Jul 4, 2005 - autoionizing resonance possesses two universal values: the width and the energy .... The summation in (3) runs over the channels with fixed.
INSTITUTE OF PHYSICS PUBLISHING
JOURNAL OF PHYSICS B: ATOMIC, MOLECULAR AND OPTICAL PHYSICS
J. Phys. B: At. Mol. Opt. Phys. 38 (2005) 2545–2553
doi:10.1088/0953-4075/38/14/017
Universal scaling of resonances in vector correlation photoionization parameters A N Grum-Grzhimailo1,2, S Fritzsche3, P O’Keeffe1 and M Meyer1 1 2 3
LURE, Centre Universitaire Paris-Sud, Bˆatiment 209D, F-91898 Orsay Cedex, France Institute of Nuclear Physics, Moscow State University, Moscow 119992, Russia Fachbereich Physik, Universit¨at Kassel, Heinrich-Plett Strasse 40, D-34132 Kassel, Germany
Received 3 May 2005, in final form 8 June 2005 Published 4 July 2005 Online at stacks.iop.org/JPhysB/38/2545 Abstract In the present work it is shown that in atomic photoionization the Fanolike behaviour of vector correlation parameters in the region of an isolated autoionizing resonance possesses two universal values: the width and the energy shift with respect to the energy of the autoionizing state. This property is confirmed by fluorescence polarimetry studies, which provide, in the particular case of Xe 4d−1 5/2 6p (J = 1) photoexcitation, the experimental determination of the alignment and the orientation of the final photoion across the resonance.
1. Introduction Autoionizing states give rise to pronounced structures in many photoabsorption spectra. The typical profile of these resonances can be described by an analytical expression containing the interference between the resonant excitation and the direct photoionization [1, 2]. The influence of the autoionizing states is also observed in the energy dependence of different vector correlation parameters providing more detailed information on the photoionization process. Experimental and theoretical studies of autoionizing resonances in vector correlation parameters, such as the anisotropy coefficient of the angular distribution of photoelectrons [3–5], the spin polarization of photoelectrons [5, 6], parameters of polarization and the angular distribution of the photoion decay products [7–9] were pioneered in 1970s and early 1980s. More recently, coincidences between photoelectrons and photoion decay products have been observed [10–13] and show the great potential of vector correlation studies in order to extract detailed information about the photoionization process on the level of a ‘complete’ experiment. The advent of new synchrotron radiation sources producing high-brilliance VUV and soft x-ray photon beams with high energy resolution and variable polarization as well as the advances in product detection techniques, provide new opportunities for studies of polarization and correlation phenomena in photoionization. In particular, it is possible now to investigate these phenomena by scanning the excitation energy over an extended region around and across the resonances, even in cases, where the direct photoionization and therefore the off-resonance 0953-4075/05/142545+09$30.00 © 2005 IOP Publishing Ltd Printed in the UK
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signal are extremely weak. Having these experimental possibilities, a more detailed study of the resonance profiles in different observable quantities is timely. In a few cases it has already been shown that resonances in the angular anisotropy parameter β and the spin polarization of photoelectrons are broader than the corresponding autoionizing resonance in the photoionization cross section [14–16]. A shift of the resonance position in β with respect to the resonance in the cross section was noted a few times [17–19] and was clearly seen in some data [20, 21]. In the present study we show that the above observations of broadening and energy shift are representative examples of a general scaling effect: the width and the shift of the resonances in the vector correlation parameters are universal quantities and are directly related to the resonance profile parameters in the integral photoionization cross section. To our knowledge, this effect has not been pointed out before. Particular cases of the above regularity have been discussed in [22, 23] for the angular anisotropy and polarization of photoelectrons and for the angular distribution of fluorescence in the dielectronic recombination. In the next section, a general theory of the scaling effect is developed and discussed in detail, in particular, for the case of a single photoion state. Experimental confirmation of the effect by means of fluorescence polarimetry in the photoionization of xenon in the region of the Xe 4d−1 5/2 6p (J = 1) resonance photoexcited by circularly and linearly polarized synchrotron radiation is presented in section 3, followed by a short discussion and a conclusion. 2. Theory 2.1. General considerations The photoabsorption cross section in the region of an isolated autoionizing state is described by the Fano formula [1, 2] (q + ε)2 (q + ε)2 , (1) σ (ε) = σb + σa 2 = σ0 1 − ρ 2 + ρ 2 ε +1 1 + ε2 where σa , σb are the interacting and not-interacting parts of the cross section, respectively, q is the asymmetry profile parameter and σ0 = σa + σb , ρ 2 = σa /(σa + σb ). These parameters are usually treated as energy-independent in the region of the resonance. The scaled energy dependence is represented by the variable ε = (E − Er )/(/2), where E is the photon energy, Er is the energy of the autoionizing state and is the decay width of this state. The correlation index ρ 2 and the profile index q contain the important spectroscopic information on the interaction of the discrete state with the continuum. The partial photoionization cross section σP (ε) into a particular ionic state P is usually expressed by the parametric formula 2C1 ε + C2 , (2) σP (ε) = σP0 1 + 1 + ε2 where σP0 is the partial cross section without the resonance and C1 , C2 are the profile parameters introduced by Starace [24]. Let us consider photoionization into a well-defined ion state P and let T denote a vectorcorrelation parameter, which is independent of the photoelectron emission angle. In this case T is expressed in terms of a bilinear combination of the dipole photoionization amplitudes Dµ in the form tµµ Dµ Dµ∗ , (3) T = N −1 µ,µ ∈P
where tµµ are the angular coupling coefficients specific for each parameter T and N = 2 µ∈P |Dµ | . The multi-index µ symbolizes different asymptotically observed photoionization
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channels, described by state vectors |µE − , where the minus sign indicates the incoming-wave boundary condition for the channel µ. The summation in (3) runs over the channels with fixed final ionic state P, usually distinguished by the orbital and total angular momenta of the photoelectron and its coupling to the ion. Equation (3) describes, for example, the parameters of spin-polarization of photoelectrons, the anisotropy parameters of the angular distribution of photoelectrons from unpolarized and polarized atoms, the alignment and the orientation of photoions, the magnetic dichroism, and others [25]. In addition, equation (3) is also valid for the vector correlation parameters in the reverse process of recombination. The photoionization amplitudes in the region of an isolated autoionizing state are expressed as [5, 24] q +i 0 , (4) Dµ = Dµ 1 + αµ ε−i where Dµ0 is the amplitude in the absence of the resonance and q is the profile parameter of the photoabsorption cross section (1). The complex parameter Vµ 2π 0 ∗ αµ = 0 D V (5) Dµ µ µ µ includes the matrix elements Vµ of the effective interaction between the discrete state and the continuum [24, 26]. The summation over µ in equation (5) runs over all photoionization channels and the factor in the square brackets is common for all αµ . The αµ parameters become real numbers in the limit of vanishing mixing between the observable channels |µE [24]. Substituting equation (4) into equation (3) and transforming into the equivalent Fano form, we obtain (q T + ε) ˜ 2 , (6) 1 + ε˜ 2 where the parameters σbT , σaT and q T are specific quantities for each individual T. The scaled energy dependence ˜ (7) ε˜ = (E − E˜ r )/(/2), T = σbT + σaT
is given in terms of a scaled width 1 ˜ = 1 + C2 − C12 2 ≡ χ and the shifted resonance energy C1 ≡ Er + . E˜ r = Er − 2 The quantities C1 and C2 are the Starace parameters of the partial cross section (2):
(8)
(9)
C1 = q ReαP − ImαP ,
(10)
C2 = (q + 1)|α| P − 2q ImαP − 2 ReαP ,
(11)
2
2
where the weighted averages αP and |α|2 P are defined by equations (30) and (31) of [24]. Equations (6)–(9) are the key results of our derivation. For any vector correlation parameter T of the form (3), they show that the width ˜ of the Fano profile is proportional to the decay width, , of the autoionizing state with a scaling factor χ , which depends only on the parameters C1 and C2 . Moreover, the scaled Fano profile (6) is shifted with respect to the energy of the autoionizing state Er by an amount , which is proportional to the decay
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|∆/Γ| 400 150 100 100
300
|q|
50
10
20
200
1
10 5
100
0.1
2
0.01
1.1 0.001
0
0.001
0.01
ρ2
0.1
1
0.001
0.01
0.1
1
ρ2
Figure 1. Contour plots of | / | and χ defined by equations (12) and (13).
width of the autoionizing state and which depends only on the parameter C1 . The values of χ and are independent of the particular meaning of T provided equation (3) is valid. To our knowledge, this universal scaling of different parameters has never been demonstrated before. Only for the particular case of a single final ion state, it was predicted for the anisotropy parameter β and the spin polarization of photoelectrons in [22]. 2.2. Particular case: a single ionic state For cases where only one final ionic state P can be reached, for example for autoionizing states lying below the second ionization threshold, the weighted averages of the α parameters obey the relation αP = |α|2 P = ρ 2 [27]. Then the coefficients C1 and C2 are directly related to the Fano parameters in photoabsorption, C1 = qρ 2 , C2 = (q 2 − 1)ρ 2 , and we obtain
(12) χ = (1 − ρ 2 )(1 + ρ 2 q 2 ) and = −qρ 2 /2.
(13)
Equations (12) and (13) have been presented in an equivalent form in [22, 23]. They provide us with a new method of determining the Fano parameters q and ρ 2 by studying the scaling factors χ and the energy shifts . The method is particularly well suited for strong and nearly symmetric (|q| 1) resonances, for which the profile indices q are difficult to extract with good accuracy from the photoabsoption profiles. The method also eliminates the problem of treating correctly possible unphysical background in the experimental spectra, which makes it sometimes very difficult or can even prohibit the accurate determination of the ρ 2 parameter from the photoabsorption cross section. Note that the sign of the shift (13) is determined solely by the sign of q and the energy E˜ r , given by equation (9) coincides with the interference minimum in the photoabsorption cross section (1). Figure 1 shows contour plots of / and χ calculated according to equations (12) and (13). For autoionizing states with asymmetric photoabsorption Fano profiles (|q| ∼ 1) the values of and ˜ hardly exceed the natural width of the resonance, . For the concrete example of the autoionizing state 3s3p6 4p1 P in Ar with q = −0.249(3), ρ 2 = 0.890(5)
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and = 76(5) meV [28], we obtain the scaling factor χ = 0.341(6) and the shift = 0.111(2) = 8.4(6) meV. This small shift can be noted in figure 1 of [29]. Nevertheless, for ‘giant’ autoionization resonances with asymmetric photoabsorption profiles, the shift ∼ can reach electronvolts and can even lead to a switch over in the ordering of the resonance structures in the vector correlation parameters as compared to the photoabsorption cross section σ [17, 18]. For nearly symmetric Lorentzian-like resonances (|q| 1) the ratio / and the scaling factor χ can be much larger. For example, calculations [22] predict ˜ 120 meV (χ 100) for the Na 2p5 3s2 2 P1/2,3/2 autoionizing states with q ∼ 300. The ‘switch over’ was also predicted for the resonances in the angular anisotropy parameters of the fluorescence after dielectronic recombination into the excited states of He [23]. For a single final ionic state, the effect of line broadening can qualitatively be understood from the following consideration. The dimensionless parameters (3) depend only on the ratios of the amplitudes for the resonant and the direct channel of the photoionization. For strong resonances the former channel dominates even in regions outside the natural width, leading in these regions to the values of the parameters around their resonant values; the values characteristic for the direct photoionization are reached only in the far wings of the resonance. Similarly, most sharp changes of the parameters due to the interference between the resonant and the direct photoionization occur in the region of the resonance minimum in the cross section, causing the shift in energy. In the case of several possible ionic states the above effects are masked due to different resonance profiles and different resonance values of the parameters in the channels related to different ionic states. A trivial case is given for a single channel µ ∈ P, when α = 1 and when the resonance structure in the parameter T vanishes (˜ = 0). The latter statement reflects the fact that for a single possible photoionization channel the vector correlation parameters reduce to algebraic values, which are independent of energy despite of resonant behaviour of the total cross section. 3. Experimental confirmation of the scaling effect In order to observe and to verify the scaling effect, we use our recent studies of alignment and orientation parameters of the Xe photoion in the region of the Xe 4d−1 5/2 6p (J = 1) resonance by means of ion fluorescence polarimetry [30–32]. The experiments were performed at the CIPO (Circular Polarization) beamline of the ELETTRA synchrotron radiation facility in Trieste (Italy). The experimental set-up and the procedure of data recording have been described in detail elsewhere [31, 32]. The present measurements differ from the former studies mainly in the fact that much higher statistics has been obtained, which allowed us to observe the fluorescence polarization while scanning the excitation energy across the resonance with both linearly and circularly polarized photon beams. A similar experiment, using linearly polarized photons and a narrower excitation region, was performed recently in order to study, for the resonant 3d-np excitation of Kr, the influence of close lying resonances on the polarization state of the photoion [33]. In our experiments, we have concentrated on two fluorescence lines at 484.43 nm and 466.85 nm corresponding to the radiative transitions 5p4 (3 P2 )6p[1]3/2 → 5p4 (3 P2 )6s[2]5/2 and 5p4 (1 D2 )6p[1]1/2 → 5p4 (3 P1 )5d[2]3/2 , respectively. The results obtained for the circular and linear polarization of the fluorescence are converted into the alignment (A20 ) and the orientation (A10 ) parameters of the corresponding ionic 5p4 6p states [30, 31]. The alignment and the orientation are particular cases of vectorcorrelation parameters described by general formula (3). Displayed in figure 2 are A20 and A10 as functions of the incident photon energy for the two ionic states, 5p4 (3 P2 )6p[1]3/2 and 5p4 (1 D2 )6p[1]1/2 (the ionic state with J = 12 cannot be
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(a)
(b)
(c)
Figure 2. Resonance Xe 4d−1 5/2 6p (J = 1) in the total ion yield (a), alignment and orientation parameters of the photoion in the (3 P2 )6p[1]3/2 state (b) and (1 D2 )6p[1]1/2 state (c). The energy dependence of the orientation and alignment parameters is fitted by equation (6) (solid curves). Arrows indicate the energy positions of the corresponding Fano profiles. Table 1. Scaled widths ˜ and shifts for ionization into Xe II 5p4 6p fine-structure states. Numbers and notations of the ionic states are given as in [30]. A10
A20
No.
Ion state
˜ (meV)
(meV)
˜ (meV)
(meV)
C1
C2
6 19
(3 P2 )6p[1]3/2 (1 D2 )6p[1]1/2
1070 ± 75 835 ± 86
+52 ± 23 −51 ± 37
1010 ± 140
+45 ± 40
−0.98 ± 0.43 +0.96 ± 0.70
101 ± 14 62 ± 13
aligned, but only oriented). For comparison, the profile of the Xe II ion yield is shown in the upper panel of the figure. Our ion yield data are in accordance with earlier measurements of the resonance energy, Er = 65.110(10) eV [34], and the decay width, = 106.3(5) meV [35], of the Xe 4d−1 5/2 6p (J = 1) state. The energy dependence of the parameters A10 and A20 has been fitted by formula (6) and the results are summarized in table 1 . As expected, the values
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of ˜ and extracted from the alignment and orientation of the 5p4 (3 P2 )6p[1]3/2 state were identical within the error bars, despite the completely different shapes of the corresponding resonance profiles. At the resonance position, the values for the alignment and orientation coincide with those given already in our earlier study [31], i.e. A20 = −0.09 and A10 = −0.40 for 5p4 (3 P2 )6p[1]3/2 and A10 = −0.29 for 5p4 (1 D2 )6p[1]1/2 . The resonance features in the parameters A20 and A10 are very broad pointing to the scaling factor χ of around 10. We would like to stress here that equation (3) for the ion orientation and alignment includes only partial-wave cross sections (for example [31]) and therefore the broadening is not related to the phase difference between the amplitudes as suggested in [15]. Furthermore, we note that the extracted energy shifts , which show opposite sign for the two final ionic states, are not large (about 50 meV), but noticeable and larger than the error bars. Finally, we can determine the Starace parameters C1 and C2 for both ionic states using ˜ and and equations (8) and (9). 4. Discussion and concluding remarks The quantitative discussion of the scaling effect is a complicated task in our case due to the large number of continua interacting with the discrete state. The resonance 4d−1 5/2 6p (J = 1) interacts with hundreds of continua built on different excited states of the Xe ion, which are producing a rich resonant Auger spectrum [36]. The numerical treatment of the parameters presented in table 1 is beyond the scope of the present paper. The direct photoionization amplitudes Dµ0 in equation (4) correspond to excitation of the Xe+ 5p4 6p satellites. Even advanced approaches do not give sufficient accuracy for the photoionization of Xe with simultaneous excitation of the 6p states in the energy region of interest [37]. Strong interchannel interactions are known to dominate the photoionization amplitudes in this energy region [38], while the Cooper minimum in the d partial cross section makes the amplitudes extremely sensitive to the model. The situation is even worse for the calculations of the αµ parameters (5), where all possible satellites have to be taken into account, including the Auger decay amplitudes Vµ of the Xe + 4d−1 5/2 6p (J = 1) state to the corresponding excited states of the Xe ion. In order to qualitatively understand the opposite signs of the shift in table 1, we consider, in accordance with equations (9) and (11), the ratio of the C1 parameters for the ionic states 6 and 19 (for q 1 and ρ 2 1 the second term in equation (11) can be neglected [27, 39]) ∗ ∗ + V3/2 D3/2 ) Re(V1/2 D1/2 Re µ∈state19 Vµ Dµ0∗ C1,state19 , (14) = = r≡ ∗ ∗ ∗ 0∗ C1,state6 Re µ∈state6 Vµ Dµ Re(V1/2 D1/2 + V3/2 D3/2 + V5/2 D5/2 ) where the subscripts 1/2, 3/2, 5/2 indicate the partial waves s1/2 , d3/2 , d5/2 of the photoelectrons, respectively. To estimate r we utilize matrix elements Vj (j = 1/2, 3/2, 5/2) for the states 6 and 19 from our recent calculations, confirmed by complete experiment on the Auger decay [31, 32], and multiconfiguration Dirac–Fock [40] amplitudes Dj . The sign of all terms in the numerator and the denominator of (14) turned out to be the same and r ≈ 1. With q ≈ 200 [41], we then obtained ≈ −300 meV for both states from table 1. However, the D3/2 and D5/2 amplitudes are drastically affected, if the interchannel interactions are taken into account, which shifts the Cooper minimum in the partial d photoionization cross section from energies around 120 eV to energies around 55 eV (figure 11 of [37]). As a result, for the photon energy of 65.1 eV, the amplitudes D3/2 and D5/2 decrease and change sign. The s1/2 channel contributes much less for the state 6 than for the state 19 [31]; therefore, the numerator of (14) keeps the sign, while it is changed for the denominator. In conclusion, we have shown that autoionizing resonance profiles in vector correlation parameters in atomic photoionization possess universal quantities, the width and the energy
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shift with respect to the energy of the autoionizing state. This statement was confirmed experimentally by means of the fluorescence polarimetry across the Xe 4d−1 5/2 6p (J = 1) resonance. We emphasize that experimentalists and theorists can each study the vector correlation parameter (3), which is the most convenient for them, i.e. not necessarily the same. The comparison can nevertheless be made in terms of the scaled width in equation (8) and the energy shift in equation (9) or, alternatively, in terms of the parameters C1 and C2 . The performances of the new high-brilliance photon sources, like third generation synchrotron facilities, make it possible now to obtain experimental results of high quality within and near to the profile of a strong absorption resonance. We hope therefore that the present study will stimulate further investigations of autoionizing resonances in the vector correlation parameters. Acknowledgments ANG gratefully acknowledges the hospitality of the LURE Laboratory and of the Universit´e Paris-Sud, and support under grant 04-02-17236 from the Russian Foundation for Basic Research. SF acknowledges support from the Deutsche Forschungsgemeinschaft (DFG). The experimental work at the ELETTRA synchrotron radiation facility was supported by the European Community under contract no. HPRI-CT-1999-00033. References [1] [2] [3] [4] [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]
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