# Validation of Transfer Map Calculation for Electrostatic Deflectors in the Code COSY INFINITY

### Abstract

The beam physics code *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 described and performed 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 numerical integrator and (2) by computing analytically and in closed form the properties of the respective elliptical orbits from Kepler theory. We compared 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*, which uses analytic formulas from a paper* by Hermann Wollnik regarding second order aberrations.

In addition to the electrostatic spherical deflector, we studied an electrostatic cylindrical deflector, where the Kepler theory is not applicable. We computed the DA transfer map by the ODE integration method (1), and we compared it with the transfer maps by (3) *COSY INFINITY*'s built-in electrostatic cylindrical deflector element
**ECL**
and (4) *GIOS*.

The transfer maps of electrostatic spherical and cylindrical deflectors obtained using the direct calculation methods (1) and (2) are in excellent agreement with those computed using (3) *COSY INFINITY*. On the other hand, we found a significant discrepancy with (4) the program *GIOS*.

* H. Wollnik, Second Order Approximation of the Three-Dimensional Trajectories of Charged Particles in Deflecting Electrostatic and Magnetic Fields,
*Nucl. Instrum. Methods* **34**, 213 (1965).

Codes discussed in this paper are

ESCPO10.fox :
COSY INFINITY program to compute all the test cases.

ESCPO10-LCS-ODE.nb :
Mathematica notebook for integration of the ODEs of motion for an electrostatic
spherical deflector in laboratory coordinates.

GIOS_sphdefl.INP :
GIOS input for the respective test case of an electrostatic spherical deflector.

GIOS_cyldefl.INP :
GIOS input for the respective test case of an electrostatic cylindrical deflector.

E. Valetov, M. Berz, K. Makino,
*Int. Journal of Modern Physics A*, in print

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