Effect of the thickness of Bi–Te compound and Cu electrode on the resultant Seebeck coefficient in touching Cu/Bi–Te/Cu composites

The resultant Seebeck coefficient α of the touching p- and n-type Cu/Bi–Te/Cu composites with different thicknesses of t _Bi–Te and t _Cu was measured as a function of t, where t _Bi–Te was varied from 0.1 to 2.0 mm, t _Cu from 0.3 to 4.0 mm and t is the lapse time after imposing the voltage. The temperature difference ΔT is produced by imposing a constant voltage of 1.70 V on two Peltier modules connected in series. The resultant α of composites was calculated from the relation α = ΔV/ΔT, where ΔV and ΔT were measured with two probes placed on both end coppers. ΔV decreases abruptly with an increase of t below t = 5 min, while above t = 7 min, it tends to saturate to a constant value. The resultant α and saturated ΔV vary significantly with changes in t _Cu and t _Bi–Te. When a composite has a combination of t _Cu = 1.0 mm and t _Bi–Te=0.1 mm, the generating powers ΔW (=(ΔV)²/4R) estimated using the saturated ΔV and calculated electrical resistance R for the p- and n-type composites have great local maximum values which are 4–5 times as large as those obtained for the conventional combination of t _Bi-Te = 2.0 mm and t _Cu = 0.3 mm. It is surprising that the generating power ΔW is enhanced significantly by sandwiching a very thin Bi–Te material between two thick coppers, unlike the conventional composition of thermoelectric modules. On the other hand, when a composite has a combination of t _Bi–Te = 0.1 mm and t _Cu = 0.3 mm, the resultant α of the p- and n-type composites exhibited great values of 711 and −755 μV/K, respectively, so that the maximum resultant ZT of the p- and n-type composites reached extremely large values of 8.81 and 5.99 at 298 K. However, the resultant ZT decreases rapidly with an increase of t _Cu or t _Bi–Te. The resultant ZT is thus found to be enhanced significantly not only in superlattice systems but also in macroscopic composites. The present enhancement in ZT is attributed to the large barrier thermo-emf generated in the Bi–Te region shallower than 50 μm from the boundary.     

Intrinsic and extrinsic mechanisms of vortex formation in superfluid3He-B

We report on the first comprehensive measurements of critical superflow velocities in³He-B which allow different mechanisms of vortex formation to be identified. As a function of temperatureT and pressureP, we measure the critical angular velocity Ω_c(T, P) at which vortices start to form in slowly accelerating rotation in a cylindrical container filled with³He-B. Owing to the long coherence length ξ(T, P)∼10–100 nm, either trapped remanent vorticity or intrinsic nucleation may dominate vortex formation, depending on the roughness of the container wall and the presence of loaded traps. NMR measurement with a resolution of one single vortex line allows us to distinguish between different processes: (1) Three extrinsic mechanisms of vortex formation have been observed. One of them is the vortex mill, a continuous periodic source which is activated in a rough-walled container well below the limit for intrinsic nucleation. (2) In a closed smooth-walled container intrinsic nucleation is the only mechanism available, with a critical velocity v_c(T, P)=Ω_c(T, P), whereR is the radius of the container. We findv _c (T, P) to be related to the calculated intrinsic stability limitv _ch (T, P) of homogeneous superflow. The existence of this connection in the form of a scaling law implies that nucleation takes place at an instability, rather than by thermal activation or quantum tunneling which become impossible because of an inaccessibly high energy barrier.     

The Dynamics of Vortex Generation in Superfluid 3He-B

A profound change occurs in the stability of quantized vortices in externally applied flow of superfluid ³He-B at temperatures ?0.6?T _c, owing to the rapidly decreasing damping in vortex motion with decreasing temperature. At low damping an evolving vortex may become unstable and generate a new independent vortex loop. This single-vortex instability is the generic precursor of turbulence. We investigate the instability with non-invasive NMR measurements on a rotating cylindrical sample in the intermediate temperature regime (0.3–0.6)?T _c. From comparisons with numerical calculations we interpret that the instability occurs at the container wall, when the vortex end moves along the wall in applied flow.     

Power generation of the touching Cu/Bi-Te/Cu composites under the periodically alternating temperature gradients

The thermo-emf ΔV of the touching p- and n-type Cu/Bi-Te/Cu composites with different thicknesses of t _Bi-Te and t _Cu was measured as a function of time by alternating the temperature difference ΔT at periods of T = 20, 60, 120, 240 and ∞ sec, where t _Bi-Te was varied from 0.1 to 2.0 mm and t _Cu from 0 to 4.0 mm. As a result, ΔV changes significantly with t _Bi-Te, t _Cu and T. The effective thermo-emf ΔV _eff increases significantly with an increase of 1/T and exhibited a local maximum at 1/T = 1/240 s⁻¹. The resultant | α | and the effective temperature difference ΔT _eff were increased significantly by optimizing t _Bi-Te and t _Cu at 1/T = 1/240 s⁻¹. The power generation ΔW _eff (= ΔV _eff²/4R _calc) estimated using the measured ΔV _eff and calculated R _calc also exhibited a local maximum at 1/240 s⁻¹ for an optimum combination of t _Bi-Te = 0.1 mm and t _Cu = 2.0 mm, so that the maxima ΔW _eff at 1/T = 1/240 s⁻¹ for the p- and n-type composites were 2.28 and 2.92 times higher than those obtained at 1/T = 0 s⁻¹. This significant increase in ΔW _eff is owing to both the increase in ΔT _eff and the increase in ZT due to the increase in |α|. The power generation was thus found to be enhanced significantly by imposing the alternating temperature gradients on touching Cu/Bi-Te/Cu composites.     

Enhancement of the generating power in Cu/Bi–Te/Cu composite thermoelectric devices

The voltage ΔV and electric current ΔI of the p- and n-type Cu/Bi–Te/Cu composite thermoelectric devices were measured as a function of ΔT for four regions of the intrinsic Bi–Te compound, Cu/Bi–Te and Bi–Te/Cu interfaces and Cu/Bi–Te/Cu composite using thermocouples set at intervals of s = 2 and 6 mm, where the lengths of Bi–Te compound and copper are 4 and 5 mm, respectively. ΔV and ΔI of all regions tended to increase linearly with an increase of ΔT. The resultant α was obtained from the relation ΔV/ΔT. The resultant α values of regions including the interface are much higher in absolute value than those of the intrinsic Bi–Te compounds, so that the barrier thermo-emf is found to occur in the forward-bias direction. It indicates that the barrier thermo-emf appears even in the semiconductor-metal junction, as in the case of the p–n junctions. The resultant α of Cu(T _H)/Bi–Te interface rich in the heat flow increases with an increase of ΔT, while that of Bi–Te/Cu(T _C ) interface poor in the heat flow decreases with an increase of ΔT. The ΔT-dependence of α of the interfaces is entirely opposite at the hot and cold sides. As a result, the resultant α of the p- and n-type Cu/Bi–Te/Cu composites remained little varied with changes of ΔT, so that the present composites have a thermal stability superior to the intrinsic Bi–Te compounds.The generating powers ΔW _Bi-Te and ΔW _Cu/Bi-Te/Cu for the p- and n-type intrinsic Bi–Te compounds and Cu/Bi–Te/Cu composites increased parabolalically with an increase of ΔT, and the ratios of ΔW _{Cu/Bi–Te/Cu} to ΔW _Bi–Te reached great values of 1.41 and 1.45 for the p- and n-type composites, respectively. It was thus found that the enhancement in the resultant α of the composite materials results in a significant improvement in the conversion efficiency for generators.     

Correlation between changes in Curie temperature and electrical resistivity during structural relaxation in Fe27Ni53P14B6 metallic glass

Changes in Curie temperature (ΔT _c, electrical resistivity (Δϱ/ϱ) and volume (ΔV/V) caused by isochronal annealing in as-quenched and pre-annealed Fe₂₇Ni₅₃P₁₄B₆ metallic glasses were compared. It was found that the ΔT _c against annealing temperature (T _a) curves in as-quenched and pre-annealed samples are very similar to the Δϱ/ϱ against T _a curves. Particularly, almost a linear relationship was observed between ΔT _c and Δϱ/ϱ in the pre-annealed sample. The results strongly suggest that the origins of both ΔT _c and Δϱ/ϱ during structural relaxation in the pre-annealed sample are attributed to identical, reversible, short-range ordering.     

New equations for enthalpy of vaporization from the triple point to the critical point

Two new equations are proposed for the enthalpy of vaporization from the triple point to the critical point. One of these equations containing four parameters is exceptionally good for fitting the data. The other equation containing three parameters is quite adequate for fitting the data but it is exceptionally suited for interpolation when the data do not cover the entire range. These equations have been tested using the enthalpy of vaporization of water from the triple point to the critical point and are compared with other equations.Nomenclature T _c Critical temperature, K - T _t Triple point, K - T _x Any particular temperature, K - T _r Reduced temperature - P _r Reduced pressure - R Gas constant - P Vapor pressure - X (T _c–T)/T _c - Y (T _c–T)/T - X _x (T _c–T)/(T _c–T _x) - X _t (T _c–T)/(T _c–T _t) - H _vt Enthalpy of vaporization at the triple point, kJ · mol^–1 - H _vx Enthalpy of vaporization at any temperature x, kJ · mol^–1 - Z _v Compressibilty factor of the saturated vapor - Z ₁ Compressibilty factor of the saturated liquid Relative deviation = 100[H_v(obs)–H_v(cal)]/H_v(obsd) Standard deviation = { [H _v(obs)–H _v(cal)]²/(No. points — No. parameters)}^0.5     

Coexistence of Itinerant Ferromagnetism and Superconductivity in ZrZn<Subscript>2</Subscript>

A microscopic coexistence of itinerant ferromagnetism and superconductivity is studied in a single- band homogeneous system, using an equation of motion method and Green’s function technique. Self-consistent equations for the superconducting order parameter Δ and magnetization m are derived. It is shown that there generally exists coexistent (Δ≠0 and m≠0) solutions to the coupled equations of the order parameters in the temperature range 0<T<min (T _c ,T _FM ), where T _c and T _FM are respectively the superconducting and ferromagnetic transition temperatures. The expressions for specific heat, density of states, free energy, and critical field are also derived. The theory is applied to explain the observations in ZrZn₂. The agreement between the theory and experimental observations is quite satisfactory.      

Electrical and photoelectric properties of GeS layered crystals grown by different techniques

Thermally stimulated currents and anisotropic electrical conductivity are studied in GeS layered crystals prepared by the Bridgman-Stockbarger, Pizzarello, and sublimation techniques. All the crystals arep-type, regardless of the growth technique, owing to the presence of Ge vacancies. The conductivity anisotropy in the melt-grown crystals is high compared to the vapor-grown GeS. The anisotropy rises exponentially with temperature. The concentrations and ionization energies of traps in GeS crystals are determined from thermally stimulated current curves. The spectral response of the photocurrent through the crystals prepared by sublimation, whose structural perfection is higher than that of the melt-grown crystals, is governed by the spectral dependence of the absorption coefficient forad ≪ 1 (near-edge region) and by the spectral dependence of reflectivity for αd > 1 (high-α region). Regardless of the growth technique, the 293-K photocurrent spectra of GeS crystals show strongly polarized peaks at 1.65 (E ∥a) and 1.78 eV (E ∥b), which are due to the Λ ₁ ^v → Λ ₁ ^c and Δ ₂ ^v → Δ ₂ ^c optical transitions. The low-temperature photoresponse athv < 1.7 eV is due to absorption by Si impurity.     

Increase in the exactness of measurement of physical parameters of high-temperature superconductor materials

A procedure for estimating the critical temperature T _c and width ΔT _c for the superconducting transition of high-temperature superconductor materials based on plotting the first derivative of the temperature correlation dependence of the electrical resistance of a sample is developed. The critical point of the derivative corresponds to T _c , and its width at half-height corresponds to ΔT _c . The procedure provides a decrease in the relative error of estimation of T _c and ΔT _c for qualitative samples by a factor of 1.7–2.     

Temperature dependence of the local Seebeck coefficient near the boundary in touching Cu/Bi–Te/Cu composites

The thermo-emf ΔV and temperature difference ΔT across the boundary were measured as functions of r and I for the touching p- and n-type Cu/Bi–Te/Cu composites composed of t _Bi–Te = 2.0 mm and t _Cu = 0.3 mm, where r is the distance from the boundary and I is a direct current producing ΔT which flows through two Peltier modules connected in series. The resultant Seebeck coefficient α across the boundary is obtained from the relation α = ΔV/ΔT. As a result, the resultant |α| of the touching p- and n-type composites have a great local maximum value at r ≈ 0.03 mm and decrease rapidly with further increase of r to approach the intrinsic |α_Bi–Te|. The maximum resultant α of the p- and n-type composites reached great values of 1,043 and −1,187 μV/K at 303 K corresponding to I = 0.8 A and of 1,477 and −725 μV/K at 360 K corresponding to I = 2.0 A. Reflecting the temperature dependence of the intrinsic α_Bi–Te, the maximum α of the p-type composite increases with an increase of T, while that of the n-type one decrease with an increase of T. Surprisingly, the maximum α of the p- and n-type composites have great gradients of 8.36 and −7.15 μV/K² in the range from 303 to 366 K, respectively, which are 21.8 and 134 times larger in absolute value than 0.383 and −0.0535 μV/K² of the intrinsic p- and n-type α_Bi–Te, so that the maximum resultant α was thus found to be much more sensitive to temperature than the intrinsic α_Bi–Te. Moreover, the local Seebeck coefficient α _l (r) derived analytically from the resultant α(r) is enhanced significantly in the narrow region below r ≈ 0.05 mm and the maximum α _l values of the p- and n-type composites were found to have extremely great values of approximately 1,800 μV/K at 360 K and −1,400 μV/K at 303 K, respectively, which are approximately 7.3 and 6.5 times higher in absolute value than the intrinsic p- and n-type α_Bi–Te at the corresponding temperatures.     

Point-Contact Spectroscopy of the Borocarbide Superconductor YNi2B2C in the Normal and Superconducting State

Point-contact (PC) spectroscopy measurements of YNi₂B₂C single crystals in the normal and superconducting (SC) state (T _c ≃ 15.4 K) for the main crystallographic directions are reported. The PC study reveals the electron–phonon interaction (EPI) spectral function with dominant phonon maximum around 12 meV and further weak structures (hump or kink) at higher energy at about 50 meV. No “soft” modes below 12 meV are resolved in the normal state. The PC EPI spectra are qualitatively similar for the different directions. Contrary, directional study of the SC gap results in Δ_{[100] ≈ 1.5 meV for the a direction and Δ_{[001] ≈ 2.3 meV along the c axis; however the critical temperature T _c in PC in all cases is near to that in the bulk sample. The value 2Δ_[001]/k _B T _c ≈ 3.6 is close to the BCS value of 3.52, and the temperature dependence Δ_[001](T) is BCS-like, while the for small gap Δ_[100](T) is below BCS behavior at T > T _c /2 similarly as in the two-gap superconductor MgB₂. It is supposed that the directional variation Δ can be attributed to a multiband nature of the SC state in YNi₂B₂C.     

LLDPE’s grown with Metallocene and Ziegler–Natta catalysts: events in the melt and FTIR analysis

LLDPE samples synthesized with Ziegler–Natta (ZN) and Metallocene (MT) catalysts have been analyzed to investigate a potential catalyst-dependent morphology and to find an explanation for the difficult processing of MT. Slow calorimetry at v = 0.02 K/min and IR at RT and in the melt are used. The differences between MT and ZN are assigned to their different composition, MT not having the linear segments, which are present in ZN. Slow calorimetry is effectively a drawing process of the melt with chain orientation followed by decay. The later event, characterized by an endotherm, ΔH _network, occurs at higher temperatures for MT, the presence of a regular distribution of methyl groups slowing down the process. The rocking, gauche, bending and stretching regions of the IR spectra are analyzed. The nascent MT has more strained bands in the rocking region. The wagging region reveals the more homogeneous environment of MT through the maximum absorbance at 1,368 cm⁻¹. Decomposition of bands is made for the rocking and wagging regions. The orthorhombic crystallinity, α_c (FTIR), measures the sum of long- and short-range orthorhombic order, the latter being obtained by α_c (FTIR)-α_c (X-rays). The values of α_c (FTIR) for MT and ZN are very similar in conditions of equilibrium. The justifications for the molecular origin of ΔH _network are presented: (i) the slow relaxation of long chains strained and oriented in the melt measured by other techniques, (ii) The correlation, for gels of a linear sample, made in different solvents, between the maximum drawability, λ_max, and ΔH _network in a slow T-ramp. The range is 80–270 for λ_max and 40–120 J/g for ΔH _network. (iii) The comparison of two traces of the same sample, between 140 °C and 270 °C, show that comparable events in the melt appear in the integrated absorbance and in the slow calorimetry signal. Analysis on thin films of the little-studied CH₂ stretching region reveals that their extinction coefficient, ε, and the shape of the bands are highly sensitive to the sample history, ε diminishing by a large factor in slowly crystallized samples. Events in the slow T-ramp, followed by a fast crystallization, on the other hand, leads to materials with standard characteristics. Slow calorimetry traces display more events (endothermic and exothermic) for MT than for ZN, a finding consistent with more flow irregularities during processing. Equilibrium conditions and better processing could be reached for MT by extending time in the melt or using higher temperatures.     

Sound Velocity Measurements in the Low and the High Field Phases of the Nuclear-Ordered bcc Solid 3He in Magnetic Fields

We have measured the temperature and magnetic-field dependences of the sound velocity for one longitudinal and two transverse waves in the low field phase (LFP) and the high field phase (HFP) of nuclear spin ordered bcc solid ³He crystals with a single magnetic domain along the melting curve. From sound velocity measurements for various crystal orientations as a function of the sound propagation direction, we determined the elastic stiffness constants, c _ij (T,B). In the LFP with tetragonal symmetry for the nuclear spin structure, we extracted six nuclear spin elastic stiffness constants Δc _ij ^ℓ (T,0.06 T) from the temperature dependence of the sound velocity at 0.06 T and Δc _ij ^ℓ (0.5 mK,B) from the magnetic-field dependence of sound velocity at 0.5 mK. In the HFP with cubic symmetry for the nuclear spin structure, we extracted three Δc _ij ^h (T,0.50 T) at 0.50 T and Δc _ij ^h (0.5 mK,B) at 0.5 mK. At the first-order magnetic phase transition from the LFP to the HFP at the lower critical field B _c1, large jumps in sound velocities were observed for various crystal directions and we extracted three . Using the thermodynamic relation between Δc _ij and the change in the internal energy for the exchange interaction in this system, ΔU _ex(T,B), Δc _ij are related to the generalized second-order Grüneisen constants Γ _ij ^X ≡∂ ²ln X/∂ ε _i ∂ ε _j as Δc _ij (T,B)=Γ _ij ^X ΔU _ex(T,B), where X represents some physical quantity which depends on the molar volume and ε _j is the j-th component of a strain tensor. In the LFP, the Δc _ij ^ℓ (T,0.06 T) were proportional to T ⁴, and Δc _ij ^ℓ (0.5 mK,B) were proportional to B ². We extracted for the spin wave velocity in the LFP, s ^ℓ , from Δc _ij ^ℓ (T,0.06 T) and for the inverse susceptibility, 1/χ ^ℓ from Δc _ij ^ℓ (0.5 mK,B). In the HFP, Δc _ij ^h (T,0.50 T) were proportional to T ⁴ and Δc _ij ^h (0.5 mK,ΔB) were proportional to ΔB(≡B−B _c1). We obtained for the spin wave velocity in the HFP, s ^h , from Δc _ij ^h (T,0.50 T) and for B _c1 from Δc _ij ^h (0.5 mK,ΔB). The values obtained for and were compared with the Multiple Spin Exchange model (MSE) with three parameters by using analytic expressions for s ^ℓ and χ ^ℓ . The three-parameter MSE does not agree with the observed Δc _ij in the LFP.      

Time-of-Flight Measurements of Vortices Emitted from Quantum Turbulence in Superfluid 4He

An oscillating obstacle generates quantum turbulence in superfluids, when vortices remained attached to obstacle surfaces or vortex rings collided with it during oscillation. Turbulence provides a source of vortices; however, the characteristics of these vortices are not clear. In the present work, we report the flight of vortices emitted from quantum turbulence in superfluid ⁴He at low temperatures, using vibrating wires as a generator and a detector of vortices. A vortex-free vibrating wire can detect only the first colliding vortex ring, though it will be refreshed after low vibration and be able to detect a vortex ring again. By measuring a period from the start of turbulence generation to the vortex detection repeatedly, we find an exponential distribution of time-of-flights with a non-detection period t ₀ and a mean detection period t ₁, suggesting a Poisson process. Both periods t ₀ and t ₁ increase with increasing distance between a generator and a detector. A vortex flight velocity estimated from period t ₀ suggests that the sizes of the emitted vortex rings distribute to a range smaller than a generator thickness or a generator vibration amplitude. Vortices are emitted radially from a turbulence region, at least in the direction of oscillator vibration.     

Spectroscopic Evidence for Competing Order-Induced Pseudogap Phenomena and Unconventional Low-Energy Excitations in High-<Emphasis Type="Italic">T</Emphasis><Subscript>c</Subscript> Cuprate Superconductors

The low-energy excitations of cuprate superconductors exhibit various characteristics that differ from those of simple Bogoliubov quasiparticles for pure d_x²-y²d_{x^{2}-y^{2}} -wave superconductors. Here, we report experimental studies of spatially resolved quasiparticle tunneling spectra of hole- and electron-type cuprate superconductors that manifest direct evidences for the presence of competing orders (COs) in the cuprates. In contrast to conventional type-II superconductors that exhibit enhanced local density of states (LDOS) peaking at zero energy near the center of field-induced vortices, the vortex-state LDOS of YBa₂Cu₃O_7−δ (Y-123) and La_0.1Sr_0.9CuO₂ (La-112) remains suppressed inside the vortex core, with pseudogap (PG)-like features at an energy larger (smaller) than the superconducting (SC) gap Δ_SC in Y-123 (La-112). Energy histograms of the SC and PG features reveal steady spectral shifts from SC to PG with increasing magnetic field H. These findings may be explained by coexisting COs and SC: For hole-type cuprates with PG above T _c, the primary CO gap (V _CO) is larger than Δ_SC and the corresponding COs are charge/pair-density waves with wave-vectors parallel to (π,0)/(0,π). For electron-type cuprates without PG above T _c, V _CO is smaller than Δ_SC and the CO wave-vector is along (π,π). This CO scenario may be extended to the ARPES data to consistently account for the presence (absence) of Fermi arcs in hole- (electron)-type cuprates. Fourier transformation of the vortex-state LDOS in Y-123 further reveals multiple sets of energy-independent wave-vectors due to field-enhanced pair- and spin-density waves. These results imply important interplay of SC with low-energy collective excitations.     

Theoretical Study of Specific Heat, Density of States and Free Energy of Itinerant Ferromagnetic Superconductor URhGe

Following the equation of motion method and Green’s function technique, the coexistence of itinerant ferromagnetism (FM) and superconductivity (SC) is investigated in a single band homogeneous system. Self-consistent equations for superconducting order parameter (Δ) and magnetic order parameter (Δ_FM) are derived. It is shown that there generally exists a coexistent (Δ≠0 and Δ_FM≠0) solutions to the coupled equations of the order parameter in the temperature range 0<T<min (T _C,T _FM) where T _C and T _FM are respectively the superconducting and ferromagnetic transition temperatures. Expressions for the specific heat, density of states and free energy are derived. The specific heat has a linear temperature dependence at low temperatures as opposed to the exponential decrease in the BCS theory. The density of states for a finite Δ_FM increases as opposed to that of a standard ferromagnetic metal. The free energy shows that the superconducting ferromagnetic state has lower energy than the normal ferromagnetic state and therefore is realized at low enough temperature. The theory is applied to explain the observations of URhGe. The agreement between theory and experimental results is quite satisfactory.      

Probing the Vorticity in a Superconductor Rapidly Quenched from the Normal State

Topological defects such as vortices subsist for some time in a rapidly frozen superconducting film (W. H. Zurek, Phys. Rep. 276, 177, 1996). We propose to use as a probe of the vortex density the finite delay T _d, which, in narrow strips, exists between a current step and the voltage response. Technically, this amounts to driving a bridge into the localized hot-spot regime by means of a pump pulse (laser or electrical). Cooling of such films as epitaxial YBCO-on-MgO, or niobium-on-sapphire, requires only a few nanoseconds at low T. A time t later, a probe pulse is applied to measure T _d. The dependence t T _d is interpreted as a witness of the fossil vorticity, long after quenching into the zero-resistance regime.     

Decay of Counterflow Quantum Turbulence in Superfluid 4He

We have simulated the decay of thermal counterflow quantum turbulence from a statistically steady state at T=1.9 K, with the assumption that the normal fluid is at rest during the decay. The results are consistent with the predictions of the Vinen equation (in essence the vortex line density decays as t ^?1). For the statistically steady state, we determine the parameter c ₂, which connects the curvature of the vortex lines and the mean separation of vortices. A formula connecting the parameter χ ₂ of the Vinen equation with c ₂ is shown to agree with the results of the simulations. Disagreement with experiment is discussed briefly.     

Tilted and crossing vortex chains in layered superconductors

In presence of the Josephson vortex lattice in layered superconductors, small c-axis magnetic field penetrates in the form of vortex chains. In general, structure of a single chain is determined by the ratio of the London [λ] and Josephson [λ _J ] lengths, α=λ/λ _J . The chain is composed of tilted vortices at large α's (tilted chain) and at small α's it consists of crossing array of Josephson vortices and pancake-vortex stacks (crossing chain). We study chain structures at the intermediate α's and found two types of phase transitions. For α≲0.6 the groud state is given by the crossing chain in a wide range of pancake separationsa≳[2−3]λ _J . However, due to attractive coupling between deformed pancake stacks, the equilibrium separation can not exceed some maximum value depending on the in-plane field and α. The first phase transition takes place with decreasing pancake-stack separation a ata≳[1−2]λ _J , and rather wide range of the ratio α, 0.4≲α≲0.65. With decreasing a, the crossing chain goes through intermediate strongly-deformed configurations and smoothly transforms into the tilted chain via the second-order phase transition. Another phase transition occurs at very small densities of pancake vortices,a ~ [20−30]λ _J , and only when α exceeds a certain critical value ∼0.5. In this case small c-axis field penetrates in the form of kinks. However, at very small concentration of kinks, the kinked chains are replaced with strongly deformed crossing chains via the first-order phase transition. This transition is accompanied by a very large jump in the pancake density.