Journal of Modern Physics, 2013, 4, 5465 doi:10.4236/jmp.2013.45B011 Published Online May 2013 (http://www.scirp.org/journal/jmp) Quantum Isomorphic Shell Model: MultiHarmonic Shell Clustering of Nuclei G. S. Anagnostatos Institute of Nuclear Physics, National Center for Scientific Research Demokritos, Greece Email: anagnos4@otenet.gr Received 2013 ABSTRACT The present multiharmonic shell clustering of a nucleus is a direct consequence of the fermionic nature of nucleons and their average sizes. The most probable form and the average size for each proton or neutron shell are here presented by a specific equilibrium polyhedron of definite size. All such polyhedral shells are closely packed leading to a shell clus tering of a nucleus. A harmonic oscillator potential is employed for each shell. All magic and semimagic numbers, g.s. single particle and total binding energies, proton, neutron and mass radii of 40Ca, 48Ca, 54Fe, 90Zr, 108Sn, 114Te, 142Nd,, and 208Pb are very successfully predicted. Keywords: Cluster Models; 40Ca, 48Ca, 54Fe, 90Zr, 108Sn, 114Te, 142Nd, 208Pb, Binding Energies; Coulomb Energies; Proton; Neutron; Mass Radii; Atomic Fermions 1. Introduction The present work supports a shell clustering of nuclear structure by taking advantage mainly of two fundamental properties of nucleons. The first is the fermionic nature of nucleons and the second is that nucleons possess finite average sizes. As will be explained shortly [see a) and b) below], the first leads to the shell structure of the nucleus and the second to the average nuclear sizes. These prop erties taken together lead to the aforementioned shell clustering of the nucleus. Indeed: a) The fermionic nature of nucleons, due to their anti symmetric wave function, makes them behave as if a repulsive force (of unknown nature) is acting among them. [1]. This repulsive property of nucleons is derived not only by this antisymmetrization, but also by the re pulsive character of the nuclear force itself, considered as a result of the quark structure of the nucleons. The exis tence of this force shifts the nuclear manybody problem to the problem of finding the equilibria of repulsive par ticles on a sphere, like the sphere of a nuclear shell. Ac cording to [2], such equilibria are possible only for spe cific numbers of particles and when these particles have most probable positions at the vertices, or middles of faces, or middles of edges, or simultaneously at these characteristic points of regular polyhedra or their deriva tive polyhedra. Such polyhedra in the present model stand for the most probable forms of nuclear shells which  taken in specific sequence, as we will see below – pre cisely reproduce the nuclear magic numbers, with no use of the strong spinorbit coupling. It is essential to empha size here that the structures of these polyhedra precisely possess the quantization of orbital angular momentum [3]. b) In addition, the finite average size of nucleons, to gether with the above most probable polyhedral structure of shells, leads to the average sizes of all nuclear shells and thus of all nuclei. This is obtained by packing the shells themselves, i.e., each polyhedral shell reaches its minimum size, that is, the bags of nucleons at its vertices come in contact with the bags of nucleons at the vertices of a previous polyhedron. However, it should be mentioned that the foundation of the shell structure of the nucleus is offered by other theories as well, where the antisymmetrization of the nucleon wave function and the nuclear interaction play an important role. The present approach, where the nuclear structure is qualitatively derived without reference to the in terparticle forces, is applicable not only in nuclear physics but also in any other branch of physics where fermions of definite average size are the constituent par ticles. Take, for example, cluster physics when the con stituent particles are atoms with halfinteger spins; thus they could be considered as atomic fermions. Thus, the present paper can guide research in other fields, as well as in nuclear physics. 2. The Model and Applications Copyright © 2013 SciRes. JMP
G. S. ANAGNOSTATOS 55 2.1. Specification of Most Probable Forms and Average Sizes of Nuclear Shells As mentioned in the introduction the fermionic nature of nucleons and their average sizes lead to the polyhedral average shell structure of a nucleus. This structure is dif ferent for neutrons and protons, since these two kinds of nucleons are considered as two different sets of particles. Indeed, this problem for one set or for two different sets of repulsive particles on a sphere or on concentric spheres (like the shapes of closed nuclear shells) was solved by Leech long ago [2]. According to this refer ence, each of these two sets should be in equilibrium by itself on a sphere or on concentric spheres and the result ing two structures should be in a new relative equilibrium as well. As aforementioned in the introduction, these equilibria are obtained at the vertices or middles of faces or mid dles of edges of regular polyhedra and their derivative polyhedra. However, when we consider the middles of faces or the middles of edges of any of the aforemen tioned polyhedra, we obtain the vertices of another regu lar or semiregular polyhedron. Thus, we see that it is sufficient to consider only the vertices of polyhedra. These polyhedra are the zerohedron, the octahedron, the hexahedron (cube), the cuboctahedron, the icosahedron, the dodecahedron, the rhombic cuboctahedron, the ico sidodecahedron, and the rhombic triacondahedron with numbers of vertices 2, 6, 8, 12, 12, 20, 24, 30, and 32, respectively. These polyhedra can appear more than one time. Among them three polyhedra can be analysed as two symmetric, equilibrium structures with smaller numbers of vertices having together the same central, angular structure as the initial polyhedron. Namely, the dodeca hedron (20 vertices) can be analysed as a cube (8 vertices) and the remaining (12 vertices), the icosidodecahedron (30 vertices) can be analysed as an octahedron (6 vertices) and the remaining (24 vertices), and the rhombic tri acondahedron (32 vertices) can be analysed as an icosa hedron (12 vertices) and a dodecahedron (20 vertices) which can further be analysed as above. Recommended books for a sufficient familiarity with regular and semi regular polyhedra are [4, 5], the former being simple and very instructive and the latter somewhat more sophisti cated. Since both the initial equilibrium polyhedra and the polyhedra derived by their aforementioned analyses are candidates to present the most probable forms of nuclear shells and subshells, what is necessary in the following is to describe how these polyhedra are concentrically and most symmetrically [6] arranged in space to present nu clear shells and subshells. First, we start with the polyhedra with smaller num bers of vertices, i.e., with 2, 6, 8, 12, and 20 vertices. Among them we assign the polyhedra with triangular faces (corresponding to stable equilibria [2] and having number of vertices 2, 6, and 12), i.e., the zerohedron, the octahedron, and the icosahedron, to the first three neu tron shells and the remaining polyhedra (corresponding to unstable equilibria [2] and having number of vertices 2, 8, and 20), i.e., the zerohedron, the hexahedron (cube), and the dodecahedron, to the first three proton shells (see Figure 1). This assignment makes each neutron shell have a corresponding proton shell with the same rota tional symmetry, i.e., a neutron zerohedron corresponds to a proton zerohedron, a neutron octahedron to a proton hexahedron, and a neutron icosahedron to a proton do decahedron. This assignment, in addition, minimizes the Coulomb energy, since the aforementioned polyhedra for protons are of a larger size than those assigned for neu trons (see Figure 1). This type of correspondence be tween neutron and proton shells is valid for the whole periodic table of nuclei. Then, their average sizes are determined with respect to the average size of a proton (rp = 0.860 fm) and that of a neutron (rn = 0.974 fm), taken as the only universal size parameters of the model. This happens by packing the candidate shells themselves and following the instruct tions of the next section for all candidate shells in Figure 1 Figure 1. Most probable forms and average sizes of nuclear shells and subshells for nuclei up to Z = 82 and N = 126. Copyright © 2013 SciRes. JMP
G. S. ANAGNOSTATOS Copyright © 2013 SciRes. JMP 56 1. In Table 1 the average shells of Figure 1 which are in contact with each other are given. Specifically, this table provides what polyhedron is in contact with a previous one. In general, the prefix Z stands for a proton polyhe dron and the prefix N for a neutron polyhedron. In Figure 1 the dodecahedra, the icosidodecahedra, and the rhombic triacondahedra are always analysed as mentioned earlier. These analyses lead to the maximum compactness. All octahedra and all cubes in the figure have parallel edges. However, the dodecahedron Z3 has parallel edges only with the dodecahedron N8 but not with dodecahedron Z5. The orientation of Z5 is derived by rotating either Z3 or N8 90° around the z axis. Simi larly, the icosahedron N3 has parallel edges only with the icosahedron N6 but not with icosahedron Z6. The orient tation of Z6 is also derived by rotating N3 or N6 90° around the z axis. Also, the orientation of icosidodecahe dron Z8 is derived from that of icosidodecahedron N4 by considering similar rotation i.e., again 90°. These rota tions lead to pairs of similar polyhedra with perpendicu lar edges. All these rotations favour the maximum com pactness of the polyhedra involved. It is worth mentioning that the three proton polyhedra Z4, Z5, and Z6 are the results of analysis of a rhombic triacondahedron maintaining the same central, angular structure , as the three neutron polyhedra N6, N7, and N8 are. Each of these two rhombic triacondahedra could be derived by rotating the one with respect to the other by 90° around the z axis. Similarly, the polyhedra N4 and N5, and the polyhedra Z7 and Z8 are the results of analy sis of two icosidodecahedra with orientations differing by 90°as well. Finally, the polyhedra N9, N10 and N11 are angularly related polyhedra. Indeed, the middles of edges of the cube N9 form the cuboctahedron N11 and the middles of faces of N11 form the rhombic cuboctahedron N10. In conclusion, each row of neutron or proton poly hedra in Figure 1 angularly refers to one and only one polyhedron which is analysed as shown in the figure. As mentioned earlier, each row in Figure 1 of neutron (pro ton) polyhedra corresponds to an opposite row of proton (neutron) polyhedra which have the same rotational symmetry. In geometrical language, polyhedra with the same rotational symmetry are called reciprocal. As we can see in Figure 1, the numbers inside brackets in the block of a polyhedron close to the central vertical line of the figure, which are equal to the sum of vertices of all previous polyhedra including this polyhedron, co incide with the magic numbers for protons and neutrons, i.e., 2, 8, 20, 50, 82, and 126. The numbers inside brack ets in the remaining blocks of the figure stand for the semimagic numbers 28 and 40 for protons, and 62, 70, 90, and 114 for neutrons. The numbers in the blocks of N4 and Z7 do not form semimagic numbers since the corresponding polyhedra are not present in all cases. As will be understood later, in many cases the octahedra N4 and Z7 remain as part of the structure of the neighbour ing icosidodecahedra N5 and Z8, respectively. The places of their vertices in the neighbouring polyhedron are marked with h (hole) and form a similar polyhedron, as happens in all other cases of analyses of polyhedra mentioned earlier. The successful reproduction of magic and semimagic numbers by the polyhedra of Figure 1 permits a tentative assignment of quantum states to the polyhedra vertices standing as average positions of nucleons. One rule is that quantum subshells are always assigned at the vertices of the same polyhedron. A test of this tentative assign ment of quantum states to the vertices of polyhedra will be made in the section of the quantum mechanical treat ment of the model below. In the future, this assignment will be further and further tested by employing more and more nuclear observables. Table 1. Initial and final radii in Fermi of polyhedra whose bags at their vertices are in contact. Initial Final Final Final Polyh. Radius Polyh. Radius Polyh. Radius Polyh. Radius N1 0.974 Z1 1.554 Z2 2.541 N2 2,511 N2 2.511 Z3 3.946 N3 3.568 N5 4.459 Z2 2.541 Z4 4.341 Z3 3.946 Z6 5.5167 N3 3.568 Z5 5.2122 Z4 4.341 N4 5.0372 N7 6.175 N5 4.459 Z7 6.293 N4 5.0372 Z8 6.8712 N10 6.748 Z5 5.2122 N6 6.6713 Z6 5.5167 N8 6.9329 N7 6.175 N9 8.123 N8 6.9329 N11 7.8688
G. S. ANAGNOSTATOS 57 In Figure 1 the highsymmetry equilibrium polyhedra [2] employed by the model to represent the most prob able forms and average sizes of all nuclear shells and subshells up to Z = 82 and N = 126 are given. Specifi cally, at the top of each block of the figure the name of the polyhedron shown in this block (left) and the quan tum states of the nucleons accommodated by this poly hedron (right) are listed, separately for protons and neu trons. In addition at the bottomleft of each block, in a black square, the numbering of this polyhedron in suc cessive order of filling, preceded with the letter Z for protons and with the letter N for neutrons, is given. Over this black square the number of the polyhedral vertices, and the number of the possible unoccupied vertices characterized as holes, h (symmetrically distributed and forming an equilibrium polyhedron themselves), are also given in parentheses. At the bottomright of each block the radius of the polyhedron in units Fermi is listed. Over this radius the cumulative number of vertices of all pre vious polyhedra and this polyhedron is also given in brackets. The pieces of information given in this section are suf ficient, if someone wants to redefine the most probable forms and the average sizes of equilibrium polyhedra of Figure 1 employed as the average forms of nuclear shells. However, if someone just wants to apply the model to any nuclear observable, it is sufficient to take the poly hedra of this figure for granted and not to get involved with its derivation. 2.2. Radii of Polyhedra and Coordinates of Their Vertices Figure 2 is employed to facilitate the present section. The sphere labelled 1 (with coordinates of its center x0, y0, z0) stands for an average nucleon position at a vertex of a polyhedron with known radius R0. The sphere la belled 2 (considered in contact with the previous sphere and with coordinates of each center x, y, z) stands for an average nucleon position at a vertex of another polyhe dron with unknown radius Rx. Apparently, among the above totally six coordinates and the distance d12 between the centers of the two spheres 1 and 2, Eq. (1) is valid (x x0)2 + (y y0)2 + (z z0)2 = d12 2, (1) On the line O1 we assume the vertex called 1 (with coordinates x1, y1, z1) of a polyhedron which has the symmetries of the polyhedron with known radius R0. On the line O2 we assume the vertex called 2 (with coordi nates x2, y2, z2) of a polyhedron which has the symme tries of the polyhedron with unknown radius Rx. It is ap parent that the coordinates x0, y0, z0 and x, y, z can be expressed with respect to the coordinates x1, y1, z1 and x2, y2, z2, and the radii R0 and Rx as follows: x0 = R0*x1/R1, y0 = R0*y1/ R1, z0 = R0*z1/ R1 x = Rx* x2/ R2, y = Rx* y2/ R2, z = Rx*z2/ R2 R0 2 = x0 2+ y0 2+ z0 2, Rx 2 = x2+y2+z2, R1 2 = x1 2+y1 2+z1 2, R2 2 = x2 2+y2 2+z2 2 (2) By substituting the above relationships to Eq.(1) finally we obtain Eq.(3) R2x  [(x1x2+y1y2+z1z2)*2 R0/( R1 R2)] Rx + [ R0 2 d12 2] = 0 (3) The quantity d12 takes on the following numerical val ues: d12 = rn + rn = 1.948 fm when two neutron bags are in contact, d12 = rn + rp = 1.834 fm when a neutron bag is in con tact with a proton bag, and d12 = 2*<r2>ch = 1.800 fm when two proton bags are in contact, where <r2>ch = 0.900 fm is the average proton charge radius from the literature [7]. By employing Eq.(3) we can determine the radius of the new polyhedron Rx with respect to the radius of the known polyhedron R0 and the coordinates x1, y1, z1 and x2, y2, z2 of the two auxiliary polyhedra employed. These coordinates for all kinds of polyhedra involved in Figure 1 are given in Table 2 together with corresponding radii R1 and R2 necessary for the application of Eq.(3). If Eq. (3) does not have a solution, then there is no contact between bag 1 of the known polyhedron and bag 2 chosen at a specific vertex of the candidate polyhedron. Working in same way, we can check whether there is a contact of bag 1 with bag 2 considered at another vertex of the same candidate polyhedron. Usually, we can fore see which vertex of the polyhedron under investigation is closer to the chosen vertex of the known polyhedron and thus one trial is enough. In general, a simple computer program can easily solve the problem employing all ver tices of the candidate polyhedron simultaneously. As mentioned, the sequences of polyhedra whose bags at their vertices are in contact with those of previous polyhedra are given in Table 1. In this table each poly hedron used as known polyhedron with radius R0 (called initial polyhedron) and the derived polyhedron with ra dius Rx (called final polyhedron) are shown together with their radii. The radius Rx is derived by taking the polyhe dron of the same row in the table as initial one and ap plying Eqs.(13). As shown in the table there are cases where one specific polyhedron used as an initial polyhe dron leads to the derivation of two or three final poly hedra of different symmetry. The described procedure strictly leads to the maximum possible density of poly hedra standing as the most probable forms and average sizes of shells and subshells for fermions. No overlap Copyright © 2013 SciRes. JMP
G. S. ANAGNOSTATOS 58 ping of bags appears among any bags at the vertices of the polyhedra in Figure 1. In general, the polyhedral average shell structure presented in Figure 1 is unique. The coordinates of all average positions for nucleons in closed shells for nuclei up Z = 82 and N = 126 are obtained by considering the proper coordinates from Ta ble 2 and then multiplying by the radius of the polyhe dron of interest from Figure 1. 2.3. Multiharmonic Treatment of a Nucleus The analytical part of the model assumes a harmonic oscillator potential for the nucleons of each proton or neutron shell (and not for all nucleons in a nucleus): V(r) = v0 +1/2 mω2r2, (4) where v0 and ω are different parameters for each proton or neutron shell. That is, the potential form is the same for all nucleons, but its parameters are different for each proton or neutron shell. Due to the two assumptions be low, however, the final number of parameters is substan tially reduced. a) The ћω, for each shell, is determined [8] according to Eq.(5) ћω = (ћ2/m)(n+3/2)/<r2>, (5) where <r2>1/2 is the average size of a specific proton or neutron shell which remains constant for all nuclei. All these sizes are given in Figure 1. The only parameters necessary in determining these sizes are the two size pa rameters rp and rn, as discussed in section 2.2. Figure 2. Determination of the radius of a polyhedron with respect to the radius of another polyhedron when nucleon bags at their vertices are in contact. Table 2. Standard coordinates and corresponding radii of all polyhedra employed by the model for nuclei up Z = 82 and N = 126 (see Figure 1). Symbol τ = (5+1)/2. Polyhedron Standard coordinates Radii Zerohedron Z1 (1, 1, 1), (1, 1, 1) 3 N1 (1, 1, 0), (1, 1, 0) 2 Cube Z2, Z4, N7, N9 (±1, ±1, ±1) 3 Octahedron N2, N5, Z7 (±1, 0, 0), (0,±1, 0), (0, 0, ±1) 1 Dodecahedron Z3, Ν8 (±τ2, ±1, 0), (±1, 0, ±τ2), (0, ±τ2, ±1), (±τ, ±τ, ±τ) τ3 Z5 (0, ±1, ±τ2),((±1, ±τ2, 0),((±τ2, 0, ±1), (±τ, ±τ, ±τ) τ3 Icosahedron N3, Ν6 (±τ, 0, ±1), (0, ±1, ±τ), (±1, ±τ, 0) 2 Ζ6, (0, ±τ, ±1), (±τ, ±1, 0), (±1, 0, ±τ) 2 Icosidodecahedron N4 (±1, ±τ, ±τ2), (±τ, ±τ2,±1), (±τ2, ±1, ±τ), 2τ (0, 0, ±2τ), (0, ±2τ, 0), (±2τ, 0,0) Ζ8 (±τ, ±1, ±τ2), (±1, ±τ2, ±τ), (±τ2, ±τ, ±1), 2τ (0, 0, ±2τ), (0, ±2τ, 0), (±2τ, 0,0) Cuboctahedron N11 (±1, 0, ±1), (0, ±1, ±1), (±1, ±1, 0) 2 Rhombic Cuboctahedron N10 N9 [± 1, ± (21), ± (21),], [ ± (21), ± (21), ± 1], [ ± (21), ±1, ±(21)] 247 Copyright © 2013 SciRes. JMP
G. S. ANAGNOSTATOS 59 b) The parameter of the depth of the potential for each proton or neutron shell is determined according to Eq.(6) Ej = vj  ћωj(nj+3/2) = vi  ћωι(ni + 3/2) = Ei, or vj = vi – ћωi(ni + 3/2) + ћωj(nj+3/2) = V0 + ћωj(nj + 3/2). (6) This assumption implies that all nucleons in a nucleus are equally bound in their own potentials (excluding Coulomb and spinorbit interactions). It is apparent from Eq.(6) that, since according to the previous assumption a) all ћω are already determined as above by applying Eq.(5) with respect only of the two size parameters rp and rn, the depth vi of the potential for each shell i can be defined with the knowledge of only one additional parameter V0 (This is the third universal parameter of the present model equal to 40.268 MeV). Solving Schroedinger’s equation for the potential of Eq.(4), we obtain [8] the following equation Lk ℓ+1/2 (z) = {[ Γ(1/2Ν + 1/2 ℓ + 1)]2 / k! Γ(ℓ +3/2)} 1F1(k ; ℓ + 3/2; z), (7) where 1F1 (α; c; z) = 1+ (α/c)z + α(α+1)z2/c(c+1) 2!+ α(α+1)(α+2)z3/c(c+1)(c+2)3! +…, with c = ℓ + 3/2, α = k, z = 2αr2, and α = mω/2ћ. The series terminates with the term (1)k[Γ(c)/Γ(c + k)]zk and the various quantum numbers involved are n = 0,1,2,…; k = 0,1,2,…; k ≤ n/2, ℓ = 0,1,2,…, ℓ ≤ n, and ℓ = n  2k [8]. The explicit forms of the first few equations of the wave functions for a harmonic oscillator potential de rived from Eq.(7) can be found in several books of Quan tum Mechanics and Nuclear Physics [8]. However, it is considered instructive for a complete list of all wave functions for nuclei up to Z = 126 and N = 184, derived from the recursion formula Eq.(7), to be shown below. R1s = (128/π)1/4α3/4 2 ar e R2s = (128/ π)1/4(3/2)1/2 α3/4 (1 4αr2/3) 2 ar e R3s = (128/π)1/4(15/8)1/2α3/4(1 8αr2/3 + 16α2r4/15) 2 ar e R4s = (128/π)1/4 (35/16)1/2 α3/4 (1 4αr2 + 16α2r4/5 – 64α3r6/105)) 2 ar e R1p = (128/π)1/4(4/3)1/2 α5/4 r 2 ar e R2p = (128/π)1/4(10/3)1/2 α5/4 r (1 – 4αr2/5) 2 ar e R3p = (128/π)1/4 (70/12)1/2 α5/4 r (1 – 8α r2/5 + 16 α2r4/35) 2 ar e R1d = (128/π)1/4 (16/15)1/2 α7/4 r2 2 ar e R2d = (128/π)1/4(56/15)1/2 α7/4 r2 (1 – 4α r2/7) 2 ar e R3d = (128/π)1/4(252/30)1/2 α7/4 r2 (1 – 8α r2/7 + 16α2r4/63) 2 ar e R1f = (128/π)1/4(64/105)1/2 α9/4 r3 2 ar e R2f = (128/π)1/4(288/105)1/2 α9/4 r3 (1 – 4α r2/9) 2 ar e R1g = (128/π)1/4(256/945)1/2 α11/4 r4 2 ar e R2g = (128/π)1/4(1408/945)1/2α11/4 r4 (1 – 4α r2/11) 2 ar e R1h = (128/π)1/4(1024/10395) 1/2 α13/4 r5 2 ar e 2 R1i = (128/π)1/4(4096/135135 )1/2 α15/4 r6 ar e R1j = (128/π)1/4 (16384/2027025)1/2 α17/4 r7 (8) 2 ar e As a test, all above wave functions have been checked for orthonormality. Due to the fact that in the model ћω is different for the different shells, the wave functions with the same orbital angular momentum quantum number ℓ are not orthogo nal. For these wave functions GramSmidth’s technique is applied [9]. It is considered instructive to give some relevant guide equations below to facilitate this orthogonalization. E2(α1, α2, r) = N2(α1, α2) [R2(α2, r) + b21(α1, α2) R1 (α1, r)] E3(α1, α2, α3, r) = N3(α1, α2, α3) [R3(α3, r) + b32(α1, α2, α3) E2(α1, α2, r) + b31(α1, α3) R1(α1, r)] E4(α1, α2, α3, α4, r) = N4(α1, α2, α3, α4) [R4(α4, r) + b 43(α1, α2, α3, α4) E3(α1, α2, α3, r) + b42(α1, α2. α4) E2(α1, α2, r) + b41(α1, α4) R1(α1, r)], (9) where N2(α1, α2) = [1  b (α1, α2) ]1/2 2 21 N3(α1, α2, α3) = [1 b( α1, α2, α3)  b( α1, α3)]1/2 2 32 2 31 N4(α1, α2, α3, α4) = [1b( α1, α2, α3, α4) b( α1, α2, α4)  b( α1,α4)]1/2, 2 43 2 42 2 41 (10) and b21(α1, α2) =  R1(α1, r) R2(α2, r) r2dr 0 b32(α1, α2, α3) =  E 2(α1, α2, r) R3(α3, r) r2dr 0 b31(α1, α3) = R1(α1, r) R3(α3, r) r2dr 0 b43(α1, α2, α3, α4) =  E 3(α1, α2, α3, r) R4(α4, r) r2dr 0 b42(α1, α2, α4) =  E2(α1, α2, r) R4(α4, r) r2dr 0 b41(α1, α4) =  R1(α1, r) R4(α4, r) r2dr (11) 0 Copyright © 2013 SciRes. JMP
G. S. ANAGNOSTATOS 60 Also, due to the fact that the model requires the or thogonalized wave functions to reproduce the relevant average radii of polyhedral shells given in Fig.1, some additional instructive equations are also given below for radii. <r2>2 (α1, α2) = E (α1, α2, r) r4dr 0 2 2 <r2>3(α1, α2, α3) = E (α1, α2, α3, r) r4dr 0 2 3 <r2>4(α1, α2, α3, α4) = E (α1, α2, α3, α4, r) r4dr 0 2 4 (12) While is known that α1 = (m/2ћ2)*ћω1, in the case of wave functions which need orthogonalization α2, α3, and α4 in the model are determined as follows. For example for the case of 2p state for neutrons, while α1 is deter mined from Eq.(5) by using the radius 3.568 fm (see N3 in Figure 1), the parameter α2 in the first equation of (12) must take on a value which leads to the radius 5.0372 fm (see N4 in Figure 1). For the case of 3p state for neutrons, the parameters α1 and α2 in the second equa tion of (12) take on the previous values, while α3 must take on a value which leads to the radius 7.8688 fm (see N11 in Figure 1). 2.3.1. Binding Energies As known, the binding energy for each quantum state in a harmonic oscillator potential is given by Eq.(13): EB = v – ћω (n + 3/2). (13) As aforementioned, in Eq.(13) all ћω come from Eq.(5) (with respect to the only two size parameters rp and rn) and all potential depths v come from Eq.(6) (with respect to the only one additional, potential parameter V0). Thus, EB from Eq.(13) is determined with respect to only 3 universal parameters. Now, since the nuclear problem basically refers to twobody forces, in order to avoid the double counting, the potential portion of the second part of Eq.(13) should be divided by two. Further, since according to the Virial theorem half of the quantity ћω (n + 3/2) is potential en ergy and half is kinetic energy, Eq.(13) takes the form of Eq.(14) EB =1/2 [v –1/2 ћω (n + 3/2)]  1/2 ћω (n + 3/2)]) = ½ v – 3/4 ћω (n + 3/2) (14) Given that the potential v, according to Eq.(6), is v = V0 + ћω (n + 3/2), (15) the final expression of binding energy for each proton or neutron state takes the form of Eq.(16). (EB)1 = ½ [V0 + ћω (n + 3/2]  3/4 ћω (n + 3/2) = ½ V0 – 1/4 ћω1 (n1 + 3/2) (16) The index 1 in Eq.(16) refers to all states where the orbital angular momentum quantum number ℓ appears for the first time. However, for a second, a third and a fourth appearance of a state with the same ℓ, in the place of the quantity ћω (n + 3/2) in Eq.(16) we should take the corresponding quantity due to the necessary orthogonalization. Thus, based on Eqs. (9) – (11), we get (EB)2 = ½ V0  ¼ N2 [ ћω2*(n2 + 3/2) + b21*ћω1(n1 + 3/2)] (EB)3 = ½ V0  ¼ N3 {ћω3*(n3 + 3/2) + b32*4 [(EB)2 + ½ V0] + b31*ћω1(n1 + 3/2)} (EB)4 = ½ V0  ¼ N4 {ћω4*(n4 + 3/2) + b43*4 [(EB)3 + ½ V0] + b42*4 [(EB)2 + ½ V0] + b41*ћω1(n1 + 3/2)} (17) Finally, the total binding energy of a nucleus in the model is given by Eq.(18) (EB)total = Σ (EB)i + ΣVLiSi  ΣECij , (18) A i1A i1Zji where: the spinorbit term is given [8] by Eq.(19) ΣVLiSi i= λ Σ [i/r*dV(r)/dr]*ℓisi A i1A i1 = λ Σ (ћωi)2/(ћ2/m)*ℓisi, (19) A i1 with λ = 0.03 (This is the fourth and the last universal parameter of the present model), and the Coulomb term is given by Eq.(20): ΣECij = Σe2/dij. (20) ZjiZji The distance dij between any two proton average posi tions in Figure 1 is calculated from the coordinates of the proton polyhedral vertices (standing for the proton average positions) in this figure (see section 2.2). An extra term in Eq.(18) due to isospin is not needed since the isospin is here taken care of by the different average shell structure between protons and neutrons as apparent from Figrue 1. In Table 3 the derived energies (including spinorbit) for all proton and neutron singleparticle states up to 208Pb are given. Specifically, in col.2 of the table for each state (col.1) the numerical value of energy for protons and underneath for neutrons are listed. In col. 4 the en ergy of all nucleons in a state (col. 3) is given again in two rows for each state as before. In the remaining cols. 511 the energies from col. 4 relevant to 40Ca, 48Ca, 54Fe, 90Zr, 108Sn, 114Te, 142Nd, and 208Pb are repeated. There, one can also see which states are considered in the framework of the model occupied by protons and neu trons in the structure of these nuclei. For all seven nuclei examined 16O is taken as a core. This means that the 16O experimental binding energy Copyright © 2013 SciRes. JMP
G. S. ANAGNOSTATOS Copyright © 2013 SciRes. JMP 61 (127.62 MeV) and its Coulomb energy (11.98 MeV de rived from the model) are assumed known. That is, the energy for the 16O core is taken equal to 139.60 MeV and is listed in the fourth row from the bottom of Table 3. In the following row of the table the calculated relevant Coulomb energies in the framework of the model are given and underneath, the predicted and experimental binding energies [10]. The agreements between these two binding energies are apparent and quite striking. This agreement is further strengthened by the fact that the model employed uses only four universal parameters. Namely, these parameters are: The two size parameters rp = 0.860 fm and rn = 0.974 fm, the one potential parame ter V0 = 40.268 MeV, and the one spinorbit parameter λ = 0.03. As obvious from Table 3 the binding energy of any nucleus examined is obtained by summing up the ener gies of the single particle states involved in this nucleus. This means that the model employed here considers ex clusively an independent particle motion for all nucleons. In other words, the model considers that for each closed shell we have a complete saturation of nuclear forces. Table 3. Single particle and total energies in MeV. Bold numbers are discussed in the text. StateEnergy# Total 40 Ca 48 Ca 54 Fe 90 Zr 108 Sn 114 Te 142 Nd 208 Pb 1d5/12.0356 72.21 72.2172.2172.2172.2172.2172.2172.2172.21 10.25861.5561.5561.55 61.5561.5561.5561.5561.55 61.55 1d3/ 11.878447.5147.5147.5147.5147.5147.5147.5147.5147.51 10.02340.0940.0940.09 40.0940.0940.0940.0940.09 40.09 2s1/12.4762 24.95 24.9524.9524.9524.9524.9524.9524.9524.95 10.60621.2121.2121.21 21.2121.2121.2121.2121.21 21.21 1f7/12.452899.62 74.7299.62 99.6299.6299.6299.62 11.924 95.3974.21 95.3995.39 95.3995.3995.3995.39 1f5/12.332673.9973.9973.9973.9973.9973.99 11.78770.7270.72 70.7256.7256.72 56.72 2p3/16.2634 65.0565.0565.0565.0565.0565.05 16.28565.1465.14 65.1465.1465.1465.14 2p1/10.162 20.3220.3220.3220.3220.3220.32 16.22732.4532.45 32.4532.4532.4532.45 1g9/9.9041099.04 99.0499.0499.0499.04 7.89878.9878.98 78.9878.9878.9878.98 1g7/ 9.721877.77   77.77 7.63561.08  61.0861.08 2d5/14.373686.24 30.3591.06 86.24 17.29103.74103.74 103.74103.74103.74 2d3/15.117460.47  60.4760.47 17.267 69.0734.54 69.0769.07 69.07 3s1/12.574225.15  25.15 10.84221.6821.6821.68 21.68 1h11/10.90912130.91 130.91 11.084133.01133.01 133.01 1h9/12.60210126.02  10.445 104.45104.45 2f7/ 8  16.311 130.49130.49 2f5/ 6  16.269 97.6197.61 3p3/ 4  17.687 70.7570.75 3p1/ 2   17.644 35.2935.29 1i13/ 14   7.436 104.1104.1 16 O core139.61139.61139.61139.61139.61139.61139.61139.61 E C 64.8164.81105.92225.8330.9 357.53448.79769.24 E B mod..342.31416.52 471.31782.98915.18961.131185.14 1636.4 E B exp.342.06416.01 471.77783.24914.66961.231185.17 1636.5
G. S. ANAGNOSTATOS 62 It is necessary to explain why in Table 3 the binding energy of neutron 1f5/2 state for 114Te, 142Nd, and 208Pb is different than that for 90Zr and 108Sn. For the nuclei of the table up to 108Sn the six neutrons of N4 are at the vertices of N5 marked with h and the binding energy of neutron 1f5/2 state is estimated as part of N5.. For the heavier nuclei the N4 is completed and the binding energy of neutron 1f5/2 state (shown in bold letters in the table) is estimated as part of this polyhedron. In Table 3 some additional binding energies are shown in bold letters. For these states some discussion is needed as for the 1f5/2 neutron state above. Specifically, in 108Sn the energy 34.54 MeV of the neutron 2d3/2 state corresponds to two neutrons and, of course, should be half the energy 69.08 MeV in column 4 (which corre sponds to four neutrons). An identical explanation is valid for the difference of energies 30.24 MeV and 60.48 MeV for the proton 2d3/2 state in the case of 114Te. In the case of 142Nd the energy 91.08 MeV of the 2d5/2 protons is calculated assuming that the six neutrons are on Z8 instead of on Z7. The Z7 is completed later, after the formation of N5. Some extra discussion is also required for the 1f7/ neutron state of 48Ca. The state binding energy shown in bold numbers in Table 3 is derived by assuming that the eight neutrons of this state fill the polyhedron Z4 which for all other seven nuclei of Table 3 is filled by protons. This is permissible according to the assumptions of the model as long this polyhedron is completely empty. The assignment of polyhedra as proton or neutron polyhedra in Figure 1 assumes that all polyhedral shells are filled as for example in the case of 208Pb. In the cases where some polyhedra are completely empty, one can consider that these polyhedra are available either for protons and or for neutrons. In the case of 48Ca the radius of Z4 is derived from that of Z2, i.e., RZ4(prime) = 2.541 + rp + rn = 4.375 fm, while previously it was RZ4 = 2.541 + 2rp = 4.341 fm. This value is smaller than the value of RN4 = 4.459 fm and thus preferable since it leads to larger bind ing energy. In general, in the case of valence protons occupying a neutron polyhedron or vice versa Eq.(3) is again applied with the proper choice of d12 as explained in section 2.2. 2.3.2. Mean Radii Since the wave functions are known in the framework of the model (section 2.3), the nuclear radii can be calcu lated straightforwardly by using these functions. How ever, due to the way these wave functions have been correlated with the size of nuclear shells via Eq.(5), av erage radii can equivalently be calculated by using sim ple formulae, as seen below. Average charge radius: <r2>ch = ΣZ ir/ Z + r– r*N/Z, (20) 1 2 i2 .. protonch 2 .neutronch where rch.proton = rp = 0.842 fm and rch..neutron =0.34 fm[7]. Average neutron radius: <r2>n =Σ r2 i/ N + r, (21) N i1 2 neutron where r= rn= 0.974 fm. neutron Average mass radius: <r2>m=[Σri 2+ Z*r + N*r] /A (22) A i1 2 p2 n If point radii are required, in the above formulae only the first term in each equation remains. All values of ri needed are included in Figure 1 (See right corner at the bottom of each block). In Table 4 the predictions of the present model for ra dii are given for all eight nuclei used as a sample in the present manuscript. In this table the predictions of the model on average charge, on average neutron and on average mass radii are given together with the difference between proton and neutron radii. The comparisons of the model predictions for charge radii with the relevant experimental data (coming from [11] and only for 54Fe from [12] which is a model prediction), in all cases, are very good. In most of the cases our predictions and the data are identical. For 114Te an experimental value has not been found. However, the comparisons of the model predictions with the available experimental data for the point neutronproton radii are good only for the first five nuclei of the table. For the next two nuclei, i.e. 114Te and 142Nd, experimental values have not been found and for the last nucleus 208Pb the comparison with experiments is problematic. For that nucleus the relevant pieces of in formation from the literature concerning neutronproton radii are given below together with comments for the other nuclei. From [13] for 40Ca we have neutronproton radius 0.20 fm till 0.40 fm and in Table 4 we write the aver age of these two values, i.e., 0.30 fm. The value 0.12 fm for 48Ca is taken from [14]. From [15] we have the values for 54Fe fm and for 90Zr 0.09 ± 0.02 fm and in Table 4 we write 0.04 and 0.09, respectively. The value 0.11fm for 108Sn in the table is estimated from the neu tron rms radius 4.615 fm from [16] minus our point pro ton rms radius 4.50 fm. Finally, the value 0.55 fm in Ta ble 4 for 208Pb is estimated by employing the ratio of neutron radius/ proton radius equal to 1.07 ± 0.03 fm from [17] and our point proton rms radius 5.46 fm, i.e., 1.1*5.46 – 5.46 = 0.55 fm. Other data somehow support ing our value 0.55 fm in Table 4 comes from [18] where the neutron radius is 6.7 fm, from [19] where this radius is 5.99 ± 0.10 fm, and from [20] where rn  rp = 0.33 ± 0.17 fm. There is a great variety of experimental and theoretical results for the neutronproton rms radius of 208Pb varying from small to large values. In general, the estimation of neutron rms radius of a nucleus is very dif ficult task. The opposite is true for the estimation of pro ton rms radius of a nucleus, due to the charge of protons. 0.06 0.08 0.04 Copyright © 2013 SciRes. JMP
G. S. ANAGNOSTATOS 63 Table 4. Average charge, neutron, point neutronproton, and mass radi. Nucleus 40 Ca 48 Ca 54 Fe 90 Zr 108 Sn 114 Te 142 Nd 208 Pb Polyh.:Radius Occupation Z1: 1.55422222222 Z2: 2.54166666666 Z3: 3.9461212121212121212 Z4: 4.341868888 Z5: 5.21221212121212 Z6: 5.516710121212 Z7: 6.29366 Z8: 6.8712224 N1: 0.97422222222 N2: 2.51166666666 N3: 3.5681212121212121212 N4: 5.037222424242424 N5: 4.459666666 N6: 6.671341212 N7: 6.1758888 N8: 6.93291212 N9: 8.1238 N10: 6.74824 N11: 7.868812 ch <r 2 > 1/2 mod. 3.483.483.72 4.274.56 4.614.95.51 ch <r 2 > 1/2 exp. 3.48 3.48 3.694.274.564.915.51 n <r 2 > 1/2 mod. 3.253.653.734.44.7 4.86 5.436.18 [ n <r 2 > 1/2  ch <r 2 > 1/2 ]mod. 0.290.120.03 0.090.1 0.220.50.64 [ n <r 2 > 1/2  ch <r 2 > 1/2 ]exp. 0.3 0.120.040.090.11  0.55 m <r 2 > 1/2 mod. 3.383.563.81 4.354.694.8 5.225.93 8 It is interesting one to remark the relation between proton and neutron rms radii of a nucleus in the present model. In the model the structures of proton shells and of neutron shells are interweaved (see Figure 1), as a result of basic properties of the constituent particles of a nu cleus, i.e., of the fermionic nature of protons and neu trons. Thus, the predictions on neutron rms radii in the present model depend on proton rms radii. The fact that the binding energies of Table 3 and the proton rms radii of Table 4 are in very good agreements with the experi mental data gives a special interest to our neutron rms predictions which should be investigated deeper. 2.3.3. Other Applications of the Model The present work follows a standard procedure starting with the Schrödinger equation. Previous versions of the model employing mainly a semiclassical approach (in the spirit of the Ehrenfest theorem [2123] that for the average values the laws of Classical Mechanics are valid), which uses the same nucleon average positions as those of Figure 1, have already been applied in many cases. Some are Σ atoms [24], nuclear momentum and density distribution [25], moments of inertia and rotating spectra [26], excited states [27], neutron nuclei [27], derivation of a twobody potential [28], and nuclear reactions [29]. 3. Conclusions The multiharmonic shell clustering of a nucleus pre nted here, employing only four universal parameters (2 size parameters rp and rn, 1 potential parameter V0, and 1 spinorbit parameter λ) has proved very successful. The predictions on magic and semimagic numbers, binding energies and radii are very satisfactory. We expect that further applications of the present approach on more ob servables will be successful as well, as are the cases of application of the semiclassical version of the model (see section 2.3.3 above). The present work apparently offers a link between collective model and independent particle model of nuclear structure. Indeed, the average sizes of shells are derived in the spirit of Independent Particle Model (see section 2.1 above) and these average forms in the spirit of Collective Model are employed to reproduce the rotational spectra [30]. While in the present work a number of nuclei (eight nuclei) are used as an example to demonstrate the model, Figure 1 could be used for the structure of the core of any nucleus. Perhaps, the most significant conclusion of the present work is that it supports a shell clustering of nuclear Copyright © 2013 SciRes. JMP
G. S. ANAGNOSTATOS 64 structure exclusively based on the fermionic nature of nucleons and their average sizes. In brief, the antisym metric wave function of nucleons makes them behave as if a repulsive force (of unknown nature) is acting among them. This force for nucleons on spherical shells, in order to obtain equilibrium of forces (which is a necessary condition for an independent particle motion of nucleons), it leads to equilibrium structures for nuclear shells, i.e., to the structures of regular polyhedra and their derivative polyhedra [2, 45]. Furthermore, these polyhedral struc tures are finally closed packed to obtain maximum nu clear density for the average positions of nucleons by taking advantage of their finite sizes. The basic computer programs employed by the present work are available on request. 4. Acknowledgements The author is deeply indebted to Dr. D. Lenis of our In stitute for his excellent computer work related to the present manuscript. REFERENCES [1] C. W. Sherwin, “Introduction to Quantum Mechanics,” Holt Rinehart and Winston, New York, 1959, p.205. [2] J. Leech, “Equilibrium of Sets of Particles on a Sphere,” Mathematical Gazette, Vol. 41, No. 36, 1957, pp. 8190. doi:10.2307/3610579 [3] G. S. Anagnostatos, “Symmetry Description of the Inde pendent Particle Model,” Lettere Al Nuovo Cimento, Vol. 29, No. 6, 1980, pp. 188192. [4] G. S. Anagnostatos, “The Geometry of the Quantization of Angular Momentum (ℓ, s, j) in Fields of Central Sym metry, Lettere Al Nuovo Cimento, Vol. 28, No. 17, 1980, pp. 573 576. [5] G. S. Anagnostatos, J. Giapitzakis, A. Kyritsis, “Rota tional Invariance of OrbitalAngularMomentum Quanti zation of Direction for Degenerate States”, Lettere Al Nuovo Cimento, Vol. 32, No. 11, 1981, pp. 332335. doi:10.1007/BF02745301 [6] H. M. Cundy and A. P. Rollett, “Mathematical Models,” 2nd Ed., Oxford University Press, 1961, p.76. [7] H. S. M. Coxeter, “Regular Polytopes”, 2nd Ed., The Macmillan Company, New York, 1963. [8] G. S. Anagnostatos, “Isomorphic Shell Model for ClosedShell Nuclei, ” International Journal of Theoretical Physics, Vol. 24, 1985, pp. 579613. doi:10.1007/BF00670466 [9] De Jager, C. W. H. de Vries, C. de Vries, “Nuclear Charge and Momentum Distribution,” Atomic Data and Nuclear Data Tables, Vol. 14, No. 56, 1974, pp. 479665. doi:10.1016/S0092640X(74)800021 [10] W. F. Hornyak, “Nuclear Structure,” Academic, New York, 1975, p 13. [11] J. D. Vergados, “Mathematical Methods in Physics,” Akourastos Giannis, Greece, 1970. [12] A. H. Wapstra and N. B. Gove, “Atomic Mass Table,” Automic Data and Nuclear Data Tables, Vol. 9, 1971, pp. 267301. doi:10.1016/S0092640X(09)800016 [13] E.G.Nadjakov, K.P.Marinova, and Yu.P.Gangrsky, “Sys tematics of Nuclear Charge Radii”, Automic Data and Nuclear Data Tables, Vol. 56, 1994, pp. 133167. doi:10.1006/adnd.1994.1004 [14] I. Angeli, “Recommended Values of R.M.S. Charge Ra dii”, Heavy Ion Phys., Vol. 8, No. 12, 1998, pp. 2329. [15] L. Ray, G. W. Hoffmann, and W. R. Coker, “Nonrelativ istic and Relativistic Descriptions of ProtonNucleus Scattering,” Physics Reports, Vol. 212, No. 5, 1992, pp. 223328. doi:10.1016/03701573(92)90156T [16] R. C. Barrett., “Colloques,” Journal de Physique, Vol. 34, 1973, pp. 2328. [17] A. Trzcinska, “Nuclear Periphery Studied with Antipro tonic Atoms”, Hyp. Inter., Vol. 194, No. 1, 2009, pp. 271276. [18] J. Terasaki, J. Engel, “SelfConsistent Description of Multipole Strength: Systematic Calculations,” Physical Review C, Vol. 74, No. 4, 2006, pp. 044301044319. doi:10.1103/PhysRevC.74.044301 [19] S. D. Schery, D. A. Lind and C. D. Zafiratos, “Radius of the Neutron Distribution in 208Pb from (p, n) Quasielastic Scattering,” Physical Review C, Vol. 9, No. 1, 1974, pp. 416418. doi:10.1103/PhysRevC.9.416 [20] G. F. Bertsch, P. F. Bortignon, R. A. Broglia, “Damping of Nuclear Excitations,” Reviews of Modern Physics, Vol. 55, 1983, pp. 287314. doi:10.1103/RevModPhys.55.287 [21] H. J. Koemer and J. P. Schiffer, “Neutron Radius of 208Pb from SubCoulomb Pickup,” Physical Review Letters, Vol. 27, No. 21, 1971, pp. 14571460. doi:10.1103/PhysRevLett.27.1457 [22] S. Abrahamyan, Z. Ahmed, H. Albataineh, K. Aniol. D. S. Armstrong, et al., “Measurement of the Neutron Radius of 208Pb through ParityViolation in Electron Scattering”, cited as: arXiv: 1201.2568v2 [nuclex], Cornell University Library, 13 Jan. 2012, journal reference: Physical Review Letters, Vol. 108, No. 11, 2012, pp. 112502112507. [23] E. Merzbacher, “Quantum Mechanics,” John Wiley and Sons, Inc. New York, 1961, p. 42 [24] C. CohenTannoudji, B. Diu, F. Laloe, “Quantum Mechanics, ” John Wiley & Sons, New York, 1977, p. 240. [25] A. Bohr, B. R. Mottelson, “Nuclear Structure,” W. Α. Benjamin, Inc., Advanced Book Program, Reading, Mas sachusetts, London, Vol. 2, 1975. [26] J. Dabrowski, J. Rozynek and G. S. Anagnostatos, “Σ Atoms and the ΣΝ Interaction,” Eur. Phys. J. A, Vol. 14, 2002, pp. 125131. [27] G. S. Anagnostatos, A. N. Antonov, P. Ginis, J. Giapitzakis and M. K. Gaidarov, “Nucleon Μomentum and Density Distributions in 4He Considering Internal Rotation,” Physical Review C, Vol. 58, No. 4, 1998, pp. Copyright © 2013 SciRes. JMP
G. S. ANAGNOSTATOS Copyright © 2013 SciRes. JMP 65 21152119. [28] M. K. Gaidarov, A. N. Antonov, G. S. Anagnostatos, S. E. Massen, M. V. Stoitsov and P. E. Hodgson, “Proton Mo mentum Distribution in Nuclei beyond 4He,” Physical Review C, Vol. 52, No. 6, 1995, pp. 30263031. [29] G. S. Anagnostatos, P. Ginis, J. Giapitzakis, “αPlanar States in 28Si, ” Physical Reviewe C, Vol. 58, No. 6, 1998, pp. 33053315 doi:10.1103/PhysRevC.58.3305 [30] P. K. Kakanis and G. S. Anagnostatos, “Persisting αPlanar Structure in 20Ne,” Physical Review C, Vol. 54, No. 6, 1996, pp. 29963013. doi:10.1103/PhysRevC.54.2996 [31] G. S. Anagnostatos, “Classical EquationsofMotion Model for HighEnergy HeavyIon Collisions,” Physical Review C, Vol. 39, No. 3, 1989, pp. 877883. doi:10.1103/PhysRevC.39.877 [32] G. S. Anagnostatos and C. N. Panos, “Semiclassical Simulation of Finite Nuclei,” Physical Review C, Vol. 42, No. 3, 1990, pp. 961965. doi:10.1103/PhysRevC.42.961 [33] G. S. Anagnostatos, “Towards A Unification of Inde pendent and Collective Models, 20th Conference of the Hellenic Nuclear Physical Society, Athens, May 2728, 2011 (PDF in: www.uoi.gr/HNPS).
