PHYSICAL REVIEW A AUGUST 1999VOLUME 60, NUMBER 2

Quantum transport theory for atomic states through solids

D. G. Arbo,1,2,3 C. O. Reinhold,1,2 P. Kurpick,1,2 S. Yoshida,1,2 and J. Burgdo¨rfer1,2,4

1Physics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-63732Department of Physics, University of Tennessee, Knoxville, Tennessee 37996-1200

3Instituto de Astronomı´a y Fısica del Espacio, C.C. 67, Succursale 28, 1428 Buenos Aires, Argentina4Institute for Theoretical Physics, Vienna University of Technology, A-1040 Vienna, Austria

~Received 14 January 1999!

We present a quantum description for the evolution of atomic states of fast projectiles traveling throughmatter. Our approach is based on the solution of a quantum Langevin equation, i.e., a stochastic time-dependent Schro¨dinger equation that describes electronic excitations of atoms during their transport throughsolids. The present description can be considered the quantized version of a previously developed classicaltransport theory. We analyze in detail the correspondence between classical and quantum transport simulations.Applications to the stripping of relativistic H2 and H through thin carbon foils and a comparison withexperimental data are presented.@S1050-2947~99!00308-X#

PACS number~s!: 34.50.Fa, 34.10.1x

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I. INTRODUCTION

Ever since the seminal work by Bohr and Lindhard onevolution of the charge state and excitation state ofatomic particles penetrating solids@1#, the investigation ofthe existence and the nature of the electronic excitation strum of swift ions has continued to draw considerable intest. A complex array of multiple scattering processes pduces a variety of excited configurations not easaccessible by other means. Despite the extensive applicaof the ion-solid interaction as a spectroscopic tool as wela ‘‘stripper’’ medium to produce high charge states, a micscopic understanding of the dynamics of the excitation pcess and of the evolution and transport of electrons accpanying fast ions is somewhat limited. The difficulties canattributed in part to the fact that perturbations of excitstates are sufficiently strong to preclude any perturbatreatment.

Some time ago, we proposed a classical transport the~CTT! @2,3# which is based on a microscopic Langevin eqution for the trajectory of projectile-centered electrons. Tclassical phase-space distributionrcl is represented by anensemble of phase-space points~test particle discretization!whose trajectories are governed by the Langevin equatAccordingly, the resulting phase-space distribution at a lapoint in time is determined by the ensemble of evolvphase-space points. In the case that only deterministic fiare present, this approach is equivalent to the classtrajectory Monte Carlo~CTMC! method@4#. The determin-istic force in the Langevin equation contains the Couloforce between the electron and the projectile nucleus, in geral modified by dynamical screening in the solid~‘‘wake’’@5,6#!, while the stochastic force represents the random mtiple scattering with electrons in the medium~in the simplestcase, the ‘‘electron gas’’! as well as atomic cores in thtarget. This prescription has yielded remarkable agreemwith a large variety of experimental data, such as forhanced mean free paths@3#, for the enhanced production ohigh-l Rydberg states and the resulting long-time tail in tdelayed x-ray spectra@7#, for the yield of convoy yields@8#,and, most recently, for the excited state distribution of H(n)

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at relativistic energies@9,10#. The latter plays a key role fothe reference design of next-generation spallation neusources~SNSs!. An H2 beam in the form of macropulsesaccelerated to;1 GeV in a linear accelerator~LINAC !, andis subsequently stripped to bare protons by transmissthrough a thin foil and injected into and stored in an accmulator ring@11,12#. The effective conversion of H2 to H1

and the underlying beam-foil interaction at the point of ijection is a crucial element for the design of high-intensSNSs. A key parameter is the population of excited neutH~n! which may get stripped by the strong magnetic fieldthe first bending magnet. The resulting protons collide wwalls and magnets leading to unacceptably high levelsradioactivity along the beam line.

Despite the remarkable success of the description of thcomplex processes within the framework of classical dynaics, the question as to the validity of the CTT has remainopen. One particularly puzzling aspect is that a fractionthe collisions suffered by the electron involve small mometum transfers for which classical dynamics is known to bredown since excitation and ionization become classically spressed@13,14#. It appears, therefore, useful to developquantum transport approach to gauge the validity and apcability of the classical transport calculations. Until recenta quantum calculation describing the electronic evolutionan atom as it moves through the solid has appeared toformidable task even when the atom carries a single electThis is due to the fact that a large number of bound acontinuum states become populated as a result of the trport process. Only now have a few contributions appearethe literature in which selected aspects of the ion transmsion problem have been tackled by numerically solvingtime-dependent Schro¨dinger equation to calculate resonacoherent excitation rates@15# and stopping of ions@16#.

In this work we present a quantum description of tranport which is based on a quantum Langevin equation. Itscribes the stochastic evolution of the quantum state unthe influence of both deterministic potentials, specificallyCoulomb potential or a dynamically screened potential, anstochastic potential which accounts for multiple scatteriThe latter is chosen to yield a stochastic force identical

1091 ©1999 The American Physical Society

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1092 PRA 60D. G. ARBO et al.

that entering the classical Langevin equation. In analogysimilar developments in quantum optics@17# and in thetheory of atom-radiation-field interaction, each stochasticaevolved wave function can be labeled as a ‘‘quantum tratory’’ or ‘‘quantum history.’’ The solution of the reducedquantum Liouville equation for the reduced density operacan be reconstructed from a Monte Carlo sampling of qutum trajectories.

Applications to be discussed in the following include ttransport of relativistic H2 and H with energies of the ordeof 1 GeV through carbon foils with applications to the injetion problem for spallation neutron sources@11,12# as well asthe recent proposal for detecting relativistic antihydrog@18# through the Lya emission after controlled excitation ifoil transmission. While the present approach is orientedwards the treatment of the ion-solid interaction, we stressapplicability of this method to a wide variety of other physcal situations such as atoms in plasma environments anthe cooler gas of a storage ring or the interaction of atowith radiation fields.

The plan of this paper is as follows. In Sec. II we giveoutline of the theoretical framework. A few technical detaof the calculations are presented in Sec. III together wnumerical results for H2 transmission. In Sec. IV we focuon the comparison between quantum and classical transin order to assess the validity of the classical approach anidentify specific quantal, i.e., nonclassical features. Atomunits are used unless otherwise stated.

II. THEORY

We shall be concerned with the transmission of an~atom! with nuclear chargeZp and velocityvW p through a thinamorphous foil of thicknessX8 ~we use primes to denotlaboratory frame variables!. All numerical examplesthroughout the paper will refer to a carbon foil. However, tmethod is, with minor modifications, applicable to other mterials. The ion, referred to in the following as the projectiis assumed to be fast compared to the Fermi velocity,vF , ofthe material (vp@vF) which corresponds to impact energigreater than hundreds of keV/u. We neglect the slowingdown or straggling of the heavy ion due to its interactiwith the solid. Typical energy losses for thin foils are a vesmall fraction of the incident energy and have a negligieffect on the evolution of projectile electrons. Thus, thelocity of the ion is treated as an approximate constantmotion. If desired, the change of the projectile velocityangular deflection can be easily taken into account, ewithin the framework of a continuous slowing down aproximation ~CSDA! and small-angle scattering theo@19,20#. The term transport refers in the following exclusively to the electronic degrees of freedom of the project

We will study the evolution of a single active electrocarried by the projectile. For ions carrying more than oelectron, additional approximations are required~e.g., see thetreatment below for transmission of H2 ions!. We decom-pose this complex system into a subsystem, the atomic sspace centered around the projectile taken to be theframe, and the environment consisting of atomic coresthe electron gas of the conduction electrons moving ataverage velocity of2vW p . The task is now to determine th

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evolution of such an open system due to the irregularly fltuating coupling to the environment, i.e., multiple scatteriof the active electron with the constituents of the solid. Sinwe focus on the transport of electrons which are closphase space correlated with the projectile, the evolutionthe electron in the rest frame of the projectile is governednonrelativistic dynamics. Relativistic effects play a role onthrough the coupling with the environment and its kinematransformation to the projectile frame.

The state of the electron during the transport procesconveniently described through its density operator

r5(j ,k

r j ,kuf j&^fku, ~2.1!

where$ufn&,n51,2, . . . , % is a complete orthonormal basset representing the open subsystem. A density-matrixscription easily allows for the treatment of initial electronstates prior to transport which are mixed states,

r i5 (a51

Ni

paua&^au, ~2.2!

wherepa are occupation probabilities of the statesua&, or, asa particular case, pure states

r i5uc i&^c i u, ~2.3!

whereuc i& is the initial wave function. Even though the projectile electron could be in a pure state prior to enteringsolid, a random transport process will produce a mixturestates.

The objective of the classical and quantum transport thries is the determination of the time evolution of the statethe electronr(t) from t50, prior to entering the solid~or, ingeneral, the time at which the initial state is formed withthe solid! up to a time T5T8A12(vp /c)2

5(X8/vp)A12(vp /c)2, which corresponds to the foil exitOur present formulation of quantum transport proceedsquantization of our classical transport theory~CTT!. For ref-erence and comparison, it is therefore useful to first revthe major steps of the CTT.

A. Classical Langevin equation

The starting point of the CTT is the time-dependent stof the open system, the active electron, which is given byreduced probability densityrcl(rW,pW ,t), whererW andpW are theposition and momentum vectors of the electron. Att50, theprobability density is given by

rcl,i5rcl~rW,pW ,0!5 (a51

Ni

parcla~rW,pW !, ~2.4!

where rcla is chosen such that it approximately mimics t

quantum stateua&^au ~see, e.g.,@21#!.The evolution of the open system due to both inter

interactions as well as the coupling to its environment~‘‘thebath’’! can be described in terms of a reduced classical Liville equation

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PRA 60 1093QUANTUM TRANSPORT THEORY FOR ATOMIC STATES . . .

]rcl

]t5$Hat,rcl%1Rclrcl

52pW •¹W rWrcl1“ rWVat•“pWrcl1Rclrcl , ~2.5!

where $ , % denote Poisson brackets andHat is the atomicHamiltonian

Hat5p2

21Vat~rW !5

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22

Zp

r1Vscr~rW !, ~2.6!

which contains the Coulomb interaction potential,2Zp /r ,possibly modified by dynamical screening~‘‘wake poten-tial’’ @5,6#! in the solid,Vscr. For the present studies involving low-charged projectiles and very high projectile speevp;c, Vat can be approximated by a pure Coulomb potetial.

In Eq. ~2.5!, Rcl is the classical relaxation superoperadescribing the dissipative interaction with the environm~the solid!. The explicit construction ofRcl is, in general, aformidable task since it involves the many-body dynamicsthe environment to which the open system couples. Inpresent case,Rcl represents the multiple scattering with thsolid. The central simplifying assumption which makes tdetermination ofRcl feasible is the validity of linear-response theory. Accordingly, the dynamics of the bath vables remain decoupled from the internal statercl of thesubsystem, or equivalently,Rcl is not of a functional ofrcl .Conversely, no restrictive assumption is made forstrength of the perturbation of the environment on the ssystem. In fact, in the present context we will consider strocoupling that precludes a perturbative treatment of the elution of the projectile electron under the influence ofRcl . Inour applicationsvp@vF and, therefore, projectile-centereelectrons, including those in the continuum, are well serated in momentum space from target electrons. In otwords, the removal of an electron from the projectile systis reasonably well-defined and there is little risk of ‘‘doubcounting.’’ Furthermore, we assume that the projectile is fenough such that collisional interactions with the solid cbe represented in terms of impulsive momentum transferqW .This approximation is based on the observation that collistimes tc.vp

21 are short compared to the orbital period ofhydrogenic electron in a hydrogenicn level, Tn52pvn

21

52pn3Zp22 . With these approximations, the classical rela

ation operator is now specified in terms of a collision integ

Rclrcl5E d3qW~vW p ,q8W !@rcl~rW,pW 2qW ,t !2rcl~rW,pW ,t !#,

~2.7!

whereW(vW p ,q8W ) represents the transition rate per unit timto change the electron momentum fromp8W to p8W1q8W ~see@9#for the transformation of the momentum transfer from tlaboratory frame,q8, to the projectile frame,q). Equation~2.7! can be viewed as a linearized Boltzmann collisiontegral. Because of the large separation in momentum spPauli blocking does not need to be included. In generalW

depends on both the momentum transferq8W and the localmomentum in the laboratory frame,p8W , as well as on thestate of the target which we consider to be in the ground s

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at zero temperature. For transport of atomic states centaround the projectile with p8W &.vW p , we assume that thetransition rates are independent ofp8W ~i.e., W depends onlyon vW p and q8W ). The transition rates are proportional to thdifferential inverse mean free paths~DIMFPs! ormomentum-differential cross sections@22–24# @i.e., W5vp(dl21/d3q8), where l is the mean free path#. Theyinclude both elastic and inelastic scattering processes ofactive projectile electron with target cores and the electgas of the medium. The DIMFPs or collision rates are cculated in linear-response theory, which treats the respoof the environment, i.e., the electron gas of the medium,to perturbation by the active electron to first order in tinteracting potential but, in general, to all orders of telectron-electron interaction of the medium~e.g., in therandom-phase approximation!.

Rather than solving Eq.~2.5! directly, the CTT proceedsby employing ‘‘test particle discretization.’’ The solution othe original classical Liouville equation for the phase-spadensityrcl is mapped onto the time evolution of a represetative ensemble of initial phase points„rW i(0),pW i(0)… chosenby Monte Carlo sampling of the initial probability density,

rcl~rW,pW ,0!51

Ntest(k51

Ntest

d@rW2rWk~0!#d@pW 2pW k~0!#, ~2.8!

whereNtest is the number of test particles whose trajectorare determined by astochastic equation of motion, theLangevin equation

dpW

dt52¹ rWVat1FW c~ t !. ~2.9!

Note that the stochastic forceFc describes a discontinuouprocess representing the momentum ‘‘jumps’’ embodiedthe collision integral@Eq. ~2.7!#. Provided thatFW c(t) gives afaithful representation of Eq.~2.7!, the solution of the origi-nal Liouville equation is given by

rcl~rW,pW ,t !51

Ntest(i 51

Ntest

d@rW2rW i~ t !#d@pW 2pW i~ t !#.

~2.10!

Since the drift term in Eq.~2.5! agrees with the deterministic part of the Langevin equation~2.9!, the only nontrivialpart in establishing a correspondence between Eqs.~2.5! and~2.9! lies in the determination of an appropriate stochasforce FW c . While such a construction is not unique, an obous strategy is to optimize the agreement with the collisoperator for a finite number of jump moments,

d

dt8^qi8

n&5E d3q8qi8nW~vW p ,q8W ! ~ i 5x,y,z!,

~2.11!

in the absence ofVat ~i.e., for free-electron transport!.We use a stochastic force given by a sequence of im

sive momentum transfers~‘‘kicks’’ !,

FW c5(k

DpW kd~ t2tk!, ~2.12!

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1094 PRA 60D. G. ARBO et al.

whereDpW k is the stochastic momentum transfer per collisiat the timetk . The determination ofFW c(t) is thereby reducedto that of a stochastic sequence of pairs (DpW k ,tk). The zerothorder of the collision kernel is given by the integrated trasition ratevpl21, wherel is the mean free path betweecollisions in the laboratory frame. This moment is automacally reproduced by choosing times in between collisioDtk85(tk82tk218 ) at random according to a Poisson distribtion

P~Dtk8!}exp@2vpDtk8/l#. ~2.13!

For each collision, the momentum transfer is sampled atdom according to a probability density

P~Dp8W k!}W~vW p ,Dp8W k!. ~2.14!

We have verified that this procedure reproduces up tosecond moments of the collision kernels for free-electtransport extremely well@3#.

Details of the collision kernels determining the mometum transfers and the times in between collisions have bextensively described elsewhere@3,9,10#. We therefore re-strict ourselves to a brief summary. We decompose thechastic sequence into two independent subsequences.sequence refers to elastic electron-target core scattewhile the other one refers to inelastic electron-electron stering leading to single-particle–single-hole excitation andcollective excitations of electrons in the medium. Elastic mmentum transfers are calculated from the differential elacross section for the scattering of electrons at the tacores. Inelastic momentum transfers are obtained from aalistic dielectric function of the foil as a function of the frequency and wave vector@23#. We consider both longitudinaand transverse excitations@5,25#. The main relativistic effectin our calculations is associated with transverse electromnetic excitations. The mean free path for these excitatidecreases for increasing velocity and this process becoincreasingly important at higher collision energies. For cbon foils, however, collisional processes are dominatedlongitudinal excitations and elastic scattering. Inelastic logitudinal collisions have the shortest mean free path, welastic collisions have the largest momentum transamong the scattering processes.

Our present quantum transport approach consists nowquantization of the classical transport theory. We startnoting that the classical Langevin equation is completequivalent to the Hamilton equations of motion associawith the stochastic Hamiltonian

H~ t !5Hat1Vc~ t !5Hat2rW•FW c~ t ! ~2.15!

involving the stochastic potential functionVc(t)52rWFW c(t)~within the framework of matter-radiation interaction, thcorresponds to the so-called length gauge!. HereVc plays therole of a stochastic variable which determines the underlystochastic process and the relaxation operator.

Moreover, the CTT solution@Eq. ~2.10!# generated by thisstochastic Hamiltonian is, for transport of a single electrcompletely equivalent to the average of various classicallutions each obtained for a particular history or random

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quence of collisionsm5$(DpW i ,t i), i 51,2, . . .% and for aparticular stateua&^au of the initial state mixture. That is,

rcl~rW,pW ,t !51

Ntraj(a51

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pa (m51

Ntraj

rclm,a~rW,pW ,t !, ~2.16!

where Ntraj is the number of random collisional sequenc~ideally, Ntraj˜`).

We note a subtle point associated with Eq.~2.16!: here wehave subjected allNtest elements of our classical ensemblethe same stochastic sequence. For a classical ensembleresenting a true many-body system, such a choice of stoctic histories would introduce artificial many-body correltions. In the present case, however, the ensemble playsrole of a classical model for the single-particle quantuwave function. Since we will use in the followingrcl only tocalculate one-particle observables, identical stochasticquences for all members of the ensemble do not affectnumerical results. The present choice is motivated bydirect analogy to the stochastic wave function describedthe next section.

B. Quantum Langevin equation

Much like for the classical system, the evolution of aopen quantum system can be formally described in terma quantum Liouville equation for the reduced density opetor r of the electron,

i]r

]t5@Hat,r#1Rr, ~2.17!

where the relaxation superoperatorR describes the dissipative interaction with the environment. After making the samsimplifying assumptions used in the classical Liouville E~2.5!, an explicit, though very cumbersome, form forR canbe given which we omit in the following since we will nodirectly solve Eq.~2.17!. Instead, we solve the quantumtransport problem by test-particle discretization of a micscopic quantum Langevin equation, in analogy to the CTThe stochastic Hamiltonian in Eq.~2.15! generates a stochastic time evolution described by a Schro¨dinger equation,

i]

]tuc~ t !&5H~ t !uc~ t !&, ~2.18!

to which we refer to in the following as the quantum Langvin equation. Due to the stochastic nature of the Hamtonian, the time-evolved stateuc(t)& is indeterministic andEq. ~2.18! is understood to describe a continuous flowbetween discontinuous jumps. Each solutionucm,a(t)& rep-resents one particular realization or history for a given rdom sequence of collisions,m, and for a given initial stateua&. While in detail quite different, a solution of the quantuLangevin equation resembles a ‘‘quantum trajectory’’ in tterminology of quantum optics@17#.

With the help of the quantum Langevin equation, the slution of the transport problem can be easily determinedterms of the Monte Carlo sampling over quantum trajectoror histories, much like the classical solution in Eq.~2.16!,i.e.,

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PRA 60 1095QUANTUM TRANSPORT THEORY FOR ATOMIC STATES . . .

r~ t !51

Ntraj(a51

Ni

pa (m51

Ntraj

rm,a~ t !

51

Ntraj(a51

Ni

pa (m51

Ntraj

ucm,a~ t !&^cm,a~ t !u, ~2.19!

where, initially,cm,a(0)5ua&.Using now the explicit form of the stochastic potential

terms of a sequence of impulses@Eq. ~2.15!#, we can writethe solution of the quantum Langevin equation for one pticular sequencem as

ucm,a~T!&5Um~T,0!uc~0!&5Um~T,0!ua&, ~2.20!

whereUm is the evolution operator after a particular randosequence of collisionsm5$(DpW k ,tk), k51,2, . . . ,M %. Due tothe d-shaped time dependence of the kicks, the time evotion operator adopts the form

Um~T,0!5e2 iHat(T2tM) )k50

M21

U~ tk11 ,tk!, ~2.21!

U~ tk11 ,tk!5eirW•DpW k11e2 iHat(tk112tk), ~2.22!

wheret050. That is, in between the timestk and tk11, theelectron evolves freely according to exp@2iHat(tk112tk)#.Subsequently, the momentum of the electron is suddeshifted inDpW k11, which is exactly described by boost opertor exp@irW•DpWk11#. In practice, upon a basis expansionterms of atomic~pseudo!states, the operators in the produ@Eq. ~2.22!# are represented by matrices. Consequently,calculation of a quantum trajectory is reduced to a sequeof matrix multiplications.

Finally, the time-evolved density operator follows froEq. ~2.19! as

r~T!51

Ntraj(a51

Ni

pa (m51

Ntraj

Um~T,0!ua&^auUm† ~T,0!

~2.23!

and the expectation value of an arbitrary observableA isgiven by ^A&5Tr„Ar(t)…. In particular, forA5uc f&^c f u,one obtains the transition probability into a final stateuc f&which is given by

Pi˜ f51

Ntraj(m50

Ntraj

(a51

Ni

paz^c f uUm~T,0!ua& z2. ~2.24!

It is worthwhile noting a few properties of the stochasevolution operatorUm . It is obviously not energy conservinwithin the atomic subsystem since it explicitly treats a timdependent external perturbation. It therefore allows forergy flow into and out of the ‘‘open’’ subsystem in accodance with

d

dt^Hat&5

d

dtTr„Hatr~ t !…5” 0. ~2.25!

A more subtle feature ofUm is that any finite-dimensionamatrix representation ofUm , unlike the operator itself, isnot unitary. The reason is that both the boost operaexp@irW•DpWk# and the free evolution operators ex@2iHat(tk112tk)# couple the subsystem to the orthogon

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complementQ of any finite-dimensional representation~seethe next section!. In simple terms, the exact electronic wavfunction evolves beyond the boundaries of the finite Hilbspace. This loss amounts to a nonunitary evolution with

d

dtTrr,0. ~2.26!

It may appear that the present method depends criticon the choice of the stochastic interaction potential@Eq.~2.15!# in terms of a sequence ofd-shaped pulses. This ishowever, not the case. The key observation is that any ational smooth time-dependent potential can be representeterms of a sequence of infinitesimal kicks as

V~ t !5 (k51

J

d~ t2tk!dpk , ~2.27!

dpk5Etk2D

tk1D

V~ t !dt, ~2.28!

in the limit D˜0, J˜`, andT5JD5const. The representation of the time-dependent perturbation in terms of aquence of infinitesimal kicks is just the essence of thecalled split-operator algorithm for integration of the timdependent Schro¨dinger equation. Therefore, any combinatioof deterministic and stochastic time-dependent potentialsbe treated according to Eq.~2.22!. This includes the effect ofexternal ac and dc fields. The only difference is thatstochastic processes with discrete jumps, the time intebetween subsequent kicks,tk112tk , remains finite and hasphysical significance in terms of a characteristic time scfor the stochastic process that determines the interactiontween subsystem and environment, while for smooth deministic processes this interval is to be taken as a findifference approximation to a continuous process and is tomade sufficiently small untilUm is independent of the lengthof the interval.

Furthermore, Eq.~2.22! permits the treatment of stochatic processes for which the jumpsDpW (t) possess an explicidependence on time or other parameters. The timcorrelation function of the jumps may correspond to ‘‘coored noise’’ with long-time correlations. The only restrictiois that the stochastic perturbation does not depend onstate of the subsystem. This is in accord with the fundamtal assumption of linear response mentioned above.

III. APPLICATIONS

A. Method of calculation

The ultimate purpose of our calculations consists of coputing population fractions of various final states after tion-solid interaction@Eq. ~2.24!#. Because typical calculations involve an average over*1000 trajectories, the keyissue for the simulation of the quantum transport is to utilan efficient and accurate method to numerically evaluateevolution operatorUm . The basic pieces for the latter havbeen developed in a series of recent papers@26,14,27#.

An explicit matrix representation of Eq.~2.22! can befound by expanding the wave function in an orthonormbasis set composed of statesufn&, n51,2, . . . ,Nmax,

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1096 PRA 60D. G. ARBO et al.

ucm~ tk11!&5 (s,i , j 51

Nmax

ufs&^fsueirW•DpW k11uf i&

3^f i ue2 iHat(tk112tk)uf j&^f j ucm~ tk!&.

~3.1!

Since the evolution operator couples states of the compHilbert space, a finite-dimensional representation of thiserator is, in general, not unitary. In other words, if a finbasis set$ufn&% is used, the norm of the wave function is npreserved. The amount of probability lost in this nonunitacalculation gives the probability flux to states outsidebasis set@see Eq.~2.26!#.

Our orthonormal basis set is constructed from a nonthonormal Sturmian basis set@28–31#, which is defined bythe complete set of functions obeying

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n

nsrD uxns&52

1

2ns2

uxns&, ~3.2!

wheren51,2, . . . , , and the parameterns defining the set iscalled the Sturmian parameter. Equation~3.2! can be solvedin spherical coordinates leading to basis statesuxn,l ,m

ns & interms of spherical quantum numbers. A complete and coable basis set for the full Hilbert space is generatedchanging the set of quasi-quantum numbersn51,2,. . . ,`,l 52(n21), . . . ,(n21), m52 l , . . . ,l . In practice, how-ever, the basis must be truncated; i.e., the basis size deby Nmax in Eq. ~3.1! is finite. In the following, we use 1<n<nmax, 0< l<(n21),2 l<m< l , which corresponds toa number of statesNmax5nmax(nmax11)(2nmax11)/6. Bydefinition, a bound hydrogenic state can be writtenun,l ,m&5uxn,l ,m

ns & if ns5n/Zp . For hydrogen (Zp51) thevalue of ns determines whether a givenn shell is exactlyreproduced by the basis~i.e., whenns5n). In general,nstakes real values and a Sturmian basis set represents ation of both bound and continuum energy levels. Undstanding which other physical energy levels can be propdescribed within a finite basis requires additional analy~see, e.g.,@27#!.

Since Sturmian functions are not orthogonal, we defithe orthonormal basis set$ufk

ns&, k51,2, . . . ,Nmax% enteringEq. ~3.1! as the set of orthonormal eigenvectors of the finSturmian representation ofHat. Namely,

ufkns&5 (

i 51

Nmax

bikuxni ,l i ,m

ns &, ~3.3!

Hatbk5EkSbk, ~3.4!

whereEk is the eigenenergy of an electron in the stateufkns&,

S is the overlap matrix (Si j 5^xni ,l i ,muxnj ,l j ,m&), andbk and

Hat are the column vector and matrix representation ofufkns&

andHat in the Sturmian basis set, respectively.Matrix elements of the boost operator, exp(iDpW•rW), can be

evaluated analytically. First, the matrix elements are calated in a basisxn,l ,m8 whose quantization axis coincides wi

te-

ye

r-

t-y

ed

s

ac--lyis

e

-

the direction ofDpW ~see, e.g.,@32#!. Subsequently, these couplings are rotated for any arbitrary directionVW of the mo-mentum transfer according to

^xn,l ,muexp~ iDpW •rW !uxn8,l 8,m8&

5 (m1 ,m2

^xn,l ,m1uexp~ iDpz!uxn8,l 8,m2

&

3Dm1ml ~VW !Dm2m8

l 8 ~VW !, ~3.5!

whereDm8mj are rotation matrix elements.

An accurate nonunitary representation of the free evotion operator in Eq.~3.1!, exp@2iHat(tk112tk)#, requires, ingeneral, a more elaborate treatment such as the use ocomplex dilation method@31#, masking function, or complexpotentials~see, e.g.,@33# for a general discussion!. In thiswork we assume that the time in between collision is shenough and that the momentum transfers are large enosuch that most of the outgoing flux of probability candescribed by the boost operator. Thus, we adopt a unimatrix representation, exp@2iHat(tk112tk)#, within the basisset. Since our basis set diagonalizesHa , exp(2iHatdt) isalso diagonal with matrix elements exp(2iEkdt) k51,2, . . . ,Nmax.

B. Transmission of H2 through thin carbon foils

In order to treat the transport of a two-electron H2 ion, anadditional approximation is required which goes beyondone-electron transport outlined above@9,10#. We reduce thecomplex problem of H2(1s,1s8)-solid interaction to twomajor steps. First, the weakly bound ‘‘outer’’ 1s8 electron iscollisionally detached, thereby leaving the ‘‘inner’’ electroin a superposition of states of the H atom. Second, thesulting H atom propagates through the solid experiencmultiple collisions, as described by the present transportproach.

The probability for destruction of H2 as a function of thefoil thicknessX is given by PH2(X8)5e2X8/lD, where theinverse mean free path~IMFP! for collisional single-electrondetachment of H2, lD

21 , is approximately given by the totaIMFP of free electrons@9#. The sudden collisional removaof the outermost electron leads to a redistribution~shake-up!of the inner electron of H2 among hydrogenicns states.Using the generalized shake-up approximation and theparameter H2 wave function of Hart and Herzberg@34#, it isfound that predominantly H(1s) and H(2s) become popu-lated @9#. The probability to find the electron in the 1s and2s states isp1s50.815 andp2s50.183, respectively. Thepopulation of highns states including those in the continuuis less than'0.2%. Within this isotropic shake approximation, states of higher angular momenta are not occupied

The shake-up process creates a coherent excitation oresidual hydrogen atom. However, since the energy andgular distribution of the detached electron remains unsolved, these coherences are partially averaged out.therefore consider two limiting cases for the initial densmatrix. When the phase-space distortion due to the detacelectron is strong, we arrive at a completely incoherent smixture,

her

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PRA 60 1097QUANTUM TRANSPORT THEORY FOR ATOMIC STATES . . .

r i5(n

ucns&pns cnsu. ~3.6!

Conversely, if the residual interaction is negligible~e.g., foran extremely fast receding electron!, a pure, fully coherentstate

c i5(n

Apnsucns& ~3.7!

would result with the relative phases directly given by tshake-up amplitude. We have verified that the numericalsults using Eqs.~3.6! and~3.7! agree with each other withinour numerical uncertainties. Most of the results presenbelow are based on Eq.~3.7!.

Having generated the initial state after collisional detament and shake-up, the evolution of the hydrogenic elecin the rest frame of the proton is governed by the nonrelaistic stochastic Hamiltonian@Eq. ~2.15!#. Figure 1 illustratesthe evolution of outgoing charge state and population frtions of n shells of hydrogen as a function of the foil thickness for a beam energy of 800 MeV. The foil thickness~ortime! at which the populations ofH(n51,2) maximize isvery different from the ones forH(n>3), indicating the ex-istence of different production mechanisms. While shakeplays a very important role in the population of then51,2shells, n>3 shells are predominantly populated throumultiple collisions. An average of eight collisions are ivolved for the largest foil thickness in the figure. Previoclassical transport calculations were found to be in reasable agreement~within a factor of 2! with experimental dataof Gulleyet al. @35# on an absolute scale~see Fig. 1!. We cannow test the classical simulation by comparing with opresent quantum transport simulation. Remarkably, the qutum and classical calculations are in very good agreemwith each other.

FIG. 1. Fractions of H2 and H(n51,2,3,4) as a function of foilthickness for transmission of a 0.8 GeV H2 beam through a carbonfoil: experimental data of Gulleyet al. @35# ~symbols!, quantumtransport~solid lines!, and classical transport~dashed lines!.

e-

d

-n-

-

p

n-

rn-nt

Clearly, any firm conclusion from this comparison rquires an assessment of the intrinsic errors. Our quantransport calculations have two sources of errors: statistand numerical. Statistical errors are associated with the finumber of quantum trajectories involved in the simulatioNtraj51000. This is obviously the origin of the fluctuationof the outgoing fraction of H2 ions when the fraction hasfallen to below'1022 and which should coincide with theclassical result in the limitNtraj˜`. Figure 2 illustrates thestatistical error of our calculation of the outgoing fractionsexcited states of hydrogen. The two results were obtaiusing a different ensemble of random collisions and theference between the curves gives directly a measure forstatistical uncertainties of our results. We therefore estimthe statistical error of our calculations to be&20%.

Numerical errors are associated with the truncation effein the solution of the time-dependent Schro¨dinger equationin a finite Hilbert space. As a convergence criterion, we ua stabilization method@27# according to which convergedresults should be stable with respect to variations of our bsize,Nmax, and the Sturmian parameter,ns . We find that anoptimal Sturmian parameter for this problem isns.3. Figure3 illustrates the convergence of our calculations with respto the basis size as it changes fromNmax5140 (nmax57) toNmax5285 (nmax59) states. The relative error of the calclations is found to be&5%. The relatively fast convergencis due to the fact that we follow the time evolution only foa relatively short time~about four orbital periods of ann52 electron and about one orbital period of ann53 elec-tron!. With increasing time or foil thickness, the error wouincrease. One physical origin for the breakdown wouldthe collisional recapture of continuum electrons which bcomes a more important process after long periods of tor, equivalently, large distances@8,10#. This would also re-quire a larger basis size to be accounted for accurately.

A more detailed test for the agreement between the csical and quantum simulations is provided by a compariof the subshell populations of the outgoing hydrogen ato

FIG. 2. Illustration of the statistical error of the quantum calclations in Fig. 1. The different calculations have been obtaineding two different random sequences of collisions.

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1098 PRA 60D. G. ARBO et al.

The time-dependent subshell populations of excited hydgenic states generated during propagation of the hydroatom through the foil probe the relative contributions of eltic and inelastic momentum transfer as well as their absovalues and direction and allow for the identification of prpensity rules favoring certain quantum numbers. Figure 4~a!displays a comparison between our classical and quancalculations and the measurements of Keatinget al. @36# forthem distributions in then54 shell. Note that the quantization axis is chosen to be perpendicular to the beam velo

vW p . The agreement between classical and quantum reand the experiment is very good. Them50 population isdrastically enhanced compared to a statistical distribu~the statistical weights ofm50,1,2,3 are 0.25, 0.375, 0.25and 0.125, respectively!. For small foil thicknesses, the propensity for populatingm50 states is primarily driven by theshake-up process following the single-electron detachmof H2. Remarkably, the preference for populatingm50states extends to all foil thicknesses and, additionally,population fractions are nearly independent of the foil thicness. This propensity is a consequence of the direction otypical momentum transfers involved in the transport pcess: for high-velocity collisions, both elastic and inelasmomentum transfers are nearly perpendicular to the baxis.

The preferential direction of the momentum transfers agive rise to the propensity to create Stark states whose spprobability densities have the largest polarization perpdicular to the beam axis. Within each Starkn manifold, themost polarized states correspond to them50 states. Thepopulation fractions of the variousun,k,m& Stark stateswithin a givenn manifold obtained theoretically and expermentally @36# are in agreement with this picture. As an eample, Fig. 4~b! shows the relative probability within then54, m50 subshell for populating Stark states withelectricquantum numberk523,21,1,3. Both experiment@36# andtheory reveal population preferences of the extreme S

FIG. 3. Relative numerical error of the calculations in Fig. 1the basis size is increased fromNmax5140 (nmax57) to Nmax

5285 (nmax59).

-en-te

m

tylts

n

nt

e-he-cm

otial-

rk

states~large-uku states! u4,23,0& and u4,3,0& by about 50%compared to theu4,21,0& and u4,1,0& states, in accordancwith the dominance of transverse momentum transfers.no strong external magnetic field is present during the traport, the population of states with the sameabsolute electricquantum number should be equal. Small deviations frequal population are a measure of the statistical error ofcalculations as well as of the experimental uncertainties.

IV. CORRESPONDENCE BETWEEN CLASSICALAND QUANTUM TRANSPORT

In order to investigate the origin of the remarkably cloclassical-quantum correspondence observed for the transproblem in more detail and identify parameter regions whdiscrepancies might occur, we present in the followingsults for the evolution of a pure hydrogenic initial sta~rather than a mixed state!. In addition, we have chosencollision energy of 8.85 GeV energy which is higher than tone in the preceding section so that the present results cadirectly used for recently proposed experiments for the extation and Lya detection of antihydrogen@18#. Note that thesame conclusions concerning classical-quantum corresdence are obtained for a beam energy of 0.8 GeV~see, e.g.,the scaling properties of the population fractions@10#!.

Figures 5 and 6 show that classical and quantum calctions for then and l distributions of outgoing excited stateof hydrogen are again in reasonably good agreement weach other. The main difference with respect to transmiss

FIG. 4. Substate distributions in then54 shell as a function offoil thickness for transmission of a 0.8 GeV H2 beam through acarbon foil: experimental data of Keatinget al. @36# ~symbols!,quantum transport~solid lines!, and classical transport~dashedlines!. ~a! Relative m distribution. ~b! Stark k distribution for m50.

nh

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PRA 60 1099QUANTUM TRANSPORT THEORY FOR ATOMIC STATES . . .

of H2 is that the fraction of then level associated with theinitial state provides the dominant source of probability ais a monotonically decreasing function of thickness. Tdominant angular momentum after excitation from H~1s! isl 51 in accordance with dipole selection rules for soft cosions. The point to be noted is that classical transportaccount for thel 51 dominance reasonably well despite tfact that dipole transitions due to virtual photon absorptare not properly accounted for within classical dynamiThe biggest differences between the classical and quancalculations are observed for the population of large angmomentum states at small thicknesses. For increasingthickness, the fractions of larger angular momentum stabecome larger, which is recognized as a signature of multscattering~e.g.,@37,38,7,39#!.

One might expect that the good agreement between csical and quantum transport is due to the benign effecaveraging over many degrees of freedom and parame~momentum transfers, collision times! inherent in transport.Conversely, the more degrees of freedom are resolvedbetter the chances to observe clear discrepancies. Genespeaking, a quantum system reaches its classical limitshort times, strong perturbations, and large quantum nbers. For the transport problem with random flight timesbetween collisions, however, the Fourier spectrum ofperturbation is broad and, consequently, a time scale forperturbation can be determined only in terms of an averaThis averaging effectively eliminates time coherences whmay otherwise lead to, for example, resonant excitation p

FIG. 5. Fractions of H(n51,2,3,4) as a function of foil thick-ness after transmission of 8.85 GeV H(1s) and H(2s) beamsthrough a carbon foil: quantum transport~solid lines! and classicaltransport~dashed lines!.

de

n

n.m

aroilesle

s-frs

heallyor

-

ee

e.h-

cesses. In addition, the transport problem involves an aage over the magnitude and the direction of the momentransfers delivered to the electron.

In order to delineate the origin of the classical-quantucorrespondence, we analyze next a simplified problem oatom subject to a train of impulses~or kicks! for which therandomness of the stochastic force is reduced by keepingdirection and strength constant~i.e., m is a constant of mo-tion!,

Htrain~ t !5Hat2zDp(i

d~ t2t i !, ~4.1!

and by constraining the time interval between clisions, Dt5t i 112t i , to be Poisson distributed,P(t i 11)}exp@2(ti112ti)/^Dt&#, where the constantDt& representsthe average time between collisions. Analysis of this prolem can provide evidence as to whether the randomnesthe collision times controlled byDt& suffices to achieve thecorrespondence between the classical and quantum resu

In Fig. 7 we compare classical and quantal ionizationan H~1s! initial state as a function ofDp after an averagenumber ofNkick510 kicks~i.e., t/^Dt&510), which is a typi-cal number for the transport problem. We display resultstwo different ^Dt& values, one much smaller and one mularger than the orbital period of the state to be ioniz(Torb52p a.u. for n51). As a point of reference, we alsshow the single kick result for which randomness in timeobviously eliminated. Note that the value^Dt&510 Torb cor-responds to the realistic transport calculations discusabove, while^Dt&5Torb/10 corresponds to the ‘‘ultrashort’

FIG. 6. Substatel distributions in then52 andn54 shells as afunction of foil thickness after transmission of a 8.85 GeV H(1s)beam through a carbon foil: quantum transport~solid lines! andclassical transport~dashed lines!.

be

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ms-

1100 PRA 60D. G. ARBO et al.

limit where the time evolution between adjacent kicks can~almost! neglected and where randomness of collision timshould play a minor role.

For a single kick, classical and quantum results are foto be in agreement with each other if the magnitude ofmomentum transfer is large enough~e.g., Dp*Dpcrit'0.5 a.u.). For small momentum transfers, ionizationcomes classically suppressed since the classical averagergy transferred to the electron@^DE&5(Dp)2/2# is muchsmaller than the level spacing nearn51. In turn, thequantum-mechanical result becomes proportional to (Dp)2

in the dipole limit ~see, e.g.,@13,14# for a more detaileddiscussion!. The results after a train of 10 kicks show that tdomain of momentum transfers for which classical and qutum results agree shifts to smallerDp. In the limit Dt˜0,this is due to the fact that the sum

DpW sum5 (i 51

Nkick

DpW i ~4.2!

rather than an individual momentum transfer has to be csidered. The effect of 10 collinear kicks becomes exacequivalent to that of a single kick with ten times larger manitude. Therefore,Dpcrit ~per kick! above which ionizationbecomes classical should scale as 1/Nkick , which is approxi-mately observed in Fig. 7. That is, the results for 10 kicand^Dt&50.1Torb are shifted inDp in about a factor of tenwith respect to the results for one kick. In turn, for^Dt&510Torb, the relative orientation ofDpW with respect to thelocal momentumpW in the orbit between different kicks irandomized. Therefore, one would expect the ‘‘effectiv

FIG. 7. Ionization probability as a function of the momentutransfer after a H(1s) atom is subject to a single momentum tranfer and to a train of constant momentum transfers~kicks! with arandom time intervalDt in between them: quantum simulatio~solid lines! and classical simulation~dashed lines!.

es

de

-en-

-

n-y-

s

’

momentum transfer determining the ionization probabilitybe Dpsum.ANkickDp. In this case, the results for 10 kickshould be shifted in;1/ANkick with respect to the results foone kick, which is also approximately observed in Fig. 7.

Figures 8 and 9 show that the conclusions found abclassical-quantum correspondence for ionization have tomodified for the excitation probability into then52 or n53 levels. The excitation probability as a function of thmomentum transfer for 10 kicks is shifted compared withone for a single kick, much like the ionization probabilitHowever, the degree of correspondence after ten randkicks is not as good. Clearly, the excitation probability intogiven level is a more delicate case than the total ionizatprobability, which includes a sum over all continuum enerlevels ~a more appropriate comparison would be the ionition into a given continuum energy!. As expected, the excitation probability is classically suppressed for small momtum transfers. However, depending on the value of^Dt&, theexcitation probability for large momentum transfers canclassically suppressed (^Dt&510) or enhanced~see@13# fora single kick, which corresponds to the limit^Dt&˜0).Therefore, not only a lower critical valueDpcrit above whichthe classical-quantum correspondence holds has to besidered, but also an upper cutoff exists. In other words,encounter the following opposing trends: correspondenceexcitation into a specific final state exists only for not tlarge and not too small values of the momentum transfe

The origin of this apparent puzzle is that classical dynaics can describe the quantum system ‘‘on the average,’’not all of its peculiarities. In general, classical-quantum dcrepancies become larger when the probabilities for the

FIG. 8. Excitation probability as a function of the momentutransfer after a H(1s) atom is subject to a single momentum tranfer and to a train of constant momentum transfers~kicks! with arandom time interval Dt in between them such thatDt&510Torb: quantum simulation~solid lines! and classical simulation~dashed lines!.

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PRA 60 1101QUANTUM TRANSPORT THEORY FOR ATOMIC STATES . . .

responding processes become smaller. For a single kickbulk of the energy levels that become populated are thwhose energy differs from the initial energy in the averaenergy transfer, DE&5(Dp)2/2 ~i.e., the so-called Betheridge!. The probability for excitation into energy levels thare far from this ridge is small and is not properly describby classical dynamics. The breakdown is also visiblemany kicks, i.e., in the time evolution of excitation~Fig. 9!.As the amount of momentum delivered to the systemcreases, the classical-quantum discrepancies increase.

Since in addition to the randomness of the collision timalso the size of the momentum transfer critically influencthe agreement between classical and quantum transporcan now explore how much randomness inDp is required to‘‘repair’’ the classical-quantum correspondence and to yithe level of agreement observed in the realistic calculatTo this end, we randomize in our model the values ofDpaccording to a uniform distribution in the interval@0.1<Dpi<1] but still keeping the direction fixed. The resultinexcitation functions are displayed in Fig. 10 and show tclassical and quantum calculations are now in reasonagreement with each other, the largest deviation forn52being about 50%. Obviously, as more averaging entersrealistic calculation, the agreement becomes even better

V. CONCLUSIONS AND OUTLOOK

In this work we have introduced a quantum transptheory to describe the evolution of the excited states of atoas they are transmitted through the solid. We have prese

FIG. 9. Fractions of H(n51,2) as a function of scaled time aftea H(1s) atom is subject to trains of three different momentutransfers with a random time interval in between them such^Dt&510Torb: quantum simulation~solid lines! and classical simu-lation ~dashed lines!.

hesee

dr

-

swe

dn.

tle

e

ts

ed

results for the transmission of relativistic H2 and H atomsthrough thin carbon foils. Our results are found to beagreement with experiment and classical transport calctions.

We have shown that the surprisingly good agreementtween classical and quantum transport is due to two effeFirst, the typical momentum transfers in ion-solid collisioare such that they lie, on the average, in the range for whclassical-quantum correspondence for a single momentransfer~‘‘kick’’ ! is expected to exist. Second, the stochasnature of the collisional interaction destroys quantum effeof time coherence and averages over a broad distributiomomentum transfers.

The present results suggest that promising candidateswhich larger discrepancies should appear would be higcharged ions as projectiles. Here, the average collisionalergy transfers would be smaller than the quantum level spings and the critical threshold for classical excitation maybe reached. Thus, the regime of ‘‘small’’ nonclassical mmentum transfers could be explored. In addition, becausthe large value of the projectile charge, dynamical screenand the fine structure of the atom may play an important roRecent experiments for fast Ar181 and Kr361 ions interactingwith carbon foils have found unexplained discrepancies wclassical simulations@40#. Work is underway to analyzethese systems using the present quantum transport forism.

ACKNOWLEDGMENTS

Support for this work has been provided by~a! ORNLDirector’s LDRD Program No. 3211-0846,~b! the U.S. De-partment of Energy, Office of Basic Energy Sciences, Dision of Chemical Sciences, under Contract No. DE-AC096OR22464 with ORNL managed by Lockheed MarEnergy Research Corporation,~c! the NSF, and ~d!CONICET.

at

FIG. 10. Fractions of H(n51,2,3,4) as a function of scaled timafter a H(1s) atom is subject to trains with random momentutransfers in the interval 0.1 a.u.,Dp,1 a.u. and with a randomtime interval in between them such that^Dt&510Torb: quantumsimulation~solid lines! and classical simulation~dashed lines!.

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1102 PRA 60D. G. ARBO et al.

@1# N. Bohr and L. Lindhard, K. Dan. Vidensk. Selsk. Mat. FyMedd.26, 12 ~1954!.

@2# J. Burgdorfer, in Proceedings of the Third Workshop on HigEnergy Ion-Atom Collisions,Vol. 294 of Lecture Notes inPhysics, edited by D. Berenyi and G. Hock~Springer-Verlag,Berlin, 1988!, p. 344.

@3# J. Burgdorfer and J. Gibbons, Phys. Rev. A42, 1206~1990!.@4# R. Abrines and I. C. Percival, Proc. Phys. Soc. London88, 861

~1966!; 88, 873 ~1966!; I. C. Percival and D. Richards, AdvAt. Mol. Phys.11, 1 ~1975!.

@5# J. Neufeld and R. Ritchie, Phys. Rev.98, 1632 ~1955!; 99,1125 ~1955!.

@6# P. Echenique, W. Brandt, and R. H. Ritchie, Phys. Rev. B33,43 ~1979!.

@7# J. Burgdorfer and C. Bottcher, Phys. Rev. Lett.61, 2917~1988!; J. Kemmler, J. Burgdo¨rfer, and C. O. Reinhold, PhysRev. A 44, 2993~1991!.

@8# C. O. Reinhold, J. Burgdo¨rfer, J. Kemmler, and P. KoschaPhys. Rev. A45, R2655~1992!.

@9# B. Gervais, C. Reinhold, and J. Burgdo¨rfer, Phys. Rev. A53,3189 ~1996!.

@10# P. Kurpick, C. O. Reinhold, J. Burgdo¨rfer, and B. Gervais,Phys. Rev. A58, 2183~1998!; C. O. Reinhold, P. Ku¨rpick, J.Burgdorfer, and S. Yoshida, Nucl. Instrum. Methods PhyRes. B146, 76 ~1998!.

@11# The NSNS Collaboration, National Spallation Neutron SouConceptual Design Report, Oak Ridge National LaboratReport No. NSNS/CDR-2, 1997~unpublished!.

@12# Outline Design of the European Spallation Neutron Souredited by I.S.K. Gardner, H. Lengeler, and G. H. Rees, 19~unpublished!.

@13# C. O. Reinhold and J. Burgdo¨rfer, J. Phys. B26, 3101~1993!.@14# C. O. Reinhold, M. Melles, H. Shao, and J. Burgdo¨rfer, J.

Phys. B26, L659 ~1993!; 29, 377 ~1996!.@15# F. J. Garcia de Abajo and P. M. Echenique, Phys. Rev. L

76, 1856~1996!.@16# G. Schiwietz and P. L. Grande, Radiat. Eff. Defects Sol

130, 137 ~1994!.@17# M. O. Scully and M. S. Zubairy, inQuantum Optics~Cam-

bridge University Press, New York, 1995!.@18# G. Blanford, K. Gollwitzer, M. Mandelkern, J. Schultz, G

Takei, G. Zioulas, D. C. Christian, and C. T. Munger, PhRev. D57, 6649~1998!.

@19# L. Landau, J. Phys.~Moscow! 8, 1 ~1944!.@20# S. Goudsmit and J. L. Saunderson, Phys. Rev.57, 24 ~1940!.@21# R. Becker and A. MacKellar, J. Phys. B17, 3923 ~1984!; J.

Pascale, C. O. Reinhold, and R. E. Olson, Phys. Rev. A42,5305~1990!; C. O. Reinhold and C. A. Falco´n, ibid. 33, 3859~1986!.

.

ey

,5

t.

s

.

@22# P. Sigmund and K. Winterbon, Nucl. Instrum. Methods119,541 ~1974!; S. Tougard and P. Sigmund, Phys. Rev. B25,4452 ~1982!.

@23# J. Ashley, C. Tung, and R. Ritchie, Surf. Sci.81, 409 ~1979!;C. Martin, E. Arakawa, T. Callcott, and J. Ashley, J. ElectrSpectrosc. Relat. Phenom.35, 307 ~1985!; J. Ashley, J.Cowan, R. Ritchie, V. E. Anderson, and J. Hoelzel, Thin SoFilms 60, 361 ~1979!; J. Ashley, J. Electron Spectrosc. RelaPhenom.28, 177 ~1982!.

@24# C. Tung, Surf. Interface Anal.11, 69 ~1986!.@25# U. Fano, Annu. Rev. Nucl. Sci.13, 19 ~1963!.@26# M. Melles, C. O. Reinhold, and J. Burgdo¨rfer, Nucl. Instrum.

Methods Phys. Res. B79, 109 ~1993!.@27# S. Yoshida, C. O. Reinhold, J. Burgdo¨rfer, B. E. Tannian, R.

A. Popple, and F. B. Dunning, Phys. Rev. A58, 2229~1998!.@28# M. Rotenberg, Ann. Phys.~N.Y.! 19, 262 ~1962!; Adv. At.

Mol. Phys.6, 233 ~1970!.@29# R. Shakeshaft, Phys. Rev. A18, 1930~1978!; 14, 1626~1976!.@30# M. Pont, D. Proulx, and R. Shakeshaft, Phys. Rev. A44, 4486

~1991!.@31# A. Maquet, S. Chu, and W. P. Reinhart, Phys. Rev. A27, 2946

~1983!.@32# Dz Belkic, J. Phys. B14, 1907~1981!; the basic integrals can

be actually found in, e.g., I. G. Gradshteyn and I. M. Ryshin Table of Integrals, Series, and Products~Academic Press,San Diego, 1980!, p. 844.

@33# S. Yoshida, S. Watanabe, C. O. Reinhold, and J. Burgdo¨rfer,Phys. Rev. A60, 1113~1999!.

@34# J. F. Hart and G. Herzberg, Phys. Rev.106, 79 ~1957!.@35# M. S. Gulley, P. B. Keating, H. C. Bryant, E. P. MacKerrow

W. A. Miller, D. C. Rislove, S. Cohen, J. B. Donahue, D. HFitzgerald, S. C. Frankle, D. J. Funk, R. L. Hutson, R.Macek, M. A. Plum, N. G. Stanciu, O. B. van Dyck, and C. AWilkinson, Phys. Rev. A53, 3201~1996!.

@36# P. B. Keating, M. S. Gulley, H. C. Bryant, E. P. MacKerrowW. A. Miller, D. C. Rislove, S. Cohen, J. B. Donahue, D. HFitzgerald, D. J. Funk, S. C. Frankle, R. L. Hutson, R.Macek, M. A. Plum, N. G. Stanciu, O. B. van Dyck, and C. AWilkinson, Phys. Rev. A58, 4526~1998!.

@37# Y. Yamazakiet al., Phys. Rev. Lett.61, 2913~1988!.@38# J. P. Gibbonset al., Phys. Rev. Lett.67, 481 ~1991!.@39# P. Nicolai, M. Chabot, J. P. Rozet, M. F. Politis, A. Chetiou

C. Stephan, A. Touati, D. Vernhet, and K. Wohrer, J. Phys23, 3609~1990!.

@40# D. Vernhet, J. P. Rozet, E. Lamour, B. Gervais, C. Fourmeand L. J. Dube, Phys. Scr.~to be published!; D. Vernhet, J. P.Rozet, I. Bailly-Despiney, C. Stephan, A. Casimi, J-P. Gradin, and L. J. Dube, J. Phys. B31, 117 ~1998!; E. Lamour,Ph.D. thesis, Universite de Caen, 1997~unpublished!.