# Light Scatter

Some time ago I was asked to compensate for light scatter when measuring the UV absorbance of protein solutions. I was intrigued because I’d never seen it applied to an analytical measurement: we’d always just accepted it as a theoretical error in an analysis of unfiltered solutions. As I looked at the maths behind it, I couldn’t get my head around how the equations we were using were related to the Rayleigh scatter equation. I’d spent most of my undergrad project looking at the maths of Raman scatter so it was like stepping back in time.

First things first – what’s happening here? In the lab we often measure the concentration of a chemical in a sample by how much light it absorbs. The more light it absorbs the more chemical there is in the sample. However, to do this experiment for colourless molecules (including proteins) you need to measure the sample using ultra-violet light. However, or so the argument goes, proteins are such large molecules that they can ‘scatter light’, meaning that the light levels you measure are lower than expected (i.e. you think your measuring light loss by absorbance, but your actually measuring light loss by absorbance and scatter).

I found two references that pointed me in right direction (1, 2) but I still couldn’t see how to get from the Rayleigh scattering co-efficient to the logarithm equation I was using. So:

$I = Io - Is = Io - Io.\beta$

where I = intensity of light leaving the solution, Io = intensity before the solution, Is = intensity of light scattered and β = Rayleigh scattering co-efficient. Such that:

$\beta = \frac{8\pi^3(n^2-1)^2}{3}\frac{\rho(h)}{N}\frac{1}{\lambda^4}$

Where n = refractive index, ρ(h) is density, N is the number of molecules and λ is wavelength.

For this derivation I will assume the n is constant at the wavelengths of interest, so:

If

$\frac{8\pi^3(n^2-1)^2}{3}\frac{\rho(h)}{N} = constant$

then

$\beta = constant\frac{1}{\lambda^4}$

Therefore the following two equations can be derived:

$I = Io - Io.constant\frac{1}{\lambda^4}$

and

$\frac{I}{Io} = - A = 1 - constant\frac{1}{\lambda^4}$

Where A is the absorbance value normally measured by spectrometers. The log (1-x) terms can be modified using the Taylor series. The Taylor series shows that:

$log(1-x) = -x - \frac{x^2}{2}-\frac {x^3}{3} ...$

By assuming that the amount of light scattered is far smaller than the light intensity before the solution (ie β <<1), then we can take the first term of the Taylor series, giving:

$A = constant\frac{1}{\lambda^4}$

This is similar to the equation is reference 2, however there are couple for differences. In paper 2, this equation is quoted:

$A = constant_1(\lambda^{-4})+constant_2$

where constant2 is a background correction factor which doesn’t appear in my derivation. I think they add the extra factor to account for the background in the experiment (and not the theory), however their equations work because they assume that the background correction is the same at all wavelengths. (I was critical of that assumption until I remembered I’d made the same one about the refractive index being the same at all relevant wavelengths!).

In the analysis I was looking at we used the logarithm to obtain an equation in the form of y = mx + c, so that:

$logA = log(constant) - 4 log\lambda$

Ta-da! (Although in the gradient, m, is fixed at a value of 4.)

Maybe in the future I could get a wee undergrad project going to look at how valid this equation really is.

Acknowledgements and notes

These equations were derived while working for the Cancer Research UK Formulation Unit, together with the Centre for Drug Development at Cancer Research UK.

The preview image is of a small sun dog photographed over Glasgow’s M8 at sunset. Sun dogs are caused by light scattering off ice crystals in the upper atmosphere.

References

1. Cox et al, An experiment to measure Mie and Rayleigh total scattering cross sections, Am. J. Phys. 70 (6), June 2002
2. Porterfield and Zlotnick, A simple and general method for determining the protein and nucleic acid content of viruses by UV absorbance, Virology 407, 281–288, (2010)

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