Radioactive decay

Alpha decay is one type of radioactive decay, in

which an atomic nucleus emits an alpha particle, and
thereby transforms (or "decays") into an atom with a
mass number decreased by 4 and atomic number
decreased by 2.

Radioactive decay (also known as

nuclear decay or radioactivity) is the
process by which an unstable atomic
nucleus loses energy (in terms of mass
in its rest frame) by emitting radiation,
such as an alpha particle, beta particle
with neutrino or only a neutrino in the
case of electron capture, gamma ray, or
electron in the case of internal
conversion. A material containing such
unstable nuclei is considered
radioactive. Certain highly excited short-
lived nuclear states can decay through
neutron emission, or more rarely, proton
emission.

Radioactive decay is a stochastic (i.e.

random) process at the level of single
atoms, in that, according to quantum
theory, it is impossible to predict when a
particular atom will decay,[1][2][3]
regardless of how long the atom has
existed. However, for a collection of
atoms, the collection's expected decay
rate is characterized in terms of their
measured decay constants or half-lives.
This is the basis of radiometric dating.
The half-lives of radioactive atoms have
no known upper limit, spanning a time
range of over 55 orders of magnitude,
from nearly instantaneous to far longer
than the age of the universe.

A radioactive nucleus with zero spin can

have no defined orientation, and hence
emits the total momentum of its decay
products isotropically (all directions and
without bias). If there are multiple
particles produced during a single decay,
as in beta decay, their relative angular
distribution, or spin directions may not be
isotropic. Decay products from a nucleus
with spin may be distributed non-
isotropically with respect to that spin
direction, either because of an external
influence such as an electromagnetic
field, or because the nucleus was
produced in a dynamic process that
constrained the direction of its spin.
Such a parent process could be a
previous decay, or a nuclear
reaction.[4][5][6][note 1]
The decaying nucleus is called the parent
radionuclide (or parent
radioisotope[note 2]), and the process
produces at least one daughter nuclide.
Except for gamma decay or internal
conversion from a nuclear excited state,
the decay is a nuclear transmutation
resulting in a daughter containing a
different number of protons or neutrons
(or both). When the number of protons
changes, an atom of a different chemical
element is created.

The first decay processes to be

discovered were alpha decay, beta decay,
and gamma decay. Alpha decay occurs
when the nucleus ejects an alpha particle
(helium nucleus). This is the most
common process of emitting nucleons,
but highly excited nuclei can eject single
nucleons, or in the case of cluster decay,
specific light nuclei of other elements.
Beta decay occurs in two ways: (i) beta-
minus decay, when the nucleus emits an
electron and an antineutrino in a process
that changes a neutron to a proton, or (ii)
beta-plus decay, when the nucleus emits
a positron and a neutrino in a process
that changes a proton to a neutron.
Highly excited neutron-rich nuclei,
formed as the product of other types of
decay, occasionally lose energy by way of
neutron emission, resulting in a change
from one isotope to another of the same
element. The nucleus may capture an
orbiting electron, causing a proton to
convert into a neutron in a process called
electron capture. All of these processes
result in a well-defined nuclear
transmutation.

By contrast, there are radioactive decay

processes that do not result in a nuclear
transmutation. The energy of an excited
nucleus may be emitted as a gamma ray
in a process called gamma decay, or that
energy may be lost when the nucleus
interacts with an orbital electron causing
its ejection from the atom, in a process
called internal conversion.
Another type of radioactive decay results
in products that vary, appearing as two or
more "fragments" of the original nucleus
with a range of possible masses. This
decay, called spontaneous fission,
happens when a large unstable nucleus
spontaneously splits into two (or
occasionally three) smaller daughter
nuclei, and generally leads to the
emission of gamma rays, neutrons, or
other particles from those products.

For a summary table showing the

number of stable and radioactive
nuclides in each category, see
radionuclide. There are 29 naturally
occurring chemical elements on Earth
that are radioactive. They are those that
contain 34 radionuclides that date before
the time of formation of the solar
system, and are known as primordial
nuclides. Well-known examples are
uranium and thorium, but also included
are naturally occurring long-lived
radioisotopes, such as potassium-40.
Another 50 or so shorter-lived
radionuclides, such as radium and radon,
found on Earth, are the products of decay
chains that began with the primordial
nuclides, or are the product of ongoing
cosmogenic processes, such as the
production of carbon-14 from nitrogen-
14 in the atmosphere by cosmic rays.
Radionuclides may also be produced
artificially in particle accelerators or
nuclear reactors, resulting in 650 of these
with half-lives of over an hour, and
several thousand more with even shorter
half-lives. [See here for a list of these
sorted by half life.]

History of discovery

Pierre and Marie Curie in their Paris laboratory, before

1907

Radioactivity was discovered in 1896 by

the French scientist Henri Becquerel,
while working with phosphorescent
materials.[7] These materials glow in the
dark after exposure to light, and he
suspected that the glow produced in
cathode ray tubes by X-rays might be
associated with phosphorescence. He
wrapped a photographic plate in black
paper and placed various
phosphorescent salts on it. All results
were negative until he used uranium
salts. The uranium salts caused a
blackening of the plate in spite of the
plate being wrapped in black paper.
These radiations were given the name
"Becquerel Rays".
It soon became clear that the blackening
of the plate had nothing to do with
phosphorescence, as the blackening was
also produced by non-phosphorescent
salts of uranium and metallic uranium. It
became clear from these experiments
that there was a form of invisible
radiation that could pass through paper
and was causing the plate to react as if
exposed to light.

At first, it seemed as though the new

radiation was similar to the then recently
discovered X-rays. Further research by
Becquerel, Ernest Rutherford, Paul Villard,
Pierre Curie, Marie Curie, and others
showed that this form of radioactivity
was significantly more complicated.
Rutherford was the first to realize that all
such elements decay in accordance with
the same mathematical exponential
formula. Rutherford and his student
Frederick Soddy were the first to realize
that many decay processes resulted in
the transmutation of one element to
another. Subsequently, the radioactive
displacement law of Fajans and Soddy
was formulated to describe the products
of alpha and beta decay.[8][9]

The early researchers also discovered

that many other chemical elements,
besides uranium, have radioactive
isotopes. A systematic search for the
total radioactivity in uranium ores also
guided Pierre and Marie Curie to isolate
two new elements: polonium and radium.
Except for the radioactivity of radium, the
chemical similarity of radium to barium
made these two elements difficult to
distinguish.

Marie and Pierre Curie’s study of

radioactivity is an important factor in
science and medicine. After their
research on Becquerel's rays led them to
the discovery of both radium and
polonium, they coined the term
"radioactivity".[10] Their research on the
penetrating rays in uranium and the
discovery of radium launched an era of
using radium for the treatment of cancer.
Their exploration of radium could be
seen as the first peaceful use of nuclear
energy and the start of modern nuclear
medicine.[10]

Early health dangers

Taking an X-ray image with early Crookes tube

apparatus in 1896. The Crookes tube is visible in the
centre. The standing man is viewing his hand with a
fluoroscope screen; this was a common way of
setting up the tube. No precautions against radiation
exposure are being taken; its hazards were not
known at the time.

The dangers of ionizing radiation due to

radioactivity and X-rays were not
immediately recognized.

X-rays

The discovery of x‑rays by Wilhelm

Röntgen in 1895 led to widespread
experimentation by scientists,
physicians, and inventors. Many people
began recounting stories of burns, hair
loss and worse in technical journals as
early as 1896. In February of that year,
Professor Daniel and Dr. Dudley of
Vanderbilt University performed an
experiment involving X-raying Dudley's
head that resulted in his hair loss. A
report by Dr. H.D. Hawks, of his suffering
severe hand and chest burns in an X-ray
demonstration, was the first of many
other reports in Electrical Review.[11]

Other experimenters, including Elihu

Thomson and Nikola Tesla, also reported
burns. Thomson deliberately exposed a
finger to an X-ray tube over a period of
time and suffered pain, swelling, and
blistering.[12] Other effects, including
ultraviolet rays and ozone, were
sometimes blamed for the damage,[13]
and many physicians still claimed that
there were no effects from X-ray
exposure at all.[12]

Despite this, there were some early

systematic hazard investigations, and as
early as 1902 William Herbert Rollins
wrote almost despairingly that his
warnings about the dangers involved in
the careless use of X-rays was not being
heeded, either by industry or by his
colleagues. By this time, Rollins had
proved that X-rays could kill experimental
animals, could cause a pregnant guinea
pig to abort, and that they could kill a
fetus.[14] He also stressed that "animals
vary in susceptibility to the external
action of X-light" and warned that these
differences be considered when patients
were treated by means of X-rays.

Radioactive substances

Radioactivity is characteristic of elements with large

atomic number. Elements with at least one stable
isotope are shown in light blue. Green shows
elements whose most stable isotope has a half-life
measured in millions of years. Yellow and orange are
progressively less stable, with half-lives in thousands
or hundreds of years, down toward one day. Red and
purple show highly and extremely radioactive
elements where the most stable isotopes exhibit
half-lives measured on the order of one day and
much less.
However, the biological effects of
radiation due to radioactive substances
were less easy to gauge. This gave the
opportunity for many physicians and
corporations to market radioactive
substances as patent medicines.
Examples were radium enema
treatments, and radium-containing
waters to be drunk as tonics. Marie Curie
protested against this sort of treatment,
warning that the effects of radiation on
the human body were not well
understood. Curie later died from
aplastic anaemia, likely caused by
exposure to ionizing radiation. By the
1930s, after a number of cases of bone
necrosis and death of radium treatment
enthusiasts, radium-containing medicinal
products had been largely removed from
the market (radioactive quackery).

Radiation protection

Only a year after Röntgen's discovery of X

rays, the American engineer Wolfram
Fuchs (1896) gave what is probably the
first protection advice, but it was not until
1925 that the first International Congress
of Radiology (ICR) was held and
considered establishing international
protection standards. The effects of
radiation on genes, including the effect
of cancer risk, were recognized much
later. In 1927, Hermann Joseph Muller
published research showing genetic
effects and, in 1946, was awarded the
Nobel Prize in Physiology or Medicine for
his findings.

The second ICR was held in Stockholm in

1928 and proposed the adoption of the
rontgen unit, and the 'International X-ray
and Radium Protection Committee'
(IXRPC) was formed. Rolf Sievert was
named Chairman, but a driving force was
George Kaye of the British National
Physical Laboratory. The committee met
in 1931, 1934 and 1937.

After World War II, the increased range

and quantity of radioactive substances
being handled as a result of military and
civil nuclear programmes led to large
groups of occupational workers and the
public being potentially exposed to
harmful levels of ionising radiation. This
was considered at the first post-war ICR
convened in London in 1950, when the
present International Commission on
Radiological Protection (ICRP) was
born.[15] Since then the ICRP has
developed the present international
system of radiation protection, covering
all aspects of radiation hazard.

Units of radioactivity
Graphic showing relationships between radioactivity
and detected ionizing radiation

The International System of Units (SI)

unit of radioactive activity is the
becquerel (Bq), named in honour of the
scientist Henri Becquerel. One Bq is
defined as one transformation (or decay
or disintegration) per second.

An older unit of radioactivity is the curie,

Ci, which was originally defined as "the
quantity or mass of radium emanation in
equilibrium with one gram of radium
(element)".[16] Today, the curie is defined
as 3.7 × 1010 disintegrations per second,
so that 1 curie (Ci) = 3.7 × 1010 Bq. For
radiological protection purposes,
although the United States Nuclear
Regulatory Commission permits the use
of the unit curie alongside SI units,[17] the
European Union European units of
measurement directives required that its
use for "public health ... purposes" be
phased out by 31 December 1985.[18]

The effects of ionizing radiation is often

measured in units of gray for mechanical
or sievert for damage to tissue.

Types of decay
Alpha particles may be completely stopped by a
sheet of paper, beta particles by aluminium shielding.
Gamma rays can only be reduced by much more
substantial mass, such as a very thick layer of lead.

Early researchers found that an electric

or magnetic field could split radioactive
emissions into three types of beams. The
rays were given the names alpha, beta,
and gamma, in increasing order of their
ability to penetrate matter. Alpha decay is
observed only in heavier elements of
atomic number 52 (tellurium) and
greater, with the exception of beryllium-8
which decays to two alpha particles. The
other two types of decay are produced by
all of the elements. Lead, atomic number
82, is the heaviest element to have any
isotopes stable (to the limit of
measurement) to radioactive decay.
Radioactive decay is seen in all isotopes
of all elements of atomic number 83
(bismuth) or greater. Bismuth, however, is
only very slightly radioactive, with a half-
life greater than the age of the universe;
radioisotopes with extremely long half-
lives are considered effectively stable for
practical purposes.
Transition diagram for decay modes of a
radionuclide, with neutron number N and atomic
number Z (shown are α, β±, p+, and n0 emissions, EC
denotes electron capture).

Types of radioactive decay related to N and Z

numbers
In analysing the nature of the decay
products, it was obvious from the
direction of the electromagnetic forces
applied to the radiations by external
magnetic and electric fields that alpha
particles carried a positive charge, beta
particles carried a negative charge, and
gamma rays were neutral. From the
magnitude of deflection, it was clear that
alpha particles were much more massive
than beta particles. Passing alpha
particles through a very thin glass
window and trapping them in a discharge
tube allowed researchers to study the
emission spectrum of the captured
particles, and ultimately proved that
alpha particles are helium nuclei. Other
experiments showed beta radiation,
resulting from decay and cathode rays,
were high-speed electrons. Likewise,
gamma radiation and X-rays were found
to be high-energy electromagnetic
radiation.

The relationship between the types of

decays also began to be examined: For
example, gamma decay was almost
always found to be associated with other
types of decay, and occurred at about the
same time, or afterwards. Gamma decay
as a separate phenomenon, with its own
half-life (now termed isomeric transition),
was found in natural radioactivity to be a
result of the gamma decay of excited
metastable nuclear isomers, which were
in turn created from other types of decay.

Although alpha, beta, and gamma

radiations were most commonly found,
other types of emission were eventually
discovered. Shortly after the discovery of
the positron in cosmic ray products, it
was realized that the same process that
operates in classical beta decay can also
produce positrons (positron emission),
along with neutrinos (classical beta
decay produces antineutrinos). In a more
common analogous process, called
electron capture, some proton-rich
nuclides were found to capture their own
atomic electrons instead of emitting
positrons, and subsequently these
nuclides emit only a neutrino and a
gamma ray from the excited nucleus
(and often also Auger electrons and
characteristic X-rays, as a result of the re-
ordering of electrons to fill the place of
the missing captured electron). These
types of decay involve the nuclear
capture of electrons or emission of
electrons or positrons, and thus acts to
move a nucleus toward the ratio of
neutrons to protons that has the least
energy for a given total number of
nucleons. This consequently produces a
more stable (lower energy) nucleus.
(A theoretical process of positron
capture, analogous to electron capture, is
possible in antimatter atoms, but has not
been observed, as complex antimatter
atoms beyond antihelium are not
experimentally available.[19] Such a decay
would require antimatter atoms at least
as complex as beryllium-7, which is the
lightest known isotope of normal matter
to undergo decay by electron capture.)

Shortly after the discovery of the neutron

in 1932, Enrico Fermi realized that
certain rare beta-decay reactions
immediately yield neutrons as a decay
particle (neutron emission). Isolated
proton emission was eventually observed
in some elements. It was also found that
some heavy elements may undergo
spontaneous fission into products that
vary in composition. In a phenomenon
called cluster decay, specific
combinations of neutrons and protons
other than alpha particles (helium nuclei)
were found to be spontaneously emitted
from atoms.

Other types of radioactive decay were

found to emit previously-seen particles,
but via different mechanisms. An
example is internal conversion, which
results in an initial electron emission, and
then often further characteristic X-rays
and Auger electrons emissions, although
the internal conversion process involves
neither beta nor gamma decay. A
neutrino is not emitted, and none of the
electron(s) and photon(s) emitted
originate in the nucleus, even though the
energy to emit all of them does originate
there. Internal conversion decay, like
isomeric transition gamma decay and
neutron emission, involves the release of
energy by an excited nuclide, without the
transmutation of one element into
another.

Rare events that involve a combination of

two beta-decay type events happening
simultaneously are known (see below).
Any decay process that does not violate
the conservation of energy or
momentum laws (and perhaps other
particle conservation laws) is permitted
to happen, although not all have been
detected. An interesting example
discussed in a final section, is bound
state beta decay of rhenium-187. In this
process, beta electron-decay of the
parent nuclide is not accompanied by
beta electron emission, because the beta
particle has been captured into the K-
shell of the emitting atom. An
antineutrino is emitted, as in all negative
beta decays.

Radionuclides can undergo a number of

different reactions. These are
summarized in the following table. A
nucleus with mass number A and atomic
number Z is represented as (A, Z). The
column "Daughter nucleus" indicates the
difference between the new nucleus and
the original nucleus. Thus, (A − 1, Z)
means that the mass number is one less
than before, but the atomic number is the
same as before.

If energy circumstances are favorable, a

given radionuclide may undergo many
competing types of decay, with some
atoms decaying by one route, and others
decaying by another. An example is
copper-64, which has 29 protons, and 35
neutrons, which decays with a half-life of
about 12.7 hours. This isotope has one
unpaired proton and one unpaired
neutron, so either the proton or the
neutron can decay to the opposite
particle. This particular nuclide (though
not all nuclides in this situation) is
almost equally likely to decay through
positron emission (18%), or through
electron capture (43%), as it does
through electron emission (39%). The
excited energy states resulting from
these decays which fail to end in a
ground energy state, also produce later
internal conversion and gamma decay in
almost 0.5% of the time.
More common in heavy nuclides is
competition between alpha and beta
decay. The daughter nuclides will then
normally decay through beta or alpha,
respectively, to end up in the same place.
Mode of decay Participating particles Daughter nucleus

Decays with emission of nucleons:

Alpha decay An alpha particle (A = 4, Z = 2) emitted from nucleus (A − 4, Z − 2)

Proton
A proton ejected from nucleus (A − 1, Z − 1)
emission

Neutron
A neutron ejected from nucleus (A − 1, Z)
emission

Double proton
Two protons ejected from nucleus simultaneously (A − 2, Z − 2)
emission

Spontaneous Nucleus disintegrates into two or more smaller nuclei

—
fission and other particles

Nucleus emits a specific type of smaller nucleus (A1, Z1) (A − A1, Z − Z1) +
Cluster decay
which is larger than an alpha particle (A1, Z1)

Different modes of beta decay:

β− decay A nucleus emits an electron and an electron antineutrino (A, Z + 1)

Positron
emission (β+ A nucleus emits a positron and an electron neutrino (A, Z − 1)
decay)

A nucleus captures an orbiting electron and emits a

Electron
neutrino; the daughter nucleus is left in an excited (A, Z − 1)
capture
unstable state

A free neutron or nucleus beta decays to electron and

antineutrino, but the electron is not emitted, as it is
captured into an empty K-shell; the daughter nucleus is
Bound state
left in an excited and unstable state. This process is a (A, Z + 1)
beta decay
minority of free neutron decays (0.0004%) due to the low
energy of hydrogen ionization, and is suppressed except
in ionized atoms that have K-shell vacancies.

Double beta
A nucleus emits two electrons and two antineutrinos (A, Z + 2)
decay

Double A nucleus absorbs two orbital electrons and emits two

electron neutrinos – the daughter nucleus is left in an excited and (A, Z − 2)
capture unstable state

Electron A nucleus absorbs one orbital electron, emits one (A, Z − 2)
capture with positron and two neutrinos
positron
emission

Double
positron A nucleus emits two positrons and two neutrinos (A, Z − 2)
emission

Transitions between states of the same nucleus:

Isomeric Excited nucleus releases a high-energy photon (gamma

(A, Z)
transition ray)

Internal Excited nucleus transfers energy to an orbital electron,

(A, Z)
conversion which is subsequently ejected from the atom

Radioactive decay results in a reduction

of summed rest mass, once the released
energy (the disintegration energy) has
escaped in some way. Although decay
energy is sometimes defined as
associated with the difference between
the mass of the parent nuclide products
and the mass of the decay products, this
is true only of rest mass measurements,
where some energy has been removed
from the product system. This is true
because the decay energy must always
carry mass with it, wherever it appears
(see mass in special relativity) according
to the formula E = mc2. The decay energy
is initially released as the energy of
emitted photons plus the kinetic energy
of massive emitted particles (that is,
particles that have rest mass). If these
particles come to thermal equilibrium
with their surroundings and photons are
absorbed, then the decay energy is
transformed to thermal energy, which
retains its mass.

Decay energy therefore remains

associated with a certain measure of
mass of the decay system, called
invariant mass, which does not change
during the decay, even though the energy
of decay is distributed among decay
particles. The energy of photons, the
kinetic energy of emitted particles, and,
later, the thermal energy of the
surrounding matter, all contribute to the
invariant mass of the system. Thus, while
the sum of the rest masses of the
particles is not conserved in radioactive
decay, the system mass and system
invariant mass (and also the system total
energy) is conserved throughout any
decay process. This is a restatement of
the equivalent laws of conservation of
energy and conservation of mass.

Radioactive decay rates

The decay rate, or activity, of a radioactive
substance is characterized by:

Constant quantities:

The half-life—t1/2, is the time taken for

the activity of a given amount of a
radioactive substance to decay to half
of its initial value; see List of nuclides.

The decay constant— λ, "lambda" the

reciprocal of the mean lifetime,
sometimes referred to as simply decay
rate.

The mean lifetime— τ, "tau" the

average lifetime (1/e life) of a
radioactive particle before decay.
Although these are constants, they are
associated with the statistical behavior
of populations of atoms. In
consequence, predictions using these
constants are less accurate for
minuscule samples of atoms.

In principle a half-life, a third-life, or even

a (1/√2)-life, can be used in exactly the
same way as half-life; but the mean life
and half-life t1/2 have been adopted as
standard times associated with
exponential decay.

Time-variable quantities:
 Total activity— A, is the number of
decays per unit time of a radioactive
sample.

Number of particles— N, is the total

number of particles in the sample.

Specific activity— SA, number of

decays per unit time per amount of
substance of the sample at time set to
zero (t = 0). "Amount of substance" can
be the mass, volume or moles of the
initial sample.

These are related as follows:

where N0 is the initial amount of active
substance — substance that has the
same percentage of unstable particles as
when the substance was formed.

Mathematics of radioactive
decay
Universal law of radioactive
decay

Radioactivity is one very frequently given

example of exponential decay. The law
describes the statistical behaviour of a
large number of nuclides, rather than
individual atoms. In the following
formalism, the number of nuclides or the
nuclide population N, is of course a
discrete variable (a natural number)—but
for any physical sample N is so large that
it can be treated as a continuous
variable. Differential calculus is used to
model the behaviour of nuclear decay.

The mathematics of radioactive decay

depend on a key assumption that a
nucleus of a radionuclide has no
"memory" or way of translating its history
into its present behavior. A nucleus does
not "age" with the passage of time. Thus,
the probability of its breaking down does
not increase with time, but stays
constant no matter how long the nucleus
has existed. This constant probability
may vary greatly between different types
of nuclei, leading to the many different
observed decay rates. However, whatever
the probability is, it does not change.
This is in marked contrast to complex
objects which do show aging, such as
automobiles and humans. These
systems do have a chance of breakdown
per unit of time, that increases from the
moment they begin their existence.

One-decay process
Consider the case of a nuclide A that
decays into another B by some process
A → B (emission of other particles, like
electron neutrinos νe and electrons e− as
in beta decay, are irrelevant in what
follows). The decay of an unstable
nucleus is entirely random and it is
impossible to predict when a particular
atom will decay. However, it is equally
likely to decay at any instant in time.
Therefore, given a sample of a particular
radioisotope, the number of decay events
−dN expected to occur in a small interval
of time dt is proportional to the number
of atoms present N, that is[20]
Particular radionuclides decay at
different rates, so each has its own

decay constant λ. The expected decay

−dN/N is proportional to an increment of
time, dt:

The negative sign indicates that N

decreases as time increases, as the
decay events follow one after another.
The solution to this first-order differential
equation is the function:

where N0 is the value of N at time t =

0.[20]

We have for all time t:

where Ntotal is the constant number of

particles throughout the decay process,
which is equal to the initial number of A
nuclides since this is the initial
substance.

If the number of non-decayed A nuclei is:

then the number of nuclei of B, i.e. the
number of decayed A nuclei, is

The number of decays observed over a

given interval obeys Poisson statistics. If
the average number of decays is <N>,
the probability of a given number of
decays N is[20]

Chain-decay processes

Chain of two decays

Now consider the case of a chain of two
decays: one nuclide A decaying into
another B by one process, then B
decaying into another C by a second
process, i.e. A → B → C. The previous
equation cannot be applied to the decay
chain, but can be generalized as follows.
Since A decays into B, then B decays into
C, the activity of A adds to the total
number of B nuclides in the present
sample, before those B nuclides decay
and reduce the number of nuclides
leading to the later sample. In other
words, the number of second generation
nuclei B increases as a result of the first
generation nuclei decay of A, and
decreases as a result of its own decay
into the third generation nuclei C.[21] The
sum of these two terms gives the law for
a decay chain for two nuclides:

The rate of change of NB, that is dNB/dt,

is related to the changes in the amounts
of A and B, NB can increase as B is
produced from A and decrease as B
produces C.

Re-writing using the previous results:

The subscripts simply refer to the
respective nuclides, i.e. NA is the number
of nuclides of type A, NA0 is the initial
number of nuclides of type A, λA is the
decay constant for A - and similarly for
nuclide B. Solving this equation for NB
gives:

In the case where B is a stable nuclide

(λB = 0), this equation reduces to the
previous solution:
as shown above for one decay. The
solution can be found by the integration
factor method, where the integrating
factor is eλBt. This case is perhaps the
most useful, since it can derive both the
one-decay equation (above) and the
equation for multi-decay chains (below)
more directly.

Chain of any number of decays

For the general case of any number of

consecutive decays in a decay chain, i.e.
A1 → A2 ··· → Ai ··· → AD, where D is the
number of decays and i is a dummy
index (i = 1, 2, 3, ...D), each nuclide
population can be found in terms of the
previous population. In this case N2 = 0,
N3 = 0,..., ND = 0. Using the above result
in a recursive form:

The general solution to the recursive

problem is given by Bateman's
equations:[22]
 Bateman's equations

Alternative decay modes

In all of the above examples, the initial

nuclide decays into just one product.[23]
Consider the case of one initial nuclide
that can decay into either of two
products, that is A → B and A → C in
parallel. For example, in a sample of
potassium-40, 89.3% of the nuclei decay
to calcium-40 and 10.7% to argon-40. We
have for all time t:

which is constant, since the total number

of nuclides remains constant.
Differentiating with respect to time:

defining the total decay constant λ in

terms of the sum of partial decay
constants λB and λC:

Notice that
Solving this equation for NA:

where NA0 is the initial number of nuclide

A. When measuring the production of
one nuclide, one can only observe the
total decay constant λ. The decay
constants λB and λC determine the
probability for the decay to result in
products B or C as follows:
because the fraction λB/λ of nuclei decay
into B while the fraction λC/λ of nuclei
decay into C.

Corollaries of the decay laws

The above equations can also be written

using quantities related to the number of
nuclide particles N in a sample;

The activity: A = λN.

The amount of substance: n = N/L.
The mass: M = Arn = ArN/L.

where L = 6.022 × 1023 is Avogadro's

constant, Ar is the relative atomic mass
number, and the amount of the
substance is in moles.
Decay timing: definitions and
relations

Time constant and mean-life

For the one-decay solution A → B:

the equation indicates that the decay

constant λ has units of t−1, and can thus

τ
also be represented as 1/ , where τ is a
characteristic time of the process called
the time constant.

In a radioactive decay process, this time

constant is also the mean lifetime for
decaying atoms. Each atom "lives" for a
finite amount of time before it decays,
and it may be shown that this mean
lifetime is the arithmetic mean of all the
atoms' lifetimes, and that it is τ, which
again is related to the decay constant as
follows:

This form is also true for two-decay

processes simultaneously A → B + C,
inserting the equivalent values of decay
constants (as given above)

into the decay solution leads to:

Simulation of many identical atoms undergoing
radioactive decay, starting with either 4 atoms (left)
or 400 (right). The number at the top indicates how
many half-lives have elapsed.

Half-life

A more commonly used parameter is the

half-life. Given a sample of a particular
radionuclide, the half-life is the time
taken for half the radionuclide's atoms to
decay. For the case of one-decay nuclear
reactions:

the half-life is related to the decay

constant as follows: set N = N0/2 and t =
T1/2 to obtain

This relationship between the half-life

and the decay constant shows that
highly radioactive substances are quickly
spent, while those that radiate weakly
endure longer. Half-lives of known
radionuclides vary widely, from more
than 1019 years, such as for the very
nearly stable nuclide 209Bi, to 10−23
seconds for highly unstable ones.

The factor of ln(2) in the above relations

results from the fact that the concept of
"half-life" is merely a way of selecting a
different base other than the natural base
e for the lifetime expression. The time

constant τ is the e 
-1
  -life, the time until
only 1/e remains, about 36.8%, rather
than the 50% in the half-life of a

radionuclide. Thus, τ is longer than t1/2.

The following equation can be shown to
be valid:
Since radioactive decay is exponential
with a constant probability, each process
could as easily be described with a
different constant time period that (for
example) gave its "(1/3)-life" (how long
until only 1/3 is left) or "(1/10)-life" (a
time period until only 10% is left), and so

on. Thus, the choice of τ and t1/2 for

marker-times, are only for convenience,
and from convention. They reflect a
fundamental principle only in so much as
they show that the same proportion of a
given radioactive substance will decay,
during any time-period that one chooses.

Mathematically, the nth life for the above

situation would be found in the same
way as above—by setting N = N0/n,
t = T1/n and substituting into the decay
solution to obtain

Example

A sample of 14C has a half-life of 5,730

years and a decay rate of 14
disintegration per minute (dpm) per gram
of natural carbon.

If an artifact is found to have

radioactivity of 4 dpm per gram of its
present C, we can find the approximate
age of the object using the above
equation:
where:

years,

years.

Changing decay rates

The radioactive decay modes of electron
capture and internal conversion are
known to be slightly sensitive to
chemical and environmental effects that
change the electronic structure of the
atom, which in turn affects the presence
of 1s and 2s electrons that participate in
the decay process. A small number of
mostly light nuclides are affected. For
example, chemical bonds can affect the
rate of electron capture to a small degree
(in general, less than 1%) depending on
the proximity of electrons to the nucleus.
In 7Be, a difference of 0.9% has been
observed between half-lives in metallic
and insulating environments.[24] This
relatively large effect is because
beryllium is a small atom whose valence
electrons are in 2s atomic orbitals, which
are subject to electron capture in 7Be
because (like all s atomic orbitals in all
atoms) they naturally penetrate into the
nucleus.
In 1992, Jung et al. of the Darmstadt
Heavy-Ion Research group observed an
accelerated β− decay of 163Dy66+.
Although neutral 163Dy is a stable
isotope, the fully ionized 163Dy66+
undergoes β− decay into the K and L
shells to 163Ho66+ with a half-life of
47 days.[25]

Rhenium-187 is another spectacular

example. 187Re normally beta decays to
187Os with a half-life of 41.6 ×
109 years,[26] but studies using fully
ionised 187Re atoms (bare nuclei) have
found that this can decrease to only
33 years. This is attributed to "bound-
state β− decay" of the fully ionised atom
– the electron is emitted into the "K-shell"
(1s atomic orbital), which cannot occur
for neutral atoms in which all low-lying
bound states are occupied.[27]

Decay rate of radon-222 as a function of date and

time of day. The color-bar gives the power of the
observed signal and represents ~4% seasonal decay
rate variation.

A number of experiments have found

that decay rates of other modes of
artificial and naturally occurring
radioisotopes are, to a high degree of
precision, unaffected by external
conditions such as temperature,
pressure, the chemical environment, and
electric, magnetic, or gravitational
fields.[28] Comparison of laboratory
experiments over the last century,
studies of the Oklo natural nuclear
reactor (which exemplified the effects of
thermal neutrons on nuclear decay), and
astrophysical observations of the
luminosity decays of distant supernovae
(which occurred far away so the light has
taken a great deal of time to reach us),
for example, strongly indicate that
unperturbed decay rates have been
constant (at least to within the
limitations of small experimental errors)
as a function of time as well.
Recent results suggest the possibility
that decay rates might have a weak
dependence on environmental factors. It
has been suggested that measurements
of decay rates of silicon-32, manganese-
54, and radium-226 exhibit small
seasonal variations (of the order of
0.1%),[29][30][31] while the decay of radon-
222 is reported to exhibit large 4% peak-
to-peak seasonal variations,[32] proposed
to be related to either solar flare activity
or the distance from the Sun. However,
such measurements are highly
susceptible to systematic errors, and a
subsequent paper[33] has found no
evidence for such correlations in seven
other isotopes (22Na, 44Ti, 108Ag, 121Sn,
133Ba, 241Am, 238Pu), and sets upper
limits on the size of any such effects.

GSI anomaly

An unexpected series of experimental

results for the rate of decay of heavy
highly charged radioactive ions
circulating in a storage ring has provoked
theoretical activity in an effort to find a
convincing explanation. The rates of
weak decay of two radioactive species
with half lives of about 40 s and 200 s
are found to have a significant oscillatory
modulation, with a period of about 7 s.[34]
The observed phenomenon is known as
the GSI anomaly, as the storage ring is a
facility at the GSI Helmholtz Centre for
Heavy Ion Research in Darmstadt,
Germany. As the decay process
produces an electron neutrino, some of
the proposed explanations for the
observed rate oscillation invoke neutrino
properties. Initial ideas related to flavour
oscillation met with skepticism.[35] A
more recent proposal involves mass
differences between neutrino mass
eigenstates.[36]

Theoretical basis of decay

phenomena
The neutrons and protons that constitute
nuclei, as well as other particles that
approach close enough to them, are
governed by several interactions. The
strong nuclear force, not observed at the
familiar macroscopic scale, is the most
powerful force over subatomic
distances. The electrostatic force is
almost always significant, and, in the
case of beta decay, the weak nuclear
force is also involved.

The interplay of these forces produces a

number of different phenomena in which
energy may be released by
rearrangement of particles in the
nucleus, or else the change of one type
of particle into others. These
rearrangements and transformations
may be hindered energetically, so that
they do not occur immediately. In certain
cases, random quantum vacuum
fluctuations are theorized to promote
relaxation to a lower energy state (the
"decay") in a phenomenon known as
quantum tunneling. Radioactive decay
half-life of nuclides has been measured
over timescales of 55 orders of
magnitude, from 2.3 × 10−23 seconds (for
hydrogen-7) to 6.9 × 1031 seconds (for
tellurium-128).[37] The limits of these
timescales are set by the sensitivity of
instrumentation only, and there are no
known natural limits to how brief or long
a decay half-life for radioactive decay of
a radionuclide may be.
The decay process, like all hindered
energy transformations, may be
analogized by a snowfield on a mountain.
While friction between the ice crystals
may be supporting the snow's weight, the
system is inherently unstable with regard
to a state of lower potential energy. A
disturbance would thus facilitate the
path to a state of greater entropy; the
system will move towards the ground
state, producing heat, and the total
energy will be distributable over a larger
number of quantum states thus resulting
in an avalanche. The total energy does
not change in this process, but, because
of the second law of thermodynamics,
avalanches have only been observed in
one direction and that is toward the
"ground state" — the state with the
largest number of ways in which the
available energy could be distributed.

Such a collapse (a gamma-ray decay

event) requires a specific activation
energy. For a snow avalanche, this
energy comes as a disturbance from
outside the system, although such
disturbances can be arbitrarily small. In
the case of an excited atomic nucleus
decaying by gamma radiation in a
spontaneous emission of
electromagnetic radiation, the arbitrarily
small disturbance comes from quantum
vacuum fluctuations.[38]
A radioactive nucleus (or any excited
system in quantum mechanics) is
unstable, and can, thus, spontaneously
stabilize to a less-excited system. The
resulting transformation alters the
structure of the nucleus and results in
the emission of either a photon or a high-
velocity particle that has mass (such as
an electron, alpha particle, or other type).

Occurrence and applications

According to the Big Bang theory, stable
isotopes of the lightest five elements (H,
He, and traces of Li, Be, and B) were
produced very shortly after the
emergence of the universe, in a process
called Big Bang nucleosynthesis. These
lightest stable nuclides (including
deuterium) survive to today, but any
radioactive isotopes of the light elements
produced in the Big Bang (such as
tritium) have long since decayed.
Isotopes of elements heavier than boron
were not produced at all in the Big Bang,
and these first five elements do not have
any long-lived radioisotopes. Thus, all
radioactive nuclei are, therefore, relatively
young with respect to the birth of the
universe, having formed later in various
other types of nucleosynthesis in stars
(in particular, supernovae), and also
during ongoing interactions between
stable isotopes and energetic particles.
For example, carbon-14, a radioactive
nuclide with a half-life of only 5,730
years, is constantly produced in Earth's
upper atmosphere due to interactions
between cosmic rays and nitrogen.

Nuclides that are produced by

radioactive decay are called radiogenic
nuclides, whether they themselves are
stable or not. There exist stable
radiogenic nuclides that were formed
from short-lived extinct radionuclides in
the early solar system.[39][40] The extra
presence of these stable radiogenic
nuclides (such as Xe-129 from primordial
I-129) against the background of
primordial stable nuclides can be inferred
by various means.

Radioactive decay has been put to use in

the technique of radioisotopic labeling,
which is used to track the passage of a
chemical substance through a complex
system (such as a living organism). A
sample of the substance is synthesized
with a high concentration of unstable
atoms. The presence of the substance in
one or another part of the system is
determined by detecting the locations of
decay events.

On the premise that radioactive decay is

truly random (rather than merely chaotic),
it has been used in hardware random-
number generators. Because the process
is not thought to vary significantly in
mechanism over time, it is also a
valuable tool in estimating the absolute
ages of certain materials. For geological
materials, the radioisotopes and some of
their decay products become trapped
when a rock solidifies, and can then later
be used (subject to many well-known
qualifications) to estimate the date of the
solidification. These include checking the
results of several simultaneous
processes and their products against
each other, within the same sample. In a
similar fashion, and also subject to
qualification, the rate of formation of
carbon-14 in various eras, the date of
formation of organic matter within a
certain period related to the isotope's
half-life may be estimated, because the
carbon-14 becomes trapped when the
organic matter grows and incorporates
the new carbon-14 from the air.
Thereafter, the amount of carbon-14 in
organic matter decreases according to
decay processes that may also be
independently cross-checked by other
means (such as checking the carbon-14
in individual tree rings, for example).

Szilard–Chalmers effect

The Szilard–Chalmers effect is defined as

the breaking of a chemical bond between
an atom and the molecule that the atom
is part of, as a result of a nuclear
reaction of the atom. The effect can be
used to separate isotopes by chemical
means. The discovery of this effect is
due to L. Szilárd and T.A. Chalmers.[41]

Origins of radioactive
nuclides
Radioactive primordial nuclides found in
the Earth are residues from ancient
supernova explosions that occurred
before the formation of the solar system.
They are the fraction of radionuclides
that survived from that time, through the
formation of the primordial solar nebula,
through planet accretion, and up to the
present time. The naturally occurring
short-lived radiogenic radionuclides
found in today's rocks, are the daughters
of those radioactive primordial nuclides.
Another minor source of naturally
occurring radioactive nuclides are
cosmogenic nuclides, that are formed by
cosmic ray bombardment of material in
the Earth's atmosphere or crust. The
decay of the radionuclides in rocks of the
Earth's mantle and crust contribute
significantly to Earth's internal heat
budget.

Decay chains and multiple

modes
The daughter nuclide of a decay event
may also be unstable (radioactive). In
this case, it too will decay, producing
radiation. The resulting second daughter
nuclide may also be radioactive. This can
lead to a sequence of several decay
events called a decay chain (see this
article for specific details of important
natural decay chains).Eventually, a stable
nuclide is produced.

Gamma-ray energy spectrum of uranium ore (inset).

Gamma-rays are emitted by decaying nuclides, and
the gamma-ray energy can be used to characterize
the decay (which nuclide is decaying to which). Here,
using the gamma-ray spectrum, several nuclides that
are typical of the decay chain of 238U have been
identified: 226Ra, 214Pb, 214Bi.

An example is the natural decay chain of

238U:

Uranium-238 decays, through alpha-

emission, with a half-life of 4.5 billion
years to thorium-234
which decays, through beta-emission,
with a half-life of 24 days to
protactinium-234
which decays, through beta-emission,
with a half-life of 1.2 minutes to
uranium-234
which decays, through alpha-emission,
with a half-life of 240 thousand years
to thorium-230
which decays, through alpha-emission,
with a half-life of 77 thousand years to
radium-226
which decays, through alpha-emission,
with a half-life of 1.6 thousand years to
radon-222
which decays, through alpha-emission,
with a half-life of 3.8 days to polonium-
218
which decays, through alpha-emission,
with a half-life of 3.1 minutes to lead-
214
which decays, through beta-emission,
with a half-life of 27 minutes to
bismuth-214
which decays, through beta-emission,
with a half-life of 20 minutes to
polonium-214
which decays, through alpha-emission,
with a half-life of 160 microseconds to
lead-210
which decays, through beta-emission,
with a half-life of 22 years to bismuth-
210
which decays, through beta-emission,
with a half-life of 5 days to polonium-
210
 which decays, through alpha-emission,
with a half-life of 140 days to lead-206,
which is a stable nuclide.

Some radionuclides may have several

different paths of decay. For example,
approximately 36% of bismuth-212
decays, through alpha-emission, to
thallium-208 while approximately 64% of
bismuth-212 decays, through beta-
emission, to polonium-212. Both
thallium-208 and polonium-212 are
radioactive daughter products of
bismuth-212, and both decay directly to
stable lead-208.

Associated hazard warning

signs
The trefoil symbol used to indicate
ionising radiation.

2007 ISO radioactivity danger symbol

intended for IAEA Category 1, 2 and 3
sources defined as dangerous sources
capable of death or serious injury.[42]
 The dangerous goods transport
classification sign for radioactive
materials

See also
Wikimedia Commons has media related
to Radioactive decay by mode.

Actinides in the environment

Background radiation
Chernobyl disaster
Crimes involving radioactive
substances
Decay chain
Decay correct
Fallout shelter
Half-life
Induced radioactivity
Lists of nuclear disasters and
radioactive incidents
Multiplicative calculus
National Council on Radiation
Protection and Measurements
Nuclear engineering
Nuclear medicine
Nuclear pharmacy
 Nuclear physics
Nuclear power
Particle decay
Poisson process
Radiation
Radiation therapy
Radioactive contamination
Radioactivity in biology
Radiometric dating
Radionuclide a.k.a. "radio-isotope"
Secular equilibrium
Transient equilibrium

Notes
1. See Wu experiment among other
counterexamples when the decaying atom
is influenced by external factors.
2. Radionuclide is the more correct term,
but radioisotope is also used. The
difference between isotope and nuclide is
explained at Isotope#Isotope vs. nuclide.

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General
 "Radioactivity" , Encyclopædia
Britannica. 2006. Encyclopædia
Britannica Online. December 18, 2006
Radio-activity by Ernest Rutherford
Phd, Encyclopædia Britannica Eleventh
Edition

External links
The Wikibook Historical Geology has a
page on the topic of: Radioactive decay

Look up radioactivity in Wiktionary, the

free dictionary.

The Lund/LBNL Nuclear Data Search

– Contains tabulated information on
radioactive decay types and energies.
Nomenclature of nuclear chemistry
Specific activity and related topics .
The Live Chart of Nuclides – IAEA
Interactive Chart of Nuclides
Health Physics Society Public
Education Website
 Beach, Chandler B., ed. (1914).
"Becquerel Rays". The New Student's
Reference Work. Chicago: F. E.
Compton and Co.
Annotated bibliography for
radioactivity from the Alsos Digital
Library for Nuclear Issues
Stochastic Java applet on the decay of
radioactive atoms by Wolfgang Bauer
 Stochastic Flash simulation on the
decay of radioactive atoms by David
M. Harrison
"Henri Becquerel: The Discovery of
Radioactivity", Becquerel's 1896
articles online and analyzed on
BibNum [click 'à télécharger' for
English version].
"Radioactive change", Rutherford &
Soddy article (1903), online and
analyzed on Bibnum [click 'à
télécharger' for English version].

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