Aligned neutron-proton pairs in N=Z nuclei

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1 Aligned neutron-proton pairs in N=Z nuclei P. Van Isacker, GANIL, France Motivation Shell-model analysis A model with high-spin bosons Experimental tests Neutron-proton correlations, UHK, Hong Kong, July 2015

2 Spin-aligned T=0 np pairs Motivation: A simple description of the N=Z nuclei 98 In, 96 Cd, 94 Ag, 92 Pd, 90 Rh. Starting point: Shell-model interpretation in terms of spin-aligned T=0 np pairs (Blomqvist). Experiments have been proposed and carried out at GANIL (Cederwall, de France, Wadsworth ). [Also, analysis of N=Z nuclei in the 1f 7/2 shell ( 42 Sc, 44 Ti, 46 V, 48 Cr, 50 Mn, 52 Fe, 54 Co).]

3 Plus ça change (1)

4 Plus ça change (2)

5 Nuclear belly dancer B. Cederwall et al., Nature 469 (2011) 68

6 A new coupling scheme? Our results reveal evidence for a spin-aligned, isoscalar neutron proton coupling scheme. [T]his coupling scheme replaces normal superfluidity (characterized by seniority coupling) in the ground and low-lying excited states of the heaviest N=Z nuclei. B. Cederwall et al., Nature 469 (2011) 68

7 Approximations Hypothesis: N=Z nuclei can be described in the (spherical) shell model, in an appropriate model space and with an appropriate interaction. Approximations: (A) Truncate shell model to a single high-j shell. (B) Truncate single-j shell space to one written in terms of aligned-spin B (J=9) pairs. (C) Replace aligned-spin B pairs by b bosons.

8 Shell-model interaction: 1g 9/2 0 T 1 Ν J,T MeV 1 2 SLGT0 T angular momentum J E.J.D. Serduke et al., Nucl. Phys. A 256 (1976) 45 H. Herndl and B.A. Brown, Nucl. Phys. A 627 (1997) 35

9 Pair analysis in the shell model Define different types of nucleon pairs: + = ( + + a j1/2 a ) JT j1/2 B JT ( ) S + : J = 0, T =1; D + : J = 2, T =1; B + : J = 9, T = 0. Calculate overlap with shell-model wave functions with the nucleon-pair shell model in an isospininvariant formulation. J.-Q. Chen, Nucl. Phys. A 562 (1993) 218; 626 (1997) 686 G.J. Fu et al., Phys. Rev. C 87 (2013)

10 B-pair analysis of 96 Cd 1.0 J 1 B 2 ; J Cd J

11 Spectrum of 96 Cd Expt SM B 2 b 2 2 b 2 3 Energy MeV Cd 48

12 B-pair analysis of 94 Ag J 1 B 3 ; J Ag J

13 Spectrum of 94 Ag Energy MeV 5 0 Expt SM B 3 b 3 2 b Ag

14 Mapping to bosons Transform to a much simpler problem in terms of interacting bosons: B + -> b + Boson energies and boson-boson interactions are derived from the shell model (i.e., not fitted). T. Otsuka et al., Nucl. Phys. A 309 (1978) 1 L.D. Skouras et al., Nucl. Phys. A 516 (1990) 255

15 Magnetic dipole moments For any state in a single-j shell g( αj) = 1 ( 2 g ν + g π ) The same result is obtained with b-ibm mapped from a single-j shell model. Magnetic dipole moments test approximation (A) but are insensitive to (B) and (C). In 46 V: In 50 Mn: " $ # %$ ( ) =1.64(3) µ N ( ) = 2.76(1) µ N µ µ ( ) ( ) 0.52 to 0.55 µ N 1f 7/ to 0.54 µ N 1g 9/2 F.C. Charlwo o d et al., Phys. Lett. B 690 (2010) 346

16 Q moment of 21 + isomer in 94 Ag Shell model in pf 5/2 g 9/2 space (M=21 -> dim=2): ( ) = 0.44 b Q Shell model in 1g 9/2 (J=21 -> dim=1): Q( + 21 ) 1 = ( e ν + e π )( ho ) b Expression in terms of b bosons: Q( b 3 21) = ( e ν + e π )( ho ) b Measurement of Q(21 + ) tests (A). Calculation confirms (B+C). A. Poves, private communication LSSM

17 Q moment of 7 + isomer in 94 Ag Shell model in pf 5/2 g 9/2 space (M=7 -> dim=37327): ( ) = 0.62 b Q Shell model in 1g 9/2 (J=7 -> dim=84): ( ) = 6.60 e ν + e π Q ( )( ho ) b Expression in terms of b bosons: Q( b 3 [ 16]7) = ( e ν + e π )( ho ) b Measurement of Q(7 + ) tests (A). Calculation confirms (B+C). A. Poves, private communication LSSM

18 A discussion between physicists What is the correct coupling scheme for these nuclei? Seniority or aligned pairs? Physicist 1: The 7 + state in 94 Ag can be written as 7 + = B 2 16 [ ] B;J = 7 Physicist 2: Not at all, this 7 + state should be written as 7 + = B 2 [ 4] B;J = 7

19 (Anti)-symmetric amnesia b 2 16 b; J 7

20 (Anti)-symmetric amnesia b 2 16 b; J 7

21 (Anti)-symmetric amnesia b 2 16 b; J 7

22 (Anti)-symmetric amnesia b 2 16 b; J 7

23 (Anti)-symmetric amnesia b 2 16 b; J 7 b 2 4 b; J 7

24 (Anti)-symmetric amnesia b 2 16 b; J 7 b 2 4 b; J 7

25 (Anti)-symmetric amnesia b 2 16 b; J 7 b 2 4 b; J 7

26 (Anti)-symmetric amnesia b 2 16 b; J 7 b 2 4 b; J 7

27 (Anti)-symmetric amnesia b 2 16 b; J 7 b 2 4 b; J 7

28 Fermion pairs versus bosons In the 1g 9/2 nucleon-pair shell model: B 2 4 [ ] B;J = 7 B 2 [ 16] B;J = 7 = In the b-ibm: b 2 [ 4] b;j = 7 b 2 [ 16] b;j = 7 = The B pair behaves as a boson.

29 Conclusions (A) Truncation of the shell model to a single-j shell. (?) (B) Truncation of the single-j shell space to one written in terms of aligned-spin B (J=7 or 9) pairs. ( ) (C) Replacement of aligned-spin B pairs by b bosons. ( ) S. Zerguine & P. Van Isacker, Phys. Rev. C 83 (2011) P. Van Isacker, Int. J. Mod. Phys. E 22 (2013)

30 Shell-model interaction: 1f 7/2 0 T 1 f Υ JT MeV Sc T angular momentum J

31 Shell-model interaction: 1f 7/2 0 T 1 f Υ JT MeV Co T angular momentum J

32 B-pair analysis of 44 Ti and 52 Fe 1.0 J 1 B 2 ; J Fe Ti J

33 B-pair analysis of 46 V and 50 Mn J 1 B 3 ; J Mn V J

34 B-pair analysis of 48 Cr 1.0 J 1 B 4 ; J Cr J

35 Structure of 96 Cd

36 Spectrum of 44 Ti 10 Expt SM B 2 b 2 2 b 2 3 Energy MeV Ti

37 Spectrum of 52 Fe 10 Expt SM B 2 b 2 2 b 2 3 Energy MeV Fe

38 Spectrum of 46 V 10 Expt SM B 3 b 3 2 b Energy MeV V

39 Spectrum of 50 Mn 10 Expt SM B 3 b 3 2 b 3 3 Energy MeV Mn

40 Spectrum of 48 Cr 10 Expt SM B 4 b 2 b 3 Energy MeV Cr

41 Spectrum of 92 Pd Expt SM B 4 b 2 b 3 Energy MeV Pd 46

42 Q moment of 5 + state in 50 Mn Experimental value: Large-scale shell model: Numerical result in the 1f 7/2 shell: ( ) = 4.2 e ν + e π Q Expression in terms of b bosons: Q b 3 [ 12]5 Parameter-free test: ( ) = 0.80(12) b Q( + 5 ) 1 = 0.58 b Q ( )( ho ) 2 ( ) = Q ; 46 V e ν + e π ( ) ( ) Q ( b 3 12 ) Q( b) Q ; 42 Sc ( )( ho ) 2 4.3( e ν + e π )( ho ) 2 [ ]5 = F.C. Charlwo o d et al., Phys. Lett. B 690 (2010) 346

43 E2 properties in b-ibm The E2 operator in the shell model: ˆ T µ F E2 ( ) = e ν r i 2 Y 2µ θ i,φ i ( ) + e π r 2 i Y ( 2µ θ i,φ ) i i ν i π The E2 operator in terms of b bosons: ˆ T µ B E2 ( ) = e b b + b ( ) µ 2 The mapping implies the relation: ( ) ( g ) 2 9 / 2 ;9 + T ˆ F ( E2) ( g ) 2 9 / 2 ;9 + = b T ˆ B ( E2) b e b = 55 ( 3π ) 2 ho 266 ( 187 e + e ) ν π

44 B(E2) values in 96 Cd

45 B(E2) values in 96 Cd A simple consequence of the aligned-pair assumption: B( E2;J + 2 J) = e 2 b 20( 2J +1) # 9 9 2& $ ' % J + 2 J 9( 2

46 B(E2) values in 94 Ag

47 B(E2) values in 94 Ag

48 B(E2) values in 92 Pd

49 Energy of 21 + isomer in 94 Ag How many independent states of three b-bosons can couple to angular momentum 21? d( υ,,j) = i 2π z =1 ( z 2J +1 1) z 2υ k =1 ( ) ( z υ +k 1) υ +J +2 z z 2 2 k =1 ( k +1 1) Answer: d (3,9,21)=2 one of which is spurious. After elimination of the spurious state, the energy of the physical 21 + state is E( 21 + ) = 3ε b ν b ν b ν b 16 P. Van Isacker, Phys. Scr. T150 (2012)

50 Energy of 21 + isomer in 94 Ag In turn, we know the boson matrix elements in terms of the shell-model matrix elements: ν b 12 = ν ν ν ν ν ν ν 9 ν b 14 = ν ν ν ν ν 9 ν 16 b = 8 17 ν 7 + 3ν ν 9

51 Energy of 21 + isomer in 94 Ag Therefore, we know the energy of the 21 + isomer in 94 Ag in terms of the shell-model interaction matrix elements: E ( b 21 + ) = ν ν ν ν ν ν ν 9

52 Energy of 21 + isomer in 94 Ag The 21 + state is unique in the 1g 9/2 shell model. Its energy is therefore known analytically: E ( f 21 + ) = ν ν ν ν ν 9 Comparison tests the reliability of the mapping: E ( f 21 + ) ν ν ν ν ν 9 E ( b 21 + ) ν ν ν ν ν ν ν 9

53 Conservation of n, J and T A unique n-particle shell-model state with angular momentum J and isospin T has energy E f ( j n JT) = a λ v λ λ The coefficients a λ satisfy 2 j a λ λ =0 2 j ( ) = n n 1 2 λ( λ +1)a λ = J J +1 λ =0 2 j 2a λ = T T +1 λ =0 even, ( ) + j j +1 ( ) n( n 2) ( ) n n 2 ( )