A first-principles order-parameter theory of the fluid-solid transition is presented in this paper. The thermodynamic potential $\ensuremath{\Omega}$ of the system is computed as a function of order parameters ${\ensuremath{\lambda}}_{i}(={\ensuremath{\lambda}}_{{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}_{i}})$ proportional to the lattice periodic components of the one-particle density $\ensuremath{\rho}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$, ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{K}}}_{i}$'s being the reciprocal-lattice vectors (RLV) of the crystal. Computation of $\ensuremath{\Omega}({{\ensuremath{\lambda}}_{i}})$ is shown to require knowing $\ensuremath{\Omega}$ for a fluid placed in lattice periodic potentials with amplitudes depending on ${\ensuremath{\lambda}}_{i}$. Using systematic nonperturbative functional methods for calculating the response of the fluid to such potentials, we find $\ensuremath{\Omega}({{\ensuremath{\lambda}}_{i}})$. The fluid properties (response functions) determining it are the Fourier coefficients ${c}_{i}(={c}_{{\stackrel{\ensuremath{\rightarrow}}{\mathrm{K}}}_{i}})$ and ${c}_{0}(={c}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}=0})$ of the direct correlation function $c(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$. The system freezes when at constant chemical potential $\ensuremath{\mu}$ and pressure $P$, locally stable fluid and solid phases [i.e., minima of $\ensuremath{\Omega}({{\ensuremath{\lambda}}_{i}})$ with ${{\ensuremath{\lambda}}_{i}}=0$ and ${{\ensuremath{\lambda}}_{i}}\ensuremath{\ne}0$, respectively] have the same $\ensuremath{\Omega}$. The order-parameter mode most effective in reducing $\ensuremath{\Omega}({{\ensuremath{\lambda}}_{i}})$ corresponds to ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{K}}}_{j}$ being of the smallest-length RLV set (${\mathrm{c}}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}}$ is largest for $|\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}|\ensuremath{\simeq}|{\stackrel{\ensuremath{\rightarrow}}{\mathrm{K}}}_{j}|$). In some cases one has to consider a second order parameter ${\ensuremath{\lambda}}_{n}$ with a RLV ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{K}}}_{n}$ lying near the second peak in ${c}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}}$. The effect of further order-parameter modes on $\ensuremath{\Omega}$ is shown to be small. The theory can be viewed as one of a strongly first-order density-wave phase transition in a dense classical system. The transition is a purely structural one, occurring when the fluid-phase structural correlations (measured by ${c}_{j}$, etc.) are strong enough. This fact has been brought out clearly by computer experiments but had not been theoretically understood so far. Calculations are presented for freezing into some simple crystal structures, i.e., fcc, bcc, and two-dimensional hcp. The input information is only the crystal structure and the fluid compressibility (related to ${c}_{0}$). We obtain as output the freezing criterion stated as a condition on ${c}_{j}$ or as a relation between ${c}_{j}$ and ${c}_{n}$, the volume change $V$, the entropy change $\ensuremath{\Delta}s$, and the Debye-Waller factor at freezing for various RLV values. The numbers are all in very good agreement with those available experimentally.

First-principles order-parameter theory of freezing

(1967). Electrons in disordered structures. Advances in Physics: Vol. 16, No. 61, pp. 49-144.

Electrons in disordered structures

The techniques of neutron diffraction and x-ray diffraction, as applied to structural studies of liquids and glasses, are reviewed. Emphasis is placed on the explanation and discussion of the experimental techniques and data analysis methods, as illustrated by the results of representative experiments. The disordered, isotropic nature of the structure of liquids and glasses leads to special considerations and certain difficulties when neutron and x-ray diffraction techniques are applied, especially when used in combination on the same system. Recent progress in experimental technique, as well as in data analysis and computer simulation, has motivated the writing of this review.

Neutron and x-ray diffraction studies of liquids and glasses

Neutron and x-ray diffraction are widely used to measure the structure of liquids and disordered solids. Using techniques such as isotope substitution or anomalous dispersion or combining neutron and x-ray data, it is sometimes possible to invert the total diffraction patterns from these materials into a set of partial structure factors, which describe the correlations between specific atom types in the material. However, even in situations where the matrix for performing this inversion appears well determined, there are significant uncertainties in the process and it is rarely possible to achieve a unique set of partial structure factors in practice. Based on the much earlier method of F. G. Edwards and J. E. Enderby [J. Phys. C 8, 3483 (1975)] and extending the reverse Monte Carlo method of McGreevy [J. Phys.: Condens. Matter 13, R877 (2001)] and others, a modified approach is developed here that allows possible atomic distribution functions, which are consistent with the measured data to be explored. The basis of the present approach is that any solution to the inversion process must be derivable from a distribution of nonoverlapping atoms or molecules as in the physical system under investigation. Solutions to the problem of inverting the measured differentialmore » cross sections to partial structure factors are then extracted assuming different levels of confidence in the data, confidence being represented by a feedback factor on a scale of 0-1. These different solutions serve to identify where ambiguities exist in the derived partial structure factors, particularly when a particular partial structure factor contributes only weakly to the total diffraction pattern. The method is illustrated using some old diffraction data on molten zinc chloride that have significant uncertainties associated with them, but that have been used extensively as the basis for a number of computer simulations of this material.« less

Partial structure factors from disordered materials diffraction data: An approach using empirical potential structure refinement

The Nanoscale Ordered MAterials Diffractometer (NOMAD) is neutron time-of-flight diffractometer designed to determine pair distribution functions of a wide range of materials ranging from short range ordered liquids to long range ordered crystals. Due to a large neutron flux provided by the Spallation Neutron Source SNS and a large detector coverage neutron count-rates exceed comparable instruments by one to two orders of magnitude. This is achieved while maintaining a relatively high momentum transfer resolution of a δ Q / Q ∼ 0.8 % FWHM (typical), and a possible δ Q / Q of 0.24 % FWHM (best). The real space resolution is related to the maximum momentum transfer; a maximum momentum transfer of 50 A−1 can be obtained routinely and the maximum momentum transfer given by the detector configuration and the incident neutron spectrum is 125  A - 1 . High stability of the source and the detector allow small contrast isotope experiments to be performed. A detailed description of the instrument is given and the results of experiments with standard samples are discussed.

The Nanoscale Ordered MAterials Diffractometer NOMAD at the Spallation Neutron Source SNS

The determination of the structure factor for liquid metals at temperatures up to 1300 °C by means of neutron diffraction techniques is described. Particular attention is given to the corrections which must be applied if reliable structure factors are to be extracted from the measurements, including those for absorption, multiple scattering and inelastic effects. It is concluded that for the heavier elements the neutron method can yield reliable data which are accurate to a few per cent over most of the range of momentum transfer.

https://iopscience.iop.org/article/10.1088/0022-3719/1/3/329/pdf

The structure factor for liquid metals I. The application of neutron diffraction techniques

The main results of a recent experimental determination of the partial structure factors for liquid Cu6Sn5 are outlined. The results are discussed in terms of a hard sphere model and some general comments on the structure of binary alloys are made.

The partial structure factors of liquid alloys

A detailed analysis of the diffraction pattern of liquid Cu6Sn5 has been carried out using neutron techniques. The scattering powers of the alloys were changed by isotopic substitution and the three structure factors required to specify a binary liquid alloy were calculated. It is found that both copper-copper and tin-tin correlations are clearly defined, and that the long-range concentration fluctuations are important in the copper-tin correlation. Some general remarks about the problem of obtaining the three structure factors for binary alloys are also made.

The partial structure factors of liquid Cu-Sn

The structure factor for liquid metals at low angles

The structure factors for four liquid metals (Zn, Tl, Sn and Bi) at temperatures close to the melting point have been determined by means of neutron diffraction and the data used to calculate the total correlation functions. In addition a detailed study has been made of liquid Pb as a function of temperature. Data are given for the structure factors, the total correlation functions and the direct correlation functions at four temperatures in the range 340 °c to 1100 °c. The total and direct correlation functions were used in the Percus-Yevick and hyper-netted chain equations to calculate the interionic potential. The structure factors were applied to the Ziman theory of electron transport in liquid metals. It is shown that the predicted temperature dependence of resistivity and thermoelectric power is in good agreement with experiment.

The structure factor for liquid metals II. Results for liquid Zn, Tl, Pb, Sn and Bi

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