CRYSTAL STRUCTURES

Matter what is available in nature can be classified into three STATES

GASEEOUS

LIQUID

SOLID

SOLIDCRYSTALLINE

SOLIDAMORPHOUS

SOLID

SOLID IN WHICH ATOMS ARE ARRANGED IN REGULAR

MANNER WITH PERFECT PERIODICITY OVER A

LONG RANGE ORDER, ARE CALLED CRYSTALLINE SOLID

ATOMS ARRANGED IN IRREGULAR MANNER,

CALLED NON-CRYSTALLINE SOLID

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

THE REGULAR ARRANGEMENT OF POINTS INSTEAD OF ATOMS IS CALLED LATTICE. IT IS AN IMAGINARY CONCEPT Eg: egg box

A GROUP OF ATOMS OR MOLECULE ATTACHED TO EACHLATTICE POINT WHICH ARE IDENTICAL IN COMPOSITIONAND ORIENTATION IS CALLED BASISEg: EGGS

CRYSTAL STRUCTURE

Don't mix up atoms withlattice points

Lattice points areinfinitesimal points in space

Lattice points do notnecessarily lie at the centreof atoms

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Crystal Structure = Crystal Lattice + Basis

UNIT CELL

IT IS A BUILDING BLOCK OF CRYSTAL STRUCTURE

IT IS A MINIMUM NUMBER OF ATOMS BY THE REPETATION OF IT IN THREE DIMENSION WE CAN CONSTRUCT THE TOTAL CRYSTAL STRUCTURE

The unit cell and, consequently,the entire lattice, is uniquelydetermined by the six latticeconstants: a, b, c, α, β and γ.These are lattice parameters

a, b, c are axial lengths; α, β andγ. Interfacial angles

Unit Cell

DEPEND UPON THE LATTICE PARAMETER CRYSTAL SYSTM CAN BE CLASSIFIED INTO SEVEN SYSTEMS THOSE ARE

1.Cubic Crystal System (SC, BCC,FCC)

2.Hexagonal Crystal System (S)

3.Triclinic Crystal System (S)

4.Monoclinic Crystal System (S, Base-C)

5.Orthorhombic Crystal System (S, Base-C, BC,FC)

6.Tetragonal Crystal System (S, BC)

7.Trigonal (Rhombohedral) Crystal System (S)

Cubic Crystals

a = b= c = = = 90º

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

Crystal Structure

Tetragonal Crystals

a = b c = = = 90º

Simple Tetragonal

Body Centred Tetragonal -BCT

Elements with Tetragonal structure → In, Sn

Orthorhombic Crystalsa b c = = = 90º

Simple Orthorhombic

Body Centred Orthorhombic

Face Centred Orthorhombic

End Centred Orthorhombic

Elements with Orthorhombic structure → Br,

Cl, Ga, I, Su

Monoclinic Crystalsa b c = = 90º

Simple Monoclinic End Centred (base centered) Monoclinic

(A/C)

Elements with Monoclinic structure → P, Pu, Po

Triclinic Crystalsa b c

• Simple Triclinic

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Trigonal/Rhombohedral Crystalsa = b = c = = 90º

• Rhombohedral (simple)

Elements with Trigonal structure → As, B, Bi, Hg, Sb, SmCrystal Structure

Crystal Structure

Hexagonal Crystals

a = b c = = 90º = 120º

Simple Hexagonal

Elements with Hexagonal structure → Be, Cd, Co, Ti, Zn

LATTICES

In 1848, Auguste Bravais demonstrated that in a 3-dimensional system there are fourteen possible lattices

A Bravais lattice is an infinite array of discrete points with identical environment

seven crystal systems + four lattice centering types = 14 Bravais lattices

Lattices are characterized by translation symmetry

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Auguste Bravais (1811-1863)

simple cubic body-centered cubic face centered cubic

Crystal Structure

BASE CENTERED

24Crystal Structure

Examples of elements with Cubic Crystal Structure

Po

n = 1n = 2 n = 4

Fe Cu

BCC FCC/CCPSC

C (diamond)

n = 8 DC

Properties of unit cell

1. Coordination Number

2. No of Atoms Per Unit Cell

3. Lattice Constant

4. Atomic Radius

5. Atomic Packing Fraction

No of Atoms Per Unit Cell

Effective no of atoms per unit cell

COORDINATION NUMBER

Coordinatıon Number (CN) : The Bravais lattice points closestto a given point are the nearest neighbours.

Because the Bravais lattice is periodic, all points have the samenumber of nearest neighbours or coordination number. It is aproperty of the lattice.

A simple cubic has coordination number 6;

A body-centered cubic lattice, 8;

A face-centered cubic lattice, 12.

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ATOMIC PACKING FACTOR

Atomic Packing Factor (APF) is defined as thevolume of atoms within the unit cell dividedby the volume of the unit cell.

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1-CUBIC CRYSTAL SYSTEM

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 to that

cell. The rest of the atom belongs to neighboring cells.

Coordinatination number of simple cubic is 6.

a- Simple Cubic (SC)

a

bc

• Rare due to low packing density (only Po has this structure)• Close-packed directions are cube edges.

• Coordination # = 6(# nearest neighbors)

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reSIMPLE CUBIC STRUCTURE (SC)

SHARING OF CORNER ATOM WITH EIGHTNEIGHBOURING UNIT CELLS

NUMBER OF ATOM PER UNIT CELL

Po

n = 1n = 2

Fe

BCC

FCC

SC

8*1/8=1 8*1/8+1=2

8*1/8+6*1/2=4

ATOMIC PACKING FACTOR OF SC

Crystal Structure

• Coordination # = 8

• Atoms touch each other along cube diagonals.

--Note: All atoms are identical; the center atom is shadeddifferently only for ease of viewing.

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BODY CENTERED CUBIC STRUCTURE (BCC)

ex: Cr, W, Fe (), Tantalum, Molybdenum

2 atoms/unit cell: 1 center + 8 corners x 1/8

B-BODY CENTERED CUBIC (BCC)

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BCC has two lattice points so BCC

is a non-primitive cell.

BCC has eight nearest neighbors.

Each atom is in contact with its

neighbors only along the body-

diagonal directions.

Many metals (Fe,Li,Na..etc),

including the alkalis and several

transition elements choose the

BCC structure.a

b c

ATOMIC PACKING FACTOR: BCC

a

APF =

4

3p ( 3 a/4 ) 3

2

atoms

unit cell atom

volume

a 3

unit cell

volume

length = 4R =

Close-packed directions:

3 a

• APF for a body-centered cubic structure = 0.68

aR

a2

a3

• Coordination # = 12

• Atoms touch each other along face diagonals.

--Note: All atoms are identical; the face-centered atoms are shadeddifferently only for ease of viewing.

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FACE CENTERED CUBIC STRUCTURE (FCC)

ex: Al, Cu, Au, Pb, Ni, Pt, Ag

4 atoms/unit cell: 6 face x 1/2 + 8 corners x 1/8

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4 (0,353a)

0.68 = V

V = APF

3

R 4 = a

cell unit

atomsBCCFCC

0,74

Atomic Packing Factor of FCC

• APF for a face-centered cubic structure = 0.74

ATOMIC PACKING FACTOR: FCC

maximum achievable APF

APF =

4

3p ( 2a/4 )3

4

atoms

unit cell atom

volume

a3

unit cell

volume

Close-packed directions:

length = 4R = 2 a

Unit cell contains:

6 x 1/2 + 8 x 1/8

= 4 atoms/unit cella

2 a

AB

C

FCC Stacking

Highlighting the faces

Highlighting

the stacking

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There are atoms at the corners of the unit cell and at thecenter of each face.

Face centered cubic has 4 atoms so its non primitive cell.

Many of common metals (Cu,Ni,Pb..etc) crystallize in FCCstructure.

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Crystal Structure

THE MOST IMPORTANT CRYSTAL STRUCTURES

Sodium Chloride Structure Na+Cl-

Cesium Chloride Structure Cs+Cl-

Hexagonal Closed-Packed Structure

Diamond Structure

Zinc Blende

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1 – SODIUM CHLORIDE STRUCTURE

Sodium chloride alsocrystallizes in a cubic lattice,but with a different unit cell.

Sodium chloride structureconsists of equal numbers ofsodium and chlorine ionsplaced at alternate points of asimple cubic lattice.

Each ion has six of the otherkind of ions as its nearestneighbours.

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SODIUM CHLORIDE STRUCTURE

If we take the NaCl unit cell and remove all the red Cl ions,we are left with only the blue Na. If we compare this with thefcc / ccp unit cell, it is clear that they are identical. Thus,the Na is in a fcc sublattice.

SODIUM CHLORIDE STRUCTURE

This structure can beconsidered as a face-centered-cubic Bravais lattice with abasis consisting of a sodium ionat 0 and a chlorine ion at thecenter of the conventional cell,

LiF,NaBr,KCl,LiI,etc

The lattice constants are in the order of 4-7 angstroms.

)(2/

zyxa

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2-CESIUM CHLORIDE STRUCTURE

CS+CL-

Cesium chloride crystallizes in acubic lattice. The unit cell may bedepicted as shown. (Cs+ is teal,Cl- is gold).

Cesium chloride consists of equalnumbers of cesium and chlorineions, placed at the points of abody-centered cubic lattice sothat each ion has eight of theother kind as its nearestneighbors.

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3–HEXAGONAL CLOSE-PACKED STR.

This is another structure that iscommon, particularly inmetals. In addition to the twolayers of atoms which form thebase and the upper face of thehexagon, there is also anintervening layer of atomsarranged such that each ofthese atoms rest over adepression between threeatoms in the base.

Crystal

Structur

e

Bravais Lattice : Hexagonal Lattice

He, Be, Mg, Hf, Re (Group II elements)

ABABAB Type of Stacking

HEXAGONAL CLOSE-PACKED STRUCTURE

a=b a=120, c=1.633a,

basis : (0,0,0) (2/3a ,1/3a,1/2c)

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A A

AA

AA

A

AAA

AA

AAA

AAA

B B

B

B

B B

B

B

B

BB

C C C

CC

C

C

C C C

Sequence ABABAB..

-hexagonal close pack

Sequence ABCABCAB..

-face centered cubic close pack

Close pack

B

AA

AA

A

A

A

A A

B

B B

Sequence AAAA…

- simple cubic

Sequence ABAB…

- body centered cubic

PACKING

4 - DIAMOND STRUCTURE

The coordination number of diamond structure is 4.

The diamond lattice is not a Bravais lattice.

Si, Ge and C crystallizes in diamond structure.

Crystal Structure

DIAMOND CRYSTAL STRUCTURE

Crystal Structure

5- ZINC BLENDE

Zincblende has equal numbers of zinc andsulfur ions distributed on a diamond latticeso that each has four of the opposite kind asnearest neighbors. This structure is anexample of a lattice with a basis, which mustso described both because of the geometricalposition of the ions and because two types ofions occur.

AgI,GaAs,GaSb,InAs,

Crystal Structure

5- ZINC BLENDE

Zinc Blende is the name given to the mineral ZnS. It has a cubic

close packed (face centred) array of S and the Zn(II) sit in

tetrahedral (1/2 occupied) sites in the lattice.