Electrical conductivity at high temperature and thermodynamic study of point defects in single crystalline cuprous oxide

In order to follow the variation of point defect interactions in relatively concentrated solutions we have studied the electrical conductivity of cuprous oxide in the range of temperature 650–1100°C and for oxygen partial pressures greater than 10^?6 atm. The P^{$\text{1}\text{π}$}_O2 dependence of the conductivity varied non linearly from about n = 8 for P_o2 close to 10^?6 atm. to lower values of n with increasing oxygen partial pressure. The activation enthalpy of conductivity determined in these ranges of temperature and oxygen partial pressure has been found to be also a function of temperature and varied between 12 and 17 kcal mol^?1.The interpretation of these results has permitted us to show that the departure from linearity of the plots of log or log $\text{σ = ?(}\text{l}\text{T}\text{)}$ excludes the existence of an ideal solution of ionized and non-ionized copper vacancies as was proposed previously in the literature. To explain these results it is possible to take into account a partially disordered distribution of these defects. It is shown that the increase of interactions and consequently the variation of the electrical conductivity as a function of the thermodynamic parameters, may be simulated by an ideal solution including new charged species of the type (V'_CuV^x_Cu).     

A thermodynamic study of nonstoichiometric cerium dioxide

Thermogravimetric measurements were performed on nonstoichiometric CeO_2?x in the temperature range 750–1500°C and from oxygen pressures of 10^?2 to 10^?26 atm. From this data the deviation from stoichiometry x = x(T, P_o2) was determined. The thermodynamic quantities Δ$\text{H}$_o2 and Δ$\text{S}$_o2 were calculated in the region 0.001? x ? 0.3 and found to be independent of temperature.In the composition region 0.001< x < 0.006, the variation of Δ$\text{S}$_o2 with x is consistent with a defect model involving randomly distributed doubly ionized oxygen vacancies. The experimental dependence of x and σ (electrical conductivity) is shown to be consistent with this model as Δ$\text{H}$_o2 (≈ -10 eV) exhibits a slight dependence on x. It is postulated that the variation in Δ$\text{H}$_o2 may result from lattice parameter increases with x, while the defects remain essentially randomly distributed.In the composition region 0.006 < x < 0.1, x ∝ with 1 < n < 5, and in the region 0.1 < x < 0.3, x ∝ with n increasing rapidly with x to n? 30. This behavior is believed to result from increasing defect interaction with increasing departures from stoichiometry. It is interesting to note that the ordered phase observed by Bevan and Kordis between CeO_1·72 and CeO_1·70 was not observed in this study at temperatures between 1300° and 1500°C.     

Electrical conductivity and chemical diffusion coefficient measurements in single crystalline nickel oxide at high temperatures

Electrical conductivity measurements on nickel oxide have been performed at high temperatures (1273 K<T< 1673 K) and in partial pressures of oxygen ranging from Po₂ = 1.89 × 10^?4 atm to Po₂ = 1 atm. The $\text{po}{}_2{}^{\mathrm{<mtext>1</mtext><mtext>n</mtext>}}$ dependence of the conductivity decreases from about $\text{1}\text{4}$ for Po₂ = 1 atm to smaller values for lower partial pressures of oxygen. The activation enthalpy for conduction increases for decreasing oxygen partial pressures (from 22.5 kcal mol^?1 at Po₂ = 1 atm to 26.0 kcal mol^?1 for Po₂ = 1.89 × 10^?4 atm). This behaviour can be explained by the simultaneous presence of singly and doubly ionized nickel vacancies, with different energies of formation.Furthermore, chemical diffusion coefficient measurements have been performed in the same temperature range, using the conductivity technique, and leading to the result:

$\text{D}\text{?}\text{= 0.244 exp (?}\text{36,600}\text{RT}\text{) cm}{}^2\text{s}{}^{\mathrm{?1}}$

.     

Conductivite electrique totale du protoxyde de cuivre en fonction de sa composition definie par des electrodes d'alliage or-cuivre

The total electrical conductivity of cuprous oxide has been measured from 800 to 1100°K as a function of its composition, especially in the range of very low oxygen pressures up to the limit of equilibrium with copper. These measurements have been carried out by means of a new method using copper-gold alloy electrodes in order to have given activities of components.The results indicate for cuprous oxide a p semi-conduction throughout its stability range.The conductivity σ (ohm^?1. cm^?1) is given as a function of the temperature T (K) and the partial pressure P_o2 (torr) of oxygen in equilibrium with the oxide by the following equation:

     

Electrical conductivity and defect structure of Ni-doped and “CO-reduced” Ni-doped SrTiO3 single crystals

The electrical conductivities of Ni-doped and “CO-reduced” Ni-doped SrTiO₃ single crystals were measured at temperatures 700–1200°C and P_o2's of 10^?7–10^?1 atm. Plots of log σ vs $\text{1}\text{T}$ at constant P_o2's were found to be linear, and the activation energies appeared to be 0.92 eV for Ni-doped SrTiO₃ and 0.50 eV for “CO-reduced” Ni-doped SrTiO₃ single crystals, respectively. Plots of log σ vs log P_o2 at constant temperature were found to be linear with an average slope of ?$\text{1}\text{4}$ for SrTiO₃:Ni and of ?$\text{1}\text{6}$ for “CO-reduced” SrTiO₃:Ni single crystals, respectively. The electrical conductivity dependencies on P_o2 indicate that a triply ionized titanium interstitial and an oxygen vacancy model are applicable to Nidoped and “CO-reduced” Ni-doped SrTiO₃ single crystals, respectively. The small polaron conduction was suggested on “CO-reduced” Ni-doped SrTiO₃ single crystal from the temperature dependence of conductivity.     

Thermal changes in the optical properties of SnSxSe2−x crystals

Optical measurements on crystals in the series SnS_xSe_2?x for 0 ? x ? 2, have yielded information on the changes in the ordinary refractive index $\text{Δn}\text{ΔT}$ and the energy gap $\text{ΔEg}\text{ΔT}$ in the temperature range 125–425 K. The coefficient $\text{Δn}\text{ΔT}$ has values +40 to +160 × 10^?6K^?1 and this confirms that covalent bonding predominantly exists in these materials. The coefficient $\text{ΔEg}\text{ΔT}$ remains fairly consistent for all values of x with an average value of -8.0×10^-4eV K^-1.     

Electrical conductivity of pure and doped α-ferric oxides

Electrical conductivities of αFe₂O₃ containing 1 2, 2 3 and 3 8 mol% CdO were measured in the temperature range of 300 to 1300°C at Po₂'s of 10^?9 to 10^?1 atm Plots of log $\text{σ}\text{vs}\text{1}\text{T}$ at constant Po₂'s were found to be linear with an inflection at temperatures around 500°C and higher Po₂'s than 1 × 10^?4 atm. The activation energies obtained by the least-squares method were 1 34 eV for the intrinsic region and 0 51 eV for the extrinsic region on αFe₂O₃ doped with 38 mol% CdO The extrinsic conductivities disappeared at lower Po₂'s than 1 × 10^-4atm, and the intrinsic conduction appeared on the specimens investigated The electrical conductivities decreased with increasing amount of CdO doping The predominant defects in this system are believed to be interstitial Fe² for the intrinsic region and oxygen vacancies for the extrinsic region.     

Diffusion and point defects in TiO2−x

The self-diffusion of ⁴⁴Ti has been measured both parallel to and perpendicular to the c axis in rutile single crystals by a serial-sectioning technique as a function of temperature (1000–1500°C) and oxygen partial pressure (10^?14 ? 1 atm). The oxygen-partial-pressure dependence of. D¹_Ti indicates that cation selfdiffusion occurs by an interstitial-type mechanism and that both trivalent and tetravalent interstitial titanium ions may contribute to cation self-diffusion. At p_o2 = 1.50 × 10^?7 atm where impurity-induced defects are unimportant,

$\text{D}{}^1{}_{\mathrm{<mtext>Ti</mtext>}}\text{(∥c)=}\text{6.50}\text{+1.33}\text{?1.11}\text{exp}\text{?}\text{(66.11±0.56}\text{kcal}\text{mole}\text{RT}\text{cm}{}^2\text{S}$

and

$\text{D}{}^1{}_{\mathrm{<mtext>Ti</mtext>}}\text{(⊥c)=}\text{4.55}\text{+1.78}\text{?1.28}\text{exp}\text{?}\text{(64.08±0.99)}\text{kcal}\text{mole}\text{RT}\text{cm}{}^2\text{S}\text{.}$

In the intrinsic region, the ratio D¹_Ti (⊥c)/D¹_Ti(∥c) was found to increase from 1.2 to 1.6 as the temperature decreased from 1500 to 1000°C. Computations based upon the defect model of Kofstad (involving the atomic defects Ti^..._iTi^...._iand V^.._o), of Marucco etal. (Ti^...._i and V^.._o), and of Blumenthal etal. (Ti^..._i and Ti^...._i) are compared with the experimental data on deviation from stoichiometry, electrical conductivity, cation self-diffusion and chemical diffusion in TiO_2?x. These comparisons provide values of the defect concentrations, cation-defect diffusivities, electron mobility and reasonable values of the correlation factor for cation diffusion by the interstitialcy mechanism. Only the model of Kofstad is inconsistent with the data.     

Mixed electrical conduction in Ce1−xCaxO2−x

The solid electrolyte Ce_1?xCa_xO_2-?x with the fluorite-type structure (Ca-doped CeO₂) is a mixed conductor. Conduction occurs predominantly by migration of O^2? ions via oxygen vacancies or by electrons, depending on the departure from stoichiometry. The ionic transference number σ_i/σ_i + σ_e was determined as a function of dopant concentration (0.07?x?0.15), temperature (400–800°C), and oxygen pressure by emf measurements with oxygen concentration cells. It is described by

The entropy term S¹(x) changes from 38.7k for x = 0.07 to 31.7k for x = 0.15; the enthalpy term, 5.42 eV, is independent of x and in excellent accord with semi-empirical calculations.     

Isotope effect for cation diffusion in CoO

The simultaneous diffusion of ⁵⁵Co and ⁶⁰Co has been measured in CoO as a function of equilibrium oxygen pressure in the range 10^?9 < po₂ < 1 atm at 1200°C. The slope of the log D¹_Co vs logpo₂ plot changes from a value of about $\text{1}\text{4}$ at high po₂ to about $\text{1}\text{5}$ at low po₂, in agreement with the extensive measurements of Dieckmann. The isotope effect is independent of po₂, which suggests that diffusion by defect clusters, interstitial Co ions and impurity-induced defects is not important in the present measurements. Conductivity, diffusion, stoichiometry and isotope-effect results are consistent with diffusion by neutral, singly charged, and doubly charged vacancies; the relative contributions from the various vacancies varies with po₂.     

Hydrogen absorption in Zr(AlxB1−x)2 (BFe,Co) laves phase compounds

The hydrogen absorption capacity of the systems Zr(Al_xFe_1?x)₂ and Zr(Al_xCo_1?x)₂ (0? x ?1) was measured at hydrogen pressure of 70 atm and room temperature and at 40 atm and liquid nitrogen temperature. The two systems present very interesting and unexpected results.A dramatic rise in the hydrogen capacity occurs for small x values similar to previous results for the systems Zr(A_xB_1?x)₂ (A  V, Cr, Mn; B  Fe, Co; 0?x?1). The maximum hydrogen content in both systems is achieved for x ≈ $\text{2}\text{12}$ at 40 atm and 80 K. Further increase of the Al content leads, however, to a steep decrease in the hydrogen capacity. This general behaviour is well described by a phenomenological model, recently proposed by us, and thus supporting the importance of short-range neighbouring effects for the hydrogen absorption capacity. The influence of Al on the hydrogen sorption properties in different intermetallic compounds is discussed.     

On the definition of the pressure of an infinite system

A microcanonical system $\text{L}$^o_Λ is considered together with a manometer $\text{M}$_Λ. The thermodynamic limit Λ → ∞ is taken for the system $\text{L}$_Λ composed of $\text{L}$^o_Λ and $\text{M}$_Λ. This yields a definition of the pressure P(v, ε) of $\text{{}\text{L}{}^{\mathrm{o}}{}_{\mathrm{\Lambda }}\text{; Λ → ∞}}$ for given values ε and v of the energy and particle densities. P(v, ε) is shown to be equal to the thermodynamic function p(z, β) derived from the grand canonical ensemble for almost all values of ε and v provided the appropriate equilibrium values of the temperature β^?1 and the chemical potential μ = β^?1 log z are inserted.     

Diffusion of oxygen vacancies in yttrium iron garnet investigated by dynamic conductivity measurements

In yttrium iron garnet, Y₃Fe₅O₁₂, the oxygen vacancy concentration at high temperatures depends on the partial oxygen pressure. Due to the electron donating nature of the vacancies, changes in the oxygen vacancy concentration can be measured by electrical conductivity measurements. We discuss a dynamic method for studying the diffusion of oxygen vacancies by measurements of the time dependence of the electrical conduction after a change in the oxygen partial pressure has taken place. It is shown that the interpretation of the measurements is straightforward if the relative change in conductivity remains small (? 10%). Measurements were performed on single crystals and on polycrystalline samples at temperatures 900–1400°C. The samples were made n-type by substitution with Si or p-type by substitution with Ca, Zn or Pb. The partial oxygen pressure was changed between 1 and 0.1 atm. For all samples the diffusion coefficient D of the oxygen vacancies can be represented by $\text{D = A}\text{exp}\text{(?}\text{Q}\text{kT}\text{)}$, where A = 8400 cm²s^?1 and Q = 2.90 eV. It is shown that the activation energy of 2.90 eV is due to the migration enthalpy of the vacancies only.     

A thermodynamic study on CaO-doped nonstoichiometric cerium dioxide

Thermogravimetric measurements were performed on CaO-doped nonstoichiometric cerium dioxide (i.e. Ce_1?yCa_yO_2?y?x) in the temperature range 800–1500°C and from oxygen pressures of 10^?1–10^?21 atm. Fr data the deviation from stoichiometry x = x(T, P_O2)_y was determined for values of y = 0.01, 0.045, 0.07, 0.10 and 0.14. The thermodynamic quantities Δ H_O2; and ΔS_O2 were calculated in the region 0.001 ? x ? 0.2 and found to be independent of temperature with the exception of those for 14 mole % CaO at large deviations from stoichiometry. p]In the composition region near stoichiometry (i.e. for x between approximately 10^?3 and 10^?2), the variation of Δ S_O2 with x is consistent with a defect model involving randomly distributed doubly ionized oxygen vacancies and electrons localized on normal cerium sites. The range of nonstoichiometry with which this defect model fits the experimental results increases with increasing CaO content. In this same region ΔH?_O2 exhibits a slight dependence on x for y = 0.01 and 0.045 and is essentially independent of x for y = 0.7,0.10 and 0.14. The experimental observation that is rationalized on the basis of the above model for the case where y > x.At large deviations from stoichiometry the dependence of x = x(T, P_O2)_y is similar to the nonstoichiometric behavior of “pure” CeO_2?x when y = 0.01 or 0.045. The dependence of ΔH_O2; and ΔS_O2 on x is also similar. for y = 0.10 and 0.14 the nonstoichiometric behavior and the dependence of ΔH_O2 on x differs significantly from the behavior of “pure” CeO_2?x.     

Characterization of LiFeCl4 and AgFeCl4 ionic conductors

LiFeCl₄ and AgFeCl₄ are obtained by direct reaction between LiCl or AgCl and FeCl₃ at 300°C and 400°C respectively. Both compounds are monoclinic with $\text{a = 7.02 (1)}\text{A}\text{?}$, $\text{b = 6.33 (1)}\text{A}\text{?}$, $\text{c = 12.72 (4)}\text{A}\text{?}$, β = 92° (30') for LiFeCl₄ and $\text{a = 10.60 (5)}\text{A}\text{?}$, $\text{b = 6.30 (5)}\text{A}\text{?}$, $\text{c = 12.34 (10)}\text{A}\text{?}$, β = 106° (1) for AgFeCl₄.LiFeCl₄ is clearly isotypic of LiAlCl₄. Magnetic measurements characterize in both cases Fe³⁺ ions in a high spin tetrahedral situation. LiFeCl₄ becomes antiferromagnetic at low temperature (T_N?10 K). AgFeCl₄ reveals a more complex situation. On contrary to the silver derivative, LiFeCl₄ is a good ionic conductor with activation energy of 0.78 eV in the solid state below 105°C, and a sharp increase in the lithium mobility at this temperature.     

Concepts of convergence for a quantum law of large numbers

We introduce the notion of uniform convergence of an operator sequence relative to a state operator, and we compare it with the notion of almost everywhere convergence, introduced by Gudder. We show that a sequence (M_n) of mean-value operators

$\text{M}{}_{\mathrm{n}}\text{=}\text{1}\text{n}\text{∑n}\text{i=1}\text{1?…?A}{}_{\mathrm{i}}\text{?1?…,}$

defined on the infinite tensor product $\text{?}\text{i}\text{H}{}_{\mathrm{i}}$, converges relative uniformly to $\text{x}\text{1}$ if and only if it converges almost everywhere to $\text{x}\text{1}$.     

The ground state of methane 12CH4 through the forbidden lines of the ν3 band

High resolution spectra of the ν₃ band of methane, ¹²CH₄, were recorded by using a “third generation vacuum Fourier interferometer”; a large pressure range (from 0.009 to 10 Torr) with a sample path fixed at eight meters was used, enabling observation of transitions with intensity ratios as low as $\text{1}\text{10 000}$. More than 350 forbidden transitions of the ν₃ band, including about 125 transitions of the Q⁺ branch, were unambiguously identified. Of the 277 transitions retained for computations, one-hundred have 11 ≤ J ≤ 16. From combination difference relations using pairs of transitions having the same upper state energy level (forbidden-allowed and forbidden-forbidden pairs were used), 276 independent differences between ground state energy levels could be determined with uncertainties of about 0.001 cm^?1.These data yielded the following values for the ground state structure constants of ¹²CH₄ along with their standard deviations (in cm^?1): $\text{β}{}^{\mathrm{o}}\text{hc}\text{=5.2410356±0.0000096}$, $\text{γ}{}^{\mathrm{o}}\text{hc}\text{=(?1±0.00074) 10}{}^{\mathrm{?4}}$, $\text{π}{}^{\mathrm{o}}\text{hc}\text{=(5.78±0.18) 10}{}^{\mathrm{?9}}$, $\text{?}{}^{\mathrm{o}}\text{hc}\text{=(?1.4485±0.0023) 10}{}^{\mathrm{?6}}$, $\text{?}{}^{\mathrm{o}}\text{hc}\text{=(1.768±0.126) 10}{}^{\mathrm{?10}}$, $\text{ξ}{}^{\mathrm{o}}\text{hc}\text{=(?1.602±0.067) 10}{}^{\mathrm{?11}}$, Thus, for the first time, the scalar constant π⁰ has been evaluated and ir values have been obtained for the two tetrahedral constants ?⁰ and ξ⁰; furthermore, these values are in very good agreement with the ones recently determined from radiofrequency data, i.e., in cm^?1: $\text{?}{}^{\mathrm{o}}\text{hc}\text{=(?1.45061±0.00014) 10}{}^{\mathrm{?6}}$, $\text{?}{}^{\mathrm{o}}\text{hc}\text{=(1.7634±0.0068) 10}{}^{\mathrm{?10}}$, $\text{ξ}{}^{\mathrm{o}}\text{hc}\text{=(?1.5432±0.0040) 10}{}^{\mathrm{?11}}$ From these values, the 276 differences can be reproduced with an overall rms deviation equal to 0.0009 cm^?1.Finally, the ground state energies of ¹²CH₄ have been calculated for J ≤ 16.     

Untersuchungen der Oxidbildung und Adsorption von Erdalkalien an einer sauerstoffvorbelegten Wolframoberfläche mit Hilfe der Oberflächenionisation

The surface ionization of alkaline-earth elements on tungsten has been studied in dependence on the temperature T and the surrounding oxygen partial pressure po₂; the values of the ionization efficiency β together with those of the change of the work function ΔΦ of the surface have been applied to get information about chemical reactions of the incident alkaline-earth atoms with adsorbed oxygen and about the adsorption of alkaline-earth elements on tungsten.Whereas in the high temperature range the tungsten surface is clean, towards lower temperatures (i.e. below ≈ 2500 K at po₂ = 1 × 10^?6 Torr or below ≈ 2000 K at po₂ = = 1 × 10^?9 Torr), an adsorption of oxygen increases the work function Φ and, consequently, the ionization efficiency β of incident metal atoms. A characteristic feature of the surface ionization of the alkaline-earth elements, however, is a rapid re-decrease of β with further decreasing temperature, which occurs at T ≈ 1400 K for Mg/W, T ≈ 1600 K for Ca/W, T ≈ 1800 K for Sr/W, and at T ≈ 2000 K for Ba/W. It is shown that this behaviour of β is caused by two different reasons: Whereas in the case of Mg/W a substantial Mg adsorption leading to a reduction of the work function is responsible for the decrease of β solely, the β values of Ca and Sr are additionally influenced by chemical reactions of the incident metal atoms with adsorbed oxygen resulting in an alkaline-earth oxide desorption. In the system $\text{Ba}\text{W}$ the decrease of the ionization efficiency β can be referred to BaO formation exclusively.Assuming a thermodynamic equilibrium between the different adparticles and using experimental values of the dissociation energy of the alkaline-earth oxides (in the gas phase), the results are in good agreement with theoretical calculations.     

A thermo-chemical study of the defect structure of barium oxide

A study of BaO has been made by use of thermogravimetric analysis, oxygen concentration analysis, and X-ray lattice parameter measurements in the temperature range 850°C ? T ? 1420°C and oxygen pressure range 7 × 10^-6 atm ? p_O2 ? 0·820 atm. Both the weight gain by the BaO samples and subsequently determined excess oxygen concentration were found to be directly proportional to . The enthalpy of incorporation oxygen in the lattice

$\text{1}\text{2}\text{O}{}_2\text{(g)=O(excess)}$

was determined to be ?0·395 ± 0·034 eV. Creation of vacancies on cation sites or of oxygen interstitials are consistent with the experimental results. As an alternative, the formation of O₂^2? ions, (as in BaO₂) as a result of incorporation of excess oxygen in the lattice, has been suggested.