User-friendly model for the energy distribution of electrons from proton or electron collisions

0735-245X/89$3 00 + 0 00 Pergamon Press plc

Nucl Tracks Radtat Meas, Vol 16. No 2/3, pp 213-218, 1989 lnt J Radtat Appl Instrum, Part D Printed m Great Britain

USER-FRIENDLY MODEL FOR THE ENERGY DISTRIBUTION OF ELECTRONS FROM PROTON OR ELECTRON COLLISIONS M EUGENE RUDD

Department of Phymcs and Astronomy, Umverslty of Nebraska, Lincoln, NE 68588-011I, U.S.A (Recewed 20 October 1988) AMtract--A model is presented which g~vescross secuons (differential m the ejected secondary electron

energy) for the ionization of atoms and molecules m proton or electron colhmons. The model is m the form of an analytical equation which holds for all pnmary and secondary energnesand for any target gas for which certmn parameters are known The accuracy is estimated to be 15-20%. The model is based on the classmal binary encounter model modified to agree wRh the Bethe theory at high energies and, for proton Impact, w~th the molecular promotion model at low energies. For multi-shell targets, the partml cross sections for the shells are added The model equation may be integrated to ymld such quantmes as the total ~onlzattoncross sectmn, the stopping cross section due to ~omzaUon,and the average secondary electron energy For proton Impact the integrations must be done numerically or by usmg approximations For electron ~mpact the equation is simpler and may easily be analyncally integrated

1. INTRODUCTION IN ANY study involving the deposmon of energy by energeuc charged parucles traversing matter, the ejecuon of secondary electrons in collisions wlth mdw~dual atoms is of central interest, since it ~s the elementary process revolving the greatest energy transfer For large energy ranges it is also the most hkely process Besides a knowledge of the total cross sections for this process as a function of energy for a variety of targets, the user often needs to know the angular and energy distnbuuon of the ejected electrons Cross secuons that are differentml m the angle and secondary energy have been measured for proton impact starting in the early 1960s and for electron impact beginning in the early 1970s. While such doubly differential cross sections (DDCS) measured at different laboratories show a generally good agreement. there are ranges of parameters for which there are large discrepancies It is especmlly difficult to make accurate measurements at low primary energies (below about 30keV for protons, or 100eV for electrons) and at low secondary energies (below about 15 eV). Because of these discrepancies, a potential user ~s faced with the task of choosing among several experimental results. A further problem is that the measurements may not have been made at the reqmred energies or for the targets needed. Ab lnltto calculaUons of these DDCS or the singly differential cross secuons (SDCS) obtained by lntegrauon over angle have been made by a number of methods These include the classical binary encounter approximation (BEA) methods (Williams, 1927,

Thomas, 1927; Gryzinski, 1965; Vriens, 1967), the Born approximation (Kuyatt and Jorgensen, 1963; Madison, 1973; Rudd and Madison, 1976), and Monte Carlo methods (Bonsen and Banks, 1971; Olson and Salop, 1977; McKenzie and Olson, 1987). Not only is the accuracy of these methods limited, but they are also useful only at high impact energies, 1.e. energies for which the projectile velocity is much greater than the orbital electron velocity. In addmon, the Born approxlmaUon reqmres a knowledge of mmal- and final-state wave functions, and th~s mformauon Is not generally available, except for the s~mplest targets. Miller and co-workers (1983, 1987), and Inokuti et al. (1987) have developed semi-empirical models for the SDCS but these, too, are useful only at Mgh energms. In the present model for proton colhsions, SDCS are gwen by a simple analytic equauon contaimng three adjustable parameters. One of these is a dimensionless parameter near umty which is independent of proton energy but is shghtly different for different targets. The other two parameters are functions of the primary energy which have been fitted by equations with four and five target parameters, respectively. The equauon for electron impact is somewhat simpler, contammg only two adjustable parameters. The funcuons describing these two parameters require a total of only four target parameters. The SDCS equauons are based on the BEA equation gwen by Williams (1927). They have been modified m such a way that they agree with Bethe's well-known treatment of the Born approximation (see Inokutl, 1971) at high energies, and they gwe the

213

214

M E RUDD

correct dependence above the kinematic cutoff. For protons of low primary energies they have been further modified to agree with the molecular promotion model (Rudd, 1979). A prehmmary version of this model was given by Rudd (1987) and later presented in more detail (Rudd, 1988) The emphasis in the present paper is on the use of the model rather than its derivation (which may be found in the other papers)

_I",L

" "

,ev !

-,31 -41- . . . . . .

"~.. .~'~..

1 60

0 -<..

',oo ev i

-

2. THE M O D E L FOR P R O T O N I M P A C T The quantities of interest m specifying the SDCS are the incident proton energy Ep. the ejected secondary energy W, and the number N of electrons in the target atom or molecule with binding energy I It ,s convenient to define the electron velocityequivalent energy as T = Ep/2 where 2 is the ratio of the proton to the electron mass. The model equation is most conveniently expressed in terms of two dimensionless quantities, the reduced secondary energy w= W/I and the reduced projectile velocity v = ( T / I ) m. The SDCS is then

a(w)

S =

-

F,+F2 w

I (1 + w)3[l + e "l...... '"]

(1)

b -6t- . . . . . . . o 0 -

"t 420 1000

keV

-5

~ 0

230O W (eV)

FIG I. Energy distributions of secondary electrons from proton collisions with hydrogen molecules. The t0 and 100 keV data are from Rudd (1979), and the 1000 keY data are from Toburen and Wilson (1972) The lines are the fits of the model

S = 4 n a ~ N ( R / l ) a, where ao is the Bohr radius and R

is the rydberg of energy (13.6 eV). The quantity w, is the kinematic cutoff energy where the cross sectmn begins to fall off exponentially. This is given by wc = 4 v 2 - 2 v - R / 4 L F j , F2, and ~ are the three adjustable fitting parameters for an electron spectrum at a gwen proton energy. Figure 1 shows the fit of the model to experimental data for protons on hydrogen at three widely separated energies. Although the shape changes with energy, the model equation fits well at all energies. Using all known SDCS data for each target, values of Ft. F2, and ~ were determined by fitting the equation to the energy spectra. The parameter ~ was found to be nearly independent of proton energy. The other two parameters varied with Ep in a fairly consistent way as shown in the example in Fig. 2 for water vapor. It was found possible to fit the energy variation of these two parameters with equations which also yielded total cross sections in agreement with the recommended values of R u d d e t al. (1985). The equations for F~ and F: are each comblnaUons of low and high energy asymptouc forms

The ten target parameters A~ . . . E~, A 2 . D , , and ~, completely specify the cross sections at all combinations of primary proton and secondary electron energies. Tables of values of these parameters for five target gases are given m Table 1. Table 2 contains the binding energies for these targets, derived from data given by Lotz (1968) and Siegbahn et al. (1969). Figure 3 shows the values of F~ and F2 calculated from equation (2) for four different targets.

3. M U L T I - S H E L L TARGETS In this model, targets with more than one shell are treated by calculating separately the cross section contribuuons from each shell and then adding them together. Usually the outermost shell contributes most of the cross section while the deep inner shells

,

I

Fi = LI + Hi

and F 2 ==L 2 H 2 / ( L 2 + H , ) ,

where H t ---- A t ln(I +

vZ)/(v 2 + BI Iv:),

o 5'

'

T/I

H 2 -~ A2/V 2 + B2/v 4, L 1 = C l v ° . / ( l +E~v ID,*4))

and L 2 = C 2v 02

'

(2)

FiG. 2 Values of the fitting parameters F~ and F2 as a function of the impact energy for H ÷ + H20 collisions The points are f r o m fitting extmrimentil d a m o f B o l o r i z a d e h a n d Rudd (1986), and Toburen and Wilson (1977).

ENERGY

DISTRIBUTION

OF ELECTRONS

Table 1 Target parameters

AI B1 CI DI EI ,4, B: C, D,

!

H,

He

Kr

H.~O

N,

Inner shells

096 26 038 023 22 104 59 1 15 02 0 87

102 24 07 1 15 07 084 60 07 005 0 87

1 44 70 03 --08 05 1 57 50 149 -10 0 77

097 82 04 --03 038 104 173 076 004 0 64

1.05 6.8 07 --03 1 15 10 59 087 --06 0 71

1 25 05 20 20 30 I 1 1.3 10 03 0 62

c o n t r i b u t e little However, the cutoff energy depends on the binding energy a n d ts therefore d~fferent for d~fferent shells At m t e r m e d m t e p r i m a r y energtes a n d at secondary energies a b o v e the cutoff for the outermost shell, the tuner shell c o n t n b u n o n s m a y dominate This is s h o w n m Fig 4 for k r y p t o n where the 3d a n d 3p shells are d o m i n a n t at high energies. T h e fit fo~ water v a p o r at three energ2es xs s h o w n m F~g 5 In b o t h Figs 4 a n d 5 the q u a n n t y log Y i s plotted on the y-axis Y ts defined as the r a n o o f the m e a s u r e d or calculated cross sectmn by the R u t h e r f o r d cross secnon, the latter being given by fi R 4 n a ~ R " / T ( W + I) 2, Plotting Y instead of the cross sectmn is a well k n o w n p r o c e d u r e whtch reduces the large range of values o n the y-axis. Smctl~ speaking, a different set o f target parameters ~s needed for each shell, b u t w~th the data presently available ~t was not possible to d e t e r m i n e so m a n y parameters. However, since the m n e r shells c o n t r i b u t e relanvely httle m m o s t cases, the following procedure was used The target electrons were d~vlded into t~,o categories, the outer or n e a r - o u t e r shells a n d the deep tuner shells. A r b n r a n l y , these werc chosen on the bas~s o f w h e t h e r the b i n d i n g energies are less t h a n or greater t h a n twice t h a t of the o u t e r m o s t shell T h e target p a r a m e t e r s for the first g r o u p were all t a k e n to be the same as for the =

%r

I~ 2 ~--J

1 (eV)

2 2

154 246 140 14 7 27 5 94 3 217 12 6 14 7 18 4 32 2 540 15 6 16 9 18 7 37 3 410

4 2 2 10 6 2 2 2 2 2 2 2 2 2

4

i

"~,.~V2

f

o.o

;

,oo

T/l, FIG 3 Values of the parameters F~ and ~ for four targets as functions of the impact energy The solid line shows hehum, the long dashed line, mtrogen, the short dashed line. hydrogen, and the dash~tot llne. krypton

o u t e r m o s t shell a n d were d e t e r m i n e d by fitting to the experimental d a t a as described a b o v e T h e tuner shells were assumed to have a different set o f parameters which were the same for all inner shells o f all targets. Only a p p r o x i m a t e values o f thts set could be determined, b u t m m o s t cases th~s was sufficient smce they usually m a k e only a small c o n t r i b u t i o n . T h e suggested set o f tuner-shell p a r a m e t e r s Is also gwen in Table 1

4. INTEGRATION OF THE MODEL CROSS SECTIONS Since a n analytical expression ts available for the SDCS, it ~s a s~mple m a t t e r to calculate a n u m b e r o f related q u a n u t i e s whxch are useful in a variety o f apphcanons

i

I

H~- Kr 150 key

1~ 0 -" "~f" ."7 - ~ - * " t \ ~ -1 "

-2 N

z

0 LtU o _

.

,Tot QI

.------%-'-,'k-.

/

20

Table 2 Binding energies H, Is He It Kr4p Kr 4p Kr 4s Kr 3d Kr 3p HzO lb, H,O 2a I H,O lb, H:O la I HzO 01 ~ N: cr~2p N: n.2p N: c;°2s N, a~2~ N: VI~

215

~

~ I

1

I

I

3

'\ I

" "-~

6

i

. . . . .

O

I(3P

_o -1 -2

I

%1

HS,-Kr 2 0 0 0 keV I

I

I

I

I

0 (w/v) ~,2 FIG 4 Energy distributions of electrons from H + + Kr collisions The points are from experimental data, at 150keV from Cheng and Rudd (1988), and at 2000keV from Manson and Toburen (1977) The dashed hnes are contributions due to individual shells, and the solid lines are totals Y ~s the rauo of the cross section to the Rutherford cross section (see text)

216

M E RUDD

The total cross section for the ejection of electrons is given by

OOl, I

o_ = I

or(w)dw

, /

,,

",~,.H2(xI0)

-7 !

(3)

0

This integration is useful for normahzmg the SDCS because the total cross sections are known relatively well. All available experamental measurements of a for proton impact were reviewed (Rudd et a l , 1985), and recommended values for many targets were given. The stopping cross section due to ionization is given by asT = (12/R)

( w + l)a(w) dw

The average ejected electron energy is calculated from Way = (f'Icr

w a ( w ) dw

)

(5)

Electrons ejected with energies greater than the ionization potential (i.e. W > I) are able to cause further ionization. The fraction of electrons with W > I is obtained from

f;

fl= ( I / a )

a(w) dw

(6)

It is equally easy to obtain the fraction of secondaries with energies greater than any other given energy These integrations may be performed for proton impact eather by numerical methods or analytically (with approximations). Rudd (I988) gives an approximate equation for cr which is very accurate at high energies, fairly good at low energies, and has about a 2 5 % error at an intermediate energy A sirnilar approximate equation is given for a,, Values of o,, for three targets are given in Fig 6. along with measured values of stopping cross sections. Ionization is expected (Wilson, 1972) to contribute 78-85% of the total stopping cross section for hydrogen above 300 keV. As seen in the figure, this model gives values which are 80-85% of the total for H2 and 80-86% for He, in excellent agreement with expectations. In the case of water vapor, however, the model yields some values in excess of the totals. Since

00

!

i

I0 ~

i

~000

'

'

~

\'

0

Ep (keY) FiG. 6. Stoppmg cross secUon as a function of impact energy for protons m three different gases. The dashed lines show the calculations of the contribution due to ionization using the present model, and the solid lines show the total stopping cross sections, for H2 and He (Whaling, 1958) and for H20 (ICRU, 1970'1

the model is based firmly on experimental SDCS, this may indicate that the total stopping cross sections from the tables (ICRU, 1970) are too low. Figure 7 shows the average energy calculated for three targets. This quantity increases with primary energy until the reduced velocity v is approximately three, after which it levels off. It might be expected that the higher the impact energy, the larger the fraction of electrons ejected with energies greater than L However, as seen in Fig. 7, this fraction rises until v = 2, after which the fraction actually decreases slightly.

5. THE MODEL FOR ELECTRON IMPACT The molecular promotion mechanism which gave rise to the factor containing the exponential in the proton model is not present in the case of electron impact, thus simplifying the result The equation for electron impact is a(w) = (S/I)[F l + F2(I + w)]/(l + w)2.

(7)

An additional small change has been made, in that F2 ts multiplied by (l + w) instead of by w as in the proton case. The primary energy dependences of F~ and F 2 are also simpler. In terms of the reduced primary energy t = T/L they may be expressed as

I

keV

FI = A 1( 1 -- e - n~, ) l n ( t ) / t

,\

o.,:,.:,,7% 0.01~ u

i

0

(4)

(8)

and '

5

(w/v)~,2 FIG 5. Energy distributions of electrons from H + + H:O collisions. The points, are from experimental data, at 15 and 70 keV from Bolonzadeh and Rudd (1986), and at 1000 keV from Toburen and Wilson (1977). The lines are the calculated values from the model

= A:/(t + BD.

(9)

In these equations only four target parameters, At, B~, A2, and B2 are needed. As in the proton case, A~ is closely related to the optical oscillator strength, and in order for the expressions for the total cross section and the stopping cross section to agree with the

ENERGY DISTRIBUTION OF ELECTRONS

1.0

I

217

I

60 /

/

01 ~ / / . / / " .

...... : ........

~

:

W,, CeV)~

:4o 20 10

" ~ ' ~

~

5

I

,,

50

I

_

500

5000

Ep (keV) FIG 7 Values,calculated from the model, of the average secondary energy W,v and of the fractton f~ of secondary electrons with energies greater than the first tonizatson potenttal of the target for proton tmpact The sohd hne shows hydrogen, the long dashed hne, hehum and the short dashed hne, water vapor

corresponding Bethe expressions at high energies, we must have Ai + A2 = 2. F~gure 8 shows experimental data for helium by several investigators plotted as Y, the ratio of the cross section to the Rutherford cross section. Also shown are the calculations from the model for A~ =0.94, B I =0.28, A2 =0.70, and B2 =4.3. Equauon (7) is easily integrated without approximauons. In the case of electrons, the upper limit of

~.~i( eV 50 e-÷ He

÷

W is T - I which makes t - 1 the upper limit of w. Some of the results are

a_ = I

a(w) dw 0

= S ( t - 1)[Fj (t + 1) + 2tF2]/2t 2

a,~ = (I2/R)

=(SI/R)[F~(1

(10)

(w + l)a(w) dw

-

l/t)+

F21n(t)].

(11)

6. CONCLUSIONS

100 eV >lc 10

50

100 500 eV

o

z o W (eV)

soo

FiG 8 Energy spectra of secondary electrons from e- + He colhslons at three energies. The hnes are calculations from the present model. Experimental data + (Crooks, 1972), A (Shyn and Sharp, 1979); • (Goodrich, 1937), [] (Rudd and DuBo]s, 1977), O (Opal et al, 1972), x (Oda, 1975). Goodnch's data have been mult~phed by 1 5

A stmple analytical equation has been presented which yields the energy distribution of secondary electrons from colhslons with incident protons or electrons At any gwen primary energy, three adjustable parameters are needed for proton impact, and two for electron smpact. The pnmary energy dependence of these parameters is further described by equat)ons with nine target parameters for protons and four for electrons. Values of these parameters for several targets are gwen, which allow easy computauon of cross sections for any combination of primary and secondary energies. By numerical or analyucal integration of the model equations, one may calculate total ionization cross sections, stopping cross sections due to ionization, average secondary electron energy, and the fraction of electrons ejected with energies above any gwen value These models should be useful m situations involvmg the transfer of energy by fast charged particles during collisions w~th atoms or molecules.

2~8

M

E. R U D D

REFERENCES Bolonzadeh M A and Rudd M E (1986) Angular and energy dependence of cross sections for ejection of electrons from water vapor It 15-150keV proton impact Phys Ret' A33, 888-892 Bonsen T F M and Banks D. (1971) Angular distribution of electrons ejected by charged particles. III Classical treatment of forward ejection J Phys B Atom molec Phvs. 4, 706-714 Cheng Wen-qm and Rudd M E (1988) Dlfferentml cross sections for ejection of electrons from rare gases by 75-150keV protons Phvs Ret, A39, 2359-2366 Crooks G B (1972) Energy and angular distributions of scattered and elected electrons produced by 50-800 eV electrons m hehum Ph D dissertation (Umverslty of Nebraska), Lincoln, Nebraska, U.S A Goodrich M (1937) Electron scattering m hehum Phvs Rel 52, 259-266 Gryzmskl M (1965)Two-particle colhsions II Coulomb colhslons m the laboratory of coordinates Phys Rer 138, A322-A335 ICRU (1970) Linear energy transport ICRU Report No 16, p 32, International Commission on Radmtlon Umts and Measurements, Washington, DC, U S.A Inokutl M (1971) Inelastic collisions of fast charged particles with atoms and molecules~the Bethe theory revisited. Rev rood Phys 43, 297-347 lnokutl M., Dillon M. A , Miller J H and Omldvar K (1987) Analytic representaUon of secondary-electron spectra J chem Phys 87, 6967--6972 Kuyatt C E and Jorgensen T Jr (1963) Energy and angular dependence of the differential cross section for product~on of electrons by 50-I00 keV protons m hydrogen gas Phys. Rev. 130, 1444-1455 Lotz W (1968) Subshell binding energies of atoms and ~ons from hydrogen to zinc J, opt Sac Am 7, 915-921 Madison D H. (1973) Angular d~stribut~ons of electrons ejected from helium by proton ~mpact Phvs Rer A8, 2449-2455 Manson S T and Toburen L H. (I 977) Energy and angular d~stnbuuon of electrons elected from Kr by I MeV proton ~mpact ~omzat~on theory and experiment In lOth Int Conf Physws Electron Atom Colhstons, abstracts of papers, p 99. Comm~ssartat fi l'Energ~e Atomique, Pans McKenz~e, M L and Olson R. E (1987) Iomzat~on and charge exchange in multiply-charged ~on-helium colhs~ons at mtermedmte energies Phvs Rev A35, 2863-2868. Mdler J H , Toburen L. H. and Manson S T (1983) Differentml cross secuons for ~omzat~on of hehum, neon, and argon by h~gh-velooty ~ons Phys Rer A27, 1337-1344 MdlerJ H , W d s o n W E . , M a n s o n S T a n d R u d d M E (1987) D~fferentml cross sections for ~omzat~on of water vapor by h~gh-velooty bare ~ons and electrons J them Phys. 86, 157-162

Oda Nobuo (1975) Energy and angular dlsmbuuons of electrons from atoms and molecules b~ electron impact Radtat Res 64, 80-95 Olson R E and Salop A (1977) Charge-transfer anc~ ~mpact-~omzat~on cross sections for fully and partmlly stripped poslUve ions colhdlng with atomtc hydrogen Phys Rev AI6, 531-541 Opal C B , Beaty E C and Peterson W K. (1972) Tables of secondary-electron-production cross sections 4tam Data 4, 209-253 Rudd M. E (1979) Energy and angular distributions of secondary electrons from 5-100 keV proton colhsions with hydrogen and nitrogen molecules Phy~ Rev A20, 787-796 Rudd M E (1987) Singly differential cross sections for producmg secondary electrons from hydrogen gas by keV to MeV proton colhslons Radzat. Res 109, 1 11 Rudd M E (1988) Dlfferenual cross sections for secondary electron productton by proton impact Ph) ~ Ret' A38, 6129-6137 Rudd M E and Madison D. H. (1976) Comparison o1 expenmental and theoreucal electron ejection cross sections m hehum by proton impact from 5 to 100 keV Phys Ret' AI4, 128-136 Rudd M E and DuBols R D (1977) Absolute doubly d~fferentml cross sections for ejection of secondary electrons from gases by electron impact. I. 100-200 eV electrons on hehum. Phys Rev. AI6, 26-32 Rudd M E , Klm Y - K . , Madison D H and Gallagher J W. (1985) Electron production m proton colhsions total cross sections Rev mad. Phys. 57, 965-994 Shyn T W and Sharp W. E (1979) Doubly differential cross sections of secondary electrons ejected from gases by electron impact' 50-300 eV on harem. Phys. Rev AI9, 557-567 Slegbahn K , Nordhng C., Johansson G , Hedman J.. Heden P F , Hamrm K , G-ehus U , Bergmark T.. Werme L O , Manne R and Baer Y (1969) ESCA Apphedta Free Malecules, North-Holland, Amsterdam Thomas L H (1927) The effect of the orbital velocity of the electrons m heavy atoms and their stopping of alphaparucles Prac Camb phil. Sac 23, 713-716 Toburen K H and Wilson W E (1972) Distributions m energy and angle of electrons ejected from molecular hydrogen by 0 3-1 5MeV protons Phys Ret, AS, 247-256 Toburen L H. and Wilson W. E (1977) Energy and angular dlstnbuuons of electrons ejected from water vapor by 0 3-1 5 MeV protons. J chem Phys. 66, 5202-5213 Vnens K (1967) Binary-encounter proton-atom collmon theory Prac phys. Sac. 90, 935-944 Whaling W (1958) The energy loss of charged particles m matter In Handbuch der Phystk (Edited by Flugge S ), Vol XXIV, pp 193-217 Spnnger, Berhn Wflhams E J (1927) The passage of alpha-rays and betarays through matter. Nature 119, 489-490 Wdson W. E (1972) Stopping power partition and mean energy loss for energetic protons m hydrogen Rad~at Res 49, 36-50