The role of symmetry in nuclear physics

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1 Journal of Physics: Conference Series OPEN CCESS The role of symmetry in nuclear physics To cite this article: Francesco Iachello 015 J. Phys.: Conf. Ser Related content - Quantum Chemistry: Molecular symmetry J Thakkar - Quantum Chemistry: Molecular symmetry J Thakkar - Symmetries and singularities in Hamiltonian systems Eva Miranda View the article online for updates and enhancements. This content was downloaded from IP address on 0/11/018 at 0:55

2 The role of symmetry in nuclear physics Francesco Iachello Center for Theoretical Physics, Sloane Laboratory, Yale University, New Haven, Connecticut , US bstract. The role of discrete symmetries in nuclear physics is briefly reviewed within the context of the algebraic cluster model (CM). The symmetries D 3 (triangle) for 3α and T d (tetrahedron) for α are discussed and evidence shown for their occurrence in 1 C(D 3)and 16 O (T d ). 1. Continuous symmetries Continuous symmetries were introduced in nuclear physics as early as the 1930 s. In 193 Heisenberg introduced isospin, SU T (), in 1937 Wigner combined isospin with spin to SU() SU T () SU S (), in the 190 s Racah developed the group theory of the shell model in ls and jj coupling, U ((l +1)(s + 1)) and U(j + 1), and in 1958 Elliott introduced the symmetry of mixed configurations, in particular the symmetry of the sd and pf shells. The use of continuous symmetries culminated in 197 with the introduction of the interacting boson model, rima- Iachello U(6). The role of these symmetries in nuclear physics is well documented and will not be discussed here.. Discrete symmetries Discrete symmetries were also introduced in the early days of nuclear physics, mostly within the framework of the α-particle model, by Wheeler in 1937 [1] and Dennison in 195 [] and later exploited by Brink [3], Robson [] and others. We have recently readdressed this problem within an algebraic description of clustering, the lgebraic Cluster Model (CM), and in this contribution some recent results will be reported. The algebraic cluster model is a description of cluster states as representations of U(3k ), where k is the number of constituents. The elements of the algebra are the bilinear products of boson creation and annihilation operators which are a bosonic quantization of the Jacobi variables and their associated momenta plus an additional s boson. For k =, there is only one Jacobi vector, ρ = r 1 r, for k=3 there are two vectors, ρ = (r 1 r )/, λ = (r 1 +r r 3 )/ 6, and for k= there are three vectors, ρ =(r 1 r )/, λ =(r 1 +r r 3 )/ 6, η =(r 1 +r + r 3 3r )/ 1, where r i are the coordinates of the constituent particles. For twobody clusters, CM was introduced years ago for applications to molecules [5] and nuclear molecules [6]. For three-body clusters CM was introduced in [7], and discussed in [8], and for four-body clusters was introduced in [9], [10] and discussed, very recently, in [11], [1]. The discrete symmetries of the α-particle model discussed in those articles and here are shown in Table 1. Content from this work may be used under the terms of the Creative Commons ttribution 3.0 licence. ny further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence by Ltd 1

3 k Nucleus U(3k ) Discrete symmetry Jacobi variables 8 Be U() Z ρ 3 1 C U(7) D 3 ρ, λ 16 O U(10) T d ρ, λ, η Table 1. Discrete symmetries of the α-particle model considered in this article. Figure 1. States of 3α with D 3 symmetry. Within the algebraic cluster model (CM) it is possible to derive many analytical results. In particular, for a rigid roto-vibrator one has explicit expressions for the energy levels in terms of the angular momentum L and the vibrational quantum numbers v i (i =1,...). These are: α(z ) E(v, L) =E ω ( v ( + 1 ) + κl(l +1) E(v 3α(D 3 ) 1,v,L)=E ω 1 v1 + 1 ) + ω (v +1) +κl(l +1) (. (1) E(v α(t d ) 1,v,v 3,L)=E ( ω 1 v1 + 1 ) + ω (v +1) +ω 3 v3 + 3 ) + κl(l +1) Spectra are characterized by the representations of the discrete group G and consist in a set of rotation-vibration bands, with specific values of the angular momentum and parity. Representations can be labeled either by G or by the isomorphic group S n (the permutation group). The conversion from G to S n is: G = Z S P, []; G = D 3 S 3, [3], E [1]; G = T d S, [], F [31], E []. Fig.1 shows the expected rotation-vibration spectra of states for 3α with D 3 symmetry. For representations, the rotational band has L P =, +, 3, ±,..., while for E representations the rotational band has L P =1,, 3 ±,... Note the unusual angular momentum and parity content of the rotational bands. This content is not the same as that of the rotational bands of a rotating ellipsoidal shape. [See, for example, the SU(3) limit of the IBM]. In Fig., the expected spectrum of a α configuration with T d symmetry is shown. For

4 Figure. States of a α configuration with T d symmetry. representations, the spectrum consists of a rotational band L P =, 3, +, 6 ±,..., fore representations L P = ±, ±, 5 ±,..., andforf representations L P =1, +, 3 ±,... The breaking of representations of U(3k ) into those of S n and thus the determination of the angular momentum content of each band is one of the novel results of CM. The occurrence of D 3 symmetry in 1 C has been confirmed by a very recent experiment [13], [1] and its evidence is shown in Figs. 3 and. The occurrence of T d symmetry in 16 O was discussed long ago by Robson [], and it has been emphasized recently in [11].The evidence is shown in Figs. 5 and 6. In addition to energy spectra one can also derive within the CM, analytic expressions for other observables, most notably electromagnetic transition rates. For example, for α configurations with T d symmetry, we have ( ) Zeβ B(EL; L P L ( (L +1) )= [+1P L 1 )] () π 3 and the form factors in electron scattering are F L ( L P ) = c L j L (qβ) (3) with c 0 =1;c 3 = 35 9 ; c = 7 3 ; c 6 = The occurrence of T d symetry in 16 O is confirmed by the B(EL) values given in Table. 3. Conclusion In conclusion, the role of discrete symmetry in cluster physics is that of providing benchmarks for energy levels and other obeservables that can be used to analyze data. In particular, by 3

5 0 5 1 C Exp 5 E* (MeV) T=1 1 + T= ± + 3± 1 ± 0 (0,0 0 ) (1,0 0 ) (0,1 1 ) E (,0 0 ) Non Cluster (0,0 0 ) (1,0 0 ) (0,1 1 ) E (,0 0 ) Figure 3. The experimental spectrum of 1 C(left) and its comparison with the theoretical spectrum with D 3 symmetry (right). The lowest non-cluster states are also shown. Reproduced with permission from [13]. E* (MeV) C Bending Band Ground State Band Hoyle Band J(J+1) Figure. Observed cluster rotational bands in 1 C. Reproduced with permission from [13].

6 Figure 5. The experimental spectrum of 16 O showing evidence for T d symmetry. The lowest non-cluster states are also shown. Figure 6. Observed cluster rotational bands in 16 O. 5

7 B(EL; L P ) Th Exp E(L P ) Th Exp B(E3; 3 ) ±10 E(3 ) B(E; + ) ±133 E( + ) B(E6; 6 + ) 85 E(6 + ) Table. B(EL) values and energies in 16 O compared with those expected from a T d symmetry. B(EL) valuesine fm L and E in kev. The theoretical energies in column 5 are calculated from E(keV ) = 511 L(L + 1). The value of β in Eq.() is extracted from the elastic form factor measured in electron scattering, β =.0fm. providing analytic expression for rotation-vibration spectra as well as electromagnetic transition rates and form factors. Strong evidence for the occurrence of D 3 symmetry in 1 CandofT d symmetry in 16 O has been presented. [There is evidence for Z symmetry in 8 Be but this has not been presented here]. The occurrence of D 3 and T d symmetry is also confirmed by microscopic calculations in the (i) molecular orbital method [15] and in (ii) lattice calculations [16]. Cluster states are instead very difficult to describe within the framework of the one-center shell model, where one needs multiparticle-multihole configurations and a model space with several ω quanta of excitation. cknowledgements This work was supported in part by U.S.D.O.E. Grant DF-FG0-91ER0608. It is dedicated to ldo Covello on the occasion of his retirement. References [1] Wheeler J 1937 Phys. Rev [] Dennison DM 195 Phys. Rev [3] Brink DM 1965 Proceedings of the International School of Physics Enrico Fermi, Course XXXVI, 7 [] Robson D 1978 Nucl. Phys ; Robson D 1979 Phys. Rev. Lett. 876; Robson D 198 Phys. Rev. C (R) [5] Iachello F 1981 Chem. Phys. Lett [6] Iachello F 1981 Phys. Rev. C 3 778(R) [7] Bijker R and Iachello F 000 Phys. Rev. C [8] Bijker R and Iachello F 00 nn. Phys. (N.Y.) [9] Bijker R 010 IP Conf. Proc [10] Bijker R 01 J. Phys.: Conf. Ser [11] Bijker R and Iachello F 01 Phys. Rev. Lett [1] Bijker R and Iachello F 01 in preparation [13] Marin-Lambarri DJ et al 01 Phys. Rev. Lett [1] Gai M 01 These Proceedings [15] Feldmeier H and Neff T 008 Proceedings of the International School Enrico Fermi, Course CLXIX, 185 [16] Epelbaum E et al 01 Phys. Rev. Lett