Beam loading issues for SNS storage ring Page: 4 of 5
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d =W,7AE
-#0-E
di /fiEk o0}(6)
where AE is the energy gain per turn, # is the beam
phase deviation in radius, V and VL represent space
charge and beam loading voltages, respectively.
For each turn of injection, the space charge
voltage is calculated using the beam line density I, as
[1],
SC Z (7)
The SNS RF system and its feedback are
currently under study [2]. The purpose of this article is
to look at the beam loading induced bunch leakage.
Therefore, we simply assume that the resistance of the
loaded cavity (with, say, one turn feedback) to be
400Q ,2000, and 1005 per ring, together with a
capacitance 500pf .
For each bunch passage, the loaded cavity sees
a beam current, and it is responded with a beam loading
voltage V on the cavity. This voltage has the
fundamental mode, and it is out of phase with the RF
voltage by about 90 degrees.
If we take R = 40092, and C = 500pf , it is
found z = 400ns, which is well below the beam
circulating period of 841 ns. Since the intensity of the
beam increases gradually and the cavity time constant is
small, the beam loading voltage increases in
proportional to the beam intensity. Thus, the transient
effect of the beam loading can be neglected, and the
static beam loading can be assumed.40
20
0
-20a
earn nera R a
Beam oin itab
2 3 4 5 6 7 8 9 10 b- V ew 8 Lng
>
1 2 3 4 5 10 11
Fig. 1. Beam Loading Voltage used in Simulation
In the simulation, for each turn, 10 sequential
beam line densities are stored and used as input to the
loaded cavity, thus to get the beam loading voltage. One
example is shown in Fig.l. The (false) transient
response is only shown up in first couple of turns in
Fig.1. Therefore, we use the response of the last turn(10th turn) as the beam loading voltage for this turn, which
is marked by bold line in Fig.1.
For the detuned cavity, similar technique is used
to get the effective RF voltage. Both the RF signal (h=1)
and the beam line density are used as input to the detuned
cavity, then the cavity response in the 10th turn is used in
the simulation. The second harmonic is not changed in the
simulation.
Total 12,000 macroparticles are used, the time
step used is one turn. The machine circumference is split
into 30 bins in calculating the space charge induced
voltage. To split into 60 and 120 bins did not show visible
improvement in the simulations. In calculating the beam
line density, minor smoothing has been used.
Results
In the first set of simulations, a fixed RF voltage
of 40 KV (20 KV for 2nd harmonic) is used. Cavity
detuning angle is set to zero, and the effective resistance of
the loaded cavity is 400n, 200n, and 10052. Resulted
particle distributions and the associated beam line densities
at the end of injection are shown in Fig.2a, 2b, and 2c,
respectively.1.2
0.8
0.6a
40 KV, 400 Ohm
0.4 r
0
-0.2
-064- - -
-0.8
-1 0 1
Phase, pi a1.2
0.8
0.6
0.4
0.2
-0.2
-0.4-
-0.6
-0.8
-140 IV, 200 Ohm
r0 1
Phase, pi b1.2
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-140 KV, 100 Ohm
-
-,0 1
Phase, pi80A
4A
0AFig.2. SNS Beam Loading Effect with V, = 40KV
Because of the beam loading induced phase shift
of the effective RF voltage, the left end of the bunch length
is 0.87, 0.78x, and 0.742, for the cases a, b, and c,
whereas the other end is the same as the injected beam,
0.67z. We also note that all beam momentum spread in
the end of injection is about 0.70%.
One way to limit the bunch leakage is to use the
RF voltage ramping during the injection. For instance, to
linearly increase the h=1 cavity voltage from 20 KV at the
start to 40 KV at the end of injection. In this way, the beam
momentum spread in ring, defined during the multiturn
injection, can be lowered. Hence, the longitudinal space
charge induced bunch lengthening can be reduced.
In Fig.3a, 3b, and 3c, the counterparts to the cases
shown in Fig.2 are shown, with the RF voltage ramping
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Weng, W. T. & Zhang, S. Y. Beam loading issues for SNS storage ring, report, August 1998; Upton, New York. (https://digital.library.unt.edu/ark:/67531/metadc698263/m1/4/: accessed April 26, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.