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. ribes the periodic nature of the atomic arrangements (units) in a X’ 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 Infinite array of discrete points arranged (and oriented) in such a way that it looks exactly the same from whichever point the array.
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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
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