The understanding of chemistry in the Solid State relies upon
information from a range of experimental sources. Different types
of experimental techniques can be used to determine different
aspects of the properties of a solid: bulk
structure, surface structure
and the electronic band structure
may all be elucidated.
X-ray Diffraction
The separation of atoms in solids is generally about
one Angstrom, 10-10m,
and this is of the order of the wavelength of x-ray radiation.
Therefore, x-rays interact with the atoms in a solid and are scattered
by the solid.
In an experiment, a sample of the solid is irradiated
with x-rays, and the direction of the scattered x-rays is recorded.
The set of angles of diffraction can tell us about the symmetry
planes present in the sample, and the separation of these planes.
From this, the positions of
the atoms and ions can be generated and the entire structure elucidated.
The scattering of the x-rays
is dominated by that from atoms in the bulk crystal, and not those
from the surface, and so x-ray diffraction is used to determine
the bulk crystal structure.
The x-rays used to irradiate the sample are normally
produced by bombarding a metal target, usually Cu or Mo, with
a beam of electrons. The electrons produce an x-ray beam with
known characteristics, and spanning a known x-ray frequency range.
The x-rays are diffracted due to reflection from
the crystal planes. If the angle of reflection is θ,
then the separation of the planes, d, can be calculated from Bragg's
Law:
| Reflection from crystalline planes |
| Bragg's Law:
(λ is the wavelength
of the incident x-rays) |
 |
| The planes are indexed by
their crossing points of the unit cell along the three
axes, h, k, and l. The separation,
dhkl, can be related to the lattice
parameter, a (in a cubic cell): |
or
|
Powder X-ray diffraction
In this method, the sample is polycrystalline,
and contains the crystalline planes oriented in all directions.
When the sample is irradiated with x-rays, a cone of scattering
from each set of planes is recorded with a range of values of
2θ, and from these the lattice
parameter, a, may
be determined using the equation above.
Applications of Powder X-ray diffraction
Routine identification
of materials: The x-ray diffraction pattern of
a sample can be compared against a library of previously recorded
diffraction patters of known materials. These library diffraction
patterns represent the fingerprints
of those materials, and so the unknown material may be identifies
by comparison of fingerprints.
Quantitative analysis of
mixtures: When a sample contains a range of phases, each
phase will contribute its own fingerprint to the overall diffraction
pattern. The relative intensities of the individual patterns
in the overall pattern can be used to determine the composition
of the sample.
Precise determination of
lattice parameters: As described above, the values
of 2θ can be used to calculate very
accurate values of the lattice parameters. This is fairly straightforward
for a cubic crystal, but for crystals of lower symmetry the
process is harder but tractable using computer analysis methods.
Systematic absences:
As seen, the diffraction pattern can be used to to give a
range of values of dhkl. Values of h, k, and l
must be assigned in order to calculate a. In some lattices,
the symmetry means that there is destructive interference
between the x-rays scattered by different atoms, and so some
values of 2θ are not observed in
the diffraction pattern, even though they may refer to a set
of crystal planes. These absences are known as systematic
absences. The presence of the systematic absences,
however, may be used to calculate the structure type, as different
structures have different systematic absences.
| Systematic Absences in Cubic lattices |
| Cell Type |
Condition for systematic absence |
| Body Centered |
(h+k+l) is odd |
| Face Centered |
h, k, and l are all odd
or
h, k, and l are all even |
Extended X-ray absorption fine structure (EXAFS)
X-ray absorption edge spectroscopy
is a method where an electron is excited from a core level of
an atom to a vacant band, and from the pattern of the excitation
energies, the density of states for the empty band can be calculated.
This gives us the electronic band structure of the solid.
The EXAFS is a modulation
of the X-ray absorption edge which extends hundreds of eV beyond
the edge to higher frequency. This is due to back scattering of
the photoelectron (which is the source of the observed x-rays)
by adjacent atoms to the emitting atom, and so is sensitive to
the distances to the adjacent atoms. These distances may be calculated,
and coordination numbers may also be obtained from the intensity
of the EXAFS signal, though these are less reliable than the distances.
An important feature of EXAFS is that signals from
atoms of different elements occur at very different frequencies,
so it can be used to examine the local structure of atoms of a
given element. For example, the structure the around iron atom
in haemoglobin may be precisely determined.
Solid-State NMR
Nuclear Magnetic Resonance (NMR) is used to determine
the magnetic susceptibilities of substances in dilute solutions.
One advantage of this technique is that very small volumes of
a solution (only 0.2 ml) are required.
A proton in a solvent molecule has a characteristic
resonance frequency, and this frequency will change when a magnetic
species is added to the solution. The mass susceptibility, which
measures the magnitude of the magnetism of the solute compound,
χ, can be determined by the shift in
frequency of the proton resonance relative to the solvent without
added solute, Δf, using the following
relationship.
| The determination
of mass susceptibility from change in chemical shift of
a solution |
 |
| Δf:
frequency difference between two lines (with and without
solute complex), in Hz
f: frequency at which the lines are being recorded
m: mass of the magnetic substance per milliliter
of solution
χ0: mass
susceptibility of the solvent
d0: density of the solvent
ds: density of the solution |
The magnetic susceptibility can be linked to the
number of unpaired electrons in the complex, and this gives
us information about the structure of the d-orbitals in a transition
metal complex, and the nature of the ligand field.
Electron Microscopy
The resolution of a standard light microscope is
limited by the fact that the wavelength of the light is much longer
than the atomic length scale. The wavelength of an electron is
less than one angstrom, so the resolution of individual atoms
is possible.
The interaction of electrons with matter is much
stronger than the interaction of x-rays (fe = 104fx,
where f is the scattering factor), and so the interpretation of
scattering patterns in electron diffraction is very complex.
One of the most important types of electron microscopy
is Scanning electron microscopy (SEM).
In this, an image is generated from the low energy electrons scattered
from the atoms at the surface of the sample, and so information
about the surface structure is recorded. This is useful for examining
the morphology of a crystal surface, and for the presence of defects
on a surface, and for investigating the arrangement of adsorbed
atoms on the surface.
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