INTRODUCTION TO CERAMIC MINERALS 1 1.7 BASIC CRYSTALLOGRAPH Y.

INTRODUCTION TO CERAMIC MINERALS1

1.7 BASIC CRYSTALLOGRAPHY

BASIC CRYSTALLOGRAPHY2

Crystallography is the experimental science of determining the arrangement of atoms in solids. In older usage, it is the scientific study of crystals. The word "crystallography" is derived from the Greek words crystallon = cold drop / frozen drop, with its meaning extending to all solids with some degree of

transparency, and graphein = write.


UNIT CELL

> a convient repeating unit of a space.

>The axial length and axial angles are the lattice constants of the unit cell.

BASIC CRYSTALLOGRAPHY4LATTICE:

>an imaginative pattern of points in which every point has an environment that is identical to that of any other point in the pattern. >A lattice has no specific origin as it can be shifted parallel to itself.


PLANE LATTICE: >A plane lattice or net represents a regular arrangement of points in two-dimensions.


SPACE LATTICE:

> a three dimensional array of points each of which has identical surrounding


BRAVAIS LATTICE: >Unique arrangement of lattice points >crystal systems are combined with the various possible lattice centering >describe the geometric arrangement of the lattice points, and thereby the translational symmetry of the crystal >also refer as SPACE LATTICES.


SEVEN STRUCTURESEVEN STRUCTURE

SYSTEM OF CRYSTALSYSTEM OF CRYSTAL

THE BRAVAIS LATTICE

TRICLINIC MONOCLINICSIMPLE

MONOCLINICBASED-CENTERED

ORTHORHOMBIC SIMPLE

ORTHORHOMBIC BASED-CENTERED

ORTHORHOMBIC BODY-CENTERED

ORTHORHOMBIC FACE-CENTERED

HEXAGONAL RHOMBOHEDRAL (TRIGONAL)

TETRAGONALSIMPLE

TETRAGONALBODY-CENTERED

CUBIC (ISOMETRIC)

SIMPLE

CUBIC (ISOMETRIC)

BODY-CENTERED

CUBIC (ISOMETRIC)

FACE-CENTERED

14 BRAVAIS LATTICE14 BRAVAIS LATTICE

SYMMETRY10

Symmetry in an object may be defined as the exact repetition, in size, form and arrangement, of parts on opposite sides of a plane, line or point.

Symmetry element is a simple geometry operation such as: translation, inversion, rotation or combinations thereof.

SYMMETRY11

Figure 34: Symmetry Elements For Cubic Form

12

Symmetry operations can include:

mirror planesmirror planes, which reflect the structure across a central plane, rotation axesrotation axes, which rotate the structure a specified number of degrees, center of symmetrycenter of symmetry or or inversion pointinversion point which inverts the structure through a central point

SYMMETRY

13Interface angle ( α ) is an angle between two crystal faces that is measured in a plane is an angle between two crystal faces that is measured in a plane

perpendicular to both of the crystal faces concerned.perpendicular to both of the crystal faces concerned. This may be done with a contact goniometer

Figure 35: (a) Contact Goniometer and (b) Measurement

0o Reference

INTERFACE ANGLE

ZONES & ZONE AXIS14

Zones the arrangement of a group of faces in such a manner that their intersection edges are mutually parallel.

• Zones axis An axis parallel to these edges is known

In Figure 36 the faces m’, a, m and b are in one zone, and b, r, c, and r’ in another. The lines given [001] and [100] are the zones axis.


MILLER INDICESMILLER INDICES


Miller Indices ~a notation system in crystallography for planes and directions in crystal (Bravais) lattices~a shorthand notation to describe certain crystallographic directions and planes in a material~ (h,k,l)

Miller-Bravais Indices A special shorthand notation to describe the crystallographic planes in hexagonal close packed unit cell

Miller Indices of directions and planes

William Hallowes Miller(1801 – 1880)

ATOMIC COORDINATE18

z

x

y0,0,0

1,1,0

0,0,1

1,0,0

1,1,1

Coordinate of Points

We can locate certain points, such as atom position in the lattice or unit cell by constructing the right-handed coordinate system

Distance is measured in terms of the number of lattice parameter we must move in each of the x,y and z coordinates to get form the origin to the point

Atom position in S.C : 000, 100, 110, 010, 001,101,111,011

Atomic coordinate

8 Cu (corner) 6 Cu (face)(0, 0, 0) (½, ½, 0)(1, 0, 0) (0, ½, ½)(0, 1, 0) (½, 0, ½)(0, 0, 1) (½, ½, 1)(1, 1, 1) (1, ½, ½)(1, 1, 0) (½, 0, ½)(1, 0, 1)(0, 1, 1)

FCC:Face Centered Cubic

Atom coordinate

(0, 0, 0)(1, 0, 0)(0, 1, 0)(0, 0, 1)(1, 1, 1)(1, 1, 0)(1, 0, 1)(0, 1, 1)(½, ½, ½)

-z

+z

+x

-x

-y

+y

BCC :Based centered cubic

Anisotropy of crystals

66.7 GPa

130.3 GPa

191.1 GPa

Young’s modulus of FCC Cu

Anisotropy of crystals (contd.)

Different crystallographic planes have different atomic density

And hence different properties

Si Wafer for computers

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MILLER INDICES OF DIRECTIONS

1. Choose a lattice point on the direction as the origin

2. Choose a crystal coordinate system with axes parallel to the unit cell edges

x

y3. Find the coordinates, in terms of the respective lattice parameters a, b and c, of another lattice point on the direction.

4. Reduce the coordinates to smallest integers.

5. Put in square brackets […]

Miller Indices of Directions

[110]1a+1b+0c

z

x

y

z

O

A1/2, 1/2, 1

[1 1 2]

OA=1/2 a + 1/2 b + 1 c

P

Q

x

y

z

PQ = -1 a -1 b + 1 c

-1, -1, 1

Miller Indices of Directions (contd.)

[ 1 1 1 ]_ _

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MILLER INDICES

FOR PLANES

6. Enclose in parenthesis

(2,0,0)

(0,3,0)

(0,0,1)

Miller Indices for planes

4. Take reciprocal

3. Find intercepts along axes

2. select a crystallographic coordinate system

1. Select an origin not on the plane

5. Convert to smallest integers in the same ratio

x

y

z

: 2 3 1

: 1/2 1/3 1

: 3 2 6

: (326)

(1/2 1/3 1) X 6

Miller Indices for planes (contd.)

origin

intercepts

reciprocals

Miller Indices

A B

C

D

O

x

y

z

E

x

y

z

ABCD

O

1 ∞ ∞

1 0 0

(1 0 0)

OCBE

O*

O*

1 -1 ∞

1 -1 0

(1 1 0)_

Plane


z

x

y

ABC

Planes in the Unit Cell

Procedure1.Identify the points at which the plane intercepts the x,y and z coordinates in terms of the number of lattice parameters. If the plane passes through thr origin, the origin of the coordinate system must be moved2.Take reciprocals of these intercepts3.Clear fractions but do not reduce to lowest integers4.Enclose the resulting umbers in parenthess ( ). Again, negative numbers should be written with a bar over the number


z

x

y

ABC

Plane A1.x=1, y=1, z=12.1/x=1, 1/y=1, 1/z=13.No fractions to clear4.Miller Indices =(111)

Plane B1.The plane never intercepts the z-axis, so x=1, y=2 and z=∞2.1/x=1, 1/y=1/2, 1/z=03.Clear fractions: 1/x=2, 1/y=1, 1/z=04.Miller Indices =(210)

Plane C1.We must move the origin, since the plane passes through 0,0,0.Let’s move the origin one lattice parameter in the y-direction. Then, x=∞, y=-1, and z=∞2.1/x=0, 1/y=-1, 1/z=03.No fraction to clear4.Miller Indices = (010)

Planes in the Unit Cell

Family of Symmetry Related Directions

x

y

z[ 1 0 0 ]

[ 1 0 0 ]_

[ 0 0 1 ]

[ 0 0 1 ]_

[ 0 1 0 ]_

[ 0 1 0 ]

Identical atomic density

Identical properties

1 0 0

1 0 0= [ 1 0 0 ] and all other directions related to [ 1 0 0 ] by symmetry

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type: <100>Equivalent directions:[100],[010],[001]

type: <110>Equivalent directions:[110], [011], [101],[-1-10], [0-1-1], [-10-1],[-110], [0-11], [-101],[1-10], [01-1], [10-1]

type: <111>Equivalent directions:[111], [-111], [1-11], [11-1]

Family of Symmetry Related Directions

Family of Symmetry Related Planes

4. (1 1 0)

_

1. ( 1 01 )

2. ( 1 0 1 )

6. ( 0 1 1 )_

3. ( 1 1 0)

5. ( 0 1 1 )

_

{ 1 1 0 }

{ 1 1 0 } = Plane ( 1 1 0 ) and all other planes related by symmetry to ( 1 1 0 )

z

y

x

[uvw] = components of a vector in the directiondirection reduced to smallest integers

(hkl)= reciprocal of intercepts of a planeplane reduced to smallest integers

<uvw>= family of symmetry related directions

{hkl}= family of symmetry related planes

Summary

INTRODUCTION TO CERAMIC MINERALS 1 1.7 BASIC CRYSTALLOGRAPH Y.

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