Volume 48, Issue 1-2

There appears to be conflicting experimental evidence on the redistribution of the electron density in the lone-pair and other regions of a molecule due to the interaction with its nearest neighbours. In some experimental as well as theoretical deformation density maps a decrease in the lone-pair density has been reported, whereas in other cases an increase has been found. It appears that two major, counteracting factors are responsible for these differences (apart from experimental errors in the diffraction studies and limited accuracy in the theoretical calculations): an increase in the lone-pair density is expected due to the polarizing influence of the neighbours, whereas simple superposition of the isolated monomer deformation densities will lead to an apparent decrease due to the overlap with the negative contours of the neighbouring atom. Depending on which of these factors is the dominant one, an increase or decrease may thus be observed. These points are illustrated by recent results on nickel sulfate hexahydrate and some other hydrogen-bonded compounds. The electron density based on the fitted deformation functions of all atoms in the structure is compared with the individual densities calculated from deformation functions of the separate monomers. In this way the effects of simple superposition of the individual densities have been studied, and a partitioning of the electrostatic and polarization contributions to the hydrogen bonds and other relatively weak bonds to the oxygen lone-pairs attempted.

X-ray diffraction data in heavy-atom compounds may be sensitive to anharmonic atomic displacements, since the large core electron densities result in appreciable scattering amplitudes at large reciprocal distances. Since bonding electron densities may also exhibit sharp features affecting high-order reflections, they may be difficult to distinguish from anharmonic effects. We have analyzed an accurate room-temperature single-crystal X-ray data set of K 2 PtCl 6 using least-squares anharmonic displacement and charge density formalisms. The Hirshfeld charge density formalism, which has successfully been applied to many light-atom structures, fails to parametrize satisfactorily the data, whereas the electron densities at K and CI are easily accounted for by an anharmonic Gram-Charlier expansion to 4th order. Densities around Pt are parametrized only by a combination of anharmonicity and charge density formalisms. If economical parametrizations of the experimental data are preferred to more complicated ones, anharmonicity may be conjectured to play an important rôle while the main bonding feature consists of a preferential occupation of the 5d-orbitals of Pt with t 2g symmetry.

The effect of anharmonic atomic vibrations on the elastic scattering of X-rays by a crystal is treated by time-independent perturbation theory. The complete expression for the anharmonic temperature factor (TF) in the high-temperature limit is presented for atoms in general positions. The phonon-phonon interactions in an anharmonic crystal may be identified with X-ray diffraction data by using various temperature and scattering vector q dependencies of different anharmonic contri-butions to the TF. The expression for the intensity of elastic scattering by an anharmonic crystal contains an additional factor -the anharmonic cross-term. This term has to be taken into account for obtaining the unbiased thermal parameters of fourth order in | q | from X-ray diffraction data.

The results of accurate X-ray diffraction studies of KTaO 3 , K 0,95 Li 0,05 TaO 3 and K 0,85 Li 0,15 TaO 3 crystals are presented. Anharmonicity of the thermal motion of the oxygen atoms and deformation electron density maps are discussed.

Owing to the close neighbourhood of Ga and As in Mendeleev's table, GaAs shows two fundamental classes of X-ray structure amplitudes distinguished by their extremely different scattering power. They are differently sensitive to the valence electron density (VED) redistribution caused by the chemical bond and must be measured by different experimental methods. Using such data, both the VED and the difference electron densities (DED) are calculated here. Comparison with theoret-ical densities shows that the VED is characterized by covalent, ionic and metallic contributions. The DED constructed from GaAs and Ge data demonstrates the electronic response caused by a "protonic" charge transfer between both f.c.c. sublattices as well as the transition from a purely covalent to a mixed covalent-ionic bond. Especially the charge-density accumulation between nearest neighbours (bond charge (BC)) depends on the distance between the bonding atoms and changes under the influence of any lattice deformation. This phenomenon is described by a BC-transfer model. Its direct experimental proof is given by measuring the variation of the scattering power of weak reflections under the influence of an external electric field. This experiment demonstrates that the ionicity of the bond changes in addition to the BC variation.

The elctron density distribution in crystalline ZnSe and ZnTe has been measured by X-ray diffraction on powder samples and is compared with the results for other zincblende-type crystals. It is shown that the data for these two compounds corroborate the linear dependence on the bond charge that has been found earlier for the effective atomic charge from infrared reflection measurements, the optical dielectric constant and the ionicity estimated from pseudopotential band theory.

The electron density and bonding in the silicates beryl, cordierite and dioptase has been investigated. The replacement of Be, Si and Al atoms in the identical structural fragments leads to a (δr-redistribution and changes the σ-bond character from slightly polar to strongly polar. The condensation of [SiO 4 ]-tetrahedra in the rings leads to the accumulation of electron density near bridge oxygens. This phenomenon is expressed especially in dioptase (pure ring silicate). Therefore, the electron density picture may be considered as a supplementary independent criterion for classi-fication of crystal structures. The experimental (δr-maps for H 2 0 molecules in dioptase are in good agreement with theoretical ones. The δr distribution of 3 d-electrons and bonding in the Jahn-Teller distorted Cu-octahedron can be interpreted from crystal-field-theory point of view.

X-ray diffraction intensities from a single crystal of the Laves phase FeBe 2 were collected at room temperature. Atomic and thermal parameters as well as a scale factor were determined by least-squares refinements of high-angle data [(sin θ)/λ ≥ 0.6Â -1 ]. The results of Fourier inversion of the data are presented in terms of maps of the total density and followed by a discussion of the "effects" of two criteria for labelling reflection intensities as "unobserved", based on their statistical accuracy.

The low-temperature molecular structure and electron density (ED) distribution of dioxane and trans-2,5-dichloro-l,4-dioxane were determined near 100 K from X-ray data. Deformation electron density maps were calculated by Fourier and multipole expansion techniques. The multipole model charge distribution of dioxane determined from experimental data was taken as a reference state, relative to which the ED of the substituted ring was analyzed. These fragment deformation density maps show quadrupolar deformations on atoms O, CI, and anomeric C only. The highly correlated orientation of the lobes at these atomic sites seems to indicate that the observed charge rearrange-ment is caused by three-center interactions.

A 20 K difTractometer for X-ray measurement is described. Electron density experiments at 15 K for oxalic acid and at 23 K for acetamide were carried through, leading to high-resolution experi-mental and multipole static density maps.

Low-temperature single-crystal neutron and X-ray diffraction data were collected for bullvalene (Figure 1). X -N Fourier and multipole static deformation electron density maps were calculated and compared. Atomic mean-square-displacement parameters were analysed in the framework of the rigid-body motion model and the residual amplitudes (U OBS - U TLS ) were graphically interpreted. The preliminary results presented here reveal either a bias in the X-ray anisotropic displacement parameters (ADPs) due to improper modelling of the electron density or unresolved errors in the neutron data.

One-bond, 13 C- 1 H coupling constants, J 1 (C-H), 'in amines, ammonium ions, and carboxylic amides correlate with structure and support the concept that the value of J 1 (C-H) is related to the charge density on the nitrogen atom; for example, amine oxides have nearly the same charge density at nitrogen as does the tetramethylammonium ion. The J 1 (C-H) values for methyls bonded to nitrogen in various amides then give an experimental estimate of the charge density at the nitrogen atom that enables an estimate of the bond order in the C-N amide-bond; the data suggest that carboxylic amides have a C-N bond order of about 1.35, that sulfonamides have an S-N bond order of about 1.45, and that phosphinamides, R 2 P(O)N(CH 3 ) 2 , have a P-N bond order of about 1.3. In contrast, aminephosphines have a P-N single bond. The value for carboxylic amides is in reasonable agreement with bond distances in amides.

The requirements for charge density studies of more complex molecules have been realized by use of the FAST area-detector diffractometer in conjunction with a rotating-anode generator, which allows to collect large quantities of diffraction data within less than two days, independent of unit cell size. Oxalic acid dihydrate has been chosen for test experiments. Data collection strategies are described and preliminary results are presented. These show that data of sufficiently high quality for charge density studies can be collected.

Incomplete and imperfect data characterize the problem of constructing electron density representations from experimental information. One fundamental concern is identification of the proper protocol for including new information at any stage of a density reconstruction. An axiomatic approach developed in other fields specifies entropy maximization as the desired protocol. In particular, if new data are used to modify a prior charge density distribution without adding extraneous prejudice, the new distribution must both agree with all the data, new and old, and be a function of maximum relative entropy. The functional form of relative entropy is s = - r In (r/t), where r and t respectively refer to new and prior distributions normalized to a common scale. Entropy maximization has been used to deal with certain aspects of the phase problem of X-ray diffraction. Varying degrees of success have marked the work which may be roughly assigned to categories as direct methods, data reduction and analysis, and image enhancement. Much of the work has been expressed in probabilistic language, although image enhancement has been somewhat more physical or geometric in description. Whatever the language, entropy maximization is a specific and deterministic functional manipulation. A recent advance has been the description of an algorithm which, quite deterministically, adjusts a prior positive charge density distribution to agree exactly with a specified subset of structure-factor moduli by a constrained entropy maximization. Entropy on an N-representable one-particle density matrix is well defined. The entropy is the expected form, and it is a simple function of the one-matrix eigenvalues which all must be non-negative. Relationships between the entropy functional and certain properties of a one-matrix are discussed, as well as a conjecture concerning the physical interpretation of entropy. Throughout this work reference is made to informational entropy, not the entropy of thermodynamics.

The nature of the bonding in Be metal was studied by investigating the MEM map, which is the electron density distribution obtained by the Maximum-Entropy Method. In order to avoid extinc-tion effects, 19 Bragg reflections were measured by a new powder-diffraction experiment that utilizes Synchrotron Radiation as an incident X-ray and an Imaging Plate as detector. The experiment was carried out at the Photon Factory BL6A2. In spite of the limited number of reflections used in the MEM analysis, the electron density distribution of Be was obtained accurately and reliably. The structure factors for unmeasured reflections were calculated and compared with the values observed by Larsen and Hansen [Acta Cryst. B40 , 169 (1984)]. The agreement is very good. Furthermore, the MEM map revealed that Be metal forms an electronic layer in the shape of a honeycomb that is parallel to the basal plane.

A procedure for calculating the electrostatic potential and the electrostatic energy of an ion in a crystal is presented. It is based on a mixed direct and reciprocal space approach, and it takes into account the detailed charge density distribution in the crystal which can be obtained from accurate X-ray diffraction measurements.

It has been shown (Z. Su and P. Coppens, Acta Cryst. A 48, 188 (1992)) that the electrostatic potential, the electric field, and the electric field gradient (EFG) can be expressed in closed forms in terms of the positions and the charge-density parameters of individual atoms, whose aspherical density is described by a pseudoatom model (e.g., N. Hansen and P. Coppens, Acta Cryst. A 34, 909 (1978)). A F o r t r a n program Molprop91 based on this method has been written (Z. Su, State Univ. of New York at Buffalo 1991). The method has been applied to the title compounds. Low-temperature X-ray diffraction data of fully deuterated benzene (G. A. Jeffrey, J. R. Ruble, Y. Yeon, and C. Lemann, private communication, 1991), /-alanine (R. Destro, R. E. Marsh, and R. Bianchi, J. Phys. Chem. 92, 966 (1988) and d,/-histidine (N. Li, Ph.D. thesis, State University of New York at Buffalo 1989) were analyzed using the least-squares deformation density refinement program Lsmol90 (a modified version of M o l l y ) . Molprop91 was subsequently used to calculate the electrostatic-potential maps in selected sections, and at the nuclear positions. For the latter, the EFGs were also evaluated. The electrostatic potentials were used to fit net atomic charges and estimate the molecular energies. Errors in the derived quantities are given.

The calculation of the electrostatic potential of a molecule removed from the crystal lattice is derived from the parameters obtained by a kappa refinement and by a Hansen-Coppens electron density model. These calculations in direct space are applied to N-acetyl-α,β-dehydrophenylalanine; deformation potentials calculated by Fourier transformation are compared to those obtained in direct space.

A total of about 37 000 diffracted intensities has been measured at 20 K for a spherical single crystal of citrinin. Using a multipole formalism to interpret the X-ray data, maps of the charge density and of its Laplacian, as well as for the electrostatic potential have been derived. A value of 7(2) D has been obtained for the magnitude of the molecular dipole moment. A study of the electric field gradient (EFG) at the nuclei has yielded the atomic quadrupole coupling constants (QCC) and asymmetry parameters (η). A topological analysis of the charge density has been performed to characterize the intramolecular covalent and hydrogen bonds.

The bonding type and characteristic chemical reactions of the sydnone ring have been rationalized by examination of the deformation electron density distribution and electrostatic potential from both experiment and theory. The net atomic charges, bond lengths and bond orders of the title compound confirm the semiaromatic bonding type for the sydnone ring. The highly negative net atomic charge of C4 suggests its susceptibility to electrophilic substitution at this position. The well known 1,3-dipolar cycloaddition reaction of the sydnone ring can be understood by the analysis of the 7 t -o r b i t a l wavefunctions, which give rise to similar deformation densities as the experimental ones. The good agreement between the experimental and theoretical (ab initio calculation using 3-21G basis sets) deformation density distribution of the title compound justifies the use of the same basis set to derive the electrostatic potential. The electrostatic potential map reveals all the possible protonation sites of the sydnone ring at O1, N2 and O6 with the deepest hole at 06 (84.6 kcal/mole).

The spin densities of the nitroxides H 2 NO, (CH 3 )HNO and (CH 3 ) 2 NO have been studied by iterative CI methods. Calculations at different geometries with various basis sets were performed. It is found that the spin distribution is delocalized within the N-O group, and substitution of the hydrogen atoms on the nitroxyl group by methyl groups changes the spin distribution significantly. Electron correlation, as well as the basis-set quality, plays an important role for the fluctuation of spin populations in the nitroxide radicals. It has been found that the spin density map can be predicted fairly well when correlation was included at certain levels and reasonable quality basis sets were employed. The spin polarization mechanism has been studied through a core excitation method. The superiority of local-spin-density (LSD) theory over the unrestricted Hartree-Fock (UHF) method in spin property studies is discussed.

The spin density in the two alkyl nitroxide compounds 4-oxo-and 4-hydroxy-2,2,6,6-tetramethyl-1-piperidinyloxy (tempone and tempol) has been determined by polarised-neutron diffraction and compared to theoretical predictions obtained in both the Unrestricted Hartree-Fock and the Local Spin Density approaches.

The spin density in two copper-nitronyl nitroxide complexes, viz. bis(hexafluoroacetylacetonato)-(2,4,4,5,5-pentamethyl-l-oxy-imidazoline 3-oxide)-copper(II) and dichloro-bis(2-phenyl-4,4,5,5-tetra-methyl-l-oxy-imidazoline 3-oxide)-copper(II), has been determined by polarised-neutron diffrac-tion. Spin populations and unpaired electron wave functions have been refined.

A polarised-neutron diffraction experiment on [Co(NH 3 ) 5 (H 2 O)][Cr(CN) 6 ] shows the unexpected features of spin populations on the cobalt ((0.5(2) e) and the "ammine" hydrogen (0.8(1) e in total) atoms. A local density approximation DV-Xα calculation on a unit that models the molecular geometry in the crystal returns features that are broadly in agreement with the experiment. The calculation also reproduces the major features of a charge density experiment on the complex, including the low (±1.6 e) charges on the ions, in place of the formal ±3 e.

Basis-set and electron-correlation effects on the appearance and disappearance of nonnuclear maxima in the electron density are examined in Li 2 , Na 2 , Na 4 and Na 5 . It is shown that nonnuclear attractors can be removed in all cases except Li 2 . The appearance of a pseudoatom in a lithium molecule correlates remarkably well with the size of the region, in an atomic calculation, of V 2 r(r) for the valence shell of the atom. This and the fact that the pseudoatom is also present in the promolecule indicate that the pseudoatoms are remnants of, or in fact are portions of, atoms that are not perturbed enough in the molecule to remove an essentially atomic characteristic.

Some consequences of the quantum fluid dynamics formulation are discussed for excited states of atoms and molecules and for time-dependent processes. It is shown that the conservation of electronic current density j(r) allows us to manufacture a gauge potential for each excited state of an atom, molecule or atom in a molecule. This potential gives rise to a tube of magnetic flux carried around by the many-electron system. In time-dependent situations, the evolution of the electronic density distribution can be followed with simple, site-dependent cellular automaton (CA) rules. The CA consists of a lattice of sites, each with a finite set of possible values, here representing finite localized elements of electronic charge and current density (since the charge density rno longer suffices to fully characterize a time-dependent system, it needs to be supplemented with information about the current density j).Our numerical results are presented elsewhere and further development is in progress.

A recently formulated strict definition of the boundary surface of molecular anions has been employed for the visualization of some simple systems. The examples included are CN - , N - 3 , NO - 3 , ClO - 4 , ClO - 3 , and BrO - 3 ions. A discussion on the anisotropy of electrostatic interaction of these anions is also presented.

A new computational scheme for electric dipole hyperpolarizabilities has been devised within the coupled Hartree-Fock method. Only the projection of second-order perturbed orbitals onto the subspace spanned by unperturbed virtual orbitals is computed. The entire molecular symmetry is exploited to reduce computational effort: a reduced two-electron integral file containing only symmetry-distinct matrix elements over the atomic basis functions is processed at each iteration. In addition, only symmetry-independent first-and second-order perturbed density matrices need to be calculated. An efficient computer program implementing the present approach has been developed.

A topographical study of the electron momentum density (EMD) of some linear molecules is presented with special emphasis on the bond-directionality principle. A new approach to the bond-directionality (BD) principle has been proposed. This is based on the analysis of curvatures of critical points at p = 0 and elsewhere on the bonding axis. The linear molecules are classified into three broad categories: those which fully obey or disobey the BD principle as well as those which satisfy it only partly. The Laplacian of the EMD at p = 0 has been associated with the quality of the wavefunction via the electron density in the tail region. Also, the similarity in the critical structure of both the EMD and the spherically averaged EMD at the origin is brought out.

We present a discussion of a number of conceptual and methodological aspects associated with the theoretical characterization and computation of charge densities and momentum distributions in solids. The main ambition has been to stress properties that both exact and approximate quantities must possess. We have also attempted to point out conceptual and computational trends which would seem to be of importance for the future of the subject.

Results of a theoretical, comparative study of the electronic properties of trans-polyacetylene and polycarbonitrile are reported. Polyacetylene consists of zigzag chains of CH units, whereas polycarbonitrile has every second CH unit replaced by an N atom. Ground-state properties (structure, electronic bonds and bands, densities of states, momentum distributions, and reciprocal form factors) of the periodic, infinite, isolated chains are studied by means of first-principles, density-functional calculations. It is demonstrated how the presence of the (nitrogen) heteroatoms in the backbone of polycarbonitrile leads to a partial localization of the electrons. In order to investigate charged chains, model calculations are subsequently performed. These indicate solitons but not polarons to be stable. In total, the analysis demonstrates how the combination of information that can be obtained from various experiments provides a detailed description of the compounds.

An overview of some of the basic concepts in the modern first-principles band theory of disordered alloys is given. The question of how the notion of Bloch energy bands and Fermi surfaces generalizes to the case of the disordered system, insofar as the average electronic spectrum is concerned, is stressed. The theory is illustrated with examples chosen from the work on binary alloys; a few examples of the very recent studies of disordered phases of the high-T c superconductors are also included. The application of the alloy theory to obtain electron and electron-positron momentum densities involved in the analysis of Compton scattering and positron annihilation (angular correlation) experiments in alloys is discussed, with a selection of relevant theoretical and experimental studies.

Accurate electron densities and X-ray form factors of Li, Be, F and their ions have been calculated. Electron correlation, crystal fields and ionic charge transfer change the form factors by up to a few percent, mainly in the range of sin θ/ λ < 1/3 Â -1 . Although electron correlation and crystal fields are small perturbations, their effects on the density and form factor are not additive. Densities or form factors of atomic and ionic systems are very similar; [Li 0 F 0 ] and [Li + F - ] procrystals differ by an effective charge transfer of not more than 0.4 e. Charge transfer and charge overlap in crystals cannot be distinguished uniquely. When the experimental data on Li 2 BeF 4 (approximately reproduced by 3/4 atomic plus 1/4 ionic procrystal) are interpreted from the atomic partial charges are as low as 0.1 e (Li + 0.1 2 Be + 0.2 F - 0.1 4 ); when interpreted from the ionic viewpoint,the charges are much higher, namely 0.7 e. Intermediate viewpoints are also possible.

A pseudopotential calculation has been performed for the charge density distribution in Si and Ge, with the core-orthogonalization terms fully included. The state of a positron subject to the Coulomb field of the resultant electron distribution in Si is also calculated. A possible improvement of the calculated distributions of electrons and positrons in momentum space is discussed.

A practical method for the calculation of Compton profiles for cubic systems with O and O h symmetry is presented that is based on a multipole expansion of the electron momentum density (EMD) in terms of cubic harmonics. The central quantities, the expansion coefficients, are deter-mined by a Gaussian-type integration (special directions) over the angular coordinates. From these coefficients the coefficients of an analogous expansion of the Compton profile can be directly calculated, establishing a transparent relationship between the electron momentum density and the Compton profile. This direct relationship offers the possibility of tracing back Compton-profile anisotropics to EMD anisotropics more easily, as demonstrated for MgO and FeAl.

Relationships among electron coordinate-space and momentum densities and the one-electron charge density matrix or Wigner function are examined. A knowledge of either or both densities places constraints on possible density matrices. Questions are approached in the context of a finite-basis-set model problem in which density matrices are elements in a Euclidean vector space of Hermitian operators or matrices, and densities are elements of other vector spaces. The maps (called "collapse") of the operator space to the density spaces define a decomposition of the operator space into orthogonal subspaces. The component of a density matrix in a given subspace is determined by one density, both densities, or neither. Linear dependencies among products of basis functions play a fundamental role. Algorithms are discussed for finding the subspaces and constructing an orthonormal set of functions spanning the same space as a linearly dependent set. Examples are presented and additional investigations suggested.

A modification of a recently developed method for the least-squares reconstruction of a one-particle reduced density matrix from experimentally accessible expectation values is applied to the test systems of atomic beryllium and neon. The improvement of the resulting matrices through inclusion of electron correlation is demonstrated. Their quality is judged by comparison of the moments of the position and momentum densities and of the spherically averaged density matrix in a suitable representation.

A recently developed method for the least-squares reconstruction of one-particle reduced density matrices from one-particle expectation values has been applied to isotropic Compton profiles of neon from the literature. The resulting densities in momentum and position space are compared with the ones obtained from ab-initio calculations.

The experimental knowledge of the dielectric response of electrons within a solid is up to now mainly gained by inelastic scattering experiments, using a plane wave as initial state of the probe particle. These experiments allow the measurement of the dynamical structure factor, related to diagonal elements of the inverse dielectric matrix ε -1 . We present a new type of scattering experiment that uses as initial state of the probe a standing wave field, thus allowing the experimental determination of non-diagonal elements of ε -1 . Comparison of the experimental results with a plasmon-band model leads to direct experimental evidence for plasmon-banding in Si.

The doubly differential cross-section for weak inelastic scattering of waves or particles by manybody systems is derived in Born approximation and expressed in terms of the dynamic structure factor according to van Hove. The application of this very general scheme to scattering of neutrons, x-rays and high-energy electrons is discussed briefly. The dynamic structure factor, which is the space and time Fourier transform of the density-density correlation function, is a property of the many-body system independent of the external probe and carries information on the excitation spectrum of the system. The relation of the electronic structure factor to the density-density response function defined in linear-response theory is shown using the fluctuation-dissipation theorem. This is important for calculations, since the response function can be calculated approximately from the independent-particle response function in self-consistent field approximations, such as the random-phase approximation or the local-density approximation of the density functional theory. Since the density-density response function also determines the dielectric function, the dynamic structure can be expressed by the dielectric function.

We report a calculation of the generalised oscillator-strength density (GOSD) of the helium atom for low momentum transfer. The calculation was performed by extraction of the searched entity from the resolvent of the complex-rotated Hamiltonian. That Hamiltonian was found using a standard ab-initio program without any change in the program code. The shape of the Bethe surface is reproduced (including resonances and the absorption edge). However, with increasing momentum transfer the magnitude of the calculated values for the Bethe ridge becomes too small. This is clearly demonstrated by an evaluation of the Bethe sum as a function of the momentum transfer. A comparison with recent additional theoretical data shows that the deviations are due to the employed basis set.

The electron momentum distribution of a given target can be extracted, under impulsive conditions, from the simple ionisation processes relevant to Compton collisions. This document classifies and investigates the physical mechanisms of double ionisation, in connection with the target electronic structure.

The double ionization of helium by fast electrons is studied by using a correlated double continuum wave function describing the two ejected electrons.

A series expansion applied to the Born propagator present in the differential cross-section allows one to obtain successive corrections to the impulse approximation. The first and second corrections explain the essential features of the Compton defects. Results for the second corrective term are presented for 1 s, 2 s, 2 p x,y , and 2 p z hydrogenic states. The decrease or increase of the scattered intensity near the maximum of the Compton profile is shown to be strongly related to geometric properties of the concerned orbital

Theoretical work on the cross-section for Compton scattering by magnetic materials is surveyed. Exact results for scattering by a free polarized electron are contrasted with corresponding results obtained perturbatively for a model of bound electrons with a finite width to the momentum distribution.

A review of achievements in the field of Compton spectrometry in the years that elapsed from the last Sagamore meeting (1988) is presented. Some physical problems that have either appeared as new or became clarified are described. Special emphasis is put to the results obtained by means of magnetic Compton scattering.

We report coincidence measurements of inelastically scattered 148 keV photons and the recoil electrons generated within thin Al and Cu foils. We have analyzed both the Doppler broadening of the scattered photon intensity and the angular correlation of the recoil electron. This corresponds to different scans through the three-dimensional electron momentum density of the target. Estimates of the triple-differential cross-section for Compton scattering are given. The experiment was performed with synchrotron radiation from a bending magnet of the DORIS storage ring at DESY, Hamburg.

Performance, resolution and spectral throughput both of the sagittally focusing monochromator and of the alternately installed two analysing systems, a Cauchois-type cylindrically bent crystal analyser for Compton-profile measurements, and a spherically bent crystal analyser for S(q, ω)-measurements, of the new Compton spectrometer at the HARWI (harter Röntgen Wiggler) of the DORIS storage ring are presented together with two typical applications.

The dynamic structure factor S(q, ω) of electrons in lithium-intercalated graphite (LiC 6 ) and potassium-intercalated graphite (KC 8 ) for momentum transfer q || c-axis has been measured by inelastic synchrotron X-ray scattering. The dielectric function ε(q, ω) was deduced from the data and compared with ε(q, ω) of pristine graphite. The main feature in the different Im [ε(q, ω)]-spectra, a more or less intense peak around 14 eV, can be assigned to maxima of the symmetry-projected joint density of occupied and unoccupied electron states near the Fermi level. Knowing the position of the occupied bands by photoemission experiments, predictions about the unoccupied π and interlayer states can be made. In the case of LiC 6 , a shift and broadening of the first Im [ε(q, ω)]-peak was found, which can be explained by a decrease of the energy of the interlayer-band because of the strong hybridization of this graphite-derived band with the Li metal 2s-band. In the case of KC 8 , a weakening of the first peak of Im [ε(q, ω)] was observed, which we attribute both to a decreasing overlap of the occupied s-derived state with the empty interlayer state and to a change of symmetry of the occupied π-derived states as well as of the empty rc-derived states.

Very-high-energy-resolution measurements using X-rays can be achieved by extreme backreflection (Bragg angle close to 90°) from perfect crystals. This technique allowed the development of the instrument INELAX for inelastic scattering experiments at the HARWI wiggler at DORIS, DESY Hamburg. Recently, a high energy resolution of 9 meV could be achieved, and the instrument proved to be an excellent tool to investigate collective excitations in condensed matter. Energy transfers from 10 meV to 12 eV and wave vectors up to about 10 Â -1 are accessible.

A new spectrometer construction based on a high-voltage W-anode x-ray tube has been used to measure the directional Compton profiles of a pyrolytic graphite crystal. Owing to the focusing geometry, the monochromatic WK α1 t beam is very well collimated, which improves the momentum resolution compared with conventional 241 Am spectrometers, operated almost at the same energy. The experimental anisotropy in the basal plane and in the direction of the c-axis is found to be very close to that of a recent calculation, based on the pseudopotential local-density-functional model.

The asymmetry of Compton Profiles (CPs) measured with a 60-keV 241 Am source on an Al monocrystal, cut along [111], is reported for three different geometrical resolutions. It is found that the geometry of the spectrometer has a substantial effect on the asymmetry of the CP. The better the geometrical resolution the smaller the asymmetry of the CPs.

New high-precision measurements of the isotropic as well as of three directional Compton spectra of lithium hydride have been carried out using 241 Am as a γ-ray source. In order to account for the extreme sensitivity of LiH powder to atmospheric moisture, the final data (i.e. the reciprocal form factor) were corrected for the LiOH content determined by titrimetric analysis. For the interpretation of the data, theoretical calculations were carried out using a Hartree-Fock program for periodic systems (CRYSTAL). Basis sets published by Dovesi et al. were used, one of which allows for polarisation of both the hydride and lithium ions. Comparison of the theoretical data with the experiment shows much better agreement of the results of complete solid-state calculations that take into account higher-order effects (polarisation and covalency) than those obtained by Löwdin orthogonalisation of free-ion wave functions (which assumes pure ionicity, neglecting all but first-order effects). The influence of further polarisation functions on the reciprocal form factor is inves-tigated and discussed. The remaining discrepancies are attributed to electron-electron correlation.

The Compton profile of hexagonal boron nitride has been measured using 59.54 keV gamma rays and a planar Ge detector. The results are compared with theoretical Compton profiles calculated for various ionic arrangements and with those available from a bond orbital model, LCAO and SC canonical HF calculations. The LCAO calculation is in relatively good agreement with the present measurement.

Compton spectra of a-oxalic acid dihydrate single-crystal slices and powder samples were measured in a helium atmosphere using 59.537 keV 241 Am radiation. In the anisotropic case, the scattering vector was oriented parallel to the shortest hydrogen bond, in the direction of the C = O bond, parallel to the C - O bond and perpendicular to the (a, c)-plane. The comparison of theoretical reciprocal form factors, calculated with a local density-approximation (LDA) method for an α-(COOH) 2 • 2H 2 O cluster in triple-zeta basis-set quality, with the experimental data has demonstrated that there is a considerable influence of intermolecular interactions on the electronic structure. Further calculations including the intermolecular interactions of the first coordination sphere by symmetrical orthogonalisation of the LDA wave functions clearly improved the agreement of experiment and theory. A theoretical analysis of the reciprocal form factor B a (s) in the direction of the three different hydrogen bonds of α-oxalic acid dihydrate on the basis of the cluster calculations has shown that in the region above s = 1.9 Å all these bonds are dominated by electrostatic attrac-tion and exchange repulsion of the donor and the acceptor system.

High-purity powder samples of lithium and sodium azide (LiN 3 , NaN 3 ), cyanate (LiOCN, NaOCN) and hydrogen fluoride (LiFHF, NaFHF) were studied by means of 59.54 keV Compton spectroscopy. The measured isotropic Compton profiles were corrected for multiple scattering and transformed to spherically averaged reciprocal form factors B a (s). The experimental results are compared with theoretical reciprocal form factors obtained from Hartree-Fock calculations with different types of basis sets (Gaussian-and Slater-type orbitals, with and without polarisation functions) both for the free ions and for several kinds of clusters. The importance of intraionic and interionic interaction for the description of chemical bonding in these compounds is pointed out and discussed.

The use of Compton scattering in the determination of electronic structure has grown considerably in the last two decades. With the advent of synchrotron radiation it has become possible, even with good resolution, to measure several single-crystal orientations to determine three-dimensional electron momentum distributions. Although most of the earlier work has been directed to low-Z materials, in the last few years medium and high-Z metals have also been investigated with this technique. In this paper we present a review of these studies on heavier metals with particular attention to the difficulties encountered. Compton profile measurements from techniques based on energetic ion beams are also considered briefly.

Directional Compton profiles have been measured along the [100], [110] and [111] crystallographic axes in lead using 412 and 59.5 keV γ-radiation from 198 Au and 241 Am radioisotope sources, respectively. Both measurements are in mutual agreement and show a small anisotropy, which is just distinguishable within the statistical error. The spherically averaged data have been compared with the predictions of relativistic Hartree-Fock (RHF) and non-relativistic Hartree-Fock (HF) free atom calculations. As expected, the experiment clearly favours the RHF rather than the HF calculation.

Theoretical estimates of the spectral distribution and total intensity of the bremsstrahlung (BS) relative to the Compton yield have been determined for photoelectrons liberated in scattering experiments on heavy metals with 60 keV radiation. Whereas the total-intensity calculations are based on the approach of Alexandropoulos et al., the spectral distribution has been computed using the Born approximation along with the Elwert and form factor corrections. These results are compared with another calculation based on the approach of Lee et al. Typical results for Mo, Ta, and W are presented. It is found that the relative contribution of BS from a W target is not trivially small, although K-shell photo-ionisation is not possible at these energies.

We show an iterative algorithm that allows to obtain accurate Compton profiles J(q) from Compton scattering spectra I 2 (ω 2 ), if the excitation radiation is not strictly monochromatic. It requires knowledge of the spectral distribution of the primary radiation I 1 (ω 1 ), validity of the impulse approximation and dominance of a monochromatic part in I 1 (ω 1 ) over the polychromatic rest. Conversely, the primary spectrum is often experimentally not directly accessible. In such a situation it is possible to evaluate the primary spectrum I 1 (ω 1 ) from the spectrum of scattered photons, I 2 (ω 2 ), with a similar iterative algorithm. We use a scattering target of high atomic number in order to ensure that the elastically scattered photons dominate the inelastically scattered ones. From the scattered spectrum we get a model for the Compton profile that allows us to separate the inelastic part of the scattered spectrum from the elastic part, which, in turn, is proportional to the spectral distribution of the primary radiation.

The simple plane wave target Hartree-Fock impulse approximation for the (e, 2e) reaction is developed. One result of the approximation is the separation of the expression for the (e, 2e) cross-section into a kinematic factor and a structure factor that contains all of the information about the target. When the target is a molecule, the structure factor can be further separated into atomic terms and a geometric term. This is illustrated for a simple one-electron homonuclear diatomic molecule. Three examples of the application of (e, 2e) spectroscopy to systems of chemical interest are given. They are borazine (inorganic benzene), the methyl siloxanes and the inorganic complex trimethylamine boron trifluoride.

The flexibility of the (e, 2e) technique or "electron momentum spectroscopy" (EMS) in obtaining information on the electronic structure of atoms, molecules, and solids is demonstrated. High-resolution EMS measurements for argon, including the first measurements of momentum profiles belonging to the 2 P 0 and 2 D e -manifolds, are used to demonstrate the technique for atomic targets. The d-wave transitions in argon are entirely due to initial-state correlations. The first (e, 2e) measurements on an excited target, and also on an oriented target, are discussed. Sodium atoms are pumped to the m l = +1 state of the excited 3p-state by σ + -light from a laser. The (e, 2e) measurements on this excited state are in excellent agreement with the momentum density given by the 3p (m l - 1) Hartree-Fock wavefunction. The recent measurements of the valence-electron momentum distributions for ethyne, as well as some earlier results for water, are used as an example of the application of EMS to the study of molecules. The application of the EMS technique to measure spectral momentum densities in condensed-matter targets is demonstrated by some recent results on amorphous carbon.

The positron annihilation technique has made well-known contributions to the study of Fermi surfaces in "classical" superconductors, including A15 phases where the definition of the Fermi surface has been questioned on the grounds of Anderson localization. In the case of the superconducting oxides, even more far out models were proposed, which made the clear imaging of the Fermi surface by positrons desirable. The difficulties due to the predicted weakness of the signal, and the large possibility for trapping have now been surmounted and the Fermi surface has been seen; what more can we learn from positrons? After presenting the actual situation with experiment, we will comment on enhancement and correlation and their effect on ACAR and lifetime studies. Then we explain the picture of Jarlborg and Singh of enhancement, with its recent tests for many substances. We conclude by asking the question of sensitivity of positrons to many-body effects. Ferromagnetism, antiferromagnetism, possibly charge density waves have been seen -superconductors, heavy fermions and spinons-holons would pose a problem. Stephan Berko (1924-1991) was interested in these problems and knew that better machines and better detectors would open new possibilities.

We discuss the application of the positron annihilation angular correlation (ACAR) spectroscopy for investigating the electronic structure and Fermiology of high-T c superconductors, with focus on the YBa 2 Cu 3 O 7 system where most of the experimental and theoretical work has to date been concentrated. Detailed comparisons between the measured 2D-ACAR positron spectra and the corresponding band theory predictions show a remarkable agreement (for the normal state), indicat-ing that the electronic structure and Fermi surface of this material is described reasonably by the conventional picture.

An introductory survey on applications of high-resolution neutron spectroscopy is presented, dealing with the motion of hydrogen in solids, namely concerning (i) random rotational motions or stationary tunneling states of NH + 4 -ions or CH 3 -groups, and (ii) diffusion of hydrogen in alloys. For the rotation of hydrogenous groups in solids, at higher temperatures rotational jumps can be found, whereas quantum states are observed by μeV-spectroscopy at temperatures below 50 K. On the other hand, hydrogen diffusion does not reveal pronounced evidence of quantum effects, except for hydrogen in a metal containing impurity atoms.

Single-particle longitudinal momentum distributions in condensed matter are now accessible to direct measurement using eV neutrons. Some systems of particular interest include a) quantum solids and fluids, formed from 3 Hie, 4 He, and H 2 , b) prototype molecular crystals, formed from noble gases, c) fluids such as H 2 and Ne, which show deviations from classical behavior and d) mixed systems, in which guest-host interactions may be important. Quantum solids can be investigated at substantially different densities, for different structures, and further, can be compared to the corresponding fluids, with which they have significant similarities in their single-particle properties such as their momentum distributions. Noble gas solids comprise a graduated family with progressively varying importance of zero-point energy, of phonon anharmonicity, and of multi-body forces. They have also been very useful for matrix isolation studies, and eV neutron scattering can yield their center-of-mass motions, not previously seen directly. Finally, noble gas fluids can be used to investigate the nature and extent of final-state effects in the neutron scatttering. Such effects are defined to be the difference between the longitudinal "neutron Compton profile" and the longitudinal momentum distribution.

The electron-volt spectrometer (EVS) at the pulsed neutron source facility (ISIS) is being developed for the study of atomic momentum distributions. Neutrons with energies in the range 1 to 100 eV are incident on the sample, and the time-of-flight (TOF) spectrum of the scattered beam is measured by an array of fixed detectors. A resonant foil difference technique is used to yield a set of TOF spectra for those neutrons scattered into a fixed energy and through fixed angles. Information on the momentum distribution of the target nuclei can be deduced within an impulse approximation in a procedure analogous to that in Compton scattering of electrons by photons. Crystalline compounds containing aligned hydrogen bonds and other hydrogenous compounds are of particular interest owing to the high cross-section of the proton at these neutron energies. With improved statistical accuracy of the data it is anticipated that deviations of the proton's potential from a harmonic potential may be determined. Non-hydrogenous systems have also been investigated. A description is given of the basic theory and interpretive method. Data obtained on numerous systems are presented and discussed.

Deep inelastic neutron scattering has been used to measure the neutron Compton profile (NCP) of a series of condensed 4 He samples at densities from 28.8 atoms/nm 3 (essentially the minimum possible density in the solid phase) up to 39.8 atoms/nm 3 using a chopper spectrometer at the Argonne National Laboratory Intense Pulsed Neutron Source. At the lowest density, the NCP was measured along an isochore through the hep, bcc, and normal liquid phases. Average atomic kinetic energies are extracted from each of the data sets and are compared to both published and new path integral Monte-Carlo (PIMC) calculations as well as other theoretical predictions. In this preliminary analysis of the data, account is taken of the effects of instrumental resolution, multiple scattering, and final-state interactions. Both our measurements and the PIMC theory show that there are only small differences in the kinetic energy and longitudinal momentum distribution of isochoric helium samples, regardless of their phase or crystal structure.

Using deep inelastic neutron scattering, direct measurements have been made, i) of the dynamic structure factor of a series of condensed argon samples over the temperature range 18 to 85 K, near melting (for wave-vector transfers 12 to 26 Ä -1) and ii) of liquid neon, near 27 K (for wave-vector transfers 10.4 to 27.6 Ä -1). Neutron time-of-flight chopper spectrometers were employed. Single-particle kinetic energies, £ k , can be obtained from the analysis of the Doppler-broadened recoil spectrum of the target particles. For argon the temperature dependence of £ k can be compared to expectations from theory, from thermodynamic data, and from previous neutron scattering measurements on collective vibrational modes. For liquid neon the wave-vector-transfer dependence of J{y), the longitudinal momentum distribution function, is being analyzed for final-state effects.

We have studied the anisotropic optical properties of binary NbO 2.5-δ (0 < δ ≤ 0.083) and ternary Nb 18-ε W 8+ε O 69 (ε = 0, 1,..., 9) compounds using the polarized regular-reflection method. We observed strong anisotropic behaviour for all reduced phases. The anisotropic effect can be related to the crystallographical structure principle and to the doping of charge carriers by the reduction of oxygen (δ) or substitution of W for Nb (ε) in the binary and ternary oxides, respectively. Our results indicate that the charge carriers are confined to the structural block units. For increasing δ and ε, metal-like properties occur (δ≥ 0.1, ε > 8) in the infinite block direction only.