Dynamical Correlation in Double Excitations of Helium Studied by ...

VOLUME 91, N UMBER 19

PHYSICA L R EVIEW LET T ERS

week ending 7 NOVEMBER 2003

Dynamical Correlation in Double Excitations of Helium Studied by High-Resolution and Angular-Resolved Fast-Electron Energy-Loss Spectroscopy in Absolute Measurements Xiao-jing Liu,1,* Lin-fan Zhu,1 Zhen-sheng Yuan,1 Wen-bin Li,1 Hua-dong Cheng,1 Yu-ping Huang,1 Zhi-ping Zhong,2 Ke-zun Xu,1 and Jia-Ming Li3 1

Laboratory of Bond-Selective Chemistry, Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui, 230027, China 2 Department of Physics, The Graduate School of the Chinese Academy of Science, P.O. Box 3908, Beijing 100039, China 3 Department of Physics, Center of Atomic and Molecular Nanosciences, Tsinghua University, Beijing 100084, China (Received 6 February 2003; published 6 November 2003) The momentum transfer dependence of fundamental double excitation processes of helium is studied by absolute measurements using an angular resolved fast-electron energy loss spectrometer with high energy resolution. It elucidates the dynamical correlations, in terms of internal correlation quantum numbers, K, T , and A. The Fano profile parameters q, fa , 2 , f, and S of doubly excited states 1 e 1 e 1 o 2 2 1; 02 S , 2 1; 02 D , and 2 0; 12 P are reported as functions of momentum transfer K . 1 e 1 e Qualitative analysis is given for the states of 2 1; 02 S and 2 1; 02 S . DOI: 10.1103/PhysRevLett.91.193203

As the simplest paradigm for studying electron correlations in atomic system, the doubly excited states of helium have attracted a lot of interest since the pioneering works of Madden and Codling [1] and Cooper et al. [2]. A detailed review was recently performed by Tanner et al. [3]. The two electrons in the doubly excited states of helium are strongly correlated and cannot be simply assigned as s, p, d, f, . . . electrons [4], and a new set of internal correlation quantum numbers, K, T , and A, have been introduced to describe the correlation of the two excited electrons [5–8]. So a doubly excited state 2S1 L with one inner electron (principal quantum number N) and one outer electron (principal quantum number n) can be represented as n K; T AN 2S1 L . The relevance of the new quantum numbers for the description of the photoexcitation or the subsequent decay processes (autoionization and radiative decay) was studied extensively by experiment [9–11] and theoretical calculations [12 –15]. As an important excitation method supplementary to photoabsorption, the angular resolved electron energy loss spectroscopy (AREELS) with fast-electron impact, although performed with a lower energy resolution, provides a more comprehensive description of the excitations, in particular, the momentum transfer dependence, and offers the possibility to study nondipole excitations [16]. However, because of the low cross sections and difficulties in achieving the required energy resolution, the experimental investigations of the double excitations of helium by this method are very limited [17,18]. There are also few theoretical investigations of this aspect. Among the experimental measurements, the energy resolution was not better than 0.5 eV [18], which is much greater than the natural linewidth of any of the investigated resonances. Here we perform absolute measurements of the double excitation processes of helium by 193203-1

0031-9007=03=91(19)=193203(4)$20.00

PACS numbers: 34.80.Dp, 32.70.Fw, 32.80.Dz

AREELS with a high energy resolution of 80 meV at an incident energy of 2.5 keV. Thus, momentum transfer dependence of the excitation cross sections, including 1 e 1 o 1 e the resonances of n 1; 0 2 S , n 0; 12 P , n 1; 02 D 1 e and other weaker resonances of 2 1; 02 S and 1 o 3 1; 02 P , can be obtained in order to elucidate dynamics of such fundamental double excitation processes. The angle-resolved electron energy loss spectrometer has been described in detail elsewhere [19,20]. The electrons are produced by a heated thorium-tungsten filament, monochromized by an electrostatic hemispherical monochromator, and collided with the target gases in the interaction chamber. After the scattered electrons passed an electrostatic hemispherical analyzer, they are energy selected and detected by a microchannel-plate based position sensitive detector. Compared with the previously used channeltron detector, it improves the measurement efficiency by about 20 times, which makes our measurements possible. The background pressure is 5 105 Pa, and the spectra from 56 to 66 eV were measured at the sample pressure of 8 103 Pa from 0 to 6 . The normalization of the spectra was made by measuring the ratio of the intensity at to that at 4 . Then the spectra at non-0 were converted to the generalized oscillator strength densities (GOSDs), and the one at 0 was converted to the optical oscillator strength densities [19,21]. The measured GOSDs are summarized in Fig. 1. It reveals the momentum transfer K 2 dependence of the overall structures from 56 to 66 eV, which consist of a series of Fano resonances. In order to show the high 1 o 1 o resolution spectra near the 2 0; 1 2 P and 3 0; 12 P resonances, after removing the ‘‘smooth ionization continuum’’ [i.e., fc E in Eq. (1)], the GOSDs at 0 , 2 , 4 , and 6 are displayed in Fig. 2. The spectra reveal 1 e 1 e the weak resonances [2;3 1; 0 2 S , 2;3 1; 02 D , 1 Se , and 1; 0 1 Po ], which will be discussed 1; 0 2 3 2 2  2003 The American Physical Society

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the excitation energy E from the resonance energy Eri scaled by the half linewidth i =2, and qi is a line profile index which represents the ratio of transition amplitude of the ‘‘modified’’ discrete state to that of the relevant continuum state. The ratio parameter 2i is defined as fai K; E 2 : (2) i EEr fc K; E For a specific resonance, the integrated GOS fi of the modified embedded discrete state is expressed as [22] fi

FIG. 1. The absolute GOSDs of the double excitations of helium from 0 to 6 .

later. From the K 2 dependence, there are clearly one 1 o strong dipole –allowed series [n 0; 1 2 P ] and two weak 1 e 1 e dipole-forbidden series [n 1; 02 S and n 1; 0 2 D ]. The resonance profiles are represented as [22,23]: X df fai jqi sini cosi j2 1 fc E dE i X qi "i 2 1 fc E; fai 1 "2i i

(1)

where fai represents the relevant continuum involving interfere with the ith resonance, fc E is the total continuum GOSD, which includes both parts that interfere and does not interfere with resonances, i is the phase parameter due to configuration interaction, "i coti E Eri =i =2 stands for the departure of

FIG. 2. The GOSDs of double excitations of helium at 0 , 2 , 4 , and 6 with smooth ionization continuum removed.

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i f q2 j ; 2 ai i EEr

(3)

since fai varies very slowly with E. Note that, for a window-type resonance, fi is 0 because qi 0, in spite of the degree of the interference between the discrete and the continuum. In order to elucidate the relevant strength involving interference between the embedded discrete state and the relevant continuum as a whole, an integrated resonance strength Si can be defined as Si

i f j q2 1: 2 ai EEr i

(4)

In order to get the parameters of the Fano resonances, the GOSDs were fitted by a least squared procedure in Eq. (1) convoluted with the instrumental function. We obtained the Fano parameters for the following three 1 o 1 e 1 e resonances: 2 0; 1 2 P , 2 1; 02 S , and 2 1; 02 D , whose natural widths are comparable to our energy resolution. With the GOSDs at 6 (particularly for dipole1 e forbidden transitions), the fitted widths of 2 1; 0 2 S and 1 o 2 0; 12 P are 122 5 and 37 3 meV, respectively, which agree well with the theoretical values [13]. Analysis of the GOSDs at other angles also confirmed the above results. Because the interval between 1 e 1 o and 2 0; 1 is only 0.24 eV, and 2 P 2 1; 02 D 1 e 1 o 1; 0 D is much weaker than 2 2 0; 12 P , it is 2 1 e difficult to obtain accurate width of 2 1; 0 2 D under present experimental conditions. So, the theoretical width 64 meV [13] is adopted, which is consistent with the experimental electric dipole-quadrupole interference value of 57 meV [24] . 1 o The fitted Fano parameters of the 2 0; 1 2 P reso2 nance are shown in Fig. 3. As K increases, the GOS f and the resonance strength S decrease, showing typical behaviors of dipole-allowed transitions. It is a resonance with constructive interference in the low energy wing and destructive interference in the high energy wing since the sign of q is negative for all K 2 . The jqj decreases slowly with K 2 . It indicates that the transition amplitude into the modified discrete state decreases more quickly than that into the relevant continuum. The 2 decreases with K 2 because the contributions of dipole-forbidden transitions increase with K 2 . It is interesting to note that in Fig. 3 there are discrepancies between our fitted 2 and those by Fan and Leung [17]. Their results were obtained by 193203-2



1 o FIG. 3. The parameters f, S, q, fa , and 2 of 2 0; 1 2 P of helium as a function of K 2 . Solid square: Present result. Open triangle: Ref. [17]. Open circle: Corrected value from Ref. [17].

adopting for their energy resolution of 0.7 eV directly. The importance of the energy convolution with the instrumental function can be understood by the energy folding integration with a Lorentzian function, i.e., ! 0 ri E 2 Z qi EE 2 1 i =2 fai 1 dE0 EEri E0 2 E0 2 1 =2 1 i =2 ! EEri 2 fai i qi i =2 1 ; (5) 2 ri i 1 EE =2 i where is the Lorentzian width. Thus, the fitted fai without taking into account the convolution will be underestimated by factor =, about 20, the same for the corresponding 2 will also be underestimated, as shown in Fig. 3. We plot the corrected values in Fig. 3, which are then consistent with our fitted 2 . The values of jqj and f, reported in [17] with the energy resolution about 0.7 eV insufficient to observe the nearby resonances 1 e and n 1; 0 2 D ], are generally smaller than ours especially at large K 2 . Such experimental discrepancies require further studies. For the Rydberg series 1 o n 0; 12 P n 2–5, the resonance profiles exhibit similar behaviors as the momentum transfer K 2 increases. 1 e The fitted Fano parameters of the 2 1; 0 2 S resonance are shown in Fig. 4. There are Rydberg series of 1 e Fano resonance [2;3 1; 0 2 S ] with negative index q (i.e., destructive interference in the high energy wing) in Fig. 2. 1 e 2 The f, S , fa , and 2 of 2 1; 0 2 S increase with K , showing typical behaviors of dipole-forbidden transitions. The jqj varies a little and smoothly with K 2 , considering the experimental uncertainty. It indicates that the transition amplitude into the modified discrete state has a similar behavior as that into the relevant 1 e continuum. The fitted Fano parameters of 2 1; 0 2 D are displayed in Fig. 5. Because of the very weak 193203-3


1 e FIG. 4. The parameters f, S, q, fa , and 2 of 2 1; 0 2 S of 2 helium as a function of K . Solid square: Present result.

1 o resonance and the strong overlap with 2 0; 1 2 P , their 1 e uncertainties are larger. The sign of q of 2 1; 0 2 D is 2 negative and jqj varies very slowly with K with a value 0:37 0:18 (f S), which is consistent with the experimental electric dipole-quadrupole interference value of 0:25 [24]. Note though that Kra¨ ssig et al. said that ‘‘the value of q cannot be unambiguously determined from electron scattering experiments because all multipoles are present.’’ It seems that, at present incident energy of 2.5 keV, the other multipoles contribute much less to the 1 e excitations. The f, S , fa , and 2 of 2 1; 0 2 D increase 2 with K , showing typical behaviors of quadrupole tran1 e sitions. The values of f of 2 1; 0 2 S are much greater 1 e than those of 2 1; 02 D , indicating that monopole transition is much stronger than quadrupole transition. Let us return to the discussion of the weak resonances in Fig. 2 to elucidate excitation dynamics according to the internal correlation quantum numbers, K, T , and A. We pay attention to the K 2 dependence of the resonances 1 e 1 e [2 1; 0 2 S and 2 1; 02 S ]. Based on Fermi’s golden rule, the transition strength is proportional to the square of the transition matrix element, i.e., the transition-operator-weighted overlap integral between the initial ground state 1s2 1 Se and the final doubly excited state 1 e 2 2 K; T 2 S . At large angles (e.g., 4 with K 0:91 a:u:), the transition matrix element results mainly from the interactions in an inner region (e.g., R q r21 r22 1 a:u:). At small angles (e.g., 0 with K 2 0:03 a:u:), the transition matrix element results mainly from the interactions in an outer region (e.g., R 5 a:u:). For the initial ground state, the wave function, in the inner region, has a broad peak at [12 180 , tan1 r1 =r2 45 ] while it, in the outer region has two ridges at 12 from 0 to 180 , 20 , and 70 , respectively, owing to the two electron ‘‘breathing’’ correlations in the hyperspherical coordinate [6]. In both inner and outer regions with respect to the initial ground state (i.e.,

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dency, which is unaccessible by photoexcitation, are elucidated. The hyperspherical wave functions of the two 1 e 1 e dipole-forbidden states of 2 1; 0 2 S and 2 1; 02 S were verified qualitatively. Theoretical calculations are needed urgently to give a more quantitative interpretation of the observed phenomena. Support for this work by National Nature Science Foundation of China (10134010, 10004010, 10176013), the Youth Foundation of the University of Science and Technology of China, and National High-Tech ICF Committee in China is gratefully acknowledged.

1 e FIG. 5. The parameters f, S, q, fa , and 2 of 2 1; 0 2 D of helium as a function of K 2 . Solid square: Present result. Open star: Ref. [24].

R about 7 a.u. because negligible contribution to the overlap integral owing to exponential decay of the ground 1 e state wave function), the wave function of 2 1; 0 2 S has a broad peak at 12 180 , 45 , while that of 1 e 2 1; 02 S has one major broad peak at 12 30 , 45 and two minor rising ridges at 12 from 0 to 180 , 20 , and 70 , with two peaks at 12 180 , respectively [6]. Thus, at small angles, the transition strength of 1 e 1 e 2 1; 02 S is stronger than that of 2 1; 02 S because of the better overlap between the wave functions of 1 e 1s2 1 Se and the 2 1; 0 2 S in the outer region. At 1 e larger angles, the transition strength of 2 1; 0 2 S is 1 e instead weaker than that of 2 1; 02 S because of the better overlap between the wave functions of the 1s2 1 Se 1 e and the 2 1; 0 2 S in the inner region. Furthermore, the 2 ) of 1; 0 1 Se is larger than the width (1;0 V1;0 2 2 2 e ) of 21; 01 width (1;0 V1;0 2 S . This can be understood as follows: Because of the doubly excited 1 e 2 state 2 1; 0 2 S with the major configuration as 2s 1 e and the doubly excited state 2 1; 02 S with the major configuration as 2p2 , the squared interaction matrix 2 element [V1;0 R0 2s; 2s; 1s; s2 ] should be larger 2 than the squared interaction matrix element [V1;0 1 2 ‘ R 2p; 2p; 1s; s ], where R 2‘; 2‘; 1s; s are Slater integrals. Figure 2 also exhibits the well-known 1 o phenomena [1,2] that the resonance 3 0; 1 2 P with A 1 o 1 is much stronger than the resonance 3 0; 1 2 P with A 1. In conclusion, absolute measurement of the doubly excited states of helium is extended to the highly excited states with very low cross sections, i.e., using the angular resolved fast-electron energy loss spectrometer with high energy resolution . Electronic correlations of dynamical processes, in particular, their momentum transfer depen-

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