4.1-3

Question 1

Binding Energy of the Nucleus

and the which letter stands for what

Answer 1

• Binding energy of nucleus = mass deficit.
  E_B = [Z · m_p + (A - Z) · m_n - m_nuc] · c²
• Z: number of protons
• A: number of nucleons
• m_p: mass of proton (≈ 940 MeV)
• m_n: mass of neutron (≈ 940 MeV)
• m_nuc: mass of nucleus.

Question 2

Binding Energy per Nucleon

Answer 2

f = E_B / A
- binding energy of nucleus divided by number of nucleons
example
- f(⁴He) = 6.6 MeV or 0.007 m_p · c²
- f(⁵⁶Fe) = 8.4 MeV or 0.009 _p · c².

Question 3

Exothermic Reactions

Answer 3

Exothermic reactions:
- fusion for Z₁ + Z₂ → Z ≤ Z(Fe)
- fission for Z → Z₁ + Z₂ ≫ Z(Fe)

• energy is released in these two kinds of nuclear reactions
• fusion of very light nuclei into somewhat heavier nuclei
• fission of very heavy nuclei into intermediate-mass nuclei

Question 4

Energy Production by Fusion Reaction

Answer 4

Nuclear Potential
- all nuclei are positively charged: repel each other
- only when brought within 10^-15m: short-range nuclear forces overcome repulsion → nuclei fuse

• r < 10^-15m:
  – Attractive strong nuclear force dominates (stronger than repulsion)
  – Minimum: E_min ≈ E_str≈ -8 (A₁ + A₂) MeV.
• r > 10^-15m:
  – Electro-static potential between two nuclei Z₁ and Z₂:
  E_es ∝ (1 / (4πε₀)) · (Z₁ · Z₂ · e²) / r
  – Maximum: E_max ≈ E_es(r ≈ 10^-15m) ≈ Z₁ Z₂ MeV.

Question 5

Problem in Energy Production by Fusion in Stars

Answer 5

• at center of sun (T≈ 10⁷K) typical kinetic energy k_BT of particle about 1 keV
• 10³ lower than electrostatic potential barrier betwen nuclei → even center of the sun not hot enough for nuclei to overcome repulsion and fuse (classically)

E_max ≫ E_kin in stars

→ particles must tunnel through electrostatic barrier.

Question 6

Calculation of Reaction Rates R

Answer 6

R = n₁ · n₂ · <σ(E) · v(E)>
- n₁, n₂: densities of reacting particles
- σ(E): cross section for reaction.
- v(E): velocity distribution of particles.
- R depends on Maxwell-Boltzmann distribution and average cross section times number of interactions.

Question 7

Maxwell-Boltzmann “kinetic energy” distribution

Answer 7

• E = mv²/ 2 (m reduced mass)
• Distribution of probability of relative kinetic energy (velocity) of two nuclei to be E:
  f(E) dE =[1/√π] · [E^1/2/(k_BT)^3/2] · exp(-E/k_BT) dE

→ reaction rate and reaction cross-section

Question 8

average cross section for fusion

Answer 8

< σ v > = ∫ σ(E) · v · f(E) · dE

Question 9

Reaction Coefficient and Tunneling Probability

Answer 9

σ(E) ∝ S(E) · e^{(-b / √E)} / E
- S(E) depends on nuclear structure
- b includes reduced mass m and the electro-static potential

b ∝ m^1/2 Z₁ Z₂ e²

Question 10

Energy Generation by Hydrogen

Answer 10

• nuclear energy generation function:
  ε = C · ρ·X₁·X₂·<σv>
• ε increases with temperature and reactions involving heavier nuclei are much less likely compared to reaction with lighter nuclei at given T
• when calculating energy generation, necessary to add up energies released in all reaction in chain
• for pp-chain the pp2 and pp3 branches are neglected
• experimentally determine the cross-section

p-p chain: ε_pp = 0.24 · ρ · X² · (10⁶/T)^-2/3 · e^{(-33.8 · (106/T)1/3)} W/kg

CNO cycle: ε_CNO = 8.7 · 10²⁰ · ρ · X · X_CNO · (10⁶/T)^-2/3 ·
e^{(-152.3 · (106/T)1/3)} W/kg

Question 11

List of Important Subatomic Particles

Answer 11

• Proton (p)
• Neutron (n)
• Electron (e^-)
• Positron (e⁺β⁺)
• Neutrino (ν)
• Anti-neutrino (ν̄)
• Alpha (α)
• Nucleon (_Z^AX).

maybe add table from script

Question 12

Nuclear Reactions in Stars (3+8)

and conservation laws

Answer 12

Must fulfill conservation laws:
- baryon number
- lepton number
- charge, etc.

Includes
- proton capture
- neutron capture
- electron capture
- fusion of two nuclei
- α-decay
- β-decays
- free neutron decay
- inverse neutron decay.

add table from script with reaction processes!