# From Femtoseconds to Hours--Measuring Dynamics over 18 Orders of Magnitude with Coherent X-rays
## 2 XPCS Concepts  
When a (partially) coherent beam is scattered by a sample that is static during the exposure, a grainy diffraction pattern, the so-called speckle pattern, is generated. This speckle pattern includes information on the position of all scattering objects in the exposed volume that can be obtained by iterative algorithms in many cases [3]. If the sample shows any dynamics during the exposure, the speckle pattern will change accordingly. The concept of XPCS is to track these changes over time by means of correlation functions from series of speckle patterns (see Figure 3). The experimental quantity in XPCS studies is the normalized intensity autocorrelation function  
\[g_{2}(q,\Delta t)=\frac{\langle I(q,t)I(q,t+\Delta t)\rangle}{\langle I(q,t) \rangle^{2}} \tag{1}\]  
that compares the intensities \(I(q,t)\) and \(I(q,t+\Delta t)\) at two different times with a lag time of \(\Delta t\). Here, \(q\) is the modulus of the wave vector transfer given by \(q\equiv|\mathbf{q}|=\frac{4\pi}{\Lambda}\sin(\theta/2)\), with wavelength \(\lambda\) and scattering angle \(\theta\). The averaging in Equation (1) is performed over both detector pixels with equivalent \(q\)-values and all times \(t\). Therefore, the \(g_{2}\)-function provides information on the average dynamics during the exposure. Strategies for measuring dynamic heterogeneities without averaging over \(t\) are presented in the next section. As a correlation function of intensity, the \(g_{2}\) function is a second-order correlation function. It relates to the first-order or field autocorrelation function \(g_{1}\) via the Siegert relation [40] to  
\[g_{2}(q,\Delta t)=1+\beta|g_{1}(q,\Delta t)|^{2}. \tag{2}\]  
Figure 2: **(a)** Publications mentioning X-ray photon correlation spectroscopy (XPCS), X-ray speckle visibility spectroscopy (XSVS) or X-ray intensity fluctuation spectroscopy (XIFS) as keyword, in the title or abstract. The solid line models a linear increase of the SCOPUS data between 1998 and 2020 with an increase rate of 1.62 publications per year. The dashed lines mark the years where new beamlines with XPCS dedication became online. **(b)** Distribution of subject areas of XPCS publications from 1995 to 2020. For each publication multiple areas are possible. Here, \(0\leq\beta\leq 1\) is the speckle contrast that is related to the degree of coherence and is further discussed in Section 2.4. Obviously, a non-zero degree coherence is needed to detect any dynamics information. The field correlation function \(g_{1}\) is also known as intermediate scattering function. It is expressed by the normalized structure factor \(S(q,\Delta t)\) that contains all dynamic information of the sample via the trajectories \(\mathbf{r}_{j}\) of all \(N\) scatterers  
\[g_{1}(q,\Delta t)=S(q,\Delta t)/S(q,0)=\frac{1}{N}\bigg{\langle}\sum_{k=1}^{N }\sum_{j=1}^{N}\exp(i\mathbf{q}\cdot[\mathbf{r}_{j}(\Delta t)-\mathbf{r}_{k}(0 )])\bigg{\rangle}. \tag{3}\]  
Depending on the underlying dynamics, \(g_{1}\) can be expressed in different forms. Some examples will be given below. The standard approach for the majority of sample systems is a Kohlrausch-Williams-Watts (KWW) function  
\[g_{1}(q,\Delta t)=\exp(-(\Gamma\Delta t)^{\gamma}). \tag{4}\]  
The characteristic relaxation time \(\tau\) is obtained as the inverse of the relaxation rate \(\Gamma=\tau^{-1}\) that shows a characteristic \(q\)-dependence \(\Gamma\propto q^{p}\). The KWW exponent \(\gamma\) is a measure of the distribution of relaxation times of individual particles. Both exponents, \(p\) and \(\gamma\), specify the type of the dynamics. For diffusion, e.g., Brownian motion of nanoparticles, one obtains an exponential decay with \(\gamma=1\) and a square-dependence of the relaxation rate, i.e., \(p=2\).