the nuclear mean field and its symmetries w. udo schröder, 2011 mean field 1
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Mean
Fie
ld
The nuclear mean field and its symmetries
W. Udo Schröder, 2011
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Mean
Fie
ld
Gross Estimates of Mean Field
W. Udo Schröder, 2011
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F ,n n F ,p pn p
; Vm m
2 3 2 4 3 2 3 2 4 32 3 2 33 3
2 2
Fermi gas kinetic energy estimates:
Light nuclei, N Z A/2:
n p F0.085 fm 38 MeV 1
V 50 MeV N Z A 11MeV r
VC(r)
R
r(r)
r
cV rF ,p V
F ,n
C asyV V V
pS
nS
V MeV50
E
E
0
0
n
p
Separation energies
S N, Z B N, Z B N , Z 8 MeV
S N, Z B N, Z B N, Z 8 MeV
1
1
For stable nuclei, Fermi energies for protons and neutrons are equal.Otherwise beta decay.
n F ,n n
Potential depth
V S 46 MeV
Coulomb Barrier
Protons somewhat less bound but confined by Coulomb barrier.
Mean
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The Nuclear Mean Field
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General properties of mean potential of nucleus (A=N+Z), radius RA
3D Square Well
Oscill
Woods Saxon
Nucleons close to center (r=0): V(r0) V0=const. NN forces are short range: V(r)0 for rR rapidly
Range of potential > RA
NN forces have saturation character: central mass density r(r0) const. for all A, V0 = const.
Total s.p. interaction: H = Hnucl+Helm+Hweak+… elm=electromagnetic, not all conservative!
2 2
20
2
0
( )
( , )
1 3
4 2 2( )
1 1
4
C
Electrostatic Coulomb Potential
for protons in nucleus A Z
e Z rr R
R RV r
e Zr R
r
VC(r)
R
r
r(r)
0
2 20
0
0
0
( )2
1 exp
Trial potentials
r RV r Infinite SquareWell
r R
V r RV r Finite SquareWell
r R
MV r r Harmonic Oscillator
VV r Saxon Woods
r R
a
M=inertiaw 0=frequency
V0 = deptha=diffuseness
V0 = depthR=range
R=range
3D Square Well
Oscill.
Woods Saxon
Mean
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The Nuclear A-Body Schrödinger problem
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1
1 1
1ˆ ˆ2
ˆ
A
i iji j i
A A
i i ij ii j i i
H T V r
T V r V r V r
Neglect residual interactions independent particlemodel
Nucleus with A =N+Z nucleons
2
1
2
: ,...., det
:
i i i i i i ii
A i i
ii
V r r rm
A body wave function r r r antisymmetric
Total energy E
3-dimensional Schrödinger problem
Symmetries Further simplifications of 3D Schrödinger problem
1 1
ˆ ˆ ˆ: :ˆ ˆ
A A
i ires ii
i ii
Single particleHamiShell Model ltonian H TH H VV rH
3A-dim. Schrödinger problem
3D Square Well
Oscill.
Woods Saxon
Mean
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Single-Particle Symmetries
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2
2
1,2...
n n nV r r rm
Mean field V r confines system discrete energy spectrum
main quantumnumbers n
3D 1D Schrödinger problem
Oscillator
Woods Saxon
Square Well
:
, ,
If central potential
Spherical symmetry rotational invarian
noan
ce
V
d
gle dependent tor
ecoupled radial a
q
n
ues conse
d angular
r V
mo
rvedr
tion
2 22
0
2
0
2
, ,
( )
&
ˆ , 1
:
,n n spin spin
m m
n
mn
n
u r
r
Integrability R r r d
Product wave functions r R r Y
finite
Spherical harmonics eigen functions to angular momentum parity operators
L Y
r u r dr
Y
Y
,
ˆ , ,
ˆ , 1 , (ˆ : )
z m m
m m spinn n
L Y m Y
Y Y Treat spin lar t rr e
3D Square Well
Oscill.
Woods Saxon
Mean
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Single-Particle Wave Functions
• See Nilsson Book
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Mean
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Multi-Particle Symmetries
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