CoulEx

Nuclear s.p. Models

The Deformed Shell Model

W. Udo Schröder, 2012

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Systematic Changes in Nuclear ShapesMøller, Nix, Myers, Swiatecki, Report LBL 1993: Calculations fit to experimental deformations.

Nuclear quadrupole moments (Q0) are large for N, Z between “magic” numbers. Close to magic numbers nuclei are spherical (Q0≈0), small and tightly bound.

e2 ~ Q0

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Deformed Potential s.p. Model

Rotational symmetry broken Conserved quantities: nucleonic angular momentum projections on symmetry axis:

L= projection of orbital S=±1/2= projection of spin = +W L S= projection of total

angular momentum

Energy level degeneracy: DW=2 (± ) W axial symmetry.

Anisotropic harmonic oscillator model Principal quantum number N=nx+ny+nz= number of oscillator quanta. (wx, wy, wz) are different fundamental frequencies)W. Udo Schröder, 2012

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Sven Gösta Nilsson (1955)

z

r

s

s

L

S

Axial SymmetryAbout z

s

s

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Anisotropic Harmonic Oscillator Model

W. Udo Schröder, 2012

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2 2 2 2 2,

2 20 2

0 , 0

3 20 ,

,

24

, 1 cos2 3

ˆ: ,

2 11 1

3 3

1 2 1

:

N z z x y x

z x y

z x y

z x y

x y z

y

mV r z x y

mV r r P

r z axis

Deformation parameter

Volume conservation

N n n n

E n n n

Each level ca

2rries up to D nucleons

, , ,, 2, 4,.....x y x y x yn n n

|W|p

|W|p(N,nz,L)

Shell Gap

Deformation Parameter

2 2 20 2

( )

4, 1 cos

2 3

Modified Nilsson Potential flatter potential

mV r r P C s D

, , ,zQuantum Numbers N n

Ground state spin determined by last (unpaired) particle.

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Shell Stabilized Deformation

Why are not all nuclei spherical? Observation: As nucleons are added to spherical (magic, closed-shell) nuclei, deformation increases microscopic origin in nuclear structure & stability.Approximate trends:

W. Udo Schröder, 2012

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,

, ,

,

,

, ,

1 1: 2

3 3

: 32

( 1 2) ( 1

32

2

)

( )

x y z x y z

x y z x y z

x y z

N z z x y x

N z x y

yE n n

E N n n

wx,y/wz

After G. Musiol et al., Kern-u. Elementarteilchenphysik, VCH 1988

Increasing d larger nz /smaller nxy levels bind more strongly. Shell mixing. New shells at integer axis ratio. Disappearance and reappearance of shells.

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End Def. Shell Model