- Email: [email protected]
Hartree-Fock calculation of superheavy magic numbers
Volume 42B, number 4
HARTREE-FOCK
PHYSICS LETTERS
CALCULATION
OF SUPERHEAVY
25 December 1972
MAGIC NUMBERS*
B. ROUBEN and J.M. PEARSON Laboratoire de Physique Nuclhaire
and
G. SAUNIER Centre de Recherches Mathimatiques Universiti de Mont&al, MontrLal, Canada
Received 24 October 1972 The Vautherin-Brink force is modified by making the two-body term much more realistic, while maintaining the fit to the known nuclei in Hartree-Fock calculations. Not only z$ 114 but also g:g tzo are found to be doubly-magic.
One way of theoretically investigating possible islands of stability is to extrapolate with a single-particle (s.p.) potential fitted to known nuclei. Several different single-particle (s.p.) calculations indicate that Z = 114 andN = 184 are magic numbers, i.e., large gaps exist between occupied and unoccupied levels for the corresponding numbers of nucleons (see ref. [ 1, 2, 31 but note also the contrary conclusion of ref. [4]). Consequently, one may expect that the doublymagic nucleus tii 1,4, which, incidentally, lies right on the line of beta-stability, should have a large barrier against spontaneous fission. The s.p. model calculations [l] and [3] indicate that this is indeed the case. Another way of making the extrapolation from the known nuclei is by means of the Hartree-Fock (HF) method. Indeed, insofar as the effective interaction used gives good result for the known nuclei, this method is to be preferred to the s.p. model, since it is more microscopic. Of course, HF calculations of the fission barriers require very extensive computing facilities, especially for the superheavy region. On the other hand, HF calculations of the superheavy magic numbers have been made for some years, since these involve only spherical configurations [ 5, 6,7] . To predict the magic numbers it is highly desirable that the s.p. spectra be correctly reproduced and to do this in HF calculations it is well known that the effective interaction should be density-dependent. How-
* Work supported by the National Research Council of Canada.
ever, of the above HF calculations only that of WB [6] fulfills this requirement. (We exclude the work of Kiihler [8] since a significant part of the problem, the spin-orbit splitting, is not treated self-consistently, but rather by the inclusion of a phenomenological s.p. potential. Thus his calculations cannot be regarded as being truly HF.) In WB the density dependence is simulated by a Skyrme-type three-body force, while the two-body part of the effective interaction is highly simplified, having a delta-function radial behaviour. An extremely good fit to the binding energies, radii and s.p. energies of all known doubly-magic nuclei has been obtained with this effective interaction: see VB [9], in which paper this interaction is known as no. 1. Using this force WB confirm that Z = 114 is magic but not N = 184: they find the magic neutron number in this region to be N = 228. However, it is rather disconcerting to note that with a slightly different force, known as no. 2, VB [9] find that Z = 114 is no longer magic. The two interactions have the same form and they differ only in the parameters; the fits to experiment are both equally good and there seems to be not a priori reason for preferring the one to the other. It would appear that the very form of their interaction is highly unstable for extrapolation; possibly this is because it is completely phenomenological, bearing no relationship at all to the real nucleon-nucleon (N-N) force. In any case, it is obvious that the more realistic the effective force is, the more reliable it will be for extrapolation, assuming, of course, a good fit to the 385
Volume
42B,
number
0
4
PHYSICS
~ 21 5/z
-xx ,4
3d 3/e t 2g7/*
3~3’2 / ,
--
---_‘/
I /13/2 /
:/,‘,
/
-’
/ ,’ r--
I2 f 7/2’
_’ -.
_ljl5/2
-.
/ii
,
/-
-’
1tl9/2-
---
IV2 /’
2g 9/i
--
-/’
-5
> ; NEUTRONS
PROTONS
---3pl/2__ -\
---3sl/2--
-_
-1
’
A ,&,3/2 ..__ 2f 512 .-
id 3(2 \ \
--lhll/2-‘*:--_ \
’
-10 ‘\
;i 13/2 \ \
\
\-
\ 2d512 \
\
\ \
\
\ \
;f 7/2 \ ’
‘\ \
lg7/2
Fig. 1. Single-particle
P- =k
\
spectrum
CALCUL
EXPERIMENTAL
of *‘*Pb.
known nuclei. We have devised, therefore, an effective interaction in which the density dependence is still simulated by a Skyrme-type three-body force, but in which the two-body component is much more realistic than in the case of VB. In the even states our effective interaction takes the form v+ = ‘;;BEP
(1)
+ ‘:HEN
where P’&BEPis the mesonic potential of Bryan and Scott [ 101 , cut off at lfm in all terms, including vector and tensor, (see ref. [ 1 l] for the precise values of the parameters we have used), and k’GHENis a short range phenomenological term of the form 1 tb$exp(-p$))+h-c] + VT/T’2 exp(--y2r2)S12
386
1972
[ D, p.&(r)p + i Wo(al + a2).P X 6(r)p] . (3)
lh9/2
CALCULATED
EXPERIMENTAL
25 December
In this latter expression S,, is the usual tensor operator 3(u 1.?) (c2 .?) - u I .02, while the entire spin dependence of the central two-body component is carried by the parameter a, the sign + being taken for triplet and singlet states respectively. The inclusion of the mesonic tail imposes a strong measure of reality on our even-state effective force, in that the separation method [ 121 indicates that the central part should be identical to the real force beyond about lfm. Sprung [ 131 has reached a similar conclusion concerning the tensor part of the effective interaction, which affects mainly the spin-orbit splitting with a negligible effect on the binding. Actually, the situation here is somewhat more equivocal than for the central component but we emphasize that our results are quite insensitive to the tail of the tensor force. The role of the short-ranged vGHEN is, of course, to simulate in first order residual effects arising in all orders from the real force; in particular, virtually the entire binding effect of the real tensor force is thrown into the central part of k&EN. Our odd-state force takes the purely phenomenological form
,I--
3*:/z 1
4.1/2
LETTERS
+ f3 h(q
- r2)6
(r2 -r3)
(2)
There is no long-range conformity to the real N-N force, since the separation method does not apply to the odd state. However, there is still some measure of reality in that the central part is fitted to the oddstate G-matrix calculated by Banerjee and Sprung [ 141 for various N-N forces: this gives for the singlet state D, = 414 MeV fm, while for the triplet state it is sufficient to set D, = 0 (note that the two VB forces have attractive odd states). The parameter Wu is determined by fitting the p-state splitting in 160: we get W0 = 130 MeV fm5. All the even-state phenomenological parameters are then determined by fitting to the binding energies and radii of the known doubly-magic nuclei, and to the s.p. energies of 208Pb. The best values obtained are t3 = 12000 MeV fm6,A= -351.5 MeV, a= 0.8442 b=-3.984fm2,a= 1.2fm-l,1-(=2.53fm2, VT = 1240 MeV frnp2, y = I.5 fm-l. The resulting fit, shown in table 1 and fig. 1, is at least as good as for the two VB forces. Being furthermore fairly realistic, we believe that our force is eminently suitable for extrapolation. A more complete discussion of this
25 December 1972
PHYSICS LETTERS
Volume 42B, number 3
r
P
P
”
P
” --__
Ip l/2 ---
3F 1”2___ 31 ,312 - 2f 5/z ---
lk17/2 _-__ _-__ --- Zhll/i Ijl3/2
lil 13/2
lpY2 --!f5/2
--_ Ijl3/2 --2hll/2
!f 7/F-
2117/Z
lkl7/2
--_
i 13/2
”
3f 712
--- 2h9/i -_3f 712
!g7/2 -_-
l/2 3d5L
Ik 17/;
2g912 --_ 1jl5/2
-2
Ih19/z
lh9/2
C
g 7/;
34312 4~112
3d
5/z 29712
lj15/2
111/z s1/2
2g
91;
>I
l/2
3s3/2 ?d 3/Z
I j 15/Z
lhll/2
29912
2d5/2
--- 3f5/2
zh9/2 ----3f 7/z
2g9/2---
2hl
---
Ij
Ii1112---
131;
4~112 3d3L
3pl/2 ----3p3/2
ljl5/2 --lkl7/2
I/;
3d3/2 4s
n
29712 --- p
2h Iill/
---
71312
3~112 ----3p3/2 4sl/2 2f 5/z
3d5L
3d3/2 3dW2
2g7/; 2f7/2
2f 512---
I jl5/; Ii1312 E 2f 712
297/Z
Ii 13/T I115/2
2g9/:
2 99/i
Iill/
d3/2
Iill/;
d5/2
Ih9/2
3~112 3p3/: 298 184
II/Z
Iill/ lg7/2
114
LPI/Z 304 184
120
lh9/2
3s 112 342 228
114
348 228
120
Fig. 2. Single-particle spectra of superheavy closed-shell nuclei. force will be presented elsewhere [ 1.51 One feature of all the HF calculations
of VB and VVB is that they neglect the “de-splitting” effect of the central-force contribution from spin-unsaturated shells and we find, using their forces in our computer programme, that this effect can on occasions be significant. They justify [ 161 this neglect on the grounds that it could be compensated by including a tensor component in the interaction. This is certainly qualitatively correct and we find in fact with our own force that the tensor component is essential if we are to obtain correct s.p. spectra in spin-unsaturated nuclei. [It is to be noted that our present tensor term is some ten times weaker that that of ref. [7]. This is not because of the absence of density dependence in the earlier force, but rather because the odd states were more repulsive, being fitted to the phase shifts rather than the G-matrix: apparently the “de-splitting” is extremely sensitive to the odd-state force.] We have performed HF calculations on the superheavy nuclei “a 184 1149 304 184 no*228342 114 and iii 120. The
binding energies and radii obtained are shown in table 1 and the s.p. spectra in fig. 2. Well-defined neutron gaps are seen to occur at N = 184 and 228. However, for N = 184 there are no large profun gaps, so like WB [6] we must call in doubt the existence of an island of stability based on 184 neutrons, albeit for different reasons. On the other hand, for N = 228 large proton gaps appear at both Z = 114 and Z = 120. This dependence of the proton gaps on the number of neutrons has already been seen in ref. [ 71 and is undoubtedly related to the tensor force. VVB [6] find the opposite effect, i.e., the proton gap becomes smaller in going from N = 184 to N = 228, showing thereby that the cancellation of the “de-splitting” term by the tensor force is not exact. We feel therefore that there is a distinct advatage to our retention of the “de-splitting” terms and explicit use of a tensor force. We thus find that both “,i’, 114 and iii 120 should be strongly doubly-magic. Using a s.p. estimate for the alpha-decay energies, we find that both of these 387
Volume
42B, number
3
PHYSICS Table 1
Binding energies per nucleon (B) and charge radii (R)of doubly-magic nuclei. Experimental values are shown in parentheses.
R (fm)
B (MeV)
1.98
(7.98)
2.13
(2.73)
8.33
(8.55)
3.48
(3.49)
8.59
(8.67)
3.51
(3.481
8.53
(8.64)
3.75
(3.84)
8.65
(8.71)
4.25
(4.27)
7.91
(7.87)
5.48
(5.501
298 184 114
7.18
6.16
342 228 114
6.13
6.39
304 184 120
7.11
6.23
348 228 120
6.79
6.45
nuclei should be alpha-stable. Furthermore, although these nuclei lie far from the line of beta-stability, we estimate that the lifetimes should be of the order of minutes or hours. (This conclusion depends critically, of course, on the s.p. spectrum and hence on the details of the interaction. In fact, VVB [6] find that the decays should be much more rapid, but it is to be noted from its behaviour in 208Pb that their force raises the occupied levels and depresses the unoccupied ones, no doubt because their three-body component is too strong.) Thus there should be no problem of detecting these nuclei in the laboratory, once formed. On the other hand, we would not expect to find these nuclei in nature. Our overall conclusion is that islands of stability are possible in the region of both elements 114 and
38%
LETTERS
25 December
1972
120 (only the former was predicted by VVB while the latter was first predicted in ref. IS]). However, these are much more likely to be associated with 228 neutrons than with the more usually considered 184. All the other doubly-magic nuclei predicted in ref. [7] have now disappeared because of the compression of the s.p. spectra due to the three-body force. Valuable discussions have been had with D. Sprung, D. Vautherin and M. Ve’ne’roni. Our thanks are due to the Centre de Calcul of the Universite’ de Montreal for its generous allocation of computer time. We are indebted to Robert Girouard and Alain Brassart for their skillful computational aid.
References
111S.G.
Nilsson et al., Nucl. Phys. A131 (1969)
1.
[21 H. Meldner, Phys. Rev. 178 (1969) 1815. [31 M. Bolsterli et al., Phys. Rev. C5 (1972) 1050. 141 E. Rost, Phys. Lett. 26B (1968) 184. I51 W.H. Bassichis and A.K. Kerman, Phys. Rev. C2 (1970) 1768, Phys. Rev. C6 (1972) 370. M. Ve’ne’roni and D.M. Brink, Phys. Lett. i61 D. Vautherin, 33B (1970) 381. [71 G. Saunier and B. Rouben, Phys. Rev. C6 (1972) 591. ISI H.S. KFhler, Nucl. Phys. A170 (1971) 88. and D.M. Brink, Phys. Rev. C5 (1972) [91 D. Vautherin 626. [lOI R.A. Bryan and B.L. Scott, Phys. Rev. 135 (1964) B434. [Ill G. Saunier and J.M. Pearson, Phys. Rev. Cl (1970) 1353. iI21 S.A. Moszkowski and B.L. Scott, Ann. Phys. (N.Y.) 11 (1960) 65. [I31 D. Sprung, Nucl. Phys. Al82 (1972) 97. [I41 P.K. Banerjee and D.W.L. Sprung, Can Jour. Phys. 49 (1971) 1899. Ll51 J.M. Pearson, B. Rouben and G. Saunier (to be submitted for publication). (private communication). [I61 M. Veneroni and D. Vautherin
HARTREE-FOCK
PHYSICS LETTERS
CALCULATION
OF SUPERHEAVY
25 December 1972
MAGIC NUMBERS*
B. ROUBEN and J.M. PEARSON Laboratoire de Physique Nuclhaire
and
G. SAUNIER Centre de Recherches Mathimatiques Universiti de Mont&al, MontrLal, Canada
Received 24 October 1972 The Vautherin-Brink force is modified by making the two-body term much more realistic, while maintaining the fit to the known nuclei in Hartree-Fock calculations. Not only z$ 114 but also g:g tzo are found to be doubly-magic.
One way of theoretically investigating possible islands of stability is to extrapolate with a single-particle (s.p.) potential fitted to known nuclei. Several different single-particle (s.p.) calculations indicate that Z = 114 andN = 184 are magic numbers, i.e., large gaps exist between occupied and unoccupied levels for the corresponding numbers of nucleons (see ref. [ 1, 2, 31 but note also the contrary conclusion of ref. [4]). Consequently, one may expect that the doublymagic nucleus tii 1,4, which, incidentally, lies right on the line of beta-stability, should have a large barrier against spontaneous fission. The s.p. model calculations [l] and [3] indicate that this is indeed the case. Another way of making the extrapolation from the known nuclei is by means of the Hartree-Fock (HF) method. Indeed, insofar as the effective interaction used gives good result for the known nuclei, this method is to be preferred to the s.p. model, since it is more microscopic. Of course, HF calculations of the fission barriers require very extensive computing facilities, especially for the superheavy region. On the other hand, HF calculations of the superheavy magic numbers have been made for some years, since these involve only spherical configurations [ 5, 6,7] . To predict the magic numbers it is highly desirable that the s.p. spectra be correctly reproduced and to do this in HF calculations it is well known that the effective interaction should be density-dependent. How-
* Work supported by the National Research Council of Canada.
ever, of the above HF calculations only that of WB [6] fulfills this requirement. (We exclude the work of Kiihler [8] since a significant part of the problem, the spin-orbit splitting, is not treated self-consistently, but rather by the inclusion of a phenomenological s.p. potential. Thus his calculations cannot be regarded as being truly HF.) In WB the density dependence is simulated by a Skyrme-type three-body force, while the two-body part of the effective interaction is highly simplified, having a delta-function radial behaviour. An extremely good fit to the binding energies, radii and s.p. energies of all known doubly-magic nuclei has been obtained with this effective interaction: see VB [9], in which paper this interaction is known as no. 1. Using this force WB confirm that Z = 114 is magic but not N = 184: they find the magic neutron number in this region to be N = 228. However, it is rather disconcerting to note that with a slightly different force, known as no. 2, VB [9] find that Z = 114 is no longer magic. The two interactions have the same form and they differ only in the parameters; the fits to experiment are both equally good and there seems to be not a priori reason for preferring the one to the other. It would appear that the very form of their interaction is highly unstable for extrapolation; possibly this is because it is completely phenomenological, bearing no relationship at all to the real nucleon-nucleon (N-N) force. In any case, it is obvious that the more realistic the effective force is, the more reliable it will be for extrapolation, assuming, of course, a good fit to the 385
Volume
42B,
number
0
4
PHYSICS
~ 21 5/z
-xx ,4
3d 3/e t 2g7/*
3~3’2 / ,
--
---_‘/
I /13/2 /
:/,‘,
/
-’
/ ,’ r--
I2 f 7/2’
_’ -.
_ljl5/2
-.
/ii
,
/-
-’
1tl9/2-
---
IV2 /’
2g 9/i
--
-/’
-5
> ; NEUTRONS
PROTONS
---3pl/2__ -\
---3sl/2--
-_
-1
’
A ,&,3/2 ..__ 2f 512 .-
id 3(2 \ \
--lhll/2-‘*:--_ \
’
-10 ‘\
;i 13/2 \ \
\
\-
\ 2d512 \
\
\ \
\
\ \
;f 7/2 \ ’
‘\ \
lg7/2
Fig. 1. Single-particle
P- =k
\
spectrum
CALCUL
EXPERIMENTAL
of *‘*Pb.
known nuclei. We have devised, therefore, an effective interaction in which the density dependence is still simulated by a Skyrme-type three-body force, but in which the two-body component is much more realistic than in the case of VB. In the even states our effective interaction takes the form v+ = ‘;;BEP
(1)
+ ‘:HEN
where P’&BEPis the mesonic potential of Bryan and Scott [ 101 , cut off at lfm in all terms, including vector and tensor, (see ref. [ 1 l] for the precise values of the parameters we have used), and k’GHENis a short range phenomenological term of the form 1 tb$exp(-p$))+h-c] + VT/T’2 exp(--y2r2)S12
386
1972
[ D, p.&(r)p + i Wo(al + a2).P X 6(r)p] . (3)
lh9/2
CALCULATED
EXPERIMENTAL
25 December
In this latter expression S,, is the usual tensor operator 3(u 1.?) (c2 .?) - u I .02, while the entire spin dependence of the central two-body component is carried by the parameter a, the sign + being taken for triplet and singlet states respectively. The inclusion of the mesonic tail imposes a strong measure of reality on our even-state effective force, in that the separation method [ 121 indicates that the central part should be identical to the real force beyond about lfm. Sprung [ 131 has reached a similar conclusion concerning the tensor part of the effective interaction, which affects mainly the spin-orbit splitting with a negligible effect on the binding. Actually, the situation here is somewhat more equivocal than for the central component but we emphasize that our results are quite insensitive to the tail of the tensor force. The role of the short-ranged vGHEN is, of course, to simulate in first order residual effects arising in all orders from the real force; in particular, virtually the entire binding effect of the real tensor force is thrown into the central part of k&EN. Our odd-state force takes the purely phenomenological form
,I--
3*:/z 1
4.1/2
LETTERS
+ f3 h(q
- r2)6
(r2 -r3)
(2)
There is no long-range conformity to the real N-N force, since the separation method does not apply to the odd state. However, there is still some measure of reality in that the central part is fitted to the oddstate G-matrix calculated by Banerjee and Sprung [ 141 for various N-N forces: this gives for the singlet state D, = 414 MeV fm, while for the triplet state it is sufficient to set D, = 0 (note that the two VB forces have attractive odd states). The parameter Wu is determined by fitting the p-state splitting in 160: we get W0 = 130 MeV fm5. All the even-state phenomenological parameters are then determined by fitting to the binding energies and radii of the known doubly-magic nuclei, and to the s.p. energies of 208Pb. The best values obtained are t3 = 12000 MeV fm6,A= -351.5 MeV, a= 0.8442 b=-3.984fm2,a= 1.2fm-l,1-(=2.53fm2, VT = 1240 MeV frnp2, y = I.5 fm-l. The resulting fit, shown in table 1 and fig. 1, is at least as good as for the two VB forces. Being furthermore fairly realistic, we believe that our force is eminently suitable for extrapolation. A more complete discussion of this
25 December 1972
PHYSICS LETTERS
Volume 42B, number 3
r
P
P
”
P
” --__
Ip l/2 ---
3F 1”2___ 31 ,312 - 2f 5/z ---
lk17/2 _-__ _-__ --- Zhll/i Ijl3/2
lil 13/2
lpY2 --!f5/2
--_ Ijl3/2 --2hll/2
!f 7/F-
2117/Z
lkl7/2
--_
i 13/2
”
3f 712
--- 2h9/i -_3f 712
!g7/2 -_-
l/2 3d5L
Ik 17/;
2g912 --_ 1jl5/2
-2
Ih19/z
lh9/2
C
g 7/;
34312 4~112
3d
5/z 29712
lj15/2
111/z s1/2
2g
91;
>I
l/2
3s3/2 ?d 3/Z
I j 15/Z
lhll/2
29912
2d5/2
--- 3f5/2
zh9/2 ----3f 7/z
2g9/2---
2hl
---
Ij
Ii1112---
131;
4~112 3d3L
3pl/2 ----3p3/2
ljl5/2 --lkl7/2
I/;
3d3/2 4s
n
29712 --- p
2h Iill/
---
71312
3~112 ----3p3/2 4sl/2 2f 5/z
3d5L
3d3/2 3dW2
2g7/; 2f7/2
2f 512---
I jl5/; Ii1312 E 2f 712
297/Z
Ii 13/T I115/2
2g9/:
2 99/i
Iill/
d3/2
Iill/;
d5/2
Ih9/2
3~112 3p3/: 298 184
II/Z
Iill/ lg7/2
114
LPI/Z 304 184
120
lh9/2
3s 112 342 228
114
348 228
120
Fig. 2. Single-particle spectra of superheavy closed-shell nuclei. force will be presented elsewhere [ 1.51 One feature of all the HF calculations
of VB and VVB is that they neglect the “de-splitting” effect of the central-force contribution from spin-unsaturated shells and we find, using their forces in our computer programme, that this effect can on occasions be significant. They justify [ 161 this neglect on the grounds that it could be compensated by including a tensor component in the interaction. This is certainly qualitatively correct and we find in fact with our own force that the tensor component is essential if we are to obtain correct s.p. spectra in spin-unsaturated nuclei. [It is to be noted that our present tensor term is some ten times weaker that that of ref. [7]. This is not because of the absence of density dependence in the earlier force, but rather because the odd states were more repulsive, being fitted to the phase shifts rather than the G-matrix: apparently the “de-splitting” is extremely sensitive to the odd-state force.] We have performed HF calculations on the superheavy nuclei “a 184 1149 304 184 no*228342 114 and iii 120. The
binding energies and radii obtained are shown in table 1 and the s.p. spectra in fig. 2. Well-defined neutron gaps are seen to occur at N = 184 and 228. However, for N = 184 there are no large profun gaps, so like WB [6] we must call in doubt the existence of an island of stability based on 184 neutrons, albeit for different reasons. On the other hand, for N = 228 large proton gaps appear at both Z = 114 and Z = 120. This dependence of the proton gaps on the number of neutrons has already been seen in ref. [ 71 and is undoubtedly related to the tensor force. VVB [6] find the opposite effect, i.e., the proton gap becomes smaller in going from N = 184 to N = 228, showing thereby that the cancellation of the “de-splitting” term by the tensor force is not exact. We feel therefore that there is a distinct advatage to our retention of the “de-splitting” terms and explicit use of a tensor force. We thus find that both “,i’, 114 and iii 120 should be strongly doubly-magic. Using a s.p. estimate for the alpha-decay energies, we find that both of these 387
Volume
42B, number
3
PHYSICS Table 1
Binding energies per nucleon (B) and charge radii (R)of doubly-magic nuclei. Experimental values are shown in parentheses.
R (fm)
B (MeV)
1.98
(7.98)
2.13
(2.73)
8.33
(8.55)
3.48
(3.49)
8.59
(8.67)
3.51
(3.481
8.53
(8.64)
3.75
(3.84)
8.65
(8.71)
4.25
(4.27)
7.91
(7.87)
5.48
(5.501
298 184 114
7.18
6.16
342 228 114
6.13
6.39
304 184 120
7.11
6.23
348 228 120
6.79
6.45
nuclei should be alpha-stable. Furthermore, although these nuclei lie far from the line of beta-stability, we estimate that the lifetimes should be of the order of minutes or hours. (This conclusion depends critically, of course, on the s.p. spectrum and hence on the details of the interaction. In fact, VVB [6] find that the decays should be much more rapid, but it is to be noted from its behaviour in 208Pb that their force raises the occupied levels and depresses the unoccupied ones, no doubt because their three-body component is too strong.) Thus there should be no problem of detecting these nuclei in the laboratory, once formed. On the other hand, we would not expect to find these nuclei in nature. Our overall conclusion is that islands of stability are possible in the region of both elements 114 and
38%
LETTERS
25 December
1972
120 (only the former was predicted by VVB while the latter was first predicted in ref. IS]). However, these are much more likely to be associated with 228 neutrons than with the more usually considered 184. All the other doubly-magic nuclei predicted in ref. [7] have now disappeared because of the compression of the s.p. spectra due to the three-body force. Valuable discussions have been had with D. Sprung, D. Vautherin and M. Ve’ne’roni. Our thanks are due to the Centre de Calcul of the Universite’ de Montreal for its generous allocation of computer time. We are indebted to Robert Girouard and Alain Brassart for their skillful computational aid.
References
111S.G.
Nilsson et al., Nucl. Phys. A131 (1969)
1.
[21 H. Meldner, Phys. Rev. 178 (1969) 1815. [31 M. Bolsterli et al., Phys. Rev. C5 (1972) 1050. 141 E. Rost, Phys. Lett. 26B (1968) 184. I51 W.H. Bassichis and A.K. Kerman, Phys. Rev. C2 (1970) 1768, Phys. Rev. C6 (1972) 370. M. Ve’ne’roni and D.M. Brink, Phys. Lett. i61 D. Vautherin, 33B (1970) 381. [71 G. Saunier and B. Rouben, Phys. Rev. C6 (1972) 591. ISI H.S. KFhler, Nucl. Phys. A170 (1971) 88. and D.M. Brink, Phys. Rev. C5 (1972) [91 D. Vautherin 626. [lOI R.A. Bryan and B.L. Scott, Phys. Rev. 135 (1964) B434. [Ill G. Saunier and J.M. Pearson, Phys. Rev. Cl (1970) 1353. iI21 S.A. Moszkowski and B.L. Scott, Ann. Phys. (N.Y.) 11 (1960) 65. [I31 D. Sprung, Nucl. Phys. Al82 (1972) 97. [I41 P.K. Banerjee and D.W.L. Sprung, Can Jour. Phys. 49 (1971) 1899. Ll51 J.M. Pearson, B. Rouben and G. Saunier (to be submitted for publication). (private communication). [I61 M. Veneroni and D. Vautherin