On the electronic structures of lattice defects in s, p, d-valence metals

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On the electronic structures of lattice defects in s, p, d-valence metals

K. Masuda-Jindo

To cite this version:

K. Masuda-Jindo. On the electronic structures of lattice defects in s, p, d-valence metals. Journal de

Physique, 1986, 47 (12), pp.2087-2094. �10.1051/jphys:0198600470120208700�. �jpa-00210402�

On the electronic structures of lattice defects in _{s, p,} d-valence metals

K. Masuda-Jindo

Department of Materials Science and Engineering, Tokyo Institute of Technology, Nagatsuta, ^Midori-ku,

Yokohama 227, Japan

(Reçu ^{le 15} juillet 1986, accept6 le 28 aogt 1986)

Résumé.

Nous étudions la structure électronique des défauts de réseau, lacunes, marches, ^{crans de surface}

et fissures de clivage, ^{dans les métaux de valence s,} p, d (nobles ^{et de} transition) ^{en utilisant une méthode de}

liaison forte améliorée. Dans ces calculs, 9 orbitales de base s, p et ^{d sont} introduites et les effets d’hybridation sp-d ^sont pris ^en compte. ^{Pour une} lacune, nous avons trouvé que les sites atomiques ^très proches ^{de celle-ci} (les ^atomes premiers voisins dans un cristal cubique à ^faces centrées) contribuent principalement à l’ énergie ^de

formation de lacune et que les contributions des sites atomiques plus éloignés de la lacune sont moins

importantes. ^Ceci justifie ^les premiers traitements théoriques ^{basés sur} l’approximation ^{du second moment.}

Les calculs de structure électronique des marches et des crans de surface sont faits pour différents ^{cas et} il est montré que les densités d’états ^sont presque indépendantes ^du type ^{de cran et} dépendent seulement de la structure cristalline, ^{en accord avec} des considérations thermodynamiques. ^{En outre,} nous avons étudié les structures électroniques de fissures de clivage dans des métaux de valence s, p, d ^et discuté la réactivité

chimique ^{de la} pointe ^{de la fissure en} conjonction ^avec le fendillement de corrosion. A l’aide de calculs de structure électronique, ^nous démontrons l’applicabilité de la méthode de liaison forte améliorée aux problèmes

liés aux défauts de réseau.

Abstract.

The electronic structures of lattice defects, single vacancy, stepped surface, surface kink and

sharp cleavage ^{crack, in s,} p, d-valence (transition ^and noble) ^{metals are} investigated using ^the improved tight- binding (TB) recursion method : in these calculations, ^nine s, p and d-basis orbitals ^are introduced and the sp-d hybridization ^{effects are} fully taken into account. For a vacancy, ^we have found that the atomic sites very close to a vacancy (e.g., ^first nearest-neighbour ^atoms in the fcc crystal) ^can primarily contribute to the vacancy formation energy and contributions coming ^{from more} distant atomic sites are less important. ^This gives ^a support for the earlier theoretical treatment based on the second moment approximation for the similar

problems. ^Surface step and surface kink electronic structure calculations are performed for the different kinds of steps ^and kinks, and it is shown that kink density ^{of states} (DOS) ^{are almost} independent of the kink type and only depend ^{on the} crystal structure, ⁱⁿ agreement ^{with the} thermodynamical consideration. In addition,

we also investigate the electronic structures of a sharp cleavage ^{crack in s,} p, d-valence metals and discuss the chemical reactivity of the crack tip ⁱⁿ conjunction ^{with the stress corrosion} cracking (SCC). ^{Due to these}

electronic structure calculations, ^we demonstrate the applicability ^{of the} improved ^TB recursion method to the various lattice defect problems.

Classification

Physics ^Abstracts

61.70

71.20

1. Introduction.

Recent advancement in the first principle ^electronic

theories [1] have enabled us to get ^{detailed informa-} tion on the cohesive properties of metals and ordered

alloys ^without introducing ^any adjustable parame- ters. Unfortunately, however, it is difficult to apply

this type of theoretical scheme directly ^{to the} important lattice defect problems ^of crystalline ^mate- rials, e.g., point defect-dislocation interaction [2],

radiation induced complex point ^defects [3], ^disloca-

tion motion [4], grain boundary segregation [5],

crack propagation [6], and other related problems.

Therefore, ^an alternative approach should be deve

loped ^for treating ^such important lattice defect

problems.

The traditional tight-binding (TB) ^{or linear combi-}

nation of atomic orbitals (LCAO) scheme has been very successful for wide variety of lattice defect

problems [7-9]. ^This type ^{of method} has the virtue of

simplicity ^{and of} physical transparence. ^In the pre- sent paper, ^{we use an} improved ^{TB scheme with s,} p and d-basis orbitals and check the applicability ^{of the}

method to the real lattice defect problems.

As a first step, ^{we will treat} simple lattice defects without allowing lattice relaxation calculations. First-

ly, ^we investigate ^the electronic structure of a

vacancy ^{in the s,} p, d-valence metals using ^the

continued fraction (recursion) technique [10-11],

and understand the physical origin (main ^contribu- tion) ^{of the formation} energy of ^a vacancy EFv. ^For

the actual numerical calculations of the electronic

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198600470120208700

2088

structures, ^{we choose a} noble metal (Cu) lattice,

since sp-d hybridization ^{effects are} particularly important in these metals.

Secondly ^we investigate the electronic structure of the stepped surface and surface kink, and show that the surface kink electronic structure does not depend

on the kink type ^and only depend ^{on the bulk} crystal

structure. This calculation is performed ^{for the} following ^{reasons : for the} vacancy problem, ^{it is} generally assumed that the number of atoms of the system ^is kept ^{constant and the} missing ^{atom is} placed ^{on a kink at} the surface. From the thermody-

namical consideration, ^{it can} be shown that putting

the atom on a surface kink is strictly equivalent ^to increasing the number of bulk atoms by ^{one. In this} respect, ^{one can} expect quite peculiar ^electronic property ^for the surface kink. Furthermore, ^{it is} known that a surface defect like ^a kink or step ^is

highly reactive and improves drastically ^the catalytic activity [12]. Therefore, it is of great significance ^to investigate ^the electronic structure of the stepped

surfaces and surface kinks.

Thirdly, ^we investigate the electronic structures of the cleavage ^{crack in s,} p, d-valence metals. In order to simulate a sharp cleavage crack, ^{we break the}

metallic bonding ^{across the} cleavage plane ^without introducing lattice distortions [13]. ^{This can be} accomplished by setting the electron transfer inte-

grals ^{across the} cleavage plane ^{to be zero. We then}

discuss the chemical reactivity ^{at the crack} tip, ⁱⁿ conjunction ^{with the stress corrosion} cracking (SCC)

of the material. This type of theoretical calculation is of quite importance ^{since it is now well} established that ductile metals (e.g., fcc metals like a-brass, ^Cu

and Ni) ^{become often brittle} and show the cleavage-

like fracture in a corrosive environment [14].

From these electronic structure calculations, ^we

demonstrate the applicability of the TB recursion method of s, p, d-basis orbitals ^to the various lattice defect problems. ^The application ^{of the TB} theory

with _{s, p and} d-basis orbitals is important ^not only

for treating noble and transition metal systems ^but also for treating many other materials. For instance,

it can be used for the study ^of sp-valence impurities

in transition metals [15] ^{and lattice} defects in the wide range of intermetallic compounds [16].

The format of the present paper is ^as follows : in section 2, ^we present ^the principle ^of calculations for the electronic structures of lattice defects using ^the

TB recursion method of s, p and d-basis orbitals.

Results of numerical calculations and related discus- sions are given in section 3, ^{and section} 4 is devoted to conclusions.

2. Principle ^of calculations.

It is now well known that the continued fraction

technique [10, 11, 17] is very powerful and efficient to calculate the electron or phonon density ^{of states} (DOS) ^for systems without translational symmetry.

In the present study, ^{we use} the continued fraction

technique ^{and the TB} electronic theory ^of s, p and d- basis orbitals. For treating ^wide variety ^{of materials}

it is highly ^{desirable to take into account the s,} p and d-basis orbitals and treat them on the same footing.

We calculate local DOS of electronic states on

atom i,

from the diagonal element of the Green’s function

where I A , i ) is the atomic orbital of type A (À

^s, px, py, ^{Pz, xy, yz, zx,} x2 - y2, ^{and 3 z2 -} r2)

centred at site Ri, ^{and H} denotes the TB Hamilto- nian of the system.

The electronic Green’s function can be given ^{as a}

continued fraction form

where the recursive coefficients an and bn ^{can be}

related to the moments of the DOS and obtained either from moments method [11] ^or directly ^from

the recursion method [10, 17].

The electronic occupancy NAi ^of the I À, Ri )

state and the structure energy U of the system ^{can be}

calculated from the local DOS P Ai ( E) ^{function as}

In general, for the calculations of NAi ^{and U, a} simple ^numerical integration ^method (e.g., Simp-

son’s method) may be used. However, ^{in order to}

obtain ^more accurate results for the calculations of

NAi ^{and U,} analytic integration ^{method is} required : Highly ^accurate energy calculations are needed for instance for obtaining interatomic force constants and atomic relaxations around a lattice defect [18, 19]. ^For this purpose, ^we rewrite the local DOS as

with

where all the recursion coefficients after n-th level

are assumed to be equal ^{to their infinite values}

aoo ^and boo.

In equation (6), ^the expansion coefficients An ^for

D2n ( E ) ^are simple functions of the recursion coeffi- cients (al,’’’’ an, aoo) ^and (bl,..., bn, boo ). The

asymptotic recursion coefficients aoo and b00 ^are

related to the bandwidth and centre of the band, ^and

can be determined by the method proposed by ^Beer

and Pettifor [20]. This termination method preserves the moment information contained in the recursion coefficients and gives fairly convergent results for the structural energy. Using equation (6), ^{the local}

DOS functions are given by ^{the form}

where Ej ^{denotes the j-th (complex) root of the} equation D2n ( E ) ^{= 0 and} Cj ^{the weight of the}

partial ^fractions 1/ ( E - Ej ) . ^{Then, the} integration

of the DOS ^can be evaluated analytically ^{in terms of} elementary integrals [18, 19], and results for NAi ^and

U are much more accurate compared ^{to those}

obtained by using ^a direct numerical integration

method.

We now discuss the band parameters required ^for

the TB electronic structure calculations in the cubic

crystals : They ^are the atomic energy levels ES, Ep,

Ed tZg ^and Ed eg , ^and two centre hopping integrals ^sso-, spa, sdcr, ppor, ppir, pdu, pd 1T, ddu,

dd 1T and dd8 (in the usual notations [21]). ^In general,

these TB band parameters ^{can be obtained} by

several different methods. They ^can be determined

roughly by using ^{the so-called} fitting procedure. ^For instance, ^{Ohta and Shimizu} [22] obtained the TB band parameters, both atomic energy levels and ^two centre hopping integrals, for bcc transition metals from the (high symmetry) eigenvalues ^{of the APW}

band structure calculations. The similar TB band

parametrization has been obtained for the fcc(Cu) crystal by ^Smith [23].

On the other hand, ^Harrison [24] gives very

simple and universal formulae for the two centre

hopping integrals ^{of the s,} p and d-bands. They ^are given ^{in terms} of materials independent ^constants (electron ^{mass m} and Planck constant h) ^{and the d-}

state radius rd, characterising ^{the d-band} materials,

and shown to have the power law dependences

Rjj-n. Specifically, n

^{2 and 5 are used} for sp- and d- bands, respectively, ^{and n}

7/2 for the sp-d hybridi-

zation matrix elements. These TB parameters ^are useful for the overall understanding of the trend in the electronic properties ^{of s,} p, d-valence materials.

The most reliable TB parameters ^{can be obtained} from the first-principles ^TB theory [25]. ^Andersen

and coworkers [25, 26] ^have succeeded in transform-

ing the minimal base of muffin-tin orbitals (MTO’s)

into a TB base. Since this transformation is exact [25], all results obtained by using the linear MTO method or augmented spherical ^wave (ASW)

method can be reproduced by the TB method. This TB scheme gives two centre hopping intervals with almost universal decay.

In the present study, ^{we use TB band} parameters determined by ^the fitting procedure [23] ^{since we do}

not perform the lattice relaxation calculations around the defects. Therefore, ^the present ^electronic

structure calculations for lattice defects are not self- consistent and the TB band parameters ^{are fixed to} those values for the perfect ^lattice. (This approxima-

tion will not change ^the essential feature of the electronic structures of lattice defects.)

As a final remark of this Section, ^’we point ^out explicitly ^the importance ^{of the} sp-d hybridization

effects on the electronic structures of _s, p, d-valence

(transition ^and noble) metals : in particular, ^we

assess actually ^the significance ^{of the} sp-d hybridiza-

tion effects in the fcc crystal ^Cu by using ^{TB band}

parameters presented ⁱⁿ table I. For this purpose,

we calculate the second moments of the d-bands both with and without sp-d hybridization ^matrix

elements. For the d-bands in the fcc crystal (with nearest-neighbour interactions), ^{second moments}

are expressed ^as

Table I.

TB parameters for ^Cu crystal ^taken from reference [23]. Energies ^are given ^{in units} of ^{e V}

2090

where t2g eg ^{denotes the symmetry for the d-}

subbands, an atomic energy levels ^are simply ^taken

to be at the origin of the energy.

From equations (9), (10) ^{and TB} parameters ⁱⁿ tablé I, ^we obtain the numerical values for the second moments in units of ( eV ) 2/atom ;

1-’2 (t2g)

2.414, 1-’2 (eg) = 1.700 and A2( t2g )

0.6061, _t2 eg

^{0.5290, for} the d-bands with and without _sp- hybridization ^matrix elements, respecti- vely. This indicates clearly ^{that the inclusion of} sp-d hybridization ^{matrix elements} is crucial for the electronic structures of _s, p, d-valence (especially ^for

noble metals) ^materials.

3. Results and discussions.

The numerical calculations on the electronic struc- ture of lattice defects are performed for noble metal Cu using ^{TB band} parameters proposed by

Smith [23], ^{which are the} leat-squares optimized

values derived from the APW band structure calcula- tions [27] ^{and are} presented in table I. We have chosen Cu crystal ^because sp-d hybridization ^effects

are particularly important for this kind of materials.

Although ^the present ^numerical calculations are

performed only ^{for Cu} crystal, ^{we can draw} general

conclusions on the electronic structures of lattice defects in s, p, d-valence (noble ^and transition) metals, since relative magnitudes ^{of the two-centre} hopping integrals ^are rather insensitive to the species

of the materials [24-26], ^{and band} shapes ^{of the}

different metals are similar. For lattice defects, ^we consider single vacancy, flat and stepped surfaces,

surface kink and sharp cleavage crack in the s, p, d- valence metals, ^{which are of} quite importance ^from

the fundamental point of view of materials science.

In figure ^{1, we} present the calculated local DOS for the perfect ^Cu crystal, including the first-nearest-

neighbour interactions only (1). The energy is given

in units of eV and energy origin ^{is taken to be} equal

to the Fermi _energy EF. ^{The continued fractions are}

terminated at various n levels (n 15 ) ^{so that one}

can easily understand the role of higher ^order

coefficients (moments) ^on the calculated DOS. The fcc cluster used in the present calculations has a

sufficient size (- 6 ⁶⁰⁰ atoms) that 15 levels

(30 coefficients _a,, a2’... aIS’ b,, b2, ... b15) ^are practically ^{correct :} Further increase of the crystallite

size does not change ^{the DOS} substantially.

One ^{can see} in figure ^{1 that the DOS of the} perfect ^fcc crystal ^{are in} general reconstructed with about 11 or 13 couples ^{of exact} coefficients. (Fortu- nately, the convergence of the structural energy U from Eq. (5) is much faster and one can obtain

essentially ^correct results of U, ^{as a} function of number of band electrons N, using the DOS with exact coefficients _up to n a 4.) .The similar conclu-

(1) ^{It is} known that _s, p, d-basis LCAO formalism works quite ^{well even for the} description of free-electron like _s, p-bands [26].

Fig. ^1.

Electronic DOS calculated for perfect ^Cu crystal. Continued fractions are terminated at various n levels: n = 4(a), 7(b), 9(c), 11(d), 13(e) ^and 15(f).

Energies ^are given in units of eV.

sion has also been obtained for the DOS calcula- tions [28] ^{based on} the d-orbital scheme without

including ^the sp-d hybridization effects. This indica- tes that the convergence of local DOS within the TB recursion method is almost independent ^{of the}

inclusion of sp-d hybridization ^effects. However, ^the shape of the d-band depends strongly ^{on the inclu-}

sion of sp-d hybridization ^effects, and the total (s, p,

d) ^{DOS for Cu} presented ⁱⁿ figure ^{1 are} considerably

different from that of the d-band DOS (see e.g.,

Fig. 1 ^{of Ref.} [28]) of the fcc crystal.

Fig. ^2.

Local DOS calculated for the first (a), ^second (b) ^{and third} (c) nearest-neighbour atomic sites around a

vacancy in the Cu crystal. Also shown in figure ^{2d is the}

DOS for the perfect crystal.

The electronic structures around a single vacancy in the Cu crystal ^are presented ⁱⁿ figure 2. The local DOS (n = 15) ^{for the first, second and third} nearest-neighbour atomic sites around ^a vacancy ^are

presented ⁱⁿ figures 2a, ^{2b and} 2c, respectively:

figure 2d shows the DOS for the perfect ^Cu crystal.

One can see in figure 2 that the local DOS for the

first-nearest-neighbour ^{site is} strongly changed ^from

the bulk _one, while those for the second ’and third

nearest-neighbour ^{atomic sites are almost} unchang-

ed. This means that the twelve nearest-neighbour

atomic sites around a vacancy ^can primarily ^contri-

bute to the vacancy formation _energy EFv ^{in the fcc} crystal, ^and contributions coming from the second and third nearest-neighbour ^{sites are less} important

(2). This conclusion gives ^{a relevant} justification ^for

the earlier theoretical treatment based ^on the second moment approximation ^for vacancy-type ^lattice defects in d-band metals [7, 8]. This theoretical

finding may ^not be changed ^{even if we take into}

account the electronic states (charges) ^{within a}

vacancy site [16, 29] : ^{for this case, the matrix}

element of the defect (vacancy) potential ^{takes finite} positive ^value (not positive infinite) ^{but the} problem

is essentially ^{the same.}

We have performed similar electronic structure calculations for surface kink and flat and stepped surfaces, ^and presented the results in figure ^{3. The}

geometry for surface kinks of the fcc crystal ^{is shown}

in figure ^{4 : two} types of surface kinks both along ^the

dense (111) ^x (010) and non-dense (100) ^x (110) steps ^are considered in the present electronic struc- ture calculations. As shown in figure 3, ^{the electronic}

structures for these surfaces and surface kinks ^are

changed drastically from that for the perfect crystal.

Fig. ^3.

Comparison of local DOS for the flat (a) ^and stepped (b) surfaces with those for the surface kinks (c) ⁱⁿ

the Cu crystal. Solid and dashed curves are for the (100)

and (111) planes, respectively. For flat and stepped surfaces, the local DOS are calculated for the atomic sites

on the first atomic layer and first atomic row of the step, respectively. ^Atomic geometry for the surface steps ^and kinks are presented ⁱⁿ figure ^4.

(2 ) The calculated vacancy formation energy (electronic contribution) ^{for Cu} crystal ^{is about 2.5 eV. This value,} however, ^{can be reduced} significantly ^{when we take into}

account the short range repulsive energy and lattice relaxation around a vacancy [7, 8].

Fig. ^4.

(a) ^Schematic drawing of the surface kink and a

microscopic view for the vacancy formation process in the

crystal. (b) ^and (c) : Two different kink atomic structures

along ^a (100) ^x (110) step (b) ^{and a} (111) ^x (010) step (c)

in an fcc lattice.

The most pronounced ^{features are} the decrease of the effective bandwidth and appearance ^{of the}

prominent peak ^{near the bulk d-levels} Ed = - 3.1 eV (exactly speaking, slightly ^{above or}

below Ed depending ^on the atomic geometry).

The kink d-bandwidth becomes much smaller than the bulk one, ^as the second moment is reduced by ^a

factor of - 2. Here, ^{it is} important ^to realize that the kink states which appear ^near the d-atomic level ^are

resonant states, ^{and not} the usual surface states.

This is due to the fact that for the surface kink, ^the symmetry ^{of the} crystal ^{is lost} completely ^{and a}

Bloch like wave (wave function in a mixed Bloch- Wannier representation) ^{can not} be defined. In contrast, ^{for flat or} stepped surfaces, ^certain crystal symmetries ^{are still} preserved and the usual surface

states appear ^near the bulk d-levels [30, 31].

One can also notice in figure 3 that the local DOS

(solid curve) for the kink along ^{the dense} (111) ^x (010) step is almost identical to that (dashed curve)

for the kink along ^{the non-dense} (100) ^x (110) ^step.

The reason for this behaviour is physically apparent

and has been discussed by ^Allan [31] ^for non-hybri-

dized d-band scheme. In _essence, this arises from the fact that the kink site is the only ^{site in} complete thermodynamic equilibrium with the bulk atoms:

the energy required ^{to remove a kink atom from the} crystal ^is just equal ^to the bulk cohesive energy and this energy does neither depend ^on the surface ^nor

on the step ^{at the} position ^{of a kink} [31, 32].

Therefore, ^this peculiar property ^{for the surface}

kink can be proved ^from

2092

where Po (E) ^and Ap(E) ^are the total bulk DOS and the variation of the DOS due to the removal of kink site atom. From equation (11), ^{one can under-}

stand that Ap ( E ) ^{does not depend on the type of}

surface kinks.

In order to understand ^the peculiar ^electronic property ^of the surface kinks in more detail, ^we present ⁱⁿ figure 5 the local DOS on the kink site by separating ^the contributions of _s, p and d-subbands.

For p and d-subbands, ^{the DOS are} summed up for three and five orbitals, respectively. ^{One can see in} figure ^{5 that} within the s, p and d-subbands, ^the

DOS for the kink along ^{the dense} (111) ^x (010) step resemble quite well those for the kink along ^{the non-}

dense (100) ^x (110) step. This indicates that surface kink chemical reactivity ^{does not} depend ^{on the} type of kinks and is the same (when the reaction rate

depends ^on the atomic orbital DOS [31]) ^{even for}

metals without d-valence electrons, i.e., non-transi- tion metals with only sp-valence ^electrons.

Finally, ^we discuss the electronic structures for

sharp cleavage crack in the fcc crystals. Although ^a

ductile material like fcc metals usually ^{do not show}

the cleavage ^fracture (it ^fails by plastic instability

and necking [6, 33]), ^the cleavage- like fracture does

occur in the materials under the corrosive environ- ment. For instance, ductile materials such as a-brass und Cu fail by ^the propagation of cracks whose

microscopic ^{features are} consistent with the occur- rence of cleavage ^fracture [34]. Usually, cleavage-

Fig. ^5.

^{Local DOS for the surface kinks in the fcc} (Cu) crystal ; Figures (a), (b) ^and (c) ^{are for the s,} p ^and d- subbands, respectively. Solid and dashed curves are the local DOS of the surface kinks along the non-,dense

(100) ^x (110) ^{and dense} (111) ^x (010) steps, respectively.

like fracture ^occurs due to accumulation of corrosion induced vacancies or divacancies, generated by ^ano-

dic dissolution, ^on prismatic {100} , {210} ^or

{ 110} planes, ^as schematically ^{shown in} figure ^6.

The prismatic {100}, {210} ^{and {110}} planes ^have

the common characteristics of low indices and low atomic density and the fcc crystal will be weakest in

cleavage ^{across such} planes, because of the fewer bonds on the plane [35].

Therefore, it is of great significance ^to investigate

the electronic structure around the cleavage ^{crack in}

fcc metals and alloys. ^{As a} typical example, ^we

consider the sharp cleavage crack in the fcc crystal having (001) ^crack plane and crack front parallel ^to 110 ) ^{direction as shown in} figure 6a. In order to

introduce the cleavage crack, ^{the electron transfer} integrals ^{across the} cleavage plane ^{are set to be zero,}

in analogy ^to the surface problem by Kalkstein and Soven [36] ^and by ^Allan [37]. ^This type ^of cleavage

crack has been in fact observed in fcc metals, _{e.g., Ni} single crystals ^with a 100 > tensile axis in gaseous

hydrogen ^{or in} liquid mercury [38].

We have calculated the electronic structures for atoms A, B and C (Fig. 6a) around the crack tip ^and presented the results in figure 7. Also shown in

figure 7d is the DOS for the flat (001) surface. One

can see in figure 7 that the electronic structure around the crack tip depends strongly ^{on the atomic}

Fig. ^6.

(a) Schematic atomic configuration ^{near the}

crack tip ^{in the fcc} crystal ; (001) ^crack plane with ^crack front parallel to ( 110 > direction. (b) ^Schematic drawing

for the SCC mechanism of fcc metals and alloys. ^V

( V2 represents ^a vacancy (di-vacancy) ^induced by

anodic dissolution.

Fig. ^7.

Local DOS calculated for atomic sites A, ^{B and} C around a sharp cleavage crack in the Cu crystal. ^The

atomic configuration around the crack tip ((001) ^crack plane with crack front parallel to ( 110 > direction) ^is

shown in figure ^6.

position ^{relative to the crack} front. The local DOS

on the atomic site C is quite ^{similar to} that for the flat (001) ^surface (Fig. 7d), ^{while the local DOS on}

the atomic site A near the crack front is similar to the bulk DOS. It is rather surprising that the local DOS at the site C (not very far from the crack front) quite resembles that of the flat surface. These theoretical findings ^indicate that the chemical reac-

tions such as the anodic dissolution or chemisorption

would preferentially ^{occur at} the atomic sites near

the site C rather than at the sites A and B in the close

vicinity of the crack front.

Therefore, ^{one can} understand that the atomic

sites quite ^{close to} the crack front is rather chemically inactive, compared ^to the atomic sites slightly ^distant

from the crack front. This is schematically ^{shown in} figure 6b. From these electronic structure calcula-

tions, ^{we come to} the conclusion that the chemical

reactivity ^{near the crack} tip (sharp cleavage crack)

would not exceed that of the flat surface : much stronger ^chemical reactivity ^{can be} expected ^for

instance at the kinks [13] ^{or at} certain atomic

irregularities around the crack tips.

4. Conclusions.

We have used the TB recursion method of _{s, p, d-} basis orbitals and investigated the electronic structu- res of various lattice defects in metals. Example

numerical calculations ^are performed for the fcc

crystal Cu, ^{with the} nearest-neighbour hopping

interactions. The electronic DOS constructed from the continued fractions ^are converged very rapidly (after ^{11 or 13} levels) ^and the method can be used for wide variety ^of lattice defect problems (vacancies,

flat and stepped surfaces, surface kinks and cleavage cracks). ^In general, ^{we have found that the} shape ^of

d-band DOS is affected quite significantly by ^includ- ing ^the sp-d hybridization ^effects, and the proper treatment of _s, p and d-basis orbitals is very impor-

tant in determining the electronic structures of lattice defects in _s, p, d-valence materials.

On the other hand, ^we have also shown that in order to obtain quantitatively ^{accurate results on the} energetics ^{of lattice} defects, analytic integration

method should be used for the local DOS functions P Ai ( E ) , instead of the direct numerical integration

method (e.g., Simpson’s method) : Highly ^accurate

energy calculations ^are required ^for obtaining ^the

atomic relaxations around the lattice defects. The present ^{TB formalism can} be used for such purpose : The short-range repulsive energies [7-9] ^{are included} straightforwardly ^{in the} present ^s, p, d-basis TB

approach ^and this allows ^us to calculate the atomic relaxations around the lattice defects.

References

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