Notes of Chemistry BSC-MSC Chapter# 2 RADIOACTIVE DECAY

CHAPTER 2     RADIOACTIVE DECAY

2.1    Radioactive decay

Radioactive decay is the process in which an unstable atomic nucleus loses energy by emitting radiation in the form of particles or electromagnetic waves. This decay, or loss of energy, results in an atom of one type, called the parent nuclide transforming to an atom of a different type, called the daughter nuclide. For example: a carbon-14 atom (the "parent") emits radiation and transforms to a nitrogen-14 atom (the "daughter"). This is a random process on the atomic level, in that it is impossible to predict when a particular atom will decay, but given a large number of similar atoms, the decay rate, on average, is predictable. 

FIG;The trefoil symbol is used to indicate radioactive material.

FIG; The danger classification sign of radioactive materials

The SI unit of radioactive decay (the phenomenon of natural and artificial radioactivity) is the becquerel (Bq). One Bq is defined as one transformation (or decay) per second. Since any reasonably-sized sample of radioactive material contains many atoms, a Bq is a tiny measure of activity; amounts on the order of TBq (terabecquerel) or GBq (gigabecquerel) are commonly used. Another unit of (radio)activity is the curie, Ci, which was originally defined as the activity of one gram of pure radium, isotope Ra-226. At present it is equal (by definition) to the activity of any radionuclide decaying with a disintegration rate of 3.7 × 10¹⁰ Bq. The use of Ci is presently discouraged by SI.

Explanation;

The neutrons and protons that constitute nuclei, as well as other particles that may approach 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 also significant, while the weak nuclear force is responsible for beta decay.

The interplay of these forces is simple. Some configurations of the particles in a nucleus have the property that, should they shift ever so slightly, the particles could fall into a lower-energy arrangement (with the extra energy moving elsewhere). One might draw an analogy with a snowfield on a mountain: while friction between the snow crystals can support the snow's weight, the system is inherently unstable with regard to a lower-potential-energy state, and a disturbance may facilitate the path to a greater entropy state (i.e., towards the ground state where heat will be produced, and thus total energy is distributed over a larger number of quantum states). Thus, an avalanche results. The total energy does not change in this process, but because of entropy effects, avalanches only happen in one direction, and the end of this direction, which is dictated by the largest number of chance-mediated ways to distribute available energy, is what we commonly refer to as the "ground state."

Such a collapse (a decay event) requires a specific activation energy. In the case of a snow avalanche, this energy classically comes as a disturbance from outside the system, although such disturbances can be arbitrarily small. In the case of an excited atomic nucleus, the arbitrarily small disturbance comes from quantum vacuum fluctuations. A nucleus (or any excited system in quantum mechanics) is unstable, and can thus spontaneously stabilize to a less-excited system. This process is driven by entropy considerations: the energy does not change, but at the end of the process, the total energy is more diffused in spacial volume. The resulting transformation alters the structure of the nucleus. Such a reaction is thus a nuclear reaction, in contrast to chemical reactions, which also are driven by entropy, but which involve changes in the arrangement of the outer electrons of atoms, rather than their nuclei.

Some nuclear reactions do involve external sources of energy, in the form of collisions with outside particles. However, these are not considered decay. Rather, they are examples of induced nuclear reactions. Nuclear fission and fusion are common types of induced nuclear reactions.

2.2 Types of Radioactive Decay

          There are two types of radioactive decays

1                    natural Radioactive decay

2                    Artificial Radioactive Decay.

Natural Radioactive Decay

            The words "radioactivity" and "radiation" come from "radius".

RADIATION means any sort of an energy form spreading out from a centre. Radio waves, light, infrared light and microwaves are all examples of radiation. These types of radiation are all related, they are members of the ELECTROMAGNETIC SPECTRUM and all travel at the speed of light, 3 x 10⁸ms^-1.

The radiation associated with radioactivity, however, is far more violent, that is, energetic. This radiation is classed as IONISING RADIATION as the radiation from the nuclei can easily destroy molecules by stripping away electrons from their atoms - ionising them. (Other Ionising Radiation includes UV light and X-rays. Both of these are members of the Electromagnetic Spectrum.)

The radiation from nuclear processes are not necessarily members of the Electromagnetic Spectrum. Two forms are actually "particles"- but come from the nucleus itself as the change occurs.

The three common natural types of radiation from nuclear decay are;

α Radiation (alpha)  - a helium atom nucleus. This is a "slow" moving particle,  with a short range in air.  Alpha particles are extremely dangerous inside the body but not very dangerous outside as they cannot penetrate the skin. The speed of αradiation is about 0.1 of the speed of light.

The radiation has two elementary positive charges and the particle has considerable mass.

 Alpha Decay

 Beta decay;

 Fig Beta-minus (β^-) decay. The intermediate emission of a W^- boson is omitted

Fig The Feynman diagram for beta decay of a neutron into a proton, electron, and electron antineutrino via an intermediate heavy W^- boson

In nuclear physics, beta decay is a type of radioactive decay in which a beta particle (an electron or a positron) is emitted. In the case of electron emission, it is referred to as "beta minus" (β⁻), while in the case of a positron emission as "beta plus" (β⁺). Kinetic energy of beta particles has continuous spectrum ranging from 0 to maximal available energy Q, which depends on parent and daughter nuclear states participating in the decay. Typical Q is of order of 1 MeV, but it can be from few keV to few tens MeV. The most energetic beta particles are ultrarelativistic, with speeds very close to the light speed.

In β⁻ decay, the weak interaction converts a neutron (n⁰) into a proton (p⁺) while emitting an electron (e⁻) and an anti-neutrino ():

.At the fundamental level (as depicted in the Feynman diagram below), this is due to the conversion of a down quark to an up quark by emission of a W^- boson; the W^- boson subsequently decays into an electron and an anti-neutrino.

In β⁺ decay, energy is used to convert a proton into a neutron, a positron (e⁺ ) and a neutrino (ν_e):

So, unlike beta minus decay, beta plus decay cannot occur in isolation, because it requires energy, the mass of the neutron being greater than the mass of the proton. Beta plus decay can only happen inside nuclei when the absolute value of the binding energy of the daughter nucleus is higher than that of the mother nucleus. The difference between these energies goes into the reaction of converting a proton into a neutron, a positron and a neutrino and into the kinetic energy of these particles.

In all the cases where β⁺ decay is allowed energetically (and the proton is a part of a nucleus with electron shells), it is accompanied by the electron capture process, when an atomic electron is captured by a nucleus with the emission of a neutrino:

.But if the energy difference between initial and final states is low (less than 2m_ec²), then β⁺ decay is not energetically possible, and electron capture is the sole decay mode.

If the proton and neutron are part of an atomic nucleus, these decay processes transmute one chemical element into another. 

Beta decay does not change the number of nucleons A in the nucleus but changes only its charge Z. Thus the set of all nuclides with the same A can be introduced; these isobaric nuclides may turn into each other via beta decay. Among them, several nuclides (at least one) are beta stable, because they present local minima of the mass excess: if such a nucleus has (A, Z) numbers, the neighbour nuclei (A, Z−1) and (A, Z+1) have higher mass excess and can beta decay into (A, Z), but not vice versa. It should be noted, that a beta-stable nucleus may undergo other kinds of radioactive decay (alpha decay, for example). In nature, most isotopes are beta stable, but a few exceptions exist with half-lives so long that they have not had enough time to decay since the moment of their nucleosynthesis. One example is ⁴⁰K, which undergoes all three types of beta decay (beta minus, beta plus and electron capture) with half life of 1.277×10⁹ years.

Some nuclei can undergo double beta decay (ββ decay) where the charge of the nucleus changes by two units. In most practically interesting cases, single beta decay is energetically forbidden for such nuclei, because when β and ββ decays are both allowed, the probability of β decay is (usually) much higher, preventing investigations of very rare ββ decays. Thus, ββ decay is usually studied only for beta stable nuclei. Like single beta decay, double beta decay does not change A; thus, at least one of the nuclides with some given A has to be stable with regard to both single and double beta decay.

Beta decay can be considered as a perturbation as described in quantum mechanics, and thus follows Fermi's Golden Rule.

Kurie plot

A Kurie plot (also known as a Fermi-Kurie plot) is a graph used in studying beta decay, in which the square root of the number of beta particles whose momentum (or energy) lie within a certain narrow range, divided by a function worked out by Fermi, is plotted against beta-particle energy; it is a straight line for allowed transitions and some forbidden transitions, in accord with the Fermi beta-decay theory.

History

Historically, the study of beta decay provided the first physical evidence of the neutrino. In 1911 Lise Meitner and Otto Hahn performed an experiment that showed that the energies of electrons emitted by beta decay had a continuous rather than discrete spectrum. This was in apparent contradiction to the law of conservation of energy, as it appeared that energy was lost in the beta decay process. A second problem was that the spin of the Nitrogen-14 atom was 1, in contradiction to the Rutherford prediction of ½.

In 1920-1927, Charles Drummond Ellis (along with James Chadwick and colleagues) established clearly that the beta decay spectrum is really continuous, ending all controversies.

In a famous letter written in 1930 Wolfgang Pauli suggested that in addition to electrons and protons atoms also contained an extremely light neutral particle which he called the neutron. He suggested that this "neutron" was also emitted during beta decay and had simply not yet been observed. In 1931 Enrico Fermi renamed Pauli's "neutron" to neutrino, and in 1934 Fermi published a very successful model of beta decay in which neutrinos were produced.

Where X and Y represent the parent and daughter nuclei respectively, (A= mass number, Z= atomic number, N= number of neutrons).

Gamma (γ) radiation

This is a very energetic form of electromagnetic radiation. Compared to light , each bit (photon) has 1 million times as much energy ( or 1 thousand times more energy than an X-ray photon).

They travel at the speed of light. They happily travel through centimetres of lead and travel easily through air. They are a danger to the human body even when not ingested due to this penetrative ability.

Gamma radiation has no charge associated with it.

Nuclear decays and reactions produce other forms of radiation as well. Some are from artificial elements, some are products of splitting atoms.

Neutrinos, antineutrinos, weird neutral particles which are emitted with electrons in beta decay. They can be ignored for the purpose of this course.

β+ ; an antielectron or positron It has all the same characteristics of a normal electron except for having the same charge as the proton and the disconcerting habit, if it meets an ordinary electron, of turning itself and the electron into gamma rays! (All "antimatter" will do this on meeting the matter counterpart!)

n; a neutron,- emitted as a side product of nuclear fission. Solo neutrons have a half life of about 10 minutes. They easily pass through steel plating and large doses finally have deadly consequences. Neutron bombs are fission bombs with less blast but more leaky neutrons. They therefore do less damage to factories but kill very satisfactorily.

All radiation can be absorbed to "negligible" levels by sufficient material and, of course, the further one is from the source, the smaller the received dosage.  (Recall that radiation means spreading out from a centre.  The further from that centre you are the more the radiation energy has spread out.)

Because of the energy of the radiation, radioactive sources can fog photographic film.  This technique is used to crudely monitor dosage for people who work with the materials

 2.3     Electron capture

Electron capture (sometimes called Inverse Beta Decay) is a decay mode for isotopes that will occur when there are too many protons in the nucleus of an atom and insufficient energy to emit a positron; however, it continues to be a viable decay mode for radioactive isotopes that can decay by positron emission. If the energy difference between the parent atom and the daughter atom is less than 1.022 MeV, positron emission is forbidden and electron capture is the sole decay mode. For example, Rubidium-83 will decay to Krypton-83 solely by electron capture (the energy difference is about 0.9 MeV).

In this case, one of the orbital electrons, usually from the K or L electron shell (K-electron capture, also K-capture, or L-electron capture, L-capture), is captured by a proton in the nucleus, forming a neutron and a neutrino. Since the proton is changed to a neutron, the number of neutrons increases by 1, the number of protons decreases by 1, and the atomic mass number remains unchanged. By changing the number of protons, electron capture transforms the nuclide into a new element. The atom moves into an excited state with the inner shell missing an electron. When transiting to the ground state, the atom will emit an X-ray photon (a type of electromagnetic radiation) and/or Auger electrons.

(Please note that it is one of the initial atom's own electrons that is captured, not a new, incoming electron as might be suggested by the way the above reactions are written.)

            Radioactive isotopes which decay by pure electron capture can, in theory, be inhibited from radioactive decay if they are fully ionized ("stripped" is sometimes used to describe such ions). It is hypothesized that such elements, if formed by the r-process in exploding supernovae, are ejected fully ionized and so do not undergo radioactive decay as long as they do not encounter electrons in outer space. Anomalies in elemental distributions are thought to be partly a result of this effect on electron capture.

Chemical bonds can also affect the rate of electron capture to a small degree (generally less than 1%) depending on the proximity of electrons to the nucleus.

Around the elements in the middle of the periodic table, isotopes that are lighter than stable isotopes of the same element tend to decay through electron capture, while isotopes heavier than the stable ones decay by (negative) beta decay. A good example of this effect would be silver, as its light isotopes use electron capture and the heavier ones decay by negative beta emission.

Common Examples

Some common radioisotopes that decay by electron capture include:

Radioisotope	Half-life
Be-7	53.28 d
Ar-37	35.0 d
Ca-41	1.03E5 a
Ti-44	52 a
V-49	337 d
Cr-51	27.7 d
Mn-53	3.7E6 a
Co-57	271.8 d
Ni-56	6.10 d
Ga-67	3.260 d
Ge-68	270.8 d
Se-72	8.5 d

2.4     Double electron capture

Double electron capture is a decay mode of atomic nucleus. For a nuclide (A, Z) with number of nucleons A and atomic number Z, double electron capture is only possible if the mass of the nuclide of (A, Z-2) is lower.

In this mode of decay, two of the orbital electrons are captured by two protons in the nucleus, forming two neutrons. Two neutrinos are emitted in the process. Since the protons are changed to neutrons, the number of neutrons increases by 2, the number of protons Z decreases by 2, and the atomic mass A remains unchanged. By changing the number of protons, double electron capture transforms the nuclide into a new element.

In most of cases this decay mode is masked by more probable modes (single electron capture etc.), but when all these modes are forbidden or strongly suppressed, double electron capture becomes the main mode of decay. There exist 35 naturally occurring isotopes that can undergo double electron capture. However, there are no confirmed observations of this process. The one reason is that the probability of double electron capture is enormously small (the theoretical predictions of half-lives for this mode lies well above 10²⁰ years). The second reason is that the only detectable particles created in this process are X-rays and Auger electrons that are emitted by the excited atomic shell. In the range of their energies (~1-10 keV), the background is usually high. Thus, the experimental detection of double electron capture is more difficult than that for double beta decay.

If the mass difference between the mother and daughter atoms is more than two masses of electron (1.022 MeV), the energy released in the process is enough to allow another mode of decay: electron capture with positron emission. It occurs simultaneously with double electron capture, their branching ratio depending on nuclear properties. When the mass difference is more than four electron masses (2.044 MeV), the third mode - double positron decay - is allowed. Only 6 naturally occurring nuclides can decay via these three modes simultaneously.

Neutrinoless double electron capture

The above described process with capture of two electrons and emission of two neutrinos (two-neutrino double electron capture) is allowed by the Standard Model of particle physics: no conservation laws (including lepton number conservation) are violated. However, if the lepton number is not conserved, another kind of the process can occur: the so-called neutrinoless double electron capture. In this case, two electrons are captured by nucleus, but neutrinos are not emitted. The energy released in this process is carried away by an internal bremsstrahlung gamma quantum. This mode of decay has never been observed experimentally, and would contradict the Standard Model if it were observed.

2.5     Modes of decay

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.

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 emission	A proton ejected from nucleus	(A–1, Z–1)
Neutron emission	A neutron ejected from nucleus	(A–1, Z)
Double proton emission	Two protons ejected from nucleus simultaneously	(A–2, Z–2)
Spontaneous fission	Nucleus disintegrates into two or more smaller nuclei and other particles	-
Cluster decay	Nucleus emits a specific type of smaller nucleus (A1, Z1) smaller than, or larger than, an alpha particle	(A–A1, Z–Z1) + (A1,Z1)
Different modes of beta decay:
Beta-Negative decay	A nucleus emits an electron and an antineutrino	(A, Z+1)
Positron emission, also Beta-Positive decay	A nucleus emits a positron and a neutrino	(A, Z–1)
Electron capture	A nucleus captures an orbiting electron and emits a neutrino - The daughter nucleus is left in an excited and unstable state	(A, Z–1)
Double beta decay	A nucleus emits two electrons and two antineutrinos	(A, Z+2)
Double electron capture	A nucleus absorbs two orbital electrons and emits two neutrinos - The daughter nucleus is left in an excited and unstable state	(A, Z–2)
Electron capture with positron emission	A nucleus absorbs one orbital electron, emits one positron and two neutrinos	(A, Z–2)
Double positron emission	A nucleus emits two positrons and two neutrinos	(A, Z–2)

Transitions between states of the same nucleus:
Gamma decay	Excited nucleus releases a high-energy photon (gamma ray)	(A, Z)
Internal conversion	Excited nucleus transfers energy to an orbital electron and it is ejected from the atom	(A, Z)

Radioactive decay results in a reduction of summed rest mass, which is converted to energy (the disintegration energy) according to the formula E = mc². This energy is released as kinetic energy of the emitted particles. The energy remains associated with a measure of mass of the decay system invariant mass, inasmuch the kinetic energy of emitted particles contributes also to the total invariant mass of systems. Thus, the sum of rest masses of particles is not conserved in decay, but the system mass or system invariant mass (as also system total energy) is conserved.

Decay chains and multiple modes

The daughter nuclide of a decay event may also be unstable (radioactive). In this case, it will also decay, producing radiation. The resulting second daughter nuclide may also be radioactive. This can lead to a sequence of several decay events. Eventually a stable nuclide is produced. This is called a decay chain.

An example is the natural decay chain of Uranium-238 which 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 the Thallium-208 and the Polonium-212 are radioactive daughter products of Bismuth-212 and decay directly to stable Lead-208.

2.6     Occurrence and applications

According to the Big Bang theory, radioactive isotopes of the lightest elements (H, He, and traces of Li) were produced very shortly after the emergence of the universe. However, these nuclides are so highly unstable that virtually none of them have survived to today. Most radioactive nuclei are therefore relatively young, having formed in stars (particularly supernovae) and during ongoing interactions between stable isotopes and energetic particles. For example, carbon-14, a radioactive nuclide with a half-life of only 5730 years, is constantly produced in Earth's upper atmosphere due to interactions between cosmic rays and nitrogen.

Radioactive decay has been put to use in the technique of radioisotopic labeling, 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-live 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 which may also be independently cross-checked by other means (such as checking the carbon-14 in individual tree rings, for example).

Radioactive decay rates

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

Constant quantities:

·         half life — symbol t_{1 / 2} — the time for half of a substance to decay.

·         mean lifetime — symbol τ — the average lifetime of any given particle.

·         decay constant — symbol λ — the inverse of the mean lifetime.

(Note that although these are constants, they are associated with statistically random behavior of substances, and predictions using these constants are less accurate for small number of atoms.)

2.7     How to Change Nuclear Decay Rates

"I've had this idea for making radioactive nuclei decay faster/slower than they normally do.  You do [this, that, and the other thing].  Will this work?"

Short Answer: Possibly, but probably not usefully.

Long Answer:

"One of the paradigms of nuclear science since the very early days of its study has been the general understanding that the half-life, or decay constant, of a radioactive substance is independent of extra nuclear considerations." (Emery, cited below.) Like all paradigms, this one is subject to some interpretation.  Normal decay of radioactive stuff proceeds via one of four mechanisms:

1.       Emission of an alpha particle -- a helium-4 nucleus -- reducing the number of protons and neutrons present in the parent nucleus by two each

2.       "Beta decay," encompassing several related phenomena in which a neutron in the nucleus turns into a proton, or a proton turns into a neutron -- along with some other things including emission of a neutrino.  The "other things", as we shall see, are at the bottom of several questions involving perturbation of decay rates

3.       Emission of one or more gamma rays -- energetic photons -- that take a nucleus from an excited state to some other (typically ground) state; some of these photons may be replaced by "conversion electrons," of which more shortly

4.       Spontaneous fission, in which a sufficiently heavy nucleus simply breaks in half.  Most of the discussion about alpha particles will also apply to spontaneous fission.

Gamma emission often occurs from the daughter of one of the other decay modes.  We neglect very exotic processes like C-14 emission or double beta decay in this analysis.

"Beta decay" refers most often to a nucleus with a neutron excess, which decays by converting a neutron into a proton:

               n ----> p + e- + anti-nu(e),

Where n means neutron, p means proton, e- means electron, and anti-nu(e) means an antineutrino of the electron type.  The type of beta decay that involves destruction of a proton is not familiar to many people, so deserves a little elaboration.  Either of two processes may occur when this kind of decay happens:

                p ----> n + e+ + nu(e), 

Where e+ means positron and nu(e) means electron neutrino; or

        p + e- ----> n + nu(e),

Where e-means a negatively charged electron, which is captured from the neighborhood of the nucleus undergoing decay.  These processes are called "positron emission" and "electron capture," respectively.  A given nucleus that has too many protons for stability may undergo beta decay through either, and typically both, of these reactions.

"Conversion electrons" are produced by the process of "internal conversion," whereby the photon that would normally be emitted in gamma decay is virtual and its energy is absorbed by an atomic electron.  The absorbed energy is sufficient to unbind the electron from the nucleus (ignoring a few exceptional cases), and it is ejected from the atom as a result.

Now for the tie-in to decay rates.  Both the electron-capture and internal conversion phenomena require an electron somewhere close to the decaying nucleus.  In any normal atom, this requirement is satisfied in spades: the innermost electrons are in states such that their probability of being close to the nucleus is both large and insensitive to things in the environment.  The decay rate depends on the electronic wavefunctions, i.e, how much of their time the inner electrons spend very near the nucleus -- but only very weakly.  For most nuclides that decay by electron capture or internal conversion, most of the time, the probability of grabbing or converting an electron is also insensitive to the environment, as the innermost electrons are the ones most likely to get grabbed/converted.

However, there are exceptions, the most notable being the the astrophysically important isotope beryllium-7.  Be-7 decays purely by electron capture (positron emission being impossible because of inadequate decay energy) with a half-life of somewhat over 50 days.  It has been shown that differences in chemical environment result in half-life variations of the order of 0.2%, and high pressures produce somewhat similar changes.  Also, a recent paper measures a 0.8% reduction in half-life for Be-7 atoms enclosed within C60 cages.  Other cases where known changes in decay rate occur are Zr-89 and Sr-85, also electron capturers; Tc-99m ("m" implying an excited state), which decays by both beta and gamma emission; and various other "metastable" things that decay by gamma emission with internal conversion.  With all of these other cases the magnitude of the effect is less than is typically the case with Be-7.

What makes these cases special?  The answer is that one or more of the usual starting assumptions -- insensitivity of electron wave function near the nucleus to external forces, or availability of the innermost electrons for capture/conversion -- are not completely valid.  Atomic beryllium only has 4 electrons to begin with, so that the "innermost electrons" are also practically the outermost ones and therefore much more sensitive to chemical effects than usual.  With most of the other cases, there is so little energy available from the decay (as little as a few electron volts; compare most radioactive decays, where hundreds or thousands of kilovolts are released), courtesy of accidents of nuclear structure, that the innermost electrons can't undergo internal conversion.  Remember that converting an electron requires dumping enough energy into it to expel it from the atom (more or less); "enough energy," in context, is typically some tens of keV, so they don't get converted at all in these cases.  Conversion therefore works only on some of the outer electrons, which again are more sensitive to the environment.

A real anomaly is the beta emitter Re-187.  Its decay energy is only about 2.6 keV, practically nothing by nuclear standards.  "That this decay occurs at all is an example of the effects of the atomic environment on nuclear decay: the bare nucleus Re-187 [i.e., stripped of all orbital electrons] is stable against beta decay [but not to bound state beta decay, in which the outgoing electron is captured by the daughter nucleus into a tightly bound orbital] and it is the difference of 15 keV in the total electronic binding energy of osmium [to which it decays] and rhenium. . . which makes the decay possible" (Emery).  The practical significance of this little peculiarity, of course, is low, as Re-187 already has a half life of over 10¹⁰ years.

Alpha decay and spontaneous fission might also be affected by changes in the electron density near the nucleus, for a different reason.  These processes occur as a result of penetration of the "Coulomb barrier" that inhibits emission of charged particles from the nucleus, and their rate is very sensitive to the height of the barrier.  Changes in the electron density could, in principle, affect the barrier by some tiny amount.  However, the magnitude of the effect is very small, according to theoretical calculations; for a few alpha emitters, the change has been estimated to be of the order of 1 part in 10⁷ (!) or less, which would be unmeasurable in view of the fact that the alpha emitters' half lives aren't known to that degree of accuracy to begin with.

All told, the existence of changes in radioactive decay rates due to the environment of the decaying nuclei is on solid grounds both experimentally and theoretically.  But the magnitude of the changes is nothing to get very excited about.

2.8     Half-life

The half-life of a quantity whose value decreases with time is the interval required for the quantity to decay to half of its initial value. The concept originated in the study of radioactive decay which is subject to exponential decay but applies to all phenomena including those which are described by non-exponential decays.

The term half-life was coined in 1907, but it was always referred to as half-life period. It was not until the early 1950s that the word period was dropped from the name.

Number of half-lives elapsed	Fraction remaining	As power of 2	As %
0	1/1	1/20	100
1	1/2	1/21	50
2	1/4	1/22	25
3	1/8	1/23	12.5
4	1/16	1/24	6.25
5	1/32	1/25	3.125
6	1/64	1/26	1.563
7	1/128	1/27	0.781
...	...	...	...
n	1 / 2n	1 / 2n	100(1 / 2n)

The table at right shows the reduction of the quantity in terms of the number of half-lives elapsed. It can be shown that, for exponential decay, the half-life t_{1 / 2} obeys this relation:

where

·         ln(2) is the natural logarithm of 2 (approximately 0.693), and

·         λ is the decay constant, a positive constant used to describe the rate of exponential decay.

The half-life is related to the mean lifetime τ by the following relation:

Examples

Constant λ can represent many different specific physical quantities, depending on what process is being described.

1.      In an RC circuit or RL circuit, λ is the reciprocal of the circuit's time constant. For simple RC and RL circuits, λ equals 1 / RC or R / L, respectively.

2.      In first-order chemical reactions, λ is the reaction rate constant.

3.      In radioactive decay, it describes the probability of decay per unit time: dN = λNdt, where dN is the number of nuclei decayed during the time dt, and N is the quantity of radioactive nuclei.

4.      In biology (specifically pharmacokinetics), from MeSH: Half-Life: The time it takes for a substance (drug, radioactive nuclide, or other) to lose half of its pharmacologic, physiologic, or radiologic activity. Year introduced: 1974 (1971).

Decay by two or more processes

Some quantities decay by two processes simultaneously (see Decay by two or more processes). In a fashion similar to the previous section, we can calculate the new total half-life T_{1 / 2} and we'll find it to be:

or, in terms of the two half-lives t₁ and t₂

i.e., half their harmonic mean.

Simple Formula

m(t) mass left depending on time:

m(0)= initial mass

t = time passed

t_{1 / 2} = half-life of the object.

Derivation

Quantities that are subject to exponential decay are commonly denoted by the symbol N. (This convention suggests a decaying number of discrete items. This interpretation is valid in many, but not all, cases of exponential decay.) If the quantity is denoted by the symbol N, the value of N at a time t is given by the formula:

where N_o is the initial value of N (at t = 0).

When t = 0, the exponential is equal to 1, and N(t) is equal to N₀. As t approaches infinity, the exponential approaches zero. In particular, there is a time such that

Substituting into the formula above, we have

Experimental determination

The half-life of a process can be determined easily by experiment. In fact, some methods do not require advance knowledge of the law governing the decay rate, be it exponential decay or another pattern.

Most appropriate to validate the concept of half-life for radioactive decay, in particular when dealing with a small number of atoms, is to perform experiments and correct computer simulations. Validation of physics-math models consists in comparing the model's behavior with experimental observations of real physical systems or valid simulations (physical and/or computer). The references given here describe how to test the validity of the exponential formula for small number of atoms with simple simulations, experiments, and computer code.

In radioactive decay, the exponential model does not apply for a small number of atoms (or a small number of atoms is not within the domain of validity of the formula or equation or table). The DIY experiments use pennies or M&M's candies. A similar experiment is performed with isotopes of a very short half-life. Of particular note, atoms undergo radioactive decay in whole units, and so after enough half-lives the remaining original quantity becomes an actual zero rather than asymptotically approaching zero as with continuous systems.

2.9   Radioactive decay rates

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

Constant quantities:

·         half life — symbol t_{1 / 2} — the time for half of a substance to decay.

·         mean lifetime — symbol τ — the average lifetime of any given particle.

·         decay constant — symbol λ — the inverse of the mean lifetime.

(Note that although these are constants, they are associated with statistically random behavior of substances, and predictions using these constants are less accurate for small number of atoms.)

Time-variable quantities:

1. Total activity — symbol A — number of decays an object undergoes per second.
2. Number of particles — symbol N — the total number of particles in the sample.
3. Specific activity — symbol S_A — number of decays per second per amount of substance. (The "amount of substance" can be the unit of either mass or volume.)

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

2.10  Activity measurements

The units in which activities are measured are: becquerel (symbol Bq) = number of disintegrations per second; curie (Ci) = 3.7 × 10¹⁰ disintegrations per second. Low activities are also measured in disintegrations per minute (dpm).

Decay timing

As discussed above, 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 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. If N is the number of atoms, then the probability of decay (– dN/N) is proportional to dt:

Particular radionuclides decay at different rates, each having its own decay constant (λ). The negative sign indicates that N decreases with each decay event. The solution to this first-order differential equation is the following function:

This function represents exponential decay. It is only an approximate solution, for two reasons. Firstly, the exponential function is continuous, but the physical quantity N can only take non-negative integer values. Secondly, because it describes a random process, it is only statistically true. However, in most common cases, N is a very large number and the function is a good approximation.

In addition to the decay constant, radioactive decay is sometimes characterized by the mean lifetime. Each atom "lives" for a finite amount of time before it decays, and the mean lifetime is the arithmetic mean of all the atoms' lifetimes. It is represented by the symbol τ, and is related to the decay constant as follows:

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. The half life is related to the decay constant as follows:

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 very nearly stable nuclides, e.g. ²⁰⁹Bi), to 10^-23 seconds for highly unstable ones.