Nilsson Model. Anisotropic Harmonic Oscillator. Spherical Shell Model Deformed Shell Model. Nilsson Model. o Matrix Elements and Diagonalization

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1 Nilsson Model Spherical Shell Model Deformed Shell Model Anisotropic Harmonic Oscillator Nilsson Model o Nilsson Hamiltonian o Choice of Basis o Matrix Elements and Diagonaliation o Examples. Nilsson diagrams

2 Spherical shell model Nuclear properties described in terms of nucleons considered as independent particles moving in an average potential create by all nucleons. Experimental evidence for shell effects: Existence of magic numbers:,8,,8,5,8,6 Large single particle separation energies Nuclei are strongly bound at shell closures Derivation of the average field from microscopic two-body forces (selfconsistent Hartree-Fock method). Assume the existence of such a potential and construct it phenomenologically Characteristics of the potential: V r V r r r r R, V r r R

3 Spherical potentials Infinite square well Harmonic oscillator Woods-Saxon potential for V r V r R for r R V r M r V r V exp / r R a Eigen-functions j kr Y m R r Y n m r / R r re L r n n numerically E n, n MR : root of j n Eigen-energies, 3/ N 3/ E n n intermediate

4 Spherical potentials Woods-Saxon potential V r V exp / r R a Harmonic oscillator V r M r Infinite square well for V r V r R for r R

5 Spherical potentials H.O. W.S. Square

6 Spherical potentials & spin-orbit V r M r C s D s j s E n n D C, 3/ SO SO SO for j for j SO

7 Deformed shell model Spherical potential well valid for closed shells Far from closed shells: deformed single particle potential Experimental evidence: Existence of rotational bands: I(I+) spectra Large quadrupole moments and quadrupole transition probabilities Single particle structure Anisotropic Harmonic Oscillator Generalied Woods-Saxon,, r R V r,, V exp a V V r p s LS,,

8 Anisotropic Harmonic Oscillator Ellipsoidal distribution: Anisotropic Harmonic Oscillator as average field m H x y m x y Frequencies are proportional to the inverse of the ellipsoid axes i i For axially symmetric shapes, we introduce the parameter x y From volume conservation x y 3 / , R a

9 Anisotropic Harmonic Oscillator Introducing dimensionless coordinates through the oscillator length b r ' r / b m we get m m 4 H x y m m 3 m 3 H 4 x y x y r r 3 5 H Axial symmetry: cylindrical basis Y N, n, n, m, m s N n n n n n m x y i x, y, n, n, m i n i n n m 3 N N n 3

10 Anisotropic Harmonic Oscillator Eigen-states characteried by Nnm m m R, n n nn mm m s s m n n L m n H n e im N

11 Anisotropic Harmonic Oscillator Energy level structure: N=3 i x, y, n, n, m i n n n m 3 N N n 3 9 N 3 nn m n N n n m n m l n deg 3 5/, 7/ /, 3/ 4 n=3 n= n= n= 3/, 5/ / 3 energy /, 3/ deformation 3 /

12 The Nilsson model: Hamiltonian Axially symmetric harmonic oscillator potential +spin-orbit term +l term H H C s D N r ry s N C D N = N N 3

13 The Nilsson model: Hamiltonian H H F F / r Y s N F U s N 3 E N f N,Z<5 5<Z<8 8<N<6 8<Z 6<N

14 The Nilsson model: Basis s nondiagonal in basis and N, n, n, m, m s s, For large deformations can be neglected: Asymptotic quantum numbers N, n, n, m, m s : Nn m For small deformations -terms can be neglected: Spherical basis N,, j, N,, m, m s Nilsson used basis Diagonal terms 3 H N,, m, m N N,, m, m s s N,, m, m s N,, m, m s

15 The Nilsson model:matrix elements Matrix elements ' m ' m 's s mm s ' m m ', m ' m s m ' s, m ' s m m m ' m ' s s, m, s, m, m m, m, s, m, m 5 4 ' ' ' m ' Y m i m ' m ' ' m m ' m s m ' s ', ' N N '

16 The Nilsson model: Matrix elements Radial matrix elements! / r n r b N e L 3 3 n r b b n b / / n '! n! n n n ' ' / n / p! n ' n n n N r N b ' '!!!! '!! p ' 3 p ' / p / N n / N, N admixtures / 3/ N r N n N N r N n n / N r N n n / N r N n n / 3/ N r N n n '

17 The Nilsson model Nilsson states: i C ; N mm i s N N mm s N / N 3/ / 3 / / N 5 / / / 5 / 3/ 3 / 3 / / N 3 7 / / 5 / 5 / 7 / 5 / / 33 5 / 5 / 3 / / 3 / / / 3 / / /

18 A ZNucleus N K Al / 5 7 Si M g / 3 5 / Na 3 3 / O / F 9 9 / Ne 9 9 / Be / Li /

19 Spherical levels split into (j+)/ levels Levels ( are twofold degenerate Asymptotic q-numbers not conserved for small deformations but useful to classify levels For positive deformations (PROLATE SHAPES), levels with lowerare shifted downwards For negative deformations (OBLATE SHAPES), levels with lowerare shifted upwards

20 N n n m + N =/ n m / N =/ n m / n m / N =3/ n m 3/ N =/ n m / N =3/ n m n m / n m / 3/ n m 3/ N =5/ n m 5/ N 3 =/ n 3 m 33 / n m 3 / n m 3 / n m 3 / N 3 =3/ n m 3 3/ N 3 =5/ n m 3 3/ n m 3 3/ n m 3 5 / n m / N 3 =7/ n m / Nnm