- Email: [email protected]
A Glauber-model approach to one-nucleon transfer reactions
.__ l!iB
a d
PHYSICS
__
ELSEVIER
Physics Reports 264 (1996)
A Glauber-model
REPORTS
39-45
approach to one-nucleon transfer reactions B.F. Bayman
Physics Department,
University of Minnesota, Minneapolis,
MN 5.5455, USA
The organizers of this meeting have honored me with this opportunity to participate in the celebration of the work of Spartak Belyaev, on the occasion of his 70th birthday. Spartak made his first strong impact on nuclear structure physics in Copenhagen, in 1958. In the few years before this, A. Bohr and B.R. Mottelson had emphasized the importance of residual interactions in their unified model of nuclear collective motion [ 11. They had used the Inglis cranking model [ 21 to show that, in the absence of these residual interactions, rotating nuclei would exhibit the rigid-body moment of inertia, which is several times larger than the inertial parameter needed to explain observed spectra of rotational nuclei. They also emphasized the so-called “energy gap”, the anomalously low density of states near the ground states of even-even nuclei, compared to the densities of states in neighboring odd-mass nuclei [ 31. Thus when David Pines came to Copenhagen in the Spring of 1958, and brought news of the BCS approach to superconductivity [ 41, in which the role of residual pairing interactions is crucial to the production of an energy gap, Bohr and Mottelson saw immediately that this was another manifestation of the same physical ideas they had struggled with. Unfortunately, BCS calculations were based on a variational wave function, and there was no obvious way to improve the zero-order calculations, or even to estimate the importance of terms that were neglected. Thus when Spartak came to Copenhagen in the Fall of 1958 and showed us how all the BCS results could be understood in terms of a well-defined, explicit transformation, in which the approximation consisted of dropping specific terms in the Hamiltonian, everyone became very excited. Every Friday morning, a meeting was held at the Niels Bohr Institute at which people discussed their current research. I have a very vivid memory of Spar&k’s first presentation. He started out very modestly, saying that he didn’t have any results to show us, only some philosophy. He emphasized that the real physics behind BCS theory was in the Cooper-pair phenomenon [ 51. Then he described for us Bogulyubov’s [ 61 inspired way of generalizing the ideas of particles and holes in a Fermi sea to a situation in which residual pairing correlations led to a diffuse Fermi surface. We all knew that the easiest way to discuss states that differed by a few particles or a few holes from a filled Fermi sea was to introduce quasi-particle creation operators, defined to be particle creation operators for states outside the Fermi sea, and particle annihilation operators for states within the Fermi sea. The new idea was to convert this into a gradual transition from pure creation operators to pure annihilation 0370-1573/96/$9.50
@
SSDZO370-1573(95)00025-9
1996 Elsevier Science B.V. All rights reserved
40
3X BaymadPhysics
Reports 264 (1996) 39-45
operators (pure particles to pure holes) to correspond to the gradual change from fully occupied to fully empty single-particle states. Because this idea was so new, we were not immediately convinced of its usefulness. I asked how one could be sure that the quasi-particle interaction, which was being ignored, didn’t contain important parts of the pairing correlation. After a somewhat confusing (to me) discussion, Ben Mottelson reminded us of the proverb “It’s always darkest before the dawn”. This turned out to be a prophetic statement, because during the next few years Spar&k showed us how almost every aspect of nuclear structure physics could be illuminated by this simple and elegant way of describing the effects of pairing calculations. In the remainder of my talk, I will describe an approach to the study of nucleon transfer in nucleusnucleus collisions at energies of 40-50 MeV/nucleon. My collaborators in this work are SM. Lenzi, A. Vitturi and F. Zardi of the University of Padova [ 71. This approach uses the traditional method of treating a direct peripheral reaction by means of the distorted-waves Born approximation. The new feature is an attempt to take advantage of the high nucleus-nucleus relative energy to justify the use of the Glauber approximation to determine the relative wave function of the two nuclei. This has the great conceptual and practical advantage of eliminating the ambiguities associated with the choice of optical potentials. The core-core relative motion is determined from measured nuclear densities and nucleon-nucleon scattering amplitudes at the corresponding energies. Let me remind you of the essential idea behind Glauber’s [8] approximation for high-energy scattering. Consider first high-energy potential scattering in one dimension. We write the Schrodinger equation -g-
+ V(z)
[ in integral form, f,&(z) =eikz +
&
1
h2k2 @(z) = -2m e(z)
fe'kI"'tV(z')~(z')dz' M
2
=e
ikz
+ m ikh2
eikz
s -m
e-‘k”cG,(z’)V(z’)dz’
+ f?+
I
e’~“~(z’)V(z’)
dz’
Z
and focus our attention on the ratio of the true solution to an unscattered
s
1 )
plane wave:
oc)
4(z)
=#(z)/eikz
= 1+
&
[
]
-cm
$(z')V(z')dz'+
e-2ikz
ezikz’q5(z’)V(z’)dz’
Z
. 1
+( z ) - 1 is thus written as a sum of two terms, and the Glauber approximation consists of neglecting the part corresponding to the back-scattering of the incident wave. This approximation can be justified [ 81 when the de Broglie wavelength is short compared to the well breadth, and the kinetic energy is large compared to the well depth, ka >> 1, Then
B.E BaymadPhysics
Reports 264 (1996) 39-45
41
b Fig. 1. Coordinates
used in the transition amplitude. A and b are the cores, between which the nucleon c is transferred.
+(zw’+&2 ]
&z’)V(z’)dz’,
-03
Wz)
_
dz
T&y$(z)V(z)
9 .
This leads to the so-called “eikonal” approximation
+(z> =
to the scattering wave function:
eikz4(z)
[( i
aexp
In three dimensions
kz-i --M
this becomes
--oo , Z
*,(x,y,z)
kz - k
=exp
I
V(x, y, z’ldz’
a good approximation for scattering angles 0 such that B2kz < 1. These wave functions for potential scattering are now used in the distorted-waves T(k,,kf)=
s
d3rid3rf~*‘-‘(kf,rf)Fif(r,r,)~“‘(ki,ri)
7
(1)
Fi,( r, r,) = ~$jY,,y’JVb~c$f#,f , $‘+‘(k. Irrr
.)=
ti(-‘*(kf,rf)=exp
exp
matrix element
(2)
[i(k.ri--&--
Kpt(hz’)dzt)]
,
(3)
v,,,(h z’)dz’ -cm
(see Fig. 1) . The two factors referring to Vopt(b, z’) can be combined
(4)
to yield
42
B.E BaymadPhysics
exp
-? [
c
;JK,,u& Z'VZ JK,,(b,z')dz'+ -m II I -4 JK,,(hz')~z' -cm .O” 1
1
h
zexp
6
Reports 264 (1996) 39-45
I
zz
03
FiLli
=exp[-&J
d2qe’4‘bpb(q)PA(4)fNN(4) . 1
Here pb( q ) and pA (q ) are the Fourier transforms of the densities of the nuclei b and A, and fNN (q) is the nucleon-nucleon scattering amplitude for momentum transfer q. The “no-recoil” approximations implicit in (5) are adequate for the short-range nucleon-nucleon interaction. The Coulomb contribution to Kpt must be treated more exactly. When this is combined with (5) and ( l)-(4), we get an almost parameter-free description of the transfer reaction. Fig. 2 shows some comparisons between experiment [9,10] and eikonalized DWBA for 40 MeVlnucleon i*C on *‘*Pb, and for “MeV/nucleon I60 on *08Pb. The curves correspond to population of low-lying single-proton and single-neutron states in 209Bi and *“Pb. Each curve is normalized to fit the data, which yields a measured spectroscopic factor. The quality of fits to the shapes of the angular distributions is comparable to or slightly better than that given by standard DWBA-without the arbitrariness associated with the choice of optical parameters. The general features of the angular distributions can be understood in terms of the conservation of linear and angular momentum [ 111. If we compare predicted angular distributions for transfer from the pl/* proton or neutron states in I60 , there are a number of interesting features. Fig. 3 shows a nucleon being transferred from a projectile to a target as the projectile moves past on either side of the target. The m-values refer to angular momentum quantization perpendicular to the reaction plane. In the situation shown in Fig. 3a, the linear momentum mismatch of the transferred nucleon is minimized if the translational linear momentum per nucleon, k, is equal to
For 40-50 MeV/nucleon projectiles, k FZ 1.4-1.6 fm-‘, so that (3) favors transfers in which ml and m2 are a large as possible. For a p nucleon from oxygen or carbon, ml is limited to 1, Thus we get better linear momentum match by populating high m2 (and thus high .!,j,) states in Bi or Pb. If the projectile passes on the other side of the target (Fig. 3b), minimum linear momentum mismatch requires highest negative values of ml, m2. This means we will not encounter a situation of appreciable interference between trajectories passing on opposite sides of the target, since they will correspond to different initial and final magnetic substates. This explains the absence of fine structure oscillations, which would be a consequence of such interference [ 121. For the situations being considered here, the Coulomb-dominated repulsive orbit turns out to be most important. Fig. 4 is a histogram showing the relative importance of different Mi, M, values in the transfer of a p-neutron from carbon to the g-shell of Pb. It is clear that the cross section is indeed dominated by transfer between maximum M-values, in this case 1 and 4. Thus the dominant M-transfer is 3. The possible L transfers are 3, 4, 5. But with an M transfer of 3, the L transfer of 3 is most likely
B.E BaymadPhysics
Reports 264 (1996) 39-45
43
zo8Pb (“C, 1‘C) “‘Pb 103 fy-T--l--i
loo tLA_LA-_1,, 0 2
4
6
o
8
e,,,(de@
3
4
4 8 cm
.
10-11 0 I. 2
2
5
6
6
8
Meg)
2oepb(1s0,'5N)20gBi
7
ClCm Meg)
Fig. 2. Calculated data.
and measured differential
cross sections. Each curved is normalized
to provide the best visual fit to the
B.E BaymmdPhysics Reports 264 (1996) 39-45
44
ia)
Z
lb)
Fig. 3. (a) Nucleon c is transferred as the cores move past each other on a Coulomb-dominated trajectory. deflection angles is the same as in (a), but the attractive part of the core-core interaction is dominant.
ML Mf
Fig. 4. Relative importance g3
:u4
-1 -4
-1 -2
-I
0
2
-I
4
0
-3
0
-,
0
,
3
I
)
0
-4
-2
I
I
I
0
2
4
of different Mi, Mf values in the transfer of a p-neutron from carbon to the g-shell of lead.
:c9=(14-141 =
-,
(b) The
33)*:
(14-141
43)*:
(14-14153)’
7
1
1
9
5
45
For ~112 to g7/2, L transfers of 3 and 4 are allowed. For pIi to g9/2, L transfers of 4 and 5 are allowed. The fact that the dominant L transfer of 3 can contribute to the population of g7/* and not to g9/2 is the reason for the enhanced g7i2 cross section. This is especially true near 8 = O”, where the extra rotational symmetry about the forward direction, combined with reflection symmetry across a plane containing the forward direction, means that the L = 4 cross section is completely forbidden.
B.E BaymadPhysics
Reports 264 (1996) 39-45
45
Thus simple conservation rules determine the general characteristics of the angular distributions of high-energy particle transfer, and Glauber theory provides a way to extract spectroscopic amplitudes that is relatively free of ambiguity.
References [l] A. Bohr and B.R. Mottelson, Kgl. Danske Videnskab. Selskab, Mat.-Fys Medd. 30, (1955) 1. [2] D.R. Inglis, Phys. Rev. 96 (1954) 1059. [3] A. Bohr, B.R. Mottelson and D. Pines, Phys. Rev. 110 (1958) 936. [4] J. Bardeen, L.N. Cooper and J.R. Schrieffer, Phys. Rev. 106 (1957) 162, 108, 1175. [5] L.N. Cooper, Phys. Rev. 104 (1956) 1189. [6] N. Bogoliubov, J. Phys. (USSR) 11 ( 1947) 23. [7] B.F. Bayman, S.M. Lenzi, A. Vitturi and F. Zardi, Phys. Rev. C 50 (1994) 2096. [ 81 R.J. Glauber, Lectures in Theoretical Physics, Vol. 1 (Wiley Interscience, New York, 1959). [9] MC. Mermaz et al., Z. Phys. A 236 ( 1987) 353. [lo] M.C. Mermaz et al., Phys. Rev. C 37 (1988) 1942. [ 1l] D.M. Brink, Phys. Lett. 40B (1972) 37. [ 121 M.J. Levine, A.J. Baltz, PD. Bond, J.D. Garrett, S. Kahana and C.E. Thorn, Phys. Rev. C 10 ( 1974) 1602; V.M. Strutinsky, Phys. Lett. 44B (1973) 245.
a d
PHYSICS
__
ELSEVIER
Physics Reports 264 (1996)
A Glauber-model
REPORTS
39-45
approach to one-nucleon transfer reactions B.F. Bayman
Physics Department,
University of Minnesota, Minneapolis,
MN 5.5455, USA
The organizers of this meeting have honored me with this opportunity to participate in the celebration of the work of Spartak Belyaev, on the occasion of his 70th birthday. Spartak made his first strong impact on nuclear structure physics in Copenhagen, in 1958. In the few years before this, A. Bohr and B.R. Mottelson had emphasized the importance of residual interactions in their unified model of nuclear collective motion [ 11. They had used the Inglis cranking model [ 21 to show that, in the absence of these residual interactions, rotating nuclei would exhibit the rigid-body moment of inertia, which is several times larger than the inertial parameter needed to explain observed spectra of rotational nuclei. They also emphasized the so-called “energy gap”, the anomalously low density of states near the ground states of even-even nuclei, compared to the densities of states in neighboring odd-mass nuclei [ 31. Thus when David Pines came to Copenhagen in the Spring of 1958, and brought news of the BCS approach to superconductivity [ 41, in which the role of residual pairing interactions is crucial to the production of an energy gap, Bohr and Mottelson saw immediately that this was another manifestation of the same physical ideas they had struggled with. Unfortunately, BCS calculations were based on a variational wave function, and there was no obvious way to improve the zero-order calculations, or even to estimate the importance of terms that were neglected. Thus when Spartak came to Copenhagen in the Fall of 1958 and showed us how all the BCS results could be understood in terms of a well-defined, explicit transformation, in which the approximation consisted of dropping specific terms in the Hamiltonian, everyone became very excited. Every Friday morning, a meeting was held at the Niels Bohr Institute at which people discussed their current research. I have a very vivid memory of Spar&k’s first presentation. He started out very modestly, saying that he didn’t have any results to show us, only some philosophy. He emphasized that the real physics behind BCS theory was in the Cooper-pair phenomenon [ 51. Then he described for us Bogulyubov’s [ 61 inspired way of generalizing the ideas of particles and holes in a Fermi sea to a situation in which residual pairing correlations led to a diffuse Fermi surface. We all knew that the easiest way to discuss states that differed by a few particles or a few holes from a filled Fermi sea was to introduce quasi-particle creation operators, defined to be particle creation operators for states outside the Fermi sea, and particle annihilation operators for states within the Fermi sea. The new idea was to convert this into a gradual transition from pure creation operators to pure annihilation 0370-1573/96/$9.50
@
SSDZO370-1573(95)00025-9
1996 Elsevier Science B.V. All rights reserved
40
3X BaymadPhysics
Reports 264 (1996) 39-45
operators (pure particles to pure holes) to correspond to the gradual change from fully occupied to fully empty single-particle states. Because this idea was so new, we were not immediately convinced of its usefulness. I asked how one could be sure that the quasi-particle interaction, which was being ignored, didn’t contain important parts of the pairing correlation. After a somewhat confusing (to me) discussion, Ben Mottelson reminded us of the proverb “It’s always darkest before the dawn”. This turned out to be a prophetic statement, because during the next few years Spar&k showed us how almost every aspect of nuclear structure physics could be illuminated by this simple and elegant way of describing the effects of pairing calculations. In the remainder of my talk, I will describe an approach to the study of nucleon transfer in nucleusnucleus collisions at energies of 40-50 MeV/nucleon. My collaborators in this work are SM. Lenzi, A. Vitturi and F. Zardi of the University of Padova [ 71. This approach uses the traditional method of treating a direct peripheral reaction by means of the distorted-waves Born approximation. The new feature is an attempt to take advantage of the high nucleus-nucleus relative energy to justify the use of the Glauber approximation to determine the relative wave function of the two nuclei. This has the great conceptual and practical advantage of eliminating the ambiguities associated with the choice of optical potentials. The core-core relative motion is determined from measured nuclear densities and nucleon-nucleon scattering amplitudes at the corresponding energies. Let me remind you of the essential idea behind Glauber’s [8] approximation for high-energy scattering. Consider first high-energy potential scattering in one dimension. We write the Schrodinger equation -g-
+ V(z)
[ in integral form, f,&(z) =eikz +
&
1
h2k2 @(z) = -2m e(z)
fe'kI"'tV(z')~(z')dz' M
2
=e
ikz
+ m ikh2
eikz
s -m
e-‘k”cG,(z’)V(z’)dz’
+ f?+
I
e’~“~(z’)V(z’)
dz’
Z
and focus our attention on the ratio of the true solution to an unscattered
s
1 )
plane wave:
oc)
4(z)
=#(z)/eikz
= 1+
&
[
]
-cm
$(z')V(z')dz'+
e-2ikz
ezikz’q5(z’)V(z’)dz’
Z
. 1
+( z ) - 1 is thus written as a sum of two terms, and the Glauber approximation consists of neglecting the part corresponding to the back-scattering of the incident wave. This approximation can be justified [ 81 when the de Broglie wavelength is short compared to the well breadth, and the kinetic energy is large compared to the well depth, ka >> 1, Then
B.E BaymadPhysics
Reports 264 (1996) 39-45
41
b Fig. 1. Coordinates
used in the transition amplitude. A and b are the cores, between which the nucleon c is transferred.
+(zw’+&2 ]
&z’)V(z’)dz’,
-03
Wz)
_
dz
T&y$(z)V(z)
9 .
This leads to the so-called “eikonal” approximation
+(z> =
to the scattering wave function:
eikz4(z)
[( i
aexp
In three dimensions
kz-i --M
this becomes
--oo , Z
*,(x,y,z)
kz - k
=exp
I
V(x, y, z’ldz’
a good approximation for scattering angles 0 such that B2kz < 1. These wave functions for potential scattering are now used in the distorted-waves T(k,,kf)=
s
d3rid3rf~*‘-‘(kf,rf)Fif(r,r,)~“‘(ki,ri)
7
(1)
Fi,( r, r,) = ~$jY,,y’JVb~c$f#,f , $‘+‘(k. Irrr
.)=
ti(-‘*(kf,rf)=exp
exp
matrix element
(2)
[i(k.ri--&--
Kpt(hz’)dzt)]
,
(3)
v,,,(h z’)dz’ -cm
(see Fig. 1) . The two factors referring to Vopt(b, z’) can be combined
(4)
to yield
42
B.E BaymadPhysics
exp
-? [
c
;JK,,u& Z'VZ JK,,(b,z')dz'+ -m II I -4 JK,,(hz')~z' -cm .O” 1
1
h
zexp
6
Reports 264 (1996) 39-45
I
zz
03
FiLli
=exp[-&J
d2qe’4‘bpb(q)PA(4)fNN(4) . 1
Here pb( q ) and pA (q ) are the Fourier transforms of the densities of the nuclei b and A, and fNN (q) is the nucleon-nucleon scattering amplitude for momentum transfer q. The “no-recoil” approximations implicit in (5) are adequate for the short-range nucleon-nucleon interaction. The Coulomb contribution to Kpt must be treated more exactly. When this is combined with (5) and ( l)-(4), we get an almost parameter-free description of the transfer reaction. Fig. 2 shows some comparisons between experiment [9,10] and eikonalized DWBA for 40 MeVlnucleon i*C on *‘*Pb, and for “MeV/nucleon I60 on *08Pb. The curves correspond to population of low-lying single-proton and single-neutron states in 209Bi and *“Pb. Each curve is normalized to fit the data, which yields a measured spectroscopic factor. The quality of fits to the shapes of the angular distributions is comparable to or slightly better than that given by standard DWBA-without the arbitrariness associated with the choice of optical parameters. The general features of the angular distributions can be understood in terms of the conservation of linear and angular momentum [ 111. If we compare predicted angular distributions for transfer from the pl/* proton or neutron states in I60 , there are a number of interesting features. Fig. 3 shows a nucleon being transferred from a projectile to a target as the projectile moves past on either side of the target. The m-values refer to angular momentum quantization perpendicular to the reaction plane. In the situation shown in Fig. 3a, the linear momentum mismatch of the transferred nucleon is minimized if the translational linear momentum per nucleon, k, is equal to
For 40-50 MeV/nucleon projectiles, k FZ 1.4-1.6 fm-‘, so that (3) favors transfers in which ml and m2 are a large as possible. For a p nucleon from oxygen or carbon, ml is limited to 1, Thus we get better linear momentum match by populating high m2 (and thus high .!,j,) states in Bi or Pb. If the projectile passes on the other side of the target (Fig. 3b), minimum linear momentum mismatch requires highest negative values of ml, m2. This means we will not encounter a situation of appreciable interference between trajectories passing on opposite sides of the target, since they will correspond to different initial and final magnetic substates. This explains the absence of fine structure oscillations, which would be a consequence of such interference [ 121. For the situations being considered here, the Coulomb-dominated repulsive orbit turns out to be most important. Fig. 4 is a histogram showing the relative importance of different Mi, M, values in the transfer of a p-neutron from carbon to the g-shell of Pb. It is clear that the cross section is indeed dominated by transfer between maximum M-values, in this case 1 and 4. Thus the dominant M-transfer is 3. The possible L transfers are 3, 4, 5. But with an M transfer of 3, the L transfer of 3 is most likely
B.E BaymadPhysics
Reports 264 (1996) 39-45
43
zo8Pb (“C, 1‘C) “‘Pb 103 fy-T--l--i
loo tLA_LA-_1,, 0 2
4
6
o
8
e,,,(de@
3
4
4 8 cm
.
10-11 0 I. 2
2
5
6
6
8
Meg)
2oepb(1s0,'5N)20gBi
7
ClCm Meg)
Fig. 2. Calculated data.
and measured differential
cross sections. Each curved is normalized
to provide the best visual fit to the
B.E BaymmdPhysics Reports 264 (1996) 39-45
44
ia)
Z
lb)
Fig. 3. (a) Nucleon c is transferred as the cores move past each other on a Coulomb-dominated trajectory. deflection angles is the same as in (a), but the attractive part of the core-core interaction is dominant.
ML Mf
Fig. 4. Relative importance g3
:u4
-1 -4
-1 -2
-I
0
2
-I
4
0
-3
0
-,
0
,
3
I
)
0
-4
-2
I
I
I
0
2
4
of different Mi, Mf values in the transfer of a p-neutron from carbon to the g-shell of lead.
:c9=(14-141 =
-,
(b) The
33)*:
(14-141
43)*:
(14-14153)’
7
1
1
9
5
45
For ~112 to g7/2, L transfers of 3 and 4 are allowed. For pIi to g9/2, L transfers of 4 and 5 are allowed. The fact that the dominant L transfer of 3 can contribute to the population of g7/* and not to g9/2 is the reason for the enhanced g7i2 cross section. This is especially true near 8 = O”, where the extra rotational symmetry about the forward direction, combined with reflection symmetry across a plane containing the forward direction, means that the L = 4 cross section is completely forbidden.
B.E BaymadPhysics
Reports 264 (1996) 39-45
45
Thus simple conservation rules determine the general characteristics of the angular distributions of high-energy particle transfer, and Glauber theory provides a way to extract spectroscopic amplitudes that is relatively free of ambiguity.
References [l] A. Bohr and B.R. Mottelson, Kgl. Danske Videnskab. Selskab, Mat.-Fys Medd. 30, (1955) 1. [2] D.R. Inglis, Phys. Rev. 96 (1954) 1059. [3] A. Bohr, B.R. Mottelson and D. Pines, Phys. Rev. 110 (1958) 936. [4] J. Bardeen, L.N. Cooper and J.R. Schrieffer, Phys. Rev. 106 (1957) 162, 108, 1175. [5] L.N. Cooper, Phys. Rev. 104 (1956) 1189. [6] N. Bogoliubov, J. Phys. (USSR) 11 ( 1947) 23. [7] B.F. Bayman, S.M. Lenzi, A. Vitturi and F. Zardi, Phys. Rev. C 50 (1994) 2096. [ 81 R.J. Glauber, Lectures in Theoretical Physics, Vol. 1 (Wiley Interscience, New York, 1959). [9] MC. Mermaz et al., Z. Phys. A 236 ( 1987) 353. [lo] M.C. Mermaz et al., Phys. Rev. C 37 (1988) 1942. [ 1l] D.M. Brink, Phys. Lett. 40B (1972) 37. [ 121 M.J. Levine, A.J. Baltz, PD. Bond, J.D. Garrett, S. Kahana and C.E. Thorn, Phys. Rev. C 10 ( 1974) 1602; V.M. Strutinsky, Phys. Lett. 44B (1973) 245.