Anomalous Hyperfine Structure of NSF 3 in the Degenerate Vibrational State v 5 = 1: Lifting of the Parity Degeneracy by the Fluorine Spin-Rotation Interaction H. Harder, S. Macholl, H. Maeder, L. Fusina, and I. Ozier

NOTATION In lowest order, the vibration-rotational levels can be labelled by kl, where k is the signed counterpart of K and l =  1. Also introduce The fluorine nuclear spin is 1/2. The total fluorine spin I T can take the values 1/2 and 3/2, corresponding to the nuclear spin symmetry Γ N = E and A 1, respectively.

klGΓ RV ITIT ΓNΓN Γ RVN p

Parity Degeneracy Lifted by F Spin Rotation Interaction The relevant matrix elements have selection rules in Δk =  1,  2. These are characterized by the spin-rotation constants c(1), c(2), respectively. H

Key to the Measurement of the Parity Splittings NSF 3 is a near-spherical top. In ν 5, at high J and low K, the levels cluster together, and several rovibrational interactions off-diagonal in kl produce severe mixing. (e.g. H 22, H 21, H 24 ) This leads to what we call a “regional resonance”. Each eigenfunction involved consists of a superposition of basis vectors, each with its own value of kl, with 3 or more having expansion coefficients that are appreciable in magnitude (e.g. larger than 0.2). This occurs over a wide range in J. In the present case, the regional resonance investigated extends in J from about 35 to 65.

Fourier Transform Microwave Spectroscopy Measurements between 8 and 26.5 GHz with FTMW waveguide spectrometers. The effective resolution is ≈ 30 kHz. R-branch spectrum for J = 2 ← 1. Q-branch spectra for J from 13 to 62. For the lower states, kl = – 3, – 2, – 1, 0, direct l-type doubling transitions (kl = +1, Γ RV = A 1 /A 2 ) ↔ (kl = – 1, Γ RV = A 2 /A 1 ) for J ≤ direct l-type resonance transitions following the selection rules (Δk = Δl =  2) for J ≤ 60 and G ≤ 3. Several series of perturbation-allowed transitions with Δ(k – l) =  3,  6.

Hyperfine Structure For Γ RV = A 1 ↔ A 2, only doublets are observed with intensities in the ratio 2:1. For Γ RV = E ↔ E, doublets are observed in most cases. However, 24 triplets were measured. The intensity ratios varied from 2:2:1 to 2:3:1. In addition, 3 quartets were observed, each with intensity ratios of about 2:1:2:1. The convention is adopted that the intensities are given from from low frequency to high.

Frequency (MHz) aaa Ψ RV (kl = +3) ↔ Ψ RV (b) J = 51 Γ RV = E ↔ E 67.5 kHz

Frequency (MHz) Ψ RV (kl = +3) ↔ Ψ RV (a) J = 51 Γ RV = E ↔ E kHz 50.6 kHz

Information on the full analysis can now be obtained from: H. Harder, S. Macholl, H. Maeder, L. Fusina, and I. Ozier, Phys. Rev. A 81, (2010).

A2A2 A1A1 A2A2 A1A1 A2A2 A2A2 A1A1 A1A1 (a)(b)(c)(e) |J,kU lU >|J,kU lU > |J,kL lL >|J,kL lL > F 1 =J+1 F 1 =J−1 F1=JF1=J F 1 =J+1 F 1 =J−1 F1=JF1=J F=F 1 +1/2 F=F 1 −1/2 F=F 1 +1/2 F=F 1 −1/2 (d) | J,  2,  1 > | J, 0, > l-type resonance example with J even

Angular Momentum Coupling Scheme Since the nitrogen spin I N = 1, F 1 = (J – 1) ; J ; (J + 1) ; (if J ≠ 0). Since the total fluorine spin I T = 1/2, F = (F 1 – 1/2) ; (F 1 + 1/2) ; (if F 1 ≠ 0).

A2A2 A1A1 A2A2 A1A1 A2A2 A2A2 A1A1 A1A1 (a)(b)(c)(e) |J,kU lU >|J,kU lU > |J,kL lL >|J,kL lL > F 1 =J+1 F 1 =J−1 F1=JF1=J F 1 =J+1 F 1 =J−1 F1=JF1=J F=F 1 +1/2 F=F 1 −1/2 F=F 1 +1/2 F=F 1 −1/2 (d)

A2A2 A1A1 A2A2 A1A1 A2A2 A2A2 A1A1 A1A1 (a)(b)(c)(e) |J,kU lU >|J,kU lU > |J,kL lL >|J,kL lL > F 1 =J+1 F 1 =J−1 F1=JF1=J F 1 =J+1 F 1 =J−1 F1=JF1=J F=F 1 +1/2 F=F 1 −1/2 F=F 1 +1/2 F=F 1 −1/2 (d)

A2A2 A1A1 A2A2 A1A1 A2A2 A2A2 A1A1 A1A1 (a)(b)(c)(e) |J,kU lU >|J,kU lU > |J,kL lL >|J,kL lL > F 1 =J+1 F 1 =J−1 F1=JF1=J F 1 =J+1 F 1 =J−1 F1=JF1=J F=F 1 +1/2 F=F 1 −1/2 F=F 1 +1/2 F=F 1 −1/2 (d) PP

A2A2 A1A1 A2A2 A1A1 A2A2 A2A2 A1A1 A1A1 (a)(b)(c)(e) |J,kU lU >|J,kU lU > |J,kL lL >|J,kL lL > F 1 =J+1 F 1 =J−1 F1=JF1=J F 1 =J+1 F 1 =J−1 F1=JF1=J F=F 1 +1/2 F=F 1 −1/2 F=F 1 +1/2 F=F 1 −1/2 (d) PP

A2A2 A1A1 A2A2 A1A1 A2A2 A2A2 A1A1 A1A1 (a)(b)(c)(e) |J,kU lU >|J,kU lU > |J,kL lL >|J,kL lL > F 1 =J+1 F 1 =J−1 F1=JF1=J F 1 =J+1 F 1 =J−1 F1=JF1=J F=F 1 +1/2 F=F 1 −1/2 F=F 1 +1/2 F=F 1 −1/2 (d)

A2A2 A1A1 A2A2 A1A1 A2A2 A2A2 A1A1 A1A1 (a)(b)(c)(e) |J,kU lU >|J,kU lU > |J,kL lL >|J,kL lL > F 1 =J+1 F 1 =J−1 F1=JF1=J F 1 =J+1 F 1 =J−1 F1=JF1=J F=F 1 +1/2 F=F 1 −1/2 F=F 1 +1/2 F=F 1 −1/2 (d)

Values Determined for the F Spin Rotation Constants To our knowledge, this is the first determination of based on frequency measurements for any symmetric top. Other methods of determining c(2) include: light drift (CH 3 F); beam maser (NH 3 ); decelerated molecular beams (ND 3 ); electric resonance molecular beams (CH 3 D). H

Acknowledgments Dr. P. Jensen and Dr. P.R. Bunker for insightful comments on the role of symmetry conventions in the derivation of the spin-rotation matrix elements.

ν ν ν F 1 =J±1 F 1 =J (a) (b) (c)

Parity Degeneracy Lifted by F Spin Rotation Interaction The relevant matrix elements have selection rules in Δk =  1,  2. These are characterized by the spin-rotation constants c(1), c(2), respectively. where the parameter c  characterizes the internal magnetic field (at a reference fluorine nucleus) in direction generated by rotation of the molecule about direction in the molecule- fixed frame. H

Examples of the Severe Mixing In eigenfunction Ψ RV (a), basis functions Ψ RV (kl) have superposition constants with magnitudes ≥ 0.2 for kl = – 3, – 1, 0, +2, and +3. For J ≤ 51, Ψ RV (kl = – 1) is dominant. For J ≥ 52, Ψ RV (kl = 0 ) is dominant. In eigenfunction Ψ RV (b), basis functions Ψ RV (kl) have superposition constants with magnitudes ≥ 0.2 for kl = – 3, – 1, 0, +2, and +3. For J ≤ 43, Ψ RV (kl = – 3) is dominant. For 44 ≤ J ≤ 51, Ψ RV (kl = 0 ) is dominant. For J ≥ 52, Ψ RV (kl = – 1) is dominant. All the basis functions in this regional resonance have Γ RV = E.