BRAVAIS LATTICE

Infinite array of discrete points arranged (and oriented) in such a way that it looks exactly the same from whichever point the array is looked at.

BL describes the periodic nature of the atomic arrangements (units) in a X’l.

X’l structure is obtained when we attach a unit to every lattice point and repeat in space

Unit – Single atoms (metals) / group of atoms (NaCl) [BASIS]

Lattice + Basis = X’l structure

2-D honey comb net

P Q

R

Not a BL

Lattice arrangement looks the same from P and R, bur rotated through 180º when viewed from Q

Primitive Translation Vectors

If the lattice is a BL, then it is possible to find a set of 3 vectors a, b, c such that any point on the BL can be reached by a translation vector

R = n1a + n2b + n3c

where a, b, c are PTV and ni’s are integers

eg: 2D lattice

a1a2

a1

a2

a1

a2

(A) (B)(C)

(D)

a1

a2

(A), (B), (C) define PTV, but (D) is not PTV

j

i

k

3-D Bravais Lattices

(a) Simple Cube

a2

a1a

PTV :

a3

z

y

x

ˆa a

ˆa a

ˆa a

3

2

1

F4

F6

F1

F2F3

F5

A

BC2

C1

All atoms are either corner points or face centers and are EQUIVALENT

Face centered cubic

a1

a2

a3

For Cube B, C1&C2 are Face centers; also F2&F3

)ˆˆ(2

a a

)ˆˆ(2

a a

)ˆˆ(2

a a

3

2

1

xz

zy

yx

PTV

(0,0,0)

(1,0,0)

(1,1,0)

(0,1,0)

(1,2,0)

(0,0,1)

(0,-1,2) (0,0,2)

(-1,0,2)(-1,-1,2)

(0,1,1)

(-1,-1,3)

a1

a2

a3

PTV

)ˆˆˆ(2

a a

ˆa a

ˆa a

3

2

1

zyx

y

x

Alternate choice of PTV

)ˆˆˆ(2

a a

)ˆˆˆ(2

a a

)ˆˆˆ(2

a a

3

2

1

yxz

xzy

zyx

a1

a2a3

Oblique Lattice : a ≠ b, α ≠ 90

Only 2-fold symmetry2

Rectangular Lattice : a ≠ b, α = 90

a

b

2mirror

Rectangular Lattice : a ≠ b, α = 90 : Symmetry operations

a

b

Hexagonal Lattice : a = b, α = 120

PUC and Unit cell for BCC

Unit Cell

Primitive Unit Cell

A is the body center

A

B is the body center

All points have identical surrounding

Body-centered cubic:2 sc lattices displaced by (a/2,a/2,a/2)

B

PUC

PUC and Unit cell for FCC

Unit Cell

PUC

PUC and Unit cell for FCC : alternate PTV

PP CP

P (Trigonal)

P I C F

PP I

IF

7 X’l Systems14 BL

ba

c

2-D Lattice

a1

a260º

A lattice which is not a BL can be made into a BL by a proper choice of 2D lattice and a suitable BASIS

A

B

The original lattice which is not a BL can be made into a BL by selecting the 2D oblique lattice (blue color) and a 2-point BASIS A-B

BCC Structure

FCC Structure

NaCl Structure

Diamond Structure

DiamondStructure

(0,0,0)

(¼, ¼,¼)

x

y

z(¾, ¾,¼)

(¼, ¾, ¾) (¾, ¼, ¾)

No. of atoms/unit cell = 8Corners – 1Face centers – 3Inside the cube – 4

Hexagonal Close Packed (HCP) Structure

HCP = HL (BL) + 2 point BASIS at (000) and (2/3,1/3,1/2)

The Simple Hexagonal Lattice

The HCP Crystal Structure

4-circle Diffractometer

Reciprocal Lattice

(000)

(000)

2π/λ

(001)

(002)

(00 -1)

(102)(202)(302)

(101)(201)(301)

(100)(200)(300)

(30 -1)

k = k´- k = G201

θ201

(201) plane

k´

Incident beam

k

a*

b*

(000) (200)(-200)

Rotaion = 0º

2π/λ

Incident beam

a*

b*

(000)(200)

(-200)

2π/λ

k

k

1001/dkΔ

Rotaion = 5º

a*

b*

(000)

(200)

(-200)

Rotaion = 10º

a*

b*(000)

(200)

(-200)

Rotaion = 20º

2π/λ

a*

b*

(000) (200)(-200)

Rotaion = 5º

2π/λ

Incident beam

2π/λ

Incident beam

Rotaion = 20º

                                                                                                                               

                                                

Schematic diagram of a four-circle diffractometer.

2θ

I

Scattering Intensities and Systematic Absence

Diffraction Intensities

• Scattering by electrons• Scattering by atoms• Scattering by a unit cell• Structure factors Powder diffraction intensity calculations– Multiplicity– Lorentz factor– Absorption, Debye-Scherrer and Bragg Brentano– Temperature factor

Scattering by atoms

• We can consider an atom to be a collection of electrons.• This electron density scatters radiation according to the Thomson

approach (classical Scattering). However, the radiation is coherent so we have to consider interference between x-rays scattered from different points within the atom

– This leads to a strong angle dependence of the scattering – FORM FACTOR.

Form factor (Atomic Scattering Factor)• We express the scattering power of an atom using a form factor (f)– Form factor is the ratio of scattering from the atom to whatwould be observed from a single electron

30

20

10

0

fCu

0 0.2 0.4 0.6 0.8 1.0

29

sinθ/λ

Form factor is expressed as a function of (sinθ)/λ as the interference depends on both λ and the scattering angle

Form factor is equivalent to the atomic number at low angles, but it drops rapidly at high(sinθ)/λ

X-ray and neutron form factor

The form factor is related to the scattering density distribution in an atoms- It is the Fourier transform of the scattering density- Neutrons are scattered by the nucleus not electrons and as the nucleus is very small, the neutron form factor shows no angular dependence

F-

C

Li+

f

sinθ/λ sinθ/λ

1H

7Li

3Heb

X-RAY

NEUTRON

Scattering by a Unit Cell – Structure Factor

The positions of the atoms in a unit cell determine the intensities of the reflections

Consider diffraction from (001) planes in (a) and (b)

If the path length between rays 1 and 2 differs by λ, the path length between rays 1 and 3 will differ by λ/2 and destructive interfe-rence in (b) will lead to no diffracted intensity

(a)

a

b

c

1

2

1

2

(b)

2

1

2

3 3

(b)(a)