Experimental and shell-model studies of the reactions 91Zr(d,p)92Zr and 90Zr(t,p)92Zr

The results of high-resolution studies of the ⁹¹Zr(d, p) reaction at E_d = 12 MeV and the ⁹⁰Zr(t, p) reaction at E_t = 11.85 MeV are presented. Absolute cross sections have been measured for both reactions and (d, p) spectroscopic factors determined. A comparison of these results with earlier data has been made, and although many of the previous assignments have been confirmed, many new features concerning the structure of ⁹²Zr have been discovered. Shell-model calculations have been performed for ⁹¹Zr and ⁹²Zr using a neutron space which includes the $\text{2}\text{d}{}_{\mathrm{<mtext>5</mtext><mtext>2</mtext>}}$, $\text{3}\text{s}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$, $\text{2}\text{d}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}$, $\text{1}\text{g}{}_{\mathrm{<mtext>7</mtext><mtext>2</mtext>}}$ and $\text{1}\text{h}{}_{\mathrm{<mtext>11</mtext><mtext>2</mtext>}}$ orbits and a proton space comprising the $\text{1}\text{g}{}_{\mathrm{<mtext>9</mtext><mtext>2</mtext>}}$ and $\text{2}\text{p}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ orbits. Realistic proton-neutron and neutron-neutron interactions based on the Sussex matrix elements were used in the calculations. Spectroscopic factors have been calculated for the ⁹⁰Zr(d, p) and ⁹¹Zr(d, p) reactions and cross sections calculated for the ⁹⁰Zr(t, p) reaction. In general, good agreement between the theoretical and the experimental results has been obtained.     

On the theory of localized spin and phonon excitations in amorphous ferromagnets

Within the framework of a perturbation theory and a quasicrystalline approximation we have solved the linearized equation of motion for the circular spin component S⁺_j = S^x_j + iS^y_j in a one-dimensional amorphous ferromagnet with periodic external excitation of the spin S⁺₀ at site j = 0. It is shown that localized spin modes of the simple form $\text{«S}{}^{\mathrm{+}}{}_{\mathrm{j}}\text{a}\text{?}\text{= S}{}^{\mathrm{+}}\text{(q0) exp[iq}{}_0\text{· R}{}_{\mathrm{j}}\text{- iω(q0) t] exp (-gk?R}{}_{\mathrm{j}}\text{?)}$ with fall-off-length κ^-1 are solutions of the ensemble-averaged equation of motion. On the other hand, we have a damping of extended spin waves according to exp(-Γt). A simple relation is derived between the fall-off-length κ^-1 of localized spin modes and the damping factor Γ of extended spin waves. Analogous results hold for phonons in amorphous materials.     

Exact approach to equilibrium for the glauber model of a one-dimensional Ising system with random ±J bonds

The (discrete-time) Glauber model is considered for a one-dimensional system of spins s_j = ±1 with nearest-neighbor Ising interaction $\text{H}\text{= -Σ}{}_{\mathrm{j}}\text{J}{}_{\mathrm{j}}\text{s}{}_{\mathrm{j}}\text{s}{}_{\mathrm{j+1}}$. The J_j = ±J are treated as random variables with an arbitrary joint probability p(J). The exact time-dependent average 〈s_j〉_t is determined, and from it the “quenched” average $\text{〈〈s}{}_{\mathrm{j}}\text{〉}{}_{\mathrm{t}}\text{〉}{}_{\mathrm{<mtext>av</mtext>}}\text{=Σ}{}_{\mathrm{<mtext>J</mtext>}}\text{p}\text{(}\text{J}\text{)〈s}{}_{\mathrm{j}}\text{〉}{}_{\mathrm{t}}$ is explicitly found.     

Study of the 92,96Zr(d, α)90,94Y reactions

The hole-hole structure of ⁹⁴Y was studied via the reaction ⁹⁶Zr(d, α)⁹⁴Y and compared to the particle-hole structure of ⁹⁰Y, which was populated by the reaction ⁹²Zr(d, α)⁹⁰Y. The deuteron beam energy was 28 MeV. Angular distributions of both reactions were obtained for the prominent lines. New states of ⁹⁴Y were observed at 0.44, 1.17, 1.39, 1.53, 1.82, 1.90, 2.17, 2.33, 2.46 and 2.77 MeV. Our data are consistent with the previously reported 2^? assignment of the ground state, and we suggest J^π = 3^? for the 0.44 MeV state, these being members of the $\text{(π2}\text{p}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{, ν2}\text{d}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}{}^{\mathrm{?1}}\text{)}$ doublet. The 1.17 state is suggested to be a member of the $\text{(πp}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}{}^{\mathrm{?1}}\text{, ν}\text{d}{}_{\mathrm{<mtext>5</mtext><mtext>2</mtext>}}{}^{\mathrm{?1}}\text{)}$ multiplet. The Q-value of the ⁹⁶Zr(d, α)⁹⁴Y reaction was measured to be 7.609 ± 0.020 MeV. The reaction ⁹⁴Zr(d, α) was performed at two angles. Several new states of ⁹²Y were observed at 0.31, 0.78, 1.03, 1.31, 1.49, 1.69 and 1.89 MeV.     

Lifetime and g-factor results for the 132− 1985 keV level in 91Nb and the 152− 2288 keV level in 91Zr

Lifetime and g-factor measurements have been made with pulsed beam-γ time-differential techniques using the ⁸⁹Y(α, 2n)⁹¹Nb and ⁸⁸Sr(α, n)⁹¹Zr reactions. A mean lifetime τ = 14.4 ± 0.5 nsec and a g-factor of 1.26 ± 0.04 were obtained for the $\text{13}\text{2}{}^{\mathrm{?}}$ 1985 keV level in ⁹¹Nb and τ = 41.9 ± 1.2 nsec and g = 0.70 ± 0.01 were obtained for the $\text{15}\text{2}{}^{\mathrm{?}}$ 2288 keV level in ⁹¹Zr. These results are compared to theoretical calculations for $\text{(π}\text{g}{}_{\mathrm{<mtext>9</mtext><mtext>2</mtext>}}\text{)}{}^2\text{(π}\text{p}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}$ and $\text{(π}\text{g}{}_{\mathrm{<mtext>9</mtext><mtext>2</mtext>}}\text{)(π}\text{p}\text{1}\text{2}\text{)(v}\text{d}{}_{\mathrm{<mtext>5</mtext><mtext>2</mtext>}}\text{)}$ configurations in ⁹¹Nb and ⁹¹Zr, respectively.     

Influence of self-fields on the Vlasov equilibrium and stability properties of relativistic electron beam-plasma systems

The influence of self-fields on the equilibrium and stability properties of relativistic beam-plasma systems is studied within the framework of the Vlasov-Maxwell equations. The analysis is carried out in linear geometry, where the relativistic electron beam propagates through a background plasma (assumed nonrelativistic) along a uniform guide field $\text{B}{}_0\text{e}\text{?}{}_{\mathrm{z}}$, It is assumed that $\text{ν}\text{γ}{}_0\text{? 1}$ for the beam electrons (ν is Budker's parameter, and γ₀mc² is the electron energy), but no a priori assumption is made that the beam density is small (or large) in comparison with the plasma density, or that conditions of charge neutrality or current neutrality prevail in equilibrium. It is shown that the equilibrium self-electric and self-magnetic fields, $\text{E}{}_{\mathrm{r}}{}^{\mathrm{s}}\text{(r)}\text{e}\text{?}{}_{\mathrm{r}}$ and $\text{B}{}_{\mathrm{\theta }}{}^{\mathrm{s}}\text{(r)}\text{e}\text{?}{}_{\mathrm{\theta }}$, can have a large effect on equilibrium and stability behavior. Equilibrium properties are calculated for beam (j = b) and plasma (j = e, i) distribution functions of the form f_b⁰(H, P_θ, P_z) = F(H ? ω_rbP_θ) × δ(P_z ? P₀)(j = b), and $\text{f}{}_{\mathrm{j}}{}^0\text{(H, P}{}_{\mathrm{\theta }}\text{, P}{}_{\mathrm{z}}\text{) = f}{}_{\mathrm{j}}{}^0\text{(H ? ω}{}_{\mathrm{rj}}\text{P}{}_{\mathrm{\theta }}\text{? V}{}_{\mathrm{j}}\text{P}{}_{\mathrm{z}}\text{?}\text{m}{}_{\mathrm{i}}\text{V}{}_{\mathrm{j}}{}^2\text{2}\text{)}$ (j = e, i), where H is the energy, P_θ is the canonical angular momentum, P_z is the axial canonical momentum, and ω_rj (the angular velocity of mean rotation for j = b, e, i), V_j (the mean axial velocity for j = e, i), and P₀ are constants. The linearized Vlasov-Maxwell equations are then used to investigate stability properties in circumstances where the equilibrium densities of the various components (j = b, e, i) are approximately constant. The corresponding electrostatic dispersion relation and ordinary-mode electromagnetic dispersion relation are derived (including self-field effects) for body-wave perturbations localized to the beam interior (r <R_b). These dispersion relations are analyzed in the limit of a cold beam and cold plasma background, to illustrate the basic effect that lack of charge neutrality and/or current neutrality can have on the two-stream and filamentation instabilities. It is shown that relative rotation (induced by self-fields) between the various components (j = b, e, i) can (a) result in modified two-stream instability for propagation nearly perpendicular to $\text{B}{}_0\text{e}\text{?}{}_{\mathrm{z}}$, and (b) significantly extend the band of unstable k_z-values for axial two-stream instability. Moreover, in circumstances where the beam-plasma system is charge-neutralized but not current-neutralized, it is shown that the azimuthal self-magnetic field $\text{B}{}_{\mathrm{\theta }}{}^{\mathrm{s}}\text{(r)}\text{e}\text{?}{}_{\mathrm{\theta }}$ has a stabilizing influence on the filamentation instability for ordinary-mode propagation perpendicular to $\text{B}{}_0\text{e}\text{?}{}_{\mathrm{z}}$.     

The Wilson-Andrei number for the SU(3) Kondo model

We give asymptotic forms for the high- and low-field magnetic susceptibility for the SU(3) linear dispersion Kondo model for T = 0. The ratio $\text{T}{}_{\mathrm{<mtext>K</mtext>}}\text{T}{}_{\mathrm{<mtext>H</mtext>}}$ is also calculated for the standard SU(2j + 1) Kondo model for general j. From these results the Wilson number W_j, defined by $\text{χ0 =}\text{Wj(gμ)}{}^2\text{j(j + 1)}\text{3kT}{}_{\mathrm{<mtext>K</mtext>}}$ where χ0 is the zero-temperature zero-field susceptibility, which has been calculated by Andrei and Lowenstein for $\text{j =}\text{1}\text{2}$, is deduced for the SU(3) model j = 1.     

A parabolic rule for the energy dependence on x = I(I + 1) for proton-neutron multiplets in odd-odd nuclei

An analytic expression for the energies of the states of the proton-neutron multiplet ($\text{j}{}_{\mathrm{<mtext>p</mtext>}}\text{j}{}_{\mathrm{<mtext>n</mtext>}}\text{)I = |j}{}_{\mathrm{<mtext>p</mtext>}}\text{-j}{}_{\mathrm{<mtext>n</mtext>}}\text{\&,…,j}{}_{\mathrm{<mtext>p</mtext>}}\text{+ j}{}_{\mathrm{<mtext>n</mtext>}}$ is derived as a quadratic function x = I(I + 1) owing to the exchange of the quadrupole and the spin-vibrational phonon. A discussion of this parabola, its consequences and illustrative applications are presented. The parabolic rule is suggested to serve as a guideline on the classification and investigation of states in odd-odd nuclei.     

General correction theorems on decomposition formulae of exponential operators and extrapolation methods for quantum Monte Carlo simulations

It is proved that the trace of the generalized Trotter formula $\text{Z(n) ≡ Tr[II exp(}\text{A}{}_{\mathrm{j}}\text{n]}{}^{\mathrm{n}}$ is an even function of n, when all A_j are symmetric, namely A^t_j = A_j, togethern with some generalizations. This yields a new extrapolatio n method of the form $\text{Z(n) ? Z(∞) + a/(n}{}^2\text{+ b)}$ for large n in quanntum Monte Carlo simulations.     

KL0-KS0 transmission regeneration on deuterons and neutrons in a momentum range of 10–50 GeV/c

The energy dependence of the K_L⁰-K_S⁰ transmission regeneration amplitudes on deuterons and neutrons in the momentum region 10–50 GeV/c is determined. The moduli of the modified transmission amplitudes are momentum dependent. These dependences are fitted by the expression A_jp^?nj, where A_j and n_j (j = d, n) are constants:

$\text{A}{}_{\mathrm{<mtext>d</mtext>}}\text{=2.88 ±0.04}\text{mb}\text{, n}{}_{\mathrm{<mtext>d</mtext>}}\text{=0.546±0.030,}\text{for deuterons}\text{,}$

$\text{A}{}_{\mathrm{<mtext>n</mtext>}}\text{=1.97 ±0.14}\text{mb}\text{, n}{}_{\mathrm{<mtext>n</mtext>}}\text{=0.530±0.019,}\text{for neutrons}\text{,}$

The amplitude phases do not depend on the kaon momentum and are equal to $\text{?}{}_{\mathrm{<mtext>d</mtext>}}\text{= (?130.9 ± 2.7)°}$?_n = (?132.3 ± 1.7)°. The mean value of the ratio of the total cross-section differences for K⁰ and K⁰ interactions with neutrons and protons is determined. The residues of the partial ω and ? amplitudes, which contribute to the kaon-nucleon interaction amplitudes, are also obtained.     

Inverse solutions to the inelastic scattering problem

Exact inverse solutions to the integral equation $\text{φ(r}{}_{\mathrm{s}}\text{|r}{}_0\text{, k) = ?}{}_{\mathrm{D}}{}^3\text{f (r, ω)g(r|r}{}_0\text{, k)g(r|r, k)d}{}^3\text{r}$, where g(r|r_j, k); j = 0 or s is the free space Green function, are derived in plane and cylindrical coordinates for fixed ω. These solutions allow an inelastic scattering potential f(r, ω) which is of compact support r ? $\text{D}$³ to be recovered from scattering data collected over the surfaces of a plane and cylinder respectively.     

Study of the three-proton three-neutron-hole nucleus 208At

Levels in ²⁰⁸At were populated in the ²⁰⁹Bi(α, 5n) reaction, and the subsequent radiation was studied using γ-spectroscopic methods including γ-ray excitation function and angular distribution, γγ(t) coincidence and γt measurements, as well as measurements of conversion electrons. The excited spectrum of ²⁰⁸At is found to consist of two almost disconnected parts which are proposed to originate from seniority-three proton and neutron cascades. Two isometric states are observed. A $\text{T}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{= 45 ± 2}\text{ns}$ state at 1090 keV is proposed to have the main configuration and J^π = 10^?. A high-spin isomer with $\text{T}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{= 1.5 ± 0.2 μs at 2276}\text{keV}$ is assigned to be the state. Shell-model arguments are used to assign configurations to most of the observed levels. Transition rates are discussed.     

Spectroscopy of 62Ni with the 61Ni(d, p) reaction

Differential cross sections, vector and tensor analysing powers have been measured for the ⁶¹Ni(d, p) reaction at a deuteron energy of 12.3 MeV. Most of the 30 transitions observed below 8.5 MeV excitation are dominated by a single j-value, which was determined from behaviour of the analysing power data. For a number of transitions it was possible to make unambiguous j-assignment relying on the established j-dependence of the T₂₂ tensor analysing power. The deduced spectroscopic factors indicate that the full strength of neutron transfer to the ($\text{2p, 1f}{}_{\mathrm{<mtext>5</mtext><mtext>2</mtext>}}$) and $\text{1}\text{g}{}_{\mathrm{<mtext>9</mtext><mtext>2</mtext>}}$ orbits was found and seven $\text{5}\text{2}{}^{\mathrm{+}}$ transitions were located above 5.3 MeV. The separated strengths of the $\text{3}\text{2}{}^{\mathrm{?}}\text{,}\text{1}\text{2}{}^{\mathrm{?}}\text{and}\text{5}\text{2}{}^{\mathrm{?}}$ transitions are compared with shell-model calculations for the low-lying states of ⁶²Ni.     

Lorentz contracted geometrical model

The model assumes that when two high energy particles collide each behaves as a geometrical object which has a Gaussian density and is spherically symmetric except for the Lorentz-contraction in the incident direction. Folding the two spatial distribution together we obtain the slope (b) of the elastic diffraction peak in terms of the c.m. velocities (β_i and β_j) and the sizes (A_i and A_j) of the two incident particles. These sizes are assumed to have the experimental s-dependence of σ_tot ∝ πA² for each reaction. The combined s-dependence of the σ_tot's and the β's gives the s-dependence of the elastic slope $\text{b}{}_{\mathrm{ij}}\text{(s) =}\text{1}\text{2}\text{(A}{}_{\mathrm{i}}{}^2\text{β}{}_{\mathrm{i}}{}^2\text{+ A}{}_{\mathrm{j}}{}^2\text{β}{}_{\mathrm{j}}{}^2\text{)}\text{σ}{}^{\mathrm{ij}}{}_{\mathrm{<mtext>tot</mtext>}}\text{(s)}\text{σ}{}^{\mathrm{ij}}{}_{\mathrm{<mtext>tot</mtext>}}\text{(∞)}$. This formula agrees with the experimental slope for p-p, $\text{p}$-p, K⁺-p, K^?-p and π^±-p elastic scattering from 3 to 1500 GeV/c, with only 3 parameters: A_π² = 6.1, A_K² = 3.3 and A_p² = 10.5 (GeV/c)^?2.     

Oscillator strengths of Cl(I) in the vacuum ultraviolet: The 2D-2P transitions

Experimental oscillator strengths are reported in absorption for the ²D_j-²P_j transitions of atomic chlorine using chemical titration for the quantitative production of Cl atoms. The closely spaced ²$\text{D}{}_{\mathrm{<mtext>5</mtext><mtext>2</mtext><mtext>,</mtext><mtext>3</mtext><mtext>2</mtext>}}\text{-}{}^2\text{P}{}_{\mathrm{<mtext>3</mtext><mtext>,2</mtext>}}$ doublet at 118.9 nm was resolved in emission using a 6.65 m normal incidence vacuum spectrometer. Multiplet emission ratios are combined with line absorption measurements to obtain the following absolute oscillator strengths: ²$\text{D}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}\text{-}{}^2\text{P}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}$ (118.875 nm), 0.0050 ± 0.0010; ²$\text{D}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}\text{-}{}^2\text{P}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ (118.877 nm), 0.068 ± 0.007; ²$\text{D}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}\text{-}{}^2\text{P}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}$ (120.135 nm), 0.102 ± 0.010.     

Spectrum and structure of the He2 molecule: Characterization of the singlet and triplet states associated with the UAO's 4s, 4dσ, 4dπ, and 4dδ

The emission spectrum of the He₂ molecule has been rephotographed in the ～4000–～5700 Å region and the $\text{4d(}{}^3\text{Σ}{}_{\mathrm{u}}{}^{\mathrm{+}}\text{,}{}^3\text{Π}{}_{\mathrm{u}}\text{,}{}^3\text{Δ}{}_{\mathrm{u}}\text{) → 2pπ}{}^3\text{Π}{}_{\mathrm{g}}$, $\text{4d(}{}^1\text{Σ}{}_{\mathrm{u}}{}^{\mathrm{+}}\text{,}{}^1\text{Π}{}_{\mathrm{u}}\text{,}{}^1\text{Δ}{}_{\mathrm{u}}\text{) → 2pπ}{}^1\text{Π}{}_{\mathrm{g}}$, $\text{4s}{}^3\text{Σ}{}_{\mathrm{u}}{}^{\mathrm{+}}\text{→ 2pπ}{}^3\text{Π}{}_{\mathrm{g}}$ and $\text{4s}{}^1\text{Σ}{}_{\mathrm{u}}{}^{\mathrm{+}}\text{→ 2pπ}{}^1\text{Π}{}_{\mathrm{g}}$ transitions analyzed. The 4dδj³Δ_u, 4dπj³Π_u, $\text{4dσj}{}^3\text{Σ}{}_{\mathrm{u}}{}^{\mathrm{+}}$ and $\text{4sh}{}^3\text{Σ}{}_{\mathrm{u}}{}^{\mathrm{+}}$ states have been characterized through v = 2 and the 4dδJ¹Δ_u, 4dπJ¹Π_u, $\text{4dσJ}{}^1\text{Σ}{}_{\mathrm{u}}{}^{\mathrm{+}}$, and $\text{4sH}{}^1\text{Σ}{}_{\mathrm{u}}{}^{\mathrm{+}}$ states for v = 0. The term levels for these perturbed and l-uncoupled states have been confirmed (a) by analyses of bands with common levels from Δv = 0, ±1 sequences and (b) by analyses of the transitions between the above states from 4d and 4s and the $\text{c}{}^3\text{Σ}{}_{\mathrm{g}}{}^{\mathrm{+}}$ and $\text{C}{}^1\text{Σ}{}_{\mathrm{g}}{}^{\mathrm{+}}$ states associated with 3pσ. Molecular constants are reported which have been partially corrected for the effects of l-uncoupling and the homogeneous perturbations between the state pairs J, H and j, h.     

Gravitational collapse in quantum mechanics with relativistic kinetic energy

It is proved that the quantum mechanical Hamiltonian $\text{H = Σ}{}_{\mathrm{i=1}}{}^{\mathrm{N}}\text{(p}{}^2\text{+ m}{}^2\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{? κ Σ}{}_{\mathrm{i>j}}\text{|x}{}_{\mathrm{i}}\text{? x}{}_{\mathrm{j}}\text{|}{}^{\mathrm{?1}}$ for bosons (resp, fermions) is bounded from below if N ≤ c_bκ^?1 (resp. $\text{N ≤ c}{}_{\mathrm{f}}\text{κ}{}^{\mathrm{<mtext>?3</mtext><mtext>2</mtext>}}$). H is unbounded from below if N ≥ c_b^lκ^?1 (resp. $\text{N ≥ c}{}_{\mathrm{f}}{}^{\mathrm{l}}\text{κ}{}^{\mathrm{<mtext>?3</mtext><mtext>2</mtext>}}$). The constants c_b and c_b^l (resp. c_f and c_f^l) differ by about a factor 2 (resp. 4).     

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.     

The neutron total and capture cross sections of 92, 94Zr

The neutron total and capture cross sections of^92,94Zr have been measured at the 80 and 40 m flight stations, respectively, on the Oak Ridge Electron Linear Accelerator (ORELA). Resonance analysis of the total cross sections gave values of s, $\text{p}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}$ and $\text{p}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ strength functions. Strong intermediate structure was observed in the $\text{p}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}$ cross section of ⁹²Zr. Significant single particle transitions identified in the p-wave capture cross section indicate the importance of the valence process for both nuclei. It is noted that the intermediate structure in the $\text{p}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}$ total cross section leads to enhanced valence effects in the capture cross section. The effects of non statistical E1 transitions following particle-hole annihilation in the compound states were not observed in either the ⁹²Zr or ⁹⁴Zr capture cross sections as they had been in that for ⁹⁰Zr.     

Nonorthogonality corrections in the method of correlated basis functions

A set of normalized linearly independent basis functions φ₁, φ₂, …, φ_j, … generates matrix representatives $\text{H}$ and $\text{N}$ of the Hamiltonian operator and the identity. An orthonormal basis $\text{φ}{}_1$, $\text{φ}{}_2$, …, $\text{φ}{}_{\mathrm{j}}$, … generated by a Löwdin transformation is characterized by the distance in Hilbert space between $\text{φ}{}_{\mathrm{j}}$ and φ_j. The choice of positive definite $\text{N}$${}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ minimizes these distances and maximizes the diagonal elements of $\text{N}$${}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$. Again for positive definite $\text{N}$${}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ and a finite basis, 1 ? j ? p, the analysis yields a general theorem on Trace $\text{N}$${}^{\mathrm{<mtext>?n</mtext><mtext>2</mtext>}}$ (? p for all positive and negative integral values of n except n = ?1 and ? p for n = ?1).Sufficient conditions are determined which permit the application of the binomial theorem to the evaluation of the transform of $\text{H}$. Approximate formulas for the energy eigenvalues through third order in nondiagonal matrix elements are presented in a compact form containing characteristic nonorthogonality corrections depending on the exterior or interior location of the matrix element in the perturbation formulas.