Effective Range Expansion Research Articles

K−pAmplitudes near theΛ(1520)

${\overline{K}}^{0}n$, $\ensuremath{\Lambda}{\ensuremath{\pi}}^{0}$, and ${\ensuremath{\Sigma}}^{0}{\ensuremath{\pi}}^{0}$ final-state cross sections and $\ensuremath{\Lambda}{\ensuremath{\pi}}^{0}$ and ${\ensuremath{\Sigma}}^{0}{\ensuremath{\pi}}^{0}$ polarizations are obtained in ${K}^{\ensuremath{-}}p$ interactions, for ${P}_{{K}^{\ensuremath{-}}}$ in the range 350-430 MeV/c, near the $\ensuremath{\Lambda}(1520)$ resonance. Two fits to the partial-wave amplitudes have been found, both assuming a standard Breit-Wigner amplitude for the $\ensuremath{\Lambda}(1520)$ resonance and a constant-scattering-length ${D}_{\frac{3}{2}}$ isospin-1 amplitude, but differing in their treatment of the $S$- and $P$-wave backgrounds. One method used a constant $K$ matrix, the other a momentum-dependent effective-range expansion for the inverse of the $K$ matrix. From the fitted partial waves, an upper limit of 10% is set for an isospin mixing between the $\ensuremath{\Lambda}{\ensuremath{\pi}}^{0}$ and ${\ensuremath{\Sigma}}^{0}{\ensuremath{\pi}}^{0}$ amplitudes.

Current Algebra, Unitarity, and the Scalar Meson

Current-algebra results for low-energy pion-pion scattering are combined with unitarity and an effective-range expansion to predict the width of a scalar meson and the $I=0$ $s$-wave phase shifts from threshold to 1 BeV.

On the anomalous effective range expansion for nucleon-deuteron scattering in the [formula omitted] state

The effective range expansion for the double nucleon-deuteron elastic scattering has a very small radius of convergence due to a pole just below threshold. It is shown in a potential model and by using analyticity arguments that the anomaly (a perturbed Ramsauer-Townsend effect) may be due to the long-range single-nucleon exchange force in the channel of the well-bound triton.

S-Wave Shape-Dependent Scattering Parameters of the Proton-Proton Interaction

The shape-dependent parameters $P$ and $Q$ of the effective-range expansion for the $S$-wave $p\ensuremath{-}p$ interaction have been obtained from experimental data between 0 and 10 MeV, including recent results at 6.141, 8.097, and 9.918 MeV. The preferred best values are $P=0.072\ifmmode\pm\else\textpm\fi{}0.005$ and $Q=0.034\ifmmode\pm\else\textpm\fi{}0.004$.

Analysis of Pion Photoproduction in the Second Resonance Region

A phenomenological analysis is made of the pion photoproduction data in the energy range $535<{E}_{\ensuremath{\gamma}}<850$ MeV. The analysis is based on the pion-nucleon phase-shift analyses using a generalized isobar model. The Watson theorem for the elastic pion-nucleon partial waves is imposed. $\ensuremath{\eta}$ photoproduction data in the same energy range is also analyzed using an effective range expansion of the $K$ matrix. The nucleon-isobar electromagnetic couplings are determined.

Interaction of Two Nucleons at Low Energies.I=1

A summary of the knowledge of the low-energy interaction of two nucleons with one unit of isobaric spin is presented. The single energy analyses of accurate low-energy proton-proton scattering data are discussed with emphasis upon the electromagnetic effects which have been included and those which should be. The phase shifts which result are presented and discussed in terms of the effective range expansion and potentials. A brief discussion of charge symmetry and independence in terms of potentials and dispersion relations is given. The equality of the neutron-neutron and proton-proton nuclear interactions, which seems to be established, may not be sufficient for understanding the problem of mirror nuclei. The question of charge independence is much more difficult.

The neutron-deuteron scattering lengths

New data on coherent and incoherent neutron-deuteron scattering permit re-evaluation of the doublet and quartet scattering lengths, with the results 4a = 6.14 ± 0.06 fm, and 2a = 0.11 ± 0.07 fm . These values imply σ free = 3.14 ± 0.06 b. A comparison has been made with the values obtained from extrapolations of the 4S and 2S phase shifts for n-d and p-d elastic scattering. Fot the n-d 2S phase shifts it is found that the effective range expansion has a pole at a negative energy ( E lab ≅ −0.1 MeV).

Effective range analysis of s- and d-wave α-α scattering

The new Q-value E r and width Γ of the 8Be ground state as determined in ref. 1) have been analysed with the low-energy α-α s-wave phase shifts δ 0 to determine the effective range expansion parameters of the phase shift δ 0. Use of these parameters gives better limits on Γ than the direct experimental ones quoted in ref. 1). The maximum of δ 0 as a function of the centre-of-momentum energy is calculated to be very close to 180° but does not exceed this value. The order of magnitude of δ 2 in this energy region is estimated on the basis on an effective-range analysis for l = 2; it is seen that δ 2 is negligible.

Analysis ofη0,X0→π+π−γwith a PossibleCViolation

The asymmetry of the ${\ensuremath{\pi}}^{+}$, ${\ensuremath{\pi}}^{\ensuremath{-}}$ energy spectra in ${\ensuremath{\eta}}^{0}$, ${X}^{0}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}\ensuremath{\gamma}$ decays is analyzed under the assumption of a possible $C$ violation in electromagnetic interactions. It is assumed that the $P$- and $D$-wave pion-pion phase shifts satisfy an effective-range expansion whose parameters are determined by the position and width of the ${\ensuremath{\eta}}^{0}$ and ${f}^{0}$ resonances. Within the validity of this assumption the upper limit for the asymmetry parameter in ${\ensuremath{\eta}}^{0}$ is found to be 1.1%, corresponding to a ratio ${R}_{\ensuremath{\eta}}=\frac{\ensuremath{\Gamma}({\ensuremath{\eta}}_{0}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{0}{\ensuremath{\pi}}^{0}\ensuremath{\gamma})}{\ensuremath{\Gamma}({\ensuremath{\eta}}^{0}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}\ensuremath{\gamma})}=0.25$; the upper limit for the asymmetry parameter in ${X}^{0}$ is 18%, corresponding to a ratio $\ensuremath{\Gamma}({X}^{0}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}\ensuremath{\gamma})=0.25$. For any given values of the decay radius and the ratio of the $C$-violating to $C$-conserving coupling constants, the ${\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ asymmetry is much larger in ${X}^{0}$ than in ${\ensuremath{\eta}}^{0}$ decay; hence the notion of maximal $C$ violation in electromagnetic processes can be suitably tested in this decay. This is, moreover, a more sensitive test than a search for the neutral mode ${X}^{0}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{0}{\ensuremath{\pi}}^{0}\ensuremath{\gamma}$, since the ${\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ asymmetry can be as large as 10% for low values of ${R}_{\mathrm{X}}=0.04$.

High-Energy Limit ofπ+π0Total Cross Section

The high-energy limit of the ${\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{0}$ total cross section $\ensuremath{\sigma}(\ensuremath{\infty})$ is estimated upon the assumption that the ${\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{0}$ forward scattering amplitude is dominated by the $S$ wave below the $\ensuremath{\rho}$ resonance and becomes purely imaginary fairly rapidly above the $\ensuremath{\rho}$ resonance. The underlying assumption is that the ${\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{0}$ forward amplitude satisfies the usual analyticity assumption. Using the experimental values of the mass and the width of the $\ensuremath{\rho}$ resonance and also assuming an effective-range expansion for the $S$ wave, we estimate $\ensuremath{\sigma}(\ensuremath{\infty})$ as a function of the scattering length and the effective range in the channel of total isospin 2. It is found that, for negative scattering length (corresponding to a repulsive interaction), $\ensuremath{\sigma}(\ensuremath{\infty})$ rises fairly steeply as the scattering length increases its magnitude for all the effective ranges. In the case of positive scattering lengeth (corresponding to an attractive interaction), however, $\ensuremath{\sigma}(\ensuremath{\infty})$ is much less dependent on the scattering length up to $0.8{\ensuremath{\mu}}^{\ensuremath{-}1}$ especially when the effective range lies between ${\ensuremath{\mu}}^{\ensuremath{-}1}$ and $2{\ensuremath{\mu}}^{\ensuremath{-}1}$ (${\ensuremath{\mu}}^{\ensuremath{-}1}$ being the pion Compton wave-length). In particular, $\ensuremath{\sigma}(\ensuremath{\infty})$ hardly decreases below 20 mb as long as the effective range is not too much greater than $2{\ensuremath{\mu}}^{\ensuremath{-}1}$. It is argued that various corrections to be considered can quite likely increase the above estimate of $\ensuremath{\sigma}(\ensuremath{\infty})$ 10% to 20% in the case of a weakly attractive $S$-wave interaction, but hardly can decrease it. It is therefore felt that the present analysis excludes the possibility of a repulsive $S$-wave interaction and also implies that the high-energy limit of the ${\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{0}$ total cross section is about the same as that for the pion-nucleon total cross section.

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.

Analysis ofp−pData Near the Interference Minimum

In the first part of this paper $p\ensuremath{-}p$ elastic scattering data at 90\ifmmode^\circ\else\textdegree\fi{}c.m. near the Coulomb-nuclear interference minimum are analyzed. The energy of the minimum of the 90\ifmmode^\circ\else\textdegree\fi{} cross section is found to be at 0.38243 \ifmmode\pm\else\textpm\fi{}0.00020 MeV. The nuclear $s$-wave phase shift, defined with respect to wave functions which solve the Coulomb plus vacuum polarization potential problem, is found to be ${{\ensuremath{\delta}}_{0}}^{E}=0.25501\ifmmode\pm\else\textpm\fi{}0.00020$, at the precise energy 0.38243 MeV. In the second part of the paper the phase shift just obtained is used, in conjunction with four very accurate phase shifts from 1.4 to 3.0 MeV, to determine the parameters of the $s$-wave effective-range expansion. If this expansion is cut off after three terms (quadratic fit) then the scattering length is found to be $a=\ensuremath{-}7.815\ifmmode\pm\else\textpm\fi{}0.008$ F; the effective range ${r}_{0}=2.795\ifmmode\pm\else\textpm\fi{}0.025$ F; and the shape-dependent parameter $P=0.028\ifmmode\pm\else\textpm\fi{}0.014$. However, it is argued that the first three coefficients in the actual power-series expansion of the effective-range function are not known this well, and in particular, $P$ may be very different from the actual coefficient occurring in the ${k}^{4}$ term.

Neutron-Neutron Scattering at Low Energies

This paper investigates the information contained in a neutron-neutron scattering experiment at low energies which could be performed by colliding beams coming from an underground nuclear explosion. The significance of such an experiment is discussed from the point of view of a check on charge symmetry and charge independence, and it is found that because of the electromagnetic complications in proton-proton scattering, and because of the proton-neutron mass difference, the knowledge of neutron-neutron scattering would be of considerable value. The functional form of the experimental data which is most convenient for analysis and the approximate relative magnitude of the terms is investigated, and it is concluded that for the kind of experiment which is envisaged (measuring cross sections to 10% from 20 keV to 2 MeV) only two parameters should be kept in the effective-range expansion. The connection between the number and distribution of energies at which the cross section is measured and the error on the individual measurements, on the one hand, and the accuracy of the effective-range parameters deduced from the experiments, on the other, is given explicitly and is found also to depend on the absolute magnitude of the scattering length. The results show that ten 10% measurements, suitably distributed between 20 keV and 2 MeV, can determine the sign of the scattering length to four standard deviations, the magnitude of the effective range to 50-70%, and the magnitude of the scattering length to about 3%. Finally, the relationship between the variation of the effective-range parameters and the corresponding variation in the parameters of the scattering potential is studied, and it is found that, while this relationship is strongly shape-dependent, a small change in the potential parameters, in any case, results in a large change in the scattering length, but a small one in the effective range. Numerical relationships show that, even in the worst case, the variation in the scattering length is about eight times the variation in the potential parameter. It is concluded that a 10% experiment at 20 energies between 20 keV and 2 MeV would be able to get information on the potential parameters sufficiently accurately so that charge-dependent or charge-symmetry violating effects could be detected.

Proton-Proton Scattering Near the Interference Minimum and the Shape Parameter

At a laboratory proton energy of 382.43 keV, $p\ensuremath{-}p$ scattering exhibits nearly complete destructive interference, and the $s$-wave phase shift may be determined from the location, ${E}_{min}$, of the minimum in the cross section for 90${\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}_{\mathrm{c}.\mathrm{m}.}$ scattering, with comparable precision. A determination of the phase shift at ${E}_{min}$ to one part in ${10}^{3}$ may be combined with recent Wisconsin results between 1.4 and 3 MeV to give information about the shape-dependent term in the effective-range expansion. Measurements have been made of the relative cross section at six energies near that of the minimum, using a scattering chamber with annular slits of 90\ifmmode^\circ\else\textdegree\fi{} vertex angle and a group of coincident pairs of silicon particle detectors. The energies of the data points were established by a radio-frequency, time-of-flight absolute velocity gauge, with an uncertainty of less than 50 eV. The scattering chamber was differentially pumped upstream and downstream. The operating pressure was 0.3 Torr, and the upstream energy loss of about 150 eV was measured directly by noting the offset of the ${\mathrm{F}}^{19}(p, \ensuremath{\alpha}\ensuremath{\gamma}){\mathrm{O}}^{16}$ resonance when hydrogen was present in the chamber between the velocity gauge and a differentially pumped C${\mathrm{F}}_{4}$ gas target. The minimum was relocated at ${E}_{min}=382.43\ifmmode\pm\else\textpm\fi{}0.20$ keV, and the corresponding $s$-wave phase shift with respect to the "electric" wave functions including vacuum polarization is found to be ${{\ensuremath{\delta}}_{0}}^{E}=0.25501\ifmmode\pm\else\textpm\fi{}0.00020$ rad. The shap-dependent parameter inferred from these results is significantly positive, $P=+0.028\ifmmode\pm\else\textpm\fi{}0.014$. This may be considered strong evidence in favor of one-pion-exchange effects in the $s$-wave interaction.

K-nucleon interactions and Y 1 * -resonance

A theoretical method is given for distinguishing between Watson’s two sets of zero effective-range -KN scattering parameter solutions, called solutionA and solutionB, which fit the K--p data quite well in the K--laboratory momentum range of (350/450) MeV/c. These two sets of solutions, which are quite similar in both the s- and d-waves but differ considerably in p-wave, are combined with the parameters, obtained by requiring Y1*-resonance at {dy1385} MeV with width 50 MeV as an I=1, p3÷2-wave, bound state of the -K-nucleon system, so as to give two new sets of energy-dependent -KN scattering parameters in I=1, p3÷2-state. It is found that the zero effective-range expansion forp-wnre -KN phase shift and the scattering parameters of Watson’s solutionB are inconsistent with the requirements of causality and of the positive deflniteness of transition probabilities.

A potential model representation of two-nucleon data below 315 MeV

An energy independent nucleon-nucleon potential model is described. The model represents the two-nucleon data below 315 MeV more faithfully than any other potential model known to date. The data include the effective range expansion parameters and the deuteron properties as well as the scattering data at higher energies. It is of special significance that the n-p data have been represented by the energy independent potential model in a satisfactory manner for the first time. For this purpose a quadratic LS potential was found essential in the T = 0 state. The T = 1 potential differs little from those previously proposed by several authors. The phase shifts predicted by the model are in fair agreement with the solutions YLAM ( T = 1) and YLAN3M ( T = 0) recently found by the Yale group.

Hard Core and Shape Parameter in the Effective Range Expansion

The shape parameter (P) in the effective range expansion for the /sup 1/ S/sub 0/ nuclear scattering phase shift is considered. Purely phenomenological nuclear potentials require that P be negative for cases in which the potential has a hard core radius>O.3 fermi; this requirement is in probable disagreement with the experimental data. A potential is shown, however, with a hard core radius of 0.4760 fermi, for which P is small but definitely positive. This singlet even parity potential is of the form (one-pion-exchange potential + short- range singular phenomenological potential). (T.F.H.)

Invalidity of the Raphael Analysis for Moderate EnergyNNScattering

Raphael's modified effective-range expansion for nucleon-nucleon scattering is usually truncated at the second term. It is explicitly shown that the resulting series is not a good representation of the predictions of reasonable potential models for moderate-energy nucleon-nucleon scattering. In particular, it can lead to the wrong sign for the shape parameter.

Proton-Proton Effective-Range Theory with Vacuum Polarization

The effect of vacuum polarization upon $p\ensuremath{-}p$ scattering is considered by first solving the problem in the Coulomb plus vacuum polarization potentials (called potential) without the nuclear potential. The nuclear phase shifts are then defined with respect to the electric wave functions, and the scattering cross section is written in terms of these phase shifts. The connection with other nuclear phase shifts (in the presence of vacuum polarization) which appear in the literature is established.The effective-range expansion for the nuclear $s$-wave phase shift is derived. An analysis of three low-energy $p\ensuremath{-}p$ scattering experiments indicates that the omission of vacuum polarization from the analysis results in a value for the shape-dependent parameter which is 0.02 smaller than the value obtained when vacuum polarization is included in the analysis. A discussion of the accuracy required to usefully delimit this parameter is included.

Tertiary and general-order collisions (II)

The basis introduced in a previous paper for describing collisions involving more than two free particles in ingoing or outgoing channels is discussed in more detail. With this formalism channel wavefunctions for many-particle channels are formally identical to the usual two-particle wavefunctions and require no special treatment. The scattering matrix referring to open channels is unitary and symmetric when many-particle channels are included; its symmetries in the complex k-plane are also discussed. Long-range effects in many-particle channels are considered, and it is shown that these can modify the threshold behaviour of matrix elements given previously. Effective range expansions, valid for many-particle channels, are given for the reactance matrix and eigenphaseshifts.