Neutron transportFrom Wikipedia, the free encyclopedia

Neutron transport is the study of the motions and interactions of neutrons with materials. Nuclear scientistsand engineers often need to know where neutrons are in an apparatus, what direction they are going, and howquickly they are moving. It is commonly used to determine the behavior of nuclear reactor cores andexperimental or industrial neutron beams. Neutron transport is a type of radiative transport.

Contents

1 Background

2 Neutron Transport Equation

3 Types of neutron transport calculations

3.1 Fixed Source

3.2 Criticality

4 Computational Methods

4.1 Discretization in Deterministic Methods

4.2 Computer Codes Used In Neutron Transport

4.2.1 Probabilistic codes

4.2.2 Deterministic codes

5 See also

6 References

7 External links

Background

Neutron transport has roots in the Boltzmann equation, which was used in the 1800s to study the kinetic theoryof gases. It did not receive large-scale development until the invention of chain-reacting nuclear reactors in the1940s. As neutron distributions came under detailed scrutiny, elegant approximations and analytic solutionswere found in simple geometries. However, as computational power has increased, numerical approaches toneutron transport have become prevalent. Today, with massively parallel computers, neutron transport is stillunder very active development in academia and research institutions throughout the world. It remains one of themost computationally challenging problems in the world since it depends on 3-dimensions of space, time, andthe variables of energy span several decades (from fractions of meV to several MeV). Modern solutions useeither discrete-ordinates or monte-carlo methods, or even a hybrid of both.

Neutron Transport Equation

The neutron transport equation is a balance statement that conserves neutrons. Each term represents a gain or aloss of a neutron, and the balance, in essence, claims that neutrons gained equals neutrons lost. It is formulated

as follows:[1]

Where:

Symbol Meaning Comments

Position vector (i.e. x,y,z)

Energy

Unit vector (solid angle) in direction ofmotion

Time

Neutron velocity vector

Angular neutron fluxAmount of neutron track length in a

differential volume about , associatedwith particles of a differential energy in about , moving in a differential solid angle

in about , at time .

Note integratingover all angles

yields scalarneutron flux

Scalar neutron flux

Amount of neutron track length in adifferential volume about , associatedwith particles of a differential energy in

about , at time .

Average number of neutrons produced per

fission (e.g., 2.43 for U-235).[2]

Probability density function for neutrons ofexit energy from all neutrons produced

by fission

Probability density function for neutrons ofexit energy from all neutrons producedby delayed neutron precursors

Macroscopic total cross section, whichincludes all possible interactions

Macroscopic fission cross section, which

includes all fission interactions in

about

Double differential scattering cross section

Characterizes scattering of a neutron froman incident energy in and direction

in to a final energy and

direction .

Number of delayed neutron precursors

Decay constant for precursor i

Total number of precursor i in at time

Source term

The transport equation can be applied to a given part of phase space (time t, energy E, location , and directionof travel ). The first term represents the time rate of change of neutrons in the system. The second terms

describes the movement of neutrons into or out of the volume of space of interest. The third term accounts for allneutrons that have a collision in that phase space. The first term on the right hand side is the production ofneutrons in this phase space due to fission, while the second term on the right hand side is the production ofneutrons in this phase space due to delayed neutron precursors (i.e., unstable nuclei which undergo neutrondecay). The third term on the right hand side is in-scattering, these are neutrons that enter this area of phasespace as a result of scattering interactions in another. The fourth term on the right is a generic source. Theequation is usually solved to find , since that will allow for the calculation of reaction rates, which are of

primary interest in shielding and dosimetry studies.

Types of neutron transport calculations

Several basic types of neutron transport problems exist, depending on the type of problem being solved.

Fixed Source

A fixed source calculation involves imposing a known neutron source on a medium and determining the resultingneutron distribution throughout the problem. This type of problem is particularly useful for shielding calculations,where a designer would like to minimize the neutron dose outside of a shield while using the least amount ofshielding material. For instance, a spent nuclear fuel cask requires shielding calculations to determine how muchconcrete and steel is needed to safely protect the truck driver who is shipping it.

Criticality

Fission is the process through which a nucleus splits into (typically two) smaller atoms. If fission is occurring, it isoften of interest to know the asymptotic behavior of the system. A reactor is called “critical” if the chain reactionis self-sustaining and time-independent. If the system is not in equilibrium the asymptotic neutron distribution, orthe fundamental mode, will grow or decay exponentially over time.

Criticality calculations are used to analyze steady-state multiplying media (multiplying media can undergo fission),such as a critical nuclear reactor. The loss terms (absorption, out-scattering, and leakage) and the source terms(in-scatter and fission) are proportional to the neutron flux, contrasting with fixed-source problems where thesource is independent of the flux. In these calculations, the presumption of time invariance requires that neutronproduction exactly equals neutron loss.

Since this criticality can only be achieved by very fine manipulations of the geometry (typically via control rods ina reactor), it is unlikely that the modeled geometry will be truly critical. To allow some flexibility in the waymodels are set up, these problems are formulated as eigenvalue problems, where one parameter is artificiallymodified until criticality is reached. The most common formulations are the time-absorption and the multiplicationeigenvalues, also known as the alpha and k eigenvalues. The alpha and k are the tunable quanitites.

K-eigenvalue problems are the most common in nuclear reactor analysis. The number of neutrons produced perfission is multiplicatively modified by the dominant eigenvalue. The resulting value of this eigenvalue reflects thetime dependence of the neutron density in a multiplying medium.

keff < 1, subcritical: the neutron density is decreasing as time passes;

keff = 1, critical: the neutron density remains unchanged; and

keff > 1, supercritical: the neutron density is increasing with time.

In the case of a nuclear reactor, neutron flux and power density are proportional, hence during reactor start-upkeff > 1, during reactor operation keff = 1 and keff < 1 at reactor shutdown.

Computational Methods

Both fixed-source and criticality calculations can be solved using deterministic methods or stochastic methods.In deterministic methods the transport equation (or an approximation of it, such as diffusion theory) is solved asa differential equation. In stochastic methods such as Monte Carlo discrete particle histories are tracked andaveraged in a random walk directed by measured interaction probabilities. Deterministic methods usually involvemulti-group approaches while Monte Carlo can work with multi-group and continuous energy cross-sectionlibraries. Multi-group calculations are usually iterative, because the group constants are calculated using flux-energy profiles, which are determined as the result of the neutron transport calculation.

Discretization in Deterministic Methods

To numerically solve the transport equation using algebraic equations on a computer, the spatial, angular, energy,and time variables must be discretized.

Spatial variables are typically discretized by simply breaking the geometry into many small regions on a

mesh. The balance can then be solved at each mesh point using finite difference or by nodal methods.

Angular variables can be discretized by discrete ordinates and weighting quadrature sets (giving rise to the

SN methods), or by functional expansion methods with the spherical harmonics (leading to the PN

methods).

Energy variables are typically discretized by the multi-group method, where each energy group represents

one constant energy. As few as 2 groups can be sufficient for some thermal reactor problems, but fast

reactor calculations may require many more.

The time variable is broken into discrete time steps, with time derivatives replaced with difference

formulas.

Computer Codes Used In Neutron Transport

Probabilistic codes

OpenMC - An MIT developed open source Monte Carlo code [3]

MCNP - A LANL developed Monte Carlo code for general radiation transport

KENO - An ORNL developed Monte Carlo code for criticality analysis

MCBEND - An ANSWERS Software Service developed Monte Carlo code for general radiation

transport

Serpent - A Finnish developed Monte Carlo neutron transport code [4]

Deterministic codes

Attila - A commercial transport code

DRAGON - An open-source lattice physics code

PHOENIX/ANC - A proprietary lattice-physics and global diffusion code suite from Westinghouse

Electric

PARTISN - A LANL developed transport code based on the discrete ordinates method

NEWT - An ORNL developed 2-D SN code

DIF3D/VARIANT - An Argonne National Laboratory developed 3-D code originally developed for

fast reactors

DENOVO - A massively parallel transport code under development by ORNL

DANTSYS

RAPTOR-M3G - A proprietary parallel radiation transport code developed by Westinghouse Electric

Company

MPACT - A parallel 3D method of characteristics code under development by the University of

Michigan

DORT - Discrete Ordinates Transport

See also

Nuclear Reactor

Boltzmann equation

TINTE

Neutron scattering

References

1. ^ Adams, Marvin L. (2009). Introduction to Nuclear Reactor Theory. Texas A&M University.

2. ^ "ENDF Libraries" (https://www-nds.iaea.org/exfor/endf.htm).

3. ^ "OpenMC" (http://mit-crpg.github.io/openmc/).

4. ^ "PSG2 Serpent" (http://montecarlo.vtt.fi/).

Lewis, E., & Miller, W. (1993). Computational Methods of Neutron Transport. American Nuclear

Society. ISBN 0-89448-452-4.

Duderstadt, J., & Hamilton, L. (1976). Nuclear Reactor Analysis. New York: Wiley. ISBN 0-471-

22363-8.

Marchuk, G. I., & V. I. Lebedev (1986). Numerical Methods in the Theory of Neutron Transport.

Taylor & Francis. p. 123. ISBN 978-3-7186-0182-0.

External links

LANL MCNP6 website (http://mcnp.lanl.gov/)

LANL MCNPX website (http://mcnpx.lanl.gov/)

VTT Serpent website (http://montecarlo.vtt.fi/)

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Categories: Neutron Nuclear physics

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