Lesson 10

Nuclear Reactions

Nomenclature

• Consider the reaction4He + 14N 17O + 1H

• Can write this asProjectile 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 + xNeutrons 32 +0=1 + y

Product is 59Ni

Conservation Laws (cont.)

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

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

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

Conservation Laws(cont.)

• Conservation of momentummv=(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

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

Center of Mass System

catkin

Kinematics

€

Q = Tx 1+mx

mR

⎛

⎝ ⎜

⎞

⎠ ⎟− TP 1−

mp

mR

⎛

⎝ ⎜

⎞

⎠ ⎟−

2

mR

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)*((effective area of one nucleus, cm2))

fraction=a/A=nx-d=nx

trans=initiale-nx

initial- trans= initial(1-e-nx)

Factoids about

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

divide 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

€

=dσ

dΩ(θ )sinθdθdφ

0

π

∫0

2π

∫

Charged Particles vs Neutrons

In reactors, particles traveling in all directionsNumber of reactions/s=Number of target atomsx x (particles/cm2/s)

I=particles/s

What if the product is radioactive?

€

dN

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

dN

dt= nσΔxφ − λN

dN

nσΔxφ − λN= dt

d(λN)

λN − nσΔxφ= −λdt

ln(λN − nσΔxφ) |0N = −λ t |0

t

λN − nσΔxφ

−nσΔxφ= e−λ t

A = λN = nσΔxφ(1− e−λ t )

Neutron Cross Sections-General Considerations

≈ ( + R r')2 = r02(AP + AT)2

€

l = r x p = pb

€

p =h

D

€

lh=hbD

€

b = lD

€

l =π l +1( )2D2 − πl 2D2

€

l =πD2(l 2 + 2l +1− l 2 )

€

l =πD2(2l +1)

€

total = σ l = πD2(2l +1) = πD2 (2l +1) = πD2(l max +1)2

l =0

l max

∑l =0

l max

∑l

∑

€

l max

€

lmax =R

D

l max +1=R + D

D

σ total = π (R + D)2

Semi-classicalQM

€

total = πD2 (2l +1)Tl

l = 0

∞

∑

The transmission coefficient

€

Tl

• Sharp cutoff model (higher energy neutrons)

€

Tl =1 for l ≤ l max

Tl = 0 for l > l max

• Low energy neutrons

€

total ∝ πD2 ε ∝ πh2

2mεε ∝

1

ε

1/v

Charged Particle Cross Sections-General Considerations

B = Z1Z2e2/R

p=(2mT)1/2 = (2)1/2(-B)1/2 = (2)1/2(1-B/)1/2

reduced mass =A1A2/(A1+A2)

€

l = rxp

l max = R 2με( )1/ 2

1−B

ε

⎛

⎝ ⎜

⎞

⎠ ⎟

1/ 2

Semi-classicalQM

€

l → lh

€

total = πD2 l max +1( )2

≈ πD2l max2 = πD2R2 2με

h21−

B

ε

⎛

⎝ ⎜

⎞

⎠ ⎟= πD2R2 1

D21−

B

ε

⎛

⎝ ⎜

⎞

⎠ ⎟

€

total = πR2 1−B

ε

⎛

⎝ ⎜

⎞

⎠ ⎟

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/RC

2)) for r > RC

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

€

Vcent(r ) =h2

2μ

l (l +1)

r 2

Rutherford Scattering

€

Fcoul =Z1Z e

2

r 2

€

PE =Z1Z2e

2

r

Rutherford Scattering

€

1

2mv2 =

1

2mv0

2 +Z1Z2e

2

d

€

v0

v

⎛

⎝ ⎜

⎞

⎠ ⎟2

= 1−d0

d

€

d0 =2Z1Z2e

2

mv2=Z1Z2e

2

Tp

mvb = mv0d )(02202 ddddvvb −=⎟⎠⎞⎜⎝⎛=

d = b cot( /2)

tan = 2b/d0

022cotdb=⎟⎠⎞⎜⎝⎛θ

Consider I0 particles/unit area incident on a plane normal to the beam.Flux of particles passing through a ring of width dbbetween b and b+db is

dI = (Flux/unit area)(area of ring)dI = I0 (2πb db)

€

dI =1

4πI0d0

2

cosθ

2

⎛

⎝ ⎜

⎞

⎠ ⎟

sin3 θ

2

⎛

⎝ ⎜

⎞

⎠ ⎟dθ

€

dσ

dΩ=dI

I0

1

dΩ=

d0

4

⎛

⎝ ⎜

⎞

⎠ ⎟2

1

sin4 θ

2

⎛

⎝ ⎜

⎞

⎠ ⎟=

Z1Z2e2

4Tpcm

⎛

⎝ ⎜

⎞

⎠ ⎟

2

1

sin4 θ

2

⎛

⎝ ⎜

⎞

⎠ ⎟

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 − 2kdkp cosθ

€

l nh = r × p = Rknh

n = Rkn

€

JA − l n −1

2

⎛

⎝ ⎜

⎞

⎠ ⎟ ≤ J

B* ≤ JA + l n +1

2

π Aπ B = (−1)l