Enhancement Factors for Positron Annihilation on Valence . . .
245
noble-gas atoms [19], with excellent agreement between the theoretical results and
experimental scattering cross sections and annihilation rates. The MBT work was
extended recently [24] to the í µí»¾-spectra for (thermal) positron annihilation on noblegas atoms. It provided an accurate description of the measured spectra for Ar, Kr and
Xe [3] and firmly established the relative contributions of various atomic orbitals to
the spectra. The calculations also yielded “exact” ab initio EF ̄
í µí»¾ nl for individual electron orbitals nl, and found that they follow a simple scaling with the orbital ionization
energy [24].
In this work we provide a more detailed analysis and report EF for annihilation of
s-, p- and d-wave positrons with momenta up to the positronium formation threshold.
We demonstrate that the EF for a given electron orbital and positron partial wave are
insensitive to the positron momentum (in spite of the strong momentum dependence
of the annihilation probability [19]). Moreover, we show that whilst the EF for the
core orbitals are almost independent of the positron angular momenta, those for the
valence subshells vary between the positron s, p and d waves. In addition to their
use in correcting IPA calculations of positron annihilation with core electrons in
condensed matter, the positron-momentum dependent EF calculated here can be used
to determine accurate pick-off annihilation rates for positronium in noble gases [39].
2 Theory of Positron Annihilation in Many-Electron Atoms
2.1 Basics
Consider annihilation of a low-energy (í µí¼ ∼ 1 eV) positron with momentum í µí°¤ in a
many-electron system, e.g., an atom. In the dominant process, the positron annihilates with an electron in state n to form two í µí»¾-ray photons of total momentum
í µí° [40]. In the centre-of-mass frame, where the total momentum í µí° is zero, the
two photons are emitted in opposite directions and have equal energies E í µí»¾ = p í µí»¾ c =
mc 2 +
1
2
(E i − E f ) ≃ mc 2 ≃ 511 keV, where E i and E f denote the energy of the initial
and final states of the system (excluding rest mass). When í µí° is non-zero, however,
the two photons no longer propagate in exactly opposite directions and their energy
is Doppler shifted. For example, for the first photon E í µí»¾ 1 = E í µí»¾ + mcV cos í µí¼, where í µí¼
is the angle between the momentum of the photon and the centre-of-mass velocity of
the electron-positron pair í µí° = í µí°∕2m (assuming that V ≪ c, and p í µí»¾ 1 = E í µí»¾ 1 ∕c ≈ mc).
The Doppler shift of the photon energy from the centre of the line then is
í µí¼ = E í µí»¾ 1 − E í µí»¾ = mc V cos í µí¼ =
Pc
2
cos í µí¼.
(1)
The typical momenta of electrons bound with energy í µí¼ n determine the characteristic
width of the annihilation spectrum í µí¼ ∼ Pc ∼
√
|í µí¼ n |mc 2 ≫ |í µí¼ n |. Hence the shift í µí¼ n ∕2
of the line centre E í µí»¾ from mc
2 = 511 keV can usually be neglected, even for the core
electrons. The í µí»¾ spectrum averaged over the direction of emitted photons (or that of
the positron momentum í µí°¤) takes the form (see, e.g., [20])
245
noble-gas atoms [19], with excellent agreement between the theoretical results and
experimental scattering cross sections and annihilation rates. The MBT work was
extended recently [24] to the í µí»¾-spectra for (thermal) positron annihilation on noblegas atoms. It provided an accurate description of the measured spectra for Ar, Kr and
Xe [3] and firmly established the relative contributions of various atomic orbitals to
the spectra. The calculations also yielded “exact” ab initio EF ̄
í µí»¾ nl for individual electron orbitals nl, and found that they follow a simple scaling with the orbital ionization
energy [24].
In this work we provide a more detailed analysis and report EF for annihilation of
s-, p- and d-wave positrons with momenta up to the positronium formation threshold.
We demonstrate that the EF for a given electron orbital and positron partial wave are
insensitive to the positron momentum (in spite of the strong momentum dependence
of the annihilation probability [19]). Moreover, we show that whilst the EF for the
core orbitals are almost independent of the positron angular momenta, those for the
valence subshells vary between the positron s, p and d waves. In addition to their
use in correcting IPA calculations of positron annihilation with core electrons in
condensed matter, the positron-momentum dependent EF calculated here can be used
to determine accurate pick-off annihilation rates for positronium in noble gases [39].
2 Theory of Positron Annihilation in Many-Electron Atoms
2.1 Basics
Consider annihilation of a low-energy (í µí¼ ∼ 1 eV) positron with momentum í µí°¤ in a
many-electron system, e.g., an atom. In the dominant process, the positron annihilates with an electron in state n to form two í µí»¾-ray photons of total momentum
í µí° [40]. In the centre-of-mass frame, where the total momentum í µí° is zero, the
two photons are emitted in opposite directions and have equal energies E í µí»¾ = p í µí»¾ c =
mc 2 +
1
2
(E i − E f ) ≃ mc 2 ≃ 511 keV, where E i and E f denote the energy of the initial
and final states of the system (excluding rest mass). When í µí° is non-zero, however,
the two photons no longer propagate in exactly opposite directions and their energy
is Doppler shifted. For example, for the first photon E í µí»¾ 1 = E í µí»¾ + mcV cos í µí¼, where í µí¼
is the angle between the momentum of the photon and the centre-of-mass velocity of
the electron-positron pair í µí° = í µí°∕2m (assuming that V ≪ c, and p í µí»¾ 1 = E í µí»¾ 1 ∕c ≈ mc).
The Doppler shift of the photon energy from the centre of the line then is
í µí¼ = E í µí»¾ 1 − E í µí»¾ = mc V cos í µí¼ =
Pc
2
cos í µí¼.
(1)
The typical momenta of electrons bound with energy í µí¼ n determine the characteristic
width of the annihilation spectrum í µí¼ ∼ Pc ∼
√
|í µí¼ n |mc 2 ≫ |í µí¼ n |. Hence the shift í µí¼ n ∕2
of the line centre E í µí»¾ from mc
2 = 511 keV can usually be neglected, even for the core
electrons. The í µí»¾ spectrum averaged over the direction of emitted photons (or that of
the positron momentum í µí°¤) takes the form (see, e.g., [20])