Background Students are expected to have a previous background in quantum physics, either at an introductory level [such as the author's text Modern Physics (New York: Wiley, 1983)] or at a more advanced, but still undergraduate, level. A brief summary of the required quantum background is given in Chapter 2.) The text is therefore designed in a “two-track” mode, so that the material that requires the advanced work in quantum mechanics, for example transition probabilities or matrix elements, can be separated from the rest of the text by skipping the sections that require such background. The second characteristic is the unashamedly experimental and phenomenological emphasis and orientation of the presentation.
UNIT I BASIC
BASIC CONCEPTS
SOME INTRODUCTORY TERMINOLOGY
The fundamental positively charged particle in the nucleus is the proton, which is the nucleus of the simplest atom, hydrogen. A one-volume summary of the properties of all known nuclides is the Table of Isotopes, edited by M.
UNITS AND DIMENSIONS
The following comprehensive nuclear physics texts provide explanations or formulations alternative to those in this book. Meyerhof, Elements of Nuclear Physics (New York: McGraw-Hill, 1967); Haro Von Buttlar, Nuclear Physics: An Introduction (New York: Academic Press, 1968).
ELEMENTS OF
QUANTUM MECHANICS
I QUANTUM BEHAVIOR
- PRINCIPLES OF QUANTUM MECHANICS
- PROBLEMS IN ONE DIMENSION The Free Particle
- PROBLEMS IN THREE DIMENSIONS The Infinite Cartesian Well
- QUANTUM THEORY OF ANGULAR MOMENTUM
- PARITY
- QUANTUM STATISTICS
- TRANSITIONS BETWEEN STATES
These functions give the angular part of the solution of the Schrödinger equation for any central potential V(r. Note also that the levels are again degenerate - since the energy depends only on t', the wave functions with different m values all have the same energy.
NUCLEAR PROPERTIES
THE NUCLEAR RADIUS
This AE is the difference between the energy of the I s state in the atom with a. Thus, a single measurement of the X-ray energy K cannot be used to calculate the nuclear radius.
MASS AND ABUNDANCE OF NUCLIDES
Neglecting corrections for the difference in the molecular binding energies of the two molecules (which are of the order of l o p 9 u), we can write Nuclide amounts The mass spectrometer also allows us to measure the relative amounts of the different isotopes of an element. A typical sample of natural krypton would consist of a mixture of the six stable isotopes with the above relative composition.
By adding the measured masses of the six stable isotopes with abundances as relative weighting factors, we can calculate the "average" atomic mass of krypton.
NUCLEAR BINDING ENERGY
We therefore delay discussing the systematics of dissociation energies until we discuss nuclear models in Chapter 5. If each nucleon attracted all the others, then the binding energy would be proportional to A (A - l), or approximately A 2 .The binding energy formula must also include the Coulomb repulsion of the protons, which also tends to make the nucleus less tightly bound.
The explanation for this effect will come from our discussion of the shell model in Chapter 5.).
NUCLEAR ANGULAR MOMENTUM AND PARITY
Measured nuclear spin values can tell us a lot about nuclear structure. Consequently, the spin of the ground state of a nucleus o d 4 must be equal to j of the odd proton or neutron. If we knew the wave function of each nucleon, we could determine the nuclear parity by multiplying the parity of each of the A nucleons together, ending up with a result of n or + or.
Like the spin I , we consider the parity 7~ to be a "global" property of the entire nucleus.
NUCLEAR ELECTROMAGNETIC MOMENTS
One important limitation on the allowed values of I comes from the consideration of the possible z-components of the total angular momentum of the individual nucleons. Another limitation on the multipole moments comes from the symmetry of the nucleus, and is directly related to the parity of the nuclear states. Each electromagnetic multipole moment has a parity, determined by the behavior of the multipole operator when r + - r.
Because of the coupling strength, we can analyze these magnetic moments to learn about the nuclear structure.
NUCLEAR EXCITED STATES
Part of the purpose of nuclear spectroscopy is to observe the possible excited states and measure their properties. For each of the following nuclei, use the semiempirical mass formula to calculate the total binding energy and the Coulomb energy: (a) 21Ne;. Obtain data for several sets of isotopes, plot the data, and compare them to the predictions of the semiempirical mass formula.
What is the most likely value of the ground state spin of 3H or 3He.
THE FORCE BETWEEN NUCLEONS
THE DEUTERON
Let's see how we can analyze this result to study the properties of the deuteron. We justify this assumption later in this section when we discuss the spin of the deuteron.). We can see in Figure 4.1 how close the deuteron is to the top of the well.
The measured spin of the deuteron is I = 1 (how this is measured is discussed in Chapter 16).
PROTON - PROTON AND NEUTRON -NEUTRON INTERACTIONS
When we square the scattered wave function to calculate the cross section, there will be a term proportional to the interference between the parts of the wave function that give scattering at 8 and at 7~ - 8. Here T is the laboratory kinetic energy of the proton of the incident (assuming the target proton is at rest), 8 is the scattering angle in the center-of-mass frame, 6, the phase shift d = 0 for pure nuclear scattering, and 77 = (e2/4nr0hc)p- ' . The comparison of pp and np scattering lengths will be discussed further in the next section.
The study of neutron-neutron scattering should be free from the effects of the Coulomb interaction that made the analysis of proton-proton scattering so complicated.
PROPERTIES OF THE NUCLEAR FORCE
Here "charge" refers to the character of the nucleon (proton or neutron) and not the electrical charge. The nucleon-nucleon interaction may also depend on the relative velocity or momentum of the nucleons. Terms of the form r p or r X p are invariant with respect to equality, but still violate time reversal.
The incident nucleon 2 has r X p outside the paper, resulting in an attractive force, and again scatters to the left.
THE EXCHANGE FORCE MODEL
A p / p where p is the momentum of the incident particle and Ap is the transverse momentum added during the collision. The exchange force model was remarkably successful in explaining the properties of the nucleon-nucleon system. Evaluate the energy of the magnetic dipole-dipole interaction in the deuteron and compare with the nuclear binding energy.
What is the most likely value of the relative orbital angular momentum of the nucleons in state B.
NUCLEAR MODELS
THE SHELL MODEL
Atomic theory based on the shell model provided remarkable explanation of the intricate details of atomic structure. When a subshell contains more than a single particle, all the particles in the subshell can contribute to the quadrupole moment. The particular application of the shell model that we have considered is known as the extremely independent particle model.
The ground state is $+, as expected for the dSl2 shell model state of the 9th nucleon.
EVEN-2, EVEN-N NUCLEI AND COLLECTIVE STRUCTURE Now let's try to understand the structure of nuclei with even numbers of protons
The energy of the first 2' excited state (figure appears to decrease fairly smoothly as a function of A (except for the regions near closed shells). The magnetic moments of the first 2+ states are predicted to be 2(Z/A), which is in the range 0.8–1.0 for the nuclei considered;.The energies of the next few states in the ground state rotational band are calculated to be.
Other properties of the excited states (for example, γ-ray emission probabilities) also help us identify the structure.
UNIT ll NUCLEAR
DECAY AND
RADIOACTIVITY
ALPHA DECAY
WHY a DECAY OCCURS
For a typical emitter 232U (72 y) we can calculate the energy release for different emitted particles from the known masses. However, we will show (Section 8.4) that the partial disintegration constant for the emission of such heavy particles is normally vanishingly small compared to that for an emission. This suggests that if a nucleus is to be recognized as an alpha emitter, it is not sufficient that decay is energetically possible.
Disintegration The constant must also not be too small, otherwise an emission will occur so infrequently that it may not be detectable.
BASIC a DECAY PROCESSES
Typically, the a particle carries about 98% of the Q value, with the much heavier nuclear fragment X' carrying only about 2%. One feature of a decay is so striking that it was noticed as early as 1911, the year Rutherford "discovered" the nucleus. The theoretical explanation of this Geiger-Nuttall rule in 1928 was one of the first triumphs of quantum mechanics.
We can compare the systematic dependence of Q on A with the prediction of the semiempirical mass formula, Equation 3.28.
The quantity f is approximately of the order of u / a where u is the relative speed of the a particle as it oscillates inside the nucleus. Thus, the result of the quantum mechanical calculation for the half-life of a decay is Although this oversimplified theory is not strictly correct, it gives us a good estimate of the decay half-life.
For this reason, it has been proposed that the decay is inhibited by changes in the nuclear structure of the initial and final states (or perhaps by a large change in the decaying angular momentum, examples of which are discussed in the next section).
The intensity depends on the wave functions of the initial and final states, and also depends on the angular momentum fa. As soon as we go above the ground state band, the a-decay intensities become very small, of the order of l o p 6 % of the total decay intensity. If we determine the spatial distribution of this decay, we can in principle determine the relative amounts of the different G values.
As an example of such an experiment, we consider the decay of 253Es into the spin-band states of the ground state of 249Bk.
The various excited states of 247Cf will rapidly decay to the ground state by emitting y-ray photons, so it is useful to have y-ray energies and intensities when constructing a decay scheme. The decaying 229Pa is assigned a + so that in this case the preferential decay to the 2+ band in the daughter has about 46% of the decay intensity. For each decay given in Problem 1, calculate the kinetic energy and velocity of the daughter nucleus after the decay.
The Q value for a decay of 203Tl is calculated to be 0.91 MeV from the masses of the initial and final nuclei.
BETA DECAY
If some of the energy is transferred to the atomic electrons, a corresponding increase in temperature should be observed. These experiments showed that the shape of the spectrum shown in Figure 9.1 is a characteristic of the decay electrons themselves and not the result of any subsequent interactions. To demonstrate the energy of P decay, we first consider free neutron decay (which occurs with a half-life of about 10 min).
So within the accuracy of the measured maximum energy (about 13 keV) we can consider the antineutrino to be massless.
