Extensions of the longitudinal envelope equation Page: 6 of 8
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momentum-dependence of particle motion through the bending magnets and depend on the
accelerator transport design. That dependence is expressed in terms of the ring momentum
compaction factor oc = 1/yT2, where yT is the "transition gamma". The equation of motion is
dz _ 1 1 >p _ 1 1 A _ A (17)
ds Y2 Y2 p Y2 Y2 Jy P y
where we have introduced the symbol TI = 1/72 - 1/yT2.
Synchrotrons have an accelerating rf voltage per turn, Vo sin($) = V0 sin(h z / R), which provides
acceleration and bunching, where $ is the rf phase, h is the rf harmonic number and 27tR is the
ring circumference. In a linearized (short-bunch) approximation, this can be decomposed into
the acceleration of the bunch center (dy/ds = e Vo sin ($,)/(27rRmc2 )) and a bunching term. The
bunching term can be written as:
dA _ eVocos($.) h
- _z = -K z (18)
ds 27tR2g3mc2 S
Developing an envelope equation in exactly the same manner as above obtains the equations:
dAS EN 3r gN (9
- = T + g -KS z
ds "Pyz3 2Py2z2 S
where we have used the symbol As = (3y/TP (dzo/ds) instead of A, to distinguish synchrotron from
linear motion. The distribution function would be the same as before, but with A, replaced by
If the acceleration term is zero, this can be simplified, as above, to:
d2z0 2 + 31rg g N -TKS z (20)
ds2 Pf2y2z3 2P 2q3z2
Nagaitsev et a13 have obtained an envelope equation for synchrotron motion but in somewhat
different coordinates(t, scaled position, and relative momentum S = Sp/p) We will transform these
equations toward that form by using dt = ds/ (3c and scaled position Z = zo /RR as variables,
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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/m1/6/: accessed February 18, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.