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NUC E 521 NUC E 521
Chapter 1:Neutron Transport Equation
K. IvanovK. Ivanov
206 206 ReberReber, 865, 865--0040, [email protected], [email protected]
IntroductionIntroduction
Two methods exist for simulating and modeling neutron transport and interactions in the reactor core:1.
Deterministic methods
solve the Boltzmann transport
equation in a numerically approximated manner everywhere throughout a modeled system
2.
Monte Carlo methods
model the nuclear system (almost) exactly and then solve the exact model statistically (approximately) anywhere in the modeled system
Although deterministic methods are fast for one-
dimensional models, both methods are slow for realistic three-dimensional problems
Deterministic MethodsDeterministic Methods
Deterministic neutronics
methods play a
fundamental role in reactor core modeling and simulation
A first-principles treatment requires solution of the linearized
Boltzmann transport equation
This task demands enormous computational resources because the problem has seven dimensions: three in space, two in direction, and one each in energy and time
Deterministic MethodsDeterministic Methods
Nowadays reactor core analyses and design are performed using nodal (coarse-mesh) two-group diffusion methods
These methods are based on pre-computed assembly homogenized cross-sections and assembly discontinuity factors obtained by single assembly calculation with reflective boundary conditions (infinite lattice)
These methods are very efficient and accurate when applied to the current low-heterogeneous Light Water Reactor (LWR) cores
Deterministic MethodsDeterministic Methods
With the progress in the Nuclear Engineering field new generations of Nuclear Power Plants (NPPs) are considered that will have more complicated rector core designs, such as cores loaded partially with mixed-oxide (MOX) fuel, high burn-up loadings, and cores with advanced fuel assembly and control rode designs
Such heterogeneous cores will have much more pronounced leakage between the unlike assemblies, which will introduce challenges to the current methods for core calculation
First, the use of pre-computed fuel-assembly homogenized cross-section could lead to significant errors in the coarse-mesh solution
Deterministic MethodsDeterministic Methods
On the other hand, the application of the two-group diffusion theory for such heterogeneous cores could also be source for large errors
New core calculation methodologies based on fine-
mesh (pin-by-pin), higher-order neutron transport treatment, multi-group methods are needed to be developed to address the difficulties the diffusion approximation meets by improving accuracy, while preserving efficiency of the current reactor core calculations
Deterministic MethodsDeterministic Methods
The diffusion theory model of neutron transport has played a crucial role in reactor theory since it is simple enough to allow scientific insight, and it is sufficiently realistic to study many important design problems
The mathematical methods used to analyze such a model are the same as those applied in more sophisticated methods such as multi-group neutron transport theory
The neutron flux (ψ) and current (J) in the diffusion theory model are related in a simple way under certain conditions
This relationship between ψ
and J is identical in form to
a law used in the study of diffusion phenomena in liquids and gases: Fick’s
law
Deterministic MethodsDeterministic Methods
The use of this law in reactor theory leads to the diffusion approximation, which is a result of a number of simplifying assumptions
On the other hand, higher-order neutron transport codes have always been deployed in nuclear engineering for mostly time-independent problems out-of-core shielding calculations
With all the limitations of two-group diffusion theory, it is desirable to implement the more accurate, but also more expensive multi-group transport approach
Deterministic MethodsDeterministic Methods
Replacing two group diffusion nodal (assembly-based) neutronics
solvers with multi-group transport (pin-
based) solutions can be performed in direct and embedded manner
While studies have been performed utilizing the two deterministic types of methods (depending on techniques to treat angular flux dependence) -
the
discrete ordinates (SN
) and spherical harmonics (PN
) approximations –
for core analysis, the majority of the
efforts have been focused on replacing diffusion approximation with Simplified P3
(SP3
) approach
Deterministic MethodsDeterministic Methods
The SP3
method has gained popularity in the last 10 years as an advanced method for neutronics
calculation
The SP3
approximation was chosen because of its improved accuracy as compared to the diffusion theory for heterogeneous core problems and efficiency in terms of computing time as compared to higher order transport approximations
The SP3
approximation is more accurate when applied to transport problems than the diffusion approximation with considerable less computation expense than the discrete ordinates (SN
) or spherical harmonics (PN
) approximations
Deterministic MethodsDeterministic Methods
Another advantage of SP3
approximation is that it can be solved by straightforward extension of the common nodal diffusion methods with little overhead
It has been shown also that the multi-group SP3
pin-by- pin calculations can be used for practical core depletion
and transient analyses
Recent results obtained elsewhere indicated the importance of consistent cross-section generation and utilization in the SP3
equations in terms of modeling the P1
scattering
Since this is an important issue for practical utilization of SP3
approximation for reactor core analysis
Stochastic MethodsStochastic Methods
Numerical methods that are known as Monte Carlo methods can be loosely described as statistical simulation methods, where statistical simulation is defined in quite general terms
It can be any method that utilizes sequences of random numbers to perform the simulation
Statistical simulation methods may be contrasted to conventional numerical discretization
methods, which
typically are applied to ordinary or partial differential equations that describe some underlying physical or mathematical system
In many applications of Monte Carlo, the physical process is simulated directly, and there is no need to even write down the differential equations that describe the behavior of the system
Stochastic MethodsStochastic Methods
The only requirement is that the physical (or mathematical) system be described by probability density functions (pdf’s)
Once the pdf’s
are known, the Monte Carlo simulation can
proceed by random sampling from the pdf’s
Many simulations are then performed (multiple “trails”
or
“histories”) and the desired result is taken as an average over the number of observations (which may be a single observation or perhaps millions of observations)
In many practical applications, one can predict the statistical error (the “variance”) in this average result, and hence an estimate of the number of Monte Carlo trials that are needed to achieve a given error
Stochastic MethodsStochastic Methods
When it is difficult (impossible, impractical) to describe a physical phenomenon via deterministic equations (balance equations, distribution functions, differential equations, etc.) Monte Carlo method becomes useful (or necessary)
In Nuclear Engineering, in the past the method was mainly used for complex “shielding”
problems and for
benchmarking of deterministic calculations
Today, however, because of the advent of faster computers and parallel computing, the technique is being used more extensively for “normal”
calculations
Stochastic MethodsStochastic Methods
But, as will be discussed later, one should not (cannot) only rely on Monte Carlo calculations because the technique is very expensive/impractical (time and money) for obtaining detail information about a physical system or for performing sensitivity/perturbation studies
In such situation, deterministic methods are necessary for aiding Monte Carlo and/or for performing the actual simulation
Using the Monte Carlo method, a model of the medium under study is set up in the computer and individual particles are thrown at the medium or are generated in it, as if coming from a source
Stochastic MethodsStochastic Methods
The particles are followed, one by one, and the various events in which they participate (collision, absorption, fission, escape, etc.) are recorded
All the events associated with one particle constitute the history of that particle
The present trend in advanced and next generation nuclear reactor core designs is towards increased material heterogeneity and geometry complexity
The continuous energy Monte Carlo method has the capability of modeling such core environments with high accuracy
Stochastic MethodsStochastic Methods
Number of coupled Monte Carlo depletion code systems have been developed and successfully used in assembly/core depletion reference calculations
Now the tendency is to couple Monte Carlo neutronic
calculation with a thermal-hydraulics code to obtain three- dimensional (3D) power and thermal-hydraulic solutions
for a reactor core
The investigations performed at PSU showed that performing Monte Carlo based coupled core steady state calculations are feasible
This conclusion indicates that in the future it will be possible to use Monte Carlo method to produce reference results at operating conditions
DeterministicsDeterministics
vs. vs. StochasticsStochastics
For reactor core modeling and simulation, deterministic methods will be used principally in the short term (3–5 years) with Monte Carlo as a benchmarking tool
In the intermediate term (5–10 years) Monte Carlo methods could be used as a hybrid tool with multi-physics coupling to deterministic neutronics
and thermal
hydraulics codes
In the long term (>10 years) multi-physics codes using non-orthogonal grids will provide complete, high-
accuracy design tools, fully integrated into reactor core design and operation
Neutron Boltzmann EquationNeutron Boltzmann Equation
Several choices are possible for describing neutron behavior in a medium filled with nuclei
A neutron is a subatomic particle called a baryon having the characteristic strong nuclear force of the standard model
Thus, a quantum mechanical description seems appropriate, leading to an involved system of Schrödinger equations describing neutron motion between and within nuclei
A neutron is also a relativistic particle with variation of its mass over time when travelling near the speed of light
Finally, a neutron possesses wave and classical particle properties simultaneously and therefore a collective description like that of Maxwell’s equations also seems appropriate
Neutron Boltzmann EquationNeutron Boltzmann Equation
In reality, a neutron displays all of the above characteristics at one time or another
When a neutron collides with a nucleus, its strong force interacts with all of the individual nucleons
However, between nuclear collisions, neutrons move ballistically
Neutrons with energies above 20 MeV
with speeds of
more than 20% the speed of light (c), exhibit relativistic motion, but most in a reactor are rarely above 0.17c
Neutron Boltzmann EquationNeutron Boltzmann Equation
The neutron wavelength is most important for ultra-low-
energy neutrons mainly existing in the laboratory
Fortunately, the classical neutral particle description with quantum mechanics describing collisions emerges as the most appropriate for the investigation of neutron motion within a nuclear reactor
We now derive the neutron Boltzmann equation, also called the neutron transport equation, to characterize a relatively small number of neutrons colliding in a vast sea of nuclei
Neutron Boltzmann EquationNeutron Boltzmann Equation
Mathematically, a neutron is a neutral point particle, experiencing deflection from or capture by a nucleus at the center of an atom
If conditions are just right, the captured neutron causes a fissile nucleus to fission producing more neutrons
The statistically large number of neutrons interacting in a reactor allows for a continuum-like description through averaging resulting in the linear Boltzmann equation
Neutron Boltzmann EquationNeutron Boltzmann Equation
Also, a statistical mechanical formulation, first attempted by Boltzmann for interacting gases, provides an appropriate description
Boltzmann’s equation, based on physical arguments, such as finite particle size, gives a more physically precise picture of particle-particle interaction than is presented here
In his description, all particles, including nuclei, are in motion with like particle collisions allowed
Because of the low density of neutrons in comparison to nuclei in a reactor however, these collisions are infrequent enabling a simplified physical/mathematical theory
Neutron Boltzmann EquationNeutron Boltzmann Equation
Several forms of the neutron transport equation exist
The integro-differential formulation, arguably the most popular form in neutron transport and reactor physics applications, is presented hereafter
However, other forms of the transport equation exist and are being used such as integral
and adjoint
forms
IntegroIntegro--Differential Differential Neutron Boltzmann EquationNeutron Boltzmann Equation
We consider the neutron as a classical interacting particle
and formulate neutron conservation in a medium
Independent variables
A neutron moving at the position denoted by the vector r:
Configuration coordinates
IntegroIntegro--Differential Differential Neutron Boltzmann EquationNeutron Boltzmann EquationIndependent variables
A second vector Ω
in a coordinate system (dotted axes) is superimposed on the neutron itself to indicate neutron direction
Velocity coordinates
The Ω-vector specifies a point on the surface of a fictitious sphere of unit radius surrounding the neutron and pointing in its direction of travel
IntegroIntegro--Differential Differential Neutron Boltzmann EquationNeutron Boltzmann Equation
Independent variables
The neutron velocity is therefore, V ≡
vΩ, where v
is the
neutron speed giving the classical neutron kinetic energy (for a neutron of mass m):
Along with time, represented by t
and measured from a
reference time, (r, Ω, E) symbolize the six independent variables of the classical description of neutron motion
IntegroIntegro--Differential Differential Neutron Boltzmann EquationNeutron Boltzmann EquationDependent variables
For convenience, we combine the position and velocity vectors into a single generalized six-dimensional vector P≡(r,V)
defining the neutron phase space
Furthermore, decomposing the velocity vector into speed and direction gives the equivalent representations, P = (r,Ω,v) = (r,Ω,E)
Note that a Jacobian
transformation is required between the
various phase space transitions
The following derivation uses the latter representation exclusively
IntegroIntegro--Differential Differential Neutron Boltzmann EquationNeutron Boltzmann Equation
Dependent variables
The derivation will be a purely heuristic one, i.e. coming from physical intuition rather than from precise mathematical rigor
In this regard, we largely follow the approach of Boltzmann
The key element of the derivation is to maintain the point particle nature of a neutron while taking advantage of the statistics of large numbers
Thus, point collisions occur in a statistically averaged phase space continuum realized by defining the phase volume element about P as shown in the next Figure E ΩrP
IntegroIntegro--Differential Differential Neutron Boltzmann EquationNeutron Boltzmann Equation
Dependent variables
Phase volume element E ΩrP
IntegroIntegro--Differential Differential Neutron Boltzmann EquationNeutron Boltzmann EquationDependent variables
Note that the unit of solid angle, ΔΩ
, is the steradian
(sr),
and that 4π
steradians
account for all possible directions on a unit sphere
The volume element of phase space ΔP
plays a central part
in the overall neutron balance
One physically accounts for neutrons entering or leaving through the boundaries of Δr
or that are lost or gained within
Δr
In addition, neutrons enter or leave the phase element through deflection into or out of the direction range ΔΩ, or by slowing down or speeding up through the energy boundaries of ΔE
IntegroIntegro--Differential Differential Neutron Boltzmann EquationNeutron Boltzmann Equation
Dependent variables
Let be the neutron density distribution:
Since vΔt
is the distance travelled per neutron of speed v,
called the neutron track length, the total track length of all neutrons in ΔP
is:
),,,( tEN Ωr
IntegroIntegro--Differential Differential Neutron Boltzmann EquationNeutron Boltzmann Equation
Dependent variables
As shown, the track length serves as the bridge between point-like and continuum particle behavior when characterizing collisions
In the above definition, it is convenient to define the following quantity as the neutron angular flux at time t
representing the total track length per second of all neutrons in ΔP
per unit of phase space
IntegroIntegro--Differential Differential Neutron Boltzmann EquationNeutron Boltzmann Equation
Dependent variables
While the neutron density and flux are associated with a phase space volume element, the neutron angular current vector, , is associated with an area
The magnitude of the angular current vector is the rate of neutrons per steradian, energy and area, in the direction Ω
that pass through an area perpendicular to Ω
at time t
Neutrons, passing through the area A
oriented with the unit
normal , as shown in the next Figure, during Δt, enter the volume element
),,,( tEJ Ωr
ntA vˆ Ωnr
IntegroIntegro--Differential Differential Neutron Boltzmann EquationNeutron Boltzmann Equation
Dependent variables
Thus, the total number of neutrons in the phase volume element is:
Neutrons passing through the area A
are moving into Δr during Δt
EtA ΩΩnP vˆ
IntegroIntegro--Differential Differential Neutron Boltzmann EquationNeutron Boltzmann Equation
Dependent variables
We then conveniently define the angular current vector to be
such that is the rate of neutrons per area passing through an area of orientation
N, Φ, and J are called dependent variables
since they depend
on the independent variables of phase space and time
EtEJ ΩΩrn ),,,(ˆn
IntegroIntegro--Differential Differential Neutron Boltzmann EquationNeutron Boltzmann EquationNuclear Data
Neutrons primarily experience scattering, capture and fission characterized through probabilities represented by microscopic and macroscopic cross sections
In the following, we assume neutrons interact with inert stationary nuclei
The macroscopic cross section for interaction of type
i
with
the nucleus of nuclide
j is:
where i = scatter
, capture
, fission
, absorption
(= c + f)•Σij
(r,E,t)
is a product of amicroscopic cross section σij
(E)
which depends on the nuclide j
and reaction type i
and the nuclear atomic density, Nj
(r,t),
which can vary spatially
IntegroIntegro--Differential Differential Neutron Boltzmann EquationNeutron Boltzmann EquationNuclear Data
The microscopic cross section nominally represents the area presented to the neutron for an interaction to occur and is a measure of the probability of occurrence
The cross section is fundamental to neutron transport theory and provides the essential element in the continuum characterization of point collisions
Since, as previously shown, the total path length of all neutrons in the element ΔP
is
the number of interactions of type i
per time, or the reaction rate for interaction i
in ΔP
with nuclide j
is simply from the
definition of cross section:
ttE PΩr ),,,(
IntegroIntegro--Differential Differential Neutron Boltzmann EquationNeutron Boltzmann EquationNuclear Data
If we assume individual interactions to be independent events, the total interaction rate of type i
is the sum over all J
participating nuclear species giving the total macroscopic cross section:
Note that the macroscopic interaction rate is independent of neutron direction and depends only on neutron motion amongst interaction centers (the nuclei)
When we next consider neutron deflection or scattering, directional information is essential, however
IntegroIntegro--Differential Differential Neutron Boltzmann EquationNeutron Boltzmann EquationNuclear Data
The law of deflection, or the differential scattering kernel, is:
We mostly require, however, the differential scattering cross section for the
jth
nuclide:
IntegroIntegro--Differential Differential Neutron Boltzmann EquationNeutron Boltzmann EquationNuclear Data
Scattering, assumed rotationally invariant, depends only on the cosine of the angle between the incoming Ω’
and
scattered Ω
neutron directions, Ω’·
Ω
We normalize the scattering kernel such that neutrons must appear in some direction and energy range:
Assuming that the speed of light is an upper limit, the integration is actually over a finite range [0, E0
]
IntegroIntegro--Differential Differential Neutron Boltzmann EquationNeutron Boltzmann EquationNuclear Data
The scattering law fsj
is generally independent of time and position
The scattering cross section for an individual nuclide, however, includes the atomic density, which can provide a time variation of the differential scattering cross section
Neutrons from fission are either prompt or delayed through subsequent nuclear decay and emission
Here, we consider only prompt neutrons
IntegroIntegro--Differential Differential Neutron Boltzmann EquationNeutron Boltzmann EquationNuclear Data
The neutrons, produced isotropically, appear in an energy and direction range according to the distribution :
with normalization:
In addition, we shall require the average number of neutrons produced per fission,
With the notation now in place, an accounting of the total number of neutrons in the phase space volume element ΔP
becomes our focus
)(E
)(E
IntegroIntegro--Differential Differential Neutron Boltzmann EquationNeutron Boltzmann EquationContributions to the Total Neutron Balance
We now account for the net number of neutrons within an arbitrary volume V
and in the partial phase space element
ΔΩΔE during a time Δt
IntegroIntegro--Differential Differential Neutron Boltzmann EquationNeutron Boltzmann EquationContributions to the Total Neutron Balance
In words, the neutron balance in the entire volume V
and
partial element ΔΩΔE
is:
All contributions to the neutron balance, which we now consider separately, are in terms of the dependent variables
IntegroIntegro--Differential Differential Neutron Boltzmann EquationNeutron Boltzmann EquationTotal Number of Neutrons
The total number of neutrons in VΔΩΔE
at times t
and Δt
is:
IntegroIntegro--Differential Differential Neutron Boltzmann EquationNeutron Boltzmann EquationScattering Gain
Neutrons from all points within V
are scattered into the
element ΔΩΔE
The total number scattered during Δt
from any element
dE’dΩ’
(at time t) is therefore the total scattering rate in V multiplied by Δt
:
IntegroIntegro--Differential Differential Neutron Boltzmann EquationNeutron Boltzmann EquationScattering Gain
Of these reach ΔΩΔE
Thus, the total number scattering into ΔΩΔE
during Δt
within
V
is the integration over all possible contributions from all differential phase space elements dE’dΩ’:
IntegroIntegro--Differential Differential Neutron Boltzmann EquationNeutron Boltzmann EquationFission Production
The number of fissions occurring within V
during Δt
in any
differential element dE’dΩ’
is:
For each fission, neutrons appear in ΔΩΔE
giving a total gain from fission in VΔΩΔE
of:
EEEΩ)'(
4)(
IntegroIntegro--Differential Differential Neutron Boltzmann EquationNeutron Boltzmann EquationLosses from Absorption and Scattering
The number of neutrons lost due to absorption in ΔΩΔE
is:
Here, absorption refers to loss independently of whether a neutron causes a fission or not
In addition, a neutron is lost when it scatters out of the direction range ΔΩ
or energy range ΔE
IntegroIntegro--Differential Differential Neutron Boltzmann EquationNeutron Boltzmann EquationLosses from Absorption and Scattering
The number lost from scattering out of VΔΩΔE
is therefore
or, from the scattering kernel normalization:
IntegroIntegro--Differential Differential Neutron Boltzmann EquationNeutron Boltzmann EquationLosses from streaming out of V
Consider the elemental area dA
on the
surface of V
with outward normal
By definition of the current vector, the number of neutrons “leaking out”
of dA
from the element ΔΩΔE
during Δt
is
Thus, over the entire surface of V, the total leakage is:
sn
Total Balance: the Total Balance: the Neutron Boltzmann EquationNeutron Boltzmann Equation
Putting all contributions together, dividing by ΔtΔΩΔE
and
taking the limit as Δt
, ΔΩ, ΔE
approach 0 gives the overall balance in any partial element dΩdE
over the entire volume at
time t:
where the total macroscopic and the differential scattering cross sections are
Total Balance: the Total Balance: the Neutron Boltzmann EquationNeutron Boltzmann Equation
Because the volume is arbitrary and assuming a continuous integrand, the above integral is zero only if the integrand is zero yielding the following neutron Boltzmann equation:
We include an external volume source rate of emission from non-flux related events, , for completeness),,,( tEQ Ωr
Total Balance: the Total Balance: the Neutron Boltzmann EquationNeutron Boltzmann Equation
The neutron Boltzmann equation is a linear integro-
differential equation and as such requires boundary and initial conditions
We will fashion these conditions after the particular benchmark considered
The benchmarks in this compilation are steady state benchmarks obtained by assuming all nuclear data to be time independent and
Total Balance: the Total Balance: the Neutron Boltzmann EquationNeutron Boltzmann Equation
Then through integration over all time, the steady state form of the neutron Boltzmann equation becomes
where
The steady state source distribution now incorporates the initial condition
),,( EQ Ωr
Total Balance: the Total Balance: the Neutron Boltzmann EquationNeutron Boltzmann Equation
The neutron transport equation given by the steady state neutron Boltzmann equation is the source of all the other transport equations in this compendium
Obviously, this equation, being an integro-differential equation in six independent variables, does not lend itself to simple analytical solutions
Therefore, we require further simplification to enable analytical solution representations for eventual numerical evaluation
Total Balance: the Total Balance: the Neutron Boltzmann EquationNeutron Boltzmann Equation
Monte Carlo and other sophisticated deterministic methods solve the steady state neutron Boltzmann equation numerically; but, as we discussed in the Introduction, these methods contain numerical and sampling errors and will not be our focus
Our objective is to solve it in the most analytically pure way possible
In the chapters to follow, we present examples of analytical solutions of zero to three dimensions and continuous to multigroup in energy
Most of the solutions come from other forms of the steady state neutron Boltzmann equation that provide the necessary simplicity for further analytical and numerical investigation
Additional Forms of the Neutron Transport EquationAdditional Forms of the Neutron Transport Equation
Before defining the benchmarks inspired by the steady state neutron Boltzmann equation , it is of interest to reformulate the equation in the various versions used to define these benchmarks
At last count, there are at least eight (and probably more) equivalent forms of it not counting Monte Carlo including:
- integral-
even/odd parity
-
slowing down kernel-
multiple collision
-
invariant embedding-
singular integral
-
Green’s function-
pseudo flux
Each form has a particular mathematical property facilitating a class of solutions
For example, the multiple collision form is appropriate for highly absorbing media
Invariant embedding is useful for half-space problems, whereas the pseudo flux form is appropriate for isotropic scattering in multidimensions
In the following sections, we consider the integral transport equation only
We also derive the monoenergetic
and multigroup
approximations in 3-D and specifically in 1-D geometies, as they enable analytical solutions and are ubiquitous in the literature
Additional Forms of the Neutron Transport EquationAdditional Forms of the Neutron Transport Equation
The Integral EquationThe Integral Equation
The integral transport equation is essentially an integration along the characteristic defining the neutron path
If a neutron travels in direction Ω, shown in the Figure below, between the points specified by vectors r’
and r, the
following relationship holds r = r’
+sΩ, where s
is the magnitude of the vector r -
r’
The Integral EquationThe Integral Equation
Since the derivative along the neutron path is
where the gradient is with respect to r’,
evaluation of the steady state neutron Boltzmann equation at r’
gives
where the collision source is
The Integral EquationThe Integral Equation
Further, if we define the optical depth, or mean free path, as
and assume we know the collision source , the
solution to , viewed as
a first order ordinary differential equations, is:
),,( Eq Ωr
The Integral EquationThe Integral Equation
For r’
= r
on the boundary of the medium with outward
normal surface it becomessn
The Integral EquationThe Integral Equation
Note that the distance to the point on the surface from the origin, , depends on the neutron direction
is the Heaviside step function representing only
incoming neutrons where is known for all incoming directions
We do not consider re-entrant boundaries
Fredholm
integral equation of the second kind for the angular
flux, is constituted by
)(Ωrs
)ˆ( ΩnΘ s
The Integral EquationThe Integral Equation
An integration of
gives
and we obtain the scalar, or angularly integrated, flux:
The Integral EquationThe Integral Equation
By substitution of the differential volume element,
, we obtain
Note that the above Equation is not an integral equation for the scalar flux since the scalar flux does not generally appear under the integral
However, we can find an integral equation for the scalar flux under certain circumstances
''' 2 dsdsd Ωr
The Integral EquationThe Integral Equation
If scattering is isotropic and the source is isotropically
emitting, then
and
with
and
The Integral EquationThe Integral Equation
The angular flux for the collision source then becomes:
If there is no flux passing through the boundary from the vacuum into the medium, , for and if the nuclear properties are uniform, an further simplification is:
0),,( Es Ωr 0ˆ Ωn s
The Integral EquationThe Integral Equation
This equation represents the fundamental neutron transport equation for a 3-D medium with energy dependence and is especially appealing in the multiple collision approximation when absorption dominates
One can also numerically solve it through collision probability methods to find fluxes for cross section homogenization
Derivative forms of the Neutron Transport EquationDerivative forms of the Neutron Transport Equation
Mathematical tractability is the key to generating analytical benchmarks
In its six-dimensional space, the steady state transport equation is anything but mathematically tractable
To make it so, we must reduce the number of independent variables leading to the two approximations in the energy variable to follow
The monoenergetic, or one-group, approximation is a special case of the multigroup approximation but is so ubiquitous in the literature and particularly physically instructive, we consider it separately
MonoenergeticMonoenergetic
ApproximationApproximation
Because of its simplicity, the monoenergetic
approximation,
also called one-group approximation, is the most widely considered neutron transport model both analytically and numerically
The model serves as a relatively simple test case to observe the efficiency of new mathematical and numerical methods
The one-group approximation retains the fundamental features of the neutron transport process yet lends itself to theoretical consideration
As with all approximations, we begin with the neutron Boltzmann equation rewritten here (next slide)
MonoenergeticMonoenergetic
ApproximationApproximation
A physically based derivation gives the one-group form directly, while the multigroup form produces the same result as a degenerate multigroup approximation
MonoenergeticMonoenergetic
ApproximationApproximation
To obtain the one-group form, we assume neutrons scatter elastically from a nucleus of infinite mass, and therefore do not lose energy
An elastic collision between a neutron and nucleus of masses m
and M
respectively is one that leaves the internal (binding)
energy of the nucleus unchanged
The collision occurs instantaneously between two mathematical point particles as shown in the Figure in the physical laboratory (L) reference frame
MonoenergeticMonoenergetic
ApproximationApproximation
The vectors r’(t)
and R’(t)
define the neutron and nucleus
positions at time t
and the vector Rcm
(t) defines the position of the center of mass in L
where A
is the mass ratio M/m
To the colliding particles of constant velocity, the center of mass remains stationary before and after collision
For a nucleus initially at rest (V’≡
0) and a neutron moving
with velocity v’
toward the nucleus, the velocity of the center of mass in the laboratory reference frame is the time derivative of Rcm
(t):
MonoenergeticMonoenergetic
ApproximationApproximation
The most convenient view of elastic scattering is from the center of mass reference frame (C) created by imposing a velocity of –Vcm
on the laboratory reference frame shown in the Figure
The particle velocities in C
before collision are therefore:
MonoenergeticMonoenergetic
ApproximationApproximation
The total linear momentum in C
is zero, since
The Figure shows the particle velocity vectors v, and V
in L
after collision
Since the total momentum after collision in C
is also zero
from conservation
MonoenergeticMonoenergetic
ApproximationApproximation
Therefore, the velocity of the nucleus in C
after collision is
which is in the exact opposite direction of the neutron
Conservation of kinetic energy in C
requires
and since implies one obtains
Thus, the particles’
speed before and after collision remain
unchanged in C
MonoenergeticMonoenergetic
ApproximationApproximation
The scattering angles made by the velocity vectors with respect to the incident neutron direction in
L
and C, θ0
and θc
, and their corresponding cosines, μ0
and μc
are also shown in the Figure
For future reference, in terms of the neutron direction before Ω’
and after collision Ω, we define
Because of the rotational invariance experienced by scattering of perfectly spherical point particles, the collision
dynamics will depend only on μ0
MonoenergeticMonoenergetic
ApproximationApproximation
To obtain a fundamental relationship between the energy before and after collision and the scattering angle in C, we form the following expression
Using gives
where the collision parameter α
is
MonoenergeticMonoenergetic
ApproximationApproximation
Since and because we can relate the scattering angle in the laboratory and center of mass frames as
after rearranging
Recalling the differential scattering kernel
with normalization
MonoenergeticMonoenergetic
ApproximationApproximation
Based on energy and momentum conservation, we express the law for scattering through an angle θ0
in the laboratory frame, and correspondingly through angle θc
in the center of mass frame as
where the delta function enforces the correct kinematics for a given scattering angle
The first factor maintains the normalization with
MonoenergeticMonoenergetic
ApproximationApproximation
From conservation of probability between the two reference frames
the scattering kernel then becomes
where
MonoenergeticMonoenergetic
ApproximationApproximation
Note that we introduce angular dependence in the laboratory representation through the transformation between C
and L
coordinates
Taking into account that when , gives
with
The above Equation confirms the intuitive notion that a neutron scatters elastically without energy loss on collision with an infinite mass
A 1α
MonoenergeticMonoenergetic
ApproximationApproximation
If, in addition, neutrons from fission and the external source appear only at energy E0
, then
After introducing the last 4 equations into the neutron Boltzmann equation and integrating over the delta function in the collision term, one obtains
MonoenergeticMonoenergetic
ApproximationApproximation
Since all scattered and fission source neutrons appear at a single energy E0
, physical consistency requires neutrons to be at that energy or expressed formally
where mathematical consistency requires
Substituting for the flux, the one-group approximation emerges
MonoenergeticMonoenergetic
ApproximationApproximation
Energy is now just a passive parameter, which we can ignore
In three dimensions, the last equation is still mathematically intractable so we need to make additional simplifications to realize analytical solutions
Multigroup ApproximationMultigroup Approximation
We initiate the multigroup approximation to the neutron Boltzmann equation by partitioning the total energy interval of interest into G
energy groups:
where g = 1
is the highest group
By expressing the energy integrals of the scattering and fission sources as sums over all groups, without loss of generality, the integrals in the neutron Boltzmann equation become (next slide)
Multigroup ApproximationMultigroup Approximation
and
Multigroup ApproximationMultigroup Approximation
By integrating the neutron Boltzmann equation over ΔEg
, we therefore have
Multigroup ApproximationMultigroup Approximation
At this point, we introduce the multigroup approximation
stating the flux and source within each group, are separable functions of energy and r, Ω
The following normalizations are also required with f(E)
and
g(E)
assumed to be piecewise smooth functions of E:
Multigroup ApproximationMultigroup Approximation
Then, with the multigroup approximation, the our multigroup neutron Boltzmann equation becomes
with the following definitions of the group parameters
Multigroup ApproximationMultigroup Approximation
In more convenient vector notation:
with the group flux and source vectors
with the group “constants”
Multigroup ApproximationMultigroup Approximation
The multigroup approximation is the basis of nearly all reactor physics codes, making it one of most widely used approximations of neutron transport and diffusion theory
For inner/outer iterative numerical algorithms, one can reformulate the multigroup approximation as a series of one-
group equations
OneOne--Dimensional Plane SymmetryDimensional Plane Symmetry
We obtain the 1-D transport equation for plane symmetry when the medium is transversely infinite (in the yz
plane)
with cross section and source variation only in the x direction
For this case, the neutron Boltzmann equation becomes
OneOne--Dimensional Plane SymmetryDimensional Plane Symmetry
After integrating over the transverse yz
plane, one obtains
Here
and the x-direction (cosine) μ
is
OneOne--Dimensional Plane SymmetryDimensional Plane Symmetry
Since the differential scattering cross section is rotationally invariant with respect to the angle of scattering
where we explicitly show the dependence on the polar and azimuthal
angles of the particle directions
Upon integration over the azimuth θ
of the scattered
direction one obtains
(next slide)
OneOne--Dimensional Plane SymmetryDimensional Plane Symmetry
with
OneOne--Dimensional Plane SymmetryDimensional Plane Symmetry
Commonly, we express the scattering kernel as in a Legendre expansion representation
where the scattering coefficients are (from orthogonality)
and thus
OneOne--Dimensional Plane SymmetryDimensional Plane Symmetry
We have applied the addition theorem for Legendre polynomials, in the form
to obtain
where the lth
Legendre moment of the angular flux is
OneOne--Dimensional Plane SymmetryDimensional Plane Symmetry
Similarly, the multigroup approximation for plane geometry is
with
and
The Neutron Kinetics EquationsThe Neutron Kinetics Equations
The neutron transport equation without delayed neutrons,
assumes that all neurons are emitted instantaneously at the time of fission
In fact, small fraction of neutrons is emitted later due to certain fission products
',E',t),rψ('dΩ,E')rdE'νE'χ(E)
,E,t),rψ()Ω'E,,E'r(σddE',E,t),r(q
,E,t),r(,E)rσ(Ωtυ
sex
ˆ
ˆˆˆ'ˆ
ˆˆ1
The Neutron Kinetics EquationsThe Neutron Kinetics Equations
To ascertain the time-dependent behavior of a nuclear reactor, one must account for the emission of these so-called delayed neutrons, for the small fraction of neutrons that are delayed make the chain reaction far more sluggish under most conditions than would first appear
To derive the neutron kinetics equations, we return to the time-
dependent equation
),,(),,ˆ,(),,ˆ,(
),,ˆ,(),(),,ˆ,(ˆ),,ˆ,(1
tErqtErqtErq
tErErtErtErt
fexs
The Neutron Kinetics EquationsThe Neutron Kinetics Equations
When delayed neutrons are considered, we express the fission contribution to the emission density as
To represent qp
, we assume that for the
ith
fissionable isotope, a fraction βi
of the fission neutrons is delayed
This delayed fraction is small, ranging from 0.0065 for 235U to about 0.0022 for 239Pu
The fraction of prompt neutrons must then be (1-
βi)
neutrons delayedfor densityemission
neutronsprompt for densityemission
dpf qqq
The Neutron Kinetics EquationsThe Neutron Kinetics Equations
The rate of prompt fission neutrons produced per unit volume from isotope i
is just
To obtain the rate at which prompt neutrons are produced we sum over the isotope index i yielding
)',()',(')1( ErErdE if
i
i
if
ipp ErErdEEtErq )',()',(')1()(),,(
The Neutron Kinetics EquationsThe Neutron Kinetics Equations
The delayed neutrons arise from the decay of variety of fission products
For neutron kinetics considerations these are usually divided into six groups, each with a characteristic decay constant λl
, for each fissionable isotope i, with a characteristic yield βi
l
, where
If the concentration of the fission product precursors of type l
is per unit volume, then the number of delayed neutrons produced by precursor type l
per unit time per unit volume will
just be
l
il
i
),( trCll
The Neutron Kinetics EquationsThe Neutron Kinetics Equations
Now suppose that the probability that the delayed neutrons from precursor type l
will have an energy between E
and E+dE
is χl
(E)dE
We find the contribution of the delayed neutrons to the emission density to be
Hence (next slide)
)',()(),,( ErCEtErq lll
ld
The Neutron Kinetics EquationsThe Neutron Kinetics Equations
)',()(
)',()',(')1()(
'ˆˆˆ',ˆ
ˆˆ1
ErCE
ErErdEE
,t),E,r()'E,,E'r(σ'ddE't),E,r(q
,E,t),r(,E)rσ(tυ
lll
l
i
if
ip
sex
The Neutron Kinetics EquationsThe Neutron Kinetics Equations
Before this equation may be solved, an additional equation must be available for each precursor concentration
Since each precursor emits one delayed neutron, the rate of production of precursor type l
is
while the decay rate is Hence the net rate of change in the precursor concentration is
i
if
il tErErdE ),',()',('
),( trCll
),(),',()',('),( trCtErErdEtrCt ll
i
if
ill
The Neutron Kinetics EquationsThe Neutron Kinetics Equations
The foregoing equations may be generalized along two lines that are worthy to mention
First, we have made the assumption that the prompt and delayed spectra and
are independent of the isotope
producing the fission
This ansatz
is invariably employed in problems where delayed
neutrons are ignored, since the neglected deviations are insignificant
ip
il
The Neutron Kinetics EquationsThe Neutron Kinetics Equations
In kinetics problems neglect of the isotope dependence of the delayed neutron spectra may sometimes lead to significant errors
It is then necessary to derive somewhat more general forms of the kinetics equations in which isotope dependent prompt and delayed spectra are used
Since this generalization causes no generic changes in computational methods, the derivation of the equations is left as an exercise
The Neutron Kinetics EquationsThe Neutron Kinetics Equations
Kinetics equations also frequently involve time-dependent cross-sections; a re-derivation of the foregoing equations would then simply introduce the time parameter t
in each of
the cross-sections appearing in the neutron kinetics equations
In kinetics calculations the time-dependence would be predetermined as a function of time
This would be the case, for example, if a control rod were to be
inserted into a reactor core at a specified rate
The Neutron Kinetics EquationsThe Neutron Kinetics Equations
More generally, changes in cross-sections resulting from thermal-hydraulic or other feedback mechanisms are induced by the fact that the heat of fission causes the temperature in the reactor to change
This in turn results in changes in the number densities and therefore the macroscopic cross-sections appearing in the kinetics equations
The inclusion of these feedback effects for so-called reactor dynamics calculations is outside the scope of this text
However, we note that in most dynamics calculations the kinetics equations are treated as a separate module into which the time-dependent cross-sections are input
The TimeThe Time--Independent Transport EquationIndependent Transport Equation
The vast majority of numerical transport equations are carried out using a form of the equations in which time dependence is not treated explicitly
For non-multiplying systems in which the external sources are time-independent and nonnegative there will always be nonnegative solutions to the time-independent form of the transport equation
if the boundary conditions are also time-independent. For multiplying systems, the situation is less straightforward, for the critical state of system must be considered.
)'ˆ()ˆˆ(')ˆ(
)ˆ()](ˆ[
,E,r'E,,E'rσddE',E,rq
,E,r,Erσ
sex
Fixed Source and Eigenvalue ProblemsFixed Source and Eigenvalue Problems
Physically, a system containing fissionable material is said to be critical if there is a self-sustaining time-independent chain reaction in the absence of external source of neutrons
That is to say, if neutrons are inserted into a critical system, then after sufficient time has elapsed for the decay of transient effects, a time-independent asymptotic distribution of neutrons will exists in which the rate of fission neutron production is just equal to the losses due absorption and leakage from the system
If such an equilibrium cannot be established, the asymptotic distribution of neutrons –
called fundamental mode –
will not
be steady state and will either increase or decrease with time exponentially as indicated in the next Figure
Fixed Source and Eigenvalue ProblemsFixed Source and Eigenvalue Problems
Time dependence of the flux for a source-free multiplying medium
Fixed Source and Eigenvalue ProblemsFixed Source and Eigenvalue Problems
If neutrons from a time-independent external source are supplied, a subcritical system will eventually come to an equilibrium state characterized by a time-independent neutron flux distribution in which the production rate of external and fission neutrons is in equilibrium with the absorption and leakage from the system
However, if the system is critical or supercritical, no such equilibrium can exist in the presence of an external source, and the neutron flux distribution will be an increasing function
of time. This flux behavior is qualitatively depicted in the next Figure
Fixed Source and Eigenvalue ProblemsFixed Source and Eigenvalue Problems
Time dependence of the flux a multiplying medium with a known source
Fixed Source and Eigenvalue ProblemsFixed Source and Eigenvalue Problems
Mathematically, a system is critical if a time-independent nonnegative solution to the source-free transport equation can be found
Hence, from the transport equation without delayed neutrons one must have a solution to
with appropriate boundary conditions, for example,
where V
is the volume of the system and Γ
is its surface
Vr',E',rd,E'rνσdE'Eχ
,E,r'E,,E'rσddE'
,E,r,Erσ
f
s
),ˆ('ˆ)()(
)'ˆ()ˆˆ('
)ˆ()](ˆ[
rn,E,r ,0ˆˆ,0)ˆ(
Fixed Source and Eigenvalue ProblemsFixed Source and Eigenvalue Problems
For a subcritical system there must exist a solution to the time-
independent form of the equation in the presence of external sources
where we may also have a known distribution of neutrons entering the system across its surface:
Vr',E',rd,E'rνσdE'Eχ
,E,r'E,,E'rσddE',E,rq
,E,r,Erσ
f
sext
),ˆ('ˆ)()(
)'ˆ()ˆˆ(')ˆ(
)ˆ()](ˆ[
rn,r,E,r ,0ˆˆ),ˆ(ˆ)ˆ(
Fixed Source and Eigenvalue ProblemsFixed Source and Eigenvalue Problems
There is a difficulty in using the source-free transport equation as it stands
In most situations the configuration under consideration will not be exactly critical until fine adjustments have been made in
composition and/or geometry
But if we try to find a solution to the source-free transport equation
and find that none exists, it tells us nothing about
whether the system is subcritical or supercritical or by how much, and therefore no useful information is gained as to how the system parameters might be adjusted to obtain criticality
Fixed Source and Eigenvalue ProblemsFixed Source and Eigenvalue Problems
For this reason criticality calculations are normally cast into the form of eigenvalue problems, where the eigenvalue provides a measure of whether the system is critical or subcritical and by how much
The two most common formulations are the time-absorption and multiplication eigenvalues, reefed to as α
and k
respectively
The α
Eigenvalue
Suppose that we look for asymptotic solutions of the source-
free transport equation in the form
where the solution satisfies the boundary conditions
]exp[)ˆ(),ˆ( t,E,rt,E,r
Fixed Source and Eigenvalue ProblemsFixed Source and Eigenvalue Problems The α
Eigenvalue
By insertion in the neutron transport equation (without delay neutrons) and setting qex
to zero, we obtain the eigenvalue equation
In general there will be a spectrum of eigenvalues for which the equation has a solution
)ˆ('ˆ)()(
)'ˆ()ˆˆ('
)ˆ()(ˆ
',E',rd,E'rνσdE'Eχ
,E,r'E,,E'rσddE'
,E,r,Erσ
f
s
Fixed Source and Eigenvalue ProblemsFixed Source and Eigenvalue Problems The α
Eigenvalue
However, at long times, only nonnegative solutions corresponding to the largest real component of α
will
predominate
From our physical definition of criticality we have, for α0
, the eigenvalue with the largest real part,
Moreover, the magnitude of Re[α0
] provides a measure of how far from criticality the system is
.subritical 0critical, 0
cal,supercriti 0Re 0
Fixed Source and Eigenvalue ProblemsFixed Source and Eigenvalue Problems
The k Eigenvalue
The k eigenvalue form of the criticality problem is formulated by assuming that ν, the average number of neutrons per fission, can be adjusted to obtain a time-independent solution to the source-free transport equation
Hence we replace ν
by ν/k
and write
)ˆ('ˆ)()(
)'ˆ()ˆˆ('
)ˆ()(ˆ
',E',rd,E'rνσdE'kEχ
,E,r'E,,E'rσddE'
,E,r,Erσ
f
s
Fixed Source and Eigenvalue ProblemsFixed Source and Eigenvalue Problems The k Eigenvalue
From our physical definition of criticality it is apparent that any chain reaction can be made critical if the number of neutrons per fission can be adjusted between zero and infinity
Thus there will be a largest value of k, for which a nonnegative fundamental mode solution can be found
Clearly the system is critical if this largest value of is k = 1
A value of k < 1
implies that the hypothetical number of
neutrons per fission, ν/k, required to make the system just critical is larger than ν, the number available in reality
Hence the system is subcritical
Fixed Source and Eigenvalue ProblemsFixed Source and Eigenvalue Problems The k Eigenvalue
Conversely, for k > 1, fewer neutrons per fission are required to make the system critical than are produced in reality, and thus the system is supercritical
To summarize,
For most reactor criticality calculations the k rather than α
eigenvalue is used for several reasons
.subritical 1critical, 1
cal,supercriti 1k
Fixed Source and Eigenvalue ProblemsFixed Source and Eigenvalue Problems The k Eigenvalue
First, as indicated by the α
eigenvalue equation, the α
eigenvalue equation has the same form as the time- independent transport equation but with α/υ
added to the
absorption cross-section
For this reason this formulation is normally referred to as the time-absorption eigenvalue
If the system is subcritical, however, α
will be negative, and
situation is likely to arise, such as in voided regions where the pseudo total cross-section that include α/υ
term will be
negative
Such negative cross-sections are problematic for many of the numerical algorithms used to solve the transport equation
Fixed Source and Eigenvalue ProblemsFixed Source and Eigenvalue Problems The k Eigenvalue
A second
drawback arises from the fact that the α
eigenvalue
causes the removal of neutrons from the system to be inversely proportional to their speed
Hence for supercritical systems α/υ
absorption will cause more
neutrons to be removed from low energies and result in a neutron energy spectrum that is shifted upward in energy compared to that of a just critical system with α
= 0
Conversely, in a subcritical system the negative value of α
causes neutrons to be preferentially added at lower energies, causing a downward shift in the energy spectrum
Fixed Source and Eigenvalue ProblemsFixed Source and Eigenvalue Problems The k Eigenvalue
In many calculations where it is desired to know not only the critical state of the system but also the spectrum that the system would have if it were critical, these spectral shifts are
undesirable
Moreover, such shifts are insignificant when the k eigenvalue formulation is used
A third
advantage of k
eigenvalue formulation lies in the
physical interpretations of the eigenvalue and the iterative technique derived from it
The technique is a powerful method for carrying out criticality calculations
Also, k
may be interpreted as the asymptotic ratio of the
number of neutrons in one generation and the number in the next
The Adjoint Transport Equation
For each of the foregoing forms of the transport equation a related adjoint equation may be formulated
Solutions for such adjoint equations may serve a variety of purposes
For fixed source problems they may be used to expedite certain classes of calculations, particularly when Monte Carlo methods are used
For eigenvalue problems probably the most extensive use of adjoint solution is in perturbation theory estimates for changes in the neutron multiplication caused by small changes in material properties
The Adjoint Transport Equation
In this section we first discuss the properties of adjoint operators and develop an adjoint equation for non-
multiplying systems
After demonstrating some uses of the resulting adjoint equation, we turn to the criticality problem in order to formulate an adjoint equation for a multiplying system and demonstrate its use for calculating perturbations in the multiplication
The Adjoint Transport Equation Non-Multiplying Systems
To begin, suppose that we have an operator H
and any pair
of functions ζ
and ζ+
that meet appropriate boundary and continuity conditions
For real-valued functions the adjoint operator H+
is defined by the identity
where <>
signifies integration over all of the independent variables
If H = H+, the operator is said to be self-adjoint; otherwise it is non-self-adjoint
HH
The Adjoint Transport Equation Non-Multiplying Systems
For the time-independent non-multiplying transport problem the transport operator may be given by
and the integration over the independent variable is
The spatial domain of the problem V
is bounded by a
surface Γ
across which we shall assume that no particles enter (i.e., vacuum boundary conditions)
)'ˆ()ˆˆ('
)ˆ()](ˆ[
,E,r'E,,E'rσddE'
,E,r,ErσH
s
dEddV
The Adjoint Transport Equation Non-Multiplying Systems
Hence the functions upon which H
operates are constrained
to satisfy
where is the outward normal
In addition, for to be finite,
ζ
must be continuous in
the direction of particle travel
To determine the form of the adjoint transport operator, we multiply the equation for the transport equation by a function and manipulate the result in order to move
to the left of the operator
rn,E,r ,0ˆˆ,0)ˆ(
n
ˆ
)ˆ( ,E,r )ˆ( ,E,r
The Adjoint Transport Equation Non-Multiplying Systems
First, we have
For the streaming term we use the fact that , and integrate the identity
over V
to obtain
where the divergence theorem has been applied to reduce the expression on the left to a surface integral
)'ˆ()ˆˆ('
)]ˆ()()ˆ(ˆ[)ˆ(
,E,r'E,,E'rσddE'
,E,r,Erσ,E,r,E,rddEdVH
s
ˆˆ
ˆˆ)ˆ(
ˆˆˆˆ dVdVnd
The Adjoint Transport Equation Non-Multiplying Systems
We note, however, that the boundary condition specified by
causes the integrant to vanish for
We reduce the function ζ+
to have the boundary condition
causing the surface integral to vanish for all
Thus,
rn,E,r ,0ˆˆ,0)ˆ(
0ˆˆ n
rn,E,r ,0ˆˆ,0)ˆ(
ˆˆ dVdV
The Adjoint Transport Equation Non-Multiplying Systems
In the collision term of the equation for we simply note that , while in the scattering term we interchange dummy variables E
and E’
as well as and to
obtain
Substituting back we have
H
'
)''ˆ()'ˆˆ'('')ˆ(
)''ˆ()ˆˆ('')ˆ(
,E,r,E,ErσdEd,E,rdEd
,E,r'E,,E'rσdEd,E,rdEd
s
s
)''ˆ()'ˆˆ'('
)]ˆ()()ˆ(ˆ[)ˆ(
,E,r,E,ErσddE'
,E,r,Erσ,E,r,E,rdEddVH
s
The Adjoint Transport Equation Non-Multiplying Systems
Hence the identity equation, , be satisfied provided we define the adjoint transport operator as
To illustrate the use of this adjoint identity suppose we solve a transport problem with a known external source distribution qex
,
where ψ
meets the boundary conditions
HH
)''ˆ()'ˆˆ'('
)ˆ()](ˆ[
,E,r,E,ErσddE'
,E,r,ErσH
s
exqH
rn,E,r ,0ˆˆ,0)ˆ(
The Adjoint Transport Equation Non-Multiplying Systems
Suppose further that we would like to know the response of a small detector with a total cross-section σd
and a volume Vd
, represented by the reaction rate localized at point ,
We define the adjoint problem as
where the adjoint flux ψ+
is required to satisfy the boundary condition of
Now we multiply by ψ+ and integrate over the
independent variables to obtain
dr
),()( ErEdEVR ddd
)( ddd rrVH
rn,E,r ,0ˆˆ,0)ˆ(
exqH
exqH
The Adjoint Transport Equation Non-Multiplying Systems
Similarly, we multiply by ψ
and integrate to
yield
Taking the difference of these two expressions yields
We replace ζ
and ζ+
with ψ
and ψ+
in the adjoint identity equation
The left-hand side of then vanishes to
yield the desired result
)( ddd rrVH
RrrVH ddd )(
RqHH ex
RqHH ex
exqR
The Adjoint Transport Equation Non-Multiplying Systems
From this illustration, we see that the detector response is just given by the volume integral of the adjoint weighted source distribution
In particular, if we assume that the particles are emitted at in direction at energy E0
at a rate of one per second,
and obtain
0r
0
)ˆˆ()()( 000 EErrqex
),ˆ,( 000 ErR
The Adjoint Transport Equation Non-Multiplying Systems
This provides us with a physical interpretation of the adjoint flux defined by
It is just the importance of particles produced at to the detector response
The concept of particle importance is elaborated elsewhere for a broad range of neutronics
problems
The foregoing methodology has widespread applications, particularly in Monte Carlo calculations
)( ddd rrVH
000 ,ˆ, Er
The Adjoint Transport Equation Non-Multiplying Systems
The foregoing formalism may be applied by slight modification to external surface source of particles, or to calculate distributions of particles leaving the problem domain
For example, suppose that qex
= 0
within V
but that particles enter the volume across Γ
Then, we would have with the boundary condition
where is known as distribution of incoming particles
Using for the adjoint, with ψ+
satisfying on the boundary, we can again
carry out the same procedure used but with qex
= 0
(next slide)
0H rnErEr ,0ˆˆ),,ˆ,(~),ˆ,( ~
)( ddd rrVH
rn,E,r ,0ˆˆ,0)ˆ(
The Adjoint Transport Equation Non-Multiplying Systems
Now, however, the boundary condition
prevents the equation of identity, ,from holding
It is clear that
Hence the desired detector response is an integral of the adjoint flux over that part of the surface for which the known distribution of incoming particles is nonzero:
RHH
rnErEr ,0ˆˆ),,ˆ,(~),ˆ,(
HH
~ˆˆ
0ˆˆ nddEdHH
n
~ˆˆ
0ˆˆnddEdR
n
The Adjoint Transport Equation Multiplying Systems
For multiplying systems a fission neutron term must also be included in the adjoint equations
For time-independent problems we define the fission operator as
The adjoint operator is then defined by
By using the same technique as that applied to the scattering operator earlier, it is easily shown that
)''ˆ(')'()( ,E,rd,ErσdE'EG f
GG
)''ˆ(')'()( ,E,rdEdE',ErσG f
The Adjoint Transport Equation Multiplying Systems
The technique of the preceding subsection may be generalized to treat subcritical systems where an external source gives rise
to a time-independent solution
One simply replaces H
and H+
by (H-G)
and (H+-G+)
respectively
A more frequent use of the adjoint equation is in estimating changes in reactivity by perturbation theory
Before illustrating the use of perturbation theory, however, we must briefly examine some properties of homogeneous equations
The Adjoint Transport Equation Multiplying Systems
For the multiplication eigenvalue problem, the k
eigenvalue
equation
may be written in operator notation as
where ψ
is taken to be the fundamental mode solution, ψ
> 0
for , obeying the vacuum boundary conditions
Likewise the adjoint equation is
where ψ+
is the fundamental mode solution ψ+
> 0
for , obeying the boundary conditions of
)ˆ('ˆ)()()'ˆ()ˆˆ('
)ˆ()(ˆ
',E',rd,E'rνσdE'kEχ,E,r'E,,E'rσddE'
,E,r,Erσ
fs
01 G
kH
Vr
01 G
kH
Vr
rn,E,r ,0ˆˆ,0)ˆ(
The Adjoint Transport Equation Multiplying Systems
If we multiply the later equation by ψ+, and the former by ψ, and integrate the difference over the independent variables, then
After simplifying this expression with identities, we have
It is easily shown that if ψ+
and ψ+ are positive, the integral on the left will not vanish
Thus for this equation to hold the eigenvalues of the transport equation and its adjoint must be equal:
011
G
kG
kHH
011
Gkk
kk
The Adjoint Transport Equation Multiplying Systems
We are now prepared to use the adjoint operators to calculate changes in reactivity, where reactivity is defined by
For small changes, δk, in k, we have
where k0
is unperturbed value
kk 1
20kk
The Adjoint Transport Equation Multiplying Systems
For the unperturbed rector we define the fundamental mode equations as
If these equations are used to represent the perturbed state, for small changes we represent the perturbations by
0100
000 G
kH
0100
000
G
kH
00
00
,,
kkkGGGHHH
The Adjoint Transport Equation Multiplying Systems
Substituting these expressions into the fundamental mode equations for the unperturbed rector, we have
For small perturbations we may disregard terms of order δ2, and thus, using
and regrouping, we obtain
0100
000
GG
kkHH
00000
111
11kk
kkkkkk
0
000
000
00002
0
111 Gk
HGk
HGk
HGk
k
The Adjoint Transport Equation Multiplying Systems
Comparing this expression with equations for unperturbed reactor, we see that first term on the right vanishes
Moreover, if we multiply by and integrate over the independent variables, we have
We may now use the properties of the adjoint operators to eliminate the last term
If we take ζ
= δψ
and ζ+ = δψ0
+, then we obtain
which must vanish
0
)()( 01
00001
0000020
GkHGkHGk
k
001
0001
000 )()( GkHGkH
The Adjoint Transport Equation Multiplying Systems
Hence
This is powerful result, for it allows us to estimate many different small reactivity changes using only two unperturbed solutions, ψ0 and ψ0
+, and the cross-section changes that appear as δH
and δG
000
01
00
20
)(
G
HGk
kk