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1. Elements

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 (OPAL-t)

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.

Illustration of a general rectangular bend (RBEND) with a positive bend angle \alpha. 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. Illustration of a general rectangular bend (RBEND) with a positive bend angle \alpha. shows an RBEND with a positive bend angle and a positive entrance edge angle.

L: Physical length of magnet (meters, see Illustration of a general rectangular bend (RBEND) with a positive bend angle \alpha.).
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 Illustration of a general rectangular bend (RBEND) with a positive bend angle \alpha.. Not supported in OPAL-t any more. Superimpose a Quadrupole instead.
E1: Entrance edge angle (radians). Illustration of a general rectangular bend (RBEND) with a positive bend angle \alpha. 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 Illustration of a general rectangular bend (RBEND) with a positive bend angle \alpha..
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. Illustration of a general rectangular bend (RBEND) with a positive bend angle \alpha. shows an RBEND3D with a positive bend angle and a positive entrance edge angle.

L: Physical length of magnet (meters, see Illustration of a general rectangular bend (RBEND) with a positive bend angle \alpha.).
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 Illustration of a general rectangular bend (RBEND) with a positive bend angle \alpha.. Not supported in OPAL-t any more. Superimpose a Quadrupole instead.
E1: Entrance edge angle (radians). Illustration of a general rectangular bend (RBEND) with a positive bend angle \alpha. 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 Illustration of a general rectangular bend (RBEND) with a positive bend angle \alpha..
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.

Illustration of a general sector bend(SBEND) with a positive bend angle \alpha 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. Illustration of a general sector bend(SBEND) with a positive bend angle \alpha 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 Illustration of a general sector bend(SBEND) with a positive bend angle \alpha), 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 Illustration of a general sector bend(SBEND) with a positive bend angle \alpha. Not supported in OPAL-t any more. Superimpose a Quadrupole instead.
E1: Entrance edge angle (rad). Illustration of a general sector bend(SBEND) with a positive bend angle \alpha shows the definition of a positive entrance edge angle.
E2: Exit edge angle (rad). Illustration of a general sector bend(SBEND) with a positive bend angle \alpha 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 Illustration of a general sector bend(SBEND) with a positive bend angle \alpha.
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 Illustration of a general rectangular bend (RBEND) with a positive bend angle \alpha. and Illustration of a general sector bend(SBEND) with a positive bend angle \alpha. 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 Illustration of a general rectangular bend (RBEND) with a positive bend angle \alpha. and Illustration of a general sector bend(SBEND) with a positive bend angle \alpha.
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 Illustration of a general rectangular bend (RBEND) with a positive bend angle \alpha.). 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 Illustration of a general rectangular bend (RBEND) with a positive bend angle \alpha.. 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:

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

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;

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

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 Illustration of a general rectangular bend (RBEND) with a positive bend angle \alpha.) 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 Figure 3 of Chapter Fieldmaps.

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 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..

Let us assume we have a correctly defined positive RBEND or SBEND element as illustrated in Illustration of a general rectangular bend (RBEND) with a positive bend angle \alpha. and Illustration of a general sector bend(SBEND) with a positive bend angle \alpha. (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 [bib.enge_elements] 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 (OPAL-t)

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.

Example:

QP1: Quadrupole, L=1.20, ELEMEDGE=-0.5265,
     FMAPFN="1T1.T7", K1=0.11;

1.6. Sextupole (OPAL-t)

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.

Example:

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

The reference system for a sextupole is a Cartesian coordinate system

1.7. Octupole (OPAL-t)

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.

Example:

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

The reference system for an octupole is a Cartesian coordinate system

1.8. General Multipole (OPAL-t)

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.

The order n is unlimited, but all components up to the maximum must be given, even if they are zero. 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 (OPAL-t).

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 (OPAL-t) 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 (OPAL-t)

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 (OPAL-cycl)

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 (OPAL-cycl)

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 (_OPAL_t)

This element only works in OPAL-t. It’s 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 flavours 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 [MV/m] (default: 0). 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.

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 SINGLE GAP 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.
DVOLT: The RF voltage error [MV/m] (default: 0). This error isn’t applied to the reference particle only to particles of the bunch. It is used to model imperfections of the machine and trajectory corrections.
DLAG: The phase lag error [rad] (default: 0). This error isn’t applied to the reference particle only to particles of the bunch. It is used to model imperfections of the machine and trajectory corrections.
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).
DESIGNENERGY: The kinetic energy that the refernce particle should reach at the exit of the cavity [MeV]. Currently only works in conjuction with the auto-phasing algorithm.

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;

Schematic of the simplified geometry of a cavity gap and parameters 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. Pillbox RF Cavity (OPAL-t)

The PILLBOX command provides an analytical model for a cylindrical RF cavity. Fringe fields aren’t supported yet. Both TM_mnp and TE_mnp modes for m \ge 0, n \ge 1 and p \ge 0 (TM) and p \ge 1 (TE) are supported. The computed field for TM_mnp is

\begin{aligned}
E_r       &= -\frac{p \pi}{L} \frac{R}{x_{mn}} E_0 J_m^\prime(k_{mn} r) \cos(m \varphi) \sin(\frac{p\pi}{L}z)\exp(\dot{\iota} \omega t)\\
E_\varphi &= -\frac{p \pi}{L} \frac{m R^2}{x_{mn}^2 r} E_0 J_m(k_{mn} r) \sin(m \varphi) \sin(\frac{p\pi}{L}z)\exp(\dot{\iota} \omega t)\\
E_z       &= E_0 J_m(k_{mn} r) \cos(m \varphi) \cos(\frac{p\pi}{L} z)\exp(\dot{\iota} \omega t) \\
B_r       &= -\dot{\iota} \omega \frac{m R^2}{x_{mn}^2 r c^2} E_0 J_m(k_{mn} r) \sin(m \varphi) \cos(\frac{p\pi}{L}z)\exp(\dot{\iota} \omega t) \\
B_\varphi &= -\dot{\iota} \omega \frac{R}{x_{mn} c^2} E_0 J_m^\prime (k_{mn} r) \cos(m \varphi) \cos(\frac{p\pi}{L}z)\exp(\dot{\iota} \omega t) \\
B_z       &= 0
\end{aligned}

where L is the length and R the radius of the pillbox, J_m is the m-th Bessel function of the first kind, x_{mn} is the n-th root of J_m , k_{mn} = \frac{x_{mn}}{R} and \omega = c\cdot\sqrt{k_{mn}^2 + \left(\frac{p\pi}{L}\right)^2}.

The computed field for TE_mnp is:

\begin{aligned}
E_r       &= \dot{\iota} \omega \frac{m R^2}{x_{mn}^{{\prime 2}} R} B_0 J_m(k_{mn}^\prime r)\sin(m \varphi) \sin(\frac{p\pi}{L} z) \exp(\dot{\iota} \omega t) \\
E_\varphi &= \dot{\iota} \omega \frac{R}{x_{mn}^\prime} B_0 J^\prime_{m}(k_{mn}^\prime r)\cos(m \varphi) \sin(\frac{p\pi}{L} z) \exp(\dot{\iota} \omega t) \\
E_z       &= 0 \\
B_r       &= \frac{p\pi}{L}\frac{R}{x_{mn}^\prime} B_0 J_m^\prime(k_{mn}^\prime r)\cos(m \varphi) \cos(\frac{p\pi}{L} z) \exp(\dot{\iota} \omega t) \\
B_\varphi &= -\frac{p \pi}{L}\frac{m R^2}{x_{mn}^{{\prime 2}} r} B_0 J_m(k_{mn}^\prime r) \sin(m \varphi) \cos(\frac{p\pi}{L} z) \exp(\dot{\iota} \omega t) \\
B_z       &= B_0 J_m(k_{mn}^\prime r) \cos(m \varphi) \sin(\frac{p\pi}{L}z) \exp(\dot{\iota} \omega t)
\end{aligned}

where x_{mn}^\prime is the n-th root of the first derivative of J_m , k_{mn}^\prime = \frac{x_{mn}^\prime}{R} and \omega = c\cdot\sqrt{k_{mn}^{\prime 2} + \left(\frac{p\pi}{L}\right)^2}.

The attributes of the PILLBOX command are:

label:PILLBOX, L=real, SCALE=real, DSCALE=real, LAG=real, DLAG=real,
               RADIUS=real, M=real, N=real, P=real, APVETO=boolean;

L: The length of the cavity (default: 0_m)
SCALE: The amplitude of the electric field (TM mode, E_0 in equation above) or magnetic field (TE mode, B_0 in equation above).
DSCALE: The error of the amplitude (default: 0). This error isn’t applied to the reference particle only to particles of the bunch. It is used to model imperfections of the machine and trajectory corrections.
LAG: The phase lag [rad] (default: 0).
DLAG: The phase lag error [rad] (default: 0). This error isn’t applied to the reference particle only to particles of the bunch. It is used to model imperfections of the machine and trajectory corrections.
RADIUS: The radius of the cavity [m] (default: 0).
M: The number of variation of field of the azimuthal variable (default: 0).
N: The number of nulls in Ez along the radial direction (default: 1).
P: The number of nodes of Ez along the longitudinal direction (default: 0).
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).

1.17. Traveling Wave Structure (_OPAL_t)

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 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.. 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 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. 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, DVOLT=real, LAG=real, DLAG=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).
DVOLT: The RF voltage error [MV/m] (default: 0). This error isn’t applied to the reference particle only to particles of the bunch. It is used to model imperfections of the machine and trajectory corrections.
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). This error isn’t applied to the reference particle only to particles of the bunch. It is used to model imperfections of the machine and trajectory corrections.
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;

1.18. SLACs Transverse Deflectinc Structure (OPAL-t)

A SLACTDS element deflects the particles in horizontal direction, see here. It’s field is computed using:

\begin{aligned}
          E_r       &= E_0 (\frac{k r}{2}^2 + \frac{k a}{2}^2) \cos(\varphi) \\
          E_\varphi &= E_0 (\frac{k r}{2}^2 - \frac{k a}{2}^2) \sin(\varphi) \\
          E_z       &= \dot{\iota} E_0 k r \cos(\varphi) \\
          B_r       &= -\frac{E_0}{c} (\frac{k r}{2}^2 - \frac{k a}{2}^2 + 1) \sin(\varphi) \\
          B_\varphi &= \frac{E_0}{c} (\frac{k r}{2}^2 + \frac{k a}{2}^2 - 1) \cos(\varphi) \\
          B_z       &= -\dot{\iota} \frac{E_0}{c} k r \sin(\varphi)
\end{aligned}

where a is the radius of the iris, k = \frac{2\pi}{\lambda} and E_0 = |E_0|\exp(\dot{\iota} k (z - \omega t)) . The frequency is fixed to 2.856 GHz, the radius of the iris 2.24 cm and the length of a cell 3.5 cm. The overall length of the RF structure can be chosen in multiples of the length of a cell. Currently no fringe fields are modelled.

A SLACTDS can be defined with the following attributes:

label:SLACTDS, NUMCELLS=real, LAG=real, DLAG=real, VOLT=real, DVOLT=real;

NUMCELLS: The number of cells of the RF structure.
VOLT: The peak RF voltage [MV/m] (default: 0)
DVOLT: The RF voltage error [MV/m] (default: 0). This error isn’t applied to the reference particle only to particles of the bunch. It is used to model imperfections of the machine and trajectory corrections.
LAG: The phase lag [rad] (default: 0).
DLAG: The phase lag error [rad] (default: 0). This error isn’t applied to the reference particle only to particles of the bunch. It is used to model imperfections of the machine and trajectory corrections.

1.19. Monitor (OPAL-t)

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.

1.20. 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.

1.20.1. 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).

1.20.2. 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).

1.21. 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).

1.22. 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.

1.23. 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).

1.24. 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;

1.25. 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.

References

[bib.tait-bryan] Tait-bryan angles.
[user-content-bib.enge_elements] J. E. Spencer and H. A. Enge, Split-pole magnetic spectrograph for precision nuclear spectroscopy, Nucl. Instrum. Methods 49, 181–193 (1967).