Nuclear Physics at BAU nuclear.bau.jo/ This course nuclear.bau.jo/Reactors

Nuclear Reactors, BAU, First Semester, 2007-2008 (Saed Dababneh).

1

Nuclear Physics at BAU Nuclear Physics at BAU http://nuclear.bau.edu.jo/

This courseThis coursehttp://nuclear.bau.edu.jo/Reactors

PrerequisitesPrerequisites• Nuclear and Radiation Physics 742

http://nuclear.bau.edu.jo/nuclear-radiation • Advanced Statistical Mechanics 761

501503747 Nuclear Reactors

Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).

2

• Review of relevant studied material in nuclear physics.• Concepts in neutron physics.• The relevant physics related to nuclear technology:

Fission chain reaction. Neutron diffusion and moderation. Heat removal from nuclear reactors. Isotope separation. …

• Components of nuclear reactors.• Nuclear reactor fuels and fuel cycles.• Nuclear reactor theory. • Basic concepts of radiation protection and nuclear safety, shielding and waste disposal.• Issues and prospects of nuclear power today and in the future.

General subjects to be covered


3

Grading

Review Test 05%Mid-term Exam 20%Projects, quizzes and HWs 25%Final Exam 50%

• Homeworks are due after one week unless otherwise announced.• Remarks or questions marked in red without being announced as homeworks should be also seriously considered!• Some tasks can (or should) be sent by email:

[email protected]


4

Review Test


5

Projects

Consider nuclear fuel cycles with emphasis on front ends.

• Work as a team. Divide and organize the job among you.• Try to explore local applicability.• Due date (for written version): December 5th. • Presentation: Will be scheduled later.

Other small projects will be announced in class.


6

Nuclear Reaction Energetics (revisited)

Conservation Laws• Charge, Baryon number, total energy, linear momentum, angular momentum, parity, (isospin??) …….

apa X

pY

pb

Y

bQTTcmcm iffi 22

+ve Q-value exoergic reaction. -ve Q-value endoergic reaction.

aYb TQTT +ve Q-value reaction possible if Ta 0. -ve Q-value reaction not possible if Ta 0. (Is Ta > |Q| sufficient?).

Conservation of momentum ……


7

• Conservation of momentum.• We usually do not detect Y.Show that:

• The threshold energy (for Ta): (the condition occurs for = 0º).

• +ve Q-value reaction possible if Ta 0.• Coulomb barriers…….!!!• -ve Q-value reaction possible if Ta > TTh.

bY

aaYYbYabaabab mm

TmmQmmmTmmTmmT

])()[(coscos 2

HW 1HW 1

abY

bYTh mmm

mmQT



8


• The double valued situation occurs between TTh and the upper limit Ta

\.

• Double-valued in a forward cone.

aY

Ya mm

mQT

\

aba

aaYYbY

Tmm

TmmQmmm ])()[(cos max

2

HW 1HW 1 (continued)(continued)

Discuss the elasticelastic and inelastic scatteringinelastic scattering

of neutronsneutrons using these relations.


9



10


What about

neutron induced

reactions?


11


What about

neutron induced

reactions?


12


What about

neutron induced

reactions?


13


• If the reaction reaches excited states of Y

58Ni(,p)61Cu

Highest proton energy

exbexYaXex EQcmEcmcmcmQ 02222 )(

less proton energy

even less ….

See Figures 11.4 in Krane What about neutron

induced reactions?


14

Neutron Interactions (revisited)

• Chadwick’s discovery.• Neutrons interact with nuclei, not with atoms. (Exceptions).

• Recall from Nuclear Physics 742:o Inelastic scattering (n,n\). Q = -E* Inelastic gammas.

Threshold?o Elastic scattering (n,n). Q = ?? (Potential and CN).

Neutron moderation?o Radiative capture (n,). Q = ?? Capture gammas.o (n,), (n,p). Q = ?? Absorption Reactions.o (n,2n), (n,3n) Q = ?? Energetic neutrons on heavy water can easily eject the loosely bound neutron.o Fission. (n,f).

HW 2HW 2 Examples of such exo- and endo-thermic reactions with Q calculations.


15

• Elastic or inelastic.• Analogous to diffraction.• Alternating maxima and minima.• First maximum at

• Minimum not at zero (sharp edge of the nucleus??)• Clear for neutrons.• Protons? High energy, large angles. Why?• Inelastic Excited states, energy, X-section and spin-parity.

31

ARR

p

h

o

R

Neutron Scattering (revisited)


16

• Probability.• Projectile a will more probably hit target X if area is larger.• Classically: = (Ra + RX)2. Classical = ??? (in b) n + 1H, n + 238U, 238U + 238U • Quantum mechanically: = 2.

• Coulomb and centrifugal barriers energy dependence of . What about neutrons?What about neutrons?• Nature of force: Strong: 15N(p,)12C ~ 0.5 b at Ep = 2 MeV. Electromagnetic: 3He(,)7Be ~ 10-6 b at E = 2 MeV. Weak: p(p,e+)D ~ 10-20 b at Ep = 2 MeV.• Experimental challenges to measure low X-sections..

CMaXaXaaX

Xa

EEmm

mm

22

Reaction Cross Section (revisited)


17

Reaction Cross Section (Simple terms)

XA (Area of the beam!!)

Monoenergetic neutrons of speed v

(cm.s-1) and density n (cm-3)

Target with N atoms.cm-3 or NAX atoms.

Position of a neutron 1 s

before arriving at target

|v|

Volume = vAcontaining nvA neutrons that hit the

“whole” target in 1 s.Beam Intensity I nvA/A = nv (cm-2s-1)

Number of neutrons interacting with target per second I, A, X and N= t I N A X


18


Number of neutrons interacting with target per second= t I N A X

Number of interactions with a single nucleus per second = t I Interpretation and units of .

nvA = IA neutrons strike the target per second, of these

tI neutrons interact with any single nucleus. Thus,

measures the probability for a neutron to hit a nucleus.

Total cross section Total number of

nuclei in the target

AAI

I tt

Effective cross-sectional area of the nucleus.

Study

examples in

Lamarsh


19


Number of neutrons interacting with target per second= t I N A X

Number of interactions per cm3 per second (Collision Density) Ft = t I N = I t

t = N t

Total cross section Volume of the

target

Macroscopic total cross

section.Probability per

unit path length.

tt

XteIXI

1

)( 0

Mean free path

Study

examples in

Lamarsh


20


Homogeneous Mixture

Molecule xmyn Nx=mN, Ny=nN

given that events at x and y are independent.

yyxxyx NN

yx nm

Study

examples in

Lamarsh


21


d,Ia

Detector for particle “b”

\NI

dRd

a

b

“X“ t

arge

t Nuc

lei /

cm2

“a” particles / s

“b” particles / scm2

Typical nucleus (R=6 fm): geometrical R2 1 b.Typical : <b to >106 b.


22


Many different quantities are called “cross section”.Krane Table 11.1

\4

),(4

),(

NI

r

d

d

drdR

a

b

Angular distribution

“Differential” cross section(,) or ( )or “cross section” …!!

Units … !

d

dddd

d

d

ddd

0

2

0

sin

sin

ddE

d

b

2

Doubly differential

Energy state in “Y”

dE

d

t for all “b” particles.


23


24

n-TOFn-TOFCERNCERN


25


26

1/v

235U thermal cross sectionsfission 584 b.scattering 9 b.radiative capture 97 b.

Fast neutrons should be moderated.

Fission Barriers

Different Features (revisited)


27

Neutron Induced Reactions (revisited)

22 nXHCCHbY IIIn X(n,b)Y

n(En)b(Q+En)

For thermal neutronsQ >> En

b(Q) constant

2

11

vE

)( nln EPvn

Probability to penetrate the potential barrier

Po(Ethermal) = 1P>o(Ethermal) = 0

vEnn

1)( Non-resonant


28

Neutron Induced Reactions (revisited)


29

bbplL

lb 222

1max, )12( lbb lll

)()(

7.656)(2

keVEub

CM HW 3HW 3

)1()12)(12(

122max aX

XaaX JJ

J

Statistical Factor (revisited)


30

Resonance Reactions (revisited)

Entrance Channela + X

ExitChannelb + YCompound

Nucleus C*

ExcitedState

ExJ

a + X Y + b Q > 0b + Y X + a Q < 0

Inverse Reaction

22 )1()12)(12(

12XaHCCHbY

JJ

JIIIaX

XaaXaX

QM StatisticalFactor ()

Identicalparticles

• Nature of force(s).• Time-reversal invariance.

22 )1()12)(12(

12YbHCCHXa

JJ

JIIIbY

YbbYbY

??bY

aX

HW 4HW 4


31

Projectile

TargetQ-value

Projectile

Q-valueTarget

Direct Capture(all energies)

Resonant Capture(selected energies with large X-section)

E = E + Q - Eex

2XaHY

Q + ER = Er

22XaHEEHE CNrrf

ba



32


33


22

2 )()(

o

fresponse

Damped OscillatorDamped Oscillator

eigenfrequency

Dampingfactor

Oscillator strength

22

2 )()()(

R

ba

EEE

0

1

t

ot


34


22

2

2

)()()1(

)12)(12(

12)(

R

baaX

XaaX EEJJ

JE

Breit-Wigner formulaBreit-Wigner formula

• All quantities in CM system• Only for isolated resonances.

a

b

e

R

aae

baR

Reaction

Elastic scattering

HW 5HW 5 When does R take its maximum value?

ba

Usually a >> b.


35


Ja + JX + l = J(-1)l (Ja) (JX) = (J)

(-1)l = (J) Natural parity.

ExitChannelb + Y

Compound Nucleus C*

ExcitedState

ExJ

Entrance Channela + X


36


Cro

ss s

ecti

on

EC a Energy

What is the “Resonance Strength” …?What is its significance?In what units is it measured?

ba

aXXa JJ

J)1(

)12)(12(

12

Charged particleradiative capture (a,)(What about neutrons?)(What about neutrons?)


37

Neutron Resonance Reactions (revisited)


38

Neutron Activation Analysis (revisited)

(Z,A) + n (Z, A+1)-

(Z+1, A+1)

(-delayed -ray)

http://ie.lbl.gov/naa !

Project 1Project 1

Documents

Nuclear Physics at BAU nuclear.bau.jo/ This course nuclear.bau.jo/Reactors