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CHAPTER 2A
The nucleus and radioactive decay
2.1 The atom and its nucleus An atom is characterized by the total positive charge in its nucleus and the atom’s mass. The positive charge in the nucleus is Ze , where Z is the total number of protons in the nucleus and e is the charge of one proton. The number of protons in an atom, Z , is known as the atomic number and dictates which element an atom represents. The nucleus is also made up of N number of neutrally charged particles of similar mass as the protons. These are called neutrons. The combined number of protons and neutrons, Z+N , is called the atomic mass number A. A specific nuclear species, or nuclide , is denoted by
ZA^ Γ 2. where Γ represents the element’s symbol. The subscript Z is often dropped because it is redundant if the element’s symbol is also used. We will soon learn that a mole of protons and a mole of neutrons each have a mass of approximately 1 g, and therefore, the mass of a mole of Z+N should be very close to an integer^1. However, if we look at a periodic table, we will notice that an element’s atomic weight, which is the mass of one mole of its atoms, is rarely close to an integer. For example, Iridium’s atomic weight is 192. g/mole. The reason for this discrepancy is that an element’s neutron number N can vary. Nuclides with different neutron numbers but the same atomic number Z are called
isotopes. Thus, Iridium has two isotopes, 19177 and , which are characterized
respectively by 114 neutrons and 116 neutrons; a mole of the former weighs 190. g and the latter weighs 192.962917 g. In naturally occurring Iridium, 37.3% of the atoms are
Ir^19377 Ir
(^191) Ir and the remaining atoms are 193 Ir. The atomic weights reported in the periodic
table represent the average masses of each element.
2.2 The mass of an atom How is the mass of an atom determined? One way to determine the mass of an atom is to use a mass spectrometer. Figure 1 shows schematically how one type of mass spectrometer works. The mass spectrometer begins with an ion source, which produces a beam of ionized atoms or molecules (in Chapter 4, we will discuss the different types of ion sources). This beam of ions passes through a velocity sector, through which only ions with a particular velocity are sampled. The velocity sector makes use of an electric ( E ) and a magnetic field ( B ). The E and B fields are oriented such that the electric and magnetic forces on the ions are equal and opposite, that is, q E = - q v×B 2. where q is the charge of the ion. Thus, only ions having a velocity equal to E/B are permitted to continue on their flight paths as ions with any other velocity would be deflected to the walls of the mass spectrometer. Once ions have been selected for their velocities, they enter a magnetic sector, where the magnetic field is uniform and perpendicular to the path of the ions. The magnetic field exerts a force q v×B acting
1
perpendicular to the ion flight path. This force is balanced by the centripetal force mv^2 /r , that is,
B B
E
collecto
rs
Ion source Velocity Sector (^) Magnetic Sector
Modified from Krane 1988
B B
E
collecto
rs
Ion source Velocity Sector (^) Magnetic Sector
Modified from Krane 1988
Figure 1. One type of mass spectrometer used for separating isotopes.
mv^2 /r = q v×B 2. where r is the radius of curvature of the ion flight path. Upon rearranging, Eq. 2.3 can be expressed in terms of r r = mv/qB 2. It can be seen from Eq. 2.4, that a magnetic sector separates ions based on their momentum mv. However, as q, B, and v are uniquely determined, each different mass appears at a different radius r. If we know the magnitude of the electric and magnetic fields precisely, then obtaining the mass of an atom is trivial. To be useful, the precision of our mass determinations must be on the order of one part in 10^6. Unfortunately, it is not possible to know the magnitude of the electric and magnetic fields to this precision. We can, however, measure relative mass differences very precisely by keeping the electric and magnetic fields constant and measuring the difference in radii of curvature, which we can do more precisely. By convention, the 12 C atom is taken to be exactly 12.000000 units on the atomic mass scale, against which all other masses are calibrated. The masses of other isotopes are determined using the “doublet” approach, which is best explained through an example. If we set up the mass spectrometer to collect the molecular ions, C 9 H 20 +^ (nonane) and C 10 H 8 +^ (naphthalene), the difference ∆ in mass units on this scale is 0.09290032±0.00000012 units^2
∆ = m ( C 9 H 20 )− m ( C 10 H 8 )= 12 m (^1 H )− m (^12 C ) 2.
where m(C 9 H 20 ) is the mass of C 9 H 20 and so forth. It follows that the mass of Hydrogen, (^1) H, is given by
(^2) We neglect molecular binding energies, which are negligible.
u
M Zmp Zme Nmn
(where mp, me, and mn refer to the masses of a proton, electron and neutron). The true
mass of 191 Ir (e.g., m ) is actually 190.960584 u, which is 1.62976 u lower than the
combined mass of the elementary particles. The reason for this decrease in mass is that the fusion of these elementary particles into the nucleus is energetically favorable, that is, energy is released when protons and neutrons come together. As energy is equivalent to mass according to the well-known equation
( A Γ )
E = mc^2 2. where E is the energy, m is the mass, and c is the speed of light, the decrease in energy is equivalent to a decrease in mass. The unit conversion between mass and energy is c^2 =931.50 MeV/u.
The energy released by combining N neutrons and Z protons to make a nucleus (^) Z
is called the binding energy (Appendix Table 1):
A Γ
E B = [ Zm p + Nmn − ( m ( A Γ)− Zme )] c 2 2.
The binding energy for electrons is small compared to the nuclear binding energies, which is why in Eq. 2.10, we subtract out the mass energy of the electrons. Eq. 2.10 can be simplified by recognizing that the most elementary nucleus is the Hydrogen nucleus, (^1) H:
Average Binding Energy Per Nucleon (MeV)
Li
Be
B
C
O Ne
Mg Si
Fe
He
(^0 4 8 12 16 20 2430 60 90 120 150 180 210 )
Mass Number A
FusionFusion^ FissionFission Average Binding Energy Per Nucleon (MeV)
Li
Be
B
C
O Ne
Mg Si
Fe
He
(^0 4 8 12 16 20 2430 60 90 120 150 180 210 )
Mass Number A
Average Binding Energy Per Nucleon (MeV)
Li
Be
B
C
O Ne
Mg Si
Fe
He
(^0 4 8 12 16 20 2430 60 90 120 150 180 210 )
Mass Number A
FusionFusion^ FissionFission
Figure 2. Binding energy per nucleon.
E B = [ Z (^1 H^ )+ Nmn − m ( A Γ)] c 2 2.
For 191 Ir, we find that its binding energy is 1478.77 MeV. Figure 2 shows the binding energy per nucleon (for 191 Ir this is 1478.77 MeV/ = 7.7423 MeV) as a function of mass number, e.g. EB/A. Except for mass numbers below ~20 u, it can be seen that EB/A is remarkably constant, lying between 7 and 10 MeV per nucleon. This feature indicates that the binding energy increases approximately linearly with mass number. Another feature of the EB/A curve is that there is maximum at 56 Fe. This means that fusion of masses lower than mass number 56 or fission of masses greater than 56 will release energy. It is this property that we take advantage of when generating nuclear energy by fusion or fission. We will see in Section 2.4 that an understanding of binding energy is fundamental to understanding why certain elements undergo radioactive decay. A nuclide will energetically tend towards decay by a particular mode (α emission, β emission, or spontaneous fission) if its atomic mass is greater than the sum of the masses of the products formed by that decay mode. This means that nuclei above A>100 are unstable towards spontaneous fission into two nuclei of approximately the same mass and that nuclei above A>140 will tend to decay by α emission.
2.3.1 A physical explanation for the binding energy Figure 3 shows a plot of Z versus N for all stable nuclei. One of the most important features is that at Z<20, the number of protons and neutrons is approximately equal. For nuclides having Z>20 , there are more neutrons than protons as shown by the deviation of the mass curve towards the neutron-rich side of the Z=N line. The stable nuclide array represents the most energetically favorable nuclides, that is, the combination of neutrons and protons that yields the highest binding energy. This array is often called the “valley of stability”. The excess of neutrons at high mass numbers can be qualitatively explained by the fact that the more protons one packs together in the nucleus, the higher the Coulombic repulsion (which lowers the binding energy) and therefore the more neutrons needed to physically increase the separation distance of the protons so that their Coulombic repulsion is decreased. In more detail, physicists have come up with a semi-empirical equation to predict the binding energy of a nuclide. The binding energy can be expressed in terms of A and Z as follows
A
A Z
E (^) B Z A avA asA acZZ A asym
2 ( , )^2 /^3 ( 1 )^1 /^3 (^2 ) 2.
We now discuss each term in sequence. The first term is an empirical relationship that describes the fact that the binding energy is approximately a linear function of the atomic mass number A. That EB ~ avA indicates that each nucleon interacts with only its nearest neighbors and that each nucleon contributes roughly the same amount to the binding energy. The constant av must be on the order of 8 MeV per nucleon. The second term accounts for the fact that nucleons on the surface of the nucleus interact less with other nucleons. As the surface area is proportional to R^2 and R ∝ A1/3 , the surface term is proportional to A2/3^ (this assumes that the nucleons are rather incompressible as is implied by the fact that EB ~ avA ). The third term accounts for the Coulombic repulsion of the protons, which decreases the binding energy. The Coulombic repulsion is directly proportional to Z(Z-1) and inversely proportional to the separation distance R , or in other words, proportional to A-1/3. The A-1/3^ proportionality allows for the Coulombic repulsion
a valley, hence the trend of naturally occurring nuclei is called the “valley of stability”. For constant A , we can calculate the minimum mass or the most energetically favorable
nuclide configuration by letting ( ∂ m ( AZ Γ)/∂ Z ) A = 0 :
[ ]
1 / 3 1
1 1 / 3 min (^28)
− −
−
aA a A
m m H aA a Z c sym
n c sym
Since ac = 0.72 MeV and asym = 23 MeV , the first two terms in the numerator are negligible and thus,
2 2 /^3
min A ac a sym
A
Z
It can be seen from Eq. 2.15 that for small A , Zmin is approximately equal to A/2. For heavy nuclides, Z/A ~ 0.4 as expected from observations of the heavy stable nuclei.
2.4 Radioactive decay If we look at the chart of nuclides, which is a plot of naturally occurring nuclides by atomic number (Z) versus neutron number (N), we see the resulting array defines the region of greatest stability, often referred to as the “valley of stability”. For nuclides of low atomic mass, greatest stability is achieved when Z ~ N. However, as atomic mass increases, the stable Z/N ratio increases to ~1.5, that is, a larger number of neutrons are needed in order to maintain stability. This can be explained by the semi-empirical mass formula (Eq. 2.12). In words, we can say that as the number of protons increases in the nucleus, extra neutrons (e.g., N>Z) are needed to offset the increase in Coulombic
Z
N
A Z N
A- Z- N-
α^
decay
A Z- N+
A Z+ N-
β+ E.C.
β−
Z
N
A Z N
A- Z- N-
α^
decay
A Z- N+
A Z+ N-
β+ E.C.
β−
Figure 4. Z versus N diagram showing how Z, A and N change during different decay schemes.
repulsion caused by the larger number of protons. It follows that nuclides, which lie away from the “valley of stability”, will be unstable, and have a tendency to decay towards the “valley of stability”. In other words, a nuclide will energetically tend towards decay by a particular mode (e.g., α emission, β emission, or spontaneous fission; Fig. 4) if its atomic mass is greater than the sum of the masses of the products formed by that decay mode. It turns out that nuclei above A>100 are unstable towards spontaneous fission into two nuclei of approximately the same mass and that nuclei above A>140 will tend to decay by α^ emission. Below, we describe various types of radioactive decays.
2.4.1 Alpha Decay At masses above ~A>140, many nuclides may decay by alpha emission because of the Coulombic repulsion between the increased number of protons being forced into the nucleus. Alpha particles are made up of 2 protons and 2 neutrons (Z=2, N=2) and therefore have the nuclear make-up of the 4 He. Alpha decay can be expressed as follows: AZ Γ→ (^) ZA −− (^42) Γ+ 24 He + Q Eq. 2.
where Q is the energy released by the decay process. The question arises as to why a radionuclide prefers to decay by alpha emission rather than by emission of a larger
particle. Assuming the parent radionuclide is at rest, conservation of energy requires
that the energy released by alpha decay is
ZA^ Γ
2 Q = ( mA − mA − 4 − m α) c Eq. 2.
It turns out that the alpha particle is one of the most tightly bound nuclides, having a low mass defect (or low binding energy). It can be seen from Eq. 2.17 that the smaller the mass defect of the emitted particles, the greater the release of energy Q and the more favorable the decay reaction is. The alpha particle, 4 He, has a mass defect of +0.002603, which is smaller than the mass defects of other low mass nuclides, such as 1 H, 2 H, 3 H and (^3) He, which means that emission of an alpha particle is an energetically more favorable
process than emission by any of these other nuclides. We leave you to verify this in the Problem Set. In general, the greater Q of alpha decay is for a given nuclide, the more likely it is to decay by alpha decay. Thus, it turns out that if we look at the half-lives of alpha decay for various radionuclides, the half-life decreases with increasing Q. There also appears to be a difference between atomic masses with even numbers of neutrons and protons from atomic masses with odd numbers of neutrons and/or protons. For a given Z and Q, the alpha decay half-lives of the latter are much longer than those of even-even nuclides. Geologically relevant alpha decay systems include the decay chain of 238 U to (^206) Pb, 235 U to 207 Pb, and 147 Sm to 143 Nd.
Alpha recoil energies We now consider the distribution of energy caused by alpha decay. When a radionuclide decays to generate an alpha particle and a product nuclide, energy Q is released. This energy is taken up as kinetic energy due to the recoil of the product particles. Assuming that the radionuclide was at rest, the following relationship must hold
2 2 2
Q = mP vP + m α v α 2.
Q β (^) −represents the kinetic energies shared by the negative beta particle and the neutrino.
Negative beta decay results in an increase in proton number at the expense of a neutron. Some geologically relevant negative beta decay systems include the decays of 87 Rb to (^87) Sr, 187 Re to 187 Os, and 176 Lu to 176 Hf.
The decay equation for positron emission is as follows
Γ→− Γ+β +ν+ Q β
A Z
A Z 1 2. In this case, the proton number decreases by one and the neutron number increases by one. An example of positron emission is the decay of 26 Al to 26
Mg, the former a very short-lived radionuclide found only as a cosmogenic nuclide or as a nucleosynthetic product of supernovas.
In the electron capture decay process, an electron reacts with a proton in the nucleus to generate a neutron. Thus, the neutron number increases at the expense of a proton, and for this reason, electron capture decay looks similar to positive beta decay
ZA Γ→^ Z −^ A 1 Γ 2. The electron that is captured into the nucleus usually comes from the lowest energy levels because such electrons are closest to the nucleus and hence have the highest probability of being captured. Once such an electron is captured, an electron vacancy appears which is then filled by an electron from a higher energy level. The transferring of a higher energy electron to a lower energy level results in the release of characteristic X-rays. The decay of 40 K to 40 Ar is an example of an electron capture decay relevant to geologists. In all of the above decays, the nucleus can be left in an excited state. This excited nuclide can then decay to a lower energy level. This process results in the release of characteristic gamma rays (or photons).
2.4.3 Spontaneous Fission Recall from our discussion of the binding energy (Fig. 2) that 56 Fe has the highest binding energy. Nuclides with mass numbers above 56 have lower binding energies and therefore it is energetically favorable for very high mass particles to separate into two nuclides with subequal masses. The release in energy caused by fission decay has been harnessed in nuclear reactors. Some nuclides, such as 235 U can be induced to fission if they are bombarded by neutrons. The fission products are not exactly equal in mass and are not always the same. There appears to be a spectrum of fission product masses. A possible neutron-induced fission reaction is (^235) U + n = 93 Rb + 141 Cs + 2n
where the product nuclides invariably fall on the neutron-rich side of the valley of stability and therefore undergo negative beta decay.
Useful units Speed of light c 2.99792458 x 10^8 m/s Charge of electron e 1.602189 x 10-19^ C Avogadro’s Number NA 6.022045 x 10^23 mole-
Particle rest masses Mass units ( u ) MeV/c^2 Electron 5.485803 x 10-4^ 0. Proton 1.00727647 938. Neutron 1.00866501 939. Alpha 4.00150618 3727.
Conversion Factors 1 eV = 1.602189 x 10-19^ J 1 u = 931.502 MeV/c^2 = 1.660566 x 10-27^ kg 1 barn = 10-28^ m^2 = 10-24^ cm^2 1 Ci = 3.7 x 10^10 decays/s
