Determination by polarization analysis of the scattering amplitudes a+ and a- and of the degree of orientation of nuclei polarized by hyperfine fields in magnetically ordered substances

Improvements of the Ito-Shull techniques are suggested for measuring the scattering amplitudes a₊ and a_- associated with the compound states $\text{I +}\text{1}\text{2}$ and $\text{I -}\text{1}\text{2}$ which are formed by the nucleus of spin I and the incident neutron. In ferro- and ferri-magnets one can increase the sensitivity by suppressing the electronic part of the magnetic scattering when polarizing the neutrons along the scattering vector. In anti-ferromagnets one can separate the nuclear magnetic part by substracting the electronic magnetic scattering with unpolarized neutrons from the total magnetic scattering (electronic + nuclear) obtained by measuring the spin-flip amplitudes U^+- or U^-+.     

Pion production and the higher resonances in pion-nucleon scattering

The reaction π + N → 2π + N has been studied in the vicinity of the higher resonances in the pion-nucleon cross section. The Low equation for the production amplitude is transformed into an integral equation by isolating the true one-meson intermediate states and discarding higher order contributions. The only part kept in the inhomogeneous term corresponds to the collision of a pion in the nucleon cloud with the incident pion in the resonant T = J = 1 state, which is simulated by an unstable vector Boson. Crossed terms are neglected and the 2π-N state is described by the static model. The terms kept in the sum over states describe the rescattering (off the nucleon) of one of the outgoing pions. The required off-the-energy-shell elastic scattering amplitude is approximated by the 3-3 resonance formula of Chew and Low. With these simplifications the Low equation for the production amplitude reduces to an easily soluble linear integral equation. The rescattering amplitude, which dominates the inhomogeneous term in the resonance region, is proportional to the 3-3 scattering amplitude of one of the outgoing pions. Although the result provides some support for the conventional isobar model, it is important to note that the largeness of the rescattering term arises from scattering far off the energy shell, rather than by “real” excitation as in the phenomenological isobar model. Quantitative calculations for the $\text{D}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}$ channel leading to a p-wave ($\text{J =}\text{3}\text{2}$) and an s-wave pion produce a maximum in the cross section near 600 Mev incident pion lab energy. For a π-π resonance energy squared S = 10, agreement with experiment is obtained with a width about one third that suggested by nucleon electromagnetic structure. In our approximation, the well known 600 Mev $\text{D}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}$ isospin $\text{1}\text{2}$ resonance occurs at the same energy as the 800 Mev $\text{D}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}$ isospin $\text{3}\text{2}$ resonance. It is assumed, but not proved, that the neglected terms are responsible for the splitting of the resonance energies. When this splitting is taken into account, the predicted charge state ratios near the second resonance agree well with existing data. The “third” resonance occurs for the state having two p-wave pions, according to the present theory, although no numerical calculations were made for this case. This point of view suggests that the $\text{F}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}$, $\text{P}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}$, and $\text{P}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ incident channels contribute to the third resonance.     

Quasi-free scattering of polarized protons

Cross section and analyzing power, energy-sharing spectra have been measured for the ¹⁶O($\text{p}$, 2p) reaction at an incident energy of 200 MeV for twenty-four different pairs of angles of the detected final-state protons. This general survey is intended to test the validity of the distorted wave impulse approximation (DWIA), particularly with respect to the predicted j-dependent analyzing power caused by the distorting optical potentials and nuclear spin-orbit coupling. Although the data in general confirm this j-dependence, agreement in detail between DWIA calculations and the analyzing power data becomes worse as the momentum of the recoiling nucleus increases. Predictably, the cross-section calculations show more sensitivity to the optical-model parameters than do the analyzing powers. Within the present limits imposed by scanty elastic scattering data, there seems to remain a choice between predicted ($\text{p}$, 2p) cross sections which either are unreasonably large or which have the incorrect energy-sharing shapes. More comprehensive elastic scattering and total reaction cross section data are clearly needed.     

The theory of electron-scattering coincidence experiments

A general expression for the electron-scattering coincidence cross section for the reaction A₁(e, e′ X) A₂ with a nuclear target is derived in the one-photon exchange approximation. The result is exact to lowest order in α, the fine-structure constant. It is expressed in terms of four kinematic factors involving the electron scattering variables in the laboratory frame, and four combinations of transition matrix elements of the nuclear current operator expressed in the center-of-momentum (COM) frame. The nuclear matrix elements are decomposed into transition amplitudes of definite angular momentum using a helicity analysis. General expressions for the angular distribution of particle X in the COM frame are then derived. The analysis is independent of the detailed structure of the nucleus and particle X and depends only on general symmetry considerations and the existence of a local electromagnetic current operator for the hadronic target. A unitary transformation from the helicity basis for the final particle X and A₂ to an LS coupling basis is relevant if X is massive and a finite number of total angular momenta J are involved in the reaction. Tables of angular correlation coefficients are given for the case where the initial nucleus A₁ has J₁^π = 0⁺. They constitute one of the most useful results of this paper.Connection is made in the “static limit,” and with the assumption that the reaction proceeds through a finite number of Breit-Wigner resonances with a corresponding factorization of the electroproduction transition matrix elements, to the familiar electromagnetic transition multipole moments involving excitation of a nuclear state J^π. The relation to previous work by de Forest and by Drechsel and Überall is discussed.Analytic expressions for the coincidence cross sections are given for spin-zero systems and some very simple, basic models of nuclear “giant resonance” excitations. It is hoped that they will be useful in obtaining insight into the coincidence cross section and in planning future experiments.Finally, a reanalysis of the recent Stanford data of Calarco et al. on ${}^{12}\text{C}\text{(e, e′ p}{}_0\text{)}{}^{11}\text{B}\text{(}\text{3}\text{2}{}^{\mathrm{?}}\text{)}$ in the vicinity of the giant dipole resonance in ¹²C is carried out using a very simple nuclear model but retaining all the terms in the coincidence cross section, some of which were previously neglected.     

Low-energy e4 compton scattering of spin-12 targets without C,P and T invariances

We have studied the low-energy e⁴ Compton scattering on $\text{spin}\text{-}\text{1}\text{2}$ targets without assuming C, P and T invariances. It is shown that the scattering amplitude has a term ω³ ln ω (ω is the energy of incident photon in the lab frame), whose coefficient is determined entirely by the charge, mass, magnetic dipole moment, and electric dipole moment of the target. To derive the result, we calculate the second-order (in ω) terms of the e² Compton scattering amplitudes. When the electric dipole moment (whose existence violates P and T but conserves PT) is set to zero, the result reduces to that obtained before by Lin assuming C, P and T invariances.     

Scattering of 1 and 2 MeV neutrons from oriented 165Ho

The “deformation effect” in the neutron-¹⁶⁵Ho differential scattering cross section, defined as the relative variation of the cross section induced by the orientation of the target, has been measured at 1 and 2 MeV incident neutron energy, in the scattering angular interval from 30° to 130°. A polycrystalline holmium target has been employed, oriented normally to the scattering plane, with a nuclear alignment parameter $\text{B}{}_2\text{B}{}_{\mathrm{<mtext>2\; </mtext><mtext>max</mtext>}}\text{= 0.24}$. The observed effect has an oscillating character as a function of the scattering angle with a maximum value of 0.12 ± 0.02 at 2 (emMeV). The effect has been theoretically calculated by an adiabatic coupled-channel calculation performed by using the optical model and deformation parameters deduced from the analysis of previous experiments. The agreement between the experimental and the theoretical results is good, particularly at 2 MeV energy where the compound nucleus contribution in the scattering is negligible.     

The post-post-Newtonian problem in classical electromagnetic theory

We find the Lagrangian to order c^?4 for two charged bodies (with $\text{e}{}_1\text{m}{}_1\text{=}\text{e}{}_2\text{m}{}_2$) in electromagnetic theory. This Lagrangian contains acceleration terms in its final form and we show why it is incorrect to eliminate these terms by using the equationsof motion in the Lagrangian as was done by Golubenkov and Smorodinskii, and by Landau and Lifshitz. We find the center of inertia and show that the potential energy term does not split equally between particles 1 and 2 as it does in the Darwin Lagrangian (Lagrangian to order c^?2). In addition to the infinite self-energy terms in the electromagnetic energy-momentum tensor, which are eliminated using Gupta's method, some new type of divergent terms are found in the moment of electromagnetic field energy and in the electromagnetic field momentum which cancel in the final conservation law for the center of inertia.     

Neutron diffraction study on the hyperfine-induced nuclear spin order in the antiferromagnet PrSn3

At several $\text{(h+}\text{1}\text{2}\text{0 0)}$ reciprocal lattice positions of PrSn₃, we observed Bragg scattering of neutrons which is purely due to the nuclear spin polarization of ¹⁴¹Pr, and could measure the temperature dependence of the polarization between 10 mK and 4.2 K. From the perfectly polarized state of the nuclear spins, the difference between the spin-dependent nuclear scattering lengths b⁺ and b^- has been determined to be b⁺ - b^- = -0.110 ± 0.006 × 10^-12 cm, which is significantly larger in magnitude than the previously reported value. An additional electronic polarization which is linearly proportional to the nuclear spin polarization has also been observed.     

Generator coordinate amplitude for scattering

In the generator coordinate method for scattering the proper boundary condition is accomplished by requiring the GC amplitude to satisfy an integral equation of the first kind. Attempts to solve this problem are first reviewed and then an improved approximation is proposed which is applicable to a wider class of scattering problems in addition to the Coulomb scattering.A better approximation is obtained in the asymptotic region, where the generator coordinate, i.e., the distance between two shell-model wells of the fragments, is larger than the touching distance of the colliding nuclei, by deriving partial differential equations of first order for the terms of an asymptotic series in $\text{1}\text{E}$, where E is the scattering energy.Extracting the information on the GC amplitude for small values of the generator parameter from the integral equation of the first kind is an ill-posed problem. It is shown that the method of statistical regularization offers a powerful and controllable procedure to uncover the GC amplitude. The unknown GC amplitude is treated as a random function with an a priori distribution of probability which is based on the assumption that the amplitude is bounded and that the errors in the input are random with zero expectation value. A useful procedure is found for fixing parameters of the a priori distribution. The solution for small values of the GC parameter is expressed in the form of a Dini series.The method is applied to the calculation of the GC amplitude for scattering of two α-particles at 15 MeV c.m. energy. The measure of the accuracy is the difference between the input wave function of relative motion and the result of folding of the GC amplitude with the kernel of the integral equation. The prescribed accuracy is reached with this method on a much larger interval than with any previously proposed method.     

Double-charge-exchange and inelastic scattering in π−+3He

The reactions π^?+³He→⁺+3n and π^?+³He→^?+³He were studied to investigate the $\text{T=}\text{3}\text{2}$ three-nucleon system. The differential cross sections were measured at scattering angles from 20 to 40 degrees. The secondary pion was momentum analyzed in a magnetostrictive-readout wire-chamber spectrometer. The double-charge-exchange reaction yielded a secondary pion energy distribution, the features of which can be due to either a $\text{T=}\text{3}\text{2}$ three-nucleon resonance or a resonace of the nucleons in the ³He nucleus. The inelasticc scaterring reaction yielded a secondary pion energy distribution peaked near threshold, consistent with resonances in both the $\text{T=}\text{3}\text{2}$ and $\text{T=}\text{1}\text{2}$ three-nucleon systems.     

Proton scattering at 1 GeV

The physics of 1-GeV proton scattering on nuclei is discussed in the light of recent calculations, and compared to the Gatchina, Los Alamos, and Saclay data. The impulse approximation (including spin-orbit effects and correlations) is reviewed, and comparison is made with other theories such as the Glauber model and the low-energy optical model. This discussion is addressed to specialists as well as nonspecialists in the field. The neutron distribution is extracted from the data and a detailed comparison is made with other determinations of this distribution and with the Hartree-Fock predictions. The neutron radii are seen to be generally larger than the proton radii. Within a given shell, they increase at a much slower rate ($\text{～A}{}^{\mathrm{<mtext>1</mtext><mtext>8</mtext>}}$) than the $\text{A}{}^{\mathrm{<mtext>1</mtext><mtext>3</mtext>}}$ rule. Except possibly for ²⁰⁸Pb, they are consistent with the Hartree-Fock predictions, but not with the values obtained from Coulomb energies. The study of inelastic scattering to collective states allows the extraction of neutron transition densities, and in particular the analog B(N, L) of the electromagnetic transition rates B(E, L) one usually considers for the protons. Neutron excitations are seen to be stronger by 20 to 40 % than proton excitation, exceeding the $\text{N}\text{Z}$ prediction of the collective model. Spin effects lead only to small changes in the cross section, but to a measurable analyzing power. The unnatural parity excitations of the lowest 2^? (T = 0) state of ¹⁶O and the 1⁺ (T = 1) state of ¹²C show that the spin-spin and tensor terms of the nucleon-nucleon amplitude are sizable. Their relative magnitudes are seen to be crucial for explaining the observed cross sections.     

An interference effect in the channel radiative neutron capture process

This study shows that the small thermal neutron radiative capture cross sections in ¹²C and neighbouring nuclides are the result of destructive interference between the potential scattering wave and the resonance scattering wave near the nuclear surface, resulting in a drastic cancellation in the radial integral. The behaviour of the scattering wave function is examined, and the general condition for the occurrence of such cancellation is discussed. The expression for the channel radiative capture cross section which has been derived has the same structure as the Lane-Lynn formula but is expressed in terms of different parameters. In addition, this investigation shows that if the optical model well depth is adjusted so that the binding energy of the $\text{p}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ orbit in ¹²C is kept at the experimental value, then the calculated results for the potential and channel radiative neutron capture cross sections are insensitive to the value of the nuclear radius.     

Antiproton-nucleus scattering with self-consistent model for medium corrections

The antiproton-nucleon t-matrix with propagation in the nuclear medium is calculated selfconsistently and applied to the construction of optical potentials for the elastic scattering of antiprotons from nuclei. We find that this treatment gives results that are sensitive to medium corrections even though the strong absorption acts to mask these corrections partially. The agreement with scattering experiments at 46.8 MeV on ¹²C is very good. We compare potentials containing medium corrections to those based on free $\text{p}\text{N}$ amplitudes for ¹²C and ⁴⁰Ca. The local approximation to the optical potential is found attractive at low energies, becoming shallower with increasing bombardment energy in the range considered here (up to about E_L = 120 MeV).     

Small-angle neutron scattering from Pb and 238U between 0.6 and 2.2 MeV

Differential cross sections for neutrons scattered from natural Pb and 99.9 % isotopically pure ²³⁸U have been measured at 0.5°, 1.0°, and 1.5°. A neutron energy continuum was produced by bombarding a thick natural lithium target with a 4 MeV, nanosecond-pulsed proton beam. Neutron energies were determined by time-of-flight techniques. Flight paths from the neutron source to the scatterer and from the scatterer to the detector were each about 5 m. For the 0.5° measurements an annular detector geometry with an angular resolution of ± 0.1° was developed to maximize detection solid angle. Data were averaged over 100 keV intervals from 0.6 to 2.2 MeV and were corrected for backgrounds, multiple scattering and inelastic scattering. Measured cross sections were compared to optical-model calculations which included electromagnetic interactions of neutrons with the nuclear Coulomb field. Inclusion of an induced neutron electric dipole moment interaction was not warranted by the data. The angular dependence of the cross section was fitted with a function $\text{A + B}\text{cot}{}^2\text{case1}\text{2}$θ at each energy. Mean values of B for ²³⁸U are in agreement with theoretical predictions. Values of B for Pb are apparently 15 % too low.     

Hartree-Fock distortion energy of the nucleus

The Hartree-Fock energy of a nucleus in an external field is expressed in terms of two auxiliary density matrices. One of them is a smoothed distribution $\text{?}$ which generates the non-selfconsistent deformed shell-model potential and is directly related to the nuclear shape. The other represents the volume polarization effect by the external field. The final expression suggests a possibility of a two-mode model for the nuclear vibrations. Renormalisation of smoothed quantities by means of phenomenological models is indicated.     

Electromagnetic waves near perfect conductors. I. Multiple scattering expansions. Distribution of modes

An expansion is established for the Green functions describing electromagnetic waves in the presence of a perfectly conducting boundary. Each term represents a process for which the wave scatters several times on the boundary and propagates freely in between. This multiple scattering expansion reduces to ray optics in the high frequency limit, and thus provides a general framework to study diffraction corrections. It also allows one to evaluate quantities averaged over the spectrum. Some symmetry properties of the expansion are exhibited, in particular, the replacement of magnetic fields by electric fields results in a change of sign at each scattering. Convergence is proven for any smooth boundary in the domain Im k ? | Re k | of complex wavenumbers k. The continuation of the convergence domain around k = 0 is shown to depend upon the topology of the boundary. The multiple scattering expansion method is applied to determine the distribution of electromagnetic eigenmodes in a conducting cavity. The density of modes ?(k) is analyzed in terms of closed classical rays, bouncing off the walls with mirror reflections. Paths of zero length yield the smooth part of ?, expanded as $\text{π}{}^{\mathrm{?2}}\text{[Vk}{}^2\text{-}\text{2}\text{3}\text{∫ dα / R}$ + $\text{O}$(k^?2)] where V is the volume of the cavity, and ∫ dα/R is the integral over the boundary of the mean curvature. Paths of finite length L yield contributions to the density ?(k), of the form Im(a exp ikL) appearing as regular oscillations in the bunching of eigenmodes. For an analytic boundary, inclusion of complex classical rays renders exact the analysis of the eigenmodes in terms of closed paths. As a consequence, the high temperature expansion for the energy of a small blackbody is obtained.     

Model independent analysis of the structure in pp scattering

An empirical study of the structure at $\text{t≈?1.1}\text{in}\text{d}\text{σ}\text{d}\text{t}$ (pp → pp), in terms of two exponential amplitudes plus interference, gives the following results. The steeper exponential shows standard Regge shrinkage, corresponding to a trajectory α ≈1+0.3 t. The other exponential slope stays constant; the overall energy dependence of this amplitude is approximately s^?1 for 10–30 GeV, leveling off possibly to a constant at ISR. The relative phase is about 2.1 rad at the lower momenta, becoming 2.5 at ISR. These results suggest a simple physical picture, in which a relatively unstructured shrinking Regge amplitude interferes with an even-signature non-Regge background. This interpretation predicts that the interference minimum in $\text{d}\text{σ}\text{d}\text{t}$ is deepest at an intermediate momentum near 175 GeV where measurements will soon be made at NAL.     

The 16O + p → 17F (T = 32) resonances at Ep = 11.26,12.71 and 14.57 MeV

Excitation functions for ¹⁶O+p reactions have been measured with high energy resolution in the region of the first, second and seventh $\text{T =}\text{3}\text{2}$ resonances in ¹⁷F at extreme backward angles. The observed resonance shapes have been analyzed with a single-level resonance formula taking the off-resonance spin-flip amplitude into account. The resonance parameters of the ¹⁷F first $\text{T =}\text{3}\text{2}$ state studied with special emphasis are E_x = 11193.3 ± 2.3 keV, Γ = 200 ± 40 eV and Γ_p0 = 19 ± 3 eV. This result and other results are compared with previous studies and theoretical predictions. The comparison with data of the mirror nucleus ¹⁷O is discussed with respect to the observed charge asymmetry of the isospin-forbidden particle decay widths.