Geometrical Symmetries in Nuclei-An Introduction

1. Geometrical Symmetries in Nuclei-An Introduction ASHOK KUMAR JAIN Indian Institute of Technology Department of Physics Roorkee, India I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

2. OUTLINE • Brief Introduction • Mean Field and Spontaneous Symmetry Breaking • Symmetries, Unitary Transformation, & Multiplets • Discrete Symmetries • Nuclear Shapes • Collective Hamiltonian, Wave-function, etc. • Spheroidal Shapes – constraints on K • Symmetric Top – Even-Even Nuclei I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

3. Basic Idea - Mean Field Basic Nucleon-Nucleon interaction remains invariant under all the basic operations like Translational invariance in space and time, Inversion in space and time, rotational invariance in space etc. When many nucleons collect to make a nucleus, it generates a mean field, which may break one or, more of these symmetries. If the mean field were also to conserve all the basic operations, we will see very little structure in the nucleus. I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

4. 2.0 6+ 7- 6- 5- 4- 3- 6- 5- 4- 3- 2- 6+ 5+ 4+ 3+ 8+ 6- MeV 4+ 8+ 6+ 8- 5- 4- 2+ 1.5 7- 3- 2- 1- 7+ 4+ 0+ 10+ 6- 5- 6+ 2+ 0+ 5+ 4- 1.0 4+ 8+ 3+ 2+ 6+ 4+ 2+ 0+ 0.5 168Er 0 I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

5. Spontaneous Breaking of the Symmetry This leads to the concept of spontaneous breaking of the symmetries by the mean field of the nucleus. Basic N-N interaction preserves all the symmetries yet the mean field may break them. An example: Transition from Shell Model to Nilsson Model As the mean field changes, so does the spectrum. Further, the mean field changes at every pretext. Change the angular mom, the excitation, the N/Z ratio, or the particle number, the mean field changes. I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

6. How to obtain the Mean Field? Theorists are always concerned about this big question. Fortunately, the gross properties, or the major phenomena come out OK for a variety of mean fields, which differ from each other only slightly. As we shall see, the main features can be understood by the application of symmetry arguments ro the mean field which in turn depends on the nuclear shapes. As most nuclei are now known to be deformed rather than being spherical, these arguments become widelyapplicable. I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

7. Unitary Transformation, Degeneracy and Multiplets Symmetry operation  Group of Unitary Transformation Û An observable Since Invariance under U implies that and I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

8. An useful Unitary transformation arising from H is exp(-i t H ), which gives for all t It implies a commutation of Q with H, and Q becomes invariant as H is a generator of time, . Similarly, rotation about the z-axis can be generated by For Jz to be invariant, I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

9. Concept of Multiplets and Degeneracy and we also have If • either is an eigenstate of both H and Jz or, • the eigenvalue E has a degeneracy. Thus, and, both are eigenstates of H with the same energy eigenvalue E. An energy eigenstate can have n-fold degeneracy if n-fold rotation of about the z-axis leaves invariant. This degeneracy will be lifted if an interaction or, deformation violates this symmetry and multiplet arises. . I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

10. An Example J is a good quantum number for spherical symmetry. A small deformation breaks the symmetry. If the deformed potential has axial symmetry about the z-axis, Jz is the only conserved quantity. The (2j+1) fold degeneracy is lifted and multiplet arises. Therefore, the quantum number  is used to label the states. 9/2 7/2 g 9/2 5/2 3/2 Ω=1/2 β 0.0 deformation I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

11. Discrete Symmetries in Nuclei • Most commonly encountered discrete symmetries in rotating nuclei are • parity P • rotation by π about the body fixed x, y, z axes, Rx (π), Ry (π,), Rz (π) • time reversal T, and • TRx (π), TRy (π), TRz (π). • These are all two fold discrete symmetries, and their breaking causes a doubling of states. • See Dobaczewski et al(2000) for complete classification I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

12. Simple rules to work out the consequences of these symmetries: • When P is broken, we observe a parity doubling of states. A sequence like I+, I+1+, I+2+, … turns into I, I+1, I+2, … . • When Rx (π) is broken, states of both the signatures occur. The two sequences like I, I+2,…etc. and I+1, I+3, …etc. having different signatures, merge into one sequence like I, I+1, I+2, I+3 … etc. • When Ry (π) T is broken, a doubling of states of the allowed angular momentum occurs. A sequence like I, I+2, I+4, … etc. becomes 2(I), 2(I+2), 2(I+4), …, each state now occurring twice (chiral doubling). • When P=Rx (π), the two signature partners will have different parity. Thus states of alternate parity occur. We obtain a sequence like I+, I+1-, I+2+, … etc. I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

13. RULE NO. 1 I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

14. RULE NO.2 I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

15. RULE NO. 3 I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

16. RULE NO. 3 I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

17. RULE NO. 4 I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

18. Part of the TABLE from Frauendorf (2000) I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

19. NUCLEAR SHAPES Radius vector of the surface of an arbitrary deformed body Different spherical harmonics have definite but different geometric symmetries and may occur in the mean field. Most common is the λ= 2 quadrupole term. Higher order terms occur in specific regions of nuclei. Permanent non-spherical shape ↔ Rotational motion I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

20. Define surface in body-fixed frame rather than space-fixed • Here time-independent parameters are used. Empirical evidence exists for the quadrupole, quad + hexadeca, quad + octupole. While axial shapes are most common,evidence exists for non-axial shapes also. • Following possibilities give rise to these many shapes • High Spin Configurations • Non-yrast configurations • Unusual N/Z ratios I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

21. Tetrahedral and Triangular shapes:Besides the usual octupole shape like Y30, predictions of Y31 and Y32 shapes also exist. Tilted Axis Rotation:A New Dimension has been provided to the whole scene by the possibility of having rotation about an axis other than one of the principal axes. This has given rise to the new types of rotational bands like the Magnetic Rotation bands and the Chiral bands. Additional types of symmetry breakings and ensuing rotational structures are expected. I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

22. The Collective Hamiltonian (Bohr and Mottleson,1975; MK Pal,1982) where I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

23. Transform to body-fixed principal axes frame Now the equation has a rotor term also. The three moments of inertia about the three principal axes appear explicitly. Quadrupole (λ=2) motion where (β,γ) parameters have been used. I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

24. Here, I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

25. Quantization of the classical H Separable in β and γ coordinates by wriiting I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

26. The β-equation is The rotor plus γ-motion equation is I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

27. For rigidity against γ-motion, only rotational motion is left, It is known that I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

28. SOME COMMON NUCLEAR SHAPES with Axial Symmetry ND Prolate Prolate + Hexadeca Prolate - Hexadeca NOT SO COMMON SHAPES Non-axial Quadru Y22 ND Oblate I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

29. Spheroidal shape – Axial symmetry about say the z-axis. I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

30. For a general ellipsoid and the coefficients of are not equal. We have Terms likeR+ R+ and R- R-, and (R+ R- + R- R+) arise. The last operator leaves unchanged. However, the first two operators change to and respectively. Thus, we have a mixing of K with K+2 and K-2 in the eigenfunction. I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

31. The eigenfunctions look like where K differ by 2. Constraints on K-values: When we go from space-fixed to body-fixed frame, an arbitrariness arises in assigning the set of parameters (β,γ) and the three Euler angles for a given set of Parameters. This is met by requiring that the wavefunction remain invariant under a set of rotation operators R1, R2, and R3: I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

32. R1 (0,π,0) x' y' y z' z' x z y' R2 (0,0,π/2) x' y x x' z R3 (π/2,π/2,π) z' y x y' I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

33. where, It puts the following conditions on the wavefunction: (i) (ii) It restricts the K to even integers only, and we obtain I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

34. The wavefunction remains invariant only if written as With K as even integers only. Axial Symmetry – Symmetric Top: Gamma motion is frozen, With K as even integer only. I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

35. Z M I R J z K I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

36. For K=0, only even-I values are allowed else the wavefunction vanishes, It can be further proved that only K=0 is the allowed K-value for the case of axial symmetry. I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

37. Even-Even Nuclei: K=0 GSB, β-bands, γ-bands GSB: K=0, Nβ=0, Nγ=0, nγ=0,I=0, 2, 4, ……etc. Gamma band: Nβ=0, Nγ=1 (one γ-phonon), K=2, I=2,3,4,…etc. Beta band: Nβ=1, Nγ=0, K=0, I=0,2,4,……etc. I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

38. I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

39. I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003