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**Nuclear Reaction:**

Nuclear reaction is the procedure whereby mass number or atomic number of target nuclei changes due to bombardment with projectile particles resulting in release of energy. Essential things for nuclear reaction are:

i) Target nucleus

ii) Projectile example ^{1}_{0}n, ^{1}_{1}H, ^{4}_{2}He,^{ 2}_{1}H

X + x → Y + y + Q or X(x,y)Y

e.g. ^{14}_{7}N(α, p) ^{17}_{7}O

^{14}_{7}N +^{ 4}_{2}He → ^{17}_{8}O + ^{1}_{1}H + Q

Kinds of Nuclear Reaction

1) F - Fission - X + x → y_{1} + y_{2}

2) I - Inelastic nuclear reaction - X + x → X^{k} +x

3) T - Transmutation - X + x → Y + y

4) E - Elastic nuclear reaction - X + x → X + x

5) C - Capture - X + x → Y^{k}

Physical qualities that are conserved in any nuclear reaction comprise:

1) Total electric change EZ = K

2) Total number of nuclei EA = K

3) Linear momentum EP = K

4) Sum of mass and energy E(mass + Ke)

5) Parity K

**Q - Value Equation:**

This is nuclear change or amount of energy released in the nuclear reaction. For nuclear reaction, the total rest mass and kinetic energy are conserved.

Example X + x → Y + y + Q

i.e [E_{X} + m_{X}C^{2}] + [E_{x} + m_{x}C^{2}]

= [E_{Y} + m_{Y}C^{2}] + [E_{y} + m_{y}C^{2}] + Q

As target nucleus X is at rest, then equation turns to

[M_{x}C^{2}] + [E_{x} + m_{x}C^{2}] = [E_{y} + M_{y}C^{2}] + [E_{y} + M_{y}C^{2}] + Q

But Q = change in energy

Hence Q = E_{x} - (E_{Y} + E_{y})

Q = [(M_{X} + m_{x}) - (M_{Y} + M_{y})]C^{2}

Q = Δmc^{2}

**Conventional Q-value Equation:**

Applying principle of conservation of momentum

M_{x}V_{x} = M_{Y}V_{Y}cosΦ + M_{y}V_{y}cosθ(x-direction)

O = M_{Y}V_{Y}sinΦ + M_{y}V_{y}sinθ(y-direction)

Form E = P^{2}/2m, p^{2} = 2mE

Thus, momentum mv = √2ME

Equation turns into:

(M_{x}E_{x})^{1/2 }= (M_{Y}E_{Y})^{1/2}cosΦ + (M_{y}E_{y})^{1/2}cosθ

O = (M_{Y}E_{Y})^{1/2} sinΦ + (M_{y}E_{y})^{1/2}sinθ

Square equations and add:

M_{x}E_{x} = M_{y}E_{Y}cos2Φ + M_{y}E_{y}cos2θ

M_{x}E_{x} = M_{Y}E_{Y} + M_{y}E_{y}

Minimum energy projectile should have before it can induce nuclear reaction is known as threshold energy

**Conservation of linear momentum:**

M_{x}V_{x} = M_{c}V_{c}

V_{c }= M_{x}V_{x}/M_{c}

But -Q = 1/2M_{x}V_{x}^{2} - 1/2M_{c}V_{c}^{2}

-Q = 1/2M_{x}V_{x}^{2 }- 1/2M_{c}(M_{x}^{2}V_{x}^{2}/M_{c}^{2})

-Q = 1/2M_{x}V_{x}^{2}[1 - M_{x}/M_{c}]

But M_{c }= M_{X} + M_{x}

-Q = 1/2M_{x}V_{x}^{2}[1 - M_{x}/(M_{X} + M_{x})]

-Q = 1/2M_{x}V_{x}2[(M_{X} + M_{x} - M_{x})/(M_{X} + M_{x})]

-Q = 1/2M_{x}V_{x}^{2}[M_{X}/(M_{X} + M_{x})]

Take 1/2M_{x}V_{x}^{2} = E_{thr}

-Q = E_{thr}[M_{X}/(M_{X} + M_{x})]

E_{thr }= -Q x [M_{X} + M_{x}/M_{X}]

E_{thr} = -Q x [1 + M_{x}/M_{X}]

**Nuclear Fission Reaction:**

This is the reaction that involves splitting of heavy nuclei in two or more lighter nuclei by bombarding heavy nuclei with thermal neutrons and it is generally accompanied with high energy released. Nuclides which can be fissioned by thermal neutrons as follow: ^{235}U, ^{237}U and ^{239}Pu.

Only ^{235}U takes place as expected while others are gotten from fertile materials. Procedure of conversion of fertile materials to fissionable material is referred to as breeder reaction. Energy released in the fission reaction is Q = Δmc^{2} = [(Σm_{i} - Σm_{f})c^{2}]

**Nuclear Fusion Reaction:**

This is a combination of two or more light nuclei to form heavier one and this involves supply of high energy. In which this energy will be able to beat coulombs force between them.

^{1}_{1}H + ^{2}_{1}H → ^{3}_{1}H + γ + Q

And in doing so they should overcome potential barrier that is equal to

V = Z_{1}Z_{2}e^{2}/4πε_{0}r

V = 0.15Z_{1}Z_{2}Mev, for r = 10^{-14}m

This type of energy (10^{9}k) can only be attained during nuclear explosions that are not practicable. Generally at this temperature, atoms exist as ions and are collected as plasma. Due to high temperature, it is also referred to as thermonuclear reaction.

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