i002Figure 2The hydrogen atom. The ten innermost orbits possible for the single electron of the atom of hydrogen are diagrammatically represented. All possible quantum transitions between the orbits are indicated as follows:—short dashes, Lyman series, terminating at a 1-quantum orbit; full lines, Balmer series, terminating at a 2-quantum orbit; long dashes, Paschen series, terminating at a 3-quantum orbit. Transfers are only possible between orbits with azimuthal quantum numbers differing by ±1.
Figure 2The hydrogen atom. The ten innermost orbits possible for the single electron of the atom of hydrogen are diagrammatically represented. All possible quantum transitions between the orbits are indicated as follows:—short dashes, Lyman series, terminating at a 1-quantum orbit; full lines, Balmer series, terminating at a 2-quantum orbit; long dashes, Paschen series, terminating at a 3-quantum orbit. Transfers are only possible between orbits with azimuthal quantum numbers differing by ±1.
Figure 2
The hydrogen atom. The ten innermost orbits possible for the single electron of the atom of hydrogen are diagrammatically represented. All possible quantum transitions between the orbits are indicated as follows:—short dashes, Lyman series, terminating at a 1-quantum orbit; full lines, Balmer series, terminating at a 2-quantum orbit; long dashes, Paschen series, terminating at a 3-quantum orbit. Transfers are only possible between orbits with azimuthal quantum numbers differing by ±1.
When the energy supply from the environment is great enough, the “outermost” (or most easily detachable) valency electron is entirely removed by the energy absorbed. In consequence the atom is superficially transformed, giving rise to a totally new spectrum, which strongly resembles the spectrum of the atom next preceding in the periodic system. Bohr’s table embodies the interpretation ofthis resemblance—the so-called Displacement Rule of Kossell and Sommerfeld[7]—which has recently been strikingly confirmed by a very complete investigation of the arc and spark (neutral and ionized) spectra of the atoms in the first long period.[8]It may be seen at once, for instance, that the removal of the outermost (or)electron from the atom of aluminum ()produces an arrangement of external electrons identical with that for magnesium (). The ionized atom produced by the complete removal of one electron gives, like the neutral atom, two kinds of line spectrum—the ultimate lines and the subordinate lines.
i003Figure 3Energy levels for the hydrogen atom. Horizontal lines represent diagrammatically the levels of energy corresponding to all the possible electron orbits up to and including those of total quantum number four. Total quantum numbers are indicated on the left margin, azimuthal quantum numbers on the right margin. Transitions are only possible between orbits which differ by ±1 in azimuthal quantum number. All such possible transitions are indicated in the diagram by heavy lines. “Forbidden jumps,” for which the difference in azimuthal quantum number is zero or greater than 1, are indicated by light lines. This diagram embodies the same relations asFigure 2, the levels representing the various orbits in that figure.
Figure 3Energy levels for the hydrogen atom. Horizontal lines represent diagrammatically the levels of energy corresponding to all the possible electron orbits up to and including those of total quantum number four. Total quantum numbers are indicated on the left margin, azimuthal quantum numbers on the right margin. Transitions are only possible between orbits which differ by ±1 in azimuthal quantum number. All such possible transitions are indicated in the diagram by heavy lines. “Forbidden jumps,” for which the difference in azimuthal quantum number is zero or greater than 1, are indicated by light lines. This diagram embodies the same relations asFigure 2, the levels representing the various orbits in that figure.
Figure 3
Energy levels for the hydrogen atom. Horizontal lines represent diagrammatically the levels of energy corresponding to all the possible electron orbits up to and including those of total quantum number four. Total quantum numbers are indicated on the left margin, azimuthal quantum numbers on the right margin. Transitions are only possible between orbits which differ by ±1 in azimuthal quantum number. All such possible transitions are indicated in the diagram by heavy lines. “Forbidden jumps,” for which the difference in azimuthal quantum number is zero or greater than 1, are indicated by light lines. This diagram embodies the same relations asFigure 2, the levels representing the various orbits in that figure.
Effectively, the ionized atom may be regarded as a new atom altogether. It reproduces the spectrum of the atom of preceding atomic number, in cases which have been fully investigated, with great fidelity, excepting that the Rydberg constant in the series formula is multiplied by four. For the twice and thrice ionized atoms the same is true, the Rydberg constant being multiplied by nine and by sixteen in the two cases. It is scarcely necessary to mention the beautiful confirmation of the theory that has been furnished by the analyses[9][10]of the spectra of Na, Mg, and Mg+, Al, Al+, and Al++, and Si, Si+, Si++, and Si+++. The attribution of the Pickering series (first observed in the spectrum ofPuppis) to ionized helium was the first established example of the displacement rule, and constituted one of the earliest triumphs of the Bohr theory.[11]The detection and resolution of the alternate components of that series, which fall very near to the Balmer lines of hydrogen in the spectra of the hottest stars, and the consequent derivation of the Rydberg constant for helium,[12]represents an astrophysical contribution to pure physics which is of the highest importance.
IONIZATION AND EXCITATION
Theionization potentialof an atom is the energy in volts that is required in order to remove the outermost electron to infinity. Theexcitation potentialcorresponding to any particular spectral series is the energy in volts that must be imparted to the atom in the normal state in order that there may be an electron in a suitable electron orbit for the absorption or emission of that series. Several different excitation potentials are usually associated with one atom. The ionization potential and the excitation potentials are collectively termed thecritical potentials.
From the astrophysical point of view, ionization and excitationpotentials are important as forming the basic data for the Saha theory of thermal ionization, with which the greater part of this work is concerned. A list of the ionization potentials hitherto determined is therefore reproduced in the following table. The first two columns contain the values obtained by the physical and spectroscopic methods, respectively. The third column contains “astrophysical estimates,” which are inserted here to make the table more complete. The derivation of the astrophysical values will be discussed[13]inChapter XI. Physical values result from the direct application of electrical potentials to the element in question, and spectroscopic values are derived from the values of the optical terms. (SeeAppendix.)
1Horton and Davies, Proc. Roy. Soc., 97A, 1, 1920.
2Mohler and Foote, J. Op. Soc. Am., 4, 49, 1920.
3A. Fowler, Report on Series in Line Spectra, 1922.
4Lyman, Phys. Rev., 21, 202, 1923.
5Horton and Davies, Proc. Roy. Soc., 95A, 408, 1919.
6Mohler, Science, 58, 468, 1923.
7D. R. Hartree, unpub.
8A. Fowler, Proc. Roy. Soc., 105A, 299, 1924.
9Payne, H. C. 256, 1924.
10Brandt, Zeit. f. Phys., 8, 32, 1921.
11Hopfield, Nature, 112, 437, 1923.
12R. H. Fowler and Milne, M. N. R. A. S., 84, 499, 1924.
13Horton and Davies, Proc. Roy. Soc., 98A, 121, 1920.
14Tate and Foote, Phil. Mag., 36, 64, 1918.
15Foote, Meggers, and Mohler, Ap. J., 55, 145, 1922.
16Foote and Mohler, Phil. Mag., 37, 33, 1919.
17Paschen, An. d. Phys., 71, 151 and 537, 1923.
18A. Fowler, Bakerian Lecture, 1924.
19Menzel, H. C. 258, 1924.
20Mohler and Foote, Phys. Rev., 15, 321, 1920.
21Duffendack and Huthsteiner, Amer. Phys. Soc., 1924.
22Horton and Davies, Proc. Roy. Soc., 102A, 131, 1922.
23Shaver, Trans. Roy. Soc. Can., 16, 135, 1922.
24Smyth and Compton, Amer. Phys. Soc., 1925.
25Russell, Ap. J., 55, 119, 1922.
26Kiess and Kiess, J. Op. Soc. Am., 8, 609, 1924.
27Catalan, Phil. Trans., 223A, 1922.
28Sommerfeld, Physica, 4, 115, 1924.
29Gieseler and Grotrian, Zeit. f. Phys., 25, 165, 1924.
30Ruark, Mohler, Foote, and Chenault, Nature, 112, 831, 1923.
31Foote and Mohler, The Origin of Spectra, 67, 1922.
32Udden, Phys. Rev., 18, 385, 1921.
33Sponer, Zeit. f. Phys., 18, 249, 1923.
34Foote, Rognley and Mohler, Phys. Rev., 13, 61, 1919.
35Catalan, C. R., 176, 1063, 1923.
36Kiess, Bur. Stan. Sci. Pap. 474, 113, 1923.
37McLennan, Br. A. Rep., 25, 1923.
38Foote and Mohler, The Origin of Spectra, 67, 1922.
39Smyth and Compton, Phys. Rev., 16, 502, 1920.
40Eldridge, Phys. Rev., 20, 456, 1922.
41Mohler and Ruark, J. Op. Soc. Am., 7, 819, 1923.
42Grotrian, Zeit. f. Phys., 18, 169, 1923.
By the use of one or other of the available methods, the data for neutral atoms are complete as far as atomic number 38, with the exception of carbon (6), fluorine (9) and germanium (32). The data for ionized atoms are also increasing, at the present time, in a very gratifying manner. The “hot spark” investigations of Millikan and Bowen,[14]which permit the estimation of the fifth and sixth ionization potentials of certain light atoms, are not included in the table. Under the conditions hitherto investigated in the stellar atmosphere, ionization corresponding to a potential of about fifty volts is the highest encountered, and accordingly ionization potentials that greatly exceed this value have no place in the present tabulation of astrophysically useful data. A knowledge of the higher critical potentials[15]is, however, of great interest in connection with the theoretical problems of the far interior of the star.
There are conspicuous gaps in the table, and it is to be feared that many of them are likely to remain unfilled. The spectra of the neutral atoms of carbon, phosphorus, and nitrogen have hitherto defied analysis, and our knowledge of the corresponding ionization potentials must therefore depend on physical methods. For carbon, silicon, and similar refractory materials, such methods are difficult of application; the same applies to the metals. It is therefore probable that the ionization potentials of the neutral atoms of several of the lighter elements, of the platinum metals, and of the rare earths, will remain unknown or uncertain for some time to come. None of the atoms thus omitted is of immediate astrophysical importance.
As shown in the table, the values for the ionized and doubly ionized light atoms O+, O++, C++, N++, S+, and S++ are deduced only astrophysically. It may be hoped that the spectra of these atoms will soon be arranged in series, so that an accurate value of the ionization potential may be available, in place of the approximate one deduced from the stellar evidence, for the corresponding absorption lines are of importance in the spectra of the hotter stars.
The spectroscopic ionization potentials have an advantage over the physical values, in that the corresponding state of the atom is known with certainty, whereas physical methods can in general onlydetectsome critical potential, without assigning it definitely to a particular transition. For example, it seems likely that in some cases the first ionization, whether caused by incident radiation or by electron impacts, corresponds to the loss of an electron by themolecule:whererepresents the atom, andthe electron. The effect of increased excitation would then be the decompositionThe first reaction would produce the ionized molecule, and the second would produce the ionized and neutral atomssimultaneously. It might thus happen that thespectrum could appear without the previous appearance of thespectrum, since all of the element was present in the formbefore ionization.
The above is only a simple illustrative example of the possible complexity in the physical determination of ionization potentials. The interpretation of four successive critical potentials for hydrogen has been discussed by Franck, Knipping and Krüger,[16]while eight have been detected by Horton and Davies[17]for the same element. Similarly Smyth[18]discusses four critical voltages for nitrogen. No explicit attempt has yet been made to use these facts for the interpretation of astrophysical data, but they may account for the unexplained absence of some neutral elements from the cooler stars. The absence is generally to be attributed, as will be shown inChapter V, to the non-occurrence of suitable lines in the part of the spectrum usually examined. Butit is possible that the persistence of the molecule has a definite significance in the case of nitrogen, where the ionization potential is as high as 16.9 volts.
i004Figure 4Relation between ionization potential and position in the periodic system. Ordinates are ionization potentials in volts, on the equal but shifted scales indicated alternately on left and right margins. Abscissae are columns of the periodic table. Physical determinations of ionization potential are indicated by open circles; dots give spectroscopic determinations, and crosses denote astrophysical estimates. Conjectural portions of the curve are indicated by broken lines, and atoms of unknown ionization potential are enclosed in parentheses.
Figure 4Relation between ionization potential and position in the periodic system. Ordinates are ionization potentials in volts, on the equal but shifted scales indicated alternately on left and right margins. Abscissae are columns of the periodic table. Physical determinations of ionization potential are indicated by open circles; dots give spectroscopic determinations, and crosses denote astrophysical estimates. Conjectural portions of the curve are indicated by broken lines, and atoms of unknown ionization potential are enclosed in parentheses.
Figure 4
Relation between ionization potential and position in the periodic system. Ordinates are ionization potentials in volts, on the equal but shifted scales indicated alternately on left and right margins. Abscissae are columns of the periodic table. Physical determinations of ionization potential are indicated by open circles; dots give spectroscopic determinations, and crosses denote astrophysical estimates. Conjectural portions of the curve are indicated by broken lines, and atoms of unknown ionization potential are enclosed in parentheses.
The increasing completeness of the table of ionization potentials suggests a re-examination of the relation recently traced by the writer[19]between ionization potential and atomic number. The original diagram, in which columns of the periodic table are treated as abscissae, and the ordinates are ionization potentials on equal but shifted scales, so that analogous elements fall one below another, is here reproduced, with the addition of data more recently obtained.
The Displacement Rule of Kossell and Sommerfeld leads us to expect a pronounced similarity between the line drawn in the diagram from the point representing one element to that representing the next, and the corresponding line for the ionized atoms of the same elements, the latter being shifted one place to the left for each electron removed. The points for once and twice ionized atoms are inserted into the diagram on this principle, and the parallelism is found to exist. The regularities of the diagram and their possible significance (such, for example, as the pairing of the valency electrons, the second being harder to remove than the first) were discussed in the original paper. All the more recent data appear to confirm the conclusion there set forth, that the relation between ionization potential and atomic number is very closely the same in each period.
DURATION OF ATOMIC STATES
In addition to the critical potentials, which give a measure of the ease with which an atom is excited or ionized, astrophysical theory requires an estimate of the readiness with which an atom recovers after excitation or ionization. It appears probable that this factor, like the critical potentials, is independent of external conditions, and depends upon something that is intrinsic in the atomic structure. The “life” of the atom has been extensively investigated in the laboratory, and has been shown to be a small fraction of a second in duration. Probably this subject of “atomic lives” is still in an initial stage, and the accuracy of the results and the range of elements discussed will be greatly increased in the near future. A summary of the material obtained up to the present time is contained in the following table. Successive columns contain the atom discussed, the deduced atomic life in seconds, the authority, and the reference.
The data are practically confined to hydrogen and mercury, and for both these elements the atomic life appears to be of the order.
Astrophysical estimates of the life of the excited calcium atom have been made by Milne,[20]who derives values of the order.This is so near to the values obtained in the laboratory that it seems permissible, in the absence of further precise data, to assume an atomic life of,as a working hypothesis, for all atoms. The same value is unlikely to obtain for all atoms; in particular it may be expected to differ for atoms in different states of ionization. But here astrophysics must be entirely dependent on further laboratory work for the determination of a quantity that is of fundamental importance.
RELATIVE PROBABILITIES OF ATOMIC STATES
The relative intensities of lines in a spectrum must depend fundamentally upon the relative tendencies of the atom to be in the corresponding states. To a subject which, like astrophysics, dependsfor its data largely upon the relative intensities of spectral lines, the theory of the relative probabilities of atomic states is of extreme importance. The question is obviously destined to become an important branch of spectrum theory. It has been discussed, from various aspects, by Füchtbauer and Hoffmann,[21]Einstein,[22]Füchtbauer,[23]Kramers,[24]Coster,[25]Fermi,[26]and Sommerfeld.[27]The comparison with observation has been made, up to the present, only for a few elements. The relative intensities of the fine-structure components of the Balmer series of hydrogen were examined by Sommerfeld,[28]and exhaustive work with the calcium spectrum has recently been carried out by Dorgelo.[29]The astrophysical application of the data bearing on relative intensities of lines in the spectrum of one and the same atom, while an essential branch of the subject, is a refinement which belongs to the future rather than to the present.
EFFECT ON THE SPECTRUM OF CONDITIONS AT THE SOURCE
(a)Temperature Class.—It is found experimentally that the relative intensities of the lines in the spectrum of a substance are altered when the temperature is changed. Some lines, notably the ultimate lines mentioned in a previous paragraph, predominate at low temperature. Other lines, which are weak under these conditions, become stronger if the temperature is raised, and lines which are the characteristic feature of the spectrum at the highest temperatures that can be attained in the furnace are often imperceptible at the outset. The effects are more conspicuous, and have been most widely studied, in the spectra of the metals, which are rich in lines and are amenable to furnace conditions. The results of such experiments, whichare chiefly the work of A. S. King, are expressed by the assignment of a “temperature class,” ranging from I to V, to each line; Class I represents the lines characteristic of the lowest temperatures, and Class V denotes the lines that require the greatest stimulation.
The temperature class of a line is intimately connected with the amount of energy required to excite the line. It may, indeed, be used as a rough criterion of excitation potential, high temperature class indicating high excitation energy. The temperature class is therefore useful in assigning series relations to unclassified lines, and is of value to the astrophysicist chiefly in this capacity of a classification criterion. King’s work on silicon shows, for instance, that 3906 is of Class II, and is therefore not an ultimate line—a fact which has considerable significance in studying the astrophysical behavior of the line.
The correlation of temperature class with excitation potential receives an immediate explanation in terms of the theory of thermal ionization. It furnishes a useful laboratory corroboration of the theory by showing that the thermal excitation of successive lines, with rising excitation potential, takes place in qualitative agreement with prediction.
The appended list shows the atoms for which the spectra have been analyzed by King on the basis of temperature class:
(b)Pressure.—In the laboratory the observed effects of pressure[30]are a widening and shifting of the lines in the spectrum—effects which differ in magnitude and direction for different lines. The phenomena are well marked under pressures of several atmospheres.
Recent developments of astrophysics, such as are summarized inChapter IIIandChapter IX, have shown that the pressures in stellar atmospheres are normally of the order of a hundred dynes per square centimeter, or less. At such pressures no appreciable pressure shifts will occur, and indeed one of the most direct methods by which these exceedingly low pressures in reversing layers have been established[31]is based on the absence of appreciable pressure effects.
(c)Zeemann Effect.—The magnetic resolution of spectral lines into polarized components[32]has, as yet, for the astrophysicist, chiefly a value as a criterion for classifying spectra. In the field of solar physics proper, a direct study of the Zeemann effect has led to important results.[33]The present study is not, however, explicitly concerned with the sun, except in comparing solar features with similar features that can also be examined in the stars.
The investigations of Landé on term structure and Zeemann effect[34]for multiplets have shown how the Zeemann pattern formed by the components into which a line is magnetically resolved can be related to the series attribution of the line. This provides a method of classifying spectra which are rich in multiplets, and which have previously defied analysis. The indirect astrophysical value of the Zeemann effect is, therefore, very great.
(d)Stark Effect.—The effect of an electric field in resolving spectral lines into polarized components was first pointed out by Stark[35]for hydrogen and helium. Several other investigators have since studied the effect for these two elements,[36]and forvarious metals.[37][38]Unlike the temperature and magnetic effects, the Stark effect has not been used as a criterion for the series relations of unclassified lines.
The Stark effect has not been detected in the solar spectrum, presumably because the concentration of free electrons prevents the formation of large electrostatic fields.
Several investigators, however, have contemplated in the Stark effect a possible factor influencing the stellar spectrum.[39][40]It does not seem unlikely that nuclear fields could operate as a sensible general electrostatic field at the photospheric level, thus producing a widening and winging of certain lines. The question has been numerically discussed by Hulburt,[41]and Russell and Stewart,[42]in an examination of Hulburt’s work, concluded that the Stark effect might possibly make some contribution (probably not a preponderant one) to the observed widths of lines in the solar spectrum. The question is not definitely settled, but it appears well to keep so important a possibility in mind.