# Physical Review

Online ISSN: 1536-6065
Publications
Article
Hulme's formula for the internal conversion of γ-radiation is simplified and used to calculate the internal conversion coefficient in the electric dipole case for electrons in the K-shell. For each of the elements Z = 69, 74, 79, 84, 89, IK is calculated for 10 values of the variable and a table obtained by interpolation is given for θ = 0.05 to θ = 1.70.

Article
The photographic emulsion method of registering the tracks of ionizing particles suffers from certain disadvantages that are mainly due to the relatively large size of grains forming the tracks and their large spacing. These disadvantages are mostly removed in new emulsions that have been prepared after numerous tests. The tracks of ionizing particles in these emulsions consist of grains of diameter d ~ 0.2 µ, which often form continuous lines.Grain spacing is no longer the main cause of apparent straggling, except below a range of 3 to 4 mm. of air. The standard deviation in determining energy from a single measurement of range is 3.25 to 3.5% for protons of 0.6 to 2.4 Mev., and 1.3 to 1.9% for α-rays of 4 to 8.5 Mev. The latter deviation is one-half to one-quarter that obtained in commercial emulsions. Ease of finding the tracks, especially of protons, is considerably increased. The tracks of fission fragments have a characteristic appearance and are easily found amidst the faint or invisible tracks of α-rays and protons. Slow electron tracks have occasionally been seen.The action of certain sensitizing and desensitizing chemicals is described and discussed in terms of the theory of the photographic process.The grain spacing is related to the diameter of the grain, the concentration of silver bromide in the emulsion, and the average probability of development of the grains touched by the ionizing particle. A quantitative theory is given for the probability of development in terms of the specific energy loss and a sensitivity parameter of the emulsion. The observed variation of the grain spacing with residual range of protons, α-rays, and fission fragments can be approximately explained by the theory.

Conference Paper
A formalism is developed to calculate radiative processes, and applied to the shift of resonance frequency due to the radiation field itself. The zeroth approximation gives the Bohr resonance condition, while the next approximation gives a shift proportional to the photon density. The first-order shift is made of two terms: electric and magnetic. They can be interpreted as second-order Stark effect and Zeeman effect due to the oscillating field, respectively. A comparison with experimental data on the Cs atom is made. A good agreement is obtained by choosing the value of parameters suitably. These values of parameters can be checked by a future experiment.

Article
Charge-exchange collisions have been studied when H+ ions and ground state H0 atoms are incident on Cs vapor. Measurements of the positive, neutral, and negative beam components after passage through the target were made as a function of the Cs target thickness at several energies between 0.5 and 20 keV. All three beam components were found to approach charge equilibrium monotonically. The maximum H- equilibrium yield is (21 ± 4)%, which occurs at an H+ energy of 0.75 keV. The H- yield decreases with increasing energy, and is 0.4% at 20 keV. At energies below 4 keV the H+ equilibrium yield is very small compared with the yield of H0 and H-. For energies greater than 10 kev the H- equilibrium yield is very small compared with the yield of H0 and H+. The cross sections σ+0, σ+-, σ0+, and σ0- were measured. The subscripts + and - refer to the H+ and H- ions, 0 as an initial subscript refers to a ground-state H0 atom, and 0 as a final subscript refers to an H0 atom in the particular states produced. The cross section σ+0 decreases with increasing energy, and ranges from (9.4 ± 2.0) × 10-15 cm2 at 1 keV to (7.5 ± 1.1) × 10-16 cm2 at 15 keV. The cross section σ+- decreases with increasing energy in the range 2-15 keV, and has the value (2.1 ± 0.6) × 10-17 cm2 at 5 keV. The cross section σ0+ increases with increasing energy, having the value (1.2 ± 0.2) × 10-17 cm2 at 2 keV, and (1.74 ± 0.26) × 10-16 cm2 at 15 keV. The cross section σ0- decreases with increasing energy, having the value (1.5 ± 0.3) × 10-16 cm2 at 2 keV and (1.4 ± 0.2) × 10-17 cm2 at 15 keV.

Article
Charge-exchange collisions have been studied when ${\mathrm{He}}^{+}$ ions with an energy in the range 1.5-25 keV are incident on Cs vapor. Measurements of the positive, neutral, and negative beam components after passage through the target were made as a function of the Cs target density at twelve energies. Over the entire energy range studied, the negative fraction was found to reach a maximum and then decrease with increasing target density. The maximum yield of ${\mathrm{He}}^{$-${}}$ ions was measured to be (1.4 \ifmmode\pm\else\textpm\fi{} 0.1)%. The maximum occurred at a ${\mathrm{He}}^{+}$ energy of 6 keV and a Cs target thickness of 5 \ifmmode\times\else\texttimes\fi{} ${10}^{14}$ atoms/${\mathrm{cm}}^{2}$. Various cross sections relevant to the charge-exchange process were determined. At 10 keV the cross section for the production of a fast ${\mathrm{He}}^{0}$ atom from a ${\mathrm{He}}^{+}$ ion incident on Cs is (6.3 \ifmmode\pm\else\textpm\fi{} 0.6) \ifmmode\times\else\texttimes\fi{} ${10}^{$-${}15}$ ${\mathrm{cm}}^{2}$, the value of the cross section for the production of a ${\mathrm{He}}^{0}$ atom when a ${\mathrm{He}}^{$-${}}$ ion is incident on Cs is (1.0 \ifmmode\pm\else\textpm\fi{} 0.3) \ifmmode\times\else\texttimes\fi{} ${10}^{$-${}14}$ ${\mathrm{cm}}^{2}$, and the value for the production of a ${\mathrm{He}}^{$-${}}$ ion when a fast ${\mathrm{He}}^{0}$ atom in the (1s) (2s) $^{3}S_{1}$ state is incident on Cs is (1.4 \ifmmode\pm\else\textpm\fi{} 0.4) \ifmmode\times\else\texttimes\fi{} ${10}^{$-${}16}$ ${\mathrm{cm}}^{2}$.

Article
Experimental results on negative-charge production in collisions of hydrogen atoms with He, Ne, Ar, and ${\mathrm{H}}_{2}$ are reported. For the three noble gases a rather smooth rise of the cross section with energy was found, absolute values ranging from about ${10}^{$-${}18}$-${10}^{$-${}16}$ ${\mathrm{cm}}^{2}$. For collisions of H on ${\mathrm{H}}_{2}$, an interesting structure appears which seems to extend structure indicated in earlier high-energy measurements. From extrapolations of our results and comparison with high-energy stripping measurements, a trend in the ratio between target ionization and projectile stripping from He to Ne to Ar is indicated.

Article
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 exp(−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, Rl and Ru, are also listed, as is the magnitude of the maximum percent error (ε) for each fit. Rl is generally ∼1.5a0(a0=0.52917 Å), while Ru∼3.5a0. The effective upper limit probably lies at ∼6−8a0. Also, with the aid of the given table and the combining rule U12≃(U11U22)12, the interaction energies of a total of 5356 heteronuclear diatoms can readily be obtained.

Article
The differential energy spectra of ${H}_{1}$ ($Z$\ge${}20$) and $H$ ($Z$\ge${}10$) nuclei were measured in a balloon flight at a time close to the solar minimum using a nuclear emulsion stack flown at an altitude of 2.7 g/${\mathrm{cm}}^{2}$ from Fort Churchill, Manitoba, Canada on 30 June 1965. $$\delta{}-ray density and residual range measurements on tracks, and measurements of change of$$\delta${}$-ray density with track length, were used to determine the charge and energy of the nuclei. The measured differential fluxes of ${H}_{1}$ ($Z$\ge${}20$) and $H$ ($Z$\ge${}10$) nuclei are 0.0013\ifmmode\pm\else\textpm\fi{}0.0006 and 0.0045\ifmmode\pm\else\textpm\fi{}0.0010 particles/(${\mathrm{m}}^{2}$ sr sec MeV per nucleon) in the energy range 300-500 MeV per nucleon and 0.0011\ifmmode\pm\else\textpm\fi{}0.0004 and 0.0049\ifmmode\pm\else\textpm\fi{}0.0008 particles/(${\mathrm{m}}^{2}$ sr sec MeV per nucleon) in the energy range 500-800 MeV per nucleon. These results are in agreement with satellite and rocket measurements made in the similiar energy range, and, when compared with low-energy measurements, indicate an energy spectrum of nucle that is flatter than those observed during earlier years.

Article
Sounding rockets carrying nuclear emulsions to study the composition and energy spectra of the lowenergy cosmic rays were flown on 17 and 23 June 1965 from Fort Churchill, Canada. The results show that the intensity of the heavy nuclei ($Z$\ge${}3$) was about the same as that in July 1964 and below that measured by other observers in May and early June 1965, before the Forbush decrease on 15 June 1965. The heliumnucleus flux between 15 and 22 MeV/nucleon was found to be significantly lower on 17 June 1965 immediately after the Forbush decrease than on 23 June 1965. The charge spectrum indicates that in the region from 40 to 90 MeV/nucleon the ratio of light- to medium-nucleus fluxes is 0.19\ifmmode\pm\else\textpm\fi{}0.07, well below the value of 0.5 found at intermediate energies (200 to 400 MeV/nucleon), and that the carbon-to-oxygen ratio is about 1, rather than 1.6 as found at higher energies. A strong preference for even nuclei relative to odd is seen for charges 9 to 14 in the 60- to 180-MeV/nucleon range.

Article
The results obtained during two balloon flights of a counter telescope designed to measure the composition of low-energy cosmic rays are presented. The telescope consists of two ¼-in.-thick plastic scintillators used to measure dEdx, followed by a 5-in.-thick crystal of NaI (Tl) to measure the residual energy E of stopping particles. From these measurements the charge and the mass of each particle are calculated. The accuracy of the measurements is sufficient to separate the isotopes of hydrogen and helium and to identify the nuclei of elements up to oxygen. The detector system was flown from Fort Churchill, Canada to a residual atmospheric depth of 2.3 g/cm2 for a total of 25 hours in July and August 1966. The differential energy spectra from 60 to 220 MeV/nucleon for protons, deuterons, He3, and He4 are presented after correcting for the effects of the overlying atmosphere. From these spectra the proton/He4 ratio is 4.3±1.0 over the full energy range, while an upper limit on the deuteron/He4 ratio of 0.25 has been set at 150 MeV/nucleon. The latter ratio corresponds to a traversal of less than 6 g/cm2 of interstellar hydrogen by the cosmic rays. The He3/(He3+He4) ratio obtained is 0.094±0.017 in the energy interval from 78-220 MeV/nucleon. The effects of the solar modulation on the proton and helium spectra have been considered by comparing the present data with previous measurements. This comparison is consistent with a modulation function proportional to either Rβ or R. The demodulated He3/(He3+He4) ratio is consistent with a traversal of 2.7±0.5 g/cm2 of interstellar gas, as calculated using a simple "slab" model, and is also consistent with a model in which a distribution of path lengths is considered.

Article
An upper limit on the continuous emission of 2.22-MeV photons (neutron-proton capture γ rays) from the sun has been determined from day and night balloon flights in May and June of 1967, from Palestine, Texas at an atmospheric depth of ∼3.8 g/cm2. The experimental apparatus consisted of a 3 in.×3 in. CsI(Tl) γ-ray spectrometer and airborne telemetry to transmit the pulse-height distribution to the receiving station, where it was recorded on magnetic tape for later analysis by a laboratory multichannel analyzer. On-board calibration sources giving lines below 2.22 MeV were always present in the spectrum and allowed a continual check on the stability of the apparatus. Comparison of the day and night average counting rates as well as the daytime absence of any line leads to a null result for any statistically significant solar flux of 2.22-MeV photons. Our resulting upper-limit flux is comparable with the value of 5×10−3 photon cm−2 sec−1 determined from the Ranger-3 experiment in January 1962 at a considerably different phase of the solar cycle.

Article
The frequency distribution of energy losses in a He-C${\mathrm{O}}_{2}$ mixture by 43.7-MeV protons was investigated by means of a proportional-counter technique. The distributions corresponding to mean energy losses of 1.64, 8.22, 16.4, and 49.5 keV were examined. Relatively good agreement was found between experimental data and Vavilov's formulation of the Landau effect for the 49.5-keV distribution. For the other distributions, corresponding to smaller mean energy losses, the discrepancy between experiment and theory increased as the mean energy loss decreased. Correction of the Vavilov distributions by use of the Blunck-Leisegang factor gave good agreement with the experimental data.

Article
The differential cross section (dσdEdΩ2) for (p,2p) reactions induced by 600-MeV protons on deuterium and helium-4 has been studied for symmetric reaction kinematics. By varying the detection angles of the two outgoing protons, we investigated the D(p,2p)n reaction for recoil momenta of the residual nucleus between 180 MeV/c antiparallel and 370 MeV/c parallel to the incident proton direction, respectively. For the He4(p,2p)H3 reaction, the corresponding numbers are 148 and 298 MeV/c, respectively. Some data for asymmetric reactions in helium-4 are also presented. Analyzing the data in terms of the plane-wave-impulse approximation, we calculate the single-proton momentum distribution in both nuclei. For deuterium, we observe a sudden change of slope in the momentum distribution around 150 MeV/c, which cannot be explained with theoretical momentum distributions obtained by Fourier transforming Hulthén or Hamada-Johnston deuteron wave functions. In helium-4, the momentum distribution is nearly Gaussian up to about 180 MeV/c; above this value, an excess of large momentum components is observed. Both the deuterium and helium-4 results seem to indicate that the plane-wave impulse approximation is not valid at this energy and for the reaction parameters investigated here. Multiple scattering effects might be at the origin of the apparent excess of large Fermi momentum components obtained when the data are analyzed with the impulse approximation.

Article
The power output of a single-mode 6328 He-Ne laser is studied as a function of cavity tuning and for different values of gas pressure. A theory due to Szöke and Javan is used to interpret the results. Two parameters γ and γ′, which characterize a homogeneous line shape in the atomic polarization to third order, are determined as a function of the total pressure P. The results for two different gas mixtures are as follows (in units of MHz/Torr): dγdP=22±7 and dγ′dP=58±13 for an 8:1 He3-Ne20 mixture; dγdP=12±4 and dγ′dP=39±9 for a 5.5:1 He3-Ne22 mixture. Linear extrapolation of γ and γ′ to zero pressure yields the radiative linewidth γN=13±12 MHz. An asymmetry Δ in the frequency dependence of the power output is determined as a function of P, with the result that dΔdP=2.3±0.6 MHz/Torr. The shift of the minimum of the power dip is toward increasing frequency for both mixtures.

Article
A singularity involving infinite densities at a finite proper time in the past is strongly suggested for the beginnings of the Universe by Einstein's general relativity theory, and is consistent with the few relevant observational data. There is no reasonable point at which to anticipate a failure of the theory, especially since a simplified quantum calculation in the accompanying paper predicts that quantum effects do not change the nature of the singularity. Therefore, we suggest that the singularity be treated as an essential element of cosmological theory, and indicate how this can be made more palatable by refining our concepts of time.

Article
Pions are usually absorbed by a pair of nucleons in complex nuclei. It is, however, possible for the pion to be absorbed by a single nucleon, leading to the emission of only one nucleon. Letourneux and Eisenberg have pointed out that this latter reaction would be useful in studying single-nucleon states within nuclei and as a test of the validity of perturbation theory for low-energy pion-nucleus interactions. Accordingly, we have searched for the reaction ${\mathrm{C}}^{12}({$\pi${}}^{+}, p){\mathrm{C}}^{11}$ at 68-MeV pion energy and 11\ifmmode^\circ\else\textdegree\fi{} proton emission angle. The observed cross section of 640\ifmmode\pm\else\textpm\fi{}130 $\mu${}b/sr for absorption by $1{p}_{\frac{3}{2}}$ neutrons is more than eight times larger than the theoretical value.

Article
Optical absorption spectra of Ni impurity in GaP due to d shell transitions, measuring crystal field parameters, noting fine structure

Article
The parametrization \sigma${}(E)=C+$\Sigma${}\stackrel{}{K}\frac{[(\frac{1}{2}{$\Gamma${}}_{K}){B}_{K}+(E$-${}{E}_{K}){A}_{K}]}{[{(E$-${}{E}_{K})}^{2}+{(\frac{1}{2}{$\Gamma${}}_{K})}^{2}]}$ for attenuation cross sections (particularly photo-ionization) is discussed, with attention to the following details: prescriptions for exact calculation of profile parameters, in which the effects of distortions are separated from the effects of multiple scatterings; the validity of assuming independent (noninterfering) resonances; the specific case of autoionizing lines; connections with alternative parametrizations; the prohibition on negative cross sections assured by unitarity; behavior at threshold; and the applicability of this parametrization to emission lines.

Article
The coupling of low-frequency surface plasmons to the ac Josephson currents in superposed thin dielectric and superconducting metal films is examined. The $I$-${}V$ characteristics for various superposed film systems are derived. The theory permits the magnitude of the parameters entering in the expression for the $I$-${}V$ characteristics, as well as their dependence on various quantities (temperature, mean free path, etc.), to be determined and compared with existing experiments. We have extended the theory of steplike structures in the $I$-${}V$ characteristics of a single Josephson junction to multiple films, and have found that more than one series of such steplike structures is possible. The theory provides a natural explanation for a three-film tunneling experiment of Giaever. Finally, the modifications of the dispersion relations for surface plasma oscillations in multiple-film systems, when admitting the presence of the ac Josephson effect, are derived.

Article
The damping of the ion acoustic waves by Coulomb collisions is studied by using the Fokker-Planck equation of Rosenbluth et al. for both the species constituting the plasma, namely, the electrons and the ions. In a plasma with weak collisions and in the absence of any external field, one finds that irrespective of the ratio $T$ of the ion temperature to the electron temperature, the characteristic frequency gets affected only by the electron-ion collisions. However, as far as the collisional damping is concerned, both the ion-ion and the electron-ion collisions play a somewhat equally important role; the electron-electron and the ion-electron contributions are negligible compared to the other two. The damping increases with increase in the collision frequency but decreases with increase in $T$.

Article
It is demonstrated phenomenologically that as a result of first-order spatial dispersion an "acoustical activity" of transverse acoustic waves, the analog of optical activity, arises in certain crystal classes. Crystals which are optically active are also found to be acoustically active. Furthermore, in the case of propagation along high-symmetry axes, the effect manifests itself, as in the case of optical activity, as a simple rotation of the plane of polarization of transverse acoustic waves. Acoustic effects of first-order spatial dispersion which have no optical analog are also treated. Acoustical activity can also be inferred from the transformation properties of the acoustic phonons, and the acoustical activity of crystals may thereby be derived from phonon-dispersion curves. An estimate of the magnitude of the acoustical activity of tellurium is so obtained. The effects of first-order spatial dispersion on transverse-optical phonons and the optical properties which arise from them are also considered. Practical problems involved in the observation of acoustical activity are discussed.

Article
This paper presents a theory of the coupling between electron-acoustic, ion-acoustic, and electromagnetic plane waves at an idealized shock front in a two-fluid, fully ionized, inviscid plasma. The equations of hydrodynamics and Maxwell's equations are used to derive the dispersion equation relating the frequency and propagation constant of possible modes of propagation in a plasma. Boundary conditions which must hold at the shock front in the presence of an incoming perturbation are then derived, and the amplitudes of predicted wave modes generated at the boundary are calculated. A generalized Poynting vector is used to calculate energy coupling ratios for the waves. In agreement with previous work on sound-shock wave interaction in neutral gases, we find enhanced ion-acoustic transmission in the present work. In addition, we also find enhanced propagation in the electron-acoustic coupled mode.

Article
The equations of the general Born-Oppenheimer separation into electronic and heavy-particle coordinates are re-examined, and the coupled equations that result for the heavy-particle motion are expressed in a particularly simple form. This is accomplished by introducing a generalized matrix operator for the effective momentum associated with the heavy particles; the matrix portion of this operator represents a coupling of the nuclear momentum with the electronic motion. The commutator between the momentum and potential matrices is a force matrix, which provides an alternative means of evaluating the momentum matrix. The momentum coupling has both radial and angular parts; the angular momentum coupling agrees with Thorson's expression. In the usual adiabatic molecular representation, the potential energy matrix is diagonalized, and all the coupling is thrown into the radial and angular momentum matrices. For collision problems it is often more important to diagonalize the radial momentum matrix, putting the radial off-diagonal coupling into the potential matrix; this generates a family of diabatic representations, the most important of which dissociates to unique separated atom states. This standard diabatic representations has the properties called for by Lichten, is uniquely defined even with the inclusion of configuration interaction, and leads immediately to the Landau-Zener-Stueckelberg limiting case under appropriate conditions.

Article
The relationship between the emitted light and decay of electron density in a pulsed helium afterglow at 1 Torr has been studied using a 0.75-meter tandem monochromator and 36-Gc/sec microwave interferometer. Substantial signal-to-noise enhancement was obtained with an automatic data-processing system enabling populations of the $n^{3}D$ levels ($n$\le${}25$) to be measured as a function of time to 10 msec in the afterglow. In addition, the populations of low quantum levels and the free-electron concentration were measured to 20 msec. It was found that the time-dependent behavior of the levels divided them naturally into three groups, only the lowest-lying of which had been measured in previous afterglow experiments. The levels of this lowest group were found to decay exponentially with the same lifetime for better than three orders of magnitude. Levels of the middle group were found to decay exponentially for 2.5 decades, the lifetimes increasing monotonically with quantum number. Levels of the third group were identified to be in Saha equilibrium with the free electrons, and uniformly had lifetimes equal to the sum of the calculated atomic and experimentally measured electron lifetimes.

Article
The time-dependent behavior of the light emitted from a pulsed helium afterglow at 3 Torr has been investigated with an improved spectrometric system of sufficient sensitivity-bandwidth ratio to resolve the behavior of the atomic light over a 50-msec time interval. The relationship between the atomic light and the time-resolved electron density, as measured with a 36-GHz free-space microwave interferometer, was found to indicate the importance of a persistent source of ionization at late times. Absorption measurements of time-dependent behavior of both the atomic $2^{3}S$ and molecular $2^{3}$\Sigma${}$ metastable species were found to strongly favor mutual ionization of two metastable molecules as being the most important source of late-time ionization at 3 Torr.

Article
The hypothesis of partially conserved axial-vector current and the current algebra are used to obtain some of the coupling parameters in a vector-meson field theory which assumes that the hadron electromagnetic current is a linear combination of neutral vector-meson fields. By use of the Ward identity, results are presented in a form that includes correction factors which take into account the use of unphysical values of the pion and kaon four-momenta. The results also take into account the most general form of vector mixing.

Article
The new spectral sum rule ∫0∞ρπ(0)(μ2)dμ2=∫0∞ρπ(1)(μ2)μ2dμ2 is derived under the assumption that there exists a unique limit in which all masses are zero for the A1ρπ and ρππ decay amplitudes. In this sum rule, ρπ(0)(μ2) is the pseudoscalar spectral function and ρπ(1)(μ2) is the axial-vector spectral function, both of which occur in the Källén-Lehmann representation of the propagator of the π-type axial-vector current. An experimental test of this sum rule is proposed. The equivalent sum rule for K-type axial-vector currents is derived from considerations of the φKK amplitude, and applications of these sum rules in conjunction with the first Weinberg sum rules are discussed. In particular, it is found that fKfπ=1 for the pion and kaon decay amplitudes.

Article
The general consequences of current algebras for may particle amplitudes are deriveed using generalized Ward-Takahashi identities. Several applications are indicated.

Article
Measurements are reported of the magnetization, resistivity and magnetoresistivity of dilute alloys of manganese in copper. At very low temperatures the magnetization is hysteretic and remanence is observed in all except the most dilute alloys. The magnetoresistance also exhibits reproducible hysteretic behavior in cases where there is hysteresis in the magnetization. Furthermore the magnetoresistance is proportional to the square of the magnetization. From these experimental results it is inferred that within any volume of dimensions comparable to the electron mean free path the net magnetization in the virgin alloy is zero. The phenomenology of one possible magnetic structure fulfilling this requirement is developed and it is shown how the low-temperature resistance anomaly and peculiar magnetoresistive behavior could result from this magnetic structure.

Article
The phonon-frequency spectrum of a mass-disordered alloy is calculated using the approximation of Elliott and Taylor for a reference lattice of mean-inverse atomic mass. The result is a relatively simple interpolation formula for the frequency spectrum which is symmetric in the two atomic types, is exact at both limits of concentration, and works qualitatively throughout the concentration range. A comparison is made with computer experiments in one and three dimensions. Good agreement is obtained in three-dimensional systems; the high-frequency structure coming from isolated clusters of atoms, which is important in one-dimensional systems, is not reproduced. The method is also used to calculate the weighted frequency spectrum from the displacement Green's function for a reference lattice of mean atomic mass. The results are contrasted with the true frequency spectrum. A comparison with inelastic neutron scattering data for 3 at.% W in Cr is made.

Article
The detailed band structure of ferromagnetic nickel is investigated in the neighborhood of the symmetry point $X$ in the Brillouin zone. The Hamiltonian is constructed in the k\ifmmode\cdot\else\textperiodcentered\fi{}p representation, taking into account exchange and spin-orbit interactions as well as the coupling of spins to the magnetic induction vector B. Group-theoretical arguments, suitably modified in the presence of ferromagnetism, are employed to determine the form of this Hamiltonian. When exchange and spin-orbit interactions are taken into account, very few accidental degeneracies are allowed. Thus the spin-orbit coupling, in general, removes accidental degeneracies between bands of opposite spin and causes a hybridization of spin-up and spin-down energy bands. In the case of ferromagnetic nickel, the resultant spin-dependent energy gaps exhibit a strong dependence on the axis of spin quantization, and consequently they permit magnetic breakdown to occur between bands of opposite spin for a select range of applied-magnetic-field directions. These effects provide an explanation for the anomalous behavior of the de Haas-van Alphen amplitude which is observed experimentally.

Article
A model introduced by Bean and extended by Kim is applied to predict voltages induced in a superconducting wire carrying low-frequency alternating current. A basic premise of Bean's model is that the magnitude of the local current density, J, is determined by the magnitude of the local magnetic induction, B. The function J(B) is assumed to have the form deduced by Kim from magnetization measurements, J=α(B+B0), with α and B0 treated as adjustable constants. Relations for voltage as a function of current have been derived for ac and for ac plus superimposed dc. Predicted V-versus-I curves are found to agree fairly well with oscilloscope tracings obtained for defect-saturated wires of Nb and various Nb alloys.

Article
Recent advances in laser technology have led to the production of ultrashort electromagnetic pulses of very high peak power. In this paper we develop a theory which describes the transient behavior of such pulses in a laser amplifier. The effects of atomic coherence and inhomogeneous broadening, necessary for a complete treatment of the problem, are included. In the present development, the degree of homogeneous and inhomogeneous broadening is variable. The two-level active atoms are characterized by a phase memory time ${T}_{2}$ and an inhomogeneous frequency distribution (which leads to a reversible decay time $T_{2}^{}{}_{}{}^{*}$). The field (Maxwell) and atomic (Schr\"odinger) equations are coupled in a self-consistent manner as in Lamb's theory of an optical maser. The resulting equations are solved analytically in certain limits, while the general case is integrated numerically. The emphasis is placed on the situation in which $T_{2}^{}{}_{}{}^{*}$ and the pulse width are much shorter than ${T}_{2}$. Thus our main effort is devoted to the physics of the inhomogeneously broadened system. The theory itself, however, contains the effects of both forms of broadening. The basic difference between inhomogeneous and homogeneous broadening is discussed. In the former case, the decay of radiation is caused by the dephasing of dipoles, and is reversible. The atomic memory is retained during the dephasing, and an internal reflection of the atomic coordinates, as is exemplified in the "photon echo" process, will cause the dipoles to rephase. The phonon interruptions or atomic collisions that bring about the homogeneous broadening are random processes which lead to an irreversible destruction of the atomic phase memory. The implications of the reversible versus irreversible decay processes are seen to have important consequences in the problem of pulse amplification.

Article
The problem of a light pulse propagating in a nonlinear laser medium is investigated. The electromagnetic field is treated classically and the active medium consists of thermally moving atoms which have two electronic states with independent decay constants γa and γb in addition to the decay constant γab describing the phase memory. The self-consistency requirement that the field sustained by the polarized medium be equal to the field inducing the polarization leads to coupled equations of motion for the density matrix, and equations of propagation for the electromagnetic field. Although the theory is developed for a Doppler-broadened gaseous medium, it may also be applied to a solid medium with inhomogeneous broadening. A unified treatment is given encompassing a wide range of pulse durations from cw signals to psec pulses. Continuous pumping is allowed, as well as any amount of detuning of the carrier frequency of the pulse from the atomic resonance frequency. The three independent decay constants γa, γb, and γab provide greater flexibility than that obtained by using 1T1 and 1T2. The equations are solved analytically in a few specialized cases and numerically in the general case. Flow charts for accomplishing the numerical integration are given. Among the special problems considered is the apparent paradox of pulses propagating faster than the velocity of light under circumstances described by Basov et al. It is shown that this contradiction with relativity arises from the use of an unphysical initial condition.

Article
Amplitude independent attenuation of longitudinal ultrasound propagating in /001/ direction in single crystal superconducting Pb compared with BCS predictions

Article
The amplitude-independent normal-state attenuation of longitudinal ultrasound propagating in the [001] crystallographic direction in single-crystal lead was investigated using the pulse-echo method at frequencies between 10 and 210 MHz in the temperature range 1 to 8°K. A method for the detection of amplitude-dependent effects is presented; the work was limited to lightly cold-worked high-purity lead and well-annealed tin- and thallium-doped lead, where the amplitude independence of the results was experimentally demonstrated. A method for the evaluation of the electron mean free path from normal-state attenuation data is presented; the electron mean free path was evaluated for the specimens used in the experiment. The phonon-limited mean free path was found to be proportional to T−a with a=4.02±0.05. Below 4.2°K, relative phonon-limited mean-free-path values derived from the acoustic data agree closely with the resistivity data of Van den Berg, but above 7.2°K the acoustic and resistivity results differ. The ql and frequency dependence of the electronic attenuation is in good agreement with the free-electron calculations of Pippard, and the magnitude of the Sommerfeld-Wilson deformation-potential constant, as derived from the present data, is |C|=(1.63±0.25) eV. In the low-ql limit, the magnetic field dependence of the electronic attenuation is not in agreement with results based on the free-electron model, nor is it consistent with the empirical behavior of the zero-field attenuation. The data do not support the electron-damped dislocation mechanism proposed by Mason to account for the anomalous superconducting-to-normal electronic-attenuation ratio that has been reported for lead. The implications of the present results as applied to Mason's theory are briefly discussed.

Article
A new method of calculating nonrelativistic scattering amplitudes is presented. The scattering amplitude is first calculated as a function of the complex energy below the scattering threshold, and the numerical results are then analytically continued to the physical region. The method is used to calculate two-body and two-channel scattering amplitudes. The numerical analytic continuation is accomplished by a rational-fraction representation similar to the Pad\'e method. Several techniques of numerical analytic continuation by rational fractions are described, and some examples are discussed.

Article
The Anderson Hamiltonian has been treated by the application of the cluster variation method for the cooperative phenomena, which was first proposed by Kikuchi and later generalized by Morita. The calculation has been carried out by retaining up to two-particle off-diagonal correlations. The self-consistent equations obtained for the number of d electrons with opposite spins are found to have similar behavior and similar self-consistent solutions to those of Anderson. The ground-state energy of this system has been calculated for the cases of infinite d-d correlation energy and for a d-d correlation energy twice the energy of the d orbital. This ground-state energy is found to be lower than the free-electron energy by an amount 2N(0)D2exp[-|εd|N(0)|V|2], where N(0) is the density of conduction states of one spin at the Fermi surface, D is the conduction bandwidth, εd is the energy of the d orbital measured from the Fermi surface, and V is the mixing strength between the conduction-electron states and the d orbital.

Article
An anharmonic-lattice-vibration theory of solid-state diffusion, based on classical theory developed previously, is presented in second-quantization form and is evaluated for Cu. Equilibrium statistical mechanics was used, and the Goldstone diagrams were isolated for the interacting-phonon events which contribute to ${D}_{0}$. A displacement transformation suitable for the inhomogeneous strain inherent in the migration mechanism was used. The results of a comprehensive theoretical analysis of the migration portion of ${D}_{0}$ were estimated numerically for Cu, using a nearest-neighbor Debye approximation. The values of ${D}_{0}$ obtained, including an experimental value of the entropy of vacancy formation, were 0.106, 0.091, and 0.078 for temperatures of 293, 793, and 1293\ifmmode^\circ\else\textdegree\fi{}K, respectively. The temperature dependence shown by ${D}_{0}$, in the classical limit, was due to the $T$ dependence of the atomic force constants, thermal expansion, and $T$-dependent anharmonic terms. At lower temperatures, quantum effects introduced terms with inverse powers of $T$ and mass. At 293\ifmmode^\circ\else\textdegree\fi{}K, which is $0.865{$\Theta${}}_{D}$ for Cu, there is a 7% decrease of ${D}_{0}$ due to quantum terms. The anharmonic terms introduced no direct effects on the mass dependence of ${D}_{0}$. If the first few anharmonic terms are included in an expansion of the activation energy, the first and most important term is linear in $T$ in the classical limit. Hence, it appears in the experimentally measured ${D}_{0}$, rather than in the activation energy. Therefore, harmonic-lattice-vibration theories or elastic theories of the activation energy should be realistic. A suggestion is made regarding the possibility of controlling the diffusion process by the artificial stimulation of phonons using laser radiation in a selected frequency range.

Article
Low-temperature anharmonicity plays an important role in ferroelectrics, high-field superconductors, light molecular solids, and other classes of materials of current interest. The Debye-Waller factor $f$, which in the harmonic approximation is a measure of the mean-square atomic displacement, is a useful gauge in the degree of low-temperature anharmonicity. The $f$ factor and its temperature dependence are discussed for several simple models in which the potentials are such that low-temperature atomic displacements are large enough to sample deviations from harmonic potential shapes. The $f$ factor is factorable in the classical temperature regime into harmonic and anharmonic terms, and also at low temperatures in certain limiting cases. Factorization facilitates the analysis of experimental data to yield a gauge of low-temperature anharmonicity.

Article
The polaron weak-coupling approximation has been applied to the problem of an electron on a spheroidal energy surface interacting with longitudinal optical phonons. The binding energy and effective masses are calculated. For SrTi${\mathrm{O}}_{3}$, where the observed masses are ${{m}_{l}}^{*}{m}_{0}=6.0$ and $\frac{{{m}_{t}}^{*}}{{m}_{0}}=1.5$, the calculated bare masses are 4.7 and 0.96, respectively, and the binding energy is 0.26 eV. It is also shown how the polaron properties may be calculated when the band edge is of the degenerate type.

Article
Galvanomagnetic measurements have been made at 4.2°K on thin zinc crystal samples, with the thin dimension parallel to the c axis. Different thicknesses in the range 70-400 were produced by electropolishing. The magnetic field H was varied between 0 and 14 kG. The longitudinal magnetoresistivity () generally shows no unusual thickness dependence. However, the transverse magnetoresistivity with the field perpendicular to the broad face () and the Hall resistivity (21) show anomalous thickness dependences. At higher fields, at a given H consistently decreases with diminishing thickness (negative size effect) and obeys an Hn law, with n nearly 2, at a given thickness. shows oscillations, probably magnetomorphic, which are superimposed on a gross Hall resistivity which is approximately proportional to H at higher fields. The gross Hall coefficient R increases rapidly from the thinnest to next thickest sample, then decreases with increasing thickness, becoming slightly negative near 400. Sondheimer's size-effect model was extended to a multiparabolic band conductor and evaluated at high fields for both compensated and uncompensated conductors. More general band models are discussed briefly. For a compensated conductor, the parabolic model predicts an inversion of the usual thickness dependence of and a maximum in the magnitude of R. The data are in qualititive agreement with these predictions. Physically, these effects are attributed to the interruption by the surface of the carrier trajectories, which prevents the cancellation effects between hole and electron carriers which normally occur at high fields in a compensated conductor. Other measurements in compensated materials are discussed briefly in light of these findings.

Article
In this paper we have continued our investigation of the properties of an electron gas in intense magnetic fields. We have obtained expressions for the thermodynamic energy density, particle density, the magnetic moment, and pair density (created in equilibrium with the system) for a gas of electrons with an anomalous magnetic moment, using the exact eigenvalues of the Dirac-Pauli equation for an electron with an anomalous magnetic moment, obtained by Ternov et al. According to the solution of Ternov et al., the lowest energy states of the electrons can be zero at certain field strengths. We have shown that pair creation does not occur spontaneously at the expense of the magnetic-field energy, but only at the expense of the thermodynamic energy of other particles of the system. Exact expressions for the pair density are given.

Article
Semiclassically, the dielectric constant of a free-electron gas in a magnetic field is highly absorptive within the cone ω=ωe±vFq. But quantum mechanically, within this region there are many large "windows" in which the absorptive component ε2−(ω, q)=0. In the first part of this paper we describe these large windows in detail. We then consider propagation within them. We show that in a nonmagnetic material there are other solutions of the dispersion relation in addition to a heliconlike mode. We next consider a ferromagnetic metal, and predict that in nickel a propagating magnon-helicon mode may exist up to microwave frequencies for wave numbers up to q≃106 cm−1, which is 103 times the q of the Kjeldaas edge. The usual microwave transmisson or reflection experiments are not suited to the observation of these effects, or the related phenomenon of giant quantum oscillations in helicon attenuation which has been previously predicted. To be successful, an experiment must be able to determine both the frequency and the wave number of the absorption process. Perhaps inelastic photon (Raman) scattering could be used for this purpose.

Article
The modes of propagation of x rays for a four-beam case in the germanium crystal are evaluated, and, after the application of the proper boundary conditions, the ratios of the exit intensities are calculated. On the assumption that pairs of beams within modes form standing waves, the anomalously transmitted beams have been obtained. These results were subjected to experimental checks which seem to justify them. In addition, the actual forms of the dispersion sheets were obtained for a three-beam case in germanium. It was also shown that the symmetry points of the dispersion sheets play a dominant role in the anomalous transmission.

Article
The angles of emission and frequency shifts of anti-Stokes radiation from small-scale trapped filaments of intense laser light in a number of liquids are examined and interpreted theoretically. For a given order of vibrational anti-Stokes radiation, a plot of angle of emission versus frequency resembles sections of nested parabolas which open towards higher frequencies, with apices a few tens of wave numbers below the unshifted anti-Stokes frequency, and very little intensity extending above this unshifted frequency. Various modulation mechanisms which smear out the Stokes and anti-Stokes frequencies are discussed, as well as the wave-vector matching conditions which appear to produce the angular distribution of radiation. Relatively simple approximations predict that the maximum extent towards lower frequencies of parabolas associated with the first anti-Stokes is one-half the vibrational frequency. The second and third anti-Stokes can extend twice and three times as far, respectively. In addition, a single set of parabolas is predicted for the first anti-Stokes radiation, but two sets for the second and third anti-Stokes. Theoretical expectations seem to agree reasonably well with observations.

Article
Antiproton scattering off $^3\He$ and $^4\He$ targets is considered at beam energies below 300 MeV within the Glauber-Sitenko approach, utilizing the $\bar N N$ amplitudes of the J\"ulich model as input. A good agreement with available data on differential $\bar p ^4\He$ cross sections and on $\bar p ^3\He$ and $\pbar ^4\He$ reaction cross sections is obtained. Predictions for polarized total $\bar p ^3$He cross sections are presented, calculated within the single-scattering approximation and including Coulomb-nuclear interference effects. The kinetics of the polarization buildup is discussed.

Article
We have solved a Fokker-Planck diffusion equation for the diffusion and acceleration of cosmic rays. A time-independent solution is obtained by assuming that the contributions to the present intensity of particles injected at a uniform rate during all past times is equivalent to summing the intensities over all future time of particles injected at a given time. Contrary to the usual dependence upon an average acceleration, we have used continuous deceleration and fluctuations in acceleration to explain the energy spectrum. This gives us a general expression for a power-law spectrum in which the exponent varies as γ=0.093ln(Em)+1.75, thus allowing a very good fit to the experimental energy spectrum. We then apply a previously developed expression for the production of antiprotons to obtain an injection spectrum. This and the above solutions are then applied to three extreme cases of possible origin. The resulting antiproton spectra depend upon whether the material encountered by primary cosmic-ray protons is mostly inside or outside the regions of acceleration. The relative ratio of antiprotons to protons with E>10 GeV is expected to be about 10−6 if the protons have passed through 2-3 g/cm2 of material.

Article
The antiproton flux in primary cosmic rays is treated by calculating the ratio of antiproton to positron production in nuclear collisions. It is shown that in the few-BeV energy range at earth, the ratio of the antiproton flux to the positron flux is given directly by their production-cross-section ratio and is independent of astronomical uncertainties. At higher energy, the equilibrium intensity ratio deviates from the production ratio due to modulation in interstellar space. The antiproton production cross section is deduced from the pion production cross section on the basis of a simple physical argument. By comparison with the observed positron intensity, we obtain an antiproton flux which is approximately ${10}^{$-${}4}$ of the proton flux for $E$\ge${}5$ BeV and falls off rapidly below 2 BeV. It is suggested that a detection of cosmic-ray antiprotons in the lower energy range would test the validity of the nuclear-interaction model used in the calculation, while extension of the measurements to higher energies, which is perhaps difficult experimentally at present, would serve as a useful probe in the study of the origin and propagation of secondary cosmic-ray particles.

Top-cited authors
• Loyola University New Orleans
• University of California, Riverside
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• Santa Fe Institute