Enhancement Factors for Positron Annihilation on Valence . . .
247
underestimates the annihilation rates. Much more accurate positron wavefunctions
(Dyson orbitals) are obtained by solving the Dyson equation which includes the nonlocal, energy-dependent positron-atom correlation potential [19, 48] (Sect. 2.2.2).
In the lowest-order approximation the annihilation amplitude is given by
A ní µí°¤ (í µí°) = ∫
e
−ií µí°⋅í µí°«
í µí¼ í µí°¤ (í µí°«)í µí¼ n (í µí°«)d
3
í µí°«,
(6)
where í µí¼ n (í µí°«) ≡ í µí¼ nlm (í µí°«) =
1
r
P nl (r)Y lm (̂ í µí°«) is the wavefunction of electron in subshell
nl. Equation (6) is equivalent to IPA. After integration over the directions of í µí° in the
spectrum (2), all positron partial waves contribute to the AMD incoherently. This
means that the annihilation amplitude can be calculated independently for each í µí³,
replacing í µí¼ í µí°¤ (í µí°«) in Eq. (6) by the corresponding positron partial wave orbital í µí¼ í µí¼ (í µí°«).
Omitting the index í µí³, we denote such amplitude A ní µí¼ (í µí°).
As described below, the main corrections to the zeroth-order amplitude originate
from the electron-positron Coulomb interaction which increases the probability of
finding the electron and positron at the same point in space.
2.2.1 The Annihilation Vertex
Figure 1 shows the amplitude A ní µí¼ (í µí°) in diagrammatic form [20, 21, 24, 49]. 4 The
total amplitude is depicted on left-hand side of the diagrammatic equation, with
the double line (í µí¼) corresponding to incoming positron that annihilates an electron in orbital n, producing two í µí»¾-rays (double-dashed line), and the circle with a
cross denoting the full annihilation vertex. The main contributions to the amplitude
are shown on the right-hand side of the equation. Diagram (a) is the zeroth-order
amplitude [IPA, Eq. (6)], diagram (b) is the first-order correction and diagram (c)
is the nonperturbative ‘virtual-positronium’ correction. This correction contains the
shaded ‘í µí»¤ -block’ which represents the sum of an infinite series of electron-positron
ladder diagrams shown in the lower part of Fig. 1.
The ladder diagrams represented by the í µí»¤ -block are important because the
electron-positron Coulomb attraction supports bound states of the positronium (Ps)
atom. To form Ps, the energy of the incident positron needs to be greater than the
Ps-formation threshold E Ps = I − 6.8 eV, where I is the ionization potential of the
atom. However, even at lower energies where the Ps can only be formed virtually, this
process gives a noticeable contribution. In practice, the í µí»¤ -block is found from the
linear equation í µí»¤ = V + Ví µí¼í µí»¤ , shown diagrammatically in the lower part of Fig. 1,
where V is the electron-positron Coulomb interaction and í µí¼ is the propagator of the
intermediate electron-positron state. Discretizing the electron and positron continua
by confining the system in a spherical cavity reduces this to a linear matrix equation,
which is easily solved numerically (see [19, 21, 48, 50] for further details).
4 It is also possible to develop a diagrammatic expansion for Z ef f [19, 20, 44, 45, 48] that enables
one to calculate the annihilation rate directly, rather than from Eq. (4).
247
underestimates the annihilation rates. Much more accurate positron wavefunctions
(Dyson orbitals) are obtained by solving the Dyson equation which includes the nonlocal, energy-dependent positron-atom correlation potential [19, 48] (Sect. 2.2.2).
In the lowest-order approximation the annihilation amplitude is given by
A ní µí°¤ (í µí°) = ∫
e
−ií µí°⋅í µí°«
í µí¼ í µí°¤ (í µí°«)í µí¼ n (í µí°«)d
3
í µí°«,
(6)
where í µí¼ n (í µí°«) ≡ í µí¼ nlm (í µí°«) =
1
r
P nl (r)Y lm (̂ í µí°«) is the wavefunction of electron in subshell
nl. Equation (6) is equivalent to IPA. After integration over the directions of í µí° in the
spectrum (2), all positron partial waves contribute to the AMD incoherently. This
means that the annihilation amplitude can be calculated independently for each í µí³,
replacing í µí¼ í µí°¤ (í µí°«) in Eq. (6) by the corresponding positron partial wave orbital í µí¼ í µí¼ (í µí°«).
Omitting the index í µí³, we denote such amplitude A ní µí¼ (í µí°).
As described below, the main corrections to the zeroth-order amplitude originate
from the electron-positron Coulomb interaction which increases the probability of
finding the electron and positron at the same point in space.
2.2.1 The Annihilation Vertex
Figure 1 shows the amplitude A ní µí¼ (í µí°) in diagrammatic form [20, 21, 24, 49]. 4 The
total amplitude is depicted on left-hand side of the diagrammatic equation, with
the double line (í µí¼) corresponding to incoming positron that annihilates an electron in orbital n, producing two í µí»¾-rays (double-dashed line), and the circle with a
cross denoting the full annihilation vertex. The main contributions to the amplitude
are shown on the right-hand side of the equation. Diagram (a) is the zeroth-order
amplitude [IPA, Eq. (6)], diagram (b) is the first-order correction and diagram (c)
is the nonperturbative ‘virtual-positronium’ correction. This correction contains the
shaded ‘í µí»¤ -block’ which represents the sum of an infinite series of electron-positron
ladder diagrams shown in the lower part of Fig. 1.
The ladder diagrams represented by the í µí»¤ -block are important because the
electron-positron Coulomb attraction supports bound states of the positronium (Ps)
atom. To form Ps, the energy of the incident positron needs to be greater than the
Ps-formation threshold E Ps = I − 6.8 eV, where I is the ionization potential of the
atom. However, even at lower energies where the Ps can only be formed virtually, this
process gives a noticeable contribution. In practice, the í µí»¤ -block is found from the
linear equation í µí»¤ = V + Ví µí¼í µí»¤ , shown diagrammatically in the lower part of Fig. 1,
where V is the electron-positron Coulomb interaction and í µí¼ is the propagator of the
intermediate electron-positron state. Discretizing the electron and positron continua
by confining the system in a spherical cavity reduces this to a linear matrix equation,
which is easily solved numerically (see [19, 21, 48, 50] for further details).
4 It is also possible to develop a diagrammatic expansion for Z ef f [19, 20, 44, 45, 48] that enables
one to calculate the annihilation rate directly, rather than from Eq. (4).