Nuclear reaction rate

The nuclear reaction rate or short reaction rate ( english reaction rate ), even occasionally burst rate , is a physical quantity of nuclear reactions . It is the quotient of the number of reactions (usually of a certain type) that take place in the space and the time span in which they are counted.

introduction

A single nuclear reaction takes place in a very limited space. It follows that all types of nuclear reactions can be counted completely independently of one another. This applies to different types of a nuclear reaction, such as elastic or inelastic neutron scattering , the capture of a neutron by an atomic nucleus or the fission of an atomic nucleus triggered by a neutron. This applies to particles such as neutrons , protons , α-particles , but also higher-energy photons , such as photons of gamma radiation . And that applies to reactions with each of the currently 3436 known nuclides (as of March 2017), which are limited to a few hundred relevant nuclides in reactor physics practice.

From the didactic point of view, i.e. the “way” to approach the conceptual world step by step, for example reactor physics , the values ​​of the nuclear reaction rate and the reaction rate density derived from it are helpful in understanding other variables such as the macroscopic cross section .

definition

The nuclear reaction rate of a reaction type is defined as ${\displaystyle R}$

${\displaystyle R={\frac {\text{Anzahl der Kernreaktionen}}{\text{Zeitintervall}}}}$.

With time interval is meant the time interval in which the number of nuclear reactions of a type has been counted. The usual unit of measurement for the rate of nuclear reactions is s ⁻¹ , in words: the number of nuclear reactions per second .

The nuclear reaction rate is an extensive quantity . It depends on the size of the area in question and can also depend on the shape. From it the intensive quantity nuclear reaction rate density (see below) can be calculated.

Nuclear reaction rates can be measured or calculated from other variables, usually from a previously calculated nuclear reaction rate density and the volume of the spatial area. Your measurement plays a subordinate role. It is important for the particle balance within the neutron diffusion theory .

Independent variable, discretization, mean

The two partners in a nuclear reaction, projectile particles and atomic nucleus, are usually inhomogeneously distributed in space. The nuclear reaction rate can therefore differ from place to place (with the same volume and the same “shape” of the spatial area under consideration). The rate of nuclear reaction also depends on the kinetic energy of the two reactants. Plus, it will change over time . These dependencies are written as an independent variable behind the symbol, ${\displaystyle E}$ ${\displaystyle t}$

${\displaystyle R=R({\vec {r}},E,t)}$.

The vector is the position vector with which the position of the spatial area is determined in a coordinate system. The size therefore has five independent variables. ${\displaystyle {\vec {r}}}$

The size and shape of the room area can vary depending on the objective. As far as the independent variable location is concerned, there are practical limits to the spatial “resolution” of the original spatial area into smaller spatial areas through discretization . In order to have a “clue”, think of the discretization of cubes, each with an edge length of one centimeter, as the lower limit value for the spatial discretization. In nuclear reactors, nuclear reactions are recorded in the energy interval (10 ⁻⁴ - 2 · 10 ⁷ ) eV of the projectile particles neutrons . The energetic resolution ranges from the entire interval to a few hundred subdivisions ("groups"). Giving guideline values ​​for the number of intervals of the time variables is problematic. In the case of a nuclear reactor in a quasi-stationary state, for example, the time step may be 5 days or longer, but only nanoseconds in the case of a nuclear weapon explosion.

If the magnitude of the nuclear reaction rate is calculated, such a discretization of the independent variables is established before this calculation. If a nuclear reaction rate is calculated or measured, it is always an average value over a spatial area, an energy interval and a time step.

example

Calculating a nuclear reaction rate can be quite simple in special cases. The nuclear reaction rate of the fission reaction can be calculated solely from the thermal output of a nuclear reactor and the energy released by the fission cycle of a reaction. The size of the thermal power is known for a nuclear reactor and results from the size equation ${\displaystyle L}$

${\displaystyle L=e\cdot R}$,

where the nuclear reaction rate, symbolize the energy released during a nuclear fission. From the equation converted according to the nuclear reaction rate ${\displaystyle R}$ ${\displaystyle e}$

${\displaystyle R={\frac {L}{e}}}$

we can calculate this.

The thermal output of the Emsland nuclear power plant is 3850 MW. It can be assumed that an energy of 200 MeV is released on average by a nuclear fission ,

${\displaystyle e=200\,{\frac {\mathrm {MeV} }{\text{Spaltung}}}}$.

The unit we can into the unit to convert: ${\displaystyle \mathrm {eV} }$ ${\displaystyle \mathrm {Ws} }$

${\displaystyle 1\,\mathrm {eV} =1{,}6022\cdot 10^{-19}\,\mathrm {Ws} \ }$.

This gives the reaction rate for the nuclear fission reaction too

${\displaystyle R={\frac {3{,}850\cdot 10^{9}\,\mathrm {W} \cdot {\text{Spaltungen}}}{2\cdot 10^{8}\cdot 1{,}6022\cdot 10^{-19}\,\mathrm {Ws} }}=1{,}20\cdot 10^{20}\,{\frac {\text{Spaltungen}}{\mathrm {s} }}}$

In words: In the power reactor, fuel cores ( ²³⁵ U or ²³⁹ Pu) are split every second . ${\displaystyle 1{,}2\cdot 10^{20}}$

The contribution of other nuclear reactions to the thermal output of the reactor is comparatively small, but is taken into account in a more precise calculation of the energy balance.

Nuclear reaction rate density

With the size of the nuclear reaction rate , we define the nuclear reaction rate density as the quotient of the nuclear reaction rate and the volume of our spatial area, ${\displaystyle R}$ ${\displaystyle r}$ ${\displaystyle V}$

${\displaystyle r={\frac {R}{V}}}$.

The nuclear reaction rate density is one of the fundamental quantities, for example in reactor physics and nuclear engineering . In cosmology , size plays a role, for example in primordial nucleosynthesis .

Nuclear reaction rate density and cross section

→ Main article : cross section

Let us assume that (resting) atomic nuclei (of one kind) and (moving) particles (also of one kind) share a space. It should be known:

• The number density of atomic nuclei, ${\displaystyle N={\frac {\text{Anzahl der Atomkerne}}{V}}}$
• the number density of the particles and ${\displaystyle n={\frac {\text{Anzahl der Teilchen}}{V}}}$
• the speed of the particles in this area of ​​space. ${\displaystyle v}$

Depending on the objective and without restricting the generality, we can always choose this spatial area so large or small that we can assume both the number density of atomic nuclei and that of the particles to be constant.

It is plausible that the frequency of collisions increases with the number density of atomic nuclei as well as with the number density of particles, and that a particle that is traveling faster hits more atomic nuclei, i.e. potential collision partners, than a slower one.

Hence, there is a direct relationship between , and . The experiment shows that this relationship is linear: ${\displaystyle N}$ ${\displaystyle n}$ ${\displaystyle v}$

${\displaystyle r\propto N\cdot n\cdot v}$.

The cross section of a type of nuclear reaction is defined as the proportionality factor, as the quotient of the nuclear reaction rate density and the other three quantities:

${\displaystyle \sigma ={\frac {r}{N\cdot n\cdot v}}}$.

The size equation for the nuclear reaction rate density is therefore ${\displaystyle r}$

${\displaystyle r=\sigma \cdot N\cdot n\cdot v}$.

Independent variable energy

Cross- sections for 6 nuclear reactions of neutron and atomic nucleus ²³⁵ U and their sum as a function of the kinetic energy of the neutrons. In the legend, z is sometimes used instead of the usual symbol n for neutron (data source: JEFF, graphic representation: core data viewer JANIS 4)

The cross section for a pair type (atomic nucleus, particle) depends very much on the (relative) speed of the two collision partners. That will go through ${\displaystyle v}$

${\displaystyle \sigma =\sigma (E)}$

symbolizes; the relationship between speed , mass and kinetic energy is at the (not relativistic ) speeds of interest here ${\displaystyle v}$ ${\displaystyle m}$ ${\displaystyle E}$

${\displaystyle E={\frac {1}{2}}\ m\ v^{2}}$.

For example, the cross section for neutrons in the energy interval (10 ⁻¹¹ to 20) MeV can differ by orders of magnitude. With one and the same projectile particle it depends strongly on the type of nuclear reaction. The figure shows the cross-sections of the six types of nuclear reactions dominating in this energy interval of neutron and atomic nucleus ²³⁵ U and the sum of these cross-sections. The cross-section of the elastic scattering changes less strongly with the neutron energy ("incident energy") than other cross-sections.

With the exception of photons as projectile particles, no cross-section can be calculated with the accuracy required by reactor physicists and nuclear technicians, for example to “construct” a nuclear reactor . The measurement of the cross-sections for neutrons in the energy interval mentioned began in the second half of the 20th century and has not yet been completed.

Nuclear reaction rate density and macroscopic cross section

The cross-section of the nuclear reaction and the number density of the atomic nuclei are properties of matter alone and do not depend on the properties of the projectile particles or their velocities. Therefore it is obvious to summarize your product in a new physical quantity, ${\displaystyle \sigma }$ ${\displaystyle N}$

${\displaystyle \Sigma =\sigma \cdot N}$.

In reactor physics, this quantity is called the macroscopic cross section .

On the other hand, the other two sizes, the particle density and the speed of the particles, do not depend on the properties of the matter in the space under consideration. They are summarized in a size that has been given the name neutron flux in reactor physics, i.e. with neutrons as projectile particles , ${\displaystyle n}$ ${\displaystyle v}$

${\displaystyle \Phi =n\cdot v}$.

The nuclear reaction rate density is thus given its final shape ${\displaystyle r}$

${\displaystyle r=\Sigma \cdot \Phi }$.

Nuclear reaction rate density and neutron flux

→ Main article : neutron flux

Of the quantities introduced above, the quantity of neutron flux is the one whose calculation requires the most computational effort for a nuclear reactor. To put it simply, the neutron flux is determined in the following steps:

• When the reactor is restarted, for example, the geometry of all components and the spaces they occupy are known from the design documents. We also know the materials the components are made of and the exact material composition. Now you cover (in principle) the entire reactor core with a discretization grid and thus create “grid boxes”. The number densities of all atomic nuclei in each lattice box can be calculated from the mass densities of the materials. This also applies to "cavities" that only occur during operation, e.g. B. be filled with water.
• The cross-sections for each individual nuclide are taken from extensive databases . The independent variable energy is usually already divided into intervals ("groups"). Depending on the task, this can be millions of individual pieces of data. ${\displaystyle \sigma (E)}$
• The macroscopic cross-sections are calculated for each individual grid box.
• A balance of the nuclear reaction rate is according to their size, including any neutron gain and loss within the box and through leakage (for each grid box English Leakage asserted) to adjacent boxes, or by the outer boundary of the reactor.
• The initially still unknown neutron flux of the grid box is a parameter within this balance.
• From a (in the simplest case) linear system of equations , in whose coefficient matrix the material data are included , the neutron diffusion equations , the neutron flux of each grid box can now be calculated. With a "fine" subdivision into grid boxes, this can mean several million unknown neutron flux values. Neutron diffusion programs are part of the “tools” of every working group on reactor physics and every analog working group in a nuclear power plant.

If the neutron flux is available for each grid box, the nuclear reaction rate and the nuclear reaction rate density for each reaction pair (nuclide, particle) and each energy (group) can ultimately be calculated for each grid box. Such neutron flux calculations were used to a large percentage on mainframes in the USA, for example, from 1950 to 1980.

Related sizes

The highest-ranking variable in the family of "nuclear reaction variables" in reactor theory is the number of nuclear reactions, followed by the equally important variables, the nuclear reaction rate and the nuclear reaction density. In the next “refinement” we come to the quantity nuclear reaction rate density, which is also defined above . The size of the next and last stage will not be discussed here. It differentiates a few nuclear reactions (e.g. elastic scattering ) according to the direction of flight , e.g. B. according to the flight direction of the neutrons that triggered the nuclear reaction. This is where the differential cross section comes into play.

Individual evidence

1. ^ ^{A b c} Samuel Glasstone, Milton C. Edlund: The elements of nuclear reactor theory . MacMillan, London 1952 (VII, 416 pp.). This monograph is fully online in the 6th print from February 1957. [1] . Full text search is possible.
2. ^ Paul Reuss: Neutron physics . EDP ​​Sciences, Les Ulis, France 2008, ISBN 978-2-7598-0041-4 (xxvi, 669, limited preview in Google book search). In the monograph by Reuss the sizes are on page 98 neutron density ( English neutron density ) neutron flux ( English neutron flux ), and nuclear reaction rate ( English Reaction rate ) defined. It should be noted: When Reuss is Reaction rate (symbol ) in this article nuclear reaction rate density (symbol called) size meant. ${\displaystyle R}$ ${\displaystyle r}$
3. ↑ WJ Huang et al. : The AME2016 atomic mass evaluation (I). Evaluation of input data; and adjustment procedures . In: Chinese Physics C . tape 41 , no. 3 , 2017, p. 30002 ( online ).
4. ↑ ^{a b} Rudi JJ Stamm'ler et al .: HELIOS Methods: Version 1.8 . Studsvik Scandpower 2003 (192 pages).
5. ↑ The mean values ​​of the reaction rates within a cube or a cylinder, for example, can be slightly different, even if they have the same volume and the center of gravity of these two figures enveloping the space is in the same place.
6. ↑ Usually the speed of the projectile particle ("particle") is much higher than the speed of the second impact partner. That is why the speed of the projectile particle is meant by speed without the addition of “relative speed”.
7. ↑ Janis 4 - Java-based Nuclear Data Information System
8. ↑ Types of nuclear reactions are provided with a globally uniform code in nuclear databases. MT = 18 means the nuclear reaction that leads to nuclear fission.
9. ↑ Since it is formed strictly according to the rules of naming physical quantities, the nuclear reaction density quantity does not need to be explicitly defined here.