10.41 ILMATRIX—An Individualized Linear Matrix for each particle for fast symplectic tracking with chromatic and amplitude-dependent effects

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 NameUnitsType Default

Description






L M double0.0

Length (used for position and time-of-flight computation)






NUX double0.0

Horizontal tune






NUY double0.0

Vertical tune






NUX1M double0.0

First chromatic derivative of the horizontal tune






NUY1M double0.0

First chromatic derivative of the vertical tune






NUX2M double0.0

Second chromatic derivative of the horizontal tune






NUY2M double0.0

Second chromatic derivative of the vertical tune






NUX3M double0.0

Third chromatic derivative of the horizontal tune






NUY3M double0.0

Third chromatic derivative of the vertical tune






NUX1AX 1∕M double0.0

First amplitude derivative of the horizontal tune wrt Ax






NUY1AX 1∕M double0.0

First amplitude derivative of the vertical tune wrt Ax






NUX1AY 1∕M double0.0

First amplitude derivative of the horizontal tune wrt Ay






NUY1AY 1∕M double0.0

First amplitude derivative of the vertical tune wrt Ay






NUX2AX 1∕M2double0.0

Second amplitude derivative of the horizontal tune wrt Ax






NUY2AX 1∕M2double0.0

Second amplitude derivative of the vertical tune wrt Ax






NUX2AY 1∕M2double0.0

Second amplitude derivative of the horizontal tune wrt Ay






NUY2AY 1∕M2double0.0

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 NameUnitsType Default

Description






NUX1AX1AY 1∕M2double0.0

Amplitude derivative of the horizontal tune wrt Ax and Ay






NUY1AX1AY 1∕M2double0.0

Amplitude derivative of the vertical tune wrt Ax and Ay






BETAX M double0.0

On-momentum horizontal beta function






BETAY M double0.0

On-momentum vertical beta function






BETAX1M M double0.0

First chromatic derivative of horizontal beta function






BETAY1M M double0.0

First chromatic derivative of vertical beta function






ALPHAX double0.0

On-momentum horizontal alpha function






ALPHAY double0.0

On-momentum vertical alpha function






ALPHAX1M double0.0

First chromatic derivative of horizontal alpha function






ALPHAY1M double0.0

First chromatic derivative of vertical alpha function






ETAX M double0.0

On-momentum horizontal eta function






ETAPX double0.0

On-momentum horizontal eta’ function






ETAY M double0.0

On-momentum vertical eta function






ETAPY double0.0

On-momentum vertical eta’ function






ETAX1 M double0.0

First chromatic derivative of horizontal eta function






ETAPX1 double0.0

First chromatic derivative of horizontal eta’ function






ETAY1 M double0.0

First chromatic derivative of vertical eta function






ETAPY1 double0.0

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 UnitsType Default

Description






ALPHAC double0.0

First-order momentum compaction factor






ALPHAC2 double0.0

Second-order momentum compaction factor






ALPHAC3 double0.0

Third-order momentum compaction factor






DS1AX double0.0

First amplitude derivative of the path length wrt Ax






DS1AY double0.0

First amplitude derivative of the path length wrt Ay






DS2AX 1∕M double0.0

Second amplitude derivative of the path length wrt Ax






DS2AY 1∕M double0.0

Second amplitude derivative of the path length wrt Ay






DS1AX1AY 1∕M double0.0

Amplitude derivative of the path length wrt Ax and Ay






TILT RADdouble0.0

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

     (                                            )
Rq =    cos2πνq + αqsin2πνq       βq sin2 πνq
            - γqsin 2πνq      cos2π νq - αq sin 2πνq
(52)

where νq is the tune in the q plane. We can expand the quantities in the matrix using

              (     )          (     )           (     )        (        )
          ∑3   ∂n-νq   δn-  ∑2   ∂nνq-  Anx   ∑2   ∂nνq-  Any     -∂2νq---
νq = νq,0 +      ∂δn    n! +      ∂An     n! +      ∂An     n! +   ∂Ax∂Ay    AxAy
          n=1        0      n=1     x  0      n=1     y  0                 0
(53)

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

          (     (    )   )
qβ = q - δ  ηq +  ∂ηq-  δ
                  ∂ δ  0
(54)

and

          (     (    ′)  )
q′ = q′ - δ η′+   ∂ηq-  δ
 β           q     ∂δ  0
(55)

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

         3          4            2  (    )         2 (     )        (        )
       ∑       n   ∑            ∑    -∂ns   Anx   ∑    ∂ns-   Any    ---∂2s--
Δs =  L    αc,nδ  +    R5nx β,n +     ∂Anx  0 n! +      ∂Any    n! +  ∂Ax ∂Ay    AxAy
       n=1         n=1          n=1               n=1        0                 0
(56)

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:

&twiss_output  
        matched = 1  
&end

IONEFFECTS