Shell Model Shortcomings of the SEMF • magic numbers for N and Z • spin & parity of nuclei • unexplained magnetic moments of nuclei • value of nuclear density The nuclear shell model • General discussion • Assumptions • choosing a potential • L*S coupling • Nuclear “Spin” and Parity Shortfalls of the shell model Magic Numbers It is based on the observation that the semi empirical mass formula, which describes the nuclear masses quite well on average, fails for certain “magic numbers”. The nuclei whose proton number Z or neutron number N is one of the magic numbers 2, 8, 20, 28, 50, 82, 126 are stable and they correspond to the noble gases in the atomic world. The doubly magic nuclei, those with both magic proton and magic neutron numbers, are exceptionally stable. These are the following nuclides: Further these magic number nuclei have higher abundance in nature. For example, helium-4 is among the most abundant (and stable) nuclei in the universe. Magic nuclei have many more stable isotopes and isotones than their neighbors do. Nuclei with N = magic number have much lower neutron absorption crosssections than surrounding isotopes. These nuclei appear to be perfectly spherical in shape; they have zero quadruple electric moments. Magic number nuclei have higher first excitation energy. Energy of alpha or beta particles emitted by magic numbered radioactive nuclei is larger than that from other nuclei. analogous to the atomic shell ment of electrons in an atom, in tability. based on the Pauli exclusion f the nucleus in terms of energy tral potential around which the ucleus. h increase with energy that orbit o describe many phenomena like e spin and parity etc.. In atoms you know you have shells, major shell chemistry is based on that you have inert gases or noble gas shells are closed and then when one more electron you p table and new period is starts right. So, you have that hyd that right, below that you have sodium, potassium. which a new shell starts and there are many things common So, there are many of the properties are comm shell and one more electron a new shell has just started. • The idea of the nuclear shell model had been proposed on the analogy of the atomic shell structure. However, there was some doubt on this idea because there were differences between the cases of atoms and nuclei. The differences are as follows: • In the case of an atom, there is a heavy nucleus at the central position of the system, and it attracts light electrons around by the Coulomb interaction. In contrast to this, there is nothing special at the nuclear center. • In the case of a nucleus, it is clear that the whole system can be described in the image of a "raindrop". Namely, the liquid drop model is valid for nuclei. However, such an image is not valid for atoms. • Moreover, the biggest problem is the difference in the magic numbers of both the systems, atoms and nuclei. Although the magic numbers in atoms are Z = 2, 10, 18, 36, 54, 86, the magic numbers in nuclei are Z or N = 2, 8, 20, 28, 50, 82, 126. • There is a big difference between these two kinds of systems, atoms and nuclei. Even if the shell structure is held in both the systems, the average potentials are presumed to be considerably different. Basic assumption of Shell Model: • • • • Each nucleon moves in potential: average of all other nucleons Net effect of nuclear motions makes potential vary smoothly Each nucleon is bound --» potential is a potential well type Each nucleon moves in 'orbit' of that potential well • The basic assumption of the nuclear shell-model relies on the point that each nucleon in a nucleus moves in an independent fashion under an average potential. • The first step will be to identify a suitable average potential for the nucleons. • Each nucleon moves independently in the nucleus uninfluenced by the motion of the other nucleons. • Each nucleon moves in a potential well which is constant from the center of the nucleus to its edge where it increases rapidly by several tens of MeV. Choosing a Potential • • • According to this model, a given nucleon moves in an effective single particle attractive potential produced by all other nucleons, so the first step is to identify a suitable average potential for the nucleons. Ideally, we would write down the Schrodinger equation for the nuclear force potential and solve it to calculate the energy levels, as done for the hydrogen atom in QM. In the beginning the spin of the nucleon was neglected and therefore we will start in spherical coordinates with a Hamiltonian corresponding to a spinless particle moving in the central field U(r), i. e. • The eigenfunction of this Hamiltonian is a product of radial and angular coordinates • total eigenfunction • the eigenvalue problem acquires the simpler, one dimensional, form, Harmonic Oscillator The eigenvalues corresponding to this potential are Energy depends only upon the number N = 2n+l where it is integer and N ≥ 0. Since ħω is a free parameter that defines the Harmonic oscillator potential, it can be chosen large enough so that there are in the spectrum a series of bands, each band labelled by the number N. The gap between bands was found to be related to the mass number A by the relation, Each value of N implies several values of (n, l) The set of (n, l) values corresponding to a particular value of N would be clustered together in bands separated by an energy ħω. Harmonic oscillator… Continue N=2n+l n l Enl Label Label as Degenercy D=2(2l + 1) ∑D n 0 1 2 3 4 1 2 3 4 5 l 0 1 2 3 4 s p d f g 0 0 0 3/2 ħω 1s 2 2 1 0 1 5/2 ħω 1p 6 8 2 0 1 2 0 7/2 ħω 1d 2s 10 2 20 3 0 1 3 1 9/2 ħω 1f 2p 14 6 40 4 0 1 2 4 2 0 11/2 ħω 1g 2d 3s 18 10 2 70 5 0 1 2 5 3 1 13/2 ħω 1h 2f 3p 22 14 6 112 Table: Energy levels corresponding to an Harmonic oscillator potential of frequency ω Harmonic oscillator… Continue Figure shows the Shell structure obtained with Harmonic oscillator potential Capacity of each level is indicated of its right Large Gaps occurs between the levels, which we associate with shell closure. Circled numbers shows the total number of nucleons at each shell closure. Neutrons and protons, being non-identical particles are counted separately so 1s state can holds 2 protons as well as 2 neutrons. The magic numbers of 2, 8, 20 emerged but the higher levels do not corresponds at all to the observed magic number. Infinite Square Well Potential • • • Square well potential more closely approximates the nuclear density, so it can be a better choice for the mean potential. Infinite square well in 1 dimension: a quick reminder Particle trapped between x=0 and x=a walls are impenetrable => ψ = 0 for x<0 and x>a Potential energy assumed to be zero when particle is inside the walls The harmonic oscillator potential as well as the infinite well potential predict the first few magic numbers. Wood Saxon Potential… An Intermediate Form The infinite square well and HO are not good approximation to the nuclear potential for several reasons: To separate a neutron or proton, we must supply enough energy to take it out of these potentials: infinite separation energy is impossible. In addition, these potentials do not have a sharp edge, but rather closely approximates the nuclear charge and matter distribution, falling smoothly to zero beyond the mean radius R. We choose a intermediate form: V r Vo 1 expr R/ a The parameters R and a give, respectively, the mean radius and skin thickness => R = 1.25A1/3and a = 0.524 fm The well depth V0 is adjusted to give the proper separation energies and is of order 50MeV. This potential is called Wood Saxon potential. Sketch of the functional form of three popular phenomenological shell model potentials: Woods-Saxon, harmonic oscillator, and the square well. Continue… Intermediate Form Intermediate potential: Filling the shells in order with 2(2l+1) nucleons, we again get the magic numbers 2, 8, 20 but the higher numbers do not emerge from calculations. Spin-Orbit potential How we can modify the potential to give proper magic numbers? Inclusion of spin-orbit potential Spin-orbit potential J2 J J J Ls 2 J L L 2L s s s 2 2 2 J L s 2L s J 2 L2 s 2 Ls 2 Ls 𝑉𝑠𝑜 𝑟 𝐿. 𝑠 J 2 j j 1 2 1 2 s 2 s s 1 2 L2 Q.M. 1 j j 1 2 1 s s 1 2 Equation (1) 𝟏 𝟏 s= ½ and Possible values of j are 𝒋 = 𝒍 + 𝟐 𝒐𝒓 𝒋 = 𝒍 − 𝟐 𝟏 Put s= ½ and 𝒋 = 𝒍 + 𝟐 in equation (1) will give < 𝐿. 𝑠 > = 𝟏 ħ2 𝑙 2 Put s= ½ and 𝒋 = 𝒍 − 𝟐 in equation (1) will give < 𝐿. 𝑠 > = − ħ2 (𝑙 2 + 1) • Consider a level such as 1f (l=3) which has degeneracy of 2(2l+1)=14, possible j values 1 5 7 are 𝑗 = 𝑙 ± 2 = 2 𝑜𝑟 2 • So levels are 1𝑓5/2 and 1𝑓7/2 The degeneracy of each level is 2j+1 • The capacity of 1𝑓5/2 state is 6 and 1𝑓7/2 state is 8. Again total 14. • 1𝑓5/2 and 1𝑓7/2 are known as spin-orbit doublets which has some energy separation. For pair of states we can find energy separation as 𝟏 𝑳. 𝒔 − 𝑳. 𝒔 = 𝟐𝒍 + 𝟏 ħ𝟐 𝟏 𝟏 𝒋=𝒍+𝟐 𝒋=𝒍−𝟐 𝟐 The energy splitting increases with l The effect of Vso(r) is negative sign so in the spin-orbit doublet larger j is pushed downward. 𝑉𝑠𝑜 𝑟 𝐿. 𝑠 To Understand…. 1g9/2 appears close to the lower major level so its capacity (10 nucleons) will be added with the lower levels. It will give magic number 50 1f7/2 appears in between 2nd and 3rd shell… (same gap) and it gives the magic number 28 A similar effect occurs at the top of each major shells so we will get all magic numbers. Next Lecture In the next lecture we will use shell model to predict the properties (e.g., parity, magnetic dipole moments) of nuclei……. Thank You