Classical Mechanics of Collinear Positron-Hydrogen Scattering
Min-Ho Lee, Chang Woo Byun, Jin-Sung Moon, Nark Nyul Choi, Dae-Soung Kim
CClassical Mechanics of Collinear Positron-Hydrogen Scattering
Min-Ho Lee, Chang Woo Byun, Jin-Sung Moon, and Nark Nyul Choi ∗ School of Liberal Arts and Teacher Training,Kumoh National Institute of Technology, Gumi 730-701, Korea
Dae-Soung Kim
Department of Global Education, Gyeonggi Collegeof Science and Technology, Siheung 429-792, Korea (7 April 2015)
Abstract
We study the classical dynamics of the collinear positron-hydrogen scattering system below thethree-body breakup threshold. Observing the chaotic behavior of scattering time signals, we in-troduce a code system appropriate to a coarse grained description of the dynamics. And, for thepurpose of systematic analysis of the phase space structure, a surface of section is introduced beingchosen to match the code system. Partition of the surface of section leads us to a surprising conjec-ture that the topological structure of the phase space of the system is invariant under exchange ofthe dynamical variables of proton with those of positron. It is also found that there is a finite setof forbidden patterns of symbol sequences. And the shortest periodic orbit is found to be stable,around which invariant tori form an island of stability in the chaotic sea. Finally we discuss apossible quantum manifestation of the classical phase space structure relevant to resonances inscattering cross sections.
PACS numbers: 34.80.Uv, 05.45.Mt, 45.50.-jKeywords: Positron scattering, Three-body Coulomb problem, Chaotic scattering, Triple collision, Symbolicdynamics, Surface of section ∗ Electronic address: [email protected] a r X i v : . [ phy s i c s . a t o m - ph ] A p r . INTRODUCTION The three-body problem of Coulombic systems is one of the most fundamental problemof theoretical physics. A large number of papers has been published which deals with theclassical mechanics as well as the quantum mechanics of electron-hydrogenic atom systems,that is two-electron atoms [1–3]. The classical motion of three bodies in a two-electron atomabove the three-body breakup threshold is known to be regular although it is nonintegrable[4], and the behavior of electron-impact ionization of hydrogenic atoms near the breakupthreshold has been treated long time [5, 6]. In contrast to the dynamics above the threshold,the classical motion below the threshold is no longer regular; the classical phase space consistsof not only reguar but also chaotic parts [7]. The chaotic dynamics below the thresholdproduces infinitely many resonances in photoelectron spectra of two-electron atoms, andthe fluctuating parts of the spectra owing to the resonances show a series of seeminglymeaningless irregular signals [8]. However, by the help of modern techniques of semiclassicalmechanics, the fluctuating parts can be characterized by the actions and stabilities of theinfinite number of closed triple collision orbits (CTCO) [3, 9].Positron-hydrogen scattering system is another three-body Coulomb problem, which hasrecently attracted a growing interest as powerful positron beams have developed since thediscovery of low energy positron beams [10–13]. The behavior positron-impact ionizationof hydrogenic atoms above the breakup threshold, whose corresponding classical dynamicsis regular as in the electron impact ionization, has been examined by both classical andquantum mechanical methods [14–17]. Considering there are a large amount of papers forthe classical dynamics of two-electron atoms for wide range of energies including belowthree-body breakup threshold, however, it is noteworthy that only a limited number ofpapers deal with classical dynamics of positron scattering by a hydrogen target below thethreshold, where the classical motion might be no longer regular and thus fluctuations inquantum scattering cross sections might occur as in the case of two-electron atoms.Stimulated by a prominent role of classical calculations in success of semiclassical inter-pretations and predictions on quantum mechanical properties of two-electron atoms [7], astudy was conducted of classical dynamics of positron-hydrogen scattering for wide rangeof energies including below as well as above the threshold [18]. However, in that work, theclassical mechanics was merely used as a calculational method for Monte Carlo simulations1ather than being analyzed for the purpose of characterization of the resonance structuresas was done in two-electron atoms. A systematic analysis of the classical phase space of thepositron-hydrogen scattering system has not been given yet for the energies relevant to theresonances, that is for energies below the breakup threshold [12, 19].Resonances lying close to the threshold are assumed to be associated with CTCOs asdemonstrated in papers deals with photoelectron spectra of two-electron atoms [3, 9, 20].By virtue of the fact that CTCOs are completely included in the collinear invariant subspace,study of classical dynamics in the subspace would give a great insight into the resonancesin positron-hydrogen scattering in the real three-dimensional world.In this paper we will report on our investigation of the classical dynamics of positronscattering by a hydrogen target in the collinear p + e − e + configuration for energies below thethree-body breakup threshold. The paper is organized as follows. In Sec. II the Hamiltonianis introduced, and a transformation is also introduced to regularize Coulomb singularity atthe binary collisions for proper numerical treatment of the classical equations of motion. InSec. III scattering time signals are given as a function of the initial state of the hydrogenatom at the instant the positron starts coming to the hydrogen atom at a fixed distance.Based on observation of the dips in the scattering time signal, a code system is introducedfor classifying orbits, and a surface of section (SOS) is also introduced for analyzing thephase space structure. In Sec IV we present our findings based on the calculational result ofthe partition of SOS showing the phase space structure. Discussions are also given in thatsection. Finally, we conclude with a summary in Sec. V. II. EQUATIONS OF MOTION
We consider the collinear p + e − e + configuration in which the proton and the positron arelocated on the opposite side of the electron. We further assume that the proton is infinitelyheavy and fixed at the origin. In atomic units, this system is described by the Hamiltonian H = 12 p − p p + p − x − x + 1 x + x (1)where x and x are respectively the distances of the proton and the positron from theelectron, and their conjuate momenta are denoted by p and p .For proper numerical treatment, the singularities at x = 0 or x = 0 are regularized by2ntroducing a fictitious time τ and a transformed Hamiltonian K such as [21] dt = x x dτ (2) K = x x ( H − E ) (3)In addition, introducing regularized coordinates Q , Q , and momenta P , P such that x = Q , x = Q , p = P Q , p = P Q (4)one can obtain the equations of motion as follows˙ Q = 14 P Q − Q Q P (5)˙ Q = − Q Q P + 12 P Q (6)˙ P = 14 Q P P + 2 Q − P Q + 2 Q Q (cid:20) E − Q ( Q + Q ) (cid:21) (7)˙ P = 14 P Q ( Q P − P ) + 2 Q + 2 Q Q (cid:20) E − Q ( Q + Q ) (cid:21) (8)where the dot notation is used for the fictitious-time derivative. III. SYMBOLIC CODE SYSTEM AND SURFACE OF SECTION (SOS)
It has been found useful to examine scattering time signals as a starting step to systematicinvestigation of the classical dyamics of three-body problems [22, 23]. In Fig. 1, a typicalsignal for scattering time is shown as a function of the angle variable ( φ ) of the action-angle variable pair of the one-dimensional hydrogen atom at time t = 0. The energy of thehydrogen atom was chosen to be E = − .
2, and the total energy of the scattering systemto be E = −
1. The positron started inward at the distance of 100 from the proton, i.e. at x + x = 100 and we computed the trajectories until the positron reached x + x = 1000after scattering. Note that the scattering time is not the fictitious time but the real timespent by the trajectory.The scattering time signasl show typical chaotic scattering signals with infinitely manypeaks and dips, and we focus on the dips as we did in two-electron atoms[23]. Note thattwo wings stretch from the primary dip located at φ = φ d ≈ .
61. as can be seen in Fig.1. The left wing stretches beyond 0 ◦ , and its end reaches a boundary (at φ ≈ ◦ ) of theregion of chaotic signals; the right wing stretches to the other bounday at φ ≈ ◦ . The3 IG. 1: (Color Online) The scattering time as a function of the angle variable ( φ ) for the totalenergy E = − E = − . wings rise infinitely at the ends. And, by examining many other dips in the region of chaoticsignals, we confirmed that every dip has its own two wings, and every wing rises infinitelyat its end.In Fig. 2, two scattering trajectories are depicted with the initial conditions slight de-viating from the primary dip; (a) one with the initial condition φ < φ d , and (b) the otherwith φ > φ d . From these plots, we can see that the trajectory with the initial conditioncorresponding to a dip would undergo the triple collision and furthermore the triple collisionseparates two qualitatively different scattering products, i.e. inelastic and exchange scatter-ing as in Figs. 2(a) and (b) respectively. By examining trajectories with initial conditionsaround many other dips in the region of chaotic signals, we confirmed that every dip is pro-duced by the trajectories ending in triple collision and every pair of wings stretching from adip are branches to different scattering products.4 IG. 2: (Color Online) Scattering trajectories evolved with the initial conditions (a) φ = 12 . < φ d and (b) φ = 12 . > φ d . Solid (red) line for electron’s position x as a function of time, and dotted(blue) line for positron’s position x + x . Insets are magnifications around the instant whentrajectories nearly approach the triple collision. Symbolic dynamics is a useful tool for a classification of orbits in nonintegrable systems[24, 25]. From the observation of the above-mentioned separation of dynamical behaviorby the triple collision, we naturally introduce a symbol code system by defining a sym-bolic assignment such as Symbolic dynamics is a useful tool for a classification of orbits innonintegrable systems [24, 25]. From the observation of the above-mentioned separation ofdynamical behavior by the triple collision, we naturally introduce a symbol code system bydefining a symbolic assignment such as E = − IG. 3: (Color Online) Partition of SOS induced from classification of orbits by symbol sequencesof length 2. The SOS consists of two sheets, x = 0 (the right half-plane) and x = 0 (the lefthalf plane). Points represented by different symbol sequences are painted with different colors.Variables Q ’s and P ’s are the regularized coordinates inroduced in Sec. II. fact that all orbits except the shortest CTCO undergo binary collisions ( x = 0 or x = 0)at least once, the set x = 0 or x = 0 composed of two sheets of binary collisions is chosenas the SOS (see Fig. 3). The Poincare mapping is then naturally defined as F ( X ) = Y if atrajectory starts at X ∈ SOS consecutively crosses at Y ∈ SOS.A partition of SOS can be induced by combining the Poincare mapping F and the symbolcode system introduced above. For an orbit starting from X ∈ SOS, a symbolic sequence offinite or infinite length can be given to it as S ( X ) = σ σ . . . σ k . . . σ k = F k ( X ) lies in the sheet x = 02 if F k ( X ) lies in the sheet x = 0 c if the orbit starting from F k − ( X ) falls in triple collisionwithout any more crossing of SOS.In Fig. 3, we present a coarse-grained partition of SOS, where the whole SOS is parti-tioned to four regions of the sequences of length 2. The borders of the regions are the stablemanifold of the triple collision which consists of orbits terminating with the symbol c. Inthe following section, we will present further finer partitions of SOS, which make it possibleto find remarkable dynamical properties of the system. IV. RESULTS AND DISCUSSION
Proceeding by three step from the Fig. 3, we obtain a finer partition of SOS in whicheach region is classified by a sequence of length 5 as shown in Fig. 4. Comparing the twosheets (a) x = 0 and (b) x = 0, we can find that they have the same geometric structure.Actually we confirmed this symmetry of geometric structure by examining the partitionclassified by symbolic sequences of length 7, a plot of which is not given here for brevity.Thus we propose a conjecture that the geometric structure of the dynamics is invariantunder exchange of the variables ( x , p ) with ( x , p ). This is a very surprising finding sincethe masses of proton and positron are quite different and thus equations of motion (5)-(7)is not invariant under the exchange.Checking the symbol sequences representing the partitioned parts in Fig. 4, we can seethat there are no trajectories representing the symbol sequences 11122, 11221, 11222, 12211,11211, and their reflections 22211, 22112, 22111, 21122, 22122. By examining the finerpartition up to symbol length 7, we conjecture that there is a finite set of forbidden patternsof symbol sequences such as 1122, 2211, 11211, 22122. Note that there are no scatteringorbits ...111222... or ...1112111...; the simplest scattering orbits are ...111212111... and...11121222... as can be seen in Fig. 2.The partition of SOS is also useful for understanding the main features of the scatteringtime signals in Fig. 1. Initial conditions evolve with time and pass through the SOS. A7 IG. 4: (Color Online) Partition of SOS by symbol sequences of length 5 repsented in two sheets(a) x = 0 and (b) x = 0. Colors are used for the same purpose as in Fig. 3.FIG. 5: (Color Online) Intersection of SOS and orbits with the initial conditions φ ∈ [5 ◦ : 15 ◦ ] and φ ∈ [150 ◦ : 240 ◦ ]. The former is denoted by A; the latter by B. The segments A and B are plottedin the SOS partitioned by symbol sequences of (a) length 5 and (b) length 7. set of initial conditions at their first passage through the SOS forms an initial line segmentin the SOS. And then the initial line segment proceeds in SOS by the Poincare map. InFig. 5, we show images of two initial line segments obtained by applying the Poincare mapseveral times. The segment A is the image of initial line segment formed from the interval φ ∈ [5 ◦ : 15 ◦ ] which includes the initial condition φ d corresponding to the dip with the widestwings in Fig. 1; the other (denoted by B) for the interval φ ∈ [150 ◦ : 240 ◦ ] enclosing theinitial conditions corresponding to the chaotic signals consisting of many peaks and dips inFig. 1. It can be seen from Fig. 5(a) that both the the line segments A and B are cut bythe stable manifold of tripl collision. However, looking into the partition of SOS by symbol8equences of length 7, we can find the number of intersection points of the segment B andthe stable manifold of triple collision increases as the partition gets finer as in Fig. 5(b).This is consistent with the feature of the scattering time signals such that there are infinitelymany dips in in the interval φ ∈ [150 ◦ : 240 ◦ ].From studies of three-body problems, it has been generally known that trajectories incaptured state can exit to uncaptured state along only the orbits falling in triple collision; thecloser the trajectory gets to triple collision, the larger the escaping velocity of the trajectorygets [4, 23]. In other words, the farther the trajectory gets from triple collision, the smallerthe escaping velocity gets. This means that orbits with initial conditions far from dips, i.e.very close to the ends of wings escape the three-body interaction zone with zero energy ofrelative motion between the hydrogen atom (positronium) and electron (proton). Thus theends of the wings rise infinitely, producing peaks in scattering time signals as was mentionedin Sec. III.In addition to the role of triple collision as an exit from captured state, the other promi-nent role of triple collision is led by time-reversal symmetry of the present system as follow:scattering trajectories can enter into captured state along only the orbits starting from triplecollision [22, 23]. Thus, in semiclassical physics, resonances in scattering cross sections canbe described in terms of contributions from CTCOs, i.e. closed orbits starting and endingin the triple collision [3, 9, 20, 27]. It can be easily seen that all CTCOs lies in the collinearconfiguration, and thus the present study of classical mechanics of collinear p + e − e + systemis relevant to the fluctuations in the scattering cross sections in the real 3-dimensional space.Therefore, as was done for two-electron atoms [3, 9], significant predictions can be made onthe total scattering cross section of positron by the hydrogen atom with the total energiesapproaching the three-body breakup threshold from below as follows: (1) fluctuation in thecross section as a function of the total energy would be chaotic, (2) the amplitude of thefluctuation decays algebraically as the energy approaches the three-body breakup threshold,and (3) the Fourier transform of the fluctuation would show distinct peaks at the positionof the actions of CTCOs.The statement (1) comes from the chaotic nature of the symbolic dynamics of the system:between any two ordinary admissible symbol sequences, we can find arbitrarily long symbolsequences with ending in triple collision. The statement (2) comes from the fact that theinstabilities of relevant part of CTCOs approach to infinity as the energy approaches the9 IG. 6: A phase portrait revealing an island of stability in the chaotic sea. The center of the toriis indicated by an arrow. breakup threshold, which is related to the irregularizability of the triple collision [3, 9]. Theexponent for algebraic decay is determined by the stability of the relevant part of CTCOs,which will be obtained in future work. Finally, the statement (3) is based on closed orbittheory [28], which is a general theory for atomic photoabsorption cross sections. However,the condition required in the theory is nothing but the restriction on the starting and endingpoint of the leading orbits to captured state: both the starting and ending point should beclose to the center of atomic system. Thus it can be applied to scattering cross sections forthe systems with CTCOs. The action values and relative instabilities of CTCOs are notcalculated in the present paper, remaining as a future work. However, we can predicted thatthere are no peaks in Fourier transform of the fluctuation which correspond to the forbiddensymbol sequences c112c, c122c, c221c, c211c, etc. This is quite important characteristicsdifferent from two-electron atoms, reflecting the phase space structure of the present system.10ooking into the partition of SOS, we get doubt if there are some islands around thepoint ( Q , P ) ≈ (1 . , .
0) in Fig. 4. In order to lift obscurity, we draw a phase portraitaround that point. As can be seen in Fig. 6, there is actually an island of stability consistingof tori in the chaotic sea. At the center of the island which is indicated by an arrow in Fig.6, there is a stable periodic orbit, which is found to be the shortest one with infinite symbolsequence 12; all orbits in the island is represented by the same symbol sequence. This isanother characteristics different from two-electron atoms of which collinear eZe subspace isknown to be fully chaotic [7, 25]. Since it has been accepted that there is no bound stateof positron-hydrogen system in the real 3-dimensional world [13], the stability of the tori inthe direction orthogonal to the collinear subspace should be investigated for semiclassicalprediction of existence or absence of bound or long-lived states. And it will be also a goodsubject of future study to understand effects of the tori through dynamical tunneing in thescattering cross sections [29].For complete study of positron-hydrogen system in future, we should mention on a scat-tering channel which is excluded in the present work, i.e. positron-electron annihilation.It is well known that the possible contribution of the positron-electron annihilation to thetotal scattering cross section is negligible except in the limit of zero positron energy [11].However,it may be necessary to consider positron-electron annihilation for total energies justbelow the three-body breakup threshold since the amplitude of fluctuating part in the totalcross section would decay to zero as the total energy approaches the breakup threhold andthus would be affected even by small perturbations owing to the annihilation channel.
V. CONCLUSION
The classical dynamics of positron scattering by the hydrogen atom in the collinear con-figurational subspace was studied for total energies below the three-body breakup threshold.A systematic analysis of the phase space structure of the system was given for the first time,resulting in important new findings: (1) the system shows chaotic scattering, (2) the topo-logical structure of the phase space is invariant under exchange of the dynamical variablesof proton with those of positron, (3) there is a finite set of forbidden patterns of symbolsequences such as 1122, 2211, 11211, 22122, (4) there are tori forming a stable island aroundthe shortest periodic orbit. And, based on the phase space structure and well-known general11ynamics near triple ollision.we explained main features of scattering time signals.Using the relevance of CTCOs to fluctuations in scattering cross sections owing to reso-nances, quantum manifestations of the classical dynamics were predicted on the total scat-tering cross section such that (1) fluctuation in the cross section as a function of the totalenergy would be chaotic, (2) the amplitude of the fluctuation decays algebraically as theenergy approaches the three-body breakup threshold, and (3) the Fourier transform of thefluctuation would reflect the phase space structure through absence of peaks representedby the symbol sequences c112c, c122c, c221c, c211c, etc. including the forbidden symbolsequences.This work is considered as a first step to full understanding of classical, semiclassical andquantum dynamics of positron-hydrogen system, and we suggested several subjects to bedone in future. In addition, it would be a kind of challenge to find a quantum mechanicalmenifestation of the structural invariance of the phase space under exchange of the dynamicalvariables of proton with those of positron.
Acknowledgments
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