Introduction to Nuclear Engineering (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
2016/9/27
Introduction to Nuclear Engineering Kenichi Ishikawa (石川顕一) http://ishiken.free.fr/english/lecture.html
[email protected] 1
Introduction to Nuclear Engineering (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
Nuclear Physics
• basic properties of nuclei • nuclear reactions • nuclear decays References • Basdevant, Rich, and Spiro, “Fundamentals in Nuclear Physics” (Springer, 2005) • Krane, “Introductory Nuclear Physics” (Wiley, 1987) • 八木浩輔「原子核物理学」(朝倉書店, 1971) • 石川顕一、高橋浩之「工学教程『原子核工学II』」（丸善、準備中）
Course material downloadable from: http://ishiken.free.fr/english/lecture.html 2
Introduction to Nuclear Engineering (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
Basic properties of nuclei
3
Introduction to Nuclear Engineering (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
A nucleus is made up of protons and neutrons
E = mc2 charge
mass (kg)
mass energy (MeV)
proton (p)
+e
1.67493 10-27
938.272
neutron (n)
0
1.67262 10-27
939.565
electron (e-)
-e
9.109 10-31
0.511
e = 1.6022 ⇥ 10
19
C
MeV = 106 eV
eV = 1.6022 ⇥ 10
×1.0014 ×1840 19
J
nucleon: proton, neutron
4
Introduction to Nuclear Engineering (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
A nucleus is labeled by atomic number and mass number
A X Z
Z : atomic number = number of protons N : number of neutrons A = Z + N : mass number
example 235 92 U
or simply
235
U “uranium-235”
N = 235 - 92 = 143 5
Introduction to Nuclear Engineering (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
nuclear binding energy and mass defect mass defect
M = Zmp + N mn proton mass
binding energy B =
mN > 0
neutron nuclear mass mass
M c2 = (Zmp + N mn max 8.7945 MeV @ 62Ni
binding energy per nucleon
B/A (MeV)
8
~ 8 MeV
6 4 2 0 0
stable unstable 50
100 150 200 mass number A 6
250
mN )c2
Introduction to Nuclear Engineering (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
Nuclear reactions
7
Introduction to Nuclear Engineering (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
Nuclear reactions free particle (photon, electron, positron, neutron, proton, …) target projectile
scattering a + X →Y + b projectile
example
nuclear reactions X(a,b)Y
target
α + 14N → 17O + p (Rutherford, 1919)
p + 7Li → 4He + α (Cockcroft and Walton, 1930) 8
Introduction to Nuclear Engineering (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
Energetics エネルギー論
a + X →Y + b
mX c2 + TX + ma c2 + Ta = mY c2 + TY + mb c2 + Tb rest mass
kinetic energy
reaction Q value
Q = (minitial mfinal )c2 = (mX + ma mY mb )c2 = TY + Tb TX Ta excess kinetic energy
Q > 0 : exothermic 発熱反応 Q < 0 : endothermic 吸熱反応 9
Introduction to Nuclear Engineering (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
Important nuclear reactions for thermal energy generation Fission（核分裂） example
235
235
U + n ! X + Y + (2 ⇠ 3) n
U + n ! 144 Ba + 89 Kr + 3n + 177 MeV
Fusion（核融合） D + T ! 4 He (3.5 MeV) + n (14.1 MeV) D + D ! T (1.01 MeV) + p (3.02 MeV)
! 3 He (0.82 MeV) + n (2.45 MeV)
D + 3 He ! 4 He (3.6 MeV) + p (14.7 MeV)
> 106 times more efficient than chemical reactions! 10
Introduction to Nuclear Engineering (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
Nuclei for fission reactors
• • •
233U, 235U, 239Pu
(fissile materials)
→ fission by thermal neutron capture Fission of 235U produces ~2.5 neutrons 238U, 232Th
(fertile materials) change to 239Pu, 232Th by neutron capture
→ fast breeder reactor 11
Introduction to Nuclear Engineering (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
Cross section 断面積
3.1 Cross-sections
Probability P proportional to
• •
109
. L
number density of target particles n target thickness dz
dz Fig. 3.1. A small particle incident on a slice of matter containing N = 6 target spheres of radius R. If the point of impact on the slice is random, the probability dP of it hitting a target particle is dP = N πR2 /L2 = σndz where the number density of scatterers is n = N/(L2 dz) and the cross section per sphere is σ = πR2 .
dP = ndz
a probability dP of hitting one of the spheres that is equal to the fraction of the surface area covered by a sphere N πR2 = σndz σ = πR2 . (3.4) L2 In the second form, we have multiplied and divided by the slice thickness dz and introduced the number density of spheres n = N/(L2 dz). The “crosssection” for touching a sphere, σ = πR2 , has dimensions of “area/sphere.” While the cross-section was introduced here as a classical area, it can be used to define a probability dPr for any type of reaction, r, as long as the probability is proportional to the number density of target particles and to the target thickness: dP =
Unit of cross section dimension of area m2, cm2 size of nucleus ~ a few fm
dPr = σr n dz .
1 barn (b) =
10-28
m2
(3.5)
The constant of proportionality σr clearly has the dimension of area/particle and is called the cross-section for the reaction r. If the material contains different types of objects i of number density and cross-section ni and σi , then the probability to interact is just the sum of the probabilities on each type: ! ni σ i (3.6) dP =
= 10-24 cm2 i
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Introduction to Nuclear Engineering (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
Differential cross section angular dependence (角度依存性を考える) Probability that the incident particle is scattered to a solid angle d
detector d target
dP for isotropic scattering (等方散乱)
d d
=
,
d ndzd = d
differential cross section (微分断面積)
4
total cross section
=
d d d
2
=
d 0
0
d ( , ) sin d d 13
Introduction to Nuclear Engineering (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
平均自由行程
反応速度
Mean free path and reaction rate 3.1 Cross-sections
109
flux F .
dF =
dF = dz
L
Fig. 3.1. A small particle incident on a slice of matter containing N = 6 target spheres of radius R. If the point of impact on the slice is random, the probability dP of it hitting a target particle is dP = N πR2 /L2 = σndz where the number density of scatterers is n = N/(L2 dz) and the cross section per sphere is σ = πR2 .
= n
macroscopic cross section (マクロ断面積) a probability dP of hitting one of the spheres that is equal to the fraction of the surface area covered by a sphere 2
2
l = 1/ n
σ = πR2 .
if there are different types of target objects (nuclei)
l = 1/
i ni
(3.5)
The constant of proportionality σr clearly has the dimension of area/particle and is called the cross-section for the reaction r. If the material contains different types of objects i of number density and cross-section ni and σi , then the probability to interact is just the sum of the probabilities on each type: ! ni σ i (3.6) dP =
reaction rate i
v i =n v l
nz
= F (0)e
z
[1/length]
1.0
also distribution of free path
(3.4)
In the second form, we have multiplied and divided by the slice thickness dz and introduced the number density of spheres n = N/(L2 dz). The “crosssection” for touching a sphere, σ = πR2 , has dimensions of “area/sphere.” While the cross-section was introduced here as a classical area, it can be used to define a probability dPr for any type of reaction, r, as long as the probability is proportional to the number density of target particles and to the target thickness: dPr = σr n dz .
F n
F (z) = F (0)e
dz
N πR mean σndz dP =free =path L
F ndz
0.8 0.6
F(z)/F(0) 0.4
1/e = 0.368 0.2 0.0
1/ n
z 14
Introduction to Nuclear Engineering (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
General characteristics of cross-sections Elastic scattering 弾性散乱 The internal states of the projectile and target (scatterer) do not change before and after the scattering. • Rutherford scattering, (n,n), (p,p), etc. Inelastic scattering 非弾性散乱
• •
(n,γ), (p,γ), (n,α), (n,p), (n,d), (n,t), etc. fission, fusion 15
Introduction to Nuclear Engineering (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
Elastic neutron scattering • relevant to (neutron) moderator in nuclear reactors
中性子減速材
• due to the short-range strong interaction 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)
resonance
1
10
2H(n,n) 6Li(n,n)
0
10
p > h/r
(p2 /2mn > 200 MeV) -1
10
-5
10
-3
10
-1
10
1
3
10 10 Energy (eV)
5
10
7
10
16
Introduction to Nuclear Engineering (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
Elastic neutron scattering 10 9
4
10
JENDL
3
8
resonance 共鳴
2
10
1H(n,n) 1
10
6.54
6 5
4.63
4 2.466 3
2
6Li(n,n)
10
6
n + Li
3
2H(n,n)
0
7.47
7.253
7
Energy (MeV)
Cross section (barn)
10
0 -5
10
-3
10
-1
10
1
3
10 10 Energy (eV)
5
10
0.478 0 0
2
4
7
10
4
H + He
1
-1
10
9.6
7
Li
6
8
10
The energy levels of 7Li and two dissociated states n-6Li and 3H-4He (t-α)
n + 6Li → 7Li* → n + 6Li 17
Introduction to Nuclear Engineering (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
Nuclear data libraries • ENDF (Evaluated Nuclear Data File, USA) • JENDL (Japanese Evaluated Nuclear Data Library, Japan)
• JEFF (Joint Evaluated Fission and Fusion file, Europe)
• CENDL (Chinese Evaluated Nuclear Data Library, China) • ROSFOND (Russia) • BROND (Russia)
http://www-nds.iaea.org/exfor/endf.htm 18
Introduction to Nuclear Engineering (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
Inelastic scattering
中性子捕獲反応
Neutron capture neutron binding energy = ca. 8 MeV
放射化
activation
発熱反応
exothermic reaction in most cases Highly excited states formed, which subsequently decay.
• Radiative capture X(n,γ) X • emits a gamma ray • Cd(n,γ) Cd ← neutron shield • Other neutron capture reactions • B(n,α) Li, He(n,p) H, Li(n,t) He • Applications: neutron detector, shield, neutron 放射捕獲（放射性捕獲） A
113
10
A+1
114
7
3
3
6
4
capture therapy for cancer
19
Introduction to Nuclear Engineering (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
Inelastic scattering
放射捕獲（放射性捕獲）
neutron radiative capture Cross section (barn)
No threshold ← exothermic, no Coulomb barrier 10
1
10
0
JENDL
10
-1
10
-2
10
-3
10
-4
10
-5
10
-6
1H(n,gamma) 6Li(n,gamma)
2H(n,gamma) JENDL
10
ENDF -5
10
-3
10
-1
10
1
10
3
10
5
10
7
Energy (eV)
discrepancy between JENDL and ENDF
E
1/v law
1/2
1/v
Energy-independent reaction rate
v 20
Introduction to Nuclear Engineering (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
Neutron capture reactions with large cross section
• • • • •
113Cd(n,γ)114Cd
: shield
157Gd(n,γ)158Gd
: neutron absorber in nuclear fuel, cancer therapy 10B(n,α)7Li
: detector, cancer therapy
3He(n,p)3H
: detector
6Li(n,t)4He
: shield, filter, detector 21
Introduction to Nuclear Engineering (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
Inelastic scattering
10B(n,α)7Li 5
10
JENDL
4
Cross section (barn)
10
10B(n,alpha)7Li
3
10
1/v law
2
10
1
10
0
10
-1
10
-5
10
-3
10
-1
10
1
3
10 10 Energy (eV)
5
10
7
10
Applications • BF3 proportional counter • Boron neutron capture therapy (BNCT) for cancer 3He(n,p)3H
• Helium-3 proportional counter 22
Introduction to Nuclear Engineering (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
光核反応
Photo-nuclear reaction
• Excitation and break-up (dissociation) through photo-absorption
• Analog of the photoelectric effect -3
0.7
2.5x10
2H
1.5
0.6
Cross section (barn)
Cross section (barn)
2.0
2H(gamma,n)1H
1.0
208Pb(gamma,n)207Pb
208Pb
0.5 0.4 0.3 0.2
0.5 0.1
0.0 0
5
10
15
20
25
30
Energy (MeV)
threshold (2.22 MeV) = binding energy of 2H
0.0 0
5
10
15
20
25
30
Energy (MeV)
giant resonance 巨大共鳴 collective oscillation of protons in the nucleus 23
Introduction to Nuclear Engineering (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
Resonance 6Li
4
10
3
(n,t)
1/v law
2
10
162
共鳴 resonance
1
10
0
-1
10
(n,p)
-3 -4
10
-5
-3
10
10
-1
10
10
1
3
5
10 10 Energy (eV)
10
7
10
9.6
9
Energy (MeV)
7
6
n + Li
6.54
−2 −4
238
3
2.466 3
2 0.478 0 2
4
7
Li
U
elastic
1
10
4
H + He
0
2
10
4
0
(n,fission) (/105 )
−2
4.63
1
U
(n,γ) (/100)
10
6 5
2
10
10
7.47
7.253
235
1
(n,gamma)
-2
10
複雑なエネルギー依存性
elastic (x10) 10
10
8
3. Nuclear reactions
(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
共鳴
−4
1 6
8
10
(n ,γ) (/10 4 ) 10
2
10
Excited states of 239U E (eV)
10
3
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
24
Introduction to Nuclear Engineering (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 Life time Decay rate
·
=
= / 1
= /
E3
: 自然幅 natural width homogeneous width
uncertainty principle 不確定性原理
0.2
-3
-2
-1
1
2
3
ドップラー幅 inhomogeneous width (E E0 )2 exp E2 25
Introduction Fundamentals to Nuclear in Nuclear Engineering Physics (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
Nuclear decays
26
Introduction to Nucleardecay Engineeringto (Kenichi for internal use only where the third mode is the unlikely radiative theISHIKAWA) ground state. In (Univ. of Tokyo) general we have ! 自然幅 Bk = 1壊変（崩壊）速度 , (4.7)
Decay rate, natural width
k
the sum of the “partial decay rates,” λk = Bk λ ! probability to λdecay in an interval dt λk = , k decay rate 壊変（崩壊）速度 dt and the sum of the = dtΓk = Bk Γ dP“partial = widths,” ! Γk = Γ . mean life time 平均寿命
(4.8)
(4.9)
k
N (t) = N (t = 0)e 4.1.2 Measurement of decay rates t1/2 = range (ln 2)from=∼ 0.693 half life 半減期 Lifetimes of observed nuclear transitions 10−22 sec number of unstable nuclei
7
Li (7.459 MeV) → n 6 Li,
to 102176yr Ge
76
Se 2e 2¯e
3
H 4 He
t1/2 = 1.78
τ = 6 × 10−21 sec
1021 yr
t/
(4.10)
> 1011 × (age of universe) !
An unstable particle has an energy uncertainty or “natural width”
=
=
=
6.58
10
22
MeV sec
Licensed to Kenichi Ishikawa 27
Introduction to Nuclear Engineering (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
壊変図
Decay diagram
half life 半減期
branching ratio 分岐比
28
Introduction to Nuclear Engineering (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
alpha decay = 42 He A ZX example 238 U
!
A 4 Z 2Y
+↵
! 234 Th + ↵ (4.2 MeV)
half life = 4.468×109 years
29
Introduction to Nuclear Engineering (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
beta decay decay +
decay
A ZN A ZN
A Z+1 N A Z 1N
+ e + ¯e
+ e+ +
e
half life = 5730 years dating 年代測定
30
GT :
Ji = Jf , Jf ± 1 Introduction Ji = Jto = 0 forbidden . (4.91) f Nuclear Engineering (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
Additionally, in both cases, the parity of the initial and final nuclei must be the same. Transitions that respect the selection rules are called “Allowed” decays. “Forbidden” decays are possible only if one takes into account the spatial dependence of the lepton wavefunctions, i.e. using (4.83) instead of (4.84) The examples of forbidden decays in Fig. 4.12 illustrate the much longer lifetimes for such transitions.
Emitted electron (positron) energy has a broad distribution 64Cu
β
_
0.2 0.6 1.0 1.4 1.8 p (MeV/c)
64Cu
β
+
0.2 0.6 1.0 1.4 1.8 p (MeV/c)
Fig. 4.14. The β− and β+ spectra of 64 Cu [44]. The suppression the of the β+ spectrum and enhancement of the β− at low energy due to the Coulomb effect is seen.
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Introduction to Nuclear Engineering (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
beta decay decay +
decay
A ZN A ZN
A Z+1 N A Z 1N
+ e + ¯e
+ e+ +
e
half life = 5730 years dating 年代測定
The existence of the neutrino was predicted by Wolfgang Pauli in 1930 to explain how beta decay could conserve energy, momentum, and angular momentum.
Pauli 32
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33
Introduction to Nuclear Engineering (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
電子捕獲（軌道電子捕獲）
Electron capture (EC) 208
radiation from the human body
4. Nuclear decays and fundamental interactions
a)
b) (A,Z)
40 19 K
(A,Z−1) k l m
l m
νe
40 18 Ar
50 1.
10
.7
49
1.277 · 10 9 a eV
EC
γ
M
2%
89
.2
8%
1. 31
4-
10
9
M
eV
β
40 20 Ca
0+
0+
c) (A,Z−1)
γ
followed by • characteristic x-ray emission 特性Ｘ線放出 • Auger effect オージェ効果
Fig. 4.15. Electron capture. After the nuclear transformation, the atom is left with an unfilled orbital, which is subsequently filled by another electron with the emission of photons (X-rays). As in the case of nuclear radiative decay, the X-ray can transfer its energy to another atomic electron which is then ejected from the atom. Such an electron is called an Auger electron.
A ZN
The decay rate is then
+e
A Z 1N
pe
fundamental process:
c (2.4GF )2 Z 3 λ = |M |2 Q2ec . π(¯ hc)4 a30
neutrino energy: E = Compared with nuclear β-decay, the Q dependence is weak, Q
2 ec
(4.98)
n
2 M (A, Z)c rather than
Q5β . This means that for small Qβ , electron-capture dominates over β+ decay, as can be seen in Fig. 2.13. The strong Z dependence coming from the decreasing electron orbital radius with increasing Z means that electron-capture
+
e
e
M (A, Z
1)c2
atomic mass (not nuclear mass)
34
Introduction to Nuclear Engineering (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
Gamma-ray emission (gamma decay) gamma ray
gamma decay
A
A+
unstable high-energy state
mA > mA
mA
energy conservation
E c
spontaneous emission
自然放出
(stable) low-energy state
mA
p=
momentum conservation
ガンマ線
mA 運動量保存
エネルギー保存
p2 = (mA E + 2mA
mA ) c2
recoil energy (energy loss) 反跳エネルギー（エネルギー損失）
E2 ER = 2mA c2
ER
E
E
mA c2
(mA
A
931.5 MeV
mA ) c2
but ER >
in general
Emitted gamma rays are not resonantly re-absorbed by other nuclei in gases 35
Introduction to Nuclear Engineering (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
内部転換
Internal conversion
An excited nucleus can interact with an electron in one of the lower atomic orbitals, causing the electron to be emitted (ejected) from the atom. s-electrons have finite probability density at the nuclear position. s軌道の電子は、原子核の位置で存在確率が有限 4 for a hydrogen atom 1s 3
The electron may couple to the excited state of the nucleus and take the energy of the nuclear transition directly, without an intermediate gamma ray.
水素原子の例
2
probability density
1 0 0 0.5
1
2
3
4
5
2s
0.4 0.3 0.2 0.1 0.0 0 0.020
1
2
3
4
interaction
5
2p
0.015 0.010 0.005 0.000 0
1
2
3
4
5
0.08
Ece
0.04 0.00 0
1
2
3
r (atomic unit)
4
オージェ効果
Energy of the conversion electron
3s
0.12
followed by • characteristic x-ray emission 特性Ｘ線放出 • Auger effect
5
(mA
mA ) c2
Eb
E
Eb
binding energy of the electron 36
Introduction to Nuclear Engineering (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
メスバウアー効果
Mössbauer effect recoil energy (energy loss) 反跳エネルギー（エネルギー損失）
E2 ER = 2mA c2
Emitted gamma rays are not resonantly re-absorbed by other nuclei in gases. but ...
Inverse transition (resonant re-absorption) possible when
• •
nuclear recoil is suppressed in a crystal (“very very large mA”) ← Mössbauer effect (discovered in 1957) the excited nucleus decays in flight with the Doppler effect compensating the nuclear recoil 37
Introduction to Nuclear Engineering (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
メスバウアー分光による寿命測定
Mössbauer spectroscopy 182
191
4. Nuclear decays and fundamental interactions
Os 191
Ir (a)
0.0417
191
191
Os
source
absorber
Ț⥺※
྾య
Ț⥺
0.129
v
Ir
γ
γ− detector
Ț⥺᳨ฟჾ
γ
v
−4
0
4
−20
0
20
v(cm/sec) 8 12
% absorption
0.2 0.4 0.6 0.8 1.0 40 ∆ E (µ eV)
Fig. 4.4. Measurement of the width of the first excited state of 191 Ir through M¨ ossbauer spectroscopy [39]. The excited state is produced by the β-decay of 191 Os. De-excitation photons can be absorbed by the inverse transition in a 191 Ir absorber. This resonant absorption can be prevented by moving the absorber with respect to the source with velocity v so that the photons are Doppler shifted out of the resonance. Scanning in energy then amounts to scanning in velocity with ∆Eγ /Eγ = v/c.
38
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メスバウアー効果
ドップラーシフト
Mössbauer effect + Doppler shift
一般相対性理論の検証
Test of Albert Einstein's theory of general relativity • •
by Pound and Rebka, 1959 Gravitational red shift of light Clocks run differently at different places in a gravitational field
Gravitational shift h(fr
gamma ray 57 Fe (14.4 keV)
fe ) = mgH
hfe = mc2 fr gH =1+ 2 fe c
gH v= c
= 7.36 ⇥ 10
fe
v c 7
H = 22.5 m
Doppler shift s fr 1 v/c = ⇡1 fe 1 + v/c
v
Jefferson laboratory (Harvard University)
fr m/s blue shift 57Fe by falling 39
