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The Algebraic Approach

Introduction The building blocks Dynamical symmetries Single nucleon description. Critical point symmetries Symmetry in n-p systems Symmetry near the drip lines. The Algebraic Approach. Lecture 1. Lecture 2. I. R. Shell Model. Geometrical Model. w. j. Single particle motion

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The Algebraic Approach

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  1. Introduction The building blocks Dynamical symmetries Single nucleon description Critical point symmetries Symmetry in n-p systems Symmetry near the drip lines The Algebraic Approach Lecture 1 Lecture 2

  2. I R Shell Model Geometrical Model w j Single particle motion Describes properties in which a limited number of nucleons near the Fermi surface are involved. Collective motion (in phase) Vibrations, rotations, deformations Describes bulk properties depending in a smooth way on nucleon number Dynamical symmetry Interacting Boson Approximation Truncation of configuration space Algebraic NUCLEAR MEAN FIELD Three ways to simplify

  3. Basic, attractive SD Interaction (2J+1)+ 6+ 4+ 2+ 0+ 0+ and 2+ lowest; separated from the rest.

  4. Pauli Principle Consider f7/2 “shell” with 6 neutrons M  7/2  5/2  3/2  1/2 Maximum seniority = 2 Maximum (d)-boson number =1 Bosons counted from nearest closed shell (i.e. particles or holes). [Eg 130Ba Z = 56 N = 74 N= 3 N= 4 ; N=7] WHY???

  5. li mi ni DYNAMICAL SYMMETRY • Describes basic states of motion available to a system - including relative motion of different constituents • Dynamical symmetry breaking splits but does not mix the eigenstates G1 G2  G3  ………. H= aC1 [G1] † bC2 [G2] † cC3 [G3] † ……

  6. Some ‘Working Definitions’ • Have states s and d with  = -2,-1,0,1,2 - 6 -dim. vector space. • Unitary transformations involving the operators s, s†, d, d† => ‘rotations that form the group U(6). • Can form 36 bilinear combinations which close on commutation, s†s, s†d, d†s, (d†d)(L) (eg: [d†s,s†s] = d†s) - these are the generators [Analogy: Angular momentum: Jx,Jy,Jz generate rotations and form group 0(3)] [For 0(3), use Jz,J ; J = Jx ± i Jy Then [J+,J-] = 2Jz ; [Jz,J] = ±J]

  7. All C’s commute and H is diagonal DYNAMICAL SYMMETRY • A Casimir operator commutes with all the generators of a group. • Eg: C1U(6) = N ; C2U(6) = N(N+5) • Now look for subsets of generators which form a subgroup. Eg: (d†d)(L) - 25 -U(5) (d†d)(1), (d†d)(3) -10 -0(5) (d†d)(1)-3 -0(3) • ie: U(6) U(5)  0(5)  0(3) - group chain decomposition • Now form a Hamiltonian from the Casimir operators of the groups. H = C1U(6) + C2U(6) C2U(5) + C2O(5) +C2O(3)

  8. M= +Jz M= -Jz J2 J1 O(2)  O(3) Example of angular momentum • For O(3), generators are Jz, J+ and J- • Then [J2, Jz] = [J2, J+] = [J2, J-] = 0 • C2O(3) = J2 • Subgroup O(2) simply Jz = C1O(2) • So H =  C2O(3) +  C1O(2) • E=  J(J+1) +  M

  9. Only 3 chains from U(6) I. “U(5)” - Anharmonic Vibrator II. “SU(3)” - Axially symmetric rotor III. “O(6)” - Gamma - unstable rotor

  10. U(5) R4/2= 2.0

  11. SU(3) R4/2= 3.33

  12. O(6) R4/2= 2.5

  13. The first O(6) nucleus ……….. Cizewski et al, Phys Rev Lett. 40, 167 (1978)

  14. and then many more….

  15. Transition Regions and Realistic Calculations • Most nuclei do not satisfy the strict criteria of any of the 3 Dyn. Symm. • Need numerical calculations by diagonalizing HIBA in s – d boson basis • Can use a very simple form of the most general H Consistent Q Formalism

  16. =0.03 MeV Z=38-82 2.05 < R 4/2 < 3.15 N.V. Zamfir, R.F. Casten, Physics Letters B 341 (1994) 1-5

  17. Summary • Algebraic approach contains aspects of both geometrical and single particle descriptions. • Dynamical symmetries describe states of motion of system • Analytic Hamiltonian is a sum of Casimir operators of the subgroups in the chain. • Casimir operators commute with generators of the group; conserve a quantum number • Each Casimir lifts the degeneracy of the states without mixing them. • Three and only three chains possible; O(6) was the surprise. • Very simple CQF Hamiltonian describes large ranges of low-lying structure

  18. ? Previously, no analytic solution to describe nuclei at the “transitional point” Evolution of nuclear shape Vibrational Transitional Rotational E = nħω E = J(J+1)

  19. V(β) Approximate potential at phase transition with infinite square well β Solve Bohr Hamiltonian with square well potential Result is analytic solution in terms of zeros of special Bessel functions Predictions for energies and electromagnetic transition probabilities γ-soft E(5) Symmetric Rotor Spherical Vibrator X(5) Critical Point Symmetries Two solutions depending on γ degree of freedom F. Iachello, Phys. Rev. Lett. 85, 3580 (2000); 87, 052502 (2001).

  20. τ = 1 Key Signatures τ = 0 E(41)/E(21) = 2.91 ξ = 2 E(02)/E(21) = 5.67 R4/2 = 2.20 E(02)/E(21) = 3.03 E(03)/E(21) = 3.59 ξ = 1 X(5) and E(5)

  21. P= NpNn Np+Nn Searching for X(5)-like Nuclei β-decay studies at Yale 152Sm R.F. Casten and N.V. Zamfir, Phys. Rev. Lett. 87, 052503 (2001). N.V. Zamfir et al., Phys. Rev. C 60, 054312 (1999).. 156Dy M.A. Caprio et al., Phys. Rev. C 66, 054310 (2002). 162Yb E.A.McCutchan et al., Phys. Rev. C 69, 024308 (2004). 166Hf Good starting point: R4/2 or P factor E.A.McCutchan. et al., Phys. Rev. C- submitted. Other Yale studies: 150Nd - R.Krücken et al., Phys. Rev. Lett. 88, 232501 (2002).

  22. Searching for E(5)-like Nuclei Ce 58 3.06 2.93 2.80 2.69 2.56 2.38 2.32 Ba 134Ba 56 2.96 2.89 2.83 2.78 2.69 2.52 2.43 2.32 2.28 Xe 54 2.33 2.40 2.47 2.50 2.48 2.42 2.33 2.24 2.16 2.04 R.F. Casten and N.V. Zamfir, Phys. Rev. Lett. 85, 3584 (2000). Te 52 2.09 2.00 1.99 2.07 2.09 2.07 2.04 2.01 1.94 1.72 Sn 50 1.54 1.67 1.75 1.81 1.79 1.68 1.84 1.85 1.87 1.88 1.86 1.80 1.71 1.63 102Pd Cd 48 2.33 1.79 2.11 2.27 2.36 2.38 2.33 2.29 2.30 2.38 2.39 2.38 46 Pd 1.79 2.12 2.29 2.38 2.40 2.42 2.46 2.53 2.56 2.58 N.V. Zamfir et al., Phys. Rev. C 65, 044325 (2002). Ru 44 1.82 2.14 2.27 2.32 2.48 2.65 2.75 2.76 2.73 42 Mo 1.81 2.09 1.92 2.12 2.51 2.92 3.05 2.92 130Xe P~2.5 Zr 40 1.60 1.60 1.63 1.51 2.65 3.15 3.23 Sr 38 1.99 2.05 3.01 3.23 Z/N 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80 Good starting point: R4/2 or P factor

  23. Symmetries and phases transitions in the IBM • Challenges for neutron-rich: • New collective modes in three fluid systems (n-skin). • New regions of phase transition • New examples of critical point nuclei? • Rigid triaxiality? D.D. Warner, Nature 420 (2002) 614

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