Introduction to Nuclear Engineering

Introduction to

Nuclear Engineering

Kenichi Ishikawa (

)

http://ishiken.free.fr/english/lecture.html

[email protected]

Course material downloadable from:

http://ishiken.free.fr/english/lecture.html References

•

•

•

•

Nuclear Physics

• basic properties of nuclei

• nuclear reactions

• nuclear decays

Basic properties of

nuclei

A nucleus is made up of protons and neutrons

charge mass (kg) mass energy 

(MeV)

proton (p)

+ e

neutron (n)

0

electron (e^-)

- e

^×1840

×1.0014

e = 1.6022 ⇥ 10

C

E = mc

MeV = 10

eV eV = 1.6022 ⇥ 10

J

nucleon: proton, neutron

by atomic number and mass number

Z : atomic number = number of protons

A Z X N : number of neutrons A = Z + N : mass number

example

235 92 U ^{or simply 235} U “uranium-235”

N = 235 - 92 = 143

nuclear binding energy and mass defect

mass defect

proton

mass neutron

mass nuclear mass

M = Zm

+ N m

m

> 0

binding energy ^{B = M c}

^{= (Zm}

+ N m

m

)c

~ 8 MeV

8 6 4 2 0

B/A (MeV)

250 200

150 100

50 0

stable unstable

max 8.7945 MeV @

Ni

binding energy

per nucleon

Nuclear reactions

Nuclear reactions

free particle (photon, electron, positron, neutron, proton, …)

scattering

nuclear reactions

α +

N →

O + p (Rutherford, 1919)

p +

Li →

He + α (Cockcroft and Walton, 1930)

projectile

target

a + X → Y + b X(a,b)Y

projectile target

example

Energetics

m

c

+ T

+ m

c

+ T

= m

c

+ T

+ m

c

+ T

a + X → Y + b

rest mass kinetic energy

reaction Q value Q = (m

m

)c

= (m

+ m

m

m

)c

= T

+ T

T

T

excess kinetic energy

Q > 0 : exothermic

Q < 0 : endothermic

235

U + n ! X + Y + (2 ⇠ 3) n

example

U + n !

Ba +

Kr + 3n + 177 MeV

Important nuclear reactions for thermal energy generation

Fission

Fusion

D + T !

He (3.5 MeV) + n (14.1 MeV) D + D ! T (1.01 MeV) + p (3.02 MeV)

!

He (0.82 MeV) + n (2.45 MeV) D +

He !

He (3.6 MeV) + p (14.7 MeV)

> 10 ⁶ times more efficient than chemical reactions!

Nuclei for fission reactors

• ^{233 U, 235 U, 239} Pu (fissile materials)


→ fission by thermal neutron capture

• Fission of ²³⁵ U produces ~2.5 neutrons

• ^{238 U, 232} Th (fertile materials) change to

239 Pu, ²³² Th by neutron capture 


→ fast breeder reactor

3.1 Cross-sections 109

L

dz

.

Fig. 3.1.A small particle incident on a slice of matter containingN= 6 target spheres of radiusR. If the point of impact on the slice is random, the probability dP of it hitting a target particle is dP=NπR²/L²=σndzwhere the number density of scatterers isn=N/(L²dz) and the cross section per sphere isσ=πR². a probability dPof hitting one of the spheres that is equal to the fraction of the surface area covered by a sphere

dP = NπR²

L² = σndz σ = πR². (3.4)

In the second form, we have multiplied and divided by the slice thickness dz and introduced the number density of spheresn=N/(L²dz). The “cross- section” 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 dPrfor 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 = σrndz . (3.5)

The constant of proportionalityσrclearly has the dimension of area/particle and is called the cross-section for the reaction r.

If the material contains different types of objectsiof number density and cross-sectionniandσi, then the probability to interact is just the sum of the probabilities on each type:

dP = !

i

niσi (3.6)

Probability P proportional to

• number density of target particles n

• target thickness dz

Unit of cross section

dimension of area m ² , cm ² size of nucleus ~ a few fm

1 barn (b) = 10 ^-28 m ² = 10 ^-24 cm ²

dP = ndz

Cross section 断面積

Differential cross section

angular dependence (

)

targetdetector d Probability that the incident particle is scattered to a solid angle d

differential cross section (微分断面積)

= d d

d =

0

d

0

d

d ( , ) sin d dP

= d

d ndzd

for isotropic scattering (等方散乱)

d

d = 4

total cross section

Mean free path and reaction rate

L

dz

.

Fig. 3.1.A small particle incident on a slice of matter containingN= 6 target spheres of radiusR. If the point of impact on the slice is random, the probability dP of it hitting a target particle is dP =NπR²/L² =σndzwhere the number density of scatterers isn=N/(L²dz) and the cross section per sphere isσ=πR². a probability dP of hitting one of the spheres that is equal to the fraction of the surface area covered by a sphere

dP = NπR²

L² = σndz σ = πR². (3.4)

In the second form, we have multiplied and divided by the slice thickness dz and introduced the number density of spheresn=N/(L²dz). The “cross- section” 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 dPrfor 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 = σrndz . (3.5)

The constant of proportionalityσrclearly has the dimension of area/particle and is called the cross-section for the reaction r.

If the material contains different types of objectsiof number density and cross-sectionniandσi, then the probability to interact is just the sum of the probabilities on each type:

dP = !

i

niσi (3.6)

flux F

dF

dz = F n

F(z) = F(0)e ^nz = F(0)e ^z

macroscopic cross section (マクロ断面積) = n [1/length]

1.0

0.8

0.6

0.4

0.2

0.0

F(z)/F(0)

z

1/e = 0.368

1/ n

mean free path also distribution

of free path if there are different types of

target objects (nuclei)

l = 1/ n

l = 1/

i

i

n

reaction rate

反応速度

v

l = n v

General characteristics of cross-sections

Elastic scattering

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

• (n,γ), (p,γ), (n,α), (n,p), (n,d), (n,t), etc.

• fission, fusion

Elastic neutron scattering

• relevant to (neutron) moderator in nuclear reactors

• due to the short-range strong interaction

中性子減速材

10^-1 10⁰ 10¹ 10² 10³ 10⁴

Cross section (barn)

10^-5 10^-3 10^-1 10¹ 10³ 10⁵ 10⁷

Energy (eV) 1H(n,n) 2H(n,n) 6Li(n,n)

JENDL flat region

el 20 b (2fm)² 0.1 b range of the strong

interaction

(p²/2mn > 200 MeV) p > h/r

resonance

10^-1 10⁰ 10¹ 10² 10³ 10⁴

Cross section (barn)

10^-5 10^-3 10^-1 10¹ 10³ 10⁵ 10⁷

Energy (eV) 1H(n,n) 2H(n,n) 6Li(n,n)

JENDL

resonance

The energy levels of ⁷Li and two dissociated states n-⁶Li and ³H-⁴He (t-α)

n + ⁶ Li → ⁷ Li ^* → n + ⁶ Li

10 9 8 7 6 5 4 3 2 1 0

Energy (MeV)

10 8

6 4

2 0

0.478 7.47 6.54

4.63 9.6

2.466 7.253

0

3H +⁴He

7Li n + ⁶Li

Nuclear data libraries

• ENDF (Evaluated Nuclear Data File, USA)

• JENDL (Japanese Evaluated Nuclear Data Library, Japan)

• JEFF (Joint Evaluated Fission and Fusion file, Europe)

•

•

•

http://www-nds.iaea.org/exfor/endf.htm

Neutron capture

• Radiative capture

• 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 neutron binding energy = ca. 8 MeV

exothermic reaction in most cases

放射捕獲（放射性捕獲）

Highly excited states formed, which subsequently decay.

A

X(n,γ)

X

activation

neutron radiative capture

10^-6 10^-5 10^-4 10^-3 10^-2 10^-1 10⁰ 10¹

Cross section (barn)

10^-5 10^-3 10^-1 10¹ 10³ 10⁵ 10⁷

Energy (eV) 1H(n,gamma)

2H(n,gamma) 6Li(n,gamma)

JENDL

JENDL ENDF

discrepancy between JENDL and ENDF

E

1/v

1/v law Energy-independent reaction rate v

No threshold ← exothermic, no Coulomb barrier

with large cross section

• ^{113 Cd(n,γ) 114} Cd : shield

• ^{157 Gd(n,γ) 158} Gd : neutron absorber in nuclear fuel, cancer therapy

• ^{10 B(n,α) 7} Li : detector, cancer therapy

• ^{3 He(n,p) 3} H : detector

• ^{6 Li(n,t) 4} He : shield, filter, detector

Applications

• ^BF

proportional counter

• Boron neutron capture therapy (BNCT) for cancer

10^-1 10⁰ 10¹ 10² 10³ 10⁴ 10⁵

Cross section (barn)

10^-5 10^-3 10^-1 10¹ 10³ 10⁵ 10⁷

Energy (eV) 10B(n,alpha)7Li

JENDL

1/v law

10 B(n,α) ⁷ Li

3

He(n,p)

H

• Helium-3 proportional counter

• Excitation and break-up (dissociation) through photo-absorption

•

threshold (2.22 MeV) = binding energy of ²H

2.5x10^-3

2.0

1.5

1.0

0.5

0.0

Cross section (barn)

30 25

20 15

10 5

0

Energy (MeV)

2H(gamma,n)1H

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0

Cross section (barn)

30 25

20 15

10 5

0

Energy (MeV)

208Pb(gamma,n)207Pb

2 H ²⁰⁸ Pb

giant resonance ^巨大共鳴

Resonance

6 Li

10^-4 10^-3 10^-2 10^-1 10⁰ 10¹ 10² 10³ 10⁴

Cross section (barn)

10^-5 10^-3 10^-1 10¹ 10³ 10⁵ 10⁷

Energy (eV)

(n,gamma) (n,n)

(n,t)

(n,p)

JENDL

resonance共鳴

1/v law ¹⁶² 3. Nuclear reactions

10 elastic

elastic (x10)

(n 4) (n,γ)(/100)

(n,fission) (/10

10

2 3

E (eV)

,γ) (/10 10−2

1 102

235U

238U

1 10 10 10

−4

−2

−4 ⁵)

10

cross−section (barn)

1

2

10

Fig. 3.26.The elastic and inelastic neutron cross-sections on²³⁵U (top) and²³⁸U (bottom). The peaks correspond to excited states of²³⁶U and²³⁹U. The excited states can contribute to the elastic cross-sections by decaying through neutron

Many excited states for heavy nuclei complicated resonance structure

重い核には多くの励起状態

複雑なエネルギー依存性

Excited states of ²³⁹U

10 9 8 7 6 5 4 3 2 1 0

Energy (MeV)

10 8

6 4

2 0

0.478 7.47 6.54

4.63 9.6

2.466 7.253

0

3H +⁴He

7Li n + ⁶Li

24

(E) A

(E E

)

+ ( /2)

-3 -2 -1 1 2 3

0.5 1.0 1.5 2.0

Lorentzian ローレンツ関数 E

full width at half maximum (FWHM)

半値全幅

long tail

Resonance

natural width homogeneous width

: 自然幅 Life time = /

Decay rate ¹ = /

ドップラー幅 inhomogeneous width

-3 -2 -1 1 2 3

0.2 0.4 0.6 0.8 1.0

Doppler effect

ドップラー効果

Nuclear decays

Introduction to Nuclear Engineering (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)

Decay rate, natural width

probability to decay in an interval dt

dP = dt

= dt

mean life time

decay rate 壊変（崩壊）速度

N (t) = N (t = 0)e

An unstable particle has an energy uncertainty or “natural width”

number of unstable nuclei

6.58 10

MeV sec

自然幅 壊変（崩壊）速度

The mean survival time is τ, justifying its name.

The inverse of the mean lifetime is the “decay rate”

λ = 1

τ . (4.4)

We saw in Sect. 3.5 that an unstable particle (or more precisely an un- stable quantum state) has a rest energy uncertainty or “width” of

Γ = ¯hλ = ¯h

τ = 6.58 × 10⁻²² MeV sec

τ . (4.5)

Since nuclear states are typically separated by energies in the MeV range, the width is small compared to state separations if the lifetime is greater than ∼ 10⁻²² sec. This is generally the case for states decaying through the weak or electromagnetic interactions. For decays involving the dissociation of a nucleus, the width can be quite large. Examples are the excited states of ⁷Li (Fig. 3.5) that decay via neutron emission or dissociation into ³H⁴He.

From the cross-section shown in Fig. 3.4, we see that the fourth excited state (7.459 MeV) has a decay width of Γ ∼ 100 keV.

It is often the case that an unstable state has more than one “decay channel,” each channel k having its own “branching ratio” B_k. For example the fourth excited state of ⁷Li has

B_n⁶_Li = 0.72 B³_H⁴_He = 0.28 B_γ⁷_Li ∼ 0.0 , (4.6) where the third mode is the unlikely radiative decay to the ground state. In general we have

!

k

B_k = 1 , (4.7)

the sum of the “partial decay rates,” λ_k = B_kλ

!

k

λ_k = λ , (4.8)

and the sum of the “partial widths,” Γ_k = B_kΓ

!

k

Γ_k = Γ . (4.9)

4.1.2 Measurement of decay rates

Lifetimes of observed nuclear transitions range from ∼ 10⁻²² sec

7Li (7.459 MeV) → n⁶Li, ³H⁴He τ = 6× 10⁻²¹ sec (4.10) to 10²¹⁷⁶yrGe ⁷⁶Se 2e 2¯e t_1/2 = 1.78 10²¹ yr

half life

t

= (ln 2) = 0.693

> 10¹¹ × (age of universe) !

Decay diagram

half life

半減期

branching ratio

分岐比

alpha decay

A Z X ! ^A _Z ⁴ ₂ Y + ↵

=

He

238

U !

Th + ↵ (4.2 MeV)

example

half life = 4.468×10

years

beta decay

decay

+

decay

dating

half life = 5730 years

AZ

N

N + e + ¯

AZ

N

N + e

+

Introduction to Nuclear Engineering (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)

Emitted electron (positron) energy has a broad distribution

binations of initial J_i and final J_f spins are possible. The Fermi term will vanishes if the angular dependences of the initial and final wavefunctions are orthogonal so we require

Fermi : J_i = J_f . (4.90)

The Gamow–Teller term can change the spin but vanishes if the initial and final angular momenta are zero:

GT : J_i = J_f, J_f ± 1 J_i = J_f = 0 forbidden . (4.91) 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.

β _ β +

p (MeV/c) p (MeV/c)

0.2 0.6 1.0 1.4 1.8 0.2 0.6 1.0 1.4 1.8

Fig. 4.14. The β⁻ and β⁺ spectra of ⁶⁴Cu [44]. The suppression the of the β⁺ spectrum and enhancement of the β⁻ at low energy due to the Coulomb effect is seen.

64 Cu

64 Cu

31

beta decay

The existence of the neutrino was predicted by Wolfgang Pauli in 1930 to explain how beta decay could conserve energy, momentum, and angular momentum.

decay

+

decay

dating

half life = 5730 years

AZ

N

N + e + ¯

AZ

N

N + e

+

Pauli

Electron capture (EC)

208 4. Nuclear decays and fundamental interactions

m l m l k (A,Z)

νe

(A,Z−1)

(A,Z−1)

a) b)

c)

γ

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.

The decay rate is then λ = c

π(¯hc)⁴

(2.4GF)²Z³

a³0 |M|²Q²ec. (4.98)

Compared with nuclearβ-decay, theQ dependence is weak,Q²_ecrather than Q⁵_β. This means that for smallQβ, electron-capture dominates overβ⁺decay, as can be seen in Fig. 2.13. The strongZ dependence coming from the de- creasing electron orbital radius with increasingZmeans that electron-capture becomes more and more important with increasingZ.

Finally, we note that nuclear decay by electron capture leaves the atom with an unfilled atomic orbital. This orbital is filled by other atomic electrons falling into it and radiating photons. The photons are in the keV (X-ray) range since the binding energy of the inner most electron of an atom of atomic numberZ is

E ∼ 0.5Z²α²mec²= 0.01Z²keV. (4.99)

followed by

•

•

オージェ効果

10.72%

1.5049 Me

V ECγ ^89.28%

1.3110 9 MeV

β ₀₊

0+

40

K

19

40

Ar

18

20

Ca

40

1.277 · 10⁹a

radiation from the human body

AZ

N + e

N +

fundamental process:

p e n

neutrino energy:

atomic mass (not nuclear mass)

E = M (A, Z )c

M (A, Z 1)c

34

gamma decay A A +

unstable high-energy state (stable) low-energy state

gamma ray ^ガンマ線

mA > mA m_A m_A m_A

momentum conservation p = E c energy conservation

運動量保存 エネルギー保存

spontaneous emission 自然放出

E + p²

2m_A = (m_A m_A)c²

recoil energy (energy loss)

反跳エネルギー（エネルギー損失）

E_R = E²

2m_Ac^{2 mAc}

2 A 931.5 MeV

E E E (m m )c² but E > in general

Internal conversion

4 3 2 1

00 1 2 3 4 5

1s

0.5 0.4 0.3 0.2 0.1

0.00 1 2 3 4 5

2s

0.020 0.015 0.010 0.005 0.000

5 4

3 2

1 0

2p

0.12 0.08 0.04

0.000 1 2 3 4 5

3s

probability density

interaction

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.

for a hydrogen atom

水素原子の例

s軌道の電子は、原子核の位置で存在確率が有限

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.

Energy of the conversion electron

Ece (m_A mA)c² Eb E Eb

binding energy of the electron

followed by

•

•

オージェ効果

Mössbauer effect

Inverse transition (resonant re-absorption) possible when

• nuclear recoil is suppressed in a crystal (“very very large m

”)

• the excited nucleus decays in flight with the Doppler effect compensating the nuclear recoil

recoil energy (energy loss)

反跳エネルギー（エネルギー損失）

Emitted gamma rays are not resonantly re-absorbed by other nuclei in gases.

E

= E

2m

c

but ...

← Mössbauer effect (discovered in 1957)

Mössbauer spectroscopy

182 4. Nuclear decays and fundamental interactions

E (µeV)

∆

0.0417 0.129 191Os

% absorption

−4 0 4 v(cm/sec)8 12

−20 0 20 40

1.0 0.8 0.6

0.4 0.2

γ γ

191Ir

v

absorber

191Ir ^γ−

detector sourceOs

191

Fig. 4.4. Measurement of the width of the first excited state of ¹⁹¹Ir through M¨ossbauer spectroscopy [39]. The excited state is produced by theβ-decay of¹⁹¹Os.

De-excitation photons can be absorbed by the inverse transition in a¹⁹¹Ir absorber.

This resonant absorption can be prevented by moving the absorber with respect to the source with velocityvso that the photons are Doppler shifted out of the reso- nance. Scanning in energy then amounts to scanning in velocity with∆Eγ/Eγ=v/c.

It should be noted that photons from the decay of free ¹⁹¹Ir have insufficient en- ergy to excite¹⁹¹Ir because nuclear recoil takes some of the energy (4.42). Resonant absorption is possible with v = 0 only if the ¹⁹¹Ir nuclei is “locked” at a crystal lattice site so the crystal as a whole recoils. The nuclear kinetic energyp²/2mA in (4.42) is modified by replacing the mass of the nucleus with the mass of the crystal.

The photon then takes all the energy and has sufficient energy to excite the original state. This “M¨ossbauer effect” is not present for photons withE >200 keV because nuclear recoil is sufficient to excite phonon modes in the crystal which take some of the energy and momentum.

メスバウアー分光による寿命測定

v

(a)

38

Test of Albert Einstein's theory of general relativity

Jefferson laboratory (Harvard University) by Pound and Rebka, 1959

gamma ray

(14.4 keV) 57

Fe

H = 22.5 m

v

f

f

Gravitational shift

Doppler shift f

f

=

s 1 v/c

1 + v/c ⇡ 1 v c h(f

f

) = mgH

hf

= mc

f

f

= 1 + gH c

gH

•

•