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P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
http://folk.uio.no/ravi/CMP2013
Prof.P. Ravindran, Department of Physics,
Central University of Tamil Nadu, India
Crystal Structures of Solids
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P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
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P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
Table of metals, metalloids, and nonmetals3
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
Solids, Liquids and Gases
1A 8A
H 2A 3A 4A 5A 6A 7A He
Li Be B C N O F Ne
Na Mg 3B 4B 5B 6B 7B –– 8B –– 1B 2B Al Si P S Cl Ar
K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr
Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe
Cs Ba La Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At Rn
Fr Ra Ac Rf Db Sg Bh Hs Mt Ds
S L G Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu
Th Pa U Np Pu Am Cm Bk Cf Es Fm Md No Lr
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P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
The Solid State1. Classification of Solid Structures
a. Crystalline Solids = regular arrangement of components in 3
dimensions
b. Amorphous Solids = disordered arrangement of components
2. Crystal Structure Basics
a. Crystal = a piece of a crystalline solid
b. Lattice = 3-dimensional system of designating where components are
c. Unit cell = smallest repeating unit of the lattice
d. Examples: simple cubic, body-centered cubic, face-centered cubic
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P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
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Early Crystallography
René-Just Haüy (1781): cleavage of calcite
Common shape to all shards: rhombohedral
How to model this mathematically?
What is the maximum number of distinguishable shapes
that will fill three space?
Mathematically proved that there are only 7 distinct
space-filling volume elements
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
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Christian Huygens - 1690
Studying calcite crystals made
drawings of atomic packing
and bulk shape.
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids8
Beryl
Be3Al2(SiO3)6
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
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Anisotropy
The physical properties of single crystals of some
substances depend on the crystallographic direction in
which measurements are taken.
For example, the elastic modulus, electrical conductivity,
and the index of refraction may have different values in
the [100] and [111] directions.
The directionality of the properties is termed anisotropy
and is associated with the atomic spacing.
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
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Isotropic
If measured properties are independent of the direction
of measurement then they are isotropic.
For many polycrystalline materials, the crystallographic
orientations of the individual grains are totally random.
So, though, a specific grain may be anisotropic, when the
specimen is composed of many grains, the aggregate
behavior may be isotropic.
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
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Single Crystals
For a crystalline solid, when the periodic and
repeated arrangement of atoms extends
throughout without interruption, the result is a
single crystal.
The crystal lattice of the entire sample is
continuous and unbroken with no grain
boundaries.
For a variety of reasons, including the distorting
effects of impurities, crystallographic defects
and dislocations, single crystals of meaningful
size are exceedingly rare in nature, and difficult
to produce in the laboratory under controlled
conditions.
Huge KDP (monopotassium phosphate)
crystal grown from a seed crystal in a
supersaturated aqueous solution at
LLNL. Below, silicon boule.
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
Single Crystal
The periodic and repeated arrangements of atoms is perfect or extends
throughout the entirety of the specimen without interruption.
All unit cells interlock in the same way and have the same orientation.
Single crystals exist in nature, but they may also produced artificially. They are
ordinarily difficult to grow, because the environment must be carefully
controlled.
Single crystals are needed for modern technologies today.
Electronic micro-chips uses single crystals of silicon and other semiconductors.
A garnet single crystal found in
Tongbei, Fujian Province, China.
If the extremities of a single crystal
are permitted to grow without any
external constraint, the crystal will
assume its geometric shape, with
flat surfaces as shown in the figure.
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P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
Polycrystalline Materials
Composed of a collection of many small crystals or grains.
Stages in the solidification
of a polycrystalline
material:
a. Crystallite Nuclei
b. Growth of the
Crystallites
c. Formation of grains
d. Microscopic view
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P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
Anisotropy
Physical properties of single crystals of some substances depend
on the crystallographic direction in which measurements are
made.
This directionality of properties is termed anisotropy, and it is
associated with the variance of atomic or ionic spacing with
crystallographic direction.
Substances in which measured properties are independent of
the direction of measurement are isotropic.
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P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
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P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
Structures
The properties of some materials are
directly related to their crystal structures.
Significant property differences exist
between crystalline and noncrystalline
materials having the same composition.
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P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids 17
• Some engineering applications require single crystals:
• Properties of crystalline materials
often related to crystal structure.
-- Ex: Quartz fractures more
easily
along some crystal planes than
others.
-- diamond
Single crystals for abrasives
-- turbine blades
Crystals as Building Blocks
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids18
Atomic Arrangement
Minerals must have a highly
ordered atomic arrangement
The crystal structure of
quartz is an example
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
Examples of silicate mineralsolivine
epidote
beryl
augite
hornblende
muscovite
quartz
Mineral pictures
from: mindat.org
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P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
Pink (Rose) : due to traces of iron, manganese or titanium.
Amethyst : May be manganese but some believe it could be organic,
iron or even aluminum.
Citrine : iron
Aventurine : inclusion of green mica (fushite)
Tiger's eye : inclusion of fiber of silicified crocidolite (variety of asbestos)
Prasiolite : Iron or copper
Milk quartz : gas and liquid inclusions
Smoky : Radioactivity on quartz containing aluminium
Blue : pressure.
Chalcedony is a variety of quartz with micro-crystals. Agate is a multicolor
variety of chalcedony and onyx is a variety of agate with parallel strips of
various nuances of black.
Quartz Varieties20
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids21
Quartz Crystals
The external
appearance of the
crystal may reflect its
internal symmetry
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids22
Quartz Blob
Or the external
appearance may show
little or nothing of the
internal structure
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids23
Building Blocks
A cube may be used to
build a number of
forms
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids24
Fluorite
Fluorite may appear as
octahedron (upper photo)
Fluorite may appear as a
cube (lower photo), in this
case modified by
dodecahedral crystal faces
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
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Crystal Growth
Ways in which a crystal can grow:
Dehydration of a solution
Growth from the molten state (magma or lava)
Direct growth from the vapor state
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
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Types of Solids
Single crsytal, polycrystalline, and amorphous, are the
three general types of solids.
Each type is characterized by the size of ordered region
within the material.
An ordered region is a spatial volume in which atoms or
molecules have a regular geometric arrangement or
periodicity.
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
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Crystalline Solid
Single Crystal
Single Pyrite
Crystal
Amorphous
Solid
Single crystal has an atomic structure that repeats
periodically across its whole volume. Even at infinite length
scales, each atom is related to every other equivalent atom in
the structure by translational symmetry
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
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Ionic Crystals
Examples include sodium chloride, cupric sulfate.
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
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Some Factors Affecting Crystalline Structure
Size of atoms or ions involved
Stoichiometry of salt
Materials involved
– Some substances do not form crystalline solids
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
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Crystals
The periodic array of atoms, ions, or molecules that form the solids is called Crystal Structure
Crystal Structure = Space (Crystal) Lattice + Basis
– Space (Crystal) Lattice is a regular periodicarrangement of points in space, and is purely mathematical abstraction
– Crystal Structure is formed by “putting” the identical atoms (group of atoms) in the points of the space lattice
– This group of atoms is the Basis
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
The definition of crystals are based on symmetry and not on
the geometry of the unit cell
Our choice of unit cell cannot alter the crystal system a
crystal belongs to
Crystals based on a particular lattice can have symmetry
equal to or lower than that of the lattice
When all symmetry (including translation) is lost the
construct is called amorphous
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P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
Ideal crystals may have perfect positional and orientational
order with respect to geometrical entities and physical
properties
In (defining) real crystals some of these strict requirements
may be relaxed:
the order considered may be only with respect to the
geometrical entity
the positional order may be in the average sense
the orientational order may be in the average sense
In addition real crystals:
are finite
may contain other defects .
Ideal versus Real crystals32
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
When we look around us many of the organic materials are non-crystalline
But, many of the common ‘inorganic’ materials are ‘usually*’ crystalline:
◘ Metals: Cu, Zn, Fe, Cu-Zn alloys
◘ Semiconductors: Si, Ge, GaAs
◘Ceramics: Alumina (Al2O3), Zirconia (Zr O2), SiC, SrTiO3
Also, the usual form of crystalline materials (say a Cu wire or a piece of
alumina) is polycrystalline and special care has to be taken to produce
single crystals
Polymeric materials are usually not ‘fully’ crystalline
The crystal structure directly influences the properties of the material(as we have seen in the Introduction chapter many additional factors come in)
Why study crystal structures?
Gives a terse (concise) representation of a large assemblage of species
Gives the ‘first view’ towards understanding of the properties of the crystal
Why study crystallography?
* Many of the materials which are usually crystalline can also be obtained in an amorphous form
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P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
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Crystallography
What is crystallography?
The branch of science that deals with the geometric
description of crystals and their internal arrangement.
Crystallography is essential for solid state physics
Symmetry of a crystal can have a profound influence on its
properties.
Any crystal structure should be specified completely, concisely and
unambiguously.
Structures should be classified into different types according to the
symmetries they possess.
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
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A basic knowledge of crystallography is essential for solidstate physicists;
– to specify any crystal structure and
– to classify the solids into different types according tothe symmetries they possess.
Symmetry of a crystal can have a profound influence on itsproperties.
We will concern in this course with solids with simplestructures.
Elementary Crystallography
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
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Crystal Lattice
What is crystal (space) lattice?
In crystallography, only the geometrical properties of the
crystal are of interest, therefore one replaces each atom by a
geometrical point located at the equilibrium position of that
atom.
Platinum Platinum surface Crystal lattice and
structure of Platinum(scanning tunneling microscope)
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
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An infinite array of
points in space,
Each point has identical
surroundings to all
others.
Arrays are arranged
exactly in a periodic
manner.
Crystal Lattice
α
a
b
CB ED
O A
y
x
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
The “unit cell” is the basic
repeating unit of the
arrangement of atoms, ions or
molecules in a crystalline solid.
The “lattice” refers to the 3-D
array of particles in a
crystalline solid. One type of
atom occupies a “lattice point”
in the array.
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P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
Examples of Unit Cells39
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
Crystal Systems
7 crystal systems
14 crystal lattices
Unit cell: smallest repetitive volume which contains the
complete lattice pattern of a crystal.
a, b, and c are the lattice constants
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P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
Unit Cell Concept
The unit cell is the smallest structural unit or building block that uniquely can describe the crystal structure. Repetition of the unit cell generates the entire crystal. By simple translation, it defines a lattice .
Lattice: The periodic arrangement of atoms in a crystal.
Lattice Parameter : Repeat
distance in the unit cell, one
for in each dimension
b
a
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P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
The Unit Cell Concept
The simplest repeating unit in a crystal is called a unit
cell.
Opposite faces of a unit cell are parallel.
The edge of the unit cell connects equivalent points.
Not unique. There can be several unit cells of a crystal.
The smallest possible unit cell is called primitive unit
cell of a particular crystal structure.
A primitive unit cell whose symmetry matches the
lattice symmetry is called Wigner-Seitz cell.
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P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
Each unit cell is defined in terms of lattice points.
Lattice point not necessarily at an atomic site.
For each crystal structure, a conventional unit cell,
is chosen to make the lattice as symmetric as
possible. However, the conventional unit cell is not
always the primitive unit cell.
A crystal's structure and symmetry play a role in
determining many of its properties, such as cleavage
(tendency to split along certain planes with smooth
surfaces), electronic band structure and optical
properties.
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P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
Crystal Systems
Units cells and lattices in 3-D:
– When translated in each lattice parameter direction, MUST fill
3-D space such that no gaps, empty spaces left.
Lattice Parameter : Repeat
distance in the unit cell, one
for in each dimension
a
bc
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P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
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The Importance of the Unit Cell
One can analyze the crystal as a whole by investigating a
representative volume.
Ex: from unit cell we can
– Find the distances between nearest atoms for
calculations of the forces holding the lattice together
– Look at the fraction of the unit cell volume filled by
atoms and relate the density of solid to the atomic
arrangement
– The properties of the periodic Xtal lattice determine the
allowed energies of electrons that participate in the
conduction process.
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
Unit cell
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P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
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The unit cell and, consequently, theentire lattice, is uniquely determinedby the six lattice constants: a, b, c,α, β and γ.
Only 1/8 of each lattice point in aunit cell can actually be assigned tothat cell.
Each unit cell in the figure can beassociated with 8 x 1/8 = 1 latticepoint.
Unit Cell
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
Unit cell: Simplest portion of the structure which is repeated
and shows its full symmetry.
Basis vectors a and b defines relationship between a unit cell
and (Bravais) lattice points of a crystal.
Equivalent points of the lattice is defined by translation vector.
r = ha + kb where h and k are integers. This constructs the entire
lattice.
a
b
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P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
By repeated duplication, a unit cell should
reproduce the whole crystal.
A Bravais lattice (unit cells) - a set of points
constructed by translating a single point in
discrete steps by a set of basis vectors.
In 3-D, there are 14 unique Bravais lattices.
All crystalline materials fit in one of these
arrangements.
In 3-D, the translation vector is
r = ha + kb + lc
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P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
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Unit Cell in 2D
The smallest component of the crystal (group of atoms,ions or molecules), which when stacked together withpure translational repetition reproduces the wholecrystal.
S
a
b
S
S
S
S
S
S
S
S
S
S
S
S
S
S
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
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Unit Cell in 2D
The smallest component of the crystal (group of atoms,ions or molecules), which when stacked together withpure translational repetition reproduces the wholecrystal.
S
S
The choice of
unit cell
is not unique.
a
Sb
S
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids52
2D Unit Cell example -(NaCl)
We define lattice points ; these are points with identical
environments
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids53
Choice of origin is arbitrary - lattice points need not be atoms - but unit cell size should always be the same.
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids54
This is also a unit cell -
it doesn’t matter if you start from Na or Cl
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids55
- or if you don’t start from an atom
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids56
This is NOT a unit cell even though they are all the
same - empty space is not allowed!
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids57
In 2D, this IS a unit cell
In 3D, it is NOT
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
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Unit Cell in 3D
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
Unit Cell in 3D
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P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids 60
Crystal Structure
Crystal structure can be obtained by attaching atoms,groups of atoms or molecules which are called basis(motif) to the lattice sides of the lattice point.
Crystal Structure = Crystal Lattice + Basis
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
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Lattice Sites in Cubic Unit Cell
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
A two-dimensional Bravais lattice with
different choices for the basis
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P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
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Crystal structure
Don't mix up atoms withlattice points
Lattice points areinfinitesimal points in space
Lattice points do notnecessarily lie at the centreof atoms
Crystal Structure = Crystal Lattice + Basis
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
Crystal Lattice
Bravais Lattice (BL)
All atoms are of the same kind All lattice points are equivalent
Non-Bravais Lattice (non-
BL)
Atoms can be of different kind Some lattice points are not
equivalentA combination of two or more BL
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P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
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Types Of Crystal Lattices
1) Bravais lattice is an infinite array of discrete points with an
arrangement and orientation that appears exactly the same,
from whichever of the points the array is viewed. Lattice is
invariant under a translation.
Nb film
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
66
Types Of Crystal Lattices
The red side has a neighbour to itsimmediate left, the blue one instead hasa neighbour to its right.
Red (and blue) sides are equivalent andhave the same appearance
Red and blue sides are not equivalent.Same appearance can be obtainedrotating blue side 180º.
2) Non-Bravais Lattice
Not only the arrangement but also the orientation must
appear exactly the same from every point in a bravais lattice.
Honeycomb
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
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Translational Lattice Vectors – 2D
A space lattice is a set of points such thata translation from any point in thelattice by a vector;
Rn = n1 a + n2 b
locates an exactly equivalent point, i.e. apoint with the same environment as P .This is translational symmetry. Thevectors a, b are known as lattice vectorsand (n1, n2) is a pair of integers whosevalues depend on the lattice point.
P
Point D(n1, n2) = (0,2)
Point F (n1, n2) = (0,-1)
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
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The two vectors a and bform a set of latticevectors for the lattice.
The choice of latticevectors is not unique.Thus one could equallywell take the vectors a andb’ as a lattice vectors.
Lattice Vectors – 2D
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
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Five Bravais Lattices in 2D
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
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Lattice Vectors – 3D
An ideal three dimensional crystal is described by 3fundamental translation vectors a, b and c. If there is alattice point represented by the position vector r, there isthen also a lattice point represented by the position vectorwhere u, v and w are arbitrary integers.
r’ = r + u a + v b + w c (1)
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
Gold (Why one naturally get Gold in pure form?
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P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
Hydrated Crystals - Water
molecules become
chemically bonded to ions in
the crystal.
Anhydrous Crystals Crystal
without water
CuSO4 • 5 H2O
a hydrated crystal 72
Hydrated and Unhydrated Crystals
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
Hydrated Crystals
Ex:
CuSO4
• 5H2O + heat CuSO
4+ 5 H
2O
(Blue) (White)
Hydrate
Anhydrous
Hydrate
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P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
Ionic solids
Group 1A (alkali metals) contains lithium (Li), sodium (Na), potassium(K),..and these combine easily with group 7A (halogens) of fluorine (F),chlorine (Cl), bromine (Br),.. and produce ionic solids of NaCl, KCl, KBr,etc.
Rare (noble) gases
Group 8A elements of noble gases of helium(He), neon (Ne), argon(Ar),… have a full complement of valence electrons and so do not combineeasily with other elements.
Elemental semiconductors
Silicon(Si) and germanium (Ge) belong to group 4A.
Compound semiconductors
1) III-V compound s/c’s; GaP, InAs, AlGaAs (group 3A-5A)
2) II-VI compound s/c’s; ZnS, CdS, etc. (group 2B-6A)
The periodic table74
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
Molecular Solids
H2O, S8, P4
Molecules occupy positions in crystal lattice
Melting points increase with size and polarity
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P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids 76
SOLID MATERIALS
CRYSTALLINE POLYCRYSTALLINEAMORPHOUS(Non-crystalline)
Single Crystal
Classification of solids
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
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Crystalline Solid is the solid form of a substance in
which the atoms or molecules are arranged in a
definite, repeating pattern in three dimension.
Crystalline Solid
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
Atomic order present in sections (grains) of the solid. Different order of arrangement from grain to grain.
Grain sizes = hundreds of m. An aggregate of a large number of small crystals or
grains in which the structure is regular, but the crystals
or grains are arranged in a random fashion.
Polycrystalline Solids78
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids 79
Polycrystal
Polycrystalline
Pyrite form
(Grain)
• Polycrystal is a material made up of an
aggregate of many small single crystals
(also called crystallites or grains).
• The grains are usually 100 nm - 100
microns in diameter. Polycrystals with
grains that are <10 nm in diameter are
called nanocrystalline
Polycrystalline Solid
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
No regular long range order of arrangement in the atoms.
Eg. Polymers, cotton candy, common window glass, ceramic.
Can be prepared by rapidly cooling molten material.
Rapid – minimizes time for atoms to pack into a more
thermodynamically favorable crystalline state.
Two sub-states of amorphous solids: Rubbery and Glassy
states. Glass transition temperature Tg = temperature above
which the solid transforms from glassy to rubbery state,
becoming more viscous.
Amorphous Solids80
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
81
Amorphous (non-crystalline) Solid is composed ofrandomly orientated atoms, ions, or molecules thatdo not form defined patterns or lattice structures.
Amorphous Solid
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
Properties of single crystalline materials vary with direction,
ie anisotropic.
Properties of polycrystalline materials may or may not vary
with direction.
If the polycrystal grains are randomly oriented, properties
will not vary with direction i.e isotropic.
If the polycrystal grains are textured, properties will vary
with direction i.e anisotropic
Single- Vs Poly- Crystal82
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
Single- Vs Poly- Crystal83
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
-Properties may/may not
vary with direction.
-If grains are randomly
oriented: isotropic.
(Epoly iron = 210 GPa)
-If grains are textured,
anisotropic.
200 m
Single- Vs Poly- Crystal84
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
85
Physical Properties
Related to Solid Structure
Density
Luster
Hardness
Electrical Properties
Melting Point
Magnetic Properties
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
Tetrakaidecahedron
(Truncated Octahedron)
Octahedron
Cube
Crystal Shape
Tetrahedron
Examples of few crystal shapes (cubic crystal system)
86
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
The shape of the crystal (Eumorphic-
well formed) will ‘reflect’ the point
group symmetry of the crystal
87
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88
Early ideas
Crystals are solid - but solids are not necessarily
crystalline
Crystals have symmetry (Kepler) and long range
order
Spheres and small shapes can be packed to
produce regular shapes (Hooke, Hauy)
?
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
Kepler wondered why snowflakes have 6 corners,
never 5 or 7. By considering the packing of
polygons in 2 dimensions, it can be shown why
pentagons and heptagons shouldn’t occur.
Empty space not allowed
89
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
Crystal Types
Three types of solids, classified according to atomic
arrangement:
(a) crystalline and (b) amorphous materials are illustrated by
microscopic views of the atoms, whereas (c) polycrystalline
structure is illustrated by a more macroscopic view of adjacent
single-crystalline regions, such as (a).
90
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
quartz
Crystal structure
Amorphous structure
91
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
Crystals
The periodic array of atoms, ions, or molecules that form the solids is called Crystal.
Crystal Structure = Space (Crystal) Lattice + Basis
– Space (Crystal) Lattice is a regular periodicarrangement of points in space, and is purely mathematical abstraction
– Crystal Structure is formed by “putting” the identical atoms (group of atoms) in the points of the space lattice
– This group of atoms is the Basis
92
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
The crystal system: Set of rotation and reflection
symmetries which leave a lattice point fixed.
There are seven unique crystal systems: the cubic
(isometric), hexagonal, tetragonal, rhombohedral
(trigonal), orthorhombic, monoclinic and triclinic.
Crystal System93
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
Bravais Lattice and Crystal System
Crystal structure: contains atoms at every lattice point.
The symmetry of the crystal can be more complicated than
the symmetry of the lattice.
Bravais lattice points do not necessarily correspond to real
atomic sites in a crystal. A Bravais lattice point may be
used to represent a group of many atoms of a real crystal.
This means more ways of arranging atoms in a crystal
lattice.
94
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
What is a lattice?
A lattice is a 3-D system of points designating the positions of
the components (atoms, ions, or molecules) that make up the
substance
Unit Cell: The smallest repeating unit of the lattice.
Eg:
simple cubic
body-centered cubic
face-centered cubic
95
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
We can pick out the smallest repeating unit…..
Lattice Example96
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
We can pick out the smallest repeating unit…called Unit Cell...
UNIT CELL97
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98
Definitions
1. The unit cell: “The smallest repeat unit of a
crystal structure, in 3D, which shows the full
symmetry of the structure”
The unit cell is a box
with:
• 3 sides - a, b, c
• 3 angles - , ,
14 possible crystal structures (Bravais lattices)
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
99
Unit Cell
Simplest (smallest) parallel piped outlined
by a lattice
Lattice: a two or three (space lattice)
dimensional array of points
Environment about all lattice points must
be identical
Unit cell must fill all space, with no “holes”
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
100
Physical Properties & Structure
Hardness and Structure
– Hardness depends on how easily structural units can be
moved relative to one another
– Molecular solids with weak intermolecular attractions are
rather soft compared with ionic compounds, where forces
are much stronger
– Covalent network solids are quite hard because of the
rigidity of the covalent network structure
– Molecular and ionic crystals are generally brittle because
they fracture easily along crystal plane
– Metallic solids, by contrast, are malleable
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
101
Physical Properties
Electrical Conductivity and Structure
– Molecular and ionic solids are generally considered
nonconductors
– Ionic compounds conduct in their molten state, as ions are
then free to move
– Metals are all considered conductors
– Of the covalent network solids, only graphite conducts
electricity
This is due to the delocalization of the resonant p
electrons in graphite’s sp2 hybridization
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
102
UNIT CELL
Primitive
Single lattice point per cell Smallest area in 2D, orSmallest volume in 3D
Conventional & Non-primitive
More than one lattice point per cell Integral multiples of the volume of
primitive cell
Body centered cubic(bcc)
Conventional ≠ Primitive cell
Simple cubic(sc)
Conventional = Primitive cell
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
103
The Conventional Unit Cell
A unit cell just fills space when
translated through a subset of
Bravais lattice vectors.
The conventional unit cell is chosen
to be larger than the primitive cell,
but with the full symmetry of the
Bravais lattice.
The size of the conventional cell is
given by the lattice constant a.
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
Primitive and conventional cells of FCC
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P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
1
2
3
1ˆ ˆ ˆ( )
2
1ˆ ˆ ˆ( )
2
1ˆ ˆ ˆ( )
2
a x y z
a x y z
a x y z
Primitive and conventional cells of BCC
Primitive Translation Vectors:
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P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
a
b c
Simple cubic (sc):
primitive cell=conventional cell
Fractional coordinates of lattice
points:
000, 100, 010, 001, 110,101, 011, 111
Primitive and conventional cells
Body centered cubic (bcc):
conventional cell
a
b c Fractional coordinates of lattice points in
conventional cell:
000,100, 010, 001, 110,101, 011, 111, ½ ½ ½
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P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
Face centered cubic (fcc):
primitive (rombohedron) cell
a
b
c
Fractional coordinates:
000, 100, 101, 110, 110,101, 011, 211, 200
Face centered cubic (fcc):
conventional cell
a
bc
Fractional coordinates:
000,100, 010, 001, 110,101, 011,111,
½ ½ 0, ½ 0 ½, 0 ½ ½ ,½1 ½ , 1 ½ ½ ,
½ ½ 1
Primitive and conventional cells
107
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
Hexagonal close packed cell
(hcp): conventional =primitive
cell
Fractional coordinates:
100, 010, 110, 101,011, 111,000,
001
points of primitive cell
a
b
c
Primitive and conventional cells-hcp
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P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
A primitive unit cell is made ofprimitive translation vectors a1 ,a2, anda3 such that there is no cell of smallervolume that can be used as a buildingblock for crystal structures.
A primitive unit cell will fill space byrepetition of suitable crystal translationvectors. This defined by theparallelpiped a1, a2 and a3. The volumeof a primitive unit cell can be found by
V = a1.(a2 x a3) (vector products)
Cubic cell volume = a3
Primitive Unit Cell and vectors
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
110
The primitive unit cell may have only one lattice point.
There can be different choices for lattice vectors , but the
volumes of these primitive cells are all the same.
P = Primitive Unit Cell
NP = Non-Primitive Unit Cell
Primitive Unit Cell
1a
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
111
Wigner-Seitz Method
A simply way to find theprimitive cell which is calledWigner-Seitz cell can be doneas follows;
1. Choose a lattice point.
2. Draw lines to connect theselattice point to its neighbours.
3. At the mid-point and normalto these lines draw new lines.
The volume enclosed is called as a
Wigner-Seitz cell.
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
112
Wigner-Seitz Cell - 3D
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
Wigner-Seitz primitive unit cell and first Brillouin zone
The Wigner–Seitz cell around a lattice point is defined as the locus of points in space that are
closer to that lattice point than to any of the other lattice points.
The cell may be chosen by first picking a lattice point. Then, lines are drawn to all nearby
(closest) lattice points. At the midpoint of each line, another line (or a plane, in 3D) is drawn
normal to each of the first set of lines.
1D case
2D case
3D case: BCCImportant
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P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
The first Brillouin zone is the Wigner-Seitz cell of the reciprocal lattice
1D
2D
Real space Reciprocal space
114
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
3D: Recall that the reciprocal lattice of FCC is BCC.
44
4
4/a
Why is FCC so important?
X = ???
115
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
The Unit Cell Concept summary
The simplest repeating unit in a crystal is called a unit cell.
Opposite faces of a unit cell are parallel.
The edge of the unit cell connects equivalent points.
Not unique. There can be several unit cells of a crystal.
The smallest possible unit cell is called primitive unit cell of a
particular crystal structure.
A primitive unit cell whose symmetry matches the lattice
symmetry is called Wigner-Seitz cell.
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P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
Each unit cell is defined in terms of lattice points.
Lattice point not necessarily at an atomic site.
For each crystal structure, a conventional unit cell,
is chosen to make the lattice as symmetric as
possible. However, the conventional unit cell is not
always the primitive unit cell.
A crystal's structure and symmetry play a role in
determining many of its properties, such as cleavage
(tendency to split along certain planes with smooth
surfaces), electronic band structure and optical
properties.
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P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
The crystal system: Set of rotation and reflection
symmetries which leave a lattice point fixed.
There are seven unique crystal systems: the cubic
(isometric), hexagonal, tetragonal, rhombohedral
(trigonal), orthorhombic, monoclinic and triclinic.
Crystal System118
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
Bravais Lattice and Crystal System
Crystal structure: contains atoms at every lattice point.
The symmetry of the crystal can be more complicated than
the symmetry of the lattice.
Bravais lattice points do not necessarily correspond to real
atomic sites in a crystal. A Bravais lattice point may be used
to represent a group of many atoms of a real crystal. This
means more ways of arranging atoms in a crystal lattice.
119
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
Geometry Of Crystals
Space Lattices
Motifs
Crystal Systems
Elementary Crystallography
M.J. Buerger
John Wiley & Sons Inc., New York (1956)The Structure of Materials
Samuel M. Allen, Edwin L. Thomas
John Wiley & Sons Inc., New York (1999)
Advanced Reading
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
The language of crystallography is one shortness
We shall consider two definitions of a crystal:
1) Crystal = Lattice + Motif
2) Crystal = Space Group + Asymmetric unit
The second definition is the more advanced one (the language of
crystallographers) and we shall only briefly consider it in this
introductory text
The second definition becomes important as the classification of crystals
(7 crystal systems) is made based on symmetry and the first definition
does not bring out this aspect
Note: Since we have this precise definition of a crystal, loose definitions
should be avoided (Though often we may live with definitions like: a 3D translationally
periodic arrangement of atoms in space is called a crystal)
Initially we shall start with ideal mathematical crystals and then slowly
we shall relax various conditions to get into practical crystals
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P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
Lattice the underlying periodicity of the crystal
Basis Entity associated with each lattice points
Lattice how to repeat
Motif what to repeat
Crystal = Lattice + Motif
Motif or Basis:
typically an atom or a group of atoms associated with each lattice point
Definition 1
Translationally periodic
arrangement of motifs
Crystal
Translationally periodic
arrangement of points
Lattice
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P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
Symmetry is perhaps the most important principle of
nature:
though often you will have to dig deeper to find this
statement
The analogous terms to symmetry are:
Symmetry Conservation Invariance
The kind of symmetry of relevance to crystallography is
geometrical symmetry
The kind of symmetry we encountered in the definition of a
lattice is TRANSLATIONAL SYMMETRY (t)
As mentioned before crystals are understood based on the language of symmetry 123
Symmetry
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids124
Auguste Bravais
Found fourteen unique lattices which satisfy the
requirements
Published Études Crystallographiques in 1849
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
Isometric (cubic) Lattices
P = primitive I = body-centered (I for German innenzentriate) F = face centered a = b = c, α = β = γ = 90 ̊
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids126
Tetragonal Lattices
a = b ≠c
α = β = γ = 90 ̊
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids127
Tetragonal Axes
The tetragonal unit cell vectors
differ from the cubic one by
either stretching the vertical
axis, so that c > a (upper image)
or compressing the vertical
axis, so that c < a (lower image)
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids128
Orthorhombic Lattice
a ≠ b ≠c
α = β = γ = 90 ̊ C - Centered: additional point in the center of each
end of two parallel faces
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids129
Orthorhombic Axes
The axes system is
orthogonal
Common practice is to assign
the axes so the the magnitude
of the vectors is c > a > b
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids130
Monoclinic Lattice
a ≠ b ≠c
α = γ = 90 ̊ (β ≠ 90 ̊)
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids131
Monoclinic Axes
The monoclinic axes
system is not orthogonal
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids132
Triclinic Lattice
a ≠ b ≠c
α ≠ β ≠ γ ≠ 90 ̊
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids133
Triclinic Axes
None of the axes are at
right angles to the
others
Relationship of angles
and axes is as shown
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
134
Hexagonal
Some crystallographers call the hexagonal group a
single crystal system, with two divisions
Rhombohedral division
Hexagonal division
Others divide it into two systems, but this practice
is discouraged
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids135
Hexagonal Lattice
a = b ≠ c
α = γ = 90 ̊
β = 120 ̊
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids136
Rhombohedral Lattice
a = b = c
α = β = γ ≠ 90 ̊
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
What are the symmetries of the 7 crystal systems?
Characteristic symmetry
Cubic Four 3-fold rotation axes
(two will generate the other two)
Hexagonal One 6-fold rotation axis
(or roto-inversion axis)
Tetragonal (Only) One 4-fold rotation axis
(or roto-inversion axis)
Trigonal (Only) One 3-fold rotation axis
(or roto-inversion axis)
Orthorhombic (Only) Three 2-fold rotation axes
(or roto-inversion axis)
Monoclinic (Only) One 2-fold rotation axis
(or roto-inversion axis)
Triclinic None
Note: translational symmetry is always present in crystals (i.e. even in triclinic crystal)
We have stated that basis of definition
of crystals is ‘symmetry’ and hence
the classification of crystals is also
based on symmetry
The essence of the required symmetry
is listed in the table
more symmetries may be part of the
point group in an actual crystal
137
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
Crystal Structures - Cubic
a
aa
Simple Face-Centered Body-Centered
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
Crystal Structures - Monoclinic
c
ab
Simple End Face-Centered
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
Crystal Structures - Tetragonal
c
a
a
Simple Body-Centered
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
Crystal Structures - Orthorhombic
c
ab
SimpleEnd
Face-Centered
Body
Centered
Face
Centered
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
For a well grown crystal (eumorphic crystal) the external shape
‘reflects’ the point group symmetry of the crystal
the confluence of the mathematical concept of point groups
and practical crystals occurs here!
The unit cell shapes indicated are the conventional/preferred
ones and alternate unit cells may be chosen based on need
It is to be noted that some crystals can be based on all possible
lattices (Orthorhombic crystals can be based on P, I, F, C
lattices); while others have a limited set (only P triclinic lattice)
Emphasis
142
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
Lattices can be constructed using translation alone
The definition (& classification) of Crystals is based on symmetry and NOT
on the geometry of the unit cell (as often one might feel after reading some
books!)
Crystals based on a particular lattice can have symmetry:
equal to that of the lattice or
lower than that of the lattice
Based on symmetry crystals are classified into seven
types/categories/systems known as the SEVEN CRYSTAL SYSTEMS
We can put all possible crystals into 7 boxes based on symmetry
Crystal system
Symmetry operators acting at a point can combine in 32 distinct ways to
give the 32 point groups
Lattices have 7 distinct point group symmetries which correspond to the
SEVEN CRYSTAL SYSTEMS
Alternate view
143
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
Crystal System Shape of Unit Cell Bravais Lattices
P I F C
1 Cubic Cube
2 Tetragonal Square Prism (general height)
3 Orthorhombic Rectangular Prism (general height)
4 Hexagonal 120 Rhombic Prism
5 Trigonal Parallopiped (Equilateral, Equiangular)
6 Monoclinic Parallogramic Prism
7 Triclinic Parallopiped (general)
Why are some of the entries missing?
Why is there no C-centred cubic lattice?
Why is the F-centred tetagonal lattice missing?
….?
14 Bravais Lattices divided into 7 Crystal Systems
P Primitive
I Body Centred
F Face Centred
C A/B/C- Centred
A Symmetry based concept ‘Translation’ based concept
144
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
Crystal System Bravais Lattices
1. Cubic P I F
2. Tetragonal P I
3. Orthorhombic P I F C
4. Hexagonal P
5. Trigonal P
6. Monoclinic P C
7. Triclinic P
?
Why so many empty boxes?
E.g. Why cubic C is absent?
P: Simple; I: body-centred;F: Face-centred; C: End-centred
145
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
End-centred cubic not in the Bravais list ?
End-centred cubic = Simple Tetragonal
2
a
2
a
146
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
14 Bravais lattices divided into seven crystal systems
Crystal system Bravais lattices
1. Cubic P I F C
2. Tetragonal P I
3. Orthorhombic P I F C
4. Hexagonal P
5. Trigonal P
6. Monoclinic P C
7. Triclinic P
147
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
THE 7 CRYSTAL SYSTEMS
148
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
1. Name of crystal system
lattice parameters and
relationship amongst them (preferred Unit Cell)
Possible Bravais lattices
Point groups belonging to the crystal system
Diagram of preferred Unit Cell
149
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
1. Cubic Crystals
a = b= c = = = 90º
m
23
m
4 432, ,3m 3m,4 23, groupsPoint
Note the 3s are in the second position
SC, BCC, FCC are lattices
while HCP & DC are crystals!
• Simple Cubic (P) - SC
• Body Centred Cubic (I) – BCC
• Face Centred Cubic (F) - FCC
Elements with Cubic structure → SC: F, O, Po ||
BCC: Cr, Fe, Nb, K, W, V||
FCC: Al, Ar, Pb, Ni, Pd, Pt, Ge
150
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
Examples of elements with Cubic Crystal Structure
Po
n = 1n = 2 n = 4
Fe Cu
BCC FCC/CCPSC
C (diamond)
n = 8 DC
151
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
Note that cubic crystals can have the shape of a cube, an octahedron, a
truncated octahedron etc.
(some of these polyhedra have the same rotational symmetry axes; noting that
cube and octahedron are regular solids (Platonic) while truncated octahedron
with two kinds of faces is not a regular solid)
The external shape is a ‘reflection’ of the symmetry at the atomic level
Point groups have be included for completeness and can be ignored by
beginners
Cubic crystals can be based on Simple Cubic (SC), Body Centred Cubic
(BCC) and Face Centred Cubic Lattices (FCC)
by putting motifs on these lattices
After the crystal is constructed based on the SC, BCC or FCC lattice, it
should have four 3-fold symmetry axes (along the body diagonals)
which crystals built out of atomic entities will usually have
if the crystal does not have this feature it will not be a cubic crystal (even
though it is based on a cubic lattice)
152
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
2. Tetragonal Crystals
a = b c = = = 90º
Simple Tetragonal
Body Centred Tetragonal -BCT
m
2
m
2
m
42m,4 4mm, 422, ,
m
4 ,4 4, groupsPoint
Note the 4 in the first place
Elements with Tetragonal structure → In, Sn
153
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
Example of an element with Body Centred Tetragonal Crystal Structure Indium
InLattice parameter(s) a = 3.25 Å, c = 4.95 Å
Space Group I4/mmm (139)
Strukturbericht notation A6
Pearson symbol tI2
Other examples with this structure Pa
Wyckoff
position
Site
Symmetryx y z Occupancy
In 2a 4/mmm 0 0 0 1
In
All atoms are In → coloured
for better visibility
[001] view
[100] views
BCT
Note: All atoms are identical (coloured differently for easy visualization)
154
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
3. Orthorhombic Crystalsa b c = = = 90º
Simple Orthorhombic
Body Centred Orthorhombic
Face Centred Orthorhombic
End Centred Orthorhombic
m
2
m
2
m
2 2mm, 222, groupsPoint
a b c
One convention
Elements with Orthorhombic structure → Br, Cl, Ga, I, S
155
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
Example of an element with Orthorhombic Crystal Structure Ga
GaLattice parameter(s) a = 2.9 Å, b = 8.13, c = 3.17 Å
Space Group Cmcm (63)
Strukturbericht notation
Pearson symbol oC4
Wyckoff
position
Site
Symmetryx y z Occupancy
Ga 4c m2m 0 0.133 0.25 1
[010] view
[001] view
Note: All atoms are identical (coloured differently for easy visualization)
156
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
4. Hexagonal Crystals
a = b c = = 90º = 120º
Simple Hexagonal
m
2
m
2
m
6 m2,6 6mm, 622, ,
m
6 ,6 6, groupsPoint
Elements with Hexagonal structure → Be, Cd, Co, Ti, Zn
157
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
Example of an element with Hexagonal Crystal Structure Mg
Note: All atoms are identical (coloured differently for easy
visualization)
158
P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Crystal Structures of Solids
5. Trigonal/Rhombohedral
Crystals
a = b = c
= = 90º
• Rhombohedral (simple)
m
23 3m, 32, ,3 3, groupsPoint
Note the 3 s are in the first position
Elements with Trigonal structure → As, B, Bi, Hg, Sb, Sm
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Example of an element with Simple Trigonal
Crystal Structure -Hg
[111] view
Wyckoff
position
Site
Symmetryx y z Occupancy
Hg 1a -3m 0 0 0 1
-HgLattice parameter(s) a = 3.005 Å
Space Group R-3m (166)
Strukturbericht notation A10
Pearson symbol hR1
Other examples with this structure -Po
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6. Monoclinic Crystalsa b c
= = 90º
Simple Monoclinic
End Centred (base centered) Monoclinic (A/C)
m
2 ,2 2, groupsPoint
Elements with Monoclinic structure → P, Pu, Po
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7. Triclinic Crystalsa b c
• Simple Triclinic
1 1, groupsPoint
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Cubic(48)
Tetragonal(16)
Triclinic(2)
Monoclinic(4)
Orthorhombic(8)
Progressive lowering of symmetry amongst the 7 crystal systems
Hexagonal(24)
Trigonal(12)
Incr
easi
ng s
ym
met
ry
Superscript to the crystal system is the order of the lattice point group
Arrow marks lead from supergroups to subgroups
Ordering the 7 Crystal Systems: Symmetry
Cubic
Hexagonal
Orthorhombic
Trigonal
Tetragonal
Monoclinic
Triclinic
Order
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Minimum symmetry requirement for the 7 crystal systems
Crystal
system
Characteric symmetry Point groups Comment
Cubic Four 3-fold rotation axes
m
23
m
4 432, ,3m 3m,4 23,
3 or 3 in the second place
Two 3-fold axes will generate the other
two 3-fold axes
Hexagonal One 6-fold rotation axis
(or roto-inversion axis) m
2
m
2
m
6 m2,6 6mm, 622, ,
m
6 ,6 6,
6 in the first place
Tetragonal (Only) One 4-fold
rotation axis
(or roto-inversion axis) m
2
m
2
m
42m,4 4mm, 422, ,
m
4 ,4 4,
4 in first place but no 3 in second place
Trigonal (Only) One 3-fold
rotation axis
(or roto-inversion axis) m
23 3m, 32, ,3 3,
3 or 3 in the first place
Orthorhombic (Only) Three 2-fold
rotation axes
(or roto-inversion axis) m
2
m
2
m
2 2mm, 222,
Monoclinic (Only) One 2-fold
rotation axis
(or roto-inversion axis) m
2 ,2 2,
Triclinic None 1 1, 1 could be present
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