Define Nuclear Reaction
Reactions between an atomic nucleus and another particle are called nuclear reactions.
A typical nuclear reaction involves two reacting particles—a heavy target nucleus and a light bombarding particle—and produces two new particles— heavier product nucleus and lighter ejected particle.
X + a ……………> Y + b ……………….(1)
The notation for a nuclear reactions is
Target(projectile, light products)heavy product(s) e.g. X(a,b)Y
Q Value in Nuclear Reaction
To complete the nuclear reaction, it must include the change in energy of the reaction also. This change in energy in a nuclear reaction is represented by Q. The complete representation of a typical nuclear reaction is-
X + a ……………> Y + b + Q ……………….(1)
Q = (Sum of the masses of reactants – Sum of the masses of products) X 931.5 MeV
Q = +Ve ; Exoergic (energy is released)
Q = -Ve ; Endoergic (energy is absorbed)
For example, for the 208Pb(7Li, γ)215At reaction, Q=- 5.589 MeV.
So, at least 5.589 MeV of energy must be converted into mass in order for the reaction to occur.
Types of Nuclear Reactions
When the target nuclei are bombarded by particles, there are some general types of interactions. In some such reactions, new nuclei are formed (nuclear transmutations); in others the original nucleus is excited to a higher energy state (inelastic scattering); in a third class, the nucleus is unchanged (elastic scattering).
Elastic Scattering
When no energy is transferred between the target nucleus and the incident particle, the process is known as elastic scattering
208Pb (n, n) 208Pb, Q = 0.
Inelastic Scattering
When energy is transferred, the process is called inelastic scattering
40Ca (a, a’) 40mCa, Q =\= 0.
where a and a’ have different kinetic energies .In cases when the incident particle is a complicated nuclide, it may also be left in excited state,
208Pb (12C, 12mC) 208mPb
This process is called mutual excitation.
Photonuclear Reactions
The absorption of gamma ray photons by atomic nuclei and the accompanying ejection
of protons p,neutrons n, or heavier particles from the nuclei. The (γ, p) and (γ, n) photonuclear reactions have been studied the most; other reactions,such as (γ, d), and (γ, t) are also known.
Fission Reactions
In nuclear fission the nucleus of an atom breaks up into two lighter nuclei producing quite a few neutrons and tremendous amount of energy.
Well-known neutron-induced fission reaction is-235U (n, 3 n) 9038Sr,14354Xe
Fusion Reactions
A process by which nuclear reactions between light elements form heavier elements (up to iron). The fusion reaction of deuterium and tritium is particularly interesting because of its potential of providing energy for the future.
T (D, n) He The vast energy potential of nuclear fusion was first exploited in thermonuclear weapons, or hydrogen bombs, which were developed in the decade immediately following World War II.
Capture Reactions
Both charged and neutral particles can be captured by nuclei inducing the emission of electromagnetic radiation, as a gamma ray.
. For example,
197Au (p, γ) 198Hg , 238U (n, γ) 239U
Neutron capture reactions are used to produce many radioactive nuclides.
Nuclear Reaction Energetics
Two quantities are important energetic considerations in evaluating a nuclear reaction:
(1) The threshold energy (Eth): The minimum projectile energy necessary to satisfy mass-energy and momentum conservation in a nuclear reaction to form products in their
ground state.
(2) The excitation energy (E*): The excess energy above the ground state for the product of a nuclear reaction.
In order to derive these quantities, mass-energy must first be conserved.
In case of endoergic reaction, the energy necessary to compensate for a negative Q-value is not sufficient for a nuclear reaction to occur. Because, all the energy of the incoming particle is not effectively used in bringing about the reaction. Some of the incident particle energy is used up in conserving momentum of the system, and over coming the repulsive Coulomb barrier that exist between the incident charged particle nucleus.
To determine the minimum incident energy required to initiate an endoergic reaction two corrections should be in consideration:
1.Momentum Correction
2.Coulomb Barrier Correction
Tunneling in Nuclear Reaction
Experimentally, it is found that nuclear reactions sometimes occur at energies less than that required by the Coulomb barrier through quantum-mechanical tunneling. This behavior is related to the wave mechanical nature of the particles involved in a nuclear reaction.
As a projectile approaches a target nucleus in a nuclear reaction, the probability that there will be overlap and hence interaction in their wave functions increases. This concept was used to explain the emission of α-particles with energies less than that required by the Coulomb barrier height. Such tunneling may also occur for projectiles approaching the nucleus from the outside. An example is provided by the reaction of protons with lithium
For this reaction the threshold energy is 1.3 MeV. However, due to tunneling the reactions begin to occur at lower proton energies. At an energy of 0.15 MeV about 0.1% of the protons penetrate the Coulomb barrier, at 0.3 MeV about 1%, and at 0.6 MeV about 20%.
Nuclear Reactions Cross Section
The probability of a certain nuclear reaction will take place depends effective size of the nucleus for that reaction. It is the probability of interaction between the target nucleus and incident panicle or a photon
The probability of reaction with a given projectile could is related to the size of the target nucleus, that is, the cross-sectional area presented by target nuclei to incident particles.
If R is the radius of the of the target nucleus then the reaction cross section-
σ = πR2
The cross-section for most nuclei, calculated in this simple way, could be on the order of 10-24 cm2 . The commonly used unit is barn (b) and
1 barn = 10-28 m2
Total Cross Section: In term of area, the total cross-section is the sum of the cross-sections due to absorption, scattering and luminescence.
σ = σA + σs + σL
Measurement of Nuclear Cross Section
Nuclear Cross section depends on number and identity of incident particles and the individuality of target nucleus.
Calculation method depends on relative thickness of the target and its effect on the intensity of the incident particles.
Measurement of Nuclear Cross Section For Thin Targets
Targets in which the incoming particles experiences no significant attenuation is categorized as thin target. If a beam of particles incident on the target, the number of nuclear reactions induced in the target per unit time (R) is given by-
R = Inxσ
Where, R = Reaction rate
I = flux = incident particles/s
x= target thickness (cm)
n= number of target nuclei/cm3
σ = cross section (cm2 )
Measurement of Nuclear Cross Section For Thick Targets
In case of thick targets incoming particles experiences significant attenuation (decrease in intensity) and this amount depends on i.Thickness of the target ii.Number of target nuclei iii.Intensity of incoming particles beam
The attenuation in a target is given by-
-dI = Inσ(dx)
Where, -dI = differential attenuation of the beam intensity
I = beam intensity = incident particles/s
x= distance into the target (cm)
n= number of target nuclei/cm3
σ = cross section (cm2 )
Excitation Functions in Nuclear Reaction
The reaction cross-section depends on the projectile energy. A plot of partial reaction cross-section as a function of projectile energy are known as excitation functions or excitation curves.
Neutron Excitation Function:
The figure shows that reaction cross section decreases with increasing neutron energy. Consequently means that neutrons with low energy have greater probability of inducing a nuclear reaction.
(i) For the low energy neutrons, nuclear cross section decreases with increasing energy and it is roughly proportional to 1/ν, where ν is the velocity of the neutron. This is referred to as the one-over-v law. On the other hand thermal neutrons have high capture cross section
(ii) The sharp peaks in the figure are known as resonance peaks, which indicates sharp increase in reaction cross section i.e. reaction probablity. These particular energies would correspond to the nuclear energy level spacings in the target nucleus that are being excited by the incoming neutron.
Charged Particle Excitation Functions:
In general, charged particle reactions are threshold reactions. As a result charged-particle cross-sections increase with increasing energy until it reaches maximum. These maxima are reached just before the threshold energy for a competing reaction channel is reached. After a maximum value is reached, the cross section
for the reaction with the lower threshold energy will decrease, and that for the next higher threshold reaction will increase until it, too, reaches a maximum. Hence the excitation function for a charged-particle reaction will show a series of curves, one for each different type of reaction that might occur. Cross sections for charged-particle reactions are usually much lower than those for neutron-induced reactions.
Figure: Excitation functions for reactions between 4He ions and 54Fe target nuclei.