The voltage-amplitude requirement of a saw-tooth
wave-form buncher is calculated to give a desired degree of bunching
for a given beam current and particle species. This calculation
includes the effect of space-charge forces with and without adjacent
beam buckets. The results are compared to TRACE-3D calculations
which do not include the space-charge effects of adjacent bunches.
It appears that TRACE-3D calculations underestimate the bunching
voltage required. The methodology and a listing of the spread
sheet that performs the analytical bunching calculation are included.
The beam consists of a series of uniform-density
charge cylinders, see Fig. 1, that are spaced D=bl
apart, with length L=Dj/180
where j
is half the total bunched-beam phase spread, b
is the average beam velocity divided by the velocity of light
(c), and l
is the fundamental-frequency-rf free-space wave length.
Space-charge forces, due to beam buckets adjacent
to the reference particle, are calculated for a reference particle
that is on the beam axis at a cylinder edge. The motion of this
reference particle models the bunching of the entire distribution.
The electrostatic potential for the reference particle is calculated
from which the axial electric field seen by this particle is determined.
The equation of motion for the axial motion of the reference particle
is solved (the on axis reference particle experiences no transverse
forces) which gives the required bunching voltage in the beam
center-of-mass frame of reference. Velocities are added nonrelativistically
to give the buncher voltage in the laboratory reference frame.
TRACE-3D uses a uniformly filled ellipsoidal
model to estimate the space-charge forces. The equation of motion
for an on-axis reference particle, using the ellipsoidal model,
is integrated and the resulting bunching voltage is compared to
TRACE and the cylinder model.
Given that I is the beam current and f
is the bunch frequency, the charge density of the beam in the
cylinder is
(1)
where Ro
is the beam radius (Ro
is assumed to remain constant during the bunching process).
We consider only electrostatic forces in the Lorentz
force equation for the time evolution of the reference particle,
ignore all image-charge forces that could exist due to a beam
pipe, and consider only axial motion. Therefore,
(2)
where m is the particle mass, e is
the electron charge, Ez
is the axial electric field due to space
charge, t is time, and v is the velocity of the
reference particle in the beam bunch rest frame. The
on-axis axial electric field due to the two bunches can be obtained
from the potential function
(3)
where eo
is the free space permitivity. The first set of integrals is for
bunch B, and the second set for bunch A. Integrating Eq. (3)
for F,
taking the derivative with respect to Zref
(Zref is
the Z coordinate of the reference particle), and setting
Zref equal
to zero to obtain Ez
gives
(4)
for the electric field seen by the reference particle
due to bunch A, and
(5)
for the electric field due to bunch B. Adding Eqs.
(4) and (5) gives the total electric field seen by the reference
particle
(6)
Because the electric field does not depend on velocity,
Eq. (2) can be integrated to give the required reference-particle
energy gain, dWcm,
due to bunching in the beam rest frame,
(7)
The initial velocity vo for an unbunched beam
(L = D) gives a final bunch length Lmin
when v = 0. The bunch length, L, and the beam radius,
Ro, are
normalized to the bunch center separation distance, D,
by defining r = Ro/D
and s = L/D. Using Eqs. (1), (4), (5), and (6) in
Eq. (7) and integrating gives
(8)
where
(9)
and
(10)
Velocities, corresponding to the center-of-mass bunching
energy-spread and the average beam velocity, are added to give
the buncher voltage required in the laboratory reference-frame.
The nominal beam-velocity in the laboratory reference frame is
where Wo
is the nominal beam energy. The velocity of the reference particle
in the beam center-of-mass reference frame is
.
Calculating the reference particle's energy in the
laboratory reference frame and subtracting the average beam energy
gives the energy gain that the buncher must supply to the reference
particle which is
. (11)
The electric field due to a uniform-charge-density-ellipsoid beam-bunch
seen on axis by a reference particle at the edge of a single bunch
is [1], [2], [3]
(12)
and g is a "form factor" which can be approximated by
[1]
. (13)
Substituting Eq. (13) into Eq. (12) and picking the reference
particle coordinates to be on axis at the beam edge (R=0,
Zref=L/2) gives for the electric
field
. (14)
Carrying through the same procedure as for the cylinder model
gives
. (15)
Combining Eqs. (11) and (15) gives the buncher voltage required
in the Lab system.
Figure 2 show the spread sheet used to calculate the buncher voltage.
The parameters are meant to be self explanatory. Figure 3 shows
a comparison of this model calculation to results obtained from
TRACE-3D, which uses an ellipsoidal beam bunch model. The TRACE
transport channel, used for the comparisons, consisted of a periodic
series of solenoid magnets with the beam channel focusing strength
set to minimize the space-charge tune depression even in the maximum
bunching case. For the maximum bunching case, the initial beam
size was increased 3% to keep the beam nearly matched (the solenoid
magnetic field strength was not varied). The
discrepancy between the TRACE calculation and the spread sheet
calculation for the ellipsoidal distribution is due to the approximation
used for the form factor in Eq. (13) where this approximation
overestimates the space-charge force by as much as 10%. The difference
between the cylindrical model and the ellipsoidal model can be
understood by comparing the ratio of the electric field calculated
in Eqs. (4) and (14). Taking the ratio of these two equations
and using Eq. (1) gives = 2 (16) Equation (16) gives a value
of close to 2 for the ratio corresponding to our examples. This
ratio is also the ratio of the center-of-mass energy spread required
for bunching. Using Eq. (11) to transform to the Lab frame shows
that the buncher voltage required for the cylinder model should
be 40% () higher than the ellipsoidal model
and is consistent with the result shown in Fig. (3). Also, the
TRACE beam distribution for j = 180o
is already bunched with a pseudo-gaussian shape which causes a
further underestimate of the buncher voltage required to obtain
the final degree of bunching. Figure 3 shows that including adjacent
bunches for calculating longitudinal space-charge effects is important
only for minimal bunching (j"150o).
[1] T. P. Wangler, "Space-Charge Limits in Linear
Accelerators," LA-8388 (1980).[2] K. R. Crandall, D. P. Rusthoi,
"TRACE 3-D Documentation," LA-UR-90-4146.[3] O. D. Kellogg,
"Foundations of Potential Theory," (1929).
*Work supported by the U.S. Department of Energy