Intermediate Coupling Theory Research Articles

Tomonaga’s intermediate coupling theory using configuration space methods

With the intermediate coupling theory — using the configuration space methods of the quantum theory of fields — we determine the state vector characterizing the real nucleon. We carry out our calculations for the case of interaction of the nucleon field described by the Dirac equation and the scalar, resp. pseudoscalar meson field. Pair creation is completely disregarded. Remaining within the frameworks of the configuration space method the recoil of the nucleon is considered. With the aid of the state vector we also calculate the mean value of some characteristic physical quantitites. The use of the configuration space method — particularly in connection with the computation of local physical quantities — makes possible to form a very clear picture about the real nucleon.

Low Even Configurations in the First Spectrum of Rhenium (Re i)

The theory of intermediate coupling with configuration interaction is used to calculate energy levels and $g$ values in the $5{d}^{5}6{s}^{2}$ and $5{d}^{6}6s$ configurations of Re i. Fifty-one observed levels are fitted with a mean deviation of \ifmmode\pm\else\textpm\fi{}325 ${\mathrm{cm}}^{\ensuremath{-}1}$ by using eleven free parameters. The $L(L+1)$ correction is important, as the mean deviation is \ifmmode\pm\else\textpm\fi{}514 ${\mathrm{cm}}^{\ensuremath{-}1}$ without it; the parameter for this correction ($\ensuremath{\alpha}=66.8\ifmmode\pm\else\textpm\fi{}13$) has a value similar to that in first and second long-period spectra. Standard deviations of the parameters are calculated, and within the limits so defined, Slater's parameters are consistent with expectations based on known behavior in first and second long-period spectra. Calculated $g$ values agree with observed values for thirty-seven levels to about \ifmmode\pm\else\textpm\fi{}0.03. Levels not previously assigned from experiment are given designations; the eigenvectors are used to illustrate qualitative features of these designations.

Photomeson Production in Intermediate Coupling

The wave functions for the ground state of a physical nucleon and for the scattering state of a physical nucleon and a pion are used to compute photopion production cross sections in an intermediate-coupling approximation. Since the mesons included in the physical nucleon suffice to account for an approximately correct value of the anomalous magnetic moment, no use is made in this theory of a correction term involving experimental values of the latter. The $p$-wave part of the photoproduction matrix element is arrived at naturally, largely from the interaction of the meson currents with the electromagnetic field. Differential cross sections are obtained for the production of ${\ensuremath{\pi}}^{+}$ and ${\ensuremath{\pi}}^{0}$ by photons of 200, 260, 335, and 440 Mev in the laboratory system. It is also pointed out that in the intermediate-coupling theory, the ambiguity in defining a gauge-invariant Hamiltonian is nontrivial. Different procedures for generalizing the meson-nucleon interaction term into a form that is gauge invariant in an extended-source theory lead to substantially different $p$-wave components of the photoproduction cross sections.

Nucleon-antinucleon forces in the intermediate coupling theory

A semi-phenomenological approach to meson theory is used in order to calculate the nucleon-antinucleon complex interaction responsible for the scattering and annihilation of antinucleons in hydrogen. It is assumed that the main part of the imaginary potential is due to processes involving virtual annihilation into an arbitrary number of mesons, all of them but one being emitted or absorbed in a p-state. An effective Hamiltonian is derived for such processes, from which a static interaction can be calculated, the nucleon and antinucleon being treated as fixed sources. The intermediate coupling theory of Tomonaga is extended to the two-body problem and applied first to the calculation of ordinary nuclear forces and of the real part of the nucleon-antinucleon interaction. The annihilation Hamiltonian is then treated approximately by the intermediate coupling method. The imaginary part of the potential is found to have a range of the order of 2 to 3.10-3 cm, beyond which it becomes a weakly oscillating function of the nucleon-antinucleon distance. The physical meaning of this result is discussed, as well as the effects which would tend to sharpen the definition of the emitted pions multiplicity. In particular, a method is proposed to investigate, in the framework of the intermediate coupling theory, the influence of the excited states of the nucleon-antinucleon system.

Small Oscillation Theory of the Interaction of a Particle and Scalar Field

We study the interaction of a nonrelativistic particle with a scalar field, with particular application to the theory of polarons. The approach is based on a general classical method for the integration of equations of motion. The Hamiltonian is transformed by successive canonical transformations, the first corresponding to describing the motion relative to special solutions of the equations of motion. This stage as applied to suitably ordered Heisenberg equations of motion is identical with intermediate coupling theory. The second transformation treats the coupled small oscillations of particle and field oscillators about the chosen special solution. This affords a natural extension of intermediate coupling theory for this problem. Differences between the classical and quantum theories arise in the ordering of operators; the differences play a crucial role in determining the effective cutoff in wave vector space.

Double Meson Production in the Intermediate-Coupling Theory, I

The intermediate-coupling theory is applied to the problem of the meson production in meson­ nucleon collisions. For simplicity, the model used is a charged scalar meson field interacting with a fixed nucleon. The amplitudes of the ordinary scattering and the meson production are obtained in the The intermediate-coupling theory has been formulated to exhibit its validity in a region of moderate coupling strength. The essential point of this theory consists in the distinc­ tion of meson configurations between bound and unbound ones. Mesons in bound configura· tion are bound to the nucleon and make the so-called meson cloud around the nucleon. The various states of the meson cloud correspond to the various isobar levels of the nucleon. Mesons in unbound configurations, which are called s·mesons , move in the outer space. When a meson collides with the nucleon, there arise the resonance-like interactions between the incident s-meson and the meson cloud. Actually in this case the one-level is allowed; that is, only one nucleon level is excited in the course of the collision process. In the scattering problems hitherto treated the presence of more than one s-meson can be neglected. It seems that the one level approximation is closely connected with the neglect of two or more s·meson configurations and therefore the process of the double meson production cannot be treated in this approximation. 1l

On the Intermediate Coupling Theory of Pseudo-Scalar Meson Field and a Nucleon

On the Intermediate Coupling Theory of Pseudo-Scalar Meson Field and a Nucleon

Scattering Problem in the Intermediate-coupling Theory, I

The intermediatc meSons in this configuration are bound to nucleon and make, so. to speak, the clothes of Rudeon. The 01,lterl unbo1,lnd configurations QJl other hand/are

Application of the Intermediate Coupling Theory to the Scattering of Pseudoscalar Mesons by a Nucleon

Application of the Intermediate Coupling Theory to the Scattering of Pseudoscalar Mesons by a Nucleon

Magnetic Moment ofK40in Intermediate Coupling

The ${\mathrm{K}}^{40}$ nucleus is of interest since its spin $I=4$ forms a notable exception to Nordheim's rule while the observed value $\ensuremath{\mu}=\ensuremath{-}1.29{\ensuremath{\mu}}_{N}$ for the magnetic moment seems to favor the $j\ensuremath{-}j$ coupling. The theory of intermediate coupling is applied to the configuration ${d}^{\ensuremath{-}1}f$ with the view of accounting for the spin and the observed magnetic moment. It is found that a small spin-orbit interaction will lead to a negative magnetic moment. A central nucleon-nucleon interaction of the form $(mP+nQ)V({r}_{12})$, where $P$ denotes the Majorana and $Q$ the Bartlett operator, is assumed and calculations have been carried out for the exponential, Yukawa and Gaussian types of potential $V({r}_{12})$ with various "ranges." For a suitable choice of the spin-orbit interaction parameter $\ensuremath{\zeta}$, the observed magnetic moment can be obtained, the exact value of $\ensuremath{\zeta}$ depending on the type of potential and range used.

On the Many-body Problem in the Intermediate Coupling Theory, I

On the Many-body Problem in the Intermediate Coupling Theory, I

A Remark on the Strong Coupling Approximation in Meson-nucleon Scattering

Tomonaga’s intermediate coupling theory of the meson-nucleon scattering is applied to the case of the strong meson-nucleon coupling, using the simplified model of the charged vector longitudinal mesons interacting with a heavy nucleon. The remarkable feature in the strong coupling limit is that, besides the ordinary levels of the nucleon, its isobars and their excited states, there exist another kind of the excited levels (the “higher levels” defined in Sec. 2) which play an important role for the scattering problem. Our result agrees with the former conclusions of the Wentzel’s strong coupling treatment. Comparison with the case of the intermediate coupling region is discussed.

Scattering Problem in the Intermediate-coupling Theory, I

The intermediatc meSons in this configuration are bound to nucleon and make, so. to speak, the clothes of Rudeon. The 01,lterl unbo1,lnd configurations QJl other hand/are

Scattering Problem in the Intermediate-coupling Theory, I

Scattering Problem in the Intermediate-coupling Theory, I

Pair Creation in Intermediate Coupling Theory

The approximations of the intermediate coupling method are applied to a model of the self-field of a single nucleon (interacting with a pseudoscalar meson field) in which only processes involving pair creation and annihilation are allowed. In order to do this a method of subtracting the vacuum before these approximations are made, is suggested. The magnetic moment of the self-field on this model is investigated, and a comparison is made with the antithetic model of the self-field in which only processes not involving pair creation or annihilation are allowed.

Calculation of Nuclear Quadrupole Effects in Molecules

Methods are given for the computation of nuclear quadrupole interactions in molecules. They apply primarily to the interpretation of microwave spectra of linear and symmetric top molecules in which either one or two nuclei on the molecular axis show quadrupole coupling to the molecular field. Tables are given of the energy values, based on Casimir's formula, for the quadrupole coupling of a single nucleus, and also of transformation coefficients for application to the case of intermediate coupling when two nuclei are involved. Spectral intensities are discussed briefly. Examples are given to illustrate various aspects of the theory. The molecules BrCN and CICN show quadrupole effects resulting from the nitrogen nucleus as well as to the halogen. In the latter case, the intermediate coupling theory is required for a complete explanation of the observed microwave-absorption lines, as there are significant deviations from the first-order linear approximation.

The Coupling ofpElectron Configurations

The structure and type of coupling of the observed ${p}^{2}$, ${p}^{3}$, and ${p}^{4}$ configurations is considered in detail in terms of the intermediate-coupling theory, in particular for the long isoelectronic sequences which have recently been analyzed. The theoretical values of the ratios of various intervals in these configurations are plotted as functions of the single parameter which specifies the type of coupling. The experimental values have been fitted to these curves and the parameters evaluated. The points fit the theoretical curves fairly well; and the departures of the experimental points of an isoelectronic sequence from these curves are sufficiently regular to enable accurate prediction of unknown levels. This method has yielded new classifications in K IV and Ca VII. For the configurations ${p}^{3}$ and ${p}^{3}s$, interval ratios are found which are predicted to be entirely independent of coupling. The parameters are plotted for the isoelectronic sequences of $\mathrm{sp}$, $p$, ${p}^{2}$, ${p}^{3}$, ${p}^{4}$, ${p}^{5}$, and ${p}^{5}s$ configurations. The electrostatic interaction parameter ${F}_{2}$ is found to be a linear function of $Z$ to a good approximation; the spin-orbit parameter ${\ensuremath{\zeta}}_{p}$ is accurately proportional to ${(Z\ensuremath{-}S)}^{4}$ for all but the first few members of each sequence. The screening constants ($S$) for ${\ensuremath{\zeta}}_{p}$ are much smaller than the corresponding screening constants for ${F}_{2}$. A complete bibliography of data for those atoms with $p$ electrons in the normal configuration is appended.

The Electric Quadrupole Moment of the Nucleus

THE hyperfine structures of many terms in the arc, as well as in the first spark, spectrum of iodine show deviations from Landie's rule1. As Murakawa has pointed out, the cause of this may be an electric quadrupole moment of the nucleus. The existence of such nuclear quadrupole moments was shown for the first time by Prof. H. Schuler and me in europium and cassiopeium2. The exact theory was given by H. Casimir3. In the case of iodine the frequent occurrence of deviations leaves no doubt that they are due to the same cause. Because the electronic structure of the terms is very complicated, most of them are not suited for a determination of the quadrupole moment. The best term is 5p4 6s 4P5/2 of I I measured by Tolansky. Its eigenfunction is approximately the product of the 6s eigenfunction and the eigenfunction of the 5p4 3P2 term of I II. The latter can be determined from the coarse structure, using the theory of intermediate coupling. By calculations analogous to those for bismuth4 one gets from Casimir's formula and Tolansky's measurements a value of about 0.2 × 1024 for the nuclear electric quadrupole moment of iodine. That the sign is negative, one may infer also from the term 5p3 (2D) 6s 3D2 measured by Murakawa.

Hyperfine Structure in Intermediate Coupling

The theory is extended to intermediate coupling and to cosine laws of force between nucleus and electron varying with different powers of their distance apart. In addition to Goudsmit's constants ${a}^{\ensuremath{'}}$, ${a}^{\ensuremath{'}\ensuremath{'}}$, there appears a third coupling constant ${a}^{\ensuremath{'}\ensuremath{'}\ensuremath{'}}$ for each electron, which gives energy matrix elements for the same $l$ and different $j$. The theory is in approximate agreement with observation. The gross structure is used to determine the parameters of the theory of intermediate coupling and hence $\stackrel{-}{(\frac{1}{{r}^{3}})}$. This together with ${a}^{\ensuremath{'}\ensuremath{'}}$ gives nuclear magnetic moments having roughly the same value for different configurations and stages of ionization. The variation of $\frac{1}{{r}^{3}}$ as determined from gross structure shows changes in screening agreeing with those supposed to exist for ${(6s)}^{2}$ of Pb I, Tl I in order to explain the isotope shift as due to differences in nuclear radii. The discrepancies between gross structure and theory are shown to be connected with similar discrepancies in hfs and the discrepancies of the ratios $\frac{{a}^{\ensuremath{'}\ensuremath{'}}}{{a}^{\ensuremath{'}}}$ are reasonably explained by perturbations due to other configurations.