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On nuclear octupole deformations
I 1.D.2 ]
Nuclear Physics A l l 2 (1968) 583w593; ( ~ North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
ON NUCLEAR OCTUPOLE DEFORMATIONS P. VOGEL t Joint Institute for Nuclear Research, Dubna, USSR Received 28 December 1967
Abstract: The dependence of the nuclear potential energy on the parameters of the quadrupole and
octupole deformation for nuclei in the 218 ~ .4 ~ 232 region is investigated. The energy is calculated in the framework of the superfluid nuclear model. The minimum of the potential energy surface is obtained for f18 = 0, i.e. there are no nuclei with non-zero equilibrium octupole deformation in the considered region. The dependence of the energy on the deformation differs essentially from the simple parabolic one, particularly a term of the Czafl~fl82 type is present in ¢(fl2, fla) leading to an asymmetric energy surface and to an interaction between E-vibrations and octupole vibrations. 1. Introduction
The problem of the dependence of the energy of the nucleus on the deformation parameters of higher multipole orders (2 > 2) has attracted much attention 1). A knowledge of this dependence is important for the interpretation of a very broad class of effects as the equilibrium deformation, vibrational spectra, fission barriers etc. In this paper we shall investigate the effect of the shell structure on deformations of the quadrupole f12 and octupole fla types. F o r nuclei in the 218 < A < 232 region, octupole I ~ = 1- states have very unusual properties. Their energies are very low, m o m e n t a of inertia large and hindrance factors for ~t-decay small 2). Such behaviour indicates, that for these nuclei the shell structure may strongly change the simple (liquid drop) dependence of the energy on the octupole deformation. Besides, this region of nuclei is also on the boundary between spherical and deformed nuclei. Therefore we have chosen this region for our calculations. Because the projection of the angular m o m e n t u m on the nuclear symmetry axis in low-lying I ~ -- 1- states is equal to zero, we shall limit ourselves to axially symmetric nuclei. Preliminary results of our calculations have been published 3). The problem of nuclear stability with respect to octupole deformations has been investigated 4-7). The restoring force parameter C3 was calculated using various approximations (usually perturbation theory). The restoring force consists of two terms of approximately the same magnitude but opposite sign. The quantity Caef characterizes the repulsion of the filled and unfilled shells which contributes to the deformation, Crest originates from the "inertia of closed shells" or volume censer1- Present and permanent address: Nuclear Research Institute, Re~, Czechoslovakia.
583
584
v. VOGEL
vation which counteracts the deformation. In refs. 4-7), the restoring force parameters were small and positive, i.e. the nuclei were "soft" but non-deformed. The purpose of this paper is to obtain more detailed information about the dependence of the potential energy on the parameters of the quadrupole and octupole deformation.
2. The method of calculating the ~(~2, ~s) function To calculate the energy of the nucleus, we shall use the method, which has already proved to be successful when considering the equilibrium quadrupole deformation (see e.g. ref. 1) and references mentioned there). According to this method, the energy of the nucleus is calculated as the energy of a system of nucleons interacting via pairing forces and moving in a deformed average field. Let the equipotential surface have the form
r = Ro k(fl2, fla) 1 + f12 Y:o( 8, q~)+ fla Yao(8, (P)--~ ~ f12fla Y1o(S, q~)
• (1)
The last term in the square brackets assures the constancy of the centre of mass, and the k(fl2, f13) function assures the volume conservation. The condition of volume conservation is a consequence of the constancy of the average nuclear density. The potential in analogy with Nilsson s) is of the form V --= ½ M ~ 2 2 r 2
E'
1+ ~-~(fli+fll)
x I1-2fl2
1
Y2o(,9, q~)-2flaYao(,9, q~)+18 V3-~nfl213aYlo(oa, tp)].
(2)
In the lowest approximation with respect to f12 and fls, potential (2) has the equipotential surfaces (1) and in the spherical limit is equal to the harmonic oscillator one. Let us note, that the octupole potential of the ~ r3yao type used in refs. 4,6) does not have surface of type (1), moreover, its equipotential surfaces have one branch extending to infinity. Therefore we use the octupole potential ~ r 2 y 3 0 following from (1). The single-particle Hamiltonian has the usual form h2 H(
2,133) =
-
--
2M
,t + v ( # 2 ,
B3)+ ct.
s
+D(t
(3)
The parameters C and D used in our calculations were 9) C , - - - 0 . 8 5 6 , Cp = -0.779, D, = -0.139, Dp = -0.253 (in MeV). The total energy (for a given level order) does not depend critically on these parameters.
NUCLEAR OCTUPOLE DEFORMATIONS
585
Hamiltonian (3) was diagonalized in the spherical oscillator basis, i.e.
(~, = ~ a~tazlNlAr,> NIAE
in the Nilsson e-representation s). In this representation, the only non-zero matrix elements of the quadrupole part of (3) are the diagonal A N = 0 ones. However, in the octupole part additional second-order terms arise VH56 1/7 72 V 3 Ylo(O, qg)) /~2/~3(40 \6-'~1~Y5o(O, tp)- ~ "~z Y3°('9' cp)+ f~ ~ which were neglected similarly as the change in the vectors ! and s. The non-zero matrix elements of the octupole part of (3) are the non-diagonal ones with A N = 1, 3, 5 . . . . We limit ourselves to elements with A N = 1 and 3. The influence of the others is negligible, because the energy differences are large and the matrix elements themselves are small. The effect of the octupole part of (3) on the energy of the nucleus is mainly connected with the matrix elements having A N = 1 between shells in the vicinity of the Fermi surface. Taking further shells placed symmetrically with respect to the Fermi surface into account as well as terms with A N = 3 increases the effect of the octupole part by about 20 ~. In our calculations, we took five oscillator shells for protons and five oscillator shells for neutrons and matrix elements with A N --- 1, 3. Note that filled shells which do not participate in the diagonalization contribute nevertheless to the stabilization of the spherical form owing to the volume conservation condition. The eigenvalues e(fl2, fla) of Hamiltonian (3) were used in the equations of the superfluid nuclear model 2
1
G
es_2)2+A 2
x/(es~2) ~ + Ae/ ,
(4)
and the quantities A and 2 for neutrons (132 < N < 140) and protons (86 < Z < 92) were calculated. The following values were used for the pairing interaction constants G n = 21/A, Gr, = 25/A (MeV). After solving the system (4), the energy d~(fl2, f13) was calculated according to e,-2] +2nN-A~/Gn
O~Bcs(fl2, f13) = E ( e s - 2 ) ( 1 -
+ ~ (et-2) ( 1 - et-).~ +)~pZ--A2/(~p, t
\
(5)
Et /
where E = ~ / ( e - 2) 2 + A ~ is the quasi-particle energy. The summation over s contains
586
P. VOGEL
all the neutron states and over t all the proton ones. For non-interacting particles, which however for each deformation fill the lowest ½N and ½Z levels, one obtains ~N
~z
gw(fl2, fla) = Z 28,(fl2, f13)+ Z 2~(fl2, f13). S=I
(6)
t=l
Let us note that formulae (5) or (6) contain implicitly various approximations, e.g. the Coulomb energy of the protons was neglected, the states were not projected onto the subspace with definite angular momentum and parity etc. However, for not too large deformations, neither of the mentioned approximations can change the results considerably. 3. Discussion
The single-particle energies ¢(fl2, f13), the quantities d(fl2 , f13) and 2(fl 2, f13) as well as the functions ~Bcs (f12, fla) and ¢ip(fl2, fla) were calculated for nuclei with 218 <__A <= 232 with the step 0.06 in f12 and fla. Between these points the parabolical interpolation was used, the control calculations show that errors connected with the interpolation are not substantial. The results are represented in the form of figures and tables and are discussed below. 3.1. THE Q U A D R U P O L E D E F O R M A T I O N
Fig. 1 gives the dependence of ~Bcs on f12 (for f13 = 0) for some selected nuclei. The curves are normalized so that the minimum energy is always equal to zero (this is true for all figures). Besides the expected shift of the minimum with increasing A towards the larger deformations, one can note a change in curvature from nucleus to nucleus. The curvature has a minimum value for nuclei with neutron number N = 136, i.e. these nuclei are "softest" with respect to quadrupole deformations. The equilibrium deformations obtained from our calculations are somewhat smaller than the experimental ones (see also table 1). 3.2. PAIRING CORRELATIONS
On fig. 2 the dependence of the square of the correlation function (energy gap) on the f12 deformation is shown for systems with various numbers of neutrons. The analogous picture for protons is shown on fig. 3. The increase of A2 with the increase of N or Z for zero deformation is related to the increase of the level density with the removal from a magic nucleus. The characteristic features of both figures is the existence of a rather narrow region of f12 values in which the energy gap does not depend on N or Z. In this region of deformations, the single-particle spectrum is close to an equidistant one. The periodical change of A2 with f12 is a consequence of the periodical change of the level density. Since the correlation energy is equal to -A2/G, one can see from the behaviour of A2 that pairing interactions promote the stabilization of the spherical form in spherical nuclei and lead to a decrease of the curvature of the potential energy for the deformed nuclei.
587
NUCLEAR OCTUPOLE DEFORMATIONS
The quantity A 2 decreases monotonously as the octupole deformation f13 increases. This qualitative feature is independent of Z or N and f12- A typical example o f the function A2(fla) is shown by the dashed lines on fig. 2 (N = 136) and fig. 3 (Z = 90). These curves were calculated for f12 = 0.2. One can conclude from the behaviour of the A2(fla) function that pairing correlations counteract the octupole deformation and increase the restoring force with respect to the octupole vibrations.
2~aRr~
10
a
/ 0 -0.1
o'.o
d.1
d2
[3
Fig. 1. T h e dependence o f 8Bcs o n fl~. /x 2
1.0 ¸
0.5 ¸
""-*-Z136
0.0 -0,1
0'.0 Fig. 2. T h e
3.3. G E N E R A L
0'.1
dn~(fl~)
13.2
[3
f u n c t i o n s (for fls = 0).
FEATURES OF THE DEPENDENCE
O F T H E E N E R G Y O N f18
The dependence of the energy of the nucleus on the octupole deformation has some general features which do not depend on the atomic number. To demonstrate them, we choose the characteristic example of the 224Ra nucleus shown on fig. 4,
588
P. VOOEL
The curves o~(f13) calculated using formulae (5) and (6) for different values of f12 are shown. The 6~(fla) curve has always a flat minimum for f13 = 0 and has no other minima. The curvature of 6"(fl3) decreases as f12 increases, i.e. the restoring force
1.0'
0.5.
0.o
-o.1
61
olo
F i g . 3. T h e Ap2(fl~)
lO ¸
d2
functions (for f18 -~ 0 ) .
/
~
132= 0.00
f32 = 0.13 132= 0.25
= 0.00
//•/132 132 = 0.13
lO-
~ EBcs
/
B2 = 025
0".1 F i g . 4.
/
0".2
133
The dependence of £Bcs and flip on fla for different fl= values.
decreases for larger quadrupole deformations. The inclusion of the pairing interaction increases the restoring force in comparison with ~ p . (Note that the energy surface Sip (f12, f13) calculated according to formula (6) corresponds to the envelope of the
NUCLEAR OCTUPOLE DEFORMATIONS
589
eners.¢ surface obtained for different configurations of independent particles.) The bottom of the o~(f13) curve is more fiat than one could expect from a simple quadratic dependence on fla3.4. PROPERTIES OF THE o#(f12,f13) SURFACE
The functions O~.cs calculated for the spherical nucleus 21SRn, for the transition nuclei 224Ra and 226Th and for the deformed nucleus 232U are shown on figs. 5-8.
Analogous figures for d~rp were shown in ref. a). 133 O.2
o.1
0.0 -
0.1
0.0
0.1
0.2
r~2
Fig. 5. The 8Bcs(fl~, flD function for ~lSRn. ~3
0.2
-o.1
o.o
o.1
C~2
Fig. 6. The drBCS(fl2,fla) function for 224Ra.
In the harmonic approximation for 8, one expects d' = E o + C 2 ( f l 2 -
flea)2 + C3 f12,
(7)
which on the figures should give a system of ellipses with axis in the flz and f13 directions. In reality, the curves on figs. 5-8 substantially differ from such a prediction.
590
P. VOGEL
O3 0.2.
0,1-
0.0 -
0.1
0.0
0.1
Q2
~2
Fig. 7. The ~BCS(fl~,fla) function for ~6Th. 0.2.
~3
0.1
0.0 --0.1
0.0
0.1
0.2
P2
Fig. 8. The ~BCS(fl2, f13) function for ~3~U. 232u ~2
228Th
0.2'
226Ra
0.1
~18Rn
0.01
0.1
0.2
~'3
Fig. 9. Points along the bottom of the "valley" in the fls direction.
NUCLEAR OCTUPOLE DEFORMATIONS
591
To demonstrate this fact more clearly we show on fig. 9 the "valleys" in the f13 direction, i.e. the curves connecting the minima of o~ for each value of f13. These lines, particularly for transition nuclei, differ considerably from the direct line f12 = const, obtained from (7). The quantities f12 and f13 will both change during vibrations with the potential energy e given on figs. 5-8. In the phonon language, this is equivalent to an interaction of octupole and quadrupole phonons. One can simply see that such an interaction leads to the lowering of the energy of octupole vibrations so as to increase their momenta of inertia. Note, that such an interaction is used often, e.g. in ref. 1o). 3.5. A P P R O X I M A T I O N O F T H E ¢(fl~,fla) F U N C T I O N BY A S I M P L E F O R M U L A
We used the least-squares method to approximate the ff(fl2, fla) function by the formula = Eo + o.)2 + + (8) The last term in (8) was added to take the phonon interaction into account. The TABLE 1 A p p r o x i m a t i o n of the ¢(fl2,
fls)
functions by formula (8)
Nucleus
C~
C3
Cz3
~18Rn 2~°Rn 222Rn
204 181 154
205 200 201
-- 500 --452 --443
C~aflealCa 0.12 0.13 0.13
0.05 0.05 0.05
flea
~Z°Ra 222Ra aURa ~2eRa ~SaRa
202 174 161 188 197
191 186 186 171 172
--465 --421 --406 --313 -- 202
0.17 0.18 0.21 0.24 0.19
0.07 0.08 0.09 0.13 0.16
224Th ~2eTh 2~STh 2S°Th
162 141 170 185
181 180 166 171
--380 -- 371 -- 287 -- 184
0.21 0.23 0.26 0.18
0.07 0.13 0.19 0.21
2a°U 83~U
154 163
165 171
--309 -- 195
0.32 0.24
0.17 0.21
results for 8 n, are given in table 1. All the coefficients are in MeV. The quantity C 2 3 f l ~ q / C a in the fifth column is the relative shift (according to the first-order perturbation theory) of the octupole phonon energy. We have used the o~w values to construct the table because they are more sensitive to the atomic number. The restoring force coefficients calculated from the O~Bcs values are in narrower limits C 2 ,.~ 100-110, C3 ~ 280-300, -C23 ~ 450-500. The other function by which we approximate the potential energy of the nucleus is taken from the liquid-drop model with the shell structure correction 11) e = e o + G exp ( - F~ fl~/c)+ F, C~fl~. 2
A
(9)
592
P. VOGEL
Th,, quantities GMs and C2M$ obtained from the experimental masses and quadrupole momenta are tabulated in ref. 11). The coefficient c in the exponential function is equal to c = A~/4rc
(;o)
In ref. 11), a/ro = 0.27, which means that the shell structure effect disappears for the deformation fl ~ 0.16. We have used the same value of c. The results of the fit for ~tP together with GMS and C2Ms values from ref. 11) are shown in table 2. The mean deviation was rather large, i.e. the function (9) describes ~ only very roughly. The deformations fl were obtained from the condition of a minimum of ¢. The parammeters G 1 and C2 have the same order of magnitude as the liquid-drop values 11), but they change much faster from nucleus to nucleus. TABLE 2 Approximation of the g(~2, fiB) function by formula (9) Nucleus
GMS
G
C2MS
Ca
Ca
flMS
fl
21aRn a2°Rn aa2Rn
--1.12 0.20 1.38
--1.42 --0.76 --0.65
30 30 31
91 72 58
145 155' 160
0.13
a2°Ra aaaRa a24Ra aa6Ra BaSRa
0.04 1.35 2.51 3.52 4.40
--0.08 0.56 0.67 3.42 5.93
29 29 29 29 29
74 59 41 29 17
155 165 170 191 220
0.12 0.19 0.21 0.22
0.20 0.25
~a4Th a~6Th aaSTh 2BaTh
2.33 3.48 4.48 5.35
1.11 1.21 3.95 7.67
22 22 22 22
36 21 9 6
172 177 198 240
0.19 0.21 0.22 0.23
0.27 0.34
4. Conclusions Let us recapitulate the main obtained results. There are no nuclei with non-zero equilibrium octupole deformations in the considered region. The potential energy surface differs substantially from a simple harmonic form, terms describing the interaction between octupole and quadrupole vibrations, as well as anharmonic terms containing higher powers of the deformation parameters are essential. The forms of the energy surface change with the atomic number A, the softest are the transition nuclei with neutron number N = 136. In conclusion I would like to express my thanks to Mrs. V. Abbrentova for her help in compiling the machine code as well as in computer calculations.
NUCLEAR OCTUPOLE DEFORMATIONS
593
References 1) S. G. Nilsson, Lectures in Varena Summer School (1967) 2) E. K. Hyde, I. Perlman and G. T. Seaborg, The nuclear properties of the heavy elements (Prentice-Hall, New Jersey, 1964) 3) P. Vogel, Phys. Lett. 2513 (1967) 65 4) K. Lee and D. R. Inglis, Phys. Rev. 108 (1957) 774 5) I. Dutt and D. Mukherjee, Phys. Rev. 124 (1961) 888 6) S. Johansson, Nucl. Phys. 22 (1961) 529 7) K. Lee, preprint (1967) 8) S. G. Nilsson, Mat. Fys. Medd. Dan. Vid. Selsk. 29, No. 16 (1955) 9) C. Gustafson et al., preprint, Lund (1966) 10) W. Donner and W. Greiner, Z. Phys. 197 (1966) 440 I1) W. D. Myers and W. I. Swiatecki, Report U C R L 11980 (1965)
Nuclear Physics A l l 2 (1968) 583w593; ( ~ North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
ON NUCLEAR OCTUPOLE DEFORMATIONS P. VOGEL t Joint Institute for Nuclear Research, Dubna, USSR Received 28 December 1967
Abstract: The dependence of the nuclear potential energy on the parameters of the quadrupole and
octupole deformation for nuclei in the 218 ~ .4 ~ 232 region is investigated. The energy is calculated in the framework of the superfluid nuclear model. The minimum of the potential energy surface is obtained for f18 = 0, i.e. there are no nuclei with non-zero equilibrium octupole deformation in the considered region. The dependence of the energy on the deformation differs essentially from the simple parabolic one, particularly a term of the Czafl~fl82 type is present in ¢(fl2, fla) leading to an asymmetric energy surface and to an interaction between E-vibrations and octupole vibrations. 1. Introduction
The problem of the dependence of the energy of the nucleus on the deformation parameters of higher multipole orders (2 > 2) has attracted much attention 1). A knowledge of this dependence is important for the interpretation of a very broad class of effects as the equilibrium deformation, vibrational spectra, fission barriers etc. In this paper we shall investigate the effect of the shell structure on deformations of the quadrupole f12 and octupole fla types. F o r nuclei in the 218 < A < 232 region, octupole I ~ = 1- states have very unusual properties. Their energies are very low, m o m e n t a of inertia large and hindrance factors for ~t-decay small 2). Such behaviour indicates, that for these nuclei the shell structure may strongly change the simple (liquid drop) dependence of the energy on the octupole deformation. Besides, this region of nuclei is also on the boundary between spherical and deformed nuclei. Therefore we have chosen this region for our calculations. Because the projection of the angular m o m e n t u m on the nuclear symmetry axis in low-lying I ~ -- 1- states is equal to zero, we shall limit ourselves to axially symmetric nuclei. Preliminary results of our calculations have been published 3). The problem of nuclear stability with respect to octupole deformations has been investigated 4-7). The restoring force parameter C3 was calculated using various approximations (usually perturbation theory). The restoring force consists of two terms of approximately the same magnitude but opposite sign. The quantity Caef characterizes the repulsion of the filled and unfilled shells which contributes to the deformation, Crest originates from the "inertia of closed shells" or volume censer1- Present and permanent address: Nuclear Research Institute, Re~, Czechoslovakia.
583
584
v. VOGEL
vation which counteracts the deformation. In refs. 4-7), the restoring force parameters were small and positive, i.e. the nuclei were "soft" but non-deformed. The purpose of this paper is to obtain more detailed information about the dependence of the potential energy on the parameters of the quadrupole and octupole deformation.
2. The method of calculating the ~(~2, ~s) function To calculate the energy of the nucleus, we shall use the method, which has already proved to be successful when considering the equilibrium quadrupole deformation (see e.g. ref. 1) and references mentioned there). According to this method, the energy of the nucleus is calculated as the energy of a system of nucleons interacting via pairing forces and moving in a deformed average field. Let the equipotential surface have the form
r = Ro k(fl2, fla) 1 + f12 Y:o( 8, q~)+ fla Yao(8, (P)--~ ~ f12fla Y1o(S, q~)
• (1)
The last term in the square brackets assures the constancy of the centre of mass, and the k(fl2, f13) function assures the volume conservation. The condition of volume conservation is a consequence of the constancy of the average nuclear density. The potential in analogy with Nilsson s) is of the form V --= ½ M ~ 2 2 r 2
E'
1+ ~-~(fli+fll)
x I1-2fl2
1
Y2o(,9, q~)-2flaYao(,9, q~)+18 V3-~nfl213aYlo(oa, tp)].
(2)
In the lowest approximation with respect to f12 and fls, potential (2) has the equipotential surfaces (1) and in the spherical limit is equal to the harmonic oscillator one. Let us note, that the octupole potential of the ~ r3yao type used in refs. 4,6) does not have surface of type (1), moreover, its equipotential surfaces have one branch extending to infinity. Therefore we use the octupole potential ~ r 2 y 3 0 following from (1). The single-particle Hamiltonian has the usual form h2 H(
2,133) =
-
--
2M
,t + v ( # 2 ,
B3)+ ct.
s
+D(t
(3)
The parameters C and D used in our calculations were 9) C , - - - 0 . 8 5 6 , Cp = -0.779, D, = -0.139, Dp = -0.253 (in MeV). The total energy (for a given level order) does not depend critically on these parameters.
NUCLEAR OCTUPOLE DEFORMATIONS
585
Hamiltonian (3) was diagonalized in the spherical oscillator basis, i.e.
(~, = ~ a~tazlNlAr,> NIAE
in the Nilsson e-representation s). In this representation, the only non-zero matrix elements of the quadrupole part of (3) are the diagonal A N = 0 ones. However, in the octupole part additional second-order terms arise VH56 1/7 72 V 3 Ylo(O, qg)) /~2/~3(40 \6-'~1~Y5o(O, tp)- ~ "~z Y3°('9' cp)+ f~ ~ which were neglected similarly as the change in the vectors ! and s. The non-zero matrix elements of the octupole part of (3) are the non-diagonal ones with A N = 1, 3, 5 . . . . We limit ourselves to elements with A N = 1 and 3. The influence of the others is negligible, because the energy differences are large and the matrix elements themselves are small. The effect of the octupole part of (3) on the energy of the nucleus is mainly connected with the matrix elements having A N = 1 between shells in the vicinity of the Fermi surface. Taking further shells placed symmetrically with respect to the Fermi surface into account as well as terms with A N = 3 increases the effect of the octupole part by about 20 ~. In our calculations, we took five oscillator shells for protons and five oscillator shells for neutrons and matrix elements with A N --- 1, 3. Note that filled shells which do not participate in the diagonalization contribute nevertheless to the stabilization of the spherical form owing to the volume conservation condition. The eigenvalues e(fl2, fla) of Hamiltonian (3) were used in the equations of the superfluid nuclear model 2
1
G
es_2)2+A 2
x/(es~2) ~ + Ae/ ,
(4)
and the quantities A and 2 for neutrons (132 < N < 140) and protons (86 < Z < 92) were calculated. The following values were used for the pairing interaction constants G n = 21/A, Gr, = 25/A (MeV). After solving the system (4), the energy d~(fl2, f13) was calculated according to e,-2] +2nN-A~/Gn
O~Bcs(fl2, f13) = E ( e s - 2 ) ( 1 -
+ ~ (et-2) ( 1 - et-).~ +)~pZ--A2/(~p, t
\
(5)
Et /
where E = ~ / ( e - 2) 2 + A ~ is the quasi-particle energy. The summation over s contains
586
P. VOGEL
all the neutron states and over t all the proton ones. For non-interacting particles, which however for each deformation fill the lowest ½N and ½Z levels, one obtains ~N
~z
gw(fl2, fla) = Z 28,(fl2, f13)+ Z 2~(fl2, f13). S=I
(6)
t=l
Let us note that formulae (5) or (6) contain implicitly various approximations, e.g. the Coulomb energy of the protons was neglected, the states were not projected onto the subspace with definite angular momentum and parity etc. However, for not too large deformations, neither of the mentioned approximations can change the results considerably. 3. Discussion
The single-particle energies ¢(fl2, f13), the quantities d(fl2 , f13) and 2(fl 2, f13) as well as the functions ~Bcs (f12, fla) and ¢ip(fl2, fla) were calculated for nuclei with 218 <__A <= 232 with the step 0.06 in f12 and fla. Between these points the parabolical interpolation was used, the control calculations show that errors connected with the interpolation are not substantial. The results are represented in the form of figures and tables and are discussed below. 3.1. THE Q U A D R U P O L E D E F O R M A T I O N
Fig. 1 gives the dependence of ~Bcs on f12 (for f13 = 0) for some selected nuclei. The curves are normalized so that the minimum energy is always equal to zero (this is true for all figures). Besides the expected shift of the minimum with increasing A towards the larger deformations, one can note a change in curvature from nucleus to nucleus. The curvature has a minimum value for nuclei with neutron number N = 136, i.e. these nuclei are "softest" with respect to quadrupole deformations. The equilibrium deformations obtained from our calculations are somewhat smaller than the experimental ones (see also table 1). 3.2. PAIRING CORRELATIONS
On fig. 2 the dependence of the square of the correlation function (energy gap) on the f12 deformation is shown for systems with various numbers of neutrons. The analogous picture for protons is shown on fig. 3. The increase of A2 with the increase of N or Z for zero deformation is related to the increase of the level density with the removal from a magic nucleus. The characteristic features of both figures is the existence of a rather narrow region of f12 values in which the energy gap does not depend on N or Z. In this region of deformations, the single-particle spectrum is close to an equidistant one. The periodical change of A2 with f12 is a consequence of the periodical change of the level density. Since the correlation energy is equal to -A2/G, one can see from the behaviour of A2 that pairing interactions promote the stabilization of the spherical form in spherical nuclei and lead to a decrease of the curvature of the potential energy for the deformed nuclei.
587
NUCLEAR OCTUPOLE DEFORMATIONS
The quantity A 2 decreases monotonously as the octupole deformation f13 increases. This qualitative feature is independent of Z or N and f12- A typical example o f the function A2(fla) is shown by the dashed lines on fig. 2 (N = 136) and fig. 3 (Z = 90). These curves were calculated for f12 = 0.2. One can conclude from the behaviour of the A2(fla) function that pairing correlations counteract the octupole deformation and increase the restoring force with respect to the octupole vibrations.
2~aRr~
10
a
/ 0 -0.1
o'.o
d.1
d2
[3
Fig. 1. T h e dependence o f 8Bcs o n fl~. /x 2
1.0 ¸
0.5 ¸
""-*-Z136
0.0 -0,1
0'.0 Fig. 2. T h e
3.3. G E N E R A L
0'.1
dn~(fl~)
13.2
[3
f u n c t i o n s (for fls = 0).
FEATURES OF THE DEPENDENCE
O F T H E E N E R G Y O N f18
The dependence of the energy of the nucleus on the octupole deformation has some general features which do not depend on the atomic number. To demonstrate them, we choose the characteristic example of the 224Ra nucleus shown on fig. 4,
588
P. VOOEL
The curves o~(f13) calculated using formulae (5) and (6) for different values of f12 are shown. The 6~(fla) curve has always a flat minimum for f13 = 0 and has no other minima. The curvature of 6"(fl3) decreases as f12 increases, i.e. the restoring force
1.0'
0.5.
0.o
-o.1
61
olo
F i g . 3. T h e Ap2(fl~)
lO ¸
d2
functions (for f18 -~ 0 ) .
/
~
132= 0.00
f32 = 0.13 132= 0.25
= 0.00
//•/132 132 = 0.13
lO-
~ EBcs
/
B2 = 025
0".1 F i g . 4.
/
0".2
133
The dependence of £Bcs and flip on fla for different fl= values.
decreases for larger quadrupole deformations. The inclusion of the pairing interaction increases the restoring force in comparison with ~ p . (Note that the energy surface Sip (f12, f13) calculated according to formula (6) corresponds to the envelope of the
NUCLEAR OCTUPOLE DEFORMATIONS
589
eners.¢ surface obtained for different configurations of independent particles.) The bottom of the o~(f13) curve is more fiat than one could expect from a simple quadratic dependence on fla3.4. PROPERTIES OF THE o#(f12,f13) SURFACE
The functions O~.cs calculated for the spherical nucleus 21SRn, for the transition nuclei 224Ra and 226Th and for the deformed nucleus 232U are shown on figs. 5-8.
Analogous figures for d~rp were shown in ref. a). 133 O.2
o.1
0.0 -
0.1
0.0
0.1
0.2
r~2
Fig. 5. The 8Bcs(fl~, flD function for ~lSRn. ~3
0.2
-o.1
o.o
o.1
C~2
Fig. 6. The drBCS(fl2,fla) function for 224Ra.
In the harmonic approximation for 8, one expects d' = E o + C 2 ( f l 2 -
flea)2 + C3 f12,
(7)
which on the figures should give a system of ellipses with axis in the flz and f13 directions. In reality, the curves on figs. 5-8 substantially differ from such a prediction.
590
P. VOGEL
O3 0.2.
0,1-
0.0 -
0.1
0.0
0.1
Q2
~2
Fig. 7. The ~BCS(fl~,fla) function for ~6Th. 0.2.
~3
0.1
0.0 --0.1
0.0
0.1
0.2
P2
Fig. 8. The ~BCS(fl2, f13) function for ~3~U. 232u ~2
228Th
0.2'
226Ra
0.1
~18Rn
0.01
0.1
0.2
~'3
Fig. 9. Points along the bottom of the "valley" in the fls direction.
NUCLEAR OCTUPOLE DEFORMATIONS
591
To demonstrate this fact more clearly we show on fig. 9 the "valleys" in the f13 direction, i.e. the curves connecting the minima of o~ for each value of f13. These lines, particularly for transition nuclei, differ considerably from the direct line f12 = const, obtained from (7). The quantities f12 and f13 will both change during vibrations with the potential energy e given on figs. 5-8. In the phonon language, this is equivalent to an interaction of octupole and quadrupole phonons. One can simply see that such an interaction leads to the lowering of the energy of octupole vibrations so as to increase their momenta of inertia. Note, that such an interaction is used often, e.g. in ref. 1o). 3.5. A P P R O X I M A T I O N O F T H E ¢(fl~,fla) F U N C T I O N BY A S I M P L E F O R M U L A
We used the least-squares method to approximate the ff(fl2, fla) function by the formula = Eo + o.)2 + + (8) The last term in (8) was added to take the phonon interaction into account. The TABLE 1 A p p r o x i m a t i o n of the ¢(fl2,
fls)
functions by formula (8)
Nucleus
C~
C3
Cz3
~18Rn 2~°Rn 222Rn
204 181 154
205 200 201
-- 500 --452 --443
C~aflealCa 0.12 0.13 0.13
0.05 0.05 0.05
flea
~Z°Ra 222Ra aURa ~2eRa ~SaRa
202 174 161 188 197
191 186 186 171 172
--465 --421 --406 --313 -- 202
0.17 0.18 0.21 0.24 0.19
0.07 0.08 0.09 0.13 0.16
224Th ~2eTh 2~STh 2S°Th
162 141 170 185
181 180 166 171
--380 -- 371 -- 287 -- 184
0.21 0.23 0.26 0.18
0.07 0.13 0.19 0.21
2a°U 83~U
154 163
165 171
--309 -- 195
0.32 0.24
0.17 0.21
results for 8 n, are given in table 1. All the coefficients are in MeV. The quantity C 2 3 f l ~ q / C a in the fifth column is the relative shift (according to the first-order perturbation theory) of the octupole phonon energy. We have used the o~w values to construct the table because they are more sensitive to the atomic number. The restoring force coefficients calculated from the O~Bcs values are in narrower limits C 2 ,.~ 100-110, C3 ~ 280-300, -C23 ~ 450-500. The other function by which we approximate the potential energy of the nucleus is taken from the liquid-drop model with the shell structure correction 11) e = e o + G exp ( - F~ fl~/c)+ F, C~fl~. 2
A
(9)
592
P. VOGEL
Th,, quantities GMs and C2M$ obtained from the experimental masses and quadrupole momenta are tabulated in ref. 11). The coefficient c in the exponential function is equal to c = A~/4rc
(;o)
In ref. 11), a/ro = 0.27, which means that the shell structure effect disappears for the deformation fl ~ 0.16. We have used the same value of c. The results of the fit for ~tP together with GMS and C2Ms values from ref. 11) are shown in table 2. The mean deviation was rather large, i.e. the function (9) describes ~ only very roughly. The deformations fl were obtained from the condition of a minimum of ¢. The parammeters G 1 and C2 have the same order of magnitude as the liquid-drop values 11), but they change much faster from nucleus to nucleus. TABLE 2 Approximation of the g(~2, fiB) function by formula (9) Nucleus
GMS
G
C2MS
Ca
Ca
flMS
fl
21aRn a2°Rn aa2Rn
--1.12 0.20 1.38
--1.42 --0.76 --0.65
30 30 31
91 72 58
145 155' 160
0.13
a2°Ra aaaRa a24Ra aa6Ra BaSRa
0.04 1.35 2.51 3.52 4.40
--0.08 0.56 0.67 3.42 5.93
29 29 29 29 29
74 59 41 29 17
155 165 170 191 220
0.12 0.19 0.21 0.22
0.20 0.25
~a4Th a~6Th aaSTh 2BaTh
2.33 3.48 4.48 5.35
1.11 1.21 3.95 7.67
22 22 22 22
36 21 9 6
172 177 198 240
0.19 0.21 0.22 0.23
0.27 0.34
4. Conclusions Let us recapitulate the main obtained results. There are no nuclei with non-zero equilibrium octupole deformations in the considered region. The potential energy surface differs substantially from a simple harmonic form, terms describing the interaction between octupole and quadrupole vibrations, as well as anharmonic terms containing higher powers of the deformation parameters are essential. The forms of the energy surface change with the atomic number A, the softest are the transition nuclei with neutron number N = 136. In conclusion I would like to express my thanks to Mrs. V. Abbrentova for her help in compiling the machine code as well as in computer calculations.
NUCLEAR OCTUPOLE DEFORMATIONS
593
References 1) S. G. Nilsson, Lectures in Varena Summer School (1967) 2) E. K. Hyde, I. Perlman and G. T. Seaborg, The nuclear properties of the heavy elements (Prentice-Hall, New Jersey, 1964) 3) P. Vogel, Phys. Lett. 2513 (1967) 65 4) K. Lee and D. R. Inglis, Phys. Rev. 108 (1957) 774 5) I. Dutt and D. Mukherjee, Phys. Rev. 124 (1961) 888 6) S. Johansson, Nucl. Phys. 22 (1961) 529 7) K. Lee, preprint (1967) 8) S. G. Nilsson, Mat. Fys. Medd. Dan. Vid. Selsk. 29, No. 16 (1955) 9) C. Gustafson et al., preprint, Lund (1966) 10) W. Donner and W. Greiner, Z. Phys. 197 (1966) 440 I1) W. D. Myers and W. I. Swiatecki, Report U C R L 11980 (1965)