Scherrer equation

Scherrer equation
Name	Scherrer equation
Field	Crystallography, Materials science, Physics, Chemistry
Introduced	1918
Inventor	Paul Scherrer
Related	Debye–Scherrer method, X-ray diffraction, Bragg's law

Scherrer equation The Scherrer equation is an empirical relation used in crystallography and materials science to estimate the average size of coherent diffracting domains (crystallites) from the broadening of diffraction peaks measured by techniques such as X-ray diffraction, neutron diffraction, and electron diffraction. Developed in 1918 by Paul Scherrer, the relation connects peak full width at half maximum with a characteristic dimension, providing a practical tool in studies conducted at institutions like University of Zurich, industrial laboratories such as Bell Labs, and research facilities including Argonne National Laboratory.

Introduction

The Scherrer equation occupies a central role in experimental work on nanocrystalline materials, thin films, and powders analyzed via instruments produced by companies like Bruker, PANalytical, and Rigaku. It complements theoretical frameworks established by Max von Laue, William Lawrence Bragg, and Paul Peter Ewald and is often used alongside methods developed by Debye and Peter Debye such as the Debye–Scherrer method. The equation is foundational in studies referenced by researchers at Massachusetts Institute of Technology, University of Cambridge, Harvard University, and research consortia including CERN and Oak Ridge National Laboratory.

Derivation and formula

The Scherrer equation is typically written as: L = K λ / (β cos θ), where L is the mean size of coherent scattering domains, K is a dimensionless shape factor, λ is the radiation wavelength, β is the peak broadening (in radians), and θ is the Bragg angle. This form derives from analysis of diffraction from finite-size crystals building on work by Max Born, Edwin Schrödinger, and the scattering theory of Ludwig Boltzmann as specialized by Paul Scherrer and contemporaries. The shape factor K often takes values near 0.9 for roughly spherical crystallites, a choice informed by studies performed at facilities like National Institute of Standards and Technology and reported in journals edited by publishers such as Elsevier and Springer Nature.

Assumptions and limitations

The Scherrer equation assumes that peak broadening arises predominantly from finite crystallite size and that crystallites are strain-free, isotropic, and roughly equiaxed. These assumptions were tested against models developed by Wilhelm Bragg and analyses by Gerhard Borrmann and later critiques by B. E. Warren. Limitations include the inability to distinguish size broadening from microstrain effects quantified via the Williamson–Hall analysis and the inability to accurately characterize anisotropic shapes noted in studies at California Institute of Technology and ETH Zurich. The equation also presumes negligible instrumental broadening, a condition addressed by calibration measurements using standards such as lanthanum hexaboride and practices recommended by International Union of Crystallography.

Practical applications

Researchers apply the Scherrer equation across disciplines—studies of catalysts at Max Planck Institute for Coal Research, thin films in groups at IBM Research, and nanoparticle synthesis at Lawrence Berkeley National Laboratory. It aids characterization in battery research involving Tesla, Inc. collaborations, corrosion studies at Bureau of Mines-affiliated labs, and geology research at United States Geological Survey. Applications extend to pharmaceuticals developed at Pfizer and Roche, where crystal size influences dissolution, and to semiconductor work performed at Intel Corporation and TSMC where grain size impacts electrical properties.

Experimental considerations and corrections

Accurate use requires correction for instrumental broadening using reference standards like silicon or LaB6 and deconvolution methods attributed to techniques refined by researchers at Brookhaven National Laboratory and Lawrence Livermore National Laboratory. Peak profile fitting often employs functions discussed by A. L. Patterson and implemented in software from TOPAS and GSAS-II developed by teams at Rigaku and Los Alamos National Laboratory. Users must account for strain broadening using methods introduced by Wilhelm Williamson and Alan Hall in the Williamson–Hall plot context, and consider size–strain separation via Fourier methods promoted by John Warren and practitioners at University of Oxford.

Extensions and related models

Extensions include size–strain analyses via the Williamson–Hall method, the Rietveld refinement approach pioneered by Hendrik Rietveld, and whole-pattern modeling techniques used at European Synchrotron Radiation Facility and Synchrotron Radiation Sources such as ESRF and Diamond Light Source. More sophisticated models draw on anisotropic broadening formalisms by H. P. Klug and L. E. Alexander and computational approaches integrating density functional theory calculations performed by groups at Princeton University and Stanford University. The Scherrer relation also interfaces with pair distribution function analysis advanced at Argonne National Laboratory and with electron microscopy correlation methods used in studies at Kavli Institute for Nanoscience.

Category:Crystallography