Nilsson Model

Spherical Shell Model

Deformed Shell Model

Anisotropic Harmonic Oscillator

Nilsson Model

o Nilsson Hamiltonian

o Choice of Basis

o Matrix Elements and Diagonalization

o Examples. Nilsson diagrams

Spherical shell modelNuclear 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: 2,8,20,28,50,82,126

• 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

( )0

0r

V rr

=

∂⎛ ⎞=⎜ ⎟∂⎝ ⎠ 0

0r R

Vr <

∂⎛ ⎞ >⎜ ⎟∂⎝ ⎠( ) 00, V r r R>

Characteristics of the potential:

Spherical potentials

Infinite square well

( ) 0 for for

V r V r Rr R

= − ≤

= +∞ >

Eigen-energies

Eigen-functions

( ) ( )mj kr Yψ Ω∼

( )

( )

22

2,2

: root of 0

n

n

E nMRj

ξ

ξ ξ

=

=

Harmonic oscillator

( ) 2 20

12

V r M rω=

( ) ( )

( ) ( )21

1/ 2 221

n m

r

n n

R r Y

R r r e L r

ψ

− +−

Ω

=

∼

( ) ( )( )

0

0

, 2 3/ 2

3/ 2

E n n

N

ω

ω

= + +

= +

Woods-Saxon potential

( ) ( )0

1 exp /VV rr R a

= −+ −⎡ ⎤⎣ ⎦

numerically

intermediate

Spherical potentials

( ) 0 for for

V r V r Rr R

= − ≤

= +∞ >

( ) 2 20

12

V r M rω=

( ) ( )0

1 exp /VV rr R a

= −+ −⎡ ⎤⎣ ⎦

Infinite square well

Harmonic oscillator

Woods-Saxon potential

Spherical potentials

SquareH.O. W.S.

Spherical potentials & spin-orbit

( )2 2 212

s j s⋅ = − −

1 for 211 for 2

SO

SO

j

j

ε

ε

= = +

= − − = −

( ) 22 20

12

V r M r C s Dω= + ⋅ +

( ) ( ) ( )0, 2 3 / 2 1 SOE n n D Cω ε= + + + + +

2 1SOεΔ = +

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+1) spectra

• Large quadrupole moments and quadrupole transition probabilities

• Single particle structure

Anisotropic Harmonic Oscillator

Generalized Woods-Saxon ( ) ( )( )

1

0

,, , 1 exp

,r R

V r Va

θ ϕθ ϕ

θ ϕ

−⎡ ⎤⎛ ⎞−

= − +⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

( )( ), ,LSV V r p sλ θ ϕ= ∇ ∧ ⋅

Anisotropic Harmonic Oscillator

( )2

2 2 2 2 2 20 2 2 x y z

mH x y zm

ω ω ω= − Δ + + +

Ellipsoidal distribution: Anisotropic Harmonic Oscillator as average field

( )1/ 60

2 30 0

4 161 , 0.953 27

ω δ ω δ δ δ β−

⎡ ⎤= − −⎢ ⎥⎣ ⎦

Frequencies are proportional to the inverse of the ellipsoid axes0

00i

i

Ra

ω ω=

From volume conservation

( )30

0x y zω ω ω ω=

For axially symmetric shapes,

we introduce the parameter δ

2 2 2 20

2 20

213

413

x y

z

ω ω ω ω δ

ω ω δ

⊥⎛ ⎞= = = +⎜ ⎟⎝ ⎠

⎛ ⎞= −⎜ ⎟⎝ ⎠

0

zω ωδ

ω⊥ −=

( ) ( )0

' /b r r bm

δω δ

= =

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( )0 2

20 2 2 2 2 2

0 0 00 0

0 0 2 2 2 2

0

0

0 2

2

20

2

2 41 12 2 3 3

2 42 2 3 3

42 3 5

m mH x y zm m m

x y z x

r r

H

y

Y

z

Hδ

ω δδ ω δ δ ω δ δ

ω ω

ω δ

ω πδ ω δ

ω

δ

δδ δ

⎡ ⎤⎛ ⎞ ⎛ ⎞= − Δ + + + + −

⎡ ⎤−Δ +⎣ ⎦

⎢ ⎥⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎣ ⎦

⎡ ⎤= − Δ + + + + + −⎢ ⎥⎣ ⎦

= −

+=

Ω

{ }, , , ,

2z s

x y z z

N n n m m

N n n n n n mρ

ρ= + + = + +

( ) ( ), ,

0

0

1 1, , 2 12 2

32 3

z i i z zi x y z

z

n n m n n n m

NN n

ρ ρε ω ω ω

ω δ

⊥=

⎛ ⎞ ⎛ ⎞= + = + + + +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎡ ⎤⎛ ⎞ ⎛ ⎞= + + −⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦

∑

Anisotropic Harmonic OscillatorIntroducing dimensionless coordinates through the oscillator length

we get

Axial symmetry: cylindrical basis

Anisotropic Harmonic Oscillator

[ ] ( )1 12

NzNn m mπ πΩ Ω = ± = −

( ) ( ) ( ) ( )

( ) ( )( ) ( )

,2z

z

z s s

z

mn

mn

im

n n m m m

n

mn

n

n

ezR

z

L

H z

ρ

ρ

ρ

ρ

ϕ

φ ψ ψ

ψ

σ χ σ

ρπ

ρ

ρ

ψ

=

∼

∼

Eigen-states characterized by

Anisotropic Harmonic Oscillator

deformation

ener

gy

nz=3nz=2nz=1nz=0

( ) ( )0 0

30 0

9 12

Nz zn n m nρε ω ω δ= = + −

( ) ( ), ,

0

0

1 1, , 2 12 2

32 3

z i z z zi x y z

z

n n m n n n m

NN n

ρ ρε ω ω ω

ω δ

⊥=

⎛ ⎞ ⎛ ⎞= + = + + + +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎡ ⎤⎛ ⎞ ⎛ ⎞= + + −⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦

∑

nz ml nρ Ω deg

0 3 0 5/2, 7/2

1 1 1/2, 3/24

1 2 0 3/2, 5/2

0 1 1/23

2 1 0 1/2, 3/2 2

3 0 0 1/2 1

2zN n n mρ= + +

Energy level structure: N=3

The Nilsson model: Hamiltonian

( )

2 2

0

0 2 22 20 20

0

0

0

0

01 1 22 2

2

N

N

C

C

H H s

r r Y

D

D

sω δ β

κμ ω

μ

κ ω

κ ω

⎛ ⎞= + ⋅ + −⎜ ⎟⎝ ⎠

⎡ ⎤ ⎡ ⎤⎛ ⎞= − Δ + − − ⋅ + −⎜ ⎟⎢ ⎥ ⎢ ⎥⎝ ⎠⎣ ⎦

=

⎣

=

⎦

−

−

Axially symmetric harmonic oscillator potential

+spin-orbit term

+l2 term

( )2 1= 32N

N N +

The Nilsson model: Hamiltonian

( )

1

0 0

0 0

2/ 6

2 3 2

2 2

0

0

2

0

20

4 1613 27

2

2

3

43

2

5 N

NU

H H

r

F

F

F

f

s

E N

Y

s

κ ω

μ

μ

ω δ

π

κ ω

δ δ δκ

η

−

= +

⎛ ⎞= − ⋅ −⎧ ⎫⎪ ⎪−⎨ ⎬⎪ ⎪⎩

−⎜ ⎟⎝ ⎠

⎛ ⎞= − ⋅ − −⎜ ⎟⎝ ⎠

⎛ ⎞= + +⎜ ⎟⎝

⎡ ⎤− −⎢ ⎥⎣

⎠

⎦ ⎭

N,Z<50 50<Z<82 82<N<126 82<Z 126<N

κ 0.08 0.0637 0.0637 0.0577 0.0635

μ 0 0.60 0.42 0.65 0.325

The Nilsson model: Basis

( )

0 0

0 0

2

3, , , , , ,2

, , , 1 , , ,

s s

s s

H N m m N N m m

N m m N m m

ω⎛ ⎞= +⎜ ⎟⎝ ⎠

= +

s⋅2

nondiagonal in basisand { }, , , ,z sN n n m mρ

For large deformations can be neglected:2

, s⋅

{ } [ ], , , , : z s zNnN n n m m mρ πΩAsymptotic quantum numbers

Nilsson used basis { }, , , sN m m

Diagonal terms

For small deformations δ-terms can be neglected:

{ }, , ,N j ΩSpherical basis

The Nilsson model:Matrix elements

' '', ' 2 ' 2

s sm m m mN N

= == ± = ±

' ' 's sm m s m m⋅

( ) ( )1, 1, , , 12

1, , , ,2

m s m m m

m s m m

± ⋅ ± = ± +

± ⋅ ± = ±

∓ ∓

'' , ' 1' 1, '

' 's s s

s s

m m mm m mm m m m

== ±= ±+ = +

Matrix elements

'20

5 2 1' ' 2 0 2 ' ' 200 2 '04 2 ' 1

m Y m i m mπ

− +=

+

The Nilsson model: Matrix elements( )

( )( )

211/ 2 2 22133

2 1 !/

1/ 2

rb

n

n rN e L r bbb n

⎛ ⎞− ⎜ ⎟ +⎝ ⎠−

− ⎛ ⎞= ⎜ ⎟⎝ ⎠Γ + +⎡ ⎤⎣ ⎦

( ) ( )( ) ( ) ( )

( )( ) ( ) ( ) ( )

( )

( )

1/ 2

'2 2' 1 ! 1 !' ' 1 ! !

' ' 1/ 2 1/ 2

1! '

1 ' 3 2

1 ! 1 ! ' 1 ! 1 !

2 1

' 1/ 2 1/ 2

n nn nN r N b

n n

n n n

p

n

N n

p

p p

σ

σσ σ σ σ μ

μ

μ

ν

ν

σ

ν

+⎡ ⎤− −= −⎢ ⎥

Γ + + Γ + +⎡ ⎤ ⎡ ⎤⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦Γ + +

×−

=

− − − + − + + − +

= −

−= + = −+

+

−−

∑

( ) ( )

( )

( )( )

( )( )

( )( )

2

2

2

2

2

2 1/ 2 3/ 2

2 2 1/ 2

2 1 1/ 2

2 2 1/ 2 3/ 2

2 2 1 2

N r N n N

N r N n n

N r N n n

N r N n n

N r N n n

= + − = +

− = + −

− = − − + −

− − = + − + −

− + = − −

Radial matrix elements

, 2N N ± admixtures

The Nilsson model

1/ 2

3 / 2

1/ 2 1/ 2

5 / 2

3 / 2 3 / 2

5 / 2 5 / 2 5 / 2

7 / 2

5 / 2 5 / 2

3 / 2

0 1/ 2 0001 3/ 2 111

1/ 2 110 1112 5 / 2 222

3/ 2 221 2221/ 2 220 200 221

3 7 / 2 3335 / 2 332 3333/ 2 331

sN N m m

NN

N

N

π

+

−

− −

+

+ +

+ + +

−

− −

−

ΩΩ

= Ω = += Ω = +

Ω = + −= Ω = +

Ω = + −Ω = + + −

= Ω = +Ω = + −Ω = +

3 / 2 3 / 2

1/ 2 3 / 2 1/ 2 1/ 2

311 3321/ 2 330 331 331 311

− −

− − − −

+ −Ω = + + − −

Nilsson states: { } { }; i si C N m mαπ

α

α αΩ

= ∑

The Nilsson model: N=0

0 1/ 2 1, 0 000+s

N nN m m= Ω = → = =

→

2000 2 000+ 0U sη μ+ − ⋅ − =

20

0 =0

000 000+ 0

0 2 0000 000+ 0

0 0 0

N

s

Y

μ= →

+ ⋅ =

⎛ ⎞+ =⎜ ⎟

⎝ ⎠∼

Diagonalization in blocks Ω,π (with or without ΔN=2 admixtures)

{ }1/ 6

2 3

0 0 0 0

0 0 0 0

220

2

4 161 43 5

3 27

2

U r

H H

Y

HF U s

δη δ δκ

π

ω ω η μκ κ

−⎡ ⎤= − −⎢ ⎥⎣

− ⋅

⎧ ⎫⎪ ⎪= −⎨ ⎬⎪ ⎪⎩

=

⎦

+ =

⎭

−+

The Nilsson model: N=11 3 / 2 1, 1

111+s

N nN m m= Ω = → = =

→

2 220 20

2

20

1 =0

110 110+ 0

4 4 2110 110+ 11 11 10 103 5 3 5 3

311 11 125 5 210 10 1200 1210 1200 1210

4 4 5

N

s

r Y r Y

r

Y

μ

π πη η η

π π

= →

+ ⋅ =

+ − = − = −

= +

= =

2 220 20

2

20

1 =01111 111+2

4 4 1111 111+ 11 11 11 113 5 3 5 3

311 11 125 5 111 11 1210 1211 1200 1210

4 4 5

N

s

r Y r Y

r

Y

μ

π πη η η

π π

= →

+ ⋅ =

+ − = − =

= +

= = −

2 1111 2 111+ 13

U sη μ η+ − ⋅ − = −

2

110+ 111-

11 2 93

13

2 13

2

2 2η η η ηλ

η

⎛ ⎞− ± + +⎜ ⎟

=⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

+

−

−

−

1 1/ 2 1, 1 110+ , 111-s

N nN m m= Ω = → = =

→

2

2

2 2110 2 110+31111 2 111

110 2 111

1

- 2

-3

U s

s

U

U

sη

η μ

η

μ

μ

η η

− − ⋅ −

+ − ⋅

+ − ⋅ − = −

−

=

= −

+

3

η

λ

The Nilsson model: N=2{ }{ }

2 2222 2 22

2 5 / 2 1, 2 2, 0

222+

2+ 23

s

U s

N n n

N m m

η μ η

= Ω = → = = = =

+ − ⋅ − +

→

= −

2

2

2

2222 2 222- 2

222

2 3 / 2 221+ , 222-

221+ 222-

3

22

1221 2 221+ 1

2 221+

3

11

2

2

32

3

s

NN m m

U s

U s

U sη

η μ η

η μ

η

μ η

η

−

+ − ⋅ − = − −

⎛ ⎞− +⎜

− ⋅ − = −

−

−

− − ⋅ − = +

+

= Ω =

→

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜

⎠

⎝

⎝

⎟

⎟

⎠

2

2

2

2

2

2

2 1/ 2 220+ , 200+ , 221-

2220 2 220+3

200 2 200+ 01221 2 221+ 13

2 2220 2 200+3

220 2 221- 6

200 2 221- 0

2

s

NN m m

U s

U s

U s

U s

U s

U s

η μ η

η μ

η μ η

η μ η

η μ

η μ

= Ω =

→

+ − ⋅ − = −

+ − ⋅ − =

− − ⋅ − = −

+ − ⋅ − =

+ − ⋅ − = −

+ − ⋅ − =

20+ 200+ 221-

2 2 2 63 3

0 0 113

η η

η

⎛ ⎞− −⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟−⎜ ⎟⎝ ⎠

3 5 / 2 332 333

1 12 63 12

N

μη μ

= Ω =

+ −

⎛ ⎞− − −⎜ ⎟

+ −⎝ ⎠

3 3 / 2 331 311 332

3 41 12 105 5

3 1 2 05

2 12

N

η μ η

η μ

μ

= Ω =

+ + −

⎛ ⎞− − − −⎜ ⎟⎜ ⎟⎜ ⎟− −⎜ ⎟⎜ ⎟−⎜ ⎟⎜ ⎟⎝ ⎠

3 =1/2 330 310 331 311

4 2 612 2 3 05 5

6 2 0 25

3 41 125 5

3 1 25

N

η μ η

η μ

η μ η

η μ

= Ω

+ + − −

⎛ ⎞− − −⎜ ⎟⎜ ⎟⎜ ⎟

− − −⎜ ⎟⎜ ⎟⎜ ⎟− + −⎜ ⎟⎜ ⎟⎜ ⎟+ −⎜ ⎟⎝ ⎠

( )

3 7 / 2 333

3 12

N

η μ

= Ω =

+

− −

The Nilsson model: N=3

000 220 200 221 440 420 400 441 4211000 0 0 0 0 0 0 0 032 2 2 6 2 2 2 2220 6 0 03 3 3 7 3 535

1200 0 0 0 7 0 0 03

1 6 1 2221 1 0 0 03 7 3 7

20 12 10440 20 0 20 021 35

22 4420 6 35 0 621 15

400 0 0 017 4441 1 20 321 7

114212

η

η η η η η

η

η η η

η μ η

η μ η

η μ η

+ + + − + + + − −

+

+ − − −

+

− − +

+ − − −

+ − − −

+

− − + −

− − 1 61η μ+ −

The Nilsson model: ΔN=2

Ω=1/2+ with ΔN=2 admixtures

AZ NNucleus K π→

73 4 1 / 2Li −→

94 5 3 / 2Be −→

198 11 3 / 2O +→

199 10 1 / 2F +→ 19

10 9 1 / 2Ne +→

2311 12 3 / 2Na +→

2512 13 5 / 2M g +→

2713 14 5 / 2Al +→

2714 13 5 / 2Si +→

• Spherical levels split into (2j+1)/2 levels

• Levels (Ωπ) are twofold degenerate

• Asymptotic q-numbers not conserved for smalldeformations but useful to classify levels

• For positive deformations (PROLATE SHAPES), levels with lower Ω are shifteddownwards

• For negative deformations (OBLATE SHAPES), levels with lower Ω are shifted upwards

2zN n n mρ= + +

[ ]+0 =1/2 0 0 000 1/ 2zN n m += Ω = =

[ ][ ][ ]

1 =1/2 1 0 110 1/ 2

0 1 101 1/ 2

1 =3/2 0 1 101 3/ 2

z

z

z

N n m

n m

N n m

− −

−

− −

= Ω = =

= =

= Ω = =

[ ][ ][ ]

2 =1/2 2 0 220 1/ 2

1 1 211 1/ 2

0 0 200 1/ 2

2 =3/2 1 1

z

z

z

z

N n m

n m

n m

N n m

+ +

+

+

+

= Ω = =

= =

= =

= Ω = = [ ][ ][ ]

211 3/ 2

0 2 202 3/ 2

2 =5/2 0 2 202 5 / 2z

z

n m

N n m

+

+

+ +

= =

= Ω = =

[ ][ ][ ]

3 =1/2 3 0 330 1/ 2

2 1 321 1/ 2

1 0 310 1/ 2

z

z

z

N n m

n m

n m

n

− −

−

−

= Ω = =

= =

= =

[ ][ ][ ][ ]

0 1 301 1/ 2

3 =3/2 2 1 321 3/ 2

1 2 312 3/ 2

0 1 301 3/ 2

3 =5/2

z

z

z

z

m

N n m

n m

n m

N

−

− −

−

−

−

= =

= Ω = =

= =

= =

= Ω [ ][ ][ ]

1 2 312 5 / 2

0 3 303 5 / 2

3 =7/2 0 3 303 7 / 2

z

z

z

n m

n m

N n m

−

−

− −

= =

= =

= Ω = =

[ ]zNn m πΩ

( ) ( ) ( ) ( ) ( ) ( ){ }2* *2 1 1

16 KI IMK M k

IK

Jk K

IIK D rDM rθ θφ φπ −

−−

+Ψ = + −

M

KΩ

I

J

R

Wave functions in the unified model ( ) ( )krφ θΨ Φ∼

Intrinsic wave functions (Nilsson) Rotation matrices

Z lab.

Z intr.

Laboratory frame

( )| 1 2 1a b a b cab c cγαβ γ

α β γ− − ⎛ ⎞

− = − + ⎜ ⎟⎝ ⎠

3-j