Sixth-order Term Research Articles

The effect of Al/Si disorder on the I $$\bar 1$$ -P $$\bar 1$$ Co-Elastic phase transition in Ca-rich plagioclase

The influence of Al/Si disorder in the anorthite tetrahedral framework upon the I $$\bar 1$$ -P $$\bar 1$$ displacive transition of that framework has been investigted at high-temperature by powder X-ray diffraction. The temperature-dependence of the order parameter in a heat-treated (disordered) anorthite and a Ca-rich plagioclase has been determined from spontaneous strain measurements. Both samples show appreciable disorder, with Q od = 0.88 in both cases. In each, the critical exponent β appears to be intermediate between values for classical tricritical (1/4) and second-order (1/2) phase transitions. This critical behaviour is consistent with a Landau potential in which the coefficient of the quartic term is positive but smaller than the coefficient of the sixth order term, corresponding to a second-order phase transition close to a tricritical point. There does not appear to be any defect strain tail near T c and inhomogeneities in Q od appear to be on rather a short length scale in these samples. The role of changing Q od appears to be more important than that of changing composition (albite component). The data are interpreted using a model of a homogeneous field due to changing Q od which renormalizes the transition temperature, T c * , and the fourth order coefficient, B eff, in the Landau expansion. The results are consistent with classical Landau behaviour, and demonstrate the care which must be taken in interpreting apparently non-classical critical exponents for phase transitions close to a tricritical point.

Phenomenological analysis of tetragonal tungsten bronze ferroelectrics

A simple Devonshire form has been derived for the phenomenological elastic Gibbs function to describe the elasto-dielectric parameters of simple proper ferroelectrics in the tungsten bronze structure family which has 4/mmm prototypic point symmetry. For the assumption that all temperature dependence is carried by the Curie-Weiss behaviour implicit in the quadratic term and that the expansion may be terminated at the first sixth-order term, reasonable agreement between calculated and derived Ps against T curves in the ferroelectric phase can be obtained for a wide range of bronze compositions. From the fitting it is clear that second and sixth rank terms are remarkably constant over a very wide range of bronze compositions. Variation in the negative fourth rank term is larger, but this is to be expected since it contains large contributions from electrostrictive and elastic terms which will depend on boundary conditions. These initial studies suggest that the phenomenological method may be used to derive expectation values for tensor parameters across the whole family of ferroelectric bronzes. The study also points up the need for more careful detailed studies of lattice strain, birefringence and permittivity as a function of temperature in model bronze compounds to provide more detailed checks of the method.

222K structural phase transition of hexagonal BaTiO3under uniaxial pressure

Abstract We present a model calculation of phenomenological theory for the structural phase transition at Ta=222 K of hexagonal BaTiO3 under uniaxial pressure. It is shown that the mm2 phase below Ta becomes unstable against formation of the 222 phase with decreasing temperature and a discontinuous phase transition from mm2 to 222 occurs at some temperature under the pressure along [100] axis because of the contribution of the sixth order terms of the soft mode amplitude.

Application of an extended Porod law to the study of the ionic aggregates in telechelic ionomers

Porod's law extended to the sixth-order term can be written I = (Kp/s4 + K6/s6)U2(s,σ) where I is the scattered intensity, s = 2(sinθ)/λ, θ being half the scattering angle and λ the wavelength used; U2(s, σ) describes the interphase profile and σ is a measure of the width of the interphase transition zone. Kp and K6 are two constants. In the same way as Kp can be related to the specific area, K6 is related to a pure number θ called here `angulosity'. For an angulous body, θ always is negative and can easily be calculated when its geometry is simple. It does not depend on the dimensions of the body. It is shown in the present paper that K6/Kp = θ/2π2S so that, in a two-phase system, the ratio K6/Kp represents the angulosity per unit area S of the interface between the phases. A least-squares analysis of the experimental small-angle X-ray scattering (SAXS) curve gives the values of Kp, K6 and σ. The method was successfully applied in the case of telechelic ionomers to characterize their ionic aggregates. These aggregates present a larger angulosity than that of a parallelepiped. Their volume is relatively small and only contains a small number of ions. The results agree with the results obtained by other techniques. It can be concluded from this that the introduction of the s−6 term into Porod's law is judicious and allows the structure of the phases to be better characterized.

Linked-cluster expansion for the single-band Hubbard model

Using the linked-cluster expansion, we have obtained the free energy, uniform susceptibility, and staggered susceptibility of the s-band Hubbard model to the sixth order in terms of βt, where t is the hopping integral to the nearest neighbors and β is the inverse of temperature. We show that a special technique based on the multiple-site Wick reduction theorem is more convenient than previous perturbation calculations. Our linked-cluster expansion result is for an arbitrary value of the on-site repulsion U. It has made corrections of previous results of several papers (up to fourth order) and achieved the sixth-order term.

The Phase Diagram of La2-xBaxCuO4

The temperature-concentration phase diagram of a high- T c superconductor La 2- x Ba x CuO 4 system is reproduced satisfactorily with the use of a Landau-type thermodynamic potential, where a sixth-order term in the transition parameters is taken into account.

The isospin independent spin-orbit interaction in the Skyrme model

It is shown that in order to obtain the correct sign for the isospin independent component of the nucleon-nucleon spin-orbit interaction in the Skyrme model, a term of sixth order in the derivatives has to be included in the lagrangian density. Such a term appears in a natural way in the version of the model, in which the anomalous baryon current is coupled to an omega meson field. With inclusion of a sixth order term the net isospin independent spin-orbit interaction is somewhat weaker than that of realistic potential models. If in addition scalar mesons are introduced as dilatons in the Skyrme model lagrangian to ensure correct QCD scaling behaviour, its strength will, however, increase substantially.

Resolution of the Q branch in the ν3 fundamental of PF5

The ν 3 fundamental of phosphorus pentafluoride, PF 5 , has been investigated with a laser sideband spectrometer. A line width of 2 × 10 −5 cm −1 was observed with the saturation technique, which was sufficient to resolve the Q -branch K structure. Four hundred sixty-seven transition frequencies up to J ″ = 76 have been measured with an accuracy of ±200 kHz. The spectrum was fitted with a symmetric rotor program including three terms of sixth order. As a result, 11 parameters have been determined, yielding an overall standard deviation of 207 kHz. With the present resolution of 600 kHz no indication of splitting due to Berry rotation was found.

Hyperon and nucleon structure in an SU(3) skyrme model

By incomparating the vector-meson-motivated sixth-order derivative term, the masses of hyperons can be described well with a reasonable value of Fπ in the framework of the exact diagonalization method of Yabu and Ando. The strangeness content of the proton is estimated to be about 18%. The πN sigma term is calculated to be 61 MeV and insensitive to the strangeness content of the proton.

Generalization of the Skyrme model for the unified theory of pions and nucleons.

Skyrme's Lagrangian is generalized within the pion field alone to include all possible terms which appear in the chiral perturbation theory up to fourth order in the field derivative and the symmetry-breaking mass. The parameters entering in the Lagrangian are fixed from the low-energy pion properties. Adding a sixth-order term to the Lagrangian for stabilization the hedgehog soliton is quantized semiclassically. Static properties of the soliton reproduce those of the nucleon with fairly good accuracy.

Nature of the smectic-A2-smectic-C2 transition: A calorimetric study.

The first high-resolution specific-heat measurement on octylphenyl-2-chloro-4-(p-cyano benzoyloxy) benzoate (${\mathrm{DB}}_{8}$ClCN) is reported. At 390.2 K, the material undergoes a second-order phase transition from a bilayer smectic-A phase (Sm-${A}_{2}$) to a bilayer smectic-C phase (Sm-${C}_{2}$). On cooling, the specific heat of ${\mathrm{DB}}_{8}$ClCN is independent of temperature in the Sm-${A}_{2}$ phase and increases at ${T}_{c}$ in a steplike fashion. In the Sm-${C}_{2}$ phase, the specific heat again becomes temperature independent. This is in striking contrast to previously reported Sm-A--Sm-C transition data, all of which conform to a Landau model with an unusually large sixth-order term.

A study of exchange and crystalline electric field interactions in Nd 2Co 14B: Comparison with Nd 2Fe 14B

The intrinsic magnetic behaviour of Nd 2Co 14B is interpreted in terms of exchange and crystal field interactions. The ‘spin-tilt’ transition away from the c axis at 37 K cannot be explained simply by a reduction in the exchange field acting on the Nd 3+ ion relative to isomorphous Nd 2Fe 14B; a significant alteration of the crystal field is required. In particular, we find evidence for an important reduction in the strength of the fourth and sixth-order crystal field terms in Nd 2Co 14B relative to Nd 2Fe 14B. A correlation between the low temperature ‘spin-tilt’ transition and occupancy of the 4c transition metal site through the Nd 2(Fe 1− x Co x ) 14B series suggests that this site in the Nd-B planes may be responsible for the changes in the Nd 3+ crystal field.

Characteristic functions and the aberrations of symmetric optical systems II Addition of aberrations

We describe a method for the calculation of the eikonal coefficients of a symmetric optical system composed of two symmetric subsystems. We consider coefficients of second to tenth order in the direction cosines. The system eikonal is seen to consist of an additive part and correction terms of sixth and higher order. We discuss the interpretation and the magnitude of the correction terms.

The E2/M1 transition ratio for γN → Δ(1232) in a modified Skyrme model

The electric quadrupole to magnetic dipole transition ratio E2/M1 for the reaction γN → Δ(1232) is calculated in a modified Skyrme model, which includes stabilizing fourth- and sixth-order terms. The coefficients of these terms are determined from vector meson couplings in the static limit; the other parameters are given by their physical values. We obtain for E2/M1 a negative sign and a magnitude which is consistent with the empirical value.

Attraction in the Skyrme model NN central interaction

An extended version of the Skyrme model lagrangian is considered, through inclusion of the symmetric fourth-order term plus a sixth-order term. We show that the Δ-polarization mechanism is able, with this lagrangian, to yield attraction in the central isoscalar nucleon-nucleon (NN) potential, over a range of parameters compatible with ππ scattering data. Also the other components of the NN interaction are consistent with the one-boson-exchange phenomenology.

Rho mesons in the skyrme model: An alternative approach

I consider an alternative to a recently proposed model by Adkins coupling the rho meson to the Skyrme soliton. Instead of a quadratic pion term a sixth-order term representing the exchange of very massive omega mesons stabilizes the soliton. All parameters in the lagrangian can be fixed by mesonic data and the static baryon properties are therefore predicted parameter-free. These predictions are a few percent better than the corresponding predictions of the massive pion Skyrme model.

Rotational spectra and structures of the Ar2–H/DF trimers

The feasibility of studying weakly bound, neutral van der Waals clusters through their microwave rotational spectra has been shown by identifying and characterizing Ar2–H/DF. Detection of the trimeric complexes was accomplished with the Flygare Mark II Fourier transform spectrometer, modified to better utilize its inherent sensitivity. Twenty-six b-dipole transitions (all ee↔oo) were observed in the 3–18 GHz region for Ar2–HF, and 19 for Ar2–DF, and their hyperfine structure analyzed. Fitting of the line centers with Kirchhoff’s NBS program gave values (all in MHz) for Watson’s determinable parameters of 3576.508(1), 1739.139(1), and 1161.054(1) for the rotational constants A″, B″, and C″ in Ar2–HF, and of 3506.791(1), 1744.056(1), and 1155.636(1) in Ar2–DF. The centrifugal distortion constants τ1, τ2, τaaaa, τbbbb, and τcccc were found to be −0.2039(3), −0.0522(1), −0.5353(5), −0.1159(1), and −0.02021(1) in Ar2–HF, and −0.2116(7), −0.0528(2), −0.4941(7), −0.1123(1), and −0.0191(1) in Ar2–DF. Three small sixth order terms were necessary to fit the data within experimental error. The trimer is found to be a planar T-shaped, highly asymmetric top (κ=−0.521) with the equilibrium position of the H/DF on the C2v figure axis (the b axis), the H/D end pointing toward the two argon atoms. The structure can be viewed as a composite of three dimers, the Ar2 dimer and two Ar–H/DF’s. In the trimer the Ar–Ar and the two Ar–H/DF distances are a bit longer (0.04 and 0.01 Å) than in the dimers, suggesting the presence of second-order effects. This picture is supported by an approximate, pseudotriatomic analysis of the force field, based on the four independent planar centrifugal distortion constants. The cluster is quite ‘‘floppy.’’ The hyperfine structure shows that torsional oscillations of the H/DF are comparable with those in the Ar–H/DF dimers, but anisotropic, being 3° larger in-plane than out-of-plane. The in-plane torsional oscillation of the Ar2 is considered.

Absence of attraction in the NN central potential derived from skyrmions

The nucleon-nucleon potential is studied using a skyrmion lagrangian containing terms of second, fourth, and sixth order in the field derivatives; the fourth-order pieces contain both an antisymmetric (repulsive) form and a symmetric (attractive) one. No attraction is found in the central NN potential V C when the full structure of the sixth-order term is retained. As yet another candidate for the source of attraction in V C , admixtures of ΔΔ components in the two-baryon wave function are considered. These slightly lower the values of V C for 1 ⩽ R ⩽ 2 fm, but by no means sufficiently to bring about a net attraction there. Thus the challenge of finding sources of attraction in the skyrmion approach remains.

The structure of the skyrmion-skyrmion interaction

The structure of the skyrmion-skyrmion interaction is investigated. Using a zero-range boson-exchange interpretation, Skyrme models can be related to models including explicit boson stabilization. The structure of the potential can be understood in terms of conventional boson-exchange pictures and corrections to the conventional Skyrme model are made to estimate mesonic effects known to be important in the nucleon-nucleon interaction. The effects of soft-pion fluctuations and a sixth order term on the skyrmion-skyrmion interaction are assessed and interpreted in terms of ω and σ mesons. The modifications to the Skyrme model produce interactions in qualitative agreement with phenomenological models of the nucleon-nucleon interaction.

Phase diagrams of cubic systems under uniaxial quadratic symmetry-breaking fields.

The effect of uniaxial quadratic symmetry-breaking fields on two- and three-component systems which possess cubic symmetry is studied. A Landau free-energy expansion which includes sixth-order terms in the order parameter is constructed and analyzed in the context of mean-field theory. A remarkably rich spectrum of complex phase diagrams featuring several ordered phases is predicted. The phase diagrams exhibit a variety of multicritical points including triple, bicritical, tetracritical, tricritical, and critical-end points. We also predict the existence of two consecutive tricritical points along phase boundaries separating ordered phases. Some of these phase diagrams may be realized in certain compounds corresponding to two- and three-component cubic systems such as ${\mathrm{Tb}}_{2}$(${\mathrm{MoO}}_{4}$${)}_{3}$ and ${\mathrm{BaTiO}}_{3}$.