Enhancement Factors for Positron Annihilation on Valence . . .
253
same for the valence and core orbitals in a given atom. It ranges from a factor of
∼2 in Helium to ∼100 in Xe, and is only weakly dependent on P. (Note that the
AMD are plotted on the logarithmic scale.) This increase is due to the action of the
attractive correlation potential ̂
í µí»´ í µí¼ on the positron (Dyson) wavefunction. It leads to
a build-up of the positron density in the vicinity of the atom and greater overlap with
the atomic electron density. This effect is stronger for the heavier, more polarizable
atoms, leading to much greater Z ef f values. In Ar, Kr and Xe the attractive positronatom potential supports low-lying virtual s levels, leading to a characteristic resonant
growth of the annihilation rates at low positron energies [19, 44, 45, 56].
When the vertex corrections are included in the annihilation amplitude (solid
curves), the AMD is enhanced above the zeroth-order (IPA) result (dashed curves).
This is due to the Coulomb attraction within the annihilating pair, which increases
the probability of finding the electron and positron at the same point in space. The
size of the enhancement is similar for the HF and Dyson positron wavefunctions. At
the same time, the enhancement is much greater for the valence electrons than for the
core electrons, as the former are more easily perturbed by the positron’s Coulomb
field. For the core electrons, the vertex correction is dominated by the first-order
diagram Fig. 1b (similar to the case of hydrogen-like ions [21]). For the valence
electrons, the nonperturbative í µí»¤ -block contribution Fig. 1c is also very important.
From Figs. 3 and 4 one can also see that the vertex enhancement is significantly
stronger at low momenta P of the electron-positron pair (which leads to narrowing
of the í µí»¾-ray spectra in comparison with those obtained with the zeroth-order amplitude [20, 24]). This can be seen most clearly in the AMD of the valence electrons.
Their high-P content is due to positron annihilation with the electron when the latter
is closer to the nucleus, and where its local velocity is higher, making it less susceptible to the positron’s attraction. (Note that for large P the calculated í µí»¤ -block vertex
corrections contain numerical errors which manifest themselves as extra oscillations
visible in AMD for valence electrons. This, however, has a negligible effect on the
annihilation spectra, since the AMD for the valence electrons at such momenta are
very small.) The vertex enhancement is considered in more detail below.
We conclude this section by noting that calculations of the corresponding í µí»¾ spectra were reported in [24, 50]. They showed excellent agreement with the measured
spectra for Ar, Kr and Xe [3], and firmly established the fraction of core annihilation
for these atoms.
4 Vertex Enhancement Factors
The enhancement of the AMD and annihilation rates due to the correlation corrections to the vertex with respect to those obtained using the zeroth-order (IPA)
approximation [cf. Eqs. (7) and (6)] can be parameterized through so-called vertex
enhancement factors. The MBT enables a direct ab initio calculation of ‘exact’ EF:
one simply compares the results obtained using the annihilation amplitude calculated
in different approximations, as described here.
253
same for the valence and core orbitals in a given atom. It ranges from a factor of
∼2 in Helium to ∼100 in Xe, and is only weakly dependent on P. (Note that the
AMD are plotted on the logarithmic scale.) This increase is due to the action of the
attractive correlation potential ̂
í µí»´ í µí¼ on the positron (Dyson) wavefunction. It leads to
a build-up of the positron density in the vicinity of the atom and greater overlap with
the atomic electron density. This effect is stronger for the heavier, more polarizable
atoms, leading to much greater Z ef f values. In Ar, Kr and Xe the attractive positronatom potential supports low-lying virtual s levels, leading to a characteristic resonant
growth of the annihilation rates at low positron energies [19, 44, 45, 56].
When the vertex corrections are included in the annihilation amplitude (solid
curves), the AMD is enhanced above the zeroth-order (IPA) result (dashed curves).
This is due to the Coulomb attraction within the annihilating pair, which increases
the probability of finding the electron and positron at the same point in space. The
size of the enhancement is similar for the HF and Dyson positron wavefunctions. At
the same time, the enhancement is much greater for the valence electrons than for the
core electrons, as the former are more easily perturbed by the positron’s Coulomb
field. For the core electrons, the vertex correction is dominated by the first-order
diagram Fig. 1b (similar to the case of hydrogen-like ions [21]). For the valence
electrons, the nonperturbative í µí»¤ -block contribution Fig. 1c is also very important.
From Figs. 3 and 4 one can also see that the vertex enhancement is significantly
stronger at low momenta P of the electron-positron pair (which leads to narrowing
of the í µí»¾-ray spectra in comparison with those obtained with the zeroth-order amplitude [20, 24]). This can be seen most clearly in the AMD of the valence electrons.
Their high-P content is due to positron annihilation with the electron when the latter
is closer to the nucleus, and where its local velocity is higher, making it less susceptible to the positron’s attraction. (Note that for large P the calculated í µí»¤ -block vertex
corrections contain numerical errors which manifest themselves as extra oscillations
visible in AMD for valence electrons. This, however, has a negligible effect on the
annihilation spectra, since the AMD for the valence electrons at such momenta are
very small.) The vertex enhancement is considered in more detail below.
We conclude this section by noting that calculations of the corresponding í µí»¾ spectra were reported in [24, 50]. They showed excellent agreement with the measured
spectra for Ar, Kr and Xe [3], and firmly established the fraction of core annihilation
for these atoms.
4 Vertex Enhancement Factors
The enhancement of the AMD and annihilation rates due to the correlation corrections to the vertex with respect to those obtained using the zeroth-order (IPA)
approximation [cf. Eqs. (7) and (6)] can be parameterized through so-called vertex
enhancement factors. The MBT enables a direct ab initio calculation of ‘exact’ EF:
one simply compares the results obtained using the annihilation amplitude calculated
in different approximations, as described here.