The Deuteron
• Deuterium (atom).
•The only bound state of two nucleons  simplest bound state
• Neither di-proton nor di-neutron are stable. Why?
• Experimentally  2.224 MeV (Recoil..!). n  H  H  
• Also inverse (,n) reaction using Bremsstrahlung (Recoil…!).
1
2
•mc2 = 2.224…??…MeV  Very weakly bound.
• Compare n-p to n-n and p-p  Charge independence of
nuclear force.
• Only ground state. (There is an additional virtual state).
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007
(Saed Dababneh).
1
The Deuteron
V(r) = -V0 r < R
=0
r>R
• Oversimplified.
HW 17
Show that V0  35 MeV.
(Follow Krane Ch.4 and
Problem 4.6), or
similarly any other
reference.
• Really weakly bound.
• What if the force were a
bit weaker…?
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007
(Saed Dababneh).
2
The Deuteron
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007
(Saed Dababneh).
3
The Deuteron
• Experiment  deuteron is in triplet state   = 1.
• Experiment  even parity.
•  = l + sn + sp
parity = (-1)l
• Adding spins of proton and neutron gives:
s = 0 (antiparallel) or s = 1 (parallel).
• For  = 1
parallel
s-state
even
parallel
p-state
odd
antiparallel
p-state
odd
parallel
d-state
even
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007
(Saed Dababneh).
4
The Deuteron
• Experiment   = 0.8574376 N  spins are
aligned…..But.?
• Direct addition  0.8798038 N.
• Direct addition of spin components assumes s-state
(no orbital component).
• Discrepancy  d-state admixture.
 = a 0 0 + a 2 2
 = a020 + a222
HW 18 In solving HW 17 you assumed an s-state.
How good was that assumption?
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007
(Saed Dababneh).
5
The Deuteron
• S-state  No quadrupole moment.
• Experiment  +0.00288 b.
HW 18
Discuss this discrepancy.
• From  and Q, is it really admixture?
• What about other effects?
• Important to know the d-state wavefunction.
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007
(Saed Dababneh).
6
Nuclear Force
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007
(Saed Dababneh).
7
Nuclear Models
• Nuclear force is not yet fully understood.
• No absolutely satisfying model, but models.
• Specific experimental data  specific model.
• Model  success in a certain range.
• Some are:
 Individual particle model. (No interaction, E. states, static properties, …).
 Liquid drop model. (Strong force, B.E., Fission, …).
 Collective model.
 -particle model.
 Optical model.
 others …..
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007
(Saed Dababneh).
8
Shell model
• Electron configuration….
1s2 2s2 2p6 3s2 3p6 4s2 3d10 4p6 ….
• Atomic magic numbers: 2, 10, 18, 36, 54, …
 Common center of “external” attraction.
 Well understood Coulomb force.
 One kind of particles.
 Clear meaning for electron orbits.
…
• Nuclear magic numbers: 2, 8, 20, 28, 50, 82,126, …
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007
(Saed Dababneh).
9
Shell model
Evidence:
1) End of radioactive series:
208Pb
thorium series
uranium series 206Pb
actinium series 207Pb
neptunium series 209Bi
2) At Z and N mn’s there are relatively large numbers
of isotopes and isotones.
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007
(Saed Dababneh).
10
Shell model
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007
(Saed Dababneh).
11
Shell model
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007
(Saed Dababneh).
12
Shell model
3) Natural abundances.
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007
(Saed Dababneh).
13
Shell model
NEUTRON CAPTURE
CROSS SECTION
4) Neutron capture cross section.
NEUTRON NUMBER
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007
(Saed Dababneh).
14
Shell model
5) Binding energy of the last neutron
(Separation Energy).
(The measured values are plotted relative to the calculations without ).
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007
(Saed Dababneh).
15
Shell model
6) Excited states.
Pb (even-A) isotopes.
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007
(Saed Dababneh).
16
Shell model
7) Quadrupole moments ….. ?
HW 19
Work out more examples for the
above evidences. For example, take
part of a plot and work on a group of
relevant nuclides.
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007
(Saed Dababneh).
17
Shell model
• Nucleons are in definite states of energy and
angular momentum.
• Nucleon orbit ?? Continuous scattering expected ..!!
• No vacancy for scattering at low energy levels.
• Potential of all other nucleons.
• Infinite square well:
rR
0
V 

rR
• Harmonic oscillator:
V 
1
m r
2
2
2
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007
(Saed Dababneh).
18
Shell model
• More realistic:
• Finite square well potential:
 V0
V 
 0
rR
rR
• Rounded well potential:
V (r)  
V0
r  R / a
V 0 ~ 57 MeV
1 e
• Correction for asymmetry (n-p has more
possibilities than n-n or p-p) and Coulomb repulsion.
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007
(Saed Dababneh).
19
Shell model
• Separation of variables:
 ( r ,  ,  )  R ( r )  ( )  (  )  R ( r ) Y l (  ,  )
m
• For a given spherically symmetric potential V(r),
the bound-state energy levels can be calculated
from radial wave equation for a particular orbital
angular momentum l.
• Notice the important centrifugal potential.
ml
ms
1s
1p
1d
2s
1f
2p
1g
2d
3s
2(2l +1)
2
6
10
2
14
6
18
10
2
Total
2
8
18
20
34
40
58
68
70
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007
(Saed Dababneh).
20
Shell model
centrifugal potential
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007
(Saed Dababneh).
21
Shell model
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007
(Saed Dababneh).
22
Shell
model
?
?
?



2(2l + 1)
accounts
correctly
for the
number of
nucleons
in each
level.
But what
about
magic
numbers?
Infinite spherical well Harmonic oscillator
(R=8F)
E   (  3 2 )  
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007
(Saed Dababneh).
E nl  ( 2 n  l 
1
2
)
23
Shell model
• So far, 2(2l + 1) accounts correctly for the number
of nucleons in each level, since we already
considered both orbital angular momentum, and
spin, but still not for closed shells.
m
m
l, m l , s, m s  Yl  s
l
Spherical
Harmonics,
Eigenfunctions of
L2 and Lz.
s
S s
 s ( s  1)   s
Szs
 m s s
2
ms
ms
2
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007
(Saed Dababneh).
ms
ms
s 1 2
 s  ms  s
24
Shell model
• 2, 8, 20 ok.
• What about other magic numbers?
• Situation does not improve with other
potentials.
• Something very fundamental about the
single-particle interaction picture is
missing in the description…..!!!!!
• Spin-orbit coupling.
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007
(Saed Dababneh).
25
Shell model
Spin-Orbit Coupling
• M. G. Mayer and independently Haxel,
Jensen, and Suess.
• Spin-Orbit term added to the Hamiltonian:
H 
p
2
2m
 V ( r )  V SO ( r ) S . L
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007
(Saed Dababneh).
26
Shell model
S .L  ( J
2
 S  L )/2
2
L
2
S
LL
antiparallel
J
J
2
J SL
jm j ls  j ( j  1) 
2
jm j ls ,
J z jm j ls  m j  jm j ls ,
L
2
jm j ls  l ( l  1) 
S
2
jm j ls  s ( s  1) 
2
jm j ls ,
2
jm j ls ,
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007
(Saed Dababneh).
UL
parallel
ls  jls
 jmj  j
l  0 ,1, 2 ,....
s 1 2
27
Shell
model
2j+1
2(2x3 + 1) = 14
1f7/2
First
time
l=3
j
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007
(Saed Dababneh).
28
Shell model
HW 20
L.S 
1
[ j ( j  1)  l ( l  1)  s ( s  1)] 
2
2
gap 
1
( 2 l  1)  ,
2
l0
2
Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007
(Saed Dababneh).
29