Space Lattice

are non-coplanar vectors in space forming a basis {

}

}∈,,|{ IzyxczbyaxL

++=

cba

,, cba

,,

One dimensional lattice

Two dimensional lattice

Three dimensional lattice

Lattice vectors and parameters

Indices of directions

Miller indices for planes

Miller indices and plane spacing

Two-dimensional lattice showing that lines of lowest indices have the greatest spacing and greatest density of lattice points

Reciprocal lattice

Illustration of crystal lattices and corresponding reciprocal lattices for a cubic system

Illustration of crystal lattices and corresponding reciprocal lattices for a a hexagonal system

*][321 hklblbkbhH

hkldH

1||

If

then and H

perpendicular to (hkl) plane

*][321 hklblbkbhH

hkldH

1||

If

then and H perpendicular

to (hkl) plane

Proof:

H•(a1/h-a2/k)

=H•(a1/h-a3/l)=0

a1/h•H/|H|=[1/h 0 0]• [hkl]*/|H|=1/|H|=dhkl

H

a1

a2

a3

Symmetry(a)mirror plane(b)rotation(c)inversion(d)roto- inversion

Symmetry operation

Crystal system

The 14 Bravais lattices

The fourteen Bravais lattices

Cubic lattices a1 = a2 = a3α = β = γ = 90o

nitrogen - simple cubic

copper - face centered cubic

body centered cubic

Tetragonal latticesa1 = a2 ≠ a3

α = β = γ = 90

simple tetragonal Body centered Tetragonal

Orthorhombic lattices a1 ≠ a2 ≠ a3 α = β = γ = 90

simple orthorhombic Base centered orthorhombic

Body centered orthorhombicFace centered orthorhombic

Monoclinic lattices a1 ≠ a2 ≠ a3 α = γ = 90 ≠ β (2nd setting)α = β = 90 ≠ γ (1st setting)

Simple monoclinic Base centered monoclinic

Triclinic latticea1 ≠ a2 ≠ a3

α ≠ β ≠ γ Simple triclinic

Hexagonal latticea1 = a2 ≠ a3

α = β = 90 , γ = 120lanthanum - hexagonal

Trigonal (Rhombohedral) latticea1 = a2 = a3

α = β = γ ≠ 90

mercury - trigonal

Relation between rhombo-

hedral and hexagonal

lattices

Relation of tetragonal C lattice to tetragonal P lattice

Extension of lattice points through space by the unit cell vectors a, b, c

Symmetry elements

Primitive and non-primitive cells Face-

centerd cubic point lattice referred to cubic and rhombo-hedral cells

All shaded planes in the cubic lattice shown are planes of the zone{001}

Zone axis [uvw]Zone plane (hkl)

then hu+kv+wl=0

Two zone planes (h1k1l1) and (h2k2l2) then zone axis [uvw]=

Plane spacing

2

1

1][

hkldl

k

h

ccbcac

cbbbab

cabaaa

hkl

Indexing the hexagonal system

Indexing the hexagonal system

Crystal structure

-Fe, Cr, Mo, V -Fe, Cu, Pb, Ni

Hexagonal close-packed

Zn, Mg, Be, -Ti

FCC and HCP

-Uranium, base-centered orthorhombic (C-centered)y=0.105±0.005

AuBe:

Simple cubic

u = 0.100

w = 0.406

Structure of solid solution (a) Mo in Cr (substitutional) (b) C in -Fe (interstitial)

Atom sizes (d) and coordination

Change in coordination

128126 124

size contraction, percent

3 3 12

A: Octahedral site,

B: Tetrahedral site

Twin

(a) (b) FCC annealing (c) HCP deformation twins

Twin band in FCC lattice,Plane of main

drawing is (1ī0)

Homework assignmentProblem 2-6 Problem 2-8 Problem 2-9 Problem 2-10

Stereographic projection

*Any plane passing the center of the reference sphere intersects the sphere

in a trace called great circle

* A plane can be represented by its great circle or pole, which is the

intersection of its plane normal with the reference sphere

Stereographic projection

Pole on upper sphere can also be projected to the horizontal (equatorial) plane

Projections of the two ends of a line or plane normal on the equatorial plane are symmetrical with respect to the center O.

Projections of the two ends of a line or plane normal on the equatorial plane are

symmetrical with respect to the center O.U

L

PP’

P

P’X

O O

• A great circle representing a plane is divided to two half circles, one in upper reference sphere, the other in lower sphere

• Each half circle is projected as a trace on the equatorial plane

• The two traces are symmetrical with respect to their associated common diameter

N

S

EW

The position of pole P can be defined by two angles and

The position of projection P’ can be obtained by r = R tan(/2)

The trace of each semi-great circle hinged along NS projects on WNES plane as a meridian

As the semi-great circle swings along NS, the end point of each radius draws on the upper sphere a curve which projects on WNES plane as a parallel

The weaving of meridians and parallels makes the Wulff net

Two projected poles can always be rotated along the net normal to a same meridian (not

parallel) such that their intersecting angle can be counted from the

net

P : a pole at (1,1)

NMS : its trace

The projection of a plane trace and pole can be found from each other by rotating the projection

along net normal to the following position

Zone circle and zone pole

If P2’ is the projection of a zone axis, then all poles of the corresponding zone planes lie

on the trace of P2’

Rotation of a poles about NS axis by a fixed angle: the corresponding poles moving

along a parallel*Pole A1 move to pole *Pole A1 move to pole A2A2*Pole B1 moves 40*Pole B1 moves 40°° to to the net end then the net end then another 20another 20°° along the along the same parallel to B1’ same parallel to B1’ corresponding to a corresponding to a movement on the movement on the lower half reference lower half reference sphere, pole sphere, pole corresponding to B1’ corresponding to B1’ on upper half sphere on upper half sphere is B2is B2

m: mirror planeF1: face 1F2: face 2

N1: normal of F1N2: normal of N2

N1, N2 lie on a plane which is 丄

to m

A plane not passing through the center of the reference sphere intersects the sphere on a small circle which also

projects as a circle, but the center of the former circle does not project as the center of the latter.

Projection of a small circle centered at Y

Rotation of a pole A1 along an inclined axis

B1:

B1B1B3 B3 B2 B2 B2 B3 B3 B1B1A1A1A1 A1 A2 A2 A3 A3 A4 A4 A4 A4

A plane not passing through A plane not passing through the center of the reference the center of the reference sphere intersects the sphere sphere intersects the sphere on a small circle which also on a small circle which also projects as a circleprojects as a circle. .

Rotation of a pole A1 along an inclined axis B1:

A1 rotate about B1 forming a small circle in the reference sphere, the small circle projects along A1, A4, D, arc A1, A4, D centers around C (not B1) in the projection plane

Rotation of 3 directions

along b axis

Rotation of 3 directions along b axisRotation of 3 directions along b axis

Rotation of 3

directions along b

axis

Standard coordinates for crystal

axes

Standard coordinates for crystal axes

Standard coordinates for crystal axes

Standard coordinates for crystal axes

Projection of a monoclinic

crystal

+C-b +b

-a

+a

xx

011

0-1-1 01-1

0-11

-110-1-10

1101-10

Projection of a monoclinic crystal

Projection of a monoclinic crystal

Projection of a monoclinic crystal

(a) Zone plane (stippled)(b) zone circle with zone axis ā, note

[100]•[0xx]=0

Location of axes

for a triclinic crystal: the

circle on net has a radius of along WE axis of the net

Zone circles corresponding to a, b, c axes of a triclinic crystal

Standard projections of cubic crystals on (a) (001), (b) (011)

d/(a/h)=cos, d/(b/k)=cos, d/(c/l)=cosh:k:l=acos: bcos: ccosmeasure 3 angles to calculate hkl

The face poles of six faces related by -3 axis that is (a) perpendicular (b) oblique

to the plane of projection