PHYSICAL REVIEW A VOLUME 40, NUMBER 9 NOVEMBER 1, 1989

Photoionization of atomic magnesium including double-electron resonances

Zikri Altun*Jesse IV. Beams Laboratory of Physics, Uniuersity of Virginia, Charlottesuille, Virginia 22901

{Received 3 April 1989)

The photoionization cross sections of the 2s, 2p, and 3s subshells of atomic magnesium and an-gular distribution asymmetry parameters for the 2p subshell have been calculated using many-bodyperturbation theory. Double-electron resonance structure near the 3s threshold has been included.At higher energies, resonance structures due to the 2s~np and 2p ~nd {ms) (n ~ 3, m ~ 4) excita-tions have also been included. The eAects of spin-orbit interactions on the 2pnd(ms) resonancesare examined. The calculated asymmetry parameter for the 2p subshell including the 2s~np reso-nance structure is compared with the calculations of Deshmukh and Manson [Phys. Rev. A 28, 209(1983)J.

I. INTRODUCTION II. THEORY

There has been considerable interest among the theor-ists and experimentalists in the photoionization of atomicmagnesium (Mg I). The cross section has been measuredby Ditchburn and Marr, ' Mehlman-Balloffet and Esteva,Esteva, Mehlman-Baloffet, and Romand, Baig and Con-nerade, Preses, Burkhardt, Garver, and Leventhal, andFiedler, Kortenkamp, and Zimmerman. These measure-ments were taken in the vicinity of the 3s threshold at7.646 eV where the double-electron resonances due to the3s ('S)~3pnd(ms)('P) (n &3, m &4) excitations dom-inate. Calculations in this energy range were carried outby Bates and Altick (configuration-interaction tech-nique), Dubau and Wells (close-coupling technique), Ra-dojevic and Johnson (multiconfiguration Tamm-Dancofftechnique), O'Mahony and Crreene' (R-matrix tech-nique), Rescigno' (complex —basis —function technique),and Moccia and Spizzo' (L technique). A calculationby Dashmukh and Manson' using the relativisticrandom-phase approximation (RRPA) covered the regionfrom the 3s threshold to 270 eV but included only thesingle-electron resonances due to the 2s ~np (n & 6) and2p ~nd (ms) (n & S, m & 10) excitations

In this paper I present a detailed calculation of thephotoionization cross section of atomic magnesium forphoton energies ranging from the 3s threshold to 300 eV.The calculations were carried out using many-body per-turbation theory (MBPT). ' ' The double- and thesingle-electron resonances are incorporated through thecoupled-equation method' which is essentially a close-coupling or K-matrix method. ' The effect of the spin-orbit interaction on the 2p nd (ms)('P) J =1 resonancesis also examined. The angular asymmetry parameter Pfor the 2p subshell was calculated including the effect ofthe 2s ~np resonances and is compared with various ex-periments and theory.

Section II includes a brief summary of the theory. Sec-tion III contains the results, and the conclusions aregiven in Sec. IV.

where Ima(co) is the imaginary part of the frequency-dependent dipole polarizability. Atomic units are usedthroughout this paper. Many-body perturbation theoryis used to calculate the electric dipole matrix elementsused in the evaluation of Ima(co). The Hamiltonian iswritten

H Ho+H

where

p2 ——+ V(r,)-2 T;

(3)

and

N N

H, = g v, —g V(r). (4)

The term v, represents Coulomb interactions betweenelectrons i and j. The single-particle potential V(r, )

represents the average interaction of the ith electron withthe remaining 2V —1 electrons, and —Z/r, is the interac-tion with the nucleus.

The dipole length matrix elements are given by

N

ZL — f zi 0Ji =1

where Po and Pf are many-particle ground and finalstates. The dipole velocity form is

v E E f d 0

The photoionization cross section in the electric dipoleapproximation is given by'

4~tr(co) = Ima(co),

4968 1989 The American Physical Society

PHOTOIONIZATION OF ATOMIC MAGNESIUM INCLUDING. . . 4969

, kp

(a) (c)

ik' 385& )gk"

k I tt

kt ~

k'

k"'

(a) (b)

i kp )kp

(e)

(c)

FIG. 1. Low-order Brueckner-Goldstone diagrams contribut-ing to the dipole matrix element for photoionization. Timeproceeds from bottom to top in these diagrams. Lines with ar-rows pointing down represent hole states and lines with arrowspointing up represent particle states. (a) Lowest-order dipole.(b) Ground-state correlation (GSC) diagram. (c) Final-statecorrelation (FSC) diagram. (d) —(k) Second-order diagrams.Dashed line ending with an isolated solid dot indicates matrixelement of z. Other dashed lines represent the electron-correlation interaction. Exchange diagrams are assumed to beincluded but are not explicitly shown.

l kp

38

PP, 28

FIG. 2. Second-order diagrams repj.'esenting the correlationsdue to the excitations of two 3s electrons.

16

where Eo and Ef are energy eigenvalues corresponding tofo and ff, respectively. The velocity and length formsare equal if the wave functions are exact eigenfunctions ofH of Eq. (2).

The lowest-order MBPT diagrams contributing to ZLor Z~ are shown in Fig. 1. In this case the transition in-volves the excitation of an electron from orbital p occu-pied in $0 to orbital k occupied in ltf. In these diagrams,lines with arrows drawn upward represent occupied ex-cited single-particle states and are called particles, andthose with arrows pointing downward represent unoccu-pied unexcited states and are called holes. The correla-tion interaction H, is represented by dashed lines be-tween particle and hole lines. The dipole interaction isrepresented by a dashed line ending with a solid dot. Theorder of a diagram corresponds to the number of interac-tions with H„and the diagrams are read from bottom totop. Diagrams in which the H, interactions precede thedipole interaction are said to contribute to the ground-state correlations (GSC). When the dipole interactionoccurs first, the diagram is said to contribute to the final-state correlations (FSC).

HF t.

coRL

IcoRV~~

HFV

07 15 19 23 27

PHOTON ENERGY (eV)

FIG. 3. Photoionization cross section calculated for the3s ~3skp ( 'P) transition not including resonance contributions.The curves labeled HFL and HFV represent the length and ve-locity cross sections in the Hartree-Fock approximation.Curves CORL and CORV are correlated length and velocity re-sults without the contribution from the double excitations oftwo 3s electrons.

4970 ZIKRI ALTUN

The Hamiltonian with spin-orbit interaction may bewritten as

N

Hso=H+ g g„, (LSJ)1, s, ,

where g„~(LSJ) are the effective spin-orbit parametersl

defined by Blume and Watson. Specifically, we examinethe effect of the spin-orbit interaction on the 2p md and2p ms states with J =1. The spin-orbit interaction mixesthe 'Pl multiplet with other +'LJ multiplets with J = 1.The mixing coefficients are obtained by diagonalizingHso using ~LSJMJ) basis states. The resultingstates are then treated as bound excited states for the 2psubshell.

The diagram of Fig. 1(c) with p=3s, k=kp,q=2s(2p), and k'=np(nd, sn) contributes to single-electron resonances in the 3s cross section in lowest or-der. The lowest-order double-electron resonances in the3s cross section are represented by the diagrams of Fig. 2.These diagrams are second order in the Coulomb interac-tion as compared with the diagram of Fig. 1(c) which isin first order in the Coulomb interaction. These diagramshave a series of simple poles at the positions of the reso-nances which may be handled by summing higher-orderdiagrams to infinite order. ' In this work we used thecoupled-equation method' to account for correlations inthe final state. In forming the coupled equations, we usednine excited configurations given by 2skp, 2p kd(ks),3skp, 3pkd (ks), 3dkf (kp), and 4skp, all with 'P final cou-pling. Here ks, kp, kd, and kf refer to both bound andcontinuum states. Solving the coupled equations is

equivalent to including the interactions among thesefinal-state channels to infinite order and is equivalent to aK-matrix calculation. ' Application of the coupled-equation method' to calculate the resonance structureeffectively results in real and imaginary shifts to thedenominators of the lowest-order FSC diagrams whichprevent singularities. The imaginary shift is related tothe full width at half maximum of the resonance stateand the real shift affects the position of the resonance. '

The expression used in the calculation of the photoelec-tron angular distribution asymmetry parameter P(co) hasbeen published elsewhere. ' '

III. CALCULATIONS AND RESULTS

The ground-state orbitals of Mg I were obtained from aself-consistent Hartree-Fock (HF) calculation and werenot changed during the rest of the calculation. All singlyexcited bound and continuum orbitals originating fromthe 2s, 2p, and 3s subshells were then calculated in thesame frozen-core HF V ' potential for a given I. Thedoubly excited continuum and bound ks, kd, kf, andkp orbitals were calculated in frozen-core Hartree-Fock V ' potentials derived from the couplings2p 3pks(kd)('P), 2p 3dkf(kp)('P), and 2p 4skp('P).The radial 3p, 3d, and 4s orbitals were calculated infrozen-core 2p Hartree-Fock V potentials. ' Foreach angular momentum, 10 bound and 42 continuum or-bitals were calculated with appropriate projection opera-tors when necessary to ensure the orthogonality of theexcited-state orbitals to ground-state orbitals of the sameangular momentum The effect of the remainder of

2.6

2.4-2.2—

2.0— Expt. I

3p4d('p)

3p3d(p)

~3p513p6d

1.8—

1.2—CA

m 1.0—O

0.8—

Expt. II

3p4s('p)

3pes

0.6-0.4—

0.2—

7.0 8.0 9.0 10.0 11.0 12.0 13.0PHOTON ENERGY ( e V)

14.0 15.0 16.0

FICx. 4. Photoionization cross section for the 3s ~3skp( P) transition involving the double-electron resonances as a function ofphoton energy. , MBPT velocity calculation (CORV), this work; ———,measurement (Expt. I) by Preses et al. , Ref. 5.;—- —.—,measurement by Ditchburn and Marr (Expt. II), Ref. 1. Double-electron resonances are labeled.

PHOTOIONIZATION OF ATOMIC MAGNESiUM INCLUDING. . . 4971

the bound excited states was taken into account by then rule' for diagrams contributing to correlations onthe ground states. For diagrams contributing to correla-tions in the final states, higher bound states were includedby explicit calculation and a back extrapolation of thecontinuum below the first K-mesh value into the boundregion. '

Calculations of the photoionization cross section forthe 3s valence shell from the threshold to 27 eV, not in-cluding e8'ects of double excitations, are given in Fig. 3.The curves labeled HFL and HFV represent the lengthand velocity forms of the cross section in the HF approxi-mation. These cross sections have a large disagreementnot only in their magnitudes but also in the positions ofCooper minima. We have studied the efFects of thecorrelations on these cross sections in two steps. In thefirst step we did not include the diagrams of Fig. 2 for thetwo 3s electrons. Diagrams included are shown in Figs.1(b)—l(g). The labelings of the hole and particle lines inthese diagrams were taken to be among the 2s, 2p, and 3sexcitations. Threshold energies used in correlating theground state are the HF frozen-core values, which for the3s, 2p, and 2s hole states are, respectively, 0.2534, 2.2882,and 3.7801 a.u. In correlating the final state, experimen-tal removal energies, which are 0.28109, 2.1258, and3.5477 a.u. ,

' were used, respectively, for these holestates. Excited bound orbital energies were taken to beHF frozen-core calculated values. Final-state correla-tions were calculated by solving the coupling equationsonly for the 2skp('P), 2p kd(ks)('P), and 3skp('P)channels. In these equations modified Coulomb gridswere used to account for diagrams 1(d), 1(e), 1(fl, and1(g). The modified Coulomb grids were obtained by firstsumming over the k" states in these diagrams and then

adding them to the Coulomb grid of the diagram in Fig.1(c). The dipole matrix elements used as input to thecoupled equations were taken to be the sum of the HF re-sults plus corrections due to the correlations in theground state as shown in Fig. 1(b). The length and veloc-ity results obtained from this first step are shown in Fig. 3in the curves labeled CORL (correlated length) andCORV (correlated velocity), respectively. There is nowconsiderable agreement between CORL and CORV cal-culations which agree with the RRPA results of Desh-mukh and Manson. ' They both have the Cooperminimum at the same position (11.63 eV) and the correla-tions brought the threshold cross sections from 14.41 Mb(HFL) and 4.06 Mb (HFV) to 6.10 Mb (CORL) and 5.20Mb (CORV). However, the results near the threshold arestill far from the measurements. '

In the next step we considered the correlations due tothe simultaneous excitation of the two 3s electrons. Weextended the coupled equations used in the first step toinclude those diagrams in Fig. 2 with the doubly excitedintermediate states 3pkd(ks), 3dkf (kp), and 4skp allcoupled to 'P. When kp, kd, and kf refer to bound excit-ed orbitals, the corresponding denominators may vanishwhen the excitation energy of the 3s ~kp transition is de-generate with the excitation energy of the doubly excitedstates. When this happens diagrams are said to contrib-ute to the double-electron resonances. In this calcula-tion, the doubly excited bound configuration used as in-termediate states are 3pnd(ms)('P), 3dnp(mf)('P), and4snp('P) (n =3—12, m =4—13). All of these states liesabove the ionization threshold and appear as autoioniza-tion resonances in the 3s cross section. In extending thecoupled equations, we used experimental thresholds forall the continuum channels and HF single-particle ener-

1 %

(I 1 I 1

II I I ~ 1 I I

t

I t 1 'I

[ )I 1 1 1

~ 2psas('p) 3s'

C)

C3UJCR

2p 4s( P)3s

2p 3d( D) 3s

2ps3d( P)3s~

~ 2$3d('P)3s

Qs

54

CORL

~ CORVa l i I~ 1}

55 56PHOTON ENERGY (eVi

57

FIG. 5. Correlated length (solid line) and velocity (dashed line) cross sections for the 3s ~3skp('P) transition in the 2p~nd, nsresonance region including spin-orbit eFects.

4972

0 4

1 ~ 2

ZIKRI ALTUN

Il

I

1

z',C)

2IJJ

C3

I—

LLJc/) 4p

Sp

6p

7p

KQ. 1

CORL ~~ CORV

0. 094 95 96

PHOTON ENERGY

105 155pHp&ON ENERGY (eV)

205

s o' '

nd the 2 threshold. The HFL and HFV curves'

1 3s hotoionization cross sections beyon t e pit

FIG. 6. Hartree-Fock and correlated partia s p oRL and CORV are correlated length and veloci ynd velocit cross sections. The curves COR anre resent the Hartree-Fock length an ve oci y c

1 d the box are due to 2s ~ np excitations.repre

h are shown in an expanded sca e insi e e oxMBPT calculations. The resonances which are s

I I I I

f

I ~ I ~I

I I I I Q I

Q. 8 I I I I IT

C3

LU

C3

I—0C3LLJ

CORV~

4p5p

6p 9p I Ip

BpI &zp

I7P

l

I

95PHOTON ENERGY

96(eV)

97

0-055

I I I

110PHOTON ENERGY

165(eV)

The HFL and HFV curves represent the'

1 2 ~ks hotoionization cross sections. ela-

FICz 7 Hartree-Fock and correlated partia p ~ s p~ ~

velocit cross sections. The curves CORL and CORV are corre a el ted length and velocity MBPT calcu a-1 'd t}1 b d to 2tions. The resonances, which are shown in an e pex andedscaeinsi e e ox, a

PHOTOIONIZATION OF ATOMIC MAGNESIUM INCLUDING. . . 4973

I I I I I I I I I 1 1 I I 1 i I 1 I I 1

I

.~Q(— ~

—~ -w. )4p Sp 6 p tp % I 2p

~p

1 Q(QJ

I

U'3

(.f)(U(2.( 0

I

)I

-/,

~ coRv~CORL

/,I

coRv ~ /

~ CORLI

Q (--9& 95 96

PI~QTQII E. NI RGY IeV I

055 105 155

PHOTON ENERGY (eV)205

I

1 I I I 'I 1 I I I I I I I I I I I I I I I I I I I I I

FIG. 8. Hartree-Fock and correlated partial 2p~kd photoionization cross section. The HFL and HFV curves represent theHartree-Fock length and velocity cross sections. The curves CORL and CORV are correlated length and velocity MBPT calcula-tions. The resonances which are shown in an expanded scale inside the box are due to the 2s ~np excitations.

gies for the bound excited states. The experimentalthresholds for the 3s ~3pkd (kf )('P), 3dkf (kp ('P), and4skp('P) transitions are 0.4438, 0.60656, and 0.59897a.u. , respectively. The diagram of Fig. 2(c) representsrelaxation and we found it to be very important for thiscalculation. The relaxation effects can also be accountedfor by calculating excited orbitals in the field of the re-laxed ionic core. The sum and integral over k"'states in the diagram of Fig. 2(b) was obtained by adifferential equation technique before it was used as an

input to the coupled equations.We compare our final correlated velocity cross section

(CORV) with measurements by Preses et al. [experimentI (Expt. I)] and Ditchburn and Marr' (Expt. II) for the3s~kp channel in Fig. 4. The resonances in the curves

0. 6

~HFV

0. 5 / HFL

6—JD

—5—0 4

w 3—CA

(AV)DCL

1

50l

100I I

150 200 250PHOTON ENERGY (e V)

300 350

0. 3LD

0. 0 I

90

CORVCORL

RRPA

140PHOTGN ENERGY

190(eV1

240

FIG. 9. Total photoionization cross section of the 2p sub-shell of magnesium as a function of photon energy. The curvelabeled CORL is correlated length result. The curve labeledRRPA is relativistic random-phase approximation result fromRef. 13.

FIG. 10. Hartree-Fock and correlated partial 2s photoioniza-tion cross section. The HFL and HFV curves represent theHartree-Fock length and velocity cross sections. The curvesCORL and CORV are correlated length and velocity MBPTcalculations. The RRPA results are from Ref. 13.

4974 ZIKRI ALTUN

12

9

(/)

C?' 3-

0 100 200PHOTON ENERGY (eV)

300

FIG. 11. Total 2s+2p +3s photoionization cross sectionfrom the first ionization threshold to 240 eV in both the length(solid line) and velocity (dashed line) forms.

(CORV, Expt. I) are due to the 3s ('S)~3pns('P) and3pnd ('P) transitions. In comparing our results withthose of Preses et al. we normalized their data to ours atthe peak of the first resonance (3p4s 'P). Using this nor-malization, our calculation is not in good agreement withthe measurement of Preses et al. from the threshold tothe minimum in the low-energy side of the 3p4s ('P) reso-nance. However, the only absolute measurement' donethus far in this region is in good agreement with our re-sult. As may be seen from Fig. 4, our threshold value forthe 3s cross section is 0.92 Mb (CORV) as compared tothe experimental values 2.55 Mb (Expt. I) and 1.20 Mb(Expt. II). Threshold values given by Bates and Altick,Dubau and Wells, Moccia and Spizzo, ' O'Mahony andGreene, ' and Rescigno" are 2.55, 1.5, 2.20, 4.3, and 2.3

Mb, respectively. The relaxation diagram of Fig. 2(c) wasfound to be the major cause of the significant reductionnear the threshold. We have not presented our correlatedlength result in Fig. 4 since it is in good agreement withthe velocity result everywhere except in the near-threshold region. The threshold value of the length crosssection is 0.57 Mb, which is about half of the experimen-tal result of Ditchburn and Marr. '

Figure 5 contains our correlated length and velocitycalculations of the 3s ~kp cross section in the 2p ~nd,ns resonance region. We calculated ten bound resonancestates for both nd and ns excitations and included thee8'ects of spin-orbit interactions on the first four of thesestates. The spin-orbit interaction mixes the 'P„P, , andD& terms of each nd resonance and 'P, and P, terms of

each ns resonance. The mixed states in which the multi-plet 2p ns('P& ) is strongest were always found to be thehighest in energy. The transition from the ground stateto the mixed states of ns resonances exhibits a 3:2 intensi-ty ratio ('P, : P, ) for ns =4s and 2:1 for ns =5s —7s.The mixed states in which the multiplets 2p nd ('P~ ) and2p nd ( P, ) dominate were found to be the highest andlowest, respectively, in energy for nd =3d —5d, with2p nd ( D& ) always in the middle. For nd =6d, the 'P~ isthe lowest and P, is the highest in energy. The transi-tions from the ground state to the mixed states of nd res-onances exhibit a 7:4:1 intensity ratio ('P: D: P) fornd =3d; 3:2:1 for nd =4d, 5d, 2:2:1 for nd =6d. The re-cent calculation of Deshmukh and Manson' also includ-ed these resonances except for the first three of the ndand four of the ns resonances. Thus far we are not awareof any experiment in this region.

In Fig. 6 we present the partial 3s ~kg cross sectionsabove the 2p threshold (58.7 eV). The resonances in the

I

Il

II

II

I—asap('r )

I

I r 1

II 1 I I C I

I

I

i ~2s5p( P)

II

I I I

w

V3

CY

094

!I I I I I I I I I I I I

95PHOTON ENERGY

I

96(eV)

s a I s

9/

FIG. 12. Total 2s +2p+3s cross section in the region of the 2s~np (n ~ 3) resonances. The solid line is the MBPT length calcu-lation. The dashed line is the MBPT velocity calculation.

PHOTOIONIZATION OF ATOMIC MAGNESIUM INCLUDING. . . 4975

correlated length and velocity cross sections are due to2s~np excitations. The resonances in the figure withinthe box represent some of these resonances in an extend-ed scale. These resonances show typical asymmetricFano line shapes. The widths of these resonances alsoinclude contributions from decay to other continuumchannels corresponding to the ionization of the 2p elec-trons. The curves labeled HFL and HFV in Fig. 6represent the length and velocity results in the HF ap-proximation.

The partial cross sections obtained for the 2s and 2psubshells from the solution of the coupled equations areshown in Figs. 7 —10. The coupled-equation length andvelocity partial cross sections for both the 2s and the 2pshells are found to be in excellent agreement in all cases.The partial photoionization cross section for the 2p ~kschannel is shown in Fig. 7. The curves labeled HFL andHFV in these figures represent the lowest-order lengthand velocity calculations. The correlated length and ve-locity cross sections in the same figures are labeledCORL and CORV, respectively. The resonances in theCORL and CORV curves are due to 2s~np excitations.The figure within the box represents some of these reso-nances in an extended scale. However, these resonancesare much stronger in the 2p ~kd cross section, as will bediscussed below. The correlations increased both theCORL and CORV cross sections compared to the HFLand HFV results.

In Fig. 8 we present the correlated length and velocity2p~kd partial cross sections (CORL, CORV) along withthe lowest-order results (HFL, HFV). The resonancesagain are due to Zs~np excitations. The figure withinthe box represents some of these resonances in an extend-ed scale. These resonances are much stronger in the2p ~kd channel than in either of the 3s ~kp and 2p ~kschannels. The correlations brought the length and veloc-ity results close to each other by pushing both of them upbelow the resonances and pushing the velocity up andlength down above the resonances compared to the HFLand HFV results.

We compare our correlated total length cross sections(CORL) for the 2p subshell with the recent calculation ofDeshmuhk and Manson' (RRPA) in Fig. 9. In compar-ing our result with the RRPA curve, we averaged overthe 2s~np resonances. The agreement between two cal-culations is good. This is not surprising because both cal-culations included the most important correlations due tothe single excitations from the 2s, 2p, and 3s shells withthe same degree of approximation in this region.

In Fig. 10 we compare our correlated 2s~kp lengthand velocity cross sections (CORL, CORV) with theRRPA calculation of Deshmukh and Manson and withHartree-Fock calculations. The agreement between themany-body calculations and the RRPA calculations is ex-cellent. The reduction in the CORL and CORV crosssections compared to the HFL and HFV curves wasfound to be mainly due to the interchannel interactionwith the 2p ~kd channel.

We present the sum of the correlated 2s, 2p, and 3spartial cross sections in both length (CORL) and velocity(CORV) approximations in Fig. 11. The resonance re-

2

CQRV CGRL

58 113 168PHOTON ENERGY (eV)

. l

223

FIG. 13. The angular asymmetry distribution parameter as afunction of photon energy for photoelectrons from the 2p sub-

shell. Correlated length (CORL) and velocity (CORV) calcula-tions using MBPT including 2s~np resonances. Dot-dashedline indicates calculation by RRPA method, Ref. 13.

gion between 90—97 eV is extended and replotted in Fig.12.

The angular distribution asymmetry parameter P(co)for the photoionization from the 2p subshell of atomicmagnesium has been calculated in both length and veloci-ty approximations. The two forms are in excellent agree-ment. Our length (CORL) and velocity (CORV) curvesare compared with the RRPA calculation of Deshmukhand Manson' in Fig. 13, and are in good agreement. Inour calculation of P(co) we used correlated dipole matrixelements obtained from our nine-channel coupled equa-tions which also includes the effects of the 2s~np reso-nances. The rapid variation in the asymmetry parameteras a function of photon energy in the near-threshold re-gion is mainly due to the strong interference between the2p ~kd and 2p —+ks channels.

IV. DISCUSSION AND CONCLUSIONS

Many-body perturbation theory has been used to treatelectron correlations including relaxation effects andsingle- and double-electron resonances in the photoion-ization of the magnesium atom over a wide energy range.This work used nine coupled final-state channels originat-ing from 2s, 2p, and 3s single excitations and the doubleexcitations of the two 3s electrons. I have included first-and higher-order correlations in the ground state andspin-orbit splitting of the lower members of the2p~nd (ns) resonances. The 3s threshold region is dom-inated by double-electron resonances, particularly3pns('P) (n «4) and 3pnd('P) (n «3). The value of the3s cross section at the threshold is 0.92 Mb (length) and0.57 Mb (velocity) compared to the previous calcula-tions ' which give values between 1.5 —4.3 Mb as dis-cussed. I also included correlations from the 2s and 2psubshells which are found to be quite significant in thiscalculation. Our threshold results for the 3s cross sectionappear low by about a factor of 2, as compared with the

4976 ZIKRI ALTUN

other calculations, " which agree very well in shapewith the recent (unnormalized) experiment by Preseset al. Our results at threshold are very sensitive to thechoice of threshold energy and in the curve presented inFig. 4 the experimental threshold energy was used.When the Hartree-Fock single-particle energy for the 3sionization energy is used, the cross section at the thresh-old is approximately 3.0 Mb, in better agreement with theother calculations and the experiment by Preses et al. 'The agreement between the length result and the absolutemeasurement by Ditchburn and Marr' near the thresholdis reasonable. However, the threshold value obtainedfrom the experiment of Preses et al. is about a factor of2.5 times larger than ours. This value is obtained by nor-malizing the experiment of Preses et ai. , either to thepeak of the 3p4s('P) resonance as calculated by Batesand Altick or by us. The relative heights of the3pss('P) and 3p6s('P) resonances given by the measure-ment of Fiedler, Kortenkamp, and Zimmermann, whichdid not include the region below the low-energy side ofthe 3p3d('P) resonance, are consistent with our calcula-

tions and the previous calculations.The total cross section does not include contributions

from the photoionization with excitation and double pho-toionization of the two 3s electrons. We plan to calculatethese cross sections in future work.

The results of this work above the double-electron res-onance region in the 3s cross section are in quite goodagreement with the RRPA calculations. ' This calcula-tion indicates that many-body theory is useful in calculat-ing photoionization cross sections and angular distribu-tion asymmetry parameters including correlations over awide energy range and is not limited to the threshold re-gion.

ACKNOWLKDGNIKNTS

I wish to thank Dr. H. P. Kelly, Dr. Daniel Frye, andMickey Kutzner for helpful discussions. We are verygrateful to the National Science Foundation for supportunder Grant No. INT-8507760 and Grant No. PHY-8705545-01.

'Permanent address: Marmara Universitesi, Fen-Ed. FakultesiIstanbul, Turkey.

'R. W. Ditchburn and G. B. Marr, Proc. Phys. Soc. London,Sect. A 66, 655 (1953).

G. Mehlman-Ballonet and J. M. Esteva, Astrophys. J. 157, 945(1969).

3J. M. Esteva, Cx. Mehlman-Balloffet, and J. Romand, J. Quant.Spectrosc. Radiat. Transfer 12, 1291 (1972).

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