Beam Lifetime

The electron beam loses electrons over time. The lifetime τ is defined as

\begin{align} \frac{1}{\tau} = \frac{1}{I}\frac{dI}{dt} \end{align}

The lifetime itself may be divided into two pieces, a vacuum lifetime and a Touschek lifetime with the total lifetime given by:

\begin{align} \frac{1}{\tau} = \frac{1}{\tau_V}+\frac{1}{\tau_T} \end{align}

The vacuum lifetime is the result of the scattering process of electrons off of the gas molecules in the beam pipe. The Touschek lifetime is the result of the scattering of electrons off of each other.
An expression valid for most synchrotron lights sources is:

\begin{align} \frac{1}{\tau} = \frac{r_0^2 cN}{8\pi \gamma^3\sigma_x\sigma_y\sigma_s\sigma_{x'}\delta_{acc}^2}F(\epsilon) \end{align}
\begin{align} \epsilon = \left(\frac{\delta}{\gamma \sigma_{x'}}\right)^2 \end{align}
\begin{align} F(\epsilon) = \frac{1}{2}\int_0^1\left(\frac{2}{u}-\ln\frac{1}{u}-2\right)e^{-\frac{\epsilon}{u}}du \end{align}

One implements the function in Matlab with
y = 0.5*quad(@(x)fun(x,xi),0,1);

function y = fun(x,xi)
y=sqrt(xi)*(1./x - 0.5*log(1./x) - 1).*exp(-xi./x);

The lifetime scales as

\begin{align} \tau\sim\frac{\sigma_s \sqrt{\epsilon_y}}{I_b} \delta_{acc}^3 \end{align}

The bunch length may be measured with a streak camera. Its value depends on the current and is given by solving the Haissinski equation, below the microwave instability.
The vertical emittance may be measured.
The energy acceptance is challenging. It may be given by the RF acceptance or by the dynamic acceptance. Computing this quantity is the work of a tracking code.

In AT, one uses the function atcalc_TouschekPM.

A more complete expression is given by Piwinski:

\begin{eqnarray} {1\over T_{\ell}} &=& \left\langle {r_0^2 c \beta_x \beta_y \sigma_h N_B \over 8\sqrt{\pi} \beta^2 \gamma^4 \sigma_{x\beta}^2 \sigma_{y\beta}^2 \sigma_s \sigma_p} \right. \nonumber \\ &\times& \int_{u_m} ^\infty \bigg[ \left( 2+{1\over u} \right)^2 \left( {u/u_m\over1+u}-1 \right) +1 \nonumber \\ & & -{\sqrt{1+u}\over \sqrt{u/u_m}} -{1\over2u} \left( 4+{1\over u}\right) \ln {u/u_m\over 1+u} \bigg] \nonumber \\ &\times & \left. e^{-B_1u} I_0(B_2u)\,{\sqrt{u}\,du\over\sqrt{1+u}} \right\rangle \end{eqnarray}


\begin{align} {1\over\sigma_h^2}\,=\,{1\over\sigma_p^2}+{D_x^2+\tilde D_x^2 \over\sigma_{x\beta}^2}+{D_y^2+\tilde D_y^2\over \sigma_{y\beta}^2} \end{align}

An important part of this calculation is the momentum acceptance.
In AT, one may use the function momentum_aperture_at, for example:
[deltamax]=momentum_aperture_at(ring,0.1,[10^-6 10^-6],0,0,-0.01,3,10,100)


Touschek Effect Calculation and its Application to a Transport Line
formula and implementation in atcollab
momentum aperture calculation in AT
Note that there are some errors in various references
F. Wang: eqn 8 missing -1
A. Streun after eqn 1D, factor of 1/2 missing

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