CBSE Class 12 Physics Nuclei Notes Set 03

1. Atomic Mass Unit and Atomic Masses

Nuclear and atomic masses are very small, so they are measured in the atomic mass unit (u). One atomic mass unit is defined as one-twelfth of the mass of a carbon-12 atom.
1 u = \( \frac{\text{mass of one } ^{12}\text{C atom}}{12} \) = 1.660539 × 10⁻²⁷ kg

(i) Mass of a proton, m_p = 1.007276 u = 1.6726 × 10⁻²⁷ kg
(ii) Mass of a neutron, m_n = 1.008665 u = 1.6749 × 10⁻²⁷ kg
(iii) Mass of a hydrogen atom, m_H = 1.007825 u
(iv) Mass of an electron = 9.11 × 10⁻³¹ kg = 0.000549 u

Atomic masses are measured accurately with a mass spectrometer. The atomic masses of many elements are not whole numbers because a natural sample is a mixture of isotopes. For example, chlorine is about 75.4% ³⁵Cl (34.969 u) and 24.6% ³⁷Cl (36.966 u), so its average atomic mass is 35.46 u.

2. Composition of the Nucleus

A nucleus is made of protons and neutrons, together called nucleons. The neutron was discovered by James Chadwick in 1932; it has no charge and a mass slightly greater than that of the proton.
(i) Atomic number Z = number of protons (equal to the number of electrons in a neutral atom)
(ii) Neutron number N = number of neutrons
(iii) Mass number A = total number of nucleons, A = Z + N
A nuclide is written as ^A_ZX. For example, ⁵⁶₂₆Fe has Z = 26, A = 56 and N = 56 − 26 = 30.

(i) Isotopes: same Z, different A (different N). Example: ¹₁H, ²₁H, ³₁H. They have identical chemical properties because they have the same number of electrons.
(ii) Isobars: same A, different Z. Example: ³₁H and ³₂He.
(iii) Isotones: same N, different Z. Example: ¹⁹⁸₈₀Hg and ¹⁹⁷₇₉Au (N = 118 for both).

3. Size of the Nucleus

Rutherford's alpha-particle scattering experiments, and later experiments with fast electrons, showed that the nucleus is extremely small: its radius is about 10⁻¹⁵ m to 10⁻¹⁴ m, while the atom is about 10⁻¹⁰ m across. Nuclear sizes are expressed in fermi: 1 fm = 10⁻¹⁵ m.

The volume of a nucleus is proportional to its mass number, so its radius is
R = R₀ A^1/3, where R₀ ≈ 1.2 fm = 1.2 × 10⁻¹⁵ m
1. Volume: V = \( \frac{4}{3}\pi R^3 = \frac{4}{3}\pi R_0^3 A \), so V is proportional to A.
2. Ratio of radii of two nuclei: \( \frac{R_1}{R_2} = \left(\frac{A_1}{A_2}\right)^{1/3} \)

Nuclear density: Mass of a nucleus ≈ A × m (m ≈ 1.66 × 10⁻²⁷ kg per nucleon), so
ρ = \( \frac{A m}{\frac{4}{3}\pi R_0^3 A} = \frac{3m}{4\pi R_0^3} \) ≈ 2.3 × 10¹⁷ kg m⁻³
1. A cancels out, so the density is (nearly) the same for all nuclei, light or heavy.
2. Nuclear matter is about 10¹⁴ times denser than water. Neutron stars have densities of this order.

4. Mass–Energy Equivalence

Einstein's special theory of relativity shows that mass is a form of energy. A mass m is equivalent to an energy
E = mc², where c = 3 × 10⁸ m s⁻¹
1. 1 kg of mass is equivalent to 9 × 10¹⁶ J.
2. 1 u is equivalent to 1.4924 × 10⁻¹⁰ J = 931.5 MeV. So 1 u = 931.5 MeV/c².
3. 1 MeV = 1.6 × 10⁻¹³ J and 1 eV = 1.6 × 10⁻¹⁹ J.
In nuclear reactions, the total of mass and energy is conserved; mass may change into energy and energy into mass.

5. Mass Defect and Nuclear Binding Energy

The mass of a nucleus is always less than the sum of the masses of its free protons and neutrons. The difference is called the mass defect ΔM.
ΔM = [Z m_p + (A − Z) m_n] − M_nucleus
Using atomic masses (the electron masses then cancel): ΔM = [Z m_H + (A − Z) m_n] − M_atom

The energy equivalent of the mass defect is the binding energy of the nucleus:
E_b = ΔM c², and with ΔM in u: E_b (in MeV) = ΔM (in u) × 931.5
1. Binding energy is the energy released when the nucleons come together to form the nucleus.
2. Equally, it is the minimum energy that must be supplied to separate the nucleus completely into free protons and neutrons.

Example (helium-4): m(⁴He) = 4.002603 u, m_H = 1.007825 u, m_n = 1.008665 u
ΔM = 2(1.007825) + 2(1.008665) − 4.002603 = 0.030377 u
E_b = 0.030377 × 931.5 ≈ 28.3 MeV, so E_b/A ≈ 7.07 MeV per nucleon.

6. Binding Energy per Nucleon

The binding energy per nucleon, E_bn = \( \frac{E_b}{A} \), is the average energy needed to remove one nucleon from the nucleus. It measures the stability of a nucleus: the larger E_bn, the more stable the nucleus.

Main features of the curve of E_bn against A:
1. E_bn is small for light nuclei (about 1.1 MeV for ²H) and rises quickly with A. ⁴He, ¹²C and ¹⁶O lie above the general trend and are especially stable.
2. For the middle range 30 < A < 170, E_bn is nearly constant at about 8 MeV. This shows that the nuclear force is short-ranged and saturates: a nucleon interacts only with its close neighbours.
3. The maximum, about 8.75 MeV, occurs near A = 56 (iron). These nuclei are the most stable.
4. For heavy nuclei E_bn falls slowly, to about 7.6 MeV for A = 238, because the Coulomb repulsion between the many protons reduces the binding.
5. Consequences: a heavy nucleus (A ≈ 240) releases energy when it splits into two middle-mass nuclei (fission), and very light nuclei release energy when they join into a heavier one (fusion). In both cases the products have a higher E_bn, i.e. are more tightly bound.

7. Nuclear Force

The force that holds nucleons together in the nucleus, against the Coulomb repulsion between protons, is the nuclear force.
1. It is the strongest force in nature, much stronger than the Coulomb force (and enormously stronger than gravity) at nuclear distances.
2. It is short-ranged: it is negligible beyond a few femtometres. Hence each nucleon is bound only to its nearest neighbours (saturation).
3. It is charge independent: the n–n, p–p and n–p nuclear forces are approximately equal.
4. It is attractive for separations greater than about 0.8 fm and strongly repulsive for separations less than about 0.8 fm, which keeps nucleons from collapsing into each other.
5. There is no simple mathematical formula (like Coulomb's law) for the nuclear force.

The graph shows the potential energy U of a pair of nucleons against their separation r. The minimum of U lies at r₀ ≈ 0.8 fm. For r > r₀ the force (−dU/dr) is attractive; for r < r₀ it is repulsive.

8. Nuclear Fission

Fission is the splitting of a heavy nucleus into two nuclei of intermediate mass, with the release of a large amount of energy. When a slow neutron is absorbed by ²³⁵U, it forms the highly unstable nucleus ²³⁶U, which splits, for example as
¹₀n + ²³⁵₉₂U → ²³⁶₉₂U → ¹⁴¹₅₆Ba + ⁹²₃₆Kr + 3 ¹₀n + energy

1. Other fragment pairs are also possible, for example ¹⁴⁴₅₄Xe + ⁹⁰₃₈Sr + 2 ¹₀n. In every case Z and A balance on both sides.
2. Energy released per fission of ²³⁵U ≈ 200 MeV. It appears mostly as the kinetic energy of the fragments and neutrons.
3. Estimate from the E_bn curve: for A ≈ 240, E_bn ≈ 7.6 MeV; for the fragments (A ≈ 120), E_bn ≈ 8.5 MeV. Energy released ≈ 240 × (8.5 − 7.6) ≈ 216 MeV, about 0.9 MeV per nucleon.
4. The energy from 1 g of ²³⁵U by fission is about 8 × 10¹⁰ J, millions of times more than burning 1 g of coal.

9. Nuclear Fusion

Fusion is the joining of two light nuclei to form a heavier nucleus, with the release of energy. Examples:
²₁H + ²₁H → ³₂He + ¹₀n + 3.27 MeV
²₁H + ³₁H → ⁴₂He + ¹₀n + 17.6 MeV

1. The energy released per fusion is smaller than per fission, but the energy released per nucleon (per unit mass of fuel) is larger: 17.6/5 ≈ 3.5 MeV per nucleon for D–T fusion.
2. Both nuclei are positively charged, so they must approach each other against strong Coulomb repulsion until they come within the range of the nuclear force. This needs very high kinetic energy, i.e. temperatures of about 10⁷ K or more. Fusion at such temperatures is called thermonuclear fusion.
3. Energy in stars: The Sun and other stars shine because of thermonuclear fusion in their cores, where the temperature is about 1.5 × 10⁷ K. Hydrogen nuclei are fused, in a series of steps, into helium; the overall result is that four hydrogen nuclei form one helium nucleus with the release of about 26.7 MeV.
4. Controlled thermonuclear fusion: Scientists are trying to produce fusion energy on Earth by heating a deuterium–tritium plasma to around 10⁸ K and confining it with strong magnetic fields. This would give a large, cleaner energy source, but it has not yet been achieved on a practical scale.

10. Quick Formula Recap