A New Theory of Optical Series

J. J. Thomson. On the Origin of spectra and Planck’s Law. Phil. Mag. 38, p. 1919.

From the standpoint of the optical properties of the atom, any theory of it must be developed in two directions.

On the one hand, it must describe the laws of serial radiation; on the other, it must give Planck’s distribution of energy in the spectrum of a black body.

Bohr’s atom describes gaseous radiation with the greatest accuracy; but up to now, without additional hypotheses, it has not been possible, on the basis of Bohr’s atom, to obtain a formula for the distribution of energy in the spectrum of a black body. Moreover, the fundamental premises of the theory named are so unusual that, from the standpoint of classical electrodynamics, it seems almost impossible to interpret them.

The work of J. J. Thomson attempts to place a dynamical foundation under Bohr’s atom. Thomson supposes that Bohr’s incomprehensible “arithmetical principles” may be excluded by a suitable generalization of Coulomb’s law. Coulomb’s law, in the author’s opinion, is valid only at distances from the nucleus greater than the dimensions of the atom; the electrical attraction between two charges approaches Coulomb’s law only asymptotically.

Therefore the electrical force acting from the nucleus on the electron, in the most general form, may be written as follows:

\[ f=\frac{Q}{r^{2}}\, f\!\left(\frac{c}{r}\right)\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ (1) \]

where \(c\) is some constant, chosen so that at distances greater than the dimensions of the atom the ratio \(\frac{c}{r}\) is very small.

\(c\) has linear dimensions and, evidently, is a quantity characteristic of a given atom; the function \(f\!\left(\frac{c}{r}\right)\) is equal to unity for small values of \(\frac{c}{r}\) (outside atomic space) and turns many times to zero inside the atom.

Understandably, there is a whole multitude of such functions; Thomson indicates one of them, namely:

\[ f\!\left(\frac{c}{r}\right)= \frac{\operatorname{Sn}\frac{c}{r}}{\frac{c}{r}} = \frac{\operatorname{Sn}x}{x} \]

where \(\frac{c}{r}=x\).

Consequently, the electrical force will be expressed by the formula:

\[ f=\frac{Qe}{c^{2}}\,x^{2}\cdot \frac{\operatorname{Sn}x}{x} = \frac{Qex}{c^{2}}\,\operatorname{Sn}x \ .\ .\ .\ .\ .\ .\ .\ .\ .\ (2) \]

This generalized Coulomb law allows J. Thomson to sketch a possible picture of gaseous radiation. The difficult-to-understand non-radiating orbits of Bohr’s atom are here naturally replaced by the equilibrium positions (zero positions) following from the generalized law of electrical attraction.

In fact, if one sets \(r=\dfrac{c}{n\pi}\), then the intensity of the electric field of the nucleus at the corresponding points vanishes and, consequently, an electron that has arrived at these places will be in equilibrium indefinitely. Once displaced from the position of equilibrium, the electron will begin to oscillate; however, the character of these oscillations is determined, according to J. Thomson, not by an electric force of the type described, but by a special magnetic field. The magnitude of the magnetic force is directed radially along the electric lines of force and is determined by the formula:

\[ H=\mu\,(a^2-r^2)\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ (3) \]

where \(a\) determines the outer boundary of the magnetic field, and \(r\) is the distance of any point from the nucleus.

Thomson evidently chooses this form of the magnetic force in order to obtain the Rydberg law of radiation. Indeed, if the outer boundary of the magnetic field is made to coincide with one of the zero positions, then for

\[ a=\frac{c}{n\pi}\quad \text{for}\quad r=\frac{c}{m\pi}. \]

Hence the magnitude of the magnetic force in a certain zero position of order \(m\) will be:

\[ H=\mu\cdot \frac{c^2}{\pi^2}\left(\frac{1}{n^2}-\frac{1}{m^2}\right)\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ (4) \]

The integrals of the differential equations of motion of an electron in a magnetic field directed along the \(z\)-axis will be:

\[ x=A\,Sn\,\frac{eH}{m}(t-t_o);\quad y=A\,Cs\,\frac{eH}{m}(t-t_o),\quad z=z_o+Bt. \]

And therefore, for the frequency of oscillation of an electron situated in some zero position of order \(m\), we obtain the formula:

\[ \nu=\frac{e}{m}\cdot \frac{c^2}{\pi^2}\cdot \frac{\mu}{2\pi} \left(\frac{1}{n^2}-\frac{1}{m^2}\right)\ .\ .\ .\ .\ .\ .\ .\ .\ .\ (5) \]

And this is the Rydberg law of radiation. If the atom contains several electrons, then, owing to the electrical interaction between them, the position of each of them will be shifted; and if the outer boundary of the magnetic field is placed at this shifted position, then, obviously, instead of the preceding formula one should write the expression:

\[ \nu=k\left[\frac{1}{(n+\delta_1)^2}-\frac{1}{(m+\delta)^2}\right]. \]

Thus, by introducing a special electric and magnetic force near the atom, we can explain the whole diversity of gaseous radiation.

Thomson, having pointed out this possibility, goes further. He tries to choose the magnetic and electric forces in such a way that, besides the explanation of the laws of gaseous radiation, the quantum law of radiation would also result. He does this in the following manner:

Let \(B\) be the flux of magnetic induction and let \(R\) be the electric force, as before of nodal character; let \(\omega\) be the cross section of a tube of flux of electric force; suppose that the drop of magnetic induction along the lines

is proportional at the given point to the fall of the flux of electric force; in other words, let there be the equality:

\[ \frac{dB}{ds}=k\,\frac{d\omega R}{ds}\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ (6) \]

Moreover \(k\) is a function of position, i.e. of \(s\). Hence, integrating, we have:

\[ B_{s_1}=\int_{s_2}^{s_1} k\,\frac{d(\omega R)}{ds}\cdot ds =\left[k\omega R\right]_{s_2}^{s_1} -\int_{s_2}^{s_1}\omega R\cdot \frac{dk}{ds}\cdot ds \]

or

\[ B_{s_1}-B_{s_2}=-\int_{s_2}^{s_1}\omega R\,\frac{dk}{ds}\,ds\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ (7) \]

Let us now choose the factor of proportionality so that the quantity

\[ \omega\,\frac{dk}{ds}=-A=Const \]

then

\[ B_{s_1}-B_{s_2}=A\int_{s_2}^{s_1}Rds\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ (8) \]

Obviously, the integral on the right-hand side expresses the work of the electric force in the transition of the electron from the zero position \(S_2\) to the position \(S_1\).

Thus

\[ B_{s_1}-B_{s_2}=AW \]

multiplying this equality by \(\dfrac{e}{m}\cdot\dfrac{1}{2\pi}\), we obtain:

\[ \nu_{s_1}-\nu_{s_2}=\frac{Ae}{m}\cdot \frac{1}{2\pi}\cdot W\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ (9) \]

In order that the left-hand side agree with the right-hand side in dimension, we must put

\[ \frac{Ae}{m}\cdot \frac{1}{2\pi}=\frac{1}{h}\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ (10) \]

where \(h\) is a constant, of the dimensions of action.
Thus:

\[ W=h\nu_{s_1}-h\nu_{s_2}\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ (11) \]

J. Thomson interprets this formula as Planck’s law of radiation; indeed, an atom of such a kind is not difficult to adapt, by the character of its radiation, to Planck’s resonator. Formula (10) in the general case may be rewritten as:

\[ \frac{Ae}{m}\cdot \frac{1}{2\pi}=\frac{1}{qh}, \]

where \(q\) is an abstract number which may be regarded as an integer.

If an electron situated in a zero position, where the magnetic force vanishes, is knocked out of this position, then in passing to a new zero position it will expend all its kinetic energy in the form of radiation.

The magnitude of this energy is determined from formula (11); and in the present case, with the introduction of the generalized constant of action, for it we shall have:

\[ W = qh\nu \ . . . . . . . . . . . . . . \tag{12} \]

The form of this formula already coincides completely with Planck’s law of radiation for a resonator. If, following J. Thomson, we wished to justify the law of the distribution of energy over the spectrum of a black body in the form in which Planck gave it, then we must evidently choose only those of them for which the transition of an electron from one position of equilibrium to another is governed by formula (12), and not by (11). Transitions of both types will give a formula more general than Planck’s formula.

If we take the relation for \(\nu_s\) in integral form, we obtain:

\[ \nu_s=\frac{Ae}{m\cdot 2\pi}\int_a^s R\,ds \ . . . . . . . . . . \tag{13} \]

\(a\)—determines the outer boundary of the magnetic field, which, as before, coincides with one of the zero positions.

In order to obtain the Rydberg law from this formula, we must generalize the Coulomb force in the appropriate way.

Let, as before:

\[ R=\frac{Qe}{r^2}f\!\left(\frac{c}{r}\right) =\frac{Qe}{c^2}x^2 f(x) =-\frac{Qe}{c^2}x^2\frac{dF(x)}{dx} \ . . . . \tag{14} \]

If this expression is inserted into the relation for \(\nu_s\), we shall have:

\[ \nu_s=\frac{Ae}{m\cdot 2\pi}\int_a^s -\frac{Qe}{c}\frac{dF(x)}{dx}\,dx = \frac{Ac}{m\cdot 2\pi}\cdot\frac{eQ}{c} \left[F(a)-F(s)\right] \]

or

\[ \nu_s=\frac{Qe}{hc}\left[F(a)-F(s)\right] \ . . . . . . . . . \tag{15} \]

Obviously, the form of the function \(F(x)\) must be chosen so that the denominator contains \(x^2\), and so that for small values of \(x\) the derivative \(\frac{dF(x)}{dx}\) would be equal to unity and, inside the atom, \(dx\) would repeatedly vanish.

However, under such conditions the problem is not solved uniquely. One can without difficulty devise a whole series of functions satisfying the indicated restrictions.

Relation (15) can be given another form, proceeding from the following considerations. The work that must be expended in order to dislodge an electron from a position of equilibrium and carry it off to infinity, on the basis of formula (4), may be written as

\[ \int_{r_0}^{\infty} eR\,dr = -\frac{Qe}{c}\int_{x_0}^{0}\frac{dF(x)}{dx}\,dx = -\frac{Qe}{c}\left[F(0)-F(x_0)\right] \]

where \(x_0\) is the magnitude determining the limiting position of the electron.

On the other hand, the same work, expressed through the ionizing potential \(V\), will be \(Ve\); consequently,

\[ \frac{Qe}{c}\left[ F(o) - Fx(x_0) \right] = Ve \]

whence

\[ c = \frac{Q}{V}\left[ F(o) - F(x_0) \right]. \]

Substituting this into formula (15), we shall have:

\[ v_s = \frac{eV}{h\left[ F(o)-F(x_0) \right]} \left[ F(a)-F(x) \right] \qquad . . . . \ (16) \]

Thomson assumes

\[ F(x)=\left\{\frac{Sn^2 x}{x^2}+Snx\,Cs^2x\right\}. \]

To find the zero positions, Thomson uses the solutions of the following transcendental equation: \(Csx=0\), which is correct only approximately. Therefore the final form of formula (16) will be as follows:

\[ v_s= \frac{eV}{h\left(1-\frac{4}{\pi^2}\right)} \left\{ \frac{1}{(P_0+0.5)^2} - \frac{1}{(P+0.5)^2} \right\}, \]

where \(P_0\) and \(P\) are integral values.

Or

\[ v_s= \frac{eV\cdot\pi^2}{h(\pi^2-4)} \left\{ \frac{1}{(P_0+0.5)^2} - \frac{1}{(P+0.5)^2} \right\} \qquad . . . . \ (17) \]

It is difficult, of course, to say what significance Thomson’s work may have for optical phenomena connected with the structure of the atom; nevertheless, despite a certain fantastic character of his considerations, here we are dealing with an interesting attempt to circumvent the cabbalism introduced into science by the theory of quanta. The effort to give a dynamic interpretation of the mysterious principles of arithmetic measurement can only be welcomed.

A. Predvoditelev.