Electrostatic ring for the pEDM madx, matrix and leapfrog tracking Mario Conte University of Genova and INFN, Italy Alfredo U. Luccio, Nicholas D’Imperio Brookhaven National Laboratory, Upton, New York September 10, 2015 0 EUCard Workshop, Mainz, Sept./10-11 2015 1
Synopsis We designed and tracked an electrostatic FODO ring for pEDM. 1. Basic design criteria for an EDM electrostatic ring; 2. Find the regions of stability and parameters for the ring through a csh script that runs a modified version of CERN Madx with electric bends and quadrupoles; 3. Track orbits using (a)Madx matrices and then (b)to higher order by differential equations kick integration with a sym- plectic code based on Leapfrog algoritm; 4. Track spin dynamics by integration of the Thomas-BMT equation with the Spink algorithm[9]. Address the issue of spin coherence; 1 EUCard Workshop, Mainz, Sept./10-11 2015
Basic design and tracking criteria 1. Design a ring lattice on the footprint of the AGS (800 m), who seemed, for awhile to be a suitable site. The design can be scaled to any other site. A large footprint asks for a small value of the electric bend field; 2. Choose a basic FODO structure where the vertical tune is much smaller than the horizontal, to optimize the measure- ment of the vertical spin component, proportional to the EDM. The FODO will not be symmetrical since cylindrical electric bends produce an intrinsic horizontal focusing; 3. Design a lattice with a positive phase slip; 4. Adopt a convenient design package (say: madx) to find tune islands of stability. Use a fast and symplectic tracking code (say: leapfrog); 5. Recur to parallel computing to calculate and optimize spin coherence for a full beam of representative particles. 2 EUCard Workshop, Mainz, Sept./10-11 2015
Orbit/Spin tracking in an electric ring Tracking of orbits and spin in an electrostatic storage ring for the EDM is important and should be done by more than one method to compare and benchmark. Tracking should be symplectic, stable in the long range and fast, because for EDM search ring turns will be in the billions. Keywords to keep in mind in tracking are accuracy and long range stability.. Codes proposed and used by various Authors for the tasks are based on 1-Runge-Kutta integration of differential equations for orbit (Lorentz) and spin (Thomas-BMT); 2-map description of machine elements (Madx) or the whole lattice (Cosy-infinity); 3-discrete kicks symplectic integration for propagation by kicks (Leapfrog, Teapot.) In this contribution we will describe a madx matrix and a Leapfrog orbit code of type 3, pllus the some features of a spin dynamics code. 3 EUCard Workshop, Mainz, Sept./10-11 2015
Example of a ring designed using Madx The code Madx distributed and maintained by CERN is a pack- age used by many accelerator designers, to optimize lattices. Madx was designed for magnetic bends. We modified it using matrices proposed by Mario Conte [1], to deal with electrostatic bends and quadruoiles. Once conceived the ring, we tested its stability by a Linux script to run Madx (by Nick D’Imperio [2]), to find islands of stability in tune space for different values of quadrupole strengths. An ex- ample table is shown. The values of β max are growing at the edge of each island while the values of the betatron tune decreases. 4 EUCard Workshop, Mainz, Sept./10-11 2015
The magic condition Spin dynamics is governed by the covariant Thomas-Bargman- Michel-Telegdi T-BMT equation d s dt = − q mγ f × s , (1) where s is the real 3-dimensional spin vector of a 1/2-spin par- ticle, and f is a function of the position and the momentum of the particle and of the electric and magnetic field encountered by the particle along its trajectory. Spin is a passenger on the orbit, that has to be calculated first at each step. In a pure electrostatic ring, e.g. with no magnets or RF cavities, f reduces to � E × v 1 � f = γ a − , (2) γ 2 − 1 c 2 with a the spin anomaly. At the magic momentum pc = mc 2 / √ a it is exactly f = 0 and the spin remains frozen in its direction at injection (longitudinal) respect to the orbit. 5 EUCard Workshop, Mainz, Sept./10-11 2015
Issues for electric accelerator lattices An electrostatic lattice behaves differently than a classical mag- netic lattice. In an e.element the kinetic energy of a particle is modulated, while in a m.element it is not, since in the Lorentz equation of motion d p /dt = e ( E + v × B ) (3) only the vector term parallel to the momentum appears while in a m.lattice it is the vector product term present, with the force perpendicular to the momentum. In the present design we adopted simple cylindrical electrodes for the bends, that produce only a radial field far from edges. While conventional no-gradient magnetic bends don’t focus the beam, electric cylindrical bends produce a small horizontal focusing, so that to produce a FODO condition the focusing and defocusing quadrupoles should be slightly different. Vertical focusing will be obtained with electrostatic quadrupoles. An option would be to provide electric bends with also a ver- tical curvature of the electrodes. For the moment we are not considering this option because it is more hard and expensive to construct with the desired accuracy. 6 EUCard Workshop, Mainz, Sept./10-11 2015
We considered a ring of 800 m length (taylored on the BNL-AGS tunnel, that was for awhile proposed as a possible site for the pEDM ring.) 72 bends of 9 m length. 80 FODO quadrupoles of 2 × 0.5 m length, 4 drifts of 2 × 9 m. Values of basic parameters are listed below 7 EUCard Workshop, Mainz, Sept./10-11 2015
The ring, with 4 × 18 bends an 8 straights Mario Conte Universit‘a di Genova, Italy Alfredo U Luccio F F F F Brookhaven National Laboratory F D D F D D D February 2015 F F D D F D D F D D F F D D F F D D F F ’y2’ pEDM electrostatic ring D D F F 72 bends− 4 double drifts D D each 9 meter F F D D F F 80 FODO quadrupoles D D F F total ring length = 800 m D D F F D D bend F F drift D D F D D F D D F F D D F F D D D D F F D F F F F 8 EUCard Workshop, Mainz, Sept./10-11 2015
Electrostatic bend El.static bend matrix � 2 − β 2 , a 1 = cos( αθ ) , a 2 = ( ρ/α ) sin( αθ ) , a 3 = − ( α/ρ ) sin( αθ ) , α = � a 1 a 2 0 0 � . a 3 a 1 0 0 M = , L b = length of bend 0 0 1 L b 0 , 0 0 1 Other matrices, say: for drifts, are Madx’s 9 EUCard Workshop, Mainz, Sept./10-11 2015
Electrostatic quadrupole gap = 2 a = 10 cm, L q = 0 . 5 m ( quad length ) β 2 γ, V 0 = a 2 mc 2 El . gradient : G E = 2 V 0 e G E e ( β 2 γ ) k a 2 , k = mc 2 2 El.static quadrupole matrices √ √ � 1 � cos( kL q ) sin( kL q ) √ M F = k √ √ √ − k sin( kL q ) cos( kL q ) √ √ � 1 � cosh( kL q ) √ sinh( kL q ) M D = k √ √ √ k sinh( kL q ) cosh( kL q ) 10 EUCard Workshop, Mainz, Sept./10-11 2015
Value (MKSA) and dimension of all ring parameters are magic proton of β = 0 . 59837912 ., γ = 1 . 24810740 , β 2 γ = 0 . 44689430 , k = 0 . 043 , 1 . 42245166 . 10 5 V, V 0 = 2 . 844903 . 10 6 V/m, E q in the quads = 2 . 54972867 . 10 6 V/m E b in the bends = Compared with the Stability Table, the above shows that the working γ of this particle is less than γ T and the vertical betatron tune can be much smaller than the horizontal, as we want, for EDM measurements. About the field in the quads, note that for the optics the quantity of importance is √ kL q with L q the length of the quadrupole. The field in the quadrupole is proportional to k . Therefore increasing the length of the √ quadrupole, but at the same time decreasing k and keeping kL q constant, can effectively reduce the field. 11 EUCard Workshop, Mainz, Sept./10-11 2015
The matrix driven tracking program We started with a first order madx matrices tracking MTRACK3 INIT GLOBAL ¨uber alles MTRACK MATRICES DBDR loop QUAD loop BCELL DRIFT RAYTRANSFER (WRITE) BEND DBDR ENERMOD FINIS flowchart 12 EUCard Workshop, Mainz, Sept./10-11 2015
Betatron oscillations by matrix tracking 13 EUCard Workshop, Mainz, Sept./10-11 2015
Orbit tracking by leapfrog Going beyond first order orbit tracking, consider canonical in- tegration of the Lorentz diff. equation of motion, Eq.(6) by leapfrog or Verlet[4] kicks, method invented for astronomy by Delambre[5] in 1792, and adapted for accelerators by Ronald Ruth[6]. It is a kick integration method that interleaves drifts, where only space coordinates are advanced, with symplectic kick bends where the momentum components are advanced. Leapfrog is an algorithm accurate to 2.nd order in time step. Teapot, by R.Talman and L.Schachinger [6] it is similarly con- structed. See also [7]. Other integration algorithms, like Runge-Kutta are accurate to 4.th order in time. However they were written with mathemat- ical accuracy in mind, while the 2.nd order Leapfrog is exactly symplectic, i.e. was written with physical accuracy in mind. Symplectic Runge-Kutta has been discussed[8]. It makes a com- puter code slower to run, which defies our goal of short computer time for tracking an EDM ring for so many turns. 14 EUCard Workshop, Mainz, Sept./10-11 2015
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