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TO the POSSIBILITY of BOUND STATES BETWEEN TWO ELECTRONS Alexander A WEPPP031 Proceedings of IPAC2012, New Orleans, Louisiana, USA TO THE POSSIBILITY OF BOUND STATES BETWEEN TWO ELECTRONS Alexander A. Mikhailichenko, Cornell University, LEPP, Ithaca, NY 14853, USA Abstract spins for reduction of minimal emittance restriction arisen from Eq. 1. In some sense it is an attempt to prepare the We analyze the possibility to compress dynamically the pure quantum mechanical state between just two polarized electron bunch so that the distance between electrons. What is important here is that the distance some electrons in the bunch comes close to the Compton between two electrons should be of the order of the wavelength, arranging a bound state, as the attraction by Compton wavelength. We attracted attention in [2,3] that the magnetic momentum-induced force at this distance attraction between two electrons determined by the dominates repulsion by the electrostatic force for the magnetic force of oppositely oriented magnetic moments. appropriately prepared orientation of the magnetic In this case the resulting spin is zero. Another possibility moments of the electron-electron pair. This electron pair considered below. behaves like a boson now, so the restriction for the In [4], it was suggested a radical explanation of minimal emittance of the beam becomes eliminated. structure of all elementary particles caused by magnetic Some properties of such degenerated electron gas attraction at the distances of the order of Compton represented also. wavelength. In [5], motion of charged particle in a field OVERVIEW of magnetic dipole was considered. In [4] and [5] the term in Hamiltonian responsible for the interaction Generation of beams of particles (electrons, positrons, between magnetic moments is omitted, however as it protons, muons) with minimal emittance is a challenging looks like problem in contemporary beam physics. With low = 2 − 2 + 2 − 2 emittance beam in hand one can arrange interactions H c (p eA/c) mc e /r (2) between the beams of particles with minimal population. with vector potential A=m⋅( σ×r)/r3, where m For an electron gas in a volume V the total number of describes the magnetic momenta of the center. electrons in all states can be estimated for uniform In [6] a three body system was considered. The third distribution as [1] heavy particle- proton served for partial compensation of Δ γ px py p S⊥lb γ ε γε γε electrical repulsion. N= dn≅2 =2 x y z , (1) π 3 π 3 Meanwhile the magnetic dipole m sets magnetic field (2 ) (2 C) 1 γε = γ Δ around as where z lb ( p / p0 ) –is an invariant longitudinal H = (3n ⋅(m ⋅ n) −m )/R3 , (3) emittance, l is the bunch length, γε and γε are the 1 1 b x y transverse horizontal and vertical invariant emittances. where n is unit vector from the center of magnetic Again, if N is close to the number of the particles in the momenta to the point of interest. In c.m. system the energy of another magnetic moment in a magnetic bunch, then the particles in the bunch close to generation m 2 condition. The beam with the number of particles N field of the first one is cannot have emittances lower than the one defined by (1), = ⋅ = ⋅ ⋅ ⋅ − ⋅ 3 H m2 H (3(m2 n) (m1 n) m1 m2 )/R . (4) γε γε γε ≥ 1 (2π )3 N . x y z 2 C Namely, this term should be added to Eq. 2. There are few So the every state is occupied by a pair of electrons basic orientations of each momentum with respect to the other one, shown in Fig. 1. (positrons) with oppositely oriented spins. As this pair represents the pure state, its full wave function is In cases a) and b), Fig. 1, the unit vector n is asymmetric with respect to spatial coordinates. As the orthogonal to the magnetic moment m associated with spin-part of the wave function is asymmetric, then the the electron, so the magnetic field at the location of the coordinate factor is symmetric, i.e. two electrons have second electron is H = −m /R3 . In cases c) and d) the nonzero probability to be identified in the same spatial = 3 point. In a view of the former comment, there is no magnetic field is H 2m /R , so this case is twice more contradiction with uncertainty condition. effective for the field value. As the energy Eq. 4 One interesting circumstance here is that the fully associated with the magnetic moment of a pair of = degenerated beam has zero polarization, as each state electrons m 2 m1 is now occupied by two electrons having oppositely oriented H =m ⋅ H = (3(m ⋅ n)2 −m 2 )/R3 , (5) 2012 byspins. IEEE – cc Creative Commons Attribution 3.0 (CC BY 3.0) — cc Creative Commons Attribution 3.0 (CC BY 3.0) c ○ In [2] and [3] there was proposed to consider the bound so the attracting force acting between two electrons due to states between two electrons with oppositely oriented theirs magnetic moments is ISBN 978-3-95450-115-1 03 Particle Sources and Alternative Acceleration Techniques Copyright 2792 A16 Advanced Concepts Proceedings of IPAC2012, New Orleans, Louisiana, USA WEPPP031 ∂E 3(m ⋅ n)2 −m 2 σ ≅ σ ≅ σ ≅ μ ≅ ⋅ 10 ≅ − ≅ , (6) x 500nm, y 3.5nm, y 200 m, N 2 10 , F 3 4 ∂R R − one can obtain from Eq. 8 δ ≅ 8 , meanwhile the 10 cm Compton wavelength is ≅ ⋅α ≅ ⋅ −11 C r0 3.8 10 cm. One peculiarity here is that these dimensions established dynamically, i.e. the particles cross the IP point in theirs transverse motion with angles up to σ′ ≅ γε γ β ≅ −4 . y ( y ) / / y 10 rad (12) Figure 1: Relative orientation of electron’s magnetic for ( (γε ) ~ 2⋅10−8 m⋅ rad, β ~ 0.2mm, γ ~ 105 [7]. moments. In cases a) and c) there is attraction by the y y magnetic force; in cases b) and d)–repulsion. The angles like Eq. 12 provide collision transverse momenta as big as which should be compared with the repulsive electric ≅ σ′ p⊥c pc y ~ 5MeV (13) ≅ − 2 2 force F e / R , so the balance is for 50GeV beam. One can see, that the transverse 3(m ⋅ n)2 −m 2 e2 momenta is big enough compared with the minimal one 3 ≅ . (7) R4 R2 Eq. 8, so the repulsion could be to overpassed in a moving Substitute for the magnetic moment of frame, so the bound states could be generated here, Fig. 2. = = electron m e / 2mc e C / 2, one can obtain, that the distance for which these two forces equalize each other is ≅ = )( ⋅ R / 2mc 3 2 C , (8) where the factor 2 in brackets corresponds to the case c). This distance was identified in [4]-[6] as well. One should agree that the orientation of spins a) and c) is stable, which could be concluded from behavior of Figure 2: Dynamical process for compression of polarized ordinary magnets known to everyone. bunch while moving towards the crossover. The energy required for bringing two electrons to the So the idea of our proposal is very simple: one should distance equal to the Compton wavelength is arrange collisions between polarized beams of electrons e2 e2 with energy more than Eq. 8 and hope that in some ≅ ⋅α ≅ α ⋅ mc2 ≅ 3.73kV , (9) collision the bonded state will emerge. As this probability C r0 is small, one should hope that accumulation of such pairs i.e. pretty small compared with the energy of transverse might be effective. One positive moment here is the pair motion at IP especially. One should remember that the of electrons bounded carries the doubled charge together energy associated with magnetic attraction with doubled mass, so the motion of such pair is possible m 2 e2 e2 α ⋅ mc2 in the ring without any problem. To register such pair one ≅ = α ≅ ≅ 0.9kV (10) can hope with registering the SR as the pair should radiate 3 2 C 4 C 4r0 4 coherently ~(2e) . might make the minimal energy required for bonding particles even lower. DAMPING RING FOR GENERATION THE BOUND STATES The average distance between particles in the bunch as it is seen by the observer in a Lab system is δ ≅ ()V / N 1/ 3 ≅ ()σ σ σ / N 1/ 3 , (11) x y z where N stands for the bunch population, σ σ -are the x , y Figure 3: The envelope functions in the colliding section transverse bunch sizes for corresponding directions. of the ring. Let us consider one example first. Substitute for estimations in Eq. 11 the parameters an electron bunch Values of envelope functions at crossover in Fig. 3 are ~ β ≅ β . available for ILC at IP, x y ~1cm 2012 by IEEE – cc Creative Commons Attribution 3.0 (CC BY 3.0) — cc Creative Commons Attribution 3.0 (CC BY 3.0) c ○ 03 Particle Sources and Alternative Acceleration Techniques ISBN 978-3-95450-115-1 A16 Advanced Concepts 2793 Copyright WEPPP031 Proceedings of IPAC2012, New Orleans, Louisiana, USA Here the cases marked a) and b) were already mentioned. Resulting momentum of state a) is zero, but the momentum of case b) is one. Meanwhile the case c) represents a claster of electrons with zero momentum also, but the number of electrons in such claster can vary. Finally the case d) describes the claster with integer momentum, equal to four etc. Once again if someone registers the clasters by SR one should immediately register the splashes associated with coherent radiation of bounded electrons. Figure 4: Damping ring.
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