Periodic Table

Radiation is all around us. In, around, and above the world we live in. It is a natural energy force that surrounds us. It is a part of our natural world that has been here since the birth of our planet. All living creatures, from the beginning of time, have been, and are still being, exposed to ionizing radiation. Ionizing radiation is generated through nuclear reactions, nuclear decay, by very high temperature, or via acceleration of charged particles in electromagnetic fields. But in general, there are two broad categories of radiation sources:

• Natural Background Radiation. Natural background radiation includes radiation produced by the Sun, lightnings, primordial radioisotopes or supernova explosions etc.
• Man-Made Sources of Radiation. Man-made sources include medical uses of radiation, residues from nuclear tests, industrial uses of radiation etc.

Special Reference: Sources and effects of ionizing radiation, Annex B. UNSCEAR. New York, 2010. ISBN: 978-92-1-142274-0.

The United Nations Scientific Committee on the Effects of Atomic Radiation (UNSCEAR) itemized types of human exposures as:

• public exposure, which is the exposure of individual members of the public and of the population in general
• occupational radiation exposure, which is the exposure of workers in situations where their exposure is directly related to or required by their work

Natural Background Radiation

All living creatures, from the beginning of time, have been, and are still being, exposed to ionizing radiation. This radiation is not associated with any human activity.  There are radioactive isotopes in our bodies, houses, air, water and in the soil. We all are also exposed to radiation from outer space.

Types of Natural Background Radiation

We divide all these natural radiation sources into three groups:

• Cosmic Radiation. Cosmic radiation refers to sources of radiation in the form of cosmic rays that come from the sun or from outer space. At ground level the muons, with energies mostly between 1 and 20 GeV, contribute about 75 % of the absorbed dose rate in free air. The remainder comes from electrons produced by the muons or present in the electromagnetic cascade. The annual cosmic ray dose at sea level is around 0.27 mSv (27 mrem). If you live at higher elevations or are a frequent airline passenger, this exposure can be significantly higher, since the atmosphere is thinner here. The effects of the earth’s magnetic field also determines the dose from cosmic radiation.
• Terrestrial Radiation. Terrestrial radiation refers to sources of radiation that are in the soil, water, and vegetation. The major isotopes of concern for terrestrial radiation are uranium and the decay products of uranium, such as thorium, radium, and radon. The average dose rate that originates from terrestrial nuclides (except radon exposure) is about 0.057 µGy/hr. The maximum values have been measured on monazite sand in Guarapari, Brazil (up to 50 µGy/hr and in Kerala, India (about 2 µGy/hr), and on rocks with a high radium concentration in Ramsar, Iran (from 1 to 10 µGy/hr). The average annual radiation dose to a person from radon is about 2 mSv/year and it may vary over many orders of magnitude from place to place. Radon is so important, that it is usually treated separately.
• Internal Radiation. In addition to the cosmic and terrestrial sources, all people also have radioactive potassium-40, carbon-14, lead-210, and other isotopes inside their bodies from birth. The concentration of potassium-40 is nearly stable in all persons at a level of about 55 Bq/kg (3850 Bq in total), which corresponds to the annual effective dose of 0.2 mSv. The annual dose from carbon-14 is estimated to be about 12 μSv/year.

Background Radiation and Health Hazard

You can not go through life without radiation. The danger of ionizing radiation lies in the fact that the radiation is invisible and not directly detectable by human senses. People can neither see nor feel radiation, yet it deposits energy to the molecules of the body.

But don’t worry, the doses from background radiation are usually very small (except radon exposure). Low dose here means additional small doses comparable to the normal background radiation (10 µSv = average daily dose received from natural background). The problem is that, at very low doses, it is practically impossible to correlate any irradiation with certain biological effects. This is because the baseline cancer rate is already very high and the risk of developing cancer fluctuates 40% because of individual life style and environmental effects, obscuring the subtle effects of low-level radiation.

Secondly, and this is crucial, the truth about low-dose radiation health effects still needs to be found. It is not exactly known, whether these low doses of radiation are detrimental or beneficial (and where is the threshold).  Government and regulatory bodies assume a LNT model instead of a threshold or hormesis not because it is the more scientifically convincing, but because it is the more conservative estimate. Problem of this model is that it neglects a number of defence biological processes that may be crucial at low doses. The research during the last two decades is very interesting and show that small doses of radiation given at a low dose rate stimulate the defense mechanisms. Therefore the LNT model is not universally accepted with some proposing an adaptive dose–response relationship where low doses are protective and high doses are detrimental. Many studies have contradicted the LNT model and many of these have shown adaptive response to low dose radiation resulting in reduced mutations and cancers. This phenomenon is known as radiation hormesis.

According to the radiation hormesis hypothesis, radiation exposure comparable to and just above the natural background level of radiation is not harmful but beneficial, while accepting that much higher levels of radiation are hazardous. Arguments for hormesis center around some large-scale epidemiological studies and the evidence from animal irradiation experiments, but most notably the recent advances in knowledge of the adaptive response. Proponents of radiation hormesis typically claim that radio-protective responses in cells and the immune system not only counter the harmful effects of radiation but additionally act to inhibit spontaneous cancer not related to radiation exposure.

 

See also: LNT Model

See also:

Radiation Protection

Cosmic Radiation

Cosmic radiation refers to sources of radiation in the form of cosmic rays that come from the Sun or from outer space. The earth has always been bombarded by high-energy particles originating in outer space that generate secondary particle showers in the lower atmosphere. Charged particles (especially high-energy protons) from the sun and outer space interact with the earth’s atmosphere and magnetic field to produce a shower of radiation (i.e. air shower), typically beta and gamma radiation. If you live at higher elevations or are a frequent airline passenger, this exposure can be significantly higher, since the atmosphere is thinner here. The effects of the earth’s magnetic field also determines the dose from cosmic radiation.

At ground level the muons, with energies mostly between 1 and 20 GeV, contribute about 75 % of the absorbed dose rate in free air. The remainder comes from electrons produced by the muons or present in the electromagnetic cascade. The annual cosmic ray dose at sea level is around 0.27 mSv (27 mrem).

Composition of Cosmic Radiation

The primary cosmic radiation consist of a mixture of high-energy protons (~87%), alpha particles (~11%), high-energy electrons (~1%) and a trace of heavier nuclei (~1%). The energy of these particles range between 10⁸ eV and 10²⁰ eV.  A very small fraction are stable particles of antimatter, such as positrons or antiprotons. The precise nature of this remaining fraction is an area of active research.

Subsequently, a large number of secondary particles, in particular, neutrons and charged pions are produced as a result of interactions between primary particles and the earth’s atmosphere. Since pions are short-lived subatomic particles, the subsequent decay of the pions result in the production of high-energy muons. At ground level the muons, with energies mostly between 1 and 20 GeV, contribute about 75 % of the absorbed dose rate in free air. The dose rate from cosmic radiation varies in different parts of the world and it depends strongly on the geomagnetic field, altitude, and solar cycle. The cosmic radiation dose rate on airplanes is so high that, according to the United Nations UNSCEAR 2000 Report, airline flight crew workers receive more dose on average than any other worker, including those in nuclear power plants.

We also have to include the neutrons at ground level. Cosmic rays interact with nuclei in the atmosphere, and produce also high-energy neutrons. According to UNSCEAR the fluency of neutrons is 0.0123 cm^-2s^–1 at sea level for a geomagnetic latitude of 45 N. Based on this, the effective annual dose from neutrons at sea level and at 50 degree latitude is estimated to be 0.08 mSv (8 mrem). Noteworthy, in the vicinity of larger heavier objects, e.g. buildings or ships, the neutron flux measures higher. This effect is known as “cosmic ray induced neutron signature”, or “ship effect” as it was first detected with ships at sea. Cosmic rays create showers in the atmosphere that include a broad spectrum of secondary neutrons, muons and protons. The secondary neutrons may be of a very high energy and may induce spallation events in materials at ground level. Therefore in the vicinity of larger heavier objects, these multiple neutrons produced in spallation events are referred to as “ship effect” neutrons.

Neutrons produced in the upper atmosphere are also responsible for generation of radioactive carbon-14, which is the best known cosmogenic radionuclide. Carbon-14 is continuously formed in the upper atmosphere by the interaction of cosmic rays with atmospheric nitrogen. On average just one out of every 1.3 x 10¹² carbon atoms in the atmosphere is a radioactive carbon-14 atom. As a result, all living biological substances contain the same amount of C-14 per gram of carbon, that is 0.3 Bq of carbon-14 activity per gram of carbon.  As long as the biological system is alive the level is constant due to constant intake of all isotopes of carbon. When the biological system dies, it stops exchanging carbon with its environment, and from that point onwards the amount of carbon-14 it contains begins to decrease as the carbon-14 undergoes radioactive decay.

Energy of Cosmic Rays

The energies of the most energetic ultra-high-energy cosmic rays (UHECRs) have been observed to approach 3 x 10²⁰ eV, about 40 million times the energy of particles accelerated by the Large Hadron Collider. The origin of the high energy particles is from outer space. It is assumed that particles with an energy up to about 10¹⁵ eV are coming from our own galaxy, whereas those with the highest energies probably have an extragalactic origin.

Classification of Cosmic Radiation

Cosmic radiation can be divided into different types according to its origin. There are three main sources of such radiation:

• Solar Cosmic Radiation. Solar cosmic radiation refers to sources of radiation in the form of high-energy particles (predominantly protons) emitted by the sun, primarily in solar particle events (SPEs).
• Galactic Cosmic Radiation. Galactic cosmic radiation, GCR, refers to sources of radiation in the form of high-energy particles originating outside the solar system, but generally from within our Milky Way galaxy.
• Radiation from Earth’s Radiation Belts (van Allen belts). Van Allen radiation belts are zones of high-energy particles (especially protons) trapped by earth’s magnetic field.

Galactic Cosmic Radiation

Galactic cosmic radiation, GCR, refers to sources of radiation in the form of high-energy particles originating outside the solar system. GCR are high-energy nuclei from which all of the surrounding electrons have been stripped away during their high-speed passage through the galaxy. The GCR incident on the upper atmosphere consist of a nucleonic component, which aggregate accounts for 98% of the total (2% are electrons). The nucleonic component then consist of a mixture of high-energy protons (~86%), alpha particles (~12%), and a trace of heavier nuclei (~1%). GCR are trapped by the galactic magnetic field, therefore they have probably been accelerated within the last few million years, and have traveled many times across the galaxy. Their acceleration mechanism is uncertain, but one of possible mechanisms is that the particle are accelerated by shock waves expanding from supernovas. The energy of these particles range between 10⁸ eV and 10²⁰ eV.  A very small fraction are stable particles of antimatter, such as positrons or antiprotons.

The precise nature of this remaining fraction is an area of active research. The GCR fluence rate varies with solar activity, being lower when solar activity is higher. At solar minimums, due to lower solar magnetic field shielding, the fluence is significantly higher than at solar maximum.

Solar Cosmic Radiation – Solar Particle Event

Solar cosmic radiation refers to sources of radiation in the form of high-energy particles (predominantly protons) emitted by the Sun, primarily in solar particle events (SPEs). The solar radiation incident on the upper atmosphere consist mostly of protons (99%), with energies generally below 100 MeV.  Solar particle events, for example, occur when protons emitted by the Sun become accelerated either close to the Sun during a flare or in interplanetary space by coronal mass ejection shocks. Note that, the Sun has an 11-year cycle, which culminates in a dramatic increase in the number and intensity of solar flares, especially during periods when there are numerous sunspots.

Solar radiation is a significant radiation hazard to spacecraft and astronauts, also produces significant dose rates at high altitudes, but only the most energetic radiation contribute to doses at ground level.  Note that, anyone who had been on the Moon’s surface during a particularly violent solar eruption in 2005 would have received a lethal dose.

Radiation from Earth’s Radiation Belts – Van Allen belts

Van Allen radiation belts are zones of high-energy particles (especially protons) trapped by earth’s magnetic field. Most of these high-energy particles originate from the solar wind, that were captured by and held around a planet by that earth’s magnetic field. The van Allen belt is formed like a torus above the equator. There are two van Allen radiation belts, an internal belt is centered at about 3,000 kilometers and an outer belt is centered at about 22,000 kilometers from the earth’s surface. It contains mainly energetic protons in the 10-100 MeV range.

Spacecraft travelling beyond low Earth orbit enter the zone of radiation of the Van Allen belts. Beyond the belts, they face additional hazards from cosmic rays and solar particle events. A region between the inner and outer Van Allen belts lies at two to four Earth radii and is sometimes referred to as the “safe zone”.

Dose Rate in Airplane – Radiation in Flight

Exposure to cosmic radiation increases rapidly with altitude. In flight there are two principal sources of natural radiation to consider: Galactic Cosmic Rays which are always present, and Solar Proton Events, sometimes called Solar Cosmic Ray (SCR) events, which occur sporadically. The dose rate from cosmic radiation varies in different parts of the world and it depends strongly on the geomagnetic field, altitude, and solar cycle. The radiation field at aircraft altitudes consist of neutrons, protons and pions. In flight, neutrons contribute 40 – 80% of the equivalent dose, depending on the geomagnetic field, altitude, and solar cycle. The cosmic radiation dose rate on airplanes is so high (but not dangerous) that, according to the United Nations UNSCEAR 2000 Report, airline flight crew workers receive more dose on average than any other worker, including those in nuclear power plants.

The ground level dose rate is on average about 0.10 μSv/h, but at the maximum flight altitude (8.8 km or 29,000 ft) it can reach about 2.0 μSv/h (or even higher values). A dose rate of 4 μSv/h may be used to represent the average dose rate for all long haul flights (due to higher altitudes). It must be added, for supersonic planes like the Concorde, that could make a transatlantic flight in 3.5 hours, the exposure rate (about 9 μSv/h) at their altitude of 18 km was increased enough to result in the same cosmic ray exposure per crossing as for conventional jets trundling along at about 8 km.

Shielding of Cosmic Radiation

Earth’s magnetic field provides a vital radiation shield of cosmic radiation. In addition to a protective atmosphere, we are also lucky that Earth has a magnetic field. Magnetic field extends several tens of thousands of kilometers into space, protecting the Earth from the charged particles of the solar wind and cosmic rays that would otherwise strip away the upper atmosphere, including the ozone layer that protects the Earth from harmful ultraviolet radiation. It shields us from the full effects of the solar wind and GCR. Without this protection, Earth’s biosphere might not exist as it does today, or would be at least limited to the subsurface.  Earth’s magnetic field provides also a radiation shield for astronauts and the ISS itself, because it is in low Earth orbit.

Calculations of the loss of carbon dioxide from the atmosphere of Mars, resulting from scavenging of ions by the solar wind, indicate that the dissipation of the magnetic field of Mars caused a near total loss of its atmosphere.

Cosmic Radiation – Is it dangerous?

We must emphasize, eating bananas, working as airline flight crew or living in locations with, increases your annual dose rate. But it does not mean, that it must be dangerous. In each case, intensity of radiation also matters. It is very similar as for heat from a fire (less energetic radiation). If you are too close, the intensity of heat radiation is high and you can get burned. If you are at the right distance, you can withstand there without any problems and moreover it is comfortable. If you are too far from heat source, the insufficiency of heat can also hurt you. This analogy, in a certain sense, can be applied to radiation also from radiation sources.

In case of radiation from cosmic rays, we are talking about so called “low doses”. Low dose here means additional small doses comparable to the normal background radiation (10 µSv = average daily dose received from natural background). The doses are very very low and therefore the probability of cancer induction could be almost negligible. Secondly, and this is crucial, the truth about low-dose radiation health effects still needs to be found. It is not exactly known, whether these low doses of radiation are detrimental or beneficial (and where is the threshold). Government and regulatory bodies assume a LNT model instead of a threshold or hormesis not because it is the more scientifically convincing, but because it is the more conservative estimate. Problem of this model is that it neglects a number of defence biological processes that may be crucial at low doses. The research during the last two decades is very interesting and show that small doses of radiation given at a low dose rate stimulate the defense mechanisms. Therefore the LNT model is not universally accepted with some proposing an adaptive dose–response relationship where low doses are protective and high doses are detrimental. Many studies have contradicted the LNT model and many of these have shown adaptive response to low dose radiation resulting in reduced mutations and cancers. This phenomenon is known as radiation hormesis.

See also:

Sources

In physics of nuclear decays, a radioactive equilibrium exists when a radioactive nuclide is decaying at the same rate at which it is being produced. The disintegrating nucleus is usually referred to as the parent nucleus and the nucleus remaining after the event as the daughter nucleus. The daughter nucleus can either be stable or radioactive. If it is radioactive, then it decays into a daughter nucleus and so on. Thus, each radioactive parent nucleus can initiate a series of decays, with each decay-product having its own characteristic decay constant.

Concentration of daughter nuclei in the radioactive equilibrium depends primarily on proportions of half-lives (or decay constants) of parent and daughter nuclei. Since the production rate and decay rate are equal, the number of atoms present remains constant over time. In any case, a radioactive equilibrium is not established immediately, but it only takes place after a transition period. This period is of the order of few half-lifes of the longest-lived nucleus in the decay chain. In case of radioactive decay chains, a radioactive equilibrium may be established between each member of the decay chain.

As was written, proportionality of half-lives is a key parameter, which determines type of radioactive equilibrium:

• Radioactive equilibrium is not established when a half-life of the parent nucleus is shorter than a half-life of the daughter nucleus. In this case the production rate and decay rate of certain member of decay chain cannot be equal.
• Secular radioactive equilibrium exists when the parent nucleus has an extremely long half-life. This type of equilibrium is particularly important in nature. Over the 4.5 billion years of the Earth’s history, especially uranium 238, uranium 235 and thorium 232 and members of their decay chains have reached radioactive equilibria between the parent nucleus and the various descendants.
• Transient radioactive equilibrium exists when a half-life of the parent nucleus is longer than a half-life of the daughter nucleus. In this case, the parent nuclide and the daughter nuclide decay at essentially the same rate.

Transient Radioactive Equilibrium

The transient radioactive equilibrium exists when a half-life of the parent nucleus is longer than a half-life of the daughter nucleus, but the concentration of parent nuclei significantly decreases in time. In this case, the parent nuclide and the daughter nuclide may decay at essentially the same rate, but both concentrations of nuclides decreases as the concentration of parent nuclei decreases. Contrary to secular equilibrium, the half-life of the daughter nuclei is not negligible compared to parent’s half-life.

An example of this type of compound decay process is a Technetium-99m generator producing technetium-99m for nuclear medicine diagnostic procedures from molybdenum-99. Technetium-99m’s short half-life of 6 hours makes storage impossible and would make transport very expensive. Therefere, for medical purposes, molybdenum-99 is used to produce technetium-99m. These two isotopes are in the transient equilibrium. The decay constant for molybdenum-99 is considerably smaller than the decay constant for technetium-99m. Although the decay constant for molybdenum-99 is smaller, the actual rate of decay is initially larger than that of molybdenum-99 because of the great difference in their initial concentrations. As the concentration of the daughter increases, the rate of decay of the daughter will approach and eventually match the decay rate of the parent. When this occurs, they are said to be in the transient equilibrium. In case of  Technetium-99m generator, transient equilibrium occurs after about four half-lives. Today, technetium-99m is the most utilized element in nuclear medicine and is employed in a wide variety of nuclear medicine imaging studies.

Also the transient equilibrium can occasionally be disrupted when one of the intermediary nuclei leaves the sample where its ancestors are confined.

Transient Radioactive Equilibrium with Source – Example

A special example of radioactive equilibrium are concentrations of iodine-135 and xenon-135 in a nuclear reactor, but in this case, the xenon burnup must be taken into account. Note that, in this special case, the half-life of the parent nucleus is shorter than the half-life of the daughter nucleus. The production and removal of xenon can be characterized by the following differential equations:

When the rate of production of iodine equals the rate of removal of iodine, equilibrium exists. This equilibrium also known as “xenon 135 reservoir”, since all of this iodine must undergo a decay to xenon. In equilibrium, the iodine concentration remains constant and is designated N_I(eq). The following equation for the iodine equilibrium concentration can be determined from the preceding equation by setting the dN_I/dt =0. Since the equilibrium iodine concentration is proportional to the fission reaction rate, it is also proportional to reactor power level.

When the rate of production of xenon 135 equals the rate of removal, equilibrium exists also for xenon. The xenon concentration remains constant and is designated N_Xe(eq). The following equation (1) for the xenon equilibrium concentration can be determined from the preceding equation by setting the dN_Xe/dt =0. For xenon 135 to be in equilibrium, iodine 135 must also be in equilibrium. Substituting the expression for equilibrium iodine 135 concentration into the equation for equilibrium xenon (1) results in the following (2).

From this equation it can be seen that the equilibrium value for xenon 135 increases as power increases, because the numerator is proportional to the fission reaction rate. But the thermal flux is also in the denominator. Therefore, as the thermal flux exceeds some value, the xenon burnup begins to dominate, and at approximately 10¹⁵ neutrons.cm^-2.s^-1, the xenon-135 concentration approaches a limiting value. The equilibrium iodine 135 and xenon 135 concentrations as a function of neutron flux are illustrated in the following figure.

See also:

Radioactive Equilibrium

In physics, the Bateman equations are a set of first-order differential equations, which describe the time evolution of nuclide concentrations undergoing serial or linear decay chain. The model was formulated by Ernest Rutherford in 1905 and the analytical solution for the case of radioactive decay in a linear chain was provided by Harry Bateman in 1910. This model can be also used in nuclear depletion codes to solve nuclear transmutation and decay problems.

For example, ORIGEN is a computer code system for calculating the buildup, decay, and processing of radioactive materials. ORIGEN uses a matrix exponential method to solve a large system of coupled, linear, first-order ordinary differential equations (similar to the Bateman equations) with constant coefficients.

The Bateman equations for radioactive decay case of n – nuclide series in linear chain describing nuclide concentrations are as follows:

Bateman Equations for Nuclear Transmutation

As was written, this model can be also used in nuclear depletion codes to solve nuclear transmutation and decay problems as well. In case of transmutation the decay constants that govern Bateman equations for a decay case are substituted by transmutation constants. By the transmutation constant λ_i,j we understand probability of the ith nuclide production per time unit from the jth nuclide destruction, as a result of nuclear interaction with the whole spectrum of interacting particles or due to the natural nuclear decay.

These equations are usually used for the exact evolution of isotopic changes in the nuclear fuel during fuel depletion. Fuel depletion is usually modelled mathematically as a set of differential equations known as evolution equations.

Special Reference: Jerzy Cetnar, General solution of Bateman equations for nuclear transmutations. Annals of Nuclear Energy 33 (2006). January 2006.

See also:

Radioactive Equilibrium

Decay Law – Equation – Formula

The radioactive decay law states that the probability per unit time that a nucleus will decay is a constant, independent of time. This constant is called the decay constant and is denoted by λ, “lambda”. This constant probability may vary greatly between different types of nuclei, leading to the many different observed decay rates. The radioactive decay of certain number of atoms (mass) is exponential in time.

Radioactive decay law: N = N.e^-λt

The rate of nuclear decay is also measured in terms of half-lives. The half-life is the amount of time it takes for a given isotope to lose half of its radioactivity. If a radioisotope has a half-life of 14 days, half of its atoms will have decayed within 14 days. In 14 more days, half of that remaining half will decay, and so on. Half lives range from millionths of a second for highly radioactive fission products to billions of years for long-lived materials (such as naturally occurring uranium). Notice that short half lives go with large decay constants. Radioactive material with a short half life is much more radioactive (at the time of production) but will obviously lose its radioactivity rapidly. No matter how long or short the half life is, after seven half lives have passed, there is less than 1 percent of the initial activity remaining.

The radioactive decay law can be derived also for activity calculations or mass of radioactive material calculations:

(Number of nuclei) N = N.e^-λt     (Activity) A = A.e^-λt      (Mass) m = m.e^-λt

, where N (number of particles) is the total number of particles in the sample, A (total activity) is the number of decays per unit time of a radioactive sample, m is the mass of remaining radioactive material.

Decay Constant and Half-Life – Equation – Formula

In calculations of radioactivity one of two parameters (decay constant or half-life), which characterize the rate of decay, must be known. There is a relation between the half-life (t_1/2) and the decay constant λ. The relationship can be derived from decay law by setting N = ½ N_o. This gives:

where ln 2 (the natural log of 2) equals 0.693. If the decay constant (λ) is given, it is easy to calculate the half-life, and vice-versa.

Bateman Equations

In physics, the Bateman equations are a set of first-order differential equations, which describe the time evolution of nuclide concentrations undergoing serial or linear decay chain. The model was formulated by Ernest Rutherford in 1905 and the analytical solution for the case of radioactive decay in a linear chain was provided by Harry Bateman in 1910. This model can be also used in nuclear depletion codes to solve nuclear transmutation and decay problems.

For example, ORIGEN is a computer code system for calculating the buildup, decay, and processing of radioactive materials. ORIGEN uses a matrix exponential method to solve a large system of coupled, linear, first-order ordinary differential equations (similar to the Bateman equations) with constant coefficients.

The Bateman equations for radioactive decay case of n – nuclide series in linear chain describing nuclide concentrations are as follows:

Example – Radioactive Decay Law

A sample of material contains 1 mikrogram of iodine-131. Note that, iodine-131 plays a major role as a radioactive isotope present in nuclear fission products, and it a major contributor to the health hazards when released into the atmosphere during an accident. Iodine-131 has a half-life of 8.02 days.

Calculate:

1. The number of iodine-131 atoms initially present.
2. The activity of the iodine-131 in curies.
3. The number of iodine-131 atoms that will remain in 50 days.
4. The time it will take for the activity to reach 0.1 mCi.

Solution:

1. The number of atoms of iodine-131 can be determined using isotopic mass as below.

N_I-131 = m_I-131 . N_A / M_I-131

N_I-131 = (1 μg) x (6.02×10²³ nuclei/mol) / (130.91 g/mol)

N_I-131 = 4.6 x 10¹⁵ nuclei

2. The activity of the iodine-131 in curies can be determined using its decay constant:

The iodine-131 has half-live of 8.02 days (692928 sec) and therefore its decay constant is:

Using this value for the decay constant we can determine the activity of the sample:

3) and 4) The number of iodine-131 atoms that will remain in 50 days (N_50d) and the time it will take for the activity to reach 0.1 mCi can be calculated using the decay law:

As can be seen, after 50 days the number of iodine-131 atoms and thus the activity will be about 75 times lower. After 82 days the activity will be approximately 1200 times lower. Therefore, the time of ten half-lives (factor 2¹⁰ = 1024) is widely used to define residual activity.

See also:

Radioactive Equilibrium

Radiometric dating (or radioactive dating) is any technique used to date organic and also inorganic materials from a process involving radioactive decay. The method compares the abundance of a naturally occurring radioactive isotope within the material to the abundance of its decay products, which form at a known constant rate of decay.

All these methods are based on the fact the rate at which radioactive nuclei disintegrate is unaffected by their environment, it can be used to estimate the age of any material sample or object which contains a radioactive isotope. Calculations of the decay of radioactive nuclei are relatively straightforward, owing to the fact that there is only one fundamental law governing all decay process.

The radioactive decay law states that the probability per unit time that a nucleus will decay is a constant, independent of time. This constant is called the decay constant and is denoted by λ, “lambda”. This constant probability may vary greatly between different types of nuclei, leading to the many different observed decay rates. The radioactive decay of certain number of atoms (mass) is exponential in time.

Radioactive decay law: N = N₀.e^-λt

Among the best-known techniques are:

• carbon-14 dating,
• potassium–argon dating,
• uranium–lead dating.

Radiometric dating methods are used in geochronology to establish the geologic time scale and can be also used to date archaeological materials, including ancient artifacts.

Carbon-14 Dating – Radiocarbon Dating

Carbon-14 dating, known also as radiocarbon dating, is a method for determining the age of an object containing organic material by using the properties of radionuclide carbon-14. Radioactive carbon-14 has a half-life of 5730 years and undergoes β− decay, where the neutron is converted into a proton, an electron, and an electron antineutrino:

In spite of this short half-life compared to the age of the earth, carbon-14 is a naturally occurring isotope. Its presence can be explained by the following simple observation. Our atmosphere contains many gases, including nitrogen-14. Besides, the atmosphere is constantly bombarded with high energy cosmic rays, consisting of protons, heavier nuclei, or gamma rays. These cosmic rays interact with nuclei in the atmosphere, and produce also high-energy neutrons. These neutrons produced in these collisions can be absorbed by nitrogen-14 to produce an isotope of carbon-14:

Carbon-14 can also be produced in the atmosphere by other neutron reactions, including in particular 13C(n,γ)14C and 17O(n,α)14C. As a result, carbon-14 is continuously formed in the upper atmosphere by the interaction of cosmic rays with atmospheric nitrogen. On average just one out of every 1.3 x 10¹² carbon atoms in the atmosphere is a radioactive carbon-14 atom.

The resulting carbon-14 combines with atmospheric oxygen to form radioactive carbon dioxide, which is incorporated into plants by photosynthesis. Consequently, all biological systems as  plants, animals and humans contain a certain level of radioactive carbon-14. As long as the biological system is alive the level is constant due to constant intake of all isotopes of carbon. When the biological system dies, it stops exchanging carbon with its environment, and from that point onwards the amount of carbon-14 it contains begins to decrease as the carbon-14 undergoes radioactive decay. On the other hand, the amount of stable carbon-12 remains unchanged. As a result, the relative concentration of these two isotopes in any organism changes after its death. The method enables datings to be made up to about 20,000 years ago with an accuracy of about ±100 years.

The technique of carbon dating was suggested initially by Willard Libby and his colleagues in 1949. In 1960, Willard Libby was awarded the Nobel Prize in chemistry for this work.

Age of the Earth – Uranium-lead Dating

The age of the Earth is about 4.54 billion years. This dating is based on evidence from radiometric age-dating of meteorite material and is consistent with the radiometric ages of the oldest-known terrestrial and lunar samples.

One of the oldest radiometric dating methods is uranium-lead dating. The age of the earth’s crust can be estimated from the ratio between the amounts of uranium-238 and lead-206 found in geological specimens. The long half-life of the isotope uranium-238 (4.51×10⁹ years) makes it well-suited for use in estimating the age of the earliest igneous rocks and for other types of radiometric dating, including uranium–thorium dating and uranium–uranium dating.

Uranium-lead dating is based on the measurement of the first and the last member of the uranium series, which is one of three classical radioactive series beginning with naturally occurring uranium-238. This radioactive decay chain consists of unstable heavy atomic nuclei that decay through a sequence of alpha and beta decays until a stable nucleus is achieved. In case of uranium series, the stable nucleus is lead-206. The assumption made is that all the lead-206 nuclei found in the specimen today were originally uranium-238 nuclei.  That means at the crust’s formation the specimen contained no lead-206 nuclei. If no other lead isotopes are found in the specimen, this is a reasonable assumption. Under this condition, the age of the sample can be calculated by assuming exponential decay of uranium-238. That is:

Uranium-lead dating method is usually performed on the mineral zircon. Zircons from Jack Hills in Western Australia, have yielded U-Pb ages up to 4.404 billion years, interpreted to be the age of crystallization, making them the oldest minerals so far dated on Earth.

See also:

Radioactive Decay

Carbon-14 dating, known also as radiocarbon dating, is a method for determining the age of an object containing organic material by using the properties of radionuclide carbon-14. Radioactive carbon-14 has a half-life of 5730 years and undergoes β− decay, where the neutron is converted into a proton, an electron, and an electron antineutrino:

In spite of this short half-life compared to the age of the earth, carbon-14 is a naturally occurring isotope. Its presence can be explained by the following simple observation. Our atmosphere contains many gases, including nitrogen-14. Besides, the atmosphere is constantly bombarded with high energy cosmic rays, consisting of protons, heavier nuclei, or gamma rays. These cosmic rays interact with nuclei in the atmosphere, and produce also high-energy neutrons. These neutrons produced in these collisions can be absorbed by nitrogen-14 to produce an isotope of carbon-14:

Carbon-14 can also be produced in the atmosphere by other neutron reactions, including in particular 13C(n,γ)14C and 17O(n,α)14C. As a result, carbon-14 is continuously formed in the upper atmosphere by the interaction of cosmic rays with atmospheric nitrogen. On average just one out of every 1.3 x 10¹² carbon atoms in the atmosphere is a radioactive carbon-14 atom.

The resulting carbon-14 combines with atmospheric oxygen to form radioactive carbon dioxide, which is incorporated into plants by photosynthesis. Consequently, all biological systems as  plants, animals and humans contain a certain level of radioactive carbon-14. As long as the biological system is alive the level is constant due to constant intake of all isotopes of carbon. When the biological system dies, it stops exchanging carbon with its environment, and from that point onwards the amount of carbon-14 it contains begins to decrease as the carbon-14 undergoes radioactive decay. On the other hand, the amount of stable carbon-12 remains unchanged. As a result, the relative concentration of these two isotopes in any organism changes after its death. The method enables datings to be made up to about 20,000 years ago with an accuracy of about ±100 years.

The technique of carbon dating was suggested initially by Willard Libby and his colleagues in 1949. In 1960, Willard Libby was awarded the Nobel Prize in chemistry for this work.

See also:

Radiometric Dating

The age of the Earth is about 4.54 billion years. This dating is based on evidence from radiometric age-dating of meteorite material and is consistent with the radiometric ages of the oldest-known terrestrial and lunar samples.

One of the oldest radiometric dating methods is uranium-lead dating. The age of the earth’s crust can be estimated from the ratio between the amounts of uranium-238 and lead-206 found in geological specimens. The long half-life of the isotope uranium-238 (4.51×10⁹ years) makes it well-suited for use in estimating the age of the earliest igneous rocks and for other types of radiometric dating, including uranium–thorium dating and uranium–uranium dating.

Uranium-lead dating is based on the measurement of the first and the last member of the uranium series, which is one of three classical radioactive series beginning with naturally occurring uranium-238. This radioactive decay chain consists of unstable heavy atomic nuclei that decay through a sequence of alpha and beta decays until a stable nucleus is achieved. In case of uranium series, the stable nucleus is lead-206. The assumption made is that all the lead-206 nuclei found in the specimen today were originally uranium-238 nuclei.  That means at the crust’s formation the specimen contained no lead-206 nuclei. If no other lead isotopes are found in the specimen, this is a reasonable assumption. Under this condition, the age of the sample can be calculated by assuming exponential decay of uranium-238. That is:

Uranium-lead dating method is usually performed on the mineral zircon. Zircons from Jack Hills in Western Australia, have yielded U-Pb ages up to 4.404 billion years, interpreted to be the age of crystallization, making them the oldest minerals so far dated on Earth.

See also:

Radiometric Dating

Radiometric dating (or radioactive dating) is any technique used to date organic and also inorganic materials from a process involving radioactive decay. The method compares the abundance of a naturally occurring radioactive isotope within the material to the abundance of its decay products, which form at a known constant rate of decay.

The radioactive decay law states that the probability per unit time that a nucleus will decay is a constant, independent of time. This constant is called the decay constant and is denoted by λ, “lambda”. This constant probability may vary greatly between different types of nuclei, leading to the many different observed decay rates. The radioactive decay of certain number of atoms (mass) is exponential in time.

Radioactive decay law: N = N₀.e^-λt

One of the oldest radiometric dating methods is uranium-lead dating. The age of the earth’s crust can be estimated from the ratio between the amounts of uranium-238 and lead-206 found in geological specimens. The long half-life of the isotope uranium-238 (4.51×10⁹ years) makes it well-suited for use in estimating the age of the earliest igneous rocks and for other types of radiometric dating, including uranium–thorium dating and uranium–uranium dating.

Uranium-lead dating is based on the measurement of the first and the last member of the uranium series, which is one of three classical radioactive series beginning with naturally occurring uranium-238. This radioactive decay chain consists of unstable heavy atomic nuclei that decay through a sequence of alpha and beta decays until a stable nucleus is achieved. In case of uranium series, the stable nucleus is lead-206. The assumption made is that all the lead-206 nuclei found in the specimen today were originally uranium-238 nuclei.  That means at the crust’s formation the specimen contained no lead-206 nuclei. If no other lead isotopes are found in the specimen, this is a reasonable assumption. Under this condition, the age of the sample can be calculated by assuming exponential decay of uranium-238. That is:

Uranium-lead dating method is usually performed on the mineral zircon. Zircons from Jack Hills in Western Australia, have yielded U-Pb ages up to 4.404 billion years, interpreted to be the age of crystallization, making them the oldest minerals so far dated on Earth.

Age of the Earth – Uranium-lead Dating

The age of the Earth is about 4.54 billion years. This dating is based on evidence from radiometric age-dating of meteorite material and is consistent with the radiometric ages of the oldest-known terrestrial and lunar samples.

 

See also:

Radiometric Dating

X-rays, also known as X-radiation, refers to electromagnetic radiation (no rest mass, no charge) of high energies. X-rays are high-energy photons with short wavelengths and thus very high frequency. The radiation frequency is key parameter of all photons, because it determines the energy of a photon. Photons are categorized according to the energies from low-energy radio waves and infrared radiation, through visible light, to high-energy X-rays and gamma rays.

Most X-rays have a wavelength ranging from 0.01 to 10 nanometers (3×10¹⁶ Hz to 3×10¹⁹ Hz), corresponding to energies in the range 100 eV to 100 keV. X-ray wavelengths are shorter than those of UV rays and typically longer than those of gamma rays.

Since the X-rays (especially hard X-rays) are in substance high-energy photons, they are very penetrating matter and are thus biologically hazardous. X-rays can travel thousands of feet in air and can easily pass through the human body.

Characteristics of X-rays – Properties

Key characteristics of X-rays are summarized in following few points:

• X-rays are high-energy photons (about 100 – 1 000 times as much energy as the visible photons), the same photons as the photons forming the visible range of the electromagnetic spectrum – light.
• X-rays are usually described by their maximum energy, which is determined by the voltage between the electrodes. It may range from about 20 kV up to 300 kV. Radiation with low voltage is called “soft” – and radiation with high voltage is called “hard”.
• Photons (gamma rays and X-rays) can ionize atoms directly (despite they are electrically neutral) through the Photoelectric effect and the Compton effect, but secondary (indirect) ionization is much more significant.
• X-rays ionize matter via indirect ionization.
• Although a large number of possible interactions are known, there are three key interaction mechanisms  with matter.
  • Photoelectric effect
  • Compton scattering
  • Rayleigh scattering
• X-rays travel at the speed of light and they can travel hundreds of meters in air before spending their energy.
• Since the hard X-rays are very penetrating matter, it must be shielded by very dense materials, such as lead or uranium.
• The distinction between X-rays and gamma rays is not so simple and has changed in recent decades.  According to the currently valid definition, X-rays are emitted by electrons outside the nucleus, while gamma rays are emitted by the nucleus.
• For X-rays generated by X-ray tube, there are two different types of X-rays spectra:
  • Bremsstrahlung
  • Characteristic X-rays
• Characteristic X-rays frequently accompany some types of nuclear decays, such as internal conversion and electron capture.

See also:

X-rays