Relativistic calculations of the K-K charge transfer and K-vacancy production probabilities in low-energy ion-atom collisions
The previously developed technique for evaluation of charge-transfer and electron-excitation processes in low-energy heavy-ion collisions [I.I. Tupitsyn et al., Phys. Rev. A 82, 042701(2010)] is extended to collisions of ions with neutral atoms. The method employs the active electron approximation, in which only the active electron participates in the charge transfer and excitation processes while the passive electrons provide the screening DFT potential. The time-dependent Dirac wave function of the active electron is represented as a linear combination of atomic-like Dirac-Fock-Sturm orbitals, localized at the ions (atoms). The screening DFT potential is calculated using the overlapping densities of each ions (atoms), derived from the atomic orbitals of the passive electrons. The atomic orbitals are generated by solving numerically the one-center Dirac-Fock and Dirac-Fock-Sturm equations by means of a finite-difference approach with the potential taken as the sum of the exact reference ion (atom) Dirac-Fock potential and of the Coulomb potential from the other ion within the monopole approximation. The method developed is used to calculate the K-K charge transfer and K-vacancy production probabilties for the Ne – F collisions at the F projectile energies keV/u and keV/u. The obtained results are compared with experimental data and other theoretical calculations. The K-K charge transfer and K-vacancy production probabilities are also calculated for the Xe – Xe collision.
pacs:34.10.+x, 34.50.-s, 34.70.+e
Collisions of highly charged ions provide a unique tool for tests of relativistic and quantum electrodynamics (QED) effects in the scattering theory (1); (2); (3). Investigations of such processes can also give an access to QED in supercritical fields, provided the total charge of the colliding nuclei is larger than the critical one, (see, e.g., Ref. (4) and references therein). One of the most attractive way for indirect observation of the supercritical field created in the collision is to investigate the dynamics of inner-shell electrons that can be rather sensitive to the collision parameters. The most favorable conditions for studying the electron dynamics in the supercritical field regime correspond to the projectile energy of about the Coulomb barrier (5). In case of UU collision this requires the energy of about MeV/u that means a low-energy collision. One of the key processes in such collisions is the charge transfer of electrons (see, e.g., Ref. (6) and references therein). A systematic approach to relativistic calculations of the charge-transfer and electron-excitation probabilities in low-energy heavy-ion collisions was developed in our previous paper (7), where the consideration was restricted to collisions of H-like ions with bare nuclei. Since the experimental study of such collisions for high- systems is presently rather problematic, an extention of the method to collisions of highly charged ions with neutral atoms, that can be studied in experiments with the current GSI and future FAIR facilities (8); (9), is needed. In this paper we present the desired extention and perform calculations for low-energy ion-atom collisions. To examine the approach we calculate the K-K charge transfer and K-vacancy production probabilities for low-energy collision of H-like F (F) and neutral Ne, the process which has been investigated both experimentally (10) and theoretically (11); (12); (13). The calculations are performed at the F projectile energies keV/u and keV/u. We also evaluate the probabilities of the K-K charge transfer and K-vacancy production in the Xe – Xe collision at the projectile energy of MeV/u. The latter processes are planned to be studied in the nearest future in experiments at GSI (8); (9).
The paper is organized as follows. In the section II.1 we describe the time-dependent one-electron equation in so-called active electron approximation (14) and the method for constructing the local Kohn-Sham potential induced by the passive electrons. The wave function of the active electron is expanded in terms of the Dirac-Fock and Dirac-Fock-Sturm basis functions, which are central-field 4-component Dirac bispinors centered at the ions. The two-center relativistic Kohn-Sham equation in the finite basis set is briefly discussed in the section II.2. The basis functions are obtained by solving numerically the atomic Dirac-Fock and Dirac-Fock-Sturm equations in an external field with a special choice of the weight function, as it was proposed in Refs. (15); (16). The external potential is spherically symmetric Coulomb-Hartree potential of the other ion (atom) taken in the monopole approximation. The basis set constructed in this way and the related calculation procedures are described in section II.3. Basic formulas for the K-vacancy production probability are given in section II.4. In section II.5 we present the results of the relativistic calculations of the Ne K-shell-vacancy production and K-K charge transfer probabilities in the Ne – F collisions as a function of the impact parameter at the projectile energies keV/u and keV/u. In this section we also present the results of the neutral Xe K-shell-vacancy production and K-K charge transfer probabilities in the Xe – Xe collision.
ii.1 Dirac-Kohn-Sham equation in the active electron approximation
In this paper we use so-called active electron approximation (14) to describe the ion-atom collision. In this approximation, we consider only the active electron which participates in the charge transfer and excitation processes, while the other passive electrons provide a screening potential. In our calculations the screening potential is defined by the density functional theory (DFT) in the local density approximation (LDA). In this approach the time-dependent wavefunction of the active electron is the solution of the relativistic time-dependent Kohn-Sham equation. In atomic units (), this equation is given by
Here is the two-center Dirac-Kohn-Sham Hamiltonian defined by
where is the speed of light and , are the Dirac matrices. and are the Hartree and exchange-correlation potentials, respectively. Both of them are the functionals of the electron density . The Hartree potential includes the electron-nucleus interaction and the electron-electron Coulomb repulsion :
and are the nuclear and electron densities, respectively. The exchange-correlation potential was taken in the Perdew-Zunger parametrization (17) including the self-interaction correction (SIC).
The electron density , obtained with the many-electron wave function which is represented by a Slater determinant, is invariant with respect to the rotations in the occupied orbitals space. For this reason, to obtain the electron density we can use atomic-like orbitals, localized on the both centers ( and ). In this case the electron density, constructed from the orthogonal localized orbitals, can be represented as a sum of densities and which are localized on the centers and . This is not the case, however, if the orbitals localized on the different centers overlap and are non-orthogonal. The electron density, derived from non-orthogonal orbitals, is given by
where are the atomic-like wavefunctions of the passive electrons and matrix is the overlapping matrix. Note that the electron density is normalized on the number of the passive electrons,
where is the total number of electrons. The electron density can be divided into three parts,
if we split the summation over indices into the sum over , the sum over and the remaining overlapping part. We can also split the overlapping density into two parts dividing the space into two regions and ,
This can be done by the plane passing through the middle of the internuclear distance (see Ref. (7) for details).
For simplicity, let us consider spherically-average values of the electron densities in each region,
This procedure does not change the normalization of the total electron density,
As a result, the potential can be approximated by the sum of the spherically symmetric potentials of the two different centers,
The overlapping densities must be taken into account especially for the short internuclear distances. Otherwise, the Pauli principle is violated and, as a result, the number of electrons on the shell of the united system can exceed .
It should also be noted that the time-dependent wave function of the active electron is orthogonalized to the wave functions of the passive electrons. This means that the transitions of the active electron to the states occupied by the passive electrons are forbidden in accordance with the Pauli principle.
ii.2 Two-center Dirac-Kohn-Sham equation
The two-center expansion of the time-dependent wave function can be written in the form
where index labels the centers, index enumerates basis functions at the given center, and is the central-field bispinor centered at the point . In what follows, the shorthand notations for states are used. The expansion coefficients of the time-dependent wave function can be obtained by solving the linear system of first-order differential equations
where indices and enumerate the basis functions of both centers, and the matrix elements of and are
The matrix elements of are given by
Obviously the matrix is non-Hermitian, if the overlapping matrix depends on time.
The functions depend on time due to two reasons. First, the basis functions centered at the target and projectile nuclei move together with the nuclei. Second, the basis functions depend parametrically on the distance between the nuclei, since their radial parts are obtained from the radial equations, where for each center the potential of the other ion (atom) is included in the so-called monopole approximation (see section II.3).
Calculations of the matrix elements , , and were considered in detail in Ref. (7).
ii.3 Basis functions
In our approach the basis set contains Dirac-Fock and Dirac-Fock-Sturm orbitals. The Dirac-Fock-Sturm orbitals can be considered as pseudo-states, which should be included in the basis to take into account the contribution of the positive- and negative-energy Dirac continuum. Both types of basis functions are central field Dirac bispinors centered at the position () of the corresponding ion,
where and are the large and small radial components, respectively, and is the relativistic angular quantum number. The large and small radial components are obtained by solving numerically the Dirac-Fock and Dirac-Fock-Sturm equations in the central field approximation. The radial Dirac-Fock equation is
where is the radial Dirac-Fock Hamiltonian of ion (), is the two-component radial wave function, and is a local external potential. The explicit form of the radial Dirac-Fock equation and the description of the corresponding computer code are presented in Ref. (18). The radial components of the Dirac-Fock-Sturm orbitals , which we denote by , are the solutions of the generalized Dirac-Fock-Sturm eigenvalue problem,
Here can be considered as the eigenvalue of the Dirac-Fock-Sturm operator and is a constant sign weight function. The energy is fixed in the Dirac-Fock-Sturm equation. If the weight function at , all Sturmian functions have the same asymptotic behavior at . It is clear that for the Sturmian function coincides with the reference Dirac-Fock orbital . In our calculations we use the following weight function
In contrast to , this weight function is regular at the origin. It is well-known that the Sturmian operator is Hermitian. It does not contain continuum spectra, in contrast to the Dirac operator. Therefore, the set of the Sturmian eigenfunctions forms a discrete and complete basis set of one-electron wave functions.
The external central-field potential in equations (17) and (18) is arbitrary, and, therefore, it can be chosen to provide most appropriate Dirac-Fock and Dirac-Fock-Sturm basis orbitals. At small internuclear distances the wave function of the atomic electron experiences also the strong Coulomb field of the other ion (atom). To the leading order this effect can be taken into account by including the Coulomb-Hartree potential of the second ion (atom) as the external potential within the so-called monopole approximation. For instance, the external central-field potential is given by
where is the spherically-symmetric part of the reexpansion of the Coulomb-Hartree potential of the ion with respect to the center and is the internuclear distance vector.
Calculations of the two-center integrals with the basis functions employed were described in detail in our previous paper (7). The time-dependent Dirac-Kohn-Sham equation for the active electron is solved using the two-center basis set expansion. The expansion coefficients are determined employing the direct evolution (exponential) operator method (7), which is more stable compared to the others, such as, e.g., the Crank-Nicholsen propagation scheme (19) and the split-operator method (20). To obtain the matrix representation of the exponential operator in the finite basis set one has to diagonalize the generalized complex Hamiltonian matrix at each time step. Since our basis set is not too large, the diagonalization procedure is not too time consuming.
ii.4 Charge-transfer and vacancy-production probabilities
The amplitudes of the charge transfer and excitations to different bound states of the projectile and target ions are calculated by projecting the time-dependent wave function of the active electron onto the atomic Dirac-Fock orbitals of the projectile and target. The corresponding calculations for collisions of H-like ions with bare nuclei were described in detail in our work (7). That is why here we restrict our consideration only to the new features of the calculation procedure that occur for atom-ion collisions within the one active electron approximation.
Consider the collision of a neutral atom (target) with a hydrogenlike ion (projectile). We assume that before the collision the active electron occupies the state of the target with spin up (in the relativistic case, with the total angular moment projection ) and the passive electron of the hydrogenlike ion occupies the state with spin down. In what follows, we are interested in two processes: the K-K charge transfer and the K-shell vacancy production.
Let is the probability to find the active electron in the state of the target after the collision, denotes the probability to find one active electron in the state of the projectile or, in other words, the probability of the K-K shell charge transfer of one electron. To obtain the probabilty of the K-shell vacancy production, we introduce the probabilities and of the events and , when a hole is created in the state of the target with spin up and down, respectively. The probabilities of these events are defined by
Assuming the events and are independent, the probability of production at least one hole in the state of the target is given by
We note that, since the sum of the probabilities and is less than 1, the vacancy production probability satisfies the condition . It should also be noted that the K-shell vacancy production defined by Eq. (22) includes the production of two holes in the target K shell.
ii.5 Results of the calculations and discussion
To test the approach we have studied the Ne – F collision for low energies where experimental and nonrelativistic theoretical results are available (11); (10); (12); (13). In this case the nuclear charge numbers are rather small and, therefore, relativistic effects are negligible. We stress, however, that our approach can be directly applied to heavier systems where the relativistic effects become stronger or even dominant.
In Fig. 1 we present the results of our calculations for the probabilities of the Ne K-shell-vacancy production (solid line) and of the K-K-shell charge transfer (dotted line) as functions of the impact parameter for the Ne-F collision at the projectile energy of keV/u. The results for the Ne K-vacancy production are compared with experimental values (circles) (10) and with theoretical results obtained by Fritsch and Lin (dashed line) (11) and by Thies et al. (dash-dotted line) (13). It can be seen that our results are in perfect agreement with the experimental ones.
The related results for the Ne-F collision at the projectile energy of keV/u are presented in Fig. 2, where the same notations as in Fig. 1 are used. We note that our theoretical results are in a good agreement with the experimental ones at small impact parameters. However, in contrast to Fig. 1, the agreement is not so good for medium and large impact parameters, although the theoretical predictions for the maximum and minimum positions agree rather well with the experimental ones.
As one can see from Figs. 1 and 2, for both energies at large impact parameters the K-vacancy production is mainly determined by the K-K-shell charge transfer, which is indicated by the dotted line. The difference between the K-vacancy production and the K-K-shell transfer probabilities at small impact parameters is due to the contribution from the charge-transfer excitation into the , , and higher vacant states of the projectile. This is in accordance with the experimental results of Ref. (21), where the K-vacancy production in the Ne – F collision was studied. The calculation of the latter process is currently under way and will be published elsewhere.
In this work we also performed the related calculations for the Xe – Xe collision at the projectile energy of MeV/u. The experimental study of this process is planned at GSI (Darmstadt) (8); (9). The probabilities of the Xe K-shell-vacancy production and of the K-K-shell charge transfer as functions of the impact parameter are plotted in Fig. 3. The solid and dotted lines represent the vacancy production and the charge transfer, respectively. For comparison, in the same figure we display the K-shell-vacancy production for the Xe – Xe collision that is indicated by the dashed line. We note again that at large impact parameters the K-vacancy production is almost completely determined by the K-K-shell charge transfer. It can be also seen that in the case under consideration the screening effect is rather small.
To investigate the role of the relativistic effects we performed the same calculations for the Xe-Xe collision in the nonrelativistic limit by multiplying the standard value of the speed of light by the factor 1000. The obtained relativistic and nonrelativistic results are presented in Fig. 4. As one can see from the figure, the oscillatory behavior of both curves is the same but the nonrelativistic curve is shifted toward higher impact parameters.
In this paper the method that was previously developed for evaluation of the electron-excitation and charge-transfer processes in collisions of H-like ions with bare nuclei has been extended to collisions of ions with neutral atoms. The extention is based on the active electron approximation, in which the interaction of the active electron with the passive electrons is accounted for by the screening DFT potential.
The method developed has been applied to evaluate the K-vacancy production and the K-K charge transfer in the low-energy Ne – F collision. The results of the calculation are compared with available experimental data and with theoretical calculations by other authors. The influence of the relativistic effects on the K-vacancy production probability is investigated for the Xe – Xe collision. It is demonstrated that the relativistic and nonrelativistic probabilities as functions of the impact parameter exhibit the same oscillatory behavior at low energies but the relativistic curves are shifted toward lower impact parameters compared to the nonrelativistic ones.
In our further investigation we plan to continue calculations of low-energy heavy-ion collisions that are of interest for current and nearest future experiments at GSI and FAIR facilities in Darmstadt. Special attention will be paid to the critical regime, when the ground-state level of the united quasimolecule dives into the negative-energy Dirac continuum.
This work was supported by DFG (Grants No. PL 254/7-1 and VO 1707/1-1), by RFBR (Grants No. 10-02-00450 and No. 11-02-00943-a), by GSI, by DAAD, by the Ministry of Education and Science of Russian Federation (Grant No. P1334). The work of A.I.B. and I.A.M. was also supported by the Dynasty foundation. Y.S.K. and A.I.B. acknowledge financial support by the FAIR–Russia Research Center.
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