⚛️Nuclear Physics Unit 6 – Nuclear Reactions and Cross Sections

Key Concepts and Definitions

• Nuclear reactions involve changes in the composition or energy of atomic nuclei through interactions with other particles or nuclei
• Cross section quantifies the probability of a specific nuclear reaction occurring when a projectile particle interacts with a target nucleus
• Differential cross section describes the probability of a reaction occurring as a function of the scattering angle and is expressed in units of area per solid angle (barns/sr)
  • Provides information about the angular distribution of reaction products
  • Useful for understanding the detailed mechanisms of nuclear reactions
• Integrated cross section represents the total probability of a reaction occurring, obtained by integrating the differential cross section over all possible scattering angles
  • Expressed in units of area (barns)
  • Determines the overall likelihood of a specific nuclear reaction
• Reaction rate quantifies the number of nuclear reactions occurring per unit time and is proportional to the cross section and the flux of incident particles
• Resonance occurs when the energy of the incident particle matches the energy required to form a short-lived excited state of the compound nucleus, leading to enhanced cross sections at specific energies

Types of Nuclear Reactions

• Elastic scattering involves the conservation of kinetic energy and momentum, with the incident particle and target nucleus remaining in their ground states (e.g., neutron-nucleus elastic scattering)
• Inelastic scattering results in the target nucleus being left in an excited state, with the incident particle losing kinetic energy (e.g., neutron-nucleus inelastic scattering)
• Capture reactions occur when the incident particle is absorbed by the target nucleus, forming a new compound nucleus (e.g., neutron capture, proton capture)
  • Radiative capture involves the emission of gamma rays as the compound nucleus de-excites to a lower energy state
  • Charged particle capture, such as proton or alpha capture, can lead to the formation of heavier nuclei
• Transfer reactions involve the exchange of nucleons between the incident particle and the target nucleus (e.g., $(d,p)$ reactions, $(\alpha,n)$ reactions)
• Fission reactions occur when a heavy nucleus splits into two or more lighter fragments, often induced by neutron capture (e.g., $^{235}U(n,f)$ in nuclear reactors)
• Fusion reactions involve the merging of two light nuclei to form a heavier nucleus, releasing energy in the process (e.g., $^2H + ^3H \rightarrow ^4He + n$ in thermonuclear fusion)
• Spallation reactions occur when a high-energy particle (usually a proton) interacts with a nucleus, causing the ejection of multiple nucleons and the formation of a residual nucleus

Nuclear Cross Sections Explained

• Cross sections are measured in units of area, typically barns ($1 \text{ barn} = 10^{-24} \text{ cm}^2$)
• Cross sections depend on various factors, including the energy of the incident particle, the type of reaction, and the properties of the target nucleus
• Microscopic cross sections describe the probability of a reaction occurring for a single target nucleus and are intrinsic properties of the nucleus
  • Determined by the underlying nuclear structure and reaction mechanisms
  • Can be calculated using theoretical models (e.g., optical model, shell model)
• Macroscopic cross sections consider the collective behavior of many nuclei in a target material and depend on the number density of the target nuclei
  • Related to the microscopic cross section by $\Sigma = N \sigma$, where $\Sigma$ is the macroscopic cross section, $N$ is the number density of target nuclei, and $\sigma$ is the microscopic cross section
  • Important for practical applications, such as reactor physics and radiation shielding
• Energy dependence of cross sections can exhibit resonances, where the cross section increases dramatically at specific energies due to the formation of compound nuclear states
  • Resonances provide information about the energy levels and structure of the compound nucleus
  • Width of the resonance is related to the lifetime of the compound nuclear state through the uncertainty principle
• Angular dependence of cross sections, described by the differential cross section, provides insights into the reaction mechanisms and the angular distribution of reaction products

Reaction Rates and Probabilities

• Reaction rate $R$ is the number of reactions occurring per unit time and is given by $R = \sigma \phi N$, where $\sigma$ is the cross section, $\phi$ is the flux of incident particles, and $N$ is the number of target nuclei
• Flux is the number of particles passing through a unit area per unit time and is typically expressed in units of particles per square centimeter per second ($\text{cm}^{-2} \text{s}^{-1}$)
• Reaction probability $P$ is the likelihood of a specific reaction occurring when a single incident particle interacts with a target nucleus and is related to the cross section by $P = \sigma / A$, where $A$ is the effective area of the target nucleus
• Mean free path $\lambda$ is the average distance a particle travels between successive interactions and is inversely proportional to the macroscopic cross section, given by $\lambda = 1 / \Sigma$
  • Determines the penetration depth of particles in a material
  • Important for designing radiation shielding and understanding the behavior of particles in matter
• Reaction rates and probabilities can be measured experimentally using various techniques, such as activation analysis, time-of-flight measurements, and coincidence counting
• Theoretical calculations of reaction rates and probabilities involve integrating the product of the cross section and the incident particle flux over the relevant energy range and angular distribution

Experimental Techniques and Measurements

• Activation analysis involves irradiating a sample with particles (e.g., neutrons) and measuring the induced radioactivity to determine the cross sections and reaction rates
  • Gamma-ray spectroscopy is used to identify the radioactive products and quantify their activities
  • Allows for the study of neutron capture cross sections and the production of radioisotopes
• Time-of-flight measurements determine the energy of particles by measuring the time it takes for them to travel a known distance
  • Used to measure the energy dependence of cross sections and to study resonances
  • Requires precise timing electronics and long flight paths to achieve good energy resolution
• Coincidence counting techniques detect multiple particles or photons emitted in coincidence from a nuclear reaction, providing information about the reaction mechanisms and angular correlations
  • Examples include $\gamma$-$\gamma$ coincidence measurements for studying nuclear energy levels and $\beta$-$\gamma$ coincidence measurements for investigating $\beta$-decay processes
• Charged particle spectroscopy uses magnetic or electric fields to separate charged particles according to their momentum or energy, allowing for the identification of reaction products and the measurement of their cross sections
  • Examples include magnetic spectrometers and silicon detectors
• Neutron detection relies on the indirect detection of neutrons through their interactions with matter, such as scintillation (e.g., $^6$Li-doped glass scintillators) or gas ionization (e.g., $^3$He proportional counters)
• Accelerator-based experiments use particle accelerators to produce high-energy beams of particles (e.g., protons, electrons) for inducing nuclear reactions and studying their cross sections
  • Examples include cyclotrons, linear accelerators, and synchrotrons
  • Enable the study of reactions at higher energies and the production of exotic nuclei

Applications in Nuclear Physics

• Nuclear energy production relies on fission reactions, where the cross sections of fissile isotopes (e.g., $^{235}$U, $^{239}$Pu) determine the criticality and power output of nuclear reactors
  • Neutron cross sections are crucial for designing reactor cores, control systems, and fuel cycles
  • Thorium-based nuclear reactors exploit the high neutron capture cross section of $^{232}$Th to breed fissile $^{233}$U
• Nucleosynthesis in stars and supernovae involves a complex network of nuclear reactions, governed by their respective cross sections
  • Proton-proton chain and CNO cycle in main-sequence stars
  • Helium burning and subsequent stages in red giant stars
  • r-process and s-process for the production of heavy elements
• Radiation therapy in medicine utilizes the cross sections of tissue and tumors for various types of radiation (e.g., photons, electrons, protons) to optimize dose delivery and minimize damage to healthy tissue
  • Neutron capture therapy uses the high cross section of $^{10}$B for thermal neutrons to selectively destroy tumor cells
  • Proton therapy exploits the Bragg peak in the proton stopping power to deliver high doses to tumors while sparing surrounding tissue
• Radioisotope production for medical imaging and research benefits from the knowledge of production cross sections and reaction rates
  • Examples include $^{99m}$Tc for SPECT imaging, $^{18}$F for PET imaging, and $^{60}$Co for gamma irradiation
• Nuclear astrophysics aims to understand the origin of elements and the evolution of stars and galaxies by studying nuclear reaction cross sections relevant to astrophysical environments
  • Big Bang nucleosynthesis and primordial abundances of light elements
  • Solar neutrino problem and the CNO cycle in the Sun
  • Neutron star mergers and the production of heavy elements through the r-process

Calculations and Problem-Solving

• Calculation of reaction rates requires integrating the product of the cross section and the incident particle flux over the relevant energy range, taking into account the energy distribution of the particles
  • For a monoenergetic beam, the reaction rate is simply the product of the cross section, flux, and number of target nuclei
  • For a continuous energy spectrum (e.g., thermal neutrons), the reaction rate involves an integral over energy, weighted by the particle energy distribution
• Determination of cross sections from experimental data involves several steps, including background subtraction, efficiency corrections, and normalization to a known standard
  • Absolute cross sections can be obtained by measuring the reaction rate and the incident particle flux independently
  • Relative cross sections are determined by comparing the reaction rates of different isotopes or reactions under the same experimental conditions
• Uncertainty analysis is essential for reporting cross section measurements and evaluating the reliability of the results
  • Sources of uncertainty include statistical fluctuations, systematic errors in detector efficiency and particle flux, and uncertainties in the properties of the target material
  • Error propagation techniques are used to combine the individual uncertainties and estimate the overall uncertainty in the cross section
• Theoretical modeling of cross sections involves various approaches, depending on the energy range and the type of reaction
  • Optical model describes the interaction of a particle with a nucleus using a complex potential, accounting for elastic scattering and absorption
  • Compound nucleus model treats the reaction as a two-step process, with the formation of an intermediate excited state (compound nucleus) and its subsequent decay
  • Direct reaction models, such as the distorted wave Born approximation (DWBA), are used for describing reactions that occur in a single step, without the formation of a compound nucleus
• Evaluation and compilation of cross section data are essential for creating reliable databases for use in various applications
  • Experimental data from different measurements are critically assessed, normalized, and combined to produce recommended cross section values
  • Theoretical calculations are used to interpolate and extrapolate the data to energy ranges where experimental measurements are scarce or unavailable

Advanced Topics and Current Research

• Surrogate reactions are used to indirectly measure cross sections of reactions that are difficult to study directly, such as neutron-induced fission of short-lived isotopes
  • Involve measuring the decay of a compound nucleus formed through an alternative reaction (e.g., $(d,p)$ reaction) and inferring the desired cross section using the Weisskopf-Ewing or Hauser-Feshbach formalism
  • Provide valuable data for nuclear reactor design and nuclear astrophysics
• Inverse kinematic reactions, where the roles of the projectile and target are reversed, are employed to study reactions with unstable or rare isotopes
  • Radioactive ion beams are used to bombard a stable target, and the reaction products are detected in coincidence
  • Enable the measurement of cross sections relevant to nucleosynthesis and the study of exotic nuclei
• Polarized beams and targets exploit the spin dependence of nuclear interactions to investigate the spin structure of nuclei and the role of spin in reaction mechanisms
  • Polarized neutron beams are used to study parity violation in compound nuclear resonances
  • Polarized targets (e.g., polarized $^3$He) are used in combination with polarized beams to measure spin-dependent cross sections
• Neutrino-nucleus cross sections are of great importance for neutrino physics and astrophysics, as they determine the interaction of neutrinos with matter
  • Coherent elastic neutrino-nucleus scattering (CE$\nu$NS) is a predicted process that is sensitive to the weak nuclear charge and can be used to study neutrino properties and search for new physics
  • Neutrino-induced reactions, such as neutrino capture and neutrino-induced fission, are relevant for understanding the role of neutrinos in nucleosynthesis and the dynamics of supernovae
• Nuclear data evaluation and uncertainty quantification are active areas of research, aiming to improve the quality and reliability of cross section databases
  • Advanced statistical methods, such as Bayesian inference and machine learning, are being developed to combine experimental data and theoretical models in a consistent and optimal way
  • Sensitivity analysis and uncertainty propagation techniques are used to assess the impact of cross section uncertainties on the predictions of nuclear models and simulations
• Ab initio calculations of nuclear reactions, based on first principles and realistic nuclear forces, are an emerging field that aims to provide a fundamental understanding of nuclear interactions and cross sections
  • Chiral effective field theory is used to derive nuclear forces from quantum chromodynamics (QCD) and to calculate few-body reaction observables
  • Coupled-cluster method and no-core shell model are applied to compute the structure and reactions of light nuclei from the ground up