01 crystals

8/8/2019 01 Crystals

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CRYSTAL STRUCTURES

Terminology

as c tructures

Symmetry Operations


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Ionic crystals often have a definite habit which gives rise to

particular crystal shapes with particular crystallographic facesdominating and easy cleavage planes present.

Single crystal of NaCl being cleaved with a razor blade


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Early observations of the regular shapes of crystals and their

built up from simple units.

s sugges on was ma e ong e ore e a om c eory o

matter was developed. For example the pictures below are

from a work on the form of cr stals from R.J.Hau in 1801.


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The unit cell shape must fill space and so there are restrictions

on e ce s apes a can e use .For example in 2-Dimensions it is not possible to have a

pentagona un t ce .


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The repeating pattern of atoms in a crystal can be used to

define a unit cell.

This is a small volume of the material that can be translated

through space to reproduce the entire crystal.

The translation of the unit cell follows the vectors iven b its

sides, these are the cell vectors (2Da &b, in 3Da,b &c).

cell.

b

a


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Space transformations. Translation symmetry.

Crystallography is largely based Group Theory (symmetry).

.

symmetry operator is unity operator(=does nothing).

(=Lattice is invariant with respect to symmetry operations)

, ,

point, r, by r=r+R.


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The unit cell is the basic repeat unit for the crystal.

If each unit cell is thought of as a single point the crystal can besimplified to a lattice.

We can always move to an equivalent point in a lattice by taking

an integer combination of the lattice vectors:

cwbvauT ++= (u,v,w) being integers.

b T (3,1)

a


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TERMINOLOGY

Lattice Point- Point that contains an atom or molecule

n e - eg on e ne y a, ,c w c w en rans a e y n egra

multiple of these vectors reproduces a similar region of the crystal

Basis Vector-A set of linearl inde endent vectors (a,b,c) which can

be used to define a unit cell


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Unit Cells

E

F

- = .


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Unit Cells

,

Unit Cell = ODEF

Primitive Unit Cell = ODEF

Primitive Basis Vectors = a,b


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Lattice Vector

R=ha+kb+lc

h,k,l are integers

er n ces

A displacement of any lattice point by R will give a

positional appearance as the original position


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Unit cell vs. primitive cell.

A primitive cell (PC) is the smallest unit which,whenrepeated indefinitely in three dimensions, will generate the

lattice.

A unit cell (UC) is the smallest unit that has all the symmetry.

C3C4

Example: BCC

Unit cell 2 atoms=1+8/8.

Symmetry: 3C4,4C3, 12C2, 6m

C2

Cn is n-fold symmetry axis.m is mirror planem

i=C2

minversion (center of symmetry).


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TERMINOLOGY

Primitive Unit Cell- The smallest unit cell, in volume, that

can e e ne or a g ven att ce

Primitive Basis Vectors- A set of linearly independentlyvec ors n a can e use o e ne a pr m ve un ce


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Sin le s ecies

A B C

D

aG

b

A - G : Primitive unit cells

All have same area

All have 1 atom/cell

a : Not a unit cell

: n ce no r m ve


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Multi S ecies

Non Primitive Primitive

,


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The positions of the atoms within one unit cell are referred to as

t e as s o t e crysta structure. o escr e t e w o e crysta werequire a lattice and a basis.

Within the unit cell the separation of two atoms will be given by:

=The crystal lattice tells us that for every pair of atoms in the

at a separation of: Trrr += 1212'

Where Tis any

=

ar1


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Definition of lattice

A spatial arrangement of atoms (S) represents a periodiclattice if this arrangement is invariant with respect to TR, where

,(=fundamental, primitive) vectors.

322211 ucnubnuanR

++= SSTR =)(n are an inte er numbers;, ,

a, band care the latticeconstants;

au1, u2, cu3are t e trans at onvectors.

In general u1, u and u3are notorthogonal. === 313221 uuuuuu

define a lattice a, b, c, , and.


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3 D Bravis Lattices

Each unit cell is such that the entire lattice can be formed

y sp ac ng t e un t ce y w t no gaps n t e

structure (close packed)

, .

5-fold (ie pentangles) cannot

In 3 dimensions there are onl 14 wa s of arran in oints

symmetrically in space that can give no gaps These arrangements are the

BRAVIS LATTICES

These can be further subdivided into 7 crystal structures


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14 Bravais Lattices


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Bravais lattices : In three dimensions there are only 14

space ng a ces, e rava s a ces.These are classified by 7

crystal systems (shapes):

triclinic :

monoclinic : ,90== cba

orthorhombic : 90=== cba

hexagonal :

90==== cba

====

rhombohedral :

,

==== cba

cubic :90===== cba


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Bravais lattices

In addition to the shape of theunit cell a label is added to

of lattice points:

P,R : the cell is not centred,

Primitive, only 1 lattice point.

C: side centred cells.

F : face centred cells.

.


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Cubic Lattices

BCC and FCC are not primitive. bcc has 4 atoms/cell, fcc has 8 atoms/cell

fcc has closest packing, then bcc then sc (for cubic) (fcc and bcc more common than sc)

r m t ve ave atom ce , ot are om o e ra r gona c e vey p


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Rhombohedral or Tri onal

Rhombohedral (R) Or Trigonal

= = = = o ,


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Monoclinic P Monoclinic (BaseCTriclinic (Primitive)a b c, 90o

a b c, = = 90o,

90o

a b c, = = 90o, 90o


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Orthorhombic

Orthorhombic (P)

Orthorhombic (BaseC)

a b c = = = 90oa b c, = = = 90o


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Hexa onal

Hexagonal (P)

a = b c, = = 90o, = 120o


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Tetra onal

Tetragonal (P)

= = = = oTetragonal (BC)

= = = = o


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Diamond Structure

, e

A

B

Tetrahedral bonding of carbon , Si and Ge

each atom bonds covalently to 4 others equally spread about atom in

.

Unit cell of resulting lattice is a double fcc

,


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o um c or e, aA face centered cubic arrangement of anions with the cations

in all of the octahedral holes

8 unit cells


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Fluorite, CaF2

The cations lie in aface centered cubic arrangement and the

anions occupy all of the tetrahedral holes

8 unit cells


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Each titanium atom is surrounded by an approximate octahedron of

Rutile, TiO2

oxygen atoms, and each oxygen atom is surrounded by an approximate

equilateral triangle of titanium atoms.

8 unit cells

S ace rou P4 mnm 136


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uar z,2

Each silicon atom is surrounded by a tetrahedron of oxygen atoms

4 unit cells

Space group P3121 (152)


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Planes

In all the structures, there are planes of atoms

extended surfaces on which lie regularly spaced atoms

These planes have many other planes parallel to them

These sets of planes occur in many orientations

BA

D


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Planes continued

Orientation of Planes are identified by

Miller Indices (hkl)

1) Take the origin at any lattice point in the crystal, and coordinate

axes in the direction of the basis vectors of the unit cell

2) Locate the intercepts of a plane belonging to the desired system

along each of the coordinate axes, and express them as integralmultiples of a,b,c along each axis

3) take the reciprocals of these numbers and multiply through by the

smallest factor that will convert them to a triad of (h,k,l) having the

same ratios.


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Intercepts (A,B,C) are at 2a, 4b and 3c

reciprocal values are 1/2 , 1/4 and 1/3

Smallest common factor is 12

(hkl) = 12 (1/2,1/4,1/3) = (6,3,4)

The inter-plane separation (dhkl) is calculated from

2

2

2

2

2

2

1

lkhd

hkl

++

=


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Miller Planes

To identify a crystal plane a set of 3 indices are used.

c The Miller indicies are

defined by taking theintercepts of the plane with

the cell vectors:

b Here the intercepts are (2,3,2).

a

2 3

intercepts : 1 1 1, ,

The Miller indices are the lowest set integers which have the same

ra o as ese nverses: .

I h h l


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c In cases where the plane

axis the plane is taken tointercept at infinity.

b

Here, intercepts are

2

, , .

Inverses are1

,0,0

a

This shows that the plane interceptinga at 2 vector lengths and

v


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Miller lattices and directions


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Miller lattices and directions.

Equivalent faces are designated by curlybrackets (braces). Thus the set of cube faces cane represente as 100 n w c{100}=(100)+(010)+(001)+(100)+(010)+(001)

001

Directions: A line is constructed throughthe origin of the crystal axis in the directionunder consideration and the coordinates ofa point on the line are determined inmultiples of lattice parameters of the unitcell.The indices of the direction are takenas e sma es n egers propor ona othese coordinates and are closed in squarebrackets. For example, suppose the

coordinates are x =3a =b and z =c/2 thenthe smallest integers proportional to thesethree numbers are 6,2 and 1 and the linehas a [621]direction.

Axis system for a hexagonal unit cell (MillerBravais


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Axis system for a hexagonal unit cell (Miller Bravaisscheme .


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Figure 14Packin of hard s heres in an fcc lattice.

Packing density.


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Packing density.

.

packed.

e .

The APF is defined as the fraction of

cellunitainatomsofvolumeAPF = vo umeceun ttota

APF =0.68

APFFCC=0.74

HCP .

Packing and interstitial sites.


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Unoccu ied interstitial site in Unoccupied interstitial site in the

the FCC structure: tetragonaland octahedral.

BCC structure: interstitial withdistorted octahedral and

oc a e ra symme ry.FCC

BCC

Interstitials are very important in formation of solid solutions. Example: C:Festeel.

Packing density and stability of the


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lattice.

nsta ty o s re ate to a ser es o

successful phase transitions in BaTiO3

BaTiO3 TiO2-anatase TiO2-rutile

01 crystals

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