Emittance Reduction between EBIS LINAC and Booster by Electron Beam Cooling; Is Single Pass Cooling Possible? Page: 3 of 19
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current, e elementary charge and A is extraction aperture area. Immediately after
extraction, the electron density is about n 1011cm 3 . Balk electrons energy spread before
extraction was about 0.1 eV. Due to kinematic compression4, energy spread of the
accelerated electrons Te (i.e. in the lab frame) is T = T *0.5(T'j , where T is thermal
spread of unaccelerated electrons. For T = 0.1 eV and U = 1 KeV, Te = 5x10-4 eV.
Assuming electron beams with parameters of the electron beam described above (e.g. 2 -
4 electron beams) overlap the ion beam completely during the whole cooling period,
cooling time t of the ion beam parallel energy spread can be calculated using the
following formulas for unmagnetized cooling.
First cooling time is calculated from Budker's laboratory frame formula (modified for
multi-charge ions)
5x102 M Y5(30)3
r ~ x10-- e(1)
m Z2ARenc
Equation 1 is in cgs units, where M, m, are ion and electron masses respectively, 0 is ion
angular spread (essentially ), Re = 2.8x10-13 cm is electron classical radius, k is the
Coulomb logarithm, and c is the speed of light. For the above parameters A 30 , and
equation 1 yields,
z ~7.68x10-10 sec
Next, cooling time is calculated from an equation derived by Poth67, who participated in
and analyzed antiproton cooling in LEAR,
6.3 + 03
6)rZ2RRecnA (2)
Equation 2 is in MKS units, a and A are velocity spreads of electrons and ions
respectively, and R; = 1.6x1013 m is gold ion classical radius. For the above parameters
(electron velocity is evaluated from the kinematically compressed temperature), equation
2 yields,
r ; 7.75x10-10 sec
Finally, cooling time is evaluated from plasma physics energy equipartition time,
r =5.56x1018 e (3)
(Mm)112 Z2An
Equation 3, which is in eV and cgs units, yields for the above parameters,
r ~7x10-10 sec
Agreement between the three cooling time values is excellent. In all three equations the
plasma physics Coulomb logarithm, which is about 30, was computed for the value of
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Hershcovitch,A. Emittance Reduction between EBIS LINAC and Booster by Electron Beam Cooling; Is Single Pass Cooling Possible?, report, April 1, 2008; United States. (https://digital.library.unt.edu/ark:/67531/metadc900790/m1/3/: accessed April 19, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.