Numerical treatment of the time-dependent Dirac equation in momentum space for atomic processes in relativistic heavy-ion collisions

A spectral method for half-integer spin fields based on spin-weighted spherical harmonics

“…This will allow for a test of the spin-weighted spectral transformations we present in the context of evolution equations. Common techniques for treating the numerical problem of the Dirac equation consist of: FD schemes formulated on a flat-lattice in configuration space [12,13], on a grid within a finite-volume in momentumspace [22] and using methods based on the split-step operator technique [7,8,20,21]. Particular to the FD approach special care must be taken so as to avoid the Fermion-doubling problem [24].…”

Electron Capture and Ionization of Pb Ions at 33 TeV

“…A theoretical description of electron capture and ionization processes has been challenging in this regime because the interaction of high-Z projectile and target species (where Za ϳ 0.5) is strong enough at small impact parameters and large g to potentially invalidate perturbation treatments. Numerous methods for treating these processes using quantum electrodynamics (QED) in the ultrarelativistic regime now exist [1][2][3][4][5][6][7][8][9][10].An ultrarelativistic ion can capture an electron via three mechanisms: (i) radiative electron capture (REC), (ii) nonradiative capture (NRC), and (iii) electron capture via e 1 e 2 pair production (ECPP), in which the e 1 e 2 pair is produced by the intense electromagnetic pulse that arises when the projectile ion passes near a target nucleus. Capture cross sections s REC , s NRC , and s ECPP scale roughly as ϳZ t ͞g, ϳZ 5 t ͞g, and ϳZ 2 t ln g, respectively, where Z t is the target atomic number [2].…”

“…A theoretical description of electron capture and ionization processes has been challenging in this regime because the interaction of high-Z projectile and target species (where Za ϳ 0.5) is strong enough at small impact parameters and large g to potentially invalidate perturbation treatments. Numerous methods for treating these processes using quantum electrodynamics (QED) in the ultrarelativistic regime now exist [1][2][3][4][5][6][7][8][9][10].…”

Staggered grid leap-frog scheme for the Dirac equation

“…Existing methods are the real-space schemes, such as the finite-difference and finite-element methods [22,23,27]. Momentum-space spectral methods and split-operator methods have been developed also [28][29][30][31]. While finite-difference and finiteelement schemes allow for an easy implementation of nonconstant coefficients they have to deal with the fermion doubling problem.…”