Radiation Damage Effects in Materials:
Damage Formation Models
A project for MANE 6960, Fall 2013
Brian Linn
The changes in material behavior caused by radiation damage is a great concern for determining the
adequacy and lifetime of materials used in nuclear applications. For this reason, developing tools which
reliably predict the affect of radiation damage on materials is crucial. For this reason, significant effort
has been put into the development of multi-scale models of material response under irradiation over
the last few years.
Introduction
The purpose of this project is to investigate the effects of primary radiation damage formation in
materials. We begin by reviewing the underlying mechanisms and process at work when a material is
exposed to radiation which lead to changes in material behavior. A brief review of recent advances
toward developing multi-scale models which attempt to predict material response to radiation is also
provided. For this project, two models were developed which illustrate the application of atomic-level
simulations with a view toward multi-scale modeling.
The simulations were carried out using the Monte Carlo transport code MCNPX and the classical
molecular dynamics code LAMMPS. As significant effort went into developing original input files for
each code, special attention is given to considerations which went into their development and
execution. The simulations presented in this work describe the irradiation of iron with monoenergetic
14.1 MeV neutrons. The simulated scenario is representative of the damage to iron in the presence of
D-T fusion, the results of which are well-known and allowed for the validation of our results. This
scenario is representative of iron utilized in applications utilizing D-T fusion, the results of which are
well-documented and were used for validation of our results.
Damage Theory
Displacement cascades can be visualized as a series of elastic collisions initiated when a high-energy
particle strikes an atom. The initially-struck atom is referred to as the primary knock-on atom (PKA).
Once displaced from its original lattice site, an atom may come to rest within the interstices of the
lattice. This results in the formation of an interstitial and is considered a point defect within the lattice.
A complementary point defect is formed if the original lattice site remains vacant, called a vacancy. In
this manner, PKAs lead to the creation of equal numbers of vacancies and interstitial. An interstitial and
vacancy which remain stable are known as a Frenkel Pair.
Cascade theories are based on the assumption that an atom in a lattice struck by a PKA must receive a
minimum amount of energy in the collision in order to be displaced from its lattice site. This minimum
energy, denoted Ed, is known as displacement threshold displacement energy. For collisions in which
the lattice atom receives energy less than Ed, the struck atom undergoes large amplitude vibrations
within its potential well but remains in its lattice site. Once initiated by the PKA atom, collisions continue
occurring until energy in excess of Ed is dissipated, typically after about 10ps. Later, we will present our
own MD simulation that illustrates the timescale over which displacement cascades occur.
The Norgett–Robinson–Torrens (NRT) model was developed as a means of predicting the number of
displacements (Nd) created by a PKA atom carrying potential energy Epka. The NRT model is governed
by the following equation:
Where Ed is the effective threshold displacement energy and \beta is known as the scattering correction
factor. The damage energy denoted (Td) in this expression beings about an important distinction. That
is, not all of a PKA atom's kinetic energy goes into a displacement-forming collision. By utilizing Ed, the
NRT model attempts to account for energy lost to electronic stopping and nuclear stopping.
Electronic stopping is the slowing down of the PKA atom via interaction with the material's electrons.
Energy losses to electronic stopping become significant for high energy PKAs, but do not create
displacements. Nuclear stopping refers to energy loss to between the a PKA atom during the elastic
collision with lattice atom and is essentially a repulsive columbic force. The appropriate value of Ed
used with the NRT equation is somewhat interpretable because the minimum energy required is
directionally specific, that is, less energy is required for displacement if applied in a principle
crystallographic direction.
I. MCNP Model: Determination of Nuclear Parameters and DPA
The generalized Monte Carlo transport code MCNPX was used to model the interaction of neutrons with
iron. In comparison to other neutron transport methods, Monte Carlo methods provide increased
accuracy in energy-physics calculations for problems involving complex geometries, but at higher
computational cost. MCNPX is specifically designed for computing accurate neutron-physics. Tracking
particles through a specified problem geometry, MCNPX utilizes continuous energy nuclear cross
section libraries to evaluate the likelihood of interaction at each point.
In this report, we use MCNPX to characterize interactions between iron and neutron emitted from the
defined source, or stemming from reactions thereof. Specific parameters calculated include the neutron
flux, fluence, and energy deposition within the target material. In a full multi-scale simulation, such
parameters would necessarily be included in subsequent Modeling, including the Molecular Dynamics
model. It is worthwhile to note that MCNPX can easily tabulate additional parameters which would be
useful in a more rigorous multi-scale model of radiation damage. Such parameters include nuclear
heating, internal gas production (He, H, etc.), photon production as well as many reaction rates of
interest.
Description of the Model
We have investigated the nuclear parameters which describe the response of iron subjected to the D-T
neutron spectra. The geometry modeled in MCNP consisted of a 1 cm x 2 cm x 2 cm slab of natural iron
(5.9% Fe-54, 91.72% Fe-56, 2.1% Fe-57, 0.28%Fe-58) surrounded by air. Our model described a
monodirectional source of 14.1 MeV neutrons incident upon the iron slab. Monodirectional source
neutrons were emitted from a 1.5 cm x 1.5 cm square surface source, placed 1.5 cm from the iron slab.
A sketch of the model is shown in Figure 1.
Results: Nuclear Parameters
Although the source defined in the model was monoenergetic, the production of neutrons through
reactions in the material is possible. In our problem, about five percent of the source neutrons modeled
resulted in underwent an (n,xn) reaction. As the neutrons produced by this reaction will affect our flux
and energy deposition tallies, and thus the calculated DPA, it is important to have an understanding of
the entire neutron spectra seen by the problem. Plots of the energy- dependent neutron flux and IRDF2002 displacement cross section for iron are given below.
The average energy deposited in the iron slab was determined using the F6 in MCNPX. From the output
file, we obtained found that the average energy-depositing reaction deposits 1.28367E-01 MeV/g .
From the modeled dimmensions of the iron slab and the density of iron, we calculated the average
reaction deposited 404 keV of energy. Note that this includes all energy depositing reactions, not just
those due to our source neutrons. Reference (Stroller slides) reports an average PKA energy of 487 keV
due to the irradiation of iron with 14.1 MeV neutrons.
The tally reported value includes energies of reactions other than PKA and does not omit energy
depositions less than the minimum displacement energy. Considering this, our result being slightly
lower than those given by Reference(Stroller slide) seems reasonable and would suggests that our
MCNPX model is fairly accurate. We also wish to note that MCNPX is capable of more sophiticated
tallies which could be used to discriminate energy deposition in future work.
Results: Calculated DPA
MCNPX was also used to calculate the dpa within the iron which we will use to qualitatively discuss the
long term radiation-induced effects. The dpa was calculated using the IRDF-2002 damage cross sections,
which are not a part of the MCNPX default cross section libraries. The IRDF-2002 cross sections are
based were developed using the NRT model described earlier, assuming values of the effective
displacement threshold energy and scattering correction factor in Iron as 40 eV and 0.8, respectively
(IRDF 2002).
The dpa was calculated in MCNPX according to the equation
The flux per volume phi was tallied by utilizing the F4 tally in MCNPX, with default units of cm^-2 s^-1
MeV^-1 cm^-3. The following FM card was applied to the this tally: Fm4 (1 2 444). The second entries
on this card tell MCNP to multiply the energy-dependent flux by the atom density (b^-1 cm^-3) and the
damage cross section for material number 2 (iron). Using this tally, MCNPX calculated the a rate of
displacements per atom of 2.28379E-01 DPA s^-1. This result agrees well with the results presented in
Reference (Mascitti).
Displacement cross sections are useful in that they may be implemented in a number of codes to give a
rough estimate of the damage produced by radiation. Traditionally, displacement cross secions in
standard cross section libraries, including IRDF-2002 utilized in this report, have been based on the
Norgett–Robinson–Torrens (NRT) displacement model.
However, it is widely recognized that the simplified approach of the NRT model results in displacements
which do not agree well with experimental data. The cross sections derived from the NRT model are
severely limited in that they cannot be applied to compound materials, lack values of
covariance/uncertainty and do not account for the recombination of defects which occurs during the
cascade evolution (Konobeyev 2011).
In recent years, advanced models utilizing MD simulations coupled with the binary collision
approximation (BCA) have been used to develop cross sections which predict displacements in better
agreement with experimental values. The DXS cross section library was developed using, in part, these
advanced models methods. Continuing efforts are underway for the development of damage cross
sections which account for primary radiation defects (IAEA TM OCT 2012).
II. LAMMPS SIMULATION OF DAMAGE CASCADE
Overview
Molecular dynamics (MD) simulations were performed to investigate damage cascades of resulting from
single neutron collisions. The collisions were analyzed as primary knock-on-atom (PKA) events , where
the energy deposited was representative of the energy profile determined by MCNP.
LAMMPS was used to simulate the displacement cascade in iron for an ~10keV collision. To represent
the lattice of iron atoms, we defined a cube which spanned 25 Fe unit cells in each of the x, y and z,
directions, thus our model contained 250000 total iron atoms. The lattice is displayed below using the
VMD viewing tool. The cube was used assumed to be a larger body by using periodic by specifying
periodic boundary conditions on all sides. While displacement cascades produce localized heating, the
length scale that they in occur typically does not affect the bulk temperature of the material overall.
Our LAMMPS model defined a 2-cell thick layer on all sides except the top in order to simulate a known
bulk temperature, yet allow for the lattice temperature in the immediate vicinity of the cascade to
fluctuate. The top layer was not included in the thermostat boundary layer since we intend to initiate
our PKA event her later, and want to avoid any effects on the energy transfer to the lattice. A Finnis–
Sinclair potential was used to describe the interatomic forces within the Fe lattice.
The typical method of simulating a PKA event using MD is to assign an instantaneous velocity and
direction to a single atom in the matrix. In order to minimize the volume modeled and minize run time,
we initiated the PKA event at an atom along the top edge and gave it a downward velocity
commensurate of a 10 keV reaction. The velocity assigned in LAMMPS was determined by using the
classical definition of kinetic energy. We choose an arbitrary downward direction, but a review the
literature reveals that consideration should of the direction be given to minimize the possibility of
channeling in the cascade response. Channeling occurs when an the PKA atom travels an excessive
distance before colliding. Such an occurrence could jeopardize the assumption of periodic boundary,
which require that the cascade does not interact with the outermost layer.
Execution in LAMMPS
The actual computations carried out by MD are done in phases. The first phase of the MD simulation
allowed our model to achieve thermal equilibrium. We initialized the velocities of all of the atoms based
on the assumed temperature, 1200 K . The system was allowed to equilibrate for 1000 steps, each step
defined as lasting 1fs. The velocities and positions of all were scaled at every step of this phase, allowing
the system to equilibrate. This step took about 14 hours to complete using a computer with 3.1 GHz
Quad core processer and 12 GB of RAM. It should be noted that at these speeds, it becomes essential
that a user specify to allow the simulation to be initiated from various points.
In the second phase of our simulation, the interior region was not scaled to the bulk temperature as it
was in 1. In this step, we allowed LAMMPS to let the internal temperature fluctuate while maintaining
the boundary layer at the bulk temperature. Phase two was run for 1000 1fs steps.
The third phase applied the initial velocity to our PKA atom. This phase should be run at the finest
timestep resolution since the highest energy reactions will take place. An intermediate fourth timestep
was defined where we backed off from the extremely refined timestep as the cascade slows down. The
final phase allowed the system to equilibrate for 10 ps, ensuring that the cascade would come to an end.
Any defects which remain after this phase can be considered long-term and can eventually alter the
material properties. These are the defects that would need to be propagated into the next phase of a
multi-scale simulation of radiation effects.
Analysis with MD Simulations
As suggested by our earlier discussion of cross section libraries , the limitations of the standard NRT
model have become increasingly apparent with the maturity of MD modeling of primary radiation
damage. The widespread use of MD simulations in the modeling of defect production have led to the
development of correction factors for NRT predictions. The OECD Nuclear Energy Agency expert group
on primary radiation damage formation, whose main objectives are stated as: (1) to assess the
limitations of the NRT-dpa standard and (2) to revisit the NRT-dpa standard and examine the possibility
of proposing a new improved standard of primary damage characteristics, recently adopted the
following form of the NRT model
MD Simulation can be used to obtain Ed, b and c in the above . Alternatively, the parameters can be
determined experimentally. It is important to note that this fit was developed using MD simulations
which neglected electronic stopping. Thus, Ed in the above equation is the damage energy for recoils
not affected by electronic stopping. In reality, a fraction of the PKA energy is dissipated by electronic
stopping. Using this fit with experimental data requires a correction of the form
This correction factor is used to obtain a quantity the group has coined 'athermal recombinationcorrected dpa' (arc-dpa).
Future research relating to MD simulations will likely refine MD results for specific applications.
Examples include the development of new interatomic potentials, the application of MD models to
explore the effect of magnetism on defect formation, and investigations of the importance of rare
events in the prediction of radiation damage incurred over long times at higher doses.
Appendix 1: MCNP Input Deck (See Attached)
Appendix 2: LAMMPS Input Deck (See Attached)