Dependence of the mechanism and regularities of energy transfer in binuclear lanthanide complexes in solutions on the nature of the anion and solvent

The effect of anions contained in solutions on the energy transfer from Tb(III) and Dy(III) ions to different Ln(III) ions is investigated in aqueous and alcohol solutions. It is shown that the regularities revealed in the energy transfer are completely determined by the ratio between the dissociation rate of the binuclear complex and the rate of energy transfer in it. The rate constant k _t of energy transfer in solutions in which labile binuclear complexes of Ln(III) ions are linked through the strong acid anions Cl^?, NO ₃ ^? , and HSO ₄ ^? depends on the nature of ions in the pairs. It is demonstrated that the energy transfer in all the systems predominantly occurs through the induction-resonance mechanism. The rate constants k _t in aqueous solutions of weak (acetic, salicylic, and carbonic) acids also depend on the nature of ions interacting in pairs but do not correlate with the Förster overlap integral of the spectra. In labile binuclear complexes, the interaction between these ions proceeds by the exchange-resonance mechanism at a distance of ≈0.4 nm. It is established that the constants k _t in alcohol solutions of Ln(III) ions are virtually independent of the nature of the pairs of the ions interacting through the acetate bridge. A comparison of the dissociation rate constants for Ln-anion complexes in alcohol solutions and the expected intracomplex rates of energy transfer in the binuclear complexes offers a satisfactory explanation of the obtained results and makes it possible to determine the association constants for binuclear lanthanide complexes in these solutions.     

Energy transfer between lanthanide ions in nanostructures of their complexes. II

The changes in the luminescence intensity and in the luminescence lifetime of Eu, Tb, and Sm complexes with some β-diketones and 1,10-phenanthroline that occur upon formation of nanostructures with complexes of triply charged ions Pr, Nd, Sm, Eu, Dy, Ho, Er, Tm, and Yb—acceptors of the electronic excitation of luminescing ions—are compared for the first time. Pairs of lanthanide ions are suggested to exist inside nanostructures, which are bound by a bridge at a distance shorter (0.4 nm) in comparison with the size of interacting complexes. For a large number of pairs of donor and acceptor ions, the averaged probabilities (w _t) of energy transfer between these ions in nanostructures in the solution of each β-diketone under study are calculated. Based on the comparison of w _t with the predictions of the theory of the inductive resonance and exchange resonance energy transfer mechanisms, the average distance R between Ln(III) ions interacting in given fragments of nanostructures is estimated for each donor-acceptor pair, and the energy transfer mechanism is identified for the first time. It is shown that energy can be transferred between Ln(III) ions in nanostructures both according to the inductive resonance and via the exchange resonance mechanisms. The type of the transfer mechanism depends on the spectral parameters of interacting ions and on the ability of complexes of given acceptor ions to form heteronuclear nanostructures, whose composition determines the distance R. The variation in the value of R revealed for different ion pairs is explained by the occurrence of exchange resonance interactions between some ions. The overestimated probabilities w _t for ion pairs characterized precisely by exchange resonance interactions can be explained by the influence of covalent bonds in nanostructures of Ln chelates on π conjugation of overlapped electronic shells of interacting particles. By using the method of energy transfer between Ln(III) ions of complexes from distant spheres of a structure, the average minimal size of nanostructures formed is estimated to be 2.6–3.4 nm.     

Investigation of the formation of nanostructures of Ln(III) salts with phosphate and carbonate anions using Eu(III) chelates as luminescent markers

The monitoring of variations in the luminescence intensity (I _lum) of nanostructures of Eu(MBTA)₃phen (MBTA is p-methoxybenzoyl trifluoroacetonate) complexes formed in aqueous solutions upon the introduction of anions is proposed as a method of analyzing the composition of Eu(III), Gd(III) and Lu(III) phosphate complexes in solutions with [PO ₄ ^3? ] < [Ln]. It is found that low-lability binuclear complexes, which rearrange within an hour or longer, are formed in these solutions. It is shown that the lability of Ln(III) carbonate complexes exceeds the lability of Ln(III) complexes with PO ₄ ^3? . An analysis of the dependence of I _lum of the solution on the concentration of Eu(III) ions and on the time from the instant of the solution preparation shows that, in aqueous solutions where the concentration of anions is higher than the concentration of Ln(III) ions, nanostructures of Eu(III) phosphate and carbonate salts are formed in the range of Ln(III) concentrations 0.5–5 μM at concentrations of anions on the order of 10 μM and at concentrations of exceeding 100 μM. The rearrangement of these nanostructures to nanostructures of Eu(MBTA)₃phen complexes is studied.     

Energy transfer between lanthanide ions in nanostructures of their complexes. I

We study the regular features of the behavior of the intensity I _lum and the luminescence decay time τ _lum of complexes of Eu and Tb ions with several β-diketones and 1,10-phenanthroline in the case where these complexes from nanostructures with complexes of lanthanide ions that are electronic excitation acceptors of these Eu and Tb ions. The composition of mixed nanostructures formed in a solution is shown to depend on the method of their preparation, on the ability of complexes to form mixed rather than homogeneous nanostructures, and on the concentration of complexes in the solution. We reveal that complexes of Yb, Tm, and Dy ions simultaneously increase I _lum and τ _lum of Eu complexes due to energy transfer through ligands of complexes and decrease the value of these quantities for Eu complexes due to energy transfer from Eu(III) ions to ions of Yb, Tm, and Dy. For all interacting complexes, the changes in I _lum and τ _lum of complexes of Eu (Tb) in the presence of complexes, energy acceptors, are shown to be determined by competition between two processes: a decrease in these quantities due to energy transfer between ions and their increase caused by an increase in the probability of nonradiative transitions in Eu (Tb) ions due to an increase in the size of structures. We propose a method of separation of these two processes.     

Luminescence of dipicolinic complexes of lanthanide ions

The luminescence decay times τ_lum of the complexes of the ions Tb(III), Eu(III), Sm(III), Dy(III), and Yb(III) with dipicolinic acid (DPA) dissolved in protonated and deuterated water, methanol, and dimethyl sulfoxide are measured. The values of τ_lum for crystals H₃[Ln(DPA)₃]·nH₂O and their aqueous solutions coincide, which points to the identity of the environment in the nearest spheres of an ion in both cases. A comparison of τ_lum of solutions of the complexes in H₂O and D₂O, as well as in CH₃OH, CH₃OD, CD₃OD, DMSO-h ₆, and DMSO-d ₆ shows that the molecular groups in the second and third spheres of an ion, exhibiting high-frequency vibrations, have a noticeable effect on the rate constants of nonradiative transitions k _nr in the ion. From this comparison, some inferences on the structure of the solvate shell of the Ln(DPA) ₃ ^3? complexes in the solvents used are made. The contributions to k _nr of Eu(III), Tb(III), Sm(III), Dy(III), Nd(III), and Yb(III) made by OH and CH groups located at different distances from the ion are estimated. It is demonstrated that the dependence of k _nr on the distance to the OH and CH groups is steeper for the Eu(III) and Tb(III) ions than for the remaining ions.     

A mechanism of nonradiative transitions in lanthanide ions and the number of water molecules in their first coordination sphere

The luminescence decay times τ_lum of the ions Sm(III), Eu(III), Tb(III), and Dy(III) in glacial acetic acid, along with τ_lum and q _lum of these ions in H₂O and D₂O in the presence of anions CO ₃ ^2? and in their absence, are measured. The number of OH groups (N _OH) in the first coordination sphere of these lanthanide ions is determined. It was shown that, for all the ions in acetic acid, N _OH≈3, while, in an H₂O+2 M Cs₂CO₃ solution, N _OH≈2.5. The experimental data on the influence of the CO ₃ ^2? anions on the rate constant of nonradiative transitions (k _nr) in the Eu(III) and Tb(III) ions are compared with calculations of k _nr performed in the dipole-dipole approximation of the inductive resonance theory. It is found that such calculations cannot correctly describe the dependence of k _nr on N _OH. The quadrupole-dipole approximation of this theory was shown to be capable of adequately describing this dependence. The criteria for applying either approximation of the theory to describe experimentally observed dependences of k _nr on N _OH are discussed.     

Regularities of energy transfer in aqueous solutions of Ln(III) ions upon formation of their bridged complexes via the CO 3 2− and OH− anions

The influence of the solvent pH and of the presence of carbonic acid anions on the energy transfer from the Eu(III) and Tb(III) ions to a large namber of Ln(III) ions, as well as on the concentration quenching of the Dy(III) ions, is studied. It is shown that, when the anions are present in solution at a concentration by 2–4 orders of magnitude lower than that of the lanthanide ions, the energy transfer rates increase by 2–4 orders of magnitude, but the luminescence decay remains exponential. It is established that the rate constant for energy transfer (k _t ) via the hydroxyl bridge increases with decreasing concentration of lanthanide ions in solution. In an alkalinized solution, (k _t ) depends weakly on the initial water pH, because the concentration of hydroxo groups is governed by hydrolysis of water under the action of the lanthanide ions introduced into it. It is found that, at the 10^?2 M concentration of lanthanide ions in solution, the values of (k _t ) change by almost two orders of magnitude depending on the choice of an ion pair; however, these changes in no way correlate with the overlap integrals of spectra, calculated by adopting the Forster mechanism, and the inclusion of an interaction of higher multipoles does not improve the agreement with the experimental data. It is shown that, when the energy is transferred via the OH^? bridge in solutions with a total concentration of Ln(III) ions of 10^?4 M, the value of (k _t ) increases up to 10⁷ M^?1 s^?1 and becomes independent of the choice of pairs of Ln(III) ions. The dependence of (k _t ) on the ratio of the dissociation constant for a binuclear complex and the probability of energy transfer between the ions within this complex are discussed. It is concluded that the change in this ratio explains the disappearance of the dependence of (k _t ) on the choice of ions as their concentration in solution decreases, but does not explain all the observed changes in (k _t ) if only the dipole-dipole mechanism of energy transfer between the ions in bridged complexes is considered.     

Energy transfer between Ln(III) ions in aqueous solutions upon formation of labile binuclear fluoride and nitrate complexes of these ions

The influence of anions on nonradiative electronic energy transfer between lanthanide ions in aqueous solutions is studied. The rate constant k _t of energy transfer was found to increase by three orders of magnitude upon addition of salts of hydrofluoric acid to the solution. This effect is caused by the formation of bridge labile binuclear complexes, an excited energy donor-fluorine anion-acceptor, which increase the encounter time between a donor and an acceptor, resulting in the increase in the probability of energy transfer. The independence of k _t of the Förster overlap integral in the bridge complex containing F^? was explained by the fact that energy transfer occurs in each event of the binuclear complex formation, the maximum rate constant of energy transfer being equal to the rate constant of the binuclear complex association. Bridge complexes formed via the NO ₃ ^? anion are also considered. These binuclear complexes are unstable, and k _t for them is proportional to the Förster overlap integral. In this case, energy transfer occurs not during each event of the complex formation. This allows us to estimate the lower limit of the dissociation constant of the binuclear complex and its stability constant. Thus, the study of influence of various anions on energy transfer represents a new efficient luminescent method for analysis of properties of labile binuclear complexes in solutions.     

Gamow Vectors and Borel Summability in a Class of Quantum Systems

We analyze the detailed time dependence of the wave function ψ(x,t) for one dimensional Hamiltonians \(H=-\partial_{x}^{2}+V(x)\) where V (for example modeling barriers or wells) and ψ(x,0) are compactly supported.We show that the dispersive part of ψ(x,t) is the Borel sum of its asymptotic series in powers of t ^?1/2, t→∞. The remainder, the difference between ψ and the Borel sum, i.e., the exponential part of the transseries of ψ, is a convergent expansion of the form \(\sum_{k=0}^{\infty}g_{k}\Gamma_{k}(x)e^{-\gamma_{k} t}\), where Γ _k are the Gamow vectors of H, and iγ _k are the associated resonances; generically, all g _k are nonzero. For large k, γ _k ～const?klog?k+k ² π ² i/4. The effect of the Gamow vectors is visible when time is not very large, and the decomposition defines rigorously resonances and Gamow vectors in a nonperturbative regime, in a physically relevant way.The decomposition allows for calculating ψ for moderate and large t, to any prescribed exponential accuracy, using optimal truncation of power series plus finitely many Gamow vectors contributions.The analytic structure of ψ is perhaps surprising: in general (even in simple examples such as square wells), ψ(x,t) turns out to be C ^∞ in t but nowhere analytic on ?⁺. In fact, ψ is t-analytic in a sector in the lower half plane and has the whole of ?⁺ a natural boundary. In the dual space, we analyze the resurgent structure of ψ.     

Energy transfer from Eu(III) and Tb(III) complexes to dyes in their mixed nanostructures. I

The formation of nanostructures that consist of complexes of β-diketones with 1,10-phenanthroline and involve dyes of the polymethine, triphenylmethane, oxazine, and xanthene series is observed in aqueous solutions. It is found that nanostructures of complexes of Ln(III) ions and dyes are reliably observed at concentrations of Ln complexes from 0.5 to 5 μM and at dye concentrations above 5 nM. Nanostructures of complexes Eu(MBTA)₃phen, Eu(NTA)₃phen, Eu(PTA)₃phen, Tb(PTA)₃phen, Gd(MBTA)₃phen, and Lu(MBTA)₃phen with dyes are studied, where MBTA is n-methoxybenzoyltrifluoroacetone, NTA is naphthoyltrifluoroacetone, PTA is pivaloyltrifluoroacetone, and phen is 1,10-phenanthroline. It is shown that nanostructures formed can contain dye molecules not only inside a nanostructure of Ln complexes but also on its outer shell. It is proved that, at a dye concentration in the solution of the order of nanomole or higher, the formation of mixed nanostructures of Eu complexes and dyes whose S ₁ level is below the ⁵ D ₀ level of Eu(III) leads to the quenching of the luminescence of Eu(III) and gives rise to the sensitized luminescence of dyes. The energy transfer efficiency from Eu(III) ions to dye molecules is determined by the ability of these molecules to incorporate into nanostructures of Eu complexes. The effect of the formation of nanostructures on the shape and position of the spectra of luminescence and absorption of dyes is studied. Comparison of the sensitized luminescence intensities of Nile blue in structures of Eu, Lu, and Gd complexes shows that the greater part of the excitation energy of Eu complexes is transferred directly from ions to dye molecules according to the inductive-resonance energy transfer mechanism rather than by means of energy migration over singlet levels of organic ligands in complexes of a nanostructure.     

Columinescence of dye molecules in nanostructures of metal ion complexes

The mechanism of columinescence (fluorescence sensitization) of dyes incorporated in nanostructures of metal complexes is studied. It is shown for the first time that the columinescence of dyes is due to the transfer of excitation energy from ligands and metal ions of complexes that form nanostructures. It is proven that the dye columinescence of rhodamine 6G (R6G) molecules incorporated into nanostructures of Al(DBM)₃phen, Al(DBM) _n (OH)_{6 ? 2n} , and Eu(DBM)₃phen (DBM is dibenzoylmethane) nanostructures is completely determined by the singlet excitation energy migration from ligands to R6G molecules. It is shown that, at small concentrations of R6G, the R6G columinescence intensity is lower in nanostructures of metal complexes with a high probability of S-T conversion and that this difference disappears at large concentrations of R6G. In the case of Nile blue (whose S ₁ level lies below the ⁵ D ₀ level of Eu(III)) incorporated in nanostructures of Eu(DBM)₃phen complexes, as well as in nanostructures of Al(DBM)₃phen and Gd(DBM)₃phen complexes with admixture of Eu complexes, we observed the S-S energy transfer from DBM to NB in addition to the delayed sensitized fluorescence of NB previously observed in nanostructures of Eu complexes, which was caused by the energy transfer from the ⁵ D ₀ level of Eu(III) to NB. At dye concentrations below 100 nM, the efficiency of NB sensitization due to the migration of singlet excitation energy from DBM is lower than in the case of the energy transfer from Eu(III) ions, while, at large concentrations of the dye, the S-S energy transfer successfully competes with the sensitization of NB by Eu(III) ions. The use of dye columinescence makes it possible to easily determine dye concentrations of 2–100 nM in solutions with standard spectrofluorimeters.     

On the energy transport velocity in a dissipative medium

An exact definition of the group velocity v _g is proposed for a wave process with arbitrary dispersion relation ω = ω′(k) + iω″(k). For the monochromatic approximation, a limit expression v _g (k) is obtained. A condition under which v _g (k) takes the form of the Kuzelev–Rukhadze expression [1] dω′(k)/dk is found. In the general case, it appears that v _g (k) is defined not only by the dispersion relation ω(k), but also by other elements of the initial problem. As applied to the dissipative medium, it is shown that v _g (k) defines the field energy transfer velocity, and this velocity does not exceed thee light speed in vacuum. An expression for the energy transfer velocity is also obtained for the case where the dispersion relation is given in the form k = k′(ω) + ik″(ω) which corresponds to the boundary problem.     

Low-Temperature Transport Properties of Lanthanum Hexaboride La<Subscript>1–<Emphasis Type="Italic">x</Emphasis></Subscript>Ce<Subscript><Emphasis Type="Italic">x</Emphasis></Subscript>B<Subscript>6</Subscript> Single Crystals

Resistivity (ρ), thermal conductivity (k) and Seebeck coefficient (S) of La_1–xCe_xB₆ single crystals with various concentrations of cerium Ce ions was measured in a wide temperature range 3?300 K. The obtained data were analyzed in the framework of the Coqblin–Shrieffer model. The contributions of scattering of carriers on magnetic ions Ce for all transport parameters ρ(T), k(T), S(T) are revealed. Strong dependence of the magnetic scattering on concentration of the cerium ions are identified. The anomalous behavior of the transport parameters ρ(T), k(T), S(T) in the region near 30 K is attributed to the Δ ~ 30 K splitting of Г₈ level.     

Cross sections of electron capture by Boron ions in gaseous media

We study the cross sections σ_{i, i?1}, σ_{i, i?2}, and σ _{i, i?3} of capture of one, two, and three electrons by boron ions with charges i=1?5 and velocities V=(1.83?5.50)V₀ in gaseous media with atomic numbers Z_t varying from 1 to 54. The oscillatory form of the Z_t dependence of electron capture cross section by boron ions, which has been established for lighter ions, is confirmed.     

High-order harmonic generation by atoms in a two-color laser field: Phase control of recombination radiation spectrum and duration

The feasibility of phase control over above-threshold tunnel ionization and subsequent recombination emission in two-frequency laser fields is studied. It is shown that, in such fields, we can control the instants of ionization t₀ (within optical cycle T) and recombination t _k . The conditions that minimize the characteristic times δt₀?T and δt _k ?T, within which effective ionization and recombination occur, were found. Phase control allows recombination radiation to be generated with the selection of a narrow spectral range, while additional high-frequency “background illumination” sets up high harmonic “amplification” conditions. It was shown that special two-frequency pumping with elliptically polarized radiation can generate coherent electromagnetic pulses of attosecond width. The width of the pulses decreases as the intensity of pumping increases and can reach subattosecond values. Experimental generation of such pulses may lead to a breakthrough in the development of new methods for femto-and attosecond diagnostics of fast processes.     

Deactivation pathways of the electronic excitation of ions of lanthanide complexes in polymers with functional groups

Complexes Eu(TTA)₃phen and Eu(MBTA)₃phen, as well as complexes Tb(MBTA)₃phen and Tb(TTA)₃phen, which do not luminesce in solutions, are shown to luminesce in polymer films (TTA is thenoyltrifluoroacetone, MBTA is n-methoxybenzoyltrifluoroacetone, and phen is o-phenanthroline). Luminescence of complexes of Eu and Tb in films of a polymer, poly(methylene-bis-anthranilamide) 1,6-hexamethylenedicarboxylic acid (PAA-5), having a high concentration of functional anthranilate groups, is studied. From the behavior of the luminescence intensity (I _lum), the luminescence decay time, and the luminescence spectra of complexes of these lanthanides in polymer films, the following regular features were revealed. (i) During the film preparation at 90°C, Ln complexes are attached to PAA-5 via anthranilate groups. (ii) Irradiation of these films in the range of the absorption band of ligands (TTA or MBTA) leads to deactivation of the electronic excitation of ions according to the diketone detachment mechanism and to further binding of complexes to polymers. In this case, I _lum(Eu(III)) decreases because the introduction of anthranilate groups of the polymer into the first coordination sphere of Eu(III) complexes enhances the nonradiative deactivation of these ions, whereas I _lum(Tb(III)) increases since the introduction of these groups suppresses the nonradiative deactivation of Tb complexes through triplet states of ligands (TTA and MBTA). (iii) Upon storage of films in the dark (20°C), complexes detach themselves from the polymer and return to their initial structure. In PAA-5 films into which Eu and Tb complexes were simultaneously introduced, the color of the emission from the irradiation spot changes from red to green.     

Extremal Theory for Spectrum of Random Discrete Schrödinger Operator. II. Distributions with Heavy Tails

We study the asymptotic structure of the first K largest eigenvalues λ _k,V and the corresponding eigenfunctions ψ(?;λ _k,V ) of a finite-volume Anderson model (discrete Schrödinger operator) \(\mathcal{H}_{V}= \kappa \Delta_{V}+\xi(\cdot)\) on the multidimensional lattice torus V increasing to the whole of lattice ? ^ν , provided the distribution function F(?) of i.i.d. potential ξ(?) satisfies condition ?log(1?F(t))=o(t ³) and some additional regularity conditions as t→∞. For z∈V, denote by λ ⁰(z) the principal eigenvalue of the “single-peak” Hamiltonian κΔ _V +ξ(z)δ _z in l ²(V), and let \(\lambda^{0}_{k,V}\) be the kth largest value of the sample λ ⁰(?) in V. We first show that the eigenvalues λ _k,V are asymptotically close to \(\lambda^{0}_{k,V}\). We then prove extremal type limit theorems (i.e., Poisson statistics) for the normalized eigenvalues (λ _k,V ?B _V )a _V , where the normalizing constants a _V >0 and B _V are chosen the same as in the corresponding limit theorems for \(\lambda^{0}_{k,V}\). The eigenfunction ψ(?;λ _k,V ) is shown to be asymptotically completely localized (as V↑?) at the sites z _k,V ∈V defined by \(\lambda^{0}(z_{k,V})=\lambda^{0}_{k,V}\). Proofs are based on the finite-rank (in particular, rank one) perturbation arguments for discrete Schrödinger operator when potential peaks are sparse.     

Impurity transport in fractal media in the presence of a degrading diffusion barrier

We have analyzed the transport regimes and the asymptotic forms of the impurity concentration in a randomly inhomogeneous fractal medium in the case when an impurity source is surrounded by a weakly permeable degrading barrier. The systematization of transport regimes depends on the relation between the time t ₀ of emergence of impurity from the barrier and time t _* corresponding to the beginning of degradation. For t ₀ < t _*, degradation processes are immaterial. In the opposite situation, when t ₀ > t _*, the results on time intervals t < t _* can be formally reduced to the problem with a stationary barrier. The characteristics of regimes with t _* < t < t ₀ depend on the scenario of barrier degradation. For an exponentially fast scenario, the interval t _* < t < t ₀ is very narrow, and the transport regime occurring over time intervals t < t _* passes almost jumpwise to the regime of the problem without a barrier. In the slow power-law scenario, the transport over long time interval t _* < t < t ₀ occurs in a new regime, which is faster as compared to the problem with a stationary barrier, but slower than in the problem without a barrier. The asymptotic form of the concentration at large distances from the source over time intervals t < t ₀ has two steps, while for t > t ₀, it has only one step. The more remote step for t < t ₀ and the single step for t > t ₀ coincide with the asymptotic form in the problem without a barrier.     

<Emphasis Type="Italic">nd</Emphasis> scattering at low energies in the two-body potential model

The S-wave phase shift δ(E) for the spin-doublet nd scattering at low energy E is calculated in the framework of the two-body approach. The effective-range-theory formula k cot δ = (1+k²/k ₀ ² )^?1(?1/α+C₂k²+C₄k⁴) is used to obtain approximate analytical results with different potentials. The corresponding coefficients C₂ and C₄ are obtained from our previous calculations of the asymptotic normalization parameter function C _t ² (aκ), where κ is the triton wave number and a is the doublet nd scattering length. The model reasonably describes δ(E), the results being quite sensitive to the choice of the effective nd potential.