2. Artificial Transmutation
This painting of an alchemist’s laboratory was
made around 1570.
For centuries, these early scientists, known as
alchemists, tried to use chemical reactions to
make gold.
The alchemists failed in their attempts to turn
lead into gold.
3. Aside
The Alchemist is a novel by Brazilian author
Paulo Coelho which was first published in
1988.
Originally written in Portuguese, it has been
translated into at least 67 languages as of
October 2009.
4. Induced (or artificial) Transformation …
The process of striking nuclei with high energy particles in
order to create (i.e. change them into) new elements
5. Nuclear Reactions in the Laboratory
In 1919, Ernest Rutherford performed the first artificial
transmutation by exposing nitrogen gas to alpha particles.
Some of the alpha particles were absorbed by the nitrogen
nuclei.
Each newly formed nucleus then ejected a proton, leaving
behind the isotope oxygen-17.
9. Rutherford’s Observations
When the very high velocity alpha particles made head-on
collision with the Nitrogen Nuclei some of them were
captured by the latter.
The composite system, which was formed as a result of
such Capture, almost immediately (within ~ 10-15 s)
disintegrated by the emission of a Proton of very high
velocity.
This was the process of Nuclear Transmutation brought
about artificially with the help of alpha-particles from a
radioactive substance, leaving a residual nucleus of
isotope 17O of oxygen.
10. Nuclear Reaction
Refers to a process which occurs when a Nuclear Particle
(e.g. a Nucleon) comes into close contact with another
during which energy and momentum exchanges takes
place.
The final products of the reaction are again some nuclear
particle or particles which leave the point of contact
(reaction site) in different directions.
The changes produced in a Nuclear reaction usually
involve strong nuclear force.
Purely electromagnetic effects (e.g. Coulomb’s
Scattering) or processes involving weak interactions (e.g.
-decay) are usually excluded from the category of
Nuclear Reaction.
Changes of Nuclear States under the influence of EM
Interactions are included.
11.
12.
13.
14. Nuclear Reaction
X + x Y + y
X Target Nucleus which is bombarded by Projectile x
Resulting Compound Nucleus breaks up almost
immediately by Ejecting a Particle y, leaving a residual
nucleus Y
x & y can be
Protons (p)
Neutrons (n)
Deuterons (d)
-particles
-rays etc.
15. Types of Nuclear Reactions
Elastic Scattering
Inelastic Scattering
Radioactive Capture
Disintegration Process
Many body Reactions
Photo-Disintegration
Nuclear Fission
Elementary Particle Reactions
Heavy Ion Reactions
16. Elastic Scattering
The ejected particle “y” is the same as the Projectile “x”
Comes out with the same energy and angular momentum
as “x” so that the residual nucleus “Y” is the same as the
target “X” and is left in the same state (ground state) as
the latter.
Process represented by X(x, x)X.
17. Inelastic Scattering
The ejected particle “y” is the same as the Projectile “x”
Comes out with the different energy and angular
momentum as “x” so that the residual nucleus “Y” is left
in an excited state.
Process can be written as X(x, y)X*
* excited state
18. Radioactive Capture
Projectile “x” is absorbed by the target Nucleus “X” to
form the excited compound nucleus C* which
subsequently goes down to the ground state by the
emission of one or more -ray quanta
Process can be written as X(x, )Y*, (Y = C)
19. Disintegration Process
Process can be represented as X(x, y)Y where X, x, Y and y
are all different either in Z or in A or in both.
First nuclear Transmutation is an example of this process
14N(, p)17O
20. Many body Reactions
When the kinetic energy of the incident particle is high,
two or more particles can come out of the compound
nucleus.
If y1, y2, y3 etc. represent these different particles then
reaction equation can be written as X(x, y1, y2, y3, …)Y
When the energy of x is very high, a very large number of
reaction products usually results (3 to 20), reactions
known as Spallation Reactions.
21. Photo-Disintegration
The target nucleus is bombarded with very high energy -
rays, so that is it raised to an excited state by the
absorption of the latter.
The compound nucleus C* = X*.
Reaction X(, y)Y
22. Nuclear Fission
When X is a heavy nucleus and y, Y have comparable
masses, the reaction is known as nuclear Fission.
238U (n, f)
23. Elementary Particle Reactions
These involve either the production of Elementary
Particles other than Nucleons or Nuclei as a result of the
Reaction or their use as Projectiles or both of these;
The reactions usually produced at extremely high energies
which may be several hundred MeV or more
24. Heavy Ion Reactions
In these reactions the target nucleus is bombarded by
Projectiles heavier than -particles.
Various types of products may be produced
The reactions usually take place at fairly high energies of
the projectile
10B(16O, 4He) 22Na
26. Conservation Laws in Nuclear Reactions
Conservation of Mass Number
Conservation of Atomic Number
Conservation of Energy
Conservation of Linear Momentum
Conservation of Angular Momentum
Conservation of Parity
Conservation of Isotropic Spin
27. Conservation of Mass Number
The total number of Neutrons and Protons in the Nuclei
taking part in a Nuclear reaction remains unchanged after
the reaction.
Thus in reaction X(x, y) Y … the sum of mass numbers of X
and x must be equal to the sum of the mass numbers of Y
and y.
In the general case of the reactions involving elementary
particles the law can be expressed by required the total
number of heavy articles (Baryons) remains uncaged in a
reaction.
28. Conservation of Atomic Number
The total number of Protons of the nuclei taking part in a
nuclear reaction remains unchanged after the reaction.
i.e. sum of the atomic number of X and x is equal to the
sum of atomic numbers of Y and y.
Z + z = Z’ + z’
29. Conservation of Energy
Q Value of a Nuclear Reaction
In nuclear physics and chemistry, the Q value for a reaction is the
amount of energy released by that reaction.
The value relates to the enthalpy (a thermodynamic quantity
equivalent to the total heat content of a system) of a chemical
reaction or the energy of radioactive decay products.
It can be determined from the masses of reactants and products.
30. Conservation of Energy
The law requires that …the total energy, including the
rest-mass energies of all the nuclei taking part in a
reaction and their kinetic energies must be equal to the
sum of the rest-mass energies and the kinetic energies of
the products.
MX, Mx, MY, My rest masses
MXc2, Mxc2, MYc2, Myc2 rest mass energies
E Kinetic Energy
MXc2 + Mxc2 + Ex = MYc2 + Myc2 + EY + Ey
Without c2 masses are expressed in energy units.
31. Conservation of Linear Momentum
pX, px, pY, py momentum vectors of the different nuclei
taking part in a reaction
px = pY + py (Laboratory frame of Reference)
In the frame of reference in which the center of mass of
the two particles before collision is at rest (C-System)
pX + px
= 0 0 = pY + py (center of mass of the product
particle is also at rest in this system)
32. Conservation of Angular Momentum
In a nuclear reaction of the type X + x Y + y the total
angular momentum of the nuclei taking part in the
reaction remains the same before and after the reaction.
IX, Ix, IY, Iy nuclear spins (total angular momentum) of
the nuclei X, x, Y, y respectively.
lX relative orbital moment (Initial State)
lY relative orbital moment (final State)
IX + Ix + lX = IY + Iy + lY
33. Conservation of Parity
Parity
Parity involves a transformation that changes the algebraic sign of
the coordinate system.
Parity is an important idea in quantum mechanics because the
wave functions which represent particles can behave in different
ways upon transformation of the coordinate system which
describes them.
34. Conservation of Parity
Parity before the reaction i should be equal to Parity
after the reaction f
X, x, Y, y intrinsic parities of the nuclei
I = X x (-1)lx at initial state
f = Y y (-1)ly at final state
X x (-1)lx = Y y (-1)ly
35. Conservation of Isotropic Spin
Isotopic spin
is a quantum number related to the strong interaction
TI = Initial isotopic spin vector
Tf = Final Isotopic spin vector
TX + Tx = TY + Ty
For Deuteron or alpha particles Tx & Ty = 0
36. Collision between Subatomic Particles
For Nuclear Reactions Certain Conservation Laws HOLD
When a nuclear Projectile is incident upon a nucleus
the particles may either suffer ELASTIC Collision
may Produce a reaction in which new particles are usually
produced.
37. Collision between Subatomic Particles
Elastic Collision
No change in the internal states of the colliding particles takes
place
Kinematics of the collision process easily analyzed by Law of
Conservation of Momentum and Kinetic Energy
Inelastic Collision
Change in the internal state of the Particles must be taken into
account
38. Collision between Subatomic Particles
Experimental Arrangements
A Beam of Mono Energetic Particles (Projectiles) is allowed to fall
on the target containing the Nuclei which are at rest
Collision is analyzed from the point of view of
An Observer at rest in the Laboratory (L-System)
An observer at rest with respect to Center of mass of the colliding
particles (C-System)
42. Elastic Collision in L-System (Non-
Relativistic)
1
'
1
1
2
'
1
2
1
2
'
2 cos
2
p
p
p
p
p
0
1
cos
2
1 1
1
'
1
1
'
1
r
E
E
r
E
E
1
1
cos
cos 2
1
2
1
1
'
1
r
r
E
E
43. Elastic Collision in L-System (Non-
Relativistic)
If the Target particle is heavier then r >>> 1
This type of scattering is observed when -particles are
scattered by nuclei as they pass through matter
The energy received by the nucleus in such collisions is
negligibly small
2
1
1
'
1
1 4
M
M
E
E
E
44. Elastic Collision in L-System (Non-
Relativistic)
If r < 1
This will yield real if
1
1
cos
cos 2
1
2
1
1
'
1
r
r
E
E
1
1
cos 2
1
2
r
0
1
4
2
1
'
1
1
r
r
E
E
E
1
1
1
2
'
1
1
1
'
1
1
4
4
E
M
E
M
E
E
r
E
E
E
45. Elastic Collision in C System (non-
Relativistic)
Velocities and Momenta in the C-System = V & P
Energies and angles of Scattering = E &
Velocities and Momenta in the L-System = v & p
Energies and angles of Scattering = e &
Particle M2 is at rest in the L-System before collision i.e.
v2 = 0.
The velocity in the L-System of the center of mass is …
47. Elastic Collision in C System (non-
Relativistic)
Generalize Equations
Center of Mass in One Dimension
Velocity of the Center of Mass
Acceleration of the Center of Mass
48. Elastic Collision in C System (non-
Relativistic)
2
1
1
1
2
1
2
2
1
1
M
M
v
M
M
M
v
M
v
M
vc
2
1
1
1
2
2
2
1
1
2
1
1
M
M
v
M
v
v
V
M
M
v
M
v
v
V
c
c
Velocities Before Collision in C-System
49. Elastic Collision in C System (non-
Relativistic)
Corresponding Momenta
1
2
1
1
2
1
2
2
2
1
2
1
1
2
1
1
1
1
v
M
M
v
M
M
V
M
P
v
M
M
v
M
M
V
M
P
2
1
2
1
M
M
M
M
Reduced Mass
50. Elastic Collision in C System (non-
Relativistic)
Two Particles have equal and opposite momenta before
collision so that their total momentum P1 + P2 = 0.
0
2
1
'
2
'
1
P
P
P
P
Law of Conservation of momentum then requires that the
total momentum of the two particles after collision is also
zero
Sum of K.E before and after collision are
2
'
2
'
2
'
2
2
2
2
2
2
2
1
2
1
'
2
'
1
2
2
2
2
1
2
1
2
1
p
M
p
M
p
E
E
p
M
p
M
p
E
E
51. Elastic Collision in C System (non-
Relativistic)
Sum of K.E before and after collision are
2
'
2
'
2
'
2
2
2
2
2
2
2
1
2
1
'
2
'
1
2
2
2
2
1
2
1
2
1
p
M
p
M
p
E
E
p
M
p
M
p
E
E
|P1|=|P2|=|P| and |P1’|= |P2’|= |P’|
Since Energy equation requires that '
2
'
1
2
1 E
E
E
E
52. Elastic Collision in C System
(non-Relativistic)
We get P’ = P so that
The momentum diagram of the particles
shows that two particles will fly apart from
the point of collision with equal and opposite
momenta so that
P1 = P2 = P1’= P2’
2
1
So K.E of the COM is
1
2
1
1
2
2
1
1
2
1
where
M
M
M
v
M
M
E c
c
53. Elastic Collision in C System
(non-Relativistic)
So K.E of the COM is
2
1
1
1
1
2
1
1
2
2
1
2
1
2
1
v
M
where
M
M
M
v
M
M
E c
c
Is the K.E of incident particle in the L-System.
54. Elastic Collision in C System (non-
Relativistic)
The Energy Ec is not available for the production of any
inelastic effect, the total amount of energy is
2
1
1
2
1
2
1
2
1
1
1
1
2
1
v
M
M
M
M
M
M
Ec
There is no change in the K.E and Momenta of the particles
after collisions in the C-System
55. Energetics of Nuclear Reactions
Exoergic Reactions
Energy Evolved
Endoergic Reactions
Absorption of Energy
Q Value
Exoergic + Endoergic
Q = EY+Ey-EX-Ex = EY+Ey-Ex
If Q > 0 Exoergic Reaction
MX + Mx > MY + My
Q < 0 Endoergic Reaction
56. Experimental Determination of Q
Q of a reaction can be determined with the help of
Q = EY+Ey-EX-Ex = EY+Ey-Ex
Where EY, Ey, Ex can be accurately determined.
Can also be estimated from the precise values of the
atomic masses of the nuclei taking part in the reaction.
Q = MX + Mx – MY - My
If one of the product Nuclei is heavy than
cos
2
y
x
y
x
Y
y
Y
y
x
Y
x
Y E
E
M
M
M
E
M
M
E
M
M
E
57. Experimental Determination of Q
If the emitted particle y is a charged particle, then one
can use a Scintillation counter, a proportional counter
(gas-filled), a solid state counter or a magnetic
spectrograph to determine its energy.
Scintillation Spectrometers can be used when the
resolving power needed is not very high.
Solid State Spectrometers the resolving power is much
better
Magnetic Spectrometers best for high resolution of
work
58. Cockroft and Walton’s Experiment
Initially Nuclear reactions were produced by using the
high energy Alpha-particles (only projectiles available)
Due to limitations of Energy and intensity, need for other
sources or projectiles was keenly felt.
Another Limitation was Alpha-Particles being doubly
charged were very strongly repelled by the positive
charge of the Nucleus could not penetrate the heavier
nuclei to produce nuclear reactions
Cockroft and Walton developed a charged particle
accelerator with the help of which a beam of protons
could be accelerated to high energy
With the help of this, they were able to produce
Disintegration of 7Li
59. Cockroft and Walton’s Experiment
Accelerator is known as Voltage Multiplier or Simply as
Cockroft Walton Generator.
700,000 Volts produced
Proton accelerated through this voltage gained a K.E of
0.7MeV
The intensity of accelerated proton beam was much
higher than was available from natural Alpha-sources
He
He
H
Li 4
2
4
2
1
1
7
3
61. Cockroft and Walton’s Experiment
P.I.Dee and E.T.S Walton produced the same reaction within
a Cloud Chamber with the help of 0.25 MeV proton beam and
obtained the photograph of the two Alpha-Particles produced
in the reaction.
Both ejected in the opposite direction from the target
Energy of 8.6 MeV each
Q of reaction was found to be 17.33 MeV (precisely)
Mathematically
Momentum of Alpha-Particle of energy 8.6 MeV = 13.5 x 10-20 kg-m/s
Momentum of proton of energy 0.25 MeV = 1.15 x 10-20 kg-m/s
Alpha-Particles have much higher Momenta each than
incident proton
62. Cockroft and Walton’s Experiment
Conservation of Momentum is possible if two alpha-
particles proceed in almost opposite directions
63. Cockroft and Walton’s Experiment
The developed Voltage Multiplier
Can generate voltage up to a maximum of about 106 Volts which
can accelerate Protons to about 1 MeV to about 2 MeV.
Can Disintegrate lighter Nuclei
For Disintegration of Higher beams Higher Energies required
64. Cross Section of Nuclear Reaction
The nuclear cross section of a nucleus is used to
characterize the probability that a nuclear reaction will
occur.
The concept of a nuclear cross section can be quantified
physically in terms of "characteristic area" where a larger
area means a larger probability of interaction.
The standard unit for measuring a nuclear cross section
(denoted as σ) is the barn, which is equal to 10−28 m² or
10−24 cm²
Cross sections can be measured for all possible interaction
processes together, in which case they are called Total
Cross Sections.
65. Cross Section of Nuclear Reaction
If a Parallel beam of N Projectiles is incident
in a given interval of time upon a target foil T
of thickness x and surface area S normally,
then the number of nuclei in T undergoing
transformation due to the reaction of the type
under consideration is proportional to the
intensity of the incident beam of projectiles
and to the total number of target nuclei
present in the foil.
Most reactions Cross Section is a few barns or
even less
For special reactions (n, ) it can be several
thousands barn
66. Partial Cross Sections
When a Nuclear Projectile x is absorbed by a target
nucleus a very short lived compound nucleus is
formed which can break up by the emission of different
types of nuclear particles (y), leaving a different residual
nucleus (Y)
May have elastic and inelastic scatterings
Each of these different reactions induced by the same
projectile x in the same target nucleus X has a different
cross section
In addition to these cross sections, the Total cross
sections can be written as t = sc + r
The cross sections for the individual types of reactions are
know as Partial Cross Sections.
X
A
Z
67. Partial Cross Sections
r sum of partial cross sections for all non elastic
processes including inelastic scattering
sc reaction cross section/ reaction channels
X + x Entrance Channel (Pair of Product Nuclei, each in
definite quantum state)
Y + y Exit Channel
For Elastic Scattering Entrance Channel = Exit Channel
y
Y
C
x
X
68. Reaction Induced by Alpha-Particles
When a nuclear reaction is induced by an alpha-particle
the compound nucleus may break up by the emission of a
Proton, a Neutron, a -ray photon etc.
(, p) reaction Rutherford's Transmutation
Endoergic reaction
Can be Exoergic for some lighter nuclei
69. Reaction Induced by Alpha-Particles
(, n) reaction
Endoergic reaction
Can be Exoergic for some lighter nuclei
70. Reaction Induced by Alpha-Particles
Example: (, n) reaction
An alpha-particle with K.E “K” initiates a nuclear reaction
with energy yield “Q”. Find the K.E if the neutron outgoing at right
angle to the motion direction of alpha-particles. Masses of alpha-
particles, neutron and Carbon are m, mn and mc respectively.
C
n
Be 12
9
)
,
(
71. Reaction Induced by Alpha-Particles
(, ) reaction radiative capture of an alpha-particle.
Exoergic reactions
72. Discovery of Induced Radioactivity
Curie and Joliot showed that when lighter elements such
as boron and aluminum were bombarded with α-particles,
there was a continuous emission of radioactive radiations,
even after the α−source had been removed.
They showed that the radiation was due to the emission of
a particle carrying one unit positive charge with mass
equal to that of an electron.
They bombarded an aluminum foil with alpha-particles
from a naturally radio active substance (Polonium) and
observed the emission on neutrons in the process.
Also positrons are emitted and their emission continued
even bombardment is stopped.
73. Proton Induced Reactions
When a high energy proton falls on the target nucleus the
compound nucleus may break up by the emission of a
Proton, a Neutron, a -ray photon etc.
(p, ) reaction Cockroft and Walton’s experiment
80. Hamiltonian
In quantum mechanics, the Hamiltonian is the operator
corresponding to the total energy of the system in most of
the cases.
It is usually denoted by H, also Ȟ or Ĥ.
Its spectrum is the set of possible outcomes when one
measures the total energy of a system.
Because of its close relation to the time-evolution of a
system, it is of fundamental importance in most
formulations of quantum theory.
81. Perturbation theory
In quantum mechanics, perturbation theory is a set of
approximation schemes directly related to mathematical
perturbation for describing a complicated quantum system
in terms of a simpler one.
The idea is to start with a simple system for which a
mathematical solution is known, and add an additional
"perturbing" Hamiltonian representing a weak disturbance
to the system.
If the disturbance is not too large, the various physical
quantities associated with the perturbed system (e.g. its
energy levels and eigen states) can be expressed as
"corrections" to those of the simple system.
82. Perturbation theory
These corrections, being small compared to the size of
the quantities themselves, can be calculated using
approximate methods.
The complicated system can therefore be studied based
on knowledge of the simpler one.
Perturbation theory is applicable if the problem at hand
cannot be solved exactly, but can be formulated by adding
a "small" term to the mathematical description of the
exactly solvable problem.
83. Cross Section of Nuclear Reaction
X(x,y)Y Transition from (X + x) (Y + y) as transition
By Dirac’s Time Dependent Perturbation Theory
Where matrix element of the transition from initial
state i to the final state f
is the density of the final states
)
(
2 2
'
E
H fi
'
fi
H
)
(E
84. Cross Section of Nuclear Reaction
The transition considered here takes place from a definite
initial state (X + x) to different possible final states (Y + y)
corresponding to the different energies of the emitted
particle y and different energy states in particle y is
produced in its ground state.
Different possible final states in the energy-range E to E +
E.
If n be the number of Levels in this Energy-Range …
(E) = dn / dE
Particles taking part in the reaction are considered free before
and after the collision (‘coz of short range character of Nuclear
Force)
85. Cross Section of Nuclear Reaction
Interaction comes into play only at the time of actual
contact between the particles
Considering the reaction to take place within a closed
volume , (E) can be calculated by computing element
of Volume in the phase space available for the particles
having momenta lying between py and (py + dpy)
The number of states in this phase volume is
Where is the volume of each cell in the phase
space.
3
2
3
2
2
4
4
y
y
y
y dp
p
h
dp
p
dn
3
2
h
86. Cross Section of Nuclear Reaction
Assumption is Residual Nucleus Y is heavy compared to the
emitted particle y so that the momentum py and the
velocity vy can be measured in the L-Frame.
The above expression for dn must be multiplied by the
statistical factor in the final states caused by the spin
orientations given by (2 Iy + 1) x (2 IY + 1) where Iy and IY
are the spins of emitted particle y and nucleus Y
respectively.
If y Photon its statistical factor is 2 because of the
two possible independent polarizations of the photons
Thus dE = vy dpy
So
87. Cross Section of Nuclear Reaction
1
2
1
2
2
1
2
1
2
2
4
)
(
3
2
2
3
2
Y
y
y
y
Y
y
y
y
y
y
I
I
v
p
I
I
dp
v
dp
p
dE
dn
E
1
2
1
2
4
2
2
'
Y
y
y
y
fi I
I
v
p
H
88. Cross Section of Nuclear Reaction
The rate of transformation can can also be deduced
from the basic definition of the reaction cross section XY
The incident particle flux which is equal to the number of
particles incident per unit area of the target foil per
second is equal to nxvx where nx is the number of particles
x per unit volume in the incident beam and vx is the
velocity of these particles relative to X.
Rate of transition per second per target nucleus X is
= XY nxvx
Taking Unit volume Enclosure nx = 1 = XY vx
89. Cross Section of Nuclear Reaction
The Matrix element is usually unknown since the nature of
the interaction between the interacting particles is not
known
If H’ is time dependent perturbation causing the transition
1
2
1
2
1
2
1
2
4
2
2
'
4
2
2
'
Y
y
x
y
y
YX
XY
Y
y
y
y
YX
x
XY
I
I
v
v
p
H
I
I
v
p
H
v
d
H
H X
Y
XY
'
*
'
90. Cross Section of Nuclear Reaction
Where ’s are time-dependent wave functions for the
initial and final states respectively.
H’ is short range strong interaction potential which is zero
outside the nucleus
Wave functions are normalized in unit volume
H’
XY is finite when wave functions are normalized else it
will be ZERO
91. Cross Section of Nuclear Reaction
IF particles are charged, they have to penetrate through
the Coulomb Potential Barrier surrounding the nuclei to
enter into and come out of X and Y respectively.
Barrier penetration probabilities are
Ux and Uy are coulomb barriers
dr
E
U
M
G
dr
E
U
M
G
y
y
y
y
x
x
x
x
2
2
2
2
92. Elastic Scattering of a Neutron
Since Neutron is Uncharged not to bother about
Coulomb Barrier Penetration
It enters into the target nucleus with a definite energy
and angular momentum to form the compound nucleus C*
and then comes out of the latter with the same energy
and angular momentum in different directions.
vx = vy = vn
|H’XY| Constant for Low Energy Neutrons
el constant independent of energy at lower
energies
t
cons
M
v
p
v
v
p
n
n
n
y
x
y
tan
2
2
2
2
93. Elastic Scattering of a Neutron
It is called Compound Elastic Scattering
The incident neutron may be scattered in different
directions by the short range potential of the nucleus
without actually entering into the nucleus
This is known as Potential Scattering
94. Exoergic Reactions by Low Energy Neutrons
(n, ), (n,p), (n, ) and (n, f)
The Q of such reactions may be of the order to few MeV
If the incident neutron energy En is low (~ 1 eV)
variation of En has practically no influence on the velocity
vy of the emitted particle
1 /vn Law
n
n
y
y
y
n
y
v
v
v
M
v
v
p 1
2
2
n
v
1
95. Inelastic Scattering of Neutrons
Neutron comes out the compound Nucleus with a lower
Energy compared to the incident energy
The residual nucleus X is left in an excited state X*
X(n, n’)X*
The reaction is endoergic and Q is numerically equal to
the excitation energy of X*
The minimum energy of the incident neutron = threshold
energy Eth
For Neutron energy slightly above Eth the incident neutron
energy En ~ constant vx = vn = constant
E’n = excess of the incident neutron energy En over the
threshold energy E’n = Eex = En - Eth
96. Inelastic Scattering of Neutrons
E’n = excess of the incident neutron energy En over the
threshold energy E’n = Eex = En – Eth
Thus
So we can write
Cross section thus increases rapidly at first just above the
threshold and then more slowly
th
n
n
y E
E
v
v
'
'
th
n
n
n
n
n
n
n
y
n
y
E
E
v
v
M
v
v
p
v
v
p
'
2
'
2
2
th
n E
E
n
n
)
,
( '
97. Endoergic reactions induced by Neutron
If emitted particle is charged Barrier Penetration factor
exp(-Gy) has also to be taken into account
The cross section increases with increasing En at first
slowly above Eth
(n, ) and (n,p) type of reactions
)
exp(
)
,
( '
y
th
n G
E
E
n
n
98. Reactions induced by charged Particle
Emitting Neutrons
If Ex is small compared to Q
Cross section increases slowly with increasing Ex and than
more rapidly
)
exp(
1
y
n
G
v
99. Principle of Detailed Balance
X + x ⇌ Y* +y
Let X + x Initial State X
Y + y Final State Y
The inverse process in the reaction initiated by the
interaction between the nuclei Y and y in exactly the
same states in which they had been produced in the
forward process yielding as products X and x also exactly
in the same states as in the forward case
C* is formed in the same state in both processes
Cross section of Inverse process
100. Principle of Detailed Balance
Cross section of Inverse process
Matrix element from X Y
1
2
1
2
4
2
2
'
X
x
x
y
x
XY
YX I
I
v
v
p
H
1
2
1
2
1
2
1
2
2
2
2
'
2
'
Y
y
X
x
y
x
XY
YX
YX
XY
I
I
I
I
p
p
H
H
101. Principle of Detailed Balance
Connecting the cross sections for the forward and inverse
processes is known as the Principle of Detailed Balance.
It plays very fundamental role in statistical theory
inherent in the mathematical structure of the Quantum
mechanics and follows from the symmetry with respect to
the reversal of the sense of time.
102. Principle of Detailed Balance
Close volume
Number of states of X within the energy range E which
are actually occupied be Nx and of Y be NY
if the number of possible states of the two types in the
energy range E at E be (Nm )x and (Nm )Y
Because of the equal probability of occupation of each
state
X + x ⇌ Y + y
Y
m
X
m
Y
X
N
N
N
N
103. Principle of Detailed Balance
E
v
P
p
N
x
x
x
X
m
3
2
2
4
E
v
P
p
N
y
Y
y
Y
m
3
2
2
4
x
y
y
x
Y
X
v
p
v
p
N
N
2
2
y
YX
Y
YX
x
XY
X
XY
v
N
v
N
)
1
2
)(
1
2
(
)
1
2
)(
1
2
( 2
2
2
2
2
2
Y
y
y
YX
X
x
x
XY
y
YX
x
XY
YX
YX
x
XY
y
YX
X
Y
YX
YX
I
I
p
I
I
p
p
p
p
p
N
N
104. Principle of Detailed Balance
The last equation is known as the reciprocity relation.
)
1
2
)(
1
2
(
)
1
2
)(
1
2
( 2
2
Y
y
y
YX
X
x
x
XY I
I
p
I
I
p
105. Partial Wave Analysis
Partial wave analysis, in the context of quantum
mechanics, refers to a technique for solving scattering
problems by decomposing each wave into its constituent
angular momentum components and solving using
boundary conditions.
106. Compound Nucleus Hypothesis
For high energy Neutrons the total cross section for neutron
absorption and scattering was of the order of R2 where R
is the nuclear radius
For higher energies Cross Section is higher limiting
value of 2 where is reduced de Broglie wavelength of
neutrons
Assumption: incident neutron moved in an average potential
well due to all the nucleons in the nucleus for an interval of
time which is of the order of
t ~ Nuclear diameter / Nuclear velocity = 10-14 / 107 = 10-21 s
greater probability of escaping from nucleus without
absorption elastic scattering cross section sc large &
Capture or reaction cross section re low
107. Compound Nucleus Hypothesis
At low energies re large (as neutron would spend a
longer time near the target nucleus and re depends on 1/v
At thermal energies both sc & re should be comparable
since both of these approach the limiting value of 2
The energy levels in this single particle potential well
should be well separated from one another of the order of 5
to 10 MeV.
There would also be some levels above the neutron
separation energy Sn (virtual levels)
The width of the level which measures the probability of its
decay would be around 1 MeV broad levels
108. Compound Nucleus Hypothesis
When the energy of the incident Neutron corresponds to the
Excitation energy of one of the Nuclear Levels, resonances
would be expected to be observed in the cross section vs.
energy graph.
These resonances should be widely spaced having large
widths
A very low energies no such resonances would be expected
since neutrons of a few electron volts energy cannot be
expected to produce resonances corresponding to levels
with gaps of several million electron volts.
In many cases the neutron absorption cross sections are
found to be very large at thermal neutron energies while
the elastic scattering cross sections are much lower
109. Compound Nucleus Hypothesis
At these very low energies, the absorption cross section is
due to the radioactive capture (n, ) type of reactions.
Resonances are observed in most nuclei for both elastic
scattering cross section and radioactive capture cross
section
These resonances mostly appear at neutron energies
between 0.1 to 20 eV closely spaced
They are also found to be sharp (widths of the order of 0.1
eV or lower)
110. Compound Nucleus Hypothesis
Devised by Neil Bohr mechanism of Nuclear Reaction
X + x C* Y + y
The incoming Projectile x quickly dissipates its energy as it
enters into the nucleus X and merges with the closely
packed nucleons in it a general random motion of all the
nucleons in the nucleus is disturbed (each electron gaining
some additional energy but not enough to escape from the
nucleus) after a relatively long time a very large
number of collisions among the nucleons have taken place
(of the order of 10 Million) enough energy may be
concentrated on one of the nucleons enabling it to escape
from the nucleus which than de-excites to ground state
can also be de-excites by emitting -rays.
111. Compound Nucleus Hypothesis
The process if emission of a nucleon (or a group of
nucleons) from the excited nucleus is thus similar to the
phenomenon of evaporation,
Analogy led to Liquid Drop Model of Nucleus
The composite system that is formed as a result of the
absorption of the incident particle x by the nucleus X is
known as the Compound Nucleus.
The mean time between Collisions is about 10-22 s
Life of compound Nucleus 10-15 s
112. Compound Nucleus Hypothesis
Inelastic Scattering
If x and y are same Y = X* target nucleus produced in a
different energy state
Compound Elastic or Resonance Scattering
Rare case
Target nucleus is same as residual
States are same
Potential Scattering
Elastic scattering may alternatively take place by the action of the
Nuclear potential on the incident particle without the entry of x into
X to form the compound nucleus.
Greater probability to occur
113. Compound Nucleus Hypothesis
Steps of Nuclear Reaction
Formation of Compound Nucleus by absorption
Disintegration of the compound nucleus in a manner which is
independent of the method of its formation into the reaction
products
Residual nucleus Y may be left in a highly excited state which may
then boil off another particle y’ to leave the nucleus Y’ leading to a
two particle emission process X(x, y’)Y’
The process may continue and another particle y” may be emitted
from the excited Y’, leaving the residual nucleus Y” in the third
stage
X + x C* Y + y C +
114. Compound Nucleus Hypothesis
The probability of decay is equal to the reciprocal of the
mean-life of the compound nucleus
The width of the level is a measure of the probability of its
decay
In the form of cross section
only one particular energy state of C* is being considered
i.e. only ONE resonance
Possible when levels are well separated and are so sharp
that they do not interfere with one another
/
)
,
( x
x
y
x
115. Discrete Levels of Compound Nucleus
Breit-Weigner one level formula
This is the Breit-Wigner formula for a single level reaction
cross-section.
For example - in the case of elastic scattering (a = b)
at the maximum of the resonance (E = E0) the cross-
section is
2
2
0
2
)
2
/
(
)
(
)
1
2
(
4
1
E
E
b
a
b
a
p
h/
where
116. Discrete Levels of Compound Nucleus
Breit-Weigner one level formula
2
2
2
)
1
2
(
a
elastic
And if no other processes to compete with (a=)
/
)
1
2
(
2
elastic
the maximum possible elastic cross-section.
117. Effect of Barrier Penetration on Reaction
Cross Section
All particles have to penetrate through the Potential
Barrier surrounding the nucleus to initiate a nuclear
reaction (except for very low energy neutrons)
Two Types of Barrier to be Penetrated
Centrifugal Barrier
Coulomb Barrier
Centrifugal Barrier
Due to orbital angular lh momentum of the incident particle x
relative to the target nucleus X when l > 0
2
2
1 2
)
1
( Mr
l
l
V
118. Effect of Barrier Penetration on Reaction
Cross Section
Coulomb Barrier
is the energy barrier due to electrostatic interaction that two
nuclei need to overcome so they can get close enough to undergo
a nuclear reaction.
This energy barrier is given by the electrostatic potential energy:
119. Effect of Barrier Penetration on Reaction
Cross Section
Radial wave function obeys the wave equation outside the
nucleus
In case of wave function inside the nucleus, an attractive short
range nuclear potential (-Vn) in place of Coulumb’s repulsive
potential will be added
For slow neutrons (l=0) both potential terms of above equation
Zero
Solution of above wave equation may be taken to be
superposition of two waves exp(ikr) and exp(-ikr) which
respectively represents a wave falling on the nuclear surface
and a wave reflected from it for r > R
0
1
2
4
2
2
2
0
2
'
2
2
2
l
l
u
r
l
l
M
r
e
ZZ
E
M
dr
u
d
n
120. Effect of Barrier Penetration on Reaction
Cross Section
As particle enters the nucleus the wave number (spatial
frequency of wave i.e. either in cycles per unit distance
or radians per unit distance) is suddenly changed to high
value because of nuclear potential Vn
Solution Ul(r) = C exp (iKr)
Where
Is inside wave number: K >> k
ikr
ikr
l Be
Ae
r
u
n
V
E
M
K
2
2
121. Effect of Barrier Penetration on Reaction
Cross Section
Wave should be partially reflected and partially
transmitted because of a sudden discontinuity.
For a step function type of one dimensional discontinuity,
we get Transmission coefficient T = 4Kk / (K + k)2
If incident particle doesn’t experience any repulsion in
entering the nucleus (zero energy neutron) all particles
intercepted by nucleus may taken to be absorbed in the
black nucleus approximation.
122. Effect of Barrier Penetration on Reaction
Cross Section
Coulomb Barrier
A positive value of U is due to a repulsive force, so interacting
particles are at higher energy levels as they get closer.
A negative potential energy indicates a bound state (due to an
attractive force).
The Coulomb barrier increases with the atomic numbers (i.e. the
number of protons) of the colliding nuclei:
where e is the elementary charge, 1.602 176 53×10−19 C, and Zi the
corresponding atomic numbers.
To overcome this barrier, nuclei have to collide at high velocities,
so their kinetic energies drive them close enough for the strong
interaction to take place and bind them together.
123. Overlapping Levels; Statistical Models
One feature of the compound nucleus model is the existence of
the resonances on the cross section when (Level-Width) << D
(Separation between the levels)
These resonances appear at very low incident energies which
indicates that the isolated levels exist at excitation energies
close to the separation energy of the neutron (Sn) from the
compound nucleus.
The total energy (En + Sn) brought in by the incident particle
may be distributed in a very large number of ways amongst the
nucleons in the nucleus, till enough energy may be
concentrated on a particular nucleon/nucleon group which
would enable it to come out.
But it might not happen ‘coz of the existence of a sharp
potential discontinuity at the nuclear surface, which causes
only a part of the incident wave to be transmitted through the
latter into the nucleus.
124. Overlapping Levels; Statistical Models
The particular configuration in which a given particle may
be able to come out with a given energy will repeat itself
many times with an average period p given by p ~ /D
where D of the mean level spacing of the compound
nucleus
After many such periods, the particle finally escapes after
a time given by = /
Since >> p & << D is the condition for the formation of
the resonance level
126. Overlapping Levels; Statistical Models
Figure a Amplitude of the wave function outside is
much larger than inside small absorption cross section
Figure c two waves have equal amplitudes En = Er
As energy of the incident particle increases the difference
between k and K decreases and hence the number of
attempts required to escape from the nucleus by a
particle becomes smaller
This makes smaller becomes larger
The compound nucleus can decay through many exit
channels corresponding to different types of Nuclei y and
Y in the final state and the different possible energy
states in which they may be produce.
127. Overlapping Levels;
Statistical Models
Here are plotted
the yields of
decay products from
the compound nucleus
64Zn
formed by
2 different routes
Note:
the relative cross-sections
for the different processes
63Cu(p,n)63Zn
and
60Ni(,n)63Zn
remain ~constant
across the
plotted energies.
128. Disintegration of the Compound Nucleus
When the excitation energy EC of the compound nucleus is
high, many exit channels are available for the
disintegration.
This is not only due to the possibility of the emission of
different types of particles e.g.
C* Y1 + y1
Y2 + y2
Y3 + y3 etc.
But also due to the emissions of any one type with
different possible energies.
Thus if C* breaks up into Y + y, these may be emitted with
different kinetic energies y leaving Y in different possible
states of excitations Ey
130. Nuclear Reaction Mechanisms in Different
Stages (by V.F. Weisskopf)
When the projectile is incident on the nucleus, it first feels the
presence of the complex optical potential, which thus
represents the first stage known as independent particle stage.
The real part of the potential V leads to shape elastic
scattering.
A part of the incident wave is absorbed due to the imaginary
part of the potential leading either to direct interaction or to
the formation of compound nucleus which corresponds to the
establishment of the statistical equilibrium of the energy
imported into the nucleus.
In between there is a region known as pre-equilibrium stage in
which a limited number of collisions with the nucleon or
nucleon groups within the nucleus takes place before statistical
equilibrium is reached.
131. Continued.
There may also be collective effects in which excitation of
some type of collective motion may occur.
This is second stage known as compound system stage.
In Final stage, production of the reaction products which
proceeds n a very short time scale is described.
The two extremes of this stage represent the direct reaction in
which a nucleon or a nucleon group comes out by the impact of
the incident nucleon, while the compound nucleon formation is
followed by the concentration of the imported energy on a
nucleon group enabling it to come out.
In Pre-equilibrium reaction, a few nucleons may come out
successively, if enough energy is received by these for emission.
The different mechanisms can be recognized by the nature of
energy and angular distributions of the emitted particles.
133. Measurement of Nuclear Reaction Life
Times
Decay of compound nucleus 10-15 seconds
Direct reactions 10-21 seconds 10-22 seconds
Technique known as Shadow Effect Method
If a crystal is exposed to a beam of Projectiles from an
accelerator, the decay products of the nuclei in the lattice sites
undergoing nuclear transformation are prevented from escaping in
all directions.
Because of the much larger concentration of the atoms in the
lattice along the crystal axes, these directions are forbidden for
the decay products.
If a photographic plate is placed behind the crystal, a pattern of
shadows will be observed which is a characteristic of the crystal.
These shadows will be formed only if the nucleus does not have
time to leave the lattice site while emitting its decay products,
which is true if it is a case of direct reaction.
134. Measurement of Nuclear Reaction Life
Times
Shadow Effect Method
In case of compound nucleus, it moves away from the lattice site
before emitting the decay products.
Hence no shadow will be formed in this case
Since compound nucleus life time is 10-15 to 10-14 s and velocity 106
m/s means it moves away from the lattice site through a distance
10-9 to 10-8 m which is much larger than the lattice dimension 10-10
m.
135. Nuclear Reactions Induced by Heavy Ions
Heavy Ions (HI) are projectiles with A > 4 (heavier than
alpha particles)
Energies 1 to 10 MeV
The products of HI induced reactions are in many cases
very short lived radioactive isotopes
Main features
Large momentum transfer
Large angular momentum transfer
Exchange of large number of nucleons
Special type of EM interactions like multiple Coulomb Excitations
etc.
136. Wave Function ………. (aside)
A wave function in quantum mechanics is a description of
the quantum state of a system.
The wave function is a complex-valued probability
amplitude, and the probabilities for the possible results of
measurements made on the system can be derived from
it.
137. Coulomb Potential Barrier
is the energy barrier due to electrostatic interaction that
two nuclei need to overcome so they can get close enough
to undergo a nuclear reaction.
138. Threshold Energy
Threshold energy for production of a particle is the
minimum kinetic energy a pair of traveling particles must
have when they collide.
The threshold energy is always greater than or equal to
the rest energy of the desired particle.
139. Electron Volt
In physics, the electronvolt (symbol eV, also
written electron volt) is a unit of energy equal to
approximately 160 zeptojoules (10−21 joules, symbol zJ)
or 1.6×10−19 joules(symbol J).
By definition, it is the amount of energy gained (or lost)
by the charge of a single electron moving across
an electric potential difference of one volt.
Thus it is 1 volt (1 joule per coulomb, 1 J/C) multiplied by
the elementary charge (e, or 1.6021766208(98)×10−19 C).
Therefore, one electronvolt is equal
to 1.6021766208(98)×10−19 J.
140. Potential Well
A region in a field of force, in particular the region in
which an atomic nucleus is situated, in which the
potential is significantly lower than at points immediately
outside it, so that a particle in it is likely to remain there
unless it gains a relatively large amount of energy.
141. Radial Wave Function
An orbital is a mathematical function called a wave
function that describes an electron in an atom.
The wave functions, ψ, of the atomic orbitals can be
expressed as the product of a radial wave function, R and
an angular wave function, Y.
Radial wave functions for a given atom depend only upon
the distance, r from the nucleus.
Angular wave functions depend only upon direction, and,
in effect, describe the shapeof an orbital.
ψ = radial function × angular function
= R × Y
142. Complex Optical Poetential
Heavy ion scattering has offered us several interesting
speculations about the resonances of composite systems.
Such systems possess so many degrees of freedom that the
scattering process becomes very complicated to describe.
The usual prescription is to use a complex potential to describe
the elastic scattering, where the imaginary part is used to lump
in all the degrees of freedom that we are unable to describe.
Because of the strong absorption, the ambiguity in the potential
is a serious problem as the elastic scattering data is usually
sensitive to only two out of the four or more parameters used
to characterize the optical potentials.
On the other hand, for lighter ions (A<40), the imaginary part of
the potential is not very large.
In principle, for this case one should be able to determine a
reasonable potential which can fit the scattering data well.
143. De Broglie wavelength
The de Broglie wavelength is the wavelength, λ,
associated with a massive particle and is related to its
momentum, p, through the Planck constant, h:
= h / p
144. The process by which a beam of radiant electromagnetic
energy is lined up to minimize divergence or convergence.
Ideally, a collimated beam is a bundle of parallel rays
perfectly lined up along an optical line-of-sight (LOS)
between a transmitter and receiver, perhaps through, and
in perfect parallel with, a waveguide.
145. Gas-discharge Ion Sources
Gas-discharge lamps are a family of artificial light sources that
generate light by sending an electrical discharge through an ionized
gas, a plasma.
Typically, such lamps use a noble gas (argon, neon, krypton, and
xenon) or a mixture of these gases.
Some include additional substances, like mercury, sodium, and metal
halides, which are vaporized during startup to become part of the gas
mixture.
In operation some of the atoms in the gas are positively ionized,
losing one of their electrons. The ions are accelerated by the electric
field toward whichever electrode is negatively charged at the time.
Typically, after traveling a very short distance the ions collide with
neutral gas atoms, which transfer their electrons to the ions.
The atoms, having lost an electron during the collisions, ionize and
speed toward the electrode while the ions, having gained an electron
during the collisions, return to a lower energy state, emitting a
photon of light of a characteristic frequency.
In this way, electrons are relayed through the gas from the negative
electrode to the positive.
