W-Band Sheet Beam Klystron Simulation Page: 4 of 17
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Beam transport will be stable in a PPM lattice so long as focussing strength exceeds
the space charge forces pushing the beam apart, and as long as the phase advance per
unit focussing cell is less than 900:
2w> Q <Z o 2k up
with km = 22r/Lm the PPM lattice wavenumber, and uo the beam velocity. For the
proposed beam parameters (given in table 2 below), this means:
2.04: Bo [kG2] : 50.3 ->1.43kG : Bo 7.lkG
Figure 1 below gives an example of a PPM focussing unit cell, assuming NbFeB
permanent magnet material in 6x3 mm (length x depth) blocks and 1008 low carbon
steel pole pieces in 4x3 mm blocks, offset 1 m toward the beam axis centerline. The on
axis field strength, by PANDIRA calculation, is 4.9 kG, or more than a factor of three
above the lower stability limit. The substantial leakage flux on the outside of the PPM
magnet stack will permit permeable shunt tuning of the lattice once assembled.
Although figure 1 (left) below clearly shows that the focussing field Bz(z) has
substantial spatial harmonic content beyond the fundamental, the klystron simulations
kept only the fundamental space harmonic, underestimating the true focal strength by
some factor less than two. Booske's form for the fields is employed, together with a
small horizontally focussing quadrupole term (which can be derived by pole tip
shaping):
Bx(x,y,z) =-Bo(kxI k,,)sinh(kxx)cosh(kyy)cos(kz) -kqy
By(x,y,z) =-Bo(ky k,,)cosh(kxx)sinh(kyy)cos(kz) -kqx
Bz(x, y, z) = Bo cosh(kxx)cosh(ky) sin(kmz)
Wiggler focussing is an alternative to PPM and has fields of the form (17):
Bx(x, y, z) = 0
By(x, y, z) = Bo f (z) cosh(ky) sin(kz)
Bz(x, y, z) = Bo f (z) sinh(k,,y)cos(k,,z)
where kv is the wiggler wavenumber, and the tapering function f(z) at the wiggler
entrance is:f (z) = sine2(k,,z /4N, )
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Colby, E. R.; Caryotakis, G.; Fowkes, W. R. & Smithe, D. N. W-Band Sheet Beam Klystron Simulation, article, September 12, 2005; Menlo Park, California. (https://digital.library.unt.edu/ark:/67531/metadc874340/m1/4/: accessed July 17, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.