Selection rules for optical transition probabilities in superlattices

Based on numerical evaluations of the transition probabilities, and using true superlattice eigenfunctions derived from the theory of finite periodic systems, new superlattice selection rules are obtained. We show that the evaluation of the large number N   of the possible recombination transitions reduces by a factor of two. We also show that taking into account these selection rules, the theoretical calculations for specific blue emitting GaN?(In_xGa_1-xN?In_yGa_1-yN)ⁿ?GaN$\text{GaN}?({\text{In}}_x{\text{Ga}}_{1-x}\text{N}?{\text{In}}_y{\text{Ga}}_{1-y}\text{N}{)}^n?\text{GaN}$ superlattices agree with those reported by Nakamura et al. [1].     

Influence of the Miniband Width on the Relaxation Time of Superlattice Electrons Scattered by Impurity Ions

Based on the Boltzmann equation, the influence of the miniband width on the relaxation time of nondegenerate electrons scattered by impurity ions in the GaAs/Al _x Ga_1–x As superlattice with doped quantum wells is numerically analyzed. The wave function being the eigenfunction of the ground state of the lower miniband of the superlattice is used to calculate the scattering probability. The dispersion of the longitudinal and transverse relaxation times versus the longitudinal wave vector is investigated.     

On the Schrödinger equation for the x2 + λx2(1 + gx2) interaction

We present an infinite set of exact solutions and eigenvalues for the one-dimensional Schrödinger equation involving the potential $\text{x}{}^2\text{+}\text{λx}{}^2\text{(1 + gx}{}^2\text{)}$. Comparison with numerical methods is made.     

Singular perturbation potentials

This is a perturbative analysis of the eigenvalues and eigenfunctions of Schrödinger operators of the form ?Δ + A + λV, defined on the Hilbert space L²(Rⁿ), where $\text{Δ = Σ}{}_{\mathrm{i=1}}{}^{\mathrm{n}}\text{?}{}^2\text{?X}{}_{\mathrm{i}}{}^2$, A is a potential function and V is a positive perturbative potential function which diverges at some finite point, conventionally the origin. λ is a small real or complex parameter. The emphasis is on one-dimensional or separable problems, and in particular the typical example is the “spiked harmonic oscillator” Hamiltonian, $\text{?d}{}^2\text{dx}{}^2\text{+ x}{}^2\text{+}\text{l(l + 1)}\text{x}{}^2\text{+ λ|x|}{}^{\mathrm{?\alpha }}$, where α is a positive constant. When this kind of perturbation is very singular, the first-order Rayleigh-Schrödinger perturbative correction, (u₀, Vu₀), where u₀ is the unperturbed eigenfunction, diverges. This analysis constructs explicitly calculable terms in a modified perturbation series to a finite order, by using linear operator theory in concert with approximation methods for differential equations. Along the way a connection between a W-K-B type approximation and Bessel functions is exploited.     

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.     

Predissociation and Λ-doubling in the even-parity Rydberg states of the nitrogen molecule

Predissociations in the y¹Π_g and $\text{x}{}^1\text{Σ}{}_{\mathrm{g}}{}^{\mathrm{?}}$ Rydberg states of N₂ (configurations $\text{1π}{}_{\mathrm{u}}{}^{\mathrm{?1}}\text{4pσ}$ and $\text{1π}{}_{\mathrm{u}}{}^{\mathrm{?1}}\text{3pπ}$, respectively) and their likely causes, are discussed. Peaking of rotational intensity at unusually low J values, without sharp breaking off, is interpreted as due to case c^? or case cⁱ predissociation. Λ doubling in the y state, attributed to interactions with the $\text{x}{}^1\text{Σ}{}_{\mathrm{g}}{}^{\mathrm{?}}$ state and with another, ¹Σ⁺, state of the same electron configuration as x, is analyzed. From this analysis the location of the (unobserved) ${}^1\text{Σ}{}_{\mathrm{g}}{}^{\mathrm{+}}$ state, here labeled x′, is obtained. It is concluded that the predissociation in the Π⁺ levels of the y state is an indirect one mediated by the interaction with x′ coupled with predissociation of x′ by a ${}^3\text{Σ}{}_{\mathrm{g}}{}^{\mathrm{?}}$ state dissociating to ${}^4\text{S +}{}^2\text{P}$ atoms: combined, however, with perturbation of the y state by the k¹Π_g Rydberg state (configuration $\text{3σ}{}_{\mathrm{g}}{}^{\mathrm{?1}}\text{4dπ}$), whose Π⁺ levels are completely predissociated.     

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.     

A search for integrable two-dimensional hamiltonian systems with polynomial potential

We report the results of a search for all integrable hamiltonian systems of type $\text{H = (}\text{1}\text{2}\text{)p}{}_{\mathrm{x}}{}^2\text{+ (}\text{1}\text{2}\text{)p}{}_{\mathrm{y}}{}^2\text{+ V(x,y)}$, where V is a polynomial in x and y of degree 5 or less and the second invariant is a polynomial in p_x and p_y of order 4 or less. Both classical and quantum integrability are discussed.     

Stereoelectronic properties of photosynthetic and related systems: Ab initio configuration interaction calculations on the ground and lower excited singlet and triplet states of magnesium chlorin and chlorin

Ab initio configuration interaction wavefunctions and energies are reported for the ground state and many low-lying singlet and triplet states of magnesium chlorin and chlorin, and are employed in an analysis of the electronic absorption spectra of these systems.In chlorin, the calculated visible spectrum consists of two ${}^1\text{(π, π}{}^1\text{)}$ states, the lower energy, y-polarized state exhibiting moderate absorption intensity in contrast to the very weak absorption of the higher energy x-polarized state. The configurational composition of both states is well described by the four-orbital model. Five ${}^1\text{(π, π}{}^1\text{)}$ states are responsible for the Soret band envelope. A moderately intense y-state lies under the low energy edge of the band envelope, while two x-polarized states of moderate and strong intensity, respectively, are responsible for the band maximum. The final two ${}^1\text{(π, π}{}^1\text{)}$ states lie at the high energy edge of the Soret band and introduce a measure of asymmetry into the band envelope. Two ${}^1\text{(n, π}{}^1\text{)}$ states of very weak oscillator strength are also found in this region of the spectrum. All the Soret states are of complex configurational composition, and several of the higher lying states contain contributions from doubly excited configurations.The calculated visible spectrum of magnesium chlorin also consists of two ${}^1\text{(π, π}{}^1\text{)}$ states, with the weakly absorbing x-polarized state lying approximately 200 cm^?1 lower in energy than the moderately intense y-polarized state. The configurational composition of both states is well described by the four-orbital model. Four ${}^1\text{(π, π}{}^1\text{)}$ states constitute the bulk of the intensity in the Soret band envelope. In distinction to chlorin, the moderately intense ${}^1\text{(π, π}{}^1\text{)}$ state at the low energy edge of the band envelope is x-polarized. Two intense ${}^1\text{(π, π}{}^1\text{)}$ states of y- and x-polarization, respectively, constitute the band maximum region, and a single x-polarized state of moderately strong intensity can be assigned to the high energy shoulder of the band envelope. Two other weakly absorbing ${}^1\text{(π, π}{}^1\text{)}$ states are also found in this region, along with another weakly absorbing state of mixed in-plane and out-of-plane polarization. No clearly defined ${}^1\text{(n, π}{}^1\text{)}$ states are observed. As was the case for chlorin, all the Soret states are of complex configurational composition, and some of the higher energy states contain significant contributions from doubly excited configurations.Chlorin and magnesium chlorin both possess three ${}^3\text{(π, π}{}^1\text{)}$ states which lie below S₁ and a single ${}^3\text{(π, π}{}^1\text{)}$ which lies slightly above S₂. All four of the low-lying ${}^3\text{(π, π}{}^1\text{)}$ states in each molecule are well described by the four-orbital model, with T₁ being essentially a single configuration in each case. The remainder of the ${}^3\text{(π, π}{}^1\text{)}$ states are clustered in the same energetic region as the comparable ${}^1\text{(π, π}{}^1\text{)}$ Soret states, with comparably complex configurational compositions.Dipole moments and charge distributions for low-lying singlet and triplet states are also reported, and are used to rationalize chemical reactivity characteristics.     

Lifetimes and g-factors of the 6− and 7− isomers in 66Ga and 68Ga

The 6^? and 7^? isomeric states in ⁶⁶Ga and ⁶⁸Ga at 1440.9 and 1229.6 keV, respectively, have been populated with the (¹³C, 2np) and (¹⁵N, n2p) reactions on natural Fe. The half-lives of these states have been measured to be $\text{T}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{(6}{}^{\mathrm{?}}\text{,}{}^{66}\text{Ga}\text{) = 57.3 ± 1.2}\text{ns}$ and $\text{T}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{(7}{}^{\mathrm{?}}\text{,}{}^{68}\text{Ga}\text{) = 64 ± 2}\text{ns}$. Using previous data on the hyperfine field of Ga in Fe, the g-factors of these states have been determined by means of the TDPAD method. The results are $\text{g(6}{}^{\mathrm{?}}\text{,}{}^{66}\text{Ga}\text{) = 0.129 ± 0.003}$ and $\text{g(7}{}^{\mathrm{?}}\text{,}{}^{68}\text{Ga}\text{) = 0.105 ± 0.003}$. These values are in very good agreement with the independent particle model if one assumes the and configurations and uses the empirical proton and neutron g-factors from odd-A neighboring nuclei instead of the Schmidt values. The large disagreements with experiment when Schmidt values are used show that core polarization effects are important in these nuclei.     

New potential energy function for alkali halide molecules

A new repulsive term in the ionic interaction potential for computing the lattice energy, the rotational constant (α₀) and the vibrational constant (w₀x₀) of alkali halide molecules has been proposed as $\text{ψ(r) = Af}{}^{\mathrm{n}}\text{r}{}^{\mathrm{-n}}\text{e}{}^{\mathrm{<mtext>?r</mtext><mtext>\lambda </mtext>}}$, which apart from being generalised, dimensionally homogenous and physically sound yields better values for lattice energy, α₀ and w₀x₀ than the previously reported potentials models.     

Inclusive K10(890) production in π−p interactions at 16 GeV/c

Inclusive $\text{K}{}^{10}$ production is studied in π^?p interactions at 16 GeV/c with x_? > 0.2. The $\text{K}{}^{10}$ is found to be pre-dominantly centrally produced with cross section $\text{σ(}\text{K}{}^{10}\text{) = (72 ± 12) μb}$ for x_? > 0.2 and compares closely to data on $\text{K}{}^{10}$ production in π⁺p interactions at the same energy.     

Analysis of (particle,xn) reactions on tantalum and gold: (II). 181Ta, 197Au(3He,xn) excitation functions and equilibrium statistical model analysis

Excitation functions for the reactions ${}^{197}\text{Au}\text{(}{}^3\text{He}\text{, x}\text{n}\text{)}{}^{\mathrm{200?x}}\text{Tl}\text{(x = 3, … 7)}$ and ${}^{181}\text{Ta}\text{(}{}^3\text{He}\text{, x}\text{n}\text{)}{}^{\mathrm{184?x}}\text{Re}\text{(x = 3, … 7)}$ were measured at bombarding energies up to 75 MeV. The data are compared to the results of an equilibrium statistical model calculation including angular momentum conservation and γ-ray competition. Satisfactory agreement was found except for the high energy tails.     

Gravitational collapse in quantum mechanics with relativistic kinetic energy

It is proved that the quantum mechanical Hamiltonian $\text{H = Σ}{}_{\mathrm{i=1}}{}^{\mathrm{N}}\text{(p}{}^2\text{+ m}{}^2\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{? κ Σ}{}_{\mathrm{i>j}}\text{|x}{}_{\mathrm{i}}\text{? x}{}_{\mathrm{j}}\text{|}{}^{\mathrm{?1}}$ for bosons (resp, fermions) is bounded from below if N ≤ c_bκ^?1 (resp. $\text{N ≤ c}{}_{\mathrm{f}}\text{κ}{}^{\mathrm{<mtext>?3</mtext><mtext>2</mtext>}}$). H is unbounded from below if N ≥ c_b^lκ^?1 (resp. $\text{N ≥ c}{}_{\mathrm{f}}{}^{\mathrm{l}}\text{κ}{}^{\mathrm{<mtext>?3</mtext><mtext>2</mtext>}}$). The constants c_b and c_b^l (resp. c_f and c_f^l) differ by about a factor 2 (resp. 4).     

The phase transition at −41°C in Cu4SnS4 as observed in electron microscopy

Evidence, by means of electron microscopy and diffraction, is presented for a phase transformation at around 232 K in Cu₄SnS₄. A superlattice is observed along $\text{q}{}_{\mathrm{1,2}}\text{= ±}\text{1}\text{2}\text{b}{}^1\text{+}\text{1}\text{2}\text{c}{}^1$, presumably associated with a displacive transformation. The first order transformation is associated with a space group change. Satellite dark field imaging revealed a domain structure composed of orientation and translation interfaces.     

Hopping conduction in the Landau level tails in GaAs-AlxGa1-xAs heterostructures at low temperatures

We have investigated the temperature dependence of the transverse conductivity σ_xx in the two-dimensional Shubnikov—de Haas oscillation minima in GaAs-Al_xGa_1-xAs heterostructures at temperatures between 10mK and 1 K. We found that for σ_xx ? σ_xx (max), where σ_xx (max) is the conductivity in the corresponding oscillation maxima, the results agree qualitatively with a theory recently published by Ono, predicting $\text{σ}{}_{\mathrm{xx}}\text{∝ (1/T) exp - (T}{}_0\text{/T)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$, but are not consistent with the estimated T₀-values.