new crystal direction and planes

7/27/2019 New Crystal Direction and Planes

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ial Sciences and Engineering,

Material Sciences and Engineering MatE271 Week 4 3

Crystallographic Planes & Directions

o Many material properties and processes vary withdirection in the crystal

o It is often necessary to be able to specify certaindirections and planes in crystals.

o Directions and planes are described using threeintegers - Miller Indices

direction plane


Indexing in 2D

o Determine in unit distances to move from onelattice point to the next in the plane (or direction).

o Put in x,y format.

(11)

(21)

(41)

(10)

(13)

Properties:

Lowest Indices - Greatest plane spacing

Lowest Indices - Greatest density of lattice points

This is true in 3-D as well


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General Rules for Lattice Directions, Planes& Miller Indices

o Miller indices used to express lattice planesanddirections

o x, y, z are the axes (on arbitrarily positionedorigin)

in some crystal systems these are notmutually

o a, b, c are lattice parameters (length of unit cellalong a side)

o h, k, l are the Miller indices for planes anddirections - expressed as (hkl) and [hkl]


o Conventions for naming

There are NO COMMAS between numbers

Negative values are expressed with a bar overthe number (-2 is expressed 2)

o Crystallographic direction:

[123]

[100]

Miller Indices for Directions


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Miller Indices for Directions

Recipe

Draw vector, define tail asorigin.

Determine length in unitcell dimensions, a, b, and c

Remove fractions bymultiplying by smallestpossible factor

Enclose in square brackets

What is ??? x = 1/2, y = 0, z = 1

[1/2 0 1] -> [1 0 2]

x

y

z

[111]

[110]

[100]

[???]

Entire lattice can bereferenced by one

unit cell!


x

y

z

x

y

z

x

y

z

x

y

z

x

y

z

x

y

z

[111] [110] [111]

[210] [010] [111]

Example - Naming Directions


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Example - Drawing Directions

o Draw [112] [111] and [222]

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Families of Directions

o Equivalence of directions

o Family of directions

e.g. [123], [213], [312], [132], [231]

(only in a cubic crystal)

In the cubic system directions having the sameindices regardless of order or sign are equivalent

[101] = [110] [101] [110]

cubic tetragonal


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Miller Indices for Planes

o (hkl) Crystallographic planeo {hkl} Family of crystallographic planes

e.g. (hkl), (lhk), (hlk) etc.

In the cubic system planes having the sameindices regardless of order or sign are equivalent

o Hexagonal crystals can be expressed in a fourindex system (u v t w)

Can be converted to a three index system using

formulas


Miller Indices for Planes

Recipe

If the plane passes through theorigin, select an equivalentplane or move the origin

Determine the intersection ofthe plane with the axes in

terms of a, b, and c

Take the reciprocal (1/ = 0)

Convert to smallest integers(optional)

Enclose by parentheses

x

y

z

(111)

Note - plane // to axis,

intercept = and

1/ = 0


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Crystallographic Planes

x

y

z

x

y

z

x

y

z

x

y

z

x

y

z

x

y

z

(111) (011)

(201) (212) (100)

(001)


X-Ray Diffraction

o Can be used to determine crystal structure(and hence identity of an unknown material)

o Diffraction occurs whenever a waveencounters a series of regularly spaced objects

that; Can scatter the wave

Have a spacing comparable to the wavelength

o X-ray wavelength ~ inter-atomic spacing andare scattered by atoms.


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Constructive & Destructive Interference

Maximum and

minimum results from

two diffracting beams

phase shifted from one

another.

Constructive

Destructive


Braggs Law

n= SQ + QT = dhklsin + dhklsin = 2 dhklsin

n = 1,2,3order of reflection

o For constructive interference, the additional path lengthSQ+QT must be an integral number of wavelengths: Realdiffraction is more complicated for non-simple cubic

X-ray Source:

Monochromatic

and in-phase

P

S TQ


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o Real diffraction is more complicated for non-simple cubicsystems because some sets of atoms (e.g. BCC centeratoms) can produce out of phase scattering at certainBragg angles . Net effectsome of the diffractedbeams, that according to Braggs Law should be present,are cancelled out.

o Example - for diffraction to occur:

o Magnitude of difference between two adjacent andparallel planes of atoms is function of Miller Indices andthe lattice parameter. For cubic symmetry:

Braggs Law, Cubic Symmetry

dhkl = a/(h2 + k2 + l2)1/2

BCC - h + k + l must be even

FCC - h, k, l must all be either even or odd


Diffractometer Technique

o Use powder (or polycrystalline) sample to guaranteesome particles will be oriented properly such that everypossible set of crystallographic planes will be available fordiffraction.

o Each material has a unique set of planar distances andextinctions, making X-ray diffraction useful in analysis ofan unknown.


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Example

o For BCC Fe, computea) the interplanar spacing

b) the diffraction angle for (220) set of planes.

The lattice parameter for Fe is 0.2866 nm and thewavelength used is 0.1790 nm. Consider 1st orderreflections only.


Reading Assignment

Shackelford 2001(5th Ed)

Read: pp 88, 101-110

Check class web site:

www.public.iastate.edu\~bastaw\courses\Mate271.html 2

new crystal direction and planes

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