Three-Dimensional Quasistatic Model for High Brightness Beam Dynamics Simulation Page: 4 of 10
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(xe + xi)2 + (ye + + (Zc + zk)2I-1/2:
[(xe -x2Nr-i+2)2 + (c + j)2 + (zc +zk)2-1/2:
[(xe +xi)2 + ( _ Y2Ny-j+2)2 + c k zk)2-1/2:
[(xe - x2Nx-i+292 + (yc -Y2N-j+2)2 + (Zc + zk)2-1/2:
[xe + xi)2 + (yc +y1)2 + (Zc - Z2N--k+2)2 -1/2[(xe -x2Nx-i+2)2 + (c + j)2 + (Zc - Z2N,-
[(xe + xi)2 + _7 Y2Ny-j+2)2 + (Zc - Z2N,-
[(xe - x2N-i+2)2 + (yc -Y2N,-j+2)2 + (Zck+2)2]-1/2:
k+2)2]-1/2:
- Z2N_-k+2)21 <i<N;1 j N,;1 k N,
N <i <2Nx;1 j N,;1 k<N,
1 <i< N <;N<j2N,;1 k N,
N <i<2Nx;N,<j 2N,;1 kN,,
1 <i<N; 1 jN,;NZ<k2N,,
N <i< 2Nx;1 jN,;N <k _2N,
1<_ i< Nx ; N,< j< 2N, ; NZ < k<5 2N,,
-1/2: Nx < i _ 2Nx;N,< j < 2N,;N< k < 2NZ.
(14)The FFT used to calculate the cyclic convolution in
Hockney's algorithm for standard Green function can be
used to calculate the potential in the field domain using the
new shifted-Green function. This avoids the requirement
that the particle domain and the field domain be contained
in one large computational domain. This leads to improved
numerical resolution for the charge densities and the re-
sulting electric fields than the conventional method, be-
cause the empty space between the charged bunches is not
included in the calculation. It is also far more efficient, in
terms of computational effort and storage, than the tradi-
tional approach of gridding the entire problem domain. To
test this algorithm, we have calculated the potential distri-
bution on the axis from the image charge of a round beam
as shown in Fig. 1. Here, the 1 nC electron charge has a 3D
waterbag distribution with 1 mm rms size (v"5 mm radius)
and located 5 mm after the cathode. The numerical solution
of the electrical potential using the shifted-Green function
method is given in Fig. 2 together with the analytical
solution. It is seen that the numerical solution and the
analytical solution agree with each other very well.
The image charge of a beam can have significant effects
on the beam quality in photoinjectors. Using the Linac
Coherent Light Sources (LCLS) S-band rf gun [23], we
did simulations with and without the image space-charge
effects. Here, the initial distribution of electron bunch is a
cold 10 ps long uniform cylinder with 1 mm radius. Thee-
z
total charge in this bunch is 1 nC. The peak acceleration
field is 120 MV/m. Figure 3 shows the transverse and
longitudinal rms sizes of the bunch with and without the
image-charge effects of the conducting photocathode in the
simulation. It is seen that the beam without image charge
has a larger initial transverse size than the beam with image
charge. The space-charge forces from the image charge of
the beam have opposite directions compared with the
space-charge forces from the beam itself. This helps to
reduce the initial beam blowup driven by the space-charge
forces.
In the above algorithms, both the Green function and the
charge density distribution are discretized on the grid. For a
beam with an aspect ratio close to 1, this algorithm works
well. However, during the emission of electrons out of the
cathode, the beam can have a very large transverse-to-
longitudinal ratio. For example, the typical transverse
size is on the order of millimeters while the longitudinal
size can be about a few tens to hundreds of microns. Under
this situation, the direct use of the Green function at each
mesh point is not efficient since it requires a large number
of mesh points along the transverse direction in order to get
sufficient resolution for the Green function along that
direction. A two-dimensional integrated Green function1300
1200-~1100
:1000
0
a 900
N 800700 .
6000
cathode
0.008 0.009 0.01 0.011
distance Z (m)0.012
0.013
FIG. 1. (Color) A schematic plot of an electron bunch in front of
the conducting photocathode together with its image charge.FIG. 2. (Color) Image-charge electric potential on the axis for a
round beam from the shifted-Green function solution together
with the analytical solution."n eica so ion -
.n c o o
.rGsc(xi, yj, Zk)
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Qiang, Ji; Lidia, S.; Ryne, R. D. & Limborg-Deprey, C. Three-Dimensional Quasistatic Model for High Brightness Beam Dynamics Simulation, article, June 19, 2006; [Menlo Park, California]. (https://digital.library.unt.edu/ark:/67531/metadc892538/m1/4/: accessed April 25, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.