**2. The Geometrical Properties of Silica**

SiO<sup>2</sup> is the chemical formula of a group of minerals constituting about 95% of the Earth's rocks and soils [23,24]. The most stable polymorph is *α*-quartz, which has a trigonal crystal structure and exists in both a right- and left-handed form. As shown in Figure 1, when *α*-quartz is heated to 573 ◦C at atmospheric pressure, it transforms into *β*-quartz and assumes a hexagonal crystal structure [25,26]. By further increasing the temperature, *β*-quartz turns into HP-tridymite (870 ◦C) and then into *β*-cristobalite (1470 ◦C). At 1713 ◦C, solid *β*-cristobalite melts into fused silica, which is a homogeneous and isotropic material characterized by a highly randomized structure. Since phase transitions are reversible, a slow cooling of the melt leads back to the crystalline form. However, if the temperature drop is rapid, the system freezes in a metastable state and eventually relaxes to silica. This non-crystalline form of SiO<sup>2</sup> is characterized by a disordered network lacking long-range order and by the presence of low-density regions.

**Figure 1.** Phase diagram of SiO<sup>2</sup> showing its main crystalline forms as well as the liquid (melted) phase.

The building block of all SiO<sup>2</sup> polymorphs is the [SiO4] 0 tetrahedron (Figure 2), that is, the three-dimensional unit formed by a silicon atom and four oxygen atoms bonded to it. Each O atom bridges two Si atoms and serves as a connection between adjacent tetrahedral units. The Si–O bond has a mixed covalent/ionic character and is described by a wavefunction with both *σ*[Si(*sp*<sup>3</sup> )–O(2*p*)] and *π*[Si(3*d*)–O(2*p*)] contributions [27,28]. This determines the shortening of the Si–O bond length (1.605 Å), the increase in the Si– O–Si angle, and the variability in the intertetrahedral angles [29]. On the other hand, the O–Si–O bond angle (*φ*) is 109.47°, indicating that the [SiO4] <sup>0</sup> unit has a perfect tetrahedral geometry [30,31].

The relative position of two corner-sharing tetrahedra is determined by the Si–O–Si bond angle (*θ*) and the Si–O–Si–O and O–Si–O–Si torsion angles. Many experimental works have attempted to estimate *θ*, but the results greatly vary from study to study. The reason for this uncertainty is that the angle is correlated with the Si–O and Si–Si distances and its value cannot directly be determined from experimental data. In fact, the interference function obtained by X-ray and neutron diffraction techniques only provides the distribution of interatomic distances and the estimation of bond and torsion angles requires the mathematical modeling of the total correlation function [32,33]. Similarly, in

solid-state nuclear magnetic resonance (NMR) spectra, the dependence of the <sup>29</sup>Si chemical shift and the <sup>17</sup>O quadrupolar coupling on *θ* requires an accurate correlation function in order to extract the bond angle distribution [34–36]. In either case, the structural modeling is generally underconstrained and the reliability of the results depends on the goodness of the starting assumptions [37]. Considering the data reported in the literature and the outcomes of their own simulations, Yuan and Cormack proposed a distribution of Si–O–Si angles with a mean value of 147° and a standard deviation of 10–13° [38]. This result was confirmed by Carpentier et al., who derived a mean value of 147° ± 11° by combining molecular dynamics with first-principles calculations of NMR parameters [33]. More recently, Malfait et al. analyzed a large set of data and located the most probable Si–O–Si angle at 149° with a full width at half maximum (FWHM) of 16° [39]. However, this value only corresponds to the peak of the mean (i.e., averaged over the four corners of each SiO<sup>4</sup> tetrahedron) Si–O–Si angle distribution derived from different NMR spectra. To calculate the individual Si–O–Si angle distribution, one should first know the statistical distribution of *θ* at the corners of the tetrahedra. Since a certain degree of correlation among these angles is inevitable, the individual Si–O–Si bond angles are not statistically independent and their distribution cannot be derived from experimental measurements.

**Figure 2.** Parameters defining the topology of silica. The angle *φ* corresponds to the O–Si–O bond angle, *θ* corresponds to the Si–O–Si bond angle, while *ω* is the dihedral angle between the planes formed by atoms O–Si–O and Si–O–Si.

Since the publication of Wright's paper [40], the three-dimensional organization of silica tetrahedra has been discussed in terms of two distinct torsion angles. Using the notation given in Figure 2, the angle *ω*<sup>1</sup> is defined as the dihedral angle between O<sup>1</sup> (or, equivalently, O<sup>2</sup> or O3), Si1, O4, Si2, while the angle *ω*<sup>2</sup> is defined by the atoms Si1, O4, Si2, O<sup>5</sup> (or, equivalently, O<sup>6</sup> or O7). However, from a geometrical perspective, the existence of two independent torsion angles is only justified if we define a conventional order to report the structure of silica (as for the primary structure of proteins). Since it is not possible to assign a univocal direction of rotation to silica rings, the two torsion angles are equivalent and carry the same information. For this reason it is sufficient to consider only one of the two torsion angles which we will call *ω*. From the analysis reported in the literature [38,41], it is found that *ω* varies as a function of the Si–O–Si bond angle and that, for *θ* in the 140–160° range, it has three maxima of around 60°, 180°, and 300°. These values correspond to the three staggered conformations of the O3Si–O–SiO<sup>3</sup> moiety, as viewed along the Si–Si axis, and are such that next-nearest-neighbor oxygen atoms are at a maximum distance.

#### **3. The Topology of Silica**

The analysis of the network topology of silica was first addressed by King, who proposed the shortest-path criterion to identify closed paths of alternating Si–O bonds [42]. According to this model, a ring is defined as the shortest closed path connecting two oxygen atoms bonded to the same Si atom. If the path is formed by *n* bonds (or, equivalently, by 2*n* atoms) the ring is called an *n*-membered ring (*n*-MR) [43,44]. By applying King's algorithm to *α*-quartz and *β*-cristobalite, both polymorphs are composed of regular six-membered rings arranged into different three-dimensional structures [45]. The former network is made up of interconnected hexagonal and trigonal helices of SiO<sup>4</sup> tetrahedra, whereas the latter consists of stacked layers of tetrahedra alternately pointing up and down [46]. In the case of amorphous silica, the flexibility of the Si–O–Si and Si–O–Si–O angles determines a high degree of structural disorder and this, in turn, leads to a quasi-random orientation of the tetrahedral units and a much broader ring-size distribution [47].

Many theoretical works have been dedicated to the statistical analysis of computermodeled *a*-SiO<sup>2</sup> structures. In Figure 3, we report the ring-size distribution of three models obtained by different computational methods. The first dataset was calculated by Rino et al. from a structural model obtained using classical molecular dynamics [31]. The second set comes from first-principle molecular dynamics calculations [48], while the last one is from a model simulated using a combination of classical and first-principles molecular dynamics [49]. The ring statistics of the three models shows that five- and six-membered rings are always the most frequent, followed by seven-membered rings. Three- and fourmembered rings, however, have different populations and their concentration depends on the simulation conditions and thermal history of the sample. The prevalence of 6-MRs can be explained by considering the volume–temperature diagram in Figure 4. In fact, when melted silica is cooled at a very slow rate, the atoms in the melt have enough time to rearrange and form crystalline *β*-cristobalite. Conversely, if the cooling is fast (quenching), the liquid cannot equilibrate with the forming solid and the result is the formation of amorphous silica [50]. Despite their different properties, the two polymorphs ideally derive from the same silica melt and their short- and medium-range topology shows a prevalence of six-membered rings.

**Figure 3.** Shortest-path ring statistics of the silica models reported by Rino et al. (grey) [31], Pasquarello and Car (red) [48], and Giacomazzi et al. (green) [49].

The presence of small-sized rings in vitreous silica is strongly correlated with the kinetics of the relaxation process. In fact, when the temperature of the melt is lowered avoiding crystallization, the liquid enters the supercooled phase and its structure continuously rearranges to

follow the system temperature. As a consequence, the first-order thermodynamic properties of the liquid (e.g., volume and enthalpy) decrease without any abrupt changes and the viscosity increases accordingly. When the viscosity becomes too high, the atomic motion slows down to the extent that the atoms cannot rearrange themselves into the volume characteristic of that temperature and pressure. At this point, the enthalpy begins to deviate from the equilibrium line and starts to follow a curve of gradually decreasing slope. Eventually, the structure of the system becomes fixed and the supercooled liquid solidifies into silica [51]. The range of temperatures over which the transition occurs is called the glass transformation range and the temperature at which the structure of the supercooled liquid is *frozen-in* in the solid state is the fictive temperature (*T*<sup>f</sup> ). Since the departure of the enthalpy from the equilibrium curve is determined by the viscosity of the liquid, i.e., by kinetic factors, a slower cooling rate allows the enthalpy to follow the equilibrium curve to a lower temperature. In this case, the fictive temperature shifts to lower values and the resulting glass has a different atomic arrangement than a more rapidly cooled one [52]. In particular, a slow quenching gives rise to a glass containing mostly medium-sized rings (6- and 7-MRs), whereas an increase in the quenching rate leads to a glass with a greater population of small- and large-sized rings [53].

**Figure 4.** Volume–temperature diagram representing the solidification of melted silica into crystalline or glassy SiO<sup>2</sup> . The supercooled liquid can be cooled at different quenching rates to produce glasses with varying fictive temperatures.

#### **4. Generation of Point Defects**

During the formation of a glass, part of the disorder characterizing the supercooled liquid is frozen-in in the solid state and the excess energy is stored as strained Si–O–Si bond angles [32,54]. The presence of distorted angles determines the formation of local highenergy structures like three- and four-membered rings. For regular planar three-membered rings, *θ* takes the value of 130.5° whereas, for regular planar four-membered rings, its value is 160.5°. In both cases, *θ* is far from its optimal value (144–155°) and, for this reason, the rings tend to release the excess energy by breaking a strained bond and turn into a bigger ring [55,56].

While in bulk silica the concentration of three- and four-membered rings is relatively low, silica optical fibers exhibit a higher concentration of small rings due to the residual stress generated during the manufacturing process [57]. In fact, the rapid cooling of optical fibers results in a *T*<sup>f</sup> higher than that measured in bulk samples [58]. This is mainly due to the influence of the drawing speed on the quenching dynamics. A higher drawing speed means a faster cooling rate which, in turn, results in a higher fictive temperature. In addition, the residual stress of the tensile load applied during the drawing process has a strong impact on increasing the concentration of defect precursors in non-irradiated samples [15,16]. This explains the higher sensitivity of OFs to radiation and the higher concentration of room-temperature stable point defects [59,60]. In the remaining part of this section, we will describe the three main intrinsic defects induced by radiation in silica-based materials.
