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Chapter 43: Problem 51

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

Short Answer

Expert verified
Reactions (a), (b), and (c) are possible.

Step by step solution

01

Charge Conservation Check

For all reactions, we need to ensure that the total charge is conserved from reactants to products. (a) Left: -1 +1 = 0; Right: +1 -1 = 0. **Conseved** (b) Left: +1 +1 = +2; Right: +1 +1 = +2. **Conserved** (c) Left: -1 +1 = 0; Right: 0 +0 +0 = 0. **Conserved** (d) Left: +1 +1 = +2; Right: 0 +0 = 0. **Not conserved** (e) Left: -1 +1 = 0; Right: +1 -1 -1 = -1. **Not conserved** Reactions (a) and (b) are charge conserved, while reactions (d) and (e) are not.
02

Baryon Number Conservation Check

The baryon number conservation must also hold, evaluated as follows: (a) Left: 0 +1 = 1; Right: 0 +1 = 1. **Conserved** (b) Left: 0 +1 = 1; Right: 0 +1 = 1. **Conserved** (c) Left: 0 +1 = 1; Right: 0 +1 +0 = 1. **Conserved** (d) Left: 0 +1 = 1; Right: 1 +0 = 1. **Conserved** (e) Left: 0 +1 = 1; Right: +1 0 0 = 1. **Conserved** All reactions conserve baryon numbers.
03

Strangeness Conservation Check

The strangeness number has to be conserved in strong interactions. (a) Left: 0; Right: +1 (-1). **Conserved** if mediated via weak interaction. (b) Left: 0; Right: +1 (+1). **Conserved** (c) Left: 0; Right: 0 (+1 -1 +0). **Conserved** if mediated via weak interaction. (d) Not applicable due to charge non-conservation. (e) Not applicable due to charge non-conservation. Reactions (a) and (b) can conserve strangeness under different scenarios, with weak interaction in (a). (c) also remains via weak interaction.
04

Electromagnetic Rule Check (if needed)

Electromagnetic interactions usually involve photons and do not change strangeness or baryon number. Since the conservation of rules already analyzed are purely centered on nuclear processes without electromagnetic considerations significantly influenced. Reactions (d) and (e) have been disqualified earlier.
05

Decision

Determine which reactions are allowed and specify interaction type if known. (a) Possible via weak interaction (b) Possible via strong interaction (c) Possible via weak interaction (d) Forbidden (charge violation) (e) Forbidden (charge violation)

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Key Concepts

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

Charge Conservation
In every physical process, whether it's a particle reaction or any other interaction, the principle of charge conservation is paramount. This principle asserts that the total electric charge before and after any process must remain constant. Consider this as the law of balancing charge books, ensuring equilibrium on both sides.
  • For instance, in reaction (a), the initial charges of \(\pi^{-}\) and \(p\) result in zero, which is mirrored in the outcome charges of \(K^{+}\) and \(\Sigma^{-}\).
  • This balance confirms the reaction can potentially occur as the charge is conserved.
  • Reactions which don't conserve charge, like (d) with a total discrepancy of two units from left to right, are fundamentally impossible under common physical laws.
Charge conservation restricts or allows reactions, acting as an initial filter before assessing other conservation laws.
Baryon Number Conservation
Baryon number conservation is another fundamental rule governing particle interactions. Every baryon, which are a class of particles including protons and neutrons, has a baryon number of one, while their antiparticles have a baryon number of minus one. All non-baryons, like mesons, hold a baryon number of zero.
  • In essence, this law demands the number of baryons to remain the same before and after a reaction.
  • Throughout our given reactions, like (a) and (b), baryon numbers on both sides of the reactions remain equivalent, upholding this conservation law.
  • Such universally conserved attributes make baryon number conservation a pivotal determinant in assessing the feasibility of particle reactions.
Strangeness Conservation
Strangeness is a property assigned to certain hadrons involving strange quarks, and it must be conserved in strong interactions. This conservation provides insight into the exhibition of reactions beyond the basic view.
  • For instance, reaction (b) conserves strangeness, indicated by equal strangeness quantity before and after, enabling a likely strong interaction.
  • However, reactions such as (a) or (c) may show strangeness imbalance, only reconcilable through weak interactions that not strictly conserve strangeness.
  • Understanding whether the interaction is strong or weak can significantly influence whether strangeness needs to remain identical pre and post-reaction.
Weak Interaction
Weak interaction is one of the four fundamental forces, responsible for processes like beta decay. Unlike strong interactions, weak interactions do not need to conserve strangeness, empowering more flexibility in particle conversion and annihilation scenarios.
  • Reactions like (a) which initially seem unconserved in terms of strangeness, can occur through the weak force where such conservation isn't a requisite.
  • Weak interactions can change the type or orientation of particles involved, distinguishing it from electromagnetic or strong force scenarios.
  • This distinction is essential in analyzing reactions that would otherwise be dismissed under the stringency of laws like strangeness conservation.
Strong Interaction
The strong interaction, or nuclear force, is immensely powerful over short distances, binding protons and neutrons within atomic nuclei. It dominates interactions involving baryons and quarks, dictating certain conservation laws.
  • One key to identifying a strong interaction is observing conservation of baryon number and strangeness, as seen in reaction (b).
  • Strong interactions are typified by their short-range but astonishingly intense bonds, far outreaching those of electromagnetic interactions.
  • The possibility of a reaction following the strong interaction pathway is contingent upon adhering to these core conservation principles.

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Most popular questions from this chapter

What are the quark combinations that can form \((a)\) a neutron, \((b)\) an antineutron, (c) a \(\Lambda^{0},(d)\) a \(\bar{\Sigma}^{0} ?\)

(II) The D \(\mathrm{D}_{S}^{+}\) meson has \(S=c=+1, B=0 .\) What quark combination would produce it?

(II) Which of the following reactions and decays are possible? For those forbidden, explain what laws are violated. (a) \(\pi^{-}+\mathrm{p} \rightarrow \mathrm{n}+\eta^{0}\) (b) \(\pi^{+}+\mathrm{p} \rightarrow \mathrm{n}+\pi^{0}\) (c) \(\pi^{+}+\mathrm{p} \rightarrow \mathrm{p}+\mathrm{e}^{+}\) (d) \(\mathrm{p} \rightarrow \mathrm{e}^{+}+\nu_{\mathrm{c}}\) \((e) \mu^{+} \rightarrow \mathrm{e}^{+}+\overline{v}_{\mu}\) (f) \(\mathrm{p} \rightarrow \mathrm{n}+\mathrm{e}^{+}+\nu_{\mathrm{e}}\)

What quark combinations produce \((a)\) a \(\Xi^{0}\) baryon and (b) a \(\Xi^{-}\) baryon?

For the reaction \(p+p \rightarrow 3 p+\bar{p},\) where one of the initial protons is at rest, use relativistic formulas to show that the threshold energy is \(6 m_{\mathrm{p}} c^{2},\) equal to three times the magnitude of the \(Q\) -value of the reaction, where \(m_{\mathrm{p}}\) is the proton mass. [Hint: Assume all final particles have the same velocity.
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