the space Fourier transform of gαβ(R) [14]. For neutron scattering, one finds, for example:
(9)
Here, bα is the neutron scattering length of particles of species α [14] and the partial static structure factors are given by
(10)
Figure 2 Neutron and X‐ray structural factors (a and b panel, respectively) for a borosilicate system in the liquid and glass state.
Source: From Ref. [6].
where N is the total number of atoms and fαβ = 1 for α = β and fαβ = 1/2 otherwise. For X‐ray scattering, the relation is
(11)
where fα(s) is the scattering‐factor function (also called form factor). Figure 2 shows that even for the same glass sample, SN(q) and SX(q) are rather different since both are a weighted sum of the partial structure factors, each of which has a multitude of positive and negative peaks. Hence, the resulting sum may show a peak for SN(q) that is basically absent in SX(q). Since interpretation of experimental data is further complicated by the fact that one has k(k + 1)/2 partial structure factors in systems with k species, computer simulations can help to interpret correctly the various peaks if they do reproduce the partial structure factors in terms of peak positions and heights. As a matter of fact, the necessary accuracy is usually ensured only by ab initio simulations.
Another important situation is that of surfaces, for which experimental scattering techniques involving grazing‐ray geometries are much less powerful to obtain microscopic information. Because effective potentials are usually developed for bulk systems, their reliability is not guaranteed for surfaces for which the local structure can be very different. One example are two‐membered rings in SiO2 that are present at the surface but absent in the bulk. Hence, ab initio simulations are to be preferred for investigating surfaces since they can deal with such heterogeneous geometries.
One interesting example of surface effects is the difference between the interatomic distance and bond‐angle distributions of water molecules in the bulk and adsorbed at the surface of amorphous silica (Figure 3). Although there are no significant differences in the positions of the O─O and O─H peaks (Figure 3a), close to the surface the fluid is less structured than in the bulk as signaled by smaller peaks. Likewise, differences concerning the angle between water dipole (i.e. the bisector of the HOH angle) and the surface normal (Figure 3b) indicate a strong change of the local geometry. Only accurate interaction potentials, such as ab initio simulations will be able to pick up these effects.
Figure 3 Structural differences between bulk (BW) and surface (SW) water in SiO2 glass [15]. (a) Water–water radial distribution function. (b) Distribution of the angle θ between the water dipole (bisecting the HOH angle) and the surface normal.
4 Vibrational Properties
The vibrational density of states (vDOS) g(ω) is the normalized distribution function of the eigenfrequencies ωp of the dynamical matrix of the system which is directly related to the second derivative of the potential with respect to the coordinates R I and R J, i.e. ∂ 2Φ/(∂R I, i ∂R J,j) for i,j ∈{x, y, z}, thus
(12)
where 3N − 3 is the number of eigenmodes with nonzero frequency. Associated with each eigenfrequency ωp of the dynamical matrix is an eigenvector e p that gives detailed information regarding the particles that oscillate with the frequency ωp. Studies of e p have allowed to gain insight into the nature of the vibrational modes of various materials such as silica and germania glasses [12, 16–18] and more complex systems ([6, 10, 11, 19], Chapter 3.4).
The structure of a solid is related to a balance of the forces between its atoms, i.e. to the first derivative of the potential since these forces must mutually compensate to ensure a mechanically stable equilibrium. In contrast, vibrational properties are related to the curvature of the potential, i.e. to its second derivatives. As a consequence, it is quite hard to find effective potentials that reproduce correctly the experimental vDOS, even for systems as simple as pure silica [20].
The true vDOS given by Eq. (12) is not directly accessible in experiments, in which only an effective vDOS, G(ω) = C(ω)g(ω), can be measured. The function C(ω) describes how the modes with frequency ω are coupled to the probing radiation (photons, neutrons, etc.). For inelastic neutron‐scattering experiments, this function can be derived, within approximations, from the eigenvectors e p of the dynamical matrix [21].
In practice, the vibrational properties of a system can be calculated in two different ways. The first is to determine the local potential minimum of the glass structure (i.e. to quench the sample to T = 0 K) and then to calculate directly the dynamical matrix, for instance, by a finite‐difference scheme for the forces. By diagonalizing this matrix, one can then obtain its eigenfrequencies and eigenvectors and, hence, g(ω). The second method consists in running a simulation in which one solves Newton’s equations of motion at low temperatures where the harmonic approximation is valid. If one measures the time autocorrelation function of the velocity of a particle, its time Fourier transform is directly proportional to the vDOS. This approach is from the computational point of view cheap, but it has the drawback of not giving any information on the eigenvectors, i.e. on the nature of the motion of the atoms.
The advantages of ab initio over effective pair‐potential simulations are illustrated by the neutron‐effective vDOS for silica (Figure 4). The effective potentials give a surprisingly good description of the structural properties of real silica [23] and also
