The convergent close-coupling method for electron-atom scattering · 2009-11-11 · Introduction Nonrelativistic CCC theory Comparison with experiment The convergent close-coupling

IntroductionNonrelativistic CCC theory

Comparison with experiment

The convergent close-coupling methodfor electron-atom scattering

I. Bray, D. V. Fursa, A. S. Kadyrov, A. T. Stelbovics

ARC Centre for Antimatter-Matter Studies, Curtin University of Technology, Perth,Western Australia

IAEA, Vienna, November, 2009

Igor Bray <[email protected]> CCC method for electron-atom scattering






Outline

1 IntroductionMotivationDistant historyRecent history

2 Nonrelativistic CCC theoryatomic target structureelectron-atom scatteringconvergence studies

3 Comparison with experimentelectron-impact excitationelectron-impact ionisationproton-impact on hydrogen




MotivationDistant historyRecent history

IntroductionMotivation

The primary motivation is to provide accurate atomiccollision data for science and industry:

AstrophysicsFusion researchLighting industryMedical and materials applications

Provide a rigorous foundation for collision theory withlong-ranged (Coulomb) potentials.





IntroductionMotivation

The primary motivation is to provide accurate atomiccollision data for science and industry:

AstrophysicsFusion researchLighting industryMedical and materials applications

Provide a rigorous foundation for collision theory withlong-ranged (Coulomb) potentials.





IntroductionDistant history

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

electron-hydrogen excitation or ionisation,electron-helium excitation or ionisation,single or double photoionisation of helium.

Consequently, we have been developing theconvergent close-coupling (CCC) theory forelectron/positron/photon/(anti)proton collisions withatoms/ions/molecules that is applicable at allenergies for the major excitation and ionisationprocesses.





IntroductionDistant history

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

electron-hydrogen excitation or ionisation,electron-helium excitation or ionisation,single or double photoionisation of helium.

Consequently, we have been developing theconvergent close-coupling (CCC) theory forelectron/positron/photon/(anti)proton collisions withatoms/ions/molecules that is applicable at allenergies for the major excitation and ionisationprocesses.





IntroductionRecent history

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

Two-body problemsThree-body problems

Consequently, have developed a surface integralapproach to scattering theory that is valid for short-and long-range potentials, see Kadyrov et al. PRL,101, 230405 (2008) and Annals of Physics 324 1516(2009).





IntroductionRecent history

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

Two-body problemsThree-body problems

Consequently, have developed a surface integralapproach to scattering theory that is valid for short-and long-range potentials, see Kadyrov et al. PRL,101, 230405 (2008) and Annals of Physics 324 1516(2009).




atomic target structureelectron-atom scatteringconvergence studies

Nonrelativistic CCC theorytarget structure and scattering

Structure: using the Laguerre basis ξ(λ)nl (r) write:

“one-electron” (H, Li, . . . , Cs) states:φ

(λ)nl (r) =

∑

n′

Cn′

nl ξ(λ)n′l (r)

“two-electron” (He, Be, . . . , Hg) states:φ

(λ)nls (r1, r2) =

∑

n′,n′′

Cn′n′′

nls ξ(λ)n′l ′(r1)ξ

(λ)n′′l ′′(r2).

Coefficients C are obtained by diagonalising thetarget (FCHF) Hamiltonian

〈φ(λ)f |HT|φ

(λ)i 〉 = ε

(λ)f δfi . (1)





Hydrogen ℓ = 0 energies for λ = 1 Laguerre bases

0.01

0.1

1

10

100

1000energy (eV)

-0.1

-1

-10

-100∞25201510

Laguerre basis size N





φ(λ)ε (r) for N = 70, λ = 2 and φ(R0)

ε (r) for R0 = 134.

-1.0

-0.5

0.0

0.5

1.0

0 40 80 120 160

φ ε(r)

radius (a.u.)

l=0 ε=1eVCCC-BCCC-L





Scattering:Electron-atom wavefunction is expanded as

|Ψ(+)i 〉 ≈ A

N∑

n=1

|φ(λ)n 〉〈φ

(λ)n |ψ

(+)i 〉 ≡ AIN |ψ

(+)i 〉. (2)

Solve for Tfi ≡ 〈k fφ(λ)f |V |Ψ

(+)i 〉 at E = ε

(λ)i + ǫki

,

〈k fφ(λ)f |T |φ

(λ)i k i〉 = 〈k fφ

(λ)f |V |φ

(λ)i k i〉

+

N∑

n=1

∑

∫

d3k〈k fφ

(λ)f |V |φ

(λ)n k〉〈kφ(λ)

n |T |φ(λ)i k i〉

E + i0 − ε(λ)n − ǫk

. (3)

Have step-function behaviour:


(λ)i k i〉 ≈ 0 for ε(λ)

f > ǫkf.






|Ψ(+)i 〉 ≈ A

N∑

n=1

|φ(λ)n 〉〈φ

(λ)n |ψ

(+)i 〉 ≡ AIN |ψ

(+)i 〉. (2)


(+)i 〉 at E = ε

(λ)i + ǫki

,



(λ)f |V |φ

(λ)i k i〉

+

N∑

n=1

∑

∫

d3k〈k fφ

(λ)f |V |φ



E + i0 − ε(λ)n − ǫk

. (3)




f > ǫkf.






|Ψ(+)i 〉 ≈ A

N∑

n=1

|φ(λ)n 〉〈φ

(λ)n |ψ

(+)i 〉 ≡ AIN |ψ

(+)i 〉. (2)


(+)i 〉 at E = ε

(λ)i + ǫki

,



(λ)f |V |φ

(λ)i k i〉

+

N∑

n=1

∑

∫

d3k〈k fφ

(λ)f |V |φ



E + i0 − ε(λ)n − ǫk

. (3)




f > ǫkf.





Define scattering amplitude ffi via

〈Φ(−)f |

←

H − E |Ψ(+)i 〉 ≈ 〈k fφ

(−)f |IN(

←

H − E)AIN|ψ(+)i 〉

= 〈φ(−)f |φ

(λ)f 〉〈k fφ

(λ)f |T |φ

(λ)i k i〉,

where εf = ε(λ)f ensured 〈φ

(−)f |IN = 〈φ

(−)f |φ

(λ)f 〉〈φ

(λ)f |.

Discrete excitation[

εf < 0, 〈φ(−)f |φ

(λ)f 〉 = 1

]

:

f (N)fi (k f ) = 〈k fφ

(λ)f |T |φ

(λ)i k i〉,

Ionisation[

εf = q2f /2, with 〈φ

(−)f | ≡ 〈q(−)

f |]

:

f (N)i (qf ,k f ) = 〈q(−)

f |φ(λ)f 〉〈k fφ

(λ)f |T |φ

(λ)i k i〉,






〈Φ(−)f |

←

H − E |Ψ(+)i 〉 ≈ 〈k fφ

(−)f |IN(

←


= 〈φ(−)f |φ

(λ)f 〉〈k fφ

(λ)f |T |φ

(λ)i k i〉,


(−)f |IN = 〈φ

(−)f |φ

(λ)f 〉〈φ

(λ)f |.


εf < 0, 〈φ(−)f |φ

(λ)f 〉 = 1

]

:


(λ)f |T |φ

(λ)i k i〉,

Ionisation[


(−)f | ≡ 〈q(−)

f |]

:

f (N)i (qf ,k f ) = 〈q(−)


(λ)f |T |φ

(λ)i k i〉,






〈Φ(−)f |

←

H − E |Ψ(+)i 〉 ≈ 〈k fφ

(−)f |IN(

←


= 〈φ(−)f |φ

(λ)f 〉〈k fφ

(λ)f |T |φ

(λ)i k i〉,


(−)f |IN = 〈φ

(−)f |φ

(λ)f 〉〈φ

(λ)f |.


εf < 0, 〈φ(−)f |φ

(λ)f 〉 = 1

]

:


(λ)f |T |φ

(λ)i k i〉,

Ionisation[


(−)f | ≡ 〈q(−)

f |]

:

f (N)i (qf ,k f ) = 〈q(−)


(λ)f |T |φ

(λ)i k i〉,





Nonrelativistic CCC theoryconvergence studies

e−-H excitation and total ionisation (S-wave model)

0.00

0.10

0.20

0.30

0.40

0.50

0.60

10 20 30 40 50 60 70 80 90 100

1s Poet 1978 CCC(30) CCC(10) CCC(5)

0.00

0.01

0.02

0.03

0.04

0.05

0.06

10 20 30 40 50 60 70 80 90 100

cross section (

πa02)

projectile energy (eV)

2s

0.00

0.01

10 20 30 40 50 60 70 80 90 100

3s

0.00

0.01

0.02

0.03

0.04

10 20 30 40 50 60 70 80 90 100

ionization






electron-impact excitation at 3 Ry

0.0001

0.001

0.01

0.1

1

10

-0.001-0.01-0.1-1

cross section (a02)

triplet CCC(10) CCC(20) CCC(30)

0.001

0.01

0.1

1

-0.001-0.01-0.1-1

energy (Ry)

singlet CCC(10) CCC(20) CCC(30)






electron-impact ionisationdσi

de(e) at 3 Ry

0.00

0.01

0.02

0.03

0.0 0.5 1.0 1.5 2.0

cross section (a02/Ry)

triplet CCC(10) CCC(20) CCC(30)

0.00

0.02

0.04

0.06

0.08

0.10

0 0.5 1 1.5 2

energy (Ry)

singlet CCC(10) CCC(20) CCC(30)

CCC(∞)






Can estimate true SDCS from σioni and

dσi

de(E/2)

0.000

0.005

0.010

0.015

0.020

0.0 1.0 2.0 3.0

cros

s se

ctio

n (π

a 02 /Ry)

secondary energy ε (Ry)

ε < k2/2 ε > k2/2

singlet FDM CCC estimate




electron-impact excitationelectron-impact ionisationproton-impact on hydrogen

Comparison with experimentelectron-impact excitation

54.4 eV e-H 2p angular correlation parameters

e-p+

-e

e-

photon







-1.0

-0.5

0.0

0.5

1.0

1.5

0 30 60 90 120 150 180

P1

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0 30 60 90 120 150 180

scattering angle (deg)

P2 Weigold, 1979 Williams, 1980







-1.0

-0.5

0.0

0.5

1.0

1.5

0 30 60 90 120 150 180

P1

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0 30 60 90 120 150 180


P2 Weigold, 1979 Williams, 1980 CCC, 1992







-1.0

-0.5

0.0

0.5

1.0

1.5

0 30 60 90 120 150 180

P1

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0 30 60 90 120 150 180


P2 Weigold, 1979 Williams, 1980 CCC, 1992 Crowe, 1996





21.8 eV e-Li 2p angular correlation parameters

superelasticelectrons 21.8 eV

incidentelectrons 20 eV

lithium beam

laserpolarization

laserbeam





21.8 eV e-Li 2p angular correlation parameters

-1.0

-0.5

0.0

0.5

1.0

1.5

0 30 60 90 120 150 180

P1

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0 30 60 90 120 150 180


P2 Teubner, 1996 CCC, 1996





20 eV e-Na spin asymmetries and spin-resolved L⊥

-0.3

-0.2

-0.1

0.0

0.1

0.2

0 30 60 90 120 150 180

Aex(3S)

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0 30 60 90 120 150 180

Aex(3P)

NIST CCC CC

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

0 30 60 90 120 150 180

L⊥0

-1.0-0.8-0.6-0.4-0.2 0.0 0.2 0.4 0.6 0.8 1.0

0 30 60 90 120 150 180


L1⊥





Comparison with experimentelectron-impact ionisation

e-H total ionisation cross section σi =∫ E

0 dεdσide (ε).

0

2

4

6

8

10

12

1 10 100 1000

cros

s se

ctio

n (1

0-17 cm

2 )

total energy E (eV)

exp, JPB (1987) CCC, PRL (1993) CCC(S=0) CCC(S=1) exp, PRA(1985)





25 eV e-H singly differential cross section dσide (ε)

0.0

0.5

1.0

0 2 4 6 8 10

cross section (10-17cm2eV-1)

secondary energy ε (eV)E

Shyn (1992)

CCC(93)

CCC(∞)





25 eV e-H doubly differential cross sections d2σidedΩ

(θ)

0

5

10

15

20

25

0 30 60 90 120 150 180

Es=2eV

0

2

4

6

8

10

12

14

16

18

20

0 30 60 90 120 150 180

cross section (10-19cm2 sr-1eV-1)


Es=3eV

0

5

10

15

20

25

0 30 60 90 120 150 180

Es=4eV

0

5

10

15

20

25

30

35

0 30 60 90 120 150 180

Es=5.7eV

Shyn (1992)

Childers (2004)

ECS

CCC





25 eV e-H EA = EB = 5.7 eV coplanar FDCS

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

-180 -120 -60 0 60 120 180

θAB=80o

Roeder

CCC

ECS

0.0

0.5

1.0

1.5

2.0

2.5

-180 -120 -60 0 60 120 180

cross section (10-19cm2/sr2/eV)

scattering angle θB

θAB=100o

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

-180 -120 -60 0 60 120 180

θAB=120o

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

-180 -120 -60 0 60 120 180

θAB=150o





e-Na EA = EB eV coplanar fully differential Xsecs

0

1

2

3

0 30 60 90 120 150 180

EA=EB=3eV

Murray CCC

0

1

2

3

0 30 60 90 120 150 180

EA=EB=5eV

0

1

2

3

0 30 60 90 120 150 180

EA=EB=10eV

0

1

2

3

0 30 60 90 120 150 180

fully differential cross section (a.u.)

EA=EB=15eV

0

1

2

3

0 30 60 90 120 150 180

EA=EB=20eV

0

1

2

3

0 30 60 90 120 150 180

scattering angle θA=-θB (deg)

EA=EB=30eV





e-H total ionisation cross section σi =∫ E

0 dεdσide (ε).

0

1

2

3

4

5

6

7

1 10 100 1000

cros

s se

ctio

n (1

0-17 cm

2 )

total energy (eV)

1S Shah et al, JPB (1987) CCC, PRL (1993)

0

20

40

60

80

100

120

140

1 10 100 1000

total energy (eV)

2S Defrance et al, JPB (1981) CCC, JPB (1997)





e-He total ionisation cross sections

0

1

2

3

4

1 10 100 1000

cross section (10-17cm2)

total energy (eV)

1-1-Sexp, JPB(1984)CCC, PRA(1995)

0

10

20

30

40

50

60

70

80

1 10 100 1000

total energy (eV)

2-3-Sexp, JPB(1976)CCC, JPB(2003)CCC(S=0)CCC(S=1)exp, PRA(1989)





Comparison with experimentelectron-impact ionisation

e-Li+ total ionisation cross sections

0

1

2

3

4

5

6

10 100 1000

cros

s se

ctio

n (1

0-18 cm

2 )


1-1-S Borovik et al, JPB (2008) CCC, JPB (2008)

0

1

2

3

4

5

6

7

8

10 100 1000


0.87 × 1-1-S + 0.13 × 2-3-S Borovik et al, JPB (2008) CCC, JPB (2008)





Comparison with experimentproton-impact on hydrogen

Electron-transfer cross section, CC(1,1) model

10-8

10-6

10-4

10-2

100

102

100 101 102 103 104 105 106

Tot

al C

ross

Sec

tion

(10-1

6 cm

2 )

Incident Energy (eV/amu)

Newman [26] Fite [27] Fite [28] Gealy [29] McClure [30] Bayfield [31] Wittkower [32] Hvelpland [33] Schwab [34] present CDW2S [12]





Concluding remarks

A general surface-integral approach to scatteringtheory has been formulated.There are (almost) no substantial discrepanciesbetween CCC and experiment for:

electron-impact excitation or ionisation of quasi one-and two-electron targetsphoton-impact single and double ionisation of He

Presently we are extending CCC topositron collisions with quasi one- and two-electrontargetsmulti-channel proton collisionsmore complicated targets, such as inert gases andmolecules.





Concluding remarks








Concluding remarks





Documents

The convergent close-coupling method for electron-atom scattering · 2009-11-11 · Introduction Nonrelativistic CCC theory Comparison with experiment The convergent close-coupling