Lattice Parameters
Why measure lattice parameters?
The physical properties of solids depend entirely upon the arrangement of the atoms that make up the solid and the distances between them.
The arrangement of the atoms in a crystal structure is a combination of the size and shape of the unit-cell and the arrangement of atoms inside the unit-cell. See our rough guide!
The arrangement of the atoms in a crystal structure is a combination of the size and shape of the unit-cell and the arrangement of atoms inside the unit-cell. See our rough guide!
Unit cell size and shape
The shape of the unit cells is described by the lattice symmetry.
The size of the unit-cell is described in terms of its unit-cell parameters. These are the edge lengths and the angles of the unit-cell.
The positions of the X-ray beams diffracted by a crystal structure are related to the size and shape of the unit-cell. We use the Huber diffractometer to measure the positions of X-ray beams diffracted from a crystal to determine the size and shape of the unit-cell.
The process of X-ray diffraction by a crystal can be thought as one of reflection of the X-rays off planes of atoms within the crystal. With this idea we can derive Bragg's Law:
The size of the unit-cell is described in terms of its unit-cell parameters. These are the edge lengths and the angles of the unit-cell.
The positions of the X-ray beams diffracted by a crystal structure are related to the size and shape of the unit-cell. We use the Huber diffractometer to measure the positions of X-ray beams diffracted from a crystal to determine the size and shape of the unit-cell.
The process of X-ray diffraction by a crystal can be thought as one of reflection of the X-rays off planes of atoms within the crystal. With this idea we can derive Bragg's Law:
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l - the wavelength of the Xrays
d - the spacing of the planes in the crystal 2q - the angle of diffraction |
The d-spacings of planes are derived from the unit-cell parameters of the crystal.
In powder diffraction the 2q angles of the reflections are the only information that we obtain.
But with a single-crystal and a four-circle goniometer to orient the crystal, we can also determine the angles through which the crystal is moved to go from one diffraction peak to the next. This information is used with the Bragg angles to determine the unit-cell parameters.
In powder diffraction the 2q angles of the reflections are the only information that we obtain.
But with a single-crystal and a four-circle goniometer to orient the crystal, we can also determine the angles through which the crystal is moved to go from one diffraction peak to the next. This information is used with the Bragg angles to determine the unit-cell parameters.
In detail.... |
The d-spacings can be expressed in terms of the reciprocal lattice vectors (a*, b* and c*) and the Miller indeces h, k, l of the reflection.
The dot product can be written in matrix form instead, using the reciprocal lattice parameters arranged as the reciprocal metric tensor denoted B. The individual reciprocal lattice parameters are related to the real lattice parameters of the crystal through equations like this one. We also have a second matrix U that describes the orientation of the crystal on the goniometer. The measured angles are used to constrain the elements of the matrix product UB, from which the unit-cell parameters are extracted. |
Lattice Parameters at High Pressure
Why high pressure?
All of our information about the deep Earth is indirect. It comes from geophysical observations. They tell us the density of the Earth at a given depth, but not which minerals are present. But the properties of the Earth's interior are dependent on the mineral structures present.
We therefore use diamond-anvil cells to measure the lattice parameters of crystals to very high pressures. This tells us the density of the mineral at high pressure, often expressed as an Equation of State. When the density of a mineral, or a combination of minerals, matches the density obtained from geophysical observations then those minerals may be present in the Earth's interior.
We simply load our crystal into a diamond-anvil cell, apply pressure, and measure the angles of the diffracted X-ray beams from the crystal. The details of the additional experimental techniques necessary for measuring structures at high pressure can be found in volume 41 of the Reviews in Mineralogy and Geochemistry, available from the MSA.
Here are two examples of high-pressure studies of the lattice parameters of a crystal.
We therefore use diamond-anvil cells to measure the lattice parameters of crystals to very high pressures. This tells us the density of the mineral at high pressure, often expressed as an Equation of State. When the density of a mineral, or a combination of minerals, matches the density obtained from geophysical observations then those minerals may be present in the Earth's interior.
We simply load our crystal into a diamond-anvil cell, apply pressure, and measure the angles of the diffracted X-ray beams from the crystal. The details of the additional experimental techniques necessary for measuring structures at high pressure can be found in volume 41 of the Reviews in Mineralogy and Geochemistry, available from the MSA.
Here are two examples of high-pressure studies of the lattice parameters of a crystal.
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The sample is spodumene, a clinopyroxene.
The crystal was loaded into a diamond-anvil cell and pressure was applied. We then measured the lattice parameters at many different pressures up to 9GPa. Each measurement takes about 24 hours. The graphs show how the various lattice parameters of spodumene vary with pressure. At ~3 GPa they all show a step due to a phase transition. Away from the phase transition the lattice parameters decrease smoothly with increasing pressure. The volume calculated from the lattice parameters can be used to determine the Equation of State of the sample. More details: Arlt and Angel (2000) Phys. Chem. Min. 27:719 Arlt and Angel (2000) Mineral. Mag. 64:237 |
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This is the equation of state of albite, which was measured by Matt Benusa as part of his senior
project on the Huber diffractometer in the Crystallography Laboratory. The small symbols are the actual data, and the estimated uncertainties are smaller than the
symbol size.
Up to about 3.5 GPa the volume variation with pressure is normal; the curve is concave upwards indicating that the structure is becoming stiffer as pressure is increased. At higher pressures the solid curve and data points turn over and the structure is clearly becoming softer again. This is very unusual behavior and is a result of the complex response of the feldspar framework to pressure. |