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Crystallographic Notation &Simple Crystal Structures
Sarah Haigh
Crystalline and amorphous states
Crystal structure - lattice and motif
Lattice directions, lattice vectors, lattice planes
Miller indices
Zones, directions lying parallel to planes Weiss zone law
Examples of crystal structures
Atomic Packing Factors
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Resources
Also crystallography software on cluster:EPS/MSCI/Eart Sci & crystallographySite licenced appl/learning & teaching/matter 2.1
Books
C Hammond The Basics of Crystallography and Diffraction OUP 2001
Barratt C and Massalski T Structure of Metals Pergamon
Phillips F An Introduction to Crystallography Oliver & Boyd
Kelly A and Groves G Crystallography and Crystal defects Longmans
H-R Wenk and A Bulakh Minerals Their constitution and Origin CUP
Websites http://www.doitpoms.ac.uk/tlplib/miller_indices/index.php
http://www.doitpoms.ac.uk/tlplib/crystallography3/index.php
http://ocw.mit.edu (and search for crystallography)
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Structure Property Relationships
strengthhardnesscorrosion propertiesfracture toughness
erosionresponse to geologicalpressures and temperatures
structures atthe atomic
level.
-X ray, neutron, electron
diffraction, electronmicroscopy
mechanical, magnetic,optical and electronic
properties.
Engineering - choice of material for design;influence of fabrication parameters (eg temperature,pressure) upon properties.
Earth Sciences - properties and behaviour of minerals, rocksand soils (erosion, leaching, the response of the earthsmantle and crust to high pressures and temperatures).
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We start by distinguishing crystalline and amorphous materials which havefundamental differences at the atomic level.
Crystalline materials such as metals, ceramics and many other importantengineering and materials
have an ordered arrangement of atoms the atoms stack together to form regular networks the atomic arrangement is often reflected in the macroscopic geometry
of crystals.
Amorphous or glassy materials such as soot, window glass and some
polymers have a more random arrangement of atoms.
Materials such as metals and ceramics are generally made up of manysmall crystals and are described as polycrystalline.
In order to differentiate between different crystalline structures, we need tounderstand the language of crystallography. Like all languages there arecertain rules, and words with specific meanings. The more clearly youunderstand these rules and the meaning of the words, the easier you willfind crystallography.
The Crystalline State
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Phases of Matter
Matter
GASESLIQUIDSLIQUIDSLIQUIDSLIQUIDS
andandandand LIQUIDLIQUIDLIQUIDLIQUID
CRYSTALSCRYSTALSCRYSTALSCRYSTALS
SOLIDSSOLIDSSOLIDSSOLIDS
Condensed Matter actually includes bothof these. Well focus on Solids!
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Gases
Gases have atoms or molecules that do not bond toone another in a range of pressure, temperature andvolume.
These molecules havent any particular order and
move freely within a container.
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7
Similar to gases, Liquids have no atomic/molecular orderand they assume the shape of the containers.
Applying low levels of thermal energy can easily breakthe existing weak bonds.
Liquid Crystals have mobile
molecules, but a type of long rangeorder can exist; the molecules have
a permanent dipole. Applying an
electric field rotates the dipole andestablishes order within the
collection of molecules.
Liquids & Liquid Crystals
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Amorphous Solids
Amorphous (Non-crystalline) Solids are composed ofrandomly orientated atoms, ions, or molecules that do
not form defined patterns or lattice structures. Amorphous materials do not have any long-range order,
but they have varying degrees of short-range order.
Examples of amorphous materials include amorphous
silicon, many polymers, & glasses.
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Periodic Arrays of Atoms
Experimental evidence of periodic structures. (See Fig. 1.)
The external appearance of crystals gives some clues to this.
Fig. 1 shows that when a crystal is cleaved, we can see thatit is built up of identical building blocks. Further, the earlycrystallographers noted that the index numbers that define
plane orientations are exact integers.
Cleaving a crystal
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Crystalline Solids
A Crystalline Solid is the solid form of a substance inwhich the atoms or moleculesare arranged in adefinite, repeating pattern in three dimensions.
Single crystals, ideally have a high degree of order, orregular geometric periodicity, throughout the entire
volume of the material.
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An infinite array of points in space in which the environment of eachpoint is identicalNOT simply a regular array of points.
Lattice Lattice
Array
Crystal Lattice
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The Mesh and Unit Mesh
The Mesh = An arrangement of lines which joins the lattice points.
The unit Mesh = One unit which, when repeated, makes up the mesh.
Primitive MeshNon-Primitive Mesh
Unit Mesh
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14
An infinite array ofpoints in space inwhich each point has
identical surroundingsto all others.
The points are arrangedexactly in a periodic
manner.
a
b
CB ED
O
A
y
x
Crystal Lattice
2 dimensional example
Lattice vectors a & b
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An Ideal Crystal An infinite periodic repetition ofidentical structural units in space.
The simplest structural unit we can imagine is a singleatom. This corresponds to a solid made up of only onekind of atom An elemental solid.
However, this structural unit could also be a group of
several atoms or even molecules.
This simplest structural unit for a given solid is called the
Basis (or motif).
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The Basis (Motif)
Crystal Lattice + Basis = Crystal Structure
The Basis (or Motif) = The repeating unit of the pattern,e.g. the arrangement of atoms (or molecules) which is placed
at each of the lattice points to obtain the crystal structure
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A two-dimensional lattice with
different choices for the basis
The same crystal is obtained wherever the motif is placed
around each lattice point; the only requirement is that the
same position is chosen every time
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Crystal Lattice + Basis (Motif) Crystal Structure
Crystalline Periodicity
Lattice
Basis
Crystal Structure
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An infinite array of points inspace.
Each point has identicalsurroundings to all others.
Points are arranged exactlyin a periodic manner.
Looks the same fromwhichever point you view it.
Lattice invariant under
translation
Crystal Lattice
a
b
CB ED
O A
y
x
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Crystal Structure
Don't mix up atoms with
lattice points
Lattice points areinfinitesimal points in
space
Lattice points do notnecessarily lie at the
centre of atoms
Crystal Structure = Crystal Lattice + Basis
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The five planar 2D lattices
; generala b 90;a b = o
; 90a b
== o
120;a b == o
Oblique p
Rectangular p Rectangular c
90;a b = o
Square pHexagonal p
p = primitive
c = centred
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The five planar 2D latticesWhy not choose a primitive cell for rectangular?
90;a b = o
Rectangular c
We can but it does not reflect the symmetry of the lattice We choose a
larger unit cell and a centred lattice because 90 angles are easiest!
Oblique?
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Unit Cell in 2D
A repeatable unit of the crystal (group of atoms, ions ormolecules), which when stacked together with pure
translational repetition reproduces the whole crystal.
S
a
b
S
S
S
S
S
S
S
S
S
S
S
S
S
S
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2D Unit Cell example -(NaCl)
We define lattice points ; these are points with identicalenvironments
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Choice of origin is arbitrary - lattice points need not beatoms - but unit cell size should always be the same.
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This is also a unit cell -it doesnt matter if you start from Na or Cl
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- or if you dont start from an atom
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This is NOT a unit cell even though they are all the same -empty space is not allowed!
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Why can't the blue triangle
be a unit cell?
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The Space Lattice and Unit Cells
Atoms, arranged in repetitive 3-Dimensional pattern, inlong range order (LRO) give rise to crystal structure.
Properties of solids depends upon crystal structure and
bonding force. An imaginary network of lines, with atoms at intersection
of lines, representing the arrangement of atoms is calledspace lattice.
Unit cell is that block ofatoms which repeats itselfto form space lattice.
Unit Cell
Space Lattice
C l S d B i
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Crystal Systems and Bravais
Lattices Only seven different types of unit cells are
necessary to create all point lattices. According to Bravais (1811-1863) fourteen
standard unit cells can describe all
possible lattice networks. The four basic types of unit cells are
Simple
Body Centered Face Centered
Base Centered
Th 14 B i L tti
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The 14 Bravais Lattices.
7 primitive Lattices
Others generated(a) placing a lattice point in
the centre of some primitiveunit cells (I)
(b) placing lattice points inthe middle of faces (F andC)
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The unit cell and, consequently,the entire lattice, is uniquely
determined by the six latticeconstants: a, b, c,, and .
Only 1/8 of each lattice point in aunit cell can actually be assignedto that cell.
Each unit cell in the figure can beassociated with 8 x 1/8 = 1 latticepoint.
Unit Cell
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A primitive unit cell is made ofprimitive translation vectors a1 ,a2,and a3 such that there is no cell of
smaller volume that can be used asa building block for crystalstructures.
A primitive unit cell will fill space byrepetition of suitable crystaltranslation vectors. This defined bythe parallelpiped a1, a2 and a3. The
volume of a primitive unit cell canbe found by
V = a1.(a2 x a3) (vector products)
Cubic cell volume = a3
Primitive Unit Cell and Vectors
2D L tti V t
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2D Lattice Vectors
a
b
R = Ua + Vb
R = -a + -2b
Notation for vectors: a or a or ar
2D L tti Di ti
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2D Lattice Directions
a
b
The direction joining the origin to the points u,v; 2u,2v defines a row ofpoints [u v] .The symbol [u v] is used for any direction parallel to this line.
[1 1]
[1 3]
[1 1]
L tti Di ti S t
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Lattice Direction Symmetry
a
b
[10]
[01]
[0-1]
[-10]
[ ]UV UV UV UV UV = = = =
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3D LATTICES
Lattice Points.A lattice point n,m,p is used to denote a lattice point at na + mb +pc
(with n, m, p necessarily integers) and where a, b, c are the lattice vectors.
Unit cell.
y
x
z
b
a
c
U it ll
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Unit cell.
Unit cell when repeated on lattice, fills 3D mesh
PrimitiveContains Only One lattice Point
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3D Lattice Directions
R = Ua + Vb + Wc defines a direction [UVW].Symmetry related directions
zy
xa
b
c
O
Q
P
L
Direction OL1. Write down the coordinates for apoint (any) along this direction e.g. Pin terms of fractions of the lengths a,b, c.The Coordinates of P are 1/2, 0, 1.
2. Express these fractions as a ratioof whole numbers and write these insquare brackets e.g. [102].
ROL = 1.a + 0.b + 2.c
Note : Any point will do e.g. Q with coordinates , 0, still gives [102]
E l f C l Di i
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210
X = 1 , Y = , Z = 0[1 0] [2 1 0]
Examples of Crystal Directions
E l f C l Di i
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X = -1 , Y = -1 , Z = 0 [110]
X = 1 , Y = 0 , Z = 0 [1 0 0]
Examples of Crystal Directions
E l f C t l Di ti
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X =-1 , Y = 1 , Z = -1/6[-1 1 -1/6] [6 6 1]
We can move vector to the origin.
Examples of Crystal Directions
C t l Pl
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Within a crystal lattice it is possible to identify sets of equally
spaced parallel planes. These are called lattice planes.
b
a
b
a
The set ofplanes in2D lattice.
Crystal PlanesWe need a similarly precise way of defining planes in a crystal.
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Equation for a Plane
http://mathworld.wolfram.com/Plane.html
Miller Indices
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Miller Indices are a symbolic vector representation for theorientation of an atomic plane in a crystal lattice and aredefined as the reciprocals of the fractional intercepts
which the plane makes with the crystallographic axes.
To determine Miller indices of a plane, take the followingsteps;
1) Determine the intercepts of the plane along each ofthe three crystallographic directions
2) Take the reciprocals of the intercepts
3) If fractions result, multiply each by the denominator ofthe smallest fraction
Miller Indices
2D Lattice Planes
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b
(11)
(10)
(01)
a
2D Lattice Planes
The set of parallel planes which intersect the x-axis at a/h and the y-axisat b/ k is written as (hk), where h and k are integers
1
1
Lattice Plane Symmetry
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Lattice Plane Symmetry
a
b
(01)
(10)
{ } ( ) ( ) ( ) ( )hk hk hk hk hk = = = =
We need a similarly precise way of defining planes in a crystal.
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Axis X Y Z
Interceptpoints 1 1 1
Reciprocals 1/1 1/ 1 1/ 1Smallest
Ratio 1 1 1
Miller ndices (111)(1,0,0)
(0,1,0)
(0,0,1)
Crystal Planes Example
We need a similarly precise way of defining planes in a crystal.
C t l Pl E l
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a
c
b
0.5
Equation of plane
2x/a + 1y/b + 1z/c = 1
Axis X Y Z
Intercept
points 1/2 1 1Reciprocals 2/1 1/1 1/1
Smallest
Ratio
2 1 1
Miller ndices (211)
Crystal Planes Example
C t l Pl E l
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a
c
b
Equation of plane
x/a + 2y/b + 2z/c = 1
Axis X Y Z
Intercept
points 1 1/2 1/2Reciprocals 1/1 2/1 2/1
Smallest
Ratio
1 2 2
Miller ndices (122)
Crystal Planes Example
C t l Pl E l
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Axis X Y Z
Interceptpoints 1
Reciprocals 1/1 1/ 1/ Smallest
Ratio 1 0 0
Miller ndices (100)(1,0,0)
Crystal Planes Example
Cr stal Planes E ample
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Axis X Y Z
Interceptpoints 1 1
Reciprocals 1/1 1/ 1 1/ Smallest
Ratio 1 1 0
Miller ndices (110)(1,0,0)
(0,1,0)
Crystal Planes Example
Crystal Planes Example
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Axis X Y Z
Interceptpoints
Reciprocals
SmallestRatio
Miller ndices (210)(1/2, 0, 0)
(0,1,0)
Crystal Planes Example
Crystal Planes Example
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Axis a b c
Interceptpoints
Reciprocals
SmallestRatio
Miller ndices ( )
Crystal Planes Example
3D Lattice Planes
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A lattice plane which intercepts the x axis at a/h, the y axis at b/k and thez axis at c/l is written as (hkl)
3 att ce a es
(100)
a/1
(110)
a/1
b/1
(111)
a/1b/1
c/1
(210)
a/2b/1
Summary
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Summary
Lattice - a 3D set of points in which every point has the identical surroundings toevery other point.
Lattice points points of the lattice
Unit cell when repeated on lattice, fills 3D mesh
Primitive unit cell - unit cell which contains only one lattice point.
Lattice vectors vectors defining sides of the unit cell - a, b, c
(always use lower case Roman) [drawn as a right-handed set]
n,m,p is the lattice point at na + mb + pc (n, m, p necessarily integers)
The angle between a and b is , b and c is a, c and a is .
The lattice direction Ua + Vb + Wc is written [UVW]
The set of directions [UVW] related by symmetry are written
The plane which cuts the x axis (// a) at a/h, the y axis (//b) at b/h andthe z axis (// c) at c/l is written (hkl)
The set of planes which are related by the symmetry are written {hkl}
hkl are the Miller indices of the plane
Cross Products
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Cross Products
http://mathworld.wolfram.com/CrossProduct.html
Plane Normals - Cubic Systems
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z
x
y
b/k a/ha/h
b/k
c/l b/k
1. In a cubic system the normal to the plane (hkl) is the direction [hkl]
[-a/h + b/k + 0c] x [0ab/k + c/l]
= [a/kl + b/hl + c/hk]
= [ha + kb + lc]
Define two vectors c/l b/k and b/k a/h inthe plane (hkl) and take their cross product.
Cubic Systems - Angles and Distances
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Cubic Systems - Angles and Distances
1. Angle between planes (h1k1l1) and (h2k2l2)
2
2
2
2
2
2
2
1
2
1
2
1
212121coslkhlkh
llkkhh
++++
++=
2. Distance between (hkl) planes in a cubic crystal withlattice parameter a
2 2 2
ad
h k l
=
+ +
Indices of a Family of Form
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Sometimes when the unit cell has rotational symmetry,
several nonparallel planes may be equivalent by virtueof this symmetry, in which case it is convenient to lump
all these planes in the same Miller Indices, but with curlybrackets.
Thus indices {h,k,l} represent all the planes equivalentto the plane (hkl) through rotational symmetry.
Thisimagecann otcurrently bedisplayed.Thisimagecann otcurrently bedisplayed.
)111(),111(),111(),111(),111(),111(),111(),111(}111{
)001(),100(),010(),001(),010(),100(}100{
y
Zone axes
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2 intersecting lattice planes form a zone
zone
axis
zone
axis
Weiss zone law: plane (hkl) belongs to zone [uvw] if hu + kv + lw = 0
if (h1 k1 l1) and (h2 k2 l2 ) in same zone, then(h
1
+h2
k1
+k2
l1
+l2
) also in same zone.
zone axis [uvw]
h1 k1 l1 h1 k1 l1h2 k2 l2 h2 k2 l2
x x x
u = k1 l2 -k2 l1v = l1 h2 -l2 h1w = h1 k2 -h2 k1
Principal Metallic CrystalStructures
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90% of the metals have either Body Centered Cubic
(BCC), Face Centered Cubic (FCC) or Hexagonal Close
Packed (HCP) crystal structure.
BCC Structure FCC Structure HCP Structure
Structures
Fractional coordinates
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1/2
Projecting from 3D to 2D
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CsCl z y
x
Al
Close-packed Structures
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Metallic materials have isotropic bonding
In 2-D close-packed spheres generate a hexagonal array
In 3-D, the close-packed layers can be stacked in all
sorts of sequences
Most common are
ABABAB..
ABCABCABC
Hexagonal close-packed
Face centred cubic close-packed
AB BABABA.A
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A at (0,0)B t (2/3 1/3)
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A
B
C
A
B at (2/3, 1/3)
C at (1/3, 2/3)
let atomic radius be R |a1| = |a2| = 2R
A
What are the unit cell dimensions?
A A2R
R
3R face diagonal = 23R = AA
A
AB = 1/3 (face diagonal)
2
3R=
ca2
a1
A B C A2
3
R
{c
2R
2 2 244
3R x R+ =
2 28
3x R=
8
3x R =
82
3c R =
Hexagonal Close-packed Structure
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|a1| = |a2| = 2R
a1 = a2 = 90; g = 120
83
c
a=ideal:
atoms per unit cell?
coordination number?
lattice points per unit cell?
atoms per lattice point?
12
2
1
2
HCP
a unit cell with only one lattice point is a primitive cell
reminder: the hexagon is not an acceptable unit cell shape
lattice type of HCP is called primitive hexagonal
Close-packed Structures
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Metallic materials have isotropic bonding
In 2-D close-packed spheres generate a hexagonal array
In 3-D, the close-packed layers can be stacked in all
sorts of sequences
Most common are
ABABAB..
ABCABCABC
Hexagonal close-packed
Face centred cubic close-packed
AB ABCABC.C
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What are the unit cell dimensions?
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face diagonal is
close-packeddirection
a
2 4a R=
2 2a R=
|a1| = |a2| = |a3|a
1
= a2
= a3
= 90
Face Centre Cubic Close-packed
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only one cell parameterto be specified
2 2R=|a1| = |a2| = |a3|
1 = 2 = 3
atoms per unit cell?
coordination number?
lattice points per unit cell?
atoms per lattice point?
12
44
1
a unit cell with more than one lattice point is a non-primitive cell
CCP structure is often simply called the FCC structure
(misleading)
lattice type of CCP is called face-centered cubic
CCP
Structure
Cubic Loose-packed Structure
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Body-centered cubic (BCC)
body diagonal is closest-packed direction
a
3 4a R=4
3a R =
|a1| = |a2| = |a3|
1 = 2 = 3 = 90
atoms per unit cell?
coordination number?
lattice points per unit cell?
8
atoms per lattice point?
2
2
1
another example of a non-primitivecell
no common name that
distinguishes lattice type from
structure type
lattice type of CLP is body-
centered cubic
Co-ordination Number
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Co-ordinaton Number (CN) : The Bravais lattice pointsclosest to a given point are the nearest neighbours.
Because the Bravais lattice is periodic, all points have
the same number of nearest neighbours or coordinationnumber. It is a property of the lattice.
A simple cubic has coordination number 6; a body-centered cubic lattice, 8; and a face-centered cubic
lattice,12.
Atomic Packing Factor
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Atomic Packing Factor (APF) is defined as thevolume of atoms within the unit cell divided by thevolume of the unit cell.
http://www.doitpoms.ac.uk/tlplib/crystallography3/packing.php
Volume of Atoms in Unit Cell
Volume of Unit CellAPF =
Simple Cubic (SC)
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Simple Cubic has one lattice point so its primitive cell.
In the unit cell on the left, the atoms at the corners are
cut because only a portion (in this case 1/8) belongs tothat cell. The rest of the atom belongs to neighbouringcells.
Co-ordinatination number of simple cubic is 6.
a
bc
Atomic Packing Factor of SC
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Body Centred Cubic (BCC)
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BCC has two lattice points so BCCis a non-primitive cell.
BCC has eight nearest neighbours.
Each atom is in contact with itsneighbors only along the body-
diagonal directions.
Many metals (Fe,Li,Na..etc),
including the alkalis and severaltransition elements choose theBCC structure. a
b c
Atomic Packing Factor of BCC
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0.68=
V
V=APF
cellunit
atoms
BCC
2 (0,433a)
Face Centred Cubic (FCC)
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There are atoms at the corners of the unit cell and atthe centre of each face.
Face centred cubic has 4 atoms so its non primitive
cell. Many of common metals (Cu,Ni,Pb..etc) crystallize in
FCC structure.
Atomic Packing Factor of FCC
7/24/2019 MATS 10221 Lecture Handout 2011
82/82
4 (0,353a)
0.68=
V
V=APF
cellunit
atoms
BCCFCC0,74