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FermiGasy
W. Udo Schröder, 2004
Ferm
iGas
Mod
el
2
Particles in Ideal 1D Box
Approximate picture: independent particles in mean field produced by interactions of all nucleons
calculate single-particle spectrum
approximate potential (first neglect VCoulomb)Average Potential
V(x)a x
2 22
2
:
1 22 2
; 0,1,2..
2 8
2( ) sin( )
n
n
n
nn
n n
Boundary Condition
stationary waves
n n ak
k n na
k hn
m ma
x k aa
1 (x)
2 (x)
3 (x)
n/
1
1 (x),
2
(x),
3
(x)
W. Udo Schröder, 2004
Ferm
iGas
Mod
el
3
Particles in Ideal Multi-D Box
3 2
3 2, ( ) ( ) ( ) sin( ) sin( ) sin( )
x y z
Dn n n x y zx y z
For a a a a
x y yx y x y z n n n
a a a a
State SpaceState Space
ni = 1,2,3,..i=x,y,z
2D
x
y
ax=ay=az=100
nx
ny
nz
2
2 2 212x y z x y zn n n
m a
214
8dV d
x
y
z
n
p na
n
pa
W. Udo Schröder, 2004
Ferm
iGas
Mod
el
4
Density of Fermi Gas States
nx
nz
3D box, side length a, volume V = a3
Every point on 3D-integer grid in p-space represents one state
n»1 continuous approximation
How many states dn in {p, p+dp} {, +d}?
x
y
z
n
p na
n
22
232
2 3 3
2 3
3 2
2 3
:
( 2 )
( 2 )
48 2
4
2 2
2
2
a
a
C
p dpdn d
a a
V p dpap dp
p dp m
x for spin
x for isospi
d
mdn V d
n
dn/d
= A nucleons
F
em
pty
W. Udo Schröder, 2004
Ferm
iGas
Mod
el
5
The Fermi Energy
dn/d
= A nucleons
F
em
pty
3 2
0 0
2 32 3
2 32
84 4
3
32
2 2
F F
a a F
FF
a
dn CA d CV dE E V
d
k Am m V
2 322 2 33
2 2F Am
A = matter density32
2
3A Fk
Fermi energy (nucleon density)2/3 Fast nucleons in dense matter
Fill all single-particle states with 4 nucleons each (spin, isospin up/down) degenerate FG
Nuclear matter: F = 37 MeV
A = 0.16 fm-3 kF = 1.36 fm-1
pF=kF =268MeV/c
Mean field potential U0 = F + B/A = 45 MeV
W. Udo Schröder, 2004
Ferm
iGas
Mod
el
6
Total Energy
3 2 5 2
0 0
5 3 5 32 3 2 3
5 35 2
5 2 5 35 2 3
0
58
4 4
23 3
2
2 42
3
F F
tot a a F
Fa
dnE d CV dE E C V
d
AVm
m r
5 32 3
3 230 5 32 3
5 2 30
38 4 2
15 205 32 4
23
totm
E r A A MeV
m r
r0 = (1.2-1.4) fm
Treat all nucleons same with 2 s x 2 qu. numbers, degenerate states
30
43a
Nuclear Volume
V r A
W. Udo Schröder, 2004
Ferm
iGas
Mod
el
7
2-Component Fermi Gas
F
Vn Vp
rr
Mix of 2 independent (n,p) gases
Protons feel Coulomb potential VCoul
In real nuclei F(N) F(Z)
Otherwise conversion ( decay) n p
Nuclei have N > Z
px
py
pF=kF
Ground state: degenerate FG (T=0)
Excited state: non degenerate FG (T≠0)
px
py
T
Ground State
W. Udo Schröder, 2004
Ferm
iGas
Mod
el
8
2 3 2 3 2 32
20
2 3 2 3 2 32
20
9( ) 38.6
4 2
9( ) 38.6
4 2
F
F
Z ZZ MeV
A Amr
N NN MeV
A Amr
r0 = 1.4 fm
3 2
2 3
2a
dn mV both spins
d
3 2 5 2
0 0
25
F F
tot a a Fdn
E d CV d C Vd
2 3 2 5 3
2 2 30
( )3 9 3( ) ( ) 0.6 ( )
10 4 5tot
tot F FE ZZ
E Z Z Z ZZmr A
2 3 2 5 3
2 2 30
( )3 9 3( ) ( ) 0.6 ( )
10 4 5tot
tot F FE NN
E N N N NNmr A
W. Udo Schröder, 2004
Ferm
iGas
Mod
el
9
Asymmetry Energy
5 3
2 3( )tot
ZE Z C
A
2 3 2
20
3 910 4
Cmr
5 3
2 3( )tot
NE N C
A
5 3 5 3
2 3 2 3( ) ( ) ( )tot tot tot
Z NE A E Z E N C
A A
5 3 5 3
2 3 2 3
2
( ) ( ) ( ) 2 22
tot tot tot
N Z A
C CE A E Z E N A A A
A
5 35 3 5 32 3
5 3 5 3
5 3 5
2
3
2
( ) ( ) ( ) 2 2
51 1 2
9( )
2 2
N Ztot to
tot
t tot
N Z A
CE A E A E A Z N A
A
CA N ZN Z NA
E AA
Z CA
5 3 25 101 1 ...
3 18Using x x x
This is the origin of the asymmetry energy in the LDM !
Seminar: Statistical Decay of Complex Systems (Nuclei)
1. Nuclear Models: The Fermi Gas2. Density of states of A-body system
Temperature concept and level density3. Spin and structure dependence of level densities
4. Weisskopf model of statistical decayExamples and applications
5. Hauser-Feshbach model6. Dynamical effects7. Pre-equilibrium decay8. Compound nucleus reactions9. Multi-fragment decay
