# Expected values and variances of Bragg peak intensities measured in a nanocrystalline powder diffraction experiment

## Abstract

A rigorous study of sampling and intensity statistics applicable for a powder diffraction experiment as a function of crystallite size is presented. Our analysis yields approximate equations for the expected value, variance and standard deviations for both the number of diffracting grains and the corresponding diffracted intensity for a given Bragg peak. The classical formalism published in 1948 by Alexander, Klug & Kummer [J. Appl. Phys.(1948),19, 742–753] appears as a special case, limited to large crystallite sizes, here. It is observed that both the Lorentz probability expression and the statistics equations used in the classical formalism are inapplicable for nanocrystalline powder samples.

- Authors:

- Brookhaven National Lab. (BNL), Upton, NY (United States). National Synchrotron Light Source II
- Columbia Univ., New York, NY (United States). Applied Physics and Applied Mathematics

- Publication Date:

- Research Org.:
- Brookhaven National Laboratory (BNL), Upton, NY (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC-22)

- OSTI Identifier:
- 1392243

- Report Number(s):
- BNL-114264-2017-JA

Journal ID: ISSN 1600-5767; JACGAR

- Grant/Contract Number:
- SC0012704; AC02-98CH10886

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Journal of Applied Crystallography (Online)

- Additional Journal Information:
- Journal Name: Journal of Applied Crystallography (Online); Journal Volume: 50; Journal Issue: 5; Journal ID: ISSN 1600-5767

- Publisher:
- International Union of Crystallography

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 73 NUCLEAR PHYSICS AND RADIATION PHYSICS; X-ray diffraction; nanocrystals; powder diffraction; sampling and intensity statistics; Lorentz factor

### Citation Formats

```
Öztürk, Hande, and Noyan, I. Cevdet.
```*Expected values and variances of Bragg peak intensities measured in a nanocrystalline powder diffraction experiment*. United States: N. p., 2017.
Web. doi:10.1107/S1600576717010494.

```
Öztürk, Hande, & Noyan, I. Cevdet.
```*Expected values and variances of Bragg peak intensities measured in a nanocrystalline powder diffraction experiment*. United States. doi:10.1107/S1600576717010494.

```
Öztürk, Hande, and Noyan, I. Cevdet. Thu .
"Expected values and variances of Bragg peak intensities measured in a nanocrystalline powder diffraction experiment". United States.
doi:10.1107/S1600576717010494. https://www.osti.gov/servlets/purl/1392243.
```

```
@article{osti_1392243,
```

title = {Expected values and variances of Bragg peak intensities measured in a nanocrystalline powder diffraction experiment},

author = {Öztürk, Hande and Noyan, I. Cevdet},

abstractNote = {A rigorous study of sampling and intensity statistics applicable for a powder diffraction experiment as a function of crystallite size is presented. Our analysis yields approximate equations for the expected value, variance and standard deviations for both the number of diffracting grains and the corresponding diffracted intensity for a given Bragg peak. The classical formalism published in 1948 by Alexander, Klug & Kummer [J. Appl. Phys.(1948),19, 742–753] appears as a special case, limited to large crystallite sizes, here. It is observed that both the Lorentz probability expression and the statistics equations used in the classical formalism are inapplicable for nanocrystalline powder samples.},

doi = {10.1107/S1600576717010494},

journal = {Journal of Applied Crystallography (Online)},

number = 5,

volume = 50,

place = {United States},

year = {Thu Aug 24 00:00:00 EDT 2017},

month = {Thu Aug 24 00:00:00 EDT 2017}

}

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