Nuclear and High Energy Particle Physics

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What is the definition of the mean free path, the equation and define all the symbols

The mean free path is the average distance a particle travels before scattering. It describes the spatial decay of the beam. λ = 1 /nσ where n is the target density and σ is the cross section.

What is the definition of the cross section, the equation and define all the symbols

The cross section represents the probability of a collision between two particles. Rate of collision per unit volume = φnσ where φ is the incident beam flux and n is target density.

What are the differences and comparisons that can be made between Rutherford’s and Mott’s cross section models for coulomb scattering?

• Rutherford’s model was developed for non-relativistic alpha particles scattering off small nuclei while Mott’s model includes quantum mechanical properties such as spin

• Rutherford’s model assumes a ‘static’ positively charged nucleus and ignores relativistic effects and spin.

• Mott’s model accounts for deviations observed at higher velocities and scattering angles​.

What is Rutherford’s cross section model suitable for?

It is suitable for low-energy collisions and small scattering angles, where simplifications for relativistic effects and spin are valid​.

What is Mott’s cross section model important for?

It is important for high-energy collisions with high projectile velocities or large scattering angles (θ→180).

What is the definition of nuclear form factor, what does it account for and what is its equation?

The nuclear form factor is a mathematical function that represents the Fourier transform of the nuclear charge distribution, describing how the internal structure of the nucleus modifies the scattering cross section. It accounts for the fact that the nucleus does not behave like a point particle but has a finite size and structure.

How do diffraction effects in scattering determine charge and mass distributions through Fourier Transformation?

Diffraction patterns arise due to the wave nature of particles interacting with the nucleus. The patterns depend on the nucleus's size and structure. The observed differential cross section is used to calculate the nuclear form factor. The inverse Fourier transform of the nuclear form factor yields the mass distribution of the nucleus.

Therefore, diffraction patterns give info about the internal structure of the nucleus and its charge and mass distributions.

How are the observed differential cross section, Mott cross section and nuclear form factor related? (equation)

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What are the limitations of the liquid drop model?

• It doesn’t explain peaks in binding energy or the stability of nuclei with magic numbers (for proton or neutron numbers)

• It doesn’t describe the non-spherical shape of some nuclei

• It doesn’t account for the quantum mechanical properties such as spin, parity, and magnetic moments

• The model relies on experimentally determined constants, making it semi-empirical rather than a fully theoretical description

• It provides limited insights into the spin pairing of protons and neutrons and how this affects nuclear stability

What is the liquid drop model?

It describes the nucleus as a ‘drop’ of incompressible nuclear fluid, with the binding energy of the nucleus calculated as a sum of 5 terms. Each term represents a different physical effect contributing to the stability of the nucleus.

What are the terms in the liquid drop model? (check image for equation, write down terms)

Volume term, surface term, coulomb term, asymmetry term, pairing term

Explain the volume term of the liquid drop model

It represents the binding energy due to the SNF. It is proportional to the total number of nucleons because the centre of the nucleus experiences max binding from surrounding nucleons.

Explain the surface term of the liquid drop model

It accounts for the reduction in binding energy for nucleons on the surface of the nucleus since they have fewer surrounding nucleons. It is proportional to the surface area of the nucleus.

Explain the coulomb term of the liquid drop model

It represents the repulsive electrostatic force between protons inside the nucleus. It is proportional to Z² (number of proton-proton interactions) and inversely proportional to the nuclear radius

Explain the asymmetry term of the liquid drop model

It comes from the Pauli exclusion principle and it minimizes the energy when the number of protons and neutrons are balanced.

Explain the pairing term of the liquid drop model

It represents the extra stability of nuclei with even numbers of protons and neutrons due to spin pairing. It is positive for nuclei with even numbers of protons and neutrons, zero for odd even combos and negative for odd numbers.

What does the liquid drop model assume?

It assumes that the nucleus is spherical and that nucleons behave like molecules in a water drop - short range attractive forces bind with shorter range repulsive forces to prevent collapsing

List the nuclear magic numbers

2, 8, 20, 28, 50, 82 and 126

List the experimental evidence for the existence of magic numbers

• Stability of Isotopes: Isotopes with neutron or proton numbers corresponding to magic numbers are often more stable than those with neighboring numbers

• Nuclear Shell Model: The model predicts that the most stable nuclei are those where nucleons fill energy levels completely, which corresponds to the magic numbers

• Odd-Even Effect: Even-even nuclei (those with both even numbers of protons and neutrons) are generally more stable than odd-odd nuclei. This effect is enhanced for nuclei near magic numbers, where the pairing of nucleons leads to greater stability.

• Neutron Rich Nuclei

• Beta Decay

What is the definition of a nuclear magic number?

They are specific numbers of nucleons in the nucleus of an atom that result in stable configurations.

What is the nuclear shell model?

It describes the arrangement of nucleons in discrete energy levels

What are the 6 principles behind the nuclear shell model?

• Magic numbers correspond to stable nuclei where shells are completely filled

• The Pauli exclusion principle applies

• Energy levels are quantized and grouped into shells, each characterised by quantum numbers

• Nucleons move in a central potential created by the collective interaction of all nucleons

• The Woods-Saxon potential better approximates the nuclear potential than a square well but cannot explain all magic numbers

• Spin-orbit coupling explains all magic numbers by splitting energy levels

What is parity and how is it determined in nuclei?

Parity tells us whether a wavefunction is symmetrical (even) or not (odd). In nuclei, parity is determined by the orbital angular momentum quantum number l (-1)^l

Calculate the spin and parity of O¹⁵₈ oxygen nucleus

Spin = ½ , Parity = -1

What is spin-orbit coupling and explain how it works (4 bullet points)

Spin-orbit coupling arises when a nucleon’s intrinsic spin interacts with its orbital motion, leading to energy level splitting.

• A moving charged nucleon (proton) generates a magnetic field due to its orbital motion

• The nucleon's intrinsic spin interacts with this field, causing a shift in energy levels

• Spin-orbit coupling is strongest at the nuclear surface, where density changes rapidly

• The potential energy of a nucleon due to spin-orbit interaction is proportional to L⋅S (the dot product of orbital and spin angular momenta)

What are the 3 effects of spin-orbit coupling on the nuclear shell model?

• Without spin-orbit coupling, the shell model failed to predict some magic numbers

• Radial-dependent spin-orbit potential causes energy levels to split which correctly explains all magic numbers

• Higher l values experience greater splitting, shifting energy levels significantly​

How to add angular momentum vectors quantum mechanically?

For integer values, (j₁ + j₂) … (j₁ - j₂)

For half integer values, (j₁ + j₂) or (j₁ - j₂)

For multiple integer values, (j₁ + j₂) … (j₁ - j₂) and ((j₁ + j₂) + j₃) … ((j₁ - j₂) - j₃)

Derive and apply the radioactive decay law

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What is half-life and its equation?

The time taken for half of the radioactive nuclei to decay T_1/2 ​= ln(2)/λ​

If the Q factor is positive, what happens?

Spontaneous decay

Explain alpha emission probability in terms of barrier tunnelling and the Gamow factor (G)

• Alpha decay occurs via quantum tunnelling through the Coulomb barrier, which classically prevents escape.

• Tunneling probability: T= e^−G, G = 2πZe²/ℏv

• Decay rate: λ = fe^−G, f ≈ v/2R

• Key factors:

  • Lower Z → Weaker Coulomb barrier → Higher decay probability.

  • Higher Q → Faster alpha particle → Increased tunneling.

• Explains why elements with high Q decay faster.

What are the limitations in the Gamow model?

• Shape of potential assumed too simple: Real Woods-Saxon potentials have softer transition to Coulomb dependence

• The model ignores orbital angular momentum effects, which modify the barrier height and tunneling probability.

• The model assumes spherical symmetry only, making it unreliable for non-spherical nuclei.

• The model assumes that alpha particles pre-exist inside the nucleus, ready to escape. However, the probability of alpha cluster formation actually varies between nuclei

What is decay probability?

The chance that an unstable nucleus will undergo radioactive decay within a certain time period

What is mean lifetime and its equation?

The average time a nucleus survives before decaying τ = 1/λ

What is the Q factor?

The energy released during decay

What are the 3 equations for decay probability?

For a large sample → P_decay = λdt

For a single nucleus → P_survival(t) = e^-λt P_decay(t) = λe^-λt

Using arguments from special relativity, briefly explain why the presence of neutrino oscillations implies that neutrinos have mass. ["solar neutrino problem"]

Neutrinos must have mass because they oscillate between flavours. If neutrinos were massless, they would move at the speed of light and time dilation would freeze any oscillations, making them impossible.

What is an idea that can solve the solar neutrino problem?

Electron neutrinos may change flavour (oscillate into muon or tau neutrinos) on their way from the Sun to Earth. Detectors sensitive only to electron neutrinos miss the oscillated neutrinos which explains the lower observed flux.

List the evidence for the existence of neutrinos from the properties of beta decay

• Beta decay electrons/positrons show a continuous range of energies

• "Missing" energy, along with differences in momentum and angular momentum, hinted at a massless, neutral particle.

• To maintain conservation laws, Pauli proposed the neutrino

Describe the basics of operation and results from the Reines and Cowan experiment

A nuclear reactor provides an intense flux of antineutrinos from neutron decay.

• Detection Method:

  • Proton-rich water doped with CdCl​ was used

  • Antineutrino + proton → positron + neutron

  • Positron annihilates with an electron → emits 2 gamma rays

  • Neutron captured by Cd → emits 2 more gamma rays

• Detection Technique:

  • Photomultiplier tubes & liquid scintillators detect gamma-ray pairs

  • Coincidence detection minimizes background noise.

• Outcome: First observation of neutrino interactions, confirmed Fermi’s beta decay theory.

Describe Čerenkov detectors (Super-Kamiokande) and how they work

• A charged particle moving faster than light in water emits Čerenkov radiation—a faint, cone-shaped blue light.

• Super-Kamiokande:

  • Located deep underground to block cosmic rays.

  • Ultrapure water as the detection medium.

  • Photomultiplier tubes capture Čerenkov light.

  • Neutrinos scatter electrons, producing light patterns that reveal energy and direction of the neutrino.

Give quantum mechanical and relativistic interpretations of neutrino oscillations

• Quantum Mechanical Interpretation:

  • Neutrino flavor eigenstates are superpositions of mass eigenstates

  • As they travel, their mass states change phases, leading to flavour oscillations

  • The probability of detecting a specific flavour depends on mass differences (Δm²), energy, and distance traveled

  • Relativistic Interpretation:

  • Neutrinos move almost at the speed of light. Tiny mass differences cause small energy shifts: (E ≈ p + m²/2p).

  • Due to phase shifts, their oscillations extend over vast distances

  • Since they change flavor, they must have mass and can't travel exactly at c

Using the liquid drop model, explain the origin of the barrier to spontaneous fission

According to the liquid drop model, the barrier to spontaneous fission arises from the competition between surface energy and Coulomb repulsion during nuclear deformation:

1. A spherical nucleus minimizes the surface area, which represents the lowest surface energy configuration.

2. As the nucleus begins to deform into a prolate (elongated) shape while maintaining the same volume, the surface area increases, causing the potential energy to rise and the binding energy to decrease.

3. This creates an energy barrier (the "fission barrier") that must be overcome for the nucleus to split.

4. Eventually, if deformation continues, the nucleus separates into two fragments, which lowers the Coulomb repulsion between protons and increases the overall binding energy.

5. For spontaneous fission to occur, the nucleus must either receive enough energy to overcome this barrier or quantum tunnel through it (which has extremely low probability for heavy nuclei).

Explain why symmetric fission is rare in terms of magic numbers

Symmetric fission is rare despite being predicted as energetically favorable by the Semi-Empirical Mass Formula (SEMF) because:

1. The shell model, which incorporates quantum mechanical effects not accounted for in the liquid drop model, predicts asymmetric fission.

2. Fission tends to produce fragments with nucleon numbers close to magic numbers (N=50, N=82, Z=50), which represent closed nuclear shells with exceptional stability.

3. In asymmetric fission of heavy nuclei like uranium-235, one fragment typically has a mass number A ≈ 80-85 (near the magic neutron number N=50) and the other has A ≈ 130-140 (near the doubly magic numbers N=82, Z=50).

4. These magic number configurations provide extra binding energy that makes asymmetric fission energetically favorable despite predictions from the simpler liquid drop model.

5. The quantum shell effects essentially modify the potential energy landscape of the fissioning system, creating valleys that guide the nucleus toward asymmetric division

What is the equation for the Yukawa potential, and how does exchange particle mass determine interaction range?

• Yukawa potential describes a force mediated by a massive exchange particle:
  V(r) = −g^2 e^(−r/R)/r where:

  • g = coupling strength,

  • R = ℏ/mc = range of interaction,

  • m = mass of the exchange particle.

• Key insight:

  • Lighter exchange particles → longer-range forces (e.g., photons in electromagnetism).

  • Heavier exchange particles → shorter-range forces (e.g., pions in the strong nuclear force).
    (​)

How do pions mediate nucleon-nucleon interactions?

• Direct potential (π⁰ exchange):

  • No charge exchange, affecting proton-proton and neutron-neutron interactions.

  • Weaker than exchange potential (not seen in nature for neutron-neutron).

• Exchange potential (π⁺/π⁻ exchange):

  • Causes proton-neutron conversion (p↔n).

  • Observed in p-n scattering experiments.

• Role of pions:

  • Act as force carriers for the residual strong force (nucleon-nucleon interaction).

  • Unlike photons, pions have mass, limiting the force range to ~1 fm.
    (​)

Why do gluon-gluon interactions determine the short range of the strong interaction?

Unlike photons, gluons carry color charge and can interact with each other. This leads to "colour confinement," where the force between quarks increases as they separate. As a result, quarks cannot be isolated, and the strong force remains short-ranged.

know that gluon-gluon interactions in hadrons ultimately determine the short range of the strong interaction

What are some key deficiencies of the Standard Model?

• Does not explain dark matter or dark energy (~96% of the universe).

• Lacks a quantum treatment of gravity.

• Cannot explain the large matter-antimatter asymmetry (~10^9:1).

What are some basic properties of the Higgs boson?

• Scalar boson (spin=0s) with mass ~125 GeV/c^2

• Carries weak isospin, interacts via the weak force.

• Couples to all massive weak bosons (W+, W-, Z0)

• Interaction strength depends on particle mass

How does the Higgs mechanism lead to mass generation?

• A Higgs field with a "Mexican hat" potential acquires a vacuum expectation value.

• This breaks electroweak symmetry, giving mass to W and Z bosons.

• Fermions gain mass via their interaction with the Higgs field.

What is CP violation in the weak interaction?

CP violation occurs when the weak interaction does not conserve the combined symmetry of charge conjugation (C) and parity (P). This means that the weak force behaves differently for particles and antiparticles.

Give an example of CP violation (check image​)

CP violation was first observed in the decay of the long-lived neutral kaon

Both decay channels occur, but with a small asymmetry, indicating CP is not perfectly conserved.

Why is CP violation important in terms of universal particle/antiparticle ratios?

CP violation is crucial for explaining why the universe is dominated by matter over antimatter. The observed matter-antimatter asymmetry suggests that CP violation must have played a role in the early universe, though the Standard Model can only account for a small part of this asymmetry.

What is a Boson? Give some examples

It is a particle with integer spin that mediates fundamental forces. Examples: photon (γ), W and Z bosons, gluons and the Higgs boson.

What is a Fermion?

It is a particle with half-integer spin that follows the Pauli exclusion principle. Example: quarks, leptons

What is a Lepton? Give some examples

It is a fundamental fermion that does not experience the strong interaction. Examples: electron, muon, tau and their associated neutrinos.

What is a Hadron and its two main types?

It is a composite particle made of quarks, bound by the strong force. Two main types: baryons and mesons.

What is a Baryon? Give 2 examples and their quark structure

It is a hadron composed of three quarks. Examples: proton (uud), neutron (udd)

What is a Meson? Give 2 examples

It is a hadron composed of a quark-antiquark pair. Examples: pions and kaons

Draw a Feynman diagram for beta minus and beta plus decay

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Draw a Feynman diagram for electron capture

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Draw a Feynman diagram for electron-proton collision

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Draw a Feynman diagram for a strong interaction between green and blue colour charged quarks generated by a gluon

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