Crystal Geometry 6up

8/12/2019 Crystal Geometry 6up

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Geometry of Crystals Chapter 2

Outline

The lattice and unit cells

Describing lines and planes in crystals The reciprocal lattice

Symmetry

Crystal Systems

Lattice centering and Bravais lattices

Indexing in the hexagonal crystal system

Crystal structures and fractional coordinates

Crystal habit (shape)

Stacking faults and twins

Introduction to the stereographic projection

Describing condensed phase structures

Describing the structure of an isolated smallmolecule is easy to do

Just specify the bond distances and angles

How do we describe the structure of a condensedphase ?

we have ~ Avogadros number of atoms to locate

we should either give up on specifying the posit ion ofevery atom or find a trick to help us out

The structure of liquids and glasses

We can use pair distribution functions to describethe structure of such systems

Crystals

Crystals are materials with a regular internalstructure

They have internal translational symmetry

Some structural motif is repeated at regular intervalsthroughout the solid

Just because a material has nice flat faces it is notnecessarily a crystal

Cut crystal glass is not a crystal!

Not all crystals have well developed faces

The structure of crystalline materials

We can use the symmetry of a crystal to reduce thenumber of unique atom positions we have to specify

The most important type of symmetry is translational

this can be described by a lattice

A lattice is just a series of points describing thetranslational symmetry of solid

The lattice points do not represent individual atoms!!

The structure associated with the lattice can be carvedup into boxes (unit cells) that pack together toreproduce the whole crystal structure


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The lattice and unit cell in 1D The lattice

Lattices and unit cells 2 D Identify a lattice and unit cell

Lattice and unit cell in 3D

We specify the lattice using the vectors a, b and c.

The vector between any two lattice points (r) satisfies the

relationship, r = na + pb + qc, where n, p and q are integers

There is no fixed relationship between the positions ofa unit cell and the lattice that is associated with it

The lattice only places constraints on the size and shape of

the unit cell, not position

Lattice and unit cell in 3D


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The unit cell

Always use a right handed axis system

Axis lengths are specified by a, b and c

Interaxial angles are specified by , and

Picking a unit cell for NaCl

Specifying points in crystals

Frequently need to specify the position of a

point, the direction of a line or the orientation

of a plane in a crystal

Can specify a point using

r = (n+u)a + (p+v)b + (q+w)c n,p,q integers

r = (na + pb +qc) + (ua + vb +wc)

u, v, w are fractional coordinates specifying a position

within a unit cell. All positions with the same u, v and w

are symmetry equivalent

Specifying the direction of a line

The direction of any line can be specified by drawing a lineparallel to the one of interest so that it goes through the unit cellorigin and picking any point u, v, w on the line Conventionally, u,v,w are multiplied by the smallest number that

produces integers u,v,w

Denote direction of line using the symbol [uvw] [uvw] are the indices of the lines direction

The use of a square bracket implies that we are talking about a direction

The notation implies that we are talking about alldirections in a crystal that are symmetry equivalent to [uvw]

For example, in a cubic material all the body diagonals of the cube aresymmetry equivalent. So all the directions [111], [-111], [1-11], [11-1], [-1-11], [-11-1], [1-1-1] and [-1-1-1] are symmetry equivalent. All of themare directions of the form .

NOTE typically negative indices are specified using a bar over the numbernot by a negative sign

The orientation of lattice planes

It is possible to describe certain directions and planeswith respect to the crystal lattice using a set of threeintegers referred to as Miller Indices

Miller indices (hkl)

Miller Indices are thereciprocal intercepts of the

plane on the unit cell axes

Identify plane adjacent to

origin can not determine for plane

passing through origin

Find intersection of plane onall three axes

Take reciprocal of intercepts

If plane runs parallel to axis,intercept is at , so Millerindex is 0


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Examples of Miller indices Families of planes

Miller indices describe the orientation and

spacing of a family of planes The spacing between adjacent planes in a

family is referred to as the d-spacing

Three different

families of planes

d-spacing between

(300) planes is one

third of the (100)

spacing

Note all

(100) planes

are members

of the (300)

family

Planes of a form

The symbol {hkl} refers to all planes that are symmetry

equivalent to (hkl). This group of equivalent planes are

referred to as planes of a form.

For the cubic system all the planes (100), (010), (001),

(-100), (0-10) and (00-1) belong to the form {100}

For a tetragonal material a=b c the form {100} wouldonly include (100), (010), (-100), and (0-10)

Planes of a zone

Planes of a zone are

planes that are all parallel

to one axis, the zone axis

Miller indices for all

planes in a zone obey the

relationship hu +kv + lw

= 0, where [uvw] is the

zone axis

Shaded planes belong to the[001] zone

d-spacing formulae

For a unit cell with orthogonal axes

(1 / d2hkl) = (h2/a2) + (k2/b2) + (l2/c2)

Hexagonal unit cells

(1 / d2hkl) = (4/3)([h2 + k2 + hk]/ a2) + (l2/c2)

Unit cells and dhkl


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Lattice point density

Low index lattice planes contain a high density of

lattice points

(010)

(110)

(120)

(210)

(210)

The reciprocal lattice

It is convenient when talking about diffraction to use

the concept of a reciprocal lattice The reciprocal lattice is related to the real space lattice

by:

a1, a2, a3 are the vectors of the real space lattice(alternatively a,b,c) and b1,b2,b3 are the vectors of thereciprocal lattice (alternatively a*,b*,c*).

Note a1.a2xa3 is the unit cell volume

321

321

aaa

aab

=

321

132

aaa

aab

=

321

213

aaa

aab

=

Properties of the reciprocal lattice

Note ai.bj=ij So a1.b1 = 1, but a1.b2 = 0 and a1.b3 = 0 etc.

This is the origin of the term reciprocal lattice.

The reciprocal lattice and real space lattice are orthonormal

Any point on the reciprocal lattice can be specified bya vector Hhkl = hb1 + kb2 + lb3 (hkl are integers)

This vector is perpendicular to the plane in real space withMiller indices (hkl)

The length of this vector Hhkl = 1/dhkl where dhkl is theinterplanar spacing in real space

We get to represent a whole family of planes in real space bya single point in reciprocal space

Geometrical relationship between real and

reciprocal space

Note reciprocal lattice

vector is always

perpendicular to the

corresponding real space plane

Only in orthogonal axis systems

are the real and reciprocal lattice

vectors parallel

Point symmetry elements

In addition to the translational symmetryassociated with a lattice, most materials haveaddition point symmetry

Point symmetry elements operate to change theorientation of structural motifs

A point symmetry operation does not alter at leastone point that it operates on

Point symmetry elements include

rotation axes

mirror planes

rotation-inversion axes

A two fold rotation


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A mirror plane An inversion center

A rotation inversion axis The symmetry elements of a cube

Identify the point symmetry elements

All combinations of point

symmetry elements are not possible

A three fold axis can not just have one two

fold axis perpendicular to it

In three dimensions the existence of twoperpendicular two folds implies the

existence of a third perpendicular two fold

The allowed combinations of point

symmetry elements are called point groups


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Point symmetry elements

compatible with 3D translations

1Center of symmetry

n (= 1,2,3,4,6)Inversion axis

n = 2,3,4,6Rotation axis

mMirror plane

SymbolSymmetry element

Point symmetry and packing

The 32 point groups

Only 32 point groups

are consistent with

periodicity in 3D

Schnflies and Hermann-

Maugin symbols for

crystallographic point

groups

Symbols for symmetry elements 1

Symbols for symmetry elements 2 Unit cell choice

There is always more than possible choice of unit cell

By convention the unit cell is chosen so that it is as

small as possible while reflecting the full symmetryof the lattice

If the unit cell contains only one lattice point is saidto be primitive. If it contains more than one lattice

point it is centered

There are seven distinct shapes of unit cell, that arereferred to as the seven crystal systems


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Unit cell choice in 2D Unit cell choice in 2D

The seven crystal systems

Four three-fold axes at 10923 to

each other

= = = 90

a = b= cCubic

One four-fold axis or one four-

fold improper-axis

= = = 90

a = b!

c

Tetragonal

One six-fold axis or one six-fold

improper axis

= = 90

= 120

a = b! c

Hexagonal

One three-fold axis = = ! 90

a = b= cTrigonal

Any combination of three

mutually perpendicular two-fold

axes or planes of symmetry

= = = 90

a !b ! cOrthorhombic

One two-fold axis or one

symmetry plane

= = 90

! 90

a !b ! c

Monoclinic

None! ! ! 90

a !b ! cTriclinic

Minimum SymmetryUnit CellSystem

Unit cells in 3D

Centering Centering operators

The location of the additional lattice pointswithin the unit cell is described by a set ofcentering operators

Body centered (I) has additional lattice point at

Face centered (F) has additional lattice points at0, 0, and 0

Side centered (C) has an additional lattice point at0


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Centering 3

Not all centering possibilities occur for each of

the seven crystal systems Only 14 unique combinations (Bravais lattices)

Some centering types are not allowed because theywould lower the symmetry of the unit cell

E.g side centered cubic is not possible as this woulddestroy the three fold symmetry that is an essentialcomponent of cubic symmetry

Some centering types are redundant

C-centered tetragonal can always be described using asmaller primitive tetragonal cell

Bravais Lattices

Indexing in the hexagonal system

In hexagonal unit cells it iscommon to refer theorientation of planes andlines to four coordinateaxes

The fourth axis a3 is just= -a2-a1 . This approachreflects the three foldsymmetry associated withthe unit cell

Properties of hexagonal indices

Indices are expressed as(hkil)

h + k = -i

All cyclic permutations ofh, k and i are symmetryequivalent

So (10-10), (-1100),(0-110) are equivalent

Describing crystal structures

The location of all the atoms in a crystalline solid canbe specified by a combination of all the symmetryelements that are appropriate and the fractionalcoordinates for a unique set of atoms (asymmetric unit)

Full symmetry of a crystal is described by its spacegroup

We specify atomic coordinates for a small number ofatoms. We then apply all the symmetry elementsincluding the lattice symmetry to build up the full 3Dstructure.

Note each lattice point may be associated with many atoms

Combining symmetry elements

For three dimensions

32 point groups

14 Bravais lattices

but only 230 space groups

For two dimensions

5 lattices

10 point groups

but only 17 plane groups


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Hexagonal close packing

Each lattice point isassociated with two atoms.One at the corner of the unitcell and one inside the unitcell

Note this structure is anABABAB. repeat ofclosepacked layers

Atomic positions in FCC structure

Can represent atoms on unit cell projection drawingwith heights marked

Can also give atomic coordinates

0,0,0 0,, ,,0 ,0,

Only need to specify these four atoms as others a producedby unit cell translational symmetry

The NaCl (rock salt) structure

Need to specify only one Cl and

Na position. The others are

produced by the lattice centering

operators 0,0,0 0,, ,,0

and ,0,

Na at 0,0,0 and Cl at 0.5,0,0

The ZnS (Zinc Blende) structure

S at 0,0,0 and Zn at 0.25,0.25,0.75

Need to specify only one Zn and S position. The

others are produced by the lattice centering

operators 0,0,0 0,, ,,0 and ,0,

AuBe structure

Structure based on primitive cubiclattice

Au and Be atoms displaced along

the cubic the fold axes of thematerial

No atoms at corners of unit cell!

No four fold symmetry!

Au at uuu, (+u)(-u)-u,-u(+u)(-u), (-u)-u(+u)

Be at www, (+w)(-w)-w,-w(+w)(-w), (-w)-w(+w)

u = 0.100 and w = 0.406

Substitutional solid solutions

The perfect long range order in a crystal can be lost bereplacing some of the atoms with others. This happenswhen solid solutions are formed.

Describe the average structure using lattice coordinatesetc rather than the actual structure


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Crystal shape

The external shape of a crystal is referred to as its

HabitNot all crystals have well defined external faces

Typically see faces on crystals grown from solution

Natural faces always have low indices (orientationcan be described by Miller indices that are smallintegers)

- Law of rational indices

The faces that you see are the lowest energy faces

- Surface energy is minimized during growth

Defects in crystals

No crystal of significant size is perfect

Most crystals contain defects Point defects missing atoms, substituted atoms,

displaced atoms etc.

Line defects dislocations

Planar defects Stacking faults, twins,

Crystallographic shear planes

Stacking faults

Stacking faults occur in wide variety of materials notjust simple metals

Consider a structure to be built up from successivelayers of atoms or other units, if the regular stacking ofthese units is interrupted we have a stacking fault

Close packed metals provide simple examples

Perfect FCC has a ABCABCABCABCABC sequence

A, B, C represent different close packed (111) layers

The sequence ABCABCBCABCABC has a stacking fault

Perfect HCP is ABABABABABABAB

ABABABCABABABABAB has a stacking fault

Faults that put two of the same layers together AA BB or CCare unlikely due to their very high energy

Twins

Sometime the regular internal structure of a crystal willbe interrupted in such a way that two pieces of thecrystal are related to one another by rotation about anaxis or reflection in a mirror plane

We then have a rotation or a reflection twin

Twin because the two components of the twin are related to each otherby a symmetry operation and have a plane of atoms in common(Composition plane)

Reflection plane is twin plane

Rotation axis is twin axis

In one piece of material this twinning process can occurmultiple times leading so several components all with a well

defined relative orientation

Twins in close packed metals

Annealing twins often form in FCC metals such as

Cu, Ni, -brass, Al that have been cold worked anannealed

As the crystal regrows a stacking fault leads to the formationof twins

Components are related by 180 rotation around

Deformation twins can occur in deformed HCP metals

such as Zn, Mg, Be and BCC metals (-Fe)

They are reflection twins

Appearance and structure of annealing twins

In (a) only two components are

present in a crystal in (b) there are

three components. B is twin band

sandwiched between components

A1 and A2 with same orientation as

each other. The atomic structure of

such as twin band in an FCC metal

is shown on the right.

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Crystal Geometry 6up