Electron impact ionization of water molecule

JOURNAL OF CHEMICAL PHYSICS

VOLUME 117, NUMBER 1

1 JULY 2002

Electron impact ionization of water molecule C. Champion,a) J. Hanssen, and P. A. Hervieux Laboratoire de Physique Molećulaire et des Collisions, Institut de Physique, 1 Boulevard Arago, Technopoˆle 2000, 57078 Metz Cedex 3, France

共Received 2 January 2002; accepted 5 March 2002兲 In the present paper, differential and total cross sections are calculated for the interaction of electrons with a water molecule. The calculations are performed in the distorted wave Born approximation framework where the incident and scattered 共fast兲 electrons are described by a plane wave function, whereas the ejected 共slow兲 electron is described by a distorted wave function. From the fivefold differential cross sections, triply and singly differential cross sections have been calculated by successive integrations. In these conditions, very good agreement is found with available experimental measurements essentially limited to triply and singly ionization cross sections. Finally, a comparison of our results with a large set of experimental data of total ionization cross sections exhibits very good agreement. © 2002 American Institute of Physics. 关DOI: 10.1063/1.1472513兴

I. INTRODUCTION

ited to singly differential and total ionization cross-section calculations. More recently, Coimbra and Barbieri14 have proposed an extension of the BEB model to calculate doubly differential cross sections 共DDCS兲 values by reducing the number of adjustable parameters from 8 共in Rudd’s model兲 to 3. In the past, quantum-mechanical studies of the vapor water ionization have been given,15–17 but they remain, in general, only in qualitative agreement with the experimental data. Thus, in absence of theoretical quantum-mechanical investigations, which were quite involved, several semiempirical approaches have been developed to overcome the limitation of ab initio theories.18 –21 Consequently, the present work appears as a detailed theoretical study of the vapor water molecule ionization by electron impact without any adjustable parameter. The obtained results are compared to a large set of experimental data, essentially limited to triply, singly differential, and total ionization cross-section measurements taken from the extensive work of Opal et al.,22 who provides triply and singly ionization cross sections for an incident energy E a ⫽500 eV in a range of ejected energies E e ⫽4.13– 205 eV and ejection angles ␪ e ⫽30° – 180°, and from the works of Bolorizadeh and Rudd23 and of Oda24 dedicated to energetic electrons with an incident energy E a ⫽500 eV. Concerning singly differential and total cross sections, the literature is more abundant: our results are compared to a large set of experimental works for a large range of incident energies.25–37 Furthermore, our theoretical approach may be easily introduced in numerical simulations such as Monte Carlo track structure code for electrons in water38 or in matter in general. Indeed, for these codes, multiple differential calculations represent useful input data to describe in detail all the ionizing events, in terms of energy deposits and angular distributions. In these conditions, we propose by the present work a detailed description of a theoretical approach developed in the distorted wave Born approximation 共DWBA兲 framework. In a first step, eightfold differential cross sections 共8DCSs兲 of

Understanding the complex interactions involved when atoms or molecules are ionized by electron impact remains, at the present time, one of the greater challenges of atomic physics. In effect, besides its fundamental aspect 共especially in the study of the electronic structure of the target兲, it is a powerful tool for the study of the ionization mechanisms, whose comprehension becomes important in many domains such as plasma physics, fusion experiments, astrophysics, and even in the study of ionizing collisions on living matter.1– 4 Indeed, the description of the interactions induced by charged particles 共electrons and ions兲 in biological cells demonstrates the need for comprehensive and useful sets of differential and total cross-section data for the scattering and emission of electrons in the biological matter, this latter being, in a first step, simulated by water.5–7 However, although there are many useful theories for electron impact ionization, few of them are extendable for complex atoms or neutral larger molecules. Concerning the water molecule ionization, the most recent study has been published by Kim et al.,8 –10 who developed a ‘‘binary-encounter-dipole 共BED兲 model’’ which combines the binary-encounter theory of Vriens11 with the dipole interaction of the Bethe theory12 for fast incident electrons. The mixing ratios for distant and close collisions, and the interference between the direct and the exchange terms, were determined by using the asymptotic behavior predicted by the Bethe theory for ionization and stoppingpower cross sections.13 The main inconvenience in the BED model is obtaining the knowledge of the optical oscillator strength data to describe the continuum, which is only available for a limited number of atoms and molecules. Kim and Rudd8 have proposed an additional approximation in the socalled binary-encounter-Bethe 共BEB兲 model. However, these two models 共the BED and BEB兲 give only a semiempirical description of the ionization process and are moreover lima兲

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198

J. Chem. Phys., Vol. 117, No. 1, 1 July 2002

Champion, Hanssen, and Hervieux TABLE I. Comparison between the calculated values 共proposed by Moccia兲 and the experimental ones concerning the geometrical and energetic properties of the water molecule. The energetic properties concern essentially the electric dipole moment ␮ and the first ionization potential 共called 1st IP兲, whereas the geometrical parameters represent the binding length O–H, the equilibrium distance H–H, and the molecular angle H–O–H. Water molecule parameter

Calculated value

Experimental value

0.8205 0.4954 1.814 2.907 106.53°

0.728 0.463 1.810 2.873 105.5°

␮ 共a.u.兲 1st IP 共a.u.兲 O–H 共a.u.兲 H–H 共a.u.兲 H–O–H FIG. 1. Reference frame of the ionizing collision of a water target. ki , ks , and ke represent the wave vectors of the incident, scattered, and ejected electrons, respectively. The corresponding polar and azimuthal angles are denoted ( ␪ s , ␸ s ) and ( ␪ e , ␸ e ), respectively.

the water molecule ionization are calculated for a particular molecular orientation defined by the Euler angles 共␣; ␤; ␥兲. Then, fivefold differential cross sections 共5DCSs兲 are deduced from the integration of the 8DCSs over all the possible molecular orientations. In a second step, 5DCSs are integrated over either the scattered or the ejected solid angle to provide triply differential cross sections 共TDCSs兲. Finally, singly differential and total cross sections 共SDCSs and TCSs, respectively兲 are numerically calculated by successive integrations and compared to an extensive set of experimental data and to other theoretical model predictions.8,17 The present paper is organized as follows: Our theoretical approach is outlined in Sec. II, and the results concerning multiple differential and total cross-section calculations are given and analyzed in Sec. III. In the first part, the calculated TDCSs are successively plotted versus the scattered angle and the ejected angle, the latter being compared to experimental measurements. In the second part, the SDCSs are presented, analyzed and compared to experimental data. Particular attention is provided to the TCSs, which are presented in the third part. Finally, a conclusion is given in Sec. IV. Atomic units are used throughout unless otherwise indicated.

II. THEORETICAL METHOD OF IONIZATION CROSS-SECTION CALCULATION

To simplify the presentation of the theoretical method, we have reported in Fig. 1 all the geometric and kinematic parameters used to describe the ionization process. The labels i, e, and s correspond to the incident, ejected, and scattered species, respectively. The water–molecule plane is defined from the (0,yz) plane thanks to the Euler angles 共␣, ␤, ␥兲 and to the molecular description taken from the work of Moccia,39 where each molecular orbital 共MO兲 wave function ⌿ j is developed in terms of Slater-type-orbital 共STO兲 functions, all centered at a common origin, namely upon the heaviest nucleus 共i.e., the oxygen atom兲. They refer to the equilibrium configurations calculated in the self-consistent field 共SCF兲 method,40 and agree very well with the experimental geometrical and energetic properties of the water molecule 共see Table I兲.

In these conditions, the nonrelativistic 8DCSs of the water molecule ionization are, in the DWBA, for a given molecular orientation 共␣, ␤, ␥兲, expressed by

冋

册

d 8␴ 共␣,␤,␥兲 d⍀ s d⍀ e dE e d ␣ d ␤ d ␥ ⫽

1 共 2␲ 兲5

N MO

兺 j⫽1

k ek s 兩关 T ab 共 ␣ , ␤ , ␥ 兲兴 j 兩 2 , ki

共1兲

where N MO is the number of molecular orbitals (N MO⫽5). The first Born transition amplitude 关 T ab ( ␣ ; ␤ ; ␥ ) 兴 j between the initial state labeled a and the final state labeled b, depends on the ionized molecular orbital and is, for a particular molecular orientation 共␣, ␤, ␥兲, given by 关 T ␣ 共 ␣ , ␤ , ␥ 兲兴 j ⫽ 具 ⌽ 共bj 兲兩 V 兩 ⌽ 共aj 兲典 ,

共2兲

where V is the perturbative interaction potential. ⌽ (aj) and ⌽ (bj) are the initial and the final wave functions, respectively. ⌽ (aj) is described as the product of the jth MO wave function ⌿ j by a plane wave function 共used to describe the incident electron兲, and ⌽ (bj) is described as the product of a plane wave function 共used to describe the scattered electron兲 by a continuum wave function defined by a distorted wave function Fk( j)(⫺) (r) of ejected momentum ke with an asymptotic e charge z e ⫽1 共used to describe the ejected electron, which is by definition the slowest particle in the final state兲. In these conditions, the amplitude transition of the jth MO can be easily written as 关 T ab 共 ␣ , ␤ , ␥ 兲兴 j ⫽

4 ␲ 共 j 兲共 ⫺ 兲具 Fke 共 r兲兩 e iq•r兩 ⌿ j 共 r兲典 , q2

共3兲

where the momentum transfer q is defined by q⫽ki ⫺ks 共see Fig. 1 for more details兲. According to the molecular description given by Moccia,39 the MO wave functions are written as Nj

⌿ j 共 r兲 ⫽

兺

k⫽1

␰

a jk ␾ n jk l

jk jk m jk

共4兲

共 r兲 , ␰

where N j is the number of Slater atomic orbitals ␾ n jk l

jk jk m jk

(r)

and a jk the corresponding weight, each atomic component being written as ␰

␾ n jk l

jk jk m jk

␰

共 r兲 ⫽R n jk l 共 r 兲 Y l jk m jk 共 rˆ兲 , jk jk

共5兲



Ionization of water molecule

199

TABLE II. List of the different coefficients and quantum numbers (n (i) ,l (i) ,m (i) ) included in the linear combination of atomic orbitals for the water molecule description. The five molecular orbitals necessary to describe the water molecule are labeled 1A 1 , 2A 1 , 3A 1 , 1B 2 , and 1B 1 , respectively. The corresponding ionization potentials are 共in atomic units兲, respectively, 20.5249, 1.3261, 0.5561, 0.6814, and 0.4954. n (i)

l (i)

m (i)

␰

1A 1

2A 1

3A 1

1B 2

1B 1

0 0 0 0 0 1 1 1 2 2 2 2 3 3 1 1 1 2 2 3 3 1 1 1 2 2 3 3

0 0 0 0 0 0 0 0 0 0 2 2 0 2 ⫺1 ⫺1 ⫺1 ⫺1 ⫺1 ⫺1 ⫺3 1 1 1 1 1 1 3

12.600 7.450 2.200 3.240 1.280 1.510 2.440 3.920 1.600 2.400 1.600 2.400 1.950 1.950 1.510 2.440 3.920 1.600 2.400 1.950 1.950 1.510 2.440 3.920 1.600 2.400 1.950 1.950

0.051 67 0.946 56 ⫺0.017 08 0.024 97 0.004 89 0.001 07 ⫺0.002 44 0.002 75 0.000 00 0.000 00 ⫺0.000 04 0.000 03 ⫺0.000 04 ⫺0.000 08 ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯

0.018 89 ⫺0.255 92 0.777 45 0.099 39 0.163 59 0.186 36 ⫺0.008 35 0.024 84 0.006 95 0.002 15 ⫺0.064 03 ⫺0.009 88 ⫺0.026 28 ⫺0.056 40 ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯

⫺0.08 48 0.082 41 ⫺0.307 52 ⫺0.041 32 0.149 54 0.799 79 0.004 83 0.244 13 0.059 35 0.003 96 ⫺0.092 93 0.017 06 ⫺0.019 29 ⫺0.065 93 ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯

¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ 0.882 70 ⫺0.070 83 0.231 89 0.254 45 ⫺0.019 85 0.045 26 ⫺0.063 81 ¯ ¯ ¯ ¯ ¯ ¯ ¯

¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ 0.720 81 0.115 32 0.248 59 0.054 73 0.004 03 0.009 35 ⫺0.026 91

1 1 2 2 2 2 2 2 3 3 3 3 4 4 2 2 2 3 3 4 4 2 2 2 3 3 4 4

␰

where the radial part R n jk l (r) is given by jk jk

␰

R n jk l 共 r 兲 ⫽

共 2 ␰ jk 兲 2n jk ⫹1/2

冑2n jk !

jk jk

r n jk ⫺1 e ⫺ ␰ jk r .

共6兲

All the coefficients (a jk , ␰ jk ) and the quantum numbers (n jk ,l jk ,m jk ) used in the STO development are reported in Table II. The calculation of the transition amplitude T ab 关Eq. 共3兲兴 is made by using the partial wave expansion method. Thus, the continuum distorted wave function Fk( j)(⫺) (r) and the e plane wave function e iq•r can be written as, respectively, ⬁

F共kj 兲共 ⫺ 兲共 r兲 ⫽ e

⫹l e

兺兺 l ⫽0 m ⫽⫺l e

⫻

e

F 共l j 兲共 k e e k er

共 j兲

共 4 ␲ 兲共 i 兲 l e e ⫺i 共 ␴ l e ⫹ ␦ l e

Y l* m 共 kê 兲 Y l e m e 共 rˆ兲 , e e

共7兲

and ⬁

e iq•r⫽

⫹l

兺兺

l⫽0 m⫽⫺l

* 共 qˆ兲 Y lm 共 rˆ兲 , 共 4 ␲ 兲 i l j l 共 qr 兲 Y lm

冋

册

1 d2 l e 共 l e ⫹1 兲 ⫺V j 共 r 兲 F 共l j 兲共 k e ;r 兲 ⫽0, 2 ⫹E e ⫺ e 2 dr 2r 2

and exhibits an asymptotic behavior given by

冉

共9兲

冊

␲ F 共l j 兲共 k e ;r 兲 ⬃sin k e r⫺l e ⫺ ␩ e ln 共 2k e r 兲 ⫹ ␴ l e ⫹ ␦ 共l j 兲 , e e 2 共10兲 where ␩ e ⫽⫺1/k e is the Sommerfeld parameter. The effective distortion potential V j (r) is calculated for each MO and is given by

兲

e

;r 兲

quantities ␴ l e and ␦ l( j) represent the Coulomb phase shift and e the short-range phase shift induced by the distortion potential, respectively. The radial function F l( j) (k e ;r) is solution of the differe ential equation

共8兲

where the quantum numbers (l e ,m e ) correspond to the ejected electron. j l (r) and Y lm (rˆ) correspond to the Bessel functions and the spherical harmonics, respectively. The

ion V j 共 r 兲 ⫽V elec j 共 r 兲 ⫹V 共 r 兲 ,

共11兲

where the first term corresponds to the electronic contribution, whereas the second term corresponds to the ionic contribution. However, since the water molecule is randomly oriented in the space, the effective distortion potential V j (r) is calculated by using the spherical average approximation, which leads to V elec j 共 r 兲⫽

1 4␲

冕

˜V elec ˆ, j 共 r 兲 dr

共12兲


200


Champion, Hanssen, and Hervieux

and

冕

1 V 共 r 兲⫽ 4␲ ion

共13兲

V 共 r兲 drˆ,

˜ ion

冋

d 5␴ d⍀ s d⍀ e dE e ⫽

where N MO

˜V elec j 共 r兲 ⫽ with Ni j⫽

再

兺

i⫽1

Nij

冕

⌿ i* 共 r⬘ 兲

1 ⌿ 共 r⬘ 兲 dr⬘ , 兩 r⫺r⬘ 兩 i

1 8␲2

if i⫽ j

1

if i⫽ j,

N

共15兲

兺

␭ˆ

兺

i⫽1

Nij

␭ r⬍ ␭⫹1 r⬎

⬁

⬘

冕

⬁

* 共 rˆ⬘ 兲 , Y ␭m ␭ 共 rˆ兲 Y ␭m ␭

0

␰

ik ik

兺兺兺 i 共 l⫺l 兲e i共 ␴ l ⫽0 m ⫽⫺l l⫽0 e

e

e

e

e

r ⬘ 2 ␰ ik R ⬘ 共 r ⬘ 兲 dr ⬘ , r ⬎ n ik ⬘ l ik ⬘

2 8 V ion共 r 兲 ⫽⫺ ⫺ , r R⬎

n l jk

e

e

respectively,

n l

Rlljk jk ⫽ e

冕

satisfy

V j (r) → ⫺1/r

le

l

0

0

0

l

冊

␮k

⫺m e

, m e⫺ ␮ k 共23兲

R n jk l 共 r 兲 j l 共 qr 兲 F 共l j 兲共 k e ;r 兲 r dr. e

共24兲

␰

jk jk

Triply differential cross sections 共TDCSs兲 are obtained by integration of the 5DCSs over either the ejected or the scattered solid angle, and are denoted ␴ (3) (E e ,⍀ s ) or ␴ (3) (E e ,⍀ e ), respectively. Whereas the first ones are obtained analytically, the second ones can only be calculated by numerical integration procedures. They are given by, respectively, 共3兲

冋

册

N MO

兺

j⫽1

⫻

and

共20兲

Thus, taking the direction of the initial momentum ki to be along the z axis 共see Fig. 1兲 and averaging the 8DCSs over all the orientations 共i.e., over the solid angle d⍀ ⫽sin ␤ d␣ d␤ d␥兲 in order to take into account the randomness of the water molecule orientations, we obtain the simplified following expression of the 5DCSs:

N

MO d 3␴ ⫽ 共 E e ,⍀ s 兲 ⫽ d⍀ s dE e j⫽1

the

V j (r) → ⫺8/r. Moreover, it is easy to check that if the ejected electron is moving in a pure Coulomb potential V j (r) ⫽⫺1/r, then ␦ l( j) ⫽0 and the Coulomb phase shift is written e as

␴ l e ⫽arg ⌫ 共 l e ⫹1⫹i ␩ e 兲 .

⬁

0

⫽

i.e.,

l jk

冊冉

le

e

r→⬁

conditions,

冉

expressed by,

l jk

l ␮

Al jkm kl ⫽ 共 ˆl jk ˆl e ˆl 兲 1/2

共19兲

potentials

共22兲

e

with Al jkm kl and the radial integration Rlljk

␴

r→0

n l

e

with ˆl ⫽2l⫹1, and

and

where R ⬎ ⫽max(r,ROH). Note that the calculated

共 j兲 le⫹ ␦ le 兲

共17兲

共18兲

boundary

共21兲

e

l ␮

e

R nik l 共 r ⬘ 兲

⬁

⫹l e

l ␮

a ik a ik ⬘ ␦ 共 m ik ⫺m ik ⬘ 兲兺 kk

⫻ ␦ 共 l ik ⫺l ik ⬘ 兲

good

兺

⫻Al jkm kl Rlljk jk Y l e m e 共 kê 兲 Y l ␮ k ⫺m e 共 qˆ兲 ,

where r ⬎ ⫽max(r,r⬘) and r ⬍ ⫽min(r,r⬘), the electronic conion tribution V elec j (r) and the ionic contribution V (r) are given by, respectively, N MO

兺

jk jk

共16兲

with 储 rOH1 储 ⫽ 储 rOH2 储 ⫽R OH⫽1.814 a.u. 共see Table I兲. Thus, by using the partial wave expansion of the Coulomb interaction

␭m ␭

l

where Sn k l is given by ␮

⫽

册

N

兺

Sn k l ⫽

4␲

冕冕冕冋

␮

2

1 8 1 ˜V ion共 r兲 ⫽⫺ ⫺ ⫺ r 兩 r⫺rOH1 兩兩 r⫺rOH2 兩

V elec j 共 r 兲⫽

兺

j⫽1

d 8␴ 共␣;␤;␥兲 d⍀ s d⍀ e dE e d ␣ d ␤ d ␥ j

jk 32 MO k s j 关 a jk 兴 2 ␮ ⫽ 4 兩S k 兩2, q j⫽1 k i k e k⫽1 ˆl jk ␮ k ⫽⫺l jk n jk l jk

jk jk

兩 r⫺r⬘ 兩

N MO

⫻d ␣ d ␥ sin ␤ d ␤

共14兲

and

1

册

冉冊

兺 l l

N MO

兺

j⫽1

and

␴ 共 3 兲共 E e ,⍀ e 兲 ⫽

兺

冉

le

l

l jk

0

0

0

冊

2 n l

关 Rlljk jk 兴 2 e

␴ 共j 3 兲共 E e ,⍀ s 兲 ,

冋

d 3␴ d⍀ e dE e

兺 j⫽1

N MO

⬅

册

d 5␴ d⍀ e d⍀ s d⍀ e dE e j

N

ˆl e ˆl

N MO

⫽

冕冋

8 ks 1 j 关a 兴2 ␲ k i k e q 4 k⫽1 jk

e

⬅

兺

兺

j⫽1

冕冋

册

共25兲

册

d 5␴ d⍀ s d⍀ s d⍀ e dE e j

␴ 共j 3 兲共 E e ,⍀ e 兲 .

共26兲




201

Singly differential cross sections 共SDCSs兲 are finally obtained by numerical integration of 共25兲 or 共26兲

冋册

冕␴ 兺冕␴

N

MO d␴ ⫽ dE e j⫽1

兺

N MO

⫽

j⫽1

共3兲 j 共Ee

,⍀ s 兲 d⍀ s

共3兲 j 共Ee

,⍀ e 兲 d⍀ e ⬅

N MO

兺

j⫽1

␴ 共j 1 兲 ,

共27兲

and the total ionization cross sections 共TCSs兲 are given by N MO

␴⫽

兺

j⫽1

冕

共 E a ⫺I P j 兲 /2

0

冋册

N

MO d␴ dE e ⬅ ␴j . dE e j j⫽1

兺

共28兲

III. RESULTS

In the following, results for calculations of TDCSs, SDCSs, and TCSs for incident energies E a ranging from 20 eV to 10 keV are presented and compared to experimental measurements. A. Triply differential cross sections

There are two manners to define TDCSs. A first one, called ␴ (3) (E e ,⍀ s ), consists of studying the role of the scattered electron, whereas a second one, called ␴ (3) (E e ,⍀ e ), concerns the distribution of the ejected electron. In what follows, we successively present these two types of TDCS calculations, the second ones being compared in Sec. B 2 to experimental data taken from Opal et al.,22 and from Bolorizadeh and Rudd.23 1. Variation with the scattering angle ␪ s

Figure 2 displays 3D plot of ␴ (3) (E e ,⍀ s ) versus the scattered angle ␪ s and the ejected energy E e for three different incident energies, namely E a ⫽100 eV, E a ⫽500 eV, and E a ⫽1 keV. For small energy transfers, the evolution of ␴ (3) (E e ,⍀ s ) is the same for the three cases and corresponds to TDCSs extremely peaked at the origin ␪ s ⫽0°. For more important energy transfers, we observe the appearance of a peak centered at a ␪ scrit value of about ␪ s ⫽40.0° and ␪ s ⫽41.4° in cases 2 共b兲 and 共c兲, respectively. In fact, for high velocities of ejection of the molecular electron 共i.e., for large ⌬E e values兲, the angular distribution does not fall off uniformly with the scattered angle but displays maximum for a critical angle ␪ scrit , which corresponds to binary collision in which the energy lost by the incident electron is completely transferred to the target molecular electron with the residual ion acting as a spectator.41,42 This is the region of the Bethe ridge which is simply defined by q 2 ⫽k 2e ,

共29兲

where the momentum transfer q is related to the scattered angle ␪ s by q 2 ⫽k 2i ⫹k s2 ⫺2k i k s cos ␪ s , which implies a

␪ scrit⫽cos⫺1

冋

␪ scrit

共30兲

value defined by

k 2i ⫹k s2 ⫺k 2e 2k i k s

册

.

共31兲

FIG. 2. 3D evolution of the TDCSs of the vapor water molecule ionization 共expressed in atomic units兲, plotted vs the scattered angle ␪ s in three different incident energy configurations: 共a兲 E a ⫽100 eV, 共b兲 E a ⫽500 eV and 共c兲 E a ⫽1 keV, for E e ranging from 0 to (E a ⫺I P 1 )/2.

This quantity depends on the ionized MO and is, for this reason, called ␪ scrit( j) in the following. In Table III we have compared the expected ␪ scrit( j) values 关given by Eq. 共31兲兴 to those observed in our TDCS calculations. Good agreements are found between these two values. 2. Variation with the ejected angle ␪ e

In Figs. 3共a兲 and 共b兲, we have compared our TDCS calculations, called ␴ (3) (E e ,⍀ e ), to the experimental results taken from Opal et al.,22 共open circles兲 and from Bolorizadeh and Rudd23 共solid circles兲 for two incident energy conditions: 共a兲 E a ⫽500 eV and 共b兲 E a ⫽1 keV. We observe reasonable agreements between the experimental and theoretical results in the two cases, except in the small angle region ( ␪ e ⬍60°), where our results overestimate the experimental


202


Champion, Hanssen, and Hervieux

TABLE III. Comparison between the calculated and observed values of the critical angles ␪ scrit corresponding to the binary peak. The analytic results deduced from Eq. 共31兲 are compared, for each molecular subshell, to the numerical values observed in our TDCS calculations.

Ionization potential 共eV兲 ␪ scrit values for: (E a ⫽500⫺E e ⫽243.3) 共eV兲 (E a ⫽1000⫺E e ⫽493.2) 共eV兲

Analytic Observed in TDCS calculations Analytic Observed in TDCS calculations

ones, maybe due to the distortion induced by the incident electron, the effect of which is not taken into account in the present work. However, for ␪ e ⬎60°, the agreement becomes very good and we theoretically observe the appearance of the binary peak located at a ␪ crit e value of about 51.0° and 63.2° for (E a ⫽500 eV,E e ⫽100 eV), and (E a ⫽1 keV,E e ⫽100 eV), respectively, which agrees well with the experimental observations. In fact, similarly to Eq. 共31兲, it is possible to show that the ␪ crit e value is simply defined by ⫺1 ␪ crit e ⫽cos

冋

k 2i ⫹k 2e ⫺k s2 2k i k e

册

,

共32兲

which leads to ␪ crit e ( j) values varying from 58.14° to 61.48° for (E a ⫽500 eV,E e ⫽100 eV) and from 68.18° to 70.30° for (E a ⫽1 keV,E e ⫽100 eV). By comparison, we have reported in Fig. 3共a兲共dashed line兲 the only available theoretical results obtained by Long et al.,17 in the framework of the density functional theory for E a ⫽500 eV and E e ⫽40 eV. We observe a good agreement between these results and ours, especially for large ␪ e values 共i.e., ␪ e ⬎60°兲.

FIG. 3. TDCSs of the vapor water molecule ionization calculated from the 5DCSs by numerical integration over the scattered solid angle ⍀ s . The theoretical results are compared to the experimental data taken from Bolorizadeh and Rudd 共solid circles兲 and from Opal et al. 共open circles兲. In the first case 共a兲, the incident energy E a ⫽500 eV and the ejection energy E e is equal to 22, 40, and 100 eV, successively, whereas, in the second case 共b兲, the incident energy E a ⫽1 keV and the ejection energy E e is equal to 20, 40, and 100 eV, respectively. The DWBA results are represented by a solid line, and we have reported in 共a兲 the theoretical results obtained by Long et al. for E e ⫽40 eV 共dashed line兲.

2A 1

1B 2

3A 1

1B 1

36.08

18.54

15.13

13.48

41.85° 45.7° 44.57° 44.1°

44.18° 40.2° 44.60° 40.0°

44.20° 38.8° 44.60° 39.9°

44.20° 38.7° 44.60° 39.8°

B. Singly differential cross sections

In the following we present SDCS calculations for a large range of incident energies E a ⫽50 eV– 10 keV. Moreover, since the two electrons coming out of the molecule are indistinguishable, one has to take into account exchange effects. In these conditions, the singly differential cross sections for an ejected energy E e with exchange, called 关 ␴ (1) j (E e ) 兴 exc , are deduced from the SDCSs without exchange by 关 ␴ 共j 1 兲共 E e 兲兴 exc⫽ ␴ 共j 1 兲共 E e 兲 ⫹ ␴ 共j 1 兲共 E i ⫺E e ⫺I P j 兲 ,

共33兲

where I P j is the jth molecular ionization potential. In Fig. 4 we have compared our theoretical results 共solid line兲 to experimental data for three incident energy conditions: 共a兲 E a ⫽100 eV; 共b兲 E a ⫽500 eV; and 共c兲 E a ⫽1 keV. We observe very good agreement between the experimental and the theoretical results, whereas large discrepancies are observed between our results and the semiempirical ones 共dashed line兲 given by Kim and Rudd8 –10 in the ‘‘binary-encounter-dipole 共BED兲 model.’’ Figure 5 displays a large comparison between experimental and theoretical results about singly differential cross sections for a large range of incident energies E a ⫽50 eV– 10 keV. The results obtained in the DWBA framework are represented by solid lines and compared to

FIG. 4. SDCSs of the water molecule ionization calculated in the DWBA framework 共solid line兲 compared to experimental data taken from Opal et al. 共open circles兲, from Bolorizadeh and Rudd 共solid circles兲 and from Vroom and Palmer 共open up triangles兲 for three different incident energies: 共a兲 E a ⫽100 eV; 共b兲 E a ⫽500 eV, and E a ⫽1 keV. The dashed line represents the semiempirical results obtained by Kim and Rudd.




203

FIG. 6. Comparison between the calculated total ionization cross sections of the vapor water molecule and experimental data taken from various sources: Bolorizadeh and Rudd 共solid circles兲, Djuric et al. 共solid down triangles兲, Schutten et al. 共solid up triangles兲, Khare and Meath 共open down triangles兲, Straub 共open up triangles兲, and Olivero 共open diamonds兲. The solid line represents the DWBA results and the dashed line the semiempirical Kim and Rudd’s results.

FIG. 5. SDCSs of the water molecule ionization calculated in the DWBA framework 共solid line兲 compared to experimental data taken from Opal et al. 共open circles兲, from Bolorizadeh and Rudd 共solid circles兲 and from Vroom and Palmer 共open up triangles兲 for different incident energies E a ranging from 50 eV to 10 keV. Multiplicative factors reported in parentheses are used for better clarity.

experimental data represented by symbols 共open circles: Opal et al.,22 solid circles: Bolorizadeh and Rudd,23 and open up triangles: Vroom and Palmer,32 respectively兲. In all these experiments, the SDCS results are obtained by numerical integration of the measured TDCSs 关i.e., the ␴ (3) (E e ,⍀ e ) presented above兴 over the ejection solid angle ⍀ e . We observe very good agreement between the experimental data and our results. C. Total cross sections

Total ionization cross sections ␴ j are calculated for each MO labeled j by numerical integration of the SDCSs 关Eq. 共33兲兴 over the energy transfers E e from 0 to (E a ⫺I P j )/2. Then, the ‘‘global’’ TCSs are obtained by summing up all the N orb individual cross sections ␴ j . In Fig. 6 the calculated TCSs of vapor water molecule ionization 共solid line兲 are compared to an extensive set of experimental data covering a large range of incident energies E a ⫽20 eV– 10 keV. The experimental ionization cross sections reported are those of Bolorizadeh and Rudd,23 Djuric et al.,25 Schutten et al.,26 Khare and Meath,28 Straub,29 and Olivero.33 Although there exists close agreement between some of the measurements of ionization cross sections over part of the energy spectrum, there is considerable variation in the range 50 eV to 1 keV. We have excluded the sets of experimental data of Gomet34 and Orient and Srivastava,36 which deviate greatly from the

other measurements. Also, we have not included results from experiments that did not provide data on an absolute scale.35 On the theoretical side, we have reported the Kim and Rudd’s results8 共dashed line兲, which are in good agreement with ours, essentially in low and high incident energy 共E a ⬍100 eV and E a ⬎1 keV, respectively兲, but display sensitive differencies for intermediate incident energies. The DWBA results are in very good agreement with the experimental data sets reported, and the overall behavior of the TCS theoretical curve is well reproduced. In particular, we observe the expected maximum located at E a ⫽120 eV, which is in good agreement with the experimental data.

IV. CONCLUSIONS

We have presented in this work a theoretical approach to calculate differential cross sections for the water molecule ionization by electron impact. The most differential cross sections 共5DCSs兲 are analytically expressed in the distorted wave Born approximation. Then, triply and singly differential cross sections are obtained by numerically integration procedures over the scattered solid angle and the energy transfers. The first ones exhibit reasonable agreement with the experimental data, especially in the large ␪ e domain since sensitive discrepancies are found for ␪ e ⬍60°, maybe due to the distortion induced by the incident electron. SDCS results are in very good agreement with experimental data and provide TCSs which agree very well with the extensive set of experimental results reported in this work. Moreover, all these calculations performed in this quantum-mechanic theory can easily be incorporated into a Monte Carlo code performed to describe the crossing of electrons through the biological matter 共simulated by water兲, which will be the subject of a forthcoming paper dedicated to the complete description of a quantum Monte Carlo track structure code for electrons in water.


204 1


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