On the structure of intruder states

Volume 173, number 4

PHYSICS LETTERS B

19 June 1986

ON THE STRUCTURE OF INTRUDER STATES G.E. A R E N A S P E R I S a n d P. F E D E R M A N

1

Departamento de Ftsiea, FCEN, Ciudad Universitaria, Nilhez, 1428 Capital Federal, Argentina

Received 15 August 1985; revised manuscript received 10 March 1986

The structure of rather complex intruder states in medium and heavy nuclei near magic Z or N is described using a simple model. The neutron-proton interaction is instrumental in the lowering of such states towards the middle of the valence nucleon shell.

The coexistence of intruder core-excited and valence-nucleon states in the low-lying spectra of nuclei throughout the Periodic Table has recently been reviewed by Heyde and collaborators [1 ]. The data show a marked preference for the vicinity of magic numbers. The microscopic description of intruder states in medium and heavy nuclei becomes rather involved. As Talmi likes to point out, just the 12 valence neutrons of l l 2 S n give raise to shell m o d e l hamiltonian matrices whose dimensions are of order 105 [2]. Even if possible (which fortunately for now they are not), such calculations are hardly desireable. This is quite obvious when one considers the propagation of even small uncertainties in the values of the two-body matrix elements. In the case of valence neutrons (protons) the lowest core-excited states are obtained by raising one or two protons (neutrons) into the valence shell. F o r example, for l l 2 S n the intruder states are obtained b y raising two protons from the 100Sn core into the valenceneutrons shell, leaving a 98Cd core. The strong and attractive interaction between neutrons and protons in the same shell results in very coherent and even collective low-lying intruder states, sometimes low enough to become the ground state [3]. The interaction of the valence nucleons with cores close to magic numbers is mostly monopole in nature, and can hardly have any influence on deformation tendencies. In such cases the shape o f intruder states is rather determined by the

strong interaction between valence neutrons and protons. Far from magic numbers the cores themselves may deform and in such cases the quadrupole interaction between valence nucleons and core may indeed dominate the picture. The main virtue of our work is that we are able to incorporate most deforming interaction effects into experimental binding energies. Let us consider first two-proton excitations in e v e n - e v e n nuclei with magic proton number Z M and let BE* denote the total (binding) energy of the coreexcited state. Then BE*(ZM,N ) = BE(Z M - 2,NM) + e(Z M - 2,NM) + V,

(1) where N M is the underlying magic neutron number. We denote by e(Z M - 2,NM) the total single-particle energy of the two protons and n = N - N M valence neutrons in the (Z M - 2,NM) core, and the interaction among them b y V. Reliable microscopic calculations of V are quite problematical since they require accurate enough intruder-state eigenfunctions and two-body interaction matrix elements. Such difficulties are avoided here by assuming that the intruder wave function is not very different from the one for two protons and n valence neutrons in a (ZM,NM) core. We can then write BE(Z M + 2 , N ) = BE(ZM,NM) + e ( Z M , N M ) + V,

(2)

and therefore Fellow of the Consejo Nacional de lnvestigaciones Cientificas y T6cnicas, Argentina. 0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

359

Volume 173, number 4

PHYSICS LETTERS B

BE*(ZM,N) = BE(Z M - 2,NM) + BE(Z M + 2 , N )

Aejv --~ 2 Vmr(Jv, je),

- BE(Z M ,NM) - axe,

(3)

where

axe = e(ZM,NM) -- e(Z M - 2,NM).

(4)

Thus, for the excitation energies of the core-excited states we obtain

E(ZM, n) =---BE(Z M , N ) - BE*(ZM,N ) = axa + Ae, (5) where aXB = BE(Z M , N ) - BE(Z M + 2, N ) + BE(ZM, N M ) -

BE(Z M - 2,NM).

Note that aXB contains most deforming tendencies. Let us now consider aXe. Denoting the occupancy of orbit] by 4 , and its single-particle energy by e/, we can write

19June1986 Ae/ --~ 2 ~r(J'zr, jc),

where V~r(Jv, ]c) is the average interaction between a proton in the ]e orbit and a neutron in the Jv orbit, and V~r~(J~, ]e) is the average interaction between protons in orbits j,~ and ]c (including the Coulomb repulsion). We are assuming that the protons in the ]c orbit couple to zero total angular m o m e n t u m also in the (Z M - 2, N M) core. Of course any Ae can be calculated using some phenomenological interaction, but relevant experimentally determined matrix elements can be scarce. If we make the simplifying assumption that A % and A % do not depend, respectively, on]~ r andjv , Ae becomes a function of only two parameters - the above interactions averaged now also over]~r and ]v - and eq. (5) becomes

E(ZM,n ) = AB + Ae = AB + 4 V ~ + 2n Vw.

(6)

Ae = A% + Aev,

Eq. (6) can be used for example in the Z = 50 region, where effective matrix elements are available only for /_~(v) = 2d5/2 and/e =1g9/2, from which [4] we obtain Vv~ = 0.37 MeV and V~r~ = - 0 . 1 5 MeV. To this we must add a Coulomb interaction contribution of about [5] - 2 5 0 keV, obtaining a total VTrTr~ - 0 . 4 M e V . Although we above formulae were obtained for twoproton excitations in nuclei with magic Z, they can easily be generalized to nuclei with any Z close to magic. For odd Z we consider one-proton excitations, obtaining in this case

where

E(Z,n)= AB + ( Z M - Z + I) ffTr. + n ( Z M - Z + I) V~

e(ZM,NM) = ~ o2 (2]n + 1) e/Tr(ZM,NM) + ~ v?v(2] v + 1) e/(ZM,NM), where the sums extend over all the orbits ]~r and ]v available to the two protons and n neutrons, respectively. Using the above expression of aXe, we obtain

A%(v ) = &(,,) ~ vlvO,) .2 (2]~r(v)

+

1) ,5ei,,(~),

(7) for Z < ZM, where &B = B E ( Z , N ) - BE(Z M + 1 , N ) + BE(ZM, NM)

Ae/.(v ) = e/~(v)(ZM,NM) -- e/n(v)(Z M - 2,NM). -

Thus, given the occupancies v2 and the single-particle energies ej in both t h e ( Z M - 2,NM) and (Z M,ArM) cores we can calculate Ae. In the case of light nuclei there is usually at least partial information on the necessary experimental single-particle energies. But this is not the case for heavier regions, like Z = 50. In such cases we can still calculate aXe in the following way. Let Jc denote the last occupied proton orbit. Since close to magic numbers e v e n even nuclei are mostly coupled to jrr = 0 +, the resulting monopole interaction gives raise to 360

B E ( Z - 1,NM) ,

and

E ( Z , n ) = A B + ( Z - Z M + 1) V~r~ + n Vv,r + V~(/2; J = O (8)

for Z > ZM, where a B = BE(Z,N) - B E ( Z + 1,N) + BE(Z~ - 1,NM)

- BE(Z M - 2,NM). For even Z and two-proton excitations we obtain

Volume 173, number 4

PHYSICS LETTERS B

E ( Z , n) = A B + 2 ( Z M - Z + 2) V.~r + n(ZM - Z + 2) Vw for Z < ZM, where

(9)

AB = BE(Z,N) - BE(Z- 2,NM) + BE(ZM,NM) -

BE(Z M + 2,N),

and

E(Z,n)=AB+2(Z-ZM+2)V~r~r+2nVv~

r

(10)

for Z ~> ZM, w h e r e AB = BE(Z,N) - BE(Z M - 2,NM) + BE(ZM,NM) - BE(Z + 2,N). Expressions ( 7 ) - ( 1 0 ) were used to calculate the ex-

19 June 1986

citation energies o f one- and t w o - p r o t o n intruder states as a f u n c t i o n of the valence n e u t r o n n u m b e r for 47 ~< Z ~< 53. F o r V ~ ( j 2 ; j = 0 ) w e use the value Vn~r(lg92/2 ; J = 0) = 1.6 MeV obtained f r o m e x p e r i m e n t [4] after adding the C o u l o m b c o n t r i b u t i o n [5 ]. The results are c o m p a r e d w i t h e x p e r i m e n t in table 1. The calculated energies reproduce the characteristic m i n i m u m towards the middle o f the valence n e u t r o n shell e x h i b i t e d b y the e x p e r i m e n t a l data. The detailed agreements are excellent, in particular for the In, Sn and Sb isotopes. This is to be e x p e c t e d , since t h e y are the closest to Z = 50. In a c o m p l e t e l y similar m a n n e r we can treat n e u t r o n intruder states near magic N. We have p e r f o r m e d calculations for isotones in the w h o l e region 79 ~< N ~< 85. The parameter V~, is estimated t h r o u g h

Table 1 Excitation energies (in MeV) of intruder states in the region around Z = 50. Experimental data not available for Z = 52. n

Nuclei 97;nAg

98 +n¢~,t 48 ~

99*n, 49111

100+n~ 50 ~n

101 +n~h 51 ~

lo2+n~ 52l e

103+n l 53l

expt. a) talc.

expt. b) calc.

expt. c) calc.

expt. b) calc.

expt. b) calc.

calc.

expt. d) calc.

6 8

1.20

2.0 e) 1.5

3.75

4.5 4.3

10

1.06

1.1

2.60

3.6

12

0.78

0.73

2.65

3.0

14

0.49

0.64

2.17

3.02

16

0.76

3.12

18

0.76

3.05

20

0.94

3.10

22

0.7

3.12

1.40 1.13 1.32 1.10 1.22 1.10 1.13 1.02 1.08 0.94 1.05 0.88 1.05 0.94 1.32

1.5 1.4

3.49

1.1

3.4 3.3

1.8

3.0

1.4

3.5

0.9 0.6

0.91

2.88

2.6

1.46

1.1

3.5

0.79

2.90

2.81

1.38

1.25

3.3

0.35

0.8

0.79

3.02

2.96

1.16

1.33

3.12

0.31

0.47

0.84

2.58

2.83

0.97

1.17

3.2

0.43

0.42

1.00

2.86

0.95

1.11

3.31

0.64

0.70

0.97

2.97

1.34

1.27

3.30

0.94

0.61

1.01 1.1 1.33

3.20 3.53 4.08 4.63

1.81 2.26 2.71

1.49 1.76 2.07 2.4

3.63 4.01 4.44 4.95

1.04

24 26 28 30

0.6 1.0 1.4

3.3 3.8 4.5

0.79 1.04 1.32 1.67

a) Centroids of the strong L = 2 transitions seen in Pd(aHe, d) Ag reactions [6]. b) Centroids of the 0 + states populated in (3He, n) reactions [6 ]. c) j n = 5/2+ states observed in one-proton transfer reactions [ 1]. d) Band-head energy of the 9/2 + intruder state from (d, 3He) reactions [1]. e) Numbers with only one decimal have large uncertainties due to the use of extrapolated binding energies [7]. 361

Volume 173, number 4

PHYSICS LETTERS B

i [BE(145Eu)_ BE(144Sm ) _ BE(143Eu ) + BE(142Sm)] or similar relations for neighbouring nuclei, obtaining Vw = 0.27 -+ 0.02 MeV. For ff~, effective-interactions systematics suggest a value of about - 0.15 MeV as before, and estimating Vvv(j2 ; J = 0) as BE(Z, 82) +BE(Z, 80) - 2BE(Z, 81) for different even values of Z we obtain V ~ ( j 2 ; j = 0) = 1.72 MeV. The results (using V ~ = 0.28 MeV for all the nuclei in this region) are listed in table 2 for comparison with experiment. The detailed agreement is again excellent. No experimental information is available for even N. Intruder states have also been observed near Z = 82, but our m e t h o d requires of some modifications for N < 126 since some of the required binding energies are not available. The consideration of holes instead o f particles eliminates the difficulties and expressions similar to eqs. ( 7 ) - ( 1 0 ) can then be obtained. As a matter of fact, all the expressions for the excitation energies of proton intruder states valid in particular cases can be summarized in a general formula, valid for all the cases:

E ( Z , N ) = AB + Ae,

(11)

where

19 June 1986

A B = B E ( Z , N ) - BE(Z + Z M - Z R , N ) + BE(ZM,NM) - BE(ZR,NM) , A6 = (Z M - Z R ) [ ( Z Z R =Z M --AZ =Z-

AZ

Z R ) Vmr + ( N - NM) Vmr],

i f Z ~>ZM, ifZ
and A Z is the number of excited protons. A completely similar expression can easily be obtained for neutron intruder states. A stringent test to the validity of the present ideas is offered by the electromagnetic transition rates. Within the framework o f the present model the rates o f transitions between intruder states should be similar to the corresponding transitions in the nucleus with the doubly magic core, where the valence nucleons form the ground bands. Unfortunately the experimental information is too scant to permit a detailed comparison. In the Sn region, B(E2) rates are known for l l 6 S n for the transitions 23 -+ 03 (26 W.u.) and 23 -+ 0~ (32 W.u.) [9]. They compare well with the value of 31 W.u.

Table 2 Excitation energies (in MeV) of intruder states in the region around N = 82. Values in parenthesis correspond to tentative J 7r assignments. Z

N 79

s2TeN s4XeN s#BaN

ssCeN ¢,oNdN 6~SmN 64GdN

81

expt. a)

calc.

2.28 (1.04)

2.19

1.45

2.27 2.01 1.79 1.60 1.05

expt. a)

83 calc.

expt. b)

85 calc.

expt. b)

80

82

84

calc.

calc.

calc.

4.37

4.4 c)

1.91 2.06 1.64 1.53

4.32 4.09 3.73 3.42

4.04 3.76 3.43 d) 3.21

3.90 3.63 3.39 3.23

1.51 1.18

3.18

3.11 2.89

3.18 3.15

calc.

2.17 1.79 (1.84) 1.42 (1.81) 1.31 1.27 (1.31)

2.24 1.88 1.71 1.51 1.28 1.28

1.82 1.63 1.55 (1.93) (1.55)

1.8 1.84 1.66 1.57 1.55 1.38

1.52 1.54

a) Energies of the intruder 7/2- states observed in one-neutron transfer reactions [ 1]. b) As in footnote a) but for 3/2+ states [ 1 ]. c) Numbers with only one decimal have large uncertainties due to the use of extrapolated binding energies [7]. d) Experimental information is available in this case. Two independent measurements [8] give 3.23 MeV and 3.30 MeV from twoneutron transfer reactions. 362

Volume 173, number 4

PHYSICS LETTERS B

observed for the 2~ ~ 0~ transition in 120Te [10] (no data are available for llSTe). The two strong B(E2) values for l l 6 S n seem to indicate an almost 5 0 % - 5 0 % fragmentation of the correlated intruder state. Summing up, we have presented a simple but accurate description of the rather complex intruder states observed in medium and heavy nuclei near magic Z or N, throughout the Periodic Table. The detailed agreements with the experimental data are remarkable in view of the simplicity o f the model and lack of free parameters. Most deforming interaction effects are taken into account b y incorporating them into experimental binding energies. We are thus left to treat explicitly only the interaction between valence nucleons and core. This interaction is mostly monopole in nature and can hardly have any influence on deformation tendencies. These come mostly from the n e u t r o n - p r o t o n interaction among valence nucleons [ 11 ]. Far from magic N and Z the cores themselves may deform and in such a case no d o u b t the quadrupole interaction between valence nucleons and core m a y dominate the picture. In a recent letter [ 12] Heyde and collaborators treat the q u a d r u p o l e - q u a d r u p o l e n e u t r o n - p r o t o n interaction in the SU(3)limit of the IBM. The region of validity o f the present description is limited b y the relative inertness o f the cores involved in the expressions of Ae contained in ( 7 ) - ( 1 0 ) , and could probably be extended b y allowing some kind of dependence of the parameters with the valence nucleon number.

19 June 1986

We would like to thank K. Heyde and S. Pittel for very helpful comments.

References [1] K. Heyde, P. Van Isacker, M. Waroquier, J.L. Wood and R.A. Meyer, Phys. Rep. 102 (1983) 291. [2] I. Talmi, Scuota lnternazionale di Fisica "Enrico Fermi". Corso LXXIX (North-Holland, Amsterdam, 1981) p. 172. [3] W.F. Piel Jr., P. Chowdhury, U. Garg, M.A. Quader, P.M. Stwertka, S. Wajda and D.B. Fossan, Phys. Rev. C31 (1985) 456. [4] J.P. Schiffer and W.W. True, Rev. Mod. Phys. 48 (1976) 191. [5] G. Wenes, P. Van Isacker, M. Waroquier, K. Heyde and J. Van Maldeghem, Phys. Rev. C23 (1981) 2291. [6] H.W. Fielding, R.E. Anderson, C.D. Zafiratos, D.A. Lind, F.E. Cecil, H.H. Wieman and W.P. Alford, Nucl. Phys. A281 (1977) 389. [7] G.T. Garvey, W.J. Gerace, R.L. Jaffe, I. Talmi and I. Kelson, Rev. Mod. Phys. 41 (1969) 51. [8] T.J. Mulligan, E.R. Flyrm, O. Hansen, R.F. Casten and R.K. Sheline, Phys. Rev. C6 (1972) 1802. J.D. Sherman, D.L. Hendrie and M.S. Zisman, Phys. Rev. C15 (1977) 903. [9] A. Biicklin, N.G. Jonsson, R. Julin, J. Kantele, M. Luontama, A. Passoja and T. Poikolainen, Nucl. Phys. A351 (1981) 490. [ 10] P.H. Stelson and L. Grodzins, Nucl. Data Tables AI (1965) 21. [11] I. Talmi, Rev. Mod. Phys. 34 (1962) 704. [12] K. Heyde, P. Van Isacker, R.F. Casten and J.L. Wood, Phys. Lett. B155 (1985) 303.

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