Towards a Complete Experiment for Auger Decay

Chapter 4 TOWARDS A COMPLETE EXPERIMENT FOR AUGER DECAY A. N. Grum–Grzhimailo Fakultät für Physik, Universität Freiburg, D-79104 Freiburg, Germany Institute of Nuclear Physics, Moscow State University, Moscow 119899, Russia

A. Dorn Fakultät für Physik, Universität Freiburg, D-79104 Freiburg, Germany

W. Mehlhorn Fakultät für Physik, Universität Freiburg, D-79104 Freiburg, Germany Abstract

1.

A new method for performing a complete experiment in the case of the Auger decay is considered. The method assumes the LSJ-coupling approximation and is based on measurements of the relative intensities and the angular distribution of the Auger electrons. The method can be used for Auger transitions from states with nonvanishing spin S and orbital angular momentum L if the fine structure levels of the final ionic state are resolved. As example, for the Auger decay the absolute ratio of decay amplitudes and their relative phase are determined.

INTRODUCTION

A complete experiment refers to the experimental determination of the quantum-mechanical amplitudes characterizing the process. Although the first complete experiment for photoionization was reported as early as 1980 by Heinzmann [1], the first "almost" complete experiments for Auger (or autoionizing) decay were performed only very recently by West et al [2], Ueda et al [3] and Grum-Grzhimailo et al [4]. We use the term "almost" because the absolute values of the amplitudes were not determined. Complete Scattering Experiments, Edited by Becker and Crowe Kluwer Academic/Plenum Publishers, New York 2001

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The Auger process will be considered as part of a two-step process: Step 1: Excitation of the Auger state by photon or particle impact

Step 2: Decay of the Auger state

The assumption of a two-step process implies also that we neglect a direct amplitude for double ionization (i.e., there is no Fano-type interference [5]) and the post-collision interaction [6] between the Auger electron and the ejected and scattered electrons of process (4.1). The complete experiment on the Auger decay relates to step 2 and the aim of the experiment is to determine the absolute values (moduli) and relative phases of the decay amplitudes. The total number N of decay amplitudes depends on the total angular momenta J and of the initial and final states of the Auger decay [7]:

if

and

if

For example, the Auger decay with and is characterized by decay amplitudes and . The complete Auger experiment with N decay amplitudes has to determine a total number of 2N – 1 moduli and relative phases. Kabachnik and Sazhina [7] have theoretically shown that modern experiments on Auger decay (i. e., angular anisotropy and spin-polarization of Auger electrons) in principle can provide enough independent experimental quantities for a complete description of the Auger process. But in praxis part of the necessary experiments are hardly feasible and/or part of the quantities to be measured are very small. Then the experimental errors prevent to extract reliable values of the moduli and relative phases of decay amplitudes. As result of this a complete experiment for Auger decay has not been performed until today by measuring the angular anisotropy and spin-polarization of Auger electrons. Instead, other kinds of complete experiments for Auger (autoionizing) decay ("almost" complete experiments) were suggested and carried out. West et al [2] and Ueda et al [3] measured the angular correlation between the autoionization electron and the subsequent fluorescence photon. Grum-Grzhimailo et al [4] proposed and performed a new approach for an "almost" complete experiment by the measurements of the intensity ratio and the angular distribution of Auger electrons. In the following section we will treat an example of a complete experiment by measuring the angular anisotropy and spin-polarization of Auger electrons. Then, in section 3. a new approach for an "almost" complete experiment as proposed by Grum-Grzhimailo et al [4] is discussed. In section 4. we

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present the performance of this "almost" complete experiment and compare the experimental results with theoretical values. Finally, in section 5. we discuss the conditions and the applicability of the new approach.

2.

EXAMPLE FOR A POSSIBLE COMPLETE EXPERIMENT We assume that the initial Auger state decays into several final states (see Fig. 4.1) and we wish to perform a complete experiment on

Figure 4.1 Diagram for Auger transitions The total decay probability and the partial decay probabilities are given by the widths and respectively.

the Auger decay to the final state . Furthermore, we assume that the Auger decay is characterized by only two decay amplitudes, and . Then the complete experiment has to determine the two moduli and and the relative phase . What independent experimental quantities are available? 1. From the ratio of integral intensities the integral intensity of transition

of transitions

and the total decay width of the initial Auger state

relative to

with

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the partial decay width due to the transition experimentally via

can be determined

Here, is the partial decay width due to the transition partial width is related to the transition amplitudes and

. The by

where the coefficients are known and depend on the particular configurations involved in the Auger transition and coupling schemes.

2. The angular intensity of the Auger electrons from transition measured by a spin-unsensitive detector and when the polarization state of the residual ion is not detected is given by [4, 8]

Here is the integral intensity of the Auger transition are the spherical harmonics and are the reduced statistical tensors (state multipoles) which describe the polarization of the decaying Auger state and depend only on the excitation process (step 1, see (4.1)). The decay parameters depend only on the dynamics of the Auger decay and are expressed as functions of and In praxis only the decay parameter is measureable, the term with is generally too small if present in Eq.(4.8). 3. A third experimental quantity to obtain information on the decay amplitudes is the spin-polarization of Auger electrons. The angular intensity of Auger electrons measured with a spin-sensitive detector is given in a general form analogous to (4.8) by

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where an additional angle characterizes the direction of the axis of the electron spin analyser. The coefficients are again functions of and The angular functions are fully determined by the geometry of the experiment and the effectiveness of the spin-sensitive detector. Note, that in contrast to Eq.(4.8) terms with both, even and odd, ranks k are present in Eq.(4.9). Therefore, spinresolved measurements are basically more informative than measurements of the ordinary angular distribution (4.8). Unfortunately, experiments with a spin-resolved detector suffer from low counting rates. Furthermore, the coefficient standing for the dynamical spin-polarization [9], is small for normal Auger transitions [10] and is therefore hardly measurerable with high enough accuracy. To measure which stands for the spin-transfer mechanism [9], one has to introduce orientation, i.e. the first rank tensor into the Auger state. Presently the only practical method to achieve this is to produce the vacancy by a circular polarized photon beam. This method requires high brilliant sources of circular polarized VUV or X-ray radiation [11]. Recently, a new method for producing Auger states with large orientation which leads also to large spin-polarization of the Auger electrons was suggested by means of laser-excited polarized atoms [12], but it has not yet been realized. A general problem for the measurements of the decay coefficients and is also to disentangle them from the statistical tensors characterizing the excitation step. Independent measurements are generally needed to solve this problem. In conclusion, for the full determination of two Auger amplitudes and (two moduli and one relative phase) we need to measure the relative integral intensities the total width of the initial Auger state and the decay parameters and . Strictly speaking, one more measurement is needed to avoid an ambiguity in the relative phase since the physical quantities depend either on or on . Up to now such a complete experiment has not been performed.

3.

PROPOSAL FOR AN "ALMOST" COMPLETE EXPERIMENT

Again we assume that the Auger decay is characterized by the two amplitudes and But instead of determining die two absolute values and we are interested here in the determination of only the ratio and of Therefore we use the term "almost" complete experiment. We further assume that the LSJ-coupling approximation is valid. This implies that the spinorbit interaction causes only a fine-structure splitting of the atomic levels, but does not change the atomic wavefunctions, including the wavefunction of the Auger electron. The Auger decay partial waves are then described by l instead

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of where l and j are the orbital and total angular momenta, respectively. This approximation is accurate enough for Auger transitions in not too heavy atoms. Introducing the LSJ-coupling approximation, in general, reduces the number of decay amplitudes. As an example we consider the decay of one fine-structure level J of multiplet LS into the two final levels and of the final multiplet (see Fig. 4.2). The Auger decay is characterized by

Figure 4.2 Diagram for Auger transitions assuming LSJ coupling. Here we consider the pair of transitions and (solid lines) or and (dashed lines).

the two decay amplitudes and where the amplitudes are now given by and are independent of J, and j. For the determination of the two quantities and we need two independent experimental quantities. Here we propose the measurements of the ratio of integral intensities of transitions and (see Fig. 4.2) and the angular distribution of Auger electrons of transition with spinunsensitive detector. The ratio R of integral intensities of transitions and is given in analogy to Eq. (4.7) by [4]

Here, due to the LSJ-coupling approximation the coefficients are pure algebraic. The general expression for the coefficient is of the form

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We used standard notations for the 6j-symbols and Clebsch-Gordan coefficients and abbreviated Since the integral intensities depend only on the squared moduli the ratio R of integral intensities of the finestructure components of Auger lines carries information on the ratio of moduli,

The angular distribution of Auger electrons of transition measured without spin-detection is given by Eq. (4.8). Here we are interested only in the first term of the anisotropy with decay parameter This parameter is given in the LSJ-coupling approximation by [4]

and depends therefore on and As can be seen, from Eqs. (4.10) and (4.12) the quantities and of the "almost" complete experiment can be evaluated from the experimental values R and Information on the ratio r of moduli can be extracted from the ratio R of integral intensities only if the quantum numbers J and are not factorizing from the sums in the nominator and denominator of Eq. (4.10). From the explicit form (4.11) for the coefficients it follows that factorization occurs in each of the following cases: Each of these conditions give therefore restrictions on the use of the proposed method to perform an "almost" complete experiment. On the other side, there is an additional advantage of the proposed method. Besides the measurement of the ratio of integral intensities of transitions and any other intensity ratio, e. g., of transitions and (dashed lines in Fig. 4.2), can be measured and used for cross checking the first result.

4.

PERFORMANCE OF AN "ALMOST" COMPLETE EXPERIMENT

We have performed an "almost" complete experiment on the following Auger decay (see Fig. 4.3) [4]:

The aim of this investigation was to obtain information on the ratio of moduli, and on The Auger state was produced via 2s-ionization of laser-excited sodium atoms by fast electron impact [13–15] (see Fig. 4.3):

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Figure 4.3 Level scheme for the excitation of the Auger states by electron impact of laser-excited atoms and the Auger decay to is excited via shake-up during the 2s-ionization process.

Besides the 2s ionization without excitation of the laser-excited electron a large fraction (about 31%) of 2s ionization is accompanied by a shake-up process [13]. The experimental procedure is briefly discussed in the following, for more details see references [13] and [14]. The laser beam (along the y-axis) and the electron beam (along the z-axis) cross the atomic beam (x-axis) (see Fig. 4.4). The Auger electrons are measured by means of a 75° sector-shaped cylindrical mirror analyser (CMA) which can be rotated in the yz plane in the angular range . The detector is either a channeltron or, at smaller target densities, a position sensitive detector (PSD). The laser beam is linearly polarized with the polarization axis parallel to the axis of the electron beam. Then the experiment has rotational symmetry around the z-axis (direction of electron beam) and the angular distribution (4.8) depends only on the angle

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Figure 4.4 Experimental setup (from ref. [14], Fig. 1).

relative to the electron beam direction. For the excitation of the sodium atoms we use an argon ion laser pumped linear CW dye laser operating at with 1 MHz bandwidth. In order to achieve a maximum relative density of the atoms the laser is operated in the two-mode version [16, 17]: both hyperfine transitions 3s, F = 2 and 3s, are excited. With a laser power of 100mW at beam diameter of 2mm the relative density has been reached. The alignment of the excited state can be determined via the degree of linear polarization of the fluorescence radiation The alignment decreases by radiation trapping for vapour densities higher than about

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[18]. For a target density of about was measured to be We performed two experiments:

the alignment

– Measurement of the ratio of integral intensities,

– measurement of the angular distribution transition parameter

and

of Auger electrons of the and determination of the decay

Experiment 1: The target density in this experiment was high (about ) and the alignment of the initial state induced by the optical laser pumping was destroyed due to radiation trapping. Ionization from the 2s shell by fast electrons does not bring additional polarization into the Auger state [12]. Therefore, all tensors in (4.8) vanish and the Auger electrons are emitted isotropically. Thus, the measurement of integral intensities can be performed at an arbitrary fixed angle, in the experiment the magic angle 7° was used. In Fig. 4.5 the electron spectrum of Auger electrons due to the decay of is shown. Here we need only the two intensities of Auger lines and The value of the ratio has been measured [15] to be . Although in the present case the fine-structure splitting of the initial components is much smaller than the width of the Auger states i. e. the fine-structure components completely overlap, contributions from different J sum incoherently into the integral intensity [21]. The branching ratio of the Auger decay to different fine-structure components of the final ion takes then the form [4]

where and are exactly known constants. In the case of resolved fine-structure J of the initial state the excitation probability P(J) cancels from (4.16) and the whole treatment contains only the decay step. In order to proceed further in the present case we have to know the relative excitation probabilities P(J) of the different fine-sructure states. These can be calculated assuming the one-particle model for the excitation process. The P(J) depend on the total angular momentum of the initial fine-structure level of laserexcited state. For one obtains and the

Towards a Complete Experiment for Auger Decay

Figure 4.5 Electron spectrum due to the decay measured at target (from ref. 15, Fig. 3b).

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and for high-density Na

final form of Eq. (4.16) is given by [15]

where . From the experimental value we evaluate So the probability for emission of an s partial wave during the Auger decay is 15 times larger than the emission of a d partial wave.

Experiment 2: In the second experiment the angular distribution of Auger electrons of the transition

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was measured. The angular intensity distribution is given by

where the decay parameter

is a function of

and

In order to obtain an anisotropic part of the angular distribution (4.19) an alignment has to be introduced into the initial Auger state This is done via the laser excitation of Na atoms with linear polarized light and low enough target density [19]; the alignment was determined via the linear polarization of fluorescence radiation to be The ionization of a 2s electron by impact of fast electrons does not bring in an additional alignment [12], i.e., The intensity distribution of Auger transition (4.18) relative to the isotropic intensity of transition is plotted in Fig. 4.6. The solid line is a fit function according to Eq.(4.19), a total anisotropy was obtained. Using the alignment value yields the decay parameter . Together with the above listed experimental value of the relative phase can be evaluated via Eq.(4.20) to be or The first value agrees well with the calculated relative phase [20, 22], therefore the second value can be ruled out. In table 4.1 we compare the experimental values of and with theoretical results obtained for different approximations of the wavefunctions [20, 22]. One can see from table 4.1 that the experimetal values agree well with the theoretical results obtained with wavefunctions which include both the core correlation and the core polarization. Especially the correlation of the core wavefunction increases strongly the ratio of squared amplitudes and brings experiment and theory in good agreement. On the other side it is interesting to note that the various approximations of wavefunctions have only very little effect on the relative phase. From this it seems that the experimental determination of the relative phase will much less stringently test its theoretical value than the ratio r of moduli of amplitudes (or the absolute values of amplitudes in case of a complete experiment). The reason for this weak dependence on the wavefunction approximation is that the relative phase is a property integrated from the inner part of the potential of the atom to infinity whereas the moduli depend directly on the quality of wavefunctions beeing active in the Auger decay.

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Figure 4.6 Intensity of transition to the isotropic transition

relative as function of angle relative to the direction of the electric vector of exciting laser beam (which is parallel to the direction of ionizing electron beam). The target density is much lower than in the case of Fig. 4.5; the alignment of laser-excited atoms is The solid line is the fit function to the experimental values (from ref. 4, Fig. 3).

The present "almost" complete experiment can be extended to a real complete experiment by measuring the total width of the decaying Auger state The total width is given by

which yields

was calculated [20] to be 9.4meV, which is much smaller than the energy resolution of the electron analyser of about 35 to 40meV in the present experiment. Therefore, the determination of the total width will be hardly feasible unless the resolution of the analyser is much improved.

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CONDITIONS AND APPLICABILITY OF THE PROPOSED METHOD

A complete experiment is complete only within the framework of the theoretical model which is used to describe it. As outlined in the introduction the basic theoretical model for the Auger decay is that the decay can be treated as step 2 of a two-step process. For the proposed method of an "almost" complete experiment we furthermore assumed (see section 3.) that LSJ coupling is valid for the initial and the final state of the Auger decay. Then several additional conditions should be fulfilled in order that the proposed method can be applied. Condition 1: In section 3. it has been shown, that the ratio of intensities of transitions and is non-statistically and depends on the ratio of moduli if the following conditions are simultaneously fulfilled: Condition 1 is therefore a rigorous condition. Condition 2: In order to measure the intensities of transitions and the fine-structure splitting should be larger than the total width of decaying state but still small enough that LSJ coupling is a good approximation. Condition 3: In order to resolve the fine-structure of transitions and the resolution of the electron energy analyser should be high enough.

Now we consider the general applicability of the proposed method. In the case of diagram Auger transitions of initially closed-shell atoms the initial Auger state has to be produced by ionization in an inner shell with (condition

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1), and the final doubly charged state after the Auger decay must have and e.g. it could be a state. Therefore the proposed method could be applicable to the following Auger transitions:

Condition 2 is well fulfilled for the Ar case of transitions (4.23). In view of the recent results on the complete experiment of the Xe 4d photoionization [23], where it has been shown that instead of the three amplitudes and the two relative phases in the relativistic approach the two amplitudes and the relative phase in the LS coupling are sufficient to describe the photoionization process, one can argue that the LSJ coupling approach is also sufficient for the above listed Auger transitions in Kr and Xe. Condition 3 is easily fulfilled for the Kr and Xe transitions: high-resolution Auger spectroscopy clearly separated the Auger lines according to their final states [24]. In the case of Ar the resolution was not high enough to resolve the lines, the fine-structure intensities were obtained by a line fitting procedure [25]. In the case of resonant Auger transitions following the photoexcitation by synchrotron radiation one faces the same restrictions as for the non-resonant Auger transitions. Therefore the proposed method is not applicable for resonant Auger transitions excited from the ground state of closed-shell atoms, but it is applicable in case of initially open-shell atoms. By utilizing the resonant Raman Auger effect the experimental width of Auger lines is mainly given by the bandpass of exciting photons (for ) and the resolution of the spectrometer. Measurements made with very high photon and electron energy resolutions allowed to resolve almost all fine-structure components in resonant Raman Auger transitions of noble gas atoms [26]. Therefore, the proposed method should be well suited for resonant Raman Auger transitions in initially open-shell atoms provided the two-step model is valid for the resonant Raman Auger effect.

6.

CONCLUSIONS We have presented a proposal for an "almost" complete experiment on

Auger decay and have performed such an experiment for the Auger decay

We determined the ratio of squared moduli, and the relative phase and compared them with calculated values. We have given the conditions which should be fulfilled that the proposed method can be applied either to normal or to resonant Auger transitions.

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Acknowledgments The authors thank O. I. Zatsarinny for sending his results on the relative phase This work is part of a joint program supported by the Deutsche Forschungsgemeinschaft (DFG) and the Russian Foundation for Basic Research (grant 96-02-00204), it is also supported by the DFG via SFB 276. ANG greatfully acknowledges the hospitality of the Albert-Ludwigs-Universität, Freiburg.

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