Complete collision data set for electrons scattering on molecular … · 2021. 4. 20. · Complete collision data set for electrons scattering on molecular hydrogen and its isotopologues

Complete collision data set for electrons scattering on molecular hydrogen and its isotopologues

D.V. FursaCurtin University, Perth, Australia

in collaboration with: L.H. Scarlett1, J.S. Savage1, U. Rehill1, N. Mori1I. Bray1, M.C. Zammit2, Y. Ralchenko3

AcknowledgementsAustralian Research CouncilLos Alamos National LaboratoryUS Air Force Office of Scientific Research Pawsey Supercomputing Centre Forrest Research Foundation

1Department of Physics and Astronomy, Curtin University, Perth, Western Australia 6102, Australia2Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA3National Institute of Standards and Technology, Gaithersburg, Maryland, USA

(1)

• MCCC method : fixed-nuclei (FN) approach

Cross sections obtained with the FN MCCC method

• MCCC method : adiabatic nuclei (AN) approach

Vibrationally-resolved cross sections from AN MCCC

Accessing MCCC collision data: MCCC database

• Example: modeling - Collisional Radiative Model for H2

• Scattering from the excited states (n=2: a 3Sg+ , c 3Pu,…)

• Isotopic and vibrational-level dependence of H2 dissociation by electron impact

• Conclusions

Overview

(2)

• Born-Oppenheimer approximation

• Fixed-nuclei approximation, R = fixed

• Diagonalization of the target Hamiltonian HT

in a Sturmian (Laguerre) basismodeling of infinite number of bound & continuum states with a finite number of pseudostates

å=

+ =YN

ntnpntpi F

1

)( )()(),( xxxx fA• N-state multi-channel expansion

€

Tfi(k f ,k i) =Vfi(k f ,k i) + d3k∫n=1

N

∑Vfn (k f ,k)Tni(k,k i)E + i0 −εn − k

2 /2

σ fi (R)∝ Tfi (R)2

MCCC method: fixed-nuclei (FN) approach

• Solve integral LS equation for the T matrix

• Cross sections

Solve for the electronic wave function→

→<latexit sha1_base64="FQ6X0eKvDzAd1s/hF9eL/T4OuVU=">AAACKnicbVDLSgMxFM3Ud31VXboJFsFVmRFFN4KPTZcVrAqdMtxJ77TBTGZIMoUy9nvc+CtuulCKWz/EdNqFWi8knJxzDzf3hKng2rju2CktLC4tr6yuldc3Nre2Kzu79zrJFMMmS0SiHkPQKLjEpuFG4GOqEOJQ4EP4dDPRH/qoNE/knRmk2I6hK3nEGRhLBZUrX4DsCqR+2uNBRJ9pPbizd/Hk1FdT9YL6fVCYai6sK6J+B4WBII/4MKhU3ZpbFJ0H3gxUyawaQWXkdxKWxSgNE6B1y3NT085BGc4EDst+pjEF9gRdbFkoIUbdzotVh/TQMh0aJcoeaWjB/nTkEGs9iEPbGYPp6b/ahPxPa2UmOm/nXKaZQcmmg6JMUJPQSW60wxUyIwYWAFPc/pWyHihgxqZbtiF4f1eeB/fHNe+05t6eVC+vZ3Gskn1yQI6IR87IJamTBmkSRl7IG3knH86rM3LGzue0teTMPHvkVzlf37nqpuQ=</latexit>

h�f |HT |�ii = "f�fi

Why electron-molecule is difficult: no central potential

à multi-channel expansion + projectile partial wave expansion lead to a very large size of

coupled equations, to solve need fast computers, effective parallelization, etc….

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• total cross section, total ionization cross section, stopping power

• elastic scattering (ICS, DCS)

• excitation cross sections (ICS, DCS) for

- b 3Su+ , a 3Sg+, c 3Pu , e 3Su+, h 3Sg+, d 3Pu, i 3Pg, j 3Du

- B 1Su+, C 1Pu, EF 1Sg+, B’ 1Su+, D 1Pu, B” 1Su+, D’ 1Pu, H 1Sg+, GK 1Sg+, I 1Pu, , J 1Dg

Cross sections for electron scattering from H2FN single-centre spherical coordinate MCCC approach

Scarlett et al., Phys. Rev. A 96 (2017) 062708Zammit et al., Phys. Rev. A 95 (2017) 022708Zammit et al., Phys. Rev. Lett. 116 (2016) 233201

Established convergence in cross sections by performing: 491, 427, 259, 92, 9-state close-coupling

Often found large disagreement with previous experimental and theoretical results

(4)

Cross sections for the b 3Su+ state

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

5 10 15 20 25

cros

s se

ctio

n (a

.u.)

incident energy (eV)

b 3Σu+

Nishimura and Danjo (1986) Khakoo and Segura (1994)

CCC (2017) recommended (2008)

Scarlett et al., Phys. Rev. A 96 (2017) 062708Zammit et al., Phys. Rev. A 95 (2017) 022708

(5)

Cross sections for the b 3Su+ state

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

5 10 15 20 25

cros

s se

ctio

n (a

.u.)

incident energy (eV)

b 3Σu+

Nishimura and Danjo (1986) Khakoo and Segura (1994)

CCC (2017) recommended (2008) Zawadzki et al. (2018)

Zawadzki et al., Phys. Rev. A 97 (2018) 050702(R)Phys. Rev. A 98 (2018) 062704

New time of flight (TOF) spectrometer at California State University, Fullerton

(6)

UKRMol+ and MCCC calculations: good agreementMeltzer et al., J. Phys. B. 53 (2020) 145204ICS

DCS

(7)

MCCC method: adiabatic nuclei (AN) approach

Tf µ ,iv (E) = dR ϕ f µ (R)Tfi (E;R) ϕiv (R)∫

σ f µ ,iv (E)∝ Tf µ ,iv (E)2

• AN T matrix

• jnµ(R) are vibrational wave functions

• vibrationally-resolved cross sections

Single-center spherical-coordinate formulation is inadequate

Spheroidal-coordinate formulation of the MCCC method

Many FN calculations have to be performed on a grid of R

Computationally efficient / feasible approach is required

↓

-1.2

-1.1

-1.0

-0.9

-0.8

-0.7

-0.6

0 1 2 3 4 5 6 7 8

X 1Σg +

b 3Σu +

B 1Σu +

En

erg

y (H

a)

Internuclear distance (a0)

MCCC(210) model ↓

• computationally expensive (many R)

• need high accuracy of FN model at large R

(8)

MCCC method: spheroidal coordinate formulation

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0 1 2 3 4 5 6 7 8

B 1Σu+

C 1Πu

B’ 1Σu+

D 1Πu

Osc

illato

r stre

ngth

Internuclear separation (units of a0)

-1.2

-1.1

-1.0

-0.9

-0.8

-0.7

-0.6

-0.5

0 2 4 6 8 10

X 1Σg +

B 1Σu + E,F 1Σg

+

C 1Πu

B’ 1Σu +D 1Πu

Ener

gy (H

a)


-1.0

-0.9

-0.8

-0.7

-0.6

0 2 4 6 8

b 3Σu +

a 3Σg +

e 3Σu +

h 3Σg +

d 3Πu

Ener

gy (H

a)


c 3Πu

PEC

PEC

OOS

Solid lines: MCCC

Dashed lines: accurate(Wolniewicz et al)

Very accurate target structure for R values spanned by first 10 H2vibrational levels

Still within 5% accuracy for larger R

Target states

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Scarlett et al., Atomic Data and Nuclear Data Tables, 137 (2021) 101361, 139 (2021) 101403

MCCC(210) fully vibrationally resolved cross sections

Data files per transition:

MCCC-el-H2-B1Su_vf=10.X1Sg_vi=0.txt

Analytic fits

Calculations have been performed for over 58,000 transitions in H2 and its 5 isotopologuesincludes dissociation cross sections

In collaboration with Yu. Ralchenko (NIST)

Initial states

Final states

Number of bound vibrational levels

Scattering on all bound levels of the 𝑋 !Σ"# state

(#)

All cross sections and fits available online

• Atom. Data Nucl. Data Tables supplementary materialsComplete collision data set for electrons scattering on molecular hydrogen and its

isotopologues:

I. Fully vibrationally-resolved electronic excitation of H2(X 1Sg+ )

Atomic Data and Nuclear Data Tables 137 (2021) 101361

II. Fully vibrationally-resolved electronic excitation of the isotopologues of H2(X 1Sg+)


• LXCat Database – lxcat.net

• IAEA hcdb: Atomic and Molecular Data for Fusion Energy Research

https://db-amdis.org/hcdb/

• MCCC Database – mccc-db.org

Accessing MCCC data

(10)

All cross sections and fits available online:

• Atom. Data Nucl. Data Tables supplementary materialsComplete collision data set for electrons scattering on molecular hydrogen and its

isotopologues:

I. Fully vibrationally-resolved electronic excitation of H2(X 1Sg+ )


II. Fully vibrationally-resolved electronic excitation of the isotopologues of H2(X 1Sg+)


• LXCat Database – lxcat.net

• IAEA hcdb: Atomic and Molecular Data for Fusion Energy Research

https://db-amdis.org/hcdb/

• MCCC Database – mccc-db.org

Accessing MCCC data

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0.0

0.5

1.0

1.5

2.0

10 100

C 1Πu

Inte

grat

ed c

ross

sec

tion

(uni

ts o

f 10−

2 a02 )

Incident energy (eV)

CCC−S(210)CC(27)

Liu et al.Borges et al.

Celiberto et al.

0.0

0.2

0.4

0.6

0.8

1.0

10 100

B 1Σu +

Inte

grat

ed c

ross

sec

tion

(uni

ts o

f 10−

2 a02 )


CCC−S(210)CC(27)

Liu et al.Borges et al.

Celiberto et al.

0

2

4

6

8

10

12

14

16

18

10 100

B’ 1Σu +

Inte

grat

ed c

ross

sec

tion

(uni

ts o

f 10−

2 a02 )


CCC−S(210)CC(27)

Liu et al.Chung et al.

Mu−Tao et al.Borges et al.

Celiberto et al.Möhlmann et al.

Vroom et al.

Dissociative excitationTapley et al., Phys. Rev. A 98 (2018) 032701

B’ 1Su+ state has the largest DE cross section

for scattering on v = 0 ground state X 1Sg+

exp.: Liu et. al, J. Phys B 45 (2012)IP: semiclassical impact-parameter method – Bari group

Celiberto et al., Atom. Data Nucl. Data Tables 77 (2001) 161FBA: Borges et al., Phys. Rev. A 57 (1998) 1025

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Collisions data & plasma physics applications: Janev & Miles comparisons

10-3

10-2

10-1

100

10 100

Inte

grat

ed c

ross

sec

tion

(a02 )


X 1Σg+ → c 3Πu

CCCJanevMiles

10-3

10-2

10-1

100

10 100

Inte

grat

ed c

ross

sec

tion

(a02 )


X 1Σg+ → C 1Πu

CCCJanevMiles

10-3

10-2

10-1

100

10 100

Inte

grat

ed c

ross

sec

tion

(a02 )


X 1Σg+ → E,F 1Σg

+

CCCJanevMiles

10-3

10-2

10-1

10 100

Inte

grat

ed c

ross

sec

tion

(a02 )


X 1Σg+ → e 3Σu

+

CCCJanevMiles

Miles, Thompson, Green, J. App. Phys. 43 (1972) 678; Janev, Reiter, Samm, JÜL-4105, Jülich, 2003

used by Ursel Fantz & Dirk Wuenderlich (Garching) in their CR model à

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Comparison of measured population densities of the d3 state to the ones calculated with the CR model Yacora H2 using cross section from either Miles et al. [25], Janev et al. [21] or the MCCC data.

from the spectroscopic measurements of the Fulcher-atransition

Further improvements:• more accurate account of cascading from

high level states• use of vibrationally resolved cross sections • use of rotationally resolved cross sections

Significant differences in the simulation results depending on the collision dataset used →

(14)

it is a metastable stateit is lower than a 3Sg

+(v=0) state by 0.0228 eV

Scattering from excited (n=2, v=0) electronic states

-0.74

-0.73

-0.72

-0.71

-0.70

-0.69

-0.68

1 2 3

a 3Σg +

c 3Πu

Tw

o-e

lect

ron

en

erg

y (H

a)


PresentRef

1 2 3-0.3

-0.2

-0.1

0.0

0.1

0.2

Osc

illa

tor

stre

ng

th

LengthVelocityRef

c 3Pu à a 3Sg+ transition is by far the largest as it is a

dipole allowed transitionsmall excitation energy means: (a) large projectile partial wave expansion is required(b) Born limit will be reached relatively fast(c) target structure accuracy is very important

Vertical excitation energies at 𝑅 = 2

Example: the c 3Pu(v=0) state

Excitation energies are relative to the c 3Pu state

(15)

• Much slower than for scattering on ground state (Lmax = 10)• Many dipole-allowed transitions unconverged even with Lmax = 25• Even with Analytical Born Completion Lmax = 20 is necessary• Agreement with recent UKRMol+ calculations (Lmax = 6)

for dipole-forbidden and spin-exchange transitions, but not dipole-allowed transitions

MCCC (AN): Scarlett et al., PRA 103 (2021) 032802

UKRmol+ (FN): Meltzer et al., J. Phys. B 53 (2020) 245203

Partial-wave convergence:

SMC (FN): Sartori et al. Phys. Rev. A 55 (1997) 3243

FBA (FN): Rescigno & Orel (from IEEE Trans. Plasma Sci. PS-11, 266 (1983).)

10−2

100

102SMC

c 3Πu (v = 0) → X 1Σg +

100

102

104

106

Inte

grat

ed c

ross

sec

tion

(uni

ts o

f a02 )

c 3Πu (v = 0) → a 3Σg +

100

102

0.01 1 100Incident energy (eV)

c 3Πu (v = 0) → b 3Σu +

10−2

100

102

104MCCC ANUKRMol+ R = 2.0

c 3Πu (v = 0) → B 1Σu +

0

10

20

30

40MCCC R = 2.0 Lmax = 6FBA

c 3Πu (v = 0) → C 1Πu

1 10 1000

2

4

6

8c 3Πu (v = 0) → EF 1Σg

+

10−2

100

102SMC

c 3Πu (v = 0) → X 1Σg +

100

102

104

106

Inte

grat

ed c

ross

sec

tion

(uni

ts o

f a02 )

c 3Πu (v = 0) → a 3Σg +

100

102


c 3Πu (v = 0) → b 3Σu +

10−2

100

102


c 3Πu (v = 0) → B 1Σu +

0

10

20

30


c 3Πu (v = 0) → C 1Πu

1 10 1000

2

4

6

8c 3Πu (v = 0) → EF 1Σg

+

10−2

100

102SMC

c 3Πu (v = 0) → X 1Σg +

100

102

104

106

Inte

grat

ed c

ross

sec

tion

(uni

ts o

f a02 )

c 3Πu (v = 0) → a 3Σg +

100

102


c 3Πu (v = 0) → b 3Σu +

10−2

100

102


c 3Πu (v = 0) → B 1Σu +

0

10

20

30


c 3Πu (v = 0) → C 1Πu

1 10 1000

2

4

6

8c 3Πu (v = 0) → EF 1Σg

+

10−2

100

102SMC

c 3Πu (v = 0) → X 1Σg +

100

102

104

106

Inte

grat

ed c

ross

sec

tion

(uni

ts o

f a02 )

c 3Πu (v = 0) → a 3Σg +

100

102


c 3Πu (v = 0) → b 3Σu +

10−2

100

102


c 3Πu (v = 0) → B 1Σu +

0

10

20

30


c 3Πu (v = 0) → C 1Πu

1 10 1000

2

4

6

8c 3Πu (v = 0) → EF 1Σg

+

10−2

100

102SMC

c 3Πu (v = 0) → X 1Σg +

100

102

104

106

Inte

grat

ed c

ross

sec

tion

(uni

ts o

f a02 )

c 3Πu (v = 0) → a 3Σg +

100

102


c 3Πu (v = 0) → b 3Σu +

10−2

100

102


c 3Πu (v = 0) → B 1Σu +

0

10

20

30


c 3Πu (v = 0) → C 1Πu

1 10 1000

2

4

6

8c 3Πu (v = 0) → EF 1Σg

+

(16)


10−1

100

101

102

103

104

B 1Σu +(v = 0) → X 1Σg

+

MCCC ANUKRMol+ R = 2.0

10−4

10−2

100

102 B 1Σu +(v = 0) → b 3Σu

+

Inte

grat

ed c

ross

sec

tion

(uni

ts o

f a02 )

0

50

100

150

200

0.01 1 100

B 1Σu +(v = 0) → c 3Πu


0

20

40

60

80B 1Σu

+(v = 0) → C 1Πu

0

20

40

60B 1Σu +(v = 0) → EF 1Σg

+

MCCC R = 2.0 Lmax = 6MCCC R = 2.5

1 10 100 1000 0

20

40

60B 1Σu +(v = 0) → a 3Σg

+

0

10

20

30

40c 3Πu (v = 0) → e 3Σu

+

0

2

4

6

8

10

12

14

Inte

grat

ed c

ross

sec

tion

(uni

ts o

f a02 )

IPc 3Πu (v = 0) → h 3Σg

+

0

5

10

15

20

25

1 10 100Incident energy (eV)

c 3Πu (v = 0) → d 3Πu

0

2

4

6


c 3Πu (v = 0) → g 3Σg +

0

5

10

15

20c 3Πu (v = 0) → i 3Πg

1 10 100 1000 0

10

20

30

40MCCC R = 2.0 Lmax = 6 c 3Πu (v = 0) → j 3∆g

IP: Laricchiuta et. al., Phys. Rev. A 69 (2004) 22706

B 1Su+

𝑅! = 2.52

(17)


CSP-ic.: Joshipura et al., J. Phys. B 43 (2010) 135207

Gryzinski: Wunderlich et al., Chem. Phys. 390 (2011) 75

101

102

103

104

Inte

grat

ed c

ross

sec

tion

(uni

ts o

f a02 )

c 3Πu (v = 0)

100

101

102

103

104

0.01 1

MCCC R = 2.0MCCC R = 2.5

B 1Σu + (v = 0)

EF 1Σg + (v = 0)

100

102

104

MCCC ANUKRMol+ R = 2.0SMCCSP−ic

a 3Σg + (v = 0)

0.01 1 100101

102

103

104


ElasticElastic

C 1Πu (v = 0)

0

20

40

60c 3Πu (v = 0)

0

20

40

60MCCC ANGryzinskiCSP−ic

a 3Σg + (v = 0)

0

20

40

60

10 100

Inte

grat

ed c

ross

sec

tion

(uni

ts o

f a02 )

B 1Σu + (v = 0)

EF 1Σg + (v = 0)

10 100 1000 0

20

40

60


IonizationIonizationC 1Πu (v = 0)

R = 2.0 used in FN UKRMol+ calculations does not accurately model scattering on more diffuse B 1Su

+ state (average R = 2.5)

Elastic scattering cross sections Ionization cross sections

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Isotopic and vibrational-level dependence of H2 dissociation by electron impact

2

ously [48] performed more detailed dissociation calculationsfor vibrationally-excited H2 including all important pathwaysto dissociation into neutral fragments from low to high inci-dent energies. These calculations can be readily extended toinclude the isotopologues in the future.

We first describe the standard treatment of dissociation,which is a straightforward adaption of the method for non-dissociative excitations. We use SI units throughout for con-sistent comparison with TT01’s formulas. For simplicity, weassume the AN approximation has been applied according tothe definitions in the review of electron-molecule collisions byLane [49], but note that the following discussion would remainvalid if the energy-balanced AN method [15] or even full elec-tronic and vibrational close-coupling was used. In terms ofthe electronic partial-wave T -matrix elements defined by Lane[49], the expression for the energy-differential cross section fordissociation of the vibrational level v into atomic fragments ofasymptotic kinetic energy Ek in the standard formulation is

d!

dEoutD "

k2in

X`0m0`m

ˇ̌h#Ek jT`0m0;`m.RI Ein/j#vi

ˇ̌2; (2)

where Ein and kin are the incident projectile energy andwavenumber, and # are the vibrational wave functions. Theconvergence of Eq. (2) with respect to the maximum ` in-cluded in the projectile plane-wave expansion is relatively fastfor the spin-exchange X 1†C

g ! b 3†Cu transition (previous

MCCC studies have found `max D 6 to be sufficient at all ener-gies [4]). However, when the partial-wave method is appliedto direct (non-exchange) transitions, larger expansions are re-quired, along with the analytical Born completion techniqueto account for the infinity of terms ` > `max [4]. The energiesof the scattered electron and dissociating fragments are relatedby

Ek D Ein ! Dv ! Eout; (3)

where Dv is the threshold dissociation energy of the vibrationallevel v [15], and Eout is the outgoing projectile energy. Thisrelationship makes it possible to treat the energy-differentialcross section as a function of either Eout or Ek. When thismethod has been applied in previous work [8–16, 18–24],Eq. (2) is not derived explicitly for the case of dissociationsince the derivation follows exactly the same steps summarizedby Lane [49] for the non-dissociative vibrational-excitationcross section. The only difference is the replacement of thefinal bound vibrational wave function with an appropriately-normalized continuum wave function #Ek.R/. In principle thecontinuum normalization is arbitrary so long as the density offinal states is properly accounted for. According to standarddefinitions [50], the density of states $ for the vibrationalcontinuum satisfies the following relation:

Z 1

0

#Ek.R/#Ek.R0/$.Ek/ dEk D ı.R ! R0/; (4)

giving a clear relationship between the continuum-wave nor-malization and density. Since the formulas presented by Lane

[49] for non-dissociative excitations are written in terms ofbound vibrational wave functions, with a density of states equalto unity (by definition), the most straightforward adaption todissociation simply replaces them with continuum wave func-tions normalized to have unit density as well. Indeed, many ofthe previous works [13, 15, 21, 24, 51, 52] which have appliedthe AN, rather than fixed-nuclei (FN), method to dissociationexplicitly state that the continuum wave functions are energynormalized, which implies unit density and the following res-olution of unity:

Z 1

0

#!Ek.R/#Ek.R

0/ dEk D ı.R ! R0/: (5)

The works which have applied the FN method also implic-itly assume energy normalization, since the FN approximationutilizes Eq. (5) to integrate over the dissociative states analyt-ically. Note that Eq. (5) implies the functions #Ek have di-mensions of 1=

penergy " length. The bound vibrational wave

functions are normalized according toZ 1

0

#!v .R/#v0.R/ dR D ıv0v; (6)

and hence they have dimensions of 1=p

length. The electronicT -matrix elements defined by Lane [49] are dimensionless,and the integration over R implied by the bra-kets in Eq. (2)cancels the combined dimension of 1=length from the vibra-tional wave functions, so it is evident that the right-hand sideof Eq. (2) has dimensions of area=energy as required (note that1=k2

in has dimensions of area).The standard approach to calculating dissociation cross sec-

tions has been applied extensively in the literature [8–24]. Itis also consistent with well-established methods for comput-ing bound-continuum radiative lifetimes or photodissociationcross sections, which replace discrete final states with disso-ciative vibrational wave functions. The latter are either energynormalized [53], or normalized to unit asymptotic amplitudewith the energy-normalization factor included explicitly in thedipole matrix-element formulas [54, 55].

TT01 criticized the standard technique, claiming that aproper theoretical formulation for dissociation did not exist,and suggested that a more rigorous derivation for the specificcase where there are three fragments in the exit channels isrequired. We have performed our own derivation followingthe ideas laid out by TT01 and found that they lead directlyto Eq. (2). However, TT01 arrived at an expression which ismarkedly different:

d!

dEoutD mH

4"3me

Ek

Ein

X`0m0`m


ˇ̌2: (7)

Here mH is the hydrogen nuclear mass, which is replacedwith the deuteron or triton mass in their later investigationinto dissociation of D2 and T2 [27]. Comparing Eqs. (2)and (7), we see that TT01’s formula is different by a factorof mHEk=2"4„2 (the T -matrix elements here are the sameas those in Eq. (2)). The distinguishing feature of TT01’s

Standard formula to calculate energy-differential cross section for dissociation (used in MCCC):

• 𝜈"! energy-normalized continuum vibrational wave functions

• Isotopic dependence only from vibrational wave functions

2



d!

dEoutD "

k2in

X`0m0`m


ˇ̌2; (2)






Z 1

0




Z 1

0

#!Ek.R/#Ek.R

0/ dEk D ı.R ! R0/: (5)




0

#!v .R/#v0.R/ dR D ıv0v; (6)






d!

dEoutD mH

4"3me

Ek

Ein

X`0m0`m


ˇ̌2: (7)


Trevisan and Tennyson (J. Phys. B 34 (2001) 2935-2949) derived a different formula:

• Explicit mass dependence – cross sections scale with mass

• Explicit dependence on the energy of the dissociating fragments Ek

• Formula was applied in R-matrix calculations of H2 dissociation cross sections which are widely used in plasma modelling applications

0

2

4

8 10 12 14

TT02

MCCC

T2

D2

H2

T2D2

Inte

gra

ted

cro

ss s

ecti

on

(u

nit

s o

f a

02)


X 1Σg

+(v = 0) → b

3Σu

+X

1Σg

+(v = 0) → b

3Σu

+

R-matrix calculations TT02: Trevisan and Tennyson, Plasma Phys. Control. Fusion 44 (2002) 1263-1276, 2217-2230

(19)

𝐸# = 𝐸$% − 𝐷& − 𝐸'()


2



d!

dEoutD "

k2in

X`0m0`m


ˇ̌2; (2)






Z 1

0




Z 1

0

#!Ek.R/#Ek.R

0/ dEk D ı.R ! R0/: (5)




0

#!v .R/#v0.R/ dR D ıv0v; (6)






d!

dEoutD mH

4"3me

Ek

Ein

X`0m0`m


ˇ̌2: (7)



2



d!

dEoutD "

k2in

X`0m0`m


ˇ̌2; (2)






Z 1

0




Z 1

0

#!Ek.R/#Ek.R

0/ dEk D ı.R ! R0/: (5)




0

#!v .R/#v0.R/ dR D ıv0v; (6)






d!

dEoutD mH

4"3me

Ek

Ein

X`0m0`m


ˇ̌2: (7)






0

2

4

8 10 12 14

TT02

MCCC

T2

D2

H2

T2D2

Inte

gra

ted c

ross

sec

tion (

unit

s of

a02)


X 1Σg

+(v = 0) → b

3Σu

+X

1Σg

+(v = 0) → b

3Σu

+


0

1

2v = 0v = 0

MCCC

TT01/02

ST98

H2 X 1Σg

+(v) → b

3Σu

+H2 X

1Σg

+(v) → b

3Σu

+

0

1

2

Inte

gra

ted c

ross

sec

tion (

unit

s of

a02)

v = 1v = 1 v = 3v = 3

0

1

2

4 8 12 16

v = 2v = 2

4 8 12 16 20Incident energy (eV)

v = 4v = 4

(20)


2



d!

dEoutD "

k2in

X`0m0`m


ˇ̌2; (2)






Z 1

0




Z 1

0

#!Ek.R/#Ek.R

0/ dEk D ı.R ! R0/: (5)




0

#!v .R/#v0.R/ dR D ıv0v; (6)






d!

dEoutD mH

4"3me

Ek

Ein

X`0m0`m


ˇ̌2: (7)



2



d!

dEoutD "

k2in

X`0m0`m


ˇ̌2; (2)






Z 1

0




Z 1

0

#!Ek.R/#Ek.R

0/ dEk D ı.R ! R0/: (5)




0

#!v .R/#v0.R/ dR D ıv0v; (6)






d!

dEoutD mH

4"3me

Ek

Ein

X`0m0`m


ˇ̌2: (7)






0

2

4

8 10 12 14

TT02

MCCC

T2

D2

H2

T2D2

Inte

gra

ted c

ross

sec

tion (

unit

s of

a02)


X 1Σg

+(v = 0) → b

3Σu

+X

1Σg

+(v = 0) → b

3Σu

+


0

1

2v = 0v = 0

MCCC

TT01/02

ST98

H2 X 1Σg

+(v) → b

3Σu

+H2 X

1Σg

+(v) → b

3Σu

+

0

1

2

Inte

gra

ted c

ross

sec

tion (

unit

s of

a02)

v = 1v = 1 v = 3v = 3

0

1

2

4 8 12 16

v = 2v = 2

4 8 12 16 20Incident energy (eV)

v = 4v = 4

10-12

10-11

10-10

10-9

4 6 8 10 12 14

LT

E r

ate

coef

fici

ent

(unit

s of

cm3/s

)

Temperature (units of 1000 K)

10-12

10-11

10-10

10-9

4 6 8 10 12 14

X 1Σg

+ → b

3Σu

+ LTE X

1Σg

+ → b

3Σu

+ LTE

TT02TT02

MCCCMCCC

ST99ST99

H2

D2

T2H2

H2

D2

T2

(21)


2



d!

dEoutD "

k2in

X`0m0`m


ˇ̌2; (2)






Z 1

0




Z 1

0

#!Ek.R/#Ek.R

0/ dEk D ı.R ! R0/: (5)




0

#!v .R/#v0.R/ dR D ıv0v; (6)






d!

dEoutD mH

4"3me

Ek

Ein

X`0m0`m


ˇ̌2: (7)


2



d!

dEoutD "

k2in

X`0m0`m


ˇ̌2; (2)






Z 1

0




Z 1

0

#!Ek.R/#Ek.R

0/ dEk D ı.R ! R0/: (5)




0

#!v .R/#v0.R/ dR D ıv0v; (6)






d!

dEoutD mH

4"3me

Ek

Ein

X`0m0`m


ˇ̌2: (7)


Which formula is correct?

?

PHYSICAL REVIEW A 103, L020801 (2021)Letter


Liam H. Scarlett ,1,* Dmitry V. Fursa ,1 Jack Knol ,1 Mark C. Zammit ,2 and Igor Bray 1

1Curtin Institute for Computation and Department of Physics and Astronomy, Curtin University, Perth, Western Australia 6102, Australia2Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA

(Received 16 November 2020; accepted 28 January 2021; published 19 February 2021)

The low-energy electron-impact dissociation of molecular hydrogen has been a source of disagreementbetween various calculations and measurements for decades. Excitation of the ground state of H2 into thedissociative b 3!+

u state is now well understood, with the most recent measurements being in excellent agreementwith the molecular convergent close-coupling (MCCC) calculations of both integral and differential crosssections [Zawadzki et al., Phys. Rev. A 98, 062704 (2018)]. However, in the absence of similar measurementsfor vibrationally excited or isotopically substituted H2, cross sections for dissociation of these species mustbe determined by theory alone. We have identified large discrepancies between MCCC calculations and therecommended R-matrix cross sections for dissociation of vibrationally excited H2, D2, T2, HD, HT, and DT[Trevisan et al., Plasma Phys. Contr. Fusion 44 1263 (2002); 44, 2217 (2002)], with disagreement in both theisotope effect and dependence on initial vibrational level. Here we investigate the source of the discrepancies,and discuss the consequences for plasma models, which have incorporated the previously recommended data.

DOI: 10.1103/PhysRevA.103.L020801

In low-temperature plasmas, electron-impact dissociationof molecular hydrogen into neutral fragments proceeds almostexclusively via excitation of the dissociative b 3!+

u state:

e− + H2(X 1!+g , v) → H2(b 3!+

u ) + e− → 2H + e−. (1)

Cross sections and rate coefficients for this process are neededto accurately model astrophysical, scientific, and technologi-cal plasmas where hydrogen is present in its molecular form[1– 3]. The importance of the reaction (1) and the relative sim-plicity it presents to theory and experiment has led to it beingone of the most studied processes in electron-molecule scat-tering. Despite this, for decades there was no clear agreementbetween any theoretically or experimentally determined crosssections. The molecular convergent close-coupling (MCCC)calculations [4] for scattering on the ground vibrational level(v = 0) of H2 were up to a factor of two smaller than therecommended cross sections, but the situation was resolvedwhen newer measurements were found to be in near-perfectagreement with the MCCC results [5,6]. Recent R-matrix cal-culations have also confirmed the MCCC results for scatteringon the v = 0 level of H2 [7].

It is also important to determine accurate cross sectionsfor dissociation of vibrationally excited and isotopically sub-stituted hydrogen molecules, due to their presence in fusionand astrophysical plasmas. For these species, however, theabsence of experimental data means it is up to theory to makerecommendations alone. The primary distinguishing factorbetween the numerous calculations of the b 3!+

u -state crosssection [8– 24] is the treatment of the electronic dynamics.There is generally a consensus in the literature that the nuclear

*[email protected]

dynamics of the dissociative transition can be treated withthe same formalism used for bound excitations, by simplyreplacing the bound final vibrational wave function with anappropriately normalized dissociative wave function. We havetaken this approach previously to study the dissociation ofvibrationally excited H+

2 and its isotopologues, finding goodagreement with measurements of both integral and energy-differential (kinetic-energy-release) cross sections [20,21].

The cross sections for dissociation of vibrationally excitedH2, HD, and D2 recommended in the well-known reviewsof Yoon et al. [25,26] come from the R-matrix calculationsof Trevisan and Tennyson [27,28] (hereafter referred to col-lectively as TT02). The results of TT02, which also includeHT, DT, and T2, have long been considered the most accuratedissociation cross sections for H2 and its isotopologues, andare widely used in applications. During our recent efforts tocompile a comprehensive set of vibrationally resolved crosssections for electrons scattering on vibrationally excited andisotopically substituted H2 [29,30], it has become apparentthat there are major discrepancies between the MCCC calcu-lations and the R-matrix calculations of TT02. Interestingly,the two methods are similar in their treatment of both theelectronic and nuclear dynamics, but differ in their funda-mental definitions of the dissociation cross section, leading toconflicting isotopic and vibrational-level dependencies in thecalculations. The formalism applied by TT02 was previouslydeveloped by Trevisan and Tennyson [17] (TT01).

Here we compare the standard method adopted in theMCCC calculations with the alternative formulation sug-gested by TT01, and determine the origin of the disagreementbetween the two approaches. It is important to note that thispaper is not concerned with analyzing the validity of theoften-used adiabatic-nuclei (AN) approximation, or variantssuch as the energy-balanced AN approximation of Stibbe and

2469-9926/2021/103(2)/L020801(7) L020801-1 ©2021 American Physical Society

• In the paper we show that Trevisan and Tennyson’s approach to deriving the dissociation cross section leads to the standard formula

• The differences in their formula are due to errors in the derivation

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• Large-scale close-coupling calculations for H2 : produced a comprehensive dataset of e--H2vibrationally resolved excitation cross sections: from the ground to n=2,3 excited electronic states, isotopically resolved dataset.

• Collisional Radiative Model for the triplet system of molecular hydrogen: using MCCC cross sections

• Produced a set of cross sections for H2 transitions between electronically excited states

• Identified & resolved a major discrepancy for the isotopic and vibrational-level dependence of H2 dissociation by electron impact

• In preparation:

- detailed data set of vibrationally resolved cross sections for H2 for transitions between electronically excited states, + isotopologues (D2, HD, …)

- rotationally-resolved electron scattering on H2 + isotopologues (D2, HD, …)rovibrationally resolved cross sections polarization of radiation in Fulcher-a band emission (d3→a3 )

- vibrational close-coupling technique with exact account of coupling to electronic degrees of

freedom• Results available from LXCat and IAEA databases and MCCC database (mccc-db.org)

Conclusions

Thank you

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Documents

Complete collision data set for electrons scattering on molecular … · 2021. 4. 20. · Complete collision data set for electrons scattering on molecular hydrogen and its isotopologues