ANGULAR CORRELATION BETWEEN K AND L X RAYS

and Gamma-Ray Spectroscopy, edited by K. Siegbahn(North-Holland, Amsterdam, 1966), Vol. II, p. 997.

A complete discussion on the various definitions of 0

found in the literature, definitions which differ from oneanother in sign, is given in H. J. Rose and D. M. Brink,

Rev. Mod. Phys. 39, 306 (1967). The values of 4 usedin the present work were calculated according toformulas given in the above-mentioned reference, p. 328.The sign convention was modified, however, when

necessary to conform to the sign convention of Ref. 14.

PHYSICAL RE VIEW A VOLUME 3, NUMBER 3 MARC H 1971

Stark Broadening of H, H, and H: A Comparison of Theory and Experiment

R. A. Hill and J. B. GerardoSandia Laboratories, Albuquerque, Net Mexico 87115

aMi

Paul C. Kepple~Department of Physics and Astronomy, University of Maryland, Co/lege Park, Maryland 20742

(Received 24 April 1970)

A precision comparison is made between the Stark-broadened profiles of H~, H~, and Hz

measured by Hill and Gerardo and the theoretical profiles calculated by Kepple and Griem. In

addition, the effects of inelastic collisions between perturbing electrons and the radiating atomare investigated. It is found that the inclusion of the inelastic collisions improves the qualita-tive agreement between theory and experiment near the line center, but that the Gaunt factorsused to estimate the broadening strength of the inelastic collisions overestimate the effect. Interms of half-widths, the best comparison to experiment was obtained using theoretical profileswhich omitted inelastic collisions. The electron densities determined from these profiles are2, 3, and 6%%up less than the interferometeric values for Hg Hy and H&, respectively.

I. INTRODUCTION

When Hill and Gerardo' published the results oftheir measurements of the Stark-broadened profilesof the lines H8, H„, and H~, the best theoreticalprofiles available for comparison were those ofGriem, Kolb, and Shen'' (GKS I, II). When Keppleand Griem (KG) published their theoretical profiles,they compared the half -intensity widths of theirprofiles to the GKS I, II profiles and thus formedonly an indirect comparison to experiment. In thispaper, these experimental and theoretical lineshapes are compared in detail.

One of the difficulties (source of errors) in com-parisons between experiment and theory is that thetheoretical profiles are usually calculated for someset of temperatures and densities which seldomcoincide with the actual temperature and density ofthe experiment. Thus, one must interpolate in thetables of computed profiles. In the case where theshape of the profile does not change much withtemperature and density (Q, for example), theinterpolation is easy and resultant errors small.However, when the shape of the line changes no-ticeably with temperature or density (as in the easeof H„) then one must be more careful or significanterrors will be introduced by the interpolation. Inorder to avoid these errors, the computed codefor Stark-broadened hydrogen lines (describedbriefly in KG, and in more detail in Ref. 5) was

rerun for the electron temperatures and electrondensities of the experiment. The only differencebetween the code as described in Ref. 5 and thatused here is the inclusion of contributions to theline broadening from inelastic collisions betweenperturbing electrons and radiating atoms as de-scribed in Sec. II. The results of the comparisonof KG theory (both with and without the contribu-tions due to inelastic collisions) with experimentappears in Sec. III.

II. INELASTIC COLLISIONS

The matrix representation of the electron broad-ening operator contains terms of the sort

where &', &", and y represent complete sets ofquantum numbers (n, l &, l2, m) and R, is a compo-nent of the electron position operator (in atomicunits). For the elastic terms (n, =n„), G reducesto the 6 of KG. Since these elastic, terms are by farthe largest, the inelastic terms (n cn„) were sim-ply omitted in GKS I, II, and KG. To investigatethe change in the line shape introduced by thisomission of inelastic terms, the quantity G

(for n, & n„) is replaced by a semiempirical Gauntfactor. With the inelastic collision thus accountedfor, the electron broadening operator is

856 HI LL, GE RARDO, AND KE P P LE

. Gr (o'~IR,.Io)(p~lR, .lo")~ ( 7~3 (i(m , .).). (o.,'Ia,.lw~) (& I~R,.ls"))

where &g (nE))„ is the thermal average of a semi-empirical Gaunt factor, and ~„„.is the differencein energy levels between hydrogenic states withprincipal quantum numbers n and n'. Since thematrix elements decrease rapidly with M, thoseelements on or near the diagonal can be approxi-mated by the term with hn= 1, with an error lessthan 40fo (less than 30/o when the Gaunt factorsare included). The relative error incurred by usingthe hn= 1 terms for the elements further off thediagonal is quite large, however, since these off-diagonal terms are much smaller than the near-diagonal terms, the actual error (the error in theelectron contribution to the profile) is negligible.Thus, since the entire inelastic contribution issmall, all terms except the term with n„=n, (n~)+1 were neglected. The effect on the individualline shapes of adding the contributions due to in-elastic collisions will be discussed in Sec. III.

III. COMPARISON OF THEORY AND EXPERIMENT

The comparison of theory and experiment wascarried out using H~, H„, and H, profiles whichwere recorded in an earlier experiment. ' High-accuracy measurements were obtained by using arapid-scan spectrometer to record the Stark-broadened line profiles, y multiple-pass laser in-terferometers (&, =1.1523 p, and &3 =0.6326 p) to

t

independently measure the electron density,and a long uniform plasma source in order to min-imize boundary-layer effects. A complete descrip-tion of the experiment is given in Ref. 1. For thiscomparison of theory and experiment, the theoret-ical profiles were computed for the particular val-ues of electron density and temperature measuredin the experiment. These theoretical profiles weremodified by the measured instrument function (in-cluding Doppler broadening as described in Ref. 1).

The electron temperature was determined fromthe H~-to-continuum ratio; an example of the datafrom which this ratio was obtained is shown inFig. 1. The interferometric value of electron den-sity for this particular record is 2.75x10" cm '.The blue wing is shown in Fig. 1(b); the red wingand central portion of the Q profile (recorded atone-fourth the sensitivity) are shown in Fig. 1(a).The large and small dashed lines represent thecontinuum intensity at the increased and decreasedsensitivity, respectively. The intensity ratio of Q0to 100 A of continuum is 24. 4 which correspondsto T, = 21000 'K. ' The half-intensity width of aprofile calculated without corrections for inelasticcollisions (IC) for an electron density of 2. 75& 10'cm including finite slit width and rise-time ef-fects is 20. 5 A. in good agreement with the measuredvalue of 20. 4 A.

1.0—S{~)

—4

St ~)

a tA) 100

. 8

(b) Sl )

.6—

0

FIG. 1, Example of an H& profileand the adjoining continuum intensityrecorded at N, = 2.75 x 10 8 cm+. Thered wing and central portion of theH~ profile (at one-fourth the sensitiv-ity) are shown in (a); the blue wing isshown in (b). The intensity ratio ofH~ to 100 A of continuum is 24. 4 whichcorresponds to T,= 21000'K.

.2—0

l

0I

-50

STARK B ROADE NING OF Hg& H7 PAND H() ~ 857

In order to determine whether the underlying in-tensity is correctly assigned to background radia-tion, a comparison of the theoretical and observedline wings was carried out. The theoretical inten-sities at a number of discrete wavelengths werecalculated without IC for 1V, = 2. 75 & 10' and T,= 21000 'K. These intensities are plotted with smallcircles in Fig. 1. In addition, theoretical inten-sities were calculated with the asymptotic expres-sion'

S~=(3.57x10 '/o ~ ) [1.90 0. 70(0. 35/o) ~ I,

4870 4840

NO IC

ooI I

4 5

(io" ')e

FIG. 2. Comparison of the half-intensity widths ofGKS II H~ profiles with the half-intensity widths of pro-files calculated with and without inelastic collisions.

where & is the reduced wavelength. These inten-sities are plotted with dots in Fig. 1. The standarddeviation between the experimental line-wing in-tensities and the calculated intensities is 5. 7'k.This agreement indicates that the method used toseparate the underlying continuum is adequate andthe Hz-to-continuum ratio should give a meaningfultemperature measurement.

In the previous analysis of the H, profiles, ' thehalf-intensity widths of the experimental profileswere found to be in excellent agreement with thehalf-intensity widths of the theoretical profiles ascalculated by GKS II. As shown in Fig. 2, the newKG theoretical profiles are somewhat broader thanthe GKS II profiles, with the result that the agree-ment between theory and experiment is not as goodwith the KG profiles. A comparison between a pro-file that was recorded at N, =2. 78~10' cm ' andKG theoretical profiles (including modification bythe instrument function) is shown in Fig. 3. Thecomparison with the theoretical profile without ICis obviously better than that which includes IC;however, the half-intensity widths in both cases arewithin 5/o of the experimental value. The compar-ison between theory and experiment is somewhatbetter at the line center with IC than without IC,

4870 4850 4840 z(A)

FIG. 3. Comparison of an observed profile forN, = 2. 7S & 10 cm (solid line) with theoretical profileswhich have been modified by the instrument functions.Calculated profiles in (a) are without inelastic collisionsand in (b) include the inelastic collisions. Narrow band

at the bottom represents the continuum intensity forT= 21 000'K.

i.e. , the central intensity minimum is not as deepif the IC are included. Even in the latter case,however, the agreement is not satisfactory. The

excess radiation at the line center may be causedby H, line radiation from the cold gas in the bound-

ary layer at the end window. This radiation would

be concentrated at the line center due to the low

electron density in the boundary layer. Qn the oth-er hand, the calculations do not include the ion dy-namics corrections to the (ion) quasistatic broad-ening (the correction to the ion quasistatic broad-ening due to the relative motion between the ionsand radiating atom) which may be non-negligibleunder some conditions of interest. One may ex-pect that the inclusion of the ion dynamics correc-tions into the code for the total line profile willsubstantially reduce the dip in the center of the HB

profile thus improving somewhat the comparisonbetween experiment and theory.

The comparison between theory and experimentfor electron densities in the range 1.5-7.5~10'cm is shown in Fig. 4. Each data point representsthe percent difference between the interferometricvalue of electron density N, (f) and the average value

of electrondensity, as inferred from the half-in-tensity widths of from 10 to 20 H, profiles, N, (S).The vertical error bars represent the standarddeviation for the average values N, (S) and do notinclude the estimated systematic error. This com-parison of theory and experiment indicates that thevalues of electron density inferred from measuredhalf-intensity widths using the theoretical profilesare 4 —8% too low when the IC are included and1 —37o too low when they are not. In both cases thisapplies for the N, range 1.5-7. 5&10 cm

In the previous analysis of the H„profiles, ' the

858 HILL, GE RARDO, AND KEPPLE

Width(A.)

Normalizedintensity

TABLE I. Comparison of theoretical linewidths

Width for N~= 2. 6 x10without IC ~

(A.)

with measured

Diffcalc —obs

(%)

widths for N, = 2. 60 x 10 8 cm+.

Width for N~=2. 36 x10with IC ~

(A)

Diffcalc —obs

(%)

0.10. 2

0.30.40.50.60.70.80. 9

61.8 + 1.145. 1+ 0.836.0+ 0.529.6+ 0.524. 4 ~ 0. 519.7 + 0.514.6+ 0.710.0 + 1.15.4+ 0.5

64. 145. 836.529. 924. 117.910.06. 03 ~ 6

3.61.51.31.0

-1.2-10.0

63.045. 036.029. 624. 319.112.87. 54. 3

1.94-0. 220. 00. 0

-0.41-3.05

These widths are for theoretical profiles which have been modified by the instrument function.

(a) WITHOUT INELASTIC COLLISIONS

0—-1—-2—-3—

EQ

I

Zl/I

IQ

-5—-6—-7—

I .. I

2 3

~'o- Hp

(&- Hy

I.-x,I I I

4 5 6

N (INTERFERCNETER) I10 cm )

Ibl WITH INELASTIC COLLISIONS

0

-2—o- Hp

a-H

, o- Hd

-8—4P

-10—

-12—

I I I I I

2 3 4 5 6

N I INTERFEROMETER) (10 cm l

FIG. 4. Over-all comparison between values of elec-tron density N, (E) as determined by the laser interferom-eter, and values N, 5) inferred from Stark-broadenedHp, H„, and H& profiles using calculated profiles (a) with-out inelastic collisions and (b) with inelastic collisions.The over-all estimated uncertainty in this comparisonis + 4%. Both comparisons are better than the estimateduncertainties in the calculated profiles (+ 10%).

half-intensity widths of the experimental profileswere 11-25% larger than the half-intensity widths

of the theoretical profiles as calculated by GKS Ifor the electron density range 1.3-V. 1& 10 cmSince the new KG theoretical profiles are broaderthan the GKS I profiles, the agreement between

theory and experiment has been improved.The signal-to-noise ratio for the recorded H„

line profiles was less than in the case of Hp be-cause of the smaller line-to-continuum ratio. In

order to smooth out this "noise" contributionseveral profiles which were recorded on consecu-tive discharges were averaged to obtain a new pro-file which was then analyzed in detail. The back-ground level was established by using the H„-to-continuum ratio for an electron temperature of21 000 'K. The average "fractional-intensity"widths of six such averaged profiles, each repre-senting an electron density of 2.60&&10 cm ', areshown in Table I along with their standard devia-tions. The analysis of these averaged data wascarried out with the aid of the calculated curves ofelectron density vs fractional-intensity width shown

in Fig. 5. These calculated fractional-intensitywidths have been modified by the instrument func-tions. The averaged fractional-intensity widthsfrom Table I are plotted on their respective curvesin Fig. 5 where the symbols 4 and x have been usedto distinguish the curves with and without IC, re-spectively. Comparing the averaged fractional-in-tensity widths with the calculated values without ICat N, =2.60&10, the measured (0.1-0.4) frac-tional-intensity widths are smaller than the calcu-lated widths while the measured (0.5-0. 9) frac-tional-intensity widths are larger than the calculatedwidths. This comparison is further illustrated inFig. 6(a) where the observed and calculated profilesare represented by solid and dashed lines, respec-tively. The percentage difference between themeasured and calculated widths are listed in TableI. While the half-intensity widths disagree by only1.2%, the central core of the calculated profile is

AND H5 ~S TARK B ROADEgING OF H~'

mECJ

29 NO IC

2.8—

2. 7—

2. 6—

0. 9 0.8 0. 7

2. &—

2.4—

2.3—WITH IC

0.80.I

3 4 g 6 7 8

0. 7

g g(A)

0 6 0 & 0 4 0.3 0. 2 0. 1

2 O. l3 004 00, 6 0. &

I

60 70]5 20

FIG 5 C lculated curve s of elec-fractional-intensitytron density

= 21 000 K

ument functions.the instrum'dths for sever-fractional-intensi y

= 2. 60 x 10i6'les recorded at Neal prof& eh the symbolcm are p

3 lotted wit ejth and without

jrcle Pfl represent~ s. Error agsline wing ~

the meas-t eh standard deviatioured wi dth shown qn Tab

reement atw. Because of the disagldb d db,which again wouthe line center w ic

the inclusion of the ion dynamics cd out with the goalison was carrie ou

1 hil dood fit in the ine w

t 1 Th lt),

e center entire y.5 with the symbo

th t th li i. 6(b), show a16of a profile calculated at N, = 2.

b rved line wingst th

bl with the o serparrofile is 5% more inf the calculated pro

r than the observed profi e.line center( . — .6) f ctional-intense y

. I.0.1-0.6 rac

w s o o parison is . o.ond to an electron dens& y

h the interferometeran served wit eas also carrie ouA comparison was a

and calculated profi es w1 which in-c . , tt d in Fig. 5 (witheluded IC. These results, plo e in

ted in Fig. 6(c),d further illustra ethe symbol 6) and efile calculated forshow that a profile c

er favorably wiith the observedstandard deviation in th 0 1-0 6

le I is 1.5 o. e cded and calcu a etween the observe

b the inclusiond at the line center yp

corresponds o anthe interferome ric value.

ison and compar-the above comparisoisons at 1.3x10 anand 7.1x10 cm ar

rison in both casesand 4(b). The comparison inb ' '

th l' i, ib fitting ectional-intensity wi

The error bars arecussed above. ewidths multip ie'

1 d by 1.5, a factordeviations in the wi

where &Xto the fact that N, —6is -' it width. is

iment indicates t atheory and experim

I

4360

I

4340

I

4320

ed H profile tNerison of an averaged 1 1 t d f'1solid line) and ca cu

16

x 10 cm without dn-

(a) N =2. 60x 10 cm

2 36x1016 th'

1 ti(b) N = 2. 50 x

' ns and (c) N~=2. 3elastic collzs1.on,lastic collisions 1sThe effect of the ine as i

th 1particularly noticeable near e

the line wings using'nferred from ee ecrofiles with and withe theoretical pro i es

respectively.~ 1

anal sis of the

i t 1 ofilidths of the experimenhalf -intensity wi sU-intensity widths owere 7/o smallerr than the ha -in e1 1 ted by GKS I fore ' rofiles as calcu a e rcore '

alp o'

the electron-density range 1.3- .

ND KEPPHI LL, GE RAR860

(bI

I

4140

of an averageed H profiled calculated profiles—2 '6 ', solid line) an ca c es(N =2 60x10 6 cm ', so '

(d hed lines) foc ' ' and Q) with ine as itic collisions an

and ex-arison eb tween theory an— ll ompf4 indica esperimen,nt shown in Fig.

H profile hasTe 21 0th intensity wid

the theoretical ~

f 51 9't and tenth-i erofil es

full hate-i tensl y . ,l . Theoretical H6pr134.6 A respective'y'-3 d modified bcalculated for ~

= 'shown with das eth instrument u

' t ity widthfunctions are

f thed with

lines in Fig. ~

t IC [Fig. 7(a)] anpro if les calculated wi o. g l ger than the ob-

ofIC [Fig. 7(b)j ate ~

p m the examPle od alue, respectiv yf the calculated

/

serve». t minimum o

(a)

the central intens yhich is in better

/

Hs~ e'f IC are included wrofiles js l.essbservation

Hagre

f H the anacement with o s

h al.ysis of theA in the example o'd of the calcu' d out with the ai

I

files was carrie o' vs fractional-in-

pro ilectron density

havelated curves «e

8. These curve~ "tensity widthinstrument functions

/

fied by the ins'dth are pl.otted on

/

measured fractio —

re the symbols ~ antheir resPec '

ti ish curves with an

I I

o ~(k have b enused to dis i~f this data in-A„analysis o

41O0 406o

t IC respe ctively.compares favor

ouobserved profile o

xdicates that the o s

d f & 2 43x10@cuiatedbl with a profile— 31x 10ie cm wi

ywd for &, =2.

c--~ without IC a

'th (0. 1-0 9)d deviation '

. 1.9/p

IC. The standarth' comparison istional-intensi y

Thus the obsewidths for t is o

erved1 with Id

b ad wi thout IC and l. 0

s to an electron density 11f'les are slightly ro -

ofile correspondsd with the inter-

S' ce the new thehe reement betw

6.5' less than the v . ith and without IC,

eoretical pro i eseen pro i

value observe wKS I profiles, the gr

th calculations wiferometer for etheory an

f'l.e h h er e

above comparison an

've discharges were av-were recorded on consecubt the contributio

t th dard

to smooth ou

4 an . error bars reprThe backgroun d level was

' dby1. 5asint e

to ein the

ed deviations 't rmined by us g

example of H„.electron temp

f2601 ctron density o

IV. DISCUSSION

file for ane ec rbserved prox10" cm is shown with a so

rofile has been re-

tion of the red wing of H, . For

2.6— 0. 9 0.8 0. 7

NO IC

0. 6 0. 5 0.4 0.3 0. 2 0. 1

2.5—

2.3—

2.2—0. 9

E2.4—

ED

I

20

0.8 0. 7

I

30

0. 6 0. 5 0.4

WITH ICI

40

ax(k)

40. 2 0. 1

I I I I I

7O 8O 90 1OO60

8. Calculated curv es of elec-"1---.',it vs fractiona -'

odifi d brofiles for

These widths have been mo ' '

functions. ectional-intensity wiged frf'1 shown in i .o epo

ith the symbolsplotted wiand without in-on the curves with an

elastic co s oe lli i ns respectively.

STARK BROADENING OF Hg, H„, AND Hq. ~ ~

electron density determined from the theoreticalprofiles withoutIC are about 2, 3, and 6 jo less thanthe interferometric values for H~, H„, and H&, re-spectively. The values of electron density deter-mined from theoretical profiles with IC are about

6, 8, and 10% less than the interferometric valuesfor H~, H„, and H„respectively. The over-all es-timated uncertainty in this comparison is + 4'fo..a 2' in establishing an average value for the electrondensity as previously discussed, ' and + 2k in anal-yzing the line profiles. While both of the abovecomparisons are better than the estimated uncertain-ties in the calculated profiles (+ 10%), the profileswith the IC are somewhat too broad, even thoughthese corrections certainly improve the compari-son in the line cores. Thus, it appears than whenthe contribution of the inelastic collisions is simplyadded to that of the elastic, the Gaunt factors over-estimate the effect. This is partly due to the factthat the strong collision term already partially ac-counts for the inelastic collisions. As mentionedbefore, the inclusion into the code of the ion dy-namics correction may improve the comparison nearthe line center.

It is interesting to note that the experimental lineprofiles are narrower than predicted by theory,and that this difference between theory and experi-ment increases as the upper quantum number in-creases. At an electron density of 2. 60&& 10' cmthe (half) half-intensity widths of H„, H„and H, are11.5, 20. 9, and 26. 0 A, respectively, and the (half)tenth-intensity widths are 31.5, 51.5, and 67. 3 A,respectively. Because the (H„—H, ) and (H, —H, )line separations are 238. 8 and 131.6 A, respective-ly, it appears that for these densities, the wings ofthe Balmer lines are beginning to overlap. Thus,it is reasonable to expect an interaction of the up-per-state energy levels which would cause a spec-tral line narrowing. This effect, which is not in-cluded in the present Stark-broadening theory,would increase as the series limit is approached.

In order to compare the results of this experi-ment with the recently published results of Beng-

ston et al. ,' it is necessary to compute the ratio

of the electron density determined from H„ to thatdetermined from H~. The mean ratio of the elec-tron density deter mined from the H~ profiles tothat determined from the interferometer is 0.978with a mean standard deviation of + 1.45 k. Themean ratio of the electron density determined fromthe H„profiles to that determined from the inter-ferometer is 0.968 with a mean standard deviationof +1.47%. Thus, the ratio of the electron densitydetermined from H„ to that determined from Hz is0. 99+0.02. This value is in satisfactory agree-ment with the value 1.01 + 0.02 reported by Bengstonet al.

Finally, it is worthwhile to investigate two of themore important experimental parameters and in

particular, the magnitude of variation in these pa-rameters necessary to show an improvement inthe comparison between theory and experiment.These quantities are the plasma length L and therelative continuum intensity. For the interfero-metric measurement of electron density, N, ~L ',where L =14.7 cm. ' In order to decrease the in-terferometric density by 4%, the plasma length Lwould have to be increased to 15.3 cm, a valuelonger than the plasma container (14.9 cm). If, on

the other hand, the continuum intensity at H& weredecreased such that the half-intensity width of theobserved profile agrees with that for a theoreticalprofile corresponding to 2. 60x10' cm, then theH~ -to-continuum ratio corresponds to an electrontemperature of 12000 K. This would require anQ-to-continuum ratio of approximately 70 whichwould imply that the continuum measurement shownin Fig. 1 is about three times too large. This re-sult is hard to accept in view of the good agreementbetween the observed and calculated Q line wingsfor a temperature of 21000 K.

ACKNOWLEDGMENT

The authors would like to thank Dr. Hans Griemfor several suggestions and helpful discussions.

*Work supported by the U. S. Atomic Energy Com-mission and the National Aeronautics and Space Admin-istration.

~Present address: U. S. Naval Research Laboratory,Washington, D. C.

R. A. Hill and J. B. Gerardo, Phys. Rev. 162, 45(1967).

H. R. Griem, A. C. Kolb, and K. Y. Shen, Phys. Rev.116, 4 (1959).

3H. R. Griem, A. C. Kolb, and K. Y. Shen, Astrophys.J. 135, 272 {1962).P. Kepple and Hans R. Griem, Phys. Rev. 173, 317

(1968).P. Kepple, University of Maryland, Center for

Theoretical Physics Report No. 831 (unpublished).

6This approach was suggested by H. Van Regemorter[Astrophys. J. 136, 906 (1967)1 and applied by Griem tothe case of isolated ion lines [Hans R. Griem, Phys.Rev. 165, 258 (1968)}.

We used the numerical values tabulated in C. W. Allen,Astrophysical Quantities, 2nd ed. (The Athelone Press,University of London, London, 1964), p. 43.

R. A. Hill and R. D. Fellerhoff, Appl. Opt. 5, 1105(1966).

R. A. Hill, Appl. Opt. 7, 2184 (1968).J. B. Gerardo and J. T. Verdeyen, Proc. IEEE 52,

690 (1964).|1J. B. Gerardo, J. T. Verdeyen, and M. A. Gusinow,

J. Appl. Phys. 36, 2146 (1965); 36, 3526 (1965).i2H. R. Griem, Plasma Spectroscopy (McGraw-Hill,

862 HI LL, GE RARDO, AND KE P P LE

New York, 1964), p. 281.H. R. Griem, Comments At. Mol. Phys. 2, 19 (1970);

J. W. Dufty, Phys. Rev. A 2, 534 (1970).4The case of overlapping lines which originate from

all the states of a degenerate level has been includedhere; the case of overlapping lines originating fromlevels of different principal quantum number has notbeen included. While a (practical) theory to treat thecomplete profile of overlapping (nondegenerate) linesdoes not exist, Baranger's one-electron approximation[M. Baranger, in Atomic and Molecular Processes,edited by D. R. Bates (Academic, New York, 1962)] canbe used to show the effect of the interference {or over-lapping) in the line wing. From Baranger's Eq. 122, theratio of the interference term to the main term is

w2(CO {djar ((d —(d)y

({d~ (d]y) ({d (dye) {d (dye

where {d&& ((di.&) is the frequency of the transition frominitial state i(i') to the final state f. It is difficult totranslate this correction, which applies only in the wings,into a meaningful correction to the half-intensity width;however, it does have the appropriate qualitative be-havior. Within a series, the nearest neighbor to a givenline lies at a higher frequency, thus, {de&&(deaf. As aresult, the entire line profile corresponding to the transi-tion i-f is shifted to the red. The effect is greater,however, for (di~&(d &(d~s~, than for (d &{di&&(d;,&, so thatthe line profile should be somewhat compressed. Themagnitude of the correction increases as the principalquantum number of the initial state increases due to thedecreased spacing between series members, and to the in-creased broadening of the higher series members.

5R. D. Bengston, M. H. Miller, W. D. Davis, and

J. R. Grieg, Astrophys. J. 157, 957 (1969).

PHYSICAL REVIEW A VOLUME 3, NUMBER 3 MARCH 1971

Relationship between Unrestricted and Extended Hartree-Fock WaveFunctions as Function of Number of Electrons

F. Martino and J. LadikPhysics Department, City College of the City University of New York, New York, New York10031

(Received 28 July 1970)

The justification of the use of unprojected many-electron wave functions for solids and

large molecules is examined from the point of view of an extended Hartree-Fock method inthe one-X approximation presented in previous papers. It is shown that, in the limit of alarge number of electrons, the Hartree-Fock operators are the same for the projected andunprojected case. It is further shown that the expectation value of purely spatial operatorsfor the projected and unprojected functions are equal in this limit. Graphs of the differencesof the relevar t quantities as a function of number of electrons are given.

H(} + + hO ~O! g (}(}i(}(}O+Se~ ()t

(la)

(lb)

Here the operators aren

H' =(Q )' (A' H" Z(A' (J —)( z)$*1

The question of the relation between the unre-stricted [ different orbitals for different spina(DODS) without spin projection ] and extendedHartree-Fock (DODS with spin projection beforevariation) methods has been of interest both inmolecular and solid-state physics. ' In the partic-ular case of the alternant molecular orbital (AMO)method ' it has been shown that the expectationvalue of the total Hamiltonian is the same for thesinglet ground state and the unprojected function ifthe number of electrons goes to infinity. In a seriesof papers (I-III hereafter) we have formulated anextended Hartree-Fock (HF) method in the one-&approximation both for molecules and solids. Thismethod provides the possibility of investigating theabove question from a new aspect.

The extended Hartree-Pock self-consistent-field(SCF) equations are [see (12a) and (12b) of I]:

~ }&A',, , [[z,"']—[tc"]}~ a' [r("]))n

h'=(A' )'(xA' H" 7 (LA' (J" J")=1

(2a)

—K" —K")~ I' [6' /("' ~ A;, (2[J"']—K')}})

(2b)

H =H (a —p), he=h (o P),

] =«(™le()(«» a yf)

[J ']=-'(J' +J ') [IP']=-'(K '+K ) . (2c)

The operator H" is the usual one-electron op-erator and the generalized 8&"6 Coulomb, EP&' (y= & or p) exchange operators are defined in I [seeEqs. (10b) and (10c) of I]. The constants h„', whichcontain the effect of spin projection are defined by

—tn;, , =Z ( —1)' )c„(s,} ~)x,"(3)

0=Q

where 2n is the number of electrons, 2s+1 is themultiplicity, the C„(s, 0+x) are the Sanibel coeffi-cients, and finally