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Field Modeling, Symplectic Tracking, and Spin Decoherence for EDM and Muon G-2 Lattices

A Dissertation

In partial fulfillment of the requirements for the degree of Doctor of Philosophy from Michigan State University.


While the first particle accelerators were electrostatic machines, and several electrostatic storage rings were subsequently commissioned and operated, electrostatic storage rings pose a number of challenges. Unlike motion in the magnetic field, where particle energy remains constant, particle energy generally changes in electrostatic elements. Conservation of energy in an electrostatic element is, in practice, only approximate, and it requires careful and accurate design, manufacturing, installation, and operational use. Electrostatic deflectors require relatively high electrostatic fields, tend to introduce nonlinear aberrations of all orders, and are more challenging to manufacture than homogeneous magnetic dipoles. Accordingly, magnetic storage rings are overwhelmingly prevalent.

The search for electric dipole moments (EDMs) of fundamental particles is of key importance in the study of C and CP violations and their sources. C and CP violations are part of the Sakharov conditions that explain the matter-antimatter asymmetry in the universe. Determining the source of CP violations would provide valuable empirical insight for beyond-Standard-Model physics. EDMs of fundamental particles have not to this date been experimentally observed. The search for fundamental particle EDMs has narrowed the target search region; however, an EDM signal is yet to be discovered.

In 2008, Brookhaven National Laboratory (BNL) had proposed the frozen spin (FS) concept for the search of a deuteron EDM. The FS concept envisions launching deuterons through a storage ring with combined electrostatic and magnetic fields. The electrostatic and magnetic fields are in a proportion that would, without an EDM, freeze the deuteron's spin along its momentum as the deuteron moves around the lattice. The radial electrostatic field would result in a torque on the spin vector, proportional to a deuteron EDM, rotating the spin vector out of the midplane.

The principle of an anomalous magnetic dipole moment (MDM) measurement using a storage ring, shared by BNL's completed E821 Experiment and the ongoing E989 Experiment operated by Fermi National Accelerator Laboratory (FNAL), requires injecting muons into a magnetic ring at the so-called magic momentum. The magic momentum, as defined in this context, would freeze the muon's spin vector along its momentum if the anomalous MDM was zero. The spin precession in the horizontal plane relative to the momentum is proportional to the anomalous MDM.

Storage rings for measurement of EDM and anomalous MDM present a new frontier in tracking code accuracy requirements. For accurate tracking of storage rings with electrostatic particle optical elements, it is necessary to model the fringe fields of such elements accurately, in particular, because not doing so provides a mechanism for energy conservation violation. However, the previous research on fringe fields tended to focus on magnetic rather than electrostatic particle optical elements. We will study and model the fringe fields of several electrostatic deflectors. Field falloffs of electrostatic deflectors are slower than exponential, and Enge functions are not suitable for accurate modeling of these falloffs. We will propose an alternative function to model field falloffs of electrostatic deflectors. We will use conformal mapping methods to obtain the main field of the Muon g-2 storage ring high voltage quadrupole, and we will calculate its fringe field and effective field boundary (EFB) using Fourier analysis.

Furthermore, we will study tracking of storage rings with electrostatic elements using map methods. We will find that, for simultaneous symplecticity and energy conservation, it is only necessary to enforce symplecticity in COSY INFINITY. We will model and track several benchmark lattices – an electrostatic spherical deflector, a homogeneous magnetic dipole, and a proton EDM lattice – in COSY INFINITY and MSURK89, our in-house eighth order Runge-Kutta-Verner tracking code. Finally, we will investigate spin decoherence and systematic errors in FS and quasi-frozen spin (QFS) lattices. Spin decoherence effects are similar in FS and QFS lattices, and spin decoherence in said lattices often remains in the same range over time, indicating the feasibility of EDM measurement using FS and QFS lattices.


E. Valetov (2017)


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