Scherrer equation

The Scherrer equation, in X-ray diffraction and crystallography, is a formula that relates the size of sub-micrometre particles, or crystallites, in a solid to the broadening of a peak in a diffraction pattern. It is named after Paul Scherrer.^[1][2] It is used in the determination of size of particles of crystals in the form of powder.

The Scherrer equation can be written as:

$\tau = \frac {K \lambda}{\beta \cos \theta}$

where:

• τ is the mean size of the ordered (crystalline) domains, which may be smaller or equal to the grain size;
• K is a dimensionless shape factor, with a value close to unity. The shape factor has a typical value of about 0.9, but varies with the actual shape of the crystallite;
• λ is the X-ray wavelength;
• β is the line broadening at half the maximum intensity (FWHM), after subtracting the instrumental line broadening, in radians. This quantity is also sometimes denoted as Δ(2θ);
• θ is the Bragg angle.

Applicability

The Scherrer equation is limited to nano-scale particles. It is not applicable to grains larger than about 0.01 to 0.02 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 instrumental effects and crystallite size; the most important of these are usually inhomogeneous strain and crystal lattice imperfections. The following sources of peak broadening are listed in reference:^[3] dislocations, stacking faults, twinning, microstresses, grain boundaries, sub-boundaries, coherency strain, chemical heterogeneities, and crystallite smallness. (Some of those and other imperfections may also result in peak shift, peak asymmetry, anisotropic peak broadening, or affect peak shape.)

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. The concept of crystallinity can be used to collectively describe the effect of crystal size and imperfections on peak broadening.

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 978-0-471-49369-3.
• B.E. Warren, X-Ray Diffraction, Addison-Wesley Publishing Co., 1969, p 251-254, ISBN 0-201-08524-0.

References

1. P. Scherrer, Göttinger Nachrichten Gesell., Vol. 2, 1918, p 98.
2. Patterson, A. (1939). "The Scherrer Formula for X-Ray Particle Size Determination". Phys. Rev. 56 (10): 978–982. Bibcode:1939PhRv...56..978P. doi:10.1103/PhysRev.56.978.
3. A.K. Singh (ed.), "Advanced X-ray Techniques in Research And Industries", Ios Pr Inc, 2005. ISBN 1586035371