On the annihilation of positrons in binary encounters withmolecules

G. Laricchia *, C. Wilkin

Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, UK

Received 29 October 1997

Abstract

A recent model which predicts, for the ®rst time, quasi-resonant enhancements in the positron annihilation proba-

bility in the vicinity of energy thresholds for inelastic processes is discussed and extended to evaluate the energy depen-

dence of this probability for a variety of atomic and molecular targets. Ó 1998 Elsevier Science B.V. All rights

reserved.

PACS: 3410; 3480G; 3610; 7165

Keywords: Virtual positronium; Virtual states; Positron annihilation cross-section; Ze�; Annihilation rates

1. Introduction

In considering binary encounters of positronswith atoms and molecules, it has generally been as-sumed that ``Except in the limit of zero velocity,. . . the annihilation cross-section is several thou-sand times smaller than the cross-section for elasticor inelastic scattering'' [1] and that because of this``although the annihilation channel is always open,its coupling to the elastic and inelastic scatteringchannels is so weak that it can be neglected whencalculating scattering parameters'' [1]. Therefore,even sophisticated variational treatments [2] of e�

± atom interactions have considered the problem

in terms of only two channels (elastic and Ps for-mation) up to the ®rst excitation threshold of thetarget, though these might be more accurate in re-producing phase shifts than the wave functionswhich are needed in the annihilation calculations.

Recently, the universal validity of the above as-sumptions have been called into question by amodel [3,4] which predicts, for the ®rst time, qua-si-resonant enhancements of the annihilationprobability in the vicinity of thresholds for inelas-tic processes. The enhanced annihilation resultsfrom the formation of a virtual state which prefer-entially leaves the positron quasi-stationary in theneighbourhood of a higher-than-average electrondensity in the remnant target. Although developedspeci®cally with respect to Ps formation, thefundamental physical mechanism proposed is ageneral one.

Nuclear Instruments and Methods in Physics Research B 143 (1998) 135±139

* Corresponding author. E-mail: g.laricchia@ucl.ac.uk; tel.:

+44 171 419 3470; fax: +44 171 380 7145.

0168-583X/98/$19.00 Ó 1998 Elsevier Science B.V. All rights reserved.

PII: S 0 1 6 8 - 5 8 3 X ( 9 8 ) 0 0 2 7 0 - 5

In this paper, the basic ideas of the model areoutlined in Section 2. New results are presentedand discussed in Section 3, whilst conclusions aredrawn in Section 4.

2. Outline of the model

The model [3] was initially developed in connec-tion with the anomalously large positron annihila-tion rates from large organic molecules [5±7]. TheDirac cross-section for positron annihilation into2c with a free electron is (in the non-relativisticlimit)

Q2c�v� � pr20c=v; �1�

with r0 being the classical electron radius, c thespeed of light and v the relative speed of the anni-hilating pair.

In the study of positron annihilation fromgaseous targets, Q2c is usually modi®ed by intro-ducing an e�ective number of electrons (Zeff )per molecule available for annihilation. Thisparameter may be de®ned through the annihilationrate as

C�v� � Q2cqev � pr20cqZeff ; �2�

where qe (the free electron density) is taken in amolecular gaseous medium to be equal to an aver-age electron density of the medium qZeff with q thetarget number density. Enhancements of Zeff overthe atomic number Z have been found [5,6] torange from 1 to 8 for the noble gases to �104 forthe heavier alkanes at room temperature. Largevalues of Zeff /Z have been interpreted as signifyingthe formation of long-lived positron-moleculebound states or resonances, possibly connectedwith vibrational excitations of the molecule [8].However recent experiments [9] with protonatedand deuterated alkanes have shown that vibration-al excitations are not crucial factors in determiningZeff .

According to our model [3], an enhanced anni-hilation may result from the formation of a virtualstate which preferentially leaves the positron qua-si-stationary in the neighbourhood of a higher-than-average electron density in the remnant tar-get, the magnitude of the associated probability

depending on the lifetime of the virtual state andthe local electron density.

A virtual particle is one which exists for a timeDt determined by Heisenberg Uncertainty Princi-ple

DE � Dt � �h; �3�where DE is the amount by which energy conserva-tion may be violated. So that, near a molecule, apositron may form a Ps atom virtually belowthreshold for a time

Dt � �hjT� ÿ Ei � 6:8=n2 eVÿ TPsj ; �4�

where T� is the incident positron energy, Ei is theionization energy of the target, (Ei ) 6.8/n2

eV)�EPs is the threshold for Ps formation in astate of principal quantum number n and TPs isthe kinetic energy of the virtual Ps atom. FromEq. (4), the duration of the virtual Ps process islongest when the atom is formed at rest (TPs� 0)and in the ground state (n� 1). Therefore, if in-volved in virtual Ps formation, the positron is ex-pected to remain close to the remnant target fora time Dt. During this time the positron may decaywithin the Ps atom either at the standard spin-av-eraged rate of ksa � 2� 109 sÿ1 or through pick-o�annihilation at a rate kpo. In this way, an expres-sion for the total annihilation rate was obtained[3] as

C � fc cPd � �1ÿ c�Pvp

� �; �5�

where Pvp � 1ÿ exp ÿ�ksa � kpo�Dtÿ �

is the totalvirtual-Ps annihilation probability and Pd �1ÿ exp�ÿkdtc� is the probability for direct annihi-lation, kd being the direct annihilation rate and tc

the duration of the collision; fc � nQcv is the colli-sion frequency (v � ������������

2E=mp

being the positronspeed) and the probability of forming virtual Psis 1ÿ exp ÿDt=tc� � � 1ÿ c.

From Eqs. (2) and (5) (where we assumed forsimplicity that one or other annihilation mode isdominant), the following expressions are obtained

Zeff � Qcvpr2

0cc 1ÿ exp ÿ kdtc� �� ��

��1ÿ c� 1ÿ exp ÿ Dt�ksa � kpo�ÿ �� � �6�

and

136 G. Laricchia, C. Wilkin / Nucl. Instr. and Meth. in Phys. Res. B 143 (1998) 135±139

Qann � Qc c 1ÿ exp ÿ kdtc� �� ����1ÿ c� 1

� ÿ exp ÿ Dt�ksa � kpo�ÿ ��

: �7�In the latter expression, the annihilation cross-sec-tion Qann is given in terms of the collision cross-section multiplied by the total annihilation proba-bility per collision.

3. Results and discussion

In order to estimate kd and kpo, we used thespin-averaged Dirac rate of pr2

0cne, took the aver-age electron density ne to be that of a gas moleculeand, again for simplicity, assumed all atoms andmolecules to be spherical with a radius a� a0,the Bohr radius. Correspondingly, we tooktc � a0=v. In the case of virtual Ps formation, wetried to incorporate, through a polarization factor

a, the enhancement of the local electron densitydue to the short range correlations arising fromthe close proximity of the positron, so that

kpo � 3r20cZa4a3

0

� �1:2� 1010Za� sÿ1: �8�

Eq. (8) implies that the spin-averaged annihilationinside the Ps atom is negligible as compared withthe pick-o� annihilation with other electrons inthe molecule.

In order to compare with experiment, the colli-sion cross section was empirically scaled [3] in allcases according to Qc � �10ÿ19a� m2 and a was ta-ken to be a dimensionless parameter equal to thestatic average electric dipole polarizability for theground-state molecules divided by 10ÿ30 m3. Addi-tionally, Z was taken to be equal to the number ofouter shell electrons since Doppler broadening da-ta for c-rays from positron annihilation indicate

Fig. 1. Comparison [3,4] between values of Zeff calculated using Eq. (6) (®lled circles) and experimental determinations (hollow circles).

Each family comprises the following atoms/molecules (in ascending order of jEi ÿ 6:84 eVjÿ1): inert (He, Ne, Ar, Kr and Xe); non-po-

lar (N2, H2, SF6, CO2, O2); alkanes (CH4, C2H6, CCl4, C3H8, C5H12, C7H16, C6H14); alkenes (C2H4, C6H12, C6H10, C6H8); ring hy-

drocarbons (C6H12, C10H20, C6H6, C10H8, C14H10); polar (CO, N2O, H2O, CCl2F2, CH3Cl, NH3, NO2, NO). Also shown are the

results of Eq. (6) with Qc� 3a(1+a)10ÿ20 m2 (squares).

G. Laricchia, C. Wilkin / Nucl. Instr. and Meth. in Phys. Res. B 143 (1998) 135±139 137

that core electrons play only a minor role [9] (al-though recent calculations [10] for C2H4 indicatethat excitations from the lowest lying state accountonly for 10% of Zeff at the lowest energies).

The results obtained in this way were comparedto the room temperature measurements of Surkoand co-workers [7] and, as shown in Fig. 1, a tol-erable description of the variation of Zeff over sixorders of magnitude was found. The correct gener-al trends were reproduced even for the alkenes,ring hydrocarbon and polar molecules where inthe latter case a slight energy shift between the ex-perimental and calculated values was noted. Morerecently, it has been suggested [11] that the cross-section would be better parametrized asQc / a�1� a�10ÿ20 m2. Indeed as shown in Fig. 1,the agreement is improved with experiment byusing this particular form especially, as expected,in the case of molecules with larger polarizability.

An important implication of Eq. (7) is that asjT� ÿ Ei � 6:8 eVj ! 0 and Dt!1, the model[3,4] predicts a resonance-type behaviour of Qann

in the vicinity of EPs. This is illustrated qualitative-ly in Fig. 2 where, for He, the contributions arisingfrom direct (or in-¯ight) annihilation and via virtu-al Ps formation are plotted relative to the elasticcollision cross-section (Qc) as a function of the in-cident positron energy. The ®gure illustrates qual-itatively the universal trends which, according tothe model, are to be expected not only for all tar-gets but also with respect to any threshold. Specif-ically, whilst the direct annihilation probability isexpected to decrease as the positron incident ener-gy is increased due to the reduced interaction time,the probability of annihilation via the occurrenceof a virtual process increases as the relevantthreshold for the process is approached. Indeedthe maximum is reached at the threshold itself with

Fig. 2. Qualitative illustration [4] of the variation of the ratio (R�Qann/Qc) of the annihilation to elastic collision cross-sections plotted

versus the incident positron for H, He, H2, Xe and C14H10. Also shown are the contributions to Qann arising from virtual Ps formation

and in-¯ight annihilation.

138 G. Laricchia, C. Wilkin / Nucl. Instr. and Meth. in Phys. Res. B 143 (1998) 135±139

the magnitude and width of the peak being targetdependent (although, as can be seen from Eq. (7),within this simple two-channel model the ratioQann/Qc equals 1 at that energy). Also shown inFig. 2 are our estimates of H, Xe, H2 and C14H10.In these estimates, the enhancement factor a hasbeen omitted from Eq. (8) in the case for H, H2

and He for which a (as de®ned above) is less than 1.The enhancement of the annihilation probabil-

ity at the Ps formation threshold has now beencon®rmed by variational calculations [12].

4. Conclusions

A new model which considers the e�ects of vir-tual Ps formation accompanied by pick-o� annihi-lation on the annihilation cross-section ofpositrons in binary encounters with atoms andmolecules has been discussed. The model predictsfor the ®rst time quasi-resonant enhancements ofQann at the threshold for Ps formation and hasbeen used for the evaluation of the energy depen-dence for a variety of targets.

It might be expected that in virtual Ps forma-tion accompanied by pick-o�, annihilation mightproceed with a non-valence electron with a higherprobability than in direct (or in-¯ight) annihilationand as such this process might be responsible formolecular dissociative sub-Ps-ionization [13]. In-deed, in some cases, the yield of particular frag-ments at incident positron energies below thelowest fragment production threshold [14], may al-low the determination of the energy dependence ofQann even above EPs.

Further studies on the importance of virtualstates in positron and indeed positronium annihi-

lation are anticipated, the latter being clearly rele-vant also for QED tests based upon precisionstudies of intrinsic Ps lifetimes [15±17].

Acknowledgements

GL wishes to thank Anand Bhatia for a helpfuldiscussion and The Engineering and Science Re-search Council for supporting positron researchat UCL under grant. no. GR/K56322.

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