Chapter 8 Structure of Finite Nuclei The nuclear shell model, developed by Mayer and Jensen in 1952, is now a very successful and highly developed microscopic theory for the structure of finite nuclei. We will explore its basic aspects and illustrate its predictive power in this chapter. We have previously seen that a successful description of nuclear matter can be obtained using a mean single particle potential and a weak residual N-N effective interaction. We will follow a similar strategy for finite nuclei. Assume that the Schr¨odinger equation can be written Hψˆ (1, 2,...A)= Eψ(1, 2,...A) (8.1) with the Hamiltonian given by Hˆ = Ti + U(i) + vij . (8.2) Xi Xi<j The single particle potential U(i) and the residual two body interaction vij are both phenomenolog- ical, to be adjusted to reproduce the properties of real nuclei. Similar to the case of nuclear matter, we hope that if we find the “best” U(i) then vij will be weak can be treated as a “perturbation”. 8.1 Magic Nuclei, Single Particle States, And Spin-Orbit Interac- tions As a first goal, we would hope to reproduce a basic systematic feature of the set of atomic nuclei: the “magic numbers”. Empirically, it is found that there are large deviations from the smooth Bethe-Weiz¨acker formula for nuclear binding energies near certain “magic” values of N and Z. The nuclei with Z, N = 2, 8, 20, 28, 50, 82, 126 have an excess of binding energy. This can be seen in Figure ??. Such a systematic feature should arise from the properties of the single particle potential U(i), so we first work on the single particle potential. Let the Hamiltonian be Hˆ = Ti + U(i) . (8.3) Xi As a first attempt, let’s explore the shell structure (or magic numbers) that result from a simple symmetric three dimensional harmonic oscillator potential. 138 8.1. MAGIC NUCLEI, SINGLE PARTICLE STATES, AND SPIN-ORBIT INTERACTIONS139 Figure 8.1: Binding energy of nuclides showing discontinuities at the magic numbers. 8.1.1 Harmonic Oscillator The potential is given by 1 U(r )= Mω2r2 (8.4) i 2 i and ω can be adjusted to give the correct nuclear radius. We label the orbitals (eigenstates) by α,β,... The energy of a particle in the orbital α is given by { } 3 E = n + ¯hω (8.5) α α 2 where nα is an integer associated with the orbital α. Each nα is the sum of three integers nα = nαx + nαy + nαz (8.6) corresponding to the harmonic oscillator index associated with the three orthogonal axes of a Cartesion coordinate system. Each nαx can take on the values 0,1,2, etc. So there are in general many different ways (each with a different orthogonal wave function) to make the index nα by using different combinations for nαx, nαy, and nαz. The number of different combinations of integers is easily shown to be (n + 1)(n + 2) α α . (8.7) 2 First write out the integers 1, 2, ... nα: 1 2 3 4 ... nα−1 nα 140 CHAPTER 8. STRUCTURE OF FINITE NUCLEI Then place 2 bars somewhere between the integers. There are nα + 1 places for the 1st bar, then nα + 2 places for the second. The number of integers to the left of the left-most bar is nαx, the number to the right of the right-most bar is nαz and the number between is nαy. The number of ways to place the bars is (n + 1)(n + 2) α α (8.8) 2 where we divide by 2 since the bars can be interchanged to give the same result. Now we fill the orbital α with the maximum number of particles, which is the degeneracy (nα + 1)(nα + 2)/2 times 4 (for spin and isospin). Then we fill all the orbitals nα from 0 up to some maximum index nmax and count the number of particles: nmax (n + 1)(n + 2) (n + 1)(n + 2)(n + 3) A = 4 α α = max max max 4 (8.9) × 2 6 × nXα=0 We also can compute the nuclear radius to evaluate ω: 3 4 nmax (n + 1)(n + 2) r2A2/3 = r2 α α (8.10) 5 0 A · h αi 2 nXα=0 ¯h 3 r2 = n + ; (virial theorem) (8.11) α Mω α 2 h i (8.12) so we obtain 42 ¯hω = MeV. (8.13) A1/3 There will be energy “gaps” between shell (orbital) fillings, when we finish filling each orbit. These occur at the values of (n + 1)(n + 2)(n + 3) N, Z = ; n = 0, 1, 2,... (8.14) 3 = 2, 8, 20, 40, 70, 112 (8.15) Although the first three numbers are a promising start, we find that this potential gives the wrong sequence of magic numbers. 8.1.2 Square Well Next we try a square well for U(~ri). This more closely approximates the nuclear density, so should be a better choice for the mean potential. U 0 r R U(r)= − 0 ≤ ≤ . (U > 0) (8.16) 0 r > R 0 The interior region Schr¨odinger equation is ~ 2φ + 2M(U E )φ = 0. (8.17) ∇ α 0 − α α 8.1. MAGIC NUCLEI, SINGLE PARTICLE STATES, AND SPIN-ORBIT INTERACTIONS141 We define k2 2M(U E ) and write the solution in the form φ = R (r)Y (Ω) where the α ≡ 0 − α α nl lm radial equation is d2R 2 dR l(l + 1) nl + nl + k2 R = 0. (8.18) dr2 r dr nl r2 nl − The solutions are the spherical Bessel ftns: Rnl = jl(knlr). (8.19) For a deep well, we require jl(knlR) = 0. (8.20) One can then find (see e.g., Abramowitz and Stegun, p. 467) the sequence of solutions shown in Table 8.1 knlR 2(2l+1) Sum π l=0 (0s) 2 2 4.49 l=1 (0p) 6 8 5.79 l=2 (0d) 10 18 2π=6.28 l=0 (1s) 2 20 6.99 l=3 (0f) 14 34 7.22 l=1 (1p) 6 40 Table 8.1: Magic numbers for the square well potential. Of course, a finite well will modify this sequence, but not drastically. So a square well cannot give the correct magic numbers either. 8.1.3 Spin-Orbit Potential Many potentials have been tried, but none in fact none give the correct sequence of magic numbers. The only successful model uses a spin-orbit potential added to the central potential. A nucleon in the central region “sees” nucleons on all sides, moving in all directions, and with all spin orienta- tions. Therefore, there would be no spin-orbit force contribution and we expect U(r)=constant, independent of ~σ, ~l. Thus, the spin orbit potential Uso is peaked at the nuclear surface, and is usually something like γ dU U (r)= cent (~l ~s)= U ~l ~s (8.21) so r dr · LS · dUcent since dr is only large at surface. The origin of this potential is associated with the exchange of vector mesons (such as the ρ meson) between nucleons. It is easy to show that the splitting between orbitals with j = l 1 is given by a relation ± 2 1 ∆ELS = E 1 E 1 = ULS l + . (8.22) j=l+ 2 j=l− 2 − h i 2 1 3 3 ∆ELS = ULS j(j + 1) l(l + 1) j(j + 1) l(l + 1) (8.23) h i2 − − 4 1 − − − 4 − 1 l+ 2 l 2 142 CHAPTER 8. STRUCTURE OF FINITE NUCLEI 1 1 3 1 1 = U l + l + l l + (8.24) 2 LS 2 2 2 2 h i − − 1 = U l + (8.25) LS 2 h i We will see that U < 0 is the choice that gives the correct sign of the splitting. h LSi Also, the Woods-Saxon potential is commonly used for the central potential: U U(r)= 0 (8.26) 1+ e(r−c)/a0 where a = surface thickness parameter and c = radius parameter A1/3. This central potential 0 ∝ with the above spin-orbit potential gives the level sequence shown in Figure 8.2 which correctly reproduces the observed magic numbers. Figure 8.2: Energy level diagram for Woods-Saxon potential with spin-orbit force. The magic numbers are indicated on the right. 8.2 Nuclear Shell Model The Woods-Saxon potential with a spin-orbit force is the mean-field that is the basis for the nuclear shell model. In addition, one can add residual interactions and correlations to produce more of the 8.2. NUCLEAR SHELL MODEL 143 details of the structure of nuclei. Here we confine ourselves to the basic features that one can study using the mean field potential. 8.2.1 Filled Orbitals States associated with filled orbitals are the “magic” nuclei where Z or N is one of the magic numbers. Many nuclei with Z or N close to magic numbers can be treated with the filled orbital plus a few particles “holes”. Consider an orbital with quantum numbers nlj which is filled with 2j + 1 particles, m = j, m = j + 1,...m = j 1,m = j. Clearly we have − − M = mi = 0. Xi The wave function can be written as a Slater determinant φ (1) φ (1) . ψ = m=−j m=−j+1 (8.27) nlj φ (2) φ (2) . −j −j+1 For each term we have Jz φmi (i) = 0. (8.28) Yi In addition, the antisymmetry of ψnlj implies that, for each i, J±(i)ψnlj = 0 so that J±ψnlj = 0.