Physical Review C

Relation between the bound state radius and the scattering length: Relativistic effects

F. Masafumi Toyama, David J. Beachey, Yukihisa Nogami, W. van Dijk

Published Jul 1, 1991• 5 citations

Abstract

Relativistic effects on the relation between the root-mean-square radius ${\mathit{r}}_{\mathit{D}}$ of the bound state and the scattering length a are examined for models which simulate the triplet S state of the two-nucleon system. Two models are considered; one is based on the one-body Dirac equation and the other on the two-body Dirac equation. The relation is expressed as (${\mathit{r}}_{\mathit{D}}$/a${)}^{2}$=(${\mathit{c}}_{0}$+${\mathit{c}}_{1}$${\mathit{x}}^{1}$+${\mathit{c}}_{2}$${\mathit{x}}^{2}$+...)/8, where x==\ensuremath{\alpha}${\mathit{r}}_{0}$; \ensuremath{\alpha} is related to the binding energy and ${\mathit{r}}_{0}$ is the effective range. This expansion is compared with the corresponding expansion of the nonrelativistic Schr\"odinger case. The first three terms are shape independent. The ${\mathit{c}}_{0}$ (=1) and ${\mathit{c}}_{1}$ (=0) are the same as those of the nonrelativistic case but a relativistic correction appears in ${\mathit{c}}_{2}$.

DOI: 10.1103/physrevc.44.67

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Physical Review C

Relation between the bound state radius and the scattering length: Relativistic effects

F. Masafumi Toyama, David J. Beachey, Yukihisa Nogami, W. van Dijk

Published Jul 1, 1991• 5 citations

Publisher

Abstract

Relativistic effects on the relation between the root-mean-square radius

{\mathit{r}}_{\mathit{D}}

of the bound state and the scattering length a are examined for models which simulate the triplet S state of the two-nucleon system. Two models are considered; one is based on the one-body Dirac equation and the other on the two-body Dirac equation. The relation is expressed as (

{\mathit{r}}_{\mathit{D}}

{)}^{2}

{\mathit{c}}_{0}

{\mathit{c}}_{1}$${\mathit{x}}^{1}

{\mathit{c}}_{2}$${\mathit{x}}^{2}

+...)/8, where x==\ensuremath{\alpha}

{\mathit{r}}_{0}

; \ensuremath{\alpha} is related to the binding energy and

{\mathit{r}}_{0}

is the effective range. This expansion is compared with the corresponding expansion of the nonrelativistic Schr"odinger case. The first three terms are shape independent. The

{\mathit{c}}_{0}

(=1) and

{\mathit{c}}_{1}

(=0) are the same as those of the nonrelativistic case but a relativistic correction appears in

{\mathit{c}}_{2}

DOI: 10.1103/physrevc.44.67

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Feb 10, 19990 citations

Relativistic effects on the relation between the root-mean-square radius

{\mathit{r}}_{\mathit{D}}

{\mathit{r}}_{\mathit{D}}

{)}^{2}

{\mathit{c}}_{0}

{\mathit{c}}_{1}$${\mathit{x}}^{1}

{\mathit{c}}_{2}$${\mathit{x}}^{2}

+...)/8, where x==\ensuremath{\alpha}

{\mathit{r}}_{0}

; \ensuremath{\alpha} is related to the binding energy and

{\mathit{r}}_{0}

is the effective range. This expansion is compared with the corresponding expansion of the nonrelativistic Schr"odinger case. The first three terms are shape independent. The

{\mathit{c}}_{0}

(=1) and

{\mathit{c}}_{1}

(=0) are the same as those of the nonrelativistic case but a relativistic correction appears in

{\mathit{c}}_{2}

Relativistic and nonrelativistic models in one dimension which simulate the deuteron and the

^{3}\mathrm{S}_{1}

nucleon-nucleon (NN) scattering are constructed and the relation between the scattering length

{a}_{t}

and the root-mean-square radius

{r}_{m}

of the simulated deuteron is examined for a variety of interactions. The linear relation between

{a}_{t}

and

{r}_{m}

, which was found by Klarsfeld et al. for realistic NN potentials, holds for a wide class of potentials, relativistic as well as nonrelativistic. Within the limitation of our one-dimensional models, this suggests that the discrepancy between theory and experiment which was pointed out by Klarsfeld et al. regarding

{a}_{t}

and

{r}_{m}

cannot be resolved by relativistic effects.

\uparrow

and

\downarrow

particles. With respect to the relation between three-body loss rate and number of nearby triplets stated in [Petrov, Salomon and Shlyapnikov, PRL 93, 090404 (2004)], the present work adds a derivation, expresses the proportionality factor in terms of the three-body parameter, and includes the general case where there are several contributions to the three-body contact and several three-body parameters.

In the limit when the two-body scattering length

a

is negative and much larger than the effective two-body interaction radius the contribution to the ground state energy due to the three-body correlations is given by the Efimov effect. For particular values of the diluteness parameter

\rho |a|^3

the three-body contribution can become the dominant term of the energy density functional. Under these conditions both Bose-Einstein and Fermi-Dirac systems could become self-bound and either boson droplets or fermion ``designer nuclei'' of various sizes and densities could be manufactured.