Lattice Function Tables

Two commands are available to generate tables of lattice functions:

TWISS,LINE=name,BEAM=name,RANGE=range,STATIC=logical,METHOD= name,
      REVBEAM=logical,REVTRACK=logical;
TWISSTRACK,BEAM=name,RANGE=range,STATIC=logical,METHOD= name,
      INIT=table-row,REVBEAM=logical,REVTRACK=logical;

Either command builds a ``twiss'' table containing the lattice functions for a range of a previously defined beam line. The first command, TWISS, uses the periodic solution for that range, while the second command, TWISSTRACK a selected line of another twiss table. The table can be referred to by its label for later manipulation. Both commands have the following read/write attributes:

LINE

The label of a previously defined beam line (no default).

BEAM

The label of a previously defined beam (default UNNAMED_BEAM). This is used to define the type and reference momentum for the particles to be sent through the line.

RANGE

If this attribute is given, the table is restricted to the range given.

STATIC

If this attribute is true, the table is filled at definition time, and it is then treated as frozen, even if parameters of the machine change.

METHOD

This attribute specifies the method to be used for filling the table. Known names are:

THIN

Use thin lens approximations.

LINEAR

Use linear map approximation (around the closed orbit). This is the default.

THICK

Use finite-length lenses.

REVBEAM

If true, the beam is assumed to run backwards through the beam line (from $s=C$ to $s=0$).

REVTRACK

If true, the calculation proceeds in the direction opposite to the beam direction. This attribute may be used in matching, when tracking lattice functions from a known position back to a position to be used to adjust an insertion.

The TWISS command has the following read-only attributes:

L

The total arc length [m].

FREQ0

The computed revolution frequency in MHz.

QX

The computed tune for mode 1.

QY

The computed tune for mode 2.

QS

The computed tune for mode 3.

U0

The computed energy loss per turn [MeV].

JX

The computed damping partition number for mode 1.

JY

The computed damping partition number for mode 2.

JE

The computed damping partition number for mode 3.

The values related to damping exist only when the flag DAMP has been set.

The TWISSTRACK can be initialised by the following attribute:

INIT

If this attribute is given, the initial position and direction is taken from the specified row of another TWISS table.

Example:

LATFUN:TWISS,LINE=CELL;

This example computes the lattice functions for CELL with zero initial conditions and saves the result under the name LATFUN.

A table generated by TWISS (not TWISSTRACK) also has some read-only attributes:

Table 7.3: Global read-only values in a TWISS table

Variable	Unit	Name
Revolution frequency	Hz	FREQ
Tune for mode 1	1	QX
Tune for mode 2	1	QY
Tune for mode 3	1	QS
Energy loss per turn	MeV	U0
Damping partition number for mode 1	1	JX
Damping partition number for mode 2	1	JY
Damping partition number for mode 3	1	JE

In each the TWISS table stores the closed orbit and the transfer map from the beginning to this position. Several column values, as well as the eigenvectors, second moments, and transfer matrices can be derived from this information These quantities are accessible with the syntax

table-name "@" place "->" column-name

Table 7.4: Column values in a TWISS table

Variable	Unit	Name
Arc length from beginning	m	S
Horizontal position for closed orbit	m	XC
Horizontal momentum for closed orbit	1	PXC
Vertical position for closed orbit	m	YC
Vertical momentum for closed orbit	1	PYC
Longitudinal position for closed orbit	m	TC
Longitudinal momentum for closed orbit	1	PTC
Dispersion for horizontal position	m	DX
Dispersion for horizontal momentum	1	DPX
Dispersion for vertical position	m	DY
Dispersion for vertical momentum	1	DPY
Dispersion for longitudinal position	m	DT
Dispersion for longitudinal momentum	1	DPT

		Plane
Courant-Snyder Functions	Unit	X	Y	T
Beta-function	m	BETX	BETY	BETT
Alpha-function	1	ALFX	ALFY	ALFT
Phase	1	MUX	MUY	MUT

Table 7.5: Some of the column values in a TWISS3 table

			Plane
Mais-Ripken Functions	Unit	Mode	X	Y	T
		1	BETX1	BETY1	BETT1
Beta-function	m	2	BETX2	BETY2	BETT2
		3	BETX3	BETY3	BETT3
		1	ALFX1	ALFY1	ALFT1
Alpha-function	1	2	ALFX2	ALFY2	ALFT2
		3	ALFX3	ALFY3	ALFT3
		1	GAMX1	GAMY1	GAMT1
Gamma-function	1/m	2	GAMX2	GAMY2	GAMT2
		3	GAMX3	GAMY3	GAMT3

The Courant-Snyder lattice functions are calculated in a way analogous to the lattice functions given by Edwards and Teng. After partitioning the eigenvector matrix into two by two blocks it can be factored as

$$E = \left( \begin{array}{lll} E_{11} & E_{12} & E_{13} \\ ... ...W_1 & 0 & 0 \\ 0 & W_2 & 0 \\ 0 & 0 & W_3 \end{array}\right),$$

where the $r_i$ are scalars and $R$ and $W$ are symplectic matrices. This implies

$$r_i = \sqrt{\vert R_{ii}\vert}, \qquad W_i = R_{ii} / r_i \rightarrow \vert W_i\vert = 1, \qquad R_{ik} = E_{ik} W_i^{-1}.$$

The lattice functions are calculated from the diagonal blocks $W_i$. Note that the matrix $R$ which defines the coupling between planes is not available for output, and that the formalism breaks down when $r_i=0$. The default column set contains the element name, the arc length, the beta, alpha and mu functions for the transverse planes, the closed orbit position, and the horizontal dispersion.

The Mais-Ripken functions have been introduced by Mais and Ripken.

The second moments (beam envelope) are defined in the sense of TRANSPORT:

$$\Sigma_{ik} = E_x ( E_{i1} E_{k1} + E_{i2} E_{k2}) + E_y (... ...{k3} + E_{i4} E_{k4}) + E_t ( E_{i5} E_{k5} + E_{i6} E_{k6}).$$

Table 7.6: Eigenvector components in a EIGEN table

Eigenvector		Mode 1		Mode 2		Mode 3
Component	Unit	real	imag	real	imag	real	imag
X-component	m	E11	E12	E13	E14	E15	E16
PX-component	1	E21	E22	E23	E24	E25	E26
Y-component	m	E31	E32	E33	E34	E35	E36
PY-component	1	E41	E42	E43	E44	E45	E46
T-component	m	E51	E52	E53	E54	E55	E56
PT-component	1	E61	E62	E63	E64	E65	E66

Table 7.7: Second moments <X1,X2> in a ENVELOPE table

Second moment	Second variable X2
First variable X1	X	PX	Y	PY	T	PT
X	S11	S12	S13	S14	S15	S16
PX	S21	S22	S23	S24	S25	S26
Y	S31	S32	S33	S34	S35	S36
PY	S41	S42	S43	S44	S45	S46
T	S51	S52	S53	S54	S55	S56
PT	S61	S62	S63	S64	S65	S66

Table 7.8: Transfer matrix elements (X2,X1) in a MATRIX table

Matrix Element	Second variable X2
First variable X1	X	PX	Y	PY	T	PT
X	R11	R12	R13	R14	R15	R16
PX	R21	R22	R23	R24	R25	R26
Y	R31	R32	R33	R34	R35	R36
PY	R41	R42	R43	R44	R45	R46
T	R51	R52	R53	R54	R55	R56
PT	R61	R62	R63	R64	R65	R66