Transcript Fermi Gas Model - Valparaiso University

Fermi Gas Model
Heisenberg Uncertainty Principle
h
dpx  dx  h  dpx 
dx
Particle in dx will have a minimum uncertainty in px of dpx
px
dx
Next particle in dx will have a momentum px
h
px  px  dpx  px 
dx
Particles with px in dpx have minimum x-separation dx

dx  dpx  h
Heisenberg Uncertainty Principle
Identical conditions apply for the y, py, and z, pz -Therefore, in a fully degenerate system of fermions,
(i.e., all fermions in their lowest energy state),
we have 1 particle in each 6-dimensionl volume --


dV ps  dx  dp x  dy  dp y  dz  dpz  h 3
dV ps  dV  dp 3
Phase space
=
volume
Spatial
Momentum

volume
volume
Heisenberg Uncertainty Principle
In some dVps the maximum number dN of unique quantum
states (fermions) is
pz
dN 
dV ps
h
dN 
p
3
py
dV  4 p 2 dp
2

3
px
Number of states in a shell in p-space
between p and p + dp
Only Heisenberg uncertainty principle; completely general


FGM for the nucleus
Treat protons & neutrons separately
Consider a simple model for nucleus-V (x)  0 ; 0  x  L
V (x)   ; x  L
V (y)  0 ; 0  y  L
V (y)   ; y  L
V (z)  0 ;
V (z)   ; z  L
0zL
V  V (x)  V (y)  V (z)


2
2M
 2  V  E
   x (x)   y (y)   z (z)
E  E x  E y  Ez
n x 
2
 x (x) 
sin  
L  L 
2
 2


n
Ex
2
2ML
x




FGM for the nucleus
E  E x  E y  Ez
2


E
Total energy eigenvalue

degenerate
ni  1,2,3,  
eigenvalues

   x (x)   y (y)   z (z)
unique states
2
2
2
n

n

n
y
z
2 x
2ML
n x 
2
 x (x) 
sin  
L  L 
Ex



2
2
n
2 x
2ML
 n x n y n z x, y,z
2
p

x

Ex 
 px 
nx
2M
L

FGM for the nucleus
   2
px 
n x  p    n x  n 2y  nz2  p n x n y nz
 L 
L


unique states
quantized momentum states
p x , p x , p x  
 n x n y n z x, y,z
h    
dp      
L   L 

3
from Heisenberg
uncertainty relation


3
3
px
pz

n x ,n y ,nz 

L
p


dp 3
p x , p x , p x 
py
FGM for the nucleus
Assume extreme degeneracy  all low levels filled up to a
maximum -- called the Fermi level (EF)
 All momentum states up to pF are filled (occupied)
We want to estimate EF and pF for nuclei -The number N of momentum states within
the momentum-sphere up to pF is -1 4  pF3
N   
8 3 dp 3
pz
p
one p-state per dp3
1/8 of sphere because nx, ny, nz > 0
px


dp 3
p x , p x , p x 
py
FGM for the nucleus
1 4  pF3
1 4  pF3
N   
pF  2ME F
N   
3
3
8 3 dp
8 3 2 
 
 L 
2 spin states
3/2


V
2ME
1  2 4
F
3
L3  V

N


N   

p
V
 2 2 
F
3
3
  
8 2  3



EF
pF 

2
  3N 2 / 3






2M

V

1/ 3
1/
3


N
2
 
3
Fermi energy
 nucleon(s)
(most energetic
Fermi momentum

V 
 (most energetic nucleon(s)
protons
N=Z
neutrons N
= (A-Z)
FGM for the nucleus
Protons
pF 
Neutrons
1/ 3
1/
3


Z
2
3 

V 

pF 
1/ 3
1/
3


A

Z
2
3 

 V 

Assume Z = N
pF 
V
1/ 3
1/
3



A
/2
2
3 
4
 R3
3
R  Ro A1/ 3

 V 

pF 
4
V   Ro3A  4.18 Ro3A
3
1/ 3
1/
3


A
/2
2
3 

 V 

 3 2 1/ 3

pF  

Ro 2  4.18 

FGM for the nucleus
Protons
pF 
Neutrons
1/ 3
1/
3


Z
2
3 

V 

pF 
1/ 3
1/
3


A

Z
2
3 

 V 

Assume Z = N
3
2 1/ 

197Mev 3
300


pF 

MeV /c


Roc 2  4.18 
Ro
pF  231 MeV /c
(Ro 1.3F)
EF
2
pF 


 28 MeV
2M
FGM potential
Test of
FGM
not FGM