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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 16 — Aug. 12, 2013
  • pp: 18899–18908
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Zero-frequency refractivity of water vapor, comparison of Debye and van-Vleck Weisskopf theory

D. Grischkowsky, Yihong Yang, and Mahboubeh Mandehgar  »View Author Affiliations


Optics Express, Vol. 21, Issue 16, pp. 18899-18908 (2013)
http://dx.doi.org/10.1364/OE.21.018899


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Abstract

We show that the zero-frequency, refractivity of water vapor calculated by the van-Vleck Weisskopf theory via a summation over all the water lines from 22.2 GHz to 30 THz can explain all of the previous measurements from 0.5 MHz to microwave, mm-waves and THz frequencies. This result removes a long standing discrepancy in comparisons of measurements and theory, and is in excellent agreement with experiments.

© 2013 OSA

Introduction

The propagation of electromagnetic waves in the atmosphere is strongly dependent on the relatively small (of the order of 0.0003) complex index of refraction n(ω) of the atmosphere. Because the index is quite small, the refractivity term [n(ω) −1] is usually discussed. Even though water vapor is only one percent of the total atmosphere, at 50% relative humidity (RH) and 20°C, water vapor can account for more than 20% of the total atmosphere refractivity. The real part of the refractivity of water vapor is composed of two parts, a slowly-varying almost frequency-independent part, applicable from kHz to MHz to microwaves up to THz, into the infrared and the visible, and a strongly frequency-dependent part together with the strong absorption, due to the imaginary part, of the water vapor rotational resonance lines from 22.2 GHz to 30 THz. Measurements of the refractivity of water vapor have been made from 545 kHz [1

1. J. D. Stranathan, “Dielectric constant of water vapor,” Phys. Rev. 48(6), 538–544 (1935). [CrossRef]

] to microwave [2

2. C. M. Crain, “The dielectric constant of several gases at a wave-length of 3.2 centimeters,” Phys. Rev. 74(6), 691–693 (1948). [CrossRef]

5

5. L. Essen and K. D. Froome, “The refractive indices and dielectric constants of air and its principal constituents at 24,000 Mc/s,” Proc. Phys. Soc. B 64(10), 862–875 (1951). [CrossRef]

], mm-wave [6

6. K. D. Froome, “The refractive indices of water vapour, air, oxygen, nitrogen and argon at 72 kMc/s,” Proc. Phys. Soc. B 68(11), 833–835 (1955). [CrossRef]

,7

7. T. Manabe, Y. Furuhama, T. Ihara, S. Saito, H. Tanaka, and A. Ono, “Measurements of attenuation and refractive dispersion due to atmospheric water vapor at 80 and 240 GHz,” Int. J. Infrared Millim. Waves 6(4), 313–322 (1985). [CrossRef]

], far-infrared (THz) [8

8. C. C. Bradley and H. A. Gebbie, “Refractive index of nitrogen, water vapor, and their mixtures at submillimeter wavelengths,” Appl. Opt. 10(4), 755–758 (1971). [CrossRef] [PubMed]

], infrared [9

9. H. Matsumoto, “The refractive index of moist air in the 3-µm region,” Metrologia 18(2), 49–52 (1982). [CrossRef]

,10

10. R. J. Hill and R. S. Lawrence, “Refractive index of water vapor in the infrared windows,” Infrared Phys. 26(6), 371–376 (1986). [CrossRef]

], and optical frequencies [11

11. R. Schödel, A. Walkov, and A. Abou-Zeid, “High-accuracy determination of water vapor refractivity by length interferometry,” Opt. Lett. 31(13), 1979–1981 (2006). [CrossRef] [PubMed]

].

A recent experiment measured the essentially constant low-frequency refractivity of water vapor, by measuring the transit time shift of a THz pulse through a 138 m path length, as a function of RH of water vapor [12

12. Y. Yang, M. Mandehgar, and D. Grischkowsky, “Time domain measurement of the THz refractivity of water vapor,” Opt. Express 20(24), 26208–26218 (2012). [CrossRef] [PubMed]

]. The time shift was measured to an accuracy of 0.1 ps, and the corresponding, essentially-constant refractivity from 0.1 to 0.9 THz was measured to be (n(0) – 1) = 70 x 10−6 ± 10% at 10 g/m3 water vapor density at 21 °C, for which (n(0) – 1) is the refractivity at zero frequency. These results were in acceptable agreement with earlier measurements [1

1. J. D. Stranathan, “Dielectric constant of water vapor,” Phys. Rev. 48(6), 538–544 (1935). [CrossRef]

8

8. C. C. Bradley and H. A. Gebbie, “Refractive index of nitrogen, water vapor, and their mixtures at submillimeter wavelengths,” Appl. Opt. 10(4), 755–758 (1971). [CrossRef] [PubMed]

]. Similar to the earlier MHz, microwave and mm-wave measurements [1

1. J. D. Stranathan, “Dielectric constant of water vapor,” Phys. Rev. 48(6), 538–544 (1935). [CrossRef]

6

6. K. D. Froome, “The refractive indices of water vapour, air, oxygen, nitrogen and argon at 72 kMc/s,” Proc. Phys. Soc. B 68(11), 833–835 (1955). [CrossRef]

], the (n(0) - 1) THz refractivity measurements were explained by a Debye type response [13

13. P. Debye, Polar molecules, 89–90 (Dover Publ. Co., New York, N.Y., 1957).

,14

14. B. R. Bean and E. J. Dutton, Radio Meteorology, Monograph #92 (National Bureau of Standards, 1966), Chap. 1.

], which considers only the permanent dipole moments of water vapor. van-Vleck Weisskopf lineshapes (v-VW) [15

15. J. H. Van Vleck and V. F. Weisskopf, “On the shape of collision-broadened lines,” Rev. Mod. Phys. 17(2-3), 227–236 (1945). [CrossRef]

,16

16. C. H. Townes and A. L. Schawlow, Microwave Spectroscopy (Dover Publ. Co., 1975).

] were used to fit the rotational resonance lines of the water vapor. However, the extension of the Debye theory up to and beyond 50 GHz for the refractivity leads to a physical inconsistency with respect to the associated very large absorption predicted by Debye theory. Consequently, the low-frequency refractivity [12

12. Y. Yang, M. Mandehgar, and D. Grischkowsky, “Time domain measurement of the THz refractivity of water vapor,” Opt. Express 20(24), 26208–26218 (2012). [CrossRef] [PubMed]

] was assigned to water dimers, trimers and clusters [17

17. A. Deepak, T. D. Wilkerson, and L. H. Ruhnke, eds., Atmospheric Water Vapor (Academic Press, 1980). This book is the Proceedings of the International Workshop on Atmospheric Water Vapor, Vail, Colorado, September 11–13, 1979.

].

Here, we show that the previously neglected (n(0) – 1) low-frequency response of v-VW theory can explain the above physical inconsistency. Using both the JPL [18

18. H. M. Pickett, R. L. Poynter, E. A. Cohen, M. L. Delitsky, J. C. Pearson, and H. S. P. Muller, “Sub-millimeter, millimeter, and microwave spectral line catalog,” JQSRT 60(5), 883–890 (1998). [CrossRef]

18. Access to specific catalog entries may be found athttp://spec.jpl.nasa.gov/.

] and HITRAN [19

19. L. S. Rothman, I. E. Gordon, A. Barbe, D. C. Benner, P. F. Bernath, M. Birk, V. Boudon, L. R. Brown, A. Campargue, J.-P. Champion, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, S. Fally, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, I. Kleiner, N. Lacome, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, S. N. Mikhailenko, C. E. Miller, N. Moazzen-Ahmadi, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. I. Perevalov, A. Perrin, A. Predoi-Cross, C. P. Rinsland, M. Rotger, M. Šimečková, M. A. H. Smith, K. Sung, S. A. Tashkun, J. Tennyson, R. A. Toth, A. C. Vandaele, and J. Vander Auwera, “The HITRAN 2008 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transf. 110(9-10), 533–572 (2009). [CrossRef]

] databases, to calculate (n(0) – 1), the experimental results from 0.5 MHz to microwave, mm-wave and THz (far-infrared) can be explained [1

1. J. D. Stranathan, “Dielectric constant of water vapor,” Phys. Rev. 48(6), 538–544 (1935). [CrossRef]

8

8. C. C. Bradley and H. A. Gebbie, “Refractive index of nitrogen, water vapor, and their mixtures at submillimeter wavelengths,” Appl. Opt. 10(4), 755–758 (1971). [CrossRef] [PubMed]

], and the associated absorption agrees with the measurements.

We can obtain mathematical agreement with Debye theory, if we set the fractional functions to 0.5 in the mathematics describing our calculations, corresponding to eliminating the rotational response. This is due to the mathematical consistency with the Debye theory as discussed in the v-VW original paper [15

15. J. H. Van Vleck and V. F. Weisskopf, “On the shape of collision-broadened lines,” Rev. Mod. Phys. 17(2-3), 227–236 (1945). [CrossRef]

], and as described by Townes and Schawlow (T&S) [16

16. C. H. Townes and A. L. Schawlow, Microwave Spectroscopy (Dover Publ. Co., 1975).

]. However, the required uniform population of all the rotational ground states Ea to conform to the Debye theory would only be possible for a temperature high enough so that Ea/kT is significantly less than one. This extreme requirement shows that for water vapor, the use of Debye theory is a good approximation, but it is not physically correct.

The earlier experiments [1

1. J. D. Stranathan, “Dielectric constant of water vapor,” Phys. Rev. 48(6), 538–544 (1935). [CrossRef]

6

6. K. D. Froome, “The refractive indices of water vapour, air, oxygen, nitrogen and argon at 72 kMc/s,” Proc. Phys. Soc. B 68(11), 833–835 (1955). [CrossRef]

], assumed the validity of Debye theory, and then used the measured value of (no (0) – 1) to calculate the magnitude of electric dipole moment of the water molecule µ = 1.839 Debye.

An alternative to the early measurements of refractivity [1

1. J. D. Stranathan, “Dielectric constant of water vapor,” Phys. Rev. 48(6), 538–544 (1935). [CrossRef]

6

6. K. D. Froome, “The refractive indices of water vapour, air, oxygen, nitrogen and argon at 72 kMc/s,” Proc. Phys. Soc. B 68(11), 833–835 (1955). [CrossRef]

] to determine the dipole moment of water µ is to measure the Stark splitting of the water vapor rotational lines [20

20. S. Golden, T. Wentink, R. Hillger, and M. Strandberg, “Stark spectrum of H2O,” Phys. Rev. 73(1), 92–93 (1948). [CrossRef]

22

22. S. A. Clough, Y. Beers, G. P. Klein, and L. S. Rothman, “Dipole moment of water from Stark measurements of H2O, HDO, and D2O,” J. Chem. Phys. 59(5), 2254–2259 (1973). [CrossRef]

], which is independent of the number density of water vapor molecules. The more recent Stark measurements at the frequencies of 22.235 GHz and 183.310 GHz [21

21. Y. Beers and G. P. Klein, “The Stark splitting of millimeter wave transitions of water,” J. Res. Natl. Bur. Stand., Sect. A 76a, 521–528 (1972).

,22

22. S. A. Clough, Y. Beers, G. P. Klein, and L. S. Rothman, “Dipole moment of water from Stark measurements of H2O, HDO, and D2O,” J. Chem. Phys. 59(5), 2254–2259 (1973). [CrossRef]

], have obtained µ = 1.848 Debye. In addition, molecular beam electric resonance spectroscopy measurements [23

23. T. R. Dyke and J. S. Muenter, “Electric dipole moments of low J states of H2O and D2O,” J. Chem. Phys. 59(6), 3125–3127 (1973). [CrossRef]

,24

24. S. L. Shostak, W. L. Ebenstein, and J. S. Muenter, “The dipole moment of water. I. Dipole moments and hyperfine properties of H2O and HDO in the ground and excited vibrational states,” J. Chem. Phys. 94(9), 5875–5882 (1991). [CrossRef]

], of the Stark splitting of the hyperfine lines in rotational states of H2O have obtained µ = 1.855 Debye, which is considered to be the most accurate value. The JPL database uses µ = 1.8546 Debye, and the HITRAN database uses 1.855 Debye.

Here, we present the van-Vleck Weisskopf theory as the more physically correct theory for the refractivity, compared to Debye theory, to compare to the early, high quality refractivity measurements. We apply v-VW theory to these same precisely measured values and obtain an excellent fit to the early experiments [4

4. L. Essen, “The refractive indices of water vapour, air, oxygen, nitrogen, hydrogen, deuterium and helium,” Proc. Phys. Soc. B 66(3), 189–193 (1953). [CrossRef]

6

6. K. D. Froome, “The refractive indices of water vapour, air, oxygen, nitrogen and argon at 72 kMc/s,” Proc. Phys. Soc. B 68(11), 833–835 (1955). [CrossRef]

], using the accepted value of the dipole moment µ = 1.855 Debye.

Comparison of van-Vleck Weisskopf theory with Debye theory

The objective of this paper is to compare v-VW refractivity theory with Debye theory at zero frequency, in order to remove the fundamental absorption discrepancy between theory and experiment [12

12. Y. Yang, M. Mandehgar, and D. Grischkowsky, “Time domain measurement of the THz refractivity of water vapor,” Opt. Express 20(24), 26208–26218 (2012). [CrossRef] [PubMed]

]. Consequently, we show that for the v-VW theory, in the limit as ω approaches zero, the absorption coefficient α(ω) approaches zero, the phase φ(ω) = Δk(ω)L approaches zero, and the refractivity at zero frequency due to the electric dipole moment of water is given by
(n(0)1)vVW=cDj2Ajπωj2.
(1)
where j refers to the water vapor resonance line with angular frequency ωj, and Aj gives the line strength [25

25. Y. Yang, M. Mandehgar, and D. Grischkowsky, “Understanding THz pulse transmission in the atmosphere,” IEEE Trans. THz Sci. Technol. 2, 406–415 (2012).

,26

26. Private communication from referee of [12].

]. The summation is to be taken over all of the spectral lines of water vapor from 22.3 THz up to 30 THz, as tabulated in the JPL and HITRAN databases [18

18. H. M. Pickett, R. L. Poynter, E. A. Cohen, M. L. Delitsky, J. C. Pearson, and H. S. P. Muller, “Sub-millimeter, millimeter, and microwave spectral line catalog,” JQSRT 60(5), 883–890 (1998). [CrossRef]

18. Access to specific catalog entries may be found athttp://spec.jpl.nasa.gov/.

,19

19. L. S. Rothman, I. E. Gordon, A. Barbe, D. C. Benner, P. F. Bernath, M. Birk, V. Boudon, L. R. Brown, A. Campargue, J.-P. Champion, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, S. Fally, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, I. Kleiner, N. Lacome, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, S. N. Mikhailenko, C. E. Miller, N. Moazzen-Ahmadi, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. I. Perevalov, A. Perrin, A. Predoi-Cross, C. P. Rinsland, M. Rotger, M. Šimečková, M. A. H. Smith, K. Sung, S. A. Tashkun, J. Tennyson, R. A. Toth, A. C. Vandaele, and J. Vander Auwera, “The HITRAN 2008 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transf. 110(9-10), 533–572 (2009). [CrossRef]

]. This v-VW refractivity at zero frequency is not well known and was not used in the recent experimental comparison [25

25. Y. Yang, M. Mandehgar, and D. Grischkowsky, “Understanding THz pulse transmission in the atmosphere,” IEEE Trans. THz Sci. Technol. 2, 406–415 (2012).

] of pulse reshaping to v-VW theory. This is because the zero frequency term only causes a pulse time shift, and consequently, has no effect on the observed pulse reshaping. We will show that the neglected low-frequency response of v-VW theory of Eq. (1) can describe all of the early measurements [1

1. J. D. Stranathan, “Dielectric constant of water vapor,” Phys. Rev. 48(6), 538–544 (1935). [CrossRef]

8

8. C. C. Bradley and H. A. Gebbie, “Refractive index of nitrogen, water vapor, and their mixtures at submillimeter wavelengths,” Appl. Opt. 10(4), 755–758 (1971). [CrossRef] [PubMed]

]. Initially, We will first describe the Debye theory as given in [12

12. Y. Yang, M. Mandehgar, and D. Grischkowsky, “Time domain measurement of the THz refractivity of water vapor,” Opt. Express 20(24), 26208–26218 (2012). [CrossRef] [PubMed]

]. We will then describe the v-VW theory, and finally compare the two theories.

Microwave and mm-wave Debye theory approach [12

12. Y. Yang, M. Mandehgar, and D. Grischkowsky, “Time domain measurement of the THz refractivity of water vapor,” Opt. Express 20(24), 26208–26218 (2012). [CrossRef] [PubMed]

]

Previously, the low-frequency experimental results up to 100 GHz have been understood in terms of the Debye theory for a molecular vapor of permanent electric dipoles [1

1. J. D. Stranathan, “Dielectric constant of water vapor,” Phys. Rev. 48(6), 538–544 (1935). [CrossRef]

6

6. K. D. Froome, “The refractive indices of water vapour, air, oxygen, nitrogen and argon at 72 kMc/s,” Proc. Phys. Soc. B 68(11), 833–835 (1955). [CrossRef]

,13

13. P. Debye, Polar molecules, 89–90 (Dover Publ. Co., New York, N.Y., 1957).

,14

14. B. R. Bean and E. J. Dutton, Radio Meteorology, Monograph #92 (National Bureau of Standards, 1966), Chap. 1.

]. For low frequencies according to Debye theory, the dipole polarization of a vapor of water molecules will be in thermal equilibrium with an applied electric field of angular frequency ω with the frequency-dependent, complex refractivity of [13

13. P. Debye, Polar molecules, 89–90 (Dover Publ. Co., New York, N.Y., 1957).

,14

14. B. R. Bean and E. J. Dutton, Radio Meteorology, Monograph #92 (National Bureau of Standards, 1966), Chap. 1.

],
(n˜(ω)1)D=(2πNDμ23kT)/(1+iωτ).
(2)
where ND is the number of molecules per cubic centimeter, µ is the permanent electric dipole moment of the water molecule, k is the Boltzmann constant, T is the absolute temperature, and τ is the relaxation time required for the external field-induced orientations of the molecules to return to a random distribution after the field is removed. The Debye refractivity at zero frequency can be obtained from Eq. (2) as,

(n(0)1)D=2πNDμ23kT.
(3)

Using Eq. (3) with the following parameters [4

4. L. Essen, “The refractive indices of water vapour, air, oxygen, nitrogen, hydrogen, deuterium and helium,” Proc. Phys. Soc. B 66(3), 189–193 (1953). [CrossRef]

6

6. K. D. Froome, “The refractive indices of water vapour, air, oxygen, nitrogen and argon at 72 kMc/s,” Proc. Phys. Soc. B 68(11), 833–835 (1955). [CrossRef]

]; water vapor density of 10 g/m3 at 20 °C (T = 293 K), corresponding to ND = 3.34 x 10 17/cm3 ; µ = 1.839 Debye = 1.839 x 10−18 StatC-cm; and k = 1.38 x 10−16 [erg/°K ], gives the value of (n (0) – 1)D = 58.5 x 10−6. In order to compare with the measurement (n(0) −1) = 61.6 x 10−6, which has been adjusted to 10 g/m3 from 10 mm of Hg, we must calculate with the multiplicative component (n(0) – 1) = (1 + б)(n(w) – 1)D. The б term has been added to account for the atomic and electric response of the water molecule that is independent of the electric dipole moment [5

5. L. Essen and K. D. Froome, “The refractive indices and dielectric constants of air and its principal constituents at 24,000 Mc/s,” Proc. Phys. Soc. B 64(10), 862–875 (1951). [CrossRef]

]. For our case б = 0.052, giving (n(0) – 1) = 61.5 x 10−6. This exceptional agreement is not surprising, because the early measurements assumed the validity of Eq. (3), and then the measured value of (no (0) – 1) was used to calculate the electric dipole moment of the water molecule. It should be noted, that using the considered to be correct value of µ = 1.855 Debye, the calculation gives 62.6, in disagreement with the measurement.

Equation (2) can now be rewritten in the form below to enable simple comparison with the measurements.
(n˜(ω)1)D=(n(0)1)D1+iωτ.
(4)
The Debye refractivity can be rewritten in terms of the real and imaginary parts as
(n˜(ω)1)D=(n(0)1)D1+(ωτ)2i(n(0)1)Dωτ1+(ωτ)2.
(5)
Initially, we are concerned with the real part of the Debye refractivity (n(0) – 1)D/ [1 + (ωτ)2 ], which falls off from the zero frequency as a Lorentzian with the half-width of ω1/2 τ = 1, equivalent to f1/2 = 1/(2πτ), where f is frequency. Note that there are only two parameters in the theory, (n(0) −1)D and τ.

The Debye absorption coefficient is given by
αD(ω)=(2ωc)(n(0)1)Dωτ1+(ωτ)2.
(6)
The real part of the Debye refractivity and the absorption can be rewritten in terms of their normalized frequency f/f1/2 dependence as,

(n(f)1)D=(n(0)1)D[11+(f/f1/2)2].
(7a)
αD(f)=4πf1/2c(n(0)1)D[(f/f1/2)21+(f/f1/2)2].
(7b)

The normalized frequency dependence in terms of (f/f1/2) for Eqs. (7a) and (7b) is shown in Fig. 1
Fig. 1 The normalized frequency dependence (f/f1/2) for the Debye refractivity (upper curve at zero) and absorption (lower curve at zero). These curves describe the frequency dependence shown in brackets in Eqs. (7a) and (7b).
. The refractivity is described by the Lorentzian curve with a unity peak at zero frequency. The associated absorption is described by the product of (f/f1/2)2 and the Lorentzian. In the high frequency limit, the refractivity curve asymtotically approaches zero, while the absorption curve monotonically converges to unity. For example, with f1/2 = 200 GHz (corresponding to τ = 0.8 ps), for the frequency f values of 50 GHz, 100 GHz, and 200 GHz, we obtain refractivity curve values of 0.94, 0.8 and 0.5, respectively, and the absorption curve values of 0.059, 0.2 and 0.5, respectively. Assuming a (n(0) – 1)D refractivity value of 58.5 x 10−6, from the refractivity curve we obtain 58.5 x 10−6 x 0.94, 0.8, and 0.5 for 50, 100 and 200 GHz, respectively. Evaluating (4π/c) f1/2(n(0) – 1) in Eq. (7b) to be 0.490/m, we obtain from the absorption curve, 0.490/m x 0.059, 0.2 and 0.5, for 50, 100 and 200 GHz, respectively. The resulting absorption coefficients of 28.9/km, 98/km, and 245/km, for 50, 100 and 200 GHz, respectively are all unphysically high and point out the inconsistency of the Debye theory. In addition, the value of f1/2 = 2000 GHz required to fit the recent refractivity measurements [12

12. Y. Yang, M. Mandehgar, and D. Grischkowsky, “Time domain measurement of the THz refractivity of water vapor,” Opt. Express 20(24), 26208–26218 (2012). [CrossRef] [PubMed]

] is much too large to assign to any known dephasing process.

Here, it is important to note, that in contrast to water vapor, liquid water is well described by the standard Debye theory with a single relaxation time of 7.9 ps corresponding to f1/2 = 20.2 GHz for frequencies of up to 100 GHz [27

27. H. J. Liebe, G. A. Hufford, and T. Manabe, “A model for the complex permittivity of water at frequencies below 1 THz,” Int. J. Infrared Millim. Waves 12(7), 659–675 (1991). [CrossRef]

]. In order to describe recent THz-TDS measurements extending from 100 to 2000 GHz [28

28. L. Thrane, R. H. Jacobsen, P. Uhd Jepsen, and S. R. Keiding, “THz reflection spectroscopy of liquid water,” Chem. Phys. Lett. 240(4), 330–333 (1995). [CrossRef]

30

30. C. Ro̸nne, L. Thrane, P.-O. Åstrand, A. Wallqvist, K. V. Mikkelsen, and S. R. Keiding, “Investigation of the temperature dependence of dielectric relaxation in liquid water by THz reflection spectroscopy and molecular dynamics simulation,” J. Chem. Phys. 107(14), 5319–5331 (1997). [CrossRef]

], a double Debye model with two relaxation times is required. The measurements are then well described by the two temperature dependent relaxation times of τ1 = 8.5 ps and τ2 = 0.2 ps (at 20 °C), corresponding to the f1/2 values of 18.9 GHz and 838 GHz, respectively [28

28. L. Thrane, R. H. Jacobsen, P. Uhd Jepsen, and S. R. Keiding, “THz reflection spectroscopy of liquid water,” Chem. Phys. Lett. 240(4), 330–333 (1995). [CrossRef]

30

30. C. Ro̸nne, L. Thrane, P.-O. Åstrand, A. Wallqvist, K. V. Mikkelsen, and S. R. Keiding, “Investigation of the temperature dependence of dielectric relaxation in liquid water by THz reflection spectroscopy and molecular dynamics simulation,” J. Chem. Phys. 107(14), 5319–5331 (1997). [CrossRef]

].

Van-Vleck Weisskopf lineshape theory

We have earlier fit our THz measurements of absorption and phase shift of water vapor [25

25. Y. Yang, M. Mandehgar, and D. Grischkowsky, “Understanding THz pulse transmission in the atmosphere,” IEEE Trans. THz Sci. Technol. 2, 406–415 (2012).

], using the causal absorption and phase shift lineshapes of the v-VW theory [15

15. J. H. Van Vleck and V. F. Weisskopf, “On the shape of collision-broadened lines,” Rev. Mod. Phys. 17(2-3), 227–236 (1945). [CrossRef]

,16

16. C. H. Townes and A. L. Schawlow, Microwave Spectroscopy (Dover Publ. Co., 1975).

,25

25. Y. Yang, M. Mandehgar, and D. Grischkowsky, “Understanding THz pulse transmission in the atmosphere,” IEEE Trans. THz Sci. Technol. 2, 406–415 (2012).

], given below as
α(ω)=DjAjπ(ωωj)2[Δωj(ωωj)2+(Δωj/2)2+Δωj(ω+ωj)2+(Δωj/2)2].
(8)
where j refers to the water vapor resonance line ωj; here, j is only a summation index and has no relationship to the total angular momentum J. The corresponding change of the wave vector (phase) is given by
Δk(ω)=Dj2Ajπωj(ωωjωj2ω2)[1Δωj28ωj(ωωj)(ωj+ω(ωjω)2+(Δωj/2)2ωjω(ωj+ω)2+(Δωj/2)2)].
(9)
For which
D=1018πN,
(10)
with N equal to the number of molecules per cubic meter, to give α(ω) in units of inverse meters. The corresponding refractivity is given by
(n(ω)1)vVW=Δkλ0/2π=Δk(c/ω).
(11)
The Aj line-strength values are from the JPL database in units of nm2 MHz [18

18. H. M. Pickett, R. L. Poynter, E. A. Cohen, M. L. Delitsky, J. C. Pearson, and H. S. P. Muller, “Sub-millimeter, millimeter, and microwave spectral line catalog,” JQSRT 60(5), 883–890 (1998). [CrossRef]

18. Access to specific catalog entries may be found athttp://spec.jpl.nasa.gov/.

], for which ωj was summed up to 10 THz involving 1,305 lines [25

25. Y. Yang, M. Mandehgar, and D. Grischkowsky, “Understanding THz pulse transmission in the atmosphere,” IEEE Trans. THz Sci. Technol. 2, 406–415 (2012).

]. The factor of 10−18 in Eq. (10) converts the Aj cross-sections into m2. The calculated absorption and phase φ = ΔkL agreed quite well with the measurements [25

25. Y. Yang, M. Mandehgar, and D. Grischkowsky, “Understanding THz pulse transmission in the atmosphere,” IEEE Trans. THz Sci. Technol. 2, 406–415 (2012).

]. Consequently, a linear dispersion theory calculation showed excellent agreement with the measured reshaped THz output pulse [25

25. Y. Yang, M. Mandehgar, and D. Grischkowsky, “Understanding THz pulse transmission in the atmosphere,” IEEE Trans. THz Sci. Technol. 2, 406–415 (2012).

].

Comparison of (n(0) – 1) of van-Vleck Weisskopf theory with Debye theory

In order to more easily compare with Debye theory, we present the Aj parameter from the JPL database [18

18. H. M. Pickett, R. L. Poynter, E. A. Cohen, M. L. Delitsky, J. C. Pearson, and H. S. P. Muller, “Sub-millimeter, millimeter, and microwave spectral line catalog,” JQSRT 60(5), 883–890 (1998). [CrossRef]

18. Access to specific catalog entries may be found athttp://spec.jpl.nasa.gov/.

], with notation similar to Townes and Schawlow [16

16. C. H. Townes and A. L. Schawlow, Microwave Spectroscopy (Dover Publ. Co., 1975).

], in more detail as
Aj=(8π33hc)νbaSxbaμx2[eEa/kTeEb/kT]Qrs.
(12)
For which a summation is made over the internal states of Ea, the lower state of the transition ωj = 2π νba and Eb, the upper state.

The JPL rotational-spin partition function Qrs is given by
Qrs=   j(2J+1)eEa/kT.
where the index j sums over all the ωj resonance lines for which Ea designates the lower state. Equation (12) can be rewritten as,
Aj=(8π33hc)νbaSxbaμx2(1e(EbEa)/kT)eEa/kTQrs.
and for frequencies up to 2 THz, an acceptable approximation is
Aj=(8π33hc)νbaSxbaμx2(hνba/kT)eEa/kTQrs,
(13)
which is needed for convergence to Debye theory. The JPL and HITRAN databases do not make this approximation. Equation (13) reduces to
Aj=(2π3c)ωbaSxbaμx2(ωba/kT)eEa/kTQrs.
(14)
Aj can now be rewritten as
Aj=Cjωj2.
and we can rewrite Eq. (1) as
(n(0)1)vVW=cDj2πCj.
(15)
for which
Cj=(2π3c)Sxbaμx2eEa/kTQrskT.
(16)
The remarkable result of Eq. (15) shows that for (n(0) −1)v-VW, below 2 THz, there is no ωj dependence and no Δωj linewidth dependence, due to the line symmetry about zero frequency of the v-VW theory. This significant situation only holds in the region for which hνab << kT, which is equivalent to vab << 5.7 THz. Consequently, it is expected that this term is reduced for frequencies f above 2 THz.

In the following discussion the 10−18 factor has been removed, because all the parameters and variables have their usual units. According to T&S in Eq. (4-28), xSba µx2 can be replaced by (2J + 1) µab2. With this substitution, Eq. (14) can be presented in more detail as
(n(0)1)vVW=cπN(2π)j(2π/3c)(2J+1)μab2eEa/kTQrskT.
which can be reduced to
(n(0)1)vVW=N(4π3kT)j(2J+1)μab2eEa/kTQrs.
(17)
The index j sums over all the ωj resonance lines for which Ea designates the lower state. Townes and Schawlow write the fractional function fa in their Eq. (4-25) as,
fa=(2J+1)eEa/kTQrs.
(18)
which gives the fraction of the total number of molecules in the lower state of the two states of ωj. We can rewrite Eq. (17) as,
(n(0)1)vVW=N(4π3kT)jfaμab2.
(19)
for which µab is the dipole moment between the upper “b” and the lower “a” of the transition ωj. This is important result shows concisely the difference between the van-Vleck Weisskopf and Debye theories. If all the Ea are set equal to zero, thereby, setting all the rotational lower Ea states to zero, fa becomes fa = 0.5, as in T&S Eq. (13-20). Using fa = 0.5, Eq. (19) becomes
[(n(0)1)vVW]Limit=N(2π3kT)jμab2.
(20)
Townes and Schawlow [16

16. C. H. Townes and A. L. Schawlow, Microwave Spectroscopy (Dover Publ. Co., 1975).

], and van-Vleck and Weisskopf [15

15. J. H. Van Vleck and V. F. Weisskopf, “On the shape of collision-broadened lines,” Rev. Mod. Phys. 17(2-3), 227–236 (1945). [CrossRef]

] have shown that
jμab2=μ2.
Thereby allowing Eq. (20) to be rewritten as,
[(n(0)1)vVW]Limit=2πNDμ23kT.
(21)
which is identical to the Debye result for a vapor of permanent dipoles, and for which we have converted to number density/cm3.

With respect to the comparison of the (n(0) – 1)v-VW with (n(0) −1)D, it is clear from Eqs. (19), (20) and (21) that the two results are significantly different from each other, because of the different fractional functions determined by the magnitude of Ea/kT. The required uniform population of all the ground states to conform to the Debye theory would only be achieved for a temperature high enough so that Ea/kT is significantly less than one.

This consideration shows that for water vapor the low frequency Debye theory may be a good approximation, but it is not physically correct. The difference can be demonstrated by calculations at the identical conditions and parameters: µ = 1.855 Debye, N = 3.340 x 1017/cm3 (10 g/m3), and T = 300 K . For this comparison (n(0) −1)D = 58.14 x 10−6 and (n(0) – 1)v-VW = 56.59 x 10−6, showing a difference of 2.56%, which would be very important in a dipole moment determination.

We now use the van-Vleck Weisskopf theory, as the more physically correct theory, to analyze the early very well done and accurate measurements [4

4. L. Essen, “The refractive indices of water vapour, air, oxygen, nitrogen, hydrogen, deuterium and helium,” Proc. Phys. Soc. B 66(3), 189–193 (1953). [CrossRef]

6

6. K. D. Froome, “The refractive indices of water vapour, air, oxygen, nitrogen and argon at 72 kMc/s,” Proc. Phys. Soc. B 68(11), 833–835 (1955). [CrossRef]

]. Consequently, we compare our v-VW calculation to their measured value of (n(0) – 1) adjusted to be 61.6 x 10−6 for a water vapor density of 10 g/m3 (instead of 10 mm Hg) corresponding to 3.34 x1017/cm3 and a temperature of 20 °C (293 K). Calculating the v-VW zero-frequency refractivity for the same number density and using the JPL database for which T = 300 K, we obtain 56.69 x 10−6, which we adjust to 293 K by the factor 1.024 to give 58.04 x 10−6, which must be multiplied by (1 + б) with б = 0.052 added to account for the atomic and electric response of the water molecule that is independent of the electric dipole moment [5

5. L. Essen and K. D. Froome, “The refractive indices and dielectric constants of air and its principal constituents at 24,000 Mc/s,” Proc. Phys. Soc. B 64(10), 862–875 (1951). [CrossRef]

], to give (n(0) – 1) = (1 + б) (n(0) – 1)v-VW = 61.06 in excellent agreement with measurement. This agreement is better than would be achieved by Debye theory in the original paper [5

5. L. Essen and K. D. Froome, “The refractive indices and dielectric constants of air and its principal constituents at 24,000 Mc/s,” Proc. Phys. Soc. B 64(10), 862–875 (1951). [CrossRef]

], if they had used the established value of µ = 1.855 Debye.

As a check on our results, we have also calculated the zero-frequency refractivity with the HITRAN database at 296 K, and obtained 57.33 x 10−6, which was adjusted to 293 K, by the factor 1.010 to give 57.92 x 10−6. This is relatively close agreement, considering that the JPL unadjusted value of 56.69 x 10−6 was adjusted to 293 K with the value of 58.04 x 10−6 involved the summation of 3085 lines up to 30 THz, and that the HITRAN calculation involved the summation of 8678 lines up to 30 THz, and shows the consistency and high accuracy of these two databases for water vapor.

As another check on our results we changed the extent of the frequency range for the summation over the water lines. For the same conditions as discussed above, the JPL calculations of (n(0) – 1) = 56.38 x 10−6 for the summation up to 10 THz with 1305 lines, and (n(0) – 1) = 47.08 x 10−6 up to 5 THz with 469 lines, and (n(0) – 1) = 18.72 x 10−6 up to 2 THz with 144 lines.

The corresponding HITRAN calculation of (n(0) – 1) = 57.03 x 10−6 for the summation up to 10 THz with 4843 lines, and (n(0) – 1) = 47.85 x 10−6 up to 5 THz with 2671 lines, and (n(0) – 1) = 19.20 x 10−6 up to 2 THz with 939 lines. The plots of (n(f) – 1) vs frequency similar to Fig. 2 for all of these different JPL and HITRAN calculations scans are quite similar, only vertically displaced by their corresponding value of (n(0) – 1).

Summary

Acknowledgments

We acknowledge a suggestion of the approach of Eq. (1) from our referee of [12

12. Y. Yang, M. Mandehgar, and D. Grischkowsky, “Time domain measurement of the THz refractivity of water vapor,” Opt. Express 20(24), 26208–26218 (2012). [CrossRef] [PubMed]

], and helpful readings of this manuscript by Joseph S. Melinger. This work was partially supported by the National Science Foundation.

Reference and links

1.

J. D. Stranathan, “Dielectric constant of water vapor,” Phys. Rev. 48(6), 538–544 (1935). [CrossRef]

2.

C. M. Crain, “The dielectric constant of several gases at a wave-length of 3.2 centimeters,” Phys. Rev. 74(6), 691–693 (1948). [CrossRef]

3.

G. Birnbaum and S. K. Chatterjee, “The dielectric constant of water vapor in the microwave region,” J. Appl. Phys. 23(2), 220–223 (1952). [CrossRef]

4.

L. Essen, “The refractive indices of water vapour, air, oxygen, nitrogen, hydrogen, deuterium and helium,” Proc. Phys. Soc. B 66(3), 189–193 (1953). [CrossRef]

5.

L. Essen and K. D. Froome, “The refractive indices and dielectric constants of air and its principal constituents at 24,000 Mc/s,” Proc. Phys. Soc. B 64(10), 862–875 (1951). [CrossRef]

6.

K. D. Froome, “The refractive indices of water vapour, air, oxygen, nitrogen and argon at 72 kMc/s,” Proc. Phys. Soc. B 68(11), 833–835 (1955). [CrossRef]

7.

T. Manabe, Y. Furuhama, T. Ihara, S. Saito, H. Tanaka, and A. Ono, “Measurements of attenuation and refractive dispersion due to atmospheric water vapor at 80 and 240 GHz,” Int. J. Infrared Millim. Waves 6(4), 313–322 (1985). [CrossRef]

8.

C. C. Bradley and H. A. Gebbie, “Refractive index of nitrogen, water vapor, and their mixtures at submillimeter wavelengths,” Appl. Opt. 10(4), 755–758 (1971). [CrossRef] [PubMed]

9.

H. Matsumoto, “The refractive index of moist air in the 3-µm region,” Metrologia 18(2), 49–52 (1982). [CrossRef]

10.

R. J. Hill and R. S. Lawrence, “Refractive index of water vapor in the infrared windows,” Infrared Phys. 26(6), 371–376 (1986). [CrossRef]

11.

R. Schödel, A. Walkov, and A. Abou-Zeid, “High-accuracy determination of water vapor refractivity by length interferometry,” Opt. Lett. 31(13), 1979–1981 (2006). [CrossRef] [PubMed]

12.

Y. Yang, M. Mandehgar, and D. Grischkowsky, “Time domain measurement of the THz refractivity of water vapor,” Opt. Express 20(24), 26208–26218 (2012). [CrossRef] [PubMed]

13.

P. Debye, Polar molecules, 89–90 (Dover Publ. Co., New York, N.Y., 1957).

14.

B. R. Bean and E. J. Dutton, Radio Meteorology, Monograph #92 (National Bureau of Standards, 1966), Chap. 1.

15.

J. H. Van Vleck and V. F. Weisskopf, “On the shape of collision-broadened lines,” Rev. Mod. Phys. 17(2-3), 227–236 (1945). [CrossRef]

16.

C. H. Townes and A. L. Schawlow, Microwave Spectroscopy (Dover Publ. Co., 1975).

17.

A. Deepak, T. D. Wilkerson, and L. H. Ruhnke, eds., Atmospheric Water Vapor (Academic Press, 1980). This book is the Proceedings of the International Workshop on Atmospheric Water Vapor, Vail, Colorado, September 11–13, 1979.

18.

H. M. Pickett, R. L. Poynter, E. A. Cohen, M. L. Delitsky, J. C. Pearson, and H. S. P. Muller, “Sub-millimeter, millimeter, and microwave spectral line catalog,” JQSRT 60(5), 883–890 (1998). [CrossRef]

Access to specific catalog entries may be found athttp://spec.jpl.nasa.gov/.

19.

L. S. Rothman, I. E. Gordon, A. Barbe, D. C. Benner, P. F. Bernath, M. Birk, V. Boudon, L. R. Brown, A. Campargue, J.-P. Champion, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, S. Fally, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, I. Kleiner, N. Lacome, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, S. N. Mikhailenko, C. E. Miller, N. Moazzen-Ahmadi, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. I. Perevalov, A. Perrin, A. Predoi-Cross, C. P. Rinsland, M. Rotger, M. Šimečková, M. A. H. Smith, K. Sung, S. A. Tashkun, J. Tennyson, R. A. Toth, A. C. Vandaele, and J. Vander Auwera, “The HITRAN 2008 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transf. 110(9-10), 533–572 (2009). [CrossRef]

20.

S. Golden, T. Wentink, R. Hillger, and M. Strandberg, “Stark spectrum of H2O,” Phys. Rev. 73(1), 92–93 (1948). [CrossRef]

21.

Y. Beers and G. P. Klein, “The Stark splitting of millimeter wave transitions of water,” J. Res. Natl. Bur. Stand., Sect. A 76a, 521–528 (1972).

22.

S. A. Clough, Y. Beers, G. P. Klein, and L. S. Rothman, “Dipole moment of water from Stark measurements of H2O, HDO, and D2O,” J. Chem. Phys. 59(5), 2254–2259 (1973). [CrossRef]

23.

T. R. Dyke and J. S. Muenter, “Electric dipole moments of low J states of H2O and D2O,” J. Chem. Phys. 59(6), 3125–3127 (1973). [CrossRef]

24.

S. L. Shostak, W. L. Ebenstein, and J. S. Muenter, “The dipole moment of water. I. Dipole moments and hyperfine properties of H2O and HDO in the ground and excited vibrational states,” J. Chem. Phys. 94(9), 5875–5882 (1991). [CrossRef]

25.

Y. Yang, M. Mandehgar, and D. Grischkowsky, “Understanding THz pulse transmission in the atmosphere,” IEEE Trans. THz Sci. Technol. 2, 406–415 (2012).

26.

Private communication from referee of [12].

27.

H. J. Liebe, G. A. Hufford, and T. Manabe, “A model for the complex permittivity of water at frequencies below 1 THz,” Int. J. Infrared Millim. Waves 12(7), 659–675 (1991). [CrossRef]

28.

L. Thrane, R. H. Jacobsen, P. Uhd Jepsen, and S. R. Keiding, “THz reflection spectroscopy of liquid water,” Chem. Phys. Lett. 240(4), 330–333 (1995). [CrossRef]

29.

J. T. Kindt and C. A. Schmuttenmaer, “Far-infrared dielectric properties of polar liquids probed by femtosecond terahertz pulse spectroscopy,” J. Phys. Chem. 100(24), 10373–10379 (1996). [CrossRef]

30.

C. Ro̸nne, L. Thrane, P.-O. Åstrand, A. Wallqvist, K. V. Mikkelsen, and S. R. Keiding, “Investigation of the temperature dependence of dielectric relaxation in liquid water by THz reflection spectroscopy and molecular dynamics simulation,” J. Chem. Phys. 107(14), 5319–5331 (1997). [CrossRef]

OCIS Codes
(010.1320) Atmospheric and oceanic optics : Atmospheric transmittance
(250.0250) Optoelectronics : Optoelectronics
(320.7160) Ultrafast optics : Ultrafast technology
(300.6495) Spectroscopy : Spectroscopy, teraherz

ToC Category:
Atmospheric and Oceanic Optics

History
Original Manuscript: May 15, 2013
Manuscript Accepted: July 23, 2013
Published: August 1, 2013

Citation
D. Grischkowsky, Yihong Yang, and Mahboubeh Mandehgar, "Zero-frequency refractivity of water vapor, comparison of Debye and van-Vleck Weisskopf theory," Opt. Express 21, 18899-18908 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-16-18899


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References

  1. J. D. Stranathan, “Dielectric constant of water vapor,” Phys. Rev.48(6), 538–544 (1935). [CrossRef]
  2. C. M. Crain, “The dielectric constant of several gases at a wave-length of 3.2 centimeters,” Phys. Rev.74(6), 691–693 (1948). [CrossRef]
  3. G. Birnbaum and S. K. Chatterjee, “The dielectric constant of water vapor in the microwave region,” J. Appl. Phys.23(2), 220–223 (1952). [CrossRef]
  4. L. Essen, “The refractive indices of water vapour, air, oxygen, nitrogen, hydrogen, deuterium and helium,” Proc. Phys. Soc. B66(3), 189–193 (1953). [CrossRef]
  5. L. Essen and K. D. Froome, “The refractive indices and dielectric constants of air and its principal constituents at 24,000 Mc/s,” Proc. Phys. Soc. B64(10), 862–875 (1951). [CrossRef]
  6. K. D. Froome, “The refractive indices of water vapour, air, oxygen, nitrogen and argon at 72 kMc/s,” Proc. Phys. Soc. B68(11), 833–835 (1955). [CrossRef]
  7. T. Manabe, Y. Furuhama, T. Ihara, S. Saito, H. Tanaka, and A. Ono, “Measurements of attenuation and refractive dispersion due to atmospheric water vapor at 80 and 240 GHz,” Int. J. Infrared Millim. Waves6(4), 313–322 (1985). [CrossRef]
  8. C. C. Bradley and H. A. Gebbie, “Refractive index of nitrogen, water vapor, and their mixtures at submillimeter wavelengths,” Appl. Opt.10(4), 755–758 (1971). [CrossRef] [PubMed]
  9. H. Matsumoto, “The refractive index of moist air in the 3-µm region,” Metrologia18(2), 49–52 (1982). [CrossRef]
  10. R. J. Hill and R. S. Lawrence, “Refractive index of water vapor in the infrared windows,” Infrared Phys.26(6), 371–376 (1986). [CrossRef]
  11. R. Schödel, A. Walkov, and A. Abou-Zeid, “High-accuracy determination of water vapor refractivity by length interferometry,” Opt. Lett.31(13), 1979–1981 (2006). [CrossRef] [PubMed]
  12. Y. Yang, M. Mandehgar, and D. Grischkowsky, “Time domain measurement of the THz refractivity of water vapor,” Opt. Express20(24), 26208–26218 (2012). [CrossRef] [PubMed]
  13. P. Debye, Polar molecules, 89–90 (Dover Publ. Co., New York, N.Y., 1957).
  14. B. R. Bean and E. J. Dutton, Radio Meteorology, Monograph #92 (National Bureau of Standards, 1966), Chap. 1.
  15. J. H. Van Vleck and V. F. Weisskopf, “On the shape of collision-broadened lines,” Rev. Mod. Phys.17(2-3), 227–236 (1945). [CrossRef]
  16. C. H. Townes and A. L. Schawlow, Microwave Spectroscopy (Dover Publ. Co., 1975).
  17. A. Deepak, T. D. Wilkerson, and L. H. Ruhnke, eds., Atmospheric Water Vapor (Academic Press, 1980). This book is the Proceedings of the International Workshop on Atmospheric Water Vapor, Vail, Colorado, September 11–13, 1979.
  18. H. M. Pickett, R. L. Poynter, E. A. Cohen, M. L. Delitsky, J. C. Pearson, and H. S. P. Muller, “Sub-millimeter, millimeter, and microwave spectral line catalog,” JQSRT60(5), 883–890 (1998).Access to specific catalog entries may be found at http://spec.jpl.nasa.gov/ . [CrossRef]
  19. L. S. Rothman, I. E. Gordon, A. Barbe, D. C. Benner, P. F. Bernath, M. Birk, V. Boudon, L. R. Brown, A. Campargue, J.-P. Champion, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, S. Fally, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, I. Kleiner, N. Lacome, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, S. N. Mikhailenko, C. E. Miller, N. Moazzen-Ahmadi, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. I. Perevalov, A. Perrin, A. Predoi-Cross, C. P. Rinsland, M. Rotger, M. Šimečková, M. A. H. Smith, K. Sung, S. A. Tashkun, J. Tennyson, R. A. Toth, A. C. Vandaele, and J. Vander Auwera, “The HITRAN 2008 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transf.110(9-10), 533–572 (2009). [CrossRef]
  20. S. Golden, T. Wentink, R. Hillger, and M. Strandberg, “Stark spectrum of H2O,” Phys. Rev.73(1), 92–93 (1948). [CrossRef]
  21. Y. Beers and G. P. Klein, “The Stark splitting of millimeter wave transitions of water,” J. Res. Natl. Bur. Stand., Sect. A76a, 521–528 (1972).
  22. S. A. Clough, Y. Beers, G. P. Klein, and L. S. Rothman, “Dipole moment of water from Stark measurements of H2O, HDO, and D2O,” J. Chem. Phys.59(5), 2254–2259 (1973). [CrossRef]
  23. T. R. Dyke and J. S. Muenter, “Electric dipole moments of low J states of H2O and D2O,” J. Chem. Phys.59(6), 3125–3127 (1973). [CrossRef]
  24. S. L. Shostak, W. L. Ebenstein, and J. S. Muenter, “The dipole moment of water. I. Dipole moments and hyperfine properties of H2O and HDO in the ground and excited vibrational states,” J. Chem. Phys.94(9), 5875–5882 (1991). [CrossRef]
  25. Y. Yang, M. Mandehgar, and D. Grischkowsky, “Understanding THz pulse transmission in the atmosphere,” IEEE Trans. THz Sci. Technol.2, 406–415 (2012).
  26. Private communication from referee of [12].
  27. H. J. Liebe, G. A. Hufford, and T. Manabe, “A model for the complex permittivity of water at frequencies below 1 THz,” Int. J. Infrared Millim. Waves12(7), 659–675 (1991). [CrossRef]
  28. L. Thrane, R. H. Jacobsen, P. Uhd Jepsen, and S. R. Keiding, “THz reflection spectroscopy of liquid water,” Chem. Phys. Lett.240(4), 330–333 (1995). [CrossRef]
  29. J. T. Kindt and C. A. Schmuttenmaer, “Far-infrared dielectric properties of polar liquids probed by femtosecond terahertz pulse spectroscopy,” J. Phys. Chem.100(24), 10373–10379 (1996). [CrossRef]
  30. C. Ro̸nne, L. Thrane, P.-O. Åstrand, A. Wallqvist, K. V. Mikkelsen, and S. R. Keiding, “Investigation of the temperature dependence of dielectric relaxation in liquid water by THz reflection spectroscopy and molecular dynamics simulation,” J. Chem. Phys.107(14), 5319–5331 (1997). [CrossRef]

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