Normalised Variables and other Derived Quantities

XN

The normalised horizontal displacement [ $\mathrm{m}^{1/2}$]: $\mathtt{XN} = x_n = \Re(E_1^T S Z)$.

PXN

The normalised horizontal transverse momentum [ $\mathrm{m}^{1/2}$]: $\mathtt{PXN} = p_{xn} = \Im(E_1^T S Z$).

WX

The horizontal Courant-Snyder invariant [m]: $\mathtt{WX} = \sqrt{x_n^2 + p_{xn}^2}$.

PHIX

The horizontal phase: $\mathtt{PHIX} = - \arctan(p_{xn} / x_n) / 2 \pi$.

YN

The normalised vertical displacement [ $\mathrm{m}^{1/2}$]: $\mathtt{YN} = y_n = \Re(E_2^T S Z)$.

PYN

The normalised vertical transverse momentum [ $\mathrm{m}^{1/2}$]: $\mathtt{PYN} = p_{yn} = \Im(E_2^T S Z)$.

WY

The vertical Courant-Snyder invariant [m]: $\mathtt{WY} = \sqrt{y_n^2 + p_{yn}^2}$.

PHIY

The vertical phase: $\mathtt{PHIY} = - \arctan(p_{yn} / y_n) / 2 \pi$.

TN

The normalised longitudinal displacement [ $\mathrm{m}^{1/2}$]: $\mathtt{TN} = t_n = \Re(E_3^T S Z)$.

PTN

The normalised longitudinal transverse momentum [ $\mathrm{m}^{1/2}$]: $\mathtt{PTN} = p_{tn} = Im(E_3^T S Z)$.

WT

The longitudinal invariant [m]: $\mathtt{WT} = \sqrt{t_n^2 + p_{tn}^2}$.

PHIT

The longitudinal phase: $\mathtt{PHIT} = + \arctan(p_{tn} / t_n) / 2 \pi$.

in the above formulas $Z$ is the phase space vector $Z = (x, p_x, y, p_y, t, p_t)^T$. The matrix $S$ is the ``symplectic unit matrix''

$$S = \left( \begin{array}{cccccc} 0 & 1 & 0 & 0 & 0 & 0 \\ ... ... & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 &-1 & 0 \end{array} \right ),$$

and the vectors $E_i$ are the three complex eigenvectors. The superscript $T$ denotes the transpose of a vector or matrix.