Lecture 4: Diffraction

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Lecture 4 Diffraction

• PHYS 430/603 material
• Laszlo Takacs
• UMBC Department of Physics

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How to determine the atomic structure of
materials?

• The distance between atoms in a solid is about
  0.25 nm while the wavelength of visible light is
  500 nm. For light, any condensed matter seems
  continuous.
• To see atoms, we need radiation with wavelength
  shorter than the atomic distance, ? lt 0.25 nm
• X-rays E hf hc/ ?? 5 keV, easy to obtain
  from characteristic radiation or from synchrotron
  light source. E.g. Cu K? 0.154 nm.
• Neutrons E p2/2m h2/2m?2 0.0131 eV. The
  corresponding temperature of random motion T
  2E/kB 304 K Room temperature.
• Electrons E h2/2m?2 24.1 eV. We usually use
  much higher energies in an electron microscope, ?
  much smaller.
• We can see atomic structure in several ways
• Using diffraction
• Atomic resolution electron microscopy
• Field ion microscopy
• Scanning microscopies

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Constructive interference is observed, if the
path difference between the two interfering rays
is an integer multiple of the wavelength.

• Youngs experiment Braggs law
• d sin(?) n? ?d sin(?) n?, usually n 1
• If this condition is not satisfied, partial
  cancellation is possible for two (or a finite
  number of) scattering slits or lattice planes.
  The cancellation is complete for an infinite
  (very large) number of slits - diffraction
  grating or lattice planes - ideal crystal.

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Debye - Scherrer method
Consider a set of lattice planes with interplanar
distance d. Braggs law specifies the
corresponding angle ?. Constructive interference
occurs only if the angle between the incoming
X-ray beam and the lattice planes is ? and it
exits so that the diffraction angle is 2? and the
incoming and diffracted beams are in the same
plane perpendicular to the lattice planes (like
for a mirror.) In a powder sample, grains with
any orientation exist with about the same
probability, thus rather than nothing (wrong ?)
or just a spot, a cone of diffracted rays is
obtained. A separate cone will exist for every
possible set of lattice planes, giving a series
of arcs on the film.
The Debye-Sherrer method used to be the workhorse
of materials science. A major disadvantage is the
use of film, that requires chemical processing
and reading the film. It also limits accuracy.
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Debye-Scherrer cameras
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Example Diffraction pattern of polycrystalline
Al using Cu K? radiation

• Al is fcc, a 0.4050 nm l 1.
• D a / sqrt (h2 k2 l2), for fcc h, k, l are
  either all even or all odd due to interference
  form atoms within a single unit cell.
• ? 0.1542 nm, the average from Cu K?1 and K?2.

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• Braggs law is sufficient to predict where (at
  what angle) diffraction can occur. It cannot
  predict the intensities for that the diffraction
  process has to be analyzed in full detail
• X-rays are scattered by electrons as point
  charges. Even the intensity of scattering on a
  single electron is angle dependent.
• Atoms contain several electrons with some
  probability density distribution. The
  inter-atomic diffraction is not fully
  constructive, it is described by the atomic form
  factor.
• Lattice planes are not ideal mirrors, the
  reflection from them is a diffraction phenomenon
  form atoms within the plane. This is described by
  another form factor. If the unit cell contains
  more than one formula unit, there are missing
  peaks, angles at which the interference is
  destructive from a unit cell.
• There are numerous additional factors, like
  absorption, finite grain size, lattice
  distortions, and imperfections of the X-ray
  optics.
• Evaluation is carried out by
• Comparing to a standard pattern - JCPDF card,
  basically fingerprinting.
• Determining the structure of an unknown material
  by solving the reverse problem. Very difficult
  from powder data.

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The spectrum of an X-ray tube with Mo anode and
the effect of filtering. K? remains a doublet,
but can be separated numerically, or a
monochromator can be used.Other frequently used
anode materials are Cu and Cr. E? hc(
1.2398, if measured in keV and nm)
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X-ray optics of a powder diffractometerThe
principle is identical to the Debye-Sherrer
method, but detection is electronically, rather
than on a film.F anode bombarded with
electrons SS Soller slits to limit axial
divergence (out of plane)Div divergence slitP
sampleR receiving slitQ anti-scatter
slitFrom here the X-ray beam enters the
detector.The diffraction angle is defined by F,
P, and R.
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Focusing geometry

• The diffraction angle is about the same from a
  large area of the sample, if the sample is
  tangent to the circle formed by the anode,
  sample, and receiving slit.

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Laue method Determination of the orientation and
symmetry of a single crystal
White X-rays are used Braggs law is always
satisfied for one wavelength or the other. Only
the orientation of the lattice planes
matters. Reflections from planes belonging to the
same zone show up as hyperbolas. Rotational
symmetry result in symmetrical pattern.
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Determination of orientation/ texture

• The wavelength is fixed. The angle from source to
  sample to detector is set to the Bragg angle
  corresponding to the distance between the desired
  set of lattice planes.
• If the sample is a single crystal, reflection is
  detected only if the orientation of the lattice
  planes satisfies the basic mirror rules - its
  normal in the plane of the beam, angles on both
  sides equal.
• For a polycrystal, the intensity is proportional
  to the fraction of properly oriented
  crystallites. Intensity as a function of
  orientation is measured.