neutron and gamma beam simulation using openmc and python
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Neutron and gamma beam simulation using OpenMCand Python’s libraries for Machine Learning
M.Eng. Norberto Sebastian Schmidt
norberto.schmidt@cab.cnea.gov.ar
Neutron Physics Department - Bariloche Atomic Center
Comision Nacional de Energıa Atomica (CNEA) - Argentina
3rd Workshop of Spanish Users on Nuclear Data on“Machine Learning in Nuclear Science and Technology Applications”
May 27, 2021
M.Eng. Norberto S. Schmidt May 27, 2021 1 / 15
Acknowledgments
Acknowledgments
• To my Master of Nuclear Engineering thesis directors: Ph.D. Jose Ignacio MarquezDamian (ESS) and Ph.D. Javier Dawidowski (CNEA).
• To M.Eng. Ariel Marquez (CNEA), Ph.D. Jose Robledo (CONICET) and B.Eng.Mauricio Debarbora (INVAP), for their helping knowledge.
• To B.Eng. Zoe Prieto (IB) and B.Eng. Inti Abbate (IB), for continuing thedevelopment of this method.
M.Eng. Norberto S. Schmidt May 27, 2021 2 / 15
Summary
Summary
1 Motivation
2 OpenMC
3 Kernel density estimation
4 Results
5 Conclusions
M.Eng. Norberto S. Schmidt May 27, 2021 3 / 15
Motivation
Motivation
Reactor block
Water
Reactor pool
Shutter
Reactor core
Neutron beamextraction tube
Collimator
BNCT filter
ShieldingsDisk chopper
Concrete walls
Beam catcherHe3 detectors
Surrounding box
RA-6 Research Reactor - Time-of-flight spectrometry experimental facility
Beam duct no 5 • Design of a time-of-flight spectrometryfacility in the RA-6 Research Reactor.
• Necessity to estimate the neutroncurrent at the exit of the duct.
M.Eng. Norberto S. Schmidt May 27, 2021 4 / 15
OpenMC OpenMC transport code description
OpenMC transport code description
BNCT
Reactorcore Reactor
pool
Reactorblock
Neutron beamextraction tube
Beam duct no5
Collimator
Shutter
Tracks surface
RA-6 Research Reactor - OpenMC model
• Open-source (https://openmc.org/).
• Modeling the RA-6 Research Reactorfrom IEU-COMP-THERM-014 NEABenchmark.
• Modification to write the particles thatcross a given surface.
• Possibility to transform track files to.mcpl format.
• Use the same track files to simulate theparticles in different codes (McStas,PHITS, etc.).
M.Eng. Norberto S. Schmidt May 27, 2021 5 / 15
OpenMC RA-6 Research Reactor radiation simulation
RA-6 Research Reactor radiation simulation
−200 −100 0 100 200x [cm]
−300
−200
−100
0
100
y[c
m]
J+n = (3.86± 0.04)× 109 [n /(cm2 s)]
Nn = 11622
RA-6 Research Reactor - Neutron flux
109
109
10 10
1011
10 12
10 13
108
109
1010
1011
1012
1013
1014
φ(x,y
)[n
/(cm
²s)
]−200 −100 0 100 200
x [cm]
−300
−200
−100
0
100
y[c
m]
J+γ = (3.09± 0.01)× 1010 [γ/(cm2 s)]
Nγ = 64441
RA-6 Research Reactor - Photon flux
10
9
109
109
10 9
10 9
109
10
9
109
10
9
10 9
10 9
109
1010
1011
10
12
1013
108
109
1010
1011
1012
1013
1014
φ(x,y
)[γ
/(cm
²s)
]
Source particles: 1× 109 n, time elapsed: 3.6× 105 s = 4.2 d, CPU: i7 8700 with 12 threads.
M.Eng. Norberto S. Schmidt May 27, 2021 6 / 15
How to generate more particles?
Example of scikit-learn kernel density estimation.
https://scikit-learn.org/stable/auto_examples/neighbors/plot_digits_kde_sampling.htm
M.Eng. Norberto S. Schmidt May 27, 2021 7 / 15
How to generate more particles?
M.Eng. Norberto S. Schmidt May 27, 2021 7 / 15
Kernel density estimation Univariate kernel density estimation
Univariate kernel density estimation
−1.00 −0.75 −0.50 −0.25 0.00 0.25 0.50 0.75 1.00u
0.0
0.5
1.0
1.5
2.0
2.5
p(u
)
Probability of u direction component
KDE (h = 0.01)
KDE (h = 0.06)
KDE (h = 0.28)
Histogram (101 bins)
Bayesian blocks
f(x, h) =1
N · hN∑i=1
K
(x−Xi
h
)
Gaussian univariate kernel function:
K(z) =e−(z2/2)
√2π
⇒ h = σ
M.Eng. Norberto S. Schmidt May 27, 2021 8 / 15
Kernel density estimation Multivariate kernel density estimation
Multivariate kernel density estimation
−5.0 −2.5 0.0 2.5 5.0x [cm]
−1.00
−0.75
−0.50
−0.25
0.00
0.25
0.50
0.75
1.00
u
Histogram (51× 51 bins)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
p(x,u
)
−5.0 −2.5 0.0 2.5 5.0x [cm]
−1.00
−0.75
−0.50
−0.25
0.00
0.25
0.50
0.75
1.00
u
KDE (hx = hu = 0.09)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
p(x,u
)
−5.0 −2.5 0.0 2.5 5.0x [cm]
−1.00
−0.75
−0.50
−0.25
0.00
0.25
0.50
0.75
1.00
u
KDE (hx = hu = 0.46)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
p(x,u
)
−5.0 −2.5 0.0 2.5 5.0x [cm]
−1.00
−0.75
−0.50
−0.25
0.00
0.25
0.50
0.75
1.00
u
KDE (hx = 0.46, hu = 0.09)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
p(x,u
)
Probability of x position coordinate and u direction component
f(x,H) =1
N · det (H)
N∑i=1
K(H−1(x−Xi)
)Gaussian multivariate kernel function:
K(z) =e−(zT z/2)
(2π)(dim(z)/2)
If H is diagonal:
f(x,H) =1
N
N∑i=1
dim(x)∏j=1
1
hjK
(xj −Xj,i
hj
)
M.Eng. Norberto S. Schmidt May 27, 2021 9 / 15
Kernel density estimation Best bandwidth selection
Best bandwidth selection
−5.0 −2.5 0.0 2.5 5.0x [cm]
−1.00
−0.75
−0.50
−0.25
0.00
0.25
0.50
0.75
1.00
u
Histogram (51× 51 bins)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
p(x,u
)
−5.0 −2.5 0.0 2.5 5.0x [cm]
−1.00
−0.75
−0.50
−0.25
0.00
0.25
0.50
0.75
1.00
u
KDE, Silverman (hx = hu = 0.46)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
p(x,u
)
−5.0 −2.5 0.0 2.5 5.0x [cm]
−1.00
−0.75
−0.50
−0.25
0.00
0.25
0.50
0.75
1.00
u
KDE, scikit-learn (hx = hu = 0.09)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
p(x,u
)
−5.0 −2.5 0.0 2.5 5.0x [cm]
−1.00
−0.75
−0.50
−0.25
0.00
0.25
0.50
0.75
1.00
u
KDE, statsmodels (hx = 0.27, hu = 0.03)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
p(x,u
)
Probability of x position coordinate and u direction component
Python’s libraries for KDE:
1 scipy: Scott’s and Silverman’s rules(depends only on dim(x) and N).
2 scikit-learn: grid searchcross-validation (same h for eachvariable).
3 statsmodels: operator cross-validation(different h for each variable).
M.Eng. Norberto S. Schmidt May 27, 2021 10 / 15
Results Sampling particles with KDE
Sampling particles with KDE
1 Run a OpenMC eigenvalue core calculation.
2 Write the variables r,Ω, E,wgt of the particlesthat cross certain surface in a file.
r = ~r = (x, y, z = z0) = (R, θ, z = z0)
Ω = Ω = (u, v, w) = (ρ = 1, ϕ, ϑ)
3 Compute the multivariate KDE usingstatsmodels CV bandwidth selection for eachvariable.
4 Write the sampled particles in a .h5 format file.
5 Run a OpenMC fixed source calculation.
x
y
z
r
Ω
u
v
w
Rθ
ϑ
φ
Coordinates ilustration.
M.Eng. Norberto S. Schmidt May 27, 2021 11 / 15
Results Sampling particles with KDE
Sampling particles with KDE
−6 −4 −2 0 2 4 6x [cm]
0.00
0.02
0.04
0.06
0.08
0.10
p(x
)
CV 1-D statsmodels
CV N-D statsmodels
CV 1-D scikit-learn
Histogram (51 bins)
Bayesian blocks
−6 −4 −2 0 2 4 6y [cm]
0.00
0.02
0.04
0.06
0.08
0.10
0.12
p(y)
CV 1-D statsmodels
CV N-D statsmodels
CV 1-D scikit-learn
Histogram (51 bins)
Bayesian blocks
2 4 6 8 10ξ = log (E0/E)
0.0
0.2
0.4
0.6
0.8
1.0
p(ξ)
CV 1-D statsmodels
CV N-D statsmodels
CV 1-D scikit-learn
Histogram (51 bins)
Bayesian blocks
−1.00 −0.75 −0.50 −0.25 0.00 0.25 0.50 0.75 1.00u
0.0
0.5
1.0
1.5
2.0
2.5
p(u
)
CV 1-D statsmodels
CV N-D statsmodels
CV 1-D scikit-learn
Histogram (51 bins)
Bayesian blocks
−1.00 −0.75 −0.50 −0.25 0.00 0.25 0.50 0.75 1.00v
0.0
0.5
1.0
1.5
2.0
2.5
p(v)
CV 1-D statsmodels
CV N-D statsmodels
CV 1-D scikit-learn
Histogram (51 bins)
Bayesian blocks
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0wgt
0
2
4
6
8
10
p(w
gt)
CV 1-D statsmodels
CV N-D statsmodels
CV 1-D scikit-learn
Histogram (51 bins)
Bayesian blocks
Best bandwidth comparison for neutron variables (in original coordinates).
M.Eng. Norberto S. Schmidt May 27, 2021 11 / 15
Results Sampling particles with KDE
Sampling particles with KDE
−6 −4 −2 0 2 4 6x [cm]
0.00
0.02
0.04
0.06
0.08
0.10
p(x
)
CV 1-D statsmodels
CV N-D statsmodels
CV 1-D scikit-learn
Histogram (51 bins)
Bayesian blocks
−6 −4 −2 0 2 4 6y [cm]
0.00
0.02
0.04
0.06
0.08
0.10
p(y)
CV 1-D statsmodels
CV N-D statsmodels
CV 1-D scikit-learn
Histogram (51 bins)
Bayesian blocks
2 4 6 8 10E [MeV]
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
p(E
)
CV 1-D statsmodels
CV N-D statsmodels
CV 1-D scikit-learn
Histogram (51 bins)
Bayesian blocks
−1.00 −0.75 −0.50 −0.25 0.00 0.25 0.50 0.75 1.00u
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
p(u
)
CV 1-D statsmodels
CV N-D statsmodels
CV 1-D scikit-learn
Histogram (51 bins)
Bayesian blocks
−1.00 −0.75 −0.50 −0.25 0.00 0.25 0.50 0.75 1.00v
0.00
0.25
0.50
0.75
1.00
1.25
1.50
1.75
p(v)
CV 1-D statsmodels
CV N-D statsmodels
CV 1-D scikit-learn
Histogram (51 bins)
Bayesian blocks
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0wgt
0
2
4
6
8
10
p(w
gt)
CV 1-D statsmodels
CV N-D statsmodels
CV 1-D scikit-learn
Histogram (51 bins)
Bayesian blocks
Best bandwidth comparison for photon variables (in original coordinates).
M.Eng. Norberto S. Schmidt May 27, 2021 11 / 15
Results Calculation of current distributions
Calculation of current distributions
0 2 4 6 8 10 12ξ = log (E0/E)
105
106
107
108
109
J(ξ
)[n
/(cm
2s)
]
Original
Sampled
Neutron current lethargy distribution J(ξ).
10−4 10−2 100 102 104 106
E [eV]
100
102
104
106
108
1010
J(E
)[n
/(cm
2s
eV)]
Original
Sampled
Neutron current energy distribution J(E).
M.Eng. Norberto S. Schmidt May 27, 2021 12 / 15
Results Calculation of current distributions
Calculation of current distributions
Total 0 eV - 0.5 eV 0.5 eV - 1 MeV 1 MeV - 20 MeV
Ori
gina
lSa
mpl
edC
ompa
riso
n
Neutron current spatial distribution J(x, y)
M.Eng. Norberto S. Schmidt May 27, 2021 12 / 15
Results Calculation of current distributions
Calculation of current distributions
−6 −4 −2 0 2 4 6x [cm]
−6
−4
−2
0
2
4
6
y[c
m]
~rC
~rU
~rD
~rL ~rR
0 45 90 135 180 225 270 315 360ϕ []
0.0
0.5
1.0
1.5
2.0
2.5
3.0
j(~r U,ϑ
0,ϕ
)[n
/(cm
2s)
]
×106
Original: Total
Sampled: Total
Original: 0 eV - 0.5 eV
Sampled: 0 eV - 0.5 eV
Original: 0.5 eV - 1 MeV
Sampled: 0.5 eV - 1 MeV
Original: 1 MeV - 20 MeV
Sampled: 1 MeV - 20 MeV
0 45 90 135 180 225 270 315 360ϕ []
0.0
0.5
1.0
1.5
2.0
2.5
3.0j(~r L,ϑ
0,ϕ
)[n
/(cm
2s)
]
×106
0 45 90 135 180 225 270 315 360ϕ []
0.0
0.5
1.0
1.5
2.0
2.5
3.0
j(~r C,ϑ
0,ϕ
)[n
/(cm
2s)
]
×106
0 45 90 135 180 225 270 315 360ϕ []
0.0
0.5
1.0
1.5
2.0
2.5
3.0
j(~r R,ϑ
0,ϕ
)[n
/(cm
2s)
]
×106
0 45 90 135 180 225 270 315 360ϕ []
0.0
0.5
1.0
1.5
2.0
2.5
3.0
j(~r D,ϑ
0,ϕ
)[n
/(cm
2s)
]
×106
Neutron current angular distribution j(ϕ)
Ω
u
v
w
ϑ
φ
M.Eng. Norberto S. Schmidt May 27, 2021 12 / 15
Results Comparison with the original tracks results
Comparison with the original tracks results
−100 −75 −50 −25 0 25 50 75 100x [cm]
−325
−300
−275
−250
−225
−200
−175
−150
−125
y[c
m]
Neutron flux with original particles10
9
10 10
104
105
106
107
108
109
1010
φ(x,y
)[n
/(cm
²s)
]−100 −75 −50 −25 0 25 50 75 100
x [cm]
−325
−300
−275
−250
−225
−200
−175
−150
−125
y[c
m]
Neutron flux with KDE sampled particles
10
6
10
6
10
6
106
10
7
10
7
10
810
9
104
105
106
107
108
109
1010
φ(x,y
)[n
/(cm
²s)
]
M.Eng. Norberto S. Schmidt May 27, 2021 13 / 15
Results Comparison with the original tracks results
Comparison with the original tracks results
−100 −75 −50 −25 0 25 50 75 100x [cm]
−325
−300
−275
−250
−225
−200
−175
−150
−125
y[c
m]
Photon flux with original particles
10
9
109
109
10 9
10 9
109
10
9
109
10
9
10 9
10 9
109
1010
104
105
106
107
108
109
1010
φ(x,y
)[γ
/(cm
²s)
]−100 −75 −50 −25 0 25 50 75 100
x [cm]
−325
−300
−275
−250
−225
−200
−175
−150
−125
y[c
m]
Photon flux with KDE sampled particles
10
6
10 6
10
6
107
10
8
10
9
104
105
106
107
108
109
1010
φ(x,y
)[γ
/(cm
²s)
]
M.Eng. Norberto S. Schmidt May 27, 2021 13 / 15
Conclusions
Conclusions
1 All this work was done using open source codes and free software tools.
2 A modification to write track files with OpenMC was generated.
3 New particles from these track files can be sampled using kernel density estimation.
4 Python has several libraries that estimate automatically the best bandwidth for theparticles’ variables.
5 Distributions of sampled particles with statsmodels resemble with originals.
M.Eng. Norberto S. Schmidt May 27, 2021 14 / 15
Thank you for your time.
Questions?
M.Eng. Norberto S. Schmidt May 27, 2021 15 / 15
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