recent ccc progress in atomic and molecular collision theory · 2020. 2. 28. · and molecular...

Introduction

Convergent close-coupling theory

Recent applications of CCC

Recent CCC progress in atomic and

molecular collision theory

I. Abdurakhmanov, J. Bailey, A. Bray∗, I. Bray, D. Fursa,

A. Kadyrov, C. Rawlins, J. Savage, and M. Zammit†

Curtin University, Perth, Western Australia∗Research School of Physics and Engineering, ANU

†Theoretical Division, Los Alamos National Laboratory, USA

Vapour Shielding CRP, IAEA, Vienna, Mar., 2019

Igor Bray <I.Bray@curtin.edu.au> Recent CCC progress in atomic and molecular collision theory

Introduction

Outline

1 Introduction

2 Convergent close-coupling theory

target structure and scattering

new approach to solving CCC equations

internal consistency

3 Recent applications of CCC

antihydrogen formation

positron and electron scattering on molecular hydrogen

heavy projectiles

Introduction

MotivationIntroduction

The primary motivation is to provide accurate atomicand molecular collision data for science and industry

Astrophysics

Fusion research

Lighting industry

Neutral antimatter formation

Medical: cancer imaging and therapy

Provide a rigorous foundation for collision theory with

long-ranged (Coulomb) potentials.

Introduction

MotivationIntroduction

The primary motivation is to provide accurate atomicand molecular collision data for science and industry

Astrophysics

Fusion research

Lighting industry

Neutral antimatter formation

Medical: cancer imaging and therapy

Provide a rigorous foundation for collision theory with

long-ranged (Coulomb) potentials.

Introduction

ChallengesIntroduction

Collisions between particles on the atomc scale are

difficult to calculate:

Governed by the Laws of Quantum Mechanics

Charged particles interact at large distances

Countably infinite discrete spectrum

Uncountably infinite target continuum

Can be multicentred (charge exchange, Ps-formation)

Introduction

History: computationalIntroduction

Prior to the 1990s theory and experiment generallydid not agree for:

electron-hydrogen excitation or ionization,

electron-helium excitation or (single) ionization,

single or double photoionization of helium.

The convergent close-coupling (CCC) theory for

electron, positron, photon, (anti)proton collisions with

atoms or molecules is applicable at all energies for

the major excitation and ionization processes.

Introduction

History: computationalIntroduction

Prior to the 1990s theory and experiment generallydid not agree for:

electron-hydrogen excitation or ionization,

electron-helium excitation or (single) ionization,

single or double photoionization of helium.

The convergent close-coupling (CCC) theory for

electron, positron, photon, (anti)proton collisions with

atoms or molecules is applicable at all energies for

the major excitation and ionization processes.

Introduction

History: formal theoryIntroduction

Prior to 2008, no satisfactory mathematicalformulation in the case of long-ranged (Coulomb)potentials for positive-energy scattering in

Two-body problems,

Three-body problems.

Surface integral approach to scattering theory is validfor short- and long-ranged potentials:

Kadyrov et al. Phys. Rev. Lett., 101, 230405 (2008),

Kadyrov et al. Annals of Physics, 324, 1516 (2009),

Bray et al. Physics Reports, 520, 135 (2012).

Introduction

History: formal theoryIntroduction

Prior to 2008, no satisfactory mathematicalformulation in the case of long-ranged (Coulomb)potentials for positive-energy scattering in

Two-body problems,

Three-body problems.

Surface integral approach to scattering theory is validfor short- and long-ranged potentials:

Kadyrov et al. Phys. Rev. Lett., 101, 230405 (2008),

Kadyrov et al. Annals of Physics, 324, 1516 (2009),

Bray et al. Physics Reports, 520, 135 (2012).

Introduction

Convergent close-coupling theorytarget structure

For complete Laguerre basis ξ(λ)nl (r), target states:

“one-electron” (H, Ps, Li,. . . ,Cs, H+2 )

φ(λ)nl (r) =

n′=1

nl ξ(λ)n′l (r),

“two-electron” (He, Be,. . . ,Hg, Ne, . . . Xe, H2, H2O)

φ(λ)nls (r1, r2) =

n′,n′′

Cn′n′′

nls ξ(λ)n′l ′(r1)ξ

(λ)n′′l ′′(r2),

Diagonalise the target (FCHF) Hamiltonian

〈φ(λ)f |HT|φ

(λ)i 〉 = ε

(λ)f δfi .

Introduction

φ(λ)nl (r) =

n′=1

nl ξ(λ)n′l (r),

φ(λ)nls (r1, r2) =

n′,n′′

Cn′n′′

(λ)n′′l ′′(r2),

〈φ(λ)f |HT|φ

(λ)i 〉 = ε

(λ)f δfi .

Introduction

φ(λ)nl (r) =

n′=1

nl ξ(λ)n′l (r),

φ(λ)nls (r1, r2) =

n′,n′′

Cn′n′′

(λ)n′′l ′′(r2),

〈φ(λ)f |HT|φ

(λ)i 〉 = ε

(λ)f δfi .

Introduction

e+-H energies for NℓH= Nℓ

Ps= 12 − ℓ, for ℓ ≤ 3

H(S) Ps(S) H(P) Ps(P) H(D) Ps(D) H(F)

-100H(S) Ps(S) H(P) Ps(P) H(D) Ps(D) H(F)

Energy levels (eV)

Introduction

two-center positron scattering

Positron-target wavefunction is expanded as

|Ψ(+)i 〉 ≈

ni〉+

NPs∑

n F Ps

ni 〉. (1)

Solve for Tfi ≡ 〈k fφf |V |Ψ(+)i 〉 at E = εi + ǫki

〈k fφf |T |φik i〉 = 〈k fφf |V |φik i〉

NT+NPs∑

d3k〈k fφf |V |φnk〉〈kφn|T |φik i〉

E + i0 − εn − k2/2. (2)

ill-conditioned, but unitary (no double counting).

Introduction

|Ψ(+)i 〉 ≈

ni〉+

NPs∑

n F Ps

ni 〉. (1)

NT+NPs∑

E + i0 − εn − k2/2. (2)

Introduction

|Ψ(+)i 〉 ≈

ni〉+

NPs∑

n F Ps

ni 〉. (1)

NT+NPs∑

E + i0 − εn − k2/2. (2)

Introduction

Use complete sets of states to isolate

′, r ′′) =∑

dkfL(kr ′)fL(kr ′′)

E + i0 − ǫn − εk

= −π

fL(knr<) (gL(knr>) + i fL(knr>)) . (3)

Eq. (2) becomes [A. Bray et al. CPC 212 55 (2017)]

NT+NPs∑

d3k〈k fφf |V′|φnk〉〈kφn|T |φik i〉. (4)

Works for kf = 0 for neutral and ionic targets.

Introduction

′, r ′′) =∑

E + i0 − ǫn − εk

= −π

NT+NPs∑

Introduction

′, r ′′) =∑

E + i0 − ǫn − εk

= −π

NT+NPs∑

Introduction

e-He+ 2s and 2p excitation

0.0000

0.0050

0.0100

0.0150

0.0200

0.0250

0.0300

0.0350

0.0400

10-4 10-3 10-2 10-1 10+0 10+1 10+2

cross section (a.u.)

singlet 2s

nGF aGF

0.0000

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

0.0007

0.0008

0.0009

0.0010

10-4 10-3 10-2 10-1 10+0 10+1 10+2

triplet 2s

nGF aGF

0.0000

0.0100

0.0200

0.0300

0.0400

0.0500

0.0600

10-4 10-3 10-2 10-1 10+0 10+1 10+2

electron energy above threshold (eV)

2p nGF aGF

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

10-4 10-3 10-2 10-1 10+0 10+1 10+2

2p nGF aGF

Introduction

Helium single and double photoionisation

σ1 Samson

aGF 0.00

σ2 nGF

Wehlitz

cross section (Mb)

photoelectron energy above threshold (eV)

σ3 nGF

Wehlitz0.000

Doerner

[A. Bray, A. Kheifets, I. Bray, PRA 95, 053405 (2017)]

[A. Kheifets, A. Bray, I. Bray, PRL 117, 143202 (2016)]

Introduction

In positron scattering there are two centres:1 target: discrete and continuous spectrum2 positronium: discrete and continuous spectrum

One-centre complete expansion:

Ps-formation is within ionization σ(1)ion

: e-loss

boundary condition problem in the extended Ore gap

Two-centre complete expansion:

explicit Ps-formation σ(2)Ps

and breakup σ(2)brk

: e-loss

ill-conditioned, but no double counting

Internal consistency:

above ionization threshold: σ(1)ion

= σ(2)Ps

+ σ(2)brk

below Ps-formation threshold: σ(1)ii = σ

Introduction

: e-loss

= σ(2)Ps

+ σ(2)brk

Introduction

: e-loss

= σ(2)Ps

+ σ(2)brk

Introduction

: e-loss

= σ(2)Ps

+ σ(2)brk

Introduction

one- and two-centre positron-hydrogen calculations

N N N Ps

ε=0 ε=0 ε=0

(2)σfi

(1) H(1) (2) (2)

Introduction

e+-H(1s) calculated with CCC(NH

lmax,NPs

lmax) for L = 0

0 20 40 60 80 100

cross section (

π a02)

projectile energy (eV)

totalCCC(309, 0)CCC(202,202)

0 20 40 60 80 100

Ps(continuum)

Ps(bound)

electron loss

[Bailey et al. Phys. Rev. A 91, 012712 (2015)]

Introduction

heavy projectiles

Ps(n ≤ 3) + p̄ → H̄ + e−

10-5 10-4 10-3 10-2 10-1 100 101

Ps energy ε (eV)

(anti)hydrogen formation CCC Ps(1s) CCC Ps(2s) CCC Ps(2p) CCC Ps(3s) CCC Ps(3p) CCC Ps(3d) UBA Ps(2s) UBA Ps(2p) UBA Ps(1s) Variational Ps(1s) Merrison 1997

[Kadyrov et al. Phys. Rev. Lett. 114, 183201 (2015)]

Introduction

heavy projectiles

Ps(n ≤ 5) + p̄ → H̄ + e−

10-5 10-4 10-3 10-2 10-1 10+0

Ps(n) energy(eV)

antiH formationPs(n

i = 1)

Ps(ni = 2)

Ps(ni = 3)

Ps(ni = 4)

Ps(ni = 5)

[Kadyrov et al. Nature Communications 8, 1544 (2017)]

Introduction

heavy projectiles

antihydrogen formation and elastic scattering

Ps(1s)+p̄ →

H̄(1s) + e−

Ps(1s) + p̄

10-5 10-4 10-3 10-2 10-1 10+0

Ps(1s) energy (eV)

Ps(1s)+p TT Ps(1s) H(1s)

[Fabrikant et al. Phys. Rev. A 94, 012701 (2016)]

Introduction

heavy projectiles

antihydrogen formation and elastic scattering

Ps(2s)+p̄ →

H̄(2s) + e−

Ps(2s) + p̄

10-5 10-4 10-3 10-2 10-1 10+0

Ps(2s) energy (eV)

Ps(2s)+p TT Ps(2s) H(2s)

[Fabrikant et al. Phys. Rev. A 94, 012701 (2016)]

Introduction

heavy projectiles

positron scattering on molecular hydrogen

e+-H2 collisions: total cross section

0.1 1 10 100 1000

ection (

units o

Incident Energy (units of eV)

GTCS X1 Σg

CCC lmax=8 N=556

Hoffman et al.Machacek et al. ang. corrected

Machacek et al.Charlton et al.Karwasz et al.

Zecca et al.

[Zammit et al. Phys. Rev. A 87, 020701 (2013)]

Introduction

heavy projectiles

e+-H2 collisions: internal consistency

10 100 1000

incident energy (eV)

CCC(142,21)

CCC(108,0)

Machacek et al.

Hoffman et al.

Charlton et al.

Karwasz et al.

Zecca et al.

10 100 1000cr

electron-loss CCC(142,21)

CCC(108,0)

Moxom et al.

Fromme et al.

[Utamuratov et al. Phys. Rev. A 92, 032707 (2015)]

Introduction

heavy projectiles

e+-H2 collisions: Ps-formation

10 100

Ps-formationCCC(141,1)

CCC(141,3)

CCC(142,3)

Biswas et al.

Zhou et al.

Fromme et al.

Machacek et al.

[Utamuratov et al. Phys. Rev. A 92, 032707 (2015)]

Introduction

heavy projectiles

electron scattering on molecular hydrogen

e−-H2 collisions: total cross section

0.1 1 10 100

ection (

units o

Incident Energy (eV)

CCCFerch et al.van Wingerden et al.Hoffman et al.Deuring et al.JonesSubramanian and KumarNickel et al.Zhou et al.

[Zammit et al. Phys. Rev. Lett. 116, 233201 (2016)]Igor Bray <I.Bray@curtin.edu.au> Recent CCC progress in atomic and molecular collision theory

Introduction

heavy projectiles

e−-H2 collisions: total ionization

50 100 150 200 250 300

ection (

units o

Incident Energy (eV)

CCCRMPS Gorfinkiel and TennysonTDCC Pindzola et al.Rapp and Englander-GoldenKrishnakumar and SrivastavaStraub et al.Lindsay and Mangan

[Zammit et al. Phys. Rev. Lett. 116, 233201 (2016)]Igor Bray <I.Bray@curtin.edu.au> Recent CCC progress in atomic and molecular collision theory

Introduction

heavy projectiles

e−-H2 collisions: b3Σ+u excitation

10 15 20 25

b 3Σu+

Nishimura and Danjo (1986) Khakoo and Segura (1994) MCCC (2017) experiment (2018)

[Zawadski et al. PRA 98, 050702R (2018)]

Introduction

heavy projectiles

proton scattering on hydrogen

p+-H collisions: internal consistency

projectile energy (keV)

experiment: e-loss QM-CCC(508, 0): e-loss QM-CCC: e-loss QM-CCC: ionization QM-CCC: e-capture

[Abdurakhmanov et al. J. Phys. B 49, 115203 (2016)]Igor Bray <I.Bray@curtin.edu.au> Recent CCC progress in atomic and molecular collision theory

Introduction

heavy projectiles

proton scattering on hydrogen

p+-H collisions: capture and ionization

tion (

electron capture

Winter Kolakowska QM-CCC

tion (

McClure Bayfield Wittkower Hvelplund

tion (

ionization

Toshima Winter Sidky Ovchinnikov Kolakowska QM-CCC

tion (

Shah 81 Shah 87 Kerby 95

[Abdurakhmanov et al. J. Phys. B 49, 115203 (2016)]

Introduction

heavy projectiles

antiproton scattering on molecular hydrogen

p−-H2 collisions: total ionization

1 10 100 1000

hr I Lu

..hr II

Lee I Lee II Ermolaev CCC

1 10 100 1000

Knudsen Hvelplund Andersen

[Abdurakhmanov et al. PRL 111, 173201 (2013)]Igor Bray <I.Bray@curtin.edu.au> Recent CCC progress in atomic and molecular collision theory

Introduction

heavy projectiles

antiproton scattering on molecular hydrogen

p−-H2 collisions: comparison with H and He

0.2 0.4 0.6 0.8 1

projectile velocity (a.u.)

Knudsen: H Hvelplund: He Knudsen: He Hvelplund: H2 Knudsen: H2

[Abdurakhmanov et al. PRL 111, 173201 (2013)]Igor Bray <I.Bray@curtin.edu.au> Recent CCC progress in atomic and molecular collision theory

Introduction

heavy projectiles

Concluding remarks

CCC method has been implemented for scattering of

electrons, positrons, photons, protons and

antiprotons on quasi one- and two-electron targets,

as well as inert gases.

Two-center problems have self-consistency checks

To-do list

Ps-H, Ps-He+, Ps-H+2

Ps-Ne+, and other inert gas ions

H-He+, H-H+2 , H-Ne+, and other inert gas ions

X2, H2O and other molecular targets

Introduction

heavy projectiles

Concluding remarks

CCC method has been implemented for scattering of

electrons, positrons, photons, protons and

antiprotons on quasi one- and two-electron targets,

as well as inert gases.

Two-center problems have self-consistency checks

To-do list

Ps-H, Ps-He+, Ps-H+2

Ps-Ne+, and other inert gas ions

H-He+, H-H+2 , H-Ne+, and other inert gas ions

X2, H2O and other molecular targets

recent ccc progress in atomic and molecular collision theory · 2020. 2. 28. · and molecular...

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