Simple analysis of scattering data with the Ornstein-Zernike equation

In this paper we propose and explore a method of analysis of the scattering experimental data for uniform liquidlike systems. In our pragmatic approach we are not trying to introduce by hands an artificial small parameter to work out a perturbation theory with respect to the known results, e.g., for hard spheres or sticky hard spheres (all the more that in the agreement with the notorious Landau statement, there is no physical small parameter for liquids). Instead of it being guided by the experimental data we are solving the Ornstein-Zernike equation with a trial (variational) form of the interparticle interaction potential. To find all needed correlation functions this variational input is iterated numerically to satisfy the Ornstein-Zernike equation supplemented by a closure relation. Our method is developed for spherically symmetric scattering objects, and our numeric code is written for such a case. However, it can be extended (at the expense of more involved computations and a larger amount of required experimental input information) for nonspherical particles. What is important for our approach is that it is sufficient to know experimental data in a relatively narrow range of the scattering wave vectors ($q$) to compute the static structure factor in a much broader range of $q$. We illustrate by a few model and real experimental examples of the x-ray and neutron scattering data how the approach works.

Figure 1

Pair correlation function for hard-sphere (squares) and LJ (circles) potentials ($x\equiv r/\sigma$ is dimensionless distance in the particle diameter $\sigma$ units.)

Figure 2

Static structure factor for PMMA spheres with attraction caused by the depletion potential. Circles present the experimental data obtained in [28]; dotted line are the results of calculation in [28]; solid line present the results of the current approach.

Figure 3

Structure factor of liquid krypton at liquid-gas coexistence. The squares present the data for the temperature 133 K; circles, 153 K; and triangles, 183 K. Our calculations are presented by the solid black line (133 K), dashed red line (153 K), and dotted blue line (183 K), correspondingly.

Figure 4

Lenard-Jones interaction potential for liquid krypton. Solid black line corresponds to 133 K, dashed red line, 153 K; and dotted blue line, 183 K ($x\equiv r/\sigma$, and $v_{\mathrm{int}}\equiv V/k_{B}T$ is the dimensionless interaction potential in $k_{B}T$ units).

Figure 5

Interaction potential for liquid krypton, rescaled to the temperature $T^{*}=153\mathrm{K}$ ($v_{\mathrm{int}}^{*}\equiv V/k_B{T}^{*}$, and $x\equiv r/\sigma$). Lines are the same as in Fig. 4.