# Diffraction Basics

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```					               Diffraction Basics
The qualitative basics:
 Coherent scattering around atomic scattering centers
occurs when x-rays interact with material
 In materials with a crystalline structure, x-rays
scattered in certain directions will be “in-phase” or
amplified
 Measurement of the geometry of diffracted x-rays can
be used to discern the crystal structure and unit cell
dimensions of the target material
 The intensities of the amplified x-rays can be used to
work out the arrangement of atoms in the unit cell
The Generalized 2D Laue Equation:

p(cos  cos  )  h

(h is the order of the diffraction, here 0 or 1)
In the specialized case where the angle of
incidence  is 90° the equation becomes:

p cos  h
For a two-dimensional lattice array of atoms, the
Laue equations are:
a(cos 1  cos 1 )   h
b(cos 2  cos  2 )  k
The Laue diffraction cones for the A and B
directions are shown below:
Diffraction will only occur when the diffraction
angles define the same direction. In the case
below this is when the cones intersect to form the
lines OX and OY
In a three-dimensional lattice array, there will be
multiple Laue diffraction cones. Below a simple
diagram shows three first order cones in ABC
space
There are now three Laue equations requiring a
simultaneous solution (i.e., there must be a
diffraction direction common to all three cones):
a(cos 1  cos 1 )   h
b(cos 2  cos  2 )  k
c(cos 3  cos  3 )  l

 A unique solution is difficult to obtain
 In Laue diffraction, the crystal is fixed and oriented
with a lattice axis parallel to the beam
  is varied by using “white” radiation
 With monochromatic radiation, movement of the
crystal is required for diffraction to occur
White           Laue: stationary single crystal
Monochromatic   Powder: specimen is polycrystalline, and therefore
all orientations are simultaneously presented to the
beam
Rotation, Weissenberg: oscillation,
De Jong-Bouman: single crystal rotates or oscillates
about chosen axis in path of beam
Precession: chosen axis of single crystal precesses
The Bragg Law
X-ray beam encounters a
3-d lattice array at left.
Assume the following:
 A third-order cone
 A second-order cone
OC

We assume these
cones intersect at a
common line satisfying
the diffraction condition.
   The rays scattered by
atoms have a path
difference of three
wavelengths
a path difference of two
wavelengths
wavelength difference
   These points of
coherent scatter define
a plane with intercepts
2a, 3b, 6c (A’’, B’’, C’’)
and a Miller index of
(321)
The Bragg Law “bottom line”:
A diffraction direction defined by the intersection of the
hth order cone about the a axis, the kth order cone about
the b axis and the lth order cone about the c axis is
geometrically equivalent to a reflection of the incident
beam from the (hkl) plane referred to these axes.

in other words:

Diffraction from a lattice array of points may be
functionally treated as reflection from a stack of planes
defined by those lattice points
On the previous diagram, the “reflected” rays combine to
form a diffracted beam if they differ in phase by a whole
number of wavelengths, that is, if the path difference AB-
AD = n where n is an integer. Therefore

d                                      d
AB              and       AD  AB cos 2        (cos2 )
sin                                   sin 

d     d
n             (cos2 )
sin  sin 
d                     d
       (1  cos 2 )        (2 sin 2  )
sin                  sin 

n  2d sin 
In the Bragg Law,   n  2d sin             , n is the order of diffraction

Above are 1st, 2nd, 3rd and 4th order “reflections” from the (111)
face of NaCl. By convention, orders of reflections are given as
111, 222, 333, 444, etc. (without the parentheses)
The Reciprocal Lattice
Problems addressed by this unusual mental exercise:

   How do we predict when diffraction will occur in a
given crystalline material?
– How do we orient the X-ray source and detector?
– How do we orient the crystal to produce diffraction?
   How do we represent diffraction geometrically in a
way that is simple and understandable?
The first part of the problem
Consider the diffraction from the (200) planes of a (cubic) LiF
crystal that has an identifiable (100) cleavage face.
To use the Bragg equation to determine the orientation required
for diffraction, one must determine the value of d200.
Using a reference source (like the ICDD database or other tables
of x-ray data) for LiF, a = 4.0270 Å, thus d200 will be ½ of a or 2.0135
Å.
From Bragg’s law, the diffraction angle for Cu K1 ( = 1.54060)
will be 44.986 2. Thus the (100) face should be placed to make an
angle of 11.03 with the incident x-ray beam and detector.
If we had no more complicated orientation problems, then we
would have no need for the reciprocal space concept.
Try doing this for the (246) planes and the complications become
immediately evident.
The second part of the problem
Part of the problem is the three dimensional nature of the
diffracting planes. They may be represented as vectors where
dhkl is the perpendicular from the origin to the first hkl plane:

While this is an improvement, the graphical representation is
still a mess – a bunch of vectors emanating from a single point
radiating into space as shown on the next slide ----
   Ewald proposed that instead of plotting the dhkl
vectors, that the reciprocal vector be plotted, defined
as:

1
d   *
hkl   
d hkl

   The units are in reciprocal angstroms and defines a
reciprocal space.
   The points in the space repeat at perfectly periodic
intervals, defining a space lattice called a reciprocal
lattice
   Figure 3.3 can now be reconstructed plotting the
reciprocal vectors instead of the dhkl vectors
   The comparison is shown in the following slides
Any lattice vector in the reciprocal lattice represents a set of
Bragg plans and can be resolved into its components:

d   *
hkl    ha  kb  lc
*         *        *

In orthogonal crystal systems, the d and d* are simple
reciprocals. In non-orthogonal systems, the reciprocals (since
they are vectors) are complicated by angular calculations

Because the angle  is not 90, the calculation of d* and a*
involve the sin of the interaxial angle.
The table below shows the relationships between axes in direct
and reciprocal space. At the bottom is a very complex
trigonometric function that defines the parameter V used in the
triclinic system.

1
V*       a * b * c * (1  cos2  *  cos2  *  cos2  * 2 cos * cos  * cos *)1/ 2
V
Figure 3.7 shows the arrangement where the (230) point is brought into
contact with the Ewald sphere.

1                         d *( 230 )
By definition   CO             and         OA 
                              2

OA d *( 230) / 2                           2 sin 
hence   sin                                          
CO    1/                                  d *( 230)

1
from the definition           d ( 230)   
of the reciprocal vector                   d *( 230)

substitution yields:         2d ( 230) sin                  The Bragg
Relationship!
The Powder Diffraction Pattern

 Powders (a.k.a. polycrystalline aggregates)
are billions of tiny crystallites in all possible
orientations
 When placed in an x-ray beam, all possible
interatomic planes will be seen
 By systematically changing the
experimental angle, we will produce all
possible diffraction peaks from the powder
   There is a d*hkl
vector
associated
with each
point in the
reciprocal
lattice with its
origin on the
Ewald sphere
at the point
where the
direct X-ray
beam exists.

   Each crystallite located in the center of the Ewald sphere has its own
reciprocal lattice with its orientation determined by the orientation of the
crystallite with respect to the X-ray beam
The Powder Camera
The Debye-
Scherrer
powder
camera
Debye
diffraction
rings from
the d*100
reflection.
Note the
1st and 2nd
order
cones, and
“back”
reflections
Some Debye-Scherrer Powder Films
The Powder Diffractometer
   Think of the diffractometer as a device for measuring
diffractions occurring along the Ewald sphere – it’s
function is to move all of the crystallites in the powder
and their associated reciprocal lattices, measuring
diffractions as they intersect the sphere

   Because of the operational geometry of diffractometers,
there must be a very large number of small crystallites
(a.k.a., “statistically infinite amount of randomly oriented
crystallites”) for the diffractometer to “see” all of the
possible diffractions

   By convention (but not by accident – note Fig 3.7)
diffraction angles are recorded as 2. Data are
commonly recorded as 2 and intensity
Next week:
Diffraction Intensity: Origin,
Variations, Extinctions and Error
Sources in diffraction experiments

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