Definition of Cross-section

nuclear cross-sections, microscopic cross-sectionsIn general, the cross-section is an effective area that quantifies the likelihood of certain interaction between an incident object and a target object. The cross-section of a particle is the same as the cross section of a hard object, if the probabilities of hitting them with a ray are the same.

For a given event, the cross section σ is given by

σ = μ/n

where

  • σ is the cross-section of this event [m2],
  • μ is the attenuation coefficient due to the occurrence of this event [m-1],
  • n is the density of the target particles [m-3].

In nuclear physics, the nuclear cross section of a nucleus is commonly used to characterize the probability that a nuclear reaction will occur. The cross-section is typically denoted σ and measured in units of area [m2]. 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 nuclear radii are of the order 10−14 m. Assuming spherical shape, nuclear radii can be calculated according to following formula:

r = r0 . A1/3

where r0 = 1.2 x 10-15 m = 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 4.5×10−30 m² for hydrogen nuclei or 1.74×10−28 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.

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

σt = σs + σi + σγ + σf + ……

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

Neutron Interactions

Neutrons are neutral particles, therefore they travel in straight lines, deviating from their path only when they actually collide with a nucleus to be scattered into a new direction or absorbed. Neither the electrons surrounding (atomic electron cloud) a nucleus nor the electric field caused by a positively charged nucleus affect a neutron’s flight. In short, neutrons collide with nuclei, not with atoms.

Neutrons may interact with nuclei in one of following ways:

See also: Neutron Nuclear Reactions

See also: Direct Nuclear Reactions

See also: Compound Nucleus Reactions

Neutron - Nuclear Reactions

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 do 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 Na 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 Na. We define the proportionality factor as the microscopic cross-section σ:

σt = C/Na.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.

CR

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.

Microscopic cross-sections constitute a key parameters of nuclear fuel. In general, neutron cross-sections must be calculated for fresh fuel assemblies usually in two-Dimensional models of the fuel lattice.

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 σγ. 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
Table of microscopic cross-sections
Table of microscopic cross-sections
Types of neutron-nuclear reactions.Types of neutron-nuclear reactions.
microscopic cross-section
Uranium 238. Comparison of cross-sections.
Source: JANIS (Java-based Nuclear Data Information Software); The JEFF-3.1.1 Nuclear Data Library
Microscopic Cross-section - Gadolinium
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
Microscopic Cross-section - Regions
Comparison of cross-sections in the 1/v and the resonance 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.

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.

See also: Nuclear Resonance

The radiative capture cross-section at energies above the resonance region drops rapidly to very small values. This rapid drop is caused by the compound nucleus, which is formed in more highly-excited states. In these highly-excited states it is more likely that one neutron obtains an energy in collision with other nucleon greater than its binding energy in the nucleus. The neutron emission becomes dominant and gamma decay becomes less important. Moreover, at high energies, the inelastic scattering and (n,2n) reaction are highly probable at the expense of both elastic scattering and radiative capture.

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 Σt (the total cross-section), the equation for the intensity of a neutron beam can be written as

-dI = N.σ.Σt.dx

Dividing this expression by I(x) gives

-dΙ(x)/I(x) = Σt.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 Σt, is:

P(x) = et.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 (~Σtdx) times the probability that the neutron will interact in the additional layer dx (i.e. ~et.x).

P(x)dx = Σtdx . et.x = Σt et.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.

mean free path - equation

whereby one can distinguish λs, λa, λf, 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:

Ni = ρi.N0 / Mi

Macroscopic Cross-section for Mixtures and Molecules

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

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).

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:
MB = 10.8
MC = 12
MMixture = 4 x 10.8 + 1×12 g/mol
NB4C = ρ . Na / MMixture
= (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

NB = 4 x 2.75×1022 atoms of boron/cm3
NC = 1 x 2.75×1022 atoms of carbon/cm3

NB10 = 0.199 x 4 x 2.75×1022 = 2.18×1022 atoms of 10B/cm3
NB11 = 0.801 x 4 x 2.75×1022 = 8.80×1022 atoms of 11B/cm3
NC = 2.75×1022 atoms of 12C/cm3

the microscopic cross-sections

σt10B = 3843 b of which σ(n,alpha)10B = 3840 b
σt11B = 5.07 b
σt12C = 5.01 b

the macroscopic cross-section

ΣtB4C = 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

λt = 1/ΣtB4C = 0.012 cm = 0.12 mm (compare with B4C pellets diameter in control rods which may be around 7mm)
λa ≈ 0.12 mm

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 MUO2 = 237.9 + 32 = 269.9 g/mol
  • the uranium density UO2 = 10.5 g/cm3

NUO2 = UO2 . NA / MUO2

NUO2 = (10.5 g/cm3) x (6.02×1023 nuclei/mol)/ 269.9
NUO2 = 2.34 x 1022 molecules of UO2/cm3

NU = 1 x 2.34×1022 atoms of uranium/cm3
NO = 2 x 2.34×1022 atoms of oxide/cm3

N235U = ω235.NA.UO2/M235U x (MU/MUO2)

N235U = 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.

See also: Doppler Broadening

Dopper broadening of resonance cross-section
Doppler broadening of resonance cross-sections.

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 the 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 φf(E), which in turn increases the absorption as the temperature increases.
Nuclear and Reactor Physics:

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

Advanced Reactor Physics:

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

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