Lesson 10
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
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
 mx 
 mp  2
1/ 2
Q  Tx 1

T
1

m
T
m
T
 P P x x  cos
 P 

 mR 
 mR  mR

m m T 
1/ 2
T  P x P
1/ 2
x
cos  {mP mxTP cos2   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

2 

0 0
d
( )sin dd
d
Charged Particles vs Neutrons
I=particles/s
In reactors, particles traveling in all directions
Number of reactions/s=Number of target atoms
x  x (particles/cm2/s)
What if the product is radioactive?
dN
 (rate of production)  (rate of decay)
dt
dN
 nx  N
dt
dN
 dt
nx  N
d( N)
 dt
N  nx
ln( N  nx ) |N0  t |t0
N  nx  t
e
nx
A  N  nx (1 e t )
Neutron Cross Sections-General
Considerations
 r x p  pb
p




b
b
    1
2
   2(

2
2

 2 1
2
2
2
)
   2(2 1)

 total   


max
0
2
(2 1)  
(2
max
2
0
1)   2(
2
1)
max

max
lmax 
max
R
1
R
 total   (R  ) 2
Semi-classicalQM

 total  

2
 (2
0
 1)T
The transmission coefficient
T
• Sharp cutoff model (higher energy neutrons)
T  1 for

max
T  0 for

max

• Low energy neutrons

 total  
2
 
2
2m

1

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)
 rx p
1/ 2
max
B 
1/ 2 
 R2  1 
  
Semi-classicalQM

 total  
2


1  
2
max
2 2
max
2  B 
B 
2 2 1 
  R 2 1   R 2 1 
  
  
2
 B 
 total  R 1 
  
2

2
Barriers for charged particle
induced reactions
Vtot(R)=VC(R)+Vnucl(R)+Vcent(R)
Vnucl(r) = V0/(1 + exp((r -R/a)))
Vcent (r ) 
( 1)
2 r 2
2
Rutherford Scattering
Fcoul 


Z1Z e
r
2
2
Z1Z2 e 2
PE 
r
Rutherford Scattering
1 2 1 2 Z1Z 2 e 2
mv  mv0 
2
2
d
v0 2
d
   1 0
 v 
d


d0 
2Z1Z2 e 2 Z1Z2 e 2

mv2
Tp
mvb = mv0d

2
v 
b   0  d 2  d (d  d 0 )
v
2
   2b
cot   
 2  d0
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 (2b db)
 
cos 
1
2 
dI  I 0 d02
d



4
sin 3  
2 
2
2 

d0 
d dI 1
1
Z1Z2 e
1


  
  cm 



d I 0 d  4 
4Tp 
4
4  

sin  
sin  
2 
2 
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
k  k  k  2kd kp cos
2
n
2
d
n
2
p
 r  p  Rkn

JA 

n
1 
   JB *  JA 
2 
 A  B  (1)
n 
1
2