Volume 43B, number 1 PHYSICS LETTERS 8 January 1973

G E N E R A L I Z E D N E U T R O N P A R T I C L E - H O L E S T A T E S

I N A N E X T E N D E D U N I F I E D - M O D E L D E S C R I P T I O N

K. HEYDE Fysisch Laboratorium, Ri]ksuniversiteit Utrecht, Netherlands and

lnsttute for Nuclear Physics, Ri]ksuniversiteit Gent, Belgium

M. WAROQUIER* and G. Vanden BERGHE Institute for Nuclear Physics, Rifksuniversiteit Gent, Belgium

P.J. BRUSSAARD Fysisch Laboratorium, Rifksuniversiteit Utrecht, Netherlands

Received 13 November 1972

Negative-parity levels for the even-even N = 82 nuclei, in the energy region 3.5 MeV <E x < 5.5 MeV, are described in an extended unified model, in terms of neutron hole states coupled to the low-lying states of the N = 83 nuclei. These states are selectively fed in the inelastic proton scattering through isobaric analogue resonances of the low-lying N = 83 levels. Application is made to the energy spectrum calculation of 14°Ce and the partial decay widths to levels in 14°Ce.

The study of the excited states in magic nuclei is of special interest to the theory of nuclear structure. The states in the N = 81 and N = 83 nuclei mat are observed in elastic proton scattering through the isobaric analogue resonances [ 1 -7 ] and neutron-transfer reactions [ 8 - 1 2 ] , have been shown to agree well with the assumption of a shell closure at N = 82. It has been possible to give a unified-model description for most of these N = 81 and N = 83 nuclei [ 13], in particular as regards the spin sequence of the low-lying states. Thus one is led to existence of rather pure neutron particle-hole (p-h) states in the even-even N = 82 nuclei, which also have been observed through (p, p')-reaction work [ 1 4 - 1 7 ] , feeding negative parity levels.

Assuming a unified-model description for the low-lying levels in the N = 83 nuclei, one can describe each state as

® + IN = 83, z ;Jf~Mj) = c#(J, 00; J') a+lMj (v)(O) + ~ c(3q, NR ;J') [~+ (v) f2f¢ R ]jMjIO) ( I ) j,N ,R

where 10") represents the vacuum state given by the physical groundstate of the N = 82, Z nucleus, alto (v) creates a neutron in the single-particle state ], m and I2~RMR creates an N-phonon state with angular momentum R and projection M R . The expansion coefficients c# (j, NR;J) are obtained after diagonalization of the particle-core in- teraction for states of sp inJ in the N = 83 nuclei. Then the isobaric analogue states of these low-lying levels can be written as

IN= 82, Z + 1;Jt3Mj) A -- (2T 0 + 1) -'/2 (cO(J, 00;J') a~My (;r)10> + ~ c¢(j, NR;J) [af (zO®I2fVR ]JMs(O) ( ],N ;~ I ,R

+ ~ { ( 2 / + 1)/(2.J + 1)} '/2 ( _ l ) f +J-I [a]+,(n) ~ [~'/,(v) *IN = 83, Z;J{J)]I]JMj ] , (2) ]',I ]

where ] has to be summed over the angular momenta of the shell-model states above the N = 82 closed shell and ] ' over the levels below the 82 closed shell; the isospin of the N = 82, Z core is denoted by T O and the time-reversed

* Aspirant of the "Nationaal Fonds voor Wetenschappelijk Onderzoek".

Volume 43B, number 1 PHYSICS LETTERS 8 January 1973

IdeV

I t.-

3=--

2-

0" ~$ 3-

T

2" ~£-

3" 5"

4" 1

3-

3!

58 - - 8 2

THEORY EXPERIMENT

(2,3)- (2,31-

(2, 3)-

(2,3)"

1- 2"

2- 3"

1- 0"

I - 2"

/~-

2"

3-

5"

4"

Fig. 1. The negative-parity levels in 14°Ce as calculated in an extended unified-model and compared with the experimental data [ 16]

annihilation operator "a/m = ( - IY '+m ai- m is used. The three different components in eq. (2) correspond to different decay modes of the isobaric analogue resonan-

ces to levels in the final even-evenN= 82,Z nucleus. The first term described decay in the proton elastic channel by emission of a proton in the single-particle orbit J,Mj. The partial decay width is given by

~jL = F~p.(2T0 + 1-1 Ic#(J, 00;J) l 2 , (3)

where r~P" denotes the width of the single-particle decay of a hypothetic~., resonance of pure single-neutron char- acter [16]. The second term describes decay to phonon excitations ~2~v R 10} of the N = 82, Z core nucleus, which can be identified d - a t east for the one-quadrupole j~r = 27 s ta te -wi th pure proton excitations [ 18]. Decay to the

+ highly collective 2 r = 21 state gives a partial decay width

lffj~l'(2~ ) = ~ . 1~/p" (2T 0 + 1) -11caq, 12;J)12. (4) 1

Volume 43B, number 1 PHYSICS LETTERS 8 January 1973

MeV

6.0

EXPERIMENT

5O

4.0

3.0 '2 keY " I - -

7/2•t ) 3/2"(11

THEORY

_ J

m i

-- '=" k e V ~

EXPERIMENT THEORY

i

EXPERIMENT

|/iO THEORY

m

EXPERIMENT THEORY EXP THEORY

m

~ k t~ .,,m-- -- . , -- kmV - ,D-- .--,~. I~/ . , , , - -

Fig. 2. Partial decay widths from the isobaric analogue resonances excited in the 14°Ce(p, p') 14°Ce*teaction as calculated in an extended unified-model and compared with the experimental data [16, 21].

The experimental inelastic (p, p')2"~ data [19] give support to the particle-core coupling model [20], in which _ +

the lowest single-neutron excitations are coupled to the j~r _ 21 quadrupole vibration of the core. Experiments at Heidelberg [16, 17], the University of Texas and ANL [13, 14] also showed a quite selective feeding of higher- lying (3.5 MeV < E x < 5.5 MeV), negative-parity levels in the inelastic proton scattering experiments. Neutron hole excitations should be present in these final states, which can be reached from the third component of eq. (2). If one assumes these states to consist of neutron-holes coupled to the low-lying levels of the N= 83 nuclei (such states are called generalized neutron particle-hole states [ 16] : GNP-H), the final states in the even-even N= 82, Z nuclei can be expanded in that basis as

bV=82, Z;IMoO= ~ c~ "' a (/ ,J~;O [a'r ( v ) ~ = 8 3 , Z ; S ~ l / M j'J,#

and the third component of eq. (2) can be rewritten as

(2T 0 + 1) - 3 ~ {(2/+ 1)/(2/+ 1)} ~ (-1)i '+Y-l~q' ,J[3;1) [a;, (r t )®lN=82,Z;Ia) l jMj . j'd,a

The partial decay width to a final level lot becomes

pijj#el.(/,-,) = ~ {(2/+ 1)/(2,/+ 1)(2T 0 + 1)} 1"~; p" [ a a ( f , J [ 3 ; 1 ) [ 2 . ]'

(5)

(6)

(7)

Volume 43B, number 1 PHYSICS LETTERS 8 January 1973

Through the measurement of angular distributions, the emission of d- and s-wave protons can be distinguised and O~ d

even the separate amplitudes a ( j , J/3;/) can be derived [21 ]. In order to derive the wave functions of the final states in the even-even N = 82 nuclei, one can construct a model

Hamiltionian, describing the coupling of neutron hole states to the lowest eigenstates of the N = 83, Z nuclei as

H=Hcore +Hp + H h + Hp.core + Hh_core + Vp. h . (8)

Here Hcore describes the collective excitations of the N = 82, Z core, Hp(Hh) the neutron single-particle (-hole) states, Hp.core(Hh.core ) the residual interaction of the particle (-hole) motion with the surface vibrations of the core and Vp. h the residual two-body particle-hole interaction. In the zero-order basis the neutron-hole states are coupled to a representation which diagonalises a part of the total Hamiltonian, i.e. the N = 83 part, and one ob- tains the basis states [/ffl, j ; IM) such that

(Hcore + Hp + Hp.core ) ~/h 1 , J/3; IM) = ~jt31jff 1 , J(3; IM).

In this basis of GNP-H states the final secular problem becomes

auqh,j[3;1) {(/;-1, j ' ~ ' ; i[Hh.core + Vp.h[]hl ,J(J;I) + 5]h],h~jj,5[3#,(6oj# + "eih ) } = E(Iot) aa(Jh,S [ j . . . . ;1).

(9)

(lO)

By the use of eq, (5) the negative-parity eigenstates of the even-even N = 82 nuclei are expanded in the GNP-H basis. As neutron particle and-hole states the orbits 2f7/2 , 2f5/2, 3P3/2, 3Pl/2, lh9/2 and 3s]-/~, 2d~/~, 2dg/1, lg~-/1,

respectively, are used. Up to three quadrupole phonons are taken into account. As neutron p-h interaction the Sussex matrix elements have been evaluated for the configuration space given. The value of the lowest unperturbed GNP-H energy separation is all even-evenN = 82, Z nuclei, equal to 6o7/2 - + ~2d-1 , was taken as Bn(N = 82, Z) -

• . 1 2

Bn(N= 81, Z), where Bn(N, Z) denotes the neutron separatton energy m the nuc~e/t]s (N,Z) obtained from the mass ta, bles of Wapstra and Gove [22]. In the application to 14°Ce all low-lying levels in 141Ce up to E x ~ 1.8 MeV that possess well determined spins, are considered. Results, as well for the energy spectra as for the partial widths for decay from the 7/21 , 3/21 , 1/21 , 5/21 and 9/21, 2 resonances, are given in figs. 1 and 2.

From the calculated wave functions one may conclude that the lowest 4 - and 5 - levels are rather pure GNP-H states 12d~/1,J,~ 7/21), whereas the 1 - , 2 - and 3 - states possess a mixed character, a fact that is fairly well reproduc- ed in the partial decay widths. This mixing results in a feeding of particular levels in 14°Ce from several resonances. A more detailed account of the formalism and calculations for the other even-even N = 82 nuclei will be published elsewhere.

The authors are grateful to Dr. A. Heusler and Dr. J.P. Wurm for correspondence and illuminating discussions on the GNP-H structure. One of the authors (K.H.) is grateful for a NATO grant which made his stay at Utrecht possible.

References

[1] P.A. Moore et al., Phys. Rev. 180 (1969) 1213, [2] N. Williams et al., Phys. Rev. C2 (1970) 1539. [3] N. Marquadt et al., Nucl. Phys. A177 (1971) 33. [4] P. Von Brentano et al., Phys. Lett. 17 (1965) 124. [5] L. Veeser, J. Ellis and W. Haeberli, Phys. Rev. Lett. 18 (1967) 1063. [6] E. Grosse et al., Nucl. Phys. A142 (1970) 345. [71 G. Clausnitzer et al., Nucl. Phys. A106 (1968) 99. [8] P.A. Moore et al., Phys. Rev. 175 (1968) 1516. [9] D. Von Ehrenstein et al., Phys. Rev. C1 (1970) 2066.

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Volume 43B, number 1 PHYSICS LETTERS 8 January 1973

[10] C.A. Wiedner et al., Nucl. Phys. A103 (1967) 433. [11] P.R. Christensen et al., Nucl. Phys. A102 (1967) 481. 12] R.K. Jolly and C.F. Moore, Phys. Rev. 145 (1966) 918. 13] G. Vanden Berghe, K. Heyde and M. Waroquier, Nucl. Phys. A165 (1971) 662. 14] P.A. Moore et al., Phys. Rev. Lett. 22 (1969) 356. 15] P.A. Moore et al., Phys. Rev. C1 (1970) 1100. 16] J.P. Wurm, A. Heusler and P. Von Brentano, Nucl. Phys. A128 (1969) 433. 17] K. Mudersbach, A. Heusler and J.P. Wurm, NucL Phys. A146 (1970) 477. 18] M. Waroquier and K. Heyde, Nucl. Phys. A164 (1971) 113. 19] H. Clement, G. Graw, W. Kretschmer and P. Schulze-Dobold, Phys. Rev. Lett. 27 (1971) 526. 20] G. Vanden Berghe, K. Heyde and M. Waroquier, Phys. Lett. 38B (1972) 467. 21] A. Heusler, H.L. Harney and J.P. Wurm, Nucl. Phys. A135 (1969) 591. 22] A.H. Wapstra and N.B. Gove, Nuclear Data Tables 9 (1971) 303.