Transcript Slide 1
Order in crystals
Symmetry, X-ray diffraction
2-dimensional square lattice
Translation
Translation
Rotation
Rotation
Point group symmetries : Identity (E) Reflection (s) Rotation (R n ) Rotation-reflection (S n ) Inversion (i) In periodic crystal lattice : (i) Additional symmetry - Translation (ii) Rotations – limited values of n
Restriction on n-fold rotation symmetry in a periodic lattice
a
q
a na (n-1)a/2
cos (180 q ) = - cos q = (n-1)/2 n 3 q o 180 Rotation 2 2 1 120 90 3 4 0 -1 60 0 6 1
Crystal Systems in 2-dimensions - 4
square oblique hexagonal rectangular
Oblique Rectangular Square Hexagonal a b, 90 o a b, = 90 o a = b, = 90 o a = b, = 120 o
Crystal Systems in 3-dimensions - 7
Cubic Tetragonal Orthorhombic Monoclinic Triclinic Trigonal Hexagonal
Bravais lattices in 2-dimensions - 5
square rectangular centred rectangular oblique hexagonal
Bravais Lattices in 3-dimensions (in cubic system)
Primitive cube (P) Body centred cube (I) Face centred cube (F)
Bravais Lattices in 3-dimensions - 14 Cubic Tetragonal Orthorhombic Monoclinic Triclinic Trigonal Hexagonal/Trigonal - P, F (fcc), I (bcc) - P, I - P, C, I, F - P, C - P - R - P
Point group operations 7 Crystal systems Point group operations + translation symmetries 14 Bravais lattices
Lattice
( o ) +
basis ( x ) = crystal structure X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X
Spherical basis Non-spherical basis C 4 C 4
Space Groups Lattice + Spherical Basis Lattice + Nonspherical Basis Point group operations 7 Crystal systems 32 Crystallographic point groups Point group operations + translation symmetries 14 Bravais lattices 230 space groups
Miller plane
z y x
a
(100)
Distance between planes = a
z y x
(010)
Distance between planes = a
z y x
(110)
Distance between planes = a/
2 = 0.7 a
z y x
(111)
Distance between planes = a/
3 = 0.58 a
Spacing between Miller planes d
hkl
= a
h
2
+k
2
+l
2
for cubic crystal system
Wavelength =
l q
Bragg’s law
q
d hkl hkl plane 2d hkl sin
q
= n
l
von Laue’s condition for x-ray diffraction k k
lattice point d d.i
-d.i
k = incident x-ray wave vector k
= scattered x-ray wave vector d = lattice vector Constructive interference condition: d.(i-i
) = m
l
(
l
d.
/2
)d.(k-k k = 2
m
) = m
l
i = unit vector = (
l
/2
)k i
= unit vector = (
l
/2
)k
K = reciprocal lattice vector
d.K = 2
k = K
n
Structure factor S hkl =
S
f n e 2
i
(
h x n +k y n +l z ) Atom position
Relates to
Atom type Intensity of x-ray scattered from an (hkl) plane I hkl
S hkl 2
Problem Set
1. Write down a set of primitive vectors for the following Bravais lattices : (a) simple cube, (b) body-centred cube, (c) face-centred cube, (d) simple tetragonal, (e) body-centred tetragonal.
2. Write down the reciprocal lattice vectors corresponding to the primitive direct lattice vectors in problem 1.
3. Prove with a simple geometric construction, that rotation symmetry operations of order 1, 2, 3, 4 and 6 only are compatible with a periodic lattice.
4. Determine the best packing efficiency among simple cube, bcc and fcc lattices.
5. In a system of close packed spheres, determine the ratio of the radius of tetrahedral interstitial sites to the radius of the octahedral interstitial sites.
6. List the Bravais lattices arising from the cubic, tetragonal and orthorhombic systems. Discuss their genesis and account for why cubic system has three, tetragonal system has two and orthorhombic system has four Bravais lattices.
7. Points on a cubic close packed structure form a Bravais lattice, but the points on a hexagonal close packed structure do not. Explain.
8. Write down the direct and reciprocal lattice vectors for diamond considering it as an fcc lattice with two atoms in the basis. Write down also the coordinates of the two atoms in the basis. Determine the systematic absences in its x-ray diffraction profile.
9. Discuss the spinel and inverse spinel structures. Give some examples of materials possessing such structures.
10. Draw schematic diagrams of the following structures : (a) rock salt, (b) cesium chloride, (c) fluorite, (d) rutile.
11. What is the difference in the structures of -graphite and -graphite ?
12. Contrast the zinc blende and wurtzite structures - give similarities and differences.
13. Diamond has a zinc blende structure - explain.
14. Draw schematic diagrams to illustrate the similarity between NiAs and CdI 2 structures.
15. Illustrate with a diagram the perovskite structure for the general oxide formula ABO 3 . What is the oxygen coordination for A and B ? Indicate this on the diagram.
(Text books of Cotton & Wilkinson, Greenwood & Earnshaw, Wells etc. give details of various structural motifs).
A more detailed presentation on x-ray diffractometry is also provided on the website