Atom scattering off a vibrating surface: An example of chaotic scattering with three degrees of freedom
AAtom scattering off a vibrating surface: Anexample of chaotic scattering with three degreesof freedom
Francisco Gonzalez Montoya a,b , Florentino Borondo c,d , andChristof Jung ba School of Mathematics, University of Bristol, Bristol, BS8 1UG,United Kingdom b Instituto de Ciencias F´ısicas, Universidad Nacional Aut´onoma deM´exico, Av. Universidad s/n, 62251 Cuernavaca, M´exico c Instituto de Ciencias Matem´aticas (ICMAT), Cantoblanco–28049Madrid, Spain. d Departamento de Qu´ımica, Universidad Aut´onoma de Madrid,Cantoblanco–28049 Madrid, Spain.
Abstract
We study the classical chaotic scattering of a He atom off a harmon-ically vibrating Cu surface. The three degree of freedom (3-dof) modelis studied by first considering the non-vibrating 2-dof model for differentvalues of the energy. The set of singularities of the scattering functionsshows the structure of the tangle between the stable and unstable man-ifolds of the fixed point at an infinite distance to the Cu surface in thePoincar´e map. These invariant manifolds of the 2-dof system and theirtangle can be used as a starting point for the construction of the sta-ble and unstable manifolds and their tangle for the 3-dof coupled model.When the surface vibrates, the system has an extra closed degree of free-dom and it is possible to represent the 3-dof tangle as deformation of astack of 2-dof tangles, where the stack parameter is the energy of the 2-dofsystem. Also for the 3-dof system, the resulting invariant manifolds havethe correct dimension to divide the constant total energy manifold. Bythis construction, it is possible to understand the chaotic scattering phe-nomena for the 3-dof system from a geometric point of view. We explainthe connection between the set of singularities of the scattering function,the Jacobian determinant of the scattering function, the relevant invariantmanifolds in the scattering problem, and the cross-section, as well as theirbehavior when the coupling due to the surface vibration is switched on.In particular, we present in detail the relation between the changes as afunction of the energy in the structure of the caustics in the cross-sectionand the changes in the zero level set of the Jacobian determinant of thescattering function.emails: [email protected], [email protected], [email protected] a r X i v : . [ n li n . C D ] M a r Introduction
The topic of this article is a 3-dof model for the inelastic scattering of a particleoffa vibrating surface, a rather complex problem of dynamics. In contrast, theelastic chaotic scattering of an atom by surfaces is a lot simpler and has beenstudied in detail for Hamiltonian systems with 2-dof [1–4]. The 2-dof Hamilto-nian systems are easier to investigate because the corresponding Poincar´e mapacts on a domain of dimension 2and is easily represented graphically. In thesemaps, the stable and unstable manifolds of the most important hyperbolic fixedpoints built up a homoclinic/heteroclinic tangle, which is the central elementof the skeleton of the whole dynamics. The stable and unstable manifolds areof dimension 1 in the domain of the map, i.e., they are also of codimension 1,and as such, they form division lines in the domain which direct the dynamicsof the map. Analogous considerations also hold in the corresponding constantenergy manifold for the flow since codimensions of stable and unstable mani-folds are equal in corresponding maps and flows. For a detailed discussion of theimportance of homoclinic/heteroclinic trajectories, see chapter 3 in [5] and [6].A pictorial illustration of the corresponding tangles is presented in chapters 13and 14 in [7].Also for systems with more dof, it is possible to apply analogous geometricand topological ideas to construct dividing surfaces. They are stable and unsta-ble manifolds of higher dimension belonging to invariant subsets also of higherdimension. But they are still of codimension 1 and they built up the corre-sponding higher dimensional homoclinic/heteroclinic tangles, see the Refs. [8,9].There has been recent progress in 3-dof chaotic scattering problems when theyare close to a symmetric system, or equivalently, close to a partially integrablesystem [10–15]. The essential idea behind this approach is that the principalphase space structures that direct the dynamics in the 3-dof system are quiterobust under perturbations of the system. Then the principal structures survivequalitatively and change only slowly under small perturbations of the symme-try. Accordingly, to investigate the nonsymmetric 3-dof system is convenient tostart with the symmetric 3-dof system, which can be considered a stack of 2-dofsystems, and then turn on smoothly the perturbations, thereby converting thestack of 2-dof systems into the irreducible 3-dof system in which we are reallyinterested.The present article aims to apply this strategy to atom scattering off os-cillating surfaces. As a central part of this work, the changes in the causticsin the doubly differential cross-section when the initial particle energy changesare studied. The model studied here is an extension of a 2-dof model proposedin [1, 2], and it includes the vibration of the surface as an additional oscillatorydof. Thereby it adds a new closed dof that exchanges energy with the particleand accordingly the chaotic scattering becomes inelastic.The important progress of the present work in comparison to previous studiesof 3-dof scattering is a detailed step by step analysis of the changes in thetopology of regions of continuity of the scattering function as function of thechanging initial particle energy and the corresponding changes in the Jacobiandeterminant of the scattering function and the resulting changes of the rainbowsingularities in the doubly differential cross-section.The organization of the paper is as follows. Section 2 is a brief review ofthe basic results of the 2-dof model for the elastic collisions of He atoms with a2tatic corrugated Cu-surface. The construction of the scattering functions of thesystem is presented. Also, the tangle between the stable and unstable manifoldsof the fixed point at an infinite distance to the Cu surface in the Poincar´e map isgiven for different values of the initial energy of the incident particle. In section3, the vibrations of the surface are turn on. In subsection 3.1 the scatteringfunction and their Jacobian determinant are studied. Subsection 3.2 explainsthe construction of the stable and unstable manifolds of the most importantinvariant subset for the 3-dof system, those dominate the inelastic He scatteringoff the vibrating Cu surface. In subsection 3.3 the implications for the doublydifferential cross-section are presented. Section 4 contains the final remarks.
The present section contains a study of the basic properties of the 2-dof versionof the model previously investigated in detail in [1–3]. This study serves aspreparation for the next section, where the 3-dof system is build up as a pile of2-dof system.Let us consider the motion of a He atom coming in from the asymptoticregion, and moving towards the interaction region with a static Cu surface. Hereand in the following, we use the expression interaction region for the region inposition space close to the surface where the interaction between the particleand the surface is important and where a strong deflection of the particle occurs.Let z and x be the spatial coordinates of the atom perpendicular and parallelto the surface, respectively ( the model considers a single tangential coordinateto the surface, i.e. in-plane scattering ). The corresponding 2-dof Hamiltonianis H ( x, z, p x , p z ) = p x m + p z m + V ( x, z ) , (1)where the function V ( x, z ) is a corrugated Morse potential with the functionalform V ( x, z ) = D (1 − e − αz ) + De − αz V c ( x ) ,V c ( x ) = (cid:88) n =1 r n cos 2 nπxa . (2)Moser coefficients D = 6.35 meV , α = 1.05 ˚ A − Fourier coefficient r = 0.03, r = 0.0004Unitary cell a = 3.6 ˚ A These potential parameters have the same numerical value as the ones in [3]for the Cu surface. Note that the potential is periodic in x direction. Thereforeit is possible to treat x as a compact variable restricted to a circle, and in thissense, this dof can be considered a closed one.In a scattering experiment, the particle starts in the asymptotic region asa free particle and moves towards the interaction region, where the particle re-mains close to the surface for some time, and finally moves back to the asymp-totic region where it again moves freely. The possible complicated behavior3f the particle for a finite time and the corresponding sensibility of its trajec-tory to initial conditions is a phenomenon called transient chaos. For detailedexplanations of this phenomenon see [16–19].The scattering functions contain essential information to analyze and com-pare the behavior of different scattering trajectories. These functions map initialasymptotic conditions to final asymptotic quantities, in analogy to the processesin a standard scattering experiment where the initial asymptotes are preparedand the final asymptotes are detected. This natural approach to obtain infor-mation on the essential structures in the phase space by a study of trajectoriesis similar in spirit to other approaches based on trajectories like for exampleLagrangian Descriptors and Fast Lyapunov Indicators, see [20–23].The main idea behind this approach to scattering systems is based on thedistinction between the trajectories running exactly on the invariant manifoldsof localized invariant subsets of the phase space and the general trajectoriesoutside of such particular submanifolds. Generic scattering trajectories comein from the asymptotic region, stay in the interaction region for a finite timeonly and return to the asymptotic region. In contrast, when a trajectory startsexactly on the stable manifold of a localized invariant subset, then it convergesto this subset and does not return to the asymptotic region in a finite time.In this sense, incoming asymptotes belonging to stable manifolds of localizedsubsets lead to singularities of scattering functions.This difference allows to identify intersections between these stable manifoldsand the domain of the scattering functions as the subset of the domain wherethe scattering functions are either infinite or are not well defined. Analogousconsiderations apply to the unstable manifolds by changing the direction of time.Now we work out these considerations in more detail for the present scatteringsystem.As a preparatory step, it is necessary to define the appropriate asymptoticvariables for the scattering system. In the incoming asymptotic region, a conve-nient set of initial conditions of a He atom are an impact parameter b measuredrelative to the x axis and an initial vectorial momentum (cid:126)p forming an angle θ with respect to the z axis. The corresponding initial energy E , which isasymptotically the kinetic energy only is fixed by the given initial momentum.In a usual incoming beam, all particles have the same initial energy E and the same initial angle θ . Equivalently the value of the asymptotic vectormomentum is constant over the beam, whereas the initial impact parameter b has a uniform random distribution over the beam. The impact parameter canbe defined in the following way: Let us imagine the system without potential.Then the entire trajectory with specific initial conditions would be a straightline. This trajectory would intersect the x -axis in a value x s . Since the potentialis periodic in x , with period P = a/
2, it is only necessary to consider x valuesmod P . In this spirit, let us define the impact parameter as b = x s mod P . Inaddition in a usual beam, the time of arrival at the surface is random. As long aswe study the nonvibrating case, this time of arrival is irrelevant. However, it willbecome essential for the vibrating case considered in the next section. Thereforelet us already consider this quantity here carefully. Think of a particle startingwith the z value z and the value p z, of p z at time t i . Then it arrives at z = 0 atthe time t s = t i − mz /p z, , where m is the mass of the particle and rememberthat p z, is negative for an incoming asymptote. Note that t s − t i = − mz /p z, is the time of flight from z = z to z = 0 for a free particle with velocity p z /m .4igure 1: Initial conditions of the scattering trajectories. The dotted line showsthe intersection for a free particle with the x -axis.As consequence we also have x s = x + p x, ( t s − t i ) /m . These asymptoticvariables are illustrated in Fig. 1. It shows as broken line a free incomingtrajectory in the x – z plane which passes the point ( x , z ) at time t i and arrivesat the point( x s ,
0) at time t s . The fat arrow represents the vector momentum (cid:126)p = ( p x, , p z, ) which is conserved along the free trajectory.The most useful scattering function in the present set up is the final scat-tering angle θ f as a function of the initial impact parameter b .The Fig. 2 shows the plot of the final scattering angle θ f as a function ofthe initial impact parameter b i in the upper panel. The initial energy is E = 2which is a typical value in the energy interval for which the system shows chaos,see Fig. 3, and the initial angle is θ = 80 ◦ . Along the b axis there is a fractalof points where the scattering function θ f is not defined. A point with thisproperty is called a singularity of the scattering function.In the neighborhood of a singularity, the scattering function shows an infinityof oscillations and an infinity of further singularities. Also, close to a singularityof the scattering function, i.e. in regions where the value of θ f changes quickly,trajectories spend more time in the interaction region than trajectories startingfar away from the singularities. For a detailed explanation of the scaling behav-ior of the scattering function and the time delay function in the neighborhoodof a singularity, see [24]. Furthermore, the z -component of the final momentumapproaches the value zero when approaching a singularity, equivalently the finalangle of inclination θ f approaches the value π/ θ f inthe b – E plane is presented, again the initial angle is θ = 80 ◦ . To each pointof the set of singularities, there belongs a trajectory that gets trapped foreverin the interaction region, and those trajectories lie on the stable manifold ofsome invariant subset, a “periodic orbit” at infinity in this case. First, notethe large region of continuity reaching up to infinite values of the energy. Inaddition, there is an infinite number of regions of continuity for small valuesof the particle energy E . In the figure, the largest inner region of continuityis marked by a yellow color, this region serves in subsections 3.1 and 3.3 as anexample region of continuity for the case when the perturbation is switched on.The yellow region in Fig. 3 corresponds to the two vertical regions of continuityin Fig. 2, which are located around the b values 0.6 and 0.72, respectively. Inthe following, the yellow region and the corresponding regions of continuity in5igure 2: The scattering functions θ f ( b ) and θ f ( φ , b ) on colour scale for theinitial energy E = 2 and the initial angle θ = 80 ◦ . The value of the angle θ f oscillates as a function of the impact parameter b , and it is constant as afunction of the phase φ associated with the vibration of the surface, which istaken here as static. The initial phase φ represents the phase shift between theparticle motion and the oscillator representing the vibrations, see the section 3.At present, the angle φ does not influence the scattering process; the surfaceis static, and the scattering is elastic. Nevertheless, this plot of the scatteringfunction θ f ( φ , b ) will be useful in the next section to compare the elastic andthe inelastic system. 6igure 3: Fractal set of singularities of the scattering function in the b – E plane,i.e. in the domain of this function for the 2-dof system. We can see the regions ofcontinuity, which are the gaps of the fractal. For every value of the initial energy E , the singularities reflect the stable manifolds W sE of the chaotic invariantset. The horizontal line at E = 2 intersects the same set of singularities as thescattering functions in Fig. 2. The region of continuity marked yellow will servein the subsections 3.1 and 3.3 as example region.the scattering functions are called the region R . In the next section, we will beinterested in a variation of E such that the number of connected componentsof the intersection between the red horizontal line and the yellow gap in Fig. 3changes from 2 to 1 to 0. This is the reason why the study is for values of E close to 2.In the general case of a chaotic dynamics, there is a small number of funda-mental periodic orbits, such that the tangle built up by the stable and unstablemanifolds of these orbits is dense in the whole chaotic invariant set. Then it issufficient to study the tangle formed by the fundamental orbits only. In mostcases, these orbits are the periodic orbits oscillating over the outermost saddlesof the potential. However, in the present case, where the potential is of theform given in Eq. (2) there is a small extra problem that the potential has anattractive asymptotic tail, and then the outermost localized orbit of the flow orthe outermost fixed point of the corresponding Poincar´e map sits at infinity.Formally this fixed point at infinity has neutral linear stability, i.e. it isparabolic. However, it is nonlinearly unstable and forms stable and unstablemanifolds and tangles (chaotic saddles) of the usual topological structure [11,12].In an appropriate Poincar´e section, the unstable manifold is obtained as thereflection of the stable manifold in the line p z = 0. This is a consequence of thetime-reversal symmetry of the Hamiltonian dynamics. To visualize the tangle,the Poincar´e map is constructed with the intersection condition x mod P = 0.By W sE and W uE are denoted the stable and unstable manifolds of the point atinfinity on the Poincar´e map, i.e the point with p z = 0 and z → ∞ . The Fig. 4displays the tangle for various values of initial energy E , and thereby showsthe development scenario of this tangle, as a function of E . The sequence of7igure 4: The tangle between the stable and unstable manifolds of the pointat infinity for different values of the initial energy E which is a conservedquantity for the 2-dof model describing the elastic scattering of He from a staticCu surface. The blue line is a segment of the stable manifold W sE , and the redline is a segment of the unstable manifold W uE . The tangle between W sE and W uE changes as E increases.plots shows the typical scenario of development for a binary Smale horseshoe.Finally, let us consider the dimension of the important geometrical objects inthe phase space relevant to the scattering process. The domain of the Poincar´emap has dimension 2, the dimension of the stable and unstable manifolds W sE and W uE of the fixed points in the map is 1, i.e. they are of codimension 1.Therefore, these manifolds divide the domain into separate pieces and they arenatural barriers that direct and channel the dynamics of the map. The dynamicsof points in a lobe delimited by segments of stable and unstable manifolds aredetermined by the structure of the tangle. The image of a lobe under thePoincar´e map is the next lobe [25]. Moreover, the iterations of the lobes rotatearound an inner fixed point in the Poincar´e map. This fixed point correspondsto a periodic orbit trapped in the periodic potential of the surface [3].The situation for the flow in the phase space restricted to a surface of con-stant energy is equivalent. Here all relevant dimensions are higher by 1, but thecodimensions remain the same. The constant energy surface has dimension 3,the stable and unstable manifolds have dimension 2, and they form a partitionof the domain. These considerations will turn out useful in the following be-cause the codimensions of the important objects that direct the dynamics staythe same when the vibration of the surface is included, and the system becomes3-dof. 8 The 3-dof model: the inelastic scattering case
In order to construct the 3-dof Hamiltonian associated with the vibrating coppersurface, we include an oscillatory term with a single frequency ω . Then, toconvert the system into a time-independent one, ωt is replaced by the phasevariable ψ of an oscillator and includes its conjugate action variable I intothe Hamiltonian. Once the system is transformed in a time-independent 3-dofversion, it is possible to apply the ideas developed in Refs. [11, 12] to study thescattering process.The construction of the 3-dof model starts from the time-independent 2-dofone. To include the surface oscillation, the variable z in the potential from Eq.2is replaced by z + B z cos ωt . After that, ωt is replaced by the phase coordinate ψ of the oscillator and next approximation is taken V ( x, z + B z cos ψ ) ≈ V ( x, z ) + B z cos ψ ∂V ( x, z ) ∂z . Thereby the new potential consists of two terms, first the old potential fromEq.2, and second an additional potential that describes the interaction betweenthe oscillator and the particle. The resulting Hamiltonian model of the completesystem is the sum of three terms: The Hamiltonian of the 2-dof model (see Eqs.(1) and (2)), a term ωI for the free oscillator, and finally an interaction betweenthe oscillator and the particle. H ( x, z, ψ, p x , p z , I ) = 12 m (cid:0) p x + p z (cid:1) + V ( x, z ) + Iω + B z cos ψ ∂V ( x, z ) ∂z . (3)The dof associated with the vibration of the surface is a closed one by con-struction. The potential V ( x, z ) is periodic in x , and accordingly, the associated x -dof can also be considered to be a closed one. In this sense, the 3-dof systemhas 1 open and 2 closed dofs. The main effect of the interaction term is anenergy transfer between the surface oscillator and the particle. Therefore, thefinal particle energy is, in general, no longer equal to the initial particle energy.Fora simple model to explain the energy transfer between a vibrating target anda scattered particle, see Refs. [26, 27].In all scattering problems, it is important to make an appropriate choice ofthe asymptotic labels. For an n -dof Hamiltonian system, the asymptotes shouldbe labeled by 2 n − p x, and p z, ,or equivalently the initial kinetic energy E and the incident angle θ betweenthe z axis and the initial vector momentum (cid:126)p . A third label consists of theimpact parameter b as defined before in section 2, the fourth label is a relativephase shift φ between the particle motion and the oscillator. The relative phaseshift φ is defined as φ = ψ i − ( t s − t i ) ω where ψ i is the phase of the oscillator9t time t i when the particle starts at the point ( x i , z i ). The quantities t s and t i are as defined before in section 2.As the fifth and last asymptotic label one can use either the initial oscilla-tor action I , or the total system energy, i.e., including that of the oscillator.However, this fifth quantity is irrelevant since the action I never enters anysignificant quantity; therefore, is not necessary to consider the value of I .In the following, E will denote the value of H , i.e., the particle energy. Itsvalue in the asymptotic region is the kinetic energy of the particle, and its valuealong an incoming asymptote is denoted E , and its value along an outgoingasymptote is denoted E f . The scattering function in the 3-dof case is again a map from the set of all possi-ble initial asymptotes to some magnitudes characterizing the final asymptotes.In the present case, the domain of this map is a 5-dimensional set. However, theinitial action I is irrelevant, and therefore it will not be considered. Moreover,also in the 3-dof case, the particles in the incoming beam have initial vectormomentum (cid:126)p i , or equivalently E and θ , and the quantities b and φ have arandom distribution with constant density. The most interesting quantities tomeasure along the outgoing asymptotes are the final energy E f of the particleand the trajectory inclination θ f . Equivalently, it is possible to the final vectormomentum (cid:126)p f .For the numerical calculations, let us consider a fixed value of the surfaceoscillation frequency ω = 0 . ω D , where ω D is Debye frecuency, and proceedsimilarly to the 2-dof system. As long as the oscillation amplitude B z = 0,the initial energy of the particle E is equal to its final value of E f . However,as soon as B z (cid:54) = 0 (for the following numerical examples, the value of theoscillation amplitude is fixed at B z = 0 . E f is different from E in general, and the scattering process becomes inelastic. A convenient quantityto characterize final asymptotes is the energy transfer ∆ E = E f − E . For thepresent system, a meaningful set of 2 scattering functions are: ∆ E ( b, φ ) and θ f ( b, φ ), both depending on b and φ while keeping E and θ fixed, like thebeam already described above.Some numerical examples, corresponding to the cases E = 1 .
9, 2 . . .
16, 2 .
175 and 2 .
45 are shown in Figs. 5 and 6, where Fig. 5 shows θ f andFig. 6 shows ∆ E . The incoming angle is always kept fixed at θ = 80 ◦ . Ascan be seen, the structure of the set of singularities of the scattering functionschanges as the value of the initial energy E is increased, and the sequence of E values used in the figures demonstrates well the fundamental pattern of thesechanges. For E = 1 .
9, two basic structures are seen: First, vertical strips alongthe φ direction, which are qualitatively similar to the strips that are observedin the uncoupled 2-dof case, i.e., for B z = 0. And second, disks around acentral point. The complement of the set of singularities is the set of regionsof continuity of the scattering functions. If the value of E is increased, thestructure of strips and disks changes. This change reflects the changes in thestructure of the associated tangle between the stable and unstable manifolds ofthe 3-dof system.In the analysis in subsection 3.3 it is necessary to consider another importantset in the domain of these scattering functions, i.e. in the b – φ plane, namely10igure 5: Scattering function θ f ( b, φ ) for the initial conditions θ = 80 ◦ andthe 6 values of the incident energy, E =1.9, 2.0125, 2.1, 2.16, 2.175, 2.45 in theparts (a), (b), (c), (d), (e) and (f) respectively. The structure of the fractal setof singularities changes when the value of E is increased. In comparison withthe unperturbed case of 2, where the scattering functions are independent of φ , here, this symmetry is broken by the perturbation, and new structures withdifferent topology appear, namely annular regions of continuity.11igure 6: Scattering function ∆ E ( b, φ ) for the initial conditions θ = 80 ◦ andthe 6 values of the incident energy, E =1.9, 2.0125, 2.1, 2.16, 2.175, 2.45 in theparts (a), (b), (c), (d), (e) and (f) respectively.12he curves along which the Jacobian determinant of these functions is zero ( forits importance see also [28] ). The Jacobian determinant is defined asdet J = (cid:12)(cid:12)(cid:12)(cid:12) ∂ ( θ f , ∆ E ) ∂ ( b, φ ) (cid:12)(cid:12)(cid:12)(cid:12) = ∂θ f ∂b ∂ ∆ E∂φ − ∂θ f ∂φ ∂ ∆ E∂b (4)In order to understand the pattern of the subset of the domain with det J = 0let us imagine first extremely small values of B z . As can be seen from Fig.2, inthis case, the derivative ∂θ f /∂b is of order 0 in B z and all the other 3 partialderivatives are of order 1. Therefore in this limit, the second product on theright-hand side of Eq. 4 is of second order and it is possible to neglect it, whilethe first product is of order one and it becomes the important one. In firstorder it is the product of the following two factors: First, ∂θ f /∂b taken fromFig.2 which has a single minimum in each interval of continuity. This leads toa single line of det J = 0 running in φ direction in each region of continuity. Inthe following, let us call such lines the vertical det J = 0 lines. And the secondfactor is the derivative ∂ ∆ E/∂φ . The functional form of the perturbation asgiven in Eq.3 makes it understandable that in the limit of extremely small B z the dependency of ∆ E on φ goes like sin φ . Then ∆ E has a relative extremaat φ = π/ φ = 3 π/
2. In the following, let us call such lines the horizontaldet J = 0 lines.Under increasing B z the vertical det J = 0 lines are relatively robust aslong as the regions of continuity remain stripes running around in φ direction.And these lines also remain lines running around in φ direction. The horizontaldet J = 0 lines are more sensitive to perturbations of the potential. It dependson the width in b direction of the regions of continuity at which value of B z thehorizontal det J = 0 lines deform strongly. The chosen value B z = 0 .
001 is theappropriate one to study the deformation of the det J = 0 lines in all details forthe region R . This is an additional reason for this choice of the numerical valueof B z in the present study.The Fig. 7 shows the numerical example for the dependence of the det J = 0curves on E . The 6 parts (a), (b), (c), (d), (e), (f) of the figure correspond tothe energy values also used in Figs. 5 and 6. Regions with det J >
J < J = 0 arethe boundary curves between white and black regions. Remember that b and φ are periodic variables and that, therefore, opposite boundaries of the figureshould be identified to turn the domain into a torus. Fig. 8 is a magnification ofthe most interesting region in Fig. 7. We are mainly interested in the region R and therefore curves in R are marked by colors. The det J = 0 curves developingout of the curves ∂θ/∂b = 0 of the unperturbed case ( i.e. the original verticalcurves ) are marked green. The det J = 0 curves developing out of the curves ∂ ∆ E/∂φ = 0 in the limit B z → R aremarked violet. First, observe the robustness of the vertical det J = 0 line butalso the robustness of the two horizontal det J = 0 lines in the largest regionof continuity in the Fig. 7. In this largest region of continuity, there are nointeresting changes in the topology of the region itself or in the det J = 0 linesfor moderate values of B z .In order to show the typical scenario for a change of the topology of regionsof continuity and related changes of the det J = 0 lines, let us focus our attention13igure 7: Signature of det J for the scattering functions. The black regions havedet J >
J < J for the scattering functions in theinterval b ∈ [0 . , . J >
J <
0. The green lines correspond to vertical det J = 0 lines in theregion of continuity R . The orange lines are the deformation of the horizontallines det J = 0 in the region R . The violet lines are the boundaries of the region R . 15n one region of continuity, namely the one which is colored yellow in Fig. 3, andwhich has already been called the region R . In Fig. 3, it is the largest region ofcontinuity besides the infinite outer region of continuity. At the lowest energy E = 1 . R consists of two components.Compare them with an intersection between a horizontal line at E = 1 . φ , but theystill have the same topology as in the unperturbed case, compare part (b) ofFig. 2. Each one of the 2 components still has its vertical det J = 0 curve(colored green), which still has the same topology as in the unperturbed case.We also observe the relatively large sensitivity of the horizontal det J = 0curves against perturbations. Only in the largest outer region of continuity, thehorizontal curves are almost the same as in the limit of extremely small B z , asexplained above. They are two almost horizontal lines near φ = ± π/
2. In con-trast, in all the smaller regions of continuity, the horizontal lines are deformedstrongly and have changed their topology such that they can no longer be rec-ognized easily as curves having developed from horizontal lines. This behaviorillustrates how the onset of strong deformations of the horizontal det J = 0 linesdepends on the size of the region of continuity; smaller regions are more sensi-tive to perturbations. We also see that the value B z = 0 .
001 is the appropriateone to see the essential behavior of the det J = 0 lines in the region R .Now let us describe in more detail the transformation of the det J = 0curves within R under a variation of the energy E . When the value of E isincreased then the gap separating the two components of R becomes smaller,and at a critical value of E , the gap breaks near φ = 3 π/
2. At this moment,the gap changes its topology from a stripe running around φ direction to adisc contractible to a point. Accordingly, also R changes its topology from twodisconnected stripes running around in φ direction to a connected set. At thesame time, a new vertical det J = 0 curve is created running between the twoextremal points in φ of the disc-shaped gap. See part (b) of the figure. With E increasing further, the gap shrinks, and the right vertical det J = 0 curveand the new middle det J = 0 curve come closer, see part (c) of the figure.At another critical value of E , these two vertical det J = 0 curves touch at φ = 3 π/
2, and change their topology as shown in part (d) of the figure. Theychange from two curves running around in φ direction to a single contractibleloop. Next, the gap disappears completely, as can be seen in part (e) of thefigure. Simultaneously with increasing E , the contractible green det J = 0curve shrinks and finally disappears, while also the right orange curve shrinksand disappears. The result is shown in part (f) of the figure. In the end, thereis a single region of continuity running around in φ direction and containinga single vertical det J = 0 curve also running around in φ direction. Therebythe scenario of the fusion of two typical regions of continuity into a single oneis finished. This scenario is the direct generalization of the 2-dof scenario of thefusion of intervals of continuity, as described in Ref. [29]. For other regions ofcontinuity similar to transformations and fusions happen under a change of theenergy. 16igure 9: The Cartesian product of the pile of 2-D tangles with the energy E as stack parameter and a circle representing the angle ψ forms the tanglebetween the stable and unstable manifolds for the 3-dof system in the Poincar´emap. Let us discuss the construction of the tangle between the stable and unsta-ble manifolds in the 4-D Poincar´e map for the 3-dof case following the ideasdeveloped in Refs. [11, 12] for systems with one open and 2 closed dof.In order to explain the construction, let us start with the case of zero oscil-lation amplitude, i.e., B z = 0. In this case, the particle energy E is conserved,and each slice, E = constant, in the 4-D Poincar´e map is invariant. The tan-gle in each one of such slices was already discussed in section 2 (see Fig. 4).Now, the higher dimensional tangle for the 3-dof system can be obtained in atwo-step process. First, we form a stack with all curves taken from the wholecontinuum of 2-dof tangles, where the value of E is the stack parameter. Thisunion creates a 3-D object.Second, we add to this object the phase variable of the oscillator by forminga Cartesian product of the 3-D stack with a circle representing the still missingvariable ψ . In this way, we end up with a tangled structure in the 4-D domainof the Poincar´e map for the 3-dof system; see Fig. 9. Notice, that this is stillonly for zero oscillator amplitude, B z = 0.In the 2-dof case and the corresponding 2-dimensional map of section 2, thetangle was created, starting from the fixed point at infinity ( z = ∞ , p z = 0), byplotting the stable and unstable manifolds associated with this point at infinity.In the stack of the 2-dimensional maps, the stack of fixed points at infinity is avertical line, and the Cartesian product of this line with the circle correspondingto the oscillator phase generates an invariant 2-dimensional cylinder at infinity.It is a 2-dimensional invariant subset of the 4-dimensional domain of the mapfor the 3-dof system.Thereby the object growing out of the outer fixed point of the 2-dof tangleis this cylindrical invariant 2-dimensional surface. If the fixed point at theinfinity of the 2-dof system were hyperbolic, then the resulting 2-D surface inthe 4-D domain would be a normally hyperbolic invariant manifold (NHIM). Inour case, this object is formally not a NHIM, but it plays an analogous role.For general information on NHIMs and their role in dynamical systems, see forexample [9, 30]. 17he stable and unstable manifolds W s/u of the invariant surface at infinityare also obtained by the same stack-and-product process as W s/u = (cid:91) E W s/uE × S . (5)Let us now check the dimension of these manifolds. The dimension of thestable and unstable manifolds in the 2-dof case for fixed energy is 2 in the flowor 1 in the map. The pile of the stable and unstable manifolds, parametrizedby the initial particle energy E , is a surface with dimension 3 in the flow ordimension 2 in the map. Then forming the product with a 1-D circle results in a4-D surface in the flow of the 3-dof system or a 3-D surface in the domain of themap. Most important: The codimension of the stable and unstable manifolds is1 in any case, and then the stable and unstable manifolds create a partition in theconstant total energy manifold of the 3-dof system, and also in the domain of thePoincar´e map. The invariant manifolds W s/u are impenetrable hypersurfacesthat direct the flow similar to what happens in the 2-dof systems. Moreover,the stable/unstable manifold is the union of the trajectories whose z componentof the momentum converges to zero for z → ∞ .So far, we have described the construction of the tangle for the 3-dof problemin the case of no coupling, i.e., for B z = 0. What happens when the coupling isswitched on? If the system would has a usual NHIM, then we could quote thetheorem of Fenichel on the persistence of NHIMs and their stable and unstablemanifolds, see, for example, Refs. [30–32]. However, in the present case, theinvariant surface at infinity is stacked up by linearly parabolic points, whichare only nonlinearly unstable. On the other hand, in this case, the interaction(see Eq. (2)) goes to zero exponentially for large values of z . Therefore, theinvariant surface itself at infinity is not affected by the interaction. The tangleof the stable and unstable manifolds coming from the points at infinity containshyperbolic components ( transverse intersections between stable and unstablemanifolds ) and in general, also nonhyperbolic components ( tangencies or situa-tions close to tangencies ) as it is generally found in usual incomplete horseshoeconstructions. The general experience indicates that for scattering processesthe hyperbolic part of the chaotic saddle dominates. The nonhyperbolic partsare less important and are influencing only very high levels of hierarchical inthe resulting fractal structure. For several examples see the references [33–39] .The hyperbolic parts are robust under small perturbations of the system, whilethe nonhyperbolic parts may change already under the smallest perturbations.In total, we can expect that the stack of tangles maintains its large-scalestructure also under moderate perturbations of the system, and therefore alsounder moderate coupling strengths between the particle dofs and the oscillatordof. As a result, the intersection of the perturbed stack with a plane surfaceshould be similar to the intersection of the unperturbed stack with a curvedsurface. In this sense, the stack construction method provides us with a valididea of the higher dimensional tangle and is also able to explain the dynamicsof the system with coupling.Now, we understand from another point of view the changes in the structureof the set of singularities in the scattering functions for the coupled 3-dof systemin Figs. 5, 6, 7. When the 3-dof system contains a perturbation, then thesymmetry in the angle φ is broken, and the manifold W s is deformed and loses18ts symmetry with respect to φ . And its intersection with the set of initialconditions change. If we change the set of initial conditions in the phase space,then the pattern of intersections is different. The plots of the various partsin Figs. 5, 6, 7 are scattering functions for different sets of initial conditions(different E ), and their singularities show the pattern of the intersection of thestable manifold.With the help of Fig. 3 and our new understanding, we can give still an-other interpretation of the change of the regions of continuity of the scatteringfunctions under perturbations. Depending on the value of φ , the value of theparticle energy E changes. Thereby the particle can change in Fig. 3 from theinitial value of E to a modified value of E , and this modification depends on φ . Imagine that the initial value lies around2, e.g., at the value 2.0125 usedin parts (b) of the Figs. 5, 6, 7. Here the region R is very close to the valuewhere it switches in Fig. 3 from having 2 components to having 1 component.For φ values around − π/ b direction R has 1 component and for φ values around π/ b direction R has 2 components. This is exactly what we observe in the parts (b)of the Figs. 5, 6, 7. Considerations of this type only hold for weak perturbationswhere the homoclinic/heteroclinic tangle still has the stack structure. Let us assume an incoming beam, as explained before. The detector shouldmeasure the distribution of the values of the outgoing particle energy E f , orequivalently and even better, the energy transfer ∆ E and the outgoing angleof inclination θ f . This detector registers neither the outgoing phase shift φ f nor the value of the outgoing impact parameter b f . The result in this typeof measurement is the doubly differential cross-section dσdθd ∆ E ( θ f , ∆ E ) for fixedvalues of the initial energy E and fixed initial angle of incidence θ .Now, let us discuss the geometrical connection between the scattering func-tion and the cross-section. Remember that the relevant scattering function forthe construction of the cross-section is the function whose domain and rangeare the b – φ and ∆ E – θ f planes, respectively. This function can be viewed as agraph in the Cartesian product of the 2-D domain with the 2-D range, i.e., inthe 4-D ( b, φ , ∆ E, θ f ) space. The incoming beam represents a constant densityon the domain, and the function maps this density into the range. The dif-ferential cross-section defined above is the resulting density in the range. Thisconnection is expressed as follows dσdθ d ∆ E ( θ f , ∆ E ) = (cid:88) i (cid:12)(cid:12)(cid:12)(cid:12) det ∂ ( θ f , ∆ E ) ∂ ( b, φ ) (cid:12)(cid:12)(cid:12)(cid:12) − , (6)where the sum runs over all preimage points in the domain, i.e., in the b – φ plane, leading to the values θ f , ∆ E in the range of the scattering function. It isclear that the cross-section has singularities where the projection of the graphinto the range is singular, i.e., where the determinant of the Jacobian matrix ofthe scattering function appearing in the previous equation is zero, i.e., they are19he lines already studied in subsection 3.1 and Fig. 7. The resulting singularitiesin the cross-section are the well-known rainbow singularities of the differentialcross-section, and they are the lines over which the number of preimages changes(in general by 2). In a general 3-dof system, there are lines along which the rankof the Jacobian matrix drops by 1, and there can also be points at which therank drops by 2. For a good explanation of rainbow singularities, see chapter 5in Ref. [40]. Note that generic rainbow singularities are of one over square roottype, and therefore the integral over them is finite. At the points where therank of the Jacobian matrix in Eq. (6) drops by 2, there might be singularitiesof another functional form, however also here the integral over the differentialcross-section is always finite. There is no violation of flux conservation.In the present system, most regions of continuity of the scattering functionare stripes running around in φ direction or annular shaped regions, see Fig. 5.However, under a change of the energy, the processes of fusion of regions ofcontinuity happen as explained in all details in subsection 3.1. In the rest ofthis subsection, we will study how the process of fusion shown in the Figs. 5, 6,7 shows up in the cross-section.If we project the graph of the scattering function of one vertical stripe oralso of one disc-shaped region of continuity into the plane ∆ E – θ f we can see acaustic like the one coming from the projection of a deformed semi-torus, thisstructure is characteristic of 3-dof systems with 1 open dof and 2 closed dofs [12],it is one normal form for these projection caustics. A more detailed explanationfor this structure is the following.First, let us consider the unperturbed case. In the inner part of a verticalstripe of continuity, the scattering function is constant in the φ direction, and wehave a kind of half semi-torus in the plot of the scattering function, rememberthat φ is a periodic variable and that the vertical stripe closes to a ring. Theprojection of this graph on the plane ∆ E – θ f is a quadrilateral. And it is a 4:1projection. If the system is perturbed, the rotational symmetry is lost, and theplot of the scattering function is deformed. The semi-torus is then deformed,and it is no longer of constant height; accordingly, the caustics change, andalso, some regions are formed where the projection is 2:1. For the perturbedcase, the caustics have the same qualitative structure for different values ofthe perturbation parameter as long as the system is in the regime of weakperturbation. Analogous considerations hold for the contractible ring-shapedregions of continuity.The complete caustic structure of the entire cross-section is a superpositionof the various basic caustic structures coming from all the different regions ofcontinuity. Of course, the various structures are shifted in their exact position,and they have different heights and different total strength. This total strengthis proportional to the area of the corresponding interval of continuity becauseit must be proportional to the incoming flux falling into this particular intervalof continuity.The Fig. 10 shows a plot of the cross-section for θ = 80 ◦ and for the sixdifferent values of the initial energy E , which also have been shown in Figs. 5,6, 7. In the plots, the basic structure of the singularities of the projection of thegraphs of the regions of continuity of the scattering function is apparent.First, let us look at part (a) of the Figs. 5, 6, 7 for E = 1.9. Here thescattering function has 3 important regions of continuity. The first one, the outerregion, which occupies the largest area in the domain and which accordingly20igure 10: Scattering cross-section for θ = 80 ◦ and for the 6values E = 1.9,2.0125, 2.1, 2.16, 2.175 and 2.45 in the parts (a) and (b), (c) and (d),(e) and(f), (g) and (h), (i) and (j), (k) and (l) respectively. The parts (a), (c),(e), (g),(i) and (k) in the left column show the cross section for a constant illuminationof the whole b – φ plane. The parts (b), (d), (f), (h), (j), (l) show it for partialillumination of a neighbourhood of the region R only.21igure 10: Continuation.22auses the strongest structure in the cross-section. It is the strong structurereaching up to rather small values of θ f . It comes close to a rectangle, whichis the result of a 4:1 projection of a semi torus. In addition, we also see strongstructures coming from the two separate components of the region R definedin subsection 3.1. Because we are mainly interested in the contributions fromregion R , we repeat in part (b) the cross-section again for E = 1.9 where,however, we have illuminated a neighborhood of the region R only. Therebythe structures caused by region R are clearly visible. They are two of theabove mentioned deformed torus-shaped structures. The structure reaching upto smaller values of θ f is the one coming from the left part of R in Figs. 5, 6, 7.Now let us proceed to E = 2.0125. The cross-sections with complete andpartial illumination of the domain of initial conditions is plotted in parts (c) and(d) of Fig. 10 respectively. When we compare parts (b) and (d) of the figure,then we see the beginning of the fusion process in the upper right part of themain structure. Parts (e) and (f) show the complete and the partial cross-sectionfor E = 2.1. Here the fusion has proceeded to large deviations compared to E = 1.9. Parts (g) and (h) show the cross-sections for E = 2.16 and parts(i) and (j) show it for E = 2.175. Here the region R has already turned intoa single vertical stripe. However, as Fig. 8 (e) shows there are still remanets ofthe additional det J = 0 curves which create the additional rainbow structuresaround θ f ≈ π/
16. Finally, the parts (k) and (l) show the cross-section for E = 2.45, where the transformation process of the region R is finished. Here therainbow structure coming from R is again just the projection of half a torus.Compare it with one of the two strong structures ( the one sitting mainly athigher values of θ f ) seen in part (b) of the figure. We see a similarity, whichindicates that the completed fusion process of two generic regions of continuityresults in a single generic region of continuity.The sequence of changes presented is the typical one for any fusion of regionsof continuity of the scattering functions. It is the 3-dof generalization of the 2-dof fusion events explained in Ref. [29]. We studied an example of chaotic scattering in a 3-dof Hamiltonian model withone open and 2 closed dofs, namely the scattering of an atom from a vibrat-ing corrugated surface. The system can be considered a small perturbation ofa partially integrable system and therefore a convenient starting point is thispartially integrable 3-dof system which can be treated as a stack of 2-dof sys-tems, where the stack parameter is the particle energy E , which is a conservedquantity of the 2-dof system.A cental object in this study is the scattering function (∆ E ( b, φ ) , θ f ( b, φ )),and its set of singularities. The set of singularities reflects the structure of thestable and unstable manifolds that divide the constant energy manifold of thetotal system and direct the dynamics. The doubly differential cross-sectionis the projection of the graph of the scattering function on its range and thesingularities of this projection are the caustics.The structures of the caustics are related to the regions of continuity ofthe scattering function and to the curves where the Jacobian determinant ofthe scattering function is zero, det J = 0. The regions of continuity generate23haracteristic types of structures on the cross-section; it is like a projection of asemi torus on the ∆ E – θ f plane. An increment in the value of the initial particleenergy creates changes in the regions of continuity of the scattering functionsand in the set of curves det J = 0, and we can see clearly how those changes arereflected in the fusion of the caustics on the cross-section.The important progress of the present work in comparison with previousinvestigations of 3-dof chaotic scattering is a detailed description of the fusionbetween two regions of continuity of the scattering function. We have dividedthe fusion process of two generic regions of continuity into a single generic regionof continuity into its 5 important steps. The process is presented by a sequenceof six plots illustrating the 5 important steps in the transformations. Also, wehave presented the corresponding steps in the changes observed in the doublydifferential cross-section.The present system helps to understand the elementary transitions in thescattering functions and the corresponding transitions in the cross-section alsoin other systems with qualitatively similar changes in the scattering function, forexample, 3-dof systems with a perturbation of partial integrability [11], [12], [41],a perturbed magnetic dipole [14], and a realistic molecular system treated in [42].The considerations of subsections 3.1 and 3.3 hold for all the above mentionedsystem in an analogous form. Note that Ref. [42] is a system with 2 open dof, andaccordingly, the structure of the vertical det J = 0 lines is more complicated.The scattering function oscillates in the interior of regions of continuity, andthis leads to a sequence of vertical det J = 0 lines in each region of continuity.Otherwise, the basic phenomena are the same.For any scattering system, there is always the interesting question of theinverse scattering problem. For chaotic scattering, one considers this problemas the problem to reconstruct information on the chaotic invariant set fromscattering data, in particular from cross-section data. So far, for 3-dof systems,there is a general strategy to do this job only for the uniformly hyperbolic case,see Ref. [43]. However, these hyperbolic cases are always far away from anystack construction; they are a kind of opposite extremal case to the partiallyintegrable case. Therefore they do not provide any clue for our present case,which is a case of mixed-phase space close to partial integrability.Finally, we offer some remarks about the implications of the classical scat-tering results for the quantum scattering problem in the semiclassical regime.Classically the cross-section is a sum over contributions from all preimages, seeEq.6. Semiclassically, the scattering amplitude is a corresponding sum over thesquare roots of the classical contributions where each contribution gets, also,a phase which is the complex exponential function of the reduced action alongthe corresponding path. The semiclassical cross-section is the absolute squareof the semiclassical scattering amplitude, and thereby it is a double sum. Ac-cordingly, the semiclassical cross-section is a sum of the classical cross-sectioncoming from the diagonal terms of the double sum and a double sum of thenondiagonal interference terms. Thereby the resulting cross-section containsinterference oscillations superimposed over the classical cross-section.In addition, a semiclassical approach should uniformize the rainbow singu-larities and remove, thereby, the infinities. In a generic rainbow line, 2 classicalcontributing paths coincide and disappear. Then the uniformized semiclassicalcontribution of these 2 classical paths to the semiclassical scattering amplitudecan be modeled by an Airy function, which is the normal form of a wave rain-24ow contribution. The contribution of a point where the rank of the Jacobiandeterminant of the scattering function drops by 2 should be described by someother appropriate catastrophe function. For more detailed information on thesemiclassical treatment of chaotic scattering see Refs. [44], [45], [46], [47]. F G acknowledges the support of CONACYT program for Postdoctoral Fel-lowship. C J acknowledges financial support by DGAPA under Grant No.IG100819. F B acknowledges the financial support from Spanish Ministry ofScience, Innovation and Universities, Gobierno de Espa˜na, under Contracts No.PGC2018-093854-BI00, the Ministerio de Econom´ıa y Competitividad (Spain)under Contracts No. MTM2015-63914-P and ICMAT Severo Ochoa ContractNo. SEV-2015-0554, and from the People Programme (Marie Curie Actions)of the European Union’s Horizon 2020 research and innovation program underGrant No. 734557.
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