
1. Elements
 1.1. Element Input Format
 1.2. Common Attributes for all Elements
 1.3. Drift Spaces (OPALt)
 1.4. Bending Magnets
 1.5. Quadrupole (OPALt)
 1.6. Sextupole (OPALt)
 1.7. Octupole (OPALt)
 1.8. General Multipole (OPALt)
 1.9. General Multipole (will replace General Multipole (OPALt) when implemented)
 1.10. Solenoid (OPALt)
 1.11. Cyclotron (OPALcycl)
 1.12. Ring Definition (OPALcycl)
 1.13. Source (_OPAL_t)
 1.14. RF Cavities (OPALt and OPALcycl)
 1.15. RF Cavities with Time Dependent Parameters
 1.16. Pillbox RF Cavity (OPALt)
 1.17. Traveling Wave Structure (_OPAL_t)
 1.18. SLACs Transverse Deflectinc Structure (OPALt)
 1.19. Monitor (OPALt)
 1.20. Collimators
 1.21. Septum (OPALcycl)
 1.22. Probe (OPALcycl)
 1.23. Stripper (OPALcycl)
 1.24. Degrader (OPALt)
 1.25. Correctors (OPALt)
 References
1. Elements
1.1. Element Input Format
All physical elements are defined by statements of the form
label:keyword, attribute,..., attribute
where
 label

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

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

normally has the form
attributename=attributevalue
 attributename

selects the attribute from the list defined for the element type
keyword
(in the exampleL
andK1
). It must be an identifier see Identifiers or Labels.  attributevalue

gives it a value see Command Attribute Types (in the example
1.8
and0.015832
).
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 heighta
, 
APERTURE
="RECTANGLE(a,b,f)" has a rectangular shape of widtha
and heightb
, 
APERTURE
="CIRCLE(d,f)" has a circular shape of diameterd
, 
APERTURE
="ELLIPSE(a,b,f)" has an elliptical shape of majora
and minorb
.
The option
SQUARE
(a,f
) is equivalent toRECTANGLE
(a,a,f
) andCIRCLE
(d,f
) is equivalent toELLIPSE
(d,d,f
). The size of the exit aperture is scaled by a factorf
. Forf < 1
the exit aperture is smaller than the entrance aperture, forf = 1
they are the same and forf > 1
the exit aperture is bigger.Dipoles have
GAP
andHGAP
which define an aperture and hence do not recogniseAPERTURE
. The aperture of the dipoles has rectangular shape of heightGAP
and widthHGAP
. 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

Xcomponent of the position of the element in the laboratory coordinate system.
 Y

Ycomponent of the position of the element in the laboratory coordinate system.
 Z

Zcomponent of the position of the element in the laboratory coordinate system.
 THETA

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

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

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

3D position vector. An alternative to using
X
,Y
andZ
to position the element. Can’t be combined withTHETA
andPHI
. UseORIENTATION
instead.  ORIENTATION

Vector of TaitBryan angles bib.taitbryan. An alternative to rotate the element instead of using
THETA
,PHI
andPSI
. Can’t be combined withX
,Y
andZ
, useORIGIN
instead.  DX

Error on xcomponent of position of element. Doesn’t affect the design trajectory.
 DY

Error on ycomponent of position of element. Doesn’t affect the design trajectory.
 DZ

Error on zcomponent 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 (OPALt)
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 OPALt. In OPALt 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 OPALt, see RBend (OPALt)), SBEND
(valid in OPALt, see
SBend (OPALt)), RBEND3D
, (valid in OPALt, see RBend3D (OPALt))
and SBEND3D
(valid in OPALcycl, see SBend3D (OPALcycl)).
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 (OPALt) and SBend (OPALt) describe the geometry and attributes of the OPALt bend elements
RBEND
andSBEND
. 
RBend and SBend Examples (OPALt) describes how to implement an
RBEND
orSBEND
in an OPALt simulation. 
SBend3D (OPALcycl) is self contained. It describes how to implement an
SBEND3D
element in an OPALcycl 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.
1.4.1. RBend (OPALt)
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

Nonbend 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
, whereE1
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}
, whereR
is the bend radius as defined in Illustration of a general rectangular bend (RBEND
) with a positive bend angle\alpha
.. Not supported in OPALt any more. Superimpose aQuadrupole
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 anRBEND
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 isFMAPN
=1DPROFILE1DEFAULT
see_Default Field Map (OPALt). The field map is used to describe the fringe fields of the magnet see1DProfile1
.
1.4.2. RBend3D (OPALt)
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

Nonbend 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
, whereE1
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}
, whereR
is the bend radius as defined in Illustration of a general rectangular bend (RBEND
) with a positive bend angle\alpha
.. Not supported in OPALt any more. Superimpose aQuadrupole
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 anRBEND3D
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 isFMAPN
=1DPROFILE1DEFAULT
see Default Field Map (OPALt). The field map is used to describe the fringe fields of the magnet1DProfile1
.
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.
1.4.3. SBend (OPALt)
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

Nonbend 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 to2 \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}
, whereR
is the bend radius as defined in Illustration of a general sector bend(SBEND
) with a positive bend angle\alpha
. Not supported in OPALt any more. Superimpose aQuadrupole
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 isFMAPN
=1DPROFILE1DEFAULT
see_Default Field Map (OPALt). The field map is used to describe the fringe fields of the magnet see1DProfile1
.
1.4.4. RBend and SBend Examples (OPALt)
Describing an RBEND
or an SBEND
in an OPALt 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 (OPALt) and SBend (OPALt), and an SBEND
has an
additional attribute E2
that has no effect on an RBEND
, see
SBend (OPALt). 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 OPALt simulation, it is
important to note the following:

Internally OPALt 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 nonzero value, theK0
andK0S
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 theDESIGNENERGY
. 
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,
1DPROFILEDEFAULT
see Default Field Map (OPALt), 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 OPALt 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 (OPALt) for
describing the magnet fringe fields see 1DProfile1
:
Bend: RBend, ANGLE = 30.0 * Pi / 180.0, FMAPFN = "1DPROFILE1DEFAULT", 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 "1DPROFILE1DEFAULT"
see Default Field Map (OPALt) 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 = "1DPROFILE1DEFAULT", ELEMEDGE = 0.25, DESIGNENERGY = 10.0E6, L = 0.5, GAP = 0.02;
The output from OPAL now reads as follows:
RBend > Reference Trajectory Properties RBend > =============================== RBend > RBend > Bend angle magnitude: 0.523599 rad (30 degrees) RBend > Entrance edge angle: 0 rad (0 degrees) RBend > Exit edge angle: 0.523599 rad (30 degrees) RBend > Bend design radius: 0.999999 m RBend > Bend design energy: 1e+07 eV RBend > RBend > Bend Field and Rotation Properties RBend > ================================== RBend > RBend > Field amplitude: 0.0350195 T RBend > Field index (gradient): 0 m^1 RBend > Rotation about x axis: 0 rad (0 degrees) RBend > Rotation about y axis: 0 rad (0 degrees) RBend > Rotation about z axis: 0 rad (0 degrees) RBend > RBend > Reference Trajectory Properties Through Bend Magnet with Fringe Fields RBend > ====================================================================== RBend > RBend > Reference particle is bent: 0.5236 rad (30.0001 degrees) in x plane RBend > Reference particle is bent: 0 rad (0 degrees) in y plane
The output is effectively identical, to within a small numerical error.
Now, let us modify this first example so that we bend instead in the negative x direction. There are several ways to do this:
1.
Bend: RBend, ANGLE = 30.0 * Pi / 180.0, FMAPFN = "1DPROFILE1DEFAULT", ELEMEDGE = 0.25, DESIGNENERGY = 10.0E6, L = 0.5, GAP = 0.02;
2.
Bend: RBend, ANGLE = 30.0 * Pi / 180.0, FMAPFN = "1DPROFILE1DEFAULT", ELEMEDGE = 0.25, DESIGNENERGY = 10.0E6, L = 0.5, GAP = 0.02, ROTATION = Pi;
3.
Bend: RBend, K0 = 0.0350195, FMAPFN = "1DPROFILE1DEFAULT", ELEMEDGE = 0.25, DESIGNENERGY = 10.0E6, L = 0.5, GAP = 0.02;
4.
Bend: RBend, K0 = 0.0350195, FMAPFN = "1DPROFILE1DEFAULT", 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 = "1DPROFILE1DEFAULT", 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 = "1DPROFILE1DEFAULT", 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 = "1DPROFILE1DEFAULT", 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 = "1DPROFILE1DEFAULT", 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 OPALt bends. In fact, by adjusting the L
attribute according to RBend (OPALt) and SBend (OPALt), 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 = "TESTMAP.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 nonzero
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 TESTMAP.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 = "TESTMAP.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 = "TESTMAP.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 TESTMAP.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 = "TESTMAP.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 (OPALt)
1DProfile1
field map described in Default Field Map (OPALt). 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 OPALt 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 midplane 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 midplane, 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 (OPALt). 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.) OPALt 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 midplane} \\
\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 OPALt 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, OPALt 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 (OPALt)
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, 1DPROFILE1DEFAULT
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 1DPROFILE1DEFAULT
is
set to 2.0 cm. This value can be overridden by the GAP
attribute of the
magnet (see RBend (OPALt) and SBend (OPALt)).
1.4.7. SBend3D (OPALcycl)
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 OPALcycl.
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 OPALcycl) 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 4.3586435e01 5.0000000e02 1.2803431e+00 0.0000000e+00 1.6214000e+00 0.0000000e+00 4.2691532e01 5.0000000e02 1.2833548e+00 0.0000000e+00 1.6214000e+00 0.0000000e+00 4.1794548e01 5.0000000e02 1.2863039e+00 0.0000000e+00 1.6214000e+00 0.0000000e+00
This is a restricted feature for OPALcycl.
1.5. Quadrupole (OPALt)
label:QUADRUPOLE, TYPE=string, APERTURE=realvector, L=real, K1=real, K1S=real;
The reference system for a quadrupole is a Cartesian coordinate system This is a restricted feature for OPALt.
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, ifB_y
is positive on the positivex
axis. This implies horizontal focusing of positively charged particles which travel in positives
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, ifB_x
is positive on the positivex
axis.
Example:
QP1: Quadrupole, L=1.20, ELEMEDGE=0.5265, FMAPFN="1T1.T7", K1=0.11;
1.6. Sextupole (OPALt)
label: SEXTUPOLE, TYPE=string, APERTURE=realvector, 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, ifB_y
is positive on thex
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, ifB_x
is positive on thex
axis.
Example:
S:SEXTUPOLE, L=0.4, K2=0.00134;
The reference system for a sextupole is a Cartesian coordinate system
1.7. Octupole (OPALt)
label:OCTUPOLE, TYPE=string, APERTURE=realvector, 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, ifB_y
is positive on the positivex
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, ifB_x
is positive on the positivex
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 (OPALt)
A MULTIPOLE
in OPALt is of arbitrary order.
label:MULTIPOLE, TYPE=string, APERTURE=realvector, L=real, KN=realvector, KS=realvector;
 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, ifB_y
is positive on the positivex
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, ifB_x
is positive on the positivex
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 (OPALt).
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 (OPALt) when implemented)
A MULTIPOLET
is in OPALt 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/0d3fc561b57e8962ed79a57cd6115e37/8FBB32A47FA14084A4A7CDDB1F949CD3_psi.ch.pdf.
label:MULTIPOLET, L=real, ANGLE=real, VAPERT=real, HAPERT=real, LFRINGE=real, RFRINGE=real, TP=realvector, 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 (nonbend 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 midplane 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 ofz
in the field expansion of vertical componentB_z
to2 \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 nonzero 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 recalculated. 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)
FrenetSerret 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
Midplane symmetry is assumed and the vertical component of the field on the midplane 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 (OPALt)
label:SOLENOID, TYPE=string, APERTURE=realvector, 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 positiveKS
and positive particle charge, the solenoid field points in the direction of increasings
.
The reference system for a solenoid is a Cartesian coordinate system Using a solenoid in OPALt 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 (OPALcycl)
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 frequencyf_{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=1.06E6;
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={1.06E6, 3.96E6,1.3E6,1.E6}, 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 (OPALt and OPALcycl) 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 trimcoil 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 PSIBFIELD, PSIPHASE or PSIBFIELDMIRRORED trim coil descriptions. The general PSIBFIELD and PSIPHASE 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 PSIBFIELDMIRRORED 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 PSIBFIELDMIRRORED)
 PHIMAX

Maximal azimuth [deg] (default 360) (not for PSIBFIELDMIRRORED)
 BMAX

Maximal B field of the trim coils [T] (for PSIBFIELD) or maximal phase shift (for PSIPHASE)
 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 PSIBFIELDMIRRORED).
 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 PSIBFIELDMIRRORED).
 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 PSIBFIELDMIRRORED).
 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 PSIBFIELDMIRRORED).
 SLPTC

Slopes of the rising edge [1/mm] (for PSIBFIELDMIRRORED type only)
Example:
tc1: TRIMCOIL, TYPE="PSIBFIELDMIRRORED", RMIN = 2022.09, RMAX = 2132.09, BMAX=2.0e4, SLPTC=1; tc15: TRIMCOIL, TYPE="PSIBFIELD", RMIN = 3000, RMAX = 4500, BMAX=13e4, 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: OPALcycl.
1.12. Ring Definition (OPALcycl)
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 midplane 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 OPALt. It’s only purpose in OPALt is to
indicate that the particle source is contained in the beamline. This is
needed to place the elements in threedimensional space when using
ELEMEDGE
. Otherwise it has no effect on the particles.
1.14. RF Cavities (OPALt and OPALcycl)
For an RFCAVITY
the three flavours have four real attributes in common:
label:RFCAVITY, APERTURE=realvector, 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 OPALt this phase is in general relative to the phase at which the reference particle gains the most energy. This phase is determined using an autophasing algorithm (see Appendix Autophasing Algorithm). This autophasing algorithm can be switched off, see
APVETO
.
1.14.1. OPALt mode
Using a RF Cavity in OPALt 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 autophased. Instead the phase of the cavity is equal toLAG
at the arrival time of the reference particle (arrival at the limit of its field not atELEMEDGE
).  DESIGNENERGY

The kinetic energy that the refernce particle should reach at the exit of the cavity [MeV]. Currently only works in conjuction with the autophasing 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=1.0e6;
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. OPALcycl mode
Using a RF Cavity (standing wave) in OPALcycl 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.
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 OPALcycl 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 softedged 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

Efold 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=9.02542e10; REAL rf_p2=1.96663e16; REAL rf_p3=2.45909e23; 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 (OPALt)
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 mth Bessel function of the first kind, x_{mn}
is the nth 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 nth 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 autophased. Instead the phase of the cavity is equal toLAG
at the arrival time of the reference particle (arrival at the limit of its field not atELEMEDGE
).
1.17. Traveling Wave Structure (_OPAL_t)
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 onaxis field of an Sband (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.
OPALt 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_{frommap}}(\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_{frommap}}(\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 OPALt.
The field of the main accelerating structure is reconstructed from the
center section of the standing wave solution shown in
The onaxis field of an Sband (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_{frommap}} (x,y,z) \cdot \cos \biggl( 2 \pi \cdot \mathrm{FREQ} \cdot t + \mathrm{LAG}+ \frac{\pi}{2} \cdot \mathrm{MODE} \Bigr) + \\
& \mathbf{E_{frommap}}(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_{frommap}}(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=realvector, 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 OPALt 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 OPALt this phase is in general relative to the phase at which the reference particle gains the most energy. This phase is determined using an autophasing algorithm (see Appendix Autophasing Algorithm). This autophasing 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 onaxis 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 autophased. Instead the phase of the cavity is equal toLAG
at the arrival time of the reference particle (arrival at the limit of its field not atELEMEDGE
).
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 LBand traveling wave structure:
lrf0: TravelingWave, L=0.0253, VOLT=14.750, NUMCELLS=40, ELEMEDGE=2.73066, FMAPFN="INLB02RAC.Ez", MODE=1/3, FREQ=1498.956, LAG=248.0/360.0;
1.18. SLACs Transverse Deflectinc Structure (OPALt)
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 (OPALt)
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 OPALt 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 OPALt.
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=realvector, L=real, XSIZE=real, YSIZE=real; label:RCOLLIMATOR,TYPE=string, APERTURE=realvector, L=real, XSIZE=real, YSIZE=real; label:FLEXIBLECOLLIMATOR, APERTURE=realvector, 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. OPALt mode
The CCOLLIMATOR
isn’t supported. ECOLLIMATOR
s and RCOLLIMATOR
s
detect all particles which are outside the aperture defined by XSIZE
and YSIZE
.
For elliptic apertures, XSIZE
and YSIZE
denote the halfaxes
respectively, for rectangular apertures they denote the halfwidth of
the rectangle.
 XSIZE

The horizontal halfaperture (default: unlimited).
 YSIZE

The vertical halfaperture (default: unlimited).
Example:
Col:ECOLLIMATOR, L=1.0E3, ELEMEDGE=3.0E3, XSIZE=5.0E4, YSIZE=5.0E4, OUTFN="Coll.h5";
The FLEXIBLECOLLIMATOR
can be used to model both simple, rectangular or elliptic collimators and more complex devices like pepperpots. 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 xdirection and shifty in the ydirection. 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.
 symmetric_difference(command1, command2)

All particles that pass either command but not both at the same time.
 intersection(command1, command2)

All particles that pass both commands at the same time.
 repeat(command, N, shiftx, shifty)

Repeats the holes that are defined by the command translating each copy successively by shiftx in xdirection and shifty in ydirection. 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.
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 pepperpot with rectangular holes can be define like this
repeat( // repeat it in ydirection repeat( // repeat it in xdirection 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
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. OPALcycl mode
Only CCOLLIMATOR
is available for OPALcycl. 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 ofPARTICLEMATTERINTERACTION
isCOLLIMATOR
, 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.
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 (OPALcycl)
This is a restricted feature for OPALcycl. 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.
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 (OPALcycl)
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.
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 (OPALcycl)
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 (OPALt)
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 (OPALt)
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=realvector, L=real, KICK=real; label:VKICKER, TYPE=string, APERTURE=realvector, L=real, KICK=real; label:KICKER, TYPE=string, APERTURE=realvector, 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
orVKICK
). If the design energy isn’t set then the actual energy of the reference particle at the position of the corrector is used. TheDESIGNENERGY
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.taitbryan] Taitbryan angles.

[usercontentbib.enge_elements] J. E. Spencer and H. A. Enge, Splitpole magnetic spectrograph for precision nuclear spectroscopy, Nucl. Instrum. Methods 49, 181–193 (1967).