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Birks' law

When particles moves through the scintillator, it loses energy, which is partially transformed into the photons that the detector collects. Let us call the response of the detector $L$, the energy of the particle $E$ and the distance traveled inside the scintillator $x$. Thus,
\[ dL = S(E) \cdot dE, \]where $S(E)$ characterizes the material of the characterizes and the circuit of the photon detection. The dependence of $S$ on $E$ arises from changes in the matter when strong interaction with the particle (high $\left|dE/dx\right|$) takes place.

Therefore, if a particle of energy $E_0$ hits the detector, the response is
\[ L = \int\limits_0^{E_0} S(E) \, dE.\]John B. Briks suggested an empirical law to describe how $S$ depends on the $E$:
\[S(E) = \frac{S_0}{1+kB \cdot \frac{dE}{dx}},\]where $S_0$ is a constant that can be measured for particles that weakly interact with the matter. The dependence of $S$ on $E$ is not explicit and is hidden in the fact that $dE/dx$ is a function of $E$. The constant $kB$ characterizes the internal dynamics of the matter and varies weakly with the type of particles.

The dependence of the response $L$ of the detector made of anthracene on the energy $E_0$ of the proton incident on the detector is provided. Also, the stopping power $dE/dx$ of the protons in anthracene as the function of their energy $E_0$ is given. Both tables can be found in Extras.

A1 Determine the value of $kB$ for anthracene.