Theory of the hyperfine structure of the <Emphasis Type="Italic">S</Emphasis> states of muonic tritium

The hyperfine structure of the energy spectrum of the S levels of muonic tritium has been calculated using the quasi-potential method in quantum electrodynamics (QED). The α⁵- and α⁶-order effects on the polarization of vacuum, the structure and recoil of the nucleus, and relativistic corrections have been taken into account. The obtained numerical values of hyperfine splittings of 239.819 meV (1S state) and 29.965 meV (2S state) can be treated as reliable estimates for comparing with future experimental data of the CREMA collaboration, and hyperfine structure interval Δ₁₂ = 8ΔE ^hfs (2S)–ΔE ^hfs (1S) =–0.100 meV can be used for verifying the QED predictions. The resultant precision values of hyperfine splitting are also important for calculating the rates of formation of (μ dt) mesomolecules in muonic catalysis reactions.     

Fine and hyperfine structure of <Emphasis Type="Italic">P</Emphasis>-wave levels in muonic hydrogen

Corrections of order α ⁵ and α ⁶ are calculated for muonic hydrogen in the fine-structure interval ΔE ^fs = E(2P _3/2) − E(2P _1/2) and in the hyperfine structure of the 2P _1/2-and 2P _3/2-wave energy levels. The resulting values of ΔE ^fs = 8352.08 μeV, ΔẼ ^hfs(2P _1/2) = 7819.80 μeV, and ΔẼ ^hfs(2P _3/2) = 3248.03 μeV provide reliable guidelines in performing a comparison with relevant experimental data and in more precisely extracting the experimental value of the (2P–2S) Lamb shift in the muonic-hydrogen atom. Original Russian Text ? A.P. Martynenko, 2008, published in Yadernaya Fizika, 2008, Vol. 71, No. 1, pp. 126–136.     

Theory of the muonic hydrogen-muonic deuterium isotope shift

Corrections of the α³, α⁴, and α⁵ orders are calculated for the Lamb shift of the 1S and 2S energy levels of muonic hydrogen μp and muonic deuterium μd. The nuclear structure effects are taken into account in terms of the charge radii of the proton r _p and deuteron r _d for one-photon interaction, as well as in terms of the electromagnetic form factors of the proton and deuteron for the case of one-loop amplitudes. The μd-μp isotope shift for the 1S-2S splitting is found to be equal to 101003.3495 meV, which can be treated as a reliable estimate when conducting the corresponding experiment with an accuracy of 10^?6. The fine-structure intervals E(1S)-8E(2S) in muonic hydrogen and muonic deuteron are calculated.     

Fine Structure of Beta Decay Strength Function and Anisotropy of Isovector Nuclear Dencity Component Oscillations in Deformed Nuclei

The experimental measurement data on the fine structure of beta-decay strength function S_β(E) in spherical, transitional, and deformed nuclei are analyzed. Modern high-resolution nuclear spectroscopy methods made it possible to identify the splitting of peaks in S_β(E) for deformed nuclei. By analogy with splitting of the peak of E1 giant dipole resonance (GDR) in deformed nuclei, the peaks in S_β(E) are split into two components from the axial nuclear deformation. In this report, the fine structure of S_β(E) is discussed. Splitting of the peaks connected with the oscillations of neutrons against protons (E1GDR), of proton holes against neutrons (peaks in S_β(E) of β⁺/EC-decay), and of protons against neutron holes (peaks in S_β(E) of β^–-decay) is discussed.     

Thermophysical study of structural phase transitions in Na<Subscript>0.95</Subscript>Li<Subscript>0.05</Subscript>NbO<Subscript>3</Subscript> solid solution

Temperature dependences of specific heat C_p(T) and coefficient of thermal expansion ;(T) for Na_0.95Li_0.05NbO₃ sodium-lithium niobate ceramic samples are investigated in the temperature range of 100–800 K. The C_p(T) and α(T) anomalies at T₃ = 310 ± 3 K, T₂ = 630 ± 8 K, and T₁ = 710 ± 10 K are observed, which correspond to the sequence of phase transitions N ? Q ? S(R) ? T2(S). The effect of heat treatment of the samples on the sequence of structural distortions was established. It is demonstrated that annealing of the samples at 603 K leads to splitting of the anomaly corresponding to the phase transition Q → R/S in two anomalies. After sample heating to 800 K, the only anomaly is observed in both the C_p(T) and ;(T) dependence. Possible mechanisms of the observed phenomena are discussed.     

Study of penetration effects in the <Emphasis Type="Italic">E</Emphasis>1 and <Emphasis Type="Italic">M</Emphasis>1 transitions in <Superscript>44</Superscript>Sc

Conversion coefficients for the E1 and M1 transitions in the ⁴⁴Ti decay have been measured with high accuracy using the sum peak and multidimensional coincidence methods. The following values have been obtained: α_K(E1) = 0.077 ± 0.003 and α_K(M1) = 0.014 ± 0.001. A significant contribution of the penetration effects for the M1 transition has been shown. The penetration parameter λ is determined to be ?260 ± 30 has been determined.     

Effect of the Lattice and Spin-Phonon Contributions on the Temperature Behavior of the Ground State Splitting of Gd<Superscript>3+</Superscript> in SrMoO<Subscript>4</Subscript>

The temperature behavior of the EPR spectra of the Gd³⁺ impurity center in single crystals of SrMoO₄ in the temperature range T = 99–375 K is studied. The analysis of the temperature dependences of the spin Hamiltonian b ₂ ⁰ (T) = b₂(F) + b₂(L) and P ₂ ⁰ (T) = P₂(F) + P₂(L) (for Gd¹⁵⁷) describing the EPR spectrum and contributing to the Gd³⁺ ground state splitting ΔE is carried out. In terms of the Newman model, the values of b₂(L) and P₂(L) depending on the thermal expansion of the static lattice are estimated; the b₂(F) and P₂(F) spin-phonon contributions determined by the lattice ion oscillations are separated. The analysis of b ₂ ⁰ (T) and P ₂ ⁰ (T) is evidence of the positive contribution of the spin-phonon interaction; the model of the local oscillations of the impurity cluster with close frequencies ω describes well the temperature behavior of b₂(F) and P₂(F).     

Hyperfine splitting in muonic hydrogen: QED corrections of the α2 order

The α² order corrections to the hyperfine splitting of the 1 s and 2 s states in muonic hydrogen have been determined. The specially normalized difference 8E _hfs(2s) ? E _hfs(1s), as well as the general situation with the theoretical calculations of hyperfine splitting in muonic hydrogen, is considered.     

An ab initio study of ion-pair states of the Br<Subscript>2</Subscript> molecule

This contribution reports an ab initio study of 18 ion-pair states of the Br₂ molecule correlating with the Br⁺(³P_J=2,1,0,¹D₂) +Br^–(¹S₀) dissociation asymptotes. Calculations were performed on the CASSCF/CASPT2 level of theory taking into account electron correlations and with inclusion of the spinorbit coupling. Ab initio results are compared with the experiment. For gerade and ungerade ion-pair potentials, the difference between calculated and experimental values of the equilibrium internuclear distance does not exceed ΔR_e = ±0.01 and ±0.05 Å, respectively. For the states correlating to the lowest dissociation asymptote Br⁺(³P₂) + Br^–(¹S₀), the accuracy of relative energies is postilions ΔT_e ± 0.02 eV. The dipole moment functions for some transitions between the ion-pair and valence states were calculated as well.     

Spectral manifestations of <Emphasis Type="Italic">d</Emphasis>-π exchange and charge transfer interactions in Cu(II) porphyrins

New absorption bands due to spin-forbidden transitions from the ground state to excited triplet states are found at 295 K in the near-IR absorption spectra of a number of Cu(II) complexes of octaethylporphyrin (OEP) that differ in the nature, number, and position of their side substituents. The frequency distribution, number, and nature of these transitions are analyzed using computer decomposition of complex contours into Gaussian components and additional data from the phosphorescence and phosphorescence excitation spectra (295-77 K). The d-π exchange integrals, determining the energy splitting ΔE = E(²T₁)–E(⁴T₁) in the compounds under study, are calculated on the basis of the experimental data obtained. It is shown that, in addition to the formally spin-allowed ²S₀ → ²T₁ transition (λ_max = 833 nm), the absorption spectrum of the nonplanar CuOETPP molecule at 295 K exhibits the low-intensity spin-forbidden ²S₀ → ⁴T₁ transition (λ_max = 937 nm). For this compound at 77 K, phosphorescence from the ⁴T₁ state is observed (λ_max = 955 nm), with a quantum yield equal to ?_Ph = 0.0015 and a decay time amounting to τ_Ph = 190 ns. For the CuOEP-Ph(o-NO₂) molecule, which contains the electron-acceptor nitro group, direct absorption from the ground state S₀ to a charge transfer (CT) state (λ_max = 845 nm) is observed at 295 K. The extinction coefficient of this absorption and the energy of the CT state are determined.     

Spectra and mean energies of prompt neutrons from <Superscript>238</Superscript>U fission induced by primary neutrons of energy in the region <Emphasis Type="Italic">E</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript><20 MeV

The time-of-flight technique is used to measure the ratios R(E, E _n )=N(E, E _n )/N_Cf(E) of the normalized (to unity) spectra N(E, E _n ) of neutrons accompanying the neutron-induced fission of ²³⁸U at primary-neutron energies of E _n =6.0 and 7.0 MeV to the spectrum N_Cf(E) neutrons from the spontaneous fission of ²⁵²Cf. These experimental data and the results of their analysis are discussed together with data that were previously obtained for the neutron-induced fission of ²³⁸U at the primary energies of E _n =2.9, 5.0, 13.2, 14.7, 16.0, and 17.7 MeV.     

Effect of the orbital structure and symmetry of electronic states on nonradiative <Emphasis Type="Italic">S-T</Emphasis> intersystem crossing: Dibenzofuran

The nonradiative S-T intersystem crossing S ₁(ππ*) ? T ₁(ππ*) in dibenzofuran (DB(O)) molecules has been theoretically investigated within the model of vibronically induced spin-orbit (VISO) coupling of electronic states, where the vibronic perturbation takes into account all out-of-plane vibrational modes of a molecule. It is established that the S-T intersystem crossing S ₁(¹ A ₁) ? T ₁(³ B ₂) involves also the intermediate (T _m )T ₂(³ A ₁) and T ₃(³ B ₂) triplet states. The calculated rate constant K _ST = (4.5–4.7) × 10⁷s^?1 of the nonradiative transition is in agreement with the known experimental data. The manifestation of approximate (belonging to the D _2h group) symmetry of singlet and triplet molecular states in VISO couplings has been studied. An effect of the heavy (oxygen) atom of a DB(O) molecule on K _ST is established.     

Antiferromagnetic Resonance in GdB<Subscript>6</Subscript>

The electron spin resonance has been measured for the first time both in the paramagnetic phase of the metallic GdB₆ antiferromagnet (T_N = 15.5K) and in the antiferromagnetic state (T < T_N). In the paramagnetic phase below T* ~ 70 K, the material is found to exhibit a pronounced increase in the resonance linewidth and a shift in the g-factor, which is proportional to the linewidth Δg(T) ~ ΔH(T). Such behavior is not characteristic of antiferromagnetic metals and seems to be due to the effects related to displacements of Gd³⁺ ions from the centrosymmetric positions in the boron cage. The transition to the antiferromagnetic phase is accompanied by an abrupt change in the position of resonance (from μ₀H₀ ≈ 1.9 T to μ₀H₀ ≈ 3.9 T at ν = 60 GHz), after which a smooth evolution of the spectrum occurs, resulting eventually in the formation of the spectrum consisting of four resonance lines. The magnetic field dependence of the frequency of the resonant modes ω₀(H₀) obtained in the range of 28–69 GHz is well interpreted within the model of ESR in an antiferromagnet with the easy anisotropy axis ω/γ = (H ₀ ² +2H_AH_E)^1/2, where H_E is the exchange field and H_A is the anisotropy field. This provides an estimate for the anisotropy field, H_A ≈ 800 Oe. This value can result from the dipole?dipole interaction related to the mutual displacement of Gd³⁺ ions, which occurs at the antiferromagnetic transition.     

Thermodynamic study of compounds of the Nd-Ba-Cu-O system

Standard enthalpies of formation for solid solutions of composition Nd_{1 + x} Ba_{2 ? x} Cu₃O _y (x = 0–0.8, y = 6.65–7.24) from oxides were determined by solution calorimetry. The heat capacity of NdBa₂Cu₃O_6.87 phase was measured in the range 5–320 K by low-temperature adiabatic calorimetry. The absolute entropy S ^o(T), the difference of enthalpies H ^o(T)-H ^o(0 K), and the reduced Gibbs energy Φ^o(T) = S ^o(T)–[H ^o(T)–H ^o(0)]/T were calculated on the basis of smoothed dependence C _p (T) in the 0–320 K range. An assessment was made for the heat capacities and the absolute entropies of solid solutions Nd_1+x Ba_2?x Cu₃O _y . The obtained set of thermodynamic parameters can be used for the calculation of phase equilibria in the Nd-Ba-Cu-O system.     

Hyperfine structure of <Emphasis Type="Italic">S</Emphasis>- and <Emphasis Type="Italic">P</Emphasis>-wave states in muonic-helium ion

Corrections of order α ⁵ and α ⁶ to the hyperfine structure of S- and P-wave energy levels of the muonic-helium ion are calculated. Electron-vacuum-polarization effects, corrections for the nuclear structure, and recoil effects are taken into account. The numerical values obtained for respective hyperfine splitting, −1334.73 meV (1S), −166.64 meV (2S), −58 712.90 μeV (2P _1/2), and −24 290.69 μeV (2P _3/2), can be viewed as a reliable estimate for a comparison with experimental data, and the hyperfine-structure interval of Δ₁₂ = 8ΔE ^hfs(2S) − ΔE ^hfs(1S) = 1.59 meV can be used to test QED predictions.     

Hyperfine magnetic anomaly in isotopic pairs <Superscript>151, 152</Superscript>Eu and <Superscript>152, 153</Superscript>Eu

High-resolution laser spectroscopy measurements of optical hyperfine splitting on the ^{151, 152, 153}Eu isotopes were performed on the atomic transition 4f ⁷6s ^{2 8} S _7/2 → 4f ⁷6s6p ⁶ P _5/2 at λ ≈ 564.58 nm. Values of the nuclear magnetic dipole and electric quadrupole moments are obtained from the measured hyperfine splitting and the magnetic hyperfine anomalies in the isotope pairs ^{151, 152}Eu and ^{152, 153}Eu are deduced. The absolute values of the hyperfine anomaly in both cases are unusually large: 5 (1)%. The possible sources causing these anomalies are discussed.     

<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.     

Nuclear-structure corrections to the energy spectra of ordinary and muonic deuterium

One-loop nuclear-structure-induced corrections of order (Zα)⁵ to the Lamb shift and to the hyperfine structure of deuterium are calculated. The contribution of deuteron-structure effects to the (ep)-(ed) and (μp)-(μd) isotopic shifts for the 1S–2S splitting is obtained with the aid of modern experimental data on the electromagnetic form factors for the deuteron. A comparison with the analogous contributions to the Lamb shift for ordinary and muonic hydrogen shows that the relative contribution of corrections associated with the nuclear structure increases as we go over from the hydrogen to the deuterium atom owing to the growth of the nuclear size.     

Determination of electronic ground state properties of a dinuclear iron(II) spin crossover complex

The dinuclear complex [(Fe(L-N₄Me₂))₂(BiBzIm)](ClO₄)₂?2EtCN (1) has been investigated by Mössbauer spectroscopy carried out in the temperature range from 5 to 150 K with externally applied magnetic fields of up to B = 5 T. By means of a consistent simulation of all experimental data sets within the Spin Hamiltonian formalism, the zero-field splitting D and the rhombicity parameter E/D of the ferrous high-spin (HS) site in this complex was determined to be D = ?15.0 ± 1.0 cm^?1 and E/D = 0.33 respectively. The sign of the quadrupole splitting of the HS site is positive which indicates that this iron site of the dinuclear complex 1 has an electronic ground state with the d_xy orbital being twofold occupied.     

Precision measurement of the rate of nuclear muon capture in the muonic hydrogen atom and the determination of the pseudoscalar form factor of the nucleon

The rate Λ _S of nuclear muon capture by a proton from the hyperfine singlet ground state of the µp atom has been measured using a new experimental method based on a time-projection chamber operating in an ultrapure hydrogen gas at a pressure of 10 atm. The capture rate has been determined from the difference between the measured lifetime of the negative muon in hydrogen and the world average lifetime of the positive muon. The analysis of 10% of the collected statistics (2 × 10¹⁰) of µe decays yields the muon capture rate Λ _S = 725.0 ± 17.4 s^?1, from which the pseudoscalar form factor of the nucleon, g _P (q _c ² = ?0.88m _µ ² ) = 7.3 ± 1.1, is determined. The further analysis of the collected experimental data should improve the precision of this measurement by a factor of 3.