Computer simulation of collision cascades in monazite

PHYSICAL REVIEW B VOLUME 27, NUMBER 9 1 MAY 1983

Computer simulation of collision cascades in monazite

Mark T. RobinsonSolid State Division, Oak Ridge Nationa/ Laboratory, Oak Ridge, Tennessee 37830

(Received 10 January 1983)

Collision cascades in the monoclinic mineral monazite have been studied with a modifiedversion of the computer-simulation program MARLowE. Most of the calculations weremade for the irradiation of polycrystalline CePO4 by normally incident 50-eV to 100-keV Aratoms. A few calculations were made for 1-keV Cu atoms recoiling from lattice sites in ahypothetical form of Cu with the monazite structure. Although there are many fewer re-

placements and substantially more channeling in the mineral, collision cascades in monaziteare generally similar to those in metals in size and shape. The dimensions of the calculatedcascades suggest that the amorphization of monazite by Ar-ion irradiation probably com-mences when collision cascades begin to overlap. The calculations give a detailed view ofthe structure of the cascades in monazite that should be useful in interpreting radiation ef-fects in proposed nuclear-waste isolation media.

I. INTRODUCTION

There has been considerable interest recently inthe effects of irradiation on the properties of com-plex materials, including certain minerals andmineral analogs that are proposed as media for theisolation of highly radioactive nuclear wastes. '

The monoclinic mineral monazite, an ore of ceriumand thorium which often contains uranium, is an in-teresting example of such materials. Unlike manyminerals containing radioactive elements, it seemsrarely to be found in nature in the partially amor-phous condition known to geologists as meta-mict. This peculiarity, added to its ability to dis-solve large amounts of the lanthanide and actinideelements, led to a suggestion of its potential as awaste-isolation medium. ' On the other hand,Cartz and his co-workers' ' have irradiated bothnatural and synthetic monazites with 3-MeV Ar+ions and find them to be quite sensitive to radiationdamage, in the sense that their x-ray diffraction pat-terns degrade significantly at ion doses around4X 10' cm . Vance and Pillay' report similarfindings from fission fragment irradiations. The in-terpretation of these and other experiments is ham-pered by a lack of quantitative theoretical under-standing of collision cascades, energy deposition,and particle ranges in such complex substances.

In a few instances, transport theoretical methods,derived from those used to study radiation effects insimple materials, ' ' have been applied to mul-ticomponent media. ' ' Such methods, based onthe Boltzmann transport equation, can be formulat-ed very generally. Analytical solutions are ob-

tained by limiting attention to structureless mediaand by making simplifying assumptions about theatomic scattering cross sections. ' Such approxi-mate treatments indicate mainly the general proper-ties of the solutions without being particularly accu-rate. Numerical solutions are often necessary' ' ' '

because of the complex nature of the problem wheninelastic energy losses, realistic atomic scattering,and target surfaces are included. As the number ofcomponents is increased, the problem rapidly be-comes more difficult and most work has been re-stricted to binary materials, with a small amount onternary ones. In no case has a transport theoreticaltreatment included the crystal structure of the tar-get.

Computer simulation in the binary-collision ap-proximation (BCA) has been a powerful techniquefor studying atomic collision processes in solids, in-cluding energetic particle ranges, light ionbackscattering, sputtering, and radiation damagein metals. It is easy to include realistic atomicscattering, inelastic energy losses, target surfaces,and crystal structure, all of which (and especiallythe last) are difficult to include in transport theoreti-cal models. Earlier studies, however, have been re-stricted either to amorphous materials or to crystal-line ones of rather simple structure and, with one ex-ception, of simple chemical composition.Nevertheless, such computations seemed a promis-ing way to study displacement damage processes,even in crystals as complex as monazite.

The collision cascade simulation program MAR-

LowE ' uses the BCA to construct the trajec-tories of energetic particles moving in crystalline

27 5347 1983 The American Physical Society

5348 MARK T. ROBINSON 27

media. The program and some of its applicationsare discussed in detail elsewhere. It has now beenmodified to permit calculations to be performed forcrystals of arbitrary structure. The modified codewas used to make a number of calculations concern-ing collision cascades produced by 50-eV—100-keVAr atoms normally incident upon polycrystallineCePO4, an idealized monazite. A brief study wasalso made of cascades produced by Cu atoms recoil-ing from lattice sites in a hypothetical form ofcopper with the monazite structure. This article isconcerned with the results of such calculations andwith a first attempt to use them to discuss the exper-imental situation. Since there is little experienceavailable upon which to base the selection of theparameters of the computational model, these arenecessarily somewhat arbitrary. It is expected thatimproved modeling can be developed after the ex-perimental situation is more clearly understood.

II. THE COMPUTATIONAL MODEL

A. Description of the target

Earlier versions of MARLOWE ~—' were

designed to treat collision cascades in crystals withorthogonal crystallographic axes. The crystals weredescribed to the program by listing (in the inputdata) the vectors from each type of lattice site to agroup of neighboring sites. For example, fcc crys-tals were often described by giving the locations ofthe 12 nearest neighbors and the six second-nearestneighbors of each lattice site. The modification con-sists of supplying a procedure for constructing therequired neighbor lists in Cartesian coordinates, us-

ing as input only standard crystallographic data(unit cell dimensions and atomic positions) on thetarget material. It was unnecessary to make majorchanges in the main parts of the program, sincethese continue to work in Cartesian coordinates asbefore. Consequently, no significant increase incomputing time occurs because of the generalizedcrystal structure. As presently constituted, the pro-gram will deal successfully with crystals of any sys-tem having up to 30 nonequivalent sites per unitcell. This limit is easily altered.

In most of the present calculations, the target waspolycrystalline CePO4. The crystal structure ismonoclinic with lattice parameters a&

——0.6777 nm,aq ——0.6993 nm, a3 ——0.6445 nm, and P=103.54.The space group is P2&/n (Cz~), with four mole-cules of Cep04 in each unit cell. (A coordinatetransformation will convert this structure to theequivalent standard space group P2&/c. ) The atom-ic positions are given by Beall et al. Each atom ofCe is surrounded by an irregular coordination shellof nine 0 atoms. 'The P atoms are centered in

slightly distorted PO4 tetrahedra. The surroundingsof each atom in the crystal were described by givingthe locations of all neighbors within 0.6a~ (0.4066nm). The number of neighbors was 19 for the Ceatoms, 24 for the P atoms, and 17, 19, 20, or 22 forthe four nonequivalent 0 atoms. This is roughly theminimum number of atoms which allows the satis-factory propagation of the monazite structure. Thestructure was assumed to be static; that is, thermaldisplacements of the atoms from their lattice siteswere ignored. To make the target polycrystalline,the crystallographic axes were rotated randomly inthe fixed-coordinate frame of the computations atthe start of each primary trajectory. The rotationwas accomplished by using standard aleatorymethods to select three uniformly distributed Eulerangles which related the crystal axes to the compu-tational frame. Each incoming Ar atom entered aCePO4 monocrystal, but no two of them encoun-tered the same orientation.

The Ar atoms were normally incident on the tar-get surface. As the collision cascades developed,some atoms passed through this surface and left thetarget. These are termed "reflected" if they were Aratoms or "sputtered" if they were target atoms.However, the latter term should not be taken tooseriously, since the proper surface binding energywas not employed, as is required in BCA sputteringcalculations. ' Consequently, it is not asserted thatthe sputtered particles in the present calculationscorrespond to those which would be observed inproper sputtering experiments.

With the unit-cell dimensions ai ——0.667271 nm,a2 ——0.688 539 nm, a3 ——0.634 582 nm, andP= 103.54', a monazite crystal has the same atomicdensity as ordinary fcc copper. By populating thesites entirely with Cu atoms, a form of copper withthe monazite structure could be constructed. A fewcalculations were performed with this "monazitecopper" to explore specifically crystal structural ef-fects on cascade development without the complica-tions of different atomic masses and other parame-ters. Neighbors within 0.6a~ (0.40036 nm) of eachsite were included in the crystal description. Inthese calculations, the primary particles were alsoCu atoms, ejected isotropically from the "Ce" latticesites in the crystal. For comparison, correspondingcalculations were also done in fcc copper, using thelattice constant a =0.36150 nm. In some of these,Cu primaries were ejected isotropically from latticesites, and in others Cu primaries were normally in-cident on polycrystalline fcc copper.

B. Slowing down process

Only the short-range repulsive part of the intera-tomic potential is used in MARLOWE. Any long-

27 COMPUTER SIMULATION OF COLLISION CASCADES IN. . . 5349

range interaction, such as an interionic Coulomb po-tential, is regarded as an essentially constant back-ground, without influence on the trajectories. TheMoliere approximation to the Thomas-Fermi poten-tial was used to describe the quasielastic interactionsof the energetic particles with the initially stationarylattice atoms. For the CePO4 calculations, thescreening lengths were those proposed by Firsov.The scattering of the projectiles was evaluated forall events with impact parameters less than 0.35a~(0.2372 nm). This distance is somewhat less thanthe smallest Ce-0 (0.2445 nm) and O-O (0.2403 nm)distances in the crystal, but rather larger than theP-0 distances (averaging 0.1527 nm). An approxi-mate correction for many-body encounters is madein MARLOWE. The impact parameter cutoff usedwas such as to produce events in which as many asfive target atoms could interact simultaneously witha projectile, although events with more than fourtargets were extremely rare. In the monazite Cu cal-culations, the screening length was 7.38 pm to corre-spond with earlier calculations on fcc Cu. ' Forthe same reason, impact parameters up to 0.22413nm (0.62a) were included in these calculations.There were no encounters involving more than fourtarget atoms.

The inelastic (electron excitation) energy losses ofall projectiles were evaluated using a "nonlocal"model based on the so-called Lindhard-Scharff-Schintt(LSS) theory. The inelastic energy losses inthis model are proportional to the velocity of theprojectile and are independent of the impact param-eters in the collisions. They are determined solely

by the energies of the particles and the lengths of theflight-path segments. The stopping power of thecompound target was found by weighting the inelas-tic stopping cross section of each kind of atom by itsconcentration(Bragg's law).

C. Energy conservation

Each collision cascade was initiated by a primaryparticle of initial kinetic energy Eo. The movingparticles in a cascade were followed collision by col-lision until they either left the target or slowed downbelow the energy E,=4.8 eV. Atoms were added tothe cascade if they received in a collision a kineticenergy in excess of E~ ——5.0 eV. Each atom so dis-placed was required to surmount an energy barrierof E~ ——0.2 eV. This small local binding energy isused to improve the agreement between the lengthsof the linear collision sequences (LCS's) calculatedwith MARLOWE and those calculated in classicaldynamical models. At each collision the fastestparticle currently in the cascade was selected as theprojectile. Since the distance between collision

points is roughly constant, this procedure simulatesthe temporal development of the cascade with someaccuracy.

When no particle remains in the cascade with akinetic energy &E„ the initial energy of the pri-mary has been partitioned into three identifiableforms, each of which may be associated either withthe atoms of a particular species or with the col-lision cascade as a whole. The three forms are thefollowing.

(i) Q, the total energy lost by inelastic processes.(ii) yEO, the total energy carried from the target

by reflected primaries and sputtered target particles.(iii) E=(1 y)EO —Q, th—e damage energy, where

y is thesputtering efficiency

The damage energy appears in MARLOWE as thesum of three terms.

(a) The kinetic energy transferred in quasielasticcollisions to target atoms which are not displaced(each contribution &Ed).

(b) The energy expended by displaced particles inovercoming binding (each contribution =Es).

(c) The final kinetic energy of each recoil which

stops within the target (each contribution &E,).

The values of E were obtained by adding togetherthese three contributions for each species of atomand for the whole cascade.

D. Replacements

In some collisions in which a target atom is dis-

placed, the projectile is left with a kinetic energy&E,. In such events, if the projectile is closer to thenow-vacant target site than it is to its own originallattice site, it occupies the target site. Such eventsare termed replacements. If, on the other hand, theprojectile is closer to its own site, and that site is stillvacant, it returns there. Such events are consideredto produce ordinary pairs, as described below. If theprojectile is closer to its own site, but that site is notvacant (because it was occupied in an earlier replace-ment, for example), it remains as an interstitialatom. In the present calculations, replacementswere permitted only when the chemical type of theentering projectile was the same as that of the recoil-ing target.

E. Defect pairs

When defect production is complete; that is, whenno particle remains in the target with a kinetic ener-

gy in excess of E„ the displaced atoms and vacant


lattice sites are classified into {Frenkel} pairs. If thelattice site nearest to an interstitial atom is vacant,the two are considered to be a close pair. If thisclosest site is occupied or previously paired, the sitesneighboring it are examined. If one of these is va-cant, a near pair is identified. If more than one suchvacant site is available, the nearest of them ischosen. The remaining vacancies and interstitialsare then paired according to their separation dis-tances such that the smallest separation is located,then the next smallest, and so on, until all possibledefect pairs have been identified. The members ofthis last group are termed distant pairs. The pairs ofall classes are considered correlated if the interstitialoriginally occupied the vacant site with which it ispaired. In the present calculations, it was requiredthat the atom in the pair be a proper occupant of thesite. This restriction is intended to account forchemical influences on defect stability, which ought,for instance, to favor the correct reconstruction ofP—0 bonds and other polar structures. The separa-tion range in close and near pairs depends somewhaton the orientation of the pair axis, so that there is anoverlap in the separations corresponding to the vari-ous classes of pairs.

In the calculations reported here, it is assumedthat all replacements, close pairs, and near pairs willbe annihilated by chemical or mechanical imbal-ances in their locales, so that they do not lead to ob-servable damage. This criterion, not a sharp dis-placement threshold energy, determines the max-imum amount of displacement damage produced ina collision cascade. Moreover, the view is taken thatthe stability of the distant pairs may be discussed interms of the separation distance between themembers of the pair. This distance is referred to asthe vacancy capture radius ry. Although this modelis certainly a greatly simplified one, it has provenuseful in discussing radiation damage production inmetals. ' '

F. Cascade generation and analysis

A CePO4 calculation involved the generation offrom 100 to 1000 cascades, each initiated by a nor-mally incident Ar atom of specified initial kineticenergy. The initial impact points of the primaryprojectiles were distributed uniformly over the targetsurface. In the monazite Cu calculations, the cas-cades were initiated by Cu atoms originating at Cesites. Their initial directions were distributed iso-tropically. The primaries in the fcc Cu calculationswere selected similarly.

Various analyses of the cascades were obtained.The results showed statistical fluctuations whichcould be reduced by increasing the sample sizes.

Where appropriate, uncertainties based on the ob-served fluctuations and the sample sizes are shownwith the calculated results. The sizes chosen were acompromise between statistical accuracy and com-puting time. It required about 42 sec on an IBM3033 computer to generate and analyze one cascadeinitiated by a 100-keV Ar atom. Such cascades in-volved as many as 6435 moving particles and aver-

aged about 22500 collisions each. Both the execu-tion time and the number of atoms participating in acascade were roughly proportional to the kinetic en-

ergy of the incident Ar particle.

III. COMPUTATIONAL RESULTS

0.35

0.30—

u 0.25

Pp 0.20O

O0.15

ZUJ

O 010

1 keV Cu Cu

MARLOWE (VERSION 11.8)NONLOCAL lNELASTlC LOSSES

fcc Cu

0.25

J

LMONAZITE Cu

0.20

0.15

0.10

0.05

100 2 4 6 8

r/g, V ECTO R RANGE

FIG. 1. Vector range distributions calculated for 1-keVCu atoms recoiling from lattice sites in fcc Cu and inmonazite Cu.

A. Monazite copper calculations

The monazite copper calculations were designedto study specifically crystal structural effects on cas-cade development without the complications ofchanging interaction potentials, atomic masses, andso on. The atomic densities of monazites are similarto those of metals: In CePO4, there are 80.822atoms/nm, compared to 84.671 atoms/nm in Cuand 60.243 atoms/nm in Al. In contrast, there are49.940 atoms/nm in Si and 37.278 atoms/nm inKC1. Thus, the spatial dimensions of collision cas-cades in monazites should be generally similar tothose in appropriately chosen metals.

Figure 1 compares the recoil ranges of 1-keV Cu


20

0llAw~1 ~

A

Vl

O 15I-KQ.Ct

UJV)

X

10—

1keV Cu Cu

UJCl

VJ

KLLJ

5—K

Q.

4JVZ'LLI

u 00

I I I I I I I

1 2 5 4 5 6ry 0 ~ FRENKEL PAIR SEPARATION

FIG. 2. Distributions of distant Frenkel pairs calculat-ed for collision cascades initiated by 1-keV Cu atomsrecoiling from lattice sites in fcc Cu and in monazite Cu.

primaries in monazite and fcc copper. The figureshows the vector range r in units of the lattice con-stant of the fcc structure. The appropriate meanvalues (r) are indicated. At this relatively low ki-netic energy, very little channeling occurs in fcccrystals when particles are emitted from lattice sites,since they must be scattered several times beforethey can enter a channel and are unlikely to survivefor so long. This is evidenced by the rather symme-trical range distribution with its small tail on thelong-range side. The monazite distribution is quitedifferent. There is a pronounced tail on the long-range side, clear evidence of increased channeling inthe unsymmetrical crystal. Thus, although the den-sities of the two crystals are identical, the detailedarrangement of the atoms in space significantlychanges the range distributions. The differences aresufficient to increase the mean range in the mona-zite Cu by about one-third over that in the fcc struc-ture.

An important feature of cascade development insimple crystal structures is the occurrence of se-quences of replacement collisions along closelypacked rows of atoms. These LCS's occur predom-inantly along the (110), (100), and (111) direc-tions in fcc crystals and are responsible for carryingthe displaced atoms to much greater distances fromtheir associated vacancies than would be the case ifthe LCS's did not occur. In monoclinic monaziteCu, however, rows of atoms appropriate to supportLCS's do not occur and the interstitial atoms are ex-

105

Ar CeP04 (POLYCRYSTALLINE)

MARLOWE (VERSION 11.8)NORMAL INCIDENCE

LSS (Ar Ar)

~ TOTAL

p Ar

o Ce

P

104

2 — ~ 0 p'-/l

i/~r"

~ ~

//P gr~ IO~ ~ ~ rrtil /rrg /rUJ 5 ~ r //r g

~ r/8 r P'

~ g y p101 — ~ «P' /r/r rp/ /

5 err)aP / p/

100 I rl r I I I I I I I

5 10 2 5 10 2 5 10 2 5 10

Ep, INC I DE NT Ar ATOM KINETIC ENE RGY (ey )

FIG. 3. Total inelastic energy loss from collision cas-cades in polycrystalline CePO4 initiated by normally in-cident Ar atoms. Contribution to the total of the inelasticenergy losses of each kind of particle is also shown.

m IO~O

5

UJ 2

pected to be generally closer to the vacancies than isthe case in fcc Cu. This behavior is illustrated inFig. 2, which shows the distant Frenkel pair distri-butions in the two cases. Although the total numberof such pairs is the same in the two calculations,there are more pairs in the fcc structure than in themonazite at all values of the pair separation distancerq/a & —1.

These calculations show two important differ-ences between collision cascades in monazite andthose in the much more symmetrical fcc structure.First, there is significantly more channeling in theformer. Other things being equal, this leads tolarger cascade volumes and a smaller number ofatoms in the cascade. Second, there are no LCS's inthe low-symmetry monazite. This also reduces thenumber of atoms in the cascade but leads to morecompact cascades and less defect survival than inthe fcc structure. The two effects may somewhatoffset each other, but the latter appears from thepresent results to be the more important. In the cal-culations on which Figs. 1 and 2 are based, the meannumber of distant pairs was about the same, but themean number of atoms participating in the cascadeswas lower by about 19%%uo in the monazite. Thus, rel-atively more defects were able to survive in themonazite case than in fcc Cu (19.6% compared to15.7%).

27MARK T. ROBINSON5352

B. Damage production in CePO4

Figure 3 shows Q, the total amount of energy ex-pended in electron excitation, as a function of the in-cident kinetic energy. The average atomic numberin CePO4 is (Z)=17.5 and the average atomicmass is (A ) =39.182 amu. The closeness of thesevalues to those of the incident Ar (Z = 18,A =39.948 amu} suggests that the latter may be re-garded as the effective "self-ion" in CePO4. The to-tal inelastic energy loss evaluated with MARLowE iscompared with an estimate based on a numerical ap-proximation to the LSS energy partition theory.This estimate assumed that the atoms of the CePOqtarget could be replaced by Ar atoms and used theFirsov screening length. The estimate is in re-markable agreement with the MARL0%E calculation.Figure 3 also shows the contribution to the total in-elastic energy loss of each kind of projectile. Thetarget atoms contribute to Q roughly, as would beexpected from their concentrations. The abundant0 atoms experience inelastic energy losses compar-able to those of the incident Ar atoms, while themuch less abundant Ce and P atoms lose much lesskinetic energy inelastically.

Figure 4 shows (n ), the mean number of recoil-ing target atoms produced in a cascade, as a func-tion of the initial kinetic energy of the Ar atoms.The number of each kind of atom is approximatelyproportional to the primary energy and the relativenumbers are in rough accord with the concentra-tions. Not all of these atoms constitute permanent

2

O~ 103

u 5

2

o 10~

O

isa 2lbX 101zz 5

X2

C

10P

I I I I I I I I I I

5 10 2 5 10 2 5 10 2 5 10Ep, INCIDENT Ar ATOM KINETIC ENERGY (eV)

FIG. 4. Mean number of recoiling atoms in collisioncascades in polycrystalline CePO4, initiated by normallyincident Ar atoms.

6 I I I

Ar Ce P04 (POLYCRYSTALLINE)


—5—Ie

z4-ii

CJU.tL,4J

ZKLLJ

2I-DO. II

1— L-"~iI

5 10 2 5 10 2 5 10 2 5 10

Ep INCIDENT Ar ATOM KINETIC ENERGY (eV )

FIG. 5. Sputtering efficiency y calculated for Ar atomsnormally incident on polycrystalline CePO4. Contributionof reflected primary particles is also shown.

displacements, however. Some of them leave thetarget altogether and others eventually come to restin close proximity to target vacancies. Because ofthe sputtering, some energy is carried away from thetarget. Figure 5 shows the energy dependence of thesputtering efficiency y and the contribution to it ofthe reflected Ar atoms. Except at low incident ener-gies, yEO is small compared to the inelastic energyloss.

E, RECOIL KINETIC ENERGY (eV)100

C) 50 30 20 15 12 10 9 8 7 6 5II ii I I illll I I I I I t I I I I I I I I I

I keV Ar CePO~ (POLYCRYSTALLINE)

MARLOWE (VERSION 11.8)350

300

250

i- 200MXLLJ

15004Jts:

OLLJ

O0

60

— 90P RECOILS

— 60

— 50

50-

40

30

0 0.05 0.10 0.15 0.20I/E, INVERSE RECOIL KINETIC ENERGY (eV 1)

FIG. 6. Modified recoil density g(E) calculated for col-lision cascades in polycrystalline CePO4 initiated by nor-mally incident 1-keV Ar atoms. Abscissa is linear in 1/E.Scale in E is shown at the top of the figure.


E, RECOIL KINETIC ENERGY (eV)100

00 50 30 20 15 12 'IO 9 8 7 6 524 I I I I I I I IIII I I I I I I I I l I I i I I I I

100 lteV Ar CeP04 (POLYCRYSTALLINE)

22 MARLOWE ( V ERS IO N & I .8 )

20

18

~i16

10I I I I I I I I I

3

Ar CeP04 (POLYCRYSTALLINE)


v) 102LLI

OZ0

Li. 2O

I-14

ZLaJo

0O4JKoUJ

U.

o 4O

IX 101

5ZZUJ 2

- 100

—g 0— I

I I I I I I I I I I I I I I I I I I I I

0 0.05 0.10 0.1 5 0.20I/E, INVERSE RECOIL KINETIC ENERGY (eV ')

FIG. 7. Modified recoil density g (E) calculated for col-lision cascades in polycrystalline CePO4 initiated by nor-

mally incident 100-keV Ar atoms. Abscissa is linear in1/E. Scale in E is shown at the top of the figure.

The energy spectrum of the target atoms as theyfirst leave their lattice sites is termed therecoil density ''" Guided by the analytical theory,it is convenient to write

n (E)dE =g (E)d ( I /E),where n (E) is the recoil density and g(E) will be re-ferred to as the modified recoil density. In theanalytical theory, ' ' g (E) is expected to be roughlyconstant. Figures 6 and 7 show g (E) for each kindof particle at two very different incident energies.The abscissas are linear in 1/E and the scales of theordinates are adjusted for the stoichiometry of thetarget. The shapes of the spectra at the two incidentenergies are quite similar. The main difference(aside from the absolute magnitudes) is that fewerparticles appear in the first channel of the histo-grams at the lower incident energy. This merely re-flects the smaller total energy available. Even at thehigher energy, very few recoils are produced with in-itial kinetic energy greater than 100 eV. This em-phasizes the well-known point that many aspectsof collision cascades are dominated by the behaviorof low-energy particles, irrespective of the initial ki-netic energy of the primary particle. At both ener-gies there are fewer Ce recoils than called for by thestoichiometry of the crystal by about one-third.

10 1

I I I I I I I I I I I

5 10 2 5 10 2 5 10 2 5 10

EO, INCIDENT Ar ATOM KINETIC ENERGY (eV)

FIG. 8. Mean number of vacancies calculated for cas-cades in polycrystalline CePO4 initiated by normally in-

cident Ar atoms. Results are shown for five different va-

cancy capture radii.

Such differences are expected in the analyticaltheory' because of differences in the atomic massesand the potential parameters. Another difference,not anticipated in the analytical theory, is betweenthe shapes of the spectra for the different species oftarget atoms. These appear to reflect the differentsurroundings of the various atoms in the crystal. Inparticular, the P atoms are at the centers of rathersmall PO4 tetrahedra. The surrounding 0 atomsshield the P atoms from approaching energetic par-ticles and also make it more difficult for a recoilingP atom to escape its lattice site. Consequently, thenumber of high-energy P recoils is reduced and thenumber of low-energy ones is increased, comparedto what is expected from the spectra of the Ce and0 recoils.

Figure 8 shows the energy dependence of (v), themean number of vacant lattice sites produced by anincident Ar atom. The close and near pairs are as-sumed to have been annihilated. Curves are shownfor five values of the vacancy capture radius ry.When none of the distant Frenkel pairs is annihilat-ed, the number of defects introduced is nearly linearin the incident energy above about 200 eV. Smallvalues of rz remove relatively more damage fromlow-energy cascades without changing the approxi-mate linearity at high energies. When all of the dis-tant pairs are annihilated, some vacant sites remainbecause of the sputtering of target atoms.


0.8

I I I I I

Ar —CeP04 (POLYCRYSTALLINE)


160

140

120—~ ~ ~

0.7

0.6Z'LLI

0.5O0J 04zO

g 0.3

0.2

Ce

P

0

RECOILS VACANCIES

(rV =0)0

~~~~ ~ ~ ~«~ ~

A A Ahk k

~~»

o 80—0ZtA 7QKZ 60—4JC)

tLJ

40O4JU.4.4J

30UJ

Ar CePO4 (POLYCRYSTALLINE)

MARLOWE ( VE RSION 11.8 )

NORMAL INCIDENCE

L~j~L,

R ECO ILS VAC AN C I ES(rv=0)

Ce oP 0O

~—— ~—0—

0.1 ~«0 ~ ~ 8 ~~~~M

0 A I I I I I I I I I I

5 10~ 2 5 10~ 2 5 104 2 5 10~

Ep INCIDENT Ar ATOM KINETIC ENERGY (eV)

FIG. 9. Fractions of particles of each kind among therecoils and among the vacancies for collision cascades inpolycrystalline CePO4 initiated by normally incident Aratoms. Dashed lines show the stoichiometric values for 0( —) and for Ce or P ( —).

2 1

These same data may be looked at in other ways.Figure 9 examines the deviations of (n) and (v)from stoichiometry for each kind of target atom.There is a slight excess of 0, both among the recoilsagd among the vacancies, although the latter disap-pears at high energies. There is a significant deficitof Ce among the recoils, presumably because of itsmass, but this is transformed into a substantial ex-cess among the vacancies. There is a small excess ofP atoms among the recoils, but this becomes a sub-stantial deficit among the vacancies. This changeseems to be a result of the relatively lower energiesof the P recoils and of their considerable energy lossin escaping from the cage of their immediate 0neighbors. Consequently, P atoms are more likelythan either Ce or 0 to form close and near pairs.This tendency is especially marked at low energiesand is responsible for the energy dependency seen inFig. 9. Above about 200 eV, both (n) and (v)show constant deviations from stoichiometry, inagreement with the predictions of the analyticaltheory. '

In simple theories of radiation damage produc-tion, '6 it is expected that (n ) and (v) will be pro-portional to E. The proportionality constant isan effective threshold energy E,h, above which mul-

25

10.5

10.0

9.5

9.0 I I I I

5 10 2 5 10 2 5 10 2 5 10

Ep INCIDENT Ar ATOM KINETIC EN ERGY (eV )

FIG. 10. Effective cascade multiplication threshold E,hfor cascades produced in polycrystalline CePO4 by nor-mally incident Ar atoms.

tiplication of the particles in a collision cascade be-gins. Figure 10 shows E,h as a function of the ini-tial Ar energy for each quantity. The effectivethreshold was evaluated as the ratio of the averagedamage energy associated with the recoiling atomsof a given species to the average number of atoms ofthat kind. The effective threshold for producingrecoils is roughly the same for all three targetspecies and is nearly constant for initial kinetic ener-gies above about 1 keV. The magnitude is, ofcourse, just what is expected from the computationalmodel parameters used, namely, E,h-2E~ ——10 eV.The effective threshold for producing vacancies,however, differs significantly from one atomicspecies to another. The value of E,h for Ce is con-stant above about 1 keV, but the values for P and 0increase steadily over the entire energy range of thecalculations. These results can be explained in partin terms of the recoil densities and in part in termsof the distributions of Frenkel-pair separations forthe different species. As noted above, there are rela-tively fewer energetic P recoils than there are Ce or0 recoils. Moreover, the P recoils are emitted fromwithin the rather compact P04 tetrahedra and ex-perience considerable energy loss in escaping. Thus,they are more likely to form close and near pair con-figurations than are the 0 and Ce recoils. The 0recoils, being the most abundant species in the crys-


Ip3

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MARLOWE (VERSION 11.8)NORMAL INCIDENCE—~—Ce PAIRS~ P PA I RS—~—0 PAIRS

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MARLOWE (VERSION 11.8 )

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~ Ar CePO4 (POLYCRYSTALLINE )

Cu Cu (POLYCRYSTALLINE)

VACANCIES ~Q ~ ~o~

1p-1 I I I I I I I I I I I I I I

P 2 4 6 8 10 12 14

r&/yr, FRENKEL PAIR SEPARATION

FIG. 11. Distributions of distant Frenkel-pair separa-tions calculated for collision cascades in polycrystallineCePO4 initiated by normally incident 100-keV Ar atoms.

I I I I I I I I I I

5 10 2 5 10 2 5 10 2 5 10

EO, INCIDENT Ar ATOM KINETIC ENERGY (eV)

FIG. 12. Average densities of recoils and of vacanciesin cascades in polycrystalline CePO4 initiated by normallyincident Ar atoms and in cascades in polycrystalline Cuinitiated by normally incident Cu atoms.

tal, are more likely than the other particles to makereplacement collisions. In addition, both the P and0 recoils may be backscattered from the heavy Ceatoms in their surroundings. This possibility alsoenhances the probability of forming close or nearpair configurations. The heavy Ce atoms, on theother hand, simply plow ahead, pushing the lighterP and 0 atoms aside.

C. Cascade structure and dimensions

The calculations give considerable insight bothinto the sizes and shapes of the cascades produced inCeP04 and into their internal structures. Figure 11shows the distributions of Frenkel pairs found incascades initiated by 100-keV Ar atoms. The distri-butions obtained at lower energies are generallysimilar. The figure shows that most displacedatoms terminate their trajectories in fairly closeproximity to vacant lattice sites of the appropriatetype and that rather small values of rz are sufficientto remove a great many defects. The effect is morepronounced for the Ce recoils than for the lighterones, especially at larger rz values. Moreover, theaverage density of defects in a collision cascade isquite small. Figure 12 shows the mean vacancy den-sity as a function of the initial primary kinetic ener-

gy, both for Ar atoms incident upon polycrystalline

10I I I I I I I I I I I I I I I I

100 keV Ar CeP04 (POLYCRYSTALLINE)

9 — MARLOWE (VERSION 11.8 )

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FIG. 13. Depth distribution of the vacancies producedin polycrystalline CePO4 by normally incident 100-keV Aratoms. Inset shows the dependence of the mean damagedepth on the vacancy capture radius.

CePO4 and for Cu atoms incident upon polycrystal-line fcc Cu. Distant Frenkel pairs and excess un-

paired vacancies are included as damage and thecascade volume is estimated from the mean numberof lattice sites at which collisions occurred. Thecurves for the two materials are similar, both inshape and in magnitude. The low overall density ofdefects (-3%) will be noted. The figure also shows


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FIG. 14. Radial distribution of the vacancies producedin polycrystalline CePO4 by normally incident 100-keV Aratoms. Inset shows the dependence of the mean damageradius on the vacancy capture radius.

&0o

the mean (spatial) density of the recoiling targetatoms in the same calculations. The two targets areagain quite similar. Comparison of the density ofrecoils with that of vacancies shows that only aboutone-eighth of the recoils eventuate in Frenkel pairs,even of rather small separation. The volumes thathave been used in these density estimates are concen-trated closely along the paths of the particles in thedeveloping collision cascades. Even so, the densitiesare small.

The overall sizes and shapes of cascades may beassessed (for the present geometry) by examining thedistributions of the vacancies in cylindrical coordi-nates, using the incident Ar direction as the axis ofthe cylinder. Figure 13 shows the distribution indepth of the vacancies generated by 100-keV Aratoms. Both the mean depth of the vacancies (zr )and the mean penetration depth of the incident par-ticles (z~, ) are indicated. As expected, the Aratoms penetrate somewhat more deeply than theaverage depth of the damage. Annihilation of thecloser pairs has rather little effect on the depth dis-tribution of the damage. As the inset in the figureshows, even a value of rz/a~ ——4 lowers the meandatnage depth by only about 8%. Figure 14 showsthe radial distribution of the same vacancies. Themean and rms values of the radial coordinate p& andthe mean radial spread of the stopped Ar atoms areindicated. The off-center maximum in the radialdistribution results from its two-dimensional nature:the distributions in any one Cartesian coordinate inthe target surface show maxima on the axis. As wasalso the case for the depth distribution, annihilationof the closer pairs does not much influence the radi-al extent of the cascades. As the inset shows, thelargest value of rv that was used reduces (pr) byonly about 12%%uo. Both Figs. 13 and 14 show evi-

Io-'5 iO 2 5 )0 2 5 10 2 5 IO

EO ~ INCIDENT Ar ATOM KINETIC ENERG& (8V)

FIG. 15. Incident-energy dependence of the mean Arpenetration depth, the mean damage depth, and the mean

damage radius for Ar atoms normally incident on po-lycrystalline CePO4. No defect recovery was included.

dence of structure in the cascades. The small groupsof particles at large values of zz or py probablyrepresent subcascades, separated from the main bo-dies of their parent cascades either by unusualscattering events or by channeling.

Figure 15 shows the dependences of the mean cas-cade dimensions on the initial Ar kinetic energy.Above about 1 keV, the mean depth and radial sizeare in essentially constant- ratio; that is, the shapes ofthese cascades do not vary much with energy.Moreover, the cascade dimensions increase with en-

ergy in about the same manner as does the incidentAr penetration depth. These properties are in ac-cord with the predictions of transport theory. Thedimensions increase roughly as Eo', a result alsoin reasonable accord with transport theory.

The cross-sectional area of a collision cascade,projected onto the target surface, might be taken asn(pr), but a more suitable quantity is n(pr),where the second factor is the square of the radiusof gyration of the collision cascade about the direc-tion of the incident atom. Figure 16 shows this fac-tor for the cascades generated in Cep04 by normallyincident Ar atoms. The crass-sectional area of acascade increases rapidly with the kinetic energy ofthe incident particles, varying approximately as ED'

above about 5 keV.

27 COMPUTER SIMULATION OF COLLISION CASCADES IN . . ~ 5357

104

1Q2

1OI

100

10-1

I I I I I I

Ar —CeP04 (POLYCRYSTALLINE)


these fluctuations are somewhat difficult to establish

reliably, however, because of the required computingtime at the higher energies. For example, at 100keV, where only 100 cascades were evaluated, themean value (pi/a, )=30 was found. No cascadehad pz/a& &10; one-fourth had pz/a& &22, one-half had 22 &py/a ~ & 34, and one-fourth hadpy/ai &34. Thus, there were rather few extremecascades and those few had large mean radii, prob-ably associated with subcascades.

Finally, although the distributions of defects ofindividual chemical species have not been presentedhere, it may be said that these do not show signifi-cant differences from the overall distributions. Atthe higher energies, the Ce and P distributions areslightly narrower than the 0 distributions, but thedifferences are slight (about 2% in the cross section-al areas).

IV. DISCUSSION

10-2 I I I I I I I I I I

5 10 2 5 10 2 5 10 2 5 10

EO I NCI DENT Ar KINETIC ENERGY (eV )

FIG. 16. Upper curve shows the mean cross-sectionalarea of collision cascades in polycrystalline CePO4 initiat-ed by normally incident Ar atoms. Absolute areas can befound by multiplying the ordinates by nLtI =1.4429 nmi.Lower curve shows the ratio of the number of sites in themean enveloping cylinder to the number of collision sitesin the average cascade.

The average cylindrical dimensions given in Figs.15 and 16 may be used to estimate the number oflattice sites in the cascade volume as Nm (pi ) (zi.),where N is the atomic density of CePO4. Thevolume is that of a cylindrical envelope withinwhich the developing collision cascade is approxi-mately contained. The ratio of the number of sitesin this cylindrical volume to the average number ofcollision sites in a cascade is shown in Fig. 16. Thisratio is the number of cascades that might be fittedinto the cylinder. It has the same energy depen-dence as the cross-sectional area of the cylinder, atleast above about 5 keV. It is probable that the ef-fective cross-sectional area of a single cascade can beestimated by dividing the cylindrical cross sectionby the number of cascades which it could contain.If this is done, the energy-independent cascade crosssection obtained in 6.65 nm . The reciprocal of thisquantity, namely, 1.50)&10' Ar atoms per cm,would be the dose at which the overlap of individualcacades becomes important.

The results presented in Figs. 13—16 are averagesover many cascades. Fluctuations from one cascadeto another occur also, of course. The statistics of

The computations which have been presented inthis article show that the collision cascades pro-duced in monazite by energetic Ar atoms are gen-erally similar to cascades in metals. Although thereare differences in detail resulting from the lowersymmetry of the mineral and from the presence ofatoms of several masses, the sizes and shapes of cas-cades in monazite and the density of defects pro-duced are generally similar to what has been foundin metallic targets. Several refinements are possiblein the calculations: first, the surface of the targethas been ignored as a potential sink for interstitialdefects. Pairing of the interstitials with surface sitescould lead to the preservation of more damage thanwas indicated here, particularly at low energies.This topic is currently being investigated. Anotherrefinement concerns the possible effects of inelasticenergy losses in producing displacement damage.Such effects are known to be important in the alkalihalides, but are widely ignored in oxides, particular-ly in ionic ones like MgO. Nevertheless, the ex-istence of particle tracks produced by energeticheavy particles shows that these effects could beimportant in at least some mineral materials. Theinclusion of appropriate analyses of the spatial dis-tribution of the electronic energy losses could beused to assess this possibility more fully. More de-tailed analysis of the local structure of the cascadeswould be usefu1. Evidence for the formation of dis-tinct subcascades was pointed out in Figs. 13 and 14.This topic needs further study. Moreover, the char-acterization of the cascade volume as a rightcylinder into which many cascades wi11 actually fitis not entirely satisfactory. A more detailed analysisof the individual cascades might lead to a more

5358 MARK T. ROBINSON

satisfying description.From the dimensions obtained for collision cas-

cades in CePO4, the ion dose at which cascade over-lap commences was estimated as about 1.5X10'Ar/cm . This value is near the lowest used in theexperiments of Cartz et al. " ' Only a few defectsneed survive from each collision cascade to providea sufficient source of defects to lead eventually tothe amorphization (metamictization) of the target.The mechanism by which amorphization comesabout cannot be deduced from the present calcula-tions, of course, but must clearly be associated withthe aggregation of defects from many cascades, notwith the collapse of individual ones.

The role of electron excitations in producing dis-placement damage in materials such as the mona-zites can be worked out experimentally by usingbombardments with ions of different masses and ve-

locities. However, effects such as track formation,while they might be significant at very low doses,are unlikely to be so at the high doses of the Cartzexperiments. A track produced by one incident ionis likely to be dispersed by the collision cascadesproduced by subsequent ions. Thus, only the dam-

age produced by the last few arriving ions could bepreserved. It is unlikely that the observations are as-sociated with such events.

Insofar as damage in nuclear-waste isolationmedia and in natural minerals is concerned, thepresent calculations are not quite appropriate. Forthese topics, it would be necessary to make analo-gous studies of the cascades produced by the properparticles, namely, a particles of about 4-MeV initialenergy and heavy recoils of about 100-keV initial en-

ergy. Such calculations are entirely feasible withMA.RI.OWE. The cascades produced by the heavyrecoils may well be much more dense than those

computed here and their effect on the structure ofthe medium may well be different. It is unlikelythat the crystal structure of the target itself would

play more than a minor role in the structure of col-lision cascades. That is, one would not expect anysystematic differences between collision cascades ina monoclinic monazite and, say, a tetragonal xeno-time of the same chemical composition. The consid-erable differences in the behavior of differentminerals during irradiation must thus be attributedmainly to the stability of the defect arrays producedin the collision cascades, not to anything occurringin the production process.

Finally, it is interesting and perhaps significantthat 20—180-keV Ar+ ions render crystalline siliconamorphous starting at doses around 2X10' cmand saturating at(1.4—2.0)&(10' cm . ' It is tobeanticipated that the collision cascades produced byAr atoms in Si will be very similar to those in

Cep04, with some allowance made for the differ-ences in density and the fact that all of the atoms ofthe semiconductor have the same mass. That mona-

zite and Si become amorphous at similar doses of Arions suggests a connection between the mechanisms

of amorphization. It is probable that modeling simi-

lar to that proposed elsewhere for Si (Ref. 41) could

be applied usefully to the amorphization of mona-

zite and other minerals.

ACKNOWLEDGMENTS

It is a pleasure to acknowledge many fruitful dis-cussions of the work presented here with O. S. Oenand F. W. Young, Jr. This research has been spon-sored by the Division of Materials Science, U.S.Department of Energy, under Contract No. W-7405-eng-26 with Union Carbide Corporation.

iAlternate Nuclear Waste Forms and Interactions in Geo-logic Media, U.S. Department of Energy Report No.CONF-8005107, 1981 (unpublished).

W. J. Weber, R. P. Turcotte, and F. P. Roberts, Ra-dioact. Waste Manage. 2, 295 (1982).

R. A. Van Konynenburg and M. W. Guinan, NuclearTechnol. 60, 206 (1983)

4A. Pabst, Am. Mineral. 37, 137 (1952).5J. Graham and M. R. Thornber, Am. Mineral. 59, 1047

(1974); 60, 734 (1975).6R. C. Ewing, Am. Mineral. 60, 728 (1975).7R. C. Ewing, Ref. 1, pp. 81—99.M. M. Abraham, L. A. Boatner, G. W. Beall, C. B.

Finch, R. J. Floran, P. G. Huray, and M. Rappaz, Ref.1, pp. 144—169.

9L. A. Boatner, G. W. Beall, M. M. Abraham, C. B.Finch, R. J. P. G. Huray, and M. Rappaz, in Manage-ment of Alpha Contaminated Wastes (International

Atomic Energy Agency, Vienna, 1981),pp. 411—422.F. G. Karioris, K. Appaji Gowda, and L. Cartz, Radiat.Eff. Lett. 58, 1 (1981)~

"L.Cartz, F. G. Karioris, and K. Appaji Gowda, Radiat.Eff. Lett. 67, 83 (1981).F. G. Karioris, K. Appaji Gowda, L. Cartz, and J. C.Labbe, J. Nucl. Mater. 108-109,748 (1982).T. C. Ehlert, K. Appaji Gowda, F. G. Karioris, and L.Cartz, Radiat. Eff. 70, 173 (1983).

' E. R. Vance and K. K. S. Pillay, Radiat. Eff. 62, 25(1982).P. Sigmund, Rev. Roum. Phys. 17, 823 (1972); 1'7, 969(1972); 17, 1079 (1972).

M. T. Robinson, in Radiation Damage in Metals, edited

by N. L. Peterson and S. D. Harkness (American So-ciety of Metals, Metals Park, Ohio, 1976), pp. 1—27.K. B.Winterbon, Nucl. Sci. Eng. 53, 261 (1974).N. Andersen and P. Sigmund, K. Dan. Vidensk. Selsk.


Mat. -Fys. Medd. 39, No. 3 (1974).C. A. Coulter and D. M. Parkin, J. Nucl. Mater. 88,249 (1980).

20M. M. R. Williams, Prog. Nucl. Energy 3, 1 (1979).T. J. Hoffman, H. L. Dodds, Jr., M. T. Robinson, andD. K. Holmes, Nucl. Sci. Eng. +8, 204 (1978).O. S. Oen, D. K. Holmes, and M. T. Robinson, J. Appl.Phys. 34, 302 (1963).M. T. Robinson and O. S. Oen, Phys. Rev. 132, 2385(1963).J. P. Biersack and L. G. Haggmark, Nucl. Instrum.Methods 174, 257 (1980).

25E. S. Mashkova, Radiat. Eff. 54, 1 (1981).M. T. Robinson, in Sputtering by Particle Bombardment

I, Vol 47 of Topics in Berlin Physics, edited by R.Behrisch (Springer, Berlin, 1981),pp. 73—144.J. R. Beeler, Jr., and D. G. Besco, J. Appl. Phys. 34,2873 (1963).

28J R Beeler Jr Phys Rev 150 470 (1966)M. T. Robinson, K. Roessler, and I. M. Torrens, J.Chem. Phys. 60, 680 (1974).M. T. Robinson and I. M. Torrens, Phys. Rev. B P, 5008(1974).

~M. T. Robinson, User's Guide to MARLOWE (Version

11), 1978 (unpublished), describes a version of the pro-

gram which is available from the National EnergySoftware Center, Argonne National Laboratory, Ar-

gonne, Illinois 60439, and from the Radiation Shielding

Information Center, Oak Ridge National Laboratory,

P.O. Box X, Oak Ridge, Tennessee 37830. The greatly

modified version of the program used for most of the

present calculations is designated Version 11.8.3 G. W. Beall, L. A. Boatner, D. F. Mullica, and W. O.

Milligan, J. Inorg. Nucl. Chem. 43, 101 (1981).M. T. Robinson, J. Appl. Phys. (in press).

340. B. Firsov, Zh. Eksp. Teor. Fiz. 33, 696 (1957) [Sov.Phys. —JETP 6, 534 (1958)].J. Lindhard and M. Scharff, Phys. Rev. 124, 128

(1961);J. Lindhard, M. Scharff, and H. E. Schist, K.Dan. Vidensk. Selsk. , Mat. -Fys. Medd. 33, No. 14

(1963).M. T. Robinson, in Nuclear Fusion Reactors (British

Nuclear Energy Society, London, 1970), pp. 364—378.37J. Lindhard, V. Nielsen, M. Scharff, and P. V. Thom-

sen, K. Dan. Vidensk. Selsk. , Mat. -Fys. Medd. 33,No. 10 (1963).P. Sigmund, Appl. Phys. Lett. 14, 114 (1969).K. B. Winterbon, P. Sigmund, and J. B. Sanders, K.Dan. Vidensk. Selsk. , Mat. -Fys. Medd. 37, No. 14

(1970).R. L. Fleischer, P. B. Price, and R. M. Walker, Nuclear

Tracks in Solids (University of California Press, Los

Angeles, 1975).J. R. Dennis and E. B. Hale, J. Appl. Phys. 4+, 1119(1978).

Documents

Computer simulation of collision cascades in monazite