Landau Model for Commensurate-Commensurate Phase Transitions in Uniaxial Improper Ferroelectric Crystals

We propose the Landau model for lock-in phase transitions in uniaxially modulated improper ferroelectric incommensurate-commensurate systems of class I. It includes Umklapp terms of third and fourth order and secondary order parameter representing the local polarization. The corresponding phase diagram has the structure of harmless staircase, with the allowed wave numbers obeying the Farey tree algorithm. Among the stable commensurate phases only those with periods equal to odd number of lattice constants have finite macroscopic polarizations. These results are in excellent agreement with experimental findings in some A2BX4 compounds.Comment: 9 pages, 5 figures, revtex, to be published in Journal of Physics: Cond. Matter as a Letter to the Edito

Landau model for uniaxial systems with complex order parameter

We study the Landau model for uniaxial incommensurate-commensurate systems of the I class by keeping Umklapp terms of third and fourth order in the expansion of the free energy. It applies to systems in which the soft mode minimum lies between the corresponding commensurate wave numbers. The minimization of the Landau functional leads to the sine-Gordon equation with two nonlinear terms, equivalent to the equation of motion for the well-known classical mechanical problem of two mixing resonances. We calculate the average free energies for periodic, quasiperiodic and chaotic solutions of this equation, and show that in the regime of finite strengths of Umklapp terms only periodic solutions are absolute minima of the free energy, so that the phase diagram contains only commensurate configurations. The phase transitions between neighboring configurations are of the first order, and the wave number of ordering goes through harmless staircase with a finite number of steps. These results are the basis for the interpretation of phase diagrams for some materials from the I class of incommensurate-commensurate systems, in particular of those for A$_2$BX$_4$ and BCCD compounds. Also, we argue that chaotic barriers which separate metastable periodic solutions represent an intrinsic mechanism for observed memory effects and thermal hystereses.Comment: 12 pages, 14 figures, LaTeX, to be published in Phys. Rev.

The Al-Pd-Mn quasicrystalline approximant ξ'-phase revisited

A detailed investigation of the Fourier space of several Al-Pd-Mn samples with composition Al-72.6 at. %, Pd-22.9 at. %, Mn-4.5 at. % is reported. In the phase diagram of the Al-Pd-Mn ternary alloy, this composition corresponds to the so-called ξ' phase which was described as an icosahedral quasicrystalline approximant. By re-examining the Fourier space by means of X-ray diffraction (powder patterns and single crystal precession patterns), complex structures in close relation with the ξ'-phase have been observed. These long-range order complex structures are described as resulting from a periodic perturbation of the ξ' structure along the c direction. Two states with periodicities $c_{\xi '}(3+\tau)$ and $c_{\xi '}(5+\tau)$ have been observed in this study (τ: golden mean). Structural models based on periodic arrangements of “defects” layers separating layers of $\xi '$ phase are proposed. These two states are certainly intermediate states between the $\xi '$ phase and the metastable decagonal quasicrystalline phase

Fine Structure of Diffuse Scattering Rings in Al-Li-Cu Quasicrystal: A Comparative X-ray and Electron Diffraction Study

A comparative X-ray and electron diffraction study has been performed on Al-Li-Cu icosahedral quasicrystal in order to investigate the diffuse scattering rings revealed by a previous work. Electron diffraction confirms the existence of rings but shows that the rings have a fine structure. The diffuse aspect on the X-ray diffraction patterns is then due to an averaging effect. Recent simulations based on the model of canonical cells related to the icosahedral packing give diffractions patterns in agreement with this fine structure effect.Nous comparons les diagrammes de diffraction des rayon-X et des électrons obtenus sur les mêmes échantillons du quasicristal icosaèdrique Al-Li-Cu. Notre but est d'étudier les anneaux de diffusion diffuse mis en évidence par un travail précédent. Les diagrammes de diffraction électronique confirment la présence des anneaux mais ils montrent aussi que ces anneaux possèdent une structure fine. L'aspect diffus des anneaux révélés par la diffraction des rayons X est dû à un effet de moyenne. Des simulations récentes basées sur la décomposition en cellules canoniques de l'empilement icosaédrique produisent des diagrammes de diffraction en accord avec ces effects de structure fine

X-ray diffraction study of NH4HSeO4 and ND4DSeO 4

This paper reports results obtained from an X-ray diffraction study of NH4HSeO4 and ND4DSeO4 single crystals in their various phases. In connection with incommensurability: a) a 3c-superstructure phase has been found both in AHSe and in ADSe, b) a 2c-superstructure has been discovered in ADSe. The non-equilibrium processes have been identified clarifying the situation about the phase diagram.Nous présentons des résultats de diffraction des rayons X obtenus dans les nombreuses phases des monocristaux de NH4HSeO4 et ND4DSeO4. En liaison étroite avec les problèmes d'incommensurabilité, nous avons découvert: — une phase de surstructure 3c présente à la fois dans AHSe et ADSe, - une phase de surstructure 2c uniquement dans ADSe. Les processus hors d'équilibre ont, grâce à cette technique, pu être identifiés et ont clarifié la situation du diagramme de phase

Structure of the modulation in thiourea. I. symmetry properties

This is the first part of a structural study of the modulation in thiourea. The modulation is described by a normal mode Qq of τ 4 symmetry along with harmonic modes Q2q, Q3q, ... Corresponding displacements are derived in view of structure refinement. In commensurate phases, these give access to the space group; in incommensurate phases, to the supersymmetry. The superspace approach to the symmetry of the modulation is compared to the Landau theory approach. Discussion of the free energy shows that a ferroelastic lock-in phase may be stable under stress and gives information as to the phases of the component modes Qq, Q2q, ..., important for structure refinement.Ceci est la première partie d'une étude structurale de la modulation dans la thiourée. La modulation est décrite par un mode normal Qq de symétrie τ4, accompagné de modes harmoniques Q 2q, Q3q... Les déplacements correspondants sont déterminés en vue de l'affinement de structure. Dans les phases commensurables, ceux-ci donnent accès au groupe d'espace ; dans les phases incommensurables, à la supersymétrie. L'approche de la symétrie grâce au supergroupe d'espace, est comparée à l'approche par la théorie de Landau. La discussion de l'énergie libre montre qu'une phase accrochée ferroélastique peut être stable sous contrainte. Elle fournit aussi de l'information sur les phases des modes Qq, Q2q, ..., importante pour l'affinement structural

A new phase transition in LiNH4SO4 : an x-ray study

X-ray studies of LiNH4SO4 between 295 K and 10 K show a phase transition at 11 °C, and a new transition between 27 K and 28 K, which seems to be 2nd order. Between 284 K and 28 K, the symmetry is shown to be P21/c. Between 27 K and 28 K, a superstructure of wave-vector q = (1/2,1/2, 0) appears, doubling the primitive cell, and leading to the symmetry Cc.Une étude par rayons X de LiNH4SO4 entre 295 K et 10 K montre l'existence d'une transition de phase à 11 °C, et une nouvelle transition entre 27 K et 28 K, qui semble du second ordre. Entre 284 K et 28 K, la symétrie est P21/c. Entre 27 K et 28 K, il apparaît une surstructure de vecteur d'onde q = (1/2, 1/2, 0), qui double la maille primitive, et donne la symétrie Cc

Icosahedral non-quasicrystalline Al-Cu-Fe

X-ray “single” crystal precession photographs of an Al$_{63.5}$Cu$_{24}$Fe$_{12.5}$ alloy of icosahedral symmetry are presented [1]. All the diffraction peak positions are interpreted within the framework of a very simple microcrystalline model in which coherent crystalline domains having particular orientational relationships restore the icosahedral symmetry on diffraction patterns. A quasicrystalline model in which phason fields are introduced is also discussed.Nous présentons des diagrammes de diffraction de rayons X obtenus à l'aide de la méthode de précession, par un alliage Al$_{63.5}$Cu$_{24}$Fe$_{12.5}$ de symétrie icosaédrique [1]. Toutes les positions des pics de diffraction sont interprétées au travers d'un modèle très simple de microcristal, dans lequel des domaines cristallins cohérents ont des relations d'orientation particulières afin de restituer la symétrie icosaédrique sur les clichés de diffraction. Un modèle de quasicristal dans lequel des champs de phasons auraient été introduits est aussi évoqué