Fermi Gas Model

Heisenberg Uncertainty Principle

†

dpx ⋅ dx ≥ h Æ dpx =hdx

†

px = px + dpx = px +hdx

Particle in dx will have a minimum uncertainty in px of dpx

dx

pxNext particle in dx will have a momentum px

Particles with px in dpx have minimum x-separation dx

†

dx ⋅ dpx = h

Heisenberg Uncertainty PrincipleIdentical conditions apply for the y, py, and z, pz --

†

dVps ≡ dx ⋅ dpx( )⋅ dy ⋅ dpy( )⋅ dz ⋅ dpz( )= h3

dVps = dV ⋅dp3

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 --

Phase spacevolume

Momentumvolume

Spatialvolume

= •

Heisenberg Uncertainty Principle

†

dN =dVps

h3

In some dVps the maximum number dN of unique quantumstates (fermions) is

px

pz

py

p

†

dN =dV ⋅ 4p p2dp

2p h( )3

Number of states in a shell in p-spacebetween p and p + dp

Only Heisenberg uncertainty principle; completely general

FGM for the nucleusTreat protons & neutrons separatelyConsider a simple model for nucleus--

†

V (x) = 0 ; 0 < x < L V (x) = • ; x ≥ LV (y) = 0 ; 0 < y < L V (y) = • ; y ≥ LV (z) = 0 ; 0 < z < L V (z) = • ; z ≥ L

†

V = V (x) + V (y) + V (z)

†

-h2

2M—2y + Vy = Ey

†

y = jx (x) ⋅ jy (y) ⋅jz (z)

†

E = E x + E y + Ez

†

jx (x) =2L

sin nxL

Ê

Ë Á

ˆ

¯ ˜

†

E x =hp( )2

2ML2 nx2

FGM for the nucleus

†

E = E x + E y + Ez

†

E =hp( )2

2ML2 nx2 + ny

2 + nz2( ) ni =1,2,3,⋅ ⋅ ⋅

Total energy eigenvalue

†

y = jx (x) ⋅ jy (y) ⋅jz (z)

†

jx (x) =2L

sin nxL

Ê

Ë Á

ˆ

¯ ˜

†

ynxnynzx,y,z( )

†

E x =hp( )2

2ML2 nx2

†

E x =px

2

2M Æ px =

hpL

nx

unique states

degenerateeigenvalues

FGM for the nucleus

†

px =hpL

nx Æ p =hpL

Ê

Ë Á

ˆ

¯ ˜ nx

2 + ny2 + nz

2 = p nxnynz( )

†

ynxnynzx, y,z( )

unique states quantized momentum states

px

pz

py

p

†

px , px , px( ) =hpL

nx ,ny ,nz( )

†

px , px , px( )

†

dp3

†

dp3 =hL

Ê

Ë Á

ˆ

¯ ˜

3=

hpL

Ê

Ë Á

ˆ

¯ ˜

3

from Heisenberguncertainty relation

FGM for the nucleus

\ All momentum states up to pF are filled (occupied)

px

pz

py

p

†

px , px , px( )

†

dp3

Assume extreme degeneracy Æ all low levels filled up to amaximum -- called the Fermi level (EF)

We want to estimate EF and pF for nuclei --

The number N of momentum states withinthe momentum-sphere up to pF is --

†

N =18

Ê

Ë Á

ˆ

¯ ˜

43

p pF3

dp3one p-state per dp3

1/8 of sphere because nx, ny, nz > 0

FGM for the nucleus

†

N =18

Ê

Ë Á

ˆ

¯ ˜

43

p pF3

2ph

LÊ

Ë Á

ˆ

¯ ˜ 3

N =18

Ê

Ë Á

ˆ

¯ ˜

22ph( )3

43

p pF3V

†

N = pV3

2MEFh2p 2

È

Î Í ˘

˚ ˙

3/2†

pF = 2MEF

†

L3 = V( )

†

EF =hp( )2

2M3NpV

È

Î Í ˘

˚ ˙

2 /3Fermi energy

(most energetic nucleon(s)

†

pF = h 3p 2( )1/3 N

VÈ

Î Í ˘

˚ ˙

1/3Fermi momentum

(most energetic nucleon(s)

protonsN = Z

neutronsN = (A-Z)

2 spin states

†

N =18

Ê

Ë Á

ˆ

¯ ˜

43

p pF3

dp3

FGM for the nucleus

†

pF = h 3p 2( )1/3 Z

VÈ

Î Í ˘

˚ ˙

1/3

Protons Neutrons

†

pF = h 3p 2( )1/3 A - Z

VÈ

Î Í ˘

˚ ˙

1/3

Assume Z = N

†

pF = h 3p 2( )1/3 A /2

VÈ

Î Í ˘

˚ ˙

1/3

†

pF = h 3p 2( )1/3 A /2

VÈ

Î Í ˘

˚ ˙

1/3

†

V =43

p R3

R = RoA1/3

†

V =43

p Ro3A = 4.18Ro

3A

†

pF =h

Ro

3p 2

2 ⋅ 4.18

Ê

Ë Á Á

ˆ

¯ ˜ ˜

1/3

FGM for the nucleus

†

pF = h 3p 2( )1/3 Z

VÈ

Î Í ˘

˚ ˙

1/3

Protons Neutrons

†

pF = h 3p 2( )1/3 A - Z

VÈ

Î Í ˘

˚ ˙

1/3

Assume Z = N

†

pF =197Mev

Roc3p 2

2 ⋅ 4.18

Ê

Ë Á Á

ˆ

¯ ˜ ˜

1/3

=300Ro

MeV /c

†

pF = 231 MeV /c (Ro =1.3F)

†

EF =pF( )2

2M= 28 MeV

FGM potential

Test ofFGM

not FGM