Matrix operators
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Matrix operators. Rotation. z. y. y'. A a. cos a. cos a. x'. a. y'. sin a. a. a. y. x. - sin a. x. a. x'. A a x --> x' A a y --> y'. Matrix operators. Rotation. z. y. y'. A a. cos a. cos a. x'. a. y'. sin a. a. a. y. x. - sin a. x. a. x'.

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Matrix operators

Rotation

z

y

y'

Aa

cos a

cos a

x'

a

y'

sin a

a

a

y

x

- sin a

x

a

x'

Aa x --> x' Aa y --> y'


Matrix operators

Rotation

z

y

y'

Aa

cos a

cos a

x'

a

y'

sin a

a

a

y

x

- sin a

x

a

x'

Aa x --> x' Aa y --> y'

x' = x cos a + y sin a

y' = - x sin a + y cos a

z' = z


Matrix operators

Rotation

z

y

y'

Aa

cos a

cos a

x'

a

y'

sin a

a

a

y

x

- sin a

x

a

x'

x' = x cos a + y sin a x' cos a sin a 0 x

y' = - x sin a + y cos a y' = – sin a cos a 0 y

z' = z z' 0 0 1 z

X' = AaX


Matrix operators

Rotation of a point

y

x’ y’ z’

x y z

a

x

Rotation of point same asx' cos a – sin a 0 x

leaving point fixed &y' = sin a cos a 0 y

rotating coord. system z' 0 0 1 z

through angle –a

X' = AaX = AaX

-1

T


Matrix operators

General transformation matrix

x3

x3’

r = (r i)i + (r j)j+ (r k)k

r = (r i')i' + (r j')j'+ (r k')k'

If r = i', j', k', in turn:

i' = (i' i)i + (i' j)j+ (i' k)k

j'= (j' i)i + (j' j)j+ (j' k)k

k'= (k' i)i + (k' j)j+ (k' k)k

x’ y’ z’

x y z

r

x2’

x2

x1

x1’


Matrix operators

General transformation matrix

x3

x3’

If r = i', j', k', in turn:

i' = (i' i)i + (i' j)j+ (i' k)k

j'= (j' i)i + (j' j)j+ (j' k)k

k'= (k' i)i + (k' j)j+ (k' k)k

(i' i), (i' j), ….. are direction

cosines lmn = cos qmn, and:

i' l11 l12l13i

j' = l21 l22l23j

k' l31 l32l33k

x’ y’ z’

x y z

r

x2’

x2

x1

x1’


Matrix operators

General transformation matrix

(i' i), (i' j), ….. are direction

cosines lmn = cos qmn, and:

i' l11 l12l13i

j' = l21 l22l23j

k' l31 l32l33k

x l11 l12l13 x' x' l11 l21l31 x

y = l21 l22l23 y' y' = l12 l22l32 y

z l31 l32l33 z' z' l13 l23l33 z


Matrix operators

General transformation matrix

Transformation matrix is then product of 3 matrices:

P = A-1R A where X' = P X

X3 ––> r = A

2. rotate a about r = R

3. r ––> X3' = A-1

x3

x3’

x’ y’ z’

x y z

r

x2’

x2

x1

x1’


Matrix operators

General transformation matrix

Transformation matrix is then product of 3 matrices:

P = A-1R A where X' = P X

P = l11 l21 l31 cos a sin a 0 l11 l12 l13

l12 l22 l32 -sin a cos a 0 l21 l22 l23

l13 l23 l33 0 0 1 l31 l32 l33


Matrix operators

General transformation matrix

Transformation matrix is then product of 3 matrices:

P = A-1R A where X' = P X

P = l11 l21 l31 cos a sin a 0 l11 l12 l13

l12 l22 l32 -sin a cos a 0 l21 l22 l23

l13 l23 l33 0 0 1 l31 l32 l33

Multiplying matrices, finally get

P = l31(1 - cos a) + cos a l32l31 (1 - cos a) + l33 sin a l31l33 (1 - cos a) - l32 sin a

l32l31 (1 - cos a) - l33 sin a l32(1 - cos a) + cos a l32l33 (1 - cos a) + l31 sin a

l33l31 (1 - cos a) + l32 sin a l32l33 (1 - cos a) - l31 sin a l33(1 - cos a) + cos a

2

2

2


Matrix operators

General transformation matrix

Transformation matrix is then product of 3 matrices:

P = A-1R A where X' = P X

P = l31(1 - cos a) + cos a l32l31 (1 - cos a) + l33 sin a l31l33 (1 - cos a) - l32 sin a

l32l31 (1 - cos a) - l33 sin a l32(1 - cos a) + cos a l32l33 (1 - cos a) + l31 sin a

l33l31 (1 - cos a) + l32 sin a l32l33 (1 - cos a) - l31 sin a l33(1 - cos a) + cos a

This matrix for rotation of basis vectors. More interesting to rotate points (xyz).

Use inverse (transpose) matrix:

P-1 = l31(1 - cos a) + cos a l32l31 (1 - cos a) - l33 sin a l33l31 (1 - cos a) + l32 sin a

l32l31 (1 - cos a) + l33 sin a l32(1 - cos a) + cos a l32l33 (1 - cos a) - l31 sin a

l31l33 (1 - cos a) - l32 sin a l 32l33 (1 - cos a) + l31 sin a l33(1 - cos a) + cos a

2

2

2

2

2

2


Matrix operators

General transformation matrix

Transformation matrix is then product of 3 matrices:

P = A-1R A where X' = P X

P-1 = l31(1 - cos a) + cos a l32l31 (1 - cos a) - l33 sin a l33l31 (1 - cos a) + l32 sin a

l32l31 (1 - cos a) + l33 sin a l32(1 - cos a) + cos a l32l33 (1 - cos a) - l31 sin a

l31l33 (1 - cos a) - l32 sin a l 32l33 (1 - cos a) + l31 sin a l33(1 - cos a) + cos a

Ex: r along z l31 = l32 = 0 l33 = 1

P-1 = cos a – sin a 0

sin a cos a 0

0 0 1

2

2

2


Matrix operators

General transformation matrix

Transformation matrix is then product of 3 matrices:

P = A-1R A where X' = P X

P-1 = l31(1 - cos a) + cos a l32l31 (1 - cos a) - l33 sin a l33l31 (1 - cos a) + l32 sin a

l32l31 (1 - cos a) + l33 sin a l32(1 - cos a) + cos a l32l33 (1 - cos a) - l31 sin a

l31l33 (1 - cos a) - l32 sin a l 32l33 (1 - cos a) + l31 sin a l33(1 - cos a) + cos a

Ex: C4 along [100] l33 = l32 = 0 l31 = 1

a = π/2 cos a = 0 sin a = 1

P-1 = 1(1-0)+0 0(1-0)-0(1) 0(1-0)+0(1) = 1 0 0

0(1-0)+0(1) 0(1-0)+0 0(1-0)-1(1) 0 0 -1

0(1-0)+0(1) 0(1-0)+1(1) 0(1-0)+0 0 1 0

(x', y', z') = (x, -z, y)

2

2

2


Matrix operators

General transformation matrix

Transformation matrix is then product of 3 matrices:

P = A-1R A where X' = P X

P-1 = l31(1 - cos a) + cos a l32l31 (1 - cos a) - l33 sin a l33l31 (1 - cos a) + l32 sin a

l32l31 (1 - cos a) + l33 sin a l32(1 - cos a) + cos a l32l33 (1 - cos a) - l31 sin a

l31l33 (1 - cos a) - l32 sin a l 32l33 (1 - cos a) + l31 sin a l33(1 - cos a) + cos a

Ex: C3 along [111] l33 = l32 = l31 = 1/ 3

a = 2π/3 cos a = -1/2 sin a = 3/2

P-1 = 1/3(3/2)-1/2 1/3(3/2)-1/2 1/3(3/2)+1/2 = 0 0 1

1/3(3/2)+1/2 1/3(3/2)-1/2 1/3(3/2)-1/2) 1 0 0

1/3(3/2)-1/2 1/3(3/2)+1/2 1/3(3/2)-1/2 0 1 0

(x', y', z') = (z, x, y)

2

2

2


Matrix operators

Unit cell transformations

Transformation of unit cell transforms:

(hkl)

reciprocal cell basis vectors

zone axes [uvw]

atom position coordinates


Matrix operators

Unit cell transformations

Transformation of unit cell transforms:

(hkl)

reciprocal cell basis vectors

zone axes [uvw]

atom position coordinates

Suppose

a2 = s11a1+ s12b1 + s13c1

b2 = s21a1+ s22b1 + s23c1

c2 = s31a1+ s32b1 + s33c1

sij also transforms (hkl) to the new basis a2, b2, c2

But a2i* = ((sij)-1)Ta1i* <–– [uvw] & (xyz) transform the same way


Matrix operators

Unit cell transformations

a2 = s11 s12 s13 a1

b2 s21 s22 s23 b1

c2 s31 s32 s33 c1

Ex: F cubic ––> P cell

aP = 1/2 1/2 0 aF

bP0 1/2 1/2 bF

cP1/2 0 1/2cF

cF

bP

cP

bF

aP

aF


Matrix operators

Unit cell transformations

a2 = s11 s12 s13 a1

b2 s21 s22 s23 b1

c2 s31 s32 s33 c1

Ex: F cubic ––> P cell

aP = 1/2 1/2 0 aF

bP0 1/2 1/2 bF

cP1/2 0 1/2cF

Typical F cubic diffraction pattern:

(111), (200), (220) …..

(111), (101), (211) for P cell

cF

bP

cP

bF

aP

aF


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