Lecture Nuclear models: The liquid dro model ermi-gas Model WS1/1: Introduction to Nuclear and Particle Physics,, Part I 1
Nuclear models Nuclear models Models with strong interaction between the nucleons Liquid dro model α-article model Shell model Models of non-interacting nucleons ermi gas model Otical model Nucleons interact with the nearest neighbors and ractically don t move: mean free ath λ << R A nuclear radius Nucleons move freely inside the nucleus: mean free ath λ ~ R A nuclear radius
I. The liquid dro model
The liquid dro model The liquid dro model is a model in nuclear hysics which treats the nucleus as a dro of incomressible nuclear fluid first roosed by George Gamow and develoed by Niels Bohr and John Archibald Wheeler The fluid is made of nucleons (rotons and neutrons), which are held together by the strong nuclear force. This is a crude model that does not exlain all the roerties of the nucleus, but (!) does exlain the sherical shae of most nuclei. It also hels to redict the binding energy of the nucleus. George Gamow (19-1968) Niels Henrik David Bohr (1885-196) John Archibald Wheeler (1911-8) 4
The liquid dro model The arametrisation of nuclear masses as a function of A and Z, which is known as the Weizsäcker formula or the semi-emirical mass formula, was first introduced in 195 by German hysicist Carl riedrich von Weizsäcker: M ( A, Z ) Zm B is the binding energy of the nucleus : + Nm n B Volum term Surface term Coulomb term Assymetry term Pairing term mirical arameters: 5
Binding energy of the nucleus Volume energy (dominant term): A - mass number V a V A Coefficient a V 16 MeV The basis for this term is the strong nuclear force. The strong force affects both rotons and neutrons this term is indeendent of Z. Because the number of airs that can be taken from A articles is A(A-1)/, one might exect a term roortional to A. However, the strong force has a very limited range, and a given nucleon may only interact strongly with its nearest neighbors and next nearest neighbors. Therefore, the number of airs of articles that actually interact is roughly roortional to A. 6
Binding energy of the nucleus Surface energy: S a S A / Coefficient a S MeV This term, also based on the strong force, is a correction to the volume term. A nucleon at the surface of a nucleus interacts with less number of nucleons than one in the interior of the nucleus, so its binding energy is less. This can also be thought of as a surface tension term, and indeed a similar mechanism creates surface tension in liquids. The surface energy term is therefore negative and is roortional to the surface area : if the volume of the nucleus is roortional to A (V4/πR ), then the radius should be roortional to A 1 / (R~A 1 / ) and the surface area to A / (S π R πα / ). 7
Binding energy of the nucleus Coulomb (or electric) energy: C a C Z A 1 / Coefficient a C.75 MeV The basis for this term is the electrostatic reulsion between rotons. The electric reulsion between each air of rotons in a nucleus contributes toward decreasing its binding energy: from QD - interaction energy for the charges q 1, q inside the ball q1q int ~ R Here R - emirical nuclear radius: R~A 1 / Thus, C ~ Z A 1 / n 8
Binding energy of the nucleus Asymmetry energy (also called Pauli nergy): Coefficient a sym 1 MeV asym a Sym ( N Z ) A An energy associated with the Pauli exclusion rincile: two fermions can not occuy exactly the same quantum state. At a given energy level, there is only a finite number of quantum states available for articles. Thus, as more nucleons are added to the nuclei, these articles must occuy higher energy levels, increasing the total energy of the nucleus (and decreasing the binding energy). Protons and neutrons, being distinct tyes of articles, occuy different quantum states. One can think of two different ools of states, one for rotons and one for neutrons. or examle, if there are significantly more neutrons than rotons in a nucleus, some of the neutrons will be higher in energy than the available states in the roton ool. The imbalance between the number of rotons and neutrons causes the energy to be higher than it needs to be, for a given number of nucleons. This is the basis for the asymmetry term. 9
Binding energy of the nucleus Pairing energy: air δ 1 / A An energy which is a correction term that arises from the effect of sin-couling. Due to the Pauli exclusion rincile the nucleus would have a lower energy if the number of rotons with sin u will be equal to the number of rotons with sin down. This is also true for neutrons. Only if both Z and N are even, both rotons and neutrons can have equal numbers of sin u and sin down articles. An even number of articles is more stable (δ< δ< for even-even nuclei) than an odd number (δ>). 1
The liquid dro model The different contributions to the binding energy er nucleon versus mass number A: The horizontal line at 16 MeV reresents the contribution of the volume energy. This is reduced by the surface energy, the asymmetry energy and the Coulomb energy to the effective binding energy of 8 MeV(lower line). The contributions of the asymmetry and Coulomb terms increase raidly with A, while the contribution of the surface term decreases. The Weizsäcker formula ( > liquid dro model) is based on some roerties known from liquid dros: constant density, short-range forces, saturation, deformability and surface tension. An essential difference, however, is found in the mean free ath of the articles: for molecules in liquid dros, this is far smaller than the size of the dro; but for nucleons in the nucleus, it is λ ~ R A. 11
Binding energy of the nucleus A grahical reresentation of the semi-emirical binding energy formula: (as a contour lot): the binding energy er nucleon in MeV (highest numbers in dark red, in excess of 8.5 MeV er nucleon) is lotted for various nuclides as a function of Z, the atomic number (on the Y-axis), vs. N, the atomic mass number (on the X-axis). The highest numbers are seen for Z 6 (iron). Z binding energy er nucleon in MeV N 1
II. ermi-gas Model 1
The basic concet of the ermi-gas model The theoretical concet of a ermi-gas may be alied for systems of weakly interacting fermions, i.e. articles obeying ermi-dirac statistics leading to the Pauli exclusion rincile Simle icture of the nucleus: Protons and neutrons are considered as moving freely within the nuclear volume. The binding otential is generated by all nucleons In a first aroximation, these nuclear otential wells are considered as rectangular: it is constant inside the nucleus and stos sharly at its end Neutrons and rotons are distinguishable fermions and are therefore situated in two searate otential wells ach energy state can be ocuied by two nucleons with different sin rojections All available energy states are filled by the airs of nucleons no free states, no transitions between the states The energy of the highest occuied state is the ermi energy The difference B between the to of the well and the ermi level is constant for most nuclei and is just the average binding energy er nucleon B /A 7 8 MeV. 14
Number of nucleon states Heisenberg Uncertainty Princile: The volume of one article in hase sace: The number of nucleon states in a volume V: (V d n ~ r ) V 4π n ~ d d V r d 4π π h V 4π ( πh) ( πh) At temerature T, i.e. for the nucleus in its ground state, the lowest states will be filled u to a maximum momentum, called the ermi momentum. The number of these states follows from integrating eq.(1) from to max : ( πh) ( πh) 6π h Since an energy state can contain two fermions of the same secies, we can have Neutrons: N ( n ) V π h max Protons: n is the fermi momentum for neutrons, for rotons d n ~ Z V (1) ( ) V π h () 15
ermi momentum Let s estimate ermi momentum : Use R R. A 1/ fm, V 4 π 4 R π R A The density of nucleons in a nucleus number of nucleons in a volume V (cf. q.()): n ermi momentum : 6π h V n ~ n V 6π h two sin states! 1/ 9πh 4A R n 4π R 1/ A 6π h 9π n 4A 1/ h R 4 A 9 R π h After assuming that the roton and neutron otential wells have the same radius, we find for a nucleus with nzn A/ the ermi momentum : 1/ n 9π h 5 MeV 8 R ermi energy: M MeV c 16 () The nucleons move freely inside the nucleus with large momentum! M 98 MeV- the mass of nucleon (4)
Nucleon otential V The difference B between the to of the well and the ermi level is constant for most nuclei and is just the average binding energy er nucleon B/A 7 8 MeV. The deth of the otential V and the ermi energy are indeendent of the mass number A: ' V + B 4 MeV Heavy nuclei have a surlus of neutrons. Since the ermi level of the rotons and neutrons in a stable nucleus have to be equal (otherwise the nucleus would enter a more energetically favourable state through β-decay) this imlies that the deth of the otential well as it is exerienced by the neutron gas has to be larger than of the n roton gas (cf. ig.), i.e. > rotons are therefore on average less strongly bound in nuclei than neutrons. This may be understood as a consequence of the Coulomb reulsion of the charged rotons and leads to an extra term in the otential: 17
Kinetic energy The deendence of the binding energy on the surlus of neutrons may be calculated within the ermi gas model. irst we find the average kinetic energy er nucleon: kin kin dn d d dn d d The total kinetic energy of the nucleus is therefore dn d d dn d d ( ) d d M d d 5 M where dn d MeV Const and d distribution function of the nucleons from (1) d Non-relativistic: kin ( ) M (5) where the radii of the roton and the neutron otential well have again been taken the same. 18
Binding energy This average kinetic energy has a minimum at N Z for fixed mass number A (but varying N or, equivalently, Z). Hence the binding energy gets maximal for N Z. If we exand (5) in the difference N Z we obtain The first term corresonds to the volume energy in the Weizsäcker mass formula, the second one to the asymmetry energy. The asymmetry energy grows with the neutron (or roton) surlus, thereby reducing the binding energy Note: this consideration neglected the change of the nuclear otential connected to a change of N on cost of Z. This additional correction turns out to be as imortant as the change in kinetic energy. 19