Effect of structural and electrochemical properties of different Cr-doped contents of Li[Ni1/3Mn1/3Co1/3]O2

Layered Li[Ni_(1−x)/3Mn_(1−x)/3Co_(1−x)/3Cr_x]O₂ materials with x = 0, 0.01, 0.02, 0.03, 0.05 are prepared by a solid-state pyrolysis method. The oxide compounds were calcined with various Cr-doped contents, which result in greater difference in morphological (shape, particle size and specific surface area) and the electrochemical (first charge profile, reversible capacity and rate capability) differences. The Li[Ni_(1−x)/3Mn_(1−x)/3Co_(1−x)/3Cr_x]O₂ powders were characterized by means of X-ray diffraction (XRD), charge/discharge cycling, cyclic voltammetry, and SEM. XRD experiment revealed that the Li[Ni_(1−x)/3Mn_(1−x)/3Co_(1−x)/3Cr_x]O₂ (x = 0, 0.01, 0.02, 0.03, 0.05) were crystallized to well layered -NaFeO₂ structure. The first specific discharge capacity and coulombic efficiency of the electrode of Cr-doped materials were higher than that of pristine material. When x = 0.02, the sample showed the highest first discharge capacity of 241.9 mAh g⁻¹ at a current density of 30 mA g⁻¹ in the voltage range 2.3–4.6 V, and the Cr-doped samples exhibited higher discharge capacity and better cycleability under medium and high current densities at room temperature.     

Microwave dielectric properties and co-firing with copper of (Bi1−xCux)(Nb1−xWx)O4 ceramics

Effect of substitution of CuO and WO₃ on the microwave dielectric properties of BiNbO₄ ceramics and the co-firing between ceramics and copper electrode were investigated. The (Bi_1−xCu_x)(Nb_1−xW_x)O₄ (x = 0.005, 0.01, 0.015, 0.02) composition can be densified between 900 and 990 °C. The microwave dielectric constants lie between 36 and 45 and the pores in ceramics were found to be the main influence. The Q values changes between 1400 and 2900 with different x values and sintering temperatures while Q_f values lie between 6000 and 16,000 GHz. The microwave dielectric losses, mainly affected by the grain size, pores, and the secondary phase, are discussed. The (Bi_1−xCu_x)(Nb_1−xW_x)O₄ ceramics and copper electrode was co-fired under N₂ atmosphere at 850 °C and the EDS analysis showed no reaction between the dielectrics and copper electrodes. This result presented the (Bi_1−xCu_x)(Nb_1−xW_x)O₄ dielectric materials to be good candidates for LTCC applications with copper electrode.     

Catalytic behavior of La–Sr–Ce–Fe–O mixed oxidic/perovskitic systems for the NO+CO and NO+CH4+O2 (lean-NOx) reactions

Mixed oxides of the general formula La_0.5Sr_xCe_yFeO_z were prepared by using the nitrate method and characterized by XRD and Mössbauer techniques. The crystal phases detected were perovskites LaFeO₃ and SrFeO_3−x and oxides -Fe₂O₃ and CeO₂ depending on x and y values. The low surface area ceramic materials have been tested for the NO+CO and NO+CH₄+O₂ (“lean-NO_x”) reactions in the temperature range 250–550°C. A noticeable enhancement in NO conversion was achieved by the substitution of La³⁺ cation at A-site with divalent Sr⁺² and tetravalent Ce⁺⁴ cations. Comparison of the activity of the present and other perovskite-type materials has pointed out that the ability of the La_0.5Sr_xCe_yFeO_z materials to reduce NO by CO or by CH₄ under “lean-NO_x” conditions is very satisfying. In particular, for the NO+CO reaction estimation of turnover frequencies (TOFs, s⁻¹) at 300°C (based on NO chemisorption) revealed values comparable to Rh/-Al₂O₃ catalyst. This is an important result considering the current tendency for replacing the very active but expensive Rh and Pt metals. It was found that there is a direct correlation between the percentage of crystal phases containing iron in La_0.5Sr_xCe_yFeO_z solids and their catalytic activity. O₂ TPD (temperature-programmed desorption) and NO TPD studies confirmed that the catalytic activity for both tested reactions is related to the defect positions in the lattice of the catalysts (e.g., oxygen vacancies, cationic defects). Additionally, a remarkable oscillatory behavior during O₂ TPD studies was observed for the La_0.5Sr_0.2Ce_0.3FeO_z and La_0.5Sr_0.5FeO_z solids.     

Mass spectra of negative ions in air-like gas mixtures at atmospheric pressure

We have obtained mass spectra of negative ions produced by rays in artificial air at atmospheric pressure (N₂: 80%, O₂: 20%, H₂O: 20–1500 ppm, CO₂: 0.2–300 ppm, NO, NO₂ 0.02 ppm). We observed two main categories: hydrates built on simple ions (O₂⁻, O₃⁻, OH⁻, CO₃⁻, CO₄⁻, HCO₃⁻, NO₂⁻, NO₃⁻), hydrates built on complex ions (NO_x⁻, HNO_γ, HCO₃⁻HNO_x, x = 2,3; Y = 2, 3). For high values of hygrometry, CO₂ content and ageing time (5 msec) we observe the disappearance of O₂⁻, O₃⁻, OH⁻ hydrates whereas the major part of the spectrum consists of complex ions.     

Voltammetric behavior of the C60-γ-cyclodextrin inclusion complex (1:2) on hmde in an aqueous solution

Electrochemical behavior of the water-soluble C₆₀-γ-CD (1:2) inclusion complex has been studied on the hanging mercury drop electrode. A one-electron reversible adsorptive electro-reduction and three irreversible adsorptive electro-reductions were detected by cyclic voltammetry. The amount of C₆₀-γ-CD adsorbed at saturation is 2.50 × 10⁻¹¹ mol cm⁻², the diffusion coefficient is 4.36 × 10⁻⁶cm²s⁻¹ and the standard rate constant of the surface reaction k_s are 0.745 s⁻¹, 0.612 s⁻¹ and 0.513s⁻¹, respectively.     

Nanosized perovskite-type oxides La1−xSrxMO3−δ (M = Co, Mn; x = 0, 0.4) for the catalytic removal of ethylacetate

Nanometer perovskite-type oxides La_1−xSr_xMO_3−δ (M = Co, Mn; x = 0, 0.4) have been prepared using the citric acid complexing-hydrothermal-coupled method and characterized by means of techniques, such as X-ray diffraction (XRD), BET, high-resolution scanning electron microscopy (HRSEM), X-ray photoelectron spectroscopy (XPS), temperature-programmed desorption (TPD), and temperature-programmed reduction (TPR). The catalytic performance of these nanoperovskites in the combustion of ethylacetate (EA) has also been evaluated. The XRD results indicate that all the samples possessed single-phase rhombohedral crystal structures. The surface areas of these nanomaterials ranged from 20 to 33 m² g⁻¹, the achievement of such high surface areas are due to the uniform morphology with the typical particle size of 40–80 nm (as can be clearly seen in their HRSEM images) that were derived with the citric acid complexing-hydrothermally coupled strategy. The XPS results demonstrate the presence of Mn⁴⁺ and Mn³⁺ in La_1−xSr_xMnO_3−δ and Co³⁺ and Co²⁺ in La_1−xSr_xCoO_3−δ, Sr substitution induced the rises in Mn⁴⁺ and Co³⁺ concentrations; adsorbed oxygen species (O⁻, O₂⁻, or O₂²⁻) were detected on the catalyst surfaces. The O₂-TPD profiles indicate that Sr doping increased desorption of the adsorbed oxygen and lattice oxygen species at low temperatures. The H₂-TPR results reveal that the nanoperovskite catalysts could be reduced at much lower temperatures (<240 °C) after Sr doping. It is observed that under the conditions of EA concentration = 1000 ppm, EA/oxygen molar ratio = 1/400, and space velocity = 20,000 h⁻¹, the catalytic activity (as reflected by the temperature (T_100%) for EA complete conversion) increased in the order of LaCoO_2.91 (T_100% = 230 °C) ≈ LaMnO_3.12 (T_100% = 235 °C) < La_0.6Sr_0.4MnO_3.02 (T_100% = 190 °C) < La_0.6Sr_0.4CoO_2.78 (T_100% = 175 °C); furthermore, there were no formation of partially oxidized by-products over these catalysts. Based on the above results, we conclude that the excellent catalytic performance is associated with the high surface areas, good redox properties (derived from higher Mn⁴⁺/Mn³⁺ and Co³⁺/Co²⁺ ratios), and rich lattice defects of the nanostructured La_1−xSr_xMO_3−δ materials.     

Fabrication and thermoelectric properties of heavily rare-earth metal-doped SrO(SrTiO3)n (n = 1, 2) ceramics

To clarify the effect of substitutional electron doping on the thermoelectric figure of merit (ZT = S²σTκ⁻¹) of Ruddlesden–Popper phase SrO(SrTiO₃)_n (or Sr_n+1Ti_nO_3n+1), measurements were conducted for several thermoelectric parameters, e.g. electrical conductivity (σ), Seebeck coefficient (S) and thermal conductivity (κ), of (Sr_1−xRE_x)_n+1Ti_nO_3n+1 (n = 1 or 2, RE (rare earth): La or Nd, x = 0.05 and 0.1) dense ceramics prepared by a conventional solid-state reaction and hot-pressing technique. Crystal structures of the resultant ceramics were represented as (Sr_1−xRE_x)_n+1 Ti_nO_3n+1 evaluated by powder X-ray diffraction followed by the Rietveld analysis. All the ceramics exhibited electrical conductivity and the σ values simply depended on the dopant concentration, indicating that both La³⁺ and Nd³⁺ ions act as electron donors. The |S| values increased with temperature due to decrease in the chemical potential. Significant reduction of the κ values was observed as compared to cubic-perovskite SrTiO₃. The ZT value increased with temperature and reached 0.15 at 1000 K for (Sr_0.95La_0.05)₃Ti₂O₇.     

Transport numbers and oxygen permeability of SrCe(Y)O3-based ceramics under oxidising conditions

Partial conductivities in the SrCe(Y)O_3−δ system have been studied in oxidising conditions in the temperature range 923–1273 K. Compositions with variable Y content (5 and 10 at.%), Sr deficiency (3 at.%), and with the addition of Fe₂O₃ as sintering aid (2 mol%) were analysed. A modified Faradaic efficiency method and oxygen permeation measurements were employed to appraise the oxide-ionic transport. Oxide-ion transference numbers in air lie in the range 0.19–0.80 and decrease with increasing temperature in the range 973–1223 K. Modelling of total conductivity as a function of oxygen partial pressure (p(O₂)) confirmed that protonic transport is minor under the studied conditions. SrCe_0.95Y_0.05O_3−δ exhibits greater oxide-ion conductivity than SrCe_0.9Y_0.1O_3−δ, indicative of dopant–vacancy association at high dopant contents. Conversely, oxygen permeability is slightly higher for SrCe_0.9Y_0.1O_3−δ as a result of faster surface-exchange kinetics. The oxygen flux through Fe-free membranes is dominated by the bulk in low p(O₂) gradients, when the permeate-side p(O₂) is higher than 0.03 atm, but surface exchange plays an increasing role with increasing p(O₂) gradient. Addition of Fe₂O₃ to SrCe(Y)O_3−δ lowers the sintering temperature by 100 K but results in the formation of intergranular second phases which block oxide-ionic and electronic transport, and thus oxygen permeation. The average thermal expansion coefficients (TECs) are (10.8–11.6) × 10⁻⁶ K⁻¹ in the temperature range 373–1373 K for all studied compositions.     

A new tantalum sulfur compound as electrode material for non-aqueous alkali metal batteries

Polycrystalline (PbS)_1.14(TaS₂)₂, a misfit layer sulfide, was used as cathodic material for lithium secondary battery. One molar LiClO₄ in propylene carbonate (PC) was used as electrolyte. The cell could be galvanostatic discharged down to x = 4.6 [Li_x(PbS)_1.14(TaS₂)₂] when the current density was 65 μA cm⁻² and the cell was cycled more than 100 times between 3.5 and 1.5 V at a current density of 260 μA cm⁻². Lattice expansion increased linearly with lithium content and was less than that reported for the Li/TaS₂ system. Chemical diffusion coefficients were determined by a modified version of the galvanostatic intermittent titration technique and they were fairly constant in the composition range 0.2 < x < 1, and an average value of 8.1 × 10⁻¹¹ cm² s⁻¹ was calculated. Sodium intercalation was also accomplished, but the uptake of this ion resulting in a significant lattice expansion compared with that observed for lithium ions. Moreover, a similar dependence of the sodium chemical diffusion coefficient on the composition was observed with an average value of 1.4 × 10⁻¹⁰ cm² s⁻¹, somewhat higher than that of lithium ion. We believe that differences in lattice expansion may be responsible for the differences found in the chemical diffusivity values.     

Aspiration below wind velocity of aerosols with sharp edged nozzles facing the wind

Aspiration coefficients of sharp-edged thin-walled nozzles have been measured at zero yaw angle in wind speeds from 77 to 309 cm sec⁻¹ and aspiration velocities from 26 to 309 cm sec⁻¹. Homogeneous liquid droplets of di-2-ethyl-hexyl sebacate of 16 and 22 μ dia. were used with Stokes numbers 0.014 < St_v < 0.64. Substantial deposition occurred inside the nozzles and in the impactors used for sampling; it was measured and allowed for when calculating the aspiration coefficients.     

Synthesis and permeation properties of ion-exchanged ETS-4 tubular membranes

Zeolite membranes, which were composed of ETS-4 with Na cations, were prepared on porous -alumina tubes by hydrothermal synthesis. The membranes, which were formed under optimized conditions, sharply rejected molecules with sizes larger than 0.4 nm. For mixtures of N₂–CO₂, N₂–O₂, N₂–Ar and N₂–CH₄ systems, N₂ permeated faster than the coexisting gas. The N₂/O₂ separation factor for an equimolar mixture was in the range of 2.3–3.5, and the N₂ permeance was in the range of (0.55–2.8)×10⁻⁸ mol m⁻² s⁻¹ Pa⁻¹ at permeation temperatures of 283–333 K. Moisture had some effect on the permeation properties for N₂–O₂ mixtures. The separation factor for the N₂/CH₄ system was larger than that of the N₂/O₂ system. When the membrane was ion exchanged with either Li⁺ or Sr²⁺, the separation factors for N₂/O₂ and N₂/CH₄ systems increased, while the permeances decreased.     

Effect of Pt on the water resistance of Co-zeolites upon the SCR of NOx with CH4

The objective of this work was to study the promotional effect of Pt on Co-zeolite (viz. mordenite, ferrierite, ZSM-5 and Y-zeolite) and Co/Al₂O₃ on the selective catalytic reduction (SCR) of NO_x with CH₄ under dry and wet reaction stream. After being reduced in H₂ at 350°C, the PtCo bimetallic zeolites showed higher NO to N₂ conversion and selectivity than the monometallic samples, as well as a combination of the latter samples such as mechanical mixtures or two-stage catalysts. After the same pretreatment, under wet reaction stream, the bimetallic samples were also more active. Among the other catalysts studied with 5% of water in the feed, (NO = CH₄ = 1000 ppm, O₂ = 2%), the NO conversion dropped to zero over Co_2.0Mor at 500°C and GHSV = 30,000 h⁻¹, whereas it is 20% in Pt_0.5Co_2.0Mor. In Pt/Co/Al₂O₃ the NO_x conversion dropped below 5% with only 2% of water under the same reaction conditions. The specific activity given as molecules of NO converted per total metal atom per second were 16.5 × 10⁻⁴ s⁻¹ for Pt_0.5Co_2.0Fer, 13 × 10⁻⁴ s⁻¹ for Pt_0.5Co_2.0Mor, 4.33 × 10⁻⁴ s⁻¹ for Pt_0.5Co_2.0ZSM-5 and 0.5 × 10⁻⁴ s⁻¹ for Pt/Co/Al₂O₃. The Y-zeolite-based samples were inactive in both mono and bimetallic samples. The species initially present in the solid were Pt° and Co°, together with Co²⁺ and Pt²⁺ at exchange positions. Co° seems not to participate as an active site in the SCR of NO_x. Those species remained after the reaction but some reorganization occurred. A synergetic effect among the different species that enhances both the NO to NO₂ reaction, the activation of CH₄ and also the ability of the catalyst to adsorb NO, could be responsible for the high activity and selectivity of the bimetallic zeolites.     

Electrosorption and inhibition properties of p-norborn2-yl phenolate ions at the mercury/solution interface

The electrosorption properties of p-norborn-2-yl phenolate ions in alkaline solutions were investigated by ac polarographic and electrocapillary measurements.

Two adsorption regions were found. At low bulk surfactant concentrations the adsorption at the positively charged electrode (−0.2 E −0.6 V) is predominant while at higher surfactant concentrations the adsorption at the negatively charged electrode (−0.6 E −1.0 V) is more pronounced. At E = −0.40 V the adsorption parameters were determined (a ≈ 2; ΔG°_A = −32.5 ± 1 kJ mol⁻¹. Between −0.6 E −1.0 V one potential of maximum adsorption for all concentrations does not exist and therefore the adsorption parameters could not be calculated.

At E = −0.40 V progressive two-dimensional nucleation with a nucleation order of 3 was observed which corresponds well with the high attraction constant.

The electrode reaction S₂O²⁻₈ + 2e⁻ → 2 SO²⁻₄ is inhibited by norborn-2-yl phenolate ions in the potential range −0.2 E −0.6 V. In the second potential range of capacity decrease the electrode process is much less retarded. At E = −0.40 V, in a similar manner as described for neutral molecules, a linear dependence of the log k_s (k_s apparent rate constant) on ln c_A and π (π = surface film pressure), respectively, has been found.     

Kinetics of the water–gas shift reaction over nanostructured copper–ceria catalysts

Water–gas shift reaction was studied over two nanostructured Cu_xCe_1−xO_2−y catalysts: a Cu_0.1Ce_0.9O_2−y catalyst prepared by a sol–gel method and a Cu_0.2Ce_0.8O_2−y catalyst prepared by co-precipitation method. A commercial low temperature water–gas shift CuO–ZnO–Al₂O₃ catalyst was used as reference. The kinetics was studied in a plug flow micro reactor at an atmospheric pressure in the temperature interval between 298 and 673 K at two different space velocities: 5.000 and 30.000 h⁻¹, respectively. Experimentally estimated activation energy, E_af, of the forward water–gas shift reaction at CO/H₂O = 1/3 was 51 kJ/mol over the Cu_0.1Ce_0.9O_2−y, 34 kJ/mol over the Cu_0.2Ce_0.8O_2−y and 47 kJ/mol over the CuO–ZnO–Al₂O₃ catalyst. A simple rate expression approximating the water–gas shift process as a single reversible surface reaction was used to fit the experimental data in order to evaluate the rate constants of the forward and backward reactions and of the activation energy for the backward reaction.     

Microstructures and mechanical properties of 8Y-FSZ ceramics with BaTiO3 additive

The microstructure and mechanical properties of 8 mol% Y₂O₃ fully stabilized zirconia (8Y-FSZ) with BaTiO₃ additive were investigated. The introduction of BaTiO₃ additive would significantly increase the density and the grain size of 8Y-FSZ ceramics. XRD, Raman spectroscopy, and dielectric measurement were performed. A rhombohedral Ba(Ti_1−xZr_x)O₃ ferroelectric phase resulted in the composite with 5 mol% additive, while for those with higher additive content, the secondary phase changes to cubic Ba(Ti_1−xZr_x)O₃. The fracture toughness of the xBaTiO₃/(1−x)8Y-FSZ composites reached a maximum and then decreased with increasing the amount of additive. The highest value reached 6.1 MPa m^1/2 for 0.05BaTiO₃/0.95(8Y-FSZ) sintered at 1475 °C for 3 h, where the piezoelectric/ferroelectric secondary phase toughening played an important role. Moreover, the fracture toughness of the composites increased firstly and then decreased with increasing sintering temperature.     

Particle motion at the wall of a circulating fluidized bed

The motion of alumina particles of mean size 74.9 μm in the region near to the wall of the 305 mm diameter riser of a cold model circulating fluidized bed has been studied using a high-speed video camera employing normal and magnifying lenses. Particles in this region were found to move predominantly downwards, against the main gas flow. High density groups or swarms of particles typically arch-shaped were observed to descend in contact with the wall at velocities in the range 0.3–0.4 m s⁻¹. Tfie distribution of swarm descent velocities was shown to be little affected by changes in superficial gas velocity over the range 3–5 m s⁻¹ and imposed mean solids mass flux over the range 2 to 80 kg m⁻² s⁻¹. A region of steady bulk downflow of solids with a velocity of approximately 1.0 m s⁻¹ was observed to appear a few millimetres from the wall at mean suspension densities greater than 5.6 kg m⁻³. Motion of particles in contact with the riser was analysed by identification of three flow forms; dilute, dense and swarm flow. Results of the analysis are linked with the observations of other workers concerning the onset of the so-called ‘similar profiles’ regime. The relationship between the measured effective particle swarm length and the cross-sectional mean suspension density was established and the implications for modelling suspension-to-wall heat transfer discussed.     

Defect formation and mechanical stability of perovskites based on LaCrO3 for solid oxide fuel cells (SOFC)

Perovskites on the basis of LaCrO₃ are of interest as ceramic interconnect materials for the development of solid oxide fuel cells (SOFCs). The interconnects are exposed to oxidising and reducing atmospheres under operating conditions. Oxygen vacancy formation was determined as a function of oxygen partial pressure between 1 and 10⁻²² bar at temperatures between 900 and 1100 °C. Different perovskite compositions made of (La,Ca/Sr)CrO_3−δ, La(Cr,Mg)O_3−δ, La(Cr,Mg/Cu/Co,Al)O_3−δ, and (La,Ca)(Cr,Al)O_3−δ were investigated. Defect models were evaluated to describe the oxygen vacancy formation and the respective thermodynamic data were determined. The results are used to explain existing literature data on the isothermal expansion of LaCrO₃ based perovskites under reducing conditions. Complementary mechanical measurements with selected perovskite compositions revealed that lower oxygen partial pressure causes higher stiffness, strength and fracture toughness. The change in properties is discussed in terms of the observed ferroelastic domains and the interaction of the domain wall motion with the oxygen vacancies.     

Reversible electrochemical insertion of anions in poly-p-phenylene from aqueous electrolytes

Poly-p-phenylene (PPP) was synthesized from benzene according to the Kovacic method. Electrodes were made from this electronic insulator by cold- or hot-pressing of the loose, brown powder, under the addition of 7.5 wt. % soot (Corax L^®, Degussa AG). The electrochemical insertion and removal of anions HSO⁻₄, ClO⁻₄ in this material in aqueous solutions of the corresponding acids was investigated by slow cyclic voltammetry.

Initially, only a surface layer of about 0.1 mm thickness takes part in the electrochemical processes, which are reversible. A maximum concentration of anions in the solid of [(−C₆H₄−)⁺₇ A⁻] is attainable. The maximum degree of insertion is equal to 0.14. The insertion potential U_I shifts strongly into the negative direction with increasing concentration c of the acid. A linear U_I/c relationship is observed as in the case of graphite, where the intercalation potential is more positive by 20–200 mV for the same electrolyte. The round trip current efficiency for the insertion/removal cycle increases with increasing acid concentration attaining 100% in 14 M H₂SO₄ or 11.3 M CHlO₄. For a given concentration, increases in the same order as with graphite (H₂SO₄ < HClO₄ < HBF₄), being somewhat lower for a given electrolyte composition. From anodic current limitation (j_lim = 5–10 mA cm⁻²), a diffusion coefficient of about D = 2 × 10⁻⁷ cm² s⁻¹ is derived for the transport of anions in the bulk of PPP. The striking similarity of our results to former findings with graphite is thoroughly discussed. Some general conclusions are derived thereof.     

Kinetics of Fischer–Tropsch synthesis on titania-supported cobalt

Rate data have been obtained for CO hydrogenation on a well-characterized 11.7% Co/TiO₂ catalyst in a differential fixed bed reactor at 20 atm, 180–240°C, and 5% conversion over a range of reactant partial pressures. The resulting kinetic parameters can be used to model precisely and accurately the kinetics of this reaction within this range of conditions. Turnover frequencies and rate constants determined from this study are in very good to excellent agreement with those obtained in previous studies of other cobalt catalysts, when the data are normalized to the same conditions of temperature and partial pressures of the reactants. Based on this comparison CO conversion and the partial pressure of product water apparently have little effect on specific rate per catalytic site. The data of this study are fitted fairly well by a simple power law expression of the form −r_CO=kP_H2^0.74P_CO^−0.24, where k=5.1×10⁻³ s⁻¹ at 200°C, P=10 atm, and H₂/CO=2/1; however, they are best fitted by a simple Langmuir–Hinshelwood (LH) rate form −r_CO=aP_H2^0.74P_CO/(1+bP_CO)² similar to that proposed by Yates and Satterfield.