Inequalities for the Schattenp-norm. IV

We prove some inequalities for the Schattenp-norm of operators on a Hilbert space. It is shown, among other things, that ifA,B, andX are operators such thatA +B ≧ |X| andA +B ≧ |X*|, then ∥AX +XB∥ _p ^p + ∥AX* +X*B∥ _p ^p ≧2 ∥X∥ _2p ^2p for 1 ≦p<∞, and max (∥AX +XB∥, ∥AX* +X*B∥) ≧ ∥X∥². Also, for any three operatorsA,B, andX, $$|| |A|X - X|B| ||_2^2 + || |A*|X - X|B*| ||_2^2 \leqq ||AX - XB||_2^2 + ||A*X - XB*||_2^2 .$$      

EPR studies on the di-nitrogen centers with nonequivalent atoms in a reddish-brown plastically deformed diamond

Two new di-nitrogen centers, which were labeled M2 and M3, were found together with known W7 and N4 centers in an unusual reddish-brown natural diamond. The following magnetic hyperfine interaction parameters (expressed in MHz) were determined for the two nitrogen atoms:A ₁ ⁽¹⁾ =117.95(5),A ₁ ⁽²⁾ =A ₁ ⁽³⁾ =84.48(5),A ₂ ⁽¹⁾ =7.1(1),A ₂ ⁽²⁾ =A ₂ ⁽³⁾ =6.6(1) for M2 andA ₁ ⁽¹⁾ =121.55(5),A ₁ ⁽²⁾ =A ₁ ⁽³⁾ =85.90(5),A ₂ ⁽¹⁾ =6.0(1),A ₂ ⁽²⁾ =5.4(1),A ₂ ⁽³⁾ =5.1(1) for M3. Hyperfine interaction tensors for the nitrogen atom N₁ with a larger interaction have axial symmetry about the <111> direction, but those for the other nitrogen atom, N₂, appear to be small, almost isotropic. Probable models of the M2 and M3 centers are suggested and discussed.     

Pulse ENDOR studies on the radical pair P 700 +· A 1 ·− and the photoaccumulated quinone acceptor A 1 ·− of photosystem I

The secondary acceptor A₁ of the electron transport chain(s) of photosystem (PS) I is a phylloquinone (vitamin K₁, VK₁). Pulse electron paramagnetic resonance and electron nuclear double resonance (ENDOR) experiments at X-band frequencies were performed on the photoaccumulated acceptor radical A ₁ ^·? and the radical pair state P ₇₀₀ ^·+ A ₁ ^·? in PS I ofThermosynechococcus elongatus. The data obtained were compared with data from the respective radical anion of VK₁ in organic solvents. The unusualg tensor magnitude of A ₁ ^·? is explained by the hydrophobic binding pocket of this radical. The hyperfine couplings and the spin (and charge) density distribution is very different for A ₁ ^.? in PS I and VK ₁ ^·? in frozen alcoholic solution. This is attributed to a rather strong one-sided hydrogen bond to A ₁ ^·? . The presence of a hydrogen bond to A ₁ ^·? has only a minor effect ong. The hyperfine coupling constants of A ₁ ^·? determined from the radical pair spectra deviate only slightly from those derived from photoaccumulated A ₁ ^·? in PS I treated with dithionite at high pH. ENDOR resonances of the proton in a H bond were detected and an estimate of the strength and geometry of this bond to A ₁ ^·? was obtained. The significance of the hydrogen bond and other (hydrophobic) interactions of A₁ with the surrounding are briefly discussed.     

Polarization observablesΣ,P y,T 1 in the reaction\vec \gamma d \to \vec pn at photon energies between 200 and 600 MeV and dibaryon resonances

The polarization observables Σ,P _y ,T ₁,P _y ^⊥ ,P _y ^∥ are derived in proton polarization measurements of the \(\vec \gamma d \to \vec pn\) reaction, using a linearly polarized photon beam of energies between 300 and 600 MeV for c.m.s. proton emission angles of 90° and 120°. A multipole analysis is performed in the framework of the gauge-invariant pole model considering theγN→πN amplitudes and the deuteron structure without dibaryon resonances and with different sets of isovector and isoscalar dibaryon resonances. It is shown that the inclusion of dibaryon resonances substantially improves agreement with experimental data.     

BFKL Pomeron in the reaction γ ⊥ * (Q 2)p→ρ ⊥ 0 p at large Q 2 and W

The reaction γ _⊥ ^* (Q ²)p→ρ _⊥ ⁰ p has been studied at large Q ² and W ²/Q ² and low momentum transfers to the nucleon, k _⊥ ² —that is, in the region where the Pomeron exchange mechanism is operative. At sufficiently large Q ², the Pomeron interacts with quarks occurring at small distances, whereby the hard component of the Pomeron is separated, so that the process is governed by the Pomeron of perturbative QCD (BFKL Pomeron). Our calculations indicate that, in vector-meson electroproduction at low k _⊥ ² , the perturbative regime cannot set in fast because, for Q ²≤100 GeV² and, accordingly, for W ²/Q ²≤10⁷, comparatively large distances of $\rho _{q\bar q} > 0.2$ fm are important.     

Low-frequency dispersion of the effective transverse conductivity in inhomogeneous media in strong magnetic field

On crystalline silicon specimens with a nonuniform carrier concentration distribution produced by an optical method, a dispersion of the effective transverse conductivity σ _⊥ ^eff (ω) is observed near the frequency ω≈ω_c=τ _⊥ ^?1 ≡ε/4πσ _⊥ ^eff . At ω<ω_c, an anomalous transverse effective conductivity is observed: σ _⊥ ^eff (ω) is greater than the transverse conductivity of a homogeneous specimen σ _⊥ ^h (ω) (in the frequency range studied in the experiment σ _⊥ ^h (ω) = const). Near ω≈ω_c, the conductivity σ _⊥ ^eff decreases, and, at ω>ω_c it coincides with σ _⊥ ^h .     

An inequality for Hilbert-Schmidt norm

For the absolute value |C|=(C*C)^1/2 and the Hilbert-Schmidt norm ∥C∥_HS=(trC*C)^1/2 of an operatorC, the following inequality is proved for any bounded linear operatorsA andB on a Hilbert space $$|| |A|---|B| ||_{HS} \leqq 2^{1/2} ||A - B||_{HS} $$ . The corresponding inequality for two normal state ? and ψ of a von Neumann algebraM is also proved in the following form: $$d(\varphi ,\psi ) \leqq ||\xi (\varphi ) - \xi (\psi )|| \leqq 2^{1/2} d(\varphi ,\psi )$$ . Here ξ(χ) denotes the unique vector representative of a state χ in a natural positive coneP ^? forM, andd(?, ψ) denotes the Bures distance defined as the infimum (which is also the minimum) of the distance of vector representatives of ? and ψ. In particular, $$||\xi (\varphi _1 ) - \xi (\varphi _2 )|| \leqq 2^{1/2} ||\xi _1 - \xi _2 ||$$ for any vector representatives ξ _j of ? _j ,j=1, 2.     

Experimental momentum spectra of identified hadrons ate + e − colliders compared to QCD calculations

Experimental data on the shape of hadronic momentum spectra are compared to theoretical predictions in the context of calculations in the Modified Leading Log Approximation (MLLA), under the assumption of Local Parton Hadron Duality (LPHD). Considered are experimental measurements ate ⁺ e ^?-colliders of ξ _p ^* the position of the maximum in the distribution of ξ _p =log(1/x _p ), wherex _p =p/p _beam. The parameter ξ _p ^* is determined for various hadrons at various centre of mass energies. The dependence on the hadron type poses some interesting questions about the process of hadron-formation. The dependence of ξ _p ^* on the centre of mass energy is seen to be described adequately by perturbation theory. A quantitative check of LPHD + MLLA is possible by extracting a value of α _s from an overall fit to the scaling behaviour of ξ _p ^* .     

Hyperfeinstruktur und Lebensdauer des 82 P 3/2-Terms von Rubidium

The hyperfine structure splitting of the 8² P _3/2 state of Rb⁸⁵ and Rb⁸⁷ has been investigated with optical double resonance. The following interaction constants have been obtained:A _8p ⁸⁵ =1.99(2) MHz,B _8p ⁸⁵ =1.98(12) MHzA _8p ⁸⁷ =6.75(3) MHz,B _8p ⁸⁷ =0.96(6) MHz. The lifetime of the 8² P _3/2 state is: τ=4.0(8) · 10^?7 sec.     

Hölder’s inequality for matrices

We prove that for arbitraryn×n matricesA ₁,A ₂,…,A _m and for positive real numbersp ₁,p ₂,…,p _m withp ₁ ^?1 +p ₂ ^?1 +…+p _m ^/?1 =1, the inequality 1 $$|Tr(A_1 A_2 ...A_m )^2 |< \mathop {II}\limits_{k = 1}^m [Tr(A_k^\dag A_k )^{p_k } ]P_k^{ - 1} $$ holds.     

Perpendicularev mass fromW decay

We point out properties of the “perpendicularev mass”, defined in terms of transverse momentap _t byM _T ² (ev)=2|p _eT | |p _vT |?2p _eT ·p _vT , that make it particularly well suited toW mass and width determinations. We give an analytic expression for its distribution inW production and subsequentW→ev decay a \(\bar pp\) colliders, accurate to order 〈p _WT ² /M _W ² 〉≈1%. A maximum likelihood fit of this formula to the five UA1 events givesM _W=80.3 _?3 ⁺⁶ GeV.     

Waves in a superlattice with anisotropic inhomogeneities

Dependences of the dispersion laws and damping of waves in an initially sinusoidal superlattice on inhomogeneities with anisotropic correlation properties are studied for the first time. The period of the superlattice is modulated by the random function described by the anisotropic correlation function K_?(r) that has different correlation radii, k _∥ ^?1 and k _⊥ ^?1 , along the axis of the superlattice z and in the plane xy, respectively. The anisotropy of the correlation is characterized by the parameter λ=1?k_⊥/k_∥ that can change from λ=0 to λ=1 when the correlation wave number k⊥ changes from k_⊥=k_∥ (isotropic 3D inhomogeneities) to k_⊥=0 (1D inhomogeneities). The correlation function of the superlattice K(r) is developed. Its decreasing part goes to the asymptote L that divides the correlation volume into two parts, characterized by finite and infinite correlation radii. The dependences of the width of the gap in the spectrum at the boundary of the Brillouin zone δν and the damping of waves ξ on the value of λ are studied. It is shown that decreasing L leads to the decrease of δν, and increase of ξ, with the increase of λ.     

Effect of the spin-orbit interaction of Ni2+ ions with a triplet orbital ground state on the magnetostriction of NiFe0.5Cr1.5O4 ferrite

The magnetization σ and the longitudinal (λ_∥) and transverse (λ_⊥) magnetostrictions of the NiFe_0.5Cr_1.5O₄ ferrite containing the tetrahedral ions Ni²⁺ with the triplet orbital ground state have been investigated for the first time at a temperature of 4.2 K in fields up to 55 kOe. It is revealed that the NiFe_0.5Cr_1.5O₄ ferrite exhibits an anomalously large magnetic anisotropy (H _c=12.5 kOe) and magnetostrictions (λ_∥≈?870×10^?6 and λ_⊥≈800×10^?6). In strong fields, the magnetostrictions λ_∥ and λ_⊥ are found to be anisotropic in character; i.e., the susceptibility Δλ_∥p and Δλ_⊥p. The conclusion is drawn that the studied compound is characterized by two paraprocesses: one paraprocess in the B sublattice has an exchange nature, and the second process in the A sublattice is due to the spin-orbit interaction of Ni _A ²⁺ ions.     

Polar smectic subphases: Phase diagrams, structures and X-ray scattering

We analyze a discrete phenomenological model accounting for phase transitions and structures of polar Smectic-C* liquid-crystalline phases. The model predicts a sequence of phases observed in experiment: antiferroelectric SmC _A ^* –ferrielectric SmC _FI1 ^* –antiferroelectric SmC _FI2 ^* (three-and four-layer periodic, respectively)–incommensurate SmC _α ^* –SmA. We find that, in the three-layer SmC _FI1 ^* structure, both the phase and the module of the order parameter (tilt angle) differ in smectic layers. This modulation of the tilt angle (and therefore of the layer spacing d) must lead to X-ray diffraction at the wave vectors Q _s =2πs/d(s=n±1/3) even for the nonresonant scattering.     

A new many-body solution of the Friedel resonance problem

It is numerically shown that the groundstate of the Friedel problem (consisting of a conduction band and a dresonance), occupied with (n+ 1) electrons, can be written as Ψ = (A a ₀ ^* +Bd*) Π _v=1 ⁿ a _v ^* Φ₀, where a ₀ ^* represents a localized conduction electron state, d* is the Friedel resonance state and Π _v=1 ⁿ a _v ⁸ Φ₀ is a Slater determinant of n single electron states a _t ^* , (Φ₀ is the vacuum state). The a _i ^* together with a ₀ ^* are part of a full ortho-normalized basis of the conductions band.     

Study of a random Heisenberg ferromagnet by the Molecular CPA method

The spin wave Green function of a random dilute Heisenberg ferromagnet is calculated by the Molecular CPA method for the bond model of the f.c.c. lattice. It is found that the ferromagnetic spin order atT=0 becomes unstable, if the concentration of the non-zero bondC _A decreases below the critical valueC _A ⁰ =0.1702, which is some-what larger than the percolation concentrationC _A ^P =0.125. In the regionC _A <C _A ⁰ , aδ function spectrum appears atE=0 corresponding to the free rotations of the total spins of the finite clusters. The peculiar structure of the spin wave state density near the critical concentration is analysed in detail and a speculation is made for the topological structure of the spin clusters.     

EPR of a fullerene-molecule-derived paramagnetic center as a mesoscopic conducting object

Discussion of theg-factor value of fullerene is based on the model of itinerant electrons restricted to the surface of the fullerene molecule C₆₀. The Ag shift, i.e., the difference between the experimentalg-factor and theg-factor of free electron Δg = g ? 2.0023 for C ₆₀ ^?1 is negative as for a very small metallic conducting particle.g-factor value is proportional to the interaction between itinerant electrons in the conduction band, thus the Δg is negative for C ₆₀ ^?1 and C ₆₀ ^?3 having less than half filled conduction band, while Δg is positive for C ₆₀ ⁺ where the conduction band is almost filled.     

Comparative study of the semiempirical three-parameter potential energy functions for diatomic molecules

A comparative study is made of four three-parameter semiempirical potential energy functions for 32 electronic states of diatomic molecules and their ions:n ₂:X¹gS _g ⁺ ,B ³π_g,A ³ gS_u,C ³ _u,B′ ³ gS_u.a ¹ π_g, a′gS _u ^? ,Ω ¹δ_u N ₂ ⁺ :X ² gS _g ^gS +A ² π,C ² gS _u ⁺ ,B ² gS _u ⁺ CO:X¹gS⁺,a ³ π, a′³ gS_u,e ³ gS^?,d ³gD₁,A ¹π CO⁺:X²gS⁺,A ² π,B ²gS⁺ O₂:X³gS _g ^? ,B ³ gS_u,c ¹ gS _u ^? ,b ¹gS _g ^s ,a ¹ δ_g,c ³ δ_u O ₂ ⁺ :X ²π_g,A ² π_g, a¹ π_g,b ⁴ gS _g ^? A program for numerically integrating the radial Schrödinger equation by the Cooley method is worked out. Certain additional units are introduced to conserve computer time. The resulting vibrational levels are compared with the experimental levels for all the electronic states studied. It is concluded on the basis of this analysis that it is not possible to describe equally well all the electronic states of various molecules on the basis of any single three-parameter potential function. A method for choosing a potential function for describing some particular electronic state of a diatomic molecule is proposed.     

IsoscalarE2 transition strengths in 10≦A≦48 nuclei

Electric quadrupole transition strengths for the 2 ₁ ⁺ ,T=1→0 ₁ ⁺ ,T=1γ-decay branches are summarized for 10≦A≦42 nuclei. In¹⁰B the corresponding branch has been remeasured by use of the⁹Be(p, γ)¹⁰B resonance reaction atE _p =320 keV; an upper limit of 0.6% is found. The variation of theE2 strengths within each individualT=1 isobaric multiplet reveals the particleor hole-character of the levels involved. The isoscalar 2 ₁ ⁺ →0 ₁ ⁺ transition strengths in 10≦A≦48 nuclei vary between 2 and 20 Weißkopf units showing drastic shell effects. Results for the ratio of neutron and proton matrix elements deduced from analogγ-decay studies and from inelastic pion scattering are compared.     

Pion photoproduction on197Au to ground and isomeric states in197Hg

The lifetime of the 331.3 keV 0 ₂ ⁺ state in¹⁰⁰Zr has been measured at the gas-filled recoil separator for fission products JOSEF. By observing the delayed coincidences between theβ-particles populating the level and theE0 conversion electrons from its decay into the ground state, a half-life of 3.37±0.30 ns has been obtained. From the measured lifetime and the relative intensities of the 0 ₂ ⁺ →0 ₁ ⁺ and 0 ₂ ⁺ →2 ₁ ⁺ transitions, values of 0.493±0.015 for theE0 strength parameterρ, and of 16 single particle units forB(E2,2 ₁ ⁺ → 0 ₂ ⁺ ) have been deduced. The enhanced nature of theE0 transitions suggests mixing of the 0 ₁ ⁺ and 0 ₂ ⁺ states which may be estimated by comparing the experimentalB(E2) values for the 2 ₁ ⁺ →0 ₁ ⁺ and 2 ₁ ⁺ →0 ₂ ⁺ transitions with the predictions of the asymmetric VMI model.