CRYSTAL SYSTEMS AND CRYSTAL STRUCTURE DETERMINATION

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CRYSTAL SYSTEMS AND CRYSTAL STRUCTURE
DETERMINATION

• Shubha Gokhale
• School of Sceinces
• IGNOU
• May 25, 2007

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CRYSTAL

• Crystal consists of 3-D periodic arrangement of
  atoms/ combination of atoms/molecules
• Each repeating 3-D unit is the UNIT CELL of that
  crystal

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3-D Unit Cell
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Different lattice systems are created by
varying the lengths a, b and c and the angles
??, ? and ?.

• To examine the point symmetry, we look for
• rotation symmetry axes,
• reflection planes and
• inversion centre.

SEVEN systems for 3-D lattices.
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1. TRICLINIC SYSTEM
All three sides different All three angles
different
a ? b ? c
? ? ? ? ?
Point symmetry Inversion
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• 2. MONOCLINIC SYSTEM
• All three sides different,
• Two right angles,
• third arbitrary
• Point symmetry
• 180? rotation about one axis
• Inversion.

a ? b ? c
a

? ? 90, ? ? 90
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• 3. ORTHORHOMBIC SYSTEM
• All three sides different,
• All three right angles
• Point symmetry
• 180? rotations about three
• mutually perpendicular axes
• 2. Inversion

a ? b ? c
? ? ? 90
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4. CUBIC SYSTEM All three sides equal, All
three right angles
a b c
? ? ? 90

• Point symmetry
• 90? rotations about three axes,
• 120? rotations about four cube diagonals
• 180? rotations about six axes
• Inversion.

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5. TRIGONAL SYSTEM All three sides equal, All
three angles equal, of arbitrary value,
a b c
? ? ? ? 90

• Point symmetry
• 120? rotation about one axis
• Inversion

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6.TETRAGONAL SYSTEM Two sides equal, All
three right angles
a b ? c
? ? ? 90
Point symmetry 1. 90? rotation about two
axis, 2. 180? rotation about one axis, 3.
inversion.
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7. HEXAGONAL SYSTEM Two sides equal, third
arbitrary, Two right angles, third angle 120?
a b ? c
? ? 90, ? 120

• Point symmetry
• 60 rotation about one axis
• Inversion

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Just like 2-D lattice, here also we can place
additional lattice points in each unit cell and
examine the translation and point symmetry of
the resulting lattice. A new translation
symmetry for the same point symmetry generates
a different type.
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A complete analysis based on mathematics and
geometry has shown that a single lattice system
can have at the most FOUR types. In all the
seven lattice systems have a total of 14
types. These are called the BRAVAIS LATTICES.
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P type
PRIMITIVE LATTICE
Lattice points only at the corners of the
parallelepiped.
BASE CENTRED LATTICE
C type
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I type
BODY-CENTRED LATTICE
Lattice points at the corners of the
parallelopiped and at the centre of the each
unit cell.
FACE- CENTRED LATTICE
F type
Lattice points at the corners of the
parallelepiped and at the centre of each face of
the unit cell.
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Cubic system has THREE types
cubic P
PRIMITIVE CUBIC LATTICE Lattice points only at
the corners of the cube. Translation symmetry
is
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BODY-CENTRED CUBIC LATTICE Lattice points at the
corners and at the centre of each
cube. Translation symmetry is
cubic I
e.g. CsCl, Ammonia
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cubic F
FACE- CENTRED CUBIC LATTICE (FCC) Lattice points
at the corners of the cube and at the centre
of each face of the cube Translation
symmetry
e.g. NaCl, KBr, MnO, Cu, CaF2
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Cubic C type is NOT POSSIBLE
By placing additional lattice points at one
pair of opposite faces of a cubic unit cell the
system becomes TETRAGONAL.
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• Seven systems divide into 14 Bravais lattices
• Triclinic P
• Monoclinic P, C
• Orthorhombic P, C, I, F
• Trigonal P
• Hexagonal P
• Tetragonal P, I
• Cubic P, I, F

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Are all the materials around us crystalline?
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• Non-crystalline materials
• amorphous
• polycrystalline

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amorphous
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Polycrystalline
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How to determine crystal structure?
Use diffraction technique
Similar to optical diffraction from a grating
Difference being the wavelength of the radiation
should be comparable with the inter-atomic
distances in the crystal!
X- rays, neutrons and electrons fit the bill!
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X-RAY DIFFRACTION
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Crystal Planes
Family of planes
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Miller index
a1 intercept is 2 a2 intercept is 3 a3 intercept
is 3
Hence Miller indices are 3,2,2 and are depicted
by ( hkl ) (322)
Calculate reciprocal of these intercepts and
reduce them to smallest three integers having
same ratio.
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Inter-planer Distance
(hkl) represent a family of planes. All parallel
crystal planes have the same Miller index. These
planes are equally spaced at distance dhkl . This
distance is defined as
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Crystal planes in a cubic unit cell
(100) dhkl a
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Crystal representation for X-Ray Diffraction
(XRD)
2-D depiction in layer forms
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X-ray diffraction Braggs Law
Rays 1 and 2 interfere constructively if Total
Path Difference is integral multiple of the
wavelength, ?
Specular Reflection
O
Total p.d. AB BC
?OAB and ?OCB are equivalent. ?ABBCdhkl sin?
dhkl
Diffraction condition is 2 dhkl sin? n ?
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We may see more than one family of planes at a
time in the diffraction pattern
dhkl different ? For same ?, different ?
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Rotating Crystal method of XRD
Monochromatic x-rays
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Powder Diffraction Method

• Similar to Rotating Crystal method.

• Monochromatic X-rays are used.

• In order to expose various planes in the
• crystal, rather than a rotating a single
  crystal,
• a powder sample is use. This exposes
  various
• crystal planes simultaneously to the
  X-rays.

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Powder Diffraction Method
Many planes are equivalent (100), (010), (001)
or (110), (101), (011) or (211), (121),
(112) etc.
Monochromatic x-rays

Each of these sets have same dhkl and hence same
?. Hence diffracted rays form a cone.
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Laue Method of XRD

• Single crystal sample
• The X- rays are polychromatic (multiple ?)
• Though ? is same here, i.e. crystal is
  fixed,
• due to different ?, different dhkl are
  diffracted
• in different directions.
• Crystal geometry can be found out but no
• crystal parameters like lattice constants.

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Comparison of XRD Methods
Rotating Crystal Powder Diffraction Laue Method
Crystal Single Crystallites Single
Method Rotating Fixed Fixed
Variable ? Orientations ?
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• Analysing X-Ray Diffraction Pattern

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Diffraction Cone
Incident beam
Reflected beam
2?
Crystal Plane
2?
Reflected beam
This cone results into arcs on the opened up
X-ray film.
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Analysis of X-ray film
2? 0
2? 180
?1
?1
?1
?1
?2
?2
?3
?2
?2
?3
?3
?3
?R
d3
d3
d3
d3
d2
d2
d2
d2
d1
d1
d1
d1
D
4?3(rad) D / R
4?3(degree) 57.296 D / R
or
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Calculating Lattice Constant
4? (degree) 57.296 D / R
If radius of the camera R 57.296 mm then, ?
(degree) D (mm) / 4
For first order diffraction, d can be calculated
by
However, (hkl) assignment is not possible here.
Hence
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Lattice Constant Calculation Contd
? d 2hkl a 2 / N where N h 2 k 2
l 2
Possible values of N can be 1,2,3,4,5,6
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Prepare a Table
? sin ? sin2 ? 1 2 3
10.83 0.1879 0.0353 0.0353 0.0177 0.0118
15.39 0.2654 0.0704 0.0704 0.0352 0.0235
18.99 0.3254 0.1059 0.1059 0.0529 0.0353
sin2 ? / N
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• Thank you!