Nick Connor

Absorbed dose is defined as the amount of energy deposited by ionizing radiation in a substance. Absorbed dose is given the symbol D. The absorbed dose is usually measured in a unit called the gray (Gy), which is derived from the SI system. The non-SI unit rad is sometimes also used, predominantly in the USA.

Examples of Absorbed Doses in grays

We must note that 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. In the following points we try to express enormous ranges of radiation exposure, which can be obtained from various sources.

• 0.05 µGy – Sleeping next to someone
• 0.09 µGy – Living within 30 miles of a nuclear power plant for a year
• 0.1 µGy – Eating one banana
• 0.3 µGy – Living within 50 miles of a coal power plant for a year
• 10 µGy – Average daily dose received from natural background
• 20 µGy – Chest X-ray
• 40 µGy – A 5-hour airplane flight
• 600 µGy – mammogram
• 1 000 µGy – Dose limit for individual members of the public, total effective dose per annum
• 3 650 µGy – Average yearly dose received from natural background
• 5 800 µGy – Chest CT scan
• 10 000 µGy – Average yearly dose received from natural background in Ramsar, Iran
• 20 000 µGy – single full-body CT scan
• 175 000 µGy – Annual dose from natural radiation on a monazite beach near Guarapari, Brazil.
• 5 000 000 µGy – Dose that kills a human with a 50% risk within 30 days (LD50/30), if the dose is received over a very short duration.

As can be seen, low-level doses are common for everyday life. The previous examples can help illustrate relative magnitudes. From biological consequences point of view, it is very important to distinguish between doses received over short and extended periods. An “acute dose” is one that occurs over a short and finite period of time, while a “chronic dose” is a dose that continues for an extended period of time so that it is better described by a dose rate. High doses tend to kill cells, while low doses tend to damage or change them. Low doses spread out over long periods of time don’t cause an immediate problem to any body organ. The effects of low doses of radiation occur at the level of the cell, and the results may not be observed for many years.

See also:

Absorbed Dose

Absorbed dose is defined as the amount of energy deposited by ionizing radiation in a substance. Absorbed dose is given the symbol D. The absorbed dose is usually measured in a unit called the gray (Gy), which is derived from the SI system. The non-SI unit rad is sometimes also used, predominantly in the USA.

Units of Absorbed Dose

• Gray. A dose of one gray is equivalent to a unit of energy (joule) deposited in a kilogram of a substance.
• RAD. A dose of one rad is equivalent to the deposition of one hundred ergs of energy in one gram of any material.

Gray – Unit of Absorbed Dose

A dose of one gray is equivalent to a unit of energy (joule) deposited in a kilogram of a substance. This unit was named in honour of Louis Harold Gray, who was one of the great pioneers in radiation biology. One gray is a large amount of absorbed dose. A person who has absorbed a whole body dose of 1 Gy has absorbed one joule of energy in each kg of body tissue.

Absorbed doses measured in industry (except nuclear medicine) often have usually lower doses than one gray, and the following multiples are often used:

1 mGy (milligray) = 1E-3 Gy

1 µGy (microgray) = 1E-6 Gy

Conversions from the SI units to other units are as follows:

• 1 Gy = 100 rad
• 1 mGy = 100 mrad

The gray and rad are physical units. They describe the physical effect of the incident radiation (i.e., the amount of energy deposited per kg), but it tells us nothing about the biological consequences of such energy deposition in living tissue.

rad – Unit of Absorbed Dose

A dose of one rad is equivalent to the deposition of one hundred ergs of energy in one gram of any material. Note that, the erg is a unit of energy and work equal to 10⁻⁷ joules. A related unit, the roentgen, is used to quantify the radiation exposure. The F-factor can be used to convert between rads and roentgens.

One rad is significantly lower dose than one gray, which is a large amount of absorbed dose. A person who has absorbed a whole body dose of 100 rad has absorbed one joule of energy in each kg of body tissue. Absorbed doses measured in industry (except nuclear medicine) often have comparable doses to one rad and the following multiples are often used:

1 mrad (millirad) = 1E-3 rad

1 krad (kilorad) = 1E3 rad

Conversions from the SI units to other units are as follows:

• 1 Gy = 100 rad
• 1 mGy = 100 mrad

The gray and rad are physical units. They describe the physical effect of the incident radiation (i.e., the amount of energy deposited per kg), but it tells us nothing about the biological consequences of such energy deposition in living tissue.

See also:

Absorbed Dose

Absorbed dose is defined as the amount of energy deposited by ionizing radiation in a substance. Absorbed dose is given the symbol D.  The absorbed dose is usually measured in a unit called the gray (Gy), which is derived from the SI system. The non-SI unit rad is sometimes also used, predominantly in the USA.

Units of absorbed dose:

• Gray. A dose of one gray is equivalent to a unit of energy (joule) deposited in a kilogram of a substance.
• RAD. A dose of one rad is equivalent to the deposition of one hundred ergs of energy in one gram of any material.

Why we deal with a radiation dose? In previous chapters, we have discussed radioactivity and the intensity of a radioactive source, measured usually in becquerels. But any radioactive source represents no biological risk as long as it is isolated from the environments. However, when people or another system (also non-biological) are exposed to radiation, energy is deposited in the material and radiation dose is delivered.

It is therefore very important to distinguish between radioactivity of a radioactive source and the radiation dose which may result from the source. Generally, the radiation dose depends on the following factors regarding to the radioactive source:

• Activity. Activity of the source directly influences the radiation dose deposited in the material.
• Type of radiation. Each type of radiation interacts with matter in a different way. For example charged particles with high energies can directly ionize atoms. On the other hand electrically neutral particles interacts only indirectly, but can also transfer some or all of their energies to the matter.
• Distance. The amount of radiation exposure depends on the distance from the source of radiation. Similarly to a heat from a fire, 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.
• Time. The amount of radiation exposure depends directly (linearly) on the time people spend near the source of radiation.
• Shielding. Finally, the radiation dose also depends on the material between the source and the object. If the source is too intensive and time or distance do not provide sufficient radiation protection, the shielding can be used.

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. The energy is transferred in small quantities for each interaction between the radiation and a molecule and there are usually many such interactions.

In nuclear power plants the central problem is to protect personal and the environment against gamma rays and neutrons, because the ranges of charged particles (such as beta particles and alpha particles) in matter are very short. On the other hand we must deal with shielding of all types of radiation, because each nuclear reactor is a significant source of all types of ionizing radiation.

See also: Gamma Ray Attenuation

See also: Neutron Shielding

 

Gray – Unit of Absorbed Dose

A dose of one gray is equivalent to a unit of energy (joule) deposited in a kilogram of a substance. This unit was named in honour of Louis Harold Gray, who was one of the great pioneers in radiation biology. One gray is a large amount of absorbed dose. A person who has absorbed a whole body dose of 1 Gy has absorbed one joule of energy in each kg of body tissue.

Absorbed doses measured in industry (except nuclear medicine) often have usually lower doses than one gray, and the following multiples are often used:

1 mGy (milligray) = 1E-3 Gy

1 µGy (microgray) = 1E-6 Gy

Conversions from the SI units to other units are as follows:

• 1 Gy = 100 rad
• 1 mGy = 100 mrad

The gray and rad are physical units. They describe the physical effect of the incident radiation (i.e., the amount of energy deposited per kg), but it tells us nothing about the biological consequences of such energy deposition in living tissue.

Absorbed Dose Rate

The absorbed dose rate is the rate at which an absorbed dose is received. It is a measure of radiation dose intensity (or strength). The absorbed dose rate is therefore defined as:

In conventional units, it is measured in mrad/sec, rad/hr, mGy/sec or Gy/hr. Since the amount of radiation exposure depends directly (linearly) on the time people spend near the source of radiation, the absorbed dose is equal to the strength of the radiation field (dose rate) multiplied by the length of time spent in that field. The example above indicates a person could expect to receive a dose of 25 millirems by staying in a 50 millirems/hour field for thirty minutes.

Examples of Absorbed Doses in grays

We must note that 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. In the following points we try to express enormous ranges of radiation exposure, which can be obtained from various sources.

• 0.05 µGy – Sleeping next to someone
• 0.09 µGy – Living within 30 miles of a nuclear power plant for a year
• 0.1 µGy – Eating one banana
• 0.3 µGy – Living within 50 miles of a coal power plant for a year
• 10 µGy – Average daily dose received from natural background
• 20 µGy – Chest X-ray
• 40 µGy – A 5-hour airplane flight
• 600 µGy – mammogram
• 1 000 µGy – Dose limit for individual members of the public, total effective dose per annum
• 3 650 µGy – Average yearly dose received from natural background
• 5 800 µGy – Chest CT scan
• 10 000 µGy – Average yearly dose received from natural background in Ramsar, Iran
• 20 000 µGy – single full-body CT scan
• 175 000 µGy – Annual dose from natural radiation on a monazite beach near Guarapari, Brazil.
• 5 000 000 µGy – Dose that kills a human with a 50% risk within 30 days (LD50/30), if the dose is received over a very short duration.

As can be seen, low-level doses are common for everyday life. The previous examples can help illustrate relative magnitudes. From biological consequences point of view, it is very important to distinguish between doses received over short and extended periods.  An “acute dose” is one that occurs over a short and finite period of time, while a “chronic dose” is a dose that continues for an extended period of time so that it is better described by a dose rate. High doses tend to kill cells, while low doses tend to damage or change them. Low doses spread out over long periods of time don’t cause an immediate problem to any body organ. The effects of low doses of radiation occur at the level of the cell, and the results may not be observed for many years.

Calculation of Shielded Dose Rate

Assume the point isotropic source which contains 1.0 Ci of ¹³⁷Cs, which has a half-life of 30.2 years. Note that the relationship between half-life and the amount of a radionuclide required to give an activity of one curie is shown below. This amount of material can be calculated using λ, which is the decay constant of certain nuclide:

About 94.6 percent decays by beta emission to a metastable nuclear isomer of barium: barium-137m. The main photon peak of Ba-137m is 662 keV. For this calculation, assume that all decays go through this channel.

Determine the primary photon dose rate, in gray per hour (Gy.h^-1), at the outer surface of a 5 cm thick lead shield. Primary photon dose rate neglects all secondary particles. Assume that the effective distance of the source from the dose point is 10 cm. We shall also assume that the dose point is soft tissue and it can reasonably be simulated by water and we use the mass energy absorption coefficient for water.

See also: Gamma Ray Attenuation

See also: Shielding of Gamma Rays

Solution:

The primary photon dose rate is attenuated exponentially, and the dose rate from primary photons, taking account of the shield, is given by:

As can be seen, we do not account for the buildup of secondary radiation. If secondary particles are produced or if the primary radiation changes its energy or direction, then the effective attenuation will be much less.  This assumption generally underestimates the true dose rate, especially for thick shields and when the dose point is close to the shield surface, but this assumption simplifies all calculations. For this case the true dose rate (with the buildup of secondary radiation) will be more than two times higher.

To calculate the absorbed dose rate, we have to use in the formula:

• k = 5.76 x 10^-7
• S = 3.7 x 10¹⁰ s^-1
• E = 0.662 MeV
• μ_t/ρ =  0.0326 cm²/g (values are available at NIST)
• μ =  1.289 cm^-1 (values are available at NIST)
• D = 5 cm
• r = 10 cm

Result:

The resulting absorbed dose rate in grays per hour is then:

If we want to account for the buildup of secondary radiation, then we have to include the buildup factor. The extended formula for the dose rate is then:

From Absorbed Dose to Equivalent Dose

As was written, each type of radiation interacts with matter in a different way. For example charged particles with high energies can directly ionize atoms. On the other hand electrically neutral particles interacts only indirectly, but can also transfer some or all of their energies to the matter. It would certainly simplify matters if biological effects of radiation were directly proportional to the absorbed dose. Unfortunately, biological effects depend also on the way in which the absorbed dose is distributed along the path of the radiation. Studies have shown that alpha and neutron radiation cause greater biological damage for a given energy deposition per kg of tissue than gamma radiation does. It was discovered, biological effects of any radiation increases with the linear energy transfer (LET). In short, the biological damage from high-LET radiation (alpha particles, protons or neutrons) is much greater than that from low-LET radiation (gamma rays). This is because the living tissue can more easily repair damage from radiation that is spread over a large area than that which is concentrated in a small area. Because more biological damage is caused for the same physical dose (i.e., the same energy deposited per unit mass of tissue), one gray of alpha or neutron radiation is more harmful than one gray of gamma radiation. This fact that radiations of different types (and energies) give different biological effects for the same absorbed dose is described in terms of factors known as the relative biological effectiveness (RBE) and the radiation weighting factor (w_R).

——–

See also:

Radiation Protection

Radiation dose is a measure of the amount of exposure to ionizing radiation. It is known, that ionizing radiation can penetrate the human body or an object. The energy absorbed from this exposure to radiation is called a radiation dose or simply a dose: absorbed, equivalent, and effective:

• Absorbed dose,
• Equivalent dose,
• Effective dose.

Why we deal with a radiation dose? In previous chapters, we have discussed radioactivity and the intensity of a radioactive source, measured usually in becquerels. But any radioactive source represents no biological risk as long as it is isolated from the environments. However, when people or another system (also non-biological) are exposed to radiation, energy is deposited in the material and radiation dose is delivered.

It is therefore very important to distinguish between radioactivity of a radioactive source and the radiation dose which may result from the source. Generally, the radiation dose depends on the following factors regarding to the radioactive source:

• Activity. Activity of the source directly influences the radiation dose deposited in the material.
• Type of radiation. Each type of radiation interacts with matter in a different way. For example charged particles with high energies can directly ionize atoms. On the other hand electrically neutral particles interacts only indirectly, but can also transfer some or all of their energies to the matter.
• Distance. The amount of radiation exposure depends on the distance from the source of radiation. Similarly to a heat from a fire, 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.
• Time. The amount of radiation exposure depends directly (linearly) on the time people spend near the source of radiation.
• Shielding. Finally, the radiation dose also depends on the material between the source and the object. If the source is too intensive and time or distance do not provide sufficient radiation protection, the shielding can be used.

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. The energy is transferred in small quantities for each interaction between the radiation and a molecule and there are usually many such interactions.

In nuclear power plants the central problem is to protect personal and the environment against gamma rays and neutrons, because the ranges of charged particles (such as beta particles and alpha particles) in matter are very short. On the other hand we must deal with shielding of all types of radiation, because each nuclear reactor is a significant source of all types of ionizing radiation.

See also: Gamma Ray Attenuation

See also: Neutron Shielding

See also:

Radiation Protection

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

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

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

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

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

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