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Chem 59-553 Planes in Lattices and Miller Indices

An essential concept required to understand the diffraction of X-rays by

crystal lattices (at least using the Bragg treatment) is the presence of

planes and families of planes in the crystal lattice. Each plane is

constructed by connecting at least three different lattice points together

and, because of the periodicity of the lattice, there will a family (series) of

planes parallel passing through every lattice point. A convenient way todescribe the orientation of any of these families of plane is with a Miller

Index of the form (hkl) in which the plane makes the intercepts with a unit

cell ofa/h, b/k and c/l. Thus the Miller index indicates the reciprocal of the

intercepts.

2-D planes

Note: If a plane does not

intersect an axis, the intercept

would be and the reciprocal is

0.

Note: If the reciprocal of the

intercept is a fraction, multiply

each of the h, k and l values by

the lowest common

denominator to so that they

become integers!


(110) planes (130) planes

a

b

(-210) planes

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(100) face

[100] vector

(100) planes

(-100) face

The orientation of planes is best represented by a vector normal to the

plane. The direction of a set of planes is indicated by a vector denoted by

square brackets containing the Miller indices of the set of planes. Miller

indices are also used to describe crystal faces.

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Chem 59-553

(hkl) denotes a set of planes

[hkl] designates a vector (the direction of the planes)

{hkl} set of faces made equivalent by the symmetry of the

system, thus:

{100} for point group 1 this refers only to the (100) face

{100} for point group -1 this refers to (100) and (-100) faces

for mmm {111} implies

(111),(11-1),(1-11),(11-1),(-1-11),(-11-1),(1-1-1),(-1-1-1)

Planes in Lattices and Miller Indices

A summary of notation that you will see in regard

to planes and/or crystal faces:


Pictures from: http://www.gly.uga.edu/schroeder/geol6550/millerindices.html

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Note that for hexagonal systems, the Miller-Bravais indices are often used

instead. These have the form (hkil), where h, k, and i are the reciprocals of

the plane intercepts for the three co-planar vectors indicated below and l is

the reciprocal for the intercept in the c direction. Note that h, k and i are not

linearly independent so the rule h+k+i = 0 must always be obeyed.

Planes in Lattices and Miller Indices

b

a

Chem 59-553 Planes in Lattices and Braggs Law

We are interested in the planes in a crystal lattice in the context of X-ray

diffraction because of Braggs Law:

n = 2 d sin()Where:

n is an integer

is the wavelength of the X-rays

d is distance between adjacent

planes in the lattice

is the incident angle of the X-

ray beam

Braggs law tells us the conditions that must be met for the reflected X-raywaves to be in phase with each other (constructive interference). If these

conditions are not met, destructive interference reduces the reflected

intensity to zero!

W.H.Bragg and son W.L.Bragg were awarded the Nobel prize in 1915.

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Chem 59-553 Simple derivation of Braggs Law

Braggs Law can be derived using simple geometry by considering the

distances traveled by two parallel X-rays reflecting from adjacent planes.

The X-ray hitting the lower plane must travel the extra distance AB and BC.

To remain in phase with the first X-ray, this distance must be a multiple of the

wavelength thus:

n = AB+BC = 2AB

(since the two triangles are identical)

The distance AB can be expressed in terms

of the interplanar spacing (d) and incident

angle () because d is the hypotenuse ofright triangle zAB shown at right.

Remember sin = opposite/hypotenuse

sin() = AB/d thus AB = d sin()

Therefore:

n = 2 d sin()

Note: d and sin() are inversely proportional

(reciprocal). This means that smaller values

of d diffract at higher angles this is the

importance of high angle data!

Chem 59-553 Diffraction of X-raysYou may wonder why to X-rays reflect in this way and what is causing them

to reflect in the first place. The actual interaction is between the X-rays and

the ELECTRONS in the crystal and it is a type of elastic scattering. The

oscillating electric field of the X-rays causes the charged particles in the

atom to oscillate at the same frequency. Emission of a photon at that

frequency(elastic) returns the particles in the atom to a more stable state.

The emitted photon can be in any direction and the intensity of the scattering

is given by the equation:

I(2) = Io [(n e4)/(2 r2 m2 c4)] [(1 + cos2(2))/2]

I(2) = observed intensity

Io = incident intensity

n = number of scattering sources

r = distance of detector from scattering source

m = mass of scattering source

c = speed of light, e = electron charge, [(1 + cos2(2))/2] is a polarization factor

Note that the mass of the scattering particle (m) is in the denominator this

means that the scattering that we see is attributable only to the electrons

(which have masses almost 2000 times less than that of a proton).

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Max von Laue derived a different set of equations describing the in phase

diffraction of X-rays by a line of scattering objects (note that the n in the

diagram below is the integer corresponding to the integer n in the Bragg

equation). Each line of objects generates cones of in phase scattering that

follow the equations:

a(cos 1 cos 1) = h (for a line in the a direction)

b(cos 2 cos 2) = k (for a line in the b direction)

c(cos 3 cos 3) = l (for a line in the cdirection)

Laues interpretation

Where is the angle between the incident beam

and the line and is the angle between the cone

and the line of scatterers. In three dimensions, a

reflection will only be observed at the intersection of

the cones in all three directions (all three equations

are satisfied).

With a little geometry (see Ladd and Palmer 3.4.3),

it can be shown that this treatment is equivalent to

Braggs law.

Chem 59-553 Summary of Diffraction by PlanesIf they interact with electrons in the crystal, incident X-rays will be scattered.

Only the X-rays that scatter in phase (constructive interference) will give

rise to reflections we can observe. We can use Braggs Law to interpret the

diffraction in terms of the distance between lattice planes in the crystal based

on the incident and diffraction angle of the reflection.

Note: The diffraction angle is

generally labeled 2 because

of the geometric relationship

shown on the left.

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Chem 59-553 The Reciprocal LatticeBecause of the reciprocal nature of d spacings and from Braggs Law, thepattern of the diffraction we observe can be related to the crystal lattice by a

mathematical construct called the reciprocal lattice. In other words, the

pattern of X-ray reflections makes a lattice that we can use to gain

information about the crystal lattice.The reciprocal lattice is constructed as

follows:

Choose a point to be the origin in the crystal

lattice.

Let the vector normal to a set of lattice planes

in the real lattice radiate from that origin point

such that the distance of the vector is the

reciprocal of the d spacing for each family of

planes. i.e. the vector for the plane (hkl) has

a distance of 1/d(hkl) (or, more generally

K/d(hkl)).

Repeat for all real lattice planes.

You can see how this works at: http://www.doitpoms.ac.uk/tlplib/reciprocal_lattice/index.php

or: http://www.xtal.iqfr.csic.es/Cristalografia/index-en.html

Chem 59-553 The Reciprocal LatticeThis procedure constructs a reciprocal lattice (RL) in which each lattice point

corresponds to the reflection that is generated by a particular family of

planes. This lattice can easily be indexed by assigning the proper (hkl) value

to each lattice point.

Note that consequence of this reciprocal relationship include:

-Large d spacings correspond to small spacings in the RL this is an

important feature that must be considered during data collection.

- Obtuse angles in the real lattice correspond to obtuse angles in the RL

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Chem 59-553 The Reciprocal LatticeFor those of you who are comfortable with vectors, here is how the reciprocal lattice is built:

Note that : a a*= 1 (etc. this is the reciprocal part)

a b*= 0 (etc. the vectors are orthogonal in this geometry)

Thus the reciprocal lattice can be represented by vectors of the form:

Rhkl= ha* + kb* + lc*,

| Rhkl| = K / dhkl

where h, k, and lare the indices of sets of planes in the crystal, and K can assume the value of

1, , or 2, depending on the user's convention (crystallography, solid-state physics, etc). In

later discussions, K will be assumed to have a value of 1. K is shown in the relations below for

completeness.

Thus the individual lattice vectors have the following definitions:

a* = K (b c) / (a (b c)) a = (b* c*) / K (a* (b* c*))

b* = K (c a) / (b (c a)) b = (c* a*) / K (b* (c* a*))

c* = K (a b) / (c (a b)) c= (a* b*) / K (c* (a* b*))

cos* = (cos coscos) /( sin sin) cos= (cos* cos* - cos*) /( sin* sin*)

cos* = (coscos- cos) /( sinsin) cos = (cos* cos* - cos*) /( sin* sin*)

cos* = (coscos - cos) /( sinsin) cos= (cos* cos* - cos*) /( sin* sin*)

V = a b c= 1/V* = abc (1 - cos2- cos2 - cos2+ 2 coscos cos)

V* = a* b* c* = 1/V = a*b*c* (1 - cos2* - cos2* - cos2* + 2 cos* cos* cos*)

Chem 59-553 The Reciprocal LatticeSome of the important relationships between the real lattice and the

reciprocal lattice (in non-vector notation) are summarized here. Note that K

= 1 in these equations.

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