Appendix C — OPAL - MADX Conversion Guide

C.1 Purpose

This appendix summarizes the unit and convention changes needed when translating between MADX and OPAL.

The legacy discussion is built around the transverse beam sigma matrix,

\[ \sigma_{\mathrm{beam}} = \begin{pmatrix} \sigma_x & \sigma_{x p_x} \\ \sigma_{x p_x} & \sigma_{p_x} \end{pmatrix} = \epsilon \begin{pmatrix} \beta & -\alpha \\ -\alpha & \gamma \end{pmatrix}, \tag{C.1}\]

with the correlation convention

\[ \sigma_{x p_x} = \delta\,\sqrt{\sigma_x\sigma_{p_x}}. \tag{C.2}\]

The relativistic quantities used throughout the appendix are

\[ \gamma = \frac{E_{\mathrm{kin}} + m_p}{m_p}, \qquad \beta = \sqrt{1 - \frac{1}{\gamma^2}}, \qquad \beta\gamma = \frac{\beta}{\sqrt{1-\beta^2}}, \tag{C.3}\]

and the magnetic rigidity is

\[ B\rho = \frac{(\beta\gamma)\,m_p\,10^9}{c} \; [\mathrm{T\,m}], \tag{C.4}\]

with m_p = 0.939277 GeV and c = 299792458 m/s.

C.2 MADX to OPAL Output Conventions

The legacy appendix lists the following quantity-by-quantity conversion rules.

Quantity MADX OPAL output Conversion
transverse momentum \bar p_x [rad] \bar p_x [\beta\gamma] \bar p_x[\beta\gamma] = \bar p_x[rad] \,(\beta\gamma)
correlation \delta \delta \delta = \sigma_{x p_x}/(\bar p_x\bar x)
emittance \epsilon_x [m\,rad] \epsilon_x [m\,\beta\gamma] \epsilon_x = \sqrt{\sigma_x\sigma_{p_x} - \sigma_{x p_x}^2}
Twiss \alpha dimensionless dimensionless \alpha = -\sigma_{x p_x}/\epsilon_x
Twiss \beta_T m/rad m/(\beta\gamma) \beta_T = \sigma_x / \epsilon_x
Twiss \gamma_T rad/m (\beta\gamma)/m \gamma_T = \sigma_{p_x} / \epsilon_x
quadrupole strength k_1 [m^{-2}] k_1 [T/m] k_1[T/m] = k_1[m^{-2}]\,B\rho

The same rules apply plane by plane in both transverse directions.

C.3 Equivalent Explicit Formulas

The original appendix writes the most frequently used conversions explicitly:

\[ \bar p_x = \sqrt{\sigma_{p_x}}, \qquad \bar x = \sqrt{\sigma_x}, \tag{C.5}\]

\[ \epsilon_x = \sqrt{\left(\bar p_x \bar x\right)^2 - \left(\delta\,\bar x\bar p_x\right)^2} = \sqrt{\sigma_x\sigma_{p_x} - \sigma_{x p_x}^2}, \tag{C.6}\]

\[ \alpha = -\frac{\delta\,\bar x\bar p_x}{\epsilon_x}, \qquad \beta_T = \frac{\bar x^2}{\epsilon_x}, \qquad \gamma_T = \frac{\bar p_x^2}{\epsilon_x}. \tag{C.7}\]

These are the formulas to use when reconstructing an OPAL beam from MADX Twiss or sigma-matrix output.

C.4 MADX Element Positions to OPAL Input

The appendix highlights one convention difference that shows up repeatedly:

  • MADX at := gives the element center.
  • OPAL ELEMEDGE expects the element entrance.

So the conversion is

\[ \mathrm{ELEMEDGE} = \text{element center} - \frac{1}{2}\text{element length}. \tag{C.8}\]

Quantity MADX OPAL input
element position at := center of element ELEMEDGE at start of element
conversion center minus half length entrance position

C.5 OPAL Output to OPAL Input Energy-Style Conversion

The appendix also gives the conversion from transverse momentum in \beta\gamma form to an energy-style input quantity in eV:

\[ p_x[\mathrm{eV}] = m_p\,10^9\left(\sqrt{\bar p_x^2 + 1} - 1\right). \tag{C.9}\]

This is the relation to use when an OPAL output quantity must be mapped back into an input-style energy convention.

C.6 Practical Use

This guide is most useful when:

  • reconstructing an OPAL beam from MADX Twiss or sigma-matrix output
  • translating focusing strengths and element positions
  • reconciling unit conventions in transverse and longitudinal phase-space descriptions