Direct Calculation of the Transfer Map of Electrostatic Deflectors, and Comparison with the Codes COSY INFINITY and GIOS

TL;DR

This paper compares direct calculation methods for electrostatic deflector transfer maps with COSY INFINITY and GIOS, using analytical, numerical, and software-based approaches for spherical and cylindrical geometries.

Contribution

It introduces a direct computation approach for transfer maps in laboratory coordinates and compares it with existing software tools, enhancing accuracy and understanding.

Findings

Analytical and numerical transfer maps agree with COSY INFINITY and GIOS results.

The methods are effective for both spherical and cylindrical deflectors.

Codes for the computations are publicly available for reproducibility.

Abstract

COSY INFINITY uses a beamline coordinate system with a Frenet-Serret frame relative to the reference particle, and calculates differential algebra-valued transfer maps by integrating the ODEs of motion in the respective vector space over a differential algebra (DA). We will describe and perform computation of the DA transfer map of an electrostatic spherical deflector in a laboratory coordinate system using two conventional methods: (1) by integrating the ODEs of motion using a 4th order Runge-Kutta integrator and (2) by computing analytically and in closed form the properties of the respective elliptical orbits from Kepler theory. We will compare the resulting transfer maps with (3) the DA transfer map of COSY INFINITY's built-in electrostatic spherical deflector element ESP and (4) the transfer map of the electrostatic spherical deflector computed using the program GIOS. In addition to the electrostatic spherical deflector, we study an electrostatic cylindrical deflector, where the Kepler theory is not applicable. We compute the DA transfer map by the ODE integration method (1), and compare it with the transfer maps by (3) COSY INFINITY's built-in electrostatic cylindrical deflector element ECL, and (4) GIOS. In addition to the code listings in the appendices, the codes to run the test cases are available at http://bt.pa.msu.edu/cgi-bin/display.pl?name=ELSPHTM17