Beam matrix tracing

Computes beam matrices at selected points of the machine from the initial beam matrix defined in the input of the operation. The equation of the six dimensional ellipsoid enclosing the beam is $X^{t}\Sigma^{-1}X = 1$

 Input  format 


BEAM matrix tracing.... (up to 80 characters)

followed by :

$$\begin{array}{cccccc} \sigma _{x}&r_{xp_{x}}&r_{xy}&r_{xp_{y... ...\\ &&&&&\sigma_{\delta} \\ mprint& [mlist]&&&& \end{array}$$

or :

$$\begin{array}{ccccc} 0&&&& \\ \beta_{x}&\alpha_{x}&\eta_{x}&... ...gma_{l}&\sigma_{\delta}&&& \\ mprint& [mlist]&&& \end{array}$$

or :

$$\begin{array}{ccccc} 0&&&& \\ 0&0&0&0&\epsilon_{x} \\ 0&0... ...gma_{l}&\sigma_{\delta}&&& \\ mprint& [mlist]&&& \end{array}$$

Parameter definitions

The relationship between the $\Sigma$ beam matrix and the input coefficients is as follows :

Let $\Sigma_{ij}$, with $i$ and $j$ taking the values $x$, $p_{x}$, $y$, $p_{y}$, $l$ and $\delta$, be the elements of the symmetric positive definite matrix $\Sigma$. Then the following formulae describe the relationship between $\Sigma$ and the input parameters :

$$\begin{array}{rcll} \sigma_{i}&=&\sqrt{\Sigma_{ii}}&{\rm =\ v... ...gma_{i}\sigma_{j}}&{\rm\ =\ correlation\ factor } \end{array}$$

$\beta \alpha$ $\parbox{9cm}{Twiss parameter values defining a beam ellipsoid}$

$\eta$ $\parbox{10 cm}{$\eta$\ -function values to define momentum dependent terms in the beam ellipsoid}$

$\epsilon_{x}$ and $\epsilon_{y}$ are the horizontal and vertical emittances. In this case, the beam is assumed uncoupled .

NOTE :$\parbox{10cm}{ when using the third input option ($\beta_{x}$\ is 0... ...om a previously run movement analysis calculation done with a matrix operation}$

mprint $\parbox{10cm}{\ \ \newline \par -2\ \ no computation is done. The op... ...ove but a table of beam envelopes is printed instead of the full beam matrix.}$

mlist $\parbox{10cm}{contains the beginning and end of all intervals in wh... ...der list of machine elements) List may contain up to mxlist pairs of numbers.}$

Units

$\sigma_{x},\ \sigma_{y},\ \sigma_{l},\ \beta_{x},\ \beta_{y}, \ \eta_{x},\ \eta_{y}$ are measured in meters

$\sigma_{p_{x}},\ \sigma_{p_{y}},\ \eta_{p_{x}},\ \eta_{p_{y}}$ are measured in radians.

$\epsilon_{x},\ \epsilon_{y}$ are measured in m-radians.

$\sigma_{\delta}$ unit is one.

The correlation coefficients $r_{ij}$ are dimensionless and vary from $-1$ to $+1$

Examples

Three examples are given. The first is extracted from demo 7 , the second and third are extracted from demo 2.

The first shows the definition of the beam using the extension $\sigma$ 's.

The second defines the transverse part of the beam by using the twiss parameters and the emittances.

The third shows a definition used in conjunction with a previous matrix analysis. It uses the twiss parameters obtained in that analysis and the value of the tranverse emittances.

BEAM MATRIX TRACING
0.00012157290 0 0 0 0 0
  0.00000246750 0 0 0 0
    0.0000826254 0 0 0
       0.0000036309 0 0
                    0.002 0
                      0.005
-1,

   beam definition
   0
   1.0 0 0 0 1.0e-06
   2.0 0 0 0 1.0e-06
   0.001 0.001
   0,

MATRIX : FIRST ORDER CELL MATRIX.
1 -1,
BEAM MATRIX
0
0 0 0 0 1.0E-06
0 0 0 0 1.0E-06
0.02 0.001
-1;