Crystal Geometry

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???? ????? ???? ???? (?????? ???? ) ????? ????
??? ?? ???? ? ?? ???? ????? ??? ????
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??????
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???? ???? ?? ???? ?? ????
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???? ???? ?? ???? ?? ???? NaCl
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?????? ?????? ???? ????(??? ?????)
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???? ????? ?? ???? ???? ??? ????
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- or if you dont start from an atom
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??? ???? ???? ????
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??? ???? ????? ???? ???? ????
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??? ???? ????? ?? ????
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?? ????? ?? ??? ????? ?? ?? ????? ????? ???? ???
?????
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1-CUBIC
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a- Simple Cubic (SC)
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Face Centered Cubic (FCC)

• 4 ??? ?? ???? ????? ???? ????
• (Cu,Ni,Pb..etc) ?????? fcc. ?????

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3 - Face Centered Cubic
Atoms are all same.
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2 - HEXAGONAL SYSTEM

• ?? ??? ?? ???? ????? ???? ????.

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2 - HEXAGONAL SYSTEM
Atoms are all same.
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3 - TRICLINIC 4 - MONOCLINIC CRYSTAL SYSTEM

• ??? ?????? ?????? ????? ????? ?? ??????

Monoclinic (Simple) a g 90o, ß ¹ 90o a ¹ b
¹c
Monoclinic (Base Centered) a g 90o, ß ¹ 90o
a ¹ b ¹ c,
Triclinic (Simple) a ¹ ß ¹ g ¹ 90 oa ¹ b ¹ c
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5 - ORTHORHOMBIC SYSTEM
Orthorhombic (FC) a ß g 90o a ¹ b ¹ c
Orthorhombic (Base-centred)a ß g 90o a ¹
b ¹ c
Orthorhombic (BC) a ß g 90o a ¹ b ¹ c
Orthorhombic (Simple) a ß g 90o a ¹ b ¹ c
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6 TETRAGONAL SYSTEM
Tetragonal (BC) a ß g 90o a b ¹ c
Tetragonal (P) a ß g 90o a b ¹ c
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7 - Rhombohedral (R) or Trigonal
Rhombohedral (R) or Trigonal (S) a b c, a
ß g ¹ 90o
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Miller Indices

• ????? ??? ???? ???????? ????? ?? ??? ????? ????
  ?? ?? ??????? ???? ?? ????
• ??? ????? ?? ?? ???? ?? ???? ?? ???? ?? ????? ??
  ?? ????? ?? ????? ????? ?? ???? ?????
• ? ???? ????????? ????? ?? ??? ?? ????? ????
  ?????? ??? ???? ???? ???? ???? ?? ????.

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Example-1
Axis X Y Z
Intercept points 1 8 8
Reciprocals 1/1 1/ 8 1/ 8
Smallest Ratio 1 0 0
Miller Indices (100) Miller Indices (100) Miller Indices (100) Miller Indices (100)
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Example-2
Axis X Y Z
Intercept points 1 1 8
Reciprocals 1/1 1/ 1 1/ 8
Smallest Ratio 1 1 0
Miller Indices (110) Miller Indices (110) Miller Indices (110) Miller Indices (110)
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Example-3
Axis X Y Z
Intercept points 1 1 1
Reciprocals 1/1 1/ 1 1/ 1
Smallest Ratio 1 1 1
Miller Indices (111) Miller Indices (111) Miller Indices (111) Miller Indices (111)
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Example-4
Axis X Y Z
Intercept points 1/2 1 8
Reciprocals 1/(½) 1/ 1 1/ 8
Smallest Ratio 2 1 0
Miller Indices (210) Miller Indices (210) Miller Indices (210) Miller Indices (210)
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Example-5
Axis a b c
Intercept points 1 8 ½
Reciprocals 1/1 1/ 8 1/(½)
Smallest Ratio 1 0 2
Miller Indices (102) Miller Indices (102) Miller Indices (102) Miller Indices (102)
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Example-6
Axis a b c
Intercept points -1 8 ½
Reciprocals 1/-1 1/ 8 1/(½)
Smallest Ratio -1 0 2
Miller Indices (102) Miller Indices (102) Miller Indices (102) Miller Indices (102)
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Miller Indices
Indices of the plane (Miller) (2,3,3)
Indices of the direction 2,3,3
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????? ??? ???? ? ?????? ????? ???? ?? ????
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????? ??? ???? ? ?????? ????? ???? ?? ????
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?????? ????? ? ????? ???? ???? ?? ???? ????? ???
???? ?????? ?????.
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Example-7
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Indices of a Family or Form
??? hkl ???? ???? ????? ??? ???? ????? ??
????? (hkl) ?? ???? ?? ??? ?? ?????? ???? ??
?????? ?? ??? ?? ????
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3D 14 BRAVAIS LATTICES AND THE SEVEN CRYSTAL
SYSTEM

• ???? 14 ???? ????? ???? ???? ?? ???? ?? ???? ??
  ?? ??????.
• ??? 14 ???? ??? ?? ??? ????? ????? ????? ???
  ??????? ?? ????.
• Cubic Crystal System (SC, BCC,FCC)
• Hexagonal Crystal System (S)
• Triclinic Crystal System (S)
• Monoclinic Crystal System (S, Base-C)
• Orthorhombic Crystal System (S, Base-C, BC, FC)
• Tetragonal Crystal System (S, BC)
• Trigonal (Rhombohedral) Crystal System (S)

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3Hexagonal Close-Packed Str.
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Hexagonal Close-packed Structure
ab a120, c1.633a,  basis (0,0,0) (2/3a
,1/3a,1/2c)
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Packing
Close pack
Sequence AAAA - simple cubic

• Sequence ABABAB..
• hexagonal close pack

Sequence ABAB - body centered cubic
Sequence ABCABCAB.. ??
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First Brillouin Zone Two Dimensional Oblique
Lattice
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First Four Brillouin Zones Square Lattice
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All Brillouin Zones Square Lattice
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Primitive Lattice Vectors BCC Lattice
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First Brillouin Zone BCC
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Primitive Lattice Vectors FCC
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Brillouin Zones FCC
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First Brillouin Zone BCC
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First Brillouin Zone FCC
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X-ray Diffraction
Typical interatomic distances in solid are of the
order of an angstrom. Thus the typical wavelength
of an electromagnetic probe of such distances
Must be of the order of an angstrom.
Upon substituting this value for the wavelength
into the energy equation, We find that E is of
the order of 12 thousand eV, which is a typical
X-ray Energy. Thus X-ray diffraction of crystals
is a standard probe.
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Wavelength vs particle energy
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Bragg Diffraction Braggs Law
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Braggs Law
The integer n is known as the order of the
corresponding Reflection. The composition of
the basis determines the relative Intensity of
the various orders of diffraction.
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Many sets of lattice planes produce Bragg
diffraction
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BRAGGs EQUATION
Deviation 2?
Ray 1
Ray 2
?
?
?
d
?
?
dSin?

• The path difference between ray 1 and ray 2 2d
  Sin?
• For constructive interference n? 2d Sin?

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Bragg Spectrometer
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Bragg Peaks
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• A beam of X-rays directed at a crystal interacts
  with the electrons of the atoms in the crystal
• The electrons oscillate under the influence of
  the incoming X-Rays and become secondary
  sources of EM radiation
• The secondary radiation is in all directions
• The waves emitted by the electrons have the same
  frequency as the incoming X-rays ? coherent
• The emission will undergo constructive or
  destructive interference with waves scattered
  from other atoms

Secondary emission
Incoming X-rays
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Sets Electron cloud into oscillation
Sets nucleus (with protons) into
oscillation Small effect ? neglected
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Oscillating charge re-radiates ? In phase with
the incoming x-rays
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von Laue Formulation of X-Ray Diffraction
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Condition for Constructive Interference
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Bragg Scattering
K
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The Laue Condition
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Ewald Construction
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Crystal structure determination
Many ?s (orientations) Powder specimen
POWDER METHOD
Monochromatic X-rays
Single ?
LAUETECHNIQUE
Panchromatic X-rays
ROTATINGCRYSTALMETHOD
? Varied by rotation
Monochromatic X-rays
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THE POWDER METHOD
Cone of diffracted rays
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POWDER METHOD
Diffraction cones and the Debye-Scherrer geometry
Different cones for different reflections
Film may be replaced with detector
http//www.matter.org.uk/diffraction/x-ray/powder_
method.htm
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Schematic X-Ray Diffractometer
Detector
X-Ray Source
Powdered sample
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Sample XRD Pattern
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strong intensity prominent crystal plane
weak intensity subordinate crystal plane
background radiation
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Determine D-Spacing from XRD patterns

• Braggs Law
• n? 2dsin?
• n reflection order (1,2,3,4,etc)
• ? radiation wavelength (1.54 angstroms)
• d spacing between planes of atoms (angstroms)
• ? angle of incidence (degrees)

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strong intensity prominent crystal plane
n? 2dsin? (1)(1.54) 2dsin(15.5 degrees) 1.54
2d(0.267) d 2.88 angstroms
background radiation
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d-spacing Intensity
2.88 100
2.18 46
1.81 31
1.94 25
2.10 20
1.75 15
2.33 10
2.01 10
1.66 5
1.71 5
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The Bragg equation may be rearranged (if n1)
from
to
If the value of 1/(dh,k,l)2 in the cubic system
equation above is inserted into this form of the
Bragg equation you have
Since in any specific case a and l are constant
and if l2/4a2 A
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Insert the values into a table and compute sin?
and sin2?. Since the lowest value of sin2? is 3A
and the next is 4A the first Entry in the Calc.
sin2? column is (0.10854/3)4 etc.
d/Å Sin? Sin2? Calc. Sin2? (h, k, I)
2.338 0.32945 0.10854 (1,1,1)
2.024 0.38056 0.14482 0.14472 (2,0,0)
1.431 0.53826 0.28972 0.28944 (2,2,0)
1.221 0.63084 0.39795 0.39798 (3,1,1)
1.169 0.65890 0.43414 0.43416 (2,2,2)
1.0124 0.76082 0.57884 0.57888 (4,0,0)
0.9289 0.82921 0.68758 0.68742 (3,3,1)
0.9055 0.85063 0.72358 0.72360 (4,2,0)
The reflections have now been indexed.