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


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+ (2S1/2) ions in He. We apply these techniques to determine transport coefficients for Ar+(2P3/2) and Ar2+ (3P2,1,0) 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.

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  1. Viehland LA (2018) Gaseous ion mobility, diffusion, and reaction. Springer Nature, Switzerland

    Book  Google Scholar 

  2. Arce L, Menéndez M, Garrido-Delgado R, Valcárcel M (2008) Trends Anal Chem 27(2):139

    Article  CAS  Google Scholar 

  3. May JC, McLean JA (2015) Anal Chem 87(3):1422

    CAS  PubMed  Google Scholar 

  4. Cumeras R, Figueras E, Davis CE, Baumbach JI, Gràcia I (2015) Analyst 140(5):1376

    Article  CAS  Google Scholar 

  5. Ben-Nissan G, Sharon M (2018) Current Opin Chem Biol 42:25

    CAS  Google Scholar 

  6. Viehland LA, Lutfullaeva A, Dashdorj J, Johnsen R (2017) Int J Ion Mobil Spectrom 20(3–4):95

    CAS  Google Scholar 

  7. Mason EA, McDaniel EW (1988) Gaseous ion mobility, diffusion, and reaction. Wiley, New York

    Google Scholar 

  8. Barnes WA, Martin DW, McDaniel EW (1961) Phys Rev Lett 6(3):110

    CAS  Google Scholar 

  9. McDaniel EW, Martin DW, Barnes WA (1962) Rev Sci Instrum 33(1):2

    CAS  Google Scholar 

  10. McAfee KB, Edelson D (1963) Proc Phys Soc 81(2):382

    Article  CAS  Google Scholar 

  11. Edelson D, McAfee KB (1964) Rev Sci Instrum 35(2):187

    Article  Google Scholar 

  12. Lovaas TH, Skullerud HR, Kristensen DH, Linhjell D (1987) J Phys D 20(11):1465

    Article  Google Scholar 

  13. Beaty E (1956) Phys Rev 104:473

    Article  Google Scholar 

  14. Beaty E (1961) Proc 5th int conf on ionization phenomena in gases 1, 183

  15. Johnsen R, Biondi MA (1978) Phy Rev A 18(3):989

    Article  CAS  Google Scholar 

  16. Helm H, Elford MT (1977) J Physics B 10(18):3849

    Article  CAS  Google Scholar 

  17. Basurto E, de Urquijo J, Alvarez I, Cisneros C (2000) Phys Rev E 61(3):3053

    Article  CAS  Google Scholar 

  18. Barata JAS, Conde CAN (2006) In: 2006 IEEE nuclear science symposium conference record, IEEE

  19. Chicheportiche A, Lepetit B, Gadéa FX, Menhenni M, Yousfi M, Kalus R (2014) Phys Rev E 89, 063102

  20. Orient OJ (1974) Acta Phys Acad Sci Hung 35(1–4):247

    CAS  Google Scholar 

  21. Ellis H, Pai R, McDaniel E, Mason E, Viehland L (1976) Atom Data Nucl Data Tables 17(3):177

    CAS  Google Scholar 

  22. URL Accessed 8 May 2019

  23. Saloman EB (2010) J Phys Chem Ref Data 39(3):1

    Google Scholar 

  24. Helm H (1978) J Phys D 11(7):1049

    CAS  Google Scholar 

  25. Siska PE (1986) J Chem Phys 85(12):7497

    CAS  Google Scholar 

  26. Gadea FX, Paidarova I (1996) Chem Phys 209:281

    CAS  Google Scholar 

  27. Cohen JS, Schneider B (1974) J Chem Phys 61:3230

    CAS  Google Scholar 

  28. Rumble J (ed) (2018) Handbook of chemistry and physics, 99th edn. Taylor & Francis, Boca Raton

    Google Scholar 

  29. Bertsekas DP, Tsitsiklis JN (2008) Introduction to probability, 2nd edn. Athena Scientific, Belmont

    Google Scholar 

  30. Viehland L, Johnsen R, Gray B, Wright T (2016) J Chem Phys 144(7):074306

    Article  Google Scholar 

  31. Nelsen RB (2006) An introduction to copulas. Springer, New York

    Google Scholar 

  32. Rank J (2007) Copulas: from theory to application in finance. Bloomberg financial. Wiley

Download references


The authors thank Dr. Rainer Johnsen for technical assistance, for useful conversations and for referring us to Ref. [24].

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Correspondence to William C. Pfalzgraff.

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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 Ts be a continuous random variable which represents the total time it takes an ion to travel from the shutter to the detector, and let fs(ts) be the probability density function for Ts. Similarly, let Tg be the time it takes an ion to pass from the gate to the detector, with probability density function fg(tg). We ignore end and energy relaxation effects, so fs(ts) and fg(tg) 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 Td represent this time. In general, the extracted f(τ) depends on the correlation between Ts and Tg, which is embodied in the joint density function fs, g(ts, tg).

For individual ions in the tube, Td = Ts − Tg. Then the cumulative distribution function is

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

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

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

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

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

where we have used the fact that fs, g is independent of τ. Eq. (13) provides a way to calculate f(τ) from fs, g. In the case where Ts is independent of Tg, fs, g is simply the product ofof the two single variable density functions, i.e., fs, g(ts, tg) = fs(ts)fg(tg). However, in general we expect that Ts and Tg 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 fs, 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 Ts and Tg. 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 Fs, g(ts, tg) = P[Ts ≤ ts, Tg ≤ tg] can be expressed in terms of its marginals Fs(ts) = P[Ts ≤ ts] and Fg(tg) = P[Tg ≤ tg] as [31],

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

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, fs, g, then can be obtained as

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

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 Tg and Ts 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))

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

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

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

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 fs, g(ts, tg) can be obtained by combining Eqs. (15) and (17) with the (normalized) experimental measurements of fs(ts) and fg(tg). 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 fs and fg are Gaussian, one can use Eq. (4) with Eqs. (15) and (17) to show that fs, g is the bivariate normal distribution with correlation coefficient ρ, i.e.,

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

where As, g is a normalization constant and

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

Using Eq. (18) for fs, g, the integral in Eq. (13) can be evaluated analytically to yield

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

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

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

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. 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. 2.

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

  3. 3.

    Fit the shutter data to obtain fs(ts) and the gate data to obtain fg(tg). Normalize fs and fg so that they integrate to unity on the interval [0,∞).

  4. 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 fs, g. This requires numerically integrating the normalized densities fs(ts) and fg(tg) 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. 5.

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

  6. 6.

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

  7. 7.

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

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Dashdorj, J., Pfalzgraff, W.C., Trout, A. et al. Determination of mobility and diffusion coefficients of Ar+ and Ar2+ ions in argon gas. Int. J. Ion Mobil. Spec. 23, 143–151 (2020).

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