Variables in MAD

EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH

Variables

For each variable the physical units are listed in square brackets.

Canonical Variables Describing Orbits

MAD uses the following canonical variables to describe the motion of particles:

X: Horizontal position x of the (closed) orbit, referred to the ideal orbit [m].
PX: Horizontal canonical momentum p_x of the (closed) orbit referred to the ideal orbit, divided by the reference momentum: PX = p_x / p₀, [1].
Y: Vertical position y of the (closed) orbit, referred to the ideal orbit [m].
PY: Vertical canonical momentum p_y of the (closed) orbit referred to the ideal orbit, divided by the reference momentum: PY = p_x / p₀, [1].
T: Velocity of light times the negative time difference with respect to the reference particle: T = - c t, [m]. A positive T means that the particle arrives ahead of the reference particle.
PT: Energy error, divided by the reference momentum times the velocity of light: PT = delta(E) / p_s c, [1]. This value is only non-zero when synchrotron motion is present. It describes the deviation of the particle from the orbit of a particle with the momentum error DELTAP.
DELTAP: Difference of the reference momentum and the design momentum, divided by the reference momentum: DELTAP = delta(p) / p₀, [1]. This quantity is used to normalize all element strengths.

The independent variable is:

S: Arc length s along the reference orbit, [m].

In the limit of fully relativistic particles (gamma >> 1, v = c, p c = E), the variables T, PT used here agree with the longitudinal variables used in [TRANSPORT]. This means that T becomes the negative path length difference, while PT becomes the fractional momentum error. The reference momentum p_s must be constant in order to keep the system canonical.

Normalised Variables and other Derived Quantities

XN: The normalised horizontal displacement
XN = x_n = Re(E₁^T S Z), [sqrt(m)].
PXN: The normalised horizontal transverse momentum
PXN = x_n = Im(E₁^T S Z), [sqrt(m)].
WX: The horizontal Courant-Snyder invariant
WX = sqrt(x_n² + p_xn²), [m].
PHIX: The horizontal phase
PHIX = - atan(p_xn / x_n) / 2 pi [1].
YN: The normalised vertical displacement
YN = x_n = Re(E₂^T S Z), [sqrt(m)].
PYN: The normalised vertical transverse momentum
PYN = x_n = Im(E₂^T S Z), [sqrt(m)].
WY: The vertical Courant-Snyder invariant
WY = sqrt(y_n² + p_yn²), [m].
PHIY: The vertical phase
PHIY = - atan(p_yn / y_n) / 2 pi [1].
TN: The normalised longitudinal displacement
TN = x_n = Re(E₃^T S Z), [sqrt(m)].
PTN: The normalised longitudinal transverse momentum
PTN = x_n = Im(E₃^T S Z), [sqrt(m)].
WT: The longitudinal invariant
WT = sqrt(t_n² + p_tn²), [m].
PHIT: The longitudinal phase
PHIT = + atan(p_tn / t_n) / 2 pi [1].

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''

and the vectors E_i are the three complex eigenvectors.

Linear Lattice Functions (Optical Functions)

Several MAD commands refer to linear lattice functions. Since MAD uses the canonical momenta (p_x, p_y) instead of the slopes (x', y'), their definitions differ slightly from those in [Courant and Snyder]. Notice that in MAD-X PT substitutes DELTAP as longitudinal variable. Dispersive and chromatic functions are hence derivatives with respects to PT. Being PT=BETA*DELTAP, where BETA is the relativistic Lorentz factor, those functions must be multiplied by BETA a number of time equal to the order of the derivative. The linear lattice functions are known to MAD under the following names:

BETX: Amplitude function beta_x, [m].
ALFX: Correlation function alpha_x, [1]:
ALFX = alpha_x = - 1/2 * (del beta_x / del s).
MUX: Phase function mu_x, [2pi]:
MUX = mu_x = integral (ds / beta_x).
DX: Dispersion D_x of x, [m]:
DX = D_x = (del x / del PT).
DPX: Dispersion D_px of p_x, [1]:
DPX = D_px = (del p_x / del PT) / p_s.
BETY: Amplitude function beta_y, [m].
ALFY: Correlation function alpha_y, [1].
ALFY = alpha_y = - 1/2 * (del beta_y / del s).
MUY: Phase function mu_y, [2pi].
MUY = mu_y = integral (ds / beta_y).
DY: Dispersion D_y of y, [m]:
DY = D_y = (del y / del PT).
DPY: Dispersion D_px of p_x, [1]:
DPY = D_py = (del p_y / del PT) / p_s.
R11, R12, R21, R22: Coupling Matrix
ENERGY: The total energy per particle in GeV. If given, it must be greater then the particle mass.

Chromatic Functions

Several MAD commands refer to the chromatic functions. (p_x, p_y) instead of the slopes (x', y'), their definitions differ slightly from those in [Montague]. Notice that in MAD-X PT substitutes DELTAP as longitudinal variable. Dispersive and chromatic functions are hence derivatives with respects to PT. Being PT=BETA*DELTAP, where BETA is the relativistic Lorentz factor, those functions must be multiplied by BETA a number of time equal to the order of the derivative. The chromatic functions are known to MAD under the following names:

Please note that this option is needed for a proper calculation of the chromaticities in the presence of coupling!

WX: Chromatic amplitude function W_x, [1]:
WX = W_x = sqrt(a_x² + b_x²),

a_x = (del beta_x / del PT) / beta_x,

b_x = (del alpha_x / del PT) - (alpha_x / beta_x) * (del beta_x / del PT).
PHIX: Chromatic phase function Phi_x, [2pi]:
PHIX = Phi_x = atan(a_x / b_x).
DMUX: Chromatic derivative of phase function mu_x, [2pi]:
DMUX = (del mu_x / del PT).
DDX: Chromatic derivative of dispersion D_x, [m]:
DDX = 1/2 * (del²x / del PT²).
DDPX: Chromatic derivative of dispersion D_px, [1]:
DDPX = 1/2 * (del²p_x / del PT²) / p_s.
WY: Chromatic amplitude function W_y, [1]:
WY = W_y = sqrt(a_y² + b_y²),

a_y = (del beta_y / del PT) / beta_y,

b_y = (del alpha_y / del PT) - (alpha_y / beta_y) * (del beta_y / del PT).
PHIY: Chromatic phase function Phi_y, [2pi]:
PHIY = Phi_y = atan(a_y / b_y).
DMUY: Chromatic derivative of phase function mu_y, [2pi]:
DMUY = (del mu_y / del PT).
DDY: Chromatic derivative of dispersion D_y, [m]:
DDY = 1/2 * (del²y / del PT²).
DDPY: Chromatic derivative of dispersion D_py, [1]:
DDPY = 1/2 * (del²p_y / del PT²) / p_s.

Variables in the SUMM Table

After a successful TWISS command a summary table is created which contains the following variables:

LENGTH: The length of the machine, [m].
ORBIT5: The T (= c t, [m]) component of the closed orbit.
ALFA: The momentum compaction alpha_p, [1].
GAMMATR: The transition energy gamma_transition, [1].
Q1: The horizontal tune Q₁ [1].
DQ1: The horizontal chromaticity dq₁, [1]:
DQ1 = dq₁ = (del Q₁ / del PT).
BETXMAX: The largest horizontal beta_x, [m].
DXMAX: The largest horizontal dispersion [m].
DXRMS: The r.m.s. of the horizontal dispersion [m].
XCOMAX: The maximum of the horizontal closed orbit deviation [m].
XRMS: The r.m.s. of the horizontal closed orbit deviation [m].
Q2: The vertical tune Q₂ [1].
DQ2: The vertical chromaticity dq₂, [1]:
DQ2 = dq₂ = (del Q₂ / del PT).
BETYMAX: The largest vertical beta_y, [m].
DYMAX: The largest vertical dispersion [m].
DYRMS: The r.m.s. of the vertical dispersion [m].
YCOMAX: The maximum of the vertical closed orbit deviation [m].
YCORMS: The r.m.s. of the vertical closed orbit deviation [m].
DELTAP: Energy difference, divided by the reference momentum times the velocity of light, [1]:
DELTAP = delta(E) / p_s c.

Notice that in MAD-X PT substitutes DELTAP as longitudinal variable. Dispersive and chromatic functions are hence derivatives with respects to PT. Being PT=BETA*DELTAP, where BETA is the relativistic Lorentz factor, those functions must be multiplied by BETA a number of time equal to the order of the derivative.

Variables in the TRACK Table

The command RUN writes tables with the following variables:

X: Horizontal position x of the orbit, referred to the ideal orbit [m].
PX: Horizontal canonical momentum p_x of the orbit referred to the ideal orbit, divided by the reference momentum.
Y: Vertical position y of the orbit, referred to the ideal orbit [m].
PY: Vertical canonical momentum p_x of the orbit referred to the ideal orbit, divided by the reference momentum.
T: Velocity of light times the negative time difference with respect to the reference particle, [m]. A positive T means that the particle arrives ahead of the reference particle.
PT: Energy difference, divided by the reference momentum times the velocity of light, [1].

When tracking Lyapunov companions (not yet implemented), the TRACK table defines the following dependent expressions:

DISTANCE: the relative Lyapunov distance between the two particles.
LYAPUNOV: the estimated Lyapunov Exponent.
LOGDIST: the natural logarithm of the relative distance.
LOGTURNS: the natural logarithm of the turn number.

hansg, January 24, 1997. Revised in February 2007.