s- and p-wave neutron spectroscopy: Part VII. Widths of Neutron Resonances

Neutron reduced widths Γ_n⁰ and Γ_n¹ are reported for about 200 resonances observed in neutron total cross sections of Ca^{40, 44}, Ti⁴⁸, Cr^{50, 52, 54}, Fe^{54, 56}, Ni^{58, 60}, Sr⁸⁸, Y⁸⁹, Sn¹²⁴, Te¹³⁰, Ba^{136, 138}, and Pb^{206, 207, 208}, in the energy region 1 to 200 ^kev. Average parameters $\text{Γ}{}_{\mathrm{n}}{}^0$, $\text{Γ}{}_{\mathrm{n}}{}^0\text{D}$, and $\text{Γ}{}_{\mathrm{n}}{}^{\mathrm{(1)}}\text{D}$ have been derived and the Wigner distribution for local spacings and the Porter-Thomas distribution for reduced widths are verified for the resonances in the even-even nuclei Ca⁴⁰, Fe⁵⁶, Ni⁵⁸, and Ni⁶⁰. A simple method of area analysis which is less tedious and time consuming than the method reported before in Part III is also described.     

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.     

Observation of proton emission following thermal neutron capture in 34.5 d 84Rb

Using a target prepared by on-line isotope separation, thermal neutron capture in ⁸⁴Rb (I^π = 2^?) has been shown to induce proton emission to the ground state (0⁺) and first excited state (2⁺) of ⁸⁴Kr. The branching ratio was measured as $\text{Γ}{}_{\mathrm{<mtext>p</mtext>}}\text{(0}{}^{\mathrm{+}}\text{)}\text{Γ}{}_{\mathrm{<mtext>p</mtext>}}\text{(2}{}^{\mathrm{+}}\text{)}\text{= 4.7 ± 0.7}$, favouring a $\text{3}\text{2}{}^{\mathrm{?}}$ assignment of the capturing state without excluding $\text{5}\text{2}{}^{\mathrm{?}}$, and the (n_th, p) cross section as 12 ± 2 b. The energy available for the process was determined to be 3.45 ± 0.01 MeV, in agreement with other mass data in the region.     

An experimental investigation of the radiative structure decay K+ → e+υγ

The decay K⁺ → e⁺υγ has been investigated. For the structure-dependent part with positive γ-helicity (SD₊) the branching ratio $\text{Γ(}\text{SD}{}_{\mathrm{+}}\text{)}\text{Γ(}\text{K}{}_{\mathrm{\mu 2}}\text{) = (2.33 ± 0.42) × 10}{}^{\mathrm{?5}}$ is obtained from 51 ± 3 events observed in the kinematical region E_e ? 235 MeV, E_γ > 48 MeV and θ_eγ > 140°. For the corresponding part with negative γ-helicity we obtain an upper limit Γ(SD_?)/Γ(SD₊) < 11 (90% CL) from the sample of electrons with energies 220 MeV ? E_e < 230 MeV and with no γ in the backward direction. This upper limit implies that the ratio of structure-dependent axial vector amplitudes lies outside the region $\text{?1.8 < a}{}_{\mathrm{<mtext>K</mtext>}}\text{υ}{}_{\mathrm{<mtext>K</mtext>}}\text{< ?0.54}$.For the decay $\text{K}{}^{\mathrm{+}}\text{→}\text{e}{}^{\mathrm{+}}\text{νν}\text{ν}$ the limit $\text{Γ(}\text{K}{}^{\mathrm{+}}\text{→}\text{e}{}^{\mathrm{+}}\text{νν}\text{ν}\text{)/Γ(}\text{K}{}_{\mathrm{e2}}\text{) < 3.8}$ 90% confidence level) was found.     

Theory of the mobility of electrons limited by charged impurities in two dimensions

The mobility μ of electrons limited by charged impurities in two dimensions is calculated, based on the proper connected diagram expansion of Kubo's formula. For a Coulomb-type interaction, the Boltzmann relaxation rate is $\text{Γ}{}^{\mathrm{(0)}}\text{= 2}\text{π}{}^2\text{h}\text{?}{}^{\mathrm{?1}}\text{n}{}_{\mathrm{I}}\text{m}{}^1\text{z}{}^2\text{e}{}^4\text{κ}{}^{\mathrm{?2}}\text{p}{}^{\mathrm{?2}}$, where n_I, repr constant. This leads to a mobility proportional to temperature T for non-degenerate electrons. The quasi-particle effect, which is self-consistently contained in the proper connected diagram expansion, yields an equation for the relaxation rate Γ. The solution of this equation yields values in general reduced relative to Γ⁽⁰⁾ at higher densities. The values of μ obtained in our calculations, are higher than those predicted by the Boltzmann theory, and deviations are pronounced at low temperatures.     

Fission and decay of excited nuclei

The statistical theory of the decay of excited nuclei including the competition between particle emission and fission is considered. Fission barriers are given by a phenomenological formula which is an analog of the known Cameron formula and approximates well experimental data on nuclei up to transuranic elements. The contribution of shell corrections and odd-even effects is discussed. The calculated ratios of evaporation and fission widths Γ_n/Γ_f agree with experiment even in the region of nuclei with Z > 90. The $\text{Γ}{}_{\mathrm{<mtext>n</mtext>}}\text{Γ}{}_{\mathrm{<mtext>f</mtext>}}$ values depend weakly on excitation energy for nuclei with Z ≈ 90, While in other regions this dependence proves rather strong. The last conclusion does not contradict known experimental data since in experiments one usually measures some effective value $\text{〈}\text{Γ}{}_{\mathrm{<mtext>n</mtext>}}\text{Γ}{}_{\mathrm{<mtext>f</mtext>}}\text{〉}$, which is only indirectly related to the real $\text{Γ}{}_{\mathrm{<mtext>n</mtext>}}\text{Γ}{}_{\mathrm{<mtext>f</mtext>}}$, values. It is shown that the calculated values of $\text{〈}\text{Γ}{}_{\mathrm{<mtext>n</mtext>}}\text{Γ}{}_{\mathrm{<mtext>f</mtext>}}\text{〉}$ depend on excitation energy either. The effects of angular momentum and uncertainty in the choice of level density parameter are discussed.     

Dye laser site selection spectroscopy on U6+ luminescent molecular centers in LiF-U3O8 crystals

The polarized excitation and emission spectra of the U⁶⁺ molecular centers giving the most intense emission lines at 18 940 cm^?1 and 19 285 cm^?1 are presented and analyzed in detáil. From the polarization experiments, it is shown that the symmetry of the centers is C_4v and that the symmetries of the observed electronic states are Γ₂, Γ₁ and Γ₅. Finally, an electronic model is proposed which associates the emission and excitation lines of the U⁶⁺ centers in the visible region to transitions between the fundamental state $\text{Γ}{}_1\text{(6p}{}^6\text{,}{}^1\text{S)}$ and the states of the excited configuration $\text{6p}{}^5\text{(}{}^2\text{P}\text{3}\text{2}\text{) 7s}$.     

Two-body hadronic decays of the 3.1 GeV resonance

Hadronic two body decays of the 3.1 GeV resonance were measured and the following branching ratios were obtained: Γ_π⁺_π^?/Γ_μμ <0.0046, Γ_K⁺_K^?/Γ_μμ <0.0084, and $\text{Γ}{}_{\mathrm{p}}\text{p}\text{/ Γ}{}_{\mathrm{\mu }}{}_{\mathrm{\mu \mu }}\text{= 0.036 ± 0.010}$ assuming the p$\text{p}$ decay distribution is ≈ + cos²θ.     

Optical absorption by neutral bound excitons

In this note we determine the oscillator strengths for the dipole absorption of neutral bound excitons in direct gap semiconductors, using our previously obtained 35-term Page and Fraser type wave function, and taking into account the detailed electronic structure as well as the electron-hole exchange interaction.The envelope part of the oscillator strengths varies considerably with the electron-hole mass ratio $\text{σ =}\text{m}{}^1{}_{\mathrm{e}}\text{m}{}^1{}_{\mathrm{h}}$, and is maximum for the (D⁰, X)- complex when σ = 0.4. For typical σ-values (σ? 0.1–0.2), . But when σ approaches zero, the overlapping of the electron and the hole envelope wave functions of the (A⁰,X)-complex decreases progressively so that the oscillator strength also decreases and tends to zero.In the case of zinc-blende materials (T_d) and positive spin-orbit coupling at $\text{k}\text{= 0}$, we confirm that the line strength for transitions to or from $\text{J =}\text{1}\text{2}\text{(Γ}{}_6\text{)}$ or $\text{J =}\text{5}\text{2}\text{(Γ}{}_7\text{+ Γ}{}_8\text{)}$ level of the (A⁰, X)-complex is equal to one quarter of the line strength to or from the $\text{J =}\text{3}\text{2}\text{(Γ}{}_8\text{)}$ level.In the case of CdS, where our computed values are only in qualitative agreement with the experimental values, we discuss the use of the phenomenological result of Rashba.     

12C(7Li,t)16O and stellar helium fusion

The reaction ¹²C(⁷Li, t)¹⁶O has been studied at $\text{E(}{}^7\text{Li}\text{) = 34}\text{MeV}$ with the LASL tandem accelerator and QDDD magnetic spectrometer. Angular distributions to levels with E_x < 11 MeV have been obtained from 0° to 90°, including 0°. The results have been analyzed with finite-range distorted-wave Born approximation theory. The α-particle spectroscopic factors and reduced widths obtained are compared with those calculated with group theory (SU(3)) and other models. The analysis of data for the 7.1 and 9.6 MeV J^π = 1^? levels, which are of great importance in stellar helium buring, yields a ratio, R, of dimensionless reduced α-widths $\text{θ}{}^2{}_{\mathrm{<mtext>a</mtext>}}\text{(7.1}\text{MeV}\text{)}\text{θ}{}^2{}_{\mathrm{<mtext>a</mtext>}}\text{(9.6}\text{MeV}\text{)}\text{= 0.35b ± 0.13}$. The observed line width of the 9.6 MeV level (Γ_c.m. = 390 ± 60 keV) is less than the accepted value (Γ_c.m. = 510 ± 60 keV) and implies θ²_a(9.6 MeV) ≈ 0.6. These results as well as data for the 6.92 MeV J^π = 2⁺ and 10.35 MeV J^π = 4⁺ “α-cluster” states indicate 0.09 < θ²_a(7.1 MeV) < 0.33 with a mean value θ²_a(7.1 MeV) = 0.14 ± 0.04. The implication for stellar helium burning is discussed.     

An experimental study of the form factor in the decay K+ → πoe+ν

The q² variation of the factor $\text{?}{}_{\mathrm{+}}\text{(q}{}^2\text{)}$ in the decay K⁺→π⁰e⁺ν has been studied using a sample of even detected in the CERN 1.1 m³ heavy-liquid bubble chamber. The data are consistent with a linear development $\text{?}{}_{\mathrm{+}}\text{(q}{}^2\text{)=?}{}_{\mathrm{+}}\text{(0) (1+λ}{}_{\mathrm{+}}\text{q/m}{}^2{}_{\mathrm{\pi }}\text{)}$ with λ₊=0.027±0.008.     

3Σu−-3Σu+ coupling in the O2B3Σu− predissociation

The role of $\text{B}{}^3\text{Σ}{}_{\mathrm{u}}{}^{\mathrm{?}}\text{-2}{}^3\text{Σ}{}_{\mathrm{u}}{}^{\mathrm{+}}$ spin-orbit mixing in the O₂ Schumann-Runge predissociation is investigated. The $\text{2}{}^3\text{Σ}{}_{\mathrm{u}}{}^{\mathrm{+}}$ state is found to cross the $\text{B}{}^3\text{Σ}{}_{\mathrm{u}}{}^{\mathrm{?}}$ state near 2.0 Å with an interaction matrix element of approximately 55 cm^?1. This state contributes to the widths of the Bv ≥ 6 levels, but introduces only small level shift perturbations. When the partial widths due to the ${}^3\text{Σ}{}_{\mathrm{u}}{}^{\mathrm{?}}\text{-}{}^3\text{Σ}{}_{\mathrm{u}}{}^{\mathrm{+}}$ interaction are added to the previously calculated widths due to the ⁵Π_u, ³Π_u, and ¹Π_u states, reasonable agreement is obtained with experimental measurements on O¹⁶O¹⁶ and O¹⁸O¹⁸. The possibility of non-Lorentzian line profiles and the dependence of the width on rotational quantum number is investigated. The approximation of the spin-orbit matrix element by its value at the crossing point is shown to be a good approximation for calculating the second difference perturbations.     

Transverse spin correlation function of the one-dimensional spin-12 XY model

The transverse spin pair correlation function p^x_n=<S^x_mS^x_m+n>=<S^x_mS^x_m+n> is calculated exactly in the thermodynamic limit of the system described by the one-dimensional, isotropic, spin-$\text{1}\text{2}$, XY Hamiltonian

$\text{H=?2J}\text{∑}\text{l=1}\text{N}\text{(S}{}^{\mathrm{x}}{}_{\mathrm{l}}\text{S}{}^{\mathrm{x}}{}_{\mathrm{l+1}}\text{+S}{}^{\mathrm{y}}{}_{\mathrm{l}}\text{S}{}^{\mathrm{y}}{}_{\mathrm{l+1}}\text{)}$

. It is found that at absolute zero temperature (T = 0), the correlation function ρ^x_n for n ≥ 0 is given by

$\text{ρ}{}^{\mathrm{x}}{}_{\mathrm{2p}}\text{=}\text{1}\text{4}\text{2}\text{π}{}^{\mathrm{2p}}\text{Π}\text{j=1}\text{p?1}\text{4j}{}^2\text{4j}{}^2\text{?1}{}^{\mathrm{2p?2j}}\text{if}\text{n=2p}$

,

$\text{ρ}{}^{\mathrm{x}}{}_{\mathrm{2p+1}}\text{=±}\text{1}\text{4}\text{2}\text{π}{}^{\mathrm{2p+1}}\text{Π}\text{j=1}\text{p}\text{4j}{}^2\text{4j}{}^2\text{?1}{}^{\mathrm{2p+2j}}\text{if}\text{n=2p+1}$

, where the plus sign applies when J is positive and the minus sign applies when J is negative. From these the asymptotic behavior as n → ∞ of |?^x_n| at T = 0 is derived to be $\text{|ρ}{}^{\mathrm{x}}{}_{\mathrm{n}}\text{| ～}\text{a}\text{n}$ with a = 0.147088?. For finite temperatures, ρ^x_n is calculated numerically. By using the results for ?^x_n, the transverse inverse correlation length and the wavenumber dependent transverse spin pair correlation function are also calculated exactly.     

An inequality for the trace of matrix products

In this paper we consider a product of n complex m×m matrices A_k (k=1,…,n) with singular values ∝^(k)_i ordered in decreasing magnitude. Using the spectral resolution for the operators $\text{A}{}^{\mathrm{dagger;}}{}_{\mathrm{k}}\text{A}{}_{\mathrm{k}}$, it is shown that $\text{|}\text{Tr}\text{A}{}_1\text{…A}{}_{\mathrm{n}}\text{|≤}\text{∑}\text{i=1}\text{m}\text{Φ}\text{i=1}\text{n}\text{α}{}^{\mathrm{(k)}}{}_{\mathrm{i}}\text{.}$This inequality is an extension of an inequality of von Neumann in the simple case that n=2. The necessary and sufficient condition for the equality sign to hold is established. Application of Hölder's inequality leads to further inequalities which can be useful in statistical mechanics.     

The spectrum of HeNe+

Discharges through mixtures of helium and neon show two band groups near 4250 and 4100 Å as first observed by Druyvesteyn. These bands, assigned to the HeNe⁺ ion by Tanaka, Yoshino, and Freeman, have been studied under high resolution and have been fairly completely analyzed. The upper state of the transition is a very weakly bound state resulting from $\text{He}{}^{\mathrm{+}}\text{(}{}^2\text{S) +}\text{Ne}\text{(}{}^1\text{S}{}_0\text{)}$. There are two lower states resulting from the two components of $\text{Ne}{}^{\mathrm{+}}\text{(}{}^2\text{P) +}\text{He}\text{(}{}^1\text{S}{}_0\text{)}$. The upper of these two (${}^2\text{Π}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$) is also very weakly bound while the lower of the two, the ²Σ⁺ ground state, has a dissociation energy of 0.69 eV and an r_e value of 1.30 Å. All bands in both band groups show four branches designated R_ff, Q_ef, Q_fe, and P_ee. From their analysis the rotational constants in the various vibrational levels of the three electronic states have been determined. While no spin splitting in the B²Σ⁺ state has been found the ground state X²Σ shows a very large spin splitting and the $\text{A}{}_2{}^2\text{Π}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ state a very large $\text{Ω-type}$ doubling. The vibrational numberings in all these states were established by the study of the spectrum of ³HeNe⁺. At the same time the hyperfine structure observed in all lines of ³HeNe⁺ confirmed the nature of the upper state B²Σ⁺ as resulting from He⁺ + Ne, i.e., by charge exchange from the ground state. The ${}^2\text{Π}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ component of the ²Π state has not been observed, presumably because of low intensity.     

Laser excitation spectra of He2

Metastable $\text{a(2sσ)}{}^3\text{Σ}{}_{\mathrm{u}}{}^{\mathrm{+}}$ He₂ molecules are produced by a dc discharge in a flowing He stream. Laser excitation downstream of the discharge produces excitation spectra for a number of He₂ states. LIF spectra are observed for the $\text{(npπ)}{}^3\text{Σ}{}_{\mathrm{g}}{}^{\mathrm{+}}$ series for n = 4–9, excepting 5 and the $\text{(npπ)}{}^3\text{Π}{}_{\mathrm{g}}$ series for n = 5–15.     

The electronic states of nitroanilines

A correlation of spectroscopic data with all-valence-electron, $\text{CNDO}\text{s-CI}$ results has been performed for a number of mono- and disubstituted benzenes containing nitro and/or amino groups. The lowest energy transitions of p-nitroaniline are predicted to be 1¹B₁ ← 1¹A₁, 2¹A₁ ← 1¹A₁, and 2¹B₁ ← 1¹A₁, in the order of increasing energy. These three transitions are assigned to (i.e., are encompassed within) the lowest energy absorption band envelope of p-nitroaniline. The lowest energy predicted transition of o- and m-nitroanilines is 2A′ ← 1A′ and is supposed to correspond to the totality of the lowest energy absorption feature in both of these systems. The orbital excitation nature of these transitions is discussed.     

The spectrum of HeAr+

Two band groups near 1450 Å, first observed by Tanaka, Yoshino, and Freeman (J. Chem. Phys.62, 4484–4496 (1975)) in discharges through mixtures of helium and argon and assigned by them to the HeAr⁺ ion, were studied under high resolution. Like the similar spectrum of HeNe⁺ previously investigated, the spectrum of HeAr⁺ is a charge transfer spectrum. The upper state B²Σ⁺ of both band groups is derived from $\text{He}{}^{\mathrm{+}}\text{(}{}^2\text{S) +}\text{Ar}\text{(}{}^1\text{S)}$ while the two lower states $\text{A}{}_2{}^2\text{Π}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ and X²Σ⁺ are derived from $\text{He}\text{(}{}^1\text{S) +}\text{Ar}{}^{\mathrm{+}}\text{(}{}^2\text{P)}$. All three states are very weakly bound, the two lower states even more weakly than the upper state. Unlike HeNe⁺ most of the HeAr⁺ bands are violet shaded. In the longward band group each band shows only three branches while in the shortward group there are four. The former observation shows that the $\text{A}{}_2{}^2\text{Π}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ state behaves like a ²Σ^? state with γ_v ≈ 0. The B, D, γ, p, and ΔG values of all states were evaluated. While the B_v values of upper and lower states are nearly equal, the D_v values are quite different and this difference accounts for the violet shading of most of the bands even when B′_v is slightly smaller than B″_v; it also accounts for some of the extraheads and linelike features in the rotational structure. As in HeNe⁺ the ${}^2\text{Π}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}$ component of ²Π was not observed.     

Structure of positive parity states in 29P and two-step processes in (p,t) reactions

From a study of (p,t) reactions on ³¹P and ³⁰Si it is suggested that in ²⁹P the states with $\text{J}{}^{\mathrm{\pi }}\text{=}\text{1}\text{2}{}_1{}^{\mathrm{+}}$ and $\text{1}\text{2}{}_2{}^{\mathrm{+}}$, the pair $\text{3}\text{2}{}_2{}^{\mathrm{+}}$, $\text{5}\text{2}{}_1{}^{\mathrm{+}}$, and the pair $\text{7}\text{2}{}_3{}^{\mathrm{+}}$, $\text{9}\text{2}{}_1{}^{\mathrm{+}}$ are related by weak coupling of a s${}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ proton with the states 0₁⁺, 0₂⁺, 2₁⁺ and 4₁⁺ respectively of ²⁸Si. Completely atypical L = 2 angular distributions have been obtained for the $\text{3}\text{2}{}_1{}^{\mathrm{+}}$ and $\text{5}\text{2}{}_2{}^{\mathrm{+}}$ states in ²⁹P and it is suggested that this is due to contribution by two-step processes.