p(p,e + ν)d reaction determines the lifetime of the sun

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2 Contents Basic properties of nuclear systems Different aspects in different time scales Symmetry breaking in finite time scale Rapidly rotating nuclei Superdeformation and new type of symmetry breaking

3 pp p(p,e + ν)d reaction determines the lifetime of the sun

4 ( ) 15 N(p,α) 12 C σ~ 0.5 b (E=2 MeV) ( ) 3 He(α,γ) 7 Be σ~ 10-6 b (E=2 MeV) ( ) p(p,e + ν)d σ~ b (E=2 MeV) 1 b = 100 fm 2

5 Bohr & Mottelson Nuclear Structure Vol.1 Bethe-Weizsäcker (Saturation) B( N, Z) = a V a a A a sym C ( N Z A 2 S 1/3 A 2/3 2 Z) A +δ ( A)

6 ρ( r) = 1 G (2π ) 3 E (q 2 )e iq r d q Pb dσ dω q [ fm 1 ]

7 Mayer-Jensen s Shell Model V ( r) = Mω r v ll 2 + magic numbers : (N,Z)=2, 8, 20, 28, 50, 82, 126 λ >> R v ls s E s 1/2 p 3/2 r p 1/2

8 25 ( = ) N = N = 126 S(2n) MeV N = 84 Sm Hf Ba Pb Sn Neutron Number From a lecture by R. Casten

9 Spin-parity of odd nuclei i f 13/2 7/2 h 9/2 82 s 1/2 d 3/2 i j 11/2 15/2 g 9/2 126 f 5/2 p 1/2 p 3/2

10 E-dep. of n-a optical potentials V 0 (E) [ MeV ]

11 ρ 0.16 fm 3 kf 1.35 fm 2 k TF = 2m 1 2 F 40 MeV k F r 1, k F a >>1 τ F R v F ~ s τ c >> τ F λ >> R quasi-particle)

12 (ev ) τ comp >> τ c ΔE > / τ c

13 [Bohr 1936, Wigner, 1955] (s ~ ev, Γ ~ mev) ( ) N.Bohr (1936) Bohigas (1989) x = s / s

14 τ coll ~ E ex τ damp ~ Γ Well-defined: τ coll < τ damp τ c > τ coll ~ τ F τ c > τ coll > τ F

15 (GQR) (sum rule value) 100% m p = n E n p n F 0 2 cf) Ring-Schuck, Nuclear Many-body problems (1980) H = 1 2B π C α 2 C ~ m 3 ~ T! GDR p z P( p) p z p y p y p x p x

16 H = λµ µ! 1 π 2 λµ + 1 2B λ 2 C $ 2 # λα λµ & " %! 1 2 π 2µ + 1 2B 2 2 C $ 2 # 2α 2µ & " % λ = 2 C 2 = π 1 a S A 2/3 3Z 2 e 2 B 2 = 1 2 Mρ 0R π R 0 λ = 2 0 +, 2 +, Bohr, Mottelson, Nuclear Structure Vol.2 B 2 (LDM) B 2 (exp) A

17 (N,Z: ) Pd (Z=46 [=50 4]) 1 (1 phonon) J π = (2 phonon) J π = 0 +, 2 +, 4 + Rowe-Wood, Fundamentals of Nuclear Models (World Scientific,2010) 2 phonon

18 ! Rotational spectra 2 phonon 2 phonon Z =

19 ! B + + " 2 + B 2 + Φ 0 + B Φ phonon (2 + ) # $ M L Φ0 2-phonon (L + ) V(β) " Φ + def cb ) 2 + exp( Φ 0 β β ~ Φ def B 2 + Φ def ~ Φ def r 2 Y 20 ( ˆr) Φ def

20 Effective potential spherical transitional deformed V(β) β Quadrupole deformation (order parameter) Spontaneous Symmetry Breaking (SSB)

21 R 4/2 E I = I(I +1) 2I R 4/2 E 4 E 2 = 20 6 = 3.33 Def. R. Casten Sph. R 4/ 2 = E E Onset of deformation!

22 R 4/2 Courtesy: R. Casten

23 B(E2:&0 + 1& &2+ 1 )& & &2+ 1& $$E2 $$0+ 1 2& 2 +& 0 +&

24 Intrinsic&Q&moment& SkM*&energy&density&func?onal& Sm Nd

25 ( ) [(TD)DFT]& & TimeGdependent&BdG&KohnGSham&eq.& E ρ q (t),τ q (t), J q (t), j q (t), s q (t), T q (t);κ q (t)! " # $ ( )!! " # $ $ % &! " # $ % & Δ Δ =!! " # $ $ % & ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( * * t V t U t h t t t h t V t U t i µ µ µ µ λ λ spin-current current spin spin-kinetic kinetic pair density

26 SkM*&func?onal& (S ~ 30 MeV, K ~ 217 MeV) Intrinsic&Q&moment& Yoshida, Nakatsukasa, PRC 83, (2011)

27 (IS,IV; L=0~3) Yoshida and TN, PRC88 (2013) hp IS-GMR IS-GDR IS-GDR IS-GMR

28 m * m = 0.8 ~ 0.9 ISGQR K = 210 ~ 230 MeV

29 Computational nuclear data tables Inakura, T.N., Yabana, PRC 84, (R) (2011); arxiv: E1 strength functions SkM* functional 3D Cartesian mesh R box = 15 fm

30 Issues in low-energy E1 strength Inakura, et al., arxiv: Does the observation constrain the neutron skin thickness and the neutron-matter EOS? Yes, but we need to go to very neutron rich! 84 Ni is much better than 68 Ni Does it influence the r-process? Significantly influence the direct neutron capture process near the neutron drip line We need calculation with a proper treatment of the continuum. Goriely, PLB436, 10

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34 Bohr, Mottelson, Nucl. Str. Vol.1 < T > «< V > < T > < V >

35 V 0 : c: Λ = 2 2Mc 2 1 V 0 Λ (H) 0.06 ( 3 He, 4 He) 0.1~0.2 (n,p) 0.5

36 v F ~ 0.3c τ SSB zeropoint motion τ SSB τ F τ SSB τ c

37 Rotating objects in the universe Nucleus is one of the fastest rotating many-body system. R0 ωrot << 1 v F Low spin Perturbative Coriolis effects R0ω v F rot ~ 1 High spin Non-perturbative Coriolis effects Structure change, band crossings

38 Cranking model Picture in the rotating frame Time-dependent Schrödinger equation i t Ψ( t) = H Ψ( t) In the uniformly rotating frame with the rotational frequency ω iωt J Ψ( t) = e Φ( t) i t Choose the rotational axis as x-axis: H ' H ω J H ' = H ω ( H ω J) Φ( ) Φ( t) = t rot J x 1-dim. cranking ω rot Cranking violates the time-reversal symmetry. However, in case of quadrupole deformation, it conserves the parity and signature symmetry: Rˆ x = e iπ J x r $ ± i, α = 1! 2 = iπα e, r = # $ 0! ± 1, α = # " " 1 Experimentally, often defined as α = I (mod2)

39 Collective and non-collective rotations Non-collective rotation Collective rotation Cranking model is applicable to both cases.

40 Quasi-particle routhians Quasi-particle eigen-energies of the cranked HFB equation & h λn ω $ % Δ rot j x Δ ( h λn ω rot j x #& U! $ )"% V #! " = e ' & U $ % V #! " Even nuclei: qp vacuum! ground band 2qp excitation Odd nuclei: 1qp exc. E =0 ' e ( ω) + e A ' ' e A ( ω), e ' B B ( ω), ( ω), QP routhians around 164 Er Negative-energy states are fully occupied in the vacuum.

41 Experimental routhians Angular momentum Rotational frequency ω rot ( I) de di E( I I ( I 2 2 I x ( I) = ( I + 1/ 2) K x + 1) E( I 1) + 1) I ( I 1) x E γ ( I I 1) I I +1 1 E γ Routhian and relative routhian ' 1 ωrot E ( ωrot ) = E( I) ωroti x( I) x x I 2 2 ' ' ' e ω ) E ( ω ) E ( ω ) ( rot rot ref rot Ground-state band as the reference band I ( ω J ω J ω x = + rot ) 0 ω rot 1 3 rot rot ' ' 1 2 Eref ( ωrot ) = I x( ω) dω = E0 J0ωrot 2 [ E( I + 1) + E( I 1) ] [ I ( I + 1) + I ( 1) ] The Harris formula parameters are determined by fitting 1 4 J ω The const. is chosen so as to make the ground state (I=0) at zero energy. E = ' 0 1 8J rot

42 Routhian analysis Bengtsson & Frauendorf, NPA327 (1979) 139. Experimental quasi-particle routhians in odd rare-earth nuclei

43 We may take, as the reference routhians, the experimental yrast routhians of neighboring even-even nuclei (or their average).

44 Back-bending phenomena Nucleus loses its angular momentum by emitting gamma rays. The 158 Er nucleus is spinning down, losing ΔI=2 each step. But, then, at I=16, it spins up, even though losing the angular momentum. Nuclear glitch ω rot This phenomenon was first discovered in 1972 using the in-beam gamma-ray spectroscopy.

45 Structure change of the yrast states At low spin, the Cooper pair is condensed in the ground state. Φ HFB & $ % k # α c c! " k + k + k N / 2 0 Coriolis force At frequency ω c, one of the Cooper pair is broken up, due to the Coriolis anti-pairing effect. + + a A a B Φ HFB ΦHFB + + a A a B ΦHFB i 10 Without increasing ω rot, the state gains about 10 units of spin values, by the alignment.

46 Cranked HFB model The cranking model successfully describes the back-bending phenomena in many nuclei, with the proper strength of the Coriolis force. Ring and Schuck, Nuclear Many-Body Problems (1980), Springer-Verlag

47 Systematic analysis Systematic analysis of the back-bending frequency (by Garrett) ω exp c 1.67Δ j max e Δ Excitation energy of the 2qp state is roughly E 2qp 2Δ where j max is the largest j near the Fermi surface. Simple picture in the right figure predicts ω c Δ j max i ~ 2 j max E 2qp ( ω c ) 0 The Coriolis force pulls down the high-j quasi-particles, to realize a ω state analogous to the gapless superconductor in condensed c matter. This can be also regarded as the band crossing between the qp vacuum (0qp) and a 2qp state; g-band and s-band. ω rot

48 Collective states under rapid rotation At low spin, many p-h pairs of spin λ contribute to the collective state. λ=even At high spin, each p-h pair is aligned by the Coriolis force, to produce an aligned phonon. J=λ J z =λ At even higher spin, one of the p-h pairs is completely aligned, by escaping from the λ-coupling. i 10 J=2j max λ=odd J=λJ z =K J=j max +j max Φ coll ' % & ph α ph $ " # + ( cpch ) Φ0 λk Φ aligned coll ' % & ph α ph $ " # + ( cpch ) Φ0 λλ i 5 ~ a A a B Φ HFB

49 Octupole vibrations in 164 Yb K=0 K=2 T.N., Act. Phys. Pol. B27 (1996) 59 K=1 K n Q3 K 0 2 $ > 200 fm! # > 100 fm! " < 100 fm 3 3 3

50 Octupole vibrations in 238 U T.N., Act. Phys. Pol. B27 (1996) 59 K=1 K=0 K n Q3 K 0 2 $ > 300 fm! # > 150 fm! " < 150 fm 3 3 3

51 In 1986, the first superdeformed band was discovered in 152 UK P.J. Twin et al, PRL 57 (1986) 811 J.D. Garret et al, Nature 323 (1986) 395. γ I+6 I+4 I+2 I Large moment of inertia Large intraband B(E2) B(E2) 2000 W.u. Major : Minor axes ~ 2:1

52 Freq. ratio Fission isomer A 190 A 150 Single-particle energy levels deformation!

53 ! Effective potential spherical transitional deformed V(β) β Quadrupole deformation (order parameter) Spontaneous Symmetry Breaking (SSB)

54 Banana-(Y 31 -type) shape phase transition in open-shell SD states Increase of valence nucleons Banana-superdeformation T.N., S.M., K.M., Prog. Theor. Phys. 87 (1992) 607. Z=80+2, N=80+N val HO+QRPA Strong pairing Weak pairing p =1.3 MeV p =1.1 MeV SSB towards banana shape

55 RPA routhians for SD 152 Dy Excitation energy in the rotating frame relative to the ground-state SD band. RPA in the rotating shell model predicted an excited SD band in 152 Dy is the K=0 octupole vibrational band. T.N., Mizutori, Matsuyanagi, Nazarewicz, PLB343 (1995) 19 Octupole band with K=0 (Y 30 ) Exp: Dagnall et al, PLB 335 (1994) MOI

56 E x Decay SD6 SD1 I Exp RPA B(E1) 10-4 W.u.

57 E x [ MeV ] More realistic calculation Nilsson+BCS+QRPA Superdeformed Dy isotopes A Shape transition to banana-super-deformation

58 Where are they? Increasing (decreasing) valence neutrons (protons) by 8-10 leads to regions near beta-stable line A 190 A Dy Fusion reaction with radioactive beam might populate these high-spin SD states near beta-stable line.

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