Neutron Cross-section | Definition & Examples

In general, the neutron cross-section is an effective area that quantifies the likelihood of certain interaction between an incident neutron and a target object. It must be noted this likelihood does not depend on real target dimensions, because we are not describing geometrical cross-section. In the vicinity of target nucleus, neutron is subjected to strong nuclear forces of target nucleons. The interaction strongly depends on many variables, such as type of target nucleus and the neutron energy. For example, the likelihood that a thermal neutron will be absorbed by xenon-135 is about a million times higher than it will be scattered.

At this point we have to distinguish between two basic types of nuclear or neutron cross-sections.

Microscopic Cross-section. The effective target area in m² presented by a single nucleus to an incident neutron beam is denoted the microscopic cross section, σ. The microscopic cross-sections characterize interactions with single isotopes and are a part of data libraries, such as ENDF/B-VII.1.
Macroscopic Cross-section. The macroscopic cross-section represents the effective target area of all of the nuclei contained in the volume of the material (such as fuel pellet). The units are given in cm^-1. It is the probability of neutron-nucleus interaction per centimeter of neutron travel. These data are commonly used by codes for reactor core analyses and design. These codes are based on pre-computed assembly homogenized macroscopic cross-sections.

nuclear cross-sections, microscopic cross-sections — Various microscopic cross-section for uranium-235 and incident thermal neutron.

Barn – Unit of Cross-section

The cross-section is typically denoted σ and measured in units of area [m²]. But a square meter (or centimeter) is tremendously large in comparison to the effective area of a nucleus, and it has been suggested that a physicist once referred to the measure of a square meter as being “as big as a barn” when applied to nuclear processes. The name has persisted and microscopic cross sections are expressed in terms of barns. The standard unit for measuring a nuclear cross section is the barn, which is equal to 10⁻²⁸ m² or 10⁻²⁴ cm². It can be seen the concept of a nuclear cross section can be quantified physically in terms of “characteristic target area” where a larger area means a larger probability of interaction.

Typical Values of Microscopic Cross-sections

Uranium 235 is a fissile isotope and its fission cross-section for thermal neutrons is about 585 barns (for 0.0253 eV neutron). For fast neutrons its fission cross-section is on the order of barns.
Xenon-135 is a product of U-235 fission and has a very large neutron capture cross-section (about 2.6 x 10⁶ barns).
Boron is commonly used as a neutron absorber due to the high neutron cross-section of isotope ¹⁰B. Its (n,alpha) reaction cross-section for thermal neutrons is about 3840 barns (for 0.025 eV neutron).
Gadolinium is commonly used as a neutron absorber due to very high neutron absorbtion cross-section of two isotopes ¹⁵⁵Gd and ¹⁵⁷Gd. ¹⁵⁵Gd has 61 000 barns for thermal neutrons (for 0.025 eV neutron) and ¹⁵⁷Gd has even 254 000 barns.

Microscopic Cross-section

The extent to which neutrons interact with nuclei is described in terms of quantities known as cross-sections. Cross-sections are used to express the likelihood of particular interaction between an incident neutron and a target nucleus. It must be noted this likelihood does not depend on real target dimensions. In conjunction with the neutron flux, it enables the calculation of the reaction rate, for example to derive the thermal power of a nuclear power plant. The standard unit for measuring the microscopic cross-section (σ-sigma) is the barn, which is equal to 10^-28 m². This unit is very small, therefore barns (abbreviated as “b”) are commonly used.

The cross-section σ can be interpreted as the effective ‘target area’ that a nucleus interacts with an incident neutron. The larger the effective area, the greater the probability for reaction. This cross-section is usually known as the microscopic cross-section.

The concept of the microscopic cross-section is therefore introduced to represent the probability of a neutron-nucleus reaction. Suppose that a thin ‘film’ of atoms (one atomic layer thick) with N_a atoms/cm² is placed in a monodirectional beam of intensity I₀. Then the number of interactions C per cm² per second will be proportional to the intensity I₀ and the atom density N_a. We define the proportionality factor as the microscopic cross-section σ:

σ_t = C/N_a.I₀

In order to be able to determine the microscopic cross section, transmission measurements are performed on plates of materials. Assume that if a neutron collides with a nucleus it will either be scattered into a different direction or be absorbed (without fission absorption). Assume that there are N (nuclei/cm³) of the material and there will then be N.dx per cm² in the layer dx.
Only the neutrons that have not interacted will remain traveling in the x direction. This causes the intensity of the uncollided beam will be attenuated as it penetrates deeper into the material.

Then, according to the definition of the microscopic cross section, the reaction rate per unit area is Nσ Ι(x)dx. This is equal to the decrease of the beam intensity, so that:

-dI = N.σ.Ι(x).dx

and

Ι(x) = Ι₀e^-N.σ.x

It can be seen that whether a neutron will interact with a certain volume of material depends not only on the microscopic cross-section of the individual nuclei but also on the density of nuclei within that volume. It depends on the N.σ factor. This factor is therefore widely defined and it is known as the macroscopic cross section.

The difference between the microscopic and macroscopic cross sections is extremely important. The microscopic cross section represents the effective target area of a single nucleus, while the macroscopic cross section represents the effective target area of all of the nuclei contained in certain volume.

Nuclear Radius

Typical nuclear radii are of the order 10⁻¹⁴ m. Assuming spherical shape, nuclear radii can be calculated according to following formula:

r = r₀ . A^1/3

where r₀ = 1.2 x 10^-15m = 1.2 fm

If we use this approximation, we therefore expect the geometrical cross-sections of nuclei to be of the order of πr² or 4.5 x 10⁻³⁰m² for hydrogen nuclei or 1.74 x 10⁻²⁸ m² for ²³⁸U nuclei.

Since there are many nuclear reaction from the incident particle point of view, but, in nuclear reactor physics, neutron-nuclear reactions are of particular interest. In this case the neutron cross-section must be defined.

Total Cross-section

In general, nuclear cross-sections can be measured for all possible interaction processes together, in this case they are called total cross-sections (σ_t). The total cross section is the sum of all the partial cross sections such as:

elastic scattering cross-section (σ_s)
inelastic scattering cross-section (σ_i)
absorption cross-section (σ_a)
radiative capture cross-section (σ_γ)
fission cross-section (σ_f)

σ_t = σ_s + σ_i + σ_γ + σ_f + ……

The total cross-section measures the probability that an interaction of any type will occur when neutron interacts with a target.

Microscopic cross-sections constitute a key parameters of nuclear fuel. In general, neutron cross-sections are essential for the reactor core calculations and are a part of data libraries, such as ENDF/B-VII.1.

The neutron cross-section is variable and depends on:

Target nucleus (hydrogen, boron, uranium, etc.). Each isotop has its own set of cross-sections.
Type of the reaction (capture, fission, etc.). Cross-sections are different for each nuclear reaction.
Neutron energy (thermal neutron, resonance neutron, fast neutron). For a given target and reaction type, the cross-section is strongly dependent on the neutron energy. In the common case, the cross section is usually much larger at low energies than at high energies. This is why most nuclear reactors use a neutron moderator to reduce the energy of the neutron and thus increase the probability of fission, essential to produce energy and sustain the chain reaction.
Target energy (temperature of target material – Doppler broadening). This dependency is not so significant, but the target energy strongly influences inherent safety of nuclear reactors due to a Doppler broadening of resonances.

Microscopic cross-section varies with incident neutron energy. Some nuclear reactions exhibit very specific dependency on incident neutron energy. This dependency will be described on the example of the radiative capture reaction. The likelihood of a neutron radiative capture is represented by the radiative capture cross section as σ_γ. The following dependency is typical for radiative capture, it definitely does not mean, that it is typical for other types of reactions (see elastic scattering cross-section or (n,alpha) reaction cross-section).

The capture cross-section can be divided into three regions according to the incident neutron energy. These regions will be discussed separately.

1/v Region
Resonance Region
Fast Neutrons Region

Charts of Cross-sections

Uranium 238. Neutron absorption and scattering. Comparison of cross-sections.

Uranium 238. Comparison of cross-sections.
Source: JANIS (Java-based Nuclear Data Information Software); The JEFF-3.1.1 Nuclear Data Library

Gadolinium 155 and 157. Comparison of radiative capture cross-sections.
Source: JANIS (Java-based Nuclear Data Information Software); The JEFF-3.1.1 Nuclear Data Library

1/v Law — For thermal neutrons (in 1/v region), absorption cross sections increases as the velocity (kinetic energy) of the neutron decreases.
Source: JANIS 4.0

1/v Region

In the common case, the cross section is usually much larger at low energies than at high energies. For thermal neutrons (in 1/v region), also radiative capture cross-sections increase as the velocity (kinetic energy) of the neutron decreases. Therefore the 1/v Law can be used to determine shift in capture cross-section, if the neutron is in equilibrium with a surrounding medium. This phenomenon is due to the fact the nuclear force between the target nucleus and the neutron has a longer time to interact.

$\sigma_a \sim \frac{1}{v}}} \sim \frac{1}{\sqrt{E}}}}} \sim \frac{1}{\sqrt{T}}}}}$

This law is aplicable only for absorbtion cross-section and only in the 1/v region.

Example of cross- sections in 1/v region:

The absorbtion cross-section for 238U at 20°C = 293K (~0.0253 eV) is:

$\sigma_a(293K) = 2.68b$ .

The absorbtion cross-section for 238U at 1000°C = 1273K is equal to:

$\sigma_a(1273K) = \sigma_a(293K) \cdot \frac {T_0}{T_1} = 2.68 \cdot \frac{293}{1273} = 0.617b$

This cross-section reduction is caused only due to the shift of temperature of surrounding medium.

Resonance Region

Compound state - resonance — Energy levels of compound state. For neutron absorption reaction on 238U the first resonance E1 corresponds to the excitation energy of 6.67eV. E0 is a base state of 239U.

The largest cross-sections are usually at neutron energies, that lead to long-lived states of the compound nucleus. The compound nuclei of these certain energies are referred to as nuclear resonances and its formation is typical in the resonance region. The widths of the resonances increase in general with increasing energies. At higher energies the widths may reach the order of the distances between resonances and then no resonances can be observed. The narrowest resonances are usually compound states of heavy nuclei (such as fissionable nuclei).

Since the mode of decay of the compound nucleus does not depend on the way the compound nucleus was formed, the nucleus sometimes emits a gamma ray (radiative capture) or sometimes emits a neutron (scattering). In order to understand the way, how a nucleus will stabilize itself, we have to understand the behaviour of compound nucleus.

ground state compound nucleus - excitation — The position of the energy levels during the formation of a compound nucleus. Ground state and energy states.

The compound nucleus emits a neutron only after one neutron obtains an energy in collision with other nucleon greater than its binding energy in the nucleus. It have some delay, because the excitation energy of the compound nucleus is divided among several nucleons. It is obvious the average time that elapses before a neutron can be emitted is much longer for nuclei with large number of nucleons than when only a few nucleons are involved. It is a consequence of sharing the excitation energy among a large number of nucleons.

This is the reason the radiative capture is comparatively unimportant in light nuclei but becomes increasingly important in the heavier nuclei.

It is obvious the compound states (resonances) are observed at low excitation energies. This is due to the fact, the energy gap between the states is large. At high excitation energy, the gap between two compound states is very small and the widths of resonances may reach the order of the distances between resonances. Therefore at high energies no resonances can be observed and the cross section in this energy region is continuous and smooth.

The lifetime of a compound nucleus is inversely proportional to its total width. Narrow resonances therefore correspond to capture while the wider resonances are due to scattering.

Macroscopic Cross-section

The difference between the microscopic cross-section and macroscopic cross-section is very important and is restated for clarity. The microscopic cross section represents the effective target area of a single target nucleus for an incident particle. The units are given in barns or cm².

While the macroscopic cross-section represents the effective target area of all of the nuclei contained in the volume of the material. The units are given in cm^-1.

A macroscopic cross-section is derived from microscopic cross-section and the atomic number density:

Σ=σ.N

Here σ, which has units of m², is the microscopic cross-section. Since the units of N (nuclei density) are nuclei/m³, the macroscopic cross-section Σ have units of m^-1, thus in fact is an incorrect name, because it is not a correct unit of cross-sections. In terms of Σ_t (the total cross-section), the equation for the intensity of a neutron beam can be written as

-dI = N.σ.Σ_t.dx

Dividing this expression by I(x) gives

-dΙ(x)/I(x) = Σ_t.dx

Since dI(x) is the number of neutrons that collide in dx, the quantity –dΙ(x)/I(x) represents the probability that a neutron that has survived without colliding until x, will collide in the next layer dx. It follows that the probability P(x) that a neutron will travel a distance x without any interaction in the material, which is characterized by Σ_t, is:

P(x) = e^-Σ_t.x

From this equation, we can derive the probability that a neutron will make its first collision in dx. It will be the quantity P(x)dx. If the probability of the first collision in dx is independent of its past history, the required result will be equal to the probability that a neutron survives up to layer x without any interaction (~Σ_tdx) times the probability that the neutron will interact in the additional layer dx (i.e., ~e^-Σ_t.x).

P(x)dx = Σ_tdx . e^-Σ_t.x = Σ_t e^-Σ_t.x dx

Mean Free Path

From the equation for the probability of the first collision in dx we can calculate the mean free path that is traveled by a neutron between two collisions. This quantity is usually designated by the symbol λ and it is equal to the average value of x, the distance traveled by a neutron without any interaction, over the interaction probability distribution.

whereby one can distinguish λ_s, λ_a, λ_f, etc. This quantity is also known as the relaxation length, because it is the distance in which the intensity of the neutrons that have not caused a reaction has decreased with a factor e.

For materials with high absorption cross-section, the mean free path is very short and neutron absorption occurs mostly on the surface of the material. This surface absorption is called self-shielding because the outer layers of atoms shield the inner layers.

Macroscopic Cross-section of Mixtures and Molecules

Most materials are composed of several chemical elements and compounds. Most of chemical elements contains several isotopes of these elements (e.g., gadolinium with its six stable isotopes). For this reason most materials involve many cross-sections. Therefore, to include all the isotopes within a given material, it is necessary to determine the macroscopic cross section for each isotope and then sum all the individual macroscopic cross-sections.

In this section both factors (different atomic densities and different cross-sections) will be considered in the calculation of the macroscopic cross-section of mixtures.

First, consider the Avogadro’s number N₀ = 6.022 x 10²³, is the number of particles (molecules, atoms) that is contained in the amount of substance given by one mole. Thus if M is the molecular weight, the ratio N₀/M equals to the number of molecules in 1g of the mixture. The number of molecules per cm³ in the material of density ρ and the macroscopic cross-section for mixtures are given by following equations:

N_i = ρ_i.N₀ / M_i

Scattering of slow neutrons by molecules is greater than by free nuclei.

Note that, in some cases, the cross-section of the molecule is not equal to the sum of cross-sections of its individual nuclei. For example the cross-section of neutron elastic scattering of water exhibits anomalies for thermal neutrons. It occurs, because the kinetic energy of an incident neutron is of the order or less than the chemical binding energy and therefore the scattering of slow neutrons by water (H₂O) is greater than by free nuclei (2H + O).

Example - Macroscopic cross-section for boron carbide in control rods

A control rod usually contains solid boron carbide with natural boron. Natural boron consists primarily of two stable isotopes,¹¹B (80.1%) and ¹⁰B (19.9%). Boron carbide has a density of 2.52 g/cm³.

Determine the total macroscopic cross-section and the mean free path.

Density:
M_B = 10.8
M_C = 12
M_Mixture = 4 x 10.8 + 1×12 g/mol
N_B4C = ρ . N_a / M_Mixture
= (2.52 g/cm³)x(6.02×10²³ nuclei/mol)/ (4 x 10.8 + 1×12 g/mol)
= 2.75×10²² molecules of B4C/cm³

N_B = 4 x 2.75×10²² atoms of boron/cm³
N_C = 1 x 2.75×10²² atoms of carbon/cm³

N_B10 = 0.199 x 4 x 2.75×10²² = 2.18×10²² atoms of 10B/cm³
N_B11 = 0.801 x 4 x 2.75×10²² = 8.80×10²² atoms of 11B/cm³
N_C = 2.75×10²² atoms of 12C/cm³

the microscopic cross-sections

σ_t^10B = 3843 b of which σ_(n,alpha)^10B = 3840 b
σ_t^11B = 5.07 b
σ_t^12C = 5.01 b

the macroscopic cross-section

Σ_t^B4C = 3843×10^-24 x 2.18×10²² + 5.07×10^-24 x 8.80×10²² + 5.01×10^-24 x 2.75×10²²
= 83.7 + 0.45 + 0.14 = 84.3 cm^-1

the mean free path

λ_t = 1/Σ_t^B4C = 0.012 cm = 0.12 mm (compare with B4C pellets diameter in control rods which may be around 7mm)
λ_a ≈ 0.12 mm

Example - Atomic number density of 235U in uranium fuel

It was written the macroscopic cross-section is derived from microscopic cross-section and the atomic number density (N):

Σ=σ.N

In this equation, the atomic number density plays the crucial role as the microscopic cross-section, because in the reactor core the atomic number density of certain materials (e.g., water as the moderator) can be simply changed leading into certain reactivity changes. In order to understand the nature of these reactivity changes, we must understand the term the atomic number density.

See theory: Atomic Number Density

Most of PWRs use the uranium fuel, which is in the form of uranium dioxide (UO₂). Typically, the fuel have enrichment of ω₂₃₅ = 4% [grams of ²³⁵U per gram of uranium] of isotope ²³⁵U.

Calculate the atomic number density of ²³⁵U (N235U), when:

the molecular weight of the enriched uranium M_UO2 = 237.9 + 32 = 269.9 g/mol
the uranium density ⍴_UO2 = 10.5 g/cm³

N_UO2 = ⍴_UO2 . N_A / M_UO2

N_UO2= (10.5 g/cm³) x (6.02×10²³ nuclei/mol)/ 269.9
N_UO2= 2.34 x 10²² molecules of UO2/cm³

N_U = 1 x 2.34×10²² atoms of uranium/cm³
N_O = 2 x 2.34×10²² atoms of oxide/cm³

N_235U = ω₂₃₅.N_A.⍴_UO2/M_235U x (M_U/M_UO2)

N_235U = 0.04 x 6.02×10²³ x 10.5 / 235 x 237.9 / 269.9 =9.48 x 10²⁰ atoms of 235U/cm³

Doppler Broadening of Resonances

In general, Doppler broadening is the broadening of spectral lines due to the Doppler effect caused by a distribution of kinetic energies of molecules or atoms. In reactor physics a particular case of this phenomenon is the thermal Doppler broadening of the resonance capture cross sections of the fertile material (e.g., ²³⁸U or ²⁴⁰Pu) caused by thermal motion of target nuclei in the nuclear fuel.

The Doppler broadening of resonances is very important phenomenon, which improves reactor stability, because it accounts for the dominant part of the fuel temperature coefficient (the change in reactivity per degree change in fuel temperature) in thermal reactors and makes a substantial contribution in fast reactors as well. This coefficient is also called the prompt temperature coefficient because it causes an immediate response on changes in fuel temperature. The prompt temperature coefficient of most thermal reactors is negative.

Self-Shielding

It was written, in some cases the amount of absorption reactions is dramatically reduced despite the unchanged microscopic cross-section of the material. This phenomena is commonly known as the resonance self-shielding and also contributes to to the reactor stability. There are two types of self-shielding.

Energy Self-shielding.
Spatial Self-shielding.

See also: Resonance Self-shielding An increase in temperature from T₁ to T₂ causes the broadening of spectral lines of resonances. Although the area under the resonance remains the same, the broadening of spectral lines causes an increase in neutron flux in the fuel φ_f(E), which in turn increases the absorption as the temperature increases.

References:

Nuclear and Reactor Physics:

J. R. Lamarsh, Introduction to Nuclear Reactor Theory, 2nd ed., Addison-Wesley, Reading, MA (1983).
J. R. Lamarsh, A. J. Baratta, Introduction to Nuclear Engineering, 3d ed., Prentice-Hall, 2001, ISBN: 0-201-82498-1.
W. M. Stacey, Nuclear Reactor Physics, John Wiley & Sons, 2001, ISBN: 0- 471-39127-1.
Glasstone, Sesonske. Nuclear Reactor Engineering: Reactor Systems Engineering, Springer; 4th edition, 1994, ISBN: 978-0412985317
W.S.C. Williams. Nuclear and Particle Physics. Clarendon Press; 1 edition, 1991, ISBN: 978-0198520467
G.R.Keepin. Physics of Nuclear Kinetics. Addison-Wesley Pub. Co; 1st edition, 1965
Robert Reed Burn, Introduction to Nuclear Reactor Operation, 1988.
U.S. Department of Energy, Nuclear Physics and Reactor Theory. DOE Fundamentals Handbook, Volume 1 and 2. January 1993.

Advanced Reactor Physics:

K. O. Ott, W. A. Bezella, Introductory Nuclear Reactor Statics, American Nuclear Society, Revised edition (1989), 1989, ISBN: 0-894-48033-2.
K. O. Ott, R. J. Neuhold, Introductory Nuclear Reactor Dynamics, American Nuclear Society, 1985, ISBN: 0-894-48029-4.
D. L. Hetrick, Dynamics of Nuclear Reactors, American Nuclear Society, 1993, ISBN: 0-894-48453-2.
E. E. Lewis, W. F. Miller, Computational Methods of Neutron Transport, American Nuclear Society, 1993, ISBN: 0-894-48452-4.

See previous:

See above:

Neutron Nuclear Reactions