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\author{Tran Hoai Nam}
\title{Answer to quizzes in \\Detectors of LHC Experiments lecture}

\begin{document}
\maketitle
\section*{Quiz 1: Number of protons  for $\mathcal{L}=10^{34}$
cm$^{-2}$s$^{-1}$}
Total luminosity of LHC:
\[
\mathcal{L} = f N_b L_{bunch\_cross} = f N_b \frac{N_p \times N_p}{4\pi\sigma_x\sigma_y}
\]
\[
\rightarrow N_p = \sqrt{ 4\pi \sigma_x\sigma_y \frac{\mathcal{L}}{fN_b}}
\]
where: $f$ is revolution frequency; 
\[
f \approx \frac{c}{l_{LHC}} 
= \frac{3\times 10^8 [ms^{-1}]}{27 \times 10^3 [m]} \approx 10^4 [s^{-1}]
\]
Substitute the numbers, number of proton in each bunch is:
\[
N_p = 
\sqrt{4\times 3.14 \times 16 \times 10^{-6} [m]\times 16 \times 10^{-6} [m]
\times
\frac{10^{34} \times 10^{4}[m^{-2}s^{-1}]}{2808 \times 10^4[s^{-1}]}}
\approx 1.07 \times 10^{-11} 
\]

\section*{Quiz 2: Characteristics of a silicon detector}
\subsection*{1) Full depletion voltage}
Let $V$ be the full depletion voltage for the 300 $\mu m$ Si detector:
\[
d = \sqrt{\frac{2\varepsilon^{'} \varepsilon_0}{q_e N_{eff}}V}
\rightarrow  V = \frac{q_e N_{eff}}{2 \varepsilon^{'} \varepsilon_0} d^2
\]
\[
V = \frac{1.6 \times 10^{-19} [C] \times 2 \times 10^{12} [cm^{-3}]}
{2\times11.6\times 8.854 \times 10^{-14} [F cm^{-1}]} (300 \times 10^{-4} [cm])^2
\approx 140.2 [V]
\]
\subsection*{2) Noise}
Capacitance of the detector when it is fully depleted:
\begin{align*}
C &= \varepsilon^{'} \varepsilon_0 \frac{S}{d} \\
 	&= 11.6 \times 8.854 \times 10^{-14} [F cm^{-1}] 
		\frac{1 [cm^2]}{300 \times 10^{-4} [cm]} \\
  &\approx 34.2 \times 10^{-12} [F] \\
	&= 34.2 [pF]
\end{align*}

Equivalent noise charge square:
\[
ENC^2 = \left(\frac{e^3}{36q_e}\right)^2 
\left[ \frac{3}{\tau} (4k_B T R_S) C^2 + \frac{5\tau}{3}(2q_eI_{leak})\right]
\]

We have:
\begin{align*}
k_B T &= 8.62 \times 10^{-5} [eV K^{-1}] \times 273.15 [K] \\
			&\approx  2.35 \times 10^{-2} [eV] \\
			&= 2.35 \times 10^{-2} \times 1.6 \times 10^{-19} [J]\\
			&= 2.35 \times 10^{-2} q_e [JC^{-1}]
\end{align*}
Thus:
\begin{align*}
ENC^2 = 
	&	\frac{2.72^6}{36^2 \times 1.6 \times 10^{-19}[C]} \times \\
	& \times \Biggl[ 
		\frac{3}{20 \times 10^{-9} [s]} 
		(4 \times 2.35 \times 10^{-2} [JC^{-1}] \times 200 [\Omega]) 
		(34.2 \times 10^{-12} [F])^{2} + \\ 
	& + \frac{5 \times 20 \times 10^{-9}[s]}{3}
		(2 \times
		 10 \times 10^{-6} [A])
	\Biggr]
\end{align*}

\begin{align*}
ENC^2 &= 
	\frac{2.72^6}{36^2 \times 1.6 \times 10^{-19}[C]} 
	 \times \Biggl[ 
	3.3 \times 10^{-12}[Js^{-1}C^{-1}\Omega F^2] + 0.3 \times 10^{-12} [C]
	\Biggr] \\
	&\approx 7.0\times 10^6
\end{align*}

(dimensionally speaking, $Js^{-1}C^{-1}\Omega F^2
= VA \times C^{-1}\times VA^{-1}\times C^2V^{-2} = C$)

\[
\rightarrow ENC \approx 2.65 \times 10^3
\]

The $ENC$ is $2.65 \times 10^3$, and it can be seen that the contribution from
capacitance is dominant.
%\begin{align*}
%ENC^2 = 
	%&\left(
		%\frac{2.72^3}{36\times 1.6\times 10^{-19}[C]}
	%\right)^2 \times \\
	%& \times \Biggl[ 
		%\frac{3}{20 \times 10^{-9} [s]} 
		%(4 \times 8.62 \times 10^{-5} [eV K^{-1}] 
		%\times 273.15 [K] \times 200 [\Omega]) 
		%(34.2 \times 10^{-12} [F])^{2} + \\ 
	%& + \frac{5 \times 20 \times 10^{-9}[s]}{3}
		%(2 \times 1.6 \times 10^{-19}[C] \times
     %10 \times 10^{-6} [A])
	%\Biggr]
%\end{align*}

%\begin{align*}
%ENC^2 = 
	%&\left(
		%\frac{2.72^3}{36\times 1.6\times 10^{-19}[C]}
	%\right)^2 \times \\
	%& \times \Biggl[ 
		%\frac{3}{20 \times 10^{-9} [s]} 
		%(4 \times 2.35 \times 10^{-2}[eV] \times 200 [\Omega]) 
		%(34.2 \times 10^{-12} [F])^{2} + \\ 
	%& + \frac{5 \times 20 \times 10^{-9}[s]}{3}
		%(2 \times 1.6 \times 10^{-19} [C]\times
     %10 \times 10^{-6} [A])
	%\Biggr]
%\end{align*}

%\begin{align*}
%ENC^2 = 
	%&\left(
		%\frac{2.72^3}{36\times 1.6\times 10^{-19}[C]}
	%\right)^2 \times \\
	%& \times \Biggl[ 
		%\frac{3}{20 \times 10^{-9} [s]} 
		%(4 \times 2.35 \times 10^{-2} \times 1.6 \times 10^{-19}[J] \times 200 [\Omega]) 
		%(34.2 \times 10^{-12} [F])^{2} + \\ 
	%& + \frac{5 \times 20 \times 10^{-9}[s]}{3}
		%(2 \times 1.6 \times 10^{-19} [C]\times
     %10 \times 10^{-6} [A])
	%\Biggr]
%\end{align*}

\end{document}
