An Individualized Linear Matrix for each particle for fast symplectic tracking with chromatic and
amplitude-dependent effects
Parallel capable? : yes
GPU capable? : no
Back-tracking capable? : no
| Parameter Name | Units | Type | Default | Description |
| L | M | double | 0.0 | Length (used for position and time-of-flight computation) |
| NUX | double | 0.0 | Horizontal tune | |
| NUY | double | 0.0 | Vertical tune |
|
| NUX1M | double | 0.0 | First chromatic derivative of the horizontal tune |
|
| NUY1M | double | 0.0 | First chromatic derivative of the vertical tune |
|
| NUX2M | double | 0.0 | Second chromatic derivative of the horizontal tune |
|
| NUY2M | double | 0.0 | Second chromatic derivative of the vertical tune |
|
| NUX3M | double | 0.0 | Third chromatic derivative of the horizontal tune |
|
| NUY3M | double | 0.0 | Third chromatic derivative of the vertical tune |
|
| NUX1AX | 1∕M | double | 0.0 | First amplitude derivative of the horizontal tune wrt Ax |
| NUY1AX | 1∕M | double | 0.0 | First amplitude derivative of the vertical tune wrt Ax |
| NUX1AY | 1∕M | double | 0.0 | First amplitude derivative of the horizontal tune wrt Ay |
| NUY1AY | 1∕M | double | 0.0 | First amplitude derivative of the vertical tune wrt Ay |
| NUX2AX | 1∕M2 | double | 0.0 | Second amplitude derivative of the horizontal tune wrt Ax |
| NUY2AX | 1∕M2 | double | 0.0 | Second amplitude derivative of the vertical tune wrt Ax |
| NUX2AY | 1∕M2 | double | 0.0 | Second amplitude derivative of the horizontal tune wrt Ay |
| NUY2AY | 1∕M2 | double | 0.0 | Second amplitude derivative of the vertical tune wrt Ay |
ILMATRIX continued
An Individualized Linear Matrix for each particle for fast symplectic tracking with chromatic and
amplitude-dependent effects
| Parameter Name | Units | Type | Default | Description |
| NUX1AX1AY | 1∕M2 | double | 0.0 | Amplitude derivative of the horizontal tune wrt Ax and Ay |
| NUY1AX1AY | 1∕M2 | double | 0.0 | Amplitude derivative of the vertical tune wrt Ax and Ay |
| BETAX | M | double | 0.0 | On-momentum horizontal beta function |
| BETAY | M | double | 0.0 | On-momentum vertical beta function |
| BETAX1M | M | double | 0.0 | First chromatic derivative of horizontal beta function |
| BETAY1M | M | double | 0.0 | First chromatic derivative of vertical beta function |
| ALPHAX | double | 0.0 | On-momentum horizontal alpha function |
|
| ALPHAY | double | 0.0 | On-momentum vertical alpha function |
|
| ALPHAX1M | double | 0.0 | First chromatic derivative of horizontal alpha function |
|
| ALPHAY1M | double | 0.0 | First chromatic derivative of vertical alpha function |
|
| ETAX | M | double | 0.0 | On-momentum horizontal eta function |
| ETAPX | double | 0.0 | On-momentum horizontal eta’ function |
|
| ETAY | M | double | 0.0 | On-momentum vertical eta function |
| ETAPY | double | 0.0 | On-momentum vertical eta’ function |
|
| ETAX1 | M | double | 0.0 | First chromatic derivative of horizontal eta function |
| ETAPX1 | double | 0.0 | First chromatic derivative of horizontal eta’ function |
|
| ETAY1 | M | double | 0.0 | First chromatic derivative of vertical eta function |
| ETAPY1 | double | 0.0 | First chromatic derivative of vertical eta’ function |
|
ILMATRIX continued
An Individualized Linear Matrix for each particle for fast symplectic tracking with chromatic and
amplitude-dependent effects
| Parameter Name | Units | Type | Default | Description |
| ALPHAC | double | 0.0 | First-order momentum compaction factor |
|
| ALPHAC2 | double | 0.0 | Second-order momentum compaction factor |
|
| ALPHAC3 | double | 0.0 | Third-order momentum compaction factor |
|
| DS1AX | double | 0.0 | First amplitude derivative of the path length wrt Ax |
|
| DS1AY | double | 0.0 | First amplitude derivative of the path length wrt Ay |
|
| DS2AX | 1∕M | double | 0.0 | Second amplitude derivative of the path length wrt Ax |
| DS2AY | 1∕M | double | 0.0 | Second amplitude derivative of the path length wrt Ay |
| DS1AX1AY | 1∕M | double | 0.0 | Amplitude derivative of the path length wrt Ax and Ay |
| TILT | RAD | double | 0.0 | Rotation angle about the longitudinal axis. |
| CROSS_RESONANCE | short | 0 | If zero, then particles that cross an integer or half-integer resonance are considered lost. |
|
| VERBOSITY | short | 0 | If nonzero, then information about particle losses is printed out. |
|
| GROUP | string | NULL | Optionally used to assign an element to a group, with a user-defined name. Group names will appear in the parameter output file in the column ElementGroup |
|
This element allows fast, symplectic tracking of transport through a periodic cell with chromatic and amplitude-dependent tunes, beta functions, and dispersion. This is done by computing a linear matrix for every particle using Twiss parameters, tunes, dispersion, etc., supplied by the user. The user can also supply selected chromatic and amplitude derivatives of these quantities, which are used to compute the individual particle’s beta functions, tune, dispersion, etc., which in turn allows computing the individual particle’s linear matrix.
The starting point is the well-known expression for the one-turn linear matrix in terms of the lattice functions
|
| (54) |
where νq is the tune in the q plane. We can expand the quantities in the matrix using
 | (55) |
where δ = (p-p0)∕p0 is the fractional momentum offset, Aq = (qβ2 + (αqqβ + βqqβ′)2)∕βq is the betatron amplitude, and the betatron coordinates are computed using
 | (56) |
and
 | (57) |
At each turn, δ, Ax, and Ay are computed for each particle. The user-supplied values of the various derivatives are then used to compute the tunes for each particle. Similar expansions are used to compute the other lattice functions. This allows computing the 2x2 transfer matrices for the betatron coordinates in the x and planes, then advancing the betatron coordinates one turn, after which the full coordinates are recomputed by adding back the momentum-dependent closed orbit.
The pathlength is computed using the expansion
 | (58) |
where αc,1 is the linear momentum compaction factor. Note that in keeping with convention the higher-order momentum compaction is expressed by polynomial coefficients, not derivatives. The terms dependent on betatron amplitude are expressed in terms of the more typical derivatives. Note the difference between the R5n terms (added in version 2019.4) and those dependent on Ax,y: the former are oscillatory while the latter will accumulate. The frequency_map command can be used to compute path-length dependence on betatron amplitude.
Using this element is very similar to using the setup_linear_chromatic_tracking command. The advantage is that using LMATRIX, one can split a ring into segments and place, for example, impedance elements between the segments.
This element was inspired by requests from Y. Chae (APS).
N.B.: There is a bug related to using ILMATRIX that will result in a crash if one does not request computation of the twiss parameters. If you encounter this problem, just add the following statement after the run_setup command:
IONEFFECTS