Lesson 11

Nuclear Reactions

Nomenclature

•  Consider the reaction 4He + 14N → 17O + 1H

•  Can write this as Projectile P + Target T →Residual Nucleus R

and Emitted Particle x •  Or

T(P,x)R 14N(4He,p)17O

Conservation Laws

•  Conservation of neutrons, protons and nucleons

59Co(p,n)? Protons 27 + 1 =0 + x Neutrons 32 +0=1 + y

Product is 59Ni

Conservation Laws (cont.)

•  Conservation of energy Consider 12C(4He,2H)14N

Q=(masses of reactants)-(masses of products)

Q=M(12C)+M(4He)-M(2H)-M14N) Q=+ exothermic Q=- endothermic

Conservation Laws(cont.)

•  Conservation of momentum mv=(m+M)V

TR=Ti*m/(m+M) •  Suppose we want to observe a reaction

where Q=-. -Q=T-TR=T*M/(M+m)

T=-Q*(M+m)/M

Center of Mass System

Center of Mass System pi=pT ptot=0

velocity of cm =Vcm velocity of incident particle = v-Vcm

velocity of target nucleus =Vcm m(v-Vcm)=MV

m(v-Vcm)-MV=0 Vcm=mv/(m+M)

T’=(1/2)m(v-Vcm)2+(1/2)MV2 T’=Ti*m/(m+M)

Center of Mass System

catkin

Kinematics

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Q = Tx 1+mx

mR

⎛

⎝ ⎜

⎞

⎠ ⎟ − TP 1−

mp

mR

⎛

⎝ ⎜

⎞

⎠ ⎟ −

2mR

mPTPmxTx( )1/ 2 cosθ

€

Tx1/ 2 =

mPmxTP( )1/ 2 cosθ ±{mPmxTP cos2θ + mR +mx( )[mRQ + (mR − mP )TP ]}

1/ 2

mR +mx

Reaction Types and Mechanisms

Reaction Types and Mechanisms

Nuclear Reaction Cross Sections

fraction of beam particles that react= fraction of A covered by nuclei

a(area covered by nuclei)=n(atoms/cm3)*x(cm)* (σ(effective area of one nucleus, cm2))

fraction=a/A=nxσ -dφ=φnxσ

φtrans=φinitiale-nxσ

φinitial- φtrans= φinitial(1-e-nxσ)

Factoids about σ

•  Units of σ are area (cm2). •  Unit of area=10-24 cm2= 1 barn •  Many reactions may occur, thus we divide σ

into partial cross sections for a given process, with no implication with respect to area.

•  Total cross section is sum of partial cross sections.

Differential cross sections

dN/d = n(d /d )dx

€

σ =dσdΩ(θ )sinθdθdφ

0

π

∫0

2π

∫

Charged Particles vs Neutrons

In reactors, particles traveling in all directions Number of reactions/s=Number of target atoms x σ x (particles/cm2/s)

I=particles/s

What if the product is radioactive?

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dNdt

= (rate of production) − (rate of decay)

dNdt

= nσΔxφ − λN

dNnσΔxφ − λN

= dt

d(λN)λN − nσΔxφ

= −λdt

ln(λN − nσΔxφ) |0N = −λt |0

t

λN − nσΔxφ−nσΔxφ

= e−λt

A = λN = nσΔxφ(1− e−λt )

Neutron Cross Sections-General Considerations

€

l = r x p = pb

€

p =h

D

€

lh =hbD

€

b = lD

€

σl = π l +1( )2D 2 − πl2D 2

€

σl = πD 2(l2 + 2l +1− l2 )

€

σl = πD 2(2l +1)

€

σ total = σl = πD 2(2l +1) = πD 2 (2l +1) = πD 2(l max +1)2l=0

lmax

∑l=0

lmax

∑l

∑

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l max

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lmax =RD

l max +1=R + D

D

σ total = π (R + D)2

Semi-classical→QM

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σ total = πD2 (2l +1)Tl

l= 0

∞

∑

The transmission coefficient

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Tl

•  Sharp cutoff model (higher energy neutrons)

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Tl =1 for l ≤ l max

Tl = 0 for l > l max

•  Low energy neutrons

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σ total ∝ πD 2 ε ∝ πh2

2mεε ∝

1ε

1/v

Charged Particle Cross Sections-General Considerations

B = Z1Z2e2/R

p=(2mT)1/2 = (2µ)1/2(ε-B)1/2 = (2µε)1/2(1-B/ε)1/2

reduced mass µ=A1A2/(A1+A2)

€

l = rxp

l max = R 2µε( )1/ 2 1−Bε

⎛

⎝ ⎜

⎞

⎠ ⎟

1/ 2

Semi-classical→QM

€

l → lh

€

σ total = πD 2 l max +1( )2 ≈ πD 2l max2 = πD 2R2 2µε

h21− B

ε

⎛

⎝ ⎜

⎞

⎠ ⎟ = πD 2R2 1

D 2 1−Bε

⎛

⎝ ⎜

⎞

⎠ ⎟

€

σ total = πR2 1− Bε

⎛

⎝ ⎜

⎞

⎠ ⎟

Barriers for charged particle induced reactions

Vtot(R)=VC(R)+Vnucl(R)+Vcent(R)

VC(r) = Z1Z2/r for r < RC

VC(r) = (Z1Z2/RC) (3/2 – (r 2/RC2)) for r > RC

Vnucl(r) = V0/(1 + exp((r-R/a)))

€

Vcent(r ) =h2

2µl(l +1)r 2

Rutherford Scattering

€

Fcoul =Z1Z e

2

r 2

€

PE =Z1Z2e

2

r

Rutherford Scattering

€

12mv2 =

12mv0

2 +Z1Z2e

2

d

€

v0v

⎛

⎝ ⎜

⎞

⎠ ⎟ 2

= 1− d0d

€

d0 =2Z1Z2e

2

mv2=Z1Z2e

2

Tp

mvb = mv0d

)( 02

202 ddddvvb −=⎟⎠

⎞⎜⎝

⎛=

d = b cot( /2)

tan = 2b/d0

0

22

cotdb

=⎟⎠

⎞⎜⎝

⎛θ

At r=d

Rearranging

where

Conservation of angular momentum

Property of a hyperbola

Consider I0 particles/unit area incident on a plane normal to the beam. Flux of particles passing through a ring of width db between b and b+db is dI = (Flux/unit area)(area of ring) dI = I0 (2πb db)

€

dI =14πI0d0

2cos θ

2⎛

⎝ ⎜

⎞

⎠ ⎟

sin3 θ2

⎛

⎝ ⎜

⎞

⎠ ⎟

dθ

€

dσdΩ

=dII0

1dΩ

=d04

⎛

⎝ ⎜

⎞

⎠ ⎟ 2 1

sin4 θ2

⎛

⎝ ⎜

⎞

⎠ ⎟

=Z1Z2e

2

4Tpcm

⎛

⎝ ⎜

⎞

⎠ ⎟

21

sin4 θ2

⎛

⎝ ⎜

⎞

⎠ ⎟

Removing the effect of Rutherford scattering

elastic scattering

Elastic (Q=0) and inelastic(Q<0) scattering

•  To represent elastic and inelastic scattering, need to represent the nuclear potential as having a real part and an imaginary part.

This is called the optical model

Direct Reactions

•  Direct reactions are reactions in which one of the participants in the initial two body interaction leaves the nucleus without interacting with another particle.

•  Classes of direct reactions include stripping and pickup reactions.

•  Examples include (d,p), (p,d), etc.

(d,p) reactions

(d,p) reactions

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kn2 = kd

2 + kp2 − 2kd kp cosθ

€

l nh = r × p = Rknhn = Rkn

€

JA − l n −12

⎛

⎝ ⎜

⎞

⎠ ⎟ ≤ JB* ≤ JA + l n +

12

π Aπ B = (−1)l

Use of (d,p) reactions

•  Surrogate for n capture •  “Negative kinetic energy neutrons” •  Spectroscopy