# Extensions of the longitudinal envelope equation Page: 2 of 8

electromagnetic force.)

In addition, there may be a bunching force due to a ramp in a longitudinal accelerating field.
This force may be written as:
FZ =-q E/(s) z (2)
where -EO'(s) is the local ramp in the longitudinal field and we have chosen to use a linear,
position-dependent bunching force. The coordinate z is position with respect to the bunch center.
In reference 4, a self-consistent longitudinal distribution with an envelope equation was
developed, consistent with these force equations.
In ref. 4, the self-consistent longitudinal distribution for a particle bunch is written as:
2 /
f(z,z',s) = 3N 1 _ - - -(z- z)2 (3)
27E z02 E2 zo
within the region in z, z' that the argument of the square root is positive (f = 0 outside). In
reference 4, position along the accelerator s is used as the independent variable (z' = dz/ds). z,
z' are particle coordinates, N is the total number of particles in the bunch. zo is the bunch half-
length (full length = 2zo) or beam envelope amplitude. EL is the longitudinal emittance
(unnormalized) in z - z' coordinates. (Lawson' includes an excellent discussion of this distribution
but unfortunately his equation 4.49 (in the current edition) contains a typo: the square root is
missing. His symbols are also somewhat different: his definition for N differs and his y" refers
to the bunching force parameter and not the second derivative of the central beam energy.) The
bunch density profile corresponding to eq. 2 is parabolic:
3N z2' (4)
X(z,s) = 1 - -
4z z0

The equation for the beam envelope amplitude zo(s) is:

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