Showing papers on "Energy (signal processing) published in 1969"

Observation of Anderson Localization in an Electron Gas

Abstract: Anderson has shown that there is no diffusion of an electron in certain random lattices, and Mott has pointed out that, for electrons in materials in which there is a potential energy varying in a random way from atom to atom, Anderson's work predicts that there should be a range of energies at the bottom of the conduction band for which an electron can move only by thermally activated hopping from one localized state to another. An energy ${E}_{c}$ will separate the energies where this happens from the nonlocalized range of energies where there is no thermal activation. Cerium sulfide, investigated some years ago by Cutler and Leavy, is a particularly suitable material testing whether this is so because, in the neighborhood of the composition ${\mathrm{Ce}}_{2}$${\mathrm{S}}_{3}$, $\frac{1}{9}$ of the cerium sites are vacancies distributed at random, and the number of free electrons can be varied with only very small changes in the number of vacancies. It is shown that the experimental results find a natural explanation in terms of this model: Conduction is by hopping when the concentration of electrons is low and the Fermi energy ${E}_{F}$ lies below ${E}_{c}$; but when the concentration is higher and ${E}_{F}g{E}_{c}$, conduction is by the usual band mechanism with a short mean free path. The thermoelectric power is examined in both ranges, and the Hall mobility in the hopping region (${E}_{F}l{E}_{c}$) seems in fair agreement with the theory of Holstein and Friedman.

Asymptotic Sum Rules at Infinite Momentum

Abstract: By combining the ${q}_{0}\ensuremath{\rightarrow}i\ensuremath{\infty}$ method for asymptotic sum rules with the $P\ensuremath{\rightarrow}\ensuremath{\infty}$ method of Fubini and Furlan, we relate the structure functions ${W}_{2}$ and ${W}_{1}$ in inelastic lepton-nucleon scattering to matrix elements of commutators of currents at almost equal times at infinite momentum. We argue that the infinite-momentum limit for these commutators does not diverge, but may vanish. If the limit is nonvanishing, we predict $\ensuremath{ u}{W}_{2}(\ensuremath{ u}, {q}^{2})\ensuremath{\rightarrow}{f}_{2}(\frac{\ensuremath{ u}}{{q}^{2}})$ and ${W}_{1}(\ensuremath{ u}, {q}^{2})\ensuremath{\rightarrow}{f}_{1}(\frac{\ensuremath{ u}}{{q}^{2}})$ as $\ensuremath{ u}$ and ${q}^{2}$ tend to $\ensuremath{\infty}$. From a similar analysis for neutrino processes, we conclude that at high energies the total neutrino-nucleon cross sections rise linearly with neutrino laboratory energy until nonlocality of the weak current-current coupling sets in. The sum of $\ensuremath{ u}p$ and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{ u}}p$ cross sections is determined by the equal-time commutator of the Cabibbo current with its time derivative, taken between proton states at infinite momentum.

Flux creep in type-ii superconductors.

Abstract: We have made measurements of the evanescent decay of the irreversible magnetization induced by magnetic cycling of solid superconducting cylinders in order to elucidate the mechanisms of Anderson's thermally activated flux-creep process. A superconducting quantum interferometer device coupled to the creep specimen by a superconducting flux transformer made possible observations of flux changes with a resolution of one part in ${10}^{9}$. The general applicability of Anderson's theory of flux creep was confirmed and the results were analyzed to show that: (1) The total flux in the specimen changed logarithmically in time, i.e., $\ensuremath{\Delta}\ensuremath{\varphi}\ensuremath{\propto}\frac{\mathrm{ln}t}{{t}_{0}}$. (2) The logarithmic creep rate $\frac{d\ensuremath{\varphi}}{d\mathrm{ln}t}$ is proportional to the critical current density ${J}_{c}$ and to the cube of the specimen radius. (3) The logarithmic creep rate appears to be only weakly temperature-dependent because a proportionality to $T$ is nearly compensated by the proportionality to ${J}_{c}$, which decreases as $T$ increases. (4) The creep process is a bulk process that is not surface-limited (in this case). (5) Flux enters and leaves the surface in discrete events containing from about one flux quantum up to at least ${10}^{3}$ flux quanta. (6) On departing from the critical state to a subcritical condition, the creep process tends to remain logarithmic in time, but the rate is decreased exponentially by decreasing $T$ and is decreased extremely rapidly by backing off of the applied field from the critical state. (7) At magnetic fields $Hl{H}_{c1}$ on the initial magnetization curve, no flux creep was observed, but the logarithmic creep rate showed a modest increase above ${H}_{c1}$ and a broad rise as $H$ approached ${H}_{c2}$. The creep process is characterized by a dimension parameter $\mathrm{VX}$ consisting of a flux bundle volume $V$ and pinning length $X$, and by an energy ${U}_{0}$, both of which are supposed to be material-sensitive parameters characteristic of the irreversible processes. These parameters were determined from the experiments. Bundle volumes $V\ensuremath{\approx}{10}^{\ensuremath{-}12}$ ${\mathrm{cm}}^{3}$ and energies ${U}_{0}\ensuremath{\approx}1$ eV were found, indicating that groups of fluxoids must be pinned and must move cooperatively. The results are found compatible with a recent model for flux pinning that includes these cooperative effects.

Phenomenological Analysis of Ground-State Bands in Even-Even Nuclei

Abstract: A variable-moment-of-inertia (VMI) model is proposed which permits an excellent fit of level energies of ground-state bands in even-even nuclei. In this model the energy of a level with angular momentum $I$ is given by the sum of a potential energy term $\ensuremath{\propto}{({\mathfrak{g}}_{I}\ensuremath{-}{\mathfrak{g}}_{0})}^{2}$ (where ${\mathfrak{g}}_{0}$ is the ground-state moment of inertia) and a rotational energy term $\frac{{\ensuremath{\hbar}}^{2}I(I+1)}{2{\mathfrak{g}}_{I}}$. It is required that the equilibrium condition $\frac{\ensuremath{\partial}E}{\ensuremath{\partial}\mathfrak{g}}=0$ be satisfied for each state. Each nucleus is described by two adjustable parameters, ${\mathfrak{g}}_{0}$ and $\ensuremath{\sigma}$ (the softness parameter), which are determined by a least-squares fit of all known levels. The calculated level energies and moments of inertia ${\mathfrak{g}}_{I}$, ${\mathfrak{g}}_{0}$, and $\ensuremath{\sigma}$ are tabulated for 88 bands, ranging from Pd to Pt and from Th to Cm. Projections of three-dimensional arrays of ${\mathfrak{g}}_{0}$ and $\ensuremath{\sigma}$ on the ($N,Z$) plane are shown. These parameters are found to vary smoothly as function of $N$ and $Z$. Breaks occur at $N=98, 104, \mathrm{and} 108$. The osmium nuclei show a pronounced maximum for ${\mathfrak{g}}_{0}$ and an equally pronounced minimum for $\ensuremath{\sigma}$ at 108 neutrons. In Pt, ${\mathfrak{g}}_{0}$ decreases steeply to 110 neutrons and then more slowly, while $\ensuremath{\sigma}$ increases correspondingly. The stable Pt nuclei with $A=190,192, \mathrm{and} 194$ still possess appreciable moments of inertia and large but "finite" softness parameters. Hence they may be characterized as "pseudospherical." For nuclei exhibiting a near-harmonic level pattern (like ${\mathrm{Xe}}^{130}$, ${\mathrm{Sm}}^{150}$, and other neutron-deficient rare-earth isotopes), ${\mathfrak{g}}_{0}$ becomes exceedingly small, but already for the 2+ state $\mathfrak{g}$ is several orders of magnitude larger. The parameters of some $K=2$ bands in even-even nuclei and of bands found in odd-odd nuclei are related to those of appropriate ground-state bands in even-even nuclei. Evidence for a rotational band in ${\mathrm{Ir}}^{194}$ is deduced from recently published experimental results. A plot of $\frac{{E}_{4}}{{E}_{2}}$ versus $A$, presented for the discussion of the region of validity of the model, namely, $2.23l\frac{{E}_{4}}{{E}_{2}}l3.33$, reveals new regularities. The empirical "Mallmann curves" ($\frac{{E}_{I}}{{E}_{2}}$ plotted versus $\frac{{E}_{4}}{{E}_{2}}$) are deduced from the VMI model within its region of validity. Graphs are presented which allow the determination of ${E}_{I}$ (for $I\ensuremath{\le}16$) and of $\ensuremath{\sigma}$ and ${\mathfrak{g}}_{0}$ for each even-even nucleus for which the first 2+ and 4+ states are known. The model suggested by Harris, which includes the next-higher-order correction of the cranking model, is shown to be mathematically equivalent to the VMI model. The recently discovered appreciable quadrupole moments of 2+ states of "spherical nuclei" are compatible with the moments of inertia of these states given by the VMI model. The relation between $\frac{B (E2)({2}^{\ensuremath{'}}\ensuremath{\rightarrow}2)}{B (E2)(2\ensuremath{\rightarrow}0)}$ and $\frac{{E}_{4}}{{E}_{2}}$ is explored.

Optical Dispersion and the Structure of Solids

Abstract: A new energy parameter ${\mathcal{E}}_{d}$ is introduced to describe dispersion of the electronic dielectric constant. This dispersion energy is found to obey an extraordinarily simple empirical relation in the more than 50 ionic and covalent crystals for which reliable refractive-index dispersion data are available. Based on this result we derive a structure-dependent electronic dielectric response function consisting of constant optical-frequency conductivity with high-and low-frequency cutoffs.

Born-Mayer-Type Interatomic Potential for Neutral Ground-State Atoms with Z=2 to Z=105

Abstract: Born-Mayer parameters are given which permit, with good accuracy (to within 6%), a greatly simplified computation of a previously derived interatomic potential, $U(R)$, based on the Thomas-Fermi-Dirac (TFD) approximation. The numerical values of $A$ and $b$ appearing in $U(R)=A \mathrm{exp}(\ensuremath{-}bR)$ are tabulated in two sets of commonly used units for 104 homonuclear pairs of neutral ground-state atoms having $Z=2$ to $Z=105$. Approximate lower and upper limits of applicability, ${R}_{l}$ and ${R}_{u}$, are also listed, as is the magnitude of the maximum percent error ($\ensuremath{\epsilon}$) for each fit. ${R}_{l}$ is generally $\ensuremath{\sim}1.5{a}_{0}({a}_{0}=0.52917 \AA{})$, while ${R}_{u}\ensuremath{\sim}3.5{a}_{0}$. The effective upper limit probably lies at $\ensuremath{\sim}6\ensuremath{-}8{a}_{0}$. Also, with the aid of the given table and the combining rule ${U}_{12}\ensuremath{\simeq}{({U}_{11}{U}_{22})}^{\frac{1}{2}}$, the interaction energies of a total of 5356 heteronuclear diatoms can readily be obtained.

Effective Potential Model for Calculating Nuclear Corrections to the Energy Levels of Hydrogen

Abstract: The present work is an attempt to re-evaluate the nuclear corrections to the energy levels of hydrogen by using an effective potential approach. The basic idea is to infer from electron-proton scattering a potential which may then be applied to the bound-state problem. In lowest order, the potential is chosen from the first-order Feynman diagram for the scattering. With this choice the Breit equation is obtained. It is then solved in an approximate way, in the non-relativistic limit of the proton, to obtain wave functions which are accurate enough for use in evaluating the effects of perturbations of the potential. The reduced mass corrections to the fine structure and the hyperfine structure levels are readily found. The effect on the hyperfine splitting of the distribution of the proton charge and magnetic moment is obtained by correcting the lowest-order potential to include the proton form factors. A further modification is needed in evaluating additional recoil corrections, of relative order $\frac{\ensuremath{\alpha}m}{M}$, to the fs and the hfs. This additional term accounts for the failure of the iteration of the lowest-order potential to reproduce the scattering obtained from the second-order Feynman diagrams. The $\frac{{\ensuremath{\alpha}}^{2}m}{M}$ contribution to the state-dependent mass corrections to the hfs is also analyzed within the context of this approach. All the corrections found are in complete agreement with previous results obtained by the Bethe-Salpeter (BS) equation, but the present method has the virtue of conceptual simplicity.

Band Structure and Fermi Surface of Re O 3

Abstract: The nonrelativistic augmented-plane-wave (APW) method has been applied to calculate the band structure of Re${\mathrm{O}}_{3}$. An important feature of this calculation is that it includes corrections to the usual "muffintin" approximation. Because of convergence difficulties, the APW calculation has been carried out only at symmetry points in the Brillouin zone. These results have been used in conjunction with the Slater-Koster linear-combination-of-atomic-orbitals interpolation scheme to determine the band structure and Fermi surface throughout the Brillouin zone. The Fermi energy occurs in the ${t}_{2g}$ manifold of the rhenium $5d$ bands. The calculation predicts a Fermi surface consisting of two closed electron sheets centered about $\ensuremath{\Gamma}$, plus a third electron sheet which is open along $〈100〉$. The tight-binding parameters, which affect both the oxygen-rhenium ($2p\ensuremath{-}5d$) energy separation and the corresponding bandwidths, have been adjusted to fit the optical and Fermi-surface data for Re${\mathrm{O}}_{3}$. Spin-orbit effects for the rhenium $5d$ bands have been included by means of a spin-orbit parameter ${\ensuremath{\xi}}_{5d}$. The optimum value for this parameter has been determined by detailed comparisons between the theoretical and experimental Fermi-surface areas. The final results agree to within 10%.

Nucleon--nucleon scattering from one-boson-exchange potentials. iii. s waves included.

Abstract: The nucleon-nucleon interaction is described over the laboratory scattering energy range 0-350 MeV by a potential used in conjunction with the Schr\"odinger equation. In momentum space the potential is a superposition of Born terms obtained from single exchanges of $\ensuremath{\omega}$, $\ensuremath{\rho}$, $\ensuremath{\pi}$, $\ensuremath{\eta}$, ${\ensuremath{\sigma}}_{0}$, and ${\ensuremath{\sigma}}_{1}$ mesons, where the ${\ensuremath{\sigma}}_{0}$ and ${\ensuremath{\sigma}}_{1}$ are hypothetical scalar mesons with isotopic spin 0 and 1, respectively. Rather than taking the usual static limit, all terms of order $\frac{{p}^{2}}{{M}^{2}}$ are retained. The inclusion of $S$ waves requires the introduction of a cutoff factor. The meson coupling constants, the masses of the ${\ensuremath{\sigma}}_{0}$ and ${\ensuremath{\sigma}}_{1}$, and a cutoff parameter are adjusted to fit the experimentally determined phase parameters. A comparison with experimental phase-shift analysis shows a good qualitative fit, on the average.

Nuclear Interaction of Oxygen with Oxygen

Abstract: Striking and regular gross structure has been observed in excitation functions for the ${\mathrm{O}}^{16}$+${\mathrm{O}}^{16}$ elastic scattering interaction in the oxygen-ion laboratory energy range from 20 to 80 MeV. Excitation functions have been measured simultaneously at five c.m. angles from 50\ifmmode^\circ\else\textdegree\fi{} to 90\ifmmode^\circ\else\textdegree\fi{}; systematic angular-distribution measurements have been carried out at narrow energy intervals spanning the gross-structure peak in the 90\ifmmode^\circ\else\textdegree\fi{} excitation function for $19\ensuremath{\le}{E}_{\mathrm{c}.\mathrm{m}.}\ensuremath{\le}22$ MeV and at intervals throughout the remainder of the energy range studied. The gross structure is reproduced surprisingly well by a Woods-Saxon optical model having an unusually shallow real well depth ($V=17$ MeV) and an energy-dependent imaginary well depth ($W=0.4 \mathrm{MeV} +0.1 {E}_{\mathrm{c}.\mathrm{m}.}$). Extensive scans have demonstrated that this is the lowest member of a discrete set of equivalent Woods-Saxon potentials; it is the only one of these, however, which provides a reproduction of the experimental data without requiring a marked and complex energy dependence of the real potential-well depth. The effects of adding a repulsive core to this lowest Woods-Saxon potential were studied in some detail; it has been found that the model predictions are quite sensitive to the core---a somewhat surprising result, perhaps reflecting the apparent long mean free path of the ${\mathrm{O}}^{16}$ ions under the present conditions. An ambiguity in the imaginary well depth has been examined in some detail; the present data do not permit resolution of this ambiguity, but throw into question the physical significance of this long mean free path. The addition of a core does not provide significantly improved fits to the experimental data. This does not preclude the the existence of the core, but suggests further study with different well parameters and shapes. A phase-shift analysis of the 28 angular distributions spanning the gross-structure peak provided no evidence for underlying resonances; no compelling explanation for the intermediate (\ensuremath{\sim}200-keV) width structure superposed on the gross-structure peak is yet available. Finer-grained excitation-function structure is present and is attributed to statistical fluctuations in the usual way. Extensive theoretical studies stimulated by, and concurrent with, these experimental studies have demonstrated that these heavy-ion data may provide unique information on the nuclear-matter problem in finite systems, including an experimental determination of the effective nuclear compressibility.

Particle Tracks in Emulsion

Abstract: A new theory of track formation in emulsion accounts for the tracks of charged particles on the basis of a theory developed earlier for the response of biological molecules and NaI(Tl) to energetic heavy ions. The probability that an emulsion grain will remain undeveloped when exposed to $\ensuremath{\delta}$ rays depositing a mean energy $\overline{E}$ is assumed to be ${e}^{\ensuremath{-}\frac{\overline{E}}{{E}_{0}}}$, where ${E}_{0}$ is the dose at which $\frac{1}{e}$ (37%) of the emulsion grains remain undeveloped, as in the one-or-more-hit cumulative Poisson distribution. The parameter ${E}_{0}$ incorporates variations in emulsion properties and processing conditions. Calculation of the spatial distribution of the ionization energy deposited by $\ensuremath{\delta}$ rays is combined with the assumed emulsion response to yield the spatial distribution of developed grains about the path of the charged particle. Calculations are in agreement with experimental data for grain counts (up to the relativistic rise), blackness profiles, and track width.

Photon Echoes in Gases

Abstract: The behavior of the photon echo produced by an inhomogeneously broadened spectral line is analyzed, with particular emphasis on the effect of spatial degeneracy of the molecular energy levels. The theoretical results are used to aid in the identification of the states involved in the echoes produced from gaseous S${\mathrm{F}}_{6}$ excited by pulses of C${\mathrm{O}}_{2}$ laser radiation near 10.6 \ensuremath{\mu}. The analysis demonstrates that for the $|{\ensuremath{\Delta}}_{j}|=1$ transitions ($P$ and $R$ branches) the echo induced by two linearly polarized pulses propagating along the same direction, with an angle $\ensuremath{\psi}$ between their electric vectors, is polarized at an angle greater than $\ensuremath{\psi}$ with respect to the first pulse, and the echo intensity decreases to a minimum, but not zero, as $\ensuremath{\psi}$ approaches $\frac{\ensuremath{\pi}}{2}$. An exception is the $j=1\ensuremath{\leftrightarrow}j=0$ transition, for which the echo is always polarized along the second pulse and the intensity varies as ${cos}^{2}\ensuremath{\psi}$. For ${\ensuremath{\Delta}}_{j}=0$ transitions ($Q$ branch) the echo polarization generally lies at an angle smaller than $\ensuremath{\psi}$, i.e., between the first two pulses, and its intensity also decreases as $|\ensuremath{\psi}|$ approaches $\frac{\ensuremath{\pi}}{2}$. The exceptions for the ${\ensuremath{\Delta}}_{j}=0$ transitions are the cases $j=\frac{1}{2}\ensuremath{\leftrightarrow}j=\frac{1}{2}$ and $j=1\ensuremath{\leftrightarrow}j=1$. In the former case, the echo is polarized at an angle $2\ensuremath{\psi}$ with respect to the electric field vector of the first pulse, and the echo intensity is independent of $\ensuremath{\psi}$; whereas in the latter case, the echo behaves like that for $j=1\ensuremath{\leftrightarrow}j=0$. The dependence of the echo behavior on the angular momentum of states involved in the transition can serve to identify the transitions when measurements of the echo polarization and intensity are possible.

Experimental Test of the Pion-Nucleon Forward Dispersion Relations at High Energies

Abstract: The small-angle differential scattering cross sections of protons for pions have been measured to high precision at the Brookhaven AGS. The range of incident momenta was 8-20 GeV/c for ${\ensuremath{\pi}}^{+}$, and 8-26 GeV/c for ${\ensuremath{\pi}}^{\ensuremath{-}}$. The real part of the pion-nucleon forward scattering amplitude was determined by observing its interference with the known Coulomb amplitude. Combining these results with precision measurements of pion-proton total cross sections over this energy range provided a critical test of the predictions of the forward dispersion relations. The results demonstrate the validity of the dispersion relations up to at least 20 GeV/c laboratory momentum. The predictions of charge independence are also verified by comparing these experimental measurements with forward charge-exchange scattering cross sections. Furthermore, if microscopic causality is violated, this occurs at "distances" less than ${10}^{\ensuremath{-}15}$ cm.

High-Energy Collision Processes in Quantum Electrodynamics. I

Abstract: We have made a systematic study of all two-body elastic scattering amplitudes in quantum electro-dynamics at high energies. In particular, we have calculated the high-energy behavior of the following processes: (1) Delbr\"uck scattering, (2) electron Compton scattering, (3) photon-photon scattering, (4) electron-electron scattering, (5) electron-positron scattering, and (6) electron-proton scattering. The processes (1) and (2) are calculated up to the sixth order in the coupling constant $e$, the process (3) up to the eighth order, and the processes (4), (5), and (6) up to the fourth order. Our calculations show that all of these amplitudes are proportional to $s$, the square of the center-of-mass energy, as $s$ becomes large. In other words, we have found that, to these orders, ${\mathrm{lim}}_{s\ensuremath{\rightarrow}\ensuremath{\infty}}\frac{d\ensuremath{\sigma}}{\mathrm{dt}}$ exists and is nonzero for all $t\ensuremath{ e}0$, where $\ensuremath{-}t$ is the square of the momentum transfer. Furthermore, we found it meaningful to assign a factor (we call it the impact factor) to each particle. More precisely, for the high-energy scattering of $a+b\ensuremath{\rightarrow}a+b$, the imaginary coefficient of $s$ for the scattering amplitude is proportional to $\ensuremath{\int}d{\mathbf{q}}_{\ensuremath{\perp}}{[{({\mathbf{q}}_{\ensuremath{\perp}}+{\mathbf{r}}_{1})}^{2}]}^{\ensuremath{-}1}{[{({\mathbf{q}}_{\ensuremath{\perp}}\ensuremath{-}{\mathbf{r}}_{1})}^{2}]}^{\ensuremath{-}1}{\mathcal{I}}^{a}({\mathbf{r}}_{1},{\mathbf{q}}_{1}){\mathcal{I}}^{b}({\mathbf{r}}_{1},{\mathbf{q}}_{\ensuremath{\perp}})$, where 2r1 is the momentum transfer, and ${\mathcal{I}}^{a}({\mathbf{r}}_{1},{\mathbf{q}}_{\ensuremath{\perp}})$ and ${\mathcal{I}}^{b}({\mathbf{r}}_{1},{\mathbf{q}}_{\ensuremath{\perp}})$ are the impact factors of particles $a$ and $b$, respectively. The integration is over the two-dimensional transverse momentum of the virtual photons. The important point is that ${\mathcal{I}}^{a} ({\mathcal{I}}^{b})$ does not depend on what particle $b (a)$ is. We have explicitly found the impact factors for the photon (up to ${e}^{4}$) and for the electron, the positron, and the proton (up to ${e}^{2}$). In the case of Delbr\"uck scattering, we have also taken care of all higher-order diagrams with an arbitrary number of photons exchanged between the virtual pair and the proton or nucleus. The coefficient of $s$ in this case can be expressed as the integral of the above-mentioned product ${\mathcal{I}}^{a} {\mathcal{I}}^{b}$ times modified photon propagators. The impact factor therefore appears to express an intrinsic property of a particle. Our result is consistent with neither the most straightforward interpretation of the Regge-pole model nor that of the droplet model. These inconsistencies are closely related to the nonplanar nature of the diagrams under consideration. Our results on Delbr\"uck scattering are also qualitatively different from those of Bethe and Rohrlich based on the impact-parameter approximation.

Measurement of the Total Cross Section for Charge Transfer into the Metastable State H(2s) for Proton Collisions with Atomic Hydrogen

Abstract: The total cross section ${\ensuremath{\sigma}}_{2s}(\mathrm{H})$ for ${\mathrm{H}}^{+}+\mathrm{H}(1s)\ensuremath{\rightarrow}\mathrm{H}(2s)+{\mathrm{H}}^{+}$ has been measured over the energy range 3-70 keV. Above 10 keV, absolute cross-section values are believed accurate to within \ifmmode\pm\else\textpm\fi{}35%. Only one maximum is found in the energy dependence of ${\ensuremath{\sigma}}_{2s}(\mathrm{H})$. The position of this maximum lies at an energy consistent with the Massey criterion. A comparison with theoretical cross sections suggests that intermediate-state coupling through the resonant charge-transfer channel ${\mathrm{H}}^{+}+\mathrm{H}(1s)\ensuremath{\rightarrow}\mathrm{H}(1s)+{\mathrm{H}}^{+}$ is unimportant at low energies. The low-energy data is consistent with a molecular picture of the collision. Calculations utilizing Sturmian wave functions are not in agreement with the data. The Born approximation cross section lies significantly below experiment between 20 and 70 keV.

Magnon Bound States in Anisotropic Linear Chains

Abstract: The recent observation of two-, three-, four-, and five-magnon bound states in the linear chains of Co${\mathrm{Cl}}_{2}$\ifmmode\cdot\else\textperiodcentered\fi{}2${\mathrm{H}}_{2}$O has prompted a theoretical examination of such states in an anisotropic linear ferromagnetic chain with $S=\frac{1}{2}$. A new method called the Ising-basis-function (IBF) method is developed. This method treats the conventional, localized Ising wave functions as Wannier functions, from which a complete, orthonormal set of Bloch functions (IBF's) is formed. Using these IBF's as basis functions, we obtain the expression for the energy of the two-magnon bound state originally found by Orbach for general longitudinal exchange anisotropy. Furthermore, we can calculate the energy of the ($ng2$)-magnon bound states for the case of strong longitudinal anisotropy. The method is also applied to describe the effect of transverse exchange anisotropy. It is shown that this anisotropy causes an interaction between bound states, particularly important near zero field, and gives rise to a finite probability of exciting the bound states by photon absorption. The generalization of this method to treat bound states in two and three dimensions and for $Sg\frac{1}{2}$ is also discussed. The method is simple and has a direct physical interpretation. As an example, a physical description of the two-magnon bound state in a general system is given. Since the IBF method automatically contains some of the magnon-magnon interactions in zero order, it should be useful in other problems where these interactions are important.

Excitation of Multiple-Magnon Bound States in Co Cl 2 ·2 H 2 O

Abstract: The results of far-infrared transmission measurements on Co${\mathrm{Cl}}_{2}$\ifmmode\cdot\else\textperiodcentered\fi{}2${\mathrm{H}}_{2}$O at helium temperatures are reported and compared with theoretical predictions. Antiferromagnetic resonance, ferrimagnetic resonance, and ferromagnetic resonance have been observed in the respective metamagnetic phases of this material. Furthermore, these magnons appear to interact with an unexpected excitation which is believed to be an optical phonon. The most striking feature of the data, however, is the appearance of absorption lines in each phase which shift with magnetic field at a rate corresponding to $g$ values of about 14, 21, 28, and even 35, as compared to the $g$ value of \ensuremath{\sim}7 for the single magnons. Furthermore, the energy of each of these $n$-fold multiple excitations is markedly less than $n$ times the energy of a onefold one. These excitations are identified as clustered spin reversals or magnon bound states, and this is the first direct observation of such states. In the simple Ising-model approximation, such clusters of $n$ adjacent spin reversals are eigen-states, and the Ising-model energies qualitatively describe the observed energy spectra in all three phases. Using the theoretical results of the preceding paper, the small non-Ising terms are included in the theory, and excellent quantitative agreement is obtained. The reasons why bound states can be observed in Co${\mathrm{Cl}}_{2}$\ifmmode\cdot\else\textperiodcentered\fi{}2${\mathrm{H}}_{2}$O are also discussed.

Photon-nucleus total cross sections

Abstract: Quantitative predictions for the energy and $A$ dependence of the cross sections for nuclear photoabsorption and inelastic electron-nucleus scattering are given. In general, the nucleons do not contribute equally to the total photon-nucleus cross section when coherent contributions of photoproduced hadrons are taken into account. At low energies (${E}_{\ensuremath{\gamma}}\ensuremath{\sim}1$ BeV), the cross sections are proportional to nuclear number $A$, but at high energies, they become proportional to the number of surface nucleons---provided that the photon interactions are mediated by hadrons of sufficiently low mass. The condition on the masses is that the momentum transfers in forward photoproduction of these states should be small compared to the reciprocals of their mean free paths in nuclear matter. In the case of $\ensuremath{\rho}$ dominance, the real-photon photoabsorption cross section has the same $A$ dependence as hadron-nucleus total cross sections for photon energies above $\ensuremath{\approx}10$ BeV, whereas the cross section for virtual photon absorption at that energy, obtained from inelastic electron scattering is nearly proportional to $A$ for spacelike momentum transfer $|{Q}^{2}|\ensuremath{\gtrsim}5$ Be${\mathrm{V}}^{2}$. We then generalize to an arbitrary spectrum of intermediate particles, and discuss the sensitivity of feasible experiments to various models in which the spectrum contains important structure beyond the $\ensuremath{\rho}$. Measurements of the photon-nucleus cross sections will provide a fundamental test of "hadron dominance" in general, and of $\ensuremath{\rho}\ensuremath{-}\ensuremath{\omega}\ensuremath{-}\ensuremath{\varphi}$ dominance in particular, as well as help to determine the basic parameters of photon-nucleon and $\ensuremath{\rho}$-nucleon interactions. We also calculate the photon-deuteron cross section and discuss the multiple scattering approach to photon-nucleus interactions. This discussion provides insight into the many-body processes which underlie the eikonal, optical-model calculations; it is also relevant to the determination of ${\ensuremath{\sigma}}_{\ensuremath{\gamma}n}$ at high energies.

Bounds for truncation error in sampling expansions of band-limited signals

Abstract: For f(t) a real-valued signal band-limited to - \pi r \leq \omega \leq \pi r (0 and represented by its Fourier integral, upper bounds are established for the magnitude of the truncation error when f(t) is approximated at a generic time t by an appropriate selection of N_{1} + N_{2} + 1 terms from its Shannon sampling series expansion, the latter expansion being associated with the full band [-\pi, \pi] and thus involving samples of f taken at the integer points. Results are presented for two cases: 1) the Fourier transform F(\omega) is such that |F(\omega)|^{2} is integrable on [-\pi, \pi r] (finite energy case), and 2) |F(\omega)| is integrable on [-\pi r, \pi r] . In case 1) it is shown that the truncation error magnitude is bounded above by g(r, t) \cdot \sqrt{E} \cdot \left( \frac{1}{N_{1}} + \frac{1}{N_{2}} \right) where E denotes the signal energy and g is independent of N_{1}, N_{2} and the particular band-limited signal being approximated. Correspondingly, in case 2) the error is bounded above by h(r, t) \cdot M \cdot \left( \frac{1}{N_{1}} + \frac{1}{N_{2}} \right) where M is the maximum signal amplitude and h is independent of N_{1}, N_{2} and the signal. These estimates possess the same asymptotic behavior as those exhibited earlier by Yao and Thomas [2], but are derived here using only real variable methods in conjunction with the signal representation. In case 1), the estimate obtained represents a sharpening of the Yao-Thomas bound for values of r dose to unity.

Energy Levels in V-47, V-49, V-51 and Mn-51, Mn-53, Mn-55 from (He-3, d) Reactions

Abstract: The ${\mathrm{Ti}}^{46,48,50}({\mathrm{He}}^{3}, d){\mathrm{V}}^{47,49,51}$ and ${\mathrm{Cr}}^{50,52,54}({\mathrm{He}}^{3}, d){\mathrm{Mn}}^{51,53,55}$ reactions are studied with high resolution at 10- and 9.5-MeV incident energies, respectively. The $l$ values and spectroscopic factors are extracted by means of the distorted-wave Born-approximation calculations. A systematic study of the ${p}_{\frac{3}{2}}$ centroid energies, of the ${d}_{\frac{3}{2}}$ and ${s}_{\frac{1}{2}}$ proton hole states, and of the splitting of the ${p}_{\frac{3}{2}}$ spectroscopic strength is presented. These quantities, and also the individual energy spectra, are compared with the expectations of the shell model and the Coriolis strong-coupling model. It is concluded that the shell model adequately describes these nuclei, while the Coriolis strong-coupling model fails to explain the observed splitting of the ${p}_{\frac{3}{2}}$ spectroscopic strength.

Magnetic Field Dependence of Laser Emission in Pb 1-X Sn X Se Diodes

Abstract: The magnetic field dependence of long-wavelength infared laser emission has been studied in ${\mathrm{Pb}}_{1\ensuremath{-}X}{\mathrm{Sn}}_{X}\mathrm{Se}$ diodes for compositions in the range $0l~xl~0.3$. For $xg0.15$ the energy of the lowest transition decreases with increasing magnetic field whereas for $xl0.15$ this energy increases. This unique observation is consistent with a theory of magnetic energy levels proposed by Baraff and also strongly supports the inversion model for the energy bands in Pb-Sn chalcogenides.

Collision Spectroscopy. II. Inelastic Scattering of He + by Ne

Abstract: The inelastic scattering of ${\mathrm{He}}^{+}$ by Ne has been studied as a function of angle $\ensuremath{\theta}$ and energy loss $\ensuremath{\Delta}E$ over the range of barycentric energies $E$ from 22 to 500 eV. Many levels of Ne are excited, including autoionizing ones, and excited ${\mathrm{He}}^{+}$ is also seen. Excitation of the $2{p}^{5}3s$ configuration of Ne at about 16.8 eV could be resolved uniquely, and was studied in detail. The process does not occur in forward scattering, but only outsides a threshold obeying the rule ${\ensuremath{\tau}}_{c}={(E\ensuremath{\theta})}_{c}\ensuremath{\cong}1035$ eV deg, typical of a curve-crossing interaction. It shows pronounced Stueckelberg interference oscillations, arising from the existence of two semiclassical trajectories inside the crossing point at ${\mathcal{r}}_{c}$, with the property that the product of the wave number $k$ and the angular spacing $\ensuremath{\Delta}\ensuremath{\theta}$ between peaks is substantially constant, i.e., $\ensuremath{\Delta}b=\frac{2\ensuremath{\pi}}{k\ensuremath{\Delta}\ensuremath{\theta}}=0.44$ a. u. These and other properties suffice to show that the upper state is attractive and to identify the transition with the outermost crossing that perturbs the elastic scattering of ${\mathrm{He}}^{+}$ by Ne, which was earlier shown to be located at ${\mathcal{r}}_{c}\ensuremath{\cong}1.9$ a.u. and at an energy of ${V}_{c}\ensuremath{\cong}13.5$ eV. The behavior of the reduced cross section $\ensuremath{\rho}=\ensuremath{\theta}sin\ensuremath{\theta}\ensuremath{\sigma}(\ensuremath{\theta}, E)$ at the first Stueckelberg peak shows that the transition probability peaks at an energy of about 25 eV, i.e., when the velocity at the crossing point is about 2.6 \ifmmode\times\else\texttimes\fi{} ${10}^{6}$ cm/sec. From this, the transition matrix element at the crossing is deduced by Landau-Zener theory to be roughly ${H}_{12}({\mathcal{r}}_{c})\ensuremath{\lesssim}1$ eV. At higher energies the maximum height of the first Stueckelberg peak shows a general falloff as ${\ensuremath{\rho}}_{0}E=3.2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}16}$ ${\mathrm{cm}}^{2}$ eV, as well as a slow oscillation whose peaks are evenly spaced in reciprocal velocity with a characteristic spacing constant ${v}_{*}=8.8\ifmmode\times\else\texttimes\fi{}{10}^{6}$ cm/sec. This slow oscillation is also seen as an amplitude modulation of the whole pattern of Stueckelberg oscillations, and we tentatively ascribe it to a second mode of dissociation of the excited state produced in the crossing, yielding ${\mathrm{Ne}}^{+}$ and $\mathrm{He}(1s2s)$, which are not detected in our experiments; the probability of producing one or the other set of products presumably depends on the time spent in passing through an interaction region, and thus on the velocity. Above 250 eV another excitation process appears with a threshold at ${\ensuremath{\tau}}_{x}\ensuremath{\cong}5000$ eV deg, arising from a crossing at ${\mathcal{r}}_{x}\ensuremath{\cong}1.23$ a.u. with a second state dissociating to the same products. Its transition matrix element is shown to be roughly ${H}_{13}({\mathcal{r}}_{x})\ensuremath{\lesssim}3$ eV. Approximate potential curves are deduced for three of the low-lying $^{2}\ensuremath{\Sigma}$ states of He${\mathrm{Ne}}^{+}$ and compared with calculations by Michels. It is shown that the available crossing may explain the measured total charge-transfer cross sections measured by Stedeford and Hasted at energies below 1 or 2 keV, but not the magnitudes near the maximum near 10 keV.

Level Densities from Excitation Functions of Isolated Levels

Abstract: Thick-target excitation functions were measured for the reactions ${\mathrm{Co}}^{59}(p,{\ensuremath{\alpha}}_{0}){\mathrm{Fe}}^{56}$, ${\mathrm{Co}}^{59}(p,{\ensuremath{\alpha}}_{1}){\mathrm{Fe}}^{56}$, ${\mathrm{Co}}^{59}(p,{\ensuremath{\alpha}}_{2}){\mathrm{Fe}}^{56}$, ${\mathrm{Mn}}^{55}(p,{\ensuremath{\alpha}}_{0}){\mathrm{Cr}}^{52}$, ${\mathrm{Mn}}^{55}(p,{\ensuremath{\alpha}}_{1}){\mathrm{Cr}}^{52}$, and ${\mathrm{Ni}}^{62}(p,{\ensuremath{\alpha}}_{0}){\mathrm{Co}}^{59}$ at two or more angles for proton bombarding energies 6-13.5 MeV. The ${\mathrm{Fe}}^{56}(\ensuremath{\alpha},{p}_{0}){\mathrm{Co}}^{59}$ reaction was studied at $\ensuremath{\alpha}$-particle bombarding energies of 12-18.5 MeV. These measurements were used to determine the parameters in three different level-density formulas which were thought to be reasonable candidates to describe the level densities of the residual nuclei. In addition, the experimental values of the cross sections to isolated levels were used in conjunction with available values of the level width $\ensuremath{\Gamma}$ from cross-section fluctuation measurements to determine level densities of compound nuclei at about 20-MeV excitation energy. The absolute values and energy dependence of the nuclear level density in the excitation energy range 0-20 MeV investigated for several nuclei agree with a back-shifted Fermi gas model. The constant-temperature model gives a reasonable fit to the level density in the energy range 0-10 MeV but fails at higher excitation energies. The conventional shifted Fermi gas model does not reproduce the energy dependence and absolute values of the various experimental level densities for any value of the level-density parameter $a$.

State-dependent effective charge in the 2p-1f shell

Abstract: Motivated by a desire to understand the electric quadrupole transition rates in $^{58}\mathrm{Ni}$, calculations of the effective charge for both neutrons and protons were carried out, first with $^{40}\mathrm{Ca}$ as a core and then with $^{56}\mathrm{Ni}$. The calculation was done in perturbation theory using the Kallio-Kolltveit interaction. Concerning state dependence, it was first observed and then proved that in the limit in which energy differences in the $2p\ensuremath{-}1f$ shell were small compared with $2\ensuremath{\hbar}\ensuremath{\omega}$ excitations, the effective charge depended on the initial and final orbital angular momentum of a given transition, but when these $l$'s were specified, it was independent of the initial and final $j$ values. The effective-charge correction was much bigger for a neutron than for a proton. This may have the effect of reducing the isovector part of the quadrupole operator and hence causing $\ensuremath{\Delta}T=1$ transitions to be inhibited. The effective charge is substantially larger with $^{56}\mathrm{Ni}$ as a core than with $^{40}\mathrm{Ca}$ as a core, but is somewhat too small to explain the $E2$ transition from the first excited $2_{1}^{}{}_{}{}^{+}$ stage to ground. The effect of state dependence in the examples considered was to change certain $E2$ rates by a factor of 1.5 to 2. In $^{58}\mathrm{N}1$, the $E2$ ratio $2_{2}^{}{}_{}{}^{+}$\ensuremath{\rightarrow}$0_{1}^{}{}_{}{}^{+}$/$2_{2}^{}{}_{}{}^{+}$\ensuremath{\rightarrow}$2_{1}^{}{}_{}{}^{+}$, if calculated with shell-model wave functions, is extremely sensitive to the two-body interaction that is used. For example, it is about sixty times smaller (and closer to experiment) if Kuo's matrix elements, which are derived from a realistic interaction, are used rather than matrix elements chosen to give a least-squares fit to the energy levels of the nickel isotopes. If only one of the matrix elements obtained from the energy fit is changed by 0.3 MeV, the ratio becomes forty-two times smaller, also closer to the experimental value. The possibility that a lowlying ${2}^{+}$ state was basically a $3p\ensuremath{-}1h$ state was examined. The lowest two such states had very weak $E2$ transitions to ground and therefore did not at all resemble the one-phonon state or the electric quadrupole state. By themselves, these states fail as candidates, not only for the $2_{1}^{}{}_{}{}^{+}$ state but also the $2_{2}^{}{}_{}{}^{+}$ state because they radiate more to the ground than to the $2_{1}^{}{}_{}{}^{+}$ state.

Double-Electron-Capture Cross Sections for Incident Protons in the Energy Range 75 to 250 keV

Abstract: The double-electron-capture cross sections were measured for protons in the energy range from 75 to 250 keV incident upon the target gases ${\mathrm{H}}_{2}$, He, Ar, Kr, ${\mathrm{N}}_{2}$, and ${\mathrm{H}}_{2}$O. A power law extrapolation of the present results for ${\mathrm{H}}_{2}$ and He joins smoothly with previously measured cross sections at higher- and lower-impact energies; however, wide discrepancies exist between measured and calculated cross section values. For the heavier target gases, a power law extrapolation of the present results is in disagreement with previous measurements. This disagreement is attributed to the power law being inappropriate at energies where electron capture from inner electron shells is possible.

Device for converting a physical pattern into an electric signal as a function of time utilizing an analog shift register

Abstract: A device for converting energy patterns in the form of pressure, heat or magnetic images into an electrical signal as a function of time where the necessity for a scanning beam or a crossed bar readout system is eliminated by cascading elements which function as both storage and energy sensitive devices and by providing circuitry for shifting the charges of the energy sensitive storage elements in a single direction along the cascaded array.

Study of the K 39 ( d , t ) K 38 Reaction at E d = 23 MeV

Abstract: The $^{39}\mathrm{K}(d, t)^{38}\mathrm{K}$ reaction has been utilized to study states in $^{38}\mathrm{K}$ below an excitation energy of 5 MeV. Spectroscopic factors have been extracted by comparing the data with distorted-wave calculations. For states for which both $l=0$ and $l=2$ transitions are allowed, attempts have been made to extract relative admixtures. Two positive-parity states, not previously reported in neutron-pickup experiments, have been observed at ${E}_{x}=3.44 \mathrm{and} 3.99$ MeV. Two weak negative-parity states have been observed at ${E}_{x}=2.64$ ($l=1$) and 4.66 MeV ($l=3$), indicating the admixture of core-excited configurations in the ground-state wave function of $^{39}\mathrm{K}$.

Energy gaps in the excitation spectrum of a superconductor.

Abstract: It is shown, for a particular model where the order parameter $\ensuremath{\Delta}(\mathrm{r})$ is a one-dimensional periodic step function, that the spectrum of the elementary excitations of a superconductor consists of disjunct energy bands, separated by forbidden intervals.