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Proton 1h112 excitations in odd-mass N = 82 nuclei
Nuclear Physics A438 (1985) 141-156 a North-Holland Publishing Company
PROTON lh, 1,2 EXCITATIONS IN ODD-MASS N = 82 NUCLEI W. ENGHARDT Zenrralinstitut ,fir Kernforschung,
and H. U. JAGER Rossendork DDR-805I
Dresden, GDR
Received 3 August 1984 Abstract: The structure of negative-parity states in odd-mass N = 82 isotones (135 $ A s 145) is investigated in the framework of a model which is based on coupling one Ihi,,* proton to ?I = + 1 states of the appropriate doubly-even isotonic core. It is shown that an effective modelspace truncation can be achieved if only core states in the vicinity of the yrast line are taken into account. These have been selected from the several hundreds or even thousands of states obtained from shell-model calculations, where the lg,,,, 2d,,,, 2d,,, and 3s,,, single-particle states are assumed to be occupied. Spectroscopic data are calculated and predictions on the structure of low-lying A = - 1 states are discussed and compared as far as possible to experimental findings. Thereby, the particle-core coupling approach is shown to be capable of describing essential properties of negative-parity levels in the N = 82 nuclei considered.
1. Introduction A large amount of the data reported on positive-parity states in the semimagic N = 82 nuclei (with 50 < Z < 64) can be explained in terms of proton wave functions of the spherical shell model le4) . It has been found that the configurations (lg,, 2d,)Z- 50, where the valence protons are distributed over the nearly degenerated shells lg; and 2d,, usually dominate in the wave functions. Only a few low-lying levels, e.g. the $: and 4: states of 143Pm or the i: and 1+ states of 14iPr 21 should arise from one-proton excitations (lg,, 2d,)Z-51(2d,, 3~~)’ to the higherlying shells 2d, or 3s+ (for the order of the shells, see table 1). The structure of the negative-parity states of N = 82 nuclei is not so well understood, since in the odd-proton nuclei only y?- isomeric states were known in 13’Cs, 139La, 141Pr, 143Pm, and 145E~ for many years 5). They are interpreted as proton excitations to the lh, shell which should lie near to, or even below, the 2d, and 3s+ shells. The 9; states were the only rz = - 1 states taken into consideration one decade ago in the structure calculations6-9), in which the hamiltonian was diagonalized in a one- and three-quasiparticle space. More recent investigations on 141Pr [refs. 3, lo)], ‘43Pm [refs.4p lo)], and 145E~ [refs. l1 - ‘“)I revealed a variety of further odd-parity states in the energy region up to E, ‘v 4.5 MeV (fig. 1). All these levels have spin J 2 9. The 145E~ states have been discussed ” - ’ 3, in ’ t erms of weak coupling of lg, and 2d, proton holes 141
142
W. Enghardt, H. U. Jiiger / Proton Ih,,,,
excitations
to J” = 3;, 5;, 7; states of 146Gd and a th, proton to ‘44Sm states. Kaczarowski et al. 15) interpreted them as members of the (lh,, 2d;‘), and (lh,, lg;‘, 2di’), multiplets, and this interpretation was supported by the comparison of experimental and calculated E2 transition rates and branching ratios. We performed 3, similar shell-model calculations for the negative-parity states of 141Pr and 143Pm in the framework of the space, (lg,, 2d,)Z- ” (lh,)‘, we were capable of handling. To achieve a consistent description of excitations of both parities in N = 82 nuclei, one should extend at least the shell-model space of ref. ‘) by taking into account the one-proton excitations to the lh, shell. But even these configurations, (lg,, 2dt)Z-51(lhy)1 and (lg,, 2d,)Z-52(2d,, 3st)‘(lhY)‘, are rather difficult to handle. In the case of 14iPr they contain, for example, 2202 states with J” = y-. The present paper deals with a reasonable truncation of this space. We construct a basis of odd-parity states of an N = 82 nucleus with mass number A by coupling a lh, proton to II = + 1 shell-model states of the core nucleus (A - 1; N = 82) but only take into account a restricted number of about 30 core states in the vicinity of the yrast line. The corresponding particle-core coupling formalism, see also ref. 16), is outlined in sect. 2. The efficiency of our space truncation is demonstrated for 145E~ (sect. 3). We calculate excitation energies, wave functions and electromagnetic properties of negative-parity states in the odd-mass isotones from 1351 to 145E~ and compare them with available data (sect. 4). Other approaches to investigate lh, excitations in N = 82 nuclei have been briefly reported. Kruse and Wildenthal 17) made shell-model calculations for the nuclei from 133Sb to 14sDy where they restricted the basis to states with seniority less than 4. For the even-mass nuclei, Scholten and Kruse ‘*) compared results of that work with their investigations performed in the generalized seniority scheme.
2. Coupling of a nucleon in an intruder orbit to shell-model core states 2.1. ENERGY
SPECTRA AND WAVE FUNCTIONS
Let us first introduce the shell-model states )(A - 1); E’J’M’) for the core nucleus with (A - 1) nucleons. They are characterized by the energy E’ and the quantum numbers J’ and M’ of the total angular momentum and its third component, respectively. All these states are supposed to have the same parity 7~‘.There is no need for specifying isospin quantum numbers because the model is applied only to nucleon excitations of one kind in semimagic nuclei. We add a nucleon to a shell (Nlj) which is empty in all core states and couple the angular momentum of this extra-shell nucleon to that of the core state : IA ; E’J’j ; JM)
=
1 (J’M’jm(JM)c~l(A M’m
- 1); EIJIMI).
(1)
W. Enghardt,
H. U. Jiiger
/ Proton
Ih,,,,
143
exciraiions
Here the symbol c+ denotes the Fermi creation operator. The states (1) form an orthonormal basis which is used to expand the wave functions (A;EJM)
= c aE,:‘,.(A ; E’J’j ; J M)
(2)
E’J’
of parity 7~= rc’(-)’ for the system with A nucleons. The level energies E and the expansion coefficients a::‘,, are determined by solving the Schrodinger equation (H, + H’)IA ; EJM)
= EIA ; EJM).
(3)
Our basis states (1) are eigenstates of the hamiltonian Ho involving the effective nucleon-nucleon interaction within the core and the single-particle energy Ed of the nucleon added: H,IA;E’J’j;JM)
= (E’+E~)(A;E’J’~;JM).
(4)
The term H’ in eq. (3) stands for the residual interaction of the extra nucleon with the nucleons active in the core states. In the shell-model technique 19), its matrix elements are evaluated using products of (one-particle) coefficients of fractional parentage. They describe the change of the core states if a nucleon inside the core is scattered by the interaction from a single-particle state k to a state i. We calculate the matrix elements of the operator H’ in such a way. We use the set of shell-model wave functions I(A - 2); E,J,M2) for the (A - 2) system which includes all configurations created by picking up one active particle from the (A - 1) core. If the wave functions for the (A - 1) and (A -2) nuclei are given, spectroscopic amplitudes ((A - 1); E’J’llc,~ll(A -2); E,J,) can be calculated and can be used to express the matrix elements of the hamiltonian H’ between our basis states : (A ; E”J”.i; JMIH’IA; = c
E’J’j; 1
c
JM) (-p+jk+y2J,
-t 1)&2J’+
l)(ZJ”+
1)
iA J, .%J,
X ((A_l);E”J”llCj~II(A_2);E,J,) X ((A-2);
E,J,llCj~ll(A-
1);E’J’).
(5)
In this equation the symbol ( ji j ; J,) VJj,j ; JI) denotes the matrix elements of the residual interaction I/ between antisymmetrized two-particle wave functions. In the present paper negative-parity states of an odd-mass nucleus (N = 82; Z) are calculated by coupling a lh, proton to states of positive parity of the (N ; Z- 1) core nucleus. All core wave functions entering the mode1 are adopted from Wildenthal’s work ‘). For the single-particle energies and the parametrization
W. Enghardt, H. U. Jtiger 1 Proton Ih, L,2 excitations
144
TABLE Single-particle
energies
and strength
parameters
I Single-particle
1
of the modified MeV)
surface delta interaction
energies
(in units
of
MSDI
lg 712
2d 5/z
2d J/Z
3s Ii2
lh
- 10.14
- 9.62
- 1.02
-7.19
- 7.25
IL,2
A,
B
0.383
0.597
of the modified surface delta interaction (MSDI) determined in ref. ‘), see table 1. We use the same interaction as a two-body force between the lh, proton and the core nucleons. Thus the only new parameter of the present approach is the single-particle energy of the lh, shell which was obtained by fitting the transition energies .E(y; +$T) and E(y; + 3:) of the odd-mass N = 82 nuclei5) with 137 $ A 5 145 (cf. fig. 1). The resulting value is also given in table 1.
2.2. ELECTROMAGNETIC
PROPERTIES
Deriving expressions for the reduced matrix elements (A ; E,J,n,IIA(A)IIA of the electromagnetic multipole operator
&(A v) =
c 0+4t
k. n
v)ln>c:c,
; EiJpci)
(6)
we have to distinguish two cases: (i) Parity-comeruing transitions (nf = xi). The particle-core coupling model is supposed to describe states where the extra particle is in a shell (Nlj) having another parity than the core shells. Both the extra nucleon and the nucleons inside the core contribute to an electromagnetic transition without parity change (7rr = q):
x (- )jm+j(jkll@)llin>
c (-
E,J,
)Ji+J2 {:
;2:.;}
X ((A- 1);E;J;llc~ll(A-2); E,J,)((A-2);
1
E~J~llcjall(A-1); EIJI) .
(7)
3!2:5/2*
,&‘:SfZ+
135 I
CALC.
712
5/2
*
+
EXP
‘\
‘37
1/2’____
f
CS
CALC.
‘\ 5n
. 7R
*
EXP
.
l
13g
La
-.y2&* ~___~ CALC
-.
I
L___ yL’.--_._ EXP
27/T!-
1292-1 ‘c
CALC
__-27/2-
‘4’ Pr
,
2
+
27,2-
+
. .
lL3Pm
__---
y2____yz EXP
7/,’
n/2-
CALC.
,
.
L
I”2-
Fig. 1. Experimental and calculated spectra of negative-parity states in odd-mass N = 82 isotones. The experimental data have been taken from The theoretical level energies of the $’ and i’ states additionally displayed have been obtained ih Wildenthal’s shell-model approach
t
t
1.0
E/MeV
refs. 3-5*10-14). ‘).
ElMeV
W. Enghardt, H. U. Jtiger / Proton Ih,,,,
146
(ii)
Parity-changing
transitions (q = -IQ).
excitations
Here,
the
transition
of the
extra
particle from the orbit (N/j) into core states or the opposite transition have to be considered. If the shell-model states (A ; EJ n = ( - )l+ ‘) are given, one can express the corresponding (A; E,J,q
transition
matrix
= -ZillA(2)llA;
elements
as
EiJi~i) = J2Ji+
l(e)”
C R$ E;J;
X C-)Ji+J~+j~jz,+r{~ jk
:!
I
j’j(j,llp(,)llj)
x (A; E,J,llc;kll(A- 1); E;J;), (A; E,J,rc,
= -n&&(2)JIA;
X ((A-1); Using the formulae and magnetic dipole described in ref. ‘O).
EiJiq)
(8)
= Ji2Ji+1)0
E;J;llcj,llA; EiJi).
(9)
(7), (8) and (9) the electromagnetic transition probabilities as well as electric quadrupole moments are calucated as
3. The effect of model-space
truncation
The particle-core coupling formalism explained enables us to truncate the model space in a reasonable way by taking into account only a certain number of lowlying core states. We study the effect of such restrictions for 145E~, which is described by coupling a lh, proton to 144Sm core states. For 144Sm the 7c = + 1 model space (lg,, 2d,)12, (lg,, 2d,)“(2d,, 3~)~ contains 255 states. If we include all these states in our particle-core calculation, we obtain, of course, the same results as in a shell-model calculation performed in the model space (lg,, 2d,)12(lh,)‘, (lg,, 2d,)“(2d,, 3st)‘(lhY)‘. We show in fig. 2 (case A) the energies and electromagnetic data obtained in such a “complete” calculation, and compare them with four other sets (B, E, C, D) of results, which are computed if the number of 144Sm core states is drastically reduced to 29, 10, 4 and one state, respectively. Most results of the particle-core coupling calculation with 29 core states (B) differ rather slightly from those of calculation A, merely a few Ml transition strengths are changed by a factor of two. We find that the non-yrast core excitations above 3 MeV make only small contributions to the 145E~ wave functions considered. The excitation energies are even found to be nearly stable against further truncations of the model space (C, E). But in this cases some Ml
W. Enghardt,
H. U. Jiiger
/ Proton
Ih,,,,
147
excitations
lo2.5
-\
> f
ci i
‘\ -\
j..
-1 -\
C
L-~23/25\ \~--
-x\-4c’ _//_
__h\ _\-
1.5
- D
-\
/w
\‘-
2.01
I’
-17/219/2-
< ‘\ --21/2-
I
13/2-15/2-
i
i DCBAEF
C
BAEF
D
A Cl
I
NIL rn
11/2_state
-Q/10efm2 11/2-state
CBAEF EF
B(M1V10-3~2 .I 11n--n/2I
I.
B(E2)/102e2fmL 15/2--11/2-
BAE C B(Ml)/lO-$2 , LI ,. 17/2-- 15/Z-
EF
0
F
8
c
A
I
I
B(E2)le2fm4 1712‘-13/2-
Fig. 2. Comparison of shell-model predictions (A) on “‘sEu (n = - 1) states with results of various particle-core coupling calculations (B, C, D, E), which differ from each other by the number of the ‘%m core excitations included. The effect of an additional space truncation by neglecting highly(F); for details, see text. The level energies are excited states of the (A-2) system ‘43Pm is illustrated given relatively to the yi’; energy obtained from the shell-model calculation (A). The electromagnetic data were calculated assuming effective coupling constants of eP = e and gzff = 0.6gv.
transition
magnitude.
strengths between low-lying II = - 1 states diminish by one order of The reason is that the contributions from (AJI = 1 transitions within
the core [cf. eq. (7)] are excluded, if only core states with even spins J” = O’, 2+, 4+, 6+, . . . are taken into account. Here we should like to add a comment on the computational labour of the present particle-core coupling model. The shell-model calculations for II = - 1 states in the odd-mass N = 82 (135 I A S 145) nuclei would require handling matrices of rank greater than 2000. In the particle-core coupling approach, large matrices (of rank 300) appear only in constructing the A = + 1 core states. Calculating the matrix elements of the residual interaction (5) and the transition matrix elements without parity change (7), one needs the spectroscopic amplitudes
148
W. Enghardr, H. U. Jciger / Proton Ih, ,,2 excitations
between the states [(A- 1); E’J’) and j(A-2); EJ,). This is a large set of data even for the investigation of 145E~. If one takes into account 10 states of ‘44Sm and all (652) positive-parity states of 143Pm, about 12000 spectroscopic amplitudes have to be handled. One expects that very high-lying states of the (A -2) nucleus give only small contributions to the sums over the states E2(J2) in eqs. (5) and (7). In order to check it we neglected highly-excited states of ‘43Pm and took into account, as an example, only 25 “/, of states with a given spin J,. The results (fig. 2, F) are comparable to those obtained with the complete set of 652 (A - 2) wave functions (E).
4. The structure of negative-parity
states in odd-mass N = 82 isotones
The application of the mode1 starts with the choice of the positive-parity states which describe the (A - 1) cores, and in the cases of 1351, i3’Cs, ‘39La, 141Pr, and 143Pm the lowest core states of each spin value possible in the rr = + 1 configuration space have been taken into account. Furthermore, the other states up to 3.5 MeV core excitation have been included. Thus, mode1 spaces of about 35 basis states have been obtained (table 2). In the present calculations we have used untruncated sets of (A - 2) states E,(J,) [cf. eqs. (5) and (7)]. 4.1. PROPERTIES
OF THE 9;
STATES
There is a large amount of experimental information on the y; levels of the odd-mass N = 82 nuclei from 137Cs up to 14’Eu including excitation energies 5), half-lives ‘), g-factors 21,4, 22), and stripping spectroscopic factors 23, 24, 25). The spectroscopic factors can be used to prove the validity of our y; wave functions (table 3), since they can be simply derived from the expansion coefficients c$~, of the particle-core coupling wave functions [cf. eq. (2)]:
sIh,,,2(J’,J)
= ((A; EJlICl+h,,,*ll(~ - 1); ~‘J’)12=
b&I’.
The excitation energies of the 9; states are well reproduced throughout the whole mass region considered (fig. 1). Although the hamiltonian used (table 1) does not depend on the mass number, the deviations of theory from experiment are only a little larger than those obtained in the quasiparticle Tamm-Dancoff approximations (QTDA) of Heyde and Waroquier 6*7, and Freed and Miles s). In table 4 we compare our results on spectroscopic factors Su,,,,z(O:, 9;) and electromagnetic data with available experimental values 3-5, “-r4* 21-26) as well as with the QTDA results of refs. ‘* 9). Generally effective proton charges e, = (1 +Z/A)e and an effective g-factor gfff = 0.6gf’ee were assumed in our
W. Enghardt,
H. U. Jiiger 1 Proton Ih, ,,2 excitations
149
150
W. Enghard!, H. U. Jiiger 1 Proton Ih, ,,2 excitations TABLE 3
The
absolute
values
of the
main components S,,,, *(.I’,J) = ~cz$F’~.~~( 2 0.04) functions of odd-mass N = 82 isotones s I,,nRV’
t, 21’ U.?I y;. . I?-.
19-,
;’ : 21’ a-. 21.
9510:) 8512:) +412:) + 514;) 7912:) + 1914:) 9014:)+4(4:)+4)6:) 7414:) + 2416:) 6616:) + 3416;)
lW6:)
137cs fl-
21’
L3-. 21 y; 11-. 21 B-. 2, a-.
:
9210:) +412:) 5~1:)+74~2:)+4~3:) 8212:)+714:)+516:) 9114:) 7614:) + 614;) + 1416:) 7616:)+ 1716:) 8716:)+ 1216;)
21’ a-. 21
‘39La
9110:)+412:) 9~1~)+52~2:)+5~2;)+8~3:) +514:)+514;)+415:)+616:)
8612:) 2914:) 7714;) 515;) 8916;)
+ 5416:) +414;) + 1416:) + 8916:) + 1016f)
14’Pr u2,
u-
21’
Is-. 21’ I,-. x1: TI w-
21’
.
q; : y; : 27-. TI 27-. TZ'
143Pm 9;. y; 9;:
. :
H-
.
4;; .z-
22 a-.
.
21’
9;:
23
23-.
9012:) 4~3:)+8~4:)+76~6:) 4314:) +44)6:) 5014:)+4416:) 4)5;)+89)6:) 5515:)+515:)+35(6;) 3115:)+6216:)
uTl
9110:)+412:) 4612;)+ 1714:)+22(6:)
.
9716:)
calculations. experimental by using the The tables
a = - 1 wave
J) [x 1001
,351 u-
in the
90(0:) +412:) 9~1:)+41~2:)+4~2:>+613:) +6\4:)+5)5:)+18(6:) 9112:) 1314:)+7116:) 4914:) +37(6:) 515:) +8916:) 3915:) + 5616:) 9816:) 3518:)+39\8;)+24110:) 63~8:)+17l8;)+20~10:) 14’Eu
9-.
21 .
y;: y-; : 11-. 2, Is-. 21’ y;: 11-.
2,. 21’
fi-
9;: a-. 21' 23-. 31'
8612:)+414:)+616:) 9210:) 7112:)+20/4:) 3712:)+52)4:)+516:) 7312:) + 1814;) +416:) 11~2:)+7~2:)+53~4:)+10~4;) +515:)+ 1216:) 713:)+9(4:)+7816:) 40(4:)+6~4;)+51~6:) 49)4:)+4416:) 515:>+9416:) 9516:)
Absolute y-transition rates Z&N(n), EY) were computed with the y-ray energies E,. All half-lives were corrected for internal conversion data of Hager and Seltzer”). 3 and 4 show that in the particle-core coupling calculations the yy
W. Enghardt,
151
H. U. Jiiger / Proion lh, ,,1 excitations TABLE 4
Electromagnetic
properties
and spectroscopic factors of the ‘I?; levels in the odd-proton with 135 < A Q 145 1351
spectroscopic factors (‘He, d) (‘Li, bHe) S Ih,,,,(Of>Y--)
EXP EXP PCC TDA “) EXP PCC
g-factors
13’Cs
1.01 0.95
0.92 0.89
0.84 l.l9(22) 0.91 0.89
1.21
1.21
I.21
half-lives [in ns]
EXP PCC PC1 TDA ‘)
B(M2, y; -+ $:) [in j~i fm’]
EXP PCC PC1 TDA ‘)
58.5 81.1
34.8 62.3 14.2
B(E3,y; + 2:) [in e2. fm6]
EXP PCC PCI TDA “)
5950 6900
4720 6210 12700
3520 5400 10700
branching
EXP PCC
0.003 0.002 0.02
0.006 0.004 0.03
ratios
PCI TDA “)
0.07 0.04 0.15
20.4 45.0 II.6
N = 82 nuclei
145E”
14’Pr
‘43Pm
0.96 0.98(15) 0.90 0.93
0.82 0.77(15) 0.9 I 0.93
0.83 0.45(7) 0.92 0.95
1.31(8) 1.21
1.14(9) I.21
1.356(g) 1.21
24.0( IO) 47.6 17.6 27.3
490(30) 1720 740 590
11.3(5) 6.5 17.7 II.4
8.4(5) 2.7 7.9 9.6
1 1000(1100) 1350 3170 3540
5000(400) 530 1680 830
4.8(2) 5.0 1.9 5.6 10.9(7) II.5 30.0 10.3 12000(800) 2350 4410 6450 0.11(l) 0.02 0.02 0.07
0.20( 1) 0.04 0.04 0.07
0.27(l) 0.09 0.13 0.04
The results of our calculations (PCC) are compared to available experimental data and to the results of the Tamm-Dancoff approximation (TDA) in a one- and three-quasiproton space. Furthermore, we show particle-core coupling results, which have been obtained by taking into account only the ground states of the even-mass cores (PCl). “) Ref. a). “) Ref. 9).
wave functions are also predicted to be of the type lhy@O:. The admixtures of excited core states amount only to a few percent. The g-factors obtained from these wave functions are obviously the single-particle values of the lh, proton, since the contributions of the core to the Ml transition matrix elements in eq. (7) are negligible. The single-particle character of the 9; states is confirmed by the spectroscopic factors and the g-factors measured. A more complicated situation is found when the reduced transition probabilities B(M2, 9; -+ 3:) and B(E3, y; -+ ST) or the half-lives of the y-; states are analysed. These quantities depend
152
W. Enghardt,
H. L;. Jijger
/ Proton
Ih, , ,2 excitorions
more sensitively on the structure of the wave functions of initial and final states. The reduced M2 transition probability in 14’Pr as well as the half-lives in 141Pr and probably in ‘39La [ref.26)] are nearly correctly predicted in our model. If another effective proton charge of e,, + 3e [see below and ref. ‘“)I was assumed. the B(E3) values and branching ratios I,(?; + $:)/I..(?; + $T ) would also be reproduced; the theoretical half-lives would be diminished in this case only by about 100, I,’ Going from r4’ Pr to 14SE~ in table 4. the differences between theory and experiment generally increase. Our model space including only one particle in the lh, orbit seems to prove inadequate for a complete description of the nuclei near to the Z = 64 subshell closure. This conjecture is supported by the mean occupation number of the lh,? proton orbit in the ground state of 144Sm which was reported z3) to amount to 1.6. Furthermore, the lh, quasiparticle energies for the odd-mass N = 82 nuclei 29), decreasing from about 1.8 MeV in r3’Cs to 0.7 MeV in 14’Eu, indicate that proton lhc configurations may already occur at 2 MeV - 3 MeV for the heavier nuclei considered. If a pure Ih, @ 0: structure was assumed for the $!; state, the excitation energies would be slightly increased (100 keV - 150 keV); for illustration see again fig. 2 (B and D). As expected. the g-factors are not changed by this model-space truncation. However. the reduced transition probabilities B(M2) and B(E3) considerably increase (table 4), since destructive contributions to the M2 and E3 transition strengths caused by configuration mixing are obviously omitted. The same effect is found, if the results of the QTDA in the one- and three-quasiparticle space are compared to those of the calculation in a one-quasiparticle basis9). The renormalization of the B(M2) values, obtained with the 0: core state only, to the results in the larger model space (table 2). where &” = 0.6gpee was assumed, requires the introduction of smaller effective y-factors. For ‘*‘Pr and ‘43Pm, as an the appreciable influence of example, we found 9:” = 0.41,95”‘. This illustrates model-space truncations to the effective electromagnetic coupling constants, which have to be used. Even smaller effective g-factors &” = 0.24g~“’ had to be taken by the experimental M2 transition rates Ejiri et ul. 2R), in order to reproduce UC el y a one-quasiparticle description. +{T)in thesen 1 b (9;
4.2. NEGATIVE-PARITY
STATES ABOVE
THE YL; ISOMERS
isomers have been observed in the Spectra of n = - 1 states on top of the y; 14’Eu [refs. 3.4. 1o-14)]. At present there is no nuclei 14’Pr, ‘43Pm, and experimental evidence for states with higher spins (J > ‘I’) in the nuclei ‘351, 13’Cs, and ‘39La. Besides the IC = - I levels actually observed, we have included in fig. 1 and table 3 excitation energies and wave functions of the lowest states belonging to each value y- s .I” 5 y-. The structure of all negative-parity states with ‘I’- 5 J” 5 y-- in the vicinity of
W. Enghardt, H. lJ. Jtiger / Proton Ih,,,,
the yrast line is essentially
characterized
153
excitations
by the ground
state and the 2:,
4:,
and
6: two-particle states of the even-even N = 82 core (cf. table 3). From the point of view of a simple weak-coupling picture one expects the yi”-;, q;, and y; states to be dominated by components which arise from aligned coupling of the lh, proton to the appropriate 2:, 4:, and 6: core states. In the most cases this assumption is confirmed by our results. However, with increasing mass number, strong admixtures of the basis states lh,@6: occur in the wave functions of the y; states. The reason for this lies in the structure of the 4: core states. The 4: wave functions are dominated by lgf contigurations in the lighter (A <: 140) and by 2di configurations in the heavier even-mass N = 82 nuclei [cf. also ref. ““)I. Considering the two-body matrix elements of the residual interaction one notices that for the lg+ proton the aligned, and for the 2d+ proton the antialigned, proton is energetically preferred. Having in mind coupling with the lh, furthermore the small energy splittings (< 200 keV) between the 4: and 6: core states, one can understand the large lhY@6: contributions to the y; states in the nuclei with A > 140. In the wave functions of the y-;, y-;, and y-; states only minor admixtures of fully aligned basis states are found, since the l:, 3:, and 5: core states lie about 0.5 MeV above the 6: level. Considering the shell-model wave functions of the core states, one obtains detailed predictions for the microscopic structure of the negative-parity states, In by lh,@lgi threethe lighter isotones the y;, yi’;, and $; states are dominated particle configurations. Due to the tilling up the lg, shell, structural changes occur around 14’Ce. In the heavier isotones the y; states have a nearly pure lh,@2d: structure, whereas with increasing mass number in the yi’; and y; wave functions the lh,@(lg,, 2d,) configurations compete more and more with the lh,@2di ones. For the $r; and 9; states the major components are lh, 0 (lg,, 2d,), and only in the lightest nuclei 1351 and i3’Cs appreciable admixtures of the lh+@lg$ configurations are found. The comparison of experimental and theoretical excitation energies in fig. 1 shows a fairly good correspondence for most of the levels of 14’Pr. In 143Pm and especially in 14’Eu the majority of the theoretical levels is predicted at too low energies, which merely reflects the deviations of the shell-model results for the corestate energies from their experimental values. However, the density and the sequence of these levels are nearly correctly reproduced. This picture is essentially disturbed by the y; levels of 141Pr and 143Pm as well as by the $7, y;, and $7 ones of 14’Eu. Here, configurations outside of our model space should become important. Low-lying collective 3; states are known in nearly all even-mass essential admixtures of neutron N = 82 isotones. Certainly they contain particle-hole excitations 31). Therefore, a description as (1s; ’ @ 3;), 5 9 or (2di’ 0 3;), 5 9 states may be more suitable. In the case of 14’Eu this picture has been successfully applied l1 -13). Similarly, one estimates the excitation energies of such JZlevels in 14’Pr and 143Pm to be 2230 keV and 2082 keV, respectively. 2
154
W. Enghurdr.
H. U. Jkger / Praron Ih, ,i2 excisions
A comparison of theoretical and experimental electromagnetic data is shown in table 5 for the IC= - 1 states with an unambigous spin-parity assignment and with theoretical excitation energies which suggest our particle-core coupling picture to be valid (see above). The result obtained for the half-life of the y; state in 14’Pr, which is deexcited by a pure Ml transition of 92 keV to the y; state, is in moderate agreement with the experiment. But, a too small lifetime is predicted for the -state of 14’Eu. Such an underestimation is caused by the reduced E2 transftiron probability B’““(E2, $; -+ ‘ii;) = 40.3 ez. fm4, which drastically TABLE5 Particle-core coupling predictions on electromagnetic properties of negative-parity states with in comparison to experimental data Experiment
J” 2 yw
Theory
half-lives [in ns] ‘4’PrB) 14’Eu “)
T,:2(y;, 3020 keV) T’,&?iL;. 2836 keV)
0.2(l) 5.5(5)
I,(J: + J;. EJ ___..-.--l&J; + J;. EJ
branching ratios
l&% -+$.--~ 301 keV) ---.... I,($ + ft;, 447 keV)
lb3PrnC)
IdsEu
I,(y;
-+ y;,
364 keV)
I,#il;
+ 9;.
447 keV)
!,(y;
-ry;,240
l&y;
-+ yLh 530 keV)
I,@;
+ff.
keV)
‘43Pm ‘)
‘*‘Eu
*)
Ref. ‘).
“1 Ref. ”
3.2
0.43
1.9
0.007
U(2)b) 4.S( 12)d) 4.6(g)‘)
0.05
22 keV)
I,($% + y;,
553 keV)
_ 0.36bl
0.01
f.,(q;
-t 9;.
262 keV)
I,(%
+ y;,
553 keV)
0.33(9) b) 1.06(21)~) l.00(28)L)
0.002
EZ/M I mixing ratios
a(J; + J;. E,)
“‘~r)
0.8 0.2
a(?; s(yP; S(y; S(y; S(f;
+ 9;. -+ y;, -+ ‘p;. + 9;. --t 9;.
92 keV) 194 keV) 301 keV) 376 keV) 240 keV)
S(y;
-* y;,
530 keV)
).
. ‘) Ref. *).
“) Ref. I’).
0 Oor O.O7cd< 0.15 Oor -0.5
‘) Ref. 13).
0.00 -0.17 - 0.08 - 0.02 -0.23 -0.23
W. Enghardt, H. U. Jtiger 1 Proton Ih, ,,2 excitations
155
exceeds the experimental value 15) of BexP(E2, y; + y-;) = 0.14&0.03 e2. fm4. Accordingly, the contributions of the lhy@2: and lh,@6: basis states to the y; and y; wave functions, respectively, may be still larger than given in table 3. This would confirm the results of Kaczarowski et al. Is). For the remaining branching and mixing ratios in 143Pm and 145Eu only a partial agreement of theory and experiment is found, indicating that for these nuclei only qualitative conclusions on the structure of the 7c = - 1 levels above the y-; isomers can be drawn from our wave functions. Owing to the better correspondence of theory and experiment for the excitation energies and for the decay properties of the yi’; and f; isomers of 14’Pr, one expects our model to provide a more reliable description in this case. Unfortunately, there do not exist any more suitable electromagnetic data for a detailed test of the 141Pr wave functions, because many of the negative-parity levels reported 3, lo) are deexcited via intensive El transitions, which are forbidden in our model space containing only the lg,, 2d+, 2d,, 3s,, and lh, orbits.
5. Summary A particle-core coupling model, where a lh, proton is coupled to a restricted set of TC= + 1 shell-model wave functions, is applied to negative-parity states in oddmass N = 82 isotones. In the particle-core coupling basis we diagonalized the same MSDI hamiltonian formerly proposed by Wildenthal ‘) for positive-parity states. This concept has been proved to yield an approximation to the results of shellmodel calculations in a large configuration space and to reduce the necessary computational labour. In this way an extension of the shell-model calculations of TC= + 1 states in N = 82 isotones ’ -4) becomes more practicable. The wave functions for the states with y- 5 J” S y- in the vicinity of the yrast line show that a stretched coupling of the lh, proton to the lowest core states (O:, 2:, 4:, 6:) is favoured. The influence of the valence proton number on the structure of the low-lying I( = - 1 states has been discussed. The spectroscopic data of the y-; states could be calculated in satisfactory agreement to the experiments. For the higher-lying states in i4rPr a fairly good correspondence with the results of measurements was obtained, too, whereas for the heavier systems ‘43Pm and 14’Eu the theory does not fit the experiments so well. However, the sequence and the density of the major part of the R = - 1 levels are also reproduced. The $;, y-;, and y; levels in 145Eu and the y; ones in 14iPr and ‘43Pm have been supposed to have a lg, @ 3; or 2d, @ 3; structure, which cannot be handled within our model space on account of the collectivity of the 3; states in even-mass N = 82 isotones. Finally we mention that the domain of applicability of the model can be extended from the odd-mass N = 82 isotones to other semimagic nuclei where single-nucleon excitations in an empty shell are expected. Calculating e.g. the
156
W. Enghardt.
H. L’. JGger / Proton lh, ,:2 excitations
- 1 spectra caused by proton excitations3’) in r:iCe,, and by neutron excitations 33) in l&Sn,,, we succeeded in predicting spectroscopic data in remarkable agreement with experiment.
I[ =
We would like to thank Drs. H. Prade, F. Stary, and G. Winter for carefully reading the manuscript and for their helpful comments. This work was supported by the “Ministerium fur Wissenschaft und Technik” and the “Akademie der Wissenschaften der DDR”.
References I) B. H. Wildenthal. Phys. Rev. Lett. 22 (1%9) 1118 2) B. H. Wildenthal, Phys. Lett. 29B (1969) 274 3) H. Prade. W. Enghardt, H. U. Jager, L. Kaubler. H.-J. Keller and F. Stary, Nucl. Phys. A370(1981) 47 4) H. Prade, L. Klubler, U. Hagemann, H. U. Jlger, M. Kirchbach, L. Schneider, F. Stary. Z. Roller and V. Paar, Nucl. Phys. A333 (1980) 33 5) C. M. Lederer and V. S. Shirley, Table of isotopes, 7th ed. (Wiley, New York, 1978) 6) M. Waroquier and K. Heyde, Nucl. Phys. Al44 (1970) 481 7) K. Heyde and M. Waroquier, Nucl. Phys. Al67 (1971) 545 8) N. Freed and W. Miles, Nucl. Phys. A158 (1970) 230 9) M. Waroquier and G. Vanden Berghe, Phys. Lett. 41B (1972) 267 IO) M. Kortelahti, M. Piiparinen, A. Pakkanen, T. Komppa and R. Komu, Phys. Scripta 24 (1981) IO I I) D. R. Haenni, H. Beuscher, R. M. Lieder, M. Miller-Veggian, A. Neskakis and C. Mayer-Boricke, Nucl. Phys. A331 (1979) I41 12) D. Bazzacco, A. M. 1. Haque, K. 0. Zell, P. von Brentano and C. Protop, Phys. Rev. C21(1980) 222 13) D. A. Rakel, R. Kaczarowski. E. G. Funk and J. W. Mihelich, Phys. Rev. Cfl (1980) 595 14) P. von Brentano, D. Bazzacco and C. Protop, Proc. Divisional Conf. of the European Physical Society - Nuclear Physics Division, Bucharest, June 1981. p. I61 15) R. Kaczarowski, E. G. Funk and J. W. Mihelich, Phys. Rev. C23 (1981) 271 I 16) W. Enghardt and H. U. Jager, ZfK-485 (1982) p. I 17) H. Kruse and B. H. Wildenthal, Bull. APS 27 (1982) 533; 725 18) 0. Scholten and H. Kruse, Phys. Lett. 12SB (1983) I13 19) A. de Shalit and 1. Talmi, Nuclear shell theory (Academic, New York, 1963) 20) A. Bohr and B. R. Mottelson, Struktur der Atomkerne, Band I (Akademie, Berlin, 1975) 21) H. Ejiri, T. Shibata and M. Takeda, Nucl. Phys. A221 (1974) 211 22) W. Klinger, R. Bohm. W. Engel, W. Sandner, R. Seebock and W. Witthuhn, Z. Phys. A290 (1979) 227 23) B. H. Wildenthal, R. Newman and R. L. Auble, Phys. Rev. C3 (1971) II99 24) P. D. Clark, T. R. Ophel, J. S. Eck, A. F. Zeller, J. Nurzynski, D. C. Weisser and D. F. Hebbard, Nucl. Phys. A349 (1980) 258 25) P. D. Clark, T. R. Ophel, J. Nurzynski, C. H. Atwood and D. F. Hebbard, Nucl. Phys. A352 (1981) 267 26) W. D. Fromm, L. Funke and K. D. Schilling, Phys. Scripta 12 (1975) 91 27) R. S. Hager and E. C. Seltzer, Nucl. Data A4 (1968) I 28) H. Ejiri, T. Shibata and M. Fujiwara, Phys. Rev. CII (1973) 1892 29) Y. Nagai, J. Styczen, M. Piiparinen, P. Kleinheinz, D. Bazzacco, P. von Brentano, K. 0. Zell and J. Blomqvist, Phys. Rev. Lett. 47 (1981) 1259 30) K. Heyde, M. Waroquier and G. Vanden Berghe, Phys. Lett. 35B (1971) 21 I 31) P. Kleinheinz, Phys. Scripta 24 (1981) 236 32) W. Enghardt, H. U. Jager. L. Kaubler, H.-J. Keller, H. Prade and F. Stary, Z. Phys. A316 (1984) 245 33) H. Prade, W. Enghardt, W. D. Fromm, H. U. JBger, L. Kaubler, H.-J. Keller, L. K. Kostov, F. Stary, G. Winkler and L. Westerberg, Nucl. Phys. A425 (1984) 317
PROTON lh, 1,2 EXCITATIONS IN ODD-MASS N = 82 NUCLEI W. ENGHARDT Zenrralinstitut ,fir Kernforschung,
and H. U. JAGER Rossendork DDR-805I
Dresden, GDR
Received 3 August 1984 Abstract: The structure of negative-parity states in odd-mass N = 82 isotones (135 $ A s 145) is investigated in the framework of a model which is based on coupling one Ihi,,* proton to ?I = + 1 states of the appropriate doubly-even isotonic core. It is shown that an effective modelspace truncation can be achieved if only core states in the vicinity of the yrast line are taken into account. These have been selected from the several hundreds or even thousands of states obtained from shell-model calculations, where the lg,,,, 2d,,,, 2d,,, and 3s,,, single-particle states are assumed to be occupied. Spectroscopic data are calculated and predictions on the structure of low-lying A = - 1 states are discussed and compared as far as possible to experimental findings. Thereby, the particle-core coupling approach is shown to be capable of describing essential properties of negative-parity levels in the N = 82 nuclei considered.
1. Introduction A large amount of the data reported on positive-parity states in the semimagic N = 82 nuclei (with 50 < Z < 64) can be explained in terms of proton wave functions of the spherical shell model le4) . It has been found that the configurations (lg,, 2d,)Z- 50, where the valence protons are distributed over the nearly degenerated shells lg; and 2d,, usually dominate in the wave functions. Only a few low-lying levels, e.g. the $: and 4: states of 143Pm or the i: and 1+ states of 14iPr 21 should arise from one-proton excitations (lg,, 2d,)Z-51(2d,, 3~~)’ to the higherlying shells 2d, or 3s+ (for the order of the shells, see table 1). The structure of the negative-parity states of N = 82 nuclei is not so well understood, since in the odd-proton nuclei only y?- isomeric states were known in 13’Cs, 139La, 141Pr, 143Pm, and 145E~ for many years 5). They are interpreted as proton excitations to the lh, shell which should lie near to, or even below, the 2d, and 3s+ shells. The 9; states were the only rz = - 1 states taken into consideration one decade ago in the structure calculations6-9), in which the hamiltonian was diagonalized in a one- and three-quasiparticle space. More recent investigations on 141Pr [refs. 3, lo)], ‘43Pm [refs.4p lo)], and 145E~ [refs. l1 - ‘“)I revealed a variety of further odd-parity states in the energy region up to E, ‘v 4.5 MeV (fig. 1). All these levels have spin J 2 9. The 145E~ states have been discussed ” - ’ 3, in ’ t erms of weak coupling of lg, and 2d, proton holes 141
142
W. Enghardt, H. U. Jiiger / Proton Ih,,,,
excitations
to J” = 3;, 5;, 7; states of 146Gd and a th, proton to ‘44Sm states. Kaczarowski et al. 15) interpreted them as members of the (lh,, 2d;‘), and (lh,, lg;‘, 2di’), multiplets, and this interpretation was supported by the comparison of experimental and calculated E2 transition rates and branching ratios. We performed 3, similar shell-model calculations for the negative-parity states of 141Pr and 143Pm in the framework of the space, (lg,, 2d,)Z- ” (lh,)‘, we were capable of handling. To achieve a consistent description of excitations of both parities in N = 82 nuclei, one should extend at least the shell-model space of ref. ‘) by taking into account the one-proton excitations to the lh, shell. But even these configurations, (lg,, 2dt)Z-51(lhy)1 and (lg,, 2d,)Z-52(2d,, 3st)‘(lhY)‘, are rather difficult to handle. In the case of 14iPr they contain, for example, 2202 states with J” = y-. The present paper deals with a reasonable truncation of this space. We construct a basis of odd-parity states of an N = 82 nucleus with mass number A by coupling a lh, proton to II = + 1 shell-model states of the core nucleus (A - 1; N = 82) but only take into account a restricted number of about 30 core states in the vicinity of the yrast line. The corresponding particle-core coupling formalism, see also ref. 16), is outlined in sect. 2. The efficiency of our space truncation is demonstrated for 145E~ (sect. 3). We calculate excitation energies, wave functions and electromagnetic properties of negative-parity states in the odd-mass isotones from 1351 to 145E~ and compare them with available data (sect. 4). Other approaches to investigate lh, excitations in N = 82 nuclei have been briefly reported. Kruse and Wildenthal 17) made shell-model calculations for the nuclei from 133Sb to 14sDy where they restricted the basis to states with seniority less than 4. For the even-mass nuclei, Scholten and Kruse ‘*) compared results of that work with their investigations performed in the generalized seniority scheme.
2. Coupling of a nucleon in an intruder orbit to shell-model core states 2.1. ENERGY
SPECTRA AND WAVE FUNCTIONS
Let us first introduce the shell-model states )(A - 1); E’J’M’) for the core nucleus with (A - 1) nucleons. They are characterized by the energy E’ and the quantum numbers J’ and M’ of the total angular momentum and its third component, respectively. All these states are supposed to have the same parity 7~‘.There is no need for specifying isospin quantum numbers because the model is applied only to nucleon excitations of one kind in semimagic nuclei. We add a nucleon to a shell (Nlj) which is empty in all core states and couple the angular momentum of this extra-shell nucleon to that of the core state : IA ; E’J’j ; JM)
=
1 (J’M’jm(JM)c~l(A M’m
- 1); EIJIMI).
(1)
W. Enghardt,
H. U. Jiiger
/ Proton
Ih,,,,
143
exciraiions
Here the symbol c+ denotes the Fermi creation operator. The states (1) form an orthonormal basis which is used to expand the wave functions (A;EJM)
= c aE,:‘,.(A ; E’J’j ; J M)
(2)
E’J’
of parity 7~= rc’(-)’ for the system with A nucleons. The level energies E and the expansion coefficients a::‘,, are determined by solving the Schrodinger equation (H, + H’)IA ; EJM)
= EIA ; EJM).
(3)
Our basis states (1) are eigenstates of the hamiltonian Ho involving the effective nucleon-nucleon interaction within the core and the single-particle energy Ed of the nucleon added: H,IA;E’J’j;JM)
= (E’+E~)(A;E’J’~;JM).
(4)
The term H’ in eq. (3) stands for the residual interaction of the extra nucleon with the nucleons active in the core states. In the shell-model technique 19), its matrix elements are evaluated using products of (one-particle) coefficients of fractional parentage. They describe the change of the core states if a nucleon inside the core is scattered by the interaction from a single-particle state k to a state i. We calculate the matrix elements of the operator H’ in such a way. We use the set of shell-model wave functions I(A - 2); E,J,M2) for the (A - 2) system which includes all configurations created by picking up one active particle from the (A - 1) core. If the wave functions for the (A - 1) and (A -2) nuclei are given, spectroscopic amplitudes ((A - 1); E’J’llc,~ll(A -2); E,J,) can be calculated and can be used to express the matrix elements of the hamiltonian H’ between our basis states : (A ; E”J”.i; JMIH’IA; = c
E’J’j; 1
c
JM) (-p+jk+y2J,
-t 1)&2J’+
l)(ZJ”+
1)
iA J, .%J,
X ((A_l);E”J”llCj~II(A_2);E,J,) X ((A-2);
E,J,llCj~ll(A-
1);E’J’).
(5)
In this equation the symbol ( ji j ; J,) VJj,j ; JI) denotes the matrix elements of the residual interaction I/ between antisymmetrized two-particle wave functions. In the present paper negative-parity states of an odd-mass nucleus (N = 82; Z) are calculated by coupling a lh, proton to states of positive parity of the (N ; Z- 1) core nucleus. All core wave functions entering the mode1 are adopted from Wildenthal’s work ‘). For the single-particle energies and the parametrization
W. Enghardt, H. U. Jtiger 1 Proton Ih, L,2 excitations
144
TABLE Single-particle
energies
and strength
parameters
I Single-particle
1
of the modified MeV)
surface delta interaction
energies
(in units
of
MSDI
lg 712
2d 5/z
2d J/Z
3s Ii2
lh
- 10.14
- 9.62
- 1.02
-7.19
- 7.25
IL,2
A,
B
0.383
0.597
of the modified surface delta interaction (MSDI) determined in ref. ‘), see table 1. We use the same interaction as a two-body force between the lh, proton and the core nucleons. Thus the only new parameter of the present approach is the single-particle energy of the lh, shell which was obtained by fitting the transition energies .E(y; +$T) and E(y; + 3:) of the odd-mass N = 82 nuclei5) with 137 $ A 5 145 (cf. fig. 1). The resulting value is also given in table 1.
2.2. ELECTROMAGNETIC
PROPERTIES
Deriving expressions for the reduced matrix elements (A ; E,J,n,IIA(A)IIA of the electromagnetic multipole operator
&(A v) =
c 0+4t
k. n
v)ln>c:c,
; EiJpci)
(6)
we have to distinguish two cases: (i) Parity-comeruing transitions (nf = xi). The particle-core coupling model is supposed to describe states where the extra particle is in a shell (Nlj) having another parity than the core shells. Both the extra nucleon and the nucleons inside the core contribute to an electromagnetic transition without parity change (7rr = q):
x (- )jm+j(jkll@)llin>
c (-
E,J,
)Ji+J2 {:
;2:.;}
X ((A- 1);E;J;llc~ll(A-2); E,J,)((A-2);
1
E~J~llcjall(A-1); EIJI) .
(7)
3!2:5/2*
,&‘:SfZ+
135 I
CALC.
712
5/2
*
+
EXP
‘\
‘37
1/2’____
f
CS
CALC.
‘\ 5n
. 7R
*
EXP
.
l
13g
La
-.y2&* ~___~ CALC
-.
I
L___ yL’.--_._ EXP
27/T!-
1292-1 ‘c
CALC
__-27/2-
‘4’ Pr
,
2
+
27,2-
+
. .
lL3Pm
__---
y2____yz EXP
7/,’
n/2-
CALC.
,
.
L
I”2-
Fig. 1. Experimental and calculated spectra of negative-parity states in odd-mass N = 82 isotones. The experimental data have been taken from The theoretical level energies of the $’ and i’ states additionally displayed have been obtained ih Wildenthal’s shell-model approach
t
t
1.0
E/MeV
refs. 3-5*10-14). ‘).
ElMeV
W. Enghardt, H. U. Jtiger / Proton Ih,,,,
146
(ii)
Parity-changing
transitions (q = -IQ).
excitations
Here,
the
transition
of the
extra
particle from the orbit (N/j) into core states or the opposite transition have to be considered. If the shell-model states (A ; EJ n = ( - )l+ ‘) are given, one can express the corresponding (A; E,J,q
transition
matrix
= -ZillA(2)llA;
elements
as
EiJi~i) = J2Ji+
l(e)”
C R$ E;J;
X C-)Ji+J~+j~jz,+r{~ jk
:!
I
j’j(j,llp(,)llj)
x (A; E,J,llc;kll(A- 1); E;J;), (A; E,J,rc,
= -n&&(2)JIA;
X ((A-1); Using the formulae and magnetic dipole described in ref. ‘O).
EiJiq)
(8)
= Ji2Ji+1)0
E;J;llcj,llA; EiJi).
(9)
(7), (8) and (9) the electromagnetic transition probabilities as well as electric quadrupole moments are calucated as
3. The effect of model-space
truncation
The particle-core coupling formalism explained enables us to truncate the model space in a reasonable way by taking into account only a certain number of lowlying core states. We study the effect of such restrictions for 145E~, which is described by coupling a lh, proton to 144Sm core states. For 144Sm the 7c = + 1 model space (lg,, 2d,)12, (lg,, 2d,)“(2d,, 3~)~ contains 255 states. If we include all these states in our particle-core calculation, we obtain, of course, the same results as in a shell-model calculation performed in the model space (lg,, 2d,)12(lh,)‘, (lg,, 2d,)“(2d,, 3st)‘(lhY)‘. We show in fig. 2 (case A) the energies and electromagnetic data obtained in such a “complete” calculation, and compare them with four other sets (B, E, C, D) of results, which are computed if the number of 144Sm core states is drastically reduced to 29, 10, 4 and one state, respectively. Most results of the particle-core coupling calculation with 29 core states (B) differ rather slightly from those of calculation A, merely a few Ml transition strengths are changed by a factor of two. We find that the non-yrast core excitations above 3 MeV make only small contributions to the 145E~ wave functions considered. The excitation energies are even found to be nearly stable against further truncations of the model space (C, E). But in this cases some Ml
W. Enghardt,
H. U. Jiiger
/ Proton
Ih,,,,
147
excitations
lo2.5
-\
> f
ci i
‘\ -\
j..
-1 -\
C
L-~23/25\ \~--
-x\-4c’ _//_
__h\ _\-
1.5
- D
-\
/w
\‘-
2.01
I’
-17/219/2-
< ‘\ --21/2-
I
13/2-15/2-
i
i DCBAEF
C
BAEF
D
A Cl
I
NIL rn
11/2_state
-Q/10efm2 11/2-state
CBAEF EF
B(M1V10-3~2 .I 11n--n/2I
I.
B(E2)/102e2fmL 15/2--11/2-
BAE C B(Ml)/lO-$2 , LI ,. 17/2-- 15/Z-
EF
0
F
8
c
A
I
I
B(E2)le2fm4 1712‘-13/2-
Fig. 2. Comparison of shell-model predictions (A) on “‘sEu (n = - 1) states with results of various particle-core coupling calculations (B, C, D, E), which differ from each other by the number of the ‘%m core excitations included. The effect of an additional space truncation by neglecting highly(F); for details, see text. The level energies are excited states of the (A-2) system ‘43Pm is illustrated given relatively to the yi’; energy obtained from the shell-model calculation (A). The electromagnetic data were calculated assuming effective coupling constants of eP = e and gzff = 0.6gv.
transition
magnitude.
strengths between low-lying II = - 1 states diminish by one order of The reason is that the contributions from (AJI = 1 transitions within
the core [cf. eq. (7)] are excluded, if only core states with even spins J” = O’, 2+, 4+, 6+, . . . are taken into account. Here we should like to add a comment on the computational labour of the present particle-core coupling model. The shell-model calculations for II = - 1 states in the odd-mass N = 82 (135 I A S 145) nuclei would require handling matrices of rank greater than 2000. In the particle-core coupling approach, large matrices (of rank 300) appear only in constructing the A = + 1 core states. Calculating the matrix elements of the residual interaction (5) and the transition matrix elements without parity change (7), one needs the spectroscopic amplitudes
148
W. Enghardr, H. U. Jciger / Proton Ih, ,,2 excitations
between the states [(A- 1); E’J’) and j(A-2); EJ,). This is a large set of data even for the investigation of 145E~. If one takes into account 10 states of ‘44Sm and all (652) positive-parity states of 143Pm, about 12000 spectroscopic amplitudes have to be handled. One expects that very high-lying states of the (A -2) nucleus give only small contributions to the sums over the states E2(J2) in eqs. (5) and (7). In order to check it we neglected highly-excited states of ‘43Pm and took into account, as an example, only 25 “/, of states with a given spin J,. The results (fig. 2, F) are comparable to those obtained with the complete set of 652 (A - 2) wave functions (E).
4. The structure of negative-parity
states in odd-mass N = 82 isotones
The application of the mode1 starts with the choice of the positive-parity states which describe the (A - 1) cores, and in the cases of 1351, i3’Cs, ‘39La, 141Pr, and 143Pm the lowest core states of each spin value possible in the rr = + 1 configuration space have been taken into account. Furthermore, the other states up to 3.5 MeV core excitation have been included. Thus, mode1 spaces of about 35 basis states have been obtained (table 2). In the present calculations we have used untruncated sets of (A - 2) states E,(J,) [cf. eqs. (5) and (7)]. 4.1. PROPERTIES
OF THE 9;
STATES
There is a large amount of experimental information on the y; levels of the odd-mass N = 82 nuclei from 137Cs up to 14’Eu including excitation energies 5), half-lives ‘), g-factors 21,4, 22), and stripping spectroscopic factors 23, 24, 25). The spectroscopic factors can be used to prove the validity of our y; wave functions (table 3), since they can be simply derived from the expansion coefficients c$~, of the particle-core coupling wave functions [cf. eq. (2)]:
sIh,,,2(J’,J)
= ((A; EJlICl+h,,,*ll(~ - 1); ~‘J’)12=
b&I’.
The excitation energies of the 9; states are well reproduced throughout the whole mass region considered (fig. 1). Although the hamiltonian used (table 1) does not depend on the mass number, the deviations of theory from experiment are only a little larger than those obtained in the quasiparticle Tamm-Dancoff approximations (QTDA) of Heyde and Waroquier 6*7, and Freed and Miles s). In table 4 we compare our results on spectroscopic factors Su,,,,z(O:, 9;) and electromagnetic data with available experimental values 3-5, “-r4* 21-26) as well as with the QTDA results of refs. ‘* 9). Generally effective proton charges e, = (1 +Z/A)e and an effective g-factor gfff = 0.6gf’ee were assumed in our
W. Enghardt,
H. U. Jiiger 1 Proton Ih, ,,2 excitations
149
150
W. Enghard!, H. U. Jiiger 1 Proton Ih, ,,2 excitations TABLE 3
The
absolute
values
of the
main components S,,,, *(.I’,J) = ~cz$F’~.~~( 2 0.04) functions of odd-mass N = 82 isotones s I,,nRV’
t, 21’ U.?I y;. . I?-.
19-,
;’ : 21’ a-. 21.
9510:) 8512:) +412:) + 514;) 7912:) + 1914:) 9014:)+4(4:)+4)6:) 7414:) + 2416:) 6616:) + 3416;)
lW6:)
137cs fl-
21’
L3-. 21 y; 11-. 21 B-. 2, a-.
:
9210:) +412:) 5~1:)+74~2:)+4~3:) 8212:)+714:)+516:) 9114:) 7614:) + 614;) + 1416:) 7616:)+ 1716:) 8716:)+ 1216;)
21’ a-. 21
‘39La
9110:)+412:) 9~1~)+52~2:)+5~2;)+8~3:) +514:)+514;)+415:)+616:)
8612:) 2914:) 7714;) 515;) 8916;)
+ 5416:) +414;) + 1416:) + 8916:) + 1016f)
14’Pr u2,
u-
21’
Is-. 21’ I,-. x1: TI w-
21’
.
q; : y; : 27-. TI 27-. TZ'
143Pm 9;. y; 9;:
. :
H-
.
4;; .z-
22 a-.
.
21’
9;:
23
23-.
9012:) 4~3:)+8~4:)+76~6:) 4314:) +44)6:) 5014:)+4416:) 4)5;)+89)6:) 5515:)+515:)+35(6;) 3115:)+6216:)
uTl
9110:)+412:) 4612;)+ 1714:)+22(6:)
.
9716:)
calculations. experimental by using the The tables
a = - 1 wave
J) [x 1001
,351 u-
in the
90(0:) +412:) 9~1:)+41~2:)+4~2:>+613:) +6\4:)+5)5:)+18(6:) 9112:) 1314:)+7116:) 4914:) +37(6:) 515:) +8916:) 3915:) + 5616:) 9816:) 3518:)+39\8;)+24110:) 63~8:)+17l8;)+20~10:) 14’Eu
9-.
21 .
y;: y-; : 11-. 2, Is-. 21’ y;: 11-.
2,. 21’
fi-
9;: a-. 21' 23-. 31'
8612:)+414:)+616:) 9210:) 7112:)+20/4:) 3712:)+52)4:)+516:) 7312:) + 1814;) +416:) 11~2:)+7~2:)+53~4:)+10~4;) +515:)+ 1216:) 713:)+9(4:)+7816:) 40(4:)+6~4;)+51~6:) 49)4:)+4416:) 515:>+9416:) 9516:)
Absolute y-transition rates Z&N(n), EY) were computed with the y-ray energies E,. All half-lives were corrected for internal conversion data of Hager and Seltzer”). 3 and 4 show that in the particle-core coupling calculations the yy
W. Enghardt,
151
H. U. Jiiger / Proion lh, ,,1 excitations TABLE 4
Electromagnetic
properties
and spectroscopic factors of the ‘I?; levels in the odd-proton with 135 < A Q 145 1351
spectroscopic factors (‘He, d) (‘Li, bHe) S Ih,,,,(Of>Y--)
EXP EXP PCC TDA “) EXP PCC
g-factors
13’Cs
1.01 0.95
0.92 0.89
0.84 l.l9(22) 0.91 0.89
1.21
1.21
I.21
half-lives [in ns]
EXP PCC PC1 TDA ‘)
B(M2, y; -+ $:) [in j~i fm’]
EXP PCC PC1 TDA ‘)
58.5 81.1
34.8 62.3 14.2
B(E3,y; + 2:) [in e2. fm6]
EXP PCC PCI TDA “)
5950 6900
4720 6210 12700
3520 5400 10700
branching
EXP PCC
0.003 0.002 0.02
0.006 0.004 0.03
ratios
PCI TDA “)
0.07 0.04 0.15
20.4 45.0 II.6
N = 82 nuclei
145E”
14’Pr
‘43Pm
0.96 0.98(15) 0.90 0.93
0.82 0.77(15) 0.9 I 0.93
0.83 0.45(7) 0.92 0.95
1.31(8) 1.21
1.14(9) I.21
1.356(g) 1.21
24.0( IO) 47.6 17.6 27.3
490(30) 1720 740 590
11.3(5) 6.5 17.7 II.4
8.4(5) 2.7 7.9 9.6
1 1000(1100) 1350 3170 3540
5000(400) 530 1680 830
4.8(2) 5.0 1.9 5.6 10.9(7) II.5 30.0 10.3 12000(800) 2350 4410 6450 0.11(l) 0.02 0.02 0.07
0.20( 1) 0.04 0.04 0.07
0.27(l) 0.09 0.13 0.04
The results of our calculations (PCC) are compared to available experimental data and to the results of the Tamm-Dancoff approximation (TDA) in a one- and three-quasiproton space. Furthermore, we show particle-core coupling results, which have been obtained by taking into account only the ground states of the even-mass cores (PCl). “) Ref. a). “) Ref. 9).
wave functions are also predicted to be of the type lhy@O:. The admixtures of excited core states amount only to a few percent. The g-factors obtained from these wave functions are obviously the single-particle values of the lh, proton, since the contributions of the core to the Ml transition matrix elements in eq. (7) are negligible. The single-particle character of the 9; states is confirmed by the spectroscopic factors and the g-factors measured. A more complicated situation is found when the reduced transition probabilities B(M2, 9; -+ 3:) and B(E3, y; -+ ST) or the half-lives of the y-; states are analysed. These quantities depend
152
W. Enghardt,
H. L;. Jijger
/ Proton
Ih, , ,2 excitorions
more sensitively on the structure of the wave functions of initial and final states. The reduced M2 transition probability in 14’Pr as well as the half-lives in 141Pr and probably in ‘39La [ref.26)] are nearly correctly predicted in our model. If another effective proton charge of e,, + 3e [see below and ref. ‘“)I was assumed. the B(E3) values and branching ratios I,(?; + $:)/I..(?; + $T ) would also be reproduced; the theoretical half-lives would be diminished in this case only by about 100, I,’ Going from r4’ Pr to 14SE~ in table 4. the differences between theory and experiment generally increase. Our model space including only one particle in the lh, orbit seems to prove inadequate for a complete description of the nuclei near to the Z = 64 subshell closure. This conjecture is supported by the mean occupation number of the lh,? proton orbit in the ground state of 144Sm which was reported z3) to amount to 1.6. Furthermore, the lh, quasiparticle energies for the odd-mass N = 82 nuclei 29), decreasing from about 1.8 MeV in r3’Cs to 0.7 MeV in 14’Eu, indicate that proton lhc configurations may already occur at 2 MeV - 3 MeV for the heavier nuclei considered. If a pure Ih, @ 0: structure was assumed for the $!; state, the excitation energies would be slightly increased (100 keV - 150 keV); for illustration see again fig. 2 (B and D). As expected. the g-factors are not changed by this model-space truncation. However. the reduced transition probabilities B(M2) and B(E3) considerably increase (table 4), since destructive contributions to the M2 and E3 transition strengths caused by configuration mixing are obviously omitted. The same effect is found, if the results of the QTDA in the one- and three-quasiparticle space are compared to those of the calculation in a one-quasiparticle basis9). The renormalization of the B(M2) values, obtained with the 0: core state only, to the results in the larger model space (table 2). where &” = 0.6gpee was assumed, requires the introduction of smaller effective y-factors. For ‘*‘Pr and ‘43Pm, as an the appreciable influence of example, we found 9:” = 0.41,95”‘. This illustrates model-space truncations to the effective electromagnetic coupling constants, which have to be used. Even smaller effective g-factors &” = 0.24g~“’ had to be taken by the experimental M2 transition rates Ejiri et ul. 2R), in order to reproduce UC el y a one-quasiparticle description. +{T)in thesen 1 b (9;
4.2. NEGATIVE-PARITY
STATES ABOVE
THE YL; ISOMERS
isomers have been observed in the Spectra of n = - 1 states on top of the y; 14’Eu [refs. 3.4. 1o-14)]. At present there is no nuclei 14’Pr, ‘43Pm, and experimental evidence for states with higher spins (J > ‘I’) in the nuclei ‘351, 13’Cs, and ‘39La. Besides the IC = - I levels actually observed, we have included in fig. 1 and table 3 excitation energies and wave functions of the lowest states belonging to each value y- s .I” 5 y-. The structure of all negative-parity states with ‘I’- 5 J” 5 y-- in the vicinity of
W. Enghardt, H. lJ. Jtiger / Proton Ih,,,,
the yrast line is essentially
characterized
153
excitations
by the ground
state and the 2:,
4:,
and
6: two-particle states of the even-even N = 82 core (cf. table 3). From the point of view of a simple weak-coupling picture one expects the yi”-;, q;, and y; states to be dominated by components which arise from aligned coupling of the lh, proton to the appropriate 2:, 4:, and 6: core states. In the most cases this assumption is confirmed by our results. However, with increasing mass number, strong admixtures of the basis states lh,@6: occur in the wave functions of the y; states. The reason for this lies in the structure of the 4: core states. The 4: wave functions are dominated by lgf contigurations in the lighter (A <: 140) and by 2di configurations in the heavier even-mass N = 82 nuclei [cf. also ref. ““)I. Considering the two-body matrix elements of the residual interaction one notices that for the lg+ proton the aligned, and for the 2d+ proton the antialigned, proton is energetically preferred. Having in mind coupling with the lh, furthermore the small energy splittings (< 200 keV) between the 4: and 6: core states, one can understand the large lhY@6: contributions to the y; states in the nuclei with A > 140. In the wave functions of the y-;, y-;, and y-; states only minor admixtures of fully aligned basis states are found, since the l:, 3:, and 5: core states lie about 0.5 MeV above the 6: level. Considering the shell-model wave functions of the core states, one obtains detailed predictions for the microscopic structure of the negative-parity states, In by lh,@lgi threethe lighter isotones the y;, yi’;, and $; states are dominated particle configurations. Due to the tilling up the lg, shell, structural changes occur around 14’Ce. In the heavier isotones the y; states have a nearly pure lh,@2d: structure, whereas with increasing mass number in the yi’; and y; wave functions the lh,@(lg,, 2d,) configurations compete more and more with the lh,@2di ones. For the $r; and 9; states the major components are lh, 0 (lg,, 2d,), and only in the lightest nuclei 1351 and i3’Cs appreciable admixtures of the lh+@lg$ configurations are found. The comparison of experimental and theoretical excitation energies in fig. 1 shows a fairly good correspondence for most of the levels of 14’Pr. In 143Pm and especially in 14’Eu the majority of the theoretical levels is predicted at too low energies, which merely reflects the deviations of the shell-model results for the corestate energies from their experimental values. However, the density and the sequence of these levels are nearly correctly reproduced. This picture is essentially disturbed by the y; levels of 141Pr and 143Pm as well as by the $7, y;, and $7 ones of 14’Eu. Here, configurations outside of our model space should become important. Low-lying collective 3; states are known in nearly all even-mass essential admixtures of neutron N = 82 isotones. Certainly they contain particle-hole excitations 31). Therefore, a description as (1s; ’ @ 3;), 5 9 or (2di’ 0 3;), 5 9 states may be more suitable. In the case of 14’Eu this picture has been successfully applied l1 -13). Similarly, one estimates the excitation energies of such JZlevels in 14’Pr and 143Pm to be 2230 keV and 2082 keV, respectively. 2
154
W. Enghurdr.
H. U. Jkger / Praron Ih, ,i2 excisions
A comparison of theoretical and experimental electromagnetic data is shown in table 5 for the IC= - 1 states with an unambigous spin-parity assignment and with theoretical excitation energies which suggest our particle-core coupling picture to be valid (see above). The result obtained for the half-life of the y; state in 14’Pr, which is deexcited by a pure Ml transition of 92 keV to the y; state, is in moderate agreement with the experiment. But, a too small lifetime is predicted for the -state of 14’Eu. Such an underestimation is caused by the reduced E2 transftiron probability B’““(E2, $; -+ ‘ii;) = 40.3 ez. fm4, which drastically TABLE5 Particle-core coupling predictions on electromagnetic properties of negative-parity states with in comparison to experimental data Experiment
J” 2 yw
Theory
half-lives [in ns] ‘4’PrB) 14’Eu “)
T,:2(y;, 3020 keV) T’,&?iL;. 2836 keV)
0.2(l) 5.5(5)
I,(J: + J;. EJ ___..-.--l&J; + J;. EJ
branching ratios
l&% -+$.--~ 301 keV) ---.... I,($ + ft;, 447 keV)
lb3PrnC)
IdsEu
I,(y;
-+ y;,
364 keV)
I,#il;
+ 9;.
447 keV)
!,(y;
-ry;,240
l&y;
-+ yLh 530 keV)
I,@;
+ff.
keV)
‘43Pm ‘)
‘*‘Eu
*)
Ref. ‘).
“1 Ref. ”
3.2
0.43
1.9
0.007
U(2)b) 4.S( 12)d) 4.6(g)‘)
0.05
22 keV)
I,($% + y;,
553 keV)
_ 0.36bl
0.01
f.,(q;
-t 9;.
262 keV)
I,(%
+ y;,
553 keV)
0.33(9) b) 1.06(21)~) l.00(28)L)
0.002
EZ/M I mixing ratios
a(J; + J;. E,)
“‘~r)
0.8 0.2
a(?; s(yP; S(y; S(y; S(f;
+ 9;. -+ y;, -+ ‘p;. + 9;. --t 9;.
92 keV) 194 keV) 301 keV) 376 keV) 240 keV)
S(y;
-* y;,
530 keV)
).
. ‘) Ref. *).
“) Ref. I’).
0 Oor O.O7cd< 0.15 Oor -0.5
‘) Ref. 13).
0.00 -0.17 - 0.08 - 0.02 -0.23 -0.23
W. Enghardt, H. U. Jtiger 1 Proton Ih, ,,2 excitations
155
exceeds the experimental value 15) of BexP(E2, y; + y-;) = 0.14&0.03 e2. fm4. Accordingly, the contributions of the lhy@2: and lh,@6: basis states to the y; and y; wave functions, respectively, may be still larger than given in table 3. This would confirm the results of Kaczarowski et al. Is). For the remaining branching and mixing ratios in 143Pm and 145Eu only a partial agreement of theory and experiment is found, indicating that for these nuclei only qualitative conclusions on the structure of the 7c = - 1 levels above the y-; isomers can be drawn from our wave functions. Owing to the better correspondence of theory and experiment for the excitation energies and for the decay properties of the yi’; and f; isomers of 14’Pr, one expects our model to provide a more reliable description in this case. Unfortunately, there do not exist any more suitable electromagnetic data for a detailed test of the 141Pr wave functions, because many of the negative-parity levels reported 3, lo) are deexcited via intensive El transitions, which are forbidden in our model space containing only the lg,, 2d+, 2d,, 3s,, and lh, orbits.
5. Summary A particle-core coupling model, where a lh, proton is coupled to a restricted set of TC= + 1 shell-model wave functions, is applied to negative-parity states in oddmass N = 82 isotones. In the particle-core coupling basis we diagonalized the same MSDI hamiltonian formerly proposed by Wildenthal ‘) for positive-parity states. This concept has been proved to yield an approximation to the results of shellmodel calculations in a large configuration space and to reduce the necessary computational labour. In this way an extension of the shell-model calculations of TC= + 1 states in N = 82 isotones ’ -4) becomes more practicable. The wave functions for the states with y- 5 J” S y- in the vicinity of the yrast line show that a stretched coupling of the lh, proton to the lowest core states (O:, 2:, 4:, 6:) is favoured. The influence of the valence proton number on the structure of the low-lying I( = - 1 states has been discussed. The spectroscopic data of the y-; states could be calculated in satisfactory agreement to the experiments. For the higher-lying states in i4rPr a fairly good correspondence with the results of measurements was obtained, too, whereas for the heavier systems ‘43Pm and 14’Eu the theory does not fit the experiments so well. However, the sequence and the density of the major part of the R = - 1 levels are also reproduced. The $;, y-;, and y; levels in 145Eu and the y; ones in 14iPr and ‘43Pm have been supposed to have a lg, @ 3; or 2d, @ 3; structure, which cannot be handled within our model space on account of the collectivity of the 3; states in even-mass N = 82 isotones. Finally we mention that the domain of applicability of the model can be extended from the odd-mass N = 82 isotones to other semimagic nuclei where single-nucleon excitations in an empty shell are expected. Calculating e.g. the
156
W. Enghardt.
H. L’. JGger / Proton lh, ,:2 excitations
- 1 spectra caused by proton excitations3’) in r:iCe,, and by neutron excitations 33) in l&Sn,,, we succeeded in predicting spectroscopic data in remarkable agreement with experiment.
I[ =
We would like to thank Drs. H. Prade, F. Stary, and G. Winter for carefully reading the manuscript and for their helpful comments. This work was supported by the “Ministerium fur Wissenschaft und Technik” and the “Akademie der Wissenschaften der DDR”.
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