Introduction to Nuclear Engineering - Freeishiken.free.fr/english/lectures/INE20160927.pdf · Introduction to Nuclear Engineering (Kenichi ISHIKAWA) for internal use only (Univ. of

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

Introduction to Nuclear Engineering

Kenichi Ishikawa ()

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

[email protected]

2016/9/27

1


Course material downloadable from:http://ishiken.free.fr/english/lecture.html

References Basdevant, Rich, and Spiro, Fundamentals in Nuclear Physics (Springer, 2005) Krane, Introductory Nuclear Physics (Wiley, 1987) (, 1971)

II

Nuclear Physics basic properties of nuclei nuclear reactions nuclear decays

2


Basic properties of nuclei

3


A nucleus is made up of protons and neutrons

charge mass (kg) mass energy (MeV)

proton (p) + e 1.6749310-27 938.272

neutron (n) 0 1.6726210-27 939.565

electron (e-) - e 9.10910-31 0.511 1840

1.0014

e = 1.6022 1019 C

E = mc2

MeV = 106 eV eV = 1.6022 1019 J

nucleon: proton, neutron

4


A nucleus is labeled by atomic number and mass number

Z : atomic number = number of protonsAZX N : number of neutrons

A = Z + N : mass number

example

23592U or simply

235U uranium-235

N = 235 - 92 = 143

5


nuclear binding energy and mass defect

mass defectprotonmass

neutronmass

nuclearmass

M = Zmp +Nmn mN > 0

binding energy B = Mc2 = (Zmp +Nmn mN )c2

~ 8 MeV8

6

4

2

0

B/A

(MeV

)

250200150100500 mass number A

stable unstable

max 8.7945 MeV @ 62Ni

binding energy per nucleon

6


Nuclear reactions

7


Nuclear reactionsfree particle (photon, electron, positron, neutron, proton, )

scatteringnuclear reactions

+ 14N 17O + p (Rutherford, 1919)p + 7Li 4He + (Cockcroft and Walton, 1930)

projectile

target

a + X Y + b X(a,b)Yprojectile target

example

8


Energetics

mXc2 + TX + mac2 + Ta = mY c2 + TY + mbc2 + Tb

a + X Y + b

rest mass kinetic energy

reaction Q value Q = (minitial mfinal)c2

= (mX + ma mY mb)c2

= TY + Tb TX Taexcess kinetic energy

Q > 0 : exothermic Q < 0 : endothermic

9


235U+ n ! X+Y+ (2 3) n

example 235U+ n ! 144Ba + 89Kr + 3n + 177MeV

Important nuclear reactions for thermal energy generation

Fission

FusionD+T ! 4He (3.5MeV) + n (14.1MeV)D+D ! T (1.01MeV) + p (3.02MeV)

! 3He (0.82MeV) + n (2.45MeV)D+ 3He ! 4He (3.6MeV) + p (14.7MeV)

> 106 times more efficient than chemical reactions! 10


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


3.1 Cross-sections 109

L

dz

.

Fig. 3.1. A small particle incident on a slice of matter containing N = 6 targetspheres of radius R. If the point of impact on the slice is random, the probabilitydP of it hitting a target particle is dP = NR2/L2 = ndz where the numberdensity of scatterers is n = N/(L2dz) and the cross section per sphere is = R2.

a probability dP of hitting one of the spheres that is equal to the fraction ofthe surface area covered by a sphere

dP =NR2

L2= ndz = R2 . (3.4)

In the second form, we have multiplied and divided by the slice thickness dzand introduced the number density of spheres n = N/(L2dz). The cross-section for touching a sphere, = R2, has dimensions of area/sphere.

While the cross-section was introduced here as a classical area, it can beused to define a probability dPr for any type of reaction, r, as long as theprobability is proportional to the number density of target particles and tothe target thickness:

dPr = r n dz . (3.5)

The constant of proportionality r clearly has the dimension of area/particleand is called the cross-section for the reaction r.

If the material contains different types of objects i of number density andcross-section ni and i, then the probability to interact is just the sum of theprobabilities on each type:

dP =

i

nii (3.6)

Probability P proportional to

number density of target particles n target thickness dz

Unit of cross sectiondimension of area m2, cm2

size of nucleus ~ a few fm

1 barn (b) = 10-28 m2 = 10-24 cm2

dP = ndz

Cross section

12


Differential cross sectionangular dependence ()

target

ddetector Probability that the incident particle

is scattered to a solid angle d

differential cross section ()

=

dd

d=

2

0d

0

d

d(, ) sin d

dP, =d

dndzd

for isotropic scattering ()d

d=

4

total cross section

13


Mean free path and reaction rate

3.1 Cross-sections 109

L

dz

.

Fig. 3.1. A small particle incident on a slice of matter containing N = 6 targetspheres of radius R. If the point of impact on the slice is random, the probabilitydP of it hitting a target particle is dP = NR2/L2 = ndz where the numberdensity of scatterers is n = N/(L2dz) and the cross section per sphere is = R2.

a probability dP of hitting one of the spheres that is equal to the fraction ofthe surface area covered by a sphere

dP =NR2

L2= ndz = R2 . (3.4)

In the second form, we have multiplied and divided by the slice thickness dzand introduced the number density of spheres n = N/(L2dz). The cross-section for touching a sphere, = R2, has dimensions of area/sphere.

While the cross-section was introduced here as a classical area, it can beused to define a probability dPr for any type of reaction, r, as long as theprobability is proportional to the number density of target particles and tothe target thickness:

dPr = r n dz . (3.5)

The constant of proportionality r clearly has the dimension of area/particleand is called the cross-section for the reaction r.

If the material contains different types of objects i of number density andcross-section ni and i, then the probability to interact is just the sum of theprobabilities on each type:

dP =

i

nii (3.6)

flux F dF = FndzdF

dz= Fn

F (z) = F (0)enz = F (0)ez

macroscopic cross section () = 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

ini

reaction rate

v

l= n v

14


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,), (p,), (n,), (n,p), (n,d), (n,t), etc. fission, fusion

15


Elastic neutron scattering relevant to (neutron) moderator in nuclear

reactors

due to the short-range strong interaction

10-1

100

101

102

103

104

Cro

ss s

ectio

n (b

arn)

10-5 10-3 10-1 101 103 105 107Energy (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

16


Elastic neutron scattering

10-1

100

101

102

103

104

Cro

ss s

ectio

n (b

arn)

10-5 10-3 10-1 101 103 105 107Energy (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-)

n + 6Li 7Li* n + 6Li

10

9

8

7

6

5

4

3

2

1

0

Ene

rgy

(MeV

)

1086420

0.478

7.47

6.54

4.63

9.6

2.466

7.253

0

3H +

4He

7Li

n + 6Li

17


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.htm18


Neutron capture

Radiative capture emits a gamma ray 113Cd(n,)114Cd neutron shield

Other neutron capture reactions 10B(n,)7Li, 3He(n,p)3H, 6Li(n,t)4He Applications: neutron detector, shield, neutron

capture therapy for cancer


neutron binding energy = ca. 8 MeV

exothermic reaction in most cases

Highly excited states formed, which subsequently decay.

AX(n,)A+1X

activation

19


neutron radiative captureInelastic scattering

10-6

10-5

10-4

10-3

10-2

10-1

100

101

Cro

ss s

ectio

n (b

arn)

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

E1/2 1/v1/v law Energy-independent reaction rate v

No threshold exothermic, no Coulomb barrier

20


Neutron capture reactions with large cross section

113Cd(n,)114Cd : shield 157Gd(n,)158Gd : neutron absorber in

nuclear fuel, cancer therapy

10B(n,)7Li : detector, cancer therapy 3He(n,p)3H : detector 6Li(n,t)4He : shield, filter, detector

21



Applications BF3 proportional counter Boron neutron capture therapy (BNCT) for cancer

10-1

100

101

102

103

104

105

Cro

ss s

ectio

n (b

arn)

10-5 10-3 10-1 101 103 105 107Energy (eV)

10B(n,alpha)7Li

JENDL

1/v law

10B(n,)7Li

3He(n,p)3H Helium-3 proportional counter

22


Photo-nuclear reaction Excitation and break-up (dissociation)

through photo-absorption Analog of the photoelectric effect

threshold (2.22 MeV) = binding energy of 2H

2.5x10-3

2.0

1.5

1.0

0.5

0.0

Cro

ss s

ectio

n (b

arn)

302520151050Energy (MeV)

2H(gamma,n)1H

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0C

ross

sec

tion

(bar

n)302520151050

Energy (MeV)

208Pb(gamma,n)207Pb2H 208Pb

giant resonance collective oscillation of protons in the nucleus

23


Resonance 6Li

10-410-310-210-1100101102103104

Cro

ss s

ectio

n (b

arn)

10-5 10-3 10-1 101 103 105 107Energy (eV)

(n,gamma)

(n,n)

(n,t)

(n,p)

JENDL

resonance1/v law

162 3. Nuclear reactions

10 elastic

elastic (x10)

(n4 )

(n,) (/100)

(n,fission) (/10

10

2 3

E (eV)

,)

(/10

210

1

210U235

U238

1 10 10 10

4

2

4 )510

cros

sse

ctio

n (b

arn)

1

2

10

Fig. 3.26. The elastic and inelastic neutron cross-sections on 235U (top) and 238U(bottom). The peaks correspond to excited states of 236U and 239U. The excitedstates can contribute to the elastic cross-sections by decaying through neutronemission. They contribute to the (n, ) cross-section by decaying by photon emissionto the ground states of 236U and 239U. In the case of 236U the states can also decayby fission so they contribute to the neutron-induced fission cross-section on 235U.

Many excited states for heavy nucleicomplicated resonance structure

Excited states of 239U

10

9

8

7

6

5

4

3

2

1

0

Ene

rgy

(MeV

)

1086420

0.478

7.47

6.54

4.63

9.6

2.466

7.253

0

3H +

4He

7Li

n + 6Li

24


Resonance line shape

(E) A(E E0)2 + (/2)2

-3 -2 -1 1 2 3

0.5

1.0

1.5

2.0

E

Lorentzian

full width at half maximum (FWHM)

long tail

Resonance

natural widthhomogeneous width

: Life time = /Decay rate 1 = /

= uncertainty principle

inhomogeneous width

-3 -2 -1 1 2 3

0.2

0.4

0.6

0.8

1.0

Doppler effect

exp (E E0)

2

E2

25


Nuclear decays

Fundamentals in Nuclear Physics (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)

26


Decay rate, natural width

probability to decay in an interval dt

dP =dt

= dt

mean life time

decay rate

N(t) = N(t = 0)et/

An unstable particle has an energy uncertainty or natural width

number of unstable nuclei

= =

=6.58 1022 MeV sec

176 4. Nuclear decays and fundamental interactions

which has the solution

N(t) = N(t = 0)et/ . (4.3)

The mean survival time is , justifying its name.The inverse of the mean lifetime is the decay rate

=1

. (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

= h =h

=

6.58 1022 MeV sec

. (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 greaterthan 1022 sec. This is generally the case for states decaying through theweak or electromagnetic interactions. For decays involving the dissociationof a nucleus, the width can be quite large. Examples are the excited statesof 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 100 keV.

It is often the case that an unstable state has more than one decaychannel, each channel k having its own branching ratio Bk. For examplethe fourth excited state of 7Li has

Bn6Li = 0.72 B3H4He = 0.28 B 7Li 0.0 , (4.6)

where the third mode is the unlikely radiative decay to the ground state. Ingeneral we have

k

Bk = 1 , (4.7)

the sum of the partial decay rates, k = Bk

k

k = , (4.8)

and the sum of the partial widths, k = Bk

k

k = . (4.9)

4.1.2 Measurement of decay rates

Lifetimes of observed nuclear transitions range from 1022 sec7Li (7.459 MeV) n 6Li, 3H 4He = 6 1021 sec (4.10)

to 1021 yr

Licensed to Kenichi Ishikawa

76Ge 76Se 2e 2e t1/2 = 1.78 1021 yr

half life t1/2 = (ln 2) = 0.693

> 1011 (age of universe) !

27


Decay diagram

half life

branching ratio

28


alpha decay

AZX ! A4Z2Y+

= 42He

238U ! 234Th + (4.2MeV)examplehalf life = 4.468109 years

29


beta decay decay

+ decay

dating half life = 5730 years

AZN AZ+1N + e + eAZN AZ1N + e+ + e

30


Emitted electron (positron) energy has a broad distribution


to the overlap integral (4.84), and the GamowTeller term proportional tothe matrix element of between initial and final state nuclei.

Just as in radiative decays, there are selection rules governing which com-binations of initial Ji and final Jf spins are possible. The Fermi term willvanishes if the angular dependences of the initial and final wavefunctions areorthogonal so we require

Fermi : Ji = Jf . (4.90)

The GamowTeller term can change the spin but vanishes if the initial andfinal angular momenta are zero:

GT : Ji = Jf , Jf 1 Ji = Jf = 0 forbidden . (4.91)

Additionally, in both cases, the parity of the initial and final nuclei must bethe same. Transitions that respect the selection rules are called Alloweddecays. Forbidden decays are possible only if one takes into account thespatial 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 muchlonger lifetimes for such transitions.

_ +

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

0.2 0.2 0.6 1.0 1.4 1.81.81.41.00.6

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


64Cu64Cu

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

AZN AZ+1N + e + eAZN AZ1N + e+ + e

Pauli32


33


Electron capture (EC)


l mmlk

(A,Z)

e

(A,Z1)

(A,Z1)

a) b)

c)

Fig. 4.15. Electron capture. After the nuclear transformation, the atom is leftwith an unfilled orbital, which is subsequently filled by another electron with theemission of photons (X-rays). As in the case of nuclear radiative decay, the X-raycan transfer its energy to another atomic electron which is then ejected from theatom. Such an electron is called an Auger electron.

The decay rate is then

=c

(hc)4(2.4GF)2Z3

a30|M |2Q2ec . (4.98)

Compared with nuclear -decay, the Q dependence is weak, Q2ec rather thanQ5. This means that for small Q, electron-capture dominates over

+ decay,as can be seen in Fig. 2.13. The strong Z dependence coming from the de-creasing electron orbital radius with increasing Z means that electron-capturebecomes more and more important with increasing Z.

Finally, we note that nuclear decay by electron capture leaves the atomwith an unfilled atomic orbital. This orbital is filled by other atomic electronsfalling 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 ofatomic number Z is

E 0.5Z22mec2 = 0.01 Z2 keV . (4.99)


followed bycharacteristic x-ray emission

Auger effect

10.72

%1.50

49 M

eV EC

89.28%1.31109 MeV 0+

0+

4-40K19

40Ar18Ca2040

1.277 10 9 a

radiation from the human body

AZN + e

AZ1N + efundamental process: p e n e

neutrino energy:

atomic mass (not nuclear mass)E = M(A,Z)c2 M(A,Z 1)c2

34


Gamma-ray emission (gamma decay)

gamma decay A A +

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

gamma ray

mA > mA mA mA mA

momentum conservation p =Ec

energy conservation

spontaneous emission

E +p2

2mA= (mA mA) c2

recoil energy (energy loss)

ER =E2

2mAc2mAc

2 A 931.5 MeV

ER E E (mA mA) c2 but ER > in general

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

35


Internal conversion

4

3

2

1

0543210

1s

0.50.40.30.20.10.0

543210

2s

0.0200.0150.0100.0050.000

543210

2p

0.12

0.08

0.04

0.00543210

3s

r (atomic unit)

prob

abili

ty d

ensi

ty

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

s

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 (mA mA) c2 Eb E Eb

binding energy of the electron

followed bycharacteristic x-ray emission

Auger effect

36


Mssbauer effect

Inverse transition (resonant re-absorption) possible when

nuclear recoil is suppressed in a crystal (very very large mA)

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

recoil energy (energy loss)

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

ER =E2

2mAc2

but ...

Mssbauer effect (discovered in 1957)

37


Mssbauer spectroscopy182 4. Nuclear decays and fundamental interactions

E ( eV)

0.0417

0.129

191Os

% a

bsor

ptio

n

4 0 4 8 12v(cm/sec)

2020 0 40

1.0

0.8

0.6

0.4

0.2

Ir191

v

absorber

191Ir

detectorsourceOs191

Fig. 4.4. Measurement of the width of the first excited state of 191Ir throughMossbauer spectroscopy [39]. The excited state is produced by the -decay of 191Os.De-excitation photons can be absorbed by the inverse transition in a 191Ir absorber.This resonant absorption can be prevented by moving the absorber with respect tothe source with velocity v so that the photons are Doppler shifted out of the reso-nance. Scanning in energy then amounts to scanning in velocity with E/E = v/c.It should be noted that photons from the decay of free 191Ir have insufficient en-ergy to excite 191Ir because nuclear recoil takes some of the energy (4.42). Resonantabsorption is possible with v = 0 only if the 191Ir nuclei is locked at a crystallattice site so the crystal as a whole recoils. The nuclear kinetic energy p2/2mA 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 originalstate. This Mossbauer effect is not present for photons with E > 200 keV becausenuclear recoil is sufficient to excite phonon modes in the crystal which take someof the energy and momentum.


v

(a)

38


Test of Albert Einstein's theory of general relativity

Mssbauer effect + Doppler shift

Jefferson laboratory (Harvard University)

by Pound and Rebka, 1959

gamma ray (14.4 keV)

57Fe

57Fe

H =

22.5 m

blue shift by falling

vfe

fr

Gravitational shift

Doppler shift

frfe

=

s1 v/c1 + v/c

1 vc

h(fr fe) = mgH

hfe = mc2

frfe

= 1 +gH

c2

v =gH

c= 7.36 107 m/s

Gravitational red shift of light Clocks run differently at different places in a gravitational field

39

Documents

Introduction to Nuclear Engineering - Freeishiken.free.fr/english/lectures/INE20160927.pdf · Introduction to Nuclear Engineering (Kenichi ISHIKAWA) for internal use only (Univ. of