Chapter 2.6.2 FAST AND SLOW COLLISIONS OF IONS, ATOMS

Fast and Slow Collisions of Ions, Atoms and Molecules 1

Chapter 2.6.2

FAST AND SLOW COLLISIONS OF IONS, ATOMSAND MOLECULESC. D. LinDepartment of Physics,Kansas State University,Manhattan, Kansas 66506-2601,U.S.A.

F. MartAnDepartamento de Qumica, C-9,Universidad Autonoma de Madrid,28049-Madrid,Spain

SCATTERING IN MICROSCOPIC PHYSICSAND CHEMICAL PHYSICS

Atomic and Molecular Scattering, Introductionto Scattering in Chemical Physics

Contents1 Introduction 12 Time-Independent Methods 2

The Hamiltonian and the Coordinate Systems 2Close-Coupling or Coupled-Channel Methods 2Molecular Orbital Expansion Method 3Hyperspherical Approach 3

3 Time-Dependent Approaches 5The Molecular Close-Coupling Method 5Direct Integration of the Time-DependentSchrodinger Equation 7Perturbation Approaches 8Relativistic IonAtom Collisions 9

4 Atomic Collisions with Multielectron Targets 105 Penning Ionisation and Associative Ionisation 116 IonMolecule Collisions 13

Quantum Mechanical Approach 14Semiclassical Approach 14

7 IonCluster Collisions 15Jellium Models 16Beyond the Jellium Approximation 17

1. Introduction

This chapter is concerned with the electronic transi-tions in two-body collisions between ions, atoms andmolecules. The collision energy covered ranges fromthe subthermal regime (less than millielectronvolts)in ion traps to the relativistic regime (greater thangigaelectronvolts) in relativistic heavy-ion colliders.Experimentally a scattering event consists of (1) thepreparation of projectiles and targets in well-definedinitial quantum states before the collision and (2) thedetermination of quantum states of the collision prod-ucts at the end of the collision.

Elementary collisions involving atoms, ions andmolecules occur in many gaseous and plasma environ-ments, such as in the upper atmosphere and in gaseousdischarges. Collisions of ions with atoms and/ormolecules provide the heating and cooling mecha-nisms for the laboratory and astrophysical plasmas.Understanding the energy transfer in collisions be-tween ions with matter and with biological moleculesis essential to cancer therapy. At the fundamentallevel, atomic collisions provide a more efficient meansfor populating excited species than photons and elec-trons. Furthermore simple atomic collisions offer afertile ground for testing the quantum mechanicalscattering theory. This chapter is addresses the appli-cation of scattering theory to solve problems in fastand slow collisions of atoms, ions and molecules.

In discussing atomic scattering theory, it is impos-sible not to mention the experimental progress. In thepast few decades, technology has improved drasticallysuch that experimentalists can now control beamsof atoms, ions and molecules with much ease, overan ever-increasing range of energies. For example,in the 1960s, multiply charged ions were producedonly at Van der Graaf accelerator or cyclotron facili-ties, which tend to have higher energies ranging fromhundreds of kiloelectronvolts per atomic mass unit tohundreds of megaelectronvolts per atomic mass unit.In the 1980s, new ion sources such as ECR and EBISproduced multiply charged ions from fractional totens of kiloelectronvolts per atomic mass unit. Sincethe 1990s, heavy-ion storage rings, combined withlaser or electron cooling, has become available forprecision experiments.

While most of the targets are prepared in theground state in a collision experiment, in many ap-plications it is desirable to prepare targets in the ex-cited states. Such excited states have been obtainedusing laser excitations and the states can be oriented

2 Fast and Slow Collisions of Ions, Atoms and Molecules

or aligned using circular- or linear-polarised light, re-spectively. Collisions with Rydberg atoms have alsobeen investigated.

For the collision products, the identification of ex-citation, charge transfer or ionisation processes can becarried out by measuring the charge states of the col-lision products. To determine the specific final states,methods such as photon emission, energy loss or en-ergy gain of projectiles and selective laser excitationsof collision products have been used. For multielec-tron systems, Auger electron emission or X-ray emis-sion can be used to identify the final states. For ion-isation processes, electron spectrometers with ever-improving resolution have been developed. In recentyears, the two-dimensional momentum distributionsof the recoil ions and of the ejected electrons havebeen measured directly. To achieve superb resolutionfor the recoil ions, the target atoms are supersonicallycooled. In the future they can be prepared by lasercooling as well.

The rest of this chapter is organised as follows.The next three sections address the basic theoreticalissues and approximations for ionatom collisions.57 discuss theories for Penning ionisation, asso-ciative ionisation and ionmolecule and ionclustercollisions.

2. Time-Independent Methods

The Hamiltonian and the Coordinate Systems

Consider a prototype ionatom collision systemwhere a bare ion A+ impinges on a one-electron atom(B+ + e). Within the nonrelativistic theory, the col-lision dynamics is governed by the time-independentSchrodinger equation

H = E, (1)

where H is the Hamiltonian and E is the total energyof the system.

The time-independent wave function is describedby six variables in the centre-of-mass frame. In such asimple collision, the inelastic process consists of targetexcitation, electron capture and ionisation processes.At low energies the ionisation process is not impor-tant. In Fig. 1 three sets of Jacobi coordinates thatcan be used to solve Eq. (1) are shown. In Jacobicoordinates, the kinetic energy operator for the threeparticles is separable

Ti = 1

2mi2ri

12i

2Ri , (2)

where i = , , . In the equation above and therest of this chapter, atomic units (me = h- = e = 1) areused throughout unless otherwise noted. The reducedmasses in this equation depend on the coordinate sys-tem used. In terms of the masses of the three parti-cles, MA, MB and m(=1 for the electron), the reducedmasses entering Eq. (2) for the set, for example, are

given explicitly by

m =MA +MB

MA +MB +1; =

MAMBMA +MB

= , (3)

where is defined to be the reduced mass of the twoheavy particles. The reduced masses for the andthe set of coordinate frames are similarly defined.If non-Jacobi coordinates are used, such as the lastsystem shown in Fig. 1, the kinetic energy operator Tbecomes

T = 1

2m2r

12

2R 1M

R r, (4)

where the last term is nondiagonal, and is known asthe mass polarisation term. In this coordinate systemwhere O is an arbitrary point, OA = pR, OB = qR withp+q = 1, the various masses are given by

m =m

m2 + , M =

, =

pMA qMBMA +MB

. (5)

The potential energy entering the Schrodinger equa-tion is very simple in general. It is the sum of thepair-wise Coulomb interactions.

Close-Coupling or Coupled-Channel Methods

A general approach to solve the multidimensionalpartial differential equation (1) is the close-couplingmethod (Bransden and McDowell, 1992; Fritsch andLin, 1991). Consider direct excitation and electroncapture processes only. The solution of Eq. (1) canbe expanded as

=M

j=1

Xj(R, r) Fj(R) +N

j=M+1

Yj(R, r) G(R), (6)

where Xj and Yj are the channel functions. They re-duce to the bound states of the two-body systems inthe asymptotic region,

Xj(R, r) R

j(r) (7)

Yj(R, r) R

j(r),

where j and j are the bound states of the (e + B+)system and the (e + A+) system, respectively. Eq. (6)when is substituted (6) into Eq. (1) and the variationalprocedure is used, a set of coupled integrodifferentialequations for Fj (R) and Gj(R) is obtained, which isexact for any three-body system except for the basisset truncation. For example, the resulting equationscan be used to study positronhydrogen atom colli-sions; see Chapter 2.6.1. The essential approximationfor ionatom collision theory is to take advantage ofthe fact that the mass of each of the heavy particles ismuch larger than the mass of the electron and to usethis fact to simplify the resulting close-coupling equa-tions. From the geometries in Fig. 1, the two radiusvectors in the set are related to the two vectors inthe set by

r = R +MB

MB +1r = R + r (8)


R =MA

MA +1R +

MA +MB +1(MA + 1)(MB +1)

r = R +1

r,

where the fact that MA and MB are much larger than1 has been used. This allows one to write

Gj(R) = Gj(R +1

r) = T(1

r) Gj(R), (9)

where T is the translation operator, T(x) = exp[i PR x]. For collisions at energies above hundreds of elec-tronvolts, Gj is a sharply peaked function in the di-rection of the incident ion at large internuclear sepa-rations, so that it is possible to approximate (9) by

Gj(R) = eivjr G j(R), (10)

where the phase factor is often called the electrontranslational factor, with vj being approximatelyequal to the velocity of the ion.

Molecular Orbital Expansion Method

Intuitively, the ionatom collision complex at low en-ergies can be approximated as a transient moleculewhere the electronic motion is separated from the nu-clear motion. The electronic wave function is ob-tained by solving

[

122r +VEn(R)

]

n(r;R) = 0 (11)

at a fixed internuclear distance R, where V is the po-tential seen by the electron. Note that the molecularframe coordinates are used (the set) in (11) but theindices have been suppressed. Each molecular orbitaln(r;R) separates into an atomic orbital on atom Aor atom B at large internuclear distances. Using thesemolecular orbitals, one can expand

= n

Fn(R ) n(r;R ). (12)

After multiplying the Schrodinger equation (HE) =0 by n and integrating over the electronic coordi-

Figure 1 Three sets of Jacobi coordinates and one non-Jacobian coordinates for three particles.

nates, one obtains a set of coupled-channel equationsfor Fn(R),

[

12

2R + En(R )E] Fn(R ) (13)

=12

k

n

2

R2

k

Fk(R )

+1

k

n |R|k RFk(R).

The brackets denote integration over the electron co-ordinates. Equation (13) is commonly known as theperturbed stationary state (PSS) approximation. Withproper generalisation to many-electron wave func-tions, the PSS approximation has been widely used fortreating ionatom and atomatom collisions since itsfirst introduction by Massey and Smith (1933). How-ever, the asymptotic wave functions in the PSS ap-proximation do not satisfy the correct boundary con-ditions. This deficiency results in a number of prob-lems in practical applications. First the coupling ma-trix elements in (13) depend on the choice of the ori-gin for the electron coordinates. Second, the couplingmatrix elements may remain nonzero as the internu-clear distance goes to infinity, suggesting that the PSSmodel may predict unphysical transitions at large in-ternuclear separations (Delos, 1981).

For collisions at higher energies, say on the orderof kiloelectronvolts per atomic mass unit, the motionof the heavy particles can be described classically. Acommonly used remedy to this approximation is tointroduce electron translational factors where eachmolecular orbital in (12) is multiplied by a phase fac-tor, exp[iv rf(R, r)], where different forms of f(R, r)have been used. The electron translational factors arenot valid in the low-energy region where inelastic col-lision requires a full quantum-mechanical treatment.One useful approach that can be used for low-energyionatom collisions without all the problems of theperturbed stationary state approximation is the hy-perspherical approach.

Hyperspherical Approach

The fundamental difficulties of the PSS model can beavoided if one formulates ionatom collisions withinthe framework of the hyperspherical approach (Lin,1995). Starting with any one of the three Jacobi coor-dinate systems as in Fig. 1, one can define two mass-weighted vectors

1 =

1

r1

2 =

2

r2, (14)

where is an arbitrary scaling mass and r1 and r2are the pair of radius vectors of any set of Jacobi co-ordinates in Fig. 1, with reduced masses 1 and 2,


respectively. Introducing the hyperradius and thehyperangle through

=

21 + 22 (15)

tan = 2/1,

the Schrodinger equation is given by(

12

2

2+

2 1422

+ V(, , )

)

= E, (16)

where is the angle between 1 and 2 and

2 = 2

2+

` 21(1)cos2

+` 22(2)sin2

(17)

is the grand angular momentum operator with i be-ing the orbital angular momentum operator associ-ated with the radius vector i. The hyperradius is invariant for all the three Jacobi coordinates, butthe remaining five hyperspherical angles = (, 1, 2)depend on the specific Jacobi coordinates. How-ever, 2() = 2() = 2(). Equation (16) can besolved in the same fashion as the BornOppenheimerapproximation with being treated as the slow vari-able,

=

F() (;), (18)

where the channel functions (;) are obtained bysolving

[

2 1422

+V(, , )

]

(;) = U() (;). (19)

The set of hyperspherical potential curves U() isanalogous to molecular potential curves in the BornOppenheimer approximation. In the limit of large ,each low-energy channel function dissociates into atwo-body bound state.

Equation (19) can be solved in the body frame sim-ilar to the BornOppenheimer approach as well. Inthe -set coordinates, define the body-frame z axisto be along 1, the y axis perpendicular to the planeformed by the three particles and x given such thatthe body-frame (x, y, z) axes form a right-handed or-thogonal set. The Euler angles of the body frame are = (1, 2, 3) with 2 = 1, 1 = 1, where (1, 1)are the spherical angles of 1. In the body frame thetotal wave function can be expanded as

(, ) =

I

FI()I (;, )DJIMJ (1, 2, 3), (20)

where D is the symmetrized Wigner D function, J isthe total angular momentum, I and MJ are the projec-tions along the body-frame and the laboratory-framequantisation axes, respectively.

This expansion is very similar to the perturbed sta-tionary state approximation in the body-frame where

the channel function FI() is the solution of(

2

2

142

2E

)

FI() + I

[

VI, I ()

+WI, I ()]

FI () = 0, (21)

where

VI, I () = I(;, )DJIMJ

2 + 2V

I (;, )DJIMJ, (22)

which includes the rotational coupling terms, and

WI, I () = 2

I(;, )

dd

I (, , )

dd

+

I(;, )

d2

d2

I (;, )

(23)

gives the radial coupling terms.It is well known from the PSS approximation that

the radial coupling terms are difficult to evaluateaccurately in the region where the potential curvesshow avoided crossings. In recent years there havebeen different proposals to deal with these sharpedavoided crossings. One is the so-called diabatic-by-sector method. In this method the hyperradius is di-vided into many small sectors and the channel func-tions I(;, ) are fixed within each sector such thatthere is no WI, I() coupling terms. Within each sec-tor the coupled differential equations are integratedfrom the beginning to the end of the sector where thetotal hyperradial wave function and its derivative arematched to the respective functions in the next sec-tor. This procedure is continued till the large R regionwhere they are matched to the asymptotic solutions inthe laboratory frame (Lin, 1995). Another method isthe so-called slow-variable method (Tolstikhin et al.,1996). These approaches are much easier to imple-ment when many channels are included in the calcu-lations.

The hyperspherical approach outlined above (Lin,1995) is useful for any three-body collision systems.It has not been widely used in ionatom collisionssince many partial waves are needed in such systems.However, it has been shown (Igarashi and Lin, 1999)recently that a procedure similar to the PSS approx-imation can be used for applying the hyperspheri-cal method to ionatom collisions; i.e., the poten-tial curves and coupling terms must only be calcu-lated once. This new observation makes the hyper-spherical approach as easy to use as the PSS model,yet without the inherent problems associated withthe PSS model. The method has been applied to ob-tain electron transfer cross sections for the processD+ + H(1s) D(1s) +H+ and the results are shown inFig. 2. Note that the cross section shows interestingstructures at low energies. We comment that the stan-dard PSS model cannot handle this process since the


isotope effect is not included in that model. However,with special treatments the PSS model can be modifiedto describe this process except close to the threshold.

Figure 2 Charge transfer cross sections for D++H(1s) D(1s)+H+ collisions at low energies calculated using the hy-perspherical approach. Taken from Igarashi and Lin (1999).

3. Time-Dependent Approaches

In the time-independent approach for describing ionatom or ionmolecule collisions it is necessary to em-ploy partial wave expansion in order to obtain thescattering cross sections. Even at thermal energies thenumber of partial waves needed for a converged cal-culation easily runs into several thousands. For colli-sion energies greater than a few kiloelectronvolts peratomic mass unit it is possible to treat the motion ofthe heavy particles classically. The electron is movingin the field of a time-dependent potential and its mo-tion is governed by the time-dependent Schrodingerequation

Hel (r;R) = i(/t) (r;R) , (24)

where Hel is the electronic Hamiltonian. Thus in thetime-dependent approach, the goal is to solve Eq. (24)to extract the electronic transition probability Pi(b)for each impact parameter b. The total cross sectionis obtained by integrating the probabilities over theimpact parameter plane.

The Molecular Close-Coupling Method

The time-dependent Schrodinger equation (24) can besolved by expanding in a basis set

(r, t) = s

as(t) s(r, t) exp(

it

Es(t )dt)

, (25)

where s has energy eigenvalues Es, and the functionsform an orthonormal complete set at each internu-clear separation R(t). From (24) and (25) one canobtain a set of coupled equations for an(t):

i an(t) = s

as(t) < n|Hel i d/dt | s

> exp{

it

(Es En) dt}

. (26)

This set of coupled equations are to be solved un-der proper initial conditions. It is desirable to choosebasis functions that reduce to the eigenstates in theseparated-atom limit. This would allow a precisespecification of the initial state and as(t) can thenbe identified as the scattering amplitudes directly.

The differential operator d/dt in (26) operates ina space-fixed collision frame (x, y, z). A convenientalternative frame is the molecular frame where z isalong the internuclear axis.

From Eq. (26) a number of analytic results can beobtained for certain two-channel models.

Two-Channel Models

The LandauZener Model In this model the Hamil-tonian takes the form

Hel =

[

H11 H12H21 H22

]

, (27)

where H12 = H21. As a function of R, H11 and H22cross at Rx, and in the coupling region, H12 is ap-proximated as a constant. Furthermore, let d(H11 H22)/dt = d(E12)/dt = , then the two-channel ver-sion of Eq. (26) is solved to obtain the probabilityfor making transitions from one channel to the other.The LandauZener probability for making such a sin-gle transition is

P = e2j, (28)

where

j = H212/ =H212

| vd (E12)/dR |(29)

and v = dR/dt is the relative radial velocity of the twoheavy particles at Rx. In a full collision there aretwo possible paths for making the transition from onechannel to the other and the probability is 2P(1P).Thus transition is most probable when P is near 1/2.When P is large the crossing can be treated as dia-batic. When P is small, the crossing is adiabatic. Ifthe interference between the two passages is includedin the LandauZener model the resulting probabilitywill show oscillations with respect to the collision ve-locity or the impact parameter. Such oscillations areknown as Stuckelberg oscillations.

The DemkovMeyerhof Model This model is usefulwhen two channels are nearly degenerate in the disso-ciation limit and the coupling between the two chan-nels is H12 = exp(t). The probability for makingthe transition between the two channels is

P = sech2{

E122

}

sin2{

H12dt}

, (30)

where E12 = H11 H22. The oscillating sine func-tion is due to the interference from the two paths.This result is known as the Demkov model. If a single


passage is used the probability for making the transi-tion is

P1 = (1+ e2x)1, (31)

where x = E12/(2). This is known as the Meyer-hof model.

The Rotational Coupling Rotational coupling ex-ists between two states when their magnetic quantumnumbers along the internuclear axis differ by one unit.If the rotational coupling matrix element is approxi-mated as a constant and the energy difference betweenthem is zero, then the transition probability is calcu-lated to be

P = sin2 0, (32)

where 0 is the scattering angle.These three simple two-channel models, together

with models like the RosenZener model, the Nikitinmodel, and the multichannel LandauZener model(Bransden and McDowell, 1992), have been used re-peatedly to interpret qualitatively or semiquantita-tively many ionatom and atomatom collisions inthe absence of detailed dynamic calculations. On theother hand, it is desirable to perform full ab initio cal-culations to obtain inelastic scattering cross sections.

Multichannel Close-Coupling Methods In Eq. (25),it is assumed that the basis functions form an orthog-onal set at each time step or at each internuclear sep-aration. The most commonly used basis functions inactual ab initio calculations consist of either molecu-lar orbitals (MO) or atomic orbitals (AO) in general.When adiabatic molecular orbitals are used, the basisfunctions are orthogonal and one must solve Eq. (26).However, as explained under Molecular Orbital Ex-pansion Method, the molecular basis functions donot satisfy the asymptotic boundary conditions andswitching functions must be introduced. By multiply-ing orbital-dependent switching functions to individ-ual adiabatic molecular orbitals, the resulting basisfunctions are not orthogonal and such nonorthog-onality must be taken into account in deriving thecoupled equations. ( Orthogonality can still be re-tained if a common translational factor is applied toall molecular orbitals.) Similarly, in calculations us-ing atomic orbitals, those orbitals belonging to thedifferent centres are not orthogonal. Thus, insteadof (25), in the general close-coupling description ofatomic collisions, the motion of the electron in config-uration space is spanned in a finite set of nonorthogo-nal basis functions k(r, t), k = 1, . . . , N (Bransden andMcDowell, 1992; Fritsch and Lin, 1991). In sucha truncated expansion the time-dependent electronicwave function is approximated as

(r, t) =N

k=1

ak(t)k(r, t) (33)

with the time-dependent coefficients ak(t). Instead of(26), the system of coupled equations is

N

k=1

Njk(t)dak(t)

dt= i

N

k=1

Mjk(t)ak(t), j = 1, . . . , N (34)

from which the amplitudes can be solved. This equa-tion contains the overlap matrix elements Mjk(t) =j | k and the coupling matrix elements Njk(t) =j | i(/t) Hel | k. In the case of the AO ex-pansion, the basis functions for the bound states inatomic centres A and B are chosen to have the correctasymptotic form

Ak (r, t) = Ak (rA)exp

(

ivA r12

it

dt v2A ieAk t)

,

Bk (r, t) = Bk (rB)exp

(

ivB r12

it

dt v2B ieBk t)

, (35)

where ik and eik (i = A, B) are the eigenstates and the

eigenenergies of the atomic states on centre i. Notethat plane-wave electron translational factors are in-cluded in each basis function to account for the factthat the electron is moving with A(B), which has avelocity vA(vB) with respect to the origin of the coor-dinate system.

In either the AO or MO expansion method there isno standard procedure to incorporate basis functionsto represent the ionisation events. In actual numeri-cal applications, pseudostates have been used. Theseare positive energy states whose wave functions havefinite range. The sum of the modulus squared of am-plitudes associated with these pseudostates gives thetotal probability for ionisation. Other methods makeuse of discretisation techniques that provide L2 inte-grable representations of continuum states with theproper delta function normalisation for any desiredvalue of energy. In the latter case, the differential (inenergy) ionisation probability can also be computed.The latter technique has been successfully used in thecase of single centre AO expansions, i.e., when all AOstates are centred either in A or B. In general, single-centre expansions are adequate at high collision ener-gies. The use of discretisation techniques in this con-text has been discussed in MartAn and Salin (1995).

In recent years elaborate close-coupling calcula-tions based on either AO or MO basis functions havebeen shown to be capable of predicting reliable excita-tion and charge transfer cross sections, as well as ion-isation cross sections, in a broad range of energies, ifthe collision system can be described by a reasonable-size basis function. As an example, in Fig. 3 thetotal charge transfer cross sections for C3+ + H col-lisions obtained by close-coupling calculations basedon the atomic and molecular orbitals are shown tocompare well with the experimental results from dif-ferent groups (Tseng and Lin, 1999). Many othersimilar collision systems have been investigated in the


past two decades and more examples can be foundin monographs (Bransden and McDowell, 1992) andin review articles (Fritsch and Lin, 1991; Kimura andLane, 1990).

Figure 3 Comparison of experimental total electron capturecross sections with results from close-coupling calculations forthe C3++ H system. Solid line is from using AO basis set, and thetwo dashed lines are from using MO basis set. See Tseng andLin, (1999).

The multichannel close-coupling method, despiteits well-known success, cannot be easily applied tocollisions involving atoms initially prepared in Ry-dberg states, or for collisions when many Rydbergstates are populated. In these cases the number ofstates needed to be included in the calculation is toolarge to be handled in the coupled-channel methods.The only theoretical approach available for such sys-tems is the classical trajectory Monte Carlo method,which treats the motion of the electron classically.

Hidden Crossing Theory In recent years a new ap-proach for ionatom collisions at low energies, calledhidden crossing theory (Solovev, 1990), has beenproposed. It is best illustrated first for the avoidedcrossing of two adiabatic potentials. Starting with adiabatic Hamiltonian as in Eq. (27) where all the ma-trix elements are real quantities, the energy eigenval-ues of this Hamiltonian is given by

E1, 2(R) =H11 +H22

2

12

(H11 H22)2 + 4H212, (36)

where E1 and E2 are the adiabatic potentials. Accord-ing to the noncrossing rule of Neumann and Wigner,these two adiabatic curves do not cross in the real-Raxis, where R is the internuclear distance. However,these two adiabatic curves do cross if R is allowed totake complex values. In fact, they cross at a complexRc satisfying

H11(Rc)H22(Rc) = 2iH12(Rc). (37)

For the Landau-Zener model the complex energy dif-ference becomes

E1(R) E2(R)

RRc. (38)

This equation shows that Rc is a branch point. Thusinstead of two adiabatic potential curves in the real-

R axis, in the complex-R plane the potential sur-face is a single surface with two sheets joined at thebranch point. The transition matrix element has apole and the transition probability can be derived us-ing Cauchys theorem by performing contour integra-tion on the complex-R plane. The transition proba-bility between the two levels is

P12 = exp(

2v

12)

, (39)

where

12 =

ImRc

Re Rc

[E1(R)E2(R)](vR/v)

(40)

with vR being the radial velocity and v is the collisionvelocity.

This simple two-channel model can be extendedto multichannel problems and used to obtain ionisa-tion cross sections at low energies. Thus for the basicH+ +H collisions, the potential surfaces for the H+2 ionare calculated to search for the branch points. Twoclasses of branch points have been identified. The firstclass belongs to the so-called S promotion, which is aseries of branch points located at essentially the samesmall real values of R. Promotion through this seriesof branch points is identified as prompt ionisation oc-curring when the two nuclei are close to each other.There is another class belonging to the T promotionwhere the branch points occur at increasingly largerreal-R values. Promotion to the continuum along thisT series is identified with electrons straddling near thetop of the potential saddle from the two nuclei. Asthe two nuclei separate, the top of the saddle is raisedand the electrons are ejected into the continuum. Thisis the saddle point mechanism for ionisation at lowenergies, analogous to the Wannier theory for elec-tron impact ionisation of atoms at threshold energies.In calculating the probability for ionisation via eachmechanism, it is taken as the product of the probabil-ities of Eq. (39) for each branch point encountered.

The hidden crossing theory provides a simple in-tuitive description of the mechanism of ionisation forionatom collisions at low energies. At present thephase information has not been included and the the-ory for treating differential cross sections is not com-plete yet.

Direct Integration of the Time-Dependent

Schrodinger Equation

With the advent of powerful vector and parallel com-puters that are becoming increasingly available, an-other approach for ionatom collisions is the di-rect numerical integration of the time-dependentSchrodinger equation on lattice grid points. Due tothe long-range nature of the Coulomb force, however,a full 3D lattice solution has been made possible only


very recently (Wells et al., 1996).

The formal solution of the time-dependentSchrodinger equation for small time intervals (t =t t0) is given by

(t) = U(t, t0) (t0). (41)

A number of different approximations may be em-ployed for the time-evolution operator U(t, t0)

U(t, t0) = exp{

iHt}

, (42)

where H = H[(t + t0)/2]. If the lattice grids are givenin Cartesian coordinates, then at each time point, thespatial wave function is represented by its values atthe discretised N = NxNyNz lattice points, where Nj isthe number of points in coordinate j. When the wavefunction is represented in the configuration space, thepotential energy operator is diagonal and the Carte-sian kinetic energy operator is represented by a simplematrix. All algorithms used are iterative in nature andbased on matrix multiplications. The lattice methodsthat have been used include the finite-difference repre-sentation, the Fourier collocation method and others.

As a common practice in the lattice-based method,the time-dependent wave function is often calculatedinside a box of finite size where errors are intro-duced. For example, an ad hoc absorbing potentialiW(x, y, z) is often employed to remove flux reflec-tions from the boundaries. Ideally one would like tomake the box as large as possible and the spacing be-tween the grids as small as possible, but memory con-straints limit a typical 3D calculation to a grid size ofabout 125125125 for calculations carried out onpresent massive parallel computers.

The lattice-based methods in principle can be ap-plied to ionatom collisions at any energies. In reality,its applications so far have been limited to simple sit-uations. It is more conveniently applied to excitationprocesses or target ionisation since the electron cloudsfor these processes are localised near the target. Elec-tron capture to the low-lying bound states can be cal-culated so long as the electron cloud for these statesdoes not lie near the box edge. While total ionisa-tion cross sections can be estimated from the flux loss,information on the ejected electron momentum dis-tributions is not available. In comparison with theclose-coupling methods, the lattice-based methods donot provide advantages so far. The best results fromboth approaches are quite comparable but the lattice-based methods in general take many more computingresources.

The time-dependent Schrodinger equation can besolved directly also in momentum space (Sidky andLin, 1998). For ionatom collisions, the solution inmomentum space appears to offer some advantages.Unlike the coordinate-space wave function, which ex-

tends over a large volume when the two nuclei are wellseparated at the end of the collision, the momentum-space wave function remains finite throughout thecollision. In fact the two nuclei are fixed in themomentum space and since high-momentum com-ponents of the wave function are small, reflectionsfrom the wall in the momentum space are negligible.While it is possible to use grid-based methods to cal-culate the momentum-space wave function, computa-tionally it is less taxing to employ the two-centre ex-pansion in the momentum space. The angular partsof the basis functions are spherical harmonics andthe radial momentum functions are expanded at lat-tice grid points. Since the integral equations in mo-mentum space are difficult to calculate, the time in-tegration is performed in the coordinate space. Byusing a least-squared fitting procedure and RungeKutta fixed time-step integration, such calculationscan be carried out on personal computers instead ofon shared supercomputers. By projecting out thebound states on each centre, excitation and chargetransfer probabilities can be extracted as in the close-coupling methods. Using this approach the momen-tum distributions of the ejected electrons can also beobtained directly. This method offers a direct non-perturbative approach for calculating the momentumdistributions of the ionisation cross sections in ionatom collisions.

Perturbation Approaches

For collisions at higher energies perturbative ap-proaches may be used to obtain the scattering crosssections. Consider the Hamiltonian of one electronin the field of a target of charge ZT and projectile ofcharge ZP again. Take the position vector of the elec-tron with respect to the target and the projectile to bex and s, respectively. One can write the Hamiltonianin the prior (i) and the post (f) form,

Hel = Hi +Vi = Hf +Vf (43)

and introduce the unperturbed initial and final statewave functions

(

Hi it

)

FBAi (r, t) = 0 (44)(

Hf it

)

FBAf (r, t) = 0.

For ionisation, which is the dominant process at highenergies,

Hi = Hf = 122r ZT/x (45)

Vi = Vf = ZP/s

and the initial and final state wave functions of (45)are given by the hydrogenic ground state and con-tinuum Coulomb functions, respectively, multipliedby the planewave phase factor. The first-order per-


turbation theory for the ionisation amplitude is thengiven by

AFBAik (b) = i+

dt

FBAk

ZPs

FBAi

, (46)

for each impact parameter b. This formulation isvalid for direct ionisation where the continuum elec-tron sees the Coulomb field of the target ion. Theexpression (46) is the semiclassical version of the firstBorn approximation. Clearly the cross section scaleswith Z2P within the first Born approximation.

The first Born approximation neglects the influenceof the Coulomb field from the projectile charge onthe initial and final state wave functions. While suchinfluence can be partially accounted for by calculatingthe higher order terms of the Born series, these higherBorn amplitudes are difficult to evaluate. Instead it ispreferable to introduce distorted wave models wheresuch influence can be included partially in the first-order theory.

For this purpose, one rewrites the Hamiltonian for-mally as

Hel = Hi +Vi = Hi +Ui +Wi = Hdi +Wi (47)

Hel = Hf +Vf = Hf +Uf +Wf = Hdf +Wf

to introduce the distortion potentials Ui and Uf forthe initial state and the final state, with Wi and Wfthe residual interactions, respectively. The distortionpotentials are chosen such that the Schrodinger equa-tions

(

Hdi it

)

+i (r, t) = 0 (48)(

Hdf it

)

f (r, t) = 0

have solutions of the form

+i (r, t) = FBAi (r, t)L

+i (r) (49)

+f (r, t) = FBAk (r, t)L

k (r). (50)

This imposes on the distortion term the condition

FBAi

(

122s

ZPs

+ iv s)

L+i

= FBAi (x lni(x) sL+i ). (51)

If the right-hand side of (51) is neglected, then theabove equation can be solved analytically to obtain

L+CDWi (r) = N()1F1(i;1; ivs+ iv s), (52)

where N(a) = exp(a/2)(1 ia), = ZP/|k v| and1F1 is the confluent hypergeometric function; i.e., (52)is a Coulomb function of momentum k v with re-spect to the projectile centre. This approximation iscalled the continuous distorted wave (CDW) approx-imation. If the Laplacian in (51) is also neglected,then

L+EISi (r) = exp(

i ln(vs+v s))

(53)

and the resulting approximation is called the eikonalinitial state (EIS) distortion. By using the CDW wavefunction for the final state and the EIS wave functionfor the initial state, the so-called CDWEIS theory hasbeen widely used to calculate the ionisation cross sec-tions, both integral and differential, for a wide rangeof collision systems. In the intermediate- to high-energy region, such calculations have been found towork well, accounting for most of the two-centre ef-fects that are not present in the first Born approxi-mation. In Fig. 4 the ionisation of H2 molecules by30-MeV C6+ ions is used to illustrate that the CDWEIS theory is capable of predicting the details of thedifferential cross sections for such systems (Tribedi etal., 1996). More details of ionisation processes can befound in the monograph by Stolterfoht et al. (1997).

Figure 4 Single differential cross sections for ionisation forC6++H2 collisions at 30 MeV and comparison to the predictionsfrom the first Born approximation and from the CDWEIS theory.Adopted from Tribedi et al . (1996).

Relativistic IonAtom Collisions

Relativistic atomic collisions involving highly chargedions with charge number up to Z=92 and energiesranging from 100 MeV/amu upwards constitute anew field where the quantum electrodynamics (QED)vacuum plays a role. For relativistic collisions, in ad-dition to the well-known atomic processes such asexcitation, electron capture and ionisation, the in-tense electromagnetic fields make the creation of freeelectronpositron pairs and other processes from theQED vacuum possible. The nonradiative electroncapture process, which is the dominant mechanismfor electron capture at lower energies, becomes lessimportant at high energies than the radiative elec-


tron capture process. At still higher energies, elec-tron capture from the pair production (CPP), whichdoes not exist at lower energies, becomes the domi-nant electron capture processes at extreme relativisiticenergies. The CPP process was first observed experi-mentally in 1992. It is a bound-free pair productionwhere the electronpositron pair is created from thevacuum with the end result that the electron is cap-tured to the bound states of the incident ion and a freepositron is ejected into the continuum. For high-Ztarget atoms, CPP cross section has been observed toincrease with increasing relativistic energies, in con-trast to the rapid decrease with increasing energiesfor the electron capture processes. Note that CPP canproceed between two bare ions colliding at relativisticenergies with the end results that the ion charge statedecreases by one unit. This mechanism has been pre-dicted to account for more than half of the beam lossfor bare ions in the Relativistic Heavy Ion Collider(RHIC) at Brookhaven, and for the bare ions in theLarge Hadron Collider (LHC) at CERN. Interestingly,this mechanism is also responsible for the productionof antihydrogen when antiprotons are bombarded onXe targets at the relativistic momentum of 26 GeV/c.

Consider the collision between a fully stripped pro-jectile ion with charge ZP on a one-electron ion withcharge ZT, in the impact parameter approximationwhere the projectile is travelling along a straight-linetrajectory with impact parameter b and velocity v, therelativistic Dirac Hamiltonian is

HD = H0 +VP(t), (54)

where H0 is the target Hamiltonian

H0 = i + ZTe2/r, (55)

with and being the Dirac matrices, and VP is thetime-dependent perturbation by the projectile

VP(r ) = ZP (1vz)/r, (56)

where r =

(xb)2 +y2 + 2(zvt)2, is the finestructure constant and is the Lorentz factor. Thisexpression is derived from the LienardWiechert po-tential of a moving charge. In relativistic theories,natural units are often used (h- = m = c = 1) as in thetwo equations above. According to the intuitive Dirachole theory, the initial state can be considered as givenby an electron in the K shell, together with a fully oc-cupied negative continuum. For describing pair pro-duction, on the other hand, the initial state is a fullyoccupied negative continuum only. For the latter, thepresence of the K-shell electron does not enter. Thissimple intuitive picture is valid if the electronelectrontwo-body interaction can be neglected and the timeevolution of each electron is independent. This inde-pendent particle picture allows the reduction of the

many-particle amplitudes in terms of single-particleamplitudes, and the calculation can be carried out inthe framework of the Dirac Hamiltonian, Eq. (54).

The simplest theoretical method for solving thetime-dependent Dirac equation is to use the lowestorder perturbation theory. The transition amplitudefrom an initial state i to a final state f is given by

Af i(b ) = iZP

dt ei(EfEi)t

d3r +f (r )1vz

ri(r ). (57)

Clearly this expression can be used to evaluate the ex-citation or the direct ionisation of the electron fromthe K shell, for example, and it would give the Z2P de-pendence of the cross sections. Using the analogy tothe ionisation process, Eq. (57) can also be used toevaluate pair production cross sections by a chargedparticle. The complication lies in the actual calcula-tion since both the initial and final states are contin-uum states. A perturbative approach for the CPP canbe obtained by treating it as an electron being trans-ferred from a negative-energy state in the target fieldto a bound state of the projectile. By employing thefirst-order theory the charge transfer amplitude is

Af i(b ) = iZT

dt

d3r +f (r)S1

1r

i(r ) ei(EitEft), (58)

where S is a Lorentz boost operator(see Eq. 4.29 ofEichler and Meyerhof (1995) to transform the final-state wave function from the projectile to the targetsystem.

For relativistic collisions by multiply chargedions, nonperturbative methods for solving the time-dependent Dirac equation are needed. As in thenonrelativistic case, close-coupling methods using theequivalent of atomic or molecular orbitals have beenused. Direct numerical solutions in the coordinatespace or in the momentum space by expanding thewave functions in lattice points have also been used.Details can be found in the book by Eichler and Mey-erhof (1995).

4. Atomic Collisions with Multielec-tron Targets

The previous two sections describe the basic scatter-ing theories that have been widely used for treatingthe different inelastic processes in atomic collisionsover a broad range of collision energies. The theorieshave been presented for the prototype systems consist-ing of one electron and two heavy particles. Most ofthe methods can be formally generalised in a straight-forward manner for many-electron targets of atoms,molecules, clusters and surfaces. However, in actualapplications for such collisions many more approxi-


mations must be implemented since representing thetime-dependent wave function of a many-electron sys-tem over the whole collision time in terms of basisfunctions or on lattice grids is essentially impossible.

There are two situations where approximate cal-culations on many-electron systems can be made us-ing ab initio scattering theories. For low-energy col-lisions, if the number of inelastic channels excited islimited, then it is possible to use many-electron molec-ular or atomic wave functions as basis set in a close-coupling expansion of the time-dependent wave func-tion.

For collisions at higher energies, multielectron pro-cesses can occur in addition to the single-electron pro-cesses. For single ionisation, single-electron captureor single-excitation processes, it is often possible toemploy the active electron model, with the other elec-trons that are not involved in the processes treatedas passive electrons. In such models the multielec-tron system is reduced to a single-electron systemwith the active electron being in the screened field ofthe target and projectile ions. For multielectron pro-cesses, for example, double ionisation or electron cap-ture accompanied by ionisation, full ab initio calcula-tions are difficult to perform. Thus the independent-electron models are often used. In this model, themany-electron wave function is written as the anti-symmetrised product of the wave functions of individ-ual electrons, so that electron correlation is neglected.In such models amplitudes for multielectron processescan be written as antisymmetrised products of single-electron amplitudes. Such independent-electron mod-els have been used to describe multielectron processesin ionatom collisions with some success. However,there are many circumstances where such simple mod-els fail. A few examples of such calculations can befound in Fritsch and Lin (1991). A step forward inincluding electron correlation in the dynamics of fastcollision processes is provided by the frozen correla-tion approximation (FCA) (MartAn and Salin, 1996).In this method, electron correlation is included inboth the initial and final states but it is frozen duringthe collision. In this way, a many-electron transitionamplitude can be written as a linear combination ofantisymmetrised products of one-electron amplitudes.

5. Penning Ionisation and Associa-tive Ionisation

Penning ionisation (PI) is the process in which anenergy donor B collides with a neutral atom ormolecule A whose ionisation energy is less than theexcitation contained in B, A looses one of its elec-trons and B remains in the ground state. It can berepresented by the reaction

A+B A+ +B +e. (59)

This process was first suggested by Frans Penning in1927. Besides PI, another process is associative ioni-sation (AI) represented by

A+B AB+(v)+ e. (60)

PI and AI are only effective at thermal energies.Therefore, the collision velocities are much smallerthan the typical electron velocities, which suggeststhe use of a quasimolecular picture in the frameworkof the BornOppenheimer (BO) approximation. Thebasic mechanism responsible for PI and AI is that ofa bound molecular state (correlated with the entrancechannel A+B) that is embedded in a molecular elec-tronic continuum (correlated with the exit channelA + B+ + e) to which the former state is coupled bythe electronelectron interaction (see Fig. 5). Thus, PIand AI can be interpreted as resonant processes sim-ilar to those observed in atomic and molecular pho-toionisation.

Excellent surveys of the different experimental andtheoretical techniques used to study reactions (59)and (60) have been published by Siska (1993) andWeiner et al. (1990), respectively. The theory of PIand AI involves both electronic structure aspects andcollision dynamics aspects. The former include eval-uation of correlated molecular states and the cou-pling with the molecular electronic continuum. In thissection only the basic formalism that connects struc-tural and dynamical aspects will be discussed. SincePI dominates AI at thermal energies (see below), ina first step one can assume that the total ionisationcross section results from reaction (59) almost exclu-sively. Then one can treat the problem semiclassicallyin this sense: the nuclei follow classical trajectoriesin the field induced by the effective interatomic po-tential, whereas the electrons are described quantummechanically. Thus, on any given nuclear trajectory,the electronic wave function is the solution of a time-dependent Schrodinger equation as that of Eq. (24).

Resonant processes are conveniently described interms of a projection operator formalism. With thepicture provided by Fig. 5 in mind, we define a basisof adiabatic states as follows. Bound electronic statesAB lying above the ionisation threshold AB+ are for-mally the solutions of

(QHelQEi)i = 0 (61)

and unbound (AB+ + e) electronic continuum statesare the solution of

(PlHelPl E)E, l = 0, (62)

where l is the angular momentum of the ejected elec-tron, and Pl and Q are projection operators that sat-isfy the conditions PlPl = l l , P = l Pl, PQ = 0 andP + Q = 1 for all R. Notice that Hel is not di-agonal in the

{

i, E, l}

basis because, in general,


Figure 5 Two-potential energy curve model for Penning ioni-sation. E is the centre-of-mass kinetic energy of collision, e0 isthe separation of the reagent and product potential asymptotes,E (R) is the (classical) local heavy-particle kinetic energy and e(R)is the kinetic energy of the Penning electron when PI occurs atseparation R.

i |QHelPl |E, l and E, l |PlHelPl |E, l are differ-ent from 0. Therefore, the

{

i, E, l}

basis is adia-batic in the sense that Eqs. (61) and (62) are ful-filled for all R. The electronic wavefunction (t) isexpanded in this basis

(t) = i

ci(t)i exp(

it

Eidt

)

+l

dEcE, l (t)E, l exp(

it

Edt

)

, (63)

where 0 represents the initial state. The initial con-dition is ci() = i0, cE, l() = 0. Substitution ofEq. (63) in the Schrodinger equation leads to the sys-tem of differential equations

iddt

ci(t) = l

dEexp(

it

(Ei E)dt

)

< i |QHelPl |E, l > cE, l(t) (64)

iddt

cE, l(t) = i

ci(t)exp(

it

(EEi)dt

)

< E, l |PlHelQ |i > + l 6=l

dEcE, l (t)

exp(

i t

(EE)dt

)

< E, l |PlHelPl |E, l > . (65)

In the latter equations, all dynamical couplingscorresponding to breakdown of the BO approxima-tion have been neglected. This is a very good ap-proximation because the dynamical couplings areproportional to the collision velocity which, in thecase of PI, is on the order of 103au. The cou-plings i |QHelPl |E, l are responsible for bound-continuum transitions while the E, l |PlHelPl |E, lcouplings represent continuumcontinuum transi-tions and, therefore, they are responsible for a redis-

tribution of the population within the continuum. Asusual, the total ionisation cross section is given by in-tegrating the probability

P(b) = limt

l

dE |cE, l(t) |2 (66)

over the impact parameter plane, where b is the im-pact parameter.

In most applications, instead of solving the systemof coupled equations (64) and (65), one makes use ofa local approximation. This approximation can beeasily derived from (64) and (65) by assuming that (i)the entrance channel is well separated in energy fromthe remaining Q states (isolated resonance approxi-mation) and (ii) all continuumcontinuum couplingswithin P subspace are 0. Using approximations (i)and (ii) in Eq. (65), integrating over time and substi-tuting the result into Eq. (64), one obtains

ddt

c0(t) = l

dE | < 0 |QHelPl |E, l > |2

t

dtc0(t)exp

(

it

t(E0 E)dt

)

. (67)

Let E be the energy interval such that < 0 |QHelPl |E, l > barely changes in the interval IE0 = [E0 E, E0 + E]. If the collision velocity v(t) = v0 1

tVdt

(where V is the interatomic potential, the reduced mass of the nuclei and v0 the initial ve-locity) is small enough such that |v| < E for all t,the integral in Eq. (67) is almost 0 outside IE0 dueto the strongly oscillatory behaviour of the exponen-tial. Then Eq. (67) can be easily integrated and thetotal ionisation probability for a given trajectory canbe written

P(b) = 1 exp(

(t)dt

)

, (68)

where

(t) = 2l

| < 0 |QHelPl |E = E0, l > |2. (69)

Equation (68) can also be written as

P(b) = 1 exp(

2

R0

(R)vb(R)

dR)

, (70)

where vb(R) is the radial velocity of the nuclei. In gen-eral, the value of the width (R) decreases exponen-tially with R and, therefore, the ionisation probabil-ity is only important at short internuclear distances.Similar results have been obtained from purely clas-sical considerations (Miller, 1970). In many experi-mental and theoretical studies, the energy donor is anoble-gas metastable atom. Figure 6 shows the en-ergy dependence of the Penning ionisation cross sec-tion in the case of the Ne(3P2) + H collision. It canbe seen that the cross sections decreases monoton-ically with impact energy, which is the typical be-haviour in most cases. To find the probability distri-bution P(b, e), so that P(b, e)de is the probability thatthe energy of the ejected electron lies in the interval


(e, e + de), one notes that the transition occurs at in-ternuclear distance R, then the energy of the ejectedelectron must be e(R) = V(R)V+(R), where V+(R) isthe potential energy curve of the residual molecularion AB+. The probabilities P(b, e) and P(R) are re-lated by P(b, e)de = P(R)dR so that

P(b, e) = i

P(R)/ |de/dR |Ri , (71)

where the sum is over those Ri (usually no more than2) that satisfy e = V(Ri)V+(Ri).

AI can arise only if the potential of the AB+ ion,V+(R), possesses a well deep enough to support boundvibrational states. If the energy carried away by theionised electron is e = V(R)V+(R) then the relativetranslational energy is E e, where E is the kineticenergy E = v20/2 of the collision. Hence, E e < 0implies that the final relative motion in the V+ poten-tial must be that of a bound state of AB+, that is, AI.The probability for AI is then given by (Siska, 1993)

PAI(b) = 2exp(

R0

(R)vb(R)

dR)

sinh(RAI

R0

(R)vb(R)

dR)

, (72)

where RAI is the (unique) root of Ee(R) = 0. A simi-lar theoretical framework has been recently proposed(MartAn and Berry, 1997) to study electron detach-ment from negative ions using excited neutrals:

A +B A+B + e (73)A +B AB(v)+ e. (74)

The processes are called, respectively, Penning detach-ment (PD) and associative detachment (AD). PD crosssections are several orders of magnitude larger thanPI cross sections. The explanation is twofold and is

Figure 6 Theoretical prediction of total and AI cross sectionsfor Ne + H. As mentioned in the text, the total ionisation crosssection is close to the PI cross section. Taken from Siska (1993).

related to the presence of a negative charge in the pro-jectile. In the first place, Stark mixing between excitedstates of B is induced by the charged projectile A.As a result of this mixing, the width (R), which rep-resents the coupling between the entrance channel andthe continuum, does not decrease exponentially, butrather R6, which is typical of dipoledipole inter-actions. In the second place, the interatomic potentialV(R) between a negative ion and a neutral is given by/R4, where is the polarisability, and may be moreattractive than between two neutrals. This attractivepotential may also lead to orbiting of the projectilearound the target.

6. IonMolecule Collisions

In this section electronic and vibrational transitionsin ionmolecule collisions are considered. The reac-tive scattering is being addressed separately in Chap-ter 2.6.4. Only collisions between ions and diatomicmolecules will be considered, including processes likecharge transfer

A+ +BC(v0) A+BC+(v), (75)

electronic and/or vibrational excitation

A+ +BC(v0) A+ +BC(v) (76)

and dissociative charge transfer

A+ +BC(v0) A+B +C+ (77)A+ +BC(v0) A+B+ +C, (78)

in the impact energy range from a few millielectron-volts to several kiloelectronvolts. The first significantattempts to investigate this problem were based on aclassical treatment of all nuclear degrees of freedomand a quantum mechanical treatment of the electronicmotion (see, e.g., the review of Kleyn et al. (1982)and the book by Bernstein (1979)). More recently,fully quantum mechanical and semiclassical methodshave become available. These methods have been ex-tensively reviewed by Sidis (1990). In this section, themost common approximations are discussed.

Although not discussed in detail in this chapter,we should mention another important area of re-search, namely that of rotational excitation withoutany vibrational or electronic transition. Studies onthis topic began in the early 1970s (see, e.g., Pack(1974) and references therein) and opened up the wayto various approximations, such as the infinite or-der sudden approximation discussed below, which iswidely used in present computations of vibronic exci-tations in ionmolecule collisions.

Quantum Mechanical Approach

After removal of the centre-of-mass motion and ne-glecting the small mass polarisation terms, the non


relativistic Hamiltonian of the A + BC system can bewritten

H = 12

2R 1

2m2

12

N

i=1

2ri +V(r, R, , ), (79)

where the set of electronic coordinates ri is called r, Nis the number of electrons, = BC (see Fig. 7), R = OA,O is the centre of mass of nuclei B and C, and themasses and M are given by = mA(mB + mC)/(mA +mB +mC), m = mBmC/(mB +mC). In general, the elec-tronic motion is described in the so-called body-fixed(BF) reference frame that accompanies the overall ro-tation of the three nuclei about the nuclear centreof mass. In contrast, the nuclear motion is usuallytreated in the laboratory reference frame. In the quan-tum mechanical approach, one solves the Schrodingerequation

H(r, R, ) = E(r, R, ) (80)

by expanding the wave functions in a basis of adia-batic states n

(r, R, ) = nCn(R, )n(r, R, , ). (81)

The n states are eigenfunctions of the electronicHamiltonian

Hel = 12

N

i=1

2ri +V(r, R, , ), (82)

so that

Heln(r, R, , ) = Enn(r, R, , ). (83)

This is similar to the molecular method in ionatomcollisions (see Molecular Orbital Expansion Method).The nuclear wave function can be expanded

Cn(R, ) = v

Fnv(R)Gnv(), (84)

where Gnv is usually written as a product of a vibra-tional function nv and a rotational wave functionYjnmjn of the isolated BC molecule

Gnv() =nv()

Yjn mjn (). (85)

The wave functions n(r, R, , ) and Gnv() formcomplete orthogonal basis sets for the variables r and. Substituting expansions (81) and (84) into theSchrodinger equation (80), and projecting into theGnvn basis leads to the system of coupled equations

Gnvn |HE |GnvnFnv(R)

=

nv

Gnvn|H |GnvnFnv (R). (86)

Solutions of (86) yields for R the scattering am-plitudes that determine the state-to-state cross sec-tions. Even with the nonadiabatic couplings at hand,evaluation of the transition amplitudes using this gen-eral formalism is a formidable task due to the largenumber of terms contained in Eq. (86). Indeed, tosolve Eq. (86), one usually writes the nuclear wave

functions Fnv as a partial wave expansion

Fnv(R) = lm

F lnv(R)R

Ylm(, ), (87)

where the Ylm spherical harmonics depend on the po-lar angles associated with the R vector (see Fig. 7).A way out of this problem appears when (i) the col-lision time is small compared to the rotational periodof the BC molecule and (ii) the radial relative motionis faster than the angular relative motion. In this casethe orientations of both the BC molecule and the Rvector can be considered as fixed in space. Assump-tions (i) and (ii) are called, respectively, the energy-sudden approximation and the centrifugal-sudden ap-proximation. Both approximations are the basis ofthe infinite order sudden (IOS) approximation, whichconsists in replacing all coupled values of j and l bycommon values j and l compatible with the value ofthe total angular momentum. As a consequence, theresulting system of coupled equations is partially di-agonalised, thus leading to a more tractable problem.

Semiclassical Approach

For collisions at higher velocities, the motion of theprojectile can be described classically. In this case,the Schrodinger equation one must solve is formallyidentical to that of Eq. (24) except that now Hel mustbe replaced by the internal Hamiltonian Hint,

Hint = Hel 1

2m2. (88)

Typically, the semiclassical method is valid for colli-sion energies above 50 eV/amu. At these collision en-ergies, the BC rotor remains practically fixed in spaceduring the collision, so that one can use the energy-sudden approximation. This is not the case for thecentrifugal-sudden approximation, which can be verypoor at high collision energies. Therefore, from thetwo conditions for the validity of the IOS approxima-tion, only the first one is fulfilled. In addition, whenthe collision energy is on the order of 100 eV/amu,the collision time is shorter than the vibrational pe-riod of the BC molecule. Consequently, the initial vi-brational wave function does not appreciably changein the time interval in which the electronic transitionstake place. This is called the vibrational-sudden ap-proximation. Thus, the wave function describing thecollisional system can be expressed as

(r, , t) = 1 Yjmj ()nv()(r, , t) (89)

with (r, , t) describing the electronic structure. Inthe context of the semiclassical approach, Eq. (89) issimply called the sudden approximation and shouldnot be confused with the IOS approximation intro-duced in the previous section. As mentioned above,the present sudden approximation is a very goodapproximation in almost the entire energy range in


Figure 7 Set of internal coordinates R, , for an A+BCcollisional system.

which the semiclassical treatment is valid. Further-more, one can use the impact parameter or eikonalmethod in which the position vector R of the incidention with respect to the centre of the diatom is assumedto follow classical straight-line trajectories with con-stant velocity v and impact parameter b: R = b + vt(Errea et al., 1997, 1999). In a close-coupling molec-ular approach the electronic wave function (r, , t)of (89) is expanded in terms of the set of (approxi-mate) eigenfunctions {i} of the BornOppenheimerelectronic Hamiltonian, Hel:

Hel i(r;R, ) = i(R, )i(r;R, ). (90)

For each value of , this expansion takes on the form

(r, , t) = j

aj(t;) j(r, , R) exp(

it

0jdt

)

. (91)

The expansion coefficients aj(t;) are then obtainedby substituting (89) and (91) into the time-dependentSchodinger equation. For a given nuclear trajectoryand fixed , one obtains that

idajdt

= k

ak j |Hel it

r, |k

exp(

it

0(k j)dt

)

. (92)

The aj coefficients are subject to the initial conditionaj(t ;) = ij. The transition probability Pf tothe {f} vibronic state is obtained by projecting thewave function (89) onto the final state, and summingover all rotational exit channels. When for a givenj all values of mj are equally probable, the sum overall initial mj values leads to an orientation-averaged

transition probability of the form

Pf(b, v) = (4)1

d

d () ()

exp(

i

0dt(f Ef)

)

af (;)

2

. (93)

The total cross sections are obtained by integratingthe transition probabilities of Eq. (93) over the im-pact parameter plane. It is generally assumed thatat high collision energies the population of the vi-brational states of the BC molecule subsequent toelectronic transitions obeys the FranckCondon (FC)principle. The FC expression of the transition proba-bilities is obtained if one assumes that the coefficientsaf(;) are slowly varying functions of , and substi-tutes them with the values at the equilibrium distanceof the BC molecule 0:

PFCf (b, v) =

d

af(;0)

2[

d () ()]2

.(94)

Calculations of ionmolecule cross sections in-volve ab initio electronic data, namely potential en-ergy surfaces, which can be accurately calculated us-ing state-of-the-art quantum chemistry programs andcouplings, which constitute the real bottleneck for ap-plications of the theory described above. An illustra-tion of the importance of the various processes thattake place in ionmolecule collisions is shown in Fig.8 for the C4+ + H2 collision. The dominant processis single-charge transfer. However, it can be seen thatdissociative charge transfer becomes competitive withthe nondissociative process at the lower impact ener-gies. In this reaction, dissociation is due to a transi-tion from the original vibrational state to the vibra-tional continuum and not to a direct transition to anelectronic dissociative state. Vibrational excitation isalso a very important process; in fact it is the dom-inant process at the lower energies. Figure 8 alsoshows results obtained within the FC approximation.The latter approximation is independent of the ini-tial vibrational state of H2 and cannot describe vibra-tional excitation.

7. IonCluster Collisions

In the past decade, there has been an increasing inter-est in the study of collisions between ions and clusters,especially metal clusters and fullerenes. The processesthat can take place in this kind of collision are simi-lar to those observed in ionmolecule collisions: theyinvolve simultaneously electronic transitions (chargetransfer, excitation, ionisation) and nuclear-core ex-citations (vibration, rotation, fragmentation). As thenumber of electronic and nuclear degrees of freedomis large, the use of fully quantum mechanical methodsis not yet possible. This complexity also explains why,in contrast with important experimental progress, the


theoretical understanding of these processes is stillmodest. Till the mid-1990s, most theoretical studieswere performed using classical mechanics with phe-nomenological two- or three-body potentials to de-scribe the interatomic forces. These purely classicalmethods are usually called molecular dynamics (MD)methods, for which there is an extensive literature andwell-documented computational packages.

Recently theoretical methods that go beyond MDby including quantum effects have begun to emerge.To date, quantum theoretical methods can be clas-sified into two different categories. In the first, forwhich most applications have been developed, onemakes use of the jellium approximation to describethe isolated cluster. In this approximation one re-places the real ionic distribution of the cluster with ahomogeneous positively charged distribution whoserole is to ensure cluster neutrality. This is a good ap-proximation for metal clusters such as sodium and ce-

Figure 8 Total cross sections for the following processesin C4+ +H2 collisions: TCT (total single-charge transfer), FC(FranckCondon result for single-charge transfer), CTB (chargetransfer to bound vibrational states), VE (vibrational excitation),TD (dissociative charge transfer transfer dissociation), VD(vibrational dissociation). Taken from Errea et al . (1997).

sium clusters. Therefore, in the dynamical treatment,the large number of nuclear degrees of freedom is re-duced to the relative position of the projectile withrespect to the centre of the cluster, R. These meth-ods have provided reasonable descriptions of elec-tronic transitions in ioncluster collisions for clustersof medium and large size. In the second category,all the nuclear degrees of freedom are included. Thishas been recently achieved (Saalmann and Schmidt,1996, 1998) through a method that combines self-consistently an MD description of the nuclear motionand a time-dependent density functional theory of theelectronic degrees of freedom. This is a very promis-ing area of research, but, to date, it has been onlyapplied to the study of very small clusters (typicallyincluding less than 10 atoms).

Jellium Models

The jellium model is a useful approximation when thecollision time is much shorter than the nuclear vibra-tional periods in the cluster. Indeed, in this circum-stance, the cluster nuclei remain fixed in space duringthe collision, so that one can neglect its detailed inter-nal structure. In addition, for short enough collisiontimes one can apply time-dependent theories wherethe projectile motion is treated semiclassically. In thiscase, R is the only nuclear degree of freedom that mustbe considered. Therefore, the semiclassical theoryof ioncluster collisions does not differ significantlyfrom that of ionatom and ionmolecule collisions.However, at variance with the latter cases, it is ex-tremely difficult to obtain a set of correlated electronicwave functions of the ioncluster compound systemto be used in close-coupling expansions of the time-dependent wave function. Even a time-dependentHartreeFock approach is far beyond the present ca-pabilities. For this reason, most methods make useof the independent electron approximation with dif-ferent degrees of sophistication. Among them are firstapproximations based on the Vlasov equations (Grossand Guet, 1995). These equations govern the timeevolution of the single-particle classical phase-spacedistribution f(r, p, t) and are obtained as the classicallimit of the time-dependent HartreeFock equations.They can be written

[t

+pr + (pV[ f ]r (rV[ f ])p] f (r, p, t) = 0, (95)

where V[ f ] is the total self-consistent potential act-ing on an electron of momentum p at position r. Theinitial phase-space distribution is taken as the Wignertransform of a realistic one-body quantal density ma-trix, e.g., the one-electron KohnSham density. Thenumerical integration of the 6-dimensional Vlasovequation is not an easy task. For this reason, sev-eral simplifying strategies must be used, such as theso-called test particle method (see Gross and Guet(1995) for details). The Vlasov equation suffers from


Figure 9 Potential energy curves for the states of the (NanH)+ systems. Full lines, H-correlated states (capture channels);full lines with circles, occupied cluster states; dashed lines, emptycluster states. Taken from MartAn et al . (1999a).

a severe deficiency: it does not take into account thePauli exclusion principle. This problem is solved inpractice by introducing a phenomenological exchangepotential analogous to that used in the local densityapproximation (LDA).

Also in the framework of the independent electronapproximation, a fully quantum mechanical descrip-tion has been recently achieved with explicit inclu-sion of the Pauli exclusion principle. In this method(MartAn et al., 1999a), the KohnSham formulationof density functional theory is used to describe theelectron density of the isolated cluster in terms ofsingle-particle orbitals. Then, from these orbitals,one obtains the corresponding one-electron poten-tials, VC, including exchange, correlation and a self-interaction correction. As a consequence of the qua-siseparability of the cluster Hamiltonian, the totalN-electron Hamiltonian H is then written as a sumof one-electron effective Hamiltonians, H = Ni=1 h(i)and thus one must only solve a set of N one-electrontime-dependent Schrodinger equations. Due to thePauli exclusion principle, the transition probability toa specific final configuration (f1, . . . , fN) = f1 . . .fNis written as a (N N) determinant, Pf1 , . . . , fN =det(nn), where nn is the one-particle density ma-trix, nn = fn | | fn, and is the density operator thataccounts for the time evolution of the spin -orbitals.Figure 9 shows the BO potential energy curves for theone-electron states of the (NanH)+ systems. Asin ionatom collisions, transitions between differentmolecular states are induced by nonadiabatic cou-plings. These couplings are particularly effective in

Figure 10 Electron capture and excitation cross sections forH+ +Nan collisions. Numbers indicate values of n. Taken fromMartAn et al . (1999b).

the vicinity of avoided crossings as those shown inFig. 9. Thus, using these diagrams, one can explainthe different electronic transitions (electron capture,excitation, ionisation, etc.) in terms of transitionsbetween different molecular states. Figure 10 showscapture and excitation cross sections for H+ +Nan col-lisions. The capture cross section is much larger thanthat observed in H+Na collisions, but the flat regionis reached much earlier in the present case. It is worthnoting that, at variance with ionatom collisions, ex-citation cross sections are only 23 times smaller thanelectron capture cross sections. Also, multiple elec-tronic transitions (multiple excitation, ionisation orcapture) are much more important, which preventsthe use of single-particle models.

More recently, some attempts have also been madein the framework of the time-dependent local densityapproximation. In this approximation, the electronicmotion is described by the time-dependent KohnSham equation in which VC is now explicitly time de-pendent to take into account the response of the clus-ter electron density to the presence of the projectile.Although this is not strictly an independent electronmodel, it does not differ too much from it. Indeed,in practice, the time-dependent correlation potentialis not known and approximate LDA forms must beused; since most of these forms are only justified fortime-independent situations, their validity relies onthe minor role played by correlation effects duringthe collision. Finally, it is worth mentioning a recenttheoretical study of Penning detachment from nega-tively charged clusters (MartAn et al., 1999b) usingthe jellium approximation. The theoretical methodis a combination of the semiclassical approach dis-cussed above and the local approximation presentedin section 5.

Beyond the Jellium Approximation

Recently, a nonadiabatic quantum molecular dy-namics method that describes classically the mo-tion of all nuclei (not only the projectile), simulta-


neously and self-consistently with electronic transi-tions, has been developed (Saalmann and Schmidt,1996, 1999). This approach has been derived on thebasis of the time-dependent density functional the-ory, whereby the KohnSham formalism within thetime-dependent local density approximation is used.As an ansatz using linear combination of atomic or-bitals for the KohnSham orbitals a set of coupleddifferential equations for the time-dependent coeffi-cients, which are used to determine the time evolu-tion of the electronic density as the consequence ofthe classical atomic motion, is obtained. Simulta-neously, Newtons equations of motion with explicittime-dependent forces must be solved reflecting thepossible energy transfer between the classical systemof ionic cores and the quantum mechanical system ofvalence electrons. Thus, the nonadiabatic quantummolecular dynamics approach can be used to studyboth charge transfer and fragmentation processes.

References

Bernstein, R. B. 1979, AtomMolecule Collision Theory.Plenum, New York.

Bransden, B. H. and McDowell, M. R. C. 1992, Charge Ex-change and the Theory of Ion-Atom Collisions. Claren-don, Oxford.

Delos, J. B. 1981, Rev. Mod. Phys. 53, 287.

Eichler, J. and Meyerhof, W. E. 1995, Relativistic AtomicCollisions. Academic Press, San Diego.

Errea, L. F., Gorfinkiel, J. D., MacAas, A., Mendez, L. andRiera, A. 1997, J. Phys. B: At. Mol. Opt. Phys. 30, 3855.

Errea, L. F., Gorfinkiel, J. D., MacAas, A., Mendez, L. andRiera, A. 1999, J. Phys. B. At. Mol. Opt. Phys. 32, 1705.

Fritsch, W. and Lin, C. D. 1991, Phys. Rept. 201, 1.

Gross, M. and Guet, C. 1995, Z. Phys. D 33, 289.

Igarashi, A. and Lin, C. D. 1999, Phys. Rev. Lett. 83, 4041.

Kimura, M. and Lane, N. F. 1990, Adv. At. Mol. Opt. Phys.26, 79.

Kleyn, A. W., Los, J. and Gislason, E. A. 1982, Phys. Rep.90, 1.

Lin, C. D. 1995, Phys. Rept. 257, 1.

MartAn, F. and Berry, R. S. 1997, Phys. Rev. A 55, 1099.

MartAn, F. and Salin, A. 1995, J. Phys. B 28, 671.

MartAn, F. and Salin, A. 1996 Phys. Rev. A 54, 3990.

MartAn, F., Politis, M. F., Zarour, B., Hervieux, P. A.,Hanssen, J. and Madjet, M. E. 1999a, Phys. Rev. A 60,4701.

MartAn, F., Madjet, M. E., Hervieux, P. A., Hanssen, J., Poli-tis, M. F. and Berry, R. S. 1999b, J. Chem. Phys. 111,8934.

Massey, H. S. W. and Smith, R. A. 1933, Proc. R. Soc. A142, 142.

Miller, W. H. 1970, J. Chem. Phys. 32, 3563.

Pack, R. T. 1974, J. Chem. Phys. 60, 633.

Saalmann, U. and Schmidt, R. 1996, Z. Phys. D 38, 153.

Saalmann, U. and Schmidt, R. 1998, Phys. Rev. Lett. 80,3213.

Sidis, V. 1990, Adv. At. Mol. Opt. Phys. 26, 161.

Sidky, E. and Lin, C. D. 1998, J. Phys. B 31, 2949.

Siska, P. E. 1993, Rev. Mod. Phys. 65, 337.

Solovev, E. A. 1990, Phys. Rev. A 42, 1331.

Stolterfoht, N., Dubois, R.D. and Rivarola, R. D. 1997,Electron Emission in Heavy IonAtom Collisions.Spinger-Verlag, Berlin.

Tolstikhin, O. I., Watanabe, S. and Matsuzawa, M. 1996,J. Phys. B 29, L389.

Tribedi, L. C., Richard, P., Ling, D., Wang, Y. D., Lin, C. D.,Moshammer, R., Kerby, G. W., Gealy, M. W. and Rudd,M. E. 1996, Phys. Rev. A 54, 2154.

Tseng, H. C. and Lin, C. D. 1999, J. Phys. B 32, 5271.

Weiner, J., Masnous-Seeuws, F. and Giusti-Suzor, A. 1990,Adv. At. Mol. Opt. Phys. 26, 209.

Wells, J. C., Schultz, D. R., Gavras, P. and Pindzola, M. S.1996, Phys. Rev. A 54, 593.

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Chapter 2.6.2 FAST AND SLOW COLLISIONS OF IONS, ATOMS