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Positron cross-field transport due to quasibound states of antihydrogen
NIM B Beam Interactions with Materials & Atoms
Nuclear Instruments and Methods in Physics Research B 261 (2007) 252–254 www.elsevier.com/locate/nimb
Positron cross-field transport due to quasibound states of antihydrogen Y. Ahat, C.E. Correa, C.A. Ordonez
*
Department of Physics, University of North Texas, P.O. Box 311427, Room 110, Denton, TX 76203-1427, United States Available online 11 April 2007
Abstract Within the past few years, experiments have been reported in which antiprotons produced at the CERN Antiproton Decelerator facility were slowed, trapped in nested Penning traps and made to interact with a positron plasma such that antihydrogen was formed. Classical trajectory simulations of the interactions between the antiprotons and positrons have been reported to indicate that positive-energy, quasibound states of antihydrogen can form at a rate that exceeds the rate of formation of stable Rydberg states. The formation of quasibound states may affect the rate of diffusion of positrons across the magnetic field that confines them in the nested Penning trap. Simulations indicate that a binary interaction associated with the formation and disintegration of a quasibound state can cause a shift of the positron’s guiding center that is much larger than the positron cyclotron radius before the interaction. A theory is presented that describes positron cross-magnetic-field diffusion due to quasibound states of antihydrogen. 2007 Elsevier B.V. All rights reserved. PACS: 52.27.Jt; 52.20.Dq; 52.20.Fs; 52.65.Cc; 52.40.Mj Keywords: Antihydrogen; Quasibound states; Magnetized positron plasma; Antiprotons
1. Introduction Nested Penning traps have recently been used to merge together groups of positrons and antiprotons such that antihydrogen atoms are produced [1–4]. A goal associated with future antihydrogen experiments is to produce and trap antihydrogen atoms in sufficient quantities for high-precision antihydrogen spectroscopy and antimatter gravity measurements. A detailed understanding of the interaction and confinement of oppositely signed particles in nested Penning traps would facilitate such experiments. A substantial knowledge base exists concerning collisional transport in single-well Penning traps [5]. However, not much research has been done associated with magnetic (cross-field) confinement in nested Penning traps. In [6], an assessment was reported of ‘‘centrifugal’’ separation of oppositely signed species due to cross-field transport in nested Penning traps. It was found that, for antihydrogen
*
Corresponding author. Tel.: +1 940 565 4860; fax: +1 940 565 2515. E-mail address: [email protected] (C.A. Ordonez).
0168-583X/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2007.04.028
experiments, centrifugal separation should not be important after the positrons and antiprotons are merged. In the work presented here, a theory is developed that describes positron cross-field diffusion due to quasibound states of antihydrogen. Such states have been predicted to form in the presence of a magnetic field and at positive energies [7,8], where the energy of the two-particle system is defined to be zero when the two particles are at rest and separated by an infinite distance. 2. Simulation of a quasibound state Fig. 1 shows the simulated trajectory of a positron that becomes temporarily bound to a motionless antiproton. The large solid dot represents the location of the antiproton. A straight line that is parallel to the magnetic field and that passes through the antiproton is used to represent the z-axis of a coordinate system, which is defined such that the antiproton is located at z = 0. Also drawn in the figure is a circle that is centered at the antiproton and located in the z = 0 plane. (The circle appears as an oval due to the
Y. Ahat et al. / Nucl. Instr. and Meth. in Phys. Res. B 261 (2007) 252–254
Fig. 1. Simulation of a quasibound state. The dimension parallel to the magnetic field (which is parallel to the straight line) is reduced by a factor of 46 relative to the other two dimensions. The parameters used are B = 1 T, T = 4 K, xg = 5 · 107 m, h = 0.4p, gx = 1, gy = 1, gz = 1 and f = 100. (Further details are given in the text.)
angle of view.) The positron approaches the antiproton from the lower left corner of the figure following a helical trajectory. The positron then becomes trapped axially within the potential energy well created by the electric field of the antiproton. During the time the positron is trapped, the positron passes through the z = 0 plane a number of times, and the guiding center of a cyclotron orbit drifts transverse to the magnetic field. The guiding center of a cyclotron orbit is defined as the positron position averaged over its cyclotron motion. Denote Kk (K?) as the portion of the positron’s kinetic energy that is associated with motion parallel (perpendicular) to the magnetic field. K? is approximately constant during the interaction, except when the positron is in close vicinity to the antiproton. If K? becomes larger while passing close to the antiproton, the positron can become trapped. The positron becomes trapped because Kk becomes smaller than the local depth of the potential energy well. The positron can only leave the vicinity of the antiproton if Kk is larger than the local well depth. In the figure, the positron leaves the vicinity of the antiproton when the positron follows the helical trajectory that extends down to the lower right corner of the figure. The circle drawn in the figure serves to show that the positron’s guiding center remains at roughly the same radial distance from the z-axis throughout the interaction. The radius of the circle is chosen to equal the positron’s guiding-center impact parameter. Hence, while the positron is in the vicinity of the antiproton, the positron motion transverse to the magnetic field is describable as a combination of cyclotron
253
motion about a guiding center and azimuthal drift motion of the guiding center at a distance from the z-axis approximately equal to the guiding-center impact parameter. The diameter of the circle in the figure is approximately 16 times larger than the initial cyclotron radius, rc. Thus, the guiding center of the positron shifts by approximately 16rc transverse to the magnetic field as a result of the binary interaction. The simulated trajectory shown in Fig. 1 was calculated by numerically solving the classical equations of motion for the positron. These types of trajectories can be extremely sensitive to the initial conditions, as pointed out in [9,10]. For the trajectory shown in Fig. 1, the initial velocity components of the positron were chosen to have values that would be typical for a positron in a plasma having a temperature of 4 K. A magnetic field of 1 T was used. These are parameters that may be used in future antihydrogen trapping experiments. In the notation of [7], the parameters used for the simulation results shown in Fig. 1 are B = 1 T, T = 4 K, xg = 5 · 107 m, h = 0.4p, gx = 1, gy = 1, gz = 1 and f = 100, where B is the magnetic field strength, T is the temperature of the positron plasma, xg is the positron guiding-center impact parameter, h is the positron’s initial polar angle about the guiding center, f is the initial axial separation between the positron and antiproton normalized by xg, gx, gy and gz are initial positron velocity components,pffiffiffiffiffiffiffiffiffiffiffiffiffiffi normalized to the positron thermal speed tth ¼ k B T =m, m is the positron mass and kB is Boltzmann’s constant. The positron time-of-travel is approximately 50 ns for the trajectory shown in Fig. 1. Fig. 2 illustrates the sensitivity of the trajectory to the initial conditions. The parameters used for the simulation results shown in Fig. 2 are the same as those used for Fig. 1, except with T = 4.1 K.
Fig. 2. Same as Fig. 1, except with T = 4.1 K.
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Y. Ahat et al. / Nucl. Instr. and Meth. in Phys. Res. B 261 (2007) 252–254
3. Cross-field diffusion Guiding-center shifts due to quasibound states are modeled as steps in a random walk diffusion process. The associated diffusion coefficient is evaluated as E 1D 2 ðDxÞ n2 rg D¼ ; ð1Þ r;t;t2 2 where Dx is the size of a step in one dimension perpendicular to the magnetic field, n2 is the antiproton density, r is the cross-section for formation of a quasibound state, g = jt t2j is the relative speed between a positron with velocity t and an antiproton with velocity t2 and hir;t;t2 represents an average taken over the cross-section for formation of a quasibound state and averages taken over the velocity distribution of each particle. The following formula has been presented for the cross-section of formation of quasibound states of antihydrogen [7]: !2=3 2 kCm r ¼ pC : ð2Þ k 2L B2 Here, C is a dimensionless constant, which was found in [7] to be C = 1.86, kC is the Coulomb force constant (kC = 1/ (4pe0) in SI units, where e0 is the permittivity of free space) and kL is the Lorentz force constant (kL = 1 in SI units). Eq. (2)pffiffiffiffiffiffiffiffi is restricted in applicability by the condition, rc << r=p, which requires the positron temperature to be much less than 290 K for a thermal positron plasma and a magnetic field of 1 T. Numerous classical trajectory simulations of the type shown in Fig. 1 were run, and they indicate that quasibound states are associated with a limitation on step sizes given by 4r ðDxÞ 6 ; p 2
ð3Þ pffiffiffiffiffiffiffiffi provided rc << r=p. Assuming n2 is uniform, taking the antiprotons to be motionless g = t, considering the positrons to have an isotropic, Maxwellian velocity distribution, and substituting Eqs. (2) and (3) into Eq. (1) gives the following upper limit for the diffusion coefficient: !4=3 kC 1=2 4 5=6 DUL ¼ 4C n2 m ð2pk B T Þ : ð4Þ k 2L B2 pffiffiffiffiffiffiffiffi Here, htit ¼ 2 2=ptth has been used. By way of example, the following parameters are considered: the positron and antiproton densities are n = n2 = 1011 m3, the positron temperature is T = 35 K, and the antiproton temperature is much less than the positron temperature. These parameters are identified in [11] as possibly suitable for future antihydrogen trapping experiments. The upper limit predicted by Eq. (4) is 4.6 · 1010 m2/s for these parameters. It is illustrative to
compare the calculated value with a diffusion coefficient for antiproton cross-field diffusion due to binary interactions that do not form quasibound states. Such a diffusion coefficient is evaluated in [11] to be 1.9 · 1010 m2/s for the same parameters. It is concluded that the positron loss rate due to cross-field diffusion associated with quasibound states should not be significantly larger than the antiproton loss rate due to collision-based cross-field diffusion for the parameters considered here and in [11]. It is also noted that the antiproton loss time scale was estimated to be about four (4) hours in [11]. 4. Concluding remarks It was speculated in [2] that cross-magnetic-field particle diffusion may be significant within the nested Penning traps used for antihydrogen experiments. Work is reported here on positron cross-field diffusion due to quasibound states. Work is reported in [6] on centrifugal separation. Work is reported in [11] on antiproton cross-field diffusion. These three studies provide theory that can be used to assess the effect of cross-field transport in nested Penning traps under conditions that are relevant to the antihydrogen experiments. The anticipated utility of Eq. (4) is for designing antihydrogen experiments for which cross-field diffusion due to quasibound states does not represent a detrimental effect. It should also be noted, however, that there exist other environments where quasibound states may form, such as the atmospheres of white dwarfs and neutron stars [9,10]. Acknowledgments This material is based upon work supported by the National Science Foundation under Grant No. PHY-0244444 and the Department of Energy under Grant No. DEFG02-06ER54883. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
M. Amoretti et al., Nature (London) 419 (2002) 456. G. Gabrielse et al., Phys. Rev. Lett. 89 (2002) 213401. M. Amoretti et al., Phys. Lett. B 583 (2004) 59. J.N. Tan et al., Nucl. Instr. and Meth. B 214 (2004) 22. C.F. Driscoll, F. Anderegg, D.H.E. Dubin, D.-Z. Jin, J.M. Kriesel, E.M. Hollmann, T.M. O’Neil, Phys. Plasmas 9 (2002) 1905. M. Amoretti, C. Canali, C. Carraro, V. Lagomarsino, A. Odino, G. Testera, S. Zavatarelli, Phys. Plasmas 13 (2006) 012308. C.E. Correa, J.R. Correa, C.A. Ordonez, Phys. Rev. E 72 (2005) 046406. C.E. Correa, S.A. Khan, J.R. Correa, C.A. Ordonez, Nucl. Instr. and Meth. B 241 (2005) 165. B. Hu, W. Horton, C. Chiu, T. Petrosky, Phys. Plasmas 9 (2002) 1116. B. Hu, W. Horton, T. Petrosky, Phys. Rev. E 65 (2002) 056212. C.A. Ordonez, J.R. Correa, Nucl. Instr. and Meth. B, these Proceedings, doi:10.1016/j.nimb.2007.04.035.
Nuclear Instruments and Methods in Physics Research B 261 (2007) 252–254 www.elsevier.com/locate/nimb
Positron cross-field transport due to quasibound states of antihydrogen Y. Ahat, C.E. Correa, C.A. Ordonez
*
Department of Physics, University of North Texas, P.O. Box 311427, Room 110, Denton, TX 76203-1427, United States Available online 11 April 2007
Abstract Within the past few years, experiments have been reported in which antiprotons produced at the CERN Antiproton Decelerator facility were slowed, trapped in nested Penning traps and made to interact with a positron plasma such that antihydrogen was formed. Classical trajectory simulations of the interactions between the antiprotons and positrons have been reported to indicate that positive-energy, quasibound states of antihydrogen can form at a rate that exceeds the rate of formation of stable Rydberg states. The formation of quasibound states may affect the rate of diffusion of positrons across the magnetic field that confines them in the nested Penning trap. Simulations indicate that a binary interaction associated with the formation and disintegration of a quasibound state can cause a shift of the positron’s guiding center that is much larger than the positron cyclotron radius before the interaction. A theory is presented that describes positron cross-magnetic-field diffusion due to quasibound states of antihydrogen. 2007 Elsevier B.V. All rights reserved. PACS: 52.27.Jt; 52.20.Dq; 52.20.Fs; 52.65.Cc; 52.40.Mj Keywords: Antihydrogen; Quasibound states; Magnetized positron plasma; Antiprotons
1. Introduction Nested Penning traps have recently been used to merge together groups of positrons and antiprotons such that antihydrogen atoms are produced [1–4]. A goal associated with future antihydrogen experiments is to produce and trap antihydrogen atoms in sufficient quantities for high-precision antihydrogen spectroscopy and antimatter gravity measurements. A detailed understanding of the interaction and confinement of oppositely signed particles in nested Penning traps would facilitate such experiments. A substantial knowledge base exists concerning collisional transport in single-well Penning traps [5]. However, not much research has been done associated with magnetic (cross-field) confinement in nested Penning traps. In [6], an assessment was reported of ‘‘centrifugal’’ separation of oppositely signed species due to cross-field transport in nested Penning traps. It was found that, for antihydrogen
*
Corresponding author. Tel.: +1 940 565 4860; fax: +1 940 565 2515. E-mail address: [email protected] (C.A. Ordonez).
0168-583X/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2007.04.028
experiments, centrifugal separation should not be important after the positrons and antiprotons are merged. In the work presented here, a theory is developed that describes positron cross-field diffusion due to quasibound states of antihydrogen. Such states have been predicted to form in the presence of a magnetic field and at positive energies [7,8], where the energy of the two-particle system is defined to be zero when the two particles are at rest and separated by an infinite distance. 2. Simulation of a quasibound state Fig. 1 shows the simulated trajectory of a positron that becomes temporarily bound to a motionless antiproton. The large solid dot represents the location of the antiproton. A straight line that is parallel to the magnetic field and that passes through the antiproton is used to represent the z-axis of a coordinate system, which is defined such that the antiproton is located at z = 0. Also drawn in the figure is a circle that is centered at the antiproton and located in the z = 0 plane. (The circle appears as an oval due to the
Y. Ahat et al. / Nucl. Instr. and Meth. in Phys. Res. B 261 (2007) 252–254
Fig. 1. Simulation of a quasibound state. The dimension parallel to the magnetic field (which is parallel to the straight line) is reduced by a factor of 46 relative to the other two dimensions. The parameters used are B = 1 T, T = 4 K, xg = 5 · 107 m, h = 0.4p, gx = 1, gy = 1, gz = 1 and f = 100. (Further details are given in the text.)
angle of view.) The positron approaches the antiproton from the lower left corner of the figure following a helical trajectory. The positron then becomes trapped axially within the potential energy well created by the electric field of the antiproton. During the time the positron is trapped, the positron passes through the z = 0 plane a number of times, and the guiding center of a cyclotron orbit drifts transverse to the magnetic field. The guiding center of a cyclotron orbit is defined as the positron position averaged over its cyclotron motion. Denote Kk (K?) as the portion of the positron’s kinetic energy that is associated with motion parallel (perpendicular) to the magnetic field. K? is approximately constant during the interaction, except when the positron is in close vicinity to the antiproton. If K? becomes larger while passing close to the antiproton, the positron can become trapped. The positron becomes trapped because Kk becomes smaller than the local depth of the potential energy well. The positron can only leave the vicinity of the antiproton if Kk is larger than the local well depth. In the figure, the positron leaves the vicinity of the antiproton when the positron follows the helical trajectory that extends down to the lower right corner of the figure. The circle drawn in the figure serves to show that the positron’s guiding center remains at roughly the same radial distance from the z-axis throughout the interaction. The radius of the circle is chosen to equal the positron’s guiding-center impact parameter. Hence, while the positron is in the vicinity of the antiproton, the positron motion transverse to the magnetic field is describable as a combination of cyclotron
253
motion about a guiding center and azimuthal drift motion of the guiding center at a distance from the z-axis approximately equal to the guiding-center impact parameter. The diameter of the circle in the figure is approximately 16 times larger than the initial cyclotron radius, rc. Thus, the guiding center of the positron shifts by approximately 16rc transverse to the magnetic field as a result of the binary interaction. The simulated trajectory shown in Fig. 1 was calculated by numerically solving the classical equations of motion for the positron. These types of trajectories can be extremely sensitive to the initial conditions, as pointed out in [9,10]. For the trajectory shown in Fig. 1, the initial velocity components of the positron were chosen to have values that would be typical for a positron in a plasma having a temperature of 4 K. A magnetic field of 1 T was used. These are parameters that may be used in future antihydrogen trapping experiments. In the notation of [7], the parameters used for the simulation results shown in Fig. 1 are B = 1 T, T = 4 K, xg = 5 · 107 m, h = 0.4p, gx = 1, gy = 1, gz = 1 and f = 100, where B is the magnetic field strength, T is the temperature of the positron plasma, xg is the positron guiding-center impact parameter, h is the positron’s initial polar angle about the guiding center, f is the initial axial separation between the positron and antiproton normalized by xg, gx, gy and gz are initial positron velocity components,pffiffiffiffiffiffiffiffiffiffiffiffiffiffi normalized to the positron thermal speed tth ¼ k B T =m, m is the positron mass and kB is Boltzmann’s constant. The positron time-of-travel is approximately 50 ns for the trajectory shown in Fig. 1. Fig. 2 illustrates the sensitivity of the trajectory to the initial conditions. The parameters used for the simulation results shown in Fig. 2 are the same as those used for Fig. 1, except with T = 4.1 K.
Fig. 2. Same as Fig. 1, except with T = 4.1 K.
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Y. Ahat et al. / Nucl. Instr. and Meth. in Phys. Res. B 261 (2007) 252–254
3. Cross-field diffusion Guiding-center shifts due to quasibound states are modeled as steps in a random walk diffusion process. The associated diffusion coefficient is evaluated as E 1D 2 ðDxÞ n2 rg D¼ ; ð1Þ r;t;t2 2 where Dx is the size of a step in one dimension perpendicular to the magnetic field, n2 is the antiproton density, r is the cross-section for formation of a quasibound state, g = jt t2j is the relative speed between a positron with velocity t and an antiproton with velocity t2 and hir;t;t2 represents an average taken over the cross-section for formation of a quasibound state and averages taken over the velocity distribution of each particle. The following formula has been presented for the cross-section of formation of quasibound states of antihydrogen [7]: !2=3 2 kCm r ¼ pC : ð2Þ k 2L B2 Here, C is a dimensionless constant, which was found in [7] to be C = 1.86, kC is the Coulomb force constant (kC = 1/ (4pe0) in SI units, where e0 is the permittivity of free space) and kL is the Lorentz force constant (kL = 1 in SI units). Eq. (2)pffiffiffiffiffiffiffiffi is restricted in applicability by the condition, rc << r=p, which requires the positron temperature to be much less than 290 K for a thermal positron plasma and a magnetic field of 1 T. Numerous classical trajectory simulations of the type shown in Fig. 1 were run, and they indicate that quasibound states are associated with a limitation on step sizes given by 4r ðDxÞ 6 ; p 2
ð3Þ pffiffiffiffiffiffiffiffi provided rc << r=p. Assuming n2 is uniform, taking the antiprotons to be motionless g = t, considering the positrons to have an isotropic, Maxwellian velocity distribution, and substituting Eqs. (2) and (3) into Eq. (1) gives the following upper limit for the diffusion coefficient: !4=3 kC 1=2 4 5=6 DUL ¼ 4C n2 m ð2pk B T Þ : ð4Þ k 2L B2 pffiffiffiffiffiffiffiffi Here, htit ¼ 2 2=ptth has been used. By way of example, the following parameters are considered: the positron and antiproton densities are n = n2 = 1011 m3, the positron temperature is T = 35 K, and the antiproton temperature is much less than the positron temperature. These parameters are identified in [11] as possibly suitable for future antihydrogen trapping experiments. The upper limit predicted by Eq. (4) is 4.6 · 1010 m2/s for these parameters. It is illustrative to
compare the calculated value with a diffusion coefficient for antiproton cross-field diffusion due to binary interactions that do not form quasibound states. Such a diffusion coefficient is evaluated in [11] to be 1.9 · 1010 m2/s for the same parameters. It is concluded that the positron loss rate due to cross-field diffusion associated with quasibound states should not be significantly larger than the antiproton loss rate due to collision-based cross-field diffusion for the parameters considered here and in [11]. It is also noted that the antiproton loss time scale was estimated to be about four (4) hours in [11]. 4. Concluding remarks It was speculated in [2] that cross-magnetic-field particle diffusion may be significant within the nested Penning traps used for antihydrogen experiments. Work is reported here on positron cross-field diffusion due to quasibound states. Work is reported in [6] on centrifugal separation. Work is reported in [11] on antiproton cross-field diffusion. These three studies provide theory that can be used to assess the effect of cross-field transport in nested Penning traps under conditions that are relevant to the antihydrogen experiments. The anticipated utility of Eq. (4) is for designing antihydrogen experiments for which cross-field diffusion due to quasibound states does not represent a detrimental effect. It should also be noted, however, that there exist other environments where quasibound states may form, such as the atmospheres of white dwarfs and neutron stars [9,10]. Acknowledgments This material is based upon work supported by the National Science Foundation under Grant No. PHY-0244444 and the Department of Energy under Grant No. DEFG02-06ER54883. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
M. Amoretti et al., Nature (London) 419 (2002) 456. G. Gabrielse et al., Phys. Rev. Lett. 89 (2002) 213401. M. Amoretti et al., Phys. Lett. B 583 (2004) 59. J.N. Tan et al., Nucl. Instr. and Meth. B 214 (2004) 22. C.F. Driscoll, F. Anderegg, D.H.E. Dubin, D.-Z. Jin, J.M. Kriesel, E.M. Hollmann, T.M. O’Neil, Phys. Plasmas 9 (2002) 1905. M. Amoretti, C. Canali, C. Carraro, V. Lagomarsino, A. Odino, G. Testera, S. Zavatarelli, Phys. Plasmas 13 (2006) 012308. C.E. Correa, J.R. Correa, C.A. Ordonez, Phys. Rev. E 72 (2005) 046406. C.E. Correa, S.A. Khan, J.R. Correa, C.A. Ordonez, Nucl. Instr. and Meth. B 241 (2005) 165. B. Hu, W. Horton, C. Chiu, T. Petrosky, Phys. Plasmas 9 (2002) 1116. B. Hu, W. Horton, T. Petrosky, Phys. Rev. E 65 (2002) 056212. C.A. Ordonez, J.R. Correa, Nucl. Instr. and Meth. B, these Proceedings, doi:10.1016/j.nimb.2007.04.035.