Combining centuries-old mathematical theorems provides efficient approach for characterizing nanoparticles' shape

Mar 15, 2012 By Gregg Gallatin

( -- Gregg Gallatin, a researcher at the NIST Center for Nanoscale Science and Technology, has shown that combining a nineteenth century flux theorem with an eighteenth century mathematical operation provides a convenient technique for using scattered light to count nanoparticles and to characterize their shapes.

This is useful both for determining how a given distribution of nanoparticle shapes affects the properties of nanoparticle functionalized materials as well as for categorizing how incorporate nanoparticles of different shapes. 

The mathematical approach, which combines Gauss’s Law with Fourier transforms, can also be used as a starting point to solve a wide variety of standard problems in mathematics and physics beyond nanotechnology. Because of the ubiquity of digital data derived from Fourier transforms, the approach is likely to find broad application to a range of physical science and engineering measurements. 

Using the technique, Gallatin demonstrates how Porod’s law, which describes how x-rays scatter from small spherically-shaped particles, can be re-derived and extended to the broader case of particles that are nonspherical, thereby providing a powerful and useful approach for determining the of using x-ray scattering.  He then demonstrates that this can be further extended to visible light scattering, which depends on the moments of the nanoparticle shape and therefore provides a more general method for measuring nanoparticle shape from scattering data. 

The technique of combining Gauss’s Law with Fourier transforms can also be applied to the classical physics problem of Fraunhofer diffraction, providing an explicit formula for the diffraction pattern of arbitrary polygonal-shaped openings in an opaque screen in terms of the vertices of the polygon.  It is also applicable to a variety of mathematics problems, including the Hopf Umlaufsatz, which states that the angle of the tangent along a simple smooth closed curve turns by 360 degrees when making a complete circuit around the curve; Stokes’ Law, which relates integrals over an area in two dimensions to the one dimensional curve bounding the area; and the isoperimetric inequality, which states that a circle is the shape that encloses the largest area for a given circumference. 

Given the simplicity and generality of this mathematical technique, Gallatin believes that it can be applied to many other problems as well.

Explore further: Stuck on you: Research shows fingerprint accuracy stays the same over time

More information: Fourier, Gauss, Fraunhofer, Porod and the shape from moments problem, G. M. Gallatin, Journal of Mathematical Physics 53, 013509-013509-13 (2012).

Related Stories

Nanoparticle imaging: A resonant improvement

Oct 28, 2011

Raman spectroscopy is a powerful technique for analyzing atomic structure based on the inelastic scatter of light from molecules, with diverse applications including medical imaging and chemical sensing. Researchers ...

Soap films help to solve mathematical problems

Jan 25, 2011

Soap bubbles and films have always fascinated children and adults, but they can also serve to solve complex mathematical calculations. This is shown by a study carried out by two professors at the University ...

Recommended for you

Another five things to know about meta-analysis

1 hour ago

Last year I wrote a post of "5 Key Things to Know About Meta-Analysis". It was a great way to focus – but it was hard keeping to only 5. With meta-analyses booming, including many that are poorly done or ...

One-third of world's people still have no proper toilets

6 hours ago

Toilets are taken for granted in the industrialized West, but still are a luxury for a third of the world's people who have no access to them, according to a report by the World Health Organization and UNICEF.

User comments : 0

Please sign in to add a comment. Registration is free, and takes less than a minute. Read more

Click here to reset your password.
Sign in to get notified via email when new comments are made.