In-line (a,n) source sampling methodology for monte carlo radiation transport simulations

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In-line (a,n) source sampling methodology for monte carlo radiation transport simulations. The new method uses a continuous-slowing-down model coupled with (a,n) cross section data to precompute the expected neutron yield over the alpha particle lifetime. This eliminates the complexity and computational cost associated with explicit charged particle transport. When combined with an integrated alpha particle decay source sampling capability.
*
Contents lists available at ScienceDirect
Nuclear Engineering and Technology
journal homepage: www.elsevier.com/locate/net
Original Article
In-line (a,n) source sampling methodology for monte carlo radiation
transport simulations
David P. Griesheimer , Andrew T. Pavlou, Jason T. Thompson, Jesse C. Holmes,
Michael L. Zerkle, Edmund Caro, Hansem Joo
Naval Nuclear Laboratory, P.O. Box 79, West Mifin, PA 15122, USA
a r t i c l e
i n f o
a b s t r a c t
Article history:
A new in-line method for sampling neutrons emitted in (a,n) reactions based on alpha particle source
Received 2 June 2017
Accepted 3 August 2017
Available online 1 September 2017
information has been developed for continuous-energy Monte Carlo simulations. The new method uses a
continuous-slowing-down model coupled with (a,n) cross section data to precompute the expected
neutron yield over the alpha particle lifetime. This eliminates the complexity and computational cost
Keywords:
Monte Carlo
alpha particle
neutron source
continuous-slowing-down
charged particle
associated with explicit charged particle transport. When combined with an integrated alpha particle
decay source sampling capability, the proposed method provides an efcient and accurate method for
sampling (a,n) neutrons based solely on nuclide inventories in the problem, with no additional user
input required. Results from several example calculations show that the proposed method reproduces
the (a,n) neutron yields and energy spectra from reference experiments and calculations.
© 2017 Korean Nuclear Society, Published by Elsevier Korea LLC. This is an open access article under the
1. Introduction
code COG [5] treats explicit alpha transport with a multistep
approach using the continuous-slowing-down approximation, but
The ability to accurately represent the emission rate and spec-
only the ability of the method to calculate total neutron emission
trum of neutrons produced by (a,n) reactions with light nuclides in
rates has been studied [7].
materials containing alpha particle emitters (such as actinides) is
In this paper, we describe a new methodology for in-line sam-
important for critical experiment analysis, reactor design, reactor
pling of neutrons emitted from (a,xn) reactions, where (a,xn) de-
protection, and shielding applications. Examples of these applica-
notes
any
alpha-particle
reaction
that
emits
neutrons.
The
tions include the modeling of articial (AmeBe, PueBe, etc.)
proposed methodology is based on a thick-target, continuous-
neutron sources used in critical assemblies, reactor startup, and
slowing-down model, resulting in a simpler and faster sampling
response checks of nuclear instrumentation; neutron production in
algorithm relative to methods that utilize explicit charged particle
fresh and irradiated fuel in both reactors and fuel cycle facilities;
transport. In addition, this paper demonstrates that many of the
and neutron shielding analyses involving spent fuel.
slowing-down parameters required for the (a,n) source calculation
Support for (a,n) sources in Monte Carlo (MC) transport codes in
can be precalculated and stored by nuclide along with other
the past [1e5] has suffered from a variety of disadvantages. First,
microscopic cross section data. The resulting method is straight-
explicit alpha transport in codes such as MCNP, FLUKA, and GEANT
forward to backt into existing continuous-energy MC codes.
is computationally expensive [1e3]. The charged particle transport
When coupled with an in-line decay source capability and
code package SRIM/TRIM is generally fast but can only be used with
suitable (a,xn) cross sections, the new methodology produces
planar geometries [4]. Additionally, there is a lack of support for
effective neutron source distributions based only on nuclide in-
elastic scattering cross sections required for explicit alpha particle
ventories in the model. This enables calculations such as subcritical
transport in some Evaluated Nuclear Data File (ENDF)-format
multiplication of highly depleted fuel, improves quality assurance
incident-alpha nuclear data libraries. In other codes, such as MCNP,
for complex models, and simplies user input.
users must manually specify the source distribution of alphas and/
or emitted neutrons from a secondary code such as SOURCES-4C [6]
owing to a lack of (a,n) information. The MC radiation transport
2. In-line (a,n) source generation
* Corresponding author.
The production rate of neutrons due to (a,xn) reactions is given
E-mail address: david.griesheimer@unnpp.gov (D.P. Griesheimer).
by the basic relationship,
1738-5733/© 2017 Korean Nuclear Society, Published by Elsevier Korea LLC. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/
0 0 0
t
y
s
0
J
I
i
m
Z
X
X
0
=dxÞ
dE
E
J J
i i
u
t
I
m
Z
i
E
J
I
i
m
B C
0 0 0
u
s
y
0
1
P
Z
i
1200
D.P. Griesheimer et al. / Nuclear Engineering and Technology 49 (2017) 1199e1210
Z
SnðEnÞ ¼ yðEaÞpðEnjEaÞSaðEaÞdEa; (1)
0
where Sn(En) and Sa(Ea) are the emission rates of neutrons and alpha
ZE pffiffiffiffi
i;jðEÞi;jðE Þ i;jðE Þ E dE :
E0;i;j
(7)
particles, respectively, as a function of energy for the corresponding
Because Eq. (7) does not depend on the density of other nuclides
particle type, y(Ea) is the total yield of neutrons from (a,xn) reactions
in the material, ti,j can be precomputed and stored along with other
resulting from an alpha particle with initial energy Ea, and p(EnjEa) is
the probability that the (a,xn) reaction induced by an alpha particle
microscopic cross section data for each nuclide.
Substituting the denition for ti,j into Eq. (5) yields
with energy Ea will produce a neutron with energy En.
In order to determine the neutron production rate during an MC
transport simulation, it is necessary to develop on-the-y methods
ymðEaÞ ¼ X XumNiti;jðEaÞ:
(8)
for determining the neutron yield and sampling from the energy
i¼1 j¼1
distribution of emitted neutrons from (a,xn) reactions.
Note that the right-hand side of Eq. (8) is similar to the de-
2.1. Determination of neutron yield
nition for a macroscopic cross section, with ti,j acting as a
microscopic cross section and umNi acting as an effective number
density. The latter term accounts for both the actual number
The expected number of neutrons created by an alpha particle
density of nuclide i, as well as the effects of the material
with initial energy Ea traveling through material m is given by
ymðEaÞ ¼ Im Ji Ni Ea yi;jðE0Þsi;jðE0ÞdE0; (2)
i¼1 j¼1 m
0;i;j
composition on the alpha-slowing-down process. Because the
factor um is based only on the density of nuclides in the local
material, it can be calculated on the y when time-in-life-
dependent number densities are retrieved for calculating macro-
scopic cross sections at each alpha particle source site. As a result,
calculation of the (a,xn) neutron yield using Eq. (8) is especially
where Im is the number of unique nuclides in material m, Ji is the
number of (a,xn) reaction types for nuclide i, Ni is the atomic
number density of nuclide i (in atoms/(barn$cm)), si,j is the
microscopic neutronproduction cross section for (a,xn) reaction j of
nuclide i, yi,j is the number of neutrons emitted in reaction j of
nuclide i, E0,i,j is the threshold energy for reaction j of nuclide i, and
e(dE0/dx)m is the stopping power (in eV/cm) for alpha particles in
material m.
easy to back t into a continuous-energy MC particle transport
code using the existing framework for calculating macroscopic
cross sections.
Given the similarity of ym(Ea) to a macroscopic cross section, it is
straightforward to dene the partial neutron yields by reaction,
ym,i,j(Ea), and by nuclide, ym,i(Ea), as
ym;i;jðEaÞumNiti;jðEaÞ; (9)
For alpha particles with energy less than 20 MeV traveling in
materials with atomic number greater than 20, the stopping power
and
in eV/cm can be well approximated by the AlsmillereEstabrook
correlation [8],
X X
ym;iðEaÞym;i;jðEaÞ ¼ mNi i;jðEaÞ;
(10)
dE0=dxÞmzXNiεiðE0Þ
i¼1
½eV=cm;
(3)
j¼1 j¼1
respectively. These partial yield values will be used in the sampling
algorithm described in the following sections.
where εi(E0) is the empirically measured function
rffiffiffiffi
εiðE0Þ ¼ 1:866  1013 E0 ½eV,barn
(4)
Before continuing, it is useful to consider some details
regarding the alpha-slowing-down approximation used in the
preceding derivation. Note that the proposed formulation of
neutron yield as a pseudo-macroscopic cross section [Eq. (10)] is
based on the AlsmillereEstabrook correlation, which relates alpha
for E0 in eV. Substituting Eqs. (3) and (4) into Eq. (2) and factoring
stopping power to the sum of slowing-down contributions from
out constant terms yields
each constituent nuclide in the material [Eq. (3)]. As a result, the
0 1
X X Z a pffiffiffiffi
ymðEaÞ ¼ m Ni@ i;jðE Þ i;jE E dE A;
(5)
factorization of the (a,n) yield expression into the macroscopic
cross section form shown in Eq. (5) is especially straightforward.
Although other approximations for alpha stopping power are
i¼1 j¼1
E0;i;j
available, few (if any) of these approximations allow such a
convenient factorization of the (a,n) neutron yield. For example,
where um is the slowing-down parameter for material m,
the BetheeBloch expression for stopping power is a function of
ln(E)/E plus additional terms (e.g., excitation potential, shell cor-
um 1:866  1013 Im Nipffiffiffiffi:
i¼1
(6)
rections) that depend on 1/E [9]. As a consequence, the resulting
factorization, where possible, would require multiple pre-
computed energy integral factors rather than just one when using
the AlsmillereEstabrook correlation.
Note that the integral in Eq. (5) is an intrinsic (microscopic)
In addition to the convenient factorization, internal testing
quantity that is proportional to the number of neutrons produced
suggests that the AlsmillereEstabrook correlation provides an ac-
by (a,xn) reaction j of nuclide i as an alpha particle with initial
curate estimate of alpha stopping power to lower energies than the
energy Ea slows down to the energy threshold for the corre-
BetheeBloch expression. In a 1973 study, Nitzki and Matzke [10]
sponding reaction. In the remainder of this paper, this quantity will
measured stopping power for alpha particles in UO2, PuO2, and
be referred to as the microscopic integral neutron production for
ThO2, and showed that the BetheeBloch approximation compares
reaction j in nuclide i, and is denoted by
well with experimental measurements down to about 4 MeV.
E
J
I
i
m
B C
0 0 0
u
s
y
0
E
X
X
E
1
E
E
0
0
;
¼  E
E dE
E
a
1
u
a
;m
D
3
x
=
a
m
m
~
~
~
D.P. Griesheimer et al. / Nuclear Engineering and Technology 49 (2017) 1199e1210
1201
However, informal studies involving UO2 indicate that the stopping
powers calculated with the AlsmillereEstabrook correlation
compare well with the 1973 experimental results from Nitzki and
Matzke down to 2 MeV. The 2e4 MeV energy range is especially
0 1
X X Za;m pffiffiffiffi
ymðEaÞ ¼ m Ni@ i;jðE Þ i;jðE Þ E dE A:
i¼1 j¼1
a;mþ1
(15)
important for in-line (a,n) calculations because many important
decay alphas are emitted with energies between 4 and 6 MeV. This
further supports the use of the AlsmillereEstabrook correlation for
these types of calculations. However, a thorough comparison be-
tween available alpha slowing-down approximations is beyond the
scope of this paper.
Furthermore, Eq. (15) can be written in terms of the microscopic
integral neutron production ti,j as
ymðEaÞ ¼ Im Ji umNiti;jEa;m  ti;jEa;mþ1: (16)
i¼1 j¼1
Note that Eq. (16) reverts tothe original form [Eq. (10)] when the
2.2. Correction for a escape
exiting alpha energy Ea,mþ1 is less than or equal to the minimum
(a,n) reaction threshold energy for the material.
The preceding derivations are based on the thick-target
In cases where the alpha particle can travel through multiple
approximation and implicitly assume that the alpha particle
will not leave the material that it was born in. In cases where the
materials during slowing down, the neutron yield in each segment
can be determined using the process outlined above. The alpha
thick-target approximation is not reasonable, the source sam-
pling methodology can be modied to account for the probability
transport calculation iterates over each material sequentially, using
the exiting alpha energy from one material as the entering alpha
the alpha particle will escape from the material that it was
originally born in. Furthermore, in volumes that contain a het-
erogeneous mixture of materials, it is possible for an alpha par-
ticle to travel through multiple materials before losing all of its
energy for the next material. In these situations, it is important to
note that the alpha particle cannot be safely terminated until it falls
below the lowest (a,n) threshold energy for any nuclide/reaction
present in the collection of materials that the alpha particle may
energy [11]. For example, this situation occurs in AmeBe neutron
source capsules that are manufactured by mixing AmO2 and Be
powders.
pass through. An example that illustrates the effects of alpha
transport in a heterogeneous material on (a,n) production is pro-
vided in Section 3.
Consider an alpha particle with energy Ea,m that enters (or is
born in) material m. Furthermore, let Dxm denote the distance
2.3. (a,n) Source sampling algorithm
that the alpha particle will travel before reaching the boundary
of material m. Note that Dxm may be computed based on the
current position and direction of the alpha particle (if the ma-
The process for sampling secondary neutrons produced via
(a,xn) reaction(s) for a given incident alpha particle consists of
terial
heterogeneity
is
explicitly
represented)
or
randomly
three basic steps. First, the expected yield of neutrons from each
sampled from a nearest-neighbor distribution (NND) (if the
nuclide and reaction type is determined and the number of sec-
material heterogeneity is modeled stochastically). Applying the
ondary neutrons is sampled in proportion to the corresponding
basic relationship between range and stopping power for a
charged particle gives
yield(s). Next, for each secondary neutron produced, the energy of
the alpha particle immediately prior to the corresponding (a,xn)
reaction is sampled. Finally, the initial energy and direction of each
Za;m  
Dxm ¼ dE0=dxÞm dE0;
a;mþ1
(11)
secondary neutron produced in the reaction is sampled based on
the target nuclide, reaction type, and energy of the incident alpha
particle at the time of the (a,xn) reaction. Note that the initial po-
sition of the neutron is assumed to be the same as the birth location
where Ea,mþ1 denotes the energy of the alpha particle when it exits
material m and enters material m þ 1.
Applying the AlsmillereEstabrook correlation for stopping po-
wer [Eq. (3)] to Eq. (11) yields
of the parent alpha particle.
For the rst step, there are several unbiased algorithms for
sampling the neutronproduction by nuclide and reaction. The most
straightforward approach is to independently sample the number
of neutrons produced for each (a,xn) reaction of each nuclide. This
Dxm ¼ um
Za;m pffiffiffiffi 2um  3=2 3=2 
3 ;m a;mþ1
(12)
sampling strategy produces a separate realization for the numberof
neutrons from each nuclide and reaction combination. However,
the disadvantage of this approach is that it requires a sampling step
Ea;mþ1
for all possible (a,xn) reactions within the material. For cases where
which gives a relationship between Dxm, Ea,m, and Ea,mþ . Note that
Eq. (12) is only satised when
alpha-induced neutron production is dominated by a few nuclides
and/or reaction types, this methodology can be inefcient because
it is performing sampling on many reactions that have a low
Dxm  23mE3=2;
(13)
probability of producing secondary neutrons.
An alternative sampling strategy is to rst sample the total
number of secondary neutrons and then determine which nuclide
indicating that the alpha particle will reach the material boundary
before losing all of its energy. If the condition in Eq. (13) is satised,
the energy of the alpha particle when it leaves material m can be
and reaction produced each neutron based on the corresponding
marginal probability distributions. In this case, the total number of
secondary neutrons produced from (a,xn) reactions is given by
determined by solving Eq. (12) for Ea,mþ1
Ea;mþ1 ¼ E3; 2  2u m2=3:
(14)
nmbymðEaÞc þ b; (17)
where ym(Ea) is the total neutron yield for an alpha particle with
energy Ea slowing down in material m and b is a realization from
The neutron yield within material m can then be determined by
the Bernoulli random variable B, with probability of success
using Ea,mþ1 as the lower limit of integration in Eqs. (2) and (5)
given by
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In-line (a,n) source sampling methodology for monte carlo radiation transport simulations. The new method uses a continuous-slowing-down model coupled with (a,n) cross section data to precompute the expected neutron yield over the alpha particle lifetime. This eliminates the complexity and computational cost associated with explicit charged particle transport. When combined with an integrated alpha particle decay source sampling capability..

Nội dung

Nuclear Engineering and Technology 49 (2017) 1199e1210 Contents lists available at ScienceDirect Nuclear Engineering and Technology journal homepage: www.elsevier.com/locate/net Original Article In-line (a,n) source sampling methodology for monte carlo radiation transport simulations David P. Griesheimer , Andrew T. Pavlou, Jason T. Thompson, Jesse C. Holmes, Michael L. Zerkle, Edmund Caro, Hansem Joo Naval Nuclear Laboratory, P.O. Box 79, West Mifflin, PA 15122, USA a r t i c l e i n f o a b s t r a c t Article history: Received 2 June 2017 Accepted 3 August 2017 Available online 1 September 2017 Keywords: Monte Carlo alpha particle neutron source continuous-slowing-down charged particle 1. Introduction A new in-line method for sampling neutrons emitted in (a,n) reactions based on alpha particle source information has been developed for continuous-energy Monte Carlo simulations. The new method uses a continuous-slowing-down model coupled with (a,n) cross section data to precompute the expected neutron yield over the alpha particle lifetime. This eliminates the complexity and computational cost associated with explicit charged particle transport. When combined with an integrated alpha particle decay source sampling capability, the proposed method provides an efficient and accurate method for sampling (a,n) neutrons based solely on nuclide inventories in the problem, with no additional user input required. Results from several example calculations show that the proposed method reproduces the (a,n) neutron yields and energy spectra from reference experiments and calculations. © 2017 Korean Nuclear Society, Published by Elsevier Korea LLC. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). code COG [5] treats explicit alpha transport with a multistep approach using the continuous-slowing-down approximation, but The ability to accurately represent the emission rate and spec-trum of neutrons produced by (a,n) reactions with light nuclides in materials containing alpha particle emitters (such as actinides) is important for critical experiment analysis, reactor design, reactor protection, and shielding applications. Examples of these applica-tions include the modeling of artificial (AmeBe, PueBe, etc.) neutron sources used in critical assemblies, reactor startup, and response checks of nuclear instrumentation; neutron production in fresh and irradiated fuel in both reactors and fuel cycle facilities; and neutron shielding analyses involving spent fuel. Support for (a,n) sources in Monte Carlo (MC) transport codes in the past [1e5] has suffered from a variety of disadvantages. First, explicit alpha transport in codes such as MCNP, FLUKA, and GEANT is computationally expensive [1e3]. The charged particle transport code package SRIM/TRIM is generally fast but can only be used with planar geometries [4]. Additionally, there is a lack of support for elastic scattering cross sections required for explicit alpha particle transport in some Evaluated Nuclear Data File (ENDF)-format incident-alpha nuclear data libraries. In other codes, such as MCNP, users must manually specify the source distribution of alphas and/ or emitted neutrons from a secondary code such as SOURCES-4C [6] owing to a lack of (a,n) information. The MC radiation transport * Corresponding author. E-mail address: david.griesheimer@unnpp.gov (D.P. Griesheimer). only the ability of the method to calculate total neutron emission rates has been studied [7]. In this paper, we describe a new methodology for in-line sam-pling of neutrons emitted from (a,xn) reactions, where (a,xn) de-notes any alpha-particle reaction that emits neutrons. The proposed methodology is based on a thick-target, continuous-slowing-down model, resulting in a simpler and faster sampling algorithm relative to methods that utilize explicit charged particle transport. In addition, this paper demonstrates that many of the slowing-down parameters required for the (a,n) source calculation can be precalculated and stored by nuclide along with other microscopic cross section data. The resulting method is straight-forward to backfit into existing continuous-energy MC codes. When coupled with an in-line decay source capability and suitable (a,xn) cross sections, the new methodology produces effective neutron source distributions based only on nuclide in-ventories in the model. This enables calculations such as subcritical multiplication of highly depleted fuel, improves quality assurance for complex models, and simplifies user input. 2. In-line (a,n) source generation The production rate of neutrons due to (a,xn) reactions is given by the basic relationship, http://dx.doi.org/10.1016/j.net.2017.08.004 1738-5733/© 2017 Korean Nuclear Society, Published by Elsevier Korea LLC. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/). 1200 D.P. Griesheimer et al. / Nuclear Engineering and Technology 49 (2017) 1199e1210 Z SnðEnÞ ¼ yðEaÞpðEnjEaÞSaðEaÞdEa; (1) 0 where Sn(En) and Sa(Ea) are the emission rates of neutrons and alpha particles, respectively, as a function of energy for the corresponding particle type, y(Ea) is the total yield of neutrons from (a,xn) reactions resulting from an alpha particle with initial energy Ea, and p(EnjEa) is the probability that the (a,xn) reaction induced by an alpha particle with energy Ea will produce a neutron with energy En. In order to determine the neutron production rate during an MC transport simulation, it is necessary to develop on-the-fly methods for determining the neutron yield and sampling from the energy distribution of emitted neutrons from (a,xn) reactions. 2.1. Determination of neutron yield The expected number of neutrons created by an alpha particle with initial energy Ea traveling through material m is given by ymðEaÞ ¼ Im Ji Ni Ea yi;jðE0Þsi;jðE0ÞdE0; (2) i¼1 j¼1 m 0;i;j where Im is the number of unique nuclides in material m, Ji is the number of (a,xn) reaction types for nuclide i, Ni is the atomic number density of nuclide i (in atoms/(barn$cm)), si,j is the microscopic neutronproduction cross section for (a,xn) reaction j of nuclide i, yi,j is the number of neutrons emitted in reaction j of nuclide i, E0,i,j is the threshold energy for reaction j of nuclide i, and e(dE0/dx)m is the stopping power (in eV/cm) for alpha particles in material m. For alpha particles with energy less than 20 MeV traveling in materials with atomic number greater than 20, the stopping power ZE pffiffiffiffi i;jðEÞ≡ i;jðE Þ i;jðE Þ E dE : (7) E0;i;j Because Eq. (7) does not depend on the density of other nuclides in the material, ti,j can be precomputed and stored along with other microscopic cross section data for each nuclide. Substituting the definition for ti,j into Eq. (5) yields Im Ji ymðEaÞ ¼ umNiti;jðEaÞ: (8) i¼1 j¼1 Note that the right-hand side of Eq. (8) is similar to the defi-nition for a macroscopic cross section, with ti,j acting as a microscopic cross section and umNi acting as an effective number density. The latter term accounts for both the actual number density of nuclide i, as well as the effects of the material composition on the alpha-slowing-down process. Because the factor um is based only on the density of nuclides in the local material, it can be calculated on the fly when time-in-life-dependent number densities are retrieved for calculating macro-scopic cross sections at each alpha particle source site. As a result, calculation of the (a,xn) neutron yield using Eq. (8) is especially easy to back fit into a continuous-energy MC particle transport code using the existing framework for calculating macroscopic cross sections. Given the similarity of ym(Ea) to a macroscopic cross section, it is straightforward to define the partial neutron yields by reaction, ym,i,j(Ea), and by nuclide, ym,i(Ea), as ym;i;jðEaÞ≡umNiti;jðEaÞ; (9) and in eV/cm can be well approximated by the AlsmillereEstabrook correlation [8], X X ym;iðEaÞ≡ ym;i;jðEaÞ ¼ mNi i;jðEaÞ; (10) ðdE0=dxÞmzXNiεiðE0Þ ½eV=cm; i¼1 j¼1 j¼1 (3) respectively. These partial yield values will be used in the sampling algorithm described in the following sections. where εi(E0) is the empirically measured function rffiffiffiffi εiðE0Þ ¼ 1:866 1013 E0 ½eV,barn (4) for E0 in eV. Substituting Eqs. (3) and (4) into Eq. (2) and factoring Before continuing, it is useful to consider some details regarding the alpha-slowing-down approximation used in the preceding derivation. Note that the proposed formulation of neutron yield as a pseudo-macroscopic cross section [Eq. (10)] is based on the AlsmillereEstabrook correlation, which relates alpha stopping power to the sum of slowing-down contributions from out constant terms yields 0 1 X X Ea pffiffiffiffi ymðEaÞ ¼ m Ni@ i;jðE Þ i;jE E dE A; (5) i¼1 j¼1 0;i;j where um is the slowing-down parameter for material m, um ≡ 1:866 1013 Im Nipffiffiffiffi: (6) i¼1 each constituent nuclide in the material [Eq. (3)]. As a result, the factorization of the (a,n) yield expression into the macroscopic cross section form shown in Eq. (5) is especially straightforward. Although other approximations for alpha stopping power are available, few (if any) of these approximations allow such a convenient factorization of the (a,n) neutron yield. For example, the BetheeBloch expression for stopping power is a function of ln(E)/E plus additional terms (e.g., excitation potential, shell cor-rections) that depend on 1/E [9]. As a consequence, the resulting factorization, where possible, would require multiple pre-computed energy integral factors rather than just one when using the AlsmillereEstabrook correlation. Note that the integral in Eq. (5) is an intrinsic (microscopic) quantity that is proportional to the number of neutrons produced by (a,xn) reaction j of nuclide i as an alpha particle with initial energy Ea slows down to the energy threshold for the corre-sponding reaction. In the remainder of this paper, this quantity will be referred to as the microscopic integral neutron production for reaction j in nuclide i, and is denoted by In addition to the convenient factorization, internal testing suggests that the AlsmillereEstabrook correlation provides an ac-curate estimate of alpha stopping power to lower energies than the BetheeBloch expression. In a 1973 study, Nitzki and Matzke [10] measured stopping power for alpha particles in UO2, PuO2, and ThO2, and showed that the BetheeBloch approximation compares well with experimental measurements down to about 4 MeV. D.P. Griesheimer et al. / Nuclear Engineering and Technology 49 (2017) 1199e1210 1201 However, informal studies involving UO2 indicate that the stopping powers calculated with the AlsmillereEstabrook correlation compare well with the 1973 experimental results from Nitzki and Matzke down to 2 MeV. The 2e4 MeV energy range is especially important for in-line (a,n) calculations because many important decay alphas are emitted with energies between 4 and 6 MeV. This further supports the use of the AlsmillereEstabrook correlation for these types of calculations. However, a thorough comparison be-tween available alpha slowing-down approximations is beyond the scope of this paper. 2.2. Correction for a escape The preceding derivations are based on the thick-target approximation and implicitly assume that the alpha particle will not leave the material that it was born in. In cases where the thick-target approximation is not reasonable, the source sam-pling methodology can be modified to account for the probability the alpha particle will escape from the material that it was originally born in. Furthermore, in volumes that contain a het-erogeneous mixture of materials, it is possible for an alpha par-ticle to travel through multiple materials before losing all of its energy [11]. For example, this situation occurs in AmeBe neutron source capsules that are manufactured by mixing AmO2 and Be powders. Consider an alpha particle with energy Ea,m that enters (or is born in) material m. Furthermore, let Dxm denote the distance that the alpha particle will travel before reaching the boundary of material m. Note that Dxm may be computed based on the current position and direction of the alpha particle (if the ma- terial heterogeneity is explicitly represented) or randomly sampled from a nearest-neighbor distribution (NND) (if the material heterogeneity is modeled stochastically). Applying the basic relationship between range and stopping power for a charged particle gives 0 1 X X Za;m pffiffiffiffi ymðEaÞ ¼ m Ni@ i;jðE Þ i;jðE Þ E dE A: (15) i¼1 j¼1 a;mþ1 Furthermore, Eq. (15) can be written in terms of the microscopic integral neutron production ti,j as ymðEaÞ ¼ X XumNiti;jEa;m ti;jEa;mþ1: (16) i¼1 j¼1 Note that Eq. (16) reverts tothe original form [Eq. (10)] when the exiting alpha energy Ea,mþ1 is less than or equal to the minimum (a,n) reaction threshold energy for the material. In cases where the alpha particle can travel through multiple materials during slowing down, the neutron yield in each segment can be determined using the process outlined above. The alpha transport calculation iterates over each material sequentially, using the exiting alpha energy from one material as the entering alpha energy for the next material. In these situations, it is important to note that the alpha particle cannot be safely terminated until it falls below the lowest (a,n) threshold energy for any nuclide/reaction present in the collection of materials that the alpha particle may pass through. An example that illustrates the effects of alpha transport in a heterogeneous material on (a,n) production is pro-vided in Section 3. 2.3. (a,n) Source sampling algorithm The process for sampling secondary neutrons produced via (a,xn) reaction(s) for a given incident alpha particle consists of three basic steps. First, the expected yield of neutrons from each nuclide and reaction type is determined and the number of sec-ondary neutrons is sampled in proportion to the corresponding yield(s). Next, for each secondary neutron produced, the energy of the alpha particle immediately prior to the corresponding (a,xn) reaction is sampled. Finally, the initial energy and direction of each Za;m Dxm ¼ ðdE0=dxÞm dE0; a;mþ1 secondary neutron produced in the reaction is sampled based on (11) the target nuclide, reaction type, and energy of the incident alpha particle at the time of the (a,xn) reaction. Note that the initial po- sition of the neutron is assumed to be the same as the birth location where Ea,mþ1 denotes the energy of the alpha particle when it exits material m and enters material m þ 1. Applying the AlsmillereEstabrook correlation for stopping po- wer [Eq. (3)] to Eq. (11) yields of the parent alpha particle. For the first step, there are several unbiased algorithms for sampling the neutronproduction by nuclide and reaction. The most straightforward approach is to independently sample the number of neutrons produced for each (a,xn) reaction of each nuclide. This Za;m pffiffiffiffi 2um 3=2 3=2 m m 3 ;m a;mþ1 Ea;mþ1 (12) sampling strategy produces a separate realization for the numberof neutrons from each nuclide and reaction combination. However, the disadvantage of this approach is that it requires a sampling step for all possible (a,xn) reactions within the material. For cases where which gives a relationship between Dxm, Ea,m, and Ea,mþ . Note that Eq. (12) is only satisfied when alpha-induced neutron production is dominated by a few nuclides and/or reaction types, this methodology can be inefficient because it is performing sampling on many reactions that have a low Dxm 23mE3=2; (13) probability of producing secondary neutrons. An alternative sampling strategy is to first sample the total number of secondary neutrons and then determine which nuclide indicating that the alpha particle will reach the material boundary before losing all of its energy. If the condition in Eq. (13) is satisfied, the energy of the alpha particle when it leaves material m can be determined by solving Eq. (12) for Ea,mþ1 Ea;mþ1 ¼ E3; 2 3Dxm2=3: (14) The neutron yield within material m can then be determined by using Ea,mþ1 as the lower limit of integration in Eqs. (2) and (5) and reaction produced each neutron based on the corresponding marginal probability distributions. In this case, the total number of secondary neutrons produced from (a,xn) reactions is given by nm≡bymðEaÞc þ b; (17) where ym(Ea) is the total neutron yield for an alpha particle with energy Ea slowing down in material m and b is a realization from the Bernoulli random variable B, with probability of success given by 1202 D.P. Griesheimer et al. / Nuclear Engineering and Technology 49 (2017) 1199e1210 evaluations for 9Be and 12C provide secondary energy distributions p ¼ ymðEaÞ bymðEaÞc: (18) for (a,n) reactions by discrete excitation level. The remaining nu- The probability that a secondary neutron was produced because of an (a,xn) reaction with nuclide i is given by the discrete proba-bility density function (PDF) p ðE Þ ¼ ym;iðEaÞ: (19) m a Similarly, the probability that a secondary neutron from an (a,xn) reaction with nuclide i was created by reaction j is given by the conditional discrete PDF pjjiðEaÞ ¼ ym;i;jðEaÞ: (20) m;i Based on Eqs. (19) and (20), selection of the target nuclide i and (a,xn) reaction type j is possible using any unbiased discrete sam-pling algorithm. Once the target nuclide and reaction type have been sampled, the energy of the alpha particle at the time of the (a,xn) reaction must be determined, which requires information on the relative probability that a partially slowed alpha particle with any energy E0a < Ea will produce a neutron via the (a,xn) reaction. Information on this slowing-down reaction probability distribution can be inferred from the ratio of the expected neutron production for an alpha particle at energy E0a to the expected total neutron produc-tion from the original energy of the alpha particle, Ea, Pm;i;j Ea0 ¼ m;i;j a ¼ i;j a ; (21) m;i;j i;j which gives the cumulative distribution function (CDF) of the desired distribution as a function of the slowing-down energy E0a, expressed in terms of the microscopic integral production for re-action j in nuclide i, ti,j. The energy of the alpha particle immediately prior to (a,xn) re- action j with target nuclide i can be sampled from the slowing-down reaction probability distribution using the CDF given in Eq. (21). The most basic sampling algorithm is to simply draw a uniformly distributed random number x between 0 and 1 and then find the correspondingalphaparticleenergyEa suchthatPðEaÞ ¼ x.However, this sampling methodology implicitly assumes that the underlying probability distribution is a histogram. If necessary, accuracy of the energysamplingmethodcanbefurtherimprovedbysamplingwithin the selected CDF bin according to the slope of the corresponding microscopic (a,xn) cross section values within the bin. The assump-tion of a linear CDF within each energy bin is reasonable, provided thatthespacingbetweentabulatedcrosssectionvaluesissufficiently small. Additional accuracycan be obtained bycalculating and storing the underlying PDF pm;i;jðEaÞ based on the raw cross section data, at the cost of additional data processing and cross section data storage. 2.4. Secondary neutron distributions Once the target nuclide, reaction type, and alpha particle reac-tion energy have been sampled, the energy and direction of the emitted neutron can be sampled from the corresponding secondary energy distribution provided in the evaluated nuclear data. The JENDL/AN-2005 library includes (a,xn) reaction data for 17 isotopes with significance in nuclear fuel cycle applications [12]. Within these evaluations, cross section data are provided for a total of 139 distinct reactions, along with secondary neutron distributions for 36 of these reactions. The difference between the number of re- actions and secondary distributions is due to the fact that only the clides combine the secondary neutron energy distributions from all (a,xn) reactions (ground state plus excited states) into a single distribution. Secondary energy distribution data in JENDL/AN-2005 is given as a coupled energy-angle distribution (ENDF File 6) using either KalbacheMann systematics [13,14] or a two-body kinematics rep-resentation (used forlevel excitation reactions for 9Be and 12C only). Methodologies for interpreting and sampling from distributions in either of these formats are published in the documentation for the ENDF format [15]. KalbacheMann systematics were originally developed to model reactions for various incident charged particles with energies significantly higher (several hundred MeV) than those of interest for this work. For nuclides examined in this work, the (a,n) neutron energy and angle distributions predicted by the JENDL/AN-2005 library evaluations using KalbacheMann systematics compared poorly to measured neutron spectra for experiments involving lower-energy alphas characteristic of transuranic nuclide decay, as demonstrated in Section 3. In particular, the predicted neutron energy spectra based on KalbacheMann systematics were consis-tently biased too low in energy. For many engineering applications, accurately representing the neutron energy distribution is much more important than the angular distribution. Neutron energies can impact detector sensi-tivity, subcritical multiplication, shielding penetration, and other physical effects. By considering the incident alpha energy, reaction Q value, compound nucleus excitation levels, and partial cross sections for ground state and level excitation reactions, the neutron energy distribution can be calculated using two-body kinematics given an assumed center-of-mass angular distribution. The energy-angle distributions predicted by JENDL/AN-2005 evaluations that use KalbacheMann systematics are inconsistent with two-body kinematics based on the provided data for level excitation en-ergies and partial cross sections. For nuclides examined in this work, neutron energy distribu-tions predicted using two-body kinematics assuming isotropic center-of-mass neutron emission compare reasonably well with experimentally measured data, as demonstrated in Section 3. Physically, angular distributions are known to be somewhat for-ward peaked. The use of a forward-peaked angular distributionwill tend to redistribute the energy spectrum toward higher energies, which may be important for certain applications such as shielding analyses, which can be sensitive to neutron energy. 3. Numerical results Aversion of the proposed in-line (a,n)source sampling capability was implemented in a developmental version of MC21 [16], an in-house continuous-energy MC radiation transport code. Processing of the raw ENDF-format (a,n) cross section data and calculation of the microscopic integral neutron production values [Eq. (7)] was performed by NDEX [16], an in-house code for creating continuous-energy cross sections for MC21. In all cases, JENDL/AN-2005 cross-section evaluations were used for (a,n)reactions, with modifications to the neutron energy-angle distributions as specified. In order to demonstrate the effectiveness of the new (a,n) source sampling method, a series of three example problems were considered and MC21 results compared with experimental mea-surements, where available, or independent calculations using the SOURCES-4C computer code. SOURCES-4C [6] is the current de facto standard for calculating the neutron yield and energy spectrum from the (a,n) and spontaneous fission (SF) reactions in neutron source materials. SOURCES-4C can calculate neutron production D.P. Griesheimer et al. / Nuclear Engineering and Technology 49 (2017) 1199e1210 1203 rates and spectra for four geometric configurations (homogenous, beam, two-region interface, and three-region interface). The (a,n) spectra in SOURCES-4C are determined assuming an isotropic neutron angular distribution in the center-of-mass frame with a library of 89 nuclide decay alpha spectra and 24 sets of product-nuclide discrete-level branching fractions. The (a,n) reaction cross sections are drawn from a variety of sources, including an evalua-tion by Geiger and van der Zwan [17] for 9Be and Perry and Wilson [18] for 17,18O. 3.1. (a,n) Source in UO2 In the first example, the in-line source methodology in MC21 was used to calculate the total neutron production due to (a,n) reactions with 17O and 18O in a thick target of UO2. This example models the UO2 thick-target measurements by Bair and Gomez del Campo [19] and West and Sherwood [20], which are two of the primary measurements used to develop the 17O and 18O evaluations in JENDL/AN-2005 [12]. In addition, the JENDL/AN-2005 evaluation report for (a,xn) reaction data provides calculated and experi-mental neutron yield results for 18O(a,n) reactions in UO2 [12], which serves as a useful benchmark for validating the in-line (a,n) source sampling method as implemented in MC21. For the initial test of the in-line (a,n) source capability, 17O and 18O (a,n) neutron production in UO2 was calculated as a function of incident alpha particle energy. In this test, a total of 14 simulations were run, with incident monoenergetic alpha particle energies ranging from 2 MeV to 15 MeV in 1-MeV increments, which covers the range of yield results provided by Boles et al. [11]. The target material for the MC21 simulation was modeled as an infinite me-dium of UO2 with 2.4% enriched uranium and natural isotopic abundances of oxygen, as shown in Table 1. Each simulation included a fixed source calculation with 10,000 total alpha source particles divided among 100 batches. The target birth weight of secondary neutrons was set to a value between 1 107 and 1 1010 in order to boost the yield of neutrons produced in (a,xn) reactions [21] so that the total number of sampled neutrons was between 1 and 100 per source alpha particle (1 104 to 1 106 secondary neutrons produced per simulation). After each simulation, the (a,xn) neutron yield was estimated from the total birth weight of secondary neutrons produced during the simulation divided by the total number of initial alpha source particles. JENDL/AN-2005 alpha cross section data was used for all simulations. The resulting neutron yield curve as a function of incident alpha particle energy from the 14 independent MC21 simulations is shown in Fig. 1. The results from this curve show excellent agree-ment with experimental works by Bair and Gomez del Campo [19] and West and Sherwood [20]. These results suggest that the pro-posed in-line (a,n) sampling method is working as intended and is capable of faithfully reproducing (a,xn) reaction rates as specified by the provided nuclear data evaluation. In order totest the coupling between the in-line alpha decayand (a,n) source capabilities, MC21 was used to calculate the naturally occurring neutron source intensity and spectrum due to (a,n) re-actions caused by alpha particles emitted during the radioactive decay of uranium isotopes in the thick-target UO2 sample. The MC21 simulation for this case used 1 106 source alpha particles sampled from the alpha decay spectra of 234U, 235U, and 238U, using the in-line decay source sampling capability in MC21 [22]. The birth weight of neutrons was set to 1 106, which increases the number of sampled secondary (a,n) neutrons by a factor of one million, as described by Griesheimer and Nease [21]. The calcula-tion used the JENDL/AN-2005 library for alpha cross section data and the JEFF 3.1.1 library for decay data. Results from the simulation indicate a naturally occurring (a,n) neutron source rate density of 6.248 104 neutrons/(s$g UO2), based on an alpha source rate density of 5.131 104 alphas/(s$g UO2). As expected, the calculated (a,n) neutron source is small compared with the SF source rate density of 1.168 102 neutrons/ (s$g UO2) for UO2. SF neutrons were included in the MC21 simu-lation to enable comparisons with SOURCES-4C (which reports SF neutron sources). The MC21 neutron source rates agree to within 2% of the corresponding values produced by an independent simulation with the SOURCES-4C code (6.248 104 (a,n) neu-trons/(s$g UO2) and 1.172 102 SF neutrons/(s$g UO2)). Fig. 2 shows the sampled alpha emission spectrum and corresponding alpha reaction rate density for the simulation, confirming that MC21 is modeling the slowing down behavior of the alpha parti- cles. Note that the resulting alpha absorption spectrum follows the characteristic shape of the 18O (a,xn), which is overlaid in Fig. 2 for comparison. Note that the cross sections for 17O are negligibly small compared to 18O and are not included in Fig. 2. Fig. 3 shows the naturally occurring (a,n) neutron emission spectrum in UO2 computed with both MC21 and SOURCES-4C. Note that the MC21 calculation using the original JENDL/AN-2005 eval-uations for 17O and 18O (utilizing KalbacheMann systematics) produces a neutron emission spectrum that is strikingly different from the corresponding spectrum calculated with SOURCES-4C. In particular, this MC21 emission spectrum is significantly shifted toward lower energies, with the most probable neutron energy occurring at 750 keV vice 2.25 MeV in the SOURCES-4C spectrum. This difference is likely due to the use of the KalbacheMann rep- resentation of the secondary neutron energy and angular distri-bution for 17O and 18O in JENDL/AN-2005. In order to test the hypothesis that the KalbacheMann repre- sentation was responsible for the shift in the neutron energy spectrum, the JENDL/AN-2005 evaluations for 17O and 18O were manually modified to use two-body kinematics with isotropic neutron emission in the center-of-mass frame. In the modified evaluations, each excited level of the recoil nucleus was repre-sented explicitly, based on the partial reaction cross sections and corresponding Q values provided in the ENDF File 3 for each (a,n) reaction. The MC21 (a,n) neutron emission spectrum using the modified reaction data for 17O and 18O (with explicit two-body kinematics) is also shown in Fig. 3. Note that the resulting spectrum shows significantly better agreement with respect to the spectrum calculated by SOURCES-4C. The absolute neutron source density Table 1 UO2 composition definition. Nuclide Nuclide density [nuclides/(b$cm)] 16O 4.5829 102 17O 1.7457 105 18O 9.6000 105 234U 4.4843 106 235U 5.5815 104 238U 2.2408 102 Isotopic abundance (at.%) 99.753 0.038 0.209 0.020 2.430 97.551 also matches SOURCES-4C to within a few percent. Experimental data by Jacobs and Liskien [23] is also overlaid for comparison. This experiment used natural UO2 and an artificial high-intensity 4.5 MeV alpha source. The alpha source intensity was sufficient to allow SF neutrons to be neglected. The MC21 spectrum using the modified JENDL/AN-2005 evaluations appear to be very consistent with the experimental spectra. The emission spectrum from the MC21 calculation using sec-ondary neutron energy and angle distributions based on two-body kinematics clearly shows the contributions of reactions that leave 1204 D.P. Griesheimer et al. / Nuclear Engineering and Technology 49 (2017) 1199e1210 Neutron yield from ( ,n) reac!ons in UO2 1.0E+00 1.0E-01 1.0E-02

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