In the calculations on page 5 we further made use of the more special assumptions, viz. that the different stationary states correspond to the emission of a different number of Planck’s energy-quanta, and that the frequency of the radiation emitted during the passing of the system from a state in which no energy is yet radiated out to one of the stationary states, is equal to half the frequency of revolution of the electron in the latter state. We can, however (see § 3), also arrive at the expressions (3) for the stationary states by using assumptions of somewhat different form. We shall, therefore, postpone the discussion of the special assumptions, and first show how by the help of the above principal assumptions, and of the expressions (3) for the stationary states, we can account for the line-spectrum of hydrogen.
§ 2. EMISSION OF LINE-SPECTRA.
Spectrum of Hydrogen.—General evidence indicates that an atom of hydrogen consists simply of a single electron rotating round a positive nucleus of charge eaz. The reformation of a hydrogen atom, when the electron has been removed to great distances away from the nucleus—e.g. by the effect of electrical discharge in a vacuum tube—will accordingly correspond to the binding of an electron by a positive nucleus considered on p. 5. If in (3) we put E = e, we get for the total amount of energy radiated out by the formation of one of the stationary states,
The amount of energy emitted by the passing of the system from a state corresponding to τ = τ 1 to one corresponding to τ = τ2, is consequently
If now we suppose that the radiation in question is homogeneous, and that the amount of energy emitted is equal to hν, where ν is the frequency of the radiation, we getWτ2 − Wτ1 = hν,
and from this
(3)
We see that this expression accounts for the law connecting lines in the spectrum of hydrogen. If we put τ2 = 2 and let τ1 vary, we get the ordinary Balmer series. If we put τ2 = 3, we get the series in the ultra-red observed by Paschenba and previously suspected by Ritz. If we put τ2 = 1 and τ2 = 4, 5, . . . , we get series respectively in the extreme ultra-violet and the extreme ultra-red, which are not observed, but the existence of which may be expected.
The agreement in question is quantitative as well as qualitative. Puttinge = 4.7 × 10−10, e/m = 5.31 × 1017, h = 6.5 × 10 −27
we get
The observed value for the factor outside the bracket in the formula (4) is3.290 × 1015.
The agreement between the theoretical and observed values is inside the uncertainty due to experimental errors in the constants entering in the expression for the theoretical value. We shall in § 3 return to consider the possible importance of the agreement in question.
It may be remarked that the fact, that it has not been possible to observe more than 12 lines of the Balmer series in experiments with vacuum tubes, while 33 lines are observed in the spectra of some celestial bodies, is just what we should expect from the above theory. According to the equation (3) the diameter of the orbit of the electron in the different stationary states is proportional to τ2. For τ = 12 the diameter is equal to 1.6 × 10−6 cm., or equal to the mean distance between the molecules in a gas at a pressure of about 7 mm. mercury; for τ = 33 the diameter is equal to 1.2 × 10−5 cm., corresponding to the mean distance of the molecules at a pressure of about 0.02 mm. mercury. According to the theory the necessary condition for the appearance of a great number of lines is therefore a very small density of the gas; for simultaneously to obtain an intensity sufficient for observation the space filled with the gas must be very great. If the theory is right, we may therefore never expect to be able in experiments with vacuum tubes to observe the lines corresponding to high numbers of the Balmer series of the emission spectrum of hydrogen; it might, however, be possible to observe the lines by investigation of the absorption spectrum of this gas (see § 4).
It will be observed that we in the above way do not obtain other series of lines, generally ascribed to hydrogen; for instance, the series first observed by Pickeringbb in the spectrum of the star ζ Puppis, and the set of series recently found by Fowlerbc by experiments with vacuum tubes containing a mixture of hydrogen and helium. We shall, however, see that, by help of the above theory, we can account naturally for these series of lines if we ascribe them to helium.
A neutral atom of the latter element consists. according to Rutherford’s theory, of a positive nucleus of charge 2e and two electrons. Now considering the binding of a single electron by a helium nucleus, we get, putting E = 2e in the expressions (3) on page 5, and proceeding in exactly the same way as above,
If we in this formula put, τ2 = 1 or τ2 = 2, we get series of lines in the extreme ultra-violet. If we put τ2 = 3, and let τ1 vary, we get a series which includes 2 of the series observed by Fowler, and denoted by him as the first and second principal series of the hydrogen spectrum. If we put τ2 = 4, we get the series observed by Pickering in the spectrum of ζ Puppis. Every second of the lines in this series is identical with a line in the Balmer series of the hydrogen spectrum; the presence of hydrogen in the star in question may therefore account for the fact that these lines are of a greater intensity than the rest of the lines in the series. The series is also observed in the experiments of Fowler, and denoted in his paper as the Sharp series of the hydrogen spectrum. If we finally in the above formula put τ2 = 5, 6, . . . , we get series, the strong lines of which are to be expected in the ultra-red.
The reason why the spectrum considered is not observed in ordinary helium tubes may be that in such tubes the ionization not so complete as in the star considered or in the experiments of Fowler, where a strong discharge was sent through a mixture of hydrogen and helium. The condition for the appearance of the spectrum is, according to the above theory, that helium atoms are present in a state in which they have lost both their electrons. Now we must assume the amount of energy to be used in removing the second electron from a helium atom is much greater than that to be used in removing the first. Further, it is known from experiments on positive rays, that hydrogen atoms can acquire a negative charge; therefore the presence of hydrogen in the experiments of Fowler may effect that more electrons are removed from some of the helium atoms than would be the case if only helium were present.
Spectra of other substances.—In case of systems containing more electrons we must—in conformity with the result of experiments—expect more complicated laws for the line-spectra those considered. I shall try to show that the view taken above allows, at any rate, a certain understanding of the laws observed.
According to Rydberg’s theory—with the generalization given by Ritzbd—the frequency corresponding to the lines of the spectrum of an element call be expressed byν = Fr(τ1 minus Fs (τ2),
where τ1 and τ2 are entire numbers, and F1, F2, F2, . . . are functions of τ which approximately are equal to K/(τ + a1)2, K/(τ + a2)2, . . . K is a universal constant, equal to the factor outside the bracket in the formula (4) for the spectrum of hydrogen. The different series appear if we put τ1 or τ2 equal to a fixed number and let the other vary.
The circumstance that the frequency can be written as a difference between two functions of entire numbers suggests an origin of the lines in the spectra in question similar to the one we have assumed for hydrogen; i.e. that the lines correspond to a radiation emitted during the passing of the system between two different stationary states. For systems containing more than one electron the detailed discussion may be very complicated, as there will be many different configurations of the electrons which can be taken into consideration as stationary states. This may account for the different sets of series in the line spectra emitted from the substances in question. Here I shall only try to show how, by help of the theory, it can be simply explained that the constant K entering in Rydberg’s formula is the same for all substances.
Let us assume that the spectrum in question corresponds to the radiation emitted during the binding of an electron; and let us further assume that the system including the electron considered is neutral. The
force, on the electron, when at a great distance apart from the nucleus and the electrons previously bound, will be very nearly the same as in the above case of the binding of an electron by a hydrogen nucleus. The energy corresponding to one of the stationary states will therefore for τ great be very nearly equal to that given by the expression (3) on p. 5, if we put E = e. For τ great we consequently get
in conformity with Rydberg’s theory.
§ 3. GENERAL CONSIDERATIONS CONTINUED
We shall now return to the discussion (see p. 7) of the special assumptions used in deducing the expressions (3) on p. 5 for the stationary states of a system consisting of an electron rotating round a nucleus.
For one, we have assumed that the different stationary states correspond to an emission of a different number of energy-quanta. Considering systems in which the frequency is a function of the energy, this assumption, however, may be regarded as improbable; for as soon as one quantum is sent out the frequency is altered. We shall now see that we can leave the assumption used and still retain the equation (2) on p. 5, and thereby the formal analogy with Planck’s theory.
Firstly, it will be observed that it has not been necessary, in order to account for the law of the spectra by help of the expressions (3) for the stationary states, to assume that in any case a radiation is sent out corresponding to more than a single energy-quantum, hν. Further information on the frequency of the radiation may be obtained by comparing calculations of the energy radiation in the region of slow vibrations based on the above assumptions with calculations based on the ordinary mechanics. As is known, calculations on the latter basis are in agreement with experiments on the energy radiation in the named region.
Let us assume that the ratio between the total amount of energy emitted and the frequency of revolution of the electron for the different stationary states is given by the equation W = f(τ) · hν, instead of by the equation (2). Proceeding in the same way as above we get in this case instead of (3)
Assuming as above that the amount of energy emitted during the passing of the system from a state corresponding to τ = τ1 to one for which −r = −r2 is equal to hν, we get instead of (4)
We see that in order to get an expression of the same form as the Balmer series we must put f(τ) = cτ.
In order to determine c let us now consider the passing of the system between two successive stationary states, corresponding to τ = N and τ = N − 1; introducing f(τ) = cτ, we get for the frequency of the radiation emitted
For the frequency of revolution of the electron before and after the emission we have
If N is great the ratio between the frequency before and after the emission will be very near equal to 1; and according to the ordinary electrodynamics we should therefore expect that the ratio between the frequency of radiation and the frequency of revolution also is very nearly equal to 1. This condition will only be satisfied if c = 1/2. Putting f(τ) = τ/2, we however, again arrive at the equation (2) and consequently at the expression (3) for the stationary states.
If we consider the passing of the system between two states corresponding to τ = N and τ = N − n, where n is small compared with N, we get with the same approximation as above putting f(τ) = τ/2,ν = nω
The possibility of an emission of a radiation of such a frequency may also be interpreted from analogy with the ordinary elecrodynamics, as in electron rotating round a nucleus in an elliptical orbit will emit a radiation which according to Fourier’s theorem can be resolved into homogeneous components, the frequencies of which are nω, if ω is the frequency of revolution of the electron.
We are thus led to assume that the interpretation of the equation (2) is not that the different stationary states correspond to an emission of different numbers of energy-quanta, but that the frequency of the energy emitted during the passing of the system from a state in which no energy is yet radiated out to one of the different stationary states, is equal to different multiples of ω/2 where ω is the frequency of revolution of the electron in the state considered. From this assumption we get exactly the same expressions as before for the stationary states, and from these by help of the principal assumptions on p. 7 the same expression for the law of the hydrogen spectrum. Consequently we may regard our preliminary considerations on p. 5 only as a simple form of representing the results of the theory.
Before we leave the discussion of this question, we shall for a moment return to the question of the significance of the agreement between the observed and calculated values of the constant entering in the expressions (4) for the Balmer series of the hydrogen spectrum. From the above consideration it will follow that, taking the starting-point in the form of the law of the hydrogen spectrum and assuming that the different lines correspond to a homogeneous radiation emitted during the passing between different stationary states, we shall arrive at exactly the same expression for the constant in question as that given by (4), if we only assume (1) that th, radiation is sent out in quanta hν and (2) that the frequency of the radiation emitted during the passing of the system between successive stationary states will coincide with the frequency of revolution of the electron in the region of slow vibrations.
As all the assumptions used in this latter way of representing the theory are of what we may call a qualitative character, we are justified in expecting—if the whole way of considering is a sound one—an absolute agreement between the values calculated and observed for the constant in question, and not only an approximate agreement. The formula (4) may therefore be of value in the discussion of the results of experimental determinations of the constants e, m, and h.
While, there obviously can be no question of a mechanical foundation of the calculations given in this paper, it is, however possible to give a very simple interpretation of the result of the calculation on p. 5 by help of symbols taken from the mechanics. Denoting the angular momentum of the electron round the nucleus by M, we have immediately for a circular orbit πM = T/ω where ω is the frequency of revolution and T the kinetic energy of the electron; for a circular orbit we further have T = W (see p. 3) and from (2), p. 5 we consequently getM = τ Mo, where
Mo = h/2π = 1.04 × 10 −27
If we therefore assume that the orbit of the electron in the stationary states is circular, the result of the calculation on p. 5 can be expressed by the simple condition: that the angular momentum of the electron round the nucleus in a stationary state of the system is equal to an entire multiple of a universal value, independent of the charge on the nucleus. The possible importance of the angular momentum in the discussion of atomic systems in relation to Planck’s theory is emphasized by Nicholsonbe.
The great number of different stationary states we do not observe except by investigation of the emission and absorption of radiation. In most of the other physical phenomena, however, we only observe the atoms of the matter in a single distinct, state, i.e. the state of the atoms at low temperature. From the preceding considerations we are immediately led to the assumption that the “permanent” state is the one among the stationary states during the formation of which the greatest amount of energy is emitted. According to the equation (3) on p. 5, this state is the one corresponds to τ = 1.
§ 4. ABSORPTION OF RADIATION
In order to account for Kirchhoff’s law it is necessary to introduce assumptions on the mechanism of absorption of radiation hich correspond to those we have used considering the emission. Thus we must assume that a system consisting of a nucleus and in electron rotating round it under certain circumstances can absorb a radiation of a frequency equal to the frequency of the homogeneous radiation emitted during the passing of the system between different stationary states. Let us consider the radiation emitted during the passing of the system between two stationary states A1 and A2 corresponding to values for τ equal to τ1 and τ2, τ1 > τ2. As the necessary condition for an emission of the radiation in question was the presence of systems in the state A1, we must assume that the necessary condition for an abs
orption of the radiation is the presence of systems in the state A2.
These considerations seem to be in conformity with experiments on absorption in gases. In hydrogen gas at ordinary conditions for instance there is no absorption of a radiation of a frequency corresponding to the line-spectrum of this gas; such an absorption is only observed in hydrogen gas in a luminous state. This is what we should expect according to the above. We have on p. 9 assumed that the radiation in question was emitted during the passing of the systems between stationary states corresponding to τ [greater than or equal to] 2. The state of the atoms in hydrogen gas at ordinary conditions should, however, correspond to τ = 1; furthermore, hydrogen atoms at ordinary conditions combine into molecules, i.e. into systems in which the electrons have frequencies different from those in the atoms (see Part III.). From the circumstance that certain substances in a non-luminous state, as, for instance, sodium vapour, absorb radiation corresponding to lines in the line-spectra of the substances, we may, on the other hand, conclude that the lines in question are emitted during the passing of the system between two states, one of which is the permanent state.
How much the above considerations differ from an interpretation based on the ordinary electrodynamics is perhaps most clearly shown by the fact that we have been forced to assume that a system of electrons will absorb a radiation of a frequency different from the frequency of vibration of the electrons calculated in the ordinary way. It may in this connexion be of interest to mention a generalization of the considerations to which we are led by experiments on the photo-electric effect, and which may be able to throw some light on the problem in question. Let us consider a state of the system in which the electron is free, i.e. in which the electron possesses kinetic energy sufficient to remove to infinite distances from the nucleus. If we assume that the motion of the electron is governed by the ordinary mechanics and that there is no (sensible) energy radiation, the total energy of the system—as in the above considered stationary states will be constant. Further, there will be perfect continuity between the two kinds of states, as the difference between frequency and dimensions of the systems in successive stationary states will diminish without limit if τ increases. In the following considerations we shall for the sake of brevity refer to the two kinds of states in question as “mechanical,” states; by this notation only emphasizing the assumption that the motion of the electron in both cases can be accounted for by the ordinary mechanics.
The Dreams That Stuff is Made of Page 9
