Magill J., Galy J. Radioactivity Radionuclides Radiation

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14 1. Origins and Discovery

Table 1.5. Filling of orbitals by electrons [1]

1s 2s 2p

H ↑ 1s

He ↑↓ 1s

Li ↑↓ ↑ 1s

Be ↑↓ ↑↓ 1s

B ↑↓ ↑↓ ↑ 1s

C ↑↓ ↑↓ ↑ ↑ 1s

N ↑↓ ↑↓ ↑ ↑ ↑ 1s

O ↑↓ ↑↓ ↑↓ ↑ ↑ 1s

F ↑↓ ↑↓ ↑↓ ↑↓ ↑ 1s

Ne ↑↓ ↑↓ ↑↓ ↑↓ ↑↓ 1s

Classiﬁcation of the Elements

Until recently, there were 114 known chemical elements. Very recently, evidence

for the existence of two new superheavy elements has been reported. Each element

is characterised by atoms containing a ﬁxed number of protons, denoted by the

atomic number Z, in the nucleus and an equal number of orbital electrons to ensure

electrical neutrality. In addition to protons, the nucleus contains a variable number

N of electrically neutral neutrons. Atoms of an element with different numbers of

neutrons but ﬁxed number of protons are known as isotopes of that element (see

inset). More than 3000 nuclides are known, but only about 10% of these are stable.

The total number of protons plus neutrons is known as the mass number A of a

nuclide. Nuclides with the same N and different Z are called isotones, and nuclides

with the same mass number A are known as isobars. In general, an atom with atomic

number Z, and neutron number N is known as a nuclide. A nuclide can be speciﬁed

by the notation:

where Z is the atomic (proton) number, N is the neutron number, A is the mass

number (N + Z), and X is the chemical element symbol.

Because of the relationships between Z, A, N (A = Z + N ) and X, a nuclide can

be uniquely speciﬁed by fewer parameters. A particular chemical element is uniquely

speciﬁed by its symbol X or the proton number Z. A nuclide is uniquely speciﬁed

by the element name X (or proton number Z) together with the mass number A.An

example is

Co which refers to the element cobalt (chemical symbol Co) with mass

number 60 (number of protons plus neutrons). A variety of ways of referring to this

nuclide are in current use i.e. Co60, Co-60, Co 60,

Co, and cobalt-60.

Synthesis of the Elements in Nature 15

Nuclide,

Refers to a particular atom or nucleus with a speciﬁc number N of neutrons and number

Z of protons. A is the mass number (= Z + N). Nuclides are either stable or radioactive.

Radioactive nuclides are referred to as radionuclides.

Atomic Number, Z

The number of positively charged protons in the nucleus of an atom.

Neutron Number, N

The number of neutrons in the nucleus of an atom.

Isotope

One of two or more atoms of the same element that have the same number of protons

(isotop

e) in their nucleus but different numbers of neutrons. Hydrogen

H, deuterium

H, and tritium

H, are isotopes of hydrogen. Radioactive isotopes are referred to as

radioisotopes.

Isotone

One of several different nuclides having the same number of neutrons (isoton

e) in their

nuclei.

Isobar

Nuclides with the same atomic mass number A(= Z + N) but with different values of N

and Z e.g.

Isomer

Atoms with the same atomic number Z and the same mass number A in different long-

lived states of excitation – the higher states being metastable with respect to the ground

state. For example, an isomer of

Tc is

99m

Tc where the m denotes the long-lived excited

state.

Synthesis of the Elements in Nature

The distribution of elements in nature is by no means uniform. Studies of stars reveal

that hydrogen is by far the most abundant element, accounting for approximately

75% of the mass of the universe. The remaining mass is mostly in the form of helium

gas with other elements contributing only to a small extent.

The hydrogen in the universe is the driving fuel for continuous change in the

chemical composition. Hydrogen is being converted to helium, which is further

changed into heavier elements.

The observed abundances of elements in the universe require the postulation of

at least eight different processes for their formation [9]. The main processes are:

• hydrogen burning (formation of helium-4 by the fusion of hydrogen nuclei),

• helium burning (formation of carbon-12 from helium-4-nuclei),

• s processes (s

low neutron capture processes, starting from iron group nuclei, close

to the stability line – see section “Nuclide Charts”, Chapter 2),

16 1. Origins and Discovery

• r processes (rapid neutron capture processes, starting from iron group nuclei,

produce nuclei far from the line of stability),

• rp processes (similar to the r-processes, but involve successive proton absorptions),

• X processes (spallation of heavier elements into lighter elements by cosmic rays).

Light elements were probably produced in the early history of the galaxy whereas

heavy elements were formed in supernovae.

According to current theory, the universe is about 10

years old and started as a

“soup” of quarks at enormous temperatures and pressures. As the universe expanded,

the temperature and pressure decreased rapidly to the point where neutrons and pro-

tons condensed out. Shortly after, nuclear fusion took place and led to the formation

of deuterons followed by tritons and onto helium-3 nuclei and alpha particles (hy-

drogen burning), thereby giving rise to the present hydrogen to helium ratio of 3:1.

In addition, some lithium-6 and lithium-7 were produced at this time. As the universe

further expanded and cooled, the fusion reactions stopped and the gases condensed

and eventually led to the formation of stars consisting of the primordial elements.

As a cloud of interstellar gas contracts and condenses to form a star, the tem-

perature will rise as the gravitational potential energy is converted to kinetic energy.

When the temperature increases to 10

K and beyond, fusion reactions will begin

to take place and further nucleosynthesis leads to the conversion of the elements

hydrogen and helium to carbon, silicon, oxygen, sulphur and heavier elements up

to iron. Fusion can proceed no further than iron since this element has the high-

Fig. 1.10. Vela supernova remnant.

photo from UK Schmidt, plates by

David Malin

Synthesis of the Elements in Nature 17

est binding energy. Fusion of heavier elements will consume energy rather than

release it.

The synthesis of elements heavier than iron require a supernova.As stated above,

the sequence of stellar burningmore or less stops when the core of the star is composed

of nuclei with mass numbers close to 56 (i.e. iron). At this point the central region of

the star has consumed all the nuclear fuel and cannot support the star’s interior against

gravitational collapse into a neutron star. In this process all protons are converted into

neutrons with the release of a large number of neutrinos within a short time. These

neutrinos interact with the outer layers of the star resulting in a large fraction of the

mass being ejected into space. In addition, large numbers of particles are produced,

in particular neutrons. It is then these neutrons, which lead to the formation of heavier

elements by transmutation through r(rapid)-process neutron capture events.

At least 24 elements are believed essential to living matter. The most abundant

of these in the human body are, in order of importance: hydrogen, oxygen, carbon,

nitrogen, calcium, phosphorus, chlorine, potassium, sulphur, sodium, magnesium,

iron, copper, zinc, silicon, iodine, cobalt, manganese, molybdenum, ﬂuorine, tin,

chromium, selenium, and vanadium.

Ninety-one elements occur naturally on earth supporting the theory that heavy el-

ements were produced during creation of the solar system. Exceptions are technetium

and promethium although these are present in stars.

In the following chapter, the properties of nuclides and their constitutents are

described in more detail.

2. Nuclear Energetics

Nuclear Units – Atomic Mass Unit and the Electron Volt

Most units used in nuclear science are based on the International System of Units or

SI units (from the French “Syst`eme International d’Unit´es”) [1]. The base SI units are

the metre (length), kilogram (mass), second (time), ampere (electric current), kelvin

(temperature), candela (luminous intensity), and the mole (quantity of substance). In

nuclear science, in addition to SI, two units are commonly used for mass and energy

i.e. the atomic mass unit and the electron volt.

Mass – The Atomic Mass Unit

Masses of atomic nuclei are generally less than 10

−21

g. For this reason, it is more

convenient to express the masses in so-called atomic mass units [2]. The atomic mass

unit (abbreviated as amu or u) is deﬁned such that the mass of a

C atom is exactly

12 u. Hence,

1u= 1.6605387 × 10

−27

kg .

Energy – The Electron Volt

Energies released in chemical reactions are of the order of 10

−19

J. For such reactions,

it is more convenientto use the energy unit electron volt, eV. By deﬁnition, the electron

volt is the kinetic energy gained by an electron after acceleration through a potential

difference V of one volt. This kinetic energy increase of eV = (1.60217646 ×

−19

C) · (1V) = 1.60217646 × 10

−19

J. Hence

1eV = 1.60217646 × 10

−19

J .

The energy equivalent of the atomic mass unit is

E = mc

= 931.494013 MeV .

Nuclear and Atomic Masses

Tables of masses always give atomic rather than nuclear masses. This implies that the

masses given include the extra-nuclear electrons.Atomic masses are more convenient

than nuclear masses since it is always atomic masses or differences between atomic

20 2. Nuclear Energetics

masses that are measured experimentally. This distinction is reﬂected in the notation

used in this book: rest mass of an atom is denoted by M





whereas the rest mass

of the nucleus is denoted by m





Notation on Masses

• rest mass of an atom is denoted by M





• rest mass of the nucleus is denoted by m





According to the above notation, the electron, neutron, and proton masses are



−1



, m





, m





respectively. For convenience, however, the notation here

is simpliﬁed to m

, m

. Since combining Z electrons with the nucleus forms a

neutral atom, the atomic and nuclear masses are related by





= m





+ Zm

−

where BE

is the energy released upon binding with the nucleus. The mass change

corresponding to the electron binding energy in the above relation (BE

for

hydrogen is approximately 10

−8

u) is negligible compared to the mass of the atom

(hydrogen mass is about 1 u) and also the electron (about 5.5 × 10

−4

u), such that,

to a good approximation [3]





∼





+ Zm

Atomic and Molecular Weights

The atomic weight A of an atom is the ratio of the atom’s mass to 1/12th of the mass

of an atom of

C in its ground state [3, 4]. The molecular weight of a molecule is

the ratio of the molecular mass of one molecule to 1/12th of the mass of one atom

C. Both the atomic and molecular weights are dimensionless. It follows that the

mass of an atom measured in atomic mass units is numerically equal to the atomic

weight of that atom. The atomic mass (in u) and hence the atomic weight of a nuclide

is almost equal to the atomic mass number A(= Z + N) such that A

∼

Most naturally occurring elements consists of two or more isotopes. The isotopic

abundance a

is deﬁned as the relative number of atoms of a particular isotope in

a mixture of the isotopes of a chemical element, expressed as a fraction of all the

atoms of the element. The elemental atomic weight A is the weighted average of

the atomic weights A

of the naturally occurring isotopes of the elements, weighted

by the isotopic abundance a

of each isotope i.e.

A =



100

Avogadro’s Number 21

Atomic Weight of Magnesium

Naturally occurring magnesium consists of three stable isotopes –

Mg,

– with isotopic abundances 78.99, 10, and 11.01 atom-percent respectively. The atomic

weights are 23.985041700 (

Mg), 24.98583692 (

Mg), and 25.982592929 (

Mg).

The atomic weight of magnesium A

is then:

+ a

100

= (0.7899 × 23.985042) + (0.1 × 24.985837) + (0.1101 × 25.982593)

= 24.3050

Avogadro’s Number

A mole of a substance is deﬁned to contain as many “elementary particles” as there are

atoms in 12 g of

C and is equal to 6.0221415×10

(2002 CODATA recommended

value). This number is known as Avogadro’s number N

. The mass in grams of a

substance that equals the dimensionless atomic or molecular weight is called the

gram atomic weight or the gram molecular weight. Hence one gram atomic weight

of a substance represents one mole and contains N

atoms or molecules.

Basic Relation between Mass and Number of Atoms

Consider a substance of mass M (in grams). This mass corresponds to M /A moles and

therefore contains (M /A ) · N

atoms. Hence the basic relation between the number of

atoms and the mass of a substance is given by:

N =

· N

Calculations usually involve an initial mass or an activity of a parent nuclide. The ﬁrst

step is then to convert this mass to a number of atoms, N, before the solutions to the decay

equations is evaluated, for example.

The activity, A, is deﬁned by A = kN where k is the decay constant ln 2/τ (s). Given an

activity, the number of atoms is given by

N =

= A ·

τ(s)

ln 2

where τ(s) is the half-life in seconds. The relationship between the mass and the activity

is given by

A ·

τ(s)

ln 2

· N

from which it follows

A =

ln 2

τ(s)

· N

M = A ·





τ(s)

ln 2

22 2. Nuclear Energetics

Mass of an Atom and Atomic Number Density

With Avogadro’s number, the mass of an individual atom can be evaluated. One mole

of a substance, with mass of A grams, contains N

atoms. Hence, the mass of an

individual atom is

M =

∼

where the atomic weight has been approximated by the atomic mass number. That

this approximation is good can be seen for the case of

208

Pb:

Approximate relation M

∼

A/N

208

Pb)

∼

208 g

6.022142 × 10

= 3.4539 × 10

−22

g .

Exact relation M = A /N

208

Pb) =

207.976652 g

6.022142 × 10

= 3.4535 × 10

−22

g .

Atomic masses are expressed in atomic mass units (u), based on the deﬁnition that

the mass of a neutral atom of

C is exactly 12 u. All other nuclides are assigned

masses relative to

C. The mass of a single atom can be computed from the fact that

one mole of any substance contains 6.0221415 × 10

atoms and has a mass equal

to the atomic mass in grams. For

C, 1 mole has a mass of 12 g, hence the mass of

one atom is given by:

C) =

12 g

6.0221415 × 10

= 1.992647 × 10

−23

g .

The atomic mass unit, u (also known as a dalton) is then

1u=

1.992647 × 10

−23

= 1.660539 × 10

−24

g .

The energy equivalent of 1 u is E = mc

= 931.494013 MeV.

Atomic Number Density

A useful quantity in many calculations is the number density of atoms. By analogy

with the basic relation between the mass and number of atoms, N = (M /A ) · N

the relation for the number density is given by

N(atoms cm

−3

) =

· N

where ρ is the mass density (in g cm

−3

Relativistic Mechanics and Einstein’s Mass/Energy Equivalence 23

Relativistic Mechanics and Einstein’s Mass/Energy Equivalence

The fundamental relation describing motion is that the force acting on a body will

result in a change of momentum i.e.

F =

where p is the linear momentum mv and F the force. The quantities F , p, and v

are all vectors. In classical mechanics, the mass of a body is constant, and the above

relation reduces to F = mdv/dt = ma, the more familiar form of Newton’s second

law. In relativistic mechanics, the mass of a body changes with its speed v such that

F =

mv = m

+ v

From conservation of energy, the work done on a particle as it moves along a path of

length s must equal the change in the kinetic energy KE from the beginning to the

end of the path. The work done by the force F on the particle as it moves through

a displacement ds is given by F · ds. The resulting change in the kinetic energy is

then given by [3]

KE =



F · ds =



· ds =



· dt =



v · dp ,

where it is assumed that initially the particle is at rest v = 0. Using the expression

for the relativistic mass

m =

(1 − v

)

1/2

and substituting in the relation for the change in kinetic energy gives

KE = m





(1 − v

)

1/2



Differentiation of the term in square brackets leads to

KE = m





(1 − v

)

1/2

(1 − v

)

3/2



= m



(1 − v

)

3/2

dv = m

(1 − v

)

1/2



(1 − v

)

1/2

− m

and ﬁnally,

KE = mc

− m

This last expression can be rewritten in the form mc

= m

+ KE. The left

hand side mc

can be regarded as the total energy of the particle consisting of the

rest-mass energy m

plus the kinetic energy of the particle. This expression for

the total energy E, i.e. E = mc

shows the equivalence of mass and energy and one of