Lesson 6 The Collective Model and the Fermi Gas Model.
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Transcript of Lesson 6 The Collective Model and the Fermi Gas Model.
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?