Download Positron annihilation spectra and core

Positron annihilation spectra and core-electron enhancement factors
D. G. Green∗ and G. F. Gribakin†
Department of Applied Mathematics and Theoretical Physics,
Queen’s University Belfast, Belfast, BT7 1NN, Northern Ireland, United Kingdom
(Dated: June 15, 2014)
γ-spectra for positron annihilation with core and valence electrons in the noble gases are calculated
using many-body theory (MBT). We show that proper inclusion of core-annihilation is crucial to
accurately describe the measured spectra [Phys. Rev. Lett. 79 39 (1997)]. We use the MBT to calculate ‘exact’ enhancement factors γ̄n` for annihilation on individual n` subshells. They parameterize
the important effects of (non-local) short-range electron-positron correlations, including the nonperturbative process of virtual positronium formation. We show that theypfollow a simple and physically motivated scaling with the subshell ionization energy In` : γ̄n` = 1+ A/In` +(B/In` )β , where
A, B and β are positive constants. We suggest that this formula can be used with relatively simple
independent-particle-approximation calculations to determine accurate core-annihilation spectra for
atoms across the periodic table and in condensed matter.
PACS numbers: 78.70.Bj, 34.80.Pa, 34.80.-i, 34.8.Uv
Low-energy positrons annihilate predominantly on the
outermost valence electrons in atoms, molecules, and
condensed matter [2–5]. Small fractions of positrons
can, however, tunnel through the repulsive nuclear potential and annihilate on the core electrons. The γrays produced in the annihilation events have a Dopplerbroadened energy spectrum that is characteristic of
the electronic states involved, e.g., annihilation on the
tightly-bound core electrons results in a distinct signature in the high-energy region of the spectrum. The core
signal in the spectra can be discerned [1, 6, 7], and the
resulting elemental specificity can be exploited to, e.g.,
identify defects in semiconductors (see e.g, [5], and references therein). Positron annihilation on core electrons
is now a key process in several important experimental techniques, e.g., the surface analytic positron-induced
Auger-electron spectroscopy (PAES) (see e.g., [7–13]),
and the novel time-resolved PAES at the new NEPOMUC reactor [14], which allows for the study of dynamics
of catalysis, corrosion, and surface alloying [15].
Interpretation of the experiments rely heavily on theoretical input. For example, for PAES one needs to know
the relative probabilities of positron annihilation with inner electrons of various atoms [11]. The positron-atom
system is highly correlated, however, and a dramatic
role is played in the annihilation process by short-range
electron-positron and positron-atom interactions. Their
effect is subshell specific, and overall, they significantly
enhance annihilation rates [16–19] and alter the shape
and magnitude of the annihilation γ-spectra [18, 20–22].
The annihilation probabilities are often computed using
the independent-particle approximation (IPA) [6, 23], or
a modified IPA framework that includes phenomenological enhancement factors that attempt to account for the
important effects the correlations [5, 7, 11, 24–30]. The
enhancement factors are usually calculated using density
functional theory methods heavily based on theoretical
considerations of the electron gas [4, 31–35]. These calcu-
lations, however, suffer from various deficiencies, as discussed in [36]. A more transparent approach is provided
by many-body theory (MBT). It can take full account of
correlations in a systematic way. Early theoretical works
applied MBT [37–45] but naturally they contained drastic approximations, and moreover were again mainly annihilation rate calculations for the electron gas. Finally,
we note that since the core electrons of condensed matter
systems are localized and atomic in nature, an alternative approach to calculating their enhancement factors is
via ab initio atomic-based calculations [41, 46].
The purpose of this Letter is twofold. First, we use
diagrammatic many-body theory (MBT), developed by
the authors, to calculate the γ-spectra for positron annihilation with the core and valence electrons of the noble
gases. Spectra for annihilation in Ar, Kr and Xe were
measured in the pioneering experiments of the Surko
group [1], but till now have eluded theoretical description. We show that the core contribution to the spectra dominates at large Doppler energy shifts, and that
by properly accounting for the correlations in both the
core and valence annihilation, the MBT provides excellent agreement with experiment. Secondly, the MBT naturally allows for the systematic inclusion of the various
correlations. With this control, we calculate ‘exact’, or
‘true’, enhancement factors γ̄n` for positron annihilation
on individual core and valence n` subshells of the noble gas atoms. We show that they follow a simple and
physically motivated scaling
with the subshell ionization
p
energy In` : γ̄n` = 1 + A/In` + (B/In` )β , where A, B
and β are positive constants. We suggest that this formula can be used in relatively simple IPA calculations to
determine accurate core-annihilation spectra for atoms
across the periodic table and in condensed matter.
Theory of γ-spectra for annihilation in atoms.— In
the dominant QED process, a positron annihilates with
an electron in a subshell n to form two γ-ray photons of
total momentum P [47, 48]. In the centre-of-mass frame,
2
P
ε
An" (P)
= ε
ε
P
P
ε
= +
n
n
(a)
ε
n
P
ν ε
+
n µ
(a)
P
P
(b)
P
ν
+
n
µ
(b)
ν1
ε
ε
+
n
ν
ν 2
µ1
Γ
Γ
µ1µ2
n
(c)
P
ν2
1
µ2
n
(c)
ν2
ν2
ν1
ν
ν2
ν1
ν2
ν2 1 ν1
ν2
ν1
ν2
+ ...
+
Γ
+
≡
FIG. 1. Amplitude for positron annihilation with an electron in state n: (a) zeroth-order
vertex (independent-particle ap+ . .µ.
Γ
≡ µ1
µ1
µ2Double
µ+
µ1
µ+
1
2
proximation); (b) first-order, and; (c) ‘Γ-block’
corrections.
linesµ2 labelled
by ε2 represent Dyson incident positron
ν1
ν2
ν1
ν1
µ1
µ1
µ2 positron
µ1 (electron)
µ2
µGreen’s
µ2
1
2
wavefunctions. Single lines labelled
νµ
(µ)
represent
functions,
lines labelled by n represent holes,
wavy lines represent Coulomb interactions and double-dashed lines represent the two γ-ray photons. The ‘Γ’-block is an infinite
ladder of electron-positron Coulomb interactions, and represents the non-perturbative process of virtual positronium formation.
the two γ-rays have equal energies mc2 = 511 keV. In
the laboratory frame, however, their energies are Doppler
shifted by = P c/2 (typically of the order of keV). The
γ-spectra for positrons of energy ε annihilating on an
electron in subshell n is [20]
Z
Z
dΩP
1 ∞
|Anε (P)|2
P dP,
(1)
wnε () =
c 2||/c ΩP
(2π)3
where Anε (P) is the annihilation amplitude. It is given
diagrammatically in Fig. 1 (see [18–20, 22, 49] for full details). Specifically, shown are the dominant contributions
to the annihilation vertex: the zeroth-order vertex, which
represents the amplitude in independent particle approximation; the first-order; and the ‘Γ-block’ corrections,
which describe attractive short-range electron positron
interactions and the non-perturbative process of virtual
positronium formation, P
respectively. As we will show, the
total spectrum wε () = n wnε (), which is probed in experiments, retains distinct characteristics of the spectra
from both individual core and valence subshells.
The fully-correlated incident positron quasiparticle
wavefunction
ψ
ε is calculated from the Dyson equation H0 + Σ̂ε ψε (r) = εψε (r) (see, e.g., [50]), where
H0 is the unperturbed Hamiltonian of the N + 1 particle system, which we take in this work to be the
Hartree-Fock Hamiltonian, and Σ̂ε is the non-local correlation potential.
It is an integral operator that acts as
R
Σ̂ε ψε (r) ≡ Σε (r, r0 )ψε (r0 )dr0 , and is equivalent to the
energy-dependent irreducible self-energy of the positron
in the presence of the atom [17–19, 22]. The calculation of Σ̂ and the method used to solve for the positron
wavefunction ψε are described in [17, 18].
In addition to the γ-spectra, of interest is also the annihilation rate. It is usually parameterized by Zeff , the
effective number of electrons that participate in the annihilation process [51]. It can be written in terms of the
γ-spectra as [20]
Z ∞
wnε () d.
(2)
Zeff (n, ε) =
−∞
In general, Zeff is larger than the number of electrons in
the subshell, owing to the enhancement from electronpositron and positron-atom correlations [2, 17–19, 52].
The annihilation amplitude and positron self-energy
diagrams, which involve summations over intermediate
excited electron and positron continuum states, are calculated numerically by employing a B-spline basis [53],
as in [17–19]. Here, we use a basis of 40 splines of order 6,
and a spherical box of radius 30 a.u. The maximum angular momentum of the intermediate states is `max =15. For
this basis the results converge rapidly in energy. Extrapolation to `max → ∞ is performed using the analytical
formulae of [54]. Full details of the numerical implementation, including the B-spline basis and the convergence
properties, are given in [49].
Results: γ-spectra of the noble gases.— Measurements of the γ-spectra for positrons annihilating with
core electrons of the noble gases Ar, Kr and Xe with isolated two-body interactions were presented in [1]. While
a simple static Hartree-Fock calculation was used successfully to identify a core electron contribution, it ignores
the state-dependent enhancement and the relative fraction of annihilations with core electrons is not predicted
accurately by this theory. In [20] many-body theory was
used to explain the qualitative features of the γ-spectra
of the noble gases, but in that work only annihilation on
the valence shells was considered, the annihilation vertex
was treated only in first-order, and positron-atom correlations were ignored: significant discrepancies between
the theoretical results and experiment were evident.
Fig. 2 shows the γ-spectra for positron annihilation on
individual subshells of Ar, Kr and Xe, calculated with
the full annihilation amplitude of Fig. 1 with the Dyson
positron of thermal momentum (k = 0.04 a.u.) [55]. The
valence spectra are narrowly peaked and dominate the
total spectra at low-Doppler energy shifts. This reflects
the fact that annihilation occurs predominantly on the
valence subshells (see Eq. 2) [19]. Compared with the
diffuse valence electrons, the tightly-bound core electrons
have a broad momentum distribution, and thus a correspondingly broad energy distribution (see Eq. 1). As the
angular momentum of the annihilated electron increases
for a given shell, the spectra become flatter at small energy shifts, behaving as w(0)−w(ε) ∝ |ε|2`n +2 for energy
shifts |ε| 1 for s-wave positrons [20]. Other features include the multiple ‘shoulders’ in individual spectra. The
3
1
10
-1
Total
-1
10
3p
2p
-2
10
Argon
-1
10
γ-spectra (arb. units)
10
γ-spectra (keV )
Argon
3s
0
2s
-3
10
-4
10
static HF
-2
10
Total
Core
-3
10
Valence
-4
10
1s
-5
10
1
Krypton
10
4s
γ-spectra (arb. units)
-1
γ-spectra (keV )
0
10
4p
-1
10
-2
3d
3p
3s
10
-3
10
-4
10
static HF
Total
-2
10
Core
-3
10
Valence
10
2s
-5
0
10
2
10
Xenon
1
10
Xenon
γ-spectra (arb. units)
Total
5s
5p
-1
γ-spectra (keV )
10
-4
2p
10
0
4d
-1
4p
4s
10
10
-2
10
-3
-1
10
static HF
Total
Core
-2
10
-3
Valence
10
3d
10
3s
0
2
-4
3p
-4
10
Krypton
-1
Total
4
6
10
8
10
Energy shift (keV)
FIG. 2.
Absolute γ-spectra for positron annihilation on
individual subshells n` in Ar, Kr and Xe: valence ns, np,
(solid black and red lines); core (n−1)s, (n−1)p, and (n−1)d
(dashed lines); inner core (n − 2)s, (n − 2)p, and (n − 2)d
(dash-dash-dotted lines); and total spectra (solid green line).
All calculated with the full annihilation vertex and Dyson
incident thermal positron.
0
2
4
6
8
Energy shift (keV)
FIG. 3. γ-spectra for positron annihilation in Ar, Kr and Xe.
Experiment: red circles. Theory (convolved with the experimental detector resolution function): valence (dashed line);
core (dash-dash-dotted line); and total (solid line), calculated
with the full annihilation vertex and Dyson incident thermal
positron. Also shown is the static HF IPA calculation of [1]
(blue dots). All curves are normalized by the experimental
Zeff .
4
(0+1+Γ)
γ̄nε ≡
Zeff
(n, ε)
(0)
Zeff (n, ε)
,
(3)
where the superscript denotes the vertex order of the amplitude (Fig. 1). Such a parameterization is common in
predictions of annihilation rates and γ-spectra (see, e.g.,
[5], and references therein). The true spectra for annihilation on a given subshell is then to be approximated as
(0)
wnε () ≈ γ̄nε wnε ().
1
10
H
+
He
γ̄ -1
number of these is determined by the number of nodes
in the electron’s radial wave function, n − `: each lobe
of non-zero wavefunction density at small (large) radii
manifests in large (small) momentum density, and thus
in large (small) energy regions of the spectra [49]. Overall, the total γ-spectra retains the characteristics of the
valence and core. Note that the contribution from subshells below the (n − 1) core is negligible in this range
of Doppler shifts. Figure 3 shows the calculated spectra after convolving with the experimental detector resolution function and normalization by the experimental
Zeff [1],[56]. For each atom the valence spectra underestimates the experimental spectrum in the high-energy
‘wings’. For Kr and Xe this underestimation is severe,
with the core contribution actually dominating at energies >
∼ 4.5 keV. In all cases inclusion of the core brings
the theoretical spectra into excellent agreement with experiment. Overall, the MBT results show unequivocally
that there is a significant contribution from annihilation
on the core, and that its proper inclusion with full account of the individual subshell enhancement is crucial
to predict accurate total spectra.
Vertex enhancement factors.— As mentioned, a dramatic role is played in the annihilation process by shortrange electron-positron and positron-atom interactions,
which act to significantly enhance annihilation rates [16–
19] and alter the shape and magnitude of the annihilation γ-spectra [18, 20–22]. Usually, the effect of these
correlations is taken into account by (phenomenological)
enhancement factors, which are calculated using various
guises of density functional theory and based heavily on
considerations of the electron gas. The many-body theory, in contrast, takes account of the subshell specific
correlations in natural, transparent and, through the expansion of the vertex (Fig. 1), systematic way. Here, we
exploit this control to calculate exact enhancement factors for annihilation on individual subshells. Specifically,
these factors parameterize the enhancement above the
IPA due to the short-range (generally non-local) interactions. We find that in practice, the correlational enhancement in the spectra is weakly dependent on the twophoton momentum, at least for the core levels [22, 49]. It
is instructive therefore to define the state-dependent enhancement factor as the ratio of the calculated true partial annihilation rate to that calculated using the zerothorder (IPA) vertex (see Eq. 2)
0
10
Li
2+
B
4+
F
8+
-1
10
1
10
2
10
3
10
IE (eV)
FIG. 4. Enhancement factors (Eq. 3) for the valence and subvalence subshells of Ar, Kr and Xe, calculated using a HF
(solid circles) and Dyson (solid squares) incident positron.
Eq. 4 is fitted to the HF factors, yielding: A = 43.6 eV
(1.60 a.u.), B = 20.5 eV (0.75 a.u.), β = 2.6 (dashed line);
and to the Dyson factors, yielding: A = 36.0 eV (1.32 a.u.),
B = 27.0 eV (0.99 a.u.), β = 2.2 (solid line). Also shown are
the equivalent enhancement factors of hydrogen: Dyson (solid
diamond) and HF (unfilled diamond), and the positive ions
He+ , Li2+ , B4+ and F8+ (plus signs), calculated in [17, 18],
and for the core 1s electron in He(1s2s 3 S e ), calculated using
the stochastic variational calculation [46] (cross).
Figure 4 shows the enhancement factors (Eq. 3) for the
core and valence subshells of the noble gases, for both
HF and Dyson incident thermal positron. Also shown
are the MBT calculated enhancement factors of hydrogen [17] and the hydrogen-like positive ions He+ , Li2+ ,
B4+ and F8+ [18], and the result of the stochastic variational calculation of [46] for annihilation with the 1s
electron in He(1s2s 3 S). For the noble gases, the valence
enhancements are in the range 2.4 < γ̄valence < 9.3. The
core enhancement factors are smaller, owing to the fact
that the tightly-bound core electrons are less susceptible to the perturbation from the interaction with the incident positron. However, they still differ significantly
from unity, with values in the range 1.3 < γ̄core < 2.0.
This shows it is crucial to take proper account of the
differing correlation effects for individual subshells when
calculating the total spectra (as done in Figs. 2 and 3).
One can estimate the strength of the short-range correlational enhancement from the first-order correction as
the ratio Vint /∆E, where Vint is the electron-positron
Coulomb interaction potential, and ∆E is a measure of
the response of the electrons to the perturbation (of the
order of the respective electron’s binding energy). For
5
positron annihilation on H-like ions, this strength scales
as 1/Z, where Z is the nuclear charge [18]. For annihilation on the core electrons of many-electron atoms√the
strength can be thought to scale analogously as 1/ IE ,
where IE is the ionization
potential of the subshell.
√
One also arrives at a 1/ IE dependence from simplifications to a calculation that explicitly includes two-particle
Green’s function of the electron-positron pair in the annihilation vertex [22]. Thus, we have fitted the function
r
γ̄n − 1 ≈
A
+
IE
B
IE
β
,
(4)
to the calculated enhancement factors of the noble gases,
where A, B and β are constants. The first term describes
the enhancement of the core electrons and is physically
motivated, as discussed above. The second term is phenomenological and accounts for the enhancement of the
valence electrons, for which higher-order corrections are
of as equal importance as the first-order one [18, 22, 49].
The solid (dashed) line in Fig. 4 is Eq. 4 fitted to the results calculated with the HF (Dyson) incident
√ positron
wavefunction (fit values in caption). The 1/ IE dependence for the core shells is evident. Given the inherent
difference between the positive ions and the neutral systems, the ions follow the trend surprisingly well.
Summary and outlook.— Many-body theory was
used to calculate the contribution of individual subshells
to the positron annihilation spectra of the noble gases.
Inclusion of core annihilation gives results in excellent
agreement with experiment. Using the MBT, exact vertex enhancement factors were calculated and shown to
follow a simple and physically motivated scaling with the
subshell ionisation energy (Eq. 4). We suggest that this
formula can be used with relatively simple independentparticle-model calculations to obtain accurate spectra for
core annihilation on atoms across the periodic table and
in condensed matter [57].
We thank Prof. C. M. Surko for valuable discussions. DGG is grateful to the Institute for Theoretical
Atomic, Molecular and Optical Physics at the HarvardSmithsonian Centre for Astrophysics, USA, where he carried out part of this work, and thanks Prof. H. R. Sadeghpour and colleagues for their generous hospitality.
∗
Correspondences to: [email protected];
Present address: Joint Quantum Centre (JQC) Durham/
Newcastle, Department of Chemistry, Durham University, South Road, Durham, DH1 3LE, UK.
†
[email protected]
[1] K. Iwata, G. F. Gribakin, R. G. Greaves, and C. M.
Surko, Phys. Rev. Lett. 79, 39 (1997).
[2] G. F. Gribakin, J. A. Young, and C. M. Surko, Rev.
Mod. Phys. 82, 2557 (2010).
[3] K. Iwata, R. G. Greaves, and C. M. Surko, Phys. Rev.
A 55, 3586 (1997).
[4] M. J. Puska and R. M. Nieminen, Rev. Mod. Phys. 66,
841 (1994).
[5] F. Tuomisto and I. Makkonen, Rev. Mod. Phys. 85, 1583
(2013).
[6] K. G. Lynn, J. R. MacDonald, R. A. Boie, L. C. Feldman, J. D. Gabbe, M. F. Robbins, E. Bonderup, and
J. Golovchenko, Phys. Rev. Lett. 38, 241 (1977).
[7] A. Eshed, S. Goktepeli, A. R. Koymen, S. Kim, W. C.
Chen, D. J. O’Kelly, P. A. Sterne, and A. H. Weiss,
Phys. Rev. Lett. 89, 075503 (2002).
[8] A. Weiss, R. Mayer, M. Jibaly, C. Lei, D. Mehl, and
K. G. Lynn, Phys. Rev. Lett. 61, 2245 (1988).
[9] T. Ohdaira, R. Suzuki, T. Mikado, H. Ohgaki, M. Chiwaki, and T. Yamazaki, Appl. Surf. Sci. 116, 177 (1997).
[10] J. Mayer, K. Schreckenbach, and C. Hugenschmidt,
Phys. Status Solidi (C) 4, 3928 (2007).
[11] A. H. Weiss, N. G. Fazleev, M. P. Nadesalingam,
S. Mukherjee, S. Xie, J. Zhu, and B. R. Davis, Radiat.
Phys. Chem. 76, 285 (2007).
[12] C. Hugenschmidt, Neutron News 18, 29 (2007).
[13] J. Mayer, C. Hugenschmidt, and K. Schreckenbach, Surface Science 604, 1772 (2010).
[14] C. Hugenschmidt, B. Lwe, J. Mayer, C. Piochacz,
P. Pikart, R. Repper, M. Stadlbauer, and K. Schreckenbach, Nuc. Instrum. Meth. A 593, 616 (2008).
[15] J. Mayer, C. Hugenschmidt, and K. Schreckenbach,
Phys. Rev. Lett. 105, 207401 (2010).
[16] K. Iwata, R. G. Greaves, T. J. Murphy, M. D. Tinkle,
and C. M. Surko, Phys. Rev. A 51, 473 (1995).
[17] G. F. Gribakin and J. Ludlow, Phys. Rev. A 70, 032720
(2004).
[18] D. G. Green and G. F. Gribakin, Phys. Rev. A 88, 032708
(2013).
[19] D. G. Green, J. A. Ludlow, and G. F. Gribakin,
“Positron scattering and annihilation on noble gas
atoms,” Submitted to Phys. Rev. A (2014).
[20] L. J. M. Dunlop and G. F. Gribakin, J. Phys. B 39, 1647
(2006).
[21] D. G. Green, S. Saha, F. Wang, G. F. Gribakin, and
C. M. Surko, New J. Phys. 14, 035021 (2012).
[22] D. G. Green, Ph.D. thesis, Queen’s University Belfast
(2011).
[23] K. O. Jensen and A. Weiss, Phys. Rev. B 41, 3928 (1990).
[24] K. O. Jensen, J. Phys.: Condens. Matter 1, 10595 (1989).
[25] M. Alatalo, H. Kauppinen, K. Saarinen, M. J. Puska,
J. Mäkinen, P. Hautojärvi, and R. M. Nieminen, Phys.
Rev. B 51, 4176 (1995).
[26] M. Alatalo, B. Barbiellini, M. Hakala, H. Kauppinen,
T. Korhonen, M. J. Puska, K. Saarinen, P. Hautojärvi,
and R. M. Nieminen, Phys. Rev. B 54, 2397 (1996).
[27] S. Daniuk, M. Šob, and A. Rubaszek, Phys. Rev. B 43,
2580 (1991).
[28] B. Barbiellini, P. Genoud, and T. Jarlborg, J. Phys.:
Condens. Matter 3, 7631 (1991).
[29] S. Kim, A. Eshed, S. Goktepeli, P. A. Sterne, A. R. Koymen, W. C. Chen, and A. H. Weiss, Phys. Rev. B 73,
014114 (2006).
[30] Z. Tang, M. Hasegawa, T. Chiba, M. Saito, H. Sumiya,
Y. Kawazoe, and S. Yamaguchi, Phys. Rev. B 57, 12219
(1998).
[31] E. Boroński and R. M. Nieminen, Phys. Rev. B 34, 3820
(1986).
6
[32] B. Barbiellini, M. J. Puska, T. Torsti, and R. M. Nieminen, Phys. Rev. B 51, 7341 (1995); B. Barbiellini, M. J.
Puska, T. Korhonen, A. Harju, T. Torsti, and R. M.
Nieminen, Phys. Rev. B 53, 16201 (1996).
[33] B. Barbiellini, M. J. Puska, M. Alatalo, M. Hakala,
A. Harju, T. Korhonen, S. Siljamki, T. Torsti, and R. M.
Nieminen, Appl. Surf. Sci. 116, 283 (1997).
[34] B. Barbiellini, M. Hakala, M. J. Puska, R. M. Nieminen,
and A. A. Manuel, Phys. Rev. B 56, 7136 (1997).
[35] A. Rubaszek, Phys. Rev. B 44, 10857 (1991);
A. Rubaszek, A. Kiejna, and S. Daniuk, J. Phys.: Condens. Matter 5, 8195 (1993); A. Rubaszek, Z. Szotek,
and W. M. Temmerman, Phys. Rev. B 58, 11285 (1998).
[36] J. Mitroy and B. Barbiellini, Phys. Rev. B 65, 235103
(2002).
[37] S. Kahana, Phys. Rev. 117, 123 (1960); Phys. Rev. 129,
1622 (1963).
[38] J. P. Carbotte, Phys. Rev. 144, 309 (1966); Phys. Rev.
155, 197 (1967).
[39] A. Salvadori and J. P. Carbotte, Phys. Rev. 188, 550
(1969).
[40] J. Arponen and P. Jauho, Phys. Rev. 167, 239 (1968).
[41] E. Bonderup, J. U. Andersen, and D. N. Lowy, Phys.
Rev. B 20, 883 (1979).
[42] D. N. Lowy and A. D. Jackson, Phys. Rev. B 12, 1689
(1975).
[43] A. J. Zuchelli and T. G. Hickman, Phys. Rev. 136, A1728
(1964).
[44] J. Crowell, V. E. Anderson, and R. H. Ritchie, Phys.
Rev. 150, 243 (1966).
[45] T. Chiba, G. B. Dürr, and W. Brandt, Phys. Status
Solidi (B) 81, 609 (1977).
[46] J. Mitroy, Phys. Rev. A 72, 032503 (2005).
[47] A. I. Akhiezer and V. B. Berestetskii, Quantum Electrodynamics (Interscience Publishers 1965, 1982).
[48] V. B. Berestetskii, E. M. Lifshitz, and L. P. Pitaevskii,
Quantum Electrodynamics - 2nd ed. - (Course of Theoretical Physics; V4) (Pergamon, Oxford, 1982).
[49] D. G. Green and G. F. Gribakin, To be submitted to
Phys. Rev. A (2014).
[50] A. A. Abrikosov, L. P. Gorkov, and I. E. Dzyalonshinkski, Methods of Quantum Field Theory in Statistical
Physics (Dover, New York, 1975).
[51] P. A. Fraser, Adv. At. Mol. Phys. 4, 63 (1968).
[52] C. M. Surko, G. F. Gribakin, and S. J. Buckman, J.
Phys. B 38, R57 (2005).
[53] C. de Boor, A practical guide to splines (Springer, New
York, 1977).
[54] G. F. Gribakin and J. Ludlow, J. Phys. B 35, 339 (2002).
[55] Thermal energy positrons are used in the experiments of
[1], and are typical for annihilation in condensed matter,
in which the thermalization timescale is much faster than
the annihilation one [4, 5].
[56] Only the positive energy shifts of the symmetric γ-ray
line is shown, since this region has greater statistical significance due to the reduced Compton scattering component [1].
[57] For positrons annihilating in condensed matter systems,
the longer-range positron-atom correlations included in
the Dyson positron wavefunction average to zero. In this
case the HF vertex enhancement factors, which neglect
these longer-range type correlations in the positron wavefunction, would be expected to be more appropriate.