Post on 18-May-2015
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