246
D. G. Green and G. F. Gribakin
w n (𝜖) =
1
c ∫ ∫
∞
2|𝜖|∕c
|A n𝐤 (𝐏)|
2 PdPd𝛺 𝐏
(2𝜋) 3 ,
(2)
where A n𝐤 (𝐏) is the annihilation amplitude, whose calculation using MBT is
described below. The quantity |A n𝐤 (𝐏)|
2 is the annihilation momentum density.
2
The annihilation rate 𝜆 for a positron in a gas of atoms or molecules with number
density n m is usually parametrized by
𝜆 = 𝜋r
2
0
cn m Z ef f ,
(3)
where r 0 = e
2 ∕mc
2 is the classical radius of the electron (in CGS units) and Z ef f
is the effective number of electrons per target atom or molecule that contribute to
annihilation [42, 43]. It is found as a sum over electron states Z ef f =
∑
n Z ef f ,n , where
Z ef f ,n = ∫
w n (𝜖) d𝜖 = ∫
|A n𝐤 (𝐏)|
2 d 3 𝐏
(2𝜋) 3
(4)
is the partial contribution due to positron annihilation with electron in state n, and
where it is assumed that the incident positron wavefunction used in the calculation
of A n𝐤 (𝐏) is normalized to a plane wave. In general, the parameter Z ef f is different
from the number of electrons in the target atom Z. In particular, positron-atom and
electron-positron correlations can make Z ef f ≫ Z [19, 24, 44–46].
2.2 Many-Body Theory for the Annihilation Amplitude
The incident positron wavefunction is taken in the form of a partial-wave expansion
3
𝜓 𝐤 (𝐫) =
4𝜋
r
√ 𝜋
k
∑
𝓁m
i
𝓁 e
i𝛿 𝓁 Y
∗
𝓁m
( ̂ 𝐤)Y 𝓁m (̂ 𝐫)P 𝜀𝓁 (r),
(5)
where 𝛿 𝓁 is the scattering phaseshift [47], Y 𝓁m is the spherical harmonic, and
where the radial function with orbital angular momentum 𝓁 is normalized by
its asymptotic behaviour P 𝜀𝓁 (r) ≃ (𝜋k) −1∕2 sin(kr − 𝜋𝓁∕2 + 𝛿 𝓁 ). In the simplest
approximation the radial wavefunctions are calculated in the static field of the
ground-state (Hartree-Fock, HF) atom. This approximation is very inaccurate for the
positron-atom problem. It fails to describe the scattering cross sections and grossly
2 Alternatively to the Doppler-shift spectrum, experiments measure the one-dimensional angular
correlation of annihilation radiation (1D-ACAR), i.e., the small angle 𝛩 between the direction of
one photon and the plane containing the other. The corresponding distribution can be obtained from
w(𝜀) using 𝛩 = 2𝜖∕mc 2 . Not also that if the positron wavefunction is constant, then the annihilation
momentum density is proportional to the electron momentum density, and the 𝛾 spectrum becomes
similar to the Compton profile [22, 23, 41].
3 In this and subsequent sections we make wide use of atomic units (a.u.).