Application of the coupled channels approach to the intermediate coupling description of photodisintegration of 1d2s-shell nuclei

NUCLEAR PHYSICS A ELSEVIER

Nuclear Physics A 653 (1999) 45-70 www.elsevie~nl/locate/npe

Application of the coupled channels approach to the intermediate coupling description of photodisintegration of ld2s-shell nuclei E.N. Golovach 1, B.S. Ishkhanov, V.N. Orlin Nuclear Physics Institute of the Moscow State University, 119899 Moscow, Russia

Received 20 January 1999; revised 22 March 1999; accepted 22 April 1999

Abstract The coupled channels approach in the shell model intermediate coupling is presented. The problems of orthonormalization of the solutions, fulfillment of boundary conditions, agreement of the effective two-body intercation with the nuclear average field and continuum discretization are considered. It is shown that in light and medium nuclei the description of the structure and decay characteristics of multipole giant resonances can be reduced to the solution of a compact algebraical system of equations. The model is applied to the description of the photonuclear reactions on the 24Mg, 28Si and 32S nuclei. The origin of the gross and intermediate structure of the dipole giant resonance (DGR) as well as the partial decay channels of the DGR are studied. It is found that the 24Mg DGR splitting results from the nuclear shape deformation. (~) 1999 Elsevier Science B.V. All fights reserved. PACS: 24.30.Cz,; 24.10.Eq; 25.20.Lj; 21.60.Cs Keywords: Coupled channels approach; Shell model intermediate coupling; Dipole giant resonance;

Photonuclear reactions; 24Mg; 28Si; 32S

1. Introduction As long ago as 1972 the continuum shell model calculation in intermediate coupling o f the photodisintegration o f 12C [ 1] was performed. It has taken into account the interaction between channels o f the type "nucleon + low-lying state o f the rest nucleus" and has given a good description o f the experimental data for the partial and total cross sections o f photonuclear reactions on 12C. The basis states of the type "nucleon + I Present address: INFN, Sezione di Genova, 1-16146 Genova, Italy. 0375-9474/99/$ - see front matter O 1999 Elsevier Science B.V. All rights reserved. PII S0375-9474(99)00166-9

E.N. Golovach et al./Nuclear Physics A 653 (1999) 45-70

46

low-lying state of the rest nucleus" have also been successfully used in the bound shell model calculations for the description of the structure and decay characteristics of the E1 and M4 resonances for some lp-shell nuclei [2] and the dipole giant resonance (DGR) for 32S [3]. The advantages of the use of a basis of this type are obvious since it allows for the influence of more complex configurations than lplh ones. Moreover, it gives the possibility to describe the decay partial channels of multipole giant resonances formed in the inner reaction region. The only thing preventing us from performing similar calculations for an extensive mass region is the difficulty of the low-lying nuclear states calculations for heavy and medium nuclei. Up to now they have been performed only for lp [4] and ld2s shells [5]. However, recently an effective method using the procedure of successive addition of nucleons was proposed for the low-lying nuclear states calculation [6]. That stimulated us to create a suffieciently simple realization for the coupled channels approach in the intermediate coupling [7]. In particular, the compact system of algebraic equations for the channels of the type "particle + core" and "hole + core" has been derived (we call any separation of a nucleus into two components a channel). In this paper the developed formalism is applied to the description of the photodisintegration for some ld2s-shell nuclei. The first part of the work deals with the description of the model (Sections 2 and 3). In the second part (Sections 4 and 5) the calculations of the photodisintegration for 24Mg, 28Si, 328 are presented.

2. Principles of the formalism Let us consider the problem of describing the nucleon-nucleus scattering states for reactions going through giant resonance excitations in light and medium open-shell nuclei (A < 100).

2.1. Intermediate coupling approximation In light and medium open-shell nuclei the giant resonance is formed rather from the configurations like "particle + (A - 1)-core in the excited state" and "hole -t- (A + l)core in the excited state" than from configurations like "particle -t- hole" because the valence particle and hole excitations produced by the initial one-particle transitions are quite quickly damped out, transferring their energy to a large number of valence nucleons (see Fig. 1). Taking this into account, we approximate nucleon-nucleus scattering states I(aB) ~+)) with the outgoing " ( + ) " or incoming " ( - ) " spherical waves at infinity as follows:

I(~B) ~±)) = ~ ~--~(~"B"I(~B)~-))I~"B")+ ~ ~--~(T"A"I(~t3)~±~)I~"A"), a"

B"

Y"

,,4"

E.N. Golovach et al./Nuclear Physics A 653 (1999) 45-70

47

ffrrF tO

m' h

F////l/tt//]//

l/l/l,

F//t///t/t/.t/l///td/l

...........

g

:::!

>

"/////////////~

r'

tO

Fig. 1. The main one-particle transitions participating in the GDR formation in light and medium nuclei. where ~ x means summation over discrete quantum numbers x and the integration is over continuous ones; a =- eae~ - eal~j~m~, ed, ce't . . . . are quantum numbers characterizing free one-particle states of the average nuclear field u ( r ) (see Fig. 1); y, y', 9/I . . . . are quantum numbers characterizing fully occupied one-particle states of the average nuclear field; [trB) = a+lB), ]a'B'), l a ' B " ) . . . are the basis states describing open (e~ > 0) and closed (e~ < 0) channels of the type "particle + core (A - 1)"; ]yA) = a_~,lA), [yqAt), ly",A') . . . are the basis states describing closed channels of the type "hole + core (A + 1 )" ( a _ r is the hole creation operator corresponding to the annihilation of a nucleon in the state I - Y ) = ( - 1 ) J ' + m r I e r , l ~ , J r , - m r ) ) ; IB), IB'), IB") .... ; I-A), IA'), ].A') .... are low-lying states for (A - 1)- and (A + 1)-nuclei, being the eigenstates of the Hamiltonian H within the configuration space of the valence shell:


(2)

It is easy to see that the intermediate coupling basis used satisfies the orthogonality conditions 1) =

( r A l r ' A') = ,~rr,6.4.~, , (aBIrA) = 0.

(3)

Indeed, the restrictions imposed on the configuration space of states IB), IA) give a , lB ) = a ~ l A ) = a+tB) = a + l . a ) = 0.

(4)

The intermediate coupling approximation (1) is a natural extension of the ordinary l p l h approach for the open-shell nuclei and, as the latter, does not account for correlations in the ground and low-lying nuclear states either (this directly follows from

E.N. Golovach et al./Nuclear Physics A 653 (1999) 45-70

48

Eqs. (4)). The states [B), [A), which are required for the realization of the approximation (1), can be calculated with the help of the method proposed in [6].

2.2. Coupled channels equations A scattering state ](aB)(±)) obeys the stationary Schr6dinger equation H I ( a B ) (+)) = (e,~ +

Ew)l(aB)(±)),

(5)

where H = Ho+Vres is the nuclear Hamiltonian, Ho is the one-particle shell model Hamiltonian with finite central symmetrical potential u(r) and V~es is the residual nucleonnucleon interaction. Substituting the expansion (1) into this equation, then, using the orthogonality conditions (3), we obtain the integral equations system (integration takes place over the continuous one-particle energies e~) for the coupled channels amplitudes (a"B" I(aB)(±)),

(~'"A" I(ate)(+~>:



(8~ + EB)>= ~ ~tt~H

+ ~

,

",/'.,4"

( ~ + eB) = ~

<~/.a'lHIo/'B">>

+ ~ <¢A'IHI¢'A"><¢'~"I(~B)<±~>.

(6)

We have

= (13' [a=,na,,,[B + ") = (B'l{a~,, In, a~,,]}lB + " ) +~a,~,,(B'IHIB")

=8~,~,,8t3,B,,(ea, +Etz,) + (B'l{a~,, [Vr~s,a~,,]}lB+ "), (T'A'IH[r"A") = 8r, r,,8.a,.a,, ( - s t , + E~,) + (A'l{a+_r,, [Vr~s,a_r,,]}lA" ) ,

(T'A'iHla'B') = (A'l{a+__r,, [ V,e,, %+,] }IB') •

(7)

(Here we used the relations (2), (3) and the fact that [a), I-T) are the eigenstates of the one-particle Hamiltonian H0.) Using (7), the coupled channels equations system can be rewritten as follows: (sa + Es - e~, - Es, ) (o/B'l (aB)(+))

= ~

(13'{{aa,, [Vr~,a+,,l}lB")(a"B"J(aB) (±~)

O/P]~ t!

+ ~ "),"..4"

(.A" I{a_r,,,+

[V~,a~,]}II3')(T"A"I(aB)(±>),+

E.N. Golovach et al./Nuclear Physics A 653 (1999) 45-70

49

(e,~ + EB + e r, -- E.A, ) (T'A'I ( a B ) ( + ) ) = ~


[V~es,a,~+,,l}lB")(d'B"l(aB)(+))

oltl ~ l l

(8)

+ ~ <-A'l{a+-e,,[V~e~,a_r,,]}l.A"><~/'.A"l(od3)(±)>. y"..4 "

2.3. Effective channels interaction The matrix elements (B'l{a~,, [ V.res,a,~,,]}lB">, + (A'l{a+r,, [V=s,a_,,,]}[.4") and (.A'[{a+r,, [ Vres, a +, ] } [B') describe the interaction between the different reaction channels. They depend on the residual effective nucleon-nucleon interaction V=s. This interaction must be taken so as not to generate an additional average nuclear field for a particle a and a hole .g-I moving in the field of the (A =F l)-core since the expansion (1) assumes that the one-particle states [a), I-9') correspond to the true average field of the nucleus. Therefore, we shall take the residual interaction Vres in the form Vres -- V -

Uadd,

(9)

where 1

V = ~ ~ ~ ~_~~ (KAIvltzP)asa+a+a~aix h

K

IX

u

is an effective nucleon-nucleon interaction particle matrix element);

((K~lVllxl,)as is

an antisymmetrized two-

Uadd=~-~(~--~(tzAlvllJA)asVa(C))a+a~ tx

A

is the average field generated for a particle (hole) by the forces V in the case the rest nucleus A T 1 is in a state IC) = [B), IB') . . . . (or I.A), I.A'). . . . ) [8]; va(C) =


is the occupation degree for a one-particle level ,~. (The symbol (...)A denotes throughout the vector coupling of rank A for spherical tensors in parentheses; na is the maximum number of particles at level ,~). Then, in view of Eqs. (4), we obtain (B'I {a,~,, [ V~es,a ,+, ] } ]B") = ~

a s

[<13']a~,a#,,lB") -6p,B,,6t~,t~,,v#,(B')],

(A'l{a+_~,, [V~s, a_r,, ] }[A")

= ~ (-~"fl'lv[-,'fl")as [-6/t,/3,,6.4,.A,,v#,(.A')]

,

E.N. Golovach et aL /Nuclear Physics A 653 (1999) 45-70

50

1 [Vres, a,~+]}lB ') = ~ Z ( f l ' - / 3 " Ivl,~, -~, , )as(.A, [a~,a_/3,,lB + + ,, ), ~'9"

(10)

where the quantum numbers fl, i f , fl" . . . . . characterize valence one-particle states of the average nuclear field u(r). One can see from Eqs. (10) that the effective interaction of coupled channels is realized through the rescattering of a particle and a hole on the valence nucleons.

2.4. Boundary conditions consideration Scattering states must obey the boundary conditions I(aB) (±)) --~ lafrl3) + outgoing (incoming) spherical waves

(ll)

at r ~ c~, where lafr) -- [(ej~j~m~)fr) is a free nucleon state. This is equivalent to the requirement that the wave packet constructed from the solutions l(~13)<+)(t)) of the Schrtdinger wave equation must turn into the wave packet of the states I(afrB) (t)) when t ~ :FOxy. (Here the upper sign corresponds to the input reaction channels " + " and the lower sign corresponds to the output reaction channels " - " . ) To allow for the correct boundary conditions we replace the homogeneous integral equations system (8) by the unhomogeneous system

(a'B'l(aB) (±)) = exp( q-iS~)8~,St3B, + x( ~

I

e~ + Et~ - ea, - Et~, + ip

(B'l{a~,, [Vr~s,a~+,,l}lB")(d'B"l(aB) ~±~)

~ll BII

[V~s,a,~+]}IB')(T".A"[(aB)(±))},

+ ~ (..4"l{a+~,,,, y" A "

(T,A,I (orB) (±)) =

1

ea + EB + er, -- E.4, 4- ip x~ ~ K

(A'l{a+_e,, [Vres, %+. ]

} ITS")(,~"t3"l (o,B)~±~)

@II BII

+ ~ (A'l{a+_~,, [V~s,a_~.]}IA"II~"~"l(~B)~±~l I, (12) T"A"

where p ~ + 0 and 6,~ is the phase shift for a nucleon scattering off the potential u(r). It is apparent that the solutions for Eqs. (12) are the solutions for Eqs. (8) at the same time. We shall prove now that such solutions satisfy the proper boundary conditions. It is sufficiently for this purpose to prove that at t ~ :FC~ the wave packet +oo

I(aB)(±)(t)))

=

f cn(e~-e~°))l(aB)~±))exp-i(e'~ --Of)

+ EB)t

(13)

51

E.N. Golovach et al./Nuclear Physics A 653 (1999) 45-70

which is constructed from the solutions of Eqs. (12) with the weight function r//cr

(14)

C~1( 8 ) -- 82 -]- 7 2 ,

transforms into the wave packet +oo

I(afrB) (t))) = f

% ( e , - e(~°))lafrB ) exp -i(e,, h+ Et~)t dea,

(15)

--OO

corresponding to the free nucleonic motion with respect to the (A - 1)-core being in the state IB). According to (12) +oo

(a'B'[(al3)(±)(t))) ~-- /

cn(e,~ - e~°))(ce'B'Ia(±)B)exp -i(e~ h+ EB)t dea

--OO +OO

, , _ ~o~ /- exp-i(e,~ + Et~)t/h + E~B,at3[e,_~, ~ c n ( e a - e ~ a ° ) ) e ~ + E t 3 _ e , , _ E t 3 , + i p

de,

(16)

--00

(r'A'l( aB) ~±) ( t ) ) ) +oo

~-- Gr,A,,aB [~= ~o)

exp -i(e,~ + Et3) t/h c n (e~ - e(,~°) ) e~ + EB + e r, - E.a, + ip de~ ,

( 17)

--OO

where Ice(±)) = exp(+16,,)la ) is the one-nucleon scattering state with the outgoing (incoming) spherical waves in the channel a (e,, > 0); F,~,B,,~B and Gr, a,,~B are functions of the continuum one-particle energy e,~ (in Eqs. (16), (17) the width r/ of the packet is supposed to be small in comparison with the widths of the resonances of these functions). When t ---* qzoo the first term (a'B'[(ce(±)B)(t))) in the right-hand side of Eq. (16) turns into (a'B'l(afrB)(t))) because its behaviour is determined by the behaviour of the wave packet la (±) (t))) formed from the one-particle scattering states Ice(±)) while the second term in (16) and the right-hand side of Eq. (17) go to 0 as exp (-~7[tl/h). Thus, solutions of the sysytem of equations (12) satisfy the proper boundary conditions.

2.5. Orthogonality of the solutions It is also easy to prove that the solutions of the unhomogeneous system of equations (12) are orthonormalized. Actually,

(((~z3) (+)(o)I (d~') (±~ (o))) = (((aB)(±)(m~)I(~'B')(±~ (m~))) = (((~±)B) (~:~)I (d(±)B ') ( T ~ ) ) ) = (((a(±) B) (0) I(a'(±)B ') (0))),

52

E.N. Golovach et al./Nuclear Physics A 653 (1999) 45-70

since, due to the unitarity of the Hamiltonians H and H0, the above scalar products do not depend on time and ](aB)(+)(t))) ~ I ( a ( ± ) B ) ( t ) ) ) for t ~ q:c~ (see previous section). But at ~7 ~ 0 the function c n ( e ~ - e~°)) ~ 8 ( e , ~ - e ~ °)) and the wave packets I(aB)(+)(0))), ](a(+)B)(0))) go to the states [(aB) (:t:)) and la(:~)B) (with e,~ = e(~°) ). Therefore,

(18)

((aB){±)l(a'B')(+))= (a(±)Bla'(±)B')= a~,SBB,. 2.6. C o n t i n u u m descretization

The integral equations (12) determine approximate continuous solutions I(a/3) (+)) obeying the proper boundary conditions for the stationary Schr6dinger equation. To be able to find them, one has to discretize the continuous nucleon energy spectrum. One way to do this is to divide the energy interval [0, emax] (emax /> 50 MeV) into small intervals de which should be smaller than the resonances widths of the scattering matrix. In this case the integral equations (12) turn into an algebraical system of equations. However, the dimension of the resulting system will be very large. Because of this, we shall apply another discretization method [ 1]. This is based on the fact that the nuclear matrix elements (/3'l{a~,, [Vr~s, a~,, + ] }1/3") and (`4']{a+z,,, [Vres, a,~+]}lB') depend only on the behaviour of the continuous oneparticle states Id), la") in the inner reaction region (r < R0 ~. 1.SA -1/3) where these states overlap with the b o u n d nucleonic configurations [B'), IB"), 1,4'). But in the finite spatial region one can expand the continuous states into a discrete one-particle basis. Using the spherical oscillator states [na) - In~l,~j,,m~) ( n is the number of oscillator quanta) as the basis, we obtain Io/> = ~--~
(19)

for r < Ro.

n'

This expansion is true for bound as well as for continuous states [a'). We employ it to approximate the nuclear matrix elements in Eqs. (12). We find

(a'B'l(at~) ~+)) = exp(4-i6,~)8,~,~,St3t~, +

1 ~ + Et3 - ea, - Et3, + ip

x{ ~ ~ y~'Jk',~%'><~'l{ak,.,, [ Vres,a.,,,,,, }+ ] nt/ottt

13tt

i I Y"

.A"

kj

It3">

kt

,

,

+

ak,~, ] } IB')
(+))~,

]

(20)

E.N. Golovach et al./Nuclear Physics A 653 (1999) 45-70

53

O,'.a' I(a13)<+)) 1

~ + EB + ~,, -- E.4, 4- ip

x { n ~ a ~-'~(.mtj{a+T,,[Vres, an,,ot,,]}113 + ,, )(n t,a ,, 13tt I(a13) (±)) I B/I

+ ~ ~A"('A/I{a+-'Y" [Vres' a-r"]}I"A")(T'"A"[ (ce13)(4-)) }'

(21)

where

(n"~"13"l (oz13) (=L)) = (13"la.,,ot,,l(~13) (±)) =

~--~(n"a"[ d')(a"13"1 (a13)~+)).

(22)

8all

Eqs. (20), (21) express the coupled channel amplitudes in terms of the finite number of values { (n"a"13"l (ot13) ~+)), (y"A"l(a13) ~±)) } characterizing the configurational composition of the scattering state in the inner reaction region. It reduces the problem of calculating the scattering states [(aB)(+)) to the calculation of their components in the inner reaction region. For the "inner" components of the scattering state, a sufficiently compact system of algebraic equations can be derived. Actually, we have almost already obtained it (see Eqs. (20), (21)). One only has to multiply Eq. (20) by the scalar product (n'a'la') and to integrate (to sum up) over an energy e~, in order to convert the left-hand side of the equation into the "inner" component (n'od13'l(a13)(:~)). Then we obtain Z n H ottt

Wn'a't3"n"°~"B"(n"°z"13"[(a13)(-F))

]3 t/

Z Wn'a't3',r"A" (T".A"I (a13)(+)) = 8a~,St~B, exp (-4-i8,~)(n'a'la), y" .A"

+ Z

Z rlll o t II

<."w'B"I (o,13)(+)>

B II

+~ ~ wr'~'.r"~"(T"A"I (a13)<~-)) = 0, 7"

..4"

where W.' a' B',n" a" B" =

~n' n" Sa' a" 813'B"

-

ZI13'l{a '.,,

[ V~es,a,,,ot,, l }113")f,,k,~, ( E - Et3, ) , +

k'

_ Z (.A"I {a+~,,,, [ Vres,a k+, ~ , , ] } J B ' ) f , , k , ~ , ( E - E B , ) , k' Wr, A,,n,,ot,,B,, = -(A'[{a+_r,,[V~es, an,,~,, + ]}113")/(E-4-%, - E.a,) , Wn,ot'B'.v',~t . . . .

(23)

E.N. Golovach et al./Nuclear Physics A 653 (1999) 45-70

54

W~,,A,,~,,,A,, =¢3~,,~,,,6A,A,, -- (A'l{a+~,, [ Vres,a-~,, ] }I.A" ) / ( E + e ~ , , - EA,) (24) are elements of the matrix W of the system of equations (23); E = e . + Et3

(25)

is the scattering state energy and

fn'k','(E-- EB,) =~-'~

(n'a'la'>(k'°gla'> y -p)

grit Smax

.

.

. . . . . _ (E - e~, - Eta, 4- tp)

d~,~ + discrete terms.

(26)

o

One can see from Eqs. ( 2 3 ) - ( 2 6 ) that in the approximation made the accounting for the continuum problem is reduced to the calculation of the Cauchy principal values of the singular integrals (26). The matrix W of the system (23) depends on the energy E of the compound nucleus but it does not depend on the channel orb the scattering state l ( a B ) (+)) corresponds to. By a suitable choice of the spherical oscillator parameters, one can limit oneself to only a few terms in the expansion (19), making the dimension of the W-matrix not too large. To find the nuclear matrix elements (B'l{ag,,~,, [V~s, a,,,a,, + ] }IB") . . . . for the known low-lying nuclear states IB'), [B"), IA'), IA") one can apply the ordinary shell model calculations,

2.7. Consideration of the angular momentum and isospin conservation To take into account the angular momentum conservation one has to couple the angular momenta of the channel components (particle and core, or hole and core) to the total angular momentum A = J. Obviously this will not change the above consideration except that quantities like (a"B"l(aB)(+)), (y"A"l(aB)(4-)), (B'l{ak, a,, [ gres, a.,,,~,, + ] }IB"), and so on, are to be substituted by ((a"B")AI(aB)CA+)), ((T"A")AI(aB)~A+)), ((Wl{ak',~')z, [Vres, (a..~,,]}IW')A), + etc. A reaction channel can also be approximately characterized by the total isospin. In this case both angular momenta and isospins of the channel components are coupled to the resulting values A = J, T. One should note that such a description of reaction channels has considerable defects, because the treatment of protons and neutrons as identical particles assumes that they move in the same potential u(r) and have identical separation energies. This obviously fails for heavy nuclei and produces some difficulties in the description of the partial reaction channels for light and medium nuclei (see Section 4).

E.N. Golovach et a l . / N u c l e a r Physics A 653 (1999) 4 5 - 7 0

55

3. Description of the photonuclear reactions The interaction of the nucleus with the electromagnetic field is usually described in terms of perturbation theory. By the absorption of y-quantum with the energy E ~ 830 MeV, one predominates the process of the electric dipole excitation of the nucleus and the isovector giant dipole resonance is formed, which decays in mainly by emitting a nucleon into the continuum. In the first order of perturbation theory the photonucleonic cross sections can be expressed through the amplitudes of the electric dipole transitions between the ground state 10} and the scattering states ](trB)(a -)} (A = J if only angular momentum conservation is assumed, or A = J, T if both angular momentum and isospin are assumed to be conserved). In the second quantization representation the dipole moment operator can be written as

D( F) = ~ Z (K[Idl[A}(a+a-a) r, k

(27)

a

where F = Jr = 1 (or F = (Jr, Tr) = (1, 1)) is the multipolarity of the one-particle transition, and d(i) = tl~(i)riYlu(qbiOi) (tlr is the isospin operator of a nucleon. To simplify the following expressions, we omit in the dipole moment operator the terms (N - Z)/A compensating for the centre of mass movement). In light and medium open-shell nuclei the one-particle operator D(F) generates basically the nucleon transitions from the valence shell to free levels and from fully occupied levels to the valence shell. In accordance with this the dipole transition amplitude from the ground state [O) to the scattering state I((aB)(-))} can be approximated as follows: ((a/3) (A-)](D(F)]O}) A= Z Z ~ (re'[ ]d] ]/3'}u(te'fl'ao; FB') a' '8' B'

x ( ( aB) ~-)l( a+.,ltY) ) a(B'l( a-'8,lO) )B,

- ~ ~ ~ (- 1)ja'-;''(¢~'lldllY')u(#'r'AO; FA') '8'

~;

.4'

×((aB)(-)[(a_r, lA'})a(A'l(a~,[O)).a,.

(28)

Here, as earlier, we used the assumption that the correlations in the ground state can be neglected; u(a'fl'AO; FB'), u(fl'y'AO; F.A') are the coefficients of recomposition of the spherical tensors [9]. In the matrix elements (ce']ld[]ff} the wave functions of the continuous state Ice'} and the bound state I/3') overlap only in the inner reaction region. Taking this into account and using the relations (19) and (22) we can transform Eq. (28) into

( ( aB) (A-)I( D( V)Io) ) A

=

n'

S , S_. or'

,8'

B'

× ((aB)
56

E.N. Golovach et al./Nuclear Physics A 653 (1999) 45-70

- ~ ~ ~--~(-1)Ja'-4' (/3'1[d113")u(/~'y'AO;/'.A') B' Y' A'

x ( ( aB) ~-)l( a-~,,lA') ) A(A'l( a~,lO) ).a,.

(29)

So, we have expressed the dipole transition amplitudes ((Otl3)(a--)[(D(I')lO))a in terms of the "inner" components of the scattering states. The latter can be found by solving the algebraic system of equations (23).

4. Application to the description of the photodisintegration of ld2s-shell nuclei The approach described above has been applied to the description of the structure and decay characteristics of the DGR for some ld2s-shell nuclei. In order to simplify the calculation, we supposed that the nuclear Hamiltonian to be invariant with respect to the charge conjugation and the isospin to be a good quantum number. In reality this assumption is violated by the fact that protons and neutrons move in different potential wells and have different separation energies. Because of this, in shell model calculations for light nuclei taking into account the isospin conservation only the proton (in the description of the (3', P) reaction) or only the neutron (in the description of the (3', n) reaction) characteristics of nuclei are usually used [ 10]. However, one should note that using the same one-particle potentials and separation energies for protons and neutrons does not allow one to correctly describe the relative probability of the emission of photoprotons and photoneutrons in the continuum. This circumstance is especially important when taking into account nuclei with N ~ Z (because of the strong influence of the neutron excess on the nucleonic separation energies). Therefore, we restricted ourselves to the consideration of the three mirror nuclei 24Mg, 28Si and 32S which are sufficiently far from the beginning and the end of the ld2s-shell to avoid the strong correlation effects in the ground state. For the given nuclei, we calculated the partial photoproton cross sections (Or(3',pi)) corresponding to the ground (i = 0), first excited (i = 1), etc., states of the final nucleus and the total El-absorption cross sections gabs ~ 2(o'(3',p0) + tr(3',pl) + o'(3',p2) + . . . ) . We did not consider the photoneutron cross sections since the allowance for the isospin conservation results in the approximate identity of the photoproton and photoneutron cross sections for the mirror nuclei.

4.1. One-particle states For the ld2s-shell nuclei, the free one-particle [cr) states correspond to the orbitals lf7/2,5/2 and 2P3/2M2; the valence states I/3) correspond to the orbitals ld5/2,3/2, 2sl/2 and the fully occupied states [3') correspond to the orbitals lp3/2,1/2. The proton one-particle potential for the free levels had been chosen in the form

E.N. Golovach et aL /Nuclear Physics A 653 (1999) 45-70

u(r) = uofo(r) +

usdfs(r) --l~r r dr

57

+ uc(r),

(30)

where u0 and Us are the amplitudes for the nuclear and spin-orbit interactions, uc is the Coulomb potential, and fo(r) and f~(r) are the Woods-Saxon form factors. The parameters of this potential have been taken from the global optical model [ 1 1 ] except for the u0 amplitude that, generally speaking, depends on the orbital momentum l. The upper limit of the amplitude u0 for the l f-states can be estimated from the data extracted from the one-nucleon transfer reactions [12] (based on the energy of the lf7/2 level). It gives uof <~ 64, 60,58 MeV for 24Mg, 28Si and 32S, respectively. We varied slightly the uof and uop amplitudes around the estimated values to get the best fit to the experimental data. The final values for uof, Uop are given in Table 1. As the one-particle states for the valence (/3) and fully occupied (y) levels, the spherical oscillator states with the parameter u = (h/moo) 1/2 = 1.005A I/6 fm (hco = 41A 1/3 MeV) have been used. The same wave functions were used in the expansion of the free shell states (]cr)) in the inner reaction region (see Eq. (19)). The energies of the Jy)-states, which are directly included in the coupled channels equations (23), (24), have been regarded as the parameters of the model. These values can be roughly estimated from the data extracted in the nucleon quasi-elastic knock-out reactions. So, the analysis of such data from Ref. [ 13] yields the following energies for 2 4 M g : elp3/2 ----22 ± 2 MeV and elp~/2 = 18 -4-2 MeV (we assumed that the maxima of the experimental strength function at the excitation energies ~ 14.4, 9.9 and 2.5 MeV of 23Na correspond to three Nilson's orbits into which the lp3/2 and lpl/2 states are split in a deformed nucleus [ 14]). The energies elp3/2 and elp,/2 were varied to get the best fit of the DGR structure. Their final values are given in Table 1. Strictly speaking, the basis described above is not orthonormalized since the oscillator states [113), [11½) approximating the bound s t a t e s [lp3/2,1/2) are not orthogonal to the continuous states 12p3/z,l/2). Therefore, action had been taken to restore the orthogonality of the basis. Namely, the ]113) and I11½) states were not included in the set of oscillator states over which the 12p3/z3/2) states were expanded in the inner reaction region. For the chosen oscillator parameter, the results converge fast in the number (N) of oscillator functions in the expansion (19). This is illustrated by Fig. 2. The figure shows the dependence of the model's photoabsorption cross section on N. The model's cross section was calculated for a truncated set of ]/3)-, IA)-states: (10,10). As can be seen from Fig. 2, a reasonable result is already achieved for N = l, and for N ~> 3 Table 1 The adjusted parameters Nucleus

24 M g 28Si 32 S

IPl/2

UOf

UOp

~"IP3/2

8

MeV

MeV

MeV

MeV

MeV

go

62 58 57

62 58 62

-- 20 -23 - 24

-- 17 -19 - 20

50 50 45

58

E.N. Golovach et al./Nuclear Physics A 653 (1999) 45-70 10

,'

8

'~

>,

;

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

U=l N=2

-

N=3

-

6

4

2

0

18

I 19

I 20

I 21

I 22

I 23

I 24

25

E (MeV)

Fig. 2. Dependence of the model's photoabsorption cross section on the number of oscillator functions used in the continuum discretization. the calculated cross sections are practically identical. We used N = 3 in the present calculations.

4.2. Low-lying states f o r A 7= 1-nuclei

The normal parity low-lying states IB> and 1.4) for the A q: 1-nuclei were calculated by means of the nucleons successive addition method [6] applied to the ld2s-shell nuclei. In these calculations, the one-particle energies and the matrix elements of the effective two-particle interaction, which are dependent on the mass number A, were taken as in the Ref. [5 ]. In the description of the photodisintegration of the 24Mg, 28Si and 32S nuclei we included all low-lying I/3> and IA> states with the excitation energies ~< 11 MeV. It allowed us to calculate the structure and decay characteristics of the DGR for the energies up to 30-32 MeV. (The maximum energy is found as the sum "11 MeV + nucleon separation energy + energy of the highest one-particle resonance in the continuum".)

4.3. Effective nucleon-nucleon interaction

The coupled channels equations (12), (23) contain neither interaction matrix elements like (see also Eqs. (10)) nor the one-particle energies of the valence levels. Therefore, it is possible to use different nuclear Hamiltonians on the one hand for the description of the low-lying states ]B), I.A), and, on the other, for the description of the scattering states I ( a B ) ( + ) ) . In this work we used Rosenfeld forces. The radial part of the forces was taken in the form of the Yukawa potential with the radius 1//.t = 1.5 fm and their amplitude Vo

E.N. Golovach et al./Nuclear Physics A 653 (1999) 45-70

59

was considered as the model parameter. It was varied to get the correct DGR energy position. The resulting values of V0 are presented in Table 1.

5. Discussion of the results The results of the calculation are presented in Figs. 3-9 and in Tables 2-6. 5.1. Intermediate structure of the DGR

Fig. 3 shows the comparison of the calculated (thin line) and the experimental (thick line) cross sections for E1 photoabsorption. Both experimental and theoretical cross sections show a considerable intermediate structure of the DGR. The creation of this structure can be related with the "embedding" of the closed channels "y.A in the continuum of the nuclear system. Indeed, as has been demonstrated in a number of works (see, e.g., Ref. [ 15] ), such discrete states must give rise to scattering resonances. 80 60 40 20

kL

0

I

[

I

I

I

I

I

I

I

I

100 'G" E io 50

0 150

-

325

~

100

50

14

16

18

20

22 E (MeV)

24

26

28

50

Fig. 3. Photoabsorption cross sections. Calculation: thin line. Experiment: for X4Mg, the photoabsorption cross section on the natural mixture of isotopes [22] (thick line) and the sum of the ( ' y , p ) [26] and ('y, n) [27] cross sections (dashed line); for 28Si, the photoabsorption cross section [23] (thick line); for 32S, the sum o f the ( ' y , p ) I24] and ('y,n) [25] cross sections (thick line).

60

E.N. Golovach et al./Nuclear Physics A 653 (1999) 45-70

The role of the closed y.A-channels corresponding to the dipole transitions lp ~ ld2s in the formation of the intermediate structure of the DGR is illustrated by Fig. 4. Here the calculation results for the DGR of 24Mg are presented in the following cases: (1) no effective interaction (dotted line); (2) the interaction is taken into account only for the configurations Ire/3) corresponding to the ld2s ~ l f 2 p transitions (dashed line); (3) interaction is taken into account between all channels (solid line). The figure demonstrates bow the residual nucleon-nucleon interaction transforms the one-particle continuum resonances laB) (dotted line) into the collective dipole state (dashed line) Table 2 Main DGR components for 24Mg at some resonance energies. Configurations are denoted as a(E) or T- l (E), where E are theoretical excitation energies for the states of (A - 1) or (A + 1) nuclei respectively E = 17.1 MeV

E = 18.6 MeV

E = 19.8 MeV

E = 24.2 MeV

E = 25.5 MeV

Config.

Config.

Config.

Config.

Config.

Frac.

% 1p~(8.80) 17.1 1p~(7.88) 11.4 1p~(7.77) 7.04 1f7/2(0.45) 5.46 1P~21(2.91) 3.60 lp~(6.01) 3.48 1p~/~(4.56) 2.95 lp~(7.62) 2.78 1p~(9.92) 2.65 1f7/2(5.51) 2.40

Frac.

% 1f7/2(0.45) lp~/~(4.56) lp~/~(6.01) 2pl/2(4.54) 2p3/2(5.51) 2p3/2(4.54) 1f7/2(5.51) lp~/~(9.92) lP~21(2.42) 1p~(2.76)

7.00 4.33 3.83 3.80 3.42 3.13 2.85 2.81 2.64 2.33

Frac.

% lp~(6.01) 2p3/2(5.51) 1p~/1(6.76) lp~/~(5.63) 2p3/2(6.09) 1f7/2(0.45) 1f7/2(5.51) -1 lP3/2(0.00) 2pl/2(2.76) 2pl/2(4.54)

5.90 4.09 4.01 3.59 3.45 3.04 2.93 2.57 2.46 2.38

Frac.

Frac.

% 1p~(7.62) lP~/21(6.76) lPu~(9.09) 1f5/2(0.45) 1p~(6.28) 1f7/2(5.80 ) 1f7/2(7.54) lp~/~(7.62) 1f7/2(8.80) lp~/~(9.20)

4.33 2.57 2.02 1.90 1.85 1.82 1.79 1.77 1.50 1.36

% lp~(10.2) 3.65 lp~/~(10.5) 2.69 lf7/2(8.80 ) 2.54 2p3/2(8.36 ) 1.92 lf7/2(8.37) 1.88 lp~(9.39) 1.84 lf7/2(8.88) 1.80 1f7/2(7.83 ) 1.79 1f7/2(8.64) 1.64 1p~(6.76) 1.63

Table 3 Main GDR components for 28Si at some resonance energies. Configurations are denoted as in Table 2 E= 18.1 MeV

E= 19.0 MeV

E= 19.7 MeV

E = 20.6 MeV

E= 21.2 MeV

Config.

Config.

Config.

Config.

Config.

Frac.

% 2p3/2(7.89) lp~(2.35) 2p3/2(4.00) lf7/2(2.72) 1f7/2(4.78) 2p3/2(4.11) 1f7/2(2.36) 2pl/2(3.65) 2p3/2(2.72) 2pl/2(1.23)

5.46 4.90 4.65 4.49 4.16 3.61 3.58 3.34 3.19 3.18

Frac.

% 1p~/~(9.52) 2p3/2(5.68) 1f7/2(0.00) 1f7/2(2.36) 2p3/2(5.49) -1 lpl/2(1.49) 2pu2(2.70) 1f7/2(6.74) 1p~(5.18) 2p3/2(2.70)

12.0 5.68 5.60 3.74 3.18 2.82 2.18 2.08 1.99 1.94

Frac.

% lf7/2(0.00) 2p3/2(6.22) lf7/2(2.72)

lp~/~(9.52)

lp~/~(lO.3) 1f7/2(6.45) lf7/2(3.15) 1p~(1.49) lp~(lO.3) 2p3/2(2.36)

6.95 3.94 3.81 3.43 3.39 2.99 2.74 2.69 2.67 2.05

Frac.

% lp~/~(ll.7) lp~(5.18) 2p3/2(6.45)

11.1 1.23 7.08 11.7 2p3/2(0.81) 7.04 lp~/~(8.40) 6.81 2p3/2(7.08 ) 8.70 2p3/2(5.68) 6.86 1f7/2(0.00) 8.40 lf7/2(6.45 ) 5.51

1p~(10.3)

Frac.

% lp~/~(ll.l) lf5/2(1.23) 2p3/2(7.08) lp~(ll.7) 2p3/2(7.04 ) 2p3/2(6.81) 2p3/2(8.70) 1p~/~(6.86) lp~(8.40) lf7/2(5.51)

3.58 1.89 1.83 1.83 1.83 1.77 1.75 1.59 1.55 1.46

E.N. Golovach et al./Nuclear Physics A 653 (1999) 45-70

61

Table 4 Main GDR components for 32S at some resonance energies. Configurations are denoted as in Table 2 E = 17.1 MeV

E = 19.1 MeV

E= 19.6 MeV

E= 21.4 MeV

E= 24.3 MeV

Config.

Frac. %

Config.

Config.

Frac. %

Config.

Frac. %

Config.

1fu2(2.49) 1f7/2(4.43) 1f7/2(5.82) 1f7/2(5.13) 2pl/2(5.96) lp~/~(5.10) 2p3/2(6.99) 2p3/2(7.00) 1f7/2(8.53) 2p3/2(6.38)

7.58 4.61 4.03 3.45 3.18 2.99 2.88 2.80 2.74 2.62

1f7/2(6.73) 2/73/2(7.55) 1f7/2(7.28) lp~(7.61) lf7/2(6.51) 2p3/2(8.41) lp~(o.oo) 2/73/2(7.61) 1fu2(2.49) 1f7/2(6.99)

5.87 4.66 3.80 3.72 2.86 2.68 2.49 2.01 2.01 1.99

lf7/2(7.55) 1f7/2(6.19) 1f7/2(8.10) 1f7/2(6.84) lp~/~(3.73) 1f7/2(5.28 ) 1f7/2(2.49 ) 1f5/2(4.71 ) 1f7/2(6.99) 1f7/2(6.50)

9.33 3.96 3.14 2.74 2.67 2.37 2.29 2.15 1.98 1.84

Frac. %

4.87 1f7/2(7.28) 4.49 1f7/2(6.50) 4.05 1f7/2(6.99) 3.74 1p~(2.17) 3.09 1f5/2(5.13) 3.04 2pu2(7.89 ) 2.75 1f7/2(6.51) 2.70 1./7/2(6.22) 2.09 2pl/2(7.00) 2.09 1f5/2(5.82 )

Frac. %

1f7/2(5.28) 4.46 1p~/~(5.82) 3.31 1p~/~(6.81) 3.25 1f5/2(7.35) 2.75 1p~/~(4.58) 2.51 1f7/2(7.50 ) 2.11 1f7/2(5.60 ) 2.07 lp~(4.67) 2.05 1p~(6.41) 1.92 1f7/2(7.78 ) 1.86

Table 5 The change in the configurational composition of the GDR for 32S in the neighbourhood of the 17.1 MeV resonance. Configurations ale denoted as in Table 2 E= 16.8 MeV

E= 16.9 MeV

E= 17.0 MeV

E= 17.1MeV

E= 17.2 MeV

Config.

Config.

Frac. %

Config.

Frac. %

Config.

Frac. %

Config.

Frac. %

7.26 4.41 4.20 3.63 3.22 3.18 2.70 2.59 2.59 2.55

1f7/2(2.49) 1f7/2(4.43 ) 2p3/2(6.19) 2p3/2(7.00) lf7/2(5.82) lp~(5.10) 2pu2(5.96 ) 2p3/2(6.38 ) 2pl/2(4.93) 1f7/2(5.13 )

6.61 4.34 3.96 3.78 3.70 3.35 3.3l 2.99 2.32 2.20

lf7/z(2.49) lf7/2(4.43 ) 1f7/2(5.82) lf7/2(5.13) 2pu2(5.96) lp~(5.10) 2p3/2(6.99) 2p3/2(7.00) 1f7/2(8.53) 2p3/2(6.38 )

7.58 4.61 4.03 3.45 3.18 2.99 2.88 2.80 2.74 2.62

2p3/2(6.99) lf7/2(2.49) lf7/2(4.43 ) 1f7/2(5.60) !f7/2(5.82 ) lf7/2(8.53) 1f7/2(5.13 ) 2p3/2(7.35) 2pl/2(5.65 ) 2pv2(6.84).

5.34 4.86 4.12 3.85 3.66 3.14 2.95 2.83 2.73 2.66

Frac. %

2p3/2(6.19) 7.09 2p3/2(6.19) 2p3/2(6.38) 6.57 2p3/2(6.38) 2pl/2(5.22) 5.14 1f7/2(2.49 ) 2p3/2(8.53) 4.60 1f7/2(4.43) 2p3/2(5.28) 3.17 2p3/2(7.00) 2p3/2(7.35) 3.09 2pu2(5.22) 2p3/2(7.00) 2.92 1f5/2(1.22 ) 1f7/2(5.13 ) 2.74 2p3/2(8.53) 2pi/2(4.93 ) 2.73 1f7/2(5.13 ) 2p3/2(6.84 ) 2.64 lpu2-j(5.10)

Table 6 The energy integrated cross sections O'int = f;0 trabsdE for the 24Mg, 28Si, 32S nuclei. The experimental data correspond to the photoabsorption cross sections presented in Fig. 3 by the bold lines Nucleus

Experiment (mbx MeV)

Exp. / 60-~

Theory (mbx MeV)

Theory / 60-~

24Mg 28Si 32S

322 379 611

0.89 0.90 1.27

497 567 682

1.38 1.35 1.42

62

E.N. Golovach et a l . / N u c l e a r Physics A 653 (1999) 4 5 - 7 0

80

60

ta 40

20

J,"l

14

16

.......- ~.....'..'..~.:.::~.:.~.:.:..:......:._.__.?.?.....................7. 18

I 20

I 22

I 24

I 26

I 28

30

E (M~V)

Fig. 4. Influence of the channels interaction on the 24Mg photoabsorption cross section. Dotted line: no channels interaction. Dashed line: only channels corresponding to the "core ~ valence shell" transitions interact. Solid line: interaction for all channels is taken into account. and enhances the strength and structure of the 24Mg DGR due to the coupling of the closed channels 3'-4 with the open channels trl3 (solid line). The oscillatory strength of the closed dipole transitions lp --~ ld2s, comprising about one half of the total dipole sum, can manifest itself in the photodisintegration cross section only due to the interaction with the open input channels, with initiation of a complex intermediate structure, as is seen in Fig. 4. The important role of the 3"-4 channels for the DGR intermediate structure formation is also clearly seen from the configurational composition of the resonance states. As noted above, having solved the system of coupled channels equations (23) one has the scalar products (n'tr'B'[(a13) ~- ~), (3"-4'1 (c~B) {-~) which determine the configurational composition of the scattering states in the inner reaction region. Using these values one can estimate the contribution of the configurations laB), lyA) to the DGR at any excitation energy E. Tables 2-4 present 10 major components of the considered DGRs at some resonance energies E. These data indicate that the intermediate structure of the DGR in the ld2s-shell nuclei has essentially a collective nature and is due to the strong mixture of the laB) and 13/.4) configurations. And from the latter, the configurations corresponding to the lpl/~ hole have the most significance. The strong mixture of the laB) and 13'.4) configurations is also illustrated by Fig. 5. It shows how the components of the considered DGRs corresponding to the laB) (dashed line) and [3'.4) (dotted line) configurations are changed with the excitation energy E. The absolute value of the 13"-4) configurations contribution decreases gradually, as one passes from 24Mg to 28Si and then to 32S, since the filling of the valence ld2s-shell blocks the lp ~ ld2s transitions. Nevertheless these configurations continue to play a considerable role in the formation of the intermediate DGR structure. It is confirmed by the data of Table 5, where the configurational composition of the DGR on 32S is given in the vicinity of the 17.1 MeV resonance. As is clearly seen from the table, the resonance

E.N. Golovach et al./Nuclear Physics A 653 (1999) 45-70

80 -

i

60 ' 40

20

ol

63

:j~

~k

i a

"f

c

....... "":.:'...... : . ' ' . . . . . . . .

I

I

I

.-'.~.:.'.:.~° -~- ~. z:'--.

I

I

28Si

100 E b

50

0 150

100

50

14

16

18

20

22 E (UeV)

24

26

28

30

Fig. 5. Configurational composition of the GDR. Solid line: the calculated photoabsorption cross section. Dotted line: the fraction corresponding to the "valence shell --* free levels" transitions. Dashed line: the fraction corresponding to the "filled levels ~ valence shell" transitions.

correlates clearly with one of the closed channels %A. When the excitation energy E comes up to the resonance energy the contribution of the a_lp,nJ.A; E.4 = 5.1 MeV) configuration to the photoabsorption cross section rises sharply.

5.2. Gross structure of the DGR The gross structure of the photoabsorption cross section on 24Mg is clearly observed. One can see two wide peaks which are located in the energy regions 16-22 and 2228 MeV. And as a consequence, the cross section has a large width of the DGR. Such a cross-section shape is usually explained by the deformation of the nuclear surface. Indeed, there are a number of experimental facts which suggest that the 24Mg is an axially symmetric prolate nucleus. The influence of the deformation on the properties of the DGR for ld2s-shell nuclei was considered in the work [ 16]. In this paper the authors, at first, calculated oneparticle states in the axially symmetrical Hartree-Fock's average field and, then, used

64

E.N. Golovach et al./Nuclear Physics A 653 (1999) 45-70

them to calculate the DGR in the frame of lplh approximation. In Ref. [ 16] it was found that dipole states in 24Mg and 28Si are divided into two energy groups: the upper one which is mainly caused by the lp ~ ld2s transitions, and the lower one which is mainly caused by the l d 2 s --* l p 2 f transitions. In addition, it was found that the upper energy group corresponds to the one-particle transitions which change the nucleon angular momentum projection on the symmetry axis by 1 (zlK = 1) while the lower energy group corresponds to the transitions with AK = 0. Therefore, it was concluded that both the configurational splitting [ 17] and the deformation splitting of the 24Mg DGR take place. Shortcomings of the cited work [ 16] are that (1) a phenomenological two-body interaction was used in the deformed average field calculation; (2) only lplh configurations were considered in the DGR description, and (3) the continuum was not taken into account. The present calculation also demonstrates that the configurations due to the lp --* ld2s transitions play an important role in the formation of the higher energy maximum in the photoabsorption cross section on 24Mg. However, as can be seen from Fig. 5, the considerable increase of the contribution of such configurations becomes apparent mainly at higher energy (E > 28 MeV) than the second maximum of DGR, where the role of the lp3/2 --* ld2s transitions become more important. This is evidence for the fact that the observed gross structure of the 24Mg DGR has a collective nature. Let us consider the possibility of DGR splitting owing to the nuclear shape deformation, which was noted above in connection with the strong collectivization of the GDR in the nuclei under consideration. In compliance with the Danos-Okamoto model [ 18], two types of collective dipole vibrations of neutrons and protons relative to each other can occur in the prolate axially symmetrical deformed nucleus: longitudinal vibrations with lower energy and transverse ones with higher energy and double intensity (owing to the degeneration of energies of the two transverse modes). These vibration modes produce the splitting of the DGR. In a unified nuclear model [ 19] longitudinal and transverse vibrations for a spheroidal even-even nucleus are described by means of the wave functions ]g'llM) and [g'~M), which can be written in the laboratory frame as IgrllM) = ~/~q+M[qt0) -- V/~(q+lgr2))IM, [a/~M) ~- vf~ql+Mla/~0)+ W/~(q+lat¢2)) 1M,

(31)

where q+M is the collective dipole vibration creation operator, Ig'0) = IO) is the ground state of the nucleus and ]g'2) is the first excited state of the nucleus with J'~ = 2 +. In the internal coordinate system the expressions (31 ) are transformed into the usual relations [q~IIM)= ~ 8 ~ 3279~0(0,) q~01g'0), [lP'~M) = V/1"~2 [~)ll ( Oi)q+l laP'O)+ 791_1 ( Oi)q + , [qSO)],

E.N. Golovachet al./Nuclear PhysicsA 653 (1999)45-70

65

because in the strong coupling approximation the relation

}~-t2M) = ~ 8 ~ D20( Oi) l~ltO) is fulfilled for the deformed even-even nuclei. Let us substitute the q+M operator by the dipole moment operator in Eqs. (31 ), taking into account that if the giant resonance is totally collectivized the operator D~M [q+M + (--1)l+Mql-M] and ql[~0) = q l [ ~ b ' 2 ) = 0 . Then we obtain

I((~t~)~-)l¢-~)l = I((aB)~A-)I¢'~>I 2

I((~B)~A-)I(D(r)I~0))A-- V2((~B)~-)I(D(r)I~,2))AI 2 IVff((aI3)~A-)I(D(F)Iq'O))A + ((aB)~A-)I(D(F)I~/'2>)AI 2" (32)

This is the ratio of the probabilities of occurrence of longitudinal and transverse dipole vibrations in a certain scattering state ](a/3)~A-)) at any energy E. So, it gives the possibility to divide the DGR into the "longitudinal" and "transverse" components. The required matrix elements ((al3)(a-)l(D(I')[qs2))A can be calculated with the help of Eq. (29), where one has to substitute the fractional parentage coefficients (]3'l(a_B,[O))A, (A'](a_/~,lO))a by (Bq(a_#,lqz2))A, (¢4'[(a-/3'l~2))A, respectively. The thus calculated "longitudinal" and "transverse" photoabsorption cross-section components are plotted in Fig. 6. One can see in the figure that the longitudinal mode of the 24Mg DGR is mainly located at the lower photoabsorption peak position while the transverse one dominates at the higher peak position. It is direct evidence of the relationship between the DGR gross structure and the deformation of 24Mg. In contrast, the "longitudinal" and "transverse" peaks of the DGR for 28Si and 32S are not separated, which indicates that these nuclei should be approximately spherical.

5.3. Integrated cross sections Fig. 3 shows that on the whole the energy location, width and structure of the dipole giant resonance on the 24Mg, 28Si and 32S nuclei are adequately described by the calculated photoabsorption sections. However, it should be pointed out that the sum of the theoretical oscillatory strengths for the E1 transitions concentrated within the energy range from the nucleon escape threshold (B) to the energy of nucleus excitation ,.~ 30 MeV in all considered instances is considerably greater than the corresponding experimental values. This is illustrated in Table 6, where the theoretical and experimental integrated cross sections O'int ---30 f~ O'absdE for the 24Mg, 28Si and 32S nuclei are presented and a comparison of them with the classical sum rule is given. It is seen from Table 6 that the experimental data are approximately within the classical sum rule in the region of formation of the dipole giant resonance (at least, it is true for 24Mg and 28Si for which experiments on direct photoabsorption were carried out), whereas the theoretical data are about 1.4 times greater than this value.

66

E.N. Golovach et al./Nuclear Physics A 653 (1999) 45-70

80-

~

60

24Mg

40

20

0

..................... ;....-,,,,..:.,...,?:::,,

~',~

_D.

1

100

2SSi

vE b 50

0 150

100

50

0

~ i : , , , ; 14

16

f .. 18

I ....... 20

"

~

22 E (MeV)

...... 1~ 24

26

t"'".'.-"-'-z.'.~ 28

30

Fig. 6. Contribution of the longitudinal and transverse vibrations into the GDR. Solid line: the calculated photoabsorption cross section. Dotted line: the longitudinal component fraction. Dashed line: the transverse component fraction.

The observed discrepancy between the predicted and experimental values may be due to the following two reasons: (1) the basis {I(a'B')A), [(~'~A')A)}, EI~,,EA, <<. 11 MeV used for the calculations is not adequately complete, and (2) the correlations in the ground state [O) are neglected. Indeed, the "doorway" [(atB~)a), I(~/A')a) states genealogically connected with the ground state of a nucleus are largely grouped in the energy region ~< 30 MeV concentrating here practically the entire dipole strength, which exceeds the classical sum rule owing to the action of exchange forces. The basis expansion, i.e. incorporation in the basis of the I(dB')a) and J(y~.A~)A) states, which are not connected genealogically with the ground state and have higher energies may lead to the transfer of a part of the excess dipole strength to the energy region > 30 MeV. On the other hand, it is well known that the standard continuum lplh calculations neglecting the correlations in the ground state of a nucleus give highly overestimated values of the dipole sum rule (see, for example, Ref. [20] ). This seems to be partly

67

E.N. Golovach et al./Nuclear Physics A 653 (1999) 45-70

1~

24Mg

°I

~,P~)

1.51

0.5 0

2.6 0.5 0 0.5 0

5.8

0.5 0 0.5 0

16

18

20

22 E (MeV)

24-

26

28

30

Fig. 7. The differential partial cross sections for the photoproton reaction on 24Mg (emission angle: 90° ). Thin line: theory. Thick line: experiment. The experimental (y, P0) cross section has been derived from (p, "Y0) cross section [28]. The other experimental data are from [26] (numbers indicate the mean energies of the final nucleus states groups). true for our calculations, where the effect of correlations in the ground state o f nuclei on the sum rule was also neglected.

5.4. Partial channels o f the D G R decay M o r e detailed information about photodisintegration properties o f nuclei is contained in partial photoproton cross sections corresponding to various states (or groups of states) o f a final nucleus. The calculated partial cross sections are shown in Figs. 7 - 9 , where they are compared with the experimental data. Sometimes the partial cross sections are used for the evaluation o f the contribution o f different shell model configurations to the giant resonance. The presented calculation shows that this is not a correct procedure. As can be seen from Table 2, not one in ten major configurations o f the D G R 24Mg contributing to the 17.1 M e V resonance corresponds to the ground state o f the final nucleus, whereas the partial cross section

E.N. Golovach et al./Nuclear Physics A 653 (1999) 45-70

68

30

28Si

20

10

0 10

0 5 0 10 5 0 lO 5 0

16

18

20

22

24

26

e (MeV)

Fig. 8. Photoproton partial cross sections for 28Si. Experimental data of three upper plots are from Ref. [29], other ones are taken from Ref. [30]. Designations are the same as in Fig. 7.

tr(y, p0) is about one half of the total photoproton cross section o - ( y , p ) ,.~ trabs/2 at the excitation energy E ~ 17.1 MeV (compare Figs. 3 and 9). The situation is similar at other energies. It demonstrates that the configurational composition of scattering states is essentially distinguished in the inner and outer reaction regions if strong collectivization takes place. Partial cross sections are much more sensitive to the details of the calculation than the total cross section. They enable one to test the model in a very rigorous way. As can be seen from Figs. 7-9, the coupled channels equations in the intermediate coupling describe in general the partial channels well enough for 24Mg, 2Ssi and 32S photodisintegration. So, we can conclude that the presented approach takes into account the main factors governing the formation and decay of the giant resonance in light and medium open-shell nuclei.

E.N. Golovach et al./Nuclear Physics A 653 (1999) 45-70

0.5

69

32S

i

(7,Po)

0 I

0.5 0 I

0.5 0 I

~0.5 t~ 13

0

0.5!0 2 1.5 1

0.5 0

16

18

20

22 24 E (MeV)

26

28

30

Fig. 9. The differential partial cross sections for the photoprotonreaction on 32S (emission angle = 90° ). Experiment: Ref. [24]. Designations are the same as in Fig. 7. 6. Conclusion We have illustrated the possibilities of the method of coupled channels in the intermediate coupling by the example of the description of photodisintegration of three ld2s-shell nuclei. The obtained results show that the model is able to describe more or less adequately the DGR formation, its energy position, width, structure and decay characteristics for the light and medium open-shell nuclei. However, there are also certain discrepancies between theory and experiment which are due to the approximations made. So, the theoretical integrated cross sections are considerably greater than the corresponding experimental values into the energy region E ~< 30 MeV. As can be seen from Figs. 3, 7-9, the calculation describes the experimental cross sections for higher excitation energies less good. This can be related to the fact that a limited number of basis configurations {I/3), ,4)} have been included in

70

E.N. Golovach et al./Nuclear Physics A 653 (1999) 45-70

the calculation. O n l y those with an excitation energy E ~< 11 M e V w e r e used. But, for the m o m e n t , it is hardly possible to calculate h i g h e r energy states, since in the energy r e g i o n E > 10-11 M e V one has to take into account the effects o f different main-shell mixture. Possibly, n e g l e c t i n g the correlation in the ground state also impairs the results ( s e e discussion in Section 5.3 and Ref. [21 ] ). In particular, this c o n c e r n s the d e f o r m e d 24Mg nucleus, w h e r e the different parity orbitals are brought closer together. H o w e v e r , to take into a c c o u n t the correlation in the ground state in the intermediate c o u p l i n g a p p r o x i m a t i o n , it is also necessary to increase the used basis.

Acknowledgements W e w o u l d like to thank Dr. V.V. Varlamov, H e a d o f the C e n t e r o f Data o f P h o t o n u c l e a r E x p e r i m e n t s o f the N P I o f M o s c o w State University, for placing in our disposal the e x p e r i m e n t a l i n f o r m a t i o n w e needed.

References [1] J. Birkhoiz, Nucl. Phys. A 189 (1972) 385. [2] N.G. Goncharova, H.R. Kissener and R.A. Eramzhyan, ECHAYA 16 (1985) 773; A.N. Goltzov, N.G. Goncharova and H.R. Kissener, Nucl. Phys. A 462 (1987) 376. [3] L. Majling, J. Risek and Yu.I. Bely et al., Nucl. Phys. A 143 (1970) 429. [4] D. Kurath, Phys. Rev. 101 (1956) 216. [5] B.H. Wildental, Prog. Part. Nucl. Phys. 11 (1984) 5. [6] E.N. Golovach and V.N. Odin, Phys. At. Nucl. 59 (1996) 1879. [7] E.N. Golovach and V.N. Odin, Phys. At. Nucl. 62 (1999) 247. [8] D.J. Rowe and C. Ngo-Trong, Rev. Mod. Phys. 47 (1975) 471. [91 H.A, Jahn, Proc. Roy. Soc. A 205 (1951) 192. [ 10] J. Raynol, M.A. Melkanoff and T. Sawada, Nucl. Phys. A 101 (1967) 369. [11] J. Rapaport, Phys. Rep. 87 (1982) 25. [12] P.M. Endt and C. Van der Leun, Nucl. Phys. A 310 (1978) 1. [13] M. Arditi, H. Doubre and M. Riou et al., Nucl. Phys. A 103 (1967) 319. [14] S.G. Nilsson, Mat.-Fys. Medd. Dan. Vid. Selsk. 29, N. 16 (1955) [ 15] F. Villars, Fundamentals in Nuclear Theory (Vienna, 1967) p. 269. [ 16] W.H. Bassichis and E Scheck, Phys. Rev. 145 (1966) 771. [17] V.G. Neudatchin and V.G. Shevchenko, Phys. Lett. 12 (1964) 18; R.A. Eramzhyan, B.S. Ishkhanov, I.M. Kapitonov and V.G. Neudatchin, Phys. Rep. 136 (1986) 229. [18] M. Danos, Nucl. Phys. 5 (1958) 23; K. Okamoto, Prog. Theor. Phys. 15 (1956) 75L [19] A. Bohr and B. Mottelson, Mat.-Fys. Medd. Dan. Vid. Selsk. 27, N. 16 (1953). [20] M. Marangoni and A.M. Saruis, Nucl. Phys. A 132 (1994) 649. [21] S. Kamerdzhiev and G. Tseplyayev, Izv. AN SSSR, Ser. Fiz. 55 (1991) 49; S. Kamerdzhiev, G. Tertychny and J. Speth, Nucl. Phys. A 569 (1994) 313. [22] B.S. Dolbilkin, V.I. Korin and L.E. Lazareva et al., Nucl. Phys. 72 (1965) 137. [23] N. Bezic, D. Jamnik and G. Kernel et al., Nucl. Phys. A 117 (1968) 124. [24] V.V. Vadamov, B.S. Ishkhanov and I.M. Kapitonov et al., Yad. Fiz. 28 (1978) 590. [25] B.I. Goryachev, B.S. Ishkhanov and V.G. Shevchenko et al., Yad. Fiz. 7 (1968) 1168. [26] V.V. Varlamov, B.S. Ishkhanov and I.M. Kapitonov et al., Yad. Fiz. 30 (1979) 1185. [27] S.C. Fultz, R.A. Alvarez and B.L. Berman et ai., Phys. Rev. C 4 (1971) 149. [28] R.C. Bearse, L. Meyer-Schutzmeister and R.E. Segel, Nucl. Phys. A 116 (1968) 682. [29] S. Matsumoto, H. Yamashita, T. Kamae, Y. Nogami, J. Phys. Soc. Japan 20 (1965) 1321. [30] V.V. Varlamov, B.S. Ishkhanov and I.M. Kapitonov et al., Izv. AN SSSR, Set. Fiz. 43 (t979) 186.