Appendix E — Benchmarks

E.1.1 TRACE 3D Units and Input

TRACE 3D uses the following phase-space conventions:

• horizontal plane: x [mm], x' [mrad]
• vertical plane: y [mm], y' [mrad]
• longitudinal plane internally: z [mm], \Delta p/p [mrad]
• longitudinal plane for input and output: \Delta\phi [deg], \Delta W [keV]

The longitudinal conversions quoted in the appendix are

\[ z = -\frac{\beta\lambda}{360}\,\Delta\phi, \tag{E.1}\]

and

\[ \frac{\Delta p}{p} = \frac{\gamma}{\gamma+1}\frac{\Delta W}{W}. \tag{E.2}\]

The benchmark input beam is defined in TRACE 3D by the parameters ER, Q, W, XI, BEAMI, and EMITI. The concrete test case used in the appendix is

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

E.1.2 TRANSPORT Units and Input

TRANSPORT uses the standard coordinates:

• horizontal plane: x [cm], theta [mrad]
• vertical plane: y [cm], phi [mrad]
• longitudinal plane: l [cm], delta [%]

The input beam can be taken from the TRACE 3D sigma matrix after changing the units with the TRANSPORT card 15 settings documented in the appendix.

E.1.3 Comparison Case

The transport line consists of:

• DRIFT 1: 0.250 m
• one SBEND or RBEND with rho = 0.250 m
• DRIFT 2: 0.250 m

The benchmark discussed in detail is the SBEND case with entrance and exit edge angles.

E.1.4 Beam-Size and Emittance Comparison

The appendix reports agreement at the end of each line element.

E.1.5 Relation to OPAL-T

The appendix then maps the TRACE / TRANSPORT setup to OPAL-T. The main code feature comparison is:

The input distribution conversion quoted by the appendix is

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

leading to the documented GAUSS distribution example.

The OPAL-T bend comparison is shown for three cases:

• SBEND without edge angles
• SBEND with edge angles
• SBEND with nonzero field index K1

E.2.1 Integration Time-Step Dependence

The appendix then studies the effect of integration time step and fringe-field range. The main conclusion is that time-step size has the larger impact on the accuracy.

E.4.1 OPAL Input

OPTION, ECHO = FALSE, PSDUMPFREQ = 10,
STATDUMPFREQ = 10, REPARTFREQ = 1000,
PSDUMPFRAME = GLOBAL, 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;

E.4.2 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            /

E.4.3 Results

The appendix treats the agreement in the following plots as a consistency check for the two space-charge implementations in a simple drift benchmark.