INTRODUCTION TO CERAMIC MINERALS 1 1.7 BASIC CRYSTALLOGRAPH Y.
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Transcript of 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.
BASIC CRYSTALLOGRAPHY3
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.
BASIC CRYSTALLOGRAPHY5
PLANE LATTICE: >A plane lattice or net represents a regular arrangement of points in two-dimensions.
BASIC CRYSTALLOGRAPHY6
SPACE LATTICE:
> a three dimensional array of points each of which has identical surrounding
BASIC CRYSTALLOGRAPHY7
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.
BASIC CRYSTALLOGRAPHY8
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.
BASIC CRYSTALLOGRAPHY15
MILLER INDICESMILLER INDICES
BASIC CRYSTALLOGRAPHY16
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
23
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 ]_ _
26
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
INTRODUCTION TO CERAMIC MINERALS29
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
INTRODUCTION TO CERAMIC MINERALS30
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
32
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