Announcements
1)Revised Lab timings: 1-3 PM (all groups)
2) Quiz 1, 28th Jan 2014, Tuesday 7:30 PM, WS 209, WS 213
Chapter 3
Crystal Geometry and
Structure Determination
RecapLattice, Motif/Basis
Crystal = Lattice + Motif
e.g. Brass, diamond, ZnS
Miller indices of direction: components of vector w.r.t to basis vector a, b and c
Miller Indices of directions and planes
William Hallowes Miller(1801 – 1880)
University of Cambridge
5. Enclose in parenthesis
Miller Indices for planes
3. Take reciprocal
2. Find intercepts along
axes in terms of respective
lattice parameters
1. Select a crystallographic
coordinate system with origin not
on the plane
4. Convert to smallest integers in
the same ratio
1 1 1
1 1 1
1 1 1
(111)
x
y
z
O
Miller Indices for planes (contd.)
origin
intercepts
reciprocalsMiller Indices
AB
CD
O
ABCD
O
1 ∞ ∞
1 0 0
(1 0 0)
OCBE
O*
1 -1 ∞
1 -1 0
(1 1 0)_
Plane
x
z
y
O*
x
z
E
Zero represents
that the plane is parallel to
the corresponding
axis
Bar represents a negative intercept
Courtesy: H Bhadhesia
Courtesy: H Bhadhesia
Courtesy: H Bhadhesia
Crystallographically equivalent planes
Miller indices of a family of symmetry related planes
= (hkl ) and all other planes related to (hkl ) by the symmetry of the crystal
{hkl }
All the faces of the cube are equivalent to each other by symmetry
Front & back faces: (100)Left and right faces: (010)
Top and bottom faces: (001)
{100} = (100), (010), (001)
{100}cubic = (100), (010), (001)
{100}tetragonal = (100), (010)
(001)
Cubic
Tetragonal
Miller indices of a family of symmetry related planes
x
z
y
z
x
y
CUBIC CRYSTALS
[hkl] (hkl)
Angle between two directions [h1k1l1] and [h2k2l2]:
C
[111]
(111)
22
22
22
21
21
21
212121coslkhlkh
llkkhh
Some IMPORTANT Results
Weiss zone law
True for ALL crystal systems
Not in the textbook
• If a direction [uvw] lies in a plane (hkl) then
• uh+vk+wl = 0
[uvw
]
(hkl)
dhkl
Interplanar spacing between ‘successive’ (hkl) planes passing through the corners of the unit cell
222 lkh
acubichkld
O
x(100)
ad 100
BO
x
z
E
2011
ad
[uvw] Miller indices of a direction (i.e. a set of parallel directions)
(hkl) Miller Indices of a plane (i.e. a set of parallel planes)
<uvw> Miller indices of a family of symmetry related directions
{hkl} Miller indices of a family of symmetry related planes
Summary of Notation convention for Indices
How do we determine the structure of a piece of crystalline solid?
You can probe the atomic arrangements by X-ray diffraction
(XRD)
Incident Beam
X-Ray Diffraction
Transmitted Beam
Diffra
cted
BeamSample
Braggs Law (Part 1): For every diffracted beam there exists a set of crystal lattice planes such that the diffracted beam appears to be specularly reflected from this set of planes.
≡ Bragg Reflection
Braggs Law (Part 1): the diffracted beam appears to be specularly reflected from a set of crystal lattice planes.
Specular reflection:Angle of incidence =Angle of reflection (both measured from the plane and not from the normal)
The incident beam, the reflected beam and the plane normal lie in one plane
X-Ray Diffraction
i
plane
r
X-Ray Diffraction
i
r
dhkl
Bragg’s law (Part 2):
sin2 hkldn
i
r
Path Difference =PQ+QR sin2 hkld
P
Q
R
dhkl
Path Difference =PQ+QR sin2 hkld
i r
P
Q
R
Constructive inteference
sin2 hkldn
Bragg’s law
Extinction Rules: Table 3.3
Bravais Lattice Allowed Reflections
SC All
BCC (h + k + l) even
FCC h, k and l unmixed
DC
h, k and l are all oddOr
if all are even then (h + k + l) divisible by 4
Diffraction analysis of cubic crystals
sin2 hkld
2sin 222 )lkh(constant
Bragg’s Law:
222 lkh
adhkl
Cubic crystals
(1)
(2)
(2) in (1) =>
)(4
sin 2222
22 lkh
a
X Ray Diffractometer
You do not get indices of plane!!
Cu target, Wavelength = 1.5418 Angstrom
2θ
44.48
51.83
76.35
92.90
98.40
121.87
144.54
Unknown sample, cubic
Determine:1)The crystal structure2)Lattice parameter
5 step program for the determination of crystal structure
1)Start with 2θ values and generate a set of sin2θ values2)Normalise the sin2θ values by dividing it with first entry3)Clear fractions from normalised column: Multiply by common number4) Speculate on the hkl values that, if expressed as h2+k2+l2, would generate the sequence of the “clear fractions” column5) Compute for each sin2θ /(h2+k2+l2) on the basis of the assumed hkl values. If each entry in this column is identical, then the entire process is validated.
2θ Sin2θ Sin2θ/Sin2θ1 Clear fractions
(hkl)? sin2θ /(h2+k2+l2)
44.48 0.143 1.00 3 111 0.0477
51.83 0.191 1.34 4 200 0.0478
76.35 0.382 2.67 8 220 0.0478
92.90 0.525 3.67 11 311 0.0477
98.40 0.573 4.01 12 222 0.0478
121.87 0.764 5.34 16 400 0.0477
144.54 0.907 6.34 19 420 0.0477
William Henry Bragg (1862–1942), William Lawrence Bragg (1890–1971)
Nobel Prize (1915)
A father-son team that shared a Nobel Prize
h2 + k2 + l2 SC FCC BCC DC
1 100
2 110 110
3 111 111 111
4 200 200 200
5 210
6 211 211
7
8 220 220 220 220
9 300, 221
10 310 310
11 311 311 311
12 222 222 222
13 320
14 321 321
15
16 400 400 400 400
17 410, 322
18 411, 330 411, 330
19 331 331 331
sin2n
dhkl
sin2 hkldn
sin2 nlnknhd
n
d
nlnknh
ad hkl
nlnknh
222,,
)()()(
Two equivalent ways of stating Bragg’s Law
1st Form
2nd Form
X-raysCharacteristic Radiation, K
Target
Mo
Cu
Co
FeCr
Wavelength, Å
0.71
1.54
1.79
1.94
2.29