Topics Covered
Introduction
What Does the ZAverage Mean?
Concluding Comments
References and Notes
About Horiba
The Zaverage calculated particle size result is often used in dynamic light scattering. This note discusses the meaning of the Zaverage value for particle size.
Figure 1. SZ100 Nanopartica Analyzer
Introduction
Dynamic light scattering (DLS) results are often expressed in terms of the Zaverage. The Zaverage arises when DLS data is analyzed by the use of the technique of cumulants (1). Since the calculation of the Zaverage is mathematically stable, the Zaverage result is insensitive to noise. And that makes it a preferred DLS size parameter. The purpose of this document is to clarify the meaning of this value.
What Does the ZAverage Mean?
The Zaverage can be expressed as the intensity based harmonic mean (2, 3) and is shown by the equation below:
D_{z} = ∑ S_{i}/∑ (S_{i}/D_{i})
Here, S_{i} is the scattered intensity from particle i and Di is the diameter of particle i. Note that the result is in the form of a harmonic mean. Since this mean is calculated from the intensity weighted distribution, leading to the statement that the Zaverage size is the the harmonic intensityweighted arithmetic average particle diameter. It is easy to understand why most people simply say “Zaverage”.
Rayleigh scatterers, S_{i} ~ D_{i}^{6}. Therefore, the Zaverage can be approximated as:
D_{z} ≈ ∑ D_{i}^{6}/∑ D_{i}^{5}
There are two points about the use of the term “Zaverage.” The first is that the use of the term Zaverage in DLS does not match the use of the term when one is using light scattering to analyze polymers. The second is that one occasionally finds other notation, such as xDLS or dDLS. Even so, “Zaverage” is the most common term.
In Figure 2 we illustrate the Zaverage with a calculation showing a lognormal size distribution. A volume weighted differential size distribution is shown in blue. A discussion of different distribution types can be found in HORIBA TN154, “Particle Size Result Interpretation: Number vs. Volume Distributions.” The example distribution’s median (D_{v,50}) size is indicated as 100 nm. From this distribution, we calculate the scattered intensity for each particle size (4). This intensity based distribution is then plotted in green. Finally, the harmonic mean of the intensitybased distribution is indicated at 97 nm. Note that the Zaverage size is close to, but does not equal D_{v,50}.
Figure 2. Hypothetical size lognormal volume weighted distribution and corresponding intensity weighted distribution showing the meaning of the Zaverage.
How is the Zaverage calculated from raw DLS data? The Zaverage size value is calculated by the methods of cumulants (1). Since this technique relies on numerically stable least squares fitting, it is relatively insensitive to experimental noise.
In cumulants analysis the baseline subtracted autocorrelation function, C, is treated as an exponential decay of the following form:
C(τ) = Aexp(2Γt + µ_{2}τ^{2} ...)
Here, C is the baseline subtracted autocorrelation function and t is delay time. Values for A, Γ, and µ_{2} can be readily obtained by a least squares fit. One then finds the intensity weighted average diffusion coefficient D_{t,avg} with the relation Γ = D_{t,avg} q^{2}. Here q is the scattering vector given by q = (4πn/λ)sin(θ/2). The refractive index of the liquid is n. The wavelength of the laser light is λ, and scattering angle, θ. Finally, one uses the StokesEinstein relation to go from D_{t} to Zaverage particle size, D_{z}.
D_{z} = k_{B}T/3πηD_{t,avg}
where

D_{z} is the hydrodynamic diameter (this is the goal: particle size!)

D_{t,avg} is the translational diffusion coefficient (by DLS)

k_{B} is Boltzmann’s constant (known)

T is thermodynamic temperature (known)

η is dynamic viscosity (known)
Unfortunately, the weighting of the average is somewhat convoluted. Recall that the decay constant is proportional to the diffusion coefficient. So, by DLS one has determined the intensity weighted diffusion coefficient. The diffusion coefficient is inversely proportional to size. Therefore, the “Zaverage size” is the intensity weighted harmonic mean size.
Concluding Comments
Despite the convoluted meaning, the Zaverage size increases as the particle size increases. Therefore it provides a reliable measure of the average size of a particle size distribution. Also, it is easily measured. For these reasons, the Zaverage size has become the accepted norm for presenting particle sizing results by DLS.
The HORIBA SZ100 also presents size measurement results as a distribution table and graph and calculated mean diameter or diameters for multimodal distributions. The methods behind those calculations are beyond the scope of this work.
References and Notes

Koppel, D.E. “Analysis of Macromolecular Polydispersity in Intensity Correlation Spectroscopy: The Method of Cumulants” J. Chem. Phys 57 (11), pp 48144820, 1972.

ISO 22412:2008 Particle Size Analysis – Dynamic Light Scattering

Thomas, J. C. “The determination of log normal particle size distributions by dynamic light scattering” J. Colloid Interface Sci. 117 (1) pp 187192 (1987)

Here the RayleighDebyeGans approach is used and a larger average particle size is chosen to illustrate the regime where the approximations in equation 2 do not apply. This calculation is for particles in water, a 532 nm laser, and a 90 degree scattering angle.
About Horiba
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This information has been sourced, reviewed and adapted from materials provided by Horiba.
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