X-ray photon correlation spectroscopy (XPCS)

XPCS is an experimental technique which can be used, for example, to study protein dynamics in solution.

Basic Principles

Figure 1: Schematic of the XPCS technique.

When a coherent beam (parallel and no frequency distribution) falls on an object with static or dynamic disorder, a random array of bright and dark spots are created by the irregular interference of scattered waves from different parts of the object. This is called speckle pattern. As the scattering parts fluctuate (due to dynamics), the speckle pattern also fluctuates. By probing such fluctuations, it is possible to obtain the information about the sample dynamics. X-ray photon correlation spectroscopy (XPCS) exploits this mechanism to study the dynamics using a high brilliance synchrotron X-ray source. In this technique, a series of images are collected over time using an area detector which records intensity fluctuations in q-space (reciprocal space, i.e. length scale, L ~ 2 π/q) with time as schematically shown in Figure 1. Such an intensity fluctuation can be quantified by the intensity auto-correlation function (ICF)

In many cases g2(q,t) can be described by an exponential decay function

Where, τ is the characteristic relaxation time, β is the scattering contrast, and γ is the stretching exponent called Kohlrausch-Williams-Watts (KWW) exponent. γ = 1 stands for a simple diffusive dynamics, and γ> 1 indicates a compressed exponential decay, and γ < 1 stands for the slower dynamics than diffusion. In the case of a non-equilibrium, such ICF can be obtained as a function of time which is presented in terms of two time correlation function (TTC) given by

where the average is over image pixels of the detector. C measures the evolution of ICF along the time tage = (t1+t2)/2. A typical TTC is shown in the left panel of Figure TTC. The width of the bright part along tage (diagonal direction) is proportional to the relaxation time at that tage. The change in width indicates the dynamical heterogeneity of the system. In this case, the heterogeneity can be quantified by a higher order correlation function, χT which is given by

χT peaks around the inflection point of g2. χT corresponding to the dynamics shown in the left panel (Figure 2) is show in the right panel of Figure 2. Here, the maximum value of χT gives the variance of relaxation time and hence the degree of heterogeneity. On the other hand, the value, t at χT(max) provides the estimation of length-scale corresponding to the spatial heterogeneity.

Figure 2: (Left): A time correlation function (TTC), and (right): χT showing dynamical heterogeneity

Experimental Setup

There are several synchrotron beamlines specialized in XPCS, e.g.

Figure 3: ESRF at night.


We studied bovine protein Immunoglobulin G (IgG, purchased from Sigma Aldrich) in the presence of polyethylene glycol (PEG). The solution was filled in a 1.5 mm quartz capillary tube. XPCS was performed using a heating stage (Linkam, UK) on the sample stage to change the sample temperature. The measurement was performed in transmission geometry with an incident beam of 8.5 keV and sample to detector distance of 21.3 m at PETRA III, P10, DESY, Hamburg, Germany. The beam dimension was 100x100 μm2. Figure 4 shows a TTC at a quench temperature of -2 °C at q = 0.0011 A-1.

Figure 4: (Left): TTC for a quench from 37 °C to -2 °C for IgG-PEG showing a slowing down of the dynamics, middle: Intensity correlation functions extracted from the diagonal cuts of the TTC, right: the relaxation time as s function of sample age. Line profiles at different sample age (tage), represents the ICF as a function of tage. The broadening of the bright regime shows a slowing down of the dynamics with sample age. The diagonal cuts of this TTC at different sample ages are shown in the middle panel of Figure 4. The colorbar shows the sample age in seconds. The relaxation time τ extracted from these ICFs are plotted as a function of tage in the right panel of Figure 4. The slowing down of the dynamics with tage can be observed here.


[1] N. Begam, A. Ragulskaya, A. Girelli, H. Rahmann, S. Chandran, F. Westermeier, M. Reiser, M. Sprung, F. Zhang, C. Gutt, and F. Schreiber. Kinetics of network formation and heterogeneous dynamics of an egg white gel revealed by coherent X-ray scattering Phys. Rev. Lett. (2021) 126 098001