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BRAVAIS LATTICE. 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.

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BRAVAIS LATTICE

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

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.

R

Q

P

2-D honey comb net

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

(B)

(A)

(C)

a2

a2

a2

a1

a1

a2

a1

(D)

a1

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

3-D Bravais Lattices

(a) Simple Cube

k

PTV :

a3

a2

a

j

a1

i

Face centered cubic

C2

PTV

B

F5

F4

F2

F3

F1

a2

A

C1

a3

F6

a1

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

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

PTV

(-1,-1,3)

(-1,-1,2)

(-1,0,2)

(0,0,2)

(0,-1,2)

a3

(0,0,1)

(0,1,1)

a2

(0,0,0)

(0,1,0)

a1

(1,1,0)

(1,2,0)

(1,0,0)

Only 2-fold symmetry

2

Oblique Lattice : a ≠ b, α ≠ 90

2

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

b

a

mirror

PUC and Unit cell for BCC

Unit Cell

Primitive Unit Cell

Body-centered cubic:

2 sc lattices displaced by (a/2,a/2,a/2)

A is the body center

B

PUC

A

B is the body center

All points have identical surrounding

P

I

P

I

F

P

I

C

F

P (Trigonal)

P

P

C

P

7 X’l Systems

14 BL

c

b

a

a2

60º

a1

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

2-D Lattice

B

A

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

No. of atoms/unit cell = 8

Corners – 1

Face centers – 3

Inside the cube – 4

(¾, ¼, ¾)

(¼, ¾, ¾)

(¾, ¾,¼)

z

(¼, ¼,¼)

y

x

(0,0,0)

DiamondStructure

(201) plane

k = k´- k = G201

k

θ201

2π/λ

Incident beam

(102)

(002)

(302)

(202)

(301)

(201)

(101)

(001)

(300)

(200)

(100)

(000)

(00 -1)

(30 -1)

a*

(-200)

(000)

(200)

b*

2π/λ

Incident beam

Rotaion = 0º

(-200)

(000)

(200)

a*

2π/λ

b*

Rotaion = 5º

a*

(-200)

(000)

(200)

b*

Rotaion = 10º

a*

(-200)

(000)

(200)

2π/λ

b*

Rotaion = 20º

(-200)

(000)

(200)

2π/λ

2π/λ

Incident beam

Incident beam

Rotaion = 5º

Rotaion = 20º

a*

b*

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 what
• would be observed from a single electron

30

29

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

20

fCu

10

0

0.6

1.0

0.8

0

0.2

0.4

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

NEUTRON

b

F-

X-RAY

3He

7Li

f

C

1H

Li+

sinθ/λ

sinθ/λ

(b)

1

3

3

2

2

(b)

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

c

(a)

b

a

1

1

2

2

(a)