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Cubic double perovskites that host heavy ions with total angular momentum $J$ = 2 can exhibit a singular magnetic state epitomized by a lone octupole and bulk ferro-type magnetism. It exists in the Chen-Balents Hamiltonian with a quadrupole interaction and competing exchange forces between the ions. Our symmetry inspired analysis mirrors the Dzyaloshinskii-Man'ko theory of latent antiferromagnetic ordering, and a 3k collinear structure. Experimental tests for the singular state include neutron or resonant x-ray Bragg diffraction.

In scattering theory the effect associated with the downward crossing of the phase-shift δ l ( k ) ( l being the orbital angular momentum and k the momentum) through δ l = π / 2 (mod π ) is called echo . In the standard nuclear theory (Breit–Wigner theory) the echo is described and evaluated in terms of scattering by an impenetrable sphere. However, this model holds only at sufficiently high energy, while it is inadequate at low energy. In this paper we show that the echo effect can be associated with two different regimes acting at low and high energy, respectively. At high energy the hard-sphere scattering model seems to describe appropriately the phenomenon. At low energy we propose a mechanism due to the exchange forces induced by the Pauli exclusion principle in the fermionic interaction, which leads to nonlocal potentials. These potentials admit for the scattering amplitude pole singularities in the fourth quadrant of the complex angular momentum plane. This paper analyzes the role played by this class of poles in the description of the low energy regime of echoes. A specific phenomenological analysis is performed, taking as typical example the α – α elastic scattering.

Physicists often claim that there is an effective repulsion between fermions, implied by the Pauli principle, and a corresponding effective attraction between bosons. We examine the origins of such exchange force ideas, the validity for them, and the areas where they are highly misleading. We propose that future explanations of quantum statistics should avoid the idea of a effective force completely and replace it with more appropriate physical insights, some of which are suggested here., Comment: 26 pages, 3 figures. Submitted to American Journal of Physics

We study propagation of electromagnetic waves in an antiferromagnetic medium by solving Maxwell equations coupled with a nonlinear spin equation for magnetization of the medium. In the case of isotropic medium, the propagation is governed by perturbed electromagnetic soliton of the perturbed modified Korteweg–de Vries equation. When anisotropy is added as a small correction to the exchange force, we do not observe any qualitative change in the propagation of EMW. However, when anisotropy is treated as a dominant force the propagation is governed by electromagnetic soliton of the derivative nonlinear Schrodinger equation. In both isotropic and anisotropic cases the magnetization of the antiferromagnetic medium is also found to exhibit solitonic behaviour (electromagnetic spin soliton) similar to the propagating electromagnetic soliton.

Effects of the three-nucleon forces on the triton binding energy are investigated using different models for which the scattering amplitudes for the off-mass-shell pions are extrapolated from the on-mass-shell data. In the independent model potential, the three-nucleon forces are constructed using the chiral symmetry through partial conservation of the axial current vectors and current algebra. On the other hand, the three-nucleon forces in the dependent model potential are derived from an effective Lagrangian which is constrained by chiral and gauge symmetry. The effect of the long range and the repulsive -exchange three-nucleon forces on the triton binding energy are investigated. The bound state triton binding energies are then calculated by solving the Faddeev equations with a Hamiltonian including the three-nucleon forces, using different nucleon-nucleon interactions. The pionic form factors are considered with different sets of the form factor cutoff, parameter . The -exchange forces with - and -wave amplitudes are found to overbind the triton binding energy by about . However, -exchange forces are found to reduce the exchange forces by about 18% of its contribution, to the triton binding energy.

The scattering of heavy ions gives clear evidence of resonances which may be grouped in families like the rotational bands. The classical Breit-Wigner theory makes use of fixed poles which describe locally the resonances, but the global character of the rotational sequences is completely lost. Furthermore, the phenomenology shows that the rotational sequences evolve into surface waves. Again the classical Breit-Wigner theory, in view of its local character, cannot describe this evolution. In this paper we describe the resonances by the use of poles of the scattering amplitude in the complex angular-momentum plane: moving poles. However, in order to interpolate a sequence of resonances belonging to the same family we must add to the poles a term which takes into account the repulsive forces due to the Pauli principle and to the hard core. This term describes the downward crossing, through {pi}/2, of the phase shifts after each resonance. At higher energies the effect of the exchange forces tends to vanish and simultaneously the resonances evolve towards diffractive effects: we have the surface waves creeping around the target. This phenomenon is described in our theory by the moving poles as the imaginary part of the angular momentum increases for increasingmore » energy. Besides a detailed study of this theory we present here an extensive analysis of the {alpha}-{sup 40}Ca elastic scattering which gives clear phenomenological support to the model. {copyright} {ital 1997} {ital The American Physical Society}« less

Tang, Toennies and Yiu have shown that despite the inherent symmetry of H+2, wavefunctions obtained from a combination of the unsymmetrized polarization expansion and the 1βR expansion can be used in the Holstein—Herring formula to calculate for large internuclear distances R the leading O (e−R) terms in the exchange energy between the lowest pair of states. However, the associated claim by Tang and Toennies that the polarization expansion of the wavefunction converges not to the gerade molecular wavefunction, but to an asymmetric function localized about a single nucleus, conflicts with other numerical and analytical results. We show by a limiting procedure that use of the infinite polarization expansion for the wavefunction in the Holstein—Herring formula provides a result that is not equal to the exact exchange energy, although it has the correct leading O (e−R) behavior and is impressively close to the exact exchange energy for large R.

We use the dispersion-theoretic approach to the calculation of long-range forces to compute the potential ${V}_{2\ensuremath{\gamma}}$ arising from two-photon exchange between a charged or neutral spin-0 particle and a charged spin-\textonehalf{} particle. We show that, with a suitable choice of one-photon-exchange potential, the leading term at large distances of the spin-independent part of ${V}_{2\ensuremath{\gamma}}$ is the same as that found previously for two spin-0 particles. We obtain explicit results for the leading terms of the spin-dependent part of ${V}_{2\ensuremath{\gamma}}$.

We discuss a new interpretation of Heisenberg’s nuclear theory which leads to a slightly different Hamiltonian function. Accordingly we treat the nuclei statistically.

We describe the large-scale collective behavior of solutions of polar biofilaments and both stationary and mobile crosslinkers. Both mobile and stationary crosslinkers induce filament alignment promoting either polar or nematic order. In addition, mobile crosslinkers, such as clusters of motor proteins, exchange forces and torques among the filaments and render the homogeneous states unstable via filament bundling. We start from a Smoluchowski equation for rigid filaments in solutions, where pairwise crosslink-mediated interactions among the filaments yield translational and rotational currents. The large-scale properties of the system are described in terms of continuum equations for filament and motor densities, polarization and alignment tensor obtained by coarse-graining the Smoluchovski equation. The possible homogeneous and inhomogeneous states of the systems are obtained as stable solutions of the dynamical equations and are characterized in terms of experimentally accessible parameters. We make contact with work by other authors and show that our model allows for an estimate of the various parameters in the hydrodynamic equations in terms of physical properties of the crosslinkers., Comment: 47 pages, 12 figures

We propose a theory of the resonance-antiresonance scattering process which differs considerably from the classical one (the Breit-Wigner theory), which is commonly used in the phenomenological analysis. Here both resonances and antiresonances are described in terms of poles of the scattering amplitude: the resonances by poles in the first quadrant while the antiresonances by poles in the fourth quadrant of the complex angular momentum plane. The latter poles are produced by non-local potentials, which derive from the Pauli exchange forces acting among the nucleons or the quarks composing the colliding particles., 30 pages, 7 figures

The symmetry of exchange interaction of charge carriers in semiconductor nanostructures (quantum wells and quantum dots) is analyzed. It is shown that the exchange Hamiltonian of two particles belonging to the same energy band can be universally expressed via pseudospin operators of the particles. The relative strength of the anisotropic exchange interaction is shown to be independent of the binding energy and the isotropic exchange constant.

Aim of this work is to extend the results obtained in a previous study on the magnetic confinement and stability of a quantum degenerate non-neutral fermion plasma. This extension consists in the inclusion in the previously set up model of the effects of the exchange forces, and generalises the Thomas-Fermi (TF) approach used in the referenced work towards a Thomas-Fermi-Dirac (TFD) statistical description. The TF model has not only been used extensively and with success in these years to study atomic, nuclear and molecular properties, or to evaluate features of matter in extreme conditions such as low temperatures and/or high densities typical of astrophysics and inertial confinement fusion experiments, but also to found hydrodynamic theories for the diffusion and stability of fermion plasmas, one component non-neutral degenerate fluids, plasmas etc. In this paper an equation for density profiles in cylindrical symmetry is found, from the semiclassical kinetic theory of quantum gases, which takes into ac...

A vortex in a Bose-Einstein condensate generates the optical analog of the Aharonov-Bohm effect when illuminated with slow light. In contrast to the original Aharonov-Bohm effect the vortex will exchange forces with the light that lead to a measurable motion of the vortex., Comment: Replacement. New title and some changes to the text. Accepted by Physical Review A

We present an accurate ab initio description of the magnetic exchange force microscopy (MExFM). As a prototypical system, the antiferromagnetic NiO(001) surface probed with a Fe tip is investigated. The tip–surface interaction is described on two levels. Short-range chemical and exchange forces between the tip apex and the surface atoms are described in the framework of spin-polarized density functional theory while long-range van der Waals forces are considered within a mesoscopic tip model. For the Ni atoms in the NiO surface as well as the Fe atoms of the tip apex, an on-site repulsion U in the transition-metal 3d shells is included. In order to understand the tip–surface interaction, we investigate the changes in the electronic structure of tip and surface versus distance. The resulting frequency shifts and MExFM images are in good qualitative agreement with experimental data.

The dependence of the potential [ital V][sub 2[gamma]] arising from two-photon exchange between charged spinless systems on the one-photon exchange potential [ital V][sub 1[gamma]] is examined. It is found that, unlike the case when at least one of the two systems is neutral, the form of [ital V][sub 1[gamma]] can play a key role in determining the large-[ital r] behavior of [ital V][sub 2[gamma]]. It is shown that if [ital V][sub 1[gamma]] is defined in a way inspired by the use of the Coulomb gauge rather than the Feynman gauge, [ital V][sub 2[gamma]] falls off as [ital r][sup [minus]3], rather than as [ital r][sup [minus]2]. This result, which removes an apparent cognitive dissonance with the related work of Spruch, does not mean that the effective potential is inherently gauge dependent. Instead it highlights the fact that in this case the concept of the asymptotic behavior of [ital V][sub 2[gamma]] is not as sharp as one might have thought. The relation of this result to an analogous one expected to hold for spinor QED and to the nonrelativistic form of the orbit-orbit interaction is also discussed.

Novel types of ground states associated with properties of heavy fermion systems are derived for crystals with covalent bonds generated by short-range exchange forces between valence electrons of atoms localized at lattice sites. It is shown that the short-range exchange forces can give rise to a narrow energy band in which electrons can exhibit an enormous effective mass. The same exchange forces provide the microscopic mechanism for spin-singlet pairing of electrons into Cooper pairs which are responsible for superconductivity in these systems. This superconductivity exhibits several different anisotropic superconducting states. The effective mass, Fermi energy, specific heat, Pauli susceptibility, critical temperatures and critical magnetic field of heavy fermion systems are calculated and compared with experimental data.

The computation of exchange forces between molecules can be performed about 20 times faster by using the Hellmann-Feynman theorem than by the usual energy methods. The accuracy of the results for He2, He-H2, (H2O)2 and (C2H4)2 is disappointing, however, particularly in the region of the Van der Waals minimum.

A simple intuitive model is used to derive the effective nuclear many-body Hamiltonian for a Van Vleck enhanced (singlet ground state) system, in preference to the use of high-order perturbation theory. It is shown that in the case of either pure dipole-dipole or exchange interactions, the Hamiltonian reduces to a formally simple expression. However, for mixed dipole-dipole and exchange forces, the situation is much more complex. Some remarks are also made concerning the Hamiltonians used by Bleaney et al. (1982) in connection with the nuclear antiferromagnet HoVO4, and Kubota and coworkers (1980) in their analysis of the properties of the nuclear ferromagnet PrNi5.

Ground state binding energies and equilibrium harmonic oscillator energies are computed for the various ground state configurations of the doubly even nucleide44Ti. Employed for the study is a variational calculation carried out in a Cartesian harmonic oscillator basis of states employing a tripartite interaction potential composed of Wigner central forces coupled to Majorana exchange forces each having Gaussian radial dependence.

The anomalous change in the low-lying spectrum from /sup 105/Pd to /sup 103/Ru in the sequence of N-italic = 59 isotones is investigated. A possible attribution of this effect to the onset of a phenomenological exchange force in the interacting boson-fermion model is investigated.

Both ac and dc magnetic susceptibilities have been measured for Eu concentrations $x=0.025, 0.05, \mathrm{and} 0.10$ over temperatures in the range from 0.007 K to 4 K. While the dc susceptibility is found to exhibit no peak down to 7 mK, the ac susceptibility shows two distinct maxima around 15 mK and 200 mK. The frequency dependence of the two-peak positions ${T}_{a}(\ensuremath{\nu})$ and ${T}_{b}(\ensuremath{\nu})$ can be adequately described by an Arrhenius law in the range of our measurements from 10-${10}^{6}$ Hz. The results for the peak at the higher-temperature ${T}_{b}$ are explained quantitatively for $x=0.025$ and $x=0.05$, without adjustable parameters, by a model of small random Eu clusters. They consist of Eu atoms coupled by strong exchange forces between nearest-and next-nearest neighbors, and experience energy barriers mainly due to the intracluster dipolar energy. One particular configuration, two nearest-neighbor Eu atoms bonded ferromagnetically, produces the anomaly near ${T}_{b}$. Thus ${T}_{b}(\ensuremath{\nu})$ arises from ordinary superparamagnetic behavior. In spite of the very low temperatures investigated, quantum-mechanical cluster reorientation is shown to be negligible due to the high-spin quantum number of Eu, even for the smallest clusters. The newly found maximum at the lower-temperature ${T}_{a}$ is interpreted as a result of intercluster dipolar energy, giving good agreement for the order of magnitude of ${T}_{a}$ and its concentration dependence. Its dependence on frequency remains to be explained.

The method of computing matrix shifts, splittings, and sideband intensities of the vibrational fundamental bands in solid N2 and CO has been extended to the calculation of resonant and off‐resonant v–v transfer rates. The contribution of mechanical anharmonicity, i.e., vibron–phonon coupling through third‐ and higher‐order terms in the potential, is found to be negligible for phonon‐assisted transfer rates, similar to the situation in ir and Raman sideband intensities. Electrical anharmonicity, i.e., the modulation of vibron resonance energies by phonon‐type motions, produces off‐resonant v–v rates which are larger than those due to mechanical anharmonicity by two orders of magnitude. Analytical expressions have been derived for the quantitative determination of rates based on static multipole–multipole interactions, and contributions by repulsive exchange forces have been calculated numerically. The calculated rates are consistent with observations in a system of CO diluted in N2 and they explain the observed temperature behavior of endothermic v–v processes in pure solid N2 where two and more phonons are involved. It is found that molecular librations dominate over translational motions in all of these phonon‐assisted rates, even in the repulsive contributions.

Previous work relating the dispersion energy to the static form factors of X-ray scattering is extended to the polarization energy. For interacting helium atoms we calculate both those contributions (without recourse to the multipole expansion), as well as the first order Coulomb and exchange forces. For large separations (R \ 8.0 au), the shape of our potential agrees well with previous, accurate results. Since we use Unsold's approximation, we discuss several choices for the mean excitation energy, Δ E , which is required to put our potential on an absolute scale. A fixed Δ E suffices for large separations. In the region of overlap, a R -dependent Δ E ( R ) is needed for accurate determination of the potential.

We consider the long-range forces, i.e., those falling off as a power of the distance, which may act between pairs of particles, one of which is neutral and spinless. It is shown that these forces may easily be calculated from the discontinuity function in the momentum transfer of the scattering amplitude for the two particles. In particular, we have investigated the two-photon exchange force between two neutral, spinless systems, and the three-photon exchange force between a charged and a neutral, spinless system. In the former case, we find that the potential behaves as ${r}^{\ensuremath{-}7}$ for large $r$, in contradiction to the London expression for the Van der Waals force, and in agreement with the result of Casimir and Polder. For the latter case, the potential is odd under charge conjugation and hence can convert a ${K}_{2}$ meson to a ${K}_{1}$ meson. We find again that the potential behaves as ${r}^{\ensuremath{-}7}$. It is found that such long-range electromagnetic interactions are presently unobservable in particle physics.

Nuclear phase shifts are calculated for the scattering of two alpha particles, using the method of the resonating group structure, and assuming central Gaussian internucleonic potentials and Gaussian wave functions. It is found that a mixture of two-thirds Serber and one-third MHWB exchange force gives good agreement with the experimentally observed phase shifts.

1. Introduction .—If plane polarised light be transmitted through a medium under the influence of a magnetic field parallel to the direction of propagation, the plane of polarisation is in general rotated. This circumstance is known as the Faraday effect, after its discoverer. For light of a given wave-length, the magnitude of the rotation per unit distance is found to be proportional to the magnetisation. The direction of rotation varies with different substances, being termed diamagnetic, or positive, if in the direction of the current producing the field, and paramagnetic, or negative, if in the opposite sense. The sign of the effect does not depend upon whether the substance is dia- or paramagnetic; negative diamagnetics are, however, infrequent. Thin films of ferromagnetics exhibit an enormous negative rotation, proportional to the magnetisation, and in the following we shall attempt to explain the origin of this by using a very simple model for the substance. We shall consider a single crystal of the metal, and, following Heisenberg, we shall choose the Heitler-London model, where each electron is considered as being attached to an atom, as a first approximation. The interaction of the electrons gives rise to the well-known exchange forces. In a ferromagnetic these exchange forces are of vital importance, and it is therefore necessary to consider states and transitions of the crystal as a whole. Further, it is well known that in a ferromagnetic the average orbital angular momentum is zero. In view of this fact, and in the interest of simplicity, we shall consider a model where each atom possesses one electron in an s -state, outside a closed shell. This, of course, does not correspond to the facts, but any other model would be very much more difficult to handle. Having chosen a model with bound electrons, we shall have no conduction (without including polar states), and shall therefore not have the typical metallic absorption of light, such as is associated with “free” electrons. With the extremely rough model used, it is obvious that we can only expect to obtain the order of magnitude of the rotation, and some idea of its variation with the intensity of magnetisation.

A Bethe-Salpeter type equation is proposed for taking into account the effects of particle exchanges between the quarks in the relativistic quark model of Feynman, Kislinger and Ravndal (see abstr. A50918 of 1971). The equation is explicitly soluble in the ladder approximation for zero mass exchange and it leads to asymptotically dipole-like electromagnetic form factors.

The properties of the forces between alpha-particles are discussed on the basis of a Heitler-London analysis augmented by a "van der Waals" calculation. The results are in some respects unfavorable to the alpha-particle model of the nucleus inasmuch as they contradict the assumptions often made when this model is applied. Thus, for instance, it is pointed out that there are no attractive van der Waals forces beyond the range of the exchange forces, the range of the second-order forces being no greater than that of the first-order forces. The forces between alpha-particles are not additive. The range of the forces is approximately equal to that of the forces between elementary nuclear particles. The Heitler-London forces are repulsive if the usual choice of a symmetric Hamiltonian is made. The details of the forces are first worked out in connection with a simple fictitious example (\textsection{}1) in terms of which these various features are most easily discussed. Calculations for the alpha-particle, for which a $\ensuremath{\psi}$-function of the Gauss type is assumed, are made in \textsection{}\textsection{}2 and 3. Finally (\textsection{}4) the results are used in an examination of the problem of the scattering of alpha-particles in helium. It appears that the scattering of the $S$ waves can be explained only partially by means of the present interaction, its fault being the failure to produce resonance at the correct energy.

Based on the assumption that the low-lying states of 52V can be described by three protons in the f 7 2 shell and one neutron in the p 3 2 , p 1 2 and f 5 2 shells, the excitation energies and the corresponding eigenfunctions were calculated. The proton-neutron interaction has been chosen as the potential of an exchange force with a Gaussian radial shape. The range and the strength as well as the exchange character were treated as free parameters. The matrix elements of the proton-proton interaction were determined using the level scheme of the neighbouring nucleus 51V. The single-particle energies are also taken from experimental data. The results of the shell-model calculation are in fairly good agreement with the experimental data not only for the excitation energies but also for the spectroscopic factors of the 51V(d, p)52V reaction, the branching ratios for M1 transitions and the lifetime of a recently found 5+ isomeric state.

We formulate an unified model for the study of the giant resonance, which takes into account the individual and collective character of this phenomenon. We perform two unitary transformations on the Hamiltonian of the nucleus, introducing two new conjugate operators α and π, with a subsidiary condition on the eigenfunctions since we have increased the number of degrees of freedom. In the new Hamiltonian, there are : a collective part in terms of α and π an intrinsic part and interaction terms. This treatment based on the success of the Bohr and Mottelson phenomenological model seems to be successful : the theoretical calculations of the energy, width and cross section of the giant resonance, agree with the experimental data. The subsidiary condition plays an important role in the calculation of the width. We study also the effect of neutron-proton exchange forces and the angular distribution of the emitted particles.

The progress of research into the behavior of the positronium in molecular substances in the last several years is reviewed. The Ps atom can be treated as a solute in a liquid. The repulsive exchange force is the dominant force which reacts against the van der Waals attractive force. One of the most useful general relationships found is the one between the pick-off quenching rate and the surface tension of the liquid. The pick-off quenching rate is found, in general, to be an additive property of the functional groups of a molecule. The quenching of theo-Ps atom by an active quenching agent in an inert solvent has been found to be primarily diffusion controlled. Some deviations from the simple Arrhenius equation have been found. For gases, more theoretical work on the collision problems of Ps with atoms or molecules is needed. For amorphous molecular solids, it is likely that the property of pick-off quenching ofo-Ps is similar to that in liquids. However, in crystalline molecular solids, the situation is quite different. A few definite cases have been found where a Ps atom does not form or exist. Further work in this area should be very interesting. The foregoing leads us to believe that positronium or positron annihilation can be useful in the study of fundamental properties of molecular liquids and solids.

The nuclear spins in solid 3He are coupled to one another as a result of the Pauli exclusion principle. The exchange energy is so small that the Pauli principle can be treated rigourously as a boundary perturbation, and the coupling of the spins is given correctly by the Heisenberg Hamiltonian. The coupling is antiferromagnetic in sign if, as is to be expected, two-particle exchange predominates, but may be ferromagnetic if three-particle exchange is more important. Only if four or more neighbouring particles can change places simultaneously must a more complicated form of the Heisenberg Hamiltonian be used. Herring's formula for the exchange energy is generalized to include exchange induced transitions between nearly degenerate levels of the system (spin-lattice coupling). It is demonstrated that coupling between the spin system and the lattice can be calculated if the exchange frequency is known as a function of strain, and a rough argument is given to show how different components of the strain tensor may affect this frequency. Temperature dependence of the exchange frequency is discussed. It is shown that at moderate temperatures thermally excited vacancies produce a new mechanism for spin exchange, which may work towards parallel alignment of the spins in the body-centred phase, but towards anti-parallel alignment in the hexagonal phase.

A wide variety of theories exist for calculating fundamental molecular collision processes such as line broadening or energy transfer rates. All of these theories involve evaluation of a ``phase'' integral or Fourier transform of the intermolecular potential. In this paper we consider two major sources of error that can occur in the evaluation of phase integrals. ``Universal'' phase integrals calculated in a straight path model are compared with calculations based on trajectories determined by solution of classical orbit equations. Serious discrepancies are pointed out and practical computational alternative is suggested. A simple empirical model for exchange forces is introduced in calculations of linewidths and V–V transfer for hydrogen halides in a CO2 atmosphere. These calculations show that exchange forces can have a large effect but that the simple procedure of adding ``a short range'' contribution to a cross section calculated from attractive multipolar forces gives a totally incorrect picture.

The triaxial shapes of the first excited state of16O and of the ground states of24Mg and32S are studied in detail. Multipartite nucleon—nucleon interaction potentials, coupling Wigner central forces to Majorana exchange forces with both having Gaussian radial dependence, are employed in a variational calculation carried out in a Cartesian harmonic oscillator basis of states.

The present calculation of the density of energy levels of a heavy nucleus is based on the statistical model of Van Vleck. As in Bethe's calculation, the particles are assumed to move in a simple potential hole, but the depth of the hole varies with the velocity of the particle. If exchange forces act, the interaction energy of a given particle with the remainder of the nucleus decreases as the velocity of the particle increases. This results in a lower density of states of the individual particles at the top of the Fermi distribution. Bethe's formula for the density of excited levels of the nucleus as a whole may be applied to the present situation if this change in the density of the individual particle states is taken into account. The spacing between the levels is over a hundred times larger than that found by Bethe, and, if one uses the Gamow value for the radius of a radioactive nucleus (\ensuremath{\sim}9\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}13}$ cm), is much too large to be reconciled with the frequent occurrence of resonance levels for the capture of slow neutrons. If one uses the new value for the radius suggested by Bethe (\ensuremath{\sim}13\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}13}$ cm), the present theory gives more reasonable values.

The mathematical problems associated with the development of a satisfactory perturbation theory for exchange forces are discussed. Different treatments give different results for the second order energy. Many of the ideas expressed are the result of a series of seminars at the University of Wisconsin in which Jansen, Epstein, Byers Brown, Certain, Buckingham, Harriman, Micha, Levine and Eliason participated.

A calculation of the exchange force between ${\mathrm{Li}}^{+}$ and He has been made with the use of simple hydrogenic state functions to represent the charge distributions of the ion and atom. The force so obtained was added to that arising from the polarizability and the van der Waals interaction between ${\mathrm{Li}}^{+}$ and He. The result was then used to calculate the mobility of ${\mathrm{Li}}^{+}$ in He gas. The theoretical value thus found, 19.4 ${\mathrm{cm}}^{2}$/sec. volt at a temperature of 18\ifmmode^\circ\else\textdegree\fi{}C, is somewhat lower than the experimental one of 25.8 ${\mathrm{cm}}^{2}$/sec. volt, reported by Hoselitz.

A one‐dimensional Schrodinger equation with the potential U(x) = (1 / 8R2)(x2 − R2)2 is considered. Asymptotic expansions for eigenvalues Eg,u = E0 ∓ 12ΔE for large R are presented. It is shown that the quantity ΔE obtained by means of the LCAO approximation exhibits incorrect behavior in the limit of large R. Comparisons of asymptotic formulae with results of numerical calculation are given.

A Heitler-London treatment of the exchange part of the mutual energy of a neutral atom and a metal is developed. The resulting interactions are evaluated on the basis of a simplified model of the metal and lead to a convenient and simple expression for the total exchange energy. In Section III this expression is applied to the interaction of ${\mathrm{H}}_{2}$ and He with metals where it is found to represent a repulsion. By adding this exchange interaction to the attractive van der Waals interaction between the molecule and the metal, a potential curve of the usual type possessing a minimum is obtained for each system. The depths of these minima are compared with observed heats of van der Waals adsorption. A discussion is given in an appendix of the effects of some of the simplifications and approximations employed.

In the conventional perturbation treatment of the many-body problem of interacting particles, the zero-order Hamiltonian corresponds to independent particles moving in a static over-all potential. A discussion of the effects of particle interactions or `correlations' shows that if one starts instead with a Hamiltonian representing noninteracting particles in a velocity-dependent over-all potential, deeper for slow, and shallower for fast particles, then part of the correlation effect is included already in zero order. In addition, a velocity-dependent over-all potential may be called for by velocity dependent interparticle forces or by exchange forces. The degree of the improvement in the convergence of a perturbation expansion based on a Hamiltonian with a velocity-dependent over-all potential is discussed and illustrated by a simple example in which the velocity dependence of the potential gives rise to a reduced "effective mass" of the particles.The many-body problem of a large, uniform system of interacting particles (e.g., the case of a heavy nucleus without surface effects) is formulated in detail in perturbation theory, starting with a velocity-dependent potential constant in space. The work involved in such a calculation turns out to be essentially the same as with a velocity-independent potential, the effect of the velocity dependence being to reduce each term in the perturbation expansion by a constant factor raised to a power equal to the number of "energy denominators" ${E}_{m}\ensuremath{-}{E}_{n}$ in the term in question. A simple equation is deduced for the optimum degree of velocity dependence of the over-all potential which ensures the most rapid convergence.

By assuming the Coulomb forces between the ions of an electrolytic solution and by combing Poisson’s equation and Boltzmann’s principle, Debye and Hükel have developed a theory of strong electrolytes, satisfactory, however, only up to moderate concentrations. The approximate character of their method has been extensively investigated by a number of authors. Proceeding from the laws of statistics, Fowler showed that only for dilute solutions is Poisson’s equation valid in the Debye-Hükel treatment, where phenomena on a molecular scale are involved, the omitted fluctuation terms becoming appreciable at stronger concentrations. Although Halpern§ claims that at no concentration of physical interest is such a neglect permissible, we may assume from more recent work|| that Fowler’s original conclusions hold. Another objection is the use of the Colulomb law of force between two ions of charge ε i and ε j F ij = ε i ε j /D r ij 2 , (1.1) Which is not justifiable for small values of r ij . In water, for example, the main part of D is contributed by the orientation of the water dipoles, and in the vicinity of an ion, a saturation effect will exist changing the value of D considerably. In addition, the polarization, van der Waals and exchange forces between the ions come into play at their close approach.

A theory of the influence of exchange forces in liquids upon the nuclear hyperfine structure in paramagnetic ``spin resonance'' spectra is presented. Formulas are derived for two limiting cases: for the exchange J much larger than the hyperfine separation A, and for A≪J. It is found that as J increases the hyperfine lines broaden (exchange broadening) and start to shift towards the ``unsplit'' Zeeman frequency. For J≈A, the hyperfine lines have coalesced into a single broad line and for J>A the line narrows (exchange narrowing) about the ``unsplit'' Zeeman frequency. These calculations are discussed in considerable detail. The effect upon the spectrum of electronic‐electronic and electronic‐nuclear dipolar interactions and the effect of the rapid molecular motions in liquids have also been considered, as well as their interactions with the exchange forces.

The properties of the light 4n doubly even nucleides ranging from 4He to 40Ca are studied in a simple restricted Hartree-Fock calculation, employing an effective three-part two-body nucleon-nucleon interaction. This interaction consists of a trio of Wigner plus Majorana exchange forces each having Gaussian radial dependence. A rectangular harmonic oscillator basis of states is utilized. Two classes of parameter sets are studied: one generated from fitting the ground state energy and root-mean-square radius of 4He and ground state energy and density of nuclear matter, and one obtained from fitting the ground state energy and root-mean-square radius of both 4He and 40Ca. The resulting parameters are used to compute the ground state binding energies and harmonic oscillator energies of seven nucleides, 3Be, 12C, 20Ne, 14Mg, 28Si, 32S, and 36Ar, treated as being axially symmetric, with two possible configurations for all but 8Be and 12C considered. Moreover, 28Si is also analysed as a spherically symmetric nucleide. Additionally, the deformations of the axial nucleides are determined.

Scaling arguments and renormalization group techniques are used in the Anderson model Hamiltonian for the nonmagnetic limit of a magnetic impurity. The range of validity of the theory is limited toU/πΔ ≪ 1, where the electron-electron collisions can already compete strongly with the electron-hole collisions. A “parquet” approximation is developed; it gives for the static susceptibility in the symmetric case χ = (πΔ)−1exp (+U/πΔ). There are some hints that the exchange force might play a crucial role in the formation of localized magnetic moment. These considerations are consistent with the electron-hole symmetry in the symmetric Anderson model.

The various types of exchange forces that are being used in current discussions of nuclear structure may all be simply expressed in terms of a formalism which attributes five coordinates to each “heavy” particle and applies the Pauli exclusion principle to all the particles in the system. The simplest assumption for the interaction law is that which implies equality of proton-proton and neutron-neutron forces and also equality with the proton-neutron forces of corresponding symmetry. This is in accord with the empirical knowledge of these interactions at present.

A secular determinant for silver has been set up, in which the ion-ion force constants have been derived from known expressions appearing in the theory of metallic cohesion. The matrix elements have been represented as the sum of three terms. One of these arises from the Coulomb interaction of the ions and has been taken from Kellermann's work on sodium chloride. The second mainly owes its origin to the exchange forces but includes any effect due to the non-spherical nature of the Fermi surface. It has been calculated from the experimental values of the shear constants after subtracting the electrostatic contribution. The third term is due to the volume forces and has been evaluated by means of two dispersion relations given respectively by de Launay and Bhatia. For this purpose, the bulk modulus for the volume forces has been calculated by Dayal & Tripathi's method. Vibration frequencies and the specific heats have been calculated separately by the use of both the dispersion relations. Both the theoretical 0-T curves agree with the experimental curve at all temperatures within the accuracy of the calculations. The knowledge of the interaction between the constituent particles of a solid is essential for the rigorous discussion of its vibrational properties. In the case of the metals, the presence of the conduction electrons introduces complicating features which accounts for the widespread use of highly simplified force models in lattice dynamics. One starts with the assumption that the selected groups of atoms interact by elastic forces, which can be determined from experimental data on the elastic constants. These short-range forces, however, form a very crude representation of the actual interaction since the Coulomb interaction between the ions is long range in character and plays an important role in determining the nature of the vibration spectrum. Under the circumstances, it seems proper that some of the main features of the modern metallic theory are directly incorporated in the lattice dynamics of metals. According to the modern theories of metallic cohesion, the binding energy can be regarded as the sum of terms which, to a useful approximation, can be calculated separately. Jaswon (I954) has divided these terms into three broad categories: (i) the energy associated with the ion cores, (ii) the energy associated with the valence electrons, and (iii) the interaction between the valence and the ion core electrons. The first of these consists of four contributions arising respectively from (a) the self-energy of the ion core, (b) the Coulomb, (c) the exchange, and (d) the van der Waals interactions of the ion cores. The first one does not enter into the calculation of the vibration frequencies, while the van der Waals energy is usually negligible. The Coulomb contribution (b) is never calculated separately and is generally combined with the Coulomb self-potential and the potential energy of the valence electrons. In a recent work Dayal & Tripathi (i962a) have shown that the shear constants of copper and alkali metals are almost completely determined by the [ 212 ]

The familiar ƒ; sum rule for the electric dipole interaction of a complex charge-current system with the electromagnetic field is known to be simply applicable even when exchange forces and other non-classical dynamical effects are present. Evaluation of the ƒ; sum for absorption from the ground state can be made knowing only the Hamiltonian and the ground state wave function. For complex systems the exact wave function is rarely known. Two methods of estimating the ƒ; sum are then attractive whenever an independent particle model is empirically favored: (i) use of the best IPM wave functions and the true Hamiltonian, or (ii) use of the same wave functions but the equivalent one-body potential found to give best energy levels for the IPM. In the first calculation, dynamical correlations are neglected; in the second, a spurious contribution to the sum arises from the velocity dependent form typical of an equivalent one-body potential. We compute the ƒ; sum by both methods in a nuclear example. The results show the differences of principle which arise, though the numerical agreement is rather close. Physical arguments are given for the origin of this close agreement, which can be expected only in particular cases. No criterion for the reliability of such IPM sum rule calculations is known.