Download A Temperature Dependence Study of Alpha/Beta Cumulative

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Computational Methods: General
A Temperature Dependence Study of Alpha/Beta Cumulative Distribution Functions Based on S(α,β) Data
Andrew T. Pavlou and Wei Ji
Department of Mechanical, Aerospace, and Nuclear Engineering,
Rensselaer Polytechnic Institute, 110 8th Street, Troy, NY 12180-3590, USA, [email protected], [email protected]
INTRODUCTION
When modeling a neutron transport process, the
probabilities of neutron interactions with target materials
must be accounted for properly. These interaction
probabilities, or cross sections, are strong functions of the
neutron energy and the target material temperature. We
usually pre-calculate and store cross sections at many
energies and temperatures to be used in neutron transport
codes. For problems involving a coupling of neutronics
and thermal hydraulics, pre-storing cross sections
becomes memory intensive because of the fine
temperature mesh needed to account for temperature
feedback. For Very High Temperature Reactors (VHTR),
many cross sections are needed on a fine temperature
mesh to resolve the TRISO particle fuel and to account
for high temperatures during accident scenarios like the
Loss of Forced Cooling (LOFC) [1].
Methods have been developed in recent years to
reduce cross section data storage in the resolvedresonance and epithermal energy ranges by generating the
temperature-dependent cross sections on-the-fly [2,3]. In
these on-the-fly methodologies, the energy distribution of
the target nuclide is assumed to follow a MaxwellBoltzmann distribution, allowing for analytical
expressions to be formed to drive the on-the-fly process.
At thermal neutron energies, however, this assumption is
not valid as chemical binding effects are dominant which
complicates the calculation of the differential scattering
cross section.
A new strategy is needed to treat the temperature
dependence of the double differential scattering data in
the thermal energy region. This paper presents an on-thefly strategy to sample a neutron’s outgoing energy and
scattering angle after a thermal scattering event for an
arbitrary temperature. A method proposed by Ballinger
[4] is introduced to construct probability density functions
(PDFs) of momentum transfer (α) and energy transfer (β).
The cumulative distribution functions (CDFs) of these
PDFs are then constructed on a temperature mesh and an
incoming energy mesh for common moderator nuclei.
Regression models are then used to fit the temperature
dependence of these CDFs. The best functional fits are
chosen based on an error analysis procedure and the
outgoing parameters are then sampled on-the-fly from the
coefficients of the chosen functional fits.
MOTIVATION
In this work, we focus our attention on the incoherent
inelastic differential scattering cross section.
The
inelastic portion of scattering involves a change in the
neutron energy after the scatter while the incoherent
portion of scattering ignores atomic plane interference
effects between the neutron and target. This cross section
is classically expressed as [4]
( E E ' , ' , T ) b
2kT
E ' /2
e S ( , , T ), (1)
E
where E and E’ are, respectively, the incident and
scattered neutron energy, Ω∙Ω’ is the scattering angle, σb
is the bound atom cross section, kT is the ambient
temperature and S(α,β,T) is the scattering law which
contains the quantum translational, rotational, and
vibrational motions of the neutron. The quantities α and β
represent, respectively, momentum and energy transfer,
' 2 EE '
,
A0 kT
(2)
' ,
kT
(3)
where A0 is the target nucleus to neutron mass ratio and μ
is the cosine of the scattering angle. Scattering law data
are generated with the nuclear data processing code
NJOY [5] and stored in ENDF thermal scattering files [6]
for select moderator materials at specific temperatures
and on an (α,β) mesh. The (α,β) mesh is chosen to allow
interpolation between values within a specified fractional
tolerance. The interpolation law used is moderatordependent. Modern Monte Carlo codes (MC21 and
MCNP6) use a continuous-in-energy representation for
the outgoing scattering parameters, requiring a very fine
energy mesh. Even for a single temperature, the S(α,β,T)
data can be large as shown in Table I for selected
moderators at room temperature from the ENDF/B-VII.1
library.
Because many temperature sets are often needed, the
memory load needed can become unnecessarily large.
The approach we take is based on CDFs of α and β from
the S(α,β,T) data. In the next section, a method to
Transactions of the American Nuclear Society, Vol. 110, Reno, Nevada, June 15–19, 2014
236
Computational Methods: General
Table I. ENDF/B-VII.1 Continuous S(α,β,T)
file sizes for selected moderator materials at
room temperature
Material
File Size [MB]
Graphite
24
Water
24.9
U in UO2
50
O2 in UO2
75
Zr in ZrH
56
H in ZrH
116
p( | E , T ) e /2 max
max
min
min
p( | , E , T ) construct these CDFs that can be used to sample the
scattered neutron energy and angle of scatter at a single
temperature is introduced. Then, these CDF forms at
different temperatures and incoming energies are
investigated for generally-used light material nuclei. The
temperature dependence of these CDFs is determined by
fitting to several regression models. The final functional
form for the temperature dependence of alpha and beta at
different values of CDFs will be constructed based on the
best fitted regression model. On-the-fly sampling of the
outgoing parameters can then be implemented using the
obtained functional forms.
max
min
S ( , , T )d
e /2 S ( , , T )d d S ( , , T )
max
min
S ( , , T )d
.
, (7)
(8)
Assuming a given initial energy E at some temperature T,
a value of beta can be sampled from Eq. (7). Then, once
this is done, Eq. (8) is used to sample a value of alpha.
From these sampled values, the scattered energy and
cosine of the scattering angle can be directly calculated.
Fig. 1 shows an example of the beta CDF generated for
graphite at an incoming energy of 1 eV at five
temperatures.
METHODOLOGY
The sampling procedure described here is based on
the direct sampling method used by Ballinger [4]. The
sampling is performed by first writing the double
differential scattering cross section, Eq. (1), in terms of α
and β,
( , , T ) b A0kT
4E
e /2 S ( , , T ).
(4)
The sampling is then performed on a PDF form of this
cross section, found by dividing by the total cross section
integrated over all alpha and beta values,
p( , | E , T ) ( , , T )
max
max
min
min
( , , T )d d . (5)
In Eq. (5), the minimum and maximum values of alpha
and beta are readily found using Eqs. (2) and (3) for
minimum and maximum energy transfer and momentum
transfer. Eqs. (4) and (5) are combined to give
p( , | E , T ) e /2 S ( , , T )
max
max
min
min
e
/2
S ( , , T )d d . (6)
This PDF is broken up into two PDFs; one conditional
PDF in alpha and the other in beta,
Fig. 1. Beta CDF for graphite at E = 1 eV for 300K,
650K, 1000K, 1500K and 2000K.
The temperature dependence of the β CDF was
examined at discrete CDF probability lines in the range
[0,1] (15 probability lines per energy) and on a mesh of
incoming neutron energies in the range [1E-5, 1] eV (71
total energies). In total, there were four variables
considered in the analysis: Ein, β, T and Pβ, where Pβ is the
β CDF probability. The α CDF temperature analysis
consists of the four variables: β, α, T and Pα, where Pα is
the α CDF probability. Although the α PDF/CDF is also
dependent on the incoming neutron energy, this variable is
ignored to reduce the data storage. Instead, the CDF is
analyzed over the entire given α mesh instead of between
αmin and αmax. Then, the α bounds are calculated as needed
for the desired incoming neutron energy and the
appropriate section of the α is sampled. Note that the β
mesh for graphite consists of 96 values in the range [0,
80].
Functional expansions based on different regression
models are examined to fit the β(T) and α(T) data at
different values of alpha and beta CDFs.
This
Transactions of the American Nuclear Society, Vol. 110, Reno, Nevada, June 15–19, 2014
237
Computational Methods: General
temperature dependence was performed on a mesh of 35
values in the range [300, 2000] K at 50 K increments for
graphite. A code was developed to build the CDFs for
beta and alpha at each temperature in the mesh from
S(α,β,T) data obtained from NJOY. The β(T) data are
then outputted at each beta and Pα in their respective
meshes. Likewise, the α(T) data are outputted at each
beta and Pα in their respective meshes. A visualization of
this procedure is given in Fig. 2 for the beta data. In the
figure, only five temperatures (300K, 650K, 1000K,
1500K and 2000K) are shown along with four probability
lines (0.2, 0.4, 0.6 and 0.8) for one incoming energy value
(1 eV). This is for visualization only. In actuality, 35
temperatures, 15 probability lines, 71 incoming energies
and 96 beta values are used for the study.
Fig. 2. Temperature dependence of the beta CDF for
graphite for E = 1 eV.
Along each CDF probability line, the temperature
dependence is examined through fitting functions. The
next section compares different temperature fitting
functions to the alpha and beta CDFs for graphite.
RESULTS
To determine the best fit for the data, eight different
regression models were considered, shown in Table II.
The an’s are the fitting coefficients and N is the order of
the functional expansion. To determine the best fit for
β(T) and α(T), the root-mean-square error (RMSE) is
calculated at every CDF probability, incoming energy,
and beta value. The RMSE is given by
NT
RMSE (
i 1
i
i ,est )2
N T ( N 1)
,
(9)
Table II. Fits for β(T) and α(T) analysis
Fit #
Fit
Fit #
N
1
a T
2
a T
n
n 0
N
n
n 0
N
n
3
a (
T )n
4
a (
T ) n
n 0
N
n0
n
n
Fit
N
n
5
a (ln T )
6
a (ln T )
7
a (
ln T ) n
8
a (
ln T ) n
n
n 0
N
n
n 0
N
n 0
N
n 0
n
n
n
n
where θi is the true value of beta or alpha at the current
temperature, θi,est is the estimated value of beta or alpha at
the current temperature from the fitting coefficients and
NT is the number of temperature values used in the
analysis. The true value of beta and alpha is found by
running the LEAPR module of NJOY at the desired
temperature to obtain the S(α,β,T) data. Then, Eqs. (7)
and (8) are used to build the true PDFs. These RMSE
values are averaged over the 15 CDF probability lines.
Next, these beta RMSE values are averaged over the
incoming energy mesh and the alpha RMSE values are
averaged over the beta mesh. In doing this, a single
averaged RMSE value is found for each functional
expansion for each expansion order. Table III compares
these averaged RMSE values for each fit for expansion
orders 1,…,4 for the β(T) data. The same is shown for the
α(T) data in Table IV. Fits 5-8 have been omitted from
the table because their values are large compared to the
other fits.
Table III. Average RMSE for β(T) for graphite.
N=1
N=2
N=3
N=4
Fit 1
0.1783
0.0793
0.0407
0.0228
Fit 2
0.0610
0.0202
0.0120
0.0092
Fit 3
0.1339
0.0489
0.0225
0.0133
Fit 4
0.0652
0.0174
0.0124
0.0091
Table IV. Average RMSE for α(T) for graphite.
N=1
N=2
N=3
N=4
Fit 1
0.5473
0.2706
0.1631
0.1197
Fit 2
0.1758
0.1165
0.0671
0.0514
Fit 3
0.4498
0.1856
0.1220
0.0888
Fit 4
0.2373
0.1271
0.0798
0.0571
The best fit is the one with the lowest average RMSE.
As the functional expansion order increases, the average
RMSE decreases, but the storage of more coefficients is
necessary. To determine which functional expansion
order is best to use, the change in the average RMSE
between adjacent expansion orders for a fit should be
small. It was decided to choose Fit 4 with N=2 for the
Transactions of the American Nuclear Society, Vol. 110, Reno, Nevada, June 15–19, 2014
238
Computational Methods: General
β(T) data and Fit 2 with N=4 for the α(T) data. The total
storage size of these coefficients is roughly 172 kB and
can be used to sample secondary parameters for any
temperature and any energy. This is a substantial
improvement from the original 24 MB storage per
temperature.
To test the goodness of these coefficients, a simple
Monte Carlo code was written to sample alpha and beta
many times using only the coefficient files. Linear
interpolation is performed to intermediate values in the
energy and beta meshes when necessary. A plot of the
energy (or beta) mesh versus the relative frequency of
each sampled beta (or sampled alpha) reproduces the beta
(or alpha) PDF. Integrating over this produces the CDF.
This is then compared to the true CDF found from the
true S(α,β,T) data at the incoming energy, beta value and
temperature. Figs. 3 and 4 show the relative errors
between the true values and the estimated values found
from the fitting coefficients for the beta CDF and alpha
CDF, respectively.
From Figs. 3 and 4, the largest relative errors occur for
high CDF probability values. This is because the standard
deviation for the number of sampled values that fall inside
the bins with CDF values greater than 0.999 is large,
resulting in large errors. To remedy this, more samples
could be run to obtain more values in these bins, lowering
the standard deviation. However, for the majority of the
CDF region, [0.001, 0.9], the largest relative error is
0.075 for beta and 0.736 for alpha.
CONCLUSIONS
Functional expansions were used to fit the
temperature dependence of the alpha and beta CDFs
constructed from S(α,β,T) data from graphite. For the
β(T) data, second-order functional expansions in T -1/2
were used to sample beta. For the α(T) data, fourth-order
functional expansions in 1/T were used to sample alpha.
Both coefficient sets gave excellent results in the majority
of the CDF range [0.001, 0.9]. The storage of coefficients
is 172 kB and can be used to sample outgoing energy and
angle for any incoming energy and any temperature onthe-fly. This is a large improvement from the current
method which requires 24 MB of storage for each
temperature. These coefficients are more compact than
the current ACE data and can be combined with other onthe-fly methods in the future to complete the modeling of
temperature effects for Monte Carlo codes.
REFERENCES
Fig. 3. Relative error in β from true and estimated CDF
for graphite, T=875K, E=0.00518 eV.
Fig. 4. Relative error in α from true and estimated CDF
for graphite, T=875K, β=1.865.
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Transactions of the American Nuclear Society, Vol. 110, Reno, Nevada, June 15–19, 2014