Effective Range Theory Research Articles

Coulombic Modified Effective-Range Theory for Long-Range Effective Potentials

A modified effective-range theory (MERT) was introduced previously to describe the scattering of a charged particle by a neutral polarizable system. The long-range components of the resultant effective one-body (generalized optical-model) potential, which cause the phase shift to have a rapidly varying energy dependence, are taken account of exactly by solving a one-body problem numerically. The short-range potential components generate only a slowly varying energy-dependent effect on the phase shift, and this effect can be accounted for by a few terms in a power series in ${k}^{2}$. The procedure is here extended to the case for which the polarizable system is itself charged. An MERT expansion is derived for the difference $\ensuremath{\delta}(k)$ between the total phase shift $\ensuremath{\eta}(k)$ and the phase shift $\ensuremath{\rho}(k)$ due to the long-range tail alone; both $\ensuremath{\eta}(k)$ and $\ensuremath{\rho}(k)$ are defined relative to pure Coulomb scattering. With the strongly energy-dependent Coulombic and other long-range effects accounted for exactly by the numerical solution of a one-body scattering problem, low-energy scattering data can be matched by the proper choice of the coefficients of just the first few terms in the MERT expansion; these terms will then determine the scattering in the (experimentally inaccessible) energy domain extending down to zero energy. For a repulsive Coulomb field, the leading term in $\ensuremath{\eta}(k)$ is determined exactly for all $L$ by the Born approximation; for $V(r)=(\frac{2\ensuremath{\mu}}{{\ensuremath{\hbar}}^{2}})(\ensuremath{-}\frac{{\ensuremath{\beta}}^{2}}{{r}^{4}})$, where $\ensuremath{\alpha}=\frac{{\ensuremath{\beta}}^{2}{\ensuremath{\hbar}}^{2}}{|\ensuremath{\mu}{e}^{2}{Z}_{1}{Z}_{2}}|={\ensuremath{\beta}}^{2}{a}_{0}$ is the electric-dipole polarizability of the target, $tan\ensuremath{\eta}(k)=\frac{{\ensuremath{\beta}}^{2}{{a}_{0}}^{3}{k}^{5}}{15}$ for $k{a}_{0}L\ensuremath{\ll}1$.

Multichannel Effective-Range Theory from theNDFormalism

An effective-range theory for systems of many coupled two-body channels is given using the $\frac{N}{D}$ formalism. The effective-range expansion is carried out in the amplitudes ${M}_{\mathrm{ij}}$ (where $M$ is essentially the matrix ${T}^{\ensuremath{-}1}$ with the right-hand cut removed). Quite in analogy with the single-channel effective-range theory, the diagonal elements ${M}_{\mathrm{ii}}$ are given by an expression quadratic in ${k}_{i}$, the relative momentum in channel $i$. The effective ranges ${R}_{\mathrm{ii}}$ are given by certain principal-value integrals which depend on the position of the left-hand singularities in the corresponding channels and can be taken to be energy-independent to the same extent as in the one-channel theory. The nondiagonal elements ${M}_{\mathrm{ij}}$, in general, have a weak energy dependence and can approximately be treated as constants. A two-channel computer experiment is performed to test these proposals in detail. Three different situations for the left-hand cut are considered: (i) a set of monopoles, (ii) a set of dipoles, and (iii) the left-hand cut produced by the exchange of scalar particles in the crossed $t$ reactions. For a large number of situations considered, the simple features proposed for the multichannel effective range theory were found to exist. The above formalism is similar to the multichannel effective-range theory of Ross and Shaw in the potential model.

Extrapolation of Cross-Section Data to Zero Energy for Long-Range Effective Potentials

The effective one-body potential representing a particle-compound-system interaction usually has a very complicated short-range behavior, but a relatively simple, and more or less known, long-range tail $U$ which can be nonlocal and energy-dependent. With $U$ present, the usual effective-range-theory (ERT) expansion for a single-channel scattering phase shift $\ensuremath{\eta}$ is inapplicable. Recent extensions of ERT include the effect of long-range potentials, but since $\ensuremath{\eta}$ then becomes very energy-dependent, one would need many terms to obtain the desired accuracy, so the results are of restricted use. We show that one can first solve, numerically, a one-body problem which includes only $U$, and then use a modified ERT expansion for the difference $\ensuremath{\delta}$ between $\ensuremath{\eta}$ and the phase shift due to $U$ alone. Since the rapid energy dependence due to $U$ is thus extracted out (exactly), terms in the expansion which depend upon the short-range potential in the presence of $U$ are more slowly varying in energy; they involve a number of adjustable parameters to be determined experimentally, but only a few are usually needed. The procedure thus makes possible the extrapolation of low-energy scattering data down to zero energy, even when effective long-range potentials are present.

Momentum-Transfer Cross Sections for Slow Electrons in He, Ar, Kr, and Xe from Transport Coefficients

Momentum-transfer cross sections for electrons in He, Ar, Kr, and Xe are obtained from a comparison of theoretical and experimental values of the drift velocities and of the ratio of the diffusion coefficient to the mobility coefficient for electrons in these gases. The theoretical transport coefficients are obtained by calculating accurate electron-energy distribution functions for energies below excitation using an assumed energy-dependent momentum-transfer cross section. The resulting theoretical values are compared with the available experimental data and adjustments made in the assumed cross sections until good agreement is obtained. The final momentum cross sections for helium is 5.0\ifmmode\pm\else\textpm\fi{}0.1\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}16}$ ${\mathrm{cm}}^{2}$ for an electron energy of 5\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}3}$ eV and rises to 6.6\ifmmode\pm\else\textpm\fi{}0.3\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}16}$ ${\mathrm{cm}}^{2}$ for energies near 1 eV. The cross sections obtained for Ar, Kr, and Xe decrease from 6\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}16}$, 2.6\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}15}$, and ${10}^{\ensuremath{-}14}$ ${\mathrm{cm}}^{2}$, respectively, at 0.01 eV to minimum values of 1.5\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}17}$ ${\mathrm{cm}}^{2}$ at 0.3 eV for Ar, 5\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}17}$ ${\mathrm{cm}}^{2}$ at 0.65 eV for Kr, and 1.2\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}16}$ ${\mathrm{cm}}^{2}$ at 0.6 eV for Xe. The agreement of the very-low-energy results with the effective-range theory of electron scattering is good.

Empirical Analysis of theH−Photodetachment Cross Section

The zero-range photoelectric cross section of Bethe and Peierls, corrected according to effective range theory, is fitted to the ${\mathrm{H}}^{\ensuremath{-}}$ photodetachment cross section. The value of $2.64 {a}_{0}$ obtained for the effective range agrees with values obtained from the wave-function calculations of Schwartz, Burke and Schey, and others. A more accurate formula for the detachment cross section is obtained by allowing for distortion in the final $p$ state by means of an approximate formula for the phase shift. This formula agrees qualitatively with the close-coupling theory phase-shift results of Burke and Schey, and provides a photodetachment cross section that fits the experimental data slightly better than the results re cently obtained by Geltman with an elaborate variational treatment. There are two adjustable parameters: the binding energy, and one parameter in the phase-shift formula. By a slightly different choice of the parameters than is required to fit the experimental data, close agreement can also be obtained with the Geltman results.

Total neutron cross section of carbon

An accurate determination of neutron total cross section of carbon has been made by the transmission method in the energy region 3 keV to 660 keV. No structure is observed in the total cross section over the whole energy region, and it is concluded that the upper limit of the width of the proposed level at 610 keV neutron energy must be 10 eV in order to escape detection in the present experiment. The observed total cross section has been analysed in terms of the effective range theory as well as the resonance theory, and the results are found to be in good agreement. The relative reduced width for the bound level at −1.86 MeV (i.e. the 3.09 MeV 2 s 1 2 state in C 13) is found to be θ 2 = 0.2, and the interaction radius is found to be 4.7 fm.

Quantum Defects and Scattering Lengths

The evaluation of quantum defects and scattering lengths is discussed and formulae relating these quantities are derived using first-order perturbation theory and effective range theory.

Application of the Multichannel Effective Range Theory to Electron-Hydrogen Scattering

The behaviour of cross sections in the neighbourhood of the 2s excitation threshold is investigated using M matrix theory. Calculations allowing for 1s-2s strong coupling are carried out for zero angular momentum in singlet, triplet and no-exchange cases. Threshold effects in all cases have the form of a cusp. A narrow resonance has been found below the threshold in the singlet case.

Effective range theory and velocity dependent potentials

For three different types of the radial dependence of the potential-square well, exponential and Yukawa potential — the shape independent potential parameters of a velocity dependent potential are evaluated as a function of thesstrength parameter of the non-static part of the potential.

Effective range theory and the low energy hyperon-nucleon interactions

The effective range theory is treated in the case of coupled channels where allowance is made for channels with higher l-waves. Also it is shown how to take the Coulomb interaction into account. A shape-independent relation connecting the volume integral of the potential and the scattering length is derived. Applying this to the analyses of the light hyperfragments leads to the estimates of the Λ-nucleon scattering lengths: a 0=− 3.6 +3.6 −1.8 f for S=0 and a 1=−(0.53±0.12)f for S=1 The low energy Σ −-proton reactions are considered. It is shown how one can take account of the kinematical effects due to the mass differences between the members of the same isomultiplet. We have calculated the effective range parameters for three different potentials corresponding to global ( g NN π = g ΣΣπ = g ΛΣπ ) and antiglobal ( g NNπ = − g ΣΣπ = − g ΛΣπ ) symmetry. These are then applied to the Σ −-proton reactions. It is shown that the global symmetry assumption is consistent with the Λ-nucleon scattering lengths, the ratio r R = Σ 0 (Σ 0 + Λ 0) obtained in the Σ −-proton reactions and with the data in the K − absorption in deuterium. Also it is shown that the same data rule out the antiglobal symmetry assumption.

Effective range theory and the low energy hyperon-nucleon interactions: J. J. de Swart and C. Dullemond. Enrico Fermi Institute for Nuclear Studies and Department of Physics, University of Chicago, Chicago, Illinois and Department of Physics and Astronomy, University of Rochester, Rochester, New York

Effective range theory and the low energy hyperon-nucleon interactions: J. J. de Swart and C. Dullemond. Enrico Fermi Institute for Nuclear Studies and Department of Physics, University of Chicago, Chicago, Illinois and Department of Physics and Astronomy, University of Rochester, Rochester, New York

On the Relation between Phase Shift and Quantum Defect

The effective range theory due to Bethe for the elastic scattering of electrons by a potential behaving asymptotically like a Coulomb field is employed to derive an exact relation between the phase shift and the quantum defect. For zero energy electrons this relation reduces to the formula obtained by Seaton.

Effective Range Theory of Multichannel Reactions in the Chain Approximation Method

The effective range theory for two- and three-channel reactions are investigated in the chain approximation method. Explicit formulas are derived for the cases in which there are two scalar particles with equal masses in the S-state, or one baryon and one meson either in the S- or P-state, in each channel. In particular the formulas are obtained by which the effective ranges for coupled systems are given in terms of the scattering lengths of coupled systems and the effective ranges for isolated channels. These formulas are expected to be tested by experiments rather unambiguously, since some of scattering lengths are known and the effective ranges for isolated channels, if defined appropriately, are determined only kinematically, independent of the strengths of interactions.

Low-Energy Scattering of a Charged Particle by a Neutral Polarizable System

In the scattering of a particle (or system) of charge $\mathrm{Ze}$ by a neutral system with an electrical polarizability $\ensuremath{\alpha}$, an electric dipole moment is induced which generates an effective potential that behaves asymptotically as $\ensuremath{-}\frac{\frac{1}{2}{Z}^{2}{e}^{2}\ensuremath{\alpha}}{{r}^{4}}$. Due to this effective long-range interaction, effective-range theory in its normal form is not applicable. Thus, for the scattering of a particle with an incident orbital angular momentum of zero, for example, the expansion of $kcot\ensuremath{\eta}(0)$ includes terms in $k$ and in ${k}^{2}\mathrm{ln}k$, in addition to the usual constant and ${k}^{2}$ terms. The effective range ${r}_{0}$ as normally defined is infinite, but one can define a quantity ${r}_{p0}$ which explicitly takes into account the long-range character of the effective potential. For $L>0$ it is ${k}^{2}cot\ensuremath{\eta}(L)$ which approaches a constant as $k$ approaches zero rather than ${k}^{2L+1}cot\ensuremath{\eta}(L)$ as for a short-range potential. The above results can have serious consequences in the scattering of electrons and of positrons by neutral spherically symmetric atoms. Some detailed consideration is given to the scattering by hydrogen atoms. The formulation of effective-range theory which is given is valid when Pauli exchange between the two colliding systems is possible. The method used for taking into account the effect of the Pauli principle (this method would be the same for long-range and short-range forces) is rather more convenient than in the usual presentation of effective range theory.

One-pion exchange and the optical model

Starting with the two-body scattering amplitude, an optical model potential for nucleon-nucleus scattering has been calculated (1) using the one-pion exchange interaction alone, (2) modifying the s-wave part of this by using effective range theory, and (3) using the modified phase shift analysis for p-p scattering supplemented by charge independence and the Gammel-Thaler potential for the n-p scattering information. Part (1) works sufficiently well to suggest that the one-pion exchange force contributes more importantly to the nucleon-nucleus interaction than to the basic two-nucleon scattering process. Part (2) produces an indifferent improvement on part (1). Part (3) represents the optical potential inferred from the most accurate two-nucleon data presently available. This last evaluation of the optical potential is considered the most reliable to date since it includes, in a good approximation, an infinite number of the high angular momentum states of the two-nucleon amplitude, which play a very essential part in the optical potential. A systematic derivation of the optical model equations is included with special emphasis on the needs of the present calculation; in addition, criteria for the validity of the model are discussed. Allowance has been made for the nuclear volume not being strictly proportional to the mass number (nuclear size effect) and for the inequality of the numbers of neutrons and protons in the nucleus (effect of neutron excess). For heavy nuclei, these effects influence the results up to 25%.

Low energy K −-p scattering and K̄N bound states

Assuming that the recently discovered Y ∗ (m Y ∗ ≈ 1385 MeV, 1 2 Γ ≈ 15 MeV) is a K ̄ N(I = 1) bound system, we formulate an effective range theory for K̄N scattering in the I = 1 state. With a large effective range for the I = 1 state and a zero effective range for the I = 0 state, we find parameters which fit the present low energy K −-p scattering data and give constructive Coulomb-nuclear interference. Our parameters indicate that the real part of the scattering length in the I = 0 state (i.e. a 0) is small. We can explain this smallness of a 0 by postulating the existence of a K ̄ N(I = 0) bound state lying below the Σπ threshold. We also expect a Σπ resonance ( I = 0) just below the K̄N threshold ( δ Σπ = 90°).

Scattering Matrix Approach to theΛ−πResonance

The pi LAMBDA resonance is studied in the energy range from m pi + m LAMBDA to mN + mK, using an effective-range theory. The effects of the three channels pi LAMBDA , pi LAMBDA , and KN are considered, taking I = 1 and assuming odd relative A SIGMA and AK parities. The scattering matrix is used to study the resonance properties of the system, and it is found that a resonance can exist, with (I,J) = (1,1/2/sup -/), that is consistent with experimental data. SIGMA LAMBDA even parity does not yield this (1,1/2-) resonance. (T.F.H.)

Multichannel effective range theory

Effective range theory is developed for systems of many coupled two-body channels with angular momenta l i . Derivatives of the amplitudes M ij (where M is essentially the inverse of the K matrix) are formed. Quite in analogy with one channel effective range theory, the diagonal elements M ii are accurately given, by an expression quadratic in the momentum k i . The coefficients R ii of k i 2 are effective range type integrals which are interpretable in terms of the range of forces and can be taken to be energy independent to the same extent as in the one channel theory. The non-diagonal elements M ij are, to a good approximation, energy independent, even for R ii greatly different from R jj and l i ≠ l j . The case of two coupled channels is studied in detail: a computer experiment was performed to test the validity of the theory; for l = 0, the properties of narrow resonances including the interactions which can lead to them are thoroughly investigated. The positions of the poles of the T matrix are briefly considered. Comparison is made between the effective range type of parametrization and Breit-Wigner theory. The present theory is contrasted to the effective range theory for the eigenphase shifts; the eigenphase shift theory is shown, in principle, to be less accurate. Some possible applications are briefly discussed.

On the distinguishability of equivalent potentials

We explore the possibility of discriminating between equivalent potentials (i.e., potentials reproducing the same phase shifts and bound state energies for the two-body problem) by making use of three-body amplitudes. Working with a simple model of a two-body target, and making approximations characteristic of a high-energy treatment, we find that the three-body amplitude can be decomposed into two parts; one sensitive to the nature of the two-body force, the other expressible in terms of two-body observables. When the two-body interaction terminates exactly beyond a certain distance a, the first part arises entirely from the overlap of the target particles if they are regarded as spheres of radius a. By consideration of effective range theory we have also given a brief discussion of potentials with infinitely long tails. For such interactions the second part of the composite amplitude is always zero. In order to avoid extensive digressions into the problem of potential construction (for potentials belonging to assigned classes) we have restricted specific discussion to the two classes for which this problem has been solved, namely, local and nonlocal separable potentials.

Some tests for compound models of elementary particles

Experiemntal means are suggested for testing the validity of some compound models of elementary particles. Asymmetry parameters of decay prcducts of hyperons are predictable from decay characteristics of the constituent particles. The effective range theory offers a second ape binding energy of a compound particle and the scattering lengths, effective ranges, and S-wave phase shifts of the constituent particles. A third method involves requiring that charge se it is possible to derive selection rules for K/sup +/, K-bar/sup 0/, K/ sup -/, K/sup 0/, K sK/sup -/, or K/sup 0/K-bar/sup 0/ annihilation into pions. These selection rules may be used in testing some compount models. Examples ofof given, and it is remarked that simular arguments may be used for baryons. (T.F.H.)