Si doped Cu0.5Tl0.5Ba2Ca2Cu3O10−δ superconductors

The Si doped (Cu_0.5Tl_0.5)Ba₂Ca₂Cu_3−ySi_yO_10−δ (y=0, 0.25, 0.5, 0.75, 1.0, 1.25) superconductor samples have been synthesized to investigate the effect of mobile carriers on the critical temperature and magnitude of diamagnetism of this family of cuprates. The Si doped samples have tetragonal structure as observed from the X-ray diffraction spectra. The c-axis length of the unit cell of (Cu_0.5Tl_0.5)Ba₂Ca₂Cu_3−ySi_yO_10−δ was increased after the increase of Si concentration, whereas the critical temperature and the magnitude of diamagnetism have been decreased. The decrease in T_c(0) and magnitude of diamagnetism is possibly due to the deficiency of carriers in CuO₂/SiO₂ planes caused by the +4 state of Si atoms. However, the post-annealing of these samples in oxygen atmosphere has increased the critical temperature and the magnitude of diamagnetism. The FTIR absorption measurements of (Cu_0.5Tl_0.5)Ba₂Ca₂Cu_3−ySi_yO_10−δ samples have shown a softening of the apical oxygen mode of the type Cu(1)O_ACu(2) and TlO_ACu(2) with increased concentration of Si in the unit cell; the softening of this mode is directly linked with the increase in the c-axis lattice parameter of the (Cu_0.5Tl_0.5)Ba₂Ca₂Cu_3−ySi_yO_10−δ superconductor.     

The zeta function of the quartic oscillator

We investigate the levels of the quartic oscillator ($\text{?}\text{d}{}^2\text{d}\text{q}{}^2\text{+ q}{}^4$) by means of the associated zeta function. By using several versions of the semiclassical approach, we prove exact results concerning the locations of the real zeros and of the poles of that function, the residues of which are essentially the coefficients of the semiclassical expansion of the Bohr-Sommerfeld quantization rule. We also compute the derivative of the zeta function at the origin, by relating it to the Fredholm determinant of the operator. These results translate as sum rules for the eigenvalues and thus are helpful for their precise computation, and a further analysis allows one to split them into separate rules for even and odd levels; the extension to higher anharmonic oscillators ($\text{?}\text{d}{}^2\text{d}\text{q}{}^2\text{+ q}{}^{\mathrm{2M}}$) is immediate. Finally we propose an asymptotic formula for large negative arguments of the zeta function, based on the nature and location of the complex singularities of the partition function of the operator $\text{(}\text{?}\text{d}{}^2\text{d}\text{q}{}^2\text{+ q}{}^4\text{)}{}^{\mathrm{<mtext>3</mtext><mtext>4</mtext>}}$.     

The gas phase infrared band contours of s-triazine and s-triazine-d3: The fundamentals of C3N3H3 and C3N3D3 and some overtone and combination bands of C3N3H3

The systematic application of band contour techniques to account for most of the observed features of the ir spectra of s-triazine and s-triazine-d₃ have been made as well as a critique as to the limitations of such methods. The experimental and computer methods used to study the gas phase infrared band contours of s-triazine and s-triazine-d₃ are out-lined. Contours of the five E′ fundamentals of s-triazine have been recorded under moderate resolution and analyzed to give the Coriolis constants $\text{ζ}{}_{\mathrm{i}}{}^{\mathrm{z}}$, i = 6–10. The effects of l-resonance are very apparent for ν₈ and ν₉, in the form of holes in the Q branches of these bands. Under the highest resolution available, ν₆ and ν₁₀ also show l-resonance effects. Values of the l-doubling constants $\text{q}{}_{\mathrm{i}}{}^{\mathrm{(+)}}$ were obtained for these four fundamentals. One of the parallel A₂″ fundamentals of C₃H₃N₃ (ν₁₂) has also been studied. It lies close to ν₁₀(E′) and an A × E type of second-order Coriolis resonance may be the cause of the intensity enhancement observed in the inner wings of the ν₁₂ and ν₁₀ bands. Hot bands of the type (ν_i + nν₁₄ ? nν₁₄) have been observed in the contours of ν₈, ν₁₀, and ν₁₂. This is felt to be responsible for the large difference between our observed zeta sum (?1.30) and the theoretical sum (?1.00).The gas phase infrared band contours of the five E′ and 2A₂″ fundamentals of C₃N₃D₃ have also been recorded under moderate resolution. From P-R separations and by computer simulation of the contours, values of the Coriolis constants $\text{ζ}{}_{\mathrm{i}}{}^{\mathrm{z}}$ have been obtained for the E′ modes. The effects of l-resonance have been observed for ν₈(E′) and ν₁₀(E′) and values of the l-doubling constants $\text{q}{}_{\mathrm{i}}{}^{\mathrm{(+)}}$ have been estimated. An extensive series of hot bands of the type (ν₁₂ + nν₁₄ ? nν₁₄) has been observed in the contour of the ν₁₂ (A₂″) fundamental. The mass effect on the Coriolis constants has been discussed.Infrared band contours of the overtone 2ν₇ and seven degenerate E′ combination bands of C₃N₃H₃ have been recorded under moderate resolution. Analysis of these contours using the P-R separation method and computer simulation of the contours has given values of $\text{ζ}{}_{\mathrm{<mtext>eff</mtext>}}{}^{\mathrm{z}}$ for these bands. Fermi resonance between 2ν₇ and ν₆ has been analyzed. The importance of considering both the observed contour as well as the observed frequency when assigning higher tone bands is illustrated.     

Nuclear incompressibility: From finite nuclei to nuclear matter

The recent increase of experimental data concerning the giant monopole resonance energy E_M gives information on the incompressibility modulus of nuclear matter, provided one can extrapolate the incompressibility of a nucleus K_A, defined by $\text{E}{}_{\mathrm{<mtext>M</mtext>}}\text{=[}\text{h}\text{?}{}^2\text{K}{}_{\mathrm{<mtext>A</mtext>}}\text{/m〈r}{}^2\text{〉]}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$, to the infinite medium. We discuss the theoretical interpretation of the coefficients of an $\text{A}{}^{\mathrm{<mtext>?1</mtext><mtext>3</mtext>}}$ expansion of K_A by studying the asymptotic behaviour of two RPA sum rules (corresponding to the scaling and the constrained model), evaluated using self-consistent Thomas-Fermi calculations. We show that the scaling model is the most suitable one as it leads to a rapidly converging $\text{A}{}^{\mathrm{<mtext>?1</mtext><mtext>3</mtext>}}$ expansion of the corresponding incompressibility K_A^s, whereas this is not the case with the constrained model. Some semi-empirical relations between the coefficients of the expansion of K_A^s are established, which reduce to one the number of free parameters in a best-fit analysis of the experimental data. This reduction is essential due to the still limited number and accuracy of experimental data. We then show the compatibility of the data given by the various experimental groups with this parametrization and obtain a value of K_n.m. = 220 ± 20 MeV, in good agreement with more microscopic analyses.     

Relaxation phenomena of binary polar liquid mixture in C6H6 from conductivity of solution measured at 10 GHz electric field

The dielectric relaxation times τ _jk ’s and dipole moments μ _jk ’s of the binary (jk) polar mixtures of N,N-dimethyl formamide (DMF) and pyridine dissolved in benzene (i) are estimated using linear slope of σ _jk ″ − σ _ijk ′ curves as well as ratio of slopes of σ _ijk ″ − w _jk and σ _ijk ″ − w _jk curves at w _jk → 0 for different mole fractions τ _jk ’s of pyridine and experimental temperatures under 9.875 GHz electric field. The measured μ _jk ’s and τ _jk ’s agree well with the reported values validating the proposed methods. The solute-solute (dimer) molecular associations are inferred from the plot of τ _jk − x _j and μ _jk − x _j curves for x _j = 0.0 to 0.5 of pyridine and thereafter solute-solvent (monomer) molecular association occurs. The theoretical μ_theo’s are calculated to predict solute-solute and solute-solvent molecular association. The μ _jk − T curves are drawn to show elongation of bond angles and bond moments with temperatures. The thermodynamic energy parameters are calculated from Eyring’s rate theory equations to know the molecular dynamics as well as to ascertain Debye-Smyth relaxation mechanism of the systems.     

Analysis of the spectra of In XII–XIV and Sn XIII–XV in the far-VUV region

The 4p ⁶4d?(4p ⁶5p + 4p ⁶4f + 4p ⁵4d ²) transitions in the spectra of In XIII-Sn XIV and the 4p ⁶4d ²?(4p ⁶4d5p + 4p ⁶4d4f + 4p ⁵4d ³) transitions in the spectra of In XII-Sn XIII are investigated. About 150 spectral lines in the wavelength range 120–160 Å, which is important for projection vacuum UV lithography, have been classified for the first time. The fine structure intervals of the ground state 4p ⁶4d ² D and the energies of eight levels of the 4p ⁶5p, 4p ⁶4f, and 4p ⁵4d ² configurations are measured in each spectrum of In XIII and Sn XIV. The energies of all levels of the ground configuration 4p ⁶4d ² and the energies of 35 levels of the interacting configurations (4p ⁶5p + 4p ⁶4f + 4p ⁵4d ³) are determined in the spectra of In XII and Sn XIII. The resonant transitions 4p ^{6 1} S ₀?4p ⁵4d ¹ P ₁ in the spectra of In XIV and Sn XV are also identified. The semiempirical parameters of the configurations under study determined by mean-square fitting of measured level energies are in agreement along both isoelectronic and isonuclear sequences of the spectra.     

Isotope dependent predissociation in the C 1 Σ + , v =0 and v =1 states of CO

Rotationally resolved spectral lines in the C - X (1, 0) band of carbon monoxide are investigated under high resolution using a coherent vacuum ultraviolet laser source, continuously tunable near 107 nm. Transition frequencies are determined by calibrating against a reference standard of iodine lines, recorded with saturation spectroscopy in the visible range, yielding an absolute accuracy of 0.003 cm^-1 in the vacuum ultraviolet. Improved molecular constants for the excited state are derived and no effects of perturbation are found at the present level of accuracy. Line broadening measurements result in information on the excited state lifetime of the C ¹ Σ ⁺ , v = 1 state for five natural isotopomers of carbon monoxide: τ( ¹² C ¹⁷ O ) = 280 ps, τ( ¹² C ¹⁸ O ) = 210 ps, τ( ¹³ C ¹⁶ O ) = 295 ps, τ( ¹³ C ¹⁷ O ) = 160 ps, and τ( ¹³ C ¹⁸ O ) = 150 ps. Within the accuracy of the present measurements no effects of J-dependent lifetimes were observed, for neither of the isotopomers. In addition direct time domain measurements of the lifetime of the C ¹ Σ ⁺ , v = 0 and v = 1 states of the main isotopomer are performed in a pump-probe experiment using a picosecond VUV-laser, yielding τ( ¹² C ¹⁶ O ) = 1780 ps for v = 0 and τ( ¹² C ¹⁶ O ) = 625 ps for v = 1. For C ¹ Σ ⁺ , v = 0 in ¹²C¹⁶O and ¹³C¹⁶O the same lifetime is found; this lifetime matches experimental values of the oscillator strength and hence supports previous results showing pure radiative decay in this state; the error margins however do not exclude some low level of predissociation. The measurements indicate that the C ¹ Σ ⁺ , v = 0 state of the ¹³C¹⁸O isotopomer is predissociated with an estimated yield of 17% (i.e. above the level of predissociation for ¹²C¹⁶O.) From the combined data predissociation yields upon excitation of the C ¹ Σ ⁺ , v = 1 state are derived, lying in the range 0.84-0.91 for the five less abundant isotopomers; for the main ¹²C¹⁶O isotopomer a strongly deviating predissociation yield of 0.65 is deduced. Received 21 December 2000 and Received in final form 26 March 2001     

Spontaneous CP violation in a Z4 horizontal symmetry model for flavor mixing

The implications of a Z₄ horizontal symmetry model of flavor mixing for CP violation are studied in the framework of minimal SU(2)_L × SU(2)_R × U(1)_{B – L} gauge theory. We show that CP violation in this model arises purely from right-handed currents. We also note that spontaneous breaking of CP symmetry requires a fine tuning of coupling parameters to the level of , which can be avoided by the inclusion of one additional singlet Higgs field, of the kind recently introduced for other purposes.     

The relationship between the transition voltage of the I–V curve of the ferroelectrics and the coercive field of the P–V hysteretic curve

The relationship between the transition voltage of the I–V curve of the ferroelectrics and the coercive field of the P–V hysteretic curve is calculated. The first mathematical analysis to explain the relation between the transition voltage V_t and the coercive voltage V_c is obtained. The origin of the interrelation between the transition voltage of the I–V curve and the coercive field is that the height of the boundary barrier is inversely proportional to the effective dielectric constant of the near-boundary region, which is dependent on a derivative of polarization on the electric field, ∂P/∂E. The term ξ(eV_t) plus the term (en_b²δ/dN_dP_s)(eV_c) equals a constant. V_t is the function of E_g, P_s, V_c, and E. There is a linear relation between V_c and V_t. This relationship will induce the matchable relations between the I–V curve and the E–P loop. As long as the V_c of the V–P loop exists, the correspondent V_t of I–V curve will certainly exist. It will be the foundation of a new ferroelectric memory, which operates by the I–V characteristics. These relations are the conditions that can enable nonvolatile memory and nondestructive readout.     

Trapped modes in a T-shaped waveguide

We study the discrete spectrum of the Laplace operator with the Dirichlet boundary conditions in a plane waveguide made of a band of unit width and half-band of width h perpendicular to it. The existence of critical width h _* ∈ (1,2] is established: in the case of h ∈ (0, h_*), the discrete spectrum σ _d ^h of the problem contains only one eigenvalur λ₁ ^h , but σ _d ^h = ⊘ at h ≥ h _*. The asymptotics of the eigenvalue λ₁ ^h at h → + 0 is found.     

Universal microstructure and conductivity relaxation of polymer-conductor composites across the percolation threshold

Micro-structural and impedance analysis of series of insulating polymer/conductor composites (PCC) as a function of frequency and volume fractions of the conductor (f_con) have been studied. Evidences of conductivity relaxation have been noticed with a correlation with the sample micro-structure. This has been understood and explained in terms of equivalent electrical circuit model of the material established through complex impedance spectroscopy (CIS) across the percolation threshold (f_c) for all the PCC. CIS analysis confirmed that PCC with f_con ≥ f_c, exhibit conductivity/interfacial relaxation due to polarization of Maxwell–Wagner–Sillars (MWS) type at f_c and the relaxation frequency increases with increase of f_con. The modulus spectroscopy analysis suggests the presence of two types of relaxations in different frequency ranges; (i) dipolar relaxation associated with the flipping of dipoles present in the pure polymer for f_con < f_c and (ii) the conductivity/interfacial relaxation due to the formation of artificial MWS dipoles at the interface of the two components. A long range dc conductivity appears at f_con ≥ f_c and Jonschers universal fractional power law is satisfied for both the regions of f_con < f_c and f_con ≥ f_c in all PCC.     

Production and decay of quark-gluon strings: inclusive distributions of hadrons in p⊥ and x

The dependence of inclusive cross sections of the production of hadrons $\text{p}\text{p}\text{→ hX}$ on p_⊥ (and also on x and √s) is calculated at high energy in the region of small p_⊥ ? 1–2 GeV. The model of production and decay of quark-gluon strings is used under the simplest assumptions about the k_⊥ dependence of the quark distributions in nucleons ～ exp(?γ₁k²_⊥) and about the form of the string fragmentation function $\text{G}\text{?}{}^{\mathrm{<mtext>h</mtext>}}\text{～}\text{exp}\text{[?γ}{}^{\mathrm{<mtext>h</mtext>}}\text{(}\text{p}{}_{\mathrm{\perp }}\text{? z}\text{k}{}_{\mathrm{\perp }}\text{)}{}^2\text{]}$ where γ₁ and γ^h are some constants. The theory reproduces all existing experimental data and yields the “seagull effect” for the dependence of 〈p_⊥〉 on x. Predictions are given for the p_⊥ dependence of the spectra of π^± mesons produced at high energies at SPS and other future colliders.     

Formation mechanism of the Si(1 1 1)7×7 reconstruction studied by scanning tunneling microscopy: Zipper-like restructuring in the sequential size changes of isolated single faulted-halves

The formation mechanism of the 7×7 reconstruction on annealed Si(1 1 1) surfaces has been demonstrated at the atomic level. In situ observations of unreconstructed regions (‘1×1’) on terraces after rapid quenching to 380 °C were done using scanning tunneling microscopy (STM) with a scanning speed of 1.7 s per frame. In the narrow ‘1×1’ regions, we imaged isolated single-faulted (F) halves of the dimer-adatom-stacking-fault (DAS) structure from “birth” to “death”. During “life”, the isolated single F-halves frequently changed their size. The size changes between odd-sized F-halves always took place through even-sized F-halves of intermediate sizes: 5×5-F^′↔6×6-F↔7×7-F^′↔8×8-F↔9×9-F^′↔10×10-F↔11×11-F^′↔12×12-F↔13×13-F^′, where the 5×5-F^′, 7×7-F^′ and so forth are irregular-type structures of the odd-sized F-halves. Even-sized F-halves and the irregular-type structures are necessary in the size changes, whereas the regular-type structures have never been involved. Lifetimes of the 10×10-F, 8×8-F, and 6×6-F at 380 °C are about 10.5, 6, and 2-3 s, respectively, which are much shorter than those of the isolated irregular-type structures of the odd-sized F-halves. With the aid of room temperature STM images of a rapidly quenched surface, we determined the atomic structures of the even-sized F-halves. We have proposed a sequential size change (SSC) model, including undiscovered parts of the size changes ‘1×1’ ↔2×2-F↔3×3-F^′↔4×4-F↔5×5-F^′, as the formation and decay mechanism of isolated single F-halves in the ‘1×1’ region. The SSC model has the following sequence: ‘1×1’ ↔ 2×2-F↔3×3-F^′↔4×4-F↔5×5-F^′↔6×6-F↔7×7-F^′↔8×8-F↔9×9-F^′↔10×10-F↔11×11-F^′↔12×12-F↔13×13-F^′. It was found by collecting statistics of size-change directions that one of two equivalent sides of the irregular-type structures, which have a mirror symmetry, is involved in the size changes thus indicating that other parts of the F-halves remain unchanged. Based on such findings, we have proposed the atomic processes for bond-rearrangements in the SSC model. The bond-rearrangements proceed along one side of a triangular F-half by breaking the existing dimers and forming new dimers like a “zipper”. Proposed atomic processes of the zipper-like restructuring are illustrated by a ball-and-stick model. The reason for the appearance of the even-sized F-halves and the irregular-type structures of the odd-sized ones is discussed in terms of the energy barrier heights along a reaction path in the size change of single F-halves.     

Correlation analysis of the shifts of the electronic spectra of anthracene, phenanthrene, and fluorene as functions of the temperature in the gas phase and of the refractive index in n-paraffin solvents

The temperature-induced shifts of the maxima of the longest-wavelength bands of the S ₁ ← S ₀ and S ₂ ← S ₀ transitions in phenanthrene and the S ₁ ← S ₀ transition in fluorene in the gas phase are measured. A correlation analysis of these data and similar data on the S ₁ ← S ₀ and S ₁ → S ₀ transitions in anthracene is fulfilled. The positions of the maxima of analogous bands of the S ₁ ← S ₀ transition in anthracene, S ₁ ← S ₀ and S ₂ ← S ₀ transitions in phenanthrene, and S ₁ ←S ₀ transition in fluorene dissolved in liquid n-paraffins are also measured and analyzed as functions of (n ² ? 1)/(n ² + 2), where n is the refractive index of the solvent. The results confirm the conclusion of the Mirumyantz-Demchuk theory about the linear temperature dependence of the wave number of a band maximum but contradict the prediction that, upon extrapolation to T = 0 K, the positions of the maxima correspond to the electronic transitions in the molecule unperturbed by out-of-plane vibrations of the aromatic core.     

Resonant two-photon ionization and mass-analyzed threshold ionization spectroscopy of the selected rotamers of m-methoxyaniline and o-methoxyaniline

We report the resonant two-photon ionization and mass-analyzed threshold ionization (MATI) spectra of m-methoxyaniline and o-methoxyaniline. The vibronic features of m-methoxyaniline are built on 34308 ± 2 and 34495 ± 2 cm⁻¹ corresponding to the origins of the S₁ ← S₀ electronic transition (E₁’s) of the cis and trans rotamers. Analysis of the MATI spectra gives the adiabatic ionization energies (IEs) of 59983 ± 5 and 60879 ± 5 cm⁻¹ for these two species. o-Methoxyaniline is found to have only one stable structure whose E₁ and IE are 33875 ± 2 and 58678 ± 5 cm⁻¹, respectively. Most of the active vibrations of m- and o-methoxyaniline in the electronically excited S₁ and cationic ground D₀ states result from the in-plane ring vibrations. Comparing these data with those of p-methoxyaniline allows us to learn about the vicinal substitution effects resulting from the relative locations of the NH₂ and OCH₃ substituents.     

Generic Diffeomorphisms with Superexponential Growth of Number of Periodic Orbits

Let M be a smooth compact manifold of dimension at least 2 and Diff ^r (M) be the space of C ^r smooth diffeomorphisms of M. Associate to each diffeomorphism f;isin; Diff ^r (M) the sequence P _n (f) of the number of isolated periodic points for f of period n. In this paper we exhibit an open set N in the space of diffeomorphisms Diff ^r (M) such for a Baire generic diffeomorphism f∈N the number of periodic points P _n f grows with a period n faster than any following sequence of numbers {a _n } _{n ∈ Z +} along a subsequence, i.e. P _n (f)>a _ni for some n _i →∞ with i→∞. In the cases of surface diffeomorphisms, i.e. dim M≡2, an open set N with a supergrowth of the number of periodic points is a Newhouse domain. A proof of the man result is based on the Gontchenko–Shilnikov–Turaev Theorem [GST]. A complete proof of that theorem is also presented. Received: 27 January 1999 / Accepted: 23 November 1999     

Effect of oxygen partial pressure on the structural and optical properties of dc reactive magnetron sputtered molybdenum oxide films

Molybdenum oxide films (MoO₃) were deposited on glass and crystalline silicon substrates by sputtering of molybdenum target under various oxygen partial pressures in the range 8 × 10⁻⁵–8 × 10⁻⁴ mbar and at a fixed substrate temperature of 473 K employing dc magnetron sputtering technique. The influence of oxygen partial pressure on the composition stoichiometry, chemical binding configuration, crystallographic structure and electrical and optical properties was systematically studied. X-ray photoelectron spectra of the films formed at 8 × 10⁻⁵ mbar showed the presence of Mo⁶⁺ and Mo⁵⁺ oxidation states of MoO₃ and MoO_3−x. The films deposited at oxygen partial pressure of 2 × 10⁻⁴ mbar showed Mo⁶⁺ oxidation state indicating the films were nearly stoichiometric. It was also confirmed by the Fourier transform infrared spectroscopic studies. X-ray diffraction studies revealed that the films formed at oxygen partial pressure of 2 × 10⁻⁴ mbar showed the presence of (0 k 0) reflections indicated the layered structure of α-phase MoO₃. The electrical conductivity of the films decreased from 3.6 × 10⁻⁵ to 1.6 × 10⁻⁶ Ω⁻¹ cm⁻¹, the optical band gap of the films increased from 2.93 to 3.26 eV and the refractive index increased from 2.02 to 2.13 with the increase of oxygen partial pressure from 8 × 10⁻⁵ to 8 × 10⁻⁴ mbar, respectively.     

Current neutralization and plasma polarization for intense ion beams propagating through magnetized background plasmas in a two-dimensional slab approximation

A two-dimensional electromagnetic Particle-in-Cell （PIC） simulation model is proposed to study the propagation of intense, ion beams with beaIn width wb small compared to the electron skin depth c/wpe through background plaslnas in tile presence of external applied magnetic fields. The effcctive eleetron gyrora,dius wge is found to be an important parameter for ion beam transport in the presence of magnetic fields,In the bealn regions, the background plasmas respond differently to the ion beanl of width wb〈wge and wb 〉 wge for the given magnetic field and beaan encrgy, For the case of beam width wb 〈 wge with relative weak external magnetic fiehts, the rotalion effects of plasma electrons a,re found to be signifieant and contributes to the signitica,nt enilaneeinent of the self electric and seif-magnetic fields. While for the ease of beam width wb 〉 wge with relative strong external magnetic fields, the rotation effects of plasma cleetrons are strongly inhibited and a, well neutraliza, tion of ion beam current can be found. Finally, the influences of diftiuent beam widths. beam energies and magnetic fields on the neutralization of ion bc, anl eurrellt are summarized for the eases of wb 〈 wge〈 c/wpe,wge 〈wb〈c/wbp and wb〈c/wpe〈wge.