We investigate the fundamental limits of using total-scattering measurements to simultaneously determine the atomic number density (ρ) and pair distribution function (g(r)) of disordered materials. Building on rigorous Fourier-transform relationships between the structure factorS(Q) andg(r), we first show analytically that even infinitely precise, noise-freeS(Q) data-spanning an unboundedQ-range-cannot uniquely specify bothρandg(r). This non-uniqueness arises from phase information loss, finite-dimensional projections inherent in one-dimensional pair distributions, and the mathematical insensitivity ofS(Q) to coordinated rescaling of density and radial distances. In addition, we highlight practical problems arising from mathematical methods aimed at extractingρvia Fourier transform of data. Direct calculation from integratingg(r)-1(Yarnell method) converges badly for high density because of long-range structure ing(r), and at low density because of a bias coming from the central atom ing(r). Indirect calculation from the slope off⋅[g(r)-1](Eggert method) depends sensitively on having good quality high-Qdata. To address these ambiguities, we introduce a density-sweep protocol using the empirical potential structure refinement (EPSR) within theab initioaugmented structure solving engine framework. By systematically varying trial densities around target values (±5%-50%) and evaluating both the internal EPSRR-factor and an externalR-factor based on finalF(Q), one can identify a clear minimum bracketing the trueρwithout reliance on external equations of state or arbitrary fitting ranges. We showcase the effectiveness of the method by application to supercritical krypton at multiple pressures, liquid D2O at 298 K and amorphous silica and reliably recover known densities within±5%.

We performed a series of neutron scattering experiments on deeply subcritical liquid nitrogen at 90 K (0.7TC). Our findings, when taken together with our previous results at 160 K (1.27TC) and 300 K (2.4TC), allow the Frenkel line phenomenon to be characterized in a reliable and consistent manner over an extremely broad temperature range, extending into the subcritical regime. Through an analysis of local order, we show how the fluid structure changes as the Frenkel line is crossed and present a new method for identifying the line. Our determination of coordination numbers shows a remarkable data collapse when plotted against density. This allows us to produce a universal relationship relating the coordination number to the density of a simple fluid, dictated by molecular/atomic size and its density on the melt line.

Physics-based first-principles pressure-volume-temperature equations of state (EOS) exist for solids and gases but not for liquids due to the long-standing fundamental problems involved in liquid theory. Current EOS models that are applicable to liquids and supercritical fluids at liquid-like density under conditions relevant to planetary interiors and industrial processes are complex empirical models with many physically meaningless adjustable parameters. Here, we develop a generally applicable physics-based (GAP) EOS for liquids including supercritical fluids at liquid-like density. The GAP equation is explicit in the internal energy, and hence links the most fundamental macroscopic static property of fluids, the pressure-volume-temperature EOS, to their key microscopic property: the molecular hopping frequency or liquid relaxation time, from which the internal energy can be obtained. We test our GAP equation against available experimental data in several different ways and find good agreement. Our GAP equation, unavoidably and similarly to solid EOS, contains a semi-empirical term giving the energy of the static sample as a function of volume only (EST(V)). Our testing includes studies along isochores, in order to examine the validity of the GAP equation independently of the validity of any function we may choose to utilize forEST(V). The only other adjustable parameter in the equation is the Grüneisen parameter for the fluid. We observe that the GAP equation is similar to the Mie-Grüneisen solid EOS in a wide range of the liquid phase diagram. This similarity is ultimately related to the condensed state of these two phases. On the other hand, the differences between the GAP equation and EOS for gases are fundamental. Finally, we identify the key gaps in the experimental data that need to be filled in to proceed further with the liquid EOS.

We have developed a new all-optical method to directly measure the pressure-volume-temperature (PVT) equation of state (EOS) of fluids and transparent solids in the diamond anvil high pressure cell by measuring the volume of the sample chamber. Our method combines confocal microscopy and white light interference with a new analysis method, which exploits the mutual dependence of sample density and refractive index: Experimentally, the refractive index determines the measured sample chamber thickness (and therefore the measured sample volume/density), yet the sample density is by far the dominant factor in determining the variation in the refractive index with pressure. Our analysis method allows us to obtain a set of values for the density and refractive index, which are mutually consistent and agree with the experimental data within error. We have conducted proof-of-concept experiments on a variety of samples (H2O, CH4, C2H6, C3H8, KCl, and NaCl) at ambient temperature and at high temperatures up to just above 500 K. Our proof-of-concept data demonstrate that our method is able to reproduce known fluid and solid EOS within error. Furthermore, we demonstrate that our method allows us to directly and routinely measure the PVT EOS of simple fluids at GPa pressures up to, at least, 514 K (the highest temperature reached in our study). A reasonable estimation of the known sources of error in our volume determinations indicates that the error is currently +/- 2.7% at high temperature and that it is feasible to reduce it to ca. +/- 1% in future work.

The mathematical form of the magnetic field due to a current loop, and the fact that it is identical to the electric field due to an electric dipole in the far field, are fundamental to our understanding of electromagnetism. While undergraduate level electromagnetism textbooks usually derive the electric field from an electric dipole, few derive the magnetic field from a current loop. Most simply state it without proof, or perform the derivation for simpler cases such as the on-axis field. Those that perform the derivation use the magnetic vector potential, a relatively advanced concept that most undergraduate students would not encounter until their final year of study, if at all. Here, a simple derivation to obtain the magnetic field due to a current loop in the far-field approximation is presented. The derivation begins from the Biot–Savart law and does not require the vector potential. The problem is set up so that only a single integration is necessary (from angle α = 0 to α = 2π around the current loop), and the result is compared with that for the electric field surrounding an electric dipole.

A comparison is made between the three principal methods for the analysis of neutron and x-ray diffraction data from noble gas fluids by direct Fourier transform. All three methods (standard Fourier transform, Lorch modification, and Soper–Barney modification) are used to analyze four different sets of diffraction data from noble gas fluids. The results are compared to the findings of a full-scale real-space structure determination, namely, Empirical Potential Structure Refinement. Conclusions are drawn on the relative merits of the three Fourier transform methods, what information can be reliably obtained using each method, and which method is most suitable for the analysis of different kinds of diffraction data. The mathematical validity of the Lorch method is critically analyzed.

The high-pressure and high-temperature phase diagram of chromium has been investigated both experimentally (in situ), using a laser-heated diamond-anvil cell technique coupled with synchrotron powder X-ray diffraction, and theoretically, using ab initio density-functional theory simulations. In the pressure-temperature range covered experimentally (up to 90 GPa and 4500 K, respectively) only the solid body-centred-cubic and liquid phases of chromium have been observed. Experiments and computer calculations give melting curves in agreement with each other that can both be described by the Simon-Glatzel equation [Formula: see text]. In addition, a quasi-hydrostatic equation of state at ambient temperature has been experimentally characterized up to 131 GPa and compared with the present simulations. Both methods give very similar third-order Birch-Murnaghan equations of state with bulk moduli of 182-185 GPa and respective pressure derivatives of 4.74-5.15. According to the present calculations, the obtained melting curve and equation of state are valid up to at least 815 GPa, at which pressure the melting temperature is 9310 K. Finally, from the obtained results, it was possible to determine a thermal equation of state of chromium valid up to 65 GPa and 2100 K.

We report the effects of high pressure, up to 10.45 GPa, on the photoluminescence of Bi-doped yttria-alumina-silica glass under 532 nm excitation. We identify three emission bands attributed to Bi3+, Bi+ and a NIR emitting Bi centre, Bi-NIR. As the pressure is increased up to similar to 6 GPa, an irreversible discontinuity in the trend for emission band energies indicates that an irreversible structural modification occurs. This irreversible discontinuity results in the peak energy of emission bands attributed to Bi+ and Bi-NIR shifting from those typical of Bi-doped oxide glasses to those observed in Bi-doped gallium-lanthanum-sulfide glass. The Bi3+ emission band can be almost eliminated at similar to 6 GPa, but its intensity increases rapidly as the pressure is further increased. The ability we report here to irreversibly modify the emission of Bi-doped glass using pressure treatment adds an extra processing technique to researchers looking to optimize the emission from Bi-doped glasses.

We have measured the melting curve of ethane from 300 to 450 K. Our results indicate that to describe all melting curve data above the triple point, it is necessary to account for a kink in the melting curve where the phase III-phase IV solid-solid transition intersects the melting curve. In the solid state, we observe some evidence for a new phase existing dose to the melting curve above 300 K. We have observed that the decomposition of ethane at high pressure and temperature can be catalyzed by ruby or samarium-doped yttrium aluminum garnet at a surprisingly low temperature of 375 K.

We have performed a series of neutron scattering experiments on supercritical krypton. Our data and analysis allow us to characterize the Frenkel line crossover in this model monatomic fluid. The data from our measurements was analyzed using Empirical Potential Structure Refinement to determine the short- and medium-range structure of the fluids. We find evidence for several shells of neighbors which form approximately concentric rings of density about each atom. The ratio of second to first shell radius is significantly larger than in any crystal structure. Modeling krypton using a Lennard-Jones potential is shown to give significant errors, notably that the liquid is overstructured. The true potential appears to be longer ranged and with a softer core than the 6-12 powerlaws permit.

This paper reviews the mechanical properties of graphene with particular attention to what is established and what is still uncertain. It clarifies the thickness and the elastic constants, and by also considering also phonon frequencies, it argues that “best values” come from graphite, when available. Properties not available from graphite include bending stiffness; this can be determined from studies of carbon nanotubes as well as graphene. In many ways, nanotubes provide access to fundamental properties of graphene, not least because they are the only form of graphene that can be unsupported (unstrained) in vacuum. Environmental effects are considered, including both interactions with substrates and with other solid and liquid media, which may affect the geometrical parameters defining graphene and associated elastic constants. Major uncertainties persist whether slipping or sticking dominates experimental observation, both between graphene and solid media, and between the layers of bilayer and multilayer graphene. The paper concludes with a short discussion of continuum and atomistic models of graphene.

We have performed a neutron scattering experiment on supercritical fluid nitrogen at 160 K (1.27 TC) over a wide pressure range (7.8 MPa/0.260 g/mL-125 MPa/0.805 g/mL). This has enabled us to study the process by which nitrogen changes from a fluid that exhibits gaslike behavior to one that exhibits rigid liquidlike behavior at a temperature close to, but above, the critical temperature by crossing the Widom lines followed by the Frenkel line on pressure (density) increase. We find that the Frenkel line transition is indicated by a transition to a regime of rigid liquidlike behavior in which the coordination number remains constant within error, in agreement with our previous work at 300 K. The Frenkel line transition takes place at approximately the same density at 160 and 300 K. The data do not conclusively show an additional transition at the location of the known Widom lines. We find that behavior remains gaslike until the Frenkel line is crossed and our data support the hypothesis that Widom line transitions are density increase-driven.

A fluid equations of state (EOS) should work at supercritical temperature in the vicinity of the critical point (where the compressibility is almost infinite as the Widom line is crossed) and in the vicinity of the melting curve (where the compressibility is virtually zero). Similar statements can be made for most other fluid properties and/or their derivatives. In addition, whilst liquids are not always miscible, liquid solutions form far more frequently than solid solutions. The Helmholtz function is not experimentally measurable, but expressions to predict many parameters that are measurable can be derived from the fundamental equation using standard thermodynamic relations. Invertibility is the ability to rearrange the equation into an analytical expression for any desired variable. The fundamental EOS is often referred to in the literature instead as the "reference EOS".

This chapter concentrates on the slightly narrower topic of established and emerging industrial applications of supercritical fluids, covering applications of liquids only where they overlap with those of supercritical fluids. It covers the applications in power generation, followed by food processing and then other applications. In recent years a massive research and development effort has been initiated into the use of CO2 as the working fluid in thermodynamic cycles as an alternative to H2O. The principal application is power generation, but there are also potential applications in the marine sector. The advantages of CO2 over H2O are twofold. Two-dimensional materials have been one of the hottest topics in physical sciences research over the past two decades.

This chapter discusses certain properties of ideal and real gases. In an ideal gas it is assumed that there are no forces exerted between the gas particles. In a real gas, however, the particles constantly exert attractive van der Waals forces on each other. This has the effect of decreasing the pressure exerted by the gas on its container, compared to the case of an ideal gas; i.e. kind of "pressure defect." An alternative to the van der Waals equation is to implement additional terms to correct the ideal gas Equation of State in the form of a power series. Molecular liquids (especially those such as water with shapes and intermolecular potentials that are a long way from being spherically symmetric) often exhibit short-range order in the orientation of molecules relative to those nearby. The heat capacity at constant volume of gases can be understood in terms of the degrees of translational, vibrational, and rotational freedom of the individual particles.

In this chapter, the authors outline the theoretical aspects of these transitions and review the experimental evidence for their existence. They begin with the Widom lines, then study the Fisher-Widom line, the various inversion curves, and finally the Frenkel line. The vapour pressure curve, marking the first order phase transition between the liquid and gas states, consists of discontinuous changes in various thermodynamic functions. It ends at the critical point. However, all functions that undergo the first order phase transition when the vapour pressure curve is crossed also undergo a narrow transition when a P,T path is followed just beyond the critical point, roughly following an extrapolation of the vapour pressure curve along an isochore. For viscosity and thermal conductivity the authors use the results of our fitting procedure to determine the conditions under which they consider the Widom line to exist.

The reductions in entropy and volume that always take place upon condensation from the gas state into the liquid state allow conclusions to be drawn about the path of the vapour pressure curve on the P,T phase diagram from first principles, in the form of the Clausius-Clapeyron equation. As a consequence of the trends as the critical point is approached, the surface energy of a liquid droplet in equilibrium with the vapour in the section of the V,P phase diagram in which the liquid and vapour phases co-exist also tends to zero. A stronger analogy to the vapour pressure curve critical point may be obtained from the case of interacting magnets. The changes in various properties that take place in the vicinity of the vapour pressure curve at significantly subcritical temperature become fundamentally different when the critical temperature is approached, for instance, the trends in the measured heat capacity for a model spherically symmetric fluid composed of neutral particles.

In this chapter, the authors study the dynamic properties of the liquid state close to the melting curve. They focus on the Maxwell and Frenkel models permitting the liquid to exhibit viscous and elastic responses to stress on different timescales and advancements in the past decade in their ability to predict-arguably-the most important fluid dynamic property using the Frenkel model. The authors describe the experimental methods utilized to study the variety of excited states existing in liquids: Raman and Brillouin spectroscopy and inelastic neutron and X-ray scattering. From use of these methods they have, in some cases, quite detailed knowledge of the sound wave propagation in liquids, vibrational and rotational excited states in molecular liquids, etc. The phonon theory of liquids is the approach to modelling the dynamic properties of liquids that begins from the assumption that their behaviour is solid-like and incorporates their ability to flow as a correction to this; building on the ideas of Maxwell and Frenkel.

We have conducted a Raman study of solid ethane (C2H6) at pressures up to 120 GPa at 300 K. We observe changes within the ν3 and ν11 Raman‐active vibrational modes providing evidence for several previously unobserved phase transitions at room temperature. These are located from 16 to 20 GPa, ~35 GPa, and ~60 GPa. We also could no longer measure the ν3 and ν11 modes from 75 GPa onward. We did not, however, observe any signs of the ethane molecule undergoing decomposition, up to the highest pressures measured. We also recorded spectra of the (2ν8, 2ν11), ν1, and ν10 modes but observed more limited changes in the behaviour of these modes.