Extensions of the longitudinal envelope equation Page: 7 of 8
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d2Z _ c 2 , 31 rg N - PheVB (21)
dt2 (R) R2 4 2Z3 2R 3 3Z2 27 mc2
We have redefined the rf voltage, using VB = -V0 sin(0), to obtain the desired sign in that term.
This equation can be simplified in form by changing emittance to Z-S units (Ez-s = EN/(R 2
expressing space charge in terms of the bunch current I = epcN/(27rR), I, = ecp4f/(37rgllPrq), A
d2Z =( -2 1PZ + I -A Z (22)
dt2 R) Z3 I Z2
This is similar to the form used in ref 3, eq. 11, but that form uses rms full-widths for bunch
length and emittance, hence differ by factors of 2 and 2 in the formulae. g is also defined
differently by a factor of two, and TI is handled differently. After these appropriate unit changes,
the expressions agree. Note that the sign of TI is not handled in ref. 3 but is handled correctly
here (I think); space charge actually bunches the beam when the beam energy is above transition
The envelope equations described here can be used as an accurate first approximation for
longitudinal motion in any accelerator. As mentioned above, experimental groups at Maryland,
IUCF, and GSI have all observed longitudinal motion and envelope oscillations in very close
agreement with the present model. The self-consistent longitudinal distribution has been used
to initiate a modal analysis as a basis for instability studies" ( as has also been done for
transverse motion, based on the Kapchinsky - Vladimirsky equation2). The dipole, quadrupole,
and sextupole modes have been observed at GSI, in close agreement with the predictions.
1. P. Liger, G. A. Krafft and D. Neuffer, Nucl. Inst. and Meth. A318, p. 290 (1992).
2. T. P. Ellison, S S Nagaitsev, M. S. Ball, D. D. Caussyn, M. J. Ellison, and B. J. Hamilton,
Phys. Rev. Lett. 70, p. 790 (1993).
3. S. Nagaitsev, T. Ellison, M. Ellison, D. Anderson, "The Investigation of Space Charge
Dominated Cooled Bunched Beams in a Synchrotron", Proc. Beam Cooling Workshop, Montreux,
4. D. Neuffer, IEEE Trans. Nucl. Sci. NS-26, 3031 (1979).
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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/m1/7/: accessed February 21, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.