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Innuclear physics, the concept of aneutron cross section is used to express the likelihood of interaction between an incidentneutron 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.^{[1][page needed]} In conjunction with theneutron flux, it enables the calculation of the reaction rate, for example to derive the thermalpower of anuclear power plant. The standard unit for measuring the cross section is thebarn, 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.

Anisotope (ornuclide) 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 eitherdecay or keep the neutron in its nucleus areneutron absorbers and will have acapture cross section for that reaction. Isotopes that undergofission arefissionable fuels and have a correspondingfission cross section. The remaining isotopes will simply scatter the neutron, and have ascatter cross section. Some isotopes, likeuranium-238, have nonzero cross sections of all three.

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

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

• the target type (hydrogen,uranium...),
• the type ofnuclear reaction (scattering,fission...).
• the incident particleenergy, 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.

The neutron cross section is defined for a given type of target particle. For example, the capture cross section of deuterium²H is much smaller than that of common hydrogen¹H.^[2] This is the reason why some reactors useheavy water (in which most of the hydrogen is deuterium) instead of ordinarylight water asmoderator: fewer neutrons are lost by capture inside the medium, hence enabling the use ofnatural uranium instead ofenriched uranium. This is the principle of aCANDU reactor.

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 sectionσ_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 sectionsσ_S andσ_A are defined and the total cross section is simply the sum of the two partial cross sections:^[3]

${\displaystyle {\sigma }_{\text{T}}={\sigma }_{\text{S}}+{\sigma }_{\text{A}}}$

If the neutron is absorbed when approaching the nuclide, the atomic nucleus moves up on the table of isotopes by one position. For instance,²³⁵U becomes^236*U with the * indicating the nucleus is highly energized. This energy has to be released and the release can take place through any of several mechanisms.

1. The simplest way for the release to occur is for the neutron to be ejected by the nucleus. If the neutron is emitted immediately, it acts the same as in other scattering events.
2. The nucleus may emit gamma radiation.
3. The nucleus may β⁻ decay, where a neutron is converted into a proton, an electron and an electron-type antineutrino (the antiparticle of the neutrino)
4. About 81% of the^236*U nuclei are so energized that they undergo fission, releasing the energy as kinetic motion of the fission fragments, also emitting between one and five free neutrons.

• Nuclei that undergo fission as their predominant decay method after neutron capture include²³³U,²³⁵U,²³⁷U,²³⁹Pu,²⁴¹Pu.
• Nuclei that predominantly absorb neutrons and then emit beta particle radiation lead to these isotopes, e.g.,²³²Th absorbs a neutron and becomes^233*Th, which beta decays to become²³³Pa, which in turn beta decays to become²³³U.
• Isotopes that undergo beta decay transmute from one element to another element. Those that undergo gamma or X-ray emission do not cause a change in element or isotope.

The scattering cross-section can be further subdivided into coherentscattering and incoherent scattering, which is caused by thespin dependence of the scattering cross-section and, for a natural sample, presence of differentisotopes of the same element in the sample.

Becauseneutrons interact with thenuclear potential, the scattering cross-section varies for differentisotopes of the element in question. A very prominent example ishydrogen and its isotopedeuterium. The total cross-section for hydrogen is over 10 times that of deuterium, mostly due to the large incoherentscattering length of hydrogen. Some metals are rather transparent to neutrons,aluminum andzirconium being the two best examples of this.

For a given target and reaction, the cross section is strongly dependent on the neutron speed. In the extreme case, the cross section can be, at low energies, either zero (the energy for which the cross section becomes significant is calledthreshold energy) or much larger than at high energies.

Therefore, a cross section should be defined either at agiven energy or should be averaged in an energy range (or group).

As an example, the plot on the right shows that thefission cross section ofuranium-235 is low at high neutron energies but becomes higher at low energies. Such physical constraints explain why most operationalnuclear reactors use aneutron moderator to reduce the energy of the neutron and thus increase the probability of fission which is essential to produce energy and sustain thechain reaction.

A simple estimation of energy dependence of any kind of cross section is provided by the Ramsauer model,^[4] which is based on the idea that theeffective size of a neutron is proportional to the breadth of theprobability density function of where the neutron is likely to be, which itself is proportional to the neutron'sthermal de Broglie wavelength.

${\displaystyle \lambda (E)=\frac{h}{\sqrt{2mE}}}$

Taking${\displaystyle \lambda }$ as the effective radius of the neutron, we can estimate the area of the circle${\displaystyle \sigma }$ in which neutrons hit the nuclei of effective radius${\displaystyle R}$ as

${\displaystyle \sigma (E)\propto \pi (R+\lambda (E){)}^2}$

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 fm, E = 10–1000 keV)${\displaystyle R}$ can be neglected. For these low energy neutrons (such as thermal neutrons) the cross section${\displaystyle \sigma (E)}$ is inversely proportional to neutron velocity.

This explains the advantage of using aneutron moderator in fission nuclear reactors. On the other hand, for very high energy neutrons (over 1 MeV),${\displaystyle \lambda }$ 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 eV–10 keV, nor the threshold energy of some nuclear reactions.

Cross sections are usually measured at 20 °C. To account for the dependence with temperature of the medium (viz. the target), the following formula is used:^[3]

${\displaystyle \sigma ={\sigma }_0{\left(\frac{T_0}{T}\right)}^{\frac{1}{2}},}$

whereσ is the cross section at temperatureT, andσ₀ the cross section at temperatureT₀ (T andT₀ inkelvins).

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-term1⁄2√π has to be included when calculating the cross-sectionEquation 38.

The Doppler broadening of neutron resonances is a very important phenomenon and improvesnuclear reactor stability. The prompt temperature coefficient of most thermal reactors is negative, owing to the nuclearDoppler 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. Theresonance 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).

Imagine a spherical target (shown as the dashed grey and red circle in the figure) and a beam of particles (in blue) "flying" at speedv (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 (volumeV). The base of the cylinder is the geometrical cross section of the target perpendicular to the beam (surfaceσ in red) and its height the length travelled by the particles during dt (lengthv dt):

${\displaystyle V=\sigma vdt}$

Notingn thenumber of particles per unit volume, there aren V particles in the volumeV, which will, per definition ofV, undergo a reaction. Notingr thereaction rate onto one target, it gives:

${\displaystyle rdt=nV=n\sigma vdt}$

It follows directly from the definition of theneutron flux^[3]${\displaystyle \mathrm{\Phi }}$ =n v:

${\displaystyle r=\sigma \mathrm{\Phi }}$

Assuming that there is not one butN targets per unit volume, the reaction rateR per unit volume is:

${\displaystyle R=Nr=N\mathrm{\Phi }\sigma }$

Knowing that the typical nuclear radiusr is of the order of 10⁻¹² cm, the expected nuclear cross section is of the order ofπ r² or roughly 10⁻²⁴ cm² (thus justifying the definition of thebarn). However, if measured experimentally (σ =R / (Φ N) ), the experimental cross sections vary enormously. As an example, for slow neutrons absorbed by the (n, γ) reaction the cross section in some cases (xenon-135) is as much as 2,650,000 barns, while the cross sections for transmutations by gamma-ray absorption are in the neighborhood of 0.001 barn (§ Typical cross sections has more examples).

The so-callednuclear cross section is consequently a purely conceptual quantity representing how big the nucleus should be to be consistent with this simple mechanical model.

Cross sections depend strongly on the incoming particle speed. In the case of a beam with multiple particle speeds, the reaction rateR is integrated over the whole range of energy:

${\displaystyle R={\int }_{E}N\mathrm{\Phi }(E)\sigma (E)dE}$

Whereσ(E) is the continuous cross section,Φ(E) the differential flux andN the target atom number.

In order to obtain a formulation equivalent to the mono energetic case, an average cross section is defined:

${\displaystyle \sigma =\frac{{\int }_E\mathrm{\Phi }(E)\sigma (E)dE}{{\int }_E\mathrm{\Phi }(E)dE}=\frac{{\int }_E\mathrm{\Phi }(E)\sigma (E)dE}{\mathrm{\Phi }}}$

WhereΦ =$\int$Φ(E) dE is the integral flux.

Using the definition of the integral fluxΦ and the average cross sectionσ, the same formulation asbefore is found:

${\displaystyle R=N\mathrm{\Phi }\sigma }$

Up to now, the cross section referred to in this article corresponds to the microscopic cross sectionσ. However, it is possible to define the macroscopic cross section^[3]Σ which corresponds to the total "equivalent area" of all target particles per unit volume:

${\displaystyle \mathrm{\Sigma }=N\sigma }$

whereN is the atomic density of the target.

Therefore, since the cross section can be expressed in cm² and the density in cm⁻³, the macroscopic cross section is usually expressed in cm⁻¹. Using the equation derivedabove, the reaction rateR can be derived using only the neutron fluxΦ and the macroscopic cross sectionΣ:

${\displaystyle R=\mathrm{\Sigma }\mathrm{\Phi }}$

Themean free pathλ of a random particle is the average length between two interactions. The total lengthL that non perturbed particles travel during a time intervaldt in a volumedV is simply the product of the lengthl covered by each particle during this time with the number of particlesN in this volume:

${\displaystyle L=lN}$

Notingv the speed of the particles andn is the number of particles per unit volume:

It follows:

${\displaystyle L=vdtndV}$

Using the definition of theneutron flux^[3]Φ

${\displaystyle \mathrm{\Phi }=nv}$

It follows:

${\displaystyle L=\mathrm{\Phi }dtdV}$

This average lengthL is however valid only for unperturbed particles. To account for the interactions,L is divided by the total number of reactionsR to obtain the average length between each collisionλ:

${\displaystyle \lambda =\frac{L}{R}=\frac{\mathrm{\Phi }dtdV}{R}}$

From§ Microscopic versus macroscopic cross section:

${\displaystyle R=\mathrm{\Phi }\mathrm{\Sigma }dt}$

It follows:

${\displaystyle \lambda =\frac{dV}{\mathrm{\Sigma }}}$

whereλ is the mean free path andΣ is the macroscopic cross section.

Because⁸Li and¹²Be form natural stopping points on the table of isotopes forhydrogenfusion, it is believed that all of the higher elements are formed in very hot stars where higher orders of fusion predominate. A star like theSun producesenergy by the fusion of simple¹H into⁴He through aseries of reactions. It is believed that when the inner core exhausts its¹H fuel, the Sun will contract, slightly increasing its core temperature until⁴He can fuse and become the main fuel supply. Pure⁴He fusion leads to⁸Be, which decays back to 2 ⁴He; therefore the⁴He must fuse with isotopes either more or less massive than itself to result in an energy producing reaction. When⁴He fuses with²H or³H, it forms stable isotopes⁶Li and⁷Li respectively. The higher order isotopes between⁸Li and¹²C are synthesized by similar reactions between hydrogen, helium, and lithium isotopes.

Some cross sections that are of importance in a nuclear reactor are given in the following table.

• Thethermal cross-section is averaged using a Maxwellian spectrum.
• Thefast cross section is averaged using the uranium-235 fission spectrum.

The cross sections were taken from the JEFF-3.1.1 library using JANIS software.^[5]

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

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

• Neutron
• Nuclear physics

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