IONIZATION COOLING IN AXIAL SYMMETRIC CHANNEL

[V. A. Lebedev]{.underline}

*Jefferson Lab*

> Muon Collaboration workshop,
>
> Fermilab,
>
> May 3-5, 2001

Talk Outline

1.  Twiss Parameters and Beam Emittances

2.  Cooling Description

3.  Canonical Angular Momentum and Second Order Moments

4.  Beta-functions for Particle Motion with Axial-symmetric Solenoidal
    Focusing

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1.  Twiss Parameters for Axial-symmetric Solenoidal Focusing

Canonical variables

> Canonical momenta: $\begin{array}{r}
> p_{x} = x^{'} - \frac{R}{2}y\ , \\
> p_{y} = y^{'} + \frac{R}{2}x\ \text{.}
> \end{array}$
>
> Relation between geometrical and canonical variables

$x = \text{Rx}$ ,

> where

$x \equiv \begin{bmatrix}
x \\
p_{x} \\
y \\
p_{y}
\end{bmatrix}\ ,\ x \equiv \begin{bmatrix}
x \\
\theta_{x} \\
y \\
\theta_{y}
\end{bmatrix}\ ,\ R = \begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & - R/2 & 0 \\
0 & 0 & 1 & 0 \\
R/2 & 0 & 0 & 1
\end{bmatrix}\ $ ,

> $R = \frac{\text{eB}_{s}}{\text{Pc}}$ - *R* is proportional to
> longitudinal magnetic field

Here and below we put a cap above transfer matrices and vectors related
to the canonical variables.

-   One can write for a single-particle phase-space trajectory along the
    beam orbit[^1]

$$\hat{x}(s) = \text{Re}\left( \sqrt{\varepsilon_{1}}{\hat{v}}_{1}(s)e^{- i(\psi_{1} + \mu_{1}(s))} + \sqrt{\varepsilon_{2}}\hat{v_{2}}(s)e^{- i(\psi_{2} + \mu_{2}(s))} \right)\ ,$$

-   vectors ${\hat{v}}_{1}(s)$and ${\hat{v}}_{2}(s)$are the
    eigen-vectors at coordinate *s*

-   *ψ*~1~ and *ψ*~2~ are the initial phases of betatron motion

-   $\sqrt{\varepsilon_{1}} \equiv I_{1}$ and
    $\sqrt{\varepsilon_{2}} \equiv I_{2}$ are the actions

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-   One can rewrite the above equations in the following form

> $$\hat{x}(s) = \hat{V}(s)a(s)$$
>
> where
>
> $\hat{V}(s) = \left\lbrack {\hat{v}}_{1^{\prime}}(s), - {\hat{v}}_{1^{''}}(s),{\hat{v}}_{2^{\prime}}(s), - {\hat{v}}_{2^{''}}(s) \right\rbrack$
> , $a(s) = \begin{bmatrix}
> \sqrt{\varepsilon_{1}}\text{cos}(\psi_{1} + \mu_{1}(s)) \\
> \sqrt{\varepsilon_{1}}\text{sin}(\psi_{1} + \mu_{1}(s)) \\
> \sqrt{\varepsilon_{2}}\text{cos}(\psi_{2} + \mu_{2}(s)) \\
> \sqrt{\varepsilon_{2}}\text{sin}(\psi_{2} + \mu_{2}(s))
> \end{bmatrix}$

-   **Eigen vector parameterization**

    -   **Edwards and Teng**

> $${\hat{v}}_{1}(s) = R^{- 1}\begin{bmatrix}
> \sqrt{\beta_{1}} \\
>  - \frac{i + \alpha_{1}}{\sqrt{\beta_{1}}} \\
> 0 \\
> 0
> \end{bmatrix}\ ,\ {\hat{v}}_{2}(s) = R^{- 1}\begin{bmatrix}
> 0 \\
> 0 \\
> \sqrt{\beta_{2}} \\
>  - \frac{i + \alpha_{2}}{\sqrt{\beta_{2}}}
> \end{bmatrix}\ ,\ R = \begin{bmatrix}
> \text{cos}\varphi & 0 & - d\text{sin}\varphi & b\text{sin}\varphi \\
> 0 & \text{cos}\varphi & c\text{sin}\varphi & - a\text{sin}\varphi \\
> a\text{sin}\varphi & b\text{sin}\varphi & \text{cos}\varphi & 0 \\
> c\text{sin}\varphi & d\text{sin}\varphi & 0 & \text{cos}\varphi
> \end{bmatrix}$$

-   **Ripken, *et al*.**

$${\hat{v}}_{1}(s) = \begin{bmatrix}
\sqrt{\beta_{1x}(s)} \\
 - \frac{i\left( 1 - u(s) \right) + \alpha_{1x}(s)}{\sqrt{\beta_{1x}(s)}} \\
\sqrt{\beta_{1y}(s)}e^{{i\nu}_{1}(s)} \\
 - \frac{\text{iu}(s) + \alpha_{1y}(s)}{\sqrt{\beta_{1y}(s)}}e^{{i\nu}_{1}(s)}
\end{bmatrix}\ ,\ {\hat{v}}_{2}(s) = \begin{bmatrix}
\sqrt{\beta_{2x}(s)}e^{{i\nu}_{2}(s)} \\
 - \frac{\text{iu}(s) + \alpha_{2x}(s)}{\sqrt{\beta_{2x}(s)}}e^{{i\nu}_{2}(s)} \\
\sqrt{\beta_{2y}(s)} \\
 - \frac{i\left( 1 - u(s) \right) + \alpha_{2y}(s)}{\sqrt{\beta_{2y}(s)}}
\end{bmatrix}\ ,$$

In the case of axial symmetric focusing the eigen-vectors are (for
simplicity we also skip dependence on *s*):

-   **Edwards and Teng**

> $${\hat{v}}_{1} = R^{- 1}\begin{bmatrix}
> \sqrt{\overset{\sim}{\beta}} \\
>  - \frac{i + \overset{\sim}{\alpha}}{\sqrt{\overset{\sim}{\beta}}} \\
> 0 \\
> 0
> \end{bmatrix}\ ,\ {\hat{v}}_{2} = R^{- 1}\begin{bmatrix}
> 0 \\
> 0 \\
> \sqrt{{\overset{\sim}{\beta}}_{2}} \\
>  - \frac{i + {\overset{\sim}{\alpha}}_{2}}{\sqrt{{\overset{\sim}{\beta}}_{2}}}
> \end{bmatrix}\ ,\ R = \frac{1}{\sqrt{2}}\begin{bmatrix}
> 1 & 0 & \overset{\sim}{\alpha} & \overset{\sim}{\beta} \\
> 0 & 1 & - \frac{1 + {\overset{\sim}{\alpha}}^{2}}{\overset{\sim}{\beta}} & - \overset{\sim}{\alpha} \\
> \overset{\sim}{\alpha} & \overset{\sim}{\beta} & 1 & 0 \\
>  - \frac{1 + {\overset{\sim}{\alpha}}^{2}}{\overset{\sim}{\beta}} & - \overset{\sim}{\alpha} & 0 & 1
> \end{bmatrix}$$

-   **Ripken, *et al*.**
    $\overset{\sim}{\alpha} = 2\alpha\ ,\ \overset{\sim}{\beta} = 2\beta$

> ${\hat{v}}_{1} = \begin{bmatrix}
> \sqrt{\beta} \\
>  - \frac{i + 2\alpha}{2\sqrt{\beta}} \\
> i\sqrt{\beta} \\
>  - i\frac{i + 2\alpha}{2\sqrt{\beta}}
> \end{bmatrix}$ , ${\hat{v}}_{2} = \begin{bmatrix}
> i\sqrt{\beta} \\
>  - i\frac{i + 2\alpha}{2\sqrt{\beta}} \\
> \sqrt{\beta} \\
>  - \frac{i + 2\alpha}{2\sqrt{\beta}}
> \end{bmatrix}$

Parameterization for Matrix V

> $$\hat{V} = \begin{bmatrix}
> \sqrt{\beta} & 0 & 0 & - \sqrt{\beta} \\
>  - \frac{\alpha}{\sqrt{\beta}} & \frac{1}{2\sqrt{\beta}} & \frac{1}{2\sqrt{\beta}} & \frac{\alpha}{\sqrt{\beta}} \\
> 0 & - \sqrt{\beta} & \sqrt{\beta} & 0 \\
> \frac{1}{2\sqrt{\beta}} & \frac{\alpha}{\sqrt{\beta}} & - \frac{\alpha}{\sqrt{\beta}} & \frac{1}{2\sqrt{\beta}}
> \end{bmatrix}$$

where we used that *u* = 1/2, ν~1~ = ν~2~ =π/2

2\. Cooling Description

-   One can write for ionization cooling due to energy loss on a thin
    absorber

$${\Delta\theta}_{} = - \theta_{}\frac{\Delta p}{p} \equiv - \theta_{}\delta$$

where we presume that the longitudinal energy is restored by
acceleration.

In matrix form that yields

$${\hat{x}}_{\text{out}} = R\begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & 1 - \delta & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 - \delta
\end{bmatrix}R^{- 1}{\hat{x}}_{\text{in}}$$

or for amplitudes

$\hat{V}a_{\text{out}} = R\begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & 1 - \delta & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 - \delta
\end{bmatrix}R^{- 1}\hat{V}a_{\text{in}}$ ⇒
$a_{\text{out}} = {\hat{V}}^{- 1}R\begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & 1 - \delta & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 - \delta
\end{bmatrix}R^{- 1}\hat{V}a_{\text{in}}$

-   Performing calculations one obtains

$$a_{\text{out}} = \left\{ \begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{bmatrix} - \delta\begin{bmatrix}
\frac{1 - R\beta}{2} & \alpha & - \alpha & \frac{1 + R\beta}{2} \\
 - \alpha & \frac{1 - R\beta}{2} & \frac{1 + R\beta}{2} & \alpha \\
 - \alpha & \frac{1 - R\beta}{2} & \frac{1 + R\beta}{2} & \alpha \\
\frac{1 - R\beta}{2} & \alpha & - \alpha & \frac{1 + R\beta}{2}
\end{bmatrix} \right\} a_{\text{in}}$$

-   Calculating values of the actions after cooling one obtains

$$\begin{array}{r}
\varepsilon_{1^{\prime}} \equiv a_{{\text{out}_{1}}^{2}} + a_{\text{out}_{2}^{2}} = \varepsilon_{1}\left\lbrack 1 - (1 - \beta R)\delta \right\rbrack + \sqrt{\varepsilon_{1}\varepsilon_{2}}\ \left\lbrack 2\alpha\text{cos}\varphi - (1 + \beta R)\text{sin}\varphi \right\rbrack\delta + O(\delta^{2}) \\
\varepsilon_{2^{\prime}} \equiv a_{{\text{out}_{3}}^{2}} + a_{\text{out}_{4}^{2}} = \varepsilon_{2}\left\lbrack 1 - (1 + \beta R)\delta \right\rbrack + \sqrt{\varepsilon_{1}\varepsilon_{2}}\ \left\lbrack 2\alpha\text{cos}\varphi - (1 - \beta R)\text{sin}\varphi \right\rbrack\delta + O(\delta^{2}) \\
\ \text{where}:\ \varphi = \mu_{1} + \mu_{2} + \psi_{1} + \psi_{2}
\end{array}$$

-   Two descriptions of the cooling

    -   After each absorber we compute

        -   new 4D phase space

        -   new partial emittances

        -   new beta-functions

    -   Or we compute everything relative to unperturbed beta-functions

        -   More productive approach although partial emittances
            (actions) depend on betatron phases

> If cooling effect of one absorber is sufficiently small one can
> perform averaging over betatron phases. That yields

$\begin{array}{r}
{\Delta\varepsilon}_{1} \approx - \varepsilon_{1}(1 - \beta R)\delta \\
{\Delta\varepsilon}_{2} \approx - \varepsilon_{2}(1 + \beta R)\delta
\end{array}$ ⇒

3\. Canonical Angular Momentum and Second Order Moments

-   Canonical momentum of a single particle

$$M = \text{xp}_{y} - \text{yp}_{x} = {\hat{x}}^{T}\begin{bmatrix}
0 & 0 & 0 & 1 \\
0 & 0 & - 1 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{bmatrix}\hat{x} = \left( \hat{V}a \right)^{T}\begin{bmatrix}
0 & 0 & 0 & 1 \\
0 & 0 & - 1 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{bmatrix}\hat{V}a = \frac{\varepsilon_{1} - \varepsilon_{2}}{2}$$

-   Second order moments of the Gaussian distribution

(note that for a single particle -
$\varepsilon_{\text{rms}} = \frac{\varepsilon}{2}$ and we use rms
emittances below)

$$\begin{array}{r}
\left\langle x^{2} \right\rangle = \left\langle y^{2} \right\rangle = \beta\left( \varepsilon_{1} + \varepsilon_{2} \right)\ , \\
\left\langle \text{xp}_{x} \right\rangle = \left\langle \text{yp}_{y} \right\rangle = - \alpha\left( \varepsilon_{1} + \varepsilon_{2} \right)\ , \\
\left\langle p_{x^{2}} \right\rangle = \left\langle p_{y^{2}} \right\rangle = \frac{1 + 4\alpha^{2}}{4\beta}\left( \varepsilon_{1} + \varepsilon_{2} \right)\ , \\
\left\langle \text{xp}_{y} \right\rangle = - \left\langle \text{yp}_{x} \right\rangle = \frac{\varepsilon_{1} - \varepsilon_{2}}{2}\ ,\  \Rightarrow \ \left\langle M \right\rangle = \varepsilon_{1} - \varepsilon_{2} \\
\left\langle \text{xy} \right\rangle = \left\langle p_{x}p_{y} \right\rangle = 0
\end{array}$$

4\. Beta-function for Particle Motion with Axial-symmetric Solenoidal
Focusing

Equation for the square root of the beta-function is similar to the
equation for Floque-function in the case of uncoupled motion:

![](media/image1.wmf){width="4.481944444444444in"
height="3.091666666666667in"}

$\frac{d^{2}\sqrt{\beta}}{\text{ds}^{2}} + \frac{R^{2}}{4}\sqrt{\beta} - \frac{1}{4\left( \sqrt{\beta} \right)^{3}} = 0$
.

The alpha-function is determined by the standard recipe:

$$\alpha = - \frac{1}{2}\frac{d\beta}{\text{ds}}$$

[^1]: **BETATRON MOTION WITH COUPLING OF HORIZONTAL AND VERTICAL DEGREES
    OF FREEDOM**, V. Lebedev, S. Bogacz, **JLab-TN-00-022 ,
    \<**http://tnweb.jlab.org/tn/2000/**\>**