DRAFT—Chapter One of “new Lamarsh book”
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updated by M. L. Adams, P. Nelson, and P. Tsvetkov
1
Review of Nuclear Physics
Those portions of nuclear physics that are particularly important in reactor theory will be reviewed briefly in this
chapter and in Chapter 2. It is presumed that the reader has already become acquainted with much of the subject
matter of these chapters through prerequisite courses in atomic and nuclear physics.
1-1
The Constituents of Nuclei
The atomic nucleus consists of Z protons and N neutrons, where Z and N are the atomic number and neutron
number, respectively. The total number of nucleons in the nucleus, that is, neutrons and protons, is equal to Z + N =
A, where A is the atomic mass number.
Nuclei having the same atomic numbers but different neutron numbers are known as isotopes.1 Oxygen, for
instance, has three stable isotopes, 16O, 17O, and 18O, and three unstable isotopes, 14O, 15O, and 19O. A table of some
of the more important isotopes encountered in nuclear engineering appears in Appendix I. More complete
tabulations are given in the references at the end of this chapter.
The rest mass of the proton is 1.67262171 .00000029
10-24 gram (g), henceforth more concisely
-24
1.67262171(29) 10 g. It carries a positive charge of 1.60217653(14) 10-19 coulombs (C), equal in magnitude
to the charge of the electron, also known as the elementary charge and often given the symbol e. The proton has
long been thought to be a stable particle, but there is some recent theoretical suggestion that it decays, albeit
extremely slowly (half-life approximately 1031 years (yr)).
The rest mass of the neutron is slightly larger than that of the proton,2 namely 1.67492728(29) 10-24 g, and it is
electrically neutral. The neutron is not stable, however, unless it is bound in a nucleus. A free neutron decays to a
proton with the emission of a β-ray and an antineutrino, a process that occurs with a half-life (see Section 1-7 below)
of approximately 10.3 minutes (min). It will be shown later in this book that the average lifetime of neutrons in a
nuclear reactor before they are absorbed or leak from the system is at most about 10-3 seconds (s). The instability of
the neutron is therefore of little importance in reactor theory.
1-2
Particle Wavelengths
All particles in nature have a split personality, behaving sometimes like individual particles and sometimes like
waves. The wavelength λ associated with a particle having momentum p is given by
where h is Planck's constant.3 It is customary to speak of a particle's reduced wavelength, denoted by “,” which is
simply its wavelength divided by 2π. Thus “” can be written as
(1-1)
where  is Planck's constant divided by 2π. For nonrelativistic neutrons, Eq. (1-1) gives
(1-2)
where E is the neutron kinetic energy.4
1
Similarly, nuclei having the same mass number but different atomic numbers are termed isobars.
See R. F. Dashen, Phys. Rev. 135, B1196 (1964), for an explanation of this difference, based on first principles.
3
A table of physical constants is given in Appendix I.
2
DRAFT—Chapter One of “new Lamarsh book”
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updated by M. L. Adams, P. Nelson, and P. Tsvetkov
1-3
Nuclear Radii
To a first approximation, the atomic nucleus can be considered to be a sphere of radius R, given by the expression
,
(1-3)
where A is the atomic mass number. The constant 1.2×10−13 is derived from various types (neutron and electron) of
scattering experiments and is somewhat sensitive to the measure adopted to define overlap between probe particle
and the nucleus. For these reasons radii computed from Eq. (1-3) are approximate, not precise. It should also be
noted that this equation is not valid for very light nuclei.5
It is often convenient to express the nuclear radius in terms of the classical radius of the electron, re, which is
defined by the formula
,
(1-4)
where ε0 is the permittivity of free space, e and me are the elementary charge and electron mass, respectively, and c
is the speed of light.6 The numerical value of re is 2.8179 × 10-13 cm, so that R can be written roughly as
.
(1-5)
Although Eq. (1-5) is not numerically exact, it is sufficiently accurate for many qualitative and semiquantitative
purposes.
The volume V of a nucleus is proportional to R3, and, in view of Eq. (1-3), V is proportional to A. Thus the
average number of nucleons per unit volume in a nucleus, that is, A/V, is roughly constant for all nuclei. Such a
uniform density of nuclear matter suggests that nuclei are similar to small drops of an incompressible fluid. This
liquid-drop model of the nucleus has been widely used in nuclear physics, especially in the early days,7 and accounts
for many properties of nuclei. This model will be discussed further in Chapter 2.
1-4
Nuclear Mass
The masses of atoms are expressed in terms of the atomic mass unit, or u. Initially this was defined as one-sixteenth
of the mass of the neutral 16O atom and denoted as amu. Since 1960 the accepted definition has been one-twelfth
the mass of the neutral 12C atom, at rest, unbound and in the ground state. The u is experimentally determined, and
the currently accepted value is 1.66053886(28) × 10−24 g. In energy units the u is equivalent to 931.494043(80)
MeV. (This is the mass multiplied by c2, the square of the speed of light.) In terms of this unit the proton and
neutron have respectively the following masses:
Compilations of nuclear masses are noted in the references at the end of this chapter. These days most available
compilations will be based on the 12C definition noted above, but some caution is advisable in using older
compilations.
4
1 electron volt = 1 eV is a unit of energy equal to approximately 1.6018 10–19 joule (J).
For further details on nuclear sizes see Chap. 3 of the text by Krane that is cited at the end of this chapter.
6
That is the “classical electron radius” re, also called the Compton radius, is defined by equating the rest energy of the electron
with the electrostatic potential energy of a sphere of charge equal to the elementary charge and radius equal to re. This is not
intended to suggest any internal structure to the electron, which thus far appears not to have shown any experimental indications
of such structure (i.e., being other than a point particle).
7
The liquid-drop model goes back to Niels Bohr; cf. N. Bohr, Nature 143, 330 (1939).
5
DRAFT—Chapter One of “new Lamarsh book”
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updated by M. L. Adams, P. Nelson, and P. Tsvetkov
1-5
Binding Energy
The masses of all nuclei are slightly less than the sum of the masses of the individual neutrons and protons contained
in them. This difference in mass is called the mass defect, Δ, and is given by
,
(1-6)
where MA is the mass of the nucleus. Equation (1-6) can also be written as
,
(1-7)
where me is the mass of an electron. The quantity Mp + me is approximately equal to the mass MH of neutral
hydrogen, while MA + Zme is approximately equal to the mass M of the neutral atom in question. The mass defect of
the nucleus is therefore
.
(1-8)
Equations (1-6) and (1-8) are not precisely equivalent because of differences in electronic binding energies, but this
is not important for most purposes, because the nuclear binding energies are several orders of magnitude greater
than the electronic binding energies.
When Δ is expressed in energy units (i.e., multiplied by c2), it is equal to the energy that is necessary to break the
nucleus into its constituent nucleons. This energy is known as the binding energy of the system, because it
represents the energy with which the nucleus is held together. On the other hand, when a nucleus is produced from
A nucleons, Δ is equal to the energy released in the process. For example, when a neutron and proton combine to
form a deuteron, the nucleus of 2H, a 2.23-MeV γ ray is emitted. Because this energy escapes as the deuteron is
formed, the mass of the deuteron in energy units is 2.23 MeV less than the sum of the masses of neutron and proton.
The neutron and proton can later be separated again, provided the binding energy is resupplied to the system. This
can be done in a number of ways; one instance is by bombarding deuterium with γ rays having an energy greater
than 2.23 MeV.
The total binding energy of nuclei generally tends to increase with increasing atomic mass number A. However,
the binding energy per nucleon is not constant, but rather tends to vary in a most interesting manner. This is most
often depicted by a binding energy curve, which is to say a plot of binding energy per nucleon vs. mass number.
One version of a binding energy curve is shown in Fig. 1-1. Note that the binding energy per nucleon varies
somewhat irregularly at low mass numbers, but above A=50 it is a generally decreasing function of A. This behavior
of the binding energy curve is particularly important in determining possible sources of nuclear energy.
Whenever it is possible to form a more stable configuration by combining two less stable nuclei, energy is
released in the process. Such reactions are possible with a great many pairs of isotopes. For instance, when two
deuterons, each with binding energy of 2.23 MeV, react to form H3, having a total binding energy of 8.48 MeV,
according to the equation
,
(1-9)
there is a net gain in the binding energy of the system of 8.48 MeV – 2 x 2.23 MeV = 4.02 MeV. In this case, this
energy appears as kinetic energy of the product nuclei 3H and 1H.
Reactions such as Eq. (1-9), in which at least one heavier, more stable nucleus is produced from two lighter, less
stable nuclei, are known as fusion reactions. Reactions of this type are responsible for both the enormous release of
energy in stellar interiors and the hope of thermonuclear power as a long-term source of energy for mankind.
Moving now to regions of large A in Fig. 1-1, it will be seen that a more stable configuration is formed when a
heavy nucleus splits into two parts. The binding energy per nucleon in U238, for instance, is about 7.5 MeV, while it
is about 8.4 MeV in the neighborhood of A = 238/2 = 119. Thus if a uranium nucleus divides into two lighter nuclei,
each with about half the uranium mass, there is a gain in the binding energy of the system of approximately 0.9 MeV
per nucleon, which amounts to a total energy release of about 238 0.9 = 214 MeV. This process is called nuclear
fission. It is the source of energy in nuclear reactors.
DRAFT—Chapter One of “new Lamarsh book”
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updated by M. L. Adams, P. Nelson, and P. Tsvetkov
Fig. 1-1. The binding energy per nucleon of the most stable isobar of mass number A. The solid circles
represent nuclei having an even number of protons and an even number of neutrons, whereas the crosses represent
odd-A nuclei. (M. A. Preston, “Physics of the Nucleus,” Addison-Wesley Publishing Company, Inc., Reading,
Mass., 1962)
Nuclei having even numbers of both neutrons and protons tend to be more stable (i.e., have higher binding
energy per nucleon) than those having odd numbers of either. Further, nuclei containing 2, 6, 8, 14, 20, 28, 50, 82,
or 126 neutrons or protons are especially stable. These nuclei are said to be magic, and their associated numbers of
nucleons are known as magic numbers. These correspond to the numbers of neutrons or protons that are required to
fill shells (or subshells) of nucleons in the nucleus in much the same way that electron shells are filled in atomic
structures. Nuclei having magic numbers of both neutrons and protons are extraordinarily stable. Calcium has two
of these doubly magic isotopes, 40Ca and 48Ca. Note that 4He (i.e., the α-particle) is a doubly magic nucleus.
The existence of magic nuclei has a number of practical consequences in nuclear engineering. For instance,
nuclei with a magic neutron number absorb neutrons to only a very small extent, and materials of this type can be
used where neutron absorption must be avoided. Zirconium, for example, whose most abundant isotope contains 50
neutrons, has been widely used as a structural material in reactors, because of its property of low absorption of
thermal neutrons, as well as its substantial resistance to heat and corrosion.8 Similarly, bismuth (in liquid form),
whose only naturally occurring isotope has 126 neutrons, has been used (in conjunction with lead) as a coolant for
reactors in Soviet nuclear-powered submarines.9
1-6
Excited States in Nuclei
The preceding sections pertain to nuclei in their ground state, that is, in the state of lowest energy. Nuclei also have
excited states or energy levels. Figure 1-2 shows the known energy levels of 12C, 28Al, and 235U, which are typical
light, intermediate, and heavy nuclei, respectively. It will be observed that in every case the density of levels
increases with increasing excitation energy. Furthermore, it should be noted that the density of levels at a given
excitation energy also increases with increasing mass of the nucleus. While it is generally true that energy levels
become increasingly dense with increasing mass number, the magic nuclei are important exceptions to this rule, and
8
Cf. http://www.chemicalland21.com/arokorhi/industrialchem/inorganic/
ZIRCONIA.htm, accessed August 29, 2004.
9
See http://www.rssi.ru/IPPE/General/lmtr.html, accessed August 29, 2004.
DRAFT—Chapter One of “new Lamarsh book”
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updated by M. L. Adams, P. Nelson, and P. Tsvetkov
their excited states tend to resemble those of the lighter nuclei. This is shown in Fig. 1-3 for the nucleus 209Bi. Such
a level scheme has little in common with those of other heavy nuclei.
It is interesting to compare the origins of excited states in nuclei and atoms. Recall that in atoms the usual
excited states are formed as the result of the continued excitation of only one electron at a time. That is, the states
are formed by raising one electron through a succession of levels until it finally escapes from the system and the
atom is ionized. More highly excited states of the atom are then formed by the excitation of a second electron until
the atom is doubly ionized, and so on. Atomic energy states arise in this way because it generally takes less energy
to raise an electron, already in an excited state, to a higher level than to place two or more electrons in excited
states.
Fig. 1-2. The energy levels of 12C, 28Al, and 235U.
The situation is quite different with nuclear energy levels. Once a nucleon has been raised to an excited level, it
very frequently requires less energy for a second nucleon to make the same or another transition than it does to raise
the first nucleon to a higher level. Thus the excited states of nuclei differ from those of atoms in that the higher
excited states of nuclei usually result from the simultaneous excitation of a number of nucleons; they are not due to
the continued excitation of a single nucleon. For this reason, excited states in nuclei can exist above the “ionization”
or binding energy of a single nucleon, a situation that is only rarely found in atomic structure. Thus an energy level
can occur at 10 MeV in a nucleus in which only 8 MeV is necessary to remove a nucleon, simply because the 10
MeV of excitation energy is shared among several nucleons.
DRAFT—Chapter One of “new Lamarsh book”
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updated by M. L. Adams, P. Nelson, and P. Tsvetkov
Fig. 1-3. The energy levels of 209Bi. The numbers indicate the energy level, in kev. The vertical arrows indicate
observed γ-ray transitions. (Extracted from the Brookhaven National Laboratory interactive data file “Nuclear
structure and decay data” (NUDAT), http://www.nndc.bnl.gov/nudat2/index.jsp, accessed August 29, 2004.)
The binding energy of the least bound nucleon in a nucleus is called the virtual energy. This is the minimum
energy that must be added to the energy levels of a nucleus in order to remove a nucleon and is entirely analogous to
the first ionization energy of an atom. Nuclear excited states above the virtual energy are called virtual states or
virtual levels, while states below the virtual energy are called bound states or bound levels. It is clearly possible for
nuclei in virtual states to decay by nucleon emission, whereas this is not possible for nuclei in bound states. It may
be noted, however, that in order for a nucleon to be emitted from a virtual state, other excited nucleons must give up
some of their own energy so that escape from the system is possible. Such a concentration of energy upon only one
nucleon can occur as the result of collisions between the nucleons in the nucleus. However, when the energy of the
excited nucleus is shared among a great many nucleons, such an unlikely distribution of energy among the nucleons
rarely occurs. On the other hand, if only a few nucleons are involved in the formation of a virtual level, the chance
that one nucleon can receive enough energy to escape from the nucleus is much greater.
1-7
Radioactivity
The spontaneous disintegration of nuclei, a process known under the (now) somewhat erroneous name
“radioactivity,” is governed by only one fundamental law, namely, that the probability per unit time that a nucleus
decays is a constant independent of time. This constant is called the decay constant and is denoted by λ.
Consider the decay of a sample of radioactive material. If at time t there are n(t) atoms that have not as yet
decayed, then in view of the definition of λ, λn(t) dt of these will decay on the average in the time dt between t and
DRAFT—Chapter One of “new Lamarsh book”
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updated by M. L. Adams, P. Nelson, and P. Tsvetkov
t + dt. The rate of decay of the sample at time t is therefore simply λn(t). This decay rate is also called the activity
of the sample, and is measured in curies. One curie is defined as 3.7 1010 disintegrations per second, with
abbreviation Ci. (One disintegration per second is termed a becquerel, with abbreviation Bq.)
The decrease in the number of undecayed nuclei in time dt is given by
.
(1-10)
This equation can be integrated to obtain
,
(1-11)
where n0 is the number of atoms at t = 0.
The time during which the activity of a radioactive sample falls by a factor of two is known as the half-life, and
is denoted by the symbol T1/2. By this definition
,
and upon substitution of this into Eq. (1-11) it is easily seen that
.
(1-12)
Consider next a sample of radioactive material containing n0 nuclei at time t = 0. In view of Eq. (1-11), there will
be n0e–λt nuclei remaining after t s. The fraction of the original nuclei that have not decayed is therefore e–λt. This
fraction can also be viewed as the probability that any nucleus will not decay in the interval from t = 0 to t = t. Now
let p(t)dt be the probability that a nucleus decays in the time dt between t and t + dt. In other words, p(t)dt is the
probability that a nucleus survives up to the time t and then decays in the interval from t to t + dt. This is evidently
equal to the probability that the nucleus has not decayed up to the time t multiplied by the probability that it does in
fact decay in the additional time dt. It follows therefore that
(1-13)
If Eq. (1-13) is integrated over all t, there is obtained
.
(1-14)
This shows that the probability that a radioactive nucleus eventually decays is equal to unity, as would be expected.
The mean life of a nucleus can now be determined by finding the average value of t over the probability
distribution p(t). Denoting the mean-life by ,
.
(1-15)
.
(1-16)
In view of Eq. (1-12), the mean life can also be written as
It is frequently necessary to consider situations in which radioactive isotopes are produced and decay within a
reactor. If they are produced at a rate of R(t) atoms/sec at the time t, the change in the number of atoms of the
isotope in the subsequent interval dt is
.
This equation can be solved, up to quadrature, by multiplying through by the integrating factor eλt. Thus
(1-17)
DRAFT—Chapter One of “new Lamarsh book”
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updated by M. L. Adams, P. Nelson, and P. Tsvetkov
,
(1-18)
where n0 is the number of atoms present at t = 0. The function n(t) can be evaluated, in principle (by numerical
means, if necessary), once the explicit form of R(t) is known.
Equation (1-18) can also be used to find the amount of an isotope at any point in a radioactive decay chain of the
type
.
Consider, for example, the accumulation of the isotope B. Because the disintegration of one atom of A gives one
atom of B, the rate of production of B is equal to the activity of A; that is, the rate of production of B is
(1-19)
If this is inserted into Eq. (1-18), and the indicated integration is performed, then there follows the result
,
(1-20)
where nA0 and nB0 are the numbers of atoms of A and B at t = 0.
1-8
The Decay of Excited States
It has been found that the fundamental law of natural radioactivity, i.e., the fact that the probability per unit time that
a system decays is a constant, also applies to the spontaneous decay of nuclei in excited states. It is customary,
however, in discussing the decay of an excited state to express the decay constant λ in terms of a new quantity, Γ,
called the level width, which is defined by the relation
.
(1-21)
Because  has units of energy × time, and λ has units of inverse time, it is evident that Γ has units of energy. In
other words, Γ is the decay constant of an excited state expressed in energy units.
The level width can be used instead of the usual decay constant to describe the decay of nuclei from excited
states. If, for example, there are n0 nuclei in a certain excited state at t = 0, t sec later there will be
(1-22)
nuclei left in this state. Formulas such as Eq. (1-22) are not often useful, however, since the decay of excited states
usually occurs so quickly that the time dependence of the process is not easily observed.
From Eqs. (1-15) and (1-21) it follows that the mean-life of a state of width Γ is given by
.
(1-23)
Thus a state of large width is short-lived; a state of small width is long-lived.
The widths of a great many excited states have been measured. For instance, the first virtual state of U239 is at
6.67 eV above the virtual energy and has a width of 27 meV (i.e., 0.027 eV). The mean life of this state is therefore
about
.
Lifetimes as short as this cannot be measured by ordinary methods, and this state appears to decay (in this case
primarily by γ-ray emission) as soon as it is formed.
DRAFT—Chapter One of “new Lamarsh book”
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updated by M. L. Adams, P. Nelson, and P. Tsvetkov
The decay of a nucleus from an excited state can frequently occur in a number of ways. If the nucleus is in a
bound state, however, nucleon emission cannot occur and, with few exceptions,10 the nucleus decays by the emission
of γ rays. On the other hand, if the nucleus is in a virtual state, then one or more nucleons, in addition to γ-rays, may
be emitted, depending on the energy of the state.
The probability per unit time of each mode of decay of an excited state is described in terms of a partial width
characteristic of each process. For instance the partial width for γ-ray emission, Γγ, which is also known as the
radiation width, is the probability per unit time (expressed in energy units) that the excited nuclei decays by γ-ray
emission. Similarly, Γn, the neutron width, gives the probability per unit time that the state decays by neutron
emission, etc. Since the total decay probability is the sum of the probabilities for all possible distinct processes, the
total width is the sum of the partial widths:
(1-24)
The relative probability that an excited state decays by a given mode is evidently the ratio of the partial width of
the particular mode to the total width. For example, the relative probability that a state decays by γ-ray emission is
Γγ / Γ, that it decays by neutron emission is Γn / Γ, and so on.
1-9
Nuclear Reactions
When two nuclear particles, that is, two nuclei or a nucleus and a nucleon, interact to produce two or more nuclear
particles or gamma radiation, a nuclear reaction is said to have taken place. If the initial nuclei are denoted by a and
b, and the product nuclei (for simplicity it will be assumed that there are only two) are denoted by c and d, the
reaction can be represented by the equation
.
(1-25)
In equations of this type, the interacting particles are usually written in terms of neutral atoms, although some of the
particles may be ionized.
In the usual experimental arrangement, one of the particles, say a, is at rest in some sort of target, and the particle
b is projected against the target. In this case, Eq. (1-25) is often written in the abbreviated form
,
or
,
(1-26)
whichever is more appropriate. For example, when oxygen is bombarded by energetic neutrons, one of the reactions
that can occur is
.
(1-27)
In abbreviated form this is
,
(1-28)
where the symbols n and p refer to the incident neutron and emergent proton, respectively.
The detailed theoretical treatment of nuclear reactions is beyond the scope of this book. For present purposes,
however, it is sufficient to note four of the fundamental laws governing these reactions:
(1)
(2)
10
Conservation of nucleons. The number of nucleons before and after a reaction must be equal.
Conservation of charge. The sum of the charges on all the particles before and after a reaction must be
equal.
The first excited state of 16O at 6.06 MeV, for example, decays by emitting an electron-positron pair; decay by β-ray emission
is also possible from certain long-lived (isomeric) states.
DRAFT—Chapter One of “new Lamarsh book”
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updated by M. L. Adams, P. Nelson, and P. Tsvetkov
(3)
(4)
Conservation of linear and angular momentum. The total linear momentum of the interacting particles
is not changed by the reaction, because no external forces act upon the particles. The same is true of
angular momentum. The importance of the conservation of momentum will be discussed further in
Chapter 2.
Conservation of energy. Energy is conserved in all nuclear reactions.
The principle of the conservation of energy can be used to predict whether a certain reaction is energetically
possible. Consider, for example, a reaction of the type given in Eq. (1-25). The total energy before the reaction is the
sum of the kinetic energies of the particles a and b plus their rest-mass energies. Similarly, the energy after the
reaction is the sum of the kinetic energies of particles c and d plus their rest-mass energies. By conservation of
energy it follows that
,
(1-29)
where Ea, Eb, etc. are the kinetic energies of particles a, b, etc. Equation (1-29) can be rearranged in the form
.
(1-30)
Thus it is seen that the change in the kinetic energies of the particles before and after the reaction is equal to the
difference in the rest-mass energies of the particles before and after the reaction.
The right-hand side of Eq. (1-30) is known as the Q-value of the reaction; that is,
.
(1-31)
From Eq. (1-30) it is clear that when Q is positive, there is a net increase in the kinetic energies of the particles.
Such reactions are called exothermic. When Q is negative, on the other hand, there is a net decrease in the energies
of the particles and the reaction is said to be endothermic. With exothermic reactions, nuclear mass is converted into
kinetic energy, while in endothermic reactions, kinetic energy is converted into mass.
The Q-value can be calculated without difficulty for any reaction involving atoms whose masses are known,
provided the product nuclei are formed in their ground states. Consider, for instance, the reaction
,
which is of considerable importance both as a laboratory source of neutrons and as a possible source of
thermonuclear power. The Q-value of the reaction is found from the following (atomic) masses:
Thus Q = 5.0301511 u − 5.0112682 u = 0.0188829 u, which is equivalent to 17.6 MeV. This means that if, for
instance, a stationary tritium (H3) target is bombarded with 1-MeV deuterons, the sum of the energies of the
emerging neutron and alpha particle will be 18.6 MeV. On the other hand, if both the deuteron and H 3 have
essentially zero kinetic energy, the emerging neutron and alpha particle share only 17.6 MeV. In this latter case, it is
easy to show that in order to conserve linear momentum, the neutron actually has an energy of 14.1 MeV while the
alpha particle has 3.5 MeV.
When a reaction leads to nuclei in excited states, the Q-value cannot be calculated directly from the masses of
the neutral atoms. In this case, either the masses of the nuclei in their excited states can be inferred from the neutral
masses and the energies of the excited states, or else the Q-value can be calculated for the reaction proceeding to the
ground states of the product nuclei and then adjusted to take into account their states of excitation.
DRAFT—Chapter One of “new Lamarsh book”
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updated by M. L. Adams, P. Nelson, and P. Tsvetkov
1-10 Summary
In the remainder of this book we will explore how the basic physics described in this chapter can be applied to create
controlled self-sustaining fission chain reactions in nuclear reactors, thereby providing compact and long-lived
sources of useful energy for a variety of applications.
References
General
BURCHAM, W. E., Nuclear Physics. New York: McGraw-Hill, 1963, Parts A and C.
EISBERG, R. M., Fundamentals of Modern Physics. New York: Wiley, 1961, Chapters 1-14,16.
ENDT, P. M., and M. DEMEUR, Nuclear Reactions, Vol. I. Amsterdam: North Holland, 1959, Chapter 2.
EVANS, R. D., The Atomic Nucleus. New York: McGraw-Hill, 1955, Chapters 1-9, 11, 12,15.
GREEN, A. E. S., Nuclear Physics. New York: McGraw-Hill, 1955, Chapters 1-6, 8, 12.
KAPLAN, I., Nuclear Physics, 2nd ed. Reading, Mass.: Addison-Wesley, 1963, Chapters 1-15,17.
KRANE, K. S., Introductory Nuclear Physics, Wiley, 1987.
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Chapters 1-3.
MAYER, M. G., and J. H. D. JENSEN, Elementary Theory of Nuclear Shell Structure. New York: Wiley, 1955.
ROHLF, J. W., Modern Physics from α to Z0, Wiley, 1994.
SEGRE, E., Editor, Experimental Nuclear Physics, Vol. III. New York: Wiley, 1959.
Data
GIBBS, R. C., and K. WAY, A Directory to Nuclear Data Tabulations. Nuclear Data Project, National Academy of
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WAY, K., Editor, Nuclear Data Tables (in several parts). National Academy of Sciences-National Research
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forthcoming data compilations.
STROMINGER, HOLLANDER, and SEABORG, “Tables of Isotopes,” Revs. Mod. Phys. 30, 585 (1958).
Atomic masses
Handbook of Chemistry and Physics, 45th ed. Cleveland: Chemical Rubber Co., 1964.
EVERLING, KONIG, MATTAUCH, and W APSTRA, “Atomic Masses of Nuclides, A≤70,” Nucl. Phys. 15, 342
(1960); 18, 529 (1960).
BHANOT, JOHNSON, and NIER, “Atomic Masses in the Heavy Mass Region,” Phys. Rev. 120, 235 (1960).
DRAFT—Chapter One of “new Lamarsh book”
12
updated by M. L. Adams, P. Nelson, and P. Tsvetkov
Nuclear energy levels
AJZENBERG-SELOVE, F., and T. LAURITSEN, “Energy Levels in Light Nuclei,” Nucl. Phys. 11, 1-340 (1959).
HELLWEGE, A. M., and K. H. HELLWEGE, Editors, “Energy Levels of Nuclei: A = 5 to A = 257,” LandoltBornstein, New Series, Group I, Volume I, Berlin: Springer Verlag (1961).
DRAFT—Chapter One of “new Lamarsh book”
13
updated by M. L. Adams, P. Nelson, and P. Tsvetkov
Problems
1-1. Verify Eq. (1-2). In what sense must the neutron be “nonrelativistic” in order for this to be a valid
approximation? Obtain the corresponding expression for the reduced wavelength in terms of (kinetic) energy (and
certain physical constants) for a relativistic neutron.
1-2. The isotopic abundances of H1 and H2, in atom percent, are 99.9851 and 0.0149 respectively. Compute the
atom densities (atoms/cm3) of H1 and H2 in ordinary water of density 1 g/cm3.
1-3. The isotopic abundance of U235 is 0.714 atom percent. Compute the atom densities of U235 and U238 in (a)
natural uranium, (b) uranium enriched to 1 atom percent in U235. [Note: In both (a) and (b) take the density of the
uranium to be 18.7 g/cm3.]
1-4. The useful substance rubber has a density of approximately 1 g/cm3 and can be represented by the chemical
formula (C5H8)x, where x depends on the degree of polymerization. Find the atom densities of carbon and hydrogen
in this material.
1-5. The density of thorium at 0°C is approximately 11.3 g/cm3. Its coefficient of linear expansion between 0°C and
100°C is constant and equal to 12.3 × 10–6/°C. Find the fractional change in its atom density as thorium is raised
from 0°C to 100°C.
1-6. A certain reactor is fueled with a mixture of UO2SO4 (uranyl sulfate) dissolved in water. The ratio of the atom
density of the uranium to the molecular density of the water is only 0.00141, so that it may be assumed that the
UO2SO4 takes up no space in the solution. Find the concentration of the UO2SO4 in grams/liter.
1-7. Show that the speed of a neutron is given by
where c is the speed of light and E is the kinetic energy of the neutron.
1-8. Beryllium has a density of 1.85 g/cm3. At what energy is the wavelength of a neutron comparable to the
average interatomic distance in this material?
1-9. Using the data in the table below, compute the fractions of the molecules of LiH that have molecular weights
of approximately 7, 8, and 9.
Isotope
Abundance, atom percent
H1
99.9851
2
H
0.0149
Li6
7.42
Li7
92.58
1-10. Using atomic mass tables, compute the binding energy of the “last neutron,” which is the energy required to
remove a single neutron, for the following nuclei.
(a) H2 (b) H3 (c) He4 (d) Be9 (e) C13 (f) Pb209 (g) U235 (h) U236 (i) U239
1-11. Using atomic mass tables, compute the average binding energy per nucleon of the following nuclei.
(a) H2 (b) He4 (c) C16 (d) Fe56 (e) Bi209 (f) U235
1-12. Complete the following reactions and determine their Q-values.
(a) Be9(α, n)
(b) Li6(p, α)
(c) C12(n,2n)
(d) N14(n, p)
232
235
236
(f) Th (n,γ)
(g) U (n,2n)
(h) U (γ , n)
(i) U238(n,3n)
(e) Al27 (d, p)
1-13. The radiation and neutron widths of the first virtual state in Xe136 are 86 meV (milli-eV) and 24 meV,
respectively, and the total width is the sum of these two. (a) What is the mean life of the state? (b) What is the
relative probability that the state decays by neutron emission?
1-14. An indium foil is irradiated to saturation in a reactor and then removed. One-half hour later its activity (which
is due to the decay of Inl16 with a half-life of 54.1 min) is measured in a device that registers 1000 counts in 1 min.
Had the activity been measured 5 min after its removal from the reactor, how many counts would the detector have
registered in 1 min?
DRAFT—Chapter One of “new Lamarsh book”
14
updated by M. L. Adams, P. Nelson, and P. Tsvetkov
1-15. In nuclear reactors a newly-formed radioactive isotope A may be transformed into another isotope B by
neutron absorption before it has had an opportunity to decay. Neutron absorption occurs at a rate proportional to the
amount of isotope A present in the system. If the proportionality constant is denoted by c, and the rate of production
(atoms of A/sec) is denoted by R(t), show that the number of atoms of isotope A present in the reactor at time t is
given by
,
where n0 is the number of atoms of A present at t = 0.
1-16. The isotope Na24 (T1/2 = 15.0 hr) can be produced by bombarding a Na23 target with deuterons. The reaction is
Na23 + H2 → Na24 + H1.
If the yield of Na24, i.e., the number of Na24 atoms produced per second multiplied by the decay constant, is 100
µCi/hr, (a) what is the activity of the Na24 after a 5-hr bombardment? (b) If the bombardment ceases after 5 hr, what
is the activity 10 hr later? (c) What is the maximum possible activity (the saturation activity) of Na24 in the target?
1-17. Many coolant materials become radioactive as they pass through a reactor. Consider a circulating liquid
coolant which spends an average of t1 sec in a reactor and t2 sec in the external circuit as indicated in Fig. 1-4. While
in the reactor it becomes activated at the rate of R atoms/cm3-sec.
Figure 1-4
(a) Show that the activity α added per cm3 of the coolant per transit of the reactor is given by
.
(b) Show that after m cycles the activity per cm3 of the coolant leaving the reactor is
.
(c) What is the maximum coolant activity at the exit?
1-18. Isotope A is produced in a reactor at the constant rate of R atoms/sec and decays by way of the chain A→B→C,
where the half-life of B is much longer than that of A. (a) If the reactor is operated for the time t0 and then shut
down, derive expressions for the activity of A and B at shutdown. (b) If t0 is large compared to TI/2(A) but small
compared to TI/2(B), at what time after shutdown is the activity of B the greatest? (c) Repeat part (b) for the case of t0
much larger than TI/2(B). What is the maximum activity of B in this case?