Transcript nuclear Shell Model

Nuclear Shell Model
• Potential between nucleons can be studied
by studying bound states (pn, ppn, pnn,
ppnn) or by scattering cross sections:
np -> np pp -> pp nD -> nD pD -> pD
• If had potential could solve Schrod. Eq.
Don’t know precise form but can make
general approximation
• 3d Finite Well with little r-dependence
(except at edge of well)
• Almost spherically symmetric (fusion can
be modeled as deformations but we’ll
skip)
• N-N interactions are limited (at high A)
due to Pauli exclusion. p + n -> p’ + n’
only if state is available
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Infinite Radial Well
• Radial part of Scrod Eq

 2 l (l  1) 
V (r )  2m
 u  Eu
2
r


u(r )  rR(r ) P(r )  4u 2
2 d 2u

2 
2m dr
• Easy to solve if l=0
u  sin kr
2 n
k
2a
p 2 (k ) 2 (hn) 2
E


2m
2m
8ma 2
• For L>0, angular momentum term goes to
infinity at r=0. Reduces effective wavelength,
giving higher energy
• Go to finite well. Wave function extends a bit
outside well giving longer effective wavelength
and lower energy (ala 1D square wells)
• In nuceli, potential goes to infinity at r=0 (even
with L=0) as that would be equivalent to
nucleon “inside” other nucleon
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Angular part
• If V(r) then can separate variables y(r,q,f) =
R(r)Y( q,f) have spherical harmonics for angular
wave function
• Angular momentum then quantized like in
Hydrogen (except that L>0 for n=1, etc)
L2y  l (l  1) 2y LZy  my
l  0,1,2 m   ll n  r  quantum#
•
•
•
•
Energy doesn’t depend on m
Energy increases with increasing n (same l)
Energy increases with increasing l (same n)
If both n,l vary then use experimental observation
to determine lower energy
• Energy will also depend on strong magnetic
coupling between nucleons
• Fill up states separately for p,n
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L,S,J Coupling: Atoms vs Nuclei
•
•
•
•
ATOMS: If 2 or more electrons, Hund’s rules:
Maximise total S for lowest E (S=1 if two)
Maximise total L for lowest E (L=2 if 2 P)
Energy split by total J (J=3,2,1 for S=1,L=2)
• NUCLEI: large self-coupling. Plus if 2 p (or 2
n) then will anti-align giving a state with J=0,
S=0, L=0
leftover “odd” p (or n) will have two possible
J = L + ½ or J = L – ½
higher J has lower energy
if there are both an odd P and an odd n (which is
very rare in stable) then add up Jn + Jp
• Atom called LS coupling nuclei called jj
• Note that magnetic moments add differently as
different g-factor for p,n
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Spin Coupling in Nuclei
• All nucleons in valence shell have same J
• Strong pairing causes Jz antiparallel (3 and -3)
spin wavefunction = antisymmetric
space wavefunction = symmetric
• This causes the N-N to be closer together and
increases the attractive force between them
• e-e in atoms opposite as repulsive force
• Can also see in scattering of polarized particles
• Even N, even Z nuclei. Total J=S=L=0 as all n,p
paired off
• Even N, odd Z or odd N, even Z. nuclear spin
and parity determined by unpaired nucleon
• Odd N, odd Z. add together unpaired n,p
• Explains ad hoc pairing term in mass formula
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Energy Levels in Nuclei
• Levels in ascending order (both p,n)
State
n L degeneracy(2j+1) sum
1S1/2
1 0
2
2***
1P3/2
1 1
4
6
1P1/2
1 1
2
8***
1D5/2
1 2
6
14
2S1/2
2 0
2
16
1D3/2 1 2
4
20***
1F7/2
1 3
8
28***
2P3/2
2 1
4
32
1F5/2
1 3
6
38
2P1/2
2 1
2
40
1G9/2 1 4
10
50***
*** “magic” number is where there is a large
energy gap between a filled shell and the next
level. More tightly bound nuclei. (all filled
subshells are slightly “magic”)
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Magic Numbers
• Large energy gaps between some filled shells
and next (unfilled) shell give larger dE/A and
more made during nucleosnthesis in stars
# protons
#neutrons
2 He
2 He-4
6 C
6 C-12
8 O
8 O-16
20 Ca
20
28 Ni
28 Cr-52(24,28)
50 Sn
50 Ni-78
82 Pb
82
126
136
• Ni-78 (2005) doubly magic. While it is
unstable, it is the much neutron rich.
• Usually more isotopes if p or n are magic. Sn
has 20 isotopes, 10 of which are stable
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Nuclear Magnetic Moments
• Protons and neutrons are made from quarks and
gluons. Their magnetic moment is due to their spin
and orbital angular momentum
N


e
   L  S 
( gl L  g S S )  N 

2mp



• The g-factors are different than electrons. orbital,
p=1 and n=0 as the neutron doesn’t have charge
• spin, g for proton is 5.6 and for neutron is -3.8
(compared to -2 for the electron; sometimes just 2).
• A proton is made from 2 up and 1 down quark
which have charge 2/3 and -1/3
• A neutron is made from 1 up and 2 down and has
“more” negative charge/moments
• No theory which explains hadronic magnetic
moments
• orbital and spin magnetic moments aren’t aligned,
need to repeat the exercise in atoms (Zeeman
effect) to get values for the z-component of the
moment
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