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Equilibrium deformations of isomers in the region 50 ≦ Z and N ≦ 82
Nuclear Physics 58 (1964) 481--490; (~) North-Holland Publishing Co., Amsterdam Not to
b© reproduced by photoprint or m i c r o f i l m without written permission from the publisher
EQUHJBRIUM DEFORMATIONS OF ISOMERS IN THE REGION 50___ z AND N--<- S2 F. DEMICHELIS and F. IACHELLO t
Istituto di Fisica Sperimentale del Politecnico, Torino Received 14 February 1964 Equilibrium deformations o f the isomers in the region 50 ~ Z and N ~ 82 are calculated using Nilsson's single-particle model. The values of some Nilsson parameters which give a predicted level sequence in agreement with the experimental one are established. Finally, a tentative interpretation is given of the singular occurrence of nearly all odd-neutron isomers in this region.
Abstract:
1. Introduction
Investigations of nuclear isomerism have provided a very important contribution to the fundamental knowledge of nuclear structure. The isomers known to date x) are shown in fig. 1. We can see that while the oddneutron, odd-proton and odd-isomers are uniformly enough distributed in the different regions of the nuclear periodic table, in the region where both proton and neutron numbers vary from 50 to 82, except for some odd isomers, nearly all are odd-neutron isomers. On the other hand this region can be considered as a region of weak deformation a). Indeed the experimental deformation 6, deduced from the measured electric quadrupole moments Qo by means of the relation
Qo = ~6ZR2o( 1+½6+ . . . ) (where Ro is the average nuclear radius), has a value 3) contained in the closed interval from 0.16 for Te tz5 to 0.07 for Ce a4°. Therefore we shall attempt a systematic analysis of the available evidence concerning the ground states and excited levels of the odd-mass isomers in this region by the application of the collective model. We have examined the nuclear equilibrium shape in terms of the independentparticle model with a deformed potential field, as considered by Nilsson 4). According to this model, the nuclear equilibrium shape corresponds to that value of 6 which makes the total energy a minimum. In order to obtain the equilibrium deformations, we have chosen the method which uses volume-conservation in the same way as Mottelson and Nilsson 5). t Undergraduate, Nuclear Dpt., Politeenico, Toting. 481
482
F.
DEMICHELI[S
AND
F.
IACHELLO
The total energy is computed from the equation
,t(~) = ~ r, ~,(~)+¼ T.,
(1)
where St(&) are single-particle eigenvalues of the total Hamiltonian H and the (Us) is a correction term which takes into account that the potential deviates from the pure oscillator form. Now the term (Us) is practically independent of the defor-
x odd o odd o
Z
even odd
N
82
~ A ~oz
50
.o~ ° o
xzxo
2C
o
0 2
8
20
50
82
126
N
Fig. I. Distribution of the isomers in the nuclear periodictable versus neutron and proton numbers. The vertical and horizontal lines represent closed shells,
mation and therefore the equilibrium deformation corresponding to a given configuration is not significantly affected by the correction term; thus, we have calculated the sum of the energies 8~(&) for neutrons and protons, respectively (assuming a definite orbital assignment for each nucleon) as a function of the deformation of the potential and have found the absolute minimum that corresponds to equilibrium. The Coulomb interaction between protons and the pairing-force effects' have been neglected in our calculations. A way to take into account the Coulomb interaction is to add to eq. (I) the Coulomb energy which, for a uniformly charged spheroid
EQUILIBRIUM DEFORMATIONS
483
of an average radius R o , is Z 2 e2
where We neglect the correction term Ec in the total energy; while the Coulomb energy affects remarkably the "envelope" curve of the Nilsson levels, it does not appreciably affect the position of the energy minimum for a particular configuration. Recently B~s and Szymanski e' 7) have calculated equilibrium deformations for the deformed nuclei in the region of the rare earths and the actinides taking into account the Coulomb interaction and the pairing force. Extending the calculations on the basis of the Nilsson model to the nuclei outside the well known deformed regions, Marshalek, Person and Sheline 2) have neglected these two correction terms. However, their results are 10 to 20 % lower than the results of B~s and Szymanski in the regions of deformation. On the other hand, according to Marshalek et al. the inclusion of Coulomb effects without pairing gives rise to excessively large deformations. In our calculations we have therefore used eq. (1) in order to establish those values of the Nilsson parameters K and # which provide good agreement between the experimental and the predicted level sequence for the various isomers in this region. Furthermore we have confirmed, also in this region, Peker's hypothesis 8). According to Peker the degrees of deformation of the different states of the isomers can differ greatly in the transition regions; e.g. the ground state can be spherical while an excited state corresponds to some stable deformation. Finally we give a tentative interpretation of the singular occurrence of isomers in this region which are nearly all odd-neutron ones. 2. Method of Calculation It is known that the interaction of a nucleon with the nuclear field, as described by Nilsson 4), is expressed by the Hamiltonian H = f-Io+Ich~oR,
where R = ~ l U - 2 1 • $--[112.
The parameter ~/is related to the deformation 6 of the nuclear potential and is ~
_ ~ 60~o(6) K
030
~
6_ [ 1 - x4~~ 2 - ~ 1w6 )v~ 3 1j- ~ • K
F r o m the diagonalization of the operator R, Nilsson obtains the eigenvalues rNa(r/); the corresponding energy eigenvalues of the total Hamiltonian then are E s a = ( N + ½)ho)o (6) + Ich~o r Na,
(2)
484
F. DEMICHELIS AND F. IACHELLO
where the quantum number N represents the total number of oscillator quanta. The three parameters in the Nilsson model are then hi5 o = 41/A ~ MeV, which is the frequency associated with the harmonic oscillator potential, the parameter r, which measures the strength of the spin-orbit potential; the parameter /~, which depends on the value of the 12 term. The values of x and/~ are chosen in order to reproduce the level sequence in spherical nuclei. For the neutrons in the N = 4 and N = 5 shells, Nilsson performed the diagonalization of R with # = 0.45; for the protons in the N = 4 shell, the calculation was performed for/~ = 0.55. The value of r was taken to be 0.05 for all levels. Mottelson and Nilsson 9) assume the value x = 0.0613 for protons in N = 4 shell. A better agreement with the empirical level spectra is obtained by assuming these last values for # and x, respectively. As noted before Marshalek et aL have calculated deformations for ground states of even nuclei in this region assuming for # and x the values suggested by Nilsson. In the case of neutrons, according to these authors, the N = 4 shell (including gl) should be shifted down by an amount at least equal to 0.30 h& 0 relatively to the original Nilsson level scheme; further the N = 5 shell except hq should be shifted, too. In the case of protons in the same region, the N = 5 shell (including hq) should be shifted down by an amount equal to 0.20 ht~ o, whereas the energies in the upper proton shell should be increased by at least 0.15 h&o. If we accept these prescriptions, however, there is no satisfactory correspondence in this region between the experimental spin assignments for the levels of the isomers and those predicted from equilibrium deformation. We have therefore adopted the following procedure. For the various odd-neutron isomers we have summed the successive orbitals of Nilsson for each value of 6 (taking into account that each orbital is filled with two nucleons corresponding to the double degeneracy of states with opposite projection of angular momentum on the nuclear axis) and for all the most probable single-particle configurations. Among the various configurations, several correspond to nuclear energies which give either no minimum or a minimum corresponding to anomalously large deformations (6 > 0.40). These configurations evidently are not taken into account. Suitable shifts of some Nilsson levels have been introduced in order to obtain the empirical spin sequence of odd-neutron isomer. These shifts were obtained by means of successive approximations. The values suggested by Nilsson were thus modified step by step in order to shift the minimum to the point of agreement with the experimental level sequence. A change in # and k gives rise to a change in the second term of the Hamiltonian H; a change in # also produces a change in R and therefore in the eigenvalues r(r/); a change in X does not alter the eigenvalues. Both variations give rise to a slight alteration in the shape of the orbitals, and, more significantly, to an alteration of the spread among the levels and hence among the energy curves for the various configurations.
EQUILIBRIUM DEFORMATIONS
485
First of all we have examined the odd-neutron isomers falling in the interval 64 < N < 82 beginning with the s oSn isomers because the number of protons corresponds to a closed shell. This analysis suggests that the shell N = 5 h q should be shifted up relatively to the original Nilsson scheme. This follows from the fact that the single-particle configuration corresponding to the [505 ~ - ] level is generally observed among the second-excited states of the odd-mass isomers in the region under consideration. The N = 5 h~ shift can correspond to a change m one or both of the Nilsson parameters # or x for N = 5. Subsequently we have examined the nuclei falling in the interval 50 ~ N < 64, beginning with the 4oZr nuclei because the number of protons corresponds to a semiclosed shell. We observe that in this region it is not necessary to introduce any shift. Indeed a change in # or in ~: does not affect appreciably the energy curves for the most probable configurations. The position of the Nilsson levels for this side of the region is not in contradiction with the observed order in the spin-sequence of the Z r nuclei. Finally we have analysed the odd-neutron isomers in the intervals 50 < Z < 64 and 64 < N < 82 using the values of/z and x obtained as described above. The conclusions just deduced for the neutrons in the range 50 -< N < 64 have been extended readily to the protons in the same range.
3. Results The independent-particle energy curves corresponding to the different configurallTm ~ 121m ~ 127m tions for 5o;~n67 , 5o;~n71 and 5obn77 are plotted in figs. 2-4, respectively.
1
1~/~ L%
/ -02
; /
f,-
402 3'/2'
34.
keY
/
j
400 I I ~
i
-6
117m
so ~ n 07
l
-4
,
keV
'1/2.---T~320
lY2---~3s0
3 / 2 - ~
10~
3/2 . . . . . . .
I/2.--
o
~
~oo
o
*
-2
Fig. 2. Equilibrium calculations corresponding to a number of intrinsic configurations in Sn llvm. The total energy E/he9 o is plotted from an arbitrary level. On the left is shown the experimental level sequence, on the right the predicted one. We see that the difference in energy between the two minima of the curves relative to the two configurations [400 ½+] and [402 3 +] is about the same as the intervals between the experimental levels. For N = 5, h ~ , one has K = 0.035,/~ = 0.30. The three neutrons outside the semiclosed shell are placed, respectively in the orbitals [402 3+] 2, [400 ½+]1; [400 ½+]2, [402 ]+]x; [400 ½+]2, [411 ½+ix; [402 3+] 2, [505 ~t-]~ and [400 ½+]~, [505 ~t-]l.
486
F. DEMICHELIS AND F. IACHELLO
The summations were performed at intervals of 6 = 0.05 for various combinations of neutron and proton numbers and the minimum is thus found with an error of +0.02. ~" 5o
121m
~1171
A
sos rv2Ll ~02
B key
. . . .
~'2+
1/2* -
bOOI ~ + ~ (~.)
-
2ooo 1920
keY
11/2._
So°
'
o
_
o
sos "/2-
-6
TI"°
4o2 3~. ~
/
A
~.5
-z,
-2
,160
I/2
[3
0.5
.2
+z,
.6
~
/~11 1/2 ÷
~
-6
-t.
-2
-2
,4
+6
=
Fig. 3. Equilibrium calculations for Sn ztlm including different intrinsic configurations. (A) The N 5, hJ/. shift is relative to the configuration [411 ~r+]; K = 0.040,/~ = 0.375. (B) The N = 5, h ¥ shift is relative to the configuration [400 ~r+]; K = 0.050,/z = 0.40. The best solution seems to be (B); indeed there is a satisfactory agreement between the experimental and the predicted levels.
C 127 50 J r 1 7 7
keY
9/2
CIy2 .).~
---12oo
7/2~
~720 o
-' Fig. 4. Equilibrium calculations for Sn127 including different intrinsic configurations. The values of K and # are the Nilsson's values.
EQUILIBRIUM DEFORMATIONS
487
From the analysis of the curves obtained for the Sn isomers (only some of which are exhibited in figs. 2-4), we can deduce the various shifts and the corresponding values o f # and i¢; these are presented in table 1. TABLE 1 Values of ,¢ and/~ for N = 5 h ~ deduced from the shifts of the h ¥ neutrons Nucleus
Shift ~o0
K
/~
Predicted level sequence
50~"67
Rnt17m
0.53
0.035
0.30
~-
RnlI9m 50-'-69
0.53
0.035
0.30
~ - - --~ }+ --~ ½+
0.43
0.040
0.30
a ~ - - - * ½+
SO~tt71R-121m
0.06
0.050
0.40
~-
--~ ~+ --~ ½+
0.32
0.040
0.375
~-
._~ ½+
0.16
0.048
0.40
~-
-+ }+ -+ ½+
0.53 0.07
0.o4o
0.35 0.40
¥-
--, ½+
0.050
}+
.-~ ½+ ,-+ ~-
0
0.050
0.45
~ - --, t + --, ½+
so-"77"~-127
0
0.050
0.45
so-"79~-129 Tet21m 50---69
0 1.01
0.050 0.018
0.45
0.18
"-* ½~/'-- --~ ~+ ~ ½+
0.77
0.025
0.25
~t-__~ ½+
~t- _~ }+
0
0.050
0.45
z~-__~ ~+
~-
50-u73R-123m R-125m
50~'75
sO_,Z79n.135m
_.~ ~+ ~ ½+
Experimental lever sequence ~t-
_+ ~+
~-
~
~
½+
~+ --+ ½+
( z ~ - ) ~ (}+) (¥-)
~ ( t +)
½+ - , ( t +) - ,
¥-
I½- -~, t -
..., ½+
... ~+
There is a satisfactory agreement between the predicted and the experimental level sequence, except for Sn l~sm. The uncertain experimental angular momenta are in parentheses. The values of K and # are lower than Nilsson's values for Sn 117m and Sn 119m and approach those values for all the other isomers. The very low values of K and/~ for To ls~= are probably caused by the presence of the protons orbitals which affect the energy curves.
1.0 431 1/2 •
j
413 5/~*
422 3~2+ 93 4 0 Z I" 53
.4047/2.~~----'j~
as
keV 1/2 + ~ - 1 5 0 0 3/2+ - 1395 ,~2 " * " 1330
Fig. 5. Equilibrium calculations for Zr ~3 including different intrinsic configurations.
488
F. DEMICHELIS AND F. IACHELLO
The values o f the parameters are functions o f the neutron number outisde the closed shell; they are smaller than the N i l s s o n values for nuclei with N far f r o m the closed shell while they approach those values near the closed shells (as one sees in table 1).
l
,o.
4 0 Z i .-g5 55 keV ?,/2*
0.5
,-270
(512,) 404
0'x . ~-~413
404 7/2./
-450
~.
0
7/;~+
Fig. 6. Equilibrium calculationslfor Zr.5 including different intrinsic configurations.
1.0
~52- e
121m 69
A
3~+ I/2" key
4111/ s~2501 ] ~
-6
-4
keV ~2880 ~2800
; --214 1/2.-- ,. o
0.5
-2
0
*2
-4
l h + ~
o
~ . _ _ V2"
key "-220 100
0
*6
Fig. 7. Equilibrium intrinsic configurations for Te x~am inducting different intrinsic configurations. (A) The N = 5, h~t shift is relative to the configurations [411 ½+]; x = 0.025,/~ = 0.25. (B) The N = 5, h¥ shift is relative to the configuration [400 ½+]; ~ = 0.018,/~ = 0.18. These conclusions could m e a n that in the far case we should use instead a pure oscillator potential, while in the near case w e approach a square well model. The energy curves for Zr 93 and Zr 95 including different intrinsic configurations are displayed in figs. 5 and 6. In figs. 7 and 8 are shown the energy curves for Te 121m and for Ba 13sin. The values o f / z and x deduced f r o m the shift m i n i m a are presented in table 1 and the predicted equilibrium deformations are listed in table 2.
489
EQUILIBRIUM DEFORMATIONS
/
\ 4o2 3h. \
1
/
// I"3
135m
56 g " 3 79
keV keV
11h---
loo
505 1~, -
-s
~
-2
o
+i
.;
,i
Fig. 8. Equilibrium calculations for Ba ls6m including different intrinsic configurations. The six protons outside the closed shell are placed in the configurations [411 ~+]~ [402 ~+]2 [420 ½+]2 for the neutron configurations [402 t +] and [505 z~-] b and in the configurations [411 t+] ~ [402 t+] z and [404 ~+]2 for the neutron configuration [505 z~-] a. TAnL~ 2 Predicted equilibrium deformations Experimental ground state spin
Predicted ground state spin
(calc.)
50~"67~-117m
{+
½+
---0.20
SO-n69~-119m 50 ~"7~-121ml ~n 123m 50~"73 50-U7s~-t25m .~_ 127 50~t177 ~_129 50~u79 $2__u69T -121m
½+ ({+) ({+)
½+ ½+ ½+
---0.08 ----0.15 -----0.25
(]~-)
~+ ~-
------0.25 .---0.25
½--
-----0.20
½+
------0.15
Nucleus
~+
gt~135m $6u~79
a~t-
--0.15
t +
1+
--0.12
~+
3+ ~+
+0.10 --0.08
~+
~+
t+
~+
~-0.18
½+
---0.22
4oZr9~
-0.20
4oZr~ 4oZrs977
0
The values of the deformation obtained from the electric quadrupole moment s) for the nuclei in this region fall in the range from 0.16 for Te 1B5to 0.07 for Ce 1~0. A comparison of these values with the values of the calculated deformation is possible for Sn 11°mand Ba xsSm only, and in this case there is a satisfactory agreement. Indeed the values of 6 obtained from the electric quadrupole moment are ~ = 0.08 for Sn ~l'm and ~ = 0.12 for Ba ~s6m, respectively. The deformations of the ground states of some Zr nuclei, calculated in order to deduce a possible shift of the orbitals dg and g~, are also listed. In some case the deformations relative to two different almost equally probable configurations are presented. This indetermination could be removed by experimental data on excited states in these nuclei.
490
F. DEMICHELIS AND F. IACHELLO
4. Conclusions F r o m the results of our equilibrium calculations for odd-neutron isomers in the region 50 ~ Z and N _~ 82 we can deduce the following conclusions: (i) The minima of the energy curves indicate a very great preference for oblate deformation. This is in agreement with the fact that quadrupole moments are negative. (ii) The deformations for which there is a minimum in the energy curves are of tht same order of magnitude as the 6 values obtained from the experimental data on the electric quadrupole moment of the nuclei in this region. (iii) The deformations corresponding to ground states are different from those corresponding to excited states; indeed, also in this region Peker's hypothesis is valid. Finally, we attempt an interpretation o f the singular occurrence of the numerous odd-neutron isomers relatively to the odd-proton isomers in this region. The nuclei belonging to this region fall in the range 64 < N < 82 and 50 < Z < 64. With odd N thus included between 64 and 82, it is possible to realize several configurations for which the energy curves each give rise to a minimum and these minima are in fact very d o s e to each other. The existence of these close minima, differing slightly in energy but greatly in angular momentum, implies a great stability and, hence, a long half-life. This, in turn, would justify the very existence of isomers. On the contrary with odd Z included between 50 and 64, few configurations give rise to energy curves with a minimum and frequently the minima are far apart. Indeed only the [402 ~r+ ] and [411 ½+] levels give the greatest contribution to the minimum of the energy curves. The existence of these far minima differing considerably in energy implies a certain instability and hence a short half-life of the excited states. This asymmetrical behaviour of Nilsson levels for the values of Z and N could justify the almost exclusive presence of odd-neutron isomers in this region. The region 82 ~ Z and N ~ 126 is similar to the region examined. An analysis of this region should confirm our tentative hypothesis, but, unfortunately, experimental data thereon are lacking at present. References 1) K. Way et ai., Nuclear Data Sheets, National Academy of Sciences (National Research Council, Washington, 1958, ft.); L. I. Rusinov, Usp. Fiz. Nauk (Soviet Physics) 4 (1961) 282; V. I. Gol'danskii and L. K. Peker, Usp. Fiz. Nauk (Soviet Physics) 4 (1961) 291 2) E. Marshalek, L. W. Person and R. K. Sheline, Revs. Mod. Phys. 35 0963) 108 3) K. Okamoto, Phys. Rev. U0 0958) 143; N. Zeldes, Nuclear Physics ? (1958) 27 4) S. (3. Nilsson, Mat. Fys. Medal. Dan. Vial. Selskab. 29, No. 16 0955) 5) B. Mottelson and S. G. Nilsson, Mat. Fys. Skr. Dan. Vial. Selsk. 1, No. 8 0959) 6) D. B~ and Z. Szymanski, Nuclear Physics 28 0961) 42 7) Z. Szymanski, Nuclear Physics 26 (1961) 63 8) L.K. Peker, Izv. Akad. Nauk SSSR (scr. fiz.) 20 (1956) 957 9) B. R. Mottelson and S. G. Nilsson, Phys. Rev. 49 0955) 1615
b© reproduced by photoprint or m i c r o f i l m without written permission from the publisher
EQUHJBRIUM DEFORMATIONS OF ISOMERS IN THE REGION 50___ z AND N--<- S2 F. DEMICHELIS and F. IACHELLO t
Istituto di Fisica Sperimentale del Politecnico, Torino Received 14 February 1964 Equilibrium deformations o f the isomers in the region 50 ~ Z and N ~ 82 are calculated using Nilsson's single-particle model. The values of some Nilsson parameters which give a predicted level sequence in agreement with the experimental one are established. Finally, a tentative interpretation is given of the singular occurrence of nearly all odd-neutron isomers in this region.
Abstract:
1. Introduction
Investigations of nuclear isomerism have provided a very important contribution to the fundamental knowledge of nuclear structure. The isomers known to date x) are shown in fig. 1. We can see that while the oddneutron, odd-proton and odd-isomers are uniformly enough distributed in the different regions of the nuclear periodic table, in the region where both proton and neutron numbers vary from 50 to 82, except for some odd isomers, nearly all are odd-neutron isomers. On the other hand this region can be considered as a region of weak deformation a). Indeed the experimental deformation 6, deduced from the measured electric quadrupole moments Qo by means of the relation
Qo = ~6ZR2o( 1+½6+ . . . ) (where Ro is the average nuclear radius), has a value 3) contained in the closed interval from 0.16 for Te tz5 to 0.07 for Ce a4°. Therefore we shall attempt a systematic analysis of the available evidence concerning the ground states and excited levels of the odd-mass isomers in this region by the application of the collective model. We have examined the nuclear equilibrium shape in terms of the independentparticle model with a deformed potential field, as considered by Nilsson 4). According to this model, the nuclear equilibrium shape corresponds to that value of 6 which makes the total energy a minimum. In order to obtain the equilibrium deformations, we have chosen the method which uses volume-conservation in the same way as Mottelson and Nilsson 5). t Undergraduate, Nuclear Dpt., Politeenico, Toting. 481
482
F.
DEMICHELI[S
AND
F.
IACHELLO
The total energy is computed from the equation
,t(~) = ~ r, ~,(~)+¼ T.
(1)
where St(&) are single-particle eigenvalues of the total Hamiltonian H and the (Us) is a correction term which takes into account that the potential deviates from the pure oscillator form. Now the term (Us) is practically independent of the defor-
x odd o odd o
Z
even odd
N
82
~ A ~oz
50
.o~ ° o
xzxo
2C
o
0 2
8
20
50
82
126
N
Fig. I. Distribution of the isomers in the nuclear periodictable versus neutron and proton numbers. The vertical and horizontal lines represent closed shells,
mation and therefore the equilibrium deformation corresponding to a given configuration is not significantly affected by the correction term; thus, we have calculated the sum of the energies 8~(&) for neutrons and protons, respectively (assuming a definite orbital assignment for each nucleon) as a function of the deformation of the potential and have found the absolute minimum that corresponds to equilibrium. The Coulomb interaction between protons and the pairing-force effects' have been neglected in our calculations. A way to take into account the Coulomb interaction is to add to eq. (I) the Coulomb energy which, for a uniformly charged spheroid
EQUILIBRIUM DEFORMATIONS
483
of an average radius R o , is Z 2 e2
where We neglect the correction term Ec in the total energy; while the Coulomb energy affects remarkably the "envelope" curve of the Nilsson levels, it does not appreciably affect the position of the energy minimum for a particular configuration. Recently B~s and Szymanski e' 7) have calculated equilibrium deformations for the deformed nuclei in the region of the rare earths and the actinides taking into account the Coulomb interaction and the pairing force. Extending the calculations on the basis of the Nilsson model to the nuclei outside the well known deformed regions, Marshalek, Person and Sheline 2) have neglected these two correction terms. However, their results are 10 to 20 % lower than the results of B~s and Szymanski in the regions of deformation. On the other hand, according to Marshalek et al. the inclusion of Coulomb effects without pairing gives rise to excessively large deformations. In our calculations we have therefore used eq. (1) in order to establish those values of the Nilsson parameters K and # which provide good agreement between the experimental and the predicted level sequence for the various isomers in this region. Furthermore we have confirmed, also in this region, Peker's hypothesis 8). According to Peker the degrees of deformation of the different states of the isomers can differ greatly in the transition regions; e.g. the ground state can be spherical while an excited state corresponds to some stable deformation. Finally we give a tentative interpretation of the singular occurrence of isomers in this region which are nearly all odd-neutron ones. 2. Method of Calculation It is known that the interaction of a nucleon with the nuclear field, as described by Nilsson 4), is expressed by the Hamiltonian H = f-Io+Ich~oR,
where R = ~ l U - 2 1 • $--[112.
The parameter ~/is related to the deformation 6 of the nuclear potential and is ~
_ ~ 60~o(6) K
030
~
6_ [ 1 - x4~~ 2 - ~ 1w6 )v~ 3 1j- ~ • K
F r o m the diagonalization of the operator R, Nilsson obtains the eigenvalues rNa(r/); the corresponding energy eigenvalues of the total Hamiltonian then are E s a = ( N + ½)ho)o (6) + Ich~o r Na,
(2)
484
F. DEMICHELIS AND F. IACHELLO
where the quantum number N represents the total number of oscillator quanta. The three parameters in the Nilsson model are then hi5 o = 41/A ~ MeV, which is the frequency associated with the harmonic oscillator potential, the parameter r, which measures the strength of the spin-orbit potential; the parameter /~, which depends on the value of the 12 term. The values of x and/~ are chosen in order to reproduce the level sequence in spherical nuclei. For the neutrons in the N = 4 and N = 5 shells, Nilsson performed the diagonalization of R with # = 0.45; for the protons in the N = 4 shell, the calculation was performed for/~ = 0.55. The value of r was taken to be 0.05 for all levels. Mottelson and Nilsson 9) assume the value x = 0.0613 for protons in N = 4 shell. A better agreement with the empirical level spectra is obtained by assuming these last values for # and x, respectively. As noted before Marshalek et aL have calculated deformations for ground states of even nuclei in this region assuming for # and x the values suggested by Nilsson. In the case of neutrons, according to these authors, the N = 4 shell (including gl) should be shifted down by an amount at least equal to 0.30 h& 0 relatively to the original Nilsson level scheme; further the N = 5 shell except hq should be shifted, too. In the case of protons in the same region, the N = 5 shell (including hq) should be shifted down by an amount equal to 0.20 ht~ o, whereas the energies in the upper proton shell should be increased by at least 0.15 h&o. If we accept these prescriptions, however, there is no satisfactory correspondence in this region between the experimental spin assignments for the levels of the isomers and those predicted from equilibrium deformation. We have therefore adopted the following procedure. For the various odd-neutron isomers we have summed the successive orbitals of Nilsson for each value of 6 (taking into account that each orbital is filled with two nucleons corresponding to the double degeneracy of states with opposite projection of angular momentum on the nuclear axis) and for all the most probable single-particle configurations. Among the various configurations, several correspond to nuclear energies which give either no minimum or a minimum corresponding to anomalously large deformations (6 > 0.40). These configurations evidently are not taken into account. Suitable shifts of some Nilsson levels have been introduced in order to obtain the empirical spin sequence of odd-neutron isomer. These shifts were obtained by means of successive approximations. The values suggested by Nilsson were thus modified step by step in order to shift the minimum to the point of agreement with the experimental level sequence. A change in # and k gives rise to a change in the second term of the Hamiltonian H; a change in # also produces a change in R and therefore in the eigenvalues r(r/); a change in X does not alter the eigenvalues. Both variations give rise to a slight alteration in the shape of the orbitals, and, more significantly, to an alteration of the spread among the levels and hence among the energy curves for the various configurations.
EQUILIBRIUM DEFORMATIONS
485
First of all we have examined the odd-neutron isomers falling in the interval 64 < N < 82 beginning with the s oSn isomers because the number of protons corresponds to a closed shell. This analysis suggests that the shell N = 5 h q should be shifted up relatively to the original Nilsson scheme. This follows from the fact that the single-particle configuration corresponding to the [505 ~ - ] level is generally observed among the second-excited states of the odd-mass isomers in the region under consideration. The N = 5 h~ shift can correspond to a change m one or both of the Nilsson parameters # or x for N = 5. Subsequently we have examined the nuclei falling in the interval 50 ~ N < 64, beginning with the 4oZr nuclei because the number of protons corresponds to a semiclosed shell. We observe that in this region it is not necessary to introduce any shift. Indeed a change in # or in ~: does not affect appreciably the energy curves for the most probable configurations. The position of the Nilsson levels for this side of the region is not in contradiction with the observed order in the spin-sequence of the Z r nuclei. Finally we have analysed the odd-neutron isomers in the intervals 50 < Z < 64 and 64 < N < 82 using the values of/z and x obtained as described above. The conclusions just deduced for the neutrons in the range 50 -< N < 64 have been extended readily to the protons in the same range.
3. Results The independent-particle energy curves corresponding to the different configurallTm ~ 121m ~ 127m tions for 5o;~n67 , 5o;~n71 and 5obn77 are plotted in figs. 2-4, respectively.
1
1~/~ L%
/ -02
; /
f,-
402 3'/2'
34.
keY
/
j
400 I I ~
i
-6
117m
so ~ n 07
l
-4
,
keV
'1/2.---T~320
lY2---~3s0
3 / 2 - ~
10~
3/2 . . . . . . .
I/2.--
o
~
~oo
o
*
-2
Fig. 2. Equilibrium calculations corresponding to a number of intrinsic configurations in Sn llvm. The total energy E/he9 o is plotted from an arbitrary level. On the left is shown the experimental level sequence, on the right the predicted one. We see that the difference in energy between the two minima of the curves relative to the two configurations [400 ½+] and [402 3 +] is about the same as the intervals between the experimental levels. For N = 5, h ~ , one has K = 0.035,/~ = 0.30. The three neutrons outside the semiclosed shell are placed, respectively in the orbitals [402 3+] 2, [400 ½+]1; [400 ½+]2, [402 ]+]x; [400 ½+]2, [411 ½+ix; [402 3+] 2, [505 ~t-]~ and [400 ½+]~, [505 ~t-]l.
486
F. DEMICHELIS AND F. IACHELLO
The summations were performed at intervals of 6 = 0.05 for various combinations of neutron and proton numbers and the minimum is thus found with an error of +0.02. ~" 5o
121m
~1171
A
sos rv2Ll ~02
B key
. . . .
~'2+
1/2* -
bOOI ~ + ~ (~.)
-
2ooo 1920
keY
11/2._
So°
'
o
_
o
sos "/2-
-6
TI"°
4o2 3~. ~
/
A
~.5
-z,
-2
,160
I/2
[3
0.5
.2
+z,
.6
~
/~11 1/2 ÷
~
-6
-t.
-2
-2
,4
+6
=
Fig. 3. Equilibrium calculations for Sn ztlm including different intrinsic configurations. (A) The N 5, hJ/. shift is relative to the configuration [411 ~r+]; K = 0.040,/~ = 0.375. (B) The N = 5, h ¥ shift is relative to the configuration [400 ~r+]; K = 0.050,/z = 0.40. The best solution seems to be (B); indeed there is a satisfactory agreement between the experimental and the predicted levels.
C 127 50 J r 1 7 7
keY
9/2
CIy2 .).~
---12oo
7/2~
~720 o
-' Fig. 4. Equilibrium calculations for Sn127 including different intrinsic configurations. The values of K and # are the Nilsson's values.
EQUILIBRIUM DEFORMATIONS
487
From the analysis of the curves obtained for the Sn isomers (only some of which are exhibited in figs. 2-4), we can deduce the various shifts and the corresponding values o f # and i¢; these are presented in table 1. TABLE 1 Values of ,¢ and/~ for N = 5 h ~ deduced from the shifts of the h ¥ neutrons Nucleus
Shift ~o0
K
/~
Predicted level sequence
50~"67
Rnt17m
0.53
0.035
0.30
~-
RnlI9m 50-'-69
0.53
0.035
0.30
~ - - --~ }+ --~ ½+
0.43
0.040
0.30
a ~ - - - * ½+
SO~tt71R-121m
0.06
0.050
0.40
~-
--~ ~+ --~ ½+
0.32
0.040
0.375
~-
._~ ½+
0.16
0.048
0.40
~-
-+ }+ -+ ½+
0.53 0.07
0.o4o
0.35 0.40
¥-
--, ½+
0.050
}+
.-~ ½+ ,-+ ~-
0
0.050
0.45
~ - --, t + --, ½+
so-"77"~-127
0
0.050
0.45
so-"79~-129 Tet21m 50---69
0 1.01
0.050 0.018
0.45
0.18
"-* ½~/'-- --~ ~+ ~ ½+
0.77
0.025
0.25
~t-__~ ½+
~t- _~ }+
0
0.050
0.45
z~-__~ ~+
~-
50-u73R-123m R-125m
50~'75
sO_,Z79n.135m
_.~ ~+ ~ ½+
Experimental lever sequence ~t-
_+ ~+
~-
~
~
½+
~+ --+ ½+
( z ~ - ) ~ (}+) (¥-)
~ ( t +)
½+ - , ( t +) - ,
¥-
I½- -~, t -
..., ½+
... ~+
There is a satisfactory agreement between the predicted and the experimental level sequence, except for Sn l~sm. The uncertain experimental angular momenta are in parentheses. The values of K and # are lower than Nilsson's values for Sn 117m and Sn 119m and approach those values for all the other isomers. The very low values of K and/~ for To ls~= are probably caused by the presence of the protons orbitals which affect the energy curves.
1.0 431 1/2 •
j
413 5/~*
422 3~2+ 93 4 0 Z I" 53
.4047/2.~~----'j~
as
keV 1/2 + ~ - 1 5 0 0 3/2+ - 1395 ,~2 " * " 1330
Fig. 5. Equilibrium calculations for Zr ~3 including different intrinsic configurations.
488
F. DEMICHELIS AND F. IACHELLO
The values o f the parameters are functions o f the neutron number outisde the closed shell; they are smaller than the N i l s s o n values for nuclei with N far f r o m the closed shell while they approach those values near the closed shells (as one sees in table 1).
l
,o.
4 0 Z i .-g5 55 keV ?,/2*
0.5
,-270
(512,) 404
0'x . ~-~413
404 7/2./
-450
~.
0
7/;~+
Fig. 6. Equilibrium calculationslfor Zr.5 including different intrinsic configurations.
1.0
~52- e
121m 69
A
3~+ I/2" key
4111/ s~2501 ] ~
-6
-4
keV ~2880 ~2800
; --214 1/2.-- ,. o
0.5
-2
0
*2
-4
l h + ~
o
~ . _ _ V2"
key "-220 100
0
*6
Fig. 7. Equilibrium intrinsic configurations for Te x~am inducting different intrinsic configurations. (A) The N = 5, h~t shift is relative to the configurations [411 ½+]; x = 0.025,/~ = 0.25. (B) The N = 5, h¥ shift is relative to the configuration [400 ½+]; ~ = 0.018,/~ = 0.18. These conclusions could m e a n that in the far case we should use instead a pure oscillator potential, while in the near case w e approach a square well model. The energy curves for Zr 93 and Zr 95 including different intrinsic configurations are displayed in figs. 5 and 6. In figs. 7 and 8 are shown the energy curves for Te 121m and for Ba 13sin. The values o f / z and x deduced f r o m the shift m i n i m a are presented in table 1 and the predicted equilibrium deformations are listed in table 2.
489
EQUILIBRIUM DEFORMATIONS
/
\ 4o2 3h. \
1
/
// I"3
135m
56 g " 3 79
keV keV
11h---
loo
505 1~, -
-s
~
-2
o
+i
.;
,i
Fig. 8. Equilibrium calculations for Ba ls6m including different intrinsic configurations. The six protons outside the closed shell are placed in the configurations [411 ~+]~ [402 ~+]2 [420 ½+]2 for the neutron configurations [402 t +] and [505 z~-] b and in the configurations [411 t+] ~ [402 t+] z and [404 ~+]2 for the neutron configuration [505 z~-] a. TAnL~ 2 Predicted equilibrium deformations Experimental ground state spin
Predicted ground state spin
(calc.)
50~"67~-117m
{+
½+
---0.20
SO-n69~-119m 50 ~"7~-121ml ~n 123m 50~"73 50-U7s~-t25m .~_ 127 50~t177 ~_129 50~u79 $2__u69T -121m
½+ ({+) ({+)
½+ ½+ ½+
---0.08 ----0.15 -----0.25
(]~-)
~+ ~-
------0.25 .---0.25
½--
-----0.20
½+
------0.15
Nucleus
~+
gt~135m $6u~79
a~t-
--0.15
t +
1+
--0.12
~+
3+ ~+
+0.10 --0.08
~+
~+
t+
~+
~-0.18
½+
---0.22
4oZr9~
-0.20
4oZr~ 4oZrs977
0
The values of the deformation obtained from the electric quadrupole moment s) for the nuclei in this region fall in the range from 0.16 for Te 1B5to 0.07 for Ce 1~0. A comparison of these values with the values of the calculated deformation is possible for Sn 11°mand Ba xsSm only, and in this case there is a satisfactory agreement. Indeed the values of 6 obtained from the electric quadrupole moment are ~ = 0.08 for Sn ~l'm and ~ = 0.12 for Ba ~s6m, respectively. The deformations of the ground states of some Zr nuclei, calculated in order to deduce a possible shift of the orbitals dg and g~, are also listed. In some case the deformations relative to two different almost equally probable configurations are presented. This indetermination could be removed by experimental data on excited states in these nuclei.
490
F. DEMICHELIS AND F. IACHELLO
4. Conclusions F r o m the results of our equilibrium calculations for odd-neutron isomers in the region 50 ~ Z and N _~ 82 we can deduce the following conclusions: (i) The minima of the energy curves indicate a very great preference for oblate deformation. This is in agreement with the fact that quadrupole moments are negative. (ii) The deformations for which there is a minimum in the energy curves are of tht same order of magnitude as the 6 values obtained from the experimental data on the electric quadrupole moment of the nuclei in this region. (iii) The deformations corresponding to ground states are different from those corresponding to excited states; indeed, also in this region Peker's hypothesis is valid. Finally, we attempt an interpretation o f the singular occurrence of the numerous odd-neutron isomers relatively to the odd-proton isomers in this region. The nuclei belonging to this region fall in the range 64 < N < 82 and 50 < Z < 64. With odd N thus included between 64 and 82, it is possible to realize several configurations for which the energy curves each give rise to a minimum and these minima are in fact very d o s e to each other. The existence of these close minima, differing slightly in energy but greatly in angular momentum, implies a great stability and, hence, a long half-life. This, in turn, would justify the very existence of isomers. On the contrary with odd Z included between 50 and 64, few configurations give rise to energy curves with a minimum and frequently the minima are far apart. Indeed only the [402 ~r+ ] and [411 ½+] levels give the greatest contribution to the minimum of the energy curves. The existence of these far minima differing considerably in energy implies a certain instability and hence a short half-life of the excited states. This asymmetrical behaviour of Nilsson levels for the values of Z and N could justify the almost exclusive presence of odd-neutron isomers in this region. The region 82 ~ Z and N ~ 126 is similar to the region examined. An analysis of this region should confirm our tentative hypothesis, but, unfortunately, experimental data thereon are lacking at present. References 1) K. Way et ai., Nuclear Data Sheets, National Academy of Sciences (National Research Council, Washington, 1958, ft.); L. I. Rusinov, Usp. Fiz. Nauk (Soviet Physics) 4 (1961) 282; V. I. Gol'danskii and L. K. Peker, Usp. Fiz. Nauk (Soviet Physics) 4 (1961) 291 2) E. Marshalek, L. W. Person and R. K. Sheline, Revs. Mod. Phys. 35 0963) 108 3) K. Okamoto, Phys. Rev. U0 0958) 143; N. Zeldes, Nuclear Physics ? (1958) 27 4) S. (3. Nilsson, Mat. Fys. Medal. Dan. Vial. Selskab. 29, No. 16 0955) 5) B. Mottelson and S. G. Nilsson, Mat. Fys. Skr. Dan. Vial. Selsk. 1, No. 8 0959) 6) D. B~ and Z. Szymanski, Nuclear Physics 28 0961) 42 7) Z. Szymanski, Nuclear Physics 26 (1961) 63 8) L.K. Peker, Izv. Akad. Nauk SSSR (scr. fiz.) 20 (1956) 957 9) B. R. Mottelson and S. G. Nilsson, Phys. Rev. 49 0955) 1615