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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).

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;
  \Deltap/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 \Deltap/p are replaced by \Delta\phi [degree] and \DeltaW [keV], respectively the displacement in phase and energy. The relationships between these longitudinal coordinates are:

and

where \beta and \gamma are the relativist parameters, \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).

In TRACE 3D, the input beam is described by the following set of parameters:

• ER: particle rest mass [MeV/c^{2}];

• 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 = \alpha_x , \beta_x, \alpha_y, \beta_y, \alpha_{\phi}, \beta_{\phi}
  

  The alphas are dimensionless, \beta_x and \beta_y are expressed in m/rad (or mm/mrad) and \beta_{\phi} in deg/keV;

• EMITI: initial total and unnormalized emittances in x-x’, y-y’, and \Delta\phi-\Delta W planes.

  EMITI = \epsilon_x , \epsilon_y, \epsilon_{\phi}
  

  The transversal emittances are expressed in \pi-mm-mrad and in \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 \sqrt{5}-times their respective RMS values.

TRACE 3D Graphic Interface

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.

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;
  \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;
  \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;
  \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).

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:

• \sqrt{\sigma_{ii}} [cm] represents one-half of the horizontal (i=1), vertical (i=3) and longitudinal extent (i=5);

• \sqrt{\sigma_{ii}} [mrad] represents one-half of the horizontal (i=2), vertical (i=4) beam divergence;

• \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 r_{ij} parameters among the 6 beam components. The correlation parameters are defined as following:

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].

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.

TRANSPORT Graphic Interface

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

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

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

1.0 1.709 0.427 1.709 0.427 0.11 0.0717 0.1148 /BEAM/ ;

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 TRANSPORT

Beam size and emittance comparison

From TRACE 3D to TRANSPORT

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].

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 [\beta_x\gamma] horizontal canonical momentum;

• vertical plane:
  Y [m] vertical position of a particle relative to the axis of the element;
  PY [\beta_y\gamma] horizontal canonical momentum;

• longitudinal plane:
  Z [m] longitudinal position of a particle in floor-coordinates;
  PZ [\beta_z\gamma] longitudinal canonical momentum;

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 \sqrt{5}: from total to RMS distribution;

• multiply for the relativistic factor \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:

At the end of this calculation, the input beam in OPAL is:

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 10^{-11} s time step. The bending magnet features of Table [Bend_Trace,Edge_Trace] have been transformed in OPAL language as:

• SBEND without edge angles:

  // Bending magnet configuration:
  K1=0.0,
  E1=0, E2=0,

  TRACE 3D and OPAL comparison: SBEND without edge angles,

  

  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,

  TRACE 3D and OPAL comparison: SBEND with edge angles

  

  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:

  K1 = \frac{1}{B\rho}\frac{\partial B_y}{\partial x},

  Section [RBend]. Instead, in TRACE 3D the field index parameter n is:

  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 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 = n/\rho^2

  • TEST 2: K1 = n

  • TEST 3: K1 = n/\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,

    TRACE 3D and OPAL comparison: SBEND with field index and default field map

    

    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.

    TRACE 3D and OPAL comparison: SBEND with field index and test field map

    

From TRACE 3D to OPAL-t

• 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].

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:

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 B_y and 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.

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. 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 -0.00000001 cm to 0.00000001 cm. The third line defines the same fringe field region for the exit edge of the dipole. The 3s 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 B_z component and its B_y component is just like the field in the middle of the dipole.

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.

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 P_x gets a kick which will result in a sudden emittance change.

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.

      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;

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 (x, x', y, y', s, \delta) phase space, where \delta = \frac{\Delta p}{p_0} and p_0 is the momentum of the reference particle, the watch point output beam distributions from the ELEGANT are list in (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.

To benchmark the CSR effect, we set up a simple beamline with 0.1m drift ] 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.

When CSR calculations are enabled for both the bending magnet and the following drift, Figure [plot-dpp-csr-on] shows the average \delta or \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 \frac{\Delta p}{p} can be found in the centroid output file (Cdelta) from ELEGANT, while in OPAL, one can calculate it using \frac{\Delta p}{p} = \frac{1}{\beta^2}\frac{\Delta \overline{E}}{\overline{E}+mc^2}, where \Delta \overline{E} is the average kinetic energy from the .stat output file.

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 \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].

One important effect to notice is that in the drift space following the bending magnet, the normalized emittance \epsilon_x(x, P_x) output by OPAL keeps increasing while the trace-like emittance \epsilon_x(x, x') calculated by ELEGANT does not. This can be explained by the fact that with a relatively large energy spread (about 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 \epsilon_x(x, P_x) = \sqrt{\langle x^2 \rangle \langle P_x^2\rangle - \langle xP_x \rangle^2} where P_x = \beta\gamma x', not the trace-like emittance which is calculated as \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.

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.