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

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

3. Matrix operators Rotation x' = x cos a + y sin a x' cos a sin a 0 xz y y' Aa
cos a
cos a a x' y'
sin a a y a x
- sin a x a x'
x' = x cos a + y sin a x' cos a sin a x
y' = - x sin a + y cos a y' = – sin a cos a y
z' = z z' z
X' = Aa X

4. Matrix operators Rotation of a pointy
x’ y’ z’
x y z a x
Rotation of point same as x' cos a – sin a x
leaving point fixed & y' = sin a cos a y
rotating coord. system z' z
through angle –a
X' = Aa X = Aa X T -1

5. Matrix operators General transformation matrix
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
x3’
x’ y’ z’
x y z r
x2’ x2 x1
x1’

6. Matrix operators General transformation matrix
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' l l l i
j' = l l l j
k' l l l k
x3’
x’ y’ z’
x y z r
x2’ x2 x1
x1’

7. Matrix operators General transformation matrix
(i' i), (i' j), ….. are direction
cosines lmn = cos qmn, and:
i' l l l i
j' = l l l j
k' l l l k
x l l l x' x' l l l x
y = l l l y' y' = l l l y
z l l l z' z' l l l z

8. 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’

9. Matrix operators General transformation matrix
Transformation matrix is then product of 3 matrices:
P = A-1R A where X' = P X
P = l11 l21 l cos a sin a l11 l12 l13
l12 l22 l sin a cos a l21 l22 l23
l13 l23 l l31 l32 l33

10. Matrix operators General transformation matrix
Transformation matrix is then product of 3 matrices:
P = A-1R A where X' = P X
P = l11 l21 l cos a sin a l11 l12 l13
l12 l22 l sin a cos a l21 l22 l23
l13 l23 l 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

11. 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

12. 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 = l33 = 1
P-1 = cos a – sin a 0
sin a cos a 2 2 2

13. 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 = l31 = 1
a = π/ cos a = 0 sin a = 1
P-1 = 1(1-0) (1-0)-0(1) (1-0)+0(1) =
0(1-0)+0(1) 0(1-0) (1-0)-1(1)
0(1-0)+0(1) 0(1-0)+1(1) (1-0)
(x', y', z') = (x, -z, y) 2 2 2

14. 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/ cos a = -1/2 sin a = 3/2
P-1 = 1/3(3/2)-1/ /3(3/2)-1/2 1/3(3/2)+1/ =
1/3(3/2)+1/2 1/3(3/2)-1/2 1/3(3/2)-1/2)
1/3(3/2)-1/ /3(3/2)+1/2 1/3(3/2)-1/
(x', y', z') = (z, x, y) 2 2 2

15. Matrix operators Unit cell transformations
Transformation of unit cell transforms:
(hkl)
reciprocal cell basis vectors
zone axes [uvw]
atom position coordinates

16. 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)T a1i* <–– [uvw] & (xyz) transform the same way

17. Matrix operators Unit cell transformations a2 = s11 s12 s13 a1
b s21 s22 s b1
c s31 s32 s c1
Ex: F cubic ––> P cell
aP = 1/2 1/ aF
bP /2 1/ bF
cP / / cF cF bP cP bF aP aF

18. Matrix operators Unit cell transformations a2 = s11 s12 s13 a1
b s21 s22 s b1
c s31 s32 s c1
Ex: F cubic ––> P cell
aP = 1/2 1/ aF
bP /2 1/ bF
cP / / cF
Typical F cubic diffraction pattern:
(111), (200), (220) …..
(111), (101), (211) for P cell cF bP cP bF aP aF