4.7823
fm
7.1850e-47
fm²
7.185030e-23
barn
7.185029940623e-47
cm²
4.7823
fm
7.1850e-47
fm²
7.185030e-23
barn
7.185029940623e-47
cm²
The Nuclear Cross Section Calculator computes the effective target area for nuclear reactions and scattering processes. The nuclear cross section $$\sigma$$ quantifies the probability that a particular nuclear reaction will occur when a projectile interacts with a target nucleus. Despite having units of area, cross sections represent interaction probability rather than physical size, and can be much larger or smaller than the geometric nuclear area.
Cross sections are fundamental to nuclear and particle physics, reactor design, radiation shielding, medical physics, and astrophysical nucleosynthesis. This calculator provides both the geometric cross section based on the nuclear radius formula $$r = r_0 A^{1/3}$$ and the experimental cross section derived from measured reaction rates. Results are given in femtometers squared, barns, and square centimeters.
The geometric (classical) nuclear cross section uses the nuclear radius formula:
$$r = r_0 \cdot A^{1/3}$$
where $$r_0 \approx 1.2\text{-}1.3$$ fm and $$A$$ is the mass number. The geometric cross section is:
$$\sigma_{\text{geo}} = \pi r^2 = \pi r_0^2 A^{2/3}$$
From experimental reaction rate measurements, the cross section is:
$$\sigma = \frac{R}{I \cdot n \cdot x}$$
where $$R$$ is the reaction rate, $$I$$ is the incident particle flux, and $$n \cdot x$$ is the target areal density (atoms per cm²). The unit conversions are:
$$1 \text{ barn} = 10^{-24} \text{ cm}^2 = 100 \text{ fm}^2$$
Real cross sections depend on energy, reaction type, and quantum mechanical effects (resonances, Coulomb barrier, etc.) and can differ from geometric estimates by many orders of magnitude.
The Nuclear Radius is the estimated size using the liquid drop model formula. The Cross Section in fm² and barn give the effective interaction area in nuclear physics units. One barn (10⁻²⁴ cm²) is roughly the geometric cross section of a medium-mass nucleus. Real cross sections vary enormously: thermal neutron capture on Xe-135 is ~2.6 million barns, while neutrino cross sections are ~10⁻⁴⁴ cm². The geometric cross section provides an order-of-magnitude reference point.
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Iron-56 has a nuclear radius of ~4.8 fm and geometric cross section of ~0.72 barns. Actual reaction cross sections depend strongly on projectile energy and type.
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Measuring 5000 reactions/s with a flux of 10¹²/cm²/s through 10²⁰ atoms/cm² gives σ = 50 barns, much larger than geometric — indicating a resonance condition.
A nuclear cross section ($$\sigma$$) is a measure of the probability that a specific nuclear reaction will occur. It has units of area (typically barns, where 1 barn = 10⁻²⁴ cm²) and can be thought of as the effective target area a nucleus presents to an incoming particle for that particular reaction.
The unit "barn" was coined during the Manhattan Project. Compared to subatomic scales, uranium nuclei seemed as easy to hit as a barn door (10⁻²⁴ cm² is actually tiny in everyday terms). The name was initially classified to obscure nuclear research. Sub-units include millibarns (mb), microbarns (µb), and nanobarns (nb).
Yes, significantly. Quantum mechanical effects, especially at resonance energies, can produce cross sections orders of magnitude larger than the geometric size. The thermal neutron capture cross section of Xe-135 is about 2.65 × 10⁶ barns — thousands of times the geometric area. Conversely, some reactions have cross sections much smaller than geometric.
Cross sections are strongly energy-dependent. For charged particles, the Coulomb barrier suppresses cross sections at low energies. Neutron cross sections often follow a 1/v law at low energies and show sharp resonance peaks at specific energies. The total variation can span many orders of magnitude across the energy range.
The total cross section ($$\sigma_T$$) includes all possible interactions. The elastic cross section ($$\sigma_{el}$$) covers scattering without nuclear change. The reaction cross section ($$\sigma_R$$) includes all non-elastic processes (absorption, fission, particle emission). They satisfy $$\sigma_T = \sigma_{el} + \sigma_R$$.
A resonance occurs when the projectile energy matches a nuclear excited state of the compound nucleus. At resonance, the cross section shows a sharp peak (Breit-Wigner shape) that can be orders of magnitude above the off-resonance value. Resonances are described by their energy, width (Γ), and spin-parity quantum numbers.
Cross sections are measured by bombarding a target of known thickness with a beam of known intensity and counting the reaction products. Sophisticated detector arrays measure the angle and energy of products. Transmission experiments measure the total cross section from beam attenuation. Time-of-flight techniques measure energy-dependent cross sections.
Reactor design depends critically on neutron cross sections. The fission cross section of U-235 (584 barns for thermal neutrons) determines fuel burnup rates. Absorption cross sections of control rod materials (B-10: 3840 barns) determine reactor control. Moderator and structural material cross sections affect neutron economy.
The constant $$r_0$$ in the nuclear radius formula $$r = r_0 A^{1/3}$$ characterizes the nuclear density. Electron scattering experiments give $$r_0 \approx 1.2\text{-}1.3$$ fm, varying slightly depending on the measurement technique (charge radius vs. matter radius). This reflects the approximately constant density of nuclear matter across all nuclei.
The astrophysical S-factor removes the dominant energy dependence (Coulomb barrier penetration) from charged-particle cross sections: $$S(E) = \sigma(E) \cdot E \cdot \exp(2\pi\eta)$$ where $$\eta$$ is the Sommerfeld parameter. S-factors vary slowly with energy, allowing extrapolation to the very low energies relevant to stellar interiors where direct measurements are difficult.
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