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

= n 1

a

+ n 2

b

+ n 3

c

where

a

,

b

,

c

are PTV and n i ’s are integers eg: 2D lattice a 2 (A) a 1 a 1 a 2 (B) a 2 a 1 (D) a 2 (A), (B), (C) define PTV, but (D) is not PTV (C) a 1

3-D Bravais Lattices (a) Simple Cube

k i

a a 1 a 3 a 2

j

PTV :  a  a  a 1 2 3  a

x

ˆ   a a

z

ˆ

y

ˆ

Face centered cubic A F4 a 2 a 3 a 1 F1 F5 F6 F2 F3 C2 C1 B a  1 PTV  a 2 (

x

ˆ  a  2  a 2 (

y

ˆ 

y

ˆ )

z

ˆ ) a  3  a 2 (

z

ˆ 

x

ˆ ) For Cube B, C1&C2 are Face centers; also F2&F3 All atoms are either corner points or face centers and are EQUIVALENT

(-1,-1,2) (-1,-1,3) (-1,0,2) (0,-1,2)

a 1 a 3

(0,0,1) (0,0,0)

a 2

(0,0,2) (0,1,0) (0,1,1) (1,1,0) (1,0,0) (1,2,0) PTV  a a  1 2   ˆa

x

ˆa

y

a  3  a 2 ˆ(

x

 ˆ

y



z

)ˆ

Alternate choice of PTV a 3 a 2 a 1  a 1  a 2 (

x

ˆ   a 2  a 2 (

y

ˆ 

y

ˆ 

z

ˆ )

z

ˆ 

x

ˆ )  a 3  a 2 (

z

ˆ 

x

ˆ 

y

ˆ )

Oblique Lattice : a ≠ b, α ≠ 90 Only 2-fold symmetry

2

Rectangular Lattice : a ≠ b, α = 90

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

2

mirror b

Hexagonal Lattice : a = b, α = 120 b a

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

PUC and Unit cell for FCC Unit Cell

PUC

PUC and Unit cell for FCC : alternate PTV

P I F P I C P I

7 X’l Systems 14 BL

F P (Trigonal) P P C P

a c b

2-D Lattice 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

2 60 º

a

1

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

BCC Structure

FCC Structure

NaCl Structure

Diamond Structure

No. of atoms/unit cell = 8 Corners – 1 Face centers – 3 Inside the cube – 4 (¼, ¾, ¾) ( ¾, ¼, ¾) (¼, ¼,¼) (0,0,0) ( ¾, ¾,¼)

DiamondStructure

z y x

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)

(302) (202) (102) 2 π/λ (002) (201) plane (301) (300) (201) (200)

k ´

(30 -1) (101) (100)

k

θ 201 (001) (000) (00 -1) Incident beam

a* b* (-200) (000) 2 π/λ (200) Rotaion = 0 º Incident beam

b* a* 2 π/λ  k  k   Δ k  1/d 100 Rotaion = 5 º

Rotaion = 10 º

Rotaion = 20 º 2 π/λ

a* b* (-200) (000) 2 π/λ (200) Rotaion = 5 º Incident beam Rotaion = 20 º

Schematic diagram of a four-circle diffractometer

.

Scattering Intensities and Systematic Absence I 2 θ

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

f Cu

20 Form factor is equivalent to the atomic number at low angles, but it drops rapidly at high (sinθ)/λ 10 0 0 0.2 0.4

0.6

sin θ/λ 0.8 1.0

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 F b X-RAY 3 He 7 Li

f

C Li+ 1 H sin θ/λ sin θ/λ

Scattering by a Unit Cell – Structure Factor

(a) 1 2

The positions of the atoms in a unit cell determine the intensities of the reflections Consider diffraction from (001) planes in (a) and (b)

c b a 1 2 2 3 (b) 1 2 3

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

(a) (b)