ISSN 1063-7761, Journal of Experimental and Theoretical Physics, 2007, Vol. 104, No. 3, pp. 357–362. © Pleiades Publishing, Inc., 2007. Original Russian Text © N.L. Asfandiarov, E.P. Nafikova, S.A. Pshenichnyuk, 2007, published in Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, 2007, Vol. 131, No. 3, pp. 401–407.

ATOMS, MOLECULES, OPTICS

Interpreting Electron Transmission Spectroscopy and Negative Ion Mass Spectrometry Data Using a Spherical Potential Well Model N. L. Asfandiarov*, E. P. Nafikova, and S. A. Pshenichnyuk Institute of Molecular and Crystal Physics, Ufa Research Center, Ural Division, Russian Academy of Sciences, Ufa, Bashkortostan, 450075 Russia * e-mail: [email protected] Received July 20, 2006

Abstract—Experimental data obtained using electron transmission spectroscopy and negative ion mass spectrometry based on resonance electron capture are interpreted within the framework of a spherical potential well model in application to a series of chloro- and bromoalkane molecules. Allowance for the scattering of a single partial p-wave of the incoming electron makes possible (i) reproduction of the ratio of a resonance peak width to the electron energy observed in the electron transmission spectra and (ii) establishment of a relation between the total cross section of electron scattering on a molecule and the dissociative electron attachment cross section. The proposed model offers a radical simplification of the approach developed previously based on the Fashbach–Fano resonance theory. PACS numbers: 34.80.Bm, 34.80.Ht DOI: 10.1134/S1063776107030028

1. INTRODUCTION Electron transmission spectroscopy (ETS) [1, 2] and the mass spectrometry of negative ions formed upon dissociative electron attachment [3, 4] provide informaexp tion about the total cross section ( σ cap ) of electron capture by molecules and the cross section (σDEA) of dissociative electron attachment (DEA) to molecules. It is quite evident that σDEA , being a part of the total electron capture cross section σcap , is related to the latter quantity as σDEA = σcaps,

(1)

where s is the ion survival factor. The process of electron capture by a molecule according to the shape resonance mechanism [3] with subsequent dissociation into fragments can be illustrated by the energy diagrams depicted in Fig. 1. Let us suppose that a molecule characterized by a negative electron affinity in the ground state captures an incoming electron (A–B transition in Fig. 1) with the formation of a negative ion in the ground electron state according to the shape resonance mechanism. This process most probably proceeds at the vertical attachment energy EVAE (energy at point B in Fig. 1). The system will remain unstable with respect to the electron ejection until reaching point C, which corresponds to the intersection of the corresponding terms. For this reason, only a small fraction of ions survive until dissociation. In a mass-spectrometric experiment,

the molecules of chloro- and bromoalkane form only stable fragment ions of halogenides (Hal–). The ratio of the total and dissociative attachment cross sections can vary within rather broad limits, depending on the survival factor s. In a quite rough approximation, this factor can be written as [5] s = exp ( – ρ ) ≈ exp ( – Γ a /Γ d ),

(2)

where Γa and Γd are the autoionization and dissociative resonance widths, respectively. It was repeatedly emphasized that combined use of the ETS and negative ion mass spectrometry (NIMS) in the investigations of electron scattering from molecules significantly increases the efficacy of each particular method. However, this combination has been used only in a few investigations [6–8]. Aflatooni et al. [6, 7] studied a broad spectrum of chlorine-substituted alkanes and revealed the following law obeyed by their electron transmission spectra. The resonance width characterized by the deep-to-peak separation ∆Edps in the spectra of chlorine-substituted alkanes is a power function of the vertical attachment energy (EVAE) of this resonance: ∆E dps = 0.51E VAE . 1.44

(3)

The electron transmission spectra are conventionally described using the following terms: Epeak , the position of a peak of the negative derivative of the electron cap-

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E, eV 5 (a)

(b)

4

4 σcap

3

3

EVAE 2 1

2

B

σDEA

EDEA

D

1

C 0 –1

A

0

Cross section Internuclear distance

Fig. 1. Schematic diagrams showing (a) a potential surface of a molecule (solid curve) and the ion (dashed curve) and (b) plots of the total (σcap) and dissociative (σDEA) capture cross sections versus electron energy.

ture cross section (–d σ cap /dE) with respect to the energy; Edip , the position of a minimum on the exp

−d σ cap /dE curve; and ∆Edps = Epeak – Edip , which is treated as the resonance peak width. It was pointed out [9] that Eq. (3) can be considered as a manifestation of the Wigner threshold law [10]: exp

Γa ∝ E

l + 1/2

,

where E is the electron energy and l is the orbital moment of the partial wave of the incoming electron. It is assumed that the main contribution to the total resonance width at not too small energies is due to the autoionization component, while the Franck–Condon gap width can be ignored [8]. Therefore, we can admit that the predominant role in the case of chlorine-substituted alkanes is played by a single p-harmonic in the expansion of the plane wave of incoming electrons into partial waves. In addition, it was established that the maximum DEA cross section obeys the following empirical relation [6]: σ DEA = 5.41 × 10 peak

2.01

– ( 16 + 0.613E VAE )

.

(4)

It should be emphasized that relation (4) only represents a function approximating a set of experimental points, rather than having a strict physical meaning. Recently, Gallup [11] attempted to find a satisfactory description of the DEA process based on the available ETS data. The potential of electron interaction with a target molecule was chosen in the form of a combination of rectangular wells modeling carbon and chlorine atoms and C–Cl bonds in a CCl4 molecule. In

addition, the polarizability of the target molecule was taken into account. This approach is essentially based on the united atom approximation, which has been used to analyze the sequence of molecular orbitals during the development of quantum chemistry methods [12]. Previously, we proposed [13] a simplified variant of the united atom approximation which took into account only a fraction of occupied orbitals in the modeled molecule. Using this approach, it is possible to evaluate the energy of the shape resonance proceeding from the electron affinity of the target molecule, the symmetry of its lower vacant orbital, and the character of the trapping center. The approach developed in [13] was intended for the investigation of molecules possessing a positive electron affinity. In contrast, the molecules considered in the present study have negative electron affinities. 2. RESULTS AND DISCUSSION Below, we use the experimental ETS data reported previously on the EVAE values and the NIMS data on the energies EDEA corresponding to the maximum yield of fragment halogenide ions Hal– formed upon the DEA to halogen-substituted alkanes [6, 9, 14–16]. Figure 2 (reproduced from [14]) shows a plot of the resonance peak width versus vertical attachment energy constructed using the available ETS data. It should be noted that the results obtained by this method for chloromethane molecules are considered as not quite reliable [6], so that the point corresponding to chloromethane in Fig. 2 is sometimes displaced close to the 1.44 approximating curve ∆Edps = 0.51 E VAE . For an analysis of the behavior of the total electron capture cross

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INTERPRETING ELECTRON TRANSMISSION SPECTROSCOPY ∆Edps, eV

section as a function of the incoming electron energy and for establishing a relation between σcap and σDEA , we use the results obtained in [11] and adopt the approximation proposed previously [13]. Ignoring details of the internal structure and neglecting the polarizability of a target molecule, we describe the scattering potential by a spherical potential well [13] as depicted in Fig. 3. The radius R0 of the potential well is chosen proceeding from the size of a substituent atom in hydrocarbons under consideration, following the approach used in [13]. According to this, the optimum values are R0(Cl) = 1.5 Å for chlorine and R0(Br) = 1.77 Å for bromine. The depth of the potential well is chosen so that the resonance in the p-wave scattering cross section would coincide with the EVAE value determined from the ETS data. However, the solution of this problem is not unique. A single bound 1s level exists in a shallow well but, as the well depth increases, another level that corresponds to the 1p state appears for the given radius. This 1p level can be interpreted as * , on which an addia vacant molecular orbital σ C–Hal tional electron is captured. Thus, the proposed approach, representing a variant of the model developed in [11], takes into account only the lowest vacant orbital. Figure 4 illustrates calculations of the cross section of the electron p-wave capture by a 2-chloropropane molecule (U0 = 12.2 eV, R0 = 1.5 Å, EVAE = 1.99 eV) for the 1s and 1p levels in the potential well occurring at −10.02 and –4.08 eV, respectively. The total cross section for the elastic scattering electrons on the given model potential was calculated using the Faxen–Holtsmark method [17] representing an analog of the classical Rayleigh scattering theory. Obviously, this approach allows us only to estimate a cross section for the elastic scattering of electrons on a molecule, since we neglect the nuclear subsystem. However, introduction of the survival factor via Eqs. (1) and (2) makes it possible to estimate the DEA cross section as well. The main equation for calculations of the electron capture cross section in the spherical potential well model is as follows [17]:

3 1 2

2

CH3Cl

1

0

1

2

3 EVAE, eV

Fig. 2. Plots of the resonance peak width in the electron transmission spectra of 36 different chlorine-substituted alkanes versus vertical attachment energy: (1) ∆Edps = 1.452

1.44

0.755 E VAE ; (2) ∆Edps = 0.51 E VAE ; points show the experimental data from [6, 9, 14–16] for (
) bromoalkanes, () monosubstituted chloroalkanes, and () polysubstituted chloroalkanes.

Energy, eV 6 l(l + 1)/r2

EVAE

RS 0

1p –6

1s –12

U0 0

dδ l τ = 2
-------, dE where τ is the electron time delay in the region of the scattering potential compared to the time t0 of free transit through the same spatial region; t0 = 2R0/v; v and E are the electron velocity and energy, respectively; and δl is the phase shift of the lth partial wave. The latter quantity is defined as

359

1 R0 2

3

4

5 r, Å

Fig. 3. A schematic diagram of the model potential of electron attraction to a chloroalkane molecule: dashed lines show the 1s and 1p energy levels in the potential well; RS is a short-lived resonance state corresponding to the EVAE energy; l(l + 1)/r2 is the term corresponding to the p-wave scattering.

where

k j 'l ( k R 0 ) j l ( K R 0 ) – K j 'l ( K R 0 ) j l ( k R 0 ) - , δ l = arctan ----------------------------------------------------------------------------------------kn 'l ( k R 0 ) j l ( K R 0 ) – K j 'l ( K R 0 )n l ( k R 0 ) JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS

2m e E k = -----------------,
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360 theor

σ cap , 10 2

–16

cm

2

σDEA, 10–19 cm2 6 (b)

(a)

EDEA 4

1 EVAE Epeak

2

0

EVAE

Edip –1

0

1

2

3

4 0 E, eV

1

2

3

4 E, eV

Fig. 4. The results of cross section calculations performed for a 2-chloropropane molecule within the framework of the spherical theor

potential well model, showing (a) the total capture cross section σ cap (solid curve), the autoionization resonance width Γa (dashed theor

curve), and the calculated electron transmission spectrum –d σ cap /dE (dash–dot curve) and (b) the calculated DEA cross section.

jl (kR0), jl (KR0), nl (kR0), nl (KR0) are the Bessel functions, and me is the electron mass; the prime indicates a derivative. The Schrödinger equation for the scattering problem and the potential function are as follows: 2

d ul l(l + 1) 2 ---------2 + k – U 0 ( r ) – ----------------u l = 0, 2 dr r 2m U 0 ( r ) = ------2-U ( r ),


⎧ U, r < R 0 , Ur = ⎨ ⎩ 0, r ≥ R 0 ,

∞

(5)

∑

4π 2 theor σ cap = -----2- ( 2l + 1 ) sin δ l , k l=0 τ l = t 0 + τ,


Γ l = ---. τl

According to [6, 7], the total capture cross section σ cap is taken equal to that for a single partial p-wave. The typical calculation results are presented in Fig. 4a. In contrast to the case of rather narrow resonances described by the Breit–Wigner formula [17], a characteristic feature of resonances in the case of potential wells of moderate depth is that the maximum scattering cross section does not coincide with the minimum of the autoionization resonance width Γa . Therefore, the behavior of the survival factor in Eq. (1) will significantly differ from that in the classical case considered in [5]. It was shown [6] that ∆Edps measured in the electron transmission spectra is determined (for not too theor

small electron energies) predominantly by the autoionization resonance width. In order to compare the results of calculations to the experimental electron transmission spectra, we contheor structed the negative derivative (–d σ cap /dE) of the cross section with respect to the energy (Fig. 4a). The results showed that the resonance width at the maximum, calculated as the function of EVAE for chlorinesubstituted alkanes obeys the following equation: ∆E dps = 0.556E VAE , theor

1.429

(6)

which is quite close to the dependence (3) observed in experiment (Fig. 5a) [6, 7]. It should be noted that the theor resonance width ∆ E dps is determined similarly to the experimental values of ∆Edps as the difference between energies at the points of maximum (Epeak) and minimum (Edip) on the –d σ cap /dE curve. The dependence of the calculated autoionization resonance width Γa at the maximum of the capture cross section for the compounds studied also obeys a power law of type (6), but with somewhat different coefficients that more significantly deviate from the values obtained in experiment [6, 7]. For the bromine-substituted alkanes, the experimental and theoretical dependences are also quite 1.452 theor 1.424 close: ∆Edps = 0.755 E VAE and ∆ E dps = 0.644 E VAE . As can be seen, the spherical potential well model provides a good description of the electron transmission spectra. The energy dependence of the resonance width theor

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∆E dps , eV 4 (a)

σDEA, 10–16 cm2 102 (b)

3

100

361

1 2 2

10–2

1

10–4

CH2Cl2

CH3Cl 0

1

2

3

4 EVAE, eV

10–6

0

1

2

3

4 EVAE, eV theor

Fig. 5. (a) Calculated plots of the resonance width for () chlorine-substituted and () bromine-substituted alkanes: (1) ∆ E dps = 1.424

theor

1.429

0.644 E VAE and (2) ∆ E dps = 0.556 E VAE ; (b) comparison of the () experimental and () calculated DEA cross sections as functions of the vertical attachment energy for chlorine-substituted alkanes.

has a rather complicated profile, but, in the region of the resonance maximum, it is well approximated by a power function that is consistent with the empirical law. Let us proceed to the second task of establishing a relation between the total and dissociative electron capture cross sections, which will be solved using the approach developed in [5]. Once the energy dependence of σcap is calculated, we have to chose a reasonable dependence of the dissociative resonance width Γd . Considering Fig. 1, we conclude that, ignoring details, the resonance width can be described as
– 0.5 Γ d = --- ∝ mE , td

(7)

where td is the time for which the nuclei separate until the system reaches point C (at which the autodetachment channel is closed), m is a constant taking into account the mass of a separated halogen atom (m = 0.17 and 0.11 for chlorine and bromine, respectively), and E is the energy of the captured electron. Substituting formulas (6) and (7) into Eq. (1), we obtain the following relation between the total an dissociative capture cross sections: Γ theor σ DEA = σ cap exp ⎛ – -----a⎞ ⎝ Γ d⎠ 1.929

0.556E VAE ⎞ theor = σ cap exp ⎛ – ------------------------- . ⎝ ⎠ m

Figure 4b shows the typical calculated profile of the DEA cross section. Figure 5b shows a comparison of the published experimental data and the calculated σDEA values plotted as functions of the vertical attachment energy EVAE . Excluding two points corresponding to chloromethane and dichloromethane (whose experimental σDEA values are insufficiently reliable), we obtain a quite satisfactory coefficient of correlation between the calculation and experiment: R2 = 0.94. It should be noted that the agreement between theory and experiment is not as impressive as surprising, since the proposed model (involving a large number of simplifications) still provides a semiquantitative agreement with experiment and gives values with a quite reasonable order of magnitude for the DEA cross sections. This result implies that the process of electron scattering from halogensubstituted alkane molecules proceeds “locally” on the C–Hal bonds and, in accordance with [6, 7], the predominant contribution to the scattering is due to the pharmonic of the incident electron wave. In order to avoid unjustified complication of the model, we have suggested that, in the case of polysubstituted derivatives, both the total and dissociative electron capture cross sections exhibit additive growth with the number of chlorine atoms in the alkane molecule. This assumption corresponds to the approximation of independent and noninteracting scattering centers. In contrast to the model proposed in [11], the potential well radius in our description was not varied from one molecule to another. However, this simplification did not hinder a

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reasonable agreement between the calculation and experiment in the series of 22 chlorine-substituted alkanes. 3. CONCLUSIONS We have demonstrated that a spherical potential well model taking into account the energy and symmetry of the lowest vacant molecular orbital of the target molecule provides adequate interpretation of the experimental data obtained using the ETS and NIMS methods. The calculated dependence of the resonance width in the series of molecules studied virtually coincides with a power law determined from the experimental data [6, 7]. The agreement between calculated cross sections of the DEA process and the published data indicates that the proposed model adequately describes the main features of the electron capture and the dissociation of a short-lived negative molecular ion formed according the mechanism of shape resonance. ACKNOWLEDGMENTS The authors are grateful to P.D. Burrow and G.A. Gallup (University of Nebraska–Lincoln, USA) and to G. S. Lomakin (Institute of Molecular and Crystal Physics, Ufa) for fruitful discussions and useful remarks. This study was supported in part by the US Civilian Research and Development Foundation (CRDF grant no. RC1-2515-UF-03) and the Russian Foundation for Basic Research (project no. 06-03-32059).

2. K. D. Jordan and P. D. Burrow, Acc. Chem. Res. 11, 341 (1978). 3. V. I. Khvostenko, Mass-Spectrometry of Negative Ions in Organic Chemistry (Nauka, Moscow, 1981) [in Russian]. 4. E. Illenberger and B. M. Smirnov, Usp. Fiz. Nauk 168, 731 (1998) [Phys. Usp. 41, 651 (1998)]. 5. T. F. O’Malley, Phys. Rev. 150, 14 (1966). 6. K. Aflatooni and P. D. Burrow, J. Chem. Phys. 113, 1455 (2000). 7. G. A. Gallup, K. Aflatooni, and P. D. Burrow, J. Chem. Phys. 118, 2562 (2003). 8. A. Modelli and M. Venuti, Int. J. Mass Spectrom. 205, 7 (2001). 9. K. Aflatooni, G. A. Gallup, and P. D. Burrow, J. Phys. Chem. A 104, 7359 (2000). 10. E. P. Wigner, Phys. Rev. 73, 1002 (1948). 11. G. A. Gallup, Phys. Rev. A 71, 022710 (2005). 12. G. Herzberg, Molecular Spectra and Molecular Structure, Vol. 3: Electronic Spectra and Electronic Structure of Polyatomic Molecules (Van Nostrand, New York, 1966; Mir, Moscow, 1969), p. 277. 13. E. P. Nafikova, N. L. Asfandiarov, A. I. Fokin, and G. S. Lomakin, Zh. Éksp. Teor. Fiz. 122, 700 (2002) [JETP 95, 605 (2002)]. 14. S. A. Pshenichnyuk, I. A. Pshenichnyuk, E. P. Nafikova, and N. L. Asfandiarov, Rapid Commun. Mass Spectrom. 20, 1097 (2006). 15. A. Modelli and D. Jones, J. Phys. Chem. A 108, 417 (2004). 16. S. A. Pshenichnyuk, N. L. Asfandiarov, I. Wnorowska, et al., Mass-spektrometriya 3, 55 (2006). 17. T.-Y. Wu and T. Ohmura, Quantum Theory of Scattering (Prentice-Hall, London, 1962; Nauka, Moscow, 1969), p. 18.

REFERENCES 1. L. Sanche and G. J. Schulz, Phys. Rev. A 5, 1672 (1972).

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