RIKEN Sept., 007 (expecting experimentalists as an audience) One-particle motion in nuclear many-body problem - 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
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. Normal-parity orbits and/or large deformation 4.. high-j orbits and/or small deformation 4.3. Nilsson diagram one-particle spectra as a function of deformation Tables 1 and Matrix-elements of one-particle operators 5. Weakly-bound and resonant neutron levels in Y 0 deformed potential 5.1. Weakly-bond neutrons 5.. One-particle resonant levels eigenphase formalism 5.3. Examples of Nilsson diagram for light neutron-rich nuclei Appendix Angular momentum projection from a deformed intrinsic state
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
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.
. Mean-field approximation to spherical nuclei.1. Phenomenological one-body potentials 3-dimensional harmonic oscillator potential 1 H = + mω r m harmonic-oscillator potential has a spectrum 3 ε = N + ω 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 = mω r + const where const = 55 MeV ω = 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,
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 168 11 70 40 0 8 184 16 8 50 8 0 8
Phenomenological finite-well potential : Woods-Saxon potential - an approximation to Hartree-Fock (HF) potential V() r V f () r = where WS 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 0 1.7 fm a 0.67 fm WS N = 51± 33 A Z MeV for + for neutrons for protons
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.
Schrödinger equation for one-particle motion with spherical finite potentials H = + + + m x y z HΨ= εψ where V() r + V s () r 1 R () () ˆ nj rx jm j r r Ψ= 1 X () rˆ = C(,,; j m, m, m ) Y (, θφχ ) s ( ) Y ( θ, φ) = ( + 1) Y ( θ, φ) jm j s j m 1/, m m, m m m s (x, y, z) (r, θ, φ) The Shrödinger equation for radial wave-functions is written as d dr ( + 1) m + ( ε 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 for m ε < 0 R() r αrh( αr) where α = ε and ε > 0 R () r cos( δ ) krj ( kr) sin( δ ) krn ( kr) δ : phase shift h j n : spherical Hankel function : spherical Bessel function : spherical Neumann function where h ( iz) j ( z) + in ( z) k m = ε
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, 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 At the nuclear surface Then, ( p s) ρ( r) P-inv T-inv ρ = ( ps θ φ ps φ θ) r 1 ρ = ( s) r r ρ() r 0 In practice, one often uses the form 1 Vc ( r) Vs () r = λ( s) r r i.e. s : P-inv & T-inv! 1 = ( ) r ( r p s ) ρ ρ() r =,0,0 r ρ 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 φ θ
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 quantum number of one-particle motion ( l, s, j, m j ) m 1 1 1 1 ( s) = { j s } = j( j+ 1) ( + 1) ( + 1) = { l 1 for j = l 1/ l for j = l + 1/ HΨ= εψ 1 () r X Ψ= R where j jm jm 1/, j j s j m m m, m r X 1 C (,, ;,, ) (, ) j m m m Y θφχ s s The radial part of the Schrödinger equation becomes d dr ( + 1) m + ( ε j V() r Vs() r ) Rj() r = 0 r
Centrifugal potential + Woods-Saxon potential + + + m x y z V() r dependence on l l = 0 1 1 1 cot ( ) r = + + θ + + V r m r r r θ θ sin θ φ 1 1 ( ) = r + V() r m r r r centrifugal potential l = l = 4 Woods-Saxon pot. centrifugal pot. W-S + centrifugal pot.
Height of centrifugal barrier + ( 1) 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 1 0.4 1.3 3.8 4 5.1 5 8.
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.
Neutron radial wave-functions l = 4 1 Ψ ( ) ( ) ( ˆ njm r = Rnj r Xjm r) r l = 0 ε = 8MeV ε = 00keV halo
For a finite square-well potential V(r) The probability for one neutron to stay inside R 0 the potential, when the eigenvalue ε nl (< 0) 0 r ε nl l 0 1 3 R 0 0 Rn ( r) dr 0 1/3 3/5 5/7 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 ε n ( ) ε n 1/4 for l = 1 finite value for l
Unique behavior of low-l orbits, as E nlj (<0) 0 Energies of neutron orbits in Woods-Saxon potentials as a function of potential radius Fermi level of neutron drip line nuclei Fermi level of stable nuclei R = radius, r 0 = 1.7 fm, V WS = 51 MeV Strength of the potential For stable nuclei (R/r 0 ) 3 A : mass number
Neutron one-particle resonant and bound levels in spherical Woods-Saxon potentials Unique behavior of l=0 orbits, both for ε nlj <0 and ε nlj >0 }One-particle resonant levels with width 1d 5/ (6) s 1/ () 1d 3/ (4) π Rj ( r) sin kr + δj mε for r and kr r width Γ dδ dε ε = ε res δ (phase shift) Strength of the potential (π/) res ε ε
One-particle resonant level in spherical finite potentials ( Coulomb potential ) For ε l > 0 and r large R () r cos( δ ) krj ( kr) sin( δ ) krn ( kr) where k m ε δ : phase shift δ l The width of the resonance; π/ ε res ε Γ dδ d ε ε= ε 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, 1966. 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)..
Resonance time delay dδ dk k =k 0 > 0 1 scattering amplitude f ( k, cos θ ) = k = For r, a wave packet in a scattering is written as dkφ ( k )exp i( k where ( k ) 0 ( + 1) e φ : sharply peaked around k = k 0 iδ sin [ ] 1 r Et) + dkφ( k ) r exp[ i( kr Et) ] δ P (cos f ( k,cosθ ) θ ) ($) Assume that at k=k 0 a sharp peak only in a given l channel. For very large t (= time), the nd term in ($) contributes only at the distance k 0 m dδ dk r t k = k 0 ) Time delay caused by the sharply changing term dδ dk > 0 for e i k k 0 dδ ik ( kr Et) k= k0 e dk iδ e in the f : dδ δ ( k ) δ ( k 0 ) + k = k ( k k 0 ) 0 dk time delay in the emergence of the scattered particles = e dδ k ik ( r+ k= k t) dk 0 m m dδ k t D = k = k 0 dk 0 dδ dk < 0 time advance!
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 s 1/ orbits.
A computer program to calculate one-neutron resonance (energy and width) in a spherical Woods-Saxon potential is available. Is there anybody who wants to have it?
Some summary of weakly-bound and positive-energy neutrons in spherical potentials (β=0) Unique role played by neutrons with 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. Only higher-l neutron orbits have one-particle resonance with small width. Change of many-body correlation, such as pair correlation and deformation in loosely bound nuclei
.. Hartree-Fock (HF) approximation self-consistent mean-field A mean-field approximation to the nuclear many-body problem with rotationally invariant Hamiltonian, H = i + m 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, δ ( 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, ϕ ( r ) i j (i and j) = 1,,.., A Variational principle δ Ψ H Ψ = 0 together with subsidiary conditions 3 ϕ i( ri) d ri = 1 leads to the HF equation. OBS. The HF solution is not an eigen function of the Hamiltonian H. Ψ
ex. HF equations for particles (a simple example!) 1 ϕ1( r1) ϕ( r1) Ψ (1, ) = ϕ ( r ) ϕ ( r ) 1 { 3 3 1 r 1 1 r 1 rvr 1 r rdr r 1 rvr 1 r 1 rdr 1 1 r 1 ( ) ( ) ( ) (, ) ( ) ( ) ( ) (, ) ( ) ( ) m ϕ + ϕ ϕ ϕ ϕ ϕ ϕ = εϕ 3 3 r r 1 rvr r 1 rdr 1 r 1 rvr r rdr r ( ) ( ) ( ) (, ) ( ) ( ) ( ) (, ) ( ) ( ) m ϕ + ϕ ϕ ϕ ϕ ϕ ϕ = ε ϕ exchange term (absent in Hartree approximation) Hartree potential V ( ) H r 1 and V ( ) H r Find the solutions, ϕ ( r ) ϕ ( r ) 1 and the above coupled equations., with ε 1 and ε, which satisfy simultaneously The usual procedure of solving the HF equation is; ϕ ( r ) V( r w.f. pot. ) w.f. ϕ ( r ) 1 1 ϕ( r 1 1 1 ) V( r ) ϕ r Find self-consistent solutions together with eigenvalues, ε 1 and ε. ( )
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.
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 3s1/ 50 50
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 For E(I) = AI(I+1), E(I=4) E(I=) = 3.33 ex. In the ground band of 168 Er 64.081 79.800 = 3.31 symmetry axis A rotational band, consisting of members with I K. K I
+ 0+ E Observed E-transition probabilities of the ground state (I=0) to the first excited + state in stable even-even nuclei. The single-particle value used as unit is 5 3 4 / 3 4 ( E) e R 0.30A e fm B sp = = 4 π 5 Bohr & Mottelson, Nuclear Structure, Vol.II, 1975, Fig.4-5
WARNING : many different definitions (and notations) of Y 0 deformation parameters δ intrinsic quadrupole moment Q 0 = 4 3 Z r k k= 1 δ uniformly-charged spheroidal nucleus with a sharp surface δ = 3 ( R ( R ) ( R ) + ( R ) 3 3) β δ osc or ε β is defined in terms of the expansion of the density distribution in spherical harmonics. radius density R( θ, ϕ) = R0 (1 + βy0 ( θ ) +...) ρ0 * ρ( r) = ρ ( r) R0 ( βy0( θ ) r * 0 +...) In the deformed harmonic oscillator model it is customary to use ω ω3 R3 R ε = δosc 3 ω + ω R 3 av To leading order, δ β δ osc, but. δn δ p for stable nuclei, but δ n < δ p possibly for neutron-rich nuclei towards the neutron-drip-line, since R > R ) R δ R δ n p n n p p
Nuclei with deformed ground state close to the β stability line All single or double closed-shell nuclei are spherical. some typical examples of deformed nuclei : 1 C 6 Oblate (pancake shape) 0 Ne 10 Prolate (cigar shape) 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. (oblate) (prolate?) 40Zr (prolate?) 7 76 36 Kr36 38 Sr 80 38 40 30 Ne 3 10 0 Mg 1 0 ) N 0 ex. ( island of inversion ) 3) N 8 ex. etc. 1 11 4 Be8 7 4 Be
Deformed ground state of N Z nuclei (proton-rich compared with stable nuclei) Coexistence of prolate and oblate shape : (Z=36) 4+ + 0+ 61 484 4+ + 0+ 538 90 76 38Sr 38 80 40 Zr40 Most probably prolate OBS. Almost all stable nuclei with N (or Z) = 40 are spherical. Ex. + 1671 4+ 1363 4+ 1469 + 169 0+ 1300 Shape of the ground state (from Coulomb excitation); oblate prolate prolate (A.Goergen, Gammapool workshop in Trento, 006) 0+ 854 + 918 + 635 0+ 0 0+ 0 74 Se 94 34 40 54 40Zr
315 (4+) 10 (4+) S(n) = 504 kev 885 + ½- 319.8 660 Strong E1 Strong E ½+ 0 0+ 0+ + 3 34 1 Mg0 1 Mg 11 4 7 β = 0.5 0.58 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! N=8 is not a magic number! (in this neutron-rich nucleus) S(n) = 5.81 4.16 MeV E(4+) E(+) = (.6) (3.1) N=0 is not a magic number! (in these neutron-rich nuclei)
Example of deformed excited states of magic nuclei 40 Ca : doubly-magic nucleus, spherical ground state 0 0 strongly-deformed band Q t = 1.80 + 0.39 0.9 eb from Doppler shift measurement β = 0.59 + 0.11 0.07 From E.Ideguchi et al., Phys.Rev.Lett. 87 (001) 501.
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.
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 structure (in the body-fixed system) of the nuclei. The notion of one-particle motion in deformed nuclei can be, in practice, much more widely, in a good approximation, applicable than that 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.
What can one learn from 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=0 rotational 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 shape. Kramers degeneracy : The levels in an odd-fermion system are at least doubly degenerate.
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 quantum spectra possibility of gaining energy by breaking away from spherical symmetry using the degeneracy 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
3.. Important deformation and quantum numbers in deformed nuclei Axially symmetric quadrupole (Y0) deformation (plus R-symmetry) - most important deformation in nuclei R x R z R (= R x = R y )< R z R (= R x = R y )> R z prolate (cigar shape) oblate (pancake shape) R y Axially-symmetric quadrupole-deformed harmonic-oscillator potential M = with V = ( ωz z + ω ( x + y )) H T + V x, ny, nz ( n z x H n = ε, n ) n, n, n where n = n x + ny y 1 3 δ ε ( n, nz ) = ( nz + ) ωz + ( n + 1) ω = ϖ N (3 nz N) + 3 where 1 = ( ω + ω ) 3 z and N = n + n + n ϖ z x y z deformation parameter ω ωz δ 3 ω + ω z Rz R R av δ > 0 δ < 0 R z R z > R < R : prolate : oblate
One-particle spectrum of Y0-deformed harmonic-oscillator potential 3 δ ε ( N, nz ) = ϖ ( N + (3nz N)) 3 degeneracy (1) At δ = 0 : spherical, 3 ε ( N ) = ϖ ( nx + ny + nz + ) = ϖ ( N + degeneracy (N+1)(N+) 3 ) () At δ 0 ε(n) splits into (N+1) levels, ε ( N, n z ) ) n z = 0, 1,,., N ω : ω 3 oblate 6 4 ND spherical symmetric prolate SD The level with ( + 1) Spin n ε ( N, nz ) ) z has degeneracy N n = n = n + n n y = 0, 1,., n x y and (3) Note closed shell appears, when ω : ωz is a small integer ratio. large degeneracy ex. 1 ω = ω z ε ( N, nz ) = ω z ( nz + n + + ) where one can have many combinations of integer (n z, n ) values that give the same value of (n z + n ).
One-particle Hamiltonian with spin-orbit potential H = T + V ( r, θ ) V ( r, θ ) = V0 ( r) + V ( r) Y0( θ ) + Vs ( r)( s) 5 Y0( θ ) = (3cos θ 1) 16π 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 ) [ f(r) Y 0 (θ), l z + s z ] = 0 and [( s), z + sz ] = 0
4. One-particle motion sufficiently bound in Y 0 deformed potential 4.1. Normal-parity orbits and/or large deformation M H 0 = T + ( ω z z + ω ( x + y )) H ' = Vs ( r)( s) V ( r, θ ) = V0( r) + V( r) Y0( θ ) + Vs ( r)( s) V( r) Y0( θ ) >> Vs ( r)( s) ε 1 ( N, n n = + ) ω + ( + 1) ω z ) ( nz z has ( n + 1) degeneracy. n = n x + n y The degeneracy can be resolved by specifying n x = 0, 1,, n for a given n. However, since [H 0, l z ] = 0, (l z : z-component of one-particle orbital angular momentum), quantum number Λ ( l z ) can be used to resolve the ( n +1) degeneracy. Possible values of Λ are Λ = ±n, ±(n -),., ±1 or 0. The basis [ n, n,λ] is useful for H ' ( s) z Including spin, Σ s z, n nzλσ H n nzλσ = ε ( n, nz ) + n nz Vs ( r) n n z ΛΣ [ n n ] z ΛΣ or N = [ N n z ΛΩ] n + n z and Ω = Λ + Σ : asymptotic quantum numbers
[ N n z ΛΩ] : approximately good quantum numbers for large Y 0 deformation ( Ω is an exact quantum-number ) Thus, in deformed nuclei it is customary to denote observed one-particle levels, or one-particle levels obtained from finite-well potentials, or HF one-particle levels etc. by [ N n z ΛΩ], in which Nn z ΛΩ> is the major component of the wave functions. Denote Ω > 0 value, though ± Ω doubly degenerate (Kramers degeneracy). ex. For deformation δ = 0.3 the proton one-particle wave-functions obtained by diagonalizing H = T + V(r,θ) with a (l s) potential are [411 3/] > = 0.96 411 3/> + = 0.418 g 9/ > -0.140 g 7/ > +0.864 d 5/ > +0.46 d 3/ > [411 1/] > = 0.900 411 1/> + = -0.163 g 9/ >+0.396 g 7/ > -0.099 d 5/ > +0.848 d 3/ > +0.97 s 1/ > [400 1/] > = 0.968 400 1/> + =0.147 g 9/ > -0.07 g 7/ > +0.539 d 5/ > -0.160 d 3/ > +0.811 s 1/ > (From A.Bohr & B.R.Mottelson, Nuclear Structure, vol.ii, Table 5-.)
V ( r, θ ) = V0 ( r) + V ( r) Y0( θ ) + V s ( r)( s) 4.. high-j orbits and/or small deformation V ( r) Y0( θ ) << Vs ( r)( s) those pushed down by potential : ( s) ex. g 9/,h 11/, i 13/, j (= one-particle angular momentum) is approximately a good quantum number. H 0 = T + V 0 + V ls (r) ( s) H = V (r) Y 0 (θ) For a single-j shell, H 0 lj > = ε 0 (lj) lj > H ljω > = ε(ljω) ljω > ε(ljω) = ε 0 (lj) + < ljω H ljω > = ε 3Ω j( j + 1) 5 j) + j V ( r) 4 j( j + 1) 4π 0( j deformation parameter spherical : (j+1) degeneracy Y 0 deformed : ± Ω degeneracy
4.3. Nilsson diagram one-particle spectra as a function of deformation [Nn z ΛΩ] Diagonalize H = T + V(r,θ) where V ( r, θ ) = V0 ( r) + V ( r) Y0( θ ) + Vs ( r)( s) π = + Levels are doubly degenerate with ± Ω. π = (π,ω) : exact quantum numbers. Levels with a given (π, Ω) interact! i.e. levels with the same (π, Ω) never cross! oblate prolate spherical symmetric
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.
Intrinsic configuration in the body-fixed system Good approximation ; (a) In the ground state of eve-even nuclei K A Ω i= 1 i = 0 Namely, ± Ω levels are pair-wise occupied. (b) In low-lying states of odd-a nuclei K A i= 1 Ω i Ω of the last unpaired particle. Low-lying states in deformed odd-a nuclei may well be understood in terms of the [Nn z ΛΩ] orbit of the last unpaired particle.
ex. The N=13 th neutron orbit is seen in low-lying excitations in 5 Mg 13 f 7/ s 1/ Note (a) I K ( I 3 ) (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. 1) Leading-order E and M1 intensity relation works pretty well Q 0 +50 fm δ 0.4 (g K g R ) 1.4 for [0 5/] etc.
ex. 11 4 Be 7 (N = 7) ½- ½+ S n = 504 kev 319.8 0 (i.e. neutron binding energy = 504 kev) The observed spectra can be easily understood if the deformation δ ~ 0.6. Indeed, the observed deformation in 1 Be(p,p ) is β ~ 0.7. N=8 is not a magic number! An additional element : weakly-bound [0 ½] major component becomes s 1/ (halo) one-particle energy is pushed down relative to p 1/ S 1/ In the spherical shell-model the above ½+ state must be interpreted as the 1-particle (in the sd-shell) -hole (in the p-shell) state, which was pushed down below the ½- state (1-hole in the p-shell) due to some residual interaction.
Table 1. Selection rule of one-particle operators between one-particle states with exact quantum numbers (N n z ΛΩ). The matrix elements between the 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!
Table. Matrix-elements of one-particle operators in (l s) j, Ω representations λ ( s) j, Ω r Yλ 0 ( 1s) j1,ω = δ ( s) j,ω Rj () r C(,1/, j; mmsω) Ym (, θφχ ) 1/, m r mm λ ( ) 1 1 Ω 1 1+ λ j1+ j+ + λ ( 1),( 1) j r j ( 1) ( 1) 1 1 C j j λ;1/, 1/,0) C j j λ; Ω,,0) ( 1 λ j r 1 j1 s j j 0 drr () r R () r r λ 1 1 ( 1 Ω s ( j1 + 1)( j + 1) 4π (λ + 1) λ ( s) j, Ω + 1 r Yλ1 ( 1s) j1, Ω = = δ ) 1+ λ λ j1 + j + 1+ λ Ω 1/ ( 1),( 1) j r j ( 1) ( 1 C j j λ;1/, 1/,0) C j j λ; Ω + 1,,1) ( 1 ( λ 1) ( 1 s) j1, Ω r Yλ 1 ( s) j, Ω + 1 λ ( s) j, Ω + r Yλ ( 1s) j1, Ω = δ 1 1 ) ( 1 Ω 1 + λ λ j1 + j + 1+ λ Ω 1/ ( 1),( 1) ) j r j ( 1) ( 1 C j j λ;1/, 1/,0) C j j λ; Ω +,,) ( 1 = λ s) j, Ω r Y ( s) j, Ω ( 1 1 λ + 1 1 ) ( 1 Ω ( j1 + 1)( j + 1) 4π (λ + 1) ( j1 + 1)( j + 1) 4π (λ + 1)
Table (continued) ( s) j, Ω + 1 s+ ( 1s) j1, Ω ( s ± = s x ± i s y etc. ) 1 + j1 + = δ, )( 1) 1/ 3( j + 1) C ( j1, 1, j; Ω,1, Ω + 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, Ω + 1 + ( 1s) j1, Ω = + j δ (, )( 1) 1/ 1 ( j1 + 1) 1( 1 + 1)(1 j 1 j1 1 + 1) C ( j1 1 j; Ω,1, Ω + 1) W ( j ;1/ 1,1) 1 j ( s) j, Ω + 1 j+ ( 1s) j1, Ω = δ ( j1, j) ( j Ω)( j + Ω + 1) j 1 j1 ( 1 j1 s) j, Ω sz ( s),ω 3( 1) 1 + j1 j1 + δ (, )( 1) 1/ 1 C ( j1,1, j; Ω,0, Ω) W ( 1/, j,1/, j1; 11) = j 1 j1 ( 1 j1 = s) j, Ω z ( s),ω + j + δ (, )( 1) 1/ 1 j1 + 1 1( 1 + 1)(1 1 + 1) C j 1 j ; Ω0 ) W ( j ;1/ 1,1) ( 1 Ω 1 j j 1 j1
Table (continued) Phase convention in wave functions - important in non-diagonal matrix-elements 1) (ls)j or (sl)j ; ( s) j = ( 1) 1 + j ( s) j Y m ( θ, φ) ) or i ( θ, φ ) Y m 3) R j (r) { > 0 (or < 0) for r 0, or > 0 (or < 0) for r very large, or output of computers
5. Weakly-bound and one-particle resonant neutron levels in Y0 deformed potential harmonic-oscillator potential
Well-bound one-particle levels in deformed potential A.Bohr and B.R.Mottelson, vol., Figure 5-1. One-particle levels in (Y 0 ) deformed harmonic oscillator potentials 4 [31 3/] [N n z Λ Ω ] asymptotic quantum numbers Parity π = (-1) N Each levels are doubly-degenerate with ±Ω 6 doubly-degenerate levels in sd-shell 3 Ω π =1/ + (l min =0) Ω π =3/ + (l min =) 1 Ω π =5/ + (l min =)}1 particles 6
5.1. Weakly-bound neutrons Radial wave functions of the [00 ½] level in Woods-Saxon potentials s 1/ d 3/ d 5/ (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 = -0.0001 MeV. Similar behavior to wave functions in harmonic osc. potentials. Wave functions unique in finite-well potentials.
W-S potential parameters are fixed except radius R. I.H., Phys. Rev. C7, 04301 (005) Woods-Saxon l min = 0 l min = l min = 1 [00 ½] [0 ½] [11 ½] potential strength (r 0 = 1.7 fm is used.)
PRC69, 041306R (004) 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.
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. three Ω π =1/ + Nilsson orbits in the sd-shell ;
5.. One-particle resonant levels eigenphase formalism 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). Resonant state with ε Ω = +100 kev Bound state with ε Ω = 0.1keV 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 + 100 kev.
Relative probability of s 1/ component inside the W-S potential P( s 1/ ) = d 5/ V ( r) d 5/ + s 1/ d 3/ V ( r) s 1/ V ( r) d 3/ + s 1/ V ( r) s 1/ In order that one-particle resonance continues for ε Ω >0, P(s 1/ ) at ε Ω =0 must be smaller than some critical value. The critical value depends on the diffuseness of the potential. One-particle shell-structure change for ε Ω (<0) 0 produces the large change of P(s 1/ ) values of respective [N n z ΛΩ] orbits as ε Ω (<0) 0. One-particle resonance
Positive-energy neutron levels in Y 0 -deformed potentials 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/,.., components l min = Ω π = 1/ p 1/, p 3/, f 5/, f 7/, h 9/,.., components l min =1 etc. The component with l = l min plays a crucial role in the properties of possible one-particle resonant levels. (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.)
For ε Ω R < 0 jω ( r) rh ( α r) b Do not restrict the system in a finite box! for r where h ( iz) j ( z) + in ( z) and α b mε Ω For ε Ω R jω where > 0 ( r) cos( δω) rj( αcr) sin( δω) rn( αcr) π sin( α + ) c r δ Ω m ε Ω αc for r δω expresses eigenphase. A.U.Hazi, Phys.Rev.A19, 90 (1979). K.Hagino and Nguyen Van Giai, Nucl.Phys.A735, 55 (004). A given eigenchannel : asymptotic radial wave-functions behave in the same way for all angular momentum components.
A one-particle resonant level with ε Ω is defined so that one eigenphase δ Ω increases through (1/)π as ε Ω increases. δ δ Ω (1/)π ε Ω ε res When one-particle resonant level in terms of one eigenphase is obtained, the width Γ of the resonance is calculated by Γ dδ Ω d εω res ε Ω = ε Ω Phys. Rev. C7, 04301 005)
I.H., Phys. Rev. C73, 064308 (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. In the limit of β 0, the definition of one-particle resonance in eigenphase formalism the definition in spherical potentials found in text books.
Variation of all three eigenphases (s 1/, d 3/ and d 5/ levels are included in the coupled channels.) eigenphase sum π/ π/ eigenphase sum No weakly-bound Nilsson level is present for this potential. A weakly-bound Nilsson level is present for this potential.
5.3. Examples of Nilsson diagrams for light 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/ ) 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
1d3/ At β=0 ; ε(s1/)-ε(1d5/) = 140 kev s1/ 1d5/ 17 C 11 (3/+) S(n) = 0.73 MeV 1p1/
(MeV)
ε(1f 5/ ) = +8.96 MeV 1f7/ p1/? At β=0 ; ε(p3/) < ε(1f7/) p3/? 33 Mg 1 (3/ ) 1d3/ S(n) =. MeV 31 Mg 19 (1/+) s1/ S(n) =.38 MeV 1d5/
(MeV)
1f5/ At β=0 ; ε(p3/) ε(1f7/) = 680 kev p1/? p3/ 1f7/ 37 Mg 5 S(n) = a few hundreds kev? 1d3/ s1/
Appendix. Angular momentum projection from a deformed intrinsic state (ex. not appropriate for including the rotational perturbation of intrinsic states) Rotational operator ) (Ω R Ω : Euler angles ),, ( γ β α z y z J i J i J i e e e R γ β α (Ω) Rotation matrix ) ( ' Ω J D MM ) ( '), ( '), ( ' ' ' ) ( ' Ω = Ω J D MM J J M J R JM δ α α δ α α Inverting the expression = Ω Ω J J D MM JM JM R α α α ' ) ( ) ( ' Multiplying by ) ( ' Ω J MM D and integrating over Ω, we obtain a projection operator Ω Ω Ω + = ) ( ) ( 8 1 R D d J JM JM P J MM J M π α α α We need to calculate the expressions Ω Ω Ω + = φ φ π φ φ ) ( ) ( 8 1 R D d J P J MM J M Ω Ω Ω + = φ φ π φ φ ) ( ) ( 8 1 HR D d J HP J MM J M φ
Appendix If φ is axially symmetric, φ φ M J z = M i J i M i e e e R y γ β α φ φ φ φ = (Ω) M i J i M i J MM e JM JM e e D y γ β α = (Ω) then, using the reduced rotation matrix ' ) ( ' JM JM e d J y i J MM θ θ = φ φ θ θ θ φ φ θ π J y i J MM J M e d d J P + = ) ( sin 1 0 + = π θ φ φ θ θ θ φ φ 0 ) ( sin 1 y J i J MM J M He d d J HP 1 for θ << 1, decreases quickly as θ larger, is symmetric about θ = π/. { φ φ θ y J e i