Karl-Erik Thylweand Patrick McCabe

1KTH-Mechanics,Royal lnstitute of Technology,S-100 44 Stockholm,Sweden

2CCDC,12 Union Road,CB2 1EZ,Cambridge,UK

1 Introduction

In an earlier publication[1]the authors showed that certain broad peaks in the total cross section could not be explained as resonance phenomena in the sense of long-lived quasi-bound states. One could see a correlation with complex-angular momentum pole positions(Regge poles)behaviors as functions of the energy(Regge trajectories).[1?2]The pole trajectories were seen to turn in the complex angular momentuml-plane,with Relhaving a maximum close to an integer value as function of energy.Regge trajectories turning in the complex angular momentuml-plane were studied in Ref.[2].

Resonances in single-channel scattering are difficult to detect and identify experimentally for collision systems with large masses;see e.g.Toennieset al.in Ref.[3].Scattering experiments with atomic particles are most sensitive to the broader resonances in a total cross section,with energies near the top of the centrifugal barrier.Such resonances may be mixed with non-resonance states of the type mentioned in Ref.[1].Sharp resonances due to tunneling through the barrier and back again give negligible contribution to the cross sections but play a dominant role in predissociation spectroscopy;see e.g.Kolos and Peek,Bernstein.[3]In ion-atom systems the number of partial waves increases in comparison with atom-atom scattering;see Konrad and Linder in Ref.[3].

Recentexperimentson electron-atom resonances,forming negative ions(anions),focus on electron affinities;see Walteret al.(2011)in Ref.[4].Laser spectroscopy techniques seem to be the primary tools(Calbreseet al.(2005)in Ref.[4]).Electron transfer via(somewhat more complex)anions is among the most fundamental of chemical reactions and features prominently in all branches of chemistry;see Bullet al.(2015)in Ref.[4].

In the present note a potential model relevant for electron-atom scattering is used. The relevant“resonance”peaks in total scattering cross section are explained in terms of scattering phase shifts.The partial-wave analysis of scattering cross sections associate the scattering dynamics with phase shifts,one for each partial wave.The phase shift can be obtained from the regular solutions of the Schr¨odinger equation.Resonance phenomena are explained in terms of phase shift behaviors in many text books.[5?8]In the low-energy limit resonant phase shifts,due to attractive potentials,typically approach positive valuesnπ(Levinson’s theorem),wherenis the maximal number of bound states existing in that potential.Typical resonances in a partial wave(integer orbital angular momentuml)are related to relatively rapid changes,or jumps,in the phase shifts,δl(E),as function of energy.In potential scattering with a single potential the phase shift de fines the so-called scattering matrix expressed asSl=exp(2iδl),and also the transition matrix element defined here asTl=Sl?1.Note that each|Tl|2is multiplied by a factor 2(2l+1)π/E,whereEis the scattering energy,in order to compute the total cross section from signi ficantl-contributions.Therefore the contribution from a single partial-wave can be large at low scattering energies,where only the few first values oflare expected to be important.[7]

The rational function Thomas-Fermi potential(RTF)used for illustration of phase-shift behaviors has two adjustable parameters which can be adjusted to con firm a couple of experimentally measurable resonances in a reasonable way. It has been applied with some success in various calculations of electron-atom total cross sections.[1?2]Several anion states,identi fied by laser spectroscopy techniques,[4]have been interpreted as scattering resonances.[2]However,relevant scattering cross section measurements showing clearly resolved sharp resonances are hard to find in the literature.

The main purpose of this note is to make clear that certain broad peaks in the total cross section as function of scattering energy may not be proper resonances.There is a similarity between such broad peaks and so-called zeromomentum(quasi-)resonances appearing in connection with Levinson’s theorem,[6?8]saying that for particular potentials the zero-energy phase shift approaches an odd(rather than an even)multiple ofπ/2.This note con firms the existence of certain peaks in the total cross-section that differ from sharp and broad resonances by having no time delays.The notion of time delay relies here on its de finition 2~dδl(E)/dE.[5?6]It turns out that these particular peaks occur at maxima of the phase shifts as functions of energy,and these maxima are close to an odd integral multiple ofπ/2.

Section 2 brie fly describes a scattering wave function satisfying the radial Schr¨odinger equation.Computations and illustrations are in Sec.3 and conclusions in Sec.4.

2 Radial Schr¨odinger Equation

By introducing a dimensionless radial variabler/a0→rwitha0being the Bohr radius,one obtains the radial Schr¨odinger equation

“~” is Planck’s constant divided by 2πandmthe reduced mass of the collision system.

The regular solution Ψlsatis fies the scattering boundary conditions that de fine the scattering matrix elementsSl,[8]i.e.

The phase shiftsδlare real and the maxima of|Tl|are given by the conditionδl=(n+1/2)π,n=0,1,2,...for whichTl=?2.

3 Computations and Illustrations

The rational function Thomas-Fermi(RTF)potential[1?2]used for illustrating an electron-atom interaction has a Coulomb attraction near the origin but adopts an attractive polarization interaction at long range.The dimensionless form of the RTF potential used here is given by[2]:

with the long-range behavior

Three sets of potential parameters are used.All of them correspond to the nuclear core chargeZ=63 and all with the same long-range polarization property de fined byab=0.015.These potentials are used to illustrate both non-resonant and resonant behaviors of phase shifts and squared moduli of the transition matrix elements

Fig.1 A typical phase shift behavior for a wide range of k-values.The RTF potential is de fined by a=0.19,with Z=63 and l=1.

Fig.2 Particular low-energy phase shifts as functions of k.The RTF potentials are de fined by a=0.15,0.19,and 0.30,with Z=63 and l=1.

A phase shift behavior in the range 0

On a more detailed level,for scattering wave numbers in the range 0

Fig.3 Comparison of the phase shifts corresponding to l=1(as in Fig.2)and l=2 and l=3 as functions of k.The RTF potential is de fined by a=0.19,with Z=63.For l=3 the phase shift passes through an odd integral multiple of π/2 in value,while for l=1,2 it does not.

By fixing the valuea=0.19 for the RTF potential and comparing the phase shifts forl=1,2,and 3,one obtains Fig.3.The corresponding behaviors of|Tl(k)|2are seen in Fig.4.Firstly,one observes that the effective potential forl=2 supports five bound states and that forl=3 supports three bound states.The phase shift curveδ2(k)does not pass through an odd integral multiple ofπ/2 in this energy range.It has a similar smooth behavior toδ1(k),but is not as close to an odd integral multiple ofπ/2 asδ1(k).

The phase shiftδ3(k)shows a typical resonance behavior.It jumps across an odd integral multiple ofπ/2 and adds to its value one unit ofπ.In this case the slope dδ3(k)/dEis comparatively large so that the semiclassical notion of time delay applies(or may apply).The energy is low(k2?1)and the partial cross section froml=3 may be signi ficantly large.The effect on|T3(k)|2in Fig.4 indicates a narrow peak and elsewhere small contributions.

Fig.4 as function of k.The RTF potential is de fined by a=0.19,with Z=63 and l=1,2,and 3.

4 Conclusions

Signi ficant,low-energy peaks in the total cross section may be observed that are not resonances.This is explained in terms ofparticular phase shift behaviors,not generally seen for arbitrary potential parameters.The responsible phase shift stays close to an odd integral multiple value ofπ/2 for a range of scattering energies.At low energies contributions froml=1 orl=2 in the cross sections may be signi ficant.In contrast,a typical resonance phase shift passes through an odd integral multiple value ofπ/2 completely,like that forl=3 in Fig.3,and in a relatively small energy region.

[1]K.E.Thylwe and P.McCabe,Eur.Phys.J.D 68(2014)323.

[2]S.M.Belov,N.B.Avdonina,Z.Fel fli,et al.,J.Phys.A 37(2004)6943;Z.Fel fli,A.Z.Msezane,and D.Sokolovski,Phys.Rev.A 79(2009)012714;S.M.Belov,K.E.Thylwe,M.Marletta,et al.,J.Phys.A 43(2010)365301.

[3]J.P.Toennies,W.Welz,and G.Wolf,J.Chem.Phys.61(1974)2461;J.P.Toennies,W.Welz,and G.Wolf,J.Chem.Phys.71(1979)614;R.B.Bernstein,Chem.Phys.Lett.25(1974)1;W.Kolos and J.M.Peek,Chem.Phys.12(1976)381;M.Konrad and F.Linder,J.Phys.B 15(1982)L405.

[4]C.W.Walter,N.D.Gibson,Y.G.Li,et al.,Phys.Rev.A 84(2011)032514;D.Calabrese,A.M.Covington,W.W.Williams,et al.,Phys.Rev.A 71(2005)042708;J.N.Bull,C.W.West,and J.R.R.Verlet,Phys.Chem.Chem.Phys.17(2015)16125.

[5]C.J.Joachain,Quantum Collision Theory,North-Holland,Amsterdam(1975).

[6]M.S.Child,Molecular Collision Theory,Dover Publications,New York(1996).

[7]H.A.Bethe and R.Jackiw,Intermediate Quantum Mechanics,3rd ed.,Benjamin/Cummings,Menlo Park(California,US)(1986).

[8]N.F.Mott and H.S.W.Massey,The Theory of Atomic Collisions,3rd ed.,Oxford University Press,London,Ch.9(1965).