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CoulEx
Nuclear s.p. Models
The Deformed Shell Model
W. Udo Schröder, 2012
Sh
ell
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els
2
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
3
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
5
,
, ,
,
,
, ,
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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ell
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els
W. Udo Schröder, 2012
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End Def. Shell Model