Determination of Crystallite 

Orientation Distribution Function 

(ODF) From Neutron Diffraction Data

August 8, 2018

Ryan Underwood1,2

Thomas Gnäupel-Herold3

1University of Central Florida (UCF)

2Summer Undergraduate Research Fellowship (SURF)

3NIST Center for Neutron Research (NCNR)
Motivation

• NIST Center for Neutron Research’s (NCNR) BT8 Stress and Texture 

Diffractometer Upgrade

• x30 increase in performance and data production

• Requires increased processing capability and quasi-real-time processing of 

experimental data

• Texture Analysis

• Prerequisite for optimal stress measurement strategy

• Allows for prediction of elastic and plastic properties of materials

• Important to material manufacturing

2
What is Texture?

• Crystallographic texture is the 

preferred orientation of grains 

or crystallites in polycrystalline 

materials

• Result of thermo-mechanical 

processing and its interaction 

with the crystal structure of the 

constituent grains

• Allows for the identification of 

the material’s anisotropic 

properties

• Represented qualitatively by 

pole figures

Normal Direction (ND)

Transverse Direction (TD)

Rolling Direction (RD)

TD

ND

RD

TD

ND

RD

www.ebsd.com

www.mirion.com

3
Pole Figure Measurement with Neutrons

T. Gnäupel-Herold, et. al., Neutron Measurements of Stresses in a Test Artifact Produced by Laser-Based Additive Manufacturing 4

Eulerian Cradle
• Graphical representation of texture 

with respect to a sample frame of 

reference

• Data collection methods

• X-ray diffraction

• Electron Backscatter Diffraction

• Neutron Diffraction

• Of all measurements methods, 

neutron diffraction samples the 

largest number of grains (106-108

grains)

Pole Figures

Stereographic Projection

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Orientation Distribution

• Density of grains in particular 

orientation

• Three-dimensional statistical 

description of the texture, f(g)

• g={ɸ1,Φ, ɸ2}

• Requires two or more pole figures

• Generalized spherical harmonics

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6
Sections Though OD Space

φ1

φ2

φ2 = 0° φ2 = 5° φ2 = 10°

Φ

Φ

φ1

https://doi.org/10.1016/j.fusengdes.2013.04.005 7
Iron (211) fibers in Euler space –

one point in a pole figure is the 

integral over the ODF intensities 

along a fiber in Euler space, with 

the intensity at each coordinate 

(φ1,Φ, φ2) representing the 

probability (in units of random 

density) of a particular grain 

orientation

PPhkkkk αα, ββ =

1

2ππ�ff gg ddφφ1ddΦddφφ2

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• Pre-existing code

• Library of 

Coefficients 

• Programs & 

Subroutines

Fortran IV

• Modernization

• Validity Check

• Optimization

Fortran 95

• Graphical User 

Interface (GUI)

• Integration into 

existing code base

Modern 

Language

Procedure

www.cyber-tec.org

www.ibm.com

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www.cyber-tec.org

H. J. Bunge, Texture Analysis 

in Material Science: 

Mathematical Methods 10
Coefficient Equations

11
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Summary

• All programs and subroutines were transcribed into text files

• Work was started in transcribing, debugging and updating Fortran IV to 

Fortran 95 syntax

• The following coefficients were confirmed from the library program output 

transcribed in Fortran 95:

• Symmetry Coefficients, ̇ BḂ

mmμμ for cubic symmetry

• Fourier Coefficients, QQmmss

• Cubic Spherical Harmonics, ̇ kk̇

mm(hkkkk)

• Fourier coefficients, aa′mmss of the associated Legendre function PPmm Φ

• Fourier coefficients, aa′mmmm of the associated Legendre function PPmm Φ

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• Center for High Resolution Neutron Scattering 

(CHRNS)

• Summer Undergraduate Research Fellowship 

(SURF)

Future Work 

• Continue debugging Fortran 95 programs and subroutines

• Optimize code

• Ensure validity of output data

• Transcribe Fortran 95 programs and subroutines into a modern language

• Produce a user-friendly GUI

• Integrate into NCNR code data base for future neutron data analysis and 

simulation

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Acknowledgements

• National Institute of Standards and 

Technology (NIST)

• NIST Center for Neutron Research 

(NCNR)