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- TL;DR Summary
- I am reading the book "Particle Accelerator Physics" (4th ed.) by Helmut Wiedemann, and facing some problems in understanding the equations describing iris electrode (section 2.2.2) and iris doublet (section 2.2.3).

**Iris electrode**The potential distribution in the vicinity of the iris electrode, denoted by ##V(r,z)## is rotationally symmetric. After some derivations, the author arrives at the following two equations:

$$\begin{align}

E_z &= - V'_0(z)\\

E_r &= \frac{1}{2}V''_0(z)r,

\end{align}$$

where ##V_0 (z) \equiv V(r = 0, z) ## and the primes denote derivatives w.r.t. ##z##.

Then, using the radial equation of motion ##m \ddot{r} = m v^2 r'' = q E_r##, where ##v## and ##q## are respectively the particle velocity and charge, the author defines an integration,

$$\begin{align}

r'_2 - r'_1 &= \dfrac{q}{mv^2} \int_{z_1}^{z_2} E_r ~ \mathrm{d}z \nonumber \\[2em]

&= - \dfrac{q}{2mv^2} \int_{z_1}^{z_2} r \dfrac{\partial E_z}{\partial z} \mathrm{d}z \label{eq:integration}

\end{align}$$

(The second step above stems from the fact that ##\nabla \cdot \mathbf{E} = 0##)

In the thin lens approximation (##r = \mathrm{const.}## and ##v = \mathrm{const.}##, see diagram above), ##\mathrm{Eqn. \eqref{eq:integration}}## becomes

$$\begin{equation}

r'_2 - r'_1 = - \dfrac{q~r_1}{2mv^2} (E_2 - E_1) \label{eq:2}

\end{equation}

$$

Substituting ##\frac{1}{2} m v^2 = q V_0## and ##E = -V'##, ##\mathrm{Eqn.}~\eqref{eq:2}## becomes

$$

r_2' - r_1' = \dfrac{r_1}{4} \dfrac{V_2' - V_1'}{V_0},

$$

and the focal length of the iris electrode is

$$\begin{equation}

\frac{1}{f} = \dfrac{V_2' - V_1'}{4V_0}, \label{eq:focal_length}

\end{equation}$$

and the transformation matrix finally becomes

$$\begin{equation}

\mathcal{M}_\mathrm{iris} = \begin{pmatrix}

1 & 0\\

\dfrac{V_2' - V_1'}{V_0} & 1

\end{pmatrix}. \label{eq:trans_mat_1}

\end{equation}$$

__Questions:__**1.**This is the first time the author mentions the transformation matrix. What is the significance of this matrix?

**2.**How is the transformation matrix constructed from the focal length?

**Iris doublet**Let me go ahead and simply quote the author:

The transformation matrices for both iris electrodes are,

$$\begin{align}

\mathcal{M}_1 &= \begin{pmatrix}

1 & 0\\

\dfrac{V_2 - V_1}{4dV_1} & 1

\end{pmatrix} \label{eq:trans_mat_iris1}\, \mathrm{and} \\[2em]

\mathcal{M}_2 &= \begin{pmatrix}

1 & 0\\

\dfrac{V_2 - V_1}{4dV_2} & 1

\end{pmatrix}. \label{eq:trans_mat_iris2}

\end{align}$$

The transformation matrix for the drift space between the electrodes can be derived from the particle trajectory

$$\begin{align}

r(z) &= r_1 + \int _0^z r'(\bar{z}) ~ \mathrm{d}\bar{z} \label{eq:some1}\\[2em]

&= r_1 + \int _0^z \mathrm{d}\bar{z} ~ \dfrac{r' ~ p_1}{p_1 + \Delta p(\bar{z})} \label{eq:some2}

\end{align}$$

The particle momentum varies between the electrodes from ##p_1 = \sqrt{2mE_\mathrm{kin}}## to ##p_1 + \Delta p(\bar{z}) = \sqrt{2m \left(E_\mathrm{kin} + q \dfrac{V_2 - V_1}{d}z \right)}## and the integral becomes

$$\begin{equation}

\int_0^d \dfrac{\mathrm{d}\bar{z}}{\sqrt{1 + \frac{V_2 - V_1}{E_\mathrm{kin}d}\bar{z}}} = \dfrac{2d \sqrt{V_1}}{\sqrt{V_2} + \sqrt{V_1}}. \label{eq:some3}

\end{equation}$$

The particle trajectory at the location of the second electrode is ##r(d) = r_2 = r_1 + \dfrac{2d \sqrt{V_1}}{\sqrt{V_2} + \sqrt{V_1}}r'_1## and its derivative ##r'_2 = r'_1 \sqrt{V_1}/\sqrt{V_2}## from which we can deduce the transformation matrix

$$\begin{equation}

\mathcal{M}_d = \begin{pmatrix}

1 & \dfrac{2d \sqrt{V_1}}{\sqrt{V_2} + \sqrt{V_1}}\\[1.5em]

0 & \dfrac{\sqrt{V_1}}{\sqrt{V_2}}

\end{pmatrix} \label{eq:trans_mat_4}

\end{equation}$$

__Questions:__**3.**How does the transformation matrix change from that in ##\mathrm{Eqn.}~\eqref{eq:trans_mat_1}## to ##\mathrm{Eqn.}~\eqref{eq:trans_mat_iris1}?## How do the primes change to unprimed variables, and how does the ##d## come in the denominator?

**4.**How do we go from ##\mathrm{Eqn.}~\eqref{eq:some1}## to ##\mathrm{Eqn.}~\eqref{eq:some2}?##

**5.**How do we get the transformation matrix in ##\mathrm{Eqn.}~\eqref{eq:trans_mat_4}?##