1. structure of metals
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Lattice
Chapter I The Structure of Metals
A latticeis an infinite array of evenly spaced pointswhich are all similarly situated. Each points are
regarded as similarly situated in the rest of the lattice
appears the same, and in the same orientation when
viewed form them. (J.F. Nye, Physical Properties of Crystals)
3 D
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Chapter I The Structure of Metals-- Ideal Crystal
An ideal crystal is defined to be a body in which theatoms are arranged in a lattice. That is:
(a) the atomic arrangement appears the same, and in thesame orientation, when viewed from all the lattice points
(b). Lattice + Basis = crystal structure(c) the form and orientation of the lattice in an idealcrystal is independent of which point in the crystal ischosen as origin.
(d) an ideal crystal is infinite in extent; real crystals arenot only bounded, but also depart from the ideal crystalsby possessing occasional imperfections
(e) Continuity
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Unit cell choices
Primitive unit cell:
A parallelepiped having lattice points at its corners only
The primitive unit cell is not unique (such A, B, C.
In 3-D, like simple cubic structure)
Multiply primitive unit cell
A unit cell which has lattice points at the
centers of its faces, or at its body center,or occasionally at other positions, in
addition to the points at its corners (like
FCC, BCC, E, D, F etc.)
Chapter I The Structure of Metals-- Unit Cell
A unit cell containing only one lattice point
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Reflection Symmetry
In crystal,
1. The symmetrical arrangement of atoms can be described
formally in terms of elements of symmetry.
2. A symmetry operation moves or transforms an object in
such a way that after transformation it coincides with itself.
Mirror line (image
line) (2D)
Mirror plane
(image plane) (3D)
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Chapter I The Structure of Metals-- Unit Cell
J.F. Nye, Physical Properties of Crystals
Continuity Ideal crystal
Discontinuity Real crystal
lattice translation vector:
a, b, and c
lattice points translation :
=u1a+u2b+u3c
u: arbitrary integer
R= R+ for a ideal
lattice frame (ideal crystal )
Translational symmetry
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Rotational symmetry:
Chapter I The Structure of Metals-- Unit Cell
2- fold
6- fold4- fold
3- fold
5-fold ??
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Chapter I The Structure of Metals-- Unit Cell
??
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Chapter I The Structure of Metals-- Unit Cell
in a cubic unit cell
3 tetrad rotation axes
4 triad rotation axes
6 diad rotation axes
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Chapter I The Structure of Metals-- Unit Cell
Cubic
Tetragonal
Orthorhombic
Monoclinic
Triclinic
Hexagonal
Trigonal
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Chapter I The Structure of Metals
Unit cell and Body-centered cubic
www.parc.xerox.com/.../ comparison_of_cubic_packing.htm
BCC unit cell
http://members.tripod.com/~EppE/jpgs/bodcubic.jpgs
lattice
Two atoms per unit cell
Coordination number 8 (# of adjacent atoms)
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Unit cell and Body-centered cubic
The unit cell of the bcc structure consists of acube with an atom at each corner and an atom in
the center of the cube
Typical metals with a bcc unit cell are
Molybdenum, Tungsten and iron ( i.e. iron atroom temperature)
Close-pack direction: diagonal direction passingthrough the centered atom, cornered atom and
the atom at opposite corner
Chapter I The Structure of Metals
(please find two more.)
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Octahedral interstices in BCC: Octahedral interstices are
bounded by 6 atoms whose centers join up to make anoctahedron (a 8-sided figures); you can find 18 oct. inter.
sites in a BCC unit cell (6 at face center, the other 12 locate
at edges.)
Chapter I The Structure of Metals
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Chapter I The Structure of Metals
tetrahedral interstices in BCC: tetrahedral interstices are
bounded by 4 atoms whose centers join up to make antetrahedron (a 4-sided figures); you can find 24 tera. inter.
sites in a BCC unit cell (there are four tetrahedral sites on
each of 6 BCC faces)
MATTER-Introduction to point defects
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Chapter I The Structure of Metals
Difference in size: Tetrahedral interstice > Octahedral interstice
MATTER-Introduction to point defects
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Chapter I The Structure of Metals
Unit cell and Face-centered cubic
www.parc.xerox.com/.../ comparison_of_cubic_packing.htm
Four atoms per unit cell
Coordination number 12 (# of adjacent atoms)
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Chapter I The Structure of Metals
Unit cell and Face-centered cubic
FCC unit cell consists of a cubic structure
with an atom at every corner of the cube and an
atom at the center of each of the six faces.
Typical examples of metals with an F.C.C unitcell include Aluminum, Silver, Gold, Nickel and
iron (i.e. iron at high temperatures). (please findtwo more)
Features of F.C.C metals are ductile and good
electrical conductors.
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Chapter I The Structure of Metals
Octahedral interstices in FCC: Octahedral interstices are
bounded by 6 atoms whose centers join up to make an
octahedron (a 8-sided figures); you can find 13 oct. inter.
sites in a FCC unit cell (one at center of the unit cell, the
other 12 locate at edges.)
MATTER-Introduction to point defectsMATTER-Introduction to point defects
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MATTER-Introduction to point defects MATTER-Introduction to point defects
Chapter I The Structure of Metals
Tetrahedral interstices in FCC: tetrahedral interstices are
bounded by 4 atoms whose centers join up to make antetrahedron (a 4-sided figures); you can find 8 tetra. inter.
sites in a FCC unit cell Diagonal passes throughthe center of the
tetrahedral site
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Difference in size:Octahedral interstice > Tetrahedral interstice
(BCC: Tetra. size > Octa. Size)
FCC
Chapter I The Structure of Metals
BCC
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Chapter I The Structure of Metals
Home work:
Draw an tetrahedron in a FCC unit cell and
index the four planes on the tetrahedron (problem
1.6 in Reed-Hill)
Calculate the packing density (efficiency) of
FCC , BCC and SC
Packing efficiency= Voccupied / Vunit cell
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The close-packed planes in FCC are the {111} set, and the
close-packed directions are the [110] set.
By moving each atom off the corner
of a FCC unit cell on e.g., [001] plane,you can see four independent slip planes
in the unit cell respectively.Arizona state university
Chapter I The Structure of Metals
Close-pack plane and close-pack direction
Unit cell and Face Centered Cubic
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Chapter I The Structure of MetalsChapter I The Structure of Metals
Unit cell and Face Centered Cubic
{200}{111}
Close-pack planeLess close-pack plane
Close-pack plane
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This arrangement provides a close packing
system of atoms as illustrated.
6 atoms per unit cell, coordination No. is
12
Zinc, Magnesium, etc four-axis coordinate system
Chapter I The Structure of Metals
Unit cell and Close-packed hexagonal (HCP, orC.P.H.)
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Magnesium (Discovered 1808)
Atomic number12
Atomic weight24.305
Periodic table group An alkaline-earth metal
A simple substance is light metals of
silver white
Hexagonal closed pack structure
lattice constant a=0.32094nm
c=0.52103nm
c/a=1.6235
Melting point650
Boiling point1107
Specific gravity1.741
Young's modulus42000MPa
Stiffness16000MPa (Poissons ratio0.38)
line coefficient of expansion
26.9410-6/(20200)
Heat conduction rate
1.55J/cmsec(Al2.88J/cmsec)
Chapter I The Structure of Metals
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Chapter I The Structure of Metals
Digital-camera EXILIM EX-S3 made by CASIO
Adopted example,
magnesium alloy
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Chapter I The Structure of Metals
Magnesium alloy- Notebook computer
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4-digit Miller indices for C.P.H. planes and directions
three axes (a1, a2, a3) along
close-packed direction on
basal plane
the fourth axis is normal to
basal plane, called C axis
the unit of measurement
along a axis is a, along c axis
is c
Chapter I The Structure of Metals
a1
-a1
-a3
a3
a2-a2
-C1
C1
a
c
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Chapter I The Structure of Metals
Axial Ratios in Close-packed-hexagonal Metals
1.601.8861.8561.6241.568c/a
TiCdZnMgBeMetal
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a1
-a1
-a3
a3
a2-a2
-C1
C1Basal plane
(u v w t)=(a1 a2 a3 c)
(a/, a/, a/, c/1c)
unit of measurement for each axis
the intersection
(0001)?
Chapter I The Structure of Metals
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Chapter I The Structure of Metals
a1
-a1
-a3
a3
a2-a2
-C1
C1
Prism planes of type I
A
C
D
B
E
F
G
H
Plane ABCD
unit of measurement for each axis
(u v w t)=(a1 a2 a3 c)
(a/-a, a/a, a/ , c/ )
the intersection
(1100)EFGH?
Prism planes of type I = {1100}
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Prism planes of type II
a1
-a1
-a3
a3
a2-a2
-C1
C1
A
D
C
B
E
F
Chapter I The Structure of Metals
Plane ABCD
unit of measurement for each axis
(u v w t)=(a1 a2 a3 c)
(a/a, a/a, a/-0.5a, c/ )
the intersection
(1120)Prism planes of type II = {1120}
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Chapter I The Structure of Metals
a1
-a1
-a3
a3
a2-a2
-C1
C1
Pyramidal planes
A B
C
D
E
Type I, order II (ABC): (1012)
Type I, order I (ABD): (1011)
Type II, order I (AED) :(1121)
Type II, order II (AEC): (1122)
(u, v, w, t)=(a1, a2, a3, t)
a1+a2=-a3
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Chapter I The Structure of Metals
Four-digit Miller indices ofdirections for CPH
C=[0001], plane normal of basal plane
Looking on basal plane
[u v w t]=[a1 a2 a3 c],
u+v= -w still holds for determining direction
Diagonal axes, type I (e.g., a1 ?)
Any component along a1?
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Chapter I The Structure of Metals
Four-digit Miller indices ofdirections for CPH
Diagonal axes, type II (e.g., A1 ?)
A1
Diagonal axes, type II (e.g., A1)
perpendicular to type I
a2
Any component along a2?
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Chapter I The Structure of Metals
Three-digit Miller indices ofdirections and planes for CPH
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Chapter I The Structure of Metals
Interstitial sites--size and shape
BCCFCC
Atoms just fit in the interstitial site, the unit cell
not being distorted or expended
In real cases, interstitial atoms (C, N, O...) largerthan the site, leading to symmetric or nonsymmetric
expansion in the lattice
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Chapter I The Structure of Metals
R: radius of atom
r: radius of maximum interstice size
J.D. Verhoeven, Found. of Phys. Metall. p. 10
the maximum sized sphere that could be placed in the voids
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www.caton.org/images/ chem/TableP.gif
Chapter I The Structure of Metals
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Chapter I The Structure of Metals
Lattice being distorted because of the interstitial atoms
symmetrical or nonsymmetrical expansion ?
excess strain due to squeezed-in interstitial atom
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Chapter I The Structure of Metals
a2
2
a2
2
regular polyhedron
For close-packed crystal structures
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Chapter I The Structure of Metals
For close-packed crystal structures (regular polyhedron)
some of the octahedral voids in FCC
some of the tetrahedral voids in FCC
some of the tetrahedral voids
in CPH
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Chapter I The Structure of Metals
Interstitial sites in BCC crystal structurea
Octa. tetra.
a23
a
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R: radius of atom
r: radius of maximum interstice size
J.D. Verhoeven, Found. of Phys. Metall. p. 10
Chapter I The Structure of Metals
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Chapter I The Structure of Metals
. AD = 2(R+r) and AB =
2R=ADcos45.
2R=2Rcos45+2rcos45 and
r/R=(1-cos45)/cos45=0.414
FCC
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Chapter I The Structure of Metals
Home work
According to the above sketches in the relation between unit cell
length (a) and atomic radius(R), please show that: (1) the tetrahedral
site in BCC is 0.291 R; (2) the octahedral site in FCC is 0.414R.
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not close-packed
close-packed
2-D packing
e.g., (111) of FCC
(0001) basal planes of CPH
Chapter I The Structure of Metals
60o
Closed-Packing sequence
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Chapter I The Structure of Metals Closed-Packing sequence
A
AA
A
A
A
A B
BB
B
B
B
B
A
AA
A
A
A
A
C
CC
C
C
C
C
Two alternativestackings on top (or
bottom) of A
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Chapter I The Structure of Metals Closed-Packing sequence
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C.R. Brooks, ASM (1982)
3-D packingB sites
C sites
Chapter I The Structure of Metals Closed-Packing sequence
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C.R. Brooks, ASM (1982)
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FCC : ABCABC.. CPH : ABAB..
Prof. L.H. Chen, class note, p. 71, NCKU
Chapter I The Structure of Metals Closed-Packing sequence
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Chapter I The Structure of Metals Closed-Packing sequence
Summary:
The FCC{111} planes look like the CPH {0001}.
The FCC system starts the same as the HCP but the third layer, ratherthan lining up with the atoms of the first layer, instead lines up with theother holes in the first layer. The fourth layer then lines up with the
atoms of the first layer, with sequence ABCABCABC.
The HCP system is formed by taking the given hexagonal array and
stacking the next layer so that it fits in one of the holes of the first layer.
The third layer then is placed so that its atoms line up with those of the
first layer, with the sequence ABABAB being repeated.
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http://www.techfak.uni-kiel.de/matwis/amat/elmat_en/kap_5/backbone/r5_1_2.html
Chapter I The Structure of Metals
Polycrystal vs. Single crystal
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Chapter I The Structure of Metals
A. Chemical discontinuity--leading to internalboundaries separating phases of different composition (or
different composition and structure)
B. Structural (crystal orientational) discontinuity--leading to internal boundaries between crystals of the
same phase, resulting from orientation difference
Polycrystal--internal boundaries within crystal
Polycrystal vs. Single crystal
General requirements
Meet eitherA orB Polycrystal
Ch I Th S f M l
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A. Chemical discontinuity
Chapter I The Structure of Metals
Al
Mg
Zn
Fe
Mn
Ch I Th S f M l
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Chapter I The Structure of Metals
B. Structural (crystal orientational) discontinuity
low angel boundary
tilt bound. twist bound. high angle boundary
Swalin, Thermo. Of Solids, Fig. 10.14, 15, 17
Ch t I Th St t f M t l
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Chapter I The Structure of Metals
grain size:27.6m grain size:12.2m
grain size:3.6m
Grain, grain size and
grain boundary
(magnesium alloy)
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Ch t I Th St t f M t l
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Chapter I The Structure of Metals
Basal Plane
(0002)
RD
ND
TD
Crystal arrangement of mass production materials
Anisotropy
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Anisotropic microstructures (e.g.. 5083 aluminum alloy)
100m
100m
100m
Anisotropy
Ch t I Th St t f M t l Th S hi P j i
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Chapter I The Structure of Metals
1 Gnomonic
2 Stereographic
3 Orthographic
Verhoeven, Fig. 1.15
the relationship between planes and directions
angles between the poles on the projections are
always the true angles between the normals of the
planes
To determine angles between planes and directions
The Stereographic Projection
Points A, B projected to different locations,
depending on projection method
----only Stereo. Projection meet the requirement
Chapter I The Str ct re of Metals
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Chapter I The Structure of Metals
Imaging the plane passing thru the
center of a hemisphere
Project the intersection of the plane
and hemisphere onto project plane
http://www.telusplanet.net/public/nstuart/proj.gif
http://www.stmarys.ca/academic/science/geology/structural/stereoalt.htm
2-D drawing of 3-D crystallographic features
A lattice plane (3D) can be represented in
projection paper (2-D) by the normal of the lattice
plane and (or) the trace of the plane on ref. sphere
Chapter I The Structure of Metals Th St hi P j tiChapter I The Structure of Metals
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Chapter I The Structure of Metals The Stereographic ProjectionChapter I The Structure of Metals
Plane(passing through
the center of the sphere)
plane
normalIntersection with ref. hemisphere
Project the Poles and traces to project plane (equatorial plane)
Chapter I The Structure of MetalsChapter I The Structure of Metals
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: plane normal of (h k l)
(hkl)
n
P P: pole of (h k l)Trace of (h k l);
a great circle
Chapter I The Structure of MetalsChapter I The Structure of Metals The Stereographic Projection
Great circle : a circle of maximum diameter, if the plane passes
through the center of the sphere
Small circle : A plane not passes through the center of the
sphere will intersect the sphere in a small circleOn a ruled globe (see next page), the longitude lines
(meridians)--Great circles; the latitude lines (except equator) --
small circles
Chapter I The Structure of Metals
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Chapter I The Structure of Metals
2o intervals
Wulff Net (stereographic projection)
Chapter I The Structure of Metals
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Chapter I The Structure of Metals
Kelly and Groves, Fig. 2.2
Gnomonic
Stereographic
Orthographic
(h k l) plane
Pole of (h k l)
Chapter I The Structure of Metals
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Stereographic projection
Fig. 2-28, Cullity
Chapter I The Structure of Metals
Chapter I The Structure of Metals Stereographic Projection
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Chapter I The Structure of Metals
2-D drawing of 3-D crystallographic features
Stereographic Projection
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Chapter I The Structure of Metals
2-D drawing of 3-D crystallographic features
Reed-Hill, Fig. 1.22, p. 20.
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Chapter I The Structure of Metals
2-D drawing of 3-D crystallographic features
Chapter I The Structure of Metals
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directions that lie in a plane
Chapter I The Structure of Metals
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Chapter I The Structure of Metals
Planes of a zone
Those planes that mutually intersect along a
common direction forms the planes of a zone.
The line of the intersection by those plane ofthe same zone is called the zone axis.
The direction of the zone axis is perpendicular
to each normal of the planes in the same zone.
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Planes of a zone
Chapter I The Structure of Metals
Reed-Hill, Fig. 1.25
zone axis [111]
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Chapter I The Structure of Metals
zone axis [111]
Reed-Hill, Fig. 1.25
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Chapter I The Structure of Metals
Reed-Hill, Fig. 1.26
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Chapter I The Structure of Metals
A plane can be represented by its pole.
The direction of the zone axis is perpendicular
to each normal of the planes in the same zone.
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C apte e St uctu e o eta s
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Angle between planes and Wulff Net--How to determine
the angle between planes or directions by means of a 2-D
stereographic projection
p
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Chapter I The Structure of Metals
A
A
B
C
B
C
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p
All meridians (longitude
lines), including the basic
circle, are great circles.
The equator is a great
circle. All other latitude
lines are small circles.
Angular between points
representing directions in
space can be measured on
the Wulff net only in the
points are made to
coincide with a great
circle of the net.
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Chapter I The Structure of Metals
Angle between 2 planes
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Chapter I The Structure of Metals
Chapter I The Structure of Metals --Traces of P1 and P2
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Chapter I The Structure of Metals Traces of P1 and P2