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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 - PowerPoint PPT Presentation
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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?