Neutron cross section

In nuclear physics, the concept of a neutron cross section is used to express the likelihood of interaction between an incident neutron and a target nucleus. The neutron cross section σ can be defined as the area for which the number of neutron-nuclei reactions taking place is equal to the product of the number of incident neutrons that would pass through the area and the number of target nuclei. 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 cross section is the barn, which is equal to 10⁻²⁸ m² or 10⁻²⁴ cm². The larger the neutron cross section, the more likely a neutron will react with the nucleus.

An isotope (or nuclide) can be classified according to its neutron cross section and how it reacts to an incident neutron. Nuclides that tend to absorb a neutron and either decay or keep the neutron in its nucleus are neutron absorbers and will have a capture cross section for that reaction. Isotopes that undergo fission are fissionable fuels and have a corresponding fission cross section. The remaining isotopes will simply scatter the neutron, and have a scatter cross section. Some isotopes, like uranium-238, have nonzero cross sections of all three.

Isotopes which have a large scatter cross section and a low mass are good neutron moderators (see chart below). Nuclides which have a large absorption cross section are neutron poisons if they are neither fissile nor undergo decay. A poison that is purposely inserted into a nuclear reactor for controlling its reactivity in the long term and improve its shutdown margin is called a burnable poison.

Parameters of interest

The neutron cross section, and therefore the probability of a neutron–nucleus interaction, depends on:

• the target type (hydrogen, uranium...),
• the type of nuclear reaction (scattering, fission...).
• the incident particle energy, also called speed or temperature (thermal, fast...),

and, to a lesser extent, of:

• its relative angle between the incident neutron and the target nuclide,
• the target nuclide temperature.

Target type dependence

The neutron cross section is defined for a given type of target particle. For example, the capture cross section of deuterium ²H is much smaller than that of common hydrogen ¹H. This is the reason why some reactors use heavy water (in which most of the hydrogen is deuterium) instead of ordinary light water as moderator: fewer neutrons are lost by capture inside the medium, hence enabling the use of natural uranium instead of enriched uranium. This is the principle of a CANDU reactor.

Type of reaction dependence

The likelihood of interaction between an incident neutron and a target nuclide, independent of the type of reaction, is expressed with the help of the total cross section σ_T. However, it may be useful to know if the incoming particle bounces off the target (and therefore continue travelling after the interaction) or disappears after the reaction. For that reason, the scattering and absorption cross sections σ_S and σ_A are defined and the total cross section is simply the sum of the two partial cross sections: which is based on the idea that the effective size of a neutron is proportional to the breadth of the probability density function of where the neutron is likely to be, which itself is proportional to the neutron's thermal de Broglie wavelength.

:<math> \lambda(E) = \frac {h} {\sqrt{2mE </math>

Taking <math> \lambda </math> as the effective radius of the neutron, we can estimate the area of the circle <math> \sigma </math> in which neutrons hit the nuclei of effective radius <math>R </math> as

:<math> \sigma(E) \propto \pi(R + \lambda(E))^2 </math>

While the assumptions of this model are naive, it explains at least qualitatively the typical measured energy dependence of the neutron absorption cross section. For neutrons of wavelength much larger than typical radius of atomic nuclei (1–10&nbsp;fm, E = 10–1000&nbsp;keV) <math>R</math> can be neglected. For these low energy neutrons (such as thermal neutrons) the cross section <math>\sigma(E)</math> is inversely proportional to neutron velocity.

This explains the advantage of using a neutron moderator in fission nuclear reactors. On the other hand, for very high energy neutrons (over 1&nbsp;MeV), <math>\lambda</math> can be neglected, and the neutron cross section is approximately constant, determined just by the cross section of atomic nuclei.

However, this simple model does not take into account so called neutron resonances, which strongly modify the neutron cross section in the energy range of 1&nbsp;eV–10&nbsp;keV, nor the threshold energy of some nuclear reactions.

Target temperature dependence

Cross sections are usually measured at 20&nbsp;°C. To account for the dependence with temperature of the medium (viz. the target), the following formula is used:

:<math> \sigma = \sigma_0 \left(\frac{T_0}{T}\right)^\frac{1}{2}, </math>

where σ is the cross section at temperature T, and σ₀ the cross section at temperature T₀ (T and T₀ in kelvins).

The energy is defined at the most likely energy and velocity of the neutron. The neutron population consists of a Maxwellian distribution, and hence the mean energy and velocity will be higher. Consequently, also a Maxwellian correction-term √π has to be included when calculating the cross section Equation 38.

Doppler broadening

The Doppler broadening of neutron resonances is a very important phenomenon and improves nuclear reactor stability. The prompt temperature coefficient of most thermal reactors is negative, owing to the nuclear Doppler effect. Nuclei are located in atoms which are themselves in continual motion owing to their thermal energy (temperature). As a result of these thermal motions, neutrons impinging on a target appears to the nuclei in the target to have a continuous spread in energy. This, in turn, has an effect on the observed shape of resonance. The resonance becomes shorter and wider than when the nuclei are at rest.

Although the shape of resonances changes with temperature, the total area under the resonance remains essentially constant. But this does not imply constant neutron absorption. Despite the constant area under resonance a resonance integral, which determines the absorption, increases with increasing target temperature. This, of course, decreases coefficient k (negative reactivity is inserted).

Link to reaction rate and interpretation

thumb|right|450px|Interpretation of the reaction rate with the help of the cross section

Imagine a spherical target (shown as the dashed grey and red circle in the figure) and a beam of particles (in blue) "flying" at speed v (vector in blue) in the direction of the target. We want to know how many particles impact it during time interval dt. To achieve it, the particles have to be in the green cylinder in the figure (volume V). The base of the cylinder is the geometrical cross section of the target perpendicular to the beam (surface σ in red) and its height the length travelled by the particles during dt (length v&thinsp;dt):

:<math> V = \sigma \, v \, dt </math>

Noting n the number of particles per unit volume, there are n&thinsp;V particles in the volume V, which will, per definition of V, undergo a reaction. Noting r the reaction rate onto one target, it gives:

:<math> r \, dt = n \, V = n \, \sigma \, v \, dt </math>

It follows directly from the definition of the neutron flux

{| class="wikitable"

|-

! colspan="2" rowspan="2" | Nucleon

! colspan="3" | Thermal cross section (barn)

! colspan="3" | Fast cross section (barn)

|-

! Scattering

! Capture

! Fission

! Scattering

! Capture

! Fission

|-

| rowspan="3" | Moderator

| ¹H

| 20

| 0.2

| -

| 4

| 0.00004

| -

|-

| ²H

| 4

| 0.0003

| -

| 3

| 0.000007

| -

|-

| ¹²C

| 5

| 0.002

| -

| 2

| 0.00001

| -

|-

| rowspan="7" | Structural <br/>materials, <br/>others

| ¹⁹⁷Au

| 8.2

| 98.7

| -

| 4

| 0.08

| -

|-

| ⁹⁰Zr

| 5

| 0.006

| -

| 5

| 0.006

| -

|-

| ⁵⁶Fe

| 10

| 2

| -

| 20

| 0.003

| -

|-

| ⁵²Cr

| 3

| 0.5

| -

| 3

| 0.002

| -

|-

| ⁵⁹Co

| 6

| 37.2

| -

| 4

| 0.006

| -

|-

| ⁵⁸Ni

| 20

| 3

| -

| 3

| 0.008

| -

|-

| ¹⁶O

| 4

| 0.0001

| -

| 3

| 0.00000003

| -

|-

| rowspan="4" | Absorber

| ¹⁰B

| 2

| 200

| -

| 2

| 0.4

| -

|-

| ¹¹³Cd

| 100

| 30,000

| -

| 4

| 0.05

| -

|-

| ¹³⁵Xe

| 400,000

| 2,000,000

| -

| 5

| 0.0008

| -

|-

| ¹¹⁵In

| 2

| 100

| -

| 4

| 0.02

| -

|-

| rowspan="3" | Fuel

| ²³⁵U

| 10

| 99

| 583

| 4

| 0.09

| 1

|-

| ²³⁸U

| 9

| 2

| 0.00002

| 5

| 0.07

| 0.3

|-

| ²³⁹Pu

| 8

| 269

| 748

| 5

| 0.05

| 2

|}

<nowiki>*</nowiki> negligible, less than 0.1% of the total cross section and below the Bragg scattering cutoff

External links

• XSPlot an online nuclear cross section plotter
• Neutron scattering lengths and cross sections
• Periodic Table of Elements: Sorted by Cross Section (Thermal Neutron Capture)

<!-- What does this mean? "Coherent scattering: In this nuclear process reemission occurs at the same frequency as the incident radiation."

-->

References