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Mathematical descriptions of axially varying penning traps Stephanie Brown AEgIS CERN University of Michigan Research Experience for Undergraduates Department of Physics and Astronomy at The Pennsylvania State University Introduc)on AEgIS is the An)hydrogen Experiment: Gravity, Interferometry, and Spectroscopy. Its major aim is to study the interac)on of an)ma?er with earth’s gravita)onal field. To maximize measurement precision we will cool the an)hydrogen while it is stored in a penning trap. The proper)es of this electromagne)c trap can be characterized mathema)cally. An)ma?er and An)hydrogen Dirac first proposed an)ma?er in 1933; however, since ma?er and an)ma?er annihilate on contact, it has proven difficult to study. Though Dirac predicted a balance between ma?er and an)ma?er, we have found that the ra)o of ma?er to an)ma?er in our universe is weighted towards ma?er. In our search to discover the cause of the ma?er/ an)ma?er imbalance, we have discovered small asymmetries. Our Trap The Penning Trap used in AEgIS is designed to contain C 2 par)cles while they are cooled. Currently, we plan to use two forms of cooling. 1. Doppler cooling: a laser is tuned to excite only molecules moving towards the laser. A photon hits the carbon, transfers its momentum, and causes a fric)onal force. 2. Evapora)ve cooling: a laser excites carbon at the high end of the thermal distribu)on. A second laser ionizes the carbon and it is radiated from the trap because the magnets no longer contain it. Simplifica)ons Fully analy)cal solu)ons to the trap are not available. However, under certain condi)ons assump)ons can be made to make the trap analy)cally tractable. 1. The trap can be modeled by a piecewise con)nuous func)on of high and low field regions 2. The radial dependence of the axial field is negligible Process The plasma can be described by two ‘constants’ of mo)on— constant only in perfect trap geometry. The Hamiltonian (single electron energy func)on) and the canonical angular momentum are both constants. Poisson’s equa)on can now be defined by the density, which is based on Boltzmann’s global thermal equilibrium. Assume that the plasma can be modeled by a piecewise con)nuous func)on. Poisson’s equa)on can then be solved separately for the high and low field regions. We began by assuming the ideals that we intend for the highfield region. • Desired density: 10 13 par)cles/m 3 • Number of Par)cles: 10 7 • The magne)c field: 0.8T • Radius: Es)mated by trap geometry • Trap length: 10 cm The en)re plasma rotates with a single ‘rigidrotator’ frequency. This connects the condi)ons for the high and low field regions. Using assump)ons from the literature Cita)on to make ini)al guesses for low field values • Low radius: • Debeye length: • An es)mate for the low field density can then be made Poisson’s equa)on can be integrated numerically, using the given condi)ons. The poten)al difference can then be used to define the separatrices of the plasma. Containment and Magne)c Traps AEgIS uses a Penning Trap to contain the C 2 plasma. Lasers can cool C 2 through Doppler cooling and evapora)ve cooling. The C 2 can then be used to cool positrons which will repel from the like charged carbon and so not annihilate. Lasers cannot cool positrons because they do not have electron transi)on states. Magne)c Mirrors Magne)c mirrors: magne)c configura)ons which reflect a charged par)cle through low and high density magne)c regions. The highly inhomogeneous fields are tailored to trap par)cles at very specific veloci)es. They are defined by a characteris)c known as the mirror ra)o which is This causes the par)cles in the plasma to bounce up and down the trap. Except, under certain condi)ons, the par)cles can become trapped in either the high or the low region. The Separatrix A func)on called the separatrix defines the difference between trapped and untrapped par)cles. There are separate separatrices for the high and lowfield regions. Par)cles with high perpendicular veloci)es cannot penetrate into the high field region. Conversely, par)cles with small perpendicular veloci)es cannot escape the high field region. H = p 2 2 m − e φ (r , z ) P θ = p θ r − ( e c ) A θ (r )r a b Figure 2. The ini)al (a) and refined model (b) used to make an ini)al es)mate of the high field radius. These are scaled for viewing. The z axis in the trap is 100x larger with respect to the radius of the trap V=0 V = V V = V z r B C 2 Plasma Figure 3. Scheme of a simple penning trap: constant magne)c field and electric poten)al. 1 r ∂ ∂r r ∂φ ∂r + ∂ 2 φ ∂z 2 = 4π en = 4π en 0 exp − 1 T [ e φ (r , z ) + m 2 ω ( eB mc − ω )] # $ % & ' ( The momenta terms of the Hamiltonian are neglected because they are negligible B h B l λ D B h B l r h Plasma frequency Low field density distribu)on poten)al separatrices constants Ideal condi)ons assump)ons B h B l = 0.8 0.45 The high and low field separatrices are shown for two cases. The shaded regions represent trapped par)cles. In our trap we would like to minimize the number of trapped par)cles. High Field Region Low Field Region a b Figure 6a. Gives the shape of the separatrices for a Penning Trap with an axially varying field. The shaded areas are par)cles trapped by the magne)c field. Figure 6b. Gives the shape of the separatrices for a tradi)onal Penning Trap. The shaded areas are par)cles trapped by the magne)c field. V || V _|_ Figure 1. A preliminary simula)on to test arrangements of permanent magnets for the penning trap. The array is a modified Halbach array of permanent magnets where the ring magnets have magne)za)on in varying direc)ons to strength the field in the trap and negate the field outside. One magnet is placed opposite to the pa?ern to lower the strength of the magne)c field in the low region. Part a shows a two dimensional image of the magnets and Part b shows a one dimensional model. Figure 5. A magne)c bo?le is an arrangement of two magne)c mirrors. This image (Gopolan 1998. Disserta)on UC Berkley.) Shows the mo)on of a par)cle through the magne)c bo?le. Proton electron Hydrogen An)Proton positron An)Hydrogen a b
