How to calculate spin-spin coupling and spin-rotation coupling strengths and their

uncertainties from spectroscopic data: Application to 6,6Li2 c(13Σg

+)

Nike DattaniOxford University

Xuan LiLawrence Berkeley

National Lab

Nature of the Problem2Σ states: For each v and each N, there are two J:

v vibrationN nuclear rotation ( Bv )J nuclear rotation + electron spin ( γv )

Nature of the Problem2Σ states: For each v and each N, there are two J:

𝐵𝑣𝑁 (𝑁+1 ) −½ γ 𝑣(𝑁+1)𝐵𝑣𝑁 (𝑁+1 )+½ γ v 𝑁

1927 Hund1929 Van Vleck

1930 Mulliken

J = N - ½

J = N + ½

Nature of the Problem

Δ

Nature of the Problem3Σ states: For each v and each N, there are three J.

v vibrationN nuclear rotation ( Bv )J nuclear rotation + electron spin ( γv ) electron spin + electron spin ( λv )

Nature of the Problem3Σ states: For each v and each N, there are three J.

𝐵𝑣𝑁 (𝑁+1 )

J = N +1 J = NJ = N -1

1929 Kramers1937 Schlapp

Nature of the Problem

Δ

Δ

Nature of the Problem

Δ

Δ

Nature of the Problem2Σ state 3Σ state

1927 Hund, 1929 Van Vleck 1937 Schlapp

Nature of the Problem2Σ state 3Σ state

1927 Hund, 1929 Van Vleck 1937 Schlapp

Each v and each N has two J( 1 energy gap )

Each v and each N has three J( 2 energy gaps )

Nature of the Problem2Σ state 3Σ state

1927 Hund, 1929 Van Vleck 1937 Schlapp

Each v and each N has two J( 1 energy gap )

Each v and each N has three J( 2 energy gaps )

spin-rotation coupling () spin-rotation and spin-spin coupling ( , )

Nature of the Problem2Σ state 3Σ state

1927 Hund, 1929 Van Vleck 1937 Schlapp

Each v and each N has two J( 1 energy gap )

Each v and each N has three J( 2 energy gaps )

spin-rotation coupling () spin-rotation and spin-spin coupling ( , )

Δ

ΔΔ

Nature of the Problem2Σ state 3Σ state

1927 Hund, 1929 Van Vleck 1937 Schlapp

Each v and each N has two J( 1 energy gap )

Each v and each N has three J( 2 energy gaps )

spin-rotation coupling () spin-rotation and spin-spin coupling ( , )

Solving for EASY ! Solving for ( , ) EASY !

Δ

ΔΔ

Nature of the Problem2Σ state 3Σ state

1927 Hund, 1929 Van Vleck 1937 Schlapp

Each v and each N has two J( 1 energy gap )

Each v and each N has three J( 2 energy gaps )

spin-rotation coupling () spin-rotation and spin-spin coupling ( , )

Solving for EASY ! Solving for ( , ) EASY !

Uncertainty in EASY ! Uncertainty in ( , ) Never done since 1937 !!

Uncertainty in Bv ???Uncertainty propagation with Newton iterations ???

Δ

ΔΔ

Application to 6,6Li2 c(13Σg+)

For v = 20-26 , N = 1 , all three J energies are seen to (+/- 0.00002 cm-1 , +/- 600 kHz)ie. λv and γv can easily be determined

We want to know their uncertainties.

We have an excellent MLR potential ie. we have the parameters and their uncertainties, for the potential

Problem 1: Uncertainty in Bv

Given the parameters of an MLR potential and their uncertainties, we can find the uncertainties of properties that come from the potential.

For Bv :

w.r.t. each parameter of the

potential

uncertainty of each parameter of the

potential Correlation matrix

Problem 1: Uncertainty in Bv

Given the parameters of the potential and their uncertainties, we can find the uncertainties of properties that come from the potential.

For Bv :

R. J. Le Roy (1998) JMS 191, 223

Problem 1: Uncertainty in Bv

(j)

Jeremy Hutson (1981) solved a similar DE, but with

POTFIT now has Tellinguisen’s implementation in CDJOEL, thanks to Bob Le Roy !

Problem 2: Uncertainty propagation

Δ

Δ

We now have ΔBv . We also have uncertainty in ΔE1 and ΔE2 from experiment.

Δλv , Δγv ?

For N =1

For N =1

For N = 1

Derivatives calculated analyticallyUncertainties in λv and γv calculated analytically

Application to 6,6Li2 c(13Σg+)

Mystery: N = 0, at B=185 GNo spin-spin or spin-rotation coupling. Three Zeeman levels.

Mystery: N = 2, at B=185 GSpin-spin and spin-rotation coupling back. Now four levels !

Problem 1: How do we calculate uncertainty in Bv given an analytic potential ?

Solution: POTFIT now readily does it (uses Hutson’s 1981 perturbation theory)

Problem 2: How do we propagate the uncertainty in Bv to get unc. in λv and γv ?

Solution: Analytic formulas now available

Problem 3: What about Zeeman interaction ?

Solution: Unknown at the moment

Conclusions

Thanks to: Prof. Bob LeRoy (discussions)Prof. Kirk MadisonMariusz SemczukWill GuntonMagnus HawJulien Witz Dr. Art MillsProf. David J. Jones

(experiments)