Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).

1

Neutron Interactions (revisited)

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

• Recall from basic Nuclear Physics: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.

Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).

2

• 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)

24

222

sin

1

4

1

4

)(

ao T

zZe

d

d

Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).

3

• 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)

Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).

4

Reaction Cross Section (Simple terms)

XA (Area of what??!!)

Monoenergetic (and unidirectional) 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

NX??

Total microscopic cross section

Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).

5

Reaction Cross Section (Simple terms)

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 (per unit area of target).

Total microscopic cross section

Total number of nuclei in the

target

AAI

I tt

Effective cross-sectional area of the nucleus.

Study

examples in

Lamarsh

Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).

6

Reaction Cross Section (Simple terms)

AAI

I tt

The probability for a neutron to hit a nucleus (per unit area of target):

Function of

what?

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

Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).

7

Reaction Cross Section (Simple terms)

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

Volume of the target

Macroscopic total cross

section.Probability per

unit path length.

tt

XteIXI

1

)( 0

Mean free path

Study

examples in

LamarshTotal

microscopic cross section

Nuclear Reactor Theory, JU, First Semester,2010-2011 (Saed Dababneh).

8

Neutron Attenuation

X

X

t

t

eXP

eXP

1)(

)(

ninteractio

ninteractiono

Recall t = N t

Probability per unit path length.

X

I0 I

Probability

mfp for scattering s = 1/s

mfp for absorption a = 1/a

…………. total mfp t = 1/t

XteIXI 0)(

Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).

9

Reaction Cross Section (Simple terms)

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

Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).

10

Reaction Cross Section

d,Ia

\

Detector for particle “b”

\\NI

dRd

a

b

“X“ t

arge

t Nuc

lei /

cm2

“a” particles / s

“b” particles / scm2

Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).

11

Reaction Cross Section

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

dEd

d 2Doubly differential

dE

d

t for all “b” particles.

Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).

12

Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).

13

n-TOFn-TOFCERNCERN

Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).

14

Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).

15

1/v

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

Fast neutrons should be moderated.

Fission Barriers

Neutron Cross Section (Different Features)