Journal of Modern Physics
Vol.4 No.5(2013), Article ID:31609,6 pages DOI:10.4236/jmp.2013.45090

Long-Range Correlation in Positron-Hydrogen Scattering System near the Threshold of Ps(n = 2) Formation

Chi Yu Hu1, David Caballero1,2

1Department of Physics and Astronomy, California State University, Long Beach, USA

2Claremont Graduate University, Claremont, USA


Copyright © 2013 Chi Yu Hu, David Caballero. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Received February 24, 2013; revised March 28, 2013; accepted April 20, 2013

Keywords: Faddeev; Resonance; Cross Section; 3-Body Scattering


This is a short report of a recently uncovered resonant phenomenon. The modified Faddeev equation that correctly includes all six open channels is used. The calculation is carried out in s-partial wave. We report a number of resonant peaks in the elastic cross sections as well as the wave amplitudes involved. This is the energy region where the Starkeffect induced electric dipole energy split in the target dominates the physics and the Long-Range behavior of the 3-body scattering system. It is found that when the center of mass collision energy in the new channels is in integer proportion to the corresponding electric dipole energy split, Bremsstrahlung photon mediated resonant scattering occurs. The corresponding wave amplitudes deform into wave-packets hundreds to thousands of Bohr radii in width. The physical implication of this phenomenon will be discussed.

1. Introduction

In a previous work [1], resonant formation of anti-hydrogen was reported for the charge-conjugate rearrangement scattering channels

However, the physical origin of these resonances was not discussed with sufficient detail. In this work, we investigate the same resonances using the elastic scattering cross sections with the intent to identify the physical mechanism responsible for these resonances.

Gailitis and Damburg [2] discussed similar resonances in electron-Hydrogen scattering systems. They conjectured that this was a special case of Levinson theorem [3]. Since the induced dipole potential well produced a number of well known “bound-states”, the Feshbach resonances [4-10] just belowed the threshold of formation, then Levinson theorem predicts the same number of phase shift oscillations exist just above the same threshold. Present calculation indeed confirms that their conjecture was correct.

2. Brief Description of the Calculation

For the present s-wave calculation, there are six open channels in this energy region. The six elastic scattering channels are:







The first (1)-(3) belong to Faddeev channel #1, the next (4)-(6) belong to Faddeev channel #2. In the figures, their cross sections are represented by while i = 1, 6 in the order listed above.

Thirty four energy points are calculated in an energy range of. The modified Faddeev equations are solved using a combination of partial wave expansion and quintic spline collocation method [1,11,12]. In this representation the s-partial wave solutions are obtained by solving a linear equation system, with N = 387936 − 488808. N is the number of linear equations.

The cut-off distance for the first Faddeev channel is 500 α0, for the second Faddeev channel is 1000 α0. α0 is the Bohr radius. The two body bound states cut-off distances are determined separately such that the grids faithfully reproduce the Hydrogen or Positronium binding energy and wave function up to.

3. Presentation of the Results

The six dimensional matrix, and all cross section matrix are calculated. is the phase shift matrix element for the incoming channel i and the outgoing channel j. The indexes i, j run from one to six according to the order listed in Equations (1)-(6).

and are shown in Figure 1. Both curves show that at a number of energy points the phase shift becomes zero from to. While between a pair of adjacent crossing, the phase shift suddenly jumps from to. This shows that the incoming particle is subjected to sudden repulsion while approaching resonant configuration.

Figures 2 and 3 show that all the elastic cross sections

Figure 1. The diagonal matrix elements and are plotted against, the energy of channel (1). Singularities occur between E1 = 0.874694 − 0.874695Ry and for both and respectively.

Figure 2. The elastic cross sections of channels 1-4 are plotted against, the energy of channel (1). All four cross sections vary little in the energy range plotted.

Figure 3. The elastic cross sections of channels 5 and channel 6 are plotted against, the energy of channel (1). On the energy range plotted, channel 5 displays one large resonant peak. Located between Two resonant peaks are found in channel (6) located at and respectively.

are relatively smooth across the energy region except and. Apparently, there is a large cross section peak associated with each K-matrix singularity at slightly shifted energy locations. The K-matrix singularity associated with the much narrower third cross section peak at the lowest energy side is too narrow to be calculated at the present energy grid.

For easy comparison with previous references, the energy of channel (1), , is used in Figures 1-3. is measured from the threshold. This energy differs slightly from the infinite proton mass values. The scattering wave function of the 3-body system is divided into two Faddeev amplitudes . are the Jacobi vectors for the and are the Jacobi vectors for the.

In the energy region considered, all elastic cross sections involving Faddeev amplitude, namely the channels (1), (2), and (3), are relatively smooth (see Figure 2). Resonances appear only in channels (5) and (6), (see Figure 3). They belong to Faddeev amplitude .

Channels (5) and (6) are the newly opened channels with energy very close to the threshold of formation. Except in the immediate neighborhood of the K-matrix singularities, they are poorly coupled with the channels in The energies of the latter channels lie orders of magnitudes higher above their respective thresholds than that of the new channels. As a result, the Faddeev amplitudes of channel (5) and (6) are three to four orders of magnitude larger than that belongs to. Numerically the total wave function

, thus only are used in the Figures below. However, Figures 1 and 3 show that the coupling of all six channels produces noticeable energy shifts between the peaks of the cross sections in Figure 3 and the corresponding K-matrix singularities in Figure 1.

Figures 4 and 5 display plots of for channels (5) and (6) respectively at an energy . This energy is near the peak of the first resonance to the right in Figure 3. Figure 6 displays a plot of for channel (5) at an energy

Figure 4. This is the graph of the wave amplitude in channel 5 at near one of the maximum of. It is noted, the part of the wave amplitude nearest to the target is much longer and wider than that of a De Broglie wave amplitude with the same wavelength displayed in Figure 5.

Figure 5. This is the graph of the wave amplitude in channel 6 at. It retains a near normal De Broglie wave structure with. However it may change at a slightly different energy.

Figure 6. This is the graph of the wave amplitude in channel 5 at with. It is noted, this wave packet is directly related to the second cross section resonance in Figure 3.

. This energy is near the peak of the second resonance. Figure 7 displays plot of for channel (6) at an energy. This energy is near the peak of the third resonance at the lowest energy end.

A systematic examination of such wave function graphs reveals a simple physical mechanism as the source of the cross section oscillation displayed in Figure 3. The Coulomb field of the incoming proton splits the degenerate energy levels of and into two energy levels, a case of the Stark effect [13], with an energy split. is the (induced) permanent electric-dipole moment of the Positronium target, is the coulomb field of the proton at the target.

Figures 1-6 suggest that resonant excitation of the target energy levels takes place when the channel energy in atomic energy unit satisfies the following quantum condition:


is the channel energy E at the peak of the mth resonance from the cross section graph Figure 3.

In the mass-scaled Jacobi coordinate system the electric Dipole moment of has the numerical value. y is the radial coordinate of the incoming proton. The meaning of is defined in the next paragraph.

Figure 1 reveals the underlying physics that enables the resonant transfer of energy from the incoming De Broglie wave to the target. When the colliding system

Figure 7. This is the graph of the wave amplitude in channel 6 at near the third cross section peak with. This wave packet is cut-off at before it reaches maximum located at ~1305.9.

approaching the resonant configuration given by Equation (7), Figure 1 reveals that sudden repulsion compels the incoming De Broglie wave to convert all its energy, , to a photon which is readily absorbed and sends the target to its excited state while the incoming “particle” is now a De Broglie wave packet centered at a distance

from the target. This photon is the quantum equivalent of the well known Bremsstrahlung Radiation generated when a charged particle is subjected to deceleration that distorts the particle’s coulombic field lines with transversal components such that an electromagnetic wave is generated.

A dimensionless form of (7) is:


is the De Broglie wave length at energy.

The wave packets presented in Figures 4, 6 and 7, provide a new way to estimate the life-time, , of the resonances. Since the De Broglie wave can not return without the disappearance of the wave packets and the return of its energy through a reverse process from that discussed above.

From Figures 4, 6 and 7, the widths of the wave packets are for the first three resonances. An estimate of the life-time of the resonances can be made using a pair of uncertainty principle. The life-times obtained are as good if not better than that obtained using traditional methods [14], Namely


From Figure 3 it is estimated that the peak of the cross section of the three resonances located at the first channel energies 0.875Ry, 0.87465Ry and 0.87454Ry for and 3 respectively. The corresponding of Equation (7) are 2.7218 × 104 a.u., 0.9718 × 104 a.u. and 0.4217 × 104 a.u. for and 3 respectively. Giving, the location of the peak of the wave packet and the life time of the resonance are calculated and listed in Table 1.

Figures 4 and 6 give a three dimensional view of the first two wave packets. It is not easy to project from this orientation. However, using a two dimensional view along the y axis, a good estimate of and is made, they agree with the numbers in Table 1 within a few percent. is located outside the range of cut-off in Figure 7. Direct measurement of is not possible. Estimated value is consistent with the value in Table 1.

4. Estimation of Upper Bounds for Higher Thresholds with n > 2

It is unlikely at the present time to calculate all possible resonances. The role these resonances play in the physical world can be better understood from the following consideration. The Stark effect no longer exists when is larger than the fine structure energy which removes the coulomb degeneracy in the energy levels. Table 2 lists the fine structure energies for where n is the principle quantum number for positronium atom. The corresponding De Broglie wave length and are the upper bounds for and respectively.

As the collision energy increases the resonances appear near each and every thresholds. As n increases, the density of energy levels increases rapidly, Tables 1 and 2. Show that increases rather rapidly as well. Equation (8) shows that the center of the wave packet increases even more rapidly due to the

Table 1. Calculated values of from (7)-(9).

Table 2. Fine structure energy and upper bounds for, and.

increase in electric dipole moments of.

Consequently, these resonances enable the three-body colliding system to take up considerable amount of physical space for a time comparable or larger than that of most atomic processes.

5. Discussion

Gailitis and Damburg [2] discussed this type of resonance fifty years ago. For ease of reference, let’s call it a Gailitis resonance. The Gailitis resonances appear only slightly above the threshold of a degenerate new channel. This is very different from the much investigated Feshbach resonances which can be found only slightly below such a threshold. As a result, the Gailitis resonances, if found occasionally [1], have been ignored as ghostly structures of numerical calculations. However, advances in supercomputers enable a complete calculation of all the properties of the resonances using the Modified Faddeev Equation without any intermediate approximations. In addition, the wave functions of individual resonances are readily rendered in 3-D graphs. From the graphs, the wave functions reveal much more interesting physics then just the energy and the width of the resonance. A comparison of the properties of these two types of resonances is summarized in Table 3.

It is shown that the Gailitis resonances span consider-

Table 3. Comparison of properties of Feshbach and Gailitis resonances.

ably larger physical space with much longer lifetimes compared to Feshbach resonances. As with Feshbach resonances, Gailitis can also be used as a similar tool to probe nature, but for studying physical spaces and timescales that are comparably larger. The sizes of Gailitis resonances are comparable to nano-particle systems. The lifetimes of Gailitis resonances are comparable to or can exceed that of most atomic processes.

These long lifetimes open up new and interesting possibilities for study. Possible processes involve interactions other than the Coulomb force or possible interactions with the environment external to the three-body system. As an example of the former, consider the threebody coulombic scattering of positron—hydrogen that produces in-flight positron—electron annihilation peaks near every Feshbach resonance as investigated in [15]. The positron-electron annihilation in flight is due to the weak interaction only. The large wave packet formed during the resonant duration is responsible for such external activities. The long lifetime is a direct result of a large wave packet, as described in Sections 3-4.

The Stark-effect is a universal phenomenon. These kind of resonances must occur whenever degeneracy exists in the target in coulombic scattering. Therefore the target is not necessarily an atom. The resonance induced longrange correlations in coulombic scattering systems encourage possible intriguing natural microscopic processes to take place during their life-time.

6. Acknowledgements

The authors thank Dr. Anand Bhatia for helpful discussions. This work is supported by a generous XSEDE resource allocation from the Texas Advanced Computer Center (TACC), at the University of Texas, Austin.


  1. C. Y. Hu, D. Caballero and Z. Papp, Physical Review Letters, Vol. 88, 2002, Article ID: 063401. doi:10.1103/PhysRevLett.88.063401
  2. M. Gailitis and R. Damburg, Journal of Experimental and Theoretical Physics, Vol. 17, 1963, p. 1107.
  3. N. Levinson, K. Dan and S. Vidensk, Matematisk-Fysiske Meddelelser Konglige Danske, Vol. 25, 1949, p. 9.
  4. A. K. Bhatia, Physical Review A, Vol. 9, 1974, p. 9. doi:10.1103/PhysRevA.9.9
  5. A. K. Bhatia and Temkin, Physical Review A, Vol. 11, 1975, p. 2018. doi:10.1103/PhysRevA.11.2018
  6. Y. K. Ho, Bhatia and Temkin, Physical Review A, Vol. 15, 1977, p. 1423. doi:10.1103/PhysRevA.15.1423
  7. Y. K. Ho, Physical Review A, Vol. 35, 1987, p. 3169. doi:10.1103/PhysRevA.35.3169
  8. T. T. Gien, Journal of Physics B, Vol. 29, 1996, p. 2127. doi:10.1088/0953-4075/29/10/020
  9. Z. C. Yan and Y. K. Ho, Physical Review A, Vol. 77, 2008, Article ID: 030701.
  10. A. K. Bhatia, Physical Review A, Vol. 86A, 2012, Article ID: 032709.
  11. C. Y. Hu, Journal of Physics B: Atomic, Molecular and Optical Physics, Vol. 32, 1999, p. 3077. doi:10.1088/0953-4075/32/12/323
  12. C.-Y. Hu, Physical Review A, Vol. 59, 1999, p. 4813. doi:10.1103/PhysRevA.59.4813
  13. L. I. Shiff, Quantum Mechanics, Vol. 156-158, 1949.
  14. C. Y. Hu, S. L. Yakovlev and Z. Papp, Nuclear Instruments and Methods in Physics Research, Vol. B247, 2006, p. 25.
  15. S. L. Yakovlev, C. Y. Hu and D. Caballero, Journal of Physics B: Atomic, Molecular and Optical Physics, Vol. 40, 2007, p. 1675. doi:10.1088/0953-4075/40/10/003