Materials Science · Radiation, vacuum and reactive environments

Displacement damage, ionisation and structural change

Follow a recoil from initial displacement to retained defects, then distinguish that structural history from electronic trapping and measured function. Derive a bounded production–recovery model and test why equal displacement estimates need not produce equal surviving states.

Useful background: Durability in real environments; Electrical transport and electronic materials; Heat transport, expansion and thermal response. Each worked case explains the quantities it uses.

Back to the chapter

After reading, you can

Explain primary knock-on atoms, Frenkel pairs, cascade relaxation, atomic mixing and subsequent defect evolution as distinct stages.
Use a stated collision model and displacement estimate without interpreting model dpa as a measured surviving-defect fraction.
Derive and solve a limited production–recombination–sink balance, including rate dependence and post-exposure recovery.
Distinguish atomic-defect annihilation from electronic carrier recombination and choose evidence connecting a retained state to a property.

Radiation can alter different parts of a material

Energetic interactions can displace atoms, producing vacancies and interstitials, or excite and ionise electronic states. A vacancy is an unoccupied lattice site; an interstitial is an atom located between the usual sites. Subsequent recombination, defect clustering and chemical reactions determine the lasting state. The initial interaction is not the complete damage history.

Imagine that a crystal has the same number of atoms before and after an exposure. Its site occupancy can still change: one atom may leave an ordinary lattice site and become a self-interstitial, leaving a vacancy. Together these constitute a Frenkel pair. An interstitial need not sit alone in the middle of a geometrical gap; split configurations such as two atoms sharing the region around a normal site are possible. Vacancy/interstitial counting here refers to a crystalline reference lattice, not a universal description of disorder in every material.

Ionisation changes electronic occupancy, whereas a displacement changes atomic occupancy or arrangement. They can occur in the same event and influence one another. A displaced atom or defect complex may introduce electronic states that capture carriers; an excited carrier can also relax without leaving a displaced atom. Calling a response electronic or structural identifies what is being described, not mutually exclusive classes of incident radiation.

A recoil initiates a cascade, not a finished damage state

The first lattice atom given substantial recoil energy is called a primary knock-on atom, or PKA. It can transfer energy to other atoms, producing secondary recoils. A displacement threshold E_d specifies the kinetic-energy requirement used for creating a lasting displacement in a stated material/model. It is not the electronic ionisation threshold. Real thresholds depend on structure and direction; a scalar average is a modelling choice.

For a nonrelativistic elastic collision of a projectile of mass m and kinetic energy E_in with a stationary target atom of mass M, the largest transferable recoil energy is T_max = [4mM/(m + M)²]E_in. In a head-on collision, momentum and energy conservation give target speed 2m v/(m + M); substituting into Mv_target²/2 gives this expression. Other scattering angles transfer less. It is an energy ceiling, not a probability that a retained Frenkel pair will form. Electronic losses and lattice binding are not included in this two-free-particle collision model.

During a dense cascade, atoms undergo many interacting movements rather than a sequence that ends at the first collision. As the disturbed region relaxes, vacancies and interstitials can recombine, and some defects survive or form clusters. An atom can finish on another regular site: atomic mixing may occur even when the final site occupancy looks perfect. Thus maximum disorder during the cascade, retained point defects and the number of atom replacements are different quantities.

Molecular-dynamics studies follow this short cascade evolution, while later diffusion and reaction require longer-time descriptions. The original schematic compares occupancy and labelled-atom identity; it does not reproduce a cascade trajectory, temperature field or a measured timescale.

Original schematic with a 5 by 3 reference lattice in each of four panels. Every panel contains 15 atoms. A has 15 occupied regular sites, with individual atoms A and B labelled on neighbouring sites. B has 14 occupied regular sites, one vacant site shown by a hollow orange square, and one extra self-interstitial shown by a green dot away from regular sites. C shows recombined occupancy: 15 occupied sites, no vacancy or interstitial. D is an independent exchange example: all 15 regular sites are occupied, but the labelled atoms A and B have exchanged sites, giving two replacements without a retained Frenkel pair. A and B are atom identities, not different elements. These are occupancy examples, not a measured cascade trajectory, crystal geometry or timescale.

Model dpa is a count normalised to atoms, not a survivor assay

An estimated displacement count is often divided by the number of reference atoms to give displacements per atom, dpa. State the convention: NRT-dpa and athermal-recombination-corrected arc-dpa are different estimators. Neither abbreviation means the electrical or mechanical property has been measured. A value above one is possible for an accumulated event count; it does not mean more than 100% of the atoms have each moved exactly once.

For the high-energy branch of the NRT damage function, N_NRT = 0.8 T_d/(2E_d), applicable when T_d exceeds 2E_d/0.8 = 2.5E_d. T_d is damage energy available to atomic displacement after the electronic energy-loss partition, rather than the full incident energy. Near threshold, the model instead has zero- and single-displacement branches; extending the high-energy line down to zero would be wrong.

For multiple recoils, sum the chosen displacement estimates and divide by the reference atom count. This supplies a model exposure coordinate. Additional information is required for the defects surviving cascade relaxation and later evolution. In-cascade recombination can lower the surviving count, while replacement sequences can mix many atoms without leaving the same number of vacancies. A fitted correction or production efficiency is material- and model-dependent, not a universal factor to copy to a new solid.

For example, the 2018 Nordlund study separates primary surviving-defect and replacement estimators using simulations and comparisons with experimental evidence in metals. It motivates keeping these outputs distinct. It does not make arc-dpa a measured long-time defect population, or establish that its fitted parameters apply to arbitrary compounds and temperatures.

Response depends on recovery and environment

Temperature can permit some defects to move or recombine while promoting other changes. Material chemistry, dose rate and interfaces affect outcomes. A displacement-damage metric does not automatically predict electrical or mechanical degradation in every material.

Once produced, mobile vacancies and interstitials may meet and annihilate as a pair. They may instead join clusters, bind impurities or reach sinks such as boundaries, dislocations and surfaces. Loss from a mobile population is not automatically restoration of the whole material: a sink can store defects or change structure, and a cluster can remain as a scattering centre or an obstacle to deformation.

Interfaces are therefore part of the kinetics, rather than a guarantee of resistance. In the copper simulations reported by Bai and colleagues, a boundary could first absorb interstitials and later emit them to remove nearby vacancies. This is a specific simulation-supported mechanism. It does not imply that every boundary is an equally effective, unbiased and unlimited sink, or that the same process controls another material.

Temperature changes mobilities and reaction rates. Faster encounter and sink removal can reduce a free-defect population, but clustering, redistribution and other processes can also change. A lower number of mobile point defects does not by itself establish improved toughness, conductivity or phase stability. Specify the retained population, spatial region and observation time.

Balance creation against encounters and sinks

Consider an original dilute, spatially homogeneous model with a fixed reference-site count. Let c_v and c_i be the numbers of mobile vacancies and mobile self-interstitials per reference lattice site. These are dimensionless concentrations, not percentages of all defect types. Each effective post-cascade creation event supplies one of each at rate G per site per hour. Equilibrium thermal production, clusters, impurity reactions and evolving sink capacity are omitted.

Assume recombination removes one vacancy and one interstitial at rate K c_v c_i. Assume linear removal of mobile defects to fixed sinks at rates s_v c_v and s_i c_i. Then dc_v/dt = G − K c_v c_i − s_v c_v and dc_i/dt = G − K c_v c_i − s_i c_i. K, s_v and s_i have units h⁻¹ with this per-site normalisation. A model written for number per cubic metre would give the encounter coefficient different units. G is a supplied surviving-pair source after the rapid cascade stage; it is not automatically a dpa rate or an absorbed dose rate.

To obtain a solvable comparison, impose equal sink coefficients s_v = s_i = s and equal initial populations. The equations then preserve c_v = c_i = c, giving dc/dt = G − Kc² − sc. This imposed symmetry is not a general assertion that vacancies and interstitials have equal mobilities or that real sinks are unbiased. K describes encounters of the mixed populations; s describes their loss to the assumed sinks.

With constant positive G, K and s, a steady state satisfies Kc_*² + sc_* = G. The nonnegative root is c_* = [√(s² + 4KG) − s]/(2K). A perturbation near it decays because the derivative of the net production rate is −2Kc_* − s < 0. In a sink-dominated limit, Kc_*² is small compared with sc_* and c_* ≈ G/s; in a recombination-dominated limit, sc_* is small compared with Kc_*² and c_* ≈ √(G/K). These are different rate dependences, so a tenfold source increase need not give a tenfold retained concentration.

For c(0) = 0, define Λ = √(s² + 4KG) and a = (Λ + s)/(2K). Separating the factored equation dc/dt = K(c_* − c)(c + a) gives c(t) = c_*[1 − exp(−Λt)]/[1 + (c_*/a)exp(−Λt)]. This finite-time solution distinguishes reaching a steady state from merely assuming one. The kinetic coefficients and source below are invented teaching inputs, not a calibration of any named material.

Equal creation totals can leave unequal populations

Use K = 10⁴ h⁻¹ and s = 1 h⁻¹. A faster history has G = 2 × 10⁻⁴ h⁻¹ for 2 h; a slower history has G = 2 × 10⁻⁵ h⁻¹ for 20 h. Each integrates to Gt_end = 4 × 10⁻⁴ creation events per site, but the source competes with removal throughout delivery. Compare each specimen at its own exposure end, not at one common time after both experiments began.

The finite-time solution gives c_fast ≈ 9.96 × 10⁻⁵ and c_slow ≈ 1.71 × 10⁻⁵. Each number is the mobile vacancy fraction and, by the imposed symmetry, also the mobile interstitial concentration per reference site. They are not total point-defect fractions; adding the two species gives 2c in this model. The graph uses cumulative creation Gt as its first horizontal axis so that equal totals can be compared without disguising the unequal delivery times.

After the source stops, set G = 0 while keeping K and s fixed. Integrating dc/dt = −Kc² − sc from an end value c₀ gives c(Δt) = c₀ exp(−sΔt)/{1 + (Kc₀/s)[1 − exp(−sΔt)]}. Δt is time since each source stopped. A higher initial c has a larger initial fractional loss rate, Kc₀ + s; recovery is not a single exponential when the quadratic term matters.

These predictions offer tests of the assumed balance: inspect rate dependence at matched creation totals and compare post-exposure decay at controlled conditions. Agreement would support this limited description, not prove that a real specimen has only two defect species. Spatial gradients, unequal sink biases, cluster formation or temperature changes can require different equations.

Original symmetric kinetic model. The populations are evaluated at each history’s own end and one hour after that end.
History	G (h⁻¹)	Delivery time (h)	c at end	c one hour later
Faster	2 × 10⁻⁴	2	9.96 × 10⁻⁵	2.25 × 10⁻⁵
Slower	2 × 10⁻⁵	20	1.71 × 10⁻⁵	5.67 × 10⁻⁶

Original symmetric homogeneous kinetic model, not measured data. Mobile vacancy and interstitial concentrations are imposed equal: c_v = c_i = c. K = 10000 per hour and s = 1 per hour. Faster source G = 0.0002 per hour acts for 2 hours; slower G = 0.00002 per hour acts for 20 hours. In the upper panel the horizontal axis is cumulative creation Gt, in units of 0.0001 pairs per reference site, not elapsed time. Both histories end at 0.0004 created pairs per site. The solid blue faster history ends at c = 0.0000996286; the dashed violet slower history at c = 0.0000170820. In the lower panel, each source is off and time is measured since its own stop. After one hour c is 0.0000224886 or 0.00000567171, respectively. Both panels show one species concentration in units of 0.0001 per reference site, not the combined defect count. Curves solve the stated fixed-coefficient model, omitting clusters, spatial gradients and biased or evolving sinks.

Connect a retained state to the measured response

Do not confuse two uses of recombination. Frenkel-pair recombination removes a vacancy and a self-interstitial from the structural defect count. Electron–hole recombination removes excited carriers; it need not remove the atomic defect that provided a recombination centre. An electronic trap can capture and later release a carrier while the underlying structural feature remains. Its charge occupancy and the time window matter.

In semiconductors, radiation-induced levels can affect carrier trapping, recombination and the electrically active charge balance. The number of mobile carriers, their mobility and a minority-carrier lifetime are different observables. In a stated uniform, one-carrier, low-field transport approximation, conductivity is σ = qnμ, with q the carrier charge magnitude, n the mobile-carrier density and μ mobility. A conductivity change alone does not separate n from μ or identify a vacancy concentration.

Build a discriminating comparison rather than assigning every conductivity change to dpa. At the same calculated displacement convention, record spectra, temperature/rate history, specimen chemistry and post-exposure time. Combine a property measurement with structural evidence for the relevant surviving populations and electrical evidence for carrier/trap behaviour. A cluster image and a transient electrical signal need not sample the same defect sizes, states or times; a feature below a method’s detection range is not proved absent.

The strongest supported conclusion identifies both mechanism and limits. A conductivity decrease with unchanged measured carrier density would motivate checking mobility within the transport model and contact errors in the measurement; a reduced minority lifetime would instead point towards carrier recombination pathways under that measurement’s conditions. Neither observation alone measures the total displacement count. The following worked comparison uses supplied n and μ to show exactly what a transport model can infer.

Worked examples

Use a collision ceiling without promising a defect

In the nonrelativistic elastic model, a projectile has E_in = 100 eV. Compare a target of mass M = 10m with one of mass M = m. For a stipulated displacement threshold E_d = 40 eV, is a retained displacement guaranteed in either case?

For M = 10m, T_max/E_in = 4 × 10/(1 + 10)² = 40/121. The maximum transfer is 33.1 eV, below the chosen 40 eV requirement. This one collision cannot supply that displacement threshold under the model.
For equal masses, the maximum-transfer fraction is one. A head-on collision can transfer up to 100 eV; other angles transfer less. Exceeding the threshold is energetically possible.
The ceiling does not specify a cross section, scattering angle, lattice direction or subsequent recombination. It therefore does not guarantee a retained pair even for the equal-mass target.

The heavy target cannot meet the stipulated threshold in this model; the equal-mass target can, but no surviving-defect probability has been supplied.

Separate a displacement estimate from surviving pair counts

A fictional region has 10⁹ reference atoms and 1000 recoils, each with damage energy T_d = 10 keV. Use E_d = 40 eV and the high-energy NRT branch. Then suppose independent modelling supplies a post-cascade pair count equal to 30% of the NRT estimate, before later recovery. Find the two normalised counts and distinguish pairs from individual defects.

Here 2.5E_d = 100 eV, well below the stipulated T_d = 10000 eV. The specified high-energy branch gives N_NRT = 0.8 × 10000/(2 × 40) = 100 estimated displacements per recoil. T_d is supplied damage energy; it is not inferred from an incident energy by ignoring electronic loss.
Summing 1000 identical recoil contributions gives 100000 estimated displacements. Dividing by 10⁹ reference atoms gives 10⁻⁴ NRT-dpa.
The separately stipulated 30% production result gives 30000 retained Frenkel pairs immediately after cascade relaxation. Each pair supplies one vacancy and one interstitial, so c_v = c_i = 30000/10⁹ = 3 × 10⁻⁵ if all are counted in these populations. The combined individual point-defect count is 60000, or 6 × 10⁻⁵ per reference atom.
The 30% is an invented input, not a universal correction or an arc-dpa calculation. Later recombination, clustering and sinks can change these populations. Replaced atoms cannot be counted from these numbers alone.

The model exposure coordinate is 10⁻⁴ NRT-dpa. The supplied post-cascade result is 3 × 10⁻⁵ pairs per reference atom, representing twice as many individual point defects before later evolution.

Calculate production–recovery competition

For the two kinetic histories in the table, calculate the steady concentrations and the finite exposure-end concentrations. Does a tenfold source rate produce tenfold more retained mobile vacancies at these chosen ends?

For the faster source, Λ = √(1 + 4 × 10⁴ × 2 × 10⁻⁴) = 3 h⁻¹. Thus c_* = (3 − 1)/(2 × 10⁴) = 10⁻⁴ and a = 2 × 10⁻⁴.
At t = 2 h, c = 10⁻⁴(1 − exp(−6))/(1 + 0.5 exp(−6)) = 9.96286 × 10⁻⁵. It is close to, but not exactly at, the steady value.
For the slower source, Λ = √1.8 = 1.34164 h⁻¹ and c_* = 1.70820 × 10⁻⁵. At 20 h the finite solution is 1.70820 × 10⁻⁵ to the shown precision.
Both histories supplied 4 × 10⁻⁴ pairs per site in total, yet their end concentrations differ by about 5.83, not ten. Substituting each c into Kc² + sc explains the competition; at the faster steady state recombination and sink removal each account for 10⁻⁴ h⁻¹.

The calculated end values are about 9.96 × 10⁻⁵ and 1.71 × 10⁻⁵, a ratio of 5.83. Equal creation integrals do not fix the retained mobile concentration.

Predict a recovery observation

Starting from those two exposure-end values, find c one hour after each source stops, retaining K = 10⁴ h⁻¹ and s = 1 h⁻¹. Would fitting c = c₀ exp(−sΔt) be exact?

Insert c₀ = 9.96286 × 10⁻⁵ into c(1 h) = c₀ exp(−1)/[1 + 10⁴ c₀(1 − exp(−1))]. This gives 2.24886 × 10⁻⁵.
For c₀ = 1.70820 × 10⁻⁵, the same expression gives 5.67171 × 10⁻⁶. The time origins are each history’s own stop; the first graph panel did not pretend the deliveries lasted equally long.
A pure sink exponential omits Kc² and would give larger remaining populations. It becomes a useful approximation only when Kc is small compared with s throughout the fitted interval.
A measured decay of one mobile population would still not prove that all clusters, trapped charge or altered properties recovered. Those were excluded from this balance.

About 2.25 × 10⁻⁵ and 5.67 × 10⁻⁶ remain after one hour. Quadratic recombination makes the chosen recovery non-exponential.

Explain equal displacement metrics with unequal conductivity

Two irradiated materials receive a similar displacement metric but show different conductivity changes. Is that impossible?

The metric does not specify surviving defect species, charge states, clustering or recovery. Even equal surviving vacancy counts would not require identical carrier trapping, mobility or contacts.
For a bounded illustration, suppose both materials initially have mobile density n₀ and mobility μ₀, with one-carrier bulk conductivity σ₀ = qn₀μ₀. After exposure, independent measurements supply n_A = 0.80n₀ and μ_A = 0.50μ₀; n_B = 0.40n₀ and μ_B = 0.90μ₀. These are chosen inputs, not values derived from dpa.
Then σ_A/σ₀ = 0.80 × 0.50 = 0.40 and σ_B/σ₀ = 0.40 × 0.90 = 0.36. Equal displacement estimates coexist consistently with different conductivity changes. Resistance measurements alone would not identify the two factors separately.
To discriminate explanations, check a suitable carrier/transport interpretation, contacts, defect-state evidence and temperature/time history. A carrier-lifetime change addresses another observable and cannot simply be renamed a vacancy count.

No. A common exposure metric does not guarantee a common property change.

Try the reasoning

Are vacancies and ionised electronic states identical? Could electron–hole recombination occur while the atomic defect responsible for a trap remains?
Show the answer and reasoning
They are distinct. Carrier recombination can occur without removing the atomic defect.
Vacancy/interstitial occupancy describes structure. Electronic occupancy describes carriers and defect charge states. Frenkel recombination annihilates a vacancy–interstitial pair; electronic recombination need not do so.
Can defects anneal during exposure? Explain why a lower free-vacancy concentration after a hotter exposure would not, by itself, prove greater overall resistance.
Show the answer and reasoning
Yes. Mobility can allow recombination or sink loss during exposure, while clustering and other changes may also occur.
A mobile-defect concentration omits defects stored in clusters or interfaces and does not directly measure a property. Identify what was counted, its time/temperature history and the relevant functional response.
Keep E_in = 100 eV and the same elastic model, but use M = 3m and E_d = 40 eV. Find T_max. Does the result establish the final defect yield?
Show the answer and reasoning
T_max = 4 × 3/(1 + 3)² × 100 = 75 eV. The threshold is energetically accessible, but the retained yield is undetermined.
The kinematic maximum alone does not supply scattering probabilities, actual transferred energy, threshold-direction information or later recovery.
A new fictional calculation has 200 recoils, each with supplied T_d = 5 keV, E_d = 50 eV and 10⁸ reference atoms. Compute the high-branch NRT-dpa. If a separate result retains 25% as pairs just after the cascades, find the vacancy and total individual point-defect concentrations.
Show the answer and reasoning
Each recoil contributes 40 NRT displacements; the total is 8000 and the coordinate is 8 × 10⁻⁵ NRT-dpa. The supplied result is 2000 pairs: 2 × 10⁻⁵ vacancies and 4 × 10⁻⁵ combined individual point defects per reference atom.
0.8 × 5000/(2 × 50) = 40 and 5000 > 2.5 × 50. The 25% belongs only to the stipulated result, not to the definition of dpa or a universal survival fraction.
In the symmetric model, let G = 3 × 10⁻⁴ h⁻¹, K = 10⁴ h⁻¹ and s = 2 h⁻¹. Find c_* and check how much production is balanced by recombination and sinks there.
Show the answer and reasoning
c_* = 10⁻⁴. Recombination is Kc_*² = 10⁻⁴ h⁻¹ and sink loss is sc_* = 2 × 10⁻⁴ h⁻¹, adding to G.
√(s² + 4KG) = √16 = 4 h⁻¹, so c_* = (4 − 2)/(2 × 10⁴). This checks the exact balance rather than using either limiting approximation without its condition.
The same pair source feeds initially equal c_v and c_i, but s_i > s_v. At the instant c_v = c_i = c > 0, what is d(c_v − c_i)/dt? May one retain the single c equation?
Show the answer and reasoning
It is (s_i − s_v)c > 0, so the populations immediately separate. The symmetric single equation is no longer valid.
Subtract the coupled balances: equal source and recombination terms cancel, leaving −s_v c_v + s_i c_i. Pair creation alone does not preserve equal mobile populations when sinks act differently.
Two one-carrier specimens have the same post-exposure σ/σ₀ = 0.50. One has n/n₀ = 1.00 and the other 0.625. Find the mobility ratios. Does equal conductivity establish equal traps or defect counts?
Show the answer and reasoning
The mobility ratios are 0.50 and 0.80, respectively. Equal conductivity does not establish equal trapping or structural states.
From σ/σ₀ = (n/n₀)(μ/μ₀), divide 0.50 by each supplied density ratio. Trapping, recombination lifetime, contact response and structural counts require their own evidence under the stated transport regime.

What to carry forward

A recoil, an in-cascade displacement count, surviving point defects and atomic replacements describe different stages or observables.
Quote the dpa convention and damage-energy assumptions; neither dpa nor an assumed survival factor is a measured property.
Creation competes with recombination and sink processes during exposure. Equal creation totals need not leave equal mobile populations.
Electronic recombination and Frenkel-pair annihilation are distinct; connect the retained state to the actual property and observation window.

References

Teaching resources and research supporting this lesson’s explanations and examples.

Appendix Three: Space Radiation Effects on Integrated CircuitsNASA Jet Propulsion LaboratoryHistorical technical guide · selected mechanismsparts.jpl.nasa.govRead source
Primary Radiation Damage FormationR. E. Stoller · Oak Ridge National LaboratoryComprehensive Nuclear Materials, 2012 · archived chapter · PDFPrimary Radiation Damage Formation ORNL publication recorddigital.library.unt.edu
Improving atomic displacement and replacement calculations with physically realistic damage modelsKai Nordlund and colleaguesNature Communications 9, 1084 (2018)nature.comRead source
Surface and Size Effects on the Behaviors of Point Defects in Irradiated Crystalline SolidsAbdurrahman Ozturk, Merve Gencturk and Karim AhmedFrontiers in Materials 8, 684862 (2021)frontiersin.orgRead source
Efficient annealing of radiation damage near grain boundaries via interstitial emissionXian-Ming Bai and colleaguesScience research abstract, 2010Efficient annealing of radiation damage near grain boundaries via interstitial emission DOE research highlight Publisher recordpubmed.ncbi.nlm.nih.gov