Other spectra exhibit several series superposed. The three types of series early recognized as occurring in the same spectrum were denoted by Schuster as the " Trunk," " Main Branch," and " Side Branch " series, but these names are now entirely superseded by the'titles: " Principal," " Sharp," and " Diffuse," originally assigned by Rydberg. A fourth type of series, called the " Fundamental " or " Bergmann " series, has since been recognized. The four chief types are closely interrelated, but apparently have a certain measure of independence. Each series may consist of singlets, doublets, or triplets.
In each series the lines converge to a definite limit, and their wave-numbers are obtained by subtracting a sequence of " terms " from the wave-number of the limit. The formulae for series in gen- eral, however, are not known with the same accuracy as for hydro- gen and enhanced helium. In some spectra, notably the arc spectra of the alkali metals, a close approximation to a series is given by such a formula as that of Hicks, 1 namely, y = A N/(m+/*+a/i) z where N has nearly the same value as for hydrogen, while M and a are constants and A is the limit of the series; as before m takes suc- cessive integral values. In some series, however, such a formula by no means gives an accurate representation of the observed lines. All that the theoretical investigator can accept with confidence at present is that the general term formula is N/[f(m)] 2 , where f(m) is a function of m whose form is known only for hydrogen and enhanced helium.
The four main sequences of terms are denoted, for brevity, by the symbols mP, mS, mD, mF, where different integral values of m correspond to the different terms in each sequence. The limit of each of the four series is the first term of one of the others, so that in the abbreviated notation, we have:
Principal series =lS mP
Sharp series = I P mS
Diffuse series =lP mD
Fundamental series =2D mF
The term iP has one, two, or three values, according as the series consists of singlets, doublets, or triplets; and, similarly, the term 2D has two or three values, in doubjet and triplet series respectively, when satellites are present in the diffuse series.
It was first shown by Ritz, and expressed in his " combination principle," that lines often occur in positions corresponding to other differences of terms besides those giving the four main series. Thus, there may be a series zS mP, iP mP, and so on. Many lines not previously included have in this way been proved to form part of general series systems.
The recognition of the importance of " terms " is a definite step towards the simplification of spectra, since the number of terms is less than the number of lines included in the series and combina- tions. Moreover, theoretical investigations indicate that the terms have a more immediate physical significance than the lines them- selves. On this account, it is of great interest to construct a " term- spectrum," in which the terms, instead of the lines, are plotted along a horizontal scale. Such a term-spectrum for the element lithium is shown in the appended diagram. For economy of space, the terms are represented horizontally by their logarithms instead of their actual values.
IITHIUM
t
i
to
>
J.
J
\
M
1
-
1
1
f
PRINCIPAL- IS -P DIFFUSE -IP-.D SHARP -1P-S FUNDAMENTAL - 2 D F
COMBINATIONS: IP-i-P. 2P-mDi 2S-2P; IP-3F,c
_ In the term-spectrum diagram, the four main sequences are dis- tinguished by the varying heights of the strokes by which their terms are represented. The highest principal term (iP) minus the sets of sharp and diffuse terms, gives the sharp and diffuse series of lines respectively, while the highest sharp term (iS) minus the set of principal terms, and the highest diffuse term (zD) minus the set of fundamental terms, give the principal and fundamental series. These four series are generally well developed, but, as already remarked, other combinations often arise. It appears, however, that all the combinations which are mathematically possible do not occur with the same frequency.
Origin of Spectra. The theory of Bohr, 2 which has already been mentioned, offers a remarkably accurate explanation of the spectra of hydrogen and enhanced helium, and gives a physical meaning to the terms which has proved very fruitful in suggesting new direc-
1 Phil. Trans. A vols. ccx., ccxii., ccxiii., ccxvii., ccxx.
- Phil. Mag., vol. xxvi., pp. 1-25; 476-502; 857-875 (1913);
vol. xxvii., pp. 506-524 (1913) ; vol. xxix., pp. 332-335 (1915).
tions of research. According to this theory, the atom of an element consists of a positively charged nucleus surrounded by an appro- priate system of electrons, such that the total negative charge of the electrons is equal to the positive charge of the nucleus. Nearly the whole of the mass of the atom is concentrated in the nucleus, which is very small in comparison with the distances separating it from the electrons. When an electron is removed from its normal position by the application of an external stimulus, it may traverse tempo- rarily one or another of certain orbits determined by quantum con- siderations. In each of these orbits it has a certain amount of energy, which is assumed to remain constant while the electron revolves in the orbit. The terms of the spectrum are then taken to be propor- tional to the respective amounts of energy. When the electron returns to its normal position, it comes to an orbit in which, for equilibrium, it must possess less energy than it had in the temporary orbit. The difference of energy, which is proportional to the difference of the corresponding terms, is emitted as a homogeneous radiation, and gives rise to a definite spectral line, while, if the elec- tron occupies successively different orbits on its return, several lines will be produced in succession. The actual spectrum at any moment is the summation of the different lines yielded by atoms in different states. The term spectrum can thus be regarded as a dia- gram of the atom, in which the nucleus is at the zero of the scale (to the right in the diagram), and the strokes are parts of the pos- sible orbits. A spectrum line appears when an electron passes from one orbit to another on its return towards its normal position in the innermost orbit.
The differences between the arc and enhanced spectra receive a simple explanation on the Bohr theory. The lines of an arc spectrum are supposed to be generated by the disturbance of a single electron and its subsequent interaction with the nucleus and remaining elec- trons. When two electrons are removed from their normal posi- tions, and one remains at a great distance, the return of the second electron generates an entirely different spectrum consisting of the enhanced lines. An atom which has lost one or more electrons is said to be " ionized."
Assuming the hydrogen atom to consist of a nucleus and a single electron, the energies of the possible orbits can be calculated, and are found to be proportional to the observed terms N/w 2 . Helium, the next lightest element to hydrogen, is believed to have two elec- trons, and the mathematical problem of determining their motion has not yet been solved. If one of the electrons is removed, how- ever, the atom is similar to that of hydrogen, except that the nucleus has a double positive charge and a greater mass. The resulting enhanced terms are therefore calculable. They again have the form N/fft *, but N has now a much larger value than it has for hydrogen.
T . , . . 2ire*E 2 mM
It is represented in both cases by the expression rj 4-M'
where e, m, and E, M, are respectively the charge and mass of the electron and nucleus, h Planck's constant and c is the velocity of light. In the case of hydrogen E = e, and when the experimental values of the various quantities are substituted in the formula, the series constant is reproduced with remarkable accuracy. The second factor increases with M, so that it will be slightly greater for helium than for hydrogen. Also, the double nuclear charge makes the first factor in the expression for N four times as great for enhanced helium as it is for hydrogen. These theoretical require- ments have been completely verified by experiment. Fowler, calcu- lating N for hydrogen and helium from the observed lines, used the theoretical expressions to calculate the value of M/m, i.e. the ratio of the masses of the hydrogen atom and the electron and obtained a result in very close agreement with that arrived at by direct measurement. Moreover, he has shown 3 that in the more complicated spectra of the alkaline earths, the enhanced line terms are also represented by formula? in which N has four times its value for arc spectra.
It has not yet been possible to calculate the theoretical terms of other spectra, on account of the mathematical difficulties con- nected with the interaction of more than two bodies. The same principles, however, are believed to apply to atoms containing many electrons, and the physical conceptions of the theory have led to valuable information regarding the order of excitation of the lines under gradually increasing stimulus.
Further developments of the theory, taking into account the variation of the mass of the electron with velocity required by the theory of relativity, have indicated that the lines of the hydrogen and enhanced helium series are complex, and under high resolution should appear to consist of several components. This has been veri- fied by Paschen, 4 who found results for helium in remarkable agree- ment with the predictions of Sommerfeld. The intensities of the several components also are in the ratio calculated by Sommer- feld by a special hypothesis.
Resonance and Ionizing Potentials. Strong support for the Bohr theory is given by experiments in which atoms are bombarded by electrons, with a view to temporary disintegration. If an electron, of charge e falls through a potential difference, v, it acquires a
1 Phil. Trans. A ccxiv., 254 (1914).
4 Ann. d. Phys., vol. 1., pp. 901-940 (1916).
