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Transcript of 59-553_Notes3
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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!
Chem 59-553 Planes in Lattices and Miller Indices
(110) planes (130) planes
a
b
(-210) planes
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Chem 59-553 Planes in Lattices and Miller Indices
Chem 59-553 Planes in Lattices and Miller Indices
(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:
Chem 59-553 Planes in Lattices and Miller Indices
Pictures from: http://www.gly.uga.edu/schroeder/geol6550/millerindices.html
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Chem 59-553 Planes in Lattices and Miller Indices
Chem 59-553 Planes in Lattices and Miller Indices
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Chem 59-553
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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Chem 59-553
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 cos- cos) /( 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.