# A Wide-Ranging NMR Probe for Multiscale Structures
## 2 Basic Principles and Models  
### Chemical Shift  
In the most established description of the chemical shift of \({}^{129}\)Xe adsorbed in a porous material, it is expressed as the sum of different contributions:  
\[\delta=\delta_{0}+\delta_{\text{Xe-Xe}}\rho_{\text{Xe}}+\delta_{\text{S}}+ \delta_{\text{SAS}}+\delta_{\text{E}}+\delta_{\text{M}} \tag{1}\]  
\(\delta_{0}\) is the chemical shift of xenon gas at zero pressure, which is conventionally set to 0 ppm and used as reference. This value can be determined by measuring the dependence of the chemical shift of free xenon gas on pressure and extrapolating to zero. The dependence of \(\delta_{0}\) on temperature can be neglected in the standard temperature operating range [32]. The second term corresponds to the chemical shift increase arising from xenon-xenon interactions. This term is directly proportional to xenon density, \(\rho_{\text{Xe}}\), and, as such, it can be neglected if the density is sufficiently small. In microporous materials, such as zeolites, this term is particularly significant: by definition, micropores have diameters below 2 nm, and, since the Xe atom has a van der Waals diameter of 0.44 nm, xenon-xenon interactions inside these pores are very relevant. Instead, in mesoporous materials, which have pore diameters between 2 nm and 50 nm, this term can generally be neglected, unless experiments are performed at very high xenon pressures. This second term is generally strongly dependent on temperature [32]. \(\delta_{\text{S}}\) derives from the interaction of xenon atoms with the surface and it is fundamental in studying porous materials and their free volume. The numerous proposed models for determining pore sizes and their distributions via \({}^{129}\)Xe chemical shift mainly focus on this term. \(\delta_{\text{S}}\) is temperature dependent as well: the mobility of xenon atoms is increased with increasing temperature and their residence time on the surfaces reduced. Thus, generally speaking, this term decreases with increasing temperature. \(\delta_{\text{SAS}}\) depends on the interaction of xenon with strong adsorption sites (SAS) such as highly charged or paramagnetic cations. The interaction of xenon with SAS can easily be recognized by analyzing the dependence of the chemical shift on xenon concentration. At very low xenon concentrations, in the presence of strong adsorption sites, the chemical shift increases significantly with decreasing xenon concentration, giving rise to a characteristic hyperbolic dependence [33]. This chemical shift increase is due to the predominance of the \(\delta_{\text{SAS}}\) term: at very low concentrations, most of the xenon atoms interact with this type of site. In the absence of SAS, this term can be neglected, and the chemical shift generally decreases with decreasing xenon concentration, even at low concentration values. \(\delta_{\text{E}}\) and \(\delta_{\text{M}}\) are contributions induced by the local electrical fields generated by charged species and by paramagnetic nuclei, respectively. These terms can be omitted in the absence of the corresponding species, greatly simplifying the equation. Equation (1) gives a quantitative relationship between the various contributions given by the surrounding environment and the xenon chemical shift. However, the xenon atoms can exchange quickly between different adsorption sites and the gas phase in the NMR time scale. The resulting chemical shift is an average of the chemical shifts of xenon atoms in different environments, weighted by the respective molar fractions. Several methods for distinguishing between different sites and obtaining the corresponding chemical shift values have been developed, as will be discussed below. Since every material has its own peculiarities and morphological characteristics, there is no general theory for this approach, and appropriate models have to be determined depending on the material under study. Alternatively, [PERSON] et al. proposed to immerse the porous material under study in a liquid or solid medium [34; 35].