Determination of mobility and diffusion coefficients of Ar⁺ and Ar²⁺ ions in argon gas

Abstract

It has recently been shown that accurate theoretical calculations can be used to calibrate a drift-tube mass spectrometer (DTMS) to measure gaseous ion mobilities accurate to within 0.6%. Here we present a new method for calibrating a DTMS instrument to obtain diffusion coefficients parallel to the electric field which are accurate to within 8%. This method is developed and verified by consideration of He⁺ (²S_1/2) ions in He. We apply these techniques to determine transport coefficients for Ar⁺(²P_3/2) and Ar²⁺ (³P_2,1,0) ions in Ar gas at 300 K, with results given as a function of E/N, the ratio of electrostatic field strength to gas number density, in the range 30–210 Td. The measured mobilities are accurate within 0.8%; for Ar⁺ they agree within 1.5% with Monte Carlo simulations, and for both the cations and dications they are in excellent agreement with previous measurements. Our method gives new diffusion coefficients that agree within 5% with quantum Monte Carlo calculations.

Appendices

Appendix A. Extracting diffusion coefficients from DTMS data

In this Appendix, we show how the flux leaving the drift tube can be extracted from a statistical analysis of the experimentally measured fluxes in the shutter and gate experiments. Throughout these Appendices, we use the notation that capital letters represent random variables and lower case letters represent the values taken by the random variable.

Let T_s be a continuous random variable which represents the total time it takes an ion to travel from the shutter to the detector, and let f_s(t_s) be the probability density function for T_s. Similarly, let T_g be the time it takes an ion to pass from the gate to the detector, with probability density function f_g(t_g). We ignore end and energy relaxation effects, so f_s(t_s) and f_g(t_g) are obtained from the normalized shutter and gate data after subtracting the background counts as described in Section 2. Our goal is to generate f(τ), the probability density function for the time it takes an ion to pass through the drift tube between the shutter and the gate. Let the random variable T_d represent this time. In general, the extracted f(τ) depends on the correlation between T_s and T_g, which is embodied in the joint density function f_{s, g}(t_s, t_g).

For individual ions in the tube, T_d = T_s − T_g. Then the cumulative distribution function is

$$ F\left(\tau \right)=P\left({T}_d\le \tau \right)=P\left({T}_s-{T}_g\le \tau \right)={\iint}_Dd{t}_g\ d{t}_s\ {f}_{s,g}\left({t}_s,{t}_g\right), $$

(11)

where the region D is defined by {(t_s, t_g)| t_s − t_g ≤ τ. and t_s, t_g ≥ 0}. This integral can be written as

$$ F\left(\tau \right)={\int}_0^{\infty }d{t}_g{\int}_0^{\tau +{t}_g}d{t}_s\ {f}_{s,g}\left({t}_s,{t}_g\right). $$

(12)

Applying the multivariable chain rule and the Fundamental Theorem of Calculus we obtain,

$$ f\left(\tau \right)=\frac{d}{d\tau}F\left(\tau \right)={\int}_0^{\infty }d{t}_g\ {f}_{s,g}\left(\tau +{t}_g,{t}_g\right), $$

(13)

where we have used the fact that f_{s, g} is independent of τ. Eq. (13) provides a way to calculate f(τ) from f_{s, g}. In the case where T_s is independent of T_g, f_{s, g} is simply the product ofof the two single variable density functions, i.e., f_{s, g}(t_s, t_g) = f_s(t_s)f_g(t_g). However, in general we expect that T_s and T_g are correlated, since ions with high velocity in the drift tube would be expected to have high velocity after passing through the gate into the detector, where the pressure is lower and there are fewer collisions. This correlation changes the form of f_{s, g} and hence a stronger correlation will give rise to a different f(τ) and a different diffusion coefficient.

Appendix B. Forming joint density functions using a copula

In this Appendix, we present a method for describing the correlation between T_s and T_g. Any correlation between random variables embodied in a multivariate density function can be fully described by an appropriately chosen copula [31]. Briefly, a copula is a function C(u, v) such that the joint cumulative distribution function F_{s, g}(t_s, t_g) = P[T_s ≤ t_s, T_g ≤ t_g] can be expressed in terms of its marginals F_s(t_s) = P[T_s ≤ t_s] and F_g(t_g) = P[T_g ≤ t_g] as [31],

$$ {F}_{s,g}\left({t}_s,{t}_g\right)=C\left({F}_s\left({t}_s\right),{F}_g\left({t}_g\right)\right). $$

(14)

The choice of the function C corresponds to an assumption about how the two random variables are correlated. Any proposed form of the correlation can be described by an appropriately chosen function C(u, v). We refer the reader to Ref. [31] for more details. The joint probability density function, f_{s, g}, then can be obtained as

$$ {f}_{s,g}\left({t}_s,{t}_g\right)=c\left({F}_s\left({t}_s\right),{F}_g\left({t}_g\right)\right)\cdotp {f}_s\left({t}_s\right)\cdotp {f}_g\left({t}_g\right), $$

(15)

where $c\left(u,\upsilon \right)=\frac{\partial }{\partial u}\frac{\partial }{\partial \upsilon }C\left(u,v\right)$ is the density of the copula [32]. Once a copula has been chosen, Eq. (15) provides a way to determine the joint density function.

In general, we do not know how the random variables T_g and T_s are correlated in a DTMS, and in this work we have made the ansatz that the correlation between the two random variables can be described by a single parameter, ρ, the Pearson correlation coefficient (c.f. Eq. (8)). This assumption corresponds to choosing a Gaussian copula for the function C [31]. The Gaussian copula is (c.f. Ref. [31], Eq. (2.3.6))

$$ C\left(u,v\right)=\frac{1}{2\pi \sqrt{1-{\rho}^2}}{\int}_{-\infty}^{\Phi^{-1}(u)} dr{\int}_{-\infty}^{\Phi^{-1}(v)} ds\times \exp\ \left[-\frac{r^2+{s}^2-2\rho rs}{2\left(1-{\rho}^2\right)}\right], $$

(16)

where Φ(x) is the standard normal cumulative distribution function. Taking the partial derivatives we have,

$$ c\left(u,v\right)=\frac{1}{2\pi \sqrt{1-{\rho}^2}}\frac{1}{\phi \left({\varPhi}^{-1}(u)\right)}\frac{1}{\phi \left({\varPhi}^{-1}(v)\right)}\mathrm{xexp}\ \left[-\frac{{\left({\varPhi}^{-1}(u)\right)}^2+{\left({\varPhi}^{-1}(v)\right)}^2-2\rho {\varPhi}^{-1}(u){\varPhi}^{-1}(v)}{2\left(1-{\rho}^2\right)}\right], $$

(17)

where ϕ is the standard normal distribution function and we have used a well known property of differentiating inverse functions, $\frac{d{\varPhi}^{-1}(x)}{dx}=\frac{1}{\phi \left({\varPhi}^{-1}(x)\right)}$ with $\phi =\frac{d\varPhi}{d x}$.

The bivariate density f_{s, g}(t_s, t_g) can be obtained by combining Eqs. (15) and (17) with the (normalized) experimental measurements of f_s(t_s) and f_g(t_g). We note that this approach can be generalized to other types of correlation by simply choosing a different copula C(u, v).

With the Gaussian copula, if the marginal probability density functions f_s and f_g are Gaussian, one can use Eq. (4) with Eqs. (15) and (17) to show that f_{s, g} is the bivariate normal distribution with correlation coefficient ρ, i.e.,

$$ {f}_{s,g}={A}_{s,g}\exp\ \left(-\frac{Z}{2\left(1-{\rho}^2\right)}\right), $$

(18)

where A_{s, g} is a normalization constant and

$$ Z=\frac{{\left({t}_g-{\overline{t}}_g\right)}^2}{\sigma_g^2}+\frac{{\left({t}_s-{\overline{t}}_s\right)}^2}{\sigma_s^2}-\frac{2\rho \left({t}_s-{\overline{t}}_s\right)\left({t}_g-{\overline{t}}_g\right)}{\sigma_s{\sigma}_g}. $$

(19)

Using Eq. (18) for f_{s, g}, the integral in Eq. (13) can be evaluated analytically to yield

$$ f\left(\tau \right)={A}_{\tau}\exp\ \left[\frac{{\left(\tau -{\overline{t}}_s+{\overline{t}}_g\right)}^2}{2\left({\sigma}_s^2+{\sigma}_g^2-2\rho {\sigma}_s{\sigma}_g\right)}\right], $$

(20)

where A_τ is a normalization constant. The variance in τ is then

$$ {\sigma}_{\tau}^2={\sigma}_s^2+{\sigma}_g^2-2\rho {\sigma}_s{\sigma}_g. $$

(21)

Appendix C: Procedure for extracting the diffusion coefficient

In this Appendix we present a step-by-step procedure for extracting the diffusion coefficient from the shutter and gate experiments.

• 1. Perform the shutter and gate experiments. Crop the data to include the signal together with some background noise portions before and after. Find the best linear fit B(t) (see Eq. (2)) to these background portions and subtract it from the data.

• 2. Choose a probability density function to fit the experimentally measured arrival time spectra, such as Eq. (3).

• 3. Fit the shutter data to obtain f_s(t_s) and the gate data to obtain f_g(t_g). Normalize f_s and f_g so that they integrate to unity on the interval [0,∞).

• 4. For a fixed value of τ, build the integrand of $f\left(\tau \right)={\int}_0^{\infty }d{t}_g\ {f}_{s,g}\left({t}_s=\tau +{t}_g,{t}_g\right)$ using Eqs. (15) and (17) to obtain f_{s, g}. This requires numerically integrating the normalized densities f_s(t_s) and f_g(t_g) to obtain the cumulative distribution functions ${F}_s\left({t}_s\right)={\int}_0^{t_s}d{t}_s^{\hbox{'}}{f}_s\left({t}_s^{\prime}\right)$ and ${F}_g\left({t}_g\right)={\int}_0^{t_g}d{t}_g^{\prime }{f}_g\left({t}_g\right)$ that appear in Eq. (refeq:bivariate-from-copula). A numerical library may be used to evaluate Φ⁻¹(u); we used the scipy.stats library in Python.

• 5. Numerically integrate Eq. (13) using the integrand built in the previous step to obtain f(τ) for a fixed value of τ.

• 6. Repeat steps 4 and 5 for as many values of τ as needed to obtain a smooth curve for f(τ).

• 7. Fit the curve obtained for f(τ) to Eq. (3) to obtain the diffusion coefficient.