Space Lattices

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GEOMETRY OF CRYSTALS

• Space Lattices
• Crystal Structures
• Symmetry, Point Groups and Space Groups

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The language of crystallography is one
succinctness
Crystal Lattice Motif
Motif or basis an atom or a group of atoms
associated with each lattice point
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Space Lattice

• An array of points such that every point has
  identical surroundings
• In Euclidean space ? infinite array
• We can have 1D, 2D or 3D arrays (lattices)

or
Translationally periodic arrangement of points in
space is called a lattice
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A 2D lattice
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Lattice
Crystal
Translationally periodic arrangement of motifs
Translationally periodic arrangement of points
Crystal Lattice Motif
Lattice ? the underlying periodicity of the
crystal Basis ? atom or group of atoms
associated with each lattice points
Lattice ? how to repeat Motif ? what to repeat
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Lattice
Motif

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Crystal
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Cells
Instead of drawing the whole structure I can draw
a representative partand specify the repetition
pattern

• A cell is a finite representation of the
  infinite lattice
• A cell is a parallelogram (2D) or a
  parallelopiped (3D) with lattice points at
  their corners.
• If the lattice points are only at the corners,
  the cell is primitive.
• If there are lattice points in the cell other
  than the corners, the cell is nonprimitive.

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Nonprimitive cell
Primitivecell
Primitivecell
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Nonprimitive cell
Primitivecell
Primitivecell
Double
Triple
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Centred square lattice Simple/primitive square
lattice
Nonprimitive cell
Primitive cell
4- fold axes
Shortest lattice translation vector ? ½ 11
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Centred rectangular lattice
Maintains the symmetry of the lattice ? the
usual choice
Nonprimitive cell
Primitive cell
Lower symmetry than the lattice ? usually not
chosen
2- fold axes
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Centred rectangular lattice
Simple rectangular Crystal

Primitive cell
Not a cell
MOTIF
Shortest lattice translation vector ? 10
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Cells- 3D

• In order to define translations in 3-d space, we
  need 3 non-coplanar vectors
• Conventionally, the fundamental translation
  vector is taken from one lattice point to the
  next in the chosen direction
• With the help of these three vectors, it is
  possible to construct a parallelopiped called a
  CELL

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Different kinds of CELLS
Unit cell A unit cell is a spatial arrangement of
atoms which is tiled in three-dimensional space
to describe the crystal. Primitive unit cell For
each crystal structure there is a conventional
unit cell, usually chosen to make the resulting
lattice as symmetric as possible. However, the
conventional unit cell is not always the smallest
possible choice. A primitive unit cell of a
particular crystal structure is the smallest
possible unit cell one can construct such that,
when tiled, it completely fills
space. Wigner-Seitz cell A Wigner-Seitz cell is a
particular kind of primitive cell which has the
same symmetry as the lattice.
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SYMMETRY

• If an object is brought into self-coincidence
  after some operation it said to possess
  symmetry with respect to that operation.

SYMMETRY OPERATOR

• Given a general point a symmetry operator leaves
  a finite set of points in space
• A symmetry operator closes space onto itself

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Symmetry operators
Translation
Takes object to same form ? Proper
Type I
Rotation
Symmetries
Roto-reflection
Mirror
Type II
Inversion
Roto-inversion
Takes object to enantiomorphic form ? improper
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Minimum set of symmetry operators required
G ? Glide reflection
R ? Rotation
S ? Screw axis
?R ? Roto-inversion
Ones with built in translation
Ones acting at a point
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Rotation Axis
If an object come into self-coincidence through
smallest non-zero rotation angle of ? then it is
said to have an n-fold rotation axis where
?180?
2-fold rotation axis
n2
?120?
n3
3-fold rotation axis
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?90?
n4
4-fold rotation axis
?60?
n6
6-fold rotation axis
The rotations compatible with translational
symmetry are ? (1, 2, 3, 4, 6)
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Point group symmetry of Lattices ? 7 crystal
systems
Symmetries actingat a point
R ? ?R
32 point groups
Along with symmetrieshaving a translation
G S
230 space groups
Space group symmetry of Lattices ? 14 Bravais
lattices
R ?R ? rotations compatible with translational
symmetry (1, 2, 3, 4, 6)
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Previously
Crystal Lattice (Where to repeat) Motif
(What to repeat)


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Crystal Space group (how to repeat)
Asymmetric unit (Motif what to repeat)
Now

Glide reflection operator
a

Usually asymmetric units are regions of space
within the unit cell- which contain atoms
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Progressive lowering of symmetry in an 1D
lattice ? illustration using the frieze groups
Consider a 1D lattice with lattice parameter a
Asymmetric Unit
Unit cell
a
mmm
Three mirror planes
The intersection points of the mirror planesgive
rise to redundant inversion centres
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Decoration of the lattice with a motif ? may
reduce the symmetry of the crystal
1
mmm
Decoration with a sufficiently symmetric motif
does not reduce the symmetry of the lattice
2
mm
Loss of 1 mirror plane
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3
mg
Presence of 1 mirror plane and 1 glide reflection
plane, with a redundant inversion centrethe
translational symmetry has been reduced to 2a
ii
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2 inversion centres
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5
m
1 mirror plane
g
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1 glide reflection translational symmetry of 2a
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No symmetry except translation
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Effect of the decoration ? a 2D example
Two kinds of decoration are shown ? (i) for an
isolated object, (ii) an object which can be an
unit cell.
4mm
Redundant inversion centre
Can be a unit cell for a 2D crystal
4mm
Decoration retaining the symmetry
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m
mm
m
30
4
No symmetry
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Lattices have the highest symmetry? Crystals
based on the lattice can have lower symmetry
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Positioning a object with respect to the symmetry
elements
mmm
Three mirror planes
The intersection points of the mirror planesgive
rise to redundant inversion centres
Left handed object
Right handed object
Object with bilateral symmetry
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Positioning a object with respect to the symmetry
elements
General site ? 8 identiti-points
On mirror plane (m) ? 4 identiti-points
On mirror plane (m) ? 4 identiti-points
Site symmetry 4mm ? 1 identiti-point
Note this is for a point group and not for a
lattice ? the black lines are not unit cells
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Positioning of a motif w.r.t to the symmetry
elements of a lattice ? Wyckoff positions
A 2D lattice with symmetry elements
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f
g
e
b
d
a
c
Number of Identi-points
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f
g
e
b
d
a
c
f
Exclude thesepoints
Exclude thesepoints
d
e
Exclude thesepoints
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Bravais Space Lattices ? some other view points

• Conventionally, the finite representation of
  space lattices is done using unit cells which
  show maximum possible symmetries with the
  smallest size.

• Considering
• Maximum Symmetry, and
• Minimum Size
• Bravais concluded that there are only 14 possible
  Space Lattices (or Unit Cells to represent them).
  
• These belong to 7 Crystal systems

Or ? the technical definition ?
There are 14 Bravais Lattices which are the space
group symmetries of lattices
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Bravais Lattice A lattice is a set of points
constructed by translating a single point in
discrete steps by a set of basis vectors. In
three dimensions, there are 14 unique Bravais
lattices (distinct from one another in that they
have different space groups) in three dimensions.
All crystalline materials recognized till now fit
in one of these arrangements. or In geometry and
crystallography, a Bravais lattice is an infinite
set of points generated by a set of discrete
translation operations. A Bravais lattice looks
exactly the same no matter from which point one
views it.
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Arrangement of lattice points in the unit cell
No. of Lattice points / cell
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14 Bravais lattices divided into seven crystal
systems

• Crystal system Bravais lattices
• Cubic P I F
• Tetragonal P I
• Orthorhombic P I F C
• Hexagonal P
• Trigonal P
• Monoclinic P C
• Triclinic P

Courtesy Dr. Rajesh Prasad
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14 Bravais lattices divided into seven crystal
systems

• Crystal system Bravais lattices
• Cubic P I F C
• Tetragonal P I
• Orthorhombic P I F C
• Hexagonal P
• Trigonal P
• Monoclinic P C
• Triclinic P

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Cubic F ? Tetragonal I
The symmetry of the unit cell is lower than that
of the crystal
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14 Bravais lattices divided into seven crystal
systems

• Crystal system Bravais lattices
• Cubic P I F C
• Tetragonal P I F
• Orthorhombic P I F C
• Hexagonal P
• Trigonal P
• Monoclinic P C
• Triclinic P

x
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FCT BCT
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Crystal system The crystal system is the point
group of the lattice (the set of rotation and
reflection symmetries which leave a lattice point
fixed), not including the positions of the atoms
in the unit cell. There are seven unique crystal
systems.
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Concept of symmetry and choice of axes
(a,b)
The centre of symmetry of the object does not
coincide with the origin
Polar coordinates (?, ?)
The type of coordinate system chosen is
not according to the symmetryof the object
Mirror
Our choice of coordinate axis does not alter the
symmetry of the object (or the lattice)
Centre of Inversion
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THE 7 CRYSTAL SYSTEMS
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N is the number of point groups for a crystal
system
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• Cubic Crystals
• a b c? ? ? 90º
• Simple Cubic (P)
• Body Centred Cubic (I) BCC
• Face Centred Cubic (F) - FCC

PyriteCube
1
1
GarnetDodecahedron
Fluorite Octahedron
1
1 http//www.yourgemologist.com/crystalsystems.h
tml
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• Tetragonal Crystals
• a b ? c ? ? ? 90º
• Simple Tetragonal
• Body Centred Tetragonal

Zircon
1
1
1
1 http//www.yourgemologist.com/crystalsystems.h
tml
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• Orthorhombic Crystals
• a ? b ? c ? ? ? 90º
• Simple Orthorhombic
• Body Centred Orthorhombic
• Face Centred Orthorhombic
• End Centred Orthorhombic

1
Topaz
1
1 http//www.yourgemologist.com/crystalsystems.h
tml
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• Hexagonal Crystals
• a b ? c ? ? 90º ? 120º
• Simple Hexagonal

Corundum
1
1 http//www.yourgemologist.com/crystalsystems.h
tml
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• 5. Rhombohedral Crystals
• a b c ? ? ? ? 90º
• Rhombohedral (simple)

1
1
Tourmaline
1 http//www.yourgemologist.com/crystalsystems.h
tml
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• Monoclinic Crystals
• a ? b ? c ? ? 90º ? ?
• Simple Monoclinic
• End Centred (base centered) Monoclinic (A/C)

1
Kunzite
1 http//www.yourgemologist.com/crystalsystems.h
tml
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• 7. Triclinic Crystals
• a ? b ? c ? ? ? ? ?
• Simple Triclinic

1
Amazonite
1 http//www.yourgemologist.com/crystalsystems.h
tml
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Concept of symmetry and choice of axes
(a,b)
The centre of symmetry of the object does not
coincide with the origin
Polar coordinates (?, ?)
The type of coordinate system chosen is
not according to the symmetryof the object
Mirror
Our choice of coordinate axis does not alter the
symmetry of the object (or the lattice)
Centre of Inversion
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Alternate choice of unit cells for Orthorhombic
lattices

• Alternate choice of unit cell for C(C-centred
  orthorhombic) case.
• The new (orange) unit cell is a rhombic prism
  with (a b ? c, ? ? 90o, ? ? 90o, ? ?
  120o)
• Both the cells have the same symmetry ? (2/m 2/m
  2/m)
• In some sense this is the true Ortho-rhombic
  cell

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z 0 z 1
z ½
Note All spheres represent lattice points. They
are coloured differently but are the same
? A consistent alternate set of axis can be
chosen for Orthorhombic lattices
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Intuitively one might feel that the orthogonal
cell has a higher symmetry? is there some reason
for this?

• The 2x and 2y axes move lattice points out the
  plane of the sheet in a semi-circle to other
  points of the lattice (without introducing any
  new points)
• The 2d axis introduces new points which are not
  lattice points of the original lattice
• The motion of the lattice points under the effect
  of the artificially introduced 2-folds is shown
  as dashed lines (---)

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Progressive lowering of symmetry amongst the 7
crystal systems
Cubic48
Increasing symmetry
Hexagonal24
Tetragonal16
Trigonal12
Orthorhombic8
Monoclinic4
Triclinic2
Superscript to the crystal system is the order of
the lattice point group
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Progressive relaxation of the constraints on the
lattice parameters amongst the 7 crystal systems
Cubic (p 2, c 1, t 1)a b c? ? ?
90º
Increasing number t
Tetragonal (p 3, c 1 , t 2) a b ? c?
? ? 90º
Hexagonal (p 4, c 2 , t 2)a b ? c? ?
90º, ? 120º
Trigonal (p 2, c 0 , t 2)a b c? ?
? ? 90º
Orthorhombic2 (p 4, c 1 , t 3) a b ? c?
? 90º, ? ? 90º
Orthorhombic1 (p 4, c 1 , t 3) a ? b ? c?
? ? 90º

• p number of independent parameters
• c number of constraints (positive ? )
• t terseness (p ? c) (is a measure of
  the expenditure on the parameters

Monoclinic (p 5, c 1 , t 4)a ? b ? c? ?
90º, ? ? 90º
Triclinic (p 6, c 0 , t 6) a ? b ? c? ? ?
? ? ? 90º
Orthorhombic1 and Orthorhombic2 refer to the two
types of cells
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Minimum symmetry requirement for the 7 crystal
systems
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REGULAR SOLIDS IN VARIOUS DIMENSIONS  
POINT
LINE SEGMENT
TRIANGLE 3
SQUARE 4
PENTAGON 5
HEXAGON 6
TETRAHEDRON 3, 3
OCTAHEDRON 3, 4
DODECAHEDRON 5, 3
ICOSAHEDRON 3, 5
CUBE 4, 3
SIMPLEX 3, 3, 3
16-CELL 3, 3, 4
120-CELL 5, 3, 3
DRP
600-CELL 3, 3, 5
HYPERCUBE 4, 3, 3
24-CELL 3, 4, 3
CRN
REGULAR SIMPLEX 3, 3, 3, 3
CROSS POLYTOPE 3, 3, 3, 4
MEASURE POLYTOPE 4, 3, 3, 3
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