... | ... | @@ -38,20 +38,28 @@ y [mm] is the displacement from the center of the beam bunch; + |
|
|
y’ [mrad] is the beam divergence;
|
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|
* *longitudinal plane:* +
|
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|
z [mm] is the displacement from the center of the beam bunch; +
|
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|
latexmath:[$\Delta$]p/p [mrad] is the difference between the particle’s
|
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|
latexmath:[\Delta]p/p [mrad] is the difference between the particle’s
|
|
|
longitudinal momentum and the reference momentum of the beam bunch.
|
|
|
|
|
|
For input and output, however, z and latexmath:[$\Delta$]p/p are
|
|
|
replaced by latexmath:[$\Delta\phi$] [degree] and latexmath:[$\Delta$]W
|
|
|
For input and output, however, z and latexmath:[\Delta]p/p are
|
|
|
replaced by latexmath:[\Delta\phi] [degree] and latexmath:[\Delta]W
|
|
|
[keV], respectively the displacement in phase and energy. The
|
|
|
relationships between these longitudinal coordinates are:
|
|
|
|
|
|
latexmath:[\[z = -\frac{\beta\lambda}{360}\Delta\phi\]]
|
|
|
[latexmath]
|
|
|
++++
|
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|
z = -\frac{\beta\lambda}{360}\Delta\phi
|
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|
++++
|
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|
|
|
|
and
|
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|
latexmath:[\[\frac{\Delta p}{p} = \frac{\gamma}{\gamma +1}\frac{\Delta W}{W}\]]
|
|
|
where latexmath:[$\beta$] and latexmath:[$\gamma$] are the relativist
|
|
|
parameters, latexmath:[$\lambda$] is the free-space wavelength of the RF
|
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|
|
|
|
[latexmath]
|
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|
++++
|
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|
\frac{\Delta p}{p} = \frac{\gamma}{\gamma +1}\frac{\Delta W}{W}
|
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|
++++
|
|
|
|
|
|
where latexmath:[\beta] and latexmath:[\gamma] are the relativist
|
|
|
parameters, latexmath:[\lambda] is the free-space wavelength of the RF
|
|
|
and W is the kinetic energy [MeV] at the beam center. This internal
|
|
|
conversion can be displayed using the _command W_ (see [Trace_man] page
|
|
|
42).
|
... | ... | @@ -63,29 +71,29 @@ TRACE 3D Input beam |
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|
In TRACE 3D, the input beam is described by the following set of
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|
parameters:
|
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|
|
|
|
* *ER*: particle rest mass [MeV/];
|
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|
* *ER*: particle rest mass [MeV/c^{2}];
|
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|
* *Q*: charge state (+1 for protons);
|
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|
* *W*: beam kinetic energy [MeV]
|
|
|
* *XI*: beam current [mA]
|
|
|
* *BEAMI*: array with initial Twiss parameters in the three phase planes
|
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|
+
|
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|
BEAMI =
|
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|
latexmath:[$\alpha_x , \beta_x, \alpha_y, \beta_y, \alpha_{\phi}, \beta_{\phi} $] +
|
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|
latexmath:[\alpha_x , \beta_x, \alpha_y, \beta_y, \alpha_{\phi}, \beta_{\phi} ] +
|
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|
+
|
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|
The alphas are dimensionless, latexmath:[$\beta_x$] and
|
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|
latexmath:[$\beta_y$] are expressed in m/rad (or mm/mrad) and
|
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|
latexmath:[$\beta_{\phi}$] in deg/keV;
|
|
|
The alphas are dimensionless, latexmath:[\beta_x] and
|
|
|
latexmath:[\beta_y] are expressed in m/rad (or mm/mrad) and
|
|
|
latexmath:[\beta_{\phi}] in deg/keV;
|
|
|
* *EMITI*: initial total and unnormalized emittances in x-x’, y-y’, and
|
|
|
latexmath:[$\Delta\phi$]-latexmath:[$\Delta W$] planes.
|
|
|
latexmath:[\Delta\phi]-latexmath:[\Delta W] planes.
|
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|
+
|
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|
EMITI = latexmath:[$\epsilon_x , \epsilon_y, \epsilon_{\phi} $] +
|
|
|
EMITI = latexmath:[\epsilon_x , \epsilon_y, \epsilon_{\phi} ] +
|
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|
+
|
|
|
The transversal emittances are expressed in latexmath:[$\pi$]-mm-mrad
|
|
|
and in latexmath:[$\pi$]-deg-keV the longitudinal emittance.
|
|
|
The transversal emittances are expressed in latexmath:[\pi]-mm-mrad
|
|
|
and in latexmath:[\pi]-deg-keV the longitudinal emittance.
|
|
|
|
|
|
In this beam dynamics code, the total emittance in each phase plane is
|
|
|
five times the RMS emittance in that plane and the displayed beam
|
|
|
envelopes are latexmath:[$\sqrt{5}$]-times their respective RMS values.
|
|
|
envelopes are latexmath:[\sqrt{5}]-times their respective RMS values.
|
|
|
|
|
|
[[ssec:T3D_graphic]]
|
|
|
TRACE 3D Graphic Interface
|
... | ... | @@ -93,14 +101,8 @@ TRACE 3D Graphic Interface |
|
|
|
|
|
An example of TRACE 3D graphic interface is shown in Figure [trace].
|
|
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|
|
|
image:figures/Benchmarks/Trace.png[TRACE 3D graphic interface where: (1)
|
|
|
input beam in transverse plane (above) and longitudinal plane (below);
|
|
|
(2) output beam in transverse plane (above) and longitudinal plane
|
|
|
(below); (3) summary of beam parameters such as input and output
|
|
|
emittances and desired value for matching function; (4) line lattice
|
|
|
with different elements and beam envelope. The color legend is: blue
|
|
|
line for horizontal plane, red line for vertical plane, green line for
|
|
|
longitudinal plane and yellow line for dispersion.]
|
|
|
.TRACE 3D graphic interface where: (1) input beam in transverse plane (above) and longitudinal plane (below); (2) output beam in transverse plane (above) and longitudinal plane (below); (3) summary of beam parameters such as input and output emittances and desired value for matching function; (4) line lattice with different elements and beam envelope. The color legend is: blue line for horizontal plane, red line for vertical plane, green line for longitudinal plane and yellow line for dispersion.
|
|
|
image:figures/Benchmarks/Trace.png[]
|
|
|
|
|
|
[[sec:TRAN]]
|
|
|
TRANSPORT
|
... | ... | @@ -127,17 +129,17 @@ standard units and internal coordinates in TRANSPORT are: |
|
|
* *horizontal plane:* +
|
|
|
x [cm] is the displacement of the arbitrary ray with respect to the
|
|
|
assumed central trajectory; +
|
|
|
latexmath:[$\theta$] [mrad] is the angle the ray makes with respect to
|
|
|
latexmath:[\theta] [mrad] is the angle the ray makes with respect to
|
|
|
the assumed central trajectory;
|
|
|
* *vertical plane:* +
|
|
|
y [cm] is the displacement of the arbitrary ray with respect to the
|
|
|
assumed central trajectory; +
|
|
|
latexmath:[$\phi$] [mrad] is the angle the ray makes with respect to the
|
|
|
latexmath:[\phi] [mrad] is the angle the ray makes with respect to the
|
|
|
assumed central trajectory;
|
|
|
* *longitudinal plane:* +
|
|
|
l [cm] is the path length difference between the arbitrary ray and the
|
|
|
central trajectory; +
|
|
|
latexmath:[$\delta$] [%] is the fractional momentum deviation of the ray
|
|
|
latexmath:[\delta] [%] is the fractional momentum deviation of the ray
|
|
|
from the assumed central trajectory.
|
|
|
|
|
|
Even if TRANSPORT supports this standard set of units [cm, mrad and %];
|
... | ... | @@ -152,19 +154,25 @@ The input beam is described in *card 1* in terms of the semi-axes of a |
|
|
six-dimensional erect ellipsoid beam. In terms of diagonal sigma-matrix
|
|
|
elements, the input beam in TRANSPORT is expressed by 7 parameters:
|
|
|
|
|
|
* latexmath:[$\sqrt{\sigma_{ii}}$] [cm] represents one-half of the
|
|
|
* latexmath:[\sqrt{\sigma_{ii}}] [cm] represents one-half of the
|
|
|
horizontal (i=1), vertical (i=3) and longitudinal extent (i=5);
|
|
|
* latexmath:[$\sqrt{\sigma_{ii}}$] [mrad] represents one-half of the
|
|
|
* latexmath:[\sqrt{\sigma_{ii}}] [mrad] represents one-half of the
|
|
|
horizontal (i=2), vertical (i=4) beam divergence;
|
|
|
* latexmath:[$\sqrt{\sigma_{66}}$] [%] represents one-half of the
|
|
|
* latexmath:[\sqrt{\sigma_{66}}] [%] represents one-half of the
|
|
|
momentum spread;
|
|
|
* p(0) is the momentum of the central trajectory [GeV/c].
|
|
|
|
|
|
If the input beam is tilted (Twiss alphas not zero), *card 12* must be
|
|
|
used, inserting the 15 correlations latexmath:[$r_{ij}$] parameters
|
|
|
used, inserting the 15 correlations latexmath:[r_{ij}] parameters
|
|
|
among the 6 beam components. The correlation parameters are defined as
|
|
|
following:
|
|
|
latexmath:[\[r_{ij}=\frac{\sigma_{ij}}{\sqrt{\sigma_{ii}gma_{jj}}}\]] As
|
|
|
|
|
|
[latexmath]
|
|
|
++++
|
|
|
r_{ij}=\frac{\sigma_{ij}}{\sqrt{\sigma_{ii}gma_{jj}}}
|
|
|
++++
|
|
|
|
|
|
As
|
|
|
explained before, with the *card 15*, it is possible to transform the
|
|
|
TRANSPORT standard units in TRACE-like units. In this way, the TRACE 3D
|
|
|
sigma-matrix for the input beam, printed out by _command Z_, can be
|
... | ... | @@ -188,11 +196,8 @@ shell written in C++ and is providing GUI type tools, which makes it |
|
|
easier to design new beam lines. A screen shot of a modern GUI Transport
|
|
|
interface [Transport_GUI] is shown in Figure [TRANSPORT].
|
|
|
|
|
|
image:figures/Benchmarks/TRANSPORT.png[GUI TRANSPORT graphic interface
|
|
|
[Tran_ex]. The continuous lines describe the beam envelope in the
|
|
|
vertical plane (above) and horizontal plane (below). The dashed line
|
|
|
displays the dispersion. The elements in the beam line are drawn as blue
|
|
|
and red rectangles]
|
|
|
.GUI TRANSPORT graphic interface [Tran_ex]. The continuous lines describe the beam envelope in the vertical plane (above) and horizontal plane (below). The dashed line displays the dispersion. The elements in the beam line are drawn as blue and red rectangles
|
|
|
image:figures/Benchmarks/TRANSPORT.png[]
|
|
|
|
|
|
[[sec:T3D_TRAN]]
|
|
|
Comparison TRACE 3D and TRANSPORT
|
... | ... | @@ -239,14 +244,14 @@ EMITI = 0.730, 0.730, 7.56 |
|
|
Thanks to the TRACE 3D graphic interface, the input beam can immediately
|
|
|
be visualized in the three phase plane as shown in Figure [Input_TRACE].
|
|
|
|
|
|
image:figures/Benchmarks/Input_Trace.png[TRACE 3D input beam in the
|
|
|
transversal plane (above) and in the longitudinal plane (below)]
|
|
|
.TRACE 3D input beam in the transversal plane (above) and in the longitudinal plane (below)
|
|
|
image:figures/Benchmarks/Input_Trace.png[]
|
|
|
|
|
|
The corresponding sigma-matrix with the relative units is displayed by
|
|
|
command Z:
|
|
|
|
|
|
image:figures/Benchmarks/TRACE_z_input.png[TRACE 3D sigma-matrix for the
|
|
|
input beam]
|
|
|
.TRACE 3D sigma-matrix for the beam
|
|
|
image:figures/Benchmarks/TRACE_z_input.png[]
|
|
|
|
|
|
Before entering the TRACE 3D sigma-matrix coefficients in TRANSPORT, a
|
|
|
changing in the units is required using the *card 15* in the following
|
... | ... | @@ -278,14 +283,13 @@ SBEND in TRACE 3D |
|
|
|
|
|
The bending magnet definition in TRACE 3D requires:
|
|
|
|
|
|
.Bending magnet description in TRACE 3D and values used in the
|
|
|
simulation
|
|
|
.Bending magnet description in TRACE 3D and values used in the simulation
|
|
|
[cols="<,<,<",options="header",]
|
|
|
|=======================================================================
|
|
|
|Parameter |Value |Description
|
|
|
|NT |8 |Type code for bending
|
|
|
|latexmath:[$\alpha$] [deg] |30 |angle of bend in horizontal plane
|
|
|
|latexmath:[$\rho$] [mm] |250 |radius of curvature of central trajectory
|
|
|
|latexmath:[\alpha] [deg] |30 |angle of bend in horizontal plane
|
|
|
|latexmath:[\rho] [mm] |250 |radius of curvature of central trajectory
|
|
|
|n |0 |field-index gradient
|
|
|
|vf |0 |flag for vertical bending
|
|
|
|=======================================================================
|
... | ... | @@ -301,22 +305,19 @@ entrance edge angle: |
|
|
|=======================================================================
|
|
|
|Parameter |Value |Description
|
|
|
|NT |9 |Type code for edge
|
|
|
|latexmath:[$\beta$] [deg] |10 |pole-face rotation
|
|
|
|latexmath:[$\rho$] [mm] |250 |radius of curvature of central trajectory
|
|
|
|latexmath:[\beta] [deg] |10 |pole-face rotation
|
|
|
|latexmath:[\rho] [mm] |250 |radius of curvature of central trajectory
|
|
|
|g [mm] |20 |total gap of magnet
|
|
|
|latexmath:[$K_1$] |0.36945 |fringe-field factor
|
|
|
|latexmath:[$K_2$] |0.36945 |fringe-field factor
|
|
|
|latexmath:[K_1] |0.36945 |fringe-field factor
|
|
|
|latexmath:[K_2] |0.36945 |fringe-field factor
|
|
|
|=======================================================================
|
|
|
|
|
|
A same configuration has been used for exit edge angle using
|
|
|
latexmath:[$\beta = {5}{^{\circ}}$]. The beam envelopes in the three
|
|
|
latexmath:[\beta = {5}{^{\circ}}]. The beam envelopes in the three
|
|
|
phase planes for this simulation are shown in Figure [Trace_env].
|
|
|
|
|
|
image:figures/Benchmarks/Trace_SBEND_edge.png[Beam envelopes in TRACE 3D
|
|
|
for a SBEND with entrance and exit edge angles. The blue line describes
|
|
|
the beam envelope in the horizontal plane, the red line in the vertical
|
|
|
plane, the green line in the longitudinal plane. The yellow line
|
|
|
displays the dispersion]
|
|
|
.Beam envelopes in TRACE 3D for a SBEND with entrance and exit edge angles. The blue line describes the beam envelope in the horizontal plane, the red line in the vertical plane, the green line in the longitudinal plane. The yellow line displays the dispersion
|
|
|
image:figures/Benchmarks/Trace_SBEND_edge.png[]
|
|
|
|
|
|
[[sbend-in-transport]]
|
|
|
SBEND in TRANSPORT
|
... | ... | @@ -324,14 +325,13 @@ SBEND in TRANSPORT |
|
|
|
|
|
The bending magnet definition in TRANSPORT requires:
|
|
|
|
|
|
.Bending magnet description in TRANSPORT and values used in the
|
|
|
simulation
|
|
|
.Bending magnet description in TRANSPORT and values used in the simulation
|
|
|
[cols="<,<,<",options="header",]
|
|
|
|=====================================================
|
|
|
|Parameter |Value |Description
|
|
|
|Card |4 |Type code for bending
|
|
|
|L [m] |30 |Effective length of the central trajectory
|
|
|
|latexmath:[$B_0$] [kG] |250 |Central field strength
|
|
|
|latexmath:[B_0] [kG] |250 |Central field strength
|
|
|
|n |0 |field-index gradient
|
|
|
|=====================================================
|
|
|
|
... | ... | @@ -340,27 +340,24 @@ parameters. In TRANSPORT, however, the fringe field is not automatically |
|
|
included with the edge angle, but it is described by a own card as
|
|
|
reported in the Table [Edge_Trans].
|
|
|
|
|
|
.Edge angle and fringe field description in TRANSPORT and values used in
|
|
|
the simulation
|
|
|
.Edge angle and fringe field description in TRANSPORT and values used in the simulation
|
|
|
[cols="<,<,<",options="header",]
|
|
|
|=================================================
|
|
|
|Parameter |Value |Description
|
|
|
|Card |2 |Type code for edge
|
|
|
|latexmath:[$\beta$] [deg] |10 |pole-face rotation
|
|
|
|latexmath:[\beta] [deg] |10 |pole-face rotation
|
|
|
|Card |16 |Type code for fringe field
|
|
|
|g [mm] |10 |half-gap of magnet
|
|
|
|latexmath:[$K_1$] |0.36945 |fringe-field factor
|
|
|
|latexmath:[$K_2$] |0.36945 |fringe-field factor
|
|
|
|latexmath:[K_1] |0.36945 |fringe-field factor
|
|
|
|latexmath:[K_2] |0.36945 |fringe-field factor
|
|
|
|=================================================
|
|
|
|
|
|
Running the Graphic TRANSPORT version, the beam envelopes in the
|
|
|
transverse phase planes for this simulation are shown in
|
|
|
Figure [Tran_env].
|
|
|
|
|
|
image:figures/Benchmarks/TRANS_SBEND_edge.png[Beam envelopes in
|
|
|
TRANSPORT for a SBEND with entrance and exit edge angles. The continuous
|
|
|
lines describe the beam envelope in the vertical plane (above) and
|
|
|
horizontal plane (below). The dashed line displays the dispersion.]
|
|
|
.Beam envelopes in TRANSPORT for a SBEND with entrance and exit edge angles. The continuous lines describe the beam envelope in the vertical plane (above) and horizontal plane (below). The dashed line displays the dispersion.
|
|
|
image:figures/Benchmarks/TRANS_SBEND_edge.png[]
|
|
|
|
|
|
[[beam-size-and-emittance-comparison]]
|
|
|
Beam size and emittance comparison
|
... | ... | @@ -370,12 +367,11 @@ In the next table, the results of the comparison between TRACE 3D and |
|
|
TRANSPORT in terms of the transversal beam sizes at the end of each
|
|
|
element in the line are summarized.
|
|
|
|
|
|
.Transversal beam size at the end of each element in the line printed
|
|
|
out by TRACE 3D and TRANSPORT
|
|
|
.Transversal beam size at the end of each element in the line printed out by TRACE 3D and TRANSPORT
|
|
|
[cols="<,<,<,<,<,<",]
|
|
|
|=======================================================================
|
|
|
|Position |z (m) |latexmath:[$\sigma_x$] (mm) |latexmath:[$\sigma_y$]
|
|
|
(mm) |latexmath:[$\sigma_x$] (mm) |latexmath:[$\sigma_y$] (mm)
|
|
|
|Position |z (m) |latexmath:[\sigma_x] (mm) |latexmath:[\sigma_y]
|
|
|
(mm) |latexmath:[\sigma_x] (mm) |latexmath:[\sigma_y] (mm)
|
|
|
|
|
|
|Input |0.000 |1.709 |1.709 |1.709 |1.709
|
|
|
|
... | ... | @@ -393,23 +389,22 @@ out by TRACE 3D and TRANSPORT |
|
|
The perfect agreement between these two codes arises immediately looking
|
|
|
at Figure [T3D_Tra_env].
|
|
|
|
|
|
image:figures/Benchmarks/T3D_Tra_SBEND_edge_env.png[Transversal beam
|
|
|
size comparison between TRACE 3D and TRANSPORT]
|
|
|
.Transversal beam size comparison between TRACE 3D and TRANSPORT
|
|
|
image:figures/Benchmarks/T3D_Tra_SBEND_edge_env.png[]
|
|
|
|
|
|
The same comparison has been performed in terms of horizontal and
|
|
|
longitudinal emittance, both expressed in latexmath:[$\pi$]-mm-mrad.
|
|
|
longitudinal emittance, both expressed in latexmath:[\pi]-mm-mrad.
|
|
|
While the vertical emittance remains constant and equal to the initial
|
|
|
value (latexmath:[$\epsilon_y = $] 0.730 latexmath:[$\pi$]-mm-mrad) ,
|
|
|
value (latexmath:[\epsilon_y = ] 0.730 latexmath:[\pi]-mm-mrad) ,
|
|
|
the horizontal and longitudinal emittances are expected growing after
|
|
|
the bending magnet. The results are reported in Table [Emittance] and in
|
|
|
Figure [T3D_Tra_emi].
|
|
|
|
|
|
.Horizontal and longitudinal emittance comparison between TRACE 3D and
|
|
|
TRANSPORT, both expressed in latexmath:[$\pi$]-mm-mrad
|
|
|
.Horizontal and longitudinal emittance comparison between TRACE 3D and TRANSPORT, both expressed in latexmath:[\pi]-mm-mrad
|
|
|
[cols="<,<,<,<,<,<",]
|
|
|
|=======================================================================
|
|
|
|Position |z (m) |latexmath:[$\epsilon_x$] |latexmath:[$\epsilon_z$]
|
|
|
|latexmath:[$\epsilon_x$] |latexmath:[$\epsilon_z$]
|
|
|
|Position |z (m) |latexmath:[\epsilon_x] |latexmath:[\epsilon_z]
|
|
|
|latexmath:[\epsilon_x] |latexmath:[\epsilon_z]
|
|
|
|
|
|
|Input |0 |0.730 |0.08 |0.730 |0.08
|
|
|
|
... | ... | @@ -424,8 +419,8 @@ TRANSPORT, both expressed in latexmath:[$\pi$]-mm-mrad |
|
|
|Drift 2 |0.631 |0.973 |0.65 |0.973 |0.65
|
|
|
|=======================================================================
|
|
|
|
|
|
image:figures/Benchmarks/T3D_Tra_SBEND_edge_emi.png[Emittance comparison
|
|
|
between TRACE and TRANSPORT]
|
|
|
.Emittance comparison between TRACE and TRANSPORT
|
|
|
image:figures/Benchmarks/T3D_Tra_SBEND_edge_emi.png[]
|
|
|
|
|
|
[[ssec:T3DtoTRAN]]
|
|
|
From TRACE 3D to TRANSPORT
|
... | ... | @@ -446,12 +441,12 @@ From TRACE 3D to TRANSPORT |
|
|
|*Vertical gap* |9 |16.5
|
|
|
|Gap |Total [mm] |Half-gap [cm]
|
|
|
|*Fringe field card* |9 |16.7 / 16.8
|
|
|
|latexmath:[$K_1$] |Default: 0.45 |Default: 0.5
|
|
|
|latexmath:[$K_2$] |Default: 2.8 |Default: 0
|
|
|
|latexmath:[K_1] |Default: 0.45 |Default: 0.5
|
|
|
|latexmath:[K_2] |Default: 2.8 |Default: 0
|
|
|
|*Bend direction* |Bend angle sign |Coord. rotation
|
|
|
|Horiz. right |Angle latexmath:[$>$] 0 |Angle latexmath:[$>$] 0
|
|
|
|Horiz. left |Angle latexmath:[$<$] 0 |Card 20
|
|
|
|Vertical bend |Card 8, vf latexmath:[$>$] 0 |Card 20
|
|
|
|Horiz. right |Angle latexmath:[>] 0 |Angle latexmath:[>] 0
|
|
|
|Horiz. left |Angle latexmath:[<] 0 |Card 20
|
|
|
|Vertical bend |Card 8, vf latexmath:[>] 0 |Card 20
|
|
|
|==============================================================
|
|
|
|
|
|
[[sec:OPAL]]
|
... | ... | @@ -464,14 +459,13 @@ TRANSPORT. The three codes support different units and require diverse |
|
|
parameters for the input beam. A summary of their main features is
|
|
|
reported in Table [Features].
|
|
|
|
|
|
.Main features of the three beam dynamics codes: TRACE 3D, TRANSPORT and
|
|
|
__OPAL__
|
|
|
.Main features of the three beam dynamics codes: TRACE 3D, TRANSPORT and __OPAL__
|
|
|
[cols="<,<,<,<",options="header",]
|
|
|
|====================================================================
|
|
|
|Code |TRACE 3D |TRANSPORT |_OPAL_
|
|
|
|*Type* |Envelope |Envelope |Time integration
|
|
|
|*Input* |Twiss, Emittance |Sigma, Momentum |Sigma, Energy
|
|
|
|*Units* |mm-mrad, deg-keV |cm-rad, cm-% |m-latexmath:[$\beta\gamma$]
|
|
|
|*Units* |mm-mrad, deg-keV |cm-rad, cm-% |m-latexmath:[\beta\gamma]
|
|
|
|====================================================================
|
|
|
|
|
|
[[ssec:OPAL_units]]
|
... | ... | @@ -484,14 +478,14 @@ three phase planes: |
|
|
* *horizontal plane:* +
|
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|
X [m] horizontal position of a particle relative to the axis of the
|
|
|
element; +
|
|
|
PX [latexmath:[$\beta_x\gamma$]] horizontal canonical momentum;
|
|
|
PX [latexmath:[\beta_x\gamma]] horizontal canonical momentum;
|
|
|
* *vertical plane:* +
|
|
|
Y [m] vertical position of a particle relative to the axis of the
|
|
|
element; +
|
|
|
PY [latexmath:[$\beta_y\gamma$]] horizontal canonical momentum;
|
|
|
PY [latexmath:[\beta_y\gamma]] horizontal canonical momentum;
|
|
|
* *longitudinal plane:* +
|
|
|
Z [m] longitudinal position of a particle in floor-coordinates; +
|
|
|
PZ [latexmath:[$\beta_z\gamma$]] longitudinal canonical momentum;
|
|
|
PZ [latexmath:[\beta_z\gamma]] longitudinal canonical momentum;
|
|
|
|
|
|
[[ssec:OPAL_input]]
|
|
|
_OPAL-t_ Input beam
|
... | ... | @@ -502,10 +496,10 @@ transferring the TRANSPORT (or TRACE 3D) input beam in terms of |
|
|
sigma-matrix coefficients, it necessary to:
|
|
|
|
|
|
* adjust the units: from mm to m;
|
|
|
* correct for the factor latexmath:[$\sqrt{5}$]: from total to RMS
|
|
|
* correct for the factor latexmath:[\sqrt{5}]: from total to RMS
|
|
|
distribution;
|
|
|
* multiply for the relativistic factor
|
|
|
latexmath:[$\beta\gamma ={0.1224}$] for 7MeV protons;
|
|
|
latexmath:[\beta\gamma ={0.1224}] for 7 MeV protons;
|
|
|
|
|
|
In case of the modified sigma-matrix in Figure [TRACE_z_Input], the
|
|
|
corresponding _OPAL_ parameters for the `GAUSS` distributions are:
|
... | ... | @@ -538,7 +532,7 @@ Comparison TRACE 3D and _OPAL-t_ |
|
|
In this section, the comparison between TRACE 3D and _OPAL-t_ is
|
|
|
discussed starting from `SBEND` definition in _OPAL-t_. The transport
|
|
|
line described in Section [T3D_TRAN] has been simulated in _OPAL_ using
|
|
|
10.000 particles and latexmath:[$10^{-11}$] s time step. The bending
|
|
|
10.000 particles and latexmath:[10^{-11}] s time step. The bending
|
|
|
magnet features of Table [Bend_Trace,Edge_Trace] have been transformed
|
|
|
in _OPAL_ language as:
|
|
|
|
... | ... | @@ -561,10 +555,9 @@ K1=0.0, |
|
|
E1=0, E2=0,
|
|
|
....
|
|
|
+
|
|
|
image:figures/Benchmarks/SBEND_noEdge_Env.png[TRACE 13D and _OPAL_
|
|
|
comparison: SBEND without edge angles,title="fig:"]
|
|
|
image:figures/Benchmarks/SBEND_noEdge_Emi.png[TRACE 13D and _OPAL_
|
|
|
comparison: SBEND without edge angles,title="fig:"]
|
|
|
.TRACE 3D and _OPAL_ comparison: SBEND without edge angles,
|
|
|
image:figures/Benchmarks/SBEND_noEdge_Env.png[width=370]
|
|
|
image:figures/Benchmarks/SBEND_noEdge_Emi.png[width=370]
|
|
|
+
|
|
|
A good overall agreement has been found between the two codes in term of
|
|
|
beam size and emittance. The different behavior inside the bending
|
... | ... | @@ -578,10 +571,9 @@ K1=0.0, |
|
|
E1=10*Pi/180.0, E2=5* Pi/180.0,
|
|
|
....
|
|
|
+
|
|
|
image:figures/Benchmarks/SBEND_Edges_Env.png[TRACE 3D and _OPAL_
|
|
|
comparison: SBEND with edge angles,title="fig:"]
|
|
|
image:figures/Benchmarks/SBEND_Edges_Emi.png[TRACE 3D and _OPAL_
|
|
|
comparison: SBEND with edge angles,title="fig:"]
|
|
|
.TRACE 3D and _OPAL_ comparison: SBEND with edge angles
|
|
|
image:figures/Benchmarks/SBEND_Edges_Env.png[width=370]
|
|
|
image:figures/Benchmarks/SBEND_Edges_Emi.png[width=370]
|
|
|
+
|
|
|
Even in this case, a good overall agreement has been found between the
|
|
|
two codes in term of beam size and emittance.
|
... | ... | @@ -589,20 +581,27 @@ two codes in term of beam size and emittance. |
|
|
+
|
|
|
The field index parameter K1 is defined as:
|
|
|
+
|
|
|
latexmath:[\[K1 = \frac{1}{B\rho}\frac{\partial B_y}{\partial x},\]]
|
|
|
[latexmath]
|
|
|
++++
|
|
|
K1 = \frac{1}{B\rho}\frac{\partial B_y}{\partial x},
|
|
|
++++
|
|
|
+
|
|
|
Section [RBend]. Instead, in TRACE 3D the field index parameter n is:
|
|
|
+
|
|
|
latexmath:[\[n = -\frac{\rho}{B_y}\frac{\partial B_y}{\partial x}.\]]
|
|
|
[latexmath]
|
|
|
++++
|
|
|
n = -\frac{\rho}{B_y}\frac{\partial B_y}{\partial x}.
|
|
|
++++
|
|
|
+
|
|
|
In order to have a significant focusing effect on both transverse
|
|
|
planes, the transport line has been simulated in TRACE 3D using
|
|
|
latexmath:[$n = 1.5$]. Since, a different definition exists between
|
|
|
latexmath:[n = 1.5]. Since, a different definition exists between
|
|
|
_OPAL_ and TRACE 3D on the field index, the n-parameter translation in
|
|
|
_OPAL_ language has been done with the following tests:
|
|
|
** TEST 1: K1 latexmath:[$=$] n/latexmath:[$\rho^2$]
|
|
|
** TEST 2: K1 latexmath:[$=$] n
|
|
|
** TEST 3: K1 latexmath:[$=$] n/latexmath:[$\rho$]
|
|
|
+
|
|
|
** TEST 1: K1 latexmath:[=] n/latexmath:[\rho^2]
|
|
|
** TEST 2: K1 latexmath:[=] n
|
|
|
** TEST 3: K1 latexmath:[=] n/latexmath:[\rho]
|
|
|
+
|
|
|
Only the TEST 2 reports a reasonable behavior on the beam size and
|
|
|
emittance, as shown in Figure [SBEND_FI] using:
|
... | ... | @@ -613,10 +612,9 @@ K1=1.5 |
|
|
E1=0, E2=0,
|
|
|
....
|
|
|
+
|
|
|
image:figures/Benchmarks/FI_SBEND_FMDef_Env_T2.png[TRACE 3D and _OPAL_
|
|
|
comparison: SBEND with field index and default field map,title="fig:"]
|
|
|
image:figures/Benchmarks/FI_SBEND_FMDef_Emi_T2.png[TRACE 3D and _OPAL_
|
|
|
comparison: SBEND with field index and default field map,title="fig:"]
|
|
|
.TRACE 3D and _OPAL_ comparison: SBEND with field index and default field map
|
|
|
image:figures/Benchmarks/FI_SBEND_FMDef_Env_T2.png[width=370]
|
|
|
image:figures/Benchmarks/FI_SBEND_FMDef_Emi_T2.png[width=370]
|
|
|
+
|
|
|
Concerning the emittances and vertical beam size, a perfect agreement
|
|
|
has been found, instead a defocusing effect appears in the horizontal
|
... | ... | @@ -626,10 +624,9 @@ shown in Figure [SBEND_FI_test], is achieved using a test field map in |
|
|
which the fringe field extension has been changed in the thin lens
|
|
|
approximation.
|
|
|
+
|
|
|
image:figures/Benchmarks/FI_SBEND_FMTest_Env_T2.png[TRACE 3D and _OPAL_
|
|
|
comparison: SBEND with field index and test field map,title="fig:"]
|
|
|
image:figures/Benchmarks/FI_SBEND_FMTest_Emit_T2.png[TRACE 3D and _OPAL_
|
|
|
comparison: SBEND with field index and test field map,title="fig:"]
|
|
|
.TRACE 3D and _OPAL_ comparison: SBEND with field index and test field map
|
|
|
image:figures/Benchmarks/FI_SBEND_FMTest_Env_T2.png[width=370]
|
|
|
image:figures/Benchmarks/FI_SBEND_FMTest_Emit_T2.png[width=370]
|
|
|
|
|
|
[[ssec:T3DtoOPAL]]
|
|
|
From TRACE 3D to _OPAL-t_
|
... | ... | @@ -651,12 +648,12 @@ From TRACE 3D to _OPAL-t_ |
|
|
|*Vertical gap* |9 |`SBEND` or `RBEND`
|
|
|
|Gap |Total [mm] |Total [m]
|
|
|
|*Fringe field card* |9 |FIELD MAP
|
|
|
|latexmath:[$K_1$] |Default: 0.45 |-
|
|
|
|latexmath:[$K_2$] |Default: 2.8 |-
|
|
|
|latexmath:[K_1] |Default: 0.45 |-
|
|
|
|latexmath:[K_2] |Default: 2.8 |-
|
|
|
|*Bend direction* |Bend angle sign |Coord. rotation
|
|
|
|Horiz. right |Angle latexmath:[$>$] 0 |Angle latexmath:[$>$] 0
|
|
|
|Horiz. left |Angle latexmath:[$<$] 0 |Angle latexmath:[$<$] 0
|
|
|
|Vertical bend |Card 8, vf latexmath:[$>$] 0 |Coord. rotation
|
|
|
|Horiz. right |Angle latexmath:[>] 0 |Angle latexmath:[>] 0
|
|
|
|Horiz. left |Angle latexmath:[<] 0 |Angle latexmath:[<] 0
|
|
|
|Vertical bend |Card 8, vf latexmath:[>] 0 |Coord. rotation
|
|
|
|==============================================================
|
|
|
|
|
|
[[sec:conclusion]]
|
... | ... | @@ -703,7 +700,7 @@ dipole definition using the default map: |
|
|
Please refer to Section [1DProfile1] for the definition of the field map
|
|
|
and the default map `1DPROFILE1-DEFAULT`. It defines a fringe field that
|
|
|
extends to 10 cm away from a dipole edge in both directions and it has
|
|
|
both latexmath:[$B_y$] and latexmath:[$B_z$] components. This makes the
|
|
|
both latexmath:[B_y] and latexmath:[B_z] components. This makes the
|
|
|
comparison between _OPAL_ and other codes which uses a hard edge dipole
|
|
|
by default,cumbersome because one needs to carefully integrate thought
|
|
|
the fringe field region in _OPAL_ in order to come up with the
|
... | ... | @@ -727,21 +724,20 @@ The proposed default map for a hard edge dipole can be: |
|
|
|
|
|
On the first line, the two zeros following `1DProfile1` are the orders
|
|
|
of the Enge coefficient for the entrance and exit edge of the dipole.
|
|
|
latexmath:[$2 cm$] is the default dipole gap width. The second line
|
|
|
latexmath:[2 cm] is the default dipole gap width. The second line
|
|
|
defines the fringe field region of the entrance edge of the dipole which
|
|
|
extends from latexmath:[$-0.00000001 cm$] to
|
|
|
latexmath:[$0.00000001 cm$]. The third line defines the same fringe
|
|
|
field region for the exit edge of the dipole. The latexmath:[$3$]s on
|
|
|
extends from latexmath:[-0.00000001 cm] to
|
|
|
latexmath:[0.00000001 cm]. The third line defines the same fringe
|
|
|
field region for the exit edge of the dipole. The latexmath:[3]s on
|
|
|
both line don’t mean anything, they are just placeholders. On the fourth
|
|
|
and fifth line, the zeroth order Enge coefficients for both edges are
|
|
|
given. Since they are large negative numbers, the field in the fringe
|
|
|
field region has no latexmath:[$B_z$] component and its
|
|
|
latexmath:[$B_y$] component is just like the field in the middle of the
|
|
|
field region has no latexmath:[B_z] component and its
|
|
|
latexmath:[B_y] component is just like the field in the middle of the
|
|
|
dipole.
|
|
|
|
|
|
image:figures/Benchmarks/report-compare-default.png[Compare emittances
|
|
|
and beam sizes obtained by using the hard edge map (_OPAL_), the default
|
|
|
map (_OPAL_), and the ELEGANT]
|
|
|
.Compare emittances and beam sizes obtained by using the hard edge map (_OPAL_), the default map (_OPAL_), and the ELEGANT
|
|
|
image:figures/Benchmarks/report-compare-default.png[]
|
|
|
|
|
|
Figure [plot-compare-default] compares the emittances and beam sizes
|
|
|
obtained by using the hard edge map, the default map and the ELEGANT.
|
... | ... | @@ -762,11 +758,11 @@ outside the fringe field regions. In Figure [plot-emit-dt], one can |
|
|
observe a discontinuity in the horizontal emittance when the hard edge
|
|
|
map is used in the calculation. This discontinuity comes from the fact
|
|
|
that _OPAL_ emittance is calculated at an instant time. Once the beam or
|
|
|
part of the beam gets into the dipole, its latexmath:[$P_x$] gets a kick
|
|
|
part of the beam gets into the dipole, its latexmath:[P_x] gets a kick
|
|
|
which will result in a sudden emittance change.
|
|
|
|
|
|
image:figures/Benchmarks/report-emit-dt.png[Horizontal and vertical
|
|
|
normalized emittances for different integration time steps]
|
|
|
.Horizontal and vertical normalized emittances for different integration time steps
|
|
|
image:figures/Benchmarks/report-emit-dt.png[]
|
|
|
|
|
|
Figure [plot-fringe-size,plot-fringe-size-zoom] examine the effects of
|
|
|
the fringe field range and the integration time step on the simulation
|
... | ... | @@ -775,11 +771,11 @@ Figure [plot-fringe-size]. We can conclude that the size of the |
|
|
integration time step has more influence on the accuracy of the
|
|
|
simulation.
|
|
|
|
|
|
image:figures/Benchmarks/report-fringe-size.png[Normalized horizontal
|
|
|
emittance for different fringe field ranges and integration time steps]
|
|
|
.Normalized horizontal emittance for different fringe field ranges and integration time steps
|
|
|
image:figures/Benchmarks/report-fringe-size.png[]
|
|
|
|
|
|
image:figures/Benchmarks/report-fringe-size-zoom.png[Zoom in on the
|
|
|
final emittance in Figure [plot-fringe-size-zoom]]
|
|
|
.Zoom in on the final emittance in Figure [plot-fringe-size-zoom]
|
|
|
image:figures/Benchmarks/report-fringe-size-zoom.png[]
|
|
|
|
|
|
[[d-csr-comparison-with-elegant]]
|
|
|
1D CSR comparison with ELEGANT
|
... | ... | @@ -815,17 +811,22 @@ The _OPAL_ dipoles all have fringe fields. When comparisons are done |
|
|
between _OPAL_ and ELEGANT [elegant] for example, one needs to
|
|
|
appropriately set the FINT attribute of the bending magnet in ELEGANT in
|
|
|
order to represent the field correctly. Although ELEGANT tracks in the
|
|
|
(latexmath:[$x, x', y, y', s, \delta$]) phase space, where
|
|
|
latexmath:[$\delta = \frac{\Delta p}{p_0}$] and latexmath:[$p_0$] is the
|
|
|
(latexmath:[x, x', y, y', s, \delta]) phase space, where
|
|
|
latexmath:[\delta = \frac{\Delta p}{p_0}] and latexmath:[p_0] is the
|
|
|
momentum of the reference particle, the watch point output beam
|
|
|
distributions from the ELEGANT are list in
|
|
|
(latexmath:[$x, x', y, y', t, \beta\gamma$]). If one wants to compare
|
|
|
(latexmath:[x, x', y, y', t, \beta\gamma]). If one wants to compare
|
|
|
ELEGANT watch point output distribution to _OPAL_, unit conversion needs
|
|
|
to be performed, i.e. latexmath:[\[\begin{aligned}
|
|
|
P_x &=& x'\beta\gamma, \\ P_y &=& y'\beta\gamma, \\ s &=& (\bar t-t)\beta c .\end{aligned}\]]
|
|
|
to be performed, i.e.
|
|
|
|
|
|
[latexmath]
|
|
|
++++
|
|
|
\begin{aligned}
|
|
|
P_x &=& x'\beta\gamma, \\ P_y &=& y'\beta\gamma, \\ s &=& (\bar t-t)\beta c .\end{aligned}
|
|
|
++++
|
|
|
|
|
|
To benchmark the CSR effect, we set up a simple beamline with 0.1m drift
|
|
|
latexmath:[$+$] 30 degree sbend latexmath:[$+$] 0.4m drift. When the CSR
|
|
|
latexmath:[+] 30 degree sbend latexmath:[+] 0.4m drift. When the CSR
|
|
|
effect is turn off, Figure [plot-emit-csr-off] shows that the normalized
|
|
|
emittances calculated using both _OPAL_ and ELEGANT agree. The emittance
|
|
|
values from _OPAL_ are obtained from the _.stat_ file, while for
|
... | ... | @@ -833,48 +834,48 @@ ELEGANT, the transverse emittances are obtained from the sigma output |
|
|
file (enx, and eny), the longitudinal emittance is calculated using the
|
|
|
watch point beam distribution output.
|
|
|
|
|
|
image:figures/Benchmarks/emit-csr-off.png[Comparison of the trace space
|
|
|
using ELEGANT and __OPAL__]
|
|
|
.Comparison of the trace space using ELEGANT and __OPAL__
|
|
|
image:figures/Benchmarks/emit-csr-off.png[]
|
|
|
|
|
|
When CSR calculations are enabled for both the bending magnet and the
|
|
|
following drift, Figure [plot-dpp-csr-on] shows the average
|
|
|
latexmath:[$\delta$] or latexmath:[$\frac{\Delta p}{p}$] change along
|
|
|
latexmath:[\delta] or latexmath:[\frac{\Delta p}{p}] change along
|
|
|
the beam line, and Figure [plot-emit-csr-on] compares the normalized
|
|
|
transverse and longitudinal emittances obtained by these two codes. The
|
|
|
average latexmath:[$\frac{\Delta p}{p}$] can be found in the centroid
|
|
|
average latexmath:[\frac{\Delta p}{p}] can be found in the centroid
|
|
|
output file (Cdelta) from ELEGANT, while in _OPAL_, one can calculate it
|
|
|
using
|
|
|
latexmath:[$\frac{\Delta p}{p} = \frac{1}{\beta^2}\frac{\Delta \overline{E}}{\overline{E}+mc^2}$],
|
|
|
where latexmath:[$\Delta \overline{E}$] is the average kinetic energy
|
|
|
latexmath:[\frac{\Delta p}{p} = \frac{1}{\beta^2}\frac{\Delta \overline{E}}{\overline{E}+mc^2}],
|
|
|
where latexmath:[\Delta \overline{E}] is the average kinetic energy
|
|
|
from the _.stat_ output file.
|
|
|
|
|
|
image:figures/Benchmarks/dpp-csr-on.png[latexmath:[$\frac{\Delta p}{p}$]
|
|
|
in Elegant and __OPAL__]
|
|
|
.latexmath:[\frac{\Delta p}{p}] in Elegant and __OPAL__
|
|
|
image:figures/Benchmarks/dpp-csr-on.png[]
|
|
|
|
|
|
In the drift space following the bending magnet, the CSR effects are
|
|
|
calculated using Stupakov’s algorithm with the same setting in both
|
|
|
codes. The average fractional momentum change
|
|
|
latexmath:[$\frac{\Delta p}{p}$] and the longitudinal emittance show
|
|
|
latexmath:[\frac{\Delta p}{p}] and the longitudinal emittance show
|
|
|
good agreements between these codes. However, they produce different
|
|
|
horizontal emittances as indicated in Figure [plot-emit-csr-on].
|
|
|
|
|
|
image:figures/Benchmarks/emit-csr-on.png[Transverse emittances in
|
|
|
ELEGANT and __OPAL__]
|
|
|
.Transverse emittances in ELEGANT and __OPAL__
|
|
|
image:figures/Benchmarks/emit-csr-on.png[]
|
|
|
|
|
|
One important effect to notice is that in the drift space following the
|
|
|
bending magnet, the normalized emittance
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latexmath:[$\epsilon_x(x, P_x)$] output by _OPAL_ keeps increasing while
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the trace-like emittance latexmath:[$\epsilon_x(x, x')$] calculated by
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latexmath:[\epsilon_x(x, P_x)] output by _OPAL_ keeps increasing while
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the trace-like emittance latexmath:[\epsilon_x(x, x')] calculated by
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ELEGANT does not. This can be explained by the fact that with a
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relatively large energy spread (about latexmath:[$3\%$] at the end of
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relatively large energy spread (about latexmath:[3\%] at the end of
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the dipole due to CSR), *an correlation* between transverse position and
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energy can build up in a drift thereby induce emittance growth. However,
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this effect can only be observed in the normalized emittance calculated
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with
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latexmath:[$\epsilon_x(x, P_x) = \sqrt{\langle x^2 \rangle \langle P_x^2\rangle - \langle xP_x \rangle^2}$]
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where latexmath:[$P_x = \beta\gamma x'$], not the trace-like emittance
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latexmath:[\epsilon_x(x, P_x) = \sqrt{\langle x^2 \rangle \langle P_x^2\rangle - \langle xP_x \rangle^2}]
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where latexmath:[P_x = \beta\gamma x'], not the trace-like emittance
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which is calculated as
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latexmath:[$\epsilon_x(x, x') = \beta\gamma\sqrt{\langle x^2 \rangle \langle x'^2 \rangle - \langle xx' \rangle^2}$]
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latexmath:[\epsilon_x(x, x') = \beta\gamma\sqrt{\langle x^2 \rangle \langle x'^2 \rangle - \langle xx' \rangle^2}]
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[prstab2003]. In Figure [plot-emit-csr-on], a trace-like horizontal
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emittance is also calcualted for the _OPAL_ output beam distributions.
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Like the ELEGANT result, this trace-like emittance doesn’t grow in the
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... | ... | @@ -888,12 +889,12 @@ _OPAL_ & `Impact-t` |
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This benchmark compares rms quantities such as beam size and emittance
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of _OPAL_ and `Impact-t` [qiang2005, qiang2006-1, qiang2006-2]. A *cold*
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10mA H+ bunch is expanding in a 1m drift space. A Gaussian distribution,
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with a cut at 4 latexmath:[$\sigma$] is used. The charge is computed by
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with a cut at 4 latexmath:[\sigma] is used. The charge is computed by
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assuming a 1MHz structure i.e.
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latexmath:[$Q_{\text{tot}}=\frac{I}{\nu_{\text{rf}}}$]. For the
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simulation we use a grid with latexmath:[$16^{3}$] grid point and open
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latexmath:[Q_{\text{tot}}=\frac{I}{\nu_{\text{rf}}}]. For the
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simulation we use a grid with latexmath:[16^{3}] grid point and open
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boundary condition. The number of macro particles is
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latexmath:[$N_{\text{p}} = 10^{5}$].
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latexmath:[N_{\text{p}} = 10^{5}].
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[[opal-input]]
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_OPAL_ Input
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... | ... | @@ -1010,10 +1011,9 @@ A good agreement is shown in the |
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Figure [plot-opal-impact1,plot-opal-impact2]. This proves to some extend
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the compatibility of the space charge solvers of _OPAL_ and `Impact-t`.
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image:figures/Benchmarks/opal-impact-1MHz-x.png[Transverse beam sizes
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and emittances in `Impact-t` and __OPAL__,title="fig:"]
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image:figures/Benchmarks/opal-impact-1MHz-y.png[Transverse beam sizes
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and emittances in `Impact-t` and __OPAL__,title="fig:"]
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.Transverse beam sizes and emittances in `Impact-t` and __OPAL__
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image:figures/Benchmarks/opal-impact-1MHz-x.png[width=370]
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image:figures/Benchmarks/opal-impact-1MHz-y.png[width=370]
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image:figures/Benchmarks/opal-impact-1MHz-z.png[Longitudinal beam size
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and emittance in `Impact-t` and __OPAL__] |
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.Longitudinal beam size and emittance in `Impact-t` and __OPAL__
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image:figures/Benchmarks/opal-impact-1MHz-z.png[] |