In dynamic light scattering (DLS) the intensity autocorrelation function of the scattered light is measured. This function is then fit to obtain a size. There are two main fitting algorithms which may confuse users:

- Cumulant (or z-average) size and polydispersity (or polydispersity index PDI) with
*one overall average*size and one overall average polydispersity. {The parameter “PDI width” in nanometers is the square root of the PDI times the z-average.} - Peak size (or distribution size) with a mean size and a width
*for each separate size peak*of the distribution

## What are the key differences between these two algorithms?

The **z-average** is determined according to an ISO method [ ISO13321:1996 or its newer pendant ISO22412:2008 ] where only the initial part of the correlation function (up to 10% of the intercept) is force fit to a single exponential decay. The decay rate is related to the overall mean size. The next order fitting term is related to the polydispersity of a Gaussian size distribution if one assumes that as the underlying particle distribution.

For the **distribution analysis** (regularization techniques such as non negative least squares NNLS, or CONTIN) the correlation function is fit to longer times (i.e. more of the data are fit, typically to 1% of the intercept). The outcome is a distribution of different contributions from the size classes or bins, and peaks can be defined with a statistical mean and standard deviation of that specific peak.

## Why aren’t the two the same numbers, trending the same way?

For a perfectly monodisperse sample, the two results should be the same, that is the z-average should be the same as the mean of the (one and only) peak in the distribution. In real applications, even for monodisperse samples, this is likely not the case and there will be small differences.

For polydisperse samples, the two can not be the same, because the z-average will still be only a single number, whereas the distribution will show two or more peaks with corresponding mean and width.

Explanations for the three potential scenarios are:

- z-average smaller than peak size: The cumulant fit only is to the initial part of the correlation function, so in a way, slightly overemphasizes the initial decay from the smaller part of the distribution. If there is an additional peak at small size due to additives or buffer components (sometimes called a “solvent peak”) this could be the cause for the average size lower than the expected peak size.
- z-average the same as peak size: Ideal, probably monodisperse sample. In very odd situations it could be a very polydisperse sample where one of the peaks happens to match the overall average size.
- z-average larger than peak size: The distribution shows a small and a large component, the average of the two is somewhere in between. Under peculiar circumstances, there may actually not be a smaller or a larger peak when the smaller peak is smaller than the lowest display cutoff (buffer components or additives) and when the largest peak is larger than the large cutoff (very large aggregates or dust)

Fortunately the quality report in the Zetasizer DTS software checks for these occurances and will point towards the correct interpretation of the encountered result. In distributions a further choice in the interpretation is discussed in an earlier blog post about intensity-volume-number distributions.

{{There is a further 4. case when there is a z-average but no peaks at all: This typically happens when the z-average is so large that any peaks found in the distribution analysis do not fall into the default display range As an example if the z-average was 20µm and the distribution analysis would find a single broad peak in the 15-25µm range then this would not be shown, instead only a flat line would appear on the distribution plot. DLS is not an ideal technique for such large particles, thus the reasonable cutoff in the display.}}

**Further Resources**

- White Paper: Dynamic light scattering – common terms defined
- Technical Note: Intensity – Volume – Number: Which Size is Correct?
- FAQ: What is the z-average?
- Post: Polydispersity what does it mean?

**Previously**

- 5 steps to improve SEC-LS data
- Demonstration of Chromatography software (3minutes)
- Which size is right: intensity volume number distributions