Several definitions of the crystal field have been used over time and their variety has lead to many misunderstandings, in both theoretical and experimental literature. Two categories of definitions can be mentioned, the first being the operators equivalents introduced by Stevens in 1952 and the second being the crystal-field operators, introduced by different authors from 1962 and expanded on the Racah spherical tensors. This paper aims at providing some clarification in this field. We first make a review of several expressions introduced in various references to compute crystal-field operators and we describe connections between them. Then, we introduce an explicit way to compute crystal-field operators, in terms of angular momentum quantum numbers $j$ and $m$ as well as in terms of $J^2$ and $J_z$ operators. We eventually give some connections between the Stevens operators equivalents and the crystal-field operators, and make usage of the coefficients of fractional parentage for the expression of the crystal-field operators for the many-body states. Various computational codes, using different crystal-field conventions, are also reviewed.

The subject of this paper are two Hopf algebras which are the non-commutative analogues of two different groups of formal power series. The first group is the set of invertible series with the group law being multiplication of series, while the second group is the set of formal diffeomorphisms with the group law being composition of series. The motivation to introduce these Hopf algebras comes from the study of formal series with non-commutative coefficients. Invertible series with non-commutative coefficients still form a group, and we interpret the corresponding new non-commutative Hopf algebra as an alternative to the natural Hopf algebra given by the co-ordinate ring of the group, which has the advantage of being functorial in the algebra of coefficients. For the formal diffeomorphisms with non-commutative coefficients, this interpretation fails, because in this case the composition is not associative anymore. However, we show that for the dual non-commutative algebra there exists a natural co-associative co-product defining a non-commutative Hopf algebra. Moreover, we give an explicit formula for the antipode, which represents a non-commutative version of the Lagrange inversion formula, and we show that its coefficients are related to planar binary trees. Then we extend these results to the semi-direct co-product of the previous Hopf algebras, and to series in several variables. Finally, we show how the non-commutative Hopf algebras of formal series are related to some renormalization Hopf algebras, which are combinatorial Hopf algebras motivated by the renormalization procedure in quantum field theory, and to the renormalization functor given by the double tensor algebra on a bi-algebra. brouder@lmcp.jussieu.fr frabetti@igd.univ-lyon1.fr kratt@euler.univ-lyon1.fr Research partially supported by the EC’s IHRP Programme, grant HPRN-CT-2001-00272, “Algebraic Combinatorics in Europe.”

New results on the local chemical environment of Cu2+ in archaeological Egyptian blue and green and also modem Egyptian green were obtained by x-ray absorption fine structure (XAFS) analysis. The information is essential for the understanding of the colouring mechanisms in both pigments. In a previous study, a clear physico-chemical characterisation of Egyptian blue and green was achieved using a complementary analytical approach with ancient and modern synthesized pigments. Electron microscopy (SEM-EDX and TEM), x-ray diffraction and micro-Raman and UV-visible spectroscopy were used to gain information about the conditions of the ancient Egyptian fabrication processes and permitted the clear distinction of both pigments. However, the exact colouring mechanisms could not be elucidated by these methods. Different Cu-bearing amorphous and crystalline phases were found in both pigments. These phases should be at the origin of the blue and turquoise colours. Using XAFS data at the Cu K-edge, new insights into the origin of the colouring mechanisms of both pigments could be obtained from the precision of the Cu speciation. In Egyptian blue, Cu2+ is mainly allocated in a square-planar site in a crystalline cuprorivaite phase, whereas in Egyptian green, Cu2+ is basically situated in a distorted octahedral site in an amorphous phase. Copyright (c) 2006 John Wiley & Sons, Ltd.

Homogenization theory is used to calculate the macroscopic dielectric constant from the quantum microscopic dielectric function in a periodic medium. The method can be used to calculate any macroscopic constitutive relation, but it is illustrated here for the case of electrodynamics of matter. The so-called cell problem of homogenization theory is solved and an explicit expression is given for the macroscopic dielectric constant in a form akin to the Clausius-Mossotti or Lorentz-Lorenz relation. The validity of this expression is checked by showing that the standard formula is recovered for cubic materials and that the average of the microscopic energy density is the macroscopic one. Finally, the general expression is applied to Bloch eigenstates.

Physikalische Methoden der Archäometrie liefern den Kunsthistorikern und Archäologen wichtige Informationen. Materialspezifische Größen geben Aufschluss über Alter, Authentizität, Herkunft und Herstellungstechniken sowie Alterserscheinungen von Gegenständen. Insbesondere die Röntgenanalyse hat in jüngster Zeit wertvolle Beiträge hierzu geliefert. Dabei spannt sich ein weiter Bogen von den Mysterien ägyptischer Augenschminke über die Herstellungstechnik römischer Gläser bis zur Authentizitätsuntersuchung mittelalterlicher Silberstiftzeichnungen.

The Hopf algebra of renormalization in quantum field theory is described at a general level. The products of fields at a point are assumed to form a bialgebra B and renormalization endows T(T(B)^+), the double tensor algebra of B, with the structure of a noncommutative bialgebra. When the bialgebra B is commutative, renormalization turns S(S(B)^+), the double symmetric algebra of B, into a commutative bialgebra. The usual Hopf algebra of renormalization is recovered when the elements of B are not renormalised, i.e. when Feynman diagrams containing one single vertex are not renormalised. When B is the Hopf algebra of a commutative group, a homomorphism is established between the bialgebra S(S(B)^+) and the Faa di Bruno bialgebra of composition of series. The relation with the Connes-Moscovici Hopf algebra of diffeomorphisms is given. Finally, the bialgebra S(S(B)^+) is shown to give the same results as the standard renormalisation procedure for the scalar field.

Chrysotile asbestos is formed by densely packed bundles of multiwall hollow nanotubes. Each wall in the nanotubes is a cylindrically wrapped layer of Mg(3)Si(2)O(5)(OH)(4). We show by experiment and theory that the infrared spectra of chrysotile present multiple ionic-plasmon resonances in the Si-O stretching bands. These collective charge excitations are universal features of the nanotubes that are obtained by cylindrically wrapping an anisotropic material. The multiple plasmons can be observed if the width of the resonances is sufficiently small as in chrysotile.

X-ray absorption spectroscopy (XAS) has proved a powerful tool to investigate the local geometry around transition metal impurities in crystals. However, the measurement of XAS in single-crystals is made difficult because of the presence of very strong diffraction peaks (glitches). Recently, a new measurement method was proposed to investigate natural circular dichroism in the X-ray range [1]. The method here is adapted to the measurement of linear dichroism. We show how isotropic spectra and linear dichroism can be extracted from the angular dependence of XAS. As an example, we present the K-edge spectrum of chromium in ruby.

The crystallographic distortion around chromium and iron has been investigated by use of both theoretical and experimental methods. Through the analysis of the isotropic and dichroic Extended X-ray Absorption Fine Structures at the Cr K-edge, the Cr-O distances have been measured in the distorted chromium coordination shell. Through ab initio molecular dynamics, the distortion of the chromium site has been independently calculated.

For the first time, commonly unaccessible local electronic structure parameters of Fe2+ and Fe3+ in minerals are derived from the calculation of the pre-edge features of X-ray absorption spectra at the Fe K edge. The Ligand Field Multiplet approach is used to calculate the eigenstates of the ions and the absolute intensities of the electric-quadrupole and dipole transitions involved in the pre-edge. For ions in tetrahedral symmetry, a new model for p-d hybridization is developed. The degree of admixture between 3d and 4p levels is derived and local structure parameters (crystal field, bond covalency) are obtained.

A new and coherent view of the local environment of Fe2+ in silicate glasses is derived from a combined study by EXAFS and Molecular Dynamics simulation. Iron is located in distorted sites, whose geometry varies continuously from a tetrahedron to a triangular bipyramid. Iron polyhedra are apex-connected to the silicate network, while edge-linked to each other, matching with a random distribution of iron in the glass. Fe2+ does not have the significance of a network-forming element. Its local structure results from the freezing of dynamical exchange processes occurring in the liquid.

In ferro- or ferrimagnctic systems x-ray magnetic circular dichroism (XMCD) can yield information on the magnetic contribution of each open orbitals for each species. in rare-earth-3d transition-metal (R-T) alloys these orbitals are of utmost interest for magnetism as they originate the magnetic moments (4f in rare earth, 3d in transition metal) or mediate the magnetic interactions (d or p bands). Despite intensive works the interpretation is still matter of discussion at the R L-2,L-3-edgcs and T K-edge. We present here a comparative study of the XMCD at the 3d metal K-edge in several R-T compounds where the 3d metal satisfies (RCo5) or not (RNi5) the Stoner criterion. The structure of the experimental XMCD spectra in the near-edge region has been observed to be sensitive to the magnetic environment of the absorbing site. Calculations of the XMCD spectra have also been performed for pure Co metal, LaCo5, TbCo5, TbCo2 and HoCo2, within the multiple scattering framework including the spin-orbit coupling. The spectra are well reproduced in the near-edge region. Results point out that the near-edge structures in the Co K-edge XMCD spectra originates almost exclusively from the spin-orbit on the R 5d states when R is magnetic.

The differential absorption and the differential change in the polarization state of an X-ray beam propagating inside a gyrotropic crystal are described using a 4 X 4 Müller matrix, the 16 elements of which are related to the anisotropic components of the multipolar polarizability tensors at the absorbing site. Analytical expressions are given up to third order for X-ray linear and circular dichroism, X-ray optical rotation and X-ray circular polarimetry in transmission. The same formalism is extended to discuss Fluorescence detected dichroism spectra with or without polarization analysis of the fluorescence. Fluorescence detected dichroism is strictly proportional to dichroism measured in the transmission geometry only for uniaxial crystals. In biaxial crystals, the tiny effects of X-ray gyrotropy are swamped by large linear dichroism signals due to the imperfect polarization transfer function of Bragg monochromators. Second order effects should also be taken into consideration. Our general formulation of linear and circular dichroism includes terms of odd parity with respect to the action of the time reversal operator: such terms cannot contribute to natural dichroism but can be activated by a magnetic field. The terms responsible for X-ray magnetic circular dichroism are well known but non-reciprocal X-ray gyrotropy effects are also predicted in magnetic crystals of appropriate symmetry.

Tektite glasses are investigated using Fe-57 Mossbauer spectroscopy. Room temperature spectra analysis is performed using two complementary analytical methods based on two-dimensional distributions of both isomer shift and quadrupole splitting. No a priori correlation between the two hyperfine parameters is considered. The first method, based on a shape independent distribution, provides the justification for the Gaussian distribution shape used in the second method. No ferric iron contribution is evidenced by Mossbauer spectra analysis in these samples, although several criteria are used. Ferrous iron sites are shown to be continuously distributed between four- and five-fold coordinated sites.

A method is presented to calculate the natural circular dichroism recently discovered in the X-ray range (XNCD). The basic formula represents XNCD as an odd second-rank tensor and leads to a sum rule that relates XNCD to the mixing of odd and even orbitals in the ground state. A multiple-scattering theory of XNCD is presented, and calculated spectra for the L-edges of iodine in LiIO3 compare favorably with the experiments.

The X-ray magnetic circular dichroism (XMCD) has been measured at the Co K edge in Co-hcp and R-Co compounds (R=La, Tb, Dy). The structure of the experimental XMCD spectra in the near-edge region has been observed to be highly sensitive to the magnetic environment of the absorbing site. Calculations of the XMCD have been carried out at the Co K edge in Co metal, LaCo$_5$ and TbCo$_5$ within the multiple-scattering framework including the spin-orbit coupling. In the three systems, the XMCD spectra in the near-edge region are well reproduced. The possibility to separate and quantitatively estimate the local effects from those due to the neighboring atoms in the XMCD cross section makes possible a more physical understanding of the spectra. The present results emphasize the major role played by the $d$ states of the Tb ions in the XMCD spectrum at the Co K edge in the TbCo$_5$ compound.

X-ray absorption spectroscopy at the L-2,L-3 edges of 3d transition metals has been used to study the electronic structure of molecule based magnets with Curie temperatures ranging from 66 K to 315 K. These magnets are bimetallic cyanides of the Prussian blue family, constructed by a three dimensional assembling of -NC-Cr-III-CN-A(II)- units. Using Ligand Field Multiplet calculations, where hybridization is mainly taken into account through configuration interaction, we have been able to reproduce nicely the spectra and to determine the exact ground state of the transition metals. We have separated covalence and charge transfer effects occurring in the bond between the 3d ions and the cyano ligand. We have also recorded X-ray Magnetic Circular Dichroism (XMCD) at nickel L-2,L-3 edges in the ferromagnet Cs-I[(NiCrIII)-Cr-II(CN)(6)]. 2H(2)O (T-C = 90 K). With Ligand Field Multiplet calculations the shape of the experimental XMCD signal can be reproduced. The application of the orbital and spin sum rules gives a too small magnetic moment on nickel. A complete calculation taking covalence into account showed that hybridization cannot be responsible for the experimental low nickel magnetic moment.