2. XXIII DGMTP, China 2005 On the Physical Significance of q Deformation in Many-body Physics J. P. Draayer, K. D. Sviratcheva, C. Bahri and A. I. Georgieva

3. XXIII DGMTP, China 2005 On the Physical Significance of q Deformation in Many-body Physics Outline  Sp(4) model ( sp (4)~ so (5)) for a description of pp+nn+pn isovector (isospin T=1) pairing correlations in light and medium mass nuclei following Helmers’s approach:  Sp(4) model ( sp (4)~ so (5)) for a description of pp+nn+pn isovector (isospin T=1) pairing correlations in light and medium mass nuclei following Helmers’s approach: Models vs. conventional seniority scheme of Racah and Flowers “quasi-spin” approach of Helmers (a)(a)  Sp q (4) model for a description of non-linear effects in quantum mechanical nuclear systems: q-deformation takes into account additional interactions between nucleons [such as higher-order many-body correlations] while preserving fundamental laws.  Sp q (4) model for a description of non-linear effects in quantum mechanical nuclear systems: q-deformation takes into account additional interactions between nucleons [such as higher-order many-body correlations] while preserving fundamental laws.

4. XXIII DGMTP, China 2005 On the Physical Significance of q Deformation in Many-body Physics Nuclei with Mass 32<A<100 20 28 50 82 126 20 28 protons (Z) (N) 50 82 Microscopic description of pairing- governed 0 + states in even-A nuclei neutrons Nuclei with Mass 32 <A<100 N=Z nuclei Stellar rapid proton capture path Stellar rapid proton capture path

5. XXIII DGMTP, China 2005 On the Physical Significance of q Deformation in Many-body Physics su2 J=0 pair of identical fermions Identical-Particle Pairing su (2) Dimension 2  =2j+1 p n  =1/2  = –1/2 Orbit j

6. XXIII DGMTP, China 2005 On the Physical Significance of q Deformation in Many-body Physics sp4 sp (4) Identical-Particle plus Proton-Neutron Pairing Add J=0 np pairs to nn and pp pairs Dimension 2  =2j+1  =1/2 p n  = –1/2 Orbit j Number of particles: Isospin:

7. XXIII DGMTP, China 2005 On the Physical Significance of q Deformation in Many-body Physics spq4 … and Many-body Interactions Identical-Particle plus Proton-Neutron Pairing sp q (4) Orbit j Dimension 2  =2j+1 p n  =1/2  = –1/2 J=0 pairs gain higher-order interactions Number of particles: Isospin:

8. XXIII DGMTP, China 2005 On the Physical Significance of q Deformation in Many-body Physics Spq4 algebra Algebra sp q (4) Algebra sp q (4) Number of particles: sp (4) q1q1 Observables remain nondeformed Isospin:

9. XXIII DGMTP, China 2005 On the Physical Significance of q Deformation in Many-body Physics Spq4 model Sp q (4) Model: Important Features  Sp q (4) model does not violate physical laws fundamental to a nuclear system: q-deformation accounts for many-body interactions without affecting the concepts of the standard quantum mechanics [no space-time geometry deformation]!  Conserves: Angular momentum J [and its algebra], hence H q transforms as a scalar under 3-dimensional rotations in real coordinate space; Total number of particles N and third projection of isospin T 0.  Makes possible the analytical modeling of a set of many-body interactions, which are in general important yet rather complicated to handle.  q-Deformation is a transformation to quasi-particles that keeps their number fixed and allows them to interact via many- body forces.  Sp q (4) model does not violate physical laws fundamental to a nuclear system: q-deformation accounts for many-body interactions without affecting the concepts of the standard quantum mechanics [no space-time geometry deformation]!  Conserves: Angular momentum J [and its algebra], hence H q transforms as a scalar under 3-dimensional rotations in real coordinate space; Total number of particles N and third projection of isospin T 0.  Makes possible the analytical modeling of a set of many-body interactions, which are in general important yet rather complicated to handle.  q-Deformation is a transformation to quasi-particles that keeps their number fixed and allows them to interact via many- body forces. same physics, but more of it

10. XXIII DGMTP, China 2005 On the Physical Significance of q Deformation in Many-body Physics Spq(4)  SUq(2)  U(1) Isospin symmetry Proton-neutron pairs p-n Like-particle pairs p-p n-n su q T (2) su q 0 (2) su q ± (2) Sp q (4)  SU q (2)  U (q) (1) T O (±) raising (lowering) operator O (0) third projection operator O (±) raising (lowering) operator O (0) third projection operator For all: u (q) (1)

11. XXIII DGMTP, China 2005 On the Physical Significance of q Deformation in Many-body Physics O (±) raising (lowering) operator O (0) third projection operator O (±) raising (lowering) operator O (0) third projection operator For all: Sp(4)  SU(2)  U(1) Isospin symmetry Proton-neutron pairs p-n Like-particle pairs p-p n-n su T (2) su 0 (2) su ± (2) Sp (4)  SU (2)  U (1) T u (1)

12. XXIII DGMTP, China 2005 On the Physical Significance of q Deformation in Many-body Physics C2(suq(2)) Isospin symmetry Proton-neutron pairs p-n Like-particle pairs p-p n-n su q T (2) su q 0 (2) su q ± (2) Sp q (4)  SU q (2)  U q (1): Casimir Operators C 2 T n +1(–1) : number of proton (neutron) pairs; n 0 : number of pn pairs. n +1(–1) : number of proton (neutron) pairs; n 0 : number of pn pairs.

13. XXIII DGMTP, China 2005 On the Physical Significance of q Deformation in Many-body Physics O2 of spq(4) Diagonal Second-order Operator O 2 ( sp q (4)) N +1(–1) : number of protons (neutrons) O 2 ( sp q (4)) = su q k (2)

14. XXIII DGMTP, China 2005 On the Physical Significance of q Deformation in Many-body Physics O2 of spq(4), gammas Diagonal Second-order Operator O 2 ( sp q (4)) 2  1  0  n +1(–1) : number of proton (neutron) pairs; n 0 : number of pn pairs. n +1(–1) : number of proton (neutron) pairs; n 0 : number of pn pairs.

15. XXIII DGMTP, China 2005 On the Physical Significance of q Deformation in Many-body Physics O2 ofspq(4) Diagonal in the q-deformed basis set Reduces to the Casimir invariant of sp (4) in the nondeformed limit Zeroth-order approximation of O 2 commutes with all the q-deformed generators Gives direct relation between the expectation values of the second-order products of the operators that build O 2 Result can be used to provide for an exact solution of a q- deformed model Hamiltonian Diagonal Second-order Operator O 2 ( sp q (4)) N +1(–1) : number of protons (neutrons) O 2 ( sp q (4)) = su q k (2)

16. XXIII DGMTP, China 2005 On the Physical Significance of q Deformation in Many-body Physics Hcl The Model(s) H  N Isovector (isospin 1) J=0 pairing interaction Symmetry breaking Diagonal isoscalar (isospin 0) pn force Symmetry term Sp(4) Dynamical Symmetry basis states  ˆ  Describe pairing-governed isobaric analog 0 + states (IAS)  Include ground states for all even-even and only some (N~Z) odd-odd nuclei  Describe pairing-governed isobaric analog 0 + states (IAS)  Include ground states for all even-even and only some (N~Z) odd-odd nuclei

17. XXIII DGMTP, China 2005 On the Physical Significance of q Deformation in Many-body Physics Hq HqHq  q N ˆ H cl The Model(s) Sp q (4) Dynamical Symmetry

18. XXIII DGMTP, China 2005 On the Physical Significance of q Deformation in Many-body Physics H cl + 3-b = H cl + 3-b +4-b+5-b… +4-b+5-b… {  q,G q,E q,F q,C q,D q }= { ,G,E,F,C,D} If: Hq HqHq  q N ˆ The Model(s) Sp q (4) Dynamical Symmetry  has the Sp q (4)  SU q (2) dynamical symmetry  contains the original Sp(4) dynamical symmetry

19. XXIII DGMTP, China 2005 On the Physical Significance of q Deformation in Many-body Physics Hq=Hcl+m-b From a Nondeformed Perspective H  N  many-body terms Sp (q) (4) Dynamical Symmetry ˆ q=1 one- and two- body terms q=e 

20. XXIII DGMTP, China 2005 On the Physical Significance of q Deformation in Many-body Physics N(N-1) C q = C From a Nondeformed Perspective Illustrative Example q=e 

21. XXIII DGMTP, China 2005 On the Physical Significance of q Deformation in Many-body Physics N(N-1) From a Nondeformed Perspective Illustrative Example may not be negligible E.g., the energy contribution of the four-body interaction can reach a magnitude of several MeV in nuclei in the upper fp +1g 9/2 shell. q=e 

22. XXIII DGMTP, China 2005 On the Physical Significance of q Deformation in Many-body Physics Parameters N D total number of nuclei; n p number of fit parameters We fix the values of the parameters so that H cl remains unchanged when deformation is introduced – this is because H cl reproduces reasonably well the overall behavior common for all the nuclei in a shell such as… 16 S 1628 Ni 28 G/G/ F/F/ 0.056 C0.190 D-0.307 S min  9.57 1.79 0.296 300.3   0.702 0.007 0.815 0.127 0.496 1.72 9.01 0.453 0.072 0.473 0.149 0.732 16.1 9.36 20 Ca 20 2411 pp+nn+pn pairing s.p. energy E/2  -0.489-1.409-1.120 symmetry energy Parameters

23. XXIII DGMTP, China 2005 On the Physical Significance of q Deformation in Many-body Physics BE Lowest Isobaric Analog 0 + State (Binding) Energy -10-5510 -200 -100 100 200 300 400 E C 0 (MeV) T0T0 A=56 A=78 A=100 E 0 (MeV) T0T0 Agree well with experiment (  ) Semi- empirical estimate(*) (*)P. Moller, J. R. Nix and K.-L. Kratz, At. Data Nucl. Data Tables 66, 131 (1997). 56 Ni 40 Ca 1f 7/2 1f 5/2 2p 1/2 2p 3/2 1g 9/2 J. Retamosa, E. Caurier, F. Nowacki and A. Poves, Phys. Rev. C 55, 1266 (1997). Coulomb correction Energy spectra of pairing- governed 0 + IAS of 319 nuclei and only 6 parameters

24. XXIII DGMTP, China 2005 On the Physical Significance of q Deformation in Many-body Physics Higher 0 Energy Spectra of Isobaric Analog 0 + States 5 10 15 20 25 30 22 Ti 2222 Ti 2422 Ti 2623 V 2323 V 25 E 0 (MeV) 24 Cr 2224 Cr 2424 Cr 26 5 10 15 20 25 30 5 10 15 20 25 30 25 Mn 2325 Mn 2526 Fe 2226 Fe 2426 Fe 26 expthexpthexpth expth 5 10 15 18 Ar 18 Very good agreement with experiment 32 S core 40 Ca core 1f 7/2 1d 3/2 Without varying parameters

25. XXIII DGMTP, China 2005 On the Physical Significance of q Deformation in Many-body Physics 2p-DripLine 33384348 -20 -10 0 10 20 30 Detailed Structure: Two-Proton Drip Line neutron number 28 50 Ge 28 Ga 29 Se 30 As 31 Kr 32 Br 33 Sr 34 Rb 35 Zr 36 Y 37 Zr 38 Y 39 Mo 40 Nb 41 Ru 42 Tc 43 Pd 44 Rh 45 Cd 46 Ag 47 S 2p =E 0 (Z)–E 0 (Z–2) (MeV) N=Z 56 Ni core Z 0.32 #E. Ormand, Phys. Rev. C 55, 2407 (1997). #P. Moller, J. R. Nix and K.-L. Kratz, At. Data Nucl. Data Tables 66, 131 (1997). #B. A. Brown, R. R. C. Clement, H. Schatz and A. Volya, Phys. Rev. C 65, 045802 (2002). 0.78 0.43 1f 5/2 2p 1/2 2p 3/2 1g 9/2 Comparison of Sp(4) model to other models:

26. XXIII DGMTP, China 2005 On the Physical Significance of q Deformation in Many-body Physics FiniteEnergyDifference Detailed Structure: N=Z Irregularities ZN Interaction between the last proton and the last neutron MeV 2E0ZN2E0ZN 56 Ni core 2E0Z22E0Z2 Non-pairing like- particle interaction

27. XXIII DGMTP, China 2005 On the Physical Significance of q Deformation in Many-body Physics FED: Ca Detailed Structure: Staggering Behavior E0nE0n 2E02n2E02n Agree well with experiment 40 Ca core 1f 7/2 n: valence number of particles

28. XXIII DGMTP, China 2005 On the Physical Significance of q Deformation in Many-body Physics FED: Ni  E 0 /  n (MeV) -10-50510 A=76 A=78 A=64 A=66 A=88 A=90 i =-2 i =-1 i =-4 i =-3 i =-6 i =-5 AT0T0 Detailed Structure: Staggering Behavior -10-50510 T0T0 A theory experiment  2 E 0 /  n  T 0 (MeV) 56 Ni core 1f 5/2 2p 1/2 2p 3/2 1g 9/2

29. XXIII DGMTP, China 2005 On the Physical Significance of q Deformation in Many-body Physics Pairing gaps I Isovector Pairing Gap,  +– 2 odd-odd  T0T0 T0T0 T0T0 (MeV)     22 +– symmetry term pairing (MeV) 40 Ca core ~ ~ Significant pp,nn and pn Interplay CORE   CORE   CORE  

30. XXIII DGMTP, China 2005 On the Physical Significance of q Deformation in Many-body Physics Pairing gaps II Pairing Gaps (MeV) 40 Ca core 1f 7/2 56 Ni core 1f 5/2 2p 1/2 2p 3/2 1g 9/2 Like-particle pairing gap

31. XXIII DGMTP, China 2005 On the Physical Significance of q Deformation in Many-body Physics Global to local The new feature is an extension of the theory to include nonlinear local deviations from the pairing solution as realized through q- deformation of the sp (4) algebra. From a Global Scale to a Local Scale http://svs.gsfc.nasa.gov/

32. XXIII DGMTP, China 2005 On the Physical Significance of q Deformation in Many-body Physics q: Physical significance The q-deformation adds to the theory, which describes quite well the overall nuclear behavior, a mean-field correction along with 2-, 3-, and many-body interactions of a local character that can be responsible for residual single-particle and many-body effects. Physical Significance of q Global behavior Local non-linearity Sp(4) Sp q (4) H = H cl  Many-body terms two-body interaction (q = 1)quantum two-body interaction (q ≠ 1)

33. XXIII DGMTP, China 2005 On the Physical Significance of q Deformation in Many-body Physics q: Physical significance Physical Significance of q H = H cl  many-body terms two-body interaction (q = 1) quantum two-body interaction (q ≠ 1) q-deformation does not influence the non-deformed two-body interaction R R1R1 Interaction strength parameters in H cl remain fixed;

34. XXIII DGMTP, China 2005 On the Physical Significance of q Deformation in Many-body Physics q: Physical significance Physical Significance of q H = H cl  many-body terms two-body interaction (q = 1) quantum two-body interaction (q ≠ 1) ‘q’ R Interaction strength parameters in H cl remain fixed; q-deformation does not influence the non-deformed two-body interaction

35. XXIII DGMTP, China 2005 On the Physical Significance of q Deformation in Many-body Physics q: Physical significance Physical Significance of q H = H cl  many-body terms two-body interaction (q = 1) quantum two-body interaction (q ≠ 1) The fundamental properties of q cannot be “mocked up” by allowing the strengths of the nondeformed interaction to absorb its effect. ‘q’ R Interaction strength parameters in H cl remain fixed; q can vary within different nuclei to reflect experimental energies.

36. XXIII DGMTP, China 2005 On the Physical Significance of q Deformation in Many-body Physics Solutions for q Solutions for q (Near-closed Shell Nuclei) 22 Ti 48 -212 1 2 3 4 5 E 0 – E 0,exp (MeV) q 27 Co 52 E 0 nondeformed energy predicted by Sp(4), q=1 Solutions for the deformation parameter: detects possible presence of local effects 40 Ca core Deviation within an individual nucleus of E 0 from the experimental value E 0,exp Near-closed Shell Nuclei q=e  

37. XXIII DGMTP, China 2005 On the Physical Significance of q Deformation in Many-body Physics Solutions for q Solutions for q (Midshell Nuclei) 22 Ti 48 -212 1 2 3 4 5 E 0 – E 0,exp (MeV) q 27 Co 52 E 0 nondeformed energy predicted by Sp(4), q=1 40 Ca core No solution: theoretical prediction closest to the experiment occurs at the nondeformed point, q=1. Midshell Nuclei q=e  

38. XXIII DGMTP, China 2005 On the Physical Significance of q Deformation in Many-body Physics Smooth behavior of q Smooth Dependence of q on Nuclear Characteristics Smooth behavior 56 Ni core N –1 Zn Ge Se 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 51015 20 0.2 0.4 0.6 0.8 1.0 Peaks at N=Z [with significant values of q] Many-body nature of the interaction is most important away from mid shell q-deformation as prescribed by the Sp q (4) model is not random in character but rather fundamentally related to the very nature of the nuclear interaction. q-deformation as prescribed by the Sp q (4) model is not random in character but rather fundamentally related to the very nature of the nuclear interaction. q=e  

39. XXIII DGMTP, China 2005 On the Physical Significance of q Deformation in Many-body Physics q: Functional Dependence Functional Dependence of q on Model Quantum Numberss N –1 Zn Ge Se 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 51015 20 0.2 0.4 0.6 0.8 1.0 Based on the discrete solutions found within each nucleus, q is assigned a functional dependence on N and T 0 : q=e    (N, T 0 )=

40. XXIII DGMTP, China 2005 On the Physical Significance of q Deformation in Many-body Physics q: Functional Dependence Functional Dependence of q on Model Quantum Numberss N –1 Zn Ge Se 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 51015 20 0.2 0.4 0.6 0.8 1.0 f 5/2 pg 9/2  1 = –2.13,  2 = 0.37,  3 = 3.07,  4 = 0.15 Fit to even-even nuclei in 1f 7/2 and 1f 5/2 2p 1/2 2p 3/2 1g 9/2 SOS  271.63 1.79 130.21 1.28 q=1q≠1 1f 5/2 2p 1/2 2p 3/2 1g 9/2  (N, T 0 )= q=e  

41. XXIII DGMTP, China 2005 On the Physical Significance of q Deformation in Many-body Physics q: uniformly superior Comparison to Nondeformed Energies The model with the local q improves the energy prediction compared to the nondeformed global model and reproduces more closely the experiment. SOS  271.63 1.79 130.21 1.28 q=1q≠1 1f 5/2 2p 1/2 2 3/2 1g 9/2 40 Ca core 1f 7/2 One reason may be that the q- deformed fermions, unlike usual quasi- particles, indeed obey the fundamental laws. E 0 – E 0,exp (MeV) (q)(q) q=e 

42. XXIII DGMTP, China 2005 On the Physical Significance of q Deformation in Many-body Physics q in mid-mass nuclei q-Parameter and Many-body Interactions 56 Ni core N –1 N +1 N T0T0 (N, T 0 ) N –1 Determined statistically,  1,  2,  3, and  4 provide an estimate for the overall significance of q-deformation within a shell. q=e  

43. XXIII DGMTP, China 2005 On the Physical Significance of q Deformation in Many-body Physics q in mid-mass nuclei q-Parameter and Many-body Interactions 56 Ni core N –1 N +1 N T0T0 (N, T 0 ) N –1 I.Significant many-body interactions are detected away from mid-shell and tend to peak at even-even N=Z nuclei where strong pairing correlations are expected. I.Significant many-body interactions are detected away from mid-shell and tend to peak at even-even N=Z nuclei where strong pairing correlations are expected. The pair formation favors the non-negligible higher-order interactions between the pair constituents that are detected via the Sp q (4) model. q=e  

44. XXIII DGMTP, China 2005 On the Physical Significance of q Deformation in Many-body Physics q in mid-mass nuclei q-Parameter and Many-body Interactions 56 Ni core N –1 N +1 N T0T0 (N, T 0 ) N –1 II.Around mid-shell the deformation adds little improvement to the q=1 theory. For these nuclei the many-body interactions as prescribed by Sp q (4) are negligible and the model is not sufficient to describe other types of local effects that may be present. II.Around mid-shell the deformation adds little improvement to the q=1 theory. For these nuclei the many-body interactions as prescribed by Sp q (4) are negligible and the model is not sufficient to describe other types of local effects that may be present. q=e  

45. XXIII DGMTP, China 2005 On the Physical Significance of q Deformation in Many-body Physics Conclusion I Microscopic description of pairing-governed 0 + states in nuclei even-A: with nuclear masses 32<A<164 with protons and neutrons occupying the same major shell even-even odd-odd even-even odd-odd Conclusion include 0 + ground states for all even-even and some odd-odd nuclei two-body interactions common nuclear properties two-body interactions common nuclear properties many-body interactions local non-linearity

46. XXIII DGMTP, China 2005 On the Physical Significance of q Deformation in Many-body Physics Conclusion Obeys standard quantum mechanics Conserves angular momentum J and number of particles N Obeys standard quantum mechanics Conserves angular momentum J and number of particles N two-body interactions common nuclear properties two-body interactions common nuclear properties many-body interactions local non-linearity

47. XXIII DGMTP, China 2005 On the Physical Significance of q Deformation in Many-body Physics Conclusion  Reasonable prediction of ground and excited pairing-governed isobaric analog 0 + state energies  Reproduction of global trends and smaller fine features in nuclear dynamics  Reasonable prediction of ground and excited pairing-governed isobaric analog 0 + state energies  Reproduction of global trends and smaller fine features in nuclear dynamics Two-proton drip line N=Z irregularities Pairing gaps Staggering behavior  Isobaric analog 0 + states possess a simple Sp(4) dynamical symmetry  Isobaric analog 0 + states possess a simple Sp(4) dynamical symmetry

48. XXIII DGMTP, China 2005 On the Physical Significance of q Deformation in Many-body Physics Conclusion  Introduction of higher-order many-body interactions: q  Physical significance of q-deformation: in the very nature of nuclear interaction  Introduction of higher-order many-body interactions: q  Physical significance of q-deformation: in the very nature of nuclear interaction Decoupling of q-deformation from the two-body interaction: strengths of the nondeformed interaction cannot absorb the effect of the deformation. Local non-linear effects (within individual nucleus) Uniformly superior q≠1 results to those in the nondeformed limit Smooth dependence on nuclear characteristics Non-negligible higher-order many-body interactions (q) in regions of dominant pairing correlations