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Isobaric analogue resonances in nuclei of closed neutron shells
I
I.E.7
I
Nuclear Physics A l l l (1968) 289--314; (~) North-HollandPublishing Co., Amsterdam N o t to be reproduced by photoprint or microfilm without written permission f r o m the publisher
ISOBARIC ANALOGUE RESONANCES IN NUCLEI OF CLOSED N E U T R O N SHELLS P. BEREGI
Central Research Institute .[or Physics, Budapest and 1. LOVAS t
Joint Institute Jor Nuclear Research, Dubna Received 7 November 1967 Abstract: The scattering amplitude of elastic proton scattering is investigated in the energy range of the isobaric analogue resonances. The scattering amplitude is expressed as .~'- = .~-opt..~_j~res, where the background term jopt is calculated by means of the standard optical model, and the resonance term is derived by the Feshbach projection-operator method. The "ideal" isobaric analogue states are introduced by applying the step-down isospin operator T_ on the wave functions of the low-lying states of the target + neutron system. Then the "target ground stated-proton" component of the ideal IAS is projected out. Such a doorway type state, the so-called "projected" IAS, is assumed to play the role of the intermediate state of the scattering process. The method is applied to the calculation of the cross section and polarization of elastic proton scattering on 13SBa,14°Ceand 14aNd. The low-lying states of the target are calculated in the framework of the quasi-particle random phase approximation, and the wave functions for the target-t--neutron system are obtained by diagonalizing the residual interaction in the quasi-boson + single-neutron basis.
1. Introduction It has been observed m a n y years ago that the energy spectra of light nuclei show a r e m a r k a b l e similarity which ca n be explained by the charge independence o f the nuclear forces. The energy levels of the isobaric nuclei can be organized into charge multiplets characterised by a given isospin. If the charge-dependent interactions are completely neglected, then the isospin is a good q u a n t u m n u m b e r , the elements of a given charge multiplet are exactly degenerate, and the states of the n e i g h b o u r i n g isobars are connected to each other by the isospin step-up a n d step-down operators T± lobTMa') = \ / T ( T +
1 ) - M r ( M r + _ 1)1~TMr± ' ) .
(1)
In reality, the charge-dependent i n t e r a c t i o r s do cxist, a n d they can cause a mixing a m o n g different isospin states; on the other hand, the degenerate charge multiplets split up. In the case of light nuclei, the mixing and the shifts are n o t too large, therefore the isospin can be used as a n approximately good q u a n t u m n u m b e r . Proceeding a l o n g the periodic table, the C o u l o m b shift is c o n t i n u o u s l y increasing and sooner or t On leave from Central Research Institute for Physics, Budapest. 289
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later it may happen that some elements of a given charge multiplet are in the discrete spectrum, the rest of them, however, are already ill the continuum. For many years, it was expected that the importance of the off-diagonal matrix elements of the chargedependent interactions giving rise to isospin mixing increase with the diagonal elements, and it was assumed that the isospin looses its meaning in the medium and heavy nuclei. This state of affairs has changed remarkably since the beginning of the sixties the isospin has started to regain its credit. In 1961, Anderson and Wong ~) discovered some narrow peaks in the yield of direct (p, n) reactions. It turned out that the observed peaks correspond to highly excited states of the residual nucleus, which are analogous with the low-lying states of the target nucleus which are shifted by the Coulomb energy AEc. The theoretical interpretation of these results was first given by Lane and Soper2). It was pointed out that Jn a direct (p, n) process, among others the proton can jump into the same single-particle state from which the neutron was knocked out. In this case, the final state has the same symmetry and the same isospin as the ground state of the target nucleus. This state, the so-called isobaric analogue state (IAS) of the target, was identified by lkeda, Fujii and Fujita 3) as a coherent superposition of monopole type (proton-particle) (neutron-hole) excitations. In 1964, Fox, Moore and Robson 4) observed some narrow resonances in (p, p) elastic scattering. As far as the quantum numbers and the energies are concerned, a surprisingly good correspondence was observed between these resonances and the low-lying states of the target+neutron system except for an over-all energy shift which can be identified with the Coulomb energy AEc. Subsequently, a large number of IAS has been observed in proton-induced reactions 5) and also a great deal of theoretical activity was devoted to the interpretation of the IAS and to the study 2,6,7) of the "isospin impurity". Te analyse the proton scattering problem, many methods were used. In Robson's approach 7), the R-matrix formalism was employed. Weidenmiiller and Mahaux s) studied different aspects of the 1AS using the tools c f the extended shell model. Many investigations were based upon Lane's equations 9). Feshbach's projection operator technique ~o), which will be applied also in this paper, was used by Kerman and Toledo de Piza 1~) and also by Stephen ~2). The starting point of the approach developed by Kerman and Toledo de Piza is the definition of the IAS IA) of the target + proton system:
IA) =
T_ 14,) [(~'i T+ 7"_I~)] ~'
(2)
where Iq~) is some low-lying state of the target + neutron system. Comparing eqs. (I) and (2), it is obvious that this definition is based upon the hypothesis of the existence of an approximate charge multiplet structure. Of course, IA) in itself cannot be the eigenstate of the system since it is embedded into the continuum and surrounded by a
ISOBARIC ANALOGUE RESONANCES
291
multitude of compound states, but it is expected that [A) is the dominant component of the scattering wave function 1~) in some energy range. In order to analyse the structure of the scattering wave function in a systematic way, it is convenient to introduce the projection operators A, p and q by the following definitions: (i)
p+q+A
(ii) p q = p A
= 1,
=qA =0,
(iii) A = IA)(AI, (iv) p projects out the "target ground state + proton" component of (1 -A)[ 7'). This kind of decomposition of the wave function is very advantageous because the coupling between the components A[~) and (p+q)l~P) can be produced only by the charge-dependent part of the Hamiltopian. Thus, in principle, we are able to calculate exactly the coupling giving rise to the width of the IAS, since the exact form of the Coulomb interaction is known. There is some drawback, however, in this decomposition, namely the "target ground state+ proton" component of the wave function is split and one part is contained already by AI~) and only the remaining part is represented by p[qJ). It is not obvious at all how to get an explicit representation for pl~P), since in AI~P) all nucleons including also the last proton are in bound states; on the other hand, in Pl 7~) the proton is in a continuum state. Because of this difficulty, we choose an other viewpoint and apply a somewhat different procedure; the main points of which can be summarized as follows: a) The projection operators P, Q and R are introduced by the following definitions: (i)
P + Q + R = 1.
(ii)
PQ = PR,=
QR = 0 .
(iii) P projects out the full target ground state+proton component of [q'). (iv) Q selects out the cemponents of [~') corresponding to such configurations in which all nucleons are in bound states. Because of the required orthogonality of the P and Q subspaces, the target ground state must be excluded from the subspace Q. (v) RI~P) is the remaining part of [7j ) which corresponds asymptotically to inelastic scattering and different fragmentation of the nuclear system. b) The elements of the Q subspace characterized predominantly by the isospin eigenvalue T = To+ ½ are picked out. Here T o stands for the isospin of the target ground state (T o = MTo). c) An equation is derived for the component PJ~) containing all the information on elastic scattering. In this equation, the coupling between PlUg) and the special components of Q]7~), labelled by T is expressed explicitly by one term of the effective Hamiltonian. d) The special components of Q I~), the so-called "projected" IAS, are expressed in terms of the wave functions of the target + neutron system.
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Using this method we calculate the scattering amplitude, the cross section and the polarization of the elastic proton scattering in the energy range of the isobaric analogue resonances. We conclude this introduction with a brief description of the contents of the following sections. In sect. 2, we give a formal derivation of the scattering amplitude in terms of effective Hamiltonians obtained by means of projection operators. By some heuristic arguments, we replace these projected Hamiltonians by model operators. In sect. 3, we formulate a simple model for the target + one-particle system, and we calculate the low-lying states of the target+neutron system. Using these model wave functions, we construct the projected IAS, and the scattering amplitude is expressed in terms of these states, In sect. 4, we outline the main steps of an application of the method for the calculation of elastic proton scattering, cross section and polarization on the N = 82 isotones. 2. Formal derivation of the scattering amplitude Multiplying the SchrSdinger equation by the projection operators P, Q and R, we get the coupled system of equations
(E--PHP)PIgJ>
= (PHQ )Q] ~> + (PHR)RIgJ>,
( E - QHQ )Q ]~u> = (QHP)P] 7~) + (QHR)R] ~P>, ( E - RHR)R] 7') = (RHP)P] 7j) + (RHQ )a ]~u).
(3)
Eliminating the components Q 1~) and R] 71), the equation for P] 71) can be written as follows: ( E - H e - HQ-- HR)PI ~> = O, (4) where
H e = PHP, 1
HQ = PHQ - QHP, E-QHQ PHR+PHQ
n R
x
[:
1 QHR1 E-QHQ d
RHR-RHQ
,
l E--~-Q-I_IQ QH R + i
]L Ho 1
E_~QH d Q H P + R H P
l
. (5)
Introducing the projection operator t which projects out the subspace characterized by the isospin T = To + ½, we separate the second term of the Hamiltonian into two components
Hq = PHQt .___~1___ tQHP + fflq, E-QHQ
(6)
ISOBARIC A N A L O G U E RESONANCES
293
where n ~ is given by
nO. = P H Q ( 1 - t )
1 tQHP E-QHO 1
+ PHQ(I - t) - - - - ( 1 - t)QUe E-QHQ 1
+ PHQt - - (1 - t)QHP. E-QHO
(7)
Now we can write eq. (4) in the following form:
( E - P f f i P ) P I ~ ) = (PHQt
1 tQHP) PI~F), E-QHQ
(8)
where P n P is defined by
P n P = H e + n O+HR.
(9)
If we wish to analyse the scattering problem in full details, then, of course, eq. (8) is completely useless because of the complexity of PnP. On the other hand, as far as approximations are concerned, the form of eq. (8) seems to be rather convenient. The formal solution of (8) can be expressed as
PI ~ ) = [~b( +)) + d ~+)PHQt[E- QHQ- tQltPd ~+)PHQt] - l tQHPltp ~+)),
(10)
where lO +) is defined as the solution of the homogeneous equation
( E - P n P ) ] O (+)) = O,
(11)
and the Green operator ( E - P n P + i e ) - t is denoted by (~+. By this formal solution, the amplitude of the elastic scattering can be written as the sum of two terms
.Y- = ~-+(~(-)IPHQt[E-QIIQ-tQHPd(+)PHQt]-'tQHPI~k(+)>,
(12)
where the first term .~" is the amplitude of the elastic scattering caused by P n P alone. We expand the second term of the scattering amplitude using the complete set of wave functions defined by the equation (e,-Qn0)lo,)
= 0.
(13)
Then
= .~'+ Y, (~b(-)lPHQtlOr)(A-t)rs(O~ltQHPitp(+)),
(14)
rs
where the matrix elements (A - 1),, can be obtained by matrix inversion of the matrix A,, defined by
A,, = (O,IE- QHQ- tQHPd (+)PHQtlO,).
(15)
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To start with a given form of the Hamiltonian H and to work out in detail all the quantities involved in the scattering amplitude is rather hopeless, therefore, we are forced to introduce some heuristic assumptions based upon some physical considerations in order to get a more practical form for the scattering amplitude. Therefore, we replace the "exact" projected operators by model operators. At the same time, we reduce our program and do not analyse the "exact" scattering amplitude in its total complexity but approximate it by an "average" scattering amplitude ( J - ) . It can be seen from the definition of PFIP that in addition to the so-called potential scattering He, the effect of the inelastic channels H R and that of some kind of intermediate states/~Q are incorporated into the amplitude J - . In most cases the number of open channels and that of the intermediate states is rather high and usually their effect can be treated properly by a complex optical potential. We shall adopt this procedure and define the average scattering amplitude (~-') replacing PfflP by the optical-model Hamiltonian H °pt of the target+proton system. In addition, we must introduce some assumptions for the projected operators QHP and PHQ; they will be identified with the two-body residual interaction. For the sake of simplicity, we shall assume that the residual Coulomb interaction is negligible, i.e. the residual interaction V commutes with the isospin operators. Adopting these assumptions, the average scattering amplitude will have the following form: ( J - ) = 3T°vt+ Z (~OV)lVtlO,)(a-1)r.,(O,ltVitP~+)) •
(16)
rs
The quantities involved in this expression are re-interpreted by the following equations: (17) (E-U°P')l~//e+)) = 0, 0,
(18)
G~+)= (E-H°pt+ia) -1,
(19)
=
h,, = ( O , I E - Q H Q - tV G~e+)VtlO~).
(20)
In eq. (18), (H°Pt) * stands for the complex conjugate of the optical-model Hamiltonian. We must emphasize that this kind of separation of the scattering amplitude into two terms is meaningful only if the coupling of states with different isospin is weak enough. On the other hand, this separation is useful only if the number of the intermediate states characterized by the isospin T is small enough compared to that of other states. If these requirements are not fulfilled, then we are forced to treat all the intermediate states on equal footing, and we have no right tc perform such a separation of HQ as in eq. (6). A great deal of experimental evidence, however, shows that the isospin of bound states is an approximately good quantum number, i.e., the inter-
ISOBARIC ANALOGUE RESONANCES
295
mediate states dominated by different isospin values are weakly coupled. On the other hand, in the energy range of interest, the density of states characterized by the isospin T = T o + 2t is very low compared to that of the "normal" isospin states (T = T o - ½). Consequently we can accept expression (16) as a good approximation for the average scattering amplitude at least in a limited energy range. It is expected that the cross section derived from eq. (16) will show intermediate structure, in addition to the broad single-particle resonances, we shall get narrow peaks due to some quasi-stationary states characterized by the isospin T = To +-12" Our next task is to establish a relationship between the states IOr) of the target + proton system and the bound states i~r) of the target + neutron system. The wave functions of the low-lying states of the target+ neutron system satisiy the equation
H [ ~ r) = Crl~,rr>
(21)
and are labelled by the isospin quantumnumbers T a n d M r (T = M r = To+½). By means of the isospin step-down operator, we get
H (T-IfI)Tr'I">)= (~Or+[H, T_]T+'t(TT.]~Tr.T~ \~/-'2To+ ~ /
(22)
2-T-~ i- - / \ # 2 T o + 1] '
or in the usual approximation Hla~rr-t> = ( ~ r + A E c ) l q ) [ r - , ) ,
(23)
where the Coulomb shift is denoted by A E c. Manipulating the projection operators, we get
(g,+ AEc-anQ)a]q ~Tr-l) = (QHP)PI'pTr-')+(QHR)R]q~r,r-').
(24)
On the right-hand side, the second term is zero according to the definition c f the projection operator R, and the first term is rather small if the neutron excess is large enough. Neglecting this term, we have
(~,+AEc-QHQ)Qi~ r-l) =0.
(2s)
Comparing eqs. (13) and (25), we see that E, = Wr+AEc,
IOr> = .-/U,QIcb~T- I >,
(26)
where ~A,', stands for a normalization constant. According to these relations, the approximate eigenstates of QHQ can be constructed from the "ideal" IAS ] ~ r r - 1 ) by the projection operator Q.
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P. B E R E G I A N D I. LOVAS
3. A simple model In this section, we formulate a simple model for the description of the target+ neutron system, and in the framework of this model we calculate the energy eigenvalues ~r and eigenfunctions lel,rr) for the low-lying states. Furthermore, using these wave functions, we calculate the matrix elements involved in the scattering amplitude. For the sake of simplicity, we restrict ourselves to such cases when the doubly even target contains only closed-neutron shells. The model Hamiltonian is H = H o + h + V,
(27)
where H o and h stand for the Hamiltonians of the target and the last odd particle, respectively, and the residual interaction is denoted by 1/". The low-lying eigenstates of the target + neutron system take the form
' " >, = X Clj]qg, znj
(28)
zj
where qb is the eigenfunction of Ho, and the single-particle wave function for the last neutron outside the closed shells is denoted by )C~i. The amplitudes C~j are obtained by diagonalization in a restricted space of the basis functions IcPtZnj). Now making use of these model wave functions, we compute the matrix elements involved in the scattering amplitude. Let us consider at first the wave function 10,) given by 10,) =
~--' .... Q T _ l ~ r r ) . v/2 To + 1
(29)
The ideal IAS as we pointed out previously has a target-ground-state+proton component. In order to get the projected IAS, we must eliminate this component. It is easy to see that in our model the target-ground-state + proton component of the ideal IAS has the form
c ,i I ooZ,j>, ,/2To+ 1
(30)
where cpo is the ground-state wave function of the target and Zpj a bound-neutron state occupied by a proton, in other words it is a neutron orbital multiplied by a proton isospin factor. Subtracting this component from the ideal IAS, we get the projected one x/2To+ 1
Iq~°Z'i) '
(31)
where the normalization factor JV', is given by r
o/if, =
1
2
--l-
(Coi) ~ "
2T~/
(32)
ISOBARIC ANALOGUE RESONANCES
297
In order to evaluate the effect of the isospin projection operator t on IO,), we must manipulate with the second term of [Or) tlq~oZpi> = t T_ 7"+ tlOoZ,~.> = ._tT_ JrPog,,i), 2To+l 2To+1 where the relation
(33)
(34)
T+ tl~ooZ;j) = I~ooZ,j.> was used. Introducing the notation
]Arr)=.A/.r{librr)
Coi [q~oZ,,j>} '
(35)
27"o+1 we get
tlOr)
= --==:, tT_
i.4[.r)"
(36)
~/ 2To + i The wave function [~(p+)) involved in the scattering amplitude can be expressed in terms of our model wave functions as
I¢~+)> = I~oon~+)>,
(37)
where ~7~+) stands for the optical-model wave function of the proton satisfying the "Coulomb distorted plane waves-outgoing waves" boundary condition. Now the relevant matrix elements can be written as follows:
=
-t
1
--"~
.......
\/2To + 1 |
\/2To + l
, TT (+) car IV14,. >,
(38)
where
14,(.+)> = I~oort(~+)>,
(39)
and r/(,+) stands for a scattering proton state occupied by a neutron. By similar manipulations, we get for Ar.~the following expression:
Ar.~ = (E-~r-AEc)3r~-Ars+{iFrs,
(40)
where
A,sF,s =
1
2To+l
Re,
2 Im 2To+l
TT
(+)
TT
(Ar~rIVG~.+'VIArr),
(41)
(42)
298
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BEREGI AND
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and G(.+) is defined by
G(.+) = T+ tG(p+)tT_.
(43)
To see the meaning of G(.+), it is enough to look at the spectral representation of the Green operater. The operator G(p+) is constructed from scattering proton states cccupied by proton, the ooerator G(.+) is constructed from the same scattering proton states occupied by neutron. The matrix elements involving the Green operator G(.+) can be written in the following form: pb
TT
-
(+)
.,
1"7"
= |jdxdx'G(+)(x, x t ) • (44) The Green function G(+)(x, x') is expressed by the regular and irregular solutions of the Schrrdinger equation
G(+)(x, x') = -7c Z q~j,m(O, gO, a)~jtm(O ', gO',a')(--1) j-m tim
x
l t vjz()E r wit ( r' ) + l"V j l ( r3 ] rr ' (vii (')E r Wjl( r) + loft " ( r) ]
--
if
r
if r > r ' .
(45)
The normalization and the asymptotic form of the regular vj~(r) and irregular ~)~(r) solutions are given by
- ~ sin ( k r - ~ - M + J j , + q In kr +a,),
(46)
wj,(r) -- V-/£-cos ( k r - ½ n l + bj1+ q In kr + th),
(47)
vj,(,') ...
where the phase shifts 6j~ are complex numbers. It is worthwhile to point out that in our case the level width F,~ in addition to the escape width due to the coupling of the "projected" IAS to the continuum states contains the damping width too, since our Hamiltonian has an imaginary part describing a coupling between the doorway-type projected IAS, and the more complicated states. Because of the factor 1/(2To+l), the level shift is quite small, consequently the spacing of the isobaric analogue resonances is quite similar to the spacing of the lowlying states of the neighbouring isobar. From the experimental point of view, this factor is very important, because to recognize the analogy among the various states would be rather difficult if the shifts of the individual levels were too large. The small widths of the observed isobaric analogue resonances are also due to the factor 1/(2T o + 1) in front of the level width formula (42).
299
ISOBARIC ANALOGUE RESONANCES
4. The cross section and polarization of the elastic proton scattering on 142Nd In this section we summarize the main points and some results of a detailed calculation for the cross section and the polarization of elastic proton scattering on 138Ba ' 14°Ce and I 4 2 N d . In these nuclei, the model outlined in the previous section is directly applicable, because the target is doubly even and contains only closed-neutron shells. We shall compare our results with the experimental data obtained by yon Brentano et ak in Heidelberg 13). We discuss in detail only the case of 1 4 2 N d because the most detailed experimental informations were available for it; on the other hand, our results obtained for the elastic proton scattering on 1aSBa and 14°Ce are qualitatively the same as in the case of 142Nd. Our first task is to work out an appropriate description for the low-lying states of the target (142Nd) and the target + neutron system (14 aNd). It seems to be justified to treat the low-lying states of the target in the framework of the microscopic collective model, because the energy of the first excited state ( E 2+ : 1.57 MeV) is appreciably less than the energy gap in this region and the observed E2 transition probability 14) (B(E2) = 0.47 e 2 10 -48 cm 4) is considerably higher than the single-particle estimate. Assuming a " p a i r i n g + q u a d r u p o l e " type interaction among the protons in the last open shell, the Hamiltonians h and H o will have the following form: h = ~ " ej bjm + bjm , jm
H0
:
~ ot, ~ i a" ~+. , a J , . - z l c~ Z jm
+ a j - m ajm (l j,+m, a j,-ra'
(48)
jmj'ra"
--½~c
~
~2u(Ot,tpl)Y~u(O2,~OE)ljlmtJ2rn2>
jtmtj2m2 j'lm'lj'2m'2
It
+ × aj,+tm, t aj'2m'2aj2m2aj~mt •
(49)
In 142Nd, l0 protons are distributed in the single-particle states lg~, 2d~, 2dl, 3s, and lh~ outside of the double magic core containing 50 protons and 82 neutrons. For the last neutron of 14aNd, the single-particle states 2f.l., 2fg., 3p~. and 3p.f are available. In order to have the correct asymptotic form for the single-particle wave functions, both the neutron operators bj+~, bjm and the proton operators aim, + aim are defined on a Saxon-Woods basis. The potential parameters are given in table I. We did not TABLE 1
The parameters of the Saxon-Woods potential
Proton Neutron
V
Vs .O.
r0
(d
(MeV)
(MeV)
(fm)
(fm)
-- 57 --47
-- 7 -7
1.25 1.25
0.65 0.65
300
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have reliable information about the location o f the relevant single-particle states; therefore we have treated the single-particle energies e~. and e~ as adjustable parameters. The effect of the pairing correlations of the p r o t o n system was taken into account by a Valatin-Bogolyubov transformation. The assumed values of the single-particle energies eiP and the strength of the pairing force G together with the calculated values o f the quasi-particle energies Ej, the transformation coefficients ui, vi, the energy gap 2A and the Fermi level 2 are given in table 2. The long-range term o f the target Hamiltonian was treated by the method of the quasi-particle r a n d o m phase approximation. TABLE 2 The parameters of the quasi-particle transformation G = 28/A = 0.197 MeV j
I gk 2d} lh~ 2d{. 3s}
2d = 2.21 MeV
}. = 0.625 MeV
ejp (MeV)
Ej (MeV)
uj
ej
0.00 0.76 2.18 2.88 3.20
1.27 1.11 1.91 2.51 2.80
0.504 0.749 0.953 0.974 0.980
0.864 0.663 0.304 0.226 0.201
Introducing the quadrupole p h o n o n creation operator as the linear combination of the quasi-proton pair creation and annihilation operators A ~ , ( j l , j 2 ) and A2u(Jt ,J2)
Q2. +
=
Y
X i l i z a 2 ,+( j 1 , j z ) - Y i u 2 A 2 _ ~ , ( j , , j 2 ) ( - I )
u,
(50)
JlJ2
the Hamiltonian is transformed into the form Ho = E o + t o ~
Q+ .Q2.,
(51)
where to stands for the energy of the o n e - p h o n o n excitation, and the ground state energy is denoted by E o. The coefficients Xj,i2, Yi~i2 and the excitation energy co are determined by the secular equation o f the Q R P A . We have chosen the value • = 4.8 • 10-3 MeV • fm - 4 for the strength of the quadrupole force in order to fit the o n e - p h o n o n energy o~ to the energy of the first excited state to = E 2÷ = 1.57 MeV.
The calculated values o f the coefficients Xi,i_, and YJu2 are given in table 3.
ISOBARIC ANALOGUE RESONANCES
301
TABLE 3 The coefficients of the quasi-particle pair creation and annihilation operators in the quadrupole phonon creation operator
Jt ½
J,
Xj, J2
½
•~. •~
~
½ ½
½ .~ 3
½
•~ ~L
½
.~
Yj, j2
o --0.0275 --0.1052 o o
o --0.0150 --0.0450 o 0
0.0275 --0.0338 0.0690 --0.1204 o
0.0150 --0.0177 0.0273 --0.0498 o
--0.1052 --0.0690 --0.5286 --0.1049 0
--0.0450 --0.0273 --0.0915 --0.0216 0
o --0.1204 0.1049 --0.3774 o
o --0.0498 0.0216 --0.0890 o
o o o o --0.1607
o o o o --0.0670
The eigenfunctions of H o denoted by ~0t(N) are labelled by the angular momentum I and the number of quadrupole phonons N. The basis for the matrix representation and diagonalization of the Hamiltonian can be constructed by vector coupling the target states tpl(N ) to the single-neutron states ;~.i" In this basis, the matrix of the Hamiltonian has the following form:
<[~P,(N)z,,y]JIHI[qh,( N')z,,j,]J> = a . , a ~ , ajj,(No~ + ~ ) + <[~,(N)z.j]Q ~"I [ ~ o , , ( ~ ') z . A " >.
(52)
In the actual diagonalization, we restrict our basis allowing only the zero- and onephonon states (N = 0, 1). Let us consider now the problem of the residual interaction V. In general, V can be written as the sum of two-body interactions
v = Z v(x, i
x,)+(,. ,,)W(x, x,).
(53)
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I,. B E R E G I A N D I.
LOVAS
For the sake of simplicity, we take into account only the lowest multipole components of V(x, x~) and W(x, xi). Assuming that the monopole component of V(x, xi) is already included in the Saxon-Woods type single-particle potential of h, we approximate V(x, xi) by a quadrupole term assuming the same form and the same strength as in the Hamiltonian H o
v(x, x,) =
Z
q,)YUO,.
(54)
#
The monopole component of W(x, xl) cannot be included completely in the single-particle potential, therefore we keep this term and neglect all the higher multipole components. Thus the residual interaction in our schematic model can be expressed as the sum of a quadrupole and a monopole term V = V(2)+
V ~°).
(55)
The relevant matrix elements of the quadrupole component can be computed easily and are given by ts, t6)
<[~Ol(N)zJJIV(2)I[tPr(N')z,f]J> = <'/.,.jllrZY2(O,~O)[IZ.~,>R.,,,,
(56)
where
RO(Oj)J O(Oj)J
= O,
R I(2j~J o(oj')J
_
~c~j,,
~
,/5(zl + 1) J,
× (uj~% Yj++uj~vj, Xj,j3, 2
l(2f)J = ( - - 1 ) J+j'+1101c j
j,
× ~ (--1)J:+J'
(P)llT.pj:>
jlj2J3
{2
2
2}(uj, uj2_vj~vj2)(Xi,j3Xj:j3+yjli3yi~j3).
X Jl
J2
J3
(57)
Here the notation <1[ [1> is used for the reduced matrix elements of the singleparticle quadrupole operator, and the 6j symbols are denoted by { ). The monopole component V(°) has only diagonal matrix elements in our restricted basis. These diagonal elements can be absorbed into the single-particle energies e~. if the same value is assumed for the zero- and one-phonon states. As we see, the monopole term is rather irrelevant from the point of view of the target+neutron problem, it will play an important role, however, in the case of the matrix elements between bound and continuum states of the target+ proton system.
ISOBARIC
ANALOGUF
I
[
I
I
tr3
303
RESONANCES
i
I
~
I
I
i
i
I
~4D
re-,
e"
I
I
I
I
t¢,0 ¢'4
i
z t.
¢~ tD
t~ d
oo
~ ~4
304
r.
BEREGI
AND
I.
LOVAS
:
We have performed the numerical diagonalization of the Hamiltonian for a few sets of single-particle energies in order to get the best agreement between the energy eigenvalues d', and the experimental spectrum of t43Nd. In this way, we have obtained the eigenfunctions for the low-lying states of the target+ neutron system
=
C,~NI[q~t(N)z,i] 7.
(5s)
The results of the diagonalization, i.e. the energy eigenvalues and eigenvectors, and the experimental energies ,7) of the low-lying states of ~43Nd are given in table 4. Our next task is to specify the optical-model parameters and to solve the opticalmodel problem, i.e. to calculate the phase shifts, the regular and irregular solutions as the function of the energy E. First of all we have calculated the average of the experimental cross section, and the optical-model parameters were obtained by fitting to this smooth cross section. The form and the parameters of the optical-model potential are given in obvious notation by the following formulae: v°pt(r) = Vc(r)+
Vcf(r)+ iW¢g(r)+ V~.o.h(r)(l. o"),
( Ze2 3 ~2Rc
v&) =
l"2 )
if r < R c
~c
=
Ze' - r-
if
r > R c,
f(r) = [1 + exp ( ( r - Rv)/av) ]- ', d ([1 +exp 9(r) = - 4 a w (1;
h(r) . . . . .
((r-Rw)/aw)]-t),
1 d ([1. +exp ((r-Rv)/av)]-') (h) 2, r dr ,,pc/
Vc = - 5 5 MeV,
Rv = 1.25A "~fro,
av = 0.65 fm,
Wc = - 1 l MeV,
Rw = 1.25A ~ fm,
aw = 0.47 fm,
V~.o. = - 7 . 5 MeV,
Rc = 1.25A ~ fm.
(59)
(60)
Using this optical potential, we have calculated the phase shifts 5j~(E), ~L(E) the regular vjz(r, E) and irregular %l(r, E) solutions in the 9.0-11.5 MeV energy range. Now that all the necessary wave functions are at our disposal, we calculate the elements of the T-matrix which in angular momentum representation can be written as follows:
-~ 2To+lV(
k;)l. P[As, )(A -, ),~(A,~ TT-II/[~.s ).
(61)
305
ISOBARIC ANALOGUE RESONANCES
The amplitude 3-~ pt can be expressed by the phase shifts, and it was computed including all the partial waves up to lma~ = 8. The second term of must be computed only for the l = 1 and l = 3 partial waves, because in our model the contribution of other partial waves vanishes. In terms of our model wave function, the matrix elements involved in the second term of are expressed as follows: (AJ~TIVI~.(s+)) -----JV'ss
l(1
1)..Coso
2To+1
x + y]
1
cS~,<[q~2(1)z.j]'lV(2)l[~oo(O)rl(.~)]s>},(62) !
where the normalization constant is given by
(cS°*s°)2]-~.
Jf/".~ = (1
(63)
2To+ 1/ Instead of assuming a specific expression for the monopole term V(°), it is enough to introduce an approximation for its "expectation value" Qpo(0)lV(°)kpo(0)>. In order to have the same radial dependence as in the case of the quadrupole component, we approximate (90(0)] V(°)kpo(0)> by a simple quadratic expression = Kr 2,
(64)
where the strength K is treated as a free parameter. Using the expression given by eq. (56), we get
..,+, >
=
{(
.A/'ss 1
1)..
27o+1.
Coso K %
× + Z C2jl 's o,(zm/~ ,~xO(OJ)JXl~njl,irZY2(,9, ~o)[iq(}~))] (65)
J
l
The calculation of the matrix elements A,~ defined by eq. (40) is quite straightforward, and it is easy to see that both the level shifts As, and level width Fs, can be obtained as a quadratic expression of the monopole strength K. Substituting the partial scattering amplitudes <.Y'j> into standard formulae, we have computed the differential cross section and the polarization of the elastic proton scattering. We have repeated the computation a few times in order to select the best value for the monopole strength K. The calculated cross sections and the experimental points measured in Heidelberg are plotted on fig. 1. The energy and angle dependence of the polarization is shown on fig. 2. It seems to be useless to describe in details the calculations for the nuclei 13SBa and ~4°Ce. It is enough to refer to fig. 3, which demonstrates the qualitative identity of the cross sections for 138Ba and 142Nd.
306
P. BEREGI AN'DI.
560
~,,~
~_ •
LOVAS
o
480
eL,48: 70" ,% ,,-~o
°
°
.
4OO 320 240
/60
#,0
9,0
g,,5
~0,0
(0,5
tt,0
#~5
Ep [MeV]
Fig. I a.
~L~a = 90
2~0
°
•".,
~
2oo
420
8O
4O
E# [MeV]
Fig. lb.
ISOBARIC ANALOGUE RESONANCES
307
gO
~~.~o"
~ ~o
\~L
""
50
\.'.
40
~/ /
"~"-
:':.:~.. .,.:I ~ - \
30
•
l\ \
,j
20
~o
~5
qao
/0,5
¢¢.0
t~,5 Ep [HEY]
Fig. lc.
E
o° o o
S~
~I~ 68
x"•C
4O
¸
24
..
-....~
8
9,0
g,5
~,o
4O,5
fro
If,5
Ep [MeV] Fig. I d.
308
P.
BEREGI
AND
I.
LOVAS
56
~LAlt ,450"
I
4O
I , '
°.,o
.
o
32 °
2z,
,
,
• ;
:
•
16
.
I
"
k'%o •
'
'
;! ",..,-,'.
I
°
Z
8
~6
9.0
~0,0
¢0,5
~{0
~5
Ep "MeV}
Fig.
le.
56
aLA8 =fZO" _
•
°,
z,O
i '\ 32
.-";
-',,,t"..
[--- J'~
2t,
"¢6 v
•..
~:
81
°
!.,'J
!/
I
9,0
g,~
~o,o
~o,5
¢t0
"..',
'ia t¢5 Ep [Net/]
Fig. la-f. The energy dependence of the cross section of the elastic proton scattering on 14~-Nd at different angles. The experimental points were measured by the Heidelberg group 13).
ISOBARIC
ANALOGUE
309
RESONANCES
,q:,z/o
o,a
9 ~ . ~ -' 5'22~
o~5
o.z.
DC-~
J',..
0.9
ks-v"
-0.2
•~ 4 .
.0,6
-C,'B
- / L ~"
Fig. 2a,
Ca'.
~ 8
-~0 °
C4
(IC' .
.
.
.
.
• ~J4
"0.6
-0.4
.*tO
.1.0
I
i
Fig. 2b.
"v",
310
P.
BEREGI
AND
I.
LOVAS
,;"b"
,7.2
L~,2
-"f2
- .5;5
i,
I
i
I
10.0
~.5
/e~
Fig. 2c.
"'?'" /'?
1
/
&;
- .~2
C.;
-J,5
Fig. 2d.
~Y ~ CIY. G,.o =t5C°
Q.b'_
/ o
,
q
_
~
I
"C3t" -(C I
i
Fig. 2e.
C/6
0~8
=¸ !7.O~°
u.,i
-JB
-/w
i
Fig. 2a-f. The energy dependence of the polarization of the elastic proton scattering on a4~Nd at different aogles.
312
P. BEREGI AND I. LOVAS
eLA8-/70*
20
,.o
/a
9s
/o0
/o5
lid
-
-
/15
°
tp [~ev]
Fig. 3. The energy dependence o f the cross section o f the elastic p r o t o n scattering on laSBa at 170 °. The experimental points were measured by the Heidelberg group la).
5. Discussion
In addition to the calculations described in the previous section, we have made a few tests in order to see what we can learn about the IAS. The first problem we have examined is the importance of the target-ground-state+proton component of the ideal 1AS. We have repeated the relevant part of the calculation using the wave functions of the ideal IAS instead of the projected ones. Comparing the results, it turned out that there is no appreciable deviation. Of course, this is not surprising at all, because the weight of the target-ground-state+proton component in the ideal IAS is less than 5 %. It is expected, however, that in the case of lighter nuclei where the neutron excess is smaller, the situation is quite different and this component is much more important. The aim of the second test was to examine the coupling of the IAS to each other via the continuum states. To see the importance of such a coupling, we have repeated the calculations neglecting the non-diagonal elements of the As, matrix, that is replacing the inverse matrix (A-1)s, by the simple diagonal matrix 1 ( A - 1)st = 6st - - - .
(66)
Arr
We have compared the results with the previous ones, and only insignificant differences were found. This shows that the coupling among the IAS is completely negligible, they act independently, and the scattering amplitude can be written in the usual Breit-
313
ISOBARIC ANALOGUE RESONANCES
Wigner form
1
~(--) TT ~
(67)
< j - s ) = ~--oPt+s 2To+--~ -- E-(g',+AEc+Ar,)+½ir,,
Von Brentano et al. 13) have performed the phenomenological analysis of the cress sections for the proton elastic scattering on the isotones N = 82 using such a representation for the scattering amplitude. We have summarized the results of this analysis and our results for 143Nd in table 5. TABLE 5
The energies, the total widths and the spins of the isobaric analogue resonances
Erc~ (MeV)
-rrcs (keV)
exp
th
9.50 10.23 10.80 11.05
9.50 10.27 10.82 11.04
exp
54 76 75 59
Y=r¢~ th
exp
th
30 86 92 44
½~9"" ~-
~~" ½6-
Comparing the spin values of the resonances obtained by this fitting procedure with our results, a contradiction cart be found in the case of the third resonance. Our model gives spin ½ instead of ~. To seek the possible reason of this contradiction, we have increased the dimension of our basis system, namely in addition to the functions I[q~o(0)Z~j]J) and [[q)2(1)Z,j]s), we have included also the functions [[¢po(2)X,s]J), 1[~02(2)7.,j]J> and I[q~4(2)X,j]s), which correspond to the two-phonon states of the target. Varying the parameters within reasonable limits, we have diagonalized the Hamiltonian. It turned out that our model is unable to produce a .7_,t, ~ spin sequence. In favour of our model perhaps it is worthwhile to mention that the angular distributions of inelastically scattered protons measured at the second and third resonances are quite different. In the case of the third resonance, the distribution is almost isotropic which is expected if the spin is ½ [ref. 18)]. On the other hand, it is obvious that our model is oversimplified, and a more sophisticated model would be able to explain the spin sequence suggested by the phenomenological analysis of the cross section. In any case it would be highly desirable to perform a direct experiment in order to get an unambiguous spin assignment fcr the third resonance. It seems to us that the best tool for this purpose is the proton polarization measurement. As can be seen on fig. 2, the sign of the polarization is opposite for J = 1+ ½and J = l - ½, and it is enough to measure only the relative sign of the polarization at the different resonances. This possibility of the unambiguous spin assignment was pointed out recently by Adams et aL 19). Finally we want to emphasize that the method discussed in this paper can be applied for more realistic residual interactions and for more sophisticated models without
314
P. BEREGIAND I. LOVAS
difficulty. Of course, m u c h more c o m p u t e r time is required, a n d to p e r f o r m such a n extensive calculation seems to be reasonable only in such case where we have more detailed a n d more reliable i n f o r m a t i o n a b o u t the circumstances, e.g. a b o u t the singleparticle energies a n d the character of the excitations of the target a n d target + n e u t r o n system. We are very m u c h indebted to Dr. P. y o n B r e n t a n o for useful discussions a n d for the experimental data. One of the authors (I.L.) is grateful to Professors A. K. K e r m a n a n d H. A. Weidenmfiller for valuable discussions. The a u t h o r s are indebted to the C o m p u t e r D e p a r t m e n t o f the Central Research Institute for Physics for r u n n i n g the p r o g r a m s on the I C T 1905 computer.
Note added in proof" After having finished this w o r k a n d s u b m i t t e d it to the I n t e r n a t i o n a l Conference o n N u c l e a r Structure, T o k y o 20), the authors learned of the e x p e r i m e n t o f G. Clausnitzer et al. 21) who performed the polarization meas u r e m e n t p r o p o s e d above. The experimental value of the polarization was f o u n d to be in a fairly good agreement with our calculations. Moreover, the spin assignm e n t o f ½- to the resonance at 10.8 MeV (ref. 21)) agrees with our result. References
1) 2) 3) 4) 5) 6) 7) 8) 9)
10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21)
J. D. Anderson and C. Wong, Phys. Rev. Lett. 7 (1961) 250 A. M. Lane and J. M. Soper, Nuclear Physics 37 (1962) 506, Nuclear Physics 37 (1962) 663 K. Ikeda, S. Fujii and F. I. Fujita, Phys. Lett. 2 (1962) 169 J. D. Fox, C. F. Moore and D. Robson, Phys. Rev. Lett. 12 (1964) 198 Conf. on isobaric spin in nuclear physics, Tallahassee (1966) L. A. Sliv and Yu. I. Kharitonov, Phys. Lett. 16 (1965) 176 D. Robson, Phys. Rev. 137 (1965) B535 H. A. Weidenmiiller, Nuclear Physics 75 (1966) 241 ; C. Mahaux and H. A. Weidenmiiller, Nuclear Physics A94 (1967) 1 T. Tamura, Phys. Lett. 22 (1966) 644; J. P. Bondorf, H. Lfitken and S. Jiigare, Phys. Lett. 21 (1966) 185; C. Mahaux and H. A. Weidenmtiller, Nuclear Physics 89 (1966) 33; E. H. Auerbach, C. B. Dover, A. K. Kerman, R. H. Lemmer and E. H. Schwarcz, Phys. Rev. Lett. 17 (1966) 1184 H. Feshbach, A. K. Kerman and R. H. Lemmer, Ann. of Phys. 41 (1967) 41; R. H. Lemmer and C. M. Shakin, Ann. of Phys. 27 (1964) 13; I. Lovas, Nuclear Physics 81 (1966) 353 A. F. R. de Toledo Piza, Ph.D. thesis MIT (1966); A. F. R. de Toledo Piza and A. K. Kerman, Ann. of Phys. 43 (1967) 363 R. O. Stephen, Nuclear Physics A94 (1967) 192 P. von Brentano, N. Marquardt, J. P. Wurm and S. A. A. Zaidi, Phys. Lett. 17 (1965) 124; J. P. Wurm, P. yon Brentano, E. Grosse, H. Seitz and S. A. A. Zaidi, Recent progress in nuclear physics with tandems, Heidelberg (1966) D. Eccleshall, M. J. L. Yates and J. J. Simpson, Nuclear Physics 78 (1966) 481 P. Doleshall and I. Lovas, Phys. Lett. 21 (1966) 546 W. Beres, Nuclear Physics 75 (1966) 255 C. L. Nealy and R. K. Sheline, Phys. Rev. 155 (1967) 1314 S. A. A. Zaidi, P. von Brcntano, D. Rieck and J. P. Wurm, Phys. Lett. 19 (1965) 45 J. L. Adams, W. J. Thompson and D. Robson, Nuclear Physics 89 (1966) 377 P. Bercgi and I. Lovas, Proc. Tokyo Conf. (1967) p. 325 G. Clausnitzer, R. Fleischrnann, G. Graw, D. Proetel and J. P. Wurm, Proc. of Tokyo Conf. (1967) p. 329
I.E.7
I
Nuclear Physics A l l l (1968) 289--314; (~) North-HollandPublishing Co., Amsterdam N o t to be reproduced by photoprint or microfilm without written permission f r o m the publisher
ISOBARIC ANALOGUE RESONANCES IN NUCLEI OF CLOSED N E U T R O N SHELLS P. BEREGI
Central Research Institute .[or Physics, Budapest and 1. LOVAS t
Joint Institute Jor Nuclear Research, Dubna Received 7 November 1967 Abstract: The scattering amplitude of elastic proton scattering is investigated in the energy range of the isobaric analogue resonances. The scattering amplitude is expressed as .~'- = .~-opt..~_j~res, where the background term jopt is calculated by means of the standard optical model, and the resonance term is derived by the Feshbach projection-operator method. The "ideal" isobaric analogue states are introduced by applying the step-down isospin operator T_ on the wave functions of the low-lying states of the target + neutron system. Then the "target ground stated-proton" component of the ideal IAS is projected out. Such a doorway type state, the so-called "projected" IAS, is assumed to play the role of the intermediate state of the scattering process. The method is applied to the calculation of the cross section and polarization of elastic proton scattering on 13SBa,14°Ceand 14aNd. The low-lying states of the target are calculated in the framework of the quasi-particle random phase approximation, and the wave functions for the target-t--neutron system are obtained by diagonalizing the residual interaction in the quasi-boson + single-neutron basis.
1. Introduction It has been observed m a n y years ago that the energy spectra of light nuclei show a r e m a r k a b l e similarity which ca n be explained by the charge independence o f the nuclear forces. The energy levels of the isobaric nuclei can be organized into charge multiplets characterised by a given isospin. If the charge-dependent interactions are completely neglected, then the isospin is a good q u a n t u m n u m b e r , the elements of a given charge multiplet are exactly degenerate, and the states of the n e i g h b o u r i n g isobars are connected to each other by the isospin step-up a n d step-down operators T± lobTMa') = \ / T ( T +
1 ) - M r ( M r + _ 1)1~TMr± ' ) .
(1)
In reality, the charge-dependent i n t e r a c t i o r s do cxist, a n d they can cause a mixing a m o n g different isospin states; on the other hand, the degenerate charge multiplets split up. In the case of light nuclei, the mixing and the shifts are n o t too large, therefore the isospin can be used as a n approximately good q u a n t u m n u m b e r . Proceeding a l o n g the periodic table, the C o u l o m b shift is c o n t i n u o u s l y increasing and sooner or t On leave from Central Research Institute for Physics, Budapest. 289
290
p. B E R E G I A N D I .
LOVAS
later it may happen that some elements of a given charge multiplet are in the discrete spectrum, the rest of them, however, are already ill the continuum. For many years, it was expected that the importance of the off-diagonal matrix elements of the chargedependent interactions giving rise to isospin mixing increase with the diagonal elements, and it was assumed that the isospin looses its meaning in the medium and heavy nuclei. This state of affairs has changed remarkably since the beginning of the sixties the isospin has started to regain its credit. In 1961, Anderson and Wong ~) discovered some narrow peaks in the yield of direct (p, n) reactions. It turned out that the observed peaks correspond to highly excited states of the residual nucleus, which are analogous with the low-lying states of the target nucleus which are shifted by the Coulomb energy AEc. The theoretical interpretation of these results was first given by Lane and Soper2). It was pointed out that Jn a direct (p, n) process, among others the proton can jump into the same single-particle state from which the neutron was knocked out. In this case, the final state has the same symmetry and the same isospin as the ground state of the target nucleus. This state, the so-called isobaric analogue state (IAS) of the target, was identified by lkeda, Fujii and Fujita 3) as a coherent superposition of monopole type (proton-particle) (neutron-hole) excitations. In 1964, Fox, Moore and Robson 4) observed some narrow resonances in (p, p) elastic scattering. As far as the quantum numbers and the energies are concerned, a surprisingly good correspondence was observed between these resonances and the low-lying states of the target+neutron system except for an over-all energy shift which can be identified with the Coulomb energy AEc. Subsequently, a large number of IAS has been observed in proton-induced reactions 5) and also a great deal of theoretical activity was devoted to the interpretation of the IAS and to the study 2,6,7) of the "isospin impurity". Te analyse the proton scattering problem, many methods were used. In Robson's approach 7), the R-matrix formalism was employed. Weidenmiiller and Mahaux s) studied different aspects of the 1AS using the tools c f the extended shell model. Many investigations were based upon Lane's equations 9). Feshbach's projection operator technique ~o), which will be applied also in this paper, was used by Kerman and Toledo de Piza 1~) and also by Stephen ~2). The starting point of the approach developed by Kerman and Toledo de Piza is the definition of the IAS IA) of the target + proton system:
IA) =
T_ 14,) [(~'i T+ 7"_I~)] ~'
(2)
where Iq~) is some low-lying state of the target + neutron system. Comparing eqs. (I) and (2), it is obvious that this definition is based upon the hypothesis of the existence of an approximate charge multiplet structure. Of course, IA) in itself cannot be the eigenstate of the system since it is embedded into the continuum and surrounded by a
ISOBARIC ANALOGUE RESONANCES
291
multitude of compound states, but it is expected that [A) is the dominant component of the scattering wave function 1~) in some energy range. In order to analyse the structure of the scattering wave function in a systematic way, it is convenient to introduce the projection operators A, p and q by the following definitions: (i)
p+q+A
(ii) p q = p A
= 1,
=qA =0,
(iii) A = IA)(AI, (iv) p projects out the "target ground state + proton" component of (1 -A)[ 7'). This kind of decomposition of the wave function is very advantageous because the coupling between the components A[~) and (p+q)l~P) can be produced only by the charge-dependent part of the Hamiltopian. Thus, in principle, we are able to calculate exactly the coupling giving rise to the width of the IAS, since the exact form of the Coulomb interaction is known. There is some drawback, however, in this decomposition, namely the "target ground state+ proton" component of the wave function is split and one part is contained already by AI~) and only the remaining part is represented by p[qJ). It is not obvious at all how to get an explicit representation for pl~P), since in AI~P) all nucleons including also the last proton are in bound states; on the other hand, in Pl 7~) the proton is in a continuum state. Because of this difficulty, we choose an other viewpoint and apply a somewhat different procedure; the main points of which can be summarized as follows: a) The projection operators P, Q and R are introduced by the following definitions: (i)
P + Q + R = 1.
(ii)
PQ = PR,=
QR = 0 .
(iii) P projects out the full target ground state+proton component of [q'). (iv) Q selects out the cemponents of [~') corresponding to such configurations in which all nucleons are in bound states. Because of the required orthogonality of the P and Q subspaces, the target ground state must be excluded from the subspace Q. (v) RI~P) is the remaining part of [7j ) which corresponds asymptotically to inelastic scattering and different fragmentation of the nuclear system. b) The elements of the Q subspace characterized predominantly by the isospin eigenvalue T = To+ ½ are picked out. Here T o stands for the isospin of the target ground state (T o = MTo). c) An equation is derived for the component PJ~) containing all the information on elastic scattering. In this equation, the coupling between PlUg) and the special components of Q]7~), labelled by T is expressed explicitly by one term of the effective Hamiltonian. d) The special components of Q I~), the so-called "projected" IAS, are expressed in terms of the wave functions of the target + neutron system.
292
P. B E R E G I
AND
I.
LOVAS
Using this method we calculate the scattering amplitude, the cross section and the polarization of the elastic proton scattering in the energy range of the isobaric analogue resonances. We conclude this introduction with a brief description of the contents of the following sections. In sect. 2, we give a formal derivation of the scattering amplitude in terms of effective Hamiltonians obtained by means of projection operators. By some heuristic arguments, we replace these projected Hamiltonians by model operators. In sect. 3, we formulate a simple model for the target + one-particle system, and we calculate the low-lying states of the target+neutron system. Using these model wave functions, we construct the projected IAS, and the scattering amplitude is expressed in terms of these states, In sect. 4, we outline the main steps of an application of the method for the calculation of elastic proton scattering, cross section and polarization on the N = 82 isotones. 2. Formal derivation of the scattering amplitude Multiplying the SchrSdinger equation by the projection operators P, Q and R, we get the coupled system of equations
(E--PHP)PIgJ>
= (PHQ )Q] ~> + (PHR)RIgJ>,
( E - QHQ )Q ]~u> = (QHP)P] 7~) + (QHR)R] ~P>, ( E - RHR)R] 7') = (RHP)P] 7j) + (RHQ )a ]~u).
(3)
Eliminating the components Q 1~) and R] 71), the equation for P] 71) can be written as follows: ( E - H e - HQ-- HR)PI ~> = O, (4) where
H e = PHP, 1
HQ = PHQ - QHP, E-QHQ PHR+PHQ
n R
x
[:
1 QHR1 E-QHQ d
RHR-RHQ
,
l E--~-Q-I_IQ QH R + i
]L Ho 1
E_~QH d Q H P + R H P
l
. (5)
Introducing the projection operator t which projects out the subspace characterized by the isospin T = To + ½, we separate the second term of the Hamiltonian into two components
Hq = PHQt .___~1___ tQHP + fflq, E-QHQ
(6)
ISOBARIC A N A L O G U E RESONANCES
293
where n ~ is given by
nO. = P H Q ( 1 - t )
1 tQHP E-QHO 1
+ PHQ(I - t) - - - - ( 1 - t)QUe E-QHQ 1
+ PHQt - - (1 - t)QHP. E-QHO
(7)
Now we can write eq. (4) in the following form:
( E - P f f i P ) P I ~ ) = (PHQt
1 tQHP) PI~F), E-QHQ
(8)
where P n P is defined by
P n P = H e + n O+HR.
(9)
If we wish to analyse the scattering problem in full details, then, of course, eq. (8) is completely useless because of the complexity of PnP. On the other hand, as far as approximations are concerned, the form of eq. (8) seems to be rather convenient. The formal solution of (8) can be expressed as
PI ~ ) = [~b( +)) + d ~+)PHQt[E- QHQ- tQltPd ~+)PHQt] - l tQHPltp ~+)),
(10)
where lO +) is defined as the solution of the homogeneous equation
( E - P n P ) ] O (+)) = O,
(11)
and the Green operator ( E - P n P + i e ) - t is denoted by (~+. By this formal solution, the amplitude of the elastic scattering can be written as the sum of two terms
.Y- = ~-+(~(-)IPHQt[E-QIIQ-tQHPd(+)PHQt]-'tQHPI~k(+)>,
(12)
where the first term .~" is the amplitude of the elastic scattering caused by P n P alone. We expand the second term of the scattering amplitude using the complete set of wave functions defined by the equation (e,-Qn0)lo,)
= 0.
(13)
Then
= .~'+ Y, (~b(-)lPHQtlOr)(A-t)rs(O~ltQHPitp(+)),
(14)
rs
where the matrix elements (A - 1),, can be obtained by matrix inversion of the matrix A,, defined by
A,, = (O,IE- QHQ- tQHPd (+)PHQtlO,).
(15)
294
P.
BEREGI AND
I.
LOVAS
To start with a given form of the Hamiltonian H and to work out in detail all the quantities involved in the scattering amplitude is rather hopeless, therefore, we are forced to introduce some heuristic assumptions based upon some physical considerations in order to get a more practical form for the scattering amplitude. Therefore, we replace the "exact" projected operators by model operators. At the same time, we reduce our program and do not analyse the "exact" scattering amplitude in its total complexity but approximate it by an "average" scattering amplitude ( J - ) . It can be seen from the definition of PFIP that in addition to the so-called potential scattering He, the effect of the inelastic channels H R and that of some kind of intermediate states/~Q are incorporated into the amplitude J - . In most cases the number of open channels and that of the intermediate states is rather high and usually their effect can be treated properly by a complex optical potential. We shall adopt this procedure and define the average scattering amplitude (~-') replacing PfflP by the optical-model Hamiltonian H °pt of the target+proton system. In addition, we must introduce some assumptions for the projected operators QHP and PHQ; they will be identified with the two-body residual interaction. For the sake of simplicity, we shall assume that the residual Coulomb interaction is negligible, i.e. the residual interaction V commutes with the isospin operators. Adopting these assumptions, the average scattering amplitude will have the following form: ( J - ) = 3T°vt+ Z (~OV)lVtlO,)(a-1)r.,(O,ltVitP~+)) •
(16)
rs
The quantities involved in this expression are re-interpreted by the following equations: (17) (E-U°P')l~//e+)) = 0, 0,
(18)
G~+)= (E-H°pt+ia) -1,
(19)
=
h,, = ( O , I E - Q H Q - tV G~e+)VtlO~).
(20)
In eq. (18), (H°Pt) * stands for the complex conjugate of the optical-model Hamiltonian. We must emphasize that this kind of separation of the scattering amplitude into two terms is meaningful only if the coupling of states with different isospin is weak enough. On the other hand, this separation is useful only if the number of the intermediate states characterized by the isospin T is small enough compared to that of other states. If these requirements are not fulfilled, then we are forced to treat all the intermediate states on equal footing, and we have no right tc perform such a separation of HQ as in eq. (6). A great deal of experimental evidence, however, shows that the isospin of bound states is an approximately good quantum number, i.e., the inter-
ISOBARIC ANALOGUE RESONANCES
295
mediate states dominated by different isospin values are weakly coupled. On the other hand, in the energy range of interest, the density of states characterized by the isospin T = T o + 2t is very low compared to that of the "normal" isospin states (T = T o - ½). Consequently we can accept expression (16) as a good approximation for the average scattering amplitude at least in a limited energy range. It is expected that the cross section derived from eq. (16) will show intermediate structure, in addition to the broad single-particle resonances, we shall get narrow peaks due to some quasi-stationary states characterized by the isospin T = To +-12" Our next task is to establish a relationship between the states IOr) of the target + proton system and the bound states i~r) of the target + neutron system. The wave functions of the low-lying states of the target+ neutron system satisiy the equation
H [ ~ r) = Crl~,rr>
(21)
and are labelled by the isospin quantumnumbers T a n d M r (T = M r = To+½). By means of the isospin step-down operator, we get
H (T-IfI)Tr'I">)= (~Or+[H, T_]T+'t(TT.]~Tr.T~ \~/-'2To+ ~ /
(22)
2-T-~ i- - / \ # 2 T o + 1] '
or in the usual approximation Hla~rr-t> = ( ~ r + A E c ) l q ) [ r - , ) ,
(23)
where the Coulomb shift is denoted by A E c. Manipulating the projection operators, we get
(g,+ AEc-anQ)a]q ~Tr-l) = (QHP)PI'pTr-')+(QHR)R]q~r,r-').
(24)
On the right-hand side, the second term is zero according to the definition c f the projection operator R, and the first term is rather small if the neutron excess is large enough. Neglecting this term, we have
(~,+AEc-QHQ)Qi~ r-l) =0.
(2s)
Comparing eqs. (13) and (25), we see that E, = Wr+AEc,
IOr> = .-/U,QIcb~T- I >,
(26)
where ~A,', stands for a normalization constant. According to these relations, the approximate eigenstates of QHQ can be constructed from the "ideal" IAS ] ~ r r - 1 ) by the projection operator Q.
296
P. B E R E G I A N D I. LOVAS
3. A simple model In this section, we formulate a simple model for the description of the target+ neutron system, and in the framework of this model we calculate the energy eigenvalues ~r and eigenfunctions lel,rr) for the low-lying states. Furthermore, using these wave functions, we calculate the matrix elements involved in the scattering amplitude. For the sake of simplicity, we restrict ourselves to such cases when the doubly even target contains only closed-neutron shells. The model Hamiltonian is H = H o + h + V,
(27)
where H o and h stand for the Hamiltonians of the target and the last odd particle, respectively, and the residual interaction is denoted by 1/". The low-lying eigenstates of the target + neutron system take the form
' " >, = X Clj]qg, znj
(28)
zj
where qb is the eigenfunction of Ho, and the single-particle wave function for the last neutron outside the closed shells is denoted by )C~i. The amplitudes C~j are obtained by diagonalization in a restricted space of the basis functions IcPtZnj). Now making use of these model wave functions, we compute the matrix elements involved in the scattering amplitude. Let us consider at first the wave function 10,) given by 10,) =
~--' .... Q T _ l ~ r r ) . v/2 To + 1
(29)
The ideal IAS as we pointed out previously has a target-ground-state+proton component. In order to get the projected IAS, we must eliminate this component. It is easy to see that in our model the target-ground-state + proton component of the ideal IAS has the form
c ,i I ooZ,j>, ,/2To+ 1
(30)
where cpo is the ground-state wave function of the target and Zpj a bound-neutron state occupied by a proton, in other words it is a neutron orbital multiplied by a proton isospin factor. Subtracting this component from the ideal IAS, we get the projected one x/2To+ 1
Iq~°Z'i) '
(31)
where the normalization factor JV', is given by r
o/if, =
1
2
--l-
(Coi) ~ "
2T~/
(32)
ISOBARIC ANALOGUE RESONANCES
297
In order to evaluate the effect of the isospin projection operator t on IO,), we must manipulate with the second term of [Or) tlq~oZpi> = t T_ 7"+ tlOoZ,~.> = ._tT_ JrPog,,i), 2To+l 2To+1 where the relation
(33)
(34)
T+ tl~ooZ;j) = I~ooZ,j.> was used. Introducing the notation
]Arr)=.A/.r{librr)
Coi [q~oZ,,j>} '
(35)
27"o+1 we get
tlOr)
= --==:, tT_
i.4[.r)"
(36)
~/ 2To + i The wave function [~(p+)) involved in the scattering amplitude can be expressed in terms of our model wave functions as
I¢~+)> = I~oon~+)>,
(37)
where ~7~+) stands for the optical-model wave function of the proton satisfying the "Coulomb distorted plane waves-outgoing waves" boundary condition. Now the relevant matrix elements can be written as follows:
=
-t
1
--"~
.......
\/2To + 1 |
\/2To + l
, TT (+) car IV14,. >,
(38)
where
14,(.+)> = I~oort(~+)>,
(39)
and r/(,+) stands for a scattering proton state occupied by a neutron. By similar manipulations, we get for Ar.~the following expression:
Ar.~ = (E-~r-AEc)3r~-Ars+{iFrs,
(40)
where
A,sF,s =
1
2To+l
Re
2 Im 2To+l
TT
(+)
TT
(Ar~rIVG~.+'VIArr),
(41)
(42)
298
p.
BEREGI AND
I.
LOVA$
and G(.+) is defined by
G(.+) = T+ tG(p+)tT_.
(43)
To see the meaning of G(.+), it is enough to look at the spectral representation of the Green operater. The operator G(p+) is constructed from scattering proton states cccupied by proton, the ooerator G(.+) is constructed from the same scattering proton states occupied by neutron. The matrix elements involving the Green operator G(.+) can be written in the following form: pb
TT
-
(+)
.,
1"7"
G(+)(x, x') = -7c Z q~j,m(O, gO, a)~jtm(O ', gO',a')(--1) j-m tim
x
l t vjz()E r wit ( r' ) + l"V j l ( r3 ] rr ' (vii (')E r Wjl( r) + loft " ( r) ]
--
if
r
if r > r ' .
(45)
The normalization and the asymptotic form of the regular vj~(r) and irregular ~)~(r) solutions are given by
- ~ sin ( k r - ~ - M + J j , + q In kr +a,),
(46)
wj,(r) -- V-/£-cos ( k r - ½ n l + bj1+ q In kr + th),
(47)
vj,(,') ...
where the phase shifts 6j~ are complex numbers. It is worthwhile to point out that in our case the level width F,~ in addition to the escape width due to the coupling of the "projected" IAS to the continuum states contains the damping width too, since our Hamiltonian has an imaginary part describing a coupling between the doorway-type projected IAS, and the more complicated states. Because of the factor 1/(2To+l), the level shift is quite small, consequently the spacing of the isobaric analogue resonances is quite similar to the spacing of the lowlying states of the neighbouring isobar. From the experimental point of view, this factor is very important, because to recognize the analogy among the various states would be rather difficult if the shifts of the individual levels were too large. The small widths of the observed isobaric analogue resonances are also due to the factor 1/(2T o + 1) in front of the level width formula (42).
299
ISOBARIC ANALOGUE RESONANCES
4. The cross section and polarization of the elastic proton scattering on 142Nd In this section we summarize the main points and some results of a detailed calculation for the cross section and the polarization of elastic proton scattering on 138Ba ' 14°Ce and I 4 2 N d . In these nuclei, the model outlined in the previous section is directly applicable, because the target is doubly even and contains only closed-neutron shells. We shall compare our results with the experimental data obtained by yon Brentano et ak in Heidelberg 13). We discuss in detail only the case of 1 4 2 N d because the most detailed experimental informations were available for it; on the other hand, our results obtained for the elastic proton scattering on 1aSBa and 14°Ce are qualitatively the same as in the case of 142Nd. Our first task is to work out an appropriate description for the low-lying states of the target (142Nd) and the target + neutron system (14 aNd). It seems to be justified to treat the low-lying states of the target in the framework of the microscopic collective model, because the energy of the first excited state ( E 2+ : 1.57 MeV) is appreciably less than the energy gap in this region and the observed E2 transition probability 14) (B(E2) = 0.47 e 2 10 -48 cm 4) is considerably higher than the single-particle estimate. Assuming a " p a i r i n g + q u a d r u p o l e " type interaction among the protons in the last open shell, the Hamiltonians h and H o will have the following form: h = ~ " ej bjm + bjm , jm
H0
:
~ ot, ~ i a" ~+. , a J , . - z l c~ Z jm
+ a j - m ajm (l j,+m, a j,-ra'
(48)
jmj'ra"
--½~c
~
~2u(Ot,tpl)Y~u(O2,~OE)ljlmtJ2rn2>
jtmtj2m2 j'lm'lj'2m'2
It
+ × aj,+tm, t aj'2m'2aj2m2aj~mt •
(49)
In 142Nd, l0 protons are distributed in the single-particle states lg~, 2d~, 2dl, 3s, and lh~ outside of the double magic core containing 50 protons and 82 neutrons. For the last neutron of 14aNd, the single-particle states 2f.l., 2fg., 3p~. and 3p.f are available. In order to have the correct asymptotic form for the single-particle wave functions, both the neutron operators bj+~, bjm and the proton operators aim, + aim are defined on a Saxon-Woods basis. The potential parameters are given in table I. We did not TABLE 1
The parameters of the Saxon-Woods potential
Proton Neutron
V
Vs .O.
r0
(d
(MeV)
(MeV)
(fm)
(fm)
-- 57 --47
-- 7 -7
1.25 1.25
0.65 0.65
300
P.
BEREGI AND
I. L O V A S
have reliable information about the location o f the relevant single-particle states; therefore we have treated the single-particle energies e~. and e~ as adjustable parameters. The effect of the pairing correlations of the p r o t o n system was taken into account by a Valatin-Bogolyubov transformation. The assumed values of the single-particle energies eiP and the strength of the pairing force G together with the calculated values o f the quasi-particle energies Ej, the transformation coefficients ui, vi, the energy gap 2A and the Fermi level 2 are given in table 2. The long-range term o f the target Hamiltonian was treated by the method of the quasi-particle r a n d o m phase approximation. TABLE 2 The parameters of the quasi-particle transformation G = 28/A = 0.197 MeV j
I gk 2d} lh~ 2d{. 3s}
2d = 2.21 MeV
}. = 0.625 MeV
ejp (MeV)
Ej (MeV)
uj
ej
0.00 0.76 2.18 2.88 3.20
1.27 1.11 1.91 2.51 2.80
0.504 0.749 0.953 0.974 0.980
0.864 0.663 0.304 0.226 0.201
Introducing the quadrupole p h o n o n creation operator as the linear combination of the quasi-proton pair creation and annihilation operators A ~ , ( j l , j 2 ) and A2u(Jt ,J2)
Q2. +
=
Y
X i l i z a 2 ,+( j 1 , j z ) - Y i u 2 A 2 _ ~ , ( j , , j 2 ) ( - I )
u,
(50)
JlJ2
the Hamiltonian is transformed into the form Ho = E o + t o ~
Q+ .Q2.,
(51)
where to stands for the energy of the o n e - p h o n o n excitation, and the ground state energy is denoted by E o. The coefficients Xj,i2, Yi~i2 and the excitation energy co are determined by the secular equation o f the Q R P A . We have chosen the value • = 4.8 • 10-3 MeV • fm - 4 for the strength of the quadrupole force in order to fit the o n e - p h o n o n energy o~ to the energy of the first excited state to = E 2÷ = 1.57 MeV.
The calculated values o f the coefficients Xi,i_, and YJu2 are given in table 3.
ISOBARIC ANALOGUE RESONANCES
301
TABLE 3 The coefficients of the quasi-particle pair creation and annihilation operators in the quadrupole phonon creation operator
Jt ½
J,
Xj, J2
½
•~. •~
~
½ ½
½ .~ 3
½
•~ ~L
½
.~
Yj, j2
o --0.0275 --0.1052 o o
o --0.0150 --0.0450 o 0
0.0275 --0.0338 0.0690 --0.1204 o
0.0150 --0.0177 0.0273 --0.0498 o
--0.1052 --0.0690 --0.5286 --0.1049 0
--0.0450 --0.0273 --0.0915 --0.0216 0
o --0.1204 0.1049 --0.3774 o
o --0.0498 0.0216 --0.0890 o
o o o o --0.1607
o o o o --0.0670
The eigenfunctions of H o denoted by ~0t(N) are labelled by the angular momentum I and the number of quadrupole phonons N. The basis for the matrix representation and diagonalization of the Hamiltonian can be constructed by vector coupling the target states tpl(N ) to the single-neutron states ;~.i" In this basis, the matrix of the Hamiltonian has the following form:
<[~P,(N)z,,y]JIHI[qh,( N')z,,j,]J> = a . , a ~ , ajj,(No~ + ~ ) + <[~,(N)z.j]Q ~"I [ ~ o , , ( ~ ') z . A " >.
(52)
In the actual diagonalization, we restrict our basis allowing only the zero- and onephonon states (N = 0, 1). Let us consider now the problem of the residual interaction V. In general, V can be written as the sum of two-body interactions
v = Z v(x, i
x,)+(,. ,,)W(x, x,).
(53)
302
I,. B E R E G I A N D I.
LOVAS
For the sake of simplicity, we take into account only the lowest multipole components of V(x, x~) and W(x, xi). Assuming that the monopole component of V(x, xi) is already included in the Saxon-Woods type single-particle potential of h, we approximate V(x, xi) by a quadrupole term assuming the same form and the same strength as in the Hamiltonian H o
v(x, x,) =
Z
q,)YUO,.
(54)
#
The monopole component of W(x, xl) cannot be included completely in the single-particle potential, therefore we keep this term and neglect all the higher multipole components. Thus the residual interaction in our schematic model can be expressed as the sum of a quadrupole and a monopole term V = V(2)+
V ~°).
(55)
The relevant matrix elements of the quadrupole component can be computed easily and are given by ts, t6)
<[~Ol(N)zJJIV(2)I[tPr(N')z,f]J> = <'/.,.jllrZY2(O,~O)[IZ.~,>R.,,,,
(56)
where
RO(Oj)J O(Oj)J
= O,
R I(2j~J o(oj')J
_
~c~j,,
~
,/5(zl + 1) J,
× (uj~% Yj++uj~vj, Xj,j3, 2
l(2f)J = ( - - 1 ) J+j'+1101c j
j,
× ~ (--1)J:+J'
(P)llT.pj:>
jlj2J3
{2
2
2}(uj, uj2_vj~vj2)(Xi,j3Xj:j3+yjli3yi~j3).
X Jl
J2
J3
(57)
Here the notation <1[ [1> is used for the reduced matrix elements of the singleparticle quadrupole operator, and the 6j symbols are denoted by { ). The monopole component V(°) has only diagonal matrix elements in our restricted basis. These diagonal elements can be absorbed into the single-particle energies e~. if the same value is assumed for the zero- and one-phonon states. As we see, the monopole term is rather irrelevant from the point of view of the target+neutron problem, it will play an important role, however, in the case of the matrix elements between bound and continuum states of the target+ proton system.
ISOBARIC
ANALOGUF
I
[
I
I
tr3
303
RESONANCES
i
I
~
I
I
i
i
I
~4D
re-,
e"
I
I
I
I
t¢,0 ¢'4
i
z t.
¢~ tD
t~ d
oo
~ ~4
304
r.
BEREGI
AND
I.
LOVAS
:
We have performed the numerical diagonalization of the Hamiltonian for a few sets of single-particle energies in order to get the best agreement between the energy eigenvalues d', and the experimental spectrum of t43Nd. In this way, we have obtained the eigenfunctions for the low-lying states of the target+ neutron system
=
C,~NI[q~t(N)z,i] 7.
(5s)
The results of the diagonalization, i.e. the energy eigenvalues and eigenvectors, and the experimental energies ,7) of the low-lying states of ~43Nd are given in table 4. Our next task is to specify the optical-model parameters and to solve the opticalmodel problem, i.e. to calculate the phase shifts, the regular and irregular solutions as the function of the energy E. First of all we have calculated the average of the experimental cross section, and the optical-model parameters were obtained by fitting to this smooth cross section. The form and the parameters of the optical-model potential are given in obvious notation by the following formulae: v°pt(r) = Vc(r)+
Vcf(r)+ iW¢g(r)+ V~.o.h(r)(l. o"),
( Ze2 3 ~2Rc
v&) =
l"2 )
if r < R c
~c
=
Ze' - r-
if
r > R c,
f(r) = [1 + exp ( ( r - Rv)/av) ]- ', d ([1 +exp 9(r) = - 4 a w (1;
h(r) . . . . .
((r-Rw)/aw)]-t),
1 d ([1. +exp ((r-Rv)/av)]-') (h) 2, r dr ,,pc/
Vc = - 5 5 MeV,
Rv = 1.25A "~fro,
av = 0.65 fm,
Wc = - 1 l MeV,
Rw = 1.25A ~ fm,
aw = 0.47 fm,
V~.o. = - 7 . 5 MeV,
Rc = 1.25A ~ fm.
(59)
(60)
Using this optical potential, we have calculated the phase shifts 5j~(E), ~L(E) the regular vjz(r, E) and irregular %l(r, E) solutions in the 9.0-11.5 MeV energy range. Now that all the necessary wave functions are at our disposal, we calculate the elements of the T-matrix which in angular momentum representation can be written as follows:
-~ 2To+lV(
k;)l. P[As, )(A -, ),~(A,~ TT-II/[~.s ).
(61)
305
ISOBARIC ANALOGUE RESONANCES
The amplitude 3-~ pt can be expressed by the phase shifts, and it was computed including all the partial waves up to lma~ = 8. The second term of
l(1
1)..Coso
2To+1
x
1
cS~,<[q~2(1)z.j]'lV(2)l[~oo(O)rl(.~)]s>},(62) !
where the normalization constant is given by
(cS°*s°)2]-~.
Jf/".~ = (1
(63)
2To+ 1/ Instead of assuming a specific expression for the monopole term V(°), it is enough to introduce an approximation for its "expectation value" Qpo(0)lV(°)kpo(0)>. In order to have the same radial dependence as in the case of the quadrupole component, we approximate (90(0)] V(°)kpo(0)> by a simple quadratic expression
(64)
where the strength K is treated as a free parameter. Using the expression given by eq. (56), we get
..,+, >
=
{(
.A/'ss 1
1)..
27o+1.
Coso K %
×
J
l
The calculation of the matrix elements A,~ defined by eq. (40) is quite straightforward, and it is easy to see that both the level shifts As, and level width Fs, can be obtained as a quadratic expression of the monopole strength K. Substituting the partial scattering amplitudes <.Y'j> into standard formulae, we have computed the differential cross section and the polarization of the elastic proton scattering. We have repeated the computation a few times in order to select the best value for the monopole strength K. The calculated cross sections and the experimental points measured in Heidelberg are plotted on fig. 1. The energy and angle dependence of the polarization is shown on fig. 2. It seems to be useless to describe in details the calculations for the nuclei 13SBa and ~4°Ce. It is enough to refer to fig. 3, which demonstrates the qualitative identity of the cross sections for 138Ba and 142Nd.
306
P. BEREGI AN'DI.
560
~,,~
~_ •
LOVAS
o
480
eL,48: 70" ,% ,,-~o
°
°
.
4OO 320 240
/60
#,0
9,0
g,,5
~0,0
(0,5
tt,0
#~5
Ep [MeV]
Fig. I a.
~L~a = 90
2~0
°
•".,
~
2oo
420
8O
4O
E# [MeV]
Fig. lb.
ISOBARIC ANALOGUE RESONANCES
307
gO
~~.~o"
~ ~o
\~L
""
50
\.'.
40
~/ /
"~"-
:':.:~.. .,.:I ~ - \
30
•
l\ \
,j
20
~o
~5
qao
/0,5
¢¢.0
t~,5 Ep [HEY]
Fig. lc.
E
o° o o
S~
~I~ 68
x"•C
4O
¸
24
..
-....~
8
9,0
g,5
~,o
4O,5
fro
If,5
Ep [MeV] Fig. I d.
308
P.
BEREGI
AND
I.
LOVAS
56
~LAlt ,450"
I
4O
I , '
°.,o
.
o
32 °
2z,
,
,
• ;
:
•
16
.
I
"
k'%o •
'
'
;! ",..,-,'.
I
°
Z
8
~6
9.0
~0,0
¢0,5
~{0
~5
Ep "MeV}
Fig.
le.
56
aLA8 =fZO" _
•
°,
z,O
i '\ 32
.-";
-',,,t"..
[--- J'~
2t,
"¢6 v
•..
~:
81
°
!.,'J
!/
I
9,0
g,~
~o,o
~o,5
¢t0
"..',
'ia t¢5 Ep [Net/]
Fig. la-f. The energy dependence of the cross section of the elastic proton scattering on 14~-Nd at different angles. The experimental points were measured by the Heidelberg group 13).
ISOBARIC
ANALOGUE
309
RESONANCES
,q:,z/o
o,a
9 ~ . ~ -' 5'22~
o~5
o.z.
DC-~
J',..
0.9
ks-v"
-0.2
•~ 4 .
.0,6
-C,'B
- / L ~"
Fig. 2a,
Ca'.
~ 8
-~0 °
C4
(IC' .
.
.
.
.
• ~J4
"0.6
-0.4
.*tO
.1.0
I
i
Fig. 2b.
"v",
310
P.
BEREGI
AND
I.
LOVAS
,;"b"
,7.2
L~,2
-"f2
- .5;5
i,
I
i
I
10.0
~.5
/e~
Fig. 2c.
"'?'" /'?
1
/
&;
- .~2
C.;
-J,5
Fig. 2d.
~Y ~ CIY. G,.o =t5C°
Q.b'_
/ o
,
q
_
~
I
"C3t" -(C I
i
Fig. 2e.
C/6
0~8
=¸ !7.O~°
u.,i
-JB
-/w
i
Fig. 2a-f. The energy dependence of the polarization of the elastic proton scattering on a4~Nd at different aogles.
312
P. BEREGI AND I. LOVAS
eLA8-/70*
20
,.o
/a
9s
/o0
/o5
lid
-
-
/15
°
tp [~ev]
Fig. 3. The energy dependence o f the cross section o f the elastic p r o t o n scattering on laSBa at 170 °. The experimental points were measured by the Heidelberg group la).
5. Discussion
In addition to the calculations described in the previous section, we have made a few tests in order to see what we can learn about the IAS. The first problem we have examined is the importance of the target-ground-state+proton component of the ideal 1AS. We have repeated the relevant part of the calculation using the wave functions of the ideal IAS instead of the projected ones. Comparing the results, it turned out that there is no appreciable deviation. Of course, this is not surprising at all, because the weight of the target-ground-state+proton component in the ideal IAS is less than 5 %. It is expected, however, that in the case of lighter nuclei where the neutron excess is smaller, the situation is quite different and this component is much more important. The aim of the second test was to examine the coupling of the IAS to each other via the continuum states. To see the importance of such a coupling, we have repeated the calculations neglecting the non-diagonal elements of the As, matrix, that is replacing the inverse matrix (A-1)s, by the simple diagonal matrix 1 ( A - 1)st = 6st - - - .
(66)
Arr
We have compared the results with the previous ones, and only insignificant differences were found. This shows that the coupling among the IAS is completely negligible, they act independently, and the scattering amplitude can be written in the usual Breit-
313
ISOBARIC ANALOGUE RESONANCES
Wigner form
1
~(--) TT ~
(67)
< j - s ) = ~--oPt+s 2To+--~ -- E-(g',+AEc+Ar,)+½ir,,
Von Brentano et al. 13) have performed the phenomenological analysis of the cress sections for the proton elastic scattering on the isotones N = 82 using such a representation for the scattering amplitude. We have summarized the results of this analysis and our results for 143Nd in table 5. TABLE 5
The energies, the total widths and the spins of the isobaric analogue resonances
Erc~ (MeV)
-rrcs (keV)
exp
th
9.50 10.23 10.80 11.05
9.50 10.27 10.82 11.04
exp
54 76 75 59
Y=r¢~ th
exp
th
30 86 92 44
½~9"" ~-
~~" ½6-
Comparing the spin values of the resonances obtained by this fitting procedure with our results, a contradiction cart be found in the case of the third resonance. Our model gives spin ½ instead of ~. To seek the possible reason of this contradiction, we have increased the dimension of our basis system, namely in addition to the functions I[q~o(0)Z~j]J) and [[q)2(1)Z,j]s), we have included also the functions [[¢po(2)X,s]J), 1[~02(2)7.,j]J> and I[q~4(2)X,j]s), which correspond to the two-phonon states of the target. Varying the parameters within reasonable limits, we have diagonalized the Hamiltonian. It turned out that our model is unable to produce a .7_,t, ~ spin sequence. In favour of our model perhaps it is worthwhile to mention that the angular distributions of inelastically scattered protons measured at the second and third resonances are quite different. In the case of the third resonance, the distribution is almost isotropic which is expected if the spin is ½ [ref. 18)]. On the other hand, it is obvious that our model is oversimplified, and a more sophisticated model would be able to explain the spin sequence suggested by the phenomenological analysis of the cross section. In any case it would be highly desirable to perform a direct experiment in order to get an unambiguous spin assignment fcr the third resonance. It seems to us that the best tool for this purpose is the proton polarization measurement. As can be seen on fig. 2, the sign of the polarization is opposite for J = 1+ ½and J = l - ½, and it is enough to measure only the relative sign of the polarization at the different resonances. This possibility of the unambiguous spin assignment was pointed out recently by Adams et aL 19). Finally we want to emphasize that the method discussed in this paper can be applied for more realistic residual interactions and for more sophisticated models without
314
P. BEREGIAND I. LOVAS
difficulty. Of course, m u c h more c o m p u t e r time is required, a n d to p e r f o r m such a n extensive calculation seems to be reasonable only in such case where we have more detailed a n d more reliable i n f o r m a t i o n a b o u t the circumstances, e.g. a b o u t the singleparticle energies a n d the character of the excitations of the target a n d target + n e u t r o n system. We are very m u c h indebted to Dr. P. y o n B r e n t a n o for useful discussions a n d for the experimental data. One of the authors (I.L.) is grateful to Professors A. K. K e r m a n a n d H. A. Weidenmfiller for valuable discussions. The a u t h o r s are indebted to the C o m p u t e r D e p a r t m e n t o f the Central Research Institute for Physics for r u n n i n g the p r o g r a m s on the I C T 1905 computer.
Note added in proof" After having finished this w o r k a n d s u b m i t t e d it to the I n t e r n a t i o n a l Conference o n N u c l e a r Structure, T o k y o 20), the authors learned of the e x p e r i m e n t o f G. Clausnitzer et al. 21) who performed the polarization meas u r e m e n t p r o p o s e d above. The experimental value of the polarization was f o u n d to be in a fairly good agreement with our calculations. Moreover, the spin assignm e n t o f ½- to the resonance at 10.8 MeV (ref. 21)) agrees with our result. References
1) 2) 3) 4) 5) 6) 7) 8) 9)
10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21)
J. D. Anderson and C. Wong, Phys. Rev. Lett. 7 (1961) 250 A. M. Lane and J. M. Soper, Nuclear Physics 37 (1962) 506, Nuclear Physics 37 (1962) 663 K. Ikeda, S. Fujii and F. I. Fujita, Phys. Lett. 2 (1962) 169 J. D. Fox, C. F. Moore and D. Robson, Phys. Rev. Lett. 12 (1964) 198 Conf. on isobaric spin in nuclear physics, Tallahassee (1966) L. A. Sliv and Yu. I. Kharitonov, Phys. Lett. 16 (1965) 176 D. Robson, Phys. Rev. 137 (1965) B535 H. A. Weidenmiiller, Nuclear Physics 75 (1966) 241 ; C. Mahaux and H. A. Weidenmiiller, Nuclear Physics A94 (1967) 1 T. Tamura, Phys. Lett. 22 (1966) 644; J. P. Bondorf, H. Lfitken and S. Jiigare, Phys. Lett. 21 (1966) 185; C. Mahaux and H. A. Weidenmtiller, Nuclear Physics 89 (1966) 33; E. H. Auerbach, C. B. Dover, A. K. Kerman, R. H. Lemmer and E. H. Schwarcz, Phys. Rev. Lett. 17 (1966) 1184 H. Feshbach, A. K. Kerman and R. H. Lemmer, Ann. of Phys. 41 (1967) 41; R. H. Lemmer and C. M. Shakin, Ann. of Phys. 27 (1964) 13; I. Lovas, Nuclear Physics 81 (1966) 353 A. F. R. de Toledo Piza, Ph.D. thesis MIT (1966); A. F. R. de Toledo Piza and A. K. Kerman, Ann. of Phys. 43 (1967) 363 R. O. Stephen, Nuclear Physics A94 (1967) 192 P. von Brentano, N. Marquardt, J. P. Wurm and S. A. A. Zaidi, Phys. Lett. 17 (1965) 124; J. P. Wurm, P. yon Brentano, E. Grosse, H. Seitz and S. A. A. Zaidi, Recent progress in nuclear physics with tandems, Heidelberg (1966) D. Eccleshall, M. J. L. Yates and J. J. Simpson, Nuclear Physics 78 (1966) 481 P. Doleshall and I. Lovas, Phys. Lett. 21 (1966) 546 W. Beres, Nuclear Physics 75 (1966) 255 C. L. Nealy and R. K. Sheline, Phys. Rev. 155 (1967) 1314 S. A. A. Zaidi, P. von Brcntano, D. Rieck and J. P. Wurm, Phys. Lett. 19 (1965) 45 J. L. Adams, W. J. Thompson and D. Robson, Nuclear Physics 89 (1966) 377 P. Bercgi and I. Lovas, Proc. Tokyo Conf. (1967) p. 325 G. Clausnitzer, R. Fleischrnann, G. Graw, D. Proetel and J. P. Wurm, Proc. of Tokyo Conf. (1967) p. 329