Extensions of the longitudinal envelope equation Page: 1 of 8
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Extensions of the Longitudinal Envelope Equation
A ccelerator Physics Group
Fermilab, PO Box 500, Batavia IL 60510
Recently, longitudinal space charge effects have become of increased importance in a variety of
dynamical situations. The CEBAF FEL injector beam dynamics shows large space-charge
effects, even at 10 MeV (y - 20).1 Space-charge dominated longitudinal motion has also been
studied in the IUCF ion storage ring23. Previously a longitudinal envelope equation with a self-
consistent phase-space distribution has been developed,4 and has been of considerable use in
analyzing the motion of these cases. Longitudinal motion in detailed agreement with this
envelope equation has been observed at the U. of Maryland Laboratory for Plasma Research,' and
at the GSI electron cooling storage ring ESR,6 as well as at the IUCF. However, the initial
presentation in ref. 4 used non-relativistic linear-accelerator bunching motion as a simplifying
approximation in order to avoid inadvertent errors and minimize misprints, and must be adapted
to include relativistic and/or synchrotron effects.
In the present note we extend the envelope equation formulae to include relativistic, synchrotron,
and acceleration effects, and define the various factors in the equations in explicit detail. The
object is to obtain a set of debugged formulae for these extended cases, with all of the various
factors defined explicitly, so that the formulae can be used as a reference without repetitive
rederivations. The usual ambiguities over emittance definitions and units and (3, y, g factors
should be resolved. The reader (or readers) is invited to discover any remaining errors,
ambiguities or misprints for removal in the next edition.
Nonrelativistic Envelope Equation
Under simplified assumptions (no transverse dependence, a round beam of radius a within a
perfectly conducting round beam chamber of radius b, and no resistive wall impedances), the
longitudinal space charge force can be written as:7
F=- qg d (my R c) (1)
47y2 dz dt
where q is the charge of the particles in the bunch (e for electrons, Ze for ions), X is the number
of particles per unit length, (3, y are the usual kinematic factors, and g = 1 + 2 ln(b/a) is "a
geometrical factor of order unity". Note that g - 3-4 in typical accelerators; also, in ref. 3 and
some other references, a g factor that is smaller by a factor of 2 is used (g3 = 1/2 + ln(b/a)).
(These expressions for the g factor ignore radial dependences; observationally Wang et al.
observe g - 2 ln(b/a).8) The force is made non-relativistic simply by taking E/m = y -> 1; that
approximation is used in reference 4. (We have used MKS units in eq.1; ref. 4 used cgs units for
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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/m1/1/: accessed January 16, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.