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Unusual Symmetries in Nuclei

Unusual Symmetries in Nuclei. ASHOK KUMAR JAIN Indian Institute of Technology Department of Physics Roorkee, India. Outline. Role of Intrinsic Wavefunction – Signature Triaxial Ellipsoid Odd Multipole Shapes – Simplex Only two planes of symmetry Only one plane of symmetry

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Unusual Symmetries in Nuclei

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  1. Unusual Symmetries in Nuclei 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 • Role of Intrinsic Wavefunction – Signature • Triaxial Ellipsoid • Odd Multipole Shapes – Simplex • Only two planes of symmetry • Only one plane of symmetry • Tetrahedral and Triangular shapes • Tilted Axis Rotation – Planar and Aplanar • Magnetic Top and Chiral Rotation I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

  3. Intrinsic Wavefunction and its consequences The total wavefunction where is the intrinsic part. Axial symmetry is assumed so that The total wavefunction must remain invariant under and acting on internal and collective coordinates with the condition that Since j is not a good q.n., I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

  4. For K=0, the following holds so that and One may also write these expressions as which leads to values =0 and =1 corresponding to r = +1 and r = -1. Both  and r are termed as the signature quantum number. I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

  5. Signature and angular momentum get connected by the relation Therefore, and the K=0 band can be classified as The GSB of even-even nuclei is the best example of r=+1 signature. A K=0 band in an odd-odd nucleus has both r=+1 and r=-1 signature. I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

  6. For K ≠ 0, the intrinsic states are two fold degenerate because of symmetry. Note that and time-reversal T the same effect on the wave-function. The time-reversed state has the negative value of , so that and The rotational wave function changes as so that a rotationally invariant wave function may be constructed as I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

  7. For odd-A nuclei, 2j odd. Now, and, together with, lead us to the classification In general, I = (α + even number). I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

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

  9. PARITY:Parity operator acts upon the intrinsic coordinates only. Since parity commutes with jz, all the states in a given band having a quantum number K, have the same parity. It is possible to have K=0 bands with π and α quantum numbers independent of each other as the two are distinct from each other. Thus, K=0 bands may have GSB and, I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

  10. Ellipsoid with D2 –symmetry – Asymmetric Top • Full D2-symmetry :invariance with respect to the three rotations by 180 about each of the three principal axes. • Finite gamma deformation – K not a good quantum number • K=0 not allowed. Only even integers K=2,4,…etc. allowed. • A typical rotational band may have I=2,4,6,….etc. as signature is still a good quantum number. I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

  11. Figure 10 I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

  12. Odd-Multipole Shapes: Simplex quantum number • Axial Symmetry – Y30 shapes only. • violates the and symmetry, but preserves . • Two minima in octupole def energy- two degenerate states arise • Denote which is conserved • K=0 band satisfy: I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

  13. V V ε3< 0 O ε3 >0 ε3< 0 O ε3 >0 3- 2+ 1- 0+ 11/2± 9/2± 7/2± 5/2± 9/2- 7/2- 5/2- 4+ 2+ 0+ 9/2+ 7/2+ 5/2+ 3- 1- E-E Odd-A E-E Odd-A I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

  14. For K≠ 0, the intrinsic states have a 2-fold Kramer’s degeneracy and We obtain, with the levels having I ≥ K allowed. I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

  15. 184 d3/2 s1/2 g7/2 d5/2 j15/2 i11/2 g9/2 Y3 126 p1/2 f5/2 p3/2 i13/2 h9/2 f7/2 Y3 82 d3/2 h11/2 s1/2 g7/2 d5/2 Y3 50 g9/2 p1/2 f5/2 p3/2 Y3 28 f7/2 20 Regions of Octupole Deformation For example, Δl=3 orbitals for both neutrons and protons come close together just beyond Pb-208 I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

  16. ε3=0 ε3≠0 ε3≠0 +tunneling even-even 4+ 3- 3- 4+ 1- 4+ 2+ K=0 2+ 2+ 1- 0+ 0+ 0+ O+ odd-A 11/2+ 11/2- 11/2± 9/2+ 9/2- K=5/2 7/2- 9/2± 7/2+ 5/2- 11/2+ 7/2± 5/2+ 9/2+ 5/2± 7/2+ 5/2+ odd-A 7/2+ S = -i 7/2+ 5/2- 7/2- 5/2- 3/2- r = i 1/2- 3/2+ ΔE 3/2+ 1/2- K=1/2 (a>0) 7/2+ 5/2+ 7/2- r = -i 5/2+ 5/2+ 1/2+ 3/2- 3/2+ 1/2+ S = i 1/2+ I (I+1) I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

  17. FIGURE 14 I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

  18. Density Distribution has two planes of symmetry • Axial symmetry is lost for μ≠ 0 • Two independent planes of symmetry for μ even. • Rotation about the long axis is possible. • For Parity doublets of odd- or, even-angular momenta arise I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

  19. FIGURE 15 I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

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

  21. Density distribution has one plane of symmetry • Odd-μ components allowed • Rotation is possible along the long axis as well as any one of the short axes • Signature is not a good quantum number. Both even and odd-spins will occur • Parity is not conserved; both parities will occur. • Rotation about long axis • Rotation about short axis • Chiral partners obtained I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

  22. FIGURE 16 I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

  23. Tetrahedral and Triangle Symmetries • Tetrahedral symmetry is related to Y-32 term • Large shell gaps at N, Z=16, 20, 32, 40, 56,58,70,90-94 • and at N=136/142 • Tetrahedral Equilibrium shapes of the order of 0.13 for • Obey simplex symmetry, parity doublets formed I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

  24. FIGURE 17 I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

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

  26. Dipole Mom. ≠0 Dipole Mom. =0 Dipole Mom.=0 Quadrupole Mom.=0 4+ 4- 4+ 4- 4+ 4- 3- 3+ 3- 3+ 3- 3+ 2+ 2- 2+ 2- 2+ 2- 1- 1+ 1- 1+ 1- 1+ 0+ 0- 0+ 0- 0+ 0- Pear-shape-like Octupole Tetraheda Pure Tetrahedra I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

  27. Triangular Shape corresponds to Y-33 def. • Has only one plane of symmetry • Possible in proton rich N=Z nuclei I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

  28. Tilted Axis Rotation • Riemann - classical rotation about an axis different than principal axes possible in ellipsoids • Planar Tilt – Axis of rotation in a principal plane • P and Ry()T are conserve . • Signature not conserved • A band like is observed • Observed in Magnetic Rotation Bands I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

  29. Figure 10 I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

  30. Aplanar Tilt: • Only parity is conserved • Four distinct situations, related by Rx() and Ry()T exist • Band obtained: • Observed in Chiral Bands I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

  31. Figure 10 I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

  32. Planar Tilt in Octupole shapes • Ry()T=P • Four distinct situations obtained by Rx() and P. • Two degenerate bands emerge having • Shown in bottom panel of Fig 15 I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

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

  34. Magnetic Top • Mean Field remains nearly isotropic • Anisotropy of currents – net magnetic dipole moment • Coupling of high-j neutron and proton, one hole and the other particle • Resultant angular momentum makes an angle with the principal axes • Magnetic effects can dominate when deformation is small • Some deformation necessary to close the blades of neutron and proton – Shears mechanism I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

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

  36. Lakshmi et al,2003 I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

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

  38. Magnetic Rotation Z=17-20, N=20-23: Cl,Ar,K,Ca Z=21-23, N=17-20 & 25-28: Sc,Ti,V Z=25-28, N=21-24: Mn,Fe,Co,Ni Z=28-40,N=42-50: Ni,Cu,Zn,Ga,Ge,As, Se,Br,Kr,Rb,Sr,Y,Zr Z=44-51,N=50-64: Ru,Rh,Pd,Ag,Cd,In,Sn,Sb N=82-88 (n-rich) Z=50-64,N=74-82: Sn,Sb,Te,I,Xe,Cs, Ba,La,Ce,Pr,Nd,Sm,Gd Z=76-82,N=126-140: Os,Ir, Pt,Au,Hg,Tl,Pb (n-rich) Z=80-86,N=110-126: Hg,Pb,Bi,Po,At,Rn I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

  39. Chiral Bands • Aplanar Tilt in Tri-axial shape • First visualised in Odd-Odd nuclei • Odd-proton aligned along the short axis • Odd-neutron aligned along the long axis • Rotational contribution along the intermediate axis • Resultant of the three ang mom is out of the three planes • Parity is conserved but Ry()T is broken • Two pairs of identical ΔI=1 bands having the same parity • Tunneling between the right- and the left-handed system gives rise to a splitting between the levels. I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

  40. 2(h11/2), 1ν(h11/2) Zhu et al, 2003 I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

  41. Observation of Chiral pair of bands in an odd-A nucleus confirms that the phenomenon is purely geometrical in nature. • One can visualise similar possibilities in even-even nuclei also. I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

  42. Dedicated to my son NaMaN The beautiful soul I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

  43. Bibiliography • General References: • A. Bohr and B.R. Mottelson, Nuclear Structure, Vol.1, Single Particle Motion, and Vol.2, Nuclear Deformations (Benjamin, 1969,1975). • M.K. Pal, Theory of Nuclear Structure, (East-West Press,1982). • P. Ring and P. Schuck, The Nuclear Many Body Problem, (Springer, 1980). • K. Heyde, Basic Ideas and Concepts in Nuclear Physics, 2nd Edition (IOP Publishing, 1999). I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

  44. L.D. Landau and E.M. Lifshitz, Quantum Mechanics, Vol.3 of Course of Theoretical Physics (Pergamon Press, 1956). • Table of Isotopes, Eighth Edition, Ed. Firestone et al. (Wiley, 1996). • Articles on Symmetries: • W. Greiner and J.A. Maruhn, Nuclear Models (Springer, 1989). • J. Dobaczewski, J. Dudek, S.G. Rohozinski, and T.R. Werner, Phys. Rev. C62, 014310, and 014311 (2000). • P. Van Isacker, Rep. Prog. Phys. 62, 1661 (1999). • S. Frauendorf, Rev. Mod. Phys. 73, 463 (2001). I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

  45. Refelction Asymmetric Shapes: • P.A. Butler and W. Nazarewicz, Rev. Mod. Phys. 68, 349 (1996). • A.K. Jain, R.K. Sheline, P.C. Sood, and K. Jain, Rev. Mod. Phys. 62, 393 (1990). • R.K. Sheline, A.K. Jain, K. Jain, I. Ragnarsson, Phys. Lett. B219, 47 (1989). • J. Gasparo, G. Ardisson, V. Barci and R.K. Sheilne, Phys. Rev. C62, 064305 (2000). • Tetrahedral and Triangular Shapes: • X. Li and J. Dudek, Phys. Rev. C49, R1250 (1994). I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

  46. S. Takami, K. Yabana, and M. Matsuo, Phys. Lett. B431, 242 (1998). • M. Yamagami, K. Matsuyanagi, and M. Matsuo, Nucl. Phys. A672, 123 (2000). • Magnetic Rotation: • S. Frauendorf, Nucl. Phys. A557, 259c (1993). • R.M. Clark and A.O. Macchiavelli, Annu. Rev. Nucl. Part. Sci. 50, 1 (2000). • Amita, A.K. Jain, and B. Singh, At. Data & Nucl. Data Tables 74, 283 (2000); revised version under publication (2003). • A.K. Jain and Amita, Pramana-J. Phys. 57, 611 (2001). • S. Lakshmi, H.C. Jain, P.K. Joshi, A.K. Jain and S.S. Malik, under publication. I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

  47. Chiral Bands: • V.I. Dimitrov, S. Frauendorf, and F. Donau, Phys. Rev. Lett. 84, 5732 (2000). • D.J. Hartley et al., Phys. Rev. C64, 031304(R) (2001). • S. Zhu et al., Phys. Rev. Lett. 91, 132501 (2003). • Critical Point Symmetries: • R.F. Casten and N.V. Zamfir, Phys. Rev. Lett. 87, 052503 (2001). • F. Iachello, Phys. Rev. Lett. 91, 132502 (2003). I.A.E.A. Workshop - I.C.T.P. Trieste, Nov.2003

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