Deduction of the quantum numbers of low-lying states of 6-nucleon systems based on symmetry

发布于:2021-06-21 11:37:03

arXiv:nucl-th/9805013v1 6 May 1998

C.G.Bao1 , Y.X.Liu2

Department of Physics,Zhongshan University, Guangzhou,510275,P.R.China.

Department of Physics, Beijing University, Beijing, 100871, P.R.China

ABSTRACT: The inherent nodal structures of the wavefunctions of 6-nucleon systems are investigated. A group of six low-lying states (including the ground states) dominated by total orbital angular momentum L=0 components are found, the quantum numbers of each of these states are deduced. In particular, the spatial symmetries of these six states are found to be mainly the {4,2} and {2,2,2}. PACS: 21.45.+v, 02.20.-a, 27.20.+n

As a few-body system the 6-body system has been scarcely investigated theoretically due to the complexity arising from the 15 spatial degrees of freedom. The existing related literatures concern mainly the ground states and a few resonances [1-5]. The study of the character of the excited states is very scarce. On the other hand, the particles of 6-body systems are neither too few nor too many. The study of them is attractive because it may lead to an understanding of the connection between the few-body theory and the modal theories for nuclei. Before solving the 6-body Schr¨dinger equation precisely, if we can have some qualitative undero standing of the spectrum, it would be very helpful. This understanding , together with the results from calculations and experiments, will lead to a complete comprehension of the physics underlying the spectrum. In [6] the qualitative feature of 4-nucleon systems has been studied based on symmetry. In this paper we shall generalize the idea of [6] to extract qualitative character of the low-lying states of 6-nucleon systems. There are two noticeable ?ndings in [6]. (i) The ground state is dominated by total orbital angular momentum L=0 component, while all the resonances below the 2n+2p threshold are dominated by L=1 components, there is a very large gap lying between them. Experimentally, this gap is about 20 MeV. This fact implies that the collective rotation is di?cult to be excited. (ii) The internal wavefunctions (the wavefunction relative to a body-frame) of all the states below the 2n+2p threshold

do not contain nodal surfaces. This fact implies that the excitation of internal oscillation takes a very large energy. Therefore, ti would be reasonable to assume that the L=0 nodeless component will be also important in the low-lying spectrum of the 6-nucleon systems. It was found in [7,8] that a speci?c kind of nodal surfaces may be imposed on the wavefunctions by symmetry. Let Ψ be an eigenstate. Let A denotes a geometric con?guration. In some cases A may be invariant to speci?c combined operations Oi ( i=1 to m). For example, when A is a regular octahedron (OCTA) for a 6-body system, then A is invariant to a rotation about a 4-fold axis of the OCTA by 90? together with a cyclic permutation of four particles. In this case we have

Oi Ψ(A) = Ψ(Oi A) = Ψ(A)


Owing to the inherent transformation property of Ψ (the property with respect to rotation, inversion, and permutation), (1) always can be written in a matrix form (as we shall see) and appears as a set of homogeneous linear algebra equations. They impose a very strong constraint on Ψ so that Ψ may be zero at A. This is the origin of this speci?c kind of nodal surfaces, they are called the inherent nodal surfaces (INS). The INS appear always at geometric con?gurations with certain geometric symmetry. For a 6-body system the OCTA is the con?guration with the strongest geometric symmetry. Let us assume that the six particles form an OCTA. Let k’ be a 4-fold axis of the OCTA, and let the particles 1,2,3, and 4 form a square k′ surrounding k’. Let Rδ denote a rotation about k’ by the angle δ (in degree), let p(1432) denotes a cyclic permutation. Evidently, the OCTA is invariant to
k O1 = p(1432)R?90


Let pij denotes an interchange of the locations of particles i and j, P denotes a space inversion. The OCTA is also invariant to O2 = p13 p24 p56 P (3)

Let i’ be an axis vertical to k’ and parallel to an edge of the above square; say, → parallel to r12 . Then the OCTA is also invariant to
i O3 = p14 p23 p56 R180 .


Let OO ′ be a 3-fold axis of the OCTA, where O denotes the center of mass. Let particles 2,5, and 3 form a regular triangle surroundung the OO ′; 1,4, and 6 form another triangle. Then the OCTA is also invariant to
oo O3 = p(253)p(146)R?120


oo” Besides, the OCTA is also invariant to some other operators, e.g., the p(152)p(364)R?120 (where OO” is another 3-fold axis). However, since the rotations about two di?erent 3-fold axes are equivalent, one can prove that this additional operator does not

introduce new constraints, and the operators O1 to O4 are su?cient to specify the constraints arising from symmetry. Let an eigenstate of a 6-nucleon system with a given total angular momentum J, parity Π, and total isospin T be written as Ψ=



where S is the total spin,


λi FLSM χλi S


λi Where M is the Z-component of L, FLSM is a function of the spatial coordinates, th which is the i basis function of the λ?representation of the S6 permutation group. λ The χS i is a basis function in the spin-isospin space with a given S and T and ? belonging to the λ ?representation, the conjugate of λ. In (7) the allowed λ are listed in Table 1, they depend on S and T [9].

S T λ 6 0 0 {1 }, {2,2,1,1},{3,3},{4,1,1} 4 1 0 {2,1 }, {3,13 }, {2,2,2}, {3,2,1}, {4,2} 2 0 {2,2,1,1}, {3,2,1} 3 0 {2,2,2} 4 3 0 1 {2,1 }, {3,1 }, {2,2,2}, {3,2,1}, {4,2} 6 1 1 {1 }, {2,14 }, 2{2,2,1,1}, {3,13 }, 2{3,2,1}, {3,3}, {4,1,1} 2 1 {2,14 }, {2,2,1,1}, {3,13 }, {2,2,2}, {3,2,1} 3 1 {2,2,1,1} Tab.1, The allowed representation λ in (7) From k’ and i’ de?ned before one can introduce a body frame i’-j’-k’. In the λi body-frame the FLSM can be expanded
λi FLSM (123456) = Q L Where αβγ are the Euler angles to specify the collective rotation, DQM is the well known Wigner function, Q are the projection of L along the k’-axis. The (123456) and (1’2’3’4’5’6’) speci?es that the coordinates are relative to a ?xed frame and the body-frame, respectively. λi Since the FLSQ span a representation of the rotation group, space inversion group, and permutation group, the invariance of the OCTA to the operations O1 to O4 leads to four sets of equations. For example, from λi λi ? λi O1 FLSQ (A) = FLSQ (O1 A) = FLSQ (A) L λi DQM (?γ, ?β, ?α)FLSQ (1′ 2′ 3′ 4′ 5′ 6′ )



λi λi where FLSQ (A) denotes that the coordinates in FLSQ are given at an OCTA, for all Q with |Q| ≤ L we have λ λi [gii′ (p(1234))e?i 2 Q ? δii′ ]FLSQ (A) = 0 i′ λ where gii′ are the matrix elements belonging to the representation λ, which are ? ? known from the textbooks of group theory (e.g., refer to [10]). From O2 and O4 , we have λi′ λ (11) [gii′ (p13 p24 p56 )Π ? δii′ ]FLSQ (A) = 0 i′
π ′


λ [(?1)L gii′ (p14 p23 p56 )δ ? Q′ i′ ? QQ λi ? δii′ δQQ′ ]FLSQ′ (A) = 0 ′


where Q= ?Q. It is noted that
j oo k R?120 = Rθ R?120 Rj ?
′ ′ ′ ′



where θ = arccos(

1 ). 3

? Thus from O3 we have
L λi L DQQ” (0, θ, 0)e?i 3 Q” DQ′ Q” (0, θ, 0)?δii′ δQQ′ ]FLSQ′ (A) = 0
2π ′

λ [gii′ [p(235)p(164)] Q′ i′ Q′′


λi Eq.(10), (11), (12), and (14) are the equations that the FLSQ (A) have to ful?lled. In some cases there is one or more than one nonzero solution(s) (i.e., not all the λi FLSQ (A) are zero) to all these equations . But in some other cases, there are no nonzero solutions. In the latter case, the ΨLS has to be zero at the OCTA con?gurations disregarding their size and orientation. Accordingly, an INS emerges and the OCTA is not accessible. Evidently, the above equations depend on and only on L, Π, and λ. Therefore the existence of the INS does not at all depend on dynamics (e.g., not on the interaction, mass, etc.). Since the search of nonzero solutions of linear equations is trivial, we shall neglect the details but give directly the results of the L=0 components in the second and fourth columns of Tab.2

0+ 0+ 0? 0? λ OCTA C-PENTA OCTA C-PENTA {6} 1 1 0 0 {5,1} 0 1 0 0 {4,2} 1 1 0 0 {3,3} 0 1 0 0 {2,2,2} 1 1 1 0 {2,2,1,1} 0 1 0 0 4 {2,1 } 0 1 0 0 {16 } 0 1 0 0 {3,2,1} 0 2 0 0 {4,1,1} 0 0 0 0 3 {3,1 } 0 0 1 0 Tab.2, The accessibility of the OCTA (regular octahedron) and the C-PENTA (regular centered-pentagon) to the LΠ = 0+ and 0? wavefunctions with di?erent spatial permutation symmetry λ. Where the ?gures in the blocks are the numbers of independent nonzero solutions. The ?gure 0 implies that nonzero solutions do not exist. The INS existing at the OCTA may even extend beyond the OCTA. For example, when the shape in Fig.1a is prolonged along k’, then the shape is called a prolonged-octahedron. This shape (denoted by B ) is invariant to O1 , O2 , and O4 , λi′ but not to O3. Hence, the FLSQ′ (B) should ful?ll only (10) to (12), but not (14). When nonzero common solutions of (10), (11), (12), and (14) do not exist, while nonzero solutions of only (10) to (12) also do not exist, the INS extends from the OCTA to the prolonged-octahedrons. An OCTA has many ways to deform;e.g., instead of a square, the particles 1,2,3, and 4 form a rectangle or form a diamond, etc.. Hence, the INS at the OCTA has many possibilities to extend. How it extend is determined by the (LΠλ) of the wavefunction. Thus, in the coordinate space, the OCTA is a source where the INS may emerge and extend to the neighborhood surrounding the OCTA. This fact implies that speci?c inherent nodal structure exists. The details of the inherent nodal structure will not be concerned in this paper. However, it is emphasized that for a wavefunction, if the OCTA is accessible, all the shapes in the neighborhood of the OCTA are also accessible, therefore this wavefunction is inherent nodeless in this domain. Another shape with also a stronger geometric symmetry is a regular centeredpentagons(C-PENTA, the particle 6 is assumed to be located at the center of mass O). Let k’ be the 5-fold axis. The C-PENTA is invariant to (i) a rotation about k’ by 2π together with a cyclic permutation of the ?ve particles of the pentagon , (ii) a 5 rotation about k’ by π together with a space inversion, (iii) a rotation about i’ by π together with p14 p23 (here i’ is the axis vertical to k’ and connecting O and particle 5). These invariances will lead to constraints embodied by sets of homogeneous equations, and therefore the accessibility of the C-PENTA can be identi?ed as also given in Tab.2.

In addition to the OCTA, the C-PENTA is another source where the INS may emerge and extend to its neighborhood; e.g., extend to the pentagon-pyramid as shown in Fig.1b with h=0. There are also other sources. For example, the one at the regular hexagons. However, among the 15 bonds, 12 can be optimized at an OCTA, 10 at a pentagon-pyramid, but only 6 at a hexagon. Therefore in the neighborhood of the hexagon (and also other regular shapes) the total potential energy is considerably higher. Since the wavefunctions of the low-lying states are mainly distributed in the domain with a relatively lower potential energy, we shall concentrate only in the domains surrounding the OCTA and the C-PENTA. When (LΠλ) =(0+{6}), (0+{4,2}), or (0+{2,2,2}), the wavefunction can access both the OCTA and the C-PENTA (refer to Table 2). These and only these wavefunctions are inherent-nodeless in the two most important domains, and they should be the dominant components for the low-lying states. All the other L=0 components must contain at least an INS resulting in a great increase in energy. From Tab.1 it is clear that the (0+{6}) component is not allowed, while the (0+{4,2}) component can be contained in [S,T]=[1,0] and [0,1] states, and the (0+{2,2,2}) component can be contained in [S,T]=[1,0], [3,0], [0,1], and [2,1] states. When [S,T]=[1,0] , the λ can be {4,2} or {2,2,2}, therefore two JΠ = 1+ partner-states with their spatial wavefunctions orthogonal to each other exist, each of them is a speci?c mixture of {4,2} and {2,2,2}. Similarly, two partner-states with [S,T]=[0,1] and JΠ = 0+ exist also. When [S,T]=[3,0] or [2,1], the λ has only one choice, therefore in each case only one state exists. Thus we can predict that there are totally six low-lying states dominated by L=0 components without nodal surfaces as listed in Tab.3, where the L,S, and λ are only the quantum numbers of the dominant component. T J Π L λ E 0 1 + 0 {4,2} and {2,2,2} 0 0 1 + 0 {4.2} and {2,2,2} 5.65 0 3 + 0 {2,2,2} 2.19 0 2 + 4.31 0 1 0 + 0 {4,2} and {2,2,2} 3.56 0 1 0 + 0 {4,2} and {2,2,2} 2 1 2 + 0 {2,2,2} 5.37 Tab.3, Prediction of the quantum numbers of low-lying states (dominated by L=0 components) of the 6-nucleon systems based on symmetry. The last column is the energies (in MeV) of the states of 6 Li taken from [11]. It is expected that these low-lying states should be split by the nuclear force. Owing to the interference of the {4,2} and {2,2,2} components, there would be an larger energy gap lying between the two partner-states of each pair. Ajzenbergselove has made an analysis on 6 Li based on experimental data [11], the results are listed in Tab.3. Although our analysis is based simply on symmetry, but the results of the two analyses are close. For the T=0 states, there are two JΠ = 1+ S 1 1 3

states ([S,T]=[1,0]) in [11] with a split, they are just the expected partners. The split is so large (5.65 MeV) that the lower one becomes the ground state while the higher one becomes the highest state of this group. There is a T=0 state in [11] at 2.19 MeV with exactly the predicted quantum numbers JΠ = 3+ . Nonetheless, there is a T=0 state in [11] at 4.31 MeV with JΠ = 2+ , which do not appear in our analysis. May be this state is dominated by L=1 component, may be there is another origin to be clari?ed. For the T=1 states, one of the expected partners with JΠ = 0+ ([S,T]=[0,1]) was found in [11] at 3.56 MeV . However, the other partner ( it would be considerably higher) has not yet been identi?ed in [11], this is an open problem. Nonetheless, if this state exists, the structure of its spatial wavefunction would be similar to the T=0 state at 5.65 MeV . The third expected T=1 state was found in [11] at 5.37MeV with exactly the predicted JΠ = 2+ . In summary we have explained the origin of the quantum numbers of the lowlying states of 6-nucleon systems. The explanation is very di?erent from that based on the shell model [12,13]. For example, according to our analysis, the JΠ = 3+ state at 2.19 MeV has S=3 and L=0. On the contrary, in the shell model the four nucleons in the 1s orbit must have their total spin zero and total isospin zero; therefore this state should have S ≤ 1 and L ≥ 2. However, it is noted that the 2+ state (having S=0 and L=2) of the 12 C lies at 4.44 MeV [14]. Since the 6 Li is 1 considerably lighter and smaller than the 12 C, the L=2 state of 6 Li should be much higher than 4.44 MeV due to having a much smaller moment of inertia. Therefore the 3+ state at 2.19 MeV is di?cult to be explained as a L ≥ 2 state. In particular, it is found that the {2,2,2} component is important; however this component is suppressed by the shell model. Thus, our analysis raises a challenge to the shell model in the case that the number of nucleons is not large enough. Evidently, much work should be done to clarify the physics underlying these systems. It has been shown that sources of INS may exist in the quantum states. Nonetheless, there are essentially inherent-nodeless components of wavefunctions (each with a speci?c set of (LΠλ)). They are the most important building blocks to constitute the low-lying states. The identi?cation of these particularly favorable components is a key to understand the low-lying spectrum. The idea of this paper can be generalized to investigate di?erent kinds of systems, thereby we can understand them in an uni?ed way. ACKNOWLEDGEMENT: This work is supported by the NNSF of the PRC, and by a fund from the National Educational Committee of the PRC. REFERENCES 1, B.S.Pudliner, V.R. Pandharipande, J.Carlson, and R.B.Wiringa, Phys. Rev. Lett. 74, 4396, (1995) 2, B.S.Pudliner, V.R. Pandharipande, J.Carlson, S.C.Pieper, and R.B.Wiringa, Phys. Rev. C56, 1720 (1997) 3, K.Varga, Y.Suzuki, Phys. Rev. C52, 2885, (1995)

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