Crystal Symmetries - New Mexico Tech: New Mexico Techljacobso/crystalslides.pdf · crystal...

METE 327 Physical Metallurgy Copyright 2008 Loren A. Jacobson 5/16/08

Crystal Symmetries


Why should we be interested?

● Important physical properties depend on crystal structure– Conductivity– Magnetic properties– Stiffness– Strength

● These properties also often depend on crystal orientation


Some Important Terms for Crystals– Crystal Structure– Bravais lattices– Symmetry operations– Basis of Close Packed

Structures– Miller Indices

● Planes● Directions

– Stereographic Projection

– Standard Projection– Principal Metal Structures

—BCC, FCC, HCP– Ionic Crystals– Diamond Structure– Twinned Crystals– Isomorphism– Polymorphism


Bravais Lattices (14 Total)– Cubic—a=b=c, all angles= 90o

– Tetragonal—a=b /=c, all angles=90o – Orthorhombic—a/=b/=c, all angles=90o – Rhombohedral—a=b=c, three equal

angles, / =90o – Hexagonal—a=b/=c, two angles =90o ,

third =120o

– Monoclinic—a/=b/=c, two angles =90o , third not.

– Triclinic-- a/=b/=c, no angles =, nor =90o


Cubic Bravais Lattices


Tetragonal and Orthorhombic


Orthorhombic, Rhombohedral, Hexagonal


Monoclinic, Triclinic


Miller Indices– A Convenient Way to Identify

Crystal Planes and Directions● For Planes, the index is the

reciprocal of the value of the intersection of the plane with a particular axis, converted to whole numbers.

● For Directions, the index is the axis coordinate of the end point of the vector,converted to nearest whole numbers.


Miller Indices (directions)– h,k,and l refer to principal axes, x,y

and z.– Directions are indicated by square

brackets, [hkl]. Families of directions are indicated by <hkl>.

– Example of a family of directions: <100> = [100], [010], [001], [-100], [0-10] and [00-1]● The first three are principal axes,

x,y, and z.


Miller Indices (planes)

– Planes are indicated by parentheses, (hkl) and families of planes by curly brackets, {hkl}.

– Example of a plane family is as follows: {100} = (100), (010), (001), (-100), (0-10), and (00-1)● These are all six faces of a cube.


Calculating Miller Index for planes

xy

z

Intercepts= 0.5, 1, 1/3

Index = (213)

Intercepts= -1, 1, 0.5

Index = (-112)


Calculating Miller Index, directions

End coordinates= 0.5,0.5,1

Index = [112]

End=0,1,0.5

Index=[021]


Hexagonal, Miller-Bravais Indices

a1

a2

a3

c Three a directions, as shown, plus c

Indices are:

(h,k,-(h+k), l)

The plane shown is:

(11-20)


Other Crystal Characterization

– It is often important to determine crystal orientation.● Single crystals.● Individual grains in a polycrystal

– If there is a preferred grain orientation, this is referred to as “texture”.

– One method is to employ the Stereographic Projection.


Stereographic Projection


Some Cubic Crystal Planes

001 Plane

110 Plane

111 Plane


Crystal Symmetries

– Translational Symmetry—a move of one cell in each of 3 axis directions restores the structure.

– Rotational Symmetry—rotation of specific angle (90o, 120o, 180o) restores the structure.

– Mirror Symmetry—reflection across a plane restores the structure.


A simple Cubic Structure(illustrating translational symmetry)

The cubic unit cell

Eight unit cells, the start of a crystal structure.


Rotational Symmetry, Cubic

Four-foldRotation

Two-FoldRotation

Three-FoldRotation


{100} Poles of a Cubic Crystal


Interplanar Angles

Points on the sphere are intersections of plane normals.


Calculating Interplanar Angles(Cubic Crystals)

Cos φ = h1h2 + k1k2 + l1l2

-------------------------- SQRT((h1

2 + k12 + l1

2 )(h22

+ k22 + l2

2 ))


Calculating Interplanar Spacing(Cubic Crystals)

1/d2 = (h2 + k2 + l2)/ a2


Standard (001) Cubic Projection


The Standard Stereographic Triangle

Useful for showing crystal axis orientations.


Metallic Bonding

– Valence, or outer electrons of metallic atoms are distributed throughout the structure. Positively charged metal ions are distributed within this “sea” of electrons.

– This allows metals to be electrical conductors.– There are second nearest neighbor interactions that influence the

crystal structure


Body Centered Cubic (BCC) Metals

α-Fe, Cr, Mo, V, β-Ti, β-Zr


Face-Centered Cubic (FCC) Metals

Cu, Al, Ni, Pb,

γ-Fe


Hexagonal Close-Packed (HCP) Metals

Be, Mg, Zn, Cd, α-Ti, α-Zr


Relation Between FCC and HCPABCABC...

ABABAB...


BCC to HCP Transformation

Close Packed BCC plane {110} becomes Close Packed HCP plane (0002).


BCC to HCP Transformation (2)1.633 a

BCC {110}

1.155 a

1.732 a

a

HCP (0002)In both cases the diagonal, where atoms touch, is of length = 2a, where a is the atom diameter. A small distortion is needed for BCC to transform to HCP. (Note that there can be 6 orientation variants of HCP.)


Types of Transformations

– Displacive, which means that atoms do not have to move very far, and often a shear displacement can lead from one crystal structure to another.

– Replacive, which means that atoms will move some distance, to their new locations and so longer range diffusion is required.

– These topics will be treated in more detail later


a

b

α

a

b

α

c = a x b

a 1

b 3b 3 = a1 x a2 a3 ( a1 x a2)

a 3a 2

c = ab sin a

c = a x b a b = ab cos a

Vector Multiplication Examples

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