Autoionization Branching Ratios for Metal Halide Molecules

Jeffrey J. KayLawrence Livermore National Laboratory

OSU Molecular Spectroscopy SymposiumSession RD (Spectroscopic Perturbations)

23 June 2011

Robert W. FieldMassachusetts Institute of Technology

Cold MoleculesCurrently there is great interest in producing dense

samples of cold molecules and ions

Low velocities, long interaction times allow investigation of unexplored areas of molecular physics:

Cold Chemistry

Control UsingExternal Fields

Precision Spectroscopic Measurements

Time Variation of mp/me, Ammonia

New States of Matter

Quantum Degeneracy

Vibrational Autoionization of MX Rydberg States

Formation of MX+ Ions in Selected Rovibronic States?

Producing MX+ by Autoionization

Most ionization methods non-state-selective

Electron bombardmentNonresonant photoionization

What about autoionization?Specific rovibronic MX Rydberg levels can be selected by laser and spontaneously eject an electron, leaving behind MX+ ion.

What is the MX+ rovibrational state distribution? MX X 2S+1ΛΩ

MXIntermediate

State

MX Rydberg State

|n,L,Λ,v,J>

MX+ + e-

IonizationContinua

J+=0 J+=1J+=2

J+=3

Production of state-selected metal halide ions?

Autoionization Branching RatiosProduct state distributions following autoionization difficult to calculate.

MX n, l ,λ,v, J → e− e,λ + MX + v+, J+

MX+ rovibronic state distribution depends sensitively on electron-ion electrostatic interactions, especially their dependence on R

The only molecule with accurate and complete predictions of autoionization rovibronic branching ratios:

H2

One state 60+ decay channels!

Quantum Defect Model for CaFRecently, we developed a complete quantum defect theory (QDT) model for CaF

Summarizes electron-ion interactions in terms of an R-, E-dependent quantum defect matrix, μ(R, E)

75 parameter model = Infinite # of Rydberg statesVibronic perturbations

Vibrational autoionizationRotational autoionization

Quantum Defect Model for CaF

A molecule summarized in 75 parametersAll spectra. All dynamics.

Electronic Structure of CaFCaF is a prototypical MX molecule

F-

Any MX molecule can be built-up from CaF by adding core-excited states and spin-orbit effects

“Sodium Atom” of diatomics:One unpaired electron outside closed shells

Ca2+

e-

Quantum Defect Theory

1ion CoulombH H H H + +

ψλN+v+

fλ E λ,m N+ ,v+ δλN+v+ ;λ'N+ 'v+ '

− KλN+v+ ;λ'N+ 'v+ '

gλ'N+ 'v+ '

E λ,m N+ 'v+ 'λ'N+ 'v+ '∑

Electron radial wave functions

Ion corewavefunctions

Reaction matrix elements

Scattering Theory: Physics Embodied in Reaction Matrix

3. Form superposition of channel functions

4. Determine: At which energies do wavefunctions satisfy boundary conditions?

2. Define “channel functions” for all energies:

1. Separate H into e-, ion, interaction terms:

The Reaction Matrix

el vib rot

Rψδ ion∑ Rψδ eλion vibion rotion∑

Short Range: Born-Oppenheimer products

Long Range:Electron-ion products

rr = rc

Division of space

r = rvr

Reaction Matrix is the Heart of a QDT Model“Frame Transformation” allows expression in terms of quantum defects

Core Short Range Long Range

Frame TransformationAt large electron-ion separation, both forms of wavefunction must be equal:

vrr r i j ion ion ioni j

A el vib rot B Ryd el vib rot

By explicitly matching wavefunctions, can express LARGE number of reaction matrix elements in terms of SMALL number of quantum defects.

Quantum Defect Matrix Elements(FEW)

Reaction Matrix Elements(MANY)

K

l v+N+ ,λ'v+ 'N+ ' Λ N+ N

Λ∑ χ

v+N+

R tanπmλλ'Λ R χv+ '

N+ ' R δR∫⎡⎣ ⎤⎦ N+ 'Λ

N

R

Rydberg States of CaF

CaF+ (1Σ+, v=0) + e-v+=0

v+=1Rydberg statesHund’s Case (b)

Ion core 1Σ+

Hund’s Case (b)

n*, l ,λ,N,M

Ionization Continuum

Hund’s Case (d) e, l N + ,v+ ,Λ+ = 0

1Σ+

1Σ+

Autoionization Branching Ratios

Γ

i→N +N ,±( ) ∝ α i , N , ±

dμdR

β j , N , ± β j , N , ± l ", N + ", N , ±,v = 0j , l∑

2

a i , N , ± = α i , N , ± l , N + , N , ±,v = 1 l , N + , N , ±,v = 1

l ,N +∑

b j , N, ± = l ', N + ', N ,v = 0 l ', N + ', N ,v = 0 β j , N , ±

l ',N + '∑

Rydberg StateHund’s Case (b)

ContinuumHund’s Case (b)

Eigenvector Decomposition(From MQDT Calculation)

Case (b) -> Case (d)

Quantum Defect Derivatives

Branching ratios can be calculated from quantum defect derivatives

Autoionization Branching Ratios: CaF

s Σ

p Σ

d Σ

f Σ

0% 10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

N+ = 0N+ = 1 N+ = 2 N+ = 3

Branching Ratios for N = 0

‘d’ Σ Rydberg series produces primarily N+ = 0 ionsAutoionization produces mostly N+ = 0, 1

MostlyN+ = 0, 1

s Σ

p Σ

d Σ

f Σ

p Π

d Π

f Π

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

N+ = 0N+ = 1 N+ = 2 N+ = 3N+ = 4

Autoionization Branching Ratios: CaFBranching Ratios for N = 1

Branching ratios broaden, shift to higher N+ (=1, 2)

MostlyN+ = 0,1,2

s Σ

p Σ

d Σ

f Σ

p Π

d Π

f Π

d Δ

f Δ

0% 10% 20% 30% 40% 50% 60% 70% 80% 90%100%

N+ = 0N+ = 1 N+ = 2 N+ = 3N+ = 4N+ = 5

Autoionization Branching Ratios: CaFBranching Ratios for N = 2

Branching ratios broaden, shift to higher N+ at higher N

MostlyN+ = 1,2,3

s Σ

p Σd Σf Σ

p Πd Πf Π

d Δf Δf Φ

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

N+ = 0N+ = 1 N+ = 2 N+ = 3N+ = 4

Autoionization Branching Ratios: CaFBranching Ratios for N = 3

Branching ratios broaden, shift to higher N+ at higher N

MostlyN+ = 2,3,4

Branching Ratios: TrendsOverall trends we observe:

1. Best rotational selectivity at low N

2. Less selectivity at high N (more open channels)

3. <N+> = N

4. Propensity rule: N+ = N, N ± 1 (due to ΔL = ±1 L-mixing)

Autoionization Branching Ratios: General MX

Rydberg statesHund’s Case (a) n*, l , s, J, M

ContinuumHund’s Case (e) e, l , s, j J + ,S+ ,v+ ,Λ+

Expect similar trends for other MX molecules.(Greatest N+-selectivity at low N; Shift to high N+ at high N)

Methodology developed here applicable to any MX

Light molecules (MgF+, TiF+): Very similar trendsHeavy molecules (BaF+, HfF+): Coupling cases change due to spin-orbit

Rydberg statesHund’s Case (b) n*, l , N , M

ContinuumHund’s Case (d) e, l N + ,v+ ,Λ+

Low Z High Z

Acknowledgments

Funding: National Science Foundation

We thank Eric Cornell and the Cornell Group (JILA/Colorado) for their interest in vibrational

autoionization.

eEDM Measurements: Cold Metal Halide Ions

Large dipole moment enables measurements of electron electric dipole moment (eEDM)

HfF+ electric field ~1010 V/cm

Easily trappable using RF traps

Metastable 3Δ1 state ideal for eEDM measurements

http://jila.colorado.edu/bec/CornellGroup/

Metal halide ions (MX+) are candidates for ultra-high-precision spectroscopic measurements

Cornell Group (JILA) eEDM SchemeHfF+: eEDM Measurements

Autoionization Branching Ratios: General MX

Γ

i→J +N ,±( ) ∝ α i , J, ±

dμdR

β j , J, ± β j , J, ± l "J + "J, ±,v = 0j , l∑

2

Initial State (Hund’s Case (a))

Final States(Hund’s Case (a))

Final States(Hund’s Case (e))

Rydberg statesHund’s Case (a) n*, l , s, J, M Continuum

Hund’s Case (e) e, l , s, j J + ,S+ ,v+ ,Λ+

Branching Ratios for High-Z MX+ Ions

R

Rydberg States of HfF

Rydberg statesHund’s Case (a)

Ion core 3Δ1

Hund’s Case (a)

n*, l , s, J, M

Ionization Continuum

Hund’s Case (e) e, l , s, j J + ,S+ ,v+ ,Λ+ = 2

HfF+ (1Σ+) + e-v+=0

v+=1

1Σ+

3Δ1

v+=03Δ1

v+=03Δ2

v+=03Δ3

(To Be Added)

Branching Ratios for HfF (3D Bar Chart)(Several values of J)

HfF+ Ion: eEDM Measurements

HfF+ Ion: eEDM Measurements

HfF+ Ion: eEDM Measurements

HfF+ Ion: eEDM Measurements

Hund’s Case (b) to Hund’s Case (d) Transformation

Ch. Jungen and G. Raseev, Phys. Rev. A 57 2407 (1998)

N + Λλ,N,M,Λ+ , π

11+δΛ,0δΛ+ ,0

11+δ

Λ+ ,0

−1 N−Λ 2N+ +1 1/2

× 1+δΛ+ ,0

−1π−q+ −N+ +λ⎡⎣ ⎤⎦

N+ λ NΛ+ Λ−Λ+ −Λ

⎛⎝⎜

⎞⎠⎟

Case (b): Good quantum numbers Λ, l , N , M ,Λ+ , p

N+,λ,N,M,Λ+, πCase (d): Good quantum numbers

Hund’s Case (a) to Hund’s Case (e) Transformation

Ch. Jungen and G. Raseev, Phys. Rev. A 57 2407 (1998)

Case (a): Good quantum numbers Λ,S,Ω, l , s, Λ+ ,S+ , J, M , pCase (e): Good quantum numbers j, J +,Ω+ ,λ,s,Λ+,S+ , J,M, π

Ω+J + j SΛΩl ,s,Λ+ ,S+ ,J , M , p( ) = −1( )S+ −Ω+ +l +Λ+ J +Ω 2S + 1( ) 2 j + 1( ) 2J + +1( )

1+ Δ1( ) 1+ Δ2( ) 1 + Δ3( )

⎡

⎣⎢⎢

⎤

⎦⎥⎥

× s S+ SΩ − Λ + Λ+ − Ω+ Ω+ − Λ+ −Ω + Λ

⎛

⎝⎜

⎞

⎠⎟

l s j

Λ − Λ+ Ω − Λ + Λ+ − Ω+ −Ω + Ω+

⎛

⎝⎜

⎞

⎠⎟

J + j J

Ω+ Ω − Ω+ −Ω

⎛

⎝⎜

⎞

⎠⎟

⎡

⎣⎢⎢

+δ Λ+ ,0 −1( )p−q−S+ − J + +l

× s S+ SΩ − Λ + Λ+ + Ω+ −Ω+ − Λ+ −Ω + Λ

⎛

⎝⎜

⎞

⎠⎟

l s j

Λ − Λ+ Ω − Λ + Λ+ + Ω+ −Ω + Ω+

⎛

⎝⎜

⎞

⎠⎟

J + j J

−Ω+ Ω + Ω+ −Ω

⎛

⎝⎜

⎞

⎠⎟

+Δ3 −1( )p−q−S− J

× s S+ S−Ω − Λ + Λ+ − Ω+ Ω+ − Λ+ Ω + Λ

⎛

⎝⎜

⎞

⎠⎟

l s j

Λ − Λ+ −Ω − Λ + Λ+ − Ω+ Ω + Ω+

⎛

⎝⎜

⎞

⎠⎟

J + j J

Ω+ −Ω − Ω+ Ω

⎛

⎝⎜

⎞

⎠⎟

Autoionization Branching Ratios

Γ

i→N +N ,±( ) ∝ α i , N , ±

dμdR

β j , N , ± β j , N , ± l "N + "N , ±,v = 0j , l∑

2

Branching ratios are calculated directly from quantum defect matrix elements, using a simple formula:

Quantum Defects (Coupling acts at short range = Case (b))

Initial State (Hund’s Case (b))

Final States(Hund’s Case (b))

Final States(Hund’s Case (d))