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'

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

Matrix operators Rotation x' = x cos a + y sin a x' cos a sin a 0 x z 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 0 x y' = - x sin a + y cos a y' = – sin a cos a 0 y z' = z z' 0 0 1 z X' = Aa X

Matrix operators Rotation of a point y x’ y’ z’ x y z a x Rotation of point same as x' 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' = Aa X = Aa X T -1

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’

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' l11 l12 l13 i j' = l21 l22 l23 j k' l31 l32 l33 k x3’ 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 l12 l13 i j' = l21 l22 l23 j k' l31 l32 l33 k x l11 l12 l13 x' x' l11 l21 l31 x y = l21 l22 l23 y' y' = l12 l22 l32 y z l31 l32 l33 z' z' l13 l23 l33 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)T a1i* <–– [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 bP 0 1/2 1/2 bF cP 1/2 0 1/2 cF 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 bP 0 1/2 1/2 bF cP 1/2 0 1/2 cF Typical F cubic diffraction pattern: (111), (200), (220) ….. (111), (101), (211) for P cell cF bP cP bF aP aF