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

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

3. 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

4. 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’

5. 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)

6. 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)

7. 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

8. 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

9. 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

10. 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/ L=4, S=3; 3.5  7F1, 8F1/2
Only few atomic spectral lines exhibit negative Zeeman sptlitting
(Stenflo et al )
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)

11. 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 = P2(9.5): geff = –0.12
SPW3, Tenerife, 2002
Berdyugina & Solanki (2001)

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

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

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

15. 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)

16. 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)

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

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

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

20. 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