Positron annihilation with core and valence electrons
-ray spectra for positron annihilation with the core and valence electrons of the noble gas atoms Ar, Kr and Xe is calculated within the framework of diagrammatic many-body theory. The effect of positron-atom and short-range positron-electron correlations on the annihilation process is examined in detail. Short-range correlations, which are described through non-local corrections to the vertex of the annihilation amplitude, are found to significantly enhance the spectra for annihilation on the core orbitals. For Ar, Kr and Xe, the core contributions to the annihilation rate are found to be 0.55%, 1.5% and 2.2% respectively, their small values reflecting the difficulty for the positron to probe distances close to the nucleus. Importantly however, the core subshells have a broad momentum distribution and markedly contribute to the annihilation spectra at Doppler energy shifts keV, and even dominate the spectra of Kr and Xe at shifts keV. Their inclusion brings the theoretical spectra into excellent agreement with the experimental -spectra across the full range of Doppler energy shifts. Additionally, the theory enables the calculation of the ‘exact’ vertex enhancement factors for individual core and valence subshells . They are found to follow a simple and physically motivated scaling with the subshell ionization energy : , where , and are positive constants. It is demonstrated that these factors can be incorporated in simple independent-particle-model calculations to successfully reconstruct the true many-body annihilation -spectra. This formula can be used to determine the enhancement factors for positron annihilation with core electrons of atoms across the periodic table, and with the localized atomic-like core electrons of condensed matter systems.
pacs: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 in condensed matter systems Iwata et al. (1997a); Puska and Nieminen (1994).
Small fractions of positrons can, however, tunnel through the repulsive nuclear potential and annihilate on the core electrons
In this work we examine, systematically and in detail, the effect of these correlations on the -spectra for positrons annihilating on individual core and valence subshells in many-electron atoms. Specifically, we use many-body theory Gribakin and Ludlow (2004); Dunlop and Gribakin (2006); Green and Gribakin (2013); Green et al. (2014) (MBT) to calculate the annihilation -ray spectra and annihilation rates for positron annihilation on individual subshells of the noble gases Ar, Kr and Xe. We show that the theory is the first to accurately reproduce the the measured spectra Iwata et al. (1997a) (see Ref. Green and Gribakin (2015) for a brief account of this part of the work). Furthermore, we establish firmly the fraction annihilation on individual subshells, and uncover a simple scaling of the correlation enhancement factors, which describe the increase in the annihilation probability beyond the independent-particle approximation (IPA), with the electron ionization energy Green and Gribakin (2015). This work, taken together with our work in Refs. Green et al. (2014) and Green and Gribakin (2015), gives a (near) complete description of the positron noble-gas system.
The ability to discern the core signal in the -spectrum was first realised experimentally in Lynn et al. (1977). The elemental specificity arising from the chemically distinct signature in the large Doppler-shift regions of the spectra corresponding to core annihilation has since been exploited Asoka-Kumar et al. (1996); Petkov et al. (1999); Alatalo et al. (1995), and measurement of the signal of individual selected core levels has been demonstrated Eshed et al. (2002). Positron annihilation on core electrons is now a key process in several established experimental techniques, e.g., positron-induced Auger-electron spectroscopy (PAES) Weiss et al. (1988); Ohdaira et al. (1997); Mayer et al. (2007); Weiss et al. (2007), used for diagnostics of industrially important materials, and the recently emerged time-resolved PAES Hugenschmidt et al. (2008), to study the dynamics of catalysis, corrosion, and surface alloying Mayer et al. (2010). Coincident measurements of the -rays and Auger electrons yields -ray spectra for individual core orbitals Eshed et al. (2002); Kim et al. (2006).
In all cases, interpretation of the measured annihilation -spectra relies heavily on theoretical input. For example, to interpret the results of PAES one needs to know the relative probabilities of positron annihilation with inner electrons of various atoms Weiss et al. (2007). These are often computed using the independent-particle approximation (IPA) Lynn et al. (1977); Jensen and Weiss (1990), or in a modified IPA framework that includes phenomenological enhancement factors that attempt to account for the important effects of electron-positron correlations Jensen (1989); Alatalo et al. (1995, 1996); Daniuk et al. (1991); Barbiellini et al. (1991); Eshed et al. (2002); Kim et al. (2006); Weiss et al. (2007); Tang et al. (1998). In most cases, the materials of interest are condensed matter systems, and the enhancement factors are usually calculated using a variety of density functional theory methods, e.g., the local density approximation (LDA) Puska and Nieminen (1994); Boroński and Nieminen (1986), the generalized gradient approximation (GGA) Barbiellini et al. (1995, 1996, 1997a, 1997b), or the weighted density approximation (WDA) Rubaszek (1991); Rubaszek et al. (1993, 1998), all of which rely heavily on theoretical considerations of the electron gas. However, due to the strong variations in the density, the LDA is not expected to work well for the core electrons, and has been found to overestimate the annihilation rates Rubaszek et al. (1998) The enhancement factor approach within the one-component and Boroński-Nieminen two-component LDA Boroński and Nieminen (1986), as well as within the GGA, have been tested and compared with bound positron-atom stochastic variational calculations in Ref. Mitroy and Barbiellini (2002), although no conclusive statements were made regarding the core enhancements.
One particular weakness of the DFT enhancement factor approach is that it obscures the true non-local many-body nature of the problem. The early theoretical works of Kahana, Carbotte and others (see, e.g., Refs. Kahana (1960, 1963); Carbotte (1966, 1967); Salvadori and Carbotte (1969); Arponen and Jauho (1968); Bonderup et al. (1979); Lowy and Jackson (1975); Zuchelli and Hickman (1964); Crowell et al. (1966); Chiba et al. (1977)) on positron annihilation were based on many-body theory, which inherently incorporates the non-local effects. Naturally however, these methods contained numerous approximations, and again were mainly annihilation rate calculations for the electron gas. Some effects of the non-local components of the amplitude have been included in phenomenological enhancement factor calculations using the WDA Rubaszek et al. (1998), and to a lesser extent in the GGA Alatalo et al. (1995, 1996). However, these methods again rely on considerations of the electron gas.
Regarding the core enhancement factors in particular, there are a number of calculations that differ in their treatment and overall, there is no definite agreement in their values. At one extreme it is assumed that the core enhancement is negligible, while at the other it is assumed to be equivalent to the valence enhancement (see, e.g., Rubaszek et al. (1998) and references therein). In reality, the core electrons are less susceptible to a perturbation from the interaction with the incident positron than the valence ones. One should therefore expect the core spectra to be enhanced by a factor 1, i.e., one should expect the correlational enhancement to be weaker for the tightly-bound core than for the valence electrons. Indeed, a recent study by the authors that investigated the correlational enhancement for annihilation of positrons on H-like ions, for which the electron binding energies span a range typical of those of core electrons in many-electron atoms, suggested that the core enhancement could be significantly greater than unity Green and Gribakin (2013).
For positron annihilation in condensed matter systems, an alternative approach to the calculation of core electron enhancement factors is suggested by the fact that the core electrons of these systems are localized and atomic in nature. Their enhancement factors are therefore open to ab initio atomic-based calculations. This was realised by Bonderup et al. Bonderup et al. (1979), who, in order to elucidate the rôle of electron-positron correlations in condensed matter systems, made a first-order perturbation theory estimate of the enhancement for positron annihilation on hydrogen-like ions. Compared to condensed matter systems, the problem of positrons annihilating in atomic systems is in principle, much simpler. For one, simplifications result from the spherical symmetry of the atom that is not present in the condensed matter systems. For the latter, increased complexity comes from the periodic crystal potential: calculating the electronic and band structure is much more involved, and from the Bloch-wave nature of the incident positron wave function, which must be properly accounted for. Despite this, for annihilation on core electrons, positron-condensed-matter studies are much more numerous than corresponding positron-atom studies.
Evidence of the core contribution to the annihilation -spectra of the noble gases was first reported by Iwata et al.Iwata et al. (1997a), in which upper bounds (2% and 3% for Ar and Kr respectively) were set on the core contribution to the total spectra. This was followed by the first experimental study of positrons annihilating with inner-shell electrons in isolated two-body positron-atom interactions Iwata et al. (1997b), in which more precise measurements of the -spectra of the noble gases Ar, Kr and Xe were reported. In that work, a static Hartree-Fock calculation was successfully used to identify the inner-shell electron contribution to the measured spectra. However, owing to the fact that it ignored the state dependent enhancement, the relative fraction of annihilations with inner-shell electrons was not predicted accurately by this theory. Almost a decade later, Dunlop and Gribakin Dunlop and Gribakin (2006) used many-body theory to explain the main features of the -spectra of the noble gases. In that work, however, only annihilation on the valence shells was considered, the short-range electron-positron interaction was treated only in first-order, and positron-atom correlations were ignored; significant discrepancies between the theoretical results and experiment were evident. Dunlop Dunlop (2005) later attempted to calculate the core contribution to the spectra via a semi-empirical fitting procedure. For this, the annihilation on the valence shells was calculated with the full incorporation of short-range electron-positron correlations, whereas the core was treated in the simple independent-particle-model (IPA) approximation. It will be shown however, that neglecting the core enhancement in this way leads to an overestimate of the core contribution to the -spectrum.
The purpose of this work is three-fold: (1) to examine in detail the effect of electron-positron correlations on the process of annihilation on tightly-bound core electrons of many-electron atoms (using the noble gases as test cases) and in doing so; (2) fully explain the measured spectra of the noble gases and establish the fraction of annihilation on individual core and valence subshells, and finally; (3) to calculate the ’true’ core-electron enhancement factors (and demonstrate that they can be used to correct relatively simple independent-particle-approximation calculations to produce accurate spectra).
The structure of the paper is as follows. In Sec. II we outline the many-body theory of annihilation -spectra, detailing the form of the fully correlated incident positron wave function and annihilation vertex. In Sec. III we briefly discuss the numerical implementation of the theory. In Sec. IV we present our calculated annihilation spectra for Ar, Kr and Xe, comparing them with the experimental results of Iwata et al. Iwata et al. (1997b). In addition, the partial are calculated for each subshell. In the final part, we calculate vertex enhancement factors and demonstrate how they can be used in independent-particle-model calculations to obtain accurate annihilation spectra. A summary is given in Sec. V.
Ii Many-body theory of annihilation -spectra
The process of electron-positron annihilation is described fundamentally by
quantum electrodynamics Akhiezer and Berestetskii (1982); Berestetskii et al. (1982).
It dictates that low-energy positron annihilation proceeds predominately via two photon production,
a process in which the total spin of the electron-positron system must be zero Akhiezer and Berestetskii (1982); Berestetskii et al. (1982).
Consider then a positron with momentum annihilating in a many-electron system with an electron in state
where is the true, fully-correlated many-particle wave function of the initial state of electrons (coordinates ) and the positron (coordinate ), and is that of the final state of system with electrons and a hole in state . At large positron-atom separations the initial particle wave function has the form
which is normalized to the positron plane wave, where is the atomic ground state wave function. In the centre-of-mass frame, where the total momentum is zero, the two photons propagate in opposite directions and have equal energies , where and denote the energy of the initial and final states (excluding rest mass). When 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 , where is the centre-of-mass velocity of the electron-positron pair and is the angle between the direction of the photon velocity and . Assuming that , and , the shift of the photon energy from the centre of the line, , is
The typical momenta of electrons bound with energy determine the Doppler width of the annihilation spectrum, . Hence the shift of the line centre from keV can usually be neglected, even for the core electrons. The annihilation -ray (Doppler) spectrum can then be written in a form similar to a Compton profile, as the projection of the annihilation probability:
For gaseous systems, however, the experimentally accessible quantity is the average of this over the direction of emission of the annihilation photons . Also, to compare with experiments that use positrons confined in a trap, one must also average over the direction of the incident positron momentum giving
where . The annihilation amplitude is therefore central to the evaluation of the annihilation -spectra. Its calculation from many-body theory is outlined below. In most experiments, including those to which we compare our results, the annihilation photons are not detected in coincidence with the final state of the atom. The observed spectrum is then given by a sum over the electronic states of the system: . As we shall see, the total spectrum still retains distinct characteristics of the individual final states.
A useful quantity that is related to the -spectra is the normalized annihilation rate parameter . It is defined as the ratio of the true annihilation rate to the spin-averaged Dirac annihilation rate on a free electron gas of number density , Fraser (1968); Pomeranchuk (1949)
where is the classical radius of the electron (in CGS units). If the electrons belong to the atoms in a gas of number density , then , where is the number of electrons per ion. The corresponding spin-averaged annihilation cross section is
where is the positron velocity. By definition, gives the effective number of electrons that contribute to the annihilation. In this way it provides a measure of the strength of the correlations. It can be determined from the annihilation -ray Doppler spectrum through the relation
The measurements to which we will compare our calculations are those of Surko and co-workers of the San Diego group Iwata et al. (1997b). These experiments used a positron trap and low gas densities. In this regime, the parameter is independent of .
ii.2 Many-body expansion of the annihilation amplitude
The many-body theory of positron annihilation -spectra was introduced in Dunlop and Gribakin (2006). In that work however, only the zeroth- and first-order correction to the annihilation vertex were discussed and the important effect of positron-atom correlations on the incoming positron wave function were neglected. In this section we give a concise overview of the theory. We present formulae for the full annihilation vertex and briefly discuss the treatment of the modified positron wave function.
Taking the non-relativistic limit of the two-photon QED annihilation amplitude leads to the second quantized operator of two-photon annihilation DeBenedetti et al. (1950); Lee (1957); Ferrell (1956); Dunlop and Gribakin (2006)
where and are the electron and positron field annihilation operators respectively
where and are the HF wave functions (of central-field type), and and are the HF field annihilation operators, of the electron and positron. In this basis the operator of annihilation becomes:
where we use the notation
The Hamiltonian of the system is
where is the Hartree-Fock Hamiltonian that satisfies , where is the -electron ground-state HF determinant wave function. The many-body expansion of proceeds through the interaction Hamiltonian , where the residual interaction is the sum of the standard two-body electron-positron and electron-electron Coulomb operators minus the one-body HF operator (for the frozen core of the -electron system).
The resultant diagrammatic expansion is shown in Fig. 1 (see the figure caption for an explanation of the diagrams). The construction of the diagrams follows standard rules. First, note that the use of HF single-particle basis ensures that only pure Coulomb operators appear in the final diagrams, i.e., there are no explicit contributions from . Furthermore, since there is a single incident positron, only terms in the expansion involving a single are non-zero Green (2011). Single-hole final-state diagrams at order in the Coulomb interaction (wavy line), where is the number of electron-positron Coulomb interactions and is the number of electron-electron Coulomb interactions, have positron propagators (single directed lines) and electron propagators, in addition to a single positron and hole wave function (labelled and respectively), and the two -rays (double-dashed lines).
The annihilation vertex
Figure 2 shows the main contributions to the annihilation vertex: the zeroth-order (IPA) vertex, as well as the non-local first-order and the positron-electron ladder series, or ’-block’, corrections. From inspection of the diagrammatic expansion, one can write the exact annihilation amplitude in the abstract form
where the first term is the zeroth-order (IPA) vertex (Fig. 2a), and is the non-local annihilation kernel. In this form, the amplitude is clearly seen to be a Fourier transform of the correlated pair wave function (the term in the brace), whose modulus squared can be referred to as the electron-positron annihilation momentum density. In the HF basis it takes the explicit form (using the notation of Eq. 13)
where the vertex operator
where the sum over and is over the excited electron and positron states. Equation 16 is written to emphasize the components common to all diagrams and is in accordance with the general diagrammatic form shown in Figs. 1 and 2: the external positron wave function (labelled ), linked through the general vertex to the annihilated electron (hole) wave function (labelled ) and the two-gamma rays . Note that the zeroth-order (IPA) result is just that of Eq. 13, i.e., . The -block is determined from the linear integral equation, arrived at trivially by inspection of its diagrammatic form (shown in Fig. 2)
where is a number close to zero. The successful computation of the -block correction to the vertex was first performed in this way in Gribakin and Ludlow (2004), in which calculations of for positron annihilation on hydrogen were reported. It was later incorporated in the calculations of the -spectra for annihilation on valence shells of the noble gases Dunlop (2005), and recently to annihilation on H-like ions Green and Gribakin (2013). It can give rise to a significant enhancement above that of the first-order correction.
The Hartree-Fock wave functions involved in the matrix elements are of central field type . The incident positron wave function is a continuum state of the form
where is normalized to a delta function of energy (in Rydbergs). The positron radial wave function is modified from the HF one by the positron-atom correlations, as explained below.
Positron ‘Dyson’ quasiparticle wave function
The correlated incident positron quasiparticle wave function can be calculated from the integral form of the Dyson equation for the positron Green’s function (see, e.g., Abrikosov et al. (1975); Fetter and Walecka (2003)):
where is the unperturbed Hamiltonian of the particle system, which we take in this work to be the Hartree-Fock Hamiltonian, and is the non-local correlation potential. It is an integral operator that acts as , and is equivalent to the energy-dependent irreducible self-energy of the positron in the presence of the atom.
The main contributions to are shown in Fig. 3. At large positron-atom distances reduces to the local dipole polarization potential , where is the dipole polarizability of the atom in the HF approximation
Formally, the quasiparticle wave function (or ’Dyson orbital’) is given by
i.e., it is the projection of the fully-correlated particle wave function (of energy ) onto the -electron atomic ground state of (of energy ), and . It can be calculated as outlined in Ref. Gribakin and Ludlow (2004), as
where is the reducible self-energy and where ’’ means the principal value of the integral. To normalise this wave function to a delta-function in energy (Rydbergs), it must be multiplied by the normalisation factor
Overall, this method of determining the positron quasiparticle wave function is equivalent to solving the Schrödinger equation with an optical potential Bell and Squires (1959).
Iii Numerical implementation
The numerical procedure for evaluating the diagrams has been successfully demonstrated previously in Refs. Gribakin and Ludlow (2004) and Dunlop and Gribakin (2006), where it was used to calculate for annihilation on atomic hydrogen, and the -spectra for annihilation on the valence shells of the noble gases. The electrons involved in these studies have relatively small ionization energies, compared to the core-electrons that are of interest here. More recently, the authors performed a study of positron annihilation on hydrogen-like ions Green and Gribakin (2013). The ions considered there had electron ionization energies similar to those of typical core electrons of many-electron atoms. Using other, variational results as a benchmark, this study provided verification of the applicability and accuracy of the numerical method to annihilation with electrons of relatively large binding energies. Full details of the numerical procedure can be found in the afforementioned works. Here, we aim to reduce repetition to the minimum compatible with the purpose of this section.
The main quantities to be calculated are the self-energy, the annihilation amplitude, and the -spectra , whose angular reduced formulae are given in Appendix A and Ref. Gribakin and Ludlow (2004). To evaluate the various many-body diagrams, one must first generate a set of single-particle Hartree-Fock basis states. For the electron, this basis consists of the atomic ground-state wave functions as well as excited electrons in the field of the -electron frozen core of the atom. For the positron, the static-field positron-atom potential is repulsive, and all states are continuum states in the field of the -electron frozen core of the atom. To evaluate these, we use a mesh in momentum space of 201 points starting from threshold with a step size of a.u.
To facilitate the calculation of the diagrams, particularly those involving the ladder block, we introduce a B-spline basis de Boor (1977); Sapirstein and Johnson (1996); Bachau et al. (2001); Gribakin and Ludlow (2004); Dunlop and Gribakin (2006); Green and Gribakin (2013). B-splines of order are piecewise polynomials of degree defined over a restricted domain (‘box’) that is divided into segments by a knot sequence of points: , where is the number of splines in the basis de Boor (1977). In this work, we use an exponential knot sequence defined as , where a.u. and is determined by the condition , the box size. It allows for the accurate description of both the ground-state atomic wave functions, which can have many oscillations over the typical size of the atom and rapidly vanish outside the atom, as well as the continuum states that extend to larger distances up to the confinement (box) radius. The choice of the exponential knot sequence also ensures rapid convergence in the summations over intermediate states in the diagrams. The choice of spline basis parameters and box size is discussed below.
By expanding the single-particle HF orbitals , where are the expansion coefficients, and is the -th spline of the basis, the Schrödinger equation can be reduced to the discrete eigenvalue problem , where and . Note that to implement the boundary conditions the first and last splines are discarded in the expansions. The solutions of the equation for a given angular momentum are a set of eigenfunctions. For the electron, the lowest energy states from this set correspond directly to the ground-state wave functions. The remaining electron states of the set describe excited electrons propagating in the frozen-core Hartree-Fock field of the ground-state electrons. The solutions for the positron are continuum states that describe the positron propagating in the field of the HF frozen core of the -electrons.
The choice of optimal spline basis parameters and box size depends on a number of competing considerations. These have been discussed in detail in Refs. Green and Gribakin (2013) and Gribakin and Ludlow (2004). For example, the vertex corrections to the annihilation amplitude involve small electron-positron distances and their accurate evaluation therefore requires an adequate minimum spatial resolution. Increased resolution can be obtained by either using a basis with a larger number of splines, or by reducing the box size, or both. For a given box size, however, increasing the number of splines in the basis reduces the density of states: a greater number of states must be included in the numerical calculations to achieve the same energy coverage and reach convergence. On the other hand, the positron-atom correlations involve are of longer-range and their accurate evaluation requires a large box size.
The calculations were performed using a basis of splines of order , and repeated using a basis of splines of order . For both spline bases, two different box sizes, a.u. and a.u., were used. We found that the , core-subshell results suffered from poor convergence, and that the best overall results were obtained using the basis of splines of order with a box size of a.u. The lowest 20 B-splines of the basis were included in the summations over intermediate states, corresponding to an energy coverage from zero to a.u., significantly greater than the energy of the valence and subvalence (core) ground-state wave functions that we are interested in (see Fig. 4). All results quoted henceforth have been calculated with these parameters.
When evaluating matrix elements involving the incident positron wave function , e.g., the self-energy matrix , or the annihilation amplitude , we use the true HF continuum states . This can be done by making further use of the spline completeness, introducing a resolution of the identity to re-write the matrix elements as, e.g.,
The insertions of , where , are made to nullify any numerical error arising due to the fact that the B-splines are zero at the boundary , whereas the Hartree-Fock states are not. In this way, the matrix elements of all quantities involving the true positron states can be evaluated as matrix elements involving individual spline basis states.
In the calculation of the positron self-energy (Fig. 3) we include only those hole states in the diagrams that correspond to the two valence electrons, owing to the negligible contribution to the dipole polarizability of the atom from the strongly-bound core electrons. Any small effect of omitting the core states from the self-energy diagrams will manifest only in the positron wave function, and should have a negligible effect on the vertex enhancement. In the diagrams of the annihilation amplitude the final hole state is, of course, taken to be any of the ground states, including the core subshells.
The finite box size leads to a cut-off in the self-energy matrix element: it is evaluated only over the domain of the box. To compensate for this, we make use of the long-range asymptotic form of the correlation potential and add to the spline-based matrix element (calculated over the domain of the box) the contribution at :
where the asymptotic radial wave functions are calculated with the HF phase shift as
where and are the spherical Bessel and spherical Neuman functions, respectively.
For the calculation of the annihilation momentum distribution we used a mesh in the -ray momentum of 101 points from 0 to 9 a.u. We also repeated the calculations with 101 points from 0 to 15 a.u., although no discernible difference was found in the resulting -spectra in the range of Doppler-shifts of interest (0–9 keV).
Finally, convergence with respect to the highest angular momentum of the intermediate excited states used in the calculation, , was achieved using the asymptotic formula Gribakin and Ludlow (2002); Dunlop and Gribakin (2006); Green and Gribakin (2013)
where, for a given , is a constant that is determined from the results of the calculations performed using different . In this work all diagrams were calculated for and .
Iv Results and Discussion
In this section we present the results of the MBT calculations. In Sec. IV.1 we discuss the effect of the correlations on the spectra of individual core and valence subshells of the noble gases Ar, Kr and Xe, and compare our calculated annihilation -spectra with the existing experimental spectra. In addition, we calculate the total, and fractional contributions to, the normalized annihilation rate parameter for each subshell. In Sec. IV.2, we use the results of the ab initio many-body theory calculations to provide state-dependent enhancement factors for the core and valence electrons that quantify the vertex enhancement to the annihilation vertex. All the results presented have been calculated using an -wave incident positron of thermal momentum ( a.u.).
iv.1 Annihilation -spectra of noble gases
Many-body calculated annihilation -spectra of valence and subvalence shells of Ar, Kr and Xe
The effects of the correlations on the -spectra are shown in Fig. 5, for the specific examples of the valence and subvalence -subshells of Ar, Kr and Xe, respectively. For each subshell, the spectra calculated using both the HF (static) and Dyson incident positron in the three different approximations of the annihilation vertex (see Fig. 2) are shown. As the spectra are symmetric about zero energy shift, only positive Doppler energy shifts are shown. Interpretation of the spectra is aided by Table 1, which gives the values of calculated as an integral of the spectra through Eq. 9.
For each atom, for a given annihilation vertex order, the spectra calculated using the Dyson positron wave function are significantly greater in magnitude than those calculated using the HF one, e.g., the spectrum of the valence subshell of Ar is increased above its Hartree-Fock result by 8 times; for Kr by times, and for Xe by as much as times. The reason for this wave function enhancement is the introduction of an attractive potential through the strong positron-atom correlations. This potential supports low-lying positron virtual levels, which give rise to the characteristic resonant behaviour: , where is the reciprocal of the scattering length , assumed here to be much larger than the radius at which annihilation occurs (typically the size of the atom) Goldanski and Sayasov (1968); Dzuba et al. (1993, 1996); Gribakin (2000). The relative strength of the wave function enhancement of Ar, Kr and Xe corresponds to the increase in their respective scattering lengths: for Ar a.u., for Kr a.u., and for Xe a.u. Ludlow (2003). Incidentally, this resonant behaviour means that drops off dramatically as increases, and thus a proper comparison with the experimental measured using thermalized positrons can only be made after performing a Maxwellian average of the calculated over . Since it is the calculation of the -spectra that is the main interest here, this fact is not of utmost concern. Nevertheless, for comparison’s sake the Maxwellian averaged of the valence shells, calculated previously using MBT Ludlow (2003), are quoted in the table.
For a given atom, deeper lying subshells experience a greater wave function enhancement. For example, for a given vertex order the Dyson spectrum for the Ar valence subshell is 8 times larger than the HF spectrum, compared to the subvalence , for which it is 13 times larger. For Xe, the valence subshell is 70 times larger than the HF spectrum, compared to the subvalence , for which it is 160 times larger. This is a result of the fact that the positron must tunnel through a repulsive potential to reach the core electrons. The additional attraction from the positron-atom correlations effectively lowers the HF repulsive potential: the wave function dies away more slowly in the tunnelling region, and consequently, the overlap of the positron wave function and the core electrons is increased. Comparatively, the overlap of the positron and valence electron wave functions is relatively unchanged.
|Subshell||-spectra FWHM (keV)|
Concentrating now on the effect of the vertex corrections, it is clear that the addition of the first and -block vertex corrections enhances the spectra above the zeroth-order IPA result. Fundamentally, this is due to the increase of the annihilation probability owing to the stronger attraction between the annihilating pair. This attraction is of particular importance, given the fact that non-relativistic annihilation occurs when the positions of the positron and electron coincide. For both the core and valence shells, the addition of the first-order correction to the IPA vertex increases the magnitude of the spectra considerably: e.g., for the valence and subvalence subshells of Kr the spectra is increased at zero energy shift respectively by times from 4.36 to 11.44 keV, and by times from 0.025 to 0.034 keV. For the valence shells, the addition of the higher order -block correction causes a further significant enhancement: e.g., for the subshell of Kr, it produces a further 2.3 times (from 11.44 to 27.6 keV) enhancement to the first-order result. For the core however, the result of adding the -block correction is much less dramatic, only raising the spectra slightly above that obtained with the first-order correction: e.g., for the subshell of Kr, the -block correction only increases the magnitude of the first-order result by a further 1.05 times (from 0.0340 to 0.0356 keV). A similar dominance of the first-order correction over the -block correction was also seen for annihilation with the tightly-bound electrons of the H-like ions in Green and Gribakin (2013). Physically, the -block describes virtual positronium formation by the tunnelling of an electron from the atom to the positron. This process is more difficult for the tightly bound spatially compact core than for the valence electrons.
From the figures it is also clear that for a given subshell, the enhancement from the vertex corrections is similar whether the HF or Dyson positron wave function is used. Physically, this is to be expected, as the vertex corrections are to a large extent independent of the positron-atom correlation. We will discuss the vertex enhancement further in Sec. IV.2, where we evaluate enhancement factors that quantify the important vertex enhancement and show how they can be used in simple IPA type calculations to calculate accurate annihilation -spectra.
The correlations, both wave function and vertex type, evidently have a dramatic effect on the magnitude of the -spectra and the annihilation rate. Their effect on the shape of the spectrum is less so, although it is still significant. Fig. 6 shows the spectrum of the Xe subshell in the different approximations to the vertex and positron wave function. For ease of comparison, all curves in the figure have been scaled to unity at zero energy shift. In addition, values of the full-width at half-maximum for all of the individual subshells calculated in the different approximations are given in Table 1. For the valence subshells, it is clear that increasing the self-energy corrections in the wave function for a given vertex order affects a broadening in the spectra: the attractive positron-atom correlations accelerate the positron towards the electron resulting in a relatively large momentum distribution, and hence a broader -spectrum. In contrast, increasing the vertex order affects a narrowing of the spectra: the first- and higher-order annihilation vertex corrections involve excited state (virtual) electrons that are relatively diffuse and therefore produce a relatively narrower momentum distribution, and hence a narrower -spectrum. The degree of cancellation that occurs between this competing narrowing and broadening leads to a relatively small change in the full-width at half-maximum (FWHM) between the simplest zeroth-order-vertex HF-wave function result and the full MBT (0+1+)-vertex Dyson-wave function result. This explains why one can obtain reasonable fractional core contributions through semi-empirical fitting of the zeroth-order HF-wave function calculated core and valence spectra to the total Iwata et al. (1997b). In contrast, the FWHM of the deeper lying subshells, e.g., the subshell, does not change noticeably. In summary, overall there is clearly some dependence of the FWHM on the approximation used, and reliable core contributions to the total spectra can therefore only be calculated by taking full account of the correlations.
Fig. 7 shows the -spectra for the valence , subvalence and shells for Ar, Kr and Xe, calculated using the full many-body theory (Dyson incident positron and full annihilation vertex) for -wave incident positron. The corresponding tabulated values are given in Appendix B. For each of the noble gases shown, the valence -subshell is the main component of the total -spectra at small Doppler energy shifts ( keV). This subshell is the most diffuse of the ground states. By considerations of the uncertainty principle, it therefore contributes to the spectrum mainly at low momenta (small energy shifts). Similarly, at small energy shifts, the magnitude of the valence shells is much greater than that of the subvalence or deeper lying shells, e.g., the valence shell of Kr is more than two orders of magnitude greater than that of the subvalence shell, which is a further three orders of magnitude greater than the next most inner shell. The need for the positron to overcome an increasing nuclear repulsion as it penetrates closer to the core causes this reduction in the probability of annihilation.
The characteristic shapes of the individual subshell spectra are governed by the properties of the annihilated electrons. As the angular momentum of the annihilated electron increases, the spectra become flatter at small energy shifts. Indeed, for an -wave positron it is easy to show from the first-order result (of the form of Eq. 13) and the small argument limit of the Bessel functions that the spectra behave as for energy shifts . Other features include the multiple ‘bell’ shapes, or ‘shoulders’, evident in the individual spectra. Looking at the subvalence subshell of Ar in particular, three distinct bell shapes are visible: the first in the region 0–2 keV, the second, with a much larger width, in the region 2–8 keV, and the third in the region keV. The number of distinct bell shapes in a given spectra is determined by the number of non-zero regions of energy density in the electron’s radial wave function, . They can be understood by appealing to the uncertainty principle. For example, the radial wave function of Ar, which has 2 nodes, has three regions of non-zero wave function density. wave function density at small radii manifests in the high-momentum (energy) regions of the spectra. The regions of non zero wave function density at larger will contribute to the small momentum (energy regions) of the spectra. A further feature is that at large Doppler shifts the shape of the spectrum of the valence -subshell, e.g., for Ar, the spectrum of the valence subshell follows that of the subvalence subshell; the requirement of orthogonality forces the and wave functions to be similar at small radii. Accordingly, their spectra are similar are large energy shifts (momenta).
Table 2 shows the contribution of individual subshells to the total spectrum as a fraction of unity. Values are given for both the simplest calculation that uses a HF incident positron with the IPA vertex, and for the full MBT calculation (full vertex with Dyson incident positron wave function). For all of the noble gases, the inclusion of the electron positron correlations result in an increased fraction of annihilation on the valence shells, compared to the core. In fact, the HF-IPA calculation overestimates the total core contribution ’ tot.’ by a factor of . This is a result of the varying strength of the state dependent enhancement, which is stronger for the valence than for the inner shells. Also shown in the table are the results of the semi-empirical fitting of Iwata et al. (1997b), in which the fractions were calculated by adjusting the ratio of HF-IPA inner to valence shell spectra in the total. For Kr and Xe this simplistic method gives fractional contributions that are of similar order to the parameterless MBT results, owing to the robustness of the shape of the spectra to the effect of correlations as discussed above (see Fig. 6). This method cannot however, produce individual spectra of accurate magnitude like those of Fig. 7. Such spectra are required to calculate state-dependent -spectra enhancement factors.
Comparison with experimental -spectra: the core-electron contribution to the spectra
where , is a normalization constant and keV is the width of the detector response, and scaling to the experimental spectra at zero energy shift. Specifically, the figures show the individual contributions of the core and valence shells to the total spectra, calculated with the full MBT, the experimental results with associated errors, and the result of a simple HF-IPA calculation. In the experimental results, the error bars represent the expected statistical variation in spectral amplitude due to counting statistics. Only the positive energy shifts of the symmetric -ray line is shown, since this region has greater statistical significance due to the reduced Compton scattering component Iwata et al. (1997b). In each figure the MBT calculated ‘valence’ spectra are the sum of the spectra of the and subshells. As was seen in Fig. 7, the contribution of the and more inner shells are negligible compared to the subvalence shell. In Figs. 8, 9 and 10 the ‘inner’ spectra are therefore calculated as just the sum of the spectra for the individual subvalence subshells: , or just for Ar. The relatively diffuse valence electrons, which have a narrow momentum distribution, dominate the spectrum at small energy shifts. This is clear from the figures, which show that the spectrum calculated from the valence shells alone agrees well with experiment at Doppler energy shifts keV, but is insufficient at larger energy shifts. In contrast, the core electrons are spatially compact, and therefore have a broader momentum distribution compared with the diffuse valence electrons. They contribute markedly to the high-energy ‘wings’ of the spectrum, and their importance in bringing the total inline with experiment is evident for all three noble gases shown. For Ar, the inclusion of the core brings theory into excellent agreement with experiment over the full range of energy shifts. For Kr the valence shell contribution alone severely underestimates the total spectrum, and the core contribution is of even greater importance than in Ar, dominating over the valence at energies keV. Again the overall agreement of the spectra with experiment is excellent, although at higher energies the theoretical total slightly underestimates the experimental spectrum. In Xe, the valence shell again severely underestimates the total spectrum. The core contribution dominates over the valence at energies keV, and is again seen to be vital in achieving agreement with experiment. Similar to the case of Kr, however, the theoretical spectrum slightly underestimates the experimental one at large energy shifts, and is ‘steeper’ than experiment. Since the theoretical results have been scaled to experiment, this suggests that the ratio of small to large momentum components is overestimated.
One possible reason for the slight discrepancy is the neglect of relativistic effects on the electron wave functions.
Indeed, Cook et al. Cook et al. (1984) have shown that the momentum distributions of the valence Xe subshells show clear and direct relativistic effects manifestations of relativistic effects.
In general, the relativistic core orbitals will be more spatially compact than their non-relativistic counterparts and will have a correspondingly larger momentum density
One further possible source of the discrepancy arises from the fact that, in order to be completely accurate, one should perform a Maxwellian average of the magnitude of the raw -spectra over the incident positron momentum , i.e., one should calculate the thermal averaged spectra given by
where and is the Boltzman constant.
This is likely to be broader than the spectra calculated at a.u.
Regardless, the discrepancy between the theoretical total and the experiment at the high energy shifts in Kr and Xe is small, and despite it the MBT results show unequivocally that: (1) the valence shells alone are insufficient to describe the total -spectra of Ar, Kr and Xe, and that; (2) there is a significant contribution from the relatively broad core spectrum to the total, and in addition, that this contribution is crucial to obtain agreement with the experimental spectra.
iv.2 Vertex enhancement factors for annihilation -spectra
It should by now be clear that the correlation corrections to the annihilation amplitude must be incorporated in an accurate calculation of the annihilation -spectra of the valence and core subshells. The full many-body theory calculation is, however, non-trivial, and its application numerically intensive. To interpret the results of, e.g., PAES experiments, one needs some way of predicting accurate core and valence annihilation spectra. Here, we appeal to the predilection of the literature and use the many-body theory to calculate enhancement factors. These factors quantify the enhancement that results from the non-local corrections to the annihilation vertex. Such factors are common in positron-atom, positronium Mitroy (2005); Gribakin and Ludlow (2004), positron-charged-ion Bonderup et al. (1979); Novikov et al. (2004) and positron-condensed-matter studies Boroński and Nieminen (1986); Puska and Nieminen (1994); Alatalo et al. (1995); Barbiellini et al. (1997a, b); Alatalo et al. (1996); Makkonen et al. (2006); Kuriplach et al. (1998); Tang et al. (1998). They are, however, usually calculated in some phenomenological way. They can be determined from an ab initio viewpoint from many-body theory, simply by calculating the required quantities in different approximations to the vertex. and can be incorporated in simple IPA calculations to obtain accurate annihilation spectra.
where is the state-dependent enhancement factor, usually interpreted as modifying the electron density at the positron (the square root is for later convenience). The calculation of from many-body theory necessitates the reduction of our non-local annihilation amplitude to the IPA form of Eq. 32. Formally, using Eq. 16 and Eq. 32, the true enhancement factor is given by
The presence of nodes in the wave functions of the denominator, however, render this quantity of limited use and we must opt for a more pragmatic approach.
The enhancement is in general dependent on the two- momentum Green and Gribakin (2013). However, we will see that to all intensive purposes this momentum dependence does not carry through to the annihilation -spectra, at least for the core levels. It is therefore instructive to define a more simplistic state-dependent enhancement factor as the ratio of the calculated true partial annihilation rate (evaluated through Eq. 9) to that calculated using the zeroth-order (IPA) vetex;
where the superscript denotes the vertex order (see Fig. 2) and labels the electron subshell. This parameterization is commonly used to predict true annihilation rates and -spectra (see, e.g., Alatalo et al. (1995); Barbiellini et al. (1997a, b); Alatalo et al. (1996); Makkonen et al. (2006); Kuriplach et al. (1998)). The true spectra for annihilation on a given subshell is then to be approximated by
This procedure therefore assumes the vertex enhancement is independent of the Doppler-energy shift.
Table 3 shows the values of for the valence and subvalence shells of Ar, Kr and Xe, calculated from the MBT using both the HF and Dyson incident positron. For the valence electrons, the vertex enhancement is dramatic, e.g., it produces a five-times increase in the spectral magnitude of the diffuse -subshell of Ar, and a nine-times increase for that of Xe (for the HF positron).
For the subvalence (core) subshells the factors satisfy 1, as one should expect. They range from 1.35 (Ar 2s) to 1.87 (Xe 4d). These values are appreciably different from unity, and highlight the necessary inclusion of correlations in the accurate calculation of the core, as well as the valence, -spectra.
Reconstruction of the true -spectra using these enhancement factors via Eq. 35 is demonstrated in Fig. 11 for the valence and subvalence subshells of Kr. Similar results are obtained for Ar and Xe Green (2011). For the subvalence core (and the valence ) subshells, the energy-independent enhancement factors are seen to successfully reconstruct the true spectra. However, the reconstructed spectrum of the valence -subshell overestimates the full MBT result at large Doppler shifts and underestimates at small ones. This is a result of the fact that both non-zero wave function density regions (‘lobes’) in the radial wave function are scaled by the energy- and momentum- independent factor equivalently. Since the core enhancement is smaller than the valence, we expect the lobe at smaller radial distances, which contributes to the large energy (momentum) regions of the spectrum, to scale by less than the lobe at larger , which contributes to the small energy regions of the spectrum.
In spite of these complications for the outermost valence-shells, the simple enhancement factors work remarkably well for the core shells. In Fig. 12 we have plotted the enhancement factors of Table 3 against the inverse ionization energy of the subshell. The enhancement factor decreases in magnitude as the binding energy increases.
As we have seen, the vertex enhancement to the core spectra is dominated by the first-order correction; higher order corrections result in a minimal amount of further enhancement (see Fig. 5 and Table 1). One can estimate the strength of the short-range correlational enhancement from the first-order correction as the ratio , where is the electron-positron Coulomb interaction potential, and is a measure of the response of the electrons to the perturbation (of the order of the respective electron’s binding energy). For positron annihilation on H-like ions, this strength scales as , where is the nuclear charge Green and Gribakin (2013). For annihilation on the core electrons of many-electron atoms the strength can be thought to scale analogously as , where is the ionization potential of the subshell. One also arrives at a dependence from simplifications to a calculation that explicitly includes two-particle Green’s function of the electron-positron pair in the annihilation vertex Green (2011). In light of this, we have fitted the function
to the calculated enhancement factors of Table 3, where , and 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 is needed to describe the enhancement of the valence electrons, for which higher-order corrections are of as equal importance as the first-order one. The solid line in Fig. 12 is Eq. 36 fitted to the HF enhancement factors yielding the values eV (1.32 a.u.), eV (0.99 a.u.), and . The dashed line is a fit of the function to the enhancement factors calculated with the Dyson incident positron wave function yielding the values eV (1.60 a.u.), eV (0.75 a.u.), and .
Finally, Fig. 13 shows the values of the enhancement factors plotted against the ionization energy of the subshell, as well as the functional form Eq. 36 extrapolated to large . The dependence for the core shells is evident. We also show the enhancements calculated for annihilation on Hydrogen and the H-like ions He, Li, B and F. Given the inherent difference between the positive ions and the neutral systems, the ions follow the trend surprisingly well. For positrons annihilating in condensed matter systems, the longer-range positron-atom correlations included in the Dyson positron quasiparticle wave function average to zero. In this case the HF vertex enhancement factors, which neglect these longer-range type correlations in the positron wave function, would be expected to be more appropriate.
V Conclusion and outlook
-spectra for positron annihilation on the core and valence subshells of the noble gases Ar, Kr and Xe has been calculated using a many-body theory framework. We have shown that the short-range electron-positron correlations, described through corrections to the zeroth-order (independent-particle-model) annihilation vertex, significantly enhance the -spectra magnitude and annihilation rate parameter . This enhancement is stronger for the valence shells, but it is still significant for the core shells (e.g., for the Xe subshell). Despite the small probability of annihilation on the core subshells, we have shown that their contribution to the total annihilation -ray spectra is vital for accurate description of experiment across the full range of Doppler-shift energies, and particularly in the high-energy ‘wings’. We also use the atomic MBT to calculate true vertex enhancement factors that take account of the non-local corrections to the annihilation vertex. We have shown that for a given subshell , these factors can be described by a simple and physically motivated function of the subshell ionization energy : , where , and are positive constants. We have demonstrated the successful reconstruction of the core spectra using these enhancement factors to augment an independent-particle-model calculation. We suggest that this formula can be used to calculate the -ray spectra for positron annihilation on the core electrons of atoms across the periodic table, and for the atomic-like core electrons of condensed matter systems. This work, taken together with Refs. Green et al. (2014) and Green and Gribakin (2015), gives a (near) complete description of the positron noble gas system.
Looking to the future, there is still much work to be done to advance the theory and application of positron annihilation -spectra.
First, development and implementation of a fully relativistic many-body theory would shed light on the discrepancies between theory and experiment in the high-energy ‘wings’ of the -spectra of Kr and Xe.
Moreover, accurate predictions of positron binding and annihilation in complex molecules are required to develop new methods of vibrational spectroscopy (Gribakin et al., 2010), to help explain the origin of the strong -ray signal from the galactic centre (Prantzos et al., 2011), and are essential to realise spectroscopic PET scanning
Appendix A Evaluation of -spectra from angular decomposition of the annihilation amplitude
The diagrammatic expansion of the annihilation amplitude has been given in Fig. 2. Its corresponding analytic form was given in Eq. 16. It is a sum of three terms, each of which is in one-to-one correspondance with the diagrams in Fig. 2. Writing the amplitude in angular decomposed form (making use of graphical angular momenta techniques Varshalovich et al. (1988)) and summing over the magnetic quantum numbers of the subshell , one can evaluate the -spectrum as
and the reduced vertex is given by
In this expression the reduced delta-function element (cf. Eq. 13) is defined by
where is a spherical Bessel function, and we use the notation . The reduced Coulomb matrix element for a particle pair propagating with coupled angular momenta is
where the reduced Coulomb matrix