Anomalous phosphine sensitivity coefficients as probes for a possible variation of the proton-to-electron mass ratio
MMon. Not. R. Astron. Soc. , 1–10 (2017) Printed 17 August 2018 (MN L A TEX style file v2.2)
Anomalous phosphine sensitivity coefficients as probes fora possible variation of the proton-to-electron mass ratio
A. Owens , , (cid:63) , S. N. Yurchenko and V. ˇSpirko , † The Hamburg Center for Ultrafast Imaging, Universit¨at Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany Center for Free-Electron Laser Science (CFEL), Deutsches Elektronen-Synchrotron DESY, Notkestrasse 85, 22607 Hamburg, Germany Department of Physics and Astronomy, University College London, Gower Street, WC1E 6BT London, United Kingdom Academy of Sciences of the Czech Republic, Institute of Organic Chemistry and Biochemistry,Flemingovo n´am. 2, 166 10 Prague 6, Czech Republic Department of Chemical Physics and Optics, Faculty of Mathematics and Physics, Charles University in Prague,Ke Karlovu 3, CZ-12116 Prague 2, Czech Republic
Accepted XXXX. Received XXXX; in original form XXXX
ABSTRACT
A robust variational approach is used to investigate the sensitivity of the rotation-vibration spectrum of phosphine (PH ) to a possible cosmological variation of theproton-to-electron mass ratio, µ . Whilst the majority of computed sensitivity coef-ficients, T , involving the low-lying vibrational states acquire the expected values of T ≈ − T ≈ − / A − A splittings in the ν /ν , ν /ν and 2 ν (cid:96) =04 / ν (cid:96) =24 manifolds of PH . A pronounced Coriolis interaction between thesestates in conjunction with accidentally degenerate A and A energy levels produces aseries of enhanced sensitivity coefficients. Phosphine is expected to occur in a numberof different astrophysical environments and has potential for investigating a driftingconstant. Furthermore, the displayed behaviour hints at a wider trend in molecules of C (M) symmetry, thus demonstrating that the splittings induced by higher-order ro-vibrational interactions are well suited for probing µ in other symmetric top moleculesin space, since these low-frequency transitions can be straightforwardly detected byradio telescopes. Key words: molecular data - infrared: ISM - submillimetre: ISM - cosmologicalparameters
Recently, the J = 2 − ) was detected in the carbon star envelope IRC +10216 (Ag´undezet al. 2014), thus confirming the presence of PH in the outflows of evolved stars but more significantly outside of thesolar system. The appearance of PH has been predicted in numerous other astrophysical environments (see the discussionby Sousa-Silva et al. (2015) and references therein), and because of prominent ‘irregularities’ displayed by its rotation-vibration spectrum, it is a promising system for investigating the cosmological variability of the proton-to-electron mass ratio, µ = m p /m e . Observing PH outside of our Galaxy is no easy feat, however, nearby Galactic molecular clouds offer a meansto constrain µ through the so-called chameleon scenario (Khoury & Weltman 2004; Brax et al. 2004) as evidenced by studiesof ammonia (Levshakov et al. 2010a,b) and methanol (Dapr`a et al. 2017).At present, the most robust constraint on a temporal variation of µ was determined from methanol absorption spectraobserved in the lensing galaxy PKS1830 −
211 (Kanekar et al. 2015). The three measured transitions possessed sensitivitycoefficients, T , ranging from − . − . µ/µ < × − yr − assuming a linear rate ofchange. This translates to no change in µ over the past ≈ . (cid:63) The corresponding author: [email protected] † The corresponding author: [email protected] (cid:13) a r X i v : . [ a s t r o - ph . C O ] A ug Owens, Yurchenko and ˇSpirko ν ν ν ν + ν ν ν ν E / cm -1 Ground State
Vibrational Energy Levels PH Figure 1.
The lowest vibrational energy levels of PH . constraint to date, which measured optical transitions in Yb + ions to derive ˙ µ/µ = (0 . ± . × − yr − (Godun et al.2014) again assuming a linear rate of change. Whilst the use of methanol has led to several astronomical constraints (Jansenet al. 2011; Levshakov, Kozlov & Reimers 2011; Bagdonaite et al. 2013a,b; Thompson 2013; Kanekar et al. 2015), it isworthwhile identifying other molecular absorbers with notable sensitivities to expand the search for a drifting µ .Due to the small difference between its rotational constants B and C , and also because of the strong x − y Coriolisinteraction between the coinciding ν /ν , ν /ν and 2 ν (cid:96) =04 / ν (cid:96) =24 states (see Fig. 1), phosphine is a potential candidatesystem for probing µ . Notably, the spectrum of PH , and presumably other molecules of C (M) symmetry, is special due tothe anomalous behaviour of the A − A splittings (Ulenikov et al. 2002). A large number of spectroscopic studies of PH havebeen reported in the literature (see M¨uller (2013) and references therein) and highly accurate data is available for the majorityof its states. Furthermore, a robust theoretical description of this molecule, which we utilize for this work, has been developedover the years (Yurchenko et al. 2003, 2005, 2006; Ovsyannikov et al. 2008a,b; Sousa-Silva, Yurchenko & Tennyson 2013;Sousa-Silva et al. 2014, 2015; Sousa-Silva, Tennyson & Yurchenko 2016), culminating in the construction of a comprehensiverotation-vibration line list applicable for elevated temperatures (Sousa-Silva et al. 2015).Model radiative transfer calculations of phosphine excitation in the envelope of IRC +10216 (Ag´undez et al. 2014;Cernicharo et al. 1999) highlighted the importance of infrared pumping from the ground to the first excited vibrational states,helping explain the presence of strong emission bands in the observed spectra. We therefore find it useful to investigate thesensitivity of the ground, fundamental, and low-lying combination and overtone vibrational states of PH (see Fig. 1) to apossible space-time variation of µ using a robust variational approach. The paper is structured as follows: In Sec. 2 we describethe variational approach used to compute sensitivity coefficients. The results for the phosphine molecule are presented anddiscussed in Sec. 3. Concluding remarks are given in Sec. 4. The sensitivity coefficient T u,l between an upper and lower state with energy E u and E l , respectively, is defined as T u,l = µE u − E l (cid:18) d E u d µ − d E l d µ (cid:19) , (1)and can be related to the induced frequency shift of a transition, or energy difference E u − E l between two states, throughthe expression ∆ νν = T u,l ∆ µµ , (2)where ∆ ν = ν obs − ν is the change in the frequency, and ∆ µ = µ obs − µ is the change in µ , both with respect to theirpresent day values ν and µ . By assuming all baryonic matter can be treated equally (Dent 2007), µ is proportional to themolecular mass. One can then perform a series of calculations with suitably scaled values for the masses of the P and H atomsand extract numerical values for the derivatives d E/ d µ using central finite differences. c (cid:13)000
The lowest vibrational energy levels of PH . constraint to date, which measured optical transitions in Yb + ions to derive ˙ µ/µ = (0 . ± . × − yr − (Godun et al.2014) again assuming a linear rate of change. Whilst the use of methanol has led to several astronomical constraints (Jansenet al. 2011; Levshakov, Kozlov & Reimers 2011; Bagdonaite et al. 2013a,b; Thompson 2013; Kanekar et al. 2015), it isworthwhile identifying other molecular absorbers with notable sensitivities to expand the search for a drifting µ .Due to the small difference between its rotational constants B and C , and also because of the strong x − y Coriolisinteraction between the coinciding ν /ν , ν /ν and 2 ν (cid:96) =04 / ν (cid:96) =24 states (see Fig. 1), phosphine is a potential candidatesystem for probing µ . Notably, the spectrum of PH , and presumably other molecules of C (M) symmetry, is special due tothe anomalous behaviour of the A − A splittings (Ulenikov et al. 2002). A large number of spectroscopic studies of PH havebeen reported in the literature (see M¨uller (2013) and references therein) and highly accurate data is available for the majorityof its states. Furthermore, a robust theoretical description of this molecule, which we utilize for this work, has been developedover the years (Yurchenko et al. 2003, 2005, 2006; Ovsyannikov et al. 2008a,b; Sousa-Silva, Yurchenko & Tennyson 2013;Sousa-Silva et al. 2014, 2015; Sousa-Silva, Tennyson & Yurchenko 2016), culminating in the construction of a comprehensiverotation-vibration line list applicable for elevated temperatures (Sousa-Silva et al. 2015).Model radiative transfer calculations of phosphine excitation in the envelope of IRC +10216 (Ag´undez et al. 2014;Cernicharo et al. 1999) highlighted the importance of infrared pumping from the ground to the first excited vibrational states,helping explain the presence of strong emission bands in the observed spectra. We therefore find it useful to investigate thesensitivity of the ground, fundamental, and low-lying combination and overtone vibrational states of PH (see Fig. 1) to apossible space-time variation of µ using a robust variational approach. The paper is structured as follows: In Sec. 2 we describethe variational approach used to compute sensitivity coefficients. The results for the phosphine molecule are presented anddiscussed in Sec. 3. Concluding remarks are given in Sec. 4. The sensitivity coefficient T u,l between an upper and lower state with energy E u and E l , respectively, is defined as T u,l = µE u − E l (cid:18) d E u d µ − d E l d µ (cid:19) , (1)and can be related to the induced frequency shift of a transition, or energy difference E u − E l between two states, throughthe expression ∆ νν = T u,l ∆ µµ , (2)where ∆ ν = ν obs − ν is the change in the frequency, and ∆ µ = µ obs − µ is the change in µ , both with respect to theirpresent day values ν and µ . By assuming all baryonic matter can be treated equally (Dent 2007), µ is proportional to themolecular mass. One can then perform a series of calculations with suitably scaled values for the masses of the P and H atomsand extract numerical values for the derivatives d E/ d µ using central finite differences. c (cid:13)000 , 1–10 H probes of a variable µ -3-2-1 0 1 2 0 100 200 300 400 500 600 700 800 T n gs ν ν Figure 2.
Sensitivity coefficients T for pure rotational transitions in the ground, ν , and ν vibrational states of PH . Here n is arunning number which counts the number of transitions. -1-0.75-0.5-0.25 0
0 100 200 300 400 500 600 ν ν ν ν ν - ν T n Figure 3.
Sensitivity coefficients T for ro-vibrational transitions from the ground to the lowest vibrational states of PH . Here n is arunning number which counts the number of transitions. Sensitivity coefficients for PH have been computed with the same variational approach as was previously employed forammonia (Owens et al. 2015b, 2016) and the hydronium cation (Owens et al. 2015a). Calculations were carried out with thenuclear motion program trove (Yurchenko, Thiel & Jensen 2007; Yachmenev & Yurchenko 2015; Yurchenko, Yachmenev &Ovsyannikov 2017) and utilized the potential energy surface (PES), dipole moment surface (DMS), and computational setupof Sousa-Silva et al. (2015), which have all undergone rigorous testing and are known to be reliable. We refer the readerto Sousa-Silva et al. (2015) for further details of the nuclear motion computations. All sensitivity coefficients, Eq. (1), havebeen determined with calculated frequencies, E u − E l , as oppose to experimental values when available. This was done forconsistency and to verify the trend in sensitivities displayed by PH , which we will discuss further in Sec. 3. In general, as shown in Table 1, Fig. 2 and Fig. 3, the majority of the calculated sensitivity coefficients for the low-lyingvibrational states acquire the expected values of T ≈ − T ≈ − / J = 2 − J = 1 − A − A doublets of PH . As is well known for a molecule with C (M) symmetry, all rotation-vibration energy levels corresponding to the same K ≡ | k | (cid:54) = 0 rotational quantum number c (cid:13) , 1–10 Owens, Yurchenko and ˇSpirko
Table 1.
Calculated frequency ν calc (in MHz), frequency difference ∆ c − e (in MHz) compared to experimental value from Belov et al.(1981), Einstein A coefficient (in s − ), and sensitivity coefficient T for vibrational ground state transitions of PH .Γ (cid:48) J (cid:48) K (cid:48) Γ (cid:48)(cid:48) J (cid:48)(cid:48) K (cid:48)(cid:48) ν calc ∆ c − e A T
Allowed A A E E A A E E E E A A A A A A E E E E A A Forbidden E E E E E E E E E
10 1 E
10 2 46404.9 27.1 0.540E-11 -0.85 E
11 1 E
11 2 46090.1 31.6 0.775E-11 -0.85 E
12 1 E
12 2 45748.3 33.5 0.108E-10 -0.86 E
13 1 E
13 2 45382.6 34.7 0.146E-10 -0.83 E
14 1 E
14 2 44995.8 37.2 0.193E-10 -0.96 E
15 1 E
15 2 44591.1 42.2 0.251E-10 -0.92 A A A A A A A A A A A A A A A
10 0 A
10 3 139307.6 81.0 0.275E-09 -0.97 A
11 0 A
11 3 138318.2 90.2 0.398E-09 -0.95 A
12 0 A
12 3 137230.0 95.3 0.557E-09 -0.93 A
13 0 A
13 3 136045.8 104.4 0.756E-09 -0.90 A
14 0 A
14 3 134750.7 109.3 0.100E-08 -0.93 E E E E E E E E E
10 2 E
10 5 326884.7 194.4 0.130E-08 -0.98 E
11 2 E
11 5 324645.2 209.3 0.199E-08 -0.95 E
12 2 E
12 5 322237.9 228.9 0.290E-08 -0.95 E
13 2 E
13 5 319665.7 247.9 0.406E-08 -0.96 E
14 2 E
14 5 316940.6 268.7 0.552E-08 -0.93 E
15 2 E
15 5 314068.6 288.1 0.731E-08 -0.92 A A A A A A A A A A A A A
10 3 A
10 6 422092.8 249.8 0.213E-08 -0.96 A
10 3 A
10 6 421984.9 247.2 0.213E-08 -0.96 A
11 3 A
11 6 419238.8 271.7 0.343E-08 -0.96 A
11 3 A
11 6 419052.9 269.8 0.343E-08 -0.96 A
12 3 A
12 6 416186.9 297.6 0.519E-08 -0.95 A
12 3 A
12 6 415878.1 294.8 0.518E-08 -0.94 A
13 3 A
13 6 412949.1 320.2 0.748E-08 -0.94 A
13 3 A
13 6 412457.5 316.6 0.747E-08 -0.95 A
14 3 A
14 6 409558.5 350.2 0.104E-07 -0.94 A
14 3 A
14 6 408797.0 341.3 0.104E-07 -0.94 A
15 3 A
15 6 406029.9 377.3 0.140E-07 -0.93 A
15 3 A
15 6 404893.7 366.7 0.139E-07 -0.92c (cid:13)000
15 6 404893.7 366.7 0.139E-07 -0.92c (cid:13)000 , 1–10 H probes of a variable µ ν cm -1 n E <--> E A <--> A -20-15-10-5 0 5 10 0 500 1000 1500 2000 2500 T n
E <--> E A <--> A Figure 4.
The wavenumbers ν (in cm − ) and sensitivity coefficients T of the ν ← ν ro-vibrational transitions of PH . Here n is arunning number which counts the number of transitions. and having overall A , A symmetry are split into doublets due to different ro-vibrational interactions (see, for example,Chen & Oka (1989)). For the nondegenerate vibrational states, the A − A splittings occur for rotational levels with K = 3 n ( n = 1 , , . . . ). For the doubly degenerate fundamental vibrational states characterized by the vibrational angular momentumquantum number (cid:96) (cid:54) = 0, the splittings occur for the K = 1 , , , , , . . . levels.In Tables 2 to 9, we have computed sensitivity coefficients for a large number of the A − A doublets for low-lyingvibrational states. The results suggest that sensitivities of the A − A splittings for non-coinciding ro-vibrational statespossess values dependent on the rotational quantum number J . For example, T ≈ − . , − , − k = 1 , ,
3, respectively(see Tables 2, 3, 4 and 5). It would be interesting to see if this trend is present in other molecules of C (M) symmetry. Forthe sensitivities corresponding to coinciding states, there is a strong and irregular dependence on the x − y Coriolis interactionthat can produce values at least one order of magnitude larger than the respective Coriolis-free predictions. This behaviouris similar to that of NH (ˇSpirko 2014; Owens et al. 2015b, 2016) and H O + (Owens et al. 2015a).A detailed study of the A − A splittings in the 2 ν (cid:96) =24 state was presented by Ulenikov et al. (2002) where it was shownthat the dependence of the splitting on J in the K = 1 rotational sub-levels was anomalous between J = 3–8. This anomalyis caused by an interaction with the closely lying 2 ν (cid:96) =04 state ( K = 0). In Fig. 5 and Table 9 we show the A − A splittingsin the 2 ν (cid:96) =24 state and corresponding sensitivity coefficients with respect to J . Aside from the J = 7 sensitivity coefficient,which greatly increases when using the experimental frequency value, there is good agreement with the work of Ulenikov et al.(2002) and the sensitivities are highly anomalous.It should be stated that for very energetically close coinciding states our variational approach may not be capable of atruly quantitative description. This is the reason why sensitivities have not been computed for certain extremely small A − A splittings. Also, where computed frequencies noticeably differ from the experimental values the resultant sensitivities shouldonly be regarded as illustrative, for example, in Table 8. We have encountered this problem before (Owens et al. 2016) andwhilst the underlying numerical derivatives are relatively stable, it is safer to regard the predicted sensitivity coefficients with c (cid:13) , 1–10 Owens, Yurchenko and ˇSpirko
Table 2.
Calculated and experimental k = 3, A − A splittings (in MHz) and their sensitivities in the ground (gs) and ν vibrationalstates of PH . J ν exp ν calc T ν exp ν calc T gs ν a b a b a b a c a c a c c c c c c c a Davies et al. (1971), b Chen & Oka (1989), c Papouˇsek et al. (1989).
Table 3.
Calculated and experimental (Ulenikov et al. 2002) k = 3, A − A splittings (in MHz) and their sensitivities in the ν and2 ν (cid:96) =24 vibrational states of PH . J ν exp ν calc T ν exp ν calc T ν ν (cid:96) =24 -30-20-100 T Calc. Exp. ( A - A ) , c m - J Calc. Exp. Figure 5.
The A – A splittings in the 2 ν (cid:96) =24 state of PH (lower panel) and the corresponding sensitivities T (upper panel). Theexperimentally determined energies by Ulenikov et al. (2002) were used in Eq. (1) to estimate the T exp values.c (cid:13)000
The A – A splittings in the 2 ν (cid:96) =24 state of PH (lower panel) and the corresponding sensitivities T (upper panel). Theexperimentally determined energies by Ulenikov et al. (2002) were used in Eq. (1) to estimate the T exp values.c (cid:13)000 , 1–10 H probes of a variable µ Table 4.
Calculated and experimental k = 1 and k = 2, A − A splittings (in MHz) and their sensitivities in the ν vibrational state ofPH . J ν exp ν calc T ν exp ν calc T k = = a a e b e c e d e d e d e d e d e d e d e d d d d d d d d a Scappini & Schwarz (1981), b Guarnieri, Scappini & Di Lonardo (1981), c Belov et al. (1983), d Tarrago, Dang-Nhu & Goldman (1981), e Papouˇsek et al. (1989).
Table 5.
Calculated and experimental (Ulenikov et al. 2002) k = 1 and k = 2, A − A splittings (in MHz) and their sensitivities in the ν vibrational state of PH . J ν exp ν calc T ν exp ν calc T k = = caution. Despite this, a large number of the computed A − A splittings are in good agreement with experiment and, moreimportantly, reside in the radio frequency region. The sensitivity of the rotation-vibration spectrum of PH to a possible variation of µ has been probed using an accuratevariational approach. Calculations utilized the nuclear motion program trove in conjunction with an established empiricallyrefined PES and ab initio DMS. The low-lying vibrational states were studied as these play an important role in phosphineexcitation in the carbon star envelope IRC +10216. Whilst the majority of computed sensitivity coefficients assumed theirexpected values, anomalous sensitivities were displayed by the A − A splittings in the ν /ν , ν /ν and 2 ν (cid:96) =04 / ν (cid:96) =24 manifolds.This behaviour arises due to strong Coriolis interactions between states and may be present in other molecules with C (M) c (cid:13) , 1–10 Owens, Yurchenko and ˇSpirko
Table 6.
Calculated and experimental (Ulenikov et al. 2002) k = 4 and k = 5, A − A splittings (in MHz) and their sensitivities in the ν vibrational state of PH . J ν exp ν calc T ν exp ν calc T k = = Table 7.
Calculated and experimental k = 4 and k = 7, A − A splittings (in MHz) and their sensitivities in the ν vibrational state ofPH . J ν exp ν calc T ν exp ν calc T k = = a a a a a c a c a c a c a c a c b c a Davies et al. (1971), b Chen & Oka (1989), c Papouˇsek et al. (1989). symmetry. The fact that molecules with highly sensitive transitions such as ammonia are already being used in advancedterrestrial experiments (Cheng et al. 2016) suggests that PH may not be a primary candidate for constraining µ in laboratorystudies. Its merit as a probe for a drifting constant is more likely to be in cosmological settings as it is a relevant astrophysicalmolecule with a well documented spectrum and a negligible hyperfine splitting (M¨uller 2013). However, it is hard to commenton the necessary conditions for its detection since its presence and formation are not well understood (see the discussion bySousa-Silva et al. (2015) and references therein). Despite this, PH as a model system shows that the splittings caused byhigher-order rotation-vibration interactions, which are essentially low-frequency transitions that can be measured using radiotelescopes, have real potential for investigating a possible variation of µ . Table 8.
Calculated and experimental (Ulenikov et al. 2002) k = 7 and k = 8, A − A splittings (in MHz) and their sensitivities in the ν vibrational state of PH . J ν exp ν calc T ν exp ν calc T k = =
12 30.55 -64 1340 743.7 -9113 1260170 1.44 18210 3414 1817 8766 0.21 3136 10565 -4.9115 2585.5 27.6 1344.4 -22.316 807.4 -1.11 1370.2 11.717 1485.1 -22.9 5056.4 -16.818 1315.0 -417 73.0 -3.1 c (cid:13)000
12 30.55 -64 1340 743.7 -9113 1260170 1.44 18210 3414 1817 8766 0.21 3136 10565 -4.9115 2585.5 27.6 1344.4 -22.316 807.4 -1.11 1370.2 11.717 1485.1 -22.9 5056.4 -16.818 1315.0 -417 73.0 -3.1 c (cid:13)000 , 1–10 H probes of a variable µ Table 9.
Calculated and experimental (Ulenikov et al. 2002) k = 1, A − A splittings (in MHz) and their sensitivities in the 2 ν (cid:96) =24 vibrational state of PH . The splitting ν = ∆ E A /A = ( E A − E A · ( − J ). The sensitivity T exp is obtained using the frequencies fromUlenikov et al. (2002) instead of the computed values. J ν calc T calc ν exp T exp . − . . − .
62 592 . − . . − .
43 653 . − . . − .
94 286 . − . . − .
75 48 . . − . .
46 31 . − . − . − .
27 150 . − . − . − .
48 370 . − . . − .
29 656 . − . . − .
210 964 . − . . − .
011 1240 . − . . − .
312 1432 . − . . − .
913 1495 . − . . − . ACKNOWLEDGMENTS
A.O. acknowledges support from the Deutsche Forschungsgemeinschaft (DFG) through the excellence cluster “The HamburgCenter for Ultrafast Imaging – Structure, Dynamics and Control of Matter at the Atomic Scale” (CUI, EXC1074). S.Y.acknowledges support from the COST action MOLIM No. CM1405. V.S. acknowledges the research project RVO:61388963(IOCB) and support from the Czech Science Foundation (grant P209/15-10267S).
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