VECTOR-QUATERNION DESCRIPTION OF MISORIENTATIONS

(1)

HAL Id: jpa-00230279

https://hal.archives-ouvertes.fr/jpa-00230279

Submitted on 1 Jan 1990

HAL

is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire

HAL, est

destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

VECTOR-QUATERNION DESCRIPTION OF MISORIENTATIONS

V. Gertsman

To cite this version:

V. Gertsman. VECTOR-QUATERNION DESCRIPTION OF MISORIENTATIONS. Journal de

Physique Colloques, 1990, 51 (C1), pp.C1-145-C1-150. �10.1051/jphyscol:1990121�. �jpa-00230279�

(2)

COLLOQUE DE PHYSIQUE

Colloque Cl, supplkent au nO1, Tome 51, janvier 1990

VECTOR-QUATERNION DESCRIPTION OF MISORIENTATIONS V.Yu. GERTSMAN

Institute of Metals Superplasticity Problems, URSS Academy of Sciences, ul.Khalturina 3 9 , Ufa 450001, U.R.S.S.

Abstract

-

A unified formal description of misorientations of crystals of any type of symmetry is suggested. A simple algorithm for determining coincidence misorientations for cubic, hexaqonal and tetragonal systems is described.

It is known that for two misoriented crystals a common sublattice may be intro- duced, i.e. coincidenceisite lattice (CSL). The concept of CSL underlies modern geometrical theory of interfaces. Misorientations which result in CSL's with a high degree of coincidence") are denominated as special misorientations.

Up to now, for calculations of special misorientations, a separate algorithm had to be developed for each lattice. The tables of special misorientations for cubic system C17 and some hexagonal lattices C23 have been calculated basing on the properties of the elements of rotation matrix. In C33 special rnisorientations of cubic crystals have been calculated on the basis of quaternion description, a similar method was suggested in C4,53 for hexaqonal crystals. Nowadays only tables of special rotations for the most wide-spread high symmetry lattices are available which can not obviously cover all possible cases, e:g. the whole variety of axial ratios for non-cubic lattices. For many applications it would be useful to have a general and simplified formalized algorithm to calculate them.

This paper suggests an economical method for description of misorientations of crystals of any type of symmetry and a simple algorithm for finding special misorientations of cubic, hexaqonal and tetragonal crystals. This algorithm is

similar to those used in C3-53 and actually generalizes them.

Of the multiple methods of mathematical description of misorientation the following two are mostly spread: in the form of an angle/axis pair and in the form of a rotation matrix (see C13). But the most economical way is to represent rotation in the form of three-dimensional Gibbs vector G since in a general case only three independent parameters are needed to present misorientation.

Thus we assume that misorientation is given by vector G = C p , q , r 3 .

-

In this case the rotation axis runs in the direction common for lattices of both the neibouring crystals and the misorientation angle is given by the length of the G ^vector

tan C 8 / 2 ) = m o d = . ( 1 )

(l) degree of coincidence is described by index

C-

the reciprocal of the density of concidence sites

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1990121

(3)

Cl-L46 COLLOQUE DE PHYSIQUE

Then, for cubic system

tan2<e/2)=p2+q2+r 2 , for hexagonal system

t a n 2 < 8 / 2 > = p 2 + q 2 - p q + < c / a > 2 r 2 , (lb) for tetragonal system

t a n 2 < e / ~ ) = p 2 + q 2 + < c / a > 2 r 2 , where c./a is an axial ratio for hexagonal and tetragonal lattices.

For the CSL to exist it is necessary and sufficient for the rotation axis to have rational indices and for the rotation angle to satisfy the following requirement K73:

t a n 2 < 0 / 2 >

=x/Y,

(2) where X and Y are integers, and

X+Y =

iz,

^{( 3 )}

where i is a certain integer.

It could be shown then that for special misorientations G = a C k , l ,m71/n,

-

where k,l ,m,n - are integers without common divisor.

==l for cubic system, = = a / c for hexagonal system, = = a / c for tetragonal system.

Thus, any.specia1 misorientation is determined by four coprime integers - quaternion

<

k, l ,m,n

>

and so, according to _{( 3 ) .}the reciprocal dencity of coincidence sites may be found in the following way:

For cubic system

X= p <

k2+12+m2+n2

>

,

where t = l

,

wien only one number out of k , l

,

^m

,

ⁿis either even or odd and the other three are of an opposite parity;F=1/2 when two numbers are even and two odd; t = 1 / 4 when all numbers of quaternion are odd.

In hexagonal and tetragonal systems for all rotations, except misorientations around the C0013 axis, CSL1s do exist. if ( c / a ) 2 =S113 , where Jl and

4

^are

integers.

For hexagonal system

2

=) C

J

^{( k 2 i}1 ' - k l ) + y (a2+3n2) J In ref.C53 it was found, that

F=

¹

^/oc~88,

where

d

is the greatest common divisor of l 2 , k2+l

-

k l

,

m2+3n2:

is the common divisor of ffc 3k, 31.

9

being a divisor of divisor of 4 ,

3

,

<

^m2

+

3n

> /d

;

8

is the common divisor of

? / l

^,^m^,^n.

s2

being a divisor of

< m 2 +3n2 )

/ d 8 .

For tetragonal system

= p ( k 2 + l Z ) + ~ ( m 2 + n 2 ) 3 (5c) Where ~ =

/&9Y,

1

$=2 when k2+l and m2+n2 are even integers, otherwise

d=1.

g

is the greatest common divisor of

r,

^{Z k ,}²¹

^,

^C^k2+12

^>

^l&!^;

$

is the greatest common divisor of

9,

2m

,

²¹¹

, <

m2 +n2 ⁾

/d .

(4)

Due to the lattice symmetry the same misorientation may be described by different equivalent ways. These equivalent misorientations are easily found in quaternion form.

The following quaternion descriptions are equivalent for cubic system:

< k , l , m , n )

<

k + 1 , k - 1 , m + n rm-rk

>

< k + m , k - m , l + n , l - - n >

< k + n , k - - n , l + m , l - - m > (6a)

< k + 1 + m + n , k + 1 - - - m - n , k - 1 - - m + n , k - - 1 + m - n >

<k+1+m-n,k+-l-m+n,k-IL+m+n,k-I-m-n) Any permutations of elements within quaternion (as well as alterations of

signs) are permitted.

Equivalent descriptions for hexagonal system:

< k p l , m , n >

< 2 k , 2 1 , r n + 3 n , m - n )

Equivalent descriptions for tetragonal system:

< k , l , m , n >

<

^k+1, k . - - 1 , m + n , m . - n >

< y m ,

y n ,

^$k.,

d 1 >

< y m + y n , y ' f n - J n . JIG+

^{$ 1 ,}

3k-

^{$ 1 )}

For hexagonal system it is only possible to change position of the first and second quaternion coefficients, but for tetragonal system also of the third and forth. Obviously, coefficients of every quaternion _{( 6 )}must be reduced to coprime form.

Now the algorithm of finding special misorientations can be easily described.

Quaternion of integers , 1 m

,

n

>

satisfying requirement (5) has to be found for the given

57

This presents no difficulty using exhaustive search executed by computer. Then one finds an equivalent description using Eqs. ( 6 ) and determines misorientation axis and angle using Eqs. ( 4 1 , (l).

Description with minimal angle usually serves as the main one. The transition from quaternion to matrix description may be found in C63.

The multiplication law for misorientations given in the vector form is as follows :

Hence, "the difference" of two misorientations, e.g. the deviation of G misorientation from special

Go:

G - G O =

-GxG-, ) /

<

1 + S O

>

,

-

(5)

and in the expanded form:

the "deviation-rotation" axis runs along the direction

The described algorithm was used to calculate special misorientations for

-

tetragonal lattice with the ratio of (c/alZ=2. Mziny ordered alloys and inter- metallic compounds with the Llo structure have such a lattice, e.g. TiAl

2 2 2 2

(c/a) =2.04, PtCr (c/a) =2.00, GaTi (c/a) =2.00, TiAg (c/a) =1.98, NaBi

2 2 2

(c/a) =1.96, PtCo (c/a) ~1.35, CuAu (c/a) =1.84 and others. Since in a disordered state such a lattice has f.c.c. structure, it is usual to employ cubic coordinate system for it, though the Bravais lattice here is a basic centered tetragonal. That is why in the table the rotation axes are given both in tetragonal and cubic coordinate systems. Among the equivalent rotations solely smallest-angle descriptions were chosen. If one value corresponds to different non-equivalent rnisorientations, they are labelled a, b, c, etc. and are given in the increasing order of

emin

as qenerally accepted C31.

Comparison of the calculated table with those given in C1,33 indicates that many special misorientations for the given lattice coincide with corresponding misorientations for cubic lattices. Particulary, any tetragonal lattice misorientation around the to011 axis is equivalent to cubic misorientations around <loo> axes. But there are also significant pecularities of special pisorientations in tetragonal lattice. Thus e.g. misorientation S 5 36.9°<100>

in cubic lattice splits into four in tetragonal lattice: 2 5 a , C5b. x ~ a , and x 1 0 b (see the table), since rotatiton axes for z 5 a and z 5 b are crystallographically non-equivalent here and the <loo> axis in tetragonal lattice is not the axis of symmetry of the 4-th order as is the corresponding

<loo> axis in cube. CSL's with 2 5 b and 2 1 0 differ in the following. In the z 5 b CSL every fifth site coincides in every (110) plane as is the case also in the corresponding CSL 2 5 in cubic lattice, but in CSL 2 1 0 such a coincidence occurs in every second (110) plane while in the-others "anti-coincidence" takes place, i.e. coinciding are the sites occupied by atoms of various species. This

"anti-coincidence" is a Feature of all CSL's with even

C .

the structure of boundaries with such misorientations is sure to contain elements of anti-phase boundaries. For example, misorientation 2 2 actually characterises the so-called C-domains.

REFERENCES

1. Grimmer, H., Bollmann, W., and Warrington, D.H., Acta Crystall.

A30

(1974) 197.

2. Warrington, D.H., J.Phys. C4 36 (1975) 87.

3. Mykura, H., In R.W.Balluffi (Ed.), Grain Boundary Structure and Kinetics", ASM-Metal Park, Ohio (1980) 445.

4 . Hagege, S. and Nouet, G., Scripta Metall.

19

(1985) 11.

5. Grimmer, H. and Warrington, D.H., Coincidence orientations of Grains in Hexaqonal m a t e r i a A EIR-Bericht No.593, Wiirenlingen (1986).

6. Korn, G.A. and Korn, T.M., Mathematical Handbook, McGraw-Hill, N.Y. et al.

7. Ranganathan, S., Acta Crystall. 21 (1966) 197.

(6)

TABLE OF CSL MISORIENTATIONS UP TO 2 = 4 9 FOR TETRAGONAL LATTICE WITH THE AXIAL RATIO

c/a=a

2 3 5a 5b 6 7 9a 9b 10a lob l la llb 13a 13b 13c 14a 14b 15a 15b 17a 17b 17c 17d

&8a 18b 19a 19b 19c 21a 21b 21c 21d 22a 22b 23a 23b 25a 25b 25c 25d 26a 26b 26c 26d 27a 27b 27c 27d 29a 29b 29c 2 9d 30a 30b 30c 30d 3 1a 31b 33a 33b 33r

Quaternion

(k,l,m,n)' Angle 90.00°

70.53O 36.87O 53.13O 60.00°

73.40°

38.94O 90. 0O0 36.87O 95.740 50.4E0 50. 48O 22.6Z0 67. 3E0 76.66O 38.21' 73. 40°

48.19' 78.46O .28.07O

28.07O 61.93' 86.63O 38.94O 67.11°

26.53O 46.83O 93.02O 44.42O 58.41°

58.41°

79.02O 62.96O 82.16O 55. 58O 85.01°

16.26O 63.90°

73.740 73.74O 22.62O 27. 80°

76.66O 92.20°

31.5g0 31.5g0 35.430 79.330 43.60°

43.60°

46.40°

76.03O 48.1g0 48.1g0 50. 70°

86.18O 52. 20°

80.72O 20.05O 58.9g0 61. 00°

Axis <urn>

in tetragonal] in cubic coordinate system

(7)

TABLE (continued) Quaternion

(k,l,m,n) Angle

---

84. 78O 91.74O 61.93O 61.93O 63.82O 93.37O 34.05O 64.62"

64.62O 66.42O 88.36O 18.9Z0 18.92O 43.14O 69.43O 91.55O 26.53O 71-59"

71.59O 73.17O 32.20°

50.13O 73.62O 73.62O 12.6E0 40.8E0 40. 88O 55. 88O 77.32O 78.75O 21.7g0 44.42O 79.02O 79.02O 80.41°

27.91°

60.77O 80.63O 80.63O 81.98O 36.87O 53.13O 53.13O 65.03O 83.62O 83.62O 40.46O 40.46O 55.58O 85.01°

43.66O 57.87O 68.80°

86.34O 49.23O 49.23O 62.01°

62. 01°

72.17O 88.83O

Axis <uvw>

in tetragonal1 in cubic coordinate system

I