250
D. G. Green and G. F. Gribakin
The positron self-energy diagrams and the annihilation amplitude contain sums
over the intermediate excited electron and positron states. In practice we calculate
them numerically using sets of electron and positron basis states constructed using
40 B-splines of order 6, in a spherical box of radius 30 a.u. We use an expotential knot sequence for the B-splines, which provides for an efficient spanning of the
electron and positron continua in the sums over intermediate states [48]. The maximum angular momentum of the intermediate states is l max =15, and we extrapolate
to l max → ∞ as in [19, 21, 48, 50].
3 Annihilation Momentum Densities for Valence and Core
Electron Orbitals in Noble Gases
Figures 3 and 4 show the AMD |A ní µí¼ (í µí°)| 2 [spherically averaged, as in Eq. (2)] for
thermal (k = 0.04 a.u.) s-wave positrons annihilating on individual core and valence
subshells of the noble gas atoms, calculated using different approximations for the
annihilation amplitude and positron wavefunction. The range of two-í µí»¾ momenta P =
0–6 a.u. corresponds to the maximum Doppler energy shift í µí¼ ≈ 11 keV.
The simplest approximation shown uses the zeroth-order (IPA) annihilation amplitude (6) with positron wavefunctions in the static field of the HF atom. Better approximations involve using the full annihilation vertex of Fig. 1 [Eq. (7)], or the best
(Dyson) positron wavefunction, or both. In general, including correlations of either
types increases the AMD and the annihilation probability.
General trends are observed throughout the noble-gas sequence. The AMD are
broader for the core orbitals for which the typical electron momenta are greater,
leading to greater Doppler shifts. The core AMD (and the core annihilation probabilities [24, 50]) also have noticeably smaller magnitudes compared with those of the
valence electrons. For all electron orbitals whose radial wavefunctions have nodes
(e.g., 2s in Ne, 2s, 3s and 3p in Ar, etc.) the AMD display deep minima related to the
nodes of the annihilation amplitude A ní µí¼ (í µí°). Their number and positions are related
to the number and positions of the nodes in the orbital’s radial wavefunction (i.e., the
radial nodes that occur closer to the nucleus result in the nodes of A ní µí¼ (í µí°) at higher
momenta). This behaviour is easy to understand from the zeroth-order amplitude (6),
which is the Fourier transform of the product of the electron and positron wavefunctions. For low positron energies, its wavefunction inside the atom decreases monotonically towards the nucleus (suppressed by the repulsive electrostatic potential at
smaller distances), and has no nodes. Hence, the nodal structure of the annihilation
amplitude is determined by the behaviour of the electron wavefunction. Inclusion of
the correlation corrections to the annihilation vertex, as described by Eq. (7), leads
only to a small shift in the positions of the nodes.
For a given approximation for the annihilation vertex, the AMD calculated using
the positron Dyson orbitals (red curves) are larger than those calculated using the
HF positron wavefunction (blue curves). The corresponding increase is nearly the
D. G. Green and G. F. Gribakin
The positron self-energy diagrams and the annihilation amplitude contain sums
over the intermediate excited electron and positron states. In practice we calculate
them numerically using sets of electron and positron basis states constructed using
40 B-splines of order 6, in a spherical box of radius 30 a.u. We use an expotential knot sequence for the B-splines, which provides for an efficient spanning of the
electron and positron continua in the sums over intermediate states [48]. The maximum angular momentum of the intermediate states is l max =15, and we extrapolate
to l max → ∞ as in [19, 21, 48, 50].
3 Annihilation Momentum Densities for Valence and Core
Electron Orbitals in Noble Gases
Figures 3 and 4 show the AMD |A ní µí¼ (í µí°)| 2 [spherically averaged, as in Eq. (2)] for
thermal (k = 0.04 a.u.) s-wave positrons annihilating on individual core and valence
subshells of the noble gas atoms, calculated using different approximations for the
annihilation amplitude and positron wavefunction. The range of two-í µí»¾ momenta P =
0–6 a.u. corresponds to the maximum Doppler energy shift í µí¼ ≈ 11 keV.
The simplest approximation shown uses the zeroth-order (IPA) annihilation amplitude (6) with positron wavefunctions in the static field of the HF atom. Better approximations involve using the full annihilation vertex of Fig. 1 [Eq. (7)], or the best
(Dyson) positron wavefunction, or both. In general, including correlations of either
types increases the AMD and the annihilation probability.
General trends are observed throughout the noble-gas sequence. The AMD are
broader for the core orbitals for which the typical electron momenta are greater,
leading to greater Doppler shifts. The core AMD (and the core annihilation probabilities [24, 50]) also have noticeably smaller magnitudes compared with those of the
valence electrons. For all electron orbitals whose radial wavefunctions have nodes
(e.g., 2s in Ne, 2s, 3s and 3p in Ar, etc.) the AMD display deep minima related to the
nodes of the annihilation amplitude A ní µí¼ (í µí°). Their number and positions are related
to the number and positions of the nodes in the orbital’s radial wavefunction (i.e., the
radial nodes that occur closer to the nucleus result in the nodes of A ní µí¼ (í µí°) at higher
momenta). This behaviour is easy to understand from the zeroth-order amplitude (6),
which is the Fourier transform of the product of the electron and positron wavefunctions. For low positron energies, its wavefunction inside the atom decreases monotonically towards the nucleus (suppressed by the repulsive electrostatic potential at
smaller distances), and has no nodes. Hence, the nodal structure of the annihilation
amplitude is determined by the behaviour of the electron wavefunction. Inclusion of
the correlation corrections to the annihilation vertex, as described by Eq. (7), leads
only to a small shift in the positions of the nodes.
For a given approximation for the annihilation vertex, the AMD calculated using
the positron Dyson orbitals (red curves) are larger than those calculated using the
HF positron wavefunction (blue curves). The corresponding increase is nearly the