Transcript Crystal structure

Department of Applied Science and
Humanities
, Gurgaon
M.D.U Syllabus
 Section A:
Crystal Structure
Quantum Physics
 Section B:
Nano-Science
Free Electron Theory
 Section C:
Band Theory of solids
Photoconductivity and Photovoltaics
 Section D:
Magnetic properties of Solids
Dr. sangeeta Negi
solids
 Non crystalline or
Amorphous solids
 Strongly bonded molecules
but no geometrical regularity
 Do not have directional
properties, so they are called
as “ isotropic” substance.
 Wide range of melting point
 Ex: Glass, plastics, rubber
 Crystalline solids
 Perfect periodicity of atomic
structure
 Has directional properties
and therefore called as
anisotropic substance.
 Sharp melting point
 Ex:metallic crystal of Cu,
Ag, Al, Mg
 Non metallic crystal:
carbon, silicon
Figures
Materials and Packing
Crystalline materials...
• atoms pack in periodic, 3D arrays
• typical of:
-metals
-many ceramics
-some polymers
crystalline SiO2
Adapted from Fig. 3.23(a),
Callister & Rethwisch 8e.
Noncrystalline materials...
Si
Oxygen
• atoms have no periodic packing
"Amorphous" = Noncrystalline
noncrystalline SiO2
Adapted from Fig. 3.23(b),
Callister & Rethwisch 8e.
6
Space lattice
 A lattice is a regular and periodic arrangement of
points in three dimensions.
 An infinite array of points in three dimensions in
which every point has surroundings identical to that of
every other points in the array.
 Figure:
Two dimensional arrays
Crystal= Lattice + Basis
 A crystal structure is formed by associating every
lattice point with unit assembly of atoms or molecules
or ions identical in composition, arrangement and
orientation. This unit assembly is called the basis.
 Figure:
Unit cell
 A unit cell is defined as a fundamental building block
of a crystal structure, which can generate the complete
crystal by repeating its own dimension in various
direction
 Figure:
 Crystallographic Axes: Unit cell consisting of three mutually perpendicular
edges OA, OB and OC. Denoted by x, y, z
 Primitives: OA, OB OC be the intercepts made by the unit cell along the
crystallographic axes. These intercepts are known as primitives. Denoted by a,
b, c
 Interaxial angle: The angle between axes . Denoted by α,β, γ.
Unit Cell
 Primitive Cell: Smallest
 Non primitive cell:
unit cell in volume
constructed by
Primitives.
 Atoms at corners of the
unit cell
 Ex: Simple cubic unit cell
 Atom at corner and other
points
 Ex: Bcc and fcc unit cell
Symbols
 Primitive lattice P: lattice points only at corner of unit
cell
 Body centered lattice BCC: I- corner + body centre of
unit cell
 Face centered lattice FCC: F- point at corner + Face
centre
 Base centered lattice : C- Corner+ top + bottom base
symmetry
 Centre of symmetry
 Plane of symmetry
 Axis of symmetry
Bravais lattices ( 1948)
 14 types of unit cells under these 7 crystal systems are
possible, they commonly called “Bravais Lattices”.
 Cubic system:i. Simple
ii. Body centered
iii. Face centered
 Tetragonal system:i. Simple
ii. Body centered
 Monoclinic system
Simple
ii. End face centered
 Orthorhombic system
i. Simple
ii. Body centered
iii. End face centered
iv. Face centered
 Triclinic
 Rhombohedral
 hexagonal
i.
Seven crystal system and bravais
lattices
Figures of crystal systems
Important parameters
 Number of atoms molecules or ions per unit cell (n).
 Coordination number (CN): It is the number of
nearest neighboring atom molecules or ions to a
particular atom.
 Atomic Radius: It is radius of an atom or half the
distance between two nearest neighboring atoms in a
crystal.
 Atomic packing factor or density of packing: ratio of
the volume occupy by the molecules atom and ion in a
unit cell to the total volume of the unit cell. APF=v/V
Simple cubic structure
 A simple cubic unit cell consists of eight corner atoms.
 The total number of atoms present in a unit cell
1/8x8=1
 Coordination number: There are four nearest
neighbors in its own plane. There is another nearest
neighbor in another plane, which lie just below this
atom. Therefore the total number of nearest neighbor
is six. Hence the CN =6
Simple Cubic Structure (SC)
• Rare due to low packing density (only Po has this structure)
• Close-packed directions are cube edges.
• Coordination # = 6
(# nearest neighbors)
21
Cont…..
 Atomic Radius: r=a/2
 Atomic packing factor APF=v/V
 v=1x4/3∏r3 V=a3 APF=∏/6
substituting r=a/2
Figure:
Atomic Packing Factor (APF):SC
APF =
Volume of atoms in unit cell*
Volume of unit cell
*assume hard spheres
• APF for a simple cubic structure = 0.52
volume
atoms
a
unit cell
R=0.5a
APF =
4
1
Adapted from Fig. 3.24,
Callister & Rethwisch 8e.
p (0.5a)
3
3
a3
close-packed directions
contains 8 x 1/8 =
1 atom/unit cell
atom
volume
unit cell
23
Body centered cubic structure
 A body centered cubic structure has eight corner




atoms and one body centered atom.
In bcc unit cell, each and every corner atom is shared
by eight adjacent unit cells
Total number of atoms contributed by the corner atom
is 1/8x8=1
Total number of atoms present in bcc unit cell 1+1=2
CN=8
Cont….
 Atomic radius r=√3/4 a
 APF=v/V, v=2x4/3∏r3, V=a3
 APF=0.68
 Figure:
Body Centered Cubic Structure (BCC)
• Atoms touch each other along cube diagonals.
--Note: All atoms are identical; the center atom is shaded
differently only for ease of viewing.
ex: Cr, W, Fe (), Tantalum, Molybdenum
• Coordination # = 8
Click once on image to start animation
(Courtesy P.M. Anderson)
Adapted from Fig. 3.2,
Callister & Rethwisch 8e.
2 atoms/unit cell: 1 center + 8 corners x 1/8
26
Atomic Packing Factor: BCC
• APF for a body-centered cubic structure = 0.68
3a
a
2a
Close-packed directions:
Adapted from
Fig. 3.2(a), Callister &
Rethwisch 8e.
R
atoms
unit cell
APF =
length = 4R =
a
4
2
p ( 3 a/4 ) 3
3
a3
3a
volume
atom
volume
unit cell
27
FACE CENTERED CUBIC STRUCTURE (FCC)
• Coordination # = 12
Adapted from Fig. 3.1(a),
(Courtesy P.M. Anderson)
Callister 6e.
• Close packed directions are face diagonals.
--Note: All atoms are identical; the face-centered atoms are shaded
differently only for ease of viewing.
Atomic Packing Factor: FCC
• APF for a face-centered cubic structure = 0.74
maximum achievable APF
Close-packed directions:
length = 4R =
2a
2a
Unit cell contains:
6 x 1/2 + 8 x 1/8
= 4 atoms/unit cell
a
Adapted from
Fig. 3.1(a),
Callister & Rethwisch
8e.
atoms
unit cell
APF =
4
4
p ( 2 a/4 ) 3
3
a3
volume
atom
volume
unit cell
29
FCC Stacking Sequence
• ABCABC... Stacking Sequence
• 2D Projection
B
A
A sites
B sites
B
C
B
C
B
B
C
B
B
C sites
• FCC Unit Cell
A
B
C
30
HEXAGONAL CLOSE-PACKED STRUCTURE
(HCP)
Ideally, c/a = 1.633 for close packing
However, in most metals, c/a ratio deviates from this value
Hexagonal Close-Packed Structure
(HCP)
• ABAB... Stacking Sequence
• 3D Projection
• 2D Projection
c
a
Top
layer
B sites
Middle layer
A sites
Bottom layer
Adapted from Fig. 3.3(a),
Callister & Rethwisch 8e.
• Coordination # = 12
• APF = 0.74
A sites
6 atoms/unit cell
ex: Cd, Mg, Ti, Zn
• c/a = 1.633
32
COMPARISON OF CRYSTAL STRUCTURES
Crystal structure
coordination #
packing factor
close packed directions

Simple Cubic (SC)
6
0.52
cube edges

Body Centered Cubic (BCC)
8
0.68
body diagonal

Face Centered Cubic (FCC)
12
0.74
face diagonal

Hexagonal Close Pack (HCP)
12
0.74
hexagonal side
Close packed crystals
A plane
B plane
C plane
A plane
…ABCABCABC… packing
[Face Centered Cubic (FCC)]
…ABABAB… packing
[Hexagonal Close Packing (HCP)]
LATTICE CONSTANT
 mass in each unit=a3
 If M is molecular weight and N the Avogadro’s
number, then
 Mass of each molecule= M/N
 Mass in each unit cell=nM/N
 A lattice constant a=(nM/N)1/3
THEORETICAL DENSITY, 
Density = mass/volume
mass = number of atoms per unit cell * mass of each atom
mass of each atom = atomic weight/avogadro’s number
STRUCTURE OF OTHER SYSTEMS
• Structure of NaCl
(Courtesy P.M. Anderson)
• Structure of Carbon
Graphite
Diamond
Sodium chloride structure
Diamond structure
Miller Indices
 Miller indices is defined as the reciprocals of the
intercepts made by the plane on the three axes.
 The orientation of planes or faces in a crystal can be
described in terms of their intercepts on the three
axes.
 Miller introduced a system to designate a plane in a
crystal.
 The set of three number is known as miller indices of
concerned plane.
Procedure for findings miller
indices
Determined the intercepts of the plane along the axes
x, y, z in terms of lattice constant a, b, c
Determine the reciprocals of these numbers
Find the least common denominators (LCD) and
multiply each by this LCD.
The result is written in paranthesis
This is called ‘Miller indices’ of the plane in the form
(hkl)
Important features of miller indices
 A plane which is parallel to any one of the coordinate
axes has an intercept of infinity. Therefore the miller
index for that axis is zero.
 A plane passing through origin is defined in terms of a
parallel plane have non-zero intercept.
 All equally spaced parallel planes have same miller
indices, the miller indices do not only defence a
particular plane but also a set of parallel planes.
Miller planes
 Figures:
CRYSTALLOGRAPHIC PLANES
 Crystallographic planes specified by 3 Miller
indices as (hkl)
 Procedure for determining h,k and l:
Z
 If plane passes through origin, translate plane





or choose new origin
Determine intercepts of planes on each of the
axes in terms of unit cell edge lengths (lattice
parameters). Note: if plane has no intercept to
an axis (i.e., it is parallel to that axis), intercept
is infinity (½ ¼ ½)
Determine reciprocal of the three intercepts (2
4 2)
If necessary, multiply these three numbers by a
common factor which converts all the
reciprocals to small integers (1 2 1)
The three indices are not separated by commas
and are enclosed in curved brackets: (hkl) (121)
If any of the indices is negative, a bar is placed
in top of that index
1/2
1/4
Y
1/2
X
(1 2 1)
THREE IMPORTANT CRYSTAL
PLANES
THREE IMPORTANT CRYSTAL PLANES
 Parallel planes are equivalent
EXAMPLE:
CRYSTAL
PLANES
 Construct a (0,-1,1) plane
Interplanar spacing
 The relation between the interplanar distance and
interatomic distance is given by
d=a/(h2+K2+l2) ½
 Angleθ between planes given by θ
bragg’s law:
 X-rays are electromagnetic waves with wavelengths
thousand times smaller than the visible light.
 To measure the wavelength, a grating of corresponding
dimensions is required and hence simple grating can
not be used.
 Bragg explains the simple explanation of the observed
angle of the diffracted beams from a crystal.
 Consider
X-RAYS TO CONFIRM CRYSTAL STRUCTURE
• Incoming X-rays diffract from crystal planes, following
Braggs law: nl = 2dsin(q)
Adapted from Fig. 3.2W,
Callister 6e.
• Measurement of:
Critical angles, qc,
for X-rays provide
atomic spacing, d.
Experimental crystal
structure
determination
Three methods of X-rays
crystallography allows for this
in the following ways
 Laue Technique: A stationary
single crystal is irradiated by a
range of x-rays wavelengths.
 Rotating crystal method: A
single crystal specimen is
rotated in a beam of
monochromatic x-rays
 Powder techniques: A
polycrystalline powder is kept
stationary in a beam of
monochromatic radiation.
Laue method
 A single crystal is mounted on a goniometer, which
enables the crystal to be rotate through known angles
in two perpendicular planes and maintain a stationary
in a beam of x-rays ranging in wavlength from about
0.2- 2.0 A0.
 Crystal select out and diffract those values of
wavelength for which plane exist of spacing d and
angle Θ.
 A flat photographic film is placed to receive either
transmitted diffracted beam or the reflected diffracted
beam.
figure
Rotating crystal method
 A small crystal is mounted on a goniometer which is
fixed to spindle so the crystal can rotate abour a fixed
axis
 The specimen is usually oriented with one of the
crystallographic axes parallel to the axis of the
rotation.
 The resulting variation in Θ brings difference lattice
planes into position for reflection and diffracted
images are recorded.
Powder diffraction method
 A monochromatic x-ray beam is allowed to fall on a
small specimen and contained in a thin walled glass
capillary tube.
 Since the orientation of the minute crystal is
completely random a certain of them will be with any
given set of lattice planes making exactly the correct
angle with the incident beam for reflection.
 After taking n=1 in bragg’s equation, there are still a
number of combinations of d and Θ that would satisfy
the bragg’s law.
figure
 For each combination of
d and θ, one cone of
reflection results, which
is coaxially with the axis
of the incident beam and
with a semi apex angleof
twice of bragg’s angle 2θ.
 There are many cones of
reflection emitted by the
powder specimen.
Imperfection
(defect) in crystals
 No crystal is perfectly regular.
 Any deviation from this perfect
atomic periodicity is an
imperfection called lattice
defect.
 Structure insensitive propertiesdensity, dielectric capacitivity,
specific heat,elastic properties
etc.
 Structure sensitive propertieselectricalresistance, diffusion
crystal growth etc.
Defects
Crystal
imperfection
Thermal
vibrations
Point defects
Line defects
Surface defects
Electronic
imperfection
Transient
imperfection
Conduction
Electron/holes
Photon/ beam
of charged
particle
Point defect
 Interstitial atom: This is an atom inserted into the
voids between the regularly occupied sites.
 Vacancies: lattice sited from which the atoms are
missing.
 Impurity atom: This is a defect in which a foreign atom
occupies a regular lattice site.
Schottky Defect
 There are irregularities of the atomic arrays in which
atoms are missing at some lattice point, such a point is
called a vacancy (also called Schottky defect).
 Derivation as per discussed in class
Frenkel defect
 When an interstitial is caused by transferring an atom
from a lattice site to an intrstitial position, a vacancy is
created. The associated vacancy and interstitital atom
is called Frenkel defect.
 Derivation as discussed in class.
Bonding in solids
 The atoms, ions or molecules in a crystal hold together by




electrostatic forces and the ability to hold is called
bonding.
Crystals are classified into following categories;
Ionic crystals: bonding by strong electrostatic attraction
between ions of opposite sign. Ex; KBr, NaCl, CsCl
Covalent crystals: bonding by sharing of electrons between
atoms, ex; C, Si, Ge
Metallic crystals: bonding by electrostatic attraction
between lattice of ion core and the free electron gas. Ex;
Cu, Na, Al
 Inert gas crystals: Bonding by vander waal’s forces ex;
solid CH4, Solid He
 Hydrogen bonded crystals: bonding by hydrogen
bonds ex; HF, H20
Ionic crystals
 Electrons are transferred from one kind of atoms to
the other kind, so that the atoms become positive and
negative ions.
 These ions arrange themselves in such a configuration
that Coulomb attraction between ion of opposite signs
is stronger than the Coulomb repulsion between ions
of the same sign.
 This electrostatic interaction of oppositely charged
ions results ‘ ionic bond’
Cohesive energy
 Energy of an ionic crystal is defined as the energy that
would be liberated in the form ation of the crystals
from the individual neutral atoms
 B= αe2/ 4∏ξo ron-1 Born Lande equation

Figure:
Properties of ionic solids
 Hard cubic crystals
 Brittle
 Transparent to visible radiation
 Poor conductors of electricity
 Easily soluable in polar liquids like water
 Ionic bonds are non directional
Covalent crystal
 In covalent crystals, one or more electron are detached
from two adjacent atoms and are shared equally by
both atoms.
 Figure;
properties
 Strong bond
 Cohesive energies of covalent crystal 6 to 12 eV/ atom
 No sharp distinction between ionic and covalent





crystal
Covalent crystal with large bond energies are very hard
High melting point and transparent to visible light
Conductivity of covalent crystals varies over a wide
range
Solids are hard, brittle and posses crystalline structure
Soluable in non-polar solvent such as benzene