Order in crystalsSymmetry, X-ray diffraction

2-dimensional square lattice

Translation

Translation

Rotation

Rotation

Point group symmetries :Identity (E)Reflection (s)Rotation (Rn)Rotation-reflection (Sn)Inversion (i)

In periodic crystal lattice :(i) Additional symmetry - Translation

(ii) Rotations – limited values of n

Restriction on n-fold rotation symmetryin a periodic lattice

a

a

na

(n-1)a/2

cos (180-) = - cos = (n-1)/2

n 3 2 1 0 -1o 180 120 90 60 0Rotation 2 3 4 6 1

Crystal Systems in 2-dimensions - 4

square

rectangular

oblique

hexagonal

Oblique a b, 90o

Rectangular a b, = 90o

Square a = b, = 90o

Hexagonal a = b, = 120o

Crystal Systems in 3-dimensions - 7

Cubic Tetragonal Orthorhombic

Trigonal HexagonalMonoclinic Triclinic

Bravais lattices in 2-dimensions - 5

square rectangular

oblique hexagonal

centred rectangular

Primitive cube (P)

Bravais Lattices in 3-dimensions(in cubic system)

Body centred cube (I)

Face centred cube (F)

Bravais Lattices in 3-dimensions - 14

Cubic - P, F (fcc), I (bcc)Tetragonal - P, IOrthorhombic - P, C, I, FMonoclinic - P, CTriclinic - PTrigonal - RHexagonal/Trigonal - P

Point groupoperations

Point groupoperations +translationsymmetries

7 Crystal systems

14 Bravais lattices

Lattice (o)

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

+ basis (x) = crystal structure

C4

C4

Spherical basis

Non-spherical basis

Lattice +Nonspherical Basis

Point groupoperations

Point groupoperations +translationsymmetries

7 Crystal systems 32 Crystallographic point groups

14 Bravais lattices 230 space groups

Lattice +Spherical Basis

Space Groups

x

y

z

(100)

Miller plane

Distance between planes = a

a

(010)

Distance between planes = a

x

y

z

(110)

Distance between planes = a/2= 0.7 a

x

y

z

(111)

Distance between planes = a/3= 0.58 a

x

y

z

ah2+k2+l2

dhkl =

Spacing between Miller planes

for cubic crystal system

dhkl

hkl plane

2dhkl sin = n

Wavelength =

Bragg’s law

von Laue’s condition for x-ray diffraction

d

k k

lattice point

k = incident x-ray wave vectork = scattered x-ray wave vectord = lattice vector

d.i -d.i

i = unit vector = (/2)ki = unit vector = (/2)k

Constructive interference condition: d.(i-i) = m

(/2)d.(k-k) = m d.k = 2m

K = reciprocal lattice vectord.K = 2n

k = K

Shkl = fn e2ihx +ky +lz ) n n n

Relates toAtom type

Atom position

Structure factor

Intensity of x-ray scattered from an(hkl) plane

Ihkl Shkl2

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 CdI2 structures.15.Illustrate with a diagram the perovskite structure for the general oxide formula ABO3. 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