PHYSICAL REVIEW A

VOLUME 42, NUMBER 11

Three-shell model for independent-electron

1

DECEMBER 1990

processes in heavy-ion —atom collisions

O. Heber, * R. L. Watson, G. Sampoll, and B. B. Bandong of Chemistry, Texas Ad'cM University, College Station, Texas 77843

Cyclotron Institute and Department

(Received 13 August 1990}

A model is developed for the treatment of multielectron ionization and exchange processes in fast ion-atom collisions under the condition that electron-electron interactions may be neglected. The model specifically takes into account ionization and capture from two electron shells of the target and one electron shell of the projectile. The predictions of the model for the case of ionization accompanied by loss of 1-4 electrons from the projectile are compared with experimental results for 40-MeV Ar + on Ar.

'

I. INTRODUCTION The independent-electron (IEA) is a approximation well-known method for describing multiple ionization in The utility of this model is a fast ion-atom collisions. result of its simplicity; the only information needed to calculate cross sections for multielectron processes is the single-electron probability p as a function of the collision impact parameter b. Under the assumption that the electron-electron interactions are negligible, binomial (or multinomial) statistics are then employed to calculate the finalprobabilities of producing various multielectron state configurations. In a binary collision between a projectile and a target, an electron (initially bound to either the target or the projectile) may be ionized, excited, or captured by one of the collision partners. Each of these processes is characterized by a different p (b) function. A variety of methods is available for the calculation of these functions, such as, for example, the classical trajectory Monte-Carlo (CTMC) method, the semiclassical approximation, and the Vlasov equation. Obviously the IEA cannot be expected to give reasonable results when electron-electron interactions are important. Indeed, because of this fact, the IEA provides a basis for deciding whether or not experimental data reflect effects associated with additional mechanisms. In many instances, such interactions only affect a few of the electrons in the collision system while the rest act inIn these cases, it is often possible to apply dependently. the IEA and treat the correlation effects separately. From the experimental data that has so far been collected on multiply ionizing collisions, it appears that the IEA works rather well over the range 0. 5 &E/A /Z where E is the projectile energy in MeV, A is its mass in A characteristic sigamu, and Z is its atomic number. nature of independent-electron behavior is a binomial probability distribution. Examples include Eo. x-ray satellite spectra resulting from multiple L-vacancy production in E-shell ionizing collisions, and recoil-ion chargestate distributions produced in coincidence with electron capture to the projectile. The IEA has most often been applied to problems involving outer-shell pure ionization, but it is also extremely useful for the analysis of much more complicated cases involving both ionization

'

(5,

42

and capture of electrons from several target shells. The object of the present work was to explore the applicability of IEA to collisions involving target ionization accompanied by electron loss from the projectile. A three-shell model is presented and simplified formulas are developed for the cases of (a) pure ionization, (b) ionization accompanied by capture to the projectile, and (c) ionization accompanied by loss from the projectile. Predictions of the model for case (c) are compared with measured charge distributions of Ar recoil ions produced in collisions with 1-MeV/arnu Ar + projectiles.

II. FORMALISM Consider a collision between a projectile ion and a target atom which results in the removal of s electrons from a particular shell of the target that contains S electrons. Two mechanisms for electron removal are of specific interest here; namely, direct ionization and electron capture. Let the probability of ionizing a single electron from electron shell S at impact parameter b be pst(b), and the probability of capturing a single electron from electron shell S (by the collision partner) be psc(b). According to the IEA, the total probability of removing s out of S electrons via all combinations of ionization and capture is

Ps, (b) =

S

S'st(b)+use(b)''j'( I &st(b)

p'sc(b)

I' '

Now let the total probability for removal of s electrons from the S shell, with t of them being captured by the collision partner, be Ps„(b) Then it follows . that

Ps, (b) =

g Ps„(b)

t=0

and applying the binomial brackets in Eq. (l),

theorem

to the first item in

s

lP»(b)+Psc(b))'=X

I=O .

t Psc(b)P;

(b),

(3)

results in the expression 6466

1990

The American Physical Society

THREE-SHELL MODEL FOR INDEPENDENT-ELECTRON

42

S Ps„(b)=

need to be included.

s

The applications to be considered here involve collisions in which the ionization and exchange (capture) of electrons are restricted to two target-atom shells and one projectile shell. A schematic drawing illustrating the processes of interest is shown in Fig. 1. Consider, for example, collisions of projectile ions having Q electrons in their outer shells with target atoms having two active shells of electrons the L shell with L electrons and the M shell with M electrons. The cross section for a multielectron process where I, m, and q electrons are removed from the L, M, and Q, shells, respectively, of which h L electrons and i M electrons are captured by the projectile, and Q electrons are captured by the target, is

—

j

o(L!h, M~;, QqJ

)=2'

I

0

Pl, „(b)PM~;(b)Pgq~(b)b db, (5)

where each of the

Ps„is given

by Eq. (4).

A. Pure ionization

Strictly speaking, the process labeled "pure ionization" includes only events for which h =i = =0. Experimentally, however, this term is generally applied to all events that leave the charge of the projectile unchanged, including those for which q =h +i. Thus a complete calculation of the cross section for pure ionization requires the summation of o(L&I, , M for ) over all combinations which q — h — i =0 and I + m — = n, where n is the final charge of the (initially neutral} target atom. As an example, consider the case of pure ionization of a neutral Ar atom, leaving it with a final charge of 6, by a projectile having four electrons in its outer shell (i.e. , L = 8, M =8, n =6, and Q =4). An all inclusive calculation of the cross section for this process would be a formidable task requiring 290 terms. Fortunately, several approximations can be applied to greatly reduce the number of terms that

j

„Q

j

.

The first approximation

of projectile electrons by the target

j

neglected. Whenever electrons are removed from inner shells, it is important to take into account vacancy multiplication via Auger decay. Since each Auger decay increases the final charge by one unit, additional terms for which l +m (n will contribute to the cross section for each n. For example, in the case under consideration, the total cross section for producing a target ion having charge 6+ by pure ionization is given by the sum of seven terms [Eq. (5)], o (I, m), for (I, m) =(0, 6), (1,5), (2,4), (3,3), (4,2), (5, 1), and (6,0). To account for Auger decay, each of these terms must be multiplied by the probability that all of the L vacancies will decay radiatively, which is approximately coL, where coL is the L-shell fluorescence yield. In addition, the following terms must be included in the calculation:

(1 —coL )o(1,4}+(1—coL

I'Ql

6467

~

is to neglect [i.e., let p&c(b)=0] Generally, this should be a good assumption for neutral targets and it eliminates all terms except those =0, leaving 65 terms remaining to be evalufor which ated. Another level of simplification is achieved by neglecting the terms for simultaneous capture and loss by the projectile. This reduces the example calculation to seven terms (for which q =h =i =0) and is usually justified by the fact that the cross-section terms for simultaneous projectile ionization and capture involve complicated products of pgr pgc pr. r Jplc pMr and pMc [see Eqs. (4} and (5)]. Therefore, when any of these probabilities is small (as must be the case for pLc and p~c when pure ionization is dominant), such terms become negligible relative to the term for t =0 or s. The electron-capture probability is a sensitive function of the projectile/electron velocity ratio. When the two target-atom shells have greatly different average electron velocities, the probability of capture from the shell whose electrons most closely match the velocity of the projectile is usually much larger than the probability of capture from the other shell. In the present formulation, it shall be assumed that the velocity regime is such that capture from the target L shell predominates and P~c can be

capture

(4)

~

)

+(1 —coL ) 0(3, 0)+coL(1 +coL(1 — col )0 (3, 2), where 1 — col is the probability

o(2, 2)+col (1 —col coL)

0(3,

)0 (2, 3)

1)—

that a single L vacancy

will undergo Auger decay.

I'MC

PQc

target

projectile

FIG. 1. A schematic diagram of the electron ionization and exchange processes included in the three-shell model.

The above procedure becomes particularly simple for light elements having Z=18 or less, where the fluorescence yields are small enough to be neglected at the level of approximation being discussed here. When this situation applies, only the terms for which 21+m =n are nonzero. However, it must be recognized that the Auger decay of electron configurations for which n ~ M is limited by the fact that not enough M electrons are left to fill all the L vacancies via Auger transitions. This condition can be imposed by including only terms for which 2l + m = n ', where n ' = 2n — M and 2n —M + 1. Thus, within the framework of the above approximations, the total cross section for producing a target ion having

O. HEBER, R. L. WATSON, G. SAMPOLL, AND B. B. BANDONG

6468

charge n by pure ionization is given by

o„(P.I) =2m

g

21+m =a

f

PLIO(b)PM

where

db,

o(b)P(200(b)b

(6) Pinko(b)

where

when n

~M

and

L

~LIo»= PM~0(b)

b p'I. I(— ~L, c b

I pLr b [

M

=

p~l(b)[

Pgoa(b) = [1 — pal(b)]2

1

]'

'

.

To obtain the cross section for producing target ions of charge n via collisions that reduce the projectile charge by k units, Eq. (5) must be summed over all combinations i =— k. This again results in a 1arge for which q — h — number of terms even for modest values of n. To greatly reduce the work involved, the same approximations as were discussed above for pure ionization genera11y may be applied. The desired cross section is then given by

g

2l+m =a

f

pL&k(b)pM

0(b)

X Phoo(b)b db

The IEA has been applied previously to the analysis of multiple electron loss from fast projectiles by Meyerhof et al. In that work, however, the charge-state distributions of the recoil ions were not measured. In this section, the IEA formulation developed in Sec. II C is used to predict cross sections for the production of the coincident projectile and target charge states. The data to be discussed here were obtained by passing a 40-MeV Ar + beam through a dift'erentially pumped gas cell containing 1 mTorr of Ar. The exit charge state of the projectile was selected by means of a magnet located 1 m downstream from the gas cell. The Ar recoil ions produced in the gas cell were extracted by an electric 6eld and their charges were determined by time-of-flight analysis. Further experimental details are given in Ref. 12. The experimental cross sections for Ar recoil-ion production accompanied by the loss of 0 —4 projectile electrons are shown in Fig. 2. Similar results have been published recently by Tonuma et al. ' The IEA analysis of this problem was performed using Eq. (8). In previous studies of pure ionization and elec+, it has been shown that the tron capture by 16-MeV one-electron probabilities can be adequately represented by the simple exponential function

0

(7)

where

p (b) =p (0)exp(

L PLIk(b)

I k PLC(b)pLI

I r

X [1 pLI(b)

C(b)]—

PI—

The above equation recently has been applied to the analysis of the charge distribution of Ar recoil ions produced in one- and two-electron capture collisions by 1' The MeV/amu Oq agreement (q =3 —8) projectiles. with experiment was found to be quite satisfactory.

C. Ionization accompanied by loss from the projectile The objective here is to calculate the cross section for producing target ions of charge n via collisions that increase the projectile charge by k units. A complete calculation must include all cross-section terms for which —h —i =k. Applying the same approximations as in q Sec. II A results in the expression

„(k,loss) =2~ y f 2l+m =a

PL!o(b)PMmo(b)PQko(b)b db

b

''

—

lr),

where p (0) is the one-electron probability at b =0 and r is a constant. In the present analysis, employment of this function form to represent the various one-electron prob-

pure ionization ~ 1 I tron loss v-— i on loss v ron loss &&--— ron loss +----

10 E

-~6

o

10

O

10 17

C3

LJ

o

(b)[1 Pgl(b)]

"

pMI— (b)]

accompanied by capture to the projectile

capture)=2m o „(k,

k Pg,

III. ANALYSIS OF CHARGE-STATE DISTRIBUTION FOR COLLISIONS INVOLVING ELECTRON LOSS FROM THE PROJECTILE

The above formulas have been applied to the analysis of cross sections for pure ionization of Ar by a wide range of oxygen and fluorine projectile charge states at 1 MeV/amu. They were found to give very good agreement with the experimental data.

B. Ionization

=

The above formula previously has not been tested, and hence it is of interest to compare its predictions with experimental data for recoil-ion production in coincidence with electron loss from the projectile.

(M

when n

a=' nn'

42

V

-&e 10

Vl

10

.

0

+

V

0

8

9

10

0

1

2

3

4

5

6

7

RECOIL — ION CHARGE

FIT+. 2. Comparison of the experimental cross sections for ionization accompanied by loss from the projectile with the predictions of the three-shell model.

42

THREE-SHELL MODEL FOR INDEPENDENT-ELECTRON

abilities needed requires six parameters. Since the target ionization in this case takes place at fairly large impact parameters, the required one-electron probabilities depend primarily on the charge of the projectile and are relatively insensitive to the projectile Z (for example, see Ref. 15). Hence, the radius parameters for pLI and ps', determined in Refs. 9 and 10 by fitting the IEA expression for the total n =1 pure ionization and one-electron capture cross sections to the experimental cross sections, and the p (0) parameters, which were calculated by the CTMC method, may be applied to the present analysis. The remaining two parameters for p&l(b) were determined by fitting the experimental total-loss cross section (summed over all recoil charges) for q = 1 and 2. The results of the model calculations are shown by the lines in Fig. 2. It is evident that the agreement with the experimental data is reasonable good for the one- and two-electron-loss cases (q =1 and 2). In this regard, however, it should be recognized that the procedure used to extract the parameters for the one-electron probabilities assures that the absolute magnitudes of the calculated q =1 and 2 cross sections summed over all recoil charges will correspond to the experimental total cross sections. Therefore, it is more valid to judge these results on the basis of how well they reproduce the relative shapes of the charge-state distributions. It is apparent that the shapes of the recoil-ion charge-state distributions are represented reasonably well for all the cases shown =0-4). (q Unfortunately, the absolute cross sections for the q = 3 and 4 cases are considerably underestimated by the model. The most likely cause of this is the neglect of ionization of the projectile L shell. An approximate calculation, which included projectile L shell ionization followed by Auger decay, was performed using the same L shell parameters as were used for the Ar target. The resulting contribution to the total cross section for q =3 (summed over all recoil-ion charges) from the ionization of one Mshell electron plus the ionization of one L-shell electron plus the ejection of one Auger electron was 2.9X10 cm, and the contribution to the total cross section for Q =4 from two M-shell plus one L-shell plus one Auger electron was 3. 6X10 ' cm . These two numbers were multiplied by the recoil-ion charge-state fractions and added to the cross sections calculated for shell ionization only, giving the results shown by the lines in Fig. 3. It is evident that this procedure has improved substantially the model predictions of the absolute cross sections, thereby demonstrating the importance of inner-shell contributions to the projectile electron-loss cross sections in this case and providing motivation for an all-inclusive four-shell calculation in the future.

I

address: Weizmann Institute of Science, Rehovot 76100, Israel. tAlso at Lawrence Livermore Laboratory, Livermore, CA 94550. J. H. McGuire and L. Weaver, Phys. Rev. A 16, 41 (1977).

Present

1

..

6469

~

0-14

10-16 K

0

I

I

pure ionization 1 electron loss ron loss ron loss ron loss

15

-17

O

-&8

Vl

10

O

10 19

10

'0 0

o

d I

I

I

I

2

3

4

5

I

6

7

8

I

I

9

10

11

RECOIL-ION CHARGE

FIG. 3. Comparison of the experimental cross sections for ionization accompanied by loss from the projectile with the results of a model calculation modified to include loss from the projectile L shell.

IV. CONCLUSION A three-shell model been for has developed independent-electron processes involving electron ionization and exchange in fast ion-atom collisions. A number of simplifying assumptions were applied to considerably reduce the effort required to compare the predictions of the model with experiment. Formulas were given specifically for the cases of pure ionization, ionization accompanied by capture to the projectile, and ionization accompanied by loss from the projectile. An analysis of experimental data for 40-MeV Ar + on Ar was performed for the case of ionization accompanied by the loss of 1-4 electrons from the projectile. The model was found to reproduce the shape of the recoil-ion charge-state distributions reasonably well, considering the complexity of the multielectron ionization and exchange process being treated. The absolute cross sections calculated for three- and four-electron loss, on the other hand, were underestimated by as much as an order of The majority of this discrepancy was asmagnitude. cribed to the neglect of projectile L-shell ionization. A rough four-shell calculation resulted in a considerable improvement in the absolute-cross-section comparison. ACKNOWLEDGMENTS

This work was supported by the Division of Chemical Sciences of the U. S. Department of Energy and the Robert A. Welch Foundation.

V. A. Sidorvich, J. Phys. B 14, 4805 (1981). 3R. E. Olson, J. Phys. B 92, 1843 (1979). 4I. Ben-Itzhak, T. J. Gray, J. C. Legg, and J. H. McGuire, Phys. Rev. A 37, 3685 (1988). 5M. Horbatsh, Z. Phys. D 1, 337 (1986).

6470

O. HEBER, R. L. WATSON, G. SAMPOLL, AND B. B. BANDONG

R. E. Olsen, in E/ectronic and Atomic Collisions, edited B. Gilbody, W. R. Newell, F. H. Read, and A. C. H.

by H. Smith

(North-Holland, Amsterdam, 1988), p. 271. ~R. L. Kauffman, J. H. McGuire, P. Richard, and C. F. Moore, Phys. Rev. A 8, 1233 (1973). A. Muller, B. Schuch, W. Groh, E. Salzborn, H. F. Beyer, and P. H. Mokler, Phys. Rev. A 33, 3010 (1986). O. Heber, G. Sampoll, B. B. Bandong, R. J. Maurer, E. Moler, R. L. Watson, I. Ben-Itzhak, J. L. Shinpaugh, J. M. Sanders, L. Hefner, and P. Richard, Phys. Rev. A 39, 4898 (1989). O. Heber, G. Sampoll, B. B. Bandong, and R. L. Watson, Phys. Rev. A 40, 5601 (1989). W. E. Meyerhof, R. Anholt, X. Y. Xu, H. Gould, B. Feinberg,

42

R. J. McDonald, H. E. Wegner, and P. Thieberger, Phys. Rev. A 35, 1967 (1987); Nucl. Instrum. Methods A 262, 10 (1987). R. J. Maurer, C. Can, and R. L. Watson, Nucl. Instrum. Methods B 27, 512 (1987); O. Heber, G. Sampoll, R. J. Maurer, B. B. Bandong, and R. L. Watson, ibid. 40, 197 (1989). T. Tonuma, H. Kumagai, T. Matsuo, and H. Tawara, Phys. Rev. A 40, 6238 (1990). R. D. DuBois and S. T. Manson, Phys. Rev. A 35, 2007 (1987). T. Tonoma, H. Shibata, S. H. Be, H. Kumagai, M. Kase, T. Kambara, I. Kohno, A. Ohsaki, and H. Tawara, Phys. Rev. A 33, 3047 (1986).

VOLUME 42, NUMBER 11

Three-shell model for independent-electron

1

DECEMBER 1990

processes in heavy-ion —atom collisions

O. Heber, * R. L. Watson, G. Sampoll, and B. B. Bandong of Chemistry, Texas Ad'cM University, College Station, Texas 77843

Cyclotron Institute and Department

(Received 13 August 1990}

A model is developed for the treatment of multielectron ionization and exchange processes in fast ion-atom collisions under the condition that electron-electron interactions may be neglected. The model specifically takes into account ionization and capture from two electron shells of the target and one electron shell of the projectile. The predictions of the model for the case of ionization accompanied by loss of 1-4 electrons from the projectile are compared with experimental results for 40-MeV Ar + on Ar.

'

I. INTRODUCTION The independent-electron (IEA) is a approximation well-known method for describing multiple ionization in The utility of this model is a fast ion-atom collisions. result of its simplicity; the only information needed to calculate cross sections for multielectron processes is the single-electron probability p as a function of the collision impact parameter b. Under the assumption that the electron-electron interactions are negligible, binomial (or multinomial) statistics are then employed to calculate the finalprobabilities of producing various multielectron state configurations. In a binary collision between a projectile and a target, an electron (initially bound to either the target or the projectile) may be ionized, excited, or captured by one of the collision partners. Each of these processes is characterized by a different p (b) function. A variety of methods is available for the calculation of these functions, such as, for example, the classical trajectory Monte-Carlo (CTMC) method, the semiclassical approximation, and the Vlasov equation. Obviously the IEA cannot be expected to give reasonable results when electron-electron interactions are important. Indeed, because of this fact, the IEA provides a basis for deciding whether or not experimental data reflect effects associated with additional mechanisms. In many instances, such interactions only affect a few of the electrons in the collision system while the rest act inIn these cases, it is often possible to apply dependently. the IEA and treat the correlation effects separately. From the experimental data that has so far been collected on multiply ionizing collisions, it appears that the IEA works rather well over the range 0. 5 &E/A /Z where E is the projectile energy in MeV, A is its mass in A characteristic sigamu, and Z is its atomic number. nature of independent-electron behavior is a binomial probability distribution. Examples include Eo. x-ray satellite spectra resulting from multiple L-vacancy production in E-shell ionizing collisions, and recoil-ion chargestate distributions produced in coincidence with electron capture to the projectile. The IEA has most often been applied to problems involving outer-shell pure ionization, but it is also extremely useful for the analysis of much more complicated cases involving both ionization

'

(5,

42

and capture of electrons from several target shells. The object of the present work was to explore the applicability of IEA to collisions involving target ionization accompanied by electron loss from the projectile. A three-shell model is presented and simplified formulas are developed for the cases of (a) pure ionization, (b) ionization accompanied by capture to the projectile, and (c) ionization accompanied by loss from the projectile. Predictions of the model for case (c) are compared with measured charge distributions of Ar recoil ions produced in collisions with 1-MeV/arnu Ar + projectiles.

II. FORMALISM Consider a collision between a projectile ion and a target atom which results in the removal of s electrons from a particular shell of the target that contains S electrons. Two mechanisms for electron removal are of specific interest here; namely, direct ionization and electron capture. Let the probability of ionizing a single electron from electron shell S at impact parameter b be pst(b), and the probability of capturing a single electron from electron shell S (by the collision partner) be psc(b). According to the IEA, the total probability of removing s out of S electrons via all combinations of ionization and capture is

Ps, (b) =

S

S'st(b)+use(b)''j'( I &st(b)

p'sc(b)

I' '

Now let the total probability for removal of s electrons from the S shell, with t of them being captured by the collision partner, be Ps„(b) Then it follows . that

Ps, (b) =

g Ps„(b)

t=0

and applying the binomial brackets in Eq. (l),

theorem

to the first item in

s

lP»(b)+Psc(b))'=X

I=O .

t Psc(b)P;

(b),

(3)

results in the expression 6466

1990

The American Physical Society

THREE-SHELL MODEL FOR INDEPENDENT-ELECTRON

42

S Ps„(b)=

need to be included.

s

The applications to be considered here involve collisions in which the ionization and exchange (capture) of electrons are restricted to two target-atom shells and one projectile shell. A schematic drawing illustrating the processes of interest is shown in Fig. 1. Consider, for example, collisions of projectile ions having Q electrons in their outer shells with target atoms having two active shells of electrons the L shell with L electrons and the M shell with M electrons. The cross section for a multielectron process where I, m, and q electrons are removed from the L, M, and Q, shells, respectively, of which h L electrons and i M electrons are captured by the projectile, and Q electrons are captured by the target, is

—

j

o(L!h, M~;, QqJ

)=2'

I

0

Pl, „(b)PM~;(b)Pgq~(b)b db, (5)

where each of the

Ps„is given

by Eq. (4).

A. Pure ionization

Strictly speaking, the process labeled "pure ionization" includes only events for which h =i = =0. Experimentally, however, this term is generally applied to all events that leave the charge of the projectile unchanged, including those for which q =h +i. Thus a complete calculation of the cross section for pure ionization requires the summation of o(L&I, , M for ) over all combinations which q — h — i =0 and I + m — = n, where n is the final charge of the (initially neutral} target atom. As an example, consider the case of pure ionization of a neutral Ar atom, leaving it with a final charge of 6, by a projectile having four electrons in its outer shell (i.e. , L = 8, M =8, n =6, and Q =4). An all inclusive calculation of the cross section for this process would be a formidable task requiring 290 terms. Fortunately, several approximations can be applied to greatly reduce the number of terms that

j

„Q

j

.

The first approximation

of projectile electrons by the target

j

neglected. Whenever electrons are removed from inner shells, it is important to take into account vacancy multiplication via Auger decay. Since each Auger decay increases the final charge by one unit, additional terms for which l +m (n will contribute to the cross section for each n. For example, in the case under consideration, the total cross section for producing a target ion having charge 6+ by pure ionization is given by the sum of seven terms [Eq. (5)], o (I, m), for (I, m) =(0, 6), (1,5), (2,4), (3,3), (4,2), (5, 1), and (6,0). To account for Auger decay, each of these terms must be multiplied by the probability that all of the L vacancies will decay radiatively, which is approximately coL, where coL is the L-shell fluorescence yield. In addition, the following terms must be included in the calculation:

(1 —coL )o(1,4}+(1—coL

I'Ql

6467

~

is to neglect [i.e., let p&c(b)=0] Generally, this should be a good assumption for neutral targets and it eliminates all terms except those =0, leaving 65 terms remaining to be evalufor which ated. Another level of simplification is achieved by neglecting the terms for simultaneous capture and loss by the projectile. This reduces the example calculation to seven terms (for which q =h =i =0) and is usually justified by the fact that the cross-section terms for simultaneous projectile ionization and capture involve complicated products of pgr pgc pr. r Jplc pMr and pMc [see Eqs. (4} and (5)]. Therefore, when any of these probabilities is small (as must be the case for pLc and p~c when pure ionization is dominant), such terms become negligible relative to the term for t =0 or s. The electron-capture probability is a sensitive function of the projectile/electron velocity ratio. When the two target-atom shells have greatly different average electron velocities, the probability of capture from the shell whose electrons most closely match the velocity of the projectile is usually much larger than the probability of capture from the other shell. In the present formulation, it shall be assumed that the velocity regime is such that capture from the target L shell predominates and P~c can be

capture

(4)

~

)

+(1 —coL ) 0(3, 0)+coL(1 +coL(1 — col )0 (3, 2), where 1 — col is the probability

o(2, 2)+col (1 —col coL)

0(3,

)0 (2, 3)

1)—

that a single L vacancy

will undergo Auger decay.

I'MC

PQc

target

projectile

FIG. 1. A schematic diagram of the electron ionization and exchange processes included in the three-shell model.

The above procedure becomes particularly simple for light elements having Z=18 or less, where the fluorescence yields are small enough to be neglected at the level of approximation being discussed here. When this situation applies, only the terms for which 21+m =n are nonzero. However, it must be recognized that the Auger decay of electron configurations for which n ~ M is limited by the fact that not enough M electrons are left to fill all the L vacancies via Auger transitions. This condition can be imposed by including only terms for which 2l + m = n ', where n ' = 2n — M and 2n —M + 1. Thus, within the framework of the above approximations, the total cross section for producing a target ion having

O. HEBER, R. L. WATSON, G. SAMPOLL, AND B. B. BANDONG

6468

charge n by pure ionization is given by

o„(P.I) =2m

g

21+m =a

f

PLIO(b)PM

where

db,

o(b)P(200(b)b

(6) Pinko(b)

where

when n

~M

and

L

~LIo»= PM~0(b)

b p'I. I(— ~L, c b

I pLr b [

M

=

p~l(b)[

Pgoa(b) = [1 — pal(b)]2

1

]'

'

.

To obtain the cross section for producing target ions of charge n via collisions that reduce the projectile charge by k units, Eq. (5) must be summed over all combinations i =— k. This again results in a 1arge for which q — h — number of terms even for modest values of n. To greatly reduce the work involved, the same approximations as were discussed above for pure ionization genera11y may be applied. The desired cross section is then given by

g

2l+m =a

f

pL&k(b)pM

0(b)

X Phoo(b)b db

The IEA has been applied previously to the analysis of multiple electron loss from fast projectiles by Meyerhof et al. In that work, however, the charge-state distributions of the recoil ions were not measured. In this section, the IEA formulation developed in Sec. II C is used to predict cross sections for the production of the coincident projectile and target charge states. The data to be discussed here were obtained by passing a 40-MeV Ar + beam through a dift'erentially pumped gas cell containing 1 mTorr of Ar. The exit charge state of the projectile was selected by means of a magnet located 1 m downstream from the gas cell. The Ar recoil ions produced in the gas cell were extracted by an electric 6eld and their charges were determined by time-of-flight analysis. Further experimental details are given in Ref. 12. The experimental cross sections for Ar recoil-ion production accompanied by the loss of 0 —4 projectile electrons are shown in Fig. 2. Similar results have been published recently by Tonuma et al. ' The IEA analysis of this problem was performed using Eq. (8). In previous studies of pure ionization and elec+, it has been shown that the tron capture by 16-MeV one-electron probabilities can be adequately represented by the simple exponential function

0

(7)

where

p (b) =p (0)exp(

L PLIk(b)

I k PLC(b)pLI

I r

X [1 pLI(b)

C(b)]—

PI—

The above equation recently has been applied to the analysis of the charge distribution of Ar recoil ions produced in one- and two-electron capture collisions by 1' The MeV/amu Oq agreement (q =3 —8) projectiles. with experiment was found to be quite satisfactory.

C. Ionization accompanied by loss from the projectile The objective here is to calculate the cross section for producing target ions of charge n via collisions that increase the projectile charge by k units. A complete calculation must include all cross-section terms for which —h —i =k. Applying the same approximations as in q Sec. II A results in the expression

„(k,loss) =2~ y f 2l+m =a

PL!o(b)PMmo(b)PQko(b)b db

b

''

—

lr),

where p (0) is the one-electron probability at b =0 and r is a constant. In the present analysis, employment of this function form to represent the various one-electron prob-

pure ionization ~ 1 I tron loss v-— i on loss v ron loss &&--— ron loss +----

10 E

-~6

o

10

O

10 17

C3

LJ

o

(b)[1 Pgl(b)]

"

pMI— (b)]

accompanied by capture to the projectile

capture)=2m o „(k,

k Pg,

III. ANALYSIS OF CHARGE-STATE DISTRIBUTION FOR COLLISIONS INVOLVING ELECTRON LOSS FROM THE PROJECTILE

The above formulas have been applied to the analysis of cross sections for pure ionization of Ar by a wide range of oxygen and fluorine projectile charge states at 1 MeV/amu. They were found to give very good agreement with the experimental data.

B. Ionization

=

The above formula previously has not been tested, and hence it is of interest to compare its predictions with experimental data for recoil-ion production in coincidence with electron loss from the projectile.

(M

when n

a=' nn'

42

V

-&e 10

Vl

10

.

0

+

V

0

8

9

10

0

1

2

3

4

5

6

7

RECOIL — ION CHARGE

FIT+. 2. Comparison of the experimental cross sections for ionization accompanied by loss from the projectile with the predictions of the three-shell model.

42

THREE-SHELL MODEL FOR INDEPENDENT-ELECTRON

abilities needed requires six parameters. Since the target ionization in this case takes place at fairly large impact parameters, the required one-electron probabilities depend primarily on the charge of the projectile and are relatively insensitive to the projectile Z (for example, see Ref. 15). Hence, the radius parameters for pLI and ps', determined in Refs. 9 and 10 by fitting the IEA expression for the total n =1 pure ionization and one-electron capture cross sections to the experimental cross sections, and the p (0) parameters, which were calculated by the CTMC method, may be applied to the present analysis. The remaining two parameters for p&l(b) were determined by fitting the experimental total-loss cross section (summed over all recoil charges) for q = 1 and 2. The results of the model calculations are shown by the lines in Fig. 2. It is evident that the agreement with the experimental data is reasonable good for the one- and two-electron-loss cases (q =1 and 2). In this regard, however, it should be recognized that the procedure used to extract the parameters for the one-electron probabilities assures that the absolute magnitudes of the calculated q =1 and 2 cross sections summed over all recoil charges will correspond to the experimental total cross sections. Therefore, it is more valid to judge these results on the basis of how well they reproduce the relative shapes of the charge-state distributions. It is apparent that the shapes of the recoil-ion charge-state distributions are represented reasonably well for all the cases shown =0-4). (q Unfortunately, the absolute cross sections for the q = 3 and 4 cases are considerably underestimated by the model. The most likely cause of this is the neglect of ionization of the projectile L shell. An approximate calculation, which included projectile L shell ionization followed by Auger decay, was performed using the same L shell parameters as were used for the Ar target. The resulting contribution to the total cross section for q =3 (summed over all recoil-ion charges) from the ionization of one Mshell electron plus the ionization of one L-shell electron plus the ejection of one Auger electron was 2.9X10 cm, and the contribution to the total cross section for Q =4 from two M-shell plus one L-shell plus one Auger electron was 3. 6X10 ' cm . These two numbers were multiplied by the recoil-ion charge-state fractions and added to the cross sections calculated for shell ionization only, giving the results shown by the lines in Fig. 3. It is evident that this procedure has improved substantially the model predictions of the absolute cross sections, thereby demonstrating the importance of inner-shell contributions to the projectile electron-loss cross sections in this case and providing motivation for an all-inclusive four-shell calculation in the future.

I

address: Weizmann Institute of Science, Rehovot 76100, Israel. tAlso at Lawrence Livermore Laboratory, Livermore, CA 94550. J. H. McGuire and L. Weaver, Phys. Rev. A 16, 41 (1977).

Present

1

..

6469

~

0-14

10-16 K

0

I

I

pure ionization 1 electron loss ron loss ron loss ron loss

15

-17

O

-&8

Vl

10

O

10 19

10

'0 0

o

d I

I

I

I

2

3

4

5

I

6

7

8

I

I

9

10

11

RECOIL-ION CHARGE

FIG. 3. Comparison of the experimental cross sections for ionization accompanied by loss from the projectile with the results of a model calculation modified to include loss from the projectile L shell.

IV. CONCLUSION A three-shell model been for has developed independent-electron processes involving electron ionization and exchange in fast ion-atom collisions. A number of simplifying assumptions were applied to considerably reduce the effort required to compare the predictions of the model with experiment. Formulas were given specifically for the cases of pure ionization, ionization accompanied by capture to the projectile, and ionization accompanied by loss from the projectile. An analysis of experimental data for 40-MeV Ar + on Ar was performed for the case of ionization accompanied by the loss of 1-4 electrons from the projectile. The model was found to reproduce the shape of the recoil-ion charge-state distributions reasonably well, considering the complexity of the multielectron ionization and exchange process being treated. The absolute cross sections calculated for three- and four-electron loss, on the other hand, were underestimated by as much as an order of The majority of this discrepancy was asmagnitude. cribed to the neglect of projectile L-shell ionization. A rough four-shell calculation resulted in a considerable improvement in the absolute-cross-section comparison. ACKNOWLEDGMENTS

This work was supported by the Division of Chemical Sciences of the U. S. Department of Energy and the Robert A. Welch Foundation.

V. A. Sidorvich, J. Phys. B 14, 4805 (1981). 3R. E. Olson, J. Phys. B 92, 1843 (1979). 4I. Ben-Itzhak, T. J. Gray, J. C. Legg, and J. H. McGuire, Phys. Rev. A 37, 3685 (1988). 5M. Horbatsh, Z. Phys. D 1, 337 (1986).

6470

O. HEBER, R. L. WATSON, G. SAMPOLL, AND B. B. BANDONG

R. E. Olsen, in E/ectronic and Atomic Collisions, edited B. Gilbody, W. R. Newell, F. H. Read, and A. C. H.

by H. Smith

(North-Holland, Amsterdam, 1988), p. 271. ~R. L. Kauffman, J. H. McGuire, P. Richard, and C. F. Moore, Phys. Rev. A 8, 1233 (1973). A. Muller, B. Schuch, W. Groh, E. Salzborn, H. F. Beyer, and P. H. Mokler, Phys. Rev. A 33, 3010 (1986). O. Heber, G. Sampoll, B. B. Bandong, R. J. Maurer, E. Moler, R. L. Watson, I. Ben-Itzhak, J. L. Shinpaugh, J. M. Sanders, L. Hefner, and P. Richard, Phys. Rev. A 39, 4898 (1989). O. Heber, G. Sampoll, B. B. Bandong, and R. L. Watson, Phys. Rev. A 40, 5601 (1989). W. E. Meyerhof, R. Anholt, X. Y. Xu, H. Gould, B. Feinberg,

42

R. J. McDonald, H. E. Wegner, and P. Thieberger, Phys. Rev. A 35, 1967 (1987); Nucl. Instrum. Methods A 262, 10 (1987). R. J. Maurer, C. Can, and R. L. Watson, Nucl. Instrum. Methods B 27, 512 (1987); O. Heber, G. Sampoll, R. J. Maurer, B. B. Bandong, and R. L. Watson, ibid. 40, 197 (1989). T. Tonuma, H. Kumagai, T. Matsuo, and H. Tawara, Phys. Rev. A 40, 6238 (1990). R. D. DuBois and S. T. Manson, Phys. Rev. A 35, 2007 (1987). T. Tonoma, H. Shibata, S. H. Be, H. Kumagai, M. Kase, T. Kambara, I. Kohno, A. Ohsaki, and H. Tawara, Phys. Rev. A 33, 3047 (1986).