Ab initio QED treatment of the two-photon annihilation of positrons with bound electrons

TL;DR

This paper develops a fully relativistic ab initio QED framework to accurately describe positron-bound electron annihilation with two-photon emission, resolving previous theoretical-experimental discrepancies for silver atoms.

Contribution

It introduces a novel ab initio QED approach for positron-electron annihilation, improving accuracy over previous approximations and resolving longstanding disagreements.

Findings

Resolved the discrepancy between theory and experiment for silver atoms.

Demonstrated significant differences from previous approximation methods.

Provided a more accurate relativistic description of the annihilation process.

Abstract

The process of a positron $\unicode{x2013}$ bound-electron annihilation with simultaneous emission of two photons is investigated theoretically. A fully relativistic formalism based on $\textit{ab initio}$ QED description of the process is worked out. The developed approach is applied to evaluate the annihilation of a positron with $K$-shell electrons of a silver atom, for which a strong contradiction between theory and experiment was previously stated. The results obtained here resolve this long-standing disagreement and, moreover, demonstrate a sizeable difference with approaches so far used for calculations of the positron $\unicode{x2013}$ bound-electron annihilation process, namely, the $\textit{Lee's}$ and $\textit{impulse}$ approximations.

Tables

Table 1. Table 1: Detector-averaged double differential angular cross section for the two-quantum annihilation of the 300 300 300 keV positron with the K 𝐾 K -shell electrons of a silver atom. The experimental value from Ref.  Nagatomo_PRL32_1158:1974 is multiplied by a factor 2 2 2 (see the text for details).

Approach	Detector-averaged DDACS (mbarn/sr2)
Impulse appr.	$240$
Lee’s appr.	$300$
Exact QED	$41\left(12\right)$
Experiment Nagatomo_PRL32_1158:1974	$15.4\left(12.8\right)$

Equations

$$\frac{d\sigma }{d{\mathbf{k}}_{1}d{\mathbf{k}}_2}=4{\alpha }^2\frac{(2\pi {)}^6}{v_i}{\left|\tau \right|}^2\delta (E_a+{\epsilon }_i-{\omega }_1-{\omega }_2),$$

$$\frac{d\sigma }{d{\Omega }_{1}d{\Omega }_2}=\int d{\omega }_{1}d{\omega }_2{\omega }_1^2{\omega }_2^2\frac{d\sigma }{d{\mathbf{k}}_{1}d{\mathbf{k}}_2}.$$

$$\tau =-\sum _n\left[\frac{⟨(-p_i{\mu }_i)\left|\mathbf{\alpha }\cdot {\mathbf{A}}_{{\mathbf{k}}_2{\lambda }_2}^{\ast }\right|n⟩⟨n\left|\mathbf{\alpha }\cdot {\mathbf{A}}_{{\mathbf{k}}_1{\lambda }_1}^{\ast }\right|a⟩}{E_a-{\omega }_1-E_n(1-i0)}+\frac{⟨(-p_i{\mu }_i)\left|\mathbf{\alpha }\cdot {\mathbf{A}}_{{\mathbf{k}}_1{\lambda }_1}^{\ast }\right|n⟩⟨n\left|\mathbf{\alpha }\cdot {\mathbf{A}}_{{\mathbf{k}}_2{\lambda }_2}^{\ast }\right|a⟩}{E_a-{\omega }_2-E_n(1-i0)}\right].$$

$${\mathbf{A}}_{\mathbf{k}\lambda }\equiv {\mathbf{A}}_{\mathbf{k}\lambda }(\mathbf{r})=\frac{{\mathbf{ϵ}}_{\lambda }{e}^{i\mathbf{k}\cdot \mathbf{r}}}{\sqrt{2\omega (2\pi {)}^3}}.$$

$${\tau }^{(\mathrm{Lee})}=\frac{i}{2}\left({\mathbf{k}}_1-{\mathbf{k}}_2\right)\cdot ⟨(-{\mathbf{p}}_i{\mu }_i{)}^{(\mathrm{SP})}\left|\left[{\mathbf{A}}_{{\mathbf{k}}_2{\lambda }_2}^{\ast }\times {\mathbf{A}}_{{\mathbf{k}}_1{\lambda }_1}^{\ast }\right]\right|a^{(\mathrm{SP})}⟩.$$

$$\frac{d{\sigma }^{(\mathrm{IA})}}{d{\Omega }_{1}d{\Omega }_2}=\frac{1}{v_i}\int d\mathbf{p}{\rho }_a(\mathbf{p})\frac{d{W}_{\mathrm{free}}}{d{\Omega }_{1}d{\Omega }_2}(\mathbf{p}).$$

Full text

Since the first observation of positrons Anderson_PR41_405:1932 , investigations of their interaction with atoms, molecules, and solids are of unaltered interest (see, e.g., Refs. Iwata_PRL79_39:1997 ; Weber_PRL83_4658:1999 ; Hunt_PRL86_5612:2001 ; Cizek_NJP14_035005:2012 and the review Surko_JPB38_R57:2005 ). Extensive investigations of the positron annihilation processes gave rise to numerous applications ranging from astrophysical researches Guessoum_AA436_171:2005 ; Lingenfelter_PRL103_031301:2009 ; Prantzos_RMP83_1001:2011 and positron-induced Auger-electron spectroscopy Weiss_RPC76_285:2007 ; Mayer_PRL105_207401:2010 to studies of the defects in metals and semiconductors Weiss ; Tuomisto_RMP85_1583:2013 , dynamics of catalysis Mayer_PRL105_207401:2010 and positron-emission tomography PET:2005 ; PET:2006 . In particular, the angular distribution of the photon pairs from the annihilation defines the spatial resolution of the defect analysis and tomography. Apart from the various applications, investigations of the positron annihilation with inner-shell electrons of heavy ionic (or atomic) targets can give a valuable insight into the kinematically analogous meson decays in quantum chromodynamics (QCD). In these decays, the QCD coupling constant is mimicked by the effective electromagnetic coupling constant being enlarged by the nuclear charge Greub_PRD52_4028:1995 ; Brodsky_HI209_83:2012 . Therefore, a quantitative understanding of the positron – bound-electron annihilation is highly requested by the on-going growth of studies considering positron – matter interaction as well as by upcoming positron facilities of a new generation, e.g., at Lawrence Livermore National Laboratory Chen_PRL102_105001:2009 ; Chen_PRL105_115003:2010 ; Piazza_RMP84_1177:2012 ; Sarri_PRL110_255002:2013 and ELI-NP Research Centre Hugenschmidt_APB106_241:2012 ; Djourelov_NIMA806_146:2016 .

The positron – bound-electron annihilation can proceed with the emission of one, two, or even more photons. More often than not, the two-quantum annihilation dominates over other channels. This process, however, has not been described rigorously within the framework of QED and with a proper account of the interaction with a nucleus yet. So far the calculations of the positron – bound-electron two-quantum annihilation were just based on two approximations: Lee’s approach Chang_ZETF33_365:1957 for ultra-slow (thermalized) and the impulse approximation for ultra-fast positrons. For slow positrons, the dominant contribution to the overall annihilation cross section with atomic targets arises from the nonrelativistic valence and outer shell electrons. These processes can be well described in the framework of the nonrelativistic Lee’s Chang_ZETF33_365:1957 approximation. On the basis of this approximation the theoretical approach which shows a remarkable agreement with related experimental studies was developed  Gribakin_PRA70_032720:2004 ; Dunlop_JPB39_1647:2006 ; Green_PRA90_032712:2014 ; Green_PRL114_093201:2015 .

On the basis of this approximation, the theoretical approach which shows a remarkable agreement with related experimental studies was developed Gribakin_PRA70_032720:2004 ; Dunlop_JPB39_1647:2006 ; Green_PRA90_032712:2014 ; Green_PRL114_093201:2015 . For ultra-fast positrons, in contrast, the impulse approximation can be applied, in which all particles are assumed to be free and where the active electron is represented by a stationary wave packet of superimposed plane-wave states. In this approximation, the annihilation process is based on the formulas which were derived almost a century ago by Dirac Dirac_ZP62_545:1930 and Tamm Tamm_ZP62_545:1930 . However, these two approximations cannot be applied to the annihilation of positrons with inner-shell electrons and for collision energies, at which the interaction with a nucleus plays a significant role. As an example, we refer to the experiment, where the two-quantum annihilation of $300$ keV positrons with $K$-shell electrons of silver was measured Nagatomo_PRL32_1158:1974 , and for which the theoretical cross sections by Gorshkov and coworkers Gorshkov_JETP45_17:1977 ; Drukarev differ by more than an order of magnitude.

Here, we develop a fully-relativistic formalism based on ab initio QED description of the two-quantum annihilation of positrons with bound electrons. In this formalism, positron- and electron-nucleus interaction is treated nonperturbatively. As the first application, we use the developed approach for the description of the two-quantum annihilation of positrons with $K$-shell electrons of silver which was studied experimentally in Ref. Nagatomo_PRL32_1158:1974 . Our double differential angular cross section of $41(12)$ mbarn/sr2 is in good agreement with the experimental value $15.4(12.8)$ mbarn/sr2, and which resolve long-standing disagreement between theory and experiment. Additionally, we compare the results of the developed exact approach with ones obtained within the Lee’s and impulse approximations as well as with result of Ref. Gorshkov_JETP45_17:1977 and discuss possible reasons of the discrepancies.

The differential cross section for the two-quantum annihilation of a positron with a bound electron in the relativistic units $\hbar=1$, $c=1$, $m=1$ is given by Akhiezer ; Berestetsky

[TABLE]

where $\alpha$ is the fine structure constant, $\varepsilon_{i}$ and $v_{i}$ are the energy and velocity of the positron, respectively, $E_{a}$ is the energy of the active electron, and $\tau$ is the amplitude whose explicit form will be specified below. In the present letter, we will consider only the double differential angular cross section defined by

[TABLE]

This cross section is assumed to be averaged over the angular momentum and spin projections of the electron and positron, respectively, and summed over the polarizations of the emitted photons. The solid angles of the emitted photons $\Omega_{1,2}$ are defined by the azimuthal $\varphi_{1,2}$ and polar $\theta_{1,2}$ angles (see Fig. 1).

Here the $x-z$ plane is spanned by the momenta of the incoming positron ${\bf p}_{i}$ and one of the emitted photons ${\bf k}_{1}$ with the $z$-axis fixed along the direction of ${\bf p}_{i}$. Here, we utilize the independent-particle approximation, in which the positron and the active electron move in an effective (Coulomb and screening) potential created by the nucleus and all the other electrons. The screening potential is induced by the Hartree charge density of these remaining electrons. Based on our prior analysis for the Rayleigh scattering of high-energetic photons volotka2016 , we expect that the independent particle approximation stays valid for the processes involving inner-shell electrons of heavy systems, where the correlation effects are suppressed by a factor $1/Z$ ($Z$ is the nuclear charge number).

The amplitude of the two-quantum annihilation of the positron with the electron in the bound $a$ state is given by two Feynman diagrams shown in Fig. 2, which correspond to the following expression Akhiezer ; Berestetsky :

[TABLE]

Here $\sum_{n}$ implies the complete summation over the whole spectrum, including the integration over the positive and negative continuum parts, $\mu_{i}$ is the helicity of the incoming positron, $\bm{\alpha}$ is the vector of Dirac matrices, and the wave function of the plane-wave photon with the polarization $\lambda$ is given by

[TABLE]

The amplitude (3) determines the differential cross section (1) uniquely and, thus, describes the two-quantum annihilation process completely. Let us turn to the details of the calculation of this amplitude.

The infinite summation $\sum_{n}$ in Eq. (3) is replaced by a sum over a quasi-complete set of the Dirac equation solutions. These solutions are obtained by using the dual-kinetic-balance finite basis set method Shabaev_PRL93_130405:2004 with basis functions constructed from $B$ splines Johnson_PRA37_307:1988 ; Sapirstein_JPB29_5213:1996 . Such an approach yields the wave functions of the quasi-states $n$, including the bound state $a$, but it can barely be applied for constructing the wave function of a positron with a given energy. The incoming positron with the four-momentum $p_{i}$ and the helicity $\mu_{i}$ is treated as an outgoing electron with the four-momentum $-p_{i}$ and the helicity $\mu_{i}$ Bjorken ; Itzykson . The explicit form of the wave function of such a particle can be found, e.g., in Refs. Rose ; Artemyev_PRA79_032713:2009 . The numerical construction of this wave function is performed with the use of the modified RADIAL package Salvat1995 . We note that the constructed wave functions of the incoming positron, quasi-states $n$, and initial bound state $a$ take into account the interaction with the effective (Coulomb and screening) potential to all orders. To calculate the matrix elements, we utilize the well-known multipole expansion technique. As a result, one gets the infinite multipole summations over the photon and positron multipoles, which are further restricted by analyzing the convergence property. More details of the developed method will be presented in a forthcoming publication.

As the first application of our ab initio approach, we calculate the double differential angular cross section (DDACS) for the process of the two-quantum annihilation of the $300$ keV positron with the $K$-shell electrons of a silver atom ($Z=47$), which was experimentally investigated in Ref. Nagatomo_PRL32_1158:1974 . In this experiment, the annihilation photons were detected at solid angles given by $\theta_{2}=100^{\circ}$, $\varphi_{2}=180^{\circ}$, and $\theta_{1}=30^{\circ}$. Fig. 3 presents the DDACS as a function of the angle $\theta_{2}$ for $\theta_{1}$ and $\varphi_{2}$ being fixed as in Ref. Nagatomo_PRL32_1158:1974 .

This figure also shows the convergence of the DDACS with respect to the number of the photon multipoles $L_{\rm max}$ that need to be taken into account in the expansion of the photons wave function. About $30$ multipoles are sufficient to obtain well-converged differential cross sections, giving rise to $60$ partial waves in the decomposition of the positron wave function. We performed all computations both, in length and velocity gauges, and obtained perfect agreement as seen from Fig. 3. Moreover, the differential cross sections differ by less than 1% if other than the Hartree screening potential is applied.

We can also compare our ab initio QED results with those from the Lee’s and impulse approximations. In Lee’s approximation Chang_ZETF33_365:1957 , which has been widely used for the description of the two-quantum annihilation of slow (thermalized) positrons Gribakin_PRA70_032720:2004 ; Dunlop_JPB39_1647:2006 ; Green_PRA90_032712:2014 ; Green_PRL114_093201:2015 , (i) the Dirac-Coulomb propagator is replaced by a free electron one, (ii) the binding energy of the initial electron $a$ and the kinetic energy of the incoming positron are assumed to be much smaller than the electron rest mass, and (iii) the Dirac electron and positron wave functions are replaced by the corresponding two-component Schrödinger-Pauli wave functions. Making use of these assumptions in Eq. (3), one can obtain the expression for the two-quantum annihilation amplitude Chang_ZETF33_365:1957

[TABLE]

Here the superscript (SP) stands for Schrödinger-Pauli wave functions. Fig. 4 compares the DDACS from this approximation with our ab initio results and shows that Lee’s approximation overestimates the DDACS by an order of magnitude when compared with the rigorous QED prediction.

This discrepancy mainly arises from the large ($300$ keV) kinetic energy of the positron and the importance of the binding and the relativistic effects for the inner-shell electrons of a silver atom.

In the relativistic impulse approximation (IA), it is assumed that the interaction with a nucleus can be neglected for high-energetic positrons and that the process can be viewed as a free positron annihilation with a stationary wave packet of superimposed plane-wave electron states. Following the derivation which has been previously described in detail for the Compton scattering Ribberfors_PRB12_2067:1975 ; Bergstrom_PRA48_1134:1993 , one can obtain the DDACS for the two-quantum annihilation in the IA

[TABLE]

Here $\rho_{a}({\bf p})$ is the momentum distribution of the initial bound-electron $a$ state and $\frac{dW_{\rm free}}{d\Omega_{1}d\Omega_{2}}$ is the double differential angular probability for the two-quantum annihilation of a free positron and a free electron with the momentum ${\bf p}$ Akhiezer . The DDACS being calculated in the IA is compared with other calculations in Fig. 4. From this figure, it is seen that the IA, like Lee’s approximation, overestimates the DDACS by an order of magnitude. This can be understood by the neglected interaction between the positron and the nucleus and, hence, an (unphysically) increased overlap of the positron and electron densities. That, in turn, leads to the growth of the cross section.

Finally, we compare the obtained results with the previous theoretical predictions by Gorshkov and coworkers Gorshkov_JETP45_17:1977 , which is displayed in Fig. 4 by a black square. These authors started from the free-particle approximation for the two-quantum annihilation and have evaluated the corrections of the first order in the interaction with the nucleus. This approach corresponds to the expansion in powers of $\alpha Z$ and $\alpha Z/v$ which in the case under investigation approximately equal $0.34$ and $0.44$, respectively. The significant deviation from the exact treatment, however, indicates that such a perturbation expansion fails to describe the DDACS of the considered process.

Fig. 4 compares the different theoretical predictions for the DDACS with the experimental value Nagatomo_PRL32_1158:1974 . First, let us note the extra factor $2$ in the denominator of Eq. (2) in Ref. Nagatomo_PRL32_1158:1974 . This factor should appear if the contributions of the same quantum states are accounted twice, which does not apply for the the DDACS. Therefore, here and below the results from Ref. Nagatomo_PRL32_1158:1974 are multiplied by a factor $2$. From Fig. 4 it is seen that all approximate theoretical results, including that of Ref. Gorshkov_JETP45_17:1977 , are by an order of magnitude away from the experimental value, and quite in contrast to our rigorous QED treatment that provides the prediction which is rather close to the experimental result.

However, the direct comparison of the calculated DDACS with the experimental value might not be fully justified in Fig. 4. This is caused by the fact that the experimental value just represents an detector-averarged DDACS. In Ref. [31], indeed, the weighted averages of the overall detector efficiencies, including the geometrical factors, $(\epsilon\Omega)_{1}$ and $(\epsilon\Omega)_{2}$ for the annihilation photons are defined to be $6.5\times 10^{-2}$ sr and $5.8\times 10^{-2}$ sr, respectively. If we assume a 100% efficiency for both detectors, we can evaluate the detector-averaged DDACS within the exact QED approach as well as within the Lee’s and impulse approximations. In the case of the exact calculation, $15$ photon multipoles were taken into account. The integration over each photon emission angle is performed by a 4-point Gauss-Legendre quadrature. A conservative error estimate for the averaged DDACS gives 30%. In Table 1 we compare the detector-averaged DDACS calculated within the exact approach, the Lee’s and impulse approximations with the experimental value Nagatomo_PRL32_1158:1974 .

From the table, one can see that after the averaging of the DDACS decreases by almost a factor 2 because the planar geometry and detector position at $\approx 180^{\circ}$ just refers to the maximum of the cross section. Any deviation from this geometry leads to the drop of the DDACS. Here it is worth noting that smaller detector efficiencies will lead to the further decrease of the DDACS. From table 1, one can also see that the results of ab initio QED approach are in good agreement with the experimental value.

In conclusion, a fully relativistic QED description of the two-quantum annihilation of a positron with a bound electron is presented for the very first time. This novel approach has been applied for the annihilation of $300$ keV positrons with the $K$-shell electrons of silver. Our result for the double differential angular cross section is in good agreement with the experimental value Nagatomo_PRL32_1158:1974 and, thus, resolves long-standing disagreement between theory and experiment. It was also shown that none of the approaches so far used for the calculation of the positron – bound-electron two-quantum annihilation can be applied in this case. We believe that the exact approach developed here can be extended to many other cases of the positron annihilation and, thus, will allow to establish more precise validity criteria for the so far employed approximations as well as to help in the interpretation of the experimental data in various applications of the positron annihilation processes.