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Relativistic calculations of charge transfer probabilities in U92+ - U91+(1s) collisions using the basis set of cubic Hermite splines

A new approach for solving the time-dependent two-center Dirac equation is presented. The method is based on using the finite basis set of cubic Hermite splines on a two-dimensional lattice. The Dirac equation is treated in rotating reference frame. The collision of U92+ (as a projectile) and U91+ (as a target) is considered at energy E_lab=6 MeV/u. The charge transfer probabilities are calculated for different values of the impact parameter. The obtained results are compared with the previous calculations [I. I. Tupitsyn et al., Phys. Rev. A 82, 042701 (2010)], where a method based on atomic-like Dirac-Sturm orbitals was employed. This work can provide a new tool for investigation of quantum electrodynamics effects in heavy-ion collisions near the supercritical regime.

In the present paper we develop an alternative approach to treatment of low-energy heavy-ion collisions. The method is based on using the cubic Hermite spline basis set on a two-dimensional uniform grid. The Dirac equation is considered in a rotating reference frame. Previously the time-dependent Schrödinger equation was treated in a rotating coordinate frame in Ref. [16]. The basis of cubic Hermite splines was used for non-relativistic time-dependent calculations [17,18] and for relativistic time-dependent calculations in the monopole approximation [19]. To our best knowledge, however, the Hermite splines have not been employed before for solving the time-dependent Dirac equation for the real two-center problem.
The theoretical methods are described in section 2. The results of our calculations of the charge-transfer probabilities for the U 92+ -U 91+ (1s) collision are presented in section 3. Atomic units ( = e = m = 1) are used throughout the paper.

Basis of the Hermite splines
In this paper a basis of cubic Hermite splines is used. This basis can be defined in the following way. The interval [a, b] is partitioned into N equal subintervals For each node x i two functions are introduced: where Since only adjacent splines overlap, the Hamiltonian and overlapping matrices are sparse. It is the significant advantage of this basis. The two-dimensional basis Hermite splines are defined as Here j = j(µ, ν, l, m) enumerates the splines φ j (x, y) in the certain way.

Solution of the time-dependent two-center Dirac equation
We consider a collision of U 92+ (target) and U 91+ (1s) (projectile). The position of the target is fixed, the projectile moves along a straight line with a constant velocity.
The electron movement is considered in the cylindrical coordinate system (ρ, z, ϕ) which is rotating around the target with the internuclear axis as z-axis. In this coordinate system the Dirac equation is written as [20] i ∂ψ ∂t =Ĥψ, where ψ is the Dirac wave function,Ĵ is the angular momentum operator, ω is the angular velocity of the reference frame, and Here α, β are the Dirac matrices, and V (r, t) is the two-center potential. In Eq. (5) we disregard some negligible terms [20].
The wave function is represented as where m is a half-integer quantum number, index k enumerates the components of ψ(ρ, z, ϕ, t) , 2M + 1 is the number of m-states under consideration, and Each component ψ k m (ρ, z, t) is expanded in the basis of cubic Hermite splines {φ i (ρ, z)} Let us introduce the new notation C j ≡ C k i,m , f j ≡ φ i Φ k m . The coefficients C j can be obtained by solving the equation Here C is the vector of the coefficients C j , H ij = f i |Ĥ|f j , S ij = f i |f j . The matrix elements S ij and H ij are calculated by numerical integration with respect to ρ and z, and analytical integration with respect to the ϕ coordinate. For sufficiently small time interval ∆t the formal solution of Eq. (11) is The exponential operator is expanded as The vector C(t) is calculated on each time step using Eq. (13) with the first four terms.

Results
The charge transfer probabilities in the U 92+ -U 91+ (1s) collisions were obtained for one (m = 1/2) and six channels with different m (see Eq. 8) at the projectile energy E lab = 6 MeV/u. The numbers of basis functions were 15360 and 92160 for one and six channels, respectively. The results are shown in Fig. 1.  Figure 1. The charge transfer probability as a function of the impact parameter. Signs "+" and "×" indicate the results of one channel and six channels calculations, respectively. The solid line corresponds to the results from Ref. [14]

Conclusion
In the present work the charge transfer probabilities in the U 92+ -U 91+ (1s) collisions were calculated for a wide range of the impact parameter. As can be seen from Fig. 1, the results of the calculations with six channels are in a good agreement with the corresponding values obtained in Ref. [14]. Since Φ m |J ω|Φ m = 0, the rotation of the internuclear axis is not taken into account in the case of the one channel calculations. As one can observe from Fig.1, the influence of this rotation is essential for the impact parameter b < 600 fm.