A remark on tensor representations

The identity $$\sum\limits_{v = 0} {\left( {\begin{array}{*{20}c} {n + 1} \\ v \\ \end{array} } \right)\left[ {\left( {\begin{array}{*{20}c} {n - v} \\ v \\ \end{array} } \right) - \left( {\begin{array}{*{20}c} {n - v} \\ {v - 1} \\ \end{array} } \right)} \right] = ( - 1)^n } $$ is proved and, by means of it, the coefficients of the decomposition ofD ₁ ⁿ into irreducible representations are found. It holds: ifD ₁ ⁿ \(\mathop {\sum ^n }\limits_{m = 0} A_{nm} D_m \) , then $$A_{nm} = \mathop \sum \limits_{\lambda = 0} \left( {\begin{array}{*{20}c} n \\ \lambda \\ \end{array} } \right)\left[ {\left( {\begin{array}{*{20}c} \lambda \\ {n - m - \lambda } \\ \end{array} } \right) - \left( {\begin{array}{*{20}c} \lambda \\ {n - m - \lambda - 1} \\ \end{array} } \right)} \right].$$      

Experimental determination of penetration parameter forM1 conversion,and theE2 admixtures for 77 and 191 keV transitions in197Au

TheL-subshell conversion for 77 keV transition andK,L ₁,L ₂-shell conversion for 191 keV transition in¹⁹⁷Au, as well asK-shell conversion transition of 158 keV in¹⁹⁹Hg were measured by means of Π√2-iron free electron spectrometer. Relative gamma-ray intensities have been determined by Ge(Li) spectrometer. From these measurements the α _K conversion coefficient value has been deduced for 191 keV transition as α_K(191 keV)=0.86±0.03. This value shows that the spin of the level at 268 keV in¹⁹⁷Au is 3/2⁺. For the penetration parameter (λ) and intensity ratio \(\left( {\delta ^2 = \frac{{\left\langle {E2} \right\rangle ^2 }}{{\left\langle {MI} \right\rangle ^2 }}} \right)\) the following values are obtained: $$\begin{gathered} \lambda = 3.4 \pm _{1.5}^{1.9} \delta ^2 = 0.11 \pm 0.03for 77 keV transition \hfill \\ \lambda = 3.2 \pm _{0.6}^{0.8} \delta ^2 = 0.17 \pm 0.04for 191 keV transition. \hfill \\ \end{gathered} $$ The agreement of these results with the predictions of De Shalit model is discussed.     

The final statesl ± K ± K ± X in jets as signatures ofB_s^0 - \bar B_s^0 mixings

Significant mixing is expected between the neutral bottom mesons \(B_s^0 - \bar B_s^0 \) in the standard model of weak interactions. We propose measurements of the processes \(\left\{ {\begin{array}{*{20}c} {e^ + e^ - } \\ {p\bar p} \\ \end{array} } \right\} \to \begin{array}{*{20}c} {b\bar b} \\ {} \\ \end{array} \to l^ + K^ - K^ - X\) as a measure of such mixing. Rates are presented for energetic bottom quark jets, produced ine ⁺ e ^? annihilation.     

Scaling for a random polymer

LetQ _n ^β be the law of then-step random walk on ?^d obtained by weighting simple random walk with a factore ^?β for every self-intersection (Domb-Joyce model of “soft polymers”). It was proved by Greven and den Hollander (1993) that ind=1 and for every β∈(0, ∞) there exist θ^*(β)∈(0,1) and such that under the lawQ _n ^β asn→∞: $$\begin{array}{l} (i) \theta ^* (\beta ) is the \lim it empirical speed of the random walk; \\ (ii) \mu _\beta ^* is the limit empirical distribution of the local times. \\ \end{array}$$ A representation was given forθ ^*(β) andµ _β ^β in terms of a largest eigenvalue problem for a certain family of ? x ? matrices. In the present paper we use this representation to prove the following scaling result as β?0: $$\begin{array}{l} (i) \beta ^{ - {\textstyle{1 \over 3}}} \theta ^* (\beta ) \to b^* ; \\ (ii) \beta ^{ - {\textstyle{1 \over 3}}} \mu _\beta ^* \left( {\left\lceil { \cdot \beta ^{ - {\textstyle{1 \over 3}}} } \right\rceil } \right) \to ^{L^1 } \eta ^* ( \cdot ) . \\ \end{array}$$ The limitsb ^*∈(0, ∞) and are identified in terms of a Sturm-Liouville problem, which turns out to have several interesting properties. The techniques that are used in the proof are functional analytic and revolve around the notion of epi-convergence of functionals onL ²(?⁺). Our scaling result shows that the speed of soft polymers ind=1 is not right-differentiable at β=0, which precludes expansion techniques that have been used successfully ind≧5 (Hara and Slade (1992a, b)). In simulations the scaling limit is seen for β≦10^?2.     

Neutral strange particle exclusive production in charged current high-energy antineutrino interactions

The cross section of the quasi-elastic reactions \(\bar v_\mu p \to \mu ^ + \Lambda (\Sigma ^0 )\) in the energy range 5–100 GeV is determined from Fermilab 15′ bubble chamber antineutrino data. TheQ ² analysis of quasi-elastic Λ events yieldsM _A=1.0±0.3 GeV/c² for the axial mass value. With zero µΛ K ⁰ events observed, the 90% confidence level upper limit \(\sigma (\bar v_\mu p \to \mu ^ + \Lambda {\rm K}^0 )< 2.0 \cdot 10^{ - 40} cm^2 \) is obtained. At the same time, we found that the cross section of reaction \(\bar v_\mu p \to \mu ^ + \Lambda {\rm K}^0 + m\pi ^0 \) is equal to \(\left( {3.9\begin{array}{*{20}c} { + 1.6} \\ { - 1.3} \\ \end{array} } \right) \cdot 10^{ - 40} cm^2 \) .     

A limit law for the ground state of Hill's equation

It is proved that the ground state Λ(L) of (?1)x the Schrödinger operator with white noise potential, on an interval of lengthL, subject to Neumann, periodic, or Dirichlet conditions, satisfies the law $$\mathop {\lim }\limits_{L \uparrow \infty } P[(L/\pi )\Lambda ^{1/2} \exp ( - \tfrac{8}{3}\Lambda ^{3/2} ) > x] = \left\{ {\begin{array}{*{20}c} {1forx< 0} \\ {e^{ - x} forx \geqslant 0} \\ \end{array} } \right.$$      

An expansion of Hartree-Fock and correlation energies,relativistic corrections and the dipole matrix element in the powers of of 1/Z

Feynman diagrammatic technique was used for the calculation of Hartree-Fock and correlation energies, relativistic corrections, dipole matrix element. The whole energy of atomic system was defined as a polen-electron Green function. Breit operator was used for the calculation of relativistic corrections. The Feynman diagrammatic technique was developed for 〈H_B>. Analytical expressions for the contributions from diagrams were received. The calculations were carried out for the terms of such configurations as 1s² 2sⁿ¹ 2pⁿ² (2 ≧n₁≧ 0, 6≧ n₂ ≧ 0). Numerical results are presented for the energies of the terms in the form $$E = E_0 Z^2 + \Delta {\rm E}_2 + \frac{1}{Z}\Delta {\rm E}_3 + \frac{{\alpha ^2 }}{4}(E_0^r + \Delta {\rm E}_1^r Z^3 )$$ and for fine structure of the terms in the form $$\begin{gathered} \left\langle {1s^2 2s^{n_1 } 2p^{n_2 } LSJ|H_B |1s^2 2s^{n_1 \prime } 2p^{n_2 \prime } L\prime S\prime J} \right\rangle = \hfill \\ = ( - 1)^{\alpha + S\prime + J} \left\{ {\begin{array}{*{20}c} {L S J} \\ {S\prime L\prime 1} \\ \end{array} } \right\}\frac{{\alpha ^2 }}{4}(Z - A)^3 [E^{(0)} (Z - B) + \varepsilon _{co} ] + \hfill \\ + ( - 1)^{L + S\prime + J} \left\{ {\begin{array}{*{20}c} {L S J} \\ {S\prime L\prime 2} \\ \end{array} } \right\}\frac{{\alpha ^2 }}{4}(Z - A)^3 \varepsilon _{cc} . \hfill \\ \end{gathered} $$ Dipole matrix elements are necessary for calculations of oscillator strengths and transition probabilities. For dipole matrix elements two members of expansion by 1/Z have been obtained. Numerical results were presented in the form P(a,a′) = a/Z(1+τ/Z).     

Exponential lower bounds to solutions of the Schrödinger equation: Lower bounds for the spherical average

For a large class of generalizedN-body-Schrödinger operators,H, we show that ifE<Σ=infσ_ess(H) and ψ is an eigenfunction ofH with eigenvalueE, then $$\begin{array}{*{20}c} {\lim } \\ {R \to \infty } \\ \end{array} R^{ - 1} \ln \left( {\int\limits_{S^{n - 1} } {|\psi (R\omega )|} ^2 d\omega } \right)^{1/2} = - \alpha _0 ,$$ with α ₀ ² +E a threshold. Similar results are given forE≧Σ.     

Energy spectrum of photons produced by neutrino (antineutrino) scattering on electrons. Effect of possible neutrino oscillations

The energy spectrum of photons emitted in neutrino (antineutrino) scattering on electrons at E_v ? m_v is calculated with the assumption that the neutral electron flow has an arbitrary (V, A) structure. The result obtained is generalized to the case of possible neutrino oscillations, \(v_e \begin{array}{*{20}c} { - \to } \\ { \leftarrow - } \\ \end{array} v_\mu , \overline v _e \begin{array}{*{20}c} { - \to } \\ { \leftarrow - } \\ \end{array} \overline v _\mu \) , at an arbitrary neutrino mixing angle. Using the Weinberg-Salam model (sin²θ_W = 0.23) estimates of the sections dσ_γ/dω and σ_γ are obtained with consideration of the reactor antineutrino flux \(\bar v_e \) . The contributions from charged and neutral lepton fluxes and their interference to dσ_γ/dω are compared.     

Eine scheinbare Abschwächung der Lokalitätsbedingung. II

If for a relativistic field theory the expectation values of the commutator (Ω|[A (x),A(y)]|Ω) vanish in space-like direction like exp {? const|(x-y ²|^α/2#x007D; with α>1 for sufficiently many vectors Ω, it follows thatA(x) is a local field. Or more precisely: For a hermitean, scalar, tempered fieldA(x) the locality axiom can be replaced by the following conditions 1. For any natural numbern there exist a) a configurationX(n): $$X_1 ,...,X_{n - 1} X_1^i = \cdot \cdot \cdot = X_{n - 1}^i = 0i = 0,3$$ with \(\left[ {\sum\limits_{i = 1}^{n - 2} {\lambda _i } (X_i^1 - X_{i + 1}^1 )} \right]^2 + \left[ {\sum\limits_{i = 1}^{n - 2} {\lambda _i } (X_i^2 - X_{i + 1}^2 )} \right]^2 > 0\) for all λ _i ≧0i=1,...,n?2, \(\sum\limits_{i = 1}^{n - 2} {\lambda _i > 0} \) , b) neighbourhoods of theX _i 's:U _i (X _i )?R ⁴ i=1,...,n?1 (in the euclidean topology ofR ⁴) and c) a real number α>1 such that for all points (x):x ₁, ...,x _n?1:x _i ∈U _i (X _r ) there are positive constantsC ⁽ⁿ⁾{(x)},h ⁽ⁿ⁾{(x)} with: $$\left| {\left\langle {\left[ {A(x_1 )...A(x_{n - 1} ),A(x_n )} \right]} \right\rangle } \right|< C^{(n)} \left\{ {(x)} \right\}\exp \left\{ { - h^{(n)} \left\{ {(x)} \right\}r^\alpha } \right\}forx_n = \left( {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ r \\ \end{array} } \right),r > 1.$$ 2. For any natural numbern there exist a) a configurationY(n): $$Y_2 ,Y_3 ,...,Y_n Y_3^i = \cdot \cdot \cdot = Y_n^i = 0i = 0,3$$ with \(\left[ {\sum\limits_{i = 3}^{n - 1} {\mu _i (Y_i^1 - Y_{i{\text{ + 1}}}^{\text{1}} } )} \right]^2 + \left[ {\sum\limits_{i = 3}^{n - 1} {\mu _i (Y_i^2 - Y_{i{\text{ + 1}}}^{\text{2}} } )} \right]^2 > 0\) for all μ _i ≧0,i=3, ...,n?1, \(\sum\limits_{i = 3}^{n - 1} {\mu _i > 0} \) , b) neighbourhoods of theY _i 's:V _i(Y _i )?R ⁴ i=2, ...,n (in the euclidean topology ofR ⁴) and c) a real number β>1 such that for all points (y):y ₂, ...,y _n y _i ∈V _i (Y _i there are positive constantsC _(n){(y)},h _(n){(y)} and a real number γ_(n){(y)∈a closed subset ofR?{0}?{1} with: γ_(n){(y)}\y ₂,y ₃, ...,y _n totally space-like in the order 2, 3, ...,n and $$\left| {\left\langle {\left[ {A(x_1 ),A(x_2 )} \right]A(y_3 )...A(y_n )} \right\rangle } \right|< C_{(n)} \left\{ {(y)} \right\}\exp \left\{ { - h_{(n)} \left\{ {(y)} \right\}r^\beta } \right\}$$ for \(x_1 = \gamma _{(n)} \left\{ {(y)} \right\}r\left( {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ 1 \\ \end{array} } \right),x_2 = y_2 - [1 - \gamma _{(n)} \{ (y)\} ]r\left( {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ 1 \\ \end{array} } \right)\) and for sufficiently large values ofr.     

Nuclear orientation and NMR/ON of205,207Po

^205,207Po have keen implanted with an isotope separator on-line into cold host matrices of Fe, Ni, Zn and Be. Nuclear magnetic resonance of oriented²⁰⁷Po has been observed in Fe and Ni, of²⁰⁵Po in Fe. The resonance frequencies for zero external field are $$\begin{gathered} v_L (^{207} Po\underline {Fe} ) = 575.08(20)MHz \hfill \\ v_L (^{207} Po\underline {Ni} ) = 160.1(8)MHz \hfill \\ v_L (^{205} Po\underline {Fe} ) = 551.7(8)MHz. \hfill \\ \end{gathered} $$ From the dependence of the resonance frequency on external magnetic field theg-factor of²⁰⁷Po was derived as $$g(^{207} Po) = + 0.31(22).$$ Using this value the magnetic hyperfine fields of Po in Fe and Ni were obtained as $$\begin{gathered} B_{hf} (Po\underline {Fe} ) = + 238(16)T \hfill \\ B_{hf} (Po\underline {Ni} ) = 66.3(4.6)T. \hfill \\ \end{gathered}$$ Theg-factor of²⁰⁵Po follows as $$g(^{205} Po) = + 0.304(22).$$ From the temperature dependence of the anisotropies ofγ-lines in the decay of^205,207Po the multipole mixing of several transitions was derived. The electric interaction frequenciesv _Q=eQV_zz/h in the hosts Zn and Be were measured as $$\begin{gathered} v_Q (^{207} Po\underline {Zn} ) = + 42(3)MHz \hfill \\ v_Q (^{207} Po\underline {Be} ) = - 70(20)MHz \hfill \\ v_Q (^{205} Po\underline {Be} ) = - 42(17)MHz. \hfill \\ \end{gathered}$$      

g-factors of rotational states in176Hf,177Hf,and180Hf

g-factors of rotational states in¹⁷⁶Hf and¹⁸⁰Hf were measured with the twelve detector IPAC-apparatus of our laboratory [1]. The natural radioactivity 3.78·10¹⁰y¹⁷⁶Lu and the 5.5 h isomer^180mHf were used which populate the ground-state rotational bands of¹⁷⁶Hf and¹⁸⁰Hf. The integral rotations ofγ-γ directional correlations in strong external magnetic fields and in static hyperfine fields of (Lu→Hf)Fe₂ and HfFe₂ were observed. The following results were obtained: $$\begin{array}{l} ^{176} Hf: g\left( {4_1^ + } \right) = + 0.334\left( {38} \right) \\ ^{180} Hf: g\left( {2_1^ + } \right) = + 0.305\left( {14} \right) \\ g\left( {4_1^ + } \right) = + 0.358\left( {43} \right) \\ {{ g\left( {6_1^ + } \right)} \mathord{\left/ {\vphantom {{ g\left( {6_1^ + } \right)} {g\left( {4_1^ + } \right)}}} \right. \kern-\nulldelimiterspace} {g\left( {4_1^ + } \right)}} = + 0.95\left( {12} \right) \\ \end{array}$$ . The hyperfine field in (Lu→Hf)Fe₂ was calibrated by observing the integral rotation of the 9/2^? first excited state of¹⁷⁷Hf populated in the decay of 6.7d¹⁷⁷Lu. Theg-factor of this state was redetermined in an external magnetic field as $$^{177} Hf: g\left( {{9 \mathord{\left/ {\vphantom {9 {2^ - }}} \right. \kern-\nulldelimiterspace} {2^ - }}} \right) = + 0.228\left( 7 \right)$$ . Finally theg-factor of the 2 ₁ ⁺ state of¹⁷⁶Hf was derived from the measuredg(2 ₁ ⁺ ) of¹⁸⁰Hf by use of the precisely known ratiog(2 ₁ ⁺ ,¹⁷⁶Hf)/g(2 ₁ ⁺ ,¹⁸⁰Hf) [2] as $$^{176} Hf: g\left( {2_1^ + } \right) = + 0.315\left( {30} \right)$$ .     

Gyromagnetic ratio of the 2+ rotational state of184W and observation of a large hyperfine anomaly with respect to183Wg in iron

Theg-factor of the 2⁺ rotational state of¹⁸⁴W was redetermined by an IPAC measurement in an external magnetic field of 9.45 (5)T as: $$g_{2^ + } (^{184} W) = + 0.289(7).$$ In the evaluation the remeasured half-life of the 2⁺ state: $$T_{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} (2^ + ) = 1.251(12)ns$$ was used. TDPAC-measurements with a sample of carrierfree¹⁸⁴Re in high purity iron gave the hyperfine fields: $$B_{300 K}^{hf} (^{184} W_2 + \underline {Fe} ) = 70.1(21)T$$ and $$B_{40 K}^{hf} (^{184} W_{2^ + } \underline {Fe} ) = 71.8(22)T.$$ A comparison with the hyperfine field known from a spin echo experiment with¹⁸³W _g Fe leads to the hyperfine anomaly: $$^{184} W_{2^ + } \Delta ^{183} W_g = + 0.145(36).$$ The hyperfine splitting observed in a Mössbauer source experiment with another sample of carrierfree^184m Re in high purity iron indicates that the smaller splitting, measured previously by a Mössbauer absorber experiment is due to the high tungsten concentration in the absorber. The new value for theg-factor of the 2⁺ state together with the result of the Mössbauer experiment allow an improved calibration for our recent investigation of theg _R -factors of the 4⁺ and 6⁺ rotational states. The recalculated values are: $$g_{4^ + } (^{184} W) = + 0.293(23)$$ and $$g_{6^ + } (^{184} W) = + 0.299(43).$$ The remeasured 792-111 keVγ-γ angular correlation $$W(\Theta ) = 1 - 0.034(4) \cdot P_2 + 0.325(6) \cdot P_4 $$ gives for the mixing ratio of theK-forbidden 792keV transition: $$\delta ({{E2} \mathord{\left/ {\vphantom {{E2} {M1}}} \right. \kern-\nulldelimiterspace} {M1}}) = - \left( {17.6\begin{array}{*{20}c} { + 1.8} \\ { - 1.5} \\ \end{array} } \right).$$ A detailed investigation of the attenuation ofγ-γ angular correlations in liquid sources of¹⁸⁴Re and^184m Re revealed the reason for erroneous results of early measurements of the 2⁺ g _R -factor: The time dependence of the perturbation is not of a simple exponential type. It contains an unresolved strong fast component.     

Lifetimes andg J factors of 6s7p and 5d6p levels of Ba I

Excited states of Ba have been investigated with optical double resonance and Hanle effect. The followingg _J factors and natural lifetimes (in 10^?9 sec) have been measured $$\begin{gathered} 6s7p\left\{ {\begin{array}{*{20}c} {^1 P_1 :g_J = 1.003(2)\tau = 13.5(6)} \\ {^3 P_1 :g_J = 1.4971(8)\tau = 85.0(8.0)} \\ \end{array} } \right. \hfill \\ 5d6p\left\{ {\begin{array}{*{20}c} {^1 P_1 :g_J = 1.004(2)\tau = 12.4(9)} \\ {^3 P_1 :g_J = 1.4847(15)\tau = 11.7(9)} \\ {^3 D_1 :g_J = 0.5064(3)\tau = 17.0(5).} \\ \end{array} } \right. \hfill \\ \end{gathered}$$ g _J is utilized to test the mixing coefficients of the wave functions in the intermediate coupling model. The lifetimes are converted into absolute transition probabilities for all the decays originating from the states investigated under the assumption that their branching ratios obtained elsewhere are correct. This assumption is not unquestionable, however.     

Spin and torsion in general relativity: I. Foundations

From physical arguments space-time is assumed to possess a connection \(\Gamma _{ij}^k = \left\{ {\begin{array}{*{20}c} k \\ {ij} \\ \end{array} } \right\} + S_{ij}^{ k} - S_{j i}^{ k} + S_{ ij}^k = \left\{ {\begin{array}{*{20}c} k \\ {ij} \\ \end{array} } \right\} - K_{ij}^{ k} \) . \(\left\{ {\begin{array}{*{20}c} k \\ {ij} \\ \end{array} } \right\}\) is Christoffel's symbol built up from the metric g _ij and already appearing in General Relativity (GR). Cartan's torsion tensor \(S_{ij} ^k = \tfrac{1}{2}(\Gamma _{ij}^k - \Gamma _{ji}^k )\) and the contortion tensor K _ij ^k , in contrast to the theory presented here, both vanish identically in conventional GR. Using the connection introduced above in this series of articles, we will discuss the consequences for GR in the framework of a consistent formalism. There emerges a theory describing, in a unified way, gravitation and a very weak spin-spin contact interaction. In section 1 we start with the well-known dynamical definition of the energy-momentum tensor σ ^ij ～ δ?/δg _ij , where ? represents the Lagrangian density of matter (section1.1). In sections1.2,3 we will show that due to geometrical reasons, the connection assumed above leads to a dynamical definition of the spin-angular momentum tensor according to τ^k _ji ～ δ?/δK _ij ^k . In section1.4, by an ideal experiment, it will become clear that spin prohibits the introduction of an instantaneous rest system and thereby of a geodesic coordinate system. Among other things in section1.5 there are some remarks about the rôle torsion played in former physical theories. In section 2 we sketch the content of the theory. As in GR, the action function is the sum of the material and the field action function (sections2.1,2). The extension of GR consists in the introduction of torsion S _ij ^k as a new field. By variation of the action function with respect to metric and torsion we obtain the field equations in a general form (section2.3). They are also valid for matter described by spinors; in this case, however, one has to introduce tetrads as anholonomic coordinates and slightly to generalize the dynamical definition of energy-momentum (sections2.4,5).     

Detection of a hexadecapole interaction in the atomic ground state3 H 6 of167Er

Using the atomic beam magnetic resonance method, precision measurements of the hyperfine structure and Zeeman interactions have been performed in the ground state 4f ¹²6s ^{2 3} H ₆ of¹⁶⁷Er. The experimental data were analyzed using an effective operator parametrized in the space of states of the ground state multiplet. It yielded eight effective hyperfine structure and Zeeman interaction constants which served to calculate the seven hyperfine separations of the ground state. The results are: $$\begin{gathered} 2F 2F' v_{FF'} (MHz) \hfill \\ 5 7 - 354.371 9409 (27) \hfill \\ 7 9 - 2{\text{78}}{\text{.231}} {\text{8263(14)}} \hfill \\ {\text{9}} 11 - 69.050 7785 (4) \hfill \\ 11 13 + 302.735 3731(12) \hfill \\ 13 15 + 866.691 3871(10) \hfill \\ 15 17 + 1,652.383 5154 (6) \hfill \\ 17 19 + 2,689.380 8050(10) \hfill \\ \end{gathered}$$ From the effective Zeeman interaction constants it was possible to determine an improvedg _I -value, uncorrected for atomic diamagnetism: $$ g_I = + 0.086 775 (19) \cdot 10^{ - 3}$$ Furthermore a hexadecapole interaction corresponding to a diagonal hexadecapole interaction constant $$A_4 = - 16 (10) Hz$$ could be established which is of the order of magnitude expected from Coulomb excitation experiments as well as theoretical calculations.     

Effekte höherer Ordnung beim erlaubten β-Zerfall von Li8

Theβ-α angular correlation of Li⁸ has been measured at electron energies of 3·5 and 7·0 Mev. Theβ-energies were selected by a magnetic lens spectrometer. Subtracting the contribution by kinematic effects we find for the coefficientb of the cos²Θ term\(b = \left( {1 \cdot 9\begin{array}{*{20}c} { + 2 \cdot 6} \\ { - 2 \cdot 3} \\ \end{array} } \right)\%\) at 3·5 Mev and\(b = \left( {4 \cdot 0\begin{array}{*{20}c} { + 2 \cdot 0} \\ { - 1 \cdot 3} \\ \end{array} } \right)\%\) at 7·0 Mev. This result is in reasonable agreement with theoretical predictions.     

Antiproton-proton annihilation at rest intoπ + π − η ′ andπ + π − η from atomicS andP states

We have measured the branching ratios for \(\bar pp\) annihilation at rest intoπ ⁺ π ^? η andπ ⁺ π ^? η′ in hydrogen gas in two data samples that have different fractions ofS-wave andP-wave initial states. The branching ratios are derived from a comparison with the topological branching ratio for \(\bar pp\) annihilations into four charged pions of (49±4)% and the branching ratio intoπ ⁺ π ^? π ⁺ π ^? π ⁰ of (18.7±1.6)%. We find a significant reduction of the branching ratios fromP-states for \(\bar pp \to \pi ^ + \pi ^ - \eta \) andπ ⁺ π ^? η′ in comparison toS-state annihilation. $$\begin{gathered} BR(S - wave \to \pi ^ + \pi ^ - \eta ) = (13.7 \pm 1.46) \cdot 10^{ - 3} \hfill \\ BR(P - wave \to \pi ^ + \pi ^ - \eta ) = (3.35 \pm 0.84) \cdot 10^{ - 3} \hfill \\ BR(S - wave \to \pi ^ + \pi ^ - \eta ') = (3.46 \pm 0.67) \cdot 10^{ - 3} \hfill \\ BR(P - wave \to \pi ^ + \pi ^ - \eta ') = (0.61 \pm 0.33) \cdot 10^{ - 3} . \hfill \\ \end{gathered} $$ In a partial wave analysis of theπ ⁺ π ^? η Dalitz plot we find the following contributions: Phase space, \(a_2^ + (1320)\pi ^ \mp \) ,ηρ⁰ andf ₂(1270)η: $$\begin{gathered} BR(S - wave \to \pi ^ + \pi ^ - \eta PS) = (6.31 \pm 1.22) \cdot 10^{ - 3} \hfill \\ BR(P - wave \to \pi ^ + \pi ^ - \eta PS) = (0.47 \pm 0.26) \cdot 10^{ - 3} \hfill \\ BR(^1 S_0 \to a_2^ \pm (1320)\pi ^ \mp ) = (2.59 \pm 0.73) \cdot 10^{ - 3} \hfill \\ BR(^3 S_1 \to a_2^ \pm (1320)\pi ^ \mp ) = (1.31 \pm 0.48) \cdot 10^{ - 3} \hfill \\ BR(P - wave \to a_2^ \pm (1320)\pi ^ \mp ) = (1.31 \pm 0.69) \cdot 10^{ - 3} \hfill \\ BR(^3 S_1 \to \rho \eta ) = (3.29 \pm 0.90) \cdot 10^{ - 3} \hfill \\ BR(^1 P_1 \to \rho \eta ) = (0.94 \pm 0.53) \cdot 10^{ - 3} \hfill \\ BR(^1 S_0 \to f_2 (1270)\eta ) = (0.083 \pm 0.086) \cdot 10^{ - 3} \hfill \\ BR(P - wave \to f_2 (1270)\eta ) = (0.64 \pm 0.26) \cdot 10^{ - 3} . \hfill \\ \end{gathered} $$ We find a 2 σ effect for the reaction \(\bar pp \to a_0^ \pm (980)\pi ^ \mp \) , \(a_0^ \pm \to \eta \pi ^ \pm \) , with a branching ratio of (0.13±0.07)·10^?3. For η' production we give a branching ratio of \(\bar pp \to \rho \eta '\) of (1.81±0.44)·10^?3 from³ S ₁. We estmate a contribution of about 0.3·10^?3 for ρη' fromP-states. The ratio of ρη and ρη' rpoduction is used to test the validity of the quark line rule. In theπ ⁺ π ^? π ⁺ π ^? γ final state we do not observe the reaction \(\bar pp \to \pi ^ + \pi ^ - \omega \) , ω→π ⁺ π ^? λ and derive an upper limit of 3·10^?3 for decay modeω→π ⁺ π ^? λ.     

Gyromagnetic ratios of collective states of166Er

The static hyperfine field ofB _hf ^4.2k (ErHo) = 739(18)T of a ferromagnetic holmium single crystal polarized in an external magnetic field of ± 0.48T at ～4.2K was used for integral perturbed γ-γ angular correlation (IPAQ measurements of the g-factors of collective states of¹⁶⁶Er. The 1,200y ^166m Ho activity was used which populates the ground state band and the γ vibrational band up to high spins. The results: $$\begin{gathered} g(4_g^ + ) = + 0.315(16) \hfill \\ g(6_g^ + ) = + 0.258(11) \hfill \\ g(8_g^ + ) = + 0.262(47)and \hfill \\ g(6_\gamma ^ + ) = + 0.254(32) \hfill \\ \end{gathered}$$ exhibit a significant reduction of the g-factors with increasing rotational angular momentum. The followingE2/M1 mixing ratios of interband transitions were derived from the angular correlation coefficients: $$\begin{gathered} 5_\gamma ^ + \Rightarrow 4_g^ + :\delta (810keV) = - (36_{ - 7}^{ + 11} ) \hfill \\ 7_\gamma ^ + \Rightarrow 6_g^ + :\delta (831keV) = - (18_{ - 2}^{ + 3} )and \hfill \\ 7_\gamma ^ + \Rightarrow 8_g^ + :\delta (465keV) = - (63_{ - 12}^{ + 19} ). \hfill \\ \end{gathered}$$ The results are discussed and compared with theoretical predictions.     

On the similarity solution of evolution equation u t =H(x,t, u,u x,u xx , ...)

Let $$\begin{gathered} u^* = u + \in \eta (x,{\text{ }}t,{\text{ }}u), \hfill \\ \hfill \\ \hfill \\ x^* = x + \in \xi (x, t, u{\text{),}} \hfill \\ \hfill \\ \hfill \\ {\text{t}}^{\text{*}} = {\text{ }}t + \in \tau {\text{(}}x,{\text{ }}t,{\text{ }}u), \hfill \\ \end{gathered}$$ be an infinitesimal invariant transformation of the evolution equation u _t =H(x,t,u,?u/?x,...,? ⁿ :u/?x ⁿ . In this paper we give an explicit expression for \(\eta ^{X^i }\) in the ‘determining equation’ $$\eta ^T = \sum\limits_{i = 1}^n {{\text{ }}\eta ^{X^i } {\text{ }}\frac{{\partial H}}{{\partial u_i }} + \eta \frac{{\partial H}}{{\partial u_{} }} + \xi \frac{{\partial H}}{{\partial x}} + \tau } \frac{{\partial H}}{{\partial t}},$$ where u _i =? ⁱ u/?x ⁱ . By using this expression we derive a set of equations with η, ξ, τ as unknown functions and discuss in detail the cases of heat and KdV equations.