Nuclear_Physics_Lecture_3.pdf
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(2) Liquid Drop Model The oldest nuclear model introduced by Bohr and Kalckar (1937) attempting to explain some gross nuclear properties like the nuclear binding energy, nuclear stability etc. Resemblances between nucleus and a liquid drop (1)Uniform density of nuclear mater ≡ Uniform density within a liquid drop (2) Constant binding energy per nucleon ≡ Latent heat of vaporisation.
(3) Liquid Drop Model Resemblances between nucleus and a liquid drop (3) Barrier potential across the nuclear surface ≡ Surface energy of the liquid drop (4) Emission of n, p, α in nuclear reaction ≡ Emission of molecules during evaporation (5) Formation of short lived compound nucleus ≡ Condensation from the vapor into the liquid phase In Liquid Drop Model, the nucleus is considered equivalent to a dense, incompressible, spherical liquid drop.
(4) Semi-empirical Mass Formula (1)Developed by Bethe - Weizsäcker (2)Partly model based (liquid drop model) (3)Partly derived empirically (4)Oldest classical model (5)Pairing term suggests some sort of shell or energy level model.
(5) Semi-empirical Mass Formula (a) Volume energy: Similar to the internal energy within a liquid drop which is proportional to the number of molecules within it The volume of a liquid drop is proportional to the number of molecules it contains The volume energy of nucleus (E1) is also proportional to the number of nucleons within it, i.e. with the mass number (A) of the nucleus. E1 ∝ A ⇒ E1 = a1 A a1 = 15.5 MeV (0.016919 u).
(6) Semi-empirical Mass Formula (b) Surface energy: The nucleus, like a liquid drop, is assumed to be spherical having radius. R = r0 A1/ 3 The force between the nucleons at the nuclear surface is similar to the surface tension of the liquid, as exits between the molecules on the surface The nucleons on the surface are acted upon by an attractive force by the nucleons inside the nucleus. There are no forces acting from the outside and that is why the nucleus assumes spherical shape.
(7) Semi-empirical Mass Formula (b) Surface energy: The surface energy (E2) corresponding to this force is proportional to the surface area. a2 = 16.8 MeV (0.019114 u) The negative sign emphasizes the fact that the surface force tends to reduce the nuclear binding energy.
(8) Semi-empirical Mass Formula (c) Coulomb energy: Coulomb repulsion between the protons tends to weaken the nuclear binding The Coulomb force acts between all the proton pairs within the nucleus and hence the Coulomb energy is proportional to (i) the number of proton pairs and (ii) inversely proportional to the average distance between the pairs which is of the order of nuclear radius: R = r0 A1/ 3.
(9) Semi-empirical Mass Formula (c) Coulomb energy: The number of proton pairs in nucleus of atomic number Z is. Coulomb energy of the nucleus is given by.
(10) Semi-empirical Mass Formula (c) Coulomb energy: Alternate derivation The nucleus can be assumed to be a uniformly charged solid sphere carrying total charge. Q = +Ze. Uniform charge density. Coulomb energy term in SEMF is equal to the total electrostatic energy of a solid sphere of radius R, equal to the nuclear radius and carrying the total charge Q = Ze.
(11) Semi-empirical Mass Formula (c) Coulomb energy: Alternate derivation The electrostatic energy can be calculated by considering the sphere to be built up layer by layer. Suppose the sphere is built up to a radius r. The charge contained within it. Hence the potential on the surface Work done to paste additional charge dq.
(12) Semi-empirical Mass Formula (c) Coulomb energy: Alternate derivation Total work done. Coulomb energy. a3 = 0.71 MeV (0.00076264 u).
(13) Semi-empirical Mass Formula (d) Asymmetry energy: This is an empirical term in SEMF Lighter nuclei have almost the same number of proton and neutron (N = Z) Coulomb repulsion between the protons becomes more important for the heavier nuclei Some extra neutrons must be present to provide additional n − n bonds to compensate for the loss in binding energy due to increased Coulomb repulsion.
(14) Semi-empirical Mass Formula (d) Asymmetry energy: The asymmetry energy (E4) proportional to (N – Z)2 = (A – 2Z)2 and inversely proportional to A. a4 = 23.7 MeV (0.02544 u).
(15) Semi-empirical Mass Formula (e) Pairing energy: Another empirical term in SEMF Stability of nuclei with even Z & even N (e – e) much higher than nuclei with odd Z & odd N (o − o) No. of stable isotopes.
(16) Semi-empirical Mass Formula (e) Pairing energy: Additional term in binding energy appears due to pairing of the same type nucleons with opposite spins The pairing energy (δ) adds to (+ve) the nuclear binding energy for e − e nuclei while it reduces (–ve) for o−o nuclei δ = for o – e nuclei. a5 = 33.5 MeV (0.036 u).
(17) Semi-empirical Mass Formula Binding Energy. Binding Fraction.
(18) Semi-empirical Mass Formula Binding Fraction.
(19) Semi-empirical Mass Formula. a1, a2, a3, a4, a5 can be determined by fitting the above formula to the observed binding fraction versus mass number curve.
(20) Semi-empirical Mass Formula SEMF predicted binding energy & binding fraction of nuclei.
(21) Semi-empirical Mass Formula SEMF / Bethe-Weizsacker formula. Applications: 1) Explanation of α – decay by heavy nuclei 2) Mass parabola and stable nucleus 3) Explanation of β – decay by mirror nuclei.
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