- Email: [email protected]
The formation of vacancy-type defect clusters by ion bombardment: A problem revisited by molecular dynamics
J
Pergamon
Ph,vc
00223697(94)00079-94
Chem.
Soltdr Vol 55. No. IO. pp. %5-953. 1994 Copyroght a’~ 1994 Elscwcr Soence Ltd Pnnted in Circa Elntam All rights reserved 0022-3697/H 17.00 + 0.00
THE FORMATION OF VACANCY-TYPE DEFECT CLUSTERS BY ION BOMBARDMENT: A PROBLEM REVISITED BY MOLECULAR DYNAMICS MA1 GHALYt tDepartment
and R. S. AVERBACKQ
of Nuclear Engineering and IDepartment University of Illinois at Urbana-Champaign.
Science and Engineering, Urbana, IL 61801, U.S.A.
of Materials
Abstract-In 1969 Professor Ball&i and students L. E. Thomas and T. Schober presented a series of three papers on the nature of defects produced by keV Au irradiation of Au (Thomas er al.. Radiut. E~crs 1, 257, 269. 279 (1969)]. They noted that vacancy clusters were produced with high efficiencies but could not find evidence for interstitial clusters. The authors interpreted their data on the basis of replacement collision sequences carrying interstitial atoms beyond the cascade core, leaving behind a vacancy rich depleted zone which subsequently collapsed into a visible dislocation loop. We have simulated this experiment by molecular dynamics and can now predict much of what these authors reported in 1969. However, 25 years later we offer a new explanation for the creation of vacancy clusters based on viscous flow of liquid metal through the irradiated surface. Keywords:
D. defects,
D. radiation
damage.
lished in 1969 [2]. In these papers, following:
1. INTRODUCTION
Defect production in energetic displacement cascades has been a fundamental concern in the field of radiation effects for over 40 years. In the early days, most interest in cascades centered on radiation
irradiation along [OOl] directions at energies between 3 and 40 keV. The probability of a single ion producing a visible cluster increased from 0.05 at 3 keV to near unity by 10 keV. (ii) At 20 keV, the number of visible defect clusters increased linearly with ion dose in the range 0.5-10 x lO’“cm’, thus establishing that the visible
intermetallic compounds, radiation-induced stress relief at surfaces, self-organization of damage structures and ion beam mixing, to name a few, but common to all of these topics is the need to understand atomic processes in displacement cas-
defects wcrc produced in individual cascades. (iii) The defects were all of vacancy character. (iv) The visible clusters were produced close to the surface with the mean depth increasing with ion
cades. In the mid-1960s transmission electron microscopy (TEM) was one of the few experimental
energy. The depth distributions of the visible defect clusters were broad, extending from the surface to about the depth of ion penetration. (v) The diameters of the clusters increased as the square root of energy, being about equal to the
methods with sufficient spatial resolution to probe the structure of radiation-induced defects [I]. It was found at this time that individual ion impacts in pure fee metals held at room temperature could lead to the formation of dislocation loops although the nature of these loops, vacancy or interstitial, the
predictions of a modified KinchinPcase model. (vi) At increasingly higher energies, the defect clusters broke up into “subcascades,” with IS keV being the critical energy for subcluster formation. (vii) Many loops were lost to the surface by glide.
threshold energy necessary to produce visible defect clusters and the role of channeling were all uncertain [I, 21. Surprisingly, these same questions are being debated 30 years later! Nevertheless, Bob Balluffi and students L. E. Thomas and T. Schober provided much insight into these questions in a series of papers
on self-ion
bombardment
the
(i) Defect clusters visible by TEM were produced in Au with relatively high probability by self-ion
damage in metals and ceramics for applications associated with atomic energy. Today these concerns have broadened to include radiation-induced amorphization and crystallization in semiconductors and
three
they showed
Considerable effort has been expended since 1969 in trying to understand the Balluffi experiments. For example, it was noted by Balluffi
of Au pub945
946
M. GHALY
and R. S. AVERBACK
and students that the number of vacancies in each loop agreed with the calculations of Kinchin and Pease. We know today that in nearly all metals the Kinchin and Pease formula overestimates defect production by a factor of three or more. The Balluffi group’s result is particularly perplexing since many
0 0 0 0 o”ooooooooooo
0
~~“~o~o~o~o~o ___h__
0
0
ONormalatom 0 Interstitialatom -Pathofprimaryparticle
vacancies in their experiments should have been lost by recombination with interstitials or by migration to the surface, since the experiments were performed at 300 K. We must thus ask why is the defect production efficiency so high in these experiments. A second question raised by this work is why no interstitial clusters were created during the irradiation. Experiments by JHger and Merkle confirmed that only vacancy loops are produced by such self-ion irradiations [3]. On the other hand, Roualt et al. reported that either vacancy or interstitial clusters could be produced in Au depending on the mass and energy of the irradiation particle. Vacancy clusters were more common when the ion’s mass was large and its energy low [4]. More recently, studies of defect clusters in Au and other metals produced by neutron irradiation indicate that interstitial as well as vacancy clusters can be produced, depending on the distance of the cluster from the irradiated surface [5]. The present paper concerns the mechanisms by which defect clusters are formed in displacement cascades. We first briefly review past ideas on this subject. We then describe recent results from computer simulations that reveal the role of local melting on cluster formation. Finally, we offer a fundamentally new picture of how defects are created in Au during heavy ion bombardment and generalize this picture to other materials through a macroscopic model.
2. OLD PICTURES OF DEFECT IN CASCADES
PRODUCTION
One of the earliest pictures of cascade processes is due to Brinkman [6]. He realized that when an energetic particle undergoes an energetic collision with a target atom, the two atoms leave the lattice site at much reduced energy, Since the collision crosssection increases rapidly with decreasing energy, the two atoms undergo almost immediate secondary collisions with neighboring atoms and these secondary recoils do the same. During this cascade of collisions, many atoms are ejected from a localized region of the material and a depleted zone is left behind, as depicted in Fig. 1. Within this picture, the formation of vacancy clusters follows naturally. If the concentration of vacancies in the depleted zone is sufficiently high, the lattice becomes unstable and collapses to a dislocation loop. Whether this would occur spontaneously or require thermal activation
Fig. 1. Brinkman’s depiction of a displacement spike and the formation of a depleted zone (after Brinkman [6]).
was long questioned,
but recent
experiments
have
shown that vacancy clusters can be produced at 10 K in some metals [7]. The Brinkman picture was refined several times in subsequent years. In 1960, Gibson et al. published their classic paper illustrating by molecular dynamics computer simulation (MD) that interstitials are efficiently separated from vacancies via replacement collision sequences [8]. A replacement sequence is generated by one atom recoiling in a close pack direction and immediately colliding with its nearest neighbor. The primary knock-on atom takes the lattice site of the neighbor while the neighbor repeats the process with the next neighbor in the row. Eventually, the recoiling atom has too little energy to replace its neighbor and becomes an interstitial atom. The vacancy is located at the initial recoil site. Thus an efficient means was identified for separating vacancies and interstitials. In parallel efforts, computer simulations using the binary collision approximation (BCA) were developed to treat high energy cascades [9, lo]. These simulations revealed that the density of defects in cascades was indeed high and that clustering of defects was possible [lo]. Wei and Seidman mapped out in atomic detail the vacancy structure in cascades produced by - 30 keV self-ions in W using field ion microscopy and found reasonable agreement with these simulations [l 11. Somewhat later, Sigmund used Boltzmann transport theory to calculate defect production and energy densities in cascades and found that defect densities were very high in materials with large atomic numbers [12, 131. He also noted that the energy densities were extremely high in these same materials and that the assumption that moving atoms only collide with stationary atoms, an assumption of the simulation
941
Vacancy-type defect clusters models (excluding MD) and transport theory, was not really justified. When these various ideas are synthesized, the following picture of defect formation in cascades ensues, a picture which was more or less universally accepted. The primary knock-on atom initiates a displacement cascade. Many atoms are ejected far from the cascade core via replacement collision sequences, leaving behind a depleted zone. In metals with high atomic number, the concentration of vacancies can be very large, large enough in less refractory metals for the lattice to become unstable and collapse to form vacancy loops. Nearly all accepted experiments on the production of defect clusters in cascades, including those of Balluffi and students, had been interpreted according to this picture. 3. THERMAL SPIKES AND CASCADE COLLAPSE:. BULK EFFECTS Calculations of energy densities within the cores of cascades in Au yield values of a few eV per atom and thus lead to the question of the role of thermal spikes 1131.A few years ago, one of the authors, along with R. Benedek and student T. Diaz de la Rubia, approached this question through molecular dynamics computer simulations [14J. They found that for a 5 keV cascade in Cu thermal spikes did exist and that
#ml 5 keV Cu
I
I
I
I
I
I
20
40
60
a0
loo
2.8 2.4 2.0 F = G
I
local melting in the cores of cascades was of fundamental importance in determining the defect structure in energetic displacement cascades. Evidence for local melting is illustrated in Fig. 2(a, b). In Fig. 2(a), a snapshot of the atoms within a cross-sectional slab of thickness a,/2 is shown, revealing the appearance of a local melt region. More detailed analysis of this region is performed in Fig. 2(b), where the pair correlation function for the atoms in that region is shown and compared with g(r) for liquid Cu. Figure 3 shows the defect structure of the 5 keV cascade in Cu. The central core of the cascade consists of a high concentration of clustered vacancies and no interstitials. The interstitial atoms are all located on the periphery of the cascade. Most were ejected from the cascade core by replacement collision sequences. One cluster of four interstitials was produced just beyond the farthest extension of the melt zone, possibly by short replacement sequences. How they clustered was unclear from this work, but probably by migration during the thermal spike under the influence of their strong mutual interaction. In closer connection to the Balluffi work, a 10 keV cascade in Au was recently simulated [15]. The evolution of this event is illustrated by a time-sequence of snapshorts of the atomic positions, in Fig. 4. Local melting is a primary feature of the cascade. Other interesting features in the dynamics are the shock wave seen propagating outward at about the speed of sound and extensive cavitation in the core of the cascade. In Fig. 4(d), the final locations of the vacancies and interstitials are projected onto a (100) plane, showing their positions relative to the once-melted cascade core. It is seen that all vacancies are located in the once-melted zone while all interstitials are located outside of this region. The spike conditions are more clearly revealed in Figs 5 and 6 through temperature and pressure profiles, respectively. The
120
5 keV WA I” Cu 0 t* I.1 pr a 1=3.6ps fW ltquid Cu (S. F&es!
-* colcuI01ed
1.6 1.2
60-
+ D-4 ,,^.Qo. *. 0 . . /’ . -. -. c/
l
q!iB215’
i5 ’ 4.5’ ’ 55’ ’
6.5
Rod&i)
Fig. 2. A 5 keV cascade in Cu: (a) locations of atoms within a (100) cross-sectional slab of thickness &2, viewed along the (100) direction, 1.1 ps after the event was initiated; (b) radial distribution function for atoms in the disordered zone at two instant of time and for liquid Cu (after Ref. [14]).
Fig. 3. Primary state of damage for a 5 keV cascade in Cu obtained by MD (after Ref. [14]).
948
M. GBALY and R. S. AVERBACK
Fig. 4. A IO keV cascade in Au: locations of atoms within a (100) cross-sectional slab of thickness a,/2, viewed along the (IOO} direction at several instants of time (after Ref. [15]).
maximum temperature after a few lattice vibrations is -6000 K, which is a few thousand K below the estimated critical temperature of Au. The maximum pressure outside the melt zone is z3dGPa. This latter quantity is of interest in regard to loop punching, which is an often discussed mechanism for producing interstitial clusters [l6, 171. Loop punching requires pressures of the order of half the shear modulus 118, 191, which for Au is 228 GPa. Hence
loop punching would not be expected in this case and it was not observed. As a result of these MD simulations, the scenario for the production of defect clusters and loops previously discussed can be refined. In the initial stages of the cascade evolution, i.e. in the so-called collisional phase, interstitials are ejected from the cascade core by short replacement collision sequences. As the primary recoil energy becomes partitioned among all
949
Vacancy-type defect clusters
8000 u^ i 1 g
6000 4000 2000
0
2
4 6 Distance (8,)
8
10
Fig. 5. Temperature as a function of distance from the amter of energy for a 10 keV cascade in Au at various instants of time (after Ref. [15J). atoms in the cascade, the cascade core undergoes local melting. All of the interstitials within this region are absorbed in the melt and quickly delocalize. This explains the reduced defect production relative to simulations employing BCA. Local fluctuations in the liquid density resulting from the vacancies may persist, depending on the time for the free volume to homogenize. In the final phase of the evolution, the melt begins to solidify, with the crystallization front moving from the surrounding, cooler solid toward the center of energy of the cascade. Free volume is dragged along with the interface since the free energy of a vacancy is much less in the liquid. Thus, similar to zone refining of impurities, vacancies are swept along with the moving interface, with the fraction being left behind depending on their concentration
and the speed of the interface. At the core of the cascade, the concentration of vacancies can be greatly enhanced owing to this zone refining process and the conditions required for cascade collapse can thereby be established. For metals which have high energy density cascades and low melting temperatures, like Au, the interface moves very slowly allowing for efficient zone refining of vacancies. In metals with higher melting temperatures and lower energy densities, the interface moves too quickly for the flux of free volume to keep up and vacancies become frozen in throughout the cascade volume. Interstitial clusters often remain on the periphery of the once melted region, although how and why the interstitials cluster just at the melt edge is uncertain. It may be that those interstitials just outside the melt
10 8 i$
6
i 4
0
2
8
10
Fig. 6. Pressure as a function of distant from the center of energy for a 10 keV cascade in Au at various instants of time (after Ref. [IS]).
M. GHALY and R. S. AVERBACK
950
are stabilized by a large pressure gradient which prevents them from recombining with “vacancies” in the melt. Moreover, since they are very mobile during the thermal spike and strongly interact with each other, close interstitials would tend to cluster. This picture of interstitial clustering is currently being examined.
0.3 ps
W
4. CASCADES AT SURFACES: EFFECTS OF CONVECTIVE FLOW A qualitative understanding of the cascade dynamics at surfaces can be gleaned from Fig. 7, where snapshots of the atomic positions are again illustrated at various instants of time [20]. It is seen that the initial stages of the cascade evolution are similar to
1.01 ps
9.6 ps
(4
Fig, 7. Locations of atoms within a (100) cross-sectional slab of thickness a&, viewed along the (lOO> direction, at various instants of time. The initial velocity of the projectile had polar and azimuthai angles of 160 and ‘W, respectively.
Vacancy-type defect clusters those in the interior of the crystal. Interstitials are ejected from the central core of the cascade, creating a depleted zone, while a shock wave follows behind. Cavitation is again a prominent feature. The onset of the thermal spike becomes evident in Fig. 7(c) with the appearance of the local melt. It is also seen in Fig. 7(c) that at this point in the cascade evolution the surface cascade begins to deviate from the one in the bulk. The interstitial atoms which were initially transported to the periphery of the melt zone are now effectively re-absorbed in the melt. Only those interstitials deposited at the ends of long replacement sequences escape this fate. In this event, there are only three. In the bulk cascade, 10 times as many interstitials remained stable on the periphery of the melt zone. Stable interstitial clusters were also observed in simulations of bulk 10 keV cascades in Cu [16], Ni [21] and Ag [22] and they appear to be a common feature in simulations of cascades in pure FCC metals. Too few events near surfaces have been simulated to judge whether interstitial clusters are, in general, unstable near surfaces, as observed in the Au event. It may be that the relief of pressure in the surface cascade, before the cascade cools, enables the interstitials to relax back into the melt. This would, of course, explain why interstitial clusters were not observed by Thomas et al. and others after keV self-ion bombardment of Au and other metals. The further development of this cascade is conveniently described in macroscopic terms. As the cascade zone heats to a few thousand Kelvin and melts, its molar volume significantly increases,
-..
.
951
%6%, creating high internal pressure. A pressure gradient is established across the melt, since the normal component of stress at the surface is zero, and liquid is thus forced onto the surface, as seen in Figs 7(c-f). About 550 atoms flowed to the surface in this event. The final stages of the cascade evolution involve cooling and resolidification as described for bulk cascades. In the bulk, the ejection of interstitial atoms from the core of the cascade was responsible for the excess vacancies. In this case, the number of interstitials was approximately one-quarter of that predicted by the Kinchin-Pease formula. For a 10 keV cascade this is x 30 defects. Far fewer interstitials are created near the surface, only three in the simulation event, but many atoms leave the core of the cascade by their convective flow to the surface. Consequently, the number of vacancies accumulated in clusters near surfaces can far exceed their corresponding number in bulk cascades. This leads to a rather extensive dislocation structure. In Fig. 8 this structure is represented by plotting the locations of atoms with high potential energy as they outline the cores of the dislocations. This mechanism of convective flow explains why TEM experiments find many more defects in loops than are believed to be created in the bulk. The thermal spike behavior observed in these simulations at the surface of Au is rather pronounced. Au, however, has a very high atomic number and a relatively low melting temperature and whether the mechanisms for loop formation outlined above apply to other materials is uncertain. We are
.:. .- ,::‘s:.. :..*.*..* . I. ..::; ....
Fig. 8. Structure of the dislocation network as illustrated by the positions of atoms with potential energies 20.40 eV. The “hole” in the surface inidicates the presence of adatoms which are not shown.
M. GHALY and R. S. AVERBACK
952
beginning MD simulations of ion bombardment in metals like Cu and Ni to address this question, but already considerable insight can be gleaned from the following thermal spike model [23]. For convenience, we assume that an energetic particle travels deep into the substrate while dissipating energy according to F&r, z) = FJ(r
= O),
(1)
where F,, is the rate of damage energy deposition along the path of the ion track normal to the surface. The energy then spreads, creating a cylinder of liquid melt. Pressure is developed in the liquid owing to the change of volume and thus liquid is forced onto the surface. The maximum radius of the liquid cylinder is obtained from conservation of energy,
where n, is the atomic density and t, is the average energy of an atom in the melt. The number of atoms passing through the surface, Q, is given by Q = nOuAr,
(3)
where A = rrr* and T is the lifetime of the melt. T is obtained from 405 = r’,
(4)
where D is the thermal diffusivity. We approximate the velocity of the fluid by Poiseuille’s equation for viscous flow, r*AP 0)
“=w)
where p is the viscosity of the liquid and AP is the drop in pressure over a distance I. The pressure drops from a maximum within the tube of liquid to zero at the surface over a distance I, which we approximate by
Q = An&,
(6)
Q =($$AP)‘“r4,
(7)
so that
where bn, is the difference in atomic densities in the melt and solid. The pressure created in the tube, far from the surface, is 1 AV -AP=
KV _
2(1+v)
I+-----
EK
where K is the compressibility of the liquid, E is the elastic modulus of the solid and v is Poisson’s ratio. Combining all terms, we obtain
For the case of 10 keV Au bombardment of Au p = 6.0 x lo**m-’ An&, = 0.06, K = 1.5 x 10’” m2N-‘, E = 78 &Pa, v = l/3, p = 5.2 x IO-* Poise and c,,,=0.3 eV. We approximate F,, x 15OOeVnm-‘. The thermal diffusivity is obtained from the MD simulations, D = 0.2 x IO’*nm* s-l so that the lifteime of the spike given by eqn (4) agrees with the lifetime deduced from the simulations. With these values, we find Q z 1000 atoms. Although this good agreement with the simulations must be considered fortuitous in view of the approximate nature of the calculation, the agreement does support the basic concept. Flow in other materials can be similarly estimated, but except for the term (F,/c,)*, differences in the various parameters are not great (or cancel). It might be assumed that viscous flow of atoms through the surface is strictly a low energy phenomenon and that at high energies the damage is created at too great a depth, but this is not true. The energy deposition along the track of the ion is described by the nuclear stopping power, S,(E) = l/n, dE/dx, which is a universal function when expressed in reduced units. The nuclear stopping power has a maximum at the reduced energy, L % 0.3. For self-ion irradiation of Cu, Ag and Au, this corresponds to ~60, 215 and 720 keV, respectively, i.e. the ratio, (F,/c,)*, is higher at higher rather than lower energies than that simulated here. Since sputtering yields also depend on (F,/c,), examination of sputtering yields provides an approximate means to determine whether viscous flow at surfaces will be an important effect. The sputtering yield for 10 keV self-ion bombardment of Au is ~20 [24]. For 70 keV Kr irradiation of Cu, the sputtering yield is z 13, and for this irradiation, about one-third of the ions produced visible dislocation loops [25]. For 30 keV self-ion irradiation of W at 10 K, the sputtering yield is a 12 and occasional loops, but more commonly highly depleted zones, were observed by field ion microscopy imaging [ 111.A question often raised in regard to the FIM experiments is why the observed number of interstitial atoms is only a small fraction of the number of observed vacancies. Several possibilities have been identified [I 11,but we add here the possibility of the loss of atoms to the surface by viscous flow.
Vacancy-type defect clusters Finally, it is of interest to note scent TEM investigations of f4 MeV neutron irradiation of Au at low temperatures [5]. In this study, specimens having a wedge geometry were irradiated at IO K and transferred into the TEM without warming above 20K. The authors report that in the very thin portions of the wedge, i.e. at depths up to 80 nm, defect clusters with vacancy character were observed, but in the thicker part of the wedge, the character of the defect cluster changed from vacancy to interstitial. The character of the clusters were determined by the so-tailed 2f- D method and the validity of this technique has been controversial. Nevertheless, the results of this study are tantalizing in view of these simulations.
953
R. Ben&k, W. Rger and F. Ehrhart are gratefully acknowied8ed. REFERENCES f . Merkle K. L., Radiation Bmage fn Metals (Edited by S. D. Harkness and N. L. Peterson), p. 58. American Society af Metals, Metals Park, OH (1976). 2. Thomas L. E., Schober T. and Ballti R. W., Radiaf. Effects 1, 257, 269, 279 (1969). 3. JIger W. and Merkle K. L., Phil. Mug. A 44,741 (1981). 4. Ruault M. O., Bemas H. and Chaumont J., Phil. Msg. A 39, 757 (1979). 5. Fukushima.F., ~~~~rnura Y., Guinan M. W., Hahn P. A. and Kiritani M.. J. NM&Marer. 179-181.943 0991% 6. Brinkman J. A., f. Appf. Phys. 2!!, 951 (1954i A& 3. Phys. 24, 246 (1956). 7. English C. A. and Jenkins M. L., Mzter. Sci. Forum, M-18, 1003(1987). 8. Gibson J. B.. Goland A. N., Milrrram M. and Vinevard G. H., P&: Rev. 120,1229 (19%0). 9. Beeler J. R. Jr and BescoD. G.., J. Am& Phvs. 35.2515 II
I
I
(1963).
10. Robinson M. T. and Torrens I. M., Phys. Rev. B 9,500s In this paper we have shown that even 25 years
after the experiments on self-ion bombardment of Au by BailufE and students, the nature of defect clusters in irradiated metals and the underlying mechanisms for their production are still controversial subjects. We have used molecular dynamics simulations to investigate this problem and have revealed a mechanism that was not considered by Balluffi et al., nor by anyone else. The impIications of this mechanism of surface melting and viscous flow are enormous, since much of our understanding of radiation damage in metals has been derived from energetic ion irradiations and all ion beam processing involves surface damage. Much work remains, however, both to verify the m~hanism suggested by the simulations through ex~riments and to explore how widespread the e&et is.
research was supported by the U.S. DOE, Basic Energy Sciences, under grant DEFGO291ER45439. Discussions with Drs T. Diaz de la Rubia, Acknowledgements-The
($974).
Il. Wei C.-Y. and Seidman D. N., Phil. &fag. A 37, 257 (1978). 12. Sigmund P., Phvs. Rev. 184,383 (1969);Phvs. Rev. 187, 76% (197.5). . 13. Sigmund P., Appi. Phys. Lett. 25, 169 (1974); Appl. Phys, Z.&t. 27, 52 (1975). 14. Diaz de la Rubia T., Averback R. S., Benedek R. and Kinn W. E.. Pkvs. Rm. L&r. 38. 1930 11987). 15. Gl& M. a&d Averback R. S. ~npubl~s~~j. 16. Diar de la Rubia T. and Guinan M. W., [email protected]. 66, 2766 (1991). 17. Van Swy8enhoven H. and Caro A., X-‘&s.&v. LRtf.70, 2098 (1993). 18. Trinkaus fi., Rudiut. Effects 78, 189 (1983). 19. Wolfer W. G.. Phil. Mae. A 58. 285 119881. 20. Ghaly M. and’Averbackk. S., dhys, lbv. btt. 72,364 (1994). 21. Zhu H. and Averback R. S. ~~~ub~s~~}. 22. Ghaly M., L&z de la Rubia T. and Averback R. S. (in press). 23. Averback R. S. and Gbaly M. (in press). 24. Andersen H. H. and Bay H. L., in Spuitering by J%&cfe 3om&ffrdmenr (Edited by R. Behrisch), Chap. 4. Springer, Berlin (198 1). 25. Kirk M. A., Robertson I. M., Jenkins M, L., English C. A., Black T. J. and Vetrano J. S., J. Nucl. Mater. 149. 21 (1987).
Pergamon
Ph,vc
00223697(94)00079-94
Chem.
Soltdr Vol 55. No. IO. pp. %5-953. 1994 Copyroght a’~ 1994 Elscwcr Soence Ltd Pnnted in Circa Elntam All rights reserved 0022-3697/H 17.00 + 0.00
THE FORMATION OF VACANCY-TYPE DEFECT CLUSTERS BY ION BOMBARDMENT: A PROBLEM REVISITED BY MOLECULAR DYNAMICS MA1 GHALYt tDepartment
and R. S. AVERBACKQ
of Nuclear Engineering and IDepartment University of Illinois at Urbana-Champaign.
Science and Engineering, Urbana, IL 61801, U.S.A.
of Materials
Abstract-In 1969 Professor Ball&i and students L. E. Thomas and T. Schober presented a series of three papers on the nature of defects produced by keV Au irradiation of Au (Thomas er al.. Radiut. E~crs 1, 257, 269. 279 (1969)]. They noted that vacancy clusters were produced with high efficiencies but could not find evidence for interstitial clusters. The authors interpreted their data on the basis of replacement collision sequences carrying interstitial atoms beyond the cascade core, leaving behind a vacancy rich depleted zone which subsequently collapsed into a visible dislocation loop. We have simulated this experiment by molecular dynamics and can now predict much of what these authors reported in 1969. However, 25 years later we offer a new explanation for the creation of vacancy clusters based on viscous flow of liquid metal through the irradiated surface. Keywords:
D. defects,
D. radiation
damage.
lished in 1969 [2]. In these papers, following:
1. INTRODUCTION
Defect production in energetic displacement cascades has been a fundamental concern in the field of radiation effects for over 40 years. In the early days, most interest in cascades centered on radiation
irradiation along [OOl] directions at energies between 3 and 40 keV. The probability of a single ion producing a visible cluster increased from 0.05 at 3 keV to near unity by 10 keV. (ii) At 20 keV, the number of visible defect clusters increased linearly with ion dose in the range 0.5-10 x lO’“cm’, thus establishing that the visible
intermetallic compounds, radiation-induced stress relief at surfaces, self-organization of damage structures and ion beam mixing, to name a few, but common to all of these topics is the need to understand atomic processes in displacement cas-
defects wcrc produced in individual cascades. (iii) The defects were all of vacancy character. (iv) The visible clusters were produced close to the surface with the mean depth increasing with ion
cades. In the mid-1960s transmission electron microscopy (TEM) was one of the few experimental
energy. The depth distributions of the visible defect clusters were broad, extending from the surface to about the depth of ion penetration. (v) The diameters of the clusters increased as the square root of energy, being about equal to the
methods with sufficient spatial resolution to probe the structure of radiation-induced defects [I]. It was found at this time that individual ion impacts in pure fee metals held at room temperature could lead to the formation of dislocation loops although the nature of these loops, vacancy or interstitial, the
predictions of a modified KinchinPcase model. (vi) At increasingly higher energies, the defect clusters broke up into “subcascades,” with IS keV being the critical energy for subcluster formation. (vii) Many loops were lost to the surface by glide.
threshold energy necessary to produce visible defect clusters and the role of channeling were all uncertain [I, 21. Surprisingly, these same questions are being debated 30 years later! Nevertheless, Bob Balluffi and students L. E. Thomas and T. Schober provided much insight into these questions in a series of papers
on self-ion
bombardment
the
(i) Defect clusters visible by TEM were produced in Au with relatively high probability by self-ion
damage in metals and ceramics for applications associated with atomic energy. Today these concerns have broadened to include radiation-induced amorphization and crystallization in semiconductors and
three
they showed
Considerable effort has been expended since 1969 in trying to understand the Balluffi experiments. For example, it was noted by Balluffi
of Au pub945
946
M. GHALY
and R. S. AVERBACK
and students that the number of vacancies in each loop agreed with the calculations of Kinchin and Pease. We know today that in nearly all metals the Kinchin and Pease formula overestimates defect production by a factor of three or more. The Balluffi group’s result is particularly perplexing since many
0 0 0 0 o”ooooooooooo
0
~~“~o~o~o~o~o ___h__
0
0
ONormalatom 0 Interstitialatom -Pathofprimaryparticle
vacancies in their experiments should have been lost by recombination with interstitials or by migration to the surface, since the experiments were performed at 300 K. We must thus ask why is the defect production efficiency so high in these experiments. A second question raised by this work is why no interstitial clusters were created during the irradiation. Experiments by JHger and Merkle confirmed that only vacancy loops are produced by such self-ion irradiations [3]. On the other hand, Roualt et al. reported that either vacancy or interstitial clusters could be produced in Au depending on the mass and energy of the irradiation particle. Vacancy clusters were more common when the ion’s mass was large and its energy low [4]. More recently, studies of defect clusters in Au and other metals produced by neutron irradiation indicate that interstitial as well as vacancy clusters can be produced, depending on the distance of the cluster from the irradiated surface [5]. The present paper concerns the mechanisms by which defect clusters are formed in displacement cascades. We first briefly review past ideas on this subject. We then describe recent results from computer simulations that reveal the role of local melting on cluster formation. Finally, we offer a fundamentally new picture of how defects are created in Au during heavy ion bombardment and generalize this picture to other materials through a macroscopic model.
2. OLD PICTURES OF DEFECT IN CASCADES
PRODUCTION
One of the earliest pictures of cascade processes is due to Brinkman [6]. He realized that when an energetic particle undergoes an energetic collision with a target atom, the two atoms leave the lattice site at much reduced energy, Since the collision crosssection increases rapidly with decreasing energy, the two atoms undergo almost immediate secondary collisions with neighboring atoms and these secondary recoils do the same. During this cascade of collisions, many atoms are ejected from a localized region of the material and a depleted zone is left behind, as depicted in Fig. 1. Within this picture, the formation of vacancy clusters follows naturally. If the concentration of vacancies in the depleted zone is sufficiently high, the lattice becomes unstable and collapses to a dislocation loop. Whether this would occur spontaneously or require thermal activation
Fig. 1. Brinkman’s depiction of a displacement spike and the formation of a depleted zone (after Brinkman [6]).
was long questioned,
but recent
experiments
have
shown that vacancy clusters can be produced at 10 K in some metals [7]. The Brinkman picture was refined several times in subsequent years. In 1960, Gibson et al. published their classic paper illustrating by molecular dynamics computer simulation (MD) that interstitials are efficiently separated from vacancies via replacement collision sequences [8]. A replacement sequence is generated by one atom recoiling in a close pack direction and immediately colliding with its nearest neighbor. The primary knock-on atom takes the lattice site of the neighbor while the neighbor repeats the process with the next neighbor in the row. Eventually, the recoiling atom has too little energy to replace its neighbor and becomes an interstitial atom. The vacancy is located at the initial recoil site. Thus an efficient means was identified for separating vacancies and interstitials. In parallel efforts, computer simulations using the binary collision approximation (BCA) were developed to treat high energy cascades [9, lo]. These simulations revealed that the density of defects in cascades was indeed high and that clustering of defects was possible [lo]. Wei and Seidman mapped out in atomic detail the vacancy structure in cascades produced by - 30 keV self-ions in W using field ion microscopy and found reasonable agreement with these simulations [l 11. Somewhat later, Sigmund used Boltzmann transport theory to calculate defect production and energy densities in cascades and found that defect densities were very high in materials with large atomic numbers [12, 131. He also noted that the energy densities were extremely high in these same materials and that the assumption that moving atoms only collide with stationary atoms, an assumption of the simulation
941
Vacancy-type defect clusters models (excluding MD) and transport theory, was not really justified. When these various ideas are synthesized, the following picture of defect formation in cascades ensues, a picture which was more or less universally accepted. The primary knock-on atom initiates a displacement cascade. Many atoms are ejected far from the cascade core via replacement collision sequences, leaving behind a depleted zone. In metals with high atomic number, the concentration of vacancies can be very large, large enough in less refractory metals for the lattice to become unstable and collapse to form vacancy loops. Nearly all accepted experiments on the production of defect clusters in cascades, including those of Balluffi and students, had been interpreted according to this picture. 3. THERMAL SPIKES AND CASCADE COLLAPSE:. BULK EFFECTS Calculations of energy densities within the cores of cascades in Au yield values of a few eV per atom and thus lead to the question of the role of thermal spikes 1131.A few years ago, one of the authors, along with R. Benedek and student T. Diaz de la Rubia, approached this question through molecular dynamics computer simulations [14J. They found that for a 5 keV cascade in Cu thermal spikes did exist and that
#ml 5 keV Cu
I
I
I
I
I
I
20
40
60
a0
loo
2.8 2.4 2.0 F = G
I
local melting in the cores of cascades was of fundamental importance in determining the defect structure in energetic displacement cascades. Evidence for local melting is illustrated in Fig. 2(a, b). In Fig. 2(a), a snapshot of the atoms within a cross-sectional slab of thickness a,/2 is shown, revealing the appearance of a local melt region. More detailed analysis of this region is performed in Fig. 2(b), where the pair correlation function for the atoms in that region is shown and compared with g(r) for liquid Cu. Figure 3 shows the defect structure of the 5 keV cascade in Cu. The central core of the cascade consists of a high concentration of clustered vacancies and no interstitials. The interstitial atoms are all located on the periphery of the cascade. Most were ejected from the cascade core by replacement collision sequences. One cluster of four interstitials was produced just beyond the farthest extension of the melt zone, possibly by short replacement sequences. How they clustered was unclear from this work, but probably by migration during the thermal spike under the influence of their strong mutual interaction. In closer connection to the Balluffi work, a 10 keV cascade in Au was recently simulated [15]. The evolution of this event is illustrated by a time-sequence of snapshorts of the atomic positions, in Fig. 4. Local melting is a primary feature of the cascade. Other interesting features in the dynamics are the shock wave seen propagating outward at about the speed of sound and extensive cavitation in the core of the cascade. In Fig. 4(d), the final locations of the vacancies and interstitials are projected onto a (100) plane, showing their positions relative to the once-melted cascade core. It is seen that all vacancies are located in the once-melted zone while all interstitials are located outside of this region. The spike conditions are more clearly revealed in Figs 5 and 6 through temperature and pressure profiles, respectively. The
120
5 keV WA I” Cu 0 t* I.1 pr a 1=3.6ps fW ltquid Cu (S. F&es!
-* colcuI01ed
1.6 1.2
60-
+ D-4 ,,^.Qo. *. 0 . . /’ . -. -. c/
l
q!iB215’
i5 ’ 4.5’ ’ 55’ ’
6.5
Rod&i)
Fig. 2. A 5 keV cascade in Cu: (a) locations of atoms within a (100) cross-sectional slab of thickness &2, viewed along the (100) direction, 1.1 ps after the event was initiated; (b) radial distribution function for atoms in the disordered zone at two instant of time and for liquid Cu (after Ref. [14]).
Fig. 3. Primary state of damage for a 5 keV cascade in Cu obtained by MD (after Ref. [14]).
948
M. GBALY and R. S. AVERBACK
Fig. 4. A IO keV cascade in Au: locations of atoms within a (100) cross-sectional slab of thickness a,/2, viewed along the (IOO} direction at several instants of time (after Ref. [15]).
maximum temperature after a few lattice vibrations is -6000 K, which is a few thousand K below the estimated critical temperature of Au. The maximum pressure outside the melt zone is z3dGPa. This latter quantity is of interest in regard to loop punching, which is an often discussed mechanism for producing interstitial clusters [l6, 171. Loop punching requires pressures of the order of half the shear modulus 118, 191, which for Au is 228 GPa. Hence
loop punching would not be expected in this case and it was not observed. As a result of these MD simulations, the scenario for the production of defect clusters and loops previously discussed can be refined. In the initial stages of the cascade evolution, i.e. in the so-called collisional phase, interstitials are ejected from the cascade core by short replacement collision sequences. As the primary recoil energy becomes partitioned among all
949
Vacancy-type defect clusters
8000 u^ i 1 g
6000 4000 2000
0
2
4 6 Distance (8,)
8
10
Fig. 5. Temperature as a function of distance from the amter of energy for a 10 keV cascade in Au at various instants of time (after Ref. [15J). atoms in the cascade, the cascade core undergoes local melting. All of the interstitials within this region are absorbed in the melt and quickly delocalize. This explains the reduced defect production relative to simulations employing BCA. Local fluctuations in the liquid density resulting from the vacancies may persist, depending on the time for the free volume to homogenize. In the final phase of the evolution, the melt begins to solidify, with the crystallization front moving from the surrounding, cooler solid toward the center of energy of the cascade. Free volume is dragged along with the interface since the free energy of a vacancy is much less in the liquid. Thus, similar to zone refining of impurities, vacancies are swept along with the moving interface, with the fraction being left behind depending on their concentration
and the speed of the interface. At the core of the cascade, the concentration of vacancies can be greatly enhanced owing to this zone refining process and the conditions required for cascade collapse can thereby be established. For metals which have high energy density cascades and low melting temperatures, like Au, the interface moves very slowly allowing for efficient zone refining of vacancies. In metals with higher melting temperatures and lower energy densities, the interface moves too quickly for the flux of free volume to keep up and vacancies become frozen in throughout the cascade volume. Interstitial clusters often remain on the periphery of the once melted region, although how and why the interstitials cluster just at the melt edge is uncertain. It may be that those interstitials just outside the melt
10 8 i$
6
i 4
0
2
8
10
Fig. 6. Pressure as a function of distant from the center of energy for a 10 keV cascade in Au at various instants of time (after Ref. [IS]).
M. GHALY and R. S. AVERBACK
950
are stabilized by a large pressure gradient which prevents them from recombining with “vacancies” in the melt. Moreover, since they are very mobile during the thermal spike and strongly interact with each other, close interstitials would tend to cluster. This picture of interstitial clustering is currently being examined.
0.3 ps
W
4. CASCADES AT SURFACES: EFFECTS OF CONVECTIVE FLOW A qualitative understanding of the cascade dynamics at surfaces can be gleaned from Fig. 7, where snapshots of the atomic positions are again illustrated at various instants of time [20]. It is seen that the initial stages of the cascade evolution are similar to
1.01 ps
9.6 ps
(4
Fig, 7. Locations of atoms within a (100) cross-sectional slab of thickness a&, viewed along the (lOO> direction, at various instants of time. The initial velocity of the projectile had polar and azimuthai angles of 160 and ‘W, respectively.
Vacancy-type defect clusters those in the interior of the crystal. Interstitials are ejected from the central core of the cascade, creating a depleted zone, while a shock wave follows behind. Cavitation is again a prominent feature. The onset of the thermal spike becomes evident in Fig. 7(c) with the appearance of the local melt. It is also seen in Fig. 7(c) that at this point in the cascade evolution the surface cascade begins to deviate from the one in the bulk. The interstitial atoms which were initially transported to the periphery of the melt zone are now effectively re-absorbed in the melt. Only those interstitials deposited at the ends of long replacement sequences escape this fate. In this event, there are only three. In the bulk cascade, 10 times as many interstitials remained stable on the periphery of the melt zone. Stable interstitial clusters were also observed in simulations of bulk 10 keV cascades in Cu [16], Ni [21] and Ag [22] and they appear to be a common feature in simulations of cascades in pure FCC metals. Too few events near surfaces have been simulated to judge whether interstitial clusters are, in general, unstable near surfaces, as observed in the Au event. It may be that the relief of pressure in the surface cascade, before the cascade cools, enables the interstitials to relax back into the melt. This would, of course, explain why interstitial clusters were not observed by Thomas et al. and others after keV self-ion bombardment of Au and other metals. The further development of this cascade is conveniently described in macroscopic terms. As the cascade zone heats to a few thousand Kelvin and melts, its molar volume significantly increases,
-..
.
951
%6%, creating high internal pressure. A pressure gradient is established across the melt, since the normal component of stress at the surface is zero, and liquid is thus forced onto the surface, as seen in Figs 7(c-f). About 550 atoms flowed to the surface in this event. The final stages of the cascade evolution involve cooling and resolidification as described for bulk cascades. In the bulk, the ejection of interstitial atoms from the core of the cascade was responsible for the excess vacancies. In this case, the number of interstitials was approximately one-quarter of that predicted by the Kinchin-Pease formula. For a 10 keV cascade this is x 30 defects. Far fewer interstitials are created near the surface, only three in the simulation event, but many atoms leave the core of the cascade by their convective flow to the surface. Consequently, the number of vacancies accumulated in clusters near surfaces can far exceed their corresponding number in bulk cascades. This leads to a rather extensive dislocation structure. In Fig. 8 this structure is represented by plotting the locations of atoms with high potential energy as they outline the cores of the dislocations. This mechanism of convective flow explains why TEM experiments find many more defects in loops than are believed to be created in the bulk. The thermal spike behavior observed in these simulations at the surface of Au is rather pronounced. Au, however, has a very high atomic number and a relatively low melting temperature and whether the mechanisms for loop formation outlined above apply to other materials is uncertain. We are
.:. .- ,::‘s:.. :..*.*..* . I. ..::; ....
Fig. 8. Structure of the dislocation network as illustrated by the positions of atoms with potential energies 20.40 eV. The “hole” in the surface inidicates the presence of adatoms which are not shown.
M. GHALY and R. S. AVERBACK
952
beginning MD simulations of ion bombardment in metals like Cu and Ni to address this question, but already considerable insight can be gleaned from the following thermal spike model [23]. For convenience, we assume that an energetic particle travels deep into the substrate while dissipating energy according to F&r, z) = FJ(r
= O),
(1)
where F,, is the rate of damage energy deposition along the path of the ion track normal to the surface. The energy then spreads, creating a cylinder of liquid melt. Pressure is developed in the liquid owing to the change of volume and thus liquid is forced onto the surface. The maximum radius of the liquid cylinder is obtained from conservation of energy,
where n, is the atomic density and t, is the average energy of an atom in the melt. The number of atoms passing through the surface, Q, is given by Q = nOuAr,
(3)
where A = rrr* and T is the lifetime of the melt. T is obtained from 405 = r’,
(4)
where D is the thermal diffusivity. We approximate the velocity of the fluid by Poiseuille’s equation for viscous flow, r*AP 0)
“=w)
where p is the viscosity of the liquid and AP is the drop in pressure over a distance I. The pressure drops from a maximum within the tube of liquid to zero at the surface over a distance I, which we approximate by
Q = An&,
(6)
Q =($$AP)‘“r4,
(7)
so that
where bn, is the difference in atomic densities in the melt and solid. The pressure created in the tube, far from the surface, is 1 AV -AP=
KV _
2(1+v)
I+-----
EK
where K is the compressibility of the liquid, E is the elastic modulus of the solid and v is Poisson’s ratio. Combining all terms, we obtain
For the case of 10 keV Au bombardment of Au p = 6.0 x lo**m-’ An&, = 0.06, K = 1.5 x 10’” m2N-‘, E = 78 &Pa, v = l/3, p = 5.2 x IO-* Poise and c,,,=0.3 eV. We approximate F,, x 15OOeVnm-‘. The thermal diffusivity is obtained from the MD simulations, D = 0.2 x IO’*nm* s-l so that the lifteime of the spike given by eqn (4) agrees with the lifetime deduced from the simulations. With these values, we find Q z 1000 atoms. Although this good agreement with the simulations must be considered fortuitous in view of the approximate nature of the calculation, the agreement does support the basic concept. Flow in other materials can be similarly estimated, but except for the term (F,/c,)*, differences in the various parameters are not great (or cancel). It might be assumed that viscous flow of atoms through the surface is strictly a low energy phenomenon and that at high energies the damage is created at too great a depth, but this is not true. The energy deposition along the track of the ion is described by the nuclear stopping power, S,(E) = l/n, dE/dx, which is a universal function when expressed in reduced units. The nuclear stopping power has a maximum at the reduced energy, L % 0.3. For self-ion irradiation of Cu, Ag and Au, this corresponds to ~60, 215 and 720 keV, respectively, i.e. the ratio, (F,/c,)*, is higher at higher rather than lower energies than that simulated here. Since sputtering yields also depend on (F,/c,), examination of sputtering yields provides an approximate means to determine whether viscous flow at surfaces will be an important effect. The sputtering yield for 10 keV self-ion bombardment of Au is ~20 [24]. For 70 keV Kr irradiation of Cu, the sputtering yield is z 13, and for this irradiation, about one-third of the ions produced visible dislocation loops [25]. For 30 keV self-ion irradiation of W at 10 K, the sputtering yield is a 12 and occasional loops, but more commonly highly depleted zones, were observed by field ion microscopy imaging [ 111.A question often raised in regard to the FIM experiments is why the observed number of interstitial atoms is only a small fraction of the number of observed vacancies. Several possibilities have been identified [I 11,but we add here the possibility of the loss of atoms to the surface by viscous flow.
Vacancy-type defect clusters Finally, it is of interest to note scent TEM investigations of f4 MeV neutron irradiation of Au at low temperatures [5]. In this study, specimens having a wedge geometry were irradiated at IO K and transferred into the TEM without warming above 20K. The authors report that in the very thin portions of the wedge, i.e. at depths up to 80 nm, defect clusters with vacancy character were observed, but in the thicker part of the wedge, the character of the defect cluster changed from vacancy to interstitial. The character of the clusters were determined by the so-tailed 2f- D method and the validity of this technique has been controversial. Nevertheless, the results of this study are tantalizing in view of these simulations.
953
R. Ben&k, W. Rger and F. Ehrhart are gratefully acknowied8ed. REFERENCES f . Merkle K. L., Radiation Bmage fn Metals (Edited by S. D. Harkness and N. L. Peterson), p. 58. American Society af Metals, Metals Park, OH (1976). 2. Thomas L. E., Schober T. and Ballti R. W., Radiaf. Effects 1, 257, 269, 279 (1969). 3. JIger W. and Merkle K. L., Phil. Mug. A 44,741 (1981). 4. Ruault M. O., Bemas H. and Chaumont J., Phil. Msg. A 39, 757 (1979). 5. Fukushima.F., ~~~~rnura Y., Guinan M. W., Hahn P. A. and Kiritani M.. J. NM&Marer. 179-181.943 0991% 6. Brinkman J. A., f. Appf. Phys. 2!!, 951 (1954i A& 3. Phys. 24, 246 (1956). 7. English C. A. and Jenkins M. L., Mzter. Sci. Forum, M-18, 1003(1987). 8. Gibson J. B.. Goland A. N., Milrrram M. and Vinevard G. H., P&: Rev. 120,1229 (19%0). 9. Beeler J. R. Jr and BescoD. G.., J. Am& Phvs. 35.2515 II
I
I
(1963).
10. Robinson M. T. and Torrens I. M., Phys. Rev. B 9,500s In this paper we have shown that even 25 years
after the experiments on self-ion bombardment of Au by BailufE and students, the nature of defect clusters in irradiated metals and the underlying mechanisms for their production are still controversial subjects. We have used molecular dynamics simulations to investigate this problem and have revealed a mechanism that was not considered by Balluffi et al., nor by anyone else. The impIications of this mechanism of surface melting and viscous flow are enormous, since much of our understanding of radiation damage in metals has been derived from energetic ion irradiations and all ion beam processing involves surface damage. Much work remains, however, both to verify the m~hanism suggested by the simulations through ex~riments and to explore how widespread the e&et is.
research was supported by the U.S. DOE, Basic Energy Sciences, under grant DEFGO291ER45439. Discussions with Drs T. Diaz de la Rubia, Acknowledgements-The
($974).
Il. Wei C.-Y. and Seidman D. N., Phil. &fag. A 37, 257 (1978). 12. Sigmund P., Phvs. Rev. 184,383 (1969);Phvs. Rev. 187, 76% (197.5). . 13. Sigmund P., Appi. Phys. Lett. 25, 169 (1974); Appl. Phys, Z.&t. 27, 52 (1975). 14. Diaz de la Rubia T., Averback R. S., Benedek R. and Kinn W. E.. Pkvs. Rm. L&r. 38. 1930 11987). 15. Gl& M. a&d Averback R. S. ~npubl~s~~j. 16. Diar de la Rubia T. and Guinan M. W., [email protected]. 66, 2766 (1991). 17. Van Swy8enhoven H. and Caro A., X-‘&s.&v. LRtf.70, 2098 (1993). 18. Trinkaus fi., Rudiut. Effects 78, 189 (1983). 19. Wolfer W. G.. Phil. Mae. A 58. 285 119881. 20. Ghaly M. and’Averbackk. S., dhys, lbv. btt. 72,364 (1994). 21. Zhu H. and Averback R. S. ~~~ub~s~~}. 22. Ghaly M., L&z de la Rubia T. and Averback R. S. (in press). 23. Averback R. S. and Gbaly M. (in press). 24. Andersen H. H. and Bay H. L., in Spuitering by J%&cfe 3om&ffrdmenr (Edited by R. Behrisch), Chap. 4. Springer, Berlin (198 1). 25. Kirk M. A., Robertson I. M., Jenkins M, L., English C. A., Black T. J. and Vetrano J. S., J. Nucl. Mater. 149. 21 (1987).