1. Developing a Mathematical Scheme to combine Single-Parameter Perturbation Results in Radiation Transport
Calculations to Predict Joined Parameter Perturbations
Caroline L. Hughes, Aakanchha, and Sunil Chirayath
Texas A&M University, Department of Nuclear Engineering, 3133 TAMU, College Station, TX, 77843-3133,
sunilsc@tamu.edu
INTRODUCTION
Radiation transport has wide applications in developing
models and predicting parameters of interest in nuclear
reactor core analysis, radiation shielding design, radiation
dose rate estimations, radiation therapy, etc. However, as in
any predictive modeling and analysis, quantifying the
uncertainty in predicted results due to the possible variations
in input parameters is extremely important. For example,
there may be two types of materials used in a model, and
varying the densities of both materials can affect radiation
transport results predicted through modeling. The state-of-
the-art Monte Carlo N-Particle transport code MCNP6 has a
built-in feature “PERT” by which the user can perturb
individual radiation transport parameter input to understand
and to assess the uncertainty in predicted results [1]. In the
case of variations in density of two materials, it is important
to combine the effects of both perturbations. Taylor series
mathematical approximation is the backbone behind the
“PERT” feature in MCNP6. Provision exists in MCNP6 code
to perturb many input parameters simultaneously and obtain
the results of each individual perturbation along with non-
perturbed parameters in one single simulation. Combining
the effects of all of the perturbations is not possible unless the
user changes the parameters entirely and performs a second
independent simulation, which is computationally expensive.
The objective of this work is to develop a mathematical
scheme to combine single-parameter perturbation results
obtained using MCNP6 radiation transport simulations to
predict joined parameter perturbation results.
METHODOLOGY
An attempt is made here to achieve this goal by
analyzing simple radiation transport problems of calculating
dose rates from point and cylindrical radiation sources. Each
source is shielded with two different materials so that the
change in dose rate (uncertainty) due to combined variations
in material density can be studied. The scheme of study is as
follows:
 MCNP6 simulation to calculate dose rate without
perturbing density of materials (D0),
 MCNP6 simulation by manually varying density of
both materials together (D1),
 MCNP6 simulation with “PERT” feature perturbing
density of each material (provides D0, D2—dose rate
due to change in first material density, D3—dose rate
due to change in second material density)
 Prediction of D1 (combined effect of density changes)
based on D0, D2, and D3 using the analytical
mathematical formulation developed as part of the
work presented here (D4). This formulation is based on
Taylor’s approximation [2]
The difference between D0 and D1 is the uncertainty in
dose rate resulting from changes in material density.
Calculating this uncertainty is a computationally expensive
procedure that requires two separate simulations and can only
test one combination of perturbations at a time. Instead, one
MCNP6 simulation with the “PERT” feature (third bulleted
item in the list above) along with the mathematical
formulation developed here (fourth bulleted item above to
predict D4) can determine uncertainty (change between D0
and D1) with significantly less computational effort. The
formulation to calculate D4 will have some limitations, which
are tested and presented here for point and cylindrical source
problems of calculating radiation dose rate (Fig. 1).
The density of each shielding material was perturbed and
the dose rate for the combined perturbation (e.g. when the
densities of lead and aluminum were perturbed by +4% and
+8% respectively) was determined using two different
methods. The perturbed dose rate was calculated using
MCNP6 perturbation results (D0, D2, and D3) and by using
the analytical mathematical formulation (D4). This was
compared to the exact dose rate (D1), which was calculated
using the brute force method of directly changing the material
densities in the MCNP6 model. Once this formula was
deemed accurate for small perturbations, additional
simulations were performed to determine the maximum
possible input parameter perturbations that can be used to
obtain a perturbed dose rate within 5% of the exact value.
Mathematical Formulation Development for Joined
Perturbations
The combination formula for perturbed dose rate was
derived analytically using Taylor expansion approximation
methodology. Results were initially based on first-order
terms and were then extended to include second-order terms
of the Taylor expansion approximation as well. The perturbed
dose rate calculated using both terms is expected to be more
accurate than that found using only the first-order terms.
Specifically, this work focuses on perturbing parameters in

2. the dose rate calculation. The simplest form of the dose rate
equation is as follows:

 enE
ratedose (1)
Where ϕ is the flux or fluence rate in photons/(cm2
∙ sec), E is
photon energy (MeV), μen is the mass energy absorption
coefficient of the medium (cm-1
), and ρ is the density of the
medium (g/cm3
). The result can be converted to units of
Gy/hr. The dose rate is considered to be a functional form of
the densities of the two shielding materials, with natural
densities of ρ1 and ρ2, which will be considered for analysis
of the perturbation effects. The dose rate calculated using the
natural densities will be denoted by D0:
 210 , fD  (2)
Perturbations of Δρ1 and Δρ2 can be applied to ρ1 and ρ2,
respectively, and the independent dose rate for each term can
be calculated using the single-variable Taylor expansion,
where D2 and D3 denote the dose rate calculated for exclusive
perturbations in ρ of the first and second shielding materials:
 
  

111
2
12
1
1
2112 ,
 

fff
fD
(3)
 
  

222
2
22
1
2
2213 ,
 

fff
fD
(4)
D2 and D3 are the values obtained from the MCNP6
perturbation method. These can either include the first term
only or both the first and second terms, depending on the
method specified when the perturbation is defined in the
MCNP6 input. The MCNP6 “PERT” feature was used to
monitor the percent contribution of the second-order terms
and verify that Equations 3 and 4 produce a valid
approximation of the dose rate for D2 and D3. The dose rate
found using the combined perturbations in the densities of the
two materials simultaneously was calculated using the Taylor
expansion in two variables:
  221122111 ,   ffffD 
      222111
2
221
2
12
1
2   fff
(5)
Equations 2-5 can be combined to find D4 in terms of D0,
D2, and D3:
0
32
4
D
DD
D  (6)
This holds true regardless of whether D2 and D3 were
calculated using only the first-order term or both the first- and
second-order Taylor expansion terms. In the case of first-
order D2 and D3, the first-order and mixed second-order term
were considered when deriving D4. In the case of first- and
second-order D2 and D3, all first- and second-order terms
were included. The dose rate was calculated just outside of
the aluminum shell for a variety of scenarios where a
parametric variation in density was applied to both materials.
The natural densities of lead (11.34 g/cm3
) and aluminum
(2.7 g/cm3
) were used in the unperturbed case and were
perturbed using both the “PERT” feature and brute force
method to find the perturbed and exact dose rates.
When MCNP performs perturbations using the Taylor
series expansion, the user has the option of obtaining
perturbed values considering only the first-order term, only
the second-order term, or both the first- and second-order
terms. The “PERT” feature is limited to these terms, so the
code developers suggest monitoring the percent contribution
(a) Point Source (b) Cylindrical Source of 60
Co
Fig. 1. Radiation Source Problems Modeled with MCNP6 to Test the Perturbation Combination Scheme Used to Calculate
Dose Rate

3. of the second-order term to the result calculated with both
terms, since it is possible that second-order term
contributions of about 20-30% can indicate an inaccurate
perturbation result. Additionally, a maximum error tolerance
of 5% was imposed on the percent difference between the
approximated (mathematical formulation of combining
multiple perturbations) and the exact value of dose rate to
ensure that the approximation is valid.
RESULTS AND DISCUSSIONS
Assessing the mathematical formulation developed for
combining multiple perturbations requires comparing the
MCNP6 prediction of D1 with the mathematical analytical
formula prediction value D4. The percent perturbation
applied to the density of each material ranged from 0 to 32%
with a step size of 4% in the case where only the first-order
Taylor expansion term was considered (Fig. 2) and 0 to 60%
with a step size of 4% in the case where the first- and second-
order Taylor expansion terms were considered (Fig. 3). These
ranges were limited by the desired percent error in the
perturbed dose rate of less than 5%. Figs. 2a-b demonstrate
the agreement between D1 and D4, obtained from the
mathematical formulation combining two perturbations
considering only the first-order term. For the point source
case, 16% change in density for each material predicted D4 to
an accuracy of greater than 95% when it was compared with
D1. For the cylindrical source case, a 95% agreement was
obtained between D4 and D1 for a density perturbation of up
to 20%. Figs. 3a-b show how considering both the first- and
second-terms extends the range of possible perturbations for
which D4 agrees with D1. For the point source case, 40%
change in density for each material predicted D4 to an
accuracy of greater than 95% when compared with D1. For
the cylindrical source case, 95% agreement was obtained
(a) Point Source (b) Cylindrical Source of 60
Co
Fig. 2. Variation in Percent Error between MCNP6 Prediction and Mathematical Approximation of Combining Perturbed
Results with First-Order Taylor Expansion Term
(a) Point Source (b) Cylindrical Source of 60
Co
Fig. 3. Variation in Percent Error between MCNP6 Prediction and Mathematical Approximation of Combining Perturbed
Results with First-Order and Second-Order Taylor Expansion Terms

4. between D4 and D1 for a density perturbation up to 48%, more
than doubling the range of allowed perturbations.
CONCLUSION AND FUTURE WORK
The results strongly suggest that the combination scheme
described in Equation 6 provides a reasonable approximation
for the dose rate when two properties are perturbed. Further
work will include incorporating more materials in the
shielding process and using Taylor expansion for three or
more variables. This analysis can be extended to a neutron
source, as well, to ensure that these results hold true for
multiple physical systems.
REFERENCES
1. D. B. PELOWITZ et al., “MCNP6 User’s Manual,
Version 1,” LA-CP-13-00634, Los Alamos National
Laboratory (2013).
2. K. A. STROUD, Engineering Mathematics: Programmes
and Problems, Macmillan, (1970).