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Lecture 1
INTRODUCTION AND PHYSICS OF THE NUCLEUS
INTRODUCTION
Isotope geochemistry has grown over the last 40 years to become one of the most important fields in the Earth Sciences as well as in geochemistry. It has two broad subdivisions: radiogenic isotope geochem- istry and stable isotope geochemistry. These subdivisions reflect the two primary reasons why the relative abundances of isotopes of some elements vary in nature: radioactive decay and chemical fractionation. One might recognize a third subdivision: cosmogenic isotope geochemistry, but this subdivision is per- haps better considered a part of radiogenic isotope geochemistry, since although cosmogenically pro- duced isotopes are not necessarily products of radioactive decay, they are products of nuclear reactions. The growth in the importance of isotope geochemistry reflects its remarkable success in attacking fundamental problems of Earth Science, as well as problems in astrophysics and physics. Isotope geo- chemistry has played an important role in transforming geology from a qualitative, observational sci- ence to a modern quantitative one. To appreciate the point, consider the Ice Ages, a phenomenon that has fascinated geologist and layman alike for the past 150 years. The idea that much of the northern hemisphere was once covered by glaciers was first advanced by Swiss zoologist Louis Agassiz in 1837. His theory was based on observations of geomorphology and modern glaciers. Over the next 100 years, this theory advanced very little, other than the discovery that there had been more than one ice advance. No one knew when these age had occurred, how long they lasted, or why they occurred. Stable and radiogenic isotopic studies in the last 50 years (and primarily in the past 30) have deter- mined the exact times of these ice ages and the exact extent of temperature change (about 3° C or so – cooler in temperate latitudes, more at the poles). Knowing the timing of these glaciations has allowed us to conclude that variations in the Earth’s orbital parameters (the Milankovitch parameters) and re- sulting changes in insolation have been the direct cause of these ice ages. Comparing isotopically deter- mined temperatures with CO 2 concentrations in bubbles in carefully dated ice cores leads to the hy- pothesis that atmospheric CO 2 plays an important role in amplifying changes in insolation. Careful U- Th dating of corals has also revealed the detailed timing of the melting of the ice sheet. Comparing this with stable isotope geothermometry shows that melting lagged warming (not too surprisingly). Other recent isotopic studies have revealed changes in the ocean circulation system as the last ice age ended. Changes in ocean circulation may also be an important feedback mechanism affecting climate. Twenty- five years ago, all this was very interesting, but not very relevant. Today, it provides us with critical in- sights into how the planet’s climate system works. With the current concern over potential global warming and greenhouse gases, this information is extremely ‘relevant’. Other examples of the impact of isotope geochemistry could be listed. The list would include such diverse topics as ore genesis, mantle dynamics, hydrology, and hydrocarbon migration, monitors of the cosmic ray flux, crustal evolution, volcanology, oceanic circulation, archeology and anthropology, envi- ronmental protection and monitoring, and paleontology. Indeed, there are few, if any, areas of geologi- cal inquiry where isotopic studies have not had a significant impact. One of the first applications of isotope geochemistry remains one of the most important: geochro- nology and cosmochronology: the determination of the timing of events in the history of the Earth and the Solar Sytem. The first ‘date’ was obtained by Boltwood in 1907, who determined the age of a ura- nium ore sample by measuring the amount of the radiogenic daughter of U, namely Pb, present. Other early applications include determining the abundance of isotopes in nature to constrain models of the nature of the nucleus and models of nucleosynthesis (the origin of the elements). Work on the latter problem still proceeds. The usefulness of stable isotope variations as indicators of the conditions of natural processes was recognized by Harold Urey in the 1940’s. This course will touch on many, though not all, of these applications. Before discussing applications, however, we must build a firm basis in the physical and chemical fundamentals.
Lecture 1
PHYSICS OF THE NUCLEUS
Early Development of Atomic and the Nuclear Theory
That all matter consists of atoms was first proposed by John Dalton, an English school teacher, in
- Prout showed in 1815 that atomic weights were integral multiples of the mass of hydrogen, something known as the Law of Constant Proportions. This observation was strong support for the atomic theory, though it was subsequently shown to be only approximate, at best. J. J. Thomson of the Cavendish Laboratory in Cambridge developed the first mass spectrograph in 1906 and provided the answer as to why the Law of Constant Proportions did not always hold: those elements not having integer weights had several “ isotopes” , each of which had mass that was an integral multiple of the mass of H. In the mean time, Rutherford, also of the Cavendish Lab, had made another important observation: that atoms consisted mostly of empty space. This led to Niels Bohr’s model of the atom, proposed in 1910, which stated that the atom consisted of a nucleus, which contained most of the mass, and electrons in orbit about it. It was nevertheless unclear why some atoms had different mass than other atoms of the same ele- ment. The answer was provided by W. Bothe and H. Becker of Germany and James Chadwick of Eng- land: the neutron. Bothe and Becker discovered the particle, but mistook it for radiation. Chadwick won the Nobel Prize for determining the mass of the neutron in 1932. Various other experiments showed the neutron could be emitted and absorbed by nuclei, so it became clear that differing numbers of neutrons caused some atoms to be heavier than other atoms of the same ele- ment. This bit of history leads to our first basic observation about the nu- cleus: it consists of protons and neu- trons.
Some Definitions and Units
Before we consider the nucleus in more detail, let’s set out some defi- nitions: N : the number of neutrons, Z : the number of protons (same as atomic number since the number of protons dictates the chemical properties of the atom), A : Mass number (N+Z), M : Atomic Mass, I : Neutron excess number (I=N-Z). Isotopes have the same number of protons but different numbers of neutrons; isobars have the same mass number (N+Z); isotones have the same number of neutrons but different number of protons. The basic unit of nuclear mass is the unified atomic mass unit (also known as the dalton and the atomic Figure 1.1 Neutron number vs. proton number for stable nu- mass unit or amu), which is based on clides.
Lecture 1
Pions and the Nuclear Force
As we noted, we can make an a priori guess as to two of the properties of the nuclear force: it must be very strong and it must have a very short range. Since neutrons as well as protons are subject to the nuclear force, we may also conclude that it is not electromagnetic in nature. What inferences can we make on the nature of the force and the particle that mediates it? Will this particle have a mass, or be massless like the photon? All particles, whether they have mass or not, can be described as waves, according to quantum theory. The relationship between the wave properties and the particle properties is given by the de Broglie Equation :
h
p
where h is Planck’s constant, λ is the wavelength, called the de Broglie wavelength , and p is momen- tum. Equation 1.2 can be rewritten as:
h
mv
where m is mass (relativistic mass, not rest mass) and v is velocity. From this relation we see that mass and de Broglie wavelength are inversely related: massive particles will have very short wave- lengths. The wavefunction associated with the particle may be written as:
c^2
!^2 "
! t^2
# $^2 " ( x , t ) =
mc
2
" ( x , t ) 1.
where ∇^2 is simply the LaPlace operator:
!^2 "
#^2
# x^2
#^2
# y^2
#^2
# z^2
The square of the wave function, ψ^2 , describes the probability of the particle being found at some point in space x and some time t. In the case of the pion, the wave equation also describes the strength of the nuclear force associated with it. Let us consider the particularly simple case of a time-independent, spherically symmetric solution to equation 1.4 that could describe the pion field outside a nucleon located at the origin. The solution will be a potential function V(r), where r is radial distance from the origin and V is the strength of the field. The condition of time-independence means that the first term on the left will be 0, so the equa- tion assumes the form:
!^2 V ( r ) = "
mc
2
V ( r ) 1.
r is related to x, y and z as:
r = x^2 + y^2 + z^2 and
! r
! x
x
r
Using this relationship and a little mathematical manipulation, the LaPlace operator in 1.4 becomes:
!^2 V ( r ) =
r
2
d
dr
r^2
dV ( r )
dr
and 1.5 becomes:
r
2
d
dr
r^2
dV ( r )
dr
mc
2
V ( r )
Two possible solutions to this equation are:
Lecture 1
We define the mass decrement of an atom as:
δ = W – M 1.
where W is the sum of the mass of the constituent particles and M is the actual mass of the atom. For example, W for 4 He is W = 2mp + 2mn + 2me = 4.034248 daltons. The mass of 4 He is 4.003873 daltons, so δ = 0.030375 daltons. Converting this to energy using Equ. 1.1 yields 28.28 MeV. This energy is known as the binding energy. Dividing by A, the mass number, or number of nu- cleons, gives the binding energy per nucleon , Eb:
Eb =
W! M
A
' c
This is a measure of nuclear stability: those nuclei with the largest binding energy per nucleon are the most stable. Figure 1.3 shows Eb as a function of mass. Note that the nucleons of inter- mediate mass tend to be the most sta- ble. This distribution of binding en- ergy is important to the life history of stars, the abundances of the elements, and radioactive decay, as we shall see. Some indication of the relative strength of the nuclear binding force can be obtained by comparing the mass Figure 1.3 Binding energy per nucleon vs. mass number.
r
exp! r
mc
and
r
exp + r
mc
The second solution corresponds to a force increasing to infinity at infinite distance from the source, which is physically unreasonable, thus only the first solution is physically meaningful. Our solution, therefore, for the nuclear force is
V ( r ) =
C
r
exp! r
mc
where C is a constant related to the strength of the force. The term mc/ has units of length-^1. It is a constant that describes the effective range of the force. This effective range is about 1.4 × 10 -^13 cm. This implies a mass of the pion of about 0.15 daltons. It is interesting to note that for a massless par- ticle, equation 1.7 reduces to
V ( r ) = C r 1.
which is just the form of the potential field for the electromagnetic force. Thus both the nuclear force and the electromagnetic force satisfy the same general equation (1.7). Because pion has mass while the photon does not, the nuclear force has a very much shorter range than the electromagnetic force. A simple calculation shows how the nuclear potential and the electromagnetic potential will vary with distance. The magnitude for the nuclear potential constant C is about 10-^18 erg-cm. The constant C in equation 1.8 for the electromagnetic force is e^2 (where e is the charge on the electron) and has a value of 2.3 × 10 -^19 erg-cm. Using these values, we can calculate how each potential will vary with distance. This is just how Figure 1.2 was produced.
Lecture 1
- Magic number nuclides are unusually abundant in nature (high concentration of the nuclides).
- Delayed neutron emission in occurs in fission product nuclei containing N+1 (where N denotes a magic number) neutrons.
- Heaviest stable nuclides occur at N=126 (and Z=83).
- Binding energy of last neutron or proton drops for N*+1.
- Neutron-capture cross sections for magic numbers are anomalously low.
- Nuclear properties (spin, magnetic moment, electrical quadrupole moment, metastable isomeric states) change when a magic number is reached.
The Shell Model of the Nucleus
The electromagnetic spectra emitted by electrons are the principal means of investigating the elec- tronic structure of the atom. By analogy, we would expect that the electromagnetic spectra of the nu- cleus should yield clues to its structure, and indeed it does. However, the γ spectra of nuclei are so Figure 1.5 Graphical illustration of total binding energies of the isobars of mass num- ber A= 81 (left) and A=80 (right). Energy values lie on parabolas, a single parabola for odd A and two parabolas for even A. Binding energies of the 'last' proton and 'last' neutrons are approximated by the straight lines in the lower part of the figure. After Suess (1987).
Lecture 1
complex that not much progress has been made interpreting it. Observations of magnetic moment and spin of the nucleus have been more useful (nuclear magnetic moment is also the basis of the nuclear magnetic resonance, or NMR, technique, used to investigate relations between atoms in lattices and the medical diagnostic technique nuclear magnetic imaging).
Table 1.1. Numbers of stable nuclei for odd and even Z and N
Z N A number of stable nuclei number of very long-lived nuclei (Z + N) odd odd even 4 5 odd even odd 50 3 even odd odd 55 3 even even even 165 11 Nuclei with magic numbers of protons or neutrons are particularly stable or ‘un- reactive’. This is clearly analogous to chemical properties of atoms: atoms with filled electronic shells (the noble gases) are particularly unreactive. In addition, just as the chemical properties of an atom are largely dictated by the ‘last’ valence electron, properties such as the nucleus’s angular momentum and magnetic moment can often be accounted for primarily by the ‘last’ odd nu- cleon. These observations suggest the nucleus may have a shell structure similar to the electronic shell structure of atoms, and leads to the shell model of the nucleus. In the shell model of the nucleus, the same general principles apply as to the shell model of the atom: possible states for particles are given by solutions to the Schrödinger Equation. Solutions to this equa- tion, together with the Pauli Exclusion principle, which states that no two particles can have exactly the same set of quantum numbers, determine how many nucleons may occur in each shell. In the shell model, there are separate systems of shells for neutrons and protons. As do electrons, protons and neu- trons have intrinsic angular momentum, called spin , which is equal to 1 / 2 ( = h /2π, where h is Planck's constant and has units of momentum, h = 6.626 x 10-^34 joule-sec). The total nuclear angular momentum, somewhat misleadingly called the nuclear spin, is the sum of (1) the intrinsic angular momentum of protons, (2) the intrinsic angular momentum of neutrons, and (3) the orbital angular momentum of nu- cleons arising from their motion in the nucleus. Possible values for orbital angular momentum are given by , the orbital quantum number, which may have integral values. The total angular momen- tum of a nucleon in the nucleus is thus the sum of its orbital angular momentum plus its intrinsic angu- lar momentum or spin: j = ± 1 / 2. The plus or minus results because the spin angular momentum vec- tor can be either in the same direction or opposite direction of the orbital angular momentum vector. Thus nuclear spin is related to the constituent nucleons in the manner shown in Table 1.2. Let’s now return to magic numbers and see how they relate to the shell model. The magic numbers belong to two different arithmetic series: N = 2, 8, 20, 40, 70, 112... N = 2, 6, 14, 28, 50, 82, 126... The lower magic numbers are part of the first series, the higher ones part of the second. The numbers in each series are related by their third differences (the differences between the differences between the differences). For example, for the first of the above series: 2 8 20 40 70 112 Difference 6 12 20 30 42 Difference 6 8 10 12 Difference 2 2 2
Table 1.2. Nuclear Spin and Odd-Even Nuclides
Number of Nucleons Nuclear Spin Even-Even 0 Even-Odd 1/2, 3/2, 5/2, 7/ ... Odd-Odd 1,
Lecture 1
Pairing Effects
In the liquid-drop model, it was necessary to add a “fudge factor”, the term δ, to account for the even- odd effect. The even-odd effect arises from a 'pairing energy' that exists between two nucleons of the same kind. When proton-proton and neutron-neutron pairing energies are equal, the binding energy defines a single hyperbola as a function of I (e.g., Figure 1.5). When they are not, as is often the case in the vicinity of magic numbers, the hyperbola for odd A splits into two curves, one for even Z, the other for even N. An example is shown in Figure 1.6. The empirical rule is: Whenever the number of one kind of nucleon is somewhat larger than a magic number, the pairing energy of this kind of nucleon will be smaller than the other kind.
Capture Cross-Sections
Information about the structure and stability of nuclei can also be obtained from observations of the probability that a nucleus will capture an additional nucleon. This probability is termed the capture- cross section, and has units of area. Neutron capture cross-sections are generally of greater use than proton capture cross sections, mainly because they are much larger. The reason for this is simply that a proton must overcome the repulsive coulomb forces to be captured, whereas a neutron, being neutral, is unaffected electrostatic forces. Neutron-capture cross sections are measured in barns, which have units if 10-^24 cm^2 , and are denoted by σ. The physical cross-section of a typical nucleus (e.g., Ca) is of the order of 5 x 10-^25 cm^2 , and increases somewhat with mass number (more precisely, R = r 0 A1/3, where A is mass number and r 0 is the nuclear force radius, 1.4 x 10-^13 cm). While many neutron capture cross sections are of the order of 1 barn, they vary from 0 (for 4 He) to 10^5 for 157 Gd, and are not simple func- tions of nuclear mass (or size). They depend on nuclear structure, being for example, generally low at
Aside: Nuclear Magnetic Resonance
Nuclear magnetic resonance (NMR) has no application in isotope geochemistry (it is, however, used in mineralogy), but it has become such an important and successful medical technique that, as long as we are on the subject of nuclear spin, a brief examination of the basics of the technique seems worthwhile. In brief, some nuclei can be excited into higher nuclei spin energy states by radio frequency (RF) radiation – the absorption of this radiation can by detected by an appropriate RF receiver and the frequency of this absorbed radiation provides information about the envi- ronment of that nucleus on the molecular level. In more detail, it works like this. As we have seen, even-odd and odd-odd nuclei have a nuclear spin. A nucleus of spin j will have 2 j + 1 possible orientations. For example, 13 C has a spin ½ and two possible orientations in space of the spin vector. In the absence of a magnetic field, all orien- tations have equal energies. In a magnetic field, however, energy levels split and those spin orien- tations aligned with the magnetic field have lower energy levels (actually, spin vectors precess around the field vector) than others. There will be a thermodynamic (i.e., a Boltzmann) distribu- tion of nuclei among energy states, with more nuclei populating the lower energy levels. The en- ergy difference between these levels is in the range of energies of RF photons (energies are of the order of 7 x 10 -^26 J, which corresponds to frequencies around 100 MHz). When a nucleus absorbs a photon of this energy, it will change its spin orientation to one having a higher energy level. The precise energy difference between spin states, and hence the precise RF frequency that must be absorbed for the transition to occur, depends on the strength of the applied magnetic field, the na- ture of the nucleus, and also the atomic environment in which that nucleus is located. The latter is a consequence of magnetic fields of electrons in the vicinity of the nucleus. Although the this ef- fect is quite small, it is this slight shift in energy that makes NMR particularly valuable as it allows a non-destructive method of probing the molecular environments of atoms. Non-destructivity is often an advantage for almost any analytical problem, but, as you can easily imagine, it is particu- larly important when the sample is a person!
Lecture 1
magic numbers of N. Capture cross- sections also dependent on the energy of the neutron, the dependence varying from nuclide to nuclide.
Collective Model
A slightly more complex model is called the collective model. It is inter- mediate between the liquid-drop and the shell models. It emphasizes the collective motion of nuclear matter, particularly the vibrations and rota- tions, both quantized in energy, in which large groups of nucleons can participate. Even-even nuclides with Z or N close to magic numbers are par- ticularly stable with nearly perfect spherical symmetry. Spherical nu- clides cannot rotate because of a dic- tum of quantum mechanics that a rota- tion about an axis of symmetry is un- detectable, and in a sphere every axis is a symmetry axis. The excitation of such nuclei (that is, when their energy rises to some quantum level above the ground state) may be ascribed to the vibration of the nucleus as a whole. On the other hand, even-even nuclides far from magic numbers depart sub- stantially from spherical symmetry and the excitation energies of their excited states may be ascribed to rotation of the nucleus as a whole. .
E!
A = 89
O O O O O
I =^15 13 11 9 7
E!
E!
E^ +
!
Rb
Nb
Y
Sr
Zr
EVEN
EVEN (^) NEUTRON ODD NEUTRON
PROTON
PROTON
ODD
E!
E!
E!
E!
g 7 / 2 or d 5 / g 7^2 / 2 or d 5 / 2 g 9 / 2 g 9 / 2
Z^36 37 38
N 53^ 52 51^50 49
Total Binding Energy
Binding Energy of the "Last" Neutron +En and of the "Last" Proton in MeV
Figure 1.6. Schematic of binding energy as a function of I, neu- tron excess number in the vicinity of N=50.
