Shape factor (X-ray diffraction)
Encyclopedia
A shape factor is used in x-ray diffraction and crystallography
to correlate the size of sub-micrometre
particles, or crystallite
s, in a solid to the broadening of a peak in a diffraction pattern. In the Scherrer
equation,
where K is the shape factor, λ is the x-ray
wavelength
, β is the line broadening at half the maximum intensity
(FWHM
) in radian
s, and θ is the Bragg angle; τ is the mean size of the ordered (crystalline) domains, which may be smaller or equal to the grain size. The dimensionless shape factor has a typical value of about 0.9, but varies with the actual shape of the crystallite. The Scherrer equation is limited to nano
-scale particles. It is not applicable to grains larger than about 0.1 μm, which precludes those observed in most metallographic
and ceramographic
microstructures.
It is important to realize that the Scherrer formula provides a lower bound on the particle size. The reason for this is that a variety of factors can contribute to the width of a diffraction peak; besides crystallite size, the most important of these are usually inhomogeneous strain and instrumental effects. If all of these other contributions to the peak width were zero, then the peak width would be determined solely by the crystallite size and the Scherrer formula would apply. If the other contributions to the width are non-zero, then the crystallite size can be larger than that predicted by the Scherrer formula, with the "extra" peak width coming from the other factors.
X-ray crystallography
X-ray crystallography is a method of determining the arrangement of atoms within a crystal, in which a beam of X-rays strikes a crystal and causes the beam of light to spread into many specific directions. From the angles and intensities of these diffracted beams, a crystallographer can produce a...
to correlate the size of sub-micrometre
Micrometre
A micrometer , is by definition 1×10-6 of a meter .In plain English, it means one-millionth of a meter . Its unit symbol in the International System of Units is μm...
particles, or crystallite
Crystallite
Crystallites are small, often microscopic crystals that, held together through highly defective boundaries, constitute a polycrystalline solid. Metallurgists often refer to crystallites as grains.- Details :...
s, in a solid to the broadening of a peak in a diffraction pattern. In the Scherrer
Paul Scherrer
Paul Scherrer was a Swiss physicist. Born in Herisau, Switzerland, he studied at Göttingen, Germany, before becoming a lecturer there. Later, Scherrer became head of the Department of Physics at ETH Zürich....
equation,
where K is the shape factor, λ is the x-ray
X-ray
X-radiation is a form of electromagnetic radiation. X-rays have a wavelength in the range of 0.01 to 10 nanometers, corresponding to frequencies in the range 30 petahertz to 30 exahertz and energies in the range 120 eV to 120 keV. They are shorter in wavelength than UV rays and longer than gamma...
wavelength
Wavelength
In physics, the wavelength of a sinusoidal wave is the spatial period of the wave—the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a...
, β is the line broadening at half the maximum intensity
Intensity (physics)
In physics, intensity is a measure of the energy flux, averaged over the period of the wave. The word "intensity" here is not synonymous with "strength", "amplitude", or "level", as it sometimes is in colloquial speech...
(FWHM
Full width at half maximum
Full width at half maximum is an expression of the extent of a function, given by the difference between the two extreme values of the independent variable at which the dependent variable is equal to half of its maximum value....
) in radian
Radian
Radian is the ratio between the length of an arc and its radius. The radian is the standard unit of angular measure, used in many areas of mathematics. The unit was formerly a SI supplementary unit, but this category was abolished in 1995 and the radian is now considered a SI derived unit...
s, and θ is the Bragg angle; τ is the mean size of the ordered (crystalline) domains, which may be smaller or equal to the grain size. The dimensionless shape factor has a typical value of about 0.9, but varies with the actual shape of the crystallite. The Scherrer equation is limited to nano
Nanotechnology
Nanotechnology is the study of manipulating matter on an atomic and molecular scale. Generally, nanotechnology deals with developing materials, devices, or other structures possessing at least one dimension sized from 1 to 100 nanometres...
-scale particles. It is not applicable to grains larger than about 0.1 μm, which precludes those observed in most metallographic
Metallography
Metallography is the study of the physical structure and components of metals, typically using microscopy.Ceramic and polymeric materials may also be prepared using metallographic techniques, hence the terms ceramography, plastography and, collectively, materialography.-Preparing metallographic...
and ceramographic
Ceramography
Ceramography is the art and science of preparation, examination and evaluation of ceramic microstructures. Ceramography can be thought of as the metallography of ceramics. The microstructure is the structure level of approximately 0.1 to 100 µm, between the minimum wavelength of visible light and...
microstructures.
It is important to realize that the Scherrer formula provides a lower bound on the particle size. The reason for this is that a variety of factors can contribute to the width of a diffraction peak; besides crystallite size, the most important of these are usually inhomogeneous strain and instrumental effects. If all of these other contributions to the peak width were zero, then the peak width would be determined solely by the crystallite size and the Scherrer formula would apply. If the other contributions to the width are non-zero, then the crystallite size can be larger than that predicted by the Scherrer formula, with the "extra" peak width coming from the other factors.
Further reading
- B.D. Cullity & S.R. Stock, Elements of X-Ray Diffraction, 3rd Ed., Prentice-Hall Inc., 2001, p 167-171, ISBN 0-201-61091-4.
- R. Jenkins & R.L. Snyder, Introduction to X-ray Powder Diffractometry, John Wiley & Sons Inc., 1996, p 89-91, ISBN 0-471-51339-3.
- H.P. Klug & L.E. Alexander, X-Ray Diffraction Procedures, 2nd Ed., John Wiley & Sons Inc., 1974, p 687-703, ISBN 978-0471493693.
- B.E. Warren, X-Ray Diffraction, Addison-Wesley Publishing Co., 1969, p 251-254, ISBN 0-201-08524-0.