Mg 3pjndj′ (j = 1,3) autoionizing series

JOURNAL OF ELEClWON SPECTROSCOPY and RelatedPhenomena

ELSEVIER

Journal of ElectronSpectroscopyand RelatedPhenomena81 (1996)33-46

Mg 3p/ndj, (j = 1,3) autoionizing series C.J. D a i Department of Physics, Zhejiang University, Hangzhou 310027 People's Republic of China

Received 10 November1995; accepted20 December1995

Abstract The spectroscopic properties of Mg 3pnd (j - 1, 3) autoionizing states have been studied by employing the combination of multichannel quantum defect theory with the R-matrix method. Energy levels of the 3pnd (] - 1) autoionizing states are obtained with resolution of fine-structure splitting. The transition lineshapes of the 3pnd (j - 3) are calculated using a theoretical model with the more complete channels. The possible impact of the higher angular momentum states on the Mg 3pnd (j - 3) autoionization spectra have been investigated. The results presented agree well with recent experimental and theoretical studies. Keywords: K matrix; Multichannel quantum-defect theory

1. Introduction For the last few years, much attention has been paid to doubly excited states of atomic magnesium [1-7]. Particularly, a series of experiments was carried out to systematically investigate spectra of both the 3pns [1] and 3pnd [1,2] autoionizing states and configuration interactions among series, as well as the angular distributions of ejected electrons, from the 3pns[3] and 3pnd[4] autoionizing states. The experiments certainly enhance our understanding on the atomic structure and the excitation dynamics of atomic magnesium. They not only enrich the atomic data but also promote further theoretical studies. However, due to various limitations encountered in the experiments (for example, difficulty in dealing with UV light in the experiments), several problems remain unresolved. Energy levels of both the 3pns and 3pnd (] = 1) states have been measured by several groups [8-11], and have been predicted by various theories [12-15]. All the studies performed, however, have not resolved the 3pj fine-structures and

have failed to provide energy levels with n greater than 9. Recently a theoretical study on the Mg 3pns states [16], where energy levels of the 3pns state with resolution of the 3pj fine structures for n up to 22 using multichannel quantum defect theory (MQDT) [1720], was reported. However, energy levels of the 3ppd (j = 3) states were measured from n = 9 up to n = 29, with a detailed MQDT analysis. However, the three 3ping (j = 3) channels have not been taken into account in the MQDT model. Although agreement with the experimental results was satisfactory, the physical mechanism involved remains to be interpreted. On the theoretical side, MQDT has evolved into a powerful tool and has enjoyed great success in analysing spectra of bound and autoionizing Rydberg states. In the early stage of its development MQDT was used as an empirical approach, which involved the empirical determination of a set of parameters that represented the interaction between channels by fitting the measured peak positions and widths of Rydberg states. More recently, the MQDT parameters have

0368-2048/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved PII S0368-2048(96)03034-4

CJ. Dai/Journal of Electron Spectroscopy and Related Phenomena 81 (1996) 33-46

34

been related to the R-matrix calculation [21-23] in order to bypass the rather cumbersome fitting procedure. Combination of MQDT with K matrices has proved remarkably successful in reproducing complicated spectra of the autoionizing series in the alkalineearth elements Mg [1,2,16], Ca [24,25], Sr [26,27] and Ba [28,29]. As we know, the radius of reaction volume r0 defines a region within which an R-matrix is calculated variationally and is critical to the R-matrix calculations, or, to the K-matrices, which are transformed from the R matrices. In the previous studies [1,2] the K matrices with r0 = 12 a.u. were used. In this work we have used the newer version of K-matrices with r0 = 20 a.u. [4] to find out the possible effects due to the change of r0. In this paper, we report energy levels of the 3pjnd; (j -- 1) state with n = 5-30, calculated with a sevenchannel MQDT model. In order to compare with the experimental results, the 3pns spectra, in which the 3pnd (j = 1) features are evident due to the heavy mixing between them, are displayed. Also, the present eight-channel MQDT model provides clear information derived from the inclusion of the 3png (j = 3) channels by comparing the present results with those previously obtained [2]. In addition, the extensive configuration interactions are demonstrated by calculating the mixing coefficients, or the wavefunctions of the final states. We add 3pnd (j = 1) and 3pnd (j = 3) spectra to obtain a resultant spectrum, which is to be compared with the experimental spectrum measured with the three, parallel linear-polarized lasers. Agreement between them implies both j = 1 and j = 3 calculations are in accordance with the experimental results. Section 2 is devoted to our theoretical formulation concerning the application of MQDT to the j = 1 and j = 3 autoionizing spectra of atomic magnesium, where channel couplings are described, and the procedures for combining the R-matrix with MQDT are summarized. Results and discussion are present in Section 3. Conclusions are made in Section 4.

2.

Theory

The procedure which combines the eigenchannel R-matrix and MQDT method is well documented,

thus only some key points and formulae are outlined here, which are useful in understanding the following sections. Detailed accounts can be found in various, papers [21-23]. As we know, different formulations of MQDT method, using different sets of parameters, can be employed to calculate the observables. In the eigenchannel MQDT formulation [19], the parameters are /~, Ui~ and D~. Once the K-matrix is known, u~ and/z~ are obtained by diagonalizing the K-matrix, D,, are the dipole matrix elements. In the phase-shifted MQDT formulation [20], the parameters are a set of phase shift, ~ and a transformed matrix with diagonal elements of zero values as well as dipole matrix elements Di. These two methods are used as a matter of choice, and have proven equivalent with each other[30]. In this article the eigenchannel MQDT formulation is, chosen to reproduce both j -- 1 and 3 spectra and to manifest the configuration interactions involved.

2.1. j = 1 case There are seven possible odd-parity j -- 1 channels below the Mg ÷ 3p states, which may be written in either LS-coupling or jj-coupling. Table 1 shows the assignments of the seven channels. Near the nucleus LS-coupling is the most physical due to the large electrostatic and exchange force, while at large distances the rate of phase accumulation depends on the energy of the core state, which indicates that the jj coupling is appropriate. The o~ eigenchannels and the i collision channels can be related by a transformation matrix U, and satisfy the following equation: det{Fi~ } = det{Ui~sin[r(vi + #~)]} = 0

(1)

where vi is the effective principal quantum number, i.e.,

oi =

(2)

Where Ii are the ionization limits (e.g., I3pl/2 =97340.33 cm -1, and I3p3/2 =97431.90 cm-1). R = 109734. 86 cm q is the Rydberg constant for atomic magnesium. The wavefunction of a final state may be written as either a linear combination of ~I'i or ~=, which are known [19]. The coefficients of the two combinations, A!p) or B~ ), characterize the admixture between

35

C.I. Dai/Journal of Electron Spectroscopy and Related Phenomena 81 (1996) 33-46 Table 1 Assignments of J = 1 channels in LS and jj couplings. Labeling Coupling

1

2

3

4

5

li > Ic~>

3sl/2epl/2 3sep IP 1

3slc:pa/2 3pns 1P1

3pv2nslcz 3pnd 1P1

3p3r2ns1;2 3sep 3P1

channels, and are determined by

6

7

3pl/2nd3/2

3pa/2nd3/2

3pns 3P1

3pnd 3P1

3p3;znds/2 3pnd 3D1

(ii) for 3pnd state

A(p) i -~ ~ Uitxc°s[Ir(°i -I- net)]Bet

(3)

t

g3pnd°CO~p.~,2 ~

i-5,6,7

D'~AI0) 2

(7)

I

where [ B ~ ) =cof(i, ot)/ [ ~

"11/2

cof2(1,a)J

(4)

The symbol cof(i,a) represents the cofactor of det IF[ in Eq. (1). In the case of Table 1, there are two open channels if one concerns the region below the 3pl/2 limit. Therefore the number of collision eigenstates is as many as that of open channels, i.e., 0 = 1, 2. Since the K matrices are precalculated at three energies of 95148.8, 97343.5 and 99538.3 cm -1 in LScoupling scheme [4], interpolation or extrapolation is necessary to obtain K matrices at all the energies we need. Furthermore, the K matrices in LS coupling may be transformed into jj-coupled matrices, viz, K jy = V K Ls V r ,

(5)

where V is an orthogonal 7 x 7 transformation matrix, whose elements may be evaluated using standard 3n-j tables [31]. Finally, one is able to obtain the cross section for the excitations from the 3sns to 3p/ns or from the 3snd 19 2 to 3pjndj, (j = 1) states, viz, (i) for 3pns state; tr3pns °c 6oo-~,2 i-~4 D-iAi(#) 2

(6)

In Eqs. (6) and (7) D i is the product of radial and angular parts of the dipole matrix element, 60 is the laser frequency for the core excitation. Variations of Oapns and O3pnagive rise to the spectra of the Mg 3pns and 3pnd autoionizing states, from which the peak positions and widths may be determined. 2.2. j = 3 case There are eight possible odd-parity j = 3 channels below the Mg + 3p states, which may be represented in either LS-coupling or jj-coupling. The assignments of channels are tabulated in Table 2. The first two channels are open if one concerns only the region below the 3pl/2 limit. The last three channels were neglected in the earlier five-channel M Q D T treatment. In this paper, we have included these three 3p/ng/channels in the M Q D T model, and attempt to give an insight into the physical mechanism of the impact on 3pnd (j = 3) spectra by a higher l (l = 4) state. The procedure is similar to the j = 1 case, and the main difference is from the K matrices, which are characterized by the j value. The K Ls matrices has to be transformed into the K ~ so that the fine-structure effects may be incorporated through the jj-LS frame transformation, denoted by V. Since the number of channels involved is eight, the V matrix now is

Table 2 Assignments ofJ = 3 channels in LS and jj couplings. Labeling

1

2

3

4

5

6

7

8

Coupling [i > Itx >

3sl/2efs:2 3pnd 3D3

3Sl/2efT:2 3s~f 1F3

3pl:znds/2 3pnd tF3

3parznd3/2 3png iF3

3pa/2nds~ 3sEf aF3

3pu2ngT/2 3pnd 3F3

3p3/2ng7:2 3prig aF3

3p3cznggcz 3png 3G3

C.J. Dai/Journal of Electron Spectroscopy and Related Phenomena 81 (1996) 33-46

36

an 8 x 8 orthogonal matrix, and is

V=

0

v~

0

0

v4-/7

0

0

0

0

~

0

0

-v~/7

0

0

0

-X/~

0

0

V/4-/9

0

0

V/-ff/15

0

0

0

0

V/2-/9

0

- V/i--/15

0

V/-~

0

0

V/-~-/45

0

V/4/15

0

0

o

o

o

v/i/3

o

o

1/2

o

o

0

1/3

o

0

x/~

0

0

0

V/5-/9

0

0

Diagonalization of the K ~ obtained by Eq. (5) with the V-matrix in Eq. (8) enables one to obtain the MQDT parameters #~ and Ui,~, and subsequently admixture coefficient A~ ) defined in Eq. (3).

-

X/ri-/45

- X/~12

(8)

-v'~ -1/6

It is worth noting that in both j = 1 and 3 cases, the radial parts of the dipole matrix elements corresponding to the two core transitions, 3Sl/2 ---"3pj, are virtually the same, which yield a constant contribution to

0

°~

A

0

v

<

~I..

0

0 96200

96400

96600

96800

97000

T e r m Energy (cm - ~ ) Fig. 1. Line shapes of the 3snd ~ 3pjndj transitions, where n is from 10 to 20. Spectral densities for the three 3pnd bound channels are calculated to assist in identifying each feature. From the top to the bottom, the spectra are for A~ to A 2, and a3pjndj,, respectively. Some peaks in the final spectra are originated from common features in several spectral densities, where j = ~1 and ~, j' = ~ and ~.

37

CJ. Dai/Journal of Electron Spectroscopy and Related Phenomena 81 (1996) 33-46

0

D

~5 v "0 e~ ¢q._

<

^

eq.~

0

,.---._A

_

95800

J

~..

95900

96000

96100

T e r m E n e r g y ( c m - 1) Fig. 2. Illustration of the classification of the features in terms of the calculated constituent spectral density. From the top to the bottom, the spectra are for 02, A ] to A~, anda3pj9dr, where j = ½and 3 j,= ~ and ~.

the final spectrum. However, the angular parts of dipole matrix elements, called overlap integrals /j, are different to j = 1 and 3 cases, i.e., O j ~"

2(vvj)2 sin[ r(vjv) ] V3/2rV 2 - 02 )

(9)

Physically, it represents the spatial overlap between the wavefunctions of the outer electron (ns or nd) before and after the core excitation. With the above knowledge,one may calculate the cross section of excitation from the 3snd 1D2 to the 3pjndy, 1F3 states, viz, o~

~

Or3pnd ('00-1,2 i - ~,4,5

D:A(p) 2 r-i

(10)

Above all, the autoionizing states not only interact with each other, producing the energy level perturbations familiar from the study of bound Rydberg states, but also interact with the degenerate continua to produce the broadening characteristic of rapidly decaying

states. From Table I and Table 2, the two continua are 3sEpj for j = 1) and 3self (for j = 3), while the rest of channels are bound. The main advantage of MQDT is that the bound and continuum channels can be treated in a unified fashion with a small set of parameters nearly independent from energy, the results of which are presented in the next section.

3. Results and discussion

Due to configuration interactions between channels, a spectrum of an autoionizing state generally contains contributions from all the possible channels, which may be evaluated from their wavefunctions. The coefficients A/~°)in the linear combination of (1), are scaled to produce a wavefunction normalized, and [A,~°)]2, usually called the spectral densities, reflect the contributions from the ith channel to the spectrum), and thus play a vital role in the final autoionization

C.J. Dai/Journal of Electron Spectroscopy and Related Phenomena 81 (1996) 33-46

38

Table 3 Admixture coefficients of Mg 3pl/2nd (J = 1) states n

E(cm-1)

A32

A~

A2

A~

A72

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

92664.76 94133.41 95008.37 95569.28 95950.82 96222.79 96411.35 96567.24 96687.54 96765.71 96844.13 96907.59 96957.45 96994.45 97032.75 97063.76 97086.96 97111.67 97132.25 97147.76 97163.78 97176.19 97188.80 97199.01 97209.32 97217.43

.00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00

.00 .00 .00 .01 .02 .08 .40 .13 .10 .00 .00 .15 .13 .00 .00 .08 .00 .06 .12 .00 .06 .00 .18 .03 .07 .00

.71 .75 .78 .81 .82 .75 .58 .76 .60 .55 .85 .70 .71 .65 .87 .79 .63 .80 .58 .85 .77 .82 .72 .77 .65 .86

.09 .03 .03 .03 .04 .03 .00 .00 .03 .04 .00 .00 .00 .00 .00 .00 .06 .00 .03 .15 .00 .00 .00 .04 .10 .00

.20 .22 .19 .16 .14 .14 .02 .11 .27 .41 .15 .15 .16 .35 .13 .13 .31 .14 .27 .00 .17 .18 .10 .16 .18 .14

spectrum. Fig. 1 and Fig. 2 show two examples of such calculations. A merged spectrum of the 3pnd (j = 1) states, with n = 10-20, is illustrated in Fig. 1, together with spectra of three spectral densities A 2, A~ A 2, corresponding to the 3pjndj, channels. From this picture over a wide spectral range, one may see that all the sharp features are due to 3pns (j -- 1) states, while several broad features may be identified as the 3pns (j = 1) states (e.g., 3p3/2 12s at 96,128. 9 cm-1). The profound 3pns features in the 3pnd spectrum are originated from heavy mixing between the two series. Furthermore, Fig. 1 and Fig. 2 allow one to divide a spectrum into several components in terms of channels to understand the underlying physics processes. It is evident that each peak in the final spectrum contains contributions from several spectral densities. In order to assign each peak in the final spectrum to a specific configuration, it is necessary to resolve both n values and the 3pj fine structures. This may be done by presenting two overlap integral squares, 02 , which are in the top panel of Fig. 2. To perform detailed

configuration assignments, many spectra for different states (n = 5-30), similar to Fig. 2 have been calculated and merged in a suitable way so that the patterns of the 3pnd profiles are apparent. Starting from the lowest 3pnd (j = 1) state where states with different n values are impossible to be confused, one may assign each peak with great confidence. In Fig. 2 the features may be classified into two groups since the 3pj ndf features are clearly located below their corresponding 02 profiles. The left one is the 3pl/2 9d3/2 State, and the right one is attributed to the 3p3/2 9dj, states, where j'= 3 and I, respectively. Furthermore, the fine-structure states can be resolved by examining the spectral densities. Although the mixing between channels is pronounced, only one channel dominates a specific peak, as shown in Table 3 and Table 4. Based on the dominance of one channel for each peak, it is possible to assign all the peaks in the final spectrum to 3pl/2ndj, or 3p3/2ndj, states. From Table 3 and Table 4 it is apparent that in each state the dominance of one specific channel is obvious. However, the assignments

C.J. Dai/Journal of Electron Spectroscopy and Related Phenomena 81 (1996) 33-46

39

Table 4 Admixture coefficients of Mg 3p3/2nd3~sn (J = 1) states n

E(cm -1)

A2

A~

A2

A~

A72

5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24 25 25 26 26 27 27 28 28 29 29 30 30

92812.33 93005.24 94205.48 94345.70 95078.44 95183.57 95641.35 95701.43 96022.90 96072.95 96295.66 96328.90 96496.24 96521.86 96647.32 96660.93 96579.22 96777.64 96857.80 96870.61 96933.02 96942.63 96994.61 97001.48 97043.12 97050.34 97084.94 97090.55 97122.44 97126.45 97150.84 97154.45 97178.24 97182.15 97196.49 97199.02 97221.32 97223.54 97238.05 97240.45 97252.87 97255.47 97267.52 97269.37 97278.67 97280.29 97289.46 97290.95 97299.67 97301.08 97308.89 97309.87

.00 .04 .00 .06 .00 .08 .00 .10 .00 .21 .00 .25 .10 .06 .00 .00 .00 .00 .00 .24 .16 .08 .00 .00 .00 .19 .00 .00 .09 .00 .00 .07 .00 .21 .23 .00 .10 .00 .00 .13 .00 .12 .00 .15 .00 .00 .00 .00 .07 .05 .06 .00

.06 .05 .00 .07 .00 .06 .00 .00 .00 .00 .00 .06 .00 .04 .00 .00 .12 .04 .00 .00 .00 .06 .08 .06 .09 .08 .00 .04 .00 .00 .00 .00 .00 .00 .00 .04 .00 .00 .00 ..00 .00 .09 .00 .00 .00 .00 .08 .14 .12 .06 .07 .00

.13 .24 .00 .19 .00 .16 .00 .12 .00 .09 .00 .11 .00 .16 .00 .32 .14 .26 .00 .08 .00 .17 .15 .23 .16 .12 .09 .21 .00 .23 .00 .14 .04 .12 .12 .16 .00 .24 .11 .10 .00 .13 .16 .09 .14 .32 .17 .11 .21 .19 .16 .17

.71 .07 .88 .07 .89 .07 .90 .08 .90 .08 .89 .05 .89 .06 .91 .07 .60 .07 .90 .07 .80 .04 .55 .04 .65 .00 .63 .05 .86 .07 .85 .07 .79 .06 .60 .04 .82 .08 .71 .08 .92 .07 .63 .08 .70 .05 .68 .02 .60 .09 .59 .16

.20 .60 .12 .61 .11 .63 .10 .70 .10 .62 .11 .63 .11 .68 .09 .61 .14 .63 .10 .61 .04 .65 .22 .67 .10 .61 .28 .70 .05 .70 .15 .72 .17 .61 .05 .76 .08 .68 .18 .69 .08 .59 .21 .68 .16 .63 .07 .70 .00 .61 .22 .67

40

C.L Dai/Journal of Electron Spectroscopy and Related Phenomena 81 (1996) 33-46

Table 5 Energy levels and quantum defects of Mg 3pjnd (J = 1) states

n

3pmnd3/2 E(cm "1)

QD

n

3pa/2nd3/2,5/2 E(cm"1

QD

5

92664.76

0.155

6

94133.41

0.150

7

95008.37

0.140

8

95569.28

0.129

9

95950.82

0.113

10

96222.79

0.091

11

96411.35

0.132

12

96567.24

0.086

13

96687.54

0.035

14

96765.71

0.181

15

96844.13

0.129

16

96907.59

0.076

17

96957.45

0.071

18

96994.45

0.188

19

97032.75

0.112

20

97063.76

0.081

21

97086.96

0.189

22

97111.67

0.093

23

97132.25

0.035

24

97147.76

0.129

25

97163.78

0.069

26

97176.19

0.144

27

97188.80

0.089

28

97199.01

0.134

29

97209.32

0.059

30

97217.43

0.119

5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24 25 25 26 26 27 27 28 28 29 29 30 30

92812.33 93005.24 94205.48 94345.70 95078.44 95183.57 95641.35 95701.43 96022.90 96072.95 96295.66 96328.90 96496.24 96521.86 96647.32 96660.93 96759.22 96777.64 96857.80 96870.61 96933.02 96942.63 96994.61 97001.48 97043.12 97050.34 97084.94 97090.55 97122.44 97126.45 97150.84 97154.45 97178.24 97182.15 97196.49 97199.02 97221.32 97223.54 97238.05 97240.45 97252.87 97255.47 97267.52 27269.37 97278.67 97280.29 92789.46 97290.95 97299.67 97301.08 97308.89 97309.87

0.126 0.021 0.168 0.037 0.172 0.014 0.171 0.037 0.175 0.014 0.173 0.026 0.170 0.019 0.174 0.070 0.228 0.049 0.175 0.118 0.169 0.024 0.159 0.033 0.200 0.041 0.216 0.070 0.169 0.046 0.241 0.113 0.201 0.039 0.410 0.293 0.172 0.051 0.208 0.059 0.242 0.061 0.163 0.163 0.239 0.097 0.244 0.098 0.192 0.038 0.132 0.013

41

C,I. Dai/Journal of Electron Spectroscopy and Related Phenomena 81 (1996) 33-46

0 °..d

~5

0

96300

96400 Term Energy (cm -1)

96500

Fig. 3. Line shape of the 3slld ~ 3pjlld i, transition. The 3pjlldj, (J = 3) spectrum are calculated with the five-channel MQDT model (the top panel), and the eight-channel MQDT model (the middle panel), respectively. The sharp dip, at 96334.4 cm -1, is corresponding to the 3p3/210g9/2 resonance, which is not resolved by the experimental spectrum [2] (the bottom panel).

given in the tables are to some extent arbitrary, because almost none of these states is a "pure" state, that may attributed exclusively to a single channel. Meanwhile, the tables indicate the fact that the configuration interactions in the 3pnd (j = 1) series are not only intensive but also extensive. For the lower n states, the peaks belonging to different n may be identified with great case, while it is not the case for the higher n states. As n increases, the spacing between successive autoionizing Rydberg states is reduced rapidly, and the spectral structures eventually coalesce into two bundles associated with the 3s --* 3pl/2 and 3s ---, 3pl/2 transitions. Therefore, for the peaks lying energetically higher, the situation becomes more complicated since the regularity of spectral patterns become ambiguous. Under these circumstances, a more reliable way to distiguish between different n is to sort peaks by their quantum defects, which are in general expected to be constant as n increases for higher-lying autoionizing states. Note

that the total area of the 3pa;zndj, bundle for the high n states (e.g., n > 20) is approximately twice that of the 3pl/2ndf state, as expected. Since the former states have twice the degeneracy of the latter, this fact may also assist one to identify the fine-structure states. All the methods described above allow one to make precise assignments of spectral features, which are shown in Table 5 together with their quantum defects. For each series in Table 5, the quantum defects (QDs) are approximately independent of energy. For the sake of comparison, we note that the quantum defects of the 3pjndj, (j = 1) states are much less than those of their j = 3 counterparts [2]. For reference, the typical values of quantum defects for 3pl/2ndj, are 0.1 (j = 1) and 1.2 (j = 3). For the 3p3/2 nd3/2 states, they are 0.2(./= 1) and 1.2 j = 3 and for the 3p3/2nds/2 states, they are 0.05 j = 1 and 1.0 (j = 3). This yields the information that the 3pjndi, (j = 1) states lying energetically higher than their j = 3 counterparts are approximately close to 3pj(n + 1)ndj,, (j = 3) states.

C.J. Dai/Journal of Electron Spectroscopy and Related Phenomena 81 (1996) 33-46

42

J

.6 t~ t~

0

Ol

I

96750

96800

I

;

96850

96900

96950

T e r m E n e r g y ( c m - ~) Fig. 4. Same as Fig. 3, except these spectra are the line shapes for the 3s15d ---, 3pinf transitions, n ranging from 13-17. The sharp dip located at 96871.8cm- 1 is due to the 3pa/214g9/2 resonance.

Tuming now to the spectra of the 3pnd (j = 3) states. Using the K matrices with r0 = 20 a.u.; we first calculated the 3pnd (j = 3) spectra with the five-channels MQDT model, and obtained virtually identical results to these with the K-matrices of r0 = 12 a.u. [2], indicating the r0 effects to the 3pnd (j -- 3) spectra are of only minor importance, although the larger K-matrix box should in principle incorporate some long-range multiple effects neglected previously. We then carried out calculations of the 3pnd (j -- 3) spectra, n ranging from 5 to 30, with the eight-channel MQDT model. The inclusion of the 3png channels in the MQDT model allows us to find out to what extent the 3png channels affect the 3pnd (j = 3) series. Fig. 3 and Fig. 4 illustrates spectra of 3pjlldj, and 3pj15d states, respectively. In both figures, the five-channel (the top panel), the eight-channel (the middle panel), and the experimental spectra [2] are shown for handy comparisons. It is seen that in general both models give satisfactory predictions to the 3pjndj, (j -- 3) spectra, although the eight-channel model provides the

better agreement with the experiment. For instance, the five-channel model gives a broader profile than the experimental one for the n = 10 state (Fig. 3) while the eight-channel model improves it significantly. The improvement is also apparent for the n = 15 state (Fig. 4). Furthermore, the eight-channel MQDT treatment allows us to describe additional weak and narrow gwave resonances, which appears as narrow windows (or dips) on the 3pnd (j = 3)spectra. It is evident that the windows in the final spectra must be due to j = 3 3png states since the features do not appear in the MQDT calculations which exclude 3png states. To understand the underlying physics involved in producing dips rather than peaks in the 3pnd (j = 3) spectra, we have classified and assigned all peaks corresponding to 3pjngj, (where j = 1/2, 3/2, j' = 7/2, 9/2) resonances. The assignments are based on many calculations, such as Fig. 5, which represents the n -10 state. In Fig. 5 from the top to the bottom panel, the spectra are of A2-A~, and of 3pl0d (j = 3) state's

43

CJ. Dai/Journal of Electron Spectroscopy and Related Phenomena 81 (1996) 33-46

&

A

A

0 0 .6

~L

0 b e., t~

0

¢,,i._

<

0 0 0

96000

96100

96200

96300

Term Energy (cm -t) Fig. 5. Illustration of the 3pjl0dj,, spectrum, both from the eight-channel MQDT calculation and from the experiment [2]. From the top to the bottom, the spectra are A 32 to m~,2 calculated 03pi, 10dj.,and the measured a3pi, 10dj.- The sharp dip at 96075.9 cm- 1is due to the 3p3/29g9/2 resonance, which is not resolved by the experiment. spectra (calculated) and (observed), which provides an overview of the sharper 3png and broader 3pnd configurations. Examination of Figs 3 - 5 reveals that the sharp dips on the three calculated spectra of 3p3/210d, 3p3/211d and 3p3/215d located at 96075.9 cm -1, 96334.4 cm -1, and 96871.8 cm -1, are 3p3/2 9g9/2, 3p3/2 10g9/2 and 3p3/2 14g9/2, respectively. The quantum defects of 3p3/2ng states are very close to zero (typically 0.004) as expected. As we know, when n > > 1, the quantum defect of a Rydberg series satisfies ~1 0¢ 1-5 scaling law [32]. Note that the sharp dips occur whenever a sharper resonance is degenerate with a broader profile. The observed spectra with dips are not due to interference in the transition amplitudes, but are due to structure in the autoionizing states themselves. Although the dips were not observed in the Mg experiment, similar phenomena are observed experimentally for Ba, on the study of 6pjnf [33], 6Ping [34], and 6pjnd [35]. In those experiments, the higher members of the

Rydberg series converging to the 6pl/2 limit are degenerate with a lower member of Rydberg series converging to the 6p3/2 limit, which is manifested as the dip in the center of the latter state. Since the fine-structure splitting of the 3p ÷ state is much smaller than that of the 6p ÷ state of Ba, (for Mg it is 92 cm -1, for Ba it is 1690 cm -1) the same phenomena are not easy to observe for Mg except for the higher n states (e.g., n > 20 states), where the higher members of the 3pl/2nd series are degenerate with lower members of 3p3/2nd series [2]. However, in the case where the 3png states are involved, the sharp dips exist in the much broader energy range, because the widths of the 3p3/2ng state are always much smaller than their 3pnd counterparts. This ensures that whenever the 3p3/2ng states are degenerate with the 3p(n + 1)d states, it will be manifested a dip in the spectrum of 3pnd state,which are demonstrated in Figs 3-5. However, since the widths of 3png resonances are expected to increase as n decreases, the windows should be

Cal. Dai/Journal of Electron Spectroscopy and Related Phenomena 81 (1996) 33-46

44

D e

|

f-%

.N

0 .6

0 96450

96500

96550

I 96600

96650

T e r m Energy (cm -~) Fig. 6. Line shape of the 3s13s ---, 3pjl3s transition. The two calculated spectra in the top panel are from the K matrices with r0 = 12 a.u. (solid line) and 20 a.u. (dashed line). The measured spectrum [1] is shown in the bottom panel for direct comparison. The sharper feature at 96567.3 cm- 1 is due to the 3pl/212d (J = 1) resonance.

resolved experimentally if the lower n states are measured (e.g., n < 8). The instrumental linewidth is unfortunately too large to allow confirmation of the sharp dips seen in the calculations for n z 10 states. Note that the experiment only measured the 3pnd (j = 3) spectra down to n = 10 [2] and the dips were not resolved (see Fig. 5) where the width of window is about 2 cm -a. One of objectives of this work is to compare the calculated 3pnd (j = 1) profiles with the experimental results. Although there have been several experimental studies on the 3pnd (j = 1) spectra with multiphoton ionization (MPI) technique by driving 3s z'l So ---"3pns (nd) tpt transitions, [5,8,11], they are only limited to the lower n states (n ,: 9) and fail to resolve the 3pj fine structures. The more complete experiments on the 3pnd states are with a three-step excitation scheme [1,2]. When all three lasers are set circularly polarized in the same sense, only the 3pnd (j = 3) state may be excited. However, if one sets all the polarizations of

the three lasers linear and parallel to each other, the j = 1 and j = 3 components will be excited simultaneously. Therefore a direct comparison between the theory and experimental results for 3pnd (j = 1) states is difficult. For these reasons, we choose to compare our j -- 1 calculations with the available experimental results indirectly with the two different approaches. Firstly, we have used the same j = 1, K matrices and MQDT model to calculate the 3pns (j -- 1) spectra, where the pronounced 3pnd features appear through heavy mixing between the two series. By comparing the calculated and experimental spectra of the 3pns states, one may easily identify the 3pnd (j ~ 1) states and make comparisons with the direct calculations of the 3pnd j = 1. An example of this approach is shown in Fig. 6. In the top panel there are two calculated spectra of the 3pjl3s states, which are from theK matrices with r0 -12 a.u. (solid line) and r0 -- 20 a.u. (dashed line). The bottom panel is the experimental spectrum of the 3pjl3s

C.L Dai/Journal of Electron Spectroscopy and Related Phenomena 81 (1996) 33-46

45

°~

0 i.4

96200

96300

96400

96500

Term Energy (cm -~) Fig. 7. Line shape of the 3sl ld ----3pjnd/(J = 1, 3). The top panel shows the resultant spectrum combining both J = 1 and J = 3 final states. The measured 3pjndj,, (J = 1, 3) spectrum following parallelly linear-polarized excitations.

state. The sharper feature at 95657.24 cm -] is corresponding to the 3plr212d (j -- 1) state, which is exactly the same as the direct calculation of the 3p1/212d (j = 1) states (see Table 2). Secondly, we have compared the j = 1 and 3 calculations with the experimental spectra of the 3pnd (j -- 1, 3) states measured with parallelly aligned polarization of all three lasers, by adding the calculated j = 1 and 3 components. An example of this approach is illustrated in Fig. 7 for the 3 p l l d (j -- 1, 3) state. Keeping the fact that the calculated 3 p l l d (] -- 3) spectrum agrees well with this experiment, the agreement of the resultant calculated spectrum with the experiment implies that the 3pnd j = 1 calculation is also in accordance with the experiment, which is the case in Fig. 7. Above all, the present calculations allow one to identify and designate all energy levels of the j = 1 3pnd states with great reliability. The transition lineshapes are also predicted with great satisfaction.

However, spectra of the lowest members (n < 10) of the 3pnd (j = 1,3) series, remain to be tested.

4. Conclusion We have presented a systematic study of the spectroscopic characteristics of the Mg 3pnd (j -- 1,3) autoionizing Rydberg series. The combination of MQDT with the R matrix enjoys great success in predicting energy levels and transition lineshapes as well as providing a detailed interpretation of physical manifestations for the series. The 3png channels produce narrow windows in the 3p (n + 1), (j = 3) resonance profiles, which are too sharp to be resolved by previous experiments. The calculations explore the extensive and intensive interchannel interactions in bothj = 1 andj = 3 3pnd autoionizing states. The 3pindj (j --- 13) energy levels predicted by the present calculations remain to be tested by further experimental study.

46

C.I. Dai/Journal of Electron Spectroscopy and Related Phenomena 81 (1996) 33-46

Acknowledgements This work was supported by the National Natural Science Foundation of China, the State Commission of Education of China, and the Science Foundation of Zhejiang University.

References [1] G.W. Schinn, C. J Dai and T.F. Gallagher, Phys. Rev., A43 (1991) 2316. [2] C.J. Dai, G.W. Sehinn and T.F. Gallagher, Phys. Rev., A42 (1990) 223. [3] M.D. Lindsay et al., Phys. Rev., A45 (1992) 231. [4] M.D. Lindsay et al., Phys. Rev., A46 (1992) 37895. [5] Y.L. Shao, C. Fotakis and D. Charalambidis, Phys. Rev., A48 (1993) 3636. [6] M.D. Lindsay et al., Phys. Rev., A50 (1994) 5058. [7] N.J. Van Druten, R. Trainham and H.G. Muller, Phys. Rev., A50 (1994) 1593. [8] G. Mehalaman-Ballofet and J.M. Esteva, Astrophys. J., 157 (1969) 945. [9] M.A. Baig and J.P. Connerade, Proc. R. Soc. London, Ser., A364 (1978) 35310. [10] D. Rasisi et al., J. Phys., B10 (1977) 2913. [11] W. Fielder, Ch. Kortenkamp and P. Zimmermann, Phys. Rev., A36 (1987) 384. [12] R. Moccia and P. Spizzo, J. Phys., B21 (1988) 1133.

[13] T.N. Chang, Phys. Rev., A34 (1986) 4554. [14] C.H. Greene, Phys. Rev., A23 (1981) 661. [15] H.C. Chi and K.N. Huang, Phys. Rev., A50 (1994) 392. [16] C.J. Dai, Phys. Rev., A51 (1995) 295117. [17] M.J. Seaton, Proc. Phys. Soc. London, 88 (1996) 801; Rep. Prog. Phys., 46 (1983) 167.. [18] K.T. Lu and U. Fano, Phys. Rev., A2 (1970) 81. [19] C.M. Lee and K.T. Lu, Phys. Rev., A8 (1973) 1241. [20] W.E. Cooke and C.L Cromer, Phys. Rev., A32 (1985) 2725. [21] C.H. Greene and Ch. Jungen, Adv. At. Mol. Phys., 21 (1985) 51. [22] C.H. Greene and M. Aymar, Phys. Rev., A44 (1991) 1773. [23] M. Aymar, J. Phys., B20 (1987) 6507. [24] V. Lange, U. Eichmann and W. Sandner, J. Phys., B22 (1989) L245. [25] F. Robicheaux and Bo Gao, Phys. Rev., A47 (1993) 2904. [26] M. Aymar and J.M. Lec.omte, J. Phys, B22 (1989) 223. [27] R.P. Wood and C.H. Greene, Phys. Rev., A49 (1994) 1029. [28] M. Teimini, M. Aymar and J.M. Lecomte, J. Phys, B26 (1993) 233. [29] D.J. Armstrong and C.H. Greene, Phys. Rev., A50 (1994) 4956. [30] C.J. Dai, S.M. Jaffe and T.F. Gallagher, J. Opt. Soc. Am., B6 (1989) 1486. [31] R.D. Cowan, The Theory of Atomic Structure and Spectra, University of California Press, Berkeley, 1981. [32] R.R. Freeman and D. Kleppner, Phys. Rev., A14 (1976) 1614. [33] R.R. Jones, C.J. Dai and T.F. Gallagher, Phys Rev., A41 (1990) 376. [34] S.M. Jaffe et al., Phys. Rev., A32 (1985) 1480. [35] F. Gounand et al., Phys. Rev., A27 (1983) 1925.