Trapping, Reflection, and Fragmentation in a Classical Model

PHYSICAL REVIEW E, VOLUME 65, 026603

Trapping, reﬂection, and fragmentation in a classical model of atom-lattice collisions

Alexander V. Plyukhin and Jeremy Schoﬁeld Chemical Physics Theory Group, Department of Chemistry, University of Toronto, Toronto, Ontario, Canada M5S 3H6 ͑Received 18 September 2001; published 17 January 2002͒ A classical one-dimensional model of the collision of an atom of mass M with a cold, semi-inﬁnite harmonic lattice comprised of identical atoms of mass m is considered. In the model, the interactions between the incident atom ͑adatom͒ and the lattice are described in terms of a truncated parabolic potential by which the adatom is harmonically bound to the lattice at short distances but evolves freely when its distance is larger than

a critical length Rc . The dynamics of the adatom colliding with an infinitely cold lattice is studied as a function of the initial velocity of the adatom. In order to determine whether the colliding atom is bound or reflected from the lattice in the asymptotic time limit, ‘‘secondary’’ collision events in which the incident atom leaves and reenters the interaction zone of the lattice are carefully considered. It is demonstrated that secondary collisions anticipated to be important for heavy adatoms (␮ϭm/MϽ1) also occur in the case of light adatoms (␮у1). It is shown that the neglect of secondary collisions leads to an underestimation of the lower energy bound for adatom reflection of roughly 10% for ␮ close to 1. By generalizing the model to allow for the breaking of lattice bonds, the phenomenon of collision-induced lattice fragmentation is investigated.

DOI: 10.1103/PhysRevE.65.026603 PACS number͑s͒: 45.10.Ϫb, 45.50.Tn

I. INTRODUCTION model, Rc is defined to be the initial length of the first link ϭ ϩ Rc a u1(0), so that the adatom is just entering the inter- The collision of an atom with polyatomic complexes or action zone at tϭ0. The interaction potential for the adatom solid surfaces is an important and complicated process, with the lattice can, therefore, be written in terms of the which generally must be described within a quantum- relative displacements of the two atoms as mechanical framework. Nevertheless, valuable insight into 1 2 Ͻ ͑ ͒ the mechanism of energy transfer in such processes can be 2 ku1 if u1 u1 0 ͓ ͔ U ϭͭ ͑2͒ gained from simplified classical models 1 . About forty 1 1 ͑ ͒2 otherwise. years ago Cabrera ͓2͔ and Zwanzig ͓3͔ investigated the pos- 2 ku1 0 sibility of the trapping of an incident atom by a cold harmonic lattice using a simple one-dimensional model. The All other pairs of nearest neighbors in the chain interact via Cabrera-Zwanzig model, hereafter referred to as the CZ harmonic potentials with force constant k. For this potential, the evolution of the system occurs in such a way that if the model, consists of a semi-infinite one-dimensional chain of ͑ ͒ classical particles ͑atoms͒ labeled iϭ1,2,3,..., beginning relative displacement u1(te) between the adatom atom 1 ͑ and the terminal atom of the chain ͑atom 2͒ exceeds its initial from the terminal atom at the free end of the chain see Fig. Ͼ 1͒. In the CZ model, the atoms iϭ2,3,... representing the value u1(0) at a time te 0, the lattice exerts no force on the lattice have mass m and are initially frozen in the equilibrium adatom that may then escape. On the other hand, if the initial ϭ 0ϭ Ϫ Ͼ ratio of kinetic and potential energies for the adatom positions, xi(0) xi a(i 1), where a 0 is the equilibrium bond length in the lattice. The corresponding initial Mv2͑0͒ velocities v (0) and displacements q (0)ϭx (0)Ϫx0 of the ␣ϭ 1 ͑ ͒ i i i i 2͑ ͒ 3 lattice atoms are taken to be zero. The terminal atom iϭ1 ku1 0 represents the incident particle ͑adatom͒ colliding with the is less than a certain threshold value ␣ , one finds that cold lattice. At tϭ0, the adatom is assumed to be moving c u (t)Ͻu (0) for all times tϾ0, which implies that the ada- towards the rest of the chain with positive initial velocity 1 1 Ͻ tom remains bound to the lattice. In the Cabrera-Zwanzig v1(0) from a position x1(0) 0 displaced from its equilib- ϭ solution of the CZ model, it was assumed that the condition rium position at the origin by q1(0) x1(0). The relative ϭ Ϫ displacements of the atoms ui qiϩ1 qi are all zero at ͒ϭ ͒ Ͼ ͑ ͒ u1͑te u1͑0 , te 0 4 tϭ0 except for that of the first link of the chain, which is ϭϪ Ͼ initially stretched, u1(0) q1(0) 0. The adatom interacts with the terminal atom of the lattice (iϭ2) via a truncated potential U1 given by

1 ͑ Ϫ Ϫ ͒2 Ϫ Ͻ 2 k x2 x1 a if x2 x1 Rc U ϭͭ ͑1͒ 1 1 ͑ Ϫ ͒2 2 k Rc a otherwise, where k is the force constant and Rc deﬁnes a cutoff length for the truncation of the harmonic potential. In the CZ FIG. 1. A schematic of the Cabrera-Zwanzig model.

guarantees the adatom will escape from the lattice and hence it does not lead to the escape of the adatom since it later be reflected in the asymptotic time limit. Under this assump- reenters the interaction region of the lattice and is trapped. tion, Eq. ͑4͒ can be used to find the critical ratio of kinetic Moreover, even when supplemented with the additional re- Ͻ and potential energies ␣ for reflection. The dynamics of the quirement that v1(te) 0 at the moment of rupture, the dy- c ͑ ͒ adatom displacements u1(t) depends on the ratio of the lat- namics described Eq. 4 may still correspond to a virtual tice and adatom masses ␮ϭm/M. For two special cases, break of the adatom-lattice bond since the adatom may move corresponding to ␮ϭ1 and ␮ϭ2, it is possible to obtain an too slowly after the bond is ruptured to escape the lattice. It will be shown that such an eventuality arises, in particular, analytical closed-form solution for u1(t) in terms of tabu- lated functions. Using Eq. ͑4͒ as the condition to determine for the cases ␮ϭ1 and ␮ϭ2 considered by Zwanzig. For ␣ ͓ ͔ ␣ ␮ϭ ␣ the threshold for reflection c , Zwanzig obtained 3 c 1, we demonstrate that the actual value c is approxi- ␣ Ϸ24.54 for the mass ratio ␮ϭ1, and a value about ten times mately 27.33, which is about 10% larger than the value c* Ϸ ␣ Ͻ␣Ͻ␣ smaller for the mass ratio ␮ϭ2. 24.54 obtained by Zwanzig. For c* c , the adatom While the CZ model is clearly too simplified for a realistic moves out of the interaction zone after the first collision with description of the collision of an atom with a surface, the the lattice for a period of time until the vibrating lattice re- model demonstrates the importance of multiphonon excita- captures the adatom. There is only one such secondary col- tions for gas-surface processes. Besides providing a qualita- lision, after which the distance between the adatom and the tively correct description of the adatom trapping, the model outermost lattice atom never reaches the critical value Rc . also predicts ͓3,4͔ a much higher transfer of energy from the In the following section we consider the case ␮ϭ1in light adatom to the lattice than what one would expect from detail, while in Sec. III, the extension of the analysis of the the interaction of the adatom with a single surface atom. reflection dynamics is extended to arbitrary values of ␮.In Although the energy transfer is overestimated in the one- Sec. IV, we discuss the behavior of the chain in which not dimensional CZ model, the enhanced energy transfer is in only the first, but all pairs of nearest neighbors interact via agreement with both experiment and more sophisticated truncated parabolic potential. This model leads to the more three-dimensional models ͓1͔. For a finite chain and M complicated scenario of atom-lattice collisions than in the →ϱ, the model is relevant to scattering of molecular clusters original CZ model and can be used to study collision- from a hard surface, which is the topic of the extensive cur- induced fragmentation. rent research ͓5͔. In this context, the model has been considered recently in Ref. ͓6͔. II. THE CASE µÄ1 Since the early work of Cabrera and Zwanzig, the CZ model has been revisited many times and generalized in a The solution of the equations of motion for a harmonic multitude of ways to describe three-dimensional ͓1͔, anhar- chain can be written as linear combinations of initial dis- monic ͓4͔, finite ͓7͔, and noncold lattices ͓8͔. In addition, placements and velocities with time-dependent coefficients more realistic potentials ͓1͔ for the adatom-lattice interac- expressed in terms of the Bessel functions Jn(t). For a semi- tions have been considered, and mass ratios ␮у1 have been infinite chain of identical atoms, the displacements and ve- investigated by McCarroll and Ehrlich using an infinite se- locities are given by ries expansion of Bessel functions ͓4͔. The behavior of the ϱ system for other mass ratios has also been discussed qualita- t q ͑t͒ϭ ͚ ͭ q ͑0͒W ͑t͒ϩv ͑0͒ ͵ dtЈW ͑tЈ͒ͮ , ͑5͒ i k ik k ik tively by Zwanzig who noted that if the adatom is heavier kϭ1 0 than the chain atoms (␮Ͻ1), the criterion for trapping ob- ϱ tained using Eq. ͑4͒ may not be appropriate since the lattice d ͑ ͒ϭ ͭ ͑ ͒ ͑ ͒ϩ ͑ ͒ ͑ ͒ͮ ͑ ͒ vibrations induced by the adatom collision may lead to recol- vi t ͚ vk 0 Wik t qk 0 Wik t . 6 ϭ dt lision and recapture events ͓3͔. k 1 The main goal of this paper is to show that actually Eq. where ͑4͒ cannot be used alone to determine the conditions for adatom reflection even when the mass of the adatom is equal to ͒ϭ ␻ ͒ϩ ␻ ͒ ͑ ͒ Wik͑t J2͉iϪk͉͑2 t J2͑iϩkϪ1͒͑2 t , 7 or less than the mass of lattice atoms, ␮у1. The inapplica- bility of Eq. ͑4͒ to unambiguously define an escape condition and ␻2ϭk/m. As this solution is less familiar than that for can readily be seen from the fact that it does not involve the the infinite chain or for the chain with both ends fixed, we adatom velocity v1 . In the Zwanzig solution of the model, it outline its derivation in the Appendix. Note that these equa- is implicitly assumed that the adatom velocity is directed tions hold for atoms located a finite distance from the free away ͑i.e., it has negative velocity͒ from the lattice at the end of the chain. moment of the break of the first link. One may anticipate that Throughout this paper, we use dimensionless displace- ␰ ␨ generally this assumption breaks down when the rupture of ments i and velocities i , which are normalized by the ini- the link occurs by a mechanism in which the velocities of the tial coordinates of the adatom adatom ͑atom 1͒ and the terminal atom of the lattice ͑atom 2͒ Ͼ ␰ ͑ ͒ϭ ͑ ͒ ͉ ͑ ͉͒ ␨ ͑ ͒ϭ ͑ ͒ ͑ ͒ ͑ ͒ are both positive and v2 v1 just before the break. We will i t qi t / q1 0 , i t vi t /v1 0 . 8 show that such a mechanism is important for large ␮. Since ␶ϭ ␻ ␰ ␶ ␨ ␶ the velocity of the adatom after leaving the interaction zone As functions of reduced time 2 t, i( ) and i( ) are ␰˙ ␶ ϭ␤␨ ␶ ␰˙ ␶ remains constant, such a break will be virtual in a sense that related by the equation i( ) i( ), where i( ) denotes

026603-2 TRAPPING, REFLECTION, AND FRAGMENTATION IN . . . PHYSICAL REVIEW E 65 026603

␰ ␶ ␤ the time derivative of i( ) and the parameter is related to This solution of the dynamics of the system remains valid the initial ratio ␣ of the kinetic and the potential energies for until the relative displacement the adatom defined in Eq. ͑3͒,by ␸ ␶͒ϭ␰ ␶͒Ϫ␰ ␶͒ ͑ ͒ 1͑ 2͑ 1͑ 18 ͒ 1 v1͑0 ␤ϭ ϭ 1 ͱ␮␣. ͑9͒ for the first link of the chain is greater than its initial value of 2␻ ͉q ͑0͉͒ 2 1 1 at which point the link breaks. ͑ ͒ It follows from Eqs. ͑5͒ and ͑6͒ that Using Eq. 16 , one finds that the first link breaks at a ␶ Ͼ ͓ ␸ ␶ ϭ ͔ ␤ time 1 0 that is, 1( 1) 1 provided the parameter ϱ ␶ ␤ Ϸ ͑ ͒ exceeds its critical value c* 2.4768. According to Eq. 9 , ␰ ͑␶͒ϭ ͚ ͭ ␰ ͑0͒W ͑␶͒ϩ␤␨ ͑0͒ ͵ d␶ЈW ͑␶Ј͒ͮ , ␣ Ϸ i k ik k ik ϭ this value corresponds to c* 24.54, in agreement with k 1 0 ͓ ͔ ␶Ͼ␶ ͑10͒ Zwanzig’s result 3 . For reduced times 1 after the break of the first link, the adatom no longer interacts with the lat- ϱ d tice and evolves freely according to ␨ ͑␶͒ϭ ͭ ␨ ͑ ͒ ͑␶͒ϩ␤Ϫ1␰ ͑ ͒ ͑␶͒ͮ i ͚ k 0 Wik k 0 Wik . ϭ ␶ ␨ ␶͒ϭ␨ ␶ ͒ϭ ͑ ͒ k 1 d 1͑ 1͑ 1 const 19 ͑11͒ ␰ ͑␶͒ϭ␰ ͑␶ ͒ϩ␤␨ ͑␶ ͒͑␶Ϫ␶ ͒. ͑20͒ ␶ 1 1 1 1 1 1 The derivative and the integral of the function Wik( ) appearing in the above equations can be expressed in terms of The evolution of the lattice atoms after the rupture, on the sums of Bessel functions noting that other hand, is given by ϱ ␶ ϱ ͵ d␶ЈW ͑␶Ј͒ϭ2͚ J ͉ Ϫ ͉ϩ ϩ ͑␶͒ ␨ ͑␶͒ϭ ͩ ␨ ͑␶ ͒ ͑⌬ ͒ ik 2 i k 2l 1 iϩ1 ͚ kϩ1 1 Wik 1 0 lϭ0 kϭ1 ϱ d ϩ ͑␶͒ ͑ ͒ ϩ␤Ϫ1␰ ͑␶ ͒ ͑⌬ ͒ͪ ͑ ͒ 2͚ J2͑iϩkϩl͒Ϫ1 , 12 kϩ1 1 ⌬ Wik 1 , 21 lϭ0 d 1

ϱ dWik 1 ϭ ͕ ͑␶͒Ϫ ͑␶͖͒ ␰ ϩ ͑␶͒ϭ ͚ ͩ ␰ ϩ ͑␶ ͒W ͑⌬ ͒ 2 J2͉iϪk͉Ϫ1 J2͉iϪk͉Ϫ1 i 1 k 1 1 ik 1 d␶ kϭ1 ϩ 1 ͕ ͑␶͒Ϫ ͑␶͖͒ ͑ ͒ ⌬ 2 J2͑iϩk͒Ϫ3 J2͑iϩk͒Ϫ1 . 13 1 ϩ␤␨ ϩ ͑␶ ͒ ͵ d⌬W ͑⌬͒ͪ , ͑22͒ k 1 1 ik In practical calculations, since the Bessel functions of large 0 index make increasingly small contributions to the sum in ϭ ⌬ ϭ␶Ϫ␶ ␰ ␶ ␨ ␶ ͑ ͒ where i 1,2,..., 1 1 , and i( 1), i( 1) are given Eq. 12 , the sum may be truncated at an index Nw deter- ͑ ͒ ͑ ͒ ␶ϭ␶ by the functions 16 and 17 at 1 . However, if one mined by the desired level of numerical accuracy. ␸ ␶ ␸ ␶ tracks the evolution of 1( ), one observes that 1( ) ﬁrst The initial conditions for the CZ model described in the ␸ ␶ increases then decreases to its initial value 1( 2) at a re- Introduction are ␶ ͑ ͒ ␤ duced time 2 see curves 1–3 in Fig. 2 provided the q ͑0͒ϭq ͑0͒␦ , v ͑0͒ϭv ͑0͒␦ , ͑14͒ parameter, determined by the initial kinetic energy of the i 1 i1 i 1 i1 ␤ Ͻ␤Ͻ␤ ␤ Ϸ adatom, falls in the interval c* c , with c 2.617. At Ͻ Ͼ ␶ with q1(0) 0 and v1(0) 0. The lattice with which the ada- time 2 , the adatom reenters the region of interaction with tom ͑atom 1͒ collides is assumed to lie on the right half of the lattice and is once again bound by the chain. The dynam- the x axis, and the initial conditions correspond to a situation ics of the chain after the adatom reenters the region of inter- ␶Ͼ␶ in which the adatom collides with the lattice from the left. In action with the lattice ( 2), is described by the dimensionless coordinates, the initial conditions assume ϱ the form d ␨ ͑␶͒ϭ ͩ ␨ ͑␶ ͒ϩ␤Ϫ1␰ ͑␶ ͒ ͪ ͑⌬ ͒ ͑ ͒ i ͚ k 2 k 2 ⌬ Wik 2 , 23 ␰ ͒ϭϪ␦ ␨ ͒ϭ␦ ͑ ͒ kϭ1 d 2 k͑0 1k , k͑0 1k , 15 ϱ ⌬ and hence from Eqs. ͑10͒ and ͑11͒, one obtains 2 ␰ ͑␶͒ϭ ͚ ͩ ␰ ͑␶ ͒W ͑⌬ ͒ϩ␤␨ ͑␶ ͒ ͵ d⌬W ͑⌬͒ͪ , i k 2 ik 2 k 2 ik ϭ ␶ k 1 0 ␰ ͑␶͒ϭϪ ͑␶͒Ϫ ͑␶͒ϩ␤ ͵ ␶͓ ͑␶Ј͒ ͑24͒ i J2͑iϪ1͒ J2i d J2͑iϪ1͒ 0 ⌬ ϭ␶Ϫ␶ ␨ ␶ ␰ ␶ ͑ ͒ where 2 2 , and i( 2), i( 2) are the functions 21 ϩ ͑␶Ј͔͒ ͑ ͒ ͑ ͒ ␶ϭ␶ ͑ ͒ ͑ ͒ J2i , 16 and 22 at 2 . Equations 23 and 24 give a relative ␸ ␶ Ͻ ␶Ͼ␶ displacement 1( ) 1 for all 2 , which implies the ada- 1 tom leaves the interaction zone only once. Rupture events of ␨ ͑␶͒ϭJ Ϫ ͒͑␶͒ϩJ ͑␶͒Ϫ ͓J Ϫ ͑␶͒ϪJ ϩ ͑␶͔͒. i 2͑i 1 2i 2␤ 2i 3 2i 1 the adatom-lattice bond leading to the reﬂection of the ada- ␤Ͼ␤ ͑17͒ tom in the asymptotic time limit occur for c , where the

026603-3 ALEXANDER V. PLYUKHIN AND JEREMY SCHOFIELD PHYSICAL REVIEW E 65 026603

␻2ϭ ␶ where 1 k/M, the memory function K( ) is given by ␶ ϭ ␻ K( ) 2J2(2 t)/t, and the ‘‘stochastic’’ force F1(t)isa linear function of initial coordinates and momenta of the lattice atoms (iϭ2,3,...).Theequations for the lattice atoms are given by ͒ϭϪ␻2 ͒Ϫ␻2 Ϫ ͒ q¨ i͑t qi͑t ͑qi qiϪ1

t ϩ␻2 ͵ Ј ͑ Ϫ Ј͒ ͑ Ј͒ϩ ͑ ͒ dt K t t qi t Fi t , 0

where the force Fi(t) is a linear function of initial coordinates and momenta of atoms iϩ1, iϩ2,... Forthe initial conditions of the CZ model ͓see Eq. ͑14͔͒, the ﬂuctuating ␶ forces Fi vanish. Then introducing the reduced time ϭ2␻t, the above equations can be written as

␮ ␮ ␶ q¨ ͑␶͒ϭϪ q ͑␶͒ϩ ͵ d␶ЈM͑␶Ϫ␶Ј͒q ͑␶Ј͒, ͑25͒ ␸ ϭ␰ Ϫ␰ 1 1 1 FIG. 2. Relative displacement 1 2 1 between the adatom 4 4 0 and the outermost lattice atom after the adatom leaves the interac- ␸ Ͻ tion zone 1 1. Curves 1, 2, and 3 correspond to situations where for the adatom, and the adatom is recaptured, while curves 4 and 5 correspond to adatom reflection. 1 1 1 ␶ ͑␶͒ϭϪ ͑␶͒ϩ ϩ ͵ ␶Ј ͑␶Ϫ␶Ј͒ ͑␶Ј͒ q¨ i qi qiϪ1 d M qi , 2 4 4 0 kinetic energy of the adatom is still large enough after one ͑ ͒ oscillation of the chain that the lattice is unable to recapture 26 ͑ ͒ the adatom see curves 4 and 5 in Fig. 2 . Thus, one can for the lattice atoms, iϭ2,3,... Here and below, the nota- conclude that the adatom is reflected by the lattice only for tion B˙ (␶) and B¨ (␶) denote the first and second derivatives of ␤Ͼ␤ , which gives a critical value ␣ Ϸ27.395, roughly c c an arbitrary variable B(␶) with respect to ␶. The memory 10% larger than the value ␣*Ϸ24.54 predicted by Zwanzig c function M(␶) in these equations takes the form ͓3͔. ␤Ͼ␤ If c and the adatom has left the scene of interaction, 2 ϭ ͑␶͒ϭ ͑␶͒ϭ 1 ͓ ͑␶͒ϩ ͑␶͔͒ ͑ ͒ the outermost lattice atom (i 2) becomes the terminal atom M ␶ J2 2 J1 J3 , 27 for the rest of the chain. If one now assumes that at this stage atom 2 begins to interact with atom 3 via a truncated para- which has the Laplace transform, M˜ (s)ϭs2Ϫsͱs2ϩ1 bolic potential, the new terminal atom can escape the lattice ϩ1/2. provided it gains enough energy from the collision of the Using the method of Laplace transformation, one can ob- adatom with the lattice. For this model, we find that the first tain the solution of Eqs. ͑25͒ and ͑26͒ in the convenient link of the remaining chain, between atoms 2 and 3, experi- iterative form ences a ‘‘virtual’’ break for a critical value of ␤Ϸ3.94, while bond-breaking events leading to the final escape of atom 2 ␶ occurs when ␤Ϸ4.35. This process can be continued to ex- q ͑␶͒ϭA͑␶͒q ͑0͒ϩͭ ͵ A͑␶Ј͒d␶Јͮ q˙ ͑0͒, ͑28͒ 1 1 1 amine how energetic adatom-lattice collisions lead to se- 0 quential fragmentation of the lattice. In Sec. IV, we examine ␶ a slightly different model of surface fragmentation in which ͑␶͒ϭ ͵ ͑␶Ϫ␶Ј͒ ͑␶Ј͒ ␶Ј у ͑ ͒ qi M qiϪ1 d , i 2, 29 all lattice atoms interact via a truncated parabolic potential, 0 thus allowing bond-breaking events to occur between any two lattice atoms at all times. where the function A(␶) has the Laplace transform

III. ARBITRARY MASS RATIOS 2 ˜A͑s͒ϭ . ͑30͒ ͱ 2 To generalize the previous discussion of adatom-lattice ͑2Ϫ␮͒sϩ␮ s ϩ1 collision dynamics to an arbitrary mass ratio ␮ϭm/M,itis ͑ ͒ ͑ ͒ instructive to consider the set of integral-differential equa- Equations 28 and 29 can be expressed in terms of the tions for atom positions, which have the form of a general- dimensionless displacements and velocities as ͓ ͔ ized Langevin equation 9,10 . For the adatom, the equation ␶ has the form ␰ ͑␶͒ϭϪ ͑␶͒ϩ␤ ͵ ͑␶Ј͒ ␶Ј ͑ ͒ 1 A A d , 31 0 t q¨ ͑t͒ϭϪ␻2q ͑t͒ϩ␻2 ͵ dtЈK͑tϪtЈ͒q ͑tЈ͒ϩF ͑t͒, 1 1 1 1 1 1 ␨ ␶͒ϭ ␶͒Ϫ␤Ϫ1 ˙ ␶͒ ͑ ͒ 0 1͑ A͑ A͑ , 32

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for the adatom, and

␶ ␰ ͑␶͒ϭ ͵ ͑␶Ϫ␶Ј͒␰ ͑␶Ј͒ ␶Ј ͑ ͒ i M iϪ1 d , 33 0

␶ 1 ␨ ͑␶͒ϭ ͵ ͑␶Ϫ␶Ј͒␨ ͑␶Ј͒ ␶ЈϪ ␦ ͑␶͒ ͑ ͒ i M iϪ1 d i,2M 34 0 ␤

for the lattice atoms, iу2. There are two cases when the inverse transform of Ã(s) in Eq. ͑30͒ has a closed form. For ␮ϭ1, one finds that ␶ ϭ ␶ ␶ϭ ␶ ϩ ␶ ͑ ͒ ͑ ͒ A( ) 2J1( )/ J0( ) J2( ) and Eqs. 28 and 29 co- incide with the general solution obtained previously in Eqs. ͑5͒ and ͑6͒ for the homogeneous chain. The second simple case is when ␮ϭ2, which yields the concise result A(␶) ϭ ␶ J0( ). To obtain A(␶) for mass ratios ␮ other than 1 and 2, one notes that Ã(s) is a two-valued function with branch points ␸ ϭ␰ Ϫ␰ ␸ FIG. 3. Relative displacements i iϩ1 i of the first ( 1), at sϭϮi. The corresponding Riemann surface, consisting of ␸ ␸ ␤ϭ third ( 3), and fifth ( 5) links of the chain for 1.6941. Positive two sheets, can be constructed making a cut along the imagi- ␸ and negative values of i correspond to stretching and compressing nary axis between the branch points. To obtain the inverse of the link, respectively. Laplace transform of Ã(s), one must choose the first sheet on which the function ͱs2ϩ1 is positive when s is real and increasingly small with large ␮, it is numerically difficult to positive since integration on the second sheet leads to a func- detect recapture events for very light adatoms. For very light ␤ Ϸ␤ tion A(␶) that behaves unphysically either in the limit ␶ adatoms, c* c , and hence the original CZ criterion to ϭ0, or when ␶→ϱ͑or in both limits͒.If␮Ͻ2, the only solve for the critical value of ␤ for adatom reflection is ap- singular points of the function Ã(s) on the first sheet are at proximately correct. On the other hand, for a relatively heavy ␮Ͻ the branch points Ϯi. The integration path of the inverse adatom ( 5), the situation is qualitatively similar to that Laplace transform for this case can be transformed into a described in the preceding section for ␮ϭ1: At the moment closed curve around the cut. For this path, one obtains the of rupture, the adatom is moving away from the lattice for ␤ϭ␤ following integral representation for A(␶) for ␮Ͻ2, c* with a small velocity and is subsequently recaptured by the vibrating lattice. 1 ␮ 1 ͱ1Ϫy 2 ͑␶͒ϭ ͵ ͑ ␶͒ ͑ ͒ A dy cos y 2 , 35 ␲ ␮Ϫ1 0 ␥Ϫy IV. GENERALIZED MODEL One rather unexpected feature of the semi-infinite, homo- where ␥ϭ(␮2/4)/(␮Ϫ1). If ␮Ͼ2, on the other hand, there geneous chain dynamics after the adatom collision is that the are also contributions to A(␶) from two simple poles lying maximal stretching of the first link of the chain may be less ϭϮ ͱ␥ on the imaginary axis at s i , and one gets than that for the subsequent links ͑see Fig. 3͒. Due to this surprising observation, first noted in Ref. ͓4͔, it is interesting 1 ␮ 1 ͱ1Ϫy 2 ␮Ϫ2 ͑␶͒ϭ ͵ ͑ ␶͒ ϩ ͑ͱ␥␶͒ to consider a generalization of the CZ model in which all A dy cos y 2 cos . ␲ ␮Ϫ1 0 ␥Ϫy ␮Ϫ1 nearest neighbor atoms in the lattice interact via the same ͑36͒ truncated potential. Since the maximal bond stretching induced by the adatom collision with the lattice does not occur Equations ͑31͒–͑36͒ give a complete description of the at the first link, one may anticipate that the result of an ada- system dynamics until the break of the first link of the chain tom collision in the generalized CZ ͑GCZ͒ model will not be ␶ at time 1 . After the chain breaks, the dynamics is described a reflection but rather a fragmentation of the chain. Restrict- by Eqs. ͑19͒–͑22͒ that, in turn, hold until the adatom is re- ing ourself to the case Mϭm, we demonstrate in this section captured by the lattice. As in the special case ␮ϭ1, one finds that the rupture events in the inner chain are virtual for low- ␤ Þ␤ ␮ ␤ ␤ that c* c for any mass ratio , where c* and c denote energy adatom collisions, yielding short-lived chain frag- the critical values of the parameter ␤ at which the first link ments that are recaptured by the lattice. As a result of the breaks and the adatom is asymptotically reflected, respec- recapture events, low-energy adatom-lattice collisions result ␮Ͼ ␤ϭ␤ tively. For a light adatom with 5at c* , the adatom in either trapping or reflection of the adatom without frag- velocity at the moment of bond rupture is positive and hence mentation of the chain. However, as the adatom energy in- directed toward the lattice, implying that the adatom is re- creases, clusters of atoms may escape the lattice in addition captured by the lattice. Such recapture events persist for ␮ to the reflected adatom. values as large as 40, and presumably occur for larger ␮. To explore the possibility of the lattice fragmentation in ␤ ␤ However, since the difference between c* and c becomes the GCZ model, one also needs the general solution of equa-

026603-5 ALEXANDER V. PLYUKHIN AND JEREMY SCHOFIELD PHYSICAL REVIEW E 65 026603 tions of motion for a free cluster in addition to the solutions 5 d ͑ ͒ ͑ ͒ ␨ ͑␶͒ϭ ͩ ␨ ͑␶ ͒ϩ␤Ϫ1␰ ͑␶ ͒ ͪ ͑⌬ ͒ϩ␨ ͑␶ ͒ 5 and 6 for the semi-inﬁnite chain. In the Appendix, it is i ͚ k 1 k 1 ⌬ Vik 1 c 1 , shown that the displacements of an atom in a free cluster kϭ1 d 1 ͑ ͒ obey 43

q ͑t͒ϭq ͑0͒ϩv ͑0͒t ⌬ ϭ␶Ϫ␶ i c c where 1 1 , and the time dependence of the atoms of N the remaining lattice is given by t ϩ ͚ ͩ q ͑0͒V ͑t͒ϩv ͑0͒ ͵ dtЈV ͑tЈ͒ͪ , k ik k ik ϭ k 1 0 ϱ

͑37͒ ␰ ϩ ͑␶͒ϭ ͚ ͩ ␰ ϩ ͑␶ ͒W ͑⌬ ͒ i 5 k 5 1 ik 1 kϭ1 where N is the number of atom in a cluster, qc(0) ⌬ ϭ Ϫ1⌺ ϭ ˙ 1 N iqi(0) and vc(0) qc(0) are the initial coordinate ϩ␤␨ ϩ ͑␶ ͒ ͵ d⌬W ͑⌬͒ͪ , ͑44͒ k 5 1 ik and the velocity of the center of mass of the cluster. The 0 ␶ ͑ ͒ function Vik( ) in Eq. 37 is given by ϱ NϪ1 1 ␨ ͑␶͒ϭ ͩ ␨ ͑␶ ͒ ͑⌬ ͒ ͑␶͒ϭ ͓ ͕ ͑ Ϫ ͒ ͖ ͑␶ ͒ iϩ5 ͚ kϩ5 1 Wik 1 Vik ͚ cos 2 i k y j cos sin y j ϭ N jϭ1 k 1 Ϫ ͕ ͑ ϩ Ϫ ͒ ͖ ͑␶ ͔͒ d cos 2 i k 1 y j cos sin y j , ϩ␤Ϫ1␰ ͑␶ ͒ ͑⌬ ͒ͪ ͑ ͒ kϩ5 1 ⌬ Wik 1 , 45 d 1 ␶ϭ ␻␶ ϭ␲ where 2 , and y j j/(2N). In the limit of large clusters, Vik(t) reduces to the corresponding function Wik(t) for the semi-infinite chain given in Eq. ͑7͒. The dimensionless where iϭ1,2,... displacements ␰ and velocities ␨ for atoms in the cluster Equipped with these results, it is straightforward to show i i ␤ Ͻ␤ obey that the bond rupture occurring at the fifth link for 1 Ͻ␤ ͑ ␤ ϭ ͒ 2 where 2 1.71... are virtual in the sense that the N d cluster leaves the interaction region of the lattice for a short ␨ ͑␶͒ϭ ͩ ␨ ͑ ͒ϩ␤Ϫ1␰ ͑ ͒ ͪ ͑␶͒ϩ␨ ͑ ͒ ␶ ␶ i ͚ k 0 k 0 ␶ Vik c 0 , time and is then captured by the lattice at time 2 . For kϭ1 d Ͼ␶ ͑ ͒ 2 , the evolution of the chain is once again described by 38 Eqs. ͑23͒ and ͑24͒, and the lattice is stable. N Similar analysis shows that neither the reflection of the ␶ ␤ ␰ ͑␶͒ϭ ͚ ␰ ͑ ͒ϩ␤␨ ͑ ͒ ͵ ␶ ͑␶͒ϩ␰ ͑ ͒ adatom nor the fragmentation of the lattice occurs for less i ͩ k 0 k 0 d ͪ Vik c 0 ␤ ϭ ␤ kϭ1 0 than the threshold value c 1.86....If is slightly larger than ␤ , the chain first experiences a virtual break at the ϩ␤␶␨ ͑ ͒ ͑ ͒ c c 0 , 39 third link, leading to a three-atom cluster, which is later recaptured by the lattice. At subsequent times, the chain is ␨ ϭ Ϫ1͚ ␨ ␰ ϭ Ϫ1͚ ␰ where c(0) N i i(0) and c(0) N i i(0). unstable and quickly ruptures at the first link. This break is ͑ The initial dynamics of the chain before a break in the not virtual and leads to the final reflection of the adatom. The ͒ chain in the GCZ model is described by the same expres- remaining lattice is then stable for all time and does not ͑ ͒ ͑ ͒ sions 16 and 17 as for the original CZ model. Using these fragment. This behavior is reminiscent of that observed in ␤ equations, one finds that the minimal for the chain rupture the tunneling mechanism of chain breaking ͓11͔ according to ␤ ϭ is 1 1.6941..., which corresponds to the break of the fifth which bond rupture results from two subsequent processes ␶ϭ␶ ϭ link at the reduced time 1 16.89,..., at which time with different time scales, the fast vitual disappearance of one bond followed by the slow collective motion of the ␸ ͑␶ ͒ϭ␰ ͑␶ ͒Ϫ␰ ͑␶ ͒ϭ1, ͑40͒ 5 1 6 1 5 1 chain. ␤ ␸ ͑␶͒Ͻ Þ ␶р␶ ͑ ͒ As increases further, the evolution of the system be- i 1, i 5, 1. 41 comes more complicated and fragmentation of the lattice is possible. To illustrate this point, consider the chronology of At time ␶ , the original chain decomposes into a cluster 1 the chain decay for ␤ϭ2.47 for which the chain first breaks of five atoms and a semi-infinite lattice. For ␶Ͼ␶ , the clus- 1 at the second link. For this value of ␤, the two-atom cluster ter evolves according to Eqs. ͑38͒ and ͑39͒ from the coordi- is not recaptured by the lattice. The remainder of the lattice nates and velocities values ␰ (␶ ) and ␨ (␶ ), i 1 i 1 vibrates after the cluster breaks free for a reduced time inter- 5 val ␦tϷ15, after which the chain experiences a virtual rup- ␰ ͑␶͒ϭ␰ ͑␶ ͒ϩ␤⌬ ␨ ͑␶ ͒ϩ ͚ ͩ ␰ ͑␶ ͒V ͑⌬ ͒ ture at the sixth-link on the remaining chain, forming a six- i c 1 1 c 1 k 1 ik 1 kϭ1 atom cluster, which is quickly recaptured. After the cluster is

⌬ recaptured, the lattice is still unstable and finally fragments 1 ϩ␤␨ ͑␶ ͒ ͵ d⌬V ͑⌬͒ͪ , ͑42͒ at the fifth link, leading to the ‘‘evaporation’’ of a five-atom k 1 ik 0 cluster from the lattice, which is henceforth stable.

026603-6 TRAPPING, REFLECTION, AND FRAGMENTATION IN . . . PHYSICAL REVIEW E 65 026603

V. CONCLUDING REMARKS ACKNOWLEDGMENT In this paper, it has been demonstrated that the Cabrera- This work was supported by a grant from the Natural Zwanzig model of the collision of an atom with a cold lattice Sciences and Engineering Research Council of Canada. has much richer behavior than previously reported. In particular, we have shown that secondary collisions in which the APPENDIX adatom reenters the interaction zone of the lattice after a In this appendix, the solution of the equations of motions virtual break are important over a wide range of mass ratios for a finite free harmonic chain and for a semi-infinite chain ␮. Due to the presence of secondary adatom-lattice collsions, are presented. The Hamiltonian of the finite harmonic chain the adatom reflection threshold energy cannot be computed of N atoms with both ends free based on a single bond-breaking event. It was demonstrated Ϫ that threshold energies calculated in this fashion provide only N p2 N 1 k ϭ i ϩ Ϫ ͒2 a lower bound for the adatom to leave the interaction zone H ͚ ͚ ͑qi qiϩ1 ϭ 2m ϭ 2 and do not guarantee that the adatom will be reflected by the i 1 i 1 lattice in the asymptotic time limit. can be reduced to the diagonal form In addition, collision-induced fragmentation was consid- Ϫ ered by generalizing the CZ model to allow for the breaking 1 N 1 ϭ ͕ 2ϩ␻2 2͖ of all lattice bonds. For the generalized CZ model ͑GCZ͒,it H ͚ P j j Q j , 2 ϭ was shown that the collision of a low energy adatom with a j 0 cold lattice produces short-lived clusters, which are quickly by means of the normal mode transformations recaptured by the vibrating lattice. It was demonstrated that Ϫ Ϫ the first fragmentation event surviving in the long time limit 1 N 1 N 1 ϭ ϭͱ corresponds to the reflection of the adatom in spite of the qi ͚ AijQ j , pi m ͚ AijP j , ͱ ϭ ϭ production of virtual clusters at early times. Although the m j 0 j 0 asymptotic phenomenology of the GCZ model is similar to with that of the CZ model for low energy collisions, the energy threshold for adatom reflection for the GCZ model was ⑀ 1/2 2͑NϪi͒ϩ1 ϭͩ j ͪ ͭ ␲ ͮ found to be of approximately half its value in the simpler Aij cos j , model, demonstrating the importance of the formation of vir- N 2N tual clusters in the energy transfer process. As the energy of ⑀ ϭ where j takes the value of 1 if j 0 and 2 otherwise. The the adatom collision with the lattice increases, the evolution frequencies of the normal modes are given by of the system becomes complicated by the evaporation of clusters, resulting in real fragmentation of the lattice. ␲ j It should be emphasized that the present considerations ␻ ϭ2␻ sinͩ ͪ , j 2N are neither general nor complete. This study focused only on the situation in which the masses of the atoms composing the where ␻ϭͱk/m. The normal modes evolve as lattice are the same and the spring constants appearing in the ͑possibly truncated͒ parabolic potentials are all equal. In ͒ϭ ͒ ␻ ͒Ϫ␻ ͒ ␻ ͒ P j͑t P j͑0 cos͑ jt jQ j͑0 sin͑ jt , principle, it is possible to consider the collision of an adatom Ϫ with nonhomogeneous lattices along the lines elaborated Q ͑t͒ϭQ ͑0͒cos͑␻ t͒ϩ␻ 1P ͑0͒sin͑␻ t͒, here, but the mathematical solution of the equations of mo- j j j j j j tion of the nonhomogeneous system becomes cumbersome. for jÞ0, and Furthermore, one may anticipate that the phenomenology of ͒ϭ ͒ϭ the collisions of an atom with a nonhomogeneous lattice P0͑t P0͑0 const would be quite similar to that reported here. In particular, ͒ϭ ͒ϩ ͒ one expects the mechanism of secondary collisions to in- Q0͑t Q0͑0 P0͑0 t crease the minimum threshold energy for adatom reflection. The present study demonstrates the importance of a care- for the mode jϭ0. The momentum of the atom i can then be ful consideration of the full dynamics of the adatom-lattice written as system after the initial bond-breaking event. The energy NϪ1 transfer from the adatom to the surviving lattice is strongly ͒ϭͱ ͒ ␻ ͒Ϫ␻ ͒ ␻ ͒ pi͑t m ͚ Aij͕P j͑0 cos͑ jt jQ j͑0 sin͑ jt ͖ influenced by the formation of virtual clusters, which are jϭ0 recaptured by the ‘‘surface’’ in secondary collisions. Such ϩͱ behavior is likely to be important in more realistic models of mAi0P0 . the collision process of an atom with a surface, which are based on other potentials of interaction, other initial condi- Since the Aij satisfy orthogonality conditions with respect to tions, such as those appropriate for studying thermal desorp- both indices, one can express P j(0) and Q j(0) in the above tion, or which involve multidimensional lattice structures for equation in terms of pi(0) and qi(0). The resulting equation the surface. has the form

026603-7 ALEXANDER V. PLYUKHIN AND JEREMY SCHOFIELD PHYSICAL REVIEW E 65 026603

N N ϭ␲ 1 d where y j j/(2N). ͑ ͒ϭ ͑ ͒ϩ ͩ ͑ ͒ϩ ͑ ͒ ͪ ͑ ͒ pi t ͚ pk 0 ͚ pk 0 mqk 0 Vik t , The case of the semi-inﬁnite chain is recovered in the N ϭ ϭ dt k 1 k 1 limit N→ϱ by converting the sum to an integral where 2 ␲/2 ͑ ͒ϭ ͵ ͓ ͑ ͉ Ϫ ͉͒ ͑ ␻ ͒ NϪ1 Vik t dy cos 2y i k cos 2 t sin y ␲ 0 ͒ϭ ␻ ͒ Vik͑t ͚ AijAkj cos͑ jt . jϭ1 ϩcos͕2y͑iϩkϪ1͖͒cos͑2␻t sin y͔͒,

After simple trigonometric manipulations, the function which is just the integral representation for the sum of two Vik(t) can be written as Bessel functions Ϫ 1 N 1 ͑ ͒ϭ ͑ ␻ ͒ϩ ͑ ␻ ͒ ͒ϭ Ϫ ͒ ␻ ͒ Vik t J2͉iϪk͉ 2 t J2͑iϩkϪ1͒ 2 t Vik͑t ͚ ͓cos͕2͑i k y j͖cos͑2 t sin y j N jϭ1 The solutions for displacements can be obtained in an analo- Ϫ ϩ Ϫ ͒ ␻ ͒ cos͕2͑i k 1 y j͖cos͑2 t sin y j ͔, gous fashion.

͓1͔ G. D. Billing, Dynamics of Molecule Surface Interactions ͓6͔ A. G. Basile and R. S. Dumont, Phys. Rev. E 61, 2015 ͑2000͒. ͑Wiley, New York, 2000͒; R. I. Masel, Principles of Adsorp- ͓7͔ H. K. Shin, J. Chem. Phys. 87, 3942 ͑1987͒. tion and Reaction on Solid Surface ͑Wiley, New York, 1996͒; ͓8͔ M. Fujinohara and C. H. Kim, J. Phys. Soc. Jpn. 61, 4381 J. Harris, in Dynamics of Gas-Surface Interactions, edited by ͑1992͒. C. T. Rettner and M. N. R. Ashfold ͑Royal Society of Chem- ͓9͔ S. A. Adelman and J. D. Doll, J. Chem. Phys. 61, 4242 ͑1974͒; istry, Cambridge, 1991͒. J. D. Doll, L. E. Myers, and S. A. Adelman, ibid. 63, 4908 ͓ ͔ ͑ ͒ 2 N. B. Cabrera, Discuss. Faraday Soc. 28,16 1959 . ͑1975͒. ͓ ͔ ͑ ͒ 3 R. W. Zwanzig, J. Chem. Phys. 32, 1173 1960 . ͓10͔ A. V. Plyukhin and J. Schoﬁeld, Phys. Rev. E 64, 041103 ͓ ͔ ͑ ͒ 4 B. McCarroll and G. Ehrlich, J. Chem. Phys. 38, 523 1963 . ͑2001͒. ͓ ͔ 5 W. Christen, U. Even, T. Raz, and R. D. Levine, J. Chem. ͓11͔ L. S. Levitov, A. V. Shytov, and A. Yu. Yakovets, Phys. Rev. Phys. 108, 10 262 ͑1998͒; T. Raz and R. D. Levine, ibid. 105, Lett. 75, 370 ͑1995͒; E. B. Kolomeisky and J. P. Straley, Phys. 8097 ͑1996͒; M. Markovic and J. B. C. Petterson, ibid. 100, Rev. B 62, 301 ͑2000͒. 3911 ͑1994͒.

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