monte carlo simulation of multiple particle spectra with...

ПРЕДПРИЯТИЕ ГОСКОРПОРАЦИИ «РОСАТОМ»

ФГУП «ВСЕРОССИЙСКИЙ НАУЧНО-ИССЛЕДОВАТЕЛЬСКИЙ ИНСТИТУТ АВТОМАТИКИ им. Н. Л. Духова»

Monte Carlo simulation of multipleparticle spectra with energy and

momentum conservationDmitry Savin and Mikhail Kosov

Dukhov Research Institute of Automatics

July 2, 2018

BackgroundGeant4

∙ A Monte Carlo toolkit for the simulation of the passage ofparticles through matter.

∙ Originally used for High Energy Physics experiments

∙ Increased precision stimulates to simulate lower energysecondary particles

∙ New applications: medical, microelectronics, neutron physics

D. Savin and M. Kosov VNIIA July 2, 2018 2 / 20

BackgroundParticleHP in Geant4

∙ Data-based model for neutrons and light charged particles

∙ Originally NeutronHP for neutrons below 20 MeV

∙ Aims to reproduce MCNP results

∙ Has a flag to correct energy by emitting additional gammas


BackgroundCHIPS-TPT physics list

∙ Built on top of CHIPS model formerly included in Geant4

∙ Light charged particles and neutrons up to 20 MeV

∙ Quantum numbers conservation in each interaction

∙ Energy and momentum conservation in each interaction


Motivation

∙ Avoids artificial fluctuations

∙ Allows to simulate correlated measurements

∙ Reproduces kinematic effects

∙ Imposes physical constrains on the data


Problem

∙ Secondary particle spectra are given uncorrelated

∙ Uncorrelated sampling of secondary particles leads to energynon-conservation

∙ Rejection of events with non-conserved energy disturbs thespectra

∙ Many ways to construct correlated particle spectra exist


Formal Problem StatementSet the order in which the particles are simulated. Assume theenergy distributions of the first and the second particles are given:

f (e) и h(e), e ∈ (0, emax)∫︁ emax

0f (e)de =

∫︁ emax

0h(e)de = 1

In the non-relativistic approximation the maximum energies(emax) are the same.

The energy distribution of the second particleg(e|e1), e ∈ (0, emax) when the energy of the first particle is e1must satisfy the conditions:∫︁ emax

0g(e|e1)de = 1∫︁ emax

0g(e|e1)f (e1)de1 = h(e)


SolutionWe find the solution in the form:

g(e|e1) =𝜃(emax − e1 − e)∫︀ emax−e10 g(e′|e1)de′

g(e),

where the integral in the denominator∫︁ emax−e1

0g(e′|e1)de′

is the normalization factor, and Riemann 𝜃 function

𝜃(x) =

{︃0 x ≤ 01 x > 0

.

represents the kinematic boundary ignoring relativistic effectsand the recoil nucleus energy.


Solution

Substituting the anzats into the condition we obtain∫︁ emax

0

g(e) · 𝜃(emax − e1 − e)1−

∫︀ emaxemax−e1 g(e

′)de′f (e1)de1 = h(e),

which easily transforms into

g(e) =h(e)∫︀ emax

ef (emax−ef )

1−∫︀ emaxef

g(e′)de′def,

where ef = emax − e1.


Solution

To calculate the cumulative distribution G =∫︀g(e)de which is

needed for the Monte Carlo simulation we transform the formulausing finite bins

ΔGi =ΔHi∑︀

j>i

ΔF j1−

∑︀k>j

ΔGk

,

where ΔGi = giΔei, ΔHi = hiΔei, ΔF i = f (emax − ei)Δei . Becausei < j < k it can be iteratively calculated starting from the upperbin where G approaches 1.


SolutionComparison with a standard solver

The integral equation is transformed to a differential equation ofthe second order:

ddeh(e)G′(e)

+f (emax − e)G(e)

= 0,

which has no solution with the boundary conditionsG(0) = 0, G(emax) = 1, but has a solution with boundaryconditions

G(𝛿) = 𝜖, G(emax − 𝛿) = 1− 𝜖,

where 𝛿 и 𝜖 - small quantities to avoid singularities on the edgesof the interval.


SolutionComparison with a standard solver

Green - example function f = h = x * (1− x) * e−5*xBlue - solution using a standard solverOrange - recursive solution

0.2 0.4 0.6 0.8 1.0

1

2

3

4

The solution using finite sums automatically avoids singularities.


SolutionThe third and subsequent particles

Using the obtained distribution G, one can evaluate thedistribution of the sum of the energies of the first two particles:

𝜑(e) =∫︁ emax−e

0

G′(e′ − e)G(emax − e′)

f (e′)de′

It can be regarded as the energy distribution of a quasi-particle tobe used to obtain the spectrum of the second particle with thesame formulae. This substitution allows to simulate an arbitrarynumber of kinematically correlated particles.


Results: The Good (Perfect)Smooth data, heavy nucleus - 109Ag(n, 2n)108Ag

E, MeV0 2 4 6 8 10

dN/dE,MeV

−1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

ratio

00.20.40.60.811.21.41.61.82

Red line - initial distribution, blue dots - simulation result, blackdash line - ratio.


Results: The Bad (Still Pretty Good)Reinterpolated data, medium nucleus 56Fe(n, 2n)55Fe

E, MeV1 2 3 4 5 6 7 8

dN/dE

,MeV

−1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

ratio

00.20.40.60.811.21.41.61.82



Results: The Ugly (Better Than Expected)Irregular data, light nucleus 9Be(n, 2n)2𝛼, CENDL

E, MeV2 4 6 8 10 12 14

dN/dE,MeV

−1

00.20.40.60.81

1.21.41.61.82

ratioе

00.20.40.60.811.21.41.61.82



Results: Correlated Spectrum9Be(n, 2n)2𝛼 at 19 MeV, CENDL

, MeV1E0 2 4 6 8 10 12 14

, MeV

2E

02

4

6

8

1012

14 -2, M

eVE2 dN2 d

2−10

1−10

1neutron energy-energy distribution


Results: Comparison With TALYS109Ag(n, 2n)108Ag at 16 MeV, TENDL2015

0 1 2 3 4 5 6E, MeV

0

0.05

0.1

0.15

0.2

0.25

0.3

dN/dE

,MeV

−1 TALYS

TPT

recoil excitation distribution

TALYS is the main program for nuclear data evaluation whichincludes complex physical models. The recoil excitation is notincluded in the evalauted data library and was simulated withTALYS and reconstructed from TPT simulation.D. Savin and M. Kosov VNIIA July 2, 2018 18 / 20

Comparison With TALYS

0 0.2 0.4 0.6 0.8 1TALYS

00.10.20.30.40.50.60.70.80.91

TPT

P-P plot

The distributions are in agreement.


Conclusion

∙ Method to sample correlated secondary neutrons

∙ Energy and momentum conservation in each scattering

∙ Reasonable reconstruction of the inclusive spectra

∙ Very simple model, little overhead

∙ Extensible to other boundary conditions


monte carlo simulation of multiple particle spectra with...

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