Chem 481 Lecture Material 1/28/09

Experimentally, nuclear volumes (Vn) are found to be proportional to A (Vn % A). Assuming a spherical nucleus of radius r,

3 3 1/3 Vn % r so r % A and r % A

1/3 -15 For many nuclei, r = r0 A where r0 • 1.4 F [note: 1Fermi(F) = 10 m = 1fm]

From scattering experiments it is known that not all (most) nuclei are not spherical nor do they have constant density or charge distribution.

Protons and neutrons, like electrons, are fermions and are associated with an intrinsic spin (s) = ±½. The total angular momentum of a nucleon (j) = l + s where l is the orbital angular momentum of the nucleon and is quantized like that for an electron. The nuclear spin (I) = Gj. The following general nuclear spins are observed:

even A I = 0 or integral odd A I = half-integral even Z, even N I = 0

odd Z, odd N if jp+jn+lp+ln = even, I = #jp-jn# if jp+jn+lp+ln = odd, I = #jp±jn#

Some properties of the nucleus suggest an analogy with a drop of liquid in which molecules interact with immediate neighbors, but not with more distant ones, and the volume of the drop is a sum of the volumes of the molecules present (since they are incompressible). The liquid drop model treats the nucleus as an incompressible fluid and gives rise to a semi-empirical binding energy equation. Nature of Radioactive Decay 1/28/09 page 2

Semi-Empirical Binding Energy Equation

aA Each of A nucleons bound only to limited number of neighbors. Assume

that each is completely surrounded and bound by energy a. Total EB = aA. bA2/3 Nucleons on surface are less tightly bound. Must subtract an amount which is proportional to surface area and r2 or A2/3.

cZ2/A1/3 Loss of binding energy due to p-p Coulomb repulsions. Each proton interacts with all others. The potential energy of Z protons uniformly distributed in a sphere of radius r is proportional to Z2/r or Z2/A1/3.

d(A-2Z)2/A A-2Z (or N-Z) represents the excess of neutrons over protons. Disregarding electrostatic effects, the lowest possible energy for a given number of nucleons is for N=Z (Pauli Exclusion Principle applies to nucleons). Thus, for an excess of one type there is a reduction in binding energy.

±e/A This is a quantitative expression of the fact that even-even nuclei (+e/A) are more stable than odd-odd nuclei (-e/A). This term is zero for even-odd or odd-even nuclei. Nature of Radioactive Decay 1/28/09 page 3

For constant A ($, EC decay processes)

The general parabolic shape of EB vs Z is due to the opposing trends of a decreasing surface energy term (higher EB) and increasing Coulomb and symmetry energy terms (lower EB) with increasing Z.

For odd A (odd Z, even N or even Z, odd N) the last term is zero and there is a single parabola (see figure below). Thus, there is only 1 stable isobar (lightest isobar on parabola) expected. One doesn’t expect adjacent isobars to be stable because they would have to have the same mass exactly. There are 103 nuclides of this type. Note that 141Pr is the stable isobar. 141Ce decays to 141Pr by $- decay with a decay energy of 0.581 Mev. Nature of Radioactive Decay 1/28/09 page 4

For even A there could be 1-3 stable isobars (see cases II-V in figure below).

For even A (odd Z, odd N) with only 1 stable isobar, there are only 4 nuclides of this type (2H, 6Li, 10B, 14N). For even A (even Z, even N) with 3 stable isobars, there are only 3 nuclides of this type (for A=136). There are 156 nuclides that are associated with even A (even Z, even N) with 1 or 2 stable isobars (see example in figure below). Nature of Radioactive Decay 1/28/09 page 5

The most stable value of Z for a given A is gotten by and solving for Z. This yields:

Since it is assumed that Z is a continuous function this will result in non-integral values for ZA. The whole number closest to ZA is the atomic number of the most stable isobar. Each isobaric binding energy parabola corresponds to a cross-sectional slice through a nuclear energy surface diagram for beta stability such a shown in the figure below.

Radionuclides that are higher on the surface are less stable and generally have shorter half lives. This is further illustrated by the Chart of the Nuclides figure below. The liquid drop model is successful in many ways, including providing an explanation for nuclear fission, but not in explaining the extra stability associated with “magic numbers” of nucleons. Nature of Radioactive Decay 1/28/09 page 6

The shell model provides an explanation for “magic numbers”. In this case each nucleon is assumed to move in a potential well which is an approximate representation of the interaction of that nucleon with all others. This is analogous to the approach used for determining the properties of atomic electrons and involves solving the Schrödinger equation for a particle moving in a particular potential-energy field. The harmonic- oscillator potential and the square-well potential, both with infinitely high walls, are commonly used. The resulting nucleon quantum states are characterized by n, the principal quantum number, related to the number of radial nodes in the wave function, and l, a measure of the orbital angular momentum of the nucleon. As for electrons, the states with l = 0, 1, 2, 3,... are designated as s, p, d, f,..., respectively. However, the nuclear definition of n is such that there are n-1 radial nodes (not n-l-1 as for electrons) and states such as 1d and 2f exist for nucleons. As for electrons, there are 2(2l+1) nucleons associated with each quantum level.

Scattering experiments indicate a well depth of ~25-30 MeV, but the minimum energy needed to remove nucleons is 7-8 MeV which indicates that nucleons, in fact, occupy energy levels above the bottom of the well (see figure below). Note that the proton energy levels are higher than the corresponding neutron energy levels as a result of electrostatic repulsions. Nature of Radioactive Decay 1/28/09 page 7

The first results of the shell model did not result in closed shells for 28, 50 and 126 nucleons. In 1949 Maria Mayer, and separately Hans Jensen, resolved this dilemma by incorporating spin-orbit coupling into the calculations. This results in each quantum level with l$1 to be split into a lower level (l+s) and upper level (l-s); note that this is opposite to the situation of spin-orbit coupling for electrons. The figure below illustrates how the shell model with spin-orbit coupling accounts for “magic numbers” of protons and neutrons. Nature of Radioactive Decay 1/28/09 page 8

The shell structure for 116Sn is shown below.

The collective model combines features of both the liquid drop and shell models and provides the best account of high-energy states of the nucleus and certain magnetic and electric properties . It considers the nucleus as having a core of nucleons in filled shells with motion resembling that of a liquid drop. A small number of neutrons (or protons) are in motion outside this core of closed shells that contains a magic number of neutrons (or protons). The "extra" nucleons move in quantized states in a potential well established by the central core.

The shell model predicts long half lives for superheavy elements (Z>104) associated with large magic numbers of protons and neutrons (eg., Z=110 and 114, N=184), resulting in islands of stability. Thus far, no long-lived superheavy elements have been found in nature nor have been synthesized in the laboratory.