One-particle motion in nuclear many-body problem

Transcription

1 RIKEN Dec., 008 (expecting experimentalists as an audience) One-particle motion in nuclear many-body problem (The 3rd lecture, V.3) Giant resonances (GR) and sum rules in stable and unstable nuclei Iuo Hamamoto Division of Mathematical Physics, LTH, University of Lund, Sweden The figures with figure-numbers but without reference, are taen from the basic reference :.Bohr and B.R.Mottelson, Nuclear Structure, Vol. I & II

2 Haraeh &van der Woude, Giant Resonances, Oxford, 001 When IVGDR was found in photo-neutron cross sections, it had a resonance shape, but the width was typically of the order of 5 MeV, which was an order of magnitude larger than the resonances nown in nuclei at that time (~ 1960 ies). Thus, it was called Giant Resonance. Photon energy resolution = several hundreds (< 500) ev.

3 The hydrodynamical model consists of incompressible neutron and proton fluids. Nuclei consist of nucleons, and the population of GR by, for example, γ-absorption is via one-particle operator. Thus, the presence of quantum-mechanical shell-structure of one-particle levels in nuclei sets a limitation on the applicability of the hydrodynamical model. If a collective mode consumes an appropriate sum-rule Possibility of being approximately described by a macroscopic hydrodynamical model Taing the simplest model for nuclei, namely harmonic-oscillator model, the above possibility exists if all one-particle excitations by a given operator have the same energy, one-frequency. ex. IVGDR is this example ; all one-particle excitations have N=1, GQR, since N=0 and. E = 1 ω 0 The hydrodynamical model is not directly applicable to spin-dependent modes.

4 GRs in heavier nuclei have often a better resonance shape than those in lighter nuclei. This may be due to the fact that in heavier nuclei ; (a) GR can be more collective, since many more 1p-1h configurations are available. (b) The spread of energies of 1p-1h excitations ( ~ 1/3 ) is smaller, while the p-h interaction which couples 1p-1h excitations with different energies has no strong -dependence. Thus, building up a collective state out of available 1p-1h configurations is easier. Lighter nuclei have less clear distinction between the surface and the inside. This maes a difference, for example, when a probe used is sensitive only to the surface or GR is of a surface type.

5 Neutron-excess gives an essential difference in charge-exchange GR, t ± GR, from the case of N=Z nuclei. ex. Some t + GR may disappear due to the Pauli principle. ex. E x (t GR) > E x (t + GR) in the presence of neutron excess. Neutron excess Excitations made by Isoscalar (i.e. isospin-independent) operators carry an isovector transition density When neutrons and protons move in the same way in nuclei with N > Z, δρ n δρ p 0.

6 Giant resonances and sum rules 7.1. Introduction 7.. Sum rules Sum rules for (1 or t z ) excitations Classical oscillator sum (= energy-weighted sum) Sum-rule in axially-symmetric quadrupole-deformed nuclei 7... Sum rules for (t ± ) charge-exchange excitations Difference, S S Giant resonances of IS or tz, of non-energy-weighted sums Isovector giant dipole resonance (IVGDR) type (excitations within the same nuclei) Isoscalar and isovector giant quadrupole resonance (ISGQR and IVGQR) Isoscalar giant monopole resonance (ISGMR) - compression mode

7 7.4. Giant resonances of charge-exchange (n p or p n ) type (excitations to the neighboring nuclei) Fermi transitions (IS) Gamow-Teller (GT) resonance (incl. magnetic giant dipole resonance) Isovector spin giant monopole resonance (IVSGMR) Isovector spin giant dipole resonance (IVSGDR) 7.5. Giant resonances in nuclei far away from the stability line ISGQR of nuclei with wealy-bound neutrons - an example of threshold strength β-decay to GTGR in drip line nuclei β decay to GTGR β + decay to GTGR + in very neutron-rich light nuclei in medium-heavy (N>Z) proton-drip-line nuclei References: M.N.Haraeh and.vander Woude, Giant Resonances, 001, Oxford.

8 7. Collective motion based on particle-hole excitations - giant resonances and sum-rules 7.1. Introduction Collective motion : Many nucleons participate coherently in the motion so that a given observable (transition) is much enhanced compared with a single-particle estimate. The best-established collective motion in nuclei is rotational motion of deformed nuclei. The properties of very low-energy collective states are sensitive both to pair correlations and to the shell-structure around the Fermi levels. Only those particles close to the Fermi levels contribute to the pair correlation. In contrast, many (if not all) particles in a nucleus participate in giant resonances (GR), so that (a) the properties of GR are almost independent of the shell-structure around the Fermi level, (b) depend on the bul properties, and (c) are expressed as a smooth function of Z, N and.

9 The total transition strength should be limited by a sum rule, which depends on the ground-state properties. Due to the collective nature, GR consumes the major part of the sum rule that is defined for respective collectivity. Then, GR may correspond to a classical picture of collective motion. Usefulness of sum-rules If an observed pea consumes the major part of the sum-rule, the pea expresses a collective mode. Moreover, there are almost no other collective excitations carrying the strength of the same operator F, while the mode created with the operator acting on the ground state is approximately an eigenstate of the Hamiltonian.

10 Examples of Giant Resonances experimentally studied in β-stable nuclei are (a) Excitations in the same nuclei (IS = Isoscalar, IV = Isovector) IS GMR* 0+ IV GQR + IV spin GR 1+ spin-parity operator observed pea energy τ r 80-1/3 MeV (for > 90) 3 IS GDR* 1 r Y1 µ ( rˆ IV GDR 1 τ ( ) ) 79-1/3 MeV (for > 50) z r IS GQR + r Y ( ˆ ) 63-1/3 MeV (for > 60) (b) Excitations to neighboring nuclei IS 0+ GT GR 1+ IV GQR + IV spin GMR* 1+ IV spin GDR 0, 1, µ r z( ) r Y µ ( rˆ ) τ z ( ) σ spin-parity operator t ± ( ) t ± ( ) σ ± ( ) r Y µ ( rˆ t ) t ± ( ) σ ( r r ) excess t ) r ( Y ( ˆ )σ ) ± ( 1 r GRs have width of several MeV (except IS) and exhaust the major part of respective sum-rule. Jπ * compression mode

11 Examples of selection rules in spherically-symmetric harmonic-oscillator potential 0 1) Operator ry 1µ (or x, y, z) N=1 (E x = N F + 1 ω ) excitations N F + 1 N F N F N F 1 Closed-shell configuration Partially-occupied N F shell 0 ω0 ) Operator r Y µ (or x, y, z ) N=0 or (E x = or ω0 ) excitations N F + N F + 1 N F + N F + 1 N F N F N F 1 Closed-shell configuration N F 1 N F Partially-occupied N F shell In realistic potentials the above selection rules do not exactly wor, but wor approximately.

12 Observed one-particle energies are not well reproduced by Hartree-Foc calculations using Syrme interactions with m* (= ( ) m ). In contrast, observed energy of ISGQR are often reproduced by RP based on the Hartree-Foc calculation with the same Syrme interaction (so-called self-consistent RP). Note that the parameters related to ISGQR are well taen care of, when Syrme parameters are determined. In this lecture we do not further go into detail of [ Syrme H.F. + RP ] calculation. Instead, we try to understand GRs, sometimes using the result of [ Syrme H.F. + RP ] calculation, but mostly using the models which are as simple as possible.

13 { Shape oscillations - typical vibrational excitations when nuclear matter is incompressible. Compression modes information on nuclear compressibility proton neutron n0 0 = 1 tr tr ρ ( r) n δ ( r r ) ( r) Y ( rˆ ) ρ λ λµ radial transition density IS - compression mode G.F.Bertsch and S.F.Tsai Physics Reports 18, (1975) 15. Tassie model ( ) dρ ( r) ρ 0( r) + r dr 3 0 IS - shape oscillation dρ r) r 0 ( dr

14 In heavier nuclei GR may show a resonance (Lorentzian?) shape and the properties can be systematic, while those of GR in medium weight and light nuclei are more individual. In very light nuclei GR strength distribution is split into several fragments. ) In lighter nuclei the collectivity is weaer, or a number of p-h configurations to contribute to GR is smaller. In lighter nuclei the difference of the relevant p-h excitation energies may be large compared with the interaction between them, Transition densities of GR with good accuracy is not experimentally available. Example of transition density of IS shape oscillation ; 3 state of 08 Pb at Ex =.61 MeV Experimental data are taen from (e, e ) in J.Heisenberg and I.Sic, P.L. 3B (1970) 49. I.H., P.L. 66B (1977) 410

15 7.. Sum rules In this lecture we treat nucleons as elementary particles, neglecting possible contributions from internal degree of freedom of nucleons. Valid in the energy interval, well below internal excitations of nucleons Sum rules for (1 or t z ) excitations Classical oscillator sum - sum of energy-weighted transition strength Then, S( Fλ ) ( Ea E0) B( Fλ ;0 aia ) a = 1 0 F, [ H, F ] 0 λ λ where H = t i + v, ij H 0 = E0 0, H a = Ea a i i< j v, ij F λ = 0 i< j Thus, = ( ) 0 a Ea E0 a F λ If v ij does not explicitly depend on the momentum of particles, vij( p ) if one-particle operator Fλ = Fλ( r, and F ) λ depends only on r SF ( λ) = 0 ( F( r λ ) ) 0 m [, ] H F λ, i ti F λ Note that the sum is expressed as a ground-state expectation value of one-body operator - insensitive to the many-body correlation in the ground state model independent. Sums with other energy weightings involve two- or many-body operators. }

16 In particular, if F = f ( r) Y ( rˆ λµ λµ ), we obtain + df f λ 1 S( Fλ ) class = + λλ ( + 1) 4π m dr r (7.1) where expresses the average per particle in the ground state of particles. For Eλ transitions with λ, motion, λ F er Y () rˆ λµ λµ neglecting the correction due to the center of mass only for protons, then, S( Eλ) class = λ(λ + 1) 4π m Ze r λ proton radial average for protons in the ground state For E0 transitions, S( E0) class Fλµ er = m Ze r only for protons, then, proton

17 For E1 transitions ; Since isoscalar dipole operator corresponds to the center of mass motion that must not create an excitation, the dipole operator which creates excitations is necessarily of isovector character. For example, electric-dipole excitation operator (in the direction of z-axis) should be ) ( p i z i e + ) ( ) ( ) ( 1 p i p j n j i z z z e + ) ( ) ( ) ( p i p j n j i z z Z e z e = ) ( ) ( p i n i z e Z z e N = spin-parity = 1 where (p) and (n) express the sum over protons and neutrons. class E S ) 1 ( µ µ µ a p i n i a ry e Z ry e N a E E ) ( ) ( ) ( ) ( ) ( Z N N Z e m π NZ e m 4 9 π = = =, Then, using (7.1), the classical oscillator sum, which should be the sum rule for IVGDR, when the interaction has, and p ) ( τ τ ) ( σ σ NZ e m Ze m π π = class p E S ery F S ) 1 ( ) ) ( ( 1 = λ 4 9 e Z m π = ; oscillator strength associated with the center of mass motion

18 ex. center of mass motion for E operator Total E moment measured with respect to the center of mass [ ] = c c c Y y X x Z z e 1 ) ( ) ( ) ( where = c z Z 1 1, etc. and = e e for proton { 0 for neutron [ ] = = y x z e 1 = = = + + c c c y e Y x e X z e Z = + c c c Y X Z e 1 ) ( Ze ( ) 1 1 y x z Z e e + = = ( ) < + + j j j j j y y x x z z Ze e e 1 (a) For a single-particle configuration e + 1 Z e Z e for proton for neutron { (b) For harmonic oscillator wave-functions and low-energy transitions ; matrix elements of (a) receive no contributions from the recoil term, namely, the correction term in (b) is exactly cancelled by the -body term in (a).

19 Distribution of S(E) in axially-symmetric deformed nuclei (1) Low-energy IS ( r Y ν ) excitations ( N 0) rotational excitation (ν = ±1 ) gamma-vibration (ν = ± ) beta-vibration (ν = 0) {(one-particle excitations) a) rotation excitation main E oscillator strength in the low-energy region { For even-even nuclei 3 I 5 16π ( E E0) B( E; K = 0, I = 0 K = 0, I = ) = e Q0 3 5 E( K0, I ) E( K0, I0) B( E; K0I0 K0I ) = e Q I 16π For odd- nuclei [ ] I 0 Observed moments of inertia S ( E) (0.05) S( E) Iobs 5Iirrot rot class comparable to low-energy + mode in spherical nuclei b) gamma vibration ( ry ± type surface vibration) taes less than a few % of S ( E) class beta vibration ( r Y0 Type surface vibration) taes less than (0.01) S ( E) class

20

21 () High-energy ( N ) excitations E strength will split depending on the quantum number ν of Y ν. The E strength of GR with ν = 0, ±1, ± is expected to be approximately 1 : : Ex Ex prolate shape oblate shape OBS. L-S doubly-closed spherical nuclei (such as 40-Ca) have only high-energy ( N ) collective quadrupole excitations. Spherical vibrating nuclei have both low- and high-energy collective quadrupole excitations. The low-energy ( N 0) IS quadrupole modes have enhanced E transitions due to the attractive quadrupole interaction, but carry less than (0.10) S(E) class.

22 ex. an extra contribution to S(E1) from an exchange interaction Increase of energy-weighted sum-rules, S(E1), from S(E1) class due to the presence of attractive Majorana space-exchange interaction ex. proton-neutron pair with -body space-exchange ( r i r j ) interaction v = f( r r ) P ij i j M P M : Majorana space-exchange operator M P = M M M [ v, z ] fp z z fp = ( z z ) fp ij i + ( τ τ ) 1+ ( σ σ ) 1 i j i j = z i : proton coordinate i i j i z j : neutron coordinate [ [ ] [ ] M z, v, z = z,( z z ) fp i ij i = = i j i M z i ( z j zi ) fp ( z j zi ) M z i ( z j zi ) fp ( z j zi ) = ( z z i j ) fp M 1 M = ( r ij ) fp 3 n attractive Majorana interaction maes an extra contribution to S(E1). z fp j M fp z M i

23 7... Sum rules for ( t ± ) charge-exchange excitations There is no sum-rule, which corresponds to the classical oscillator strength for IS operators. Instead, model-independent and non-energy weighted sum-rules, for the difference between t and t + transitions. Isospin of nucleon, t n = p t = t z n = n t z p = p t + p = n t = 0 t p = 0 + n t ± t x ± ity t + ( tz( ) N, Z tz( i) N, Z = N Z j i ), t ( j) = 3 σ µ = 1 µ 3 4 s µ 4( s 1 1 ( = = ) = 4 +1 ) = 3 µ = 1 : Basic formulas (a) charge-exchange non-spin-flip excitations : Oˆ ± = t ± ( ) f ( r ) m O 0 n O+ m ˆ n ˆ 0 = N ( f ( r)) Z n ( f ( r)) p (b) charge-exchange spin-dependent excitations : m O 0 n Oˆ + m O ˆ t ( ) ( ) f ( ) ± = ± ˆ 0 = 3 N ( f ( r) ) Z ( f ( r) ) n n µ p σ µ r

24 7.3. Giant resonance of isoscalar (IS) or t z type Isovector giant dipole resonance (IVGDR) : the oldest and best nown Giant Resonance Systematics of observed IVGDR frequency Observed 79-1/3 MeV is in good agreement with the value estimated in the harmonic-oscillator model. For light nuclei with < 50 a deviation from the systematics is observed. --- Other types of GR show the same tendency. Note that unpertrubed I π =1 p-h energies in realistic potentials are approximately degenerate and close to 41-1/3 MeV (!) also in drip-line nuclei! Fig of.bohr & B.R.Mottelson, Nuclear Structure, vol.ii

25 Well-established IVGDR observed pea(s) in photo absorption cross section Photo absorption cross section of 197 u Figs.6-18 and 6-6 of.bohr & B.R.Mottelson, Nuclear Structure, vol.ii dditional dipole strength is observed on the high-energy side, that appears to be associated with short-range (velocity-dependent) correlations between nucleons. 140MeV σ 0 photoabs γ pion threshold de S( E1) class Photo absorption cross section of 16 O 5MeV σ photoabsdeγ S(E1) class 0 where 9 NZ S( E1) class = e 4π M S(E1) obs = S(E1) class (1 + x) x comes from the velocity- and ( τ τ ) - dependent terms in the nucleonic interactions.

26 IVGDR is the giant resonance, of which the semi-classical picture is possible. Goldhaber-Teller model Steinwedel-Jensen model n p Neutron and proton fluids are oscillating within a sphere, eeping the total density constant. For simplicity, assuming [IVGDR ~ a standing wave in a nucleus with a fixed boundaries], the frequency ω R -1 Then, in contrast to one-pea structure in spherical nuclei, in axially-symmetric quadrupole-deformed nuclei ωz 1 R z N=90 ω 1 R the strength distribution would have two peas corresponding to an oscillation of neutrons vs protons along the long and short axes, as observed in 150 Nd 90. N=8

27 For a prolate (oblate) shape the integrated cross section associated with the vibration along the symmetry axis (= longer (shorter) axis), which has lower (higher) frequency, should be about a half of the one along two shorter (longer) axes. Ex Ex prolate shape oblate shape The energy splitting is proportional to the deformation ω ωz ω R δ R δ Thus, the ground state of 150 Nd 90 is prolately deformed!

28 Harmonic oscillator potential N x, y, z N 0 f i only for N f = N i ±1 one-particle energy N i + 1 N i ω 0 unoccupied occupied ll excitations are from the last-filled major shell to the next major shell, with excitation energies, E= ω0 41 1/3 MeV. Many degenerate particle-hole (p-h) excitations, especially in heavier nuclei. schematic model : (for IVGDR) G.E.Brown, Unified Theory of Nuclear Models, p.9-3 and p a collective state many degenerate p-h excitations ground state ground state fter including a separable repulsive interaction between the p-h excitations, only one collective state is pushed up, while all other states remain at the unperturbed excitation-energy, and the collective state absorbs all transition strength, if one taes separable interaction relevant transition operator

29 Taing the strength of IV dipole-dipole interaction from the symmetry term of the phenomenological nuclear one-body potential, in the harmonic oscillator potential model we obtain unperturbed p-h energy, ω 0 = 41 1/3 MeV 1/3 80 MeV for the excitation energy of the collective state, (= IVGDR) in agreement with the observed systematics in medium-heavy nuclei. [ ] 3 p-h energies, 41 1/ MeV collective IVGDR at due to the repulsive p-h interaction 80 1/ 3 MeV, which consumes the major part of E1 strength. means, e p eff ( E1) and ( E1) for low-energy E1 transitions are much smaller e n eff than the values of (N/) e and (Z/) e, respectively (see Sect. 6.1).

30 In the self-consistent calculations plus RP for spherical nuclei, the strength of IVGDR is split into several peas even for heavier nuclei. The transition density of lower-lying peas appears to be closer to the Steinwedel-Jensen prediction, while that of higher-lying peas loos more lie the Goldhaber-Teller one. Radial transition density : Goldhaber-Teller model Steinwedel-Jensen model ρ ρ tr SW r) dρ ( r) dr tr 0 GT ( ( r) rρ0( r) Examples of self-consistent calculations plus RP ( IVGDR ISGDR ) Ca0 8 Pb16 I.H., H.Sagawa and X.Z.Zhang, PRC 57, R1064 (1998)

31 ex. Pigmy dipole resonances observed at much lower energy than IVGDR of the 140 region consume less than 1 % of S(E1) class. obtained from by folding with a Lorentzian with a width of 500 ev ( from. Zilges, 007 )

32 7.3.. Isoscalar and isovector giant quadrupole (ISGQR and IVGQR) resonance Operator spin-parity observed pea energy ISGQR ry ( ) /3 MeV ˆ µ r τ + ˆ z( ry ) µ ( r) IVGQR (130-1/3 MeV?) schematic model : (for ISGQR) many degenerate p-h excitations a collective state ground state ground state fter including a separable attractive interaction between the p-h excitations, only one collective state is pushed down, while all other states remain at the unperturbed excitation-energy, and the collective state obtains all transition strength, if one taes separable interaction relevant transition operator

33 Taing the strength of IS quadrupole-quadrupole interaction from the self-consistent condition that the eccentricity of the potential is the same as that of the density, in the harmonic oscillator potential model we obtain.bohr and B.R.Mottelson, Nuclear Structure, vol.ii, p.509 unperturbed p-h energy, ω 0 = 8-1/3 MeV ω 0 = 58-1/3 MeV for the excitation energy of the collective state (= ISGQR) In stable nuclei the estimate based on the above h-o potential model wors well, because most one-particle levels in the major shell (N i + ) one-particle energy are narrow resonances in realistic potentials. 0 N i + N i + 1 N i unbound levels } bound levels unoccupied occupied in stable nuclei 7 to 10 MeV In the schematic harmonic oscillator model N x, y, z f N i 0 N i -1 only for N f = N i ± or N f = N i

34 Using (7.1), 50 4π m λ class = r Energy Weighted Sum Rule (EWSR) S ( IS, = ) The classical sum-rule for IS giant resonances should wor when the interaction is p In the harmonic-oscillator potential model, N x, y, z 0 only for N f =N and N i f = N i ± f N i Thus, in contrast to IVGDR, the quadrupole operator has N N matrix elements, in addition to N N+ matrix elements. nd, the IS (attractive) coupling between the two inds of modes, N = 0 and, shifts some transition strength to lower-energy modes. In open-shell nuclei the N N transitions are possible within the last filled major shell, while in medium-heavy nuclei the transitions are present even in the closed-shell nuclei, due to the presence of the spin-orbit splitting. For example, in the doubly-closed shell nucleus 8 Pb 16 one finds 4 low-energy excitations; proton-excitations, 1h 11/ 1h 9/, f 7/ and neutron-excitations, 1i 13/ 1i 11/, g 9/. Nevertheless, since the sum-rule considered here is the energy-weighted sum-rule, the observed IS lower-lying quadrupole excitations exhaust only up till 15 percent of EWSR.

35 Some summary of the observed properties of ISGQR of medium-heavy nuclei M.N.Haraeh &. van der Woude, Giant Resonances, 001 Observation of the IS giant quadrupole resonance (ISGQR) - one of the first observations of a giant resonance other than the well-nown IVGDR R.Pitthan, Z. Phys. 60 (1973) 83

36 Experimental information on isovector giant quadrupole resonance (IVGQR) is very limited. The reason for this can be ; (a) Due to the high frequency mode, large bacground and possible overlap with many other excitations; (b) Large width and relatively small excitation cross section; (c) Lac of a selective experimental tool to excite IVGQR Some experimental evidence : D.Sims et al., Phys.Rev.C55 (1997) 188; interference (E1/E) effects in reactions involving photons. T.Ichihara et al., Phys.Rev.Lett. 89 (00) 14501; 60 Ni ( 13 C, 13 N) 60 Co reaction For reference, the result of a self-consistent HF+RP calculation is shown below. 0 Ca 0 is a stable nucleus, while 0 Ca 40 is possibly a neutron-drip-line nucleus. In both nuclei ISGQR appears as a clean collective pea, while IVGQR spreads over several peas with varying form factors. The threshold strength in 60 Ca comes from the presence of wealy-bound neutrons in the ground state, which are not present in stable nuclei. I.Hamamoto, H.Sagawa and X.Z.Zhang, Nucl.Phys. 66 (1997) 669. threshold strength

37 [ ] 8 1/ p-h energies, 3 3 MeV collective ISGQR at 58 1/ due to the attractive p-h interaction MeV, which consumes the major part of IS quadrupole strength. means ; ISGQR maes a considerable amount of positive contribution to (E) e pol of low-energy E transitions.

38 Isoscalar giant monopole resonance (ISGMR) - compression mode In 08 Pb, observed ISGMR ( E ISGMR 14 MeV, Γ 3 MeV ) exhausts about 100 % of the energy-weighted sum rule. Observed properties of ISGMR D.H.Youngblood, H.L.Clar and Y.W.Lui, RIKEN Review No.3 (July, 1999) 159.

39 Examples of experimental data of ISGMR S.Shlomo and D.H.Youngblood, PRC 47, 59 (1993)

40 Measured energy of ISGMR (= breathing mode ), E ISGMR, information on the compressibility of nuclear matter ( K nm ). Nuclear compressibility is an important information on the equation of state of nuclear matter. ex. shape of the density distribution, values of the radii, the strength of shoc wave following the collapse of supernovae, etc. However, the relation, E ISGMR K nm is model-dependent! K nm is defined by K nm d ( E / ) dρ 9ρ 0 ρ = ρ n effective compression modulus, K, for a nucleus with mass number in terms of E ISGMR () is defined by m( EISGMR( )) r m K = where r m is the mean-square mass radius. Writing 1/ 3 N Z Z K = Kvol + Ksurf + Ksym + KCoul + 4 / 3 ($) { lim K = K vol = Knm lim K = (7 /10)K nm from a Hartree-Foc calculation with a constraint on the r.m.s. radius. 0 if the mode corresponds to a radial scaling of the ground-state density.

41 Moreover, various K i values in ($) are poorly determined, since the variations of K with N and Z are very small for available nuclei. Thus, some experts state (for example, Blaizot et al., NP 591 (1995) 435) : Phenomenological expansion ($) using measured E ISGMR () values cannot be used to obtain K nm. Microscopic calculations remain the most reliable tool for determining K nm from measured E ISGMR () values. K nm = 10 ± 30 MeV

42 Comparison of calculated ISGMR using self-consistent Hartree-Foc calculations plus RP with various Syrme interactions, which have different K nm values. K nm = 17, 56 and 355 MeV for SM*, SGI and SIII, respectively. I.H., H.Sagawa & X.Z.Zhang, PRC 56, 311 (1997) 40 Ca Pb 8 16 Calculated ISGMR in medium weight and light nuclei usually does not have a clean one-pea shape. Calculated ISGMR in heavy nuclei is obtained as a well defined resonance and exhausts the sum rule. (In the above calculation the particle decay width of GR is fully taen into account, while the spreading width, coming from the coupling to p-h configurations, is not included.)

43 The effective charge of E0 transitions, really been studied. eeff ( E0), for low-energy E0 transitions has not In heavier nuclei self-consistent calculations plus RP produce a relatively clean resonance pea. Nevertheless, the calculated pea energy is not so different from averaged unperturbed p-h energies in the potential based on harmonic-oscillator. Calculated values of E0 polarization charge, e ( E0) transitions due to ISGMR may not be large and may depend sensitively on the models and parameters used. n 1 ex. recent information on eeff ( E0) from the data on 8 pol 4 Be, for low-energy E0 S.Shimoura et al., Phys. Lett. B654 (007) 87; I.H. and S.Shimoura, J. of Phys. G34 (007) Measured partial life + + ( E0) 01 + τ ( g. s.) 0 e n eff r = 0.87 e fm = 40 ± 16 ns + 01 (E0) = (E0) = e e n eff e n pol The presence of wealy-bound neutrons in the deformed potential is duly taen into account. OBS. The polarization charge for E0 transitions obtained from subtracting the center of mass motion is analogous to that of E transitions described in Sect and is e n eff (E0) = ( Z / ) e 1 = (0.08) e for Be

44 Comparison of IS and IVGR in 08 Pb calculated by self-consistent Hartree-Foc plus RP using SKM* interaction. Particle decay width is fully taen into account, though spreading width coming from the coupling to p-h configurations is not included. I.Hamamoto., H.sagawa and X.Z.Zhang, J.Phys.G 4 (1998)1417. r r 3 Y 1µ r Y µ

45 7.4. Giant Resonances of charge-exchange type ( T z = ±1 ) Various isospin states, which can be excited by acting an isovector excitation operator on a nucleus with T = T z 0. Excitation strengths [ C(T 0 1 T ; T 0, T z, T 0 + T z ) ] T (T 0 + 1) (T 0 + 1) T T T 0 1 T T T 0 T 0 T 0 1 T T 0 1 T T z = +1 T z = 0 T z = 1 T 0 Z N Z 1, N+1 Z+1, N 1 N Z N Z N Z T z = + 1 T 0 = T z = T z = 1

46 E X (t + GR) < E X (t GR) in the presence of neutron excess p-h excitation energy E X measured from the ground state of mother nuclei t + E 0 t - E 0 E E Z N Z N E X (t + )= E 0 E E X (t )= E 0 + E { E 0 ~ α ħω 0 ~ 1/3 E ~ /3 for neutron-excess excess in stable neutron-rich rich nuclei E X (t + ) becomes monotonically smaller as larger. This relation, E X (t + GR) < E X (t GR), is present in all charge-exchange (t ± ) GRs. In N>Z nuclei towards neutron-drip-line E x (t + GR) << E x (t GR) E x in respective final nuclei

47 Some expected features unique in spin-isospin ( σ µ t ± ) Giant Resonances 1) [ t ± σ µ ] Not almost all strength under the GR pea. Instead, a considerable amount of high-energy tail above the pea is expected, with the tensor correlation responsible for the highest energy components. Dependence of the high-energy tail on respective GRs? ) [ σ µ ] Relatively large width (or large spread) of GR ) a) Unperturbed 1p-1h excitations have already an energy spread of E s where the spin-orbit splitting of high-j orbit is expressed by except for GTGR and some IVSGDR where the spread b) Due to the same sign of the couplings to a particle and a hole; the coupling of 1p-1h to p-h configurations is strong, in contrast to spin-independent modes. E E s s ( 7-9 MeV), and

48 Examples of charge-exchange Giant Resonances studied in β-stable nuclei spin-isospin modes * compression mode spin-parity operator Oˆ ( ± ) IS 0+ GT GR 1+ IV GQR + IV spin GMR* 1+ IV spin GDR 0, 1, t ± ( ) t ± ( ) σ ± ( ) r Y µ ( rˆ t ) t ± ( ) σ ( r r ) excess t ) r ( Y ( ˆ )σ ) ± ( 1 r Jπ Direct and systematic experimental data are available only for IS and GTGR. IS = Isobaric nalogue State IV = IsoVector GMR = Giant Monopole Resonance GDR = Giant Dipole Resonance

49 t ± ( ) σ ) t± llowed β decay; F ± Fermi transitions : ( ( ) GT ± 1, j = ± (, j) p Gamow-Teller transitions : p, j = ± 1 n and (, j) n, j = 1 n Isospin of nucleon, t n = p t = t z n = n t z p = p t + p = n t = 0 t p = 0 + n 40 ex. In the L-S doubly-closed-shell N=Z nucleus, Ca 0 0, one expects t ( ) gr 0 ) t ( ) gr ± σ µ ( ± and 0 F ± operators are raising and lowering operators of the z-component of total isospin (T) without changing the total isospin, T=0 ; t ± ) = T± ( T T, T = T, T ± 1 In particular, T T = T = 0 = 0 ± z z ± z GT ± operators may change the total isospin, T = 1, 0, +1, but T=0 T=0

50 β-decay can populate only the states with Ex Q β± in daughter nuclei. g.s. β ± Q β± That means, in β-stable nuclei β-decays of ground states can populate only the low-energy tail of GTGR in daughter nuclei. Thus, those β-decays are considerably hindered. g.s. mother nucleus daughter nucleus In contrast, using charge-exchange reactions on mother nuclei, (p, n), ( 3 He, t) for t (n p in target nuclei) (n, p), (d, He), (t, 3 He) for t + (p n in target nuclei) the response is obtained up till high excitation energy in daughter nuclei. The price which one must pay is ; the analysis of data to obtain nuclear matrix elements is much more complicated than in β decays. In those charge-exchange reactions, Gamow-Teller Giant Resonance (GTGR) was found!

51 Fermi transitions ; (F ± = ) O ˆ ± = t ± ( ) spin-parity of the operator = 0 + m moˆ 0 noˆ 0 + n ( S S + ) = (N Z) ex. For N>Z The sum rule for Fermi transitions is usually exhausted by the transition to the Isobaric nalogue State (IS), which has a very narrow width. T = (N-Z)/ g.s. F IS g.s. T = (N-Z)/ T = (N-Z)/ 1 IS = T ± 0 (N,Z) (N-1, Z+1) That means, Isospin is a good quantum number, in general, in both light nuclei and medium-heavy nuclei with neutron excess. Isospin of the ground state is maximum broen for N=Z nuclei with Z large. T z = (N-Z)/ T z = (N-Z)/ 1 In this example F T = T = ( N ) / = 0 ) T = ( N Z) / T z S = ( N Z) / + 1 S + + z Z = N Z cannot have component.

52 7.4.. Gamow-Teller resonance ; (GT ± = ) Oˆ t ( ) σ ( ) ± = ± µ 3 3 moˆ 0 ˆ no+ 0 µ = 1 m µ = 1 n ( S S + ) = 3( N Z) spin-parity of the operator = 1 + Some experimental observation In order to observe GTGR, the incident energy of proton or 3 He beams must be chosen carefully. (The population of spin-isospin modes relative to excitations of other types depends on the incident energy.) C.Gaarde et al., Nucl.Phys.369 (1981) 58. The 0 71 Ga( 3 He,t) 71 Ge spectrum at 450 MeV. M.Fujiwara et al., Nucl.Phys.599 (1996)3c. J.Rapaport and E.Sugarbaer, nn.rev.nucl.part.sci., 44 (1994) 109.

53 ( 3 He, t) Pb16 15 ex. Observed properties of IS and GTGR in with = 450 MeV Bi ) 83 E(3 He (T = ) (T 1) IS GTGR Ex (MeV) Width (MeV) Sum rule (%) 100 ~ 60 H.imune et al., PRC 5, 604 (1995). IS = T - gr of 08 Pb> GTGR = GT - gr of 08 Pb> From the ( 3 He,t) reaction; only the GTGR pea region is included and S + = 0 was assumed due to Pauli blocing. Missing (GT) strength used to be a problem in 1980s. 1) Bac-ground subtraction problem ; - broad GT bump is located on top of a continuum. Including this continuum or not maes a large difference in the extracted strength. - GTGR has a clean resonance shape? ) S + may not be negligible even for medium-heavy nuclei. C.Gaarde, Niels Bohr centennial Conf., ) Possible missing GT strength is carried by the excitation, [nucleon resonance at 13 MeV]?

54 The direct measurement of S - and S +, performing both (p,n) and (n,p) reactions; K.Yao and H.Saai et al., Phys.Lett. B615 (005) Zr (p,n) E p = 95 MeV 90 Zr (n,p) E n = 93 MeV S + was carefully measured! multipole decomposition technique was applied to extract the GT component from the continuum. GT quenching factor extracted from Ex < 50 MeV : Q S - S + 3(N Z) = 0.88 ± 0.06 IVSM = IsoVector Spin Monopole modes are expected around the place indicated. re IVSM or IVSM + modes populated in these reactions? 1) The coupling to non-nucleonic degrees of freedom (ex. -resonance!?) in nuclei is presumably very small. ) n appreciable amount of GT strength is found in the energy region much higher than the pea energy of GTGR. prediction by G.F.Bertsch and I.H., PRC 6 (198) 133 ; Due to the spin-isospin character of GT operator, some unperturbed 1p-1h GT strength is shifted to the higher-lying (10-45 MeV) p-h states, with the tensor correlation responsible for the highest energy components.

55 K.Yao and H.saai et al., Phys.Lett. B615 (005) 193. T=6 T=5 T 0 = { ECoul ( Nb) ECoul ( Y ) } np = =.3 MeV T 0 = = 16.9 M ( Nb) M ( Y ) + np = 5.4 MeV M ( 90 Nb) M ( 90 Zr) + np = 6.9 M ( 90 Y ) M ( 90 Zr) np = 1.5 T 0 =5 90 Nb 90 Zr 90 Y ( M = 8.66) ( M = 88.77) ( M = 86.49) (b.e.= ) (b.e.= ) (b.e.=78.400) 1) The T=6 part of GTGR is only a fraction, 1/66, of the total GTGR strength. ) ll GTGR + strength in 90 Y has T=6, which is however expected to be very small. 3) The total strength S + of IVSM + (all with T=6) on 90 Zr is not small and about 70 % of that S of IVSM on 90 Zr. T=4 T=6 T=4 T=6 = = 1 np [ M( n) M( H)] c ( mn mp me ) c 0.78 MeV

56 σ τ The energy of GTGR is pushed up from unperturbed (proton-hole) (neutron) or (proton) (neutron-hole) energies, due to the repulsive interaction in the channel. Effective GT operator, (GT) eff ( ) (GT) free Spin-dependent part of magnetic dipole (M1) operator is approximately (M 1) ) O 3 µ = 1 t ( ) σ ( ) z µ 3 e 4π mc ) ± M1GR 3 ± µ = 1 = [ T z = 0] part of GT operator, (GT = t ( ) σ ( ) ( g + g ) g 1 0 ( M1, µ ) = µ ssµ = g 1 { s = p n 1 p n ( µ + ( gs + gs ) sµ ) ( µ + ( gs gs sµ ) τ z µ + gssµ = ) g 5.58 for proton { 3.8 for neutron µ In heavy nuclei the strength of M1 GR is highly fragmented. 1 Cf. In 6 6C (S p =15.96, S n =18.7 MeV) E x (1 +, T=0) = 1.7, E x (1 +, T=1) = 15.1 MeV ex. 08 Pb (a j-j closed shell nucleus) { neutron : proton : 1 ( i 13 / i11/ ) ( h 11 / h9 / ) 1 + ε p h ε p h = 5.57 MeV = 5.85 MeV M1 strength for E x < S n (= 7.37 MeV) measured by 08 Pb( γ, γ ) using highly polarized tagged photons Giant M1 resonance centered around 7.3 MeV, with a full width of about 1 MeV. eff g s (0.7) free gs for low-energy M1 transitions. R.M.Laszewsi et al., PRL 61, (1988) 1710

57 IsoVector Spin Giant Monopole Resonance (IVSGMR) ; m O 0 n O+ m ˆ ˆ ( ) = 3 N r Z r n n p ˆ ± = t ± ( ) σ ( ) r O µ spin-parity of the operator = 1 + This IVSM operator has the same spin, isospin and parity as those of GT operator, though IVSM mode is a compression mode while GT is not. Moreover, the GT strength extends to the continuum energy region much higher than that of the main pea, in the high energy region it may be experimentally difficult to differentiate IVSM strength from higher-lying GT strength. Taing into account the orthogonality to GT operator, theoretically one needs to use Oˆ IVSM ( r < r ) t ( ) ( ) > = ± σ µ in order to obtain only the strength of IV Spin Monopole mode. I.H. and H.Sagawa, PRC 6 (000) However, IVSM mode has a form factor quite different from that of GT transitions. ( 3 He, t) with appropriate incident energies may excite IVSMR more easily than (p,n)? The dependence of cross sections on incident energies or a comparison of (p,n) with ( 3 He,t) may differentiate the strength of IVSM from that of GT.

58 In nuclei with a larger neutron excess E x (GR ) > E x (GR + ) less (if not zero) GT + strength is expected due to Pauli blocing (namely, the neutron level in p n by GT + operator is already occupied). Excitation energy of IVSGMR + in daughter nuclei becomes considerably lower, compared with that of IVSGMR in daughter nuclei. [ Maximum energy of relevant p-h configurations estimated from the ground state of mother nuclei] (The collective pea may appear just above the max p-h energy, when unperturbed p-h excitations are spread over a broad energy region, compared with the strength of relevant p-h interactions.) For stable nuclei ( N 3 5/ 3 Z) β stable 6 10 n p ( N F N F ) ω / 3 MeV IVSM + p-h excitations. IVSM p-h excitations. n+1, j=l-1/ n+1, j=l 1/ E l s n+1, j=l+1/ E ls n+1, j=l+1/ ħω 0 ħω / /3 n, j=l+1/ n, j=l+1/ Z N Z N /3 Ex( IVSM+ ) = ω E s /3 Ex( IVSM ) = ω E s [ Ex (IVSGMR ) E x (IVSGMR + ) ] > x /3, since IVSGMR is more collective than IVSGMR + due to the neutron excess. /3 = 6.4 MeV for 1 ω 0 08 Pb

59 The relation [E x (t + GR ) < E x (t GR )] in nuclei with neutron excess is valid for all types of t ± GRs, though the actual energy difference depends also on the collectivity of modes. In nuclei which are much more neutron-rich than β-stable nuclei, one has 1) 3 5/ 3 ( N Z) > 6 10 ) The ground state of t + daughter nuclei becomes much higher than that of mother nuclei. Then, possible IVSGMR + may have even lower E x in daughter nuclei. Or, some appreciable 1 + strength may be found at lower E x, when GT + transitions should be forbidden. One may try reactions such as (n,p) or (t, 3 He) on such neutron-rich nuclei in the inverse inematics, and find out the lower-lying spin-dependent strength? Some comments: 1) Knowing that even the simplest compression mode, ISGMR, has not a simple resonance shape in the light-medium mass region, IVSM strength may not be concentrated on one collective resonance. In the schematic harmonic oscillator model ; unperturbed p-h excitations for ISGMR are totally degenerate at ω0 1/3 those for IVSGMR are spread over ω0 ± E s 80 ± 8 MeV., while ) Similar to GTGR or the GT strength distribution, IVSGMR may have a considerable amount of strength tail at the energy higher than the major pea, since it is also a spin-isospin mode.

60 I.H. and H.sagawa, PRC 6, (000) Ex. of calculated charge-exchange spin monopole (t ± SMR) modes 08 Pb Tl 08 Pb Bi t + mode t mode HF plus TD with a Syrme interaction Response functions Radial part of transition density of IVSGMR ± (compression modes!) t + t E x is measured from the ground state of the mother nucleus Possible high-energy tail of the strength is not obtained in this ind of calculations (namely, [HF plus TD] or [HF plus RP]).

61 IsoVector Spin Giant Dipole Resonance (IVSGDR) ; (There are a considerable amount of experimental data.) Oˆ ± = t± ( ) r ( Y ( rˆ ) σ ( ) ) Jπ Defining S± m t± ( ) r ( Y1 ( rˆ ) σ ( ) ) 0 Jπ where Jπ = 0, 1 and, one obtains J = 0,1, m J J J + 1 S S+ = N rn Z rp 4π J J 9 ( S S+ ) = N r Z r 4π n p (&) 1 Jπ ex. Using experimental data from 90 Zr(p,n) and 90 Zr(n,p) on the l.h.s. of (&), the difference between K.Yao, H.Sagawa and H.Saai, Phys.Rev.C 74, (R) (006) r n r p and can be obtained, if observed charge radius. r p is nown from the (Haraeh & Woude, Giant Resonances, 001) 90 The ground state of 40 Zr has 50 T T0 T z = ( = ) = (50 40) / = 5 IVSD : T = 4, 5, 6 IVSD + : T = 6 neutron sin thicness : r n r p in in Zr ( p, n) Nb Zr50 ( n, p) 39Y51 multipole decomposition analysis at θ=4.6 º (= max of SD mode) was performed, and the SD strengths up to 40 MeV in the left figure were included. N r Z = 07 ± 17 fm n r p = 0.07 ± 0.04 fm

62 7.5. Giant resonances in nuclei far away from the stability line drip-line nuclei very different N/Z ratio, compared to stable nuclei with the same, in addition to the presence of wealy-bound nucleons. β stable nuclei proton-drip-line nuclei neutron-drip-line nuclei V(r) V(r) V(r) r r r r r r protons neutrons protons neutrons protons neutrons Since the Fermi levels for protons and neutrons are very different in drip line nuclei, this binding energy difference of least-bound protons and neutrons will produce interesting phenomena in charge-exchange reactions or β decays.

63 ISGQR of nuclei with wealy-bound neutrons (an example of wealy-bound neutrons threshold strength) Ex. Calculated GQR of β-stable nuclei Calculated GQR of neutron-drip-line nuclei threshold strength 50 Increase of energy-weighted sum-rules, S( IS, λ = ) class = r, by the threshold strength 4π m extra contribution by wealy-bound neutrons in the ground state to r. Threshold strength couples very little with other p-h configurations threshold strength contributes very little to e pol (E).

64 60 Ex. ISGQR of a possibly neutron-drip-line nucleus with wealy-bound neutrons, 0Ca40 (calculated results only) Compared with ISGQR in β-stable nuclei, the frequencies of possible } neutron p-h configurations are lower, while the frequencies of proton { ISGQR has p-h configurations remain nearly the same or become larger. lower frequency broader width However, collective correlation structure transition density } are similar to those of β-stable nuclei. Hartree-Foc potentials and one-particle energy levels Unperturbed neutron response to r Yµ ħω 0 proton response I.H., H.Sagawa and X.Z.Zhang, PRC 64, (001).

65 7.5.. β-decay to GTGR in drip line nuclei N=Z β + decay to GTGR + β decay to GTGR

66 1) β decay in nuclei with N > Z H.Sagawa, I.H. and M.Ishihara, PLB 303, 15 (1993) β stable nuclei very neutron-rich light (Z < 7) nuclei T 0 g.s. (p,n) GTGR T 1, T 0 0, T IS T 0 E coul np The relative energy between IS and GTGR is a function of (N-Z)/. The larger (N-Z)/, the lower GTGR. g.s. (p,n) IS f ( N-Z ) β β GTGR g.s. T 0 1 M(Z,N+1) M(Z+1,N) np β (Z, N+1) (Z+1, N) g.s. T 0 = N+1 Z (Z, N+1) (Z+1, N) np = ( M(n) M( 1 H)) c = 0.78 MeV E Coul (Z+1) = E Coul (Z+1) E Coul (Z) ((Z+1) Z ) 1/3 Z 1/3 Energy difference of different T states in a given nucleus E(, T + 1, M T = T ) E(, T, M T = T ) 4b sym T + 1/ N Z

67 Dependence of the energy difference between IS and GTGR on (N Z)/ E(GT) E(IS) = N Z Note N Z 4 O C K.Naayama,.Pio Galeao and F.Krmpotic, PLB114, 17 (198) C He Pb 0.115

68 ) β + decay in nuclei with N > Z F.Fris, I.H. and X.Z.Zhang, PRC 5 (1995) 468. β stable nuclei medium-heavy proton-drip-line nuclei (Z > 50) T T GTGR + T E Coul np (n,p) T 0 g.s. β + g.s. T g.s. β + GTGR + (Z+1, N) (Z, N+1) β + M(Z+1,N) M(Z,N+1) + np p T 0 = N Z 1 g.s. (Z+1,N) (Z, N+1) (Z-1, N+1) S p The mass difference, M(Z+1,N) M(Z,N+1), increases rapidly, as stable proton-drip-line nuclei. GTGR+ comes easily into the scope of β + decays, namely below the ground state of mother nuclei.