REFERENCEIC/TT/T6

INTERNATIONAL CENTRE FOR

THEORETICAL PHYSICS

LATTICE DISTORTION

DUE TO OXYGEN AID NITROGEN DI-INTERSTITIAL CLUSTERS

IN NIOBIUM AND VANADIUM

INTERNATIONALATOMIC ENERGY

AGENCY

UNITED NATIONSEDUCATIONAL,

SCIENTIFICAND CULTURALORGANIZATION

C.E. Laciana

A.J. Pedraza

and

E.J. Savino

1977 MIRAMARE-TRIESTE

IC/T7/T6

International Atomic Energy Agency

and

United Nations Educational Scientific ajid Cultural Organization

INTEHNATIONAL CENTRE FOE THEORETICAL PHYSICS

LATTICE DISTORTION DUE TO OXYGEH AND NITROGEN DI-INTERSTTTIAL CLUSTERS

IN UIOBIUM AND VANADIUM *

C.E. Laciana, A.J. Pedraza **

Departamento de Materiales,Comision Nacional de Energia Atomica,

Buenos Aires, Argentina,

and

E.J. Savino ***

International Centre for Theoretical Physics, Trieste, Italy.

MIRAMAEE - TRIESTE

August 1977

* To be submitted for publication.

** Present address: Universidade Federal de Minas Gerais, Esccla deEngenhariavRua Espxrito Santo 35, 30000 Belo Horizonte, Brazil.

*** On leave of absence from Departaraento de Materiales, Comision Nacionalde Energia Atc5mica, Buenos Aires, Argentina.

ABSTRACT

The lattice distortion and strain energy due to oxygen and nitrogen

di-interstitials located at neighbour octahedral sites are calculated in

niobium and vanadium. The lattice static Green function method is used.

The technique is extended in order to include in the calculation the lattice

induced relaxation of the impurities. Effective impurity-host atom inter-

atomic potentials are obtained from the single interstitial internal friction

and volume expansion data. It is concluded that the configuration with the

interstitials located at third neighbour sites is the most stable one. A

mechanism of migration of this pair towards a geometrically equivalent

configuration is proposed.

I. INTRODUCTION

The Snoek internal friction peaks [l] measured in bcc metals are

generally wider than those related to a single relaxation process. This

broadening effect is attributed to the tendency of the interstitialB to form

clusters. The works of Powers and Doyle [2], Keefer and Wert [3,U], GIbala

and Wert [5] and Ahmad and Szkopiak [6] appear to confirm that assumption.

Their measurements provide information about the cluster dlpole tensor and

activation energy for rotation, let, little is known with certainty about

the arrangement of the interstitial atoms forming the cluster. Fisher [7] and

Yang [8] applied elastic models to study the distortion due to ga3 interstitial

clusters in bcc metals. Johnson et al. [9] studied di-interstitial clusters

for carbonin a iron and nitrogen in vanadium. If the octahedral occupancy

by the gas interstitials is assumed,the elastic calculations [7] indicate

stability of the third and fifth neighbour pairs (Fig.l), -while the computer

simulation ones [9] favour the third neighbour as the most rtafcle. The binding

energies calculated1 by Johnson et al. [9] fer carton in iron are 0.13, 0.11

and 0.08 eV for the third, fourth and fifth neighbour pair, respectively (Fig.l).

On the other hand, it has been suggested that the structure of some tantalum

alloys may indicate a preference for the fourth nearest neighbour pair in the

solid solution [10]. The relatively large measured values of the ratio among

the di-interstitial and the mono-interatitial shape factors have led to the

conclusion that the distortions produced by each interstitial of the pair must

be additive [1]. This condition is fulfilled by the third, fourth and eighth

neighbour configuration among those shown in Fig.l.

The purpose of this work is to perform a theoretical study of the

configurations of the interstitial pairs in bcc metals in the hope of clarifying

some of the uncertainties mentioned above. A3 the discrete character of the

lattice cannot ie neglected in the calculation of the distortions owing

to the defects, tvo different calculation techniques are appropriate, the

computer simulation technique [9] and the lattice static Green|techniqufi [11].

The first, though very refined, requires a great deal of computer time and,

in view of the uncertainty in the determination of the interatomic potentials,

the advantages of including the lattice anharmonicity far from the defect's

nearest neighbour distance remains a subject of controversy. The second

technique,adopted here, rests on the static Green function method recently

reviewed by Tewary [ll]. This technique requires a minimum of computational

effort and fully allows for the discrete character of the lattice. While the

harmonic approximation is accepted for those atoms outside the defects core,

anharmonic effects,i.e. displacements beyond haraonlcity, are accounted for

Tevary [l2],Savino and Tewary'113] and Laciana et al. [lk] have cal-

culated the distortion oving to single gas impurities in a bcc lattice.

Effective gas—host atom interatomic forces were derived by these authors from

the measured values of the impurity dipole tensor. The effect of local an-

hannonicity was therefore included to all orders in those forces and effective

interatomic potentials could be fitted. Those potentials and the calculation

technique vere extended by Savino and Tewary [15] to calculate the impurities

activation energy for migration. The success of that extension [1^,15]

rendered further support to the validity of the effective potentials used.

In the present paper effective impurity-host atom interaction potentials

are deduced for the oxygen and nitrogen mono-interstitials in niobium and

vanadium by assuming octahedral occupancy [l2].-[jj»] and used for calculating

the distortion due to di-interstitial clusters (oxygen-oxygen, nitrogen-

nitrogen) at neighbour octahedral sites. In general no direct interaction

among the impurities is allowed for-The lattice static Green function values

calculated by Schober et al. [l6] for niobLum and those obtained by Fourier

inversion of the force constant matrix adopted by Tewary [12] for vanadium

are used in the calculation. For calculating the Green function values in

vanadium,6U000 points of the first Brillouin zone are included. On account of

the loas of symmetry with respect to the single interstitial case, the Green

function method as discussed by Tewary [ll] must be modified. The corresponding

calculation technique is discussed in the next section.

II. THEORY

The lattice static Green function method is adopted in order to

calculate the distortionrEo an interstitial impurity cluster in an otherwise

perfect lattice. Far from the defect .'s core the interaction among lattice

atoms is assumed to be harmonic. The total energy of the impurity cluster-

host lattice atom Bystem may thus be written in the a&iabatic approximation as

0r

S 5

where JJ denotes the atomic displacement, SL points towards the atom in the

host lattice and n towards the impurity. The first term in (l), (j>0 ,

corresponds to the energy of a reference state, where both the lattice and the

cluster are in equilibrium. The second, V , relates to the energy stored in the

bonds between atoms X and n as well as the anharmonie contributions toif n/

the energy in the defect space, where anhamonie displacements are allowed for.

The third and fourth terms in Eq.(l) are, respectively, the lattice and cluster

distortion energies in the harmonic approximation, 41 being the force constant

matrix. The equilibrium configuration can be calculated by imposing over the

energy (l) the minimization conditions:

= O for a l l {2)

These yield the system of equations

t> ~

C3b)

which has the formal solutions

where

s f i , r ) - ? ( i . r j "

'< •-- { rt, ( « ' ) ) .

2 is therefore the perfect l a t t i c e Green function and i t can be calculated

by s ta t ing Born-Von Karman boundary conditions ei ther in a f in i t e [11,12]

or in an effectively inf in i te l a t t i c e [16]. Fa* a small cluster, y can be

calculated explicit^ by matrix Inversion. Under the assumption of non-

direct interaction between the impurities, y is singular and (1+b) reduces to

V V = 0n (5)

If the impurlty-faOBt atom interaction is assumed as given by an effective pair

potential, i.e.

16)2It

the effective forces between the cluster and the lattice are

(7a)

la) , (7b)

where the derivatives are taken at the equilibrium configuration and these

forces may then explicitly depend on the displacements. Eqs.(!*a) and (5) can

now be written as

and

(8a)

r<J ( - > (8b)

If a perfect lattice Green function is generated for a cell of N

atoms, vith cyclic boundary conditions, the dimensions of y(2) and F(£)

are 3N and those of 5(1,1.') and <$>(ltl'} are 3H x 3M • These dimensions

tend effectively to infinite for a Green function obtained by a q space

integration near the origin, as proposed by Sehoter et al. [l6]. Howe.ar,

assuming that the forces Fill,) are not null only for a sub space of c*

atoms (the above-mentioned defect space), the lattice distortion car. be cal-

culated from Eqs.(8). These constitute a system of implicit equations for the

- 5 -

forces and displacements of the <yf*atoms that can be solved numerically. The

use of the lattice symmetry- allows in general for a further reduction in the

number of equations.

III. APPLICATION TO PAIES OF GJiS MTERSTITIALS

Eq.s.(8) must be solved in order to find the displacement of either

the individual gas atoms that constitute the pair or the host lattice atoms.

Eq.(6) can now be written aa

where v(£si) states the effective Interaction potential of a single interstitial

located at an i site.

The procedure devised by Tewary [12] In order to derive an effective

impurity-host atom interaction potential is adopted. In a bcc host crystal

with diluted gas impurities, the principal values of the stress dipole tensor

can be deduced from internal friction and volume expansion measurements if

the octahedral or tetrahedral site occupancy is assumed. This tensor Gb

is related to the effective forces exerted by an impurity over its neighbour

host lattice atoms

}(10)

If the direct interaction of the impurity with atoms further away than Its

second neighbours is neglected, those forces F(£) can fee calculated from the

dipole tensor values. By replacing ohose forces in Eq.(8a) the corresponding

displacements of the host atoms can be obtained and an effective Interaction

potential may be derived from these forces and displacements. This procedureus

only alldwsjjto fit a discrete number of derivatives of the potential function

to tl?3 proposed values of the forces, while the shape of the potential curve

remains rather arbitrary. In .rder to assert the validity of the calculation,

two different potentials are ixr'd. A Born-Mayer

(11)

- 6 -

and a 6,12 Lennard-Jonee

(12)

where a is the la t t i ce parameter and r = | i . - i | . The parameters deducedfor oxygen and nitrogen in niobium and vanadium by assuming an octahedralin t e r s t i t i a l occupancy are reported in Table I . The dipole tensor valuesreported by Buchholz [IT] and Tewary [12] are used for the calculation exceptfor the V-0 system where the potentials deduced by Laciana et a l . [lit] areadopted. The existence of a small difference hetveen our parameters for theBorn-Mayer potentials and those of Tewary [12], when the same dipole tensorand Green function values - see Hef.lt - have been uaed in both calculations,can only be attributed to a different value of the host crystal la t t iceparameter. The value reported by Pearson [18] for niobium and that ofHef.19 for vanadium have been chosen by us.

I V . HESULTS

The di-interstitial configurations to be studied are schematically

shown in Fig.l. The first impurity is assumed to be located at an octahedral

interstice taken ae the origin. The second is located at the (001) site,

configuration (l) and, respectively, at (001), (2); (111), (3); (200), {h);

(020), (lt')i (501), (5); (211), (6) and (202), (8). The positions are

oeasured in units of "a/£", a being the lattice parameter, and the con-

figurations are numbered according to the square of the distance between

interstitials. For the above initial location of the impurities the system

of Eqs.(8) is numerically solved in order to obtain the displacement of the

host atoms and the impurities. The latter are allowed to displace under

the constraints of preserving the symmetry of the defect-host lattice system

and of keeping the distance among the impurities larger than their covalent

diamater [20], If this separation is reached,the rigid ball approximation

for the direct interaction between the impurities is adopted. This last

constraint becomes effective only for some cases of configuration (l), where

the lattice exerts forces over the two interstitials pushing them towards the

intermediate tetrahedral site. Under the previously mentioned constraints,

EqB.(8) do not necessarily constitute a compatible set, i.e. for the con-

figuration (2) no solution has been found. This implies that, assuming the

validity of the approximations leading to those equations, that configuration

cannot exist at equilibrium.

The lattice strain energy induced by the defect is

(13)

The elastic interaction energy among the impurities which constitute the pair

can be calculated by using Eshelby's approach [21] in the discrete lattice

approximation as discussed by Savino [22j. la first order this energy

can be obtained by multiplying the effective force exerted by one impurity

on a neighbour atom jl , assuming no second impurity exists , times the

displacement of this seme atom if only the second defect is at its

interstitial site

(HO

These energy values for the systems studied are reported in Table II. However,

Eq.(lh) is only valid for calculating the interaction energy when the sources

of internal stress of either defect do not affect the same lattice atom. In

the configurations studied, a more realistic value of the lattice induced

interaction energy can be obtained by subtracting from the lattice strain

energy induced by the defect pair, twice the one corresponding to a

single impurity

II

e U5)

These energy values assuming either a Bo: n-Mayer or a Lennard-Jones potential

for the single interstitial are reported in Table II for the configurations

mentioned above. The relative stability of a given cluster will later be

discussed in relation to this energy.

The non-null displacements of the impurities from the initial, un-

distorted, octahedral sites for some pair configurations ' are reported in

Table III. The largest displacements occur at the configurations (l) and (k1).

Th-5 full tables of the displgctjr.er>" of the impurities and the values of

iipole tensor are available en request.

-8-

The feasibility of anelastic relaxation due to the pair of defects

is determined by its point symmetry [lj, whereas the strength of the phenomenon

depends or the dipole tensor - Eq.(lO) - associated with the pair. In our cal-

culation the local lattice symmetry at the defect's core can be destroyed and,

then, the tensor defined "by (10) is not symmetric. Hovever, as the resulting

configuration is stable against rotation, a symmetric dipole tensor can be

defined at the final, relaxed,position of the host lattice atoms

M.') (16)

*)For some of the octahedral pairs studied J the principal values and axes,

in terms of the crystallographic axes, of this tensor are reported in Table IV.

For the sake of comparison the dipole tensor of the single interstitial,

assuming octahedral occupancy by the defect, is also reported there. The

destruction of the local lattice symmetry implies that the eigenvectors of

the dipole tensors at the relaxed configuration - Eq..(l6)- must not necessarily

agree with a lattice symmetry direction. The corresponding separations from

a lattice direction are then reported in Table IVB.

V. DISCUSSION AND CONCLUSIONS

A systematic study of di-interstitial oxygen and nitrogen clusters in

niobium and vanadium was performed. In this section, the validity of the

calculation will first be discussed and the main assumptions summarized. After-

wards, the stability of the pair clusters will be studied and the result con-

cerning the most stable among them compared with experimental findings reported

in the literature. Finally a feasible relaxation mode of that configuration

H external stress is

The pair of Impurities were assumed to be located at neighbour octa-

hedral sites and the lattice static Green function method was used for the

calculation. The validity of the results rests on the perfect lattice Green

function used - see [12] and [lM for a discussion on this point - and on the

assumption of an octahedral occupancy by the single interstitial. This

assumption on the single defect symmetry is most relevant to determine the

effective impurity-host atom interaction potential. In addition this potential

i s assumed not to he affected by the presence of a second interstitial nearby.

•) See footnote on page 8.

Also, the direct interaction among interstitials is neglected except when

they move one towards the other to distances smaller than their covalent

radius. This approximation of considering the main interaction among gas

atoms being induced by the large lattice distortions owing to the defects

seems a sensible one In view of the relatively small covalent radius of these

atoms. The electronic states of a given interstitial can,however,be modified

by the presence of the other, the eventual effect of this modification is

neglected here.

As the derivatives of the effective potentials have been fitted to the

measured values of the single defect dipole tensor, the«s.-potentials give an

accurate representation of that defect. However, in the calculation of the

displacements owing to the pair of defects, separations among defect and

lattice atoms relatively far from those used for fitting the potential are

also important. More than one potential shape should then be used for the

calculations in order to test their validity. A 6,12 Lennard-Jones and a Born-

Mayer potential were used in this paper. No major qualitative differences

were found for the two different potentials. However , the comparison of

the quantitative values is useful in giving some estimate of the uncertainties

due to imprecision in our knowledge of the force laws. A most relevant and

apparently potential independent conclusion from our calculation is that the

impurity pair cannot remain bound in configuration (2).

The stability of the pair can be discussed in relation to - i t s binding

energy. Two methods for calculating this energy have been proposed In this

paper, an elastic first-order one based on Eshelby and Savino's approach

[21,22] - Eg,, (lit) - and a more accurate one which Includes the anharmonic

relaxation in the vicinity of the cluster - Eq.(15). Similar resuLts arise

from both methods, the main difference between them is that the first-order

calculation predicts the configuration (l) to be the moat stable sviiile i t is•ut^alj unstable in tUe -more accurate anharmciTiic a rproacfo. This difference

must \>e taken aa a proof that ariharmonic effects cannot >e neep-ectea. ?oT VtAa

configuration. This is further shown in the calculated large dleplacementa

of the impurities from the interstitial site (Table I I I ) . The proximity

of one impurity to the other and the absence of a lattice trapping barrier

at th* initial interstitial location determine either the impurities being

attracted towards the intermediate tetrahedral site or s t r o n g repelled from

it The occurrence of one or the other displacement depends on the intera-

ction potential used for the calculation (see Table III} but, for either

potential, there is a strong lattice distortion and therefore a high strain

energy associated with this configuration. Similarly this energy is also

high, for the configuration ( V ) , wher.e both interstitials share a common first

neighbour atom. TTie symmetry or the configuration does not allow this atom

to be displaced from the perfect lattice position and, therefore, a large

distortion c:1 the remaining lattice must arise in order to accommodate the

impurities.

Prom Table II and the discussion above, it is concluded that the most

stable configuration is (3) for all the pairs studied. The anharmonic

calculation predicts the configurations (h), (5) and (6) also to be stable,

whereas in the first-order approach only (6) is stable among these for

all the pairs studied. On the other hand, the configuration (8) appears as

a very stable one in the vanadium lattice {Table III).

Our calculations then predict that the observed gas in ters t i t ia l pair

is located at the configuration (3). This conclusion is consistent with the

experimental findings. Gibala and West [5] have reported a binding energy ofthe

0.0J eV for^oxygen-oxygen cluster in niobium, this value approximately agrees

with the one resulting from our calculation in the anharmonic approximation.

The calculated dipole shape factor (Table IV) of the configuration (3) shows

that the principal distortions produced by each component of the di- interst i t ia l

cluster are nearly additive in agreement with the experiments [5]. On the

other hand, the calculated binding energies approximately agree with the

difference between the relaxation energy of the cluster and that of the

single in ters t i t ia l , as assumed by Szkopiak [23].

In order to study the kinetics of the anelastic relaxation owing to

the inters t i t ia l psir the defect reaction among different configurations must

be included. In that sense Nowick [2k] has developed a general theory and

has solved as an illustrative example the present case of the in ters t i t ia l -

inters t i t ia l pair in a bcc la t t ice . However, this author has considered the

pair being bound in configurations (1), (2) and (3)- Out of these three our

calculation predicts (3) to be stable, whereas the configuration (l) is highly

unstable and (2) does not exist. These findings suggest a completely

different pattern of relaxation from that proposed by Nowicic [2lt], The two

impurities can be considered as never reaching,in the lattice, a separation

among them smaller than a corresponding to the configuration (3). The

*) This comparison can only "be a qualitative one because the dipole tensorhave a

of the pair does not/fully tetragonal symmetry - see {1] - and, in addition,

it is calculated at the relaxed position of the lattice atoms, while the one

corresponding to the single defect is evaluated at the perfect lattice

position.

stress induced relaxation from one configuration (3) to another can always

be achieved by allowing an intermediate configuration (6) to be formed. This

is constituted by an individual jump which is followed by a second one of

the other impurity in order to reach a new configuration (3). The relaxation

process is schematically shown in Fig.2, where the lattice relaxes from a

configuration (3) to a stress-favoured (3) * • This process seems to be

the most consistent with our results and it has been previously proposed by

Johnson et al, [9] on a similar basis. The feasibility of the pair attaining

other configurations in addition to those mentioned can be Included in

Nowick's general theory [2U]but the previously discussed restrictions to

the movement of the pair must be taken into account.

ACKNOWLEDGMENTS

This work was performed under the auspices of the Comisio'n

de Energia Atimica, Buenos Aires, Argentina. One of us (E.J.S.) vould like

to thank Professor N.H. March .for critical reading of the manuscript,and

Professor Abdus Salam, the International Atomic Energy Agency and UNESCO

for hospitality at the International Centre for Theoretical Physics, Trieste.

KEFEKEHCES

[l] A.S.Hoviek and B.S. Berry, Anelastic Relaxation in Crystalline

Solids (Academic Press, Hew York and London 19T2).

[2] R.W.Powers and M.V. Doyle, Trans. Met. Soc. AIME 21£, 655 U959)•

[3] D. Keefer and C. ¥ert, Acta. Met. 11,, 1*89 (1963).

[It] D. Keefer ana C. Wert, J . Phys. Soc. (Japan) Suppl.III 16,, 110 (1963).

[5] B. Gibala and C.A. Wert, Acta Met. .U, 1095 (1966).

[6] M.S. Ahmad and Z.C. Szkopiak, J. Phys. Chem. Solids 31., 1799 (1970).

[7] J.C. FiBher, Acta Met. 6_, 13 (.1958).

[8] Yang Cheng Chu, Report FTD-H.T-23-325-6'8 (AD-68U-989) , Air Force Systems

Command Wright-Patterson AFB, 0hio(l968).

[9] H.A. Johnson, G. Dienes and A.C. Damask, Acta Met. 12,, 1215 (1961*)-

[10] A.S. Howtck and VuR. Heller, Adv. Fhya. 12., 215 (1963).

[11] V.K. Tevary, Adv. Fhye. 22., 757 (1973).

[12] V.K. Tevary, J. Phys. F 3_, 1515 (1973).

in[13] E.J. Savino and V.K. Tewary,^"Internal Friction and ultrasonic

attenuation in crystalline solids", Proceedings of an International

Conference, Eda. D. Lenz and K. Liicke (Springer-Verlag, Berlin,

Heidelberg, Hew York 1975) . P-196.

[lit] C.E. Laciana, A.J. Pedraza and E.J. Savino, submitted to Phys. Stat. Sol.

[15] E.J. Savino and V.K. Tewary, J. Phys. F 3., 1910 (1973).

[16] H.R. Sehober, M. Mostoller and P.H. Dederichs, Phys. Stat. Sol, (b),

6h_, 173 (197!*).

[17] J . Buchholz.lProe. In t . Meeting on Hydrogen in Metals, Vol.11, p.51*1*

(Julieh 1972).

[18] W.B. Pearson, A Handbook of Lattice Spacings and Structures of Metals

and Alleys, Vol.2 (Pergamon Press, 1967).

[19] ASTM Publication PD1S-191 (1969).

[20] L. Pauling, The Chemical Bond (Cornell University Press, Ithaca,

Hew York 1967).

[21] J.D. Eshelby, Solid State Fhys. 3_, 79 (1956).

[22] E.J. Savino, to "be published in Phil. Mag.

[23] Z.C. Satopiak, Suppl. J. de Physique 32, C2-1 (1971).

[2U] A.S. Howick, J. Phys. Chem. Solid! £1, l8l9 (1970).

Taole I

Parameters for the impurity-host atom effective potentials used in

the calculations.

* " " ^ - - ^ _ ^ jystemj-'otential *^~^~~-^_^

Born-Mayer Vfl[eV]

P

Z. r.p,-.rd-Jones P[eV)

Q[eV]

Nb-N

0.1171*

13.7253

29.8687

It.l*l6&

Nb-0

0.2lt5lt

8.1635

9.251*9

-1.1*181*

V-N

0,0799

15.6055

31-3508

5.6015

V-0

0.0593

ie.ltO69

1*0.6987

8.3196

- l i t -

Table II

Interaction energy 'between the impurities in units of eV.

System

TXbS

DTb-0

V-H

V-0

-—-^Configuration

AH1

L-J

B-M

A E oB >

L-J

B-+I

A E ; )

&E

L-J

B-M

AK ** '

L - J

B-M

( 1 )

-0.67

0.U6

O.I16

-0.60

0.17

0.07

-0.55

0.23

0.5S

-0.52

0.67

0.661

(3)

-0.30

-0.15

-0.17

-0.24

-0.10

-0.0S

-0.26

-0.07

-0.16

-0.27

-0.0a

-0.E1

(It)

0.03

-0.03

-0.02

0.03

-0.03

-0.03

0.02

-0.02

-0.02

0.02

-0.03

-0,03

(If)

1.28

0.97

1.03

1.06

0.60

0.53

1.12

0.98

1.05

1.16

1.05

1.17

(5)

0.09

-0.03

-0.03

0.12

-0.05

-o.oi*

-0.01

-0.06

-O.07

-0.01

-0.06

-0.07

(6)

-o.oi*

-0.03

-0.03

-O.C*

-0.03

-0.03

-0.02

-0.01

-0.01

-0.02

-0.01

-0.01

(S)

0.11

O.Olt

0-0U

0.10

0.01

0.00

-0.05

-0.05

-0.05

-0.05

-0.06

-0.05

Evaluated through Eq.(llt) in the text.

••) Evaluated through Eq..(.15) in the text; L-J: 6,12 Lennard^Jones potential

and B-M: Born-Mayer potential,used for the calculation.

Tahle III

Non-null displacements of the impurities from the octahedral site (inunits of "a/2") for configurations ( l ) , (3) and [It1}. The interstit ial atthe origin is indicated with a (0) and the second that constitutes the pairvith a ( l ) .

Configuration

Symmetry of thedisplacements(see Fig.l)

ffb-N L-J

B-M

Nb-0 L-J

B-M

V-K L-J

B-M

V-0 L-J

B-M

(1)

-0.1*6

-0.39

-0.35

o.io011

O.O39CR

-0.39

-0.5lt

-0.39

. .

(3)

uy ~ ^ y

0.03

0.01

-0.00

0.00

0.07

0.02

0.06

0.01

X Z

UX "B

-0.09

-0.07

0.00

-0.02

-0.13

-0.07

-0.12

-0.05

(1.0

-0.30

-0.30

-0.26

-0.26

-a.30

-0.31

-0.31

-0.31

CR: L'he rigia ball approximation is adopted for the interstitials' interaction,the . ir.purities are located at the covalent diameter distance.

-15--16-

Table IV(b)

• >

M

h CM

id usft -

3 o-H•r* -P4J TH

£ P

.£ Op -p

(J<*H (I)O >

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Deviations of some of the eigenvectors of Table IV{a) from a

lattice symmetry direction, in units of "a/2".

-puaav

I

om

a

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a

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Configuratioti

Nb-N

Nb-0

V-N

V-0

e 1 L-J

B-M

e L-J

B-M

Ej L-J

B-M

e3 L-J

B-M

e l L " J

B-M

C3 L-J

B-M

Z1 L-J

B-M

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(3)

(0.077, 0.006,

(0.01*8, 0.002,

{-o.ooit, 0.105

(-0.002, 0.068

(0.021, 0.000,

{0.018, 0.000,

C-0.000, 0.029

C-0.000, 0.025

C0.11B, o.Qi.1*,

(0.01(3. 0.002.,

(-0.010,0.167,

(-0.001, 0.061

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Co.027, 0.001,

C-oiio 0.168,

(-0.001, 0.038

0.077)

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Co.o, 0.008, 0

C-0.000, O.OlS

Co.001, 0.0, 0

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C-0.000, 0.012

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, 0.011)

-18-

Fig.l A schematic representation of the interstitial pairs configuration

external

stress

FiR-2 Stress induced relaxation from a configuration (3) to a (3)*.

CURRENT ICTP RREPRIHTS ADD INTERNAL DEPORTS

IC/76/99 W. WADIA and A.O. EL-SHAL: Hartree-Fock procedure for bosons with appl icat ionto the alpha p a r t i c l e model for -^C nucleus.

IC/77/27 I . AHMAD: Scat ter ing of high-energy a - p a r t i c l e s on ""C.

IC/77/32 H. INAGAKI: Quantization of scalar-spinor instanton.

IC/77/33* M.S. SAGOO: Recombination of electrons with spherical droplets in aIHT.REP. co l lo ida l plasma.

IC/7T/31! C.H. LEE: Coupled g rav i ta t iona l and electromagnetic per turbat ion equationswith the source terms.

IC/77/35 Z. HORVATH and L. PALLA: Extended raonopoles in Kauge f ie ld t h e o r i e s .

IC/77/37

JC/77/38"INT.REP.

IC/77/39*IHT.REP.

IC/77AO

IC/7TA1

JC/77A2*INT.REP.

SHAFIQUE AHMED: Relativistic two-fermion equations vith form factors andanomalous magnetic moment interactions.

SHAFIQUE AHMED; Relativistic two-body equation and superpositroniumresonance s ta tes .

V.P. FROLOV: Massive vector fields and black holes.

A.N. MITRA: Dynamics of a six-quark deuteron: Form factor and structurefunctions.

M.S. BAAKLINI: Bon-linear realizations and Higgs mechanism for extendedaupersyiEmetry.

A. SHAFEE and S. SHAFEE: The swallow-tail catastrophe and ferromagnetichysteresis.

A.H. CHAMSEDDItrE: Massive supergravity from spontaneously breakingorthosymplectic gauge symmetry.

IC/7TA3

IC/7T./JM- U.S. BAAJE^HI: Supersywmetry and classical solutions

IC/77/52* U.S. BAAKLIBI: Cartan's geometrical s t ruc ture of aupergravity.INT.REP.

IC/77/54* H. IHAGAKI: Zero modes of quantum f luctuat ions for the ins tanton.IMT.KEP.

IC/77/55 S. FERRASA, M. KAKU, P. van NIEUWENHUIZEN and P.K. TOVffiSEND: Gaugingthe graded conformal group with uni tary in te rna l symmetries.

IC/77/57* A. AUBILIA and D. CHEISTODOULOU: Str ing dynamics in an external f i e l d :IHT.REP. A family of exact so lu t ions .

IC/77/58* N.S. BAAKLINI, S. FERRARA'and P. van NIEUWENHUIZEN: Classical solut ionsIHT.REP. in supergravity.

IC/TTA5 E. NADJAKOV: Quantum system l i f e t imes and mtittBurc-ment pe r turt iat iona .

IC/77/1+7* P. CAMPBELL: Pro jec t ive geometry,IMT.BEF.

uri'I twia to r theoi^y.

1C/77/U9* G. VIDOSSICH: CtutplyKin's method i s NewLon'E irINT.REP.1C/T7/50 A.M. MITRA, S. CHAKRABAI1TY and 6. JOOi): A «etis;

r a d i a t i v e decays of cc laesucis.VMD mechanism fur

. : ; : > . . ; ' ; ' r . , : :

J , ^ ' ; ^ : . - L i i i i •. t .^ ; ' 1 : . t r l L . t . : - .

L: kr,~e, A . ' / - T : J ^ - : . 1 . *{•.-:.•: ".at: i ui i i .P . . i r i' w r : < K I . J K : . •;,• v ; • .<.:••]•