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After a brief introduction with a discussion of the concept of the correlation function needed for the formulation of the problem, a progress report is given concerning our knowledge of microdynamic processes in liquids covering the fields of neutron scattering, IR-absorption and lightscattering, dielectric and nuclear magnetic relaxation as well as studies of diffusion and electric conductance. 1. Einleitung wie Ladung, magnetisches Moment etc. zugeordnet sind: die Massenpunkte mIissen ,,beziffert" sein gemiin ihrer Position im Periodensystem (siehe Abb. 1). Jeder Massenpunkt hat einen Ort ri und eine Geschwindigkeit ui = dri/dt als Augenblickswert. Das mikroskopische Augenblicksbild einer FWsigkeit k8nnen wir als ein System vieler Massenpunkte im Raum ansehen, denen je nach Umstiinden weitere physikalische Eigenschaften Ber. Bunsenges. Phys. Chem. 85, 992-lo05 (1981) -0 Verlag Chemie GmbH, D-6940 Weinheim, 1981.

The computer simulation and inelastic neutron scattering studies of simple, monatomic liquids are reviewed, together with the theory appropriate for their interpretation. All three aspects of the dynamic properties of monatomic liquids began a new era of development around 1965, and the review is concerned chiefly with the experiments and theory reported in the intervening decade.Computer simulation (molecular dynamics) studies have featured strongly in the development of non-equilibrium statistical mechanics techniques for the calculation of correlation functions that enter the various measured susceptibilities. In fact molecular dynamics studies of systems interacting via continuous potentials have developed to the level of sophistication where they set a standard for neutron scattering experiments. The neutron experiments are difficult to perform with high accuracy, yet they offer a unique means of investigating the dynamics of simple liquids in the domain of wavevectors and frequencies larger than about 0.05 A-1 and 5 x 1011 s-1 respectively. The problems faced in analysing neutron data to the level required today are reviewed in detail, with particular attention to multiple scattering corrections. A review of neutron experiments on monatomic liquids shows that surprisingly few bear close scrutiny today. Theory has developed through the rigorous generalization of the Markovian theory of fluctuations set out by Landau and Lifshitz (1959). The development has taken the form of a generalized Langevin equation, proposed by Zwanzig (1961) and Mori (1965a), which provides a framework within which neutron and molecular dynamics data may be interpreted. The technique is reviewed, related to linear response theory, and used to summarize the recent theories.

Orbital currents are proposed to be the order parameter of the pseudo-gap phase of cuprate high-temperature superconductors. We used resonant x-ray diffraction to observe orbital currents in a copper-oxygen plaquette, the basic building block of cuprate superconductors. The confirmation of the existence of orbital currents is an important step toward the understanding of the cuprates as well as materials lacking inversion symmetry, such as magnetically induced multiferroics. Although observed in the antiferromagnetic state of cupric oxide, we show that orbital currents can occur even in the absence of long-range magnetic moment ordering.

A brief survey of theory designed to interpret observations on magnetic materials using neutron and X‐ray scattering techniques is presented. Emphasis is placed on X‐ray scattering because it remains an emerging subject and neutron scattering is thoroughly reviewed in several places.

A theory of neutron scattering by magnetic materials is reviewed with emphasis on the use of electronic multipoles that have universal appeal, because they are amenable to calculation and appear in theories of many other experimental techniques. The conventional theory of magnetic neutron scattering, which dates back to Schwinger (1937) and Trammell (1953), yields an approximation for the scattering amplitude in terms of magnetic dipoles formed with the spin (S) and orbital angular momentum (L) of valence electrons. The so-called dipole-approximation has been widely adopted by researchers during the past few decades that has seen neutron scattering develop to its present status as the method of choice for investigations of magnetic structure and excitations. Looking beyond the dipole-approximation, however, reveals a wealth of additional information about electronic degrees of freedom conveniently encapsulated in magnetic multipoles. In this language, the dipole-approximation retains electronic axial dipoles, S and L. At the same level of approximation are polar dipoles -called anapoles or toroidal dipoles -allowed in the absence of a centre of inversion symmetry. Anapoles are examples of magneto-electric multipoles, time-odd and parity-odd irreducible tensors, that have come to the fore as signatures of electronic complexity in materials. PrologueThe interaction between neutrons and electrons was explored in the 1930s, followed by a definitive study in 1953 by George Trammell that later was reformulated in a more compact format. Why re-visit these calculations after six decades during which time magnetic neutron scattering has become the method of choice for determining motifs of magnetic dipoles and their excitations, e.g., spin-waves? The principal motivation is to complete available calculations by including electronic operators that change sign when space coordinates are inverted, whereas the magnetic dipole is unchanged by inversion.By way of orientation, consider the amplitude for the magnetic scattering of neutrons by electrons in the limit of small scattering angles, i.e., a small scattering wavevector k. The scattering amplitude is Q ⊥ = [k x (Q x k)]/k 2 , in which an intermediate operator,can be justified for small k. The first contribution to Q is the magnetic dipole moment; the magnetic axial vector Q ≈ (1/2){2S + L} is likely to be a familiar approximation to all who use the neutron scattering technique to study magnetic materials (an atomic form factor in Q is approximated by unity for the moment). By contrast, the second contribution {i k x D} with D both magnetic (time-odd) and polar (parity-odd) is most likely not expected. The dipole D has no matrix elements different from zero if magnetic ions that contribute to it occupy sites that are centres of inversion symmetry. In the absence of a centre of symmetry a matrix element such as 〈3dD4p〉 for a 3d-transition ion can be different from zero. Many magnetic materials use sites that are deprived of a centre of inversion

We present a theoretical analysis of resonant x-ray Bragg diffraction data from multiferroic TbMnO 3 presented by Mannix et al. ͓Phys. Rev. B 76, 184420 ͑2007͔͒ and Voigt et al. ͓Phys. Rev. B 76, 104431 ͑2007͔͒. We have chosen an approach that does not rely on knowledge of the low-temperature phase space group of the sample, which is not precisely known. Results show that the low-temperature satellite reflections originate from dipole-dipole ͑E1-E1͒ and dipole-quadrupole ͑E1-E2͒ events. Presence on quadrupole-quadrupole ͑E2-E2͒ events can be excluded. The physical origin of the data is discussed in terms of atomic multipoles ͑expectation value of an operator equivalent͒ that represent magnetization, lattice distortions, and magnetoelectric properties of the Tb and Mn ions. A handed ͑chiral͒ cycloid of atomic multipoles, traced out in the b-c plane, is shown to be a plausible model of the Tb electron structure within a multiferroic modification that exists in the temperature interval 7 K Ͻ T Ͻ 28 K. Appendixes A-F record universal expressions for unit-cell structure factors appropriate to all four polarization channels ͑Ј-, Ј-, Ј-, Ј-͒ of E1-E1, E1-E2, and E2-E2 resonance events.