3.3 Cross-section concept

Cross-sections are crucial in nuclear physics, measuring the probability of nuclear interactions. They quantify the likelihood of specific reactions when particles bombard target nuclei, essential for understanding nuclear systems in various applications.

Cross-sections are typically measured in barns, with 1 barn equaling 10^-24 cm^2. They can be microscopic (for individual nuclei) or macroscopic (for bulk materials). Different types include total, elastic scattering, inelastic scattering, absorption, and fission cross-sections.

Definition of cross-section

• Fundamental concept in nuclear physics measures probability of nuclear interactions
• Quantifies likelihood of specific nuclear reactions occurring when particles bombard target nuclei
• Crucial for understanding and predicting behavior of nuclear systems in various applications

Geometric vs nuclear cross-section

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• Geometric cross-section represents physical size of target nucleus
• Nuclear cross-section extends beyond physical dimensions includes quantum mechanical effects
• Effective area for nuclear interactions often larger than geometric size due to long-range forces
• Quantum tunneling allows interactions even when particles don't directly collide

Units of cross-section

• Typically measured in barns (b) where 1 barn equals $10^{-24} \text{ cm}^2$
• Chosen for convenience in nuclear physics scale of atomic nuclei
• Subunits include millibarns (mb) and microbarns (μb) for smaller cross-sections
• Sometimes expressed in $\text{cm}^2$ or $\text{m}^2$ for macroscopic calculations

Microscopic vs macroscopic cross-section

• Microscopic cross-section (σ) refers to individual nucleus interaction probability
• Macroscopic cross-section (Σ) accounts for material density and atomic number
• Relationship given by $\Sigma = N \sigma$ where N represents number density of target nuclei
• Macroscopic cross-section used in bulk material calculations (reactor design, shielding)

Types of cross-sections

• Various cross-section types describe different nuclear interaction processes
• Each type corresponds to specific outcomes of particle-nucleus collisions
• Understanding different cross-sections essential for predicting nuclear reaction rates

Total cross-section

• Sum of all possible interaction probabilities between incident particle and target nucleus
• Includes elastic scattering, inelastic scattering, and absorption processes
• Measured by beam attenuation experiments
• Provides overall picture of interaction likelihood without specifying reaction type

Elastic scattering cross-section

• Describes probability of incident particle bouncing off nucleus without energy transfer
• Nucleus remains in ground state after collision
• Important for neutron moderation in nuclear reactors
• Dominant process for low-energy neutrons interacting with light nuclei (hydrogen, deuterium)

Inelastic scattering cross-section

• Represents likelihood of incident particle exciting target nucleus
• Nucleus transitions to higher energy state then decays by emitting gamma rays
• Significant for fast neutrons in reactor cores
• Used in nuclear spectroscopy to study energy levels of nuclei

Absorption cross-section

• Probability of incident particle being captured by target nucleus
• Includes radiative capture (n,γ) and charged particle reactions (n,p), (n,α)
• Critical for neutron poison behavior in reactors (boron, cadmium)
• Basis for neutron activation analysis in material science and archaeology

Fission cross-section

• Likelihood of incident particle causing nuclear fission in target nucleus
• Key parameter for nuclear reactor design and operation
• Varies significantly with neutron energy and target isotope
• Exhibits resonance peaks at specific energies for fissile nuclei (U-235, Pu-239)

Factors affecting cross-sections

• Cross-sections not constant vary based on multiple factors
• Understanding these dependencies crucial for accurate nuclear calculations
• Allows prediction of nuclear behavior under different conditions

Energy dependence

• Cross-sections often strongly dependent on incident particle energy
• Low-energy region shows 1/v behavior for many absorption reactions
• Resonance region exhibits sharp peaks due to nuclear energy level structure
• High-energy region generally smoother varies with reaction type

Target nucleus properties

• Atomic number (Z) and mass number (A) influence cross-section values
• Nuclear structure (magic numbers, deformation) affects interaction probabilities
• Isotopic composition important for natural elements with multiple isotopes
• Spin and parity of target nucleus impact selection rules for reactions

Incident particle characteristics

• Type of incident particle (neutron, proton, alpha, etc.) determines possible reactions
• Charge of particle affects Coulomb barrier penetration probability
• Spin of incident particle influences angular momentum transfer in reactions
• Particle wavelength relative to nuclear size impacts scattering patterns

Measurement techniques

• Accurate cross-section measurements essential for nuclear data libraries
• Various experimental methods used depending on reaction type and energy range
• Continuous improvement in techniques leads to more precise nuclear data

Transmission experiments

• Measure total cross-section by detecting attenuation of particle beam through target
• Use of collimated beam and thin targets to minimize multiple scattering effects
• Time-of-flight techniques employed for energy-dependent measurements
• Require corrections for finite geometry and background radiation

Scattering experiments

• Determine differential cross-sections by detecting scattered particles at various angles
• Use of position-sensitive detectors to cover wide angular range
• Coincidence measurements for specific reaction channels (inelastic scattering)
• Challenges include background subtraction and detector efficiency calibration

Activation analysis

• Measure absorption cross-sections by analyzing induced radioactivity in target
• Suitable for reactions producing radioactive products with suitable half-lives
• Gamma spectroscopy used to identify and quantify activation products
• Requires careful sample preparation and irradiation conditions control

Cross-section data libraries

• Comprehensive collections of evaluated nuclear data for various applications
• Essential resource for nuclear engineers, physicists, and radiation protection specialists
• Continuously updated with new experimental measurements and theoretical calculations

ENDF format

• Evaluated Nuclear Data File standard format for storing nuclear data
• Hierarchical structure with different sections for various data types
• Includes cross-sections, angular distributions, and decay data
• Machine-readable format allows easy integration with nuclear codes

Major nuclear data libraries

• ENDF/B (United States) widely used in reactor physics and shielding calculations
• JEFF (Joint Evaluated Fission and Fusion) European library for nuclear applications
• JENDL (Japan) focuses on fast reactor and fusion reactor data
• CENDL (China) provides evaluated data for Chinese nuclear program

Cross-section uncertainties

• Quantification of uncertainties crucial for reliability of nuclear calculations
• Covariance matrices used to represent correlations between different data points
• Propagation of uncertainties through nuclear codes important for safety analysis
• Ongoing efforts to reduce uncertainties through improved measurements and evaluations

Applications of cross-sections

• Cross-section data fundamental to numerous nuclear science and engineering fields
• Accurate cross-sections essential for safe and efficient design of nuclear systems
• Enables optimization of nuclear processes and radiation protection measures

Reactor physics calculations

• Neutron transport calculations rely heavily on cross-section data
• Criticality analysis determines reactor core behavior and control rod effectiveness
• Burnup calculations predict fuel depletion and isotope production over time
• Transient analysis assesses reactor response to operational changes and accidents

Radiation shielding design

• Cross-sections used to calculate attenuation of radiation through materials
• Optimization of shield thickness and composition for various radiation types
• Important for nuclear power plants, medical facilities, and space radiation protection
• Monte Carlo simulations often employed for complex geometries

Nuclear medicine dosimetry

• Accurate cross-sections crucial for calculating radiation dose in diagnostic and therapeutic procedures
• Positron emission tomography (PET) relies on annihilation cross-sections
• Boron neutron capture therapy (BNCT) effectiveness depends on boron capture cross-section
• Internal dosimetry calculations for radiopharmaceuticals use decay data and interaction cross-sections

Theoretical models

• Complement experimental measurements in understanding nuclear interactions
• Provide predictions for unmeasured cross-sections and extrapolations to new energy ranges
• Essential for interpreting experimental results and guiding future measurements

Optical model

• Describes elastic scattering and absorption using complex potential
• Treats nucleus as partially transparent sphere for incident particles
• Parameters adjusted to fit experimental data
• Provides smooth background for other reaction models

Compound nucleus model

• Assumes formation of intermediate excited state before decay
• Applicable to low-energy reactions with many open channels
• Uses statistical methods to calculate branching ratios for different decay modes
• Explains resonance structure in cross-sections

Direct reaction model

• Describes fast processes where incident particle interacts with single nucleon
• Important for high-energy reactions and light nuclei
• Includes stripping, pickup, and knockout reactions
• Uses distorted wave Born approximation (DWBA) for calculations

Cross-section calculations

• Combine theoretical models with numerical methods to predict cross-sections
• Essential for filling gaps in experimental data and understanding reaction mechanisms
• Continuous improvement in computational techniques enhances predictive power

Numerical methods

• Finite difference and finite element methods for solving transport equations
• R-matrix theory for resonance region calculations
• Coupled-channel methods for direct reactions
• Hauser-Feshbach statistical model for compound nucleus reactions

Monte Carlo simulations

• Probabilistic approach to modeling particle transport and interactions
• MCNP (Monte Carlo N-Particle) widely used code for neutron transport
• GEANT4 toolkit for simulating passage of particles through matter
• Variance reduction techniques employed to improve efficiency

Cross-section software tools

• TALYS code for nuclear reaction calculations
• EMPIRE system for nuclear reaction modeling
• NJOY for processing evaluated nuclear data files
• SAMMY for R-matrix analysis of experimental data