Highly Sensitive Ammonia Probes of a Variable Proton-to-Electron

Letter pubs.acs.org/JPCL

Highly Sensitive Ammonia Probes of a Variable Proton-to-Electron Mass Ratio Vladimír Špirko* Academy of Sciences of the Czech Republic, Institute of Organic Chemistry and Biochemistry, Flemingovo nám. 2, 166 10 Prague 6, Czech Republic ABSTRACT: The mass sensitivity of the vibration−rotation− inversion energy levels of ammonia is probed using the nonrigid inverter theory. It is shown that the sensitivity exhibits non-negligible centrifugal distortion dependence, which is currently disregarded. The centrifugal distortion effects are especially important in the case of the Δk = ±3 “forbidden” transitions involving accidentally coinciding roinversional states |a,J,K = 3⟩ and |s,J,K = 0⟩ of the ν2 vibrational state of 14NH3. The energy differences of these states exhibit very anomalous mass sensitivities (see Jansen et al. J. Chem. Phys. 2014, 140, 010901), thus appearing as new highly sensitive probes of the cosmological variability of the protonto-electron mass ratio.

SECTION: Spectroscopy, Photochemistry, and Excited States

A

counterparts. Therefore, as the wavelengths of the vibrational and rotational transitions of the hydrogen molecule are too short to be monitored in the region of radio waves, it is recommendable to rely on other, possibly more suitable molecular particles, for instance, on ammonia, as proposed by Flambaum and Kozlov7 in 2007 (note that this issue was first brought forward by van Veldhoven et al.8). This approach, the so-called ammonia method, is based on comparing the frequency shifts of the purely inversion transition of 14 NH3(1,1) and rotational transitions of other molecules, closely tracing the ammonia spatial distribution. Not only is the inversion frequency ideally suitable for monitoring in the required frequency region, but it also exhibits half of an order higher sensitivity to the μ variation (Tinv = −4.4) than the rotational transitions (Trot ≈ −1). Not surprisingly, the ammonia method thus made it possible to establish immediately a much stricter absolute limit (Δμ/μ) = (0.6 ± 1.9) × 10−6 at the red shift z = 0.6847 and very soon afterward9 to set it to |(Δμ/μ)| < 1.8 × 10−6. The success of the method has also triggered a very active search for other molecular probes. The search was soon crowned by a long series of different molecular transitions possessing higher sensitivities than that of the NH3(1,1) transition (see, e.g., refs 10 and 11 and the references therein). As particularly suitable, the internal

possible space−time variation of the fundamental constants of nature is a principal problem that has been discussed for a long time.2,3 One of the ways of probing this variation consists of relating the wavelengths λi of the cosmological (red-shifted) atomic and molecular spectral lines to their rest wavelengths λ0i by ⎛ ΔΦ ⎞⎟ λi = λi0(1 + z)⎜1 + Ti ⎝ Φ ⎠

(1)

where ΔΦ = Φ0 − Φ is the variation of the probed fundamental constant Φ, z is the red shift of the probed cosmological absorber/emitter, and Ti, the so-called sensitivity coefficient, is a constant that can be derived from the first principles. Concerning the proton-to-electron mass ratio μ = (mp/me), all of the studies in the first 3 decades after 1975 (ref 4) relied on the vibronic spectrum of molecular hydrogen. So far, although there were some claims of marginal detections of the variation sought (for instance, Reinhold et al.6 obtained (Δμ/ μ) = (+2.6 ± 0.6) × 10−5 from two absorbers at the red shift z ≈ 2.6−3.0), none of them have been confirmed. As a matter of fact, a recent critical analysis of Bagdonaite et al.5 has yielded a “molecular hydrogen limit” constraint of |(Δμ/μ)| ≤ 10−5 for z ≈ 2.0−3.0. Undoubtedly, this “limit” could be, and will be, improved by improving the sensitivity and spectral resolution of the available optical facilities that are suitable for monitoring molecular hydrogen. It is also known that the sensitivity coefficients of the molecular vibronic transitions are 2 orders of magnitude smaller than their purely vibrational or rotational © 2014 American Chemical Society

Received: January 24, 2014 Accepted: February 20, 2014 Published: February 20, 2014 919

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Figure 1. The sensitivities, T, and Einstein coefficients, A (taken from ref 23), of the inversion transitions in the ground vibrational state of 14NH3. FK corresponds to the reference value of Flambaum and Kozlov.7

Table 1. Inversion Frequencies (ν), Einstein Coefficients (A), and Sensitivities (T) of Interstellar 14NH3 in the Ground Vibrational Statea J 7 10 6 9 8 6 7 6 5 5 8 9 7 10 6 11 4 5 4 5 4 6 a

K 3 7 1 6 5 2 4 3 1 2 6 7 5 8 4 9 1 3 2 4 3 5

ν/MHz 18017.3 18285.4 18391.6 18499.4 18808.5 18884.7 19218.5 19757.5 19838.3 20371.5 20719.2 20735.5 20804.8 20852.5 20994.6 21070.7 21134.3 21285.3 21703.4 22653.0 22688.3 22732.4

A/s−1 0.2371 0.6837 0.3735 0.6362 0.5806 0.1615 0.5096 0.4150 0.6540 0.2828 0.1111 0.1211 0.1005 0.1315 0.8821 0.1428 0.1182 0.7239 0.5114 0.1546 0.1311 0.1742

× × × × × × × × × × × × × × × × × × × × × ×

−7

10 10−7 10−8 10−7 10−7 10−7 10−7 10−7 10−8 10−7 10−6 10−6 10−6 10−6 10−7 10−6 10−7 10−7 10−7 10−6 10−6 10−6

T

J

K

ν/MHz

−4.640 −5.345 −4.858 −5.186 −5.037 −4.826 −4.914 −4.795 −4.700 −4.546 −4.949 −5.075 −4.842 −5.205 −4.727 −5.364 −4.568 −4.634 −4.545 −4.592 −4.514 −4.655

3 7 2 8 9 1 2 3 4 10 5 6 7 8 9 10 11 12 13 14 18

2 6 1 7 8 1 2 3 4 9 5 6 7 8 9 10 11 12 13 14 18

22834.2 22924.9 23098.8 23232.2 23657.5 23694.3 23722.5 23870.1 24139.4 24205.3 24533.0 25056.0 25715.2 26519.0 27477.9 28604.7 29914.5 31424.9 33156.8 35134.3 46123.3

A/s−1 0.9902 0.1926 0.5123 0.2118 0.2332 0.1657 0.2216 0.2538 0.2797 0.2580 0.3053 0.3338 0.3675 0.4083 0.4584 0.5204 0.5978 0.6948 0.8170 0.9721 0.2179

× × × × × × × × × × × × × × × × × × × × ×

10−7 10−6 10−7 10−6 10−6 10−6 10−6 10−6 10−6 10−6 10−6 10−6 10−6 10−6 10−6 10−6 10−6 10−6 10−6 10−6 10−5

T −4.464 −4.734 −4.413 −4.839 −4.942 −4.365 −4.385 −4.419 −4.471 −5.053 −4.509 −4.571 −4.640 −4.714 −4.801 −4.894 −4.991 −5.104 −5.226 −5.359 −5.983

The frequencies and Einstein coefficients have been taken from refs 20 and 23, respectively.

new avenue by highlighting a record-breaking sensitivity (T = −938) of the Δk = ±3 “forbidden” transition between the |a,J = 3, K = 3⟩ and |s,J = 3, K = 0⟩ accidentally degenerated roinversional states of 14NH3 in its ν2 vibrational state (for details, see ref 1). Like the original ammonia method of Flambaum and Kozlov (FK),7 the procedure used to calculate the latter sensitivity is based on a simple (one-dimensional) inversion Hamiltonian that disregards small-amplitude vibrational effects (centrifugal distortion effects). This approximation appears as fairly reasonable for the whole class of metastable (J = K) inversion transitions in the ground vibrational state;15 for some other transitions, however, it may become inaccurate. This especially concerns the whole class of the so-called Δk = ±3 transitions,16,17 which are of principal importance for this study. One way of overcoming this problem may consist of replacing the “rigid inverter” sensitivities by their analogues obtained

rotation transitions of methanol with sensitivity coefficients an order of magnitude higher than that of 14NH3(1,1) appear (see ref 12). As expected, the methanol data have allowed the determination of new limits on (Δμ/μ), namely, (Δμ/μ) = (0.0 ± 1.0) × 10−7 (see ref 13). Obviously, the results appear not to be decisive. Nevertheless, as there is always some space for reducing observational errors and as there are still unprobed transitions that exhibit higher mass sensitivities than those used for evaluating the temporary limits, there is still some space for improvements. Interestingly, thanks to a major extension of the probed measurements, Bagdonaite et al.14 have just recently pushed the methanol limit to its last version (Δμ/μ) = (−1.0 ± 0.8stat ± 1.0sys) × 10−7. Even more recently, in a review that proposes a set of molecular species that might improve the present limit, Jansen, Bethlem, and Ubachs (JBU) have opened a possibly 920

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Table 2. Rotation−Inversion Frequencies (ν), Einstein Coefficients (A), and Sensitivities (T) of Interstellar 14NH3 in the Ground Vibrational Statea

a

p′

J′

K′

υ2′

p″

J″

K″

υ2″

ν

A

T

s a a

1 2 2

0 0 1

0 0 0

a s s

0 1 1

0 0 1

0 0 0

572498 1214859 1215245

0.1561 × 10−2 0.1791 × 10−1 0.1344 × 10−1

−0.862 −1.063 −1.064

The frequencies and Einstein coefficients have been taken from refs 21 and 23, respectively.

Consequently, although not negligible, the predicted variation of the sensitivities of the observed 14NH3 ground vibrational state transitions cannot lead to a substantial change of the present (Δμ/μ) limit. However, a fairly different situation is to be anticipated in the case of the ν2 vibrational state, for which the inversion splitting (∼30 cm−1) is on the order of magnitude of the rotational spacings, thus allowing for close energy coincidences (resonances) and spectral irregularities1,16,17 (see Figure 2). Indeed, as can be seen in Table 3, both of the astronomically so-far observed ro-inversional transitions in the ν2 state exhibit sensitivities profoundly differing from the reference FK sensitivity. Very importantly, the magnitudes of the predicted sensitivities differ strongly, hence allowing for ammonia analyses exclusively without any reference molecule. Apparently, such analyses should be much less sensitive to the assumption of the FK method that the species being compared reside in the same location and are thus at the same red shift. The discussed transitions exhibit not only sizable mass sensitivities but also frequencies that qualify them for detection by the Atacama large millimeter/submillimeter array.30 Therefore, both of the transitions are very promising candidates for observations that may lead to a substantial revision of the present (Δμ/μ) limit. As can be seen in Table 4, although not so “irregular” as their ro-inversional analogues, the sensitivities of the rovibrational transitions are found in a certain disharmony with their purely vibrational limit of −1/2, thus also providing a reason for future infrared monitorings. The proximity of the a(1,1) and s(2,1) states, which is behind the anomalous sensitivity of the corresponding transition a(1,1) → s(2,1), is not a singular case in the ν2 state. As can be seen in Figure 2, there is a sequence of the s(J,0) and a(J,3) states, which exhibit a much higher degree of coincidence. Consequently, although the mass sensitivities of these states do not differ markedly from those of the other ν2 states, the sensitivities of the corresponding combination differences s(J,0) − a(J,3) (J = 3,5) are very high (see Table 2), thus being even more promising tools for probing (Δμ/μ) than the a(1,1) → s(2,1) transition. Certainly, it should be emphasized that the transitions involved in the latter combination differences have not been observed extraterrestrially yet, and their determination is hampered by the fact that it requires a detection of relatively weak Δk = ±3 forbidden transitions. However, at least for J = 3, these transitions have Einstein coefficients that are comparable to those of the allowed

Figure 2. The vibration−rotation−inversion levels of 14NH3 in the ν2 state.

using the “nonrigid inverter” theory.18,19 For the ground vibrational state, as can be seen in Figure 1 and Table 1, the latter theory predicts the sought sensitivities with only moderate centrifugal distortion dependence; the largest deviation of the calculated sensitivities of the astronomically observed transitions from the reference value, T1,1 = −4.4, has been calculated for the (18,18) inversion transition, T18,18 = −6.0. Similarly, as can be seen in Table 2, the sensitivities of the rotation−inversion transitions are in a semiquantitative agreement with their purely rotational value of −1. As a matter of fact, much larger deviations predicted for some Δk = ±3 are associated with transitions possessing Einstein coefficients, which are too small to allow for their detections (see Figure 1).

Table 3. Rotation−Inversion Frequencies (ν), Einstein Coefficients (A), and Sensitivities (T) of Interstellar 14NH3 in the ν2 Vibrational Statea p′

J′

K′

υ2′

p″

J″

K″

υ2″

ν

A

T

obs. ref.

s a

2 0

1 0

1 1

a s

1 1

1 0

1 1

140142.1(140142) 466245.5(466244)

0.1474 × 10−4 0.1824 × 10−2

16.92b −6.409

24,25 26

a

The frequencies and Einstein coefficients have been taken from refs 22 and 23, respectively. The frequencies derived from astronomical observation are given in parentheses. bThe JBU sensitivity coefficient reaches a value of 18.8 (see ref 1). 921

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Table 4. Transition Wavenumbers (ν), Einstein Coefficients (A), and Sensitivities (T) of the ν2 State of Interstellar 14NH3a p′

J′

K′

υ2′

p″

J″

K″

υ2″

ν

a a a a a a

6 2 1 2 2 3

6 2 1 1 0 3

0 0 0 0 0 0

s s s s s s

6 2 2 1 1 3

6 2 1 1 0 3

1 1 1 1 1 1

927.3230(927.3232) 931.3333(931.3334) 971.8821(971.8822) 891.8820 892.1567 930.7571

A 0.1316 0.1030 0.5238 0.6795 0.9054 0.1158

× × × × × ×

102 102 10 10 10 102

T

ref.

−0.356 −0.366 −0.394 −0.339 −0.339 −0.481

27 27 27 28 28 28

a

The wavenumbers and Einstein coefficients have been taken from refs 22 and 23, respectively. Wavenumbers derived from astronomical observations are given in parentheses.

Table 5. Vibration−Rotation−Inversion Transitions Associated with the |a,J,K = 3, υ2 = 1⟩ − |s,J,K = 0, υ2 = 1⟩ Resonances (frequencies in MHz)a

a

p′

J′

K′

υ2′

p″

J″

K″

υ2″

ν

A

T

obs. ref.

a s a a s a a s a a s a s a s a

3 3 3 3 3 3 3 3 3 5 5 5 5 5 5 5

3 0 3 3 0 3 3 0 3 3 0 3 0 3 0 3

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

s s s a a s s s s s s s s a a s

3 3 3 2 2 3 3 3 3 5 5 5 5 4 4 5

3 3 0 0 0 0 3 3 0 3 3 3 3 0 0 0

0 0 1 1 1 1 1 1 1 0 0 1 1 1 1 1

29000313.7 28997430.0 2883.7 772594.9 769710.2 2884.7 1073050.7 1070166.6 2884.1 28971340.5 29050552.5 979649.1 1058861.1 1956241.1 2035453.1 79212.0

0.1176 × 102 0.2025

−0.481 −0.396 −853.1b −0.986 2.21 −853.1 −3.411 −1.122 −853.1 −0.482 −0.401 −3.552 −1.116 −0.998 0.170 29.02

29 29 29 16 16 16 16 16 16 22 22 22 22 22 22 22

0.6018 × 10−4 0.3471 × 10−2 0.1634 × 10−1 0.2765 × 10−3 0.4692 0.2147 0.5141 0.3714 0.4129 0.7023

× × × × × ×

10 10−2 10−2 10−5 10−4 10−1

The Einstein coefficients have been taken from ref 23. bThe JBU sensitivity coefficient reaches a value of −938 (see ref 1).

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transitions (compare Tables 1, 2, and 5) and, as a matter of fact, have already been observed terrestrially.16,29 Therefore, it is not unthinkable to expect their astronomical detection by means of such new astronomical installations as the Atacama large millimeter/submillimeter array or very long baseline interferometry30 or China’s VLBI (very long baseline interferometry) system.31 Undoubtedly, observing these transitions would allow for a truly decisive testing of the constancy of the protonto-electron mass ratio. The centrifugal distortion effects probed in this study are substantial and should be respected in any quantitative analysis of the observed data, including also the Galactic measurements.32

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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REFERENCES

ACKNOWLEDGMENTS

The work was a part of the Research Project RVO:61388963 (IOCB) and was supported by the Czech Science Foundation (grant P208/11/0436) and by the Ministry of Educationi, Youth and Sports of the Czech Republic (KONTAKTII(LH11022)). 922

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