semiclassical theory of alignment effects in near-resonant energy
TRANSCRIPT
PHYSICAL REVIEW A, VOLUME 63, 052711
Semiclassical theory of alignment effects in near-resonant energy-transfer collisionsof rare-gas atoms with aligned Rydberg atoms
Eric G. LaytonJoint Institute for Laboratory Astrophysics, University of Colorado, Boulder, Colorado 80302
Michael A. Morrison*Department of Physics and Astronomy, University of Oklahoma, Norman, Oklahoma 73019-0225
~Received 29 September 1999; published 17 April 2001!
Recent experiments@E. M. Spainet al., J. Chem. Phys102, 24 ~1995!# discovered alignment effects in crosssections for near-resonant energy-transfer collisions of Xe atoms with Ca Rydberg atoms at a single meanrelative velocity. A collaborative quantum-mechanical study@W. Isaacs and M. A. Morrison, Phys. Rev. A57,R9 ~1998!# confirmed these findings and discovered pronounced oscillations in the velocity dependence ofstate-to-state cross sections. Collisions corresponding to the 17dm0
→18pm transitions in the Ca-He system arehere analyzed semiclassically. This analysis shows that the origin of these oscillations is a phase interferenceprocess unique to Rydberg target states. We further demonstrate the importance of retaining the energy defectand of using quantum-defect phase-shifted radial functions in calculations of alignment cross sections forRydberg states.
DOI: 10.1103/PhysRevA.63.052711 PACS number~s!: 34.60.1z
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I. INTRODUCTION
In conventional low-energy electron-atom scatteringperiments, with the target gas in a cell or beam, atoms ininitial staten0 l 0 are uniformly distributed among magnetsublevelsm052 l 0 ,...,1 l 0 . These experiments determinaverages over initial and sums over final sublevels, sucthe level-to-level integral cross section
sn0 l 0→nl51
2l 011 (m052 l 0
l 0
(m52 l
l
sn0 l 0 m0→nlm . ~1!
Such experiments yield no information about the role ofmagnetic quantum numberm in the scattering process. Oncannot determine, for example, whether energy transfeone ‘‘m channel’’ (m0→m) is more or less efficient thananother. Nor can one ascertain whether the interactionserves or obliterates information about the initial distributiof target states among magnetic sublevels. Such insightquire an initial state with ananisotropicdistribution of mag-netic sublevels. With the advent of pulsed lasers, the preration of such states became feasible, and studiescollisions involving laser-excited initial states of atoms bgan to proliferate@1–11#. Alignment phenomena have generated great interest because of the detailed insight theyvide into fundamental mechanisms that influencedynamics and properties of colliding particles@12–16#.
Two types of nonstatistical distributions can be realizan aligned state, in which the probabilityp(m0) of findingan atom in sublevelm0 depends on the magnitude but notthe sign ofm0 , i.e., p(2m0)5p(1m0), and anorientedstate, in which the population depends on both the mag
*Electronic address: [email protected];www.nhn.ou.edu/ morrison
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tude and sign ofm0 , i.e., p(2m0)Þp(1m0) ~see, for ex-ample, Ref.@17# and references therein!. Here the magneticquantum numbers refer to a quantization axis chosen parto the relative velocity of the projectile and target, i.e., tcollision axis. In this paper, we are concerned with collisioinvolving aligned states, which can be prepared via absotion induced by linearly polarized lasers@8,10#.
Nearly all experiments on collisions of projectile atomwith aligned target atoms prepare the target in alow-lyingexcited state. Prototypical are the experiments by Leone acollaborators on spin changing and fine-structure changnear-resonant energy-transfer collisions of calcium with vaous nonreactive rare-gas atoms@1–11#. In these experimentsone, two, or three linearly polarized lasers are used to exCa to low-lying alignedp, d, or f excited states, respectivelyfor d and f states, the relative orientation of the electric fievectors of the lasers can be adjusted to produce initial stwith quite different spatial distributions of electron densitAlignment curves are obtained by varying the angleb be-tween the relative velocityv and the electric fieldEL of oneof the lasers. The variations of these cross sections witbsignify alignment effects; the intensity of these variatioquantifies the strength of these effects. Experimentally, vaing b changes the admixture of magnetic substates ininitial state, altering the ‘‘shape’’ of the electron probabilidensity. Alignment phenomenon signal that, in effect, ttarget ‘‘remembers’’ its initial alignment characteristicthrough the collision. In addition to demonstrating such phnomena, data from these experiments has revealed thetive efficiency of various pathways for energy transfer, dtinguished by the magnitude of the initial magnetic quantnumber@4,6#.
These data have been interpreted in terms of ‘‘orblocking and following’’ models, which are predicated on thformation during the collision of a transient quasimolecuelectronic state@2,18,19#. In this picture the orbital of theexcited electron temporarily couples to the internuclear a
©2001 The American Physical Society11-1
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ERIC G. LAYTON AND MICHAEL A. MORRISON PHYSICAL REVIEW A 63 052711
of the quasimolecule within a few tens of bohr of the codinate origin. Consequently, depending on the distancewhich the orbital ‘‘locks’’ and on the symmetry of the resulting electronic state, cross sections may exhibit alignmeffects of varying degree. Theoretically, avoided crossingrelevant potential-energy curves are invoked to determwhich symmetry of the quasimolecular state is primarilysponsible for energy transfer. Except for the work of Hicman on Ca–Xe collisions@20,21#, applications of this modehave been rendered qualitative by the lack of accupotential-energy curves for systems of interest. Neverthelanalyses in terms of ‘‘orbital locking and following’’ theorhave added significantly to our understanding of the dynaics of energy transfer@12,13#.
A quite different physical situation occurs if the initiastate of the target atom is a Rydberg state. In this caseelectron’s extremely diffuse probability density and the higenergy density of accessible bound states make problema molecular~Born–Oppenheimer! description of the dynamics @22,23#. Indeed, orbital locking theories predict that nalignment effects will appear if the initial aligned statehighly excited, for molecular bond formation is not an important collision mechanism. To explore this new physirealm, a joint experimental-theoretical investigation was itiated, with measurements of alignment effects in thermstate-changing collisions involving aligned Ca Rydbergoms and rare-gas atoms by Leone and collaborators@24,25#and a dual theoretical program consisting of complemenquantum@26,27# and semiclassical calculations.
The experimental and quantum-mechanical componof this project recently reported the unexpected presencpronounced alignment effects in cross sections for neresonant energy-transfer collisions of Xe with Rydbergatoms. Measurements by Spainet al. @25# revealed unam-biguous alignment effects in cross sections for the 1d→18p at a single mean relative velocity; e.g., a marked prerence for end-on encounters when the Ca atom is prepin a dz2 state.
Quantum calculations by Isaacs and Morrison@26,27#confirmed these results and, by exploring a wide rangerelative velocities, uncovered oscillatory structures in crsections for this and other transitions in both Ca–Xe aCa–He collisions.The theoretical formulation used in thquantal calculations explicitly precluded the formation ofquasimolecular state. So the origin of the alignment effectsthe oscillations, and their dependence on the initial and fimagnetic quantum numbers of the electron remainemystery.
In the present paper we seek to resolve these mysterieanalyzing alignment effects in near-resonant Ca–He cosions using the semiclassical impact parameter method~Apreliminary account of this paper has appeared in Ref.@28#.!Our theoretical formulation allows easy identification of apects of the system that are crucial to the observed feat~e.g., the role of the energy defect and the phase shift inelectron’s radial functions due to quantum defects!. More-over it leads to an explanation of the aforementioned oslations as due to an interference process unique to Rydtarget states. It explains the disappearance of these os
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tions at large relative velocities~greater than a few thousanmeters per second!. Finally, it elucidates the striking dependence of these oscillations on the initial and final magnequantum numbers.
Section II summarizes key properties of alignment phnomena and defines relevant quantities. Our theoreticalmulation is described in Sec. III. Although the emphasisthis paper is on the Ca–He cross sections in Sec. IV B,first present results forl mixing in Na–He collisions: in SecIV A we use this well-studied problem@29–32# to test ourformalism, its underlying approximations, and our numeriapplication. In Sec. IV we also show that two approximtions widely used in previous studies of conventional ragas atom—Rydberg atom collisions—neglect of the enedefect and use of hydrogenic wave functions—produce sous errors in alignment studies. Finally, Sec. V offers ointerpretation of the oscillations under discussion.
II. ALIGNMENT EFFECTS
The preparation of an initially aligned state by a lineapolarized laser introduces into the dynamics the directionthe electric fieldEL . ~If more than one laser is used, e.g.,prepare an alignedd state, thenEL is usually chosen as thelectric field of the first laser.! The subsequent collision defines a second direction, the initial relative velocityv, whichis an axis of rotational symmetry for the collision. This vetor further defines thez axis of the collision frame. Thealignment-selected cross sectionssn0 l 0→nl(b;v) that aremeasured in the ‘‘two-vector correlation experiments’’Leone and collaborators are near-resonant energy-trancross sections for the transitionn0 l 0→nl as a function of theangleb betweenv andEL . A least-squares fit to these dayields magnetic sublevel cross sectionss um0u(v). This crosssection is the sum over all final magnetic sublevels ofstate-to-state cross sections,
s um0u~v !5 (m52 l
l
sn0 l 0 m0→nlm~v !. ~2!
One such cross section is obtained for each initial magnquantum numberm0 .
The fundamental theoretical quantities for describithese collisions are the scattering amplitudesf (K ,nlm←K0 ,n0 l 0 m0) for initial and final relative momentaK0 andK and for all allowed initial and final magnetic substatesm0andm. In the collision frame, with thez axis coincident withthe initial relative velocity, K5(u,w) are the scatteringangles. From these amplitudes~or from the scattering matrixin the collision frame! one can compute the magnetsublevel cross sectionss um0u(v).
These cross sections are independent of the sign ofm0because of reflection symmetry in the scattering plaformed byK0 andK . They are related to alignment-selectecross sections by the diagonal elements of the density mafor the initial state,
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SEMICLASSICAL THEORY OF ALIGNMENT EFFECTS . . . PHYSICAL REVIEW A 63 052711
sn0 l 0→nl~b;v !5 (m052 l 0
l 0
rm0 ,m0
~ l 0!~b!s um0u~v !. ~3!
The density-matrix elementrm0 ,m0
( l 0) (b), appropriately nor-
malized@17#, is the probability for finding the Rydberg electron in a state with quantum numbersl 0 andm0 . For a purestate, this element is given by the reduced Wigner maelement,p(um0u)5rm0 ,m0
( l 0) (b)5dm,m0
( l 0) (b). Equation ~3! ef-
fectively deconstructs the alignment cross section into ‘‘nematic quantities’’rm0 ,m0
( l 0) (b), which fully account for the
alignment properties of the particular initial state, and ‘‘dnamical quantities,’’ the magnetic sublevel cross sectis um0u. Absent alignment effects, the latter cross sectionsindependent ofum0u, and the sum in Eq.~3! is independentof b. So the extent to which each cross sectionsum0u(v) de-pends onum0u at a particular relative velocityv is a measureof the strength of the alignment effect at that velocity; if thecross sections are independent ofum0u, then no such effectare present and the collision has obliterated all informaticoncerning the initial alignment of the Rydberg electron@9#.Thus the magnetic sublevel cross sections fully descalignment phenomena; they are the data we present inIV B.
III. THEORY OF STATE-TO-STATE CROSS SECTIONS
To calculate eitherl-mixing or magnetic sublevel crossections we require amplitudes for the state-to-state trations a05(n0 l 0 m0)→a5(nlm) for all magnetic quantumnumbersm0 andm allowed by the orbital angular momenl 0 and l of the excitation (n0 l 0)→(nl). As our theoreticaltreatment is the same for both types of cross sections,here describe this common methodology, leaving detailsparticular scattering quantities to the relevant subsectionSec. IV.
Figure 1 shows the geometry of the collision and relevvariables. The collision frame is defined by its origin—tnucleus of the Ca1 core—and itsz axis ~with unit vectorez!,which is parallel to the initial relative velocityv. We adoptthe ‘‘quasifree-electron model’’@22,33,34# in which the corefunctions as a ‘‘spectator,’’ a point particle that does nparticipate in the collision. Rather, its role is to support tinitial and final states of the Rydberg electron. In the cosion frame, then, the relative velocity reduces to the protile velocity. The system wave function depends on the stial variablesr of the Rydberg electron and on those of trare-gas projectileR. In the semiclassical approximationhowever, the latter variables are treated classically aR5R(t). So the system wave function reduces to the Rydbelectron wave function and thus depends only onr . The evo-lution of the system is described by a set of coupSchrodinger–Hamilton equations.
For the collisions considered here, the inelasticity is vsmall compared to the relative kinetic energy. So the clacal trajectory of relative motion should be quite undisturband well approximated as rectilinear@35,36#. Identifyingb asthe impact parameter of the projectile, we write
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where we have exploited the axial symmetry of the collisito choose a trajectory that intersects thex axis ~with unitvector ex! at b. We describe the interaction of the rare-gprojectile with the Rydberg electron as a binary encounrepresented by a point contact potential discussed below.describe the quantum states of the Rydberg electron astionary angular momentum eigenstates labeled by quannumbersn, l, andm. Within this model we calculate finalstate transitions amplitudes numerically using first-ordtime-dependent perturbation theory@37–40#.
The semiclassical impact parameter method has a losuccessful history in research on atomic collisions in gen@35# and Rydberg atom-rare-gas collisions in particu@20,23#. Of particular relevance to the present study is tsemiclassical analysis of inelasticn, l changing and quasielastic l-mixing collisions of rare-gas projectiles witRydberg-atom targets by Lebedev and Fabrikant@36# ~seealso Sec. 4.3 and 6.3 of Ref.@41# and references therein!.These authors combined the impact-parameter methodnormalized perturbation theory to derive elegant analyticpressions and simple scaling formulas for the probabilitand cross sections for such collisions.
Treating the relative motion of the rare-gas projectile athe atomic core classically is a good approximation providthe de Broglie wavelength associated with this motionsmall compare to the size of the target atom@42#. This re-quirement is easily met for collisions of interest here; tradius of the Rydberg atom is huge, and the projectile-csystem has a large reduced mass and thus a small de Brwavelength.
In our investigations the projectile is helium. Becausefirst excitation threshold of He~'20 eV! is well above thebinding energies of the Rydberg electron, and the polaability of He ~0.2501 Å3! is small @43#, we treat this projec-tile as a structureless point mass. Since the Rydberg elec
FIG. 1. Geometry and coordinates for semiclassical descripof collisions of a rare-gas atom by a Rydberg electron in the cosion frame.
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ERIC G. LAYTON AND MICHAEL A. MORRISON PHYSICAL REVIEW A 63 052711
is distributed over a large volume of space and its probabdensity is correspondingly small, the electron–He interactis weak and acutely nonadiabatic, consisting primarily ofimpulse from the projectile to the electron. We model thinteraction using the Fermi contact potential
V~ t !52pAS \2
meD d~r2R!, ~5!
where A51.19a0 is the electron-He scattering leng@44,45#. Originally introduced by Fermi in studies of presure shifts in the spectra of alkali Rydberg atoms perturby rare buffer gases@33#, this model has been widely usedsubsequent research on Rydberg atom-rare-gas collis@20,23#. For rare-gas perturbers heavier than helium, effearising from the perturbation of the projectile’s ground-stelectron density by the Rydberg target electron becomeportant. Lebedev and Fabrikant@36# have shown how tomodify the simple scattering length approximation used heaccording to which the electron-rare-gas scattering amplitequals2A, to accommodate these effects.
We expand the semiclassical wave function of the Rberg electron in a basis of bound states as
C~r ,t !5(a
aa~ t !ca~r !e2 i enlt/\, ~6!
whereaa(t) is a transition amplitude andenl is the quantum-defect shifted energy in atomic units~hartree!
enl521
2~n2dnl!2 . ~7!
For the Rydberg states of Ca considered here, the quandefectsdnl are d17d50.9043 andd18p51.8721. For colli-sions with Rydberg atoms, this expansion is more approate than one in quasimolecular states, which yields a comcated set of coupled equations and requires accupotential-energy curves. For collisions in which the targatom is not in a Rydberg state, however, an expansionquasimolecular states would be suitable@46,47#; in such acollision the target is more ‘‘opaque,’’ i.e., its interactiowith the projectile is stronger, in part due to increased tarelectron density.
Neglecting spin-orbit and spin-spin interactions rendthe Rydberg atom core spherically symmetric, and wewrite its Hamiltonian eigenfunctions in terms of radial funtions unl(r ) as
ca~r !51
runl~r !Ylm~u,w!. ~8a!
Here Ylm(u,w) is a spherical harmonic@48# whose axis ofquantization is thez axis of the collision frame
Ylm~u,w!5NlmPlumu~cosu!eimw, ~8b!
with normalization constant
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Nlm5~21!mA~2l 11!
4p
~ l 2m!!
~ l 1m!!. ~9!
The radial function can either be approximated analyticaas hydrogenic@48#, calculated in the JWKB approximatio@36#, or computed numerically in the Coulomb approximtion @49,50#. The latter approach, used here and describelow, introduces a phase shift in each radial function frits pure hydrogenic form, the amount of the shift being tquantum defect for the state. This approach thus incorpor~semiempirically! the effects of the core electrons on the intial and final states of the Rydberg electron—inclusionwhich is essential to alignment phenomena@see Sec. IV B#.
The evolution of the system is described by the Sch¨-dinger equation
i\]
]tC~r ,t !5H~ t !C~r ,t ! ~10!
and the rectilinear trajectoryR(t) of Eq. ~4!. The Hamil-tonianH(t) is the sum of the Rydberg electron Hamiltoniaand the Fermi potential~5!. Upon inserting the expansion~6!into Eq. ~10!, we obtain the usual initial value problem
d
dtaa~ t !52
i
\ (a8
aa8~ t !eiDet/\^cauV~ t !uca8&, ~11!
where we note that the transition amplitudesaa(t) are intrin-sically collision frame quantities. The energy defectDe[en8,l 82enl embodies the inelasticity of the transition. Uing the Fermi potential~5! to evaluate the coupling matrixelements transforms these equations to
d
dtaa~ t !522p iAS \
mD(a8
aa8~ t !eiDet/\Pa8a@R~ t !#,
~12!
where we have introduced thetransition density
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SEMICLASSICAL THEORY OF ALIGNMENT EFFECTS . . . PHYSICAL REVIEW A 63 052711
aa~ t→1`!5aa~ t→2`!2 i ~2pA!S \
meD
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aa8~ t→2`!E2`
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~14!
To evaluate these amplitudes, we change the variable otegration from t to z. Using cylindrical coordinates@b,w,z(t)#, with z(t)5vt, and introducing the radial function via Eq. ~8a! and the associated Legendre polynomPl
umu(cosu) in the spherical harmonic~8b!, we obtain theexplicit form
aa~ t→1`!5aa~ t→2`!2 i4pA
v S \
meD
3(a8
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3E0
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PlumuS z
RD Plum8uS z
RDunl~R!
3un8 l 8~R!J~z;v,De!dz, ~15!
whereR25z21b2. The function
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ERIC G. LAYTON AND MICHAEL A. MORRISON PHYSICAL REVIEW A 63 052711
strate in Sec. IV B, the energy defect plays a crucial roleexplaining the oscillatory behavior of magnetic sublevcross sections. This point is evident from Eq.~19b!; in thelimit that the argument of the exponential vanishes, the trsition probability and therefore the state-to-state crosstion exhibit a simple inverse square dependence on velo
The second difference concerns our choice of the axispatial quantization. Using thez axis, as in the collisionframe adopted here, yields symmetry properties that provinsight and simplify cross section calculations. For exampin the limit where De/v→0, only transitions for whichl1m1 l 81m8 is even~or zero! are allowed. Although thisapproximate symmetry is broken when the limit is nottained, it does explain the high-energy behavior of the stto-state cross sections calculated using the quantmechanical impulse approximation@26,27#.
Third, many prior studies of Rydberg-atom rare-gas sctering approximate the radial functionunl(r ) by those ofatomic hydrogen in the pure-Coulomb approximation. Frare-gas collisions with aligned Rydberg atoms, this appromation grossly underestimates state-to-state cross secand misrepresents the alignment effects~see Sec. IV B!. Tosome extent, this failure results from the comparatively laquantum defects for the states of Ca of interest. The hydgenic approximation is adequate for studies ofl mixing insodium.
IV. RESULTS AND DISCUSSION
We begin in Sec. IV A withl-mixing collisions of He withNa Rydberg atoms. We do so to validate our model, to veour numerical procedures and codes, and to explore valuethe principal quantum numbern of the Rydberg electron awhich the binary encounter mechanism is valid. We thturn to alignment phenomena in Ca–He collisions in SIV B.
A. Angular momentum mixing processes
Gallagher and co-workers@51–55# performed a series oexperiments in which rare buffer gases were introduceda cell of alkali atoms in an excited configurationn0 l 0 atabsolute temperatureT. They observed a lengthening of thlifetime of the excited atoms, which they attributedcollision-induced transitions to configurations with samen0but differentl 0 . Studying the process
Na~n0d!1He→Na~n0l !1He ~ l .2! ~21!
for values ofn0 from 5 to 15, they found that forn0<10, thel-mixing cross sectionsn0 ,l 0
( l mix)(v) increases withn0 as n04,
while beyondn0510 this cross section declines.The l-mixing cross section is just the sum overl . l 0 of
the level-to-level cross sections of Eq.~1!,
sn0 ,l 0~ l mix!~v !5 (
l . l 0sn0 l 0→n0l~v !. ~22!
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To compare with measured data, this cross section musaveraged over the Maxwell–Boltzmann distribution of retive velocities appropriate to the gas temperatureT,
sn0 ,l 0~ l mix!~v !5
^sn0 ,l 0~ l mix!~v !v&
^v&'
mv2
2kBTsn0 ,l 0
~ l mix!~v !, ~23!
wherem is the reduced mass~3.409u for Na-He! andkB isBoltzmann’s constant. Here we have adopted the apprmate form of Derouard and Lombardi@39#. The conditionsfor validity of this form are well-satisfied for the process~21!because the quantum defects forl .1 states of Na are sma@56#. Note, however, that unlike Derouard and Lomarbwho were primarily interested in more highly excited stafor which De is quite small, we do not set the energy defeto zero.
Several theoretical papers have addressed this phenenon @29,31,36,37,39,40,57–59#. Gersten @37#, using amodel based on the semiclassical impact parameter mewith a quasifree-electron interaction between the rare-and the Na target, reported results forn054 – 7. Gersten ne-glected the energy defects and, because the assumptiohis model are invalid at small atomic radii forn0<10, used ahard collision approximation for projectile impact parameters less than a critical valueb0 . In semiclassical time-dependent perturbation theory, the contribution to the ingral in Eq.~18! over impact parameterb from small impactparameters tends to overestimate the transition probabHowever, the agreement in Fig. 2 between quantal and seclassical cross sections at all but the lowest relative velocisuggests that in the present application this effect maynegligible except at these low velocities. Gersten@37# pro-posed correcting for this overestimation, a consequencthe increased coupling at small impact parameter betwthe initial and final target Rydberg states, by setting the tr
FIG. 2. Partial magnetic cross sections for 17d→18p transitionsin Ca–He collisions. The present semiclassical results~points! arecompared to the quantum mechanical cross sections of IsaacsMorrison @27# ~lines! for um0u50 ~solid line and closed circles!, 1~long dash line and open triangles!, and 2~short dash line and opensquares!.
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SEMICLASSICAL THEORY OF ALIGNMENT EFFECTS . . . PHYSICAL REVIEW A 63 052711
sition probability at some valueb0 equal to an arbitrary constant; Lebedev and Fabrikant@36# have discussed this stratagem and applied it to studies of heavy-rare-gas atcollisions with Rydberg Rb atoms~see also Sec. 6.2 of Re@41#!. Because the choice of this constant is ambiguousbecause the comparison in Fig. 2 show that the existencecharacteristics of the oscillations in the present state-to-scross sections do not hinge on this point, we have not immented this correction.
Derouard and Lombardi@39# followed a methodologysimilar to Gersten but did not invoke the hard collision aproximation. Because they neglect energy defects, their csections depend inversely on the square ofv. Their velocityaveragedl-mixing cross sections withn058 – 14 began toagree with experiment at aboutn0510. They also were ableto fit their cross sections to ann0
4 dependence, consistenwith experimental findings.
Table I compares our velocity-averagedl-mixing crosssections with those of Derouard and Lombardi and withexperimental data of Gallagheret al. All results correspondto T5430 K. Unlike the data, neither theoretical resshows an initial increase withn0 . Our cross sections, whicinclude the energy defect, are smaller than those of Derouand Lombardi, who setDe to zero. The effect of this approximation onsn0 ,l 0
( l mix)(v) is far less important than on the Ca-H
alignment cross sections of Sec. IV B.@For purposes of direccomparison, we also computedl-mixing cross sections in thezero-defect approximation. For 10D and 11D, respectivewe found the values~in units of 1000 Å2! 3.84 and 2.98,respectively, in good agreement with those of DerouardLombardi.# Moreover, the effect of the energy defect dimiishes very slowly with increasingn0 , roughly asn0
21.The failure of both theories forn0,10 reflects the quite
different physical mechanisms at work in collisions withtarget in a low-lying excited state, as opposed to one iRydberg state. Belown0510, the Na target is relativelyopaque to He. The electron density of its excited~valence!electron is sufficiently large that first-order perturbatitheory and the binary encounter approximation fails. Hereapproach that models the transient complex formed bysystem, such as expansion in quasimolecular electronic s@46#, appears necessary.
Above n0510, however, the valence electron density bcomes sufficiently Rydberg-like that perturbation theory a
TABLE I. Cross sections forl mixing in units of 1000 Å2.
Level Theorya Theory ~present! Experimentb
6D 8.94 0.38~7!
9D 5.15 4.42 1.04~20!
10D 3.93 3.51 2.2~8!
11D 3.00 2.80 1.85~20!
12D 2.37 2.25 1.69~30!
13D 1.93 1.84 1.65~30!
14D 1.59 1.51 1.33~20!
aDeouard and Lombardi@39#.bGallagheret al. @51,53#.
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the binary encounter approximation succeed, the interacof the Rydberg atom with the rare-gas projectile being suthat a quasifree interaction model adequately describescollision. In general, the onset of this high-n regime willdepend on the nature of the Rydberg atom~its quantum de-fects and the importance of configuration interaction! and onthe extent to which the internal structure of the projectparticipates in the collision. Our study ofl-mixing collisionsof He with excited Na atoms confirms that forn0.10 andabsent significant configuration interaction, one may codently model collisions of Rydberg species with He usingbinary encounter approximation with a Fermi contact pottial interaction and solving the scattering equations via firorder perturbation theory.
B. Alignment and oscillations in Ca–He collisions
Figure 2 shows quantum and semiclassical magnetic slevel cross sectionss um0u(v) for the 17d→18p transition inCa–He scattering. The results of the present calculatiagree very well with those of the fully quantum-mechanicimpulse approximation@27#, even at relative velocities alow as several hundred m/s. Both theories predict significalignment effects over the entire velocity range, the relatefficiency of a particularum0u channel depending significantly on velocity. For these channels,s0 manifests the mospronounced oscillations, with peaks spaced asv21. The os-cillations in s1 are equally evident, although their peaks amore widely spaced than those ins0. In sharp contrast,s2
shows no oscillations, varying smoothly with velocity; folargev, this variation conforms to thev22 prediction of Eq.~19a!.
Figure 3 deconstructs these magnetic sublevel crosstions into their state-to-state constituents. For clarity, thare separated into groups according to the initial magnquantum numberum0u. Symmetry with respect to inversiothrough the origin and reflection through the azimuthal~xy!plane, imply, in the semiclassical theory, that cross sectifor m0→m andm0→2m are equal. In the quantum theorthis symmetry is formally approximate; for this system, hoever, it holds to graphical accuracy@26,27#. In both theories,cross sections form0→m and for 2m0→2m are equal.Hence we show only non-negative values ofm0 andm. Fig-ure 3 highlights the dependence of the oscillations ins um0u
on the initial quantum numberum0u and shows that for casewhere s um0u does oscillate, these structures arise fromcontributing state-to-state cross sections. We further seeasv increases, the variation with velocity of all state-to-stacross sections reduces to thev22 dependence expected fromEq. ~19b!.
The vital role of the energy defectDe for the oscillationsis illustrated by the comparisons in Fig. 4 of magnetic sulevel cross sections from semiclassical calculations withDe51.69 cm21, the value for the 17d→18p transition in Ca, tothose withDe50. ~See Ref.@36# for an example of the sig-nificance of the energy defect forn, l-changing and quasielastic rare-gas atom Rydberg-atom collisions.! The zero-defect results manifest the familiarv22 dependence at alvelocities. TheDe.0 results, however, show this behavi
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ERIC G. LAYTON AND MICHAEL A. MORRISON PHYSICAL REVIEW A 63 052711
only as v→`, a limit that the cosine factor in Eq.~19b!shows to be equivalent toDe→0.
Semiclassical and quantal calculations for the 23d→22 ftransition ~not shown!, for which the energy defect is 0.2cm21, confirm this dependence onDe. For this transition,none of the magnetic sublevel cross sections exhibit osctions for 500<v<3000 m/s. Alignment effects do, howeveappear forv&1000 m/s, wheres2 exceedss0 and s1,which are of comparable magnitude.
Almost as important as including the correct energyfect is properly accounting for the shift in phase of the radfunction due to the core,unl(r ) of Eq. ~8a!. Many prior stud-ies of rare-gas collisions with Rydberg atoms have appromated these functions by analytic hydrogenic forms. Whave found this to be a very poor approximation for aligment effects in Ca-He cross sections.
In the Coulomb approximation@50#, the potential energyof the Rydberg electron is replaced by the pure Coulo
FIG. 3. State-to-state cross sections for 17d→18p transitionswith various initial magnetic quantum numbersum0u: ~upper! 0→0 and 0→1, ~middle! 1→0 and 1→1, and ~lower! 2→0 and2→1. The solid lines corresponds to final magnetic quantum nuber m50 and the dash lines tom51.
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potentialV(r )5Zc /r , whereZc5Z2N11 with Z andN thenuclear charge and number of electrons, respectively.radial Schro¨dinger equation is solved using the quantudefect shifted energy of Eq.~7!. Since this energy is not aneigenvalue of the pure-Coulomb Hamiltonian, radial funtions that decay asymptotically (r→`) blow up asr→0unless they are cutoff at some radiusr c comparable to thesize of the atom. We chooser c to be the inner classicaturning point. For a Rydberg state, the spatial extent ofradial function is so great that the region 0,r ,r c is negli-gible. ~For the states of Ca considered here,r c'3a0 , whilethe mean radii of these states is approximately 400a0 .! Fol-lowing Bates and Damgaard@49#, we calculate radial func-tions from a truncated series approximation to the asymptcally decaying radial function. The most evident differenbetween the Bates–Damgaard radial functions and hygenic functions is a shift in their phase; we therefore refethem as quantum-defect phase shifted radial functions.
Figure 5 compares magnetic sublevel cross sectionsculated with these quantum-defect phase shifted radial futions@49# to values obtained using hydrogenic functions. Tmagnitudes of the hydrogenic cross sections are consistebelow values obtained using phase shifted radial functioAnd, although the hydrogenic cross sections do predict
FIG. 4. Partial cross sections for 17d→18p transitions in Ca-Hecollisions calculated with energy defectDe51.69 cm21 ~curves!and in the zero-defect approximation~symbols! for um0u50 ~solidcurve!, 1 ~long-dash curve!, and 2~short-dash curve!.
FIG. 5. Partial cross sections for 17d→18p transitions in Ca-Hecollisions calculated with quantum-defect phase-shifted radial futions ~thick curves! and with hydrogenic radial function~thincurves! for um0u50 ~solid curves!, 1 ~long-dash curves!, and 2~short-dash curves!.
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SEMICLASSICAL THEORY OF ALIGNMENT EFFECTS . . . PHYSICAL REVIEW A 63 052711
cillatory alignment effects, these predictions are qualitativincorrect over much of the velocity range studied, in socases incorrectly designating the order of these crosstions. For example, at about 1400 m/s, the cross sections0
calculated using hydrogenic functions lies betweens1 ands2, whereas the more accurates0 calculated with quantumdefect phase shifted functions clearly exceeds thesecross sections.
The reason for this sensitivity is evident from the coparison of hydrogenic and quantum-defect phase shifteddial functions in Fig. 6. The hydrogenic functions showsignificant phase difference between the 17d and 18p states.This difference is all but eliminated when the quantum dfects for these states are taken into account. Similar bredowns in the hydrogenic approximation appear for ottransitions@26#.
V. INTERPRETATION
Equation ~19b! explains the oscillations in the state-tstate and magnetic sublevel cross sections as interferphenomena resulting from the spatial distribution of Rydbelectron wave functions@28#. The functions in the radia
FIG. 6. Radial functions for the 17d ~solid curves! and 18p~dash curves! states of Ca as calculated in the hydrogenic appromation ~upper! and using the quantum defects of Ca~lower!.
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product u17d(R)u18p(R), which appears in the transitiodensity factors in Eq.~19b!, identify certain regions of spacwhere Rydberg electron probabilities are higher than injacent regions. Figure 7 shows the resulting structurein theradial product for the 17d→18p transition densities alongrare-gas atom trajectoriesz(t) for contributing impactparametersb. For each transitionm0→m, this radialproduct is modulated by a different angular factP2
um0u(cosu)P1umu(cosu), where along a trajectory cosu5z/R.
For the 0→0 case, for example, the resulting transition desity manifests the rich landscape of peaks and valleys seethe density plots in part~a! of Fig. 8. An incident rare-gasatom finds several distinct regions along its trajectory whit can induce the transition 17d0→18p0 .
The phase of the cosine function in Eq.~19b! is crucial tothe probability that such a transition will actually occur. Ffixed velocityv, the integrand in this equation will contribute significantly to the transition probability only for valueof z andz8 such that~a! the separationz2z8 is an integralmultiple of the wavelength 2pDe/\v, and~b! the transitiondensity factorsPa8a(z,b) andPa8a(z8,b) are appreciableatboth z and z8. One can understand the variation withv of thetransition probability by~equivalently! varying the spacingz2z8 in the phase. As this spacing changes, the prodPa8a(z,b)Pa8a(z8,b) for the 0→0 transition in part~a! ofFig. 8 varies dramatically, as the pointsz andz8 pass throughpeaks and valleys in the transition density. In effect, the ftors Pa8a(z,b) and Pa8a(z8,b) represent two ‘‘opportuni-ties’’ or ‘‘paths’’ whereby the Rydberg electron can be ecited to statea8. The cosine factor results in interferencoscillations between these two excitation paths, i.e., betw~acutely nonadiabatic! interactions atz andz8. It is not sur-prising, then, that these oscillations disappear asDe→0, asin Fig. 4, or asv→`, as in Fig. 3. Each state-to-state crosection, of course, entails an integral over impact parame
i-
FIG. 7. Radial factorsu17d(r )u18p(r ) in atomic units forr alongthe rare-gas atom trajectory.
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ERIC G. LAYTON AND MICHAEL A. MORRISON PHYSICAL REVIEW A 63 052711
This integration complicates the picture but does not alterqualitative predictions of Eq.~19b!.
The differentangular functions in the transition densitiefor different m0→m transitions—the aforementioned proucts of associated Legendre polynomials—are responsfor the dependence of state-to-state and magnetic subcross sections on magnetic quantum number. Considertwo extreme cases, 0→0 and 2→1. For 0→0, the angulardensityP2
0P10 is localized primarily along thez axis, parallel
to the trajectory of the rare-gas atom. Together, the variaof the radial and angular products produces the rich arraregions of~comparatively! high transition probability in part~a! of Fig. 8, allowing plenty of opportunities for interference. By contrast, for 2→1 the angular densityP2
2P11 is lo-
calized primarily in thexy plane. The resulting transitiondensity for this case, shown in part~b! of Fig. 8, offers farless variation as the separationz2z8 ~equivalently, the ve-locity! varies. The result is the smooth dependence withv of
FIG. 8. Transition densities for the 0→0 ~upper figure! and 2→1 ~lower figure! state-to-state 17dm→18pm8 transitions in Ca-Hecollisions. Light regions correspond to large values, dark to smvalues.
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the 2→1 cross section in Fig. 3.To clarify further the origin of the oscillations, we con
sider the extreme model of a Rydberg state in whichelectron presents to the rare-gas atom only two ‘‘planes’’transition density, one atz1 and one atz2 , i.e., we approxi-mate the transition density~13! by
Pa8a†R~ t !‡5ca8* @R~ t !#ca* @R~ t !#$d@z~ t !2z1#
1d@z~ t !2z2#%. ~24!
Since in this model, transitions can occur only at interactiotimest1 or t2 , wherez(t1)5z1 andz(t2)5z2 , the transitionprobability ~19b! reduces to
Pa0→a~v;b!5S 2pA\
meD 2 1
v2 S Pa8a2
~z1 ,b!1Pa8a2
~z2 ,b!
12 cosFDe
\v~z22z1!G
3Pa8a~z1 ,b!Pa8a~z2 ,b! D . ~25!
All three appearances of the transition densityPa8a(z,b) inthis result contribute to the dependence of the magnetic slevel cross sectionss um0u(v) on initial magnetic quantumnumber um0u. But only the third term, a consequencequantum-mechanical interference between excitation pobilities at planes atz1 andz2 , can induce oscillations—andthen only if the planes atz1 andz2 are at or near peaks in thtwo transition densities. Figure 9 shows the resulting crsectionsin this modelfor the extreme cases 0→0 and 2→1 with planes atz15150a0 andz25500a0 . These resultsare typical. Extensive tests~not shown! demonstrated thatprovidedz1 andz2 are far enough apart (Dz*150a0), theirparticular values do not matter. One can easily find plathat induce oscillations in the 0→0 cross section, while thereare no planes that cause structure in the 2→1 cross section.The variations in Fig. 3 withv and withm0 andm of crosssections determined from the actual transition densitiesFig. 8 reflect the more distributed nature of these densitiecompared with this two-plane model.
ll
FIG. 9. Cross sections for a ‘‘two-plane’’ model of semiclascal 17dm→18pm8 transitions in Ca-He collisions. For purposesillustration, planes were located atz15150a0 andz25500a0 .
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SEMICLASSICAL THEORY OF ALIGNMENT EFFECTS . . . PHYSICAL REVIEW A 63 052711
VI. CONCLUSION
The primary results of this study are three. First, our stato-state cross sections calculated with quantum~impulse ap-proximation! and semiclassical~rectilinear impact parameterfirst-order perturbation! theories are in close agreement ovmost of the velocity range studied. Second, the semiclasstheory provides an interpretation of the oscillations in manetic sublevel cross sections for Rydberg-atom-rare-gaslisions as due to quantum-mechanical path interferencefects. And third, our study shows that previously usapproximations—neglect the energy defect, and the hydgenic approximation for Rydberg electron radial functionsare inappropriate for the study of alignment effects. The pmary theoretical results are Eq.~19b!, which reveals theorigin of oscillations in these cross sections, and Eqs.~20!,which greatly facilitates their calculation.
The cross sections in Figs. 2 and 3 show that despite qdifferent approaches to the dynamics, the quantal impuand present semiclassical theories produce nearly idenmagnetic sublevel and state-to-state cross sections for nresonant energy transfer collisions of rare-gas atoms wRydberg atoms. Only at the lowest relative velocities dothis agreement break down. These two studies share onlyFermi contact potential, the spectator model of the core,the representation of Rydberg electron radial functionsquantum-defect phase shifted radial functions. Implemetion of both theories is of comparable~modest! difficulty, sothe choice of theory can be based on the physical quantof greatest interest. The present semiclassical approach hlights interpretation. If one seeks cross sections for collisiinvolving a rare-gas projectile that is more polarizable thHe ~e.g., Xe!, then one may prefer the impulse approximtion, where it is much easier to include the long-ranelectron-atom polarization interaction than in the pressemiclassical treatment. For He, however, inclusion of poization in no way alters the qualitative behavior of any of tcross sections presented here@26#.
The importance of the energy defect illustrated in Figimplies that experiments to probe~or exploit! oscillations ins um0u should emphasize transitions with the largest-enedefects, within the range of true Rydberg states and subto experimental constraints. Although partial magnetic cr
s.
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sections for lower-lying excited states manifest quite simstructures~as seen in the Ca-Xe calculations of Hickm@20#!, the equations of the present theory are not direcapplicable to transitions between such states.
The comparison in Fig. 5 between magnetic sublecross sections calculated with quantum-defect phase shradial functions and with hydrogenic functions argustrongly for the use of the former in future studies of rare-gatom collisions with Rydberg atoms. While the methodBates and Damgaard@49#, which uses a truncated series aproximation to the asymptotically decaying phase-shifCoulomb function, works well for low-lying Rydberg statewith small D l 5 l 2 l 0, it encounters severe numerical difficulties for higher-n and/or largerDl states.
The present calculations confirm the findings of recexperimental and quantal investigations that alignmentfects do occur in near-resonant energy transfer collisioeven when the target state is a Rydberg atom. Both theoical studies were designed to explicitly preclude the formtion of a transient quasimolecular state during the collisiInstead, both treat the collision as a three-body processvolving the Rydberg electron, the rare-gas atom, and the cof the Rydberg atom, whose role is reduced to that ospectator. Both studies reveal identical oscillations, mpronounced in magnetic sublevel cross sections of lowum0u,which depend strikingly on the relative velocity, the enerdefect, and the initial magnetic quantum number.
In closing, we note that, while the measurements of Spet al. corroborate the presence of alignment effects in neresonant energy transfer cross sections for Rydberg atrare gas collisions, the predicted oscillations in these crsections await confirmation or refutation by experiment. Whope the present findings will stimulate such experiment
ACKNOWLEDGMENTS
We acknowledge useful conversations with Dr. J. DelDr. N. F. Lane, Dr. A. P. Hickman, Dr. I. Fabrikant, Dr. GA. Parker, Dr. K. Mullen, Dr. F. B. Dunning, Dr. J. P. Driessen, Dr. N. Shafer-Ray, and Dr. S. R. Leone and the supof the National Science Foundation under Grant No. PH9722055.
em.
, J.
. P.
hys.
ep.
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