Continuum shell-model investigation of the photoexcited giant dipole resonance in the non-magic nucleus 13C

Nuclear Physics A330 (1979) 109-124: (~) North-Holland Publishino Co., Amsterdam Not to be reproduced by photoprint or microfilmwithout written permission from the publisher

CONTINUUM SHELL-MODEL INVESTIGATION OF THE PHOTOEXCITED GIANT DIPOLE RESONANCE I N T H E N O N - M A G I C N U C L E U S 13C J. H O H N t

Technische Universitdt Dresden, 8027 Dresden, DDR

and H. W. BARZ and 1. ROTTER

Zentralinstitut Jffr Kern[orschun9, Rossendor/~ 8051 Dresden, DDR

Received 4 July 1978 (Revised 5 January 1979) Abstract: The influence of 3p-2h configurations on the properties of the giant dipole resonance (GDR)

in 13C has been investigated. An extended formulation Of the continuum shell model has been used to analyse the photoabsorption process. The results are compared with experimental data and earlier calculations. The 3p-2h configurations are found to be important for the higher excitation energies in the GDR. The particle widths estimated in a R-matrix formulation are discussed in the light of the present results.

I. Introduction

T h e c o n t i n u u m shell m o d e l ( C S M ) is a p r o p e r tool for the investigation of giant r e s o n a n c e p h e n o m e n a . In recent years, the lp-1 h m o d e l has been a p p l i e d successfully to s t u d y the giant d i p o l e r e s o n a n c e s ( G D R ) in m a g i c nuclei. F o r n o n - m a g i c nuclei m o r e c o m p l i c a t e d c o n f i g u r a t i o n s t h a n lp-1 h are excited by the a b s o r p t i o n o f T - q u a n t a b e c a u s e of the c o m p l e x s t r u c t u r e o f the target state. A f o r m u l a t i o n of the C S M which allows one to t a k e into a c c o u n t c o m p l i c a t e d s h e l l - m o d e l c o n f i g u r a t i o n s has been d e v e l o p e d by us 1). A c c o r d i n g to the c o n c e p t i o n of this m o d e l , all r e s o n a n c e p h e n o m e n a are g e n e r a t e d by s h e l l - m o d e l wave functions c o n s t r u c t e d in a discrete basis of single-particle states. M o s t of these m a n y - p a r t i c l e states lie a b o v e the particle t h r e s h o l d s a n d a r e c o u p l e d to the o n e - p a r t i c l e c o n t i n u u m via the residual interaction. In such a way, r e s o n a n c e states o c c u r a n d their decay is t r e a t e d by c o u p l e d c h a n n e l s m e t h o d s . T h e direct n u c l e o n emission c o m p e t i n g with the r e s o n a n c e r e a c t i o n yields only a small c o n t r i b u t i o n in the p h o t o a b s o r p t i o n process. O u r m o d e l allows a s e p a r a t e t r e a t m e n t of the o n e - p a r t i c l e c o n t i n u u m a n d of the discrete c o n f i g u r a t i o n space. This m e t h o d has the a d v a n t a g e that we a r e able t Present address: Staatliches Amt fiir Atomsicherheit und Strahlenschutz, DDR - 1157 Berlin. 109

I l0

J. HOHN

et al.

to extract the structure components of the resonance states and their resonance parameters. In this manner the method allows a unified description of structure and reaction aspects. Here, we have applied our model to investigate the G D R in 13C. The measured photoabsorption cross sections show a broad distribution of dipole strength between 18 MeV and about 35 MeV. The older experimental data 2.3) as well as the results of Patrick et al. 4) show a rather smooth energy dependence of the photoexcited G D R without any fine structure. McKenzie's data 5) and the data of Jury et al. o) for the photoneutron cross section carried out with higher resolution indicate however a pronounced structure in the G D R region which is supported very well by the measurements of Koch et al. 7). Investigating theoretically the G D R in 13C, bound-state shell-model calculations have been carried out by e.g. Kissener et al. 9) taking into account all l h~o excitations. Marangoni et al. 10.11) have investigated the G D R in 13C in the CSM. But these calculations have been performed in the 2p-lh approximation. The calculations by Albert et al. 8) performed in a bound-state shell model use also the 2p-lh approximation. But it is difficult to draw conclusions on the relative importance of complex configurations and the continuum by comparing the results of these different calculations. Our aim is to study the influence of the one-particle continuum by comparing the results of our calculations with and without the continuum using the same discrete space and the same residual interaction. Further we investigate the contribution of the 3p-2h configurations to the cross section and its detailed resonance structure as well as to the large spreading of the GDR. The influence of the 3p-2h configurations on the high energy tail of the G D R (below 35 MeV) which up to now has not been analysed in the CSM is investigated with special care.

2. Basic equations The model used is presented in detail in ref. 1) (hereafter referred to as I). Here we give a brief outline only and emphasize the main features which arise in photonuclear reactions. The electromagnetic interaction Hi, t between photons and the nucleus can be treated by a first-order Born approximation. Then the nucleon emission process, which takes place after the photon is absorbed by the target ground state q~x, is described by the Schr6dinger equation

(E-H)tP = Him~,

(2.1)

with the nuclear Hamiltonian H. The source term Hint q~V describing the absorption of a photon with wave vector k and spin polarisation vector ep reads HintqbT -- x/2rchzcx/kep exp ( i k . r ~ T ,

(2.2)

where the operator j describes the current density of the nucleons. The photon

CONTINUUM SHELL MODEL

111

density is assumed to be unity and ~ = ~3v is the Sommerfeld constant. The wave function 7' obeys boundary conditions with pure outgoing waves only. To realize the conception of our model we use the projection operator technique in order to divide the model space into two subspaces Q and P. The Q-space contains the states in which all nucleons occupy discrete single-particle orbits. To describe all resonances by the Q-space the narrow single-particle resonance states have been discretized by means of a cut-off technique and are included into the set of the bound single-particle states. As basic wave functions we choose the QBSEC ~i which are obtained by diagonalizing the full Hamiltonian H in the discrete subspace of the antisymmetrisized products of single-particle states. Hence the projection operator Q is given by Q = ~i ~)
= G~+)Hint~T A- ~ ~"~(R +) R

<~+)*]Hi.,Iq~T>.

(2.3)

E - ~'R(E)

The propagation operator GCp+) varies smoothly with energy E. The wave function defined by = G(p+)HintqbT,

(2.4)

describes the direct emission of nucleons into the continuum due to the absorption of the photon. The second term of eq. (2.3) represents the contributions of the resonance states R, the wave functions of which are defined by ~(R+) ----(1 + G ( + ) H p Q ) ~ R .

(2.5)

Thereby the functions ~R are the eigenfunctions of an effective Hamiltonian of the Q-space

(HeQ + HQpG(p+)Hpe)(~ R = ~R(E)(~R.

(2.6)

The effective Hamiltonian contains the coupling of the QBSEC 4~i to the continuum due to the residual interaction HvQ of the P- and Q-space. The eigenvalue ~R(E) = ER(E)-liFR(.E) of the nonhermitean operator in eq. (2.6) is energy dependent and defines the position E R = ER(ER) and width F R = PR(ER) of the resonance state. The matrix elements in eq. (2.3) give the amplitude for the excitation of the resonance state R. After a lifetime of h/FR the state R decays into the one-particle continuum.

J. HOHN etal.

112

This decay is described by G~e+)Ht,Q~R of the resonance state wave function ~R in eq. (2.5). The wave function ~ is expanded into a set of wave functions ~, of the residual nuclei with A - 1 nucleons and spin-angular momentum wave function ?~j,, of the emitted particle which are coupled together to the total angular momentum J:

~ 1 ~ ,.l,~J) <,?r)

=

f

•

Yt mM

(J I'

M, i]

JM)

~'glimOC

(2.7)

The operator s/antisymmetrisizes the wave function. The radial part ~,i(r) in each channel c = tlj satisfies the following integro-differential equation resulting from eq. (2.4):

2

¢'

(E6c~,- Hc~,)~/'(r) =

£1J)(r) - ~ tl

u,mi(r)|dr ' Ur,rtj(r ' )[~ H,q,'~J~(r')+£1Jl(r')]. J

(2.8)

c'

The Hamiltonian Hcc, has the form

/-/co =

-2m

dr 2

r2

and contains the energy E, of the residual nucleus and the shell-model potential %j(r) of a Woods-Saxon type, where r means the third component of the isospin of the particle. The coupling potential V , has been generated by the use of a zero-range interaction. The single-particle wave functions u~j(r) with node number n are eigenfunctions of the shell-model Hamiltonian in the square bracket of eq. (2.9). The integral terms in eq. (2.8) ensure that the channel wave functions ~,~(r) are orthogonal to all discrete single-particle wave functions u,~l~(r) from which the Q-space is constructed. Therefore the Pauli principle for the many-particle wa,ve function in eq. (2.3) is fulfilled. The electromagnetic interaction (2.2) enters into eq. {2.8) via the source term f,, which reads for electric dipole absorption

L~"~(r) = hcx/~rr~k;'(- )" +"- ~ M, × ~ ~P"~{T)il

, {,: 1

Itl

p

;}(j ½ j; ;) ( 1 [TJTB

rt~+ 2 -

ru"e~"t°J"(r)"

(2.10)

Here, the photon moves along the z-axis. The symbols ( ] ) and { } are Clebsch-Gordan coefficients and six-j symbols, respectively, while ] means x/2j + 1. The sum runs over all single-particle orbits fl from which a nucleon can be excited into lj continuous orbits. These contributions are proportional to the spectroscopic amplitude .~r) ~,6't representing the probability amplitude to find the single-particle state fl and the residual nucleus ~, in the initial target state q~T (see eq. (3.22) of I). The resonance wave function t) R in eq. (2.5) is generated in a similar manner as described in detail in I. F r o m the amplitude of the outgoing waves the cross section can be calculated.

CONTINUUM

SHELL MODEL

113

3. Model space for the photonuclear reaction on ~3C 3.1. C A L C U L A T I O N

1N T H E Q - S P A C E

First we carry out a usual bound state shell-model calculation in order to generate the QBSEC 4~i in the target nucleus 13C. The single-particle basis used to construct the 4~i is generated by a single-particle Hamiitonian which handles the single-particle resonance state d~ like a discrete state. The state dependent single-particle potential v tj(r) in eq. (2.8) has the form

v~tj(r) = Vtf(r)+4[fm]ZV~]'°)s " 1 1r drr d f(r) + (z + 2) Vc(r)

(3.1)

with f ( r ) = [ l + e x p ( ( r - R ) / a ) ] -1, R = ro(A-1)~ and a Coulomb potential Vc corresponding to a homogeneous charged sphere of radius R. To reproduce the experimental single-particle energies used by Marangoni et al. lo) we have taken the potential parameters given in table 1. For the 2s, l d single-particle resonance TABLE 1 P o t e n t i a l p a r a m e t e r s a n d c o r r e s p o n d i n g single-particle energies e.~j (in MeV) for the A = 12 and 13 nuclei (r o = 1.25 fm, a = 0.53 fm)

1

r

0

2L - ½

58.30 60.36

½ - ½

54.48 54.62

19.82 19.25

- 15.95 - 18.72

½

56.36 57.38

4.85 4.87

1.54 - 1.12

1

2 1

Vet [ M e V ]

V~ '°'1 [ M e V ]

%~,+1,.2

e'o~n 1/2

-31.13 - 36.03

EI~/I+l.2

-

0.42 1.86

1.94 -4.95 -

6.32 3.42

states a cut-off radius of Re. t = 10 fm has been used. The diagonalization has been carried out with a zero-range force Vo(a+bP~2)f(r 1 - r 2 ) whereby the parameters V0 = 700 MeV. fm 3, a = 0.73 and b = 0.27 are taken from ref. 11). The parameter V0 has been fixed by best fitting the energy of the main peak of the experimental photoreaction data in 13C using 2p-lh configurations only 11). The same parameters are used also here in order to show the influence of the 3p-2h configurations on the position and width of the GDR. The QBSEC 4~i which form the G D R built on the ground-state have spin and isospin values j~ = ~1+, ~3+ with T = ~1 and 3, respectively. In order to construct them the configurations lp~+21s~-llp~ n and lp](2s, ld)lp~" with n = 0, 1 and 2 have been used. The QBSEC in the excitation energy region between 10 MeV and 26 MeV consist mainly of 2p-lh configurations wherein the admixture of 3p-2h configurations varies from 10 ~o up to 40 ~o. This rather strong admixture of 3p-2h configurations has been found also for states with large B(E1) values which are

114

J. H O H N et al.

'72L B(E1)I

,

~3C T = 1/2

1.00 075 0.50 0.25 i I

0 B(E1)

[e2mb]

ill

i

3.5O

I i

13c

, :J,~ i!

3,2.= 3.0C

1

T,3~

+

0.5C ]

02~

I

15

2o

,

I,JLI ~i 25

i, 1

30 35 ET [NleM]

Fig. 1. B(E1) values for the QBSEC with different isospin ( J " =

1 +

40

45

[Er ~v] full line, J'~ = ~ +

broken line).

shown in fig. 1. In the excitation energy region between 26 MeV and 31 MeV there are seven QBSEC, consisting mainly of 3p-2h configurations with a 2p-lh admixture up to 40 %. Between 31 and 35 MeV excitation energy the admixture of 2p-lh components does not exceed l0 ~o. The configurations with one hole in the ls shell are distributed over a wide energy region. Their admixtures become important at energies above 35 MeV. A group is concentrated at about 45 MeV corresponding to the 2p-lh configuration with one hole in the l s shell and two particles in the l p½ shell. The more complicated configurations with one hole in the ls shell and one or two additional holes in the lp~ shell lie at still higher excitation energies. The complex nature of the Q B S E C requires the inclusion of complicated configurations into the residual target states too. In the shell-model calculations for the A = 12 systems, configurations lp~lp~-" with n = 0, 1 and 2 are taken into account. We find that the 12C ground state is mainly formed by the n = 0 configuration (92 %) and the excited states predominantly by n = 1. The main contribution to the ground state of 12B comes from n = 1 (92 ~o) and the first excited J ~ T = 2* 1 state is given by a nearly pure lP½1P~ 1 configuration. No elimination of the centre-of-mass spurious states has been tried because of our truncated basis.

CONTINUUM SHELL MODEL

115

3.2. COUPLED CHANNELS CALCULATION

The solution of the coupled equations (2.8) has been performed taking into account seven channels with the following target states of 12C" 0+0 (0 MeV), 2+0 (4.43 MeV), 1+0 (12.71 MeV), 1+1 (15.11 MeV), 2+1 (16.11 MeV) and 12B: 1+1 (0 MeV), 2+1 (0.95 MeV). These channels are known from experiment to be important for the nucleon decay. The partial waves have been restricted to l < 4 and the sum in eq, (2.8) runs over the nodes n of the discrete single-particle states. The maximal rank of the coupled inhomogeneous differential equation system solved in the present investigation was 22. The 13C ground state consists of 1-",+~lp~" configurations with n = 0, 1, 2. The P~ inclusion of the one-particle continuum lowers the energy of the ground state of 13C by 2.7 MeV. This fact leads to an enlargement of the energy of the giant resonance in comparison with that obtained by Marangoni 10,1~) where neither the complicated configurations nor the influence of the continuum in the ground state of 13C have been taken into account. To guarantee the opening of the channels at the correct energy the experimental energies of the states of the residual nuclei have been taken.

et al.

4. Results and comparison with the experimental data 4.1. PROPERTIES OF THE RESONANCE STATES

To investigate the photoexcited GDR in 13C we have taken into account 60 QBSEC in the energy region between 8 MeV and 38 MeV with B(E1) values larger than 0.01 e 2 • mb. The coupling of the QBSEC to the one-particle continuum leads to an energy shift. In general, the positions of the resonance states have been found to be at lower energies (up to 2 MeV) compared with the position of the dominating QBSEC component. The widths of the resonances vary between 30 keV and 2 MeV whereby their average width is about 600 keV. Since the average distance between the resonances amounts to approximately 500 keV, the resonances overlap in most cases. In order to characterize the strength of the electric dipole transition we use the reduced transition probability _

B½(EI)

e

1

x/2Jrt+~l(~+'*, i(z+ ~ - Z)rY1 [~T),

(4.1)

which depends on the structure of the resonance state ~+) in eq. (2.5) and of the target state q~T"The distribution of the B(E1) values is shown in fig. 1 from which the formation of the GDR between 15 MeV and 35 MeV can be observed. The nuclear structure of the state at 46 MeV contains 4 0 ~ of the (ls½)-l(2P½) 2 components. The energy-integrated dipole absorption cross section is approximately given by

fdEtr.abs ~

~ Re ~R(ER)' R

(4.2)

116

J. HOHN et al.

where the dipole strength is defined by 1693~

~R = ~ 9e

2J R+ 1

E 2I~+IB(E1)

(4.3)

with the photon energy E . Eq. (4.2) has been derived from eq. (2.3) under the assumption that D R and F R are energy independent and the direct contribution is neglected. From eq. (4.2) we obtain 225 M e V . mb for the resonances used here. This value overestimates the classical Thomas-Reiche-Kuhn rule by about 25 °,,i while the experimental cross section 4) gives 90 ..... /o" 4.2. CROSS SECT1ONS

The experimental photoreaction data on z3C available up to now differ remarkably concerning the structure as well as the magnitude of the cross sections. The experiments by Cook 2) and Patrick et al. 4) show a broad and smooth distribution of the dipole strength between 10 MeV and 35 MeV. A pygmy resonance is located in the region near 13 MeV. The G D R has a rather large width of about 10 MeV with two main peaks near 20 MeV and 27 MeV, respectively. However photoneutron cross sections measured by McKenzie 5) and recently by Jury et al. 6) and also by Koch et al. 7) show much more and pronounced structure and the magnitude of the cross section is reduced by up to a factor of two. Similar differences have been found for

O[r.b] 6O

13C(~,n ~-p) 50

40

30

1

20

10

-....

........................................

Z

Fig. 2. Total photoabsorption cross section compared with the experimental data (broken line) of Patrick et al. 4). The dotted line represents the direct nucleon emission.

CONTINUUM

SHELL M O D E L

117

the (7, P) cross section in the data of Cook 2), Patrick et al. 4) and Denisov et al. 3). In fig. 2 the total 13C(7, p + n) cross section is shown in comparison with the experimental data of Cook 2). The calculated value of the cross section is overestimated, on average by about a factor of 1.5. This failure might be corrected by taking into account an imaginary part in the potential in order to simulate neglected channels and other similar effects. In our formalism, an absorptive potential cannot be introduced in the P-space alone, but has to be taken into account also in the Q-space

,

~3C(%,n)

3O

0 ' ~ : - ~ J ....

~o

~s

" " !.......................................................................

2o

25

3o

Er [MeV]

Fig. 3. Total photoneutron absorption cross section in comparison with the experimental data of Cook 2) (dash-dotted line), Patrick et al. 4) (broken line), McKenzie 5) (dot-dot-dashed line) and Jury et aL 6) (vertical bars). The 13C(), ' 2n) cross section has been found to be negligible compared to the (7, n) cross section 4). The dotted line gives the direct photoneutron emission.

20

13C( T ' P )

10

o

! ./-~

2o

3o

EEM,V

Fig. 4. Total photoproton absorption cross section compared with the experimental data of Cook 2) (dash-dotted line) and Patrick et al. 4) (broken line). The dotted line represents the direct photoproton emission.

J. HOHN etal.

118

producing the effect of the neglected channels on the total widths of the resonance states. We shall not deal with this question in more detail. The gross structure and the width of the G D R are in good agreement with the experimental data. The low energy peak of the G D R near 20 MeV is reproduced in the calculation by two peaks at 18.3 MeV and 20 MeV, respectively. The next groups of resonances form the main peak of the G D R from which the first group centred around 23 MeV is indicated by the experimental data z) too. The main contrast to the experiment of Cook 2) arises from the significant structure obtained in the calculated absorption cross section. An interesting result concerns the rather large contribution of direct photonucleon emission to the total absorption cross section (see also figs. 3 and 4 as well as table 2). Such a large direct reaction part is not observed for the magic nucleus 160 because the nucleons in 160 are more strongly bound than in 13C. TABLE 2

Integrated cross sections S = ~dEa up to E max = 35 MeV in units of M e V . mb

Scalculated

Scale Smca ~

S. . . . . . d

Channel res. and direct (~t, no) (7, n i ) (7, nz) (7, n3) (7, n4) (7, Po) (7, Pl) (7, n) (7, P) (7, n + p)

28.75 50.00 17.50 27.50 50.00 27.50 41.25 173.75 68.75 242.50

direct only

10.00 15.00 25.00

Patrick etal. 4)

Cook 2)

Patrick etal. 4)

25.00 8.75

1.15 5.71

30.0 42.50 23,75 28,75 110.00 56.25 171.25

0.92 1.18 1.16 1.43 1.64 1.27 1.47

117.50 60.25

Cook 2)

1.53 1.08

In fig. 3 the calculated photoneutron cross section is represented together with the recent experimental data 4- 6) as well as the older ones 2). The calculations have been performed up to 35 MeV since we are interested in the contribution that the 3p-2h configurations give to the giant resonance. At higher energies, the 2p-lh configurations connected with one hole in the ls shell become important (see fig. 1). Moreover, the number of channels taken into account in the calculation is restricted. Therefore, the widths of the single resonance states are too small in comparison with more realistic calculations. This effect becomes the more important the more channels are open. The presence of a pronounced structure in the cross section is well supported by the calculation in the G D R region. There is a shift of about 2 MeV of the calculated structure as compared with the recent experimental data by Jury et al. 6) Such a

CONTINUUM SHELL MODEL

119

I

o[mb]10

s

r~(l, n0)

o(o ~MeV) o

DIS . . . . . . . . . . . . . . . A0-0.5~ v

"

.

~

~

~,._,

~

O[mb]10 5 A 2 0.5

Ao -0.5 O[mb]10

4-

I÷ I(

*

1

11/2

5

4

~-

~

A2 0

;F.o-O

O[mb]10 5 ~

0.5 -0.5 [~110 5

13C('I"n~') 2" I (~.11 MeV)

4

~c(~.po)

. . . . ....

4-

~-

4

nA

A_Z 05

AO -0.5 O[~]W 5 ~0 0.5 -0.5

10 Fig. 5. Partial photoabsorption cross sections and A2/A o ratios for the decay of resonance states into seven channels. The experimental data (broken lines) are taken from Patrick et al. 4).

120

J. H O H N

et al.

shift may be connected with the different influence of the one-particle continuum on the position of the resonance states. The 13C ground state has a simple nuclear structure and is shifted by 2.7 MeV due to the one-particle continuum while the main resonance states of the G D R are shifted by only about 800 keV and the narrower resonance states by less than 100 keV. These different shifts obtained in our calculations arise obviously from the neglect of channels with more complicated nuclear structure [see ref. 16) where shifts with a larger number of channels are calculated]. In fig. 4 the (7, P) cross section is shown where in the experimental (7, P) data 2.4) no indication for a splitting up of the G D R into two groups of peaks is observed. To discuss the decay properties in more detail we have analysed the partial cross sections for the decay of the resonance states underlying the G D R into five neutron and two proton channels. A pygmy resonance seen experimentally in both channels is indicated only in the (7, no) channel in the calculation by a J ~ T = 73 +7 resonance shifted to higher energy at 16.85 MeV with a dipole strength of Re ~ = 0.31 MeV. mb. The low energy peak of the G D R at 20 MeV is observed in the (7, no) channel only and is reproduced by the calculation. But the theoretical (7, nl) partial cross section fails to explain the experimental data. This lack occurs probably because of the insufficient description of the U T = 2+0 state at 4.43 MeV in 12C. Up to 26 MeV excitation energy the 7"< resonances have mainly (2s, ld)lp~ ~ structure. Their decay into the (7, n2) channel, the threshold of which is near 18 MeV, is weak and therefore not seen in the experimental data available up to now. The T = 1 residual target states can be reached by the decay of the T = ~ (7">) resonances. The gross structure of the (7, n3) channel is described by three resonances at 21.65 MeV (71+ 7), 3 23.11 MeV '2(~+~]2' and 26.76 MeV 't-3+-3~ 2 2'" F o r the analog states U T = 2+1 in 12C and 12B one observes experimentally nearly the same value of the partial cross sections. The calculation supports this result whereby the dominant contribution comes from the decay of the J = T = 53+37 resonance at 26.76 NIeV which exhausts the largest part of the dipole strength (Re ~ = 78 M e V . mb). The peak observed in the experimental (7, P0) partial cross section near 23 MeV is reproduced in the calculation. There is however no indication in the experimental data for the calculated peak around 27 MeV. The integrated photo cross sections are given in table 2. The ratios of calculated and measured integrated cross sections up to E 7. . . . = 35 MeV are about one with the exception of the (7, nl) partial cross section. In fig. 5 the calculated anisotropy ratio A2/A o of the angular distribution is shown, where A o and A 2 are the zeroth- and second-order coefficients in a Legendre polynomial expansion of the differential cross section. The A 2 / A o ratio has a negative value if the resonance states with JR = ~z decay onto channel states of the residual nuclei with I~ = 0 + and 1 +, respectively. A positive value of the anisotropy ratio will be obtained for the decay o f J R = ~ resonances onto I , = 2 + states of the residual nuclei. However, the sign of the A 2 / A 0 ratio is determined mainly by the interference due to the decay of JR = 1 and 23-resonances and different partial waves as a rule.

CONTINUUM SHELL MODEL

121

4.3. ROLE OF THE 3p-2h CONFIGURATIONS IN THE GDR

Nearly 30 of all the 60 QBSEC taken into account in the coupled channels calculation have a dominant 3p-2h structure. The inclusion of complicated configurations of the 3p-2h type into the shell-model calculation additionally to the 2p-1 h configurations has the following influence on the properties of the GDR. First of all the photoabsorption cross section shows more structure especially, in the double humped main peak of the G D R around 23 MeV and 28 MeV, respectively [see refs. 10.11) for comparison]. The resonance states in the G D R which exhaust the largest part of the dipole strength have mainly 2p-lh structure corresponding to the lp~ ~ ld transitions. The admixture of 3p-2h excitations amounts here to 15 oj~,as a rule, in some cases even up to 40 ~,;. In all resonance states lying at excitation energies higher than 30 MeV the 2p-lh admixture does, however, not exceed I0 jo~;.Therefore, the high energy tail of the G D R is nearly completely represented by complicated configurations. The resonance states with 3p-2h structure carry about 20'~, of the dipole strength (40 MeV- mb) and g~ve rise to the large spreading of about 10 MeV for the G D R observed experimentally. This could not be reproduced in the 2p-lh approximation ~o). No remarkable reduction of the magnitude of the cross section has been obtained by the inclusion of more complicated configurations. The reason may be that the ground state correlations used in ~3C are too small. Kissener et al. 9) using CohenKurath wave functions obtain a 25 ''j contribution of lp~lp~ 2 configurations in the ground state. 4.4. 1SOSPIN EFFECTS

In the B(E1) values for the resonance states with different T-values one observes the known isospin splitting of the G D R in ~aC. There is a relatively clear separation in energy with a concentration of the dipole strength of the 7"< resonances around 21 MeV and of the 7"> resonances near 28 MeV. In fig. 6 we have represented the photoabsorption cross sections for two different calculations obtained by taking into account QBSEC only with T = ½ or with T = -~, respectively. The comparison with the experimental estimates 4) shows a shift of the centre of gravities for the dipole strength concentration of different isospin of about 2 MeV to lower energies but nearly the same energy separation of 7 MeV. This value is in good agreement with the 60 MeV law 12) which predicts E -- 6.9 MeV. Further, our calculations agree with earlier predictions 9.11) that the T< states carry roughly one-third of the dipole strength and the T~ states carry about two-thirds of the strength whereby a large overlap has been found. In our model the QBSEC with different isospin can mix via the continuum. To estimate the amount of this T-mixture we have calculated the partial widths for the T-forbidden decay of T> resonances to T = 0 states of the residual nucleus 12C. As a characteristic example we have chosen the lowest T> state (J = 1 + ) at 21.75 MeV

J. HOHN et al.

122

r

T- splitting

30

~T Lrq~J

Fig. 6. lsospin splitting of the G D R in 13C. The calculated cross sections for excited T = 21-(broken lines) and T = ~ states (full lines) are represented together with the corresponding estimated experimental cross sections from Patrick el al. 4) (thick lines).

with a dominating 2p-lh structure (86 0~,). Its decay into the n o, n 1 and n 2 channels is T-forbidden. The sum of the corresponding partial widths is 3.3 keV being smaller than 2 of the total width. This value has the same order of magnitude as the value obtained experimentally 13) for the isospin-forbidden decay of the 3 - 3~ level at 15.1 MeV in 13C. Thus, it seems to be that the partial widths of resonance states in 13C to isospin-forbidden channels are independent of whether the T-allowed channels are open or closed.

4.5. C O M P A R I S O N OF FORMULATION

PARTICLE

WIDTHS

USING

CSM

AND

AN

R-MATRIX "

Combining the usual shell model with the R-matrix theory the width of an isolated resonance state R is expressed in terms of the spectroscopic factors .~(a~-'by means of F R M . R = 21,2 Y" ~

S(R)2 P ik a nlj,

t

I~

c

i

c y~

(4.4)

¢

where c stands for the channel quantum numbers t, l, j. Eq. (4.4) follows from the R-matrix theory assuming that all the single-particle wave functions have the same amplitude at the channel radius a c and that only one main quantum number n contributes for given ! and j values. The amplitudes are generally estimated by the 2 where m is the reduced mass. Wigner limit providing the value of 7 2 = i3~2n / m a c, The penetrability factor P depends sensitively on the product of the channel wave number k c and the channel radius a cThe widths calculated with eq. (4.4) in conjunction with standard shell-model wave

CONTINUUM SHELL MODEL

123

functions are in reasonable agreement with the widths obtained in the CSM following from eq. {2.6) if the overlap between the resonance wave function and the channel wave function is large. This has been shown numerically 14) for resonance states with i p - l h structure in 160 • For small ~~,tR~ the widths calculated in both models /~t may differ strongly. 100, o

o

0

0

0

o

0o

%o

10

°0

0 0 ~"

o

l

0

0

o

01~ 0

0

OCD 0

0

0 0 0

x

1

0

0

0

0 o

0 0

0 0 ~ 0 0 o 0

E'[MeV] Fig. 7. Ratio X of calculated particle width by the R-matrix theory and the CSM.

In fig. 7 we have represented the ratio X = FRM/FCSM for the resonance states contained in the present investigation. The channel radius has been chosen to be a = 4.5 fm. The ratio X decreases for resonance states with increasing excitation energy which likely arises not from the different energy dependence in eqs. (2.6) and (4.4). The most important influence on the ratio X is surely due to the structure of the resonance states and the states of the residual nuclei, respectively. The resonance states for excitation energies larger than 30 MeV have a dominant 3p-2h structure but the states of the residual nuclei are mainly of l p-1 h structure. Thus, the overlaps between the QBSEC and the states of the residual nuclei decrease strongly with increasing excitation energy. When these overlaps are reduced for some channels the effect of coupling to other channels becomes important. This effect is not considered in the standard shell model but is accomodated in the CSM by coupling the continuum states to the QBSEC via the residual force. Therefore, the ratio X decreases as the overlaps get smaller. In extended shell-model calculations this decrease may partly be compensated by taking into account more channels with complicated structure.

124

J. HOHN et al.

5. Conclusions In the present c a l c u l a t i o n we have investigated the influence of c o m p l i c a t e d c o n f i g u r a t i o n s on the p h o t o e x c i t e d G D R in the nucleus ~3C. T h e results extend o u r k n o w l e d g e a b o u t the relative influence of the c o n t i n u u m a n d of the c o m p l i c a t e d c o n f i g u r a t i o n s on the p r o p e r t i e s of the G D R . W e c o n c l u d e that for higher energies the inclusion of m o r e c o m p l e x c o n f i g u r a t i o n s is of g r e a t e r i m p o r t a n c e than the inclusion of the o n e - p a r t i c l e c o n t i n u u m in o r d e r to explain the large s p r e a d i n g of the G D R . T h e 2 p - l h c o n f i g u r a t i o n space, used by M a r a n g o n i et al. lo. 1~) for the r e s o n a n c e states, has been e n l a r g e d by 3p-2h excitations. This a n d the inclusion of 2p-2h c o n f i g u r a t i o n s in the target states allows one to e x p l a i n the large width of the G D R of a b o u t 10 MeV. F o r e x c i t a t i o n energies between 30 M e V a n d 35 M e V the r e s o n a n c e states are m a i n l y g e n e r a t e d by 3p-2h c o n f i g u r a t i o n s which c a r r y a b o u t 20 i~i of the d i p o l e strength a n d r e p r o d u c e the high energy tail of the average p h o t o a b s o r p t i o n cross section in sufficient a g r e e m e n t with the e x p e r i m e n t a l data. F u r t h e r , the p r o n o u n c e d s t r u c t u r e in the high r e s o l u t i o n p h o t o n e u t r o n d a t a 5.6) in the m a i n p e a k of the G D R for higher energies than 25 M e V are well s u p p o r t e d by the present calculation. T o a n a l y s e the s t r u c t u r e c o m p o n e n t s of the G D R in m o r e detail, imp r o v e d m e a s u r e m e n t s of the e x c i t a t i o n functions as well as of the a n g u l a r d i s t r i b u t i o n s are desirable. O u r investigation of the widths o b t a i n e d in the C S M a n d in the shell m o d e l with R - m a t r i x t h e o r y using the s a m e c o n f i g u r a t i o n space a n d the s a m e residual interaction shows t h a t the ratio between the widths c a l c u l a t e d in the two m o d e l s is d e p e n d e n t on the s t r u c t u r e of the r e s o n a n c e states. W e are grateful to Prof. V. V. B a l a s h o v for s t i m u l a t i n g the investigation of the relations between C S M a n d R - m a t r i x theory.

References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16)

H. W. Barz, I. Rotter and J. H6hn, Nucl. Phys. A275 (1977) I 11 B. C. Cook, Phys. Rev. 106 (1957) 300 V. P. Denisov, A. V. Kulikov and L. A. Kulchetskij, JETP (Sov. Phys.) 19 (1964) 1007 B. H. Patrick, E. M. Bowey, E. J. Winhold, J. M. Reid and E. G. Muirhead, J. of Phys. G! (1975) 874 E. D. McKenzie, M. Sc. thesis, University of Melbourne (1974) J. W. Jury, B. L. Berman, D. D. Paul, P. Meyer, K. G. McNeill and J. G. Woodworth, Lawrence Livermore Laboratory, UCRL-77470 (1978) R. Koch and H. H. Thies, Nucl. Phys. A272 (1976) 296 D. J. Albert, A. Nagl, J. George, R. F. Wagner and H. ~berall, Phys. Rev. C16 (1977) 503 H. R. Kissener, A. Aswad, R. A. Eramzhian and H. U. J/iger, Nucl. Phys. A219 (1974) 601 M. Marangoni, P. L. Ottaviani and A. M. Saruis, Nucl. Phys. A277 (1977) 239 M. Marangoni, P. L. Ottaviani and A. M. Saruis, Phys. Lett. 49B (1974) 253 R. Leonardi and E. Lipparini, Phys. Rev. C l l (1975) 2073 R. E. Marrs, E. G. Adelberger and K. A. Snover, Phys. Rev. C16 (1977) 61 H. W. Barz, J. Birke, H. U. J/iger, H. R. Kissener, I. Rotter and J. H6hn, Yad. Fis. 24 (1976) 508 A. M. Lane and R. G. Thomas, Rev. Mod. Phys. 30 (1958) 257 H. W. Barz, I. Rotter and J. H6hn, Nucl. Phys" A307 (1978) 285