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Chapter 7: Bulk Nuclear Properties and Nuclear Matter

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Chapter 7: Bulk Nuclear Properties and Nuclear Matter Chapter 7 Bulk Nuclear Properties and Nuclear Matter The universe contains a remarkably wide variety of atomic nuclei, with mass numbers A (the sum of the numbers of proton Z and the neutron N) up to 250. While there are many interesting properties details that differentiate these nuclei from each other, there is also a powerful set of systematic trends and general properties that provide an important and useful framework for understanding the basic structure of nuclei. There properties are essentially determined by the so-called mean-field approach, in which one nucleon experiences a field which is the sum of the interaction with many other nucleons. This mean field property is a reflection that the density of the nucleons are relatively low and the interaction between the nucleons are relatively week. As such, the nucleon-nucleon correlation is small. The Hartree-Fock Mean field theory is main theoretical tool for dealing with systems with little correlations. Beyond that, the nucleon-nucleon correlations can be calculated using Bethe-Goldstone equations. These equations differ from free-space Schrodinger equation in that many-body Pauli blocking effects are taken into account. These theoretical tools are best illustrated in the example of nuclear matter in which the Coulomb potential is turned off and there are equal number of protons and neutrons. The finite size effect of the nuclear system will be taken up in the next Chapter. 7.1 Nuclear Radii and Densities One of the first relevant properties of the nucleus was determined by Rutherford: the radius is only of order a few fm. The next major step was the use of electron scattering to accurately characterize the charge distribution of nucleons and nuclei. This technique was pioneered by Hofstadter in the 1950’s. In Chapter 4, we considered the elastic scattering of electrons from nucleons. For spinless (i.e., J = 0) nuclei, elastic electron scattering is even simpler. The nuclear matrix element for this process is just 0 Jˆµ 0 , where J µ is the electromagnetic current operator. For elastic scattering h | | i from a J = 0 nucleus, we have 0 Jˆµ 0 = δµ0 0 ρˆ 0 , (7.1) h | | i h | | i so that only the charge operator contributes. The electron scattering cross section is then dσ = σ F (~q) 2 ; (J = 0 J = 0) (7.2) dΩ Mott · | | → 114 7.1. NUCLEAR RADII AND DENSITIES 115 where σMott is the cross section on a pointlike nucleus. An example of this cross section is shown in Figure 7.1. Figure 7.1: Cross section for elastic electron scattering from lead as a function of momentum transfer q along with a theoretical calculation. The form factor F (q) is directly related to the Fourier transform of the nuclear charge distri- bution. First define ρ(q) ρ(x) e−i~q·~xd3~x, (7.3) ≡ h i Z where for a spin 0 nucleus ρ(x) is spherically symmetric. Then use the identity h i ∞ −i~q·~x l e = (2l + 1)i jl(qr)Pl(cos θ) (7.4) Xl=0 and the definition sin qr j (qr)= . (7.5) 0 qr Only the l = 0 term survives due to the spherical symmetry of ρ(x) h i 4π ∞ ρ(q)= ρ(r) sin(qr)rdr . (7.6) q Z0 116 CHAPTER 7. BULK NUCLEAR PROPERTIES AND NUCLEAR MATTER We can then expand sin(qr) and obtain the relation: 4π 1 ρ(q) = ρ(r) qr q3r3 + ... rdr (7.7) q − 6 Z 1 = 4π ρ(r)r2dr q2 r2ρ(r)4πr2dr + ... (7.8) − 6 Z Z 1 = Ze(1 q2 r2 + ...) (7.9) − 6 h i We should note two important properties of ρ(q): lim ρ(q)= Ze (total charge) (7.10) q→0 dF 1 dρ 1 lim = lim = r2 (7.11) q2→0 dq2 Ze q2→0 dq2 −6h i or dF r2 = 6 (mean square charge radius) (7.12) 2 2 h i − dq q =0 Figure 7.2: Charge density distributions for nuclei as determined from elastic electron scattering along with theoretical calculations. 7.2. FERMI GAS MODEL 117 The general strategy is to measure ρ(q) and Fourier transform to obtain ρ(r). As shown in Figure 7.2, one finds a nuclear charge distribution that is well described by a Wood-Saxon form ρ ρ(r)= 0 . (7.13) 1+ e(r−R)/a The basic properties of this distribution are the following. (a)ρ 0.08 efm−3 • 0 ∼ 1 (b)c = r A 3 ; r = 1.2fm. • 0 0 ∼ Therefore, as one adds nucleons the nuclear volume simply grows with the number of nucleons in such a way that the density of nucleons (per unit volume) is constant. That is, nuclei seem to behave like an incompressible fluid of constant density. This property is very different than that of atoms. The elastic electron scattering probes the charge distribution of the protons. The neutron distribution in the nucleus can be very different in principle. However, if one assumes that the strong interactions dominate the nuclear structure and the isospin symmetry is good, the neutron and proton distributions shall not be too different. The neutron distribution in a nucleus can be measured in principle by parity-violating electron scattering with Z-boson exchanges, because the neutral current charge of the proton almost vanishes. 7.2 Fermi Gas Model The fact that the nucleon density is approximately constant throughout the nuclear interior implies that it is not energetically more favorable to be in one location compared to any other. Thus, we expect that the mean potential will be approximately constant throughout the nuclear volume. We may then approximate the nucleus as a free Fermi gas (i.e., non-interacting fermions) confined to a well which approximates the nuclear volume. The kinetic energies associated with localization to the nuclear volume ( few MeV) are large compared to room temperature T = 1 eV, so it ∼ room 40 is safe to use a degenerate Fermi gas (T = 0) for temperatures below about 109 ◦K. (Such large ∼ temperatures are in fact encountered in stellar environments. We are allowed to place 4 particles (spin-isospin) in each orbital. The density of available states is then given by the expression d3p dN = 4 Ω (7.14) × (2π¯h)3 4πp2dp = 4 Ω (7.15) × (2π¯h)3 where Ω is the nuclear volume. If we integrate to the highest filled orbital at momentum pF , we then count the total number of nucleons 16πΩ pF 2Ω 2Ω A = p2dp = p3 = k3 . (7.16) 3 3 2 3 F 3π2 F 8π ¯h Z0 3π ¯h Then the nuclear density is A 2 ρ = = k3 . (7.17) Ω 3π2 F 118 CHAPTER 7. BULK NUCLEAR PROPERTIES AND NUCLEAR MATTER We can compute the average kinetic energy of the nucleons as follows. pF 4 5 1 0 p dp 1 pF /5 T = pF 2 = 3 (7.18) h i 2MN R0 p dp 2MN pF /3 3 p2 = F R (7.19) 5 2MN We can numerically determine the Fermi momentum pF by using the known value of nuclear density (determined from electron scattering charge densities). Recall that the nuclear radius is given by R = 1.2fm A1/3, which implies 4π Ω= (1.2)3A . (7.20) 3 Thus we may evaluate the nuclear density as 3 ρ = fm−3 (7.21) 4π(1.2)3 = .14 fm−3 (7.22) = 2.34 1014 g/cm3 . (7.23) × We then use 2 1/3 3π ρ −1 kF = fm (7.24) 2 ! to obtain pF = ¯hkf = 250 MeV/c . (7.25) (This value agrees very well with the analysis of F (pz) distributions determined from quasielastic electron scattering measurements!) We can now numerically evaluate the average kinetic energy as 3 (¯hk )2 3 (197.3 1.3)2 T = f = × (7.26) h i 5 2M 5 2 938 n × = 21 MeV (7.27) and the maximum kinetic energy is then 5 T = T = 35 MeV. (7.28) max 3h i We note that, under laboratory conditions, the zero temperature limit is certainly valid. Clearly the depth of the well has to be larger than 35 MeV. If a nucleon has binding energy of order 15 MeV or so, as in nuclear matter, the depth of the well will be about 15 + 35 = 50 MeV. 7.3 Independent Particles and Quasi-Elastic Electron Scattering In probing the nuclear structure with electromagnetic probes, the spatial resolution of the virtual photon is governed by the momentum transfer ~q. For momentum transfers up to several hundred MeV, elastic and inelastic scattering from the nucleus can be used to study properties of the ground 7.3. INDEPENDENT PARTICLES AND QUASI-ELASTIC ELECTRON SCATTERING 119 state (such as the charge distribution) and low energy excitations. At higher momentum transfers up to 1-2 GeV/c, the dominant process involves the interaction of the virtual photon with an individual nucleon in the nucleus. (The amplitude for coherent interaction with several or many nucleons decreases rapidly in magnitude at these higher momentum transfers.) The simplest process of this type involves the quasifree knockout of a nucleon, where the struck nucleon is not internally excited but rather recoils elastically. This process is known as “quasielastic” scattering and is a widely used tool for the study of the properties of the nucleon in the nuclear medium. The major effect of the nuclear medium on the struck nucleon is the momentum associated with the localization of the initial state. Thus we consider the case of electron scattering from an initial state nucleon which is in motion in the laboratory frame.
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