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Nuclear cross section
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The nuclear cross section of a nucleus is used to describe the probability that a nuclear reaction will occur.[1][2] The concept of a nuclear cross section can be quantified physically in terms of "characteristic area" where a larger area means a larger probability of interaction. The standard unit for measuring a nuclear cross section (denoted as ) is the barn, which is equal to 10−28 m2, 10−24 cm2 or 100 fm2. Cross sections can be measured for all possible interaction processes together, in which case they are called total cross sections, or for specific processes, distinguishing elastic scattering and inelastic scattering; of the latter, amongst neutron cross sections the absorption cross sections are of particular interest.
In nuclear physics it is conventional to consider the impinging particles as point particles having negligible diameter. Cross sections can be computed for any nuclear process, such as capture scattering, production of neutrons, or nuclear fusion. In many cases, the number of particles emitted or scattered in nuclear processes is not measured directly; one merely measures the attenuation produced in a parallel beam of incident particles by the interposition of a known thickness of a particular material. The cross section obtained in this way is called the total cross section and is usually denoted by a or .
Typical nuclear radii are of the order 10−15 m. Assuming spherical shape, we therefore expect the cross sections for nuclear reactions to be of the order of or 10−28 m2 (i.e., 1 barn). Observed cross sections vary enormously: for example, slow neutrons absorbed by the (n, ) reaction show a cross section much higher than 1,000 barns in some cases (boron-10, cadmium-113, and xenon-135), while the cross sections for transmutations by gamma-ray absorption are in the region of 0.001 barn.
Microscopic and macroscopic cross section
[edit]Nuclear cross sections are used in determining the nuclear reaction rate, and are governed by the reaction rate equation for a particular set of particles (usually viewed as a "beam and target" thought experiment where one particle or nucleus is the "target", which is typically at rest, and the other is treated as a "beam", which is a projectile with a given energy).
For particle interactions incident upon a thin sheet of material (ideally made of a single isotope), the nuclear reaction rate equation is written as:
where:
- : number of reactions of type x, units: [1/time⋅volume]
- : beam flux, units: [1/area⋅time]
- : microscopic cross section for reaction , units: [area] (usually barns or cm2).
- : density of atoms in the target in units of [1/volume]
- : macroscopic cross-section [1/length]
Types of reactions frequently encountered are s: scattering, : radiative capture, a: absorption (radiative capture belongs to this type), f: fission, the corresponding notation for cross-sections being: , , , etc. A special case is the total cross-section , which gives the probability of a neutron to undergo any sort of reaction ().
Formally, the equation above defines the macroscopic cross-section (for reaction x) as the proportionality constant between a particle flux incident on a (thin) piece of material and the number of reactions that occur (per unit volume) in that material. The distinction between macroscopic and microscopic cross-section is that the former is a property of a specific lump of material (with its density), while the latter is an intrinsic property of a type of nuclei.
See also
[edit]References
[edit]- ^ Younes, Walid; Loveland, Walter (2021). An Introduction to Nuclear Fission. Springer. pp. 10, 25–26, 56–58. ISBN 9783030845940.
- ^ Rhodes, Richard (1986). The Making of the Atomic Bomb. New York: Simon & Schuster Paperbacks. pp. 333–334, 282–287. ISBN 9781451677614.
- Nuclear Reactor Analysis by James J. Duderstadt and Louis J. Hamilton - Published by John Wiley & Sons, Inc.
- Perkins, Donald H. (1999). Introduction to High Energy Physics. Cambridge University Press. ISBN 978-0-521-62196-0.
- Mubarakmand, Samar; Masud Ahmad; M. Anwar; M.S. Chaudhry (1977). "Cross-section measurements with a neutron generator". The Nucleus. 42 (1–2). Nilore, Islamabad: PINSTECH: 115–185.
Nuclear cross section
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Fundamental Concepts
Definition and Physical Interpretation
The concept of the nuclear cross section emerged in the early 1930s within the burgeoning field of nuclear physics, as researchers sought a quantitative measure for the probability of interactions between incident particles and atomic nuclei.[5] This term drew from earlier analogies in scattering theory, adapting the idea of an effective interaction area to describe nuclear reactions.[6] Physically, the nuclear cross section represents the effective or apparent area presented by a target nucleus to an incoming particle for a specific type of interaction, such as scattering or capture. Despite the actual geometric radius of a typical nucleus being on the order of m (1 femtometer), which yields a classical cross-sectional area of approximately m², measured nuclear cross sections are generally much larger, often around m² or 1 barn.[7][8] This discrepancy arises because nuclear interactions are probabilistic quantum mechanical processes, not strictly geometric collisions; the cross section thus quantifies the likelihood of a reaction occurring per incident particle, influenced by factors like nuclear forces and wave nature of particles.[9] Intuitively, one can analogize the cross section to the silhouette of a target in a classical projectile experiment: if a beam of particles is directed at many such targets, the fraction that "hits" is proportional to the target's effective area relative to the beam's path. In nuclear terms, this extends to quantum events, where determines the reaction probability for a flux of incident particles. The fundamental relation governing the expected number of reactions is given by where is the number of target nuclei, and is the incident particle flux (particles per unit area per unit time).[9] This formulation underpins both microscopic cross sections, which apply to individual nuclei, and macroscopic ones, which describe bulk material behavior.[7]Units and Notation
The primary unit for expressing nuclear cross sections is the barn (b), defined as m², which corresponds to an effective interaction area on the scale of nuclear dimensions.[10] This unit originated during the Manhattan Project in the 1940s, when physicists Charles P. Baker and Marshall G. Holloway humorously described unexpectedly large cross sections as being "as big as a barn," leading to its adoption as a standard measure.[11] Subdivisions using SI prefixes are commonly employed for finer resolution, such as the millibarn (mb = b) and microbarn (µb = b), particularly in contexts requiring precision for low-probability reactions.[1] Alternative units are occasionally used to align with broader particle physics conventions or computational preferences; for instance, the square femtometer (fm²) is equivalent, with 1 b = 100 fm², reflecting the femtometer scale of nuclear radii.[10] In some older literature or engineering calculations, cross sections may be reported in square centimeters (cm²), where 1 b = cm², to facilitate integration with macroscopic transport equations.[1] Standard notation distinguishes between the microscopic cross section, denoted by the Greek letter σ, which represents the interaction probability per target nucleus, and the macroscopic cross section, denoted by Σ, which incorporates the number density of targets in a material. Energy dependence is explicitly indicated as σ(E), where E is the energy of the incident particle, underscoring that cross sections vary significantly with kinematics; angular or isotopic dependencies may be similarly subscripted, such as σ_{th}(^{235}U) for thermal neutrons on uranium-235.[12] Reporting conventions emphasize cross sections as functions of energy, scattering angle, or specific isotopes to enable predictive modeling in reactors or experiments, often averaged over thermal (≈0.025 eV) or fast neutron spectra. For example, the thermal neutron radiative capture cross section on ^{235}U is approximately 99 b, illustrating the scale for absorption processes in fissile materials.[12]Microscopic Cross Section
Reaction Probability and Target Analogy
The microscopic cross section quantifies the probability of a specific nuclear interaction occurring between an incident particle and a single target nucleus. It is defined through the relation , where is the reaction rate (interactions per unit volume per unit time), is the number density of target nuclei (nuclei per unit volume), and is the incident particle flux (particles per unit area per unit time).[3] This formulation arises from the overall interaction rate , which scales linearly with both the density of targets and the incoming beam intensity.[13] Conceptually, represents an effective interaction probability per target nucleus per incident particle, rendered dimensionless when normalized against the beam's geometric spread, though it carries units of area to reflect the spatial scale of the interaction.[3] A useful analogy for understanding the microscopic cross section is that of a target presenting an effective "cross-sectional area" to the incoming particles, akin to aiming projectiles at a physical disk or sphere. In the simplest classical model of hard-sphere scattering, where the incident particle interacts only upon direct contact with the target nucleus of radius , the cross section equals the geometric area .[2] This area determines the fraction of incident trajectories that result in a collision, directly tying to the interaction probability. However, real nuclear cross sections often exceed this geometric limit by factors of 10 or more, due to quantum mechanical effects such as wave diffraction and tunneling, which allow interactions even for non-contact trajectories.[2] The value of thus varies significantly with the reaction type—scattering versus absorption, for instance—and is typically expressed in barns (1 barn = m²), a unit chosen to match the approximate scale of nuclear areas.[3] The probability encoded in the cross section is strongly energy-dependent, often exhibiting sharp peaks at resonance energies where the incident particle's energy matches a quasi-bound state in the compound nucleus. Near such a resonance, the cross section follows the Breit-Wigner form , a Lorentzian profile that captures the enhanced interaction probability.[14] Here, is the resonance energy at which the peak occurs, and is the resonance width, quantifying the energy range over which the effect persists and inversely related to the lifetime of the resonant state via .[14] This energy variation underscores how cross sections can fluctuate dramatically, from millibarns at off-resonance energies to thousands of barns at peaks, influencing applications like neutron capture in reactors. Cross sections also exhibit isotopic specificity, differing markedly between nuclides due to underlying nuclear structure effects such as nucleon pairing. For instance, nuclei with an odd number of neutrons often display higher reaction cross sections compared to their even-neutron neighbors, as the absence of pairing for the unpaired neutron lowers the energy threshold for certain interactions and enhances peripheral sensitivity.[15] This odd-even staggering, observed in isotopes like neon and magnesium, arises from pairing correlations that stabilize even-even systems more effectively, reducing their effective interaction radius in some cases.[15] Such variations highlight the role of quantum shell structure in modulating interaction probabilities beyond simple geometric considerations.Partial and Total Cross Sections
In nuclear physics, microscopic cross sections are often decomposed into partial cross sections, denoted as , which quantify the probability of specific reaction channels occurring when an incident particle interacts with a target nucleus. These partial cross sections include elastic scattering (), where the incident particle is deflected without energy loss to the target; inelastic scattering (), involving excitation of the target nucleus; radiative capture (), leading to the emission of a gamma ray and formation of a compound nucleus; and fission (), where the nucleus splits into fragments. Each partial cross section contributes to the overall reactivity of the nucleus, with their magnitudes depending on the incident particle's energy, the nuclear structure, and the reaction mechanism.[16] The total cross section, , represents the sum of all partial cross sections over every possible interaction channel: . This quantity measures the overall probability of any interaction and is typically determined experimentally through transmission experiments, where the attenuation of a beam of incident particles passing through a target is analyzed. Conceptually, the optical theorem provides a fundamental relation linking to the forward scattering amplitude : , where is the wave number of the incident particle; this theorem arises from the unitarity of the S-matrix and conservation of probability in quantum scattering theory.[17][9][18] Quantum mechanical unitarity imposes bounds on , ensuring that the total interaction probability does not exceed the maximum allowed by conservation principles, with each partial wave contribution limited such that . For example, the total cross section for neutron-proton scattering is approximately 20 barns at 1 MeV incident energy. Threshold effects further influence partial cross sections: for channels below the reaction's Q-value (the minimum energy required), with the cross section rising sharply once the threshold is surpassed due to the opening of new kinematic possibilities.[19][20]Macroscopic Cross Section
Relation to Microscopic Cross Section
The macroscopic cross section, denoted as Σ, extends the microscopic cross section σ to describe interactions in bulk materials by incorporating the density of target nuclei. Specifically, Σ is defined as the product of the atomic number density n (number of atoms per unit volume) and the microscopic cross section σ, yielding Σ = n σ.[21] This relation scales the probability of interaction from a single nucleus to the collective behavior within a volume of material.[22] The microscopic cross section σ, which has units of area such as barns (1 barn = 10^{-24} cm²), represents the effective target size for an individual nucleus.[21] In materials composed of multiple isotopes or elements, such as compounds or alloys, the macroscopic cross section accounts for the ensemble average. For a homogeneous mixture, Σ is the sum over all atomic species: Σ = ∑_i n_i σ_i, where n_i is the number density of the i-th species and σ_i is its corresponding microscopic cross section (averaged over isotopes if present within a species).[21] This can also be expressed in terms of mass fractions w_i for practical computation in nuclear engineering: Σ = ∑_i (ρ w_i N_A / A_i) σ_i, where ρ is the material density, N_A is Avogadro's number, and A_i is the atomic mass of species i; however, the form simplifies to the number density sum in uniform media.[23] Such averaging ensures Σ reflects the overall interaction probability without resolving individual contributions unless specified for partial cross sections. For absorption cross sections relevant to shielding materials, the macroscopic absorption cross section Σ_abs (in cm⁻¹) is calculated similarly using the microscopic absorption cross section σ_abs (in barns). The number density for each element is n = (ρ w / A) N_A, where ρ is density (g/cm³), w is mass fraction, A is atomic mass (g/mol), and N_A is Avogadro's number (6.022 × 10^{23} mol⁻¹). Then, Σ_abs = ∑i n_i σ{abs,i} × 10^{-24}, converting barns to cm².[23][24] For isotropic media, like liquids or gases with random atomic arrangements, the macroscopic cross section Σ is independent of the incident particle direction, simplifying transport calculations.[23] In contrast, highly ordered anisotropic crystals may exhibit direction-dependent Σ due to lattice effects, though this is less common in typical nuclear applications.[22] A representative example is the elastic scattering macroscopic cross section in water (H₂O) for thermal neutrons (~0.025 eV), which serves as a moderator in light-water reactors. Here, Σ_{elastic} ≈ 3.5 cm^{-1}, primarily derived from the hydrogen contribution. Due to molecular binding effects, the effective bound scattering cross section for hydrogen in water is ≈50 barns (higher than the free-atom value of ~20 barns), with n_H ≈ 6.7 × 10^{22} cm^{-3}, dominating over oxygen (n_O ≈ 3.3 × 10^{22} cm^{-3}, σ_O ≈ 4 barns), yielding the summed value for the compound at standard density (1 g/cm³).[23][25] This illustrates how microscopic probabilities aggregate to macroscopic properties essential for neutron moderation efficiency.[23] For thermal neutron absorption in shielding materials, consider pure lead (Pb, density ρ = 11.34 g/cm³, atomic mass A = 207.2 g/mol, σ_abs = 0.171 barns). The number density n = (11.34 / 207.2) × 6.022 × 10^{23} ≈ 3.29 × 10^{22} cm^{-3}, yielding Σ_abs ≈ 3.29 × 10^{22} × 0.171 × 10^{-24} ≈ 5.63 × 10^{-3} cm^{-1}. Similarly, for pure boron (B, density ρ = 2.34 g/cm³, A = 10.81 g/mol, σ_abs = 767 barns), n ≈ 1.30 × 10^{23} cm^{-3}, yielding Σ_abs ≈ 1.30 × 10^{23} × 767 × 10^{-24} ≈ 0.100 cm^{-1}. These values highlight the effectiveness of materials like boron for efficient thermal neutron absorption in shielding applications.[26][24]Attenuation Coefficient and Mean Free Path
The attenuation of a beam of particles, such as neutrons, through a material follows an exponential decay law, analogous to the Beer-Lambert law in optics. The intensity at a distance into the material is given by where is the initial intensity and is the total macroscopic cross section, representing the probability of interaction per unit path length.[27] This macroscopic cross section arises from the atomic density of the target nuclei and the microscopic cross section , via . The exponential form assumes a homogeneous medium and neglects secondary effects like scattering buildup.[27] The mean free path , defined as the average distance a particle travels before undergoing an interaction, is the reciprocal of the total macroscopic cross section: . In scenarios involving multiple isotropic scatterings, the relaxation length—the effective distance over which the particle flux or dose attenuates to of its value—is approximately .[27][28] The value of varies significantly with particle energy and material composition. For thermal neutrons, absorbers like cadmium exhibit a very short mean free path of approximately 0.01 cm due to their high absorption cross sections, enabling efficient thermal neutron capture. In contrast, moderators such as graphite have a longer mean free path of about 2.5 cm, reflecting lower interaction probabilities per unit volume dominated by scattering rather than absorption, which facilitates neutron slowing down with minimal loss.[29][25][30] These parameters are essential in practical applications, particularly radiation shielding design, where the exponential attenuation law predicts beam reduction but requires corrections via buildup factors to account for non-exponential behavior from scattered particles that increase effective dose behind the shield.Types of Cross Sections
Scattering Cross Sections
Scattering cross sections quantify the probability of neutron deflection by a nucleus without absorption, encompassing both elastic and inelastic processes. These contribute to the total cross section as partial components, where the scattering cross section σ_s = σ_el + σ_inel, with elastic and inelastic denoting the preservation or loss of the neutron's kinetic energy relative to the center-of-mass frame.[31] Elastic scattering, denoted σ_el, involves no change in internal nuclear energy, conserving the total kinetic energy in the center-of-mass system while altering the direction of the incident neutron. This process dominates at low incident energies, where other reactions are negligible, and is crucial for neutron moderation in reactors. For instance, the neutron-proton elastic scattering cross section is approximately 20 barns at thermal energies (around 0.025 eV).[32] In heavier nuclei, σ_el typically ranges from a few to tens of barns at low energies, decreasing with increasing energy due to the finite nuclear size.[28] Inelastic scattering, σ_inel, occurs when the incident neutron excites the target nucleus, leading to subsequent de-excitation via gamma emission or particle ejection, with the neutron emerging at lower energy. This process has a threshold determined by the excitation energy of the first nuclear level, often around 0.05–0.1 MeV for heavy nuclei and higher for lighter ones.[31] Above threshold, σ_inel rises rapidly and typically peaks around 10 MeV for heavy nuclei, reflecting contributions from direct and compound nuclear reactions, before plateauing toward the geometric limit of πR² (where R is the nuclear radius).[33] The angular distribution of scattered neutrons is described by the differential cross section dσ/dΩ(θ), which gives the probability per unit solid angle as a function of scattering angle θ. The total cross section for a process is obtained by integration: For charged particle scattering dominated by Coulomb interactions, the Rutherford formula provides a classical benchmark: This yields strong forward peaking, though nuclear forces modify it for neutrons at short distances.[34] In transport applications, such as Monte Carlo simulations of neutron behavior, the transport cross section σ_tr accounts for momentum transfer efficiency by weighting the differential cross section: This reduces the effective scattering for small-angle events, which minimally alter direction, and is essential for accurate modeling of diffusion and shielding.[35]Absorption and Reaction Cross Sections
The absorption cross section, denoted as , quantifies the probability that an incident particle, typically a neutron, is absorbed by a target nucleus, leading to reactions such as radiative capture or charged particle emission without elastic scattering. It is the sum of partial cross sections for specific absorption processes, expressed as , where represents the radiative capture cross section (e.g., (n, )) involving gamma-ray emission from the excited compound nucleus, and includes inelastic processes like charged particle emission (e.g., (n, p) or (n, )).[36][37][38] For thermal neutrons, the absorption cross section of B is exceptionally high at approximately 3840 barns, making it a key material for neutron detection and control applications.[39] Typical thermal neutron absorption cross-sections (in barns) for common elements used in shielding materials, based on natural isotopic abundances at ~0.025 eV, are listed below:[26]| Element | σ_a (barns) |
|---|---|
| H | 0.332 |
| C | 0.0035 |
| O | 0.00019 |
| Pb | 0.171 |
| W | 18.3 |
| U | 7.57 |
| Si | 0.171 |
| Al | 0.232 |
| Ca | 0.43 |
| Fe | 2.56 |
| B | 767 |
| Na | 0.53 |