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Chapter 32: Problem 46

Which of the following reactions are possible, and by what interaction could they occur? For those forbidden, explain why. (a) \(\pi^- + p \rightarrow K^0 + p + \pi^0\) (b) \(K^- + p \rightarrow \Lambda^0 + \pi^0\) (c) \(K^+ + n \rightarrow \Sigma^+ + \pi^0 + \gamma\) (d) \(K^+ \rightarrow \pi^0 + \pi^0 + \pi^+\) (e) \(\pi^+ \rightarrow e^+ + \nu_e\)

Short Answer

Expert verified
(b), (c), and (e) are possible; (a) and (d) are forbidden due to strangeness conservation violation.

Step by step solution

01

Examine Reaction (a)

We need to check conservation laws for the reaction \( \pi^- + p \rightarrow K^0 + p + \pi^0 \). Check each:- **Baryon Number**: Initial baryon number (1) is equal to final baryon number (1). Conserved.- **Charge**: Initial charge is 0 (\( \pi^- \) is -1, \( p \) is +1), and final charge is also 0 (\( K^0 \), \( p \) is +1, \( \pi^0 \) is 0). Conserved.- **Strangeness**: Initial strangeness is 0 (none for \( \pi^- \) or \( p \)), whereas the final strangeness would be -1 due to \( K^0 \). This violates strangeness conservation in strong and electromagnetic interactions.The reaction is forbidden due to strangeness conservation violation.
02

Examine Reaction (b)

Assess the reaction \( K^- + p \rightarrow \Lambda^0 + \pi^0 \).- **Baryon Number**: Initial (1, \( p \)) is equal to final (1, \( \Lambda^0 \)). Conserved.- **Charge**: Initial charge is 0 (\( K^- \) is -1, \( p \) is +1), final charge is also 0 (\( \Lambda^0 \), \( \pi^0 \) are neutral). Conserved.- **Strangeness**: Initial is -1 (\( K^- \)), final is -1 (\( \Lambda^0 \)). Conserved.This reaction is allowed and could occur via a strong interaction.
03

Examine Reaction (c)

Check conservation for \( K^+ + n \rightarrow \Sigma^+ + \pi^0 + \gamma \).- **Baryon Number**: Initial is 1 (\( n \)), final is 1 (\( \Sigma^+ \)). Conserved.- **Charge**: Initial charge is +1 (\( K^+ \)), final charge is +1 (\( \Sigma^+ \)). Conserved.- **Strangeness**: Initial is +1 (\( K^+ \)), final is +1 (\( \Sigma^+ \)). Conserved.The reaction is allowed and could proceed via a strong or electromagnetic interaction.
04

Examine Reaction (d)

Analyze \( K^+ \rightarrow \pi^0 + \pi^0 + \pi^+ \).- **Baryon Number**: All particles are mesons, so this is not applicable.- **Charge**: Initial charge is +1 (\( K^+ \)), final charge is +1 (\( \pi^+ \)). Conserved.- **Strangeness**: Initial strangeness is +1 (\( K^+ \)), while final strangeness is 0, violating conservation.The reaction is forbidden due to strangeness conservation violation in any interaction.
05

Examine Reaction (e)

Evaluate \( \pi^+ \rightarrow e^+ + u_e \).- **Baryon Number**: All are leptons and mesons, so this is not applicable.- **Charge**: Initial charge is +1 (\( \pi^+ \)), final charge is +1 (\( e^+ \)). Conserved.- **Lepton Number**: Conserved separately for electron flavors (no external \( u_e \) pair production).This reaction is allowed and occurs via a weak interaction known as beta decay.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Conservation Laws
In particle physics, conservation laws are fundamental principles. They state that certain properties of an isolated system remain constant over time. These include:
  • Baryon Number: Represents the total number of baryons (like protons and neutrons) minus the total number of antibaryons. This number must remain the same before and after a reaction.
  • Charge Conservation: The total electric charge is conserved in any reaction, meaning the sum of charges before the reaction must equal the sum after the reaction.
  • Strangeness: A quantum number related to the presence of strange quarks. In strong and electromagnetic interactions, strangeness must be conserved.
These conservation laws are like checklists. Particles need to "pass" each one for a reaction to occur. If any law isn't met, the reaction is forbidden within that interaction type.
Baryon Number
Baryon number is a special count in reactions involving baryons, which include protons and neutrons. Each baryon carries a baryon number of +1, while antibaryons have -1.
  • This number serves as a safeguard for processes, ensuring matter isn't simply disappearing or popping into existence without a partner.
  • In reactions like those in the original exercise, initial baryon numbers must match final baryon numbers.
  • In typical calculations, you'd tally the baryon numbers of all particles both before and after a reaction to confirm whether this conservation rule holds.
Observing conservation of baryon number is critical in particle reactions, which helps us understand why certain processes can happen while others cannot.
Strangeness
The quantum number known as strangeness helps us categorize particles involving strange quarks, like kaons or lambda particles. This characteristic isn't immediately noticeable but is crucial in particle interactions.
  • In strong and electromagnetic processes, strangeness is conserved. This means the sum of strange quarks has to be the same at both the beginning and end of a reaction.
  • The strange quark has a strangeness of -1, with particles containing these quarks also contributing to the strangeness count.
  • In the exercise, strangeness conservation plays a key role in determining whether some reactions can occur, as certain forbidden reactions fail to conserve strangeness.
Violations of strangeness conservation often indicate weak interactions, where this number isn't strictly maintained, offering insight into the forces at play in a given reaction.
Charge Conservation
Charge conservation ensures the sum of all electrical charges in a closed system remains unchanged, playing a paramount role in particle physics analyses.
  • Each particle carries an intrinsic charge: neutral particles have 0, while others have positive or negative values depending on their nature.
  • In a reaction, the algebraic total of all initial charges must match the total of final charges for the reaction to be allowed.
  • In the original exercise, careful tracking of charges helps validate which of the listed reactions are theoretically possible under known physical laws.
Charge conservation is fundamental across all known interactions. Adhering strictly to this law ensures particles interact consistently with electrodynamics, helping maintain universal stability.

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Most popular questions from this chapter

Calculate the \(Q\)-value for the reaction \(\pi^- + p \rightarrow \Lambda^0 + K^0\), when negative pions strike stationary protons. Estimate the minimum pion kinetic energy needed to produce this reaction. [Hint: Assume \(\Lambda^0\) and \(K^0\) move off with the same velocity.]

If \(2 \times 10^{14}\) protons moving at , \(\upsilon \approx c\) with \(KE = 4.0 TeV\), are stored in the 4.3-km-radius ring of the LHC, (a) how much current (amperes) is carried by this beam? (b) How fast would a 1500-kg car have to move to carry the same kinetic energy as this beam?

(II) A cyclotron with a radius of 1.0 m is to accelerate deuterons \((^{2}_{1}H)\) to an energy of 12 MeV. (a) What is the required magnetic field? (b) What frequency is needed for the voltage between the dees? (c) If the potential difference between the dees averages 22 kV, how many revolutions will the particles make before exiting? (d) How much time does it take for one deuteron to go from start to exit? (e) Estimate how far it travels during this time.

(II) Draw a Feynman diagram for the reaction \(n + \nu_ \mu \rightarrow p + \mu^-\).

Which of the following reactions are possible, and by what interaction could they occur? For those forbidden, explain why. (a) \(\pi^- + p \rightarrow K^+ + \Sigma^-\) (b) \(\pi^+ + p \rightarrow K^+ + \Sigma^+\) (c) \(\pi^- + p \rightarrow \Lambda^0 + K^0 + \pi^0\) (d) \(\pi^+ + p \rightarrow \Sigma^0 + \pi^0\) (e) \(\pi^- + p \rightarrow p + e^- + \overline{\nu}_e\)
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