Extensions of the longitudinal envelope equation Page: 4 of 8
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3N z2 zo AP2 8
f(z, A, s) = 1- - - - (A- z) (8)
N\ Zo2 E z0
where EN is the emittance in z-A coordinates and is invariant (normalized). Here zo(s) is the
envelope amplitude and Ar(s) = fy3(d zo/ ds). The resulting density profile is parabolic:
X(z,s)= N 1 - , (9)
which implies a linear space-charge force.
The phase-space distribution is a self-consistent solution of the Vlasov equation:
+zla +A0 =0 (10)
s az BA
if z0(s), Ar(s) are solutions of the envelope equations:
ds2 y (11)
dA EN 3r gN
P - + g - KA(s)zo
ds jy3z3 2jy2zo
These can be combined into a single envelope equation (found also in Smithl0):
d 3dz0 2 3Nrg
- 3 = N__ q -KA(s)z (o12)
ds ds jy3zo 2 1y2z2
Note that the normalized (z-A) emittance EN is related to the (z-z') emittance EL by EN = REL'
If the central beam energy is unchanged (R, y are constant), we can return to unnormalized
coordinates, obtaining the envelope equation:
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Neuffer, David. Extensions of the longitudinal envelope equation, report, April 30, 1997; Batavia, Illinois. (digital.library.unt.edu/ark:/67531/metadc624405/m1/4/: accessed January 16, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.