K-shell ionization by heavy particles: Comparison between theoretical cross sections and experiments

NUCLEAR

INSTRUMENTS

AND

METHODS

169 ( 1 9 8 0 )

249-258:

(~

NORTH-HOLLAND

PUBLISHING

CO.

K-SHELL IONIZATION BY HEAVY PARTICLES: COMPARISON BETWEEN THEORETICAL CROSS SECTIONS AND EXPERIMENTS H. PAUL

Institut fiir Physik, Johannes-Kepler-Universitiit Lmz. A-4040 Linz-AuhqL Austria

Graphical comparisons are presented for proton- and alpha-induced K-shell ionization cross sections on various targets, taken from the literature. These are compared to each other and to the theoretical calculations by Basbas, Brandt and Laubert; Ford, Fitchard and Reading; and Laegsgaard, Andersen and Lund. A comparison of the basic assumptions underlying the various theories is given in tabular form. The theory by Basbas et al. gives the best overall agreement with the data, except for high Z 2 . The theory by Laegsgaard et al. can be used only for medium to large Z2, where it gives satisfactory agreement with the data. The theory by Ford et al. gives slightly lower results than Basbas et al. due to the more realistic wave functions used.

Recently several refined calculations ~-4) for inner shell ionization cross sections by heavy particles have become available, and an extensive compilation of experimental data has also been publishedS). This paper is an attempt at graphical comparisons between data and theories. Table 1 gives an overview of available semi-classical (SCA) and Born approximation (PWBA) calculations. These calculations refer to "direct" ionization, i.e., without electron capture by the projectile. From eq. (51) of Basbas et al. 3) it can be seen that the ratio of electron capture to direct ionization cross sections varies, very roughly, as (ZI/Z2) 3, where Z~ refers to the projectile and Z2 to the target atomic number. We therefore consider here only cases where Z~ ~ Z2.

The SCA calculation 2) used here (cf. appendix) is based on an analytical representation of the cross section. The binding correction is introduced simply by replacing the atomic number of the target (Z2) by the atomic number of the united atom (Z~ +Z2). To correct for the Coulomb repulsion, the deceleration of the projectile is taken into account. Relativistic effects are taken into account by a correction factor2); they may be qualitatively understood by noting that the maximum energy transfer to an electron (2 my 2 for a free electron) increases when the electron becomes more massive. The corrected PWBA cross sections used here are calculated according to ref. 3. The "universal function" is obtained by interpolation from the table by Rice et al.l°). The correction for binding and polar-

TABLE | SCA and PWBA theories for direct K-shell ionization by heavy particles. (Cf. also table 2 in the contribution by Kocbach et al. to these proceedings.) Trajectories

SCA : BH 6)

Order

Electronic wave function

/f Screening

1st

hydrog.

0

LAL 2)

straight hyperbolic (straight)

1st

hydrog.

all

PT 4) FFR 1)

hyperbolic straight

1st 2nd (exact)

hydrog. Hartree-Fock

all all

PWBA : ML 7) KCM 8) BBL 3, 9)

plane waves

I st

hydrog.

all

(same as above)

for bound electrons for bound electrons no implicit

yes (all electrons)

Corrections for : Coulomb Binding/Pol.

Relativity

yes

no

no

yes (Deceleration) yes no

yes

yes

no implicit

no no

no

no

no

yes

yes

no

250

H. PAUL

ization is taken from eq. (45) of ref. 3, using ck = 1.5 and the analytical expressions in eqs. (39) and (27) of that paper. Finally, eq. (49) of ref. 3 is used to describe the Coulomb correction. The higher order SCA calculation with Hartree-Fock wave functions by Ford et al. ]) has been used by plotting the results for Z2 = 13, 22, and 28 given explicitely in the paper. (The projectile energies of these calculations are rather high, so that the neglect of Coulomb deflection is of no consequence.) In a few cases, eqs. (14) and (16) of ref. 1 have been used to extend the results to other targets and projectiles. Fig. 1 shows, for protons on A1, how the cross section is changed if the screened hydrogenic wave function is replaced by one obtained from a selfconsistent field. The curves below 0.6 MeV are taken from fig. 4 of ref. 9 and indicate that the hydrogenic wave function is almost as good as the Hartree-Slater wave function. The curve above 0.6 MeV, on the other hand, taken from table 2 of ref. 1, shows that the Hartree-Fock wave functions used by Ford et al. give somewhat lower cross sections than hydrogenic ones. This seems surprising. The dashed curve in fig. 1, also taken from table 2 of ref. 1, shows the influence of the second order correction. At 0.75 MeV, e.g., this correction is much more important than the binding correction. The data It) presented in the following figures are taken, with a few exceptions, from table 1 of ref. 5. X-ray or Auger cross sections were converted to

TABLE 2 References for experimental data.

AK74 BA77

BE73 GE79 HO75A

HO75B KB76 KL76 KH75 KN77 LA76 LE75 LO78

MD75A M176 MK74

MK76 R179 RN76 RS76

fHF(2)

SO76

~'HF

ST75

1Sp// / f ~ _6HS _

~HF

~'hydro

(Dhydro

ionization cross sections using the fluorescence yields (fitted values) by Bambynek et al. ~2). For figs. 2, 3, 5b and 6, the newer compilation by Krause 13) was used instead. To convert the K~ data by BE73 to K~+B ), the fitted experimental K/~/K~ values due to Salem et al. TM)were used (cf. table 2 for measurement codes). The experimental ionization energies were taken from Bearden et al.~5). With the exception of figs. 2, 3, 5b and 6, the theoretical curves and experimental points were plotted by means of an on-line computer program written for the HP-2000 computer at North Texas State University'6).

11_ 10. 09_

.o "~

o.8.

OE

R. Akselsson and T.B. Johansson, Z. Physik 266 (1974) 245. T. Badica, S. Ciortea, S. Dima, A. Petrovici, 1. Popescu and V. Neascu, X-ray Spectr. 6 (1977) 90. R.C. Bearse, J.A. Close, J.J. Malanify and C.J. Umbarger, Phys. Rev. A7 (1973) 1269. M. Geretschl~ger, private communication. F. Hopkins, R. Brenn, A. R. Whittemore, J. Karp and S. K. Bhattacherjee, Phys. Rev. A l l (1975) 916. F. Hopkins, R. Brenn, A. R. Whittemore, N. Cue and V. Dutkiewicz, Phys. Rev. A l l (1975) 1482. N. Kobayashi, N. Maeda, H. Hori and M. Sakisaka, J. Phys. Soc. Japan 40 (1976) 1421. E. Koltay, D. Berenyi, 1. Kiss, S. Ricz, G. Hock and J. Bacs6, Z. Physik A278 (1976) 299. N. A. Khelil and T.J. Gray, Phys. Rev. A I1 (1975) 893. B. Knaf, G. Presser and J. St~ihler, Z. Physik A 282 (1977) 25. A. Langenberg and J. van Eck, J. Phys. B9 (1976) 2421. R. Lear and T.J. Gray, Phys. Rev. A 8 (1973) 2469. J. S. Lopes, A. P. Jesus, G. P. Ferreira and F. B. Gil, J. Phys. B II (1978) 2181. F. D. McDaniel, T. J. Gray and R. K. Gardner, Phys. Rev. A It (1975) 1607. M. Milazzo and G. Riccobono, Phys. Rev. A 13 (1976) 578. R. H. McKnight, S. T. Thornton and R. R. Karlowicz, Phys. Rev. A 9 (1974) 267. R.H. McKnight and R.G. Rains, Phys. Rev. A 14 (1976) 1388. R. Rice, private communication (1979). R.R. Randall, J.A. Bednar, B. Curnutte and C L. Cocke, Phys. Rev. A 13 (1976) 204. Md. Rashiduzzaman Khan, D. Crumpton and P.E. Francois, J. Phys. B 9 (1976) 455. C. G.Soares, R. D. Lear, J. T. Sanders and H. A. Van Rinsvelt, Phys. Rev. A 13 (1976) 953. N. Stolterfoht and D. Schneider, Phys. Rev. A 11 (1975) 721.

L

p --~AI

I

1

I

2 E I(MeV)

I

I

3

l.

Fig. 1. The ratio of ionization cross sections, calculated by first order Born approximation theories for protons on A1, is shown versus proton energy (full curves). Below 0.6 MeV, the Hartree-Slater result 9) is shown divided by the uncorrected hydrogenic cross section; above 0.6 MeV, the Hartree-Fock result I ) is shown similarly normalized. The dashed curve shows the second order cross section 1) divided by the first order cross section I).

0

I.

2-

3-

~

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~b

el,

o

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o

o

.~

o

,f

¢

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o

o

\

T

o

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.~

~

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o

o

o o

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~

~

a

0

o

.2-

[0-

12"

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xlon b

0

10

o

K - S h e l l Ionization Cross

r~

o

o

Atorr~ Nun~r Z2

o

2MeV Protons

Sectioforn

Fig. 2. K-shell ionization cross section for 2 MeV protons vs target atomic number, Z ; . The theoretical curves are from ref. 1 (FFR), ref. 2 (LAL), and ref. 3 (BBL). (a) a I multiplied by Z 2 J0. (b) a I multiplied by Z 2 6.

~x

¢_

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K - SI~II Ionization for 21~leV Pr~ons

° °oo

o

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-]

>

z

r-

o

,o

o

io

b

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o oo

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o °o

Fig. 3. Same as fig. 2, for 1 MeV protons.

o,,

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:6

~ Mean value

o MI 76

oBET3

K- Shelt Ionization Cross Sectionfor IMeV Protons

~o

~o

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7-

9-

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11-

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o

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K-Shell Ionization Cross Section for l l'4eVProeons

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253

IONIZATION

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:2

~

o

I

;

~

,'0

a

o

I 0.2

EI(MeV}

I

I

0.4 0.6 EI(MeV)

Fig. 4. K-shell ionization cross section for protons on Ne, (a) for E I < 10 MeV, (b) for E 1 < 1 MeV.

/.0. x 10 z o

32_

o

BBL

2/,. 16.

~

8.

I

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0 2

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8

K-Shell l~isation Cross

b MeV3 -

10

~cti~

for Protons on Ti

0R$76

I °+24,;,

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/ +/ \_X\o ~_\\o

:+ ;r;, GE~

00I~0° WOO° °00 0

0LO

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e1

Fig. 5. K-shell ionization cross section for protons on Ti, (a) a I plotted directly, (b)

a[/E~ is

plotted.

I

I

08

I0

254

H.

Figs. 2 and 3 give an overview of all the ionization cross sections for protons as a function of Z 2, at two fixed energies. In order to make the discrepancies more visible, linear scales have been used, and the cross section has been multiplied by Z~° or Z~ to show different parts of the total range more clearly. For Cu and Ag targets, weighted averages are also shown with error bars. The points in figs. 2 and 3 are taken from many different authors; for a few of these, special symbols are used. The measurement RN76 and possibly some of the points due to BE73 appear somewhat low compared to the others. For low Z2, and around Z2=60, rather large discrepancies between different measurements appear. The theoretical curve due to Ford et al. (FFR) 1) is lower than that due to Basbas et al. (BBL)3), presumably because of the difference in wave functions. For Z2<~35, the data are well represented by BBL; for high Z2, this curve is apparently too low because of relativistic effects. For low Z2, the curve due to Laegsgaard et al. (LAL)2) seems low; this is to be expected 2) since the united-atom binding correction

~.

+~ ~ + / o ~

•

0 0

//o

PAUL

is not a good approximation for ~>~0.25. Hence, LAL should be used only for Z2 > 57 at 1 MeV, and for Z2 > 76 at 2 MeV ! Fig. 4 (for protons on Ne) shows good overall agreement between the six sets of experimental data available (which include both X-ray and Auger measurements), and with BBL, over a large range of energy. Since ~>0.25 almost over the whole range shown, LAL should not be used. Fig. 5a shows the available data for protons on Ti together with the three theoretical curves (LAL should not be used, however, above 0.1 MeV). FFR is slightly below BBL, as expected. For better comparison, part of the cross section data are shown again in fig. 5b, divided by E~, where E1 is the projectile energy. The data sets RS76, AK74, RI79, LO78 agree well with each other, whereas BE73 and KN77 appear slightly low with respect to the others. The (relative) data set GE79 was normalized at 0.8 MeV. Fig. 6 (protons on Cu) looks rather similar. The sets LE75, HO75A, RS76, AK74 agree with each other, whereas MI76, BE73, RN76 appear low and

K- She# Ionization Cross Section for Protons on Cu

\

x 0

o × + O ¢ • v

LE?5 HO75A HOFSB KL F6 MI?6 RS?6 BE?3 AK ?4 o RNF6 GE 79 (normalized) M mean value

0

o .

x

eI

i0

7

v

Proton Energy E t

]Fig. 6. K-shell ionization cross section for protons on Cu, divided by E~.

K-SHELL

40.

Fig. 8 shows very good overall agreement of the LAL curve with experiment, although this curve should be valid only below EL = 2.2 MeV

32. ^.^

BL

24_ ..o

w

16_ 8_

0

0

I

2

I

I

110

I

z, 6 EI(MeV)

8

o o

08_ o

~ b-

255

I O N I Z A T I O N

0.6_

o o

° o /

8

(

= 0.25).

The remaining figures show alpha particle cross sections. Here, similar statements concerning the validity of LAL apply. In fig. 9 (alphas on carbon) the X-ray measurement (LA76) and the Auger measurement (ST75) agree with each other, but are much higher than KB76. Similarly, in fig. 10 (alphas on nitrogen), LA76 and MK76 agree with each other but are much higher than KB76. Here, MK76 and KB76 are both Auger measurements and hence directly comparable. The data by KB76 are low by a factor 10 due to a mispri,nt in the original publication. Fig. 11 (alphas on Fe) shows good overall agreement between the data sets (MK74, MD75A, SO76), and with BBL.

o,'-.

0.2_

~

BBL

5_ x10 6 BBL

/

l,_ OZ,

08 li2 E1(MeV)

116

3_

Fig. 7. Similar to fig. 4, for protons on Ag.

,a E

part of KL76 appears high. As in fig. 5b, BBL fits the data best. The GE79 data are again normalized. In fig. 7 (protons on Ag) it can be seen that BBL represents the overall trend of the data quite well, even though the curve is too low below 2 MeV because of relativistic effects.

2_

I_ ~

0

;" I

I

0,4

16

o BBL

x 105

3.2_

20_ L

o

15_ --

1.6_

o o

o

o

10_

BBL

LAL

5_ o I 1

I 2

I 3

i 4

1 5

E 1(MeV) Fig. 8. K-shell ionization cross section for protons on Au.

o I

0,8 1.2 EI(MeV)

25_ o

2,4_

o I

o I

2.0

Fig. 9. K-shell ionization cross section for alphas on C. The low points (due to KB76) should be multiplied by a factor ]0.

,.o_

x 102

~Z)--

o

210

0 0

i

i

i

I

2

3

E I (MeV)

Fig. 10. S a m e as fig. 9, for a l p h a s o n N.

256

H. PAUL

20_ 16_

AL ~

may have to be revised later. Also, a rather small fraction of the data from the compilation by Gardner and Gray s) has been discussed here. Clearly, a more complete treatment will eventually be necessary.

o o

L 12_

.o ~-- 8_

Appendix

BBL

The SCA calculation

0 0

I 0.~

^ ^

I

0.8

11.6

1!2

For definiteness, we give here the formulas used in calculating the corrected SCA cross section, all taken from ref. 2. The cross section, corrected for binding, is given by

21.0

EI(MeV) F i g . 11. K-shell ionization cross section for alphas on F e .

_

aB In fig. 12 (alphas on Cu), a l E 4 has been plotted vs. energy. Three sets (MK74, MD75A, SO76) agree well with each other, whereas KL76 appears relatively high. The curve by Pauli and Trautmann (PT) is calculated as in ref. 4, but with relativistic wave functions ~7). This curve seems rather high, probably because of the lack of a binding correction. In conclusion, it should be stressed that many of the statements to the effect that certain results are "high" or "low" are of a subjective nature, and

2127Z

2 2

-1~8

45 RyZ 1 rK~EBu

x

x (1+0.0563~+1.380~2+0.2191~3) -4, where Ry is the Rydberg energy, rKu = ao/(Z1 +Z2K)

(2)

is the K-shell radius of the united atom, EBu is the experimental binding energy of the united atom, a0. is the Bohr radius, Z2K = Z 2 - 0 . 3 , ~

(3)

(4)

Fad/FKU~

b

~-Th

K-Shell Ionization Cross S@C~I~for ot on Cu

O8"

07

06

x

x

x

4- SO~

'~~4~

\

05X

X

"

x 0.~-

~

L

~

I

~

x x

x

x .

03-

02-

/

/

/

_

B~.

o

Oo"

.

.OOo

T "i- _j_ar~

A

01-

oo

0

F i g . 12. a x l E 4 for alphas o n Cu.

(1)

i~V

F~'ojectRe En~gy Ef

K-SHELL IONIZATION tad = hvl/EBu,

(5)

and v; is the laboratory velocity o f the projectile. To calculate the C o u l o m b correction, one finds the reduced velocity v~' = v 1 ~¢/( 1 - b / R ) ,

(6)

where R = N"/[R2 k rain + r2d) Rml n :

½b

+ ~/(¼ b 2 +

(7) ya2),

b = Z 1 Z2 e2/EcM,

(8)

(9)

and raa is taken u n c h a n g e d f r o m eq. (5). The corrected cross section is calculated from eq. (1), but after replacing eqs. (4) and (5) by:

rKvEsu.

(10)

257

7) E. Merzbacher and H . W . Lewis, in Encyclopedia of physics (ed. S. FliJgge; Springer Verlag, Berlin, 1958) vol. 34, pp.

166-192. 8) G. S. Khandelwal, B. H. Choi and E. Merzbacher, At. Data 1 (1969) 103. 9) G. Basbas, W. Brandt and R. Laubert, Phys. Rev. A7 (1973) 983. 10) R. Rice, G. Basbas and F. D. McDaniel, At. Data Nucl. Data Tables 20 (1977) 503. ll) I am very grateful to Dr. Gray for sending me his table in the form of punched cards. 12) W. Bambynek et al., Rev. Mod. phys. 44 (1972) 716. 13) M.O. Krause, J, Phys. Chem. Ref. Data 8 (1979) 307. 14) S. I. Salem, S. L. Panossian and R. A. Krause, At. Data Nucl. Data Tables 14 (1974) 91. 15) j.A. Bearden and A.F. Burr, Rev. Mod. Phys. 39 (1967) 125. 16) I am grateful to the members of the Physics Dept. at NTSU for their hospitality. 17) D. Trautmann, private communication. 18) E. Laegsgaard, private communication. I am very grateful to E. Laegsgaard for sending us a copy of his program.

Finally, the relativistic correction is applied as a factor

c [~r(2~)~(1 +~-2)~2-, ×

~R O'NR

x sin (2 7 arctan ~- 1)]2,

Discussion (11)

where

=

,,/I-1_~2 (Z2K..I_ Z1)2]

,

(12)

ac is the fine-structure constant, and ~ is calculated from eqs. (4) and (5), not (10). The additional factor 18) C, given by C = 2 ( 4 ~ - 3 ) / F ( 4 ~ - 1)

(13)

is not contained in ref. 2. It a m o u n t s to a 5% correction, e.g., for protons o n Ag. T h e " s y m m e t r i z a t i o n c o r r e c t i o n " which replaces the projectile e n e r g y E by E - 2! AE, is also discussed in ref. 2, but has not been included in our calculations.

References l) A. L. Ford, E. Fitchard and J.F. Reading, Phys. Rev. A16 (1977) 133. 2) E. Laegsgaard, J. U. Andersen and M. Lund, 10th Int. Conf. on Physics of Electronic and atomic collisions, (Paris), (ed. G. Watel; North-Holland Publ. Co., Amsterdam, 1978) p. 353. 3) G. Basbas, W. Brandt and R. Laubert, Phys. Rev. AI7 (1978) 1655. 4) M. Pauli and D. Trautmann, J. Phys. Bll (1978) 667. 5) R. K. Gardner and T. J. Gray, At. Data Nucl. Data Tables 21 (1978) 515. 6) j. Bang and J. M. Hansteen, Kgl. Dan. Vid. Selsk., Mat.-Fys. Medd. 31 (1959) no. 13.

Reading: I w a n t to m a k e two remarks on the discrepancy between the H a r t r e e - F o c k calculations and the hydrogenic calculations which has c o m e out. First we all k n o w that to m e a s u r e absolute crosssections is a very difficult problem and we all stop checking and m e a s u r i n g w h e n we agree with the norm. N o w there are a few m o r e theoretical n o r m s to check against. Since publication o f our calculations I have noticed a t e n d e n c y for experimental results to be reported which lie closer to our theoretical calculations. There is a need for unbiased absolute m e a s u r e m e n t s . T h e second point is that you s h o w e d a curve where the H a r t r e e - F o c k calculation gave a smaller result than the hydrogenic. T h e difference is due to the non-locality o f the potential. Because o f antis y m m e t r y , the electron is necessarily excluded f r o m the center o f the atom, and if you do a non-local calculation o f the b o u n d state wave functions and the scattering wave functions y o u always get a cross section less than from the simple h y d r o g e n i c m o d el. Most o f this difference should also be obtained using the Siater wave function which puts in a repulsive potential to suppress the wave function in the center. But there m a y be a small effect due to the latent non-locality even b e y o n d t h a t ; this is called the Percy effect and is well k n o w n in nuclear physics.

258

H. PAUL

Brandt: Could you give us a simple physical feeling for this effect? Reading: It is simply that the L-electrons leak s o m e w h a t into the region of the nucleus, and because of the presence of these L-shell-electrons near the nucleus there is some part of the K-wave function which is not allowed to be present due to the Pauli principle. Presser: I want to m a k e a c o m m e n t on the m e a s u r e m e n t s of absolute cross sections. It is difficult to take averages from m e a s u r e m e n t s of different groups if you are not really sure that they have different calibration systems. Otherwise the errors are mainly scaling errors and not statistical. Just comparing calibrated sources we have found discrepancies of about five or six standard deviations. Paul: You are right, the statistical problem is a formidable one. So in principle one would have to look very exactly at the data, but in the Gardner and Gray tables there are about 3000 points, so m y only excuse is that hopefully you have m a n y different groups and m a n y different calibration sources and there should be s o m e statistics in the way the p e o p l e have chosen their calibration sources. Presser: The second thing is that one should not forget the scaling errors which are in most cases in the order of 2 0 - 3 0 %. So if you take into account these errors then all theories agree. Rrsel: Now I want to ask Mr. Reading: Did you m a k e your H a r t r e e - F o c k calculations with a local potential or a non-local one? Reading: With a non-local potential. Rrsel: You know the problem with the exchange potential ?

Reading: I know the problem because we have used a non-local exchange potential. Rrsel: There are no calculations with a non-local potential; rather you take the H a r t r e e - S l a t e r formulation and then you approximate the non-locality by a local potential. Reading: That is the incorrect thing to do, we do not do it. Our calculations include a n t i s y m m e t r y fully. Rrsel: If you would like to have an exact binding energy you can take a local potential and you choose for the exchange coefficient a factor of 0.6. If you like to have a good wave function you choose the exchange coefficient 1 and the binding energy then differs by 1 or 2 eV and you have a very good bound-state wave function so I think it is not necessary to take the non-locality into account. Reading: It is necessary in the sense that we get a slightly different result. Rrsel: Another question: Do you take the same potential in the final state and in the initial state? Reading: The answer is yes. Rrsel: Then you have s o m e problems. In the final state, you have an ionized atom, but you calculate the wave function in a neutral atom. Reading: W e did it in both ways and we found the discrepancies to be 5 - 7 % . Rrsel: How would you calculate H a r t r e e - F o c k wave functions with a hole? Reading: We did not. But you can first do the H a r t r e e - F o c k calculation with all states occupied. T h e n you get the states. Then you can r e m o v e one electron and recalculate the potential. That is how we arrived at our estimated error.