Solid Parahydrogen Thickness Revisited

Abstract

We report updated infrared (IR) absorption measurements on vapor-deposited cryogenic parahydrogen (pH₂) solids that indicate a ≈10% systematic error in our previous approach for determining a pH₂ solid's thickness (S. Tam and M.E. Fajardo. Appl. Spectrosc. 2001. 55(12): 1634-1644). We provide corrected values for the integrated absorption intensities of the Q₁(0)+S₀(0) and S₁(0)+S₀(0) bands calculated over the 4495–4520 cm⁻¹ and 4825–4855 cm⁻¹ regions, respectively. New polarized IR absorption spectroscopy data demonstrate the insensitivity to polarization effects of the peak intensity of the Q_R(0) phonon sideband near 4228 cm⁻¹. This feature provides an even quicker way for determining the thickness of a pH₂ solid than via the integrated absorptions.

Keywords

Solid parahydrogen pH2 rapid vapor deposition matrix isolation spectroscopy infrared absorption IR polarized IR absorption spectroscopy sample thickness determination

Introduction

We previously promulgated a method for determining the thicknesses of cryogenic parahydrogen (pH₂) solids from their infrared (IR) absorption spectra.¹ We have also espoused the advantages of solid pH₂ as a “host” for matrix isolation spectroscopy in which impurity “dopants” are deliberately introduced into the matrix host.² For example, the excellent optical clarity of pH₂ solids produced by rapid vapor deposition (RVD)^3,4 permits quantitative absorption measurements on even millimeters-thick samples. These long optical path lengths enable working at a few parts-per-million (ppm) dopant concentrations, for which dopant–dopant interactions are minimized. Systematic studies of phenomena such as dopant clustering and intramolecular energy transfer can then be performed by methodically increasing the dopant concentrations. Extracting accurate dopant concentrations from absorption spectra via Beer's Law^5,6 requires accurate knowledge of the matrix host thickness.

In the course of our solid pH₂ studies, we upgraded to a different Fourier transform IR (FT-IR) spectrometer and discovered that the method presented by Tam and Fajardo¹ systematically over-predicts pH₂ sample thicknesses by ≈10%. In what follows, we discuss the nature of this error, report corrected values for the intrinsic intensities of the relevant solid pH₂ absorptions, and present an even simpler sample thickness determination approach based on the peak intensity of a nearby solid pH₂ absorption.

Experimental

We produce our pH₂ solids by direct in-vacuum deposition, onto a liquid helium cooled cryogenic substrate, of pre-cooled pH₂ gas flowing from an ortho-topara (O/P) hydrogen converter.^4,7 We typically operate the O/P converter near T ≈ 14 K, i.e., just above the pH₂ triple point, resulting in residual oH₂ concentrations below ≈100 ppm.

The experiments described herein were performed in two separate laboratories: one at Edwards Air Force Base (AFB), California, and the other at Eglin AFB, Florida. They are distinguished primarily by the exclusive use of optically transparent deposition substrates (e.g., BaF₂ and CsI windows) in the Edwards AFB experiments^1–4 versus the exclusive use of a reflective (gold-coated copper mirror) deposition substrate in the Eglin experiments.^7–9 Accordingly, the Edwards AFB IR measurements all utilize a simple single-pass transmission geometry (IR beam incident along the substrate surface normal), whereas the Eglin spectra are acquired in a reflection/absorption geometry (IR beam incident at 45° to the substrate surface normal) which double-passes the pH₂ sample.

We report our IR data as decadic absorbance spectra versus wavenumber, $\bar{ν}$ (cm⁻¹)

A_{10} (\bar{ν}) \equiv - lo g_{10} [\frac{I (\bar{ν})}{I_{0} (\bar{ν})}]

(1)

where the I₀ single-beam intensity data are obtained with a cold substrate just prior to sample deposition. Accordingly, we state Beer's Law as^5,6

2.303 A_{10} (\bar{ν}) = α X (\bar{ν}) C_{X} ℓ

(2)

in which: α_X(

\bar{ν}

) is a molecular absorption coefficient (cm²/mol), C_X is the concentration (mol/cm³) of species X, and ℓ is the optical path length through the sample (cm).

The sample thickness measurements reported by Tam and Fajardo¹ were based on IR spectra recorded at Edwards AFB using a Mattson Nova Cygni FT-IR spectrometer equipped with a liquid nitrogen (lN₂) cooled HgCdTe detector. As discussed by Tam and Fajardo,¹ this detector was overdriven by the high-incident IR intensity, raising concerns about the effects of nonlinear detector response.^10,11 This prompted supplemental measurements on the ν₁+ν₄ combination band of liquid CHCl₃ near 4200 cm⁻¹, which reassuringly showed a linear absorption response versus sample cell path length. Unfortunately, this well-intentioned approach failed to detect a ≈10% systematic spectrometric error. As noted without supporting data by Fajardo,² this error was discovered years later during the analysis of solid pH₂ spectra obtained with a Bruker IFS120HR FT-IR spectrometer equipped with an lN₂ cooled InSb detector with superior linearity characteristics.

A horizontal section through the Edwards sample deposition chamber is shown in Fig. 1 of Tam and Fajardo.¹ The delivery tube for the cold pH₂ gas makes a 45° angle to the deposition substrate surface normal and terminates ≈3 cm from the substrate center. Most of the pH₂ gas condenses on the “front” side of the cold deposition window facing the delivery tube; however, a small fraction condenses on the opposite “back” side. The total solid pH₂ thickness interrogated by the IR beam is thus the sum of the thicknesses of these two deposits

ℓ_{Edwards} = d_{total} = d_{front} + d_{back}

(3)

Figure 1.

IR absorption spectrum of an as-deposited pH₂ solid at T = 2.2 K; recorded in the Eglin reflection geometry and presented at 0.1 cm⁻¹ resolution. The net optical path length through the nominally 8 ppm CH₄/pH₂ solid is ℓ_Eglin = 1.2 mm.

Both of these deposits grow to produce non-planar (e.g., convex or wedged) solids. However, as shown in Fig. 7 of Tam and Fajardo,¹ for very thin samples the accreting pH₂ surfaces remain sufficiently parallel to the deposition window to produce transmission interference fringes in the IR spectra. These features can be analyzed to yield the interferometric film thicknesses, d_Int, using⁶

d_{Int} = m / {2 {n_{pH}}_{2} cos (θ) [{\bar{ν}}_{1} - {\bar{ν}}_{2}]}

(4)

in which m is the number of fringes observed over the

\bar{ν}

₁−

\bar{ν}

₂ wavenumber range, θ = 0 for normal incidence, and n_{pH
₂} is the index of refraction of solid pH₂. To remain consistent with Tam and Fajardo¹ and Fajardo,² we retain the value of n_{pH
₂} = 1.14; however, we note that more recent determinations^12,13 of n_{pH
₂} have reported values closer to 1.13.

A perspective sketch of the Eglin sample deposition chamber is shown in Fig. 3 of Molek et al.⁷ In this case, the pH₂ flow is incident from just below the surface normal of the vertically oriented reflective substrate. The incident and reflected IR beams make 45° angles to the surface normal, in a horizontal plane parallel to the optical table top. We use matched pairs of removable grid-type polarizers in the incident and reflected IR beams to perform Polarized IR Absorption Spectroscopy (PIRAS) measurements.^2,7–9 Filters oriented to pass horizontally polarized light (i.e., in the plane of reflection) correspond to the so-called p polarization of the IR beam, whereas vertical polarization corresponds to s polarization.¹⁴ We construct s-p PIRAS spectra as the difference between the decadic absorbances measured for the s and p polarizations

A_{10} (\bar{ν}) [s - p] = A_{10} (\bar{ν}) [s] - A_{10} (\bar{ν}) [p]

(5)

Figure 3.

Correlation plots of $\int A_{Q + S} (\bar{ν}) d \bar{ν}$ vs. $\int A_{S + S} (\bar{ν}) d \bar{ν}$ for samples produced in the Edwards (open circles) and Eglin (open diamonds) experimental geometries.

This PIRAS capability has proven valuable in resolving problems arising from unanticipated polarization effects in nominally “unpolarized” IR measurements.⁸ Our RVD pH₂ samples condense as polycrystalline solids, including metastable face-centered cubic (fcc) and thermodynamically stable hexagonal close-packed (hcp) regions.² Annealing to T > 4 K for a few minutes results in the irreversible conversion of the fcc regions to the hcp structure, and to a considerable degree of alignment of the hcp crystallites' unique c-axes with the substrate surface normal.² Since even a nominally “unpolarized” IR beam is transversely polarized with respect to its propagation direction, the Edwards geometry results in a strong bias against detecting absorptions with transition moments parallel to the hcp crystallites' c-axes. This caused us to not observe the crystal field splittings of several rovibrational lines of water monomers isolated in pH₂ solids, an error we resolved using PIRAS data.⁸ We anticipate related difficulties in comparing solid pH₂ spectra obtained in different laboratories employing different optical arrangements, and so use the PIRAS capability to identify robust absorption features with minimal polarization dependences.

In the Eglin apparatus, the incident IR beam is refracted at the vacuum:solid pH₂ interface causing it to bend towards the surface normal to an internal angle, θ_i, given by Snell's Law¹⁴

θ_{i} = si n^{- 1} {(n_{0} / n_{{pH}_{2}}) sin [θ_{0}]} \approx 38.3 \deg

(6)

where θ₀ = 45° is the angle of incidence and n₀ ≡ 1.00 is the index of refraction of vacuum. The optical path length through the sample is related to the solid pH₂ thickness, d, by

ℓ_{Eglin} = 2 d / cos (θ_{i}) \approx 2.550 d

(7)

As before, IR spectra of the thinnest samples show transmission interference fringes, and the interferometric thickness can be calculated using θ = 38.3° in Eq. 4.

Results and Discussion

Figure 1 shows a survey spectrum of the relevant solid pH₂ IR absorptions, which originate from intermolecular interactions between the pH₂ molecules.^15–17 The standard spectroscopic notation “L_v(J)” labels rovibrational transitions to vibrational level v, originating from v = 0 and rotational level J; with L = Q and S for ΔJ = 0 and +2, respectively.¹⁸ Three different classes of absorption features are apparent: (1) sharp zero-phonon “single transitions” such as the S₁(0) peak near 4486 cm⁻¹, (2) somewhat broader zero-phonon “double transitions” including our featured Q₁(0)+S₀(0) and S₁(0)+S₀(0) bands, and (3) broad blue-degraded phonon sidebands such as the S_R(0) band associated with the S₁(0) peak (here the “R” subscript indicates the creation of one or more phonons in the absorption process).

As discussed by Tam and Fajardo,¹ in highly symmetrical environments the net intermolecular interactions leading to the single transitions are subject to the “cancellation effect,” and so the observed intensities of these features depend strongly on the symmetries of the sites occupied by the pH₂ molecules. This makes them unsuitable for sample thickness determinations in our variable mixed composition fcc/hcp pH₂ solids.

The double transitions are immune to the cancellation effect, and the intensities of the Q₁(0)+S₀(0) and S₁(0)+S₀(0) bands depend mainly on the number density of pH₂–pH₂ pairs in the solid. Since the bulk densities of fcc and hcp pH₂ solids are the same, and pH₂ molecules in the two structures share the same number of nearest neighbors (i.e., 12), the double transitions are well suited for sample thickness determinations. We abbreviate the Q₁(0)+S₀(0) and S₁(0)+S₀(0) bands as Q+S and S+S, respectively. Because they both overlap the S_R(0) phonon sideband, we define their integrated band intensities using sloping line segment baselines joining the integration endpoints. For the Q+S and S+S features, these integration regions are: 4495–4520 cm⁻¹ and 4825–4855 cm⁻¹, respectively. We further abbreviate the integrated absorption intensities as: $\int A_{Q + S} (\bar{ν}) d \bar{ν}$ and $\int A_{S + S} (\bar{ν}) d \bar{ν}$ .

The broad feature marked “Q_R(0)” is the phonon sideband of the IR-silent Q₁(0) single transition. The Q₁(0) peak is absent in both fcc and hcp pH₂ crystals due to the cancellation effect. We define the peak height of the Q_R(0) feature as

Δ A_{10} [Q_{R} (0)] \equiv A_{10} (4228.5 c m^{- 1}) - A_{10} (4100 c m^{- 1})

(8)

Figure 2 shows PIRAS spectra of the Q_R(0) phonon sideband for an annealed pH₂ solid, which consists primarily of hcp crystallites aligned with the substrate surface normal. The Q_R(0) s-p PIRAS spectrum for the same as-deposited sample (not shown) exhibits negligible polarization dependence, with A₁₀( $\bar{ν}$ )[s-p] = 0.000 ± 0.002 over the 4100 to 4400 cm⁻¹ region. Coincidently, the s-p PIRAS spectrum of the Q_R(0) band for the annealed sample shows a near-zero polarization dependence at the 4228.5 cm⁻¹ peak. This means that the peak height, ΔA₁₀[Q_R(0)], should be particularly insensitive to the pH₂ sample's annealing history and to details of the optical arrangement used to record the IR absorption spectra.

Figure 2.

Polarization dependence of the Q_R(0) phonon sideband in an annealed pH₂ solid at T = 2.2 K; recorded in the Eglin reflection geometry and presented at 1.0 cm⁻¹ resolution. The net optical path length through the nominally 6 ppm NO/pH₂ solid is ℓ_Eglin = 4.1 mm.

Figure 3 shows correlation plots of the integrated intensities of the Q+S and S+S bands of as-deposited pH₂ samples containing < 1000 ppm dopant concentrations, and limited to spectra with

\int A_{Q + S} (\bar{ν}) d \bar{ν}

< 25 cm⁻¹. These low-dopant concentrations make completely negligible any intensity corrections for the slightly reduced number density of pH₂–pH₂ pairs (as discussed for large oH₂ concentrations by Gush et al.¹⁵). The slopes of the unweighted least squares best-fit straight lines constrained to pass through the origin are 12.93 and 12.70 for the Edwards and Eglin data, respectively. The unweighted means (μ) of the quotients

\int A_{Q + S} (\bar{ν}) d \bar{ν} / \int A_{Q + S} (\bar{ν}) d \bar{ν} \int A_{S + S} (\bar{ν}) d \bar{ν} \int A_{S + S} (\bar{ν}) d \bar{ν}

calculated over the 30 Edwards and 18 Eglin data are μ_Edwards = 12.92(0.24) and μ_Eglin = 12.86(0.12), respectively (95% confidence). We judge the differences between the Edwards and Eglin results to be negligible. We summarize these results in Table 1 as a recommended value of 12.9(0.2) for the quotient, in excellent agreement with the value of 13.0(0.4) from Tam and Fajardo,¹ and fair agreement (δ ≈ 6%) with the value of 13.7 derived by inspection of Fig. 1 by Kettwich et al.¹³

Table 1.

Parameters for solid pH₂ thickness determinations.

Quantity	Tam and Fajardo¹	Recommended
$\int A_{Q + S} (\bar{ν}) d \bar{ν} / \int A_{Q + S} (\bar{ν}) d \bar{ν} \int A_{S + S} (\bar{v}) d \bar{v} \int A_{S + S} (\bar{v}) d \bar{v}$	13.0 ± 0.4	12.9 ± 0.2
$\int A_{Q + S} (\bar{ν}) d \bar{ν} / \int A_{Q + S} (\bar{v}) d \bar{v} d_{total} d_{total}$	82 ± 3 cm⁻²	90 ± 2 cm⁻²
$\int A_{S + S} (\bar{ν}) d \bar{ν} / \int A_{S + S} (\bar{v}) d \bar{v} d_{total} d_{total}$	6.3 ± 0.3 cm⁻²	7.0 ± 0.2 cm⁻²
ΔA₁₀[Q_R(0)]/d_total	n/a	0.90 ± 0.03 cm⁻¹

Note: Uncertainties given as 95% confidence limits. Q+S = Q₁(0)+S₀(0); 4495–4520 cm⁻¹. S+S = S₁(0)+S₀(0); 4825–4855 cm⁻¹. ΔA₁₀[Q_R(0)] ≡ A₁₀(4228.5 cm⁻¹)–A₁₀(4100 cm⁻¹).

Figure 4 shows a correlation plot of the integrated intensity of the Q+S band versus the net optical path length through both front and back side pH₂ deposits produced in the Edwards transmission geometry. Included are data from the six pH₂ samples for which both d_front and d_back can be determined directly from transmission interference fringes in the IR spectra (cf. Fig. 7b in Tam and Fajardo¹). The slope of the unweighted least squares best-fit straight line constrained to pass through the origin is 0.00902 cm⁻¹/μm. The unweighted mean of the quotient $\int A_{Q + S} (\bar{ν}) d \bar{ν} / \int A_{Q + S} (\bar{ν}) d \bar{ν} d_{total} d_{total}$ is μ = 0.00898(0.00020) cm⁻¹/μm. We summarize these results as a recommended value of 0.0090(0.0002) cm⁻¹/μm = 90(2) cm⁻², which is ≈10% greater than the value of 82(3) cm⁻² from Tam and Fajardo¹, and which constitutes one of the major findings of this study. If we combine the results: $Int [Q + S] dv / Int [S + S] dv$ = 12.9(0.2) with $\int A_{Q + S} (\bar{ν}) d \bar{ν} / \int A_{Q + S} (\bar{ν}) d \bar{ν} d_{total} d_{total}$ = 90(2) cm⁻², we arrive at $\int A_{S + S} (\bar{ν}) d \bar{ν} / \int A_{S + S} (\bar{ν}) d \bar{ν} d_{total} d_{total}$ = 7.0(0.2) cm⁻², where the 95% confidence limits have been propagated in quadrature.

Figure 4.

Correlation plot of $\int A_{Q + S} (\bar{ν}) d \bar{ν}$ vs. d_total for as-deposited pH₂ samples produced in the Edwards transmission geometry.

We can also analyze the six experiments represented in Fig. 4 to determine the ratio of the thicknesses of the front and back side deposits. The unweighted mean of the quotient d_front/d_back is μ = 6.9(0.3). This is a considerable change from the value of “11.5” reported by Tam and Fajardo¹ for the Mattson FT-IR setup, which we attribute to sampling a different (offset) optical path through the wedged pH₂ deposits with the Bruker FT-IR setup. We can use this ratio to estimate d_front (e.g., for use in Beer's Law dopant concentration calculations) from interferometric measurements of d_back and from spectroscopic measurements of d_total using

d_{front} = 6.9 d_{back} = (\frac{6.9}{7.9}) d_{total}

(9)

Finally, Fig. 5 shows a correlation plot of d_total calculated from $\int A_{Q + S} (\bar{ν}) d \bar{ν}$ versus ΔA₁₀[Q_R(0)]; all measured in the Edwards transmission geometry. The slope of the unweighted least squares best-fit straight line constrained to pass through the origin is 1.11 cm. The unweighted mean of the quotient d_total/ΔA₁₀[Q_R(0)] for the 50 data points is μ = 1.12(0.02) cm. We summarize these results as a recommended value of ΔA₁₀[Q_R(0)]/d_total = 0.90(0.03) cm⁻¹, where the 95% confidence limits have been combined in quadrature.

Figure 5.

Correlation plot of d_total vs. ΔA₁₀[Q_R(0)] for as-deposited pH₂ solids in the Edwards transmission geometry.

Conclusion

We identify and correct a ≈10% systematic error in our previous approach¹ for determining a pH₂ solid's thickness based on integrating the Q₁(0)+S₀(0) and S₁(0)+S₀(0) absorption features. We identify the height of the Q_R(0) peak absorption as an alternative convenient proxy for the pH₂ thickness, and one likely to be robust against polarization-dependent measurement artifacts. Thus, we provide three approaches for calculating the total thickness of solid pH₂ in the optical path

d_{total} (cm) = \int A_{Q + S} (\bar{ν}) d \bar{ν} (c m^{- 1}) / 90 (c m^{- 2})

(10)

d_{total} (cm) = \int A_{S + S} (\bar{ν}) d \bar{ν} (c m^{- 1}) / 7.0 (c m^{- 2})

(11)

d_{total} (cm) = Δ A_{10} [Q_{R} (0)] / 0.90 (c m^{- 1})

(12)

with estimated errors less than ±3% (95% confidence).

Footnotes

Acknowledgments

MEF thanks his coworkers on various solid pH₂ projects—at Edwards AFB: Mr. S. Tam, Ms. M.E. DeRose, and Dr. M. Macler; at Eglin AFB: Dr. C.M. Lindsay and Dr. C.D. Molek. Approved for Public Release; Distribution Unlimited (96TW-2019-0165).

Conflict of Interest

The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author received no financial support for the research, authorship, and/or publication of this article.

References

Tam

Fajardo

M.E.

“Single and Double Infrared Absorptions in Rapid Vapor Deposited Parahydrogen Solids: Application to Sample Thickness Determination and Quantitative Infrared Absorption Spectroscopy”. Appl. Spectrosc. 2001. 55(12): 1634–1644.

M.E. Fajardo. “Matrix Isolation Spectroscopy in Solid Parahydrogen: a Primer”. In: L. Khriachtchev, editor. Physics and Chemistry at Low Temperatures. London, UK: World Scientific, 2011. Pp 167–202.

Fajardo

M.E.

Tam

“Rapid Vapor Deposition of Millimeters Thick Optically Transparent Parahydrogen Solids for Matrix Isolation Spectroscopy”. J. Chem. Phys. 1998. 108(10): 4237–4241.

Tam

Fajardo

M.E.

“Ortho/para Hydrogen Converter for Rapid Deposition Matrix Isolation Spectroscopy”. Rev. Sci. Instrum. 1999. 70(4): 1926–1932.

P.F. Bernath. Spectra of Atoms and Molecules. Oxford, UK: Oxford University Press, 2005. 2nd ed.

P.R. Griffiths, J.A. de Haseth. “Quantitative Analysis”. Fourier Transform Infrared Spectroscopy. New York, USA: John Wiley and Sons, 1986. Chap. 9, Pp. 197–224.

Molek

C.D.

Lindsay

C.M.

Fajardo

M.E.

“A Combined Matrix Isolation Spectroscopy and Cryosolid Positron Moderation Apparatus”. Rev. Sci. Instrum. 2013. 84(3): 035106.

Fajardo

M.E.

Lindsay

C.M.

“Crystal Field Splitting of Rovibrational Transitions of Water Monomers Isolated in Solid Parahydrogen”. J. Chem. Phys. 2008. 128(1): 014505.

Fajardo

M.E.

Lindsay

C.M.

Momose

“Crystal Field Theory Analysis of Rovibrational Spectra of Carbon Monoxide Monomers Isolated in Solid Parahydrogen”. J. Chem. Phys. 2009. 130: 244508.

10.

T. Hirschfeld. “Quantitative FT-IR: A Detailed Look at the Problems Involved”. In: J.R. Ferraro, L.J. Basile, editors. Fourier Transform Infrared Spectroscopy. New York, USA: Academic Press, 1979. Vol. 2.

11.

Chase

D.B.

“Nonlinear Detector Response in FT-IR”. Appl. Spectrosc. 1984. 38(4): 491–494.

12.

Perera

Tom

B.A.

Miyamoto

, et al. “Refractive Index Measurements of Solid Parahydrogen”. Opt. Lett. 2011. 36(6): 840–842.

13.

Kettwich

S.C.

Anderson

D.T.

Walker

M.A.

, et al. “The Infrared Dielectric Function of Solid Para Hydrogen”. Mon. Not. R. Astron. Soc. 2015. 450: 1032–1041.

14.

F.L. Pedrotti, L.S. Pedrotti. Introduction to Optics. Englewood Cliffs, New Jersey, USA: Prentice-Hall, 1987.

15.

Gush

H.P.

Hare

W.F.J.

Allin

E.J.

, et al. “The Infrared Fundamental Band of Liquid and Solid Hydrogen”. Can. J. Phys. 1960. 38: 176–193.

16.

J. Van Kranendonk. Solid Hydrogen. New York, USA: Plenum Press, 1983.

17.

Rao

K.N.

Gaines

J.R.

Balasubramanian

T.K.

, et al. “Infrared Spectrum of Hydrogen in the Condensed Phase”. Acta Phys. Hung. 1984. 55(1–4): 383–393.

18.

Mulliken

R.S.

“Report on Notation for Spectra of Diatomic Molecules”. Phys. Rev. 1930. 36(4): 611–629.