%*************  PHY 5846  Fall 1998  -- Homework  ***************8
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\begin{document}
\centerline {\bf PHY5846C: INTRODUCTION TO EXPERIMENTAL TECHNIQUES}
\medskip
\centerline {ACCELERATORS}
\bigskip
\centerline {\bf ***** Homework due 5 November 1998 *****}

\begin{enumerate}
\item  The luminosity $\Lum $ of a colliding beam machine is defined by
$$ N = \Lum \sigma ~,$$
where $N$ interactions occur per second for cross section $\sigma $. For a machine
with $n_1, n_2$ particles per bunch in the two beams with $n_B$ bunches per beam,
and a bunch revolution frequency $f$, show that
\begin{itemize}
\item[(a)] the luminosity is given by 
$$ \Lum = {{n_1 n_2}\over {A}} f n_B ~,$$
for uniform cylindrical bunches of cross sectional area $A$,
\item[(b)] while for bunches with gaussian profiles perpendicular to the beam
direction, with rms widths $\sigma_x , \sigma_y $, the luminosity becomes
$$ \Lum = {{n_1 n_2}\over {4\pi \sigma_x \sigma_y }} f n_B ~.$$ 
\item[(c)] Calculate the luminosity for an $e^+e^-$ collider of circumference
2.4km with two bunches in each beam and beam currents 10mA each, for 
$\sigma_x = 500\mu $m, $\sigma_y = 50 \mu $m. How many events are accumulated
per day for a cross section of 1nb?
\item[(d)] Calculate the luminosity for a proton synchrotron with a cycle time
of 50s delivering $10^{11}$ particles per burst to a fixed liquid hydrogen target
of length 50cm.
\end{itemize}
(See refs. [1], [2a], [2b]).

\item 
In a collider, the particle beams are stored in the machine for many hours.
Throughout this time, for each turn, the two beams must pass close to each other 
 at the focal points corresponding to the 
interaction regions where detectors are located. Furthermore, the energy loss
due to synchrotron radiation must be compensated by supplying power to the
accelerating RF cavities.
\begin{itemize}
\item[(a)] Calculate the angular accuracy required to maintain the relative
alignment of the two beams to within $100\mu $m for ten hours, and compare it
with the angular accuracy required to direct a space craft to within 20km
at the orbit of Uranus (at $\approx 3\cdot 10^9 $km from the Sun).
\item[(b)] Calculate: 
\begin{itemize}
\item the energy lost due to synchrotron radiation by one 
particle per revolution 
\item the average power that the RF cavities must transfer to the beams in order to
compensate for the energy loss by synchrotron radiation
\begin{itemize}
\item[(b1)] in PETRA (circumference 2.4km, bending radius 192m) at 15 GeV,
           for 10mA current in each beam;
\item[(b2)] in LEP (circumference 26.66km, bending radius 3096m) at 45 GeV,
           for 3mA current in each beam;
\item[(b3)] in LEP at 80 GeV, for 3mA current;
\item[(b4)] in the TeVatron collider (circumference 6.28km, bending radius 754m)
              at 800 GeV, for 3mA proton and 1.5mA antiproton current.
\end{itemize}
\end{itemize}
\end{itemize}

(See refs. [3], [2a], [2d], [2e]).
\end{enumerate}
\bigskip
References
\begin{enumerate}
\item  Landau-Lifshitz: Theory of classical fields, chapter 2\\
   (for def. of cross section and relativistically invariant formulation of flux
factor) 
\item    Particle Data Group: Review of particle properties, European Physical
    Journal C
{\bf 3} (1998) 1 - 794.\\
\tab  (a) p. 69 for physical constants,\\
\tab  (b) p. 76 for properties of materials used in detectors,\\
\tab (c) p. 138 for physics of colliders\\
\tab (d) p. 141 for properties of colliders,\\
\tab  (e) p. 78 for useful electromagnetic relations,\\
\tab (f) p. 186 for useful formulae of relativistic kinematics.
\item  J.D. Jackson: Classical Electrodynamics (sect. 14.2 for synchrotron
radiation).
\end{enumerate}
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