Displacement cascades in the ordered compound CuTi studied by molecular-dynamics simulations

Nuclear Instruments and Methods in Physics Research B 95 (1995) 25-33

B e a m Interactions with Materials & Atoms

ELSEVIER

Displacement cascades in the ordered compound CuTi studied by molecular-dynamics simulations Huilong Zhu a,b Nghi Q. Lam b,, a Materials Research Laboratory, Universi~. of Illinois at Urbana-Champaign, Urbana, IL 61801, USA b Materials Science DiL,ision, Argonne National Laboratory, Argonne, IL 60439, USA

Received 12 May 1994; revised form received 12 September 1994 Abstract The properties of displacement cascades of up to 5 keV in energy in the ordered intermetallic compound CuTi were investigated by molecular-dynamics simulations. Various aspects of the cascade evolution were examined, including the production of Frenkel pairs, "pure" replacements, and antisite defects, as well as the anisotropy of the displacement threshold energy. The minimum displacement threshold energy (15 eV) is found for the (100) recoil directions. The average threshold energy for displacement initiated by a Ti primary-knock-on atom (78 eV) is ~ 1.7 times larger than that by a Cu primary-knock-on atom (47 eV). The damage function was analyzed, based on the average number of stable Frenkel pairs generated by both kinds of primary knock-on atoms in 18 directions. Multiple defect production is found for cascade energies >_ 500 eV. Around this energy, ~ 25 replacements are created for each stable Frenkel pair. Planar cascades occur in the (100), (010) and (110) planes under certain conditions, producing significantly more Frenkel pairs than in the case of three-dimensional cascades. Melting of the core of a 5 keV cascade during the first 5 ps causes efficient, local atomic mixing, After recrystallization at the end of the event, the impact region shows a high degree of chemical disorder, characterized by a chemical short-range order parameter of 0.49. The efficiency of Frenkel-pair production by a 5 keV recoil is estimated to be 0.14.

1. Introduction

Information about the dynamic evolution of displacement cascades is essential for the understanding of damage accumulation and property changes in irradiated materials. Molecular dynamics simulations have been extremely useful in providing this information (e.g., Ref. [1]). However, due to the lack of realistic interatomic potentials for alloys and compounds, most of previous molecular dynamics studies were performed on pure metals [1-15]. Only a limited number of simulations has been carried out for binary systems. The first molecular dynamics simulation of displacement cascades in an ordered compound, Fe3A1, was undertaken by Jackson et al. [16] in 1972, who used a combination of pairwise Morse and Born-Mayer potentials to

* Corresponding author, tel. + 1 708 252 4953, fax + 1 708 252 4798.

model the interactions between the alloying elements. Simulations utilising pair potentials derived from the pseudopotential theory were reported by Chudisov et al. for the structural evolution of the cascade region during the cooling phase in the ordered compounds Nb3Sn [17] and Mo3Si [18] and in a dilute A1-Fe alloy [19]. More recently, displacement cascades in [3-SIC [20] and other intermetallic compounds [21-24] were also studied by molecular dynamics simulations. In the present paper, we report the results of our recent study of displacement cascades in the ordered intermetallic compound CuTi. The cascades were generated by both Cu and Ti primary knock-on atoms (PKA) of various energies. The production and spatial distribution of Frenkel-pairs, "pure" replacements and antisite defects, the anisotropy of displacement threshold energy, and the damage function were examined. These properties were analyzed and compared with those of pure metals, based on the information obtained from numerous simulated events. In addition, the thermal spike behavior of high-energy cascades was investigated by examining the dynamics of a 5 keV cascade event.

0168-583X/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSD1 0168-583X(94)00341-6

H. Zhu, N.Q. Lam / Nucl. Instr. and Meth. in Phys. Res. B 95 (1995) 25-33

26

2.2. Interatomic potentials

2. Model and computational procedure

2.1. Molecular-dynamics model The primary simulation technique used in the present work was a molecular-dynamics scheme based on a modified, vectorized version of the code DYNAMO [25]. The vectorized link-cell method [26] was also implemented in order to speed up the simulation of elevated-energy cascades involving a large number of atoms. The model lattice that represents the ordered CuTi compound has a B l l structure, as illustrated in Fig. 1. The simulated crystallite was rectangular, containing from 1944 to 128 000 atoms depending on the PKA energy, which ranges from 10 to 5000 eV. Periodic boundary conditions were used to provide an infinite extension of the simulation cell. Interactions between atoms in the system were governed by an appropriate combination of potentials - the Ziegler et al.'s potentials [27] for short distances of approach, and embbeded-atom potentials [28] for near-equilibrium separations (further details are given in Section 2.2). The lattice was initially maintained at 0 K. The integration time step was not constant; it was periodically adjusted such that the fastest atom would move no more than 0.006 nm. The largest time step was 5 × 10 -as s, used near the end of the cooling phase. The displacement threshold energies, E d, were calculated for both Cu and Ti PKAs in 13 directions by finding the minimum value of the recoil energy that produced one stable Frenkel pair at the end of the cooling phase. The damage function was estimated by averaging the numbers of Frenkel pairs over 18 directions for both PKAs. A number of simulation runs was also made, in which the velocities of the recoil atoms were generated in some given directions in order to examine the relationship between the spatial distribution of the cascade and the PKA direction.

•

Ti

C) cu

The potentials (~)EAMderived from the embbeded-atom method (EAM) [29,30] for the C u - T i system [28] were used to describe the interactions between atoms at nearequilibrium separations. These potentials correctly predicted the stable structures, atomic volumes, and heats of formation of several intermetallic compounds of this alloy system [28,31], as well as yielded reasonable values for the energetics of point defects [32]. For interactions at smaller separations (less than ~ 0.15 nm), the potentials of Ziegler et al. [27], @z, were utilized, To bridge these two kinds of potentials in a continous manner, a transitional potential, @n, was needed. For T i - T i interactions, this potential has the form 5

@B(Rij) = E ckRkij

(1)

k=0

where Rij is the separation distance between atoms i and j, and c k are splining constants determined by the following constraints: @ z ( r l ) = @B(rl),

¢bEAM(r2) = @B(r2),

(2a)

~(rl)

qb~AM(r2) = qS~(r2),

(2b)

qb~AM(r2) = q0;(r2),

(2C)

= ~(rl),

q)~(rl) = qb;(ra),

with r 1 and r 2 being respectively the upper and lower limits of the potential region to be bridged, chosen so that the forces are continuous functions of Rij when an atom moves from one potential regime to another. For C u - C u and Cu-Ti interactions, on the other hand, the transitional potential is represented by •

4

~;-

~ ck e x p ( - - b k R i J a ) , k=l

(3) where Z i and Zj are the respective atomic numbers, e is the electron charge, a is the screening length, Cl, c2, bl and b e are constants associated with the potential of Ziegler et al. [27], and c3, c4, b3, and b 4 are adjustable parameters. This assumed transitional potential must also satisfy, for given y and R*, the above constraints, i.e.,

~ z ( r l ) = 't'B(rl, ~'),

~'E~(r2) = ~B(r2, ~'), (4a)

• i ( r , ) = ~ ; ( r , , ~,),

~ E ~ ( r ~ ) = ' ~ ( r ~ , ~,). (4b)

2.649 A

.... I ..... (

..........

" ~ 3.108A,-~

Fig. 1. Structure of the ordered compound CuTi.

The nature of the fit depends on the choice of the base function (R*/Rij) ~ and on r 1 and r 2. To obtain a good fit, we used R * = r I for qbEAM(rz) > ~bz(r2) , and R * = r 2 for ~EAM(r 2) < qbz(r2). In addition, by comparing various transitional potentials (e.g., dPtB(Rq, y'), subject to the

H. Zhu, N.Q. Lam /Nucl. Instr. and Meth. in Phys. Res. B 95 (1995) 25-33 constraints (4a) and (4b) at r'1 ( > r 1) and q)n(Rij, y) and m i n i m i z i n g the function

50

r 2) with

.........O....... Antisitereplacements .... ..El--- All replacements 40

0 -2 = LI2((I)B(Rij,]/)

-- ~2)tB(Rij,~/)) 2

dRij ,

(5) 30

the value of the parameter "y that yields the best fit can be derived. The interactions between atoms in the system can n o w be completely described by the composite potential

x

Frenkelpairs

lx,d]~.a.l~l~' "13 t~:l~3

Ti P K A

20

10

¢(Rij): ffl)z( Rij

27

0

for Rij <-rl,

3O

@(Rij)=

qOB(Rq, Ymin) f o r r l < R i j < r 2 , ClgEAM(Rij )

for r 2 <

(6)

Cu P K A

Rij < rc, 20

where r c is the cutoff distance for the E A M potential. The parameters used for the bridge functions are given in Table 1.

E

[~ill~

13 o.oo6

10

3. Results and discussion 0 .01

3.1. Evolution of low-energy cascades

Table 1 Parameters for the bridge potentials For Ti-Ti interactions (eq. (1))

For Cu-Cu interactions (eq. (3))

For Cu-Ti interactions (eq. (3))

co = c1 c2 = c3 = c4 = c5 =

3' = 0.10 c 1 - 0.1818 c 2 = 0.5099 c 3 = 0.2694 X 10 2 c a = 0.3772 b 1 = - 3.2 b 2 = - 0.9423 b 3 = 0.4297 b4 = 0.3887

3' = - 0 . 2 5 c 1 - 0.1818 c 2 = 0.5099 c 3 = 0.4052 X 103 c 4 = 1.553 b 1 = - 3.2 b 2 - - 0.9423 b 3 = 0.3325 b4 = 0.3870

3702.10997 -9452.2213 8644.8354 - 2916.2690 - 12.3446 130.3155

1

......

0

Time (picosecond)

In general, the temporal development of a cascade in the ordered c o m p o u n d CuTi, like in pure metals [2,6], consists of a collisional phase followed by a cooling phase. During the collisional phase (0 to ~ 0.25 ps), the numbers of Frenkel pairs and replacements (including " p u r e " replacements and antisite defects) increased rapidly with time, as shown in Figs. 2a and 2b for a 500 eV cascade. At the end of this phase, the n u m b e r of Frenkel pairs reached a m a x i m u m . The time scale of the collisional phase is practically the same as that found for elemental metals [2,6,11,15], but is almost an order of magnitude larger than the value determined for 13-SIC which has a stiffer crystal binding potential [20]. The duration of the subsequent cooling phase was found to depend on the P K A energy;

=

.1

r I [,~] = 0.88326

r, [,&]= 0.99020

r 1 [,~] = 0.88677

r e [,~] = 1.76652

r e [,~] = 1.48531

r2 [~,] = 1.77355

Fig. 2. Time evolution of the numbers of Frenkel pairs, replacements (all "'pure" and antisite replacements) and antisite defects generated by Ti PKA (a) and Cu PKA (b) in a 500 eV event.

for example, it was ~ 2.5 ps for recoil energies < 500 eV and ~ 12 ps for 5 keV. During this phase, the n u m b e r of Frenkel pairs decreased monotonically to a steady-state value ( ~ 2 Frenkel pairs), which was only ~ 10% of the peak value. All vacancies were in the Cu sub-lattice, in agreement with the results of defect calculations by Shoemaker et al. [32]. The numbers of " p u r e " replacements and antisite defects, on the other hand, first decreased in the early stage of the cooling phase, and then increased again toward steady-state levels. The early decrease, more pronounced for Cu P K A s (Fig. 2b) than for Ti P K A s (Fig. 2a), was caused by the inward relaxation of atoms from the peripheral " c o m p r e s s e d " regions. At the end of this phase, the total n u m b e r of replacements induced by a Ti P K A was considerably higher than that created by a Cu PKA; however, the numbers of antisite defects produced by either P K A were practically equal. Approximately, a 500 eV cascade event generated 25 replacements for each stable Frenkel pair. A n example of the spatial distributions of vacancies, interstitials, " p u r e " replacements, and antisite defects, produced by a Cu PKA, at the end of the collisional and cooling phases is given in Fig. 3. About half of the total n u m b e r of replacements, which had been produced in the collisional phase, returned to their original lattice sites in the cooling phase. Long linear replacement collision sequences, observed (although rarely) in pure metals [6,8], were not found for CuTi because of the complex structure

28

11. Zhu, N.Q. Lam /Nucl. Instr. and Meth. in Phys. Res. B 95 (1995) 25-33

~X

'°

~0 ×

"

~'°'~.Q ~°i

ts.o

(a)

(b)

Fig. 3. Spatial distributions of defects, produced in 500 eV cascade with a Cu PKA in the (111) direction, at 0.2 ps (a) and 2.5 ps (b). Vacancy: [2, interstitial: O, "pure" replacement: O, antisite replacement: ×.

and atomic-mass difference between the two alloying species. In short linear replacement sequences, the terminal interstitials can push the replacements back to their original sites, leaving no defects and replacements at the end of event. As a result, decreases in the numbers of " p u r e " replacements and antisite defects were found at the beginning of the cooling phase (see Fig. 2). Under certain conditions, e.g., at relatively low PKA energies ( < 500 eV) and with favorable PKA directions, planar cascades were observed to form in the (100), (010) or (110) planes. For example, Fig. 4 shows the spatial

distribution of defects produced by a 500 eV cascade event which took place in a (010) plane. The PKA velocity vector lay in this plane, as indicated by the arrow in Fig. 5 which shows the trajectories of atoms involved during the collisional phase. At the end of the cooling phase, a stable, planar vacancy-rich cascade core surrounded by interstitial atoms was found in the same plane (Fig. 6). A total of 8 stable Frenkel pairs remained, a number which is approximately 4 times higher than that found for the case of three-dimensional cascades (see Figs. 3b and 3d). The duration of the cooling phase of a planar cascade was noticeably shorter than that of a three-dimensional cascade.

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Fig. 5. Trajectories of atoms involved in 500 eV cascade between 0 and 0.2 ps. The direction of the Cu PKA is shown by the arrow. Movement of cascade atoms is confined within the (010) plane. The open and filled squares denote the initial and final positions of Cu atoms, and open and filled circles indicate those of Ti atoms, respectively.

H. Zhu, N.Q. Lam /Nucl. Instr. and Meth. in Phys. Res. B 95 (1995) 25-33 This is due to the fact that all energetic atoms in the planar cascade are readily in contact with "cold" atoms on both sides of the cascade. It is the more efficient cooling of the planar cascade that leads to a larger number of defects at the end of the event (see Fig. 6). No planar cascades were observed in the (001) planes. The mechanism and implication of planar-cascade development are the subject for further studies.

29

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•

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3.2. Threshold energy 0 The displacement threshold energy, E d, is defined as the minimum recoil energy required to produce a stable Frenkel pair. This energy strongly depends on recoil direction, as shown in Fig. 7 where the values of E d calculated for 13 directions are plotted. For Ti PKAs, the directional dependence of E d was similar to that in pure Cu, i.e., a local minimum in E d is located in the vicinity of the low-index directions [6]. For Cu PKAs, this behavior was not observed in the (110) and (112> directions. The minimum E d was found to be ~ 15 eV, significantly

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I

I

I

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<100> <110>

(degree)

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smaller than that found for pure Cu ( ~ 25 eV) [6], because stable Frenkel pairs could be created with shorter replacement sequenses. The same minimum value was also obtained for E d in Ni3A1 with a Ni PICA [24]. In general, the values of E a determined with a Ti PKA, Ed(Ti) , are larger than those with a Cu PKA, Ed(Cu) , in almost all directions; on the average, Ed(Ti)= 78 eV and Ea(Cu)= 47 eV. The effective threshold energy, calculated by averaging over all the directions and two kinds of PKA, is thus E~ff -- 63 eV. It should be mentioned that the concept of an effective displacement threshold energy may not be physically meaningful because of the large variations in E d for defect production with different recoil directions. 3.3. Damage function The damage function, u(T), is defined as the average number of stable Frenkel-pairs remaining at the end of the cooling phase, resulting from a recoil of energy T:

O •

:O.o :o.o:o:o:o. o iii O • O

(7)

where, in the binary CuTi system, i stands for Cu and Ti PKAs, and the partial damage function vi(T) is calculated by

0

0

:o:o:o.o.o

I

i

o: o:o. o. o: o. o: o. o..o. o: o:o O

<100>

0

o : o , o , o•o: ooo: o:o: o: o 5p $ o

I 45

,,(T) = ½E ",(O,

0

o

I

2(].0

z

C

I 45

O

O

Z Fig. 6. Atom distribution in the (010) plane at the end of the collisional phase, 0.22 ps, and after the event, 2.5 ps. The open and filled circles denote Ti and Cu atoms, respectively.

n(r)

= fn(n,

T) d n / 4 - r r ,

(8)

with Y2 being the solid angle. In order to estimate ~,(T), simulations were performed for 18 PKA directions and two kinds of PKA in the energy range 10 < T < 500 eV. The average numbers of Frenkel pairs found at the end of the collisional phase ( ~ 0.2 ps) and at the end of the cooling phase ( ~ 2.5 ps) are plotted in Fig. 8 as a function of recoil energy. Also shown for comparison in this figure is the Kinchin-Pease damage function [33], calculated with E d = 15 eV which is the minimum PKA energy required to produce one Frenkel pair near the {100) direction (see Fig. 2). The damage function u(T), calcu-

H. Zhu, N.Q. Lam /Nucl. Instr. and Meth. in Phys. Res. B 95 (1995) 25-33

30

3.4. Thermal spike behavior of high-energy cascades

/ t 2.0

dr i 1.0

O

End of Collision Phase (Cu PKA)

•

End of Cooling Phase (Cu & Ti PiCAs)

- - •

0.0 - -

I

100

200

Kinchin-Pease

i

Function i

300

400

i

500

600

Recoil Energy (eV)

Fig. 8. Average number of Frenkel pairs as a function of recoil energy at the end of the collisional (0.2 ps) and cooling (2.5 ps) phases• The Kinchin-Pease function (dashed line) calculated with E d = 15 eV is shown for comparison•

lated by averaging over 18 directions and two types of PKAs, is shown by full circles. Unlike the Kinchin-Pease relation, u(T) varies very slowly with recoil energy. For example, in order for u(T) = 1, the PKA energy T has to be ~ 200 eV, and multiple defect production ( u ( T ) > 2) only occurs when T > 500 eV. This behavior is similar to that found for pure Cu [6]. However, the plateau at u(T) = 0.5 extending from ~ 30 to 125 eV observed on the damage function for Cu [6] was not found in the present case.

The dynamics of a 5 keV cascade event is illustrated in Figs. 9 and 10. First, Fig. 9 shows the temperature distribution from the cascade center at different times. Here, the temperature To(r) was calculated by equating the average kinetic energy Ek(r) in a spherical shell of radius r with 3kBTc(r) (k B being the Boltzmann constant). As can be seen, at the end of the collisional phase, the temperature within a sphere of radius ~ 7% (with a o = 3.108 A) was brought far above 1258 K, which is the melting temperature of the CuTi compound [34]. Local melting thus occurred in this central region. With increasing time, resolidification took place at the periphery of the molten zone; after a few ps, for example, the radius of this zone was reduced to ~ 5a o. Resolidification was completed at t > 5 ps. Fig. 10 shows "snapshots" of the atom configuration within a (010) cross-sectional slab of thickness 0.5a o through the cascade center at different times. The topologically-disordered region first grew in size, attaining a maximum diameter of ~ 4.5 nm at ~ 0.2-0.5 ps, and then shrank subsequently. At the end of the cooling phase, i.e. after ~ 12 ps, the damaged zone recovered its crystallinity, but contained significant chemical disorder. About 1690 replacements were observed, and only 19 Frenkel pairs survived annihilation (14 interstitials, 1 diinterstitial, and 1 triinterstitial; 8 single vacancies 1 divacancy, 1 tetravacancy, and 1 pentavacancy). Deviding the number

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A _2.ops

~

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10

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R (a o = 3.108 A)

Fig. 9. Time evolution of the temperature distribution from the cascade center. The melting temperature Trn = 1258 K of the ordered compound CuTi is indicated.

H. Zhu, N.Q. Lam /Nucl. lnstr, and Meth. in Phys. Res. B 95 (1995) 25-33

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of stable Frenkel pairs by 133, which is the total number of Frenkel pairs predicted by the modified Kinchin-Pease expression [35] using E d = 15 eV, one obtains a defectproduction efficiency e = 0.14 1 for 5 keV recoil in CuTi. Experimentally, the relative efficiency for producing longrange migrating defects in alloys by 5 keV recoils was ~ 0.2 [36], consistent with the value of e. The present finding is also in agreement with recent molecular-dynamics simulation results [8,23]. For example, using 30 and 40 eV as values of E d, Diaz de la Rubia et al. [23] estimated the defect-production efficiencies for 5 keV cascade in Cu3Au and Ni3A1 to be 0.26 and 0.15, respectively.

3.5. Chemical disorder in high-energy cascades Chemical disorder in the 5 keV cascade was characterized by a chemical short-range order (CSRO) parameter. To determine the CSRO parameter for the B l l structure,

one has not only to identify the 8 nearest neighbors for each atom, but also to distinguish to which layer (for example, a and 13 for atom ,~' as shown in Fig. 1) these neighbors belong. Otherwise, the traditional CSRO definition, q = ( 8 P A B - 4 ) / 4 {with PAB being the probability for an A atom to have a B atom as its nearest neighbor, A ¢ B}, gives no distinction between the perfect and totally chemically-disordered lattices. Since the difference between the distances of the first and second nearest neighbors is small in the B l l structure, one has to consider the angular distribution of the neighboring atoms to locate the true nearest neighbors [37]. To focus on the region of the cascade, we only calculate the CSRO parameter in the vicinity of the replacement atoms. For instance, if atom ~.~ is a replacement atom at (x, y, z), then its first and second nearest neighbors, 14 atoms in total, can be found easily. In order to determine the true nearest neighbors in the c~ plane, the following criterion is used:

(Z i --Z) > A,

(9)

( x i - x ) 2 + (Yi-Y)~- > A2, This value may be better considered as the lower limit for the defect-production efficiency because the minimum E a is used in the modified Kinchin-Pease expression. If, instead, the "commonly-used" value E d = 30 eV is taken, one obtains c - 0.29. The effective value E d = 63 eV, on the other hand, yields e = 0.60.

where (x i, Yi, Zi) represents the position of the /th nearest neighbor and A = Co/8 (see Fig. 1). Similarly, the true nearest neighbors in the [3 plane can be identified by replacing - A for A in Eqs. (9).

32

H. Zhu, N.Q. Lam /Nucl. Instr. and Meth. in Phys. Res. B 95 (1995) 25-33

Letting "0¢j ( ( = a , 13, and j = Cu, Ti) be the number of j-type first nearest neighbors of atom .9~ in the ~ plane of a B l l lattice, we define the CSRO parameter in a cascade as 1

Acknowledgements

NR [ max(rl~c u,T]uTi) -- 2

q ' = ~-~R i= --~1

2

+ m a x ( ~ c u,~13Ti) 2

--

2]

tion, a defect-production efficiency of 0.14 was estimated for 5 keV recoil in CuTi.

(10)

J

where N R is the total number of replacement atoms found at the end of the cooling phase. For the perfectly-ordered and completely-disordered states, q ' = 1 and 0, respectively. A value of q ' = 0.49 was obtained for CSRO in a 5 keV cascade in CuTi. This value is significantly larger than that found recently for a 10 keV cascade in NiA1 ( q ' = 0.27) [37]. The difference is due to the fact that replacement atoms in the B l l structure have a lower efficiency to create chemical disordering than those in the B2 structure of NiA1. As a matter of fact, when an atom exchanges with one of its 8 nearest neighbors in a perfect B l l lattice, the probability of producing chemical disorder is only 50%, whereas a similar exchange definitely causes chemical disordering in the perfect B2 structure.

4. Summary Molecular-dynamics simulations were performed to investigate the basic properties of displacement cascades in the ordered intermetallic compound CuTi. Both Cu and Ti recoils of energies ranging from 10 to 5000 eV were considered. The production of Frenkel pairs and replacements as well as the anisotropy of the displacement threshold energy were examined. The minimum displacement threshold energy (15 eV) was found for the (100) recoil directions. The average threshold energies for displacement initiated by Ti and Cu primary-knock-on atoms were 78 and 47 eV, respectively. The damage function, calculated by averaging the number of stable Frenkel pairs over 18 PKA directions, was analyzed. Multiple-defect production, i.e., a damage function of > 2, occurred at cascade energy > 500 eV. Near 500 eV, ~ 25 replacements were created for each stable Frenkel pair. Occasionally, planar cascades were found in the (100), (010) and (110) planes, producing a much larger number of Frenkel pairs than in the case of three-dimensional cascades. At a higher energy, 5 keV, the cascade core melted during the first 5 ps, giving rise to efficient atomic mixing. As a result, although the cascade region recrystallized at the end of the event, significant, local chemical disorder remained (characterized by a chemical short-range order parameter of 0.49). With respect to the modified Kinchin-Pease rela-

The authors would like to thank Drs. R. Benedek, R. Devanathan and M.J. Sabochick for many helpful discussions and valuable assistance in the early stage of the present simulations. This work was supported by the US Department of Energy, BES-Materials Sciences, under Contract W-31-109-Eng-38. It greatly benefited from an allocation of computer time on the Cray sytem at the National Energy Research Supercomputer Center (Lawrence Livermore National Laboratory).

References [1] R.S. Averback, T. Diaz de la Rubia and R. Benedek, Nucl. Instr. and Meth. B 33 (1988) 693. [2] M.W. Guinan and J.H. Kinney, J. Nucl. Mater. 103/104 (1981) 1319. [3] J.H. Kinney and M.W. Guinan, Philos. Mag. 46 (1982) 789. [4] V.I. Protasov and V.G. Chudinov, Radiat. Eft. 66 (1982) 1. [5] D.E. Harrison and R.P. Webb, Nucl. Instr. and Meth. 218 (1983) 727. [6] W.E. King and R. Benedek, J. Nucl. Mater. 117 (1983) 26. [7] V.I. Protasov and V.G. Chudinov, Radiat. Eff. 83 (1984) 185. [8] T. Diaz de la Rubia, R.S. Averback, R. Benedek and W.E. King, Phys. Rev. Lett. 59 (1987) 1930. [9] T. Diaz de la Rubia, R.S. Averback, R. Benedek and H. Hsieh, J. Mater. Res. 4 (1989) 579. [10] T. Diaz de la Rubia and M.W. Guinan, J. Nucl. Mater. 174 (1990) 151. [11] C.A. English, W.J. Phythian and A.J.E. Foreman, J. Nucl. Mater. 174 (1990) 135. [12] S.P. Chou and N.M. Ghoniem, Phys. Rev. 13 43 (1991) 2490. [13] F. Rossi and N.V. Doan, Nucl. Instr. and Meth. B 61 (1991) 27; N.V. Doan, F. Rossi and L. Boulanger, J. Nucl. Mater. 191-194 (1992) 1106. [14] K. Morishita, N. Sekimura, T. Toyonaga and S. Ishino, J. Nucl. Mater. 191-194 (1992) 1123. [15] A.F. Calder and D.J. Bacon, J. Nucl. Mater. 207 (1993) 25. [16] R.O. Jackson, H.P. Leighly Jr. and D.R. Edwards, Philos. Mag. 25 (1972) 1169. [17] V.G. Chudisov, V.P. Gogolin, B.N. Goshchitskii and V.I. Protasov, Phys. Status Solidi A 80 (1983) 349. [18] V.G. Chudisov, N.V. Moseev, B.N. Goshchitskii and V.I. Protasov, Phys. Status Solidi A 85 (1984) 435. [19] V.G. Chudisov, B.N. Goshchitskii, N.V. Moseev and V.V. Andreev, Phys. Status Solidi A 119 (1990) 437. [20] A. E1-Azab and N.M. Ghoniem, J. Nucl. Mater. 191-194 (1992) 1110; Radiat. Eft. Def. Solids (in press). [21] A. Caro, M. Victoria and R.S. Averback, J. Mater. Res. 5 (1990) 1409.

H. Zhu, N.Q. Lam /Nucl. Instr. and Meth. in Phys. Res. B 95 (1995) 25-33 [22] H. Zhu, N.Q. Lam, R. Devanathan and M.J. Sabochick, Mat, Res. Soc. Syrup. Proc. 235 (1992) 533. [23] T. Diaz de la Rubia, A. Caro, M. Spaczer, G.A. Janaway, M.W. Guinan and M. Victoria, Nucl. Instr. and Meth. B 80/81 (1993) 86. [24] F. Gao and D.J. Bacon, Philos. Mag. A 67 (1993) 289. [25] M.S. Daw, M.I. Baskes and S.M. Foiles, private communication. [26] M.W. Finnis, MOLDY6-A Molecular Dynamics Program for Simulation of Pure Metals, Harwell Report AERE-R-13182 (1989). [27] J. Ziegler, J.P. Biersack and U. Littmark, The Stopping and Range of Ions in Solids, vol. 1 (Pergamon, New York, 1985). [28] M.J. Sabochick and N.Q. Lam, Phys. Rev. B 43 (199t) 5243. [29] M.S. Daw and M.I. Baskes, Phys. Rev. B 29 (1984) 6443.

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[30] D.J. Oh and R.A. Johnson, J. Mater. Res. 3 (1988) 471. [31] N.Q. Lam, P.R. Okamoto, M.J. Sabochick and R. Devanathan, J. Alloys Comp. 194 (1993) 429. [32] J.R. Shoemaker, R.T. Lutton, D. Wesley, W.R. Wharton, M.L. Oehrli, M.S. Herte, M.J. Sabochick and N.Q. Lam, J. Mater. Res. 6 (1991) 473. [33] G.H. Kinchin and R.S. Pease, Rep. Prog. Phys. 18 (1955) 1. [34] J.L. Murray, ed., Phase Diagrams of Binary Titanium Alloys (ASM International, Metals Park, Ohio, 1987) pp, 80-95. [35] P. Sigmund, Appl. Phys. Lett. 14 (1969) 114. [36] L.E. Rehn and P.R. Okamoto, Mater. Sci. Forum 15-18 (1987) 985. [37] H. Zhu, R.S. Averback and M. Nastasi, Philos. Mag. (in press).