Nuclear Collective Motions

Transcription

1 Nuclear Collective Motions Takashi Nakatsukasa Theoretical Nuclear Physics Laboratory, RIKEN Nishina Center th Chris Engelbrecht Summer School in Theoretical Physics Nuclei and Nucleonic Systems: Exotic nuclei, halos, nuclear synthesis and institute for Theoretical Physics at Stellenbosch Institute for Advanced Study, Stellenbosch, Western Cape, South Africa

2

3 Contents Liquid-drop, shell, unified models, cranking model Nuclear structure at high spin and large deformation Sum-rule approaches to giant resonances Basic theorem for the Time-dependent densityfunctional theory (TDDFT) Linearized TDDFT (RPA) and elementary modes of nuclear excitation Theories of large-amplitude collective motion Anharmonic vibrations, shape coexistence phenomena

4 Quarks, Nucleons, Nuclei, Atoms, Molecules atom nucleon nucleus q q N e molecule N q α α Strong Binding clustering deformation rotation vibration Strong Binding rare gas cluster matter Weak binding Weak binding

5 Separation energy (Ionization potential) Ionization potential of atom O O O+ O2+ O3+ O4+ O5+ O6+ O7+.47 ev Rapid decrease Neutron separation energy Abbas, Mod Phys Lett A 20, (2005), 2553 Gradual decrease (roughly constant) Nuclear clustering and deformation

6 Saturation Binding energy B/A 8 MeV Density ρ 0.4 fm-3 d 2 fm Liquid drop model Bethe-Weizsäcker mass formula B( N, Z ) = av A as A2 / 3 ( N Z )2 asym A Z2 ac / 3 + δ ( A) A Bohr & Mottelson, Nuclear Structure Vol.

7 Liquid drop picture 0+, 2+, Liquid drop vibrator and rotor Surface vibrations in spherical nuclei H = π + C α 2 2B 5-dim. quadrupole oscillator 4 λµ 2 λµ λ 2 λµ 4+ λ + I k' H= k =, 2, 3 2ℑ k ( β, γ ) g-band 4 + V ( β, γ ) β + 2 sin 3γ 4 2 B 2β β β β sin 3γ γ γ Nuclear fission (Bohr-Wheeler) EC Z2 x= ~ 2 ES A Nucleus is liquid? β-band γ-band

8 Electronic single-particle motion in atom V(r) r Single-particle levels in Coulomb pot. Magic number 2s,2p Ionization Potential s Electrons as free gas trapped by the Coulomb potential Bohr & Mottelson, Nuclear Structure Vol.

9 Nucleonic single-particle motion in nucleus Neutron Separation energy Neutron # N Bohr & Mottelson, Nuclear Structure Vol.

10 Proton Separation energy Bohr & Mottelson, Nuclear Structure Vol. Proton # Z

11 Mayer-Jensen s Shell Model Harmonic oscillator potential + spin-orbit force V (r ) = Mω r + vll + vls s 2 Correct magic numbers. Nucleons as free gas in the potential. Nucleus is gas? λ free > > R0

12 Collective (Unified) Model by Bohr & Mottelson Nucleons are independently moving in a potential that slowly changes. Collective motion induces oscillation/rotation of the potential. The fluctuation of the potential changes the nucleonic single-particle motion. H = H vib + H part + H coupl H vib = π 2 Bλ λµ 2 λµ + Cλ α 2 2 λµ In modern approaches, microscopic construction p i2 H part = + V0 ( ri ) i 2M H coupl = { V ( ri ; α λ µ ) V0 ( ri )} k (ri ) α i i λµ * λµ Yλ µ (ϑ i, ϕ i )

13 Weak coupling case 3 phonon + particle (h9/2 proton) Hcoupl is small Multiplet (3/2+,, 5/2+) Hamamoto, PR0 (974) 63. Bi 209 Pb 208

14 Weak coupling case 2+ phonon + (quasi-)particle (g9/2 proton) Shibata, Itahashi, Wakatsuki, NPA237 (975) 382. Precursor phenomenon of the quadrupole instability Anomalous coupling states 45 Rh 9/2+ 7/2+

15 Intermediate coupling More protons make coupling of 2+ phonon to the intruder g9/2 proton, more important Anomalous coupling Phonon+QP Dressed 3QP mode ( 3QP-QRPA ) Pauli effects favor I=j- state (Different sign for 6j coef.) Kuriyama, Marumori, Matsuyanagi, Prog. Theor. Phys. Suppl. 58 (975). uv2+vu2 uu2 vv2 Higher order effects, such as mode-mode coupling and anharmonicity, demand a systematic treatment. Nuclear Field Theory (D.Bes, Prog. Theor. Phys. Suppl. 74&75 (983).)

16 Toward strong coupling Effective potential spherical transitional deformed V(β) β Quadrupole deformation (order parameter) Spontaneous Symmetry Breaking (SSB)

17 Nambu-Goldstone modes SSB Degenerate ground state [J, H ] = 0, e iε J 0 0 Then, zero-energy modes of excitation, which connect different ground states exist. (The deformation defines the orientation, then, the rotational motion is generated.) In finite systems, this is intimately related to the separation of collective d.o.f. H = H coll (q, p ) + H ' (ξ, π ), E = Ecoll + E ', Ψ = Φ (H Φ coll ' = H H coll ' The ground state of H is degenerate. P2 H = H 2 AM Φ J2 H = H 2ℑ Φ ' ' ) loc = ck K Φ coll cj J Φ coll K def = J ' 0 ' 0

18 Strong coupling Hamiltonian (Axially symmetric quadrupole deformation) def H = H rot + H Coriolis + H rec + H part + H vib + H rot - vib + H part - vib H rot = [ I 2 I 32 2ℑ ( β 0 ) H Coriolis = H rec = H def part = j I = [ I + j + I j+ ] ℑ 2ℑ j2 2ℑ ( β 0 ) i ] Rotation-particle coupling Renormalized in the Hpart : Coriolis attenuation p i2 ( ) + V r ; β 0 i 0 2M i p i2 ˆ ( ) + V r k ( r ) β Y ( r ) 0 i i 0 20 i 2M Case that Hvib, Hrot-vib, and Hpart-vib are not present. Particle-Rotor Model

19 Strong coupling limit In case that the Coriolis effect is weak, the K-quantum number is approximately a good quantum number. R Strong-coupling wave function (with R-symmetry) IKM Φ I ( Θ ) + ( ) I + K Φ DMK nk nk DMI K ( Θ ) K Energy spectrum with ΔI = EnK ( I ) = ε nk + ( I ( I + ) K 2 ) 2ℑ Energy spectrum for K=/2 band with ΔI = EnK = / 2 ( I ) = ε nk + a= Φ nk = / 2 2ℑ j+ Φ I + / 2 I ( I + ) + ( ) a I nk = / 2 Decoupling parameter j 239 Pu

20 Decoupling limit (Aligned band) In the opposite limit to the strong coupling case that the Coriolis force is much stronger than the energy splitting in different K states. R H Coliolis = [ I j ] I i ℑ ℑ i j R I i : must be even integer Stephens et al, PRL29 (972) 438. Energy spectrum with ΔI= 2 [ I ( I + ) 2 I i ] 2ℑ = const. + R( R + ) 2ℑ E ( I ; i ) = const. + Rotational spectrum of a favored band (maximally aligned with i=j ) shows a striking similarity to that of the neighboring even nucleus. j = / 2, (h/ 2 neutron)

21 Coriolis attenuation and cranking model Analysis with the particle-rotor model reveals that we need an additional factor to weaken the Coriolis interaction. H Coriolis ρ H Coriolis, ρ = 0.4 ~ 0.8 The recoil term j2 2ℑ ( β 0 ) H rec = Effect of the recoil term H rot + H Coriolis + H rec = = ω= R I i ℑ ℑ I is proportional to j2 K2 : K-dependence [ ] I 2 j2 R j 2ℑ ( β 0 ) ℑ [ ] I 2 j2 ω j 2ℑ ( β 0 ) ρ = i I The attenuation is important at low spin. The Cranking Model automatically takes account of the attenuation factor. H def part = i p i2 ( ) + V r ; β ω j 0 i 0 c 2M Uniform rotation ωc : c-number

22 Cranking model Picture in the rotating frame Time-dependent Schrödinger equation i Ψ (t ) = H Ψ (t ) t ωrot In the uniformly rotating frame with the rotational frequency ω Ψ (t ) = e iω t J Φ (t ) i Φ (t ) = ( H ω J ) Φ (t ) t Choose the rotational axis as x-axis: H ' H ω J -dim. cranking H ' = H ω rot J x 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 = e iπ α ± i, α = 2, r= 0 ±, α = Experimentally, often defined as α = I (mod 2)

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

24 Cranked shell model Single-particle motion in the rotating frame Cranked shell model Hamiltonian ' def h = i p i2 + V0 ( ri ; β 0 ) ωrot ( j x ) i 2 M The signature quantum number Eigenvalues of h ' ( ω rot ) ϕ = e' ( ω rot ) ϕ hdef As a function of ω called routhian diagram r = e iπ α r = ± i for α = 2 Alignment de ' i ϕ jx ϕ = dω rot Cranked Nilsson s.p. spectra (ε2=0.26) Solid (π,α)=(+,+/2) Dotted (π,α)=(+, /2) Dashed (π,α)=(,+/2) Dashdotted (π,α)=(+,+/2) signature splitting

25 Toward the high spin Inglis (954) introduced the cranking Hamiltonian to treat the cranking term as a perturbation (low-spin limit). I (ω rot ) = 2 ω 0 ω ph rot ℑ Inglis lim Φ n Jx Φ 2 0 En E0 Inglis moment of inertia Including the pairing effect, this is qualitatively consistent with experimental data. However, the cranking model becomes the most valuable tool to investigate the high-spin states: Back-bending phenomena Quasi-particle routhians Superdeformed rotational bands ℑ irr < ℑ exp < ℑ rig

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

27 Quasi-particle routhians Quasi-particle eigen-energies of the cranked HFB equation h λ N ω rot j x U U = e ' (h λ N ω rot j x ) V V Even nuclei: qp vacuum ground band 2qp excitation Odd nuclei: qp exc. QP routhians around 64Er Negative-energy states are fully occupied in the vacuum. E =0 e 'A (ω ) + eb' (ω ), e 'A (ω ), eb' (ω ),

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

29 Routhian analysis Bengtsson & Frauendorf, NPA327 (979) 39. Experimental quasi-particle routhians in odd rare-earth nuclei

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

31 Back-bending phenomena Nucleus loses its angular momentum by emitting gamma rays. The 58Er nucleus is spinning down, losing ΔI=2 each step. But, then, at I=6, it spins up, even though losing the angular momentum. Nuclear glitch ωrot This phenomenon was first discovered in 972 using the in-beam gamma-ray spectroscopy.

32 Structure change of the yrast states At low spin, the Cooper pair is condensed in the ground state. Φ HFB α k ck+ ck+ k N /2 0 Coriolis force At frequency ωc, one of the Cooper pair is broken up, due to the Coriolis anti-pairing effect. + + a A ab Φ Φ HFB a A+ ab+ Φ HFB HFB i 0 Without increasing ωrot, the state gains about 0 units of spin values, by the alignment.

33 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 (980), Springer-Verlag

34 Systematic analysis Systematic analysis of the back-bending frequency (by Garrett) ω cexp.67 jmax where jmax is the largest j near the Fermi surface. Simple picture in the right figure predicts ωc e Excitation energy of the 2qp state is roughly Δ E2 qp 2 i ~ 2 jmax E2 qp (ω c ) 0 jmax The Coriolis force pulls down the high-j quasi-particles, to realize a ωc state analogous to the gapless superconductor in condensed matter. This can be also regarded as the band crossing between the qp vacuum (0qp) and a 2qp state; g-band and s-band. ωrot

35 Band crossings γ-vibrational band also shows a back-bending behavior. Crossing between the collective γ-band and an aligned 2qp band

36 Pairing Phase Transition Mottelson & Valatin (960) predicted the pairing phase transition, from superconducting (low spin) to normal (high spin) phases. I ( I + ) I ( I + ) Epair + > 2ℑ super 2ℑ normal This is expected to occur around I=30~40 in the rare-earth region However, the quantal size effects (QSE) are important in nuclei: For instance, the size of the Cooper pair ξ0 = v F π k F ~ 0.7 fm, ξ 0 ~ 7 fm > R0 ; Epair ~ 2 MeV, ktc = 2 ~ MeV 3.54 Fluctuation is important: Beyond mean field 4qp The back-bending provides an example of stepwise phase transition (super to normal). s-band (2qp) g-band (0qp)

37 Microscopic structure of collective modes of excitations The ground state is assumed to be a Slater determinant. Φ 0 = c+ c A+ 0 In case of free particles (no interaction), excited states are Ω + ph Ω + pp 'hh ' + p h = c c, Φ ph = c +p c +p 'ch ch ', = Ω Φ + ph Φ pp 'hh ' Φ ph = Ω + ph Φ 0 = Ω + pp 'hh ' Φ 0 However, there is no collective state. All the excitations are individual, or not coherent. For the collective excitations, the (residual) interaction plays an essential role. 0

38 Problem : Prove the followings: The Hamiltonian H = H 0 + Vres, Vres = κ Q Q Suppose all the p-h excitations have the same excitation energy and that all the matrix elements among p-h states are the same: H0 Φ Φ p 'h ' ph = ε Φ Vres Φ ph ε = e p eh ph = v ( = κ q02 ) Diagonalize the Hamitonian matrix in the subspace of p-h states. Then, the following state has an eigenenergy of Φ coll = ph Φ N ph Ecoll = ε + N ph v ph and the all the rests have Enon -coll = ε If v<0, low-lying collective states, such as nuclear surface vibrations If v>0, high-lying collective states, such as (isovector) giant resonances

39 Problem 2: With the same Hamitonian H = H 0 + Vres, Vres = κ Q Q Suppose all the p-h states have the same matrix element of Q Φ ph QΦ 0 = q0 Then, show that the collective state Φ coll = ph Φ N ph ph has a transition matrix element: Φ coll and all the rests have zero transitions: QΦ Φ 2 0 = N ph q02 non -coll QΦ 2 0 = 0 The state is called collective because the state has the coherence to the operator Q. The collectivity should be defined by a certain (one-body) operator Q. The number of p-h states contained in the state is sometimes misleading. Φ non -coll = N ph / 2 ph Φ N ph ph N ph / 2 p 'h ' Φ N ph p 'h '

40 Φ coll = C ph Φ ph ph + C php 'h ' Φ php 'h ' + php 'h ' The random-phase approximation (RPA) includes a part of these higher-order terms Ω + n = {X ph n } ( ph)c +p ch + Yn ( ph)ch+ c p, Φ n = Ω + n Φ 0 This can be regarded as a harmonic approximation around the ground state. Φ ph = Ω + ph Φ 0

41 Collective states under rapid rotation At high spin, each p-h pair is aligned by the Coriolis force, to produce an At even higher spin, one of aligned phonon. the p-h pairs is completely aligned, by escaping from the λ-coupling. At low spin, many p-h pairs of spin λ contribute to the collective state. λ=even J=2jmax J=λ Jz=λ J=λJ z=k λ=odd J=jmaxj+j max Φ Φ coll α ph i 0 ph ( c +p ch λ K Φ ) 0 aligned coll α ph ph (c c ) + p h λλ Φ 0 i 5 ~ 0 a A+ ab+ Φ HFB

42 Band crossing between collective bands and non-collective 2qp bands E 2Δ E2 qp 2 The lowest mode of excitation: Ecor 2 Ecoll Collective surface vibration aligned 2qp band Ecoll < 2 2 for quadrupole icoll λ = 3 for octupole i2 qp 2 jmax ~ 0 ωc ωc E cor 2 Ecoll ~ i2qp iλ i2 qp λ ωrot

43 Quasi-particle random-phase approximation in the rotating frame Marshalek, NPA266 (976) 37 Shimizu & Matsuyanagi, Prog. Theor. Phys. 79 (983) 44 T.N., Matsuyanagi, Mizutori, Shimizu, PRC53 (996) 223 Collective vibrational states are calculated with the QRPA based on the cranked shell model H = hdef + Γ pair ω H res = rot J x + H res '' '' κ Q Q λ λ λ 2 λ = 2,3 [H, Ω ] + n QRPA = ω n Ω + n

44 Octupole vibrations in 64Yb T.N., Act. Phys. Pol. B27 996) 59 K=0 K=2 K= K n Q3 K > 200 fm > 00 fm 3 < 00 fm 3

45 Octupole vibrations in 238U T.N., Act. Phys. Pol. B27 996) 59 K= K=0 K n Q3 K > 300 fm > 50 fm 3 < 50 fm 3

46 In 986, the first superdeformed band was discovered in 52Dy. γ I+6 I+4 I+2 I UK P.J. Twin et al, PRL 57 (986) 8 J.D. Garret et al, Nature 323 (986) 395. Large moment of inertia Large intraband B(E2) B(E2) 2000 W.u. Major : Minor axes ~ 2:

47 We have studied elementary modes of excitation in SD states. S.Mizutori, Y.R.Shimizu, K.Matsuyanagi, Prog. Theor. Phys. 83 (990) 666; 85 (99) 559; 86 (99) 3. TN, SM, YRS, KM, Prog. Theor. Phys. 87 (992) 607. TN, KM, SM, W.Nazarewicz, Phys. Lett. B 343 (995) 9. TN, KM, SM, YRS, Phys. Rev. C 53 (996) 223. Single-particle orbitals with different Nosc Quadrupole collectivity is weak. Octupole modes of excitation are dominant in low-lying spectra.

48 Excitation energy in the rotating frame relative to the ground-state SD band. RPA routhians for SD 52Dy RPA in the rotating shell model predicted an excited SD band in 52Dy is the K=0 octupole vibrational band. T.N., Mizutori, Matsuyanagi, Nazarewicz, PLB343 (995) 9 Octupole band with K=0 (Y30) MOI Exp: Dagnall et al, PLB 335 (994)

49 Determination of spin and parity Spin and parity of SD bands have been determined by observation of decay gamma rays. 94 Hg, 92,94Pb, 52Dy, etc. Lauritsen, PRL 88 (2002) 04250

50 Ex SD6 SD Decay I Exp RPA B(E) 0-4 W.u.

51 In A=90 region, we assign most of excited SD bands as Y32 octupole vibrational bands. QRPA in the rotating shell model predicts the K=2 (Y32) octupole vibration as the lowest excitation modes for even-even SD nuclei. This assignment systematically reproduces moments of inertia. TN, KM, SM, YRS, Phys. Rev. C 53 (996) 223

52 Confirmed by experiments Linking transitions B(E) 0-5 W.u.

53 Octupole dominance has been confirmed by many experiments. 994 Crowell et al (PLB 333, 320; PRC 5, R599) 996 Wilson et al (PRC 54, 559) 997 Hackman et al (PRL 79, 400) 997 Amro et al (PLB 43, 5) 997 Bouneau et al (ZPA 358, 79) 200 Rossbach et al (PLB 53, 9) 200 Korichi et al (PRL 86, 2746) 200 Prévost et al (EPJA 0, 3) 2002 Lauritsen et al (PRL 89, 04250)

54 Strong octupole correlation is established in SD states. Then, what else can we expect to see? Freq. ratio Fission isomer A 90 Single-particle energy levels A 5 0 If we can extend the exploring area of SD states with radioactive beam, deformation Anything new in open-shell SD states?

55 Shape phase transition phenomena (sph. to def.) for open-shell nuclei are well known in the ground states. spherical transitional deformed Effective potential V(β) Increasing neutrons β Quadrupole deformation (order parameter) Spontaneous symmetry breaking (SSB)

56 What happens if we add neutrons to closed-shell SD nuclei? Effective potential V(β) (I+7) (I+6) + (I+5) (I+4) + (I+2) + β I + SD 52Dy What kind of parameter? (I+3) (I+) (I+6)+ (I+4)+ (I+2)+ I+ 62 Dy

57 Qualitative analysis with harmonic oscillator potential T.N., S.M., K.M., Prog. Theor. Phys. 87 (992) 607. Freq. ratio Possible p-h excitations Spherical nuclei unocc occ Single-particle energy levels r3y30 r3y3 r3y32 r3y33 K= (Y3) mode of excitation is analogous to quadrupole modes in spherical nuclei. Driving to shape transition in open-shell configuration. deformation cf) Bohr & Mottelson s textbook (975). r2y20,22

58 Banana-(Y3-type) shape phase transition in open-shell SD states T.N., S.M., K.M., Prog. Theor. Phys. 87 (992) 607. Increase of valence nucleons Z=80+2, N=80+Nval HO+QRPA Strong pairing Weak pairing p=.3 MeV p=. MeV Banana-superdeformation SSB towards banana shape

59 More realistic calculation Nilsson+BCS+QRPA Ex [ MeV ] Superdeformed Dy isotopes ε=0.59, Δ=0.5 MeV, κ=.05κho A Shape transition to banana-super-deformation

60 Where are they? Increasing (decreasing) valence neutrons (protons) by 8-0 leads to regions near beta-stable line A 90 A Dy Fusion reaction with radioactive beam might populate these high-spin SD states near beta-stable line.