Physics NUCLEUS FUSION AND FISSION

Binding Energy

It is found that the rest mass of the stable nucleus of a stable atom is always less than the sum of the masses of constituent nucleons. The difference is called the `text(mass defect)` `D_m`. The energy equivalent of 0 m (i.e., `D_mc^2`) is utilized in keeping the nucleons bound together. This energy is known as the `text(binding energy)` In order to break the nucleus into its constituent nucleons, an amount of energy equal to this binding energy has to be supplied to the nucleus. The mass defect per nucleon i.e., (D_m)/A = P, is called the `text(packing fraction)` of the nucleus.
If the mass of the nucleus `text( )_zX^A` is `M` , then the mass defect

`D_m = [Zm_p + (A-Z)m_n -M]`

where `m_p` and `m_n` are the masses of the proton and neutron
respectively

and Binding energy `= D_m . c^2 = [Zm_p + (A- Z)m_n - M]c^2`

The binding energy per nucleon `= B/A = ([Zm_p +(A -Z)m_n-M]c^2)/A`

`A` graph between `B/A` and `A` is given in figure. A study of the curve indicates following points :

(i) Elements having atomic weights between `40` to `120` have roughly a constant value of `B/A` around `8.5` `MeV` . In fact the maximum value of `B/A` is `8.79MeV` in case of `text( )_56Fe`.

(ii) The average value of `B/A` of nuclei having atomic weightless than 40 and greater than 120 is smaller than 8.5 MeV. For heavier nuclei the value of `B/A` decreases to around `7.5 MeV`.

(iii) There exists a cyclic occurrence of maximum value of `B/A` for elements having atomic weight multiple of four in the low atomic weight region.

(iv) The nuclei having more B/ A ratio are more stable.

Figure suggests that energy can be liberated from the nucleus in two different ways. If a heavy nucleus splits into two lighter nuclei, energy can be released because binding energy per nucleon is greater for the two lighter nuclei than it is for the original nucleus. This process is called `text(nuclear fission)`. Similarly two lighter nuclei can combine to form a heavier nucleus releasing energy as the binding energy per nucleon for the heavier nucleus is greater than that of the two lighter nuclei. This process is known as `text(nuclear fusion)` .

`text(Remarks)`

Binding energy of electrons `(eV)` are ignored as it is smaller compared to the rest mass energy of a nucleon `(10^3 MeV).`

Nuclear Reactions

A radioactive substance breaks up by emitting radiation. The daughter nucleus, left behind, has different physical and chemical properties and is assigned a new place in the periodic table. Thus, radioactivity is the phenomenon by which a substance gets converted into another one. This change can be brought about by artificial method, by bombarding a given nucleus with some radiation. The particles constituting the incident radiation must possess sufficient kinetic energy so as to penetrate into the given nucleus. As they enter the given nucleus, a compound nucleus is formed which is generally unstable. The compound nucleus then breaks up to produce nucleus by emitting radiation. The process is, schematically represented as

`I`(Incoming Particle) + T (Target nucleus) `->` C(Compound nucleus) `->` P(Product nucleus) + O(Outgoing radiation).............`(1)`

This is a reaction in which only the nuclei take part. Orbital electrons have no contribution in it. The reaction is known as a nuclear reaction.

When nitrogen is bombarded with a -particle following reaction takes place :

`text( )_2He^4 + text( )_7N^14 -> text( )_9F^(18) -> text( )_8O^(17) + text( )_1H^1`

Thus `text( )_7N^(14)` gets converted into `text( )_8O^(17)` due to bombardment with a- particles.

Laws Governing Nuclear Reactions

Following laws are obeyed in the course of a nuclear reaction :

(i) Law of conservation of charge : The electric charge involved in a nuclear reaction must be same before and after the reaction.

(ii) Law of Conservation of number of nucleon : The total number of nucleons involved in the nuclear reaction must be same before and after the reaction.

(iii) Law of conservation of energy: The total energy (rest + K.E.) of the reacting particles must be equal to the total energy of the product particles.

(iv) Law of conservation of linear momentum : The total linear momenta of the reacting particles must be equal to the total linear momenta of the product particles.

Classification of Nuclear Reactions

Nuclear reactions can be classified into the following categories.

(i) `text(Elastic scattering) :` The incident particle gets deflected without any change in its energy, i.e.,

`text( )_2He^4 + text( )_70Au^(197) -> text( )_70Au^(197) + text( )_2He^4`

The bombarding particle passes sufficiently large distance away from the target nucleus so to as get repulsion which changes its direction of motion without any change in its energy.

(ii) `text(Inelastic Scattering :)` If the bombarding particle passes close to target it gets deflected. Due to strong repulsion the target particle also acquires some energy. So, the energy left with scattered particle is less than that it had initially,

`text( )_1H^1 +text( )_3Li^7 -> text( )_3Li^7 + text( )_1H^1`

means existence of `text( )_3Li^7` in one of its excited states.

(iii) `text(Simple capture)` : The incoming particle is captured by the target nucleus. The product nucleus which is generally in the excited state decays to the ground state by emitting g - ray `text( )_1 H^1 + text( )_6 C^(12) -> text( )_7 N^(13) -> text( )_7 N^(13) +h nu`

(iv) Disintegration : The intermediate compound nucleus breaks up and results in a product nucleus and an outgoing particle. The product nucleus has different chemical properties as compared to the target particle. Majority of nuclear reactions belong to this category.

(a) Disintegration by `alpha` - particles

`text( )_2He^(4) +text( )_5B^(10) -> text( )_7N^(14) -> text( )_6C^(13) +text( )_1H^1`

(b) Disintegration by protons

`text( )_1H^1 + text( )_5B^(12) -> text( )_6N^(13) -> text( )_4Be^(9) + text( )_2He^4`

(c) Disintegration by neutrons

`text( )_0n^1 + text( )_5B^(10) -> text( )_5B^(11) -> text( )_3 Li^7 +text( )_2He^4`

(d) Disintegration by deuterons

`text( )_1H^2 +text( )_5B^(10) -> text( )_6C^(12) -> text( )_2He^4 + text( )_2 He^4 + 2 He^4 `

(e) Photon-disintegration

`g + text( )_4B^9 -> text( )_4Be^9 -> text( )_4Be^8 +text( )_0n^1`

Q-Value of a Nuclear Reaction

Consider a nuclear reaction, schematically, represented by equation .

I(Incoming particle) + T (Target nucleus) `->` C(Compound nucleus) `->` P(Product nucleus) +O(Outgoing radiation)

Let `KE_I , KE_P` and `KE_O` be the kinetic energies associated with `I` , `P` and `O` respectively while the target `T` is at rest initially.

`Q` � Value of a nuclear reaction is given by

`Q = KE_P + KE_O - KE_I`

Let `m_I, m_T , m_P` and `m_O` respectively, be the masses of `'I', 'T' , 'P'` and `'O'`

Before reaction, Energy of `'I' = m_Ic^2 + KE_I`

Energy `T = m_TC^2`

Total energy of the system `= m_Ic^2 + KE_I + m_Tc^2`

After reaction, Energy of `'P' = m_Pc^2 + KE_P`

Energy of `'O' = m_Oc^2 + KE_O`

Total energy of the system `= m_Pc^2 + KE_P + m_Oc^2 + KE_O`

According to the law of conservation of energy

`m_Ic^2 + KE_I + m_Tc^2 = m_Pc^2 + KE_P + m_Oc^2 + KE_O`

`Q = KE_P + KE_O- KE_I = [(m_I + m_T)- (m_P + m_O)]c^2`

or `Q = D_mc^2`

where `' D_m'` is the mass defect between initial and final particles.

`textCase(1)` A reaction is said to be `text(exothermic if Q is positive)`. Q is positive if `(m_P + m_O ) < (m_I + m_T )` The part of mass which disappears gets converted into the energy in accordance with Einstein's mass-energy relations.

`textCase(2)` A reaction is said to be `text(endothermic if Q is negative)` .
`Q` is negative if `(m_P + m_O ) > (m_I + m_T )` i.e., the sum of the masses of product particles is greater than that of reactant particles. For this reaction to proceed, the incoming particle mass must posses kinetic energy ( from centre of mass frame ) equivalent or greater to the mass defect.

`text(NOTE : Depending on how one rakes i.e)`. `D_m =` `text((final - initial)) or text((initial - final)) quadtext(the sign convention of Q would change.)`

The Q - Equation

The analytical relationship between the kinetic energy of the product, outgoing particle and nuclear disintegration energy Q is called as the `text(Q-equation)`.

Let us consider nuclear reaction represented as(fig-a)

`I + T -> P + O + Q.........................(2)`

where I= Incoming particle

T =Target nucleus

P = Product nucleus

O= Outgoing particle

The momentum triangle is indicated below(fig-b)

we get that `P_P^2= P_I^2+P_O^2-2P_IP_O cos theta`

We know that `KE = P^2/(2m) => P^2 = 2mKE`

`=> 2 M_pKE_P = 2M_IKE_I + 2m_OKE_O-2 xx 2 sqrt(m_IKE_Im_OKE_O) costheta`

or `KE_P = (m_IKE_I)/M_P + (m_OKE_O)/M_P - (2sqrt(m_Im_OKE_O))/M_P cos theta.................(3)`

Also, from the definition of Q-value. we have

`Q = KE_P + KE_O-KE_I.......................(4)`

From equation (3) and (4), we get

`Q = KE_O(1+ m_O /M_P) - KE_I(1+m_I/M_P) - (2sqrt(m_Im_OKE_O))/M_P cos theta`

This is known as `text(Q-equation)`

The kinetic energies `KE_I ,KE_O` and `q` are all measured in the laboratory coordinate system. If outgoing particle is scattered at `90�` with respect of the line of motion of incoming particle then `theta= P//2` and Q�value equation reduces to

`Q = KE_O(1+ m_O /M_P) - KE_I(1+m_I/M+P) `

`text(NOTE)` : - Since the Q-equation is based on mass-energy conservation in a nuclear reaction, it holds for all types of reactions.