Understanding Diffraction

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Understanding Diffraction

• Understanding Diffraction.(Using some
  illustrative but crude analogies!).
• Understanding Crystallite size broadening.(The
  basis of Scherers formula).

Analogies are very brittle? do not stretch them!
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Diffraction

• Diffraction sometimes seems a mysterious
  phenomenon, which is difficult to understand.
• We have noted that electromagnetic waves (light,
  X-rays etc.), water waves (i.e. elastic waves in
  a solid or fluid), matter waves (electrons,
  neutrons) etc. can be diffracted.? In fact it is
  best to start understading diffraction using
  water waves with a single slit.
• Diffraction can be thought of as a special case
  of constructive and destructive interference (a
  case where there is a large number of
  scatterers).? What are these scatterers? A
  Any entity which impedes (partially and
  redirects) the path of wave can be conceived as
  a scatterer. Scatterers has to be understood in
  conjunction with the wave being considered i.e.
  an entity may be a scatter for one kind of
  waves, but not for another (e.g. an array of
  atoms is a scatterer for X-rays, but it is not a
  scatterer for water waves). ? In a periodic
  array they can be entities of the motif i.e. a
  geometrical entity (atoms, ions, blocks of wood),
  physical property (e.g. aligned spins) or a
  combination of both.
• Experiments have been conducted where matter
  waves have been diffracted from a crystal made
  of electromagnetic radiation (waves)! (Atoms
  diffracted from a Laser lattice).
• We will use some crude analogies and some
  schematic cartoons to get a hang of this
  phenomenon ? these should not be taken literally.

Usually in a periodic array. Though other
simple configurations may be envisaged.
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What is the minimum I need to see
diffraction/interference?

• The bare minimum is one edge. Two edges forming
  a single slit is better to get a better picture.
• An experiment with two slits (e.g Youngs double
  slit experiment), is very fascinating in many
  respects.

• The phenomenon of diffraction is often stated as
  bending of waves around edges.
• When a wave encounters a slit, this aspect leads
  to intensity in the geometrical shadow region of
  the slit (i.e. the interference pattern observed
  on the screen consists of peaks and valleys, with
  some intensity pattern in the geometrical shadow
  region as well).
• One method to understand this is the Huygens
  wavelet construction. (Wherein, each point in the
  unobstructed wave-front is considered as a source
  of wavelets. These wavelets propagate to the
  screen and lead to a interference pattern).

Which blocks part of the wave.
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Let us start by throwing some balls on a wide
slit.
Geometrical shadow region of zero intensity
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Obstacle
Near the edges the intensity will be different
(but we will ignore this for now)
Intensity on screen
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If we shine incoherent light we will get a
similar intensity distribution.
Screen
The balls in the gap pass right through in a
straight line (well most of the ones!), while
the ones blocked by the obstruction reflect back
(reflection not shown).
Intensity no. of balls/area/time
Warning these cartoons do not depict
diffraction- they are a way to start visualizing
the issues!
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What about the ones hitting the edge?
More cartoons on network
0
Altered intensity pattern (this is not one peak
but a broad diffuse one as the way the balls hit
the barrier edge will send them off in different
angles)
Intensity on screen
Obstacle
1
Screen
Centre of mass near edge.
Glancing angle collision.
This is not what happens in diffraction. This is
to tell you that watch out for sharp corners!!
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What if the slit width is of the order of the
ball size?
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There is intensity in the geometrical shadow
region as well!
Region of geometrical shadow

• So we have seen that even with macroscopic balls
  it is possible to get intensity in the region
  of the geometrical shadow.
• For this effect to be prominent we have noticed
  that the slit width has to be of the order of the
  size of the ball.

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Another crude analogy to understand diffraction

• Consider a series of speed breakers (bumps) on
  the road (say sinusoidal bumps of amplitude A
  ?/2). Let a vehicle (with tire diameter d)
  arrive at a velocity v.
• Three cases can be considered here (i) d gtgt ?
  (large dump trucks, with tire diameter twice the
  height of a human) , (ii) d ? (a scooter tire),
  (iii) d gtgt ? (a toy car).
• In the first and last scenarios the vehicle does
  not feel any jerk. In the case of the scooter
  tire, we feel a lot of jerk and thrown about a
  lot in the z-direction.
• This analogy helps us visualize the statement
  if the wavelength is of the order of the scatter
  spacing then we are in the diffraction dominated
  regime.

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Understanding constructive and destructive
interference

• In these set of slides we will try to visualize
  how constructive and destructive interference
  take place (using the Braggs view of diffraction
  as reflection from a set of planes).
• It is easy to see as to how constructive
  interference takes place however, it is not that
  easy to see how rays of the Bragg angle go
  missing.

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(press page down button to see the successive
graphics)
Constructive Interference
Here we see waves scattered from two successive
planes interfering constructively. (A phase
shift of ? occurs at Y on scattering)
Ray-1
Ray-2
Plane-1
d(hkl)
Plane-2

• Note the phase difference of ? introduced during
  the scattering by the atom at Y.
• The extra path length travelled by Ray-2 (as
  compared to Ray-1) is ?/2 ?/2 ?.
• In general for constuctive interference the path
  difference has to be an integral multiple of ?
  1?, 2?, 3?, 4?,...

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The rays ray-1 and ray-2 arrive in phase in the
X-B plane.
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Assuming that path difference of ? gives
constructive interference Similar to the path
difference of ?, path difference of 2?, 3? n?
also constructively interfere.
Path difference between Ray-1 Ray-2
1st order reflection
All Constructively interfere
2nd order reflection
3rd order reflection
Also to be noted is the fact that if the path
difference between Ray-1 and Ray-2 is ? then the
path difference between Ray-1 and Ray-3 is 2? and
Ray-1 and Ray-4 is 3? etc.
Going across planes
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Destructive Interference
Exact destructive interference (between two
planes, with path difference of ?/2) is easy to
visualize. The angle is not Braggs angle (let us
call it ?d ).
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Destructive Interference
At a different angle ? the waves scattered from
two successive planes interfere (nearly)
destructively
Warning this is a schematic
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• In the previous example considered ? was far
  away (at a larger angular separation) from
  ?Bragg and it was easy to see the (partial)
  destructive interference.
• In other words for incidence angle of ?d (couple
  of examples before) the phase difference of ? is
  accrued just by traversing one d.
• If the angle is just away from the Bragg angle
  (?Bragg), then one will have to go deep into the
  crystal (many d) to find a plane (belonging to
  the same parallel set) which will scatter out of
  phase with this ray (phase difference of ?) and
  hence cause destructive interference.
• In the example below we consider a path
  difference of ?/10 between the first and the
  second plane (hence, we will have to travel 5
  planes into the crystal to get a path difference
  of ?/2).

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• If such a plane (as mentioned in the page before)
  which scatters out of phase with a off Bragg
  angle ray is absent (due to finiteness of the
  crystal) then the ray will not be cancelled and
  diffraction would be observed just off Bragg
  angles too ? line broadening!(i.e. the
  diffraction peak is not sharp like a ??-peak in
  the intensity versus angle plot)
• Line broadening can be used to calculate
  crystallite size (grain size).
• This is one source of line broadening. Other
  sources include residual strain, instrumental
  effects, stacking faults etc.

Click here to know more about peak broadening.