Threshold displacement energy
Encyclopedia
The threshold displacement energy 
is the minimum kinetic energy
that an atom in a solid
needs to be permanently
displaced from its lattice site to a
defect
position.
It is also known as "displacement threshold energy" or just "displacement energy".
In a crystal
, a separate threshold displacement
energy exists for each crystallographic
direction. Then one should distinguish between the
minimum
and average 
over all
lattice directions threshold displacement energies.
In amorphous solids it may be possible to define an effective
displacement energy to describe some other average quantity of interest.
Threshold displacement energies in typical solids are
of the order of 10 - 50 eV
.
of materials.
The maximum energy
that an irradiating particle can transfer in a
binary collision
to an atom in a material is given by (including relativistic
effects)
where E is the kinetic energy and m the mass of the incoming irradiating particle and M the mass of the material atom. c is the velocity of light.
If the kinetic energy E is much smaller than the mass
of the irradiating particle, the equation reduces to
In order for a permanent defect to be produced from initially perfect crystal
lattice, the kinetic energy that it receives
must obviously be larger than the formation energy of a Frenkel pair.
However, while the Frenkel pair formation energies in crystals are typically around 5–10 eV, the average threshold displacement energies are much higher, 20–50 eV. The reason for this apparent discrepancy is that the defect formation is a complex multi-body collision process (a small collision cascade
) where the atom that receives a recoil energy can also bounce back, or kick another atom back to its lattice site. Hence, even the minimum threshold displacement energy is usually clearly higher than the Frenkel pair formation energy.
Each crystal direction has in principle its own threshold displacement energy, so for a full description one should know the full threshold displacement surface
for all non-equivalent crystallographic directions
[hkl]. Then
and
where the minimum and average is with respect to all angles in three dimensions.
An additional complication is that the threshold displacement energy for a given direction is not necessarily a step function, but there can be an intermediate
energy region where a defect may or may not be formed depending on the random atom displacements.
The one can define a lower threshold where a defect may be formed
,
and an upper one where it is certainly formed
.
The difference between these two may be surprisingly large, and whether or not this effect is taken into account may have a large effect on the average threshold displacement energy.
.
It is not possible to write down a single analytical equation that would relate e.g. elastic material properties or defect formation energies to the threshold displacement energy. Hence theoretical study of the threshold displacement energy is conventionally carried out using either classical
or quantum mechanical
molecular dynamics
computer simulations. Although an analytical description of the
displacement is not possible, the "sudden approximation" gives fairly good approximations
of the threshold displacement energies at least in covalent materials and low-index crystal
directions
An example molecular dynamics simulation of a threshold displacement event is available in http://www.youtube.com/watch?v=ZWu5Qf8y6iQ. The animation shows how a defect (Frenkel pair
, i.e. an interstitial
and vacancy) is formed in silicon when a lattice atom is given a recoil energy of 20 eV in the 100 direction. The data for the animation was obtained from density functional theory
molecular dynamics
computer simulations.
Such simulations have given significant qualitative insights into the threshold displacement energy, but the quantitative results should be viewed with caution.
The classical interatomic potential
s are usually fit only to equilibrium properties, and hence their predictive capability may be limited. Even in the most studied materials such as Si and Fe, there are variations of more than a factor of two in the predicted threshold displacement energies. The quantum mechanical simulations based on density functional theory
(DFT) are likely to be much more accurate, but very few comparative studies of different DFT methods on this issue have yet been carried out to assess their quantitative reliability.
extensively with electron irradiation
experiments. Electrons with kinetic energies of the order of hundreds of keV
s or a few MeV
s can to a very good approximation be considered to collide with a single lattice atom at a time.
Since the initial energy for electrons coming from a particle accelerator is accurately known, one can thus
at least in principle determine the lower minimum threshold displacement
energy by irradiating a crystal with electrons of increasing energy until defect formation is observed. Using the equations given above one can then translate the electron energy E into the threshold energy T. If the irradiation is carried out on a single crystal in a known crystallographic directions
one can determine also direction-specific thresholds
..
There are several complications in interpreting the experimental results, however. To name a few, in thick samples the electron beam will spread, and hence the measurement on single crystals
does not probe only a single well-defined crystal direction. Impurities may cause the threshold
to appear lower than they would be in pure materials.
at temperatures where defects are mobile and can recombine. At such temperatures,
one should consider
two distinct processes: the creation of the defect by the high-energy
ion (stage A), and subsequent thermal recombination effects (stage B).
The initial stage A. of defect creation, until all excess kinetic
energy has dissipated in the lattice and it is back to its
initial temperature T0, takes < 5 ps. This is the fundamental
("primary damage") threshold displacement energy, and also the one
usually simulated by molecular dynamics
computer simulations.
After this
(stage B), however, close Frenkel pairs may be recombined
by thermal processes. Since low-energy recoils just above the
threshold only produce close Frenkel pairs, recombination
is quite likely.
Hence on experimental time scales and temperatures above the first
(stage I) recombination temperature, what one sees is the combined
effect of stage A and B. Hence the net effect often is that the
threshold energy appears to increase with increasing temperature,
since the Frenkel pairs produced by the lowest-energy recoils
above threshold all recombine, and only defects produced by higher-energy
recoils remain. Since thermal recombination is time-dependent,
any stage B kind of recombination also implies that the
results may have a dependence on the ion irradiation flux.
In a wide range of materials, defect recombination occurs already below
room temperature. E.g. in metals the initial ("stage I") close Frenkel
pair recombination and interstitial migration starts to happen already
around 10-20 K .
Similarly, in Si major recombination of damage happens already
around 100 K during ion irradiation and 4 K during electron irradiation
Even the stage A threshold displacement energy can be expected
to have a temperature dependence, due to effects such as thermal
expansion, temperature dependence of the elastic constants and increased
probability of recombination before the lattice has cooled down back to the
ambient temperature T0.
These effects, are, however, likely to be much weaker than the
stage B thermal recombination effects.
amount of defects
produced by higher energy irradiation using the Kinchin-Pease or NRT
equations
which says that the number of Frenkel pairs produced
for a nuclear deposited energy
of
is
for any nuclear deposited energy above
.
However, this equation should be used with great caution for several
reasons. For instance, it does not account for any thermally activated
recombination of damage, nor the well known fact that in metals
the damage production is for high energies only something like
20% of the Kinchin-Pease prediction.
The threshold displacement energy is also often used in
binary collision approximation
computer codes such as SRIM
to estimate
damage. However, the same caveats as for the Kinchin-Pease equation
also apply for these codes (unless they are extended with a damage
recombination model).
Moreover, neither the Kinchin-Pease equation nor SRIM take in any way
account of ion channeling
, which may in crystalline or
polycrystalline materials reduce the nuclear deposited
energy and thus the damage production dramatically for some
ion-target combinations. For instance, keV ion implantation
into the Si 110 crystal direction leads to massive channeling
and thus reductions in stopping power .
Similarly, light ion like He irradiation of a BCC metal like Fe
leads to massive channeling even in a randomly selected
crystal direction.
is the minimum kinetic energyKinetic energy
The kinetic energy of an object is the energy which it possesses due to its motion.It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes...
that an atom in a solid
Solid
Solid is one of the three classical states of matter . It is characterized by structural rigidity and resistance to changes of shape or volume. Unlike a liquid, a solid object does not flow to take on the shape of its container, nor does it expand to fill the entire volume available to it like a...
needs to be permanently
displaced from its lattice site to a
defect
Crystallographic defect
Crystalline solids exhibit a periodic crystal structure. The positions of atoms or molecules occur on repeating fixed distances, determined by the unit cell parameters. However, the arrangement of atom or molecules in most crystalline materials is not perfect...
position.
It is also known as "displacement threshold energy" or just "displacement energy".
In a crystal
Crystal
A crystal or crystalline solid is a solid material whose constituent atoms, molecules, or ions are arranged in an orderly repeating pattern extending in all three spatial dimensions. The scientific study of crystals and crystal formation is known as crystallography...
, a separate threshold displacement
energy exists for each crystallographic
Crystallography
Crystallography is the experimental science of the arrangement of atoms in solids. The word "crystallography" derives from the Greek words crystallon = cold drop / frozen drop, with its meaning extending to all solids with some degree of transparency, and grapho = write.Before the development of...
direction. Then one should distinguish between the
minimum
and average over alllattice directions threshold displacement energies.
In amorphous solids it may be possible to define an effective
displacement energy to describe some other average quantity of interest.
Threshold displacement energies in typical solids are
of the order of 10 - 50 eV
Electronvolt
In physics, the electron volt is a unit of energy equal to approximately joule . By definition, it is equal to the amount of kinetic energy gained by a single unbound electron when it accelerates through an electric potential difference of one volt...
.
Theory and simulation
The threshold displacement energy is a materials property relevant during high-energy particle radiationParticle radiation
Particle radiation is the radiation of energy by means of fast-moving subatomic particles. Particle radiation is referred to as a particle beam if the particles are all moving in the same direction, similar to a light beam....
of materials.
The maximum energy
that an irradiating particle can transfer in abinary collision
Binary collision approximation
The binary collision approximation signifies a method used in ion irradiation physics to enable efficient computer simulation of the penetration depth and...
to an atom in a material is given by (including relativistic
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
effects)
where E is the kinetic energy and m the mass of the incoming irradiating particle and M the mass of the material atom. c is the velocity of light.
If the kinetic energy E is much smaller than the mass
of the irradiating particle, the equation reduces toIn order for a permanent defect to be produced from initially perfect crystal
Crystal
A crystal or crystalline solid is a solid material whose constituent atoms, molecules, or ions are arranged in an orderly repeating pattern extending in all three spatial dimensions. The scientific study of crystals and crystal formation is known as crystallography...
lattice, the kinetic energy that it receives
must obviously be larger than the formation energy of a Frenkel pair.However, while the Frenkel pair formation energies in crystals are typically around 5–10 eV, the average threshold displacement energies are much higher, 20–50 eV. The reason for this apparent discrepancy is that the defect formation is a complex multi-body collision process (a small collision cascade
Collision cascade
A collision cascade is a set of nearby adjacent energetic collisions of atoms induced by an energetic particle in a solid or liquid....
) where the atom that receives a recoil energy can also bounce back, or kick another atom back to its lattice site. Hence, even the minimum threshold displacement energy is usually clearly higher than the Frenkel pair formation energy.
Each crystal direction has in principle its own threshold displacement energy, so for a full description one should know the full threshold displacement surface
for all non-equivalent crystallographic directions
Miller index
Miller indices form a notation system in crystallography for planes and directions in crystal lattices.In particular, a family of lattice planes is determined by three integers h, k, and ℓ, the Miller indices. They are written , and each index denotes a plane orthogonal to a direction in the...
[hkl]. Then
and
where the minimum and average is with respect to all angles in three dimensions.
An additional complication is that the threshold displacement energy for a given direction is not necessarily a step function, but there can be an intermediate
energy region where a defect may or may not be formed depending on the random atom displacements.
The one can define a lower threshold where a defect may be formed
,and an upper one where it is certainly formed
.
The difference between these two may be surprisingly large, and whether or not this effect is taken into account may have a large effect on the average threshold displacement energy.
.
It is not possible to write down a single analytical equation that would relate e.g. elastic material properties or defect formation energies to the threshold displacement energy. Hence theoretical study of the threshold displacement energy is conventionally carried out using either classical
or quantum mechanical
molecular dynamics
Molecular dynamics
Molecular dynamics is a computer simulation of physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a period of time, giving a view of the motion of the atoms...
computer simulations. Although an analytical description of the
displacement is not possible, the "sudden approximation" gives fairly good approximations
of the threshold displacement energies at least in covalent materials and low-index crystal
directions
An example molecular dynamics simulation of a threshold displacement event is available in http://www.youtube.com/watch?v=ZWu5Qf8y6iQ. The animation shows how a defect (Frenkel pair
Frenkel defect
The Frenkel Defect is shown by ionic solids. The smaller ion is displaced from its lattice position to an interstitial site. It creates a vacancy defect at its original site and an interstitial defect at its new location.-Definition:...
, i.e. an interstitial
Interstitial defect
Interstitials are a variety of crystallographic defects, i.e. atoms which occupy a site in the crystal structure at which there is usually not an atom, or two or more atoms sharing one or more lattice sites such that the number of atoms is larger than the number of lattice sites.They are generally...
and vacancy) is formed in silicon when a lattice atom is given a recoil energy of 20 eV in the 100 direction. The data for the animation was obtained from density functional theory
Density functional theory
Density functional theory is a quantum mechanical modelling method used in physics and chemistry to investigate the electronic structure of many-body systems, in particular atoms, molecules, and the condensed phases. With this theory, the properties of a many-electron system can be determined by...
molecular dynamics
Molecular dynamics
Molecular dynamics is a computer simulation of physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a period of time, giving a view of the motion of the atoms...
computer simulations.
Such simulations have given significant qualitative insights into the threshold displacement energy, but the quantitative results should be viewed with caution.
The classical interatomic potential
Molecular dynamics
Molecular dynamics is a computer simulation of physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a period of time, giving a view of the motion of the atoms...
s are usually fit only to equilibrium properties, and hence their predictive capability may be limited. Even in the most studied materials such as Si and Fe, there are variations of more than a factor of two in the predicted threshold displacement energies. The quantum mechanical simulations based on density functional theory
Density functional theory
Density functional theory is a quantum mechanical modelling method used in physics and chemistry to investigate the electronic structure of many-body systems, in particular atoms, molecules, and the condensed phases. With this theory, the properties of a many-electron system can be determined by...
(DFT) are likely to be much more accurate, but very few comparative studies of different DFT methods on this issue have yet been carried out to assess their quantitative reliability.
Experimental studies
The threshold displacement energies have been studiedextensively with electron irradiation
Particle radiation
Particle radiation is the radiation of energy by means of fast-moving subatomic particles. Particle radiation is referred to as a particle beam if the particles are all moving in the same direction, similar to a light beam....
experiments. Electrons with kinetic energies of the order of hundreds of keV
Kev
Kev can refer to:*Kev Hawkins, a fictional character.*Kevin, a given name occasionally shortened to "Kev".*Kiloelectronvolt, a unit of energy who symbol is "KeV".* Krefelder Eislauf-VereinKEV can refer to:...
s or a few MeV
MEV
MeV and meV are multiples and submultiples of the electron volt unit referring to 1,000,000 eV and 0.001 eV, respectively.Mev or MEV may refer to:In entertainment:* Musica Elettronica Viva, an Italian musical group...
s can to a very good approximation be considered to collide with a single lattice atom at a time.
Since the initial energy for electrons coming from a particle accelerator is accurately known, one can thus
at least in principle determine the lower minimum threshold displacement
energy by irradiating a crystal with electrons of increasing energy until defect formation is observed. Using the equations given above one can then translate the electron energy E into the threshold energy T. If the irradiation is carried out on a single crystal in a known crystallographic directions
Miller index
Miller indices form a notation system in crystallography for planes and directions in crystal lattices.In particular, a family of lattice planes is determined by three integers h, k, and ℓ, the Miller indices. They are written , and each index denotes a plane orthogonal to a direction in the...
one can determine also direction-specific thresholds
..There are several complications in interpreting the experimental results, however. To name a few, in thick samples the electron beam will spread, and hence the measurement on single crystals
does not probe only a single well-defined crystal direction. Impurities may cause the threshold
to appear lower than they would be in pure materials.
Temperature dependence
Particular care has to be taken when interpreting threshold displacement energiesat temperatures where defects are mobile and can recombine. At such temperatures,
one should consider
two distinct processes: the creation of the defect by the high-energy
ion (stage A), and subsequent thermal recombination effects (stage B).
The initial stage A. of defect creation, until all excess kinetic
energy has dissipated in the lattice and it is back to its
initial temperature T0, takes < 5 ps. This is the fundamental
("primary damage") threshold displacement energy, and also the one
usually simulated by molecular dynamics
Molecular dynamics
Molecular dynamics is a computer simulation of physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a period of time, giving a view of the motion of the atoms...
computer simulations.
After this
(stage B), however, close Frenkel pairs may be recombined
by thermal processes. Since low-energy recoils just above the
threshold only produce close Frenkel pairs, recombination
is quite likely.
Hence on experimental time scales and temperatures above the first
(stage I) recombination temperature, what one sees is the combined
effect of stage A and B. Hence the net effect often is that the
threshold energy appears to increase with increasing temperature,
since the Frenkel pairs produced by the lowest-energy recoils
above threshold all recombine, and only defects produced by higher-energy
recoils remain. Since thermal recombination is time-dependent,
any stage B kind of recombination also implies that the
results may have a dependence on the ion irradiation flux.
In a wide range of materials, defect recombination occurs already below
room temperature. E.g. in metals the initial ("stage I") close Frenkel
pair recombination and interstitial migration starts to happen already
around 10-20 K .
Similarly, in Si major recombination of damage happens already
around 100 K during ion irradiation and 4 K during electron irradiation
Even the stage A threshold displacement energy can be expected
to have a temperature dependence, due to effects such as thermal
expansion, temperature dependence of the elastic constants and increased
probability of recombination before the lattice has cooled down back to the
ambient temperature T0.
These effects, are, however, likely to be much weaker than the
stage B thermal recombination effects.
Relation to higher-energy damage production
The threshold displacement energy is often used to estimate the totalamount of defects
Crystallographic defect
Crystalline solids exhibit a periodic crystal structure. The positions of atoms or molecules occur on repeating fixed distances, determined by the unit cell parameters. However, the arrangement of atom or molecules in most crystalline materials is not perfect...
produced by higher energy irradiation using the Kinchin-Pease or NRT
equations
which says that the number of Frenkel pairs produced
for a nuclear deposited energy
Stopping power (particle radiation)
In passing through matter, fast charged particles ionize the atoms or molecules which they encounter. Thus, the fast particles gradually lose energy in many small steps. Stopping power is defined as the average energy loss of the particle per unit path length, measured for example in MeV/cm...
of
isfor any nuclear deposited energy above
.However, this equation should be used with great caution for several
reasons. For instance, it does not account for any thermally activated
recombination of damage, nor the well known fact that in metals
the damage production is for high energies only something like
20% of the Kinchin-Pease prediction.
The threshold displacement energy is also often used in
binary collision approximation
Binary collision approximation
The binary collision approximation signifies a method used in ion irradiation physics to enable efficient computer simulation of the penetration depth and...
computer codes such as SRIM
Stopping and Range of Ions in Matter
Stopping and Range of Ions in Matter is a group of computer programs which calculate interaction of ions with matter; the core of SRIM is a program Transport of ions in matter . The programs were developed by James F. Ziegler and Jochen P. Biersack around 1983 and are being continuously upgraded...
to estimate
damage. However, the same caveats as for the Kinchin-Pease equation
also apply for these codes (unless they are extended with a damage
recombination model).
Moreover, neither the Kinchin-Pease equation nor SRIM take in any way
account of ion channeling
Ion implantation
Ion implantation is a materials engineering process by which ions of a material are accelerated in an electrical field and impacted into another solid. This process is used to change the physical, chemical, or electrical properties of the solid...
, which may in crystalline or
polycrystalline materials reduce the nuclear deposited
energy and thus the damage production dramatically for some
ion-target combinations. For instance, keV ion implantation
into the Si 110 crystal direction leads to massive channeling
and thus reductions in stopping power .
Similarly, light ion like He irradiation of a BCC metal like Fe
leads to massive channeling even in a randomly selected
crystal direction.
See also
- Threshold energyThreshold energyIn particle physics, the threshold energy for production of a particle is the minimum kinetic energy a pair of traveling particles must have when they collide. The threshold energy is always greater than or equal to the rest energy of the desired particle...
- Stopping power (particle radiation)Stopping power (particle radiation)In passing through matter, fast charged particles ionize the atoms or molecules which they encounter. Thus, the fast particles gradually lose energy in many small steps. Stopping power is defined as the average energy loss of the particle per unit path length, measured for example in MeV/cm...
- Crystallographic defectCrystallographic defectCrystalline solids exhibit a periodic crystal structure. The positions of atoms or molecules occur on repeating fixed distances, determined by the unit cell parameters. However, the arrangement of atom or molecules in most crystalline materials is not perfect...