244
D. G. Green and G. F. Gribakin
MOmentum Correlation (AMOC) experiments (see, e.g., [14–16]), in which the í µí»¾
spectra are measured as a function of the positron “age” (i.e., time after emission
from source). AMOC enables study of positron and positronium cooling (and, more
generally, transitions between positron states, e.g., for different trapping states, or
via chemical reactions) [14–16].
Interpretation of the experiments relies heavily on theoretical input, e.g., in PAES
one requires accurate relative annihilation probabilities for core electrons of various
atoms [17]. Such quantities, however, are not easy to calculate, as the annihilation
process is strongly affected by short-range electron-positron and long-range positronatom correlations. These effects significantly enhance the annihilation rates [18, 19]
and alter the shape and magnitude of the annihilation í µí»¾ spectra [20–24], compared
to independent-particle approximation (IPA) calculations.
A powerful method that allows for systematic inclusion of the correlations in
atomic systems is many-body theory (MBT). MBT enables one to calculate the socalled enhancement factors (EF), which quantify the increase of the electron density at the positron due to the effect of correlations. The EF can be used to correct
the IPA annihilation probabilities and í µí»¾-spectra [2, 17]. They are particularly large
(∼10) for the valence electrons, but are also significant for the core electrons [25]. 1
EF were introduced in early MBT works involving positron annihilation in metals
that were based on considering positrons in a homogeneous electron gas [28, 29].
Subsequently, density functional theories (DFT) were developed to describe positron
states and annihilation in a wider class of condensed-matter systems [30, 31]. These
methods usually rely on some input in the form of the correlation energy and EF for
the positron in electron gas from MBT [32]. When applied to real, inhomogeneous
systems, position-dependent EF can lead to spurious effects in the spectra [5], and
show deficiencies when benchmarked against more accurate calculations [33].
A recent study of a model system of eight electrons and one positron confined
in a harmonic potential [34] highlighted significant discrepancies between the annihilation momentum densities (AMD) and EF obtained using exact diagonalization
and those found using common DFT approaches. The best agreement for the shapes
of AMD was observed for position-dependent EF [35, 36] calculated in the Kahana
formalism [36, 37], though there was a factor-of-two difference for the total annihilation rate. It was also suggested in [34] (see also [38]) that the EF could be defined
rigorously using natural geminals. These quantities are electron-positron pair wavefunctions which diagonalise the two-body reduced density matrix, and which can be
extracted from the accurate many-particle wavefunction. It would be very useful if
the natural geminals could be used without the knowledge of the total wavefunction,
and it is possible that this can be done using MBT.
In the context of the positron-atom problem, the MBT calculations provided an
accurate and essentially complete picture of low-energy positron interaction with
1 Positron annihilation with core electrons is also affected by exchange-assisted tunnelling [26, 27].
This is a manifestation of electron exchange, which increases the wavefunctions of inner electrons in
the range of distances of the valence electrons. For this effect to be properly included in a calculation,
one needs to use true nonlocal exchange potentials, e.g., at the Hartree-Fock level, as is the case in
the present calculations.
D. G. Green and G. F. Gribakin
MOmentum Correlation (AMOC) experiments (see, e.g., [14–16]), in which the í µí»¾
spectra are measured as a function of the positron “age” (i.e., time after emission
from source). AMOC enables study of positron and positronium cooling (and, more
generally, transitions between positron states, e.g., for different trapping states, or
via chemical reactions) [14–16].
Interpretation of the experiments relies heavily on theoretical input, e.g., in PAES
one requires accurate relative annihilation probabilities for core electrons of various
atoms [17]. Such quantities, however, are not easy to calculate, as the annihilation
process is strongly affected by short-range electron-positron and long-range positronatom correlations. These effects significantly enhance the annihilation rates [18, 19]
and alter the shape and magnitude of the annihilation í µí»¾ spectra [20–24], compared
to independent-particle approximation (IPA) calculations.
A powerful method that allows for systematic inclusion of the correlations in
atomic systems is many-body theory (MBT). MBT enables one to calculate the socalled enhancement factors (EF), which quantify the increase of the electron density at the positron due to the effect of correlations. The EF can be used to correct
the IPA annihilation probabilities and í µí»¾-spectra [2, 17]. They are particularly large
(∼10) for the valence electrons, but are also significant for the core electrons [25]. 1
EF were introduced in early MBT works involving positron annihilation in metals
that were based on considering positrons in a homogeneous electron gas [28, 29].
Subsequently, density functional theories (DFT) were developed to describe positron
states and annihilation in a wider class of condensed-matter systems [30, 31]. These
methods usually rely on some input in the form of the correlation energy and EF for
the positron in electron gas from MBT [32]. When applied to real, inhomogeneous
systems, position-dependent EF can lead to spurious effects in the spectra [5], and
show deficiencies when benchmarked against more accurate calculations [33].
A recent study of a model system of eight electrons and one positron confined
in a harmonic potential [34] highlighted significant discrepancies between the annihilation momentum densities (AMD) and EF obtained using exact diagonalization
and those found using common DFT approaches. The best agreement for the shapes
of AMD was observed for position-dependent EF [35, 36] calculated in the Kahana
formalism [36, 37], though there was a factor-of-two difference for the total annihilation rate. It was also suggested in [34] (see also [38]) that the EF could be defined
rigorously using natural geminals. These quantities are electron-positron pair wavefunctions which diagonalise the two-body reduced density matrix, and which can be
extracted from the accurate many-particle wavefunction. It would be very useful if
the natural geminals could be used without the knowledge of the total wavefunction,
and it is possible that this can be done using MBT.
In the context of the positron-atom problem, the MBT calculations provided an
accurate and essentially complete picture of low-energy positron interaction with
1 Positron annihilation with core electrons is also affected by exchange-assisted tunnelling [26, 27].
This is a manifestation of electron exchange, which increases the wavefunctions of inner electrons in
the range of distances of the valence electrons. For this effect to be properly included in a calculation,
one needs to use true nonlocal exchange potentials, e.g., at the Hartree-Fock level, as is the case in
the present calculations.