Collision-induced absorption spectra (CIA) for monatomic gas mixtures of Helium-Xenon

Abstract

The lineshapes of collision-induced absorption (CIA) at room temperature are computed quantum mechanically for gaseous binary mixtures of helium with xenon using theoretical induced dipole moment and interatomic potentials. Empirical and literature models of the induced dipole moments which reproduce the experimental spectra and the first few spectral moments are given. Good agreement between the computed and experimental lineshapes of absorption is obtained when the potential parameters which are fitted well to the vibrational energy levels, thermophysical and transport properties are used.

Keywords

Collision-induced absorption spectral moments of absorption interatomic potential induced dipole moment Helium-Xenon

1. Introduction

In a recent paper, we calculated at room temperature, the collision-induced light scattering (CILS) spectra of krypton (Kr) gas using an empirical induced trace and anisotropy polarizability [1]. Also, in a previous works, we calculated at various temperatures, the collision-induced absorption (CIA) spectra of gaseous mixtures Ar-Xe and He-Ar using an empirical induced dipole moment and interatomic potentials [2, 3]. The derivation of these quantities is based on classical physics and relies on the first three even moments of scattering and absorption [1, 2, 3]. Due to the observed agreement for scattering and absorption spectra of these systems, we extend this treatment of absorption to He-Xe pairs. We use the available literature and experimental thermophysical, transport properties and vibrational energy levels for this mixture to construct the parameters of the relevant interaction potential and induced dipole moment.

In the far infrared region of the spectrum, mixtures of dense phase noble gases show an absorption band which is due to the induced dipole moment arising from the deformation of the electronic clouds during the collision of two unlike atoms [4, 5]. As the induced dipole moment depends on the distance between the colliding atoms, the translational state of the system can change owing to the interaction of the induced dipole with the electromagnetic field, resulting in a translational absorption band. From the early observation of this phenomenon in spectra of mixtures of noble gases, substantial effort has been dedicated to their study, because the parameters involved in the construction of the interatomic potential and the induced dipole moment may be deduced from measurements of the translational band [5, 6, 7].

For He-Xe mixture, no accurate potential is available. We calculated an approximate interatomic potential using mostly the methods outlined in a previous papers [2, 3]. As the relevant details and references were given therein [2, 3], we will only restate the equations when it is necessary for the sake of continuity. To reiterate, the basic strategy in this paper is to include collision-induced (CIA) absorption data in addition to mixtures viscosity, diffusion, thermal conductivity coefficients, isotopic thermal factors and second pressure virial coefficients data at various temperatures, to fit the simple functional form of modified Tang-Toennies model (MTT) and Barker, Fisher and Watts (BFW) interatomic potentials for the He-Xe interaction.

The thermophysical, transport and collision-induced absorption properties used in the fitting are complementary. For this mixture, viscosity, thermal conductivity, isotopic thermal factor and diffusion data are most sensitive to the wall of the potential from $r_{m}$ inward to a point where the potential is repulsive [8], while the second pressure virial coefficients reflect the size of $r_{m}$ and the volume of the attractive well [9]. The measured collision-induced absorption (CIA) at various temperatures is most sensitive to the repulsive part of the potential [10].

We present in this paper at room temperature a new analysis of the translational band of collision-induced absorption spectra (CIA) of He-Xe, based on fitting the spectral moments of the profiles of the measurements. Spectral profiles are calculated with the help of a quantal computer program numerically and compared with the measured spectra. The comparison of calculated and measured spectra provides valuable insights concerning the quality of existing models of both the interatomic potential and induced dipole moments. Analysis of spectral moments of CIA to determine the parameters of the induced dipole moments adopted is given in Section 2. The theoretical method for calculating the intensity of the lineshapes and the associated spectral moments of absorption are briefly given in Section 3, together with the computational implementation. The calculations of the different properties using the present interatomic potentials are presented in Section 4. Results are presented and discussed in Section 5 and the concluding remarks are given in Section 6.

2. Analysis of spectral moments of CIA to determined the induced dipole moments

In order to calculate the spectral line profiles of absorption and the associated zeroth, first and second moments one needs the induced dipole moment. Results with different models can be compared with experiment to assess the quality of the induced dipole moment. In this paper and for the sake of comparison and discussion, we considered six models of the induced dipole moment which are the dispersion type [11, 12], the exponential function models [13, 14, 15], the analytical dipole moment models [16, 17], the higher-order polynomial model [18], the empirical dipole model [12], the ab initio SCF, MP ${}_{2}$ , B3LYP and B3PW91 models [19].

Below, we shall use the analytical dipole moment model to see if the induced dipole moment can be approximated by such a simple model. Particularly in the case of the He-Xe gas mixture, for which the fundamental theory is at present of limited value, this model will be seen to provide a useful empirical basis description of the interaction-induced dipole.

Over a broader range of separations [16, 17], it has been argued that an analytical dipole model like

$\displaystyle\mu(r)=\mu_{0}\exp\left(-\frac{r-\sigma}{r_{1}}-\left(\frac{r-% \sigma}{r_{2}}\right)^{2}\right)-\frac{D_{7}}{r^{7}}$ (1)

should be expected to approximate the dipole moment more closely.

Suitable values for the coefficients $\mu_{0}$ , $D_{7}$ , $r_{1}$ and $r_{2}$ were obtained via an economical least-squares routine fitting the lowest three spectral moments of absorption at room temperature to the values computed from the experimental moments [7]. This solution was then checked against the lineshapes.

The method of detailed analysis of the first three even moments of the polarized and depolarized light scattering spectrum (CILS) has been used by Meinander et al. [20] and El-Kader [1, 21] for the determination of the extra-dipole-induced dipole (DID) contribution to the pair-polarizability trace and anisotropy of inert gases and CH ${}_{4}$ and mercury. This consists of establishing an appropriate parameterized model form for the trace and the anisotropy, subsequently then searching for the sets of parameters that are consistent with the experimental values of the polarized and depolarized spectral moments.

In order to proceed, it is convenient to rewrite the induced dipole moment $\mu(r)$ of the Eq. (1) in terms of the reduced variable $x=r/r_{m}$ where $r_{m}$ is the separation at the minimum of the interatomic potential $V(r)$ . In this case, one has:

$\displaystyle\mu(x)=\mu_{0}\exp\left(-\left(\frac{x-\sigma^{*}}{x_{1}}\right)-% \left(\frac{x-\sigma^{*}}{x_{2}}\right)^{2}\right)-\frac{D_{7}^{*}}{x^{7}}$ (2)

where $\sigma^{*}=\sigma/r_{m}$ ; $x=r/r_{m}$ , $x_{1}=r_{1}/r_{m}$ , $x_{2}=r_{2}/r_{m}$ and $D_{7}^{*}=D_{7}/r_{m}^{7}$ .

Substituting Eq. (2) into the moment expressions of absorption spectrum [22] make it feasible to rewrite them in the form of quadratic equations for the unknown $\mu_{0}$ and $D_{7}$ with coefficients as parametric functions of $r_{1}$ and $r_{2}$ . The Newton-Raphson matrix-equation solver is applied in order to solve the multi-property optimization problem to obtain the values of the unknowns mentioned above. This can be done by introducing the equation system by means of the Jacobian matrix [23]:

$\displaystyle\textbf{D}(\textbf{X})=\left|\begin{array}[]{ccc}{\displaystyle% \frac{\partial M_{0}}{\partial\mu_{0}}}&{\displaystyle\frac{\partial M_{0}}{% \partial r_{1}}}&{\displaystyle\frac{\partial M_{0}}{\partial D_{7}}}\\ {\displaystyle\frac{\partial M_{1}}{\partial\mu_{0}}}&{\displaystyle\frac{% \partial M_{1}}{\partial r_{1}}}&{\displaystyle\frac{\partial M_{1}}{\partial D% _{7}}}\\ {\displaystyle\frac{\partial M_{2}}{\partial\mu_{0}}}&{\displaystyle\frac{% \partial M_{2}}{\partial r_{1}}}&{\displaystyle\frac{\partial M_{2}}{\partial D% _{7}}}\end{array}\right|$ (3) $\displaystyle\textbf{X}=\left(\begin{array}[]{c}\mu_{0}\\ r_{1}\\ D_{7}\end{array}\right)$ (4) $\displaystyle\textbf{F}(\textbf{X})=\left(\begin{array}[]{c}\Delta M_{0}\\ \Delta M_{1}\\ \Delta M_{2}\end{array}\right)$ (5)

where $\Delta M_{n}=M_{n}-M_{n,\text{exp}}\ (n=0,1,2)$ , and $M_{n}$ , $M_{n,\text{exp}}$ are the calculated and experimental spectral moments, respectively.

$\displaystyle\textbf{X}_{i+1}=\textbf{X}_{i}-\left.\textbf{D}^{-1}\right|_{% \textbf{X}_{i}}\textbf{F}(\textbf{X}_{i})$ (6)

at any step $i$ of the iteration procedure. Once convergence is obtained, the column vector solution X has to satisfy $\textbf{F}(\textbf{X})=$ 0. For the specific functional form of $\mu(r)$ given by Eq. (1), the nine elements of the matrix D are determined and given in the Appendix [2, 24].

Table 1

Comparison of experimental and theoretical spectral moments of HeXe mixture at $T=$ 298 K

$M_{0}*$ 10 ${}^{61}$ (erg cm ${}^{6}$ )		$M_{1}*$ 10 ${}^{49}$ (erg cm ${}^{6}$ /s)		$M_{2}*$ 10 ${}^{35}$ (erg cm ${}^{6}$ /s ${}^{2}$ )
Cal.	Exp.	Cal.	Exp.	Cal.	Exp.
(1.08) ${}^{\text{a}}$ , (1.4) ${}^{\text{b}}$	1.32[7]	(6.554) ${}^{\text{a}}$ , (7.46) ${}^{\text{b}}$	7.46[7]	(5.90) ${}^{\text{a}}$ , (6.41) ${}^{\text{b}}$	6.41[7]
(1.76) ${}^{\text{c}}$ , (1.72) ${}^{\text{d}}$		(8.225) ${}^{\text{c}}$ , (8.69) ${}^{\text{d}}$		(6.94) ${}^{\text{c}}$ , (7.45) ${}^{\text{d}}$
(1.78) ${}^{\text{e}}$ , (1.653) ${}^{\text{f}}$		(8.6) ${}^{\text{e}}$ , (8.395) ${}^{\text{f}}$		(7.3) ${}^{\text{e}}$ , (7.19) ${}^{\text{f}}$
(1.8) ${}^{\text{g}}$ , (2.16) ${}^{\text{h}}$		(8.99) ${}^{\text{g}}$ , (10.25) ${}^{\text{h}}$		(7.69) ${}^{\text{g}}$ , (8.71) ${}^{\text{h}}$
(1.4) ${}^{\text{i}}$ , (1.627) ${}^{\text{j}}$		(7.07) ${}^{\text{i}}$ , (7.95) ${}^{\text{j}}$		(6.0) ${}^{\text{i}}$ , (6.73) ${}^{\text{j}}$
(1.23) ${}^{\text{k}}$ , (1.24) ${}^{\text{l}}$		(6.20) ${}^{\text{k}}$ , (6.63) ${}^{\text{l}}$		(5.26) ${}^{\text{k}}$ , (5.65) ${}^{\text{l}}$
(1.59) ${}^{\text{m}}$ , (1.9) ${}^{\text{o}}$		(7.95) ${}^{\text{m}}$ (9.19) ${}^{\text{o}}$		(6.757) ${}^{\text{m}}$ , (7.8) ${}^{\text{o}}$
(1.2) ${}^{\text{q}}$ , (0.94) ${}^{\text{r}}$		(5.3) ${}^{\text{q}}$ , (5.63) ${}^{\text{r}}$		(4.69) ${}^{\text{q}}$ , (5.01) ${}^{\text{r}}$
(1.2) ${}^{\text{s}}$ , (1.38) ${}^{\text{t}}$		(6.84) ${}^{\text{s}}$ , (7.15) ${}^{\text{t}}$		(6.05) ${}^{\text{s}}$ , (6.3) ${}^{\text{t}}$

${}^{\text{a}}$ Values calculated using the present BFW interatomic potential and induced dipole moment Eq. (1) with $D_{7}=$ 0, ${}^{\text{b}}$ Values calculated using the present BFW interatomic potential and induced dipole moment Eq. (1), ${}^{\text{c}}$ Values calculated using the present MTT interatomic potentialand induced dipole moment Eq. (1), ${}^{\text{d}}$ Values calculated using LJ (12-6) interatomic potential [25] and induced dipole moment Eq. (1), ${}^{\text{e}}$ Values calculated using (m-8-6) interatomic potential [26] and induced dipole moment Eq. (1), ${}^{\text{f}}$ Values calculated using ESMSV interatomic potential [27] and induced dipole moment Eq. (1), ${}^{\text{g}}$ Values calculated using LJ (12-6) interatomic potential [27] and induced dipole moment Eq. (1), ${}^{\text{h}}$ Values calculated using Exp-6 interatomic potential [27] and induced dipole moment Eq. (1), ${}^{\text{i}}$ Values calculated using the TT interatomic potential [28] and induced dipole moment Eq. (1), ${}^{\text{j}}$ Values calculated using the HFD-DK interatomic potential [29] and induced dipole moment Eq. (1), ${}^{\text{k}}$ Values calculated using the HFD-BC interatomic potential [30] and induced dipole moment Eq. (1), ${}^{\text{l}}$ Values calculated using the MSV interatomic potential [29] and induced dipole moment Eq. (1), ${}^{\text{m}}$ Values calculated using the HFD-B interatomic potential [31] and induced dipole moment Eq. (1), ${}^{\text{o}}$ Values calculated using HFD-B interatomic potential [29] and induced dipole moment Eq. (1), ${}^{\text{q}}$ Values calculated using the present BFW interatomic potential and B3PW91 induced dipole moment [19], ${}^{\text{r}}$ Values calculated using the present BFW interatomic potential and B3LYB induced dipole moment [19], ${}^{\text{s}}$ Values calculated using the present BFW interatomic potential and MP2 induced dipole moment [19], ${}^{\text{t}}$ Values calculated using the present BFW interatomic potential and SCF induced dipole moment [19].

The advantage of this method to obtain $\mu_{r}$ is the speed of computation and the avoidance of the trial-and-error approach. The method of moments analysis has been applied to the collision-induced light scattering (CILS) spectra of inert gases, mercury and methane [1, 20, 21]. Because the correction to the first-order DID expressions is quite small over most of the interatomic separations probed by the atomic motions, and the two correction terms cancel out to a large extent the effects of each other, the numerical values derived for the parameters $\mu_{0}$ and $D_{7}$ of the induced dipole moment, are quite sensitive to the input values of the moments.

Table 1 gives the numerical results of the spectral moments at room temperature while Fig. 1 shows the different regions of the dipole moment behavior which are consistent with the experimental results. Here the remaining theoretical and the ab initio induced dipole models [19] are reported as well.

Table 2

Parameters of the different interatomic potentials and the associated values of $\delta$

Compound	Pots.	$\varepsilon/k_{B}$ (K)	$r_{m}$ (Å)	$\xi$	$A_{0}$	$A_{1}$	$A_{2}$	$A_{3}$	$A_{4}$	$A_{5}$	$A$	Bet
He ${}_{2}$	BFW	10.9	2.97	8.57	0.87866	$-$ 5.07669	7.38142	$-$ 46.0	150.0	$-$ 36.0	–	–
	MTT	11.0	2.968	–	–	–	–	–	–	–	11.6572	1.38316
		$\delta_{B_{12}}$	$\delta_{\eta}$	$\delta_{\lambda}$	$\delta_{D}$	$\delta_{\text{Iso}}$	$\delta_{O}$
	BFW	0.19	0.37	0.28	0.48	0.59	0.41
	MTT	0.23	0.41	0.49	0.57	0.66	0.49
Xe ${}_{2}$	BFW	283.4	4.36	14.2	1.08279	$-$ 0.016262	$-$ 11.0443	$-$ 64.0	277.0	$-$ 103.0	–	–
	MTT	284.0	4.355	–	–	–	–	–	–	–	68.94	1.0031
		$\delta_{\textit{vl}}$	$\delta_{B_{12}}$	$\delta_{\eta}$	$\delta_{\lambda}$	$\delta_{D}$	$\delta_{\text{Iso}}$	$\delta_{O}$
	BFW	0.75	0.86	0.62	0.59	0.87	0.93	0.78
	MTT	0.77	0.89	0.66	0.63	0.91	0.97	0.82
He-Xe	BFW	27.9	3.97	9.025	1.091	$-$ 4.94843	18.0169	$-$ 53.0	33.0	$-$ 150.0	–	–
	MTT	27.4	3.995	–	–	–	–	–	–	–	6.42672	1.05684
		$\delta_{\textit{vl}}$	$\delta_{B_{12}}$	$\delta_{\eta_{\text{mix}}}$	$\delta_{\lambda_{\text{mix}}}$	$\delta_{D_{\text{mix}}}$	$\delta_{\text{Iso}_{\text{mix}}}$	$\delta_{O}$
	BFW	0.2	0.73	0.56	0.49	0.76	0.82	0.63
	MTT	0.2	0.79	0.61	0.53	0.82	0.87	0.68

Whatever the atoms considered, del $=$ 0.01. Besides, $\delta_{j}$ is defined by $\delta_{j}^{2}=(1/n_{j}\sum_{i=1}^{n_{j}}{\Delta_{ji}^{-2}(P_{ji}-p_{ji})^{2}})$ , where $P_{ji}$ and $p_{ji}$ are, respectively, the calculated and experimental values of property $j$ at point $i$ and $\Delta_{ji}$ is the experimental uncertainty of property $j$ at point $i$ . The subscripts vl, $B_{12}$ , $\eta$ , $\lambda$ , $D$ , and Iso refer, respectively, to the vibrational energy levels, interaction second pressure virial coefficient, viscosity, thermal conductivity, diffusion coefficient and thermal diffusion factor. The overall rms deviation was obtained from $\delta_{O}=\sqrt{\frac{1}{N}(\sum_{j=1}^{N}{\delta_{j}^{2}})}$ .

The empirical pair potentials BFW and MTT with the different parameters are given in Table 2 for the considered mixture. The moment analysis program was easily modified to calculate the moments of the absorption spectra as functions of $\mu_{0}$ and $D_{7}$ using Eq. (1). The experimentally determined values of the moments, with error limits, now each define a range of acceptable values for the parameters. This approach to the construction of an empirical model for $\mu(r)$ is acceptable if all moments define a common range of $\mu_{0}$ , $D_{7}$ and $r_{1}$ values. As one can see from Table 1 and Fig. 2, the agreement between the experimental values and the theoretical ones using our empirical potential BFW is excellent and we have verified that it remains acceptable for $\mu_{0}=$ 0.022555 a.u., $r_{1}=$ 0.0418 nm, $r_{2}=$ 0.215 nm and $D_{7}=$ 8380 a.u.

3. Theory of lineshapes of collision-induced absorption

The quantum mechanical calculations for collision-induced absorption (CIA) are described. The atomic wavefunctions, which enter the computation of the matrix elements, are obtained by numerical integration of the radial Schrödinger equation [32] using the energy density normalization.

From quantum mechanical theory, collision-induced absorption spectra (CIA) can be computed if the interaction potential is known along with a suitable model of the collision-induced dipole moment [33]. The absorption coefficient $\alpha(\omega)$ is related to the product of volume $v$ and the so-called spectral function, $G(\omega)$ , according to [33]:

$\displaystyle\alpha(\omega)=\frac{4\pi^{2}}{3\hbar c}n_{1}n_{2}\omega(1-\exp(-% \hbar\omega/k_{B}T))vG(\omega)$ (7)

Here, $\omega$ designates angular frequency; $\hbar$ is the Planck’s constant; $c$ is the speed of light in vacuum; $n_{1}$ and $n_{2}$ are the number densities of the mixed gases; and $vG(\omega)$ is the spectral density, which may be calculated if the induced dipole $\mu(r)$ and potential functions are given. The spectral moments are defined as:

$\displaystyle M_{n}=\int_{-\infty}^{\infty}{\omega^{n}vG(\omega)\mathrm{d}\omega}$ (8)

for $n=$ 0, 1, 2, … These moments can be compared with values calculated directly from the sum rules [22], where quantum corrections were made for the pair distribution functions $g(r)$ , $g_{e}(r)$ and $g_{m}(r)$ [34].

Figure 1.

Empirical induced dipole moment $\mu(r)$ of He-Xe with ab initio models [19].

Figure 2.

Comparison between the calculated translational collision-induced absorption spectra of He-Xe at $T=$ 298 K using the present BFW interatomic potential with the parameters given in Table 2 and empirical and ab initio induced dipole moments with the experimental ones.

It is often inconvenient to use tabular data in spectral moments and line shape computations. Therefore, we have obtained an analytical model of the exchange or overlap dipole in the range of interest (near 0.3 nm) by a least mean squares fit. It is of the form Eq. (1).

Range and strength are the parameters determined by the analysis, with the results 0.0075 $\leqslant\mu_{0}\leqslant$ 0.0082 a.u., and 0.0365 $\leqslant r_{1}\leqslant$ 0.038 nm for He-Xe gas mixture.

As a first step and for He-Xe gas mixture, we used the exponential dipole, Eq. (1) with $D_{7}=$ 0 and the parameters given above. The star curve in Fig. 2 gives the calculated line shape for comparison with the measurements [7]. The agreement is less than perfect. Near the peak of absorption, the calculated absorption is too weak and the high frequency wing is not well represented at all, in spite of the fact that the lowest three spectral moments of the measurements agree very closely with those of the computed line shape. Therefore, we add a dispersion part $D_{7}>$ 0 to the exponential dipole Eq. (1).

Since an accurate determination of these spectral integrals requires knowledge of the absorption coefficient $\alpha(\omega)$ at low and high frequencies, which is not available, it is best to approximate the spectral function $vG(\omega)$ by an analytical model profile with three parameters, the so-called $K_{0}$ model [35], which was chosen to provide a remarkably close representation of virtually all line shapes arising from exchange and dispersion force induction. These parameters have been determined by fitting the experimental spectrum, using a least mean squares procedure. From the fit, the three lowest spectral moments of the measurements at room temperature for the considered mixture are readily obtained with 0.019 $\leqslant\mu_{0}\leqslant$ 0.023 a.u., 0.0395 $\leqslant r_{1}\leqslant$ 0.0423 nm, 0.198 $\leqslant r_{2}\leqslant$ 0.224 nm and 8000 $\leqslant D_{7}\leqslant$ 8400 a.u.

4. Multi-property analysis and the interatomic potentials

The interatomic potentials we provide here are obtained through the analysis of the vibrational energy levels [36, 37, 38, 39, 40, 41, 42, 43, 44], pressure second virial coefficients [25, 26, 27, 28, 29, 30, 31, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74] and a set of gaseous transport properties [25, 29, 30, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110].

For the analysis of all these experimental data, we consider the following potentials:

A. MTT potential

The modified Tang-Toennies model (MTT) which in the whole range of interactions can be represented by the formula [111]:

$\displaystyle V(r)=Ar^{\textstyle\left(\frac{7}{2\text{Bet}}-1\right)}e^{-2% \text{Bet}\,r}-\sum\limits_{n=3}^{\infty}\left(\frac{C_{2n}f_{2n}(br)}{r^{2n}}\right)$ (9)

where $f_{2n}(br)$ is the appropriate damping functions, given by the expression derived by Tang and Toennies [112]:

$\displaystyle f_{2n}(br)=1-e^{-br}\sum\limits_{k=0}^{2n}{\frac{(br)^{k}}{k!}}$ (10)

and

$\displaystyle b=2\text{Bet}-\frac{{\displaystyle\frac{7}{2\text{Bet}}}-1}{r}$ (11)

B. BFW potential

The empirical Barker, Fisher and Watts (BFW) interatomic potential [113] is represented by the following formula:

$\displaystyle V(r)=\varepsilon\left(\sum\limits_{i=0}^{5}{A_{i}(r/r_{m}-1)^{i}% \exp(\xi(1-r/r_{m}))}-\sum\limits_{n=3}^{\infty}{\frac{C_{2n}}{(x^{2n}+\text{% del})}}\right)$ (12)

where $x=r/r_{m}$ is the reduced distance, $\varepsilon$ is the potential depth, $r_{m}$ is the distance at the minimum potential and the rest are fitting parameters.

Even at the present (MTT) level, there are seven free parameters ( $\varepsilon$ , $r_{m}$ , $A$ , Bet, $C_{6}$ , $C_{8}$ , $C_{10}$ ) and in the BFW model, there are really thirteen parameters ( $\varepsilon$ , $r_{m}$ , $A_{0}$ , $A_{1}$ , $A_{2}$ , $A_{3}$ , $A_{4}$ , $A_{5}$ , $\xi$ , del, $C_{6}$ , $C_{8}$ , $C_{10}$ ) which are far too many to determine from the present data. Accordingly, we proceeded as follows: the coefficients $A$ , Bet, $A_{0}$ and $A_{1}$ , are determined from the conditions of continuity. The long-range dispersion coefficients $C_{6}$ , $C_{8}$ , $C_{10}$ were taken from atomic data of Jiang et al. [114] and from theoretical calculations of Thakkar et al. [115], leaving rest parameters in these models that were varied to fit the vibrational energy levels $\delta_{\textit{vl}}$ . This minimization is further supported by calculating $\delta_{B}$ , $\delta_{\eta}$ , $\delta_{\lambda}$ , $\delta_{D}$ and $\delta_{\text{Iso}}$ , the rms deviations calculated from second pressure virial coefficients, mixture viscosity, thermal conductivity, diffusion coefficients and thermal diffusion factor, respectively. This decision leads to the potentials parameters in Table 2 as our best estimate of He ${}_{2}$ , Xe ${}_{2}$ and He-Xe interatomic potentials.

4.1 Vibrational energy levels

With the above obtained interaction potentials, the vibrational energy levels can be calculated by solving the radial one-dimensional Schrödinger equation. In the present paper, this equation is solved numerically. The interaction potentials of homonuclear and hetronuclear rare-gas dimers with the internuclear separation between 3.0 a.u. and 200 a.u. are used to do the calculation. The size of the grid points is 1600. Table 2 presents the calculation results for He ${}_{2}$ , Xe ${}_{2}$ and He-Xe dimers. It is seen from Table 3 that the agreement of the two sets of spacings predicted by the MTT and BFW is good.

The experimental and other theoretical results are also listed in the Table 3 for comparison [36, 37, 38, 39, 40, 41, 42, 43, 44]. It is gratifying to find that the spacings of these dimers predicted by the MTT and BFW potential models are in excellent agreement with the experimental results if the experimental error bar are taken into consideration.

4.2 Analysis of second pressure virial coefficients

Table 3
Comparison of our computed vibrational energy levels for He ${}_{2}$ , Xe ${}_{2}$ and He-Xe gases with experimental and other theoretical data. All values are given in cm ${}^{-1}$

		BFW	MTT	Sheng [36]	Xie [37]	Hellmann [38]	Aziz [39]	Dham [40]	Slviak [41]	Hanni [42]	Lit. results [43, 44]
He ${}_{2}$	0–1	–	–	–	–	–	–	–	–	–	–
Xe ${}_{2}$	0–1	19.67	20.02	19.369	19.5	19.31 $\pm$ 0.08	19.61	19.61	18.69	19.39	19.90 $\pm$ 0.3
	1–2	18.47	18.75	18.214	18.37	18.15 $\pm$ 0.08	18.41	18.45	17.53	18.23	18.55 $\pm$ 0.3
	2–3	17.27	17.49	17.06	17.23	16.99 $\pm$ 0.09	17.21	17.28	16.37	17.08	17.20 $\pm$ 0.3
	3–4	16.08	16.24	15.92	16.08	15.83 $\pm$ 0.09	16.02	16.09	15.21	15.93	16.17 $\pm$ 0.3
	4–5	14.90	14.99	14.81	14.93	14.68 $\pm$ 0.09	14.83	14.89	14.07	14.79	14.63 $\pm$ 0.3
	5–6	13.71	13.77	13.65	13.78	13.54 $\pm$ 0.09	13.66	13.71	12.92	13.65	13.70 $\pm$ 0.3
	6–7	12.54	12.57	12.34	12.62	12.41 $\pm$ 0.09	12.50	12.54	11.79	12.52	12.63 $\pm$ 0.3
	7–8	11.39	11.39	11.11	11.48	11.29 $\pm$ 0.09	11.35	11.40	10.68	11.41	11.33 $\pm$ 0.3
	8–9	10.26	10.21	10.28	10.35	10.19 $\pm$ 0.09	10.23	10.27	9.58	10.32	10.15 $\pm$ 0.3
	9–10	9.16	9.11	9.604	9.24	9.12 $\pm$ 0.09	9.14	9.16	8.51	9.25	8.95 $\pm$ 0.3
He-Xe	0–1	7.69	7.68	–	–	–	–	–	–	–	7.6688 $\pm$ 0.54

An effective means for checking the validity of the different potential parameters can be made using second pressure virial coefficient data [25, 26, 27, 28, 29, 30, 31, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74] at different temperatures. The interaction second pressure virial coefficient $B_{12}$ at temperature $T$ was calculated classically with the first three quantum corrections from [116]:

$\displaystyle B_{12}(T)=B_{\text{cl}}(T)+\lambda B_{\text{qm},1}(T)+\lambda^{2% }B_{\text{qm},2}(T)+\lambda^{3}B_{\text{qm},3}(T)$ (13)

where

$\displaystyle B_{\text{cl}}(T)=2\pi N_{0}\int_{0}^{\infty}{[1-\exp(-V(r)/k_{B}% T)]r^{2}\mathrm{d}r}$ (14)

and the first three quantum corrections $B_{\text{qm},1}(T)$ , $B_{\text{qm},2}(T)$ and $B_{\text{qm},3}(T)$ are given in [116], with $\lambda=\hbar^{2}/(12mk_{B}T)$ , $\hbar=h/2\pi$ , $m$ and $N_{0}$ are the atomic mass and Avogadro’s number. The calculated $B_{12}$ was compared with the experimental results [45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74] using the present BFW, MTT and different intermolecular potentials [25, 26, 27, 28, 29, 30, 31, 74]. As it may be clearly seen in Figs 3–5 and Table 2, the interatomic BFW and MTT potentials give the best agreement with the experimental values over a high range of temperatures.

Figure 3.

Temperature dependence of the Helium gas interaction pressure second virial coefficients $B_{12}$ in cm ${}^{3}$ mol ${}^{-1}$ versus temperature in K. Comparison is made with previously available experimental and theoretical results [45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59]. The calculations were performed using the present BFW interatomic potential.

Figure 4.

The same as in Fig. 3 but for Xenon.

Figure 5.

The same as in Fig. 3 but for He-Xe mixture.

4.3 Analysis of traditional transport properties

An additional check on the proposed potential consists of the calculation of the transport properties, i.e., mixture viscosity $\eta_{\text{mix}}(T)$ , mixture diffusion coefficient $D_{\text{mix}}(T)$ , mixture thermal conductivity $\lambda_{\text{mix}}(T)$ and mixture isotopic thermal factor $\text{Iso}_{\text{mix}}(T)$ at different temperatures $T$ of He-Xe. These are obtained via the formulae of Monchick et al. [117] and their comparison to the accurate experimental and theoretical results [25, 29, 30, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110] which are clear by calculating the associated values of $\delta_{a}$ as shown in Table 2. The agreements for system under consideration are excellent in the whole temperature range.

In addition to the inversion of spectroscopic observations and bulk properties, there are other sources for the determination of intermolecular forces. These are: 1) quantum mechanical calculations (ab initio method) and 2) molecular-beam scattering. In this work, we restricted our efforts to the extraction of information about the interatomic potential energy from the transport properties. In this respect, according to the kinetic theory of gases at low density and the Chapman-Enskog solution of the Boltzmann equation, the transport properties can be expressed with the help of a series of collision integrals that depend on the intermolecular potential energy, and are defined as [118]:

$\displaystyle\theta=\pi-2b\int_{r_{0}}^{\infty}{\frac{\mathrm{d}r}{r^{2}(1-(b/% r)^{2}-(V(r)/E))^{1/2}}}$ (15) $\displaystyle Q^{(l)}(E)=2\pi\left({1-\frac{1+(-1)^{l}}{2(1+l)}}\right)^{-1}% \int_{0}^{\infty}{(1-\cos^{l}\theta)b\,\mathrm{d}b}$ (16) $\displaystyle\Omega^{(l,s)}(T)=\left({(s+1)!(k_{B}T)^{s+2}}\right)^{-1}\int_{0% }^{\infty}{Q^{(l)}(E)\exp(-E/k_{B}T)E^{s+1}\,\mathrm{d}E}$ (17)

where $\theta$ is the scattering angle, $Q^{(l)}(E)$ the transport collision integral, $b$ the impact parameter, $E$ the relative kinetic energy of colliding atoms and $r_{0}$ the closest approach of two atoms. Thus, three successive numerical integrations are required to obtain a collision integral.

The reduced collision integral is defined by:

$\displaystyle\Omega^{*(l,s)}(T)=\frac{\Omega^{(l,s)}(T)}{\pi\sigma^{2}}$ (18)

and $\sigma$ is the length scaling factor, such that $V(\sigma)=$ 0.

The potential energy would serve as the input information required in calculating the collision integrals, and consequently the transport properties. Kinetic-theory expressions for the transport properties (viscosity, thermal conductivity and diffusion coefficient) in terms of the collision integrals for the binary gas mixtures are given by the following equations [25]:

$\displaystyle\frac{1}{\eta_{\text{mix}}}=\frac{X_{\eta}+Y_{\eta}}{1+Z_{\eta}}$ (19) $\displaystyle X_{\eta}=\frac{x_{1}^{2}}{\eta_{1}}+\frac{2x_{1}x_{2}}{\eta_{12}% }+\frac{x_{2}^{2}}{\eta_{2}}$ (20) $\displaystyle Y_{\eta}=\frac{3}{5}A_{12}^{*}\left(\frac{x_{1}^{2}}{\eta_{1}}% \left(\frac{M_{1}}{M_{2}}\right)+\frac{2x_{1}x_{2}}{\eta_{12}}\left(\frac{(M_{% 1}+M_{2})^{2}}{4M_{1}M_{2}}\right)\left(\frac{\eta_{12}^{2}}{\eta_{1}\eta_{2}}% \right)+\frac{x_{2}^{2}}{\eta_{2}}\left(\frac{M{}_{2}}{M_{1}}\right)\right)$ (21) $\displaystyle Z_{\eta}=\frac{3}{5}A_{12}^{*}\left(x_{1}^{2}\left(\frac{M_{1}}{% M_{2}}\right)+2x_{1}x_{2}\left(\frac{(M_{1}+M_{2})^{2}}{4M_{1}M_{2}}\left(% \frac{\eta_{12}}{\eta_{1}}+\frac{\eta_{12}}{\eta_{2}}\right)-1\right)+x_{2}^{2% }\left(\frac{M_{2}}{M_{1}}\right)\right)$ (22)

where $x_{i}$ , $M_{i}$ , $\eta_{i}$ and $T^{*}$ are mole fractions, molecular weights in gms, viscosity in $\mu P$ at the mixture temperature of species $i$ ( $i=$ 1, 2) and reduced temperature, respectively.

In the above expressions the interaction viscosity $\eta_{12}$ is given by:

$\displaystyle\eta_{12}=\frac{5}{16}\left(\frac{(2M_{1}M_{2})k_{B}T}{\pi(M_{1}+% M_{2})}\right)^{1/2}\frac{1}{\sigma^{2}\Omega^{*(2,2)}(T^{*})}$ (23) $\displaystyle A_{12}^{*}=\frac{\Omega^{*(2,2)}(T^{*})}{\Omega^{*(1,1)}(T^{*})}$ (24)

In addition, the mixture diffusion coefficient in (m ${}^{2}$ /s) and interaction thermal conductivity in units of (mW/m $\cdot$ K) may be written as:

$\displaystyle D_{\text{mix}}=\frac{3}{8}\left(\frac{(M_{1}+M_{2})k_{B}^{3}T^{3% }}{2\pi M_{1}M_{2}}\right)^{1/2}\frac{1}{P\sigma^{2}\Omega^{*(1,1)}(T^{*})}$ (25) $\displaystyle\lambda_{12}=\frac{75}{64}\left(\frac{(M_{1}+M_{2})k_{B}^{3}T}{2% \pi M_{1}M_{2}}\right)^{1/2}\frac{1}{\sigma^{2}\Omega^{*(2,2)}(T^{*})}$ (26)

with the pressure $P$ in atm and the temperature $T$ in K.

The expressions of the mixture thermal conductivity and isotopic thermal factors of pure gases and mixtures are defined in details in Appendix C of [25].

In order to calculate the transport properties of the mixtures considered, the viscosity and thermal conductivity of pure noble gases are determined using the present BFW and MTT interatomic potential models with the different parameters are given in Table 2.

5. Results

In this section, we present the results of the analysis for noble gas mixtures and the empirical models for $\mu(r)$ are compared to the ab initio results of Maroulis et al. [19]. Good agreement between the ab initio models and our empirical ones is generally observed. The profiles and relevant spectral moments obtained via our models for the dipole moment agree with the measurements Fig. 2 and Table 1 display the quantum profiles of the collision-induced light absorption spectra and their moments at room temperature using the empirical models of the induced dipole moment compared with experiments [7].

Summarizing our analysis, we find that He-Xe inert gas mixtures develop an incremental induced dipole moment during collisions, besides the exponential one, which contributes substantially at intermediate-range distances and can be ascribed to other mechanisms of electron cloud distortion, such as overlap and electron-correlation effects.

6. Conclusion

We have developed a model for the induced dipole moment $\mu(r)$ with adjustable parameters fitted to the spectral profiles of collision-induced absorption at room temperature using quantum mechanics and to the first few spectral moments of the measured spectra using classic mechanics.

It is clear that the two methods are complementary and could be used jointly to reduce the computational cost and allow the maximum of information to be extracted from measurements.

The present study further demonstrate that the present BFW and MTT models yield reliable approximations of the interatomic potential of He-Xe gas mixtures The treatment proposed in this study represents an improvement on the model from thermodynamical and transport properties over a wider temperature range. Also, it is interesting to note that the empirical model derived for the induced dipole agrees reasonably well with the ab initio results of Maroulis et al. for this mixture and produces lineshapes in good agreement with experiment.

Footnotes

Acknowledgments

We are grateful to Dr. J. Borysow, Dr. L. Frommhold and Dr. G. Birnbaum for making available their published Fortran code with the different results of the collision induced absorption (CIA) for various systems.

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Collision-induced absorption spectra (CIA) for monatomic gas mixtures of Helium-Xenon

Abstract

Keywords

1. Introduction

2. Analysis of spectral moments of CIA to determined the induced dipole moments

4.2 Analysis of second pressure virial coefficients

Table 3 Comparison of our computed vibrational energy levels for He 2 , Xe 2 and He-Xe gases with experimental and other theoretical data. All values are given in cm - 1

6. Conclusion

Footnotes

Acknowledgments

References

Table 3
Comparison of our computed vibrational energy levels for He ${}_{2}$ , Xe ${}_{2}$ and He-Xe gases with experimental and other theoretical data. All values are given in cm ${}^{-1}$