nuclear physics the fermi-gas model (8th...
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Nuclear Physics
(8th lecture) Content
• Nuclear models #2: The Fermi-gas model (contd.)
The total kinetic energy
The asymmetry energy
The surface energy
• Nuclear models #3: The nuclear shell model
Experimental indication
What characterizes a shell?
Shells in atomic physics (reminder)
The nuclear single-particle shell model
The 1D and 3D harmonic oscillator case
2
The Fermi-gas model Since the nucleons are fermions, we can try to set up a model of
fermion „gas”.
Assumption: the fermions are „closed” in a spherical potential
well, but they move „freely” inside .
f
f
pp
pppn
if ,0
if ,1
p pf
1
The number of states in the phase-space:
ph
pp
d
dd2
2
dd2d
3
23
3
33
rpr
N
Vhp
p
hVf
f NN
8
3
3
42 33
3
3
From this follows:
The highest kinetic
energy at the Fermi-level:
MeV 332
2
M
pE
f
f
31
0
31
232
9
A
Z
r
hp fp
Ef
-40
33
7 R E
The depth of the
nuclear potential
is ~ 40 MeV for every
nucleus!
Approximating: 2
1
A
N
A
Z
But: ArRV 3
0
3
3
4
3
4
3
The total kinetic energy of the nucleons:
5
4d4
2
2dd
22
12
5
3
0
22
3
332
3
f
p
k
p
Mh
Vppp
Mh
Vpn
M
pE
f
pr
The total kinetic energy of the
protons and the neutrons:
35
232
23
10
3Z
MVE p
k
35
232
23
10
3N
MVE n
k
The asymmetry energy:
We note that , and a Taylor-expansion yields:
...2
1
9
5
2
1
3
5
2
1 312
32
35
353
5
35
35
A
ZN
A
ZNA
A
ZN
A
NAZ
2
1
A
Z
A
N
A
ZN
...2
1
9
5
2
1
3
5
2
1 312
32
35
353
5
35
35
A
ZN
A
ZNA
A
ZN
A
ZAN
31
8
3
Vhp f
N
4
In the second part only the second order
terms remain (the linear terms cancel):
A
ZN
A
ZNA
22
This way we get an asymmetry term, similar to the Weizsäcker
formula’s!
const.4
33
0
rV
A
3
53
523
223
10
3NZ
MVEEE n
k
p
kk
The total kinetic energy:
3
13
123
53
523
2
2
2
1
2
1
9
5
2
1
2
13
10
3
A
ZNA
MV
AEk
const.63,02
12
3
5
const.,522
2
12
3
1
The first part contributes to the volume term:
Note: it has positive sign, therefore it decreases the binding energy!
(unlike in the Weizsäcker formula)
A
Note: the asymmetry term also has a positive sign! This means
that it increases the kinetic energy, and therefore decreases the
binding energy (like in the Weizsäcker formula).
2
5
The surface energy:
For simplicity consider now a cube
(instead of a sphere)
zkykxkV
zyx sinsinsin23
rThe wave function:
,nL
kx
,m
Lk y
Lkz
and 2222
zyx kkk k
The number of states in the phase space :
33
33
4
8
12
3
4
22
LKKVAN
where K = kf
So the total number of nucleons: from where: 3KA 31
AK
Volume of the 1/8 sphere in the
space of (n, m, l) quantum numbers
where
3LV
6
For n = 0, or m = 0, or l = 0 there is no wave function!
However, they were also included in the volume of 1/8 sphere!
A correction needed for the 3 quart-circle surface:
23
4
3
3
4
8
12
KLKLAN
Obviously K’>K, i.e the kinetic energy
will be increased!
Since the number of the states
in the volume >> on the surface,
at first approximation → the correction term 31
AK 32
A
The „surface energy” term can also be explained – at least
qualitatively – in the Fermi-gas model.
Note: the attractive „nuclear forces”, however, are not included
and explained (they are introduced only as an „external”
potential, holding together the Fermi-gas)!
7
The main assumption of the Fermi-gas model was, that the
nucleons move „independently” – without interacting with each-
other – in an outside potential well.
Is this not a crazy assumption for strongly interacting particles?
f
f
pp
pppn
if ,0
if ,1
p pf
1
Answer:
In the ground state of the nucleus
every level is occupied up to the
Fermi-level. Particles could scatter
out only over the Fermi-level.
If a particle gets higher energy and momentum (outside the
Fermi-level) in a scattering process, energy and momentum
conservation would mean that the „other” particle should get
lower energy/momentum → no empty state there, forbidden!
This shows also the validity of the Fermi-gas model:
mainly the ground state properties 8
Nuclear shell model 1) Experimental indications
a) Neutron capture cross sections
b) Binding energy difference of
the last neutron
3
9
c) Excitation energy of the first excited states:
Conclusion: nuclei with the following Z and/or N numbers are
extremely stable (in comparison with their neighbours):
2, 8, 20, 28, 50, 82, 126 These are the „magic numbers”.
Nuclei with magic Z or N numbers are called „magic nuclei”
If Z and N both are magic numbers, they are „double magic nuclei”
The following nuclei are double magic:
He,4
2 O,16
8Ca,40
20 Ca,48
20Ni,48
28 Ni,78
28Pb208
82 10
2) What characterizes a „shell”?
„Shell is a set of quantum-mechanical states with the same main
quantum number” True?
Remember, in atomic physics:
The noble gases are
(experimental fact):
„Magic numbers” in atomic physics should be: 2, 10, 18, 36, 54, 86.
The nth „shell” can hold theoretically 2n2 electrons. Therefore the
theoretical magic numbers would be: n 2n2 Magic number
1 2 2 (☺)
2 8 10 (☺)
3 18 28 (??)
4 32 60 (??)
5 50 110 (??)
There is a discrepancy between
theory and experiment!
Conclusion: The magic numbers in
Nature are not defined by the rule
above!
2He, 10Ne, 18Ar, 36Kr, 54Xe, 86Rn
11
A quantum-mechanical system has a
set of states with different energies,
available for its constituents.
The excitation occurs if a particle goes
into a non-occupied state at higher
energy. If this state is „close” in
energy, then the excitation is easy
(requires small amount of energy).
If the next empty state is „far” in
energy, then the excitation is difficult.
Shell is a set of quantum-mechanical states with similar energy.
Shells are separated by larger energy gaps.
E
12
Shells in atomic physics (reminder):
22
0
2
4 1
8 nh
meE
H-atom (Bohr-model)
Wave function:
,,,,,
m
l
n
mln Yr
rRr r
r
What is the relation between n in the Bohr-model (defining the
energy-shells), and the 3 quantum numbers of the wave-function?
lml
l
nr
...3,2,1,0
...3,2,1,0
...3,2,1 n
2
2
2
0 nme
hr
The quantum-mechanical treatment (Schrödinger-equation):
,,,,2
22
rErrVm
This determines the
„energy-shells”!
(spherical coordinates)
4
13
1 lnn r
n nr l 2(2l+1)
1 0 0 1s 2
2 1 0 2s 2
0 1 2p 6
3 2 0 3s 2
1 1 3p 6
0 2 3d 10
4 3 0 4s 2
2 1 4p 6
1 2 4d 10
0 3 4f 14
This determines the „shells”!
Why are the shells different in the Periodic System?
2
10
28
60
18
36
Shells in atomic physics (reminder):
Bohr Quant. mech. No. of particles
But only in the H-atom!
(„True” Coulomb potential of
a point-like nucleus)
2
10
14
2
2
2
0 nme
hr
Bohr:
The „size” gets
larger!
Why are the shells different in the Periodic System?
The „screening”
of the occupied
inner states
deforms the
potential!
Splitting of the
states occur
according to l !
The shell parameters depend on the shape of the potential!
15
This is the „single-particle shell model”
Every particle moves independently in a mean potential
3) The nuclear shell model
„The shell parameters depend on the shape of the potential!”
How is the „nuclear potential”?
ji
jiji
A
i
i
i
Vm
H rr ,2
ˆ,
1
22
The nuclear Hamiltonian:
single-particle
operators
particle-particle
interactions A „trick”:
ji
A
i
ijiji
A
i
ii
i
VVVm
H1
,
1
22
~,
~
2ˆ rrrr
„mean” potential
(for single-particles) „residual” interaction
(will be neglected)
16
The shape of the „mean” nuclear potential.
• Central → depends only on the absolute value of ri: V(ri)
• Similar shape as the nuclear density
(since the nucleons create it)
The nuclear shell model (contd.)
0
0
rVrV
R
It is difficult to solve the
Schrödinger-equation for this shape.
Two different approximations:
square
well
harmonic oscillator
5
17
The nuclear shell model (contd.)
Case #1: square well (infinite)!
The Schrödinger equation in the interior region in spherical
coordinate system is: ,,,
20
2
rErVrM
rkjr nln ,, Solution: Bessel functions:
0 if ,
if ,0
0
V
Rr
R,rVrV
We get for the radial part:
0122
,202
,
2
,
2
n
nn
rEV
M
dr
d
rdr
d
where
,02,
2 2nn EV
Mk
Since the potential is central,
the wave-function can be separated:
m
lYr
rr
,
18
kR 2(2L+1) Sum
L= 0 (1s) 2 2
4,49 L= 1 (2p) 6 8
5,79 L= 2 (3d) 10 18
2 L= 0 (2s) 2 20
6,99 L= 3 (4f) 14 34
7,22 L= 1 (3p) 6 40 ?
2
8
20
The functions have to go to 0 at r = R
→ only roots of the Bessel functions
are allowed → kn,l → discrete values
→ En,l discrete values too.
Second column: angular momentum
quantum number (L) and the atomic
physics notation of the state.
The nuclear shell model (contd.) units 2
in 2
2
MR
L = 0
L = 1
L = 2
L = 0
L = 3
L = 1
EV 0
Note: inside a shell the
higher L states go lower!
19
The nuclear shell model (contd.)
Case #2 : the harmonic oscillator:
Wave function
(in one dimension):
xm
Hem
nx n
xm
nn
2
41 2
!2
1
where 22
d
d1 x
n
xn
n ex
exH
are the Hermite-polinomials
The energy spectrum:
2
1nEn
The 3D harmonic oscillator (in Descartes coordinates):
zm
Hym
Hxm
Hezyxzyxzyx nnn
zyxm
nnn
2
,,
222
,,
2
3
2
3nnnnE zyxn where zyx nnnn
,...2,1,0n
22
02
1rmVrV
Easiest solution is in Descartes coordinates
20
The nuclear shell model (contd.)
a) The parity of the states:
Since and
xm
Hexxx n
xm
n
2
2
xHxHx
x
x n
n
n 1
We get: xxx
x
x n
n
n 1
In 3D: zyxzyxzyx
zyx
zyx nnn
nnn
nnn ,,1,, ,,,,
rr n
n
n 1Finally:
b) The number of degeneration of the states
zyx nnnn How many different combinations of nx, ny, nz
delivers the same n?
n
n
x
x
nnnng
0 2
211
The possible number of particles in a state: 212 nng
6
21
The nuclear shell model (contd.)
The 3D harmonic oscillator (in spherical coordinates):
Wave function:
,,,,,
m
l
n
mln Yr
rRr r
r
lml
l
nr
...3,2,1,0
...3,2,1,0
,,,,2
22
rErrVm
The idea for finding the correspondence: functions should be
chosen to follow
• the same parity behaviour and
• the same number of degenerations!
Question:
What is the relation between n in the Descartes-states (defining the
energy-shells), and the 3 quantum numbers of the wave-function?
22
n (n+1)(n+2) nr l 2(2l+1)
0 2 0 0 1s 2
1 6 0 1 2p 6
2 12 1 0 2s 2
0 2 3d 10
3 20 1 1 3p 6
0 3 4f 14
4 30 2 0 3s 2
1 2 4d 10
0 4 5g 18
Remember:
rr n
n
n 1
212 nng
The nuclear shell model (contd.)
lnn r 2
Descartes spherical
2
8
20
40
70
Magic numbers:
2, 8, 20, 28, 50, 82, 126
The harmonic oscillator is a good „first” approximation,
but something more should be included!