MICROCOPY RESOl UTION TEST CHART NATIONAL BUREAU OF STANDARDS STANDARD REFERENCE MATERIAL 1010a •SSWW8M333 £*JfeA ~ RT/T1B/86/3
G. MAINO, A. VENTURA, L. ZUFFI, P. VAN ISACKER
GAMMA-RAY ABSORPTION AND SCATTERING IN TRANSITIONAL REGIONS OF THE INTERACTING BOSON MODEL
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COMITATO NAZIONALE PER LA RICERCA E PER LO SVILUPPO DELL'ENERGIA NUCLEARE E DELLE ENERGIE ALTERNATIVE ISSNX>3»»«333
COMITATO NAZIONALE PER LA RICERCA E PER LO SVILUPPO DELL'ENERGIA NUCLEARE E DELLE ENERGIE ALTERNATIVE
GAMMA-RAY ABSORPTION AND SCATTERING IN TRANSITIONAL REGIONS OF THE INTERACTING BOSON MODEL
G. Maino, A. Ventura ENEA-Dipartimento Tecnologie Intersettoriali di Base, Centro ricerche energia «Ezio Clementel», Bologna
L. Zuffi Dipartimento di Fisica dell'Università di Milano-INFN Sezione di Milano
P. Van Isacker School of Mathematical and Physical Sciences University of Sussex, Brighton BN1 9QH (England)
RT/TIB/86/3 Testo pervenuto net febbraio 1986
Reprinted from the Proceedingn of the Fourth International Conference on Nuclear Reaction Mechanisms, Varenna (Italy), Junf 198L, edited by E. Gadioli, Milano (198?.), pp.423-432.
I contenuti lecnico-wienli'ki del rapporti tecnici dell'Knca rispecchiano l'opinione degli autori e non necessariamente quelh dell'ente 3 K
SUMMARY
The present work describes, in the frame of the interacting boson model, the processes of absorption, elastic and inelastic scattering of 8-20 MeV photons from even neodymium and osmium isotopes, corresponding to the U(5)—*- SU(3) and SU(3) —*• 0(6) transitional regions of the model.
RIASSUNTO
Il presente lavoro descrive, nel quadro del modello a bosoni interagenti, i processi di assorbimento e diffusione elastica ed inelastica di fotoni di energia fra 8 e 20 MeV da catene di isotopi pari del neodimio e dell'osmio, corrispondenti, rispettivamente, alle transizioni U(5) -> SU(3) e SU(3) -*• 0(6) fra le simmetrie limiti del modello. 5
GAMMA-RAY ABSORPTION AND SCATTERING IN TRANSITIONAL REGIONS
OF THE INTERACTING BOSON MODEL
G. Maino and A. Ventura
ENEA, Divisione Fisica e Calcolo Scientifico 1-40138 Bologna (Italy)
L. Zuffi
Dipartiaento di Fisica dell'Università di Milano and INFN, Sezione di Milano 1-20133 Milano (Italy)
P. Van Isacker
Charles de Kerchovelaan 3 B-9000 Gent (Belgiua)
1. INTRODUCTION
After many successful applications to low-lying collective levels and related electromagnetic transitions of non-magic medium-heavy nuclei, the interacting boson model (IBM) has in recent years proved to be suitable for the description of high-lying collective 2 3 4 states as well, in particular giant dipole resonances (GDR) ' ' .
Calculations of gamma-ray «osorption and scattering from nuclei through the excitation of GDR were first carried out at the limit 3 4 < symmetries of the model, in particular the SU(3) limit ' , describing axisymmetric rotors, then extended through the diagonalization of
• * Address after October 1, 1985:
School of Mathematical and Physical Sciences
University of Sussex
Brighton BN1 9QH (England) 6 a more general IBM Hamiltonian to transitional nuclei in the lanthanide 5,6 region . The purpose of the present paper is to show r.ew computational results for nuclei in the U(S)- SU(3) and SU(3) - 0(6) transitions of the model, with special emphasis on inelastic scattering, where good agreement with experimental data had already been obtained 3,7 for well-deformed nuclei
2. THE MODEL
A simultaneous description of both low-lying positive-parity and giant dipole states in an even-even nucleus can be given in terms of coupled s,d (L =0 ,2 ) and p (L =1 ) bosons.
Following ref. 3, we choose basis states, ]yj>, of the form:
|>|»>= (S+f (d+MpV |0> . (1)
Here, m+nsN , the usual boson number, and q=0, or 1.
While s and d bosons represent pairs of like particles, or holes in the valence shell, the p boson is here considered a collective one-particle one-hole excitation across a major shell closure. A different viewpoint had been taken in refs. 2,5,7, where m+n+q=N .
The two parametrizations, which correspond to different microscopic interpretations of the dipole states, will be compared in a more extended paper.
The complete IBM Hamiltonian can be written in the form:
A * A A H = H(s,d) • H(p) • H(s,p,d). (2)
A Here, H(s,d) is the usual s-d boson Hamiltonian in the multipole expansion form:
H(s,d) -1 n. •aJ*!p)*«JL.L)+a0(Q.Q)*ajr.T Wa (T .T ), (3) ddv » i. J 3 3 4 4 4 7 where the multipole operators are defined as in ref. 1. A H(») is the free p-boson Hani1tonian:
H(p) « L", ; (4)
A H(s,p,d) couples s, d and p bosons:
H(s,p,d) « b (d xd) .(p xp) • b (d xd) .(p xp) o 1 (5) • b [(s xd +d xs) +%{à xd> J.(p xp)
The spherical tensors and their scalar products in formulae (3-5) have the usual definition and the boson annihilators are a* S (-1) 'a , where L=0,l,2 for s,p,d bosons, respectively. The electric dipole transition between a high-lying l" and a low-lying 0* , or 2* state is given by the matrix elements of the operator :
£(1) _ , + ~. ,,. D = D (p +p) (6)
The decay width, P , of the giant dipole states cannot be evaluated within the framework of the model, but is taken to be energy- dependent as follows:
T(E) ««(E^ , (7) where oL and (i are to be adjusted on experimental data. The numerical diagonalization of the coupling Hamiltonian (5) in the basis (1), with q«l, provides us with the energies, E , of n the giant dipole states, which are to be inserted, together with the reduced matrix eleaents of the operator (6) and the decay widths (7), 8 into the formulae of nuclear polarizabilities , P L 8
V*LW—^^ —* f<4iiScl,iii><»;ii5cl)«o;>. L ^ [3(2Lfl)]1/2 (*c)2 " ° " *
L ? (8) f 1 (-1) . t r J5(2e) Xl ,- 6 E*E«*ir/2* E -E-tr/2 °Lo LLf m 2 * n n n n * ABMC
Here, L is the nuclear angular momentum, Lf the angular momentum of the final state, E(E') the incident (scattered) photon energy, and |l ^ the n dipole state at energy E , characterized by a n n width P . The last tern on the right-hand-side of (8) is the Thoason amplitude, with rn^. $he atomic mass unit. Formula (8) is valid for photon wavelengths greater than the nuclear radius, a condition satisfied by the photon energy range and the nuclear radii considered in this work.
The absorption and scattering cross sections for unpolarized photons are expressed in terms of P :
Or , ^ ^ I«P (9) a /a E o which corresponds to the optical theorem, and:
|p , cw S§ ».M*4>-r Aif «- . cio) where & is the scattering angle. The functions g of interest here, with L • 0,2, are written as:
g (9) - \iUco92*) ; gj&) « T;(13 • cos2fr). (11) O D 2 XZ
The incident photon energy, E, and the scattering angle,
&, are so chosen that Raylefgh and Delbrtick contributions to elastic scattering are negligible.
The computational procedure assumes that the parameters of the t-d boson Haailtonian (3) are known, having already been adjusted 9 on the low-energy spectrum and related electromagnetic transitions.
The p-boson energy, £ , roughly follows the empirical rule: P
-1/3 t (MeV) = 77.5 A . (12) P
The leading term of the coupling Hamiltonian (5) is the
quadrupole-quadrupole interaction responsible for the CDR splitting:
its strength, b , is adjusted on the experimental photoabsorption
to reproduce the main characteristics of the cross section. The monopole-monopole interaction, with strength b in formula (5), has o the main effect of shifting the centroid of energy. The dipole-dipole
term, with strength b , has been kept equal to zero for simplicity
and the V parameter of the s-d quadrupole operator in (5) is already
contained in the Q operator in (3). Finally, the D0 coefficient of the dipole operator (6) is taken as a normalization constant to
reproduce the experimental photoabsorption peak. In this way the
elastic and inelastic scattering cross sections are evaluated without
adjustable parameters. It is, therefore, scattering, rather than
photoabsorption, that can provide a good test of validity of the model.
3. RESULTS AND COMMENTS
Gamma-ray absorption and scattering through QOR excitation
have been evaluated for neodymium and osmium isotopes, as examples
of transitional regions around A « 150 and A « 190, respectively.
The neodymium chain can be described in the frame of the model 144 as • U(5)^SU(3) transition: Nd is, in fact, an enharmonic vibrator
and Nd an axial rotor.
The osmium chain is an example of SU(3)-* 0(6) transition, from 186 192 axial rotors ( 0s) to Jf-soft nuclei ( 0s).
The IBM parameters for the nuclei at the beginning and end
of both chains are given in Table I. The p-boson parameters are 10
TABLE I
IBM parameters for Nd and Os isotopes
144 150. 186. 192 Nd Nd Os Os
N 6 9 11 8
£ (MeV) 0.6163 0.3478 d 0.0 0.0
a (MeV) 0.08326 0.0131 o 0.0 0.0
a1(MeV) 0.0050 0.0006 0.0080 0.0125
a2(MeV) -0.01155 -0.0161 -0.0340 -0.0360
a3(MeV) 0.0500 0.0144 0.0 0.0
a (MeV) 0.100 0.0398 0.0 O.O 4 X -0.60 -1.10 -0.29 -0.13
£ (MeV) 14.84 14.59 13.60 13.40 P
b (MeV) 0.40 0.40 0.30 0.30 o
b^MeV) 0.0 0.0 0.0 0.0
b2(MeV) 0.65 0.65 0.35 0.35
OttMeV1"^) 0.06 0.06 0.08 0.09 P 1.60 1.60 1.40 1.42 D (fra) 7.94 8.15 o 8.92 9.36
The IBM parameters in the s-d boson Hamiltonian of formula (3)
are taken from ref.5 for Nd and ref.9 for Os isotopes. 11 practically constant within each chain, £ excepted, it following P the law (12). The coefficient b is smaller in the higher-mass region and 7 would be further reduced for heavier nuclei, like actinides .
Calculated absorption and scattering cross sections are shown in Figs. 1 and 2 for neodymium and osmium, respectively.
In the neodymium case the transition from a single to a two- humped shape of the absorption cross section is reproduced, although 150 the calculated first hump in Nd is somewhat flatter than the 10 experimental result
As for elastic and inelastic scattering, a comparison with data is not possible. However, the ratio of the excitation functions for the 2 state and for the ground state, which increases with nuclear deformation, could be measured with good accuracy with existing facilities, thus providing a test of a transitional region of the model, which has already been shown to work well for deformed nuclei i• 3'7
The shape transition is less evident in the absorption cross sections of the osmium isotopes, but the experimental data are well reproduced.
The low-lying levels of osmium nuclei and their coupling to
giant dipole states are here described by a broken 0(6) symmetry.
A more detailed discussion of the present results, as well as the study of other transitional isotope chains are left to a more extended paper.
ACKNOWLEDGEMENTS
We are grateful to Prof. F. Iachello for valuable comments,
Prof. R.F. Casten for the s-d boson parameters of the osmium isoto pes and Dr. S.S. Dietrich for the photoabsorption data of the same nuclei. 12
REFERENCES
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10 i. r—— T- ••i-i r *Nd e-utf
10 12 U 16 10 12 14 I» It ?0 Ey(MeV) Ev (MéV)
900 *>v *°Nd 0.140°
.-4 10 • " > i i i i i X) 12 14 « 10 12 14 16 IS 20 E (MeV) Ey(MeV) r
rig. 1. Photon absorption and scattering by 144 ' 150Nd . Absorption data are from ref. 10; the straight-line segments represent the dipole stren gths, S -tó"l\D(1,ll ot>|2 in arbitrary units. The scattering n n " l ' cross sections, evaluated at a scattering angle &• 140°, are label- led with the spin and parity of the final nuclear state. 14
K) 12 14 16 EytMeV) Ey(MeV)
500 10° —"" ' ' ^Nk 'l'I'. : 192- ,92 450 0s /\ ~ o i;-o: : : e-140 f 1 : 400 a> \ _i ^ 101 r ^ ^ - .550 • JS ^\ : $ 2 300 1 // v' < E ;~* .-.-2 / w ?b0 t-r- 10 T • / , - - « ~ a r A • . i . i i / i i i i i t \ .1 i . i X K*IV 0 X) 12 14 16 11 10 12 14 16 18 2 0 E (MeV) E,,(M <3V ) y 186 192 Pig. 2. Photon absorption and scattering by ' Os. Absorption data from ref. 11. Dipole strengths and scattering cross sections are evalua ted as in Pig. 1. Edito dall'ENEA, Direzione Centrale Relazioni. Viale Regina Margherita !25, Roma. Finito di stampare nell'aprile 1986 Fotoriproduzione e Slampa Arti (irafichc S. Marcello - Roma