Question

Radiation Therapy with ${\pi }^{-}$Mesons. Beams of${\pi }^{-}$mesons are used in radiation therapy for certain cancers. The energy comes from the complete decay of the ${\pi }^{-}$to stable particles.(a) Write out the complete decay of a${\pi }^{-}$meson to stable particles. What are these particles?(b) How much energy is released from the complete decay of a single ${\pi }^{-}$meson to stable particles? (You can ignore the very small masses of the neutrinos.)(c) How many ${\pi }^{-}$mesons need to decay to give a dose of $\mathbf{50}.\mathbf{0}\mathbf{Gy}$to $\mathbf{10}.\mathbf{0}\mathbf{g}$of tissue?(d) What would be the equivalent dose in part (c) in $\mathbf{Sv}$and in rem? Consult Table 43.3 and use the largest appropriate RBE for the particles involved in this decay.

Short answer

1. The complete decay of a ${\pi }^{-}$meson to a stable particle is $\pi \to e+{\overline{\nu }}_{\mu }+{\nu }_{\mu }+{\overline{\nu }}_e$, and the particles are:

$e^{-}$-stable electron

${\mu }^{-}$- Muon

${\overline{\nu }}_{\mu }$- Muon neutrino

${\overline{\nu }}_e$-Muon electron neutrino

b. The energy released from the complete decay of a single${\pi }^{-}$meson to the stable particle is $139.5\text{MeV}.$

c. $2.24\times {10}^{10}$particles ${\pi }^{-}$meson needs to decay a dose of $50\text{Gy}$to $10\text{g}$of tissue.

d. The equivalent dose is$75.0\text{Sv}$or $7500\text{rem}.$

Step-by-step solution

Define the Absorbed dose and Equivalent dose

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.

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

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

Determine the complete decay and energy released

a)

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

$$\begin{gathered} \pi \to {\mu }^{-}+{\overline{\nu }}_{\mu } \\ {\pi }^{-}\to e+{\overline{\nu }}_{\mu }+{\nu }_{\mu }+{\overline{\nu }}_e\left[\because {\mu }^{-}=e+{\nu }_{\mu }+{\overline{\nu }}_e\right] \end{gathered}$$

Where, the particles are:$e^{-}-$ stable electron Muon, ${\overline{\nu }}_{\mu }-$Muon neutrino and ${\overline{\nu }}_e-$Muon electron antineutrino.

b)

The rest energy $E_r\left({\pi }^{-}\right)$of ${\pi }^{-}$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${\pi }^{-}$particles when it is at rest.

So,

$Q=E_r\left({\pi }^{-}\right)-E_r\left(e^{-}\right)$

Substituting the values, and we get,

$$\begin{gathered} Q=140-0.511 \\ =139.5\text{MeV} \end{gathered}$$

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

Determine the number of particles and equivalent dose.

c)

Given that,

$$\begin{gathered} \text{Asorbed dose}=50.0\text{Gy} \\ m=0.10\text{g} \end{gathered}$$

The number of ${\pi }^{-}$mesons required;

$$\begin{gathered} N=\frac{E}{Q} \\ =\frac{\left(\text{absorbed dose}\right)\times m}{Q} \\ =\frac{\left(50\right)\left(0.010\right)}{139.5\left(1.60\times {10}^{-13}\right)} \\ =2.24\times {10}^{10}\text{particle} \end{gathered}$$

Hence, $2.24\times {10}^{10}$particle ${\pi }^{-}$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 $\text{Sv}$:

$$\begin{gathered} \text{Equivalent dose}=\left(1.50\right)\times 50 \\ =75.0\text{Sv} \end{gathered}$$

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 $7500\text{rem}.$