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University Physics Volume 3

11.2 Particle Conservation Laws

University Physics Volume 311.2 Particle Conservation Laws
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  1. Preface
  2. Unit 1. Optics
    1. 1 The Nature of Light
      1. Introduction
      2. 1.1 The Propagation of Light
      3. 1.2 The Law of Reflection
      4. 1.3 Refraction
      5. 1.4 Total Internal Reflection
      6. 1.5 Dispersion
      7. 1.6 Huygens’s Principle
      8. 1.7 Polarization
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    2. 2 Geometric Optics and Image Formation
      1. Introduction
      2. 2.1 Images Formed by Plane Mirrors
      3. 2.2 Spherical Mirrors
      4. 2.3 Images Formed by Refraction
      5. 2.4 Thin Lenses
      6. 2.5 The Eye
      7. 2.6 The Camera
      8. 2.7 The Simple Magnifier
      9. 2.8 Microscopes and Telescopes
      10. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
    3. 3 Interference
      1. Introduction
      2. 3.1 Young's Double-Slit Interference
      3. 3.2 Mathematics of Interference
      4. 3.3 Multiple-Slit Interference
      5. 3.4 Interference in Thin Films
      6. 3.5 The Michelson Interferometer
      7. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    4. 4 Diffraction
      1. Introduction
      2. 4.1 Single-Slit Diffraction
      3. 4.2 Intensity in Single-Slit Diffraction
      4. 4.3 Double-Slit Diffraction
      5. 4.4 Diffraction Gratings
      6. 4.5 Circular Apertures and Resolution
      7. 4.6 X-Ray Diffraction
      8. 4.7 Holography
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
  3. Unit 2. Modern Physics
    1. 5 Relativity
      1. Introduction
      2. 5.1 Invariance of Physical Laws
      3. 5.2 Relativity of Simultaneity
      4. 5.3 Time Dilation
      5. 5.4 Length Contraction
      6. 5.5 The Lorentz Transformation
      7. 5.6 Relativistic Velocity Transformation
      8. 5.7 Doppler Effect for Light
      9. 5.8 Relativistic Momentum
      10. 5.9 Relativistic Energy
      11. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
    2. 6 Photons and Matter Waves
      1. Introduction
      2. 6.1 Blackbody Radiation
      3. 6.2 Photoelectric Effect
      4. 6.3 The Compton Effect
      5. 6.4 Bohr’s Model of the Hydrogen Atom
      6. 6.5 De Broglie’s Matter Waves
      7. 6.6 Wave-Particle Duality
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
    3. 7 Quantum Mechanics
      1. Introduction
      2. 7.1 Wave Functions
      3. 7.2 The Heisenberg Uncertainty Principle
      4. 7.3 The Schrӧdinger Equation
      5. 7.4 The Quantum Particle in a Box
      6. 7.5 The Quantum Harmonic Oscillator
      7. 7.6 The Quantum Tunneling of Particles through Potential Barriers
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    4. 8 Atomic Structure
      1. Introduction
      2. 8.1 The Hydrogen Atom
      3. 8.2 Orbital Magnetic Dipole Moment of the Electron
      4. 8.3 Electron Spin
      5. 8.4 The Exclusion Principle and the Periodic Table
      6. 8.5 Atomic Spectra and X-rays
      7. 8.6 Lasers
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
    5. 9 Condensed Matter Physics
      1. Introduction
      2. 9.1 Types of Molecular Bonds
      3. 9.2 Molecular Spectra
      4. 9.3 Bonding in Crystalline Solids
      5. 9.4 Free Electron Model of Metals
      6. 9.5 Band Theory of Solids
      7. 9.6 Semiconductors and Doping
      8. 9.7 Semiconductor Devices
      9. 9.8 Superconductivity
      10. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    6. 10 Nuclear Physics
      1. Introduction
      2. 10.1 Properties of Nuclei
      3. 10.2 Nuclear Binding Energy
      4. 10.3 Radioactive Decay
      5. 10.4 Nuclear Reactions
      6. 10.5 Fission
      7. 10.6 Nuclear Fusion
      8. 10.7 Medical Applications and Biological Effects of Nuclear Radiation
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    7. 11 Particle Physics and Cosmology
      1. Introduction
      2. 11.1 Introduction to Particle Physics
      3. 11.2 Particle Conservation Laws
      4. 11.3 Quarks
      5. 11.4 Particle Accelerators and Detectors
      6. 11.5 The Standard Model
      7. 11.6 The Big Bang
      8. 11.7 Evolution of the Early Universe
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
  4. A | Units
  5. B | Conversion Factors
  6. C | Fundamental Constants
  7. D | Astronomical Data
  8. E | Mathematical Formulas
  9. F | Chemistry
  10. G | The Greek Alphabet
  11. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
  12. Index

Learning Objectives

By the end of this section, you will be able to:
  • Distinguish three conservation laws: baryon number, lepton number, and strangeness
  • Use rules to determine the total baryon number, lepton number, and strangeness of particles before and after a reaction
  • Use baryon number, lepton number, and strangeness conservation to determine if particle reactions or decays occur

Conservation laws are critical to an understanding of particle physics. Strong evidence exists that energy, momentum, and angular momentum are all conserved in all particle interactions. The annihilation of an electron and positron at rest, for example, cannot produce just one photon because this violates the conservation of linear momentum. As discussed in Relativity, the special theory of relativity modifies definitions of momentum, energy, and other familiar quantities. In particular, the relativistic momentum of a particle differs from its classical momentum by a factor γ=1/1(v/c)2γ=1/1(v/c)2 that varies from 1 to ,, depending on the speed of the particle.

In previous chapters, we encountered other conservation laws as well. For example, charge is conserved in all electrostatic phenomena. Charge lost in one place is gained in another because charge is carried by particles. No known physical processes violate charge conservation. In the next section, we describe three less-familiar conservation laws: baryon number, lepton number, and strangeness. These are by no means the only conservation laws in particle physics.

Baryon Number Conservation

No conservation law considered thus far prevents a neutron from decaying via a reaction such as

ne++e.ne++e.

This process conserves charge, energy, and momentum. However, it does not occur because it violates the law of baryon number conservation. This law requires that the total baryon number of a reaction is the same before and after the reaction occurs. To determine the total baryon number, every elementary particle is assigned a baryon number B. The baryon number has the value B=+1B=+1 for baryons, –1 for antibaryons, and 0 for all other particles. Returning to the above case (the decay of the neutron into an electron-positron pair), the neutron has a value B=+1,B=+1, whereas the electron and the positron each has a value of 0. Thus, the decay does not occur because the total baryon number changes from 1 to 0. However, the proton-antiproton collision process

p+pp+p+p+p,p+pp+p+p+p,

does satisfy the law of conservation of baryon number because the baryon number is zero before and after the interaction. The baryon number for several common particles is given in Table 11.2.

Particle name Symbol Lepton number (Le)(Le) Lepton number (Lμ)(Lμ) Lepton number (Lτ)(Lτ) Baryon number (B) Strange-ness number
Electron ee 1 0 0 0 0
Electron neutrino υeυe 1 0 0 0 0
Muon μμ 0 1 0 0 0
Muon neutrino υμυμ 0 1 0 0 0
Tau ττ 0 0 1 0 0
Tau neutrino υτυτ 0 0 1 0 0
Pion π+π+ 0 0 0 0 0
Positive kaon K+K+ 0 0 0 0 1
Negative kaon KK 0 0 0 0 –1
Proton p 0 0 0 1 0
Neutron n 0 0 0 1 0
Lambda zero Λ0Λ0 0 0 0 1 –1
Positive sigma Σ+Σ+ 0 0 0 1 –1
Negative sigma ΣΣ 0 0 0 1 –1
Xi zero Ξ0Ξ0 0 0 0 1 –2
Negative xi ΞΞ 0 0 0 1 –2
Omega ΩΩ 0 0 0 1 –3
Table 11.2 Conserved Properties of Particles

Example 11.1

Baryon Number Conservation Based on the law of conservation of baryon number, which of the following reactions can occur?

(a)π+pπ0+n+π+π+(b)p+pp+p+p(a)π+pπ0+n+π+π+(b)p+pp+p+p

Strategy Determine the total baryon number for the reactants and products, and require that this value does not change in the reaction. Solution

For reaction (a), the net baryon number of the two reactants is 0+1=10+1=1 and the net baryon number of the four products is 0+1+0+0=1.0+1+0+0=1. Since the net baryon numbers of the reactants and products are equal, this reaction is allowed on the basis of the baryon number conservation law.

For reaction (b), the net baryon number of the reactants is 1+(−1)=01+(−1)=0 and the net baryon number of the proposed products is 1+1+(−1)=1.1+1+(−1)=1. Since the net baryon numbers of the reactants and proposed products are not equal, this reaction cannot occur.

Significance Baryon number is conserved in the first reaction, but not in the second. Baryon number conservation constrains what reactions can and cannot occur in nature.

Check Your Understanding 11.1

What is the baryon number of a hydrogen nucleus?

Lepton Number Conservation

Lepton number conservation states that the sum of lepton numbers before and after the interaction must be the same. There are three different lepton numbers: the electron-lepton number Le,Le, the muon-lepton number Lμ,Lμ, and the tau-lepton number Lτ.Lτ. In any interaction, each of these quantities must be conserved separately. For electrons and electron neutrinos, Le=1;Le=1; for their antiparticles, Le=−1;Le=−1; all other particles have Le=0.Le=0. Similarly, Lμ=1Lμ=1 for muons and muon neutrinos, Lμ=−1Lμ=−1 for their antiparticles, and Lμ=0Lμ=0 for all other particles. Finally, Lτ=1,−1Lτ=1,−1, or 0, depending on whether we have a tau or tau neutrino, their antiparticles, or any other particle, respectively. Lepton number conservation guarantees that the number of electrons and positrons in the universe stays relatively constant. (Note: The total lepton number is, as far as we know, conserved in nature. However, observations have shown variations of family lepton number (for example, Le)Le) in a phenomenon called neutrino oscillations.)

To illustrate the lepton number conservation law, consider the following known two-step decay process:

π+μ++νμμ+e++νc+νμ.π+μ++νμμ+e++νc+νμ.

In the first decay, all of the lepton numbers for π+π+ are 0. For the products of this decay, Lμ=−1Lμ=−1 for μ+μ+ and Lμ=1Lμ=1 for νμ.νμ. Therefore, muon-lepton number is conserved. Neither electrons nor tau are involved in this decay, so Le=0Le=0 and Lτ=0Lτ=0 for the initial particle and all decay products. Thus, electron-lepton and tau-lepton numbers are also conserved. In the second decay, μ+μ+ has a muon-lepton number Lμ=−1,Lμ=−1, whereas the net muon-lepton number of the decay products is 0+0+(−1)=10+0+(−1)=1. Thus, the muon-lepton number is conserved. Electron-lepton number is also conserved, as Le=0Le=0 for μ+μ+, whereas the net electron-lepton number of the decay products is (−1)+1+0=0(−1)+1+0=0. Finally, since no taus or tau-neutrons are involved in this decay, the tau-lepton number is also conserved.

Example 11.2

Lepton Number Conservation Based on the law of conservation of lepton number, which of the following decays can occur?

(a)np+e+νe(b)πμ+νμ+νμ(a)np+e+νe(b)πμ+νμ+νμ

Strategy Determine the total lepton number for the reactants and products, and require that this value does not change in the reaction.

Solution For decay (a), the electron-lepton number of the neutron is 0, and the net electron-lepton number of the decay products is 0+1+(−1)=00+1+(−1)=0. Since the net electron-lepton numbers before and after the decay are the same, the decay is possible on the basis of the law of conservation of electron-lepton number. Also, since there are no muons or taus involved in this decay, the muon-lepton and tauon-lepton numbers are conserved.

For decay (b), the muon-lepton number of the ππ is 0, and the net muon-lepton number of the proposed decay products is 1+1+(−1)=11+1+(−1)=1. Thus, on the basis of the law of conservation of muon-lepton number, this decay cannot occur.

Significance Lepton number is conserved in the first reaction, but not in the second. Lepton number conservation constrains what reactions can and cannot occur in nature.

Check Your Understanding 11.2

What is the lepton number of an electron-positron pair?

Strangeness Conservation

In the late 1940s and early 1950s, cosmic-ray experiments revealed the existence of particles that had never been observed on Earth. These particles were produced in collisions of pions with protons or neutrons in the atmosphere. Their production and decay were unusual. They were produced in the strong nuclear interactions of pions and nucleons, and were therefore inferred to be hadrons; however, their decay was mediated by the much more slowly acting weak nuclear interaction. Their lifetimes were on the order of 10−1010−10 to 10−8s,10−8s, whereas a typical lifetime for a particle that decays via the strong nuclear reaction is 10−23s.10−23s. These particles were also unusual because they were always produced in pairs in the pion-nucleon collisions. For these reasons, these newly discovered particles were described as strange. The production and subsequent decay of a pair of strange particles is illustrated in Figure 11.4 and follows the reaction

π+pΛ0+K0.π+pΛ0+K0.

The lambda particle then decays through the weak nuclear interaction according to

Λ0π+p,Λ0π+p,

and the kaon decays via the weak interaction

K0π++π.K0π++π.
Figure a shows a photograph with a black background and a white pattern of swirls and lines on it. There is a bright white spot on the top left. Figure b shows the same pattern as a line drawing. It is labeled in various places with names of particles.
Figure 11.4 The interactions of hadrons. (a) Bubble chamber photograph; (b) sketch that represents the photograph.

To rationalize the behavior of these strange particles, particle physicists invented a particle property conserved in strong interactions but not in weak interactions. This property is called strangeness and, as the name suggests, is associated with the presence of a strange quark. The strangeness of a particle is equal to the number of strange quarks of the particle. Strangeness conservation requires the total strangeness of a reaction or decay (summing the strangeness of all the particles) is the same before and after the interaction. Strangeness conservation is not absolute: It is conserved in strong interactions and electromagnetic interactions but not in weak interactions. The strangeness number for several common particles is given in Table 11.2.

Example 11.3

Strangeness Conservation (a) Based on the conservation of strangeness, can the following reaction occur?

π+pK++K+n.π+pK++K+n.

(b) The following decay is mediated by the weak nuclear force:

K+π++π0.K+π++π0.

Does the decay conserve strangeness? If not, can the decay occur?

Strategy Determine the strangeness of the reactants and products and require that this value does not change in the reaction.

Solution

  1. The net strangeness of the reactants is 0+0=0,0+0=0, and the net strangeness of the products is 1+(−1)+0=0.1+(−1)+0=0. Thus, the strong nuclear interaction between a pion and a proton is not forbidden by the law of conservation of strangeness. Notice that baryon number is also conserved in the reaction.
  2. The net strangeness before and after this decay is 1 and 0, so the decay does not conserve strangeness. However, the decay may still be possible, because the law of conservation of strangeness does not apply to weak decays.

Significance Strangeness is conserved in the first reaction, but not in the second. Strangeness conservation constrains what reactions can and cannot occur in nature.

Check Your Understanding 11.3

What is the strangeness number of a muon?

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