The collision probability method in 1D – part 2...The collision probability method in 1D – part...

The collision probability method in 1D –part 2

Alain Hebert

[email protected]

Institut de genie nucleaire

Ecole Polytechnique de Montreal

ENE6101: Week 9 The collision probability method in 1D – part 2 – 1/25

Content (week 9) 1

Calculation of 1D collision probabilities

plane 2D geometry

cylindrical 1D geometry

spherical 1D geometry


The collision probability method 1

Consider an infinite lattice of unit cells, each of them represented as⋃

i

Vi

V ∞i represents the infinite set of regions Vi belonging to all the cells in the lattice

The sources of secondary neutrons are uniform and equal to Qi on each region Vi.

The collision probability Pij,g is the probability for a neutron born uniformally and

isotropically in any of the regions Vi of the lattice to undergo its first collision in

region Vj of a unit cell or assembly.

If the total cross section Σ(r) is constant and equal to Σj in region Vj , reduced CPs are

pij =Pij

Σj=

1

4π Vi

∫

V ∞

i

d3r′∫

Vj

d3re−τ(s)

s2(1)

where the optical path τ(s) is given by

τ(s) =

∫ s

0ds′ Σ(r − s′ Ω) .(2)


Plane 2D geometry 1

We will now study the case of a 2D geometry defined in the x–y plane and homogeneous

along the z–axis. A cylinder or tube of infinite length is an example of such a geometry. In

this case, it is possible to integrate analytically the integral transport equation along the axial

angle θ. This angle is use to define the direction cosine of the particle relative to the z–axis.

(x,y)

θ

ε

x

y

z

Ω

r = (x,y,z)

The direction of the particle is

Ω = sin θ cos ǫ i+ sin θ sin ǫ j + cos θ k.

Integration over θ is possible by taking the

projection of each particle free path on the

x–y plane. We write

d2Ω = dθ dǫ sin θ

τ(ρ) = τ(s) sin θ

dρ = ds sin θ(3)

where ρ is the projection of s on the x–y

plane.


Plane 2D geometry 2

We will limit ourself on the integration of the integral transport equation, defined on the

domain of a unique unit cell. We use the relations r = r′ + sΩ and d3r = s2 d2Ω ds and

obtain

pij =1

4π Vi

∫

Vi

d3r′∫

4πd2Ω

∫

Ij

ds e−τ(s)(4)

where the quantity Ij is the set of points belonging simultaneously

to the half straight line of origin r and direction −Ω;

to volume Vj of the unit cell.

With the help of figure, Eq. (4) can be rewritten as

pij =1

4π Vi

∫ 2π

0dǫ

∫

Vi

d2r′∫ π

0dθ

∫

Ij

dρ e−

τ(ρ)sin θ(5)

where Vi now represents a surface in the x–y plane and Ij is the set of points belonging

simultaneously

to the half straight line of origin r′ and direction ǫ;

to surface Vj of the unit cell.


Bickley functions 1

Bickley functions are defined as

Kin(x) =

∫ π/2

0dθ sinn−1 θ e−

xsin θ =

∫ π/2

0dθ cosn−1 θ e−

xcos θ(6)

with n ≥ 1.

0 0.5 1 1.5 2 2.50

0.5

1

1.5

Ki1(x)

Ki2(x)

Ki3(x)

Ki4(x)

Ki5(x)

x

The following equations are relating Bickley

functions of different orders:

∫ x′

xdu Kin(u) = Kin+1(x)−Kin+1(x

′) ,

andd

dxKin(x) = −Kin−1(x)

They can be evaluated efficiently using the matlab script akin.


Plane 2D geometry 1

Equation (5) can be simplified by introducing the Bickley functions. We obtain

pij =1

2π Vi

∫ 2π

0dǫ

∫

Vi

d2r′∫

Ij

dρKi1[τ(ρ)] .(7)

Sets of integration lines, referred as tracks, are draw over the complete domain. Each set is

characterized by a given angle ǫ and contains parallel tracks covering the domain. Tracks in

a set can be separated by a constant distance ∆h or can be placed at optimal locations, as

depicted in sub-figures (a) and (b), respectively. Each track is used forward and backward,

corresponding to angles ǫ and −ǫ.

∆h

(a) Rectangular quadrature (b) Gaussian quadrature

Three-point formula

Three-point formula

Three-point formula


Plane 2D geometry 2

In the case of convex volumes i and j, Eq. (7) can be rewritten as

pij =1

2π Vi

∫ 2π

0dǫ

∫ hmax

hmin

dh

∫ ℓi

0dℓ′

∫ ℓj

0dℓ Ki1(τij + Σi ℓ

′ + Σj ℓ) if i 6= j(8)

where τij is the optical path of the materials between regions i and j and

pii =1

2π Vi

∫ 2π

0dǫ

∫ hmax

hmin

dh

∫ ℓi

0dℓ′

∫ ℓi

ℓ′dℓ Ki1[Σi (ℓ− ℓ′)] .(9)

x

y

ε

ViVj

h

r

r'

ll ij

l i

l j

l'


Cylindrical 1D geometry 1

In the case of 1D cylindrical geometry, the

tracks are identical for any value of the

angle ǫ.

The tubular volumes are concave, making

extra terms to appear. The corresponding

geometry is depicted in figure. Two tracks

are represented, both contributing to

collision probability components pij with

i < j and pii. Track 1© corresponds to the

integration domain where region i is concave

and track 2© corresponds to the integration

domain where it is convex. In this figure, τij

and τii are the optical paths of the materials

located between regions i and j or between

two instances of region i.

h

Vi

Vj

ri-1/2 ri+1/2

rj-1/2 rj+1/2

τij

τii

li

lj

li

li

lj

lj

lj

track 2

1track



In this case, Eq. (8) can be rewritten as

pij =2

Vi

∫ ri−1/2

0dh

∫ ℓi

0dℓ′

∫ ℓj

0dℓ

[

Ki1[(τij + τii +Σi ℓi) + Σi ℓ′ + Σj ℓ]

+ Ki1(τij +Σi ℓ′ +Σj ℓ)

]

+

∫ ri+1/2

ri−1/2

dh

∫ ℓi

0dℓ′

∫ ℓj

0dℓ Ki1(τij +Σi ℓ

′ +Σj ℓ)

if i < j(10)

where ri±1/2 are the radii bounding region i and Eq. (9) can be rewritten as

pii =2

Vi

∫ ri−1/2

0dh

∫ ℓi

0dℓ′

[

2

∫ ℓi

ℓ′dℓ Ki1[Σi (ℓ− ℓ′)] +

∫ ℓi

0dℓ Ki1[τii + Σi (ℓ

′ + ℓ)]

]

+

∫ ri+1/2

ri−1/2

dh

∫ ℓi

0dℓ′

∫ ℓi

ℓ′dℓ Ki1[Σi (ℓ− ℓ′)]

(11)

where the first and second terms are the contributions from tracks 1© and 2©, respectively.



These equations can be simplified by introducing the following definitions:

Cij(τ0) =

∫ ℓi

0dℓ′

∫ ℓj

0dℓ Ki1(τ0 + Σi ℓ

′ + Σj ℓ)(12)

and

Di =

∫ ℓi

0dℓ′

∫ ℓi

ℓ′dℓ Ki1[Σi (ℓ− ℓ′)](13)

so that

pij =2

Vi

∫ ri−1/2

0dh [Cij(τij + τii +Σi ℓi) + Cij(τij)] +

∫ ri+1/2

ri−1/2

dh Cij(τij)

(14)

if i < j, and

pii =2

Vi

∫ ri−1/2

0dh [2Di + Cii(τii)] +

∫ ri+1/2

ri−1/2

dhDi

.(15)



Integration in ℓ and ℓ′ is next performed, leading to the following relations:

a) Σi 6= 0 and Σj 6= 0 :

Cij(τ0) =1

Σi Σj

[

Ki3(τ0)−Ki3(τ0 +Σi ℓi)−Ki3(τ0 + Σj ℓj)

+ Ki3(τ0 +Σi ℓi +Σj ℓj)]

b) Σi = 0 and Σj 6= 0 :

Cij(τ0) =ℓi

Σj[Ki2(τ0)−Ki2(τ0 +Σj ℓj)]

c) Σi 6= 0 and Σj = 0 :

Cij(τ0) =ℓj

Σi[Ki2(τ0)−Ki2(τ0 + Σi ℓi)]

d) Σi = Σj = 0 :

Cij(τ0) = ℓi ℓj Ki1(τ0)



e) Σi 6= 0 :

Di =ℓi

Σi−

1

Σ2i

[Ki3(0)−Ki3(Σi ℓi)]

f) Σi = 0 :

Di =πℓ2i

4.



The above relations are implemented in Matlab using the following two scripts:

function f=cij_f(tau0,sigi,sigj,segmenti,segmentj)

if sigi ˜= 0 && sigj ˜= 0

f=(akin(3,tau0)-akin(3,tau0+sigi*segmenti)- ...

akin(3,tau0+sigj*segmentj)+ ...

akin(3,tau0+sigi*segmenti+sigj*segmentj))/(sigi*sigj) ;

elseif sigi == 0 && sigj ˜= 0

f=(akin(2,tau0)-akin(2,tau0+sigj*segmentj))*segmenti/sigj ;

elseif sigi ˜= 0 && sigj == 0

f=(akin(2,tau0)-akin(2,tau0+sigi*segmenti))*segmentj/sigi ;

else

f=akin(1,tau0)*segmenti*segmentj ;

end

function f=di_f(sig,segment)

if sig ˜= 0

f=segment/sig-(akin(3,0)-akin(3,sig*segment))/sigˆ2 ;

else

f=pi*segmentˆ2/4 ;

end


Tracking 1

The tracking of 1D cylindrical and spherical geometries is similar and can be

generated with the same algorithm.

A matlab script f=sybt1d(rad,lgsph,ngauss) is presented to produce a tracking

object in these cases.

The figure presents an example with I = 2 regions and K = 2 tracks per regions.

Tracks 1© and 2© have two segments, identified as ℓ1 and ℓ2. Tracks 3© and 4© have

only one segment, identified as ℓ2.

h

1

2

3

4

φ

l1

l2

l1

l2

l2

l2

h1 h3h2 h4


Tracking 2

A K point Gauss-Jacobi quadrature is

used

The four tracks are obtained using

rad=[0.075 0.15] ;

track=sybt1d(rad,false,2) ;

track(i,j) is a 2D cell array

containing I ×K elements, one per

track. The i index refers to the tracks

located in ri−1/2 < h < ri+1/2. The

(i, j)–th cell is also a cell array

containing three elements:

the weight wk of the track,

the product wk cosφ,

a 1D array of dimension I − i+ 1

containing the track segments.

The ℓ1 segment length of track 2© is

l2=track1,23(1)

leading to l2 = 0.07186585.

Any length trackik,il3(1)

corresponding to a segment that cross

the h axis must be multiplied by 2.

h

1

2

3

4

φ

l1

l2

l1

l2

l2

l2

h1 h3h2 h4


Tracking 3

The following script will perform a numerical integration of p11 in 1D cylindrical tube of radius

rad and macroscopic total cross section sigt:

function f=p11_cyl(rad,sigt)

% compute the p11 component in 1D cylindrical geometry

track=sybt1d(rad,false,5) ;

val=0 ;

for i=1:5

segment=2*track1,i3(1) ;

if sigt ˜= 0

d=segment/sigt-(akin(3,0)-akin(3,sigt*segment))/sigtˆ2 ;

else

d=pi*segmentˆ2/4 ;

end

val=val+track1,i1*d ;

end

f=val/(pi*radˆ2) ;


Boundary conditions 1

The external boundary condition of a 1D cylindrical geometry is introduced by

choosing an albedo β+ set to zero for representing a voided boundary, or set to one

for representing reflection of particles.

The voided boundary condition is similar to the vacuum condition used for 1D slab

geometry.

The reflecting boundary condition is implemented in the context of the Wigner-Seitz

approximation for lattice calculations in reactor physics. The Wigner-Seitz

approximation consists of replacing the exact boundary with an equivalent cylindrical

boundary, taking care to conserve the amount of moderator present in the cell. The

correct condition in this case is the white boundary condition in which the particles are

reflected with an isotropic angular distribution.

Fuel Fuel



The application of a white boundary condition is based on the transformation of the reduced

collision probability matrix P = pij , i = 1, I and j = 1, I corresponding to a vacuum

boundary condition. We first compute the escape probability vector P iS = PiS , i = 1, I,

a column vector whose components are obtained from

PiS = 1−I

∑

j=1

pij Σj(16)

representing the probability for a neutron born in region i to escape from the unit cell without

collision.

The probability for a neutron entering in the unit cell by its boundary, with an isotropic

angular distribution, to first collide in region i is represented by the component PSi. It is

possible to show that

pSi =PSi

Σi=

4Vi

SPiS(17)

where S is the surface of the boundary.



The transmission probability PSS is therefore given as

PSS = 1−I

∑

i=1

pSi Σi .(18)

The reduced collision probability matrix P = pij , i = 1, I and j = 1, I corresponding to a

white boundary condition can now be obtained using

P = P+ P iS

[

β+ + (β+)2 PSS + (β+)3 P 2SS + (β+)4 P 3

SS + . . .]

p⊤Sj

or, using a geometric progression, as

P = P+β+

1− β+ PSSP iS p⊤

Sj .(19)

The collision probability matrix P can be scattering-reduced and used in the matrix flux

equation of the CP method.


Spherical 1D geometry 1

Computing the tracking and collision probabilities in 1D spherical geometry is similar to

the cylindrical geometry case.

The tracking is obtained by the sybt1d Matlab script

The expression of reduced collision probabilities is written

pij =1

Vi

∫

Vi

d3r′∫

Ij

ds e−τ(s)(20)



which can be rewritten as

pij =2π

Vi

∫ ri−1/2

0dh h

∫ ℓi

0dℓ′

∫ ℓj

0dℓ

[

e−[(τij+τii+Σi ℓi)+Σi ℓ′+Σj ℓ]

+ e−(τij+Σi ℓ′+Σj ℓ)

]

+

∫ ri+1/2

ri−1/2

dh h

∫ ℓi

0dℓ′

∫ ℓj

0dℓ e−(τij+Σi ℓ′+Σj ℓ)

if i < j(21)

where ri±1/2 are the radii bounding region i and

pii =2π

Vi

∫ ri−1/2

0dh h

∫ ℓi

0dℓ′

[

2

∫ ℓi

ℓ′dℓ e−[Σi (ℓ−ℓ′)] +

∫ ℓi

0dℓ e−[τii+Σi (ℓ′+ℓ)]

]

+

∫ ri+1/2

ri−1/2

dh h

∫ ℓi

0dℓ′

∫ ℓi

ℓ′dℓ e−[Σi (ℓ−ℓ′)]

(22)

where the first and second terms are the contributions from tracks 1© and 2©, respectively.



These equations can be simplified by introducing the following definitions:

Cij(τ0) =

∫ ℓi

0dℓ′

∫ ℓj

0dℓ e−(τ0+Σi ℓ′+Σj ℓ)

(23)

and

Di =

∫ ℓi

0dℓ′

∫ ℓi

ℓ′dℓ e−Σi (ℓ−ℓ′)

(24)

so that

pij =2π

Vi

∫ ri−1/2

0dh h [Cij(τij + τii +Σi ℓi) + Cij(τij)] +

∫ ri+1/2

ri−1/2

dh h Cij(τij)

(25)

if i < j and

pii =2π

Vi

∫ ri−1/2

0dh h [2Di + Cii(τii)] +

∫ ri+1/2

ri−1/2

dh hDi

.(26)



Integration in ℓ and ℓ′ is next performed, leading to the following relations:

a) Σi 6= 0 and Σj 6= 0 :

Cij(τ0) =1

Σi Σj

[

e−τ0 − e−(τ0+Σi ℓi) − e−(τ0+Σj ℓj) + e−(τ0+Σi ℓi+Σj ℓj)]

b) Σi = 0 and Σj 6= 0 :

Cij(τ0) =ℓi

Σj

[

e−τ0 − e−(τ0+Σj ℓj)]

c) Σi 6= 0 and Σj = 0 :

Cij(τ0) =ℓj

Σi

[

e−τ0 − e−(τ0+Σi ℓi)]

d) Σi = Σj = 0 :

Cij(τ0) = ℓi ℓj e−τ0



e) Σi 6= 0 :

Di =ℓi

Σi−

1

Σ2i

[

1− e−Σi ℓi]

f) Σi = 0 :

Di =ℓ2i

2.


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