elements.asciidoc



Back to Main Page


Table of Contents


1. Elements

1.1. Element Input Format
1.2. Common Attributes for all Elements
1.3. Drift Spaces

1.4. Bending Magnets

1.4.1. RBend (OPAL-t)
1.4.2. RBend3D (OPAL-t)
1.4.3. SBend (OPAL-t)
1.4.4. RBend and SBend Examples (OPAL-t)
1.4.5. Bend Fields from 1D Field Maps (OPAL-t)
1.4.6. Default Field Map (OPAL-t)
1.4.7. SBend3D (OPAL-cycl)


1.5. Quadrupole
1.6. Sextupole
1.7. Octupole
1.8. General Multipole
1.9. General Multipole (will replace General Multipole when implemented)
1.10. Solenoid
1.11. Cyclotron

1.12. Ring Definition

1.12.1. Local Cartesian Offset
1.12.2. Local Cylindrical Offset


1.13. Source

1.14. RF Cavities (OPAL-t and OPAL-cycl)

1.14.1. OPAL-t mode
1.14.2. OPAL-cycl mode


1.15. RF Cavities with Time Dependent Parameters

1.15.1. Time Dependence
1.15.2. Fringe Field


1.16. Traveling Wave Structure
1.17. Monitor

1.18. Collimators

1.18.1. OPAL-t mode
1.18.2. OPAL-cycl mode


1.19. Septum (OPAL-cycl)
1.20. Probe (OPAL-cycl)
1.21. Stripper (OPAL-cycl)
1.22. Degrader (OPAL-t)
1.23. Correctors (OPAL-t)
1.24. Beam Stripping (OPAL-cycl)
1.25. References


1. Elements


1.1. Element Input Format

All physical elements are defined by statements of the form


label:keyword, attribute,..., attribute


where


label

Is the name to be given to the element (in the example QF), it is an
identifier see Identifiers or Labels.

keyword

Is a keyword see Identifiers or Labels, it is an element type keyword (in
the example QUADRUPOLE),

attribute

normally has the form


attribute-name=attribute-value


attribute-name

selects the attribute from the list defined for the element type
keyword (in the example L and K1). It must be an identifier
see Identifiers or Labels.

attribute-value

gives it a value see Command Attribute Types (in the example 1.8 and
0.015832).


Omitted attributes are assigned a default value, normally zero.


Example:


QF: QUADRUPOLE, L=1.8, K1=0.015832;


1.2. Common Attributes for all Elements

The following attributes are allowed on all elements:


TYPE

A string value see String Attributes. It specifies an "engineering
type" and can be used for element selection.

APERTURE

A string value see String Attributes which describes the element
aperture. All but the last attribute of the aperture have units of
meter, the last one is optional and is a positive real number.
Possible choices are


APERTURE="SQUARE(a,f)" has a square shape of width and
height a,


APERTURE="RECTANGLE(a,b,f)" has a rectangular shape of width
a and height b,


APERTURE="CIRCLE(d,f)" has a circular shape of diameter d,


APERTURE="ELLIPSE(a,b,f)" has an elliptical shape of major
a and minor b.

The option SQUARE(a,f) is equivalent to RECTANGLE(a,a,f) and
CIRCLE(d,f) is equivalent to ELLIPSE(d,d,f). The size of the
exit aperture is scaled by a factor 
f. For
f < 1 the exit aperture is smaller than the entrance
aperture, for f = 1 they are the same and for
f > 1
 the exit aperture is bigger.


Dipoles have GAP and HGAP which define an aperture and hence do
not recognise APERTURE. The aperture of the dipoles has rectangular
shape of height GAP and width HGAP. In longitudinal direction it
is bent such that its center coincides with the circular segment of
the reference particle when ignoring fringe fields. Between the
beginning of the fringe field and the entrance face and between the
exit face and the end of the exit fringe field the rectangular shape
has width and height that are twice of what they are inside the
dipole.


Default aperture for all other elements is a circle of 1.0m.


L

The length of the element (default: 0m).

WAKEF

Attach wakefield that was defined using the WAKE command.

ELEMEDGE

The edge of an element is specified in s coordinates in meters. This
edge corresponds to the origin of the local coordinate system and is
the physical start of the element. (Note that in general the fields
will extend in front of this position.) The physical end of the
element is determined by ELEMEDGE and its physical length. (Note
again that in general the fields will extend past the physical end of
the element.)

PARTICLEMATTERINTERACTION

Attach a handler for particle matter interaction, see Chapter Particle Matter Interaction.

X

X-component of the position of the element in the laboratory
coordinate system.

Y

Y-component of the position of the element in the laboratory
coordinate system.

Z

Z-component of the position of the element in the laboratory
coordinate system.

THETA

Angle of rotation of the element about the y-axis relative to the
default orientation,
\mathbf{n} = \left(0, 0, 1\right)^{\mathbf{T}}
.

PHI

Angle of rotation of the element about the x-axis relative to the
default orientation,
\mathbf{n} = \left(0, 0, 1\right)^{\mathbf{T}}


PSI

Angle of rotation of the element about the z-axis relative to the
default orientation,
\mathbf{n} = \left(0, 0, 1\right)^{\mathbf{T}}


ORIGIN

3D position vector. An alternative to using X, Y and Z to
position the element. Can’t be combined with THETA and PHI. Use
ORIENTATION instead.

ORIENTATION

Vector of Tait-Bryan angles bib.tait-bryan. An alternative to rotate
the element instead of using THETA, PHI and PSI. Can’t be
combined with X, Y and Z, use ORIGIN instead.

DX

Error on x-component of position of element. Doesn’t affect the design
trajectory.

DY

Error on y-component of position of element. Doesn’t affect the design
trajectory.

DZ

Error on z-component of position of element. Doesn’t affect the design
trajectory.

DTHETA

Error on angle THETA. Doesn’t affect the design trajectory.

DPHI

Error on angle PHI. Doesn’t affect the design trajectory.

DPSI

Error on angle PSI. Doesn’t affect the design trajectory.


All elements can have arbitrary additional attributes which are defined
in the respective section.


1.3. Drift Spaces


label:DRIFT, TYPE=string, APERTURE=string, L=real;


A DRIFT space has no additional attributes. Examples:


DR1:DRIFT, L=1.5;
DR2:DRIFT, L=DR1->L, TYPE=DRF;


The length of DR2 will always be equal to the length of DR1. The
reference system for a drift space is a Cartesian coordinate system. This
is a restricted feature of OPAL-t. In OPAL-t drifts are implicitly
given, if no field is present.


1.4. Bending Magnets

Bending magnets refer to dipole fields that bend particle trajectories.
Currently OPAL supports the following different bend elements: RBEND, (valid
in OPAL-t, see RBend (OPAL-t)), SBEND (valid in OPAL-t, see
SBend (OPAL-t)), RBEND3D, (valid in OPAL-t, see RBend3D (OPAL-t))
and SBEND3D (valid in OPAL-cycl, see SBend3D (OPAL-cycl)).


Describing a bending magnet can be somewhat complicated as there can be
many parameters to consider: bend angle, bend radius, entrance and exit
angles etc. Therefore we have divided this section into several parts:


RBend (OPAL-t) and SBend (OPAL-t) describe the geometry and attributes of the
OPAL-t bend elements RBEND and SBEND.


RBend and SBend Examples (OPAL-t) describes how to implement an RBEND or
SBEND in an OPAL-t simulation.


SBend3D (OPAL-cycl) is self contained. It describes how to implement
an SBEND3D element in an OPAL-cycl simulation.


Figure 1 illustrates a general rectangular bend (RBEND) with a positive bend angle 
\alpha
. The entrance edge angle, E_{1}
, is positive in this example. An RBEND has parallel entrance and exit pole faces, so the exit angle, E_{2}
, is uniquely determined by the bend angle, \alpha
, and E_{1}
 (E_{2}=\alpha - E_{1}
). For a positively charge particle, the magnetic field is directed out of the page.


Figure 1. Illustration of a general rectangular bend (RBEND) with a positive bend angle \alpha
.


1.4.1. RBend (OPAL-t)

An RBEND is a rectangular bending magnet. The key property of an
RBEND is that it has parallel pole faces. Figure 1 shows an
RBEND with a positive bend angle and a positive entrance edge angle.


L

Physical length of magnet (meters, see Figure 1).

GAP

Full vertical gap of the magnet (meters).

HAPERT

Non-bend plane aperture of the magnet (meters). (Defaults to one half
the bend radius.)

ANGLE

Bend angle (radians). Field amplitude of bend will be adjusted to
achieve this angle. (Note that for an RBEND, the bend angle must be
less than 
\frac{\pi}{2} + E1
, where E1 is the entrance
edge angle.)

K0

Field amplitude in y direction (Tesla). If the ANGLE attribute is
set, K0 is ignored.

K0S

Field amplitude in x direction (Tesla). If the ANGLE attribute is
set, K0S is ignored.

K1

Field gradient index of the magnet,
K_1=-\frac{R}{B_{y}}\frac{\partial B_y}{\partial x},
where R
 is the bend radius as defined in Figure 1.
Not supported in OPAL-t any more. Superimpose a Quadrupole
instead.

E1

Entrance edge angle (radians). Figure 1 shows the definition of
a positive entrance edge angle. (Note that the exit edge angle is
fixed in an RBEND element to
\mathrm{E2} = \mathrm{ANGLE} - \mathrm{E1}
).

DESIGNENERGY

Energy of the reference particle (MeV). The reference particle travels
approximately the path shown in Figure 1.

FMAPFN

Name of the field map for the magnet. Currently maps of type
1DProfile1 can be used. The default option
for this attribute is FMAPN = 1DPROFILE1-DEFAULT
see_Default Field Map (OPAL-t). The field map is used to
describe the fringe fields of the magnet see 1DProfile1.


1.4.2. RBend3D (OPAL-t)

An RBEND3D3D is a rectangular bending magnet. The key property of an
RBEND3D is that it has parallel pole faces. Figure 1 shows an
RBEND3D with a positive bend angle and a positive entrance edge angle.


L

Physical length of magnet (meters, see Figure 1).

GAP

Full vertical gap of the magnet (meters).

HAPERT

Non-bend plane aperture of the magnet (meters). (Defaults to one half
the bend radius.)

ANGLE

Bend angle (radians). Field amplitude of bend will be adjusted to
achieve this angle. (Note that for an RBEND3D, the bend angle must
be less than 
\frac{\pi}{2} + E1
, where E1 is the
entrance edge angle.)

K0

Field amplitude in y direction (Tesla). If the ANGLE attribute is
set, K0 is ignored.

K0S

Field amplitude in x direction (Tesla). If the ANGLE attribute is
set, K0S is ignored.

K1

Field gradient index of the magnet,
K_1=-\frac{R}{B_{y}}\frac{\partial B_y}{\partial x},
where R
 is the bend radius as defined in Figure 1.
Not supported in OPAL-t any more. Superimpose a Quadrupole
instead.

E1

Entrance edge angle (radians). Figure 1 shows the definition of
a positive entrance edge angle. (Note that the exit edge angle is
fixed in an RBEND3D element to
\mathrm{E2} = \mathrm{ANGLE} - \mathrm{E1}
).

DESIGNENERGY

Energy of the reference particle (MeV). The reference particle travels
approximately the path shown in Figure 1.

FMAPFN

Name of the field map for the magnet. Currently maps of type
1DProfile1 can be used. The default option
for this attribute is FMAPN = 1DPROFILE1-DEFAULT
see Default Field Map (OPAL-t). The field map is used to
describe the fringe fields of the magnet 1DProfile1.


Figure 2 illustrates a general sector bend(SBEND) with a positive bend angle 
\alpha
. In this example the entrance and exit edge angles E_{1}
 and E_{2}
 have positive values. For a positively charge particle, the magnetic field is directed out of the page.


Figure 2. Illustration of a general sector bend(SBEND) with a positive bend angle \alpha


1.4.3. SBend (OPAL-t)

An SBEND is a sector bending magnet. An SBEND can have independent
entrance and exit edge angles. Figure 2 shows an SBEND with a
positive bend angle, a positive entrance edge angle, and a positive exit
edge angle.


L

Chord length of the bend reference arc in meters (see Figure 2),
given by: 
L = 2 R \sin\left(\frac{\alpha}{2}\right)


GAP

Full vertical gap of the magnet (meters).

HAPERT

Non-bend plane aperture of the magnet (meters). (Defaults to one half
the bend radius.)

ANGLE

Bend angle (radians). Field amplitude of the bend will be adjusted to
achieve this angle. (Note that practically speaking, bend angles
greater than 
\frac{3 \pi}{2} (270 degrees) can be
problematic. Beyond this, the fringe fields from the entrance and exit
pole faces could start to interfere, so be careful when setting up
bend angles greater than this. An angle greater than or equal to
2 \pi
 (360 degrees) is not allowed.)

K0

Field amplitude in y direction (Tesla). If the ANGLE attribute is
set, K0 is ignored.

K0S

Field amplitude in x direction (Tesla). If the ANGLE attribute is
set, K0S is ignored.

K1

Field gradient index of the magnet,
K_1=-\frac{R}{B_{y}}\frac{\partial B_y}{\partial x},
where R
 is the bend radius as defined in Figure 2.
Not supported in OPAL-t any more. Superimpose a Quadrupole
instead.

E1

Entrance edge angle (rad). Figure 2 shows the definition of a
positive entrance edge angle.

E2

Exit edge angle (rad). Figure 2 shows the definition of a
positive exit edge angle.

DESIGNENERGY

Energy of the bend reference particle (MeV). The reference particle
travels approximately the path shown in Figure 2.

FMAPFN

Name of the field map for the magnet. Currently maps of type
1DProfile1 can be used. The default option
for this attribute is FMAPN = 1DPROFILE1-DEFAULT
see_Default Field Map (OPAL-t). The field map is used to
describe the fringe fields of the magnet see 1DProfile1.


1.4.4. RBend and SBend Examples (OPAL-t)

Describing an RBEND or an SBEND in an OPAL-t simulation requires
effectively identical commands. There are only slight differences
between the two. The L attribute has a different definition for the
two types of bends sees RBend (OPAL-t) and SBend (OPAL-t), and an SBEND has an
additional attribute E2 that has no effect on an RBEND, see
SBend (OPAL-t). Therefore, in this section, we will give several
examples of how to implement a bend, using the RBEND and SBEND
commands interchangeably. The understanding is that the command formats
are essentially the same.


When implementing an RBEND or SBEND in an OPAL-t simulation, it is
important to note the following:


Internally OPAL-t treats all bends as positive, as defined by
Figure 1 and Figure 2. Bends in other directions within the x/y plane are
accomplished by rotating a positive bend about its z axis.


If the ANGLE attribute is set to a non-zero value, the K0 and
K0S attributes will be ignored.


When using the ANGLE attribute to define a bend, the actual beam
will be bent through a different angle if its mean kinetic energy
doesn’t correspond to the DESIGNENERGY.


Internally the bend geometry is setup based on the ideal reference
trajectory, as shown in Figure 1 and Figure 2.


If the default field map, 1DPROFILE-DEFAULT
see Default Field Map (OPAL-t), is used, the fringe fields will
be adjusted so that the effective length of the real, soft edge magnet
matches the ideal, hard edge bend that is defined by the reference
trajectory.


For the rest of this section, we will give several examples of how to
input bends in an OPAL-t simulation. We will start with a simple
example using the ANGLE attribute to set the bend strength and using
the default field map see Default Field Map (OPAL-t) for
describing the magnet fringe fields see 1DProfile1:


Bend: RBend, ANGLE = 30.0 * Pi / 180.0,
         FMAPFN = "1DPROFILE1-DEFAULT",
         ELEMEDGE = 0.25,
         DESIGNENERGY = 10.0,
         L = 0.5,
         GAP = 0.02;


This is a definition of a simple RBEND that bends the beam in a
positive direction 30 degrees (towards the negative x axis as if
Figure 1). It has a design energy of 10 MeV, a length of 0.5 m, a
vertical gap of 2 cm and a 0
^{\circ} entrance edge angle.
(Therefore the exit edge angle is 30^{\circ}.) We are
using the default, internal field map "1DPROFILE1-DEFAULT"
see Default Field Map (OPAL-t) which describes the magnet fringe
fields see 1DProfile1. When OPAL is run, you will get the
following output (assuming an electron beam) for this RBEND
definition:


RBend > Reference Trajectory Properties
RBend > ===============================
RBend >
RBend > Bend angle magnitude:    0.523599 rad (30 degrees)
RBend > Entrance edge angle:     0 rad (0 degrees)
RBend > Exit edge angle:         0.523599 rad (30 degrees)
RBend > Bend design radius:      1 m
RBend > Bend design energy:      1e+07 eV
RBend >
RBend > Bend Field and Rotation Properties
RBend > ==================================
RBend >
RBend > Field amplitude:         -0.0350195 T
RBend > Field index (gradient):  0 m^-1
RBend > Rotation about x axis:   0 rad (0 degrees)
RBend > Rotation about y axis:   0 rad (0 degrees)
RBend > Rotation about z axis:   0 rad (0 degrees)
RBend >
RBend > Reference Trajectory Properties Through Bend Magnet with Fringe Fields
RBend > ======================================================================
RBend >
RBend > Reference particle is bent: 0.523599 rad (30 degrees) in x plane
RBend > Reference particle is bent: 0 rad (0 degrees) in y plane


The first section of this output gives the properties of the reference
trajectory like that described in Figure 1. From the value of
ANGLE and the length, L, of the magnet, OPAL calculates the 10 MeV
reference particle trajectory radius, R. From the bend geometry and
the entrance angle (0
^{\circ} in this case), the exit
angle is calculated.


The second section gives the field amplitude of the bend and its
gradient (quadrupole focusing component), given the particle charge
(
-e in this case so the amplitude is negative to get a
positive bend direction). Also listed is the rotation of the magnet
about the various axes.


Of course, in the actual simulation the particles will not see a hard
edge bend magnet, but rather a soft edge magnet with fringe fields
described by the RBEND field map file FMAPFN
see 1DProfile1. So, once the hard edge bend/reference
trajectory is determined, OPAL then includes the fringe fields in the
calculation. When the user chooses to use the default field map, OPAL
will automatically adjust the position of the fringe fields
appropriately so that the soft edge magnet is equivalent to the hard
edge magnet described by the reference trajectory. To check that this
was done properly, OPAL integrates the reference particle through the
final magnet description with the fringe fields included. The result is
shown in the final part of the output. In this case we see that the soft
edge bend does indeed bend our reference particle through the correct
angle.


What is important to note from this first example, is that it is this
final part of the bend output that tells you the actual bend angle of
the reference particle.


In this next example, we merely rewrite the first example, but use K0
to set the field strength of the RBEND, rather than the ANGLE
attribute:


Bend: RBend, K0 = -0.0350195,
         FMAPFN = "1DPROFILE1-DEFAULT",
         ELEMEDGE = 0.25,
         DESIGNENERGY = 10.0E6,
         L = 0.5,
         GAP = 0.02;


The output from OPAL now reads as follows:


RBend > Reference Trajectory Properties
RBend > ===============================
RBend >
RBend > Bend angle magnitude:    0.523599 rad (30 degrees)
RBend > Entrance edge angle:     0 rad (0 degrees)
RBend > Exit edge angle:         0.523599 rad (30 degrees)
RBend > Bend design radius:      0.999999 m
RBend > Bend design energy:      1e+07 eV
RBend >
RBend > Bend Field and Rotation Properties
RBend > ==================================
RBend >
RBend > Field amplitude:         -0.0350195 T
RBend > Field index (gradient):  0 m^-1
RBend > Rotation about x axis:   0 rad (0 degrees)
RBend > Rotation about y axis:   0 rad (0 degrees)
RBend > Rotation about z axis:   0 rad (0 degrees)
RBend >
RBend > Reference Trajectory Properties Through Bend Magnet with Fringe Fields
RBend > ======================================================================
RBend >
RBend > Reference particle is bent: 0.5236 rad (30.0001 degrees) in x plane
RBend > Reference particle is bent: 0 rad (0 degrees) in y plane


The output is effectively identical, to within a small numerical error.


Now, let us modify this first example so that we bend instead in the
negative x direction. There are several ways to do this:


1.


Bend: RBend, ANGLE = -30.0 * Pi / 180.0,
         FMAPFN = "1DPROFILE1-DEFAULT",
         ELEMEDGE = 0.25,
         DESIGNENERGY = 10.0E6,
         L = 0.5,
         GAP = 0.02;


2.


Bend: RBend, ANGLE = 30.0 * Pi / 180.0,
         FMAPFN = "1DPROFILE1-DEFAULT",
         ELEMEDGE = 0.25,
         DESIGNENERGY = 10.0E6,
         L = 0.5,
         GAP = 0.02,
         ROTATION = Pi;


3.


Bend: RBend, K0 = 0.0350195,
         FMAPFN = "1DPROFILE1-DEFAULT",
         ELEMEDGE = 0.25,
         DESIGNENERGY = 10.0E6,
         L = 0.5,
         GAP = 0.02;


4.


Bend: RBend, K0 = -0.0350195,
         FMAPFN = "1DPROFILE1-DEFAULT",
         ELEMEDGE = 0.25,
         DESIGNENERGY = 10.0E6,
         L = 0.5,
         GAP = 0.02,
         ROTATION = Pi;


In each of these cases, we get the following output for the bend (to
within small numerical errors).


RBend > Reference Trajectory Properties
RBend > ===============================
RBend >
RBend > Bend angle magnitude:    0.523599 rad (30 degrees)
RBend > Entrance edge angle:     0 rad (0 degrees)
RBend > Exit edge angle:         0.523599 rad (30 degrees)
RBend > Bend design radius:      1 m
RBend > Bend design energy:      1e+07 eV
RBend >
RBend > Bend Field and Rotation Properties
RBend > ==================================
RBend >
RBend > Field amplitude:         -0.0350195 T
RBend > Field index (gradient):  -0 m^-1
RBend > Rotation about x axis:   0 rad (0 degrees)
RBend > Rotation about y axis:   0 rad (0 degrees)
RBend > Rotation about z axis:   3.14159 rad (180 degrees)
RBend >
RBend > Reference Trajectory Properties Through Bend Magnet with Fringe Fields
RBend > ======================================================================
RBend >
RBend > Reference particle is bent: -0.523599 rad (-30 degrees) in x plane
RBend > Reference particle is bent: 0 rad (0 degrees) in y plane


In general, we suggest to always define a bend in the positive x
direction (as in Figure 1) and then use the ROTATION attribute
to bend in other directions in the x/y plane (as in examples 2 and 4
above).


As a final RBEND example, here is a suggested format for the four bend
definitions if one where implementing a four dipole chicane:


Bend1: RBend, ANGLE = 20.0 * Pi / 180.0,
              E1 = 0.0,
              FMAPFN = "1DPROFILE1-DEFAULT",
              ELEMEDGE = 0.25,
              DESIGNENERGY = 10.0E6,
              L = 0.25,
              GAP = 0.02,
              ROTATION = Pi;

Bend2: RBend, ANGLE = 20.0 * Pi / 180.0,
              E1 = 20.0 * Pi / 180.0,
              FMAPFN = "1DPROFILE1-DEFAULT",
              ELEMEDGE = 1.0,
              DESIGNENERGY = 10.0E6,
              L = 0.25,
              GAP = 0.02,
              ROTATION = 0.0;

Bend3: RBend, ANGLE = 20.0 * Pi / 180.0,
              E1 = 0.0,
              FMAPFN = "1DPROFILE1-DEFAULT",
              ELEMEDGE = 1.5,
              DESIGNENERGY = 10.0E6,
              L = 0.25,
              GAP = 0.02,
              ROTATION = 0.0;

Bend4: RBend, ANGLE = 20.0 * Pi / 180.0,
              E1 = 20.0 * Pi / 180.0,
              FMAPFN = "1DPROFILE1-DEFAULT",
              ELEMEDGE = 2.25,
              DESIGNENERGY = 10.0E6,
              L = 0.25,
              GAP = 0.02,
              ROTATION = Pi;


Up to now, we have only given examples of RBEND definitions. If we
replaced "RBend" in the above examples with "SBend", we would still
be defining valid OPAL-t bends. In fact, by adjusting the L
attribute according to RBend (OPAL-t) and SBend (OPAL-t), and by adding the
appropriate definitions of the E2 attribute, we could even get
identical results using `SBEND`s instead of `RBEND`s. (As we said, the
two bends are very similar in command format.)


Up till now, we have only used the default field map. Custom field maps
can also be used. There are two different options in this case
see 1DProfile1:


Field map defines fringe fields and magnet length.


Field map defines fringe fields only.


The first case describes how field maps were used in previous versions
of OPAL (and can still be used in the current version). The second
option is new to OPAL OPALversion 1.2.00 and it has a couple of
advantages:


Because only the fringe fields are described, the length of the
magnet must be set using the L attribute. In turn, this means that the
same field map can be used by many bend magnets with different lengths
(assuming they have equivalent fringe fields). By contrast, if the
magnet length is set by the field map, one must generate a new field map
for each dipole of different length even if the fringe fields are the
same.


We can adjust the position of the fringe field origin relative to
the entrance and exit points of the magnet see 1DProfile1.
This gives us another degree of freedom for describing the fringe
fields, allowing us to adjust the effective length of the magnet.


We will now give examples of how to use a custom field map, starting
with the first case where the field map describes the fringe fields and
the magnet length. Assume we have the following 1DProfile1 field map:


1DProfile1 1 1 2.0
 -10.0  0.0  10.0 1
  15.0  25.0 35.0 1
  0.00000E+00
  2.00000E+00
  0.00000E+00
  2.00000E+00


We can use this field map to define the following bend (note we are now
using the SBEND command):


Bend: SBend, ANGLE = 60.0 * Pi / 180.0,
             E1 = -10.0 * Pi / 180.0,
             E2 = 20.0  Pi / 180.0,
             FMAPFN = "TEST-MAP.T7",
             ELEMEDGE = 0.25,
             DESIGNENERGY = 10.0E6,
             GAP = 0.02;


Notice that we do not set the magnet length using the L attribute.
(In fact, we don’t even include it. If we did and set it to a non-zero
value, the exit fringe fields of the magnet would not be correct.) This
input gives the following output:


SBend > Reference Trajectory Properties
SBend > ===============================
SBend >
SBend > Bend angle magnitude:    1.0472 rad (60 degrees)
SBend > Entrance edge angle:     -0.174533 rad (-10 degrees)
SBend > Exit edge angle:         0.349066 rad (20 degrees)
SBend > Bend design radius:      0.25 m
SBend > Bend design energy:      1e+07 eV
SBend >
SBend > Bend Field and Rotation Properties
SBend > ==================================
SBend >
SBend > Field amplitude:         -0.140385 T
SBend > Field index (gradient):  0 m^-1
SBend > Rotation about x axis:   0 rad (0 degrees)
SBend > Rotation about y axis:   0 rad (0 degrees)
SBend > Rotation about z axis:   0 rad (0 degrees)
SBend >
SBend > Reference Trajectory Properties Through Bend Magnet with Fringe Fields
SBend > ======================================================================
SBend >
SBend > Reference particle is bent: 1.0472 rad (60 degrees) in x plane
SBend > Reference particle is bent: 0 rad (0 degrees) in y plane


Because we set the bend strength using the ANGLE attribute, the magnet
field strength is automatically adjusted so that the reference particle
is bent exactly ANGLE radians when the fringe fields are included.
(Lower output.)


Now we will illustrate the case where the magnet length is set by the
L attribute and only the fringe fields are described by the field map.
We change the TEST-MAP.T7 file to:


1DProfile1 1 1 2.0
 -10.0  0.0  10.0 1
 -10.0  0.0  10.0 1
  0.00000E+00
  2.00000E+00
  0.00000E+00
  2.00000E+00


and change the bend input to:


Bend: SBend, ANGLE = 60.0 * Pi / 180.0,
             E1 = -10.0 * Pi / 180.0,
             E2 = 20.0  Pi / 180.0,
             FMAPFN = "TEST-MAP.T7",
             ELEMEDGE = 0.25,
             DESIGNENERGY = 10.0E6,
             L = 0.25,
             GAP = 0.02;


This results in the same output as the previous example, as we expect.


SBend > Reference Trajectory Properties
SBend > ===============================
SBend >
SBend > Bend angle magnitude:    1.0472 rad (60 degrees)
SBend > Entrance edge angle:     -0.174533 rad (-10 degrees)
SBend > Exit edge angle:         0.349066 rad (20 degrees)
SBend > Bend design radius:      0.25 m
SBend > Bend design energy:      1e+07 eV
SBend >
SBend > Bend Field and Rotation Properties
SBend > ==================================
SBend >
SBend > Field amplitude:         -0.140385 T
SBend > Field index (gradient):  0 m^-1
SBend > Rotation about x axis:   0 rad (0 degrees)
SBend > Rotation about y axis:   0 rad (0 degrees)
SBend > Rotation about z axis:   0 rad (0 degrees)
SBend >
SBend > Reference Trajectory Properties Through Bend Magnet with Fringe Fields
SBend > ======================================================================
SBend >
SBend > Reference particle is bent: 1.0472 rad (60 degrees) in x plane
SBend > Reference particle is bent: 0 rad (0 degrees) in y plane


As a final example, let us now use the previous field map with the
following input:


Bend: SBend, K0 = -0.1400778,
             E1 = -10.0 * Pi / 180.0,
             E2 = 20.0  Pi / 180.0,
             FMAPFN = "TEST-MAP.T7",
             ELEMEDGE = 0.25,
             DESIGNENERGY = 10.0E6,
             L = 0.25,
             GAP = 0.02;


Instead of setting the bend strength using ANGLE, we use K0. This
results in the following output:


SBend > Reference Trajectory Properties
SBend > ===============================
SBend >
SBend > Bend angle magnitude:    1.0472 rad (60 degrees)
SBend > Entrance edge angle:     -0.174533 rad (-10 degrees)
SBend > Exit edge angle:         0.349066 rad (20 degrees)
SBend > Bend design radius:      0.25 m
SBend > Bend design energy:      1e+07 eV
SBend >
SBend > Bend Field and Rotation Properties
SBend > ==================================
SBend >
SBend > Field amplitude:         -0.140078 T
SBend > Field index (gradient):  0 m^-1
SBend > Rotation about x axis:   0 rad (0 degrees)
SBend > Rotation about y axis:   0 rad (0 degrees)
SBend > Rotation about z axis:   0 rad (0 degrees)
SBend >
SBend > Reference Trajectory Properties Through Bend Magnet with Fringe Fields
SBend > ======================================================================
SBend >
SBend > Reference particle is bent: 1.04491 rad (59.8688 degrees) in x plane
SBend > Reference particle is bent: 0 rad (0 degrees) in y plane


In this case, the bend angle for the reference trajectory in the first
section of the output no longer matches the reference trajectory bend
angle from the lower section (although the difference is small). The
reason is that the path of the reference particle through the real
magnet (with fringe fields) no longer matches the ideal trajectory. (The
effective length of the real magnet is not quite the same as the hard
edged magnet for the reference trajectory.)


We can compensate for this by changing the field map file TEST-MAP.T7
file to:


1DProfile1 1 1 2.0
 -10.0  -0.03026  10.0 1
 -10.0  0.03026  10.0 1
  0.00000E+00
  2.00000E+00
  0.00000E+00
  2.00000E+00


We have moved the Enge function origins see 1DProfile1 outward
from the entrance and exit faces of the magnet see 1DProfile1
by 0.3026 mm. This has the effect of making the effective length of the
soft edge magnet longer. When we do this, the same input:


Bend: SBend, K0 = -0.1400778,
             E1 = -10.0 * Pi / 180.0,
             E2 = 20.0  Pi / 180.0,
             FMAPFN = "TEST-MAP.T7",
             ELEMEDGE = 0.25,
             DESIGNENERGY = 10.0E6,
             L = 0.25,
             GAP = 0.02;


produces


SBend > Reference Trajectory Properties
SBend > ===============================
SBend >
SBend > Bend angle magnitude:    1.0472 rad (60 degrees)
SBend > Entrance edge angle:     -0.174533 rad (-10 degrees)
SBend > Exit edge angle:         0.349066 rad (20 degrees)
SBend > Bend design radius:      0.25 m
SBend > Bend design energy:      1e+07 eV
SBend >
SBend > Bend Field and Rotation Properties
SBend > ==================================
SBend >
SBend > Field amplitude:         -0.140078 T
SBend > Field index (gradient):  0 m^-1
SBend > Rotation about x axis:   0 rad (0 degrees)
SBend > Rotation about y axis:   0 rad (0 degrees)
SBend > Rotation about z axis:   0 rad (0 degrees)
SBend >
SBend > Reference Trajectory Properties Through Bend Magnet with Fringe Fields
SBend > ======================================================================
SBend >
SBend > Reference particle is bent: 1.0472 rad (60 degrees) in x plane
SBend > Reference particle is bent: 0 rad (0 degrees) in y plane


Now we see that the bend angle for the ideal, hard edge magnet, matches
the bend angle of the reference particle through the soft edge magnet.
In other words, the effective length of the soft edge, real magnet is
the same as the hard edge magnet described by the reference trajectory.


1.4.5. Bend Fields from 1D Field Maps (OPAL-t)


Figure 3. Plot of the entrance fringe field of a dipole magnet along the mid-plane, perpendicular to its entrance face. The field is normalized to 1.0. In this case, the fringe field is described by an Enge function see Enge function with the parameters from the default 1DProfile1 field map described in Default Field Map (OPAL-t). The exit fringe field of this magnet is the mirror image.


So far we have described how to setup an RBEND or SBEND element, but
have not explained how OPAL-t uses this information to calculate the
magnetic field. The field of both types of magnets is divided into three
regions:


Entrance fringe field.


Central field.


Exit fringe field.


This can be seen clearly in [fig_rbend_field_profile].


The purpose of the 1DProfile1 field map see 1DProfile1
associated with the element is to define the Enge functions
(Enge function) that model the entrance and exit fringe fields.
To model a particular bend magnet, one must fit the field profile along
the mid-plane of the magnet perpendicular to its face for the entrance
and exit fringe fields to the Enge function:


  F(z) = \frac{1}{1 + e^{\sum\limits_{n=0}^{N_{order}} c_{n} (z/D)^{n}}}


where 
D is the full gap of the magnet,
N_{order}
 is the Enge function order and z
is the distance from the origin of the Enge function perpendicular to
the edge of the dipole. The origin of the Enge function, the order of
the Enge function, N_{order}, and the constants
c_0
 to c_{N_{order}} are free parameters
that are chosen so that the function closely approximates the fringe
region of the magnet being modeled. An example of the entrance fringe
field is shown in Figure 3.


Let us assume we have a correctly defined positive RBEND or SBEND
element as illustrated in Figure 1 and Figure 2. (As already stated, any
bend can be described by a rotated positive bend.) OPAL-t then has the
following information:


\begin{aligned}
B_0 &= \text{Field amplitude (T)} \\
R &= \text{Bend radius (m)} \\
n &= -\frac{R}{B_{y}}\frac{\partial B_y}{\partial x} \text{ (Field index, set using the parameter } \mathrm{K1} \text{)} \\
F(z) &= \left\{
\begin{array}{lll}
    & F_{entrance}(z_{entrance}) \\
    & F_{center}(z_{center}) = 1 \\
    & F_{exit}(z_{exit})
\end{array}
\right.\end{aligned}


Here, we have defined an overall Enge function, 
F(z), with
three parts: entrance, center and exit. The exit and entrance fringe
field regions have the form of Enge function with parameters
defined by the 1DProfile1 field map file given by the element
parameter FMAPFN. Defining the coordinates:


\begin{aligned}
y &\equiv \text{Vertical distance from magnet mid-plane} \\
\Delta_x &\equiv \text{Perpendicular distance to reference trajectory (see Figures)} \\
\Delta_z &\equiv \left\{
\begin{array}{lll}
    & \text{Distance from entrance Enge function origin perpendicular to magnet entrance face.} \\
    & \text{Not defined, Enge function is always 1 in this region.} \\
    & \text{Distance from exit Enge function origin perpendicular to magnet exit face.}
\end{array}
\right.\end{aligned}


using the conditions


\begin{aligned}
\nabla \cdot \vec{B} &= 0 \\
\nabla \times \vec{B} &= 0
\end{aligned}


and making the definitions:


\begin{aligned}
F'(z) &\equiv   \frac{\mathrm{d} F(z)}{\mathrm{d} z} \\
F''(z) &\equiv  \frac{\mathrm{d^{2}} F(z)}{\mathrm{d} z^{2}} \\
F'''(z) &\equiv \frac{\mathrm{d^{3}} F(z)}{\mathrm{d} z^{3}}
\end{aligned}


we can expand the field off axis, with the result:


\begin{aligned}
B_x(\Delta_x, y, \Delta_z) &= -\frac{B_0 \frac{n}{R}}{\sqrt{\frac{n^2}{R^2} +  \frac{F''(\Delta_z)}{F(\Delta_z}}} e^{-\frac{n}{R} \Delta_x} \sin \left[ \left( \sqrt{\frac{n^2}{R^2} + \frac{F''(\Delta_z)}{F(\Delta_z)}} \right) y \right] F(\Delta_z) \\ \\
B_y(\Delta_x, y, \Delta_z) &= B_0 e^{-\frac{n}{R} \Delta_x} \cos \left[ \left( \sqrt{\frac{n^2}{R^2} + \frac{F''(\Delta_z)}{F(\Delta_z)}} \right) y \right] F(\Delta_z) \\ \\
B_z(\Delta_x, y, \Delta_z) &= B_0 e^{-\frac{n}{R} \Delta_x} \left\{\frac{F'(\Delta_z)}{\sqrt{\frac{n^2}{R^2} + \frac{F''(\Delta_z)}{F(\Delta_z)}}} \sin \left[ \left( \sqrt{\frac{n^2}{R^2} + \frac{F''(\Delta_z)}{F(\Delta_z)}} \right) y \right] \right. \\ \\
&- \frac{1}{2 \sqrt{\frac{n^2}{R^2} + \frac{F''(\Delta_z)}{F(\Delta_z)}}} \left(F'''(\Delta_z) - \frac{F'(\Delta_z) F''(\Delta_z)}{F(\Delta_z)} \right) \left[ \frac{\sin \left[ \left( \sqrt{\frac{n^2}{R^2} + \frac{F''(\Delta_z)}{F(\Delta_z)}} \right) y \right]}{\frac{n^2}{R^2} + \frac{F''(\Delta_z)}{F(\Delta_z)}} \right. \\ \\
&- \left. \left. y \frac{\cos \left[ \left( \sqrt{\frac{n^2}{R^2} + \frac{F''(\Delta_z)}{F(\Delta_z)}} \right) y \right]}{\sqrt{\frac{n^2}{R^2} + \frac{F''(\Delta_z)}{F(\Delta_z)}}} \right] \right\}\end{aligned}


These expression are not well suited for numerical calculation, so, we
expand them about 
y
 to O(y^2)
 to obtain:


In fringe field regions:


\begin{aligned}
B_x(\Delta_x, y, \Delta_z) &\approx -B_0 \frac{n}{R} e^{-\frac{n}{R} \Delta_x} y \\
B_y(\Delta_x, y, \Delta_z) &\approx B_0 e^{-\frac{n}{R} \Delta_x} \left[ F(\Delta_z) - \left( \frac{n^2}{R^2} F(\Delta_z) + F''(\Delta_z) \right) \frac{y^2}{2} \right] \\
B_z(\Delta_x, y, \Delta_z) &\approx B_0 e^{-\frac{n}{R} \Delta_x} y F'(\Delta_z)
\end{aligned}


In central region:


\begin{aligned}
B_x(\Delta_x, y, \Delta_z) &\approx -B_0 \frac{n}{R} e^{-\frac{n}{R} \Delta_x} y \\
B_y(\Delta_x, y, \Delta_z) &\approx B_0 e^{-\frac{n}{R} \Delta_x} \left[ 1 -  \frac{n^2}{R^2} \frac{y^2}{2} \right] \\
B_z(\Delta_x, y, \Delta_z) &\approx 0
\end{aligned}


These are the expressions OPAL-t uses to calculate the field inside an
RBEND or SBEND. First, a particle’s position inside the bend is
determined (entrance region, center region, or exit region). Depending
on the region, OPAL-t then determines the values of
\Delta_x
, y
 and \Delta_z, and
then calculates the field values using the above expressions.


1.4.6. Default Field Map (OPAL-t)

Rather than force users to calculate the field of a dipole and then fit
that field to find Enge coefficients for the dipoles in their
simulation, we have a default set of values we use from [2] that are
set when the default field map, 1DPROFILE1-DEFAULT is used:


\begin{aligned}
  c_{0} &= 0.478959 \\
  c_{1} &= 1.911289 \\
  c_{2} &= -1.185953 \\
  c_{3} &= 1.630554 \\
  c_{4} &= -1.082657 \\
  c_{5} &= 0.318111\end{aligned}


The same values are used for both the entrance and exit regions of the
magnet. In general they will give good results. (Of course, at some
point as a beam line design becomes more advanced, one will want to find
Enge coefficients that fit the actual magnets that will be used in a
given design.)


The default field map is the equivalent of the following custom
1DProfile1 (see 1DProfile1 for an explanation of the field
map format) map:


1DProfile1 5 5 2.0
 -10.0 0.0 10.0 1
 -10.0 0.0 10.0 1
  0.478959
  1.911289
 -1.185953
  1.630554
 -1.082657
  0.318111
  0.478959
  1.911289
 -1.185953
  1.630554
 -1.082657
  0.318111


As one can see, the default magnet gap for 1DPROFILE1-DEFAULT is
set to 2.0 cm. This value can be overridden by the GAP attribute of the
magnet (see RBend (OPAL-t) and SBend (OPAL-t)).


1.4.7. SBend3D (OPAL-cycl)

The SBend3D element enables definition of a bend from 3D field maps.
This can be used in conjunction with the RINGDEFINITION element to
make a ring for tracking through OPAL-cycl.


label: SBEND3D, FMAPFN=string, LENGTH_UNITS=real, FIELD_UNITS=real;


FMAPFN

The field map file name.

LENGTH_UNITS

Units for length (set to 1.0 for units in mm, 10.0 for units in cm,
etc).

FIELD_UNITS

Units for field (set to 1.0 for units in T, 0.001 for units in mT,
etc).


Field maps are defined using Cartesian coordinates but in a polar
geometry. The following conventions have to be fulfilled:


3D Field maps have to be generated in the vertical direction (z
coordinate in OPAL-cycl) from z = 0 upwards. Maps cannot be generated
symmetrically about z = 0 towards negative z values.


The field map file must be in the form with columns ordered as follows:
[
x, z, y, B_{x}, B_{z}, B_{y}
].


Grid points of the position and field strength have to be written on
a grid in (
r, z, \theta) with the primary direction
corresponding to the azimuthal direction, secondary to the vertical
direction and tertiary to the radial direction.


Below two examples of a SBEND3D which loads a field map file named “90degree_Dipole_Magnet.out” defining a hard edge model of 90 degree dipole magnet with homogenous magnetic field. The first 8 lines are presumed to be header material and are ignored. The first 8 lines in the field map are ignored. Positions have units of m and fields units of Tesla. The corresponding 3D magnetic field map is shown in the following figure in the Cartesian coordinate system (x, y, z). A horizontal cross section of the 3D magnetic field map when z = 0 is also shown.


Dipole: SBEND3D, FMAPFN="90degree_Dipole_Magnet.out", LENGTH_UNITS=1000.0, FIELD_UNITS=-10.0;


The first few lines of the field map file are as follows:


	4550000	4550000	4550000	1
X [LENGTH_UNITS]
Z [LENGTH_UNITS]
Y [LENGTH_UNITS]
BX [FIELD_UNITS]
BZ [FIELD_UNITS]
BY [FIELD_UNITS]
0
4.3586435e-01   5.0000000e-02   1.2803431e+00   0.0000000e+00   1.6214000e+00   0.0000000e+00
4.2691532e-01   5.0000000e-02   1.2833548e+00   0.0000000e+00   1.6214000e+00   0.0000000e+00
4.1794548e-01   5.0000000e-02   1.2863039e+00   0.0000000e+00   1.6214000e+00   0.0000000e+00


This is a restricted feature for OPAL-cycl.


Figure 4. A hard edge model of 90
 degree dipole magnet with homogeneous magnetic field. The right figure is showing the horizontal cross section of the 3D magnetic field map when z = 0


1.5. Quadrupole


label:QUADRUPOLE, TYPE=string, APERTURE=real-vector,
      L=real, K1=real, K1S=real;


The reference system for a quadrupole is a Cartesian coordinate system
This is a restricted feature for OPAL-t.


A QUADRUPOLE has the following real attributes:


K1

The normal quadrupole component
K_1=\frac{\partial B_y}{\partial x}. The default is 0
\mathrm{Tm^{-1}}. The component is positive, if
B_y
 is positive on the positive x-axis.
This implies horizontal focusing of positively charged particles which
travel in positive s
-direction.

K1S

The skew quadrupole component.
K_{1s}=-\frac{\partial B_x}{\partial x}. The default is 0
\mathrm{Tm^{-1}}. The component is negative, if
B_x
 is positive on the positive x
-axis.

DK1

The normalised quadrupole coefficient error.

DK1S

The normalised skew quadrupole coefficient error.


Example:


QP1: Quadrupole, L=1.20, ELEMEDGE=-0.5265, K1=0.11;


1.6. Sextupole


label: SEXTUPOLE, TYPE=string, APERTURE=real-vector,
       L=real, K2=real, K2S=real;


A SEXTUPOLE has the following real attributes:


K2

The normal sextupole component
K_2=\frac{\partial{^2} B_y}{\partial x^2}. The default
is 0 \mathrm{T m^{-2}}. The component is positive, if
B_y
 is positive on the x
-axis.

K2S

The skew sextupole component
K_{2s}=-\frac{\partial{^2}B_x}{\partial x^{2}}. The
default is 0 \mathrm{T m^{-2}}. The component is negative, if
B_x
 is positive on the x
-axis.

DK2

The normalised sextupole coefficient error.

DK2S

The normalised skew sextupole coefficient error.


Example:


S:SEXTUPOLE, L=0.4, K2=0.00134;


The reference system for a sextupole is a Cartesian coordinate system


1.7. Octupole


label:OCTUPOLE, TYPE=string, APERTURE=real-vector,
      L=real, K3=real, K3S=real;


An OCTUPOLE has the following real attributes:


K3

The normal octupole component
K_3=\frac{\partial{^3} B_y}{\partial x^3}. The default
is 0 \mathrm{Tm^{-3}}. The component is positive, if
B_y
 is positive on the positive x
-axis.

K3S

The skew octupole component
K_{3s}=-\frac{\partial{^3}B_x}{\partial x^{3}}. The
default is 0 \mathrm{Tm^{-3}}. The component is negative, if
B_x
 is positive on the positive x
-axis.

DK3

The normalised octupole coefficient error.

DK3S

The normalised skew octupole coefficient error.


Example:


O3:OCTUPOLE, L=0.3, K3=0.543;


The reference system for an octupole is a Cartesian coordinate system


1.8. General Multipole

The MULTIPOLE element defines a thick multipole.
If the length is non-zero, the strengths are per unit
length. If the length is zero, the strengths are the
values integrated over the length.
With zero length no synchrotron radiation can be calculated.


A MULTIPOLE in OPAL-t is of arbitrary order.


label:MULTIPOLE, TYPE=string, APERTURE=real-vector,
      L=real, KN=real-vector, KS=real-vector;


KN

A real vector see Arrays, containing the normal multipole
coefficients, 
K_n=\frac{\partial{^n} B_y}{\partial x^n}.
(default is 0 \mathrm{Tm^{-n}}). A component is positive, if
B_y
 is positive on the positive x
-axis.

KS

A real vector see Arrays, containing the skew multipole
coefficients,
K_{n~s}=-\frac{\partial{^n}B_x}{\partial x^{n}}.
(default is 0 \mathrm{Tm^{-n}}). A component is negative, if
B_x
 is positive on the positive x
-axis.

DKN

A real vector see Arrays, containing the normal
normalised multipole strength errors. (default is 0 
\mathrm{Tm^{-n}}
).

DKS

A real vector see Arrays, containing the skew
normalised multipole strength errors. (default is 0 
\mathrm{Tm^{-n}}
).


The number of poles of each component is (
2 n + 2
).


Superposition of many multipole components is permitted. The reference
system for a multipole is a Cartesian coordinate system


The following example is equivalent to the quadruple example in
Quadrupole.


M27:MULTIPOLE, L=1, ELEMEDGE=3.8, KN={0.0,0.11};


A multipole has no effect on the reference orbit, i.e. the reference
system at its exit is the same as at its entrance. Use the dipole
component only to model a defective multipole.


1.9. General Multipole (will replace General Multipole when implemented)

A MULTIPOLET is in OPAL-t a general multipole with extended
features. It can represent a straight or curved magnet. In the curved
case, the user may choose between constant or variable radius. This
model includes fringe fields. The detailed description can be found at:
https://gitlab.psi.ch/OPAL/src/uploads/0d3fc561b57e8962ed79a57cd6115e37/8FBB32A4-7FA1-4084-A4A7-CDDB1F949CD3_psi.ch.pdf.


label:MULTIPOLET, L=real, ANGLE=real, VAPERT=real, HAPERT=real,
      LFRINGE=real, RFRINGE=real, TP=real-vector, VARRADIUS=bool;


L

Physical length of the magnet (meters), without end fields. (Default:
1 m)

ANGLE

Physical angle of the magnet (radians). If not specified, the magnet
is considered to be straight (ANGLE=0.0). This is not the total
bending angle since the end fields cause additional bending. The
radius of the multipole is set from the LENGTH and ANGLE attributes.

VAPERT

Vertical (non-bend plane) aperture of the magnet (meters). (Default:
0.5 m)

HAPERT

Horizontal (bend plane) aperture of the magnet (meters). (Default: 0.5
m)

LFRINGE

Length of the left fringe field (meters). (Default: 0.0 m)

RFRINGE

Length of the right fringe field (meters). (Default: 0.0 m)

TP

A real vector see Arrays, containing the multipole
coefficients of the field expansion on the mid-plane in the body of
the magnet: the transverse profile
 T(x) = B_0 + B_1 x + B_2 x^2 + \ldots  is set by
TP=B_0
, B_1
, B_2 (units:
 T \cdot m^{-n}). The order of highest multipole
component is arbitrary, but all components up to the maximum must be
given, even if they are zero.

MAXFORDER

The order of the maximum function 
f_n used in the field
expansion (default: 5). See the scalar magnetic potential below. This
sets for example the maximum power of z in the field
expansion of vertical component B_z to
2 \cdot \text{MAXFORDER} 
.

EANGLE

Entrance edge angle (radians).

ROTATION

Rotation of the magnet about its central axis (radians,
counterclockwise). This enables to obtain skew fields. (Default 0.0
rad)

VARRADIUS

This is to be set TRUE if the magnet has variable radius. More
precisely, at each point along the magnet, its radius is computed such
that the reference trajectory always remains in the centre of the
magnet. In the body of the magnet the radius is set from the LENGTH
and ANGLE attributes. It is then continuously changed to be
proportional to the dipole field on the reference trajectory while
entering the end fields. This attribute is only to be set TRUE for a
non-zero dipole component. (Default: FALSE)

VARSTEP

The step size (meters) used in calculating the reference trajectory
for VARRARDIUS = TRUE. It specifies how often the radius of curvature
is re-calculated. This has a considerable effect on tracking time.
(Default: 0.1 m)


Superposition of many multipole components is permitted. The reference
system for a multipole is a Cartesian coordinate system for straight
geometry and a 
(x,s,z) Frenet-Serret coordinate system for
curved geometry. In the latter case, the axis \hat{s} is
the central axis of the magnet.


The following example shows a combined function magnet with a dipole component
of 2 Tesla and a quadrupole gradient of 0.1 Tesla/m.


M30:MULTIPOLET, L=1, RFRINGE=0.3, LFRINGE=0.2, ANGLE=PI/6, TP={2.0, 0.1}, VARRADIUS=TRUE;


The field expansion used in this model is based on the following scalar potential:


 V = z f_0(x,s) + \frac{z^3}{3!} f_1(x,s) + \frac{z^5}{5!} f_2(x,s) + \ldots


Mid-plane symmetry is assumed and the vertical component of the field on the mid-plane is given by the user under the form of the transverse profile 
T(x)
. The full expression for the vertical component is then


B_z = f_0 = T(x) \cdot S(s)


where 
S(s)
 is the fringe field. This element uses the Tanh model for the end fields, having only three parameters (the centre length s_0
 and the fringe field lengths \lambda_{left}
, \lambda_{right}
):


 S(s) = \frac{1}{2} \left[ \tanh \left( \frac{s + s_0}{\lambda_{left}} \right) -
 \tanh \left( \frac{s - s_0}{\lambda_{right}} \right) \right]


Starting from Maxwell’s laws, the functions 
f_n
 are computed recursively and finally each component of the magnetic field is obtained from V
 using the corresponding geometries.


1.10. Solenoid


label:SOLENOID, TYPE=string, APERTURE=real-vector,
      L=real, KS=real;


A SOLENOID has two real attributes:


KS

The solenoid strength
K_s=\frac{\partial B_s}{\partial s}, default is 0
\mathrm{Tm^{-1}}
. For positive KS and positive particle
charge, the solenoid field points in the direction of increasing
s
.


The reference system for a solenoid is a Cartesian coordinate system
Using a solenoid in OPAL-t mode, the following additional parameters
are defined:


FMAPFN

Field maps must be specified.


Example:


SP1: Solenoid, L=1.20, ELEMEDGE=-0.5265, KS=0.11,
     FMAPFN="1T1.T7";


1.11. Cyclotron


label:CYCLOTRON, TYPE=string, CYHARMON=int,
      PHIINIT=real, PRINIT=real, RINIT=real,
      SYMMETRY=real, RFFREQ=real, FMAPFN=string;


A CYCLOTRON object includes the main characteristics of a cyclotron,
the magnetic field, and also the initial condition of the injected
reference particle, and it has currently the following attributes:


TYPE

The data format of field map, Currently the following formats are implemented:
CARBONCYCL, CYCIAE, AVFEQ, FFA, BANDRF and default PSI format. For
the details of their data format, please read
Field Maps.

CYHARMON

The harmonic number of the cyclotron 
h
.

RFFREQ

The RF system 
f_{rf} (unit:MHz, default: 0). The
particle revolution frequency f_{rev} =
f_{rf}
 / h
.

FMAPFN

  File name for the magnetic field map.
BSCALE:
  Scale factor for the magnetic field map.

SYMMETRY

Defines symmetrical fold number of the B field map data.

FMLOWE

Minimal energy [MeV] the fieldmap can accept. Used in GAUSSMATCHED distribution.

FMHIGHE

Maximal energy [MeV] the fieldmap can accept. Used in GAUSSMATCHED distribution.

RINIT

The initial radius of the reference particle (unit: mm, default: 0)

PHIINIT

The initial azimuth of the reference particle (unit: degree, default:
0)

ZINIT

The initial axial position of the reference particle (unit: mm,
default: 0)

PRINIT

Initial radial momentum of the reference particle
P_r=\beta_r\gamma
 (default : 0)

PZINIT

Initial axial momentum of the reference particle
P_z=\beta_z\gamma
 (default : 0)

MINZ

The minimal vertical extent of the machine (unit: mm, default :
-10000.0)

MAXZ

The maximal vertical extent of the machine (unit: mm, default :
10000.0)

MINR

Minimal radial extent of the machine (unit: mm, default : 0.0)

MAXR

Minimal radial extent of the machine (unit: mm, default : 10000.0)


During the tracking, the particle (
r, z, \theta) will be
deleted if MINZ < z <
 MAXZ or MINR < r <
MAXR, and it will be recorded in the HDF5 file <inputfilename>.h5
(or ASCII if ASCIIDUMP is true).
Example:


ring: Cyclotron, TYPE="RING", CYHARMON=6, PHIINIT=0.0,
      PRINIT=-0.000240, RINIT=2131.4, SYMMETRY=8.0,
      RFFREQ=50.650, FMAPFN="s03av.nar",
      MAXZ=10, MINZ=-10, MINR=0, MAXR=2500;


If TYPE is set to BANDRF, the 3D electric field map of RF cavity will be
read from external H5Hut file and the following extra arguments need to specified:


RFMAPFN

The file name(s) for the electric field map(s) in H5Hut binary format.

RFPHI

The initial phase(s) of the electric field map(s) (rad)

RFFREQ

The frequencies of the electric field maps. 0 indicates a constant field.

ESCALE

The scale factor(s) for the electric field map(s)

SUPERPOSE

An option whether the electric field map(s) is superposed (see also below).


Example for single electric field map:


COMET: Cyclotron, TYPE="BANDRF", CYHARMON=2, PHIINIT=-71.0,
PRINIT=pr0, RINIT=r0, SYMMETRY=1.0, FMAPFN="Tosca_map.txt",
RFPHI=Pi, RFFREQ=72.0,  RFMAPFN="efield.h5part",
ESCALE=1.06E-6;


We can have more than one RF field maps.


Example for multiple RF field maps:


COMET: Cyclotron, TYPE="BANDRF", CYHARMON=2, PHIINIT=-71.0,
PRINIT=pr0, RINIT=r0, SYMMETRY=1.0, FMAPFN="Tosca_map.txt",
RFPHI={Pi,0,Pi,0}, RFFREQ={72.0,72.0,72.0,72.0},
RFMAPFN={"e1.h5part","e2.h5part","e3.h5part","e4.h5part"},
ESCALE={1.06E-6, 3.96E-6,1.3E-6,1.E-6}, SUPERPOSE={true,false,false,true};


If SUPERPOSE is set to true and if a particle is located in the field region,
the field is always applied.
If SUPERPOSE is set to false, then only one field map with SUPERPOSE false is applied,
the one which has highest priority, is used to do interpolation for the particle tracking.
The priority ranking is decided by their sequence in the list of RFMAPFN
argument, i.e., "e1.h5part" has the highest priority and "e4.h5part"
has the lowest priority.


Another method to model an RF cavity is to read the RF voltage profile
in the RFCAVITY element see RF Cavities (OPAL-t and OPAL-cycl) and make a momentum kick
when a particle crosses the RF gap. In the center region of the compact
cyclotron, the electric field shape is complicated and may make a
significant impact on transverse beam dynamics. Hence a simple momentum
kick is not enough and we need to read 3D field map to do precise
simulation.


In addition, a trim-coil field model is also implemented
to do fine tuning on the magnetic field. The trimcoils can be added with:


TRIMCOIL

Array of the trim coil names


A TRIMCOIL object can be defined in two ways:


TYPE

Type specifies PSI-BFIELD, PSI-PHASE or PSI-BFIELD-MIRRORED trim coil descriptions.
The general PSI-BFIELD and PSI-PHASE descriptions are based on rational functions with polynomials in the nominator and the denominator.
The function describes the magnetic field [T] resp. the phase shift as function of the radius [mm].
Separate functions can be given for the radial and azimuthal direction.
These functions are multiplied together for the function.
If a function in a direction is not specified it is the identity 1.
The PSI-BFIELD-MIRRORED type is described in http://accelconf.web.cern.ch/AccelConf/ipac2017/papers/thpab077.pdf

RMIN

Inner radius of the trim coil [mm]

RMAX

Outer radius of the trim coil [mm]

PHIMIN

Minimal azimuth [deg] (default 0) (not for PSI-BFIELD-MIRRORED)

PHIMAX

Maximal azimuth [deg] (default 360) (not for PSI-BFIELD-MIRRORED)

BMAX

Maximal B field of the trim coils [T] (for PSI-BFIELD) or maximal phase shift (for PSI-PHASE)

COEFNUM

Coefficients of the numerator for the radial direction,
first coefficient is zeroth order.
If COEFNUMPHI is not specified, the numerator is 1
(not for PSI-BFIELD-MIRRORED).

COEFDENOM

Coefficients of the denominator for the radial direction,
first coefficient is zeroth order.
If COEFDENOM is not specified, the denominator is 1, and the description will be a normal polynom
(not for PSI-BFIELD-MIRRORED).

COEFNUMPHI

Coefficients of the numerator for the azimuthal direction,
first coefficient is zeroth order.
If COEFNUMPHI is not specified, the numerator is 1.
(not for PSI-BFIELD-MIRRORED).

COEFDENOMPHI

Coefficients of the denominator for the azimuthal direction,
first coefficient is zeroth order.
If COEFDENOMPHI is not specified, the denominator is 1, and the description will be a normal polynom
(not for PSI-BFIELD-MIRRORED).

SLPTC

Slopes of the rising edge [1/mm] (for PSI-BFIELD-MIRRORED type only)


Example:


tc1:  TRIMCOIL, TYPE="PSI-BFIELD-MIRRORED", RMIN = 2022.09, RMAX = 2132.09, BMAX=2.0e-4, SLPTC=1;
tc15: TRIMCOIL, TYPE="PSI-BFIELD",          RMIN = 3000,    RMAX = 4500,    BMAX=13e-4,
      COEFNUM   = {-0.426038643356, 0.311242287271, -0.0484487029431},
      COEFDENOM = {19.3541404562, -22.2057165548, 9.99489842329, -2.00909633025, 0.14942099903};

Ring: CYCLOTRON, TYPE="RINGCYC", CYHARMON=6, PHIINIT=0.0, PRINIT=0.0,
      RINIT=2131, SYMMETRY=8.0, RFFREQ=50.65, BSCALE=1, FMAPFN="s03av.nar",
      TRIMCOIL={tc1, tc15};


This is a restricted feature: OPAL-cycl.


1.12. Ring Definition


label: RINGDEFINITION,
       RFFREQ=real, HARMONIC_NUMBER=real, IS_CLOSED=string, SYMMETRY=int,
       LAT_RINIT=real, LAT_PHIINIT=real, LAT_THETAINIT=real,
       BEAM_PHIINIT=real, BEAM_PRINIT=real, BEAM_RINIT=real;


A RingDefinition object contains the main characteristics of a
generalized ring. The RingDefinition lists characteristics of the
entire ring such as harmonic number together with the position of the
initial element and the position of the reference trajectory.


The RingDefinition can be used in combination with SBEND3D, offsets
and VARIABLE_RF_CAVITY elements to make up a complete ring.


RFFREQ

Nominal RF frequency of the ring [MHz].

HARMONIC_NUMBER

The harmonic number of the ring - i.e. number of bunches in a single
pass.

SYMMETRY

Azimuthal symmetry of the ring. Ring elements will be placed
repeatedly SYMMETRY times.

IS_CLOSED

Set to FALSE to disable checking for ring closure.

LAT_RINIT

Radius of the first element placement in the lattice [m].

LAT_PHIINIT

Azimuthal angle of the first element placed in the lattice [degree].

LAT_THETAINIT

Angle in the mid-plane relative to the ring tangent for placement of
the first element [degree].

BEAM_RINIT

Initial radius of the reference trajectory [m].

BEAM_PHIINIT

Initial azimuthal angle of the reference trajectory [degree].

BEAM_PRINIT

Transverse momentum 
\beta \gamma for the reference
trajectory.


In the following example, we define a ring with radius 2.35 m and 4
cells.


ringdef: RINGDEFINITION, HARMONIC_NUMBER=6, LAT_RINIT=2350.0, LAT_PHIINIT=0.0,
         LAT_THETAINIT=0.0, BEAM_PHIINIT=0.0, BEAM_PRINIT=0.0,
         BEAM_RINIT=2266.0, SYMMETRY=4.0, RFFREQ=0.2;


1.12.1. Local Cartesian Offset

The LOCAL_CARTESIAN_OFFSET enables the user to place an object at an
arbitrary position in the coordinate system of the preceding element.
This enables drift spaces and placement of overlapping elements.


END_POSITION_X

x position of the next element start in the coordinate system of the
preceding element [m].

END_POSITION_Y

y position of the next element start in the coordinate system of the
preceding element [m].

END_NORMAL_X

x component of the normal vector defining the placement of the next
element in the coordinate system of the preceding element [m].

END_NORMAL_Y

y component of the normal vector defining the placement of the next
element in the coordinate system of the preceding element [m].


1.12.2. Local Cylindrical Offset

The LOCAL_CYLINDRICAL_OFFSET enables the user to place an object at an
arbitrary position in the coordinate system of the preceding element in cylindrical coordinates.
This enables drift spaces and placement of overlapping elements.


THETA_IN

Angle between the previous element and the displacement vector [rad].

THETA_OUT

Angle between the displacement vector and the next element [rad].

LENGTH

Length of the offset [m].


1.13. Source

The SOURCE element only works in OPAL-t. Its only purpose in OPAL-t is to
indicate that the particle source is contained in the beamline. This is
needed to place the elements in three-dimensional space when using
ELEMEDGE. Otherwise it has no effect on the particles.


1.14. RF Cavities (OPAL-t and OPAL-cycl)

For an RFCAVITY the three modes have four real attributes in common:


label:RFCAVITY, APERTURE=real-vector, L=real,
      VOLT=real, LAG=real;


L

The length of the cavity (default: 0 m)

VOLT

The peak RF voltage (default: 0 MV). The effect of the cavity is
\delta E=\mathrm{VOLT}\cdot\sin(2\pi(\mathrm{LAG}-\mathrm{HARMON}\cdot f_0 t))
.

LAG

The phase lag [rad] (default: 0). In OPAL-t this phase is in general
relative to the phase at which the reference particle gains the most
energy. This phase is determined using an auto-phasing algorithm
(see Appendix Auto-phasing Algorithm). This auto-phasing algorithm can be
switched off, see APVETO.

DLAG

The phase lag error [rad] (default: 0).


1.14.1. OPAL-t mode

Using a RF Cavity in OPAL-t mode, the following additional parameters
are defined:


FMAPFN

Field maps in the T7 format can be specified.

TYPE

Type specifies STANDING [default] or SINGLEGAP structures.

FREQ

Defines the frequency of the RF Cavity in units of MHz. A warning is
issued when the frequency of the cavity card does not correspond to
the frequency defined in the FMAPFN file. The frequency of the cavity
card overrides the frequency defined in the FMAPFN file.

APVETO

If TRUE this cavity will not be auto-phased. Instead the phase of
the cavity is equal to LAG at the arrival time of the reference
particle (arrival at the limit of its field not at ELEMEDGE).


Example standing wave cavity which mimics a DC gun:


gun: RFCavity, L=0.018, VOLT=-131/(1.052*2.658),
     FMAPFN="1T3.T7", ELEMEDGE=0.00,
     TYPE="STANDING", FREQ=1.0e-6;


Example of a two frequency standing wave cavity:


rf1: RFCavity, L=0.54, VOLT=19.961, LAG=193.0/360.0,
     FMAPFN="1T3.T7", ELEMEDGE=0.129, TYPE="STANDING",
     FREQ=1498.956;
rf2: RFCavity, L=0.54, VOLT=6.250, LAG=136.0/360.0,
     FMAPFN="1T4.T7", ELEMEDGE=0.129, TYPE="STANDING",
     FREQ=4497.536;


1.14.2. OPAL-cycl mode

Using a RF Cavity (standing wave) in OPAL-cycl mode, the following
parameters are defined:


FMAPFN

Name of file which stores normalized voltage amplitude curve
of cavity gap in ASCII format. (See data format in
RF field)

VOLT

Peak value of voltage amplitude curve in MV.

TYPE

Defines Cavity type, SINGLEGAP represents cyclotron type cavity.

FREQ

Frequency of the RF Cavity in units of MHz.

RMIN

Radius of the cavity inner edge in mm.

RMAX

Radius of the cavity outer edge in mm.

ANGLE

Azimuthal position of the cavity in global frame in degree.

PDIS

Perpendicular distance (impact parameter) of cavity from center of cyclotron in mm.
If its value is positive, the radius increases clockwise (larger radius has smaller azimuthal angle).

GAPWIDTH

Set gap width of cavity in mm.

PHI0

Set initial phase of cavity in degree.


Example of a RF cavity of cyclotron:


rf0: RFCavity, VOLT=0.25796, FMAPFN="Cav1.dat",
     TYPE="SINGLEGAP", FREQ=50.637, RMIN = 350.0,
     RMAX = 3350.0, ANGLE=35.0,   PDIS = 0.0,
     GAPWIDTH = 0.0, PHI0=phi01;


Figure 5 shows the simplified geometry of a cavity gap
and its parameters.


Figure 5. Schematic of the simplified geometry of a cavity gap and parameters


1.15. RF Cavities with Time Dependent Parameters

The VARIABLE_RF_CAVITY element can be used to define RF Cavities with
Time Dependent Parameters in OPAL-cycl mode. Variable RF Cavities must
be placed using the RingDefinition element.


FREQUENCY_MODEL

String naming the time dependence model of the cavity frequency,
f
 [MHz].

AMPLITUDE_MODEL

String naming the time dependence model of the cavity amplitude,
E_0
 [MV/m].

PHASE_MODEL

String naming the time dependence model of the cavity phase offset,
\phi
 [rad].

WIDTH

Full width of the cavity [m].

HEIGHT

Full height of the cavity [m].

L

Full length of the cavity [m].


The field inside the cavity is given by


\mathbf{E} = \big(0, 0, E_0(t)\sin[2\pi f(t) t+\phi(t)]\big)


with no field outside the cavity boundary. There is no magnetic field or
transverse dependence on electric field.


1.15.1. Time Dependence

The POLYNOMIAL_TIME_DEPENDENCE element is used to define time
dependent parameters in RF cavities in terms of a third order
polynomial.


P0

Constant term in the polynomial expansion.

P1

First order term in the polynomial expansion [ns
^{-1}
].

P2

Second order term in the polynomial expansion [ns
^{-2}
].

P3

Third order term in the polynomial expansion [ns
^{-3}
].


The polynomial is evaluated as


g(t) = p_0 + p_1 t + p_2 t^2 + p_3 t^3.


An example of a Variable Frequency RF cavity of cyclotron with
polynomial time dependence of parameters is given below:


1.15.2. Fringe Field

It is possible to model a soft-edged RF cavity with time dependent parameters using the VARIABLE_RF_CAVITY_FRINGE_FIELD element. This will place a full cavity including the field body and fringe fields.  VARIABLE_RF_CAVITY_FRINGE_FIELD must be placed using the RingDefinition element.


FREQUENCY_MODEL

String naming the time dependence model of the cavity frequency,
f
 [MHz].

AMPLITUDE_MODEL

String naming the time dependence model of the cavity amplitude,
E_0
 [MV/m].

PHASE_MODEL

String naming the time dependence model of the cavity phase offset,
\phi
 [rad].

WIDTH

Full width of the cavity [m].

HEIGHT

Full height of the cavity [m].

L

Full length of the cavity bounding box [m].

CENTRE_LENGTH

Length of the cavity field flat top [m].

END_LENGTH

E-fold Length of the cavity field ends [m].

CAVITY_CENTRE

Position of the centre of the cavity relative to the start [m].

MAX_ORDER

Maximum power in vertical coordinate z to which the field will be evaluated.


REAL phi=2.*PI*0.25;

REAL rf_p0=0.00158279;
REAL rf_p1=9.02542e-10;
REAL rf_p2=-1.96663e-16;
REAL rf_p3=2.45909e-23;

RF_FREQUENCY: POLYNOMIAL_TIME_DEPENDENCE, P0=rf_p0, P1=rf_p1,   P2=rf_p2, P3=rf_p3;
RF_AMPLITUDE: POLYNOMIAL_TIME_DEPENDENCE, P0=1.0;
RF_PHASE: POLYNOMIAL_TIME_DEPENDENCE, P0=phi;

HARD_RF_CAVITY: VARIABLE_RF_CAVITY,
           PHASE_MODEL="RF_PHASE", AMPLITUDE_MODEL="RF_AMPLITUDE",
           FREQUENCY_MODEL="RF_FREQUENCY", L=0.100, HEIGHT=0.200, WIDTH=2.000;

SOFT_RF_CAVITY: VARIABLE_RF_CAVITY_FRINGE_FIELD,
           PHASE_MODEL="RF_PHASE", AMPLITUDE_MODEL="RF_AMPLITUDE",
           FREQUENCY_MODEL="RF_FREQUENCY", L=0.200, HEIGHT=0.200, WIDTH=2.000
           CENTRE_LENGTH=0.1, END_LENGTH=0.01, CAVITY_CENTRE=0.1, MAX_ORDER=4;


1.16. Traveling Wave Structure


Figure 6. The on-axis field of an S-band (2997.924 MHz) TRAVELINGWAVE structure. The field of a single cavity is shown between its entrance and exit fringe fields. The fringe fields extend one half wavelength (\lambda/2
) to either side.


An example of a 1D TRAVELINGWAVE structure field map is shown in
Figure 6. This map is a standing wave solution generated
by Superfish and shows the field on axis for a single accelerating
cavity with the fringe fields of the structure extending to either side.
OPAL-t reads in this field map and constructs the total field of the
TRAVELINGWAVE structure in three parts: the entrance fringe field, the
structure fields and the exit fringe field.


The fringe fields are treated as standing wave structures and are given
by:


\begin{aligned}
    \mathbf{E_{entrance}}(\mathbf{r}, t) &= \mathbf{E_{from-map}}(\mathbf{r}) \cdot \mathrm{VOLT} \cdot \cos \left( 2\pi \cdot \mathrm{FREQ} \cdot t
        + \phi_{entrance} \right) \\
    \mathbf{E_{exit}}(\mathbf{r}, t) &= \mathbf{E_{from-map}}(\mathbf{r}) \cdot \mathrm{VOLT} \cdot \cos \left( 2\pi \cdot \mathrm{FREQ} \cdot t
        + \phi_{exit} \right)
  \end{aligned}


where VOLT and FREQ are the field magnitude and
frequency attributes (see below).
 \phi_{entrance}= \mathrm{LAG}, the phase attribute
of the element (see below).  \phi_{exit}  is dependent
upon both LAG and the NUMCELLS attribute (see below) and is calculated
internally by OPAL-t.


The field of the main accelerating structure is reconstructed from the
center section of the standing wave solution shown in
Figure 6 using


\begin{aligned}
    \mathbf{E} ( \mathbf{r},t ) &= \frac{\mathrm{VOLT}}{\sin (2 \pi \cdot \mathrm{MODE})} \\
    & \times \Biggl\{ \mathbf{E_{from-map}} (x,y,z) \cdot \cos \biggl( 2 \pi \cdot \mathrm{FREQ} \cdot t + \mathrm{LAG}+ \frac{\pi}{2} \cdot \mathrm{MODE} \Bigr) + \\
    & \mathbf{E_{from-map}}(x,y,z+d) \cdot \cos \biggl( 2 \pi \cdot \mathrm{FREQ} \cdot t + \mathrm{LAG} + \frac{3 \pi}{2} \cdot \mathrm{MODE} \Bigr) \Biggr\}
  \end{aligned}


where d is the cell length and is defined as
\text{d} = \lambda \cdot \mathrm{MODE} . MODE is an
attribute of the element (see below). When calculating the field from
the map (\mathbf{E_{from-map}}(x,y,z)), the longitudinal
position is referenced to the start of the cavity fields at
\frac{\lambda}{2} (In this case starting at
z = {5.0}cm). If the longitudinal position advances past
the end of the cavity map (\frac{3\lambda}{2} = {15.0}cm
in this example), an integer number of cavity wavelengths is subtracted
from the position until it is back within the map’s longitudinal range.


A TRAVELINGWAVE structure has seven real attributes, one integer
attribute, one string attribute and one Boolean attribute:


label:TRAVELINGWAVE, APERTURE=real-vector, L=real,
      VOLT=real, LAG=real, FMAPFN=string,
      ELEMEDGE=real, FREQ=real, NUMCELLS=integer,
      MODE=real;


L

The length of the cavity (default: 0 m). In OPAL-t this attribute is
ignored, the length is defined by the field map and the number of
cells.

VOLT

The peak RF voltage (default: 0 MV). The effect of the cavity is
\delta E=\mathrm{VOLT}\cdot\sin(\mathrm{LAG}- 2\pi\cdot\mathrm{FREQ}\cdot t)
.

LAG

The phase lag [rad] (default: 0). In OPAL-t this phase is in general
relative to the phase at which the reference particle gains the most
energy. This phase is determined using an auto-phasing algorithm
(see Appendix Auto-phasing Algorithm). This auto-phasing algorithm can be
switched off, see APVETO.

DLAG

The phase lag error [rad] (default: 0).

FMAPFN

Field maps in the T7 format can be specified.

FREQ

Defines the frequency of the traveling wave structure in units of MHz.
A warning is issued when the frequency of the cavity card does not
correspond to the frequency defined in the FMAPFN file. The frequency
defined in the FMAPFN file overrides the frequency defined on the
cavity card.

NUMCELLS

Defines the number of cells in the tank. (The cell count should not
include the entry and exit half cell fringe fields.)

MODE

Defines the mode in units of 
2\pi, for example
\frac{1}{3}
 stands for a \frac{2 \pi}{3}
structure.

FAST

If FAST is true and the provided field map is in 1D then a 2D field
map is constructed from the 1D on-axis field, see Fieldmaps Types and Format.
To track the particles the field values are interpolated from this map
instead of using an FFT based algorithm for each particle and each
step. (default: FALSE)

APVETO

If TRUE this cavity will not be auto-phased. Instead the phase of
the cavity is equal to LAG at the arrival time of the reference
particle (arrival at the limit of its field not at ELEMEDGE).


Use of a traveling wave requires the particle momentum P and the
particle charge CHARGE to be set on the relevant optics command before
any calculations are performed.


Example of a L-Band traveling wave structure:


lrf0: TravelingWave, L=0.0253, VOLT=14.750,
      NUMCELLS=40, ELEMEDGE=2.73066,
      FMAPFN="INLB-02-RAC.Ez", MODE=1/3,
      FREQ=1498.956, LAG=248.0/360.0;


1.17. Monitor

A MONITOR detects all particles passing it and writes the position,
the momentum and the time when they hit it into an H5hut file.
Furthermore the exact position of the monitor is stored. It has always a
length of 1 cm consisting of 0.5 cm drift, the monitor of zero length and
another 0.5 cm drift. This is to prevent OPAL-t from missing any
particle. The positions of the particles on the monitor are interpolated
from the current position and momentum one step before they would passe
the monitor.


OUTFN

The file name into which the monitor should write the collected data.
The file is an H5hut file.


If the attribute TYPE is set to TEMPORAL then the data of all
particles are written to the H5hut file when the reference particle hits
the monitor.


This is a restricted feature for OPAL-t.


1.18. Collimators

Four types of collimators are defined:


ECOLLIMATOR

Elliptic aperture,

RCOLLIMATOR

Rectangular aperture.

FLEXIBLECOLLIMATOR

Description of shape and location of holes can be provided

CCOLLIMATOR

Radial rectangular collimator in cyclotrons


label:ECOLLIMATOR, TYPE=string, APERTURE=real-vector,
      L=real, XSIZE=real, YSIZE=real;
label:RCOLLIMATOR,TYPE=string, APERTURE=real-vector,
      L=real, XSIZE=real, YSIZE=real;
label:FLEXIBLECOLLIMATOR, APERTURE=real-vector,
      L=real, DESCRIPTION=string, FNAME=string, OUTFN=string;


Each type has the following attributes:


L

The collimator length (default: 0 m).

OUTFN

The file name into which the monitor should write the collected data.
The file is an H5hut file.


Optically a collimator behaves like a drift space, but
during tracking, it also introduces an aperture limit. The aperture is
checked at the entrance. If the length is not zero, the aperture is also
checked at the exit and at every timestep. Lost particles are saved in an H5hut file defined by OUTFN. The ELEMEDGE defines the location of the collimator and L
the length.


The reference system for a collimator is a Cartesian coordinate system.


1.18.1. OPAL-t mode

The CCOLLIMATOR isn’t supported. ECOLLIMATORs and RCOLLIMATORs
detect all particles which are outside the aperture defined by XSIZE
and YSIZE.
For elliptic apertures, XSIZE and YSIZE denote the half-axes
respectively, for rectangular apertures they denote the half-width of
the rectangle.


XSIZE

The horizontal half-aperture (default: unlimited).

YSIZE

The vertical half-aperture (default: unlimited).


Example:


Col:ECOLLIMATOR, L=1.0E-3, ELEMEDGE=3.0E-3, XSIZE=5.0E-4,
    YSIZE=5.0E-4, OUTFN="Coll.h5";


The FLEXIBLECOLLIMATOR can be used to model both simple, rectangular or elliptic collimators and more complex devices like pepper-pots. The configuration of holes can be described with a special language. This language knows the following commands


rectangle(width, height)

A rectangle that is centered at the origin of the 2D coordinate system. The arguments width and heigth can be mathematical expressions.

ellipse(width, height)

An ellipse that is centered at the origin of the 2D coordinate system. The arguments width and heigth can be mathematical expressions.

polygon(x_0, y_0; x_1, y_1; x_2, y_2[; x_3, y_3[;… x_N, y_N]])

A polygon with with vertices (x_0, y_0), (x_1, y_1), (x_2, y_2), …, (x_N, y_N). The first vertex doens’t have to be repeated, instead (x_N, y_N) is connected with (x_0, y_0). The polygon is then triangulized for a fast detection of stopped particles. In order for the triangulization to work the edges of the polygon may not cross each other. All arguments of the command polygon can be mathematical expressions.

mask('filename.pbm', width, height)

A black and white bitmap file (Portable Bitmap format) can be provided to describe the collimator. White pixels stop particles. The first argument is the path to the pixmap file, the second and third are the width and height of the mask in meters. The arguments width and height can be mathematical expressions.

translate(command, shiftx, shifty)

Translates the holes that are define by the command by shiftx in the x-direction and shifty in the y-direction. The arguments shiftx and shifty can be mathematical expressions.

rotate(command, angle)

Rotates the holes that are defined by the command about the origin of the 2D coordinate system. The argument angle can be a mathematical expression.

union(command1, command2 [, command3 [, command4 […]]])

Collects the holes that are defined the by the commands.

difference(command1, command2)

All particles that pass command1 and not command2 pass the difference.


Figure 7. Illustration of a difference between to circles


symmetric_difference(command1, command2)

All particles that pass either command but not both at the same time.


Figure 8. Illustration of a symmetric difference between to circles


intersection(command1, command2)

All particles that pass both commands at the same time.


Figure 9. Illustration of a intersection between to circles


repeat(command, N, shiftx, shifty)

Repeats the holes that are defined by the command translating each copy successively by shiftx in x-direction and shifty in y-direction. The arguments shiftx and shifty can be mathematical expressions.

repeat(command, N, angle)

Repeats the holes that are defined by the command rotating each copy successively. The argument angle can be a mathematical expression.


The supported mathematical constants and functions are listed in the following table.


Table 1. Mathematical constants and functions


e
pi
abs(x)
acos(x)


acosh(x)
asin(x)
asinh(x)
atan(x)


atanh(x)
cbrt(x)
ceil(x)
cos(x)


cosh(x)
deg2rad(x)
erf(x)
erfc(x)


exp(x)
exp2(x)
floor(x)
isinf(x)


isnan(x)
log(x)
log2(x)
log10(x)


rad2deg(x)
round(x)
sgn(x)
sin(x)


sinh(x)
sqrt(x)
tan(x)
tanh(x)


tgamma(x)
atan2(y,x)
max(x,y)
min(x,y)


A simple elliptic collimator with major and minor axis of 4 cm and 3 cm respectively can be defined using


ellipse(0.04, 0.03)


A regular pepper-pot with rectangular holes can be define like this


repeat( // repeat it in y-direction
       repeat( // repeat it in x-direction
              translate(
                        rotate(
                               rectangle(
                                         0.002,
                                         0.002
                                        ),
                               0.78539
                              ),
                        -0.028,
                        -0.028
                       ),
              16,
              0.004,
              0.0
             ),
       16,
       0.0,
       0.004
      )


The latter example will produce a holes as in the following picture


Figure 10. Pepper-pot with rectangle holes


In the FLEXIBLECOLLIMATOR command the description of the holes can be provided as a string (using DESCRIPTION; the string may not contain comments and newlines) or in a separate file (using FNAME; comments and newlines are allowed).


1.18.2. OPAL-cycl mode

Only CCOLLIMATOR is available for OPAL-cycl. This element is radial
rectangular collimator which can be used to collimate the radial tail
particles. When a particle hits this collimator, it will be absorbed
or scattered. The algorithm is based on the Monte Carlo method. Please
note when a particle is scattered, it will not be recorded as the lost
particle. If this particle leaves the bunch, it will be removed during
the integration afterwards, so as to maintain the accuracy of space
charge solving.


XSTART

The x coordinate of the start point. [mm]

XEND

The x coordinate of the end point. [mm]

YSTART

The y coordinate of the start point. [mm]

YEND

The y coordinate of the end point. [mm]

ZSTART

The minimum vertical coordinate [mm]. Default value is
-100mm.

ZEND

The maximum vertical coordinate. [mm]. Default value is
-100mm.

WIDTH

The width of the collimator. [mm]

OUTFN

The file name into which the collimator should write the collected data.

PARTICLEMATTERINTERACTION

PARTICLEMATTERINTERACTION is an attribute of the element. Collimator
physics is only a kind of particlematterinteraction. It can be applied
to any element. If the type of PARTICLEMATTERINTERACTION is
COLLIMATOR, the material is defined here. The material "Cu",
"Be", "Graphite" and "Mo" are defined until now. If this is not
set, the particle matter interaction module will not be activated. The
particle hitting collimator will be recorded and directly deleted from
the simulation.


Figure 11. Collimator


Example:


REAL y1=-0.0;
REAL y2=0.0;
REAL y3=200.0;
REAL y4=205.0;
REAL x1=-215.0;
REAL x2=-220.0;
REAL x3=0.0;
REAL x4=0.0;
cmphys:particlematterinteraction, TYPE="Collimator", MATERIAL="Cu";
cma1: CCollimator, XSTART=x1, XEND=x2,YSTART=y1, YEND=y2,
      ZSTART=2, ZEND=100, WIDTH=10.0, PARTICLEMATTERINTERACTION=cmphys ;
cma2: CCollimator, XSTART=x3, XEND=x4,YSTART=y3, YEND=y4,
      ZSTART=2, ZEND=100, WIDTH=10.0, PARTICLEMATTERINTERACTION=cmphys;


The particles lost on the CCOLLIMATOR are recorded in the HDF5 file
<inputfilename>.h5 (or ASCII if ASCIIDUMP is true).


1.19. Septum (OPAL-cycl)

This is a restricted feature for OPAL-cycl. The particles hitting on the
septum is removed from the bunch. There are 5 parameters to describe a
septum.


XSTART

The x coordinate of the start point. [mm]

XEND

The x coordinate of the end point. [mm]

YSTART

The y coordinate of the start point. [mm]

YEND

The y coordinate of the end point. [mm]

WIDTH

The width of the septum. [mm]

OUTFN

The file name into which the septum should write the collected data.


Figure 12. Septum


Example:


eec2: Septum, xstart=4100.0, xend=4300.0,
ystart=-1200.0, yend=-150.0, width=0.05;


The particles lost on the SEPTUM are recorded in the HDF5 file
<inputfilename>.h5 (or ASCII if ASCIIDUMP is true).


1.20. Probe (OPAL-cycl)

The particles hitting on the probe is recorded. There are 5 parameters
to describe a probe.


XSTART

The x coordinate of the start point. [mm]

XEND

The x coordinate of the end point. [mm]

YSTART

The y coordinate of the start point. [mm]

YEND

The y coordinate of the end point. [mm]

STEP

The step size of the probe (for histogram and peak finder output). Default: 1 [mm]

OUTFN

The file name into which the probe should write the collected data.


Figure 13. Probe


Example:


prob1: Probe, xstart=4166.16, xend=4250.0,
ystart=-1226.85, yend=-1241.3;


The particles probed on the PROBE are recorded in the HDF5 file
<inputfilename>.h5 (or ASCII if ASCIIDUMP is true).
Please note that these particles are not deleted
in the simulation, however, they are recorded in the "loss" file.


The radius of the particles recorded in the PROBE is recorded in the histogram ".hist" and peak ".peaks" file.
The histogram file contains data as recorded in actual probe measurements.
The corresponding peaks file contains the peaks found in the probe histogram by the same peak finder used for the PSI measurements.
Note that for probes in multiple quadrants the histogram and peaks file is often not meaningful since the absolute radius is stored.


1.21. Stripper (OPAL-cycl)

A stripper element strip the electron(s) from a particle. The particle
hitting the stripper is recorded in the file, which contains the time,
coordinates and momentum of the particle at the moment it hit the
stripper. The charge and mass are changed. It has the same geometry as
the PROBE element. Please note that the stripping physics is not
included yet.


There are 9 parameters to describe a stripper.


XSTART

The x coordinate of the start point. [mm]

XEND

The x coordinate of the end point. [mm]

YSTART

The y coordinate of the start point. [mm]

YEND

The y coordinate of the end point. [mm]

OPCHARGE

Charge number of the outcoming particle. Negative value represents
negative charge.

OPMASS

Mass of the outcoming particles. [
\mathrm{GeV/c^2}
]

OPYIELD

Yield of the outcoming particle (the number of outcoming particles per
incoming particle), the default value is 1.

STOP

If STOP is true, the particle is stopped and deleted from the
simulation; Otherwise, the outcoming particle continues to be tracked
along the extraction path.

OUTFN

The file name into which the stripper should write the collected data.


Example: 
H_2^+
 particle stripping


prob1: Stripper, xstart=4166.16, xend=4250.0,
ystart=-1226.85, yend=-1241.3,
opcharge=1, opmass=PMASS, opyield=2, stop=false;


No matter what the value of STOP is, the particles hitting on the
STRIPPER are recorded in the HDF5 file
<inputfilename>.h5 (or ASCII if ASCIIDUMP is true).


1.22. Degrader (OPAL-t)

Elliptical degrader with an overall length L.


XSIZE

Major axis of the transverse elliptical shape, default value is 1e6.

YSIZE

Minor axis of the transverse elliptical shape, default value is 1e6.


Example: Graphite degrader of 15 cm thickness.


DEGPHYS: PARTICLEMATTERINTERACTION, TYPE="DEGRADER", MATERIAL="Graphite";

DEG1: DEGRADER, L=0.15, ELEMEDGE=0.02, PARTICLEMATTERINTERACTION=DEGPHYS;


1.23. Correctors (OPAL-t)

Three types of correctors are available:


HKICKER

A corrector for the horizontal plane.

VKICKER

A corrector for the vertical plane.

KICKER

A corrector for both planes.


They act as


label:HKICKER, TYPE=string, APERTURE=real-vector,
      L=real, KICK=real;
label:VKICKER, TYPE=string, APERTURE=real-vector,
      L=real, KICK=real;
label:KICKER, TYPE=string, APERTURE=real-vector,
      L=real, HKICK=real, VKICK=real;


They have the following attributes:


L

The length of the closed orbit corrector (default: 0 m).

KICK

The kick angle in rad for either horizontal or vertical correctors
(default: 0 rad).

HKICK

The horizontal kick angle in rad for a corrector in both planes
(default: 0 rad).

VKICK

The vertical kick angle in rad for a corrector in both planes
(default: 0 rad).

DESIGNENERGY

Fix the magnitude of the magnetic field using the given DESIGNENERGY
and the angle (KICK, HKICK or VKICK). If the design energy isn’t
set then the actual energy of the reference particle at the position
of the corrector is used. The DESIGNENERGY is expected in MeV.


A positive kick increases 
p_{x}
 or p_{y}
respectively. Use KICK for an HKICKER or VKICKER and HKICK and
VKICK for a KICKER. Instead of using a KICKER or a VKICKER one
could use an HKICKER and rotate it appropriately using PSI.


Correctors don’t change the reference trajectory. Otherwise they are
implemented as RBEND with 
\mathrm{E1} = 0 and without
fringe fields (hard edge model). They can be used to model earth’s
magnetic field which is neglected in the design trajectory but which has
a noticeable effect on the trajectory of a bunch at low energies.


Examples:


HK1:HKICKER, KICK=0.001;
VK3:VKICKER, KICK=0.0005;
KHV:KICKER, HKICK=0.001, VKICK=0.0005;


The reference system for an orbit corrector is a Cartesian coordinate
system.


1.24. Beam Stripping (OPAL-cycl)

Beam stripping represents an abstract element that includes the necessary
parameters to consider the interactions with the residual gas and
the magnetic field of the cyclotron. When the particle interacts,
it is recorded in the file, which contains the time,
coordinates and momentum of the particle at this moment.
The particle could produce a new particle, changing
the charge and mass.


There are 7 parameters to describe beam stripping.


PRESSURE

The average pressure of the residual gas in the cyclotron. [mbar]

TEMPERATURE

Temperature of residual gas. [K]

PMAPFN

File name of the mid-plane pressure map.
The pressure data is stored in a sequence shown in 2D field map on the median
plane with primary direction corresponding to the azimuthal direction,
secondary direction to the radial direction
(same file structure as Cyclotron TYPE=CARBONCYCL).
If PMAPFN is specified, the PRESSURE parameter is ignored.

PSCALE

Scale factor for the pressure field map (default: 1.0).

GAS

Type of gas for residual vacuum: H2 or AIR

STOP

If STOP is true, the particle is stopped and deleted from the
simulation. Otherwise, the outcoming particle continues to be tracked
as SECONDARY particle (default: true).

PARTICLEMATTERINTERACTION

PARTICLEMATTERINTERACTION is an attribute of the element. Beam stripping
physics is only a kind of particlematterinteraction.


Example: Beam stripping by 
H_2
 residual gas.


bstp_phys:particlematterinteraction, TYPE="BEAMSTRIPPING";
bstp: BEAMSTRIPPING, PRESSURE=1E-8, TEMPERATURE=300,
      GAS="H2", STOP=true, PARTICLEMATTERINTERACTION=bstp_phys;


No matter what the value of STOP is, the particles stripped are recorded in the HDF5 file
<elementname>.h5 (or ASCII if ASCIIDUMP is true).


1.25. References

 [1] Tait-bryan angles.


 [2] J. E. Spencer and H. A. Enge, Split-pole magnetic spectrograph for precision nuclear spectroscopy, Nucl. Instrum. Methods 49, 181–193 (1967).