NUCLEAR AND PARTICLE PHYSICS -1

Nucleus: Atomic nucleus is a bound quantum system. It can exist in different quantum states characterised by energies, angular etc. The lowest energy state is called the ground state and its properties in ground state are termed as static properties. The dynamic of nuclei are exhibited in the process of nuclear decay, nuclear reactions, fission, fusion.

The nucleus of an atom is made of the elementary particles, protons and neutrons. The protons and neutrons are held together by a very strong short range attractive force as nuclear interaction.

The nucleus is identified by number of protons (Z = atomic number) and number of nucleons (A = mass number). The number of neutrons N = A-Z. (^𝐴_𝑍𝑋_𝑁)

Isotopes: Nuclides having same number of Z but different N and hence A. ex-¹₁𝐻 and ²₁𝐻

Isotones: Nuclides having same number of neutrons N but different Z and A. ¹³₆𝐶 and ¹⁴₇𝑁 are isotones with N=7.

Isobars: Nuclides having same number of A (mass number) but different Z and N. ¹⁴₆𝐶 and ¹⁴₇𝑁 are isobars.

Nuclear mass: In Nuclear Physics, we are concerned with the masses of the nuclei, the experimental determination using mass spectroscopes yield the atomic masses. The nuclear mass M_n is defined from the atomic mass M by subtracting the masses of Z orbital electrons from the latter. M_n = M-Zm_e. This expression is not exact because it does not take into the account of binding energies of the electrons in atoms. Hence the mass of electrons is very small and the error is very small.

Atomic mass unit: u (unified atomic mass unit)- 1/12 of the mass of ¹²C (taken to be exactly 12 units)

Earlier, amu (atomic mass unit) is taken as 1/16 of the mass of ¹⁶O.

1 u = 931.5 Mev

Mass Defect: The difference of atomic mass M(A, Z) from the mass number A

 Δ= Mass defect = M(A, Z) – A

Packing fraction: f =(M(A, Z) – A)/A

The packing fraction is defined as the mass defect per nucleon

_M – 1 = f      ∴M(A, Z) = A(1+f)

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Binding Energy: The mass of the nucleus is approximately less than the sum of the masses of its component nucleons.

The binding energy of a nucleus is the energy required to separate the nucleons composing the nucleus.

For a nucleus of mass M(A, Z) composed Z protons and A-Z neutrons, the binding energy is

E_b = [ ZM_p + (A-Z)M_n – M(A, Z)] х c²

Average binding energy per nucleon = -E_b/A

Mass number multiples of 4 hike peaks are observed ⁴He, ⁸Be, ¹²C, ¹⁶O have equal number of protons and neutrons. They have binding energy more than their neighbours…………….

Definition of nuclear radius

The nuclear radius or nuclear force radius represents the distance from the centre of nucleus at which an external uncharged nucleon first feels its nuclear influence.

A nucleus has no clearly defined boundary. Nuclear radius has an arbitrary boundary. Rutherford model of atom

Rutherford’s scattering of alpha particles gave a model of the nuclear atom in which the entire positive charge and almost entire mass of the atom is concentrated within a sphere of radius  10^-10m

At the point of closest approach (d_m), the kinetic energy (K_α) of the _α particle is entirely converted to electrostatic potential energy.

K_α = (1/4πε_0){(𝑍𝑒)(2𝑒)/ 𝑑_{𝑚  }}    d_m =(1/4πε_0){(𝑍𝑒)(2𝑒)/ K_α }  

                𝑑_𝑚                               

K_α = 5.486 Mev, d_m = 4.15 х 10^-11 m for gold (Z = 79)

                               D_m = 247 х 10^-14 m for silver (Z = 74)

Nuclear radius is much smaller than atomic radius of the order of 10⁴ and 10⁵

Measurement techniques found that radius of atomic nuclei lies in the range of 10^-14 to 10^-15 m.

Experimental results show that nuclear radius is proportional to A^1/3 (where A is the mass number) R = r₀A^1/3 where r₀ = is a constant The accepted value  of  r₀ is 1.07 * 10^-15 m .

(R) Radius is the radius of nuclear mass distribution. One can define radius of nuclear charge distribution. Mass radius and charge radius are expected to be nearly same.

Nuclear density: The volume of the nucleus is proportional to the number of nucleons it consists of. Hence the volume is 4/3πR³ = 4/3πr₀³A = 1.12 * 10^-45A m³

The average density of nuclear matter is

 ρ= 𝑀/𝑉 = ( 1.66*10^-27 A/1.12 * 10^-45A)kg/m³ = 1.49 * 10¹⁸kg/m³

The density of nucleus is constant and independent of A. The density of nucleus is almost 10¹⁵ times greater than the density of matter in bulk. The matter in the bulk is empty since most of the mass is concentrated in the nuclei.

The density of the nuclear matter varies with the distance from the centre of the nucleus. The density is almost constant for a considerable distance from the centre and gradually decreases to zero near the surface of the nucleus.