. It is clear that if we are free to assign this value to the N which is denned by equation (794), then Bohr's theoretical formula (793) will contain Balmer's observational formula as the special series of lines obtained on taking t2 = 2. We are not free to assign any arbitrary value to the N of equation (794) since the value of every quantity which enters into N is known. The values of e and m have already been given (§ 28); for hydrogen E is equal to e, and the value of h can be obtained from a study of the spectrum of black body radiation and in a variety of other ways. The best determinations of h give

h = 6-545 xlO-27,

and on substituting these values for h, e and m, the value of iV given by equation (794) is found to be

N= 3-294 xlO18,

which agrees, to within the errors in the determination of e and h, with the observed value N** 3'290 x 1015.

It is, then, clear that N has precisely the required value in equation (793), and that Bohr's theoretical formula (793) includes Balmer's observational formula (795) as a special case. It was this success of Bohr's theory that brought about its immediate acceptance by the majority of physicists. The theory, however, requires that Balmer's series should be only one of an infinite number. Other series are obtained by putting t2 equal to 1, 3, 4, 5, ... oo in equation (793), and, if Bohr's theory is correct, these series ought equally to appear in the hydrogen spectrum. The majority of the lines of these series were unknown when Bohr's theory was first published, but all the predicted lines have been found which lie in the region of the spectrum which is acces- sible to observation.

717. According to this theory the spectrum of positively charged helium is the same as that of hydrogen except that E in equation (794) must be replaced by 2e instead of by e, as for hydrogen. Or, if we denote the value of N for hydrogen by NH, the value of N for helium will be ^NH and the spectrum will be given by

"***&-&■ ■

In this formula even values for both t2 and X] give values of v which are identical with the whole system of values of v given by equation (793) for the hydrogen spectrum. The spectrum of ionised helium ought accordingly to shew all the lines of the normal hydrogen spectrum and, in addition, the

638 The Electrical Structure of Matter [en. xxi

various lines which are obtained by giving odd values to t2 or t2 or both in the above formula. This is in actual fact found to be the case, except for a reservation which must now be explained.

In deducing formula (793) we assumed the nucleus to be so massive that it could be treated as a fixed centre of force. In actual fact the nucleus of the hydrogen atom has about 1844 times the mass of the negative electron, so that this assumption leads to an error of the order of one in 1844 in the value of N for hydrogen. For helium the ratio of the two masses is about 7300 to one, and the predicted value of N for helium is in error by about one part in 7300. When allowance is made for these errors, the value of iV for helium is not exactly four times the value for hydrogen, and the difference is shewn spectroscopically by the hydrogen spectrum not coinciding exactly with the corresponding lines of the helium spectrum. By measuring the distance between corresponding lines Fowler deduced the value 1836 for the mass-ratio hydrogen nucleus and the negative electron. Allowing for the relativity correction and other refinements Paschen subsequently amended this to 1843-7, a value which is in excellent agreement with the values of this ratio determined by other methods.

718. So far we have considered only circular orbits. The orbit of an electron about a nucleus is, however, in no way restricted to being circular; as the law of force is that of the inverse square the orbit may be elliptic, parabolic or hyperbolic. But, just as the radius of a circular orbit is restricted to having certain definite values, so the eccentricity of an elliptic orbit, as well as its major axis, is restricted to having certain values.

An adequate discussion of the manner of calculating these restrictions would carry us too far outside the scope of the present book. The following will, however, suffice for the immediate purpose in hand.

Let q1} q2) q3, ... be the generalised coordinates of any dynamical system, defined as in § 548, and let ply p2, p3, ... be the corresponding momenta defined by

!>! = !? etc (796),

oqx

where E is the energy expressed as a function of q1} q2, q3> ••• and q1} q2, q3) ....

For certain dynamical systems it is possible to deduce, by ordinary mechanics,

a number of equations of motion such that only one coordinate and the

corresponding momentum, e.g., q1 and p1} enter in each. As we shall at once

see, equation (784) is an equation of this type involving only the momentum

corresponding to the coordinate 0, and equation (787) is another, for it involves

only the coordinate r and the corresponding momentum mr. The solution of

such an equation will be the same as if the whole system had only the one

degree of freedom corresponding to this one coordinate, so that either the

coordinate will increase or decrease beyond limit, or will oscillate repeatedly

717, 718]

Bohr's Theory

639

between two constant extreme values. In the latter case it is found that the proper restriction to apply to the motion of any coordinate qx is that given by

JPldqi = Th (797),

where the integration extends throughout a whole oscillation in the value of q, h is the constant already denned, and r is any integral number.

For instance if qx is the coordinate 6 in the orbit of an electron about a nucleus, the momentum, as denned by equation (796), is px = mr26. Equation (785) may accordingly be written in the form

px = constant,

which is of the required form, since it contains no coordinates or momenta other than px and qx. A complete oscillation extends from qi = 0 to q1 = 27r, so that equation (797) assumes the form

2irp1 = rh, or mr'2 6 = rhj^ir,

which is identical with our equation (786).

Similarly, if q2 is the coordinate r of the same orbit, p2 = mr so that equation (787) is of the required form. The first integral of this equation is equation (788), and this may be written in the form

eE

= -w,

£i + — (-

m r-m \2i

where W is constant, the negative energy of the orbit. This equation gives p£ as a quadratic function of 1/r ; it may be written in the form

*■' = \h) U " r) [r ~ rj '

where r1} r2 are determined by

(t)~r^ S)'S+S-* w

In the course of a complete oscillation r varies from rx to r2 and then back to rx. Thus the path of integration in equation (797) may be taken to be twice the range from rx to r2, and the equation assumes the form

rh\ [r* 2WJ,,

dr = r'h

.(799),

1/1 1\1

r) \r r2j __

where r is a new integer. Evaluating the integral by the transformation

we readily obtain 2

• = -cos20 + -sin26>, r rx r2

r,

1

dr =

IT

V(n^)

[Vr2 - VrJ2.