EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH

Known Differences to Other Programs

Definitions

As from Version 8.13, MAD8 uses an additional constant momentum error delta_s in all optical calculations. The transfer maps contain the exact dependence upon this value; therefore the tunes for large deviations can be computed with high accuracy as opposed to previous versions.

The choice of canonical variables in MAD still leads to slightly different definitions of the lattice functions. In MAD the Courant-Snyder invariants in [Courant and Snyder] take the form

W_x = gamma_x x² - 2 alpha_x x p_x + beta_x p_x²

Comparison to the original form

W_x = gamma_x x² - 2 alpha_x x x' + beta_x x'²

shows that the orbit functions cannot be the same. A more detailed analysis, using

x' = p_x / (1 + delta)

shows that all formulas can be made consistent by defining the MAD orbit functions as

beta_xM = beta_xC * (1 + delta), alpha_xM = alpha_xC, gammaa_xM = gamma_xC / (1 + delta),

For constant delta_s along the beam line and delta = 0, the lattice functions are the same. In a machine where delta varies along the circumference, e.g. in a linear accelerator or in an electron-positron storage ring, the definition of the Courant-Snyder invariants must be generalised. The MAD invariants have the advantage that they remain invariants along the beam line even for variable delta.

With the new method this problem occurs in Twiss module only for non-constant delta.

Treatment of Energy Error in TWISS