Electron energy deposition in helium

Physica 85C (1977) 219-222 © North-Holland Publishing Company

ELECTRON ENERGY DEPOSITION IN HELIUM DAYASHANKAR

Division of Radiological Protection, Bhabha Atomic Research Centre, Bombay 400 085, India Received 13 February 1976 Revised 5 July 1976

The excitation yields for 2 ls, 21p and 23p states and the ionization yields of helium for electron impact energies up to i keV have been calculated in the continuous slowing-down approximation. The calculations utilize recent available informa tion on the energy spectrum of secondary electrons and the energy loss function. The results are presented as the energy deposition efficiencies for the excited states and the mean energy expended per ion pair formed.

1. Introduction

been computed by a few authors [3-6] using their semi-empirical expressions for the secondary-electron spectrum. However, it should be noted that these expressions were tentative in nature.

An energetic electron passing through a gas loses most of its energy through excitations and ionizations of the atoms or molecules. The products of these primary processes being precursors of all observed consequences of radiation action, a quantitative study of the yields of these products is of great value in the interpretation of radiation action. For computing these yields one has to consider not only the effect of primary electrons, but also that of the secondary electrons (of all generations) energetic enough to produce additional excitations and ionization. To take into account the effect of these secondaries it is essential to have knowledge of the energy spectrum of secondary electrons. Until a few years ago, reliable information on the secondary electron spectrum at low energies was not available. Recently Opal et al. [1] have reported experimental data on the energy spectrum of secondary electrons over a wide range of energies for a number of targets. Making use of these data, Dayashankar [2] has recently constructed the energy loss function (stopping cross section) for electrons in helium which represents a significant improvement at low energies over the values reported earlier. In this work, the problem of the electron energy deposition in helium has been investigated by using the experimentally determined secondary electron spectrum and the constructed energy loss function. Prior to this work, some of the yields in helium have

2. Method of calculation we use the continuous slowing-downapproximation for considering the energy degradation of a

primary electron of energy E 0 and all the resulting secondaries in the helium gas of uniform density. The number of excitations to a given excited s t a t e / d u e to the energy degradation of the primary alone is given by

NO/(EO) = ~ [o](E)/L(E)] dE,

(1)

w~ where I¢l is the excitation energy of the state ] and of(E) is the excitation cross section. L(E) is the energy loss function. The number of secondary electrons of energy e per unit energy interval produced by the primary electron of energy E 0 is given by Eo

n(Eo, e) =

f [Q(E, e)/L(E)] dE, 2e "+I

where Q(E, e) is the differential ionization cross 219

Dayashankar/Electron energy deposition in helium

220

section (Le. the energy spectrum of secondary electrons of energy e ejected in collisions by electrons of energy E). I is the ionization potential (24.586 eV for helium). The number of excitations to the state / due to the energy degradation of secondary electrons and that due to the tertiary electrons are respectively given by

~(Eo-I) n(Eo, e)No/(e ) de, N1/(Eo) = f

(2)

w~ and ~(Eo-/)

Nv(Eo) =

/

n(EO, e) N1/(e) de.

(3)

2W/+I

The limits of integration in the above equations follow from the convention that the slower of the two electrons that emerge after a collision is referred to as secondary electron. Analogous expressions can be given for the ionization yields. Calculations were performed for the excitation yields of 21S, 21p and 23p states of helium for incident electron energies from the thresholds to 1 keV. The ionization yields were calculated for the energy range from 50 eV to 1 keV. The integrals were evaluated numerically. For Q(E, e) we used the experimental data of Opal et al. [1] up to a secondary electron energy of 200 eV, which can be represented as

where a 0 is the Bohr radius and R is the Rydberg energy. This formula adequately describes the energy spectrum of the high energy secondaries. For the energy loss function, L(E), in the energy range from 50 eV to 1 keV we have used the data recently constructed by Dayashankar [2] who utilized the available information on the ionization and excitation cross sections and the energy spectrum of secondary electrons. The input cross sections for the construction of the loss function being measured ones, the loss function may be considered to be fairly reliable. It may also be mentioned that the constructed loss function is in excellent agreement with the wellknown Bethe formula in the region of its validity. For energies lower than 50 eV the same approach was followed and it was assumed that the energy spectrum at these energies is also given by eq. (4). The loss function used in the calculations is shown in fig. 1. The excitation energies [9] and the sources of the excitation cross-section data [10-14] for the excited states are given in table I.

22

(4)

t

i

~

i

i

i i i

O

18

- -

Z

16 --

~ z

14 ~

LL

i

12

/

(.n

where n = 2.1 and ~'= 15.8 eV for helium. C(E) is a normalization parameter given by

I I llJ

~ o 20 -

o_

c(E) Q(E, e ) , 1 + (e/-Ef

I

°10 8

C(E) = Qi(E)/ff~ arctan

[(E -

w Z

I)I2E],

"'

where Qi(E) is the total ionization cross section. For Qi(E) we have used the data of Rapp and EnglanderGolden [7]. For secondary electron energies higher than 200 eV, for which experimental data are not available, we employed the Mort formula [8] which is given by

E

e(E-e)

(E

e)2 '

6

-/ I

20

I IIII

50

I

100

INCIDENT ELECTRON

I

I

t

I lJ

1000 ENERGY ( e V )

Fig. 1. Energy loss function for electrons in helium. Solid part o f the curve is taken from ref. 2. Dotted part is obtained by extending the work o f ref. 2.

221

Dayashankar/Electron energy deposition in helium

Table I Excited states, excitation energies and sources of excitation cross-section data Excited state

21S

Sources of the cross-section data

Excitation energy [9] (eV) 20.612

21p 23p

Up to 80 eV from Rice et al. [10]. From 90 to 150 eV the values interpolated between the data of Rice etal. [10] and Vriens et al. [ 11 ]. From 200 to 400 eV from Vriens et al. [ 11 ] and for higher energies the calcuiated values from Bell et al. [12]. Donaldson etal. [13]. Jobe and St. John [14].

21.214 20.960

0.12

-

0.11

-

0.10

>U Z I.U

_u

0.09

!

iltl

r

/i

i

i

i

I I I

\

a.

z O m

LI. LI,. LU

0.07

Z O

0.06

60

0.05

50 tu

0.04

40 z

0.03

30 ~

t/3 O 0. Ill r'~.

70 ,.,.

-

1.I.1 O..

ew Z laJ

tlJ Z LII

0.02

-

20

0.01

-

10 I

i

t

lilJ

50

I

100

=

=

I

i

~l

0

000

INCIDENT ELECTRON ENERGY ( e V )

0.06

---,

90 e:

--80

20 I

1

100;

0.08

0

i

,,

110~

Fig. 3. Variation of the energy deposition efficiency for 21p state of helium and of the mean energy per ion pair with incident electron energy.

23p

>. 0.05 CJ Z

u2_ u_ tu

3. Results and discussion 0.04

Z

o_ I-.

0.03

~er" W

z 0.02 -I / uJ I / I/ O.Ol

/

_1 I I/ I/ u

0

7

I

20 INCIDENT

I

I I Ill,

I

I

I

I

I

I I

100 ELECTRON

1000 ENERGY

(eV)

Fig. 2. Variation of the energy deposition efficieneies for 21S and 23P states of helium with incident electron energy.

The results o f the excitation yields can be conveniently expressed as the energy deposition efficiency, P/, defined as the fraction o f the incident electron energy transferred to -the state/'. That is, P~ = (N//E0)N/, where A~ is the total number o f excitations to the state/'. Similarly, the results o f the ionization yields can be expressed as the mean energy expended per ion pair formed (w). The numerical results o f the yield calculations reveal that the contribution due to higher generation secondaries decreases so rapidly that one can neglect the contribution beyond the tertiary degradation. The energy deposition efficieneies for 21S and 23p states as a function o f incident electron energy are shown in fig. 2, and that for 21p state is shown in fig. 3. The variation o f w with incident electron energy is also shown in fig. 3. As expected, the w-value decreases rapidly with increasing incident

222

Dayashankar/Electron energy deposition in helium

energy and reaches a constant value o f 49 eV for incident energies higher than about 500 eV. This constant value is in good agreement with the experim e n t a l value o f 46 eV [15].

Acknowledgements Thanks are due to Drs. U. Madhvanath and G. Venkataraman for helpful discussions.

References [1] C. B. Opal, W. K. Peterson and E. C. Beaty, J. Chem. Phys. 55 (1971) 4100. [2] Dayashankar, Physica 81C (1976) 409. [3] W. F. Miller, Ph.D. Thesis, Purdue University, 1956. [4] L. R. Peterson and A. E. S. Green, J. Phys. B (Proe. Phys. Soc.) 1 (1968) 1131.

[5] L. R. Peterson, Phys. Rev. 187 (1969) 105. [6] B. L. Jhanwar and S. P. Khare, J. Phys. B (Atom. Molec. Phys.) 6 (1973) 462. [7] D. Rapp and P. Englander-Golden, J. Chem. Phys. 43 (1965) 1464. [81 M. Inokuti, Rev. Mod. Phys. 43 (1971) 297. [9] C. E. Moore, Atomic Energy Levels VoL I, Nat. Stand. Ref. Data Ser., Nat. Bur. Stand. (U.S.) 35/V.I (Dec. 1971). [10] J. K. Rice, D. G. Truhlar, D. C. Cartwright and S. Trajmar, Phys. Rev. A 5 (1972) 762. [ 11 ] L. Vriens, J. A. Simpson and S. R. Mielczarek, Phys. Rev. 165 (1968) 7. [12] K. L. Bell, D. J. Kennedy and A. E. Kingston, J. Phys. B (Atom. Molec. Phys.) 2 (1969) 26. [13] F. G. Donaldson, M. A. Hender and J. W. McConkey, J. Phys. B (Atom. Molec. Phys.) 5 (1972) 1192. [14] J. D. Jobe and R. M. St. John, Phys. Rev. 164 (1967) 117. [15] A. Dalgarno, Atomic and Molecular Processes, D. R. Bates, ed. (Academic Press, New York, 1962), ch. 15.