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:toc:
[[chp:opalcycl]]
:stem: latexmath
:sectnums:
[[chp:benchmarks]]
Benchmarks
----------
[[sec:T3D]]
_OPAL-t_ compared with TRANSPORT & TRACE 3D
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
[[trace-3d]]
TRACE 3D
^^^^^^^^
TRACE 3D is an interactive beam dynamics program that calculates the
envelopes of a bunched beam, including linear space-change forces
[Trace_man]. It provides an instantaneous beam profile diagram and
delineates the transverse and longitudinal phase plane, where the
ellipses are characterized by the Twiss parameters and emittances (total
and unnormalized).
[[ssec:T3D_units]]
TRACE 3D Units
^^^^^^^^^^^^^^
TRACE 3D supports the following internal coordinates and units for the
three phase planes:
* *horizontal plane:* +
x [mm] is the displacement from the center of the beam bunch; +
x’ [mrad] is the beam divergence;
* *vertical plane:* +
y [mm] is the displacement from the center of the beam bunch; +
y’ [mrad] is the beam divergence;
* *longitudinal plane:* +
z [mm] is the displacement from the center of the beam bunch; +
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
[keV], respectively the displacement in phase and energy. The
relationships between these longitudinal coordinates are:
latexmath:[\[z = -\frac{\beta\lambda}{360}\Delta\phi\]]
and
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
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).
[[ssec:T3D_input]]
TRACE 3D Input beam
^^^^^^^^^^^^^^^^^^^
In TRACE 3D, the input beam is described by the following set of
parameters:
* *ER*: particle rest mass [MeV/];
* *Q*: charge state (+1 for protons);
* *W*: beam kinetic energy [MeV]
* *XI*: beam current [mA]
* *BEAMI*: array with initial Twiss parameters in the three phase planes
+
BEAMI =
latexmath:[$\alpha_x , \beta_x, \alpha_y, \beta_y, \alpha_{\phi}, \beta_{\phi} $] +
+
The alphas are dimensionless, latexmath:[$\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$] planes.
+
EMITI = latexmath:[$\epsilon_x , \epsilon_y, \epsilon_{\phi} $] +
+
The transversal emittances are expressed in latexmath:[$\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}$]-times their respective RMS values.
[[ssec:T3D_graphic]]
TRACE 3D Graphic Interface
++++++++++++++++++++++++++
An example of TRACE 3D graphic interface is shown in Figure [trace].
image:figures/Benchmarks/Trace.png[TRACE 3D graphic interface where: (1)
input beam in transverse plane (above) and longitudinal plane (below);
(2) output beam in transverse plane (above) and longitudinal plane
(below); (3) summary of beam parameters such as input and output
emittances and desired value for matching function; (4) line lattice
with different elements and beam envelope. The color legend is: blue
line for horizontal plane, red line for vertical plane, green line for
longitudinal plane and yellow line for dispersion.]
[[sec:TRAN]]
TRANSPORT
^^^^^^^^^
TRANSPORT is a computer program for first-order and second-order matrix
multiplication, intended for the design of beam transport system
[bib:transport]. The TRANSPORT version for Windows provides a graphic
beam profile diagram, as well as a sigma matrix description of the
simulated beam and line [Transport_GUI]. Differently from TRACE 3D, the
ellipses are characterized by the sigma-matrix coefficients and the
Twiss parameters and emittances (total and unnormalized) are reported as
output information.
[[ssec:TRAN_units]]
TRANSPORT Units
^^^^^^^^^^^^^^^
At any specified position in the system, an arbitrary charged particle
is represented by a vector, whose components are positions, angles and
momentum of the particle with respect to the reference trajectory. The
standard units and internal coordinates in TRANSPORT are:
* *horizontal plane:* +
x [cm] is the displacement of the arbitrary ray with respect to the
assumed central trajectory; +
latexmath:[$\theta$] [mrad] is the angle the ray makes with respect to
the assumed central trajectory;
* *vertical plane:* +
y [cm] is the displacement of the arbitrary ray with respect to the
assumed central trajectory; +
latexmath:[$\phi$] [mrad] is the angle the ray makes with respect to the
assumed central trajectory;
* *longitudinal plane:* +
l [cm] is the path length difference between the arbitrary ray and the
central trajectory; +
latexmath:[$\delta$] [%] is the fractional momentum deviation of the ray
from the assumed central trajectory.
Even if TRANSPORT supports this standard set of units [cm, mrad and %];
however using *card 15*, the users can redefine the units (see page 99
on TRANSPORT documentation [bib:transport] for more details).
[[ssec:TRAN_input]]
TRANSPORT Input beam
^^^^^^^^^^^^^^^^^^^^
The input beam is described in *card 1* in terms of the semi-axes of a
six-dimensional erect ellipsoid beam. In terms of diagonal sigma-matrix
elements, the input beam in TRANSPORT is expressed by 7 parameters:
* latexmath:[$\sqrt{\sigma_{ii}}$] [cm] represents one-half of the
horizontal (i=1), vertical (i=3) and longitudinal extent (i=5);
* latexmath:[$\sqrt{\sigma_{ii}}$] [mrad] represents one-half of the
horizontal (i=2), vertical (i=4) beam divergence;
* latexmath:[$\sqrt{\sigma_{66}}$] [%] represents one-half of the
momentum spread;
* p(0) is the momentum of the central trajectory [GeV/c].
If the input beam is tilted (Twiss alphas not zero), *card 12* must be
used, inserting the 15 correlations latexmath:[$r_{ij}$] parameters
among the 6 beam components. The correlation parameters are defined as
following:
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
directly used as input beam in TRANSPORT. An example of TRACE 3D
sigma-matrix structure is shown in Figure [trace_z].
.Sigma-matrix structure in TRACE 3D[Trace_man]
image:figures/Benchmarks/TRACE_z.png[]
From the sigma-matrix coefficients, TRANSPORT reports in output the
Twiss parameters and the total, unnormalized emittance. Even in this
case, a factor 5 is present between the emittances calculated by
TRANSPORT and the corresponding RMS values.
[[ssec:TRAN_graphic]]
TRANSPORT Graphic Interface
+++++++++++++++++++++++++++
An improved version of TRANSPORT has been embedded in a new graphic
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]
[[sec:T3D_TRAN]]
Comparison TRACE 3D and TRANSPORT
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
This study has been done following the same trend of the Regression Test
in _OPAL_ [AMAS], replacing the electron beam with a same energy proton
beam. Due to the different beam rigidity, the bending magnet features
have been redefined with a new magnetic field.
The simulated beam transport line contains:
* drift space (DRIFT 1): 0.250 m length;
* bending magnet (SBEND or RBEND): 0.250 m radius of curvature;
* drift space (DRIFT 2): 0.250 m length.
Keeping fixed the lattice structure, many similar transport lines have
been tested adding entrance and exit edge angles to the bending magnet,
changing the bending plane (vertical bending magnet) and direction
(right or left). In all the cases, the difficulties arise from the
non-achromaticity of the system and an increase in the horizontal and
longitudinal emittance is expected. In addition, the coupling between
these two planes has to be accurately studied.
In the following paragraph, an example of Sector Bending magnet (SBEND)
simulation with entrance and exit edge angles is discussed.
[[input-beam]]
Input beam
++++++++++
The starting simulation has been performed with TRACE 3D code. According
to Section [T3D_input], the simulated input beam is described by the
following parameters:
....
ER = 938.27
W = 7
FREQ = 700
BEAMI = 0.0, 4.0,0.0, 4.0, 0.0, 0.0756
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)]
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]
Before entering the TRACE 3D sigma-matrix coefficients in TRANSPORT, a
changing in the units is required using the *card 15* in the following
way:
....
15. 1. 'MM' 0.1 ; //express in mm the horizontal and vertical beam size
15. 5. 'MM' 0.1 ; //express in mm the beam length
....
At this point, the TRANSPORT input beam is defined by *card 1*:
....
1.0 1.709 0.427 1.709 0.427 0.11 0.0717 0.1148 /BEAM/ ;
....
using exactly the same sigma-matrix coefficients of
Figure [TRACE_z_Input]. Other two cards must be added in order to use
exactly the TRACE 3D R-matrix formalism:
....
16. 3. 1863.153; //proton mass, as ratio of electron mass
22. 0.05 0.0 700 0.0 /SPAC/ ; //space charge card
....
[[sbend-in-trace-3d]]
SBEND in TRACE 3D
+++++++++++++++++
The bending magnet definition in TRACE 3D requires:
.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
|n |0 |field-index gradient
|vf |0 |flag for vertical bending
|=======================================================================
The edge angles are described with another type code and parameters
which include also the fringe field. They must be added before and after
the bending magnet if entrance and exit edge angles are present and if
the fringe field has to be taken into account. In particular for the
entrance edge angle:
.Edge angle description in TRACE 3D and values used in the simulation
[cols="<,<,<",options="header",]
|=======================================================================
|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
|g [mm] |20 |total gap of magnet
|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
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]
[[sbend-in-transport]]
SBEND in TRANSPORT
++++++++++++++++++
The bending magnet definition in TRANSPORT requires:
.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
|n |0 |field-index gradient
|=====================================================
As for TRACE 3D, the edge angles are described with another card and
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
[cols="<,<,<",options="header",]
|=================================================
|Parameter |Value |Description
|Card |2 |Type code for edge
|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
|=================================================
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-size-and-emittance-comparison]]
Beam size and emittance comparison
++++++++++++++++++++++++++++++++++
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
[cols="<,<,<,<,<,<",]
|=======================================================================
|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
|Drift 1 |0.250 |1.712 |1.712 |1.712 |1.712
|Edge |0.250 |1.712 |1.712 |1.712 |1.712
|Bend |0.381 |1.638 |1.587 |1.638 |1.587
|Edge |0.381 |1.638 |1.587 |1.638 |1.587
|Drift 2 |0.631 |1.206 |1.264 |1.206 |1.264
|=======================================================================
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]
The same comparison has been performed in terms of horizontal and
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$]-mm-mrad) ,
the horizontal and longitudinal emittances are expected growing after
the bending magnet. The results are reported in Table [Emittance] and in
Figure [T3D_Tra_emi].
.Horizontal and longitudinal emittance comparison between TRACE 3D and
TRANSPORT, both expressed in latexmath:[$\pi$]-mm-mrad
[cols="<,<,<,<,<,<",]
|=======================================================================
|Position |z (m) |latexmath:[$\epsilon_x$] |latexmath:[$\epsilon_z$]
|latexmath:[$\epsilon_x$] |latexmath:[$\epsilon_z$]
|Input |0 |0.730 |0.08 |0.730 |0.08
|Drift 1 |0.250 |0.730 |0.08 |0.730 |0.08
|Edge |0.250 |0.730 |0.08 |0.730 |0.08
|Bend |0.381 |0.973 |0.65 |0.973 |0.65
|Edge |0.381 |0.973 |0.65 |0.973 |0.65
|Drift 2 |0.631 |0.973 |0.65 |0.973 |0.65
|=======================================================================
image:figures/Benchmarks/T3D_Tra_SBEND_edge_emi.png[Emittance comparison
between TRACE and TRANSPORT]
[[ssec:T3DtoTRAN]]
From TRACE 3D to TRANSPORT
++++++++++++++++++++++++++
.Bending magnet features in TRACE 3D and TRANSPORT
[cols="<,<,<",options="header",]
|==============================================================
|Parameter |Trace 3D |Transport
|*Bend card* |8 |4
|Angle |Input parameter [deg] |Output information [deg]
|Magn. field |Calculated. [T] |Input parameter [kG]
|Radius of curv. |Input parameter [mm] |Output information [m]
|Field-index |Input parameter |Input parameter
|Effect. length |Calculated [mm] |Input parameter [m]
|*Edge card* |9 |2
|Edge angle |Input parameter [deg] |Input parameter [deg]
|*Vertical gap* |9 |16.5
|Gap |Total [mm] |Half-gap [cm]
|*Fringe field card* |9 |16.7 / 16.8
|latexmath:[$K_1$] |Default: 0.45 |Default: 0.5
|latexmath:[$K_2$] |Default: 2.8 |Default: 0
|*Bend direction* |Bend angle sign |Coord. rotation
|Horiz. right |Angle latexmath:[$>$] 0 |Angle latexmath:[$>$] 0
|Horiz. left |Angle latexmath:[$<$] 0 |Card 20
|Vertical bend |Card 8, vf latexmath:[$>$] 0 |Card 20
|==============================================================
[[sec:OPAL]]
Relations to _OPAL-t_
^^^^^^^^^^^^^^^^^^^^^
In _OPAL_, the beam dynamics approach (time integration) is hence
completely different from the envelope-like supported by TRACE 3D and
TRANSPORT. The three codes support different units and require diverse
parameters for the input beam. A summary of their main features is
reported in Table [Features].
.Main features of the three beam dynamics codes: TRACE 3D, TRANSPORT and
__OPAL__
[cols="<,<,<,<",options="header",]
|====================================================================
|Code |TRACE 3D |TRANSPORT |_OPAL_
|*Type* |Envelope |Envelope |Time integration
|*Input* |Twiss, Emittance |Sigma, Momentum |Sigma, Energy
|*Units* |mm-mrad, deg-keV |cm-rad, cm-% |m-latexmath:[$\beta\gamma$]
|====================================================================
[[ssec:OPAL_units]]
_OPAL-t_ Units
^^^^^^^^^^^^^^
_OPAL-t_ supports the following internal coordinates and units for the
three phase planes:
* *horizontal plane:* +
X [m] horizontal position of a particle relative to the axis of the
element; +
PX [latexmath:[$\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$]] horizontal canonical momentum;
* *longitudinal plane:* +
Z [m] longitudinal position of a particle in floor-coordinates; +
PZ [latexmath:[$\beta_z\gamma$]] longitudinal canonical momentum;
[[ssec:OPAL_input]]
_OPAL-t_ Input beam
^^^^^^^^^^^^^^^^^^^
For the input beam, a `GAUSS` distribution type has been chosen. For
transferring the TRANSPORT (or TRACE 3D) input beam in terms of
sigma-matrix coefficients, it necessary to:
* adjust the units: from mm to m;
* correct for the factor latexmath:[$\sqrt{5}$]: from total to RMS
distribution;
* multiply for the relativistic factor
latexmath:[$\beta\gamma ={0.1224}$] for 7MeV protons;
In case of the modified sigma-matrix in Figure [TRACE_z_Input], the
corresponding _OPAL_ parameters for the `GAUSS` distributions are:
....
T3D SIGMA OPAL -T
-------------------------------------------------------
1.7088 mm SIGMAX = 1.7088/sqrt(5)e-3 m
0.4272 mrad SIGMAPX = 0.4272/sqrt(5)*0.1224e-3
1.7088 mm SIGMAY = 1.7088/sqrt(5)e-3 m
0.4272 mrad SIGMAPY = 0.4272/sqrt(5)*0.1224e-3
0.1092 mm SIGMAZ = 0.1092/sqrt(5)e-3 m
0.0717 % SIGMAPZ = (0.0717*10)/sqrt(5)*0.1224e-3
....
At the end of this calculation, the input beam in _OPAL_ is:
....
D1: DISTRIBUTION, TYPE=GAUSS,
SIGMAX = 0.7642e-03, SIGMAPX= 0.0234e-03, CORRX= 0.0,
SIGMAY = 0.7642e-03, SIGMAPY= 0.0234e-03, CORRY= 0.0,
SIGMAZ = 0.0488e-03, SIGMAPZ= 0.0392e-03, CORRZ= 0.0, R61= 0.0,
INPUTMOUNITS=NONE;
....
[[sec:T3D_OPAL]]
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
magnet features of Table [Bend_Trace,Edge_Trace] have been transformed
in _OPAL_ language as:
....
Bend: SBEND, ANGLE = 30.0 * Pi/180.0,
K1=0.0,
E1=0, E2=0,
FMAPFN = "1DPROFILE1-DEFAULT",
ELEMEDGE = 0.250, // end of first drift
DESIGNENERGY = 7E+06, // ref energy eV
L = 0.1294,
GAP = 0.02;
....
* *SBEND without edge angles:*
+
....
// Bending magnet configuration:
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:"]
+
A good overall agreement has been found between the two codes in term of
beam size and emittance. The different behavior inside the bending
magnet for the horizontal emittance is still undergoing study and it’s
probably due to a diverse coordinate system in the two codes.
* *SBEND with edge angles:*
+
....
// Bending magnet configuration:
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:"]
+
Even in this case, a good overall agreement has been found between the
two codes in term of beam size and emittance.
* *SBEND with field index:*
+
The field index parameter K1 is defined as:
+
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:[\[n = -\frac{\rho}{B_y}\frac{\partial B_y}{\partial x}.\]]
+
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
_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:[$\rho$]
+
Only the TEST 2 reports a reasonable behavior on the beam size and
emittance, as shown in Figure [SBEND_FI] using:
+
....
// Bending magnet configuration:
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:"]
+
Concerning the emittances and vertical beam size, a perfect agreement
has been found, instead a defocusing effect appears in the horizontal
plane. These results have been obtained with the default field map
provided by _OPAL_. However, a better result, only in the beam size as
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:"]
[[ssec:T3DtoOPAL]]
From TRACE 3D to _OPAL-t_
+++++++++++++++++++++++++
.Bending magnet features in TRACE 3D and __OPAL-t__
[cols="<,<,<",options="header",]
|==============================================================
|Parameter |Trace 3D |_OPAL-t_
|*Bend card* |8 |`SBEND` or `RBEND`
|Angle |Input parameter [deg] |Input/Calc. parameter [rad]
|Magn. field |Calculated. [T] |Input/Calc parameter [T]
|Radius of curv. |Input parameter [mm] |Output information [m]
|Field-index |Input parameter |Input parameter
|Length |Calculated [mm] |Input/Calc parameter [m]
|Length type |Effective |Straight
|*Edge card* |9 |`SBEND` or `RBEND`
|Edge angle |Input parameter [deg] |Input parameter [rad]
|*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 |-
|*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
|==============================================================
[[sec:conclusion]]
Conclusion
^^^^^^^^^^
* *TRACE 3D and TRANSPORT:* +
- a perfect agreement has been found between these two codes in
transversal envelope and emittance; +
- changing the TRANSPORT units, the input beam parameters, in terms of
sigma-matrix coefficients, can directly be imported from TRACE 3D file.
* *TRACE 3D and _OPAL-t_:* +
- a good agreement has been found between these two codes in case of
sector bending magnet with and without edge angles; +
- the default magnetic field map seems not working properly if the field
index is not zero +
- an improvement of the test map used is needed in order to match the
TRACE 3D emittance see Figure [SBEND_FI_test].
[[hard-edge-dipole-comparison-with-elegant]]
Hard Edge Dipole Comparison with ELEGANT
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
[[opal-dipole]]
_OPAL_ Dipole
^^^^^^^^^^^^^
When defining a dipole (`SBEND` or `RBEND`) in _OPAL_, a fringe field
map which defines the range of the field and the Enge coefficients is
required. If no map is provided, the code uses a default map. Here is a
dipole definition using the default map:
....
bend1: SBEND, ANGLE = bend_angle,
E1 = 0, E2 = 0,
FMAPFN = "1DPROFILE1-DEFAULT",
ELEMEDGE = drift_before_bend,
DESIGNENERGY = bend_energy,
L = bend_length,
WAKEF = FS_CSR_WAKE;
....
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
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
integrated fringe field value (FINT in ELEGANT) that usually used by
these codes, e.g. the ELEGANT and the TRACE3D. So we need to find a
default map for the hard edge dipole in _OPAL_.
[[map-for-hard-edge-dipole]]
Map for Hard Edge Dipole
^^^^^^^^^^^^^^^^^^^^^^^^
The proposed default map for a hard edge dipole can be:
....
1DProfile1 0 0 2
-0.00000001 0.0 0.00000001 3
-0.00000001 0.0 0.00000001 3
-99.9
-99.9
....
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
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
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
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]
Figure [plot-compare-default] compares the emittances and beam sizes
obtained by using the hard edge map, the default map and the ELEGANT.
One can see that the results produced by the hard edge map match the
ELEGANT results when FINT is set to zero.
[[integration-time-step]]
Integration Time Step
^^^^^^^^^^^^^^^^^^^^^
When the hard edge map is used for a dipole, finer integration time step
is needed to ensure the accurate of the calculation.
Figure [plot-emit-dt] compares the normalized emittances generated using
the hard edge map in _OPAL_ with varying time steps to those from the
ELEGANT. 0.01ps seems to be a optimal time step for the fringe field
region. To speed up the simulations, one can use larger time steps
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
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]
Figure [plot-fringe-size,plot-fringe-size-zoom] examine the effects of
the fringe field range and the integration time step on the simulation
accuracy. Figure [plot-fringe-size-zoom] is a zoom-in plot of
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]
image:figures/Benchmarks/report-fringe-size-zoom.png[Zoom in on the
final emittance in Figure [plot-fringe-size-zoom]]
[[d-csr-comparison-with-elegant]]
1D CSR comparison with ELEGANT
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1D-CSR wake function can now be used for the drift element by defining
its attribute `WAKEF = FS_CSR_WAKE`. In order to calculate the CSR
effect correctly, the drift has to follow a bending magnet whose CSR
calculation is also turned on.
....
bend1: SBEND, ANGLE = bend_angle,
E1 = 0, E2 = 0,
FMAPFN = "1DPROFILE1-DEFAULT",
ELEMEDGE = drift_before_bend,
DESIGNENERGY = bend_energy,
L = bend_length,
WAKEF = FS_CSR_WAKE;
....
....
drift1: DRIFT, L=0.4, ELEMEDGE = drift_before_bend +
bend_length, WAKEF = FS_CSR_WAKE;
....
[[benchmark]]
Benchmark
^^^^^^^^^
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
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
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 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
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
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__]
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
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
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
from the _.stat_ output file.
image:figures/Benchmarks/dpp-csr-on.png[latexmath:[$\frac{\Delta p}{p}$]
in Elegant and __OPAL__]
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
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__]
One important effect to notice is that in the drift space following the
bending magnet, the normalized emittance
latexmath:[$\epsilon_x(x, P_x)$] output by _OPAL_ keeps increasing while
the trace-like emittance latexmath:[$\epsilon_x(x, x')$] calculated by
ELEGANT does not. This can be explained by the fact that with a
relatively large energy spread (about latexmath:[$3\%$] at the end of
the dipole due to CSR), *an correlation* between transverse position and
energy can build up in a drift thereby induce emittance growth. However,
this effect can only be observed in the normalized emittance calculated
with
latexmath:[$\epsilon_x(x, P_x) = \sqrt{\langle x^2 \rangle \langle P_x^2\rangle - \langle xP_x \rangle^2}$]
where latexmath:[$P_x = \beta\gamma x'$], not the trace-like emittance
which is calculated as
latexmath:[$\epsilon_x(x, x') = \beta\gamma\sqrt{\langle x^2 \rangle \langle x'^2 \rangle - \langle xx' \rangle^2}$]
[prstab2003]. In Figure [plot-emit-csr-on], a trace-like horizontal
emittance is also calcualted for the _OPAL_ output beam distributions.
Like the ELEGANT result, this trace-like emittance doesn’t grow in the
drift. However, their differences come from the ELEGANT’s lack of CSR
effect in the fringe field region.
[[opal-impact-t]]
_OPAL_ & `Impact-t`
~~~~~~~~~~~~~~~~~~~
This benchmark compares rms quantities such as beam size and emittance
of _OPAL_ and `Impact-t` [qiang2005, qiang2006-1, qiang2006-2]. A *cold*
10mA H+ bunch is expanding in a 1m drift space. A Gaussian distribution,
with a cut at 4 latexmath:[$\sigma$] is used. The charge is computed by
assuming a 1MHz structure i.e.
latexmath:[$Q_{\text{tot}}=\frac{I}{\nu_{\text{rf}}}$]. For the
simulation we use a grid with latexmath:[$16^{3}$] grid point and open
boundary condition. The number of macro particles is
latexmath:[$N_{\text{p}} = 10^{5}$].
[[opal-input]]
_OPAL_ Input
^^^^^^^^^^^^
....
OPTION, ECHO = FALSE, PSDUMPFREQ = 10,
STATDUMPFREQ = 10, REPARTFREQ = 1000,
PSDUMPLOCALFRAME = FALSE, VERSION=10600;
TITLE, string="Gaussian bunch drift test";
REAL Edes = 0.001; // GeV
REAL CURRENT = 0.01; // A
REAL gamma=(Edes+PMASS)/PMASS;
REAL beta=sqrt(1-(1/gamma^2));
REAL gambet=gamma*beta;
REAL P0 = gamma*beta*PMASS;
D1: DRIFT, ELEMEDGE = 0.0, L = 1.0;
L1: LINE = (D1);
Fs1: FIELDSOLVER, FSTYPE = FFT, MX = 16, MY = 16, MT = 16, BBOXINCR=0.1;
Dist1: DISTRIBUTION, TYPE = GAUSS,
OFFSETX = 0.0, OFFSETY = 0.0, OFFSETZ = 15.0e-3,
SIGMAX = 5.0e-3, SIGMAY = 5.0e-3, SIGMAZ = 5.0e-3,
OFFSETPX = 0.0, OFFSETPY = 0.0, OFFSETPZ = 0.0,
SIGMAPX = 0.0 , SIGMAPY = 0.0 , SIGMAPZ = 0.0 ,
CORRX = 0.0, CORRY = 0.0, CORRZ = 0.0,
CUTOFFX = 4.0, CUTOFFY = 4.0, CUTOFFLONG = 4.0;
Beam1: BEAM, PARTICLE = PROTON, CHARGE = 1.0, BFREQ = 1.0, PC = P0,
NPART = 1E5, BCURRENT = CURRENT, FIELDSOLVER = Fs1;
SELECT, LINE = L1;
TRACK, LINE = L1, BEAM = Beam1, MAXSTEPS = 1000, ZSTOP = 1.0, DT = 1.0e-10;
RUN, METHOD = "PARALLEL-T", BEAM = Beam1, FIELDSOLVER = Fs1, DISTRIBUTION = Dist1;
ENDTRACK;
STOP;
....
[[impact-t-input]]
`Impact-t` Input
^^^^^^^^^^^^^^^^
....
!Welcome to Impact-t input file.
!All comment lines start with "!" as the first character of the line.
! col row
1 1
!
! information needed by the integrator:
! step-size, number of steps, and number of bunches/bins (??)
!
! dt Ntstep Nbunch
1.0e-10 700 1
!
! phase-space dimension, number of particles, a series of flags
! that set the type of integrator, error study, diagnostics, and
! image charge, and the cutoff distance for the image charge
!
! PSdim Nptcl integF errF diagF imchgF imgCutOff (m)
6 100000 1 0 1 0 0.016
!
! information about mesh: number of points in x, y, and z, type
! of boundary conditions, transverse aperture size (m),
! and longitudinal domain size (m)
!
! Nx Ny Nz bcF Rx Ry Lz
16 16 16 1 0.15 0.15 1.0e5
!
!
! distribution type number (2 == Gauss), restart flag, space-charge substep
! flag, number of emission steps, and max emission time
!
! distType restartF substepF Nemission Temission
2 0 0 -1 0.0
!
! sig* sigp* mu*p* *scale p*scale xmu* xmu*
!
0.005 0.0 0.0 1. 1. 0.0 0.0
0.005 0.0 0.0 1. 1. 0.0 0.0
0.005 0.0 0.0 1. 1. 0.0 0.0462
!
! information about the beam: current, kinetic energy, particle
! rest energy, particle charge, scale frequency, and initial cavity phase
!
! I/A Ek/eV Mc2/eV Q/e freq/Hz phs/rad
0.010 1.0e6 938.271998e+06 1.0 1.0e6 0.0
!
!
! ======= machine description starts here =======
! the following lines, which must each be terminated with a '/',
! describe one beam-line element per line; the basic structure is
! element length, ???, ???, element type, and then a sequence of
! at most 24 numbers describing the element properties
! 0 drift tube 2 zedge radius
! 1 quadrupole 9 zedge, quad grad, fileID,
! radius, alignment error x, y
! rotation error x, y, z
! L/m N/A N/A type location of starting edge v1 <B0><B0><B0> v23 /
1.0 0 0 0 0.0 0.5 /
....
[[results]]
Results
^^^^^^^
A good agreement is shown in the
Figure [plot-opal-impact1,plot-opal-impact2]. This proves to some extend
the compatibility of the space charge solvers of _OPAL_ and `Impact-t`.
image:figures/Benchmarks/opal-impact-1MHz-x.png[Transverse beam sizes
and emittances in `Impact-t` and __OPAL__,title="fig:"]
image:figures/Benchmarks/opal-impact-1MHz-y.png[Transverse beam sizes
and emittances in `Impact-t` and __OPAL__,title="fig:"]
image:figures/Benchmarks/opal-impact-1MHz-z.png[Longitudinal beam size
and emittance in `Impact-t` and __OPAL__]