Unusual Symmetries in Nuclei ASHOK KUMAR JAIN Indian

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 1

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 2

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 3

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 4

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 5

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 6

For odd-A nuclei, 2 j 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 7

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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 9

Ellipsoid with D 2 –symmetry – Asymmetric Top • Full D 2 -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 10

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

Odd-Multipole Shapes: Simplex quantum number • Axial Symmetry – Y 30 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 12

V V O ε 3< 0 ε 3 >0 3 - 11/2± 2+ 9/2± 7/2± 5/2± 10+ E-E Odd-A O ε 3< 0 ε 3 >0 9/2 - 4+ 2+ 0+ 9/2+ 31 - E-E I. A. E. A. Workshop - I. C. T. P. Trieste, Nov. 2003 7/2+ 7/25/2 - 5/2+ Odd-A 13

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 14

184 d 3/2 s 1/2 g 7/2 d 5/2 j 15/2 i 11/2 126 Regions of Octupole Deformation Y 3 g 9/2 p 1/2 f 5/2 p 3/2 i 13/2 h 9/2 f 7/2 82 50 d 3/2 h 11/2 s 1/2 g 7/2 d 5/2 Y 3 For example, Δl=3 orbitals for both neutrons and protons come close together just beyond Pb-208 Y 3 g 9/2 p 1/2 f 5/2 p 3/2 Y 3 28 f 7/2 20 I. A. E. A. Workshop - I. C. T. P. Trieste, Nov. 2003 15

ε 3=0 ε 3≠ 0 +tunneling even-even 3 - 4+ K=0 1 - 3 - 4+ 4+ 2+ O+ 0+ 2+ 2+ 10+ 0+ odd-A 11/2+ 11/2± 9/2+ K=5/2 9/27/2 - 9/2± 7/2+ 5/2 - 11/2+ 7/2± 5/2+ 9/2+ 5/2± 7/2+ 5/2+ odd-A S = -i 7/2+ 5/2 - r=i 1/2 - 3/2+ 5/2 - ΔE 1/2 - K=1/2 (a>0) 7/2+ 5/2+ 7/2 - r = -i 1/2+ 7/2 - 3/2 - 5/2+ 3/2 - S =- i. I. C. T. P. 1/2+ Workshop I. A. E. A. Trieste, Nov. 2003 I (I+1) 1/2+ 3/2+ 16

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

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 18

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

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Density distribution has one plane of symmetry • Odd-μ components allowed • Rotation is possible along the long axis as well as the short axes any one of • Signature is not a good quantum number. Both even and oddspins 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 21

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

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 23

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

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Dipole Mom. ≠ 0 4+ Dipole Mom. =0 Quadrupole Mom. =0 44+ 4 - 3 - 3+ - 3 2+ 2 - 1 - 1+ 0+ 0 - Pear-shape-like Octupole Tetraheda I. A. E. A. Workshop - I. C. T. P. Trieste, Nov. 2003 3+ Pure Tetrahedra 26

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 27

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 28

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

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 30

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

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 32

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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 34

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

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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, 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 38

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 39

2 (h 11/2), 1ν(h 11/2) Zhu et al, 2003 I. A. E. A. Workshop - I. C. T. P. Trieste, Nov. 2003 40

• 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 41

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

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, 2 nd Edition (IOP Publishing, 1999). I. A. E. A. Workshop - I. C. T. P. Trieste, Nov. 2003 43

• 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. C 62, 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 44

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, I. Ragnarsson, Phys. Lett. B 219, 47 (1989). • J. Gasparo, G. Ardisson, V. Barci and R. K. Sheilne, Phys. Rev. C 62, 064305 (2000). Tetrahedral and Triangular Shapes: • X. Li and J. Dudek, Phys. Rev. C 49, R 1250 (1994). I. A. E. A. Workshop - I. C. T. P. Trieste, Nov. 2003 45

• S. Takami, K. Yabana, and M. Matsuo, Phys. Lett. B 431, 242 (1998). • M. Yamagami, K. Matsuyanagi, and M. Matsuo, Nucl. Phys. A 672, 123 (2000). Magnetic Rotation: • S. Frauendorf, Nucl. Phys. A 557, 259 c (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, I. A. E. A. Workshop - I. C. T. P. 46 under publication. Trieste, Nov. 2003

Chiral Bands: • V. I. Dimitrov, S. Frauendorf, and F. Donau, Phys. Rev. Lett. 84, 5732 (2000). • D. J. Hartley et al. , Phys. Rev. C 64, 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 47