Molecular size effects on diffraction resonances in positronium formation from fullerenes
Paul-Antoine Hervieux, Anzumaan R. Chakraborty, Himadri S. Chakraborty
aa r X i v : . [ phy s i c s . a t m - c l u s ] J u l Molecular size effects on diffraction resonances in positronium formation fromfullerenes
Paul-Antoine Hervieux, ∗ Anzumaan R. Chakraborty, † and Himadri S. Chakraborty ‡ Universit´e de Strasbourg, CNRS, Institut de Physique et Chimie des Mat´eriaux de Strasbourg, 67000 Strasbourg, France Department of Natural Sciences, D.L. Hubbard Center for Innovation,Northwest Missouri State University, Maryville, Missouri 64468, USA (Dated: July 4, 2019)We previously predicted [P.A. Hervieux et al. Phys. Rev. A , 020701 (2017)] that owing topredominant electron capture by incoming positrons from the molecular shell, C acts like a spher-ical diffractor inducing resonances in the positronium (Ps) formation as a function of the positronimpact energy. By extending the study for a larger C fullerene target, we now demonstrate thatthe diffraction resonances compactify in energy in analogy with the shrinking fringe separation forlarger slit size in classical single-slit experiment. The result brings further impetus for conducting Psspectroscopic experiments with fullerene targets, including target- and/or captured-level differentialmeasurements. The ground states of the fullerenes are modeled in a spherical jellium frame of thelocal density approximation (LDA) method with the exchange-correlation functional based on thevan Leeuween and Baerends (LB94) model potential, while the positron impact and Ps formationare treated in the continuum distorted-wave final state (CDW-FS) approximation. PACS numbers: 34.80.Lx, 36.10.Dr, 61.48.-c
I. INTRODUCTION
Formation of an exotic and quasi-stable electron-positron bound-pair, positronium (Ps) – a pure leptonicatom, by shooting positrons at matter is a fundamen-tal process in nature. The Ps formation channel coversa large portion of the positron scattering cross sectionfrom simple atoms and molecules [1], while exhibitingeven higher success rates on thin films and surfaces [2].Besides probing material structure and reaction mecha-nism, applied interests in the Ps formation are aplenty.Ps perishes via a unique electron-positron annihilationpathway [3, 4] with astrophysical [5, 6], materials [7],and pharmaceutical [8] interests. Efficient Ps forma-tion is the precursor of the production of dipositroniummolecules [9] and antihydrogen atoms [10, 11] requiredto study the effect of gravity on antimatter [12, 13]. Pos-sible production of Bose-Einstein condensate of Ps hasalso been predicted [14, 15], besides the importance ofPs in diagnosing porous materials [16] as well as in prob-ing bound-state QED effects [17]. Furthermore, similar-ities in the formalism for Ps formation in matter withthe exciton theory in quantum dots have recently beenshown [18].There is an abundance of theoretical investigationsin the literature to calculate Ps formation from a widevarieties of target. This includes atomic targets: (i)the hydrogen atom using variants of coupled-channelmethods [19, 20] and multichannel Schwinger’s princi-ple [21]; (ii) noble gas atoms in the distorted-wave [22], ∗ [email protected] † Present address: Department of Physics, Missouri University ofScience and Technology, Rolla, Missouri 65409, USA ‡ [email protected] the boundary-corrected Born [23], and the relativistic op-tical potential method [24]; and (iii) alkali metal atomsin the optical potential approach [25] and in the clas-sical trajectory Monte Carlo method when the targetsare in Debye plasma environments [26]. Among thesestudies, ab initio close-coupling calculations, pioneeredby Walters and collaborators [27], have in general beenvery successful [28]. Relatively limited calculations withmolecular targets include: (i) the molecular hydrogen byutilizing the convergent close-coupling theory [28] and amodel coupled-channel formalism [29]; and (ii) the wa-ter molecule in the continuum distorted-wave final-stateapproximation [30].Precision experimental techniques to measure Ps for-mation signals have also been achieved by impingingpositrons into varieties of materials, such as, atomicand molecular gases [31, 32], polyatomic molecules [33],molecular solids [34], liquids and polymers [35], zeo-lites [36], metal surfaces and films [37, 38], metal-organic-frameworks [39, 40], and embedded mesostructures [41].To facilitate precision measurements of gravitational freefall of antimatter as well as the optical spectrum of Ps,Doppler-corrected Balmer spectroscopy of Rydberg Pshas been applied [42]. Recently high yields of laser as-sisted production of low-energy excited Ps is achieved inthe interaction of cold-trapped positrons with Rydbergexcited Cs atom [10].In spite of such broad landscape of Ps research, stud-ies of Ps formation by implanting positrons in vapor orsolid phase nanoparticles are rather scarce. On the otherhand, clusters and nanostructures straddle the bound-ary between atoms and condensed matters which enablethem to exhibit hybridized properties of both domainsoften revealing remarkable behaviors with unusual spec-troscopy [43]. A lonely theoretical study of Ps forma-tion using the Na cluster targets was made about twodecades ago [44]. Recently, however, pilot studies of Psformation from the C fullerene has been published byus [45, 46]. It has been shown that the formation of a gasof delocalized electrons within a finite nanoscopic regionof more defined short-range boundary at the C shell,in contrast of a long-range, diffused Coulombic decay ofatomic and molecular electron densities, ensures predom-inant electron capture from local regions in space. Thisleads to diffraction in the capture amplitude, particularlyat positron energies that cannot excite plasmon modes.Indeed, Ref. [45] revealed a series of diffraction resonancesin Ps formation from C that may be observed in theexperiment both in ground and excited state Ps forma-tion.In general, the Ps formation from fullerenes can besingularly attractive due to fullerene’s eminent symmetryand stability in room-temperature, and its previous trackrecord of success in photo-spectroscopic experiments [47].In this communication, we extend our study of the Ps for-mation to a larger fullerene C . Ps(1 s ) formation forcaptures from various C molecular orbitals also showdiffraction resonances as a function of positron impactenergy. What is particularly noticeable going from C to C is an appropriate reduction of the energy sep-aration between the resonances due to the increase ofthe molecular size as a strong signature of the underly-ing diffraction process. In fact, Fourier transforms of theresonant signals expressed in the target recoil momentumscale map the fullerene radii very well. The following sec-tion presents methods applied to carry out the numericalcalculations. Section III presents and discusses the mainresults, while the final section concludes the article withsome words to encourage future experiments. II. DESCRIPTION OF THE METHODSA. LDA to model fullerene ground states
The details of the method follow the framework asdescribed in Ref. [48]. The jellium potentials, V jel ( ~r ),representing 60 and 240 C ions, respectively for C and C , are constructed by smearing the total positivecharge over spherical shells with radius r c and thickness∆. r c is taken to be the known radius of each molecule:3.54 ˚A (6.7 a.u.) for C [47] and 7.14 ˚A (13.5 a.u.)for C [49]. A constant pseudopotential V is addedto the jellium for quantitative accuracy [50]. The Kohn-Sham equations for systems of 240 and 960 electrons,made up of four valence (2 s p ) electrons from each car-bon atom, are then solved to obtain the single electronground state orbitals in the local density approximation(LDA). The parameters V and ∆ are determined by re-quiring both charge neutrality and obtaining the exper-imental value [51] (for C ) and the known theoreticalvalue [52] (for C ) of the first ionization thresholds.Consequently, the values of ∆ are found to be 1.30 ˚Aand 1.50 ˚A respectively for C and C . Using the single-particle density ρ ( ~r ) the LDA poten-tial can be written as, V LDA ( ~r ) = V jel ( ~r ) + Z d~r ′ ρ ( ~r ′ ) | ~r − ~r ′ | + V XC [ ρ ( ~r )] , (1)where the 2nd and 3rd terms on the right are the directHartree and the basic exchange-correlation (xc) compo-nents. This basic xc functional V XC is parametrized di-rectly from ρ ( ~r ) by the following formula [53]: V XC [ ρ ( ~r )] = − (cid:18) ρ ( ~r ) π (cid:19) / − . " . (cid:18) πρ ( ~r )3 (cid:19) / , (2)in which the first term on the right is exactly derivableby a variational approach from the HF exchange energyof a uniform electron system with a uniform positivelycharged background and the second term is the so calledcorrelation potential, a quantity not borne in HF for-malism. The xc-functional that utilizes Eq. (2) is thenfurther refined by adding a parametrized potential [54]in terms of the reduced density and its gradient ∇ ρ asfollows, V LB = − β [ ρ ( ~r )] / ( ξX ) βξX sinh − ( ξX ) , (3)where β = 0 .
05 is empirical and X = [ ∇ ρ ] /ρ / . Theparameter ξ is a factor arising in transition from the spin-polarized to spin-unpolarized form [55]. This method ofgradient-correction to the xc-functional, termed as LB94,is more built into the theory and leads to a considerableimprovement in the asymptotic behavior of the electronby comparing well with the exact Kohn-Sham potentialscalculated from correlated densities.We show the ground state radial potentials and bandsof C and C in Fig. 1(b). C produced bands of six π (one radial node) and ten σ (nodeless) states. Amongthese the homo and homo-1 levels are of 7 h ( ℓ = 5) and 6 g ( ℓ = 4) π character respectively – a result known from thequantum chemical calculations [56] supported by directand inverse photoemission spectra [57], and from energy-resolved electron-momentum density measurements [58].The homo-2 level is of 10 l ( ℓ = 9) σ character. TheLDA radial wavefunctions for these three outer statesare shown in Fig. 1(a). Linear response type calculationsusing this ground state basis well explained measuredphotoemission response of C at the plasmon excitationenergies [48, 59]. Similar calculations at higher energiesalso supported an effective fullerene width accessed inthe photoemission experiment [47]. These general groundstate properties also hold good for C that producedbands of nineteen σ and eleven π states, while its homoand homo-2 are of 19 w ( ℓ = 18) and 18 v ( ℓ = 17) σ character respectively with homo-1 being 12 m ( ℓ = 10) π (Fig. 1). -0.6-0.4-0.200.20.40.60.8 R a d i a l o r b it a l ( a . u . )
19w (homo): C
12m (homo-1)18v (homo-2)
Radial co-ordinate (Angstrom) -50-40-30-20-100 R a d i a l po t e n ti a l ( e V ) C C C σ C π C σ C π
7h (homo), C
6g (homo-1)10 l (homo-2) 1.3 1.5 (a)(b) FIG. 1. (Color online) (a) Ground state radial wavefunctionsfor homo, homo-1, and homo-2 of C and C calculated inLDA. (b) Corresponding radial potentials are shown and shellwidths are identified. Energy bands of σ and π characters (seetext) are illustrated B. CDW-FS to model Ps formation
We consider an incoming positron of momentum ~k i which captures an electron from a C N fullerene boundstate φ i ( ~r − ) to form a Ps state φ f ( ~ρ ). As illustratedin Fig. 2, the positron and electron position vectors, ~r + and ~r − respectively, originate from the center of the C + N ion so that ~ρ = ~r + − ~r − is their relative position vector. ~k +( − ) denote positron (electron) outgoing momenta in Psthat are equal, resulting ~k β = 2 ~k +( − ) to be the momen-tum of Ps itself. To be exact, all the momenta are re-duced momenta. Since we access energies above fullereneplasmon resonances, the many-body effect is not impor-tant, justifying the use of mean-field LDA wavefunctionsand potentials for C N (subsection IIA) in the frameworkof independent particle model. In that frame, the prior form of the Ps formation amplitude can be given in thecontinuum distorted-wave final-state (CDW-FS) approx-imation [44, 60] as, T − αβ ( ~k i ) ∼ Z d~r − F ( − ) ∗ ~k − ( ~r − ) W ( ~r − ; ~k i ) φ i ( ~r − ) , (4)in which W ( ~r − ; ~k i ) = Z d~r + F ( − ) ∗ ~k + ( ~r + ) φ ∗ f ( ~ρ ) × (cid:20) V sci ( r + ) − ρ (cid:21) F (+) ~k i ( ~r + ) , (5)with, F (+) ~k i = N + ν ′ α exp( i~k i · ~r + ) × F ( − iν ′ α ; 1; − i~k i · ~r + + ik i r + ) (6a) and F ( − ) ~k − F ( − ) ~k + = N − β − N − β + exp( i ( ~k − · ~r − + ~k + · ~r + )) × F ( − iβ − ; 1; − i~k − · ~r − − ik − r − ) × F ( iβ + ; 1; − i~k + · ~r + − ik + r + ) . (6b) e + e + e + ee Ps Ps d D Q d ~1/ D Q k i k b r + r – homo/homo-1 r r b FIG. 2. (Color online) A schematic diagram of the diffractionmechanism in the Ps formation from C . Position vectors ofpositron, electron, and Ps from the center of the molecularcation, and the electron-positron relative position vector ~ρ in Ps are schematically shown. The incoming and outgoingmomentum vectors are indicated. Diffraction resonances inthe ratio between Ps formation cross sections for C homoand homo-1 captures are included. We have defined [60], β + ≃ β − = ( Z + 1) µ β k β = ( Z + 1) µ α k ± (7) ν ′ α = Zµ α k i , (8)where Z is the net charge of the target (here Z = 0), andthe reduced masses are µ α ≃ µ β ≃
2, so that N − β ± = Γ(1 ∓ iβ ± ) exp( ∓ π β ± ) (9) N + ν ′ α = Γ(1 + iν ′ α ) exp( − π ν ′ α ) . (10)The positron scattering potential in Eq. (5) is given by, V sci = V sri ( r + ) + 1 r + , (11)where V sri is the short-range part of the positron-residualtarget interaction associated to the fullerene orbital la-beled i so that V sri ( r + ) = − V jel ( r + ) − N orb X k =1; k = i V H [ ρ k ( ~r )] − ( Z + 1) r + , (12)in which N orb is the number of fullerene orbitals (see sub-section IIA), and V H and V jel are respectively the Hartree and the jellium potential as in Eq. (1).We, thus, write the prior version of the CDW-FS am-plitude Eq. (4) as [60, 61] T − αβ = N + ν ′ α N −∗ β + N −∗ β − Z d~r + d~r − exp n i~k i · ~r + − i~k + · ~r + − i~k − · ~r − o F ( − iν ′ α ; 1; − i~k i · ~r + + k i r + ) φ i ( ~r − ) × (cid:18) V sri ( r + ) + 1 r + − ρ (cid:19) φ ∗ f ( ~ρ ) F ( − iβ + ; 1; i~k + · ~r + + ik + r + ) F ( iβ − ; 1; i~k − · ~r − + ik − r − ) . (13)In order to evaluate the amplitude, a partial wave expan-sion technique introduced in [60] has been employed.The initial fullerene orbital is φ i ( ~r − ) = R n t ℓ t ( r − ) Y ℓ t ,m t (ˆ r − ) , (14)where n t , ℓ t and m t are the quantum numbers. The finalwavefunction is given by φ f ( ~ρ ) = 1 √ − ρ/ Y , (ˆ ρ ) ≡ ˜ R s ( ρ ) Y , (ˆ ρ ) , (15)since the ground state, 1 s , of the Ps atom is consideredin the present work. The angle differential cross section(DCS) for the capture then reads (cid:20) dσd Ω (cid:21) n t ℓ t m t = 14 π k β k i µ α µ β (cid:12)(cid:12)(cid:12) T − αβ (cid:12)(cid:12)(cid:12) (16)with (cid:12)(cid:12)(cid:12) T − αβ (cid:12)(cid:12)(cid:12) = (4 π ) ( k i k + k − ) ˆ l t × (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X l i L i l i e iδ li ˆ l i ˆ L / ( − L S l i L Y L,m t (ˆ k β ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (17)where the notation b l = 2 l + 1 has been used. Moreover,we have defined S l i L = X ll ′ l f i − l − l f e i ( δ l + δ lf ) ( − l ′ A ll ′ l f l i L R ll ′ l i l f (18)with A ll ′ l f l i L = ˆ l ˆ l ′ ˆ l f (cid:18) ℓ t l l ′ (cid:19) (cid:18) l i l ′ l f (cid:19) (cid:18) l l f L (cid:19) × (cid:18) l i L ℓ t − m t m t (cid:19) (cid:26) l i L ℓ t l l ′ l f (cid:27) , (19) R ll ′ l i l f = Z ∞ F l i ( k i r + ) V ll ′ ( r + ) F l f ( k + r + ) dr + , (20) V ll ′ ( r + ) = Z ∞ r − R n t ℓ t ( r − ) J l ′ ( r − ; r + ) F l ( k − r − ) dr − , (21) J l ′ ( r − ; r + ) = 12 Z +1 − ˜ R s ( ρ ) (cid:18) V sri ( r + ) + 1 r + − ρ (cid:19) × P l ′ ( u ) du , (22)and ρ = (cid:0) r − + r − r − r + u (cid:1) / . (23)The functions F l ( k ± r ) and F l ( k i r ) are the Coulomb ra-dial wave functions with the Sommerfeld parameters η = β ± and η = ν ′ α , respectively. The phase shifts δ l are the usual Coulomb phase shifts δ l = arg Γ( l + 1 + iη ). P l indicates the Legendre polynomial of degree l .Upon averaging Eq. (16) over m t and denoting theelectron occupancy number of the fullerene ( n t ℓ t ) stateby occ( n t ℓ t ), we obtain (cid:20) dσd Ω (cid:21) n t ℓ t = occ( n t ℓ t )2(2 ℓ t + 1) × X m t (cid:20) dσd Ω (cid:21) n t ℓ t m t . (24)Finally, the angle-integrated cross section is evaluated as,[ σ ] n t ℓ t = Z π sin( θ ) dθ Z π dϕ (cid:20) dσd Ω (cid:21) n t ℓ t = occ( i ) × πµ α µ β k β k i ( k + k − ) X l i L ˆ l i ˆ L ˜ S l i L ˜ S ∗ l i L , (25)where ( θ, ϕ ) are the angles of ~k β (with respect to theincoming positron direction defined by ~k i and which isconsidered to be along the z -axis), and˜ S l i L = X ll ′ l f i − l − l f e i ( δ l + δ lf ) ( − l ′ ˜ A ll ′ l f l i L R ll ′ l i l f , (26)˜ A ll ′ l f l i L = ˆ l ˆ l f (cid:18) ℓ t l l ′ (cid:19) (cid:18) l i l ′ l f (cid:19) (cid:18) l l f L (cid:19) × (cid:26) l i L ℓ t l l ′ l f (cid:27) . (27) III. RESULTS AND DISCUSSIONS
We compute Ps(1 s ) formation cross sections, Eq. (25),for the capture from various fullerene levels as a functionof the positron impact energy in LDA + CDW-FS schemediscussed above. Cross sections show trains of shaperesonances. These resonances emerge from a diffractionmechanism based on the fullerene molecular structure.An elegant analytic interpretation of this diffraction ef-fect is given in Ref. [45] which we briefly review here.This treatment assumed plane waves, instead of threedistorted Coulomb continuum waves in Eqs. (6). It alsoassumed Ps formations in the forward direction, whichwas found to be the most dominant direction in bothearlier [62] and contemporary experiments [63]. It wasfurther noted: (i) Large values of V sci [Fig. 5(a)] at themolecular shell indicate the shell to be the dominant zoneof repulsive positron-fullerene interactions; (ii) The shapeof the Ps(1 s ) radial wavefunction ˜ R s ( ρ ) as a functionof electron-positron separation ρ = | ~r + − ~r − | justifiesthe maximum Ps probability density at r − = r + ; (iii)The radial wave functions of fullerene i -th levels of cap-ture [Fig. 1(a)] ensure that electrons to form Ps are onlyavailable at the shell zone. The analytic simplificationto interpret the exact numerical results then followed anapproximation of the radial integration to obtain the am-plitude for a π (number of radial node η r = 1) or a σ ( η r = 0) state capture in the following form: T − αβ ( ~k i ) ∼ S ( ~k i ) − k − q sin( Qr c − ℓ t π/ η r π/ × Z dr − A ( r p ) sin( Qα ( r p )) , (28)where the momentum transfer vector ~q = ~k + − ~k i and therecoil momentum Q = k i − k ± for the Ps formation inthe forward direction. S ( ~k i ) is the contribution of inte-grations over 1 /ρ in Eq. (5) which is assumed weak in res-onance structures. r p are the radial positions of dominantcontributions representing ρ = 0 where Ps(1 s ) wavefunc-tions have their transient maxima at distances α fromthe molecular radius r c . Obviously, the integral in theabove equation spatially dephases the sin( Qα ) modula-tion, since α varies with r − , retaining maxima in the am-plitude only via the term cos( Qr c − ℓ t π/ η r π/ Q : when an odd integer multiple of the half-wavelengthof effective continuum wave as a function of Q fits a dis-tance r p , a rather complicated diffraction pattern in theenergy domain is formed from the constructive interfer-ence. Several such fringe systems due to the variable r p cumulatively overlap to finally result into a centroid fringe pattern of more uniform peaks (bright-spots) via adephasing mechanism in the integration described above.Since these diffraction peaks appear in the energy (mo-mentum) domain, they are characteristically diffractionresonances.Resonances, however, must scale differently in thecross sections, which are derived from the squared modu-lus of the amplitude via Eq. (16). Therefore, squaring ofEq. (28) results in doubling the argument of the trigono-metric function to obtain cos( Qd c − ℓ t π + η r π ) for approx-imate resonance positions in the cross section, where d c is -3 -2 -1 P s ( s ) c r o ss s ec ti on ( a . u . )
7h (homo)6g (homo-1)10 l (homo-2)6h0.5 1 1.5 2 2.5Recoil momentum in forward direction (a.u.)10 -3 -2 -1 P s ( s ) c r o ss s ec ti on ( a . u . )
19w (homo)12m (homo-1)18v (homo-2)11mH(1s) target (a)(b) C C FIG. 3. (Color online) (a) Ps(1 s ) formation cross sections forcaptures from four outer levels of C (a) and C (b) asa function of the recoil momentum ( Q ) in forward direction.The corresponding result of hydrogen 1 s capture is also shownon panel (a) for the comparison. the fullerene diameter; note that besides d c , the positionsalso depend on phase-shifts based on initial state infor-mation ℓ t and η r . In any case, at the cross section level,one may draw an analogy with the single-slit experimentof classical wave optics: Ps formation amplitudes fromdiametrically opposite sites of a fullerene molecule quan-tum mechanically interfere to produce fringe patterns inthe momentum domain, as schematically shown in Fig. 2.Consequently, the resonance pattern must shrink for alarger fullerene due to increased slit width as we discussbelow.Fig. 3 presents the exact numerical Ps(1 s ) cross sec-tions as a function of the forward-emission recoil mo-mentum Q that displays series of broad resonances fora set of four capture levels of C and C . The rangeof Q corresponds to the electron excitation energy fromroughly 50 eV, which is above the plasmon excitations,to 270 eV, which is below the K -shell of atomic carbon(in order to validate the jellium modeling). Note thatthe Ps(1 s ) cross section in Fig. 3(a) for the capture fromthe 1 s level atomic hydrogen is flat, since no diffraction ispossible when the electron is captured from “everywhere”in a Coulomb system. We also note in Fig. 3 that, asexpected, the non-resonant background strength of thecross sections is proportional to the number of electronsthat fills the level, while for a fixed occupancy number(same ℓ t ) a π level produces stronger cross section thana σ likely due to larger spatial spread of π wavefunctions[Fig. 1(a)].Considering the resonances in Fig. 3, we first note ageneral trend: the resonances at low Q for the capturefrom high angular momentum states, such as, homo-2 ofC and homo, homo-2 of C , are significantly wide.This is likely a direct consequence of stronger distortionsof continuum waves for higher ℓ t . For the relative posi-tions of the resonances of varying capture states severalobservations can be made. As seen in Fig. 3(a), the res-onances for captures from C homo (7 h ) versus homo-1(6 g ) are positioned out-of-phase, since, even though bothlevels are of π character ( η r =1), their angular quantumnumber ℓ t differs by one unit resulting in a 180 o relativephase-shift in cos( Qd c − ℓ t π + η r π ). This comparisonis however more complicated between homo (19 w ) andhomo-1 (12 m ) captures of C [Fig. 3(b)], since, not onlythat these levels are of respectively σ and π nodal charac-ters, but also their ℓ t values are vastly different. However,an out-of-phase offset between homo and homo-2 (18 v )resonances of C is roughly noted at least at higher Q , since both are σ levels but their ℓ t differs by one.Furthermore, comparing the results between C homoand an inner σ level 6 h of the identical angular charac-ter, out-of-phase resonance locations are seen – a patternwhich is obviously due to their π versus σ characters ac-counting for a half-cycle shift. Likewise, the same reasonexplains why the homo-1 compared to the inner 11 m cap-ture in C displays out-of-phase resonances. All theseobservations generally indicate that electronic structuralinformation can be accessed spectroscopically by level-differential Ps formation from fullerenes.The variations and similarities in the shape of the res-onances are most spectacularly illustrated by consideringthe cross section ratios, shown in Fig. 4, which neutral-ize the non-resonant background decays. Accessing theseratios in experiments by the Ps formation spectroscopymay improve the accuracy by minimizing experimentalnoise from cancellations. It is important to note againthat even though we are attempting to use plane wave de-scriptions in our analysis to interpret the key results, theexact character of the resonances in Fig. 4 are far morediverse. While the ratios of capture levels of π charac-ters, both the combinations chosen for C [Fig. 4(b)] andhomo-1/11 l , 3 p /2 s for C [Fig. 4(a)], produce reason-ably uniform structures, the resonances are dramaticallycomplex for homo/homo-1 of C where it is a π -to- σ ratio with vastly different angular symmetry. The mostnoticeable general distinction between the distribution ofthe resonances of C versus C comes from the molec-ular size. The underpinning of the diffraction process isevidenced in a nearly halfway shrinkage of the fringe pat-terns for roughly doubly larger C . This clearly upholdsthe single-slit analogy. A more quantitative analysis isshown below.A powerful approach to bring out the connection ofdiffraction resonances with the fullerene diameter is toevaluate the Fourier spectra of the cross sections as C r o ss - s ec ti on r a ti o l
3p / 2s C r o ss - s ec ti on r a ti o
7h (homo) / 6g (homo-1)3p / 2s C (a)(b) ∆ Q~ 0.49 ∆ Q~ 0.43 C ∆ Q~0.22 ∆ Q~0.25 ∆ Q~0.23
FIG. 4. (Color online) Ratios of selected combinations of crosssections, for C (a) and for C (b), illustrate resonances.Typical separations (∆ Q ) between some of these resonancesare marked. a function of Q . To generate the input signals forthe Fourier transform of the resonances on a flat, non-decaying background, we considered ratios of the resultsof two consecutive angular levels of π electrons for boththe fullerene molecules. Fourier magnitudes of these ra-tios are calculated by using the fast Fourier transformalgorithm after applying an appropriate window func-tion to reduce spurious structures. While such window-ing adds some extra width to the “frequency” peaks, itpractically does not compromise the peak positions. Theresults are presented in Fig. 5 in reciprocal (radial) coor-dinates. All the curves in Fig. 5(b) exhibit strong peakslocated around the diameter d c for each of C and C ,as expected from our model equation, Eq. (28), that in-cludes the function cos( Qd c − ℓ t π + η r π ); the transformmagnitudes are insensitive to phase-shifts connected to ℓ t and η r . To guide the eye, LDA radial potentials andpositron scattering potentials for homo captures are plot-ted in Fig. 5(a). Notice, the small, systematic offset of thepeaks towards lower values with the increasing angularmomentum. This is even another signature of the factthat the continuum waves are Coulomb distorted andso are more complicated than simple plane waves usedin our model analysis. In fact, we could generally an-ticipate this variation by noting in Fig. 4(a) the typicalseparations ∆ Q of 0.22 a.u. , 0.25 a.u. and 0.23 a.u. re-spectively for homo/homo-1, homo-1/11 l and 3 p /2 s ratioand then determining their Fourier conjugate (2 π/ ∆ Q )values of 28.5 a.u. , 25.1 a.u. and 27.3 a.u. being some-what different but close to the radius of 27 a.u. of C .But in this case, homo and homo-1, being of σ and π nodal characters, should be left out in explaining thetrend in all- π Fourier spectra [Fig. 5(b)]. For C , theratios considered in Fig. 4(b) are all of π symmetry andtherefore conform with the Fourier spectra trend. In-deed, the quoted ∆ Q values are 0.49 a.u. and 0.43 a.u.respectively for homo/homo-1 and 3 p /2 s ratio of C cor-respond to slightly increasing Fourier conjugate values of12.8 a.u. and 14.6 a.u., yet being close to the moleculardiameter of 13.4 a.u. In summary, these Fourier recip-rocal spectra unequivocally support the theme that thehost of broad resonances are indeed the fringe patternsin the energy domain for a Ps formation channel wherethe Ps-emission diffracts in energy off the shell – a spher-ical slit. The comparison demonstrates the resonances tobe more compact in energy for C , which is a largerdiffractor. Our calculations (not shown) for the forma-tion of excited Ps(2 s ) have also produced similar generaltrends. R a d i a l po t e n ti a l s ( a . u . ) V LDA (r − ) V (r + ) F ou r i e r m a gn it ud e ( a r b . u . )
7h / 6g6g / 5f5f / 4d4d / 3p3p / 2s (a)(b)r c d c C C r c d c V LDA (r − ) V (r + )
12m / 11 l
10k / 9j8i / 7h6g / 5f4d / 3p
FIG. 5. (Color online) (a) LDA radial potentials and thepositron scattering potentials for the homo capture from bothC and C ; molecular radii are also pointed out. (b) TheFourier transform magnitudes of Ps(1 s ) cross section ratiosfor captures from various π levels of both the fullerenes as afunction of the radial coordinate × IV. CONCLUSION
In conclusion, we extend our previous calculation [45]and compare the Ps formation cross sections in the CDW- FS method among electron captures from various elec-tronic levels as well as between C and C . The molec-ular ground state structures are modeled by a simple butsuccessful LDA methodology that used LB94 exchange-correlation functional. Hosts of strong and broad shaperesonances in the Ps formation are found that can ac-cess electronic structure information of the targets. Theresonances engender from a diffraction effect in the Psformation process localized on the fullerene shell whichis further established by comparing results of two differ-ent fullerene “spherical slits”. Application of a Fourieranalysis technique to the Ps spectra has facilitated theanalysis. The success of an analytic model based on for-ward emissions suggests that the effect is likely predom-inant in the forward direction of Ps formation.As an additional future motivation, the Ps(2 p ) channelis attractive too, since it can be monitored optically [64].Also, the effect discussed in this paper should be univer-sal for Ps formation from nanosystems, including metal(alkaline earth, noble, coinage) clusters, carbon nan-otubes, or even quantum dots that, like fullerenes, con-fine finite-sized electron gas. The work further motivatesa new research direction to apply Ps formation spec-troscopy to gas-phase nanosystems which began with ourearlier published research [45, 46], since fullerenes cur-rently enjoy significant attraction in precision measure-ments. Fullerenes [65] and metallic nanoparticles [43, 66]are nowadays available in gas-phase. However, prob-ing the target-state differential Ps-signals is still chal-lenging for current techniques [67]. But accessing thiswill be beneficial in general and, in particular, since thepredicted resonances, having a target angular-state andradial-structure dependent momentum-shift [Eq. (28)],will largely flatten out in the total Ps measurement. Thetechnique to measure the recoil momentum of the cationsmay be improved by using a supersonic gas jet to increasethe overlap with the positron beam. Resolving the Pslevel may not be so critical (may be done by laser spec-troscopy of a dense Ps gas), since Ps(1 s ) signal shouldlargely dominate. We hope that this theoretical effortwill help add further to the motivation in measuring dif-ferential Ps production at least within a narrow forwardangle. ACKNOWLEDGMENTS
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