91Ó°ÊÓ

The ratio of total energy delivered to the mass of the tissue is known as the absorbed dose.

$\text{Absorbed dose}=\frac{\text{Total energy delivered}}{\text{Mass of the tissue}}$

$=\frac{E}{m}$

The unit of absorbed dose is Gray $\left(Gy\right)$ and $1\text{Gy}=1.0\text{J/kg}.$

The product of the absorbed dose and the RRB of the radiation is known as the biological equivalent dose.

$\text{Equivalent dose}\left(S_v\right)=PBE×\text{Ansprbed dose}\left(Gy\right)$

Also, $1.0\text{rem}=100\text{Sv}.$

a)

The complete decay of ${\mathrm{Ï€}}^{-}$meson to the particle is:

$$\begin{gathered} \mathrm{Ï€}→{\mathrm{μ}}^{-}+{\stackrel{¯}{\mathrm{ν}}}_{\mathrm{μ}} \\ {\mathrm{Ï€}}^{-}→e+{\stackrel{¯}{\mathrm{ν}}}_{\mathrm{μ}}+{\mathrm{ν}}_{\mathrm{μ}}+{\stackrel{¯}{\mathrm{ν}}}_e\left[âˆ\mu {\mathrm{μ}}^{-}=e+{\mathrm{ν}}_{\mathrm{μ}}+{\stackrel{¯}{\mathrm{ν}}}_e\right] \end{gathered}$$

Where, the particles are:$e^{-}-$ stable electron Muon, ${\stackrel{¯}{\mathrm{ν}}}_{\mathrm{μ}}-$Muon neutrino and ${\stackrel{¯}{\mathrm{ν}}}_e-$Muon electron antineutrino.

b)

The rest energy $E_r\left({\mathrm{Ï€}}^{-}\right)$of ${\mathrm{Ï€}}^{-}$is $140\text{MeV}$, and the rest energy$E_r\left(e^{-}\right)$of the electron is $\text{0.511}\text{MeV}.$

Assume that, the decay ends up with the electron, and decay starts with${\mathrm{Ï€}}^{-}$particles when it is at rest.

So,

$Q=E_r\left({\mathrm{Ï€}}^{-}\right)-E_r\left(e^{-}\right)$

Substituting the values, and we get,

Hence, the energy released from the complete decay of a single ${\mathrm{Ï€}}^{-}$ meson to the stable particle is $139.5\text{MeV}.$

c)

Given that,

The number of ${\mathrm{Ï€}}^{-}$mesons required;

Hence, $2.24×{10}^{10}$particle ${\mathrm{π}}^{-}$meson needs to decay a dose of $50\text{Gy}$to $10\text{g}$of tissue.

d)

The largest value of RBE for the electron is $1.50\text{Sv/Gy}.$

The equivalent dose in $\text{Sv}$:

Now, since $1.0\text{rem}=100\text{Sv}$

Therefore, $\text{Equivalent dose}=7500\text{rem}.$

Hence, the equivalent dose is $75.0\text{Sv}$ or $7500\text{rem}.$