Jakubaßa-Amundsen, Doris H.

English

Universität München: Elektronischen Publikationen

Z. Physik A 293, 281-285 (1979) Zeitschrift  
Physik A 
9 by Springer-Verlag 1979 
 -Electrons from K- and L-Shell Ionization in Relativistic Systems* 
D.H. Jakubassa 
Institute of Physics, University of Aarhus, Aarhus, Denmark 
Received June 22, Revised Version August 10, 1979 
Inner-shell ionization in adiabatic heavy-ion collisions is calculated within the monopole 
approximation by using the relativistic united-atom representation for the wave func- 
tions. As an example, ionization probability and energy distribution of the f-electrons is 
given for the (Pb, Pb) system and compared with experiment. 
1. Introduction 
The spectroscopy of electrons emitted in slow heavy- 
ion collisions is one of the tools to study the proper- 
ties of strongly bound quasi-atomic states. High- 
energy g-electrons can be attributed to the high com- 
ponents of the momentum distribution of the initially 
bound electrons, and provide thus information on 
their energy and wave function. When using very 
heavy collision partners the electronic states of 
superheavy atoms transiently formed during the col- 
lision, can be studied. 
In the recent years there have been a number of 
experimental and theoretical approaches to this prob- 
lem. Starting with the bombardment  of heavy ions 
such as Pb, with lighter projectiles [1] one has now 
also used heavy projectiles [2] and detected electrons 
with an energy much higher than the K-shell binding 
energy of the corresponding united atom. Theoreti- 
cally, the ionization probability for slow collisions 
has been estimated in the framework of the atomic 
model, using Born approximation [3] and adjusting 
the wave functions to the destorted nuclear field [43. 
This method has been applied for asymmetric sys- 
tems. When the nuclear charges are of equal magni- 
tude the molecular description is preferred, where 
one chooses the wave functions as eigenstates to the 
two-center field while the transitions are caused by 
the change of the field in time. Since the momentum 
transfer to the electron is large in heavy systems, it 
has been shown in the case of K-shell ionization that 
the dominant part of the transition is induced by the 
monopole expansion term V o of the two-center poten- 
tial [5]. Thus the two-center problem can be reduced 
* This work was partially supported by the Deutsche Forschungs- 
gemeinschaft 
to a one-center problem by choosing the wave func- 
tions as eigenfunctions to the potential V0, so that 
one avoids the problem of calculating two-center 
continuum states. Another way to get rid of the two- 
center problem in slow collisions, is to keep the full 
potential, but take the wave functions as eigenstates 
to the united atom [6] which is a good approxima- 
tion since the inner-shell ionization occurs mainly at 
very small internuclear distances R. Thereby one can 
include transitions of higher multipolarity which may 
become important for the excitation of higher p- 
states. 
In our work we combine the two approximations, i.e. 
we use united-atom wave functions and a monopole 
potential. Instead of V o we take, however, a one- 
center monopole potential with time-dependent 
charge. This is justified because the radius of the electron 
orbit is much larger than R (in the ionization region) 
such that from the electron point of view the two 
nuclei are well described by one charge-changing 
nucleus. Further, the calculations are thereby sim- 
plified considerably. In Sect. 2 we derive the cross 
section for K- and L-shell ionization, and compare in 
Sect. 3 with other theories and experiment. 
2. Theory 
In symmetric or near-symmetric collisions the domi- 
nant contribution to K-shell ionization comes from 
the 2p~r molecular state which correlates to the gll- 
shell of the united atom, and only a minor fraction 
comes from 1 s a ionization. 
In the adiabatic perturbation theory the amplitude 
for direct transitions is given by 
0340-2193/79/0293/0281/$01.00 
282 D.H. Jakubassa: 6-Electrons from K- and L-Shell Ionization 
ay i = dt (0yl I tPi) ~ ei/~ !dt(e,-e,). (2.1) 
- :e g f  - - / 5 i  
When one approximates the wave functions 0 and 
energies E of initial and final state by their time- 
independent values in the united atom limit [7] (the 
so-called Briggs model [6]), one obtains by means of 
partial integration 
1 ~ d t ( @ l  VIOl) e ~I~(E~-~)t. (2.2) 
a f i = ~  -oo 
We replace the two-center potential V by a monopole 
field [8] 
Z(R)  e a 
V(t) - (2.3) 
r 
and determine the time-dependent charge by a fit to 
the 1 s a level (obtained from a two-center calculation 
[5] for fixed internuclear distance R) and by means of 
the relation Z(E) between energy and nuclear charge. 
In the case of the (Pb, Pb) quasimolecular system, we 
take 
84-0.485R + 3.5 • 10-3R 2 
R<=45 fm 
779 + 29 • 10- 3 R) 
Z(R)  = Z 2 + 
45 fm<R-<60 fm 
104.9/R0.1 e-7.006 • 10-4R 
R > 6 0  fm (2.4) 
where Z 2 is the target charge. For lighter systems one 
may obtain Z(R) by means of a variational principle 
[8]. 
For the wave functions ~p we take relativistic func- 
tions belonging to an atom with charge Z. In order to 
obtain the right impact-parameter dependence of the 
ionization probability with the potential (2.3) it is 
important to take the charge Z that enters into 
from (2.4) at the distance of closest approach R(b, 
t=0). In the case of high-relativistic united atoms 
where the point-nucleus Dirac wave function is no 
longer valid, one may introduce an effective charge 
Z~ff<137 into the Dirac function. This Z~u results 
from a fit of the K-shell ionization probability calcu- 
lated with Dirac functions (to Neff) to a calculation 
using Hartree-Fock wave functions for extended nu- 
clei [9]. For the P b + P b  united atom (Z=164) for 
example, one finds Zaf  = 134.5. In this high-Z region, 
Z~ff and thus the wave functions vary slowly with Z. 
The energy El (also to be taken at R(b, t=O))  is 
obtained from a two-center calculation [5]. Actually, 
since it is only needed for small R, it may also be 
found from united-atom perturbation theory. R(t) is 
determined by a Rutherford trajectory9 
The advantage of the potential (29 together with the 
use of time-independent energies and wave functions 
is the splitting of the transition amplitude (2.2) into a 
time integral Tyi(b ) and a space integral 
e2(~o  ~ t) 
a f i  -- i h dt Z(R)  e i/~(Es-E~) 
e 2 
- i h Tzi(b) MYi(Y3~m(f2k') e) (29 
which can be evaluated independently9 The exact 
two-center potential factorizes only in momentum 
space [6], thus involving an extra integral (over mo- 
mentum). 
Using the partial wave expansion of the continuum 
Dirac functions [10], the angular part of the ejected 
electron is represented by the spherical harmonic 
spinor Yj~m projected on the polarization direction e 
of the electron. My~ is the remaining part of the space 
integral and can be evaluated analytically. The 
angular integration is trivial since only monopole 
transitions occur9 We take Landau's definition [10] 
of the relativistic Coulomb waves and normalize 
them to plane waves 9 
f, g(ky, r)= 2 ~ n -  ~ (1 +_ m c 2 / E f )  -~ 
9 e~V/2 IF(y + 1 + i v)[ 
F(27+1) ( k f r ) ' - i  
9 Im, Re(e iker +i~- 1F i (32 - i v, 2 ]2 + 1, - 2 i k f  r)). 
Then, the radial integrals involved are of the follow- 
ing type 
oo 
I(n) = ~ dr r 2v-" e -~~ eik*" 
0 
9 , F l ( Y - i v ,  2 7 + l  , - 2 i k r r )  
= F ( 2 7 _ n +  l ) ( c % _ i k , ) - 2 ~ + .  1 
[ - 2 i k y ]  
9 2F, ~32-iv, 2 3 2 - n + l ,  2 3 2 + l , ~ _ - i ~  ! (2.6) 
where the hypergeometric function 2/71 simplifies for 
n = 0  to 
[1+ 2 ikr  ~-v+i~ 2Fl(t/=0) = 
~o - i k ~ /  
In the case of lst/2 and 2p1/2 initial states, 32 
= 1 / 1 -  (Zoff eZ/h c) 2, ~o is the inverse shell radius (e~ 
= Z e f f / a o ,  ~ = c ~ ( 2 + 2 7 )  -t/2, ao=h2/me2),  k f  and ET 
the momentum and energy of the ejected electron, 
D.H. Jakubassa: &Electrons from K- and L-Shell Ionization 283 
respectively, and v=Z~ff e 2 Ef/((h c) 2 ky). The space 
integral Myi is given by the sum of the matrix ele- 
ments of the large and small component of ~b: 
M f, l s,/~ = N i  ' N j -  [1/(1 + y)(1 + m c2/Ef) 
9 Im(e  Ir I , (n  = 1)) 
-1/ (1  - 7)(1 - m c2/E:) Re(e i~" Is(n = 1))] 
M:,  2v,/2 = N~' N: {1/(1 + g2/2)(1 + m c2/Ef)  
9 [(g2 - 2 )  Im(e ~r Iv(n = 1)) 
2(g z - 1)a~ Im(eir p Iv(n = 0))] 
2 7 + 1  ] 
- 1/(1 - g2/2)(1 - m c2/Er [g2 Re( eiep Iv(n = 1)) 
2(g 2 -  1)ag Re(eir ~ Ip(n=O))]~ (2.7) 
2 2 + 1  J )  
where g 2 = ( 2 + 2 7 )  1/2. In (2.7) enters the real and 
imaginary part of I(n) from (2.6), evaluated with c~ 
and c~, respectively. 
exp(i ~) = ((7 + i v)/( - 1 + i v m c2/E:)) ~, 
exp(i ~p) = ((y + i v)/(1 + i v m c2/Ef)) 89 
and the normalization constants are 
,_[ 1] ,  
N~ - (2c~) 2~+a2F(27+1)  
N , =  [(2 eg)2,+l 27 +'1 ]~ 
4F(27 + 1) gz(g2-  1) 
N r  ~-~ IF(?+ 1 +iv)J 
where F is the gamma function. 
The transition probability differential in electron mo- 
mentum is given by 
dP(b) _ 2 k}  ~ df2k: Z la:*J 2. (2.8) 
dkr 
The factor 2 accounts for the 2 electrons in the initial 
state (=sum over the spin states). The integral over 
electron angle is trivial since the monopole approxi- 
mation corresponds to isotropically distributed elec- 
trons (as long as the initial state is not polarized). We 
get from (2.5) 
dP(b) = 2 k}(e2/h T:,(b) Mfl) 2. (2.9) 
dkr 
From this we obtain the double differential cross 
section by multiplying it with the elastic Rutherford 
cross section 
d 2 a dP(b) d~ R 
d Q d k :  d k f  dO" 
(2.10) 
3. Results and Comparison with Experiment 
The total probability P(b) for inner-shell ionization 
calculated from (2.9) is shown in Fig. 1 for the 1 s G 
and 2 p g  initial states. As an example, the system 
(Pb, Pb) was chosen at a projectile energy of 4.7 MeV/ 
amu corresponding to v/c=O.1 which is much 
smaller than the orbiting velocity of the inner-shell 
electron such that the adiabatic description is valid. 
For comparison, Fig. 1 shows also the results from a 
two-center calculation [5]. The 1 s • ionization prob- 
ability can be reproduced very well within our mono- 
pole model while in the 2p o- case only the relative 
dependence on impact parameter is the same but the 
absolute values are a factor of 3 too low. The better 
agreement in the 1 s a case is due to the choice of the 
potential (2.3) which is deduced from ground state 
properties. For higher states, the two-center character 
of the potential becomes more important even if one 
only retains its monopole expansion term as done in 
[5]. 
The experiments were performed with the Unilac 
accelerator in Darmstadt. Both ionization probability 
. I l I I I 
10 -1 _ %~%~ 
10 -2 " " ~ "  2po 
10-3 
\ \ \  1SO 
[ I I I I 10"40 20 /,0 60 80 100 
b(fm) 
Fig. 1. Ionization probability for the Pb + Pb system (4.7 MeV/amu 
projectile energy) versus impact parameter. Full lines correspond 
to the ionization of 1 sa and 2p a states calculated in the monopole 
approximation, dashed lines are the calculations from [5]. Experi- 
mental data (sum of l sa  and 2pa) are taken from [11] 
284 D.H. Jakubassa: &Electrons from K- and L-Shell Ionization 
Ef (MeV) 
0.8 1.0 1.2 1./, 1.6 
3 /, 5 6 7 8 
kf (10 -3 fm -1 ) 
Fig. 2. Differential cross section for ~-electron emission in 4.7 MeV/ 
amu (Pb, Pb) collisions as a function of electron momentum k s. 
The data are from [12]. Theory is normalized to experiment 
1 i i t i i I i i i 
9 " ~ k f  = 30o - 5 0 ~  
\ ' X  
10 ~ 9 Okf = 30'= 
E 
N 1 
'~ "'."". (so.2,o 
10-  I i i I i i i i i i ~ 
0.8 1.0 1,2 1. z 1.6 IB 
Ef  (MeV) 
Fig. 3. Differential cross section for 3-electrons emitted in 4.7 MeV/ 
ainu (Pb, Pb) collisions as a function of total electron energy E s. 
The full lines are calculations within the monopole approximation, 
and the dashed line is the sum of l sa  and 2pa direct excitation 
from a two-center calculation (Soft). The experimental data are 
from Kienle et aI. (m) [12] and from Walcher et al. (o) [-13]. 
Theory is normalized to experiment (see text) 
[11] and high-energy electron spectra in coincidence 
with K-X-rays [2] were measured. The shape of the 
impact-parameter dependence of the ionization prob- 
ability is well described by theory (Fig. 1). However, 
aside from the direct ionization, there may also be 
contributions from transitions to higher bound states 
which are emptied previous in the collision, or from 
multistep processes that could account for the dis- 
crepancy in absolute value. Also experimental back- 
ground effects cannot be excluded. 
Figure 2 shows the differential cross section for elec- 
tron emission obtained from a coincidence experi- 
ment where besides the electron momentum k I and 
emission angle ,gk• also the scattering angle 01a b is 
measured. The theoretical curve is the sum of the 
1 s a and 2p o- ionization calculated from (2.10) and 
normalized to experiment at E I =0.8 MeV. 
In Fig. 3 the differential cross section is summed over 
all scattering angles and the experimental data are 
compared with the results from our monopole model 
(multiplied by a factor of 34.4) and from the two- 
center calculation (multiplied by 13.8). We find a 
good agreement in the energy dependence of the 
ejected electrons. Also shown are the separate contri- 
butions from the 1 s a and 2p cr ionization. While for 
lower electron energies the l s a / 2 p a  fraction is 
around 0.2-0.3, the 1 s a ionization becomes of equal 
importance for small impact parameters (b<20 fm) 
when Er exceeds ~I .2MeV.  Contributions from 
other initial states are neglected since they would 
involve multistep processes to create a K-vacancy. 
The energy dependence of the b-electrons mirrors 
thus mainly the behaviour of the 2p o- electrons and 
can be used to determine their momentum distribu- 
tion. If one is interested in the 1 s a state, one must 
either choose asymmetric systems where the spacing 
between the projectile and target 1 Sl/a state is large 
enough to prevent vacancy sharing, while another 
possibility would be to measure at very small impact 
parameters and high electron energy and use only 
slightly asymmetric systems (to avoid the symmetri- 
zation between projectile and target). 
It has also been attempted to measure the b-electron 
spectrum at different electron angles 0kz. Theoreti- 
cally, one expects a nearly isotropic distribution since 
monopole transitions are dominant for the j = l / 2  
states in K and L shell. Further, in the united-atom 
representation, dipole transitions cancel for symmet- 
ric systems. There may be an anisotropy originating 
from the fact that the electron cannot adjust com- 
pletely to the two-center field, thus leading to a finite 
dipole transition. Since the ratio of v and vp, the 
orbiting velocity of the united atom 2 Pl/2 electron, is 
below 0.3, this effect should be small. A relativistic 
investigation of this problem is in progress. 
I would like to thank C. Kozhuharov and P. Kienle for the insight 
in their experiments, and P.A. Amundsen and G. Soft for interest- 
ing discussions. 
References 
1. Kozhuharov, C., Kienle, P., Jakubassa, D.H., Kleber, M.: Phys. 
Rev. Lett. 39, 540 (1977) 
D.H. Jakubassa: &Electrons from K- and L-Shell Ionization 285 
Bosch, F., Krimm, H., Martin, B., Povh, B., Traxel, K., Wal- 
cher, Th.: MPI-Report, Heidelberg (1978) 
2. Berdermann, E., Bokemeyer, H., Bosch, F., Clemente, M., 
Giittner, S., Kienle, P., Kozhuharov, C., Krimm, H., Martin, 
B., Povh, B., Traxel, K., Walcher, Th.: DPG Verhandlungen, 
Spring Meeting Gent (1979) 
3. Choi, B.H., Merzbacher, E.: Phys. Rev. 177, 233 (1969) 
4. Andersen, J.U., L~egsgaard, E., Lund, M., Moak, C.D.: Nucl. 
Instrum. Methods. 132, 507 (1976) 
5. Soft, G., Betz, W., Miiller, B., Greiner, W., Merzbacher, E.: 
Phys. Lett. 65A, 19 (1978) 
6. Amundsen, P.A.: J. Phys. Bl l ,  L737 (1978) 
7. Briggs, J.S.: J. Phys. B8, L 485 (1975) 
8. Jakubassa, D.H.: Phys. Lett. 58A, 163 (1976); Z. Physik A285, 
249 (1978) 
9. Anholt, p~.: Phys. Rev. AI7, 976 (1978) 
10. Achieser, A.I., Berestezki, W.B.: Quantenelektrodynamik, p. 96. 
Frankfurt/Main: Deutsch 1962 
Landau, L.D., Lifshitz, E.M.: Relativistic quantum theory. Vol. 
4a, p. 110. Oxford: Pergamon Press 1971 
11. Behnke, H.H., Liesen, D., Hagmann, S., Mokler, P.H., Arm- 
bruster, P.: Z. Physik A288, 35 (1978) 
12. Kozhuharov, C., Kienle, P.: Private communications (1978) 
13. Walcher, Th.: Private communication (1979) 
Doris H. Jakubassa 
Institute of Physics 
University of Os/o 
Oslo 3, Norway 


Gamma-electrons from k- and l-shell ionisation in relativistic systems

Open Access LMU

Z. Physik A 293, 281-285 (1979) Zeitschrift Physik A 9 by Springer-Verlag 1979  -Electrons from K- and L-Shell Ionization in Relativistic Systems* D.H. Jakubassa Institute of Physics, University of Aarhus, Aarhus, Denmark Received June 22, Revised Version August 10, 1979 Inner-shell ionization in adiabatic heavy-ion collisions is calculated within the monopole approximation by using the relativistic united-atom representation for the wave func- tions. As an example, ionization probability and energy distribution of the f-electrons is given for the (Pb, Pb) system and compared with experiment. 1. Introduction The spectroscopy of electrons emitted in slow heavy- ion collisions is one of the tools to study the proper- ties of strongly bound quasi-atomic states. High- energy g-electrons can be attributed to the high com- ponents of the momentum distribution of the initially bound electrons, and provide thus information on their energy and wave function. When using very heavy collision partners the electronic states of superheavy atoms transiently formed during the col- lision, can be studied. In the recent years there have been a number of experimental nd theoretical approaches to this prob- lem. Starting with the bombardment of heavy ions such as Pb, with lighter projectiles [1] one has now also used heavy projectiles [2] and detected electrons with an energy much higher than the K-shell binding energy of the corresponding united atom. Theoreti- cally, the ionization probability for slow collisions has been estimated in the framework of the atomic model, using Born approximation [3] and adjusting the wave functions to the destorted nuclear field [43. This method has been applied for asymmetric sys- tems. When the nuclear charges are of equal magni- tude the molecular description is preferred, where one chooses the wave functions as eigenstates to the two-center field while the transitions are caused by the change of the field in time. Since the momentum transfer to the electron is large in heavy systems, it has been shown in the case of K-shell ionization that the dominant part of the transition is induced by the monopole xpansion term V o of the two-center poten- tial [5]. Thus the two-center problem can be reduced * This work was partially supported by the Deutsche Forschungs- gemeinschaft to a one-center problem by choosing the wave func- tions as eigenfunctions to the potential V0, so that one avoids the problem of calculating two-center continuum states. Another way to get rid of the two- center problem in slow collisions, is to keep the full potential, but take the wave functions as eigenstates to the united atom [6] which is a good approxima- tion since the inner-shell ionization occurs mainly at very small internuclear distances R. Thereby one can include transitions of higher multipolarity which may become important for the excitation of higher p- states. In our work we combine the two approximations, i.e. we use united-atom wave functions and a monopole potential. Instead of V o we take, however, a one- center monopole potential with time-dependent charge. This is justified because the radius of the electron orbit is much larger than R (in the ionization region) such that from the electron point of view the two nuclei are well described by one charge-changing nucleus. Further, the calculations are thereby sim- plified considerably. In Sect. 2 we derive the cross section for K- and L-shell ionization, and compare in Sect. 3 with other theories and experiment. 2. Theory In symmetric or near-symmetric collisions the domi- nant contribution to K-shell ionization comes from the 2p~r molecular state which correlates to the gll- shell of the united atom, and only a minor fraction comes from 1 s a ionization. In the adiabatic perturbation theory the amplitude for direct transitions is given by 0340-2193/79/0293/0281/$01.00 282 D.H. Jakubassa: 6-Electrons from K- and L-Shell Ionization ay i = dt (0yl I tPi) ~ ei/~ !dt(e,-e,). (2.1) - :e g f  - - /5 i  When one approximates the wave functions 0 and energies E of initial and final state by their time- independent values in the united atom limit [7] (the so-called Briggs model [6]), one obtains by means of partial integration 1 ~ dt (@l  VIOl) e ~I~(E~-~)t. (2.2) af i=~ -oo We replace the two-center potential V by a monopole field [8] Z(R) e a V(t) - (2.3) r and determine the time-dependent charge by a fit to the 1 s a level (obtained from a two-center calculation [5] for fixed internuclear distance R) and by means of the relation Z(E) between energy and nuclear charge. In the case of the (Pb, Pb) quasimolecular system, we take 84-0.485R + 3.5 • 10-3R 2 R<=45 fm 779 + 29 • 10- 3 R) Z(R) = Z 2 + 45 fm<R-<60 fm 104.9/R0.1 e-7.006 •10-4R R>60 fm (2.4) where Z 2 is the target charge. For lighter systems one may obtain Z(R) by means of a variational principle [8]. For the wave functions ~p we take relativistic func- tions belonging to an atom with charge Z. In order to obtain the right impact-parameter dependence of the ionization probability with the potential (2.3) it is important o take the charge Z that enters into from (2.4) at the distance of closest approach R(b, t=0). In the case of high-relativistic united atoms where the point-nucleus Dirac wave function is no longer valid, one may introduce an effective charge Z~ff<137 into the Dirac function. This Z~u results from a fit of the K-shell ionization probability calcu- lated with Dirac functions (to Neff) to a calculation using Hartree-Fock wave functions for extended nu- clei [9]. For the Pb+Pb united atom (Z=164) for example, one finds Zaf = 134.5. In this high-Z region, Z~ff and thus the wave functions vary slowly with Z. The energy El (also to be taken at R(b,t=O)) is obtained from a two-center calculation [5]. Actually, since it is only needed for small R, it may also be found from united-atom perturbation theory. R(t) is determined by a Rutherford trajectory9 The advantage of the potential (29 together with the use of time-independent energies and wave functions is the splitting of the transition amplitude (2.2) into a time integral Tyi(b ) and a space integral e2(~o ~ t) af i  -- i h dt Z(R) e i/~(Es-E~) e 2 - i h Tzi(b) MYi(Y3~m(f2k') e  (29 which can be evaluated independently9 The exact two-center potential factorizes only in momentum space [6], thus involving an extra integral (over mo- mentum). Using the partial wave expansion of the continuum Dirac functions [10], the angular part of the ejected electron is represented by the spherical harmonic spinor Yj~m projected on the polarization direction e of the electron. My~ is the remaining part of the space integral and can be evaluated analytically. The angular integration is trivial since only monopole transitions occur9 We take Landau's definition [10] of the relativistic Coulomb waves and normalize them to plane waves 9 f, g(ky, r)= 2 ~ n- ~ (1 +_ m c2/Ef )  -~ 9 e~V/2 IF(y + 1 + i v)[ F(27+1) (k f r ) ' - i  9 Im, Re(e iker +i~- 1F i (32 - i v, 2 ]2 + 1, - 2 i kf r)). Then, the radial integrals involved are of the follow- ing type oo I(n) = ~ dr r 2v-" e -~~ eik*" 0 9 ,F l (Y - iv ,  27+l  , -2 ik r r )  =F(27_n+ l ) (c%_ik , ) -2~+. 1 [ -2 iky ]  9 2F, ~32-iv, 232-n+l ,  232+l ,~_ - i~  ! (2.6) where the hypergeometric function 2/71 simplifies for n=0 to [1+ 2ikr  ~-v+i~ 2Fl(t/=0) = ~o - i k~/  In the case of lst/2 and 2p1/2 initial states, 32 =1/1 -  (Zoff eZ/h c) 2, ~o is the inverse shell radius (e~ =Zef f /ao ,  ~=c~(2+27)  -t/2, ao=h2/me2), kf  and ET the momentum and energy of the ejected electron, D.H. Jakubassa: &Electrons from K- and L-Shell Ionization 283 respectively, and v=Z~ff e 2 Ef/((h c) 2 ky). The space integral Myi is given by the sum of the matrix ele- ments of the large and small component of ~b: M f, l s,/~ = N i  ' N j -  [1/(1 + y)(1 + m c2/Ef) 9 Im(e Ir I ,(n = 1)) -1/(1 - 7)(1 - m c2/E:) Re(e i~" Is(n = 1))] M:, 2v,/2 = N~' N: {1/(1 + g2/2)(1 + m c2/Ef) 9 [(g2 -2 )  Im(e ~r Iv(n = 1)) 2(g z - 1)a~ Im(eir p Iv(n = 0))] 27+1 ] - 1/(1 - g2/2)(1 - m c2/Er [g2 Re( eiep Iv(n = 1)) 2(g 2 -  1)ag Re(eir ~ Ip(n=O))]~ (2.7) 22+1 J )  where g2=(2+27) 1/2. In (2.7) enters the real and imaginary part of I(n) from (2.6), evaluated with c~ and c~, respectively. exp(i ~) = ((7 + i v)/( - 1 + i v m c2/E:)) ~, exp(i ~p) = ((y + i v)/(1 + i v m c2/Ef)) 89 and the normalization constants are ,_[ 1],  N~ - (2c~) 2~+a2F(27+1) N ,= [(2 eg)2,+l 27 +'1 ]~ 4F(27 + 1) gz(g2- 1) Nr ~-~ IF(?+ 1 +iv)J where F is the gamma function. The transition probability differential in electron mo- mentum is given by dP(b) _ 2 k} ~ df2k: Z la:*J 2. (2.8) dkr The factor 2 accounts for the 2 electrons in the initial state (=sum over the spin states). The integral over electron angle is trivial since the monopole approxi- mation corresponds to isotropically distributed elec- trons (as long as the initial state is not polarized). We get from (2.5) dP(b) = 2 k}(e2/h T:,(b) Mfl) 2. (2.9) dkr From this we obtain the double differential cross section by multiplying it with the elastic Rutherford cross section d 2 a dP(b) d~ R dQdk:  dkf  dO" (2.10) 3. Results and Comparison with Experiment The total probability P(b) for inner-shell ionization calculated from (2.9) is shown in Fig. 1 for the 1 s G and 2pg initial states. As an example, the system (Pb, Pb) was chosen at a projectile energy of 4.7 MeV/ amu corresponding to v/c=O.1 which is much smaller than the orbiting velocity of the inner-shell electron such that the adiabatic description is valid. For comparison, Fig. 1 shows also the results from a two-center calculation [5]. The 1 s • ionization prob- ability can be reproduced very well within our mono- pole model while in the 2p o- case only the relative dependence on impact parameter is the same but the absolute values are a factor of 3 too low. The better agreement in the 1 s a case is due to the choice of the potential (2.3) which is deduced from ground state properties. For higher states, the two-center character of the potential becomes more important even if one only retains its monopole expansion term as done in [5]. The experiments were performed with the Unilac accelerator in Darmstadt. Both ionization probability . I l I I I 10 -1 _ %~%~ 10 -2 " " ~ "  2po 10-3 \ \ \  1SO [ I I I I 10"40 20 /,0 60 80 100 b(fm) Fig. 1. Ionization probability for the Pb + Pb system (4.7 MeV/amu projectile nergy) versus impact parameter. Full lines correspond to the ionization of 1 sa and 2p a states calculated in the monopole approximation, dashed lines are the calculations from [5]. Experi- mental data (sum of lsa and 2pa) are taken from [11] 284 D.H. Jakubassa: &Electrons from K- and L-Shell Ionization Ef (MeV) 0.8 1.0 1.2 1./, 1.6 3 /, 5 6 7 8 kf (10 -3 fm -1 ) Fig. 2. Differential cross section for ~-electron emission in 4.7 MeV/ amu (Pb, Pb) collisions as a function of electron momentum k s. The data are from [12]. Theory is normalized to experiment 1 i i t i i I i i i 9 "~kf  = 30o - 50~ \ ' X  10 ~ 9 Okf = 30'= E N 1 '~ "'."". (so.2,o 10- I i i I i i i i i i ~ 0.8 1.0 1,2 1. z 1.6 IB Ef (MeV) Fig. 3. Differential cross section for 3-electrons emitted in 4.7 MeV/ ainu (Pb, Pb) collisions as a function of total electron energy E s. The full lines are calculations within the monopole approximation, and the dashed line is the sum of lsa and 2pa direct excitation from a two-center calculation (Soft). The experimental data are from Kienle et aI. (m) [12] and from Walcher et al. (o) [-13]. Theory is normalized to experiment (see text) [11] and high-energy electron spectra in coincidence with K-X-rays [2] were measured. The shape of the impact-parameter dependence of the ionization prob- ability is well described by theory (Fig. 1). However, aside from the direct ionization, there may also be contributions from transitions to higher bound states which are emptied previous in the collision, or from multistep processes that could account for the dis- crepancy in absolute value. Also experimental back- ground effects cannot be excluded. Figure 2 shows the differential cross section for elec- tron emission obtained from a coincidence xperi- ment where besides the electron momentum k I and emission angle ,gk• also the scattering angle 01a b is measured. The theoretical curve is the sum of the 1 s a and 2p o- ionization calculated from (2.10) and normalized to experiment at E I =0.8 MeV. In Fig. 3 the differential cross section is summed over all scattering angles and the experimental data are compared with the results from our monopole model (multiplied by a factor of 34.4) and from the two- center calculation (multiplied by 13.8). We find a good agreement in the energy dependence of the ejected electrons. Also shown are the separate contri- butions from the 1 s a and 2p cr ionization. While for lower electron energies the l sa /2pa  fraction is around 0.2-0.3, the 1 s a ionization becomes of equal importance for small impact parameters (b<20 fm) when Er exceeds ~I.2MeV. Contributions from other initial states are neglected since they would involve multistep rocesses to create a K-vacancy. The energy dependence of the b-electrons mirrors thus mainly the behaviour of the 2p o- electrons and can be used to determine their momentum distribu- tion. If one is interested in the 1 s a state, one must either choose asymmetric systems where the spacing between the projectile and target 1 Sl/a state is large enough to prevent vacancy sharing, while another possibility would be to measure at very small impact parameters and high electron energy and use only slightly asymmetric systems (to avoid the symmetri- zation between projectile and target). It has also been attempted to measure the b-electron spectrum at different electron angles 0kz. Theoreti- cally, one expects a nearly isotropic distribution since monopole transitions are dominant for the j= l /2  states in K and L shell. Further, in the united-atom representation, dipole transitions cancel for symmet- ric systems. There may be an anisotropy originating from the fact that the electron cannot adjust com- pletely to the two-center field, thus leading to a finite dipole transition. Since the ratio of v and vp, the orbiting velocity of the united atom 2 Pl/2 electron, is below 0.3, this effect should be small. A relativistic investigation of this problem is in progress. I would like to thank C. Kozhuharov and P. Kienle for the insight in their experiments, and P.A. Amundsen and G. Soft for interest- ing discussions. References 1. Kozhuharov, C., Kienle, P., Jakubassa, D.H., Kleber, M.: Phys. Rev. Lett. 39, 540 (1977) D.H. Jakubassa: &Electrons from K- and L-Shell Ionization 285 Bosch, F., Krimm, H., Martin, B., Povh, B., Traxel, K., Wal- cher, Th.: MPI-Report, Heidelberg (1978) 2. Berdermann, E., Bokemeyer, H., Bosch, F., Clemente, M., Giittner, S., Kienle, P., Kozhuharov, C., Krimm, H., Martin, B., Povh, B., Traxel, K., Walcher, Th.: DPG Verhandlungen, Spring Meeting Gent (1979) 3. Choi, B.H., Merzbacher, E.: Phys. Rev. 177, 233 (1969) 4. Andersen, J.U., L~egsgaard, E., Lund, M., Moak, C.D.: Nucl. Instrum. Methods. 132, 507 (1976) 5. Soft, G., Betz, W., Miiller, B., Greiner, W., Merzbacher, E.: Phys. Lett. 65A, 19 (1978) 6. Amundsen, P.A.: J. Phys. Bl l ,  L737 (1978) 7. Briggs, J.S.: J. Phys. B8, L 485 (1975) 8. Jakubassa, D.H.: Phys. Lett. 58A, 163 (1976); Z. Physik A285, 249 (1978) 9. Anholt, p~.: Phys. Rev. AI7, 976 (1978) 10. Achieser, A.I., Berestezki, W.B.: Quantenelektrodynamik, p. 96. Frankfurt/Main: Deutsch 1962 Landau, L.D., Lifshitz, E.M.: Relativistic quantum theory. Vol. 4a, p. 110. Oxford: Pergamon Press 1971 11. Behnke, H.H., Liesen, D., Hagmann, S., Mokler, P.H., Arm- bruster, P.: Z. Physik A288, 35 (1978) 12. Kozhuharov, C., Kienle, P.: Private communications (1978) 13. Walcher, Th.: Private communication (1979) Doris H. Jakubassa Institute of Physics University of Os/o Oslo 3, Norway 

Gamma-electrons from k- and l-shell ionisation in relativistic systems

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