Group-theoretical description of domain and phase boundaries in crystalline solids

The theory is reviewed of domains and domain boundaries arising in phase transitions accompanied by symmetry breaking. Conclusions concerning the number, the crystallographic type and the spatial orientation of coherent interfaces between crystals of the same structure (domain boundaries) and between different structures of the same material (interphase boundaries) are presented in terms of the space group theory and of the Landau theory of phase transitions. The application of the two-dimensional space groups and the diperiodic groups in three dimensions to the discussed objects is described. The conditions for the coexistence of domains and phases without macroscopic stress are given. An example of the group-theoretical analysis of domain structure is given for a real material: NaO₂.     

Phase transitions in two-dimensional uniformly frustratedXY models. II. General scheme

For two-dimensional uniformly frustratedXY models the group of symmetry spontaneously broken in the ground state is a cross product of the group of two-dimensional rotations by some discrete group of finite order. Different possibilities of phase transitions in such systems are investigated. The transition to the Coulomb gas with noninteger charges is widely used when analyzing the properties of relevant topological excitations. The number of these excitations includes not only domain walls and traditional (integer) vortices, but also vortices with a fractional number of circulation quanta which are to be localized at bends and intersections of domain walls. The types of possible phase transitions prove to be dependent on their relative sequence: in the case the vanishing of domain wall free energy occurs earlier (at increasing temperature) than the dissociation of pairs of ordinary vortices, the second phase transition is to be associated with dissociation of pairs of fractional vortices. The general statements are illustrated with a number of examples.     

Domain structure of the antiferromagnetic insulating state in Nd0.5Sr0.5MnO3

Optical reflectivity studies of the ferromagnetic metal (FMM) to antiferromagnetic insulator (AFI) phase transition were performed on Nd_0.5Sr_0.5MnO₃ single crystals in a wide temperature and magnetic field range. The formation of a domain structure in the AFI state was observed. On the basis of the experimental results and symmetry analysis we conclude that these domains are crystal twins, confirming the symmetry lowering of the crystal at this phase transition. The twin domain structure of the AFI state in the Nd_0.5Sr_0.5MnO₃ is visible in reflected unpolarized light due to a different tilting of the surface of the domains. We find no evidence for intermediate FMM+AFI stripe phase.     

MS-Windows software for space groups: Application to domain structure symmetry analysis

An integrated package of programs has been developed for IBM-Compatible PCs to investigate the structures and representations of crystallographic space groups. The package is implemented as a Microsoft Windows application using Borland Delphi with user code in Object-Pascal.

Parts of this software have been adapted to assist in the symmetry analysis of domain structures. For a given phase transition the software identifies all domain states and finds, e.g. (i) symmetry groups of all domain states, (ii) all operations that transform a given domain state into another domain state, (iii) classes of crystallographically equivalent domain pairs with similar domain distinction, (iv) symmetries of ordered and unordered domain pairs, (v) twinning groups of domain pairs and associated minimal permutable sets of domain states, (vi) intermediate groups of the inverse twinning problem.

As an illustrative example of the use of the software we consider the symmetry analysis of domain structures in the 2H polytype TaSe₂.     

Cosmological baryon-number domain structure and the first order phase transition of a vacuum

If CP-nonconservation arises from spontaneous symmetry breaking in the very early universe, the universe will have a domain structure of baryon number. We propose a model of the early universe in which domains are stretched exponentially and the radius of the domains is much greater than that of the horizon of the standard big bang model, provided that the grand unified theory undergoes a first order phase transition. If the size of the stretched domains is sufficiently big to avoid pair annihilations of baryon and antibaryon domains, the difficulties of the baryon symmetric universe may be removed.     

Deflected light structure in NaNH4SeO4·2H2O crystals

Deflection of light studies in function of temperature in NaNH₄SeO₄·H₂O crystal is presented. At 180?K, this compound undergoes a para-ferroelectric/ferroelastic phase transition of the second order. It changes the symmetry from the orthorhombic symmetry class 222 (space group P2₁2₁2₁) to the monoclinic symmetry class 2 (space group P2₁). A distinct deflection pattern, resulting from the ferroelastic domain structure, occurs in the low temperature phase. The intensity of deflected beams varies considerably with temperature. Detailed studies revealed the structure of deflected spots during cooling–heating cycle. These spots change intensity, become more dispersed and finally split with decreasing temperature. Moreover, a variation of deflection angles was observed. A possible explanation of these phenomena is given in a framework of crystals optics.     

Coupling of ferroelectric and antiferromagnetic order parameters in hexagonal RMnO3

Second-harmonic generation (SHG) is used as a probe for the coexisting ferroelectric and antiferromagnetic orders in hexagonal RMnO₃. SH contributions coupling to the electric and/or magnetic order parameters are identified on the basis of their spectral dependence and the symmetries of the corresponding order parameters. The SH signals from the ferroelectric and antiferromagnetic orders were employed to image the electric and magnetic domain structures separately. The transformation properties of electric and magnetic domains are discussed with respect to the transformation properties of the corresponding order parameters. An investigation of the mutual coupling between the coexisting electric and magnetic orders reveals apparently independent domain structures, which contradicts the symmetry and transformation properties of electric-dipole-induced SHG in this ferroelectromagnetic group of compounds. Apart from higher-order multipole contributions to SHG, interface contributions from clamped electric and magnetic domains can solve the contradiction. Received: 16 October 2001 / Published online: 24 April 2002     

Strain at junctions in multidomain configurations

This article deals with the determination of multidomain configurations in ferroics taking into account the elastic compatibility of the neighbouring domains. The proposed method is based on transformation matrices, which describe the spacial relationship of two crystal lattice bases for coherent domain pairs. The matrices are calculated using the symmetry operations between domains and lattice parameters. The principle point of this method is the comparison of the product of transformation matrices and the unity matrix. For example, by using this method it was shown that the junction of three and five orientation domains is strained whereas the encounter of four domains is strain free in three different ferroelastic perovskite phases. The derived allowed four-domain configurations are confirmed by experimental data.     

High-frequency excitations of stripe-domain lattices in magnetic garnet films

Magnetic garnet films of composition (Y,Bi)₃(Fe,Al)₅O₁₂ have been grown by liquid phase epitaxy on [111] and [110] oriented substrates of gadolinium gallium garnet. The domain wall resonance and the two branches of the domain resonance of periodic stripe domains are measured as function of the bias induction applied in the film plane parallel to the stripes. Resonance frequencies up to 7.5 GHz are observed. An improved version of the hybridization model is developed to describe these resonances. It turns out that hybridization of the domain resonance branches is determined by the cubic anisotropy for [111] oriented films, while for [110] oriented films coupling of the domain resonances is mainly caused by the orthorhombic anisotropy. The theoretical model is in excellent agreement with experiments, no fitting parameters are used. It is also used to derive the phase relation between the precessing magnetizations of neighbouring domains.     

A new behaviour of the interstitial element H or D in solution in rare-earth metal

The study of the resistivity of the Lu-(H or D) system makes clear certain characteristics of the α phase and indicates a change for the interstitial element in the transition range 170–400°C. On both sides of this domain an appreciable difference is observed in the properties of the solid solution. The α1 phase of the domain below 170°C has a higher resistivity and a maximum hydrogen concentration independent of the temperature. For the phase α of the domain above 400°C, the expansions both with the concentration growth and with the rise in temperature are isotropic contrary to the α1 phase.The resistivity and X-ray diffraction studies, taking into account the lattice symmetry, allow us to conclude that a change may occur in the type of interstices filled by the hydrogen atoms, tetrahedral for the α phase and octahedral with H atoms associating in pairs for the α1 phase. Among the lanthanide metals, we find that only lutetium and thulium, in uniting with H or D, have the low temperature α1 phase.     

Structure of the α-phase of solid methanol

The crystal structure of α-methanol at 15K has been determined from neutron powder diffraction measurements. The structure is orthorhombic, space group P2₁2₁2₁. The molecular geometry is found to be very similar to that in the gas phase, but the methyl group no longer has ideal 3-fold symmetry. The crystal is formed by infinite hydrogen-bonded chains of molecules with adjacent chains ‘pointing’ in opposite directions. The O-H … O hydrogen bonds are almost linear. No phase intermediate between the low temperature α-phase and the high temperature β-phase was found, but a new, metastable phase was discovered.     

Plastic domain and ionic conductivity of (C4H9)4NI

Unusual ionic transport in tetrabutylammonium iodide (C₄H₉)₄NI (TBAI) was observed by high resolution proton NMR spectroscopy accompanied by diffusion experiments. Some parts of the TBA cations are diffusing not only in the plastic phase II, but in the low temperature ordered phase I as well. The diffusing TBA ions are considered to be inside the remaining plastic domains dispersed in the ordered crystal matrix. The volume fraction of the plastic domain is evaluated from the observed narrow component of the proton NMR spectra. The observed unusual temperature dependence of the diffusion coefficient can be explained by assuming a manifold decrease of the dimension of the plastic domain in phase I. The differences in degree of crystallinity at the interface of plastic domains with respect to its surroundings led to the surface enhancement of diffusivity in phase I. Finally, the observed ionic conductivity is compared with the estimation from the diffusion coefficient and mobile cation densities. The results suggest that only a fraction of the cations take part in the observed ionic conductivity.     

High-pressure synthesis, structural and Raman studies of a two-dimensional polymer crystal of

Two-dimensional polymerisation of a C₆₀ single crystal has been obtained under high-pressure high temperature conditions (700 K - 2 GPa). Crystalline order is preserved but the crystal splits into variants (orientational domains). The analysis of X-ray diffraction and Raman spectroscopy data reveals that the polymer crystal is primarily tetragonal with some admixture of rhombohedral phase. Furthermore, Raman spectroscopy gives evidence for additional C₆₀-C₆₀ dimers, which are probably disordered. For the tetragonal phase, it is shown that successive polymer layers are rotated by about the stacking axis, according to the P4₂/mmc space group symmetry. The structure of the rhombohedral phase is also clarified. The role of the interlayer interactions in stabilising the two-dimensional polymer phases of C₆₀ is discussed. Received 8 October 1999     

SEM,SAED, and TEM investigations of domain structure in PZT ceramics at morphotropic phase boundary

SEM investigations of ferroelectric domain structure in PbZr_0.53Ti_0.47O₃ are consistent with a model of spatial domain configuration for the piezoelectric ceramics previously proposed for BaTiO₃. TEM and SAED results revealed not only the twinning relation of adjacent tetragonal 90° domains but also the simultaneous presence of the ferroelectric rhombohedral phase. A succession model of ferroelectric domains T₁RT₂RT₁... which needs a smaller energy for the rotation of the polarization vector due to the coexistence of a R domain between the two T 90° domains is proposed. This model is also confirmed by the estimated value of elastically stored energy in the mixed wall and by the dependence upon the sintering temperature of T and R unit cell distortions previously measured by x-ray diffraction.     

Tweed microstructures: Experimental observations and some theoretical models

Experimental observations, mainly by transmission electron microscopy, of tweed microstructures in minerals (K-feldspar and cordierite), high-T _c superconductors (Co-doped YBCO) and metals are reviewed. Structural models of the local structure within the tweed are compared and evaluated with respect to spectroscopic measurements in a number of systems. It is concluded that in tweed microstructures which result from symmetry changes driven by cation-ordering, the local degree of order is not itself modulated. A model based on twin-related domains of the low symmetry form is consistent with experimental observations. Mechanisms of coarsening of the tweed microstructure to form the lamellar twinning observed in the fully ordered structure are briefly reviewed.     

Structural phase transition in a perovskite-type NH3(CH2)3NH3CuCl4 crystal – X-ray and optical studies

X-ray powder studies and optical studies (polarized microscopic observation and linear birefringence studies) of the crystal NH₃(CH₂)₃NH₃CuCl₄ are presented. The X-ray powder studies revealed a change of symmetry from orthorhombic room-temperature phase to monoclinic phase above 434 K. A reversible phase transition of the first order at 434 K on heating and 432 K on cooling was observed in birefringence studies. Optical polarized microscopic observation revealed monodomain and multidomain states in the room-temperature orthorhombic phase with domain walls in (110) and (1-10) planes. The hypothetical prototypic phase is expected to be tetragonal. The change of symmetry from orthorhombic to monoclinic and expected domain structure was found above 434 K in the (010) plane.     

Permutation-inversion symmetry of C70 fullerite in the high-temperature phase

The permutation-inversion symmetry group of C₇₀ fullerite in its high-temperature phase is constructed with allowance for the rotation of its constituent molecules, and the local symmetry group of a rotating molecule in the crystal is identified. Irreducible representations of these groups are constructed that are compatible with the principle of wave-function symmetry with respect to permutations of identical nuclei. A group-theoretic classification is made of the quantum states of a rotating molecule and of the crystal in the high-temperature phase of C₇₀ fullerite. Selection rules are derived for electronic, vibrational, and rotational spectra in terms of irreducible representations of the permutation-inversion symmetry group of the crystal. Fiz. Tverd. Tela (St. Petersburg) 39, 1895–1901 (October 1997)     

Coupling of orientational and translational modes in solid C60 and C70

The orientational phase transitions in solid C₆₀ and C₇₀ are accompanied by quite different anomalies in the crystalline strains. In solid C₆₀ the phase transition Fm3m→Pa3 is primarily an orientational effect (antiferro-rotational), which is driven by the condensation of orientational modes belonging to X₅ ⁺ irreducible representation (irreps) of Fm3m. These modes are the primary order parameters (oops) and their number is equal to the number of irreps of T_2g and T_1g symmetry within the manifolds under consideration. Taking into account irreps up to the manifold 1=12, we have studied the rotation-rotation-translation (RRT) coupling between the oops and the lattice displacements. We have investigated the resulting lattice contraction and the change of the elastic constant c₁₁ at the phase transition. In solid C₇₀ (fcc-phase) we investigate the bilinear coupling of orientational fluctuations of T_2g symmetry to transverse acoustic lattice displacements. This coupling is the driving mechanism for the ferroelastic phase transition Fm3m → R3m. Finally we investigate the transition from the rhombohedral phase to a low temperature monoclinic phase. This transition in antiferro-rotational.     

Tensor distinction of domains in ferroic crystals

Ferroic crystals contain two or more domains and may be distinguished by the values of components of tensorial physical properties of the domains. We have extended Aizu’s global tensor distinction by magnetization, polarization, and strain of all domains which arise in a ferroic phase transition to include distinction by toroidal moment, and from phases invariant under time reversal to domains which arise in transitions from all magnetic and non-magnetic phases. For determining possible switching of domains, a domain pair tensor distinction is also considered for all pairs of domains which arise in each ferroic phase transition.