Molecules in Magnetic Fields Svetlana Berdyugina Kiepenheuer Institut für Sonnenphysik, Freiburg, Germany Hot Molecules in Exoplanets and Inner Disks

Content Coupling of angular momenta Atomic Zeeman effect Molecular Zeeman effect Molecular Paschen-Back effect SPW3, Tenerife, 2002

Coupling of angular momenta A moving charge creates a magnetic field. The magnetic field associated with the orbital motion of the electron can be idealized as that of an infinitesimal dipole with a magnetic moment: Analogously, another magnetic moment is associated with the electron internal angular momentum (spin): Coupling of angular momenta is the interaction of their magnetic moments LS-coupling leads to the total angular momentum with the quantum number J: J L S

Atomic Zeeman effect M=0 () M=±1 () J=0 or g=g’ J=J’ & g≠g’   M=0 () M=±1 () J=0 or g=g’ J=J’ & g≠g’ J>J’ & g<g’ J>J’ & g>g’

Molecular Zeeman effect (a) If a molecule possesses a magnetic moment, it interacts with an external magnetic field: μ = μL + μS = μ0 (L+2S) ΔEH = –μ • H = – μ0 (L+2S) • H Strong spin coupling – Hund’s case (a) splitting is symmetrical, it is larger for low J numbers and rapidly decreases as J increases Landé factors can be positive and negative SPW3, Tenerife, 2002 Berdyugina & Solanki (2002)

Molecular Zeeman effect (b) Weak spin coupling – Hund’s case (b) splitting is symmetrical and for large J values becomes independent of quantum numbers, =μ0H Landé factors can be positive and negative Intermediate spin coupling – Hund’s case (ab) splitting is symmetrical but depends also on the rotational and spin-orbit coupling constants (perturbation calculations) Landé factors can change their sign with J (Berdyugina & Solanki, 2002) SPW3, Tenerife, 2002 Berdyugina & Solanki (2002)

Molecular Zeeman patterns Strong spin coupling – Hund’s case (a) if ΔΩ=0, the effective Landé factors for R and P branches are zero, lines of Q branches exhibit a simple Zeeman triplet splitting if ΔΩ0, the effective Landé factors for R and P branches are of opposite sign, which is determine by the sign of ΔΩ ΔJ=0 ΔJ=1 ΔJ=–1 Q R P SPW3, Tenerife, 2002

Molecular Zeeman patterns Weak spin coupling – Hund’s case (b) if ΔΛ=0, all branches have non-zero effective Landé factors, which are of opposite sign for different multiplet sub-branches if ΔΛ0, the effective Landé factors for R and P branches are of opposite sign, which is determine by the sign of ΔΛ ΔJ=0 ΔJ=1 ΔJ=–1 Q R P SPW3, Tenerife, 2002

Molecular Zeeman patterns Intermediate spin coupling – Hund’s case(ab) absolute values of the effective Landé factors can increase as J increases the sign of the effective Landé factors can change within a given rotational band ΔJ=0 ΔJ=1 ΔJ=–1 Q R P SPW3, Tenerife, 2002

Negative Landé Factors The effective Landé factor of an atomic or molecular line: geff = ½ (gu +gl) + ¼ (gu –gl) [Ju(Ju+1) – Jl(Jl+1)], geff < 0 if gu< 0 and gl < 0 The Landé factor of an atomic level (LS coupling): g = 3/2 + [S(S+1) – L(L+1)] / [2J(J+1)], J=L+S, ..., |L – S| g < 0 if L(L+1) – S(S+1) > 3J(J+1), ½(L+1) < S < L for J =L – S, L=3, S=2.5  6D1/2 L=4, S=3; 3.5  7F1, 8F1/2 Only few atomic spectral lines exhibit negative Zeeman sptlitting (Stenflo et al. 1984) The Landé factor of a molecular level (Hund’s case a): g = (Λ+Σ)(Λ+2Σ) / J(J+1), J = |Λ+Σ|, |Λ+Σ|+1, |Λ+Σ|+2, ... g < 0 if –Λ < Σ < ½ Λ , Λ=1, Σ= –½, 0  2Π1/2, 3Π 1, ... Λ=2, Σ= –1, –½, 0, ½  3Δ1, 2Δ3/2, 3Δ2, 2Δ5/2, ... Negative Landé factors are common in molecular lines (Berdyugina & Solanki, 2002) SPW3, Tenerife, 2002 Berdyugina & Solanki (2002)

OH Zeeman patterns g2 = –(2J–1) / [(J+1)(2J+1)] Landé factors of OH levels (Hund’s case b): g1 = (2J+3) / [J(2J+1)] g2 = –(2J–1) / [(J+1)(2J+1)] Effective Landé factors of the OH lines obtained by perturbation calculations: P1(10.5): geff = 0.11 P2(9.5): geff = –0.12 SPW3, Tenerife, 2002 Berdyugina & Solanki (2001)

Puzzling OH in sunspots -doublets with positive and negative g-factors Ro-vibrational OH (2,0) (Berdyugina & Solanki 2001): P1(10.5): geff = 0.11 P2(9.5): geff = –0.12 Pure rotational OH (0,0) (Jennings et al.): R1(25.5): geff = 0.038 R2(24.5): geff = –0.039 Mol Stokes tool

First successful data modeling Berdyugina et al. (2000) TiO A3 - X3: Zeeman regime MgH A2 - X2: Paschen-Back regime

Perturbed FeH in sunspots Strong, unknown perturbation from higher electronic states Lande factors: Calibrated quantum-mechanical model observations model Afram et al. (2007, 2008)

Molecular Paschen-Back effects Theory for arbitrary molecular electronic states PBE, full PRT, magnetic splitting, intensity, polarization, scattering (numerically) PBE on the fine structure PBE on the fine and rotational structure Berdyugina et al. (2005)

Molecular Paschen-Back effect Main branch Satellite branch Forbidden branch Peculiarities due to the PBE Stokes profile asymmetries  Net polarization across line profiles Wavelength shifts depending on the magnetic field strength Polarization sign changes Weakening of main branch and strengthening of satellite and forbidden transitions Berdyugina et al. (2005)

PBE in MgH MgH A2 - X2 system : main+satellite(or forbidden) lines Berdyugina et al. (2005)

PBE in CaH CaH A2 - X2 system: sunspots Berdyugina et al. (2006) observations model Berdyugina et al. (2006)

Kuzmychov & Berdyugina (2013) PBE in CrH CrH A6 - X6 Kuzmychov & Berdyugina (2013)

Conclusions Molecular lines are magnetically sensitive: Strong Zeeman signal is expected for transitions with intermediate case spin coupling (e.g. TiO, FeH) Negative Landé factors are common for molecular lines Unusual Stokes V signatures of infrared OH lines is the natural outcome of molecular properties Stokes polarimetry with the pairs of negative and positive Zeeman-split OH lines offers advantages for assessing instrumental effects (I,Q,U to V cross-talk) or magnetooptic effects in the solar atmosphere Molecular Pachen-Back effect leads to higher magnetic sensitivity: Stokes profile asymmetries  Net polarization across line profiles Wavelength shifts depending on the magnetic field strength Polarization sign changes Weakening of main branch and strengthening of satellite and forbidden transitions SPW3, Tenerife, 2002