Theoretical Study of Trapping Sites for Muon in the Heme Group of Cytochrome c and Associated Shifts in Muon Spin Relaxation Frequencies

Muon Spin Relaxation (μSR) experiments are currently being conducted on the important electron transfer molecule cytochrome c (cyt c) with the goal to find out about microscopic details of the path the moving electron is taking. Simultaneously we are using the Unrestricted Hartree–Fock Cluster Procedure to determine the trapping sites for muon (μ⁺) and muonium (Mu) in the heme unit of cyt c and the associated hyperfine interactions. For the trapping sites with the highest binding energies for μ⁺, namely the nitrogen and the carbon of the pyrrole rings, we have used the available magnetic susceptibility data together with our calculated hyperfine fields to predict the Knight-shifts. At room-temperature we found 88.4 ppm and 79.0 ppm for the most attractive N- and adjacent C-site, respectively. At 150 K, these shifts increase to 172.7 ppm for the N-site and 153.7 ppm for the C-site. This revised version was published online in September 2006 with corrections to the Cover Date.     

Theory for Relative Strengths of Trapping of He+ Ions in Solid, Liquid and Gaseous Hydrogen

The study of trapping of He⁺ ion in solid hydrogen is important both as a problem in solid state physics and also as an applied physics problem in the field of muon catalyzed fusion (μCF). In μCF, He⁺ ion acts as a trap for μ⁻, interrupting the chain reaction aspect of the catalytic role of μ⁻ in producing fusion of deuteron and triton and of triton and triton in solid hydrogen composed of ²H–³H and ³H–³H molecules, respectively. Using the Hartree–Fock procedure, combined with procedures for including many-body effects, as well as relaxation effects associated with the He⁺–H₂ distances and the adjustment of the H–H separation, we have investigated the trapping of He⁺ in gaseous and solid state environments. For the former, the environment of He⁺ is simulated by a single hydrogen molecule and for the solid by clusters appropriately chosen to represent the hexagonal close-packed structure. Our results for the gaseous state indicate that the trapping is rather strong with a binding energy of 8.5 eV, with almost equal binding energy in the linear and triangular configurations with respect to the H–H direction. For the solid, both the likely sites for He⁺ trapping, namely the tetrahedral and octahedral interstitial sites, are also found to provide deep traps (8.6 eV) of almost equal strength, independent of the orientations of the neighboring molecules, showing that the trapping is not influenced by the orientational disorder in the surrounding hydrogen molecules. Further, the influence of next nearest neighbor hydrogen molecules is found to enhance the trapping energy for He⁺ substantially, by 0.6 eV, with the incorporation of the third nearest neighbors having a much smaller added effect, demonstrating the convergence of our results with respect to the size of the cluster chosen to simulate the solid. The substantial influence on the He⁺ trapping energy found for the neighbors beyond the nearest ones provides an explanation of the greater accumulation of helium in the solid state of hydrogen in μCF experiments as compared to the liquid. Suggestions are made regarding the possible reasons for the almost negligible accumulation of helium in the liquid state. This revised version was published online in August 2006 with corrections to the Cover Date.     

Calculation of open p-shell atoms in the algebraic approach of the Hartree–Fock method

Highly precise calculations of analytical Hartree–Fock orbitals and energies have been performed within the limits of the Roothaan–Hartree–Fock atomic theory (Roothaan–Bagus method) for all open p-shell atoms of the Periodic Table. They were calculated in an algebraic approach using Slater-type atomic orbitals (AOs) as basis functions. Nonlinear parameters (orbital exponents) of AOs were optimized with exceptional accuracy by second-order methods. As a result, it was possible to satisfy exactly the virial relation (10^–14–10^–17) with calculated atomic term energies being close to the Hartree–Fock limit.     

Use of combined Hartree–Fock–Roothaan theory in evaluation of lowest states of K[Ar]4s03d1 and Cr?+?[Ar]4s03d5 isoelectronic series over noninteger n-Slater type orbitals

By using noninteger n-Slater type orbitals in combined Hartree–Fock–Roothaan method, self-consistent field calculations of orbital and lowest states energies have been performed for the isoelectronic series of open shell systems K[Ar]4s ⁰3d ¹ (² D) (Z = 19–30) and Cr⁺[Ar]4s ⁰3d ⁵ (⁶ S) (Z = 24–30). The results of the calculations for the orbital and total energies obtained by using minimal basis-sets of noninteger n-Slater type orbitals are given in the tables. The results are compared with the extended-basis Hartree–Fock computations. The orbital and total energies are in good agreement with those presented in the literature. The results can be useful in the study of various properties of heavy atomic systems when the combined Hartree–Fock–Roothaan approach is employed.     

Determination of Easy Axis in a Chemical Ferromagnet, 4-(p-Chlorobenzylideneamino)–TEMPO (TEMPO=2,2,6,6-Tetramethylpiperidin-1-Oxyl)

The trapping sites for muon and muonium in ferromagnetic p-Cl–Ph–CH=N–TEMPO [(4-(p-chlorobenzylideneamino)–TEMPO (TEMPO = 2,6,6-tetramethyl-piperidin-1-yloxyl)] and the hyperfine interaction tensors for these sites are obtained using first-principles Unrestricted Hartree–Fock theory. The calculated hyperfine interactions are used to compare the calculated zero field muon spin rotation (μSR) frequencies for different choices for the easy axis and the observed frequency. It has been concluded that the two trapping centers that can best explain the observed μSR frequency are a trapped singlet muonium near the radical oxygen and a trapped muon site near the chlorine. The direction of the easy axis also is determined to be the b-axis of the monoclinic lattice. This direction for the easy axis is confirmed by determining the direction of the distributed magnetization in the molecular solid which leads to a minimum dipole–dipole interaction energy. The consequences of this agreement for the easy axis direction by two independent procedures are discussed. This revised version was published online in September 2006 with corrections to the Cover Date.     

Theoretical Investigation of Muon and Muonium Trapping and Hyperfine Interactions in the Chemical Ferromagnet p-NPNN (β-phase)

The trapping sites for muon and muonium in β-phase ferromagnetic p-NPNN have been determined by the first-principles Unrestricted Hartree–Fock procedure. Four trapping sites are found for the muon near the two nitrogen and two oxygen atoms of the two NO groups. For the singlet state of trapped muonium, two trapping sites are found near the two oxygens of two NO groups and for the triplet state two trapping sites are found near the two oxygens of the NO₂ group. The observed μSR signal at zero field with frequency 2.1 MHz is assigned to the singlet muonium sites near the two oxygens of the two NO groups and the high frequency signal ascribed to an isotropic hyperfine constant of 400 MHz is assigned to the two trapped muon sites near the two nitrogen atoms of the two NO groups. This revised version was published online in September 2006 with corrections to the Cover Date.     

Theory of Isotope Effects on He+ Trapping in Solid Hydrogen

We are currently investigating the influence of vibrational effects on the strength of trapping of He⁺ in solid hydrogen. Such effects can lead to an isotope dependence of the trapping energy associated with the hydrogen molecules and He⁺ ion. At the present time, our focus is on the isotope effect for ³He⁺ and ⁴He⁺, which we are studying through the vibrational motions of the trapped He⁺ ions in the potential they experience as they move about their equilibrium positions. The potential governing the vibrations has been obtained from Hartree–Fock cluster calculations of the total energy of the cluster composed of the He⁺ ion and up to the third nearest neighbor hydrogen molecules as a function of the displacement of the He⁺ ion from its trapped position. The energy eigenvalues for the ground vibrational states of ³He⁺ and ⁴He⁺ in this potential come out as 1.29 and 0.96 meV, respectively, leading to corresponding reductions by these amounts in the binding energy of 8.6 eV for both ions without vibrational effects. The difference of these reductions can be considered as an isotope shift, its value for this case being 0.33 meV. From the analysis for these results, it is suggested that isotope shift effects for deuteron and triton in solid D–T would have the same order of magnitude. A procedure for more accurate investigations of the isotope shifts is discussed. This revised version was published online in August 2006 with corrections to the Cover Date.     

Muon/muonium surface interactions

The interactions of muonium (μ ⁺ e ⁻, Mu) with the surfaces of fine silica powders have been extensively studied using zero, longitudinal and transverse field μSR techniques. These studies indicate diffusion and trapping behavior of the Mu atoms on the silica surface, which is strongly influenced by the surface hydroxyl (OH) concentration. Specifically, the presence of the surface OH groups is observed to inhibit the surface mobility of the Mu atoms at low temperatures. Information provided by zero and longitudinal field data suggest a random anisotropic distortion of the Mu hyperfine interaction (RAHD) as the principal relaxation mechanism. A recently developed RAHD spin relaxation theory is used to interpret these data. Additional investigations, using platinum loaded silica, have yielded the first observed surface reaction of Mu. Studies of the interactions of positive muons with surfaces have been also extended to single crystals, where low energy (<10 eV)μ ⁺ andMu ⁻ ions are observed to be reemitted from some materials (e.g., the <100> surface of lithium fluoride). Future applications of these emission phenomena toward the development of a slow^847-3 (or Mu⁻) beam are considered.     

<Emphasis Type="Italic">Ab initio</Emphasis> calculation of band absolute intensities in IR spectra

Ab initio calculations in a harmonic approximation of absorption band absolute intensities in infrared spectra were carried out for 3 hydrocarbons and 14 halogenated hydrocarbons. The calculated data were compared with experimental values of the absolute absorption intensities. It is shown that a Hartree–Fock calculation method overestimates significantly (by an average of 66%) the integrated absolute intensities of the fundamental bands in the region 575–4000 cm^–1. The deviation is reduced to 32% in the case of the MP2 method of accounting for electron correlations. Most of the overestimation occurs for bands corresponding to vibrations involving halogen atoms.     

Final states in Si and GaAs via RF μSR spectroscopy

The ionization of muonium centers in Si and GaAs have been studied using radio frequency (RF) resonant techniques. In Si all three muonic centers are detectable by RF. No evidence was found for delayed Mu and Mu^* states at any temperature. However, our results on the diamagnetic final state (μ _f ⁺ ) show that it is composed of prompt fractions (as seen by conventional μSR) and delayed fractions arising from the ionization of Mu^* and Mu. We observe a full μ _f ⁺ fraction at 317 K when the Mu relaxation rate is above 10 μs⁻¹. GaAs differs from the situation in Si in that we observed only a partial conversion of Mu^* and Mu to a μ⁺ final state up to 310 K in spite of the fact that the transverse field relaxation rates become very high at 150 and 250 K respectively.     

Ab Initio Investigation of Helium Bubble Formation from Trapped He+ Ions in Solid Hydrogen

We are involved in a program aimed at a first-principles examination of the possibility of helium bubble formation in solid hydrogen. Our procedure is based on an extension of the Hartree–Fock cluster approach that we have been using for study of individual He⁺ ions in solid hydrogen. The preliminary results of our cluster investigations of a particular configuration of two He⁺ ions in neighboring tetrahedral interstitial positions are presented here. It has been found that the hydrogen molecules common to the two He⁺ ions lead to a substantial attractive force between the ions that almost overcomes their inherent repulsion, leading to a binding energy for this configuration only slightly less than the binding energy for two well separated trapped He⁺ ions. The analysis of our results have suggested future investigations of different pairs of trapping sites which appear to be more favorable for providing higher binding energies for the He⁺ pairs. This revised version was published online in August 2006 with corrections to the Cover Date.     

Hyperfine interactions of 19O in TiO2 and quadrupole moments of 13,19O

The electric quadrupole interaction of ¹⁹O(I^π=(5/2)⁺, T_1/2=27.0 s) in TiO₂ single crystal was studied in detail by means of the β-NQR to determine the electric quadrupole moments Q of short-lived β-emitting nuclei ¹⁹O and ¹³O(I^π=(3/2)^-, T_1/2=8.6 ms). Two implantation sites were found for the implanted O nucleus and the quadrupole coupling constants of ¹⁹O at these sites were determined. We observed FT-NMR of the enriched stable isotope ¹⁷O in TiO₂ and obtained the electric field gradient (EFG) at the oxygen substitutional site. With this knowledge, we have determined Q(¹³O)=11.0 ± 1.3 mb and Q(¹⁹O)=3.7 ± 0.4 mb. The present results are compared with the theoretical values calculated by the shell model code, OXBASH and by the Hartree–Fock calculation with the realistic potential. This revised version was published online in August 2006 with corrections to the Cover Date.     

Phase structure of the linear sigma model with the standard symmetry breaking term

The linear sigma model at finite isospin chemical potential μ and temperature T is systematically studied by means of the Cornwal–Jackiw–Tomboulis (CJT) effective potential calculated in the improved Hartree–Fock (HF) approximation, where the Goldstone theorem and the thermodynamic consistency are respected. It results that in the chiral limit, for μ=0 the chiral phase transition is second order as expected from the general universality arguments, and for μ≠0 the phase diagram for the pion condensation in the (T,μ) plane exhibits a tricritical point which is crossover from first-order to second-order phase transitions. In the physical world, where the chiral symmetry is explicitly broken, the pion condensation occurs at μ=m _π , the pion mass in vacuum, and its phase diagram is basically in agreement with those found from the chiral perturbation theory. The chiral symmetry gets restored at high values of T for fixed μ and of μ for fixed T.     

MSR Studies in the Progress Towards Diamond Electronics

The recent development of device quality synthetic diamond dramatically increases the potential of diamond as a wide band gap semiconductor. A remaining obstacle is the lack of shallow n-type dopants. Molecular dopant systems have been shown theoretically to lead to the shallowing of levels in the band gap. Some of these systems involve defect-hydrogen complexes. This, and other phenomena, motivate the study of the chemistry and dynamics of hydrogen in diamond. Much information on this topic has been obtained from Muon Spin Rotation (MSR) experiments. These experiments view the muonium (Mu ≡ μ⁺ e ⁻) atom as a light chemical analogue of hydrogen. Data on isolated muonium in diamond is reviewed, and evidence on formation of N-Mu-N (a shallow dopant candidate), the trapping of Mu at B-dopants, and fast quantum diffusion of muonium are discussed.     

Direct patterning of gold oxide thin films by focused ion-beam irradiation

For direct writing of electrically conducting connections and areas into insulating gold oxide thin films a scanning Ar⁺ laser beam and a 30 keV Ga⁺ focused ion beam (FIB) have been used. The gold oxide films are prepared by magnetron sputtering under argon/oxygen plasma. The patterning of larger areas (dimension 10–100 μm) has been carried out with the laser beam by local heating of the selected area above the decomposition temperature of AuO_x (130–150 °C). For smaller dimensions (100 nm to 10 μm) the FIB irradiation could be used. With both complementary methods a reduction of the sheet resistance by 6–7 orders of magnitude has been achieved in the irradiated regions (e.g. with FIB irradiation from 1.5×10⁷ Ω/□ to approximately 6 Ω/□). The energy-dispersive X-ray analysis (EDX) show a considerably reduced oxygen content in the irradiated areas, and scanning electron microscopy (SEM), as well as atomic force microscopy (AFM) investigations, indicate that the FIB patterning in the low-dose region (10¹⁴ Ga⁺/cm²) is combined with a volume reduction, which is caused by oxygen escape rather than by sputtering. Received: 30 May 2000 / Accepted: 31 May 2000 / Published online: 13 July 2000     

Spin dynamics of the muon in muonium in normal metals

The question of the charge state of the proton (the positive muon) in metals is of fundamental importance for the theory of metal hydrides. The theory developed here permits determination of the charge state of μ ⁺ in normal metals. The experimental possibilities of the observation of Mu atoms in metals at various strengths of the external magnetic field and various temperatures are analyzed. Zh. éksp. Teor. Fiz. 111, 730–736 (February 1997)     

Mean Field Magnetic Phase Diagrams for the Two Dimensional <Emphasis Type="Italic">t</Emphasis> — <Emphasis Type="Italic">t</Emphasis>′ — <Emphasis Type="Italic">U</Emphasis> Hubbard Model

We study the ground state phase diagram of the two dimensional t — t′ — U Hubbard model concentrating on the competition between antiferro-, ferro-, and paramagnetism. It is known that unrestricted Hartree–Fock- and quantum Monte Carlo calculations for this model predict inhomogeneous states in large regions of the parameter space. Standard mean field theory, i.e., Hartree–Fock theory restricted to homogeneous states, fails to produce such inhomogeneous phases. We show that a generalization of the mean field method to the grand canonical ensemble circumvents this problem and predicts inhomogeneous states, represented by mixtures of homogeneous states, in large regions of the parameter space. We present phase diagrams which differ considerably from previous mean field results but are consistent with, and extend, results obtained with more sophisticated methods. PACS: 71.10.Fd, 05.70.Fh, 75.50.Ee     

Identification ofμ +-sites in the 1-2-3 compoundin the 1-2-3 compound

Possibleμ ⁺ sites have been identified by a comparison of measured spontaneous internal fields in YBa₂Cu₃O_6+δ and REBa₂Cu₃O_7−δ and of measured second moments and width of transverse field powder spectra with the corresponding calculated quantities. In the YBa₂Cu₃O₇ system only one possibleμ ⁺ site emerges at a distance of ∼1.05 A from a chain oxygen O(4) at the position (0.15(1), 0.44(1), 0.071(1)). In the system YBa₂Cu₃O₆ the only possibleμ ⁺ sites are near to an oxygen O(1) with z/c=0.133 and the same distance as above. The analysis leads also to improved valuesμ _Ho = 2.2(1)μ _B andμ _Cu = 0.67(4)μ _B and to a determination of the latters direction:μ _Cu ‖〈110〉.     

Hartree–Fock self-consistent field method for atoms with two open shells of identical symmetry

A generalization of the Roothaan–Bagus method (Roothaan–Hartree–Fock atomic theory) on atoms with open shells of identical symmetry is given. Using orbital exponents of Slater-type atomic orbitals optimized with high accuracy by methods for the minimization of the first and second orders, energy values for atoms with two open s-type shells are calculated within the limits of the Roothaan–Hartree–Fock atomic theory.     

Hartree–Fock versus quantum Monte Carlo study of persistent current in a one-dimensional ring with single scatterer

We calculate the persistent current of interacting spinless electrons in a one-dimensional ring containing a single δ barrier. We use the self-consistent Hartree–Fock method and the quantum Monte Carlo method which gives fully correlated solutions. Our Hartree–Fock method treats the non-local Fock term in a local approximation and also exactly (if the ring is not too large). Treating the Fock term exactly we attempt to support our previous Hartree–Fock result obtained in the local approximation, in particular the persistent current behaving like I∝L^-1-α, where L is the ring length and α>0 is the power depending only on the electron–electron interaction. Finally, we use the Hartree–Fock solutions as an input for our quantum Monte Carlo calculation. The Monte Carlo results exhibit only small quantitative differences from the Hartree–Fock results.