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ELECTRON CAPTURE SHAKEOFF : VARIATIONAL APPROACH TO INCLUSION OF CORRELATION
AND SCREENING EFFECTS
R. Intemann, J. Law, A. Suzuki
To cite this version:
R. Intemann, J. Law, A. Suzuki. ELECTRON CAPTURE SHAKEOFF : VARIATIONAL AP-
PROACH TO INCLUSION OF CORRELATION AND SCREENING EFFECTS. Journal de
Physique Colloques, 1987, 48 (C9), pp.C9-555-C9-558. �10.1051/jphyscol:1987993�. �jpa-00227414�
ELECTRON CAPTURE SHAKEOFF : VARIATIONAL APPROACH TO INCLUSION OF CORRELATION AND SCREENING EFFECTS
R.L. INTEMANN, J. LAW' and A. SUZUKI"
Department of Physics, Temple University, Philadelphia, PA 19122, U.S.A.
' ~ u e l p h Waterloo Program for Graduate Work in Physics, Guelph Campus, University of Guelph, Guelph, Ontario, NIG 2W1, Canada ' * ~ e p a r t m e n t of Physics, Science University of Tokyo,
Tokyo 162, Japan
ABSTRACT
K-electron shakeoff probabilities during K capture have been calculated using a relativistic correlation-split variational wave function for the initial two-electron state and a numerical solution to the Dirac equation for the scattering state. The effects of screening by L,M,
...
shell electrons on both electronic states are included through the use of an effective central potential determined self-consistently. Results are presented for several nuclides of interest and comparison is made with earlier theoretical results and with representative experimental data.
INTRODUCTION
Inner-shell ionization during orbital electron capture has been the subject of many investi- gations. Of particular interest is the case in which, during an allowed K-capture transition, the remaining K-shell electron is ionized (K-electron shakeoff). For this process the total ionization probability per K-capture event is independent of all nuclear matrix elements, but sensitive to correlation effects between the two initial K electrons and to the effects of screening by the other electrons of the atom. Indeed, it is this sensitivity which makes the process a valuable probe of atomic inner-shell structure.
For the study of this process there have evolved two theoretical approaches, distinguished by the way in which the interaction between the K electrons is treated. In the variational approach - employed most recently by Suzuki and Law (1) - the electron-electron interaction is included in the unperturbed Hamiltonian whose eigenstates are then obtained by means of a self-consistent-field method. In the propagator approach - the most recent results of which are those of Intemann [2] - the electron-electron interaction is regarded as a perturbation on the nuclear Coulomb interaction, and the initial two-electron wave function is obtained by means of conventional perturbation theory. However, each approach has certain weaknesses, and neither has yielded results fully in accord with the substantial body of available experimental data.
In an effort to achieve better agreement with experiment, we have developed a hybrid ap- proach based on a variational method and fully relativistic. In it, we assume an initial state in which two K electrons are moving under the influence of the n-iclear Coulomb potential, their own mutual interaction, and an effective central potential due to the other (passsive) orbital electrons, generated self-consistently. To determine the two-electron wave function, we adopt a saddle-point variational method employing a Coulomb-constrained, correlation-split trial func- tion based on hydrogenic forms. The final state is assumed to consist of a continuum electron moving in the Coulomb field of the daughter nucleus and a self-consistently derived central
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987993
C9-556 JOURNAL DE PHYSIQUE
potential due to the remaining electrons in frozen orbitals; it is obtained through numerical so- lution of the Dirac equation for this case. The K-shell ionization probability is then calculated by the overlap method.
THEORY
The theory of K shakeoff in K-electron capture is based upon the sudden approximation.
Owing to the change of nuclear charge by the orbital (K-shell) electron capture, the whole atomic system undergoes a rapid transition from the ground state of the parent atom to the final state, ejecting to the continuum - along with a neutrino - the other K electron and leaving the daughter atom in a double-K-vacancy state. In describing this phenomenon, only those leptons which participate in the rearrangement process are included in the characterization of the leptonic states. All other electrons are regarded as inert; it is assumed that their influence on the process, largely one of shielding, can be taken into account through a suitable choice of forms for the wave functions of the participating electrons. In this approximation, the transition amplitude (for an allowed EC transition) is of the form,
where
46)
and4y),
labeled by the appropriate eigenvalues of the nuclear charge number operator, and4,
are the normalized wave functions representing the two initial K electrons, the ejected electron with momentum3,
and the neutrino, respectively, and B is a 4 x 4 matrix containing the operative nuclear matrix elements.To construct a suitable form for the two-K-electron initial state, one which takes adequate account of the correlation effects associated with the interaction between the two K electrons, we employ the saddle-point variational method [3]. For our trial function we adopt the form,
where X, represents a Pauli singlet state. Known as a correlation-split (CS) wave function, this form has been used extensively in correlation studies on He-like ions where its effectiveness, especially when screened hydrogenic forms are used for
&,
has been well demonstrated [4].Indeed, for this reason we choose for $* the relativistic hydrogenic forms,
with
where i = ?/r,
r
denotes the gamma function, ui are the Pauli matrices, andx
is a two- component Pauli spinor normalized to (x, X) = 1.This trial function provides six free parameters to be determined by applying the saddle-point variational method to the energy functional E[4] = (4, H4)/(4,4) for which the Hamiltonian, given by (in atomic units)
consists of a sum of Dirac-Coulomb Hamiltonians for each electron, an effective external potential energy for each characterizing its interaction with L,M,
...
shell electrons, and a term describing the interaction between the two initial electrons, including the magnetic contribution.To obtain an appropriate representation for VLM(~), we have performed a Dirac-Fodr cal- culation for the ground state of the parent atom using the program of Grant et al. [5]. From
taken
-
in numerical form - as the remainder of this SCF potential energy.The variational calculation must be carried out numerically, and considerable computer time is required due to the number of free parameters. For this reason we have chosen to constrain the parameters 6* and -y* by means of the analytical relationships that they satisfy for the case of a single (positive energy) electron moving in a nuclear Coulomb field,viz.,
an approximation which appears quite reasonable in view of the largely Coulombic nature of the interactions. In fact, with these constraints in effect, the energy functional possesses a true minimum at some positive energ, and the two remaining free parameters
Z*
are determined by the requirement that the trial function4
minimize<
H>.
The results obtained for several nuclides of interest are listed in Table I. For the purpose of comparison we have also listed Zg, the two free parameters calculated for the caseV'
= 0.For the CS wave function, the total probability per K-capture event for the shakeoff of the remaining K electron is readily calculated;/l/ the result is
where $K is the hydrogenic wave function for a K electron orbiting the daughter nucleus, the K-shell binding energies for the parent and daughter atoms are B K ( ~ ) and Bx(f), respectively, and W =
d m .
For a nucleus of uniform density, the nuclear average of $* is given by<
$* > N = ~ * 3 1 1 ~ * * / ( 3-
6 4 whereRN
is the nuclear radius.To determine
&,
the continuum wave function of the ejected electron, we assume - in the spirit of the sudden approximation - that the orbitals of the passive electrons remain frozen.In this case, the potential seen by the outgoing electron consists of a Coulomb potential due to the daughter nucleus and the same effective potential due to the L,M,
...
electrons as de- scribed above. Hence, we obtain +d by solving the Dirac equation for this potential using an angular momentum expansion and standard numerical methods. In turn, PKK is determined by numerical integration of (5).RESULTS AND DISCUSSION
Preliminary to undertaking this elaborate numerical calculation, we have found it useful to study the special case corresponding to VLM = 0. Here one may use for 4d the well known analytical form appropriate to an attractive Coulomb field. Such a calculation not only pro- vides a valuable point of reference from which to judge the more refined but heavily numerical calculation based on a SCF potential, but it also affords an opportunity to make an assessment of the effectiveness of our trial function in modeling K-K correlation and screening effects, at least to the extent that they are reflected in the resulting values of PKK. The results obtained are displayed in Table I under the heading CSO. For comparison we also have listed the previ- ous theoretical results of the semi-relativistic propagator (SRP) theory [2] and of the theory of Suzuki and Law (SL) [I] along with representative experimental data, the references for which may be found in [I] and [2].
Also of interest are the very early predictions of Law and Campbell (LC) [6] which were obtained by employing hydrogenic Dirac wave functions for both electronic states, completely neglecting all correlation and screening effects. These latter results are extremely large - much larger than the experimental data
-
and the dramatic reduction in PKK which results when the hydrogenic Dirac wave function for the initial state is replaced by the CS wave function is especially striking. Finally, we have listed in Table I under the heading CS the results for PKK obtained from implementation of the full numerical procedure described above.C9-558 JOURNAL DE PHYSIQUE
Table I. Results of various models for K-shell shakeoff probability in K-electron capture.
PKK ( X lo-')
Nuclide Z ZO Z+ Z- SRP C S O CS SL LC Expt.
"AT 16.18 19.18 15.85 19.30 25.84 28.41 27.63 52.94 115.5 44 3~ 8
37 f 9
"Mn 22.90 26.45 22.46 26.64 11.25 13.21 11.42 24.3 36 f 3
"Fe 23.86 27.49 23.41 27.68 9.42 10.95 9.04 20.06 48.36 12 f 4 10.1 =k 2.7
"Zn 27.72 31.62 27.22 31.85 8.10 6.82 15.3 22 f 2 ' l ~ e 29.65 33.69 29.13 33.93 5.08 6.24 4.62 11.84 29.83 13.3 f 1.4
13 f 5 12 ' = S r 35.45 39.87 34.86 40.15 3.38 4.58 3.14 9.38 6.0 f 0.5 loSPd 43.21 48.09 42.55 48.42 1.74 2.56 1.32 6.03 3.13 f 0.31 lo9Cd 45.16 50.14 44.48 50.48 0.34 0.368 0.19 0.89 15.2 f 2.4
2.8 f 0.7 1.02 f 0.36 It is immediately apparent from a comparison of the CS and C S O results that the inclusion of the screening potential V L ~ has only a very small effect on PKK for light atoms Iike Ar; but its influence grows rapidly with Z, accounting for almost a factor of two in the case of Cd for which Z = 48. It is also evident from the table that the CS results generally lie much closer to the results of the SRP theory than to those of the SL theory. But whether or not they represent an improvement in the theoretical situation we consider it premature to judge. It remains to be determined how sensitive the CS results are to the particular approximation used in determining VLM
-
in particular, the neglect of exchange effects - and whether they remain stable when the constraints on the variational parameters 6* andr*
are relaxed.REFERENCES
[I] SUZUKI, A. and LAW, J., Phys. Rev. C 25 (1982) 2722.
[2] INTEMANN, R.L., Phys. Rev. C 31 (1985) 1961.
[3] FRANKLIN, J. and INTEMANN, R.L., Phys. Rev. Lett. 54 (1985) 2068.
[4] See, for example, KILLINGBECK, J. Molee. Phys. 23 (1972) 907 and 23 (1972) 921.
[5] GRANT, I.P., MCKENZIE, B.J., NORRINGTON, P.H., MAYERS, D.F., and PYER, N.C., Comput. Phys. Commun. 21 (1980) 207.
[6] LAW,J. and CAMPBELL, J.L., Nucl. Phys. A 199 (1973) 481.
