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The limit ${Z}_{3}=0$ is studied in the Zachariasen model using dispersion theory techniques. The connection between bound states and elementary particles is demonstrated in this limit and it is shown how Castillejo-Dalitz-Dyson ambiguities are removed.

The calculation of the partftion function for a simple onedimensional order-disorder model is reduced to the solution of a certain functional equation. This equation is rigorously and it is shown that in the limit of indefinitely long-range interactions the model exhibits a finite discontinuity in the specific heat. (auth)

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Precise measurements of the critical field of superconducting tin films were made as a function of temperature and thickness. Particular attention was paid to measurements near the transition temperature where the London limit holds. It was found that near the transition temperature the critical field could be expressed in the form ${H}_{c}={(\ensuremath{\Delta}t)}^{\frac{1}{2}}\ensuremath{\gamma}(1+\ensuremath{\epsilon}\ensuremath{\Delta}t)$, where $\ensuremath{\gamma}$ and $\ensuremath{\epsilon}$ are independent of temperature. In terms of the Ginzburg-Landau theory modified to include the lower temperatures, the constant $\ensuremath{\gamma}$ determines the penetration depth and the constant $\ensuremath{\epsilon}$ is different for different modifications of the field-independent free energy.Penetration depths determined in this way were found to be a function of thickness. Assuming that the coherence length $\ensuremath{\xi}$ in the film is determined by the thickness and using the expression for the thickness dependence of the penetration depth given by Tinkham, one can obtain a bulk coherence length ${\ensuremath{\xi}}_{0}$ of approximately 2100 \AA{}, as well as a bulk penetration depth of 510 \AA{}, from the data. This way of determining ${\ensuremath{\xi}}_{0}$ is quite different from the high-frequency method of Pippard, and it is proposed as an independent and alternative method.Two particular modifications of the Ginzburg-Landau theory are considered. The first, which is the original theory, predicts a value of $\ensuremath{\epsilon}=0.31$. The second, which is the Gorter-Casimir modification proposed by Bardeen, predicts a value of $\ensuremath{\epsilon}=\ensuremath{-}0.19$. Experimentally, $\ensuremath{\epsilon}$ was determined to have an average value of 0.14\ifmmode\pm\else\textpm\fi{}0.10.

The strong reaction d + d - min gave no sig He/sup 4/ + rature of the d /sup o/, which is forbidden by isotopic spin conservation, was looked for with a 460Mev deuteron beam from the Berkeley 184-in. cyclotron. Preliminary results were reported elsewhere. A more comprehensive analysis of this same experiment was recently completed. From this analysis, an upper limit for the differential cross section for the reaction in the c.m. system at about F /sub cm/ - = 90 nif- is iound to be (0.097 ction in DDT-C/ 0.027) x 10/ sup -33/ cm/sup 2//sr. The data are consistent with no rature of the d /sup o/ production. From a comparison of this upper limit with the theoretical prediction of the cross section for this reaction in isotopic spin need not be conserved, it is concluded that isotopic spin is at least (99.74 ction in DDT-C/ 0.08)% conserved in this strong interaction. (auth)

Upper and lower bounds for the spacelike pion form factor are given in terms of a dispersion integral of the modulus of its timelike form factor. Similar bounds are obtained for its phase (in the timelike region) which in turn provide information on the low-energy P-wave pion-pion scattering. The upper and lower bounds approach the same limit, and hence become equalities, if the zeros of the form factor are far from the physical region of interest.

A relativistic current operator is given in the $O(4, 2)$ description of the H atom whose matrix elements reduce to the correct Galilean form in the limit $c\ensuremath{\rightarrow}\ensuremath{\infty}$. This current is used to obtain a relativistic wave equation which contains the H-atom states as solutions and in which the electromagnetic field is coupled minimally.

Double-pion photoproduction is studied with the help of the equal-time commutation relations and the hypothesis of the partially conserved axial-vector current. In the present model we disperse both the pions and the photon, and hence the pions are treated symmetrically. Further, we work in the off-mass-shell limit [(pionfour−momentum)2 → O] in contrast to the soft-pion limits used by earlier authors. Certain difficulties connected with gauge invariance, which arise from this limiting procedures, are noted and discussed; also, the present calculation is expected to be valid only in the low-energy range close to threshold. The results obtained are in good agreement with experiment.

Scintillation counters have been applied to the study of the yields of photoprotons from twenty elements. The relative yields increase by a factor of approximately one hundred as $Z$ increases from 4 to 28, and then decrease to a value, at $Z=50$, of about one-tenth that of the maximum at $Z=28$. An upper limit is placed on the yields from several heavy elements. The absolute values are compared with the neutron data of Price and Kerst, and the sum of the two yields is compared with theory.

Using a method introduced in a previous paper, we define covariant functions corresponding to the retarded product and the equal-time commutator. From these we obtain general sum-rule identities, employing three independent infinite limits. These identities considerably limit the form of the equal-time commutator. The simplest form consistent with the general covariance properties is discussed.

The validity of certain cutoff procedures, which have lately been employed in the treatment of termwise divergent perturbative expansions in nonrenormalizable field theories, is investigated in the context of nonrelativistic scatterirg from strongly singular repulsive potentials. For the cases considered the heuristic cutoff prescription indeed yields the correct expression for the weak-coupling limit of the phase shift. (auth)

A study is made of the constraints placed upon the couplings and resonance spectrum required to saturate finite-energy sum rules consistently in the region around t=0 . It is argued that the probable existence of crossing trajectories and cuts places a fundamental limit on the usefulness of finite-energy sum rules. The demand that the spin of the resonances increase monotonically with mass implies that an infinite number of asymptotically parallel daughter trajectories in the resonance region are required for consistency. This result is independent of any statement about the energy dependence of the leading trajectory.

The on-shell electromagnetic mass difference of kaons is studied using the four-point-function identities developed by Gerstein and Schnitzer. The mass difference, which is logarithmically divergent, remains so even in the soft-kaon limit because of the inapplicability of Weinberg's generalized second sum rule. It is shown that the correct sign of the ${K}^{+}\ensuremath{-}{K}^{0}$ mass difference can be obtained at sufficiently high cutoff even without using the $\ensuremath{\sigma}$-term.

It is shown for systems of strongly interacting particles that in the limit of very many particles a transformation exists leading to an alternative problem which can be solved by a self-consistent field method. The transformed wave function describes "particles" moving in a collectively determined uniform potential; the transformation relating the transformed and original wave functions determines the amount of incoherence and correlation in the original function. The method appears to be more useful for fermion systems and is illustrated by applications to some aspects of nuclear structure.

Some apparent discrepancies in the definition or calculation of the Coulomb T matrix are investigated in an approach that uses shielded wave functions. It is found that the screened Coulomb T matrix behaves anomalously in the neighborhood of the energy shell and is in fact discontinuous in the limit of zero screening. A closed-form expression for the T matrix, which has been derived previously, is shown to be essentially correct off the energy shell. (auth)

The Kadanoff-Baym transport theory leads to an expression for polaron mobility which becomes exact for low, but finite, temperature, ${e}^{\frac{\ensuremath{\Theta}}{T}}\ensuremath{\gg}1$, but not necessarily $\frac{\ensuremath{\Theta}}{T}\ensuremath{\gg}1$. This expression is evaluated exactly in the weak-coupling limit. Comparison with previous variational calculations shows that the latter have not predicted even qualitatively valid corrections to the known asymptotic zero-temperature result. Simple arguments allow the present weak-coupling expression to be extended to finite coupling. These arguments are verified by a quantitative calculation using the Feynman model. It is found that the corrections due to finite temperature are at least as important as those due to a finite coupling constant.

Several different cosmologies have been proposed in which the strength of the gravitational interaction is variable. Also, it has been suggested that the gravitational interaction may play a significant role in determining the structure of elementary particles, and in particular that the value of the fine structure constant may depend on the strength of the gravitational interaction. It is shown that these two effects taken together would lead to observable discrepancies in the ages of terrestrial rocks and meteorites as determined by different radioactive decay schemes. Analysis of the geophysical data leads to an upper limit of about 3 parts in 10/sup 10/ per year on the rate of change of the fine structure constant. If the assumed relation between gravitation and particle structure were valid, this would correspond to a limit on variations in the strength of the gravitational interaction of 2 parts in 10/sup 11/ per year. This upper limit is one-fifth of the size of the variations expected according to Dirac's cosmology, and roughly as big as the variation to be expected according to the BransDicke cosmology. It is concluded that either the assumed connection between gravitation and elementary particles does not exist, or if the connectionmore » does exist, that the geophysical data provide a significant limit on possible variations in the strength of the gravitational interaction. (auth)« less

It is shown that the interference particle parameter, b/sub nu / (ML,EL'), appearing in the directional correlation of internal-conversion electrons, has the opposite sign from that given in the literature. It is also shown that in the high-energy limit, the directional-correlation pattern for conversion electrons is identical with that of the corresponding gamma rays. (auth)

In order to put Zwaan's scheme for deriving the Kramers connection formulas on a rigorous basis, differential equations are set up governing the variation in the coefficients used to fit a linear combination of the B. W. K. type approximation functions to an exact solution of Schr\"odinger's equation in one dimension. Approximate solutions of the differential equations are worked out which lend themselves to the setting up of the connection formulas and give definite upper bounds to the errors involved in their use. The method is also used to set up the Sommerfeld phase integral quantum condition independently of the connection formulas. An upper limit to the error in the energy is worked out. A similar treatment of the problem of the transmission of matter waves through rounded potential barriers is formulated.

Spectral sensitivity measurements between 5000 and 2300A have been made on barium surfaces prepared by fractional distillation in a gettered vacuum. The resulting yields are compared with those predicted by Mitchell's square-top barrier theory which is here modified to a form that facilitates comparison with experimental data. From 5000 to 3000A relative values of the observed yields are in good agreement with the modified theory. At 2967A there is an abrupt rise in the experimental curve which continues to the limit of the measurements. This is attributed to the onset of the volume photoelectric effect. The theoretical threshold for the volume effect calculated from the rough formulae of Tamm and Schubin agrees well with the experimental value.

The hyperfine structure of the four lowest atomic levels of ${\mathrm{Ni}}^{61}$ has been studied by the atomic-beam magnetic-resonance technique. Agreement between the results and the theory of the hyperfine interaction is good. The importance of taking account of the substantial departure of the atom from the $\mathrm{LS}$ limit in evaluating the electric quadrupole moment $Q$ of the ${\mathrm{Ni}}^{61}$ nuclear ground state is discussed. Values of $Q$ are determined from two different electron configurations; the values obtained differ by only 3%. The result is in agreement with the previous theoretical prediction of Cohen et al.

We argue that the expression for the energy-momentum tensor as a quadratic function of currents given in the Sugawara model must be reinterpreted as the limit of the corresponding expression, with the currents given a small separation minus its vacuum expectation value as the separation goes to zero, if the expression is to have any chance at all of making sense in a quantum field theory. We calculate this limit for a wide class of free Dirac theories with the usual currents in both two and four dimensions. In many (though not all) cases, we find that the limit yields the usual energy-momentum tensor. In calculating the limit in four dimensions, we have to multiply by the square of the separation to ensure that the limit exists; this is a reflection of the divergence of the Schwinger terms in the equal-time commutators of currents.

An earlier study of variational solutions of the Liouville equation for simple fluids is extended to include high-frequency thermal excitations. In the long-wavelength limit, variational eigenvalues are determined in correspondence with dispersion relations derived from linearized Navier-Stokes equations which contain frequency-dependent viscous and thermal transport coefficients. Two distinct sets of high-frequency longitudinal excitations arise. In the limit that mechanical moduli are much larger than thermal moduli, one of these excitations simplifies to a quasiphonon composed only of velocity fluctuations and associated time derivatives. On the other hand, in the case of vanishing viscosity, the excitation is seen to be composed only of energy fluctuations and time derivatives of energy fluctuations and, as such, may be the classical analog of high-frequency second sound in liquid He II. The various eigenfunctions are discussed in relation to results of recent neutron-scattering experiments.

This note is about the construction of energy-independent potentials from the Tamm-Dancoff equations, by a systematic method of Klein's. It is shown that, when the method is applied to the new Tamm-Dancoff equation of Dyson, the difficulty of the spurious singularities can be overcome. It is proved further that the old and new Tamm-Dancoff methods give the same potential in the adiabatic limit.

A model of plasma oscillations in the presence of small-angle collisions is presented which admits of exact analytic solution. Certain features of the true collision terms are preserved. Namely, the effect of collisions is represented by a diffusion in velocity space, which makes the distribution function tend to the Maxwell distribution, and which conserves the number of particles. In the limit of infrequent collisions the results of Landau are recovered.

The importance of magneto-hydrodynamic phenomena to different parts of cosmic physics is becoming more and more evident. This fact and the extreme mathematical difficulties involved in an exact treatment of many problems make it necessary to consider the possibility for experimental investigations. In this paper the fundamental equations are solved for cylindrically limited waves, a case which seems to be best adapted for experiments. It is shown that an "ideal" magneto-hydrodynamic wave in a liquid with finite conductivity can exist only in a certain frequency interval. The upper limit is set by the conductivity and the magnetic field strength. Above this value the waves degenerate into skin-waves. The lowest frequency is determined by the geometrical dimensions and by the conductivity. Waves in different liquids are compared, and liquid sodium is found to be the best medium for an experiment. A brief account is given for a preliminary experiment and other investigations are proposed.

We illustrate in detail the summation of multiphoton exchange diagrams for $\mathrm{ee}$, $\ensuremath{\gamma}e$, and $\ensuremath{\gamma}\ensuremath{\gamma}$ elastic scattering amplitudes at the high-energy limit using the infinite-momentum technique reported earlier. These diagrams are shown to give rise to amplitudes proportional to $s$, the center-of-mass energy squared, multiplied by simple combinations of the eikonal forms of high-energy scattering.

The 15.1-Mev scattering resonance in carbon is used to study the yield of photons at the high-frequency limit of the bremsstrahlung spectrum. A survey of the yield at 0 deg was carried out for several elements. The 0 deg yield at the limit increases strongly with the atomic number in agreement with recent calculations. Measurements of the 15.1Mev isochromats from a thin thorium (Z = 90) target for several angles are presented and compared with available calculations. A value of 1.61 plus or minus 0.18 mb is obtained for (k/Z/sup 2/ ) (d sigma /dk), the cross section integrated over angle. This is in reasonable agreement with theoretical values of 1.96 mb based on the inverse photoeffect and 1.74 mb as obtained from a direct evaluation of the bremsstrahlung matrix element in the extreme relativistic limit. (auth)

The general theory of the settling of small particles in a fluid is used to determine an upper limit for the time required to establish the steady, exponential distribution of density. It is shown that the steady state will, in in any case in which the initial distribution of particles is uniform, be sensibly attained in that length of time required by one of the particles, moving with the velocity given by Stoke's law, to fall two tube lengths. An examination of Burton's experiment (which sought to prove that the exponential distribution law holds for only very small depths below the surface) indicates that the steady state should have been reached in about 37 days. His failure to find any evidence of an exponential distribution of density after 50 days cannot be ascribed to an inadequacy in the time allowed for settling.

The fourth frequency moments of the longitudinal and transverse momentum correlation functions in a simple, classical liquid have been derived. Numerical results in the $k=0$ limit are given for a Lennard-Jones potential.

The mass defect of ${\mathrm{Li}}^{6}$ has been calculated with the use of a symmetric Hamiltonian. Perturbation theory was employed in the main part of the work, and an upper limit for the contribution of all functions to the first and second order perturbation energies was obtained. This upper limit is 10 percent smaller than the experimental binding energy. It was then shown on the basis of variational calculations that perturbation theory, carried to this stage, over-shoots the mark, and the upper limit to the binding energy is reduced to 26 Mev, which is 6 Mev short of the experimental value. These figures are based on a set of nuclear constants which favor large mass defects. The perturbational effect of all doubly excited configurations upon the $^{3}D$ state of ${\mathrm{Li}}^{6}$ was investigated and compared with the effect of similar functions upon the $^{3}S$ state. The result is that the first order $^{3}D\ensuremath{-}^{3}S$ splitting, which is about 2 Mev, is enlarged to 6 Mev, both levels being depressed.

The non-linear interaction of helicons (whistlers) is found to depend radically on whether the plasma has diffuse or sharp boundaries. The conventional picture of three-wave interaction obtained by perturbing the normal mode amplitudes is valid for the diffuse case. However, for a plasma with sharp boundaries, e. g., a solid state plasma including collisions, surface currents play an important role in the limit of large fi r . It is shown that the normal mode method breaks down at even very small amplitudes. A new strong-coupling method is developed which predicts an interaction rate in the large ft T limit that never exceeds the order of the wave linear damping, but should be experimentally observable.

Author(s): Bander, M; Bjorken, JD | Abstract: New sum rules from current algebra valid in the limit of large negative four-momentum squared of the current are obtained; some of these depend upon commutators of space components of the current densities. They contain convergence factors which make their validity plausible on most models of high-energy behavior, with the Regge-pole model being one of them. With one exception, their utility consists of placing constraints upon models of hadrons designed to saturate the current-algebra scheme. © 1968 The American Physical Society.

The Feinberg-Pais theory of summing the uncrossed ladder graphs is re- examined. A single regularization of the W propagator is used throughout. Instead of the iterative procedure used previously, the leading terms of the individual graphs are summed exactly. The result of Feinberg and Pais for energies much below 300 Bev is obtained for all energies in the limit of infinite regulator mass. (auth)

The formation of radioactive ${\mathrm{Be}}^{10}$ by the ($d,p$) reaction from beryllium has been observed. The assignment of the activity to beryllium is the result of very careful chemical separations. ${\mathrm{Be}}^{10}$ emits negative electrons with an upper limit of 560\ifmmode\pm\else\textpm\fi{}50 kev (end point in Al = 180\ifmmode\pm\else\textpm\fi{}20 mg/${\mathrm{cm}}^{2}$). The half-life is very long; data are given from which its value could be computed if the yield were known.

Pair production in high-energy electron-electron collisions is studied with special attention given to pion pair production. A method of calculation is formulated which yields results with reasonable directness in the relativistic limit. The orders of magnitude of counting rates for various experimental settings are ascertained. A complete result is obtained for the case in which two pions emerge with equal energies and opposite momenta.

A variational principle is used to "solve" an approximate integral equation for the screening functions of the electron gas. It is shown that the simplest possible trial function gives a solution to the equation for the dielectric constant which is exact in both the limits of small and large momentum transfers. The results are compared with other calculations. It is shown that the approximation developed recently by Kleinman is quite good in the static large-$k$ limit, but otherwise incorrect. The dielectric constant derived from the variational calculation is used to derive an expression for the ground-state energy; this expression is similar in its essential features to the interpolation schemes of Hubbard and of Nozi\`eres and Pines, even though the Hubbard approximation considerably underestimates the exchange enhancement of the vertex function at large $k$. Finally, it is suggested that similar variational principles may have other uses, as in the paramagnon problem.

Equations of motion are derived for generating functions of field correlations and detector counting probabilities for an arbitrary system of electromagnetic fields in the presence of their sources. For a system with linear constitutive relations, the equations are solved exactly. Such systems are shown to be always Gaussian, and some representative counting distributions are computed. A perturbative treatment of a system with a cubic nonlinearity in its constitutive relation is made to find the counting distribution to first order in the nonlinear coefficient. A comparison is made with the below-threshold limit of a recent laser-fluctuation theory, and a possible application to measurement of phonon-phonon interactions is mentioned.

A formulation is given whereby the grand partition function of a many- body system satisfying BoseEinstein or Fermi-Dirac statistics is expressed in terms of certain U functions defined ior the same system with Boltzmam statistics. lt is then shown that these U functions can be evaluated in successive approximations in terms of a binary kernel B which can be computed from a solution of the two-body problem. The approach to the limit of infinite volume is studied. The example of a hard sphere interaction is discussed in some detail. (auth)