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Analysis of R-Matrix in Einzel Lens: Theory, Calculations, and Simulations

This study explores the transformation matrix R in an einzel lens for charged particle optic systems. The R-matrix, a rotation matrix, is determined and tested for various initial conditions. Simulations are conducted using fixed and variable parameters to analyze the behavior of the optic element. Calculations demonstrate the impact of different input parameters on the R-matrix values. Conclusions highlight the accuracy and limitations of the theoretical model, emphasizing the importance of considering higher-order aberrations for precise results in einzel lens applications.

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Analysis of R-Matrix in Einzel Lens: Theory, Calculations, and Simulations

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  1. Determination of R – matrix Supervisors: Prof. NikosTsoupas Prof. ManolisBenis Sándor Kovács MuratYavuz Alkmini-VasilikiDagli

  2. Abstract • Inourworkwecalculatedthetransformation matrix of a linearchargedparticleopticselement. • y coordinates and theelevationanglesatthefinalpositionwereanalysedrelatedtoinitialcondition • Linearity of thechoseneinzellenswas tested • R-matrixwas tested forarbitraryinitialconditions

  3. Theory • R-matrixis a transformation matrix (rotationmatrix), whichtransportsabeam of particlesfromtheentrance of theelectrostatic (ormagnetostatic) elementtotheexit. • Transportmatrixcan be determinedinanycase of lineardevices. R-matrix is a numericalmodel of an opticalelement

  4. Einzellens • 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 opticsto focus ions in flight which is accomplished through manipulation of the electric field in the path of the ions.

  5. Parameters • Inthesimulationsweuseda simionexample(examples/einzel.iob) • Fixed parameters: • geometricalparameters of thelens • energy of theelectrons (50eV) • initialplane of theelectrons (x=-50mm) • datarecordingplaneafterparticlespassesthroughthedevice (x=140mm) • Potentialonelectrode 2 (V2=110V) • Variableparameters: • Initial y coordinates • Elevationangle

  6. Calculations Fromthepreviousequations: Ifweset=0° and y0>0mm then R11 and R21can be calculated. If we sety0=0 then>0°R12and R22can be calculated.

  7. We chosethevariables: y0 = 1 mm; 0= 0° Thenwegot a log file withthevalues: X(-50 mm) Y(1 mm) Elv(0 deg) X(140 mm) Y(0.0515507 mm) Elv(-0.554364 deg) UsingthepreviousequationsR11 = 0,051551 and R21= -0,554364 werecalculated. Changingthepreviousvariablesas: y0 = 0 mm; 0 = 1° X(-50 mm) Y(0 mm) Elv(1 deg) X(140 mm) Y(1.81902 mm) Elv(0.079828 deg) Followingthesamemethodwecalculated R12= 1,819020 and R22 = 0,079828.

  8. Finallythetransformation matrix (R) is found: Accordingtothetheory: det R = 1 Inordertoconfirmthevalidity of our R – matrixwehavetocalculatethedeterminant of it: det R = R11· R22 - R12 · R21 = 1,012514 ≈ 1 The calculatedR-matrixelementswere tested severaltimeswitharbitrary input parametersintherange of min=-5° tomax=5° and y0 min=-1 to y0 max=1.

  9. Conclusions • The difference from the theoretical value (det R = 1) is 1,25%! • The reason of thiserror is thatthereshould be higherorderaberrations. • Thisfirstorderapproximation is validonlyunder 1° elevationangle. • Linearitywascheckedatseveral y coordinatesinsidethelens and maximum deviationfromtheparaxialaxiswassmallerthan 10% of thelensradius. • Electrostaticlensesareverysensitivetoaberrations.

  10. Thankyouforyourattention!

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