X-ray%20diffraction%20and%20minerals

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X-ray diffraction and minerals
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Diffraction bending of wavefront past an
obstacle.
Two adjacent sources of waves produce a
diffraction pattern as waves interfere
constructively (i.e. add their amplitudes).
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Source of X-rays a filament is heated to boil
off electrons. Electrons are accelerated towards
a metallic target.
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X-rays are generated by dislodging inner-shell
electrons in the metallic target. Higher-shell
electrons drop in the empty energy level.
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The target, hit by electrons, emits a broad
spectrum of X-rays of various wavelengths. Most
of it is block by a filter, and only the highest
intensity, of a nearly unique wavelength, is
kept.
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• Diffraction pattern
• - incoming X-ray hits the mineral
• the X-rays excite electrons of atoms in the
  mineral being investigated.
• inner-shell electrons scatter back the X-rays as
  they undergo transitions among energy levels

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• The Laue diffraction experiment (1912)
• central broad spot is the incident X-ray beam
• smaller spots are beams diffracted by the crystal

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X-ray pattern from a single crystal with c axis
parallel to the X-ray beam. You can see its
3-fold symmetry, perhaps evidence of 3m.
Laue photograph named after the first scientist
who, in 1912, showed that X-rays are diffracted
by crystals.
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Braggs law for constructive interfence to
occur, the path difference (ABCD) among waves
scattered by a set of lattice planes must equal
a whole number (n1, 2, ...) of wavelengths.
Only some angles theta will give you this result.
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E this tube is the X-ray source. Inside it,
there is a 40,000 volt difference between a
tungsten filament and a copper target.
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Early X-ray diffraction powder camera. Film is
rolled around the inner rim and records myriad of
diffracted beams as semi-circular sections of
cones.
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X-ray spectra used to be recorded on film strips
rolled up within a round chamber.
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The distance from the center of each line to
the center of the hole (where X-rays entered the
chamber) is proportional to the angle 2-theta.
The intensities of the lines were originally
estimated by a human eye, on a scale of 1 to 100,
before electronic detectors became routine.
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The first information we get from XRD is whether
or not the solid being investigated is
crystalline or amorphous. Silica glass, for
example, has SiO4 tetrahedra like
quartz. Synthetic (i.e. human-made) quartz is
indistinguishable from natural quartz by XRD if
their structure (space group) and composition are
the same.
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The powder X-ray diffraction pattern of an
amorphous solid
- No sharp peak - Broad hump
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The trick is to relate each diffracted peak to
the right family of planes (hkl). The job gets
easier in minerals with a high degree of
symmetry, because there is only a relatively
small number of possible interplanar spacings.
d(100) d(010) d(001) So all these
planes diffract at the same theta angle.
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Braggs law n? 2 d sin ? 1) What do we
know? ?, i.e. the wavelenth of the X-ray
radiation 2) What do we assume? n 1 (Peaks for
higher n are weaker.) 3) What do we want to
know? d, i.e. the interplanar spacings of the
lattice
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(1 angstrom 10 8 cm)
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The unit cell is described as being the smallest
regular repeat unit in a crystalline
lattice. These cells are defined by three unit
lengths (a, b, c) along the crystallographic
axe,s and the three interaxial angles (?, ?, ?).
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The patterns with higher symmetry (and more nodes
per unit cell) produce a lesser number of
diffracted peaks, even if the interplanar
spacings (100), (010) and (001), which describe
the unit cell, are unchanged.
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Indices of diffracted X-ray peaks are usually
written without parentheses. 111, 222 and 333
correspond to the 1st, 2nd and 3rd order
reflections of the (111) planes. 222 is produced
when the X-rays of successive planes have a path
difference of 2wavelength (two lambdas).
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In non-primitive cells there are additional
lattice planes, usually half-way between the
usual lattice planes. They are offset by
translations, but they have the same atomic
pattern, so they will diffract X-rays just like
the other planes. However, the phase difference
will lead to negative interference, i.e. they
will be a half-wavelength behind the X-rays
diffracted by other sets of planes.
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• Top and bottom planes diffract in phase
• the crests of waves are lined up
• The middle plane diffracts out of phase. Its
  trough cancels the crest of the plane above.
• negative interfence no diffracted beam

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This cancellation of diffraction peaks is called
a systematic absence. The path difference at
the usual theta-angle value is exactly half of
one wavelength. When waves are exactly out of
phase, you get negative interference. Each
lattice plane cancels the peak diffracted by the
next lattice plane.
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The patterns with higher symmetry (and more nodes
per unit cell) produce a lesser number of
diffracted peaks, even if the interplanar
spacings (100), (010) and (001), which describe
the unit cell, are unchanged.
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Powder X-ray diffraction is a routine technique
to measure the amount of crystalline SiO2
(quartz) present in mineral dust or soil. A
chemical analysis will not distinguish the SiO2
of quartz from the silicate portion present in
the structure of clays and many other minerals.
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Even when a single mineral is present, a chemical
analysis may not tell you what that mineral is....
This Anglo-Saxon brooch contains an inlay of
CaCO3, but is it calcite or aragonite (2 common
polymorphs)?
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Braggs law predicts at which angles the peaks
will be diffracted, but not their
intensities. Diffraction intensities are
influenced by the atomic number (Z) of the atoms
in the structure, by the shape and size of the
specimen, and by other factors related to the
machine. We use the peak intensities to determine
where the atoms are in the unit cell.
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Because each mineral is different from all others
in either its chemistry or the geometric pattern
of its atomic arrangement (space group), each
powder XRD pattern is a fingerprint. Often, the
three most intense peaks and their theta-angle
are all that is needed to fingerprint a mineral,
even in a mixture. A database is searched by a
fast computer to match known patterns to the
peaks measured from an unknown.
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• , 200, 300 are n1, 2, 3... in Braggs law
• But they all come from the (100) planes.

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Single-crystal work is used for specialized
purposes. One is to determine the space
group. You need to use all the information
available to orient your crystal along the axes
of symmetry. You then check how much symmetry is
present. Unfortunately, XRD always adds a center
of symmetry to the pattern.
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This four-circle diffractometer is used to mount
a single crystal, and rotate it in space. A
detector moves around it to measure the position
(theta-angle) and intensity of diffracted peaks.
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How to solve crystal structures?
The electron density ( ) at a point X, Y, Z in a
unit cell of volume V is (X,Y,Z) 1/V Fhkl
cos 2 (h ? X k ? Y l ? Z) - Therefore if
we know Fhkl and (for each h, k, l) we can
compute for all values of X, Y, and Z and plot
the values obtained to give a three-dimensional
electron density map. Then, assuming atoms to be
at the centres of the electron density peaks, we
would have the entire structure.
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• The presence or absence of a center of inversion
  is usually determined from properties such as
• presence of polar forms (e.g. pyramids,
  monohedra) which indicate that the end of a
  crystallographic axis is different from the -
  end.
• piezoelectricity which can only exist in
  crystalline structures having at least one polar
  axis.

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Large spots aluminum. Small spots silicon.
Laue photographs are used to study the epitaxial
relationships between thin films and the material
on which they are grown.
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When detecting twinning matters ! Piezoelectric
crystals may not display that property if they
are twinned. Twinning can show up in - external
forms - re-entrant angles (non-convex morphology)
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Ion order-disorder can be detected by X-ray
diffraction. This is very different from the
lack of order found in an amorphous solid.
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A cathode filament is heated so that it boils off
electrons. A large voltage (20-100kV) is
maintained between the filament and the target (a
metal such as Mo, Cu, Co, Fe or Cr). The
electrons are accelerated and hit the target
metal.
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Structures with lighter elements can be studied
using neutron diffraction. Neutrons are
scattered by the nucleus, and their scattering
varies less from element to element. whereas
X-rays are scattered by the electron cloud, and
light elements barely re-emit them.