Nuclear photon scattering below and above the giant resonances

Nuclear Physics A530 (1991) 267-282 North-Holland

UC

SCATTERING N NC N. BARON1,2, G. BAUR' and M. SCHUMACHER2 2

' Institut für Kernphysik, Forschungszentrum Jülich, D-5170 Jülich.. Germany IL Physikalisches Institut, Universität Göttingen, D-3400 Göttingen, Germany Received 6 December 1990 (Revised 19 February 1991)

Abstract: Compton scattering on nuclei is studied from the general point of view in terms of matrix elements of microscopic nuclear operators in the low-energy region. The scattering amplitude is approximated by a consistent expansion up to second order in the photon energy where special care is taken about the low-energy Thomson limit. To extend the validity into the giant resonance region an explicit nuclear model isintroduced to get analyticexpressions for the scatteringamplitude without any further approximation. The consistent treatment of the matrix elements leads to the occurrence of form-factor effects . An interpretation of scattering data on 2" up to 50 MeV is given in terms of this model calculation . Additional excitation mechanisms are included in a phenomenological way .

The scattering of light is a powerful method to investigate the properties of systems like nuclei. Especially nuclear Compton scattering is a well-developed method in low-energy nuclear physics. In the h t 10 years several measurements have been performed' -4) mainly on '2C and 2oarb . Recent measurements on 208 Pb [ref. ,)] up to 100 MeV have focused interest in the medium-energy range. From the theoretical point of vievii titere is, a twofold way to get cross sections for nuclear Compton scattering. One is to use experimental data on the total photo-absorption cross section crabs of the nucleus under consideration as an input for the well-established dispersion relations. Another way is to assume a well-chosen nuclear model and to calculate explicitly the general expression for the scattering amplitude 6). In this paper we study nuclear Compton scattering up to around 50 eV on the basis of a simple but yet realistic model to have an understanding of the physics. Since the main role will be played by high-lying collective states, i.e. the giant resonances, we choose an extended Brink model 7,8), which allows a very simple and clear separation ofthe wavefunction in terms ofa collective variable and internal variables. With this method we are able to achieve a very clear separation of form-factor effects. After a general discussion of the Compton scattering amplitude and their lowenergy behaviour we present our theoretical model used to carry out a calculation 0375-9474/9l/$031 .50 © 1991 - Elsevier Science Publishers B.V. (North-Holland)

N. Baron et al. / Nuclear photon scattering

268

of the amplitude without any further approximation (sect. 2) . In sect. 3 we apply our model calculation to 208 Pb and compare our results to experimental data. To extend our interpretation of Compton scattering we treat in addition photo absorption on `' °s b even i energy regions above 50 eV. This is done in sect. 4. Our conclusions are given in sect. 5. t

sc

e

o nuclei

e consider the scattering of a real photon with definite energy and momentum by a nucleus, which is initially at rest and in the ground state. In the following we want to treat only the elastic channel of the reaction . In general such a process is described in very good appâoximation by the two well-known eyn an graphs of second order'). Since we assume the nucleus to be a system of nonrelativistic interacting nucleons, the amplitude itself has to be formulated nonrelativistically. e limiting process leads to the occurrence of three terms. One of the arises from the negative eigenstates, called seagull amplitude, wile the other two, called resonance amplitude, arise from the positive ones') . Fig. graphically shows the three contributions to the nonrelativistic scattering amplitude. e describe the initial and final photon by a vector potential in the transverse gauge )=

where , is

~` exp { i -,f - r} , (2.1 2 e well-Jr.- nown spherical basis and u = -:E 1 denotes the possible helicities . I0> W f' k f

f sea

10

ci f,kf

fR

Fig. 1. Symbolic picture of the three contributions to the scattering amplitude coming from the nonrelativistic reduction of the Feynman graphs . The first term is th;; seagull amplitude fsea, while the two other terms are put together in the resonance amplitude fR .

N. Baron et al. / Nuclear photon scattering

269

The scattering amplitude for elastic scattering on a spherical spin-zero nucleus, which is in general connected with the T matrix via 9 ) f = -(2v) 2 kikf Tr, ,

(2.2)

can be written aS b' 10) felast -

®)

+7 n

(®I

f

d3 rAf -,jn)(nl

f

dar

E -Ei - wi - te

d3 rAi ' .il n)(nl

J

P 10)

d3

2 wiwf ~ - 1 M (01

d ar

(2.3)

E' - Ei +wf+ie

Here p (r) is the charge density of the nucleus and j(r) the current density, which obeys the continuity equation - i[H,

.1 =

The energy denominators are given by lo) E, - Ei - toi

_ ( _

n

(

n

n

k; !Ä

(2.4)

p]

)

(')

k2

(

0

FA

-

)2

2AM P

(2 .5)

where we explicitly consider the recoil energies, while the index (i) refers to intrinsic energies . Following Ericson and Hüfner'°) the seagull amplitude reads in c.m. coordinates fsea = - f* .

;~'

Ze2

W

(01

exp 1i(

i-

f) -

xl'0) = - 9w ' ~,'

Ze 2 Fo .

(2.6)

Note that Foo is a form factor depending on the scattering angle O. 2.1 . LOW ENERGY EXPANSION (RE 'ISITED)

for To obtain a good approximation for the general scattering amplitude eq. (2.3) 11) - the small energies, we expand - following Ericson, Hüfner' °) and Petrunkin amplitude ir,: powers of the photon energy around zero. First we note that the seagull amplitude eq; (2.6) in lowest order is given by fsea - - ~ *

CA 'Ze2

/

,

(2 .7)

?7C)

aron et al. / Nuclear photon scattering

orison limit '2,13 ). To treat the remaining is is evidently different from the two arts of the resonance amplitude the energy denominator as well as the e occurrence of only even powers of k in the u erator has to be expanded . expansion of the matrix elements in the numerator is due to parity selection rules for the intermediate states . i e in the se of the seagull amplitude we point out the lowest approximation or the resonance amplitude : (2 .8)

e follow ricson, fner' o) for the calculation of the other contributions to the resonance amplitude. e list below the single contributions of the relevant combina tions of numerator d denominator order of expansion for an overall expansion u to second order in energy. sea ® -

°

(2 .9a)

~,° e2

(2.9b) (2.9c) (2 .9d)

a

(2.9e)

(2 .9f) ~° R,®) - - ~ .

1 1VZe2 (X2)k2 .

(2 .9g)

3 A

e upper index (,) denotes the order of energy dependence of the numerator and denominator, respectively. ( stands for mixed order in the expansion ofthe vector potentials in the numerator.) e dipole operator is defined by: (2.10) e sum of these contributions is the complete scattering amplitude up to terms of second order in energy: fIL

.

Il°

f

2 A Ad (1-3k2(2))-adv2

f* X

+

i°X]~-

P

2

(2 .11)

ere a is the electric polarizability , 1(O1 ZIn)12 a=2y nE (') - E(') n n

0

( 2.12 )

N. Baron et al. / Nuclear photon scattering

271

Xp is the paramagnetic susceptibility X =2V n

I(OIMIZIn)e2 - E(') E(>) n O

(2 .13)

and %D is the diamagnetic susceptibility 1 Zee

Xn=_6

(x2)+

6A

'

e particularly emphasize that the well-known Thomson limit 12.13) holds in every case and that it originates from the lowest approximation of the seagull amplitude as well as from the one of the resonance amplitude: ~,+ NZe2/A )g; _ - ~* - ~.'(~)2/A f~mom = (-Ze 2/

-

(2.15)

We further note that the single contributions to the Thomson amplitude from the seagull and resonance terms, respectively, strongly depend on the model under consideration. In fact, if we include quark degrees of freedom, the single contributions are shifted 14) . For that reason a splitting of the overall amplitude in a sum 1s) of amplitudes due to special parts of the scattering process need not be invariant. 2.2. EXTENDED BRINK MODEL

Since our aim is to analyze scattering data up to energies about 50 eV, it is not sufficient to use the low-energy expansion of the scattering amplitude (eq. (2.11)). We use an explicit model for the most important degrees of freedom to evaluate eq. (2.3) directly. In order to evaluate the matrix elements of the electromagnetic operators it is in general not sufficient to use the long wavelength approximation. The use of e.g. RPA wave functions would lead to rather lengthy calculations. To show the main features, we use an extended Brink model"') to describe the high-lying giant resonances . In this model we can separate the wave functions into a part which describes the motion of the c.m. of all protons against the one of all neutrons and an internal motion of all protons and neutrons, respectively . Since the charge operator (we neglect the effects due to the magnetic moments here) acts only on the protons, we do not need to consider the internal motion of the neutrons explicitly here (see fig. 2). In this model we have the convenient separation of the total wave function z,si )=X( z) çb (si),

(2.16)

where X and 0 are both of the well-known three-dimensional harmonic-oscillator 16) type, which has the form u(r) = r' exp { -2Ar2}F(-n,1+2 ; Ar2)Yrm (09 0) .

(2.17)

N. Baron et A / Nuclear photon scattering

272

I

z

dig. 2. The choice of coordinates for the variable separation in the Brink model is shown. The neutrons are put together in their c.m . R ,,, while the proton coordinates s, with respect to their c.m. R are considered explicitly. For the calculations the c.m. frame of all nucleons is used.

j..

ere F( a, h; c) is the confluent series, Y is the spherical harmonic depending on Ge angula- part of r and A = (M/ N) W ne corresponding energy eigenvalues are given by E = (o(2n + W) .

(118)

2

ith these rather simple but realistic wave functions we can evaluate analytically e electromagnetic matrix elements fully including the retardation. This is done explicitly for 2°81'b in sect. 3. Tie interaction operator is given by (neglecting spin currents) . z

Aliplying variable separation on eq. (2.19), a matrix element of the interaction operator reads in general: (

'Pexcl

I 'PO)

__1

z +- I (0,,x E Z j=1

('Oexcl

Z-1 z Z

i=1

e

I(A,)(x,,xr eile Ra gu

IXO)

j e"

e ik-Rzl X0)

s~

I

00)(Xexci

e giant resonance states correspond to a wave function of the form

.

(2.20)

N. Baron et al. / Nuclear photon scattering

273

The O-part in the second term of eq. (2.20) then is the ground-state expectation value of the interaction operator in si coordinates. From the Wigner-Eckart theorem and the transversality condition for real photons, one deduces for every expectation value:

(00''I

eik-s~

2.22

== 0 -0

The O-part in the first term of eq. (2.20) is easily interpreted as the form factor for the proton charge distribution : z d3 s pe(s) e "% -s = FPe . çb çb = Y, ( oI e'* s'l o)

(2.23)

Z i-1

The whole matrix element then reads (

excI ®It I Vro> = (Xexcl

e

Z

2.24

RZIXO) FPe '

This leads to a transparent interpretation of the physics. The matrix element in z coordinates indicates the collective nuclear excitation, while the strongly energydependent form factor FPe takes the role of a weight considering that at hi er energies noncollective modes suppress the giant resonance. We can safely neglect the recoil for heavy nuclei and treat the energy denominators as lorentzians using the location and the width of the resonance as an input. The whole amplitude in forward direction then reads: Ze2

.f = -

M

Sw~ . % 1.'F00 + F2Pe

(X0l ®t..(

()k OI ®P, ( Z)IX.)(X.I ®P,"( n 1 iFn

Z)IX.)(X.IOp,'( Z)IXO) E R+ E,, + 2 lin

'

)IXO)

2.25

where Fo0 denotes the angular-dependent form factor of the seagull amplitude (eq. (2.6)) . With the explicit expressions for the wave functions (eq. (2.17) and for the interaction operator (eq. (2 .19)), the amplitude can be calculated without any further approximation. The differential cross section is derived from eq. (2 .25) by taking the magnitude squared and multiplying with the specific angular distribution fac'7) after summing and averaging over the incoming and outgoing photon tors polarizations. Since the photon emission from the giant-resonance states is dominated by the direct decay mechanism, the amplitudes for different processes have to be added coherently . [We note that in the model of multistep-compound 1:)rocesses, the contributions from levels with different spins and parities are added incoherently, 'g).] This also applies to Coulomb excitation and subsequent (photon) see e.g. ref. decay of the giant-resonance states [see e.g. ref. '9)], since it can be viewed in first order as a scattering of a spectrum of equivalent photons . In principle, interference effects between different multipolarities will also show up in this case.

274

N. Baron et al. / Nuclear photon scattering

licatio to

208

208 Pb up to 501VieV [ref. 5)], we assume 'o analyze Co pton scattering data on that the main contribution to the amplitude arises from the one-phonon excitation . e normalized wave functions in external z coordinates for the ground state and the first excited state are explicitly given by

e

,yo( z)=

3®4 (,k) -

-9ZAR22

~r

ze

-là,

2

,

1v(

za

(3 .2)

these wave functions the interaction matrix element eq. (2.24) can be evaluated anal ically 20,21) without using any approximation like the long wavelength limit: o)= -

e

5

2

e®

2/aA

Iy=t1

(3 .3)

were =( / ) an A =1.9277( / ) 2®3fm -2 [see e.g. ref. 22)] refers to the characteristic length scale of e proton c.m. separation against the neutron c.m.. e point out, that from e use of the extended Brink model the characteristic ty A is completely determined through the location of the GDR and the T s e which is exhausted in our model. note that ifone wound formally use the long wavelength limit for the evaluation e z integrals one would obtain an approximation up to second order in energy for e matrix element. e factor of the quadratic term of this approximation does on e lace in the exact calculation where the asymptotic expression for essel functions is inserted; inconsistencies in the order ofexpansion are possible . er e previous assumptions the sum over intermediate states in the resonance litude eq. (2 .25) reduces to one term and the whole amplitude can be expressed in terms of eq. (3 .3). n the energy denominators we take the values for the width of e resonance as well as for the location of the resonance from refs. 23° 5 ), so that a close expression depending on two parameters for the resonance amplitude is obtained . ith the ground-state wave function `eq. `3.1)) for the external coordinate and the analogous ground-state wave function for the internal coordinates the form factor of the seagull amplitude depending on the scattering angle reads explicitly (®je;(k;-kd-

.10)

= e- k2sin22®(1/11+1/Â)

3.4

where  denotes the inverse squared characteristic length of the proton distribution . Since /A << 1/1, the main contribution to the form factor arises from the internal

part of the complete wave function (eq. (2.16)). The magnitude of  is determined trough the form factor Fpe of the resonance amplitude, which is connected with the charge distribution via Fourier transform. As a good approximation we write

N. Baron et al. 1 Nuclear photon scattering

275

the form factor FPe as : .(k2) = e

-k2/4a

(3.5)

.

We fix the parameter 1 of the gaussian by demanding that it reproduces the well-known mean square nuclear charge radius . For the 2o8Pb system we have A = 44.4 fm -2 and Jl = 5.01 x 10-2 fm -2 . For the other parameters used to determine the energy denominators see table l . Since the characteristic length scale of the collective motion is much smallerthan the nuclear dimensions, one could in principle A use, to a rather good approximation, the long wavelength approximation for O,,(RZ) in eq. (2.25). After summing and averaging over the photon polarizations and multiplying with the angular distribution, a closed expression for the E1 part of the differential cross section is obtained. To get realistic values for the differential cross section up to energies about 50 MeV one has to take into account exchange contributions to the excitation mechanism which are disregarded in deriving eqs. (2.3) and (2.6). This is usually done by 24) : introducing an enhancement factor K which is defined through the GGT sum rule mA

fo

dEtrabs(E)=(1+K)

NZ S .

(3 .6)

Here S is the classical dipole sum 2 . e2/ M. With the optical theorem [see ref. 25)] and the dispersion relation for the forward scattering amplitude [see ref. 26)] the enhancement factor K can be introduced in 2o8Pb system we choose a value of K = 0.35 the whole scattering amplitude. For the 27)]. which is consistent with other calculations [see e.g. ref. TABLE 1

Numerical values for the parameters used to determine the Compton scattering amplitude for 2013 Pb in an extended Brink model for O =60° and ® =150 °. The choice of parameters for the resonance is in accordance with ref. 5) while the system parameters are derived from basic relations [see for example ref. 31 )] 2ospb

0 =150°, 0 = 60°

A Â D E El

44.0411 fm-2 5.01 x 10-2 fm2 0.72 (MeV - fm)1/2 13.5 MeV 3.5 MeV 0.35

R rEl KEl

276

N. Baron et al. / Nuclear photon scattering

V)

e

Fig. 3. The differential cross section da/dfl for =150° including only El contributions from the extended Brink model is plotted against the photon energy . The full line corresponds to our calculation, the points from Mile data an ref. 5). Discrepancies for low energies may be due to neglected compound states .

Ery (M eV) Fig. 4. The same plot as in fig. 3 for a scattering angle of 19 = 60". Again the calculation is compared to data from ref. 5) .

N. Baron et al. / Nuclear photon scattering

277

Figs . 3 and 4 show the results for the differential cross section for Compton scattering on 208 Pb as a function ofthe photon energy. The calculations are compared with the data from ref. 5). In fig. 3 the differential cross section is shown for the scattering angle ® =150°. In the low-energy region (below 5 eV) the differential cross section behaves nearly as a constant which is the Thomson limit with respect to the scattering angle. The relative minimum between 5 and 10 MeV is due to interference effects between the resonance and the seagull amplitude. In the rising of the cross section to the pea value of the GDR at 14 MeV our calculation underestimates the data. The authors of ref. 5) consider inelastic photons to be in the exit channel in this energy region, which they could not distinguish from the elastic ones in this experiment. On the other hand measurements ofBertrand and Beene 28) ofthe electromagnetic excitation and decay mechanism in heavy-ion collisions with 2°8Pb in this energy region indicate contributions from compound states, which are not included in our calculation. e GDR peak itself and the data up to 50 MeV are very well described in terms of our model calculation. Also the differential cross section for the scattering angle ®= 60® in fig. 4 shows the same general behaviour. The only significant deviations from the data are again in the energy region between 5 and 10 MeV. The change of the peak magnitude in comparison with fig. 3 is mainly caused by the different angular distribution factors. In addition the angular dependence of the form factor in the seagull amplitude gives rise to changes in the calculation . All parameters required for the calculation of the differential cross sections for both scattering angles are shown in table 1. e independence ofthe scattering angle shows the consistency ofour model calculation. We want to point out that the very good agreement between our calculation based 208pb in the on the extended Brink model and the scattering data up to 50 MeV for region above the resonance is mainly due to the complete inclusion ofthe retardation. . Additional excitation mechanisms: Photoabsorption on 'Ph The differential cross section for Compton scattering on 208 Pb in our extended Brink model does not show any significant deviations from the data between 20 and 50 MeV which may be due to the neglection of additional excitation mechanisms. On the other hand, every interpretation of elastic photon scattering should be consistent with the photoabsorption of the same system. The relation between the Compton scattering amplitude and the total photoabsorption cross section is given by the optical theorem 25 ). Because ofthe form factor (eq. (3 .5)) the photoabsorption cross section calculated from eq. (2 .25) shows a very strong decrease for energies above 30 MeV. This is in contrast with the photoabsorption data from ref. 29) which show a smooth transition between the GDR peak and a very broad peak structure located at about 75 MeV. This structure is interpreted as absorption through the, isovector E2 and the quasi-deuteron excitation . (The isoscalar E2 contribution is,

N

anon et al. / Nuclear photon scattering

are with the isovector one, negligible since it is covered by the dominant eak.) b ac i®ve consistency with the photoabso tion it seems to be imperative to include these excitation contributions in the Compton scattering amplitude. From our general model we could obtain neither a realistic 2 strength nor a description of the quasi-deuteron excitation, so that we treat these parts of the ampïitude e lly. s a empirical ansa for a fit of both contributions we use refit- igner curves so that the contributions to the photoabso tion cross section is given by 5'30) $ - ,,) +(,i) are the strength, the location and the width of the contributions, (

ere c1' p, à an respectively.

à

(E,) is only applied in the quasi-deuteron case an has the form s ) ) ® 1- e -( E~-Etj,,)>,E , y, ( .2) chr ( .2) ensures a smooth decrease of the cross section to the threshold energy of the proton-neutron pair emission at 15 eV. Fig. S sows the calculated photoabso tion cross sections for 208 Pb as a function of the photon energy in e region between 2 and 140 eV in comparison with

s

" y I ~` -

30

40

" '

~

5v

s e .

0

v0

e e e e

f

e e e

70

e

1

80

e

e

e

s

~

90

~r)

e

e

e

~

e

100

e

e

e

1

e

110

e

~

120

" ~ .

130

140

Fig. ~. e total photoabsorption cross section for 2°8Pb is plotted against the photon energy in the region between 20 and 140 MeV in comparison with data points from ref. 29) . The dash-dotted line corresponds to a calculation based only on the extended Brink model. The dashed line shows the effect of the inclusion of a quasi-deuteron contribution alone while in the calculation of the full line an isovector E2 contribution is considered on top of this . The parameters for the additional contributions are shown in table 2.

N. Baron et al. / Nuclear photon scattering

279

the data from ref. 29). The dash-dotted line corresponds to the cross section resulting from eq. (2.25) with the parameters of table 1 via the optical theorem. Evidently this calculation is not able to describe the photoabsorption data above 30 eV. In the calculation for the dashed line we added a quasi-deuteron contribution with the parameters from table 2 while the full line corresponds to a calculation with an additional E2 contribution whose parameters are also shown in table 2. From fig. 5 it is obvious that a quasi-deuteron and an isovector 2 contribution is needed to describe the photoabsorption data above 30 MeV. To include these contributions in our calculation of the differential cross section for Compton scattering we use the optical theorem and the dispersion relation to extract the forward scattering amplitude from eq. (4.1) for both cases (for the E2 case we explicitly use the fact that the whole amplitude vanishes for zero energy). Taking into account the different angular distributions the resulting Compton scattering amplitude is the coherent sum of the three single contributions so that the differential cross section for Compton scattering is given by 17,30) du _

dû

with

-o

d~ ) d~

E1

d~ d~l

E2

_ lfÉ

E1,E2,Int

d£I

EA

(4.3) 9

+fQn o l 2(i(1+co52®)) ~

(4.4)

= VE°I 2 (2(1 - 3 cos2 ®+4 cos4 ®)) ,

d~

d~

du

®=o

3

®-o

®=o:~

Int

(4.5) (4.6)

It is clear from eq. (4.4) that the interference between the amplitudes fEl and fQD is, independently of the scattering angle, only determined through the different phases ofthe amplitudes . In the energy region between 20 and 50 eV they interfere TABLE

2

Numerical values for the parameters to fit the photoabsorption cross section for energies above 30 MeV by l'sreit-Wigner curves . The choice is mainly in accordance with ref. 5) 2oapb

v~ E; T; A Ethr

®E

QD-effect

E2-resonance

15.0 mb 75.0 MeV 180 .0 MeV

25.0 mb 20.5 MeV 9.0 MeV 44.0411 fm-2

14.9 MeV 5.0 MeV

28 0

N. Baron et ad. / Nuclear photon scattering

Ery (

eV)

Fig. 6. The same plot as in fig. 3. The full line corresponds to the one in fig. 3 while the dashed line represents the calculation of the differential cross section with the additional quasi-deuteron and isovector E2 contributions.

destructively because this region is located above the GDR but below the QD resonance. The EIE2 interference contribution of eq. (4 .6) is strongly angular ependent. the set of parameters from tables I and 2 we are able to calculate again the differential cross section for Compton scattering as a function of the photon energy for a fixed scattering angle. This is done in fig. 6 for 09 =150°. As in fig . 3 the full line corresponds to the calculation based only on eq. (2 .25) while the dashed line shows the full calculation from e4. (4.3). Fig. 6 clearly indicates that the modifications from the additional %;xcitation contributions on the differential cross section for energies up to 50 MeV are not significant. (An analogous calculation for 19 = 60" shows the same behaviour.) Even if in the energy region between 20 and 50 MeV the GDR itself is negligible, the cross section for Compton scattering is dominated by the amplitude which we denote by fE, This is due to the scattering on the charge distribution of the system which is described by the seagull amplitude. This leads to a purely real contribution to fEj . Since the energy dependence of the seagull amplitude is given only by the form factor eq. (3 .4) this contribution dominates the additional ones for energies up to 50 MeV. 5. IC®nclusio We studied nuclear Compton scattering on a spherical spin=zero nucleus in terms of a transparent nuclear model. After revisiting the low-energy expansion, a clear

N. Baron et al. / Nuclear photon scattering

28 1

separation of collective and single-particle variables was obtained in an extended Brink model. This led to a simple and realistic analytical evaluation of nuclear matrix elements, with full inclusion of retardation effects. We applied our model to 208 Pb, where we first only included El contributions to the intermediate states from our extended Brink model. With this model we were able to give a good interpretation of the Compton scattering data up to 50 eV. To achieve consistency with an interpretation of photoabsorption data on 208 it was necessary to include isovector E2 and quasi-deuteron excitations which was done empirically via an ansatz. The inclusion of these additional excitation mechanisms did not lead to significant changes ofthe differential cross sections for Compton scattering. This is due to the dominance of the scattering on the charge distribution ofthe system in this energy region which does not contribute to the photoabsorption. In conclusion we got a clear interpretation of Compton scattering data on 208 only in terms of the scattering on the charges of the system and through the GDR as an intermediate state. Up to 50 eV additional excitation mechanisms like the isovector E2 and the quasi-deuteron excitation are of less importance for the elastic scattering channel. One of the authors (N.B.) would like to thank the Studienstiftung des deutschen Volkes for the support of his studies. This work was partly supported by the DFG. References 1) R. Leicht, K.-P. Scheelhaas, M. Hammen, J. Ahrens and B. Ziegler, Nucl. Instr. Meth. 179 (1981) 131 2) R Leicht, M. Hammen, K.-P. Scheelhaas and B. Ziegler, Nucl. Phys. A362 (1981) 111 3) M. Sanzone-Arenh6vel, K.-P. Scheelhaas and B. Ziegler, Proc. Int . School of intermediate energy nuclear physics, ed. R. Bergere, S. Costa and C. Schaerff (World Scientific, Singapore, 1982) p.29 4) E. Hayward and B. Ziegler, Nucl. Phys. A414 (1984) 333 5) K.-P. Scheelhaas, J.M. Henneberg, M. Sanzone-Arenh6vel, N. Wieloch- Laufenberg, U. Zurmiihl, B. Ziegler, M. Schumacher and F. Wolf, Nucl. Phys. A489 (1988) 189 6) A.J. Akhiezer and V.B. Berestetskii, Quantum electrodynamics, monographs and texts in physics and astronomy vol. XI (Wiley, New York, 1965) . 7) D.M. Brink, Nucl. Phys. 4 (1957) 215 8) J.P. Elliott and T.H.R. Skyrme, Proc. Roy . Soc. A232 (1955) 561 9) C.J. Joachain, Quantum coiiision theory (North-Holland, Amsterdam, 1975) 10) T.E.O. Ericson and J. Hüfner, Nucl. Phys. B57 (1973) 604 11) V.A. Petrunkin, Nucl . Phys . 55 (1964) 197 12) F.E. Low, Phys. Rev. 96 (1954) 1428 13) M. Gell-Mann and M.L. Goldberger, Phys. Rev . 96 (1954) 1433 14) N. Baron, G. Baur and M. Schumacher, to be published 15) M. Schumacher, P. Rullhusen and A. Baumann, Nuovo Cim . Al00 (1988) 329 16) S. Flügge, Rechenmethoden der Quantentheorie (Springer, Berlin, 1965) p.100ff 17) H. Arenh6vel, M. Danos and W. Greiner, Phys. Rev . 157 (1967) 1109 18) H. Feshbach, A. Kerman and S. Koonin, Ann . Phys. 125 (1980) 429 19) F.E. Bertrand and J.R. Beene, Proc.1989 nuclear physics coif. Sao Paulo, Brazil, August 20-26, vol. 2 20) A.R. Edmonds, Angular momentum in quantum mechanics (Princeton Univ. Press Princeton, 1960)

282

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