Superspace Approach Research Articles

When minerals become complex: an elementary introduction to superspace crystallography to describe natural-occurring incommensurately modulated structures

Received: April 2007 / Accepted: December 2007 - © Springer-Verlag 2008 Abstract Aperiodic crystals are compounds with perfect long-range order of their atomic structures, but without three-dimensional translational symmetry. In the last thirty years, incommensurate crystal structures have been discov- ered in many compounds from synthetic inorganic and organic compounds to proteins. With the introduction of the superspace theory it was realized that any crystal structure requiring more than three integers to index its diffraction pat- tern can be described as a periodic object in superspace with dimension equal to the number of the required integers. The structure observed in physical space is a three-dimensional intersection of the structure described as periodic in superspace. In recent years, the use of CCD and imaging plate systems con- siderably changed the sensitivity of data collection for modulated structures and, therefore, there was a need for further improvement of the methods. To- day, several computer programs are able to solve and refine incommensurately modulated structures using the superspace approach. In nature, it is uncommon to find minerals having strong and sharp in- commensurate satellites that could be used for a multidimensional refinement. As a very clear example, the case of natural akermanite, Ca2MgSi2O7 ,w hich belongs to the melilite-group of minerals, will be shown in detail. The five- dimensional refinement carried out on this crystal shows that the displacive modulation of the atoms is mainly related to a variation of the Ca cation coor- dination. Keywords Superspace crystallography, Incommensurate structure, Modula-

Synthesis, Crystal Structure, and Magnetic Properties of Sr1.31Co0.63Mn0.37O3: A Derivative of the Incommensurate Composite Hexagonal Perovskite Structure

We report the synthesis, structural investigation, and magnetic property studies of Sr1.31Co0.63Mn0.37O3 that adopts an incommensurate composite hexagonal perovskite-related structure. The crystal structure has been solved using a (3 + 1)-dimensional superspace approach from powder X-ray and neutron diffraction data (SSG R3̅m(00γ)0s, a = 9.5548(1) Å, c = 2.5599(1) Å, q = 0.65581(4)c*, RB = 0.041, RP = 0.059). The structure consists of face-sharing chains of octahedra and trigonal prisms, wherein the trigonal prismatic sites are preferentially occupied by Co with some cation disorder. A combination of electron diffraction and high-resolution electron microscopic analysis has demonstrated that the compound possesses a complicated microstructure related to the formation of domains with slightly different lengths of the modulation vector. X-ray absorption near-edge spectroscopic (XAS) studies clearly indicate the presence of Mn in the 4+ and Co in the 3+ oxidation state. While the magnetic susceptibility data indicates the presence of antiferromagnetic correlations in the system, the calculation of effective paramagnetic moment (μcal = 3.561 μB), assuming the metal oxidation states as obtained from XAS and the cation distribution as obtained from neutron refinement, is in agreement with the value obtained experimentally (μexp = 3.676 μB).

Hierarchy of [formula omitted] mechanics models

Using the N = 4 superspace approach in one dimension (time), we construct general N = 8 supersymmetric mechanics actions for the multiplets ( b , 8 , 8 − b ) classified in hep-th/0406015, with the main focus on the previously unexplored cases of ( 8 , 8 , 0 ) , ( 7 , 8 , 1 ) and ( 6 , 8 , 2 ) , as well as on ( 5 , 8 , 3 ) for completeness. N = 8 supersymmetry of the action amounts to a harmonicity condition for the Lagrangian with respect to its superfield arguments. We derive the generic off-shell component action for the “root” multiplet ( 8 , 8 , 0 ) , prove that the actions for all other multiplets follow from it through automorphic dualities and argue that this hierarchical structure is universal. The bosonic target geometry in all cases is conformally flat, with a unique scalar potential (except for the root multiplet). We show that the N = 4 superfield constraints respect the full R-symmetry and find the explicit realization of its quotient over the manifest R-symmetry on superfields and component fields. Several R-symmetric N = 4 superfield Lagrangians with N = 8 supersymmetry are either newly found or reproduced by a simple universal method.

Commensurate and incommensurate “5M” modulated crystal structures in Ni–Mn–Ga martensitic phases

It is well known that the composition of ferromagnetic shape memory Ni–Mn–Ga Heusler alloys determines both temperature of martensitic transformations and the structure type of the product phase. In the present work we focused our attention on the structural study of the so-called “5M” modulated structure. In particular, the structure of Ni 1.95Mn 1.19Ga 0.86 martensitic phase is analysed by powder X-ray diffraction (PXRD) and compared with that of the stoichiometric Ni 2MnGa martensite. The study of the diffraction data reveals the occurrence of commensurate (C) structural modulation in Ni 1.95Mn 1.19Ga 0.86; this contrasts with Ni 2MnGa, where an incommensurate (IC) structural modulation was evident. The two phases also differ in the symmetry of the fundamental martensitic lattice. In fact, the incommensurate modulation is related to an orthorhombic basic structure, while the commensurate variant presents a monoclinic symmetry. The commensurate modulated structure has been investigated by using the superspace approach already adopted to solve the structure of Ni 2MnGa martensite. The structure has been determined by Rietveld refinement of PXRD data.

Five-dimensional [formula omitted] AdS superspace: Geometry, off-shell multiplets and dynamics

As a step towards formulating projective superspace techniques for supergravity theories with eight supercharges, this work is devoted to field theory in five-dimensional N = 1 anti-de Sitter superspace AdS 5 | 8 = SU ( 2 , 2 | 1 ) / SO ( 4 , 1 ) × U ( 1 ) which is a maximally symmetric curved background. We develop the differential geometry of AdS 5 | 8 and describe its isometries in terms of Killing supervectors. Various off-shell supermultiplets in AdS 5 | 8 × S 2 are defined, and supersymmetric actions are constructed both in harmonic and projective superspace approaches. Several families of supersymmetric theories are presented including nonlinear sigma-models, Chern–Simons theories and vector–tensor dynamical systems. Using a suitable coset representative, we make use of the coset construction to develop an explicit realization for one half of the superspace AdS 5 | 8 as a trivial fiber bundle with fibers isomorphic to four-dimensional Minkowski superspace.

-supersymmetric quantum mechanics in a scenario with Lorentz-symmetry violation

Abstract We show in this Letter that the dynamics of a non-relativistic particle with spin, coupled to an external electromagnetic field and to a background that breaks Lorentz symmetry, is naturally endowed with an N = 1 -supersymmetry. This result is achieved in a superspace approach where the particle coordinates and the spin degrees of freedom are components of the same supermultiplet.

Superspace description of the crystal structures of Ca n (Nb,Ti) n O3n + 2 (n = 5 and 6)

The crystal structures of two members of the homologous series Ca(n)(Nb,Ti)(n)O(3n + 2), with n = 5 and 6, are presented within the superspace formalism. A common (3 + 1)-dimensional superspace model is used to describe the crystal structures of both compositions within a particular homologous series, where the primary modulation wavevector and the width of the atomic domains vary systematically with composition. The two crystal structures are characterized as commensurately modulated structures consisting of discontinuous atomic domains described by occupational crenel functions. The displacive modulation functions for the two compounds exhibit similarities, but they also show that the idea of a unified superspace model does not extend toward the precise atomic positions. For n = 6, the centrosymmetric (3 + 1)-dimensional superspace symmetry provides a natural explanation for the pseudo-symmetries that are present in the non-centrosymmetric (three-dimensional) superstructure of this compound. The efficiency of the superspace approach is demonstrated by structure refinements in (3 + 1)-dimensional superspace and by comparing these results with the refinements in their three-dimensional superstructures.

Gauging [formula omitted] supersymmetric mechanics II: [formula omitted] models from the [formula omitted] ones

Exploiting the gauging procedure developed by us in hep-th/0605211, we study the relationships between the models of N = 4 mechanics based on the off-shell multiplets ( 4 , 4 , 0 ) and ( 1 , 4 , 3 ) . We make use of the off-shell N = 4 , d = 1 harmonic superspace approach as most adequate for treating this circle of problems. We show that the most general sigma-model type superfield action of the multiplet ( 1 , 4 , 3 ) can be obtained in a few non-equivalent ways from the ( 4 , 4 , 0 ) actions invariant under certain three-parameter symmetries, through gauging the latter by the appropriate non-propagating gauge multiplets. We discuss in detail the gauging of both the Pauli–Gürsey SU ( 2 ) symmetry and the Abelian three-generator shift symmetry. We reveal the ( 4 , 4 , 0 ) origin of the known mechanisms of generating potential terms for the multiplet ( 1 , 4 , 3 ) , as well as of its superconformal properties. A new description of this multiplet in terms of unconstrained harmonic analytic gauge superfield is proposed. It suggests, in particular, a novel mechanism of generating the ( 1 , 4 , 3 ) potential terms via coupling to the fermionic off-shell N = 4 multiplet ( 0 , 4 , 4 ) .

Effective action of softly broken supersymmetric theories

We study the renormalization of (softly) broken supersymmetric theories at the one loop level in detail. We perform this analysis in a superspace approach in which the supersymmetry breaking interactions are parametrized using spurion insertions. We comment on the uniqueness of this parametrization. We compute the one loop renormalization of such theories by calculating superspace vacuum graphs with multiple spurion insertions. To perform this computation efficiently we develop algebraic properties of spurion operators, that naturally arise because the spurions are often surrounded by superspace projection operators. Our results are general apart from the restrictions that higher super covariant derivative terms and some finite effects due to noncommutativity of superfield dependent mass matrices are ignored. One of the soft potentials induces renormalization of the Kaehler potential.

Superspace approach to crystallographic shear structures

A systematic method for the superspace description of crystallographic shear (CS) structures is proposed. A (3 + 1)-dimensional [(3 + 1)D] superspace model for ideal CS structures is constructed in connection with the shear operation in 3D space, which is defined by a shear plane and a shear vector. All structural parameters in the model are related to those of the parent structures by this operation. This leads to inherently discontinuous and sawtooth-like modulation functions.

First (3 + 2)-dimensional superspace approach to the structure of levyclaudite-(Sb), a member of the cylindrite-type minerals

The structure of synthetic levyclaudite-(Sb), approximately (Pb(1 - y)Sb(y)S)(1.357)[Sn(1 - x)(Cu(2))(x)S(2)], has been determined by single-crystal X-ray diffraction on the basis of the (3 + 2)-dimensional superspace approach. This misfit-layer compound, of the cylindrite type, results from the combination of two heavily modulated triclinic Q and H subsystems with a common q wavevector and only one shared reciprocal axis (stacking direction). The Q pseudo-tetragonal layer, approximately (Pb(0.70)Sb(0.30)S), derived from the NaCl archetype, is positively charged; the H pseudo-hexagonal layer, approximately (Sn(0.85)Cu(0.30)S(2)), derived from the CdI(2) archetype, is negatively charged, owing to the replacement of Sn(4+) in an octahedral coordination by Cu(+) pairs in an opposite triangular coordination. The analysis shows a strong transverse displacive modulation of the two layers, referred to as a 'mondulation', correlated to a maximal Sb site occupation factor in the concavity of the Q layer undulation. The wavelength control of the ;mondulation' obeys the vernier principle (14c(Q) congruent with 13c(H)), which would correspond to an energy minimization through a charge transfer density modulation wave, common to all two-dimensional misfit-layer inorganic compounds.

Five-dimensional supersymmetric Chern–Simons action as a hypermultiplet quantum correction

Abstract Building on the covariant supergraph techniques in 4D N = 2 harmonic superspace, we develop a manifestly 5D N = 1 supersymmetric and gauge covariant formalism to compute the one-loop effective action for a hypermultiplet coupled to a background vector multiplet. As a simple application, we demonstrate the generation of a supersymmetric Chern–Simons action at the quantum level, both in the Coulomb and the non-Abelian phases. These superfield results are in agreement with the earlier component considerations of Seiberg et al. Our analysis suggests that similar calculations in terms of hybrid 4D superfields or within the 5D projective superspace approach may allow one to extract suitable formulations for the non-Abelian 5D supersymmetric Chern–Simons theory.

(3+2)-dimensional superspace approach of the structure of lévyclaudite-(Sb), a member of the cylindrite-type minerals

(3+2)-dimensional superspace approach of the structure of lévyclaudite-(Sb), a member of the cylindrite-type minerals

Superspace embedding of sheelite-like structures

JANA2006 is a new version of crystallographic compu-ting system JANA [1] originally developed for incom-mensurately modulated structures. During more than 20 years of development it became widely used unified system for structure analysis of periodic and aperiodic crystals with emphasis on solving difficult structures. This unifying trend is followed further in JANA2006. The main improvement of the new system consists in complete generalization of multiphase and multisource crystallographic data input. Thus the new system can combine for instanceX-ray single crystal data with neutron powder data even for samples containing several phases in different ratios. An example utilizing combination of data will be given in the presentation. The Rietveld refinement option as already implemented into JANA2000 offers the standard profile functions and anisotropic tensors for strain profile broadening. Fundamental approach to profile fitting [2] is an important new feature in JANA2006 giving more exact physical sense to profile parameters and connecting them better with the real experimental arrangement. Support for TOF data was added, too. JANA2006 will support refinement of magnetic structures. The superspace approach to handle mutually magnetic and structural modulations is under development. Charge flipping generalized for superspace [3] will be integral part of the program as an alternative solutionmethod especially useful formodulated structures. New graphical interface attempts bringing multilevel menu-driven control to more intuitive one level interface. Dynamical elements were added like updating of Fourier contour plots or powder profiles during refinement. The automatic tasks like (super)space group determination and graphical way of adding hydrogen atoms were improved.Work with simple structure is also better supported, especially creation of CIF output conforming better to current requirements of Acta Crystallographica E,C. From the technical point of view JANA2006 differs significantly of the previous versions by possibility to allocate memory dynamically (it uses FORTRAN95) and by using precompiled binaries for LINUX and MAC OS X operating systems. All JANA programs are freeware. JANA2006 is currently available in beta version by request; the first publicly available beta version is expected at the end of this year.

Modulated structure of potassium sodium strontium barium niobates (KNSBN): harmonic solution

The structures of two specimens of the (K(1 - y)Na(y))(2F - 2)(Sr(x)Ba(1 - x))(2 - F)Nb2O6 family (KNSBN) have been solved and refined (x(I) = 0.6, y(I) = 0.45, F(I) = 1.07 and x(II) = 0.72, y(II) = 0.45, F(II) = 1.08). The KNSBN compounds appear to be incommensurately modulated, as described in a five-dimensional superspace approach with the tetragonal P4bm (pp1/2, p - p1/2) superspace group. Their modulation wavevectors are almost independent of the composition. The description of the modulation is restricted to first-order harmonics in order to limit the number of refined parameters. The modulation mainly affects the positions of the O atoms and their displacements depend on the Sr/Ba ratio. The structural models for the two compounds are not only almost identical, but also analogous to the structural model of the Sr(x)Ba(1 - x)Nb2O6 parent compound: doping with K+ and Na+ cations has no significant influence on the incommensurate modulation, contrary to the strong dependence observed for the physical properties.

Description of Ba1 + x Ni x Rh1 − x O3 with x = 0.1170 (5) in superspace: modulated composite versus modulated-layer structure

The structure of the compound Ba(1 + x)Ni(x)Rh(1 - x)O3 [x = 0.1170 (5)] has been analyzed at room temperature within the (3 + 1)-dimensional superspace approach using single-crystal X-ray diffraction data. Two different models are presented, the compound is refined as modulated composite as well as modulated-layer structure. In both models discontinuous atomic domains are applied to describe the structural modulations. While the first approach stresses the pseudo-one-dimensional constitution, the latter highlights the layered character of these structures.

Getting more out of an incommensurately modulated structure: the example of K5Yb(MoO4)4

A method based on the superspace approach is presented with the aim of generating a family of modular structures from a single incommensurately modulated structure. This approach based on the variation of the modulation vector q is applied to the generation of the K5Yb(MoO4)4, potassium ytterbium tetramolybdate, family of modular structures. The beta coefficient of the modulation vector q = betab* is a temperature-dependent variable which determines the modification. Our method gives a unified frame to describe and explain the three temperature-dependent phases of K5Yb(MoO4)4. Phase beta can be represented as a polytypic modification with 1/2b* < or = q < or = 2/3b*; phases gamma (q = 1/2b*) and alpha (q = 1b*) are the lowest and the highest temperature members of the K5Yb(MoO4)4 family, respectively.

Construction of one-loop Script N = 4 SYM effective action in the harmonic superspace approach

We develop a systematic approach to construct the one-loop ${\cal N}=4$ SYM effective action depending on both ${\cal N}=2$ vector multiplet and hypermultiplet background fields. Beginning with the formulation of ${\cal N}=4$ SYM theory in terms of ${\cal N}=2$ harmonic superfields, we construct the one-loop effective action using the covariant ${\cal N}=2$ harmonic supergraphs and calculate it in ${\cal N}=2$ harmonic superfield form for constant Abelian strength $F_{mn}$ and corresponding constant hypermultiplet fields. The hypermultiplet-dependent effective action is derived and given by integral over the analytic subspace of harmonic superspace. We show that each term in the Schwinger-De Witt expansion of the low-energy effective action is written as integral over full ${\cal N}=2$ superspace.

Dynamical Equations from a First-Order Perturabtive Superspace Formulation of 10D, N=1 String-Corrected Supergravity

Utilizing a first-order perturbative superspace approach, we derive the bosonic equations of motion for the 10D, = 1 supergravity fields. We give the lagrangian corresponding to these equations derived from superspace geometry. Moreover, the equivalence of this lagrangian to the first-order perturbative component level lagrangian of anomaly-free supergravity is proven. Our treatment covers both the two-form and six-form formulations.

Synthesis and study of Sr-substituted misfit layer sulfides

Two series of [(Ln 1/3Sr 2/3S) 1.5] 1.15NbS 2 (1.5Q/1H) and (Ln 1− x Sr x S) 1+ y NbS 2 (1Q/1H) misfit layer sulfides have been synthesised and studied by X-ray powder and single crystal diffraction, EDX-analysis and magnetic measurements. For the early lanthanides (Ln = La, Ce) only the (Ln 1− x Sr x S) 1+ y NbS 2 ( x < 0.40–0.45, y = 0.15–0.17) compounds were formed whereas for late lanthanides (Ln = Pr, Nd, Sm, Gd–Er, Yb, Lu), and yttrium, both types of phases can be obtained. The crystal structure of (Pr 0.55Sr 0.45S) 1.15NbS 2 has been refined on the basis of X-ray single crystal data using the superspace approach. It consists of double layers [Pr 0.55Sr 0.45S] of NaCl-type (Q-part: a = 5.799(3) Å, b = 5.810(2) Å, c = 23.331(9) Å, z = 4) and NbS 2-sandwiches (H-part: a = 3.332(3) Å), z = 8, yielding a q-vector q = ( α 0 0), α = 0.74) alternating along the c-direction. The superspace group pair is Fm2 m ( α 0 0): Fm2 m ( α 0 0) (No. 42.7). The refinement converged to R w(obs) = 0.069. According to the structure refinement and EDX-analysis data in the [Pr 0.55Sr 0.45S]-slab almost half of the Pr-atoms are randomly replaced by Sr. The minimal formal value of charge transfer from the Q- to the H-part of the structure necessary to stabilise the misfit Nb-based layer sulfides has been estimated as about 0.6 ē per Nb atom.