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1. OPAL - MADX Conversion Guide

1. OPAL - MADX Conversion Guide

We note with \alpha,\beta and \gamma the Twiss parameters.

\begin{aligned}
\sigma_{beam} &=& \begin{pmatrix}\sigma_{x} & \sigma_{x p_{x}}\\\sigma_{x p_{x}} &  \sigma_{ p_{x}}\end{pmatrix}
= \begin{pmatrix} \sigma_{x} & \delta\cdot\sqrt{\sigma_{x}\sigma_{ p_{x}}}\\\delta\cdot\sqrt{\sigma_{x}\sigma_{ p_{x}}} & \sigma_{ p_{x}}\end{pmatrix}
= \begin{pmatrix} <x^{2}> & <x p_{x}>\\<x p_{x}> & < p_{x}^{2}> \end{pmatrix} \\
&=& \begin{pmatrix}
  \frac{1}{N}\sum_{i=1}^{N}x_{i}^{2} & \frac{1}{N}\sum_{i=1}^{N}x_{i} p_{x_{i}}\\
  \frac{1}{N}\sum_{i=1}^{N}x_{i} p_{x_{i}} & \frac{1}{N}\sum_{i=1}^{N} p_{x_{i}}^{2}
 \end{pmatrix}
= \epsilon\cdot\begin{pmatrix} \beta & -\alpha\\ -\alpha & \gamma\end{pmatrix}
\end{aligned}

\begin{aligned}
\bar{p}_{x} & = & \sqrt{\frac{1}{N}\sum_{i=1}^{N} p_{x_{i}}^{2}} & = & \sqrt{\sigma_{ p_{x}}} & & \bar{x} & = & \sqrt{\frac{1}{N}\sum_{i=1}^{N}x_{i}^{2}} \\
\bar{p}_{y} & = & \sqrt{\frac{1}{N}\sum_{i=1}^{N}p_{y_{i}}^{2}} & = & \sqrt{\sigma_{p_{y}}} & & \bar{y} & = & \sqrt{\frac{1}{N}\sum_{i=1}^{N}y_{i}^{2}} \\
\\
\\
\gamma & = & \frac{E_{kin}+m_{p}}{m_{p}} & & & & \beta & = & \sqrt{1-\frac{1}{\gamma^{2}}} = \frac{v}{c} \\
(\beta\gamma) & = & \frac{E_{kin}+m_{p}}{m_{p}}\cdot\sqrt{1-\frac{1}{\gamma^{2}}}& = & \frac{\beta}{\sqrt{1-\beta^{2}}} & & \text{B}\rho & = & \frac{\left(\beta\gamma\right)\cdot m_{p}\cdot 10^{9}}{c}~\left[\text{T m}\right] \\
m_{p} & = & 0.939277 [GeV] & & & & c & = & 299792458 [m/s]
\end{aligned}

Quantity MADX Conversion OPAL-Output

Quantity	`MADX`	Conversion	OPAL-Output
Momenta	`\bar{p}_{x}`	[rad]	`\bar{p}_{x}`[`\beta\gamma`]	=	`\left(\bar{p}_{x}\left[\text{rad}\right]\right)\cdot\left(\beta\gamma\right)`	`\bar{p}_{x}`	[`\beta\gamma`]
Correlation of `\bar{x}`,`\bar{p}_{x}`	`\delta`	[1]	`\delta`	=	`\left(\sigma_{x p_{x}}\left[\text{m }\text{rad}\right]\right)/\left(\left(\bar{p}_{x}\left[\text{rad}\right]\right)\cdot\left( \bar{x}\left[\text{m}\right]\right)\right)`	`\delta`	[1]
				=	`\left(\sigma_{x p_{x}}\left[\text{m }\text{rad}\right]\right)/\sqrt{\left(\sigma_{x}\left[\text{m}^{2}\right]\right)\cdot\left(\sigma_{ p_{x}}\left[\text{rad}^{2}\right]\right)}`
Emittance	`\epsilon_{x}`	[m rad]	`\epsilon_{x}`[m `\beta\gamma`]	=	`\sqrt{\left( \bar{p}_{x}\left[ \beta\gamma \right] \right) ^{2} \cdot \left(\bar{x}\left[\text{m}\right]\right)^{2} - \left(\delta \cdot \left(\bar{x}\left[\text{m}\right]\right) \cdot \left(\bar{p}_{x}\left[\beta\gamma\right]\right)\right)^{2}}`	`\epsilon_{x}`	[m `\beta\gamma`]
				=	`\sqrt{\left( \sigma_{{p}_{x}}\left[ \left(\beta\gamma\right)^{2} \right] \right)\cdot \left(\sigma_{x}\left[\text{m}^{2}\right]\right) - \left(\delta \cdot \sqrt{\left(\sigma_{x}\left[\text{m}^{2}\right]\right)\cdot\left(\sigma_{p_{x}} \left[\left(\beta\gamma\right)^{2}\right]\right)}\right)^{2}}`
				=	`\sqrt{\left( \sigma_{{p}_{x}}\left[ \left(\beta\gamma\right)^{2} \right] \right)\cdot \left(\sigma_{x}\left[\text{m}^{2}\right]\right) - \left(\sigma_{x p_{x}}\left[\text{m} ~\beta\gamma\right]\right)^{2}}`
Twiss Parameter `\alpha`	`\alpha`	[1]	`\alpha\left[1\right]`	=	`-\delta\cdot\left(\bar{x}\left[\text{m}\right]\right)\cdot\left(\bar{p}_{x}\left[\beta\gamma\right]\right)/\left(\epsilon_{x}\left[\text{m}~\beta\gamma\right]\right)`	`\alpha_{T}`	[1]
				=	`-\delta\cdot\sqrt{\left(\sigma_{x}\left[\text{m}^{2}\right]\right)\cdot\left(\sigma_{ p_{x}}\left[\left(\beta\gamma\right)^{2}\right]\right)}/\left(\epsilon_{x}\left[\text{m}~\beta\gamma\right]\right)`
Twiss Parameter `\beta_{T}`	`\beta_{T}`	[m/rad]	`\beta_{T}\left[\text{m}/\beta\gamma\right]`	=	`\left(\bar{x}\left[\text{m}\right]\right)^{2}/\left(\epsilon_{x}\left[\text{m}~\beta\gamma\right] \right)`	`\beta_{T}`	[m/`\beta\gamma`]
				=	`\left(\sigma_{x}\left[\text{m}^{2}\right]\right)/\left(\epsilon_{x}\left[\text{m}~\beta\gamma\right] \right)`
Twiss Parameter `\gamma_{T}`	`\gamma_{T}`	[rad/m]	`\gamma_{T}\left[\beta\gamma/\text{m}\right]`	=	`\left(\bar{p}_{x}\left[\beta\gamma\right]\right)^{2}/\left(\epsilon_{x}\left[\text{m}~\beta\gamma\right]\right)`	`\gamma_{T}`	[`\beta\gamma`/m]
				=	`\left(\sigma_{p_{x}}\left[\left(\beta\gamma\right)^{2}\right]\right)/\left(\epsilon_{x}\left[\text{m}~\beta\gamma\right]\right)`
Focusing strength	`k_{1}`	`\left[\text{m}^{-2}\right]`	`k_{1}\left[\text{T}/\text{m}\right]`	=	`\left(k_{1}\left[\text{m}^{-2}\right]\right)\cdot\left(\text{B}\rho\left[\text{T m}\right]\right)`	`k_{1}`	`\left[\text{T}/\text{m}\right]`
Quantity	`MADX`	Conversion	`OPAL-Input`
Element Position	`at :=`	`\left[\text{m}\right]`	`ELEMEDGE`	=	(Center of the element) - (Length of the element)/2	`ELEMEDGE =`	`\left[\text{m}\right]`
	Center of the element					Begin of the element
Quantity	`OPAL-Output`	Conversion	`OPAL-Input`
Momenta	`\bar{p}_{x}`	`\left[\beta\gamma\right]`	`p_{x}\left[\text{eV}\right]`	=	`m_{p}\cdot10^{9}\cdot\left(\sqrt{\left(\bar{p}_{x}\left[\beta\gamma\right]\right)^{2} +1}-1\right)`	`\bar{p}_{x}`	[eV]

Momenta

\bar{p}_{x}

[rad]

\bar{p}_{x}[\beta\gamma]

\left(\bar{p}_{x}\left[\text{rad}\right]\right)\cdot\left(\beta\gamma\right)

\bar{p}_{x}

[\beta\gamma]

Correlation of \bar{x},\bar{p}_{x}

\delta

[1]

\delta

\left(\sigma_{x p_{x}}\left[\text{m }\text{rad}\right]\right)/\left(\left(\bar{p}_{x}\left[\text{rad}\right]\right)\cdot\left( \bar{x}\left[\text{m}\right]\right)\right)

\delta

[1]

\left(\sigma_{x p_{x}}\left[\text{m }\text{rad}\right]\right)/\sqrt{\left(\sigma_{x}\left[\text{m}^{2}\right]\right)\cdot\left(\sigma_{ p_{x}}\left[\text{rad}^{2}\right]\right)}

Emittance

\epsilon_{x}

[m rad]

\epsilon_{x}[m \beta\gamma]

\sqrt{\left( \bar{p}_{x}\left[ \beta\gamma \right] \right) ^{2} \cdot \left(\bar{x}\left[\text{m}\right]\right)^{2} - \left(\delta \cdot \left(\bar{x}\left[\text{m}\right]\right) \cdot \left(\bar{p}_{x}\left[\beta\gamma\right]\right)\right)^{2}}

\epsilon_{x}

[m \beta\gamma]

\sqrt{\left( \sigma_{{p}_{x}}\left[ \left(\beta\gamma\right)^{2} \right] \right)\cdot \left(\sigma_{x}\left[\text{m}^{2}\right]\right) - \left(\delta \cdot \sqrt{\left(\sigma_{x}\left[\text{m}^{2}\right]\right)\cdot\left(\sigma_{p_{x}} \left[\left(\beta\gamma\right)^{2}\right]\right)}\right)^{2}}

\sqrt{\left( \sigma_{{p}_{x}}\left[ \left(\beta\gamma\right)^{2} \right] \right)\cdot \left(\sigma_{x}\left[\text{m}^{2}\right]\right) - \left(\sigma_{x p_{x}}\left[\text{m} ~\beta\gamma\right]\right)^{2}}

Twiss Parameter \alpha

\alpha

[1]

\alpha\left[1\right]

-\delta\cdot\left(\bar{x}\left[\text{m}\right]\right)\cdot\left(\bar{p}_{x}\left[\beta\gamma\right]\right)/\left(\epsilon_{x}\left[\text{m}~\beta\gamma\right]\right)

\alpha_{T}

[1]

-\delta\cdot\sqrt{\left(\sigma_{x}\left[\text{m}^{2}\right]\right)\cdot\left(\sigma_{ p_{x}}\left[\left(\beta\gamma\right)^{2}\right]\right)}/\left(\epsilon_{x}\left[\text{m}~\beta\gamma\right]\right)

Twiss Parameter \beta_{T}

\beta_{T}

[m/rad]

\beta_{T}\left[\text{m}/\beta\gamma\right]

\left(\bar{x}\left[\text{m}\right]\right)^{2}/\left(\epsilon_{x}\left[\text{m}~\beta\gamma\right] \right)

\beta_{T}

[m/\beta\gamma]

\left(\sigma_{x}\left[\text{m}^{2}\right]\right)/\left(\epsilon_{x}\left[\text{m}~\beta\gamma\right] \right)

Twiss Parameter \gamma_{T}

\gamma_{T}

[rad/m]

\gamma_{T}\left[\beta\gamma/\text{m}\right]

\left(\bar{p}_{x}\left[\beta\gamma\right]\right)^{2}/\left(\epsilon_{x}\left[\text{m}~\beta\gamma\right]\right)

\gamma_{T}

[\beta\gamma/m]

\left(\sigma_{p_{x}}\left[\left(\beta\gamma\right)^{2}\right]\right)/\left(\epsilon_{x}\left[\text{m}~\beta\gamma\right]\right)

Focusing strength

k_{1}

\left[\text{m}^{-2}\right]

k_{1}\left[\text{T}/\text{m}\right]

\left(k_{1}\left[\text{m}^{-2}\right]\right)\cdot\left(\text{B}\rho\left[\text{T m}\right]\right)

k_{1}

\left[\text{T}/\text{m}\right]

Quantity

MADX

Conversion

OPAL-Input

Element Position

at :=

\left[\text{m}\right]

ELEMEDGE

(Center of the element) - (Length of the element)/2

ELEMEDGE =

\left[\text{m}\right]

Center of the element

Begin of the element

Quantity

OPAL-Output

Conversion

OPAL-Input

Momenta

\bar{p}_{x}

\left[\beta\gamma\right]

p_{x}\left[\text{eV}\right]

m_{p}\cdot10^{9}\cdot\left(\sqrt{\left(\bar{p}_{x}\left[\beta\gamma\right]\right)^{2} +1}-1\right)

\bar{p}_{x}

[eV]