Multi-step damage accumulation in irradiated crystals

Appl Phys A (2009) 97: 147–155DOI 10.1007/s00339-009-5294-z

Multi-step damage accumulation in irradiated crystals

Jacek Jagielski · Lionel Thomé

Received: 8 January 2009 / Accepted: 10 June 2009 / Published online: 4 July 2009© Springer-Verlag 2009

Abstract This article presents a model of damage accumu-lation in irradiated crystals. This model is based on the as-sumption that the damage accumulation occurs through aseries of structural transformations triggered by the destabi-lization of the current structure of crystals. Formal equationsdescribing the damage accumulation build-up and experi-mental assessment of the model are presented and discussedin the framework of the actual knowledge of radiation ef-fects in oxide crystals (yttria-stabilized zirconia (YSZ) andmagnesium-aluminate spinel (MAS)), silicon carbide crys-tals and zirconia implanted nickel crystals.

PACS 61.80.Az · 61.72.Cc · 61.43.-j

1 Introduction

The evaluation of the damage generated in crystalline solidslocated in radiative media is a major challenge in many tech-nological domains connected to electronic, space and nu-clear industries. Ion beams delivered by various types of low,medium and high-energy accelerators provide very efficient

J. Jagielski (�)Institute for Electronic Materials Technology, Wolczynska 133,01-919 Warsaw, Polande-mail: [email protected]

J. JagielskiThe Andrzej Soltan Institute for Nuclear Studies,05-400 Swierk/Otwock, Poland

L. ThoméCentre de Spectrométrie Nucléaire et de Spectrométrie de Masse,CNRS/IN2P3, Université Paris-Sud, Bât. 108, 91405 Orsay,France

tools for the simulation of the different types of interactions(nuclear and electronic) involved during the slowing-downof energetic particles. A rather broad panoply of experimen-tal techniques can then be implemented to monitor the dam-age build-up and to characterize the nature of the defectscreated. Most widely used for that purpose are Rutherfordbackscattering and channelling (RBS/C), transmission elec-tron microscopy (TEM), X-ray diffraction in grazing inci-dence (GXRD), Slow Positron Implantation Spectroscopy(SPIS) and optical measurements.

A huge number of articles published during the last threedecades rely on the determination, using one of the above-mentioned techniques, of the disorder build-up (i.e. the vari-ation of the damage as a function of the ion fluence orof an equivalent dynamic parameter) in ion-irradiated ma-terials (see examples in [1–15]). A large variety of kinet-ics (going from an exponential rise to more or less abruptsigmoids or even more complicated behaviours) was re-ported. The shape of the observed build-up is mostly inde-pendent of the nature of the damage created at saturation(point or extended defects, dislocation loops, polygoniza-tion, amorphization, . . . ).

Phenomenological descriptions were most often based onthe consideration that the overall damage results from theoverlapping of a given number of ion impacts in the ana-lyzed volume [16], leading to theoretical damage kineticsoften rather far from experimental ones. The main weak-ness of these models relies in the fact that they assumethat the mechanisms controlling the damage accumulationprocess start from the beginning of the irradiation process.In other words all current models are based on the assump-tion that damage accumulation occurs via continuous single-step processes. Another general conjecture of these modelsis to consider only processes occurring in the spatial scaleof collision cascades. This implies that the damage build-up

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148 J. Jagielski, L. Thomé

should be independent of the macroscopic properties of ma-terials. However, it was frequently observed that damage ac-cumulation strongly depends on global characteristics, suchas the sample thickness or the grain size. This feature is illus-trated by the different results often obtained in in-situ TEMexperiments performed on thin samples and in RBS/C stud-ies carried out on massive specimens of the same material.

The aim of the present paper is to revisit the research fieldby proposing a new model of damage accumulation, whichis based on the hypothesis that the damage build-up resultsfrom multiple processes of atomic reorganization occurringin successive steps. This multi-step damage accumulationmodel (referred in the following to as MSDA1) is then usedto account for experimental data recorded in ceramic crys-tals irradiated with ions in both nuclear (at low energy) andelectronic (at high energy) stopping power regimes.

2 Damage accumulation models

2.1 Current models

A quite large number of phenomenological models were de-veloped in the past to account for the damage accumula-tion data recorded in irradiated solids. The first and moregeneral description of the damage build-up was provided byGibbons, which assumed that the variation of the amount ofdamage, fD, accumulated during irradiation as a function ofthe ion fluence, Φ , follows the general equation [16]

fD = fD(∞)

[1 −

n−1∑k=0

(σΦ)k

k! exp(−σΦ)

](1)

where fD(∞) is the value of fD measured at saturation, i.e.for very large ion fluences (for instance fD = 1 in the casewhere irradiation leads to amorphization of the irradiatedlayer), σ is the disordering cross-section, and n is the num-ber of ion impacts necessary to create permanent damage inthe material. Actually the Gibbons description mainly relieson the value of the parameter n. The two extreme cases areschematized in Fig. 1. In the direct ion impact (DI) descrip-tion (n = 1), each incident ion creates permanent disorderin a given volume of the target, and the damage build-upis controlled by the probability that further ions hit an un-perturbed volume of the crystal. In the defect accumulation(DA) process (n � 1), the damage results from the accu-mulation of a large number of ion impacts in the same vol-ume of the target, according to a percolation-like mecha-nism [16].

The results of experiments performed in a low-energyrange (up to hundreds of keV) clearly showed that the dam-age accumulation process generally occurs in several steps

1A brief description of the model has been given in [17].

Fig. 1 Selected models of damage accumulation

[15, 18]. This observation led Hecking et al. to propose amodel based on the assumption that sub-threshold cascadesmay lead to the growth of amorphous regions. The Heck-ing model also takes into account the effects related to anincrease of the irradiation temperature.

Later on, more sophisticated models were established [7]to account for more complex experimental damage kineticsthan those predicted by (1). They are generally based on acombination of DI and DA descriptions, with the possibilityof considering additional processes such as cascade overlap(CO), interface-controlled (IC) and defect-stimulated (DS)mechanisms (the most representative models are shown inFig. 1). Nevertheless, all these models suffer the same red-hibitory drawback: they predict that the amount of accumu-lated damage versus the irradiation fluence exhibits rathersimple shape and cannot therefore take into account build-upinvolving more than one step (as it is the case of, e.g., zirco-nia, spinel and GaN crystals irradiated with low-energy ions[9, 11, 18]). Furthermore, the most advanced Hecking andDI/DS models assume that the damage build-up is propor-tional to the ion fluence, what means that all mechanismsmust operate from the very beginning till the very end ofthe irradiation process. Consequently, these models are un-able to account (for instance) for the case of amorphizationdriven by chemical effects where the concentration of im-planted atoms should be high enough to stabilize the amor-phous phase at high fluences.

2.2 MSDA model

The MSDA model firstly relies on the consideration thatthe damage build-up results from a series of successiveatomic reorganizations which are triggered by microscopicor macroscopic solicitations. Each new step occurs whenthe current structure of the irradiated material is destabi-lized by accumulated damage, i.e. becomes less energeti-cally favorable than other possible atomic configurations.

Multi-step damage accumulation in irradiated crystals 149

It is worth noting that the energy of damaged structures isalways higher than that of a perfect crystal. Nevertheless,among various available defect configurations, one of themis characterized by a free energy which is lower than that ofany other. Thus we assume that each ion impact leads to thetransformation of a given volume of the crystal into a newatomic configuration which, despite of being metastable, hasthe lowest free energy in the current conditions. The con-cept of amorphization driven by free-energy decrease wasalready invoked to discuss experimental results related tothe crystal-to-amorphous transition [3, 19]. In the presentwork this idea is simply extended to all intermediate struc-tural transformations occurring during ion irradiation, notonly those leading directly to amorphization. It is also worthnoting that a destabilized structure fulfills the conditions re-quired for the “nucleation site” often mentioned in articlesdescribing damage accumulation processes [8, 12]. Actuallya “nucleation site” has the physical meaning of a volume ofthe material ready to transform into a new structural arrange-ment. Such a process is controlled by the probability thateach ion hits an unmodified volume of the material, so that itcan be described in the framework of a direct-impact mech-anism. It is likely that the volume transformed by a singleion impact is comparable to that of a collision cascade forlow-energy ions or to that of a damaged track for swift ions.However, short-range migration of defects and atoms mayalso influence significantly the transformed volume.

According to the above hypotheses, the equations de-scribing the sequences of damage accumulation in the mostcomplex cases, i.e. when multi-step processes occur, can bewritten in the following way. The first step of damage accu-mulation describes the transformation from a perfectly crys-talline to a partially damaged structure, via a direct-impactmechanism, and can therefore be accounted for by the fol-lowing equation:

fd,1 = f satd,1

[1 − exp(−σ1Φ)

]for 0 < Φ < Φ1 (2a)

where f satd,1 is the level of damage at saturation in the first

step, σ1 is cross-section for damage formation and Φ is theirradiation fluence.

Damage induced by incoming ions generally leads to fur-ther destabilization of the crystal structure, which leads toa next step of structural transformation. The only differencewith the former transformation is that the parameter describ-ing the amount of accumulated damage (fD) varies fromf sat

D,1 to f satD,2, i.e. from the saturation at the end of stage 1

to the saturation at the end of stage 2. The correspondingequation may thus be written as

fd,2 = f satd,1 + (

f satd,2 − f sat

d,1

)[1 − exp

(−σ2(Φ − Φ1))]

for Φ1 < Φ < Φ2 (2b)

According to the above considerations, the followingequation describes the transformation from stage i − 1 tostage i:

fd,i = f satd,i−1 + (

f satd,i − f sat

d,i−1

)[1 − exp

(−σi(Φ − Φi−1))]

for Φi−1 < Φ < Φi (2c)

The final equation which accounts for multi-step transfor-mations is then the sum of the individual steps m describedby (2a)–(2c):

fd =m∑

i=1

(f sat

d,i − f satd,i−1

)G

[1 − exp

(−σi(Φ − Φi−1))]

(3)

where G is a function which transforms negative values into0 and leaves positive values unchanged.

A comparison between MSDA and earlier models basedon the Gibbons approach clearly shows strong differences.First of all, n and m parameters have not the same meaningin both descriptions. In MSDA m is the number of steps re-quired for the development of the entire sequence of damageaccumulation, whereas for Gibbons n is the number of ionimpacts needed to create a permanent disorder in a givenregion of the crystal. Despite the fact that no mechanismswere proposed to explain the difference between the first,second and, e.g., tenth impacts in the Gibbons model, thisparameter may reach surprisingly large (and certainly notreliable) values for crystals where a sharp increase of thedamage fraction is observed at high irradiation fluences [20].In this respect the MSDA model in its present developmentis rather straightforward to apply since the successive trans-formations are always described by a single impact mecha-nism, which is controlled by the probability that an ion hitsa non-transformed volume of the material. It is worth notingthat more complicated kinetics could be considered, but theyare generally not necessary to fit the experimental data (seenext section). Secondly, the shapes of the build-up are quitedifferent in both models. In MSDA the damage accumu-lation kinetics in each step reveals a sublinear dependenceconfirmed by experimental results. In models based on theGibbons approach (with the only exception of the DI/DSmodel), when the number of impacts n exceeds 1, the kinet-ics is supralinear at low irradiation fluences. Thirdly, in thenew approach the key parameters of the relevant processes,i.e. the triggering forces for structural transformations andthe material structure at saturation in each step, can be exper-imentally identified and related to the model. Finally, in con-trast to previous descriptions, in the MSDA model fD doesnot vary with Φ but with (Φ −Φi ). This feature implies thatthe damage formation is a discontinuous process in whichdifferent build-up mechanisms are operative at the variousstages of irradiation, i.e. above threshold fluences (Φi ).


3 Application of the MSDA model to selected casesof damage accumulation

This section is devoted to the presentation of selected ex-amples where the MSDA model has been used to reproducethe damage accumulation build-up in ion-irradiated crystals.These examples mostly concern oxide ceramics (zirconia,spinel) investigated by RBS/C. However, data obtained onother materials by various experimental techniques able toquantify the amount of damage introduced in a crystal byion irradiation (like TEM, XRD, PAS, SPIS, . . . ) may alsobe considered. For the sake of clarity and in order to progressfrom simple to more complicated situations, this section issplit into several subsections starting from data illustratingsingle-step damage accumulation build-up, and then exam-ining build-up involving two, and more than two disorderingsteps.

3.1 Single-step damage accumulation

The simplest case for the application of the MSDA model toion-irradiated crystals concerns situations where the dam-age build-up may be accounted for by a single-step process.This case is typical for crystals irradiated with swift heavyions. Actually, as demonstrated by many experiments, thehigh electronic energy deposition (Se) due to the passageof a swift ion induces the formation of an electrostaticallyunstable cylinder of ionized atoms and the emission of elec-trons [21]. In consequence, each ion creates a permanentlydamaged track that corresponds to a single-step damage ac-cumulation process, i.e. m = 1 in (3). In the cases wherecontinuous tracks are formed, σ is simply the area of theperpendicular section of the track. In this geometry, fits tothe experimental data provide a value of the track radius:r = (σ/π)1/2. Figure 2 shows that the damage build-ups ob-tained by RBS/C experiments are satisfactorily representedwith (3) in which m = 1. Based on the fit of spinel data, atrack diameter of 2.4 nm was calculated, in good agreementwith the results of TEM experiments [22]. It is worth notingthat in this case the equations proposed by the MSDA modelare identical to the Gibbons description with n = 1.

3.2 Two-step damage accumulation

The case where crystalline ceramics are irradiated with low-energy ions is more complex since the topology of radi-ation defect formation is less simple. Damage is not cre-ated in cylindrical tracks but results from the production andoverlapping of defect cascades with a geometry determinedby the mass and energy of irradiating ions. Thus the dam-age build-up generally occurs in multiple-step processes,and it is accounted for by (3) with a value of m dependingon irradiated materials and on irradiation parameters (na-ture and mass of incident ions, irradiation temperature. . . ).

Fig. 2 Accumulated damage (fD) in MAS (a) and YSZ (b) crystalsirradiated at RT with 450 MeV Xe ions. Solid lines are fits to RBS/Cdata using the MSDA model with m = 1 (single-step process)

Figure 3 presents the typical case of yttria-stabilized zirco-nia (YSZ) irradiated with various low-energy noble-gas ions[11] where a two-step process is evidenced (m = 2 in (3)).It is worth noting that all damage accumulation curves arewell reproduced with a single fit based on the MSDA modelby using a dpa horizontal scale (dpa values have been calcu-lated with the SRIM computer code [23]).

For YSZ irradiated at room temperature with low-energynoble-gas ions the crystallinity of irradiated materials waspreserved despite the very high fluences used (up to 5 ×1016 cm−2). This result is evidenced by the values of fD

reached at saturation (extracted from RBS/C experiments)which are much lower than unity, demonstrating that a long-range order still exists in irradiated crystals. Such a con-clusion is confirmed by TEM experiments which do notshow the presence of amorphous or polycrystalline phases.However, in some cases a two-step damage accumulationprocess may lead to complete amorphization. An example ofsuch a transformation is shown in Fig. 4, which presents thedamage accumulation build-up obtained by RBS/C in ion-irradiated SiC crystals [24]. It is worth noting that, even in


Fig. 3 Accumulated damage (fD) in YSZ crystals irradiated at RTwith various low-energy noble-gas ions: 80 keV Ne, 160 keV Ar,300 keV Kr, 450 keV Xe. The solid line is a fit to RBS/C data us-ing the MSDA model with m = 2 (two-step process). The conversionfactors for fluence (expressed in 1015 cm−2) to dpa scales are: 0.45 forNe, 0.71 for Ar, 2.13 for Kr and 3.36 for Xe ions

Fig. 4 Accumulated damage (fD) in SiC crystals irradiated at RT withlow-energy ions (150 keV Cs ions). The solid line is a fit to RBS/C datausing the MSDA model with m = 2 (two-step process). The dashed lineis a fit to the same data using the Gibbons equation with n = 1

the case of radiation sensitive materials (as SiC), the damageaccumulation build-up is far better reproduced by a two-stepmechanism than by a single-step one (Fig. 4). The results ofcalculations using the MSDA model are corroborated by dis-cussions concerning the amorphization mechanisms in SiC,which point out the need to exceed a given defect concen-tration threshold to start amorphization [25].

A two-step damage accumulation build-up is also ob-served in metals implanted with chemically active atomswhen the matrix-dopant pair forms an amorphizable com-pound. This is frequently the case for metal crystals im-planted with selected metal or metalloid ions. An example

Fig. 5 Accumulated damage (fD) in Ni single crystals implanted atRT with Zr ions. The solid line is a fit to RBS/C data using the MSDAmodel with m = 2 (two-step process)

of such a dependence is shown in Fig. 5 where the resultsobtained for nickel implanted with zirconium ions are pre-sented. Note that, also in this case which can be labeled as“two-step delayed” (i.e. when a sharp growth of accumu-lated damage is observed only for very high implantationfluences), the MSDA model is able to correctly reproducedamage accumulation kinetics.

3.3 Damage accumulation involving more than two steps

Processes involving a larger number of steps (3 or more)were generally observed for compound crystals irradiatedeither at very low temperature or with ion species which arechemically active. In the former case radiation damage isfrozen so that defect recombination or defect migration donot occur, whereas in the latter case the presence of chemi-cally active impurities above a concentration threshold leadsto the stabilization of radiation-induced disorder. In both sit-uations irradiation may induce amorphization of the crys-talline structure, even in radiation-resistant materials. Fig-ure 6 illustrates this feature with data recorded on YSZand magnesium-aluminate spinel (MAS) irradiated with Csions [26]. As compared to Fig. 2, a third step is evidencedat high fluence, so that the disordering build-up can be de-scribed by (3) with m = 3. Note that the plateaus in damageaccumulation curves shown in Fig. 6 appear at the same lev-els (∼0.05 and ∼0.7) as in GaN irradiated with low-energyions [18]. This effect indicates that the damage accumula-tion process follows a similar sequence in very different ma-terials. Solid lines in Fig. 6 are fits to the experimental dataaccording to the MSDA model. The good agreement be-tween the experimental data and calculations confirms theability of the MSDA model to correctly reproduce the mostcomplex damage accumulation kinetics. It is worth noting


Fig. 6 Accumulated damage (fD) in YSZ and MAS crystals irradiatedat RT with low-energy (160 keV) Cs ions. Solid lines are fits to RBS/Cdata using the MSDA model with m = 3 (three-step process). The con-version factors for fluence (expressed in 1015 cm−2) to dpa scales are:2.45 for MAS and 3.46 for YSZ crystals

that, from the fits made with the MSDA model, one can ex-tract values of the cross-sections for each step of damage ac-cumulation. These values are directly related to the volumeof material transformed by a single ion impact. This possi-bility constitutes an interesting tool for the investigation ofstructural and phase transformations in irradiated materials.

4 Relationship between the mechanisms of structuraltransformation and the MSDA model

Besides the fact that the disordering build-ups obtained ex-perimentally are better reproduced by the MSDA model thanby previous descriptions, one of the main advantages of thismodel is the possibility to divide the damage kinetics intoseveral, clearly identified steps. Each step of damage accu-mulation corresponds to a given type of defects in the irra-diated crystal, and the model provides the fluences at whichdifferent physical mechanisms lead to the respective trans-formations. Thus, a complete analysis of the overall dam-age accumulation may be regarded as the identification ofthe defect structures created at each step and of the physi-cal forces triggering the transformations from step i to stepi + 1. Various processes may be invoked to account for thedestabilization of the structure of a crystal, which are mostlyrelated to the accumulation of stress or to the increase ofthe concentration of foreign atoms acting as damage sta-bilizers. These processes may lead to various transforma-tions of the crystalline structure, such as the coalescence ofpoint defects in defect clusters, the creation of dislocations,the subdivision of the irradiated layer in small polycrystals

(polygonization), the formation of new crystallographicallyordered (or disordered) phases, partial or complete amor-phization, etc.

The simplest case of damage accumulation process is thesituation where each incoming ion creates a permanentlydamaged volume in the crystal. The resulting damage build-up, which is typical of crystals irradiated with swift heavyions, is described by a single-step mechanism (see Sect. 3.1).A single-step damage accumulation is also often observed inself-irradiated metals where efficient defect annihilation oc-curs, leading to a saturation of radiation damage at a lowlevel.

In most cases, low-energy ion irradiations lead to a dam-age accumulation process evolving in several steps. Owingto the general knowledge of materials under irradiation, thefirst step of the damage accumulation process occurring atlow irradiation fluences may be regarded as due to the accu-mulation and aggregation of point defects. Taking into ac-count the high mobility of self-interstitial atoms, even at lowtemperatures (RT and below) [27], it is likely that most ofthem migrate towards the crystal surface. The saturation ofthe backscattering yield observed in RBS/C experiments ata low level (RBS/C probes mainly the presence of intersti-tial atoms) justifies this assumption. The dominant defectsat low irradiation fluences are thus vacancies, as confirmedby the combination of RBS and positron annihilation stud-ies [28]. The MSDA model was used to fit the variation ofthe S parameter (which provides a measurement of the va-cancy concentration) extracted from SPIS experiments per-formed on YSZ irradiated with low-energy noble-gas ions.Fig. 7 shows that a good agreement is obtained between ex-perimental data and model predictions. The fact that stepsin the vacancy production and amount of displaced atoms(S and fD, respectively) occur at the same fluences con-firms the link which exists between the damage accumula-tion process and the structural transformations of the mater-ial.

The accumulation of vacancies leads to the formation offree volume within the crystal structure, which generatescompressive stresses. One may thus expect that the secondstructural transformation (step 2) is triggered by radiation-induced stresses. The corresponding threshold fluence maythus be regarded as the irradiation dose causing a sufficientlyhigh level of stress in the crystal lattice. This assumption waschecked by experiments performed on thin (∼130 µm) MAScrystals, where the damage accumulation build-up upon Arion irradiation was determined by RBS/C and the evolutionof the stress in the irradiated layer was measured with theStoney method (Fig. 8) [29]. One can note a sharp increaseof the stress at low irradiation fluences, which reaches amaximum at the onset of the second step of damage accu-mulation. The results show that the stress in the surface layerincreases up to a value which cannot be exceeded without


Fig. 7 Accumulated damage(fD) measured by RBS/C (fullcircles) and S parametermeasured by SPIS (open circles)in YSZ crystals irradiated at RTwith low-energy (320 keV) Arions. Both sets of data are fittedwith the MSDA model (m = 2)

Fig. 8 Accumulated damage(fD) in MAS crystals irradiatedat RT with low-energy noble-gasions (300 keV Ar). The solidline is a fit to RBS/C data usingthe MSDA model with m = 2(two-step process). The dashedline is the stress level measuredon the same samples

destruction of the original structure. Therefore a structuraltransformation takes place in order to relax the radiation-induced stress. This interpretation is corroborated by resultsof molecular dynamic calculations focused on the studiesof the relations between either pressure and amorphization[30] or radiation-induced stress and amorphization [31]. Thecomparison of damage accumulations kinetics recorded forMAS crystals of different thicknesses points to interestingobservation: in thick samples the onset of step 2 was ob-served at 1 dpa, whereas in thin samples the same step ap-peared at 2 dpa, i.e. at twice the irradiation fluence [32]. Therationalization of this effect in the framework of models tak-ing into account ballistic considerations only is doubtful. Onthe other hand, the MSDA model provides a rather straight-forward interpretation of this experiment. In thin samplesthe stress leading to the destabilization of the damaged struc-ture characteristic of the low-fluence range is partially re-leased due to sample bending. Thus a higher irradiation flu-ence is needed to obtain the same effect. Such a dependenceof the critical amorphization fluence on the sample thicknesswas also observed for MAS by, e.g., Wang et al. [33]. It is

worth noting that in some cases a sufficiently high level ofstress may lead directly to the amorphization of materials,even in the absence of irradiation, if the free energy of theequilibrium alloy is increased above that of its amorphouscounterpart [34].

The following transformation (step 3) is generally causedby the presence of foreign atoms acting as damage stabiliz-ers. This feature is confirmed by the comparison of dam-age accumulation kinetics recorded for samples irradiatedwith noble gases and chemically active elements. These re-sults are shown in Figs. 3 (noble-gas irradiation) and 6 (ir-radiation with Cs ions). One can note that noble-gas irra-diation leads to incomplete destruction of the crystallinestructure (two-step process), whereas irradiation with Csions causes an additional third step of damage accumulationleading to complete amorphization of the crystal. TEM ex-periments performed on Cs-irradiated samples in their finalstate showed that irradiation leads to the creation of a fullyamorphous layer at a depth where the ions were implanted,as far as the concentration of those species exceeds a giventhreshold. In this case the threshold fluence is defined by


the lowest impurity concentration required to stabilize theamorphous phase.

In summary, low-energy ion irradiation of oxide crys-tals generally leads to the following sequence of events:point defect accumulation at low fluences; stress-inducedstructural modifications leading to heavily damaged (yetstill crystalline) structure at moderate fluences; stabilization(e.g. impurity-related) of an amorphous structure at high flu-ences. The interpretation of the results in the framework ofthe MSDA model allows for: (i) a correct reproduction ofdamage accumulation kinetics with a simple equation, and(ii) the establishment of straightforward relationships be-tween the steps of damage accumulation and the mecha-nisms responsible for the structural transformations to oc-cur. Furthermore, it should be noted that in all the casesdiscussed in this paper, a good reproduction of damage ac-cumulation kinetics is obtained by assuming direct-impacttransformations with no need to use more complicated de-scriptions.

The comparison of the kinetics of damage accumulationat different ion energies for a given material points to a cru-cial question: why, as shown in Sect. 3, swift heavy ion bom-bardment usually leads to a single-step process, whereas thedamage induced by low energy irradiation occurs in severalsteps? This discrepancy could be due to differences in boththe topologies and the processes of energy deposition byswift or slow ions. At high energy the electronic excitation-induced melting of ionized matter in ion tracks leads to thedirect transformation of the melt volume into a new structure(single-step process). The volume of cylindrical ion tracks islarge enough to prevent the quenching material to restore theoriginal crystalline structure still existing in the surround-ing good crystal. Conversely, the volume of cascades cre-ated by low energy ions is rather small so that the struc-ture of the cascade region may relax towards a less disor-dered state. Obviously the cascade collapse is to a large ex-tent influenced by the crystalline structure of the virgin sur-rounding crystal leading to very efficient dynamic annealingprocesses. During low-energy irradiation the forces trigger-ing the destabilization of the structure, e.g. stress fields orchemical effects, accumulate progressively upon irradiation.The whole volume of the irradiated layer must be destabi-lized for an irreversible transformation to occur; then onlyeach ion impact may cause the transformation of the cas-cade core towards a new structure. The critical fluence re-quired to trigger the transformation depends on the stabilityof the structure; it is thus dependent on macroscopic para-meters like sample thickness, stress fields, temperature, etc.This process is repeated at each step of the damage build-upprocess, leading to the multi-step character of damage accu-mulation.

5 Summary and conclusions

A new phenomenological description of the damage ac-cumulation occurring during ion irradiation of crystallinesolids is proposed. This description is based on the assump-tion that the damage accumulation process occurs in sev-eral steps (MSDA), each step being triggered by the desta-bilization of the current structure of the material. The struc-tural transformations represented by the MSDA model areaccounted for by a direct-impact mechanism. Such an ap-proach describes in a simple and elegant way a completesequence of transformations from a perfect crystal to a to-tally amorphous structure for various damage accumulationsequences observed till now: single step (Fig. 2), two steps(Figs. 3, 4 and 7), two-step delayed (Fig. 5) and three steps(Fig. 6). The model was tested in various experimental sit-uations using data recorded on MAS, YSZ and some othercrystals irradiated with low- and high-energy ions. An ex-cellent agreement between experimental results and the pre-dictions of the model is obtained in all cases.

The MSDA model presents several advantages whencompared with previous ones. First of all, it is based on arather simple description of each step of structural transfor-mations, namely a direct-impact process. Secondly, it is ableto reproduce complex damage accumulation kinetics, typi-cal of crystals irradiated with low-energy ions, which canhardly be accounted for by former models. Thirdly, it pro-vides values of the cross-sections of structural transforma-tions occurring during the various steps of the damage accu-mulation which may be used to evaluate the diameter of iontracks or the size of damage cascades. It allows an identifi-cation to be made of the steps occurring during the damagebuild-up and, consequently, of the critical fluences at whichstructural transformations take place. Finally, it provides astraightforward explanation of the differences in damage ac-cumulation kinetics observed in samples of the same mater-ial having different thicknesses or different grain sizes.

The MSDA model in its present version constitutes aversatile tool for the description of damage accumulationprocesses in irradiated crystals. In contrast to former mod-els, the damage build-up is described as a discontinuousprocess composed of several steps governed by mechanismsoperating at distinctive phases of transition, from perfectlycrystalline to heavily disordered or amorphous structures.It may thus be helpful to researchers involved in scientificfields where radiation effects are of great importance, suchas the development of advanced materials for space or nu-clear industries.

Acknowledgements This work was partially financed by the French-Polish cooperation program COPIN, by the Groupement de RechercheNOMADE and Polish Ministry of Science and Higher Education (re-search grant N507 043 31/1128).


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Multi-step damage accumulation in irradiated crystals