P. JANSEN, W. UBACHS, H. L. BETHLEM SENSITIVITY OF TRANSITIONS IN INTERNAL ROTOR MOLECULES TO A POSSIBLE VARIATION OF THE PROTON-TO-ELECTRON MASS RATIO P. JANSEN, W. UBACHS, H. L. BETHLEM Institute for Lasers, Life and Biophotonics, VU University, Amsterdam, The Netherlands I. KLEINER Laboratoire Interuniversitaire des Systèmes Atmosphériques, CNRS et Universités Paris Diderot et Paris Est, Créteil, France L-H. XU Department of Physics and Centre for Laser, Atomic and Molecular Sciences, University of New Brunswick, Saint John, Canada a=e2/hc4pe0

General Objectives: Predict spatio- temporal variations of the constants of nature detecting a possible drift of a fundamental constant on a cosmological time scale : possible variation of the fine-structure a constant by Webb et al (PRL 1999) possible variation of the proton-to-electron mass ratio μ (Ubachs,Rheinhold et al (PRL 2006) variation of μ ----> shift in the position of a spectral line ----> change in the spectrum of atoms and molecules . Mu= 1836.15267245(75). Not all lines will shift in the same amount or direction. The response of a transition to a variation of α or μ is characterized by its sensitivity coefficient, Kμ or Kα , which is defined as the proportionality constant between the fractional frequency shift of the transition, Dν/ν, and the fractional shift in α or μ number with a value of about 1/137, the constant dictates the strength of the electromagnetic force and, hence, determines the exact wavelengths of light an atom will absorb 2

Sensitivity coefficient Km The response of a transition to a variation of μ is characterized by its sensitivity coefficient, Km : proportionality constant between the fractional frequency shift of the transition, Dν/ν, and the fractional shift in μ Varshalovich and Levshakov (1993) Laboratory measurements Astronomical measurements m = 1836.15267245(75).

Previous Cosmological observations: 1) comparing optical transitions of molecular hydrogen (H2) in high-redshifted objects (QUASARS) with accurate laboratory measurements  limit of Dμ/μ < 10−5 for look-back times of 12 billion years, Km = −0.05 to +0.02 (Rheinhold et al PRL 2006) 2) Transitions between inversion levels of NH3 (Km = −4.2 ) Astronomical observations of NH3 in microwave  stringent constraints around (1.0 ± 4.7) × 10−7 (Flambaum and Kozlov PRL 2007) 3) Torsional-rotational transitions in CH3OH (Km = −88 to 330) : Jensen et al , Levshakov,Koslov and Reimers 2011 4 transitions used to constrain Dμ/μ at (11.0 ± 6.8) × 10−8 at a look-back time of 7 billion years (Bagdonaite, jensen, Bethleem, Ubachs, Henkel, Menten Science 2013)

The spectroscopic point of view: which molecules (and which type of transitions) have to highest Km ? Sensitivity of Internal rotation transition To calculate K, the energy levels and their dependance in m (the mass) has to be known : Numerical calculations (precise) « Toy » model (physically simple) Sensitivity is increased for transitions between nearly degenerate energy levels showing different dependances in m : internal rotation can fulfill this

What is internal rotation ? Ts symmetry s = 0 : A species s= ±1, E species vt: torsional quantum number J = |K| = 1 ± K, s= +1 K, s= -1 Xu et al Methanol, JMS 2008

HOW TO MODEL INTERNAL ROTATION? For one C3v top, and a frame with a plane of symetry Cs HRAM = Htor + Hrot + Hd.c + Hint 1) Diagonalization of the torsional part of the Hamiltonian in an axis system where torsion-rotation coupling is minimal (Rho Axis Method, RAM), Kirtman et al, Lees and Baker , Herbst et al: Htor= F (pa - r.Jz)2 + V(a) F: internal rotation constant r depends on Itop/Imolecule Eigenvalues = torsional energies 2) Eigenvectors are used to set up the matrix of the rest of the Hamiltonian: Hrot = ARAMJa2 + BRAMJb2 +CRAMJc2 + Dab(JaJb + JbJa) Hd.c usual centrifugal distorsion terms Hint higher order torsional-rotational interactions terms : cos3a et pa and global rotational operators like Ja, Jb , Jc

Theoretical Model: the global approach RAM = Rho Axis Method (axis system) for a Cs (plane) frame HRAM = Hrot + Htor + Hint + Hd.c. Torsional operators and potential function V(a) Constants 1 1-cos3a p2a Japa 1-cos6a p4a Jap3a V3/2 F r V6/2 k4 k3 J2 (B+C)/2* Fv Gv Lv Nv Mv k3J Ja2 A-(B+C)/2* k5 k2 k1 K2 K1 k3K Jb2 - Jc2 (B-C)/2* c2 c1 c4 c11 c3 c12 JaJb+JbJa Dab or Eab dab Dab dab6 DDab ddab Rotational Operators Rotational constants give the structure. Using the diagonalization of the inertial tensor, one can get the orientation angles of the CH3 group relative to the PAS = angle of torsion, r = couples internal rotation and global rotation, ratio of the moment of inertia of the top and the moment of inertia of the whole molecule Kirtman et al 1962 Lees and Baker, 1968 Herbst et al 1986 Hougen, Kleiner, Godefroid JMS 1994

How to scale internal rotation contants in m How to scale internal rotation contants in m ? we assume that the neutron mass has a similar time variation as the proton mass V3, V6, .. : no mass dependance (B.O approx.): m0 A, B, C, Dab, F, direct dependance in the mass: scale as 1/m r depends on Itop/Imolecule: scales as m0 M=mp/me if m changes the mass(neutron)/me changes the same way ! we assume that the neutron mass has a similar timevariation as the proton mass.

Generating the molecular constants, using the scale relation in m calling Belgi as a « slave » program … 3) Calculated the energy levels in terms of m Xu et al JMS 2008 Methanol energy levels, Dm =0

some Transitions of CH3OH 51  60 A 6668.6 MHz Km = -42.1 32  31 E 24928.7 MHz Km =18. 22  21 E 24934.4 MHz Km =18. 20 ->3-1 E 12178.6 MHz Km = -32.8

What about other internal rotors present in interstellar objects? Molecules V3 F s r Jmax Kmax vtmax Nlines Range Npar Std cm-1 cm-1 =4V3/9F Xu et al JMS 2008 Kleiner et al JMS 2009 Ilyushin et al, JMS 2009 Ilyushin et al JPCRD 2008 Ilyushin et al , JMS 2004 CH3SH methyl mercaptan 441 15.0 13 0.66 40 15 3 19700 MW-THz-FIR 78 1.1 Xu et al JCP 2012

The « toy » model The m dependance of a state:(neglecting vibration-rotation-torsion interactions) Where Is the sensitivity of a vibrational level Is the sensitivity of a rotational level Neglecting the vibrational part: With DERot = E’rot- E’’rot DEtor= E’tor - E’’tor

How to estimate DErot and DEtor ? Rotational energy (symmetric top) A, B, C are inversionaly proportional to moments of inertia (mass) , scale like 1/m Torsional energy (Lin and Swalen 1959) F internal rotation constant, scales like 1/m For a DK =+1 transition: s = 4V3/9F A1, B1, C1 : coefficients

Difference in energy DErot and DEtor for the A J,K J+1, K-1 transitions in CH3OH The highest sensitivity coefficients occur when DEtor ≈ -DErot

hn (exp. Data) = DErot + DEtor n, F, g assumed 1 GHz x Km

Maximum Km Km Km max when n = 1GHz DEtors max when ?

Thank you for your attention

Sensitivity Constants Although implicit in previous work, Varshalovich and Levshakov (1993) explicitly developed the sensitivity constant which for a line i is defined as The rest frame wavelengths are related to the observed wavelengths by where m0 is the proton-to-electron mass ratio in the present epoch, at zero redshift, and mz the mass ratio for the absorbing cloud at high redshift. Each line has a unique sensitivity constant Ki which can be slightly negative, zero or positive. The higher the vibrational quantum number the larger the sensitivity constant.

The spectroscopic point of view: which molecules (and which type of transitions) have to highest Km ? 1) Vibrational energy Sensitivity of an vib. Level: w is the vibrational frequency, m is the reduced mass The rest frame wavelengths are related to the observed wavelengths by 2) Rotational energy

How to estimate DErot and DEtor ? Rotational energy (symmetric top) A, B, C are inversionaly proportional to moments of inertia (mass) , scale like 1/m Torsional energy (Lin and Swalen 1959) F rotation constant Interne , scale like 1/m For a DK =+1 transition: s = 4V3/9F

How to estimate DErot and DEtor ? Torsional energy (Lin and Swalen 1959) 1) 2) Fit a0,, a1 r=0.8 s = 4V3/9F 3) Fit r=0.3 r=0.07

F scales as 1/m

Coefficients a1 in function of s Lin and Swalen (1959)

Calcul of Km using the toy model hn (exp. Data) = DErot + DEtor