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Package
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Trak Charged Particle Toolkit
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Input files
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ApertureLens.MIN, ApertureLens.EIN, ApertureLensIn.DST, ApertureLens.TIN
Download ApertureLensEffect.zip
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Description
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A accelerated charged-particle beam is defocused when it is extracted through an aperture. The top figure, a charged particle gun with a circular aperture, shows the cause. Equipotential lines bulge into the aperture resulting in transverse electric field components. In the example, a low-current beam of protons is accelated to energy 10 keV. The source and aperture radius are 2.0 cm. The accelation gap and drift space each have length 5.0 cm. The entrance electrode has potential +10 kV.
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Results
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An approximate formula for the focal length treating the aperture as a thin lens is
f = 4*Ti/(E2 - E1),
where T is the kinetic energy (in eV) of a non-relativistic particle at the aperture and E1 and E2 are the electric fields in the upstream and downstream regions. If the aperture electrode is at 0.0 V and the protons are accelerated from rest, then Ti = 10000.0 eV, E2 = 0.0 V/cm and E1 = 2000.0 V/m. The predicted focal length is f = -20.0 cm. We can infer the effective focal length from the particle file generated by the calculation at the exit plane. The lower figure shows the (r,r') phase space plot. A particular paraxial particle has position r = 0.636 cm and r' = 0.0271 rad in the exit. A linear back projection has length 23.47 cm and intersects the axis at a point 18.47 cm upstream from the lens. The calculated focal is therefore f = -18.47 cm, close to the prediction. If the protons start with energy Ti = 5000.0 eV at the entrance plane, then the predicted focal length is f = -30.0 cm. In this case, a sample paraxial trajectory has r = 0.670 cm and r' = 0.0197 rad at the exit plane, implying a focal length f = -29.0 cm. Determing f from the exit distribution when E2 is nonzero is difficult because particles are accelerated in the second gap with a change in radial angle. One approach to investigate scaling with electric field is to use a narrower gap and to define a diagnostic plane close to the lens output.
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Comments
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The equation is derived in Sect. 6.5 of the text Principles of Charged Particle Acceleration, available for download at https://www.fieldp.com/cpa.html. In comparison to the simple formula, numerical calculations give highly accurate results, even for large apertures. Codes also determine non-linear focusing focusing forces that increase emittance, as in the lower illustration.
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