smc-3704-30-Nuclear Transformation

Consider the possibility of an electron and a positron colliding in a particle accelerator to produce a proton and an antiproton, as shown in the equation below.

\(\text{electron}+ \text{positron}\ \rightarrow \ \text{proton}+ \text {antiproton}\)

Which statement makes the correct conclusion about the possibility of such a reaction, and provides a plausible reason for this conclusion?

The reaction is impossible because electrons and positrons will combine to produce a single neutral particle.
The reaction is possible because the masses of the proton and antiproton are the result of their relativistic velocities.
The reaction is possible because the masses of the proton and antiproton come mainly from energy supplied by the accelerator.
The reaction is impossible because protons are much more massive than electrons and hence the reaction violates the law of conservation of mass.

Show Answers Only

\(C\)

Show Worked Solution

Option \(C\) is correct:

The electron and positron have very small rest masses compared to protons and antiprotons.
The particle accelerator supplies enormous kinetic energy to the electron and positron by accelerating them to speeds close to the speed of light.
This kinetic energy can be converted into mass to create the heavier proton-antiproton pair according to Einstein’s \(E = mc^2\) equation.
This means conservation of energy is satisfied, even though the products have much greater rest mass than the reactants.

Other options:

The reaction is possible – see above (eliminate \(A\) and \(D\)).
Option \(B\) is incorrect. The proton and antiproton’s mass must come from the reactants’ energy (electron + positron + accelerator energy), not from their own relativistic velocities.

The simplified model below shows the reactants and products of a proton-antiproton reaction which produces three particles called pions, each having a different charge.

\(\text{p}+\overline{\text{p}} \rightarrow \pi^{+}+\pi^0+\pi^{-}\)

There are no other products in this process, which involves only the rearrangement of quarks. No electromagnetic radiation is produced. Assume that the initial kinetic energy of the proton and antiproton is negligible.

Protons consist of two up quarks \(\text{(u)}\) and a down quark \(\text{(d)}\) . Antiprotons consist of two up antiquarks \((\overline{\text{u}})\) and a down antiquark \((\overline{\text{d}})\). Each of the pions consists of two quarks.

The following tables provide information about hadrons and quarks.

Table 1: Hadron Information

\begin{array} {|l|c|c|}
\hline
\rule{0pt}{2.5ex} \quad \quad \ \ \textit{Particle} & \ \ \textit{Rest mass} \ \ & \quad \textit{Charge} \quad \\
& \left(\text{MeV/c}^2\right)&\\
\hline
\rule{0pt}{2.5ex} \text {proton (p)} \rule[-1ex]{0pt}{0pt} & 940 & +1 \\
\hline
\rule{0pt}{2.5ex} \text {antiproton}(\overline{\text{p}}) \rule[-1ex]{0pt}{0pt} & 940 & -1 \\
\hline
\rule{0pt}{2.5ex} \text {neutral pion }\left(\pi^0\right) \rule[-1ex]{0pt}{0pt} & 140 & \text{zero} \\
\hline
\rule{0pt}{2.5ex} \text{positive pion }\left(\pi^{+}\right) \rule[-1ex]{0pt}{0pt} & 140 & +1 \\
\hline
\rule{0pt}{2.5ex}\text {negative pion }\left(\pi^{-}\right) \rule[-1ex]{0pt}{0pt} & 140 & -1\\
\hline
\end{array}

Table 2: Quark charges

\begin{array} {|l|c|}
\hline
\rule{0pt}{2.5ex} \quad \quad \ \ \textit{Particle} \rule[-1ex]{0pt}{0pt} & \quad \textit{Charge} \quad \\
\hline
\rule{0pt}{2.5ex} \text {down quark (d)} \rule[-1ex]{0pt}{0pt} & -\dfrac{1}{3} \\
\hline
\rule{0pt}{2.5ex} \text {up quark (u)} \rule[-1ex]{0pt}{0pt} & +\dfrac{2}{3}\\
\hline
\rule{0pt}{2.5ex} \text {down antiquark}(\overline{\text{d}}) \rule[-1ex]{0pt}{0pt} & +\dfrac{1}{3}\\
\hline
\rule{0pt}{2.5ex} \text{up antiquark }(\overline{\text{u}}) \rule[-1ex]{0pt}{0pt} & -\dfrac{2}{3} \\
\hline
\end{array}

Identify the quarks present in the \(\pi^{-}, \pi^{+}\)and the \(\pi^0\) particles. (2 marks)

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The energy released in the reaction is shared equally between the pions.
Calculate the energy released per pion in this reaction. (2 marks)

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Calculation of the pions' velocities using classical physics predicts that each pion has a velocity, relative to the point at which the proton-antiproton reaction occurred, which exceeds 3 × 10\(^8\) m s\(^{-1}\).
Explain the problem with this prediction and how it can be resolved. (3 marks)

--- 6 WORK AREA LINES (style=lined) ---

Show Answers Only

a. \(\pi^{-}:\ \overline{\text{u}}\text{d}\)

\(\pi^{+}:\ \text{u}\overline{\text{d}}\)

\(\pi^{0}:\ \text{u}\overline{\text{d}}\)

b. \(\text{Initial mass}\ = 940 + 940 = 1880\ \text{MeV/c}^{2} \)

\(\text{Final mass}\ = 3 \times 140 = 420\ \text{MeV/c}^{2} \)

\(\Delta \text{Mass (per pion)}\ = \dfrac{1460}{3} = 487\ \text{MeV/c}^{2} \)

\(\text{Using}\ \ E=mc^2:\)

\(\text{Energy released (per pion)}\ = 487\ \text{MeV}\)

c. Problem with prediction:

The calculation shows the pions moving faster than light speed (3 × 10\(^{8}\) m/s), which can’t happen in reality.

Resolving the problem:

Since these pions are moving at extremely high speeds, we need to account for relativity.
Relativity means that pions’ mass actually increases as they get faster, which prevents them from ever reaching light speed.
Part of the energy given to the pions goes into increasing their mass rather than just increasing their velocity.

Show Worked Solution

a. \(\pi^{-}:\ \overline{\text{u}}\text{d}\)

\(\pi^{+}:\ \text{u}\overline{\text{d}}\)

\(\pi^{0}:\ \text{u}\overline{\text{d}}\)

b. \(\text{Initial mass}\ = 940 + 940 = 1880\ \text{MeV/c}^{2} \)

\(\text{Final mass}\ = 3 \times 140 = 420\ \text{MeV/c}^{2} \)

\(\Delta \text{Mass (per pion)}\ = \dfrac{1460}{3} = 487\ \text{MeV/c}^{2} \)

\(\text{Using}\ \ E=mc^2:\)

\(\text{Energy released (per pion)}\ = 487\ \text{MeV}\)

♦ Mean mark (b) 41%.

c. Problem with prediction:

The calculation shows the pions moving faster than light speed (3 × 10\(^{8}\) m/s), which can’t happen in reality.

Resolving the problem:

Since these pions are moving at extremely high speeds, we need to account for relativity.
Relativity means that pions’ mass actually increases as they get faster, which prevents them from ever reaching light speed.
Part of the energy given to the pions goes into increasing their mass rather than just increasing their velocity.

♦ Mean mark (c) 43%.

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