# Pair distribution function and structure factor of spherical particles

## Abstract

The availability of neutron spallation-source instruments that provide total scattering powder diffraction has led to an increased application of real-space structure analysis using the pair distribution function. Currently, the analytical treatment of finite size effects within pair distribution refinement procedures is limited. To that end, an envelope function is derived which transforms the pair distribution function of an infinite solid into that of a spherical particle with the same crystal structure. Distributions of particle sizes are then considered, and the associated envelope function is used to predict the particle size distribution of an experimental sample of gold nanoparticles from its pair distribution function alone. Finally, complementing the wealth of existing diffraction analysis, the peak broadening for the structure factor of spherical particles, expressed as a convolution derived from the envelope functions, is calculated exactly for all particle size distributions considered, and peak maxima, offsets, and asymmetries are discussed.

- Authors:

- Materials Science and Technology Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 (United States)
- (United States)

- Publication Date:

- OSTI Identifier:
- 20787950

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Physical Review. B, Condensed Matter and Materials Physics; Journal Volume: 73; Journal Issue: 9; Other Information: DOI: 10.1103/PhysRevB.73.094107; (c) 2006 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY; ASYMMETRY; CRYSTAL STRUCTURE; DISTRIBUTION; DISTRIBUTION FUNCTIONS; GOLD; NANOSTRUCTURES; NEUTRON DIFFRACTION; NEUTRONS; PARTICLE SIZE; PARTICLES; POWDERS; SOLIDS; SPHERICAL CONFIGURATION; STRUCTURE FACTORS

### Citation Formats

```
Howell, Rafael C., Proffen, Thomas, Conradson, Steven D., Lujan Neutron Scattering Center, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, and Materials Science and Technology Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545.
```*Pair distribution function and structure factor of spherical particles*. United States: N. p., 2006.
Web. doi:10.1103/PHYSREVB.73.0.

```
Howell, Rafael C., Proffen, Thomas, Conradson, Steven D., Lujan Neutron Scattering Center, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, & Materials Science and Technology Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545.
```*Pair distribution function and structure factor of spherical particles*. United States. doi:10.1103/PHYSREVB.73.0.

```
Howell, Rafael C., Proffen, Thomas, Conradson, Steven D., Lujan Neutron Scattering Center, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, and Materials Science and Technology Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545. Wed .
"Pair distribution function and structure factor of spherical particles". United States.
doi:10.1103/PHYSREVB.73.0.
```

```
@article{osti_20787950,
```

title = {Pair distribution function and structure factor of spherical particles},

author = {Howell, Rafael C. and Proffen, Thomas and Conradson, Steven D. and Lujan Neutron Scattering Center, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 and Materials Science and Technology Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545},

abstractNote = {The availability of neutron spallation-source instruments that provide total scattering powder diffraction has led to an increased application of real-space structure analysis using the pair distribution function. Currently, the analytical treatment of finite size effects within pair distribution refinement procedures is limited. To that end, an envelope function is derived which transforms the pair distribution function of an infinite solid into that of a spherical particle with the same crystal structure. Distributions of particle sizes are then considered, and the associated envelope function is used to predict the particle size distribution of an experimental sample of gold nanoparticles from its pair distribution function alone. Finally, complementing the wealth of existing diffraction analysis, the peak broadening for the structure factor of spherical particles, expressed as a convolution derived from the envelope functions, is calculated exactly for all particle size distributions considered, and peak maxima, offsets, and asymmetries are discussed.},

doi = {10.1103/PHYSREVB.73.0},

journal = {Physical Review. B, Condensed Matter and Materials Physics},

number = 9,

volume = 73,

place = {United States},

year = {Wed Mar 01 00:00:00 EST 2006},

month = {Wed Mar 01 00:00:00 EST 2006}

}