Coupling constants for several light nuclei from a dispersion analysis of nucleon and deuteron scattering amplitudes

A forward dispersion relation cannot be applied to charged particle scattering amplitudes unless the influence of the Coulomb interaction is explicitly considered. Earlier studies have shown how Coulomb effects can be taken into account when direct (s-channel or bound-state) poles are investigated. In this paper we extend the Coulomb modification to include I = 0 exchange (u-channel) processes as well. We then apply a forward dispersion relation to empirical d + α, p + d and n + d elastic scattering amplitudes which contain both direct and exchange poles with and without Coulomb effects. We obtain detailed and model-independent information on the following vertices: ⁶Li-α-d (S- and D-state) ⁴He-d-d, ³He-d-p, ³H-d-n and d-p-n. From the coupling constants we calculate the asymptotic normalization (spectroscopic factors) C²₁ of the corresponding cluster wave functions, which become: $\text{C}{}^2{}_0\text{(}{}^6\text{Li}\text{, α}\text{d}\text{) = 4.62 ± 0.23}$, $\text{C}{}^2{}_2\text{(}{}^6\text{Li}\text{, α}\text{d}\text{) = (1 ± 6) × 10}{}^{\mathrm{?4}}$, C²₀(α, dd) < 2, $\text{C}{}^2{}_0\text{(}{}^3\text{He, dp}\text{) = 3.5 ± 0.4}$, $\text{C}{}^2{}_0\text{(}{}^3\text{H, dn}\text{) = 2.6 ± 0.3}\text{and}\text{C}{}^2{}_0\text{(}\text{d, np}\text{) = 1.66 ± 0.1}$.     

Binding energies of λλ hypernuclei and the λλ interaction

The binding energies of the ΛΛ hypernuclei ¹⁰_ΛΛBe and ⁶_ΛΛHe are calculated variationally with a 2α + 2Λ and with an α + 2Λ model, respectively. For ¹⁰_ΛΛBe the integrations were made with Monte Carlo techniques while for ⁶_ΛΛHe direct numerical methods were used. A wide range of phenomenological ΛΛ potentials based on meson-exchange models was considered. An approximately universal linear relation between the calculated values of $\text{B}{}_{\mathrm{\Lambda \Lambda }}\text{(}{}^{10}{}_{\mathrm{\Lambda \Lambda }}\text{Be}\text{)}$ and of $\text{B}{}_{\mathrm{\Lambda \Lambda }}\text{(}{}^6{}_{\mathrm{\Lambda \Lambda }}\text{He}\text{)}$ is obtained. For the experimental value of $\text{B}{}_{\mathrm{\Lambda \Lambda }}\text{(}{}^{10}{}_{\mathrm{\Lambda \Lambda }}\text{Be}\text{) = 17.7 ± 0.1}\text{MeV}$ this relation predicts a much too small value of $\text{B}{}_{\mathrm{\Lambda \Lambda }}\text{(}{}^6{}_{\mathrm{\Lambda \Lambda }}\text{He}\text{)}$, well below the lower limit of the quoted experimental value of 10.9 ± 0.6 MeV. For ΛΛ potentials with repulsive cores the relations, for a given $\text{B}{}_{\mathrm{\Lambda \Lambda }}\text{(}{}^{10}{}_{\mathrm{\Lambda \Lambda }}\text{Be}\text{)}$, between the low-energy ΛΛ scattering parameters a, r₀ and the intrinsic range b are both approximately universal and independent of the detailed potential shape. For the experimental value of $\text{B}{}_{\mathrm{\Lambda \Lambda }}\text{(}{}^{10}{}_{\mathrm{\Lambda \Lambda }}\text{Be}\text{)}$ the preferred values are 2.5 ? ?a ? 3.5 fm, 2.6 ? r₀ ? 3.1 fm. The large and negative values of a correspond to conditions not too far from a bound ¹S₀ ΛΛ state and could indicate a 6-quark dibaryon state with the same quantum numbers and above the ΛΛ threshold.     

Study of the (α, 2α) reaction at 700 MeV

The results of a measurement of the cross section $\text{d}{}^3\text{σ/}\text{d}\text{Ω}{}_{\mathrm{\alpha 1}}\text{d}\text{Ω}{}_{\mathrm{\alpha 2}}\text{d}\text{E}$ for the (α, 2α) reaction on ⁶Li, ¹⁰B, C, O, Na, Al, Fe, Cu and Pb, are presented. Values for the effective cluster number n_0eff are deduced for these targets excluding lead, using the PWIA. It is observed that the PW1A value of n_eff is strongly dependent upon the theoretical shape assumed for the internal momentum distribution of the α-cluster. Using a S-state pole approximation for the momentum distribution leads to a value of the virtual decay amplitude g for A → α + (A ? 4) which is the same from ⁶Li to Fe.     

Molecular three-body approach to Λ9Be and Σ9Be hypernuclei

The dependence of the low-lying spectra of ${}_{\mathrm{\Lambda ,\; \Sigma }}{}^9\text{Be}$ hypernuclei on hyperon-α interaction in the molecular α + α + Λ(Σ) scheme has been studied. A suggestion is made for obtaining the strengths of both p-wave and spin-orbit Λ-α interactions from the experimental ${}_{\mathrm{\Lambda }}{}^9\text{Be}$ spectrum. A relation between the Σ⁰-binding energies $\text{B}{}_{\mathrm{\Sigma }}\text{(}{}_{\mathrm{\Sigma }}{}^9\text{Be}\text{)}$ and $\text{B}{}_{\mathrm{\Sigma }}\text{(}{}_{\mathrm{\Sigma }}{}^9\text{He}\text{)}$ has been established and on its basis a prediction is made about a possible binding energy of ${}_{\mathrm{\Sigma }}{}^5\text{He}$.     

The irreducible Raman scattering tensor operator for the 32 crystallographic point groups and its application to the resonance and electronic Raman effect

The irreducible components of the Raman scattering tensor operator $\text{α}\text{?}{}_{\mathrm{\gamma }}{}^{\mathrm{<mtext>\Gamma </mtext>}}\text{(}\text{Γ}{}_{\mathrm{ks}}\text{Γ}{}_{\mathrm{k\prime s\prime }}\text{)}$ under the symmetry of a general point group are calculated. The unitary transformations U(Γ_γΓ_ksΓ_k′s′, ρσ) from the Cartesian $\text{α}\text{?}{}_{\mathrm{\rho \sigma }}$ and spherical $\text{α}\text{?}{}_{\mathrm{Q}}{}^{\mathrm{K}}$ components, respectively, to the irreducible components $\text{α}\text{?}{}_{\mathrm{\gamma }}{}^{\mathrm{<mtext>\Gamma </mtext>}}\text{(}\text{Γ}{}_{\mathrm{ks}}\text{Γ}{}_{\mathrm{k\prime s\prime }}\text{)}$ for the 32 crystallographic point groups are collected in tables. As an example the unitary transformation U(Γ_γΓ_ksΓ_k′s′, ρσ) is used to discuss the behavior of the scattering tensor in a resonance Raman experiment. With the help of the general formalism the scattering tensor for electronic Raman transitions of transition metal ions is calculated. As an example the scattering tensors of electronic Raman transitions within the ⁵T₂ state of the high-spin trigonal distorted octahedral Fe²⁺ are calculated and the refinement of the selection rules is discussed.     

Competition cusps in (α, γ) reactions

Absolute cross sections are presented for the reactions ³⁷Cl(α, γ)⁴¹K for 2.90 MeV ≦ E^lab_α ≦ 5.23 MeV, ${}^{62}\text{Ni}\text{(α, γ)}{}^{66}\text{Zn}\text{for}\text{5.07}\text{MeV}\text{≦ E}{}_{\mathrm{\alpha }}{}^{\mathrm{?2}}\text{lab}\text{≦ 8.64}\text{MeV}\text{,}{}^{62}\text{Ni}\text{(α,}\text{n}\text{)}{}^{65}\text{Zn}$ for energies near the (α, n₀) threshold at $\text{E}{}_{\mathrm{\alpha }}{}^{\mathrm{<mtext>lab</mtext>}}\text{= 6.90}\text{MeV up to}\text{8.76}\text{MeV}\text{,}{}^{64}\text{Ni}\text{(α, γ)}{}^{68}\text{Zn for}\text{4.50 ≦ E}{}_{\mathrm{\alpha }}{}^{\mathrm{<mtext>lab</mtext>}}\text{≦ 7.45}\text{MeV}$, and for ⁶⁴Ni(α, n)⁶⁷N from E_α^lab = 5.30 MeV up to E_α = 7.45 MeV. Substantial competition cusps were observed in the excitation function for all three (α, γ) reactions. The data were found to be in reasonable agreement with the predictions of the current versions of global Hauser-Feshbach models used in nucleosynthesis calculations. Including width fluctuation corrections and realistic neutron strength functions generally improves the ability of the models to predict the depth of the (α, γ) competition cusps; the depths of the predicted (α,γ) cusps are insensitive to the degree of isospin mixing. Taken together with studies of competition effects in proton-induced reactions, the present data confirm the importance of width fluctuation and strength function effects, and indicate essentially complete isospin mixing between T^< and T^> states in the compound nucleus.     

How to continue the predictions of perturbative QCD from the spacelike region where they are derived to the timelike regime where experiments are performed

The predictions of perturbative QCD are derived in the deep euclidean region, whereas the physical region for most observables is timelike. The confrontation of these predictions with experiment thus necessitates an analytic continuation. This we find introduces large higher order corrections in terms of α_s(|Q²|), the usual choice ofperturbative expansion parameter. These corrections are naturally absorbed by changing to the expansion parameter $\text{a}\text{(Q}{}^2\text{) = |α}{}_{\mathrm{<mtext>s</mtext>}}\text{(Q}{}^2\text{)|(}\text{Re}\text{α}{}_{\mathrm{<mtext>s</mtext>}}\text{(Q}{}^2\text{)/|α}{}_{\mathrm{<mtext>s</mtext>}}\text{(Q}{}^2\text{)|)}{}^{\mathrm{<mtext>(n?2)</mtext><mtext>3</mtext>}}$, where α_s(Q²)ⁿ is the leading term in the spacelike region. For the intermediate range of Q² experimentally accessible at present, where $\text{a}\text{(Q}{}^2\text{)}$ is significantly smaller than α_s(|Q²|), we find the resulting phenomenology is improved. In particular, we demonstrate how the values of $\text{Λ}{}_{\mathrm{<mtext>MS</mtext>}}$ obtained from analyses of quarkonium decays become consistent.     

Extending the QCD leading logs up to reggeon field theory

The deep inelastic structure function D(ω, q²) is calculated in the leading log approximation for $\text{(}\text{Nβ}{}_2\text{π}{}^2\text{)α}{}^2{}_{\mathrm{<mtext>S</mtext>}}\text{(q}{}_0{}^2\text{) 1}\text{n}\text{ω < 0.84 1}\text{n}\text{(}\text{1}\text{α}{}_{\mathrm{<mtext>S</mtext>}}\text{(q}{}^2\text{))}$. For larger ω up to ( the influence of reggeon cuts proves to slow down the growth of the structure function. A reggeon diagram technique is developed, and D is calculated up to a pre-exponent O(1), leading to D(ω, q²) ∝ q² for $\text{(}\text{Nβ}{}_2\text{π}{}^2\text{)α}{}^2{}_{\mathrm{<mtext>S</mtext>}}\text{(q}{}^2{}_0\text{) 1}\text{n}\text{ω ? 0.42}\text{α}{}^2{}_{\mathrm{<mtext>S</mtext>}}\text{(q}{}_0{}^2\text{)}\text{α}{}_{\mathrm{<mtext>S</mtext>}}{}^2\text{(q}{}^2\text{)}$. By assuming the reggeon diagrams when ω is still greater, one can expect to obtain a strong coupling behaviour: D(ω, q²) ∝ q²(ln ω)^η (η <2).     

The (6Li, 6He) spin-isospin-flip reaction

A formulation of the (⁶Li, ⁶He) spin-isospin-flip scattering problem is presented. The method is applicable to quasielastic collisions between⁶Li described in pureLS coupling scheme and any target nucleus which can be described in a shell model or an adiabatic deformed-nucleus model picture. The effective interaction responsible for the (⁶Li, ⁶He) reaction involves an integration of the (σ₁ · σ₂) (ρ₁ · ρ₂) compo nent of the central part and the (τ₁· τ₂) tensor part of the nucleon-nucleon interaction over the internal wave function of the projectile. The ⁴⁸Ca(⁶Li,⁶He)⁴⁸Sc reaction populating the ($\text{πf}{}_{\mathrm{<mtext>7</mtext><mtext>2</mtext>}}\text{νf}{}_{\mathrm{<mtext>7</mtext><mtext>2</mtext>}}{}^{\mathrm{?1}}$) multiplet in ⁴⁸Sc is analysed. It that the tensor force makes some contribution to the transitions to high spin states.     

Two-electron,one-photon transitions in nickel

The two emission lines, Kα₁α₃^h and Kα₂α₃^h resulting from the two-electron transitions $\text{1s}{}^{\mathrm{?2}}\text{→ 2s}{}^{\mathrm{?1}}\text{2p}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}{}^{\mathrm{?1}}$ and $\text{1s}{}^{\mathrm{?2}}\text{→ 2s}{}^{\mathrm{?1}}\text{2p}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}{}^{\mathrm{?1}}$ were resolved for elemental nickel. Their measured energies agree well with calculations. Their relative intensity $\text{I(Kα}{}_1\text{α}{}_3{}^{\mathrm{h}}\text{)/I(Kα}{}_2\text{α}{}_3{}^{\mathrm{h}}\text{) ≈}\text{3}\text{4}$ and their intensity relative to that of the Kα diagram lines is about 10^?4. This is some 10⁴ times larger than both theoretical results and the results of ion-atom collision experiments.     

A remarkable duality in one-particle quantum mechanics between some confining potentials and (R+L∞ϵ) potentials

The equivalence between the bound-state problem for the spatial harmonic oscillator and for the Coulomb potential is reexamined and extended: a similar duality is displayed between large classes of confining potentials and (R+L^∞_?) potentials, i.e., potentials for which the Schrödinger operator has among other properties a discrete spectrum bounded below, in (-∞, 0). The involved confining potentials U(r) must satisfy: $\text{r}{}^{\mathrm{-\alpha }}\text{U(r)}\textcolor{red}{}\text{g>0}{}_{\mathrm{r\to +\infty }}\text{, r}{}^{\mathrm{<mtext>-2\alpha </mtext><mtext>(2+\alpha )</mtext>}}\text{U (r}{}^{\mathrm{<mtext>2</mtext><mtext>(2+\alpha )</mtext>}}\text{) ? g}\text{is R}\text{+}\text{L}{}^{\mathrm{\infty }}{}_{\mathrm{?}}$.     

“Forbidden” rotational spectra of symmetric-top molecules: PH3 and PD3

A millimeter-wave spectrometer having a sensitivity of 4 × 10^?10 cm^?1 in the 2-mm region has been constructed for observation of extremely weak millimeter-wave spectra of gases. It has been used to measure J → J, K = 0 ← 3 transitions in PH₃ and J → J, K = 0 ← 3 as well as K = ±1 ← ±4 transitions in PD₃. The B₀ and C₀ spectral constants (in MHz) are: for PH₃, B₀ = 133 480.15 ± 0.12 and C₀ = 117 488.85 ± 0.16; for PD₃, B₀ = 69 471.10 ± 0.03 and C₀ = 58 974.37 ± 0.05. The effective ground-state values obtained for the bond angle and bond length are: for PH₃, $\text{r}{}_0\text{(}\text{A}\text{?}\text{) = 1.420}{}_0$ and $\text{α}{}_0\text{(}{}^{\mathrm{o}}\text{) = 93.34}{}_5$; for PD₃, $\text{r}{}_0\text{(}\text{A}\text{?}\text{) = 1.417}{}_6$ and $\text{α}{}_0\text{(}{}^{\mathrm{o}}\text{) = 93.35}{}_9$. The corresponding zero-point-average values were calculated to be: for PH₃, $\text{r}{}_{\mathrm{z}}\text{(}\text{A}\text{?}\text{) = 1.4269}{}_9\text{± 0.0002}$ and $\text{α}{}_{\mathrm{z}}\text{(}{}^{\mathrm{o}}\text{) = 93.228}{}_7$; for PD₃, $\text{r}{}_{\mathrm{z}}\text{(}\text{A}\text{?}\text{) = 1.4226}{}_5\text{± 0.0001}$ and $\text{α}{}_{\mathrm{z}}\text{(}{}^{\mathrm{o}}\text{) = 93.256}{}_7\text{± 0.004}$. For both species, the equilibrium values are $\text{r}{}_{\mathrm{e}}\text{(}\text{A}\text{?}\text{) = 1.4115}{}_9\text{± 0.0006}$ and $\text{α}{}_{\mathrm{e}}\text{(}{}^{\mathrm{o}}\text{) = 93.32}{}_8\text{± 0.02}$.     

On the super symmetry charges in the theory of scalar and spinor free fields

It is shown that for spinorial charges Q^(L))_α (α = 1, 2, L = 1, …, S) satisfying the commutation relations

$\text{{Q}{}^{\mathrm{(L)}}{}_{\mathrm{\alpha }}\text{, Q}{}^{\mathrm{(M)}}{}_{\mathrm{\beta }}\text{} = ε}{}_{\mathrm{\alpha }}\text{βa}{}^{\mathrm{LM}}\text{Q,}$

$\text{{Q}{}^{\mathrm{(L)}}{}_{\mathrm{\alpha }}\text{, Q}{}^{\mathrm{(M)+}}{}_{\mathrm{\beta }}\text{} = cσ}{}^{\mathrm{\mu }}{}_{\mathrm{\alpha \beta }}\text{P}{}_{\mathrm{\mu }}\text{δ}{}^{\mathrm{LM}}\text{,}$

$\text{[Q}{}^{\mathrm{(L)}}\text{)}{}_{\mathrm{\alpha }}\text{, P}{}^{\mathrm{\mu }}\text{] = 0,}$

where Q is a scalar charge commuting with the spinor charges as well aswith the energy- momentum vector Pμ, there can exist several different multiplets for free massive scalar and spinor fields.     

Search for parity-nonconservation effects in deep-inelastic μ-N interaction

The difference of the cross sections for deep inelastic scattering of muons with average momenta 21 GeV/c with right and left helicity at large angles, i.e., with large momentum transfer, has been measured. No statistically-significant dependence of cross sections on the longitudinal polarization of muons has been found, i.e. no parity-nonconservation effects in deep inelastic μN interaction have been observed. The magnitude of the cross-section asymmetry $\text{R =}\text{[〈σ}{}_{\mathrm{<mtext>R</mtext>}}\text{〉 ? 〈σ}{}_{\mathrm{<mtext>L</mtext>}}\text{〉]}\text{[〈σ}{}_{\mathrm{<mtext>R</mtext>}}\text{〉+ + 〈σ}{}_{\mathrm{<mtext>L</mtext>}}\text{〉]}$ may be represented as R = β〈Q²〉 = (? 4 ± 6) × 10^?3 〈Q², (GeV/c)²〉. The limitations G_o^(μ) = (+ 6 ± 10)G have been obtained for the constant G_o^(μ) of vector-axial interaction $\text{(}\text{G}{}_{\mathrm{<mtext>o</mtext>}}{}^{\mathrm{(\mu )}}\text{2}\text{) [}\text{μ}\text{γα(1 + γ}{}_5\text{)μ] J}{}_{\mathrm{\alpha }}{}^{\mathrm{<mtext>o</mtext>}}$ (hadron, V-A).     

Light composite fermions in a dynamical subquark model of pregauge interactions

The masses of composite leptons and quarks are discussed in a “dynamical subquark model of pregauge interactions”. In this model, the leptons and quarks are made of a spinor and scalar subquark with equal mass, M, and the gauge bosons and Higgs scalar of the SU(3)_c×SU(2)_L×U(1)_Y model are made of a subquark-antisubquark pair. The SU(2)_L×U(1)_Y symmetry is spontaneously broken by the composite Higgs scalar and the (scalar) subquark mass parameter is in turn bounded as $\text{M > 5.4}\text{TeV}\text{(=2π(}\text{2}\text{G}{}_{\mathrm{<mtext>F</mtext>}}{}^{\mathrm{?1}}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{where}\text{G}{}_{\mathrm{<mtext>F</mtext>}}$ is the Fermi coupling constant). The spontaneously generated mass of a lepton or quark, m_i⁽ⁿ⁾ (i = 1, 2; n = 1 ～ N_g), is calculated to be: $\text{m}{}_{\mathrm{i}}{}^{\mathrm{(n)}}\text{= r}{}_{\mathrm{i}}{}^{\mathrm{(n)}}\text{= r}{}_{\mathrm{i}}{}^{\mathrm{(n)}}\text{× (4+3N}{}_{\mathrm{<mtext>g</mtext>}}\text{)α}{}_{\mathrm{<mtext>e.m.</mtext>}}\text{(}\text{2}\text{G}{}_{\mathrm{<mtext>F</mtext>}}{}^{\mathrm{?1}}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{/3}\text{6}\text{(=0.35r}{}_{\mathrm{i}}{}^{\mathrm{(n)}}\text{(4+3N}{}_{\mathrm{<mtext>g</mtext>}}\text{)}\text{GeV}\text{),}\text{where}\text{r}{}_{\mathrm{i}}{}^{\mathrm{(n)}}$ are the parameters satisfying that 0 ? r_i⁽ⁿ⁾ ? 1 and Σ (r_i⁽ⁿ⁾)² = 1;N_g is the total number of generations of the leptons and quarks; α_e.m. is the fine structure constant. The appearance of light composite fermions is related to a specific mechanism of generating global chiral symmetries of the leptons and quarks. Global symmetries of scalar subquarks yield chiral symmetries of the leptons and quarks. Our model turns out to satisfy 't Hooft's anomaly conditions on massless composite fermions.     

Strong coupled-channels effects in the 9Be(α, t)10B reaction

Differential cross sections were measured for the reactions ${}^9\text{Be}\text{(α, α')}{}^9\text{Be}\text{,}{}^9\text{Be}\text{(α,}\text{t}\text{)}{}^{10}\text{B and}{}^9\text{Be}\text{(α,}{}^3\text{He}\text{)}{}^{10}\text{B at}\text{E}{}_{\mathrm{\alpha }}\text{= 65}\text{MeV}$ for angles ranging from θ_lab = 6° to 48°. Optical-model analysis was performed for elastic α-scattering from ⁹Be at E_α = 48, 65 and 104 MeV, and DWBA and CC calculations were done for the inelastic α-scattering at E_α = 65 MeV. DWBA calculations for the ⁹Be(α, ³He) reactions do not fit the transfer data so well and extracted spectroscopic factors are in disagreement with those of Cohen and Kurath and with values obtained from other reactions. Full CRC calculations assuming a band structure for the low-lying states of ¹⁰B and employing a modified set of Cohen and Kurath spectroscopic factors yield globally better fits both in shape and in absolute cross section for differential cross sections to low-lying states in ¹⁰B obtained in ⁹Be(α, t)¹⁰B at $\text{E}{}_{\mathrm{\alpha }}\text{= 65}\text{MeV and}{}^9\text{Be}\text{(}{}^3\text{He, d}\text{)}{}^{10}\text{B at}\text{E}{}_{\mathrm{<mtext>d</mtext>}}\text{= 17}\text{MeV}$. In general, strong coupled-channel effects mainly affecting the distorted waves are observed both in entrance and exit channels.     

The tensor polarization of 5Li (32−) in the reaction α + d → α + p + n

At the Hamburger Isochronous Cyclotron the formation of the particle unstable ground state of ${}^5\text{Li(}\text{3}\text{2}{}^{\mathrm{?}}$ has been investigated in the reaction α + d → α + p + n at E_α = 28.3 MeV. The neutrons were detected in two liquid scintillation counters, while a large Si (Li) detector was used to observe the α-particles. In a first analysis, the tensor polarization parameters of the five-nucleon system ⁵Li are determined from the experimental data. The results are in quite good agreement with theoretical predictions of Hackenbroich et al.     

Alpha capture to the giant dipole resonances of 42Ca, 44Ca and 52Cr

Differential cross sections for the ${}^{38}\text{Ar}\text{(α, γ}{}_0\text{)}{}^{42}\text{Ca}\text{,}{}^{40}\text{Ar}\text{(α, γ}{}_{\mathrm{0,\; 1}}\text{)}{}^{44}\text{Ca and}{}^{48}\text{Ti}\text{(α, γ}{}_{\mathrm{0,\; 1}}\text{)}{}^{52}\text{Cr}$ reactions were measured at 90° to the beam direction in 50 or 100 keV steps over the bombarding energy ranges 6.0–15.0 MeV, 5.5–11.1 MeV and 6.0–12.0 MeV respectively. Gamma-ray angular distributions were measured at forty bombarding energies. These show that the (α, γ₀) reaction proceeds through 1^? levels and to a lesser extent 2⁺ levels, whereas the (α, γ₁) reaction most probably proceeds through 1^? and 3^? levels. It is deduced that 〈Γ〉/〈D〉 ≦ 1 for the ⁴⁰Ar(α, γ)⁴⁴Ca. reaction whereas the fine structure observed in the ⁴⁸Ti(α, γ)⁵²Cr reaction is probably due to fluctuations. From a comparison with other data it is shown that the (α, γ) reaction is most probably statistical in nature. Using Hauser-Feshbach theory it is deduced that the ³⁶Ar(α, γ)⁴⁰Ca. reaction is inhibited by isospin selection rules and an estimate is made of isospin mixing in the ⁴⁰Ca giant dipole resonance. The ${}^{38}\text{Ar}\text{(α, γ)}{}^2{}^{42}\text{Ca and}{}^{40}\text{Ar}\text{(α, γ)}{}^{44}\text{Ca}$ data are considered with respect to theories of isosopin splitting of the giant dipole resonance.     

Generalized renormalization group equation for period-doubling bifurcations

Given a mapping $\text{x → ?(λ,x)}$, whose bifurcation points accumulate at λ = Λ_∞, it is shown that the iterates $\text{(?α)}{}^{\mathrm{p\times }}\text{[?}{}^{\mathrm{2p}}\text{(Λ}{}_{\mathrm{\infty }}\text{?μ/δ}{}^{\mathrm{p}}\text{, X}{}_{\mathrm{\infty }}\text{+y/(?α)}{}^{\mathrm{p}}\text{)?X}{}_{\mathrm{\infty }}\text{]}$ converge to a function ψ(μ,y)asp→∞, where X_∞ maximizes $\text{?(Λ}{}_{\mathrm{\infty ,x}}\text{)}$. The function ψ is universal up to scaling in μ and y, and satisfies ψ(μ/δ,ψ(μ/δ,?y/α)) = ?α^?1ψ(μ,y). This result generalizes the well-known Feigenbaum universal function g(y) with g(g(?y/α)) = ?g(y/α, which is the special case for μ = 0.