CTMC Investigation of Capture and Ionization Processes in P + H(1s) Collisions in Strong Transverse Magnetic Fields

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1 Commun. Theor. Phys. 63 (2015) Vol. 63, No. 4, April 1, 2015 CTMC Investigation of Capture and Ionization Processes in P + H(1s) Collisions in Strong Transverse Magnetic Fields HE Bin (Ê), 1, WANG Jian-Guo ( Á), 1 and R.K. Janev 2 1 Data Center for High Energy Density Physics Research, Institute of Applied Physics and Computational Mathematics, Beijing , China 2 Macedonian Academy of Sciences and Arts, P.O. Box 428, 1000 Skopje, Macedonia (Received October 27, 2014; revised manuscript received January 8, 2015) Abstract The collision processes of proton with H(1s) atoms, which is embedded in strong transverse magnetic fields perpendicular to the initial velocity of the projectile, are studied with the classical trajectory Monte Carlo method in the energy range 25 kev/u 2000 kev/u and B 10 4 T. It is found that the charge exchange cross section is decreased while the ionization cross section is increased significantly. The physics of magnetic field effects is analyzed by the time evolution of electron energy and trajectories, and it is found that these effects are induced by the diamagnetic term in the interaction, continuum electron trapping in the target regions and the Lorentz force. The velocity distributions of the ionized electrons, significantly influenced by the applied fields, are also presented. PACS numbers: Fa Key words: classical trajectory Monte Carlo, transverse magnetic fields, capture, collision 1 Introduction The study of atomic collision processes in magnetic fields may be applied in astrophysics and the fundamental physics, for example, the strong magnetic field may affect the formation of anti-hydrogen in the collision of antiproton-positronium. [1] Usually the magnetic field affects both the electronic structure of colliding system [2 6] and its collision dynamics. The ion-atom collisions may be nontrivial when a diamagnetic energy of a high excited atom, exerted by a typical laboratory field with the strength of a few Tesla, becomes comparable to the Coulomb energy. [7 8] The ionization mechanism in ion-rydberg atom collisions becomes much more complex, [9 10] and even the distributions of the momentum and energy for ionized electrons become quite different when a magnetic field is applied. [11] Numbers of theoretical works have been devoted to study the ion-atom collisions in magnetic fields in the quantal or classical methods. The quantal studies include the atomic orbital close-coupling (AOCC) method [12 13] and the molecular orbital close-coupling (MOCC) method, [14 15] and these works were focused on the magnetic field effects upon the electron capture process. The classical trajectory Monte Carlo method (CTMC) method has been used to study both the capture and ionization processes in He 2+ + H(1s) system [16] for strong magnetic fields, but no significant magnetic effects were found. Later, the collisions involved with the high-rydberg atoms was investigated by the CTMC method [9 11] and a significant change of the capture cross section has been found. These studies are restricted in the case of longitudinal magnetic field, where the field is parallel with the projectile velocity, so that some symmetry properties are reserved. But for transverse magnetic field the symmetry properties are destroyed, which makes it difficult to study the related processes in quantum theory. In a recent work, [17] we studied the charge transfer, excitation and ionization processes in the He 2+ +H and C 6+ +H collisions in both the longitudinal and transverse magnetic fields. We found the transverse magnetic field leads to the significant reduction for the charge transfer process and significant increasing for the excitation and ionization processes. Especially we found that the magnetic field significantly changes the ionization mechanism and the momentum distribution of the ionized electrons. Hydrogen is the most abundant element in the universe and its isotopes are the fusion materials in laboratory, and meanwhile different from the He 2+ +H and C 6+ +H collision systems, the H + +H collision is a symmetric system, it is interesting to study the difference in ionization mechanism for the asymmetric and symmetric systems and it is also important to provide the H + +H collision data in the strong magnetic field for the applications in astrophysical and laboratory environments. In this work we will investigate the ionization and capture process in the collision of proton with H(1s) in detail in an external strong magnetic field by CTMC method. The related cross sections will be calculated and the elec- Supported by National Natural Science Foundation of China under Grant Nos , , , , , the Natoinal Basic Research Programm of China under Grant No. 2013CB922200, and the Science and Technology Foundation of Chinese Academy of Engeering Physics under Grant No. 2014B hebin-rc@163.com c 2015 Chinese Physical Society and IOP Publishing Ltd

2 500 Communications in Theoretical Physics Vol. 63 tron trajectories will be analyzed to explore the possible ionization mechanism formed by the field. We will see that the longitudinal Lorentz force and the diamagnetic term in the interaction potential have important influence upon the ionization process. The paper is organized as follows. The next section devotes a brief account of the computational method. In Sec. 3, the related cross sections are presented and the influence of the field upon the cross sections will be analyzed. In Sec. 4, the detailed ionization processes will be studied by the time evolution of electron trajectories to have better understanding of the relevant mechanisms. In Sec. 5 the results on energy and momentum distributions of emitted electrons are presented and discussed. Finally some conclusions are summarized. Atomic units (e = m e = = 1) are used in this work unless otherwise explicitly indicated. 2 A Brief Description of CTMC Method It is well known that the CTMC method has been widely and successfully utilized to study the heavyparticle collisions from intermediate to high projectile energy range, in which all the competitive processes can be explicitly and simultaneously taken into account. In the present work this method is still used to do the relevant research. For convenience to discuss the problems let us introduce the parameter γ = B/B 0 to measure the strength of magnetic field B, where B 0 = T is the field for which the electron-cyclotron frequency of the electron is equal to the atomic unit of frequency. [16] In our simulation the homogeneous magnetic field B is set to be in the x-direction and the initial projectile velocity is in the z-direction. The CTMC method simulates the ion-atom collision by sampling trajectories computed from a large ensemble of initial projectile-target configurations (see, e.g., Refs. [19 20]). The collision processes are considered as a three-body problem, that is, the incident proton, target nucleus H + and the active electron in target initially. The initial electronic orbits on the target are prepared in such a way as to mimic the quantum momentum distributions. In our simulation for γ 0.2, the magnetic shift of the energy of H(1s) state is smaller than 2% (see, e.g., Refs. [3], [13]) and, as argued in Ref. [10], the electron wave function is only weakly perturbed by such fields. This means that for γ 0.2 the classical electron momentum distribution remains close to the quantum-mechanical one. [18] So we still adopt the field-free micro-canonical distribution to represent the initial electron state. We also notice that recently a revised procedure to produce the classical initial state micro-canonical distribution of a bound atomic state in the presence of magnetic fields of considerably higher strength [9 10] was proposed. Initially the projectile moves in a longitudinal direction (or z-direction) and the origin of the coordinate system is at the initial position of the nuclei of H(1s) atom. The motion of all the particles is then determined by an iterative solution of Hamilton s equations of motion. The integration of the motion equations is carried out from time t = 0 up to the time when the distance between the projectile and the nuclei of the target R pt is sufficiently large to ensure that their interaction is negligible. At the end of each trajectory, the relative classical binding energies, the total electron energy relative to the target (E et ) and to the projectile (E ep ), are calculated to judge if a reaction (charge transfer when E et > 0 and E ep < 0, ionization when E et > 0 and E ep > 0, elastic scattering, etc.) occurred. Here the total energies E et and E e are expressed as E et = (1/2)v 2 e 1/r et and E ep = (1/2)( v e v) 2 1/r ep, where v e is the electron velocity, v is the projectile velocity, and the indices e, p and t refer to the electron, projectile and the target proton, respectively. In the calculations, the total number of trajectories is taken to be at least and the maximum of the impact parameter was chosen high enough to get converged results and minimize the statistical error (to less than 5%). In the present work the impact parameter varies between 0.0 and Total Cross Sections The CTMC results of the charge transfer and ionization cross section for different magnetic fields γ x (the subscript x in γ means that the transverse magnetic field) are shown in Figs. 1(a) 1(b), together with the field-free (γ = 0.0) case. The two figures obviously indicate that the cross sections are greatly altered by the transverse magnetic field especially for the low projectile energies. On the one hand the charge exchange cross section is about 4 or 5 times reduced with increasing the field strength in the entire energy range investigated; on the other hand the ionization (Fig.1 (b)) cross section increases with the increasing of field strength, such as at E = 25 kev/u, the ionization cross section for γ x = 0.2 case is about 4 times of the field-free cross section. As the projectile energy increases, the magnetic effect on total ionization cross sections decreases gradually. The excitation process, which is not shown here, is similar to the ionization process. In addition, the decreasing of the charge transfer cross section is about equal to the sum of the increasing of the ionization and excitation cross section. When E 80 kev/u, it becomes very small for the influence of the magnetic field upon total ionization cross section. This is understandable that in this energy region the capture cross section is much smaller than the ionization cross section so that its reduction could not lead to the significant increasing of the ionization cross section. The great change of the related cross sections at low projectile energy can be understood qualitatively if we notice the electron potential involved with the external transverse magnetic field V ( r, R, B) = 1 r 1 r R γ xl x γ2 x (y2 + z 2 ), (1)

3 No. 4 Communications in Theoretical Physics 501 where r and R are the electron and projectile radius vectors with respect to target nucleus, respectively, l x is the projection of electron angular momentum on the field direction and independent on the electron coordinates. If we take the coordinate of R as (b, 0, z 0 ), the potential (1) for γ x = 0.2 can be plotted, which is shown in Fig. 2 for b = 6, z 0 = 30, and y = 0. Obviously a high potential barrier, caused by the term of diamagnetic potential, is formed between the two Coulomb centers and the height of this barrier obviously is proportional to γ 2 x. When the projectile moves towards to the target, the electron is accelerated. The magnetic field can make the electron confined easily between the two protons and accelerated by the projectile. If the electron can be captured by the projectile, it must have high energy enough to stride over the barrier. The barrier becomes higher and higher rapidly when the distance between the projectile and the target gets farther and farther away. Since the barrier is too high, most electrons could not stride over the barrier but they can be easily ionized or excited finally. This leads to a dramatic reduction of electron capture cross section and the increasing of the ionization and excitation cross sections with the increasing of the transverse magnetic field. Also it makes the decreasing of the charge transfer cross section close to the sum of the increasing of the ionization and excitation cross section. barrier due to (1/8)γ 2 x y2 corresponding to large impact parameters. The collisions with large impact parameters give dominant contribution to the electron capture process at low energies in the field-free case. Now due to the transverse magnetic field the barrier (1/8)γ 2 xy 2 is formed and must be stridden over by the captured electrons. Therefore, these captured electrons must have a high velocity component in y-direction (v ey ). Such a high v ey would be restrained by the longitudinal Lorentz force γ x v ey. This force transfers the transverse motion into longitudinal motion, which hampers the electron to be close further to the projectile. So it reduces the possibility for the electron to be captured, which will be shown more in next section. In summarize the huge barrier caused by the transverse magnetic field results in the great suppression of the capture process and the increasing of ionization process. Fig. 2 3-D plots of the potential V (r; B α) with the transverse (α = x) magnetic fields for the P + H(1s) collision system for l α = 0, y = 0, b = 6.0, and z 0 = Fig. 1 Total cross sections for electron capture (a) and ionization (b) in P + H(1s) collisions in transverse (γ x = 0.05, 0.1, 0.2) external magnetic fields and in the field-free case (γ = 0.0). Besides the high potential barrier term due to (1/8)γ 2 xz 2, it is easy to see from Eq. (1) there is also a 4 Time Evolution of Inelastic Events in a Magnetic Field In order to know more about the dynamics of ionization processes in an external magnetic field, a lot of electron trajectories are analyzed, which include the time evolution of electron energy relative to the target (E et ) and projectile (E ep ) as well as the electron distance from the target (r et ) and the projectile (r ep ). Three typical trajectories which correspond to three ionization mechanisms are found: direct ionization (the electron rapidly becomes free and escapes from the target field), trapping by the target for some time and suddenly escape from that region in the field (x-) direction, and Lorentz ionization of captured electron. Here due to the longitudinal Lorentz force caused by the transverse magnetic field, two relevant mechanisms playing an important role in the case of longitudinal magnetic field disappears, which include the long-time trapping by the projectile and the saddle-point ionization. [11] The trajectories corresponding to the three mechanisms mentioned above are shown in Figs. 3, 4 and

4 502 Communications in Theoretical Physics Vol. 63 5, respectively. All these are calculated at E = 25 kev/u and γ x = 0.2, and they belong to capture events if γ = 0.0. Fig. 3 Time evolutions of electron energies E et, E ep (a), electron distances r et, r ep (b) and the components of Coulomb and Lorentz force (c) of a direct ionization event at E = 25 kev/u and γ x = 0.2 with electron trajectory trapped in the projectile region. Here F L-P is the component of Lorentz force, acting on the electron, which points from the electron to the projectile, and F L-N is the component that is perpendicular to F L-P. Similar definitions of F C-P and F C-N are made for Coulomb force. The direct ionization is shown in Fig. 3: the electron becomes free when the projectile approaches to the target (Fig. 3(a)) and it moves far away from the target nucleus and the projectile immediately (Fig. 3(b)). After the electron is free, E ep still oscillates due to the oscillation of v ez caused by the longitudinal Lorentz force. Since the ionization happens around t = 20.0, the evolutions of the Coulomb force (F C-P, F C-N ) and Lorentz force (F L-P, F L-N ) are plotted in Fig. 3(c) within the time range: 10.0 t Here P and N mean the component of the force acting on the electron, which points from the electron to the projectile and is perpendicular to that direction. When time is close to 20.0, F C-N is not smaller than F C-P since both r et and r ep are small. Meanwhile F L-N is much larger than F L-P, which means that totally speaking the Lorentz force is not good for the capture of the electron although sometime it has a component towards the projectile. When t > 20.0, F L-P is in the reverse direction of F C-P and F L-N is much larger than F L-P, which are unfavorable to the capture of the electron. When t > 30.0, both F C-P and F C-N become very small because both r et and r ep are very large. Finally it becomes an ionization event due to the large F L-N generated by the transverse magnetic field. Figures 4 illustrate the ionization mechanism with electron trapped by the target. Although the energy of ejected electron E et is 0.5 (cf. Fig. 4(a)), the electron stays in the target region for a long time inter v al (t , cf. Fig. 4(b)) with its motion affected by both the target nucleus and the magnetic field. Electron trajectory trapping near the projectile is not possible in this case since the Lorentz force drives the continuum electrons to escape along the x-direction. Similar evolutions of the trajectory with Fig. 3(c) are shown in Fig. 4(c) within the important ionization time range: 10.0 t When the time is close to 20.0, F L-N is much larger than F L-P although both F L-P and F C-P are in the same direction. When time > 20.0, F L-P is in the reverse direction of F C-P, and F L-N is much larger than F L-P. All these are in favor of ionization instead of the electron capture. The Lorentz ionization mechanism is demonstrated in Fig. 5. The electron is captured by the projectile at about t 20 (Figs. 5(a), (b)), then it is driven away by the Lorentz force and rapidly ionized at t 300 which moves in a constant value of v ex (of 0.45, which is not shown here). For the process the evolutions of the Coulomb and Lorentz force are plotted in Fig. 5(c). When time is close to 20.0, F L-N is much larger than F L-P although both F L-P and F C-P are in the same direction. Here in most time F L-P is in the reverse direction of F C-P and F L-N is close to F C-P. Once r ep becomes larger when t > 300.0, F C-P is decreased and smaller than F L-N so that the electron is no longer captured and becomes ionized. So it is the Lorentz force, FL = v B, ionizing the loosely bound captured electrons. By analyzing about 2000 ionization trajectories the direct, trapped trajectory and Lorentz ionization mechanisms are found to share about 22.5%, 74.1% and 3.4% in the total ionization, respectively. It is noted that these values are equal to 14%, 82%, and 4%, respectively for the collision of He 2+ + H(1s). [17]

5 No. 4 Communications in Theoretical Physics 503 Fig. 4 Same as in Fig. 3, but for an ionization event with electron trajectory trapping in the target region. 5 Velocity Distributions of Emitted Electrons in the Field Direction In order to know more deeply about the ionization under the external magnetic field, let us see the velocity distributions of emitted electrons, particularly of the component in the field direction. The distributions of the velocity components f(v ex ), f(v ey ), and f(v ez ) that will be discussed in the present section are all defined as f(a) = N(a) N(a)da, (2) where N(a) is the number of events with the quantity a having values between a and a+ a, and a is sufficiently small. A similar normalization is also done for the 2-D distribution f(v ex, v e-norm ), where v e norm = vey 2 + vez 2 is used. Fig. 5 Same as in Fig. 3, but for Lorentz ionization of a captured electron. Figure 6 plots the distribution of v ex (panel (a)) and v e-norm (panel (b)) velocity component of the emitted electrons with γ x = 0.2 together with its distribution in the field-free case. Three distinct features of the distribution can be seen in these figures caused by the strong magnetic field: (i) strong increasing of backward ejected electrons (v ez < 0); (ii) the disappearance of the peak in the v ez distribution close to the projectile velocity and (iii) appearance of a dip in the distribution around v x 0 in Fig. 6(a). All these are understandable. Usually because of the magnetic field the trapped electrons must have many collisions with the target nucleus before they escape from it (see Fig. 4). Meanwhile the escaped electron velocity perpendicular to the magnetic field always oscillates because of the magnetic field. All these result in the symmetry distribution of v ez and v ey around 0.0.

6 504 Communications in Theoretical Physics Vol. 63 The longitudinal Lorentz force restrains the trapping of the electrons by the projectile and it makes v ez and v ey transfer into each other. This leads to the observed second feature. The electron velocity component in the direction of the magnetic field should not be small for the electron to overcome the Coulomb barrier of the target and escape away, which leads to the formation of the observed dip. Fig. 6 Electron velocity distributions of ionized electrons in a transverse magnetic field of strength γ x = 0.2 ejected in the field direction (a) and in the direction perpendicular to the field (panel (b)) at E = 25 kev/u. Corresponding distributions in the field-free case (γ = 0.0) is also shown in panel (a). 6 Conclusions So far the capture and ionization processes in collisions of proton with hydrogen atom in external transverse magnetic fields γ x 0.2 have been investigated by CTMC. The related cross sections are presented and three typical trajectories are analyzed including the velocity distributions of ionized electrons. The following conclusions can be drawn: (i) With the increasing of the transverse magnetic field the total electron capture cross section is highly reduced associated with the dramatic enhancement of the ionization cross section. All these are because of the effective diamagnetic potential barrier between the Coulomb potential wells of the nuclei as well as the Lorentz force. (ii) Time evolution of lots of trajectories reveals the transient formation of a double potential well for the electron motion at low collision energies, in conjunction with the electron cyclotron motion, facilitates the ionization process. Trapping of continuum electrons by the target nuclei due to the cyclotron motion becomes the main ionization mechanism in the transverse magnetic field. (iii) The strong transverse magnetic field has significant influence upon the distributions of velocity components of ionized electrons. The backward emitted electrons are increased and distinct dips around zero for v x appear which is absent in the field-free case. References [1] J. Liu, E.Y. Sidky, Z. Roller-Lutz, and H.O. Lutz, Phys. Rev. A 68 (2003) [2] R.H. Garstang, Rep. Prog. Phys. 40 (1977) 105. [3] W. Rösner, G. Wunner, H. Herold, and H. Ruder, J. Phys. B: At. Mol. Phys. 17 (1984) 29. [4] D.M. Larsen, Phys. Rev. A 25 (1982) [5] G. Wunner, H. Herold, and H. Ruder, Phys. Lett. A 88 (1982) 344. [6] U. Wille, Phys. Rev. A 38 (1988) [7] U. Fano, F. Robicheaux, and A.R.P. Rau, Phys. Rev. A 37 (1988) [8] E.A. Solov ev, Sov. Phys. JETP 55 (1982) [9] S. Bradenbrink, E.Y. Sidky, Z. Roller-Lutz, and H.O. Lutz, J. Phys. B: At. Mol. Opt. Phys. 30 (1997) L161. [10] S. Bradenbrink, E.Y.Sidky, Z. Roller-Lutz, and H.O. Lutz, Phys. Rev. A 55 (1997) [11] S. Bradenbrink, H. Reihl, Z. Roller-Lutz, and H.O. Lutz, J. Phys. B: At. Mol. Opt. Phys. 30 (1997) [12] S. Bivona, B. Spagnolo, and G. Ferrante, J. Phys. B: At. Mol. Phys. 17 (1984) [13] S. Bivona and M.R.C. McDowell, J. Phys. B: At. Mol. Phys. 20 (1987) [14] U. Wille, Phys. Lett. A 125 (1987) 52. [15] S. Suzuki, N. Shimakura, and M. Kimura, J. Phys. B: At. Mol. Opt. Phys. 29 (1996) [16] T.P. Grosdanov and M.R.C. McDowell, J. Phys. B: At. Mol. Phys. 18 (1985) 921. [17] B. He, J.G. Wang, and R.K. Janev, Phys. Rev. A 79 (2009) [18] V.S. Popov, B.M. Karnakov, and V.D. Mur, JETP 88 (1999) 902. [19] R. Abrines and I.C. Percival, Proc. Phys. Soc. A 88 (1966) 873. [20] R.E. Olson and A. Salop, Phys. Rev. A 16 (1977) 531. [21] R.E. Olson, Phys. Rev. A 27 (1983) 1871.