Discovery of nucleus:
In 1911, Rutherford and his teams discovered the atomic nucleus from α scattering experiment.
Constituents of nucleus:
All atomic nuclei, are made up of protons and neutrons except H-atom. Proton has +ve charge while the electrons are negatively charged. The neutrons are electrically neutral. The collective forms of protons and neutrons in an atom are called neucleons.
An atom is denoted by [latex]_Z X^A[/latex], where, Z is the atomic number and A is the mass no. The no. of neutron is given as: N = A-Z.
General properties of nucleus:
1. Size:
According to Rutherford, radius of atomic nucleus is of the order of [latex]10^{-14}[/latex] m to [latex]10^{-15}[/latex] m, while an atom has radius of about [latex]10^{-10}[/latex] m. Thus, nucleus is about 10000 times smaller than atom.
2. Charge:
A nucleus consists of protons and neutrons. A proton has positive charge, while a neutron is chargeless. So, the nuclei are positively charged and has charge of Ze.
3. Mass:
Since, a nucleus consists of protons and neutrons, the mass of nucleus is equal to sum of masses of nucleons.
∴ mass of nucleus = Zmp + Nmn, where, Z = atomic no., N = neutron no., mp = mass of proton and mn = mass of neutrons.
4. Nuclear density:
The mass per unit volume of a nucleus is called nuclear density. i.e.
Nuclear density, [latex]\rho = \frac{Nuclear , Mass}{nuclear , volume} = \frac{Am_N}{\frac{4}{3}\pi R^3} = \frac{Am_N}{\frac{4}{3}\pi (r_o A^{\frac{1}{3}})^3} = \frac{3m_N A}{4\pi r_o^3 A} = \frac{3m_N}{4\pi r_o^3}[/latex]
Where, mn = mass of nucleon
R = [latex]r_oA^{1/3}[/latex], empirical formula for nuclear radius, A = mass no. and [latex]r_o = 1.2 \times 10^{-15}[/latex] m.
So, [latex]\rho = 2.3 \times 10^{17} Kg/m^3[/latex]. Since, nuclear density, is independent of mass no. (A) it is almost constant for all nuclei.
5. Nuclear Spin and magnetic moment:
Protons and neutrons are in continuous motion in the orbits. So, this orbital motion produces the nucleons with angular momentum. Besides this, nucleons have internal angular momentum known as spins. So, they possess magnetic moment associated with their orbital motion and spin.
Atomic mass unit:
The mass is usually measured in unified atomic mass unit (u). The atomic mass unit is defined as the [latex]1/12^{th}[/latex] of the mass of carbon atoms ([latex]_6C^{12}[/latex]).
We know, the number of molecules in 1 mole of carbon is [latex]6.02 \times 10^{23}[/latex] and mass of 1 mole of carbon is 12 gram.
So, we can write, the mass of [latex]6.023 \times 10^{23}[/latex] C-atoms = 12 gram.
The mass of 1 C-atom = [latex]\frac{12}{6.023 \times 10^{23}}[/latex] gm.
By the definition of atomic mass unit (U)
1 U = [latex]\frac{1}{12}[/latex] x mass of 1 carbon atom.
= [latex]\frac{1}{12} \times \frac{12}{6.02 \times 10^{26}}[/latex] Kg.
∴ 1 U = [latex]1.66 \times 10^{-27}[/latex] kg.
Relation between the atomic mass unit (U) and MeV:
From mass energy relation, we have, E = [latex]mc^2[/latex]
So, the energy related to the mass 1U is:
E = ([latex]1.66 \times 10^{-27}) \times (3 \times 10^8)^2[/latex] J
= [latex]1.66 \times 10^{-27}\times 9 \times 10^{16}[/latex] J
∴ E = [latex]\frac{9 \times 1.66 \times 10^{-27+16}}{1.6 \times 10^{-13}}[/latex] MeV [∵ 1MeV = [latex]1.6 \times 10^{-13}[/latex] J]
= 931 MeV
∴ 1U = 931 MeV.
Einstein’s Mass Energy Relation:
Earlier to Einstein, mass and energy was considered two separate entities and hence, there were two distinct laws for their conservation. But, in 1905, Einstein from his theory of relativity showed that mass and energy are interconvertible to each other. So, only a single mass energy conservation law is considered.
The change in energy ∆E is related to the change of mass ∆m, by the equation, [latex]\Delta E = \Delta m c^2[/latex] where, c = velocity of light. i.e. c = [latex]3 \times 10^8[/latex] m/s.
For the change of mass 1 kg, the equivalent energy is: E= 1 x ([latex]3 \times 10^8)^2 = 9\times 10^{16}[/latex] J
In nuclear physics, energy is usually expressed in electron volt. We have, the electron volt is the energy gained by a charge equal to that of an electron in moving through a p.d. of one volt.
1 eV = [latex]1.66 \times 10^{-19}\ C \times 1V = 1.66 \times 10^{-19}[/latex] J
For the larger unit, 1 MeV = [latex]10^6[/latex] eV
∴ 1 MeV = [latex]1.66 \times 10^{-13}[/latex] J.
Types of nuclei:
i. Isotopes:
The atoms of an element which have the same atomic number but different mass no. are called isotopes. In isotopes, the no. of protons inside the nucleus or the no. of electrons outside the nucleus is the same but the no. of neutrons inside the nucleus is not same. The chemical properties of isotopes are same. For example, [latex]_8O^{16}, _8O^{17}, _8O^{18}\ and\ _6C^{14}, _6C^{12}[/latex] are the isotopes of oxygen and carbon respectively.
ii. Isobars:
The atoms which have same mass no. but different atomic no. are called isobars. Isobars have same no. of nucleons (neutrons and protons) in their nuclei and hence they have same atomic weight. The chemical properties of isobars are different. For example: [latex]_18Ar^{40}\ and\ _20Ca^{40}[/latex].
iii. Isotones:
Isotones are the nuclei, which have same neutron no. For example, i) =[latex]_2He^4\ \&\ _1H^3[/latex] (N = 2), ii) [latex]_1H^2\ and\ _2He^3[/latex](N = 1).
Binding energy and Mass Defect:
Binding energy:
It has been found that the rest mass of the nucleus of a stable atom is always less than the sum of the masses of its constituent particles i.e. protons and neutrons. The difference between the sum of the masses of nucleons and the rest mass of the nucleus is called mass defect. The energy equivalent of this mass defect is called binding energy of the nucleus.
Hence, binding energy may be measured as the work required to be done to separate the nucleus on infinite distance apart, so that they no longer interact with each other.
If ∆m is the mass defect, then according the mass energy relation:
BE = [latex]∆mc^2[/latex]
If M is the rest mass of a nucleus, [latex]_ZX^A[/latex], [latex]m_n\ and\ m_p[/latex] are the masses of neutron and proton, then
B.E. = [Zmp + (A-Z)mn – M][latex]c^2[/latex]
In the terms of amu,
BE = [Zmp + (A-Z)mn – M] x 931.5 MeV
BE per nucleons curve:
Binding energy per nucleons is the ratio of total B.E. to the mass no. i.e.
[latex]\overline{B.E.} = \frac{Total , B.E.}{A}[/latex]
It is also known as average B.E. It determines the measure of stability of the nucleus. Greater the value of B.E. per nucleons, greater the stability. The B.E. per nucleon with the different mass no is shown in the graph. From the graph, it is clear that the curve first rises sharply and then gradually until it reaches the maximum value of 8.79 MeV corresponding to iron. Then it drops slowly to 7.6 MeV for Uranium. It is clear that the intermediate nuclei are more stable than the lighter or lower nuclei.
Nuclear Reaction:
i. Nuclear fission:
The splitting up of a heavy nucleus into two nuclei of nearly equal or comparable masses with liberation of energy is called nuclear fission.
Bohr Wheeler theory of nuclear fission:
Nuclear fission was first observed by Fermi and his co-workers and was explained by Neil Bohr and J.A. Wheeler. They explained the nuclear fission on the basis of liquid drop model of nucleus. When the neutron strikes [latex]_{92}U^{235}[/latex] nucleus (which behaves like a liquid drop and owing to surface tension it is perfectly spherical in shape), some energy called excitation energy is imparted to the nucleus. The energy due to surface tension tries to keep the shape of the nucleus spherical but the excitation energy tries to deform it. As a result, the oscillations are set up inside the compound nucleus, and they come to a dumbbell shape. Then, Coulomb’s repulsive force between protons separate the two dumbbells into two spherical shape nuclei.
In this way, nuclear fission occurs. It can be explained by the following nuclear reaction:
[latex]_{92}U^{235} + _0n^1 → _{92}U^{236} → _{56}Ba^{141} + _{36}Kr^{92} + 3_0n^1 + Q[/latex]
In the above reaction [latex]_{92}U^{235}[/latex] is bombarded with neutron thus forming a compound nucleus [latex]_{92}U^{236}[/latex] which disintegrates into two nuclei of [latex]_{56}Ba^{141}\ and\ _{36}Kr^{92}[/latex] with release of three neutrons and a lot of thermal energy Q.
Energy released in Fission:
The initial mass of nucleus is greater than the sum of masses of the products and the difference in mass (∆m) is converted into energy according to Einstein’s mass-energy relation, i.e. ∆E = [latex]∆mc^2[/latex]. We have,
Mass of neutron, [latex]_0n^1[/latex] = 1.008665 u.
Mass of [latex]_{92}U^{235}[/latex] = 235.045733 u.
∴ Total initial mass = 236.054398 u.
Mass of [latex]_{56}Ba^{141}[/latex] = 140.91770 u.
Mass of [latex]_{36}Kr^{92}[/latex] = 91.8854 u.
Mass of 3[latex]_0n^1[/latex] = 3.0260 u.
∴ Total final mass = 235.829100 u.
∴ Mass difference (∆m) = 236.054398 – 235.829100
= 0.2253 u.
∴ Energy released per fission = 0.2253 x 931 MeV
≈ 200 MeV
Which is very large amount of energy.
Nuclear fusion:
When two or more than two light nuclei fuse together to form heavy nucleus with the liberation of energy, the process is called nuclear fusion.
E.g., [latex]_1H^2 + _1H^2 → _2He^4[/latex] + 24 MeV
The above nuclear reaction is energetically possible only if the mass of [latex]_2He^4[/latex] nucleus is less than sum of masses of the two deuteron nuclei. The difference in the initial mass and the final mass of the product nuclei is liberated as the energy of fusion reaction.
Energy released in Fusion:
The energy released in this reaction can be calculated as:
Mass of deuteron = 2 x 2.014102 u = 4.028204 u.
Mass of [latex]_2He^4[/latex] = 4.002604 u.
∴ Decrease in mass = 0.025600 u.
∴ Energy released = 0.0256 x 931 MeV ≈ 24 MeV.
Thus, the energy released in fusion = 24 MeV.
Chain reaction:
A nuclear reaction initiated is capable of triggering the nuclear reaction and the process goes on until all the atoms of the target material are bombarded. This is known as chain reaction.
When a neutron bombards an atom of Uranium in a piece of uranium, three neutrons are produced along with the release of energy of about 200 MeV. These three neutrons can further bombard other three Uranium atoms producing 9 neutrons and the process goes on until all the uranium atoms are bombarded.
Types of chain reaction:
i. Controlled chain reaction:
The no. of neutrons and energy is controlled to a desired level in controlled chain reaction. This principle is used in nuclear reaction.
ii. Uncontrolled chain reaction:
The neutrons are allowed to multiply independently in this reaction. Hence, the energy is released all the time. This is the principle behind atom bomb.
Multiplication factor:
It is defined as the ratio of secondary neutrons produced to the initial number of neutrons. It is denoted by K. i.e.
K = [latex]\frac{\text{No. of neutrons in any generations}}{\text{No. of neutrons in previous generation}}[/latex]
For K = 1, the reaction is steady or critical.
For K > 1, the reaction is building up or super-critical.
For K < 1, the reaction is dying down or sub-critical.
Packing fraction:
The packing fraction of the isotope is defined as the ratio of its mass defect to its mass number. i.e.
Packing fraction = [latex]\frac{\text{mass defect}}{\text{mass number}}[/latex]
Creation and Annihilation:
A subatomic particle having same mass as another particle but opposite value of charge is called antiparticle. For eg. the antiparticle of electron is positron (<sub>+1</sub>e<sup>o</sup>), which has same mass as that of electron but has a positive charge equal in magnitude to that of electrons.
The antiparticle of proton is antiproton. Particles form matter but antiparticles form antimatter. When an energetic [latex]\gamma[/latex] – ray photon falls on a heavy substance, it is absorbed by the nucleus of the substance and its energy gives rise to the production of an electron and a positron. This phenomenon, in which energy is converted into mass is called creation or pair production and is represented by the equation:
[latex]h\nu[/latex] = [latex]_{-1}e^o + _{+1}e^0[/latex]
([latex]\gamma[/latex] – photon) (electron) (positron)
When a particle and antiparticle meet each other, the destruction of particle and antiparticle occur with the emission of energy. This is called annihilation of matter. For example, when electron meets positron, there is emission of energy and [latex]\gamma[/latex] – ray. i.e.
[latex]e^- + e^+ → \gamma[/latex] + 1.02 MeV