INFLUENCE OF RELATIVISTIC EFFECTS ON ELECTRON-LOSS CROSS SECTIONS OF HEAVY AND SUPERHEAVY IONS COLLIDING WITH NEUTRAL ATOMS
I. Yu. Tolstikhina''1'. I. I. Tupitsyn€, S. N. Andreev", V. P. Shevelko"*
" l.i In ih r Physical Institute, Russian Academy of Sciences 119991, Moscow, Russia
b Moscow Institute of Physics and Technology 141700, Dolgoprudny, Moscow Region, Russia
'"Department of Physics, St. Petersburg State University 198504, Saint-Petersburg, Russia
Received January 10, 2014
The influence of relativistic effects, such as relativistic interaction and relativistic wave functions, on the electron-loss cross sections of heavy and superheavy atoms and ions (atomic number Z > 92) colliding with neutral atoms is investigated using a newly created RICODE-M computer program. It is found that the use of relativistic wave functions changes the electron-loss cross section values by about 20-30% around the cross-section maximum compared to those calculated with nonrelativistic wave functions. At relativistic energies E > 200 MeV/u, the relativistic interaction between colliding particles leads to a quasiconstant behavior of the loss cross sections v'el ~ const, to be compared with the Born asymptotic law <r§£ ~ luE/E.
DOI: 10.7868/S0044451014070013
1. INTRODUCTION
The relativistic effects, i.e., relativistic wave functions and relativistic interaction, already become important in atoms and ions with the nuclear charges Z > 30 (see, e.g., fl 4]) and are taken into account in calculation of the radiative atomic characteristics such as binding energies, oscillator strengths, transition probabilities, etc.
As concerns the collision properties, the influence of relativistic effects 011 excitation, radiative recombination, and electron-capture and electron-loss cross sections for heavy ions colliding with neutral atoms are discussed in various review articles and books [5 11].
Here, we investigate the influence of relativistic effects 011 the electron-loss cross sections of heavy and superheavy (Z >92) many-electron ions colliding with neutral atoms, i.e., for the reactions
Xq+ + A ^ xiq+1)+ + SA + e~, (1)
* E-mail: shevfflsd.lebedev.ru
where Xq+ denotes the incident projectile ion with the charge q and A is the target atom; EA indicates that the outgoing target atom A can be excited or ionized.
Together with another charge-changing process electron capture electron loss plays a key role in many fields of atomic, accelerator, and plasma physics: in production of long-lived ion beams in accelerators and requirements for vacuum conditions [12], ion thermonuclear fusion program [13], particle tumor therapy [14], heavy-ion probe beam (HIPB) diagnostics [15], etc.
Another problem showing the importance of charge-changing processes is closely related to nuclear physics. Recently, a detection of superheavy elements with atomic numbers up to Z = 118 in nuclear fusion evaporation reactions became possible using the gas-filled separators based on charge-state equilibrium phenomena (see, e.g., [16,17]). Properly setting the magnetic rigidity of the separators requires an accurate knowledge of ion velocity-to-charge ratio v/q, where q is the average (equilibrium) ion charge after the separator. In the atomic approach, the average charge and equilibrium charge-state fractions can be expressed in terms
of the electron-loss and capture cross sections [18] calculations of which in the case of superheavy elements require accounting for the relativistic effects.
In our previous papers devoted to the calculation of electron-loss cross sections (see, e.g., [19,20]), we used the RICODE computer program, which employs the relativistic interaction between colliding particles but nonrclativistic radial wave functions for the bound and continuum states of the projectile active electron. This code provides a reasonable agreement with experimental electron-loss cross sections for heavy many-electron positive ions (up to uranium ions) colliding with neutral atoms at ion energies 1 MeV/'u < E < 100 GeV/'u. In the case of few-electron heavy projectiles (H- and He-like ions), it was found that the influence of relativistic effects on the wave functions is strong and leads to a severalfold reduction of the electron-loss cross sections (see, e. g., [21, 22]).
The aim of this paper is to investigate the influence of the relativistic effects on the electron-loss cross sections for heavy and superheavy many-elect ron ions colliding with neutrals using a newly created RICODE-M program. It is found that in the vicinity of the cross-section maximum, the influence of the relativistic effects on the wave functions is large for neutral and low-charged atoms and ions, and at relativistic energies E > 200 MeV/'u, the influence of the relativistic ion atom interaction plays a major role and significantly changcs the cross section dependence on the collision energy compared with the Born asymptotic behavior. A description of the RICODE-M program is also given.
2. THE RICODE-M COMPUTER PROGRAM
The RICODE-M program (Relativistic Ionization CODE Modified) is created to calculate single-electron loss cross sections for reaction (1) and is based on the relativistic Born approximation [23]. The general structure of RICODE-M is similar to that of the previous RICODE program [19] but with one important difference: RICODE-M generates relativistic radial wave functions for both bound and continuous states of the projectile active electron, while RICODE applies nonrclativistic wave functions. In both codes, the relativistic (magnetic) interaction between colliding particles is used.
2.1. Core potentials, wave functions, and binding energies
In RICODE-M, the radial wave functions P(r) of the active electron in the bound and continuum states
are calculated by numerically solving the Schrodingcr-type equation with relativistic central-symmetric potential Uc(r) of the atomic core and a given energy e:
lil
2 dr2
1(1 + 1) 1
2 r2
Ur
u)
P(r)=eP(r), (2)
where the scaling factor ui is an eigenvalue of Eq. (2). The radial wave functions P(r) of the projectile active electron in the bound and continuous states are normalized as
Pn,(r)dr = 1,
OO
J P£\(r)P£>\{r) dr = 7rô(e — e')
(3)
where n and I denote the principal and orbital quantum numbers of the bound electron, and e and A are the energy and orbital quantum numbers of the ionized electron.
The core potential Uc(r) in (2) is a relativistic local potential constructed based on the density functional theory (DFT) in the local density approximation (LDA). The potential Uc(r) is created by the projectile nucleus and the rest electrons and consists of three parts:
uc(r) = Unucl (r) + Ucoui(r) + r, .,.,(/•) The nuclear potential Unuci(r) is given by
TT t \ I Pnuclir)
L'?ur/(r) = J IT^r
(4)
(5)
where expression for the nuclear density pnuci(r) depends on the nuclear model to be used in RICODE-M (a point, volume, or Fermi models); Ucoui(r) and i(/•) denote the Coulomb and exchange interaction potentials between core electrons.
For bound states, the binding energies e < 0 are found from the subroutine HFDD of the RICODE-M program by solving the relativistic Dirac Fock radial equations (see [24]):
+ Q„n(r)
dr r '
U„ucl(r)
Y„.n(r)
XQ (r)
x P (r) — r P (r) - "rK '
^ 1 nnK1 I — "-nn1 nnK1 I
Unucl(r)+^1-2c2
Xp (r)
Table 1. Calculated relativistic (srei) and non-relativistic (enonre\) binding energies of a neutral Rg atom (Z — 111) having the electronic configuration
1s22s2 ... 5d106s26/5/146d97s2. srel, enonrei — the result by the RICODE-M program, e — relativistic Hartree-Fock calculations [2]
Shell £ [2], a.u. £rei, a. u. £nonrel 1 U.
7*1/2 0.4276 0.4278 0.2441
64/2 0.4119 0.4118 0.6477
6(/3/2 0.5172 0.5171 0.6477
6i>3/2 2.2765 2.2764 2.3227
6P1/2 3.8476 3.8477 2.3227
6S1/2 5.3549 5.3564 3.8094
5/7/2 3.0226 3.0224 2.7984
5/5/2 2.7986 2.7984 2.7984
54/2 10.1982 10.1979 11.0519
5d3/2 11.2795 11.2791 11.0519
5i>3/2 17.2870 17.2865 16.6413
5^1/2 24.6863 24.6874 16.6413
5*1 /2 28.7578 28.7645 19.7148
1*1/2 6898.68 6900.22 5481.18
1 2 3 4 5 6 7 8
r, a.u.
Fig. 1. Calculated electron density P2S (r) of the Is orbital in a neutral Rg atom, Z = 111. Solid curve — the fully relativistic calculation [25]; dashed curve — the nonrelativistic wave function but the relativistic binding energy, a RICODE result; solid curve with open circles — the relativistic wave function and relativistic binding energy, a RICODE-M result; dotted curve — the fully nonrelativistic calculation
Here, c is a speed of light, k = (-l)/+j+1/2(j + 1/2) is the relativistic quantum number, nlj are the principal, orbital, and total angular quantum numbers, PnK(r) and QnK(r) are the respective large and small components of the Dirac radial wave function, YnK(r)/r is the Hartree-Coulomb potential, and /r are
the inhomogeneous parts of the differential Dirac-Fock equations due to the nonlocal exchange interaction and nondiagonal Lagrange multipliers.
For continuous states, the energies £ > 0 and the radial wave functions are found from Eq. (2) with the same relativistic core potential as for the initial bound state, i.e., the relativistic core potential Uc(r) is represented in the local form for both bound and continuum states. Although the binding energies found in RICODE-M do not completely correspond to the local potential C/c(r), the scaling factor cu corrects this small inconsistence; the calculated cj values are usually close to unity: cj ~ 1.
As an example of the calculated radial wave function, electron densities of the 7s orbital in superheavy Rg atoms (Z = 111, configuration Qd97s2) are presented in Fig. 1. The dotted line represents a fully nonrelativistic calculation, i.e., the one with the nonrelativistic binding energy and the nonrelativistic core
potential Uc(r). The dashed line is a result of the RICODE program with the relativistic binding energy but with a nonrelativistic potential C/c(r), and
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