Enhancement Factors for Positron Annihilation on Valence . . .
255
(see, e.g., [5, 61–66]). The true spectrum for annihilation on a given subshell for a
given positron momentum can then be approximated by
w nl (í µí¼) ≈ ̄
í µí»¾ nl w
(0)
nl
(í µí¼)
(14)
where w
(0)
nl
is the í µí»¾ spectrum calculated using the zeroth-order vertex. Accurate
reconstruction of the true spectra for s-wave thermal positrons using Eq. (14) has
been demonstrated for noble-gas atoms in [50].
In a recent paper [24] we showed that for thermal s-wave positrons (k = 0.04 a.u.)
̄
í µí»¾ nl follows a near-universal scaling with the orbital ionization energy I nl ,
̄
í µí»¾ nl = 1 +
√
A∕I nl + (B∕I nl )
í µí»½
,
(15)
where A, B and í µí»½ are constants. 6 The second term on the RHS of Eq. (15) describes
the effect of the first-order correction, Fig. 1b. Its scaling with I nl was motivated
by the 1∕Z scaling for positron annihilation in hydrogen-like ions [21]. The third
term is phenomenological and describes the effect of the í µí»¤ -block correction that is
particularly important for the valence subshells.
Here we extend the calculations of the enhancement factors to s-, p- and d-wave
positrons with momenta up to the positronium-formation threshold. At small, e.g.,
room-temperature, thermal positron momenta k ∼ 0.04 a.u., the contributions of the
positron p and d waves to the annihilation rates are very small, owing to Z ef f (k) ∝ k 2í µí³
low-energy behaviour. (This is a manifestation of the suppression of the positron
wavefunction in the vicinity of the atom by the centrifugal potential í µí³(í µí³ + 1)∕2r
2 .)
However, for higher momenta close to the Ps-formation threshold, the s-, p- and dwave contributions to the annihilation rates become of comparable magnitude (see
Fig. 16 and Tables III–VII in Ref. [19]).
Figures 5–9 show the enhancement factors for positron annihilation with electrons in the valence (np and ns) and core [(n − 1)s, (n − 1)p, (n − 1)d, as applicable]
orbitals of He, Ne, Ar, Kr, and Xe, as functions of the positron momentum.
The EF for positron annihilation with 1s electrons in He (Fig. 5) are 2.6–3.0 for
the s-wave, 3.8–4.1 for the p-wave, and 5.2–5.9 for the d-wave. They show only
a weak dependence on the positron momentum, which is a typical feature of all
the data. There is also little difference between the EF obtained with the staticfield (HF) positron wavefunctions (dashed lines) and those found using the positron
Dyson orbitals (solid lines). This is in spite of the fact that the use of the correlated
Dyson positron wavefunctions increases the AMD (and the annihilation rates [19])
by almost an order of magnitude for s-wave positrons (Fig. 3).
The weak dependence of the EF on the positron energy and the type of positron
wavefunction used is related to the nature of the vertex enhancement. The intermediate electron and positron states in diagrams Fig. 1b and c that describe the
6 For HF positron wavefunctions the values of the parameters are A = 1.54 a.u. = 42.0 eV, B =
0.92 a.u. = 24.9 eV, and í µí»½ = 2.54. For Dyson positron wavefunctions the values are A = 1.31 a.u. =
35.7 eV, B = 0.83 a.u. = 22.7 eV, and í µí»½ = 2.15 [24].
255
(see, e.g., [5, 61–66]). The true spectrum for annihilation on a given subshell for a
given positron momentum can then be approximated by
w nl (í µí¼) ≈ ̄
í µí»¾ nl w
(0)
nl
(í µí¼)
(14)
where w
(0)
nl
is the í µí»¾ spectrum calculated using the zeroth-order vertex. Accurate
reconstruction of the true spectra for s-wave thermal positrons using Eq. (14) has
been demonstrated for noble-gas atoms in [50].
In a recent paper [24] we showed that for thermal s-wave positrons (k = 0.04 a.u.)
̄
í µí»¾ nl follows a near-universal scaling with the orbital ionization energy I nl ,
̄
í µí»¾ nl = 1 +
√
A∕I nl + (B∕I nl )
í µí»½
,
(15)
where A, B and í µí»½ are constants. 6 The second term on the RHS of Eq. (15) describes
the effect of the first-order correction, Fig. 1b. Its scaling with I nl was motivated
by the 1∕Z scaling for positron annihilation in hydrogen-like ions [21]. The third
term is phenomenological and describes the effect of the í µí»¤ -block correction that is
particularly important for the valence subshells.
Here we extend the calculations of the enhancement factors to s-, p- and d-wave
positrons with momenta up to the positronium-formation threshold. At small, e.g.,
room-temperature, thermal positron momenta k ∼ 0.04 a.u., the contributions of the
positron p and d waves to the annihilation rates are very small, owing to Z ef f (k) ∝ k 2í µí³
low-energy behaviour. (This is a manifestation of the suppression of the positron
wavefunction in the vicinity of the atom by the centrifugal potential í µí³(í µí³ + 1)∕2r
2 .)
However, for higher momenta close to the Ps-formation threshold, the s-, p- and dwave contributions to the annihilation rates become of comparable magnitude (see
Fig. 16 and Tables III–VII in Ref. [19]).
Figures 5–9 show the enhancement factors for positron annihilation with electrons in the valence (np and ns) and core [(n − 1)s, (n − 1)p, (n − 1)d, as applicable]
orbitals of He, Ne, Ar, Kr, and Xe, as functions of the positron momentum.
The EF for positron annihilation with 1s electrons in He (Fig. 5) are 2.6–3.0 for
the s-wave, 3.8–4.1 for the p-wave, and 5.2–5.9 for the d-wave. They show only
a weak dependence on the positron momentum, which is a typical feature of all
the data. There is also little difference between the EF obtained with the staticfield (HF) positron wavefunctions (dashed lines) and those found using the positron
Dyson orbitals (solid lines). This is in spite of the fact that the use of the correlated
Dyson positron wavefunctions increases the AMD (and the annihilation rates [19])
by almost an order of magnitude for s-wave positrons (Fig. 3).
The weak dependence of the EF on the positron energy and the type of positron
wavefunction used is related to the nature of the vertex enhancement. The intermediate electron and positron states in diagrams Fig. 1b and c that describe the
6 For HF positron wavefunctions the values of the parameters are A = 1.54 a.u. = 42.0 eV, B =
0.92 a.u. = 24.9 eV, and í µí»½ = 2.54. For Dyson positron wavefunctions the values are A = 1.31 a.u. =
35.7 eV, B = 0.83 a.u. = 22.7 eV, and í µí»½ = 2.15 [24].