Mean

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The nuclear mean field and its symmetries

W. Udo Schröder, 2011

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Mean

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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

W. Udo Schröder, 2011

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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

W. Udo Schröder, 2011

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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

W. Udo Schröder, 2011

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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

W. Udo Schröder, 2011

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Mean

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Multi-Particle Symmetries

W. Udo Schröder, 2011

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