1. Determination of R – matrix Supervisors: Prof. Nikos Tsoupas Prof. Manolis Benis Sándor Kovács Murat Yavuz Alkmini-Vasiliki Dagli

2. In our work we calculated the transformation matrix of a linear charged particle optics element. y coordinates and the elevation angles at the final position were analysed related to initial condition Linearity of the chosen einzel lens was tested R-matrix was tested for arbitrary initial conditions

3. R-matrix is a transformation matrix (rotation matrix), which transports a beam of particles from the entrance of the electrostatic (or magnetostatic) element to the exit. Transport matrix can be determined in any case of linear devices. R-matrix is a numerical model of an optical element

4. An einzel lens is a charged particle lens that focuses without changing the energy of the beam. It consists of three or more sets of cylindrical or rectangular tubes in series along an axis. It is used in ion optics to focus ions in flight which is accomplished through manipulation of the electric field in the path of the ions.

5. In the simulations we used a simion example (examples/einzel.iob) Fixed parameters: o geometrical parameters of the lens o energy of the electrons (50eV) o initial plane of the electrons (x=-50mm) o data recording plane after particles passes through the device (x=140mm) o Potential on electrode 2 (V 2 =110V) Variable parameters: o Initial y coordinates o Elevation angle

8. X(-50 mm) Y(1 mm) Elv(0 deg) X(140 mm) Y(0.0515507 mm) Elv(-0.554364 deg) Using the previous equations R 11 = 0,051551 and R 21 = -0,554364 were calculated. X(-50 mm) Y(0 mm) Elv(1 deg) X(140 mm) Y(1.81902 mm) Elv(0.079828 deg) Following the same method we calculated R 12 = 1,819020 and R 22 = 0,079828.

9. Finally the transformation matrix (R) is found: According to the theory: det R = 1 In order to confirm the validity of our R – matrix we have to calculate the determinant of it: det R = R 11 · R 22 - R 12 · R 21 = 1,012514 ≈ 1

10. The difference from the theoretical value (det R = 1) is 1,25%! The reason of this error is that there should be higher order aberrations. This first order approximation is valid only under 1° elevation angle. Linearity was checked at several y coordinates inside the lens and maximum deviation from the paraxial axis was smaller than 10% of the lens radius. Electrostatic lenses are very sensitive to aberrations.