Nuclei Atoms in nature generally are electrically neutral, as they have equal numbers of protons in the
nucleus and orbiting electrons. However, within the nucleus there are other particles called neutrons, which are electrically neutral but have about the same
mass as protons. There are two numbers used to characterize a nucleus:
Z , the atomic number, which equals the number of protons;
A , the mass number, which equals the number of
nucleons (protons and neutrons). An element
X is
defined by the atomic number
Z , while
A denotes the particular
isotope of that element. The usual notation for an element
X is AZ
X - for example, there are four common isotopes of Carbon: 116
C , 126
C , 136
C , and 146
C , with 126
C being the most abundant ( > 98% ). It is convenient in some circumstances to measure masses in terms of the
unified mass unit, u, which is defined so that 12
C has a mass of 12 u exactly; in SI units, 1
u = 1.66 x 10- 27
kg . (1) There is also another convenient unit of mass which arises from Einstein''s Special Theory of Relativity. We have not covered this in this course, so we simply quote the relevant (well-known) relation:
E =
mc 2, which associates an energy
E to a mass
m , with
c = 3.0 x 108 m/s being the speed of light. Thus, dimensionally,
E/
c 2 is a unit of mass. It is customary to express this unit in terms of MeV/c 2, where 1 MeV=10 6 eV= 1.6 x 10- 13 J - note that ``c'''' here is considered part of the unit, and one
does not substitute the numerical value of 3.0 x 108 m/s in it.One important illustration of the equivalence of mass and energy has to do with what is called the
binding energy of the nucleus. It is observed that the mass of any nucleus is always less than the sum of the masses of the individual constituent nucleons which make it up. This ``loss'''' of mass which then results when nucleons form a nucleus is attributed to a ``binding energy'''', and is a measure of the strength of the
strong force holding the nucleons together.
Measuring
Radioactivity The radioactivity of a substance is measured by how many decays per unit time occur. Two popular units of the
activity are used: the
curie (Ci), defined as 1
Ci = 3.7 x 1010
decays/
s (8) and the
becquerel (Bq), defined as 1
Bq = 1
decay/
s (9) Note that, by themselves, these units
do not measure accurately how dangerous a given amount of radiation might be for humans - for medical purposes other units of radioactivity reflecting this aspect are more appropriate. It is found that, if a given radioactive substance at a certain time contains
N nuclei, then at a
short time
t later a certain number
N have decayed which is given by
N = -
Nt, (10) where is called the
decay constant. Note that has units of inverse time, or s - 1. The
decay rate,
R , is defined as the number of decays per unit time:
R = =
N. (11) For finite times it is found that the number of
Nuclei present after a time
t is given by
N =
N0
e - t (12) where
N0 is the number of nuclei present at
t = 0 and
e = 2.71828... is the base of the ``natural'''' logarithms (compared to the base 10 of the ``common'''' logarithms). It is sometimes convenient to introduce the
half-life,
T1/2 , of a substance, defined as the time over which exactly one half of a substance remains.Nuclear
Reactions The reactions considered so far occur spontaneously, and involve a conversion of mass into (mainly) kinetic energy of the decay products. There is another class of
Nuclear reactions which can be induced, an example of which is bombarding a nitrogea particle: 42
He + 147
N 178
O + 11
H .In this case the sum of the masses of the elements on the left-hand-side of this equation is less than the sum of the masses of the decay products on the right-hand-side. In order for the reaction to proceed some initial kinetic energy must be supplied to the left-hand-side reactants; such energy, again through Einstein''s relation of gets converted into mass of the products. There are two general types of such nuclear reactions:
fusion, where lighter elements ``fuse'''' into heavier ones. Fusion reactions are the main source of energy for the sun, and are also used in certain types of nuclear weapons.
fission, where heavier elements are broken into lighter ones.Such a
chain reaction has important consequences in the design of nuclear reactors, as well as nuclear weapons.However, given the risks of runaway chain reactions, as well as the presence of dangerous, long-lived radioactive reactants and products in fission reactors, much research is being invested into this form of energy production.
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