Amorphous pair-correlation function

Unlike the solid state, the liquid state cannot be characterized by a static description. In a liquid, bonds break and refomi continuously as a fiinction of time. The quantum states in the liquid are similar to those in amorphous solids in the sense that the system is also disordered. The liquid state can be quantified only by considering some ensemble averaging and using statistical measures. For example, consider an elemental liquid. Just as for amorphous solids, one can ask what is the distribution of atoms at a given distance from a reference atom on average, i.e. the radial distribution function or the pair correlation function can also be defined for a liquid. In scattering experiments on liquids, a structure factor is measured. The radial distribution fiinction, g r), is related to the stnicture factor, S q), by... 

As noted earlier, the diffraction of X-rays, unlike the diffraction of neutrons, is primarily sensitive to the distribution of 00 separations. Although many of the early studies 9> of amorphous solid water included electron or X-ray diffraction measurements, the nature of the samples prepared and the restricted angular range of the measurements reported combine to prevent extraction of detailed structural information. The most complete of the early X-ray studies is by Bon-dot 26>. Only scanty description is given of the conditions of deposition but it appears likely his sample of amorphous solid water had little or no contamination with crystalline ice. He found a liquid-like distribution of 00 separations at 83 K, with the first neighbor peak centered at 2.77 A. If the pair correlation function is decomposed into a superposition of Gaussian peaks, the area of the near neighbor peak is found to correspond to 4.23 molecules, and to have a root mean square width of 0.50 A. 

Fig. 7 g. Atom pair correlation functions for amorphous solid and liquid D2O. Dotted curves are weighted sums of contributions from OO, OD and DD interactions. Broken curves show contributions from OO interactions alone. Solid curves represent nearly evenly weighted sums of OD and DD atom pair correlation functions. (From Ref. 27>)... 

FIGURE 20.14 Pair correlation function for the case of a liquid, of glass, and/or amorphous solid. (From Gcpel, W. and Wiemhcfer, H.-D.,Statistische Thermodynamik, Spektrum Akademischer Verlag, Heidelberg, Berlin (2000).)... 

For an amorphous mass of particles there is no correlation between particles that are far apart. The joint probability of finding particles at r and ri is simply the product of the individual probabilities. Let us define a configurational pair-correlation function g(ri, r2) as... 

Amorphous ice has been studied in some detail by both X-ray and neutron diffraction [738-740]. The O- -O pair-correlation functions are similar to those of liquid water, except that on condensing on very cold surfaces, i.e., 10 K, there is an extra sharp peak at 3.3 A. This indicates some interpenetration of the tetrahedral disordered ice-like short-range structures. It appears that none of the many proposed atom-atom potential energy functions can simulate a structure for liquid water that predicts pair-correlation functions which are a satisfactory fit to the experimental data [741, 742]. Opinions seem to differ as to whether the discrepancy is in the theory or the experiments. 

An alternative approach to this question is to investigate the effect in the disordered solid phase where the spatial correlations are more pronounced and higher argon concentrations can be achieved. Fig 5a shows the two curves for a sample of codeposited amorphous ice with 8 atomic per cent of argon. A difference analysis yields a composite pair correlation function in the form... 

In section 5 the structure or atomic arrangement associated with amorphous alloys is defined in terms of the pair correlation functions and the radial distribution... 

Viscous flow sets in only above about 0.6 of the glass transition temperature 7g which for many technical interesting alloys is between 400 and 600°C. Even though amorphous metals have an atomic structure (usually described by the pair correlation function) that is similar to the corresponding alloy in the liquid state, their viscous flow resembles rather the behavior of crystalline metals than those of liquids. 

Neutron diffraction scattering with isotope substitution (NDIS) is considered today the only means by which we can extract direct information about the site-site pair correlation functions of liquids and amorphous materials [143-146]. What makes this approach possible is the justifiable assumption that light (H2O) and deuterated D2O) water exhibit the same structural features [147], and allow the extraction of... 

In order to describe the static structure of the amorphous state as well as its temporal fluctuations, correlation functions are introdnced, which specify the manner in which atoms are distributed or the manner in which fluctuations in physical properties are correlated. The correlation fimctions are related to various macroscopic mechanical and thermodynamic properties. The pair correlation function g r) contains information on the thermal density fluctuations, which in turn are governed by the isothermal compressibility k T) and the absolute temperature for an amorphous system in thermodynamic equilibrium. Thus the correlation function g r) relates to the static properties of the density fluctuations. The fluctuations can be separated into an isobaric and an adiabatic component, with respect to a thermodynamic as well as a dynamic point of view. The adiabatic part is due to propagating fluctuations (hypersonic soimd waves) and the isobaric part consists of nonpropagating fluctuations (entropy fluctuations). By using inelastic light scattering it is possible to separate the total fluctuations into these components. 

In this chapter we present results of neutron diffraction and conputer simulation studies on hydrogenated and deuterated CuxTi. x(0.35 < x < 0.67) amorphous alloys. Ihe first part will serve as an illustration of the above considerations and presents the evolution of the interference functions and pair-correlation functions as a function of alloy composition and hydrogen or deuterium content -. In the second part we will use these eiqperlmental results to build up a computer model of these alloys. The last part will present the thermal evolution of a Cuq 50 TIq 50 Hq 5 amorphous alloy as followed by large-angle and small-angle neutron scattering ... 

Fig. 2 Reduced pair-correlation functions for amorphous (a) CUgyTi33, (b) cusoTiso and (c) CuasTtes alloys.

The above treatment assumes a single atomic component system whereas the rare earth transition metal amorphous materials (R-T) are two component and thus three pair correlation functions, G(r), exist, one each for the possible R-R, R-T, and T-T combinations. These are lumped together to produce the observed scattered intensity function, but may be separated by experiments on isotopically substituted alloys which have different neutron scattering amplitudes, or as in the case of Co-P by separating the magnetic components using polarized neutrons (Bletry and Sadoc, 1975). 

Fig. 11.9 Decomposition of partial pair correlation functions g, (r) into neighbor distributions in amorphous As2Se3 and liquid (450 K) GeTc4. The insets show the positions (first moments) of the neighbor distributions and their standard deviations (second moments represented as error bars)...

The structure factors, S q), and pair correlation functions, g r) of Sb2Tc3 and Sb2Te in the liquid and amorphous phases are shown in Fig. 18.2. If the overall agreement between experiment and simulation is reasonable, it is not as good as in the case of other chalcogenide glasses, such as GeSc2 [22]. It was established [23] that the discrepancy is mostly due to an over coordination of Te atoms in DFT calculations. However, AIMD simulation reproduce all trends observed experimentally. 

Fig. 18.2 Structure factors (lefi) and pair correlation functions (right) of Sb2Te and Sb2Te3 in their liquid and amorphous state. The experimental data (to be published) were recorded by neutron scattering and are shown with symbols. The thin black line corresponds to the AIMD simulation data, for which the total g(r) are computed with proper weighting of the partials by the elements neutron scattering lengths...

The pair correlation functions (PCFs) provide local structural information of essential interest for amorphous materials. The peaks in these functions describe the average distance of neighboring atoms from a central atom. Since amorphous materials do not possess long range order, gfr) -> 1 as r oo. For crystalline structures, g r) is a sum of delta functions, with each term coming from a coordination shell. 

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