Scherrer's equation

The Scherrer equation (after the Swiss physicist Paul Scherrer ) offers the possibility of experimentally determining the crystal size in X-ray diffraction .

In general, the diffraction pattern of X-ray diffraction can be described by the Bragg equation . The prerequisite for this, however, is that the crystals examined have a certain thickness and thus a sufficient number of parallel lattice planes with a distance d _hkl are present. With powder diffractometry and other powder methods such as the Debye-Scherrer method , the crystals should therefore have a grain size of at least 0.1 μm. When analyzing the crystal structure of single crystals , the crystals are usually 50–500 μm in size. ${\ displaystyle n \ lambda = 2d \, \ sin (\ theta)}$

schematic representation of the Bragg reflection

If the crystals are very small (crystal size ), this results in a broadening of the X-ray reflections, which is described by the Scherrer equation: ${\ displaystyle L <100 \ dots 200 \, \ mathrm {nm}}$

{\ displaystyle L = {\ frac {K \ cdot \ lambda} {\ Delta (2 \ theta) \ cdot \ cos \ theta _ {0}}}}

It is

${\ displaystyle \ Delta (2 \ theta)}$ the full width at half maximum of the reflex, measured in radians
${\ displaystyle K}$ the Scherrer form factor with a value of approximately 1
${\ displaystyle \ lambda}$ the wavelength of the X-rays
${\ displaystyle L}$ the expansion of the crystal perpendicular to the lattice planes of the reflex
${\ displaystyle \ theta _ {0}}$ the Bragg angle (sometimes referred to as ). ${\ displaystyle 2 \ theta / 2}$

literature

P. Scherrer: Determination of the size and the internal structure of colloidal particles by means of X-rays . In: News from the Society of Science in Göttingen, Mathematical-Physical Class . Weidmannsche Buchhandlung, Berlin 1918, p. 98-100 ( uni-goettingen.de ).
A. Guinier: X-Ray Diffraction in Crystals, Imperfect Crystals, and Amorphous Bodies. Dover Publications, New York 1994, ISBN 0-486-68011-8 , Chapter 5.
U. Holzwarth and N. Gibson: The Scherrer Equation versus the 'Debye-Scherrer Equation'. In: Nature Nanotechnology . Volume 6, No. 534, 2011, p. 534, doi: 10.1038 / nnano.2011.145 .