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INSTITUTE OF PHYSICS PUBLISHING PHYSICA SCRIPTA Phys. Scr. T125 (2006) 1–7 doi:10.1088/0031-8949/2006/T125/001 Orbits, shapes and currents Stefan Frauendorf Department of Physics, University of Notre Dame, Notre Dame, IN 46556, USA and IKH, FZ-Rossendorf, PF 510119, 01314 Dresden, Germany E-mail: [email protected] Received 14 July 2005 Accepted for publication 1 September 2005 Published 28 June 2006 Online at stacks.iop.org/PhysScr/T125/1 Abstract Shapes and current distributions of nuclei and alkali clusters are discussed in terms of the single particle motion of fermions in the average potential. For small particle number, an interpretation in terms of the lowest spherical harmonics is presented. For large particle number, an interpretation in terms of classical periodic orbits is presented. PACS numbers: 21.10.Re, 21.10.Dr, 23.20.Lv, 24.10.Pa, 36.40.Cg, 73.22.−f, 73.23.Ra 1. Introduction Sven Gösta Nilsson’s work [1] that is commemorated at this meeting treated for the first time the motion of nucleons in a realistic spheroidal potential generated by all other nucleons. The famous Nilsson diagrams, which show the single particle energies as functions of the deformation parameter, gave us a first quantitative estimate of the nuclear shape. This seminal paper comprises in an ingeniously simple way, the concept of the mean field approach: Z δ e[ρ(Er )] dτ → {t + V [ρ(Er )] − ωl z }ψi = i ψi → ρ(Er ). (1) The average potential V [ρ(Er )] is generated by variation of some energy density functional with respect to the nucleon density ρ(Er ). Finding the energies i and wavefunctions ψi of the nucleons in the potential V [ρ(Er )] permits us to calculate the density ρ(Er ). Making the potential self-consistent with density determines the shape of the density distribution and the potential, which is called the nuclear shape. Inclusion of the cranking term −ωl z permits us to calculate the current E r ) of a rotating nucleus that carries a finite distribution j(E amount of angular momentum. Since Nilsson’s paper, nuclear shapes have been studied in great detail using a variety of mean field approaches. They all gave very similar shapes. Moreover, the shapes of alkali atom clusters were calculated in a mean field approximation similar to that of nuclei. Here, one calculates the energies and wavefunctions of one electron in the average potential generated by all other delocalized conduction electrons and the positive ions, which are treated as a ‘jellium’ of positive charge. As discussed later, the cluster shapes are very similar 0031-8949/06/125001+07$30.00 to nuclear shapes. These examples illustrate the general observation that the shapes of all finite fermion systems that have near constant density in the interior which goes to zero within a thin surface layer, are essentially the same. These shapes reflect the quantized motion of the fermions in a leptodermic potential. The relation between this quantal motion and the shape that the system takes will be discussed. There are two simple situations. For small particle number the shape of the system reflects the shape of the lowest quantal orbitals. Since the shape of the molecules reflects the shape of the s- and p-valence orbitals, it is called the chemical regime. For large particle number, the combined geometry of many single particle orbitals determines the shape of the system. If one is interested only in the gross structure, the shapes can be discussed in terms of periodic classical orbits of the fermions in the average potential. There is a certain analogy to the construction of a concert hall. In order to avoid echoes, one studies how sound impulses bounce from the walls and travel through the interior (periodic orbits). It is called the acoustic regime. Clusters are simpler than nuclei, because only the delocalized conduction electrons determine the shape. In nuclei, both the protons and the neutrons do this. If both prefer the same shape, the resulting shape is similar to those of clusters. If the protons and neutrons prefer different shapes, the result will be a compromise. A comparisons of systematic calculations for clusters, as shown in figure 1, with those in e.g. [3] clearly reveals this simple interplay of protons and neutrons. Hence, only one kind of fermion as in the case of clusters may be considered. The current distribution of rotating nuclei and alkali clusters exposed to a magnetic field is also largely determined by the quantization of the fermionic motion. The two © 2006 The Royal Swedish Academy of Sciences Printed in the UK 1 Stefan Frauendorf 2. The chemical regime Figure 1. The deformation parameters of heavy alkali clusters as calculated in [2] using the shell correction method. The deformation parameters αl approximately correspond to the standard deformation parameters βl introduced by a multipole expansion of the surface. phenomena are related as follows. The external magnetic field induces electric currents in a cluster that are carried by the conduction electrons. These currents generate a magnetic moment. A nucleus does not rotate like a rigid body. The currents that generate the total angular momentum consist of two parts E r ) = jErig (E j(E r ) + jEsh (E r ), jErig (E r ) = ρ(E r )ωE × rE. (2) The term jErig (E r ) is the current of a rigid body that rotates with the angular velocity ω. E The deviation jEsh (E r ) is caused by quantized motion of the fermions. It represents the residual current in the rotating frame of reference that is fixed to the body. This residual currents correspond to the electric currents in a cluster, where the magnetic field should be expressed by the Larmor frequency. 2 In the case of molecules the direction of the chemical bonds is determined by the density distribution of the valence s-orbital (isotropic spherical harmonic Y00 ) and the valence p-orbitals (the anisotropic spherical harmonics Y1m ). The interaction with the partners in the bonds may cause a strong mixing of the valence s- and p-orbitals, which is called hybridization. The hybrid orbitals have new shapes, like the three bonds of the sp2 hybrid, which lie in a plane with a mutual angle of 120◦ . In the lightest fermion systems only the s-, p- and dorbitals are occupied. The shape of the system reflects the shape of these orbitals. Figure 2 shows the shapes of the lightest alkali clusters and nuclei. In most cases they closely resemble the shapes of the density distributions obtained by filling the s-, p- and d-orbitals in a spherical potential. The nodal structure of the first eigenmodes in a compact potential has to be similar to these orbitals, which means a similar density distribution. The admixture of higher shells modifies the radial profile of the orbitals such that the density changes from the interior value to zero within a thin surface layer. Such a profile minimizes the total energy of the energy density functional, which has a minimum at the saturation density. As a consequence, the P shapes (equidensity surfaces) resemble the surfaces R = l,m n lm |Ylm |2 , which are also shown. Moreover, the energy density functionals favour a small surface area because of the positive surface energy. Since the surface tension of nuclei is larger than that of clusters, the nuclear surface follows less well the shapes of the valence orbitals, because a more rounded shape has a lower surface energy. If the state l, √m 6= 0 is only half filled, the hybrid orbital (Ylm + Yl−m )/ 2 is formed, which has a lower surface energy. The hybridization is also driven by the tendency to clusterization. The N = 10 system is an example. As seen in figure 3, the calculated shape is somewhat reflection asymmetric. The asymmetry can be generated by some admixture of the harmonics Y2±1 to the Y1±1 and of Y30 to Y20 (see figure caption). The mixing coefficients of this hybridization are determined by the tendency to form the two particularly stable subsystems, N = 8 and N = 2. The energy gain in forming two closed shell clusters has to compete with the loss due to the larger surface of the two fragments. The shape is a compromise, showing partial clusterization. As seen in figure 2, the shapes of the N = 12, 16 and 18 systems also differ somewhat from the reflection-symmetric shapes generated by filling the spherical orbitals. The asymmetry is caused by a similar hybridization mechanism as discussed for N = 10 (see [4]). The rotational and magnetic response reflects the current distribution generated by combining the lowest orbitals such that the cranking term −ωl z is taken into account. Figure 4 shows the velocity field in the rotating frame of E r )/ρ(E reference vEsh (E r ) = j(E r ) − ωE × rE for the N = 4 system. The presence of strong vortices demonstrates the dramatic deviation of the current from rigid rotation. The pattern is dominated by the transition current between the l = 1, m = 0like and l = 1, m = 1-like orbitals without radial nodes. It is combined with some irrotational current. The example is a harmonic oscillator potential at equilibrium deformation as discussed in [5]. In this case, the moment of inertia takes Orbits, shapes and currents (l = 0, m = 0) n = 2 = 00 2 008 20 2 00 2101 00 210 2 008 20 2 (21 + (2 − 1)) 2 008 20 2 212 00 210 2 (11 + (1 − 1))1 00810 −2 00810 −1 008 008 201 00 20 20 −2 (21 + (2 − 1)) −2 00 20 20 −1 (21 + (2 − 1)) −1 00 20 00830 2 Figure 2. The shapes of the lightest alkali clusters and N = Z nuclei as calculated P in [4] using the ultimate jellium energy density functional. Shown are the surfaces of half density. In addition, the surfaces R = l,m n lm |Ylm |2 are shown, where the occupation number n lm of each orbital is given as (lm)n . Figure 3. The shape of the N = 10 cluster as in figure 2. The upper panel shows the surfaces generated from spherical harmonics, where the configuration 002 102 112 1(−1)2 202 is left and the hybridized configuration 002 102 (11 + 0.5 × 21)2 (1(−1) +0.5 × 2(−1))2 (20 + 0.23 × 30)2 is right. The lower panel shows the shape calculated in [4]. the value for rigid rotation Jrig , i.e. the contributions of the vortices to the total angular momentum cancel. In the case of a more realistic leptodermic potential, the contributions Figure 4. The velocity field in the rotating frame of four fermions in a spheroidal harmonic oscillator potential, the axis ratio of which is chosen to satisfy the condition for equilibrium deformation (see [5]). from the currents in the body fixed frame do not cancel in general. The deviations of the moment of inertia from the rigid body value will be discussed for large systems. They are stronger than for small systems, because the potential deviates 3 Stefan Frauendorf more from the oscillator one. However, it is stressed that the flow pattern deviates strongly from rigid rotation, even if the moment of inertia is close to the rigid body value. 3. The acoustic regime Now the case of large particle number is discussed, which corresponds to a mesoscopic scenario. The level density, binding energy, and other quantities may be considered as a sum of a smooth part that represents the properties of a small but macroscopic droplet and a shell part that describes the consequences of the quantal motion of the fermions near the Fermi surface. The shapes are the result of a compromise between the liquid drop energy, which prefers a spherical shape to reduce the surface energy, and the shell energy, which seeks a shape for which the level density near the Fermi surface is low. The gross dependence of shapes on the particle number is discussed, which corresponds to some average of adjacent clusters or nuclei. The gross shell structure is described by means of the periodic orbit theory (POT), a detailed presentation of which was given in [6]. The energy is decomposed into a smooth part Ẽ, which is the energy of the droplet, and an oscillating part E sh , which describes the shell energy, X E = Ẽ + E sh , E sh = Eβ , (3) Figure 5. Classical periodic orbits in a spheroidal cavity at moderate deformation. β where β labels the periodic orbits that generate the shell energy. The gross shell structure is given by the few shortest orbits. For simplicity, the case of a cavity, i.e. an infinite potential step at the surface is discussed. Then the classical orbits are composed of straight lines and specular reflection on the surface. The energy e is conveniently expressed by √ the wavenumber k = 2me/h̄. For the spherical cavity, the periodic orbits are equilateral polygons. The upper panel of figure 5 shows the triangle and the square, which are the simplest. Each polygon belongs to a threefold degenerate family, which corresponds to the possible orientations of the polygon in space that are generated by the three Euler angles. All orbits of a family have the same length L, the circumference of the polygon. As seen in figure 5, there are two types of orbits in the case of a cavity with moderate spheroidal deformation. The equator orbits are the regular polygons in the equatorial plane. They belong to a onefold degenerate family generated by the possible rotations of the polygon in the plane. The meridian orbits are the polygons in a plane that goes through the symmetry axis. They belong to a twofold degenerate family, which is generated by the rotation of the meridian plane about the symmetry axis and a shift of the reflection points on the surface within the meridian plane. The contribution from each family to the shell energy is given by 2 h̄ kF L β , Eβ = Aβ (kF ) sin(L β k + νβ )D (4) τβ γ Ro where Ro = ro N 1/3 is the radius of a sphere with the same volume as the spheroidal cavity. The period of revolution √ of a particle moving with the Fermi momentum h̄kF = 2meF on the orbit is denoted by τβ . Each term in the sum oscillates 4 Figure 6. Shell energy for a cavity-like spheroidal potential. For α = 0, the shape√is spherical, for α = 0.5, the shape is prolate with an axis ratio of 3, √ and for α = −0.5, the shape is oblate with an axis ratio of 1/ 3. The dashed and full lines correspond to sin(L 4 kF + ν4 ) = 1, −1, respectively. The figure is relevant for alkali clusters. Taken from [7]. with the frequency kF as a function of the length L β of the orbital. The Maslov index νβ is a constant phase (see [6]). The amplitude Aβ depends on the degeneracy of the periodic orbit: the more symmetries a system has, the greater the degeneracy, and the more pronounced are the oscillations of the shell energy. The damping factor D(k L β /γ Ro ) is a decreasing function of its argument, which filters out the shortest orbits that are responsible for the gross shell structure. 3.1. Shapes For a qualitative discussion of the shapes, it is sufficient to consider only the family of tetragons. Figure 6 shows the shell energy of N fermions in a spheroidal cavity with deformation α. The volume of the cavity is constant, V = 4 πro3 N . The figure includes the lines of constant length of 3 the tetragons L 4 kF + ν4 = π(2n + 1/2) (dashed) and L 4 kF + ν4 = 2π(2n − 1/2) (full), which correspond to the maxima and minima of the sin-function in equation (4), respectively. At spherical shape, the shell energy oscillates as function of the particle number N , because L 4 ∝ N 1/3 . The minima at N = 58, 92, 136 are the spherical shell closures. When the cavity takes a prolate deformation (α > 0), the meridian orbits become longer. In order to keep the length constant, the size of the cavity must decrease, which corresponds to a smaller particle number N . Hence, the lines of constant Orbits, shapes and currents length of the meridian orbits are down sloping. The equator orbits become shorter at prolate deformation. In order to keep the length constant, N must increase, and the lines of constant length are up sloping. Figure 6 shows the interference pattern between the meridian and equator orbits. There is a system of down sloping valleys and ridges emaneting from the minima and maxima at spherical shape, which follow the lines of constant length of the meridian orbits. Superimposed is a system of up sloping valleys and ridges that follows the lines of constant length of the equator orbits. The contribution from the meridian orbits is stronger because they are twofold degenerate, whereas the equator orbits are only onefold degenerate. On the oblate side, the meridian orbits become shorter and the equator orbits longer, and the particle number N must increase and decrease respectively, in order to keep the length constant. The landscape for the shell energy of the realistic nuclear potential is similar to the cavity case, except that the spin–orbit term changes the magic numbers for spherical shell closure to N = 50, 82, 126. Hence, the following interpretation applies to nuclei as well if the appropriate degree of shell filling is considered. The shape is spherical (α = 0) for closed shells (N = 58, 92, 136). Taking particles away, the equilibrium shape is located in the valley on the prolate side that is generated by the meridian orbits. The deformation gradually increases in the valley. If N decreases further, the equilibrium shape moves over the saddle on the ridge that is generated by the meridian orbits. The deformation decreases abruptly after the saddle. Figure 1 shows the smooth increase of the deformation α with decreasing N and its sudden decrease. This N -dependence of the quadrupole deformation is also experimentally well known for nuclei. Reference [8] first pointed out that it reflects the down sloping meridian valleys. The meridian ridges and valleys are less down sloping on the oblate side than on the prolate side. The different slope can be explained in terms of geometry. In the lower panel of figure 5, consider the meridian square orbit that has two of its sides perpendicular and two parallel to the symmetry axis of the cavity. If a is the length of the symmetry semi-axis and b the length of the other two axes, then the length of the orbit L 4 ∝ a + b. The volume of the cavity is V = 4πab2 /3. It has to be the same as for spherical shape V = 4π Ro3 /3. This means L 4 ∝ b/Ro + 1/(b/Ro )2 . This function has a larger negative slope for b/Ro < 1 (prolate) than for b/Ro > 1 (oblate). In fact, the slope becomes positive for b/Ro > 21/3 . The system tries to avoid the mid-shell mountain at spherical shape by taking a deformed shape. It is energetically favourable to go to the prolate side, because the argument of the sin-function changes more rapidly. This is the explanation of [9] for the preponderance of prolate over oblate shapes, which can be seen in figure 1 and which is well known for nuclei. The weaker valley-ridge structure generated by the equator orbits is less important for the quadrupole deformation. However, it generates the hexadecapole deformation. As seen in figure 6, the meridian and equator valleys cross in the middle of the shell at N ≈ 76, α ≈ 0.3 and N ≈ 114, α ≈ 0.25. The constructive interference generates the local minimum, which corresponds to the favoured shape. Figure 1 shows that these are the quadrupole deformation parameters α of Figure 7. Shapes of heavy alkali clusters as calculated in [2] using the shell correction method. The arrows have the same length. mid-shell clusters, and that the hexadecapole deformation α4 ≈ 0 for them. Moving along the meridian valley to larger or smaller N , brings the system out of the equator valley, which makes the shell energy less negative. It is energetically favourable to change the shape such that the equator diameter remains the same as in the middle of the shell. Then the length of the equator orbits does not change, and their contribution to the shell energy remains at the most negative value. The hexadecapole deformations α4 in figure 1 are positive below and negative above the middle of the shell. Figure 7 shows that the equator diameter of the equilibrium shapes is approximately constant indeed. The measured nuclear shapes follow the same sequence through a shell: lemon-like in the lower part of the shell, spheroidal in the middle, and barrellike in the upper part of the shell. 3.2. Currents and moments of inertia Here, the case of slow rotation is considered, which means that the periodic orbits are close to the discussed polygons (for details concerning this simplification see [7]). The important quantity is the rotational flux through the orbit. It is defined in the same way as the magnetic flux, which takes its role in the case of clusters in a magnetic field, h̄8(θ ) = 2m Aβ ω cos θ. (5) Here, Aβ is the area enclosed by the orbit, and θ is the angle between the normal of its plane and the axis of rotation. The rotation manifests itself in the appearance of an additional modulation factor M in the expression for the shell energy, X X Eβ → Mβ E β . (6) β β The modulation factor is an oscillating function of 8β , which is the flux through the orbit perpendicular to the rotational axis. These oscillations determine the shell energy at high spin, which is discussed in [7]. Only the first term (quadratic) of the expansion into powers of 8β is considered, which gives an expression for the moment of inertia, X J = Jrig + Jsh , Jsh = Jβ , (7) β 5 Stefan Frauendorf Figure 8. The experimental ground state shell energy (upper panel) and the shell moment of inertia of unpaired nuclei (lower panel) as functions of the neutron number N . The different symbols give the proton number Z . Taken from [7]. where Jrig is the rigid body value. For the discussion, only one term in the sum over the periodic orbits is kept. Then the moments of inertia are given by Jshk = h̄ 2 2 (k Ak )2 E shk , eF2 F Jsh⊥ = h̄ 2 2 (k A⊥ )2 E sh⊥ , 2eF2 F (8) where E(kF )k is the contribution of the equator orbits to the ground-state shell energy (4), and E(kF )⊥ the corresponding contribution of the meridian orbits. The areas of the respective orbits are Ak and A⊥ . Figure 8 shows the experimental ground state shell energies of heavy nuclei as function of the neutron number, which are obtained in the usual manner by subtracting the energy of a spherical droplet from the experimental ground state energies. In the lower panel the experimental shell contribution to the moment of inertia (‘shell m. o. i.’) of unpaired nuclei is shown. It is obtained by fitting the expression I 2 /2Jexp to the experimental yrast energies above spin I = 20h̄, where the pair correlations are essentially quenched. The experimental shell moment is the difference Jsh = Jexp − Jrigid (for details see [7]). The deviations of the moments of inertia from the rigid body value are strong. Their 6 N -dependence is similar to the one of the ground state shell energy, which is expected from the relation (8). Equation (8) gives a relative scale of Jsh ≈ (h̄ 2 /1000 MeV2 )A4/3 E sh , which is correct as the comparison of the two panels shows. However, concerning POT, there is one important difference between ground state energies and moments of inertia. Whereas all orbits contribute to the ground state shell energy, only the orbits that carry rotational flux contribute to the shell moment of inertia. That is, only the meridian orbits contribute if the rotational axis is perpendicular to the rotational axis, and only the equator orbits contribute if the rotational axis is parallel to the rotational axis. This is indicated by the subscripts k and ⊥ for the parallel and perpendicular orientation of the rotational axis. On the other hand, both the meridian and equator orbits contribute to the ground state shell energy, i.e. E sh = E shk + E sh⊥ . The two orientations of the rotational axis can be easily distinguished. If it is perpendicular to the symmetry axis, one observes a regularly spaced rotational band. If it is parallel to the symmetry axis, the yrast line becomes an irregular sequence of states that contains many high-K isomers. The strongly negative value of Jsh at the magic number N = 126 reflects the shell closure. The POT interpretation is the same as discussed above for E sh . The dip at N = 82 is less deep, because Z is mid shell. As for E sh , the positive value of Jsh around N = 90 is caused by the meridian ridge. Nuclei in this region rotate about an axis perpendicular to the symmetry axis and show rapid alignment of nucleon angular momentum with this axis (back bending). The bump in Jsh demonstrates that the meridian orbits are responsible, because only they are perpendicular to the axis of rotation and thus carry rotational flux. The bump of Jsh around N = 110 is caused by the equator ridge. Nuclei in this region rotate about the symmetry axis, which is reflected by the appearance of many highK isomers. This demonstrates that the equator orbits must be responsible, because only they are perpendicular to the rotational axis and carry rotational flux.1 The equator orbits generate only a shoulder in E sh , because the meridian orbits make a negative contribution that increases with N . Figure 9 shows the velocity field of a nucleus within the group around N = 90, which have a positive shell moment of inertia. As expected, there is a substantial current in the body fixed frame, which has the same direction as the rotation. It causes the positive shell moment of inertia. The velocity field resembles the one generated by nucleons moving on the meridian orbits shown in the insert. As discussed in [10], nuclei outside this region have a similar current distribution in the rotating frame. However, the velocities are opposite to the rotation, which is consistent with a negative shell moment of inertia. A more quantitative relation between the currents and the motion of particles on classical orbits remains to be established.2 The negative shell energy that causes super deformation of rotating nuclei is substantial. At a first glance one might expect that this would correspond to a negative shell moment of inertia. However, superdeformed nuclei are known to 1 For most of the yrast states the rotational axis is not completely parallel to the symmetry axis. That is why Jsh does not become positive. It does become positive in microscopic calculations that assume parallel rotation [7]. 2 Rotation about the symmetry axis was not studied in [10]. Orbits, shapes and currents 4. Conclusion Figure 9. The velocity field of 162 Yb in the laboratory system (left) and the body fixed system (right) at angular momentum of about 15h̄. Taken from [10]. The inset shows the tetragonal orbits that are responsible for the current in the rotating frame. The arrows indicate that more particles run anti-clockwise than clockwise. The shapes of nuclei and alkali clusters reflect the quantized motion of the fermions at the Fermi surface in the average potential. They are not sensitive to the interaction that generates the potential, i.e. their properties are universal to all leptodermic potentials. The relation between the quantal single particle motion and the shapes and currents is relatively transparent for small and large particle number. For small particle number, the pattern originates from the density and current distributions of the first (hybridized) spherical harmonics. For large particle number, the pattern is generated by the shortest classical orbits in the potential. The orbit length controls the deformation. Constant length of the orbits in the meridian plane determines the quadrupole deformation, and constant length of the orbits in the equator plane determines the hexadecapole deformation. The current pattern is universal as well. In the absence of pair correlations, the currents of rotating nuclei strongly deviate from rigid flow. In heavy nuclei, this is reflected by the moments of inertia at high spin differing from the rigid body value. The difference can be related to the classical periodic orbits. It suggests a strong magnetic response of alkali clusters. Acknowledgments This work was supported by the US Department of Energy under contract DE-FG02-95ER40934. References Figure 10. Classical orbits in a superdeformed cavity. have moments of inertia that are close to the rigid body value. The reason is that super deformation is caused by other types of orbits [11], examples of which are shown in figure 10. Superdeformed nuclei rotate about an axis perpendicular to the symmetry axis. Therefore the equator orbits do not contribute to the shell moment of inertia. The meridian orbits are of the butterfly type. They do not carry flux because the contributions of the two wings compensate each other. (The motion is clockwise in one wing and anticlockwise in the other.) Since none of the orbits carries rotational flux, the moment of inertia takes the rigid body value. [1] Nilsson S G 1955 Mat.-Fys. Medd. K. Dan. Vidensk. Selsk. 29 1 [2] Frauendorf S and Pashkevich V V 1996 Ann. Phys., Lpz. 5 34 [3] Möller P, Nix J R, Myers W D and Swiatecki W J 1995 At. Data Nucl. Data Tables 59 185 [4] Koskinen M, Lipas P O and Manninien M 1995 Z. Phys. D 35 285 [5] Bor A and Mottelson B 1975 Nuclear Structure II (New York: Benjamin) [6] Brack M and Bhaduri R K 1997 Semiclassical Physics (Reading: MA: Addison-Wesley) [7] Deleplanque M A, Frauendorf S, Pashkevich V V and Chu S Y Z. Phys. A 283 269 [8] Strutinsky V M, Magner A G, Ofengenden S R and Dossing T 1977 Z. Phys. A 283 269 [9] Frisk H 1990 Nucl. Phys. A 511 309 [10] Fleckner J, Kunz J, Mosel U and Wüst E 1980 Nucl. Phys. A 339 227 [11] Yamagami M and Matsuyanagi K 2000 Nucl. Phys. A 672 123 7