JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

Estimation of Ionization Potentials by Comparison with Neighboring ElementsHENRY NoRuus RUSSELL

Princelon University Observatory, Princeton, New Jersey(Received June 14, 1950)

If a spectrum contains an unperturbed series of at least three members, the ionization potential can befound with fair accuracy from its limit. If but two members are available,. serious errors occur if the Ritzcorrection is ignored. This correction can often be closely estimated by comparison with elements of neigh-boring atomic numbers. This method is familiar, but published estimates have not been made on a uniformsystem. Such estimates for spark spectra from Ca ii to Zn ii give ionization potentials differing from asmooth formula by 40.02 volt, while the differences in the three cases where the Ritz correction was notapplied, average :1:0.39 volt.

THE analyses of the spectra of an element in suc-cessive stages of ionization fix with great preci-

sion the relative energy levels in each stage. Connectionof those in one and the next demands the determinationof the limit of a series. One accurate limit locates allthe levels of the second relative to those of the firstand determines the ionization potential, even thoughthis may correspond to the limit of some other series,or to a double electron jump.

An unperturbed series of at least four members givesa value of its limit reliable to a few cm-1, or better,when an extended Ritz formula is used:

N 2R

where T is the term-value and N-1 the degree ofionization. A remarkably simple method of calculationis given by Shenstone.' If but three consecutive mem-bers are known, one may set =0 and obtain a goodapproximation. But when only two members areavailable, the assumption that a= 0 (that is, of a simpleRydberg formula) leads to serious errors-which, sincea is almost always positive, make the calculated limitand ionization potential come out too high.

This error may be greatly reduced by comparison ofthe series constants with those for elements of neigh-boring atomic numbers. If x is the fractional part of thedenominator

X1 -X2 = (T 1 - T2) { a+ (Tl+ T2)}.Now, for an electron of the same type, e and , andconsequently x1 -x 2, vary smoothly with the atomicnumber, and values can be.estimated for interveningelements if well-determined ones are not too far apart.This method has been familiar for more than twentyyears, and has been used to find ionization potentialsfor almost all the elements in which series of only twomembers are available. The methods of estimation ofx1-x 2 for different elements varied considerably in theearly work and it appears worth while to discuss someresults of a uniform treatment.

This method is properly applicable only when the''running" electron of the series is "external" and notpart of a shell in process of formation, and only spectra

LA. G. Shenstone, Phil. Trans. Roy. Soc. London A235, 199(1936).

corresponding to successive steps in the formation ofsuch a shell are comparable.

It is most useful in the "iron group" (and its homo-logs) where sequences of eleven spectra, with shells of0 to 10 d-electrons added to the same base, exist.

If n is the number of electrons outside the argonshell in the neutral atom, there is one such sequence,3dm-2 ms in the second spectra, and two in the first,3dn-1 ms and 3dn-2 4s s. (Though 3dn- 2 4 2 does notproperly belong to the last series, the values of x1 -x 2between this and 3dn-2 4s 5s may often be used.)

A considerable obstacle in practice arises from per-turbations of the terms of a given series by neighboringterms. For example, the addition of a p electron usuallyproduces two triads of terms of multiplicity differing by2, which may be perturbed by members of triads fromother limits, and by one another. When the limit is an Sterm, each electron gives only two series and longer andless perturbed series are often observed.

These complications are worse in arc than in sparkspectra-at least in the iron group-since the number ofpossible perturbing terms is greater.

The second spectra of the elements of this group formtherefore a good example. The series arising from theaddition of s electrons to the lowest level of the thirdspectrum are best known and least likely to be per-turbed. Among the eleven spectra to be considered,series of more than two members have been observed inonly four-Ca II, Mn II, Cu ii, and Zn II. In three ofthese the limit is an S term; for Cu II Shenstone'sunusually complete analysis is available. Series of twomembers are known in all the rest except V II, in whichno terms arising from 5s were found, despite a thoroughanalysis-some other laboratory source being doubtlessrequired for their production.

The calculations here described are based on thedata given in Vol. I of Atomic Energy Levels and addi-tional material prepared for Vol. II and generouslyfurnished by Mrs. Sitterly. The essential elements aregiven in Table I. The first column contains the absolutevalue L of the ground state of each second spectrum(II) referred to the ground state of the third spectrum(III). The terms of lower multiplicity have as limit thenext component of this ground term in the third spec-trum, which is higher by L.

The second column gives the types and energy levels618

VOLUME 40, NUMBER 9 SEPTEMBER, 1950

ESTIMATION OF IONIZA

TABLE I.

Spec-trum

L Final Final Diff. MeanAL Level X2-Xl AL x2-XI D M

Call IS1 0 095748 52167 +0.0326 +0.0326 +0.0000 +0.0326

Sc II104000

+197

3Dl 057551

'D2 254158252

Ti II 4F1j 0110000 62180

+1842F21 4629

63168

V iI 5F 1 2605114600 -

+1453F2 8640

Cr GD1 11962136000 82692

+60 4D1 19528

84208

Mn 11 7S3 0126147 74560

06S2 9473

76375

Fe I 6D41 0130978 77861

+4364D3i 7955

79439

Co II 5FI 3350137150 84012

+8413F4 9813

85479

Ni 11 4F4I 8394149200 91798+1361

2F3J 1355093526

Cu II 3D3 21929163666 108015+2072lD2 26265

110366

Zn II 2S1 0144891

0 88437

+0.0197

+0.0130

+0.0321

+0.0163

-0.0117

-0.0314

+0.0516

+0.0300

+0.0441

+0.0204

+0.0629

-760+0.0376 +0.0065 +0.0344

+0.0311

-494+0.0434 +0.0155 +0.0356

+0.0279

[ +3600]

-2940+0.0482 +0.0183

+0.0299

0+0.0516

+0.0300

-454+0.0533

+0.0296

+422

+0.0406

-0.0003-2792

-0.0158

0+0.0487

+0.0414

+0.0543

+0.0320

+0.0518

+0.0362

+0.0216 +0.0408

+0.0237 +0.0414

+0.0223 +0.0432

+0.0156 +0.0440

+0.0487 +0.0073 +0.0450

+0.0414

0

+0.0478 +0.0478 0.0000 +0.0478

of the lowest components of the series terms-the upperentry of each pair corresponding to 4s and the lowerentry to 5s. Subtracting these, according to the multi-plicity, from L or L+ 6L, the term values and denomi-nators are found and so the differences x2 -x1 . Forexample, in Fe ii, L+bL= 131414. Subtracting thegiven 4D3] levels from this gives term-values 123459 for4s, and 51975 for 5s. The corresponding denominatorsare 1.8856 and 2.9060, whence x2 -x 1 =+0.0204.

The differences D of the computed values of x2-xlfor the two multiplicities in a given spectrum are almostindependent of the assumed series limit. These runsmoothly-indicating that perturbations are small. Themeans M of the two, which are sensitive to the adoptedlimit, are ragged. The well-determined values (Ca, Mn,Cu, Zn), however, are represented by M= 0.0401

+0.00145(Z-25) with a maximum residual of 0.0009.By trial and error, it is easy to find the corrections ALto each assumed limit which will make the resultingvalues of M agree with this empirical formula.

TION POTENTIALS 619

TABLE I. Second ionization potentials for the iron group.

Ionization Seriesenergy limit

Sp cm-l Volts, volts O-C Old I.P. Corr.

Ca 95748 11.87 11.87 +0.02 11.87 0.00Sc ii 103240 12.80 12.80 +0.03 12.89 -0.09Ti II 109506 13.57 13.57 -0.03 13.63 -0.06V II (118200) (14.65) (14.33) (-0.01) 14.2 (+0.45)Cr ii 133060 16.49 15.01 -0.01 16.86 -0.37)Mn ii 126147 15.64 15.64 +0.01 15.64 0.00Fe ii 130524 16.18 16.18 -0.00 16.23 -0.05Co ii 137572 17.05 16.64 -0.03 17.0 +0.05Ni ii 146408 18.15 17.11 -0.02 18.5 -0.35Cu ii 163666 20.29 17.57 +0.02 20.29 0.00Zn ii 144891 17.96 17.96 +0.01 17.96 0.00

The factor of proportionality is taken as 0.00012395.

These are given in the fourth column of Table I andfollowed by the resulting values of x2 -xi, D, and M.The resulting values of L+AL give a system of ioniza-tion potentials which have been consistently, thoughempirically, adjusted.

These appear in Table II. The second and thirdcolumns give the adopted energy difference betweenthe ground limits in the second and third spectrum, incm-l and volts, and the fourth that between thelowest term 3dn-2 4s and its limit. The next columngives the residual of this from the empirical expression11.850+0.610 (Z-20)+0.029 (Z-20) (30-Z)+0.0013(Z-20) (25-Z) (30-Z). These residuals are remarkablysmall, run smoothly, and indicate again that theperturbations are probably small.

From the run of these residuals it has been estimatedthat the missing value for V II is not far from -0.01,which leads to the energy levels given in parentheses.

The last two columns of the table give the values ofthe I.P. taken from the literature, and the correctionsresulting from the present work.

The four values depending on long series were leftunaltered. In the four cases where a previous attemptwas made to apply a Ritz correction the change aver-ages 4A0.06 volt-probably significant, but of no seriousastrophysical importance. Where no such attempt wasmade, the corrections average =t0.39 volt. The twouncorrected Rydberg series give results much too high,while the rough estimate for V ii was too low. Thepresent results indicate that, under the most favorableconditions, careful estimates of the Ritz correction givevalues for the ionization potential correct within afraction of one percent. It would be of great interest tocheck this conclusion, and obtain really reliable values,by the discovery of further series members. It may wellrequire specially designed light-sources to provide therequired lines with measurable intensity.

This communication presents an example of themethod-not a complete survey of its applications.Series involving three or more s-electrons are known inthe arc spectra of nine of the twelve elements of the irongroup-excepting Sc i, V i, and Co i. The data for thehomologous elements Rb to Cd are still incomplete.

Hearty thanks are due to Mrs. Sitterly for her kind-ness in checking the numerical data.