Nuclear Physics Formulas

1. Nuclear force

A strong force acts among the nucleons in the nucleus. This force is called nuclear force. The range of this force is of the order of 10^-14 m. Nuclear force does not depend on charge, depends on spin.

2. Size of nucleus

Nucleus is assumed spherical. Its radius and mass number are related as
R = R_o A^1/3 = 1.25 × 10^-15 A^1/3 m

3. Atomic mass unit (amu)

One amu is equal to \(\left(\frac{1}{12}\right)\) of the mass of one atom of ₆C¹².
1 amu = 1u = 1.66 × 10^-27 kg
Energy of 1 amu = 931.5 MeV

4. Mass defect and nuclear binding energy

In reality the mass of the nucleus is always less than the total mass of its components. The difference is called mass defect. The energy equivalent to this mass defect is called nuclear binding energy.
(a) Mass defect = sum of masses of all nucleons – mass of the nucleus >
╬öm = (Zm_p + Nm_n) – M
= [Zm_p + (A – Z)m_n – M]

(b) Packing fraction = \(\frac{\text { Mass defect }}{\text { Mass number }}\)
or f = \(\frac{\Delta \mathrm{m}}{\mathrm{A}}=\left(\frac{\mathrm{M}-\mathrm{A}}{\mathrm{A}}\right)\)

(c) Nuclear binding energy = (╬öm)c² = ╬öm (a.m.u) ├ù 931.5 MeV

(d) Binding energy per nucleon
= \(\frac{\text { Nuclear binding energy }}{\text { Mass number }}\)
= \(\frac{(\Delta \mathrm{m}) \mathrm{c}^{2}}{\mathrm{A}}\)
\(\overline { B.E } \) = \(\frac{\left[\mathrm{Zm}_{\mathrm{p}}+(\mathrm{A}-\mathrm{Z}) \mathrm{m}_{\mathrm{n}}-\mathrm{M}\right] \mathrm{c}^{2}}{\mathrm{A}}\)

Binding energy curve:

5. Nuclear reactions

Nuclear reactions take place when the nucleus of an atom is bombarded with high energy protons, neutrons, ╬▒-particles, deuterons, ╬│-rays etc.
Following conservation laws are obeyed in the nuclear reactions:

• Mass-energy conservation law.
• Linear momentum conservation law.
• Angular momentum conservation law.
• Charge conservation law.
• Nucleon conservation law.

Q-value- The equivalent energy corresponding to the net loss of mass in a reaction is called ΓÇÿQΓÇÖ value.
As in _ZX^A → _Z1X^A1 + _Z2X^A2 + Q
Q = (M_x – M_Y1 – M_Y2)c²
= (A₁E_b1 + A₂E_b2 – AE_b)
Where E_b is binding energy per nucleon.

6. Neutrons

Neutron was discovered by Chadwick in 1932. Neutron is a neutral particle. Its mass is slightly greater than that of proton. Mass of neutron is 1.675 × 10^-27 kg.

7. Nuclear fission

The process in which a heavy nucleus breaks-up into two nearly equal nuclei and a large amount of energy is released is called nuclear fission
₉₂U²³⁵ + ₀n¹ ΓåÆ (₉₂U²³⁶) ΓåÆ ₅₆Ba¹⁴¹ + ₃₆Kr⁹² + 3₀n¹ + Q
Q Γëê 200 MeV per fission
Atomic bombs based on the principle of nuclear fission of uncontrolled chain reaction.

8. Controlled chain reaction and atomic reactor

In controlled chain reaction the rate of fissions is controlled. Therefore, this j process is slow. This process is used for the production of electricity in an I atomic reactor. Main parts of an atomic reactor are:

• Nuclear fuel as U²³⁵, Pu²³⁹
• Moderator as D₂O (Heavy water), graphite.
• Coolant as water, air, CO₂, liquid Na.
• Controller as cadmium.
• Shield as thick concrete wall or thick lead sheet

9. Nuclear fusion

When two light nuclei combine to form a heavy nucleus, the process is called nuclear fusion. This process is possible only at very high temperatures ~10⁷ K. Energy produced in sun and stars is due nuclear fusion.
4₁H¹ ΓåÆ ₂He⁴ + 2╬▓⁺ + 2v + Q
Q Γëê 24.7 MeV per cycle.

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