1. No category

Fundamentals in Nuclear Physics (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
2014/5/22
Fundamentals in
Nuclear Physics
原子核基礎
Kenichi Ishikawa (石川顕一)
http://ishiken.free.fr/english/lecture.html
[email protected]
2014.5/22
1
Fundamentals in Nuclear Physics (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
Nuclear reactions (2)
2014/5/22
2
Fundamentals in Nuclear Physics (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
Resonance
4
10
3
(n,t)
1/v law
2
10
162
共鳴
resonance
1
10
3. Nuclear reactions
0
10
-1
10
(n,p)
-3
10
-4
10
-5
10
-3
-1
10
10
1
3
10
10
Energy (eV)
5
10
3.1 Cross-sections
7
10
117
10
E (MeV)
7.459
6.668
−2
−4
(n,fission) (/105 )
238
2
U
elastic
1
−2
3 H 4 He
7 Li
Fig. 3.5. The energy levels of 7 Li and two dissociated states n −6 Li and 3 H − 4 He.
6
(n,γ) (/100)
10
10
0
U
10
4.630
0.477
2
10
10
n 6 Li
235
1
(n,gamma)
-2
10
複雑なエネルギー依存性
elastic (x10)
(n,n)
10
重い核には多くの励起状態
Many excited states for heavy nuclei
complicated resonance structure
JENDL
cross−section (barn)
Cross section (barn)
10
6Li
共鳴
−4
1
(n ,γ)
(/10 4 )
10
10
2
Excited states of 239U E (eV)
10
3
2014/5/22
Fig. 3.26. The elastic and inelastic neutron cross-sections on 235 U (top) and 238 U
(bottom). The peaks correspond to excited states of 236 U and 239 U. The excited
3
Fundamentals in Nuclear Physics (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
Resonance line shape
Resonance
(E)
A
E0 )2 + ( /2)2
(E
2.0
1.5
full width at half
maximum (FWHM)
半値全幅
Doppler effect
1.0
1.0
ドップラー効果
0.8
long tail
0.5
0.6
0.4
-3
-2
-1
1
2
Lorentzian ローレンツ関数
E3
0.2
-3
Life time
Decay rate
·
=
: 自然幅
natural width
= /
homogeneous width
uncertainty principle
= /
1
不確定性原理
-2
-1
1
2
3
ドップラー幅
inhomogeneous width
(E E0 )2
exp
E2
2014/5/22
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Fundamentals in Nuclear Physics (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
Time-dependent wave function of an excited state
(r, t) = (r)e
2
2
| (r, t)| = | (r)|
iE0 t/
does not decay
To be consistent with the exponential decay law
2
2
| (r, t)| = | (r)| e
t/
(r, t) = (r)e
iE0 t/
e
t/2
Energy spectrum (by Fourier transform)
2
P (E)
eiEt/ e
iE0 t/
e
t/2
dt
0
1
(E
E0 )2 + ( /2)2
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核子-原子核散乱の量子力学的取り扱い
Quantum treatment of
nucleon-nucleus scattering
4
10
JENDL
3
Cross section (barn)
10
2
10
1H(n,n)
1
10
2H(n,n)
6Li(n,n)
0
10
-1
10
-5
10
-3
10
-1
10
1
3
10
10
Energy (eV)
5
10
7
10
2014/5/22
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Fundamentals in Nuclear Physics (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
isotropic scattering
V(r)
R
angular momentum 角運動量
1.2A1/3 fm
R
等方散乱
r
L = kR
kR
1
for neutron scattering 中性子散乱
( c)2
2mn c2 R2
p2
E=
2mn
V0
Schrödinger equation
eikr
r
z
eikz
2
2m
k (r)
2
シュレーディンガー方程式
+ V (r)
= eikz
13 MeV
A2/3
f eikr
+
r
2 2
k
k (r) =
2m
k (r)
(r > R)
2014/5/22
7
Fundamentals in Nuclear Physics (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
eikz =
(2l + 1)il jl (kr)Pl (cos )
l=0
10.48
spherical Bessel function
Legendre polynomial
球ベッセル関数
ルジャンドル多項式
Graphs
263
P0 (cos ) = 1
P1 (cos ) = cos
3
P2 (cos ) = cos2
2
Figure 10.48.1: jn (x), n =l 0(1)4, 0  x  12.
jl (kr)
(kr) /(2l + 1)!!
118
1
2
Figure 10.48.2: yn (x), n = 0(1)4, 0 < x  12.
3. Nuclear reactions
5
800 MeV (x50)
5
5
2
5
2
5
dσ /dcos θ (barn)
sin z
j0 (z) =
z
sin z
cos z
j1 (z) = 2
p
z + y (x), 0  x  12.
Figure 10.48.3:zj (x), y (x), j (x)
1H
1MeV
100MeV (x20)
0
5
0.1 MeV
9Be
1 MeV
10 MeV
0
Figure
10.48.4: j05 (x), y50 (x),
5
12.
q
j05 2 (x) + y50 2 (x), 0  x 
0.1 MeV
208Pb
1 MeV (x0.5)
0
−1
10 MeV (x0.2)
cosθ
1
Fig. 3.6. The differential cross-section, dσ/d cos θ = 2πdσ/dΩ, for elastic scattering of neutrons on 1 H, 9 Be and 208 Pb at incident neutron energies as indicated [30].
At low incident momenta, p < ¯
h/Rnucleus , the scattering is isotropic whereas for
2014/5/22
8
Fundamentals in Nuclear Physics (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
k (r)
=
sin kr
e
kr
anisotropic
ikz
+
sin kr f eikr
+
kr
r
isotropic
等方
非等方
sin kr f eikr
+
kr
r
isotropic
k (r)
uk (r) = r
(r > R)
near the boundary
kR
1
ポテンシャルの境界近く
isotropic also at r < R
ポテンシャル内でも等方
k (r)
2
2 2
k
d2 uk
+ V (r) uk (r) =
uk (r)
2
2m dr
2m
uk (r) =
sin kr
+ f eikr
k
(r > R)
2014/5/22
9
Fundamentals in Nuclear Physics (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
Solution at r < R
uk (r) = A sin Kr
(r < R)
Boundary condition uk(r) and uk’(r) continuous at r = R
境界条件
kR
f =R
1
tan KR
KR
1
2mV0
K
2
low-energy scattering
Cross section
= 4 |f |2 = 4 R2
Scattering length
a=
tan KR
KR
2
1
散乱長
f (k = 0)
(k
0) = 4 a2
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Fundamentals in Nuclear Physics (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
= 4 |f |2 = 4 R2
tan KR
KR
2
1
R2
30
25
20
15
10
5
1
2
3
4
5
6
KR
R
2mV0
2
~ depth of attractive potential
kR = /2
infinite scattering length
R2
no scattering
“Ramsauer-Townsend effect”
ラムザウアー・タウンゼント効果
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Fundamentals in Nuclear Physics (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
Nucleon-nucleon effect
核子-核子散乱
4
10
flat region
20 b
el
JENDL
3
Cross section (barn)
10
range of the strong
interaction
(2fm)2
0.1 b
2
10
1H(n,n)
kR
1
10
6Li(n,n)
0
-5
10
-3
10
-1
10
1
3
10
10
Energy (eV)
5
10
7
10
|u|
2
33
deutron
u(r) = sin kr
-1
10
V0R2~109 MeV fm2
1.4 Nuclear forces and interactions
2H(n,n)
10
/2
u(r) = exp(−κ r)
重陽子
r=R
V(r)
V(r)=−V0
V(r)=0
r
Fig. 1.10. A square-well potential and the square of the wavefunction ψ(r) =
u(r)/r. The depth and width of the well are chosen to reproduce the binding energy
and radius of the deuteron. Note that the wavefunction extends far beyond the
effective range of the s = 1 nucleon–nucleon potential.
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the deuteron is bound. For the s = 0 system, V0 R is slightly less than
Fundamentals in Nuclear Physics (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
109 MeV fm2 so there is no bound state. In the scattering problem considered here, the quarter wavelength leads to a scattering cross-section that is
much larger than 4πR2 since the tangent in (3.197) is large.
Nucleon-nucleon effect
Table 3.3. The low-energy nucleon–nucleon scattering amplitudes and effective
ranges taken from the compilation [34]. The last two columns give the potential
parameters derived from the deuteron binding energy and the scattering formula
(3.197). Note that f ≫ R for the s = 0 amplitudes.
核子-核子散乱
f
(fm)
R
(fm)
V0
(MeV)
V0 R 2
(MeV fm2 )
n–p (s=1, T=0)
+5.423 ± 0.005
1.73 ± 0.02
46.7
139.6
n–p (s=0, T=1)
−23.715 ± 0.015
2.73 ± 0.03
12.55
93.5
p–p (s=0, T=1)
−17.1 ± 0.2
2.794 ± 0.015
11.6
90.5
n–n (s=0, T=1)
−16.6 ± 0.6
2.84 ± 0.03
11.1
89.5
The cross-section for 3the scattering
neutrons on unpolar1on unpolarized
2
2
=
4
|f
|
+
4
|f
|
20
p
s=1
s=0
ized protons is thenweighted
in theb(s = 0) and (s = 1)
4 sum of the cross-section
4
state. Since there are three spin-aligned states and only one anti-aligned state
we have
σn−p = (3/4)4π|fs=1 |2 + (1/4)4π|fs=0 |2 = 20.47 b .
(3.199)
This corresponds to the low-energy limit of the neutron–proton cross-section2014/5/22
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