Crystallographic Powder Diffraction Analysis with GSAS-II

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Edition 0.9.1 – Preview version

Brian H. Toby
Advanced Photon Source
Argonne National Laboratory

Use of the Advanced Photon Source, an Office of Science User Facility operated for the U.S. Department of Energy (DOE) Office of Science by Argonne National Laboratory, was supported by the U.S. DOE under Contract No. DE-AC02-06CH11357.

This book was typeset using LaTeX software on the overleaf.com website using the Science Textbook format from Richard R. Masters (https://github.com/ironmeld/science-textbook-template).
This work is copyright © 2026 Brian H. Toby and is made available under the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0, https://creativecommons.org/licenses/by-nc-sa/4.0/deed.en) license.

Dedication

In memory of Sidney Toby,
and honoring Diane, Cassie and Josh Toby.

Thanks

I have benefited from many mentors over my years. I want to thank Joseph Potenza and Harvey Schugar, my undergraduate advisors at Rutgers, who started my journey in crystallography. Many other members of the Chemistry Department were also great mentors – too many to mention. Alas, too few who who are still with us. While my subsequent years at Caltech were a time of great personal growth, perhaps the less said of my academic experience there the better, but I did have the opportunity to spend some time with and T.A. for Dick Marsh, from whom I learned much; he was always a pleasure to talk with. After Caltech, while employed by Union Carbide, Corp. (rest in pieces), I was introduced to high-resolution powder diffraction, neutron diffraction and Rietveld by David Cox at Brookhaven National Lab, and first met Takeshi Egami, then at the University of Pennsylvania. Both changed my life for the better, in ways too great to explain here. I also got to meet Larry Finger during his experiments at the NSLS. He showed me that one can be a great scientist, as well as a computer hardware and software expert. During my years at Penn working for Takeshi, I learned much about TOF diffraction and its analysis from Jim Jorgenson and Frank Rotella while making measurements Argonne’s IPNS neutron source. It was around then that I also started asking questions of Bob Von Dreele and Allen Larson, then both at LANSCE in Los Alamos. At some point Bob entrusted me with the VAX source code for GSAS and I eventually even had the temerity to provide modified code back. My few years at Air Products, Inc. were good ones and APCI bought me a $60K Silicon Graphics workstation that forced me out of my VAX comfort zone. At NIST, I benefited greatly by being able to work with and learn from Judy Stalick – very much missed. The users and staff at the NCNR were a great privilege to work with, but I want to single out Tony Santoro, Ted Prince and Jack Rush; all no longer with us, alas.

Bob Von Dreele has been a mentor, teacher and work partner, for so many years that I do not know where to begin in thanking him. Working together, we have been recognized for a couple of awards, so it has been quite a partnership. He even had to put up with me as a supervisor twice. Bob and I had the good fortune to land at the kind shores of Argonne, at a time when we were both in need, and that allowed us to collaborate much more closely. I thank Peter Lee, Mark Beno and Gabrielle Long for that. Over the years, I have supervised a few postdocs and interacted with many more, as well as even more students – they should not think that learning went in only one direction.

It seems that many members of the crystallographic community marry into the mental health system, and I am no exception. My partner in every aspect of my life, psychologist Diane Pies Toby, Ph.D., has helped me grow my career in so many ways, not to mention reading and critiquing much of what I write. Our children, Cassie and Josh, have also helped me grow and are always a joy (even when bringing tsouris) in my life.

My mother, Frina, introduced me to research computing and my father, Sidney, showed me how much satisfaction that research and teaching bring. I was actually able to coauthor a paper with both of them during my Rutgers undergrad days. I hope my sister, Carole, has forgiven me for, no doubt, some very boring dinner conversations.

1 An Introduction to Rietveld Fitting
1.1 History
1.2 The role of GSAS-II in this book
1.3 What is Rietveld Analysis?
1.4 Getting Started
I  Prerequisites
2 X-ray and Neutron Scattering
2.1 Terminology: 2theta and Q
2.2 The physics of scattering
2.3 Scattering Probabilities
2.4 Resonant scattering
2.5 Diffraction from materials
2.6 Absorption and Extinction
2.7 For More Information
3 Crystallographic Symmetry
3.1 The lattice
3.2 The unit cell and fractional coordinates
3.3 Fourier Space
3.4 The reciprocal lattice
3.5 The structure factor equation
3.6 The electron density equation
3.7 Site Occupancies
3.8 Ideal vs. real crystals
3.9 Centering and the 14 Bravais lattices
3.10 Crystallographic Symmetry Elements
3.11 Space groups
3.12 Systematic Extinctions
3.13 Site symmetry and the asymmetric unit
3.14 Space groups in GSAS-II
3.15 Why does GSAS-II allow non-standard space groups?
3.16 Polar space groups
3.17 For More Information
4 Powder Diffraction
4.1 History
4.2 Diffraction by Polycrystalline Materials
4.3 For More Information
II  Basic Information
5 X-ray Diffractometers
5.1 Debye-Scherrer
5.2 Bragg-Brentano Diffractometer
5.3 Synchrotron-based instrumentation
5.4 X-ray detection and counting statistics
5.5 X-ray absorption in Debye-Scherrer Geometry
5.6 For More Information
6 Neutron Diffraction
6.1 Reactor-based instrumentation
6.2 Accelerator-based (TOF) instrumentation
6.3 Neutron absorption
6.4 Neutron detection and counting statistics
6.5 For More Information
7 Least-squares Minimization
7.1 Least-Squares
7.2 A Quick Introduction to Linear Algebra
7.3 Linear Least-Squares
7.4 Gauss-Newton Minimization
7.5 The Hessian etc.
7.6 Covariance Matrix
7.7 Steepest Descent Minimization
7.8 Marquardt damping
7.9 Model Failure
7.10 For More Information
8 An Introduction to the GSAS-II GUI
8.1 Installing GSAS-II & Web Resources
8.2 The GSAS-II GUI
8.3 Terminology
8.4 GSAS-II Projects
8.5 For More Information
9 R-factors
9.1 Profile R-factors
9.2 Reduced Chi-squared
9.3 Bragg R-factors
9.4 For More Information
III  Pattern Fitting
10 Overview: Fitting diffraction patterns
11 Fitting Background
11.1 Background Computation Modes
11.2 Approaches to Fitting Background
11.3 Defining Fixed Background Points
11.4 The Auto-Background Feature
11.5 Negative Uiso values
12 Fitting Peak Positions
12.1 Ensuring partial peak alignment
12.2 Sample displacement parameters
12.3 Visual inspection tips
12.4 Lattice Parameter Precision and Accuracy
13 Fitting Peak Intensities
13.1 Parameters that Affect Computed Intensities
13.2 The intensity fitting process
13.3 How to know when the fit is good enough?
14 Fitting Profile functions
14.1 Peak broadening contributions convolute
14.2 Sources of instrumental broadening
14.3 Sample broadening
14.4 Approaches for peak fitting functions
14.5 Physically-Inspired Peak Shapes
14.6 GSAS-II Profile Implementation
14.7 GSAS-II Sample Broadening Terms
14.8 GUI Access to Sample Broadening Terms
14.9 GSAS-II Sample Broadening Visualization
14.10 Determining Instrumental Profile Terms
14.11 FPA Generation of Instrumental Terms
14.12 How to Refine Peak Profiles for a Sample
14.13 Constraints on broadening terms
14.14 The Warren-Averbach Method
14.15 Parting words on Profile Fitting
IV  Using GSAS-II
15 Importing and Exporting Data
15.1 Supported import data types
15.2 Supported export data types
16 Overview: GSAS-II Capabilities
17 Peak fitting
17.1 Peak Parameters Definitions
17.2 Creating a Peak List
17.3 Peak Width modes
17.4 The Peak Fitting Process
17.5 Extra Peak mode
17.6 Sequential Peak Fitting
18 Structural Graphics in GSAS-II
18.1 Phase “Atoms” tab
18.2 Phase “Draw Atoms” tab
18.3 Phase “Draw Options” tab
19 GSAS-II Utility Computation Tools
19.1 Absorb: Compute X-ray Absorption Coefficient
19.2 Fprime: Compute atomic scattering
19.3 PlotXNFF: Plot scattering power curves
20 Intensity Extraction & Fourier Maps
20.1 Rietveld’s Intensity Extraction Method
20.2 Fourier Maps
20.3 Pawley Fitting
20.4 Le Bail Fitting
20.5 For More Information
21 Sequential fitting
21.1 What is a sequential fit?
21.2 Preparing for a sequential fit
21.3 Starting a sequential fit
21.4 Sequential fit results table
21.5 Parametric fitting
21.6 Other types of sequential fits
21.7 Really large sequential fits
21.8 Sequential fits: the future
22 GSAS-II and CIF
22.1 Introduction to CIF
22.2 GSAS-II CIF Imports
22.3 GSAS-II CIF Exports
22.4 For More Information
V  Simplifying Structural Models
23 Overview: Simplifying Models
24 Constraints
24.1 GSAS-II Parameter Naming
24.2 GSAS-II Constraint Types
24.3 Constraint and New Variable Processing
24.4 Typical Constraint Use
25 Restraints
25.1 Bond Distance Restraints
25.2 Angle Restraints
25.3 Plane Restraints
25.4 Chiral Restraints
25.5 Chemical Composition Restraints
25.6 Magnetic Moment Restraints
25.7 General Restraints
25.8 Choosing the Restraint Weighting
25.9 Testing Restraints
25.10 Restraints not provided by GSAS-II
26 Rigid Body Constraints
26.1 Vector Rigid Bodies
26.2 Residue Rigid Bodies
26.3 Rigid Body Axes and Origin
26.4 Defining Residue Rigid Body Examples
26.5 Inserting Rigid Bodies
26.6 Orienting Rigid Bodies: Quaternions
26.7 Rigid Bodies and Crystallographic Symmetry
26.8 Example: Inserting rigid bodies
26.9 TLS Displacement Parameters for Rigid Bodies
VI  Advanced Concepts
27 Advanced Concepts
28 Transformations of Symmetry
28.1 Subgroup and Supergroup Relationships
28.2 Representational Analysis
28.3 Symmetry tools in GSAS-II
28.4 The Bilbao Crystallographic Server
28.5 The BYU ISOTROPY Software Suite Web Server
28.6 For More Information
29 Indexing Powder Diffraction Patterns
29.1 Autoindexing
29.2 Manual indexing
29.3 Testing for supercells
29.4 Electron diffraction
29.5 For More Information