Nuclear Reactions, Physics tutorial

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Nuclear Reactions, Physics tutorial

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 ¹₀n, ¹₁H, ⁴₂He,²₁H

X + x → Y + y + Q or X(x,y)Y

e.g. ¹⁴₇N(α, p) ¹⁷₇O

¹⁴₇N +⁴₂He → ¹⁷₈O + ¹₁H + Q

Kinds of Nuclear Reaction

1) F - Fission - X + x → y₁ + y₂

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_XC²] + [E_x + m_xC²]

= [E_Y + m_YC²] + [E_y + m_yC²] + Q

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

[M_xC²] + [E_x + m_xC²] = [E_y + M_yC²] + [E_y + M_yC²] + Q

But Q = change in energy

Hence Q = E_x - (E_Y + E_y)

Q = [(M_X + m_x) - (M_Y + M_y)]C²

Q = Δmc²

Conventional Q-value Equation:

Applying principle of conservation of momentum

M_xV_x = M_YV_YcosΦ + M_yV_ycosθ(x-direction)

O = M_YV_YsinΦ + M_yV_ysinθ(y-direction)

Form E = P²/2m, p² = 2mE

Thus, momentum mv = √2ME

Equation turns into:

(M_xE_x)^1/2= (M_YE_Y)^1/2cosΦ + (M_yE_y)^1/2cosθ

O = (M_YE_Y)^1/2 sinΦ + (M_yE_y)^1/2sinθ

Square equations and add:

M_xE_x = M_yE_Ycos2Φ + M_yE_ycos2θ

M_xE_x = M_YE_Y + M_yE_y

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

Conservation of linear momentum:

M_xV_x = M_cV_c

V_c= M_xV_x/M_c

But -Q = 1/2M_xV_x² - 1/2M_cV_c²

-Q = 1/2M_xV_x²- 1/2M_c(M_x²V_x²/M_c²)

-Q = 1/2M_xV_x²[1 - M_x/M_c]

But M_c= M_X + M_x

-Q = 1/2M_xV_x²[1 - M_x/(M_X + M_x)]

-Q = 1/2M_xV_x2[(M_X + M_x - M_x)/(M_X + M_x)]

-Q = 1/2M_xV_x²[M_X/(M_X + M_x)]

Take 1/2M_xV_x² = 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: ²³⁵U, ²³⁷U and ²³⁹Pu.

Only ²³⁵U 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 = Δmc² = [(Σm_i - Σm_f)c²]

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.

¹₁H + ²₁H → ³₁H + γ + Q

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

V = Z₁Z₂e²/4πε₀r

V = 0.15Z₁Z₂Mev, for r = 10^-14m

This type of energy (10⁹k) 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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