Neutron diffusion

The neutron diffusion ( lat. Diffundere , pour ',' scattered ',' propagate '), a special case in the mathematical treatment of the general neutron transport , is mainly important in the calculation of nuclear reactors . Diffusion means here, too, a process that occurs without any external influence, but not only the compensation of differences in number density; The processes involved in neutron diffusion are more diverse, because free neutrons can be created anew through nuclear reactions and disappear through absorption .

Simplified neutron diffusion equations

A spatial area is homogeneously filled with material that can both generate neutrons through nuclear fission and absorb neutrons.

The neutron flux can be obtained by solving the differential equation of neutron diffusion:

${\displaystyle {\text{Zeitliche Änderung}}={\text{Produktion}}-{\text{Leckage}}-{\text{Absorption}}}$

${\displaystyle {\frac {\partial n}{\partial t}}=S+D\ \Delta \Phi -\Sigma _{a}\ \Phi }$

It mean

character	unit	designation
${\displaystyle n}$	1 / cm³	Neutron number density
${\displaystyle t}$	s	time
${\displaystyle S}$	1 / cm³s	Local neutron source
${\displaystyle \Phi }$	1 / cm²s	Neutron flux
${\displaystyle D}$	cm	Neutron diffusion coefficient
${\displaystyle \Delta }$	1 / cm²	Laplace operator
${\displaystyle \Sigma _{a}}$	1 cm	Macroscopic absorption cross section

The neutron diffusion equations are size equations , i.e. they are independent of units. But there are common units in reactor physics , which are given in the second column of the table. Each term of the above differential equation system that is formed as the product of the macroscopic cross section and the neutron flux, e.g. B.

${\displaystyle \Sigma _{a}\ \Phi =\{\Sigma _{a}\ \Phi \}\cdot [\Sigma _{a}\ \Phi ]}$,

has the unit as a physical quantity

${\displaystyle [\Sigma _{a}\ \Phi ]=\mathrm {\frac {1}{cm^{3}\ s}} }$.

This is the unit of a nuclear reaction rate density .

One-dimensional neutron diffusion equations

In a medium that is limited by two parallel surfaces with an infinite surface, the so-called plate reactor, the simplified neutron diffusion equation results

${\displaystyle {\frac {\partial n}{\partial t}}=S+D\ {\frac {\partial ^{2}\Phi }{\partial x^{2}}}-\Sigma _{a}\ \Phi }$.

In cylindrical geometry, a reactor in the shape of a cylinder of infinite length (polar coordinates), the simplified neutron diffusion equation reads

${\displaystyle {\frac {\partial n}{\partial t}}=S+D\ \left({\frac {\partial ^{2}\Phi }{\partial r^{2}}}+{\frac {1}{r}}\ {\frac {\partial \Phi }{\partial r}}\right)-\Sigma _{a}\ \Phi }$.

Correspondingly in spherical geometry

${\displaystyle {\frac {\partial n}{\partial t}}=S+D\ \left({\frac {\partial ^{2}\Phi }{\partial r^{2}}}+{\frac {2}{r}}\ {\frac {\partial \Phi }{\partial r}}\right)-\Sigma _{a}\ \Phi }$.

In the steady state is the change in the neutron number density zero . We also proceed from this assumption for the stationary multi-group neutron diffusion equations ( see below ). ${\displaystyle {\tfrac {\partial n}{\partial t}}=0}$

Time-independent neutron diffusion equation

In a reactor, the source term S is described by in the steady state . ${\displaystyle S=k_{\infty }\ \Sigma _{a}\ \Phi }$

${\displaystyle \Delta \Phi +(k_{\infty }-1)\ {\frac {\Sigma _{a}}{D}}\ \Phi =0}$.

The neutron is the multiplication factor in the infinitely expanded medium. ${\displaystyle k_{\infty }}$

${\displaystyle L^{2}={\frac {D}{\Sigma _{a}}}}$

is the square of the diffusion length . Since it has to be in the critical reactor , you can determine the size ${\displaystyle L}$ ${\displaystyle k_{\infty }>1}$

${\displaystyle B^{2}={\frac {k_{\infty }-1}{L^{2}}}}$

introduce; In reactor physics , it is called buckling , or river arching in German . The simplified form of the neutron diffusion equation is thus

${\displaystyle \Delta \Phi +B^{2}\ \Phi =0}$.

This equation is of the Helmholtz equation type .

Stationary multi-group neutron diffusion equations

The real case of a heterogeneous reactor is described by the stationary multi-group neutron diffusion equations. The stationary neutron flux for the energy group at the location satisfies the homogeneous, time-independent multi-group neutron diffusion equations ${\displaystyle \Phi ^{g}({\vec {r}})}$ ${\displaystyle g}$ ${\displaystyle {\vec {r}}}$

${\displaystyle -\nabla (D^{g}({\vec {r}})\nabla \Phi ^{g}({\vec {r}}))+\Sigma _{r}^{g}({\vec {r}})\Phi ^{g}({\vec {r}})=Q^{g}({\vec {r}})}$

with the swell rate density , a sum of cleavage rate and scatter swell densities, in the form ${\displaystyle Q^{g}({\vec {r}})}$

${\displaystyle Q^{g}({\vec {r}})={\frac {\chi ^{g}}{k_{\text{eff}}}}\sum _{h=1}^{G}\nu \Sigma _{f}^{h}({\vec {r}})\Phi ^{g}({\vec {r}})\;+\sum _{h=1,\ h\neq g}^{G}\Sigma _{s}^{gh}({\vec {r}})\Phi ^{g}({\vec {r}})}$

For ${\displaystyle g=1,\ 2,\ldots ,G}$

and all locations in space for which these differential equations are to be solved. ${\displaystyle {\vec {r}}}$

These equations form a system of partial, elliptical differential equations of the 2nd order . In the form shown here, the continuous energy variable has already been divided into intervals, into energy groups. The so-called group constants , which are included in the coefficients of the system of equations, are (with a few exceptions) dependent on the material, location and energy. Before one can start with the solution of the multi-group neutron diffusion equations, these coefficients must have been calculated with a cell program and numerically available as input data. ${\displaystyle G}$

It mean

character	unit	designation
${\displaystyle {\vec {r}}}$	cm	Location, the coordinates of a point in the solution area
${\displaystyle \Phi ^{g}({\vec {r}})}$	1 / cm²s	Neutron flux of the energy group at the location , the fundamental eigenfunction of the differential equation system ${\displaystyle g}$ ${\displaystyle {\vec {r}}}$
${\displaystyle k_{\text{eff}}}$	1	Effective multiplication factor, the eigenvalue that belongs to the fundamental eigenfunction
${\displaystyle g}$ and ${\displaystyle h}$	1	Energy group number
${\displaystyle G}$	1	Number of energy groups
${\displaystyle \nabla }$	1 cm	Nabla or gradient operator
${\displaystyle D^{g}({\vec {r}})}$	cm	Neutron diffusion coefficient of the group in place ${\displaystyle g}$ ${\displaystyle {\vec {r}}}$
${\displaystyle \Sigma _{r}^{g}({\vec {r}})}$	1 cm	Macroscopic total loss cross section of the group at the site , also known as macroscopic removal cross section (hence index r ) ${\displaystyle g}$ ${\displaystyle {\vec {r}}}$
${\displaystyle \nu \Sigma _{f}^{h}({\vec {r}})}$	1 cm	Macroscopic neutron production cross-section of the group on site . This is a product of the mean number of neutrons per fission and the macroscopic fission cross-section (index f of fission ). ${\displaystyle h}$ ${\displaystyle {\vec {r}}}$ ${\displaystyle \nu }$ ${\displaystyle \Sigma _{f}^{h}({\vec {r}})}$
${\displaystyle \Sigma _{s}^{gh}({\vec {r}})}$	1 cm	Macroscopic scattering cross-section from the group to the local group . Note that the matrix indices often in the form be written (index s of scattering or scattering ) ${\displaystyle h}$ ${\displaystyle g}$ ${\displaystyle {\vec {r}}}$ ${\displaystyle h\to g}$
${\displaystyle \chi ^{g}}$	1	Gap spectrum of the group . Is generally independent of location ${\displaystyle g}$

Each of the individual equations of the system is the differential form of a conservation equation for the number of neutrons in space at the location whose energies lie in the interval that is determined by the boundaries of the energy group . ${\displaystyle G}$ ${\displaystyle {\vec {r}}}$ ${\displaystyle g}$

The system of differential equations is completed by two continuity conditions and one condition for all points that lie on outer surfaces. If the system is symmetrical, for example if it has mirror planes, then special boundary conditions apply to these planes.

Individual evidence

1. ^ Samuel Glasstone, Milton C. Edlund: The elements of nuclear reactor theory . MacMillan, London 1952 (VII, 416 pp.). Like no other, this monograph shaped the younger generation of reactor physicists in West and East and the later textbook writers. Your 6th print from February 1957 is fully available online. [1] . Full text search is possible. Neutron diffusion is discussed on p. 106.
2. ↑ CWJ McCallien: SNAP, Multigroup 3-D Neutron Diffusion in XZ, R-Theta-Z, Hexagonal-Z, Triangular-Z Geometry . AEA-RS-1214, 1993.
3. ^ Paul Reuss: Neutron physics . EDP ​​Sciences, Les Ulis, France 2008, ISBN 978-2-7598-0041-4 (xxvi, 669, limited preview in Google book search). Reuss deals with neutron diffusion a. a. on p. 650.