Gamma -ray spectra in the positron-annihilation process of molecules at room temperature
Lin Tang, Xiaoguang Ma, Jipeng Sui, Meishan Wang, Chuanlu Yang
aa r X i v : . [ phy s i c s . a t m - c l u s ] M a r Gamma -ray spectra in the positron-annihilation process ofmolecules at room temperature
Lin Tang, Xiaoguang Ma, ∗ Jipeng Sui, Meishan Wang, and Chuanlu Yang
School of Physics and Optoelectronic Engineering, Ludong University,Yantai, Shandong 264025, People’s Republic of China (Dated: March 20, 2020)
Abstract
In this study, a fully self-consistent method was developed to obtain the wave functions of thepositron and electrons in molecules simultaneously. The wave function of a positron at roomtemperature , with a characteristic energy of approximately 0 . eV , was used to analyse theexperimental results of its annihilation in helium, neon, hydrogen, and methane molecules. Theinteractions between the positron and molecule provide a significant correction in the gamma-rayspectra of the annihilating electron–positron pairs. It was also observed that high-order correlationsoffered almost no correction in the spectra, as the interaction between the low-energy positron andelectrons cannot drive the electrons into excited electronic states. More accurate studies, whichconsider the coupling of the positron–electron pair states and vibration states of nuclei, must beundertaken. PACS numbers: 78.70.Bj, 82.30.Gg, 36.10.DrKeywords: positron–electron annihilation; positron–electron pair; positron wave function ∗ Electronic address: [email protected] . INTRODUCTION Gamma-ray spectra in low-energy positron annihilation, usually at room temperature,were measured extensively in many gas-phase molecules earlier[1, 2]. In recent years, sig-nificant theoretical studies have also improved our understanding of the positron–electronannihilation process in atoms and small molecules[3–7]. However, the theoretically predictedtotal profiles of gamma-rays for most of the molecules were only within approximately 70%agreement with the experimental results[2]. The annihilation spectra obtained in the recentlow-energy plane-wave positron (LEPWP) approximation [4, 5] were always broader thanthose measured experimentally[1].The broadening of the gamma-ray spectra is mostly related to the momentum distri-bution of the positron–electron pairs in the small momentum region, i.e. the electronsdistributed in the lowest momentum region and the low-energy positron play an importantrole in the annihilation process. However, the understanding of low-energy positrons andtheir behaviour in molecules is still incomplete as compared to the more familiar subjectof electrons. One of the reasons is that an accurate positron wave function in the anni-hilation process in molecular systems is difficult to obtain. In the present study, a fullyself-consistent method was developed to obtain the wave functions of the positron and elec-trons in molecules simultaneously. This low-energy positron wave function can account forpart of the positron–electron correlations. The corresponding gamma-ray spectra agree verywell with the recent experimental results.
II. THE WAVE FUNCTION AND ANNIHILATION OF POSITRON–ELECTRONPAIRS
Generally, there are four different orbitals: α − , β − electrons and α + , β + positrons inpositron-molecule systems under the unrestricted Hartree-Fock (UHF) self-consistent fieldapproximation[8, 9]. The Fock operators F = H + J − K for electrons and positrons aredefined by the single-electron Hamiltonian H i = Z φ ∗ i (1) Hφ i (1) d~r , (1)2oulomb integrals J ij = Z Z φ ∗ i (1) φ ∗ j (2) 1 r φ i (1) φ j (2) d~r d~r , (2)and exchange integrals K ij = Z Z φ ∗ i (1) φ ∗ j (2) 1 r φ j (1) φ i (2) d~r d~r , (3)between the i th and j th molecular orbitals. In most of the actual implementations, themolecular orbitals are expansions of the atomic orbital basis functions φ i = P r χ r C ri , where χ r is a set of atomic orbitals and C ri s are the weight coefficients calculated by a self-consistentfield procedure.In this form, the Fock matrices for the α − and β − electrons are F α − = H − + J − − K α − and F β − = H − + J − − K β − , respectively. The elements between the atomic orbital basisfunctions s and r for electrons are h H − i rs = Z χ ∗ r (1) {− ∇ − X N Z N | r − R N | } χ s (1) d~r (4) h J − i rs = X tu ( p α − tu + p β − tu − p α + tu − p β + tu ) Z Z χ ∗ r (1) χ ∗ t (2) 1 r χ s (1) χ u (2) d~r d~r (5) h K α − i rs = X tu p α − tu Z Z χ ∗ r (1) χ ∗ t (2) 1 r χ u (1) χ s (2) d~r d~r (6)for the α − electrons, and h K β − i rs = X tu p β − tu Z Z χ ∗ r (1) χ ∗ t (2) 1 r χ u (1) χ s (2) d~r d~r (7)for the β − electrons, respectively. The Coulomb interaction between electrons and positronsis considered in Eq.(5).The elements of the Fock matrices F α + = H + + J + − K α + and F β + = H + + J + − K β + for the α + and β + positrons are easily found between the positron basis functions as givenbelow: h H + i rs = Z χ ∗ r (1) {− ∇ + X N Z N | r − R N | } χ s (1) d~r (8) h J + i rs = X tu ( p α + tu + p β + tu − p α − tu − p β − tu ) Z Z χ ∗ r (1) χ ∗ t (2) 1 r χ s (1) χ u (2) d~r d~r (9) h K α + i rs = X tu p α + tu Z Z χ ∗ r (1) χ ∗ t (2) 1 r χ u (1) χ s (2) d~r d~r (10)3or the α + positrons, and h K β + i rs = X tu p β + tu Z Z χ ∗ r (1) χ ∗ t (2) 1 r χ u (1) χ s (2) d~r d~r (11)for the β + positrons, respectively. Eq.(9) considers the correlation of the electrons withpositrons.The density matrices for the electrons and positrons are given by an iterative calculationof the weight coefficients i.e. p α − tu = n α − P i =1 C ∗ ti C ui , p β − tu = n β − P i =1 C ∗ ti C ui , p α + tu = n α + P i =1 C ∗ ti C ui , and p β + tu = n β + P i =1 C ∗ ti C ui . Thus, a set of molecular orbitals for electrons and positrons is obtainedby the solution of Roothaan equations[8]. The total α − or β − -electron wave function Φ α − orΦ β − , respectively, satisfying the Pauli principle, is built up as an anti-symmetrised productof all the above molecular electronic orbitals[9]. In the same way, the total α + or β + -positronwave function Φ α + or Φ β + , respectively, obeying the Pauli principle, is expanded in terms ofan anti-symmetrised product of one-positron orbitals. As a result, the positron–electron pairorbital is written as a product i.e. Φ = Φ α − Φ β − Φ α + Φ β + . The Slater-determinant ensuresthat the motion of electrons or positrons with parallel spin is correlated. Moreover, thedistinguishability of the electrons and positrons is represented in the Hartree product formwhich signifies the simultaneous probability of finding the electrons and positrons at thesame point, i.e. the probability of existence of a positron–electron pair.In the annihilation process, the momentum ~p of the emitting photons is equal to themomentum of the positron–electron pair, i.e. ~p = ~k − + ~k + . Then, the momentum distributionof photons is obtained by a Fourier transform from the positron–electron pair wave function A ( ~p ) = Z Φ − ( ~r )Φ + ( ~r ) e − i~p · ~r d~r (12)Alternatively, according to the convolution theorem, the momentum distribution of photonsis written as A ( ~p ) = φ − ( ~k − ) ⊗ φ + ( ~k + ) = Z + ∞−∞ φ + ( τ ) φ − ( ~p − τ ) dτ (13)where, φ − ( ~k − ) = Z Φ − ( ~r ) e − i ~k − · ~r d~r (14)and φ + ( ~k + ) = Z Φ + ( ~r ) e − i ~k + · ~r d~r (15)are the wave functions of the electrons and positrons in momentum space, respectively.4he momentum of the electron-positron pair is rotationally averaged in the gas or liquidexperiments. Hence, the theoretical momentum distribution must be spherically averaged[1]to enable comparison with the experimental measurements. The radial distribution functionin momentum space is defined by D ( p ) = Z π dθ Z π dφP sin θ | A ( ~p ) | (16)where, P, θ, φ are the spherical coordinates [7]. Hence, the theoretical spherically averagedmomentum distribution is given by σ ( p ) = D ( p )4 πp (17)which signifies the averaged probability to encounter the electron-positron pair on the surfacewith momentum | P | .The gamma-ray spectra in the annihilation process undergo a Doppler shift in energyowing to the longitudinal momentum component of the positron–electron pair[7]. Hence,integration over a plane perpendicular to p must be performed to obtain the total probabilitydensity at the momentum p = 2 ǫ/c . Then, the gamma-ray spectra for the positron–electronpair are given by Ω( ǫ ) = 1 c Z ∞ ǫ/c σ ( p ) pdp. (18)The Doppler shift from the centre ( mc = 511 keV) is given by ǫ . III. APPLICATION AND DISCUSSION
Fig. 1 shows a comparison of the present theoretical gamma-ray spectra of helium in thepositron–electron annihilation process with other calculated and measured values. A widthof 2.63 keV was obtained by measurement of the annihilation radiation [10] and the widths2.50 keV and 2.31 keV were fitted by one and two Gaussian functions, respectively, for thesame measurements[1]. In recent studies, the width 2.31 keV was considered to be relativelyaccurate and is taken as the reference value to be compared with our theoretical value. TheLEPWP approximation gives the width of 3.06 keV[4] indicated by the short blue dashedline. This approximation considered only the electron wave function of atoms or moleculesand regarded the positron wave function as a plane wave with zero energy. When the energy5 A nn i h il a t i on -r a y s pe c t r a ( a r b . un i t s ) Energy shift (keV)
FIG. 1: Gamma-ray spectra in the positron–electron pair annihilation process in helium comparedwith experimental and theoretical values. of the incoming positron tends to zero, the plane wave is negligible. Therefore, Eq.(12) usesonly the electron wave function to calculate the photon momentum[1] as given below: A ( ~p ) = < ~p | δ | µν > = Z ψ µ ( ~r ) ψ ν ( ~r ) e − i~p · ~r d~r = Z ψ µ ( ~r ) e − i~p · ~r d~r (19)where, ψ µ ( ~r ) and ψ ν ( ~r ) are the electron and positron wave functions, respectively. Accordingto previous studies, neglecting the wave function of the positron results in an overestimationof the width by 30 percent.In many cases the positron was considered to be thermalised before annihilation and themomentum of the positron–electron pair was approximately equal to that of the electron.However, when the positron was close to the atom, the attraction between them increasedthe speed of the positron slightly; the speed of the electron was also modified to some extent6s it was attracted to the positron. Therefore, the momentum distribution of the positron–electron pair was not exactly equivalent to the momentum distribution of the electron in theundisturbed atom. Many approximations, such as polarisation orbit approximation (2.45keV[11]), S-wave phase shift (2.20 keV[12]), and an atmospheric pressure condition (2.01keV[13]) were considered in previous studies. As the positrons were disturbed, the electronswere attracted and moved closer to the positrons and farther from the nucleus, therebyincreasing the probability of positrons pairing with low momentum electrons. The largediscrepancy between the LEPWP line and reference value is probably due to the absence ofpositron wave functions in Eq.(19).The long red dashed line with the width 2.54 keV is the spectra obtained using thepositron wave function determined by considering the zero-order approximation. As men-tioned in Eqs.(1-13), we used the ab initio self-consistent method to solve the Schrodingerequations of the positron and electron at the same time and obtained their self-consistentwave functions. These wave functions were used to determine the momentum distributionof the photon pair. The influence of the positron on the wave function of the electron wasconsidered in the self-consistent process and the wave function of the positron was obtainedafter considering the influence of the atom on it, hence the gamma-ray spectra are moreaccurate. In comparison with the LEPWP method, the discrepancy in the short blue linereduced by 22 percent.Figure 2 shows the schematic diagram of the positron and electron density distributionnear the helium atom. The positron density distribution was almost perfectly sphericalon a large scale, as shown in Fig. 2(a), whereas on a smaller scale (Fig. 2(b)) it was nolonger spherically symmetric. The density of the positron increased noticeably at a distanceof several Bohr from the centre. The polarising effect of electrons on the positron densitydistribution is evident. The wave function of the positron condensed in some directionsexhibiting some components of a p-wave characteristic due to the influence of electrons.The electron of the helium atom also showed p-wave components due to the interactionwith the incoming positrons as shown in Fig. 2(c). The above analysis indicates that itis necessary to consider the variation and polarisation of the positron and electrons in thetheoretical calculation which was confirmed by the results of our calculation.The black solid line in Fig. 1 represents the gamma-ray spectra obtained by consideringthe excitation processes of positron and electrons induced by the interaction between the7 IG. 2: Positron and electron density distribution near helium atom (The unit of the scale on thegraph is Bohr). (a) positron, on a radius scale of 1000 Bohr; (b) positron, on a radius scale of 10Bohr; (c) electron, on a radius scale of 10 Bohr. positron and helium atom. As mentioned in reference[3],the first-order correction for theinteraction with positrons and electrons is as follows[3]: − X µ,ν < ~p | δ | µν >< µν | V | nǫ >ǫ − ǫ ν − ǫ µ + ǫ n (20)where, < µν | V | nǫ > = Z ψ ∗ µ ( ~r ) ψ ∗ ν ( ~r ) 1 | ~r − ~r | ψ µ ( ~r ) ψ ν ( ~r ) (21)However, the correction of the first term was not apparent as it was only 4 percent approx-imately. This is because the low-energy positrons did not have sufficient energies to drivethe electrons to the excited states, and most of the effects were due to polarisation and notexcitation . The width as determined in the present study was 2.44 keV. The widths 2.50keV[16] and 2.53 keV[1] were obtained by calculations using the variational method, whichconsidered the polarisation effect and 2.40 keV[15] and 2.45 keV[11] were obtained by thepolarisation orbit approximation as well as consideration of virtual positive ion (Ps) bubbles.Thus, the present theoretical results agree well with those of previous studies.Helium has only two electrons that are pulled out easily by the positron hence, the wavefunction of the positron and correlation effects are of great significance in gamma-ray spectra.As the interaction between the positron and electrons was considered in the calculation of thegamma-ray spectrum, the accuracy of theoretical calculation was vastly improved. From themany-body theory of zero-order and first-order approximations, the widths obtained were2.54 keV and 2.44 keV, respectively, for helium. The self-consistent electron and positron8 A nn i h il a t i on -r a y s pe c t r a ( a r b . un i t s ) Energy shift (keV) Expt. (3.28 keV) LEPWP (5.22 keV) MBT-0th (3.83 keV) MBT-1st (3.76 keV)
FIG. 3: Theoretical gamma-ray spectra in neon in comparison with other theoretical and experi-mental values. wave functions were considered by the zero-order approximation, which provided a 17%correction on the plane-wave approximation. The first-order approximation accounted forthe excitation processes of the electrons and positrons, which yielded a width of 2.44 keVwith a 20% correction on the plane-wave approximation. However, the first-order term onlyhad a 4% correction on the zero-order terms.Fig. 3 shows the theoretical and measured gamma-ray spectra in a neon atom. Thewidth of the experimental gamma-ray spectra fitted by two Gaussian functions and usedas reference was 3.28 keV[3]. It is seen that the width of the gamma-spectra produced bythe annihilation of electrons and positrons in neon atom are generally larger than that ofthe helium atom. The value measured by the annihilation radiation was 3.19 keV[9] andthe data fitted by one Gaussian function was 3.36 keV[1]. The theoretical value of 2.04 keV9
IG. 4: Positron and electron density distribution after polarisation, where (a) and (b) representthe density of positrons, (c) represents the density of 2s electrons, (d), (e) and (f) represent thedensity of 2p x , 2p y , and 2p z electrons, respectively. was calculated by Doppler widening at an atmosphere cite9, indicating that the positronwave function was dominant in this annihilation process. A spectral width of 3.28 keV wasdetermined by the variational method[1]. Although 3.32 keV[14] and 3.73 keV[15] were bothcalculated by using the polarised orbit approximation, 3.32 keV was the width for the solidneon atom. This unusual phenomenon may be due to the larger distribution range of relativemomentum of electrons and positrons in the neon atom than that in the helium atom.As shown in Fig. 4, the wave function of the positron is no longer spherical. Thisdeformation may be due to the influence of the p electron. The wave functions of the 2selectrons under the influence of positrons were polarised to a large extent. The shape ofthe wave functions of 2p x , 2p y , and 2p z electrons were basically unchanged; however, theirdirections were altered after being polarised by the positron, i.e. the p x wave became a p z wave, p y wave became a p x wave, and p z wave became a p y wave. Hence, compared to theresults of LEPWP, the errors in the results of the MBT-0th method were reduced by a large10 A nn i h il a t i on -r a y s pe c t r a ( a r b . un i t s ) Energy shift (keV)
FIG. 5: Gamma-ray spectra in hydrogen molecule compared with the experimental and othertheoretical values. percent bringing it close to the reference value. Therefore, it is very important to considerthe interaction between the positron and electron in the calculation of gamma-ray spectra.However, the first-order correction was less evident than in the MBT-0th method, hence,the excitation process of the positron and electron may not play an important role in theabove annihilation process.Fig. 5 shows the gamma-ray spectra of positron–electron annihilation in the hydrogenmolecule. The experimental value, which was fitted by two Gaussian functions as reference,to compare with the theoretical value, was 1.64 keV[1]. The theoretical calculated gamma-ray spectrum is shown by the three curves in the figure. The short blue dashed line representsthe calculation by the LEPWP method, with a width of 2.09 keV[4] and error of 27%. Thespectrum calculated by the MBT-0th method is shown as the long red dashed line with11
IG. 6: Positron and electron density distribution in hydrogen. a width of 1.82 keV and an error of 11%. In comparison with the former two cases ofhelium and neon, the correction effect of the zeroth order approximation of the positron–electron interaction for the hydrogen molecule was not particularly evident. The spectralline calculated by the MBT-1st method is represented by the solid line in the graph, with awidth of 1.80 keV, which was the same as the spectral line in the zero order approximation,and the error was reduced only by 1%. Therefore, the excitation process of positron–electronannihilation can be considered to be negligible. Comparatively, the data measured by a two-dimensional angular association (2D-ARCR) of annihilation radiation was 1.56 keV[17]. Awidth of 1.66 keV was measured by the annihilation radiation (ARCR) for liquid H2[18] and1.71 keV was the width by one Gaussian function fitting[1]. The theoretical width of 1.93keV was calculated by the static HF method and did not include positron correlation[19].The width obtained by the polarisation orbit approximation for liquid H2[20] was 1.70 keV.Hence, the coupling of the positron–electron pair and vibration must be considered in futurestudies.Fig. 6 shows the schematic diagram of the electron wave functions in positron andhydrogen molecule. The wave function of the positron near the hydrogen molecule is shownon a large scale. It is a perfect spherical wave, except at the origin where it is nearly zero. It12 A nn i h il a t i on -r a y s pe c t r a ( a r b . un i t s ) Energy shift (keV) Expt. (2.09 keV) LEPWP (2.98 keV) MBT-0th (2.38 keV) MBT-1st (2.38 keV)
FIG. 7: Comparison of theoretical gamma-ray spectrum of methane with experimental results. is seen that the wave function of the positron is clearly polarised near the hydrogen moleculeand its density on the side of the hydrogen-hydrogen bond is higher than that on other parts.The density near the hydrogen atom is the lowest and slightly higher between two hydrogenatoms.For methane, shown in Fig. 7, the reference value of 1.64 kev[1] was the most recent mea-surement. The theoretical values calculated by LEPWP, MBT-0th, and MBT-1st methodswere 2.98 keV[4], 2.38 keV, and 2.38 keV, with errors of 43%, 14%, and 14%, respectively.Therefore, compared to the LEPWP method without considering the positron effect, therewas a significant improvement in the accuracy of the results of the MBT-0th method. How-ever, as the positrons did not have sufficient energy to excite the electrons in the methanemolecules, the MBT-1st method showed no evident improvement in the calculation as com-pared to the MBT-0th method. Fig. 8 shows the positron and electron density distributions13
IG. 8: Positron and electron density distribution in methane. in methane. On a larger scale, the positron density distribution is a perfect spherical wave.The details are seen on a smaller scale, where the positron is seen to be evidently polarised.The first-order correction had little effect on the final results as the excitation of positronsand electrons did not occur in the annihilation process.
IV. CONCLUSIONS
The present work is the first step to obtain an accurate wave function of the positronin the annihilation process of many-atomic molecules. The fully self-consistent method wasdeveloped to study the positron annihilation process in molecules. The results showed thatthe present method incorporates the wave functions of the electrons and positron appropri-ately. The interaction and polarisation between the incoming positron and electrons wereconsidered precisely in this method. In the present scheme, the one-body term includedthe correlation potential which accounted for the polarisation and orientation effects in thepositron–electron annihilation process. The wave function of the positron determined bythe present scheme provided a significant correction in the gamma-ray spectra (nearly 30percent). The present theoretical method showed that an accurate wave function of the14ositron is very important to explain the gamma-ray spectra in molecules. As the positronin the experiments was cooled below the formation threshold of positronium, the excita-tion processes for higher-order interactions had very little effect and this was proved in thepresent work.
V. ACKNOWLEDGEMENT
This work was supported by the National Natural Science Foundation of China undergrants No. 11674145 and Taishan Scholars Project of Shandong province (Project No.ts2015110055). [1] Iwata K, Greaves R G and Surko C M 1997
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