distribution of the atoms on grain boundaries and in the bravais cell of the coincidence site...

Naumm AN and R. HERRMANN: Distribution of the Atoms on Grain Boundaries 691

phys. stat. sol. (b) 118, 691 (1983)

Subject classification: 1.1; 1.4; 21

Xektion Physik der Humboldt- Universitat zu Berlin1)

Distribution of the Atoms on Grain Boundaries and in the Bravais Cell of the Coincidence Site Lattice BY NGUYEN AN and R. HERRMANN

The distribution of the atoms on symmetrical and asymmetrical coincidence grain boundaries in s.c., f.c.c., b.c.c., diamond, and zinc blende lattices is calculated. A general method is developed which allows to determine the number of atoms and the structure of the two-dimensional lattice of any grain boundary plane and of the planes next to grain boundary.

Es wird die Verteilung der Atome auf symmetrische und asymmetrische Koinzidenz-Korngrenzen im p.k.-, k.f.z.-, k.r.z.-, Diamant- und Zinkblende-Gitter berechnet. Eine allgemeine Methode wird entwickelt, die es erlaubt, die Zahl der Atome und die Struktur des zweidimensionalen Gitters in einer beliebigen Korngrenzenebene zu bestimmen sowie in den der Korngrenze benach- barten Ebenen.

1. Introduction

I n order to describe the geometrical distribution of atoms in grain boundaries on the basis of the coincidence site lattice (CSL) concept only the knowledge of the distribution of the atoms in the Bravais cell of the CSL is necessary.

Usually the Bravais cell of the CSL contains a large number of atoms. Therefore, it is useful to consider the distribution of the atoms on layers parallel to the planes of the Bravais cell, i.e. the distribution of the atoms on the planes of the so-called displace- ment shift complete lattice (DSCL). There are many papers considering the distribution of atoms on grain boundaries in metals [l to 41 and in semiconductors [5 , 61. But these works deal only with low-indices rotation axis (i.e. [loo] or [llo]). I n these cases the distribution of the atoms in the grain boundaries is relatively simple. I n more complicated cases, if the indices of the rotation axis become high, an analytical method for the determination of the atoms a t the grain boundaries must be used.

It is the purpose of this paper to present such an analytical method to explain the distribution of atoms in grain boundaries and to conipute the number and structure of the planes in the Bravais cell of the CSL for s.c., f.c.c., b.c.c., and diamond lattices. The single groups of atoms, i.e. the atoms a t the corners of the cube, a t the centres of the planes, in the centre of the body, and a t the body diagonals have been distinguished and separately treated. Therefore, the zincblende lattice is included in these considera- tions.

As an example the distribution of the atoms in the Bravais cell of the CSL in a bicrystal with the rotation axis [110] and the rotation angle 6 = 26.53' is discussed.

2. Characterization of the Atoms and Their Lattice Points

The conventional cubic cell contains one lattice point for s.c., two lattice points for b.c.c., four lattice points for f.c.c., and eight lattice points for the diamond lattices (Fig. 1).

__ 1) Hessische Str. 2, DDR-1040 Berlin, GDR.

692 NGUYEN AN and R. HERRMANN

Big. 1. Distribution of the different groups of atoms in a crystal Bravais cell. 0 atoms of the type P, x F, A D, I

I n order to describe the distribution of the atoms a t the grain boundaries it is useful to characterize each single group of atoms of the different bases in the conventional cubic cell by a vector. We choose as the basis of the conventional cubic cell %, a2, a3 and

xl, x2, x3 are the coordinates of the atoms. xl = x l q , x2 = x2a2 , x3 = x,a3; (1)

2.1 The single cubic lattice

The atoms attached to every corner of the cubic cell are denoted by P. The position of the atoms is written as

3

i = l P = mla,+ m2a, + m3a3 = C m i a s . (2)

On the figures these atoms are indicated by dark points (0 ) if their position is on the plane under consideration and by open points (0) if their position is above the plane.

2.2 The face-centred cubic lattice

Beside the atoms P a t the corner of the cubic cell the f.c.c. lattice containsatoms a t the face centres denoted by F. The position of the atoms is written as

3

2=1 Fl = C [mi + +(l - ai . (3)

On the figures these atoms are indicated by crosses ( x ) if their position is on the plane under consideration and by edged crosses (@) if their position is above the plane.

2.3 The body-centred cubic lattice

Beside the atoms P a t the corner of the cubic cell the b.c.c. lattice contains atoms in the centre of the body denoted by I.

The position of these atoms is written as 3

2 = 1 I = ,x (mi + +) ai . (4)

The atoms are indicated by dark squares (m) if their position is on the plane under consideration and by open squares (0) if their position is above the plane.

Distribution of the Atoms on Grain Boundaries and in the Bravais Cell 693

2.4 The diamond lattice

Beside the atoms P a t the corners of the cubic cell and the atomsF a t the face centres the diamond lattice contains the atomsD. The position of these atoms is displacedfrom the atoms P by one quarter of a body diagonal,

and from the atoms F1 respectively displaced: 3

i=l D1 = C [mi + f(3 - SSli)] ai (5b)

These atoms are indicated by dark triangles (A) if their position is on the plane under consideration and by open triangles ( A ) if their position is above the plane (Fig. 1). The diamond lattice may be viewed as two f.c.c. structures, one with the atoms P and F and the other with the atoms D. The zincblende lattice results when one kind of atoms (i.e. In) is placed on one f.c.c. lattice (i.e. P and F) and another kind of atoms (i.e. Sh) is placed on the other f.c.c. lattice D.

3. Transition from the Crystal Lattice to the CSL

In order to estimate the number and the kind of atoms in the Bravais cell of theCSL and on the planes of the Bravais cell it is necessary to consider the atoms in the CSL.

Before the transformation from the crystal lattice to the CSL will be done some preli- minary remarks are necessary. I n [7] we have used the following characteristics of the CSL: c as rotation axis with the crystallographic direction [uvw], perpendicular to the plane (uvw); a as the normal of the grain brounday plane (hukaZa), and b, which deter- mines the direction of the grain boundary in the (uvw) plane, followed from a = b x C.

a, b. and c are the basis vectors of the Bravais cell of the CSL. All three planes of the Bravais cell (hukaZo), ( h b k b l b ) , and (uvw) are possible grain boundary planes. Therefore, the further consideration will be done in a general form.

We choose

x; = x;a , X; = XHb , X; = X ~ C . (6) xi, 24, xi are the coordinates of the atoms in the CSL. In the crystal coordinate system a, b, and c are written as

a == [ h a k ~ l u l = biiui2u131 , c = [hckc lc ] = [u31u32fi33]

b = [hbkblbl = [ u 2 1 ~ 2 2 ~ 2 3 1 > (7 b)

as crystallographic directions.

the crystal lattice follows as

PJ - 3

The directional cosinus aij between the vectors a, 6 , c of the CSL and q, a2, u3 of

(8) uij a.. -

( c U t ) l / Z j=1

uil are (u l j , u 2 j , u3j correspond to a, b, c with j = 1, 2, 3) the components of the basis vectors of the CSL Bravais lattice. The transformation of the coordinates of the lattice 45 pliysica (h) l l S p

694 N G U Y E N AN and R. HERRnIANlV

points is given by 2

x: - a,.%. 8 - v 7

j = 1

4. The Number of Atoms in the Bravais Cell and on the Planes of this Cell

I n [7] we have chosen

z - vcsL Vcryst *

ITcsL is the volume of the Bravais cell of the CSL and Vcryst the volume of the conventional cubic cell. For the s . ~ . lattice 2 is the number of atoms in theBravais cell of the CSL. For the other lattices this number is

Q = 2'E. (10) (Here s = 0 for s.c., s = 1 for b.c.c., s = 2 for f.c.c., and s = 3 for the diamond or the zincblende lattice, respectively.)

Our further consideration is directed towards the distribution of the atoms on the planes of the Bravais cell and on the layers parallel to these planes.

4.1 The atoms P of the S.C. lattice

If the origins of the crystal system and the CSL Bravais system coincide, then the position of atoms P is always the corner of the CSL cell. Therefore, the atonis P of the S . C . lattice are attached to the corners of the cubic cell. Their distance to a lattice plane ( U ; ~ U ? L U ~ ~ ) (i = 1 , 2, 3) is given by

3

j = 1 C mjuij

(11) A P = ( c u p 2 j = l

The distance between the planes (uiluizui3) is

Between two lattice points of the CSL N atom layers are distributed on the distance of 3

xi = ( c *L;) l /z , j = 1

The number of atoms a t one layer, Q*, follows from the number of atoms in the Bra- vais cell,

(14) z N

Q* =-. With (29) from [7] it follows

3 3 E = C uyj = R2 C u & ,

j=l j=1


where 3

R2 = C u'$. j = 1

The atoms a t the planes perpendicular to a, 6 , and c are

Qfa = 1 for (ullu12u13) 3

Qp"e = ~2 for ( ~ 3 1 ~ 3 2 ~ 3 3 ) .

Q& = R2 for ('21u22u23) 7

z

4.2 The atoms F in the plane centres

The f.c.c. lattice contains the atoms P and F. The distance of an atom F from a lattice plane (uiluizui~) is given by

3

C Imj + f(1 - &)I uij

( c Ut. )1 /2

AF - j = 1 1 - 3

j = l

If one lattice index uij = uiL is an even number and the other two are odd (i.e. a (211) plane, L = l ) , or conversely if one coefficient uiL is odd and the other two are even (i.e. a (122) plane, L = l), corresponding to No. l a , , l b , 2a, and 2 b in Table 1, then it follows that APl=L = AF, is a multiple of Aui,

AF, = n Au, , and AF1,L is a multiple of Au,/2,

The atonis FL are attached to the planes n and the atoms F1+L to the planes (n + $). If all three coefficients are odd, corresponding to No. 3a and 3 b in Table 1, then

AF, (I = 1 ,2 ,3 ) are multiples of Aui, i.e. all atonis F, are attached to the planes n.

AFl+& = (n + $) Aui (n = 0, 1,2, ... , N ) .

4.3 The atoms D in the diamond lattice

The distance of an atom D from a lattice plane (uilui2ui3) is for I = 1, 2 , 3 3 c [mg + + (3 - %j)] uij

AD - j = 1 2 - 3

( c U t ) l / Z j=1

j = 1 4 j=1 uiz 2 ) 3 3 = ( miuij + - uij - - hu i

and 3 3

AD4 = ( 2 mjuij + + C u i j ) Aui , j=1 j=1

respectively. 45

696 KGUYEN AN and R. HERRMANN

Table 1 Distribution of the different types of atoms F,, P, D, in various CSL Bravais planes.

uzl , u22, u23 are the indices of the i-thCSLBravais plane. 2 M = C u2, or 2M -+ 1 = C uU.

2' is measured in units of Au,. o means odd, e means even

3 3

3 =1 J--1

No. ui l uj2 ui3 M d F1 F, F3 P D, D, D, D,

l a 0 o e e O x x x x

x x

l b 0 o e o o x x x x 1 4 -.

~~ ; x x 3 4 -

2tL e e o e O x x 1 4 x x ~~

-~ ; x x 3 4 -

2b e e o o O x x 1 4 -

-; x x d 4 x x -~

x x

x x

x x

3a 0 o o e o x x x x 1

1 2 3 4

4 x x x x _.

..

-

3b 0 o o o o x x x x 1 4 1 2 3

_.

_ _

4 x x x x -~

If one coefficient ui j = uiL is even ( u ~ L = 2 N L ) and the other two are odd (uil = = 2Nl+l)) corresponding to No. 1 a and 1 b in Table 1 (i.e. for a (211) plane, L = l ) , then it follows

3 2 ujj = 2M

j - 1 and


If M (No. 1 a, Table 1) is even, then from (19) it follows

ADz+L = (n + +) Aui , AD, = AD, = n Aui

(n = 0 , 1 , 2 , ... , N ) , i.e. the atoms Dz+L are attached to the planes (n. + f ) and the atonis DL, 0, to the planes n.

If M is odd (No. 1 b, Table l), then D ~ + L are attached to the planes n and DL, D4 to the planes ( n + a).

If, however, one coefficient uil is odd (uiL = 2NL + 1) and the other two coefficients are even, (uiz = 2N2+1) , i.e. (No. 2a, 2b, Table I ) ,

2 uij = 2M + 1 , j=1

then it follows 3

AD, = [x mpij + $(M - 1 ) - N L + 41 Aui , j=1

3

j = 1 AD,+, = ( C mluij + $M - N z + 41 Aui

even for 1 = L ,

j = 1

If M (No. 2a, Table 1) is even, AD,+, = ( n + $) 4 u i and the Dl+L are attached to the planes (n + f ) , ADi=L = AD, = (n + +) Aui and DL, D, are attached to the planes (n + $).

If M is odd (No. 2b, Table l), ADibL = (n + $) Aui and AD,,, = AD, = = ( 7 ~ + $) Aui, i.e. the atonis D1, are attached to the planes (n + $) and DL, 0, to ( n t f).

If all three coefficients are odd, again it follows (20) and

3 AD, = ( 2 mluij +

j=1

If M (No. 3a, 3b, Table 1) is even, then AD, = AD, = ( n + f ) Au,, i.e. the atoms D are attached to the planes (n + f ) . If M is odd, AD, = AD, = (n + $) Au, and the atoms D are attached to the planes (n + +). Table 1 contains the possible distributions of the atoms P, Fl and Dz, D,, belonging to different layers.

If a plane ( ~ , 1 u i 2 ~ , 3 ) contains the atoms P and F,, then these atoms are equivalent and their number is equal. The atoms P and D,, Fl and D,, F, and D,, F, and D,, respectively, in the same sense are equivalent. If both atoms of such a pair are attached to a plane, then their number is equal. From Table 1 it follows that for No. 5 the atoms F,, F,, F,, P in the diamond lattice are attached to the layer z’ = 0, but the atoms D,, D,, D,, D, to the layer z’ = +AuL. The layers x’ = +Aui and z’ = $Au, do not contain atoms. For NO. 6 atoms exist a t z’ = 0 and x’ = +Au, only. As a result it follows that the layers perpendicular to [uvw] with threeoddindices,(e.g. ( l l l ) , (311)) are double layers. I n the other two directions the sequence of layers is equal.

698 NQUYEN AN and R,. HERRMANK

5. Calculation of the Distribution of the Atoms at the Planes of the Bravais Cell of the CSL and in Parallel Layers

The coordinates of the atoms belonging to aBravais cell of the CSL are limited by the conditions

O s x ; s l u l , O s x i s l b l , O ~ x ~ s l c l . (23) The coordinates x i are given by the transformation (9).

layer is under consideration, the conditions (23) are If the distribution of the atoms in a plane of the CSL, i.e. for xi t= 0 or in a parallel

X ; = n Au, , 0 s X ; 5 161 , 0 5 X ; 5 IcI . (24) The coordinates of the atoms in the Bravais cell can be computed from xl, x2, x3, which are estimated for the different lattice types by (1) to (5) using (9) and (23). There is the possibility that the CSL vectors U , b, c and their combinations A, B, C, D [8] have the double length of P or F1. I n these cases corresponding atoms P or Fl are attached to the middle of these vectors.

The inequalities (23) deliver not only one resolution, but a resolution region. There- fore, different combinations of the rn3 as resolutions are possible. These combinations include all coordinates of the atoms belonging to the Bravais cell.

The computation of the distribution of the atoms a t the planes of the Bravais cell will be shown for a concrete example. A bicrystal with the rotation axis c = [IlO] and the rotation angle 0 = 26.53" is estimated by the Bravais cell vectors

u = [ ~ I B I , b = [ M I , c = [ i i o ] .

Table 2 Coordinates of the atoms in the plane (116). a) xi = 0; b) xi =,$. xt is measured in units of Au; (Au, = m, Au, = YE, Au, = YT). The notations Fi, DE are explained in Fig. 2

a) x; = O

0 0 19 19 0 19 - 19 4

19 4 19 4

- -

b) X; =


The distribution of the atoms on the symmetrical grain boundary plane (b, c ) , i.e. (116) plane, is given if xi = 0. This plane is estimated by No. l a in Table 1. The atoms P, F,, D,, D, are attached a t this plane xi = 0. Parallel layers are xi = n Au, and xi = (n + +) Au,. On the latter planes the atoms F,, F,, D,, and D, are attached: Au, = 38-lI2, N = 38. From (15) it follows that only one atom P exists in the plane (116) and therefore only one atom F,. The rotation axis [110] = c has the double length of a vector Pl (3). Hence it follows that one atom F, is attached to the middle of the edge [110] of the Bravais cell. The position of the atoms D,, D, follows from conditions (24). Here xi, x.;, xi are the coordinates of all atoms a t the grain boundary plane (116). The coordinates of all these atoms are shown in Table 2a. Their distribution a t the (116) plane a t xi = 0 is shown in Fig. 2a.

I n the layers xi = (n + $) Au, and with n = 0 for the xi = f Au, on the layer next to the grain boundary plane the atoms F,, F,, D,, D, are attached (cf. Table 1).

Due to the periodicity of the layer structures the relative positions of the atoms in all planes with different n are the same. The coordinates of these atoms and their distributions in the plane next to the grain boundary are shown in Tahle 2 b and Fig. 2b. The distribution of the atoms on the other two planes of the Bravais cell and on the parallel layers, respectively, follows from Table 1, too. The plane (a, c ) , i.e. the (331) plane (No. 3 b, Table 1) and the parallel layers are given by

xi = n Au, , xi = ( n + $) Au, . The planes estimated by x; = n Au, contain two atoms of P, F,, F,, and F,. The vectors [110] and [116] are double vectors of F3. There are other atoms F, in the middle of these edges. [110! + [1I6] is a double vector of P and respectively there is an atom P in the centre of the plane. + ([110] + [116]) is a double vector of F, and + ([110] - [116]) is a double vector of I?,. Respectively atoms F, and F, are attached to the diagonals of the plane. The distribution of the atoms in the (a, C) plane x' = 0 and x' = + Au, is shown in Table 3 and in Fig. 3 a and 3 b.

b

Fig. 3. Distribution of the atoms on the CSL Bravais plane (331). a) On the atom layer zi = 0; b) on the atom layer xk = -:- Au,

7 00 NCUYEN AN and R . HERRMANN

Table 3 Coordinates of the atoms in the plane (331). a)

a) xi = 0

= 0; b) x,; = +

~ ______ ~~~

m1 m2 m 3 X1 x2 x;

0 1 0 i 0 0 i 0 0 0 0 i 0

0 0 6 6 3 2 5 2 5 0 3 6 3

0 0 0 0

38 0 38 0

2 0 4 0 4 0

4 0 4 0 0 0

La. 0 38 0

2 0

3 8

- 38

38.3

38

38.3

38 -

0 2 2 0 1 3 2 1 2 1 2 3 2

-

..

~

-

1 2 1 0

m1 m2 ~~

i

i

i i

0

0

0 0 0 0

3 3 6 i 4 Fi i 6 i 4

0 2 1 3 2 1 2

_.

-.

0 1 2 1 2 .i 2

_ _ ~

There are alternating sequences of P-, F-atom layers with D-atom layers. For a zincblende lattice, i.e. InSb, these’layers contain I n atoms or Sb atoms only. The distribution of the atoms a t the (a, b) plane, i.e. the (110) plane, is given by

x$ = n Au, and xi = (n + $) Au, . c = [ l l O ] is the rotation axis and therefore it is possible to construct the “selected” lattice (4a) and (5a) of [7] where p = [OOf] and q = [ I l O ] . The coordinates of the points of this lattice and their distribution in the planes xh = 0 is shown in Table 4 and in Fig. 4. Fig. 5 shows the general distribution of the atoms.


Table 4

Coordinates of the atoms in the plane perpendicular to the rotation axis c = [110]

0 1 1 0 0 1 3 4 3 4

-

-

0 0 0 0 2 0 2 0 1 0 1 0 1

5 2 0 4 0

-

-

I n conclusion is it necessary to emphasize that for symmetrical grain boundaries the CSL and the distribution of the atoms on this grain boundaries are for both crystals always the same. But for asymmetrical and pseudosymmetrical grain boundaries the distribution of atoms on the grain boundary differs in both crystals. I n an asymmetrical or pseudosymmetrical grain boundary the boundary is constituted by two different crystal planes in which the distribution of the atoms is different. Beside the CSL sites additional atoms exist from the plane with lower indices (see, e.g., Fig. 4a in [7]).

P’

Fig. 4 Fig. 5

Fig. 4. Distribution of the atoms on the CSL Bravais plane (110) a t

Fig. 5. Distribution of the atoms on the CSL Bravais planes (liB), (33I) perpendicular to the plane ( 1 10)

== 0

7 02 NGUYEN AN and R. HERRMANN: Distribution of the Atoms on Grain Boundaries

References 11 B. BAROUX, M. BISCONDI, and C. Goux, phys. stat. sol. 38, 415 (1970). 21 V. VITEX, D. A. SMITH, and R. C. POND, Phil. Nag. 41, 649 (1980). 31 D. A. SMITH, V. VITEX, and R. C. POND, Acta metall. 25, 475 (1977). 41 G. A. BRUGGEMAN and G. H. BISCHOP, J. appl. Phys. 44,4468 (1973). 51 H. J. MOLLER, Phil. Mag. A43, 1045 (1981). 161 H. J. MOLLER, J. Physique C1, 33 (1982). 71 NGUYEN AN, R. HERRMANN, and G. WORM, phys. stat. sol. (b) 116, 501 (1983). [8] NGUYEN AN, G. WORM, and R. HERRMANN, phys. stat. sol. (b) 114,349 (1982).

(Received N a y 12, 1983)

distribution of the atoms on grain boundaries and in the bravais cell of the coincidence site...

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