Impact-parameter dependence of K-shell ionization

Nuclear Instruments and Methods 192 (1982) 79-101 North-Holland Publishing Company

79

I M P A C T - P A R A M E T E R D E P E N D E N C E OF K - S H E L L I O N I Z A T I O N J.U. A N D E R S E N , E. L A E G S G A A R D

and M. L U N D Institute of Physics, Universityof Aarhus, DK-8000 Aarhus C, Denmark

The impact-parameter-dependent probability of K-shell ionization by protons has been measured in the energy range 0.5-2.5 MeV for copper, silver, and gold targets. The data are used to test the scaling relations predicted from semiclassical calculations in first Born approximation, with unperturbed projectile trajectory and nonrelativistic hydrogenic electron wave functions. A systematic procedure is described for correcting such calculations for the inadequacy of these approximations, and the correction scheme is tested by comparison partly with experiments, partly with more complete theoretical treatments. The relation between ionization by scattered particles and by particles emitted in nuclear decays or absorbed in compound-nucleus reactions is discussed, and quantitative estimates are derived for the monopole term, which dominates for projectile velocities much lower than the K-shell electron velocity.

1. Introduction This p a p e r presents results of a detailed study of the i m p a c t - p a r a m e t e r d e p e n d e n c e of K-shell ionization by energetic protons. T h e d e p e n d e n c e on impact p a r a m e t e r is of direct importance in connection with ionizing processes, for which the impact p a r a m e t e r has s o m e h o w been specified. As important examples, we may first mention ionization in nuclear reactions [1-3] or in nuclear decays [4]. For such processes, the impact p a r a m e t e r is zero on the atomic scale, and the ionization probability is closely related to that obtained for large-angle elastic scattering. Second, for particles channeled in a crystal, collisions with small impact p a r a m e t e r s are prevented, and knowledge about the relative contributions from different impact p a r a m e t e r s to the ionization cross section is necessary for a detailed description of the observed dips in Xray yield [5]. In addition to such direct applications, differential m e a s u r e m e n t s are important for our general understanding of innershell ionization since they provide a m o r e detailed test of the theoretical descriptions than is obtained from m e a s u r e m e n t s of total cross sections. T h e pioneering work on the i m p a c t - p a r a m e t e r d e p e n d e n c e of inner-shell ionization by heavy, charged particles was the semiclassical treatment by Bang and H a n s t e e n [6] of K-shell ionization in the limit of low projectile velocities. T h e y showed that a semiclassical, first-order pertur-

bation calculation (SCA) with unperturbed projectile path and nonrelativistic hydrogenic electron wave functions leads to a d e p e n d e n c e on impact p a r a m e t e r of the probability for K-shell ionization, which is determined by a unique function of the reduced p a r a m e t e r x = b/r~d, where b is the impact p a r a m e t e r and r~d = hv/EB is the adiabatic distance for ionization of an electron with binding energy EB by a projectile with velocity v. T h e function has its m a x i m u m at x = 0 and falls off rapidly for x > 1. A generalization of this scaling law to higher projectile velocities has been derived by Kocbach [7]. On the basis of the same approximations, it was shown that the ionization probability is approximately proportional to a function of two reduced variables, x = b/r,d and ~ = r~drK, where rK is the K-shell radius. T h e function approaches the f o r m given by Bang and H a n s t e e n for ~--, 0. "In a previous publication of some aspects of the present work [8] (in the following referred to as I), these scaling laws were discussed with the main emphasis on a qualitative explanation of the somewhat surprising result that for low velocities, i.e., for r,~,~rK, only impact p a r a m e t e r s much smaller than the K-sheU radius contribute to the total cross section for K-shell ionization. Through comparison with measurements covering a large range of the reduced parameters, we shall here examine the validity of the scaling laws for the shape of the ionizationprobability function P(b). In I, we also discussed the very large devia-

0029-554X/82/0000-0000/$02.75 O 1982 North-Holland

gO

J.U. Andersen et al. / Impact-parameter dependence

tions from predicted ionization cross sections, which signal a breakdown of the simplifying assumptions in the S C A calculations, and it was shown that some of the deviations may be accounted for by simple corrections to the scaling parameters. The main emphasis was on corrections to total cross sections, but the prescriptions were based on the studies of the impactparameter dependence of K-shell ionization, which are presented in detail here. In our discussion of the corrections, we shall try to supplement the arguments given in I and emphasize the impact-parameter dependence of the corrections. In particular, we shall try to obtain reliable estimates of the ionization probability for small impact parameters and the related probability of ionization in nuclear reactions and decays. The experimental results presented here [9] represent a systematic extension of earlier measurements [10,11]. The ionization probability as a function of impact parameter P ( b ) has been measured for proton-induced K-shell ionization of target materials of low, medium, and high atomic number (Cu, Ag, Au) for energies in the range 0.5-2.5 MeV. Total cross sections in the same energy range and for a larger variety of target materials have been reported separately [12]. Also more detailed double-differential measurements such as those reported in ref. 13 have been performed, but they will not be discussed here. Some of the results presented here have been used previously in connection with studies of special aspects of K-shell ionization by chargedparticle impact [8,11].

2. Experimental method 2.1. Principles o f m e a s u r e m e n t

In measurements of the differential ionization probability P ( b ) , the impact parameter b must be determined for the ionizing collisions. When classical-orbital pictures may be applied to the deflection of the projectile, b is related to the center-of-mass (CM) scattering angle O through Rutherford's scattering law, t g ~ = bo/2b,

(1)

where b0 is the minimum distance of approach. bo

2Z122e2

=

Mov2

,

(2)

M2)being the reduced mass and Z2 the target nuclear charge. The scattering angle ~0 in the laboratory is given by Mo = M 1 M 2 / ( M 1 3 -

ME sin O tg ~o = M~ + ME cos O'

(3)

which for small scattering angles reduces to ~o = OMo/M1 = bl/b,

(4)

where bl is defined as b0 but with M0 replaced by M1. It is the minimum distance of approach for M2~>M~, and thus scattering through small angles may be calculated as if the target atom were infinitely heavy. From relation (4), the impact parameter may be obtained from a measured scattering angle q~. The condition for applying this classical relation was derived by Bohr [14], r = bo/~ = 2Z1Z2e2 > 1 ,

(5)

where ~t is the CM de Broglie wavelength X = h/Mov.

When K-shell ionization by protons is considered, r may conveniently be expressed as K

=

2(VK/V),

(6)

where VK is the K-shell electron velocity. Condition (5) is fulfilled for projectile velocities smaller than vK, which is precisely the velocity regime considered in this paper. It may be noted that eq. (5) is a condition only for the definition of an impact parameter from the scattering angle, not for a semiclassical treatment of the ionization, which is valid under much less restrictive conditions. Also at velocities v > vK, where the inequality (5) breaks down, an impact parameter or a range of impact parameters may be selected experimentally; examples were mentioned in sect. 1. Even at velocities v < VK, where eq. (5) is fulfilled, the finite magnitude of r implies some uncertainty in the impact-parameter definition.

J.U. Andersen et al. / Impact-parameter dependence

The magnitude is approximately given by [10]

(ab/b)

--- 2/K.

(7)

Although small for most of our measurements, this uncertainty should be kept in mind when measurements and calculations are compared. Simple estimates based on a diffusion-like picture indicate that for our smallest K values, a smearing corresponding to eq. (7) could modify the width of the measured ionization function P(b) by - 1 0 % . More detailed estimates of the accuracy of the semiclassical approximation have recently been derived on the basis of DWBA calculations [15].

2.2. Experimental technique When the ionization probability is large, ionization events may be identified either by the inelastic energy loss of the scattered particle, or by a coincidence between an X-ray and a scattered particle. For measurements of K-shell ionization by low-velocity protons, for which the probability is very low, it is an advantage to combine the two methods. This was done with the setup shown in fig. 1. K X-rays from the

MAGNETDETEC10R

x 0EEc01 COLLIMATOR ~. P~F.~ TON ~

l

~

SCATTERING ANGLE THINTARGETFOIL

.O'TTERINGOETECTO.

Fig. 1. Experimental setup for measurements of ionization probabilities.

81

target were detected by an X-ray detector at 90° to the beam axis and scattered protons by a magnetic spectrometer at a variable angle (-5 °, +60 °) to the beam axis. Protons scattered through a fixed angle - 9 0 ° were simultaneously detected by a separate detector. This detector was used together with the X-ray detector for measurements of total ionization cross sections. In connection with the differential measurements, it could be used to analyze the target composition since energy signals from particles scattered off different light elements were resolved. The incident beam was provided by the Aarhus University 2 MV Van de Graaff accelerator with an absolute energy calibration of - 1 % in the energy range 0.5-2.5 MeV. Measurement of small scattering angles requires a narrow parallel beam. For the differential measurements, the beam divergence was reduced to less than 0.075 ° by two collimators 4 m apart. Slit scattering was suppressed by a third collimator with a diameter slightly larger than that of the second collimator and at a distance of 20 cm from this and 7 cm from the target. The beam-spot area was 0.5 × 0.5 mm 2. The target foils were made by vacuum evaporation. Since self-supporting foils cannot be made very thin, some of the foils were evaporated onto standard carbon-foil backing. For measurements of total cross sections, only the requirements of small self-absorption of the X-rays and small projectile energy loss in the target set an upper limit to the foil thickness. For differential measurements and, in particular, for small scattering angles, the most restrictive condition is that scattering into the forward detector should not be influenced by multiple scattering. Furthermore, the targets should be thin enough for contributions from ionization of other target atoms to be negligible. This requirement, which is important also for large scattering angles, may be formulated as trNt~P(b), where tr is the total cross section, N is the atomic density, and t is the target thickness. To meet these requirements, gold and silver targets were made exclusively as self-supporting foils of approximate thickness, 1000-2000~ and 400500/~, respectively. The copper foils were made on carbon backing, 100/~ on 2 p,g/cm 2 and 400/~ on 4/~g/cm2. For each foil, the multiple-scattering distribution was checked to ensure that for

82

J.U. Andersen et al. / Impact-parameter dependence

the angular range covered by the measurement, the scattering was determined by single scattering. The foils were generally of high purity, but traces of carbon and oxygen were found. Also, during time-consuming measurements, appreciable amounts of light impurities, stemming from pump oil, etc., could build up on the beam-spot area. It was found that the rate of build-up could be strongly reduced by heating the target to about 100°C, and this was utilized for the silver and gold measurements. Since for small scattering angles, the elastic energy loss is too small to allow energy resolution of particles scattered from different elements, the elastic-scattering yield for the 'forward detector had to be corrected for the scattering off impurities. Hence the foil composition had to be checked regularly by means of the 90 ° detector. Both particle detectors were standard silicon surface-barrier detectors. The solid angle of the 90 ° detector was calibrated with the ot particles from an 24~Am source. Prior to detection in the forward counter, the particles were momentum analyzed in a magnetic spectrometer (see fig. 1). This consisted of an entrance collimator system, a 90 ° double-focusing sector magnet (resolution A E / E ~ 10-3), an exit collimator system, and a large-area particle detector placed at the focal plane of the magnet. The exit collimator could be selected either as a narrow slit for differential energy measurements or as a wider slit for collection of particles over a broad energy range. The entrance collimator system consisted of nine different-size collimators on a wheel, the smallest being circular and the others rectangular, with a width-to-height ratio of 1:5. For each scattering angle, the largest collimator was selected which fulfilled the condition that the definition of the scattering angle be better than __-5%. A NaI scintillation counter was used for all measurements, and the effective solid angle for gold K X-rays was calibrated with an 24~Am source, using the 59.9 keV 3/line, and with a 2°7Bi source, using the lead K X-rays. For copper and silver, the solid angle was determined from the total cross section measured separately with a silicon X-ray detector. Absorbers were generally used to discriminate against low-energy X-rays. Coincidences between X-rays and scattered protons were recorded by employing standard

constant-fraction timing and time-to-pulse-height conversion and accumulating the time spectra in a multichannel analyzer. When the scattered particles are energy-analyzed, accidental coincidences are usually no problem, and no attempt was made to optimize the time resolution, which was generally in the range 25-125 ns. When structure in the time spectrum of accidental coincidences was observed, it was removed by tuning the rf ion source [16].

2.3. Experimental procedure When no energy analysis of the scattered particles is performed, the probability for ionizing the K shell at scattering angle ~ may be obtained as

P@)

=

(N0/N,)(47r/OtoK),

(8)

where Nc is the number of p a r t i c l e - X-ray coincidences, Nt is the total number of detected scattered particles,/2~ is the effective solid angle of the X-ray detector, and tot is the fluorescence yield. The number of coincidences may be calculated from the recorded time spectrum as the sum of the counts within a narrow window around the peak of real coincidences with a correction for accidental coincidences based on the flat part of the spectrum outside the peak. This correction may be substantially reduced by discriminating against elastically scattered particles through energy analysis. When the ionization energy is large compared to the detector resolution, the discrimination may be performed on the energy signals from the particle detector. This was done for the gold measurements at large scattering angles, for which a large solid angle for particle detection was needed. The remaining measurements were performed with the setup shown in fig. 1. Energy discrimination with a spectrometer has several advantages. Firstly, the energy resolution is better than that obtained with a solid-state detector. In fact, the main limitation of the resolution is energy-loss straggling in the foil. Secondly, the elastically scattered particles can be excluded from the particle counter, which allows a substantial increase in the rate of data accumulation; the elastic yield must then be measured separately. This was done consecutively for all

J.U. Andersen et al. / Impact-parameter dependence

scattering angles covered by the experiment by normalizing to the X-ray yield. The number of elastic events Nt corresponding to the number of coincidences Nc could then be obtained by scaling with the ratio of the X-ray yields. This procedure has the advantage of eliminating possible changes in effective solid angle of the X-ray counter during the time-consuming inelastic measurement. Similarly, the influence on this measurement of dead-time losses in the X-ray detector is cancelled. Also, the measured angular dependence of the scattering yield may be used to check the assumption of single scattering. The elastic-scattering yield must be corrected for scattering by impurities or by the carbon backing. These corrections were estimated from the energy spectrum recorded with the 90 ° detector. For scattering through small angles, the differential cross section is to a good approximation given by the Rutherford cross section for all elements, but at 90°, strong deviations occur in the energy range of our experiments. This is shown in fig. 2 for protons scattered through 90° by carbon. Based on these results, the deviations from Rutherford scattering could be taken into account for the major correction, i.e., scattering from carbon. The impact parameters corresponding to the measured scattering angles were calculated from eq. (3) or (4). At the largest impact parameters, a small correction for screening of the nuclear Coulomb potential was included. The screening may be estimated with reasonable accuracy from a Thomas-Fermi potential, and the corrections I

6

I I

9

PROTONS SCATTERED ON THROUGH 92.5" ( L A B )

C

5

!

t.u 4

,9 ,,=,

!

:3[: 3 5g~ ,-J W

I

%

&

,i0

,is

210

2.5

ENERGY [ MeV ]

Fig. 2. Measured cross section at 92.5°(!ab) for proton scattering in C relative to the Rutherford cross section.

83

were calculated from b ~ b(1.0031 - O.O0386b/a),

(9)

where a is the screening distance, a = 0.8853a0(Z 2/3+ Z~3) -la. For the present measurements, the correction never exceeded 5%.

3. Experimental results Results of the measurements on copper and silver are shown in fig. 3. To observe the scaling relations in the data, it was found useful to represent the individual impact-parameter distributions by analytical fits. On theoretical grounds, the ionization function P(b) is expected to have zero derivative at b = 0 and to decrease exponentially at large b. Both conditions are met by the simple function, F(b) = A cosh-B(Cb),

(10)

which has been used to fit the data in fig. 3. The parameter A reflects the absolute magnitude of the probability, A = F(0), while the product of the parameters B and C is determined by the exponential slope at large impact parameters, F(b) oc exp(-BCb) for large b. The shape of the function in the intermediate region is then adjusted by a variation of the relative magnitude of B and C. As seen from fig. 3, the fits represent the measured data extremely well. The parameters obtained from least-squares fitting are shown in table 1. Because of the strong correlation between B and C, their product is much better determined than their individual values, and it is therefore given in the tables instead of C. The total cross section may be obtained by integration, o" = 2~ f b dbF(b), and this value is given in the table together with the directly measured total cross section. Such a comparison is an important check on the absolute accuracy of the differential measurements since the total cross-section determinations are much simpler and thus more reliable. From fig. 3 it is seen that the width of the ionization function P(b) is decreasing systematically with decreasing projectile velocity, and

84

J.U. Andersen et al. / Impact-parameter dependence 5 r

s~

H

2 . ~te~

on

! 0 -3

•

• .

• ~.

5

~L

Cu t • • *~t*

] 0 t •

t

o j

"re

5

, t

,~

o

" ~ •

,, *

' I~

'

t

2

.

,

t

t t

I

2. 0 0 0

I0 -s JF

1 ,~Jq

s ~

1

i

% %*,

" ~ '~ "

*,

; .ooo

i

10 -6 i

.

'

'

t

t

~,

~L

!

'

..

". O. 707

i. t

'~,,

ft

j

,

• '~

2

l

t~

,~

t

t

!

10 -5

2

,qg

on

5

~ *

I I

2 L

5

H

t

2.500

ft

. 2

10

1 0 -7

6

000

! I .414

0.500 5

s

1. 000

O. 7 0 7 2 @ .0

0,5

'..O b

! .S

• 2.0

• 2.S

2 [ 0.0

O.S

1 .0

b

"rCm:

I .5

2.0

2.5

[pm]

Fig. 3. Experimental results for the K-shell ionization of Cu and Ag by protons. The data points are shown together with the empirical fits given by eq. (10) and table' 1. For each distribution, the proton energy is given in MeV. The absolute uncertainty of the data is - 1 5 % .

also the exponential tail at large b is steeper at the lower energies. The shape may be characterized by two parameters, the half-width rl/2 and the logarithmic derivative 1/l at a large impact parameter, which here is chosen as b = 2r~, where r.d is the calculated adiabatic distance.

These quantities may be obtained from the fits as

raa : F(r,/2) = ~F(0), l:

1/l=-~lnF(b),

b=2r~d.

Table 1 Parameters obtained from least-squares fitting of eq. (10) to the experimental data of Cu and Ag. The total cross sections obtained from integration of the fitted results o.~nt are shown together with the directly measured values o'K. The ionization probabilities and cross sections for Cu and Ag have been multiplied by factors of 0.828 and 0.906, respectively, relative to the original data [9] to account for the average discrepancy from the more accurate results of ref. 12. The absolute accuracy of the parameter A is expected to be better than 15%. E (MeV)

Z2

0.500 0.707 1.000 1.414 2.000 2.500 0.707 1.000 1.414 2.000 2.500

29 29 29 29 29 29 47 47 47 47 47

A (10 4)

B

0.547 1.45 3.19 6.51 11.15 14.61 0.029 0.108 0.298 0.738 1.160

2.27 2.31 1.22 1.53 1.45 0.87 2.65 2.45 1.79 1.83 1.38

BC (pm -~)

(Tint

OrK

(b)

(b)

2.75 2.30 1.71 1.59 1.36 1.12 5.91 5.09 4.12 3.65 3.05

1.37 5.34 14.1 38.4 86.4 123.0 0.018 0.083 0.28 0.90 1.71

1.72 5.71 16.1 41.4 95.1 147.2 0.021 0.088 0.29 0.90 1.81

(ll)

J.U. Andersen et al. / Impact-parameter dependence

First, we may then compare the experimental values of these parameters with the theoretical predictions for the adiabatic limit [6], according to which both lengths should scale with the adiabatic distance. Such a comparison is shown in fig. 4, where the reduced parameters rl/2/r~ and l/r,d are given. The broken lines correspond to the values obtained by fitting the theoretical results [eq. (24)] with the function [eq. (10)] in the impact-parameter region 0-3rad. It is seen from fig. 4 that the experimental points fall considerably below the theoretical curves, the deviations being largest for the higher projectile velocities. We may next investigate whether the discrepancies can be explained by the more general SCA treatment. According to this, the ionization probability should be proportional to a function of two reduced parameters x = b/r~ and ~ = rad/rK. The reduced-shape parameters should

1.Z4

1.2

L~i.0

85

then be a function of ~ only, and the values indicated by small circles in fig. 4 have been obtained from fits with the function [eq. (10)] to tabulated values [19] of the ionization function. The solid curve has been derived from the parametrization of these theoretical results, given in sect. 4.4. The trend of the data is consistent with the generalized scaling law, but their magnitude is - 2 0 % below theory. Possible explanations for these discrepancies are discussed in sects. 4 and 5, where also the absolute magnitude of the ionization, represented by the parameter A in eq. (10), is compared to theoretical results. Finally, in sect. 7, the data are compared in detail with SCA calculations corrected according to the prescriptions developed in sect. 5. For the gold measurements, the corrections are extremely large. The results have therefore not been included in the above discussion of scaling but are given separately in table 2. With the scintillation counter, it was not possible to distinguish the K X-rays from the 77 keV 3'-rays resulting from nuclear Coulomb excitation of 197Au. At a proton energy of 2 MeV, the total 3' yield is only - 1 % of the total K X-ray yield, but for the differential measurements at very small impact parameters, the relative 3' plus K-con-

L-0.8 0.6

o Rg

Table 2 T h e experimental ionization probability for 2 M e V protons on Au. For the smallest impact parameters, a correction for nuclear C o u l o m b excitation h a s been applied, as described in the text. T h e quoted uncertainty is statistical, and the absolute accuracy is expected to be better than 15%.

D.B

j07 ~"

1D.6 13.5 13.4

0.0

0.2

.4

0.6

0.8

1.0

.2

~ = ~ d / r " ~. Fig. 4. Characteristic parameters [eq. (11)] obtained from the fits to the experimental data, compared to theoretical values. Broken lines represent the theoretical expectations in the adiabatic limit [eq. (24)]. T h e small circles are values derived from fits to tabulated values of t h e ionization probability [19], while the solid curves are derived from a parametrization of these fits [eqs. (10), (25)].

b

~o

P(b)

(pm)

(deg)

(10 -6)

0.012 0.027 0.049 0.062 0.078 0.102 0.121 0.128 0.154 0.207 _ 0.251 0.315 0.424

135 93 60 50 40 31.1 26.4 25.0 20.9 15.6 12.9 10.3 7.7

3.88 ---0.36 3.53 ± 0.25 3.93 ---0.43 2.07 ± 0.61 2.91 ± 0.24 2.04 ---0.12 2.00 ± 0.07 2.05 ± 0.17 1.33 ± 0.07 0.86 _+0.03 0.65 ---0.02 0.34 _+0.02 0.10 - 0.01

86

J.U. Andersen et al. / Impact-parameter dependence

version fraction rises to - 2 5 % of the K-ionization probability. The gold data given in table 2 have been corrected for this nuclear contribution from the 77, 269, and 279 keV levels of 197Au by applying the theoretical values given by Alder and Winther [17]. These corrections are negligible at impact parameters larger than -0.07 pm. The results given in tables 1 and 2 deviate slightly from those given earlier [8-11,16]. This is partly caused by a change of the accepted fluorescence yield used to obtain ionization probabilities from measured X-ray yields. The values used here are 0.445, 0.83, and 0.964 for copper, silver, and gold, respectively, as given by Bambynek et al. [18]. Also a recent, precise determination of total cross sections [12] allowed a slight renormalization of the ionization probabilities since, as described in sect. 2.2, the absolute scale for the differential measurements is tied to an independent measurement of the total cross section.

4. Theory Our calculations of the impact-parameter dependence of K-shell ionization are based on the semiclassical approximation (SCA), as developed in recent years by the Bergen group in particular. We shall briefly summarize the formalism below, discuss screening of the target atomic potential and the scaling laws obtained with the screening corrections usually adopted in SCA calculations, and finally introduce a convenient parametrization of the comprehensive tabulation of SCA results given by Hansteen et al. [191.

4.1. SCA formalism The projectile with charge Zle and velocity v is described as moving along a straight-line, classical trajectory, Rb(t) = (b, O, vt), with impact parameter b relative to the target nucleus at the origin. The development of the wave function of a target K electron is governed by a Hamiltonian,

For simplicity, we use a non-relativistic formulation with an unscreened static potential, but we shall later discuss modifications separately. The differential probability for transition of the electron from the initial state li) to a final state If) with energy Ef is obtained from first-order time-dependent perturbation theory as dP(b) dEf-

i l-h- f

dt e i°/gp

li) 2

-Zle2

.

'

(13)

-wa

where

co is the

transition

frequency,

co=

( E l - Ei)/h. The time integration may be performed explicitly [6], and one obtains dP(b) dEf = 2i(~)([IK°(qp)eiqZli)l

2

(14)

where K0 is a modified Bessel function. The symbol p denotes the distance between the electron coordinate and the projectile path, t92= (x - b) 2 + y2, and hq is the momentum transfer in the z direction, q = co/v. This expression was discussed in detail in I, and it was shown that one may understand the restriction of the ionization for low projectile velocities to impact parameters smaller than the adiabatic distance b <<-1/q as a combination of the restriction by the K0 function to p values p <<-1/q and the selection by the exponential factor of high z momenta in the initial wave function, which are confined to small distances in the x - y plane of the electron from the nucleus. For the evaluation of formula (14), it is convenient to expand the interaction in multipole components. A particularly simple derivation of the following formulas was given by Kocbach [20]. Here we formulate the results with a slightly different notation [16]. For K-shell ionization, the angular quantum numbers l and m may be associated with the final state, and one obtains (for two K electrons) dP(b) = 2 ~ IMb(Et, l, m)l 2 , dEf t,m

(15)

o¢

h2

H = - 2m Vz

Z2e 2 r

Zle 2 Ir

-

Rdt)l "

(12)

Mb(Ef, 1, m ) = Zle~ hv i l f dsA~ (s, b, oJ)F[f(s) 0

(16)

J.U. Andersen et al. / Impact-parameter dependence

An important simplification in eq. (16) is the separation of the properties of the perturbation and of the atomic system into the two functions A t and F~, given by

87

mental binding energy,

EB = OZ~ [Ryl.

(21)

For high Z2, relativistic effects are important, and it is more reasonable to define 0 as the ratio of the experimental binding energy to the relativistic hydrogenic binding energy,

A t ( s , b, to) = 7r-3/2 f dO, Y~(g2s) oc

X f V d t e it'°'-s'Rb0)l ,

(17)

E ~ a = (1 - y)mc 2 (22)

-ac

Y = (1 - Z~ol2)m ,

where Y~,, is a spherical harmonic function, and

F~f(s) = f r Edrjt(sr)Rf(r)Ri(r),

(18)

0

where ]l is a spherical Bessel function, and R~, RI are the radial parts of the initial- and final-state wave functions. According to Kocbach [7], eq. (17) reduces to

A t ( s , b, to) = 2 ((2• + ( l - m)!~ 1/2 s 1 ) ~ ] P'P(q/s)

×Jm[b'X/(s 2_ q2)], 0,

for s > q,

(19)

for s < q,

for a straight-line path. Here PT' is an associated Legendre polynomial and J,, is a Bessel function.

4.2. Screening The problems involved in introducing screening of the atomic potential by the other target electrons were discussed in some detail in I (see also ref. 21). Here we shall discuss only the procedure conventionally applied in SCA calculations. Inner screening is represented by a change of the effective atomic number of the target, Z2 -~ Zs = Z2 - 0.3.

(20)

It affects both the wave function and the energy of the initial state. Outer screening affects only the binding energy and is introduced through an empirical factor 0 determined from the experi-

where a is the fine-structure constant, a -~ 1/137. With this definition, 0 is approximately constant for high Z2. Also for low Z2, the interpretation of 0 defined in eq. (21) as representing the outer screening is questionable since the experimentally observed binding energy has a contribution from atomic relaxation. The main contribution is from the other K electron, A0 - 0.6/Z2, which is not negligible for small Z2, and the justification for the inclusion of this effect in the screening factor is not obvious. The effect of screening on the final-state wave function is a more delicate problem. In SCA calculations, mostly unscreened hydrogenic wave functions have been used [19]. Calculations with Hartree-Fock-Slater wave functions [22] have shown that this modification mainly affects higher multipole contributions to the ionization probability [eq. (15)]. Since the monopole contribution dominates for low particle velocities, the effect is not too important in this region. The dipole contribution, however, will often be important for large impact parameters, and it appears that final-state screening may explain the experimental observation of lower tails of P(b) than predicted by the normal SCA calculation (cf. sect. 3).

4.3. Scaling A very important aspect of SCA calculations with the simple screening corrections discussed above is an approximate scaling law [7],

P(b) =

~, x ) ,

(23)

88

J.U. Andersen et al. / Impact-parameter dependence

with ~ = rad/rK and x = b/rad. Here we have introduced the adiabatic distance for Ef = 0, tad = l/qo, and the K-shell radius rK = ao/Zs. This scaling law allows the comprehensive tabulation of ionization probabilities [19], on which the theoretical estimates in the present paper are based. As argued in I, the scaling law may be understood qualitatively from inspection of eq. (14). In the adiabatic limit, an even simpler result obtains, as established already by Bang and Hansteen [6]. The expression (23) may for ~:,~ 1 be separated into a product of functions of sc and x, respectively, such that the impact-parameter dependence scales with the adiabatic distance rad. For sc approaching' unity, also the distance rK becomes important, and the shape as well as the magnitude of P ( b ) will depend on sc.

10o s

2 -!

I0 \

5

XCu ORg

2

×,k

I

\,

i

""

i

10-2 I.B

1.4

1.0

4.4. Parametrization of S C A results J

0.6

The asymptotic result obtained by Bang and Hansteen for the differential probability [eq. (13)] could be expressed analytically through a modified Bessel function. Only for b = 0 is the integration over Ef simple, but a convenient and very accurate approximation has been given by Brandt et al. [23], P ( b ) ~ (1 + 1.96x+~-~Trx2) e -2x ,

1.446},

C = r;d max{(0.516 + 0.7905s¢), 0.737}.

0.2

0.4

O.B

0.8

1.0

1.2

Fig. 5. Parametrization of tabulated ionization probabilities [19] with the function eq. (10). The values extracted from the fits are shown as small circles, and the solid curves are the approximations to these results given by eqs. (25) and (28). The experimentally determined values of A/Aaa for Cu and Ag are also shown. The value A,d for the adiabatic limit is given by eq. (26).

(24)

with the parameter x defined in connection with eq. (23). It turns out that also the tabulated results [19] based on the more general scaling law [eq. (23)] may be represented in a convenient way. The tabulated impact-parameter dependence P ( b ) has been fitted with the function given in eq. (10), and the resulting parameters are given in fig. 5 as functions of sc. Asymptotically, for ~ 0 , they should approach their respective values obtained by fitting the expression (24). These are indicated by horizontal lines in fig. 5. G o o d approximations to the values of B and C are obtained from B C = r;d max{(1.307 + 0.4320~),

0,0

(25)

The transition to the adiabatic limit should be

smoother than given by eq. (25), but the error committed will for most purposes by negligible. The parameter A in eq. (10), which represents the ionization probability at zero impact parameter, A = P(0), is in the adiabatic limit given by [6]

Aad--

27 e 2 Z~ ,/:6 7 aoEB ~ "

(26)

In fig. 5, the A values obtained from the fits are normalized to this limiting expression, and the horizontal line at unity then corresponds to the adiabatic limit. The dashed curve, given by A/Aad = (1 + 1.72~:2)-4 ,

(27)

was derived [23] from P W B A total cross sections combined with the asymptotic b dependence of the ionization probability, eq. (24). Since for total cross sections, the P W B A and SCA cal-

J.U. Andersen et al. / Impact-parameter dependence

culations agree (cf. I), the discrepancy in fig. 5 between the dashed curve and the tabulated results reflects the deviation at finite values of from the asymptotic shape of P(b). The SCA results are fitted extremely well by the expression A(~) = A,d(1 + 0.062~ + 1.238(2) -' ,

(28)

as shown by the solid curve in fig. 5. With the parameters obtained from eqs. (25) and (28), the tabulated values of P(b) are in the parameter range ~ < 2 and x < 3 reproduced to within 10%, and for most cases, the accuracy is much better. The shape of the calculated ionization function P(b) was compared to measurements'in sect. 3, and it was found that the scaling was well fulfilled, while the measured dependence on impact parameter was systematically steeper than calculated. As mentioned in sect. 4.2, this deviation can be largely ascribed to screening effects. The absolute magnitude of the ionization probability, on the other hand, does not follow the predicted scaling very well. This is shown in fig. 5 by the comparison between measured and calculated values of the A parameter. The discrepancies, as we shall see in sect. 7, can be explained by the corrections described in the following section. For our set of measurements, mainly the absolute value of P(b) is affected by these corrections.

5. Corrections to first-order Born approximation

At high projectile velocities, a first-order Born calculation of K-shell ionization is justified by the smallness of the perturbation. As seen from eq. (14), a dimensionless measure of the perturbation is given by Bohr's K for a collision between an electron at rest and the projectile,

Ke = 2Zle2/(hv).

(29)

For low projectile velocities, this quantity becomes large, but the transition probabilities decrease with decreasing velocity because the ionizing collisions become more and more adiabatic. Thus in this region, a first-order perturbation treatment is justified not by the small magnitude of the perturbation but by its slow

89

variation with time. In principle, the treatment should therefore be based on adiabatic perturbation theory or on the perturbed-stationarystate method. The adiabatic relaxation of the initial electron state is the first correction we shall consider. A modification of the electronbinding energy was first introduced by Brandt et al. [24], and the correction is usually denoted the binding correction. Secondly, the perturbation of the projectile motion by the Coulomb repulsion from the target nucleus becomes important at low velocities. The main effect of the repulsion is a reduction of the projectile velocity in the vicinity of the nucleus, where the ionization takes place, and the magnitude of the correction is therefore governed by the ratio of two lengths, bo/rad, where b0 is the collision diameter [eq. (2)] for the projectile scattering by the nucleus and r~d the adiabatic distance for the ionization process defined in connection with eq. (23). The correction was first treated by Bang and Hansteen [6] in the impact-parameter formulation of firstorder Born approximation. They also considered the additional correction of the projectile velocity due to the inelastic energy loss in the ionizing collision. Finally, for decreasing projectile velocity, relativistic effects on the electronic wave functions become increasingly important, in particular, of course, for high target atomic numbers. Relativistic effects were first incorporated in a P W B A description of K-shell ionization by Jamnik and Zupan~i~ [25]. All of these effects have been discussed extensively in the literature and are basically well understood. The simple scaling laws discussed in the previous section do not hold when the corrections are large and detailed evaluations will be limited to special cases. In I a correction procedure was described, which from comparisons with the more detailed calculations appears generally to be reasonably accurate. It was attempted to base the procedure on simple physical arguments and to maintain conceptual as well as technical simplicity. The corrections for binding and Coulomb repulsion were introduced as modifications of the scaling parameters for tabulated SCA results. A detailed justification was given in I, and we shall here give only a brief summary of the procedures with a few sup-

9{)

J.U. Andersen et al. / Impact-parameter dependence

plementary arguments. For relativistic effects, on the other hand, the discussion in I was focused on simple corrections to total cross sections, and we shall discuss in more detail the impactparameter dependence of this correction. Also relativistic modifications of the correction procedure for the binding effect, which were omitted in I, will be discussed here. 5.1. B i n d i n g (nonrelativistic )

The nonrelativistic impact-parameter-dependent binding correction was discussed in I. First, the presence of a charge Z~e at a distance R from the target nucleus modifies the binding energy of a ls state (hydrogenic) by an amount, which to first order (unrelaxed wave function) is given by Zle 2

a E B = - - i f - [1 - (1 +

R/rK) exp(--2R/rK)].

(30)

Second, the relaxation of the wave function may be estimated [11] from a variation of the K-shell radius rk to maximize the total binding energy,

ionization process. An indication of the approximate validity of eq. (32) even for ~ 1 is obtained from a classical estimate of the displacement of an electron in a collision at impact parameter b = 2rK with a projectile at velocity v = VK/2 corresponding to ~:= 1 (~ = 2V/VK0). The relative displacement of the electron at the moment of closest approach is approximately Ab/b = bo/2b ~-Z1/Z2 when we assume that the collision diameter b0 corresponds to a relative electron-projectile velocity of X/2t,. This is exactly the result obtained from adiabatic relaxation. The distance R in eq. (31) should depend on impact parameter, and the simplest choice is the minimum distance of approach to the nucleus, R min(b) = bo

with b0 given by eq. (2). Since the characteristic distances along the path in the perturbation integral eq. (13) are of order rad, we have chosen instead to evaluate the binding correction at a distance R = [R2i,(b) + r2~]1/2 .

EB(rk) -

(33)

(34)

e2

e2a° ~- Z s - - + AEB(R, r k ) - AE ...... 2r~:2 rk "

(31) The first term represents the kinetic energy, a0 being the Bohr radius, the second term is the potential energy due to the attraction to the target nucleus, corrected for inner screening. The third term is given by eq. (30), with rK replaced by rk, and the last term represents the screening by target atomic electrons, calculated for the united atom, A E . . . . . = E ~ d ( Z 2 + Z , - 0.3)

-

E~xP(Z2 + Z , ) .

(32)

This choice leads to the correct (united-atom) limit of the binding energy (and K-shell radius) for R - o 0 but does of course not reproduce the target-atom binding for R ~ . In the velocity region considered, corresponding to ~: < 1, the important distances .E are always smaller than the radius of the L shell, rL = 2rK, and therefore all outer electrons will be contracted during the

This choice has the advantage that the correction disappears for increasing ~: even for b = 0, which is asymptotically correct, but, on the other hand, detailed calculations with the distortion approximation [26,27] support the results of Basbas et al. [28], indicating that for intermediate velocities, the binding correction may be even larger than obtained from evaluation at R = Rm~n. For ~:,~ 1, however, there is not much difference between the results obtained with eqs. (33) and (34), and as we shall see, eq. (34) is consistent with the choice of distance for the Coulomb repulsion, which is much more critical. The variational calculation is easily performed by iteration [ll] but is still more complicated than the simple binding correction, eq. (30). However, as argued in I, the relaxation is very important for small ~ since the decrease of rK increases the high-momentum content of the wave function, i.e., F~f(s) for large s [cL eqs. (16), (18), (19)]. Although the influence of relaxation on the binding energy is of second order in Z1/Z2 (an increase!), the effect on the binding cor-

91

J.u. Andersen et al. / Impact-parameter dependence

rection is a reduction of the first-order term. Without relaxation, the first-order correction of the total cross section is overestimated by ~50% for # ~ 1 [cf. eqs. (III.11) and (II1.12) of I]. 5.2. Binding (relativistic)

Relaxation of the K-electron function is of particular importance for evaluation of the relativistic enhancement of ionization probabilities, discussed in sect. 5.4, but for high Z2 values, where such effects are large, also the binding correction should be evaluated relativistically. The relativistic analogue of eq. (30) has been derived by Amundsen [29],

AEB=Z'e-~2[ 2(-~-1)

effective values of Z ' and E~ may be obtained for the impact-parameter-dependent distance R in eq. (34). It should be noted that the accuracy of such a one-parameter variation is more doubtful for the relativistic case because the relativistic modifications of the wave function are very sensitive to the shape of the potential at small distances. An indication of the accuracy of the binding energy obtained from eq. (37) is obtained from the comparison shown in fig. 6 with results of calculations based on the two-centre Dirac equation [30] for 53I on 79Au. This system is too symmetric for our correction scheme to be valid for predictions of the K-shell ionization, but the estimates of binding energies agree fairly well*. The results from the nonrelativistic procedure

x,r/(2y, 2RIrK)IF(2y + 1) - 2 ( 2 R)2"-' e x p ( - 2 R ) / F ( 2 y

+ 1)],

tO0

(35)

150 @ -E

where y is the incomplete gamma function,

0

200

//~

• m 250 Ld

~(a, x)

=i

e-tta-1 dr.

(36)

300

0 [Note the misprint in ref. 29: a factor 2 on the last two terms in eq. (35) is missing.] The symbol y was defined in eq. (22). In order to perform a variational calculation, we must once again, as in eq. (31), separate the dependence of the binding energy on the effective nuclear charge Zse and on the parameter Zs in the wave function. The total binding energy given in eq. (22) may be split into kinetic and potential contributions, using the result (l/r) = (yr~:) -1, and we obtain Eh = (1 - y ' ) m c 2 - ( Z ' - Zs)e21v'rk + AEB(R, Z') - AE~.~n,

(37)

where we have indicated with a prime which quantities have to be calculated with the variable wave-function parameter Z'. In analogy to eq. (32), the screening correction should be calculated for the united atom according to eq. (22). From a variation of Z ' to maximize E~, the

1.0

/ 1// 0.9

///

L~O.B L" 0.7 0.6

°'Slo-~ ~

~ 1'o-~ ~

~ i'o° ~ ~

Io'

Fig. 6. Comparison for 531 on 79Au of the relativistic (solid curves) and nonrelativistic (broken curves) results for the effective binding energy and radius of the K shell. For decreasing internuclear separation, the nonrclativistic results converge much faster towards the united-atom values than do the relativistic results. Shown as circles are results of calculations based on the two-center Dirac equation [30], shifted by 58 k e V to account for outer screening. * In ref. 30, also the case CI-~ Pb was considered, but it has later been discovered [31] that the results could not be trusted for such asymmetric cases.

'42

J.U. Andersen et al. / Impact-parameter dependence

are for this case clearly inadequate. Note, however, that the errors in binding energy and in effective K-shell radius tend to compensate each other. By inspection, we find that the nonrelativistic procedure is sufficiently accurate for Z 1 + Z 2 ~ 60 with an error of ~< 10% for ZI = 20, and the choice of procedure is of course less important for smaller Z~.

_5.3. C o u l o m b repulsion a n d energy loss

As argued in the introduction to this section, the deceleration of the projectile in the target Coulomb field will be important when the adiabatic distance rad is not very much larger than the collision diameter bo. This is reflected in the approximate correction derived by Bang and Hansteen [6], /

1

. .

x

crK ~ O'K e x p t - ~Iroo/ rad) .

(38)

As discussed in I, this is an underestimate of the reduction due to Coulomb repulsion, and instead, a simple velocity correction was suggested, ECM--~ ECM -- Z 1 Z 2 e 2 / R ,

(39)

with R given by eq. (34). By inspection, this prescription was found to agree closely with detailed numerical calculations for hyperbolic projectile trajectories [32]. When the correction is very large, e.q. (39) leads to a slight underestimate of the reduction. Note that the introduction of an effective velocity during the ionization is of importance for the relativistic corrections to be discussed in sect. 5.4. For very low velocities, where the ionization energy is not negligible compared to the projectile energy, also a correction to the projectile motion due to the interaction with the electron becomes important. It has been suggested [6] that this effect may be included approximately by the replacement ECM --> EcM - A E / 2 ,

(40)

in analogy to the prescription usually applied to nuclear Coulomb excitation [17]. The importance of the effect depends on the ratio

EBI2EcM = A/rad ,

(41)

where A is the de Broglie wavelength of the reduced-mass projectile. For low velocities (v < VK), from eqs. (5) and (6), this quantity is seen to be small compared to the ratio bo/rad, which governs the magnitude of the correction for Coulomb repulsion. Thus the energy-loss effect is important only when the ionization probability is already strongly reduced owing to nuclear Coulomb repulsion. 5.4. Relativistic effects

For atoms with high atomic numbers, relativistic effects are important for the binding energy, as discussed in sect. 5.2. The relativistic modifications of the wave functions for innershell electrons may, however, be important in Coulomb ionization even for low-Z atoms. The reason is that for low projectile velocities, the probability of ionization depends on the magnitude of the high-momentum tail of the electron-momentum distribution [cf. eqs. (16)-(19)], which even for low-Z atoms is modified by relativistic effects. Since the minimum momentum transfer hqo is inversely proportional to projectile velocity, relativistic corrections are expected to increase strongly with decreasing velocity, as demonstrated for total cross sections by Amundsen et al. [33]. We have based the relativistic corrections to the ionization probability P ( b ) on the more detailed treatment by Amundsen and Kocbach [34]. The monopole contribution to P ( b ) may be obtained by integration over final electron energy Ef of the formulas given in eqs. (12) and (13) of this reference. [Note that the normalization factor Df given in eq. (8) should be multipled by 2~-1.] For ~: approaching unity, also dipole and perhaps quadrupole contributions become significant, in particular for large b. Evaluation of the expressions given by Amundsen [35] requires considerable computational effort, and we have chosen instead to apply the relativistic correction for the monopole term also to the higher-multipole contributions. Thus our procedure consists in evaluating the relativistic correction factor to P ( b ) as the ratio between the relativistic and nonrelativistic monopole contributions as obtained from formulas (12) and (13)

J.U. Andersen et al. / Impact-parameter dependence

93

of Amundsen and Kocbach [34] after integration over El. (The nonrelativistic result is obtained by setting the fine-structure constant a = 0.) This correction factor is applied to the value P(b) obtained from the parametrization given in sect. 4.4 of tabulated nonrelativistic results, which include higher-multipole contributions. This procedure appears to be quite accurate for most cases, the reason being that when higher-multipole contributions are important, i.e., for ~ ~ 1 and mainly for large b, the relativistic correction is not very large nor very different for the different multipole contributions. Technically, the relativistic correction is much more complicated than those discussed in the previous sections, and we have therefore made an attempt to express the correction factor as a simple function of y and ~, along the lines suggested by Amundsen et al. [33]. As they suggest, the impact-parameter dependence of the correction may be approximated by the relation

represents the full calculations for b = 0 extremely well, with errors less than 10% for <-0.8 and Z2-< 100. For b > 0, the accuracy of eqs. (42) and (44) is not as high, especially not for large values of ~ and Z2, and the errors approach 20% at b=rad and 40% at b=2r~d within the restricted parameter ranges sc -< 0.5 and Z2--- 100. The simple relativistic correction factor developed here may also be used to improve the accuracy of the relativistic correction of total cross sections discussed in I. Integration over impact parameters of eq. (42) leads to

pR(b) = f(3,, ~)Pm~(b/3,),

We have now discussed prescriptions for the three major corrections to tabulated K-shell ionization probabilities at low projectile velocities, for binding, for Coulomb repulsion and energy loss, and for relativistic modifications of wave functions. It should be emphasized that when the corrections are substantial, the order in which they are applied is essential. The velocity correction, accounting for Coulomb repulsion and energy loss, as given in eqs. (39) and (40), is very sensitive to the magnitude of the distance R defined in eq. (34), and this distance depends through rad on the correction of the binding energy, obtained by variation of eq. (31) or (37). The distance R also enters into these equations, but since the binding energy varies more slowly with R, we have used the unmodified value of rad in eq. (34) in this connection. The relativistic corrections, as given in eqs. (42) and (44), depend on both the effective velocity and on the effective atomic number Z', derived with the prescription for the binding correction. A comparison of one result derived with differential measurements of the ionization probability was given already in I. In sect. 7 we shall make a more complete comparison with the data described in sect. 3, and we shall also compare with a few sets of data obtained by other groups, which are of particular relevance to the

(42)

where the indices R and NR refer to relativistic and nonrelativistic results, respectively. This relation is based on the approximate scaling with 3' of the dimensions of the wave function, reflected in the exact result (r -1) = (3'rK)-1. The scaling breaks down at large impact parameters, b>r~, where the nonrelativistic exponential slope is approached. For ~ ~ 1 and b = 0, it is possible to extract the dominating terms of the relativistic expression for the monopole contribution to the ionization probability, and the factor f(y, ~) in eq. (42) is for b = 0 obtained as

R3,, ~)-- g(3,)3,-~:"-'

x [ ~ sin ~ry+ h(y)sin,r(y-½)] 2 ,

(43)

where g(y) and h(3,) are complicated functions of y but not very different from unity. This rather simple expression compares well with the full calculations, but to improve the accuracy for approaching unity, a third term was included in the last factor. The function f(3,, $) = 3,-2(1 - 3" + 3,s)~,,-4 [½(~:-1+ $) sin w,/ 1 + ~(y + 1) sin ~'(y - ½)]2,

(44)

trR/o.NR __ 3,Zf(3,, ~),

(45)

and eq. (44) should now be evaluated with the values of Z2, EB, and v obtained as discussed in I but with the relativistic binding correction described in sect. 5.2.

5.5. Summary of corrections

~4

J.U. Andersen et al. / Impact-parameter dependence

evaluation of the accuracy and limitations of our procedure.

A~-= (At,.+)*. The Kronecker symbol 6,,,0 indicates that only final states with m = 0 contribute and the O function that the real part vanishes for s < q. The function

6. Ionization in nuclear reactions and decays !

As mentioned in sect. 1, an important aspect of this study of the impact-parameter dependence of K-shell ionization is the application to nuclear reactions and decays, where the impact parameter is zero on the atomic scale. In ref. 16 we discussed the related problems with main emphasis on the dipole contribution to the ionization probability [l = 1 in eq. (15)], which gives an angular dependence of the probability for large scattering angles, and later we have studied in detail the K-shell ionization in a decay of polonium isotopes [36]. A comprehensive nonrelativistic theoretical treatment has been given by Ciocchetti and Molinari [37], who derived an analytical expression for the monopole-ionization amplitude for ionization in ot decay. A discussion of this result, including corrections of some serious misprints in their formula, is given in ref. 38. We shall here in the SCA formulation given in sect. 4, discuss the monopole contribution, which dominates at low particle velocities, with particular emphasis on relativistic corrections to the ratio of probabilities for ionization in particle scattering and in nuclear decays or compound-nucleus reactions. These corrections turn out to be very large for many cases of interest such as nuclear a decay.

Or(z)

1 d u Pt(u) =2~ f Z-U

(47)

1

is a Legendre function of the second kind [39]. (Note that the factor ~ was missing in the definition in ref. 16.) For the monopole case (l = 0), the imaginary term in eq. (46) vanishes in the bombardment case, independent of the scattering angle. However, as discussed by Ciocchetti and Molinari [37], the function (46) does not correctly represent a nuclear decay. The perturbation is the potential from the emitted particle moving away from the nucleus minus the potential from the same particle remaining at rest in the nucleus. The corresponding correction for a reaction with formation of a compound nucleus is the potential from the absorbed particle remaining at rest in the nucleus for t > 0. Apart from the complex conjugation, the two cases then deviate only by the difference in nuclear charge of the unperturbed system, and this difference disappears at low velocities when the correction for adiabatic relaxation is included (sect. 5.1). The correction affects only the imaginary part of the function (46), and we obtain for 1 = 0

2i A°+(s, w) = sl O(s - q) + ~ss

[O0(s)-q

S

l

,48,

6. l. Path function The starting point is formula (16), in which the dependence of the ionization amplitude on the trajectory is separated out in the function A~. For half a trajectory, corresponding to nuclear decay, formula (19) for b = 0 is modified to [16] ,+

,

Am (s, to) = s (2l + 1)m6m.oPt

O(s - q)

The function Q0 is given by

Qo(x) = ~1 log I ~1 + x ,

(49)

and the correction term in eq. (48) therefore cancels the imaginary part for small values of s,

s~q. (46)

Here we have indicated by a + sign that the trajectory is outgoing. For an incoming trajectory, the amplitude is complex-conjugated,

The ionization amplitude corresponding to the imaginary part of eq. (48) is sometimes called the shake-off contribution [40], and it dominates for large particle velocities, e.g., for/3 decay [38]. In ref. 16, it was argued that because of the partial cancellation of the two imaginary terms in eq. (48), the ionization probability for a decay (small

J.U. Andersen et al. / Impact-parameter dependence

v and large q) would be dominated by the real part. As we shall see, this is not correct, and, in fact, the imaginary part is completely dominating in the nonrelativistic result [37] for the K-shell ionization in the a decay of 21°po.

6.2. Radial matrix element According to eq. (16), the ionization amplitude is obtained by convolution of A°+(s, to) with the radial matrix element F°(s) defined in eq. (18). The function may be expressed analytically for nonrelativistic unscreened hydrogenic wave functions [6], 8 X/m# Im [# + i(s k)] ~"-' [# + i(s + k)] i"+~' hs

F°f(s)

(50)

-

#=

with r~ 1 and ~7 = - # / k , where hk is the momentum of the ejected electron. This function for E l = 0 (i.e., for k ~ 0 ) is displayed in fig. 7 together with the real and 0.5

~°,(s/q)

o.4q=2 / ~ /,,

/,,

.o

\',

imaginary parts of the function sA °÷ with a logarithmic abscissa scale. This is convenient since the natural variable of integration in eq. (16) is log(s/q). A variation of the variable q only displaces the function F ° on this scale. If the dependence of F~(s) on the final-state energy Ef can be ignored, it is therefore very easy by numerical integration to calculate the monopole contribution to the differential ionization probability [eqs. (15), (16)] and to integrate over final-state energy. This procedure is quite accurate for low particle velocities (~ = #/qo < 1) since the high-momentum tail (s > #) of F°(s) depends only slightly on El. This simplification is particularly valuable when the relativistic function F°~(s) is introduced. A general expression for s > # was derived by Amundsen and Kocbach [34] on the basis of the formulas given by Jamnik and Zupan~i~ [25]. The power series in 1/s, however, converges very slowly, and furthermore, convolution with the function in eq. (48) requires definition of F°(s) for all values of s. We have therefore reexamined the formulae in ref. 25, and it turns out that for k ~ 0 (Ef=0), a relatively simple expression may be derived, ~/m (2#)2"%/(1 + 3') F°E'=°(s)= hs V[23'F(23')]

/,,

~-, (n + 2)(-2¢)" x Im ~ (1 - is) 2~ "n !(23' + n)"

0.1 ~ 0.0

1.o_ _ ~

95

Re(sR~")

(51)

This series converges rapidly for all values of s. It may alternatively be expressed as

Vm (2¢y" F°~"°(s)= hs V[23'F(23')]

O 0 oo

x ~7 (n + 2)(-2~)" sin[(23' + n)arctg(s/#)] n !(23' + n)(1 + s2) y+n/2

(52)

-I .0

-2.0 0.I

0.2

0.5

I .0

2.0

5.0

IO

s/q Fig. 7. Illustration of the radial monopole matrix element F~. The values of y correspond to the nonrelativistic case and to the relativistic case for Z2 =' 82, respectively. In the lower half of the figure are shown the real and imaginary parts of the straight-line path function sA~+ corresponding to particle emission in, e.g., a decay.

We have checked that this series reproduces the two first terms in the power series in 1/s given in ref. 34, and a numerical comparison shows complete agreement for s > #. Another check is provided by the nonrelativistic expression in eq. (50), which agrees with eq. (52) for 3' = 1. It may also be worth noting that the variation with 3' of the first term in eq. (52) nearly reproduces the relativistic correction factor suggested in eq. (8)

96

J.U. Andersen et al. / Impact-parameter dependence

of ref. 33. This factor was based on a simple estimate of the relativistic enhancement of F°(s) for s = q0 and was used in I to obtain estimates of total cross sections for K-shell ionization.

I

~.0t

i

t i

6.3. Ionization probabilities According to eq. (16), the monopole ionization amplitudes corresponding to the real and imaginary parts of A °÷ in eq. (48) are now given approximately as

'

°°'

Z~e 2 Ma+e(Ef) = -~v f ds Re[A°+(s)]F°e,=o(S), 0

(53)

oo

1.O

Ml+m(Et) = Z'e2 f ds hv J 0

and the ionization probability is obtained by integration over Ef of the sum of the squares of these amplitudes, according to eq. (15). The trajectory of a particle scattered with impact parameter b---0 is composed of two halftrajectories of incidence and emission, respectively. As noted above, the imaginary terms in the corresponding path functions A °- and A °÷ have opposite signs. For monopole excitation, there is no dependence on direction, and hence the amplitude for ionization is obtained as

M°o(E,, 0, 0)

=

(54)

2M.R'(E,),

independent of scattering angle. We may therefore calculate the ratio between the probabilities for ionization in scattering and decay from the amplitudes in eq. (53),

4 f dErIMR+e(E,)I2 P(O) =

o

(55)

PdecayidEIIMRe(Ef)I2"~/

dEflMl+m(Ef)12

o

o

Numerical results are given in fig. 8 as functions of ~: for fixed y. With the approximation in eq. (53), the scaling with ~: = (/qo is exact when unscreened hydrogenic wave functions are used [eq. (52)]. The nonrelativistic result (y = 1) is in the sc region included in fig. 8 virtually identical to that obtained

0.[3 0.02

0.05

O.t

0.2

0.5

I .0

2.0

~'r~dlr K Fig. 8. Ratio of ionization probabilities for scattering and decay for several values of y. These results are calculated in the straight-line monopole approximation. Also shown are experimental results for ot scattering/decay for 84Po, where y - 0.8. Shown as circles are a few results from ref. 37. These nonrelativistic values include the dipole contribution.

on the basis of Chiocchetti and Molinari's analytical expression [38], and this confirms the approximation in eq. (53) for small s~. The only experimental information on the ratio in eq. (55) is from measurements of K-shell ionization in a decay and helium scattering. We have included our previously obtained results [36] for a decay of polonium isotopes in fig. 8. The corresponding sc values have been corrected according to the prescription discussed in sect. 5. The value of y is close to 0.8, and the results are in fair agreement with the corresponding curve. As seen from the more detailed calculations [40,41], one should not expect better accuracy of such a simple estimate. The dipole contribution, for example, is not negligible ( - 1 0 % ) . The most serious limitation of these results is in fact the neglect of the dipole term. For sc > 1, the dipole contribution to K-shell-ionization cross sections is large, and even for s~ < 1, the imaginary part of the path factor A~*(s) [eq. (46)] may play an important role. It then becomes

J.U. Andersen et al. / Impact-parameter dependence

necessary to specify the scattering angle in eq. (55) since this term gives an angular dependence of P ( b = 0 ) [16]. However, this feature also makes it possible to check experimentally the magnitude of the imaginary dipole contribution [3]. It is also worth noting that the strong relativistic enhancement of the real part of the monopole term [eq. (53)] reduces the relative importance of the dipole terms for small values. Some information about the influence of the dipole terms in the nonrelativistic case may be obtained from the work of Ciocchetti and Molinari. First, the relative magnitude of the monopole and dipole contributions in particle decay may be observed in fig. 2 of ref. 37. For ~ < 1 (~ = 2vp/Z in their notation), the dipole contribution is seen to be small. Second, in fig. 9 of ref. 37, a few numerical examples are given, from which the ratio in eq. (55) may be evaluated, including dipole contributions. These results are indicated in fig. 8 and are seen not to be too different from the estimates based on the monopole terms alone.

7. Comparison with experiment In this section, we shall compare the data described in sect. 3 to the results of semiclassical calculations (sect. 4) modified according to the correction scheme discussed in in sect. 5. Furthermore, we shall discuss the merits and limitations of this scheme through comparisons of the calculations with a few other sets of data.

7.1. Present data The data for protons on copper and silver were used earlier to test the scaling relations predicted by the SCA calculations (figs. 4 and 5), and while the shape of the ionization function P(b) agreed with the predicted scaling, albeit with a discrepancy in magnitude of the width, the absolute magnitude of the ionization probability violated the scaling relation. Comparisons for copper and silver with the modified SCA calculations are shown in figs. 9 and 10 for two proton energies. The unmodified SCA results, as obtained from the parametrization in sect. 4.4, are indicated by the dashed

97

2.4 0.707 'I o l .5

MeV

H

on

Cu

\\

7~ ~_0 .fl

~.

0.0 1.2

-

\

' i .0

"~'-'~"~]'x

* ~+' ~ ' ~ " - ~ ' - ~ 2.0 2.000

--~

MeV H o n

3.0 Cu

0.B

3

.

~_0.4

0.0

0.0

++

".. ++ ~

' I .0

'

2.0 b

' 3.0

[pm]

Fig. 9. Comparison of selected Cu data with results at various stages of the correction procedure. The curves of short dashes are unmodified SCA results as obtained from the parametrization [eqs. (25) and (28)]. The curves of long dashes represent values after the binding correction [eq. (37)], and the curves of long dash-short dash also include both the Coulomb repulsion and energy-loss corrections [eqs. (39) and (40)]. Finally, the solid curves include all corrections with the relativistic effects being given by eq. (43).

curve, while the results obtained after successive corrections for binding, for Coulomb repulsion and energy loss, and for relativistic effects, are indicated by long dashes, long and short dashes, and a solid line, respectively. The corrections are not very large and tend to cancel, in particular for the case with the largest corrections, 1 MeV protons on silver. The tendency of a more rapid decrease than predicted of P(b) with increasing b is still observed (screening effect, see sect. 4.2), but the discrepancy noted in fig. 5 for the absolute magnitude of P(b) has disappeared and thus can be attributed to the combined effects of binding, Coulomb repulsion, and relativistic corrections. For the gold measurements, the corrections are very large, as can be seen from fig. 11. The relativistic corrections leading to the solid curve were here as well as for the previous figures evaluated from the simple procedure [eqs. (42)

J.U. Andersen et al. / Impact-parameter dependence

98

½

1.2

:

0.5 l.OO0

MeV H on Rg

!

0.4

~

2.ODD MeV H on

Au

o_0.4

~?0.3 0.0

'

0.5

1.2

[ _ ~

! .O

'

I .5

2.0

2.500 MeV H on Fig

o0.B p - " ' , + \ 121

o. 4

0.0

0.0

"~+'\

'

0.5

! .O b [pm]

! .5

~--

2.0

Fig. 10. Comparison of selected Ag data to results of the correction procedure. T h e various stages are shown with the same notation as in fig. 9.

and (44)]. Results from the more accurate procedure based on expressions in ref. 34 are indicated by the dotted curve, which fits the measurements slightly better. This curve may be compared to the result of Amundsen [42] based on a full SCA calculation with relativistic wave functions and a hyperbolic projectile path, and apart from the region of very small impact parameters, b < 0 . 1 p m , the two curves agree extremely well. For b - 0, the reduction due to Coulomb repulsion is somewhat larger [P(0) lower by -25%] in the complete calculation. This reduction is compensated for by a contribution from the imaginary dipole term at large scattering angles, and the good agreement at small b values in fig. 11 is therefore somewhat fortuitous. Still, the agreement for the gold case with a more detailed and accurate evaluation is an important check of our corrections for Coulomb repulsion and relativistic effects.

Z2. Applicability of correction scheme For an evaluation of the accuracy of the correction procedure, such a comparison with more

0.0 0,0

0.I

0.2 b

0.3

0.4

0.5

Epm]

Fig. 11. Comparison of ionization probabilities for 2 M e V protons on Au. T h e notation is the same as in fig. 9, but here a dotted curve is included, representing the results obtained with the more accurate (monopole) relativistic correction. T h e comparison between this curve and the solid curve indicates the accuracy of the approximation given by eq. (43).

detailed calculations is obviously to be preferred to comparison with experiment. As discussed in sect. 5, also the individual corrections for Coulomb repulsion and for relativistic effects have been tested in this way. The binding correction is more difficult to include in ab-initio calculations. A comparison with some of our earlier measurements [11] is shown in fig. 12. The curves include all the corrections, but the effects of Coulomb repulsion and relativistic corrections are small (cf. fig. 9). The good agreement indicates that the simple binding correction largely accounts for the observed deviation from the Z~ proportionality of P(b), obtained in a simple SCA treatment. In ref. 11, these data were used to demonstrate the necessity of including the wave-function relaxation in the binding correction. Without this effect, the correction is much too large. As argued in I, this is in general the case when the binding correction is important. In ref. 11, also measurements for lower velo-

J.U. Andersen et al. / Impact-parameter dependence

cities were made for oxygen on copper, and for these, the ionization probability tends to be somewhat lower than calculated. It appears from the detailed study in ref. 27 that these discrepancies can be explained as a reduction due to a variation of the binding energy during the collision, and that such a small reduction generally should be present for low projectile velocities, ~ ,~ 1. On the other hand, for large velocities, ~ - 1 , the variation of the binding energy during the collision will result in an increase of P(b) above the estimate based on a constant (modified) binding energy. While for the individual corrections, our scheme appears to be fairly accurate, one might expect larger discrepancies in cases where all corrections are large. In particular, the evaluation of the relativistic correction with a modified Z parameter in the wave function and with reduced projectile velocity is of uncertain accuracy. One indication of a surprisingly high accuracy was obtained in connection with the measurements for protons on gold discussed above. A comparison for an even more extreme

99

case, chlorine on lead [43], is shown in fig. 13. Also the binding correction is large here. The measured total cross sections for ionization are compared to calculations, including the detailed b-dependent corrections (solid line), or the simple corrections suggested in I for total cross sections (dashed line). The relativistic correction in this simple prescription, however, was replaced by the more accurate procedure suggested at the end of sect. 5.4. The prescription is seen then to reproduce the results of the more detailed procedure very well, and also the agreement with the data is excellent, considering the very large magnitude of all the corrections (one to two orders of magnitude). In fig. 13 is also shown a comparison between measured and calculated probabilities of ionization for b - 0 (backscattering). Here, the discrepancies are much larger although still small compared to the magnitude of the corrections. At 48MeV, the binding and repulsion cor2

t0 ° 5

2

//y+z/__

I0 -I 10 -3

i0 -2 5

on

Pb

,9/+

=/

5

2

Cl

il/l/J+

2

10 -3

2

P(O)

J3

+

+

s

8-

2

10 -4

10-4 5

2

i O -s 2O

2

0,0

2.0

1.0 b

3.0

[pm]

Fig. 12. The K-shell ionization probability for various 2 MeV/amu projectiles incident on Cu [11]. The solid curves include all corrections, but for the heavier ions, binding corrections are dominating.

i

40

i

J

h

60 80 100 E [ MeV ] Fig. 13. Total K-shell-ionization cross section (upper part) and ionization probability for b ~ 0 (lower part) for CI on Pb [43]. The solid curves are from resuRs of the impactparameter-dependent correction scheme. The dashed curve represents the total cross section as obtained from the simple corrections described in ref. 8, except for the change in the relativistic correction discussed at the end of sect. 5.4.

100

J.U. Andersen et al. / Impact-parameter dependence

rections reduce P(0) by factors of 14 and 10, respectively, while the relativistic effects lead to an increase by a factor of 400! In view of the large magnitude of these partly cancelling corrections, the discrepancies should be interpreted with caution. Since there is a fairly large discrepancy even for the highest energy where the repulsion correction is small (a factor of 1.8), it appears, however, that the discrepancy may originate in the evaluation of the relativistic correction, which is large even at 100 MeV (a factor of 85). It was perhaps also to be expected that the simple one-parameter relaxation of the wave function (sect. 5.2) would overestimate the highmomentum content of the wave function and hence the ionization probability. But obviously, more differential measurements on systems of this kind are needed for comparison before any definite conclusions can be drawn. Chlorine on lead is an example, where the simple prescriptions may be used to obtain quantitative estimates of the importance of the different corrections. Of course, these estimates will be more accurate when the corrections are small and, as discussed in ref. 12, the correction scheme may for such cases be used to eliminate the influence of the effects contained in the scheme and thus perhaps disclose a systematic trend in the data. This is illustrated in fig. 14,

1.50

125 o

I 0 x~

~J 0

o + <> ×

0.75

0.50

0

0.5

1

Ti Cu Ag Pb 15

= lad IrK

Fig. 14. Ratio of experimental K-ionization data to theoretical values obtained with the b-dependent corrections. This figure is equivalent to fig. 5 of ref. 12, which was in error, mainly for the heavier targets. The theoretical results are here based on the full relativistic correction with the formulae of ref. 34 (see sect. 5.4), which is slightly different from that given by eqs. (42) and (44).

which is equivalent to fig. 5 of ref. 12. The theoretical cross section used for that figure contained an error in the b-dependent relativistic correction discussed in sect. 5.4, which we discovered during the preparation of the present paper. In effect, the relativistic correction factor was too low by a factor of -~,. With the corrected factor, the measurements for different target materials are more consistent (except for lead), and the values of o'e~p/o'th cluster in the region of 0.8-0.9 for low ~c values. This reduction below unity corresponds to the narrowing of the P(b) functions due to the effect of screening on final-state wave functions. We would like to express our gratitude to P.A. Amundsen, J.M. Hansteen, and L. Kocbach, members of the Bergen group, for the very great stimulation we have had through numerous discussions and visits. The idea and the work presented here have developed over the last ten years through collaboration with many visitors and colleagues at the Institute of Physics in Aarhus, and we would like to acknowledge the important contribution by L.C. Feldman, C.D. Moak, and K. Taulbjerg in particular.

References [1] P.C. Gugelot, in Direct interactions and nuclear reaction mechanisms, eds., E. Clementel and C. Villi (Gordon and Breach, New York, 1962) p. 382. [2] J.F. Chemin, S. Andriamonje, J. Roturier, B. Saboya, J.P. Thibaud, S. Joly, S. Plattard, J. Uzureau, H. Laurent, J.M. Maison and J.P. Shapira. Nucl. Phys. A331 (1979) 407. [3] S. R/Shl, S. Hoppenau and M. Dost, Phys. Rev. Lett. 43 (1979) 1300. [4] M.S. Freedman, Ann. Rev. Nucl. Sci. 24 (1974) 2119. [5] J.U. Andersen and J.A. Davies, Nucl. Instr. and Meth. 132 (1976) 179. [6] J. Bang and J.M. Hansteen, Mat. Fys. Medd. Dan. Vidensk. Selsk. 31 (1959) no. 13. [7] L. Kocbach, J. Phys. B9 (1976) 2269. [8] E. Laegsgaard, J.U. Andersen and M. Lund, Proc. 10th Int. Conf. on The physics of electronic and atomic collisions, ed., G. Watel (North-Holland, Amsterdam, 1978) p. 353. [9] M. Lurid, Thesis (University of Aarhus, 1974) unpublished. [10] E. Laegsgaard, J.U. Andersen and L.C. Feldman, Proc. Int. Conf. on Inner-shell ionization phenomena, Atlanta, Georgia (1972), eds., R.W. Fink et al. (USAEC, Oak Ridge, Tenn., USA, 1973) p. 1019.

J.U. Andersen et al. / Impact-parameter dependence [11] J.U. Andersen, E. Laegsgaard, M. Lund and C.D. Moak, Nucl. Instr. and Meth. 132 (1976) 507. [12] E. Laegsgaard, J.U. Andersen and F. H.Ogedal, Nuel. Instr. and Meth. 169 (1980) 293. [13] E. Laegsgaard, J.U. Andersen and M. Lund, Phys. Fenn. 9, suppl. 1 (1974) 52. [14] N. Bohr, Mat. Fys. Medd. Dan. Vidensk. Selsk. 18 (1948) no. 8. [15] D. Trautmann and F. R6sel, Nucl. Instr. and Meth. 169 (1980) 259. [16] J.U. Andersen, L. Kocbach, E. Laegsgaard, M. Lund and C.D. Moak, J. Phys. B9 (1976) 3247. [17] K. Alder and A. Winther, Electromagnetic excitation (North-Holland, Amsterdam, 1975). [18] W. Bambynek, B. Craseman, R.W. Fink, H.U. Freund, H. Mark, C.D. Swift, R.E. Price and P.V. Rao, Rev. Mod. Phys. 44 (1972) 716. [19] J.M. Hansteen, O.M. Johnsen and L. Kocbach, At. Data. Nucl. Data Tables 15 (1975) 305. [20] L. Kocbach, Z. Physik A279 (1976) 233. [21] L. Kocbach, J.M. Hansteen and R. Gundersen, Nucl. Instr. and Meth. 169 (1980) 281. [22] O. Aashamar and L. Kocbach, J. Phys. B10 (1977) 869. [23] W. Brandt, K.W. Jones and H.W. Kraner, Phys. Rev. Lett. 30 (1973) 351. [24] W. Brandt, R. Laubert and I. Sellin, Phys. Lett. 21 (1966) 518. [25] D. Jamnik and C. Zupan~i/~, Mat. Fys. Medd. Dan. Vidensk. Selsk. 31 (1957) no. 2. [26] L. Kocbach, private communication. [27] E.C. Goldberg and V.H. Ponce, Phys. Rev. A22 (1980) 399. [28] G. Basbas, W. Brandt and R. Laubert, Phys. Rev. A7 (1973) 983.

[29] [30] [31] [32] [33] [34] [35] [36]

[37] [38] [39]

[40]

[41] [42] [43]

101

P.A. Amundsen, Phys. Lett. 55A (1975) 79. B. Miiller and W. Greiner, Z. Naturforsch, 31a (1976) 1. B. Miiller, private communication. L. Kocbach, Phys. Norv. 8 (1976) 187. P.A. Amundsen, L. Kocbach and J.M. Hansteen, J. Phys. B9 (1976) L203. P.A. Amundsen and L. Kocbach, J. Phys. B8 (1975) L122. P.A. Amundsen, J. Phys. B9 (1976) 971. M. Lund, E. Laegsgaard, J.U. Andersen and L. Kocbach, Abstracts of Papers 10th Int. Conf. on The physics of electronic and atomic collisions (Commissariat ~t l'Energie Atomique, Paris, 1977) p. 40; M. Lund, Ph.D. Thesis (University of Aarhus, 1978) unpublished. G. Ciocchetti and A. Molinari, Nuovo Cim. B40 (1965) 69. M. Lund, J.U. Andersen, E. Laegsgaard and L. Kocbach, to be published. M. Abramowitz and I.A. Stegun, Handbook of mathematical functions (NBS Applied Mathematics Ser, 55, 1964). L. Kocbach, Nordic Spring Symp. on Atomic inner-shell phenomena (Dept. of Physics, Univ. of Bergen, Norway, 1978) vol. 2, p. 65; L. Kocbach, J.U. Andersen, E. Laegsgaard and M. Lund, Abstract of Papers 10th Int. Conf. on The physics of electronic and atomic collisions (Commissariat ~ l'Energie Atomique, Paris, 1977) p. 42. L. Kocbach, J.U. Andersen, E. Laegsgaard and M. Lund, to be published. P.A. Amundsen, J. Phys. B10 (1977) 2177; P.A. Amundsen, J. Phys. B l l (1978) 3197. D. Burch, W.B. Ingalls, H. Wieman and R. Vandenbosch, Phys. Rev. A10 (1974) 1245.