Photoreactions in 13C in the 2p-1h continuum shell model

Nuclear Phyatea A277 (1977) 239-269 ; © North-Holland Publiahlng Co., Arnaterdant Not to be reproduced by photoprint or microfilm without wrlttm permiaion from the publhher

PHOTOItEACTIONS IN ' 3C IN THE 2p-16 CONTINUUM SHELL MODEL M . MARANGONI, P. L . OTTAVIANI and A . M . SARUIS Centro di Calcolo del CNEN, Bologna, Italy Received 19 July 1976 (Revised 18 October 1976) AbatraM : The decay properties of the "C giant dipole resonance have been discussed in the frame of the continuum shell model in the 2p-1h approximation . The theoretical photoreaction cross sections and angular distributions have bcen compared with experiment.

1. Intr+odactioo Although the main properties of the giant dipole resonance in t 3 C are fairly known in the frame of the bound state shell model, there is as yet a lack oftheoretical discussion on the dynamical aspects of the photonuclear reactions in the giant resonance region . In addition, the large variety of experimental data, available at present on the photodisintegration cross sections and angular distributions, provides reliable information on the decay properties of the giant dipole resonance being theoretically discussed and interpreted. We have therefore analysed the dipole photoreactions of t3 C in the 2p-lh approximation of the continuum shell model. So far, the shell model including the continuum continues to stand as a valid framework for studying giant resonances in closed-shell nuclei'_ a). In the present paper we have extended the coupled~hannel method derived for closed shell nuclei a-6 ) to nuclei with one particle outside a closed shell (valence-particle nuclei). The method has then been applied in a simplified scheme to ' 3C . The photodisintegration of ' 3C has been investigated with both bound and continuum calculations, in order to clear up the decay properties as well as the underlying structure of the dipole states . Preliminary results have been published in a previous letter ~). In sect. 2, we give the main equations and definitions ofthe continuum shell model for valence-particle nuclei in the 2p-lh approximation. We derive the coupled-channel equations for ' 3 C. This approach is then applied to the calculation of the dipole photoreaction cross sections and angular distributions. Sect. 3 contains the results, as well as the discussion of the ' 3 C photonuclear reactions. 239

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M . MARANGONI et al.

2. Contmam~ shell model for valence-particle nadei: The ?,p-16 approxnoatbn for' 3C For the purpose of clarifying the computational methods the basic equations a. 9 ) and the underlying assumptions will be first summarized. Then, on the assumption of a limited 2p-lh basis space consistent with the observed reaction channels in' 3C, we derive a set of manageable coupled-channel integro-dif%rential equations in the r-representation . The properties of the adopted 2p-lh basis space are discussed in the framework of a bound state calculation. Finally, we treat in a schematic way the coupling to the omitted configurations . 2.1 . THE BASIC EQUATIONS

Let us consider a system of A nucleons. We separate the shell-model nuclear Hamiltonian in the usual form : H = Ho +V.

(2 .1)

Here, Ho is the independent particle Hamiltonian, and V is the associated residual interaction. Assuming a single-particle potential of finite depth, the single-particle Hamiltonian has both a discrete and a continuous spectrum ; we use ~a to denote its eigenfunctions defined by r

w

(2.2)

where b stands for the various quantum numbers needed to specify the state 8 =- (ea, la, .la~ Ta~ ma) --_ (d, ma)~

(2.3)

We also write S -_ (d, - rna). The radial wave functions ua(r, ed) are eigenfunctions with eigenvalues ea of the dü%rential operator _ _ %Z d2 _ la(la + 1) where va(r) is the single-particle potential operator. From now on we shall write e instead of e, in the continuous spectrum . The functions ud are normalized in the usual way e) and satisfy the following completeness relation a

ua(r, eah
J

(2 .5)

In the general formalism ofthe shell model with one nucleon in the continuum s ' 9~ the eigenstetes I~B) of the A-nucleon system at energy E, with an incoming wave in channel c, satisfying the equation

PHOTOREACTIONS IN "C

24 1

are expanded as follows

where the states I~É), Iii) and IXâ) are assumed to have definite total angular momentum J and its projection M, with a given parity n. The I~,) states are manyparticle wave functions where all nucleons occupy bound shell-model orbitals and are assumed to form an orthonormal set of Ho eigenfunctions : <~tl~~)

= a~r

(2.8a)

The IXé) states represent a nucleon in a continuum state at energy e coupled to a bound eigenstate of the residual (A-1}nucleon system . Here the (A-1}nucleon system target eigenstetes are described in terms of configurations with all nucleons in bound shell-model orbitals . Let us write I nlM) to denote the eigenfunctions of the (A -1}nucleon system with total angular momentum 1 and its projection M~, the index n denoting any further quantum number necessary to specify the state completely . We have (nlpMIn~l.M.) =

5

1111~(/rwrw " SA/w/1(w.f


(2.9a) (2.9b)

Let hj~T~ be the quantum numbers ofthe nucleon in the continuum. In this notation the channel index c stands for the set of quantum numbers {hj~sr nl } . From the definition of the IXâ) states and from eqs. (2.9) we may write
(2.10)

where VB is that part of the residual interaction which refers to nucleons in the target 9). The functions I~,) and IX~) also satisfy the orthogonality relations

Later on we use the expression "subspace 1" to refer to the subspace spanned by the I~,) states and "subspace 2" for the subspace spanned by the IXâ) states. Substituting eq. (2.~ in eq. (2.6) and using relations (2.8), (2.10) and (2.11) yields

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M. MARANGONI et al.

the following system of basic equations for the coefficients bB(f) and a`a~E', c~ (E~-~~(i)+ E b~.i)<~tlVl~,>+ E (~ + E". - ~`r(E', c') + ~ bB(j) + ~ 1

~-

J

J

~'~`~E',?)<~,I~Ixâ > = o,

(2.12a)

dE"a`~(E~ c~~) = 0. (2.12b)

2.2. THE lp AND 2p-lh STATES

Let us describe the valence-particle nuclei in terms of the model space of the one particle (lp) and 2p-lh configurations . The uncorrelated ground state of the A-1 closed shell nucleus is assumed to be the physical vacuum and the excited states of the (A-1}nucleon system are described in the lp-lh Tamm-Dancoff approximation. In the formalism of the second quantization we introduce the fermion operators a8, as creating and destroying a nucleon in a shell-model state ~~. In what follows y stands for a shell-model state above the Fermi surface which may be bound or unbound, while a and ß are simply the bound shell.-model states above and below the Fermi surface, respectively . The eigenfunctions of Ho in subspace 1 are written as aq lo> I~k> - ~~

Ci~m,.~~I~M> E <.iaidn~pIIM~~(-~'-~°ar aa a~lo>

(lp state) (2p-lh states), (2.13)

where j0> is the physical vacuum and yo and y denote bound states. The 2p-lh states thus obtained are in general not mutually orthogonal and linearly independent. We have to select a set oflinearly independent ones and orthononmalize them by the Schmidt method to obtain the I~,i of eqs. (2.8). The functions IXii are written as

where y now denotes an unbound shell-model state. The target states InlM) in the lp-lh Tamm-Dancoff approximation are given by

InhM~> _ ~ ~" ~ Cia.ir~~pl1,.M~>(-~°-~°aâaplo~, ~~)
where âNw are the particlo-hole configuration mixing coefficients. For the ground state I~~`°) we assume a lp uncorrelated state.

(2.15)

PHOTOREACTIONS IN "C

243

2.3. CHOICE OF SINGLE-PARTICLE POTENTIALS AND BASIC CONFIGURATIONS

The calculations have been performed assuming a shell-model Woods-Saxon potential, with a spin-orbit and a Coulomb term, of the following form : a ° _ - yi [l +et'-x") /a~ _ t + V1o.(f
(2.16)

Here, fiilm,~c is the Compton wavelength of the n-meson, while R° = r°(A-1)} . The Coulomb potential is that given by a spherically symmetrical charge distribution . TAHLE 1

Parameters of the single particle potentials p+ i :B

"C protons

"C neutrons

n!'

ep iaN (MeV)

V (MeV)

Vi'. (MeV)

nl. ~

lpiiz siiz dsiz da~z

-1.94 0.42 1 .57 6.28

55.00 59.48 57.00 57.00

19 0 4.85 4.85

1 Pi~z 2s~~z lds~z dan

Ipaiz lsi/z

15.95 31 .36

55.90 59.48

19 0

lpa~z lsl~z

E,sP

ii

B

iaC (MeV)

Env

n+ `zC

v,~

(MeV)

(MeV)

-4.95 -1 .86 - 1 .1 3.39 e "C

54.66 60.73 57.77 57.77

19 0 4.85 4.85

18.72 36.24

54.95 60.73

19 0

ro ~ 1.25 fm, a = 0,53 fm.

Table 1 lists the set of single-particle states in the Woods-Saxon well for neutrons and protons with the corresponding parameter values. These parameters are chosen to fit the experimental single-particle energies inferred from the level schemes of the ' 3 C, t 3N and t t C, t'B nuclei t°). The set given also satisfactorily reproduces the experimental nuclear charge radius of t3 C, rib = 2.32 fin of ref. tt). Since the t 'C ground state has J~ _ ~-, we are interested in the Jx = i+, Z+ states of the compound system, for dipole transitions. In the light of the preceding subsections we must now define the oonfigurational subspaces 1 and 2. In our scheme subspace 2 for the ` 3C nucleus is spanned by the configurations with one nucleon in the continuum coupled to a t ~C or t ~B state described in the lp-lh Tamm-Dancoff approximation. The target states and corresponding reaction channels c considered are given in fig. 1 . The t Z C and t ZB states are assumed to have a good isospin quantum number T ; their excitation energies e, are taken from experiment in order to describe the threshold behaviour correctly. We should point out that the residual nucleus is described in terms only of configurations with all nucleons in bound shell-model orbitals. This in general involves a restriction when it comes to giving a suitable

244

M. MARANGONI et al . 13` ,

One - Particle Continuum 1p and 2p-1h confifiurationa 30

E~(MeV)

R

2i .os

R

10

- .

R -

R -

R

s

: .

0 Io £ Target

Te 1P-1h

0+

QS izC 0

2*

4.43 i2C 0 1p, 1pa !( 3

1+

~+

~*

2*

iz C

l3S

15 .11

0.95

16.11

iz B

iz C

iz B

rzC

0

1

1

1

12 .73

)_,

~,1p~~3~, , 1p, (1Pa),1p~rp 1p, (1Pa 3 3 3 F ~ 3) 3 ~

2+

1

~~ (1p~~_ i

3

~L~rb~ ~ ~ 1 ~ 2 3 4 5 6 7 9 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

Fig. 1 . Energies of the 1pand 2p-lh basis confgurations with the one-particle continuum for J` _ }+, ~ +

description of the target . In otu case however we may be sure of a fairly good approximation in representing the target states of fig. 1 as pure lp~{lp.t) -t configurations, in fact this choice does not differ appreciably from the Tamm-Dancoff results of ref. tz~ which give a satisfactory description of tZC except for the first 2 +, T = O level. We are confident however that a more sophisticated treatment of this level would not destroy the strong dominance of the lp~{lp~) - ' component in the wave function . In fig. 1 the horizontal full lines limiting the shaded area represent the threshold energies for nucleon emission. The horizontal dashed lines correspond either to bound (B) or to resonant (R) single-particle states for the odd nucleon v. As far as subspace 1 is concerned we have considered all the lp and 2p-lh configurations corresponding to a 1 fiiw excitation with all nucleons in bound shell-model orbitals. In bound configurations, and as far as the particle-hole pair is concerned, we have treated isospin as a good quantum number. As outlined in sect. 2 we have selected two basic sets, with J~ _ ~+ and i +, of2p-lh linearly independent eigenstetes of Ho. The oorresponditlg oonfigt>
PHOTOREACTIONS IN "C

245

Teem 2

The lp and 2p-lh basic configurations and mixing amplitudes of the strongest dipole states of "C Configurations

Mixing amplitudes ~`, T =

v

lp-lh

2s ;,z ld,fz ~i~z ld ;~ z ld~~ z 2s;~z ld ;~z ld 3 z ~~iz ld,~z ld,~z 2s ;~ z lds~z ldj~z ~~~z lds~z ld3,z ~iiz Id3~ z ld,~ z lPin lPin lPin ~

lPi~z(lPa~z) - ' lp,iz(lpa~z) - ` lp,~z(lpa/z) -' lp,iz(lpan) - ' 1Piiz(lPa~z) - ' ` lp,~z(lpaiz)'- ' lPiiz(lPa~z) lp,iz(Ipaiz) - ' lp,~z(Ipa~z) -` lp,~z(lpa/z) -` 1Piiz(lPa/z)- ' lp,iz(lp~iz) -_ ` lPiiz(lPan) ' lp,iz(lpniz) -` lp,iz(lpniz) -` lPin(lPaiz) -' lp,~z(lpaiz) - ' lp,iz(lpsiz) - ` lPiiz(ls~n) - ' lPirz(lsy)~ ' lPnz(ls,iz) - `

I

Tf

Tj,

0+ 0+ 2+ 2+ 2+ 1+ l+ 1+ l+ 1+ l+ 1+

0 0 0 0 0 0 0 0 i 1 1 1 1 1 1 1 1 1 1 I 0 I 1

0 0 0 0 0 0 0 0 -l -1 -1 0 0 0 -1 -1 -1 0 0 0 0 0 -1

1+ 1+ 2++ 2 2+ 2+ 2+ 2+ l l1

I+, I

I+, I

I+, I

~ +, I

I+, I

1+, ~

E= 14 .2 E= 14.3 E= 19 .6 E=24.0 E=24.8E=25 .2 MeV MeV MeV MeV MeV MeV 0.409 0.074 -0.141 0.615 -0.012 0.463 -0.093 -0.040 0.192 -0 .040 0.028 -0 .136 0.028 -0 .047 0.308 -0 .021 0.033 -0 .218 0.015

0.090 -0.432 0.319 0.444 0.006 0.173 0.064 -0.122 -0.045 0.550 -0 .031 -0 .389 0.022

0.154 0.034 -0 .147 0.044 0.051 -0 .776 0.130 -0 .014 0.259 -0.094 0.010 -0.183 0.066 -0.048 0.374 -0.040 0.034 -0 .265 0.029

0.005 0.290 -0.127 0.007 0.410 -0.179 -0.061 0.478 0.024 -0 .086 0.677 0.034

0.035

-0 .025

0.423 0.049

0.183 -0 .036

0.598

0.259

0.224 -0.321

-0.531 -0.133

0.317 -0 .454

-0 .750 -0 .188

possible linearly independent sets ; our choice is motivated by the fact that each basic configuration corresponds to only one reaction channel considered in fig. 1, except for the ls} hole configurations. Furthermore the bound configurations of fig. 1 corresponding to the dashed lines (B) are the same as those given in table 2. If the ls } hole subspace can be neglected, the above correspondence allows an easy treatment of the basic equations (2.12) in the r-representation, as we shall show in the next subsections. 2.4. DISCRETE CALCULATIONS

In order to analyse the incidence of the 1s~ hole subspace on the dipole spectrum and strengths, we have performed a set of discrete calculations, taking into account all the configurations of 1 >~tcu excitation. In table 2 we have listed the basic configurations assumed which, with the exception of the ls.~ hole ones, correspond to the

246

M. MARANGONI et al.

~+~ ~+ Jx =

orthonormal basis states. We point out that, excluding the ls~ hole subspace, the above configurations are precisely the channel configurations of fig. 1 corresponding to bound or resonant single-particle states for the odd nucleon. We have diagonalized the Hamiltonian in the following cases : (a) in the space of table 2, (b) in the space of table 2 without the 1 s~ hole configurations, (c) in the space of the bound configurations of table 2 with the exclusion of the ls~ hole ones. We refer to the eigenstates resulting from this diagonalisation as the quasi-bound 9) states of our model space. In the calculation we have assumed that isospin is a good quantum number : we have taken for the protons the single-particle energies and wave functions as for the neutrons. The single-particle resonances have been "discretized" using the method of ref. 13). In the matrix elements the contributions which refer to nucleons in the target have been replaced by the experimental excitation energies of the ' ZC states in order to compare the discrete calculation with the continuum calculation. In case (a) the c.m . motion spurious states have been removed with the ElliottSkyrme method. In fig. 2 we give the energies and the integrated absorption cross section ia) for each bound state obtained in the diagonalisations (a), (b) and (c). A zero range force, with a Super mixture, has been used [eq. (1)] . We find that the ls.~ hole configurations have non-negligible components only in the states with energies above 29 MeV. Besides, we find that neglecting these configurations does not appreciably affect the spectrum and the dipole strengths below this energy, as can also be seen from a comparison of figs . 2a and b. The small differences between calculations (a) and (b) below 15 MeV are due to the c.m. motion spurious states, not removed in case (b). The calculations have also been performed assuming for the residual interaction a finite range force of Gaussian form with range parameter ~ = 1.7 fin [ref.'2)]. The main features of the spectra and of the corresponding ground-state dipole transition probabilities have been found to be similar to the delta force ones. We have also performed a discrete calculation with the Coulomb term in the singleparticle potential. It has been found that the Coulomb tenor does not mix the isospin below 20 MeV, whereas at higher energies the amount of mixing is only a few per cent ( < 4 ~) . 2.5 . COUPLED-CHANNEL METHOD

In order to calculate photoreaction cross sections we need to know the function Then we have to solve the system ofeqs. (2.12) to obtain the coefficients

~Fx everywhere.

The basis configuration chosen lead to a very complicated system of coupled integro-differential equations in the r-representation, with singularities at the eigenvalues of the matrix <~,~H~~~) . Owing to our assumption of pure lp-lh

PHOTOREACTIONS IN "C

247

E~(AAeV) Fig . 2 . Theoretical integrated absorption cross sections for "C obtained in the discrete calculations showing the T = } (dashed lines) and T = ~ (full lines) contributions . (a) In the space of table 2. (b) In the spar of table 2 without the ls,~~ hole configurations. (c) In the space of the bound configurations of table 2 with the exclusion of the ls,~ s hole ones .

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M . MARANGONI et al.

target states we may obtain, however, more manageable equations with a convenient restriction of subspace 1. In the discrete calculations of subsect. 2.4 it has been shown that in the energy range 0-29 MeV subspace 1 can be reduced with a good approximation by eliminating the ls.t hole configurations. Consequently we assume that a similar space reduction can also be performed in the continuum calculations. The remaining bound configurations are all orthonormal and each one corresponds to only one channel (fig. 1). In this case, the expansion of the wave function ili`E can be performed as follows, by associating each bound component with its corresponding continuum component I~E) _ ~ ~ ~ . bSlEe', C71~,)+ dE'R~e E', c'

4' > ip

J

~)Ix~') },

(2.1 ~

where I~`.~.) represents a nucleon in a bound single-particle (state above the Ferrai surface (eF is the Ferrai energy) at energy e~. with quântum numbers 1~.j~'zc' coupled to the same target eigenstate as in IXé.). It should be noted that expansion (2.1~ is not equal to expansion (2.'7), because the I~`~i states, in contrast to the I~r) states, are not eigenfunctions of Ho : since they are defined in the same way as the I,li) states, they satisfy equations similar to eq. (2.10). However, owing to the above mentioned assumptions, the linear spaces spanned by the two sets of bound states are identical. A set of basic equations are obtained for the coefïicients 68(Ec', c') and a`E(E', c') by replacing in eqs. (2.12) E, by e~. + E, and V by V - VS in the bound-bound and boundcontinuum matrix elements. We now derive the coupled-channel equations in the r-representation. We write eq. (2.1 ~ in a more compact form as (2.18) where ~ denotes a summation over the discrete and an integration over the continuum above the Ferrai surface, while I;Y`~,) denotes a I~,) or a IX:, ) state. Let us introduce the function s) J$(r, ~ _ ~ bElEd, C~~d(r, Ed)+ dE'a`El~, C~I~d(r, E~) ~,>s~, J

This is the radial wave function of the projection of Illt`E) on the target state In'1.M.) corresponding to channel c'. From the equations for the coeffcients 6°E(e~', c') and ~fE(8', c~, and using the definition of the operator D given by (2.4) and the completeness relation (2.5~ we obtain the following system of coupled integro-

PHOTOREACTIONS IN "C

249

differential equations for the unknown functions fÉ(r, c') :

_ - ~ { dri~b(r-ri)- ~ u~'(r, e~'h~~'(ri, e~')11~t°~(r,)fâ(rv c") &, <~ . J u~ (r~ Ec'l~c'(Tl~ $~')~ME~..(rh r2)f8(r2~ ci ~ -~ drldr2[a(r - ri) J 7~ 4'<6F

(2.20)

The coupling matrix elements M°.(r) and MK.(r, rl) are defined by
J

dru~(r, e~)MK'(r)u~'(r, e~.) - drdrlu~(r, E~)MM'(r, ri~~.(rh e~ .) .

J

(2.21)

Their expressions are wen in appendix A. It should be noted that eqs. (2.20) can only be derived on the assumption of pure lp-lh target states . These equations represent an extension of the equations derived for even nuclei in the lp-lh model space s) to the valence-particle nuclei in the model space of lp and 2p-lh. The solutions to system (2.20) are found by imposing the usual asymptotic boundary conditions 6). The expressions for the photonuclear cross sections are derived in appendix B. 2.6 . APPROXIMAI~ TREATMENT OF THE OMITTED CONFIGURATIONS

In the present subsection we show that if configurations with higher complexity than lp and 2p-lh are treated approximatively in a statistical way, it leads to the replacement of the real single-particle potentials with complex potentials, in eqs. (2.20). Let us write the total wave function in the following form : where IAA stands for the subspaces 1 and 2 ofthe bound and unbound lp and 2p-lh configurations, defined in subsect. 2.1, and IBS is the subspace of the bound and unbound configurations of higher complexity . Following the procedure outlined in ref. a~ we write the Schrôdinger oquation (2.6) for the wave function (2.22) and we perform the elimination of subspace B. In the notation of reî ~, we obtain where

(Hc~ -E)IA) = 0, H~ = H~+V~°

yeA . 1 E-H°°

(2.23) (2.24)

250

M . MARANGONI et a1 .

We can notice that eq. (2.23) is formally equivalent to the coupled equations (2.20). The second term in the r.h.s. of eq. (2.24) represents the coupling of space A to space B. Later on, we shall refer to situation in which the coupling term in eq . (2.24) can be treated approximatively in a statistical way e). Let us denote by I s), E~ T, the eigenstetes of HBB, their energies and widths, respectively . The eigenstetes of H" have a width T, because space B includes part of the continuum. It should be noted that the restricted Hamiltonian HBB has eigenvalues of practically the same density as those of the complete Hamiltonian~ H, since the eigenstetes of H in space A cover only the class of lp and 2p-lh states. We can write eq. (2.24) as a relation between matrix elements VIs) _
(2.25)

where In) stands for a configuration of space A. If the spacings D of levels E, are such that l', ~ D, then eq. (2.25) is a smooth function of E and an averaging of the matrix elements _
PHOTOREACTIONS IN ' 3C

25 1

3. Results and discaesion The calculations have been carried out with a zero-range two-body residual interaction V(w rz) = Vogt - rzXao+atmet ' Qz~

(3 .1)

with a Soper mixture : ao = 0.865 and at = 0.135 . The strength Vo = 700 MeV ~ fm3 has been fixed by best-fitting the energies and the widths of the main peaks observed experimentally in the cross sections . The choice ofthe S-function interaction simplifies the numerical treatment of the coupled integro~ifferential equations (2.20). Besides we have noted in the discrete calculation that a finite range does not appreciably change the results obtained by a zero range force, which leads us to expect the same for the continuum calculation. The imaginary single-particle potential discussed in subsect. 2.6, has been assumed to have a Woods-Saxon derivative form whereas its depth has been faced bycomparing the calculated cross sections with experiments. That is - iW4et'-R olia~l + el'-xol~a~ - z (3.2)

Fig. 3. Total theoretical cross sections for neutron emission to the ground 4.44 MeV 2+, T = 0 and 12.73 MeV 1 +, T = 0 states in''C calculated with theaddition of the Coulomb potential term (full curve) and without the Coulomb potential term (dashed curve). The absorptive potential W has bxn set equal to zero .

252

M . MARANGONI et al.

where W(MeV) = 0.0712E~(MeV)-0.352.

(3.3)

In the present section, we first discuss the Coulomb eRects on the cross sections.

a E

b

Fig . 4 . Theoretical cross sections for neutron emission to the 15 .11 MeV ( *, T = 1 state in '=C and for proton emission to the ground state 1 *, T = 1 in '=B . Same convention as in 6g. 3.

PHOTOitEACTIONS IN "C

253

Then, by comparingour results with the experimental data, we can infer the configurational structure of the observed resonances . Finally, we evaluate the usual sum rules and deduce the isospin splitting of the GDR.

Fig. 5 . Theoretical cross sections for neutron emission to the 16 .11 MeV 2 + , T a 1 state in "C and for proton emission to the 0 .95 MeV 2+, T ~ 1 state in "B. Some convention as in fig. 3.

254

M . MARANGONI et al.

3 .1 . COULOMB EFFECTS

In order to discuss the effects arising from the presence of the Coulomb term in the Woods-Saxon potential (2.16), we should compare calculation (p), where the Coulomb term is present in the proton single-particle potential, with calculation (n) when identical single-particle states are assumed for protons and neutrons. The results of calculations (p) and (n) are plotted in figs. 3-5. When the Coulomb term is present in the Woods-Saxon potential the independentparticle Hamiltonian Ho does not commute with the total isospin operator TZ, so that some isospin mixing due to the Coulomb term can be observed. In the discrete calculation of sect. 2 the isospin mixing has been found to be rather small and each state retains a qualitatively good isospin quantum number. In the continuum calculation however the Coulomb term has an appreciable effect on the strength distribution between (y, n) and (y, p), as will be shown later on. In order to analyse the features exhibited by the cross sections of figs . ~5 we refer to the shell-model one-level (one quasi-bound state), many channels approximation, discussed in ref. 16). We assume that near resonance energies ER, the cross section in channel c displays a Lorentzian shape given in the notation of ref. 16) by (3.4) where I'1 is the total dipole width, I'1~ the partial decay width in channel c and l'lY the dipole decay width. They are defined by

where Q is the electric dipole operator and c~, co are the proton and neutron channels respectively . To distinguish case (n) we shall insert an upper index (n) in expressions (3.4}{3.'7). According to (3.4~ the ratio of the cross sections in channel c at E = Eß in the two calculations (p) and (n) is a(Y, ~) _ _rio rir l~~i1~~. (3.8) ~°~Y, ~) ~i~ ~ii ri In order to evaluate the size of the r.h .s. of eq. (3.8), some comments are needed : (i) From definition (3.~ we infer that the ratio l'i r/l'li; is equal to unity or even probably somewhat smaller than unity, because the proton wave functions extend beyond the neutron wave functions. (ü) The ratio I'i~II'~i~ is equal to unity when c is a neutron channel. When c is a proton channel, then it is somewhat larger than one. It is usually argued that

PHOTOREACTIONS IN "C

25 5

the effect of the Coulomb barrier is compensated for by the larger value of kinetic energy in the proton channel. This implies that the penetration factor for a proton channel in a model with neutron wave functions is smaller than the penetration factor for a proton channel in a model with the Coulomb term . (iü) The ratio Iii>lrl is equal to unity when only the neutron channels are open. It becomes smaller than unity when the proton channels are open since rl ~p > IKlP. Obviously, it becomes increasingly smaller than unity as the number of open proton channels increases. According to points (i}{iü~ the size ofthe ratio (3.8)can now be evaluated. We have to distinguish between three cases for channels c : (a) For channels c leading to the target states 0+ , 2 + and 1 +, T = 0 of 12C (neutron emission), the proton widths in eq. (3.~ are practically equal to zero over all the energy range so that,with rl x I~i~ the ratio (3.8) is practically equal to unity. This is the main reason why the results ofcalculations (p) and (n) in fig. 3 are fairly similar. (b) For channels c leading to the states 1 + and 2+, T = 1 of 1ZC, we have 1, whereas IKl~/rl < 1 because of the presence of the proton widths. We riJ~i~ actually observe that the (y, n~ cross sections with a Coulomb term given in figs. 4 and 5 are reduced by about 40 ~ at resonance compared with results (n). (c) For channels c leading to the states 1 + and 2 +, T = 1 of 12B (proton emission) we have the two competitive effects r~CP

> 1,

rcl~

< 1. r1 - ICP Because of the presence of the neutron widths in the total width, we have . hc°> > hco~ i i~ P

(3.9) (3.10)

Therefore the ratio (3.8) tends to be higher than unity, in spite of the square in the last factor on the r.h.s., this is shown by the results given in figs . 4 and 5. 3.2. COMPARISON WITH EXPERIMENT

Although many measurements are available on the photodisintegration of 13C, they suffer from a fairly large uncertainty and inconsistency . Cook's 13C(y, xn) measurement l') does not separate the reaction channels for multiple nucleon emission. For the total 13C(y, n)1ZC cross section Cook has estimated a lower safety limit above 23 MeV. This limit seems to be too low inthe light ofmore recent measuromeats 1 e.' ~ which show that the main part ofthe cross section above the two partide threshold arises from the (y, ny~ process. The 13C(y, xn) cross section up to 29 MeV has also been obtained with good resolution by McKenzie 2°} McKenzie's crosssection values are smaller than Cook's measurements, but the broad katures of these data are similar. Fig. 6 presents the total iaC(y, n)12C calculated cross-section compared with the data of refs. "~ ~~.

Fig. 6. Measured and calculated ' s C(y, n) suss sections. The data are from Cook ") (solid triangles) and McKentie =q. The calculation (full curve) has been performed with the addition of the Coulomb potential term and the absorptive potential has been assumed as W(MeV) = 0.0712E~(MeV)-0 .352.

Fig. 7. Total ' sC(y, p)' = i3 cross section compared with the data of ref. ") (solid triangles) and ref. ' `) (solid circles) . Same calculation as in 6g. 6.

Several measurements are available on the t3C(y, p)t2H cross section, but again with some inconsistency between them. Indood Cook's results ") disagree in absolute value both with the data of Patrick et al. t9) and with the measurements of Denisov et al. 2tß In fig. 7 we give.our total ta C(y, p) t2B computed cross suction (full curve) and the data of refs. t7, 2tß Ta+o sets of data on t3 C(Y, P)t2B cross section at 90° are available, one obtained by Kosiek et al. ss) and the other deduced by Shin et al. 2s)

257

PHOTOREACTIONS IN "C

Fig. 8. Calculated "C(y, p)'2B cross section at 90'. The data are from Kosiek et a/. ZZ) (histogram) and from Shin et al. ~') (solid circles) . Same calculation as in fig. 6.

0.8 0.4

_________ 'ac~~ .

OZ

P )°B

â o _0,2 -0.4 -0.8 1-J ._ ~_ ~ . . _-1 .-

18

20

22

24 E~( MeV)

28

28

30

Fig. 9. The A2 coefficients of the expansion daJdl~ _ (a°/4n)[1+~ ;_,A P(cos B)] for the reaction 'a C(y, p)` =B relative to ground-state proton emission (full curve) and both ground and first-excited state proton emission (dashed curve). The data are from ref. za). Same calculation as in fig. 6.

from the t Sae, e'p) reaction. Both data are deduced from the assumption that all transitions lead to the ground state, although non-ground-state protons are present in these meastuements . For this reason the data of refs. 22, Zs) are compared in fig. 8 with our calculated tsC(y, p) t2B cross section at 90° for the decay to both the ground Pnd first~xcited state of t2 B. Fig. 9 shows the AZ coefficients of the Legendre polynomial expansion da/dl) _ 23). ~QO/4n)[1 +~=tA P.~oos B)] for the reaction 13 C(Y, P)` Z B. The data are from ref. The theoretical full curve corresponds to the ' 3 C(y, Po)t ZB reaction, the theoretical

258

M . MARANGONI et al.

dashed curve corresponds to the is C(y, p)tZB reaction where the ground state and the first~xcite~} state proton emission are included. The measurements on the i s C(y, ny~ and t 3C(y, py') cross sections of refs . t e. t 9) give a clear indication of the residual nucleus levels which are left by the emitted nucleons. In refs . t a' t 9) the excited levels identified in the residual nuclei are the 4.44 MeV 2+, T = 0 and 15.11 MeV 1 +, T = 1 states of 12C and the 0.95 MeV 2+ , T = 1 state of t ZB. The set of levels experimentally observed in the photodisintegration of t 3C are all taken into account in our calculations. In fig. 10 we give the theoretical partial cross sections (full curves) for proton and neutron emission to the considered states ofthe residual nuclei, compared with the experimental cross sections of ref. l 9 ) (dashed curves) and ref. a4) (dashed and dotted curve). l:n fig. 11 we report the measured tZ C(p, yo)t3N cross section at 90° of Berghofer 6 nC( ~, p~')°B 4

1'T"1 QS .

2'T"1 0.>i611AaV

i

~

..

,)~

i

i i

i

i

0 4

b

i

1'T"1 16.11IIArN

~

2

i

_~`

Ô'Ô v ``

2'T" 1 16.11 AAW `,

i 2'T"O 4.44 AiW

0

1'T"0 12,7 AAa1l

O'T"O ßS .

2

0

Fig . 10 . Calculated partial cross sections (full curves) compared with the experimental results of ref. ' ~ (dashed curves) and of ref. ") (darshed and dotted curves) . Same calculation as in fig. 6.

PHOTOREACTIONS IN "C

259

E~( MeV)

Fig. 1 l. Calculated '~C(n, yu)"C cross section at 90°. The data are from the "C(p, yu)"N cross section at 90' of Berghofer et al. ~`) (solid circles), Fisher et al. 2') (solid squares) and HasinofF et al . a°) (dashed curves). Same calculation as in fig. 6.

et al. za), Fisher et al. and Hasinoff et al. ss) to be compared with our t2C(n, yo)t3C

theoretical cross section at 90°. This comparison is meaningful as far as the structure is concerned, because it has been shown in subsect. 3.1 that the Coulomb effects do not change the strong features ofthe cross sections such as resonannces. The data of 13erghofer et al. Za) are in disagreement with those of Fisher et al. ss) and of Ferroni et al. ae) especially concerning the strength of the peak at 20.5 MeV. 3.3 . DISCUSSION

3.3.1. Decay properties of the GDR. The t3C(y,xn) experimental cross section given in fig. 6 shows a "pygmy" resonance in the region of 10-16 MeV and a giant resonance between 20 and 27 MeV. In Cook's results the pygmy resonance is centred at 13.5 MeV and the giant resonance displays two broad peaks at 22 and 26 MeV. McKenzie's more recent results show more structure in the energy range considered here . The pygmy main strength is again at 13.5 MeV, whereas two well-pronounced peaks at 23.5 and 25.5 MeV and a lower peak at 20.5 MeV are present. As noted in subsect. 3.2, McKenzie's aoss-section values are lower than Cook's, as can also be seen from the cross section integrated up to 29 MeV (table 3). Our taC(y, n) cross section well reproduces the main features of the t Say, xn) experimental cross section.

260

M. MARANGONI

et al.

Theoretically the pygmy resonance and the first peak in the giant resonance region are fotmd 1 MeV lower in energy than in ref. 2 ~. The main peak observed by McKenzie at 23.5 MeV is well reproduced, whereas the 25.5 MeV peak has not been found. The is C(y, p) tZ B calculated cross section, shown in fig. 7, satisfactorily reproduces both the strength and the structure of the experimental data of Denisov et al. 21); a lower peak and a higher peak were found at 20.0 and 23 .5 MeV respectively. The agreement obtained with the (y, n) and (y, p) experimental strengths is also due to the presence of the Coulomb term in the optical potential ; indeed, as shown in subsect. 3.1, the Coulomb term does not change the cross-section structure, but modifies the strength distribution between (y, n) and (y, p) above 20 MeV energy, reducing the neutron channel strength and increasing that of the proton channels. An equally good agreement has. been obtained with the experimental (y, p) cross section at 90°. Our calculated cross section predicts the main resonance 1 MeV lower in energy than in refs. ~a, 23) . We note the discrepancy in the giant resonance position between the data of ref. 2') and of refs. ss, za). From the analysis ofthe theoretical partial cross sections in the photodisintegration Tee~ 3

Integrated cross sections Integrated soss sections (MeV ~ mb) Reaction

' ~N(y, p) '~C(y, n)

`a C(y, n) ''C(y, xn) ''C(y, xn) "C(Y. P) 'a C(y, P)

Final state

ground state ground state 4.44 MeV 12.73 MeV 15.11 MeV 16.11 MeV total total total ground state 0.95 MeV total

exp

talc

E~=29

E~=38

19

23

8

8

24 [26]

30 [48]

Ref.

~=29 E~=38 17 13 4 8 19

17 14 6 11 23

61

71

licrghofer etal. ") plus Fisher et al. _') Patrick et al . ' ~ Patrick et al. `~ Patrick et al. `~

62 93

117

Mckenzie =q Cook ")

[22] 19

[25] 30

14 26

17 33

Patrick et al. '~ Patrick et al. `~

44

60

40

50

Denisov et al . ")

of t3C, shown in fig.10 and table 3, we can deduce that the "pygmy and the 20.5 MeV resonanoes in the total (y, n) cross section are due to neutron emission to the ground and first 2* excited states of l'C . Resides, we find that the ground-state transition is slightly stronger than the transition to the 4.44 MeV 2* level, in agreement with ref. Za) and in disagreement with the theoretical result of Kissener et al . Z') . The

PHOTOREACTIONS IN "C

26 1

theoretical cross section for decay to the 12.73 MeV 1 +, T = 0 of t Z C is found to be very small, in agreement with the lack of experimental evidence for this transition. We also deduce that the main giant resonance in the (y, n) cross section decays almost exclusively to the 15.11 MeV 1 + , T = 1 and 16.11 MeV 2 + , T = 1 states of tzC with a distribution close to like 1 : 2. Similarly the (y, p) resonance is found to decay to the 1 +, T = 1 ground state and to the 2+, T = 1 first excited state of t 2 B with a feeding of the latter about twice as strong as the former . The good agreement _ with the (y, p) experimental data of refs. zt ~3) suggests that the main part of the strength is due to these transitions alone, which have been experimentally observed by Patrick et ai. t 9). However their measured or deduced data do not agree very well with our results. Our continuum calculation fails to explain the peak around 32 MeV which has been observed ~ ~~ tzC(P~ Yo)t3N reaction at 90° [ref. Zs)] (see fig. 11). This broad resonance can be related to the T = i, 34.6 MeV state found in our discrete calculation (a~ due to the ls.t hole configurations which are not taken into account in our continuum calculation. 3.3.2. Configurational structure of the GDR. In order to study the continuum effects and identify the isospin components of the ' 3 C giant dipole resonace let us look at the discrete calculations shown in fig. 2 and table 4 and at the continuum total T~i.e 4

Dipole sums Calculations discrete (s) discrete (b) discrete (c) continuum

T _~

T=}

Total

ao (MeV ~ mb)

a_, (mb)

ao (MeV ~ mb)

a_, (mb)

ao (MeV ~ mb)

a_, (mb)

96 70 49

5.2 4.9 3.4

171 155 143

6.9 6.5 6.0

267 225 192 250

12.1 11 .4 9.4 11 .5

photoreaction cross section, given in fig. 12. These are all obtained by assuming isospin to be a good quantum number ; we recall that the Coulomb term does not modify the salient features of the calculation. The continuum cross section of fig. 12 is computed with the absorptive potential W equâl to zero. We recall that fig. 2c shows the energies and the dipole strengths of the quasi-bound states of our model space, while fig. 2b shows the dipole discrete spectrum obtained with essentially the same basic space assumption as for the continuum calculation (fig. 12) through a discretisation ofthe energy continuum nearthe single-particle resonanoes . Spectrum (b) is also reported in fig. 12. For the same calculation table 2 gives the 2p-lh configurational mixing coefficients for the states with higher dipole strengths. From calculations (b) and (c) we note that inclusion of the d~ discretized single-

Fig. 12. Total photoreaction cross section for "C (full curve) compared with the integrated absorption cross sections obtained in the discrete calculations (b) . The dashed lines represent the T = ~ components and the full lines represent the T = ~ components . In the calculations the isospin has been assumed to be a good quantum number and the absorptive potential W has been set equal to zero .

particle resonance does not modify the quasi-bound state energies and does not substantially change the T, state strength, whereas it increases the T~ strength in the pygmy region and arotmd 20 MeV. The structure observed in the continuum cross section of fig . 12 has a one-to-one correspondence with the discrete spectrum (b~ Te ar r?

Dipole sums Theory Shell model (disaete talc.)')

ao(~ (MeV ~ mb)

co(T+1) (MeV ~ mb)

ao (MeV ~ mb)

a o(T+1) ao

95 .99

171 .08

267 .07

0 .64

5 .22 5 .66

Sum rules (model dependent values) Sum rules (model independent values)

a_,(T) (mb)

194 = 60 NZlA')

5 0 .61 ~

r, = 2.80 fm, Rm = 2 .38 fm, I~ = 2.34 fm, R, = 2 .41 fm (Weoretical values). r~ = 2 .32 f 0 .02 fm Ref. "). ~ Ref. 3 "). ~ Ref. ") . °) Ref. ") . 7 Ref. ").

263

PHOTOREACfIONS IN "C

We note that as usual the presence of the "continuum" shifts the dipole state to lower energies, the amount of the displacement being state dependent. We deduce that about 75 ~ of the strength arises from the quasi-bound states, as can be seen by comparing the integrated absorption cross section Q° and the bremsstrahlung weighted cross section Q_ 1 [ref. t°)] given in table 4. For below 20 MeV our discrete calculations almost exclusively predict a T~ strength in agreement with the decay to only the t zC T = 0 states ; in the main resonance region near 23 .5 MeV they predict a T, strength, like previous theoretical 2p-lh calculations z7-z9). As a consequence our calculations ascribe the pygmy resonance and the peak near 20 MeV to T = i states, and the giant peak at 23.5 MeV to T = i states . 3.3.3. Sum rules and the GDR isospin splitting. Because ofthe close correspondence between the structure observed between the continuum and the discrete results, respectively, we can limit ourselves to the discrete calculation in order to evaluate the total strengths of the GDR isospin components and their related sums. We discuss case (a), because of the completeness of its basis space. The results are given in table 5. For the notation we refer to ref. 14). In the first row we give the dipole sum values obtained from diagonalisation and in the second row the sum rules'° -sz ) related to the assumed shell-model ground state. In the third row we give some model independent numerical estimates of the sum rules ta " so-3s). As expected from a shell-model evaluation, the integrated absorption cross section Q° is higher than the classical sum rule value as well as the experimental ones given in table 3. The calculated value of the bremsstrahlung weighted cross section Q_ t (first row) is 8 ~ lower than the "model dependent» sum rule value (second row), this diûerence being due to the Woods-Saxon wave functions which, unlike the oscillator wave functions, do not exactly exhaust the dipole strength in our discrete space configurations. The missing amount results from the contributions of the 3hca configurations omitted in the bound calculation and present in the continuum. However for the ratios Q_ t (T+ 1)/Q_,(T) and Q_,(T + 1)/6_ 1 , we find the values expected from sum rule evaluation. s.

ana sum rules a_, (mb)

6.86

12 .08 = 0.40 A`~'

1 .32

0.57

1 .784

7.56

13 .22 = 0.43 A4 ~'

1 .34

0.57

1 .878

1 .22 °) 1 .36 ~

0.55 °)

0.36 A4~'

')

[mf. ")] (experimental value).

a_,(T+1) ä_,(7~

a_,(T+1) a_,

U a_,(7~ -Ta_,(T+1)(mb) (MeV)

a_,(T+1) (mb)

dE (MeV)

56

6.5

60 °) ~ 31 b)

6.9 ') 5 3.6,°)

264

M. MARANGONI et al.

The "effective Lane potential» U defined in refs. aa, ae) ~ Qo(T+ 1)

tro(T) - T + 1

turns out to be 56 MeV from our calculation. This value of U agrees very well with the estimate of ref. ash but disagrees with the 31 MeV upper limit of ref. 3a) . Our calculated ratio tro(T + 1)ltro exceeds the model independent upper limit of ref. aa) by 5 ~. Assuming this upper limit for the vo(T + 1)/QO ratio, along with our calculated values for tro, tr_ 1 and tr°_ 1 = 2[Q_ 1(T)-TfJ_ 1(T + 1)], we obtain for the Lane potential the upper limit of 38 MeV. This shows the strong dependence of the Lane potential on the tro(T + 1 )/QO values. In table 5 we give the rms radii Rm, R~, R, of nuclear matter, and proton and neutron distributions calculated from our shell-model ground state. We also give the isovector r.m.s. radius rp defined in ref. a°) as rn = Rp + Ns/2T, where e = R~ -Rp. The rms radius Rp is in good agreement with the radius of the experimental charge distribution ofref. 11). We note that the lower values of the tr°_ 1 and Ugiven in ref. ") have been obtained with the assumption e = 0, not valid in our shell-model calculation . The computer programme for the IBM 380/168 has been written by Dr. F. Fabbri . We would like to thank Dr. F. Fabbrißor her assistance on the computational work. Appendix A COUPLING MATRIX ELEMENTS

In the second quanti~tion formalism, the Hamiltonian (2.1) is expressed using the basis set of the single-particle wave functions (2.2) as A = ~ Etai ai+~ ~ <11~V I kl)ai ai at~r iJkt i

(A.1)

where V is the residual two-body interaction
J

dr ldr Z~*(r1~j(rz)V(rv rz~t~rl~e(rz)~

(A .2)

We assume here an approximate self-consistency of our single-particle model, i.e. we exclude in the following any self~nergy terms assumed to be already included lII Et. The residual interaction is written as where

V(rv r2) = Vo(ao+a1Q1 ' ez~~rl -rz~~ a0 = a00+a01T1 - T2~

al - a10+allsl ~ t2"

(A.3)

PHOTOREACTIONS IN "C

265

Using a multipole expansion for the coordinate dependence of V(rl, rz), i.e. 9(Iri - rzD = ~9c(ri, rz)Yc~Pi)Yi~(~z), ca

(A .4)

the residual interaction can be put in the form V(rv rz) _ ~9c(rv rz)Uc( 1+ 2~ L

(A.5)

where Uc(1, 2) depends on the angular variables and spin and isospin operators. Let us calculate the matrix elements of eq. (2.21). With the assumption of pure lp-lh target states, with a good isospin quantum number, the IXLU ) functions are written, using the notation of sect. 2, as Iz`~N)

_ ~ < 1~.~yM.I~M)

~
where T and T=~ are the isospin of the target and its z-projection . By the index co we denote the channel in which the 1 zC nucleus lies in the ground state 10). In this case we may write, from eq. (A.6~ It can easily be verified that

I~â~) = aY 10).

(A.7)

For orthogonal 2p-lh states (A.6~ when c, c' ~ co, we find, using the expansion (A.4) ôMl~

_~,
X

~3~`a-TeIT.T=~)(-)~+L°

(
_ ~ CAi(co~ c, L) J ~rluro(r~ ELO~Wrv En)9c(r, r1hL(r, E~h~,(ri. Ea) - Ai(co, c, L) drdrlu~(r, Erobi~(ri, Eb).9c(r, ri~L(rv E~hi~(r, e~ , J l

(A.9)



X ~
~ <.la'.)b'm. .mß.h.M,.)(-~'-~°' ~ ~~iT~'-T6'IT.T:." )

266

M . MARANGONI et al. X

(-)~+t°"Laßp"(CYa~ V~Y~a~)-
_ ~ Ca66"

CCl(C,

C~,

L)

J

drdrlu~(r, Ec/~alrl+ Eadl:(r, r il~d(r, Ec'l~u'(rl, En")

- C2(C, C~, L) drdrlu~(r, Ecl~a~rl, EaA'1L(r, r11~c "(rl+ Ee " l~su" (r, Ea") J -8 " ~Di(c, c',

L) drdrlu~(r, e~)ub.(rl, Eb"J~9c(r, ri~~"(r, E~"~b(rv Eb) J

-~(c, ~, L) drdrlu~(r, e~h~b "(ri, Ee"~9c(r, rih<<" (ri, e~"h~~(r, Eb) , IJ J

(A.10)

where now a and b stand for the quantum numbers nceded to specify the singleparticle states except for sa and T 6. The factors A A Z , C 1 , C 2 , D 1 and DZ are geometrical coefficients depending on the angular moments and isospin of the wave functions and on the spin and isospin parameters of the residual interaction (A.3). From a comparison ofexpression (2.21) and expressions (A.9~ (A.10) it follows that: (i) For the direct term M°a(r) _ ~ Ai(co, c, drlub(rv Eb~9c(r, ri~.(ri, E~, c L)J

(A.11)

with c ~ cm and when c, c' # co Mn"(r) =

8~" ~ ~(c,

c~, L) driu,(ri, EaIYL(r, rll~a~rl+ Ea') J -8" ~ D;(c, c~,

c

L)J ~iub"(rv 8n"~9c(r, riht~rv eb) .

(A.12)

(ü) For the exchange term M~(r, rl) _ ~ Ai(co, ~ L~n(ri, ca~c(r, ri~.(r, E~,

(A .13)

with c ~ co, and v~hen c, c' $ co Mâ (r, ri) = ar,a" ~ ~(~ ~, L~.(rv E~9c(r, ri~. "(r, eo ") L

-

a r ~ ~(~ c~, L~n" (ri, Eb" ~9c(r,Ti~e(r, s~ c

(A.14)

PHOTOREACTIONS IN "C

267

Appendix B PHOTONUCLEAR CROSS SECTIONS

In calculating the cross section for the electric dipole capture y-rays one needs the transition matrix element of the dipole operator Qi describing the transition to the ground state from the state of the A-nucleon system which asymptotically contains a nucleon of spin s, z-projection Q, incident along the ~c direction on a target state with spin IM z-projection M. In our notation this state is given by ~~(k~; sas, nlM)) =

4n

~

e~,`Yk"(~~)
k~ n~~~w

(B.1)

The ~âM are the solutions of the Schrôdinger equation (2.12) for total angular momentum numbers J and M with an incoming wave in channel c. The Q k is the Coulomb phase shift. For the transition matrix element we get Qi(kc; sa~s, nlMp .iMf) _ ~~'`Y'~Qi~~(k~ ; sQt, nlM,~) =

4n

"'( ~ e~`°Y, kJ
k~ n~1~r

(B~)

In the second quantization formalism, the electric dipole operator Qi is represented by Qi = ~
(B.3)

where

with e denoting the electric charge . The ground state is written, using the notation of sect. 2, as ~~rxr)

= a~~0),

with ja~ = J~,

ma~ = Mf.

(B.5)

We use the expression (2.18 the definitions (A.6), (A.'7), (B.5) and the expression (B.2) to obtain ~~'N'~SG1~WS~) _ ee, > ar

~ ~S(EcdCO)\aj~L'1~y0) sro > s~.

(j~ "I, .m,, "M" ~JM) ~
x ~ <~k." -T6" IT~" Ts,. .)(- )}+""(ga," <~`IQila~)-aa,a"<~`IQiIY'))~

(B.6)

We now replace bar, " in the second term of the r.h.s. of the above expression by

268

M. MARANGONI et al .

the scalar product «a,i~q 'i . Finally, using eq. (2.19 we get ~~'M'IQiI~~i = ~-)"i-t(2.If+1)-~~J1M-v~JfMf~ x { E~af, co) drue,(r, E,,)rfÉ(r, co) J

+ E

~" *M

[alAl -

aT~taT=~0 te'ta,lSle "!~6(af, e') drub.(r, En"hua '(r, E6.) J

x drua,(r, Ec,)fE(r, c')- 8,,a'E~(af, C') drub .(r, E6'N fE(r, c )]~ " J J

(B.7)

Here a, b and af stand for the quantum number needed to specify the single-particle states except for the isospin quantum number. The factors Eô(af, co d EB(a , c') and f E~(aß c') are geometrical coefficients . The first term in the r.h.s. of eq. (B.7) is the contribution due to the transition of the odd nucleon from its ground-state level to higher levels, without core excitation, whereas the second and third terms arise from core excitations. The differential capture cross section for y-rays emitted in the direction kr for nucleons incident along k~ is given by 37)

N,p vr'

Qi(k~ ; sa~, nlMp fMf~;'(k~ ;soT, nlMb iMf)Xi(k,)X i~(~~),

(B.8)

where kq = E/i~c, v is the velocity of the incident nucleon, and X;(~Y) is the vector spherical harmonic. The differential photodisintegration cross section for emission of a nucleon of spin shaving the residual nucleus inthe state nl, is related to the above cross section by ~~n~,~

1) 2) 3) 4) 5) 6) 7) 8) 9)

.1{kr

kc)/~ _ (2s+ 1x27 + 1) daG~ ~I. a. j{k r kc)/d11. u2Jf + 1) References

(B.9)

J. Birkholz, Phys. Rev. Cll (1975) 1861 S. Krewald and J. Speth, Phys . Lett. 32B (1974) 295 S. Shlomo and G. Bertach, Nucl . Phys. A243 (1975) 507 M. Marangoni, P. L. Ottaviani and A. M. Saruis, Report CNEN, RT/FI (76) 10 H. Buck and A. D. Hill, Nud. Phys. A9S (1%7) 271 M. Marangoni and A. M. Saruis, Phys . Lett . 24B (1%7) 218 ; Nucl . Phys . A132 (1969) 649 M. Marangoni, P. L. Ottaviani and A. M. Saruis, Phys. Lett . 19B (1974) 253 C. Bloch, Proc. 34th Int. School of Physics Enrioo Ferrai (Academic Pras, New York, 1966) C. Mahaux and H. A. weidenmOller, Shell-model approach to nuclear reactions (North-Holland, Amsterdam, 1969)

PHOTOREACTIONS IN "C 10) 11) 12) 13)

269

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