10.4 BGGEXP—A magnetic field element using generalized gradient expansion.

A magnetic field element using generalized gradient expansion.
Parallel capable? : yes
GPU capable? : no
Back-tracking capable? : no

A magnetic field element using generalized gradient expansion.

A magnetic field element using generalized gradient expansion.

This element simulates transport through a 3D magnetic field specified in terms of a generalized gradient expansion [50]. After reconstructing the field, it simply integrates the equations of motion based on the Lorentz force equation in cartesian coordinates.

The generalized gradients are provided in SDDS files. In addition to several columns describing the gradients, the file must contain a parameter:

• m — The multipole index, using the convention where m = 0 is solenoid, m = 1 is dipole, m = 2 is quadrupole, etc. N.B.: this convention conforms with [50] but is not the usual one used by elegant. This should be stored as a short integer.

The files may also include optional parameters xCenter and yCenter giving the center of the expansion in meters.

In the original implementation, which is still supported, only normal field components were included In that case, the user should use the FILENAME field to provide a file with the following floating-point columns:

• z — Longitudinal coordinate. Units should be “m”.
• Cnmn — The n^th generalized gradient of the m^th harmonic, where n = 0,2,4,.... There is no preset limit to the number of generalized gradients. Units are ignored, but should be SI.
• dCnmn/dz — The longitudial derivative of the n^th generalized gradient, for the m^th harmonic, where n = 0,2,4,.... The number of derivatives must match the number of generalized gradients Cnmn.

The field expansion in this case is

(30)

where it is understood that the expansion is about the xCenter and yCenter values, if given.

Note that there is potential confusion between the xCenter parameter in the input files and the DXEXPANSION parameter in the element definition. These provide similar functionality and only one is needed. Both give the position of the horizontal center of the expansion relative to the magnetic field coordinate system.

In version 2020.5 and later, both normal and skew expansions are supported. In this case, the user may provide filenames via the NORMAL_FILENAME and SKEW_FILENAME fields. In this, case, the files must contain the following floating-point columns:

• z — Longitudinal coordinate. Units should be “m”.
• CnmSn (normal) or CnmCn (skew) — The n^th generalized gradient of the m^th harmonic, where n = 0,2,4,.... There is no preset limit to the number of generalized gradients. Units are ignored, but should be SI. Note that the “S” in the name for the normal components make be confusing. It refers to the fact that these terms appear in the potential with a factor of sinmϕ, whereas the skew terms have cosmϕ factors (hence the “C”).
• dCnmSn/dz (normal) or dCnmCn/dz (skew) — The longitudial derivative of the n^th generalized gradient, for the m^th harmonic, where n = 0,2,4,.... The number of derivatives must match the number of generalized gradients Cnmn.

The field expansion in this case is

(31)

where it is understood that the expansion is about the xCenter and yCenter values, if given. Users should note that the skew field sign convention used by [50] and BGGEXP differs from that used in elegant. In particular, to convert a normal field to a skew field while conforming to elegant’s conventions, one must use C_m,s^p →-C_m,c^p and dC_m,s^p∕dz →-dC_m,c^p∕dz.

Data for use with BGGEXP can be prepared with the programs computeCBGGE (section 8.3) and computeRBGGE (section 8.4), which are distributed with elegant.

Synchrotron radiation can be included by setting SYNCH_RAD=1 for classical radiation only and also ISR=1 for incoherent (quantum) effects. This will impact the results of moments_output calculation as well as tracking.

Important notes and limitations:

1. The calculations of twiss_output, including radiation integrals, are at this point not affected, nor is the setup of rf cavities for storage rings via the rf_setup command.
2. The symplectic integrator, in addition to being symplectic, is typically more accurate than the non-symplectic integrator. It is also considerably slower. However, at minimum, users should use the symplectic integrator to verify that the accuracy of the non-symplectic integrator is adequate.
3. The BX and BY parameters allow imposing uniform horizontal and vertical magnetic fields on the device. This can be helpful if the terminal trajectory deviates from the expected value, e.g., an on-axis particle ends up off-axis. This may happen if the device has a dipolar field that is truncated at the ends before it has decayed sufficiently. Note that these values are multiplied by the STRENGTH factor before being applied to the beam.

In addition to the STRENGTH factor, there are five parameters that can be used to scale multipoles of different orders: FACTOR0, FACTOR1, FACTOR2, FACTOR3, and FACTOR4 scale the solenoidal, dipolar, quadrupolar, sextupolar, and octupolar fields, respectively. The BXFACTOR, BYFACTOR, and BZFACTOR allow multiplying the indicated field components by the given factors. The the exception of the STRENGTH factor, these scaling parameters may result in unphysical fields.

If IS_BEND is non-zero, the magnet is assumed to be a bending magnet, in which case additional parameters are required.

• ZVERTEX, XVERTEX — Coordinates of the vertex point in coordinate frame of the field data. For a symmetric dipole, ZVERTEX is typically zero, while XVERTEX would be the displacement of the vertex point from the cylinder axis.
• ZENTRY, XENTRY — Coordinates of the nominal entry plane.
• ZEXIT, XEXIT — Coordinates of the nominal exit plane.