CRYSTAL SYMMETRIES

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CRYSTAL SYMMETRIES

• SHUBHA GOKHALE
• SCHOOL OF SCIENCES
• IGNOU
• May 25, 2007

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Symmetry is a way of life.
Some simple patterns
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Symmetries in Nature

• These are all 2-D patterns.

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We also have regular 3-D patterns
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What are crystals ?
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Crystals are a special class of solids which have
a regular arrangement of atoms in 3-D
Quartz crystal
Salt crystal
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Points to ponder upon

• Nature of crystals
• Classification of crystals on the basis of their
  structure
• Identification of crystal structure

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Lets begin with 2-D
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This is an example of a 2-D lattice
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Unit cells
LATTICE
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LATTICE

• Infinite repetition of identical structural units
  in space forms a LATTICE

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The base of any tree in this grove can be reached
by a vector defined as follows
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(No Transcript)
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3- D lattice
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(No Transcript)
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Basis comprises more than one element
( 1/4,1/4)
( 0, 0)
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A STRUCTURE is a combination of a LATTICE
and a BASIS
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When each lattice point is associated with
one or more atoms of the same or different
chemical substance in one, two or three
dimensions we get a CRYSTAL
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A lattice has various SYMMETRIES

• Translation

• Reflection

• Rotation

• Inversion

Let us begin with some 2-D lattices
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Displacement through a translation vector
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Reflection symmetry
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Square Lattice
Quarter circle rotation
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Rotation symmetry axis

If the angle of rotation can be written as
We have an n fold rotation axis
In case of square lattice we have 4-fold symmetry
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Possible rotational symmetries
cos ?n (PX/PQ) but PQ a ?PXSY a cos ?n

• PS ma 2a cos ?n ra
• m 2 cos ?n r
• cos ?n (m r)/2 N/2

P
S
a
a
N can be -2, -1, 0, 1, and 2 corresponding to
?n 180,120, 90, 60, and 360 (0)
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The possible symmetry axes are
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Inversion
(0,0)
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In Rotation, Reflection and Inversion at least
one point of the lattice remains FIXED
Inversion origin remains unchanged
Rotation all points on the rotation axis remain
unmoved
Reflection all points on the reflection plane
remain unmoved
POINT SYMMETRIES
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2-D lattice structures

• Oblique lattice
• Rectangular lattice
• Square lattice
• Hexagonal lattice

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OBLIQUE unit cell
a ? b ? ? 90
f
POINT symmetry 2-fold ROTATION
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OBLIQUE lattice
POINT symmetry 2-fold ROTATION
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RECTANGULAR unit cell
a ? b ? 90
f
POINT symmetry 2-fold ROTATION
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Rectangular Lattice
2 reflection planes, and inversion symmetry
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SQUARE unit cell
a b ? 90
f
POINT symmetry 4-fold ROTATION
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SQUARE Lattice
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SQUARE Lattice
4 reflection planes, and inversion symmetry
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HEXAGONAL unit cell
?
a b ? 120º
POINT symmetry 6-fold ROTATION
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HEXAGONAL unit cell
6 reflection planes, and inversion symmetry
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A 2-D lattice generated with any set of unit
vectors a, b and the angle ? between them can
have only one of the following four combinations
of point symmetries

• OBLIQUE 2-fold rotation
• RECTANGULAR 2-fold rotation reflection
• SQUARE 4-fold rotation reflection
• HEXAGONAL 6-fold rotation reflection

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By definition a lattice has TRANSLATIONAL
symmetry
Can we place additional points in each unit cell
and still maintain the LATTICE CHARACTER?
What is the POINT SYMMETRY of the resulting
lattice?
What is its TRANSLATIONAL SYMMETRY?
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Add lattice points at the centre of each unit
cell of a square lattice
POINT symmetry is still the SAME 4-fold
ROTATION, REFLECTION in 4 planes and
INVERSION
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Add lattice points at the centre of two opposite
sides of each unit cell
POINT symmetry is now DIFFERENT 2-fold
ROTATION, REFLECTION in 2-planes and
INVERSION.
This is a RECTANGULAR lattice.
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• To identify the type to which a lattice belongs,
  we look at its TRANSLATIONAL symmetry.
• To identify the system to which it belong we look
  at its POINT symmetries.

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To be continued 3-D systems
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Aknowledgment

• Thanks to
• Dr. Subhalakshmi Lamba
• for the help in preparing these slides

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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 parallelopiped.
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.
F type
FACE- CENTRED LATTICE Lattice points at the
corners of the parallelopiped 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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Aknowledgment

• Thanks to
• Dr. Shubhalakshmi Lamba
• for the help in preparing these slides