Crystallographic Powder Diffraction Analysis with GSAS-II

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Edition 0.9.3 – 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 wish I could again 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 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 able to spend time at Brookhaven National Lab and there was introduced to high-resolution powder diffraction, neutron diffraction and Rietveld by David Cox and first met Takeshi Egami, then at the University of Pennsylvania. Both of them changed my life for the better, in ways too great to explain here. At BNL, I also got to meet Larry Finger. 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. So many 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. Przemek Klosowski was my computing guru when I was at the NCNR and he got me started with Linux, scripting languages and GUI programming, and changed my professional life.

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.

I was very fortunate that 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 first undergrad year at Rutgers, but I do owe an apology to my sister, Carole, for what were – no doubt – some very boring dinner table conversations during that period.

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 Diffraction measurement corrections
2.8 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 Details of the GSAS-II Data Tree
15.1 The GSAS-II Data Tree Contents
15.2 Powder diffraction 2D image (IMG) child entries
15.3 Powder diffraction (PWDR) child entries
15.4 For More Information
16 Importing and Exporting Data
16.1 Supported import data types
16.2 Supported export data types
17 Overview: GSAS-II Capabilities
18 Peak fitting
18.1 Peak Parameters Definitions
18.2 Creating a Peak List
18.3 Peak Width modes
18.4 The Peak Fitting Process
18.5 Extra Peak mode
18.6 Sequential Peak Fitting
19 Structural Graphics in GSAS-II
19.1 Phase “Atoms” tab
19.2 Phase “Draw Atoms” tab
19.3 Phase “Draw Options” tab
20 GSAS-II Utility Computation Tools
20.1 Absorb: Compute X-ray Absorption Coefficient
20.2 Fprime: Compute atomic scattering
20.3 PlotXNFF: Plot scattering power curves
21 Intensity Extraction & Fourier Maps
21.1 Rietveld’s Intensity Extraction Method
21.2 Fourier Maps
21.3 Pawley Fitting
21.4 Le Bail Fitting
21.5 For More Information
22 Sequential fitting
22.1 What is a sequential fit?
22.2 Preparing for a sequential fit
22.3 Starting a sequential fit
22.4 Sequential fit results table
22.5 Parametric fitting
22.6 Other types of sequential fits
22.7 Really large sequential fits
22.8 Sequential fits: the future
23 GSAS-II and CIF
23.1 Introduction to CIF
23.2 GSAS-II CIF Imports
23.3 GSAS-II CIF Exports
23.4 For More Information
V  Simplifying Structural Models
24 Overview: Simplifying Models
25 Constraints
25.1 GSAS-II Parameter Naming
25.2 GSAS-II Constraint Types
25.3 Constraint and New Variable Processing
25.4 Typical Constraint Use
26 Restraints
26.1 Bond Distance Restraints
26.2 Angle Restraints
26.3 Plane Restraints
26.4 Chiral Restraints
26.5 Chemical Composition Restraints
26.6 Magnetic Moment Restraints
26.7 General Restraints
26.8 Choosing the Restraint Weighting
26.9 Testing Restraints
26.10 Restraints not provided by GSAS-II
27 Rigid Body Constraints
27.1 Vector Rigid Bodies
27.2 Residue Rigid Bodies
27.3 Rigid Body Axes and Origin
27.4 Defining Residue Rigid Body Examples
27.5 Inserting Rigid Bodies
27.6 Orienting Rigid Bodies: Quaternions
27.7 Rigid Bodies and Crystallographic Symmetry
27.8 Example: Inserting rigid bodies
27.9 TLS Displacement Parameters for Rigid Bodies
VI  Advanced Concepts
28 Advanced Concepts
29 Transformations of Symmetry
29.1 Subgroup and Supergroup Relationships
29.2 Representational Analysis
29.3 Symmetry tools in GSAS-II
29.4 The Bilbao Crystallographic Server
29.5 The BYU ISOTROPY Software Suite Web Server
29.6 For More Information
30 Indexing Powder Diffraction Patterns
30.1 Autoindexing
30.2 Manual indexing
30.3 Testing for supercells
30.4 Electron diffraction
30.5 For More Information