Chemical diffusion in nonstoichiometric chromium sulfide,Cr2+yS3

Using the re-equilibration kinetic method the chemical diffusion coefficient in nonstoichiometric chromium sesquisulfide, Cr_2+yS₃, has been determined as a function of temperature (1073–1373 K) and sulphur vapour pressure (10?10⁴ Pa). It has been found that this coefficient is independent of sulphur pressure and can be described by the following empirical equation: $\text{D}\text{?}\text{Cr}{}_{\mathrm{2+y}}\text{S}{}_3\text{=50.86}\text{exp(-39070 cal/mole/}\text{RT)}\text{(cm}{}^2\text{s}{}^{\mathrm{?1)}}$. It has been shown that the mobility of the point defects inCr_2+yS₃ is independent of their concentration and that the self-diffusion coefficient of chromium in this sulfide has the following function of temperature and sulphur pressure: . (cm²s^?1).     

Transverse spin correlation function of the one-dimensional spin-12 XY model

The transverse spin pair correlation function p^x_n=<S^x_mS^x_m+n>=<S^x_mS^x_m+n> is calculated exactly in the thermodynamic limit of the system described by the one-dimensional, isotropic, spin-$\text{1}\text{2}$, XY Hamiltonian

$\text{H=?2J}\text{∑}\text{l=1}\text{N}\text{(S}{}^{\mathrm{x}}{}_{\mathrm{l}}\text{S}{}^{\mathrm{x}}{}_{\mathrm{l+1}}\text{+S}{}^{\mathrm{y}}{}_{\mathrm{l}}\text{S}{}^{\mathrm{y}}{}_{\mathrm{l+1}}\text{)}$

. It is found that at absolute zero temperature (T = 0), the correlation function ρ^x_n for n ≥ 0 is given by

$\text{ρ}{}^{\mathrm{x}}{}_{\mathrm{2p}}\text{=}\text{1}\text{4}\text{2}\text{π}{}^{\mathrm{2p}}\text{Π}\text{j=1}\text{p?1}\text{4j}{}^2\text{4j}{}^2\text{?1}{}^{\mathrm{2p?2j}}\text{if}\text{n=2p}$

,

$\text{ρ}{}^{\mathrm{x}}{}_{\mathrm{2p+1}}\text{=±}\text{1}\text{4}\text{2}\text{π}{}^{\mathrm{2p+1}}\text{Π}\text{j=1}\text{p}\text{4j}{}^2\text{4j}{}^2\text{?1}{}^{\mathrm{2p+2j}}\text{if}\text{n=2p+1}$

, where the plus sign applies when J is positive and the minus sign applies when J is negative. From these the asymptotic behavior as n → ∞ of |?^x_n| at T = 0 is derived to be $\text{|ρ}{}^{\mathrm{x}}{}_{\mathrm{n}}\text{| ～}\text{a}\text{n}$ with a = 0.147088?. For finite temperatures, ρ^x_n is calculated numerically. By using the results for ?^x_n, the transverse inverse correlation length and the wavenumber dependent transverse spin pair correlation function are also calculated exactly.     

EPR of V2+ in SrCl2

The X-band EPR spectrum of SrCl₂:V has been measured at liquid nitrogen temperature. A signal associated with V²⁺ in a site of trigonal symmetry is observed. The EPR data have been explained using the spin hamiltonian $\textcolor{red}{}\text{= μ}{}_{\mathrm{\beta }}\text{H}\text{g}\text{?}\text{S}\text{+ D[S}{}^2{}_{\mathrm{z}}\text{?}\text{1}\text{3}\text{S(SH)] +}\text{S}\text{A}\text{?}\text{I}$, with D ? hv, g_∥ = 1.957 ± 0.004, g₆ = 1.954 ± 0.004, A_∥ = 230 ± 5 MHz, A₆ = 235 ± 5 MHz. This V²⁺ defect is similar to those previously reported in fluoride crystals with the fluorite structure.     

Kerr-effect and dielectric tensor elements of magnetite (Fe3O4) between 0.5 and 4.3 eV

The complex polar Kerr effect (rotation and ellipticity) of magnetite single crystals has been measured at room temperature between 0.5 and 4.3 eV. From the magneto-optical data and the optical constants, the off-diagonal elements (?_xy) of the dielectric tensor has been derived. Three well separated magneto-optical transitions have been indentified. At about 0.75 eV one strong magneto-optical structure with a diamagnetic line shape is assigned to a $\text{3d}{}^6\text{→3d}{}^5\text{(}{}^6\text{A}{}_{\mathrm{1g}}\text{) 4s}$ transition from Fe²⁺ in octahedral sites. Two other structures with paramagnetic line shapes near 1.85 and 2.90 eV are assigned to the orbital promotion processes $\text{3d}{}^6\text{(Fe}{}^{\mathrm{2+}}{}_{\mathrm{oct}}\text{)→3d}{}^5\text{(}{}^4\text{T}{}_{\mathrm{1g}}\text{) 4s}$ and $\text{3d}{}^5\text{(Fe}{}^{\mathrm{3+}}{}_{\mathrm{tet}}\text{)→3d}{}^4\text{(}{}^5\text{T}{}_2\text{) 4s}$, respectively, which take into account Fe 3d^n?1 final state effects.     

Electron exchange between Fe2+ and Fe3+ ions on octahedral sites in spinels studied by means of paramagnetic Mössbauer spectra and susceptibility measurements

Mössbauer spectra were obtained of the paramagnetic spinels $\text{Zn}{}^{\mathrm{2+}}\text{|}\text{Zn}{}^{\mathrm{2+}}{}_{\mathrm{<mtext>(1?x)</mtext><mtext>2</mtext>}}\text{Ti}{}^{\mathrm{4+}}{}_{\mathrm{<mtext>(1+x)</mtext><mtext>2</mtext>}}\text{Fe}{}^{\mathrm{3+}}{}_{\mathrm{<mtext>(1?</mtext><mtext>x)</mtext>}}\text{Fe}{}^{\mathrm{2+}}{}_{\mathrm{x}}\text{|}\text{O}{}_4$ and susceptibilities were measured. The strong difference between the paramagnetic Fe²⁺ and Fe³⁺ spectrum, due to the different quadrupole splitting, is used for the distinction between the two species. At 300 K a superposition of the Fe³⁺ and the Fe²⁺ spectra is found for most of the iron and, in addition, some continuous absorption. The latter is strongest for equal Fe³⁺ and Fe²⁺ concentration $\text{(x =}\text{1}\text{2}\text{)}$ while it disappears towards the end members (Fe³⁺ only or Fe²⁺ only) as well as with decreasing temperature (between 78 and 200 K). From this it is concluded that it arises from thermally activated electron exchange, the frequency of which passes a “critical” value of ～10⁸ sec^?1 for increasing temperature. Paramagnetic susceptibilities are found to obey a Curie-Weiss law down to low temperatures. From the dependence of the asymptotic Curie temperature on the composition the magnetic interaction parameters J₁₁ = ?1.4 K, J₂₂ = ?3.3 K and J₁₂ = + 1.6 K for the Fe³⁺Fe³⁺, Fe²⁺Fe²⁺ and Fe³⁺Fe²⁺ interactions are derived. The experimental results are discussed in terms of a hopping model with an activation energy q ～- 0.12eV and a non-equivalence of the octahedral sites expressed by a varying potential energy difference U₀ between neighbouring sites. The continuous absorption at 300 K for $\text{x =}\text{1}\text{2}$ is attributed to about 17% of the iron on sites with U₀ running from 0 to ??0.06 eV. The ferromagnetic Fe³⁺, Fe²⁺ interaction (J₁₂) is attributed to electron transfer from localized Fe²⁺ ions to Fe³⁺ neighbours via a transfer integral b of the order of 0.05 eV. The magnitudes of J₁₂ and b are tentatively explained.     

Temperature dependence of the ESR spectra of Mn2+ in iron pyrite (FeS2)

The first observation of the ESR spectra of Mn²⁺, entering substitutionally for Fe²⁺ in the Van Vleck paramagnet FeS₂ (polycrystals), is reported. The data from 5 to 295 K fits the spin-Hamiltonian $\text{h}{}_{\mathrm{s}}\text{= gβ}\text{H}\text{·}\text{S}$ + $\text{[S}{}^{\mathrm{<mtext>2</mtext>}}{}_{\mathrm{z}}\text{?}\text{1}\text{3}\text{S(S + 1)] + A}\text{S}\text{·}\text{I}$, with g = 2.000 ± 0.001, A/β = ?95.0 ± 0.5 Oe and D/β varying from 50 Oe (5K) to 59 Oe (295 K). The temperature dependence of D can be described in terms of a single phonon-mode with frequency ? 145 cm^?1.     

Analysis of the (ν1 + ν2) and (ν2 + ν3) combination bands of ozone

The fine structures of the (ν₁ + ν₂) and (ν₂ + ν₃) combination bands of ozone in the 5.7-μm region have been recorded and analyzed. The two vibrational states are coupled through Coriolis and second-order distortion terms. The interaction has been treated by the numerical diagonalization of the secular determinant for the two coupled states. With the centrifugal distortion parameters fixed to the ground state values, the following constants have been obtained: ν₁ + ν₂ = 1796.26₆, A₁₁₀ = 3.610₄, B₁₁₀ = 0.4414₅, $\text{C}{}^1{}_{110}\text{= 0.3902}{}_9$, ν₂ + ν₃ = 1726.52₆, A₀₁₁ = 3.553₇, B₀₁₁ = 0.4398₂, $\text{C}{}^1{}_{011}\text{= 0.3884}{}_4$, Y₁₃ = ?0.46₆, and X₁₃ = ?0.01₀ cm^?1. In addition, the following anharmonic constants have been obtained: x₁₂ = ?7.82₁ and x₂₃ = ?16.49₄ cm^?1. The value of the dipole moment ratio, $\text{R =}\text{〈011|μ}{}_{\mathrm{z}}\text{|0〉}\text{〈110|μ}{}_{\mathrm{x}}\text{|0〉}$, is 1.30 ± 0.10.     

The A-dependence of J/ψ-particle inclusive distributions

The differential cross sections $\text{d}\text{σ}\text{d}\text{x}$ and $\text{d}\text{σ}\text{d}\text{p}{}_{\mathrm{t}}{}^2$ of inclusive J/ψ production by 43 GeV/c π^? off Be, Cu and W nuclei have been measured. Fitting $\text{d}\text{σ}\text{d}\text{p}{}_{\mathrm{t}}{}^2\text{～ A}{}^{\mathrm{\alpha }}\text{(p}{}_{\mathrm{t}}{}^2\text{)}$ we observed the increase of α with p_t².     

Thioformaldehyde: Vibrational analysis of the Ã1A2-X̃1A1 visible absorption system

The electronic absorption spectra of thioformaldehyde and thioformaldehyde-d₂ have been obtained. A vibrational analysis of the discrete band system in the 6100-4400-Å region is reported. The type A origin bands are at $\text{16 394}\text{16 484}\text{cm}{}^{\mathrm{?1}}$ for $\text{CH}{}_2\text{S}\text{CD}{}_2\text{S}$, and are magnetic dipole allowed. The electronic transition is $\text{A}\text{?}{}^1\text{A}{}_2\text{-}\text{X}\text{?}{}^1\text{A}{}_1$ under the C_2v point group. Most of the intensity of the system is in type B bands, and is due to vibronic mixing with higher ¹B₂ states when the inversion mode ν₄ is excited. The molecule in the excited ¹A₂ state is “floppy-planar,” having a broad potential function with a barrier of the order of 20 cm^?1 to the inversion motion.     

Analytical properties of conformal invariant four point functions

On the basis of the known group theoretical structure of the conformal invariant four point functions in the case of identical scalar fields ?(x) of scale dimension d,

$\text{<0|?(x}{}_1\text{)?(x}{}_2\text{)?(x}{}_3\text{)?(x}{}_4\text{)|0〉 = [(x}{}_1\text{?x}{}_4\text{)}{}^2\text{(x}{}_2\text{?x}{}_3\text{)}{}^2\text{]}{}^{\mathrm{-d}}\text{g(A,B),}$

the analytical properties of g(A, B) as a function of the harmonic ratios A and B are investigated. By imposing the conditions of spectrality and locality, and using invariance under complex dilatations, it is shown that the function g(A, B) must be homomorphic in the whole complex A-plane and B-plane with exception of the values A=0 and A=∞, B arbitrary, and B=0 and B=∞, A arbitrary.     

Bounded diffusion in solid solution electrode powder compacts. Part II. The simultaneous measurement of the chemical diffusion coefficient and the thermodynamic factor in LixTiS2 and LixCoO2

Two electrochemical methods - involving the application of a long-time galvanostatic current pulse and a small potentiostatic voltage step to a M/M_xSSE cell - are presented. From the overvoltage, respectively current response the chemical diffusion coefficient ($\text{D}{}_{\mathrm{M}}{}^{\mathrm{+}}$) and the thermodynamic factor (? ln a/? ln c) are obtained. The methods have been applied to the cells: Li/1M·LiClO₄ in propylenecarbonate/Li_xTi_1.03S₂ 0.05 < x < 0.95, T = 20°C; and Li_xCoO₂ 0.10 < × < 1, T = 20°C. From the application of the current pulse/voltage decay method it followed: $\text{D}{}_{\mathrm{<mtext>Li</mtext>}}{}^{\mathrm{+}}\text{(}\text{Li}{}_{\mathrm{x}}\text{Ti}{}_{1.03}\text{S}{}_2\text{) = 1?4 × 10}{}^{\mathrm{?8}}\text{cm}{}^2\text{s}{}^{\mathrm{?1}}$, with a slight tendency to increase with decreasing x; $\text{D}{}_{\mathrm{<mtext>Li</mtext>}}\text{C}\text{(}\text{Li}{}_{\mathrm{x}}\text{CoO}{}_2\text{) = 2?40 × 10}{}^{\mathrm{?9}}\text{cm}{}^2\text{s}{}^{\mathrm{?1}}$, decreasing with decreasing x. These values are among the highest found for solid state Li⁺-ion diffusion, and will be closely evaluated and compared with data reported by other workers. The x-dependence of the thermodynamic factor, determined from kinetic data, for Li_xTi_1.03S₂ (x = 0.05-0.95) and Li_xCoO₂ (x = 0.60-1.00) is in accordance with a simple thermodynamic model. Unlike for Li_xTi_1.03S₂, the thermodynamic factor for Li_xCoO₂, determined from the EMF-x relation, cannot be accounted for by this model. Furthermore, a fast, but crude method to determine the average chemical diffusion coefficient in Li_xTi_1.03S₂ and Li_xCoO₂ is discussed.     

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.     

Theory of the spin depolarisation rate of μ+ SR in pure Fe

The diffusion constant (D) and the spin depolarisation rate (λ_S) of the positive muon in the ferromagnetic b.c.c. lattices such as Fe have been calculated on the basis of the small polaron (SP) model with particular attention to the hopping and the coherent motions. They have been obtained in the forms: $\text{D=}\text{d}{}^2\text{w}{}_{\mathrm{D}}\text{6}$ and $\text{λ}{}_{\mathrm{S}}\text{=}\text{4γ}{}^2{}_{\mathrm{\mu }}\text{H}{}^2{}_{\mathrm{d}}\text{3w}{}_{\mathrm{S}}$, where d is a jump length, and H_d is a magnitude of internal dipolar field. At high temperatures, the activated-type hopping motion gives a dominant contribution to w_D, and w_S coincides with w_D. With decreasing temperature, w_D turns to be increasing owing to the coherent motion. Whereas w_S deviates from w_D, after passing the minimum, to have a maximum, and then decreases in inverse proportion to a lifetime (_B) of the SP band states. Because of the cell periodicity of the internal fields, the spin in an SP band state feels a definite magnetic field corresponding to the state, and sees various fields if scattered to other states. These scattering processes succeeding in a time shorter than (γ_μH_d)^-1 result in the motional narrowing of the spectra, with λ_{S ?} γ²_μH_d_B.     

Connections between thermodynamic quantities and vacancy and diffusion characteristics in binary metallic solid solutions

With the use of the similarity of interatomic potentials relations concerning vacancy and diffusion characteristics in disordered regular solid solutions have been derived. It has been shown that the vacancy concentration is constant along ifT_c(x) = T_A + (T_B ? T_A)x + 2ΔT_x(1?x), (T_A and T_B are the melting points of pure components A and B respectively, and ΔT is proportional to the excess enthalpy of mixing, x is the concentration of the atoms B) which is proportional to the binding energy of the crystal. The validity conditions of several empirical rules known in the literature are also analyzed. It has been found that the generalization of the well-known rule for self- and impurity diffusion in pure metals has the following form In $\text{D}{}_0{}^{\mathrm{z}}\text{(x) ～ p}\text{Q}{}^{\mathrm{z}}\text{(x)}\text{T}{}_{\mathrm{c}}\text{(x)}$ (Z = AorB) where p is a constant for alloys having identical structures (D₀^z(x) and Q^z(x) denote the preexponential factors and the activation energies respectively). The results calculated from the relations derived were compared with experimental data for tracer diffusion in the systems AgAu, CuNi (having slight deviation from regularity), Pb-Tl (showing ordering phenomena) and AlZn (clustering effects) and a good agreement was found.     

Determination of molecular constants of nitric oxide from (1-0), (2-0), (3-0) bands of the 15N16O and 15N18O isotopic species

The (1-0), (2-0), and (3-0) transitions of ¹⁵N¹⁶O and ¹⁵N¹⁸O are investigated. The wavenumbers of the rotation-vibration lines are reported for the overtone bands and the ${}^2\text{Π}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}\text{-}{}^2\text{Π}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ (1-0) subband. It is shown that in the data reduction it is advantageous to calculate first merged spectroscopic constants ignoring the Λ-type doubling. The vibrational constants ω_e, ω_ex_e, ω_ey_e and the vibrational dependence of the rotational constants are determined. The study of ¹⁵N¹⁸O allows the determination of the equilibrium values of the centrifugal distortion correction A_De to the spin-orbit constant and of the spin-rotation constant γ_e from the isotopic invariance of the ratios $\text{A}{}_{\mathrm{<mtext>De</mtext>}}\text{B}{}_{\mathrm{<mtext>e</mtext>}}$ and $\text{γ}{}_{\mathrm{<mtext>e</mtext>}}\text{B}{}_{\mathrm{<mtext>e</mtext>}}$. It is found that $\text{A}{}_{\mathrm{<mtext>De</mtext>}}\text{B}{}_{\mathrm{<mtext>e</mtext>}}\text{= (?3.9 ± 1.3) × 10}{}^{\mathrm{?6}}$ and $\text{γ}{}_{\mathrm{<mtext>e</mtext>}}\text{B}{}_{\mathrm{<mtext>e</mtext>}}\text{= (?4.00 ± 0.05) × 10}{}^{\mathrm{?3}}$.     

Polarization spectroscopy of the E0g+-B0u+ band system of Br2

The E-B (0_g⁺-0_u⁺) band system of Br₂ has been investigated at Doppler-limited resolution using polarization labeling spectroscopy. Merged E state data for the three naturally occurring isotopes in the range v_E = 0–16, expressed in terms of the constants for ⁷⁹Br₂, are (in cm^?1) Y_0,0 = 49 777.962(54), Y_1,0 = 150.834(22), Y_2,0 = ?0.4182(28), Y_3,0 = 6.6(11) × 10^?4, Y_0,1 = 4.1876(28) × 10^?2, Y_1,1 = ?1.607(16) × 10^?4, and Y_0,2 = 1.39(39) × 10^?8. The bond distance is $\text{r}{}_{\mathrm{<mtext>e</mtext>}}\text{= 3.194}\text{A}\text{?}$, and the diabatic dissociation energy to $\text{Br}{}^{\mathrm{+}}\text{(}{}^3\text{P}{}_2\text{) +}\text{Br}{}^{\mathrm{?}}\text{(}{}^1\text{S}{}_0\text{)}\text{is 34 700 cm}{}^{\mathrm{?1}}$.     

Overtone bands of 14N16O and determination of molecular constants

The wavenumbers of the rotation-vibration lines of ¹⁴N¹⁶O are reported for the (2-0) and (3-0) bands. The full set of spectroscopic constants for the three bands (1-0), (2-0), and (3-0) has been determined with the method developed by Albritton, Schmeltekopf, and Zare for merging the results of separate least-squares fits. The vibrational constants ω_e, ω_ex_e, ω_ey_e, and the vibrational dependence of the rotational constants have been deduced. The apparent spin-orbit constant $\text{A}\text{?}{}_{\mathrm{<mtext>v</mtext>}}$ and its centrifugal correction $\text{A}\text{?}{}_{\mathrm{<mtext>D</mtext>}}$ (including the spin-rotation constant) have a vibrational dependence of the following form: $\text{A}\text{?}{}_{\mathrm{v}}\text{=}\text{A}\text{?}{}_{\mathrm{e}}\text{? α}{}_{\mathrm{A}}\text{(v +}\text{1}\text{2}\text{) + γ}{}_{\mathrm{A}}\text{(v +}\text{1}\text{2}\text{)}{}^2$ and $\text{A}\text{?}{}_{\mathrm{<mtext>D</mtext><mtext>v</mtext>}}\text{=}\text{A}\text{?}{}_{\mathrm{<mtext>D</mtext><mtext>e</mtext>}}\text{? β}{}_{\mathrm{A}}\text{(v +}\text{1}\text{2}\text{) + δ}{}_{\mathrm{A}}\text{(v +}\text{built+1}\text{2}\text{)}{}^2$; the values of the constants in these two equations have been determined.     

Casimir energy of a scalar field with a space-dependent mass distribution

The Casimir energy is evaluated for a free scalar field that has a mass term m²(x₁), depending on one space coordinate x₁. The formalism for evaluating the Casimir energy is developed for the case of m²(x₁) finite everywhere in d-dimensional space-time. The case with $\text{m}{}^2\text{(x}{}_1\text{) = m}{}_0{}^2\text{θ(}\text{1}\text{2}\text{L}\text{-|x}{}_1\text{|) + m}{}_{\mathrm{\infty }}{}^2\text{θ (|x}{}_1\text{| -}\text{1}\text{2}\text{L}$) is explicity evaaluated for any value of 1197 1568 V m₀ and m_∞ without any approximation. The result consists of valume energy terms, a surface term, and a non-leading term. Most of the UV divergences are in the volume energy terms and renormalize the coupling constants of the underlying theory. The surface energy term is finite for d ? 4 and divergent for d ? 5 due to the boundaries being sharp. A closed finite expression is obtained for the non-leading term. Our results are shown to reproduce the known Casimir energies for the limiting cases, 1197 15 m₀ → ∞ andm_∞ → ∞.     

Microwave optical double resonance and continuous wave dye laser excitation spectroscopy of NO2: Rotational assignment of the K = 0−4 subbands of the 593 nm band

A rotational assignment of approximately 80 lines with K_a′ = 0, 1, 2, 3, and 4 has been made of the 593 nm ${}^2\text{A}{}_1\text{→}{}^2\text{B}{}_2$ band of NO₂ using cw dye laser excitation and microwave optical double-resonance spectroscopy. Rotational constants for the ²B₂ state were obtained as A = 8.52 cm^?1, B = 0.458 cm^?1, and C = 0.388 cm^?1. Spin splittings for the K_a′ = 0 excited state levels fit a simple symmetric top formula and give $\text{(?}{}_{\mathrm{bb}}\text{+ ?}{}_{\mathrm{cc}}\text{)}\text{2}\text{= ?0.0483}\text{cm}{}^{\mathrm{?1}}$. Spin splittings for K_a′ = 1 (N′ even) are irregular and are shown to change sign between N′ = 6 and 8. Assuming that the large inertial defect of 4.66 amu Ų arises solely from A, a structure for the ²B₂ state is obtained which gives $\text{r}\text{(NO) = 1.35}\text{A}\text{?}$ and an ONO angle of 105°. Alternatively, weighting the three rotational constants equally gives $\text{r = 1.29}\text{A}\text{?}$ and θ = 118°.     

Inclusive pion distributions in e+e− annihilation from sum rules and duality

The threshold behaviour of pion production presented in our earlier work is successfully compared with the new SPEAR data. By using duality and sum rules we derive F_T^(π+)(x) ≈ F_L^(π+)(x) ≈ F_T^(π0)(x) ? F_L^(π0)(x) for x near 1. An accompanying results is $\text{σ}{}_{\mathrm{<mtext>\pi </mtext><mtext>A</mtext>}}{}_2\text{(s) ≈ 2σ}{}_{\mathrm{\pi \omega }}\text{(s) ≈ 4σ}{}_{\mathrm{\pi \pi }}\text{(s) ≈ 9(m}{}_{\mathrm{?}}{}^2\text{/s)}{}^3\text{σ}{}_{\mathrm{<mtext>\mu </mtext><mtext>\mu </mtext>}}$ for large s.