Computation of Doubly-Excited 1S<SUP>2</SUP>3l3l′<SUP>1</SUP>P° and <SUP>1</SUP>F° States in Beryllium-Like Ions

Ind.anJ.Phys. 82(4) 387-401(2008)

#

Computation of doubly-excited ls

3l3l

P° and

F° states in beryllium-like ions

Utpal Roy¹* and YK Ho²

'Department of Computer and System Sciences, Siksha-Bhavan, Visva-Bharati Santiniketan 731 235, West Bengal, India

institute of Atomic and Molecular Sciences, Academia Sinica, PO Box 23-166, Taipei, Taiwan 106, ROC E-mail uroym@yahoo co in

Received 16 August 2007, accepted 15 January 2008

Abstract : We present here an implementation of the stabilization method to calculate the resonance energies and widths of the doubly-excited 1s23/3/'1F° and 1s23ft/'1P° states of beryllium-like ions by calculating the density of resonance states These systems have been treated formally as a two-electron problem by using a suitable model potential to describe the effect of the 1s2 core Slater orbitals are used to represent the two electron wave functions. Our results are compared with those available in the literature

Keywords Doubly excited states, stabilization method, resonance energy and width, density of states, slater orbits

PACS Nos. : 32.30.-r, 32.80.Dz, 32.80.Hd, 32.70.Jz

1. Introduction

During the collision of slow multiply-charged ions with atoms and molecules, the doubly excited states of some positive ions are frequently observed. These doubly- excited states where the two electrons are excited from the ground state configuration, are formed either by double electron capture processes, where two electrons are transferred to the projectile ion in single collision [1-4], or by transfer excitation processes where one of the projectile electrons is excited together with the capture of an electron from the target. These doubly excited states are identified by Auger electron spectroscopy or through their radiative decays. One major difficulty in studies of such electron capture and transfer excitation collisions is the lack of the knowledge about the energy levels of the doubly' excited states for most atoms and ions.

Extensive calculations for the energies and widths of doubly excited states have been

'Corresponding Author © 2008 IACS

carried out for helium-like atoms and ions [5-7]. For doubly excited states, for

example, of beryllium like ions, which have been observed in many laboratories, the knowledge of the energy levels and the autoionization widths for such doubly-excitea states will play an important role in the study of collision dynamics.

The stabilization method proposed by Mandelshtam et al [8] is one of the simplest and most powerful tools to study atomic and molecular resonances. The complex co-ordinate rotation method is difficult to apply for larger systems. Moreover to have even the flavor about the resonance energy Er and the width r scattering methods are believed to be more difficult. The stabilization method does not use complex analytic continuation [9] nor does it require the asymptotic behavior of the continuum wave functions [10,11]. Interestingly, the method avoids the use of imaging technique and complex potential. The simple stabilization method needs only the diagonalization of the real matrix elements repeatedly with the varied box sizes L Then the stabilization diagram can be obtained by plotting the different eigenenergies En(L) vs. the box size L The stabilization diagram provides the information for the large region of energy spectrum as well as presents some flat energy region in the vicinity of an avoided-crossing that corresponds to the occurrence of a resonance. The physical origin of the flat region indicates the fact that the resonance scattering wave function is localized within the short range, and as such, its energy is stabilized. To obtain the resonance energy and the width, one needs to calculate the density of resonance states. According to Mandelstham et al [8] density of resonance states pn(E) can be calculated in the energy region of interest from which the resonance energy and the width can easily be extracted through a fitting procedure.

Since the pioneering work of Hazi and Taylor [12] on the stabilization method, a considerable amount of theoretical work has been carried out in order to calculate resonance parameters with square integrable functions (see for example [13-15]) Muller et al [16] applied this method to calculate the positions and widths of He 'S*

resonances in the Rydberg manifolds. Martin and Politis [17] applied such a procedure to study the resonance charge-transfer in atom surface interactions, and Bachau and Martin [18] extracted the resonance parameters of H and H~ in static electric fields Recently, Ho and co-workers have applied the method to extract the resonance energy and total width for doubly excited states of H" [19] and also for 'D9 [20] and 1Ge [21]

states of the four-electron beryllium-like ions. Fang and Ho [22] have applied it successfully for the Mg atom. Roy and Ho [23,24] have employed this method to deduce resonance parameters in positron collisions with lithium atoms.

Here we apply the method to investigate the 1 s23 / 3 / '1F ° and 1 s23 / 3 / '1P ° doubly-excited states of beryllium-like B+, C2+, N3+, 04+, F5* and Ne6+ ions below the N = 3 threshold of the three-electron systems. Furthermore, we use a model potential to represent the interaction between the inner core and the outer active electrons.

Products of Slater orbitals are used to represent the doubly-excited two-electron wave

Computation of doubly-excited 1s23l3l' 1P°and 1F° states in beryllium-like ions 389 functions. Basis sets of N = 851 and N = 561 terms are used respectively for these

systems to calculate the 1 s23 / 3 /; 1P and 1s23/3// 1P states For the lower angular momentum states, the highly correlated Hylleraas-type wave functions are more appropriate. However, the use of extensive Hylleraas functions to construct the stabilization plots would require a major computational effort. It is outside the scope of the present investigation. The use of the products of Slater orbitals should be able to provide accurate results for such high angular momentum states.

2. The stabilization method

The method is based on the density of resonance states that depends on the value of L, the box size. The density of states has contributions from two regions which we symbolize by writing p(E) = pQ{E) + />P(E), where P and Q refers to the open and close spaces, respectively, in the Feshbach projection formalism. The pp(E) is the smooth function of energy £, p°(E) is the result of complex poles of the Green's function.

The spectral density of resonance pole is given by

P°(E)

- - l m E ¹

(E-Ek) + i^

(1)

f

where Ek- / — is the /c-th complex pole of Green's function. Mandelshtam et al [8]

have sown that p°(E) can be obtained by

,°(E) = - — ! — t PL{E)dL (2)

where

PL(E) = J2HEJ(L)-E)> 0) and L is the size of the box in which Hamiltonian is diagonalised. For atomic systems,

we can replace the 'hard wall' by a 'soft wall* (Muller et al [16]), and eq. (2) can be replaced by the following

.i "max

p°(E) = -1 f p

(E)da,

nmm

where a is a nonlinear parameters in the wave function (to be discussed later).

Equation (3) becomes

P

JE) = J:H

^)-

)-

By using the equation

/ " * ( / - f ( x ) ) d x = ( 6 )

eq. (2) can be evaluated as

For an isolated resonance pQ{E) can be derived from eq. (1) (Bowman [25]), with

''°'

> = *

-4- n2 / ffil

(E

-Ef

iy

The p°(E) can be calculated from the stabilization graph using eq. (7), and the resonance parameters (Er and F) can be obtained by fitting p°(E) to eq. (8).

3. The Hamiltonian and model potential

We use a model potential to represent the interaction between the inner core electrons with the outside valence electrons. Recently, Bachau et al [26] have proposed a generalization of the Feshbach method to study the (Is^ln'/') doubly excited states of beryllium-like systems. The effect of 1s2 core is represented by a model potential l/^with

Vm = f - f ( l + / * r ) ea' \ (9)

where fi is the effective charge seen by the valance electrons described by the 1s orbital. In this work, the Z dependence of ft is given as p = 0.6845Z - 0.3944 [26].

The parameter was determined by fitting the exact energy of the lowest 1 &2s 2S state of the corresponding three-electron ion. The use of the model potential would lead to an estimated [27] error of about 0.0004 Ryd (more bound) for the 2s2S state. For the N = 3 states, such errors are estimated to be 0.0032, 0.0053 and 0.0004 Ryd, for the 3s 2S, 3p 2P and 3d 2D states, respectively. For two-electron systems, the estimated differences in the energy are approximately doubled.

The Hamiltonian for the four-electron atomic system (in Rydberg unit) is given by

Computation of doubly-excited 1$²3I3I' ¹P°and ¹F° states in beryllium-like ions 391

HM = H°(1) + H°(2) + Vm0) + VJ2) + f - , ( 1 0)

where W°(1) and /7°(2) are the hydrogenic Hamiltonians of charge Z a n d ~ is the

'12

interaction potential between two optical electrons out side the 1s2 core Here in eq (10)

H ° - - V2- — , (11)

and Vm{\) and Vm(2) are the model potentials for two outer electrons described above 4. The wave functions

We use products of Slater orbitals to represent the two-electron functions The product of Slater orbitals are the following

* =

E E

Wh, (riK(r

)Vjfft 2)SK <r

), (12)

where

,?a (r) - rn a' e x p Ka /) (13)

In eq (12), A is the antisymmetnzing operator, S is a two particle spin eigen function, // are individual Slater orbitals and the Y is the eigen-function of the total angular momentum L,

^ 0 . 2 ) - = ^ 5 : c ( /a, /6, L , mvm/^ M ) // a m / f l( 1 ) V , ^ ( 2 ) , (14) where C is the Clebsch-Gordan coefficients

5. Calculations and results

For 1F° and 1P° states, we use respectively the expansion length of N = 851 and N = 561 to develop the stabilization plots The basis sets are constructed by using the Slater orbitals of 11 s-type, 11 p-type, 10 cf-type, 9 Mype, 8 g-type, 7 />type, 6 /-type, 5 Mype, 4 Mype, 3 m-type, 2 n-type and 1 otype To establish the 'soft wall', we multiply the exponents of the Slater orbitals (eq. (13)) by a scaling factor a. By changing a, the diffuseness of the wave function is changed, thereby changing the range of the potential in which the wave function expands. Figure 1 shows the stabilization plots of energy eigenvalues with the scaling factor a for the 1P° states of the Ne6+ (Z = 10) ion We compute the energy eigenvalues for 181 different box sizes (a) within the range 1.5 to 2.5 (atomic unit) in the interval of 0 005. But for & (Z = 5) and C2+ (Z = 6) we cover the range of a from 1 3 (atomic unit) to 2.4 (atomic unit)

to avoid some noises in the region of the energy of interest. The Figure 1 represents the stabilization plot of Ne⁶⁺ (Z = 10) for 29th to 39th eigen values in the energy range of -16.0 Ryd to -11.0 Ryd. It is evident from the figure that eigenvalues near E = -13.8 Ryd and E = -12.8 Ryd show stabilization character. These two energies

"O

-15

Figure 1. Stabilization plot (E vs. a) for the 1 s23/3/'1P° states for Z = 10.

represent the two resonances for the 1 s23 / 3 // 1P° states. To extract the accurate resonance energies (E^r) and widths ( r ) from this stabilization diagram for Ne⁶⁺ (Z = 10) we need to investigate the 35th, 36th and 37th set of energy eigenvalues. The density of resonance states can be evaluated with the help of the formula

f>n(E) £ n K + l ) - £nK - l )

r/+1 a ^{/ 1} £„<<>,)-£, ⁽¹⁵⁾

where the index / is the Mh varied a value, i.e. ar. Also in eq. (15), oM and a/+1 are (/+1)-th and (/+1)-th varied a values next to a„ respectively. After calculating the density of resonance states with the above formula (eq. (15)) we fit to the following Lorentzian form that gives the accurate resonance energy Er and the total width f, with

p„(E)

r

(*-*,) + r

Computation of doubly-excited 1s²3l3l' ¹P° and ¹F° states in beryllium-like ions 393

where

y0 = the baseline offset,

A = the total area under the curve from the baseline, E^r = the center of the peak of the curve,

r = the full width of the peak of the curve at half height.

We follow the similar procedure to extract the resonance energy Er and the r tor the resonance states 1 s²3 / 3 /^{/ 1}P ° and 1 s²3 / 3 /^{/ 1}F ° for different Z values In this article we have evaluated two resonances for the 1s23/3/1P° states and one resonance tor the 1s23/3// 1F° state for each Z values except for Z = 5, in which one clean resonance in the stabilization diagram for the 1s23/3// 1P° state was observed.

In Figures 2 and 3 we show the nature of pn(E) with varying eigenenergy E for the ¹P° states of the Ne⁶⁺ (Z = 10) ion The solid circles are the results of actual calculations of the density of resonance states pn(E) using the eq. (15). The solid line in each figure is the fitted Lorentzian form of the corresponding pn(E) From the fitting of Figure 2 we obtain Er= -13.6468 Ryd and r = 0.01784 Ryd for the 1P°(1) state.

Similarly from Figure 3 we obtain Er = -12.7863 Ryd and r = 0.00989 Ryd for the P°(2) state These are the accurate resonance energy E^r and the width F for 1 s23 / 3 // 1P ° states of Ne6* (Z = 10). Like Figure 1, Figure 4 shows the stabilization diagram of 1 s23 / 3 // 1F ° states for the C24 (Z = 6) ion. A close look to the stabilization diagram helps us to understand that, with many other narrow resonances, a clean as well as huge resonance has occurred near the energy region E = -3.0

(/)

sate

0}

resonanc

*o

Density

J O I

3CH

i

2 5 H 20-J

1 5H

J

1 ( H 0 5 -|

OOH

-14 2 -14 0 -13 8 -13 6 -13 4 -13 2 E(Ryd)

figure 2. The fitting of the density of resonance states (solid circles) to the Lorentzian form of eq. (16) for the

z= 10, ^s?3l^lnP0 (1) state. The full curve is the fit.

c/>

</)

Q>

O

c

«J c o o

CO

c O

30 -

25 -

20 -

15 -

10 -

5 -

0 -

, 1 1 1 1 T " t '" • i

- 1 3 2 - 1 3 0 - 1 2 8 E(Ryd )

- 1 2 6 - 1 2 4

Figure 3. The fitting of the density of resonance states (solid circles) to the Lorentzian form of eq (16) for the Z = 10, 1 s23 / 3 / '1P ° (2) state The full curve is the fit

- 2 8

^ - 3 0 - 1

- 3 2 H

Figure 4. Stabilization plot (E vs a) for the 1s238/'1F° states for Z = 6,

(Ryd). By calculating the density of resonance states with the help of the eq. (15), and by fitting it to the Lorentzian form (16), one can easily extract the resonance energy Er and the resonance width r. In the same manner we also extract the Er and r for

Computation of doubly-excited 1s23l3l' 1P° and ' P states in beryllium-like ions 395 the resonance of lsz3l3lnF° states for other different Z values. Figure 5 represents one such resonance of the 1s23/3/"F° state for the C2 +(Z = 6) ion. The obtained values of E, and r are -2.96723 Ryd and 0.01138 Ryd, respectively.

0)

0)

S

CO

8?

c

3

24-

20-

16-

12-

8 -

4 -

o -

|

iL. i

- 3 1 -3 0 - 2 9

E(Ryd)

-2 8

Figure 5. The fitting of the density of resonance states ( solid circles) to the Lorentzian form of eq (16) for the Z= 6,1 s23 / 3 / " F ° state The full curve is the fit

6. Results and discussions

To represent the characteristics of the resonance states for the whole isoelectronic sequence, we draw their variation with the effective nuclear charge (Z - 2). We express the energy of the doubly-excited state as the sum of the energy for the inner electron (with N = 3) and the outer valence electron moving in a Coulomb field of charge (Z - 3), with an effective orbital quantum number N'. The doubly excited energy E can be expressed as

E = - ( Z - 2 )

( Z - 3 )

N2

(N'T

We further define E* = -(A/*)"2, and obtain

£ +

E* =

(Z-3f

(18) Figure 6 displays the plot of E* vs. ( Z - 2)"1 for the 1s23/3/,1F° states. In Figure 6 the obtained results have been compared with those of Bachau et al [28]. For all Z

- 0 120

-0125H

-0 130H

-0 135H

- 0 140

- 0 145 A

Z=10 2=9 z = 8 z = 7

Z=5

0 10 0 15 0 20 0 25 1/(Z-2)

0 30 0 35

Figure 6 Plot of E* vs 1 /(Z~ 2) and comparison with the results of Bachau et al [28] for the 1 S23I3I'1 Fl state with Z = 5 to 10

values our results is somewhat lower than those of Bachau et al [28] But in high Z values the results show better agreement than those for the low Z values The first column of the Table 1 represents the results of resonance energy Er and resonance width r of the 1 s23 / 3 // 1F ° states for Z = 5 to 10, and the second column shows the comparison of the obtained results with those of Bachau et al [28] For all Z values our results are somewhat higher than those of Bachau et al [28] Basically Bachau et al [28] employed procedure to calculate the eigenvalues of QHQ, in the language of Feshbach formalism However they have not calculated the Feshbach shifts, the interaction between the open channel components of the wave function Pi and the close channel components of the wave function Q& On the other hand we employ the procedure to calculate the resonance directly, without dividing the wave functions into different open and close channel components It should be mentioned that Bachau et al [28] have also used a model potential to represent the core and the outer valence electrons The third column of the table represents the resonance width r, converted from the values of nonradiative decay rates of Vacek and Hansen [27]

obtained by configuration interaction calculations. These values are always higher than our present calculations, and those of Bachau et al [28] Nakamura et al [30] have also calculated the widths for some of the doubly excited states of the 04 + ion The value of r obtained from their Auger width is higher than the results of Bachau et al [28] and of the present calculations Figure 7 presents the plot of the resonance width r with (Z - 2)~¹ for the results obtained by the present calculations together with those obtained by Bachau et al [28], and by Lin [29], for the 1s23/3// 1F° states. The curve obtained from the present calculation lies a bit higher than that of Bachau et al [28]

Computation of doubly-excited 1s23l3l' 1P° and 1F° states in beryllium-like ions

Table 1 . Comparison of resonance energy Ert and the width / of the 1 s23 / 3 / '1 F° states for Z = 5 to 10 with the results of Bachau et al [28] and of others The quoted values are in Ryd

397

10

Present (Ryd) Bachau et al [28]

E, = -1 57964 T= 0 00859 Er = - 2 96723

T = 0 0 1 1 3 8

Er= - 4 8008 T = 0 01584 Ef = - 7 07861

T= 0 01933

E, = - 9 80185 / = 0 02216 Er = -12 9684

T= 0 02368

Er = -1 5746 r= 0 00610 Er = - 2 9612

7 = 0 01041

Er = - 4 7934 7^=0 01400 E^f = - 7 0712

7 = 0 0 1 6 8 7

E^r = - 9 7946 7 = 0 01921 Er = -12 9638

/ = 0 02112 (a) N Vacek and J E Hansen [27]

(b) C D Lin [29]

(c) N Nakamura et al [30]

Others

T = 0 0207*

7 = 0 0283*

T = 0 0237a

7 = 0 0260^a r = 0 0259b

7 = 0 0326^c

7 = 0 0292*

cc 0 028 0 024 0 020 -I 0 016 -|

0 012 A

0 008 H

0004 4

0 10

C D Lin

Vacek and Hansen

Z=10

Figure 7. Plot of T vs 1 / ( Z - 2) and companson with the results of Bachau et al [28] and of others for the 1 s*ZI3ifiF° states with Z = 5 to 10

We got the results only for two values of Z (Z = 8 and Z = 6) from the reference of Lin [29] Quoted results for these Z values lie much higher than our present results and those of Bachau et al [28] The nature of the results of Vacek and Hansen [zr are also presented in the figure

Table 2 shows the comparison of the resonance energy E^r and the resonance width r o f the 1 s23 / 3 // 1P ° (1) and of 1s23/3// 1P° (2) states with those obtained by

Table 2. Comparison of resonance energy Er and the width / of 1 s23/3/'1 P° states for Z= 5 to 10 with the results of Bachau et al [28] and of others The quoted values are in Ryd

z

5 E, = - 1 76924 / = 0 01206 6 Er = - 3 2525 r= 0 01053

7 £r = - 5 1848 / = 0 01185 8 E, = - 7 5629 / = 0 01383

9 Er = - 1 0 3840 1 = 0 01509 10 E, = - 1 3 6468

T = 0 01784

Bachau et al [28]

£,= - 1 7640 / = 0 01751 E,= - 3 2508 T = 0 01974

E, = - 5 1846 r = 002117 Ef= - 7 5646 r = 002210

Er = - 1 0 3910 7 = 0 0229 Er = - 1 3 6636

r = 0 02348 (a) N Vacek and J E Hansen [27]

(b) C D Lin [29]

(c) N nakamura et al [30]

Others

/ = 0 0277*

r = 0 0293b

/ = 0 0281*

T = 0 0285"

/" = 0 0289"

T = 0 0245c

I = 0 0291*

]P°(2)

Er= - 2 8779 / = 0 00722

Er = - 4 6877 / = 0 00876 Er = - 6 9424 T = 0 00892

Ef= - 9 6427 T = 0 01031 Ef = - 1 2 7863

T = 0 00989

Bachau et al [28]

Ef= - 2 8708 / = 0 00679

Er = - 4 6782 / = 0 00798

£ , = - 6 9322 r= 000888

Er = - 9 6324 T = 0 00958

£ , = - 1 2 7786 T = 0 0 1 0 1 4

Others

/ = 0 0082"

1 = 0 0093*

J = 0 0102*

/ = 00119c

r=oon3

Bachau et al [28] for Z = 5 to 10. From the table it can be observed that the resonance energies Er of both the states 1s23/3// 1P° (1) and of 1s23/3// 1P° (2) for Z = 6 to 10 agree quite well with those obtained by Bachau et al [28]. The resonance widths r obtained from our calculation for Z = 6 to 10 for the states 1 s23 / 3 // 1P ° 0 ) are a bit smaller, whereas, the values for the state 1 s²3 / 3 /^{/ 1}P ° (2) are slightly higher (except for Z = 9) than the results obtained by Bachau et al [28]. For Z = 5, our calculated resonsnce width r does not maintain the trend with the results obtained for Z = 6 to 10. Like the results of the Table 1 and the values of Vacek and Hansen [27] always lie higher than those of the present calculations and of Bachau et al [28]

Computation of doubly-excited 1s2 3131' 1P° and 1F° states in beryllium-like ions 399

Figure 8 and Figure 9 show the plots of r vs. (Z - 2)~1 for the 1s23/3// 1P° (1) and

0f i s23 / 3 // 1P ° (2) states, respectively. In both the figures results have been compared with those of Bachau et al [28] and of Vacek and Hansen [27].

0.040 0 035 0.030 A 0 025 A

£ 0 020 -I 0 0 1 5 0 0 1 0 -I 0 005 0 000

Vacek and Hansen

H Bachau et al

Present calculation 2=10

Z=9 Z=8 Z=7

_L

Z=6 Z=5

•' t • • i » i i

0 12 0 16 0 20 0 24 0 28 0 30 V ( Z - 2 )

Figure 8. Plot of F vs 1 / ( Z - 2) and comparison with the results of Bachau et al [28] and of others for the 1 s?3/3/'1P° (1) states with Z = 5 to 10

0 0 1 2

0 0 1 0 i

0 008

CC 0 006

0 004 1

0 002

0 000

Vacek and Hansen

H Bachau et al

Present cai julation

Z =i n z=9 z=8

I i

Z=7 Z=6

T 1 1 1 p T . r-

0 12 0 14 0 16 0 18 0 20 0 22 0.24 0 26

V(Z-2)

figure 9. Plot of F vs. 1 / ( Z - 2) and comparison with the results of Bachau et al [28] and [27] for the 1 s23131'1P°

(2) states with Z = 6 to 10.

The use of the stabilization method in the calculations of resonances is based on the simple observation that localized states of an atomic system, in contrast to the unbound scattering states, will not be affected if the system is put into a box of sufficiently large but otherwise arbitrary radius. We have seen that we can obtain resonance parameters (energy and width) if we look closely for the horizontal avoided crossing patterns in the energy eigenvalues as a function of the box radius ft, and then calculate the density of resonance states which depends only on the parameter R We have studied here the calculations of the resonance energy Er and the width r for doubly-excited 1s23/3//1P° and 1s23/3//1F° states of beryllium-like ions for Z = 5 to 10, with the use of model potential of eq. (9), for a nonrelativistic two- electron system in the field of 1s2 core. This approach which employs only L2-type wave functions, together with the use of a suitable model potential, is now believed to have obtained accurate results for both resonance energy and width for doubly excited states in two-electron atomic systems outside a fixed core.

Acknowledgment

Financial support by the National Science Council of Taiwan, ROC, is gratefully acknowledged

References

[1] R Mack PhD Thesis Utrecht University (unpublished)

[2] M Mack J H Nijland, P van der Straten, A Niehaus and R Morgenstern Phys Rev A39 3846 (1989) [3] A Bordenave-Montesquieu, P Benoit-Cattin, P Boudjema, A Gleizes and H Bachau J Phys B20 L695

(1987)

[4] R Mann Phys Rev A35 4988 (1987) [5] Y K Ho J Phys B12 387 (1979) [6] D H Oza J Phys B20 L13 (1987) [7] C D Lin Phys Rev A39 4355 (1989)

[8] V A Mandelshtam, T R Ravurr and H S Taylor Phys Rev Lett 70 1932 (1993) [9] Y K Ho Phys Rep 99 1 (1983)

[10] T N Chang and L 2hu Phys Rev A51 374 (1995)

[11] T K Fang, B I Nam, Y S Kim and T N Chang Phys Rev A55 433 (1997) [12] A U Hazi and H S Taylor Phys Rev A1 1109 (1970)

[13] A U Hazi J Phys B11 L259 (1978)

[14] C A Nicolaides, N A Piangos and Y Komnmos Phys Rev A48 3578 (1993) [15] A Riera J Phys Chem 97 1558 (1993)

[16] J Muller, X Yang and J Burgdbrfer Phys Rev A49 2470 (1994) [17] F Martin and M F Pohtis Surf Sci 356 247 (1996)

[18] H Bachau and F Martin J Phys B29 1451 (1996) [19] S S Tan and Y K Ho Chin J Phys 35 701 (1997) [20] W J Pong and Y K Ho J Phys B37 2177 (1998)

Computation of doubly-excited 1s

3l3l'

P° andEstates in beryllium-like ions 401

[21 ] W J Pong anf Y K Ho Physica Scripta 61 690 (2000) [22] T K Fang and Y K Ho J. Phys. B32 3863 (1999) [23) Utpal Roy and Y K Ho J. Phys B35 1875 (2002)

[24] Utpal Roy and Y K Ho Nuclear Inst. Methods B221 36 (2004) [25) J M Bowman J. Phys. Chem 90 3492 (1986)

[26] H Bachau, P Galan and F Martin Phys. Rev A41 3534 (1990) [27] N Vaeck and J E Hansen J Phys B22 3137 (1989)

[28] H Bachau, P Galan, F Martin, A Riera and M Yanez At Data Nuc! Data Tables 44 305 (1990) [29] C D Lin Review of Fundamenta Processes and Applications of Atoms and Ions (ed) C D Lin (Singapore

World Scientific) p390 (1993)

[30] N Nakamura, T Nabeshima, F J Currell, Y Kanai, S Kitazawa, M Koide, H A Sakaue, H Ida, Y Matsui, K Wakiya, T Takayanagi, T Kambara, Y Awaya, H Suzuki, S Ohtani and UI Safronova J Phys B27 L785 (1994)