CHAPTER VIICRITICAL DISCUSSION OF IONIZATION THEORY

THE theory of thermal ionization, of which the preceding chapter contains an illustrative discussion, may be treated from two points of view—the sufficiency of the analytical treatment, and the nature of the underlying physical assumptions. Actually the two questions are merely two different ways of regarding the validity of the theory, but they divide the discussion conveniently into a section dealing with the analytical treatment and a section dealing with the physical assumptions.

The original treatment by Saha[370]was based on the Law of Mass Action, and the application to stellar atmospheres raised questions of a physical rather than of an analytical nature. These questions are fundamental not only to the Saha treatment, but also to the more recent development of the theory, and they will be discussed in the second half of the present chapter. The first half will be devoted to the analytical formulae.

MARGINAL APPEARANCE AND MAXIMUM

Saha’s discussion was based on the observation of “marginal appearance”—the spectral class at which a particular absorption line is at the limit of visibility. The use of this quantity as a criterion for the temperature scale has certain practical drawbacks. Marginal appearance depends directly on relative abundance, since a more abundant element will give visible lines at a lower “fractional concentration,” that is to say, when a smaller fraction of the element is contributing to the lines in question. Further, in estimating the intensities of lines in stellar spectra, difficulty is experienced when the lines are faint, and the spectral class at which they are first or last seen depends on their width and definition, the intensityof the continuous background, the presence of other lines, and the dispersion used. All of these factors are subject to variation, and in particular the intensity distribution in the continuous background changes with the temperature. The statistical theory of Fowler and Milne has, therefore, a great advantage in that it leads to an estimate of the temperature at which a given line attains maximum. A maximum, unlike a marginal appearance, can be determined without ambiguity from homogeneous material, whatever the dispersion. In the cooler stars the estimates may be made difficult by blending, but the uncertainty can generally be removed by examining the maxima of several related lines. Of the observational factors enumerated above as affecting the estimation of marginal appearance, the changing intensity distribution of the continuous background with the temperature is the only one that may prove serious for the method of maxima.

THEORETICAL FORMULAE

The theory developed by Fowler and Milne has been exhaustively discussed by these authors in several papers, and it appears unnecessary to reproduce the analysis in detail. The ionization and excitation curves have been treated diagrammatically in the previous chapter. The detailed formulae follow.

“Ifis the ionization potential of the atom,the (negative) energy of a given excited state, then for a given partial electron pressureof free electrons, the temperatureof maximum concentration of atoms in the given excited stateis given by[371]

“The temperature at which the concentration of singly ionized atoms reaches a maximum is given by[372]

The two formulae that have been quoted assume, in effect, that at any stage of ionization the number of atoms, in stages other than those whose constants appear in the formulae, is negligible. When two ionizations occur in very close succession, this assumption no longer holds, and the equations, as modified to embody the necessary correction, are as follows.[373]

For the subordinate series of a neutral atom,

“This equation must be used to calculatewhenever the ionization potential of the stage in question is closelyfollowedby the ionization potential of the succeeding stage.”

For the subordinate series of the ionized atom,This equation “must be used wherever the ionization potential of the stage in question is closelyprecededby the ionization potential of the preceding stage. The corrections ... result in making the maxima for the lines of the two stages occur farther apart in the temperature scale. If we express the correction in the form of a factor ()... thenis of the order.Sincevaries roughly asor,we see that the importance of the correction is determined by the closeness ofto 1.”

The values of,the fractional concentration of the atom in question, are obtained through the application of the ordinary methods of statistical mechanics to the equilibrium between atoms and electrons in the reversing layer. The values ofat the maximum are obtained by differentiating the expression forwith respect to,and equating to zero, since the maximum of absorption will occur whenis at a maximum.

The analytical treatment calls for no comment. Its basis has been fully discussed by R. H. Fowler[374]in a series of papers. The weights ()of the atomic states employed were based on the work of Bohr[375]on the relative values of the a priori probabilities of the different stationary states for hydrogen. On this view,for all atoms excepting those of H and He+, for which it is equal to 2. The convergence of the series forwas not established by Fowler and Milne, but the authors regard the subsequent investigation by Urey[376]as justifying their assumption that “for physical reasons one must suppose the series effectively cut off after a certain number of terms. Usually the series then reduces (as regards its numerical value) practically to its first term.”

PHYSICAL CONSTANTS REQUIRED IN THE FORMULAE

The application of the equations will of course depend upon an accurate knowledge of the constants involved. The quantities,,,andrequire no comment. The symmetry numberof the neutral atom is in effect the number ofspectroscopicvalency electrons given in Bohr’s table,[377]for the atoms for which it is known. In all the applications made by Fowler and Milne the quantity was equated to 1 or 2, and it is very probable that this number is not in any case exceeded. For carbon, where the chemical valency is equal to 4, the value ofis still 2, as has been shown by Fowler’s analysis of the spectrum of ionized carbon.[378]The value ofis not known for atoms in the long periods, but in the present work it is assumed to be 1 and 2 for atoms with arc spectra which show even and odd multiplicities, respectively. The uncertainty in the value ofintroduces only a relatively small error into the result, sincedepends on the first power of,and in no case considered canexceed five.

The most important factor involved in the theory is,the partial pressure of electrons in the reversing layer. By assumingconstant at about,and treatingas the unknown, a temperature scale which agrees substantially with those derived from measurements of radiation may be deduced from the observed positions of the maxima. The first discussion of the data then available was made by Fowler and Milne in their original paper.[379]Subsequent investigations of the positions of maxima have been published by Menzel[380]and by the writer.[381]These observations, and the scale derived from them, will be discussed in the two following chapters.

The value ofhas been recently shown by several kinds of investigation to be at least as low as was assumed by Fowler and Milne, so that their assumption that a uniform mean pressure can be used, as a first approximation, in deriving a temperature scale from their formula appears to be justified. Milne[382]points out that “on whatever specific assumptions” the theory rests, “the mean pressure for a maximum of intensity in an absorption line is found to depend on the absolute value of the absorption coefficient. In fact ... it is clearthat the greater the absorbing power of the atoms in question, the more opaque is the stellar atmosphere in the frequency concerned, and so the greater the height and the smaller the pressure at which the line originates.” That the absorption coefficient in the stellar atmosphere is very high is suggested by the reorganization times (“lives”) of such atoms as have been investigated,[383]and Milne’s discussion of the life of the excited calcium atom from astrophysical data lends weight to the suggestion. A high absorption coefficient leads at once to low pressures in the reversing layer, and theory has gone far towards indicating that pressures of the order ofare to be expected on a priori grounds.[384]

The observational evidence bearing on pressures in the reversing layer will be found[385]inChapter III. The case appears to be a strong one, resting on evidence of many different kinds—notably pressure shifts, line sharpness, and series limits. Russell and Stewart,[386]in their exhaustive discussion of the question, conclude that “all lines of evidence agree with the conclusion that the total pressure of thephotospheric gasesis less than 0.01 atmosphere, and that the average pressure in thereversing layeris not greater than 0.0001 atmosphere.”

The observational evidence gives thetotalpressure, but the partial electron pressure will not differ greatly from this. Although even in the hottest stars three ionizations is the greatest number observed, most of the elements that constitute the stellar atmosphere are appreciably ionized at temperatures greater than 4000°, so that the partial electron pressure is at least half the total pressure.

PHYSICAL ASSUMPTIONS

The method applied by Saha to stellar atmospheres was borrowed from physical chemistry. The Law of Mass Action, and the theory of ionization in solutions which is based upon it, have in general been very well satisfied in dilute solution.[387]The ionization consideredby chemical theory is the separation of amoleculein solution into charged radicals. The essential point is the acquisition of a charge at dissociation, and this is the only feature that the chemical ionization has in common with the thermal ionization, where theatomis separated into a positively charged ion and an electron which constitutes the negative charge.

The step from the theory first formulated for solutions to the theory of gaseous ionization is a long one, and its legitimacy has been questioned.[388]It appears, however, that the step is justified.[389]The stellar conditions are certainly simpler than those in a solution, and if the requisite dilution obtains, the law may be expected to hold with considerable closeness. Saha contemplated pressures of the order of an atmosphere, and it may be shown that under such conditions the volume concentration would be too great and the theory would be invalid. At pressures of,however, the effect of concentration is just becoming inappreciable, and the theory probably holds with fair exactness.

LABORATORY EVIDENCE BEARING ON THE THEORY

(a)Ultimate Lines[390]—The physical tests of the Saha theory that have been made in the laboratory have all supported it strongly. The fact that the ultimate lines of an atom are the lines normally absorbed by the cold vapor has long been familiar. Indeed it is this fact that is tacitly assumed in the identification of lines of zero excitation potential in the laboratory with lines which are strongest in the low-temperature furnace spectrum. De Gramont[391]designated the ultimate lines “raies de grande sensibilité” for the detection of small quantities of a substance, because they are the last to disappear from the flame spectrum when the quantity of the substance is decreased.

(b)Temperature Class.—The effect, upon the absorption spectrum of a substance, of raising the temperature has also long been recognized as an increase in the strength of lines associated with the higher excitation potentials. The use of A. S. King’s “temperature class” in assigning series relations[392]involves a tacit admission of the validity of the theory of thermal ionization in predicting the relative numbers of atoms able to absorb light corresponding to different levels of energy.[393]

(c)Furnace Experiments.—King’s explicit investigation[394]of the effects of thermal ionization in the furnace has contributed valuable positive evidence for the theory. For example, the production of the subordinate series of the neutral atoms of the alkali metals by raising the temperature was an experimental proof of the principle mentioned in the last paragraph; and the suppression of the enhanced lines of calcium by the presence of an excess of free electrons, derived from the concurrent ionization of potassium, with an ionization potential 1.77 volts lower than that of calcium, and the similar results obtained for strontium and barium, fulfill the predictions of ionization theory in a striking fashion.

(d)Conductivities of Flames.—The conductivity of a flame may be used as a measure of the ionization that is taking place at the temperature in question, and the available data on flame conductivities have been discussed by Noyes and Wilson[395]from the standpoint of the theory of thermal ionization. The calculations based upon the conductivities imparted to a flame by the different alkali metals, and leading to an estimate of the ionization constant, were in satisfactory agreement with the theoretical predictions of the ionization constant from the known critical potentials. The theory of thermal ionization is, therefore, strongly supported by all the laboratory investigations which have so far been undertaken in testing it.

SOLAR INTENSITIES AS A TEST OF IONIZATION THEORY

Before proceeding to discuss the stellar intensity curves, it is proposed to review some of the solar evidence, which can be treated as an observational test of the predictions of the theory relating to the distribution of atoms among the possible atomic states at a given temperature.

In two papers, Russell[396]has given a discussion of the solar and sunspot spectra, showing that ionization theory offers a very satisfactory interpretation of most of the observed phenomena. Attention was called to the anomalous behavior of barium and lithium,[397]and it was suggested that the theory of thermal ionization, while taking account of the temperature of the reversing layer, omitted to consider the effect of the absorption of photospheric radiation. This omission might cause a deviation such as is observed for barium, but appears inadequate to account for the behavior of lithium. In the case of lithium, low atomic weight, and a consequent high velocity of thermal agitation, has been suggested as the cause of the anomaly. The question of the absorption of photospheric radiation has more recently been discussed by Saha,[398]in the form of a correction to his own ionization equations. It has been pointed out by Woltjer[399]that the correction introduced by Saha and Swe may also be derived from considerations advanced by Einstein[400]and Milne.[401]The correction can be evaluated, but appears in every case to be rather small. The effect of the photospheric radiation is certainly one that must be included in a satisfactory theory, but at present, observation is probably not of sufficient accuracy to demand such a refinement.

The work just quoted was qualitative. A more quantitative test of ionization theory in the solar spectrum can also be made[402]by comparing the intensities of solar lines corresponding to different excitation potentials, but belonging to the same atom. Theatoms which give a large number of lines in the solar spectrum are those of the first long period of the periodic table, and these, as is well known, consist of multiplets, with components of very different intensities. It appears to be legitimate to select the strongest line associated with any energy level for the comparison; the strength of this line probably represents fairly well the tendency of the atom to be in the corresponding state.

The atoms for which there are enough known lines of different excitation energies in the solar spectrum are those of calcium, chromium, titanium, and iron. The correlation between the excitation potential associated with a given line and the intensity of the line in the solar spectrum is illustrated by the preceding tabulation. Successive columns give the atom, the excitation potential, thecomputed fractional concentration, expressed logarithmically, and the observed intensity, taken from Rowland’s table.

It will be seen that the correlation is very marked, and that it appears to furnish good evidence that the theory of thermal ionization predicts correctly the relative tendencies of the atoms to absorb the different frequencies. The fractional concentrations are of course not absolute values, as the number of atoms in a state of high excitation is a definite fraction, not of thewholenumber of atoms, but of thenumber left overfrom the lower excitations. Neither are the intensities given by Rowland absolute, and therefore the comparison appears sufficient to show the strong correlation between excitation potential and solar intensity.

FOOTNOTES:[370]Proc. Roy. Soc., 99A, 136, 1921.[371]M. N. R. A. S., 83, 403, 1923;ibid., 84, 499, 1924.[372]M. N. R. A. S., 83, 403, 1923.[373]M. N. R. A. S., 84, 499, 1924.[374]R. H. Fowler, Phil. Mag., 45, 1, 1923.[375]Bohr, Mem. Ac. Roy. Den., 4, 2, 76, 1922.[376]Ap. J., 59, 1, 1924.[377]Chapter I,p. 9.[378]Proc. Roy. Soc., 103A, 413, 1923.[379]M. N. R. A. S., 83, 404, 1923.[380]H. C. 258, 1924.[381]H. C. 252, 256, 1924.[382]Phil. Mag., 47, 209, 1924.[383]Chapter I,p. 21.[384]Proc. Phys. Soc. Lond., 36, 94, 924.[385]P. 45.[386]Ap. J., 59, 197, 1924.[387]See, for instance, H. J. H. Fenton, Outlines of Chemistry, 128, 1918.[388]Lindemann, quoted by Milne, Observatory, 44, 264, 1921.[389]Milne, Observatory, 44, 264, 1921.[390]Chapter VI,p. 94.[391]C. R., 171, 1106, 1920.[392]Russell, Ap. J., in press.[393]A. S. King, Mt. W. Contr. 247, 1922.[394]A. S. King, Mt. W. Contr. 233, 1922.[395]Ap. J., 57, 20, 1923.[396]Mt. W. Contr. 225, 1922.[397]Mt. W. Contr. 236, 1922.[398]Saha and Swe, Nature, 115, 377, 1925.[399]Nature, 115, 534, 1925.[400]Phys. Zeit., 18, 121, 1917.[401]Phil. Mag., 47, 209, 1924.[402]Payne, Proc. N. Ac. Sci., 11, 197, 1925.

[370]Proc. Roy. Soc., 99A, 136, 1921.

[371]M. N. R. A. S., 83, 403, 1923;ibid., 84, 499, 1924.

[372]M. N. R. A. S., 83, 403, 1923.

[373]M. N. R. A. S., 84, 499, 1924.

[374]R. H. Fowler, Phil. Mag., 45, 1, 1923.

[375]Bohr, Mem. Ac. Roy. Den., 4, 2, 76, 1922.

[376]Ap. J., 59, 1, 1924.

[377]Chapter I,p. 9.

[378]Proc. Roy. Soc., 103A, 413, 1923.

[379]M. N. R. A. S., 83, 404, 1923.

[380]H. C. 258, 1924.

[381]H. C. 252, 256, 1924.

[382]Phil. Mag., 47, 209, 1924.

[383]Chapter I,p. 21.

[384]Proc. Phys. Soc. Lond., 36, 94, 924.

[385]P. 45.

[386]Ap. J., 59, 197, 1924.

[387]See, for instance, H. J. H. Fenton, Outlines of Chemistry, 128, 1918.

[388]Lindemann, quoted by Milne, Observatory, 44, 264, 1921.

[389]Milne, Observatory, 44, 264, 1921.

[390]Chapter VI,p. 94.

[391]C. R., 171, 1106, 1920.

[392]Russell, Ap. J., in press.

[393]A. S. King, Mt. W. Contr. 247, 1922.

[394]A. S. King, Mt. W. Contr. 233, 1922.

[395]Ap. J., 57, 20, 1923.

[396]Mt. W. Contr. 225, 1922.

[397]Mt. W. Contr. 236, 1922.

[398]Saha and Swe, Nature, 115, 377, 1925.

[399]Nature, 115, 534, 1925.

[400]Phys. Zeit., 18, 121, 1917.

[401]Phil. Mag., 47, 209, 1924.

[402]Payne, Proc. N. Ac. Sci., 11, 197, 1925.

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