Physical Review A

Hyperspherical close-coupling study of hyperfine transitions in low-energy p+pμ and e±+Ps scattering

Akinori Igarashi, Isao Shimamura, Nobuyuki Toshima

Published Aug 1, 1998• 25 citations

Abstract

We apply the hyperspherical close-coupling method to the elastic and spin-flip processes in $p+p\ensuremath{\mu}$ and ${e}^{\ensuremath{-}}+\mathrm{Ps}$ (or equivalently, ${e}^{+}+\mathrm{Ps})$ scattering at low collision energies. The spin-spin interactions ${V}_{S}$ are included explicitly in the Hamiltonian, producing the correct hyperfine-splitting energy $\ensuremath{\Delta}\ensuremath{\epsilon}$ in the separated-atom limit. A comparison is made with a simpler method in which the effect of ${V}_{S}$ is represented only by the symmetry of the wave functions with respect to the interchange of the two identical particles. Explicit inclusion of ${V}_{S}$ is found to be of vital importance for the calculation of the spin-flip cross sections for collision energies lower than $\ensuremath{\Delta}\ensuremath{\epsilon}$ for both $p+p\ensuremath{\mu}$ and ${e}^{\ensuremath{-}}{(e}^{+})+\mathrm{Ps}$ collisions. The elastic cross sections are less sensitive to ${V}_{S},$ but those for $p+p\ensuremath{\mu}$ are still affected by ${V}_{S}$ for collision energies lower than $\ensuremath{\Delta}\ensuremath{\epsilon}$.

DOI: 10.1103/physreva.58.1166

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Muon transfer and elastic scattering int+dμ collisions at low energies

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Physical Review ADec 1, 199430 citations

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Physical Review A

Hyperspherical close-coupling study of hyperfine transitions in low-energy p+pμ and e±+Ps scattering

Akinori Igarashi, Isao Shimamura, Nobuyuki Toshima

Published Aug 1, 1998• 25 citations

Publisher

Abstract

We apply the hyperspherical close-coupling method to the elastic and spin-flip processes in

p+p\ensuremath{\mu}

and

{e}^{\ensuremath{-}}+\mathrm{Ps}

(or equivalently,

{e}^{+}+\mathrm{Ps})

scattering at low collision energies. The spin-spin interactions

{V}_{S}

are included explicitly in the Hamiltonian, producing the correct hyperfine-splitting energy

\ensuremath{\Delta}\ensuremath{\epsilon}

in the separated-atom limit. A comparison is made with a simpler method in which the effect of

{V}_{S}

is represented only by the symmetry of the wave functions with respect to the interchange of the two identical particles. Explicit inclusion of

{V}_{S}

is found to be of vital importance for the calculation of the spin-flip cross sections for collision energies lower than

\ensuremath{\Delta}\ensuremath{\epsilon}

for both

p+p\ensuremath{\mu}

and

{e}^{\ensuremath{-}}{(e}^{+})+\mathrm{Ps}

collisions. The elastic cross sections are less sensitive to

{V}_{S},

but those for

p+p\ensuremath{\mu}

are still affected by

{V}_{S}

for collision energies lower than

\ensuremath{\Delta}\ensuremath{\epsilon}

DOI: 10.1103/physreva.58.1166

Related papers

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We apply the hyperspherical close-coupling method to the elastic and spin-flip processes in

p+p\ensuremath{\mu}

and

{e}^{\ensuremath{-}}+\mathrm{Ps}

(or equivalently,

{e}^{+}+\mathrm{Ps})

scattering at low collision energies. The spin-spin interactions

{V}_{S}

are included explicitly in the Hamiltonian, producing the correct hyperfine-splitting energy

\ensuremath{\Delta}\ensuremath{\epsilon}

in the separated-atom limit. A comparison is made with a simpler method in which the effect of

{V}_{S}

is represented only by the symmetry of the wave functions with respect to the interchange of the two identical particles. Explicit inclusion of

{V}_{S}

is found to be of vital importance for the calculation of the spin-flip cross sections for collision energies lower than

\ensuremath{\Delta}\ensuremath{\epsilon}

for both

p+p\ensuremath{\mu}

and

{e}^{\ensuremath{-}}{(e}^{+})+\mathrm{Ps}

collisions. The elastic cross sections are less sensitive to

{V}_{S},

but those for

p+p\ensuremath{\mu}

are still affected by

{V}_{S}

for collision energies lower than

\ensuremath{\Delta}\ensuremath{\epsilon}

Low-energy elastic and inelastic scattering in the Ps(

1s

)-Ps(

2s

) channel is treated in a four-body hyperspherical coordinate calculation. Adiabatic potentials are calculated for triplet-triplet, singlet-singlet, and singlet-triplet spin symmetries in the spin representation of coupled electrons and coupled positrons, with total angular momentum

L=0

and parity equal to

+1

. The

s

-wave scattering lengths for the asymptotic Ps(

1s

)-Ps(

2s

) channel are calculated for each spin configuration. Results obtained for the

s

-wave scattering lengths are ${a}_{\mathrm{TT}}=7.3(2){a}_{0}\ensuremath{-}i0.02(1){a}_{0}, {a}_{\mathrm{SS}}=13.2(2){a}_{0}\ensuremath{-}i0.9(2){a}_{0}$, and

{a}_{\mathrm{ST}}=9.7(2){a}_{0}

for each spin configuration. Spin recoupling is implemented to extract the scattering lengths for collisions of Ps in different spin configurations through properly symmetrized unitary transformations. Calculations of experimentally relevant scattering lengths and cross sections are carried out for Ps atoms initially prepared in different uncoupled spin states.

1

mK to thermal temperatures. We show that spin-flipping transitions are induced by the splitting of the $b {}^{3}{\ensuremath{\Sigma}}_{u}$ Born-Oppenheimer potential via the long-range magnetic interactions among electrons. We find that the spin-flipping cross sections in the tritium dimer are about a magnitude larger than that predicted by mass scaling the H-H cross sections. For the former, we show that the spin-exchange cross sections are several magnitudes larger, at cold temperatures, than that of the hydrogen system. We compare the results of the multichannel approach with those obtained using approximate methods such as the degenerate internal-state, the elastic, and Born approximations and discuss their respective range of validity.

We investigate the elastic and inelastic scattering of positrons by sodium atoms in both the ground state, Na(

3s

), and excited states, Na*(

3p

4s

3d

), using the hyperspherical coordinate method with a model potential to represent the atomic core. The threshold behavior of positronium (Ps) formation cross sections is analyzed as the positron impact energy

E

approaches zero. Within this framework, we derive a generalized expression for partial-wave Ps-formation cross sections at low positron energies, which applies to both ground-state and excited-state sodium targets. Our results confirm that the total threshold behavior follows the expected power-law dependence:

σ_{\text{Ps}} \propto E^{-1/2}

for exothermic reactions and

σ_{\text{Ps}} \propto E^{a}

for endothermic reactions, where

a > 0

. Furthermore, we find that Ps-formation cross sections for positron scattering from excited Na states are significantly larger than those from the ground state in the low-energy region. A notable enhancement of Gailitis-Damburg oscillations is observed above the Ps(

n=2

) threshold, which may account for the increase observed in experimental data. Incorporating contributions from excited sodium targets improves agreement with experimental results and may help resolve discrepancies between theoretical predictions and measurements.