Transcript Columbus13-fond-const-25mai13.ppt

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
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)

mp
me
variation of μ ----> shift in the position of a spectral line
----> change in the spectrum of atoms and
molecules
Sensitivity coefficient K
 The response of a transition to a variation of μ is characterized
by its sensitivity coefficient, K : 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
 = 1836.15267245(75).
Previous Cosmological observations:

1) comparing optical transitions of molecular hydrogen (H2) in highredshifted objects (QUASARS) with accurate laboratory measurements 
limit of Dμ/μ < 10−5 for look-back times of 12 billion years,
K = −0.05 to +0.02 (Rheinhold et al PRL 2006)

2) Transitions between inversion levels of NH3 (K = −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 (K = −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 K ?
Sensitivity of Internal rotation transition
To calculate K, the energy levels and their dependance in 
(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  : internal rotation can fulfill this
Ts symmetry
s = 0 : A species
s= ±1, E species
What is internal rotation ?
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)
Rotational Operators
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
dab
dab6
DDab
ddab
1
Kirtman et al 1962
Lees and Baker, 1968
Herbst et al 1986
a = 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
Hougen, Kleiner, Godefroid JMS 1994
How to scale internal rotation contants in  ?
we assume that the neutron mass has a similar time variation as the proton mass
V3, V6, .. : no mass dependance (B.O approx.): 0
A, B, C, Dab, F, direct dependance in the mass: scale as 1/
r depends on Itop/Imolecule: scales as 0
1) Generating the molecular constants, using the scale relation in 
2) calling Belgi as a « slave » program …
3) Calculated the energy levels in terms of 
Xu et al JMS 2008
Methanol energy levels, D =0
some
Transitions of CH3OH
51  60 A 6668.6 MHz
K = -42.1
32  31 E 24928.7 MHz
K =18.
22  21 E 24934.4 MHz
K =18.
20 ->3-1 E 12178.6 MHz
K = -32.8
What about other internal rotors present in interstellar objects?
Molecules

V3
F
s
cm-1
cm-1
=4V3/9F
r
Jmax Kmax
vtmax
Nlines Range Npar Std
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
Xu et al JCP 2012
441
15.0
13
0.66 40
15
3
19700 MW-THz-FIR 78 1.1
The « toy » model
The  dependance of a state:(neglecting vibration-rotation-torsion interactions)
Where
K vib  -1 / 2
Is the sensitivity of a vibrational level
K rot  -1
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/
Torsional energy (Lin and Swalen 1959)
F internal rotation
constant, scales like
1/
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
K
Maximum K
K
K 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
Ki 

d ln i  d i
 dn i

d ln  i d 
ni d
The rest frame wavelengths are related to the observed wavelengths by
i
i0
 (1 + z )(1 + K i D )

 where 0 is the proton-to-electron mass ratio in the present epoch, at zero
redshift, and z 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 K ?
dEup dElo
 di

K 

(
)
i d Eup - Elo d
d
1) Vibrational energy
Evib  ( v + 1 / 2)
 
1
2c
k
m
2) Rotational energy
E rot  BJ ( J + 1) + ( A - B ) K 2
h2
h2
A  2 ,B  2
8 I a
8 I b
Sensitivity of an vib. Level:
w is the vibrational frequency,
m is the reduced mass
K rot  -1
K vib  -1 / 2
How to estimate DErot and DEtor ?
Rotational energy (symmetric top)
A, B, C are inversionaly proportional to moments of inertia (mass) ,
scale like 1/
Torsional energy (Lin and Swalen 1959)
F rotation constant
Interne , scale like
1/
For a DK =+1 transition:
s = 4V3/9F
How to estimate DErot and DEtor ?
Torsional energy (Lin and Swalen 1959)
1)
2) Fit
r=0.8
a0,, a1
s = 4V3/9F
3) Fit
r=0.3
r=0.07
F scales as 1/
Coefficients a1 in function of s
Lin and Swalen (1959)
Calcul of K using the toy model
K rot  -1
hn (exp. Data) = DErot + DEtor