Strong absorption and the distribution of zeros of the S-matrix

The only distributions normally included in a discussion of the statistical theory of nuclear resonance reactions are the distributions of the widths (Γ) and spacings (D) of the levels of the compound nucleus. However, as the usual Hauser-Feshbach theory makes clear, $\text{Γ}$ and $\text{D}$ alone are not sufficient to determine the ratio $\text{σ}{}_{\mathrm{cc\prime }}{}^{\mathrm{<mtext>dir</mtext>}}\text{σ}{}_{\mathrm{cc\prime }}{}^{\mathrm{<mtext>fl</mtext>}}$. In an attempt to determine what further statistical information is sufficient to determine this ratio, in the special limit that it tends to zero for all cc′, c ≠ c′ (the “strong-absorption” limit), we study several “picket fence” S-matrix models, as well as a random-residue model exhibiting Ericson fluctuations. These models indicate that the strong-absorption limit $\text{(}\text{S}{}_{\mathrm{cc\prime }}\text{= 0,}\text{all}\text{cc′)}$ is directly related to the distribution of the zeros of S_cc′(E) in the upper half of the complex E-plane, and that strong absorption is reached only if these zeros are distributed with a high density in the region E → + i ∞. As a by-product, we obtain a generalization of the theorem of Moldauer and Simonius $\text{[|}\text{det}\text{S}\text{| =}\text{exp}\text{(?π}\text{Γ}\text{/}\text{D}\text{)]}$. Our generalization applies to individual optical S-matrix elements (and so to direct-reaction cross sections) rather than just to their determinant.     

Nonexponential decay law

The decay of a nonstationary state usually starts as a quadratic function of time and ends as an inverse power law (possibly with oscillations). Between these two extremes, the familiar exponential decay law may be approximately valid. The main purpose of this paper is to find the conditions which must be satisfied by the Hamiltonian and by the initial state, for the exponential law to have a significant domain of validity. It is shown that the evolution of a nonstationary state is governed by a nonnegative function W(E), having the dimensions of an energy. Among its properties are: the energy uncertainty is given by $\text{(ΔH)}{}^2\text{= ?W(E)dE}$, and the inverse lifetime by Γ = 2πW(E₀), where E₀ is the expectation value of H. The detailed shape of W(E) defines two characteristic times between which the exponential decay law is a good approximation: roughly speaking, the smoother W(E), the larger the domain of validity of the exponential law. For instance, if W(E) is very smooth ($\text{|}\text{dW}\text{dE}\text{| ? 1}$) except for a sharp threshold at E = E_thr, the transition from quadratic to exponential decay occurs for $\text{t ?}\text{1}\text{(E}{}_0\text{? E}{}_{\mathrm{<mtext>thr</mtext>}}\text{)}$, and the transition from exponential to inverse power law when $\text{Γt ?}\text{log}\text{[}\text{(E}{}_0\text{? E}{}_{\mathrm{<mtext>thr</mtext>}}\text{)}\text{Γ}\text{]}$.     

Three-nucleon results for separable potentials modelled on the Reid soft-core potential

Using a solution to the inverse scattering problem we have generated phase-equivalent separable potentials in the ${}^1\text{S}{}_0$ and ${}^3\text{S}{}_1\text{?}{}^3\text{D}{}_1$ states, which have nearly the same singlet UPA form factors and deuteron parameters (E_D, P_D, Q_D, A_S and $\text{A}{}_{\mathrm{D}}\text{A}{}_{\mathrm{S}}$) as the Reid soft-core potential. We compare our results for the binding energy of the triton and the neutron-deuteron doublet scattering length with the corresponding values for the Reid soft-core potential.     

Activation energy of field emission flicker noise power due to adsorbed potassium on tungsten

The temperature dependence of the field emission flicker noise spectral density functions has been investigated for potassium adsorbed on tungsten (112) planes by a probe hole technique. By integration of the spectral density functions $\text{W(?) = B}{}_{\mathrm{i}}\text{?}{}^{\mathrm{?ge}}\text{i}$ the noise power $\text{(δn}{}^2{}_{\mathrm{\Delta ?}}$ for different frequency intervals Δ? is obtained. From the exponential temperature dependence of $\text{(δn}{}^2{}_{\mathrm{\Delta ?}}$ noise power “activation energies” q_Δ? are determined. Plots of these energies versus coverage show a similar “oscillating” behaviour as recently found for $\text{W(?}{}_{\mathrm{j}}\text{)}$ or which indicates phase transitions of the adsorbed potassium submonolayers. The noise activation energies are discussed in terms of existing models and a comparison is made between the experimental q values and surface diffusion energies E_d as determined by conventional methods.     

Measurement of the background energy content of Nd-glass picosecond pulses with saturable absorbers

The background energy content of a mode-locked Nd-glass laser is determined with photodetectors and saturable absorbers. By comparing the signal height of a fast detector with the readout of an integrating energy meter the noise energy E_u2 outside the rise-time of the fast detector is measured. The background energy E_u1 within the rise-time is analysed by transmission measurements through two subsequent absorber cells. The obtained mean background to pulse energy (and intensity) ratios of ( and indicate a high degree of mode-locking.     

Consistent MFA correlation functions of Ising models for all temperatures

A correct calculation of the Ising model correlation function C(q) = 〈(S(q) ? 〈S(q)〉) (S(-q) ? 〈S(-q)〉)〉 in the MFA results in

$\text{C(q)=}\text{〈S}{}^2\text{〉?〈〉}{}^2\text{1?(〈S}{}^2\text{〉?〈S〉}{}^2\text{βJ(q}\text{1}\text{N}\text{∑}\text{q}\text{1}\text{1?〈S〉}{}^2\text{βJ(q)}{}^{\mathrm{?1}}\text{C(q) fulfills the exact sum rule N}{}^{-1}\text{Σ}{}_{\mathrm{q}}\text{C(q) = 〈S}{}^2\text{〉 ? 〈S〉}{}^2$

. Previous literature supposed a violation of this sum rule to be a characteristic disadvantage of this approximation.     

Hydrodynamical analysis of non-linear sputtering yields

A shock wave model is proposed to explain non-linear sputtering yields in heavy-ion bombardments on high-Z materials. A derived formula for the non-linear yields is approximately proportional to $\text{S}{}_{\mathrm{<mtext>n</mtext>}}\text{(E)}{}^{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}$, not to S_n(E), where S_n(E) is the nuclear stopping power. Experimental data show this tendency clearly and the formula reproduces well the energy dependence of experimental yields. The reaction process is divided into the two subprocesses of the development of collision cascade and the propagation of shock wave. The present theory includes one free parameter which is the energy density of cascading atoms in a local region near the target surface. This may be regarded as a function of the compressibility of the target material, but is independent of the projectile energy and mass. It is possible to obtain various quantities at the initial and final stages of the shock wave propagation, such as temperature, compressed density and pressure.     

Nature of the transition in nonlinear spin-S Ising models (half-integer spin)

We consider the spin-S Hamiltonian $\text{H}\text{=-JΣ}{}_{\mathrm{〈i,j〉}}\text{Q}{}_{\mathrm{S}}\text{(S}{}_{\mathrm{i}}\text{)Q}{}_{\mathrm{S}}\text{(S}{}_{\mathrm{j}}\text{)?λΣ}{}_{\mathrm{i}}\text{Q}{}_{\mathrm{S}}\text{(S}{}_{\mathrm{i}}\text{)}$ where Q_S(y) is a polynomial in y of degree 2S whose eigenvalues are ±1 and rigorously show that except for a multiplicative constant, the partition functions for all half-integer values of S are identical.     

Neutron scattering from 26Mg

Elastic and inelastic cross sections have been measured for 24 MeV neutrons incident on ²⁶Mg using the time-of-flight technique. These cross sections and existing proton data were analyzed in terms of a Lane-consistent optical potential. Distorted-wave and coupled-channel calculations are presented. Deformation parameters for the first 2⁺ state (1.81 MeV) and the second 2⁺ state (2.94 MeV) derived from both the (n, n') and (p, p') data are used to obtain the ratio of neutron and proton transition matrix elements M_n(E2) and M_p(E2) for these states. Comparison of results for $\text{M}{}_{\mathrm{<mtext>n</mtext>}}\text{(}\text{E}\text{2)}\text{M}{}_{\mathrm{<mtext>p</mtext>}}\text{(}\text{E}\text{2)}$ with those obtained from pion work, from electromagnetic (EM) rates and theoretical evaluations are presented.     

High accuracy energy-level calculations of the rotational-vibrational weakly bound states of ddμ and dtμ mesic molecules

Calculations are performed of the nonrelativistic energies $\text{E}{}_{\mathrm{JV}}$ of rotational-vibrational weakly bound states J = ν = 1 of ddμ and dtμ mesic molecules: $\text{E}{}_{11}\text{(}\text{dd}\text{μ) = ? 1.956}\text{eV}$ and $\text{E}{}_{11}\text{(}\text{dt}\text{μ) = ? 0.656}\text{eV}$ with an accuracy of 0.001 eV. With the relativistic effects and nuclear finite size corrections taken into account the result is: $\text{E}{}_{11}\text{(}\text{dd}\text{μ) = ? 1.946}\text{eV}$ and $\text{E}{}_{11}\text{(}\text{dt}\text{μ) = ? 0.634}\text{eV}$.     

The E-B band system of diatomic iodine

Sequential pump and probe pulses are used to excite state-selected E ← B ← X transitions in I₂ vapor, and the E ← B bands recorded by polarization-labeling spectroscopy at relatively high dispersion. Molecular constants and Dunham expansion coefficients Y_n,0 (n = 0–7), Y_n,1 (n = 0–3), and Y_n,2 (n = 0, 1) for the E state are obtained in the range 0 ≤ v_E ≤ 96, based on a global least-squares fit of 1050 assigned rotational transitions. Definite evidence is cited to show that the EO_g⁺ ion-pair state correlates diabatically with the ground state $\text{I}{}^{\mathrm{?}}\text{(}{}^1\text{S}{}_0\text{) + I}{}^{\mathrm{+}}\text{(}{}^3\text{P}{}_2\text{)}$ of the separated ions.     

τ → δν decay induced by light quark mass difference

The decay constant of scalar δ^±(980) induced by up and down quark mass difference is determined to be $\text{?}{}_{\mathrm{\delta }}{}^{\mathrm{<mtext>I</mtext>}}\text{= 2.1}\text{MeV}$ and hence BR(τ → δν) = 1.6 × 10^?5. A detection of a significantly larger branching ratio would imply non-conser vation of the vector current.     

A further contribution to the theory of the transient hot-wire technique for thermal conductivity measurements

We study the density of states of a one-dimensional tightbinding electron model with random hopping elements. The Hamiltonian is $\text{H = -∑}{}_{\mathrm{i}}\text{J}{}_{\mathrm{<mtext>i+</mtext><mtext>1</mtext><mtext>2</mtext>}}\text{(a}{}^{\mathrm{+}}{}_{\mathrm{i}}\text{a}{}_{\mathrm{i+1}}\text{+a}{}^{\mathrm{+}}{}_{\mathrm{i+1}}\text{a}{}_{\mathrm{i}}\text{)}$, where the $\text{J}{}_{\mathrm{<mtext>i+</mtext><mtext>1</mtext><mtext>2</mtext>}}$'s are independent identically distributed random variables. It is proved that the single particle density of states D(E) diverges near E = 0 as $\text{1}\text{|(E}\text{log}{}^3\text{|E|)|}$.     

Two theorems on the level order in potential models

We study the potentials of the form U(r)=?r^?1+λV(r), $\text{(}\text{d}\text{d}\text{r}\text{)(}\text{r}{}^2\text{d}\text{V}\text{d}\text{r}\text{)?0}$, and show that the energy levels satisfy the inequalities $\text{E(N}{}_{\mathrm{c}}\text{, l)?E(N}{}_{\mathrm{c}}\text{, l+1)}$ to first order in λ, where N_c denotes the coulombic principal quantum number and l the angular momentum. Similarly for potentials $\text{U(r)=r}{}^2\text{+λV(r), (}\text{d}\text{d}\text{r}{}^2\text{)}{}^2\text{V(r)?0}$, we prove to first order in λ that $\text{E}\text{?}\text{(N}{}_{\mathrm{H}}\text{,}\text{l}\text{)?}\text{E}\text{?}\text{(N}{}_{\mathrm{H}}\text{,}\text{l}\text{+2)}$, where NH denotes the harmonic oscillator quantum number. In the latter case, we give also quantitative restrictions on the relative positions at the lth and (l+1)th states.     

Luminescence quenching of F centers due to nonradiative de-excitation during lattice relaxation

F centers have no luminescence in some crystals. This occurs only in cases intermediate between the adiabatic and the nonadiabatic limits, when we adopt a reasonable damping rate of the configuration coordinate whose oscillation is triggered by photon excitation at an energy E. When $\text{(}\text{EE}{}_1\text{)}\text{E}{}_1\text{> 1 ? 4Λ}$, with E₁  absorption-peak energy, the luminescence quantum efficiency decreases steeply. Here Λ  S/E₁ represents Bartram and Stoneham's parameter with S  lattice-relaxation energy. Otherwise, the nonradiative de-excitation requires a thermal activation, but the activation energy is much smaller than that required at thermal equilibrium.     

Interference effects in 12C(p, γ)13N and direct capture to unbound states

Excitation functions of the capture reaction ¹²C(p, γ₀)¹³N have been obtained at θ_γ = 0° and 90° and E_p = 150–2500 keV. The results can be explained if a direct radiative capture process, E1(s and d → p), to the ground state in ¹³N is included in the analysis in addition to the two well-known resonances in this beam energy range $\text{[E}{}_{\mathrm{p}}\text{= 457(}\text{1}\text{2}{}^{\mathrm{+}}\text{)}$ and 1699 $\text{(}\text{3}\text{2}{}^{\mathrm{?}}\text{) keV]}$. The direct capture component is enhanced through interference effects with the two resonance amplitudes. From the observed direct capture cross section, a spectroscopic factor of C²S(l = 1) = 0.49 ± 0.15 has been deduced for the $\text{1}\text{2}{}^{\mathrm{?}}$ ground state in ¹³N. Excitation functions for the reaction ${}^{12}\text{C(p,γ}{}_1\text{p}{}^1\text{)}{}^{12}\text{C}$ have been obtained at θ_γ = 0° and 90° and E_p = 610–2700 keV. Away from the 1699 keV resonance the capture γ-ray yield is dominated by the direct capture process E1 (p → s) to the $\text{2366 (}\text{1}\text{2}{}^{\mathrm{+}}\text{) keV}$ unbound state. Above E_p = 1 MeV, the observed excitation functions are well reproduced by the direct capture theory to unbound states (bremsstrahlung theory). Below E_p = 1 MeV, i.e., E_p → 457 keV, the theory diverges in contrast to observation. This discrepancy is well known in bremsstrahlung theory as the “infrared problem”. From the observed direct capture cross sections at $\text{E}{}_{\mathrm{p}}\text{? 1 MeV}$, a spectroscopic factor of C²S(l = 0) = 1.02 ± 0.15 has been found for the $\text{2366 (}\text{1}\text{2}{}^{\mathrm{+}}\text{) keV}$ unbound state. A search for direct capture transitions to the $\text{3512 (}\text{3}\text{2}{}^{\mathrm{?}}\text{)}$and $\text{3547 (}\text{5}\text{2}{}^{\mathrm{+}}\text{) keV}$ unbound states resulted in upper limits of C²S(l = 1) ≦ 0.5 and C²S(l = 2) ? 1.0, respectively. The results are compared with available stripping data as well as shell-model calculations. The astrophysical aspect of the ¹²C(p, γ₀)¹³N reaction also is discussed.     

Temperature dependence of ferromagnetic anisotropy energy in cubic crystals

The magnetic anisotropy constant K₁ for cubic ferromagnetic crystals has been discussed based on the general expressions derived by Yang for hexagon crystals. By matching the experimental data, we obtained

$\text{K}{}_1\text{(T,H)}\text{K}{}_1\text{(0,0)}\text{=?6.14}\text{I}\text{?}{}_{\mathrm{<mtext>5</mtext><mtext>2</mtext>}}\text{(T,H) + 3.36}\text{I}\text{?}\text{9}\text{2}\text{(T,H) + 4.88m}{}^2\text{(T,H) ?1.10[}\text{I}\text{?}\text{5}\text{2}\text{(T,H)]}{}^2$

for nickel, and $\text{K}{}_1\text{(T,H)/K}{}_1\text{(0,0)=I}{}_{\mathrm{<mtext>9</mtext><mtext>2</mtext>}}$ for iron, where Î_{$\text{5}\text{2}$} and Î_{$\text{9}\text{2}$} are the hyperbolic bessel functions of order 2 and 4 respectively and m is the reduced magnetization. Both expressions have a theoretical basis.     

Paзлoжeниe пo 1Z для чapтpи-фoкoвcкoй и кoppeляциoннoй энepгии, peлятивиcтcкич пoпpaвoк, дипoльнoгo мaтpичнoгo элeмeнтa

Energies and dipole matrix elements have been calculated for He, Li, Be, B, C, N, O, F, and Ne-like ions (configurations 1s²2sⁿ¹2pⁿ²?1s²2s^n1?12pⁿ²⁺¹). The Hartree-Fock energy, the correlation energy, and relativistic corrections were taken into account. Relativistic corrections were obtained by computing the entire quantity H_B. Numerical results are presented for energies of the terms in the form

$\text{E=E}{}_0\text{Z}{}^2\text{+ ΔE}{}_1\text{Z + ΔE}{}_2\text{+}\text{1}\text{Z}\text{ΔE}{}_3\text{+}\text{α}{}^2\text{4}\text{(E}{}_0{}^{\mathrm{p}}\text{Z}{}^4\text{+ ΔE}{}_1{}^{\mathrm{p}}\text{Z}{}^3\text{)}$

, and for the fine structure of the terms in the form

$\text{〈1s}{}^2\text{2s}{}^{\mathrm{n1}}\text{2p}{}^{\mathrm{n2}}\text{LSJ|H}{}_{\mathrm{Б}}\text{|1s}{}^2\text{2s}{}^{\mathrm{n1\prime }}\text{2p}{}^{\mathrm{n2\prime }}\text{L′S′J〉=(?1)}{}^{\mathrm{L+S\prime +J}}\text{LSJ}\text{S′L′1}\text{×}\text{α}{}^2\text{4}\text{(Z?A)}{}^3\text{[E}{}^{\mathrm{(0)}}\text{(Z ? B)+}\text{E}{}_{\mathrm{c0}}\text{]+(?1)}{}^{\mathrm{L\; +\; S\prime \; +\; J}}\text{LSJ}\text{S′L′2}\text{α}{}^2\text{4}\text{(Z?A)}{}^3\text{E}{}_{\mathrm{cc}}$

. Dipole matrix elements are required for calculation of oscillator strengths or transition probabilities. For the dipole matrix elements, two terms of the expansion in $\text{1}\text{Z}$ have been obtained. Numerical results are presented in the form P(a, a′) = (a/Z)[1 + (τ/Z)].     

Nuclear orientation study of the excited states of 129I populated in the decay of 129gTe and 129mTe

From the angular distributions of γ-rays emitted by oriented ^129gTe and ^129mTe nuclei implanted in iron by isotope separator, unique spin assignments could be made for the excited states of ¹²⁹I at 487.4 keV $\text{(}\text{5}\text{2}{}^{\mathrm{+}}\text{)}$, 696.0 keV $\text{(}\text{11}\text{2}{}^{\mathrm{+}}\text{)}$, 729.6 keV $\text{(}\text{9}\text{2}{}^{\mathrm{+}}\text{)}$, 768.9 keV $\text{(}\text{7}\text{2}{}^{\mathrm{+}}\text{)}$, 1050.4 keV $\text{(}\text{7}\text{2}{}^{\mathrm{+}}\text{)}$ and 1111.8 keV $\text{(}\text{5}\text{2}{}^{\mathrm{+}}\text{)}$. In addition, E2/M1 amplitude ratios for the following ¹²⁹I γ-rays (energies are in keV) are derived: δ(459.6) = ?(0.076^+0.037_?0.148); δ(487.4) = 0.50^+0.17_?0.10 or δ^? = 0.35^+0.15_?0.09; δ(556.7) = 0.06±0.02 or δ^? = ?(0.10±0.02); δ(624.4) = 0.10±0.26 or δ^? > 0.4; the 696.0 keV γ-ray is pure E2; δ(729.6) = ?(0.34±0.06) or δ^?1 = 0.55±0.05; δ(741.1) = ?(0.27±0.10) or δ^?1 = ?(0.43±0.12); δ(817.2) = 0.46±0.04 or δ^?1 =0.20±0.03 if $\text{I}{}^{\mathrm{\pi }}\text{(845}\text{keV}\text{) =}\text{7}\text{2}{}^{\mathrm{+}}$; δ(1022.6) = ?(0.02 ±0.02) or δ^?1 = ?(0.23±0.02); $\text{δ(1084) = 0.56}{}^{\mathrm{+0.04}}{}_{\mathrm{?0.14}}$; δ(1111.8) = 0.06±0.05 or δ^?1 = ?(0.08±0.05). The anisotropy of the 531.8 keV γ-ray excludes $\text{1}\text{2}{}^{\mathrm{+}}$ as a possible spin assignment for the 559.6 keV level, so that no $\text{1}\text{2}{}^{\mathrm{+}}$ level is fed in the decay from ¹²⁹Te. Anisotropies for the 209, 250.7, 278.4 and 281.1 keV γ-rays are also measured. Comparison of the level scheme is made with theoretical predictions from both the pairing-plus-quadrupole model and the intermediate coupling unified model.     

Observation of interference between Iu=12 and Iu=32 baryon exchange amplitudes in 4 GeV/c backward inelastic scattering

Interference between the $\text{I}{}_{\mathrm{<mtext>u</mtext>}}\text{=}\text{1}\text{2}$ and $\text{I}{}_{\mathrm{<mtext>u</mtext>}}\text{=}\text{3}\text{2}$ baryon exchange amplitudes is observed in the reaction π^?p → pπ^?π⁰, with the proton produced forward with $\text{cos}\text{θ}{}_{\mathrm{p}}{}^1\text{>0.8}$. The Dalitz plot shows that the reaction is dominated by the quasi two body final states ρ^?p(δ exchange) and $\text{N}{}^{10}\text{(1670)π}{}^0\text{(}\text{N exchange}\text{)}$, with δ(1238), N¹(1520) and higher mass N¹'s also produced. The relative phase between the ρ and the N¹(1670) production amplitudes is measured to be 135° ± 10° and is compared with the Regge pole signature factor phase predictions.