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Name:DISTRIBUTION, TYPE = DISTRIBUTION_TYPE,
                   ATTRIBUTE #1 =,
                   ATTRIBUTE #2 =,
                   .
                   .
                   .
                   ATTRIBUTE #N =;

The internal units used by OPAL-t and OPAL-cycl are described in Section [variablesopalt,variablesopalcycl]. When defining a distribution, both OPAL-t and OPAL-cycl use meters for length and seconds for time. However, there are different options for the units used to input the momentum. This is controlled with the INPUTMOUNITS attribute, defined in Table [distattrinputmounits].

Once the distribution type is chosen, the next attribute to specify is the EMITTED attribute, which is defined in Table [distattremitted]. The EMITTED attribute controls whether a distribution is injected or emitted. An injected distribution is placed in its entirety into the simulation space at the start of the simulation. An emitted beam is emitted into the simulation over time as the simulation progresses (e.g. from a cathode in a photoinjector simulation). Currently, only OPAL-t supports emitted distributions. The default is an injected distribution.

Depending on the EMITTED attribute, we can specify several other attributes that do not depend on the distribution type. These are defined in Section [universaldistattributes,injecteddistattributes,emitteddistattributes] in Table [distattruniversal,distattrinjected,distattrsemitted].

1.2.1. Universal Attributes

1.2.2. Injected Distribution Attributes

1.2.3. Emitted Distribution Attributes

The most versatile distribution type is to use a user generated text file as input to OPAL. This allows the user to generate their own distribution, if the built in options in OPAL are insufficient, and have it either injected or emitted into the simulation. In Table [distattrfromfile] we list the single attribute specific to this type of distribution type.

An example of an injected FROMFILE distribution definition is:

an example of an emitted FROMFILE distribution definition is:

The text input file for the FROMFILE distribution type has slightly a slightly different format, depending on whether the distribution is to be injected or emitted. The injected file format is defined in Table [fromfileinjfileformat]. The emitted file format is defined in Table [fromfileemitfileformat].

Note that for an emitted FROMFILE distribution, all of the particle’s time, t, coordinates will be shifted to negative time (if they are not there already). The simulation clock will then start at t = 0 and distribution particles will be emitted into the simulation as the simulation progresses. Also note that, as the particles are emitted, they will be modified according to the type of emission model used. This is defined by the attribute EMISSIONMODEL, which is described in Section [emissionmodel]. A choice of NONE for the EMISSIONMODEL (which is the default) can be defined so as not to affect the distribution coordinates at all.

To maintain consistency N and NPART from the BEAM command in Chapter [beam] must be equal.

As the name implies, the GAUSS distribution type can generate distributions with a general Gaussian shape (here we show a one-dimensional example):

where \bar{x} is the average value of x. However, the GAUSS distribution can also be used to generate an emitted beam with a flat top time profile. We will go over the various options for the GAUSS distribution type in this section.

1.4.1. Simple GAUSS Distribution Type

Name:DISTRIBUTION, TYPE = GAUSS,
                   SIGMAX = 0.001,
                   SIGMAY = 0.003,
                   SIGMAZ = 0.002,
                   SIGMAPX = 0.0,
                   SIGMAPY = 0.0,
                   SIGMAPZ = 0.0,
                   CUTOFFX = 2.0,
                   CUTOFFY = 2.0,
                   CUTOFFLONG = 4.0,
                   OFFSETX = 0.001,
                   OFFSETY = -0.002,
                   OFFSETZ = 0.01,
                   OFFSETPZ = 1200.0
                   USEEV = TRUE;

W = {1.2}{MeV}

1.4.2. GAUSS Distribution for Photoinjector

\begin{aligned}
  \mathrm{TRISE} = t_{R} &= \left(\sqrt{2 \ln(10)} - \sqrt{2 \ln \left(\frac{10}{9} \right)} \right) \sigma_{R}\\
  & = 1.6869 \sigma_{R} \\
  \mathrm{TFALL} = t_{F} &= \left(\sqrt{2 \ln(10)} - \sqrt{2 \ln \left(\frac{10}{9} \right)} \right) \sigma_{F}\\
  & = 1.6869 \sigma_{F}\end{aligned}

\mathrm{TPULSEFWHM} = \mathrm{FWHM}_{P} = t_\mathrm{flattop} + \sqrt{2 \ln 2} \left( \sigma_{R} + \sigma_{F} \right)

\begin{aligned}
  t_{E}(\mathrm{CUTOFFLONG}) &= \mathrm{FWHM}_{P} - \frac{1}{2} (\mathrm{FWHM}_{R} + \mathrm{FWHM}_{F})
  + \mathrm{CUTOFFLONG} (\sigma_{R} + \sigma_{F}) \\
  &= \mathrm{FWHM}_{P} + \frac{\mathrm{CUTOFFLONG} - \sqrt{2 \ln 2}}{1.6869} (\mathrm{TRISE} + \mathrm{TFALL})\end{aligned}

1.4.3. Correlations for GAUSS Distribution (Experimental)

\sigma= \left[
\begin{array}{cccc}
1    &c_x  &r51    &r61\\
c_x  &1    &r52    &r62\\
r51  &r52  &1      &c_t\\
r61  &r62  &c_t    &1
\end{array}
\right]

\sigma= \left[
\begin{array}{cccc}
1      &0.756  &0.023    &0.496\\
0.756  &1      &0.385    &-0.042\\
0.023  &0.385  &1        &-0.834\\
0.496  &-0.042 &-0.834   &1
\end{array}
\right]

Dist:DISTRIBUTION, TYPE = GAUSS,
                   SIGMAX = 4.796e-03,
                   SIGMAPX = 231.0585,
                   CORRX = 0.756,
                   SIGMAY = 23.821e-03,
                   SIGMAPY = 1.6592e+03,
                   CORRY = -0.999,
                   SIGMAZ = 0.466e-02,
                   SIGMAPZ = 74.7,
                   CORRZ = -0.834,
                   OFFSETZ = 0.466e-02,
                   OFFSETPZ = 72e6,
                   R61 = 0.496,
                   R62 = -0.042,
                   R51 = 0.023,
                   R52 = 0.385;

The FLATTOP distribution type is used to define hard edge beam distributions. Hard edge, in this case, means a more or less uniformly filled cylinder of charge, although as we will see this is not always the case. The main purpose of the FLATTOP is to mimic laser pulses in photoinjectors, and so we usually will make this an emitted distribution. However it can be injected as well.

1.5.1. Injected FLATTOP

1.5.2. Emitted FLATTOP

Dist:DISTRIBUTION, TYPE = FLATTOP,
                   SIGMAX = 0.001,
                   SIGMAY = 0.002,
                   TRISE = 0.5e-12,
                   TFALL = 0.5e-12,
                   TPULSEFWHM = 10.0e-12,
                   CUTOFFLONG = 4.0,
                   NBIN = 5,
                   EMISSIONSTEPS = 100,
                   EMISSIONMODEL = ASTRA,
                   EKIN = 0.5,
                   EMITTED = TRUE;

1.5.3. Transverse Distribution from Laser Profile (Under Development)

1.5.4. GUNGAUSSFLATTOPTH Distribution Type

1.5.5. ASTRAFLATTOPTH Distribution Type

The BINOMIAL type of distribution is based on [JohoDist]. The shape of the binomial distribution is governed by one parameter m. By varying this single parameter one obtains the most commonly used distributions for our type of simulations, as listed in Table [binomdist].

When emitting a distribution from a cathode, there are several ways in which we can model the emission process in order to calculate the thermal emittance of the beam. In this section we discuss the various options available.

1.7.1. Emission Model: NONE (default)

Dist:DISTRIBUTION, TYPE = FLATTOP,
                   SIGMAX = 0.001,
                   SIGMAY = 0.002,
                   TRISE = 0.5e-12,
                   TFALL = 0.5e-12,
                   TPULSEFWHM = 10.0e-12,
                   CUTOFFLONG = 4.0,
                   NBIN = 5,
                   EMISSIONSTEPS = 100,
                   EMISSIONMODEL = NONE,
                   EKIN = 0.5,
                   EMITTED = TRUE;

1.7.2. Emission Model: ASTRA

\begin{aligned}
    p_{total} &= \sqrt{\left(\frac{\mathrm{EKIN}}{mc^{2}} + 1\right)^{2} - 1} \\
    p_{x} &= p_{total} \sin(\phi) \cos(\theta)) \\
    p_{y} &= p_{total} \sin(\phi) \sin(\theta)) \\
    p_{z} &= p_{total} |{\cos(\theta)}|
  \end{aligned}

\phi = 2.0 \arccos \left( \sqrt{x} \right)

1.7.3. Emission Model: NONEQUIL

Dist:DISTRIBUTION, TYPE = GAUSS,
                   SIGMAX = 0.001,
                   SIGMAY = 0.002,
                   TRISE = 1.0e-12,
                   TFALL = 1.0e-12,
                   TPULSEFWHM = 15.0e-12,
                   CUTOFFLONG = 3.0,
                   NBIN = 10,
                   EMISSIONSTEPS = 100,
                   EMISSIONMODEL = NONEQUIL,
                   ELASER = 6.48,
                   W = 4.1,
                   FE = 7.0,
                   CATHTEMP = 325,
                   EMITTED = TRUE;

It is possible to use multiple distributions in the same simulation. We do this be using a distribution list in the RUN command see Chapter [track]. Assume we have defined several distributions: DIST1, DIST2 and DIST3. If we want to use just one of these distributions in a simulation, we would use the following RUN command to start the simulation:

If we want to use all the distributions at the same time, then the command would instead be:

In this second case, the first distribution (DIST1) is the master distribution. The main consequence of this is that all distributions in the list will be forced to the same EMITTED condition as DIST1. So, if DIST1 is to be emitted, then all other distributions in the list will be forced to this same condition. If DIST1 is to be injected, then all other distribution is the list will also be injected.

The number of particles in the simulation is defined in the BEAM command see Chapter [beam]. The number of particles in each distribution in a distribution list is determined by this number and the WEIGHT attribute of each distribution (Table [distattruniversal]). If all distributions have the same WEIGHT value, then the number of particles will be divided up evenly among them. If, however we have a distribution list consisting of two distributions, and one has twice the WEIGHT of the other, then it will have twice the particles as its partner. The exception here is any FROMFILE distribution type. In this case, the WEIGHT attribute and the number of particles in the BEAM command are ignored. The number of particles in any FROMFILE distribution type is defined by the text file containing the distribution particle coordinates. (Section [fromfiledisttype]).