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
 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
    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
   Radiation                          Method
White           Laue: stationary single crystal
Monochromatic   Powder: specimen is polycrystalline, and therefore
                all orientations are simultaneously presented to the
                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
                about beam direction
The Bragg Law
            X-ray beam encounters a
               3-d lattice array at left.
               Assume the following:
             A third-order cone
               about OA
             A second-order cone
               about OB
             A first-order cone about

            We assume these
            cones intersect at a
            common line satisfying
            the diffraction condition.
   The rays scattered by
    adjacent atoms on OA
    atoms have a path
    difference of three
   Those about OB have
    a path difference of two
   About OC, one
    wavelength difference
   These points of
    coherent scatter define
    a plane with intercepts
    2a, 3b, 6c (A’’, B’’, C’’)
    and a Miller index of
      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
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

                       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
   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.

V*       a * b * c * (1  cos2  *  cos2  *  cos2  * 2 cos * cos  * cos *)1/ 2
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)

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

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

 Powders (a.k.a. polycrystalline aggregates)
  are billions of tiny crystallites in all possible
 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
                                                                   with each
                                                                   point in the
                                                                   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-
rings from
the d*100
Note the
1st and 2nd
cones, and
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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