Trovacenyl-based organometallic triradicals: Spin frustration, competing interactions and redox splitting

Two new triradicals based on trovacene [η⁷-tropylium)vanadium(η⁵-cyclopentadienyl)], 1,3,5-tri([5]trovacenyl) benzene4 and 1,3,5-tri([5]trovacenyl)-6-methoxybenzene5 were prepared and their magnetic properties were studied by continuous-wave X-band electron paramagnetic resonance (EPR) spectroscopy and by temperature-dependent magnetic susceptometry. The EPR spectra of4 and5 in liquid toluene solution demonstrate that the three unpaired electrons localized on the vanadium atoms interact with each other in both complexes. The data from magnetic susceptometry revealed that the electron spins in both triradicals are antiferromagnetically coupled despite themeta-phenylene bridge. The exchange coupling constants are equal in the C₃-symmetrical triradical4 (J=J′=?0.68 cm^?1), which leads to a twofold degenerate spin ground state (spin frustration). The symmetry lowering by methoxy substitution of the benzene spacer in5 results in the effect of c ompeting interactions (J=?1.83 cm^?1 andJ′=?2.38 cm^?1). In addition to magnetocommunication, the effect of ring substitution on electrocommunication is also discernable. It manifests itself indisparate redox splittings δE _1/2 (0/?, ?/2?) and δE _1/2 (?/2?, 2?/3?) for5, while these parameters are equal for the C₃-symmetrical trinuclear complex4.     

Localization in the ground state of the ising model with a random transverse field

We study the zero-temperature behavior of the Ising model in the presence of a random transverse field. The Hamiltonian is given by $$H = - J\sum\limits_{\left\langle {x,y} \right\rangle } {\sigma _3 (x)\sigma _3 (y) - \sum\limits_x {h(x)\sigma _1 (x)} } $$ whereJ>0,x,y∈Z ^d, σ₁, σ₃ are the usual Pauli spin 1/2 matrices, andh={h(x),x∈Z ^d} are independent identically distributed random variables. We consider the ground state correlation function 〈σ₃(x)σ₃(y)〉 and prove:

1. Letd be arbitrary. For anym>0 andJ sufficiently small we have, for almost every choice of the random transverse fieldh and everyx∈Z ^d, that $$\left\langle {\sigma _3 (x)\sigma _3 (y)} \right\rangle \leqq C_{x,h} e^{ - m\left| {x - y} \right|} $$ for ally∈Z ^d withC _{x h} <∞.
2. Letd≧2. IfJ is sufficiently large, then, for almost every choice of the random transverse fieldh, the model exhibits long range order, i.e., $$\mathop {\overline {\lim } }\limits_{\left| y \right| \to \infty } \left\langle {\sigma _3 (x)\sigma _3 (y)} \right\rangle > 0$$ for anyx∈Z ^d.

     

Coordination preference and magnetic properties of FeII assemblies with a bis-azole bearing 1,2,4-triazole and tetrazole

With a new bis-azole molecular fragment (Htt) bearing 1,2,4-triazole and tetrazole, a mononuclear complex [Fe(tt)₂(H₂O)₄]·2H₂O (1), a trinuclear complex [Fe₃(tt)₆(H₂O)₆]·2H₂O (2) and a 1D coordination polymer [Fe(tt)(Htt)₂]BF₄·2CH₃OH (3) were obtained by varying reaction conditions. Htt acts either as an anionic or neutral ligand depending upon the reaction medium and pH. Thermal variation of spin states of 1–3 were investigated in the range 77–300?K by ⁵⁷Fe M?ssbauer spectroscopy. 1 totally remains in high-spin state over the entire temperature range whereas no spin crossover was evidenced in 2. Nearly 1:1 high-spin and low-spin population ratio is found in 3, which remains constant over the entire temperature range investigated.     

Excited quark-quark interactions in a bag model

We calculate the colour magnetic and electric fields,B ^a andE ^a , in the static cavity approximation of the MIT bag model. This is done rigorously for all direct and exchange four-currents of quarks in theS _1/2,P _1/2 andP _3/2 bag states. We then find the quark-quark colour interaction for direct and exchange diagrams. The one-pion exchange diagrams due to chiral symmetry are also calculated.     

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.     

Large solutions for harmonic maps in two dimensions

We seek critical points of the functionalE(u)= \(\mathop \smallint \limits_\Omega\) |βu|², where Ω is the unit disk in ?² andu:Ω→S ² satisfies the boundary conditionu=γ on ?Ω. We prove that if γ is not a constant, thenE has a local minimum which is different from the absolute minimum. We discuss in more details the case where γ(x, y)=(R _x,R _y, \(\sqrt {1 - R^2 }\) ) andR<1.     

Aging effect in CaLaBa{Cu1???xFex}3O7???�� with 0 �� x �� 0.07 studied by M?ssbauer spectroscopy

In this work, we study the long-term aging effect caused by Fe atoms in the superconductor CaLaBa{Cu_1???xFe_x}₃O_7????? with 0 ?? x ?? 0.07. XRD confirms that this system has a YBCO-like structure. The critical temperature (T_c) is strongly affected by aging and depends on the amount of Fe in the structure. Room temperature Mössbauer spectroscopy reveals the presence of the typical species A, B?CB ??, C and new species E ?? and F. Interestingly; A, which corresponds to the Fe^3?+? atom located in the Cu(1) of the chains with spin S _z = 3/2, shows a drastic reduction which means migration to the species B, B ?? and C. Species B and B ?? correspond to the Fe^3?+? in the Cu(2) site forming planar quasi-octahedral and planar square pyramidal, while the C specie is a square pyramidal with O(5) respectively (spin S_z = 3/2 in all these cases). Aging causes loss of superconductivity in the samples with 5 and 7% of iron content.     

High spin states in205Pb and a precision test of the nuclear shell model for three nucleons

Using the²⁰⁴Hg(α, 3n) reaction withα-particles of about 40 MeV, we have proved by applying nowadays conventionalγ-ray spectroscopy in-beam technique, that there are two isomeric states in²⁰⁵Pb at the excitation energies 5,161.3 and 3,195.5 keV having the half-lives 71±3 and 217±5 ns, respectively. These isomeric states have spins and parities 33/2⁺ and 25/2^? and are mainly due to thei ₁₃ ^2/?3 andi ₁₃ ^2/?2 p ₁ ^2/?1 configurations, respectively. This conclusion is supported by the experimentalg-factors of these states being ?0.159±0.008 and ?0.0676±0.0011, respectively. It is furthermore shown that theE2 effective neutron charge is the same forE2 transitions from the 33/2⁺ state in²⁰⁵Pb and from the 12⁺ state in²⁰⁶Pb as required by the assumption that the²⁰⁸Pb core is responsible for the totalE2 strength of the neutron holes, and that these states are due to thei ₁₃ ^2/?3 andi ₁₃ ^2/?2 configurations. The calculatedB(E3) values ofE3 transitions from isomeric states in²⁰⁵Pb and²⁰⁶Pb agree reasonably well with the experimental values as expected from the assumption that theE3-strength should come from particle coupling to the octupole states of the²⁰⁸Pb core. The energies of the six most well established excited states in²⁰⁵Pb with angular momenta in the region 19/2–33/2 were calculated using empirical single-particle energies, empirical two-particle interactions and angular momentum algebra. The average deviation between experimental and calculated energies is ?3 keV and the root mean square deviation 6 keV as compared to the uncertainty ± 5 keV in the nuclear masses used in the calculation. For the orbits concerned the shell model is thus valid with an extremely high precision. The contribution of effective three-particle interaction in these orbits must consequently be less than about 5 keV.     

Secular-free solution up to third order of a model nonlinear equation for dispersive waves

The perturbation method of Lindstedt is applied to study the non linear effect of a nonlinear equation $$\nabla ^2 {\rm E} - \frac{1}{{c^2 }}\frac{{\partial ^2 {\rm E}}}{{\partial t^2 }} - \frac{{\omega _0^2 }}{{c^2 }}{\rm E} + \frac{{2v}}{{c^2 }}\frac{{\partial {\rm E}}}{{\partial t}} + E^2 \left[ {\frac{{\partial {\rm E}}}{{\partial t}} \times A} \right] = 0,$$ where (A. E)=0 andA,c, ω ₀ andν are constants in space and time. Amplitude dependent frequency shifts and the solution up to third order are derived.     

Strongly positive semigroups and faithful invariant states

Let (?, τ, ω) denote aW*-algebra ?, a semigroupt>0?τ _t of linear maps of ? into ?, and a faithful τ-invariant normal state ω over ?. We assume that τ is strongly positive in the sense that $$\tau _t (A^ * A) \geqq \tau _t (A)^ * \tau _t (A)$$ for allA∈? andt>0. Therefore one can define a contraction semigroupT on ?= \(\overline {\mathcal{M}\Omega } \) by $$T_t A\Omega = \tau _t (A)\Omega ,{\rm A} \in \mathcal{M},$$ where Ω is the cyclic and separating vector associated with ω. We prove 1. the fixed points ?(τ) of τ are given by ?(τ)=?∩T′=?∩E′, whereE is the orthogonal projection onto the subspace ofT-invariant vectors, 2. the state ω has a unique decomposition into τ-ergodic states if, and only if, ?(τ) or {?υE}′ is abelian or, equivalently, if (?, τ, ω) is ?-abelian, 3. the state ω is τ-ergodic if, and only if, ?υE is irreducible or if $$\mathop {\inf }\limits_{\omega '' \in Co\omega 'o\tau } \left\| {\omega '' - \omega '} \right\| = 0$$ for all normal states ω′ where Coω′°τ denotes the convex hull of {ω′°τ _t } _t>0. Subsequently we assume that τ is 2-positive,T is normal, andT* _t ?₊Ω \( \subseteqq \overline {\mathcal{M}_ + \Omega } \) , and then prove 4. there exists a strongly positive semigroup |τ| which commutes with τ and is determined by $$\left| \tau \right|_t \left( A \right)\Omega = \left| {T_t } \right|A\Omega ,$$ 5. results similar to 1 and 2 apply to |τ| but the τ-invariant state ω is |τ|-ergodic if, and only if, $$\mathop {\lim }\limits_{t \to \infty } \left\| {\omega 'o\tau _t - \omega } \right\| = 0$$ for all normal states ω′.     

General lower bounds for resonances in one dimension

Lower bounds are derived for the magnitude of the imaginary parts of the resonance eigenvalues of a Schrödinger operator $$ - \frac{{d^2 }}{{dx^2 }} + V(x)$$ on the line, depending only on the support and bounds ofV and on the real part of the resonance eigenvalue. For example, if the resonance eigenvalue is denotedE +i?, then there existC and ?₀ depending only on ‖E‖_∞ andE such that if the support ofV is contained in an interval of length ? > ?₀, then $$\left| \varepsilon \right| > \frac{{m^3 \sqrt E }}{{(m + \sqrt E )^2 }}\exp ( - m\ell )(1 - C\ell ^{ - 1} ),$$ wherem‖V(x)?E? _∞ ^1/2 .     

Widths of the 7.74 MeV analogue state in39K

The³⁸Ar(p, p)³⁸Ar and³⁸Ar(p, γ)³⁹K reactions have been investigated in the analogue resonance region at E_p=1.39 MeV. For the 7.74 MeV state of³⁹K branching ratios, unique spin and parity, \(J^\pi = \tfrac{3}{2} - \) , proton and total width, Γ_p=Γ=1.1±0.2 keV, have been determined.     

Measurement of internal conversionK/L 3 andK/L ratios forE2 transition

Internal conversionK/L ₃ ratio forE2 transitions was measured for nuclei with mass numbers from 152–198, and the energy interval of 80–500keV. The results of all the 11 transitions measured are in good agreement with theoretical values ofSliv andBand, Pauli, Bhalla andHager andSeltzer.¹⁵²Eu,¹⁶⁰Tb,¹⁶⁶Ho,¹⁸²Ta,¹⁸⁶Re,¹⁸⁸Re, Radioactivity¹⁹²Ir,¹⁹⁸Au/all (n, γ)/measuredc c ratios.     

First-order phase transition from an antiferromagnetic ferroelectric to a cycloidal multiferroic with weak ferromagnetism during the joint action of applied magnetic and electric fields

The thermodynamics of the phase transition in a perovskite-like multiferroic, in which an antiferromagnetic ferroelectric transforms into a new magnetic state where a spiral spin structure and weak ferromagnetism can coexist in applied magnetic field H, is described. This state forms as a result of a first-order phase transition at a certain temperature (below Néel temperature T _N ), where a helicoidal magnetic structure appears due to the Dzyaloshinskii-Moriya effect. In this case, the axes of electric polarization and the helicoid of magnetic moments are mutually perpendicular and lie in the ab plane, which is normal to principal axis c. Additional electric polarization p, which decreases the total polarization of the ferroelectric P, appears in the ab plane. The effect of applied magnetic and electric fields on the properties of a multiferroic with a helicoidal magnetic structure is described. An alternating electric field is shown to cause a field-linear change in magnetic moment m, whose sign is opposite to the sign of the change of electric field E. The detected hysteretic phenomena that determine the temperature ranges of overheating and supercooling of each phase are explained. A comparison with the experimental data is performed.     

Spinaustauschstreuung von Kalium an Caesium

The differential spin exchange cross sectionI _ex(?, E) for the system K-Cs has been measured as a function of angle for 2°???40° and collision energy for 1.2 · 10^?13 erg ?E?2.1 · 10^?13erg. Confirming previous experiments it exhibits a maximum neargj≈10°. The angular position of maximum changes with collision energy. Increasing the energy lowers the maximum angle. The analysis of the cross section with regard to information about the interaction of the collision partners turns out to be very complicated. A first few model calculations are presented and discussed. The most obvious, qualitative relations between the singlet and triplet interaction potentials,V ₀ andV ₁, and the differential spin exchange cross sectionI _ex(?) appear to be: 1. The average magnitude ofI _ex is markedly different below and above the rainbow angle \(\vartheta _{R_1 }\) of the shallow triplet potential V₁. From the observed intensity drop towards large angles we estimate a lower bound for the triplet potential well depth? ₁?1.4 · 10^?14 erg. 2. The expected relation between the integral spin exchange cross sectionσ _ex and the difference potentialΔV(R)=V ₁-V ₀ at large distances is also borne out by the differential spin exchange cross section I_ex(?) below \(\vartheta _{R_1 }\) . Relative to the intensity beyond \(\vartheta _{R_1 }\) the cross section increases if range and strength ofΔV is increased.     

On aC*-algebra approach to phase transition in the two-dimensional Ising model. II

We investigate the statesφ _β on theC*-algebra of Pauli spins on a one-dimensional lattice (infinitely extended in both directions) which give rise to the thermodynamic limit of the Gibbs ensemble in the two-dimensional Ising model (with nearest neighbour interaction) at inverse temperature β. We show that if β _c is the known inverse critical temperature, then there exists a family {v _β :β≠β _c } of automorphisms of the Pauli algebra such that $$\phi _\beta = \left\{ {_{\phi _\infty \circ v_\beta ,}^{\phi _0 \circ v_\beta ,} } \right. _{\beta > \beta _c .}^{0 \leqslant \beta< \beta _c } $$ .     

Multipole-analysis in π0-photoproduction on protons belowE=234 MeV

Taking ReM _1? as a parameter ReE ₀₊, ReE ₁₊ and ReM ₁₊ are determined by a phenomenological analysis using the measurements ofHitzeroth andGovorkov et al. Taking for ReM _1? the present theoretical values ReE ₁₊ turns out to be negative. ¦ReE ₀₊¦ is smaller than former theoretical calculations suggest. The influence of the imaginary parts and of the higher multipoles is discussed.     

Two-dimensional representation of position, velocity and acceleration by PFG-NMR

A novel view on the presentation of pulsed field-gradient nuclear magnetic resonance experiments to encode position and translational displacements is given. A conventional diffusion or flow experiment employing two magnetic field gradients of effective areak ₁, andk ₂ separated by a time interval Δ can formally be expressed as a means to probek space in a two-dimensional way. While for most applications, a full coverage of the [k ₁,k ₂] space is not necessary, an experiment withk ₁ = ?k ₂ can be regarded as a sampling of the secondary diagonal in [k ₁,k ₂] space. Likewise, the main diagonal is represented by the conditionk ₁ =k ₂ and encodes position. Thus, the [r ₁,r ₂] space conjugate to [k ₁,k ₂], which is obtained by Fourier transformation, can be transferred into a position/displacement correlation plot simply by rotation of the coordinate system by an angle of 45°. While displacementR =r ₂ ?r ₁ corresponds to an average velocity? =R/Δ, an extension towards higher-order derivations such as acceleration is straightforward by modification of the pulse sequence. We discuss this new concept in a general way, treating both the magnetic field and the particle position by Taylor expansions with respect to space and time, respectively, and present examples for fluid flowing through capillary systems in the light of the suggested interpretation.