symmetrical grain boundaries with special angles and their coincidence site lattices
TRANSCRIPT
NGIJYEN AN et al.: Symmetrical Grain Boundaries with Special Angles 349
phys. stat. sol. (b) 114, 349 (1982)
Subject classification: 1.1; 1.4
Sektion Physik der Humholdt- Universitcit zu Berlin')
Symmetrical Grain Boundaries with Special Angles and Their Coincidence Site Lattices
BY NGUYEN AN^), G. WORM, and R. HERRMA"
A simple method for an explicite calculat,ion of the coincidence site lattices is presented. This method based on clear geometrical conditions is convenient for the interpretation of experimental results. The indices of the grain boundary planes for symmetrical rotation axes are estimated. The type and the orientation of the Bravais lattice of the coincidence site lattices in cubic crystals are obtained.
Es wird eine einfache Methode fur die explizite Berechnung der Koinzidenzplatzgitter vorgestellt. Diese Methode, die auf rein geometrischen Bedingungen begrundet ist, ist fur die Interpretation der experimentellen Ergebnisse geeignet. Die Indizes der Korngrenzenebenen fur symmetrische Drehachsen werden berechnet. Der Typ und die Orientierung des Bravaisgitters der Koinzidenz- platzgitter in kubischen Kristallen werden bestimmt.
1. Introduction
For the determination of the coincidence site lattices (CSL) and their special angles, Grimmer et al. [l] used a theory-of-number method giving the possibility to determine the coincidence site lattices explicitly for cubic lattices. Andreeva and Fionova [2, 31 chose a matrix method for the determination of the coincidence site lattices and their Bravais lattices for special grain boundaries in cubic lattices. Fortes [4] and Bonnet and Durand [5] determined the special angles for grain boundaries of hexagonal lattices and other geometries, respectively. Summarized remarks can be found in the work of Orlov et al. [6].
In the present paper a convenient method for an explicit calculation of the co- incidence site lattices, their special angles, and their Bravais lattices, with the basic vectors belonging to them, for cubic lattices as s.c., f.c.c., and b.c.c. and the diamond structures are presented, a simple method based on clear geometrical conditions.
2. Determination of the Grain Boundary Planes and the Special Angles
Two space lattices of identical structure and with a common lattice point 0 oriented relative to each other - the [100]1 axis of the first points in the 03 direction and the [lo0311 of the other in the 0% direction - can be brought to full coincidence of all lattices sites by
a) using a rotation which translates the axis OB iuto the axis OA and then b) using a rotation around this common axis translating the second lattice to full
coincidence (Fig. 1) with the first.
l) Hessische Str. 2, DDR-1040 Berlin, GDR. ?) Permanent address: University of Hanoi, Vietnam.
350 NGUYEN AN, G. WoRnr, and R. HERRMANN
Fig. 1. Arrangement of two crystals (1) and (2) arbitrarily oriented along the directions 0: and 0%. S represents the rotation axis around which the transformation RAB is carried out
Both rotations can be expressed by apure transformation CP-1 ([uvw] means thedirec- tion of the rotation axis and 8 the rotation angle). In the further consideration we special ize to symmetrical tilt grain boundaries.
Two crystallographic planes with the same Miller indices, each of them belonging to one of the two crystals, are symmetrical with respect to a plane which is deter- mined by the rotation axis [uvw] and the half rotation angle 812. This gives the possibility for the construction of a bicrystal with a symmetrical tilt boundary in the following way: A single crystal will be cut parallel to the axis [uvw] in the (uvw) plane along the planes Pl(hkZ)I and P2(hkZ)II penetrating each other at the angle 8, symmetrical with respect to a third plane P,, the mirror plane. After that the part between Pl and Pz will be removed from the crystal. By a rotation around the cutting line [uvw] of both planes the planes PI and P2 may be translated into each other [7] (Fig. 2).
This leads to a bicrystal with a tilt angle 8 between the planes PI and Pz which contains the tilt axis as i t is given for symmetrical tilt grain boundaries.
If a crystal contains a grain boundary the crystal energy is increasing. This energy increase is localized a t the grain boundary. It will decrease partially if for special grain boundaries a part of the lattice sites of both crystals in the grain boundary coincides. I f we consider two interpenetrating translation 'lattices and rotate one lattice around a common axis of both lattices, the pattern of coincidence sites of both lattices a t special rotation angles represents the coincidence site lattice. The proportion of the ordinary lattice points to those of the coincidence site lattice is given
Fig. 2. Schematic construction of a bicrystal from crystal I and 11. ( h k l ) ~ and ( h k l ) ~ ~ are the indices of the crystal planes penetrating at the grain boundary. P,(WLE)I and P,(hkZ)11 are the cutting lines, along them the crystal will be separated. 8 is the angle, around which the cutting lines PI and P,, will be transformed into each other
Symmetrical Grain Boundaries with Special Angles 35 1
Fig. 3. Relative position of the symmetrical planes (OlO), (110) of the symmetrical axes [OOI], [IlO], [111] and of the axes [uOl] and [l Iw], respectively, lying in the symmetrical planes
by 2. It is a measpre of the density of the coincidence site lattices. For symmetrical grain boundaries 2 = h2 + k2 + 12. Constructing the 0-lattice Bollmann [8] has derived an equation for the 0-points based on a transformation of single lattice points and hence single vectors into one another. The problem would be essentially simpler if we transform whole lattice planes into one another.
This is possible if there is a symmetrical mirror plane over which we can transform each lattice plane of the first lattice into the corresponding plane of the other lattice.
For cubic lattices these are the (100) and the (110) planes. For all rotation axes in these symmetric planes we can determine all crystallographic planes, which will be translated one into another. The indices and the special tilt angles 8, of these planes follow from two geometrical conditions :
1. The indices of the planes are symmetric relative to the half angle plane and there is the transforniation
i k = k , OI1 A is the transition matrix of the symmetric planes and (hkl)I are the indices of the grain boundary plane with respect t o the first, (hkl)II with reepect to the second plane.
2. All planes (hkl) contain the rotation axis [uuw] and therefore the rotation axis and the plane normal are perpendicular to each other,
[hkl]. [UVW] = 0 . (2) Fig. 3 shows the symmetric planes { 100) and { 110) which contain the rotation axes (loo), (110), ( l l l ) , and (112). The latter is not a symmetric axis [9] but it will be included in the consideration. From conditions (1) and (2) we can calculate the analytical expressions for the special angles and the indices { hkl} for any symmetrical tilt grain boundary.
Taking some examples of three-dimensional cubic lattices we examine the orienta- tions for which coincidence site lattices exist and we investigate the structure of the grain boundary planes.
a) The tilt grain boundary [ O l l ] (100) (tilt axis [ O l l ] , symmetry plane (100)): The transit,ion matrix for the mirror transition over the (100) plane is
352 NGUYEN AN, G . WORM, and R. HERRMANN
Fig. 4. Relation between the tilt angles 0, end n - 0, corresponding to the symmetrical grain boundaries [011] (011) and [ O l l ] (100)
From (1) and (2) the indices of the crystal planes of both crystals a t the grain boundary for all coincidence site lattices in this orientation follow:
_ _ Ch@I ; (hkk)II ( 4 4
for the first grain and the second grain, respectively. The special angle3 of the co- incidence site lattice are determined by
--h2 + 2k2 h2 + 2k2 . cos e O -
b) The tilt grain boundary [ O l l ] (011): The transition matrix for the mirror transi- tion over the (011) plane has the form
A011,= (A;;) (5)
and for the indices of the crystallographic plane in the grain boundary, follow respec- tively
The special angles of the coincidence site lattices are determined by
(hkx)I and (hkk)II. ( 6 4
A simple comparison of (4b) and (6b) shows that the following condition is valid: ~O(100) = 7c - eo(o11) ( 7)
c) The coincidence site lattices of the tilt grain boundary [Ool] (100) are represented as shown in Fig. 4.
by (hkO)I and (hk0)II ( 8 4
for the indices of the crystallographic planes of the grains in the boundary and by
for the special angles. d) The coincidence site lattices of the t,ilt grain boundary [Ool] (1 10) are represented
by (hkO), and (khO)II
Symmetrical Grain Boundaries with Special Angles 3 53
for the indices of the crystallographic planes of the grains in the boundary and by
2hk cos e - ~
- h2 + k2
for the special angles. Froin this follows 80(llo) = n/2 - 80(100).
e) The coincidence site lattices of the tilt grain boundary [ l l l ] (110) are represented by
(hl& +%)I and (khh + ~ ) I I ( 1 0 4
for the indices of the crystallographic planes of the grains in the boundary and by
2hk + ( h + k)2 C 0 8 8 -
- h 2 + k2 + (hm
for the special angles.
f ) The coincidence site lattices of the tilt grain boundary [112] (110) are represented by
( h k - T ) I h + k and ( L h T ) h + k I1
for the indices of t,he planes of the grains in the boundary and by
cos =
for the special angles. The special angles for some (hkl) are represented in Table 1. These results correspond to the results of Bollmann [9].
Tab le 1 Planes and tilt angles of some symmetrical grain boundaries
[uvw] plane of 2 (hkl) 8 [uvw] plane of 2 (hkE) 8 symmetry symmetry
[ O l l l (100) 3 (111) 9 (122)
11 (311) 19 (133) 27 (511) 33a (144) 33b (455)
[Ool] (100) 1 (110) 5 (210)
70.63" [lll] (011) 38.94" 50.48' 26.53' 31.69" 20.05" 59.00"
63.10" 90.00" [112] (011)
3 (211) 60.00' 7 (123) 21.79"
13 (134) 32.20" 19 (235) 13.17' 21 (146) 38.31" 31 (156) 42.10'
7 (132) 44.42' 35 (153) 57.12"
37 (347) 9.43"
This table contains a selection of symmetrical tilt boundaries. I n columns 1 and 2 we find the rotation axis and the symmetrical plane of the boundary, respectively, the grain boundary plane is represented by Z = h2 + k2 + Z2 (columns 3 and 4), 8 is the tilt angle
354 NQUYEN AN, G . WORM, and R. HERRMANN
3. Estimation of the Bravais Lattices of the Coincidence Site Lattices
For the interpretation of the physical properties of the grain boundaries it is necessary to estimate the Bravais lattices of the coincidence site lattices, because the symmetry of the coincidence site lattices is determined by this Bravais lattice.
Andreeva has estimated the Wigner-Seitz cell, and from this the Bravais Iattice was calculated [3].
Now we use a simple geometrical method based on the symmetry properties of the crystal lattice only. If, for two adjacent crystals with a common tilt axis [uvw], the plane perpendicular to the tilt axis (uvw) is a coincidence site lattice plane for special angles O,, always taking place for cubic lattices, the basic vectors of the Bravais lattice of the coincidence site lattices can be chosen as follows: c parallel to the tilt axis [uvw] and the other two vectors a and b in the plane (uvw) perpendicular to the tilt axis (see also Fig. 5).
Connecting the corresponding lattice sites on both sides of a symmetrical boundary we obtain the direction parallel to the surface normal of the grain boundary. This surface normal is the basic vector a. The intersection line of the grain boundary and the coincidence site lattice plane is the third basic vector of the Bravais lattice. This vector results from the vect,or product of the other two
b = c x a . (12) Summarizing we obtain for the basic vectors of the Bravais cells of the coincidence site lattice an orthogonal system if the initial lattice is a cubic lattice under con- sideration. Generally la1 + I bl + IcI and the Bravais lattices are orthogona.1 systems represented by
a = [hkl] , b = [vl - wk wh - ul uk - vh] , c = [uvw] . (13) The form of these vectors for different lattice types is as follows:
1. For the S.C. lattice the vectors a, b, c have the form
[mJI (14) in the lattice of one of the two crystals and they are the shortest lattice vectors in their direction (m, n, b are integers).
2. If we can describe the vectors a, b, c of the f.c.c. lattices in the form
i ( m + $ n + f I > (15)
( m + + n + f 2 ) . (16)
(i is an integer too) the shortest lattice vectors in the corresponding directions are
I Fig. 5. Construction of the grain boundary Bravais lattice containing tilt axis c = [rvzu], vector b directed along the cutting line between boundary plane and CSL-plane, and normal vector of the grain boundary a
Symmetrical Grain Boundaries with Special Angles 355
3. For the b.c.c. lattice the shortest lattice vectors are
respectively. Therefore, i t is possible to determine the Bravais lattice of the coincidence site
lattices of any grain boundary plane if the tilt axis [uvw] is known. This is valid for any tilt axis, lying in the symmetry planes, e.g., (loo), (llo), (lll), ( l l w ) , and (low).
For the fourfold axis (100) follows I a1 = I b[ =j= I C I and the corresponding coincidence site lattices belong to the tetragonal system.
For the threefold axis (111) one can choose the three basic vectors perpendicular to the rotation axis having the same length 141 = laal = [%I =+ IcI and their point symmetry operation is a rotation by 120" around c. The coincidence site lattices belong to the hexagonal (or to the orthohexagonal with the basis vectors U, = a, b = c x a, c) system.
For the twofold axis (110) follows la1 + I bI + IcI and the corresponding coincidence site lattices belong to the rhombic system. The same result follows if the rotation axis is not a symmetry axis. This is given for all ( l l w ) axes in the (110) planes and all ( low) axes in the (100) planes. I n addition to the considered lattice types the ele- mentary cell of the coincidence site lattices can be face-centered, base-centered, or body-centered.
To describe this situation we define some additional vectors
I n dependence on whether the elementary cell of the coincidence site lattices is face- centered, base-centered, or body-centered the vectors A , B, C are double translations vectors, one of them is a double translations vector or D is a double translation vector and there are special conditions for these vectors.
For a S.C. lattice the elementary cell of the coincidence site lattices is centered if one of the vectors A, B, C, or D is a double translation vector or if all these vectors are translation vectors and can be represented in the form [mnl].
For the f.c.c. crystal lattice the elementary cell of the coincidence site lattices is centered if the vectors A, B, C, and D are double translation vectors with the form [mnl] and/or [m + f n + f Z].
For the b.c.c. crystal lattice the elementary cell of the coincidence site lattices is centered if the vectors A , B, C , and D or one of them are a double translation vectors with the form [mnl] and/or [m + f n + f I + f].
For a diamond lattice the conditions for a centered coincidence site lattice are the same as for the f.c.c. lattice. Additionally the vectors A, B, C, and D can be double translation vectors given by
The type of the Bravais lattice depends on the indices u, v, w of the tilt axis only. 2 determines the parameters I Q J , J b J , (cI. In Table 2 the coincidence site lattice types, their Brevais lattices, and the lattice
parameters are represented for different rotation axes. Additionally results are de-
Tab
le 2
CS
L B
rava
is l
attic
es o
f so
me
sym
met
rical
gra
in b
ound
arie
s
[uvw
] B
rava
is
Z
a b
C
b+
c
a+
c
a+b
a +
b +
c B
rava
is la
ttice
CSL
la
ttice
[Oll]
P F
I D
P
F
I D
2[12
0]
[ 120
1 2[
120]
[1
201
[345
] [3
51]
a [34
5]
~3
21
orth
ogon
al
orth
ogon
al
orth
ogon
al
orth
ogon
al
orth
ogon
al
ort,h
ogon
al
orth
ogon
al
orth
ogon
al
orth
ogon
al
orth
ogon
al
orth
ogon
al
orth
ogon
al
orth
ogon
al
orth
ogon
al
orth
ogon
al
orth
ogon
al
B
A
B
A’
B
I F
I A
P
F
P
C
C
C
C’
5 3
Tabl
e 2
give
s so
me
info
rmat
ion
on th
e B
rava
is la
ttice
s of
sym
met
rical
gra
in b
ound
arie
s. I
n co
lum
n 2
are
liste
d th
e ty
pes
of t
he B
rava
is la
ttice
s 3
belo
ngin
g to
the
corr
espo
ndin
g cr
ysta
l lat
tice,
in c
olum
n 11
thos
e of t
he g
rain
bou
ndar
y C
SL. S
ymbo
ls P
, F, I
, D re
pres
ent t
he 8.
c.-, f
.c.c
.-, b.
c.c.
- and
di
amon
d la
ttice
, re
spec
tivel
y. A
, B
, C
are
thos
e ba
se c
ente
red
CSL
plan
es b
eing
for
med
by
thei
r el
emen
tary
vec
tors
(b,
c), (
c, a) an
d (a
, b),
resp
ectiv
ely.
B’
and
C’ a
lso
corr
elat
e to
the
elem
ent,a
ry ve
ctor
s (c
, a)
and (a
, b), b
ut a
long
thei
r ax
es c
and
a th
ere
exis
ts a
n ad
ditio
nal l
attic
e
ug @
plac
e ly
ing
Icl/4
and
lal/4
apa
rt fr
om th
e or
igin
?j
Symmetrical Grain Boundaries with Special Angles 357
Fig. 6. Position of two additional lattice points in ‘;filol the plane B(a, c) of the Bravais lattice cell of
the grain boundary CSL with c = [112], Z = 7. The lattice pointa are arranged along [310] direc- tion (0%) a t D’ and D” having the equal distance f [310] towards the origin of the cell and to each other, respectively. D is a Bravais lattice point, too
D
c. 0 li=f42J;I
livered in Table 2. With the described simple geometrical method not only the crystal system of the Bravais lattice of the coincidence site lattices in the grain boundary can be determined, but the lengt,h and direction of the vectors of the elementary cell, too. Moreover it follows that the elementary cell of the Bravais lattice of the coincidence site lattices in special cases has additional non-centered lattice points.
For the rotation axis [112] with ,Z = 7 the coincidence site lattice is a S.C. lattice and the €3 (a, c) plane is not centered. However, in this plane along the diagonal b + c two additional equidistant lattice points exist. If the grain boundary coincides with this plane, two additional lattice points per elementary cell in this special grain boundary exist (Fig. 6).
For the rotation axis [Oll] with Z = 3 the Bravais lattice is basis centered in the A (c, b) plane. I n the diamond lattice, however, we obtain an additional lattice point in the [l l l] direction a t + [l l l] and the two-dimensional lattice contains a double layer of lattice points.
Assuming the grain boundary energy is proportional to the inverse density of the coincidence sites, the double layer in the grain boundary along the [ill] direction will decrease the grain boundary energy to half its value.
References [l] H. GRIMMER, W. BOLLMANN, and D. T. WARRINGTON, Acta cryst. A30, 197 (1974). [2] A. V. ANDREEVA and L. K. FIoNovA.~iz. Metallov i Metallovedenie 44, 395 (1977). [3l A. V. ANDREEVA, Fiz. Metallov i Metallovedenie 49, 706 (1980). [4] M. A. FORTES, phys. stat. sol. (b) 82, 377 (1977). [S] R. BONNET and F. DLTRAND, Scripta metall. 9, 935 (1975). [S] A. N. ORLOV, V. N. PEREVESENZEV, and V. V. RIBIN, Grain Boundaries in Metals, Metallurgiz-
dat, Moscow 1980 (in Russian). “71 J. H O R N s m A , Physica 26, 409 (1959). 181 W. BOLLYANN, Surface Sci. 31, 1 (1972). 191 w. BOLLMANN, Crystal Defects and Crystalline Interfaces, Springer-Verlag, Berlin 1970.
(Received July 2, 1982)