Transcript Shell model I - Evidence

The Shell Model of the Nucleus
1. Evidences
[Sec. 5.1 and 5.2 Dunlap]
Alpha Particle Decay Q
What is causing this bump
around Z=82, N=126, A=208
The fission barrier on the SEMF
To calculate the
height of the fission
barrier using the
SEMF is fairly
complex, but can
be done as seen in
this study – Fig12.3
Dunlap.
The dotted lines
show variations that
are understood on
the shell model.
Note that the
barrier is only small
~3MeV for A>250.
Atomic Shell Model
By the end of the 1920s, the laws of quantum mechanics had been worked out.
They had been applied to the hydrogen atom. They had also been extended to
the MULTI-ELECTRON ATOM. This gave the first full understanding of the
PERIODIC TABLE OF THE ELEMENTS.
Atomic Shell Model
Starting with the Solution of the Schrodinger Equation for the HYDROGEN ATOM
  2c 2 2

  V (r )  E

2
 2mc

The natural coordinate system to use is spherical
coordinates (r, , ) – in which the Laplacian operator is
1  2 
1 
 
1
2
  2 r
 2
 sin 
 2 2
r r  r  r sin  
  r sin   2
2
and the central potential being “felt” by the
electron is the Coulomb potential
V (r ) 
e2
(40 )r
Atomic Shell Model
Angular solutions of the 3D Schrodinger Eqn. are the spherical harmonic
functions Yl,m(,). l is the angular momentum quantum number, m is
called the magnetic quantum number. l=0
l=1
l=2
l=3
m
+3
+2
+1
0
-1
-2
-3
Every l state has (2l+1) magnetic substates
Atomic Shell Model
Radial solutions of the hydrogen atom wavefunction are complicated
a
functions involving the associated Laguerre function L b (x)
 2Z
Rn ,l (r )   
 na0



3
n  (l  1) !
3
2n(n  l )! 
1/ 2

e
Zr
na0
 2Zr  2l 1  2Zr 
.Ln l 

.
 na0 
 na0 
n
nr
0
1
1
2
0
2
3
1
0
Principle Quantum No = (nr  l )  1
Atomic Shell Model
The amazing thing about
the 1/r potential is that
certain DEGENERGIES
(same energies) occur for
different principal quantum
no “n” and “l”.
i.e. when n=2, l=0 and l=1
have the same energies
When n=3, l=0,1 and 2
have the same energy.
2
Z
En    E
n
Atomic Shell Model
However when we extend the model to MULTI-ELECTRON atoms the
degeneracy is lost.
The potential each electron moves in is now more complicated.
e2
V (r ) 
r
Ze 2
V (r ) 
r
The potential seen
by the electron
changes from
these two
extremes as it
Occupancy moves about the
nucleus.
Atomic Shell Model
2
10
18
36
54
86
Atomic Shell Model
Single electron separation energy
Atomic
Atomic Shell
Shell Model
Model
Covalent
Radius
Atomic Radius
Evidence for Nuclear Shells
Single neutron separation energy
Evidence for Nuclear Shells
The famous binding energy per nucleon (B/A) as predicted by the SEMF –
does not get it quite right. There are ripples and bumps which occur at the
nuclear MAGIC NUMBERS, 28, 50, 82 and 126
Evidence for Nuclear Shells
Another evidence for EXTRA STRONG NUCLEAR BINDING at the
special “MAGIC NUMBERS” is that the frequency of ISOTONES is
greatest when N=20, 28, 50 and 82.
THE NUCLEAR MAGIC NUMBERS are: 2, 8, 20, 28, 50, 82, 128, 184
Reason for Nuclear Shells
ATOM
NUCLEUS
Type of particles
Fermions
Fermions
Indentity of particles
electrons
neutrons + protons
Charges
all charged
some charged
Occupancy considerations
PEP
PEP
Interactions
EM
Strong + EM
Shape
Spherical
Approximately spherical
The atom and nucleus have some differences – but in some essential
features (those underlined) they are similar and we would expect similar
quantum phenomenon - i.e. some kind of SHELL STRUCTURE.