A revised version of the lecture slides given in Sept., 2007.

Transcription

1 RIKEN June, 010 (expecting experimentalists for audience) A revised version of the lecture slides given in Sept., 007. One-particle motion in nuclear many-body problem (V.1) - from spherical to deformed nuclei - from stable to drip-line - from static to rotating field - from particle to quasiparticle - collective modes and many-body correlations in terms of one-particle motion Ikuko Hamamoto Division of Mathematical Physics, LTH, University of Lund, Sweden The figures with figure-numbers but without reference, are taken from the basic reference : A.Bohr and B.R.Mottelson, Nuclear Structure, Vol. I & II

2 1. Introduction. Mean-field approximation to spherical nuclei - well-bound, weakly-bound and resonant one-particle levels.1. Phenomenological one-body potentials (harmonic-oscillator, Woods-Saxon, and finite square-well potentials).. Hartree-Fock approximation self-consistent mean-field 3. Observation of deformed nuclei 3.1. Rotational spectrum and its implication 3.. Important deformation and quantum numbers in deformed nuclei 4. One-particle motion sufficiently-bound in Y 0 deformed potential 4.1. Limits of small and large deformation 4.. Nilsson diagram one-particle spectra as a function of deformation Table 1 Deformed one-particle wave-functions in terms of spherical basis Tables and 3 Matrix-elements of one-particle operators 5. Weakly-bound and resonant neutron levels in Y 0 deformed potential 5.1. Weakly-bound neutrons 5.. One-particle resonant levels eigenphase formalism 5.3. Examples of Nilsson diagram for lighter neutron-rich nuclei Appendix Angular momentum projection from a deformed intrinsic state

3 1. Introduction Mean-field approximation to many-body system The study of one-particle motion in the mean field is the basis for understanding not only single-particle mode but also many-body correlation. Mean field Hartree-Fock approximation Self-consistent potential = Hatree-Fock potential Phenomenological one-body potential (convenient for understanding the physics in a simple terminology and in a systematic way) Harmonic-oscillator potential Woods-Saxon potential Note, for example, the shape of a many-body system can be obtained only from the one-body density mean-field approximation

4 Harmonic-oscillator potential is exclusively used, for example, the system with a finite number of electrons bound by an external field ( = a kind of NANO structure system). This system is a sufficiently bound system so that harmonic-oscillator potential is a good approximation to the effective potential. Another finite system to which quantum mechanics is applied is clusters of metalic atoms shell-structure based on one-particle motion of electrons. In this system a harmonic-oscillator potential is also often used.

5 . Mean-field approximation to spherical nuclei.1. Phenomenological one-body potentials 3-dimensional harmonic oscillator potential 1 H m r m harmonic-oscillator potential has a spectrum 1g, d, 3s 3 N 1s 1p 1d, s 1f, p where N nx ny nz ( n r 1) in rectilinear coordinates in polar coordinates = N, N-, 0 or 1 In the above figure 1 V () r m r const where const = 55 MeV = 8.6 MeV Degeneracy of the major shell with a given N ( 1) spin = (N+1)(N+) leads to the magic numbers (l = even for N=even, odd for N=odd), 8, 0, 40, 70, 11, 168,

6 One-particle levels for β stable nuclei ( S n S p 7-10 MeV ) Modified harmonic-oscillator potential can often be a good approximation. (surface) Spin-orbit finite-well h.o. Large energy gap in one-particle spectra Magic number N, Z = 8, 0,8,50,8,16, Nuclei with magic number : spherical shape Normal-parity orbits majority in a major shell of medium-heavy nuclei High-j orbits, 1g 9/, 1h 11/, 1i 13/, 1j 15/, which have parity different from the neighboring orbits do not mix with them under quadrupole (Y μ ) deformation and rotation. One-particle motion in the mean-field shell structure (= bunching of one-particle levels) nuclear shape

7 Phenomenological finite-well potential : Woods-Saxon potential - an approximation to Hartree-Fock (HF) potential V() r V f () r WS where 1 f() r r R 1 exp a V(r) R a r a : diffuseness R : radius R = r 0 A1/3 A : mass number V WS standard values of parameters V r fm a 0.67 fm WS N 5133 A Z MeV for + for neutrons for protons

8 Woods-Saxon potential vs. harmonic-oscillator potential Harmonic-oscillator potential cannot be used for weakly-bound or unbound (or resonant) levels. For well-bound levels; Corrections to harmonic-oscillator potential are; a) repulsive effect for short and large distances push up small l orbits b) attractive effect for intermediate distances push down large l orbits higher l one-particle wave-functions only l=0 one-particle wave-funcions In the above figure the parameters are chosen so that the root-mean-square radius for the two potentials, are approximately equal.

9 Schrödinger equation for one-particle motion with spherical finite potentials H mx y z H where V() r V () s r 1 R () () ˆ nj rx jm j r r X 1 () r ˆ C (,,;,, ) (, ) s j m m m Y jm j s j m 1/, m m, m ( ) Ym(, ) ( 1) Ym(, ) s (x, y, z) (r, θ, φ) The Shrödinger equation for radial wave-functions is written as d dr ( 1) m nj V() r Vs() r Rnj () r 0 r ($) For example, for neutrons eq.($) should be solved with the boundary conditions; at r = 0 R () r 0 at r large (where V(r) = 0) for 0 R () r rk ( r) for 0 where R () r cos( ) krj ( kr) sin( ) krn ( kr) : phase shift k j n m : Modified spherical Bessel function of the third kind (See p.443 of Abramowitz & Stegun, Handbook of mathematical functions) : spherical Bessel function : spherical Neumann function where k m

10 One-body spin-orbit potential in phenomenological potentials : surface effect! In the central part of nuclei the density, ρ(r) = const. Then, the only direction, which nucleons can feel is the momentum, From the two vectors, p and the spin s, of nucleons one cannot make P-inv (i.e. reflection-invariant) and T-inv (i.e. time-reversal invariant) quantity linear in the momentum. For example, ( p s ) ( p s ) s P-inv T-inv At the nuclear surface () r 0 Then, ( p s) ( r) ( ps ps ) r 1 ( s) r r In practice, one often uses the form 1 Vc ( r) Vs () r ( s) r r i.e. : P-inv & T-inv! 1 ( r p) s r () r,0,0 r r p in polar coordinate (r,θ,φ) r (,0,0) r r p (0, rp, rp ) where λ=const. and V c (r) is one-body central potential such as the Woods-Saxon potential

11 In the presence of spin-orbit potential V ls (r) ( ( s) ), the total angular momentum of nucleons 1 j s j becomes a good quantum-number. ( s), z 0 ( s), s z 0 ( s), z s z 0 H V() r m quantum number of one-particle motion ( l, s, m l, m s ) H V() r V s() r m quantum number of one-particle motion ( l, s, j, m j ) ( s) j s j( j1) ( 1) ( 1) = l 1 for j = l 1/ l for j = l + 1/ H 1 1 R () r X where X C (,, j ; m, m, m ) Y (, ) 1/, j jm j r jmj s j m m m, m s s The radial part of the Schrödinger equation becomes d dr ( 1) m j V() r Vs() r Rj() r 0 r

12 Centrifugal potential + Woods-Saxon potential mx y z V() r dependence on l l = cot ( ) r V r mr r r sin 1 1 ( ) = r V() r mr r r centrifugal potential l = l = 4 Woods-Saxon pot. centrifugal pot. W-S + centrifugal pot.

13 ( 1) Height of centrifugal barrier R h where R h > r 0 A 1/3 The height : higher for smaller nuclei higher for larger l orbits ex. For the Woods-Saxon potential with R=5.80 fm, a=0.65 fm, r 0 =1.5 and V WS = 50 MeV ; l height of centrifugal barrier 0 0 MeV

14 Height of centrifugal barrier ; 1) well-bound particles are insensitive. ) affects eigenenergies and wave-functions of weakly-bound neutrons, especially with small l 3) affects the presence (or absence) of one-particle resonance, resonant energies and widths.

15 Neutron radial wave-functions 1 ( r) R ( r) X ( rˆ ) r njm nj jm l = 4 l = 0 ε = 8MeV ε = 00keV halo

16 For a finite square-well potential V(r) The probability for one neutron to stay inside the potential, when the eigenvalue ε nl (< 0) 0 ε nl R 0 r l l ( 0) R 0 0 ( ) Rn r dr 0 1/3 3/5 5/7 (l 1)/(l+1) Root-mean-square radius, r rms, of one neutron ; rrms r In the limit of ε nl (<0) 0 1/ r rms ( ) for l = 0 n ( ) 1/4 n for l = 1 finite value for l

17 Spherical nuclei : unique behavior of low-l orbits, as ε l (<0) 0 Energies of neutron orbits in Woods-Saxon potential ( R = r 0 A 1/3 with r 0 = 1.7 fm is varied. ) Change of shell structure 1d 3/ 1d 5/ 1d 3/ s 1/ s 1/ 1d 5/ s 1/ 1d 3/ 1d 5/ Strongly bound Stable sd-shell nuclei Very weakly bound in finite-well potential There is no neutron l=0 (s) resonance!

18 One-particle resonant level in spherical finite potentials ( Coulomb potential ) For ε l > 0 and r large R () r cos( ) krj ( kr) sin( ) krn ( kr) where k : phase shift m The width of the resonance; π/ ε res d d res The resonance energy ε res is defined so that the phase shift δ l increases with energy ε as it goes through π/ (modulo π). For example, see ; R.G.Newton, SCATTERING THEORY OF WAVES AND PARTICLES, McGraw-Hill, At ε res ; (1) a sharp peak in the scattering cross section; () a significant time delay in the emergence of scattered particles; (3) the incoming wave (i.e. particles) can strongly penetrate into the system; (4)..

19 d Resonance time delay k k 0 0 dk scattering amplitude For r, a wave packet in a scattering is written as dk ( k )exp i( k where ( k ) f ( k, cos ) 1 r Et) dk( k ) r expi ( kr Et) : sharply peaked around k k 0 k 1 0 ( 1) e i sin P (cos f ( k,cos ) ) ($) Assume that at k=k 0 a sharp peak only in a given l channel. For very large t (= time), the nd term in ($) contributes only at the distance r k 0 m t d ) k k 0 dk for k k0 e i d dk ik ( kret) kk0 e d ( k ) ( k 0 ) k k ( k k 0 ) 0 dk = e d k ik ( r kk t) dk 0 m Time delay caused by the sharply changing term d dk 0 i e in the f : t D m k time delay in the emergence of the scattered particles 0 d dk k k 0 d dk 0 time advance!

20 β-stable nuclei One-particle levels which contribute to many-body correlations V(r) r neutron drip line nuclei role of continuum levels and weakly-bound levels V(r) V(r) Continuum levels r Γ r Importance of one-particle resonant levels with small width Γ in the many-body correlations. Obs. no one-particle resonant levels for l = 0 (i.e. s) orbits.

21 Some summary of weakly-bound and positive-energy neutrons in spherical potentials (β=0) Unique role played by neutrons in small l ; s, p orbits (a) Weakly-bound small-l neutrons have appreciable probability to be outside the potential; ex. For a finite square-well potential and ε nlj (<0) 0, the probability inside is 0 for s neutrons 1/3 for p neutrons Thus, those neutrons are insensitive to the strength of the potential. Change of shell-structure (b) No one-particle resonant levels for s neutrons. For a given ε l j > 0, higher-l orbits have one-particle resonance with smaller width. One-particle neutron resonance with lower-l disappears at a smaller ε l j (> 0) value. Change of many-body correlation, such as pair correlation and deformation in nuclei with weakly-bound neutrons

22 .. Hartree-Fock (HF) approximation self-consistent mean-field A mean-field approximation to the nuclear many-body problem with rotationally invariant Hamiltonian, H m i i i j v ij effective two-body interaction phenomenology! Popular effective interaction, v ij,is so-called Skyrme interaction many different versions exist, but in essence, ( r r ) interaction i plus density-dependent part that simulates the 3-body interaction. j The total wave function Ψ is assumed to be a form of Slater determinant consisting of one-particle wave-functions, ( r ) i j (i and j) = 1,,.., A Variational principle H 0 together with subsidiary conditions 3 i( ri) d ri 1 leads to the HF equation. OBS. The HF solution is not an eigen function of the Hamiltonian H.

23 ex. HF equations for particles (a simple example!) 1 1( r1) ( r1) (1, ) ( r ) ( r ) r 1 1 r 1 rvr 1 r rdr r 1 rvr 1 r 1 rdr 1 1 r 1 ( ) ( ) ( ) (, ) ( ) ( ) ( ) (, ) ( ) ( ) m 3 3 r r 1 rvr r 1 rdr 1 r 1 rvr r rdr r ( ) ( ) ( ) (, ) ( ) ( ) ( ) (, ) ( ) ( ) m exchange (Fock) term (absent in Hartree approximation) Hartree potential V ( ) H r 1 and V ( ) H r Find the solutions, ( r ) ( r ) 1 and the above coupled equations., with ε 1 and ε, which satisfy simultaneously The usual procedure of solving the HF equation is; ( r ) V( r ) V( r 1( r 1) w.f. pot. w.f. ) ( r ) 1 1 ( r 1 ) Find self-consistent solutions together with eigenvalues, ε 1 and ε.

24 drip-line nuclei very different N/Z ratio, compared to stable nuclei with a given A β stable nuclei V(r) proton-drip-line nuclei V(r) neutron-drip-line nuclei r r r r r r V(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. Weakly-bound one-proton motion in medium-heavy nuclei may not be so different from the well-bound one, due to the high Coulomb barrier.

25 Hartree-Fock potential and one-particle energy levels V N (r) : neutron potential, V P (r) : proton nuclear potential, V P (r)+v C (r) : proton total potential A typical double-magic β-stable nucleus 08 Pb 8 16 One of Skyrme interactions ; SkM* See : J.Bartel et al., Nucl. Phys. A386 (198) 79.

26 Hartree-Fock potentials and one-particle energy levels V N (r) : neutron potential, V P (r) : proton nuclear potential ex. of neutron-drip-line nuclei ex. of proton-drip-line nuclei 3s 1/ 50 50

27 3. Observation of deformed nuclei 3.1. Rotational spectrum and its implication Some nuclei are deformed --- axially-symmetric quadrupole (Y0) deformation Observation : 1) rotational spectra E(I) AI(I+1) ) large quadrupole moment or large E(I I-) transition probability rotation of deformed nuclei time-dependent electric field strong el-mag radiation For E(I) = AI(I+1), E(I=4) E(I=) = 3.33 ex. In the ground band of 168 Er = 3.31 I A rotational band with a given K consists of members with I K. K sym axis

28 + 0+ E Observed E-transition probabilities from the ground state (I π =0 + ) to the first excited + state in stable even-even nuclei. The single-particle value used as unit is / 3 4 B sp ( E) e R 0.30A e fm 4 5 Bohr & Mottelson, Nuclear Structure, Vol.II, 1975, Fig.4-5

29 WARNING : many different definitions (and notations) of Y 0 deformation parameters δ intrinsic quadrupole moment Q Z r k k1 uniformly-charged spheroidal nucleus with a sharp surface 3 ( R ( R ) ( R ) ( R ) 3 3) β or β δ osc or ε β is defined in terms of the expansion of the density distribution in spherical harmonics. radius density R(, ) R0(1 Y0 ( )...) 0 * ( r) ( r) R0 ( Y0( ) r * 0...) In the deformed harmonic oscillator model it is customary to use 3 R3 R ε = osc 3 R 3 av To leading order, δ β δ osc, but. possibly for neutron-rich nuclei towards the neutron-drip-line, since R R ) Rn n Rp p : displacement of the surface n p for stable nuclei, but n p n p

30 Nuclei with deformed ground state close to the β stability line All stable single or double closed-shell nuclei are spherical. line of β stability some typical examples of deformed nuclei : 1 C 6 4 Mg 1 Oblate (pancake shape) Prolate (cigar shape) Region of deformed even-even nuclei rare-earth nuclei with 90 N 11 mostly prolate Some new region of deformed ground-state nuclei away from β stability line; 1) N Z 38 ex. ) N 0 ex. 3) N 8 ex. (oblate?) (prolate?) 40Zr (prolate?) Kr36 38 Sr Ne Ne Mg Mg Be7 1 4Be Be 10 ( island of inversion ) etc.

31 Deformed ground state of N Z nuclei (proton-rich compared with stable nuclei) Coexistence of prolate and oblate shape : (Z=36) ( 4+ ) Sr Zr Most probably prolate 40 OBS. Almost all stable nuclei with N (or Z) = 40 are spherical. + Ex Shape of the ground state (from Coulomb excitation); oblate prolate prolate (A.Goergen, Gammapool workshop in Trento, 006) Se Zr

32 315 (4+) 10 (4+) S(n) = 504 kev ½- ½ Strong E Strong E Be Mg0 1 Mg β = The spin-parity of the ground state, ½+, as well as the small energy distance between the ½ - and ½+ levels, 30 kev, is easily explained, If the nucleus is deformed! S(n) = MeV E(4+) E(+) = (.6) (3.1) N=8 is not a magic number! N=0 is not a magic number! (in this neutron-rich nucleus) (in these neutron-rich nuclei)

33 Example of deformed excited states of magic nuclei 40 Ca : doubly-magic nucleus, spherical ground state 0 0 strongly-deformed band Q t = eb from Doppler shift measurement β = From E.Ideguchi et al., Phys.Rev.Lett. 87 (001) 501.

34 Implication of rotational spectra : (1) Existence of deformation (in the body-fixed system), so as to specify an orientation of the system as a whole. () Collective rotation, as a whole, and internal motion w.r.t. the body-fixed system are approximately separated in the complicated many-body system. Classical system : An infinitesimal deformation is sufficient to establish anisotropy. Quantum system : [zero-point fluctuation of deformation] << [equilibrium deformation], in order to have a well-defined rotation. Indeed, collective rotation is the best established collective motion in nuclei.

35 For some nuclei Hartree-Fock (HF) calculations with rotationally-invariant Hamiltonian end up with a deformed shape! spherical shape HF solutions for closed-shell nuclei deformed shape HF solutions for some nuclei exhibit rotational spectra Deformed shape obtained from HF calculations is interpreted as the intrinsic shape in the body-fixed system. The notion of one-particle (or one-quasiparticle if pair correlation is included) motion in deformed nuclei can be much more widely, in a good approximation, applicable than in spherical nuclei. ) The major part of the long-range two-body interaction is already taken into account in the deformed mean-field. Thus, the spectroscopy of deformed nuclei is often much simpler than that of spherical vibrating nuclei.

36 What can one learn from observed rotational spectra? (a) Quantum numbers of rotational spectra symmetry of deformation ex. Parity is a good quantum number space reflection invariance, K is a good quantum number Axially-symmetric shape ( E(I) I(I+1) ), where K is the projection of angular momentum along the symmetry axis. The K π =0 + ground band has only I = 0,,4, shape is R- invariant, Kramers degeneracy time reversal invariance, etc. (b) rotational energy, E(I) - E(I-) E transition probability } size of deformation R-invariant shape : in addition to axially-symmetry, the shape is further invariant w.r.t. a rotation of π about an axis perpendicular to the symmetry axis. (If a shape is already axial symmetric, reflection invariance is equivalent to R-inv.) ex. Y 0 deformed shape is R-invariant, but not Y 30 deformed (pear) shape. Kramers degeneracy : The levels in an odd-fermion system are at least doubly degenerate.

37 Pause: Appearance of only even (or odd) angular momenta Coming from the R-symmetry of Y 0 deformation ( or Y λ0 deformation with even λ ); The ground-state (pairwise levels, Ω and Ω, are occupied) of Y 0 deformed even-even nuclei (K π = 0 + ) has r (eigenvalue of R-operation) = +1 and, thus, the rotational-band has members with I π = 0 +, +, 4 +, 6 +,. The rotational band based on excited K π = 0 configurations may have either r = 1 with members I π = 1, 3, 5,. that has been often observed in medium-heavy deformed nuclei or r = +1 with members I π = 0,, 4,. OBS. The rotational bands based on intrinsic configurations with K π =1 have members I π = 1,, 3, 4, 5,... Coming from the (Fermion or Boson) statistics ; The two-fermion system in a j-shell (j : half integer) has only even total angular momentum, ( j ) J where J is an even integer. The two-phonon system with the same phonon ( momentum, ( ) J where J is an even integer. : integer) has only even total angular

38 Why are some nuclei deformed? Usual understanding ; Deformation of ground states (ND, R : R z 1 : 1.3) Jahn-Teller effect Many particles outside a closed shell in a spherical potential near degeneracy in one-particle quantum spectra possibility of gaining energy by breaking away from spherical symmetry using the degeneracy. Weakly-bound nuclei possible change of shell-structure near degeneracy in one-particle spectra at neutron numbers different from stable nuclei. Superdeformation (SD, R : R z 1 : ) at high spins in rare-earth nuclei or fission isomers in actinide nuclei new shell structure (and new magic numbers!) at large deformation

39 3.. Important deformation and quantum numbers in deformed nuclei Axially symmetric quadrupole (Y 0 ) deformation (that has R-symmetry) - most important deformation in nuclei R x R z z-axis = symmetry axis R y R (= R x = R y )< R z prolate (cigar shape) z R (= R x = R y )> R z oblate (pancake shape) z

40 One-particle Hamiltonian with spin-orbit potential H T V ( r, ) V ( r, ) V0( r) V ( r) Y0( ) Vs ( r)( s) 5 Y0( ) (3cos 1) 16 where θ is polar angle w.r.t. the symmetry axis ( = z-axis) Quantum numbers of one-particle motion in H (1) Parity π = (-1) l where l is orbital angular momentum of one-particle. () Ω l z +s z ) [ V (r) Y 0 (θ), l z + s z ] = 0 and ( s), z s 0 z

41 4. One-particle motion well-bound in Y 0 deformed potential 4.1. Limits of small and large deformation Small deformation and/or high-j orbits r, ) V0( r) V ( r) Y ( ) Vs ( r)( s) ( 0 those pushed down by ( s) potential : ex. g 9/,h 11/, i 13/, Single j-shell : j (= one-particle angular momentum) is (approximately) a good quantum number. V V ( r) Y0( ) Vs ( r)( s) Ω = ± j, ± (j 1),.., ±½ : symmetry-axis component of j j Y j only for In the linear order of β, the Y 0 deformation only shifts ( β) the energy of doubly-degenerate one-particle states with ± Ω. One-particle wave-functions remain the same for β 0. spherical : (j+1) degeneracy Y 0 deformed : ± Ω degeneracy oblate spherical sym prolate

42 Large (or realistic) deformation V( r) Y0( ) V s( r)( s) Many j-shells coupled by Y 0 deformation 1 j11 f ( r) Y0 j 0 That means, in a Y 0 [H, Ω π are good quantum-numbers. for ( 1) 1 ( 1) 1 j 1 j deformed mean field with spin-orbit potential (One-particle angular-momentum j is not a good quantum-number. [ H, j ] 0 ) : projection of one-particle angular-momentum along the symmetry axis, and a good quantum-number : parity (1) J z ] = [Y 0, J z ] = 0 J Ω z sym axis Ω J z = L z + s z For example, Ω π = 1/ + s 1/, d 3/, d 5/, g 7/,g 9/,.., components l min = 0 Ω π = 3/ + d 3/, d 5/, g 7/, g 9/, i 11/.., components l min = Ω π = 1/ p 1/, p 3/, f 5/, f 7/, h 9/,.., components l min = 1 Ω π = 3/ p 3/, f 5/, f 7/, h 9/, h 11/,.., components l min = 1

43 Y 0 deformed harmonic-oscillator potential M Vdef. h. o. z z ( x y ) One-particle energy ( ε(n) at δ=0 splits into (N+1) levels ). ) nz = 0, 1,.., N where N n 1 ( n, nz ) nz z n 1 = 3 N (3nz N) 3 δ β V( r) Y0( ) V s( r)( s) z n where n n x n y 1 ( z ) 3, n ) has ( n 1) ( n z N n x n y n z 3 degeneracy. z z Rz R R av N= spin n x 0,1,..., n N=1 N=0 Denote one-particle energies by [N n z n x ] or [N n z Λ] [n x n y n z ] or (cf. [ Vdef. h. o., Lz ] 0 Λ ( L z ) n, ( n ),.., ±1 or 0 ( ) z oblate spherical sym prolate ( ) z Asymptotic quantum numbers [ N n z Λ Ω] where Λ > 0, Ω Lz sz including spin (= ½) = Λ ±½ > 0

44 z z ( x = 3 y ) 1 x y z z ( x y ) 1 z z 3 f(r) + c δ r Y 0 (θ) up till O(δ) r 16 r Y 0( ) 5 h.o. potential : no surface maybe applicable for strongly bound system no spin-orbit potential maybe applicable for large deformation J One-particle levels in axially-symmetric quadrupole (Y 0 ) deformation (prolate or oblate shape) are denoted by ( J z ) z (sym axis) [N n z Λ Ω] : asymptotic quantum-numbers. ( L z ) with Ω 0 and Λ 0 where Ω and parity π = (-1) l = (-1) N is a good quantum-number. One-particle energy is degenerate for ±Ω One-particle levels for a finite quadrupole deformation β are often denoted by [N n z Λ Ω], since for β >0.3 the wave functions are approximately expressed by [N n z Λ Ω], except high-j orbits.

45 4.. Nilsson diagram one-particle spectra as a function of deformation Ex. [ harmonic-oscillator + surface effect + spin-orbit ] potential = [modified oscillator] potential Doubly-degenerate (± Ω) one-particle levels are denoted by asymptotic quantum numbers [N n z Λ Ω] with Λ, Ω > 0, which become good quantum numbers for very large β. N = n z + n Λ(>0) = n, n -,,1 or 0 N= n z =0 For β (~ δ) > 0 (prolate shape) 1) [prolate] For a given N : ε Ω lower for n z larger ex. Nilsson levels in sd-shell (N=) N= n z =1 ε Ω π = 1/ + Ω π = 3/ + Ω π = 5/ + [00 1/] [11 1/] [0 1/] [0 3/] [11 3/] [0 5/] [110 ½] [101 1/] ) [surface] For given {N, n z } : ε Ω lower for Λ larger 3) [spin-orbit] For given {N, n z, Λ} : ε Ω lower for Ω larger [101 3/] oblate prolate At large β (~ δ), ε ( 3n z N ) i.e. ε as a function of β depends only on β(3n z N) spherical

46 Proton orbits in prolate potential (50 < Z < 8). g 7/, d 5/, d 3/ and s 1/ orbits, which have π = +, do not mix with h 11/ by Y 0 deformation. [N=4, n z =0] Levels are doubly degenerate with ± Ω. h 11/ orbit = high-j orbit with π = At small δ and h 11/ orbit, ε δ (3Ω j(j+1) ) At large δ, ε δ(3n z N) [N=4, n z =] At δ > 0.3 for prolate shape quantum numbers [Nn z ΛΩ] work well, except for high-j orbits.

47 Table 1. Ex.1. Deformed one-particle wave-functions denoted by the asymptotic quantum numbers [N n z ΛΩ] are expanded in terms of spherical basis. (A modified-oscillator Hamiltonian was diagonalized.) parity of the states: N ( 1) ( 1) OBS. Various computer programs are at present publicly available if one is satisfied with the diagonalization of modified-oscillator Hamiltonian.

48 Table 1. (continued) Ex.. Using Tables 5-a and 5-b in A.Bohr and B.R.Mottelson, Nuclear Structure, vol.ii, one obtains, for example, Normal-parity orbits; [411 3/] > = /> + = g 9/ > g 7/ > d 5/ > d 3/ > [411 1/] > = /> + = g 9/ > g 7/ > d 5/ > d 3/ > s 1/ > [400 1/] > = /> + = g 9/ > 0.07 g 7/ > d 5/ > d 3/ > s 1/ > High-j orbits; [53 5/] > = /> + = 0.88 h 11/ > h 9/ > 0.44 f 7/ > 0.06 f 5/ > for proton one-particle wave-functions at deformation δ = 0.3, which are obtained by diagonalizing a modified-oscillator Hamiltonian, H = T + V(r,θ) plus (l s) potential OBS. [411 3/] > : states obtained by diagonalization 411 3/ > : bases states exactly expressed by the quantum numbers N n z ΛΩ

49 Intrinsic configuration in the body-fixed system Good approximation ; (a) In the ground state of eve-even nuclei K A i1 i 0 Namely, ± Ω levels are pairwise occupied. (b) In low-lying states of odd-a nuclei K A i1 i Ω of the last unpaired particle. Low-lying rotational bands in deformed odd-a nuclei may well be classified in terms of one-particle orbit [Nn z ΛΩ] occupied by the last unpaired particle. The rotational band based on [N n z ΛΩ] (a) I K ( I 3 )= Ω (b) the bandhead state has I=K. Exception may occur for K=1/ bands. (c) some irregular rotational spectra are observed for K=1/ bands.

50 ex. The N=13 th neutron orbits observed as low-lying excitations in 5 Mg 13 - a textbook example [N n z ΛΩ] Bohr & Mottelson, Nuclear Structure, Vol.II f 7/ S(n) = 7.3 MeV (oblate) (prolate) Nilsson levels : double (±Ω) degeneracy The above interpretation of the data works quantitatively : measured large E transitions within the bands β 0.4 observed E- and M1-intensity relations g eff s = ( ) g free s

51 Some selection rules in terms of exact quantum numbers, I, K, or N n z ΛΩ ( More complete formulas are given in Table. ) 1) Between states ( I, K) and ( I', K' ) E E or M transitions are forbidden for I I' or transitions are forbidden for K K', even if I I' M ( K-selection rule ) ) Noting that M1 or s, and GT s ( x i y ) z Nnz N, n z 1, 1, 1 or N, n z 1, 1, 1 Nnz = Nn z ( sx isy ) sz Nn z = Nnz = Nn z 1/ N, n,, 1 z where s z How well K-selection rule of 1) works in reality exhibits how good axial symmetry is. The selection rules in ) approximately work for transitions between realistic one-particle states, [N n z ΛΩ] and [N n z Λ Ω ], if β > 0.3

52 Table. Selection rule of one-particle operators between one-particle states with exact quantum numbers, N n z ΛΩ. The matrix elements between realistic levels with the assigned asymptotic quantum numbers, [N n z ΛΩ], can be obtained, to leading order, from this table. E1 operator E operator M1 operator Gamow-Teller operator From J.P.Boisson and R.Piepenbring, Nucl. Phys. A168(1971)385. If you use this kind of tables, you must be careful about the sign of the non-diagonal matrix elements, which depends on the phase convention of wave functions!

53 Table 3. Matrix-elements of one-particle operators in (l s) j, Ω representations ( s) j, r Y 0 ( 1s) j1, = ( s) j, 1 Rj () r C(,1/, j; mms) Ym (, ) 1/, m r mm j1j ( 1),( 1) j r j ( 1) ( 1) 1 1 C j j ;1/, 1/,0) C j j ;,,0) ( 1 j r 1 j1 s j j 0 drr () r R () r r 1 1 ( 1 s j1 1)( j 1) 4 ( 1) ( ( s) j, 1 r Y1 ( 1s) j1, = = 1 j1 j 1 1/ ( 1),( 1) j r j ( 1) ( 1 C j j ;1/, 1/,0) C j j ; 1,,1) ( 1 ( s) j, r Y ( 1s) j1, = = ( 1) ( 1s) j1, r Y 1 ( s) j, ) ( 1 1 j1 j 1 1/ ( 1),( 1) j r j ( 1) ( 1 C j j ;1/, 1/,0) C j j ;,,) ( 1 ( 1s) j1, r Y ( s) j, 1 1 ) ( 1 j1 1)( j 1) 4 ( 1) ( ( j1 1)( j 1) 4 ( 1)

54 Table 3 (continued) ( s) j, 1 s ( 1s) j1, ( s ± = s x ± i s y etc. ) 1 j1 =, )( 1) 1/ 3( j 1) C ( j1, 1, j;,1, 1) ( 1/, j,1/, j1; 11) ( 1 1 j 1 j1 j j 0 drr () r R () r 1 1 W j 1 j1 ( s) j, 1 ( 1s) j1, = j (, )( 1) 1/ 1 ( j1 1) 1( 1 1)(1 j 1 j1 1 1) C ( j11 j;,1, 1) W ( j ;1/ 1,1) 1 j ( s) j, 1 j ( 1s) j1, = ( j1, j) ( j )( j 1) j 1 j1 ( 1 j1 s) j, sz ( s), 3( 1) 1 j1 j1 = (, )( 1) 1/ 1 C ( j1,1, j;,0, ) W ( 1/, j,1/, j1; 11) j 1 j1 ( 1 j1 = s) j, z ( s), j (, )( 1) 1/ 1 j1 1 1( 1 1)(1 1 1) C j 1 j ; 0 ) W ( j ;1/ 1,1) ( 1 1 j j 1 j1

55 Table 3 (continued) Phase convention in wave functions - important in non-diagonal matrix-elements 1) The coupling order of spin and orbit angular momentum ; (ls)j or (sl)j ; ( s) j ( 1) 1 j ( s) j ) Angular part of one-particle wave functions is defined by ; Y (, ) or i (, ) m Y m 3) The phase convention of R j (r) ; R j (r) > 0 (or < 0) for r 0, or > 0 (or < 0) for r very large, or output of computers

56 5. Weakly-bound and resonant neutron levels in Y 0 deformed potential harmonic-oscillator potential 5.1. Weakly-bound neutrons [N n z ΛΩ] Remember the Nilsson diagram based on modified oscillator Hamiltonian for the sd-shell 6 doubly-degenerate levels in sd-shell 6 3 Ω π =1/ + (l min =0) Ω π =3/ + (l min =) 1 Ω π =5/ + (l min =)}1 particles A.Bohr and B.R.Mottelson, vol., Figure 5-1.

57 As ε Ω (<0) 0, the structure of one-particle wave-functions may deviate from [N n z ΛΩ], even for β large. Nevertheless, one-particle levels are denoted by original [N n z ΛΩ]. Ω π = 1/ + one-particle level has l min = 0 component. ex. Radial wave functions of the [00 ½] level in Woods-Saxon potentials. (The radius of potentials is adjusted to obtain respective eigenvalues ε Ω.) Bound state with ε Ω = -8.0 MeV. Bound state with ε Ω = MeV. s 1/ d 3/ d 5/ Similar behavior to wave functions in harmonic osc. potentials. Wave functions unique in finite-well potentials.

58 For ε 0, the s-dominance will appear in all Ω π =1/ + bound orbits. However, the energy, at which the dominance shows up, depends on both deformation and respective orbits. ex. Calculated s ½ probability in three Ω π =1/ + Nilsson orbits in the sd-shell as a function of energy eigenvalue ε Ω.

59 Deformed halo nuclei l min =0 Ω π =1/ + neutron orbit s 1/, as ε Ω 0. deformed core, irrespective of the size of deformation and the kind of one-particle orbits. The rotational spectra of deformed halo nuclei must come from the deformed core.

60 Ex. Ω π = 1/ and 3/ one-particle levels have l min = 1 component. The p-components increase as ε Ω 0, but the probability at ε Ω = 0 respective levels, deformations, and the diffuseness of potentials. depends on Calculated probabilities of (l j) components of one-particle [N n z Λ Ω] levels in the pf shell as a function of energy eigenvalue ε Ω. [330 1/] orbit [31 3/] orbit

61 5.. One-particle resonant levels in deformed potential eigenphase formalism Woods-Saxon l min = 0 l min = l min = 1 1) The majority of bound (ε Ω < 0) neutron Ω π = 1/ + levels do not continue to oneparticle resonant levels for ε Ω > 0. Even the resonant levels surviving for very small ε Ω > 0 die out at small ε Ω, if an appreciable amount of l min = 0 component is contained in wave functions. [0 ½] [00 ½] ) The l min value in the components of deformed wave functions is crucial for both the width of one-particle resonant levels and up till which value of ε Ω ( > 0) the resonant level can survive. [11 ½] 3) One-particle levels die out at smaller ε Ω ( >0) values, for the potential with a larger diffuseness. potential strength W-S potential parameters are fixed except radius R. (r 0 = 1.7 fm is used.)

62 Radial wave functions of the [00 ½] level s 1/ d 3/ d 5/ The potential radius is adjusted to obtain respective eigenvalue (ε Ω < 0) and resonance (ε Ω > 0). Bound state with ε Ω = 0.1keV Resonant state with ε Ω = +100 kev Existence of resonance d component Width of resonance s component OBS. Relative amplitudes of various components inside the potential remain nearly the same for ε Ω = 0.1keV kev.

63 Neutron resonant levels in deformed potential One-particle resonant levels in deformed potentials are defined using eigenphase formalism : One eigenphase δ Ω increases through π/ as ε Ω increases one-particle resonance in deformed potential I.H., Phys.Rev. C7, (005); C73, (006) R.G.Newton, Scattering Theory of Waves and Particles (McGraw-Hill, New York, 1966) (Among an infinite number of positive-energy one-particle levels, one-particle resonant levels are most important in the construction of many-body correlations of nuclear bound states.) In the limit of β 0 the definition of one-particle resonance in the eigenphase formalism the definition in spherical potentials in terms of phase shift.

64 For 0 R () r rk ( r) j where For 0 R j where k b b ( r) m b for ( r) cos( ) rj( cr) sin( ) rn( cr) for r sin( ) m c c r : eigenphase common to all Do not restrict the system in a finite box! r is the modified spherical Bessel function of the third kind, and j channels Due to the axially-symmetric deformation, radial wave-functions R j (r) for a given Ω but different lj values are coupled. All (lj) components of a solution of coupled-channel equations have a common eigenphase. A given eigenchannel : asymptotic radial wave-functions behave in the same way for all (lj) components.

65 One-particle resonant level in a deformed potential : one of eigenphases δ Ω increases through π/ as ε Ω increases. δ Ω π/ ε Ω res When one-particle resonant level in terms of one eigenphase is obtained, the width Γ Ω of the resonance in the intrinsic system is calculated by d d res : intrinsic width

66 I.H., Phys. Rev. C73, (006) Some comments on eigenphase ; 1) For a given potential and a given ε Ω there are several (in principle, an infinite number of) solutions of eigenphase δ Ω. ) The number of eigenphases for a given potential and a given ε Ω is equal to that of wave function components with different (l,j) values. 3) The value of δ Ω determines the relative amplitudes of different (l,j) components. 4) In the region of small values of ε Ω ( > 0), only one of eigenphases varies strongly as a function of ε Ω, while other eigenphases remain close to the values of nπ.

67 I.H., PRC 76 (007) ; I.H., J. Phys. G, 37 (010) Examples of Nilsson diagrams for lighter neutron-rich nuclei 1. ~ 17 C 11 (S(n) = 0.73 MeV, 3/ + ). ~ 31 Mg 19 (S(n) =.38 MeV, 1/ + ) ~ 33 Mg 1 (S(n) =. MeV, 3/ ) 3. ~ 31 Ne 1 (S(n) = 0.9 ± 1.64 MeV, halo structure) 4. ~ 37 Mg 5 (S(n) = a few hundreds kev?) 5. ~ 41 Si 7 (S(n) = 1.34 ± 0.57 MeV) 6. ~ 45 S 9 (S(n) =.86 ± 0.77 MeV) 7. A ~ 75 region Near degeneracy of some weakly-bound or resonant levels in spherical potential, unexpected from the knowledge on stable nuclei -the origin of deformation and. Jahn-Teller effect

68 One-particle neutron energies as a function of quadrupole deformation β [N n z ΛΩ] N ~ 8 region 1d3/ At β=0 ; ε(s 1/ )-ε(1d 5/ ) = 140 kev s1/ 1d5/ 17 C 11 (3/+) S(n) = 0.73 MeV 1p1/ 11 Be 7 (1/+ ) ½- ½+ S(n) = 0.50 MeV 0.3 β ~ in 1 Be from (p,p ) In μ calc g R (for β 0) eff free g g s s are used. [1] H.Ogawa et al., Eur.Phys.J. A13(00) 81 H.Ueno et al., N.P.A738 (004) 11 [13] W.Geithner et al., PRL 83 (1999) 379 [11] K.Asahi et al., N.P.A704 (00) 88c

69 potential strength (MeV)

70 One-particle neutron energies as a function of quadrupole deformation β ε res (1f 5/ ) = MeV [N n z ΛΩ] Resonant states at β=0 ; ε res (p 3/ ) < ε res (1f 7/ ) N ~ 0 region p1/? 33 Mg 1 S(n) =. MeV p3/? [31 3/] [330 1/] 31 Mg 19 S(n) =.38 MeV or are excluded by the sign of measued magnetic moments. In μ calc g R g (0.7) g eff s (for β 0) free s are used. Then, calc( f7 / ) 1. 3N calc( d3/ ) N [16] D.T.Yordanov et al., PRL 99 (007) 1501 [14] G.Neyens et al., PRL 94 (005) 0501 I.H., J. Phys. G, 37 (010) 05510

71 potential strength (MeV)

72 One-particle neutron energies as a function of quadrupole deformation β [N n z ΛΩ] p1/? p3/? 31 Ne 1 S(n) = 0.9 ± 1.64 MeV T.Nakamura, N.Kobayashi, Y.Kondo, Y.Satou, et al., PRL 103, 6501 (009), Coulomb breakup of 31Ne halo structure The 1st neutron cannot be placed on in [0 3/ ] because of observed halo structure. 3/ from [330 1/] for p( 1) halo I π = 3/ + from [0 3/] for / - from [31 3/ ] for no halo p( 1) ( ) ) min halo 1/ + from [00 1/] for s( 0) halo

73 One-particle neutron energies as a function of quadrupole deformation β [N n z ΛΩ] In the case of very weak binding N=8 is not a magic number! 1f5/ At β=0 ; ε(p 3/ ) ε(1f 7/ ) = 680 kev p1/? p3/ 1f7/ [31 5/] [31 ½] 37 Mg 5 S(n) = a few hundreds kev? 1d3/ s1/ I π = 5/ from [31 5/] for / from [31 1/] for no halo p( 1) ( ) 3 halo ) min

74 One-particle neutron energies as a function of quadrupole deformation β N ~ 8 region [N n z ΛΩ] At β=0 ; ε(p 3/ ) ε(1f 7/ ) = 1.0 MeV 41 Si 7 S(n) = 1.34 ± 0.6 MeV In μ calc g R g (0.7) g eff s (for β 0) free s are used. Then, calc( p ) 1/ 0. 4 N calc( p ) 3/ 1. 3 N calc( f7 / ) 1. 3 N Cf. In 43 S 7 the 30 kev isomeric state has 7/ from g-factor measurement The ground state is deformed?

75 One-particle neutron energies as a function of quadrupole deformation β N ~ 8 region [N n z ΛΩ] 45 S 9 S(n) =.86 ± 0.7 MeV In μ calc g R g (0.7) g eff s (for β 0) free s are used. Then, calc( p ) 1/ 0. 4 N calc( p ) 3/ 1. 3 N calc( f7 / ) 1. 3 N

76 One-particle neutron energies as a function of quadrupole deformation β N ~ 50 region A ~ 75 region ε res (1h 11/ ) = MeV ε res (1g 7/ ) = MeV at β=0 In the case of very weak binding At β=0 ; ε(3s 1/ ) < ε(d 5/ ) 51 st neutron 75 ex. 4Cr 51? Neutron-drip-line nuclei with N=51have a good chance to have the ground or very low-lying I π = 1/ + state, irrespective of spherical or deformed shape.

77 In lecture V.3 a way of calculating observed quantities in lab system using deformed intrinsic wave functions is shown, which can be of practical use though it involves an approximation. The set of basis wave functions given in V.3 are useful also for including rotational perturbation. Here, we show ; Appendix. Angular momentum projection from a deformed intrinsic state (not applicable when rotational perturbation of intrinsic states has to be included.) Rotational operator R() Ω : Euler angles (,, ) R() e ij z e ij J Rotation matrix ( ) JM D MM ' y e ij J R( ) ' J ' M ' (, ') ( J, J ') '( ) Inverting the expression J R ) JM D MM ( ) JM ' ( ' J J D MM ' ( Multiplying by ) P J M JM JM z D MM and integrating over Ω, we obtain a projection operator J 1 J dd ( ) R( ) MM 8 We need to calculate the expressions J J 1 J PM dd ( ) R( ) MM 8 J J 1 J HPM dd ( ) HR( ) MM 8

78 Appendix (continued) If is axially symmetric, M J z R() e im e ij y e im D J MM () e im JM e ij y JM e im then, using the reduced rotation matrix d J MM ' ( ) JM e i J y JM ' which describes the collective rotational motion, one obtains P J M J 1 J d sindmm ( ) 0 e i J y HP J M J 1 J ij y d sind MM ( ) He 0 where the overlap functions : i e J y 1 for θ << 1, decreases quickly as θ larger (at least in heavier deformed nuclei), is symmetric about θ = π/.