- Email: [email protected]
Angular distribution of ejected electrons from Ba 6pns (J=1) autoionizing states
Journal of Electron Spectroscopy and Related Phenomena 128 (2003) 135–140 www.elsevier.com / locate / elspec
Angular distribution of ejected electrons from Ba 6pns (J51) autoionizing states Y. Zhang*, C.J. Dai, S.B. Li, J. Lu State Key Laboratory of Modern Optical Instrumentation and Physics Department, Zhejiang University, Hangzhou 310027, China
Abstract The asymmetry parameters of Ba 6p j ns ( j51 / 2, 3 / 2 J51) autoionizing states, which characterizes the angular distribution with the form (s / 4p )[1 1 b P2 (cos u )], have been investigated, using multichannel quantum defect theory, combined with R-matrix method. Furthermore, energy dependence of asymmetry parameters of electrons ejected from Ba 6p j ns autoionizing states to the 6s 1 / 2 , 5d 3 / 2 , 5d 5 / 2 , and 6p 1 / 2 ion states has been studied. The results indicate the 6p 1 / 2 ns and 6p 3 / 2 ns series have different characteristics in their angular distributions of ejected electrons. Our theoretical results not only agree well with the previous experimental ones, but also predict some new phenomena. 2002 Elsevier Science B.V. All rights reserved. Keywords: Angular distribution; Autoionizing series; Multichannel quantum defect theory
1. Introduction Angular distributions of ejected electrons from autoioninzing states of alkaline–earth atoms have become a powerful tool for studying atomic structure and the various interactions between atomic configurations. For the past few years, autoionizing Rydberg series of the alkaline–earth atoms have been extensively studied, most of which have been focused on the line shapes of transitions and the interactions between different autoionizing series in the excitation spectra [1,2]. Since the ions produced from autoionization process provide no information about the final states of ion core, it is necessary to explore the properties of atoms further by studying the ejected electrons. *Corresponding author. E-mail address: [email protected] (Y. Zhang).
Since angular distributions of ejected electrons from an autoionizing state represent the differential cross sections decaying from an autoionizing state to different ionic states, they are not only related to the magnitudes of transition matrix elements, but also related to their phases. Up to now, numerous experiments of angular distributions have been carried out by populating autoionizing states using the isolatedcore excitation (ICE) scheme [3–5]. Although the angular distributions of Ba 6p j ns autoionizing Rydberg states were studied [6], their theoretical analysis was not accurate enough, because the fitting procedure employed there ignored many channels, and a lot of information could not be obtained. Here we have investigated the angular distributions of ejected electrons from 6pns (J51) autoionizing states in the alkaline–earth Ba atom, using multichannel quantum defect theory (MQDT), combined with R-matrix method [7,8]. Since all the channels
0368-2048 / 02 / $ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S0368-2048( 02 )00266-9
136
Y. Zhang et al. / Journal of Electron Spectroscopy and Related Phenomena 128 (2003) 135–140
are taken into account, the full information can be obtained in this work.
DJcs l J 5 i 2l exp(isl, Jc ) 3
O O k(J s)J , luJ , (sl)jlA c
r
cs
c
kr
k
3 exp(iptr )k r iDiJ0 l,
2. Treatment
(4)
In order to construct a theoretical model for the problem, it is necessary to briefly review the experiment with the ICE scheme, the first two dye lasers excite one electron from the ground state of Ba atom to a bound 6sns 1 S0 Rydberg state just below the ionization potential, then the third dye laser excites the inner electron to have a transition 6sns → 6pns, producing the autoionizing state. When all of the lasers are linearly polarized, the photoionization process of Ba can be expressed as
where sl, Jc is the Coulomb phase-shift, 2 ptr and A k r refer to the phase-shift of the collision eigenchannel and the mixing coefficient of the dissociation channel, respectively. Traditionally, the MQDT parameters were obtained by adjusting them to satisfy experimental data, using a simplified MQDT model. Here we used a set of MQDT parameters obtained from the R-matrix calculation, which successfully explained our earlier experiments on autoionization spectra [2,9]. On the other hand, the angular part of dipole matrix element can be evaluated as
Ba(6sns, J0 50)1g (Jg 51)→Ba(6p1 / 2,3 / 2 ns, J51)
k[(Jc s)Jcs , l]Ju[Jc , (sl)j]Jl 5 (21) Jc 11 / 21l 11
→ Ba 1 (Jc ) 1 e 2 (s 5 1 / 2, l, j)
(1)
Here J0 is the total angular momentum of the 6sns 1 S0 Rydberg states and Jg refers to the incoming photon g ; Jc is the angular momentum quantum number of the final ionic state and (l, j) refer to the outgoing electrons. The angular momentum and parity satisfy the equations
3 [Jcs , j] 1 / 2
5
Jc l
1 ] 2 1
Jcs j
6
With the above information, one is able to derive the 1 asymmetry parameters bJc for the Ba 6s 1 / 2 , 5d 3 / 2, 5d 5 / 2 and 6p1 / 2 ionic states [10,11]. The results are
J0 1 Jg 5 Jc 1 s 1 l 5 Jcs 1 l
2uD011u 2 2uD111u 2 b6s 1 / 2 5 ]]]]] uD011u 2 1uD111u 2
p0 pg 5 pc (21)l .
b5d 3 / 2
The asymmetry parameter b describing the differential cross section satisfies the equation dsJc (u ) sJc ]] 5 ] [1 1 bJc P2 (cos u )], 4p dV
(6)
] uD211u 2 1 4uD231u 2 2 5uD111u 2 2 6Œ6 Re(D211 D * 231 ) 5 ]]]]]]]]]]]]]] 2 2 2 5 f uD211u 1uD231u 1uD111u g (7)
(2)
where sJc is the partial cross section of autoionization to the final ionic state; u is the angle between the third laser polarization and the momentum of ejected electrons; P2 (cos u ) is the second-order Legendre polylnomial; bJc is the asymmetry parameter for the given Jc ionic state, and is closely related to the dipole matrix element DJcs l J 5 k(Jc s)Jcs , l, J 2 iDiJ0 l,
(5)
(3)
which can be derived from MQDT. It can be further deduced as
b5d 5 / 2 ] uD211u 2 1 4uD231u 2 2 5uD331u 2 2 6Œ6 Re(D211 D * 231 ) 5 ]]]]]]]]]]]]]] 5 f uD211u 2 1uD231u 2 1uD331u 2 g (8) ] * ) uD121u 2 2 2Œ2(D101 D 121 ]]]]]]] b6p 1 / 2 5 21D 2 D u 101u u 121u
(9)
For the particular case dealing with Ba 6pns (J5 1) autoionizing series, a thirteen-channel model is used to study the asymmetry parameter. These channels are 6s 1 / 2 np 1 / 2 , 6s 1 / 2 np 3 / 2 , 5d 3 / 2 np 1 / 2 ,
Y. Zhang et al. / Journal of Electron Spectroscopy and Related Phenomena 128 (2003) 135–140
137
5d 3 / 2 np 3 / 2 , 5d 3 / 2 nf 5 / 2 , 5d 5 / 2 np 3 / 2 , 5d 5 / 2 nf 5 / 2 , 5d 5 / 2 nf 7 / 2 , 6p 1 / 2 ns 1 / 2 , 6p 1 / 2 nd 3 / 2 , 6p 3 / 2 ns 1 / 2 , 6p 3 / 2 nd 3 / 2 and 6p 3 / 2 nd 5 / 2 , which are converging to the 6s 1 / 2 , 5d 3 / 2 , 5d 5 / 2, 6p 1 / 2 and 6p 3 / 2 thresholds, and they are labeled as i51–13. We have also taken a further step to study the periodicity by calculating the spectrum density A 2i . In MQDT, the spectral density corresponding to the ith dissociation channel is determined by
OA ,
A 2i 5
r
2 ir
(10)
where r refers to the collision eigenchannel, A i r is the coefficient in the wavefunction of collision eigenchannel and can be expressed as 1 A ir 5 ] Nr
OU a
ia
cos f p (ni 1 ma ) g B (ar ) ,
(11)
where ni is the effective quantum number for a closed channel and is replaced with 2 tr for an open channel; Nr is the normalization constant; ma is the eigenchannel quantum defect relevant to the a th close-coupled channel, and Ui a are the matrix elements of the unitary transformation U between the collision channels and the close-coupled channels; B (ar ) is the coefficient of the wavefunction wa in the a th close-coupled eigenchannel, which can be obtained by solving the MQDT equations [12].
Fig. 1. Comparison between (a) the theoretical asymmetry parameters of Ba 6p 3 / 2 ns autoionizing states above the 6p 1 / 2 ionic limit decaying to the 5d 3 / 2 the ionic states, (b) the spectral density of the 11th 6p 3 / 2 ns 1 / 2 channel, and (c) the corresponding spectra.
3. Results and discussion With the above formulae, we have investigated the angular distribution of ejected electrons from Ba 6p j ns (J51) autoionizing states. For the 6p 3 / 2 ns states above 6p 1 / 2 ionic limit, the values of the asymmetry parameters are found to be independent of the principal quantum numbers, and the angular distributions of ejected electrons are periodic with n. At the central resonance energies, the values of asymmetry parameters are almost identical. An example is shown in Fig. 1 containing the results of the asymmetry parameters b5d 3 / 2 , the spectral density A 211 and the corresponding spectra of Ba 6p 3 / 2 ns autoionizing states. The fact that the asymmetry parameters do not depend on the effective quantum number indicates these autoionizing states are not perturbed by other series in this range. Since the
asymmetry parameters measure the relative amplitudes of the open 6s´ p 3 / 2 and 6s´ p 1 / 2 waves, they are independent of the effective quantum number. Furthermore, it is obvious that the asymmetry parameters, the spectral density and the corresponding spectra have the same period although they have different phases in their variation with n. This is because the main peaks in the 6p 3 / 2 ns series are due to the contribution from the 11th 6p 3 / 2 ns 1 / 2 channel, while the contributions from the 6p 3 / 2 nd 3 / 2 and 6p 3 / 2 nd 5 / 2 channels only lead to the subsidiary peaks on both sides. In order to test our theoretical approach, we have made comparisons with the previous experiments. An example is shown in Fig. 2 containing the result of the asymmetry parameter of Ba 6p 3 / 2 50s autoionizing state comparing with previous experimen-
138
Y. Zhang et al. / Journal of Electron Spectroscopy and Related Phenomena 128 (2003) 135–140
Fig. 2. Comparison between the experimental (a–c) and the theoretical asymmetry parameters (d–f) of Ba 6p 3 / 2 50s autoionzing states decaying to the 6s 1 / 2 , 5d 3 / 2 and 5d 5 / 2 ionic states, respectively. The positions of the 6p 3 / 2 48d, 6p 3 / 2 49d and 6p 3 / 2 50s are indicated, and the theoretical curves have been calculated using MQDT, combined with R-matrix method.
tal result [6]. The asymmetry parameters are periodic with the effective quantum number, and resonances are found close to the 6p 3 / 2 48d and 6p 3 / 2 49d autoionizing states, which are caused by the configuration interactions between 6p 3 / 2 nd and 6p 3 / 2 ns autoionizing Rydberg states. The fact that the present theoretical model agrees well with the experimental results encourages one to take a further step for prediction of those new phenomena. Fig. 3 gives such an example, in which the asymmetry parameters of Ba 6p 3 / 2 ns autoionizing states decaying to the 6p 1 / 2 ion limit have been investigated, which are very difficult to obtain in experiments. This is true because the ejected electrons from the 6p 3 / 2 ns states are very slow due to the narrow energy interval between 6p 1 / 2 and 6p 3 / 2 ion limits. The result in Fig. 3 shows that the asymmetry parameters b6p 1 / 2 are also periodic and vary in a narrow range. We have also studied 6p 3 / 2 ns autoionizing states below the 6p 1 / 2 ionic limit, which can not decay to the 6p 1 / 2 ion limit. It is found that the physical situation in this energy regime is more complicated than that above the 6p 1 / 2 ionic limit because of many bound 6p1 / 2 nl states interacting with the 6p 3 / 2 ns
Fig. 3. Comparison between (a) the theoretical asymmetry parameters of Ba 6p 3 / 2 ns autoionizing states above the 6p 1 / 2 ionic limit decaying to the 6p 1 / 2 the ionic states, (b) the spectral density of the 11th 6p 3 / 2 ns 1 / 2 channel, and (c) the corresponding spectra.
states. An example is shown in Fig. 4 containing the asymmetry parameters b6s 1 / 2 of 6p 3 / 2 11s and 6p 3 / 2 12s autoionizing states comparing with their spectra. It is obvious that the b parameter becomes very irregular with energy because of the complexity of their corresponding spectra. Therefore, the periodicity in b parameter disappears for this energy regime. Now let us turn to the results of the 6p 1 / 2 ns states, whose situations are rather different. An example of such investigations is shown in Fig. 5 containing the asymmetry parameters of Ba atom decaying to 6s 1 / 2 , 5d 3 / 2 and 5d 5 / 2 states, respectively. The periodicity is destroyed by the additional interactions with the 6p 3 / 2 ns states below the 6p 1 / 2 ionization limit. The
Y. Zhang et al. / Journal of Electron Spectroscopy and Related Phenomena 128 (2003) 135–140
139
Fig. 4. Comparison between (a) the theoretical asymmetry parameters of Ba 6p 3 / 2 11s and 6p 3 / 2 12s autoionizing states below the 6p 1 / 2 ionic limit decaying to the 6s 1 / 2 the ionic states and (b) the corresponding spectra.
configuration interactions result in a complicated energy dependence of the asymmetry parameters. A systematic study shows that unlike the case in the 6p 3 / 2 ns states, the asymmetry parameters for different 6p 1 / 2 ns show a great variety of energy dependence. A comparison with experimental results is shown in Fig. 6 for the Ba 6p 1 / 2 24s autoionizing state, from which one may conclude that they are in good agreement within experimental uncertainty. Especially, the minimum of the asymmetry parameter b6s 1 / 2 can be obtained from the theory, while it was not measured in the experiment.
4. Conclusions Using MQDT, the asymmetry parameters of 6pns (J51) autoionizing states in the alkaline–earth Ba atom are systematically studied in our work. For the 6p 3 / 2 ns series above the 6p 1 / 2 ionic limit, the
Fig. 5. The asymmetry parameters of Ba 6p 1 / 2 ns autoionizing states decaying to the (a) 6s 1 / 2 , (b) 5d 3 / 2 and (c) 5d 5 / 2 ionic states, and (d) the corresponding spectra.
asymmetry parameters are periodic with the effective quantum numbers, which are close related to the spectral density, and for the 6p 3 / 2 ns series below the 6p 1 / 2 ionic limit, the asymmetry parameters are irregular with energy. Whereas for the 6p 1 / 2 ns series, the energy dependence of asymmetry parameters is very complicated because of the configuration interactions with the 6p 3 / 2 ns states. For most cases, the asymmetry parameter varies over its full range from 21 to 2, and the theoretical results agree well with the previous experimental results. Furthermore, this work has also investigated the property of the b6p 1 / 2 , which has not been measured so far. A further study of the angular distributions of ejected electrons from Ba 6pnd (J51,3) autoionizing states is in progress.
140
Y. Zhang et al. / Journal of Electron Spectroscopy and Related Phenomena 128 (2003) 135–140
National High-tech ICF Committee of China, and by the Natural Science Foundation of Zhejiang Province, China (Grant No.101055).
References
Fig. 6. Comparison between the experimental (a–c) and the theoretical asymmetry parameters (d–f) of Ba 6p 1 24s autoionzing states decaying to the 6s 1 / 2 , 5d 3 / 2 and 5d 5 / 2 ionic states, respectively. The calculated curves are obtained using the same MQDT analyses as the theoretical investigation shown in Fig. 2.
Acknowledgements This work is supported by the National Science Foundation of China (Grant No.10174065), by the
[1] J. Lu, C.J. Dai, C.Q. Li et al., Chin. Phys. Lett. 16 (1999) 560. [2] S.B. Li, C.J. Dai, W. Sun, P. Xue, J. Phys. B 34 (2001) 2124. [3] M.D. Lindsay, L.T. Cai, C.J. Dai, Phys. Rev. A 45 (1992) 231. [4] M.D. Lindsay, C.J. Dai, L.T. Cai, Phys. Rev. A46 (1992) 3789. [5] H. Lagadec, B. Carre, D. Porterat, J. Phys. B: At. Mol. Opt. Phys. 29 (1995) 471. [6] H. Hieronymus, M. Kohl, J. Neukammer, Phys. Rev. A 41 (1990) 1477. [7] M. Aymar, J. Phys. B: At. Mol. Opt. Phys. 23 (1990) 2697. [8] V. Lange, M. Aymar, U. Eichmann, J. Phys. B: At. Mol. Opt. Phys. 24 (1990) 91. [9] J. Lu, C.J. Dai, C.Q. Li et al., Chin. Phys. Lett. 61 (1999) 339. [10] C.J. Dai, Phys. Rev. A 53 (1996) 3237. [11] C.J. Dai, Opt. Commun. 132 (1996) 527. [12] C.M. Lee, K.T. Lu, Phys. Rev. A 8 (1973) 1241.
Angular distribution of ejected electrons from Ba 6pns (J51) autoionizing states Y. Zhang*, C.J. Dai, S.B. Li, J. Lu State Key Laboratory of Modern Optical Instrumentation and Physics Department, Zhejiang University, Hangzhou 310027, China
Abstract The asymmetry parameters of Ba 6p j ns ( j51 / 2, 3 / 2 J51) autoionizing states, which characterizes the angular distribution with the form (s / 4p )[1 1 b P2 (cos u )], have been investigated, using multichannel quantum defect theory, combined with R-matrix method. Furthermore, energy dependence of asymmetry parameters of electrons ejected from Ba 6p j ns autoionizing states to the 6s 1 / 2 , 5d 3 / 2 , 5d 5 / 2 , and 6p 1 / 2 ion states has been studied. The results indicate the 6p 1 / 2 ns and 6p 3 / 2 ns series have different characteristics in their angular distributions of ejected electrons. Our theoretical results not only agree well with the previous experimental ones, but also predict some new phenomena. 2002 Elsevier Science B.V. All rights reserved. Keywords: Angular distribution; Autoionizing series; Multichannel quantum defect theory
1. Introduction Angular distributions of ejected electrons from autoioninzing states of alkaline–earth atoms have become a powerful tool for studying atomic structure and the various interactions between atomic configurations. For the past few years, autoionizing Rydberg series of the alkaline–earth atoms have been extensively studied, most of which have been focused on the line shapes of transitions and the interactions between different autoionizing series in the excitation spectra [1,2]. Since the ions produced from autoionization process provide no information about the final states of ion core, it is necessary to explore the properties of atoms further by studying the ejected electrons. *Corresponding author. E-mail address: [email protected] (Y. Zhang).
Since angular distributions of ejected electrons from an autoionizing state represent the differential cross sections decaying from an autoionizing state to different ionic states, they are not only related to the magnitudes of transition matrix elements, but also related to their phases. Up to now, numerous experiments of angular distributions have been carried out by populating autoionizing states using the isolatedcore excitation (ICE) scheme [3–5]. Although the angular distributions of Ba 6p j ns autoionizing Rydberg states were studied [6], their theoretical analysis was not accurate enough, because the fitting procedure employed there ignored many channels, and a lot of information could not be obtained. Here we have investigated the angular distributions of ejected electrons from 6pns (J51) autoionizing states in the alkaline–earth Ba atom, using multichannel quantum defect theory (MQDT), combined with R-matrix method [7,8]. Since all the channels
0368-2048 / 02 / $ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S0368-2048( 02 )00266-9
136
Y. Zhang et al. / Journal of Electron Spectroscopy and Related Phenomena 128 (2003) 135–140
are taken into account, the full information can be obtained in this work.
DJcs l J 5 i 2l exp(isl, Jc ) 3
O O k(J s)J , luJ , (sl)jlA c
r
cs
c
kr
k
3 exp(iptr )k r iDiJ0 l,
2. Treatment
(4)
In order to construct a theoretical model for the problem, it is necessary to briefly review the experiment with the ICE scheme, the first two dye lasers excite one electron from the ground state of Ba atom to a bound 6sns 1 S0 Rydberg state just below the ionization potential, then the third dye laser excites the inner electron to have a transition 6sns → 6pns, producing the autoionizing state. When all of the lasers are linearly polarized, the photoionization process of Ba can be expressed as
where sl, Jc is the Coulomb phase-shift, 2 ptr and A k r refer to the phase-shift of the collision eigenchannel and the mixing coefficient of the dissociation channel, respectively. Traditionally, the MQDT parameters were obtained by adjusting them to satisfy experimental data, using a simplified MQDT model. Here we used a set of MQDT parameters obtained from the R-matrix calculation, which successfully explained our earlier experiments on autoionization spectra [2,9]. On the other hand, the angular part of dipole matrix element can be evaluated as
Ba(6sns, J0 50)1g (Jg 51)→Ba(6p1 / 2,3 / 2 ns, J51)
k[(Jc s)Jcs , l]Ju[Jc , (sl)j]Jl 5 (21) Jc 11 / 21l 11
→ Ba 1 (Jc ) 1 e 2 (s 5 1 / 2, l, j)
(1)
Here J0 is the total angular momentum of the 6sns 1 S0 Rydberg states and Jg refers to the incoming photon g ; Jc is the angular momentum quantum number of the final ionic state and (l, j) refer to the outgoing electrons. The angular momentum and parity satisfy the equations
3 [Jcs , j] 1 / 2
5
Jc l
1 ] 2 1
Jcs j
6
With the above information, one is able to derive the 1 asymmetry parameters bJc for the Ba 6s 1 / 2 , 5d 3 / 2, 5d 5 / 2 and 6p1 / 2 ionic states [10,11]. The results are
J0 1 Jg 5 Jc 1 s 1 l 5 Jcs 1 l
2uD011u 2 2uD111u 2 b6s 1 / 2 5 ]]]]] uD011u 2 1uD111u 2
p0 pg 5 pc (21)l .
b5d 3 / 2
The asymmetry parameter b describing the differential cross section satisfies the equation dsJc (u ) sJc ]] 5 ] [1 1 bJc P2 (cos u )], 4p dV
(6)
] uD211u 2 1 4uD231u 2 2 5uD111u 2 2 6Œ6 Re(D211 D * 231 ) 5 ]]]]]]]]]]]]]] 2 2 2 5 f uD211u 1uD231u 1uD111u g (7)
(2)
where sJc is the partial cross section of autoionization to the final ionic state; u is the angle between the third laser polarization and the momentum of ejected electrons; P2 (cos u ) is the second-order Legendre polylnomial; bJc is the asymmetry parameter for the given Jc ionic state, and is closely related to the dipole matrix element DJcs l J 5 k(Jc s)Jcs , l, J 2 iDiJ0 l,
(5)
(3)
which can be derived from MQDT. It can be further deduced as
b5d 5 / 2 ] uD211u 2 1 4uD231u 2 2 5uD331u 2 2 6Œ6 Re(D211 D * 231 ) 5 ]]]]]]]]]]]]]] 5 f uD211u 2 1uD231u 2 1uD331u 2 g (8) ] * ) uD121u 2 2 2Œ2(D101 D 121 ]]]]]]] b6p 1 / 2 5 21D 2 D u 101u u 121u
(9)
For the particular case dealing with Ba 6pns (J5 1) autoionizing series, a thirteen-channel model is used to study the asymmetry parameter. These channels are 6s 1 / 2 np 1 / 2 , 6s 1 / 2 np 3 / 2 , 5d 3 / 2 np 1 / 2 ,
Y. Zhang et al. / Journal of Electron Spectroscopy and Related Phenomena 128 (2003) 135–140
137
5d 3 / 2 np 3 / 2 , 5d 3 / 2 nf 5 / 2 , 5d 5 / 2 np 3 / 2 , 5d 5 / 2 nf 5 / 2 , 5d 5 / 2 nf 7 / 2 , 6p 1 / 2 ns 1 / 2 , 6p 1 / 2 nd 3 / 2 , 6p 3 / 2 ns 1 / 2 , 6p 3 / 2 nd 3 / 2 and 6p 3 / 2 nd 5 / 2 , which are converging to the 6s 1 / 2 , 5d 3 / 2 , 5d 5 / 2, 6p 1 / 2 and 6p 3 / 2 thresholds, and they are labeled as i51–13. We have also taken a further step to study the periodicity by calculating the spectrum density A 2i . In MQDT, the spectral density corresponding to the ith dissociation channel is determined by
OA ,
A 2i 5
r
2 ir
(10)
where r refers to the collision eigenchannel, A i r is the coefficient in the wavefunction of collision eigenchannel and can be expressed as 1 A ir 5 ] Nr
OU a
ia
cos f p (ni 1 ma ) g B (ar ) ,
(11)
where ni is the effective quantum number for a closed channel and is replaced with 2 tr for an open channel; Nr is the normalization constant; ma is the eigenchannel quantum defect relevant to the a th close-coupled channel, and Ui a are the matrix elements of the unitary transformation U between the collision channels and the close-coupled channels; B (ar ) is the coefficient of the wavefunction wa in the a th close-coupled eigenchannel, which can be obtained by solving the MQDT equations [12].
Fig. 1. Comparison between (a) the theoretical asymmetry parameters of Ba 6p 3 / 2 ns autoionizing states above the 6p 1 / 2 ionic limit decaying to the 5d 3 / 2 the ionic states, (b) the spectral density of the 11th 6p 3 / 2 ns 1 / 2 channel, and (c) the corresponding spectra.
3. Results and discussion With the above formulae, we have investigated the angular distribution of ejected electrons from Ba 6p j ns (J51) autoionizing states. For the 6p 3 / 2 ns states above 6p 1 / 2 ionic limit, the values of the asymmetry parameters are found to be independent of the principal quantum numbers, and the angular distributions of ejected electrons are periodic with n. At the central resonance energies, the values of asymmetry parameters are almost identical. An example is shown in Fig. 1 containing the results of the asymmetry parameters b5d 3 / 2 , the spectral density A 211 and the corresponding spectra of Ba 6p 3 / 2 ns autoionizing states. The fact that the asymmetry parameters do not depend on the effective quantum number indicates these autoionizing states are not perturbed by other series in this range. Since the
asymmetry parameters measure the relative amplitudes of the open 6s´ p 3 / 2 and 6s´ p 1 / 2 waves, they are independent of the effective quantum number. Furthermore, it is obvious that the asymmetry parameters, the spectral density and the corresponding spectra have the same period although they have different phases in their variation with n. This is because the main peaks in the 6p 3 / 2 ns series are due to the contribution from the 11th 6p 3 / 2 ns 1 / 2 channel, while the contributions from the 6p 3 / 2 nd 3 / 2 and 6p 3 / 2 nd 5 / 2 channels only lead to the subsidiary peaks on both sides. In order to test our theoretical approach, we have made comparisons with the previous experiments. An example is shown in Fig. 2 containing the result of the asymmetry parameter of Ba 6p 3 / 2 50s autoionizing state comparing with previous experimen-
138
Y. Zhang et al. / Journal of Electron Spectroscopy and Related Phenomena 128 (2003) 135–140
Fig. 2. Comparison between the experimental (a–c) and the theoretical asymmetry parameters (d–f) of Ba 6p 3 / 2 50s autoionzing states decaying to the 6s 1 / 2 , 5d 3 / 2 and 5d 5 / 2 ionic states, respectively. The positions of the 6p 3 / 2 48d, 6p 3 / 2 49d and 6p 3 / 2 50s are indicated, and the theoretical curves have been calculated using MQDT, combined with R-matrix method.
tal result [6]. The asymmetry parameters are periodic with the effective quantum number, and resonances are found close to the 6p 3 / 2 48d and 6p 3 / 2 49d autoionizing states, which are caused by the configuration interactions between 6p 3 / 2 nd and 6p 3 / 2 ns autoionizing Rydberg states. The fact that the present theoretical model agrees well with the experimental results encourages one to take a further step for prediction of those new phenomena. Fig. 3 gives such an example, in which the asymmetry parameters of Ba 6p 3 / 2 ns autoionizing states decaying to the 6p 1 / 2 ion limit have been investigated, which are very difficult to obtain in experiments. This is true because the ejected electrons from the 6p 3 / 2 ns states are very slow due to the narrow energy interval between 6p 1 / 2 and 6p 3 / 2 ion limits. The result in Fig. 3 shows that the asymmetry parameters b6p 1 / 2 are also periodic and vary in a narrow range. We have also studied 6p 3 / 2 ns autoionizing states below the 6p 1 / 2 ionic limit, which can not decay to the 6p 1 / 2 ion limit. It is found that the physical situation in this energy regime is more complicated than that above the 6p 1 / 2 ionic limit because of many bound 6p1 / 2 nl states interacting with the 6p 3 / 2 ns
Fig. 3. Comparison between (a) the theoretical asymmetry parameters of Ba 6p 3 / 2 ns autoionizing states above the 6p 1 / 2 ionic limit decaying to the 6p 1 / 2 the ionic states, (b) the spectral density of the 11th 6p 3 / 2 ns 1 / 2 channel, and (c) the corresponding spectra.
states. An example is shown in Fig. 4 containing the asymmetry parameters b6s 1 / 2 of 6p 3 / 2 11s and 6p 3 / 2 12s autoionizing states comparing with their spectra. It is obvious that the b parameter becomes very irregular with energy because of the complexity of their corresponding spectra. Therefore, the periodicity in b parameter disappears for this energy regime. Now let us turn to the results of the 6p 1 / 2 ns states, whose situations are rather different. An example of such investigations is shown in Fig. 5 containing the asymmetry parameters of Ba atom decaying to 6s 1 / 2 , 5d 3 / 2 and 5d 5 / 2 states, respectively. The periodicity is destroyed by the additional interactions with the 6p 3 / 2 ns states below the 6p 1 / 2 ionization limit. The
Y. Zhang et al. / Journal of Electron Spectroscopy and Related Phenomena 128 (2003) 135–140
139
Fig. 4. Comparison between (a) the theoretical asymmetry parameters of Ba 6p 3 / 2 11s and 6p 3 / 2 12s autoionizing states below the 6p 1 / 2 ionic limit decaying to the 6s 1 / 2 the ionic states and (b) the corresponding spectra.
configuration interactions result in a complicated energy dependence of the asymmetry parameters. A systematic study shows that unlike the case in the 6p 3 / 2 ns states, the asymmetry parameters for different 6p 1 / 2 ns show a great variety of energy dependence. A comparison with experimental results is shown in Fig. 6 for the Ba 6p 1 / 2 24s autoionizing state, from which one may conclude that they are in good agreement within experimental uncertainty. Especially, the minimum of the asymmetry parameter b6s 1 / 2 can be obtained from the theory, while it was not measured in the experiment.
4. Conclusions Using MQDT, the asymmetry parameters of 6pns (J51) autoionizing states in the alkaline–earth Ba atom are systematically studied in our work. For the 6p 3 / 2 ns series above the 6p 1 / 2 ionic limit, the
Fig. 5. The asymmetry parameters of Ba 6p 1 / 2 ns autoionizing states decaying to the (a) 6s 1 / 2 , (b) 5d 3 / 2 and (c) 5d 5 / 2 ionic states, and (d) the corresponding spectra.
asymmetry parameters are periodic with the effective quantum numbers, which are close related to the spectral density, and for the 6p 3 / 2 ns series below the 6p 1 / 2 ionic limit, the asymmetry parameters are irregular with energy. Whereas for the 6p 1 / 2 ns series, the energy dependence of asymmetry parameters is very complicated because of the configuration interactions with the 6p 3 / 2 ns states. For most cases, the asymmetry parameter varies over its full range from 21 to 2, and the theoretical results agree well with the previous experimental results. Furthermore, this work has also investigated the property of the b6p 1 / 2 , which has not been measured so far. A further study of the angular distributions of ejected electrons from Ba 6pnd (J51,3) autoionizing states is in progress.
140
Y. Zhang et al. / Journal of Electron Spectroscopy and Related Phenomena 128 (2003) 135–140
National High-tech ICF Committee of China, and by the Natural Science Foundation of Zhejiang Province, China (Grant No.101055).
References
Fig. 6. Comparison between the experimental (a–c) and the theoretical asymmetry parameters (d–f) of Ba 6p 1 24s autoionzing states decaying to the 6s 1 / 2 , 5d 3 / 2 and 5d 5 / 2 ionic states, respectively. The calculated curves are obtained using the same MQDT analyses as the theoretical investigation shown in Fig. 2.
Acknowledgements This work is supported by the National Science Foundation of China (Grant No.10174065), by the
[1] J. Lu, C.J. Dai, C.Q. Li et al., Chin. Phys. Lett. 16 (1999) 560. [2] S.B. Li, C.J. Dai, W. Sun, P. Xue, J. Phys. B 34 (2001) 2124. [3] M.D. Lindsay, L.T. Cai, C.J. Dai, Phys. Rev. A 45 (1992) 231. [4] M.D. Lindsay, C.J. Dai, L.T. Cai, Phys. Rev. A46 (1992) 3789. [5] H. Lagadec, B. Carre, D. Porterat, J. Phys. B: At. Mol. Opt. Phys. 29 (1995) 471. [6] H. Hieronymus, M. Kohl, J. Neukammer, Phys. Rev. A 41 (1990) 1477. [7] M. Aymar, J. Phys. B: At. Mol. Opt. Phys. 23 (1990) 2697. [8] V. Lange, M. Aymar, U. Eichmann, J. Phys. B: At. Mol. Opt. Phys. 24 (1990) 91. [9] J. Lu, C.J. Dai, C.Q. Li et al., Chin. Phys. Lett. 61 (1999) 339. [10] C.J. Dai, Phys. Rev. A 53 (1996) 3237. [11] C.J. Dai, Opt. Commun. 132 (1996) 527. [12] C.M. Lee, K.T. Lu, Phys. Rev. A 8 (1973) 1241.