University of Tokyo (Hongo) Nov., Ikuko Hamamoto. Division of Mathematical Physics, LTH, University of Lund, Sweden

Transcription

1 University of Tokyo (Hongo) Nov., 006 One-particle motion in nuclear many-body problem - from spherical to deformed nuclei - from stable to drip-line - from particle to quasiparticle picture kuko 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. &

2 . ntroduction. Mean-field approximation to spherical nuclei.. Phenomenological one-body potentials (harmonic-oscillator and Woods-Saxon potentials).. Hartree-Fock approximation self-consistent mean-field 3. Observation of deformed nuclei 3.. Rotational spectrum and its implication 3.. mportant deformation and quantum numbers in deformed nuclei 3.3. Axially-symmetric quadrupole-deformed harmonic-oscillator potential 4. One-particle motion sufficiently-bound in Y 0 deformed potential 4.. 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 5. Back to the laboratory system (with Y 0 deformed intrinsic shape) 5.. Total (= intrinsic x rotational) wave functions and consequence of symmetry 5.. Rotational energy and intensity relations 5.3. nterpretation of experimental data

3 6. [Seminar] : Weakly-bound and positive-energy neutrons 6.. n spherical nuclei 6.. Weakly-bound neutrons in Y 0 deformed nuclei 6.3. Positive-energy neutrons in Y 0 deformed nuclei 7. Some topics of proton-rich and neutron-rich nuclei 7.. Proton emission in nuclei beyond proton drip-line 7.. Density distribution in neutron-rich nuclei - neutron halo and neutron skin 8. Pairing interaction and pair correlation 8.. Occupation-number representation and second quantization 8.. Paired particles and resulting correlation in spherical nuclei 8.3. BCS approximation 8.4. BCS in deformed nuclei 8.5. Hartree-Fock-Bogoliubov (HFB) theory 9. Collective motion based on particle-hole excitations - giant resonances and sum-rules 9.. sovector giant dipole resonance 9.. sovector and isoscalar giant quadrupole resonances 9.3. Giant resonances of charge-exchange type

4 . ntroduction 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

5 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 n this system a harmonic-oscillator potential is also often used.

6 . Mean-field approximation to spherical nuclei.. Phenomenological one-body potentials 3-dimensional harmonic oscillator potential H = + mω r m harmonic-oscillator potential has a spectrum 3 ε = N + ω where N = nx + ny + nz = ( n r ) + in rectilinear coordinates in polar coordinates = N, N-, 0 or n the above figure V () r = mω r + const where const = 55 MeV ω = 8.6 MeV Degeneracy of the major shell with a given N ( + ) spin = (N+)(N+) (l = even or odd) leads to the magic numbers, 8, 0, 40, 70,, 68,

7 One-particle levels for β stable nuclei ( S n S p 7-0 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, 0,8,50,8,6, Nuclei with magic number : spherical shape Normal-parity orbits majority in a major shell of medium-heavy nuclei High-j orbits, g 9/, h /, i 3/, j 5/, which have parity different from the neighboring orbits do not mix with them under quadrupole deformation and rotation. One-particle motion in the mean-field shell structure (= bunching of one-particle levels) nuclear shape

8 Phenomenological finite-well potential : Woods-Saxon potential - an approximation to Hartree-Fock (HF) potential V() r V f () r = where WS f() r = r R + exp a V(r) R a r a : diffuseness R : radius R = r 0 A/3 A : mass number V WS standard values of parameters V r 0.7 fm a 0.67 fm WS N = 5± 33 A Z MeV for + for neutrons for protons

9 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 n the above figure the parameters are chosen so that the root-mean-square radius for the two potentials, are approximately equal.

10 Centrifugal potential + Woods-Saxon potential dependence on l m x y z V() r cot ( ) r = + + θ + + V r m r r r θ θ sin θ φ ( ) = r + V() r m r r r centrifugal potential Woods-Saxon pot. centrifugal pot. W-S + centrifugal pot.

11 Height of centrifugal barrier + ( ) R h where R h > r 0 A /3 The height is higher for smaller nuclei and for orbits with larger l. ex. For the Woods-Saxon potential with R=5.80 fm, a=0.65 fm, r 0 =.5 and V WS = 50 MeV ; l height of centrifugal barrier 0 0 MeV

12 Schrödinger equation for one-particle motion with spherical finite potentials H = m x y z V() HΨ= εψ Ψ= U () r Y m r (x, y, z) (r, θ, φ) ( ) Y m( θ, φ) = ( + ) Y m( θ, φ) d ( + ) r + + V () r U () r = ε U () r mrdr mr (#) Writing U () r = R () r, the radial equation (#) can be written as r d dr ( + ) m + ( ε V() r ) R () 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 and ε > 0 α = ε h ( iz) j ( z) + in ( z) R () r cos( δ ) krj ( kr) sin( δ ) krn ( kr) δ : phase shift where k m = ε

13 One-body spin-orbit potential in phenomenological potentials : surface effect! n 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 ρ = ( s) r r ρ() r 0 n practice, one often uses the form Vc ( r) V s () r = λ( s) r r i.e. s : P-inv & T-inv! = ( ) 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

14 n the presence of spin-orbit potential V ls (r) ( ( s) ), the total angular momentum of nucleons 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 ( s) = { j s } = j( j+ ) ( + ) ( + ) = { l for j = l / l for j = l + / HΨ= εψ () r X Ψ= R where j jm jm /, j j s j m m mm, r X C(,, j; m, m, m ) Y ( θφχ, ) s s The radial part of the Schrödinger equation becomes d dr ( + ) m + ( ε j V() r V s() r ) R j() r = 0 r

15 .. 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) =,,.., A Variational principle δ Ψ H Ψ = 0 together with subsidiary conditions 3 ϕ i( ri) d ri = leads to the HF equation. OBS. The HF solution is not an eigen function of the Hamiltonian H. Ψ

16 ex. HF equations for particles (a simple example!) ϕ( r) ϕ( r) Ψ (, ) = ϕ ( r ) ϕ ( r ) { 3 3 r r rvr r rdr r rvr r rdr r ( ) ( ) ( ) (, ) ( ) ( ) ( ) (, ) ( ) ( ) m ϕ + ϕ ϕ ϕ ϕ ϕ ϕ = εϕ 3 3 r r rvr r rdr r 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 V ( ) H r and Find the solutions, ϕ ( r ) ϕ ( r ) and the above coupled equations., with ε 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 ) ϕ( r ) V( r ) ϕ r Find self-consistent solutions together with eigenvalues, ε and ε. ( )

17 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 6 One of Skyrme interactions ; SkM* See : J.Bartel et al., Nucl. Phys. A386 (98) 79.

18 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/ 50 50

19 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, SCATTERNG THEORY OF WAVES AND PARTCLES, McGraw-Hill, 966. At ε res ; () 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)..

20 3. Observation of deformed nuclei 3.. Rotational spectrum and its implication Some nuclei are deformed. Observation : ) rotational spectra E() A(+) ) large quadrupole moment or large (E; -) transition probability For E() = A(+), E(=4) E(=) = 3.33 ex. n the ground band of 68 Er = 3.3 symmetry axis A rotational band, consisting of members with K. K

21 + 0+ E Observed E-transition probabilities of the ground state (=0) to the first excited + state in stable even-even nuclei. The single-particle value used as unit is / 3 4 ( E) e R 0.30A e fm B sp = = 4 π 5 Bohr & Mottelson, Nuclear Structure, Vol., 975, Fig.4-5

22 WARNNG : many different definitions (and notations) of Y 0 deformation parameters δ intrinsic quadrupole moment Q 0 = 4 3 Z r k k= 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 ( + βy0 ( θ ) +...) ρ0 * ρ( r) = ρ ( r) R0 ( βy0( θ ) r * ) n 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

23 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 : C 6 Oblate (pancake shape) 0 Ne 0 Prolate (cigar shape) rare-earth nuclei with 90 N mostly prolate Some new region of deformed ground-state nuclei away from β stability line; ) N Z 38 ex. (oblate) (prolate?) 40Zr (prolate?) Kr36 38 Sr Ne Mg 0 ) N 0 ex. ( island of inversion ) 3) N 8 ex. etc. 4 Be8 7 4 Be

24 Deformed ground state of N Z nuclei (proton-rich compared with stable nuclei) Coexistence of prolate and oblate shape : (Z=36) Sr Zr40 Most probably prolate 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

25 35 (4+) 0 (4+) S n = 504 kev ½ Strong E Strong E ½ Be Mg0 Mg 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, f the nucleus is deformed! N=8 is not a magic number! (in this neutron-rich nucleus) β = E(4+) E(+) =.6 3. N=0 is not a magic number! (in these neutron-rich nuclei)

26 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.deguchi et al., Phys.Rev.Lett. 87 (00) 50.

27 mplication of rotational spectra : () 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. ndeed, collective rotation is the best established collective motion in nuclei.

28 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.

29 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() (+) ), where K is the projection of angular momentum along the symmetry axis. The K=0 rotational band has only = 0,,4, shape is R- invariant, Kramers degeneracy time reversal invariance, etc. (b) rotational energy, E() - E(-) 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. (f 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 a odd-fermion system are at least doubly degenerate.

30 3.. mportant deformation and quantum numbers in deformed nuclei Axially symmetric quadrupole (Y0) deformation (plus R-symmetry) - most important deformation in nuclei R x R (= Rx = Ry )< Rz prolate (cigar shape) R z R (= R x = R y )> R z oblate (pancake shape) R y One-particle Hamiltonian H = T + V ( r, θ ) V ( r, θ ) = V0 ( r) + V ( r) Y0( θ ) + V s ( r)( s) 5 Y0( θ ) = (3cos θ ) 6π where θ is polar angle w.r.t. the symmetry axis ( = z-axis) Quantum numbers of one-particle motion in H () Parity π = (-) l where l is orbital angular momentum of one-particle. () Ω l z +s z ) [ f(r) Y0 (θ), l z + s z ] = 0 and [( s), z + sz ] = 0

31 Why are some nuclei deformed? Usual understanding ; Deformation of ground states (ND, R : R z :.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 : ) at high spins in rare-earth nuclei or fission isomers in actinide nuclei new shell structure (and new magic numbers!) at large deformation

32 3.3. Axially-symmetric quadrupole-deformed harmonic-oscillator potential ) )( ( s r V s )) ( ( y x z M z + + ω ω ) ( ) ( ) ( ), ( 0 0 θ θ Y r V r V r V + = V T H + = z y x z z y x n n n n n n n n H,, ), (,, = ε y n x n n where + = = ω ω ε ) ( ) ( ), ( n n n n z z z 3 (3 ) 3 z N n N δ ϖ = + ) ( 3 + = ω ω ϖ z where deformation parameter av z z z R R R + ω ω ω ω δ 3 z ω z R ω R and since + + = ), ( z z y x M r V ω ω ω θ x y z N n n n = + + δ > 0 : prolate > R R z δ < 0 : oblate < R R z

33 One-particle spectrum for axially-symmetric quadrupole-deformed harmonic-oscillator potential 3 δ ε ( N, nz ) = ϖ ( N + (3nz N)) 3 degeneracy () At δ = 0 : spherical, 3 ε ( N ) = ϖ ( nx + ny + nz + ) = ϖ ( N + degeneracy (N+)(N+) 3 ) () At δ 0 ε(n) splits into (N+) levels, ε ( N, n z ) ) n z = 0,,,., N 6 4 The level with ε ( N, n ) has degeneracy z ( + ) Spin n ) N n = n = n + n z n y = 0,,., n x y and ω : ω 3 oblate ND spherical symmetric prolate SD (3) Note closed shell appears, when ω : ωz is a small integer ratio. large degeneracy ex. 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 ).

34 4. One-particle motion sufficiently bound in Y 0 deformed potential 4.. Normal-parity orbits and/or large deformation M H 0 = T + ( ω z z + ω ( x + y )) H ' = V s ( r)( s) V ( r, θ ) = V0( r) + V( r) Y0( θ ) + V s ( r)( s) V( r) Y0( θ ) >> V s ( r)( s) ε ( N, n n = + ) ω + ( + ) ω z ) ( nz z has ( n + ) degeneracy. n = n x + n y The degeneracy can be resolved by specifying n x = 0,,, 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 +) degeneracy. Possible values of Λ are Λ = ±n, ±(n -),., ± or 0. The basis [ n, n,λ] is useful for H ' ( s) z ncluding spin, Σ s z, n nzλσ H n nzλσ = ε ( n, nz ) + n nz V s ( r) n n z ΛΣ [ n ] n or [ N n z ΛΩ] : asymptotic quantum numbers z ΛΣ N = n + n z and Ω = Λ + Σ

35 [ 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) are [4 3/] > = /> + = 0.48 g 9/ > g 7/ > d 5/ > d 3/ > [4 /] > = /> + = g 9/ > g 7/ > d 5/ > d 3/ > s / > [400 /] > = /> + =0.47 g 9/ > g 7/ > d 5/ > d 3/ > +0.8 s / > (From A.Bohr & B.R.Mottelson, Nuclear Structure, vol., Table 5-.)

36 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 ) << V s ( r)( s) those pushed down by potential : ( s) ex. g 9/,h /, i 3/, 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 + ) 5 j) + j V ( r) 4 j( j + ) 4π 0( j deformation parameter spherical : (j+) degeneracy Y 0 deformed : ± Ω degeneracy

37 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 ) + V s ( 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

38 Proton orbits in prolate potential (50 < Z < 8). g 7/, d 5/, d 3/ and s / orbits, which have π = +, do not mix with h / by Y 0 deformation. [N=4, n z =0] Levels are doubly degenerate with ± Ω. h / orbit = high-j orbit with π = At small δ and h / orbit, ε δ (3Ω j(j+) ) 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.

39 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. E operator E operator M operator Gamow-Teller operator From J.P.Boisson and R.Piepenbring, Nucl. Phys. A68(97)385. f 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!

40 ntrinsic configuration in the body-fixed system Good approximation ; (a) n the ground state of eve-even nuclei K A Ω i= i = 0 Namely, ± Ω levels are pair-wise occupied. (b) n low-lying states of odd-a nuclei K A i= 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.

41 5. Back to the Laboratory system (with Y 0 deformed intrinsic shape) From now on: (,, 3) : body-fixed system (x, y, z) : laboratory system z (= axis fixed in the laboratory) 3 (= symmetry axis fixed in the body) K 3 M z J R 3 (sym axis) Total angular momentum = R + J R : angular momentum of collective rotation J : intrinsic angular momentum K = Ω axially-sym shape K ( 3 ) = Ω ( J 3 ) No collective rotation about symmetry axis ; R 3 = 0 ( OBS. No collective rotation in spherically-symmetric nuclei )

42 5.. Total (= intrinsic x rotational) wave functions and consequences of symmetry f the intrinsic and rotational parts of the Hamiltonian are separated, the eigenstates of the Hamiltonian are the product form Ψ α, = Φ α (q) φ α, (ω) Φ α (q) : intrinsic wave-function φ α, (ѡ) : rotational wave-function where α : quantum number specifying intrinsic states, q : intrinsic variable, ω : angular variables specifying the orientation of the deformed body with respect to the laboratory system, : angular-momentum quantum-numbers. Rotational wave functions ( α ) ; () n -dimensional rotation (a rotation about a fixed axis) φ α, (ω) ~ exp ( imθ ) () n 3-dimensional rotation φ α, (ω) ~ D MK (ω) ω θ M ω 3 Euler angles (Φ,θ,ψ), to specify the orientation 3 quantum numbers: of the body. ( ), M ( z ), K ( 3 )

43 D = ( + ) D z MK D = MK = MD MK 3D MK KDMK π 0 MK ω = ( φθψ,, ) dω sinθdθ dφ dψ π π * 8π sinθdθ dφ d D '( ) D '( ) (, ) ( M, M) ( M ', M' ) 0 ψ 0 MM ω MM ω = δ δ δ + [ x, y ] i z x, y, z [ x, y, z,,, 3] = 0,, 3 [, ] = i3 = : referred to the lab.system : referred to the body-fixed system ( ( + ) M ( M ) ) /, M ± i, M = x y ( ( + ) K( ) /, K ± ± i, K ± = K )

44 Rotational degrees of freedom restricted by the symmetry of deformation ex. Spherically symmetric nuclei No collective rotation ex. Axially-symmetric deformed nuclei No collective rotation about the symmetry axis ex. R-invariant axially-symmetric deformation rotation R (π) ( rotation π about the axis symmetry axis) must not be included in the rotational degrees of freedom Correspondingly, the form of total wave function (in general, a sum of products of intrinsic and rotational wave-functions) is governed by the symmetry of deformation.

45 Total wave function for Y 0 deformed intrinsic shape (a) axially-symmetric shape no collective rotation about the sym axis (=3-axis) K ( 3 ) = Ω ( J 3 ) Ψ KM = Φ K ( q) D MK ( φ, θ, ϕ) + 8π J 3 3 R (π) R (π) rotation π about the -axis (b) R-invariant shape, in addition to axial symmetry Ψ KM = + 6 π K { Φ ( q) D ( φ, θ, ψ ) + ( ) + Φ ( q ) D ( φ, θ, ) } K MK M, K ψ K ( q) Φ ( q) Rotation by R (π) does not belong to collective rotation (quantum effect!). Φ K K ($) ( K > 0 ) i.e. from the two intrinsic states with K and K, only a single rotational state can be formed for a given. Obs. The st and nd term in ($) can be connected by the operator with K = K. ( ) dependent term in observables ex. For K=/ bands the term ( ( ) ) in rotational energy For a Hamiltonian with a coupling between intrinsic and rotational motion, a set of wave functions ($) can be used as a basis for diagonalization. ex. particle-rotor model (Bohr &Mottelson, vol., Chap. 4A).

46 Ψ KM = + 6π K { Φ ( q) D ( φ, θ, ψ ) + ( ) + Φ ( q) D ( φ, θ, )} K MK M, K ψ K ω ( φθψ,, ) R-inv shape the cross term of the first and second terms in the above { } can produce ; ex. ( ) dependent term in the expectation value of the operator j ± ( ) Φ ( qd ) ( ω ) j Φ ( qd ) ( ω ) + K K M, K ± K MK ( ) + K Φ ( ) ( ), ( ) K q j K q dωd ± Φ M K ω DMK( ω) ( ~ Coriolis coupling) that is non-zero only for K=/. ) j ± and change K-value by ±. ( ) dependent term in the rotational energy of K=/ bands. ex. dependent part of matrix elements of the operator Θ λ λ D λ µ = Θν µν( ω) ( ) ( ) Φ ( qd ) ( ω) Θ D ( ω) Φ ( qd ) ( ω) + K λ λ K M, K ν µν K MK ν + K λ λ K q ν K q d DM, K Dµν DMK ν ( ) Φ ( ) Θ Φ ( ) ω ( ω) ( ω) ( ω) can be non-zero for ν = K. ν λ Θ µ λ Θ ν : operator in the lab system : operator in the intrinsic system For example, in B(M) values for K=/ bands, in B(E) for K= bands, but in B(E) for K=/ bands. λ = and ν for M λ = and ν for E

47 R-invariance : deformation is invariant under R (π) ( rotation π about the -axis) Then, R (π) is not included in collective rotational degrees of freedom. R R (π) can be expressed as R R e (π), rotation π of the lab system (x, y, z) about the -axis R i R (π), rotation π of the body about the -axis Ψ : total wave-function R e Ψ = R Ψ is determined by i R Ψ = R Ψ R R Ψ =Ψ e Then, for i i R R ( + R R ) Ψ= ( R R + ) Ψ i e i e i e e and Ψ, Ψ' ( + Ri Re) Ψ satisfies ReΨ ' = RiΨ' ( + R i R ) Φ e K ( q) D MK ( φ, θ, ψ ) = Φ K ( q) D MK ( φ, θ, ψ ) + ( ) + K Φ K ( q) D M K ( φ, θ, ψ ) ($) ) R e D MK iπ + K ( φ, θ, ψ ) = e D ( φ, θ, ψ ) = ( ) D ( φ, θ, ψ ) MK M K Φ K ( q) R i Φ K ( q) n fact, Φ (q ) K = T Φ (q K ) : ntrinsic state with K, which is degenerate with Φ K (q) where T : time reversal operator R K K since R i inverts the direction of the 3-axis. R-inv Total wave function is a definite combination of two degenerate states with K and K.

48 K=0 band Ψ 4π 0, M = DM, K = 0( φ, θ, ψ ) Φ K = 0( q) = YM ( θ, φ) K 0( q) + K = Φ = R Y e M ( θ, φ) = Y ( π θ, φ + π ) = ( ) Y ( θ, φ) M inverts the direction of the sym axis (=3-axis) M R e R (π), rotation π of lab system (x, y, z) about the -axis = equivalent to invert the 3-axis for the fixed lab system (x, y, z) R i Φ r K = 0 = Φ K =0 R e Ψ = R Ψ i ( ) = r = even for r = + = odd for r = - The ground state of even-even nuclei has K=0 and r = + (Pairwise-occupied (±Ω) nucleon states have r = +.) ) Φ(,) = ( φω () φ () φ () φω ()) where φ R Ω Ω Ω i φω = RiφΩ R i Φ(,) = ( φ () φω () + φω () φ ()) = Φ(, ) Ω Ω for Ω = half integer. ( or Ri φω = φω ) This explains: the ground-band of even-even nuclei has only π = 0 +, +, 4 +,..

49 5.. Rotational energy and intensity relations Some observable quantities in case of K (q) Φ : one-particle configuration [N n 3 ΛΩ] in Y 0 deformed potential (K = Ω) (a) Within a given band Rotational energy : E K, ) A ( + ) + a( ) + rot( decoupling parameter ( + ) δ ( K, ) a [ Nn3ΛΩ] j+ [ Nn3ΛΩ] K0 K [ Nn3ΛΩ] E [ Nn3ΛΩ] B E; K K ) = = ( B( M; K, K, = ± where Q 0 : intrinsic quadrupole moment ) = 5 e 6π = 0, if Ω ½ Q 0 K0 K + K K K [ Nn3ΛΩ] M[ Nn3ΛΩ] + ( ), K,, K K [ Nn3ΛΩ] M[ Nn3ΛΩ] 0 3 e = ( gk gr) K K0 K for K > ½ 4π Mc e > + ( gk gr) + ( ) 3 6π Mc b 0 for K = ½ where b ( = magnetic decoupling parameter) is defined by 3 e ( gk gr) b = [ Nn3ΛΩ = / ] M [ Nn3ΛΩ = / ] 4π Mc g K K = [ Nn3ΛΩ] g 3 + gss3 [ Nn3ΛΩ] g R : g-factor of the even-even core Z/A > denotes the greater of and

50 ( One-particle ) M operator in the intrinsic (= body-fixed) system M g R R + g + g s s 3 e ( M) ν = ( gr R ν + g ν + gs s ν) 4π Mc = R + + s rotational angular momentum of the even-even core = 3 e ( grν + ( g gr) ν + ( gs gr) sν) 4π Mc magnetic moment M transition ) The operator ν does not make any transitions.

51 Rotational spectra unique in the intrinsic configuration with Ω=/ E rot K = Ω = = ( + ) + a( ) +, ( + For one-particle in a single j-shell ( high-j shell) 3/ 9/ decoupling parameter ) lowest of the Ω=/ band j / a = ( ) ( j + ) = + for j=/ / - for j=3/ 3/ +3 for j=5/ / -4 for j=7/ 3/ +5 for j=9/ 5/ -6 for j=/ 3/ and 7/ +7 for j=3/ 5/ 5/ / / 7/ 3/ M.E.Bunker and C.W.Reich,Rev.Mod.Phys.43 (97)348. n rotational bands with high-j configuration [ = j mod ] levels are pushed down relative to [ = j- mod ] levels, also after including the full Coriolis coupling. a= jm= j jm= = jm= j jm= j /, / +, / ( ), / +, / j / = ( ) ( j + )

52 (b) Transitions between bands with one-particle configurations [Nn 3 ΛΩ] and [N n 3 Λ Ω ] ( Ω=K and Ω =K ) B( M; K K' ' ) =, K,, K' K ' K' [ N' n ' Λ' Ω' ] M[ Nn ΛΩ] 3 3 (assume K ½, K ½) kinematical factor intrinsic matrix element, common in all transitions = 0 for - > or K K > + M The ratio of B(M) values between given two bands is obtained from the Clebsch-Gordan coefficients,;, K,, K' K ' K' - B(M) : B(M) : B(M) K [ N ' n3' Λ' Ω' ] K [ Nn3ΛΩ] K, K' K +, K' : K, K' K K' : K, K' K, K' K = Ω K = Ω n reality, observed B( M; K K' ' ) values for but K K > are very small but 0. K-forbidden transitions (observed non-zero value gives a measure of the deviation from axially-symmetric shape.)

53 5.3. nterpretation of experimental data ex. The N=3 th neutron orbit is seen in low-lying excitations in 5 Mg 3 f 7/ s / Note (a) K ( 3 ) (b) the bandhead state has =K. Exception may occur for K=/ bands. (c) some irregular rotational spectra are observed for K=/ bands. ) Leading-order E and M intensity relation works pretty well Q fm δ 0.4 (g K g R ).4 for [0 5/] etc.

54 ex. 4 Be 7 ½- ½+ S n = 504 kev (i.e. neutron binding energy = 504 kev) The observed spectra can be easily understood if the deformation δ ~ 0.6. ndeed, the observed deformation in 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 / (halo) one-particle energy is pushed down relative to p / n the spherical shell-model the above ½+ state must be interpreted as the -particle (in the sd-shell) -hole (in the p-shell) state, which was pushed down below the ½- state (-hole in the p-shell) due to some residual interaction.