mats 10221 lecture handout 2011

7/24/2019 MATS 10221 Lecture Handout 2011

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Crystallographic Notation &Simple Crystal Structures

Sarah Haigh

[email protected]

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]


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.


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)


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)


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



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


(0,1,0)




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Axis X Y Z

Interceptpoints

Reciprocals

SmallestRatio

Miller ndices (210)(1/2, 0, 0)

(0,1,0)




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Axis a b c

Interceptpoints

Reciprocals

SmallestRatio

Miller ndices ( )


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





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




8


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


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4 (0,353a)

0.68=

V

V=APF

cellunit

atoms

BCCFCC0,74

mats 10221 lecture handout 2011

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