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Electron scattering in the interacting boson model
Nuclear Physics A358(1981)89c-112~. 0 Nonh-HollandPublishing Co., Amsterdam Not to lx reproduced by photoprmt or microfh without written permission from the publisher.
ELECTRON SCATTERING IN THE INTERACTING BOSON MODEL F. IACHELLO Kernfysisch Versneller
Instituut, University of Groningen, The Netherlands and
Physics Department, Yale University, New Haven, Ct. 06520 Abstract: I discuss a new method for analyzing elastic and inelastic electron scattering experiments in medium mass and heavy nuclei.
1. Introduction Medium mass and heavy nuclei are characterized by the occurrence of lowlying collective states. A large amount of experimental information has been accumulated in the last years on static properties (energies, electromagnetic transition rates, quadrupole and magnetic moments, etc.) of these states. The experimental evidence indicates that although the major features of the observed properties can be understood in terms of the concept of a deformation, the actual way in which nuclear properties appear is rather complex and it changes from nucleus to nucleus. Since electron scattering is capable of exploring the spatial dependence of the nuclear degrees of freedom, it may provide new and important information. In the past, electron scattering has mostly been used to extract information on ground state properties and on properties of a selected number of excited states in light- en closed-shell heavy nuclei. With the advent of high resolution electron facilities, a study of low-lying collective states in medium mass and heavy nuclei has become feasible. There are several possible ways in which electron scattering data can be analyzed: one can perform an ab initio microscopic calculation. For example, one could perform a Hartree-Fock (HF) calculationl).This can be done, either in a spherical or in a deformed, axially symmetric, basis. If done in a deformed basis, the HF calculation must be followed by a projection of the angular momentum. In this way, one can calculate elastic scattering cross sections in spherical nuclei, and cross sections for the excitation of the ground state band in rotational nuclei. In order to calculate inelastic excitations to other states, one must then do a random phase calculation (RPA) of the appropriate degrees of freedom. Although this is a completely microscopic method, it suffers from the fact this it is very difficult to apply it to nuclei which are neither spherical nor axially symmetric deformedl). a second alternative21 is that of introducing a collective Hamiltonian, H, written in terms of some collective variables au, and their conjugate momenta, nu. The parameters appearing in this Hamiltonian are determined by fitting the energy levels. The solutions of the eigenvalue equation HJl=EJI then provide the wave functions of the states. In addition to the choice of a collective Hamiltonian, one then assumes a charge density, written in terms of the collective variables,au, as
I
= pbW9,$))
,
(1.1)
with R(e,g)
= c(I+a0&5
fa(2)xY(2)(8,41
l(O)> .
(1.2)
9oc
F. IACHELLO
The quantity a00 describes monopolevibrations, and is related to the o 's by volume conservation. The transition densities can then be obtained by u expanding (1.1) in powers of o,,'s and taking matrix elements between the eigenstates of H. This method suffers from the fact that it is difficult to make a connection with the microscopic theory, and moreover, one needs many terms in the expansion, before reasonable agreement with experiment can be obtained in rotational nuclei. In this talk, I will describe a third method for analyzing electron scattering experiments, which shares same properties with both methods discussed above, since it contains both microscopic and collective features. This method is based on the interacting boson model of collective nuclear states, recently introduced by Arima and myself3). I will begin my talk with a brief review of the model, Sect. 2, and then proceed to a discussion of the transition densities from a phenomenological point of view, Sects. 3-7. In Sect. 8 a brief outline of the microscopic calculations will be given.
2. Review
of the interacting
boson model
Ihe basic assumptions of the interacting boson model are4): we start from the spherical shell-model and consider, for simplicity, only the degrees of freedom of particles outside the major closed shells at 28, 50, 82 and 126, the "valence" particles; 12 (ii) this is still a formidable problem, with dimensionality of the order of IO for a typical medium mass nucleus (A = 150), clearly outside the reach of today's computers. Therefore, in the next step, we truncate the large shellmodel to a smaller one (dimension of the order 102), by considering only the degrees of freedom of low-lying, correlated pair states. Of these states we take into account the two lowest ones, with total angular momentum J=O and J=2; (iii) we replace the pair states by boson-like degrees of freedom denoted by sand d- respectively. Thus, the shell-model problem for 'itXeC4 for example, is replaced by a problem for a coupled system of NV=2 proton bosons and N,=7 neu :ron bosons, as shown in Fig. 1.
(i)
(a)
zXeG4
7r
Fig.
‘&%4
(b
u
1.a. A schematic representation of the shell-model problem for ':tXes4. b. The boson problem which replaces the shell-model problem for
ELECTRON
SCATTERING
91c
The number of pairs (bosons) is counted from the nearest closed shell, i.e. Nlr(" is the number of proton (neutron) particle pairs up to the middle of the she11 and the number of nroton (neutron) hole oairs from there on. In discussing properties of nuclei; the first step is that of providing the Hamiltonian H = Hr + Hv + Vxv .
(2. I)
In (2.11, Ha(“) is the proton (neutron) boson Hamiltonian, and V,, is the proton-neutron boson interaction. The parameters appearing in (2.1) can either be obtained by a fit to the experimental energy levels or, in a more ambitious r approach, can be calculated from the shell model=). These parameters have now been mapped for large regions of the periodic table and calculations of a large number of nuclei are already available. Examples are shown in Figs. 2, 3 and 4, where a comparison between experimental and theoretical energy levels of the Bt,Xe,sBBaand BBCeisotopeS is shown6). As a result of the diagonalization one also obtains the wave functions of the states, to be denoted in the following by IN,, N,, Ji', where i = 1,2,3,... indicates the first, second, third,...state of angular momentum J. Having obtained the wave functions IN,, N,, Ji>, one can calculate all other nuclear properties by specifying the appropriate operators. For electron scattering, the appropriate operators are the electromagnetic transition operators of multipolarity Q,
,(Q) = ,(a) + Tip) . 71
(2.2)
These operators describe the coupling of the proton (neutron) bosons to the external electromagnetic field, Fig. 5. In previous analyses, when only static properties were considered, the coupling constants were taken as numbers. In studying electron scattering, they must be replaced by form factors, describing the spatial dependence of the coupling (boson form factors). The most general form of the transition operators can be written down explicitly by introducing creation (d,,, + st) and annihilation (d,,,s)operators for bosons. If, in first approximation, one assumes that the transition operators
Fig. 2. Calculated energy spectra in B4Xe. The circles, squares and triangles are the experimental values.
92c
F. IACHELLO
Fig. 3. Calculated energy spectra in 56Ba. The circles, squares and triangles are the experimental values.
Fig. 4. Calculated energy spectra in 58Ce. The circles, squares and triangles are the experimental values.
KVI 2054 I
Fig. 5. Schematic representation of the coupling of bosons to the external electromagnetic field.
ELECTRON
SCATTERING
93c
are one-body boson operators, the explicit form is
,@I u (r)
= At2 CL(f)(r)
[d+xs+s+xd](;)+ $t)(r) [d+x;l](;)+
+6
+ q(z)(r)
LO
; u=‘TI,v.
(2.3)
Here dP = (-yd_u, and the square brackets denote angular momentum couplings. It is interesting to note that, within the limits of this approximation, i.e. that T(E)(r) is a one-body operator and only s- and d-bosons are included, no multipole higher than 11=4is possible. I now proceed to a discussion of each multipole P.separately.
3. Scalar densities When &=O, three terms contribute to the transition operators, a term containing the operator [d+xd](O), a term containing the operator [s+xs]("), and a term containing the unit operator2 1. It is convenient to introduce the number operators for d-bosons, nd = ~!5[d+xd](O) and s-bosons, n = [s+xG](O).Since in the interacting boson model, the total number of bosons must be conserved, one has N = nd + ns, and the two terms containing [d+xd] and [s+xi](O) can be combined together to give
g(‘)(r)
JS
nd + ;(O)(r)
i+O)()
ns = v(')(r) N + [.+
- q(')(r)] nd.
(3.1)
As a result, the 9_=0proton transition operator can be written as T:')(r) = n:')(r) + yiz)(r)N,,+ y,!f)(r)N,+ Bii)(r)nd + giz)(r)nd 'TI V
(3.2)
and the P.=Oneutron transition operator as T:')(r) = n:')(r) + yii)(r)N,,+ Y$)(r)NV + Bzt)(r)nd + Bit)(r)nd IT V
(3.3)
where the coefficients n(')(r), v(')(r), g(')(r) are either equal to ?(O)(r), y(O)(r) and 'j(o)(r)or a linear combination of them. Taking matrix elements of (3.2) and (3.3) between eigenstates of the Hamiltonian (2.1), INx, NV, OT>, one can obtain the diagonal densities
p (0)
(r) = n:')(r) + Y~~)(r)Nv+Y~~)(r)Nv+g~~)(r) + x,ofM$ (r)
V
+
(3.4)
F. IACHELLO
94c
(0) (r)
P
u
=
,o;+ot
ns”
1
+8~~)(r)B~"~i
,
and the transition
(3.5)
+ B~~)(r)B~"~i
,
densities
(0) P
(r) T,ot+o;
= S$
(r)+6~~)(r)
E B(O)(,)B(O) + 6~~)(r)B~"~i 71ll a,li 9 (0) P V,oj:rr
:
(3.6)
,
= 6~~)(r)+B~~)(r)
: B~~)(r)B~~ii
i
.
+ 6~~)(r)B~~~i
(3.7)
B(o) . contain the nuclear structure inforThe coefficients Bs .i, 'i' mation and can be f",~cul~~~~ for*&'n&lL& using the program NPBOS written by B(o) Otsuka. The diagonal densities (3.4) and (3.5) and the transition densities (3.6) and (3.7) can be used to calculate electron scattering cross sections. They can also be used to calculate static properties. These properties will involve only certain integrals over the densities (3.4)-(3.7). For example, proton mean-square radii are given by m 4 o(')(r)+
(3.8)
dr,
T,OfO, if the diagonal
densities
are normalized
as
m
I 0
A similar expression written as 2
=
holds
4*r2 p(O) (r) dr = 2. lT,0++0;
for. Accordingly,
proton mean-square
(3.9)
radii
(o’o)+ y(‘)N + y(O) ll
can be
(3.10)
IlT 1T
(0) ,
From
is the mean-square radius of the closed-shell and y!:), yVu are integrals over vi:)(r), v$$)(r), S!:)(r) and S!:)(r). (3.10) one can calculate isotope shifts. These are given by
ELECTRON
(N,&+l)
A$> i
SCATTERING
-
(N,,NJ
2
95c
> +
a OI
=
(3.11) where the superscripts indicate the values of Nx and N, for which the mean square radii are calculated. By constructing in a similar way the diagonal densities for states /Nx,Nv,2f>,one can also calculate the isomer shift. This is given by
&J,
E
IT
(:n’Nv) _ (~~~NJ = x2 1TO 1
1
= 3(C) ,(0)(~n,N,)_,(O)(N,,Nv) llr x,11 x,11
(3.12)
where
,(Of $$&J x,11
=
I
R(O)(;,,N,) =
(3.13)
(3.14)
A detailed account of the calculation of isotope and isomer shifts in medium mass and heavy nuclei will be given in Ref. 7. Here 1 confine myself to showing the isoto e shift for the 54Xe, gBa and 5&e isotopes. The ~~~~~~1~~1~~~~~ ;$ and R$$ are /v$)/ = 0.062 fmI , Bzt) = 0,112 fm2, Bit) = 0, Fig. 6.
Fig. 6. Isotopes shifts") in 54Xe, s6Ba and 5sCa. The error bar on the experimental points does not include systematic errors.
!+6c
F. IACHELLO
From the discussion given above, it is clear that electron scattering may provide information not only on the integrals (3.8) but on the boson density themselves. For example, measurements of elastic cross sections in a series of isotopes provides information on the quantity
(3.15)
Similarly, a measurement of the inelastic cross section for the excitation of the state 0; provides information on
P
(O)(r)= B::)(r)
Bi"i2 + 8iz)(r) BiOi2
,
71,0;+0;
,
(3.16)
Since, in this approach, there are only three independent quantities, y (O)(r) S&$>(r) and 8~~)(r) a combined and systematic measurement will provide,n"firs;, a determination of these functions, and, second, a test of the approximation. Up to now, we have explicitly kept apart proton contributions and neutron contributions. In a less accurate analysis, one may re lace proton and neutron densities by an average density. In this approximationg ), sometimes called interacting boson model-l or IBA-13), the appropriate scalar transition operator is T(O)(rl = u(')(r) + v(')(r)N + 8(')(r)nd.
(3.17)
From this, one can calculate average densities by taking matrix elements of (3.17) between eigenstates of the IBA-I Hamiltonian, \N,OI>,
(r) = n(')(r) + v(')(r)N + 8(')(r) G E n(O)(r) + v(')(r)N + 8(')(r)Bly) ,
(3.18)
and
p(t) opo.+(r)
=
8(O) (r) %
E
p
J
(r)B(*) ij
’
i#j
(3.19)
(0) and B!?' contain the nuclear structure information and can The coefficients B.. be calculated usini'the proi$am PHINT written by Scholten. The nuclear mean-square radius is now given by
= .r’>(*’ f y(‘+.J + B(“)B(o) 11
’
(3.20)
ELECTRON
97c
SCATTERING
and the isotope shift by A,r2>(y)
O1
(
= y(O) + B(O) B(0)(N+l) 11
_ B(O)(N) 11
= Y(O) + 8(C) A.
= > (3.21)
It is interesting to compare (3.21) with the corresponding expression in the liquid drop model. Here, one usually parametrizes the nuclear density in the form p(r/R(R,+)) with R(B,$) given by (1.2). After introducing the intrinsic variable, 6, the density becomesq)
+6 Y20 PG)= P (r/ I1-&62
I).
(3.22)
The corresponding mean square radius is =
3 2 Introducingst =?RoA
(I+& e2)St'
(3.23)
213
, one has an isotope shift
2 A 213 A<82, 4d- - + G3 R. A = 5 Al/3
(3.24)
Eq. (3.24) can be directly compared with (3.21), since it has been shown by several authors") that the classical limit of the expectation value is proportional to ~8~>. Although the structure of (3.21) and (3.24) is identical there are however, differences. Microscopic calculations of the coefficients v(Ol and 8(Oj indicate that they are, quite often, very different from the coefficients appearing in (3.24). It is interesting to note that the experimental values, Fig. 6 are also considerably lower than those given by (3.24) (415 RifAl/3 a 0.234 fm2 for A a 120). In addition, difference may arise due to the fact that the proton distribution may not follow the neutron distribution. These effects are taken into account in the interacting boson model by retaining explicitly proton and neutron degrees of freedom, as done above. In order to take into account these effects within the framework of the liquid drop model, one must introduce explicitly a proton and a neutron deformation. To what extent these two deformations are different can be elucidated by careful measurements of meansquare radii in several isotopic chains. The comparison between (3.21) and (3.24) also clarifies the physical meaning of the various terms appearing in (3.2). Here v(O)(r) and v@)(r) describe the proton monopole polarizabilitiesdue to the add!:ion of proton and neutron pairs, while 8$8)(r) and 8@)(r) describe the effects of the proton and neutron deformation on the proton distribution (proton quadrupole polarizabilities).In the microscopic approach, these polarizabilities depend on the particular shellmodel state occupied by the protons and neutrons and thus may change from one region to another of the periodic table. The results shown in Fig. 6 indicate that these functions are (approximately)the same for all isotopes discussed there, 54Xe, 56Ba and 5sCe, but this may not be the case for other isotopic chains. Systematic electron scattering studies of isotopic chains in medium and heavy mass nuclei are not available yet. A preliminary analysisll) of some data on the s28m isotopes in terms of the IBA-I parametrization (3.18) yields the function y(C)(r) shown in Fig. 7. From measurements of ground state densities alone it is difficult to extract the function 8(C)(r). However, as one can see from Eq. (3.19). a measurement of the form factor for the inelastic excitation
98~
F. IACHELLO
I 0
2
4
8
6
10
12
rtfmf -Fig. 7. The boson scalar density r' v(*)(r) extracted from a fit") state densities in 62Sm.
to the ground
of the state 0; would provide direct information on b (O)(r), Fig. 8. It would be interesting to perform this experiment, at least in one or two nuclei, This measurement would also give the monopole transition moment
4
4 -CO) 4~ o 2f po+x+(r)
mli(EO) =
IL-
dr
,
(3.25)
R2 2 213 where R2 is some convenient normalization of m(EO), usually taken as R 2=RO A (RO=I.2 fm).
KVI 205
Fig. 8. Schematic representation of electron scattering leading to 01 (i = 1,2) states.
ELECTRON
99c
SCATTERING
In IBA-I,
p)
B(o)
.
mli(EO) = ----IiR2
4.
(3.26)
Quadrupole densities
Quadrupole densities can be discussed in a similar way. One begins by introducing transition operators Tz2)(r) = ai2)(r) [dtxs+stxd]i2) + Bi2)(r)[dtxd]i2),
(4.1)
Ti2)(r) = c~:~)(r)[dtxs+st~d]S2) + 6~2)(r)[dtxd]~2).
(4.2)
The related transition densities can be obtained by taking matrix elements of (4.1) and (4.2) between initial and final states. For applications to electron scattering, the interesting transition densities are
p
(r)
=
alj2)(r)
- (29 1 I[d+xs+s+xd] N,,JQO;’
+
s,of2+ 1 i + 812)(r)
Z ai2)(r) Ai2ii + Ri2)(r) Bi2ii
(4.3)
,
f
and
p
=
( [d+xs+s+xd](v2) 1 (NT,~V,Oi> t
aS2)(r)
v,o;-t2+ 1 + BJ2)(r) 5
= ai2)(r)
As2ii + SS2)(r) Bi2ii ,
(4.4)
,
,
from which one can form the total transition density
. (4.5)
In (4.5At27i2) and eA2) are the boson effective quadrupole charges. The numbers A(2) (u=n,v) which contain the nuclear structure information can be ogtAiAed"&iiefore using the program NPBOS. From the transition densities (4.5) one can obtain B(E2) values. First, one computes the integral m .
u~2)A~f~i+S~2)B~f~i
(4.6)
0
The B(E2) values are then just the square of the integrals (4.6) m
(2) B(E2;OI&?f) = [ r p + (5) dr I* . 4
o,+2i
(4.7)
F. IACHELLO
loot
An example of these calculations'*) is shown in Fig. 9 for the isotopic chains s6Ba and 5sCe. 54Xe, The transition densities (4.5) can be used directly to calculate electron scattering cross section. In (4.5) there are four unknown form factors, two for protons, aA2)(r), B$*)(r) and two for neutrons, crS2)(r), SJ2)(r). In electron scattering, these form factors are weighted with the effective charges in e62) and e$2). It is interesting to note that the samf form factors appear in the description of the inelastic excitation of the 2i states with other projectiles. The only difference is that the weighting factors ei2) and eA2) are replaced by others, appropriate to the projectile under consideration. For example, in they are replaced by the pion-proton and pion-neutron t-matrix If these are sufficiently different from ei2), e$2), a combined measurement of inelastic excitation of the same states with electrons and pions will provide a way to determine the relative proton and neutron contribution . If one neglects differences between protons and neutrons, (IBA-I), the expressions (4.1) to (4.6) are modified accordingly. There is now an average transition operator
T(2)(,) = p and an average
transition
(r) [dtxs+stxd](2)
+ B(2)(r)[d'xd](2)
,
(4.8)
density .(*)(,)A(*) li
+ B(*) (r)Bif)).
(4.9)
In this case, there are only two unknown form factors CY(2)(r) and B(2)(r), and the nuclear structure information, are given the numbers A(?), B(2), conta?&g by the prograiiPHIN*f B(E2) values are obtained by calculating
(4. IO)
and using (4.7) with o(2) replaced by p(2). It has been found empirically that the approximation in which one neglects differences between proton and neutron transition densities is often sufficient to describe B(E2) values. An interesting question is to what extent this approximation is sufficient to describe electron scattering data. One consequence of (4.9) is that, since there are only two form factors, the cross sections for inelastic excitations of the first three 2+ states should be related, Fig. 10. Experimental tests of this sort are being performed at BATES on ls4Gd B4 So. A failure of (4.9) to describe the data may indicate that either the use of an average density is not appropriate or that higher order (two-body) boson terms should be added to the transition operators. From a microscopic analysis5) these terms are expected to be of the order of 1% of the leading terms. Thus, one expects them to be important only for very weak transitions. The experimental knowledge of form factors for the inelastic excitation of , provides a determination of the boson quadruensities extracted from a fit 14) to in Fig. 11. The density a(2)(r) is surface peaked, while the density S(2)(r) ($nows a more complex structure. In extracting the densities o(2)(r) and S (r) it should be noted that the inelastic excitation of the 2; state is dominated by the a (*)(r) term, while both terms a(2)(r) and S(2)(r) contribute to the inelastic excitation of the states 22 + and 2; (Table I). Another interesting question is whether or not the boson densities change from nucleus to nucleus. This question can be answered by analyzing the inelastic excitation of 21 states in an isotopic chain. A preliminary analysis'l) indicates that large changes do not occur.
ELECTRON
I,
50
1,
64
I,
58
I,,
62
NECK
SCATTERING
,
66
I,,
70
,
14
?‘8
I
82
NUMBER
Fig. 9. Comparison between calculated and experimental B(E2; 0; + 2;) values'*) in54Xe, 56Ba and 5&e.
Fig. IO- Schematic representation of electron scattering leading to 2f states.
(i=1,2,3)
102c
F. IACHELLO
0
I
i
I
I
I
2
4
6
8
IO
r(ftn) Fig. II. The boson quadrupole densifies+a(2)(r) and S(*)(r) extracted from a fit14) to the 0:+2; and 0,+23 transition densities in 150Nd 60 9,,.
+ + (2) and B::), Table I. Structure coefficients Ali and B(E2) values for 0, + 2. 1 transitions in 150Nd
li
B(2) Ii
B(E2)calc
+ OI + 5+
10.02
-2.26
27200
27200
+ + Ol + 22
1.68
1.30
134
76
+ + Ol + 23
2.73
1.45
629
690
Transition
A(2)
B(E2 )exp[e2fmq
For comparison, it may be worthwhile discussing how the analysis of the measurements would proceed in the rotational model. Here one parametrizes the density as in (1.1) and (1.2), with a fixed deformation 8, '(') = I+expT(r-R)/ij’
(4.11)
R = "[l-h
(4.12)
B2+6 Y,,l ,
and constructs transition densitiesIs)
=(2)(r) P
0";2;
=
dhl YLo(“) i
P
(:I
(4.13)
ELECTRON
;‘+,
= 6’
c
103c
SCATTERING
dfl YLo(“) Y20(“)
oi'2;
I G)1 ap
$-band
a~
R=c[l- &
13~+BY20]
(4.14)
y-band =c[1-~B2+8Y201 (4.15)
where $' and 8" are arbitrary numbers which measure the amplitude of the 8 and y oscillations. By expanding the density (4.11) in powers of 8, one can rewrite the transition densities as
zc2)(r) = ai o;+ 2;
p'(r-c) + bi p”(r-c)
(4.16)
where ai and bi are linear combinations of the various parameters 8, fi',etc. and p’(r-c)
=
p”(r-c)
-JL dr =
I+exp[yr-c)/t]
-& p’ (r-c)
(4.17)
(4.18)
By comparing (4.9) with (4.16), one can observe the close similarity between the p'(r-c) replaces the boson density a (2)(r) of (4.9) zi ~s~:o~Sh~~,1:2e~4~t~~~r).However, one also observes some differences since, in principle, all coefficients ai, bi in (4.16) are allowed to vary independently, while in (4.9) the coefficients A(?), B(?) are determined from the diagonalization of the collective Hamiltonian. Frk thii'point of view, the analysis in terms of the density (4.9) should be rather compared with that in terms of the generalized rotation-vibrationmodel of Ref. 2, in which the coefficients ai, bi are also determined from a diagonalization of the collective Hamiltonian.
5. Hexadecapole densities The discussion of other multipole excitations in the interacting boson model proceeds as in the previous cases of L=O and L=2. For hexadecapole excitations, one introduces the hexadecapole transition operators Ti4)(r) = 8i4)(r) [dtxd]i4) ,
(5.1)
TJ4)(r) = 8i4)(r) [dtxd]A4) .
(5.2)
The related transition densities are p(4)(r) = 8:")(r): + s,o++4 1 i (5.3)
l&k
F. IACHELLO
and Nn,Ny,Of> =
(5.4)
from which one can form the total transition density p(4)(r) =ie(4) B(4) ~ (r) Brii II f 0)4f
+ e54) f3:4)(r)Bi4ii .
,
(5.5)
In (5.5), et4) and eC4' are the boson effective hexadecapole charges. The numbers Bc4). and ' B(4). cxn be obtained as before, from the program NPBOS. From the t?&ition denzic$es fS.5) one can'calculate B(E4) values B(E4;Of + 4;) =
r6pC4'(r) dr '. C! 0;+4', 1
(5.6)
If differences between neutrons and protons can be neglected, the expressions (5.1) to (5.5) are modified to TC4)(r) = B(4)(r)[dtxg](4)
(5.7)
and p(4)(r) = eC4) BC4)(r) BC4' . li ) Of-$ (
(5.8)
In this case there is only one unknown form factor B (4)(r) and B(E4) values are given in terms of only one parameter, Bc4), appearing in the integral m 6 -(4) (r) dr e et4) B(4) BC4) (5.9) r P Ii ' 0;+4; 6 Thus, in this approximation, all cross sections for inelastic excitations of 4+ states should be given in terms of the same form factor, Fig. 12.
Fig. 12, Schematic representation of electron scattering leading to 4f (i=1,2,3) states.
ELECTRON
SCATTERING
105c
The restriction of the model space to J=O and J=2 (S and D) pairs is a very severe truncation of the shell model. An interesting question is to what extent other pairs are important in the description of the low-lying spectrum. In particular, contributions of J=4 coupled pairs (G-pairs) may be large. This problem has been emphasized recently by Bohr and Mottelson16). A measurement of inelastic electron scattering to several 4+ states in the same nucleus or in a series of isotopes will help clarifying this point. In fact, the difference of the form factors for the inelastic excrtations of the 4+ states will provide a direct test of the J=4 pair components in the wave functions. Within the framework of the interacting boson model, this component is described by introducing a g-boson. The explicit introduction of this degree of freedom modifies the E4 transition operator which now becomes
?(4)(r) = Bc4)(r)[dtxilc4) + ac4)(r)[ gtxs+stxi](4) + g4) (r) [gtxz+dtxglc4) + nc4)(r)[ gtx,lc4)
+
(5. IO)
.
Finally, it should be noted that no multipole higher than !L=4is allowed in the interacting boson model with J=O,2 pairs only. This has the consequence that, for example, the inelastic excitation of states with 5x=6+ comes only through higher order terms. An E6 transition operator can be obtained either as
+(6+,)
= &6)
(r)
[dtxg+gtxd](6)
(5.11)
or, directly, through J=6 coupled pairs (i-bosons).From this point of view, measurements of inelastic cross sections for states with large J, will provide considerable information on the limitations of the interacting boson model with J=O and J=2 pairs only.
6. Magnetic dipole densities The magnetic dipole transition operators can be written as T(')(r) = S:')(r) [d+xd]L'),
(‘5.1)
,?I) v (r) = B:')(r) [d+xd]$').
(6.2)
These operators have up to now only been used to calculate magnetic moments. In order to do this, one first constructs diagonal densities. For 2 states, for example, one considers the densities p(')(r) = @i')(r) I r,222;
(6.3) and
p
(r) = 8: I)(r)
(6.4) From (6.3) and (6.4), one can form the total density p(')(r) = gill 232;
B:‘)(r)
Bi:ii + g\Il) BsI)B:lii
(6.5)
F. IACHELLO
106c
(1) (1) where g,(l) and 8;') are the effective boson magnetic moments and B~,ii, Bv,ii are obtained from the program NPBOS, as before. Magnetic moments are then grven in terms of the integrals
!
(r)
dr
= g(l) II
@(‘) II
B(l) a,ii + g;') Bsl) Billi.
+ Similarly, Ml transition rates between states 2f and 2j can be calculated the non-diagonal transition densities p (') (r). 2;+2+ J Once more, it is interesting to study the case in which differences between proton and neutron distributions are neglected. The appropriate transition operator is
T(‘)(r) = 6(‘) (r)
(6.6) using
[d+xa](‘).
(6.7)
[d'xd]") IS ’ proportional to the angular momentum operator k (1) , and is diagonal in any coupling scheme, -no Ml transition is possible in :k~~eck~:! Furthermore , $ g-factors -are equal. Introducing theaverage densities However
one has
/i-
(1) p
B(1)
g2 ;=4xg
(6.9)
ii
(1‘) are inwhere B(') is obtained from 6 (l)(r) as in (6.6), and the numbers Bii dependent of the index i. Again, it is worthwhile comparing this result with that provided by the liquid drop model. Here also all Ml transitions vanish and the g-factors are all equal to + = Z/A g2i The observed Fig. 13.
g-factors -
in many nuclei I-
(6.10)
show large deviations I
I
.
I
from the values
KVI 2C I
g2’
(nm) 1
--.__
0.4
--_
Z/A
ti
0.3
w 0.2
’ t
0.1
62Sm 32
I
I
I
I
86
90
94
96
NEUTRON NUMBER Fig.
13. g-factors
of 2; states")
in seSm
Z/A,
ELECTRON
SCATTERING
107c
This indicates that protons and neutrons donotparticipateequally to the collective motion in nuclei. Electron scattering may help clarifying this point. In fact, one important consequence of introducing explicitly proton and neutron degrees of freedom is the occurrence of I+ collective bands4). These bands arise from the coupling of the collective quadrupole degrees of freedom of protons and neutrons. The low-lying states of a nucleus correspond to symmetric coupling (maximum Fspin)4). However, the coupling can be only partially symmetric (F-spin less than maximum), leading to states with J*=I+, Fig. 14. There is at-present no experimental information available on collective bands with Jr=]+. Electron scattering may provide this information. The transition densities appropriate for the inelastic excitations of I+ states are o(‘)(r) lT,0p1;
= 8i1)(r)
N,,,Nv,O;>
:
(6.11) and
p
B\l’)(r)
(r) = v,o++1+ I i
f BiI)(r) From (6.11)
and (6.12)
one can
p(‘)(r) 0;+1; B(MI)
values
=
piI)
form
BAlii
can be obtained cc
from
r * o(‘)(r) OfI;
dr = gi’)
(6.12)
total
density
BJ1ji+ gA1)
B:‘)(r)
B:‘)(r)
,
(6.13)
by first
6i’)Bi’ii
B:‘ii.
(6.13)
Bi1ii
(6.14)
,
computing
+ pi’)
,
0 and then
5
*
,
the
N~,N~,o;>
Bil)
,
co
B(MI ;O;
(6.15)
If the magnetic moments of the low-lying 2+ states are known, one can calculate, using (6.14) and (6.15) the expected B(MI) value for the inelastic excitation Of Ii states. It would be interesting to locate these states through inelastic electron scattering experiments.
7. Magnetic M3 transitions
and the
can be treated
transition
densities
oi3L$:r+ ‘I
octupole in
the
densities
same way.
The appropriate
operators
are
Ti3) (r)
= 8i3) (r) [dtxd ]h3),
(7.1)
T:3) (r)
= 6i3) (r) [d+xd ]i3),
(7.2)
for
= 6i3)(r)
excitations
of
3
+
states
are
N=,Nv,Oi>
:
i (7.3)
108~
F. IACHELLO
KVI 2047
3+,i*
-
?r -
-2+ 4+,2+,0+
V
=
aD -o+
0+ Qp
Fig.
14.
/ I , I [
-2+
-2+
2+
j
Irl
i
Schematic representation of the coupling of proton and neutron bosons in the vibrational limit4).[2] denotes the totally symmetric represen[II] denotes the antisymmetric representation tation of U(6),u n, while z.
(7.4) From these,
one can o(3)
form (r)
ot_3; E(M3) values
the
total
= g(3) 7T
can be obtained
density
b(3) 71 (r)
by first dr = g(3) n
B:3ii ,
computing e(3) II
and then
the
+353)(r)
Bi3iiS I
(7.5)
integral
(3) g(3) n , i + gv
@(3) v
B(3) v,i
(7.6)
m B(M3;Of
one may consider Again, treated on the same footing.
transition
-+ 3;)
=
(7.7)
the approximation in which protons One then considers the operators $3)
and the
+ gi3)
(r)
= 8(3)(r)
Id+xd ](3)
and neutrons
are
(7.8)
densities
(7.9)
ELECTRON
Eq. (7.6)
SCATTERING
109c
is then replaced by (3) g(3) b(3) li '
(7.10)
For M3 transitions, only one form factor, 8 (3)(r), enters, in this approximation. Once more, measurements of M3 inelastic excitations of collective states can provide considerable information on details of the collective motion in heavy nuclei. As in the previous case of electric multipoles, no magnetic multipole higher than &=3 is possible in the interacting boson model with S and D pairs only.
8. Microscopic calculations In the preceding sections, I have discussed the phenomenologicalaspects of the interacting boson model. These aspects are those which have been most extensively investigated. However, it is worthwhile mentioning that microscopic calculations are also possible5) and they are being presently performed. An outline of the microscopic calculations is the following. In order to construct boson densities, one first constructs the bosons as correlated fermion pairs 2 IS> = r CL. Ij ;J=O> 3 3
(8.1)
and (D> = j$, B.., Ijj';J=2> II
(8.2)
where the sums go over the active shells. The boson form factors are thenobtained by evaluating the graphs shown in Fig. 15. This evaluation requires, in addition to a knowledge of the expansion coefficients oj and b.*r also the single particle wave functions for the states j. These wave functionsJ&e taken, in first approximation, as single particle wave functions in a harmonic oscillator potential. More sophisticated calculations would assume wave functions calculated in a Woods-Saxon potential.
Fig. 15. Schematic representation of the coupling of the fermion pair states (8.1) and (8.2) to the external electromagnetic field.
lloc
F. IACHELLO
Since eleetron scattering provides directly the boson densities, it will give a very stringent test of the microscopic aspects of the interacting boson model. It will be interesting to see to what extent the simple assumption of bosons as correlated fermion in the valence shells, is able to describe the observed densities.
9. Conclusions I have presented here a framework within which electron scattering experiments on medium mass and heavy nuclei can be analyzed. In this framework, one constructs the spectrum in terms of some elementary modes of excitation (s- and d-bosons). From this point of view, the approach to nuclear spectra provided by the interacting boson model is very similar to the approach to hadronic spectra provided by the quark model, Fig. 16. Electron scattering experiments on heavy nuclei may thus provide information on the way in which these elementary (boson) constituents couple to the external electromagnetic field, Fig. 5, much in the same way in which electron scattering experiments provide information on the way in which quarks couple to the electromagnetic field. In addition, and independently from the interacting boson model, our knowledge of collective states in medium mass and heavy nuclei is, to a large extent, limited to static properties (energies, B(ER) and B(ML) values and static moments). A detailed study of properties of collective states as a function of momentum transfer, may reveal new and unexpected features. For example, it may clarify the role played by protons and neutrons in generating collective states. To this end, it is not sufficient to measure the elastic and inelastic excitation of the ground state band. In order to understand important aspects of collective states in nuclei, it is essential to measure cross sections for inelastic excitations of other states, in particular Cl;,25, 23, 4; and It, 3;. My discussion has been confined to positive parity states in even-even nuclei. An analysis similar to that presented here can be performed for collective states with negative parity in even-even nuclei, and for states in odd-even nuclei. Collective states with negative parity in even-even nuclei can be generated by introducing an octupole or f-boson'). This is, once more, similar to what occurs in hadron physics where, in order to generate charmed hadrons, it is necessary to introduce, another, charmed or c-quark, Fig. 17. For odd-even nuclei, one must NUCLEAR PHYSICS FW?llClE PHYSICS
I:rl~l~,
I-
0-
-I
i -d
J=2
-a
J=O
-8 -u,d
tI
1 BOSONS
T=O T=$- 0
1 QUARKS
Fig. 16. Building blocks of collective low-lying states in nuclei (s,d bosons) and of low-lying states in hadrons (u,d,s quarks).
ELECTRON
lllc
SCATTERING
NUCLEAR __- PHYSICS PARTICLE PHYSIC+ ---r---I I
I 21
I
,I,
-I-c
1
I
-d
J=2
-I
J=O
T=O
j I -r
-,,dT=@O
Fig. 17. New degrees of freedom of nuclei (f,g?,...bosons) and of hadrons (c,....quarks) at higher excitation energies
introduce the degree6 of freedom of the last, unpaired article. A model to ! ). A detailed study of discuss odd-even nuclei has been recently developed18-' collective states in odd-even heavy nuclei is another of many areas where electron scattering can provide new and interesting information.
References 1. J.W. Negele and G. Rinker, Phys. Rev. Cl5 (1977) 1499. E. Borie, D. Drechsel and K. Lesuo, Nux Phys. A211 (1973) 393. See for example, "Interacting Bosons in Nuclear Physics", F. Iachello ed. (Plenum Press, New York, 1979). 4. A. Arima, T. Otsuka, F. Iachello and I. Talmi, Phys. Lett. 66B (1977) 205. T. Otsuka, A. Arima, F. Iachello and I. Talmi, Phys. Lett. 76B (1978) 141. 5. T. Otsuka, Ph.D. Thesis, University of Tokyo, Japan, (1979); T. Otsuka, A. Arima and F. Iachello, Nucl. Phys. A309 (1978) I. 89B 6. F. Iachello, G. Puddu, 0. Scholten, A. Arima and T. Otsuka, Phys. Lett. (1979) I. 7. A.E.L. Dieperink and F. Iachello, to be published. a. A. Arima and F. Iachello, Ann. Phys. (N.Y.) 99 (1976) 253: A. Arima and F. Iachello, Ann. Phys. (N.Y.) El (1978) 201; A. Arima and F. Iachello, Ann. Phys. (N.Y.) 123 (1979) 468. 9. A. Bohr and B.R. Mottelson, "Nuclear Structus, vol. 1 (W.A. Benjamin, Inc., New York, 1969), p. 164. 10. J.N. Ginocchio and M.W. Kirson, Phys. Rev. Lett. 44 (1980) 1744; A.E.L. Dieperink, 0. Scholten and F. Iachello, PhF. Rev. Lett. 44 (1980) 1747. II. M.A. Moinester, G. Azuelos, J. Alster and A.E.L. Dieperink, to bFpublished. 12. G. Puddu, 0. Scholten and T. Otsuka, Nucl. Phys. A348 (1980) 109. 13. T.S. H. Lee, to be published. 14. A.E.L. Dieperink, F. Iachello, A. Rinat and C. Creswell, Phys. Lett. -76B (1978) 135; A.E.L. Dieperink, in "Interacting Bosons in Nuclear Physics", F. Iachello ed. (Plenum Press, New York, 1979), p.129. 15. J.W. Lightbody, Jr., in Proc. of the Int. Conf. on Photonuclear Reactions and 2. 3.
112c
16. 17. 18. 19. 20.
F.lACHELLO
Applications", Asilomar,l973, ed. B.C. Berman (Lawrence Livermore Laboratory, University of California, Livermore, 1973). A. Bohr and B.R. Mottelson, to be published. H.W. Kugel, R.R. Borchers and R. Kalish, Nucl. Phys. Al86 (1972) 513. F. Iachello and 0. Scholten, Phys. Rev. Lett. 43 (1979) 679. 0. Scholten, Ph. D. Thesis, University of GronKgen, The Netherlands (1980). F. Iachello, Nucl. Phys. A347 (1980) 51.
ELECTRON SCATTERING IN THE INTERACTING BOSON MODEL F. IACHELLO Kernfysisch Versneller
Instituut, University of Groningen, The Netherlands and
Physics Department, Yale University, New Haven, Ct. 06520 Abstract: I discuss a new method for analyzing elastic and inelastic electron scattering experiments in medium mass and heavy nuclei.
1. Introduction Medium mass and heavy nuclei are characterized by the occurrence of lowlying collective states. A large amount of experimental information has been accumulated in the last years on static properties (energies, electromagnetic transition rates, quadrupole and magnetic moments, etc.) of these states. The experimental evidence indicates that although the major features of the observed properties can be understood in terms of the concept of a deformation, the actual way in which nuclear properties appear is rather complex and it changes from nucleus to nucleus. Since electron scattering is capable of exploring the spatial dependence of the nuclear degrees of freedom, it may provide new and important information. In the past, electron scattering has mostly been used to extract information on ground state properties and on properties of a selected number of excited states in light- en closed-shell heavy nuclei. With the advent of high resolution electron facilities, a study of low-lying collective states in medium mass and heavy nuclei has become feasible. There are several possible ways in which electron scattering data can be analyzed: one can perform an ab initio microscopic calculation. For example, one could perform a Hartree-Fock (HF) calculationl).This can be done, either in a spherical or in a deformed, axially symmetric, basis. If done in a deformed basis, the HF calculation must be followed by a projection of the angular momentum. In this way, one can calculate elastic scattering cross sections in spherical nuclei, and cross sections for the excitation of the ground state band in rotational nuclei. In order to calculate inelastic excitations to other states, one must then do a random phase calculation (RPA) of the appropriate degrees of freedom. Although this is a completely microscopic method, it suffers from the fact this it is very difficult to apply it to nuclei which are neither spherical nor axially symmetric deformedl). a second alternative21 is that of introducing a collective Hamiltonian, H, written in terms of some collective variables au, and their conjugate momenta, nu. The parameters appearing in this Hamiltonian are determined by fitting the energy levels. The solutions of the eigenvalue equation HJl=EJI then provide the wave functions of the states. In addition to the choice of a collective Hamiltonian, one then assumes a charge density, written in terms of the collective variables,au, as
I
= pbW9,$))
,
(1.1)
with R(e,g)
= c(I+a0&5
fa(2)xY(2)(8,41
l(O)> .
(1.2)
9oc
F. IACHELLO
The quantity a00 describes monopolevibrations, and is related to the o 's by volume conservation. The transition densities can then be obtained by u expanding (1.1) in powers of o,,'s and taking matrix elements between the eigenstates of H. This method suffers from the fact that it is difficult to make a connection with the microscopic theory, and moreover, one needs many terms in the expansion, before reasonable agreement with experiment can be obtained in rotational nuclei. In this talk, I will describe a third method for analyzing electron scattering experiments, which shares same properties with both methods discussed above, since it contains both microscopic and collective features. This method is based on the interacting boson model of collective nuclear states, recently introduced by Arima and myself3). I will begin my talk with a brief review of the model, Sect. 2, and then proceed to a discussion of the transition densities from a phenomenological point of view, Sects. 3-7. In Sect. 8 a brief outline of the microscopic calculations will be given.
2. Review
of the interacting
boson model
Ihe basic assumptions of the interacting boson model are4): we start from the spherical shell-model and consider, for simplicity, only the degrees of freedom of particles outside the major closed shells at 28, 50, 82 and 126, the "valence" particles; 12 (ii) this is still a formidable problem, with dimensionality of the order of IO for a typical medium mass nucleus (A = 150), clearly outside the reach of today's computers. Therefore, in the next step, we truncate the large shellmodel to a smaller one (dimension of the order 102), by considering only the degrees of freedom of low-lying, correlated pair states. Of these states we take into account the two lowest ones, with total angular momentum J=O and J=2; (iii) we replace the pair states by boson-like degrees of freedom denoted by sand d- respectively. Thus, the shell-model problem for 'itXeC4 for example, is replaced by a problem for a coupled system of NV=2 proton bosons and N,=7 neu :ron bosons, as shown in Fig. 1.
(i)
(a)
zXeG4
7r
Fig.
‘&%4
(b
u
1.a. A schematic representation of the shell-model problem for ':tXes4. b. The boson problem which replaces the shell-model problem for
ELECTRON
SCATTERING
91c
The number of pairs (bosons) is counted from the nearest closed shell, i.e. Nlr(" is the number of proton (neutron) particle pairs up to the middle of the she11 and the number of nroton (neutron) hole oairs from there on. In discussing properties of nuclei; the first step is that of providing the Hamiltonian H = Hr + Hv + Vxv .
(2. I)
In (2.11, Ha(“) is the proton (neutron) boson Hamiltonian, and V,, is the proton-neutron boson interaction. The parameters appearing in (2.1) can either be obtained by a fit to the experimental energy levels or, in a more ambitious r approach, can be calculated from the shell model=). These parameters have now been mapped for large regions of the periodic table and calculations of a large number of nuclei are already available. Examples are shown in Figs. 2, 3 and 4, where a comparison between experimental and theoretical energy levels of the Bt,Xe,sBBaand BBCeisotopeS is shown6). As a result of the diagonalization one also obtains the wave functions of the states, to be denoted in the following by IN,, N,, Ji', where i = 1,2,3,... indicates the first, second, third,...state of angular momentum J. Having obtained the wave functions IN,, N,, Ji>, one can calculate all other nuclear properties by specifying the appropriate operators. For electron scattering, the appropriate operators are the electromagnetic transition operators of multipolarity Q,
,(Q) = ,(a) + Tip) . 71
(2.2)
These operators describe the coupling of the proton (neutron) bosons to the external electromagnetic field, Fig. 5. In previous analyses, when only static properties were considered, the coupling constants were taken as numbers. In studying electron scattering, they must be replaced by form factors, describing the spatial dependence of the coupling (boson form factors). The most general form of the transition operators can be written down explicitly by introducing creation (d,,, + st) and annihilation (d,,,s)operators for bosons. If, in first approximation, one assumes that the transition operators
Fig. 2. Calculated energy spectra in B4Xe. The circles, squares and triangles are the experimental values.
92c
F. IACHELLO
Fig. 3. Calculated energy spectra in 56Ba. The circles, squares and triangles are the experimental values.
Fig. 4. Calculated energy spectra in 58Ce. The circles, squares and triangles are the experimental values.
KVI 2054 I
Fig. 5. Schematic representation of the coupling of bosons to the external electromagnetic field.
ELECTRON
SCATTERING
93c
are one-body boson operators, the explicit form is
,@I u (r)
= At2 CL(f)(r)
[d+xs+s+xd](;)+ $t)(r) [d+x;l](;)+
+6
+ q(z)(r)
LO
; u=‘TI,v.
(2.3)
Here dP = (-yd_u, and the square brackets denote angular momentum couplings. It is interesting to note that, within the limits of this approximation, i.e. that T(E)(r) is a one-body operator and only s- and d-bosons are included, no multipole higher than 11=4is possible. I now proceed to a discussion of each multipole P.separately.
3. Scalar densities When &=O, three terms contribute to the transition operators, a term containing the operator [d+xd](O), a term containing the operator [s+xs]("), and a term containing the unit operator2 1. It is convenient to introduce the number operators for d-bosons, nd = ~!5[d+xd](O) and s-bosons, n = [s+xG](O).Since in the interacting boson model, the total number of bosons must be conserved, one has N = nd + ns, and the two terms containing [d+xd] and [s+xi](O) can be combined together to give
g(‘)(r)
JS
nd + ;(O)(r)
i+O)()
ns = v(')(r) N + [.+
- q(')(r)] nd.
(3.1)
As a result, the 9_=0proton transition operator can be written as T:')(r) = n:')(r) + yiz)(r)N,,+ y,!f)(r)N,+ Bii)(r)nd + giz)(r)nd 'TI V
(3.2)
and the P.=Oneutron transition operator as T:')(r) = n:')(r) + yii)(r)N,,+ Y$)(r)NV + Bzt)(r)nd + Bit)(r)nd IT V
(3.3)
where the coefficients n(')(r), v(')(r), g(')(r) are either equal to ?(O)(r), y(O)(r) and 'j(o)(r)or a linear combination of them. Taking matrix elements of (3.2) and (3.3) between eigenstates of the Hamiltonian (2.1), INx, NV, OT>, one can obtain the diagonal densities
p (0)
(r) = n:')(r) + Y~~)(r)Nv+Y~~)(r)Nv+g~~)(r)
V
+
(3.4)
F. IACHELLO
94c
(0) (r)
P
u
=
,o;+ot
ns”
1
+8~~)(r)B~"~i
,
and the transition
(3.5)
+ B~~)(r)B~"~i
,
densities
(0) P
(r) T,ot+o;
= S$
(r)
E B(O)(,)B(O) + 6~~)(r)B~"~i 71ll a,li 9 (0) P V,oj:rr
:
(3.6)
,
= 6~~)(r)
: B~~)(r)B~~ii
i
.
+ 6~~)(r)B~~~i
(3.7)
B(o) . contain the nuclear structure inforThe coefficients Bs .i, 'i' mation and can be f",~cul~~~~ for*&'n&lL& using the program NPBOS written by B(o) Otsuka. The diagonal densities (3.4) and (3.5) and the transition densities (3.6) and (3.7) can be used to calculate electron scattering cross sections. They can also be used to calculate static properties. These properties will involve only certain integrals over the densities (3.4)-(3.7). For example, proton mean-square radii are given by m 4 o(')(r)+
(3.8)
dr,
T,OfO, if the diagonal
densities
are normalized
as
m
I 0
A similar expression written as 2
=
holds
4*r2 p(O) (r) dr = 2. lT,0++0;
for
proton mean-square
(3.9)
radii
can be
(3.10)
IlT 1T
(0) ,
From
is the mean-square radius of the closed-shell and y!:), yVu are integrals over vi:)(r), v$$)(r), S!:)(r) and S!:)(r). (3.10) one can calculate isotope shifts. These are given by
ELECTRON
(N,&+l)
A$> i
SCATTERING
-
(N,,NJ
2
95c
> +
a OI
=
(3.11) where the superscripts indicate the values of Nx and N, for which the mean square radii are calculated. By constructing in a similar way the diagonal densities for states /Nx,Nv,2f>,one can also calculate the isomer shift. This is given by
&J,
E
IT
1
= 3(C) ,(0)(~n,N,)_,(O)(N,,Nv) llr x,11 x,11
(3.12)
where
,(Of $$&J x,11
=
I
R(O)(;,,N,) =
(3.13)
(3.14)
A detailed account of the calculation of isotope and isomer shifts in medium mass and heavy nuclei will be given in Ref. 7. Here 1 confine myself to showing the isoto e shift for the 54Xe, gBa and 5&e isotopes. The ~~~~~~1~~1~~~~~ ;$ and R$$ are /v$)/ = 0.062 fmI , Bzt) = 0,112 fm2, Bit) = 0, Fig. 6.
Fig. 6. Isotopes shifts") in 54Xe, s6Ba and 5sCa. The error bar on the experimental points does not include systematic errors.
!+6c
F. IACHELLO
From the discussion given above, it is clear that electron scattering may provide information not only on the integrals (3.8) but on the boson density themselves. For example, measurements of elastic cross sections in a series of isotopes provides information on the quantity
(3.15)
Similarly, a measurement of the inelastic cross section for the excitation of the state 0; provides information on
P
(O)(r)= B::)(r)
Bi"i2 + 8iz)(r) BiOi2
,
71,0;+0;
,
(3.16)
Since, in this approach, there are only three independent quantities, y (O)(r) S&$>(r) and 8~~)(r) a combined and systematic measurement will provide,n"firs;, a determination of these functions, and, second, a test of the approximation. Up to now, we have explicitly kept apart proton contributions and neutron contributions. In a less accurate analysis, one may re lace proton and neutron densities by an average density. In this approximationg ), sometimes called interacting boson model-l or IBA-13), the appropriate scalar transition operator is T(O)(rl = u(')(r) + v(')(r)N + 8(')(r)nd.
(3.17)
From this, one can calculate average densities by taking matrix elements of (3.17) between eigenstates of the IBA-I Hamiltonian, \N,OI>,
(r) = n(')(r) + v(')(r)N + 8(')(r)
(3.18)
and
p(t) opo.+(r)
=
8(O) (r)
E
p
J
(r)B(*) ij
’
i#j
(3.19)
(0) and B!?' contain the nuclear structure information and can The coefficients B.. be calculated usini'the proi$am PHINT written by Scholten. The nuclear mean-square radius is now given by
= .r’>(*’ f y(‘+.J + B(“)B(o) 11
’
(3.20)
ELECTRON
97c
SCATTERING
and the isotope shift by A,r2>(y)
O1
(
= y(O) + B(O) B(0)(N+l) 11
_ B(O)(N) 11
= Y(O) + 8(C) A
= > (3.21)
It is interesting to compare (3.21) with the corresponding expression in the liquid drop model. Here, one usually parametrizes the nuclear density in the form p(r/R(R,+)) with R(B,$) given by (1.2). After introducing the intrinsic variable, 6, the density becomesq)
+6 Y20 PG)= P (r/ I1-&62
I).
(3.22)
The corresponding mean square radius is
3 2 Introducing
(I+& e2)
(3.23)
213
, one has an isotope shift
2 A 213 A<82, 4d- - + G3 R. A
(3.24)
Eq. (3.24) can be directly compared with (3.21), since it has been shown by several authors") that the classical limit of the expectation value
98~
F. IACHELLO
I 0
2
4
8
6
10
12
rtfmf -Fig. 7. The boson scalar density r' v(*)(r) extracted from a fit") state densities in 62Sm.
to the ground
of the state 0; would provide direct information on b (O)(r), Fig. 8. It would be interesting to perform this experiment, at least in one or two nuclei, This measurement would also give the monopole transition moment
4
4 -CO) 4~ o 2f po+x+(r)
mli(EO) =
IL-
dr
,
(3.25)
R2 2 213 where R2 is some convenient normalization of m(EO), usually taken as R 2=RO A (RO=I.2 fm).
KVI 205
Fig. 8. Schematic representation of electron scattering leading to 01 (i = 1,2) states.
ELECTRON
99c
SCATTERING
In IBA-I,
p)
B(o)
.
mli(EO) = ----IiR2
4.
(3.26)
Quadrupole densities
Quadrupole densities can be discussed in a similar way. One begins by introducing transition operators Tz2)(r) = ai2)(r) [dtxs+stxd]i2) + Bi2)(r)[dtxd]i2),
(4.1)
Ti2)(r) = c~:~)(r)[dtxs+st~d]S2) + 6~2)(r)[dtxd]~2).
(4.2)
The related transition densities can be obtained by taking matrix elements of (4.1) and (4.2) between initial and final states. For applications to electron scattering, the interesting transition densities are
p
(r)
=
alj2)(r)
- (29 1 I[d+xs+s+xd] N,,JQO;’
+
s,of2+ 1 i + 812)(r)
Z ai2)(r) Ai2ii + Ri2)(r) Bi2ii
(4.3)
,
f
and
p
=
( [d+xs+s+xd](v2) 1 (NT,~V,Oi> t
aS2)(r)
v,o;-t2+ 1 + BJ2)(r)
= ai2)(r)
As2ii + SS2)(r) Bi2ii ,
(4.4)
,
,
from which one can form the total transition density
. (4.5)
In (4.5At27i2) and eA2) are the boson effective quadrupole charges. The numbers A(2) (u=n,v) which contain the nuclear structure information can be ogtAiAed"&iiefore using the program NPBOS. From the transition densities (4.5) one can obtain B(E2) values. First, one computes the integral m .
u~2)A~f~i+S~2)B~f~i
(4.6)
0
The B(E2) values are then just the square of the integrals (4.6) m
(2) B(E2;OI&?f) = [ r p + (5) dr I* . 4
o,+2i
(4.7)
F. IACHELLO
loot
An example of these calculations'*) is shown in Fig. 9 for the isotopic chains s6Ba and 5sCe. 54Xe, The transition densities (4.5) can be used directly to calculate electron scattering cross section. In (4.5) there are four unknown form factors, two for protons, aA2)(r), B$*)(r) and two for neutrons, crS2)(r), SJ2)(r). In electron scattering, these form factors are weighted with the effective charges in e62) and e$2). It is interesting to note that the samf form factors appear in the description of the inelastic excitation of the 2i states with other projectiles. The only difference is that the weighting factors ei2) and eA2) are replaced by others, appropriate to the projectile under consideration. For example, in they are replaced by the pion-proton and pion-neutron t-matrix If these are sufficiently different from ei2), e$2), a combined measurement of inelastic excitation of the same states with electrons and pions will provide a way to determine the relative proton and neutron contribution . If one neglects differences between protons and neutrons, (IBA-I), the expressions (4.1) to (4.6) are modified accordingly. There is now an average transition operator
T(2)(,) = p and an average
transition
(r) [dtxs+stxd](2)
+ B(2)(r)[d'xd](2)
,
(4.8)
density .(*)(,)A(*) li
+ B(*) (r)Bif)).
(4.9)
In this case, there are only two unknown form factors CY(2)(r) and B(2)(r), and the nuclear structure information, are given the numbers A(?), B(2), conta?&g by the prograiiPHIN*f B(E2) values are obtained by calculating
(4. IO)
and using (4.7) with o(2) replaced by p(2). It has been found empirically that the approximation in which one neglects differences between proton and neutron transition densities is often sufficient to describe B(E2) values. An interesting question is to what extent this approximation is sufficient to describe electron scattering data. One consequence of (4.9) is that, since there are only two form factors, the cross sections for inelastic excitations of the first three 2+ states should be related, Fig. 10. Experimental tests of this sort are being performed at BATES on ls4Gd B4 So. A failure of (4.9) to describe the data may indicate that either the use of an average density is not appropriate or that higher order (two-body) boson terms should be added to the transition operators. From a microscopic analysis5) these terms are expected to be of the order of 1% of the leading terms. Thus, one expects them to be important only for very weak transitions. The experimental knowledge of form factors for the inelastic excitation of , provides a determination of the boson quadruensities extracted from a fit 14) to in Fig. 11. The density a(2)(r) is surface peaked, while the density S(2)(r) ($nows a more complex structure. In extracting the densities o(2)(r) and S (r) it should be noted that the inelastic excitation of the 2; state is dominated by the a (*)(r) term, while both terms a(2)(r) and S(2)(r) contribute to the inelastic excitation of the states 22 + and 2; (Table I). Another interesting question is whether or not the boson densities change from nucleus to nucleus. This question can be answered by analyzing the inelastic excitation of 21 states in an isotopic chain. A preliminary analysis'l) indicates that large changes do not occur.
ELECTRON
I,
50
1,
64
I,
58
I,,
62
NECK
SCATTERING
,
66
I,,
70
,
14
?‘8
I
82
NUMBER
Fig. 9. Comparison between calculated and experimental B(E2; 0; + 2;) values'*) in54Xe, 56Ba and 5&e.
Fig. IO- Schematic representation of electron scattering leading to 2f states.
(i=1,2,3)
102c
F. IACHELLO
0
I
i
I
I
I
2
4
6
8
IO
r(ftn) Fig. II. The boson quadrupole densifies+a(2)(r) and S(*)(r) extracted from a fit14) to the 0:+2; and 0,+23 transition densities in 150Nd 60 9,,.
+ + (2) and B::), Table I. Structure coefficients Ali and B(E2) values for 0, + 2. 1 transitions in 150Nd
li
B(2) Ii
B(E2)calc
+ OI + 5+
10.02
-2.26
27200
27200
+ + Ol + 22
1.68
1.30
134
76
+ + Ol + 23
2.73
1.45
629
690
Transition
A(2)
B(E2 )exp[e2fmq
For comparison, it may be worthwhile discussing how the analysis of the measurements would proceed in the rotational model. Here one parametrizes the density as in (1.1) and (1.2), with a fixed deformation 8, '(') = I+expT(r-R)/ij’
(4.11)
R = "[l-h
(4.12)
B2+6 Y,,l ,
and constructs transition densitiesIs)
=(2)(r) P
0";2;
=
dhl YLo(“) i
P
(:I
(4.13)
ELECTRON
;‘+,
= 6’
c
103c
SCATTERING
dfl YLo(“) Y20(“)
oi'2;
I G)1 ap
$-band
a~
R=c[l- &
13~+BY20]
(4.14)
y-band =c[1-~B2+8Y201 (4.15)
where $' and 8" are arbitrary numbers which measure the amplitude of the 8 and y oscillations. By expanding the density (4.11) in powers of 8, one can rewrite the transition densities as
zc2)(r) = ai o;+ 2;
p'(r-c) + bi p”(r-c)
(4.16)
where ai and bi are linear combinations of the various parameters 8, fi',etc. and p’(r-c)
=
p”(r-c)
-JL dr =
I+exp[yr-c)/t]
-& p’ (r-c)
(4.17)
(4.18)
By comparing (4.9) with (4.16), one can observe the close similarity between the p'(r-c) replaces the boson density a (2)(r) of (4.9) zi ~s~:o~Sh~~,1:2e~4~t~~~r).However, one also observes some differences since, in principle, all coefficients ai, bi in (4.16) are allowed to vary independently, while in (4.9) the coefficients A(?), B(?) are determined from the diagonalization of the collective Hamiltonian. Frk thii'point of view, the analysis in terms of the density (4.9) should be rather compared with that in terms of the generalized rotation-vibrationmodel of Ref. 2, in which the coefficients ai, bi are also determined from a diagonalization of the collective Hamiltonian.
5. Hexadecapole densities The discussion of other multipole excitations in the interacting boson model proceeds as in the previous cases of L=O and L=2. For hexadecapole excitations, one introduces the hexadecapole transition operators Ti4)(r) = 8i4)(r) [dtxd]i4) ,
(5.1)
TJ4)(r) = 8i4)(r) [dtxd]A4) .
(5.2)
The related transition densities are p(4)(r) = 8:")(r)
l&k
F. IACHELLO
and Nn,Ny,Of> =
(5.4)
from which one can form the total transition density p(4)(r) =ie(4) B(4) ~ (r) Brii II f 0)4f
+ e54) f3:4)(r)Bi4ii .
,
(5.5)
In (5.5), et4) and eC4' are the boson effective hexadecapole charges. The numbers Bc4). and ' B(4). cxn be obtained as before, from the program NPBOS. From the t?&ition denzic$es fS.5) one can'calculate B(E4) values B(E4;Of + 4;) =
r6pC4'(r) dr '. C! 0;+4', 1
(5.6)
If differences between neutrons and protons can be neglected, the expressions (5.1) to (5.5) are modified to TC4)(r) = B(4)(r)[dtxg](4)
(5.7)
and p(4)(r) = eC4) BC4)(r) BC4' . li ) Of-$ (
(5.8)
In this case there is only one unknown form factor B (4)(r) and B(E4) values are given in terms of only one parameter, Bc4), appearing in the integral m 6 -(4) (r) dr e et4) B(4) BC4) (5.9) r P Ii ' 0;+4; 6 Thus, in this approximation, all cross sections for inelastic excitations of 4+ states should be given in terms of the same form factor, Fig. 12.
Fig. 12, Schematic representation of electron scattering leading to 4f (i=1,2,3) states.
ELECTRON
SCATTERING
105c
The restriction of the model space to J=O and J=2 (S and D) pairs is a very severe truncation of the shell model. An interesting question is to what extent other pairs are important in the description of the low-lying spectrum. In particular, contributions of J=4 coupled pairs (G-pairs) may be large. This problem has been emphasized recently by Bohr and Mottelson16). A measurement of inelastic electron scattering to several 4+ states in the same nucleus or in a series of isotopes will help clarifying this point. In fact, the difference of the form factors for the inelastic excrtations of the 4+ states will provide a direct test of the J=4 pair components in the wave functions. Within the framework of the interacting boson model, this component is described by introducing a g-boson. The explicit introduction of this degree of freedom modifies the E4 transition operator which now becomes
?(4)(r) = Bc4)(r)[dtxilc4) + ac4)(r)[ gtxs+stxi](4) + g4) (r) [gtxz+dtxglc4) + nc4)(r)[ gtx,lc4)
+
(5. IO)
.
Finally, it should be noted that no multipole higher than !L=4is allowed in the interacting boson model with J=O,2 pairs only. This has the consequence that, for example, the inelastic excitation of states with 5x=6+ comes only through higher order terms. An E6 transition operator can be obtained either as
+(6+,)
= &6)
(r)
[dtxg+gtxd](6)
(5.11)
or, directly, through J=6 coupled pairs (i-bosons).From this point of view, measurements of inelastic cross sections for states with large J, will provide considerable information on the limitations of the interacting boson model with J=O and J=2 pairs only.
6. Magnetic dipole densities The magnetic dipole transition operators can be written as T(')(r) = S:')(r) [d+xd]L'),
(‘5.1)
,?I) v (r) = B:')(r) [d+xd]$').
(6.2)
These operators have up to now only been used to calculate magnetic moments. In order to do this, one first constructs diagonal densities. For 2 states, for example, one considers the densities p(')(r) = @i')(r)
(6.3) and
p
(r) = 8: I)(r)
(6.4) From (6.3) and (6.4), one can form the total density p(')(r) = gill 232;
B:‘)(r)
Bi:ii + g\Il) BsI)B:lii
(6.5)
F. IACHELLO
106c
(1) (1) where g,(l) and 8;') are the effective boson magnetic moments and B~,ii, Bv,ii are obtained from the program NPBOS, as before. Magnetic moments are then grven in terms of the integrals
!
(r)
dr
= g(l) II
@(‘) II
B(l) a,ii + g;') Bsl) Billi.
+ Similarly, Ml transition rates between states 2f and 2j can be calculated the non-diagonal transition densities p (') (r). 2;+2+ J Once more, it is interesting to study the case in which differences between proton and neutron distributions are neglected. The appropriate transition operator is
T(‘)(r) = 6(‘) (r)
(6.6) using
[d+xa](‘).
(6.7)
[d'xd]") IS ’ proportional to the angular momentum operator k (1) , and is diagonal in any coupling scheme, -no Ml transition is possible in :k~~eck~:! Furthermore , $ g-factors -are equal. Introducing theaverage densities However
one has
/i-
(1) p
B(1)
g2 ;=4xg
(6.9)
ii
(1‘) are inwhere B(') is obtained from 6 (l)(r) as in (6.6), and the numbers Bii dependent of the index i. Again, it is worthwhile comparing this result with that provided by the liquid drop model. Here also all Ml transitions vanish and the g-factors are all equal to + = Z/A g2i The observed Fig. 13.
g-factors -
in many nuclei I-
(6.10)
show large deviations I
I
.
I
from the values
KVI 2C I
g2’
(nm) 1
--.__
0.4
--_
Z/A
ti
0.3
w 0.2
’ t
0.1
62Sm 32
I
I
I
I
86
90
94
96
NEUTRON NUMBER Fig.
13. g-factors
of 2; states")
in seSm
Z/A,
ELECTRON
SCATTERING
107c
This indicates that protons and neutrons donotparticipateequally to the collective motion in nuclei. Electron scattering may help clarifying this point. In fact, one important consequence of introducing explicitly proton and neutron degrees of freedom is the occurrence of I+ collective bands4). These bands arise from the coupling of the collective quadrupole degrees of freedom of protons and neutrons. The low-lying states of a nucleus correspond to symmetric coupling (maximum Fspin)4). However, the coupling can be only partially symmetric (F-spin less than maximum), leading to states with J*=I+, Fig. 14. There is at-present no experimental information available on collective bands with Jr=]+. Electron scattering may provide this information. The transition densities appropriate for the inelastic excitations of I+ states are o(‘)(r) lT,0p1;
= 8i1)(r)
N,,,Nv,O;>
:
(6.11) and
p
B\l’)(r)
(r) = v,o++1+ I i
f BiI)(r) From (6.11)
and (6.12)
one can
p(‘)(r) 0;+1; B(MI)
values
=
piI)
form
BAlii
can be obtained cc
from
r * o(‘)(r) OfI;
dr = gi’)
(6.12)
total
density
BJ1ji+ gA1)
B:‘)(r)
B:‘)(r)
,
(6.13)
by first
6i’)Bi’ii
B:‘ii.
(6.13)
Bi1ii
(6.14)
,
computing
+ pi’)
,
0 and then
5
*
,
the
N~,N~,o;>
Bil)
,
co
B(MI ;O;
(6.15)
If the magnetic moments of the low-lying 2+ states are known, one can calculate, using (6.14) and (6.15) the expected B(MI) value for the inelastic excitation Of Ii states. It would be interesting to locate these states through inelastic electron scattering experiments.
7. Magnetic M3 transitions
and the
can be treated
transition
densities
oi3L$:r+ ‘I
octupole in
the
densities
same way.
The appropriate
operators
are
Ti3) (r)
= 8i3) (r) [dtxd ]h3),
(7.1)
T:3) (r)
= 6i3) (r) [d+xd ]i3),
(7.2)
for
= 6i3)(r)
excitations
of
3
+
states
are
N=,Nv,Oi>
:
i (7.3)
108~
F. IACHELLO
KVI 2047
3+,i*
-
?r -
-2+ 4+,2+,0+
V
=
aD -o+
0+ Qp
Fig.
14.
/ I , I [
-2+
-2+
2+
j
Irl
i
Schematic representation of the coupling of proton and neutron bosons in the vibrational limit4).[2] denotes the totally symmetric represen[II] denotes the antisymmetric representation tation of U(6),u n, while z.
(7.4) From these,
one can o(3)
form (r)
ot_3; E(M3) values
the
total
= g(3) 7T
can be obtained
density
b(3) 71 (r)
by first dr = g(3) n
B:3ii ,
computing e(3) II
and then
the
+353)(r)
Bi3iiS I
(7.5)
integral
(3) g(3) n , i + gv
@(3) v
B(3) v,i
(7.6)
m B(M3;Of
one may consider Again, treated on the same footing.
transition
-+ 3;)
=
(7.7)
the approximation in which protons One then considers the operators $3)
and the
+ gi3)
(r)
= 8(3)(r)
Id+xd ](3)
and neutrons
are
(7.8)
densities
(7.9)
ELECTRON
Eq. (7.6)
SCATTERING
109c
is then replaced by (3) g(3) b(3) li '
(7.10)
For M3 transitions, only one form factor, 8 (3)(r), enters, in this approximation. Once more, measurements of M3 inelastic excitations of collective states can provide considerable information on details of the collective motion in heavy nuclei. As in the previous case of electric multipoles, no magnetic multipole higher than &=3 is possible in the interacting boson model with S and D pairs only.
8. Microscopic calculations In the preceding sections, I have discussed the phenomenologicalaspects of the interacting boson model. These aspects are those which have been most extensively investigated. However, it is worthwhile mentioning that microscopic calculations are also possible5) and they are being presently performed. An outline of the microscopic calculations is the following. In order to construct boson densities, one first constructs the bosons as correlated fermion pairs 2 IS> = r CL. Ij ;J=O> 3 3
(8.1)
and (D> = j$, B.., Ijj';J=2> II
(8.2)
where the sums go over the active shells. The boson form factors are thenobtained by evaluating the graphs shown in Fig. 15. This evaluation requires, in addition to a knowledge of the expansion coefficients oj and b.*r also the single particle wave functions for the states j. These wave functionsJ&e taken, in first approximation, as single particle wave functions in a harmonic oscillator potential. More sophisticated calculations would assume wave functions calculated in a Woods-Saxon potential.
Fig. 15. Schematic representation of the coupling of the fermion pair states (8.1) and (8.2) to the external electromagnetic field.
lloc
F. IACHELLO
Since eleetron scattering provides directly the boson densities, it will give a very stringent test of the microscopic aspects of the interacting boson model. It will be interesting to see to what extent the simple assumption of bosons as correlated fermion in the valence shells, is able to describe the observed densities.
9. Conclusions I have presented here a framework within which electron scattering experiments on medium mass and heavy nuclei can be analyzed. In this framework, one constructs the spectrum in terms of some elementary modes of excitation (s- and d-bosons). From this point of view, the approach to nuclear spectra provided by the interacting boson model is very similar to the approach to hadronic spectra provided by the quark model, Fig. 16. Electron scattering experiments on heavy nuclei may thus provide information on the way in which these elementary (boson) constituents couple to the external electromagnetic field, Fig. 5, much in the same way in which electron scattering experiments provide information on the way in which quarks couple to the electromagnetic field. In addition, and independently from the interacting boson model, our knowledge of collective states in medium mass and heavy nuclei is, to a large extent, limited to static properties (energies, B(ER) and B(ML) values and static moments). A detailed study of properties of collective states as a function of momentum transfer, may reveal new and unexpected features. For example, it may clarify the role played by protons and neutrons in generating collective states. To this end, it is not sufficient to measure the elastic and inelastic excitation of the ground state band. In order to understand important aspects of collective states in nuclei, it is essential to measure cross sections for inelastic excitations of other states, in particular Cl;,25, 23, 4; and It, 3;. My discussion has been confined to positive parity states in even-even nuclei. An analysis similar to that presented here can be performed for collective states with negative parity in even-even nuclei, and for states in odd-even nuclei. Collective states with negative parity in even-even nuclei can be generated by introducing an octupole or f-boson'). This is, once more, similar to what occurs in hadron physics where, in order to generate charmed hadrons, it is necessary to introduce, another, charmed or c-quark, Fig. 17. For odd-even nuclei, one must NUCLEAR PHYSICS FW?llClE PHYSICS
I:rl~l~,
I-
0-
-I
i -d
J=2
-a
J=O
-8 -u,d
tI
1 BOSONS
T=O T=$- 0
1 QUARKS
Fig. 16. Building blocks of collective low-lying states in nuclei (s,d bosons) and of low-lying states in hadrons (u,d,s quarks).
ELECTRON
lllc
SCATTERING
NUCLEAR __- PHYSICS PARTICLE PHYSIC+ ---r---I I
I 21
I
,I,
-I-c
1
I
-d
J=2
-I
J=O
T=O
j I -r
-,,dT=@O
Fig. 17. New degrees of freedom of nuclei (f,g?,...bosons) and of hadrons (c,....quarks) at higher excitation energies
introduce the degree6 of freedom of the last, unpaired article. A model to ! ). A detailed study of discuss odd-even nuclei has been recently developed18-' collective states in odd-even heavy nuclei is another of many areas where electron scattering can provide new and interesting information.
References 1. J.W. Negele and G. Rinker, Phys. Rev. Cl5 (1977) 1499. E. Borie, D. Drechsel and K. Lesuo, Nux Phys. A211 (1973) 393. See for example, "Interacting Bosons in Nuclear Physics", F. Iachello ed. (Plenum Press, New York, 1979). 4. A. Arima, T. Otsuka, F. Iachello and I. Talmi, Phys. Lett. 66B (1977) 205. T. Otsuka, A. Arima, F. Iachello and I. Talmi, Phys. Lett. 76B (1978) 141. 5. T. Otsuka, Ph.D. Thesis, University of Tokyo, Japan, (1979); T. Otsuka, A. Arima and F. Iachello, Nucl. Phys. A309 (1978) I. 89B 6. F. Iachello, G. Puddu, 0. Scholten, A. Arima and T. Otsuka, Phys. Lett. (1979) I. 7. A.E.L. Dieperink and F. Iachello, to be published. a. A. Arima and F. Iachello, Ann. Phys. (N.Y.) 99 (1976) 253: A. Arima and F. Iachello, Ann. Phys. (N.Y.) El (1978) 201; A. Arima and F. Iachello, Ann. Phys. (N.Y.) 123 (1979) 468. 9. A. Bohr and B.R. Mottelson, "Nuclear Structus, vol. 1 (W.A. Benjamin, Inc., New York, 1969), p. 164. 10. J.N. Ginocchio and M.W. Kirson, Phys. Rev. Lett. 44 (1980) 1744; A.E.L. Dieperink, 0. Scholten and F. Iachello, PhF. Rev. Lett. 44 (1980) 1747. II. M.A. Moinester, G. Azuelos, J. Alster and A.E.L. Dieperink, to bFpublished. 12. G. Puddu, 0. Scholten and T. Otsuka, Nucl. Phys. A348 (1980) 109. 13. T.S. H. Lee, to be published. 14. A.E.L. Dieperink, F. Iachello, A. Rinat and C. Creswell, Phys. Lett. -76B (1978) 135; A.E.L. Dieperink, in "Interacting Bosons in Nuclear Physics", F. Iachello ed. (Plenum Press, New York, 1979), p.129. 15. J.W. Lightbody, Jr., in Proc. of the Int. Conf. on Photonuclear Reactions and 2. 3.
112c
16. 17. 18. 19. 20.
F.lACHELLO
Applications", Asilomar,l973, ed. B.C. Berman (Lawrence Livermore Laboratory, University of California, Livermore, 1973). A. Bohr and B.R. Mottelson, to be published. H.W. Kugel, R.R. Borchers and R. Kalish, Nucl. Phys. Al86 (1972) 513. F. Iachello and 0. Scholten, Phys. Rev. Lett. 43 (1979) 679. 0. Scholten, Ph. D. Thesis, University of GronKgen, The Netherlands (1980). F. Iachello, Nucl. Phys. A347 (1980) 51.