Lesson 6

The Collective Model and the Fermi Gas Model

Evidence for Nuclear Collective Behavior

• Existence of permanently deformed nuclei, giving rise to “collective excitations”, such as rotation and vibration

• Systematics of low lying 2+ states in many nuclei

• Large transition probabilities for 2+0+ transitions in deformed nuclei

• Existence of giant multipole excitations in nuclei

Deformed Nuclei

• Which nuclei? A=150-190, A>220• What shapes?

How do we describe these shapes?

[ ]

( )22

20

2

1

53

4

),(1),(

baR

R

ab

YRR

avg

avg

avg

+=

−=

+=

πβ

φθβφθ

Rotational Excitations

• Basic picture is that of a rigid rotor

)31.01(

1

5

25

2

2

)1(

2

2

2

β+=ℑ

=ℑ

ℑ+

=

mR

mR

where

JJE

ellipsoid

sphere

roth

Are nuclei rigid rotors?

• No, rigidirrot ℑ<ℑ<ℑ exp

rigidirrotational

Backbending

Rotations in Odd A nuclei

• The formula on the previous slide dealt with rotational excitations in e-e nuclei.

• What about odd A nuclei?• Here one has the complication of

coupling the angular momentum of the rotational motion and the angular momentum of the odd nucleon.

Rotations in Odd A nuclei (cont.)

€

E(I) =h2

2ℑI(I +1) −K(K +1)[ ]

For K 1/2

I=K, K+1, K+2, ...

For K=1/2

€

EK=1/ 2(I) =h2

2ℑI(I +1) −α (−1)I +1/ 2(I +1/2)[ ]

I=1/2, 3/2, 5/2, ...

Nuclear Vibrations

• In analogy to molecules, if we have rotations, then we should also consider vibrational states

• How do we describe them?• Most nuclei are spherical, so we base

our description on vibrations of a sphere.

Shapes of nuclei

shaded areas are deformed nuclei

Types of vibrations of spherical nuclei

Formal description of vibrations

€

R(θ, t) = R0 1+ α λ ,0(t)Yλ ,0(cosθ)λ= 0

∞

∑ ⎡

⎣ ⎢

⎤

⎦ ⎥

Suppose =0

€

R(θ, t) = R0 1+3

4πα 1,0(t)cosθ

⎡

⎣ ⎢

⎤

⎦ ⎥€

R(θ, t) = R0 1+α 0,0(t) / 4π[ ]

Suppose =1

Suppose =2

€

R(θ, t) = R0 1+α 2,0(t)3

2cos2θ −

1

2

⎛

⎝ ⎜

⎞

⎠ ⎟

⎡

⎣ ⎢

⎤

⎦ ⎥

Nuclear Vibrations

Levels resulting from these motions

You can build rotational levels on these vibrational levels

€

E =h2

2ℑJ(J +1) −K 2

[ ]

Net Result

Nilsson model

• Build a shell model on a deformed h.o. potential instead of a spherical potential

Superdeformednuclei

Hyperdeformednuclei

Successes of Nilsson model

Nuclide

Z N Exp. Shell Nilsson

19F 9 10 1/2+ 5/2+ 1/2+

21Ne 10 11 3/2+ 5/2+ 3/2+

21Na 11 10 3/2+ 5/2+ 3/2+

23Na 11 12 3/2+ 5/2+ 3/2+

23Mg 12 11 3/2+ 5/2+ 3/2+

Real World Nilsson Model

Fermi Gas Model

• Treat the nucleus as a gas of non-interacting fermions

• Suitable for predicting the properties of highly excited nuclei

DetailsConsider a box with dimensions Lx,Ly,Lz in which particle states are characterized by their quantum numbers, nx,ny,nz. Change our descriptive coordinates to px,py,pz, or their wavenumbers kx,ky,kz where ki=ni/Li=pi/

The highest occupied level Fermi level.

3

2

222222

222

2222

22

2222

3

4

8

1

sinIm2

)(22

),,(

⎟⎠

⎞⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛=

=++==

++=

++=

kLN

conditionsonquantizatigpomL

nkkk

mmk

nnnE

nnnLk

kkkk

states

zyxzyx

zyxf

zyxf

hhh

€

h

Making the box a nucleus

MeVA

ZAE

MeVA

ZE

MeVm

kE

Then

A

Z

r

Z

L

N

Lk

neutronsf

protonsf

ff

statesf

3/1

3/1

22

3/1

0

3/13/1

53

53

322

3

2

3

2

3

2

⎟⎠

⎞⎜⎝

⎛ −=

⎟⎠

⎞⎜⎝

⎛=

≈=

⎟⎠

⎞⎜⎝

⎛=⎟⎠

⎞⎜⎝

⎛=⎟⎠

⎞⎜⎝

⎛=

h

ππ

ππ

ππ

Thermodynamics of a Fermi gas

• Start with the Boltzmann equationS=kBln

• Applying it to nuclei

€

S = k lnΩ = k ln ρ(E*)[ ] − ln ρ (0)[ ]{ }

S =dE *

T0

T

∫

E* = a(kT)2

dE* = 2ak 2TdT

S = 2ak 2 dT = 2ak 2T = 2k(aE*)1/ 2

0

T

∫

ρ(E*)

ρ(0)= exp

S

k

⎛

⎝ ⎜

⎞

⎠ ⎟= exp 2(aE*)1/ 2

[ ]

Relation between Temperature and Level Density

• Lang and LeCouteur

• Ericson10/

])(2exp[)(

1

12

1)(

2

2/14/5

4/12

Aa

TaTE

aETEa

E

=−=

+⎟⎟⎠

⎞⎜⎜⎝

⎛=

ρ

10/

])(2exp[)(

1

12

1)(

2

2/14/5

4/12

Aa

aTE

aEEa

E

==

⎟⎟⎠

⎞⎜⎜⎝

⎛=

ρ

What is the utility of all this?