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 (t1/2) and the decay constant λ. Half-Life and Decay Constant – Calculation
Half-Life and Decay Constant
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”.
One of the most useful terms for estimating how quickly a nuclide will decay is the radioactive half-life (t1/2). The half-life is defined as the amount of time it takes for a given isotope to lose half of its radioactivity.
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 (t1/2) and the decay constant λ. The relationship can be derived from decay law by setting N = ½ No. 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.
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:
The number of iodine-131 atoms initially present.
The activity of the iodine-131 in curies.
The number of iodine-131 atoms that will remain in 50 days.
The time it will take for the activity to reach 0.1 mCi.
Solution:
The number of atoms of iodine-131 can be determined using isotopic mass as below.
NI-131 = mI-131 . NA / MI-131
NI-131 = (1 μg) x (6.02×1023 nuclei/mol) / (130.91 g/mol)
NI-131 = 4.6 x 1015 nuclei
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 (N50d) 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.
Robert Reed Burn, Introduction to Nuclear Reactor Operation, 1988.
U.S. Department of Energy, Nuclear Physics and Reactor Theory. DOE Fundamentals Handbook, Volume 1 and 2. January 1993.
Paul Reuss, Neutron Physics. EDP Sciences, 2008. ISBN: 978-2759800414.
See also:
Half-Life
We hope, this article, Half-Life and Decay Constant – Calculation, helps you. If so, give us a like in the sidebar. Main purpose of this website is to help the public to learn some interesting and important information about radiation and dosimeters.
Half-Life – Calculation – Example. Iodine-131 has a half-life of 8.02 days. Calculate the activity of the iodine-131 in curies. Radiation Dosimetry
One of the most useful terms for estimating how quickly a nuclide will decay is the radioactive half-life (t1/2). The half-life is defined as the amount of time it takes for a given isotope to lose half of its radioactivity. As was written, radioactive decay is a random process at the level of single atoms, in that, according to quantum theory, it is impossible to predict when a particular atom will decay. In other words, a nucleus of a radionuclide has no “memory”. A nucleus does not “age” with the passage of time. Thus, the probability of its breaking down does not increase with time, but stays constant no matter how long the nucleus has existed.
Therefore, the rate of nuclear decay can be also measured in terms of half-lives. Each radionuclide has its own particular half-life that never changes, regardless of the quantity or form of the material (i.e., solid, liquid, gas, element or compound) or its past history. 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). After five half-lives have elapsed, only 1/32, or 3.1%, of the original number of atoms remains. After seven half-lives, only 1/128, or 0.78%, of the atoms remains. The number of atoms existing after 5 to 7 half-lives can usually be assumed to be negligible.
Half-Life and Radioactivity – Example
The relationship between half-life and the amount of a radionuclide required to give an activity of one curie is shown in the figure. This amount of material can be calculated using λ, which is the decay constant of certain nuclide:
The following figure illustrates the amount of material necessary for 1 curie of radioactivity. It is obvious, that the longer the half-life, the greater the quantity of radionuclide needed to produce the same activity. Of course, the longer lived substance will remain radioactive for a much longer time. As can be seen, the amount of material necessary for 1 curie of radioactivity can vary from an amount too small to be seen (0.00088 gram of cobalt-60), through 1 gram of radium-226, to almost three tons of uranium-238.
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:
The number of iodine-131 atoms initially present.
The activity of the iodine-131 in curies.
The number of iodine-131 atoms that will remain in 50 days.
The time it will take for the activity to reach 0.1 mCi.
Solution:
The number of atoms of iodine-131 can be determined using isotopic mass as below.
NI-131 = mI-131 . NA / MI-131
NI-131 = (1 μg) x (6.02×1023 nuclei/mol) / (130.91 g/mol)
NI-131 = 4.6 x 1015 nuclei
The activity of the iodine-131 in curies can be determined using its decay constant:
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 (t1/2) and the decay constant λ. The relationship can be derived from decay law by setting N = ½ No. 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.
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 (N50d) 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.
Robert Reed Burn, Introduction to Nuclear Reactor Operation, 1988.
U.S. Department of Energy, Nuclear Physics and Reactor Theory. DOE Fundamentals Handbook, Volume 1 and 2. January 1993.
Paul Reuss, Neutron Physics. EDP Sciences, 2008. ISBN: 978-2759800414.
See also:
Half-Life
We hope, this article, Half-Life – Calculation – Example, helps you. If so, give us a like in the sidebar. Main purpose of this website is to help the public to learn some interesting and important information about radiation and dosimeters.
In general, the biological half-life is the time taken for the amount of a particular element in the body to decrease to half of its initial value due to elimination by biological processes alone, when the rate of removal is roughly exponential. Radiation Dosimetry
Biological Half-Life
In general, the biological half-life is the time taken for the amount of a particular element in the body to decrease to half of its initial value due to elimination by biological processes alone, when the rate of removal is roughly exponential. The biological half-live (tbiological) can be defined for metabolites, drugs, and other substances. It is also very important in radiation protection, when considering internal exposure.
If the source of radiation is inside our body, we say, it is internal exposure. The intake of radioactive material can occur through various pathways such as ingestion of radioactive contamination in food or liquids, inhalation of radioactive gases, or through intact or wounded skin. Most radionuclides will give you much more radiation dose if they can somehow enter your body, than they would if they remained outside. The biological half-life depends on the rate at which the body normally uses a particular compound of an element. Radioactive isotopes that were ingested or taken in through other pathways will gradually be removed from the body via bowels, kidneys, respiration and perspiration. This means that a radioactive substance can be expelled before it has had the chance to decay.
As a result, the biological half-life significantly influences the overall dose from internal contamination. If a radioactive compound with radioactive half-life (t1/2) is cleared from the body with a biological half-life tb, the “effective’’ half-life (te) is given by the expression:
As can be seen, the biological mechanisms always decreases the overall dose from internal contamination. Moreover, if t1/2 is large in comparison to tb, the effective half-life is approximately the same as tb. For example, tritium has the biological half-life about 10 days, while the radioactive half-life is about 12 years.
Biological Half-life of Iodine
For example, the biological half-life for iodine inside the human body is about 80 days (according to ICRP). Iodine in food is absorbed by the body and preferentially concentrated in the thyroid where it is needed for the functioning of that gland. When 131I is present in high levels in the environment from radioactive fallout, it can be absorbed through contaminated food, and will also accumulate in the thyroid. 131I decays with a half-life of 8.02 days with beta particle and gamma emissions. As it decays, it may cause damage to the thyroid. The primary risk from exposure to high levels of 131I is the chance occurrence of radiogenic thyroid cancer in later life. For 131I, ICRP has calculated that if you inhale 1 x 106 Bq, you will receive a thyroid dose of HT = 400 mSv (and weighted whole-body dose of 20 mSv).
Biological Half-life of Strontium
Sr-90, Ra-226 and Pu-239 are radionuclides known as bone-seeking radionuclides. These radionuclides have long biological half-lives and are serious internal hazards. Once deposited in bone, they remain there essentially unchanged in amount during the lifetime of the individual. The continued action of the emitted alpha particles can cause significant injury: over many years they deposit all their energy in a tiny volume of tissue, because the range of the alpha particles is very short.
Biological Half-life of Tritium
Tritium is a byproduct in nuclear reactors. Most important source (due to releases of tritiated water) of tritium in nuclear power plants stems from the boric acid, which is commonly used as a chemical shim to compensate an excess of initial reactivity.
Note that, tritium emits low-energy beta particles with a short range in body tissues and, therefore, poses a risk to health as a result of internal exposure only following ingestion in drinking water or food, or inhalation or absorption through the skin. The tritium taken into the body is uniformly distributed among all soft tissues. According to the ICRP, a biological half-time of tritium is 10 days for HTO and 40 days for OBT (organically bound tritium) formed from HTO in the body of adults. As a result, for an intake of 1 x 109 Bq of tritium (HTO), an individual will get a whole-body dose of 20 mSv (equal to the intake of 1 x 106 Bq of 131I).
It was also shown that the biological half-time of HTO depends strongly on many variables and varies from about 4 to 18 days. During the warmer months, the average half-life is lower, which is attributed to increased water intake. As well as, drinking larger amounts of alcohol will reduce the biological half-life of water in the body.
Robert Reed Burn, Introduction to Nuclear Reactor Operation, 1988.
U.S. Department of Energy, Nuclear Physics and Reactor Theory. DOE Fundamentals Handbook, Volume 1 and 2. January 1993.
Paul Reuss, Neutron Physics. EDP Sciences, 2008. ISBN: 978-2759800414.
See also:
Half-Life
We hope, this article, Biological Half-Life, helps you. If so, give us a like in the sidebar. Main purpose of this website is to help the public to learn some interesting and important information about radiation and dosimeters.
In general, the effective half-life is the time taken for the amount of a specific radionuclide in the body to decrease to half its initial value from both elimination by biological processes and radioactive decay. Radiation Dosimetry
In general, the effective half-life is the time taken for the amount of a specific radionuclide in the body to decrease to half its initial value from both elimination by biological processes and radioactive decay. This variable is widely used in radiation therapy. An effective half-life of the radioactive substance will involve a decay constant that represents the sum of the biological and radioactive decay constants, as in the formula:
Besides the radioactive half-life, the effective half-life is determined by the biological half-life, which is the time taken for the amount of a particular element in the body to decrease to half of its initial value due to elimination by biological processes alone, when the rate of removal is roughly exponential. The biological half-life (tbiological) can be defined for metabolites, drugs, and other substances. It is also very important in radiation protection, when considering internal exposure.
If the source of radiation is inside our body, we say, it is internal exposure. The intake of radioactive material can occur through various pathways such as ingestion of radioactive contamination in food or liquids, inhalation of radioactive gases, or through intact or wounded skin. Most radionuclides will give you much more radiation dose if they can somehow enter your body, than they would if they remained outside. The biological half-life depends on the rate at which the body normally uses a particular compound of an element. Radioactive isotopes that were ingested or taken in through other pathways will gradually be removed from the body via bowels, kidneys, respiration and perspiration. This means that a radioactive substance can be expelled before it has had the chance to decay.
As a result, the biological half-life significantly influences the effective half-life and the overall dose from internal contamination. If a radioactive compound with radioactive half-life (t1/2) is cleared from the body with a biological half-life tb, the effective half-life (te) is given by the expression:
As can be seen, the biological mechanisms always decreases the overall dose from internal contamination. Moreover, if t1/2 is large in comparison to tb, the effective half-life is approximately the same as tb. For example, tritium has the biological half-life about 10 days, while the radioactive half-life is about 12 years. On the other hand, radionuclides with very short radioactive half-lives have also very short effective half-lives. These radionuclides will deliver, for all practical purposes, the total radiation dose within the first few days or weeks after intake.
Effective Half-life of Iodine
For example, a biological half-life for iodine inside the human body is about 80 days (according to ICRP). Iodine in food is absorbed by the body and preferentially concentrated in the thyroid where it is needed for the functioning of that gland. When 131I is present in high levels in the environment from radioactive fallout, it can be absorbed through contaminated food, and will also accumulate in the thyroid. 131I decays with a half-life of 8.02 days with beta particle and gamma emissions. As it decays, it may cause damage to the thyroid. The primary risk from exposure to high levels of 131I is the chance occurrence of radiogenic thyroid cancer in later life. As a result, the effective half-life of 131I is about 7.5 days, which is given primarily by radioactive half-life. For 131I, ICRP has calculated that if you inhale 1 x 106 Bq, you will receive a thyroid dose of HT = 400 mSv (and weighted whole-body dose of 20 mSv).
Effective Half-life of Strontium
Sr-90, Ra-226 and Pu-239 are radionuclides known as bone-seeking radionuclides. These radionuclides have long biological half-lives and are serious internal hazards. Once deposited in bone, they remain there essentially unchanged in amount during the lifetime of the individual. The continued action of the emitted alpha particles can cause significant injury: over many years they deposit all their energy in a tiny volume of tissue, because the range of the alpha particles is very short.
Effective Half-life of Tritium
Tritium is a byproduct in nuclear reactors. Most important source (due to releases of tritiated water) of tritium in nuclear power plants stems from the boric acid, which is commonly used as a chemical shim to compensate an excess of initial reactivity.
Note that, tritium emits low-energy beta particles with a short range in body tissues and, therefore, poses a risk to health as a result of internal exposure only following ingestion in drinking water or food, or inhalation or absorption through the skin. The tritium taken into the body is uniformly distributed among all soft tissues. According to the ICRP, an effective half-time of tritium is 10 days for HTO and 40 days for OBT (organically bound tritium) formed from HTO in the body of adults. Both half-lives are given primarily by biological half-life. As a result, for an intake of 1 x 109 Bq of tritium (HTO), an individual will get a whole-body dose of 20 mSv (equal to the intake of 1 x 106 Bq of 131I).
It was also shown that the biological half-time of HTO depends strongly on many variables and varies from about 4 to 18 days. During the warmer months, the average half-life is lower, which is attributed to increased water intake. As well as, drinking larger amounts of alcohol will reduce the biological half-life of water in the body.
Robert Reed Burn, Introduction to Nuclear Reactor Operation, 1988.
U.S. Department of Energy, Nuclear Physics and Reactor Theory. DOE Fundamentals Handbook, Volume 1 and 2. January 1993.
Paul Reuss, Neutron Physics. EDP Sciences, 2008. ISBN: 978-2759800414.
See also:
Half-Life
We hope, this article, Effective Half-Life, helps you. If so, give us a like in the sidebar. Main purpose of this website is to help the public to learn some interesting and important information about radiation and dosimeters.
Alpha decay or α-decay represents the disintegration of a parent nucleus to a daughter through the emission of the nucleus of a helium atom. Alpha decay is a quantum tunneling process. In order to be emitted, the alpha particle must penetrate a potential barrier. Radiation Dosimetry
Alpha decay (or α-decay and also alpha radioactivity) represents the disintegration of a parent nucleus to a daughter through the emission of the nucleus of a helium atom. This transition can be characterized as:
As can be seen from the figure, alpha particle is emitted in alpha decay. Alpha particles are energetic nuclei of helium. Alpha particles consist of two protons and two neutrons bound together into a particle identical to a helium nucleus. Alpha particles are relatively large and carry a double positive charge. They are not very penetrating and a piece of paper can stop them. They travel only a few centimeters but deposit all their energies along their short paths.
In practice, this mode of decay has only been observed in nuclides considerably heavier than nickel, with the lightest known alpha emitters being the lightest isotopes (mass numbers 106–110) of tellurium (element 52). In nuclear reactors alpha decay occurs for example in the fuel (alpha decay of heavy nuclei). Alpha particles are commonly emitted by all of the heavy radioactive nuclei occuring in the nature (uranium, thorium or radium), as well as the transuranic elements (neptunium, plutonium or americium).
Theory of Alpha Decay – Quantum Tunneling
Among the variety of channels in which a nucleus decays, alpha decay has been one of the most studied. The alpha decay channel in heavy and super heavy nuclei has provided information on the fundamental properties of nuclei far from stability, such as their ground state energies and the structure of their nuclear levels.
Alpha decay is a quantum tunneling process. In order to be emitted, the alpha particle must penetrate a potential barrier. This is similar to cluster decay, in which an atomic nucleus emits a small “cluster” of neutrons and protons (e.g. 12C).
The height of the Coulomb barrier for nuclei of A « 200 is about 20-25 MeV. The alpha particles emitted in nuclear decay have typical energies of about 5 MeV. On the one hand an incoming 5 MeV alpha particle is scattered from a heavy nucleus and it cannot penetrate the Coulomb barrier and get sufficiently close to the nucleus to interact via the strong force. On the other hand, a 5 MeV alpha particle bound in a nuclear potential well is able to tunnel that same Coulomb barrier.
By 1928, George Gamow (and independently by Ronald Gurney and Edward Condon) had solved the theory of alpha decay via quantum tunneling. They assumed that the alpha particle and the daughter nucleus exist within the parent nucleus prior to its dissociation, namely the decay of quasistationary states (QS). A quasistationary state is defined as a long-lived state that eventually decays. Initially, the alpha cluster oscillates in the potential of the daughter nucleus, with the Coulomb potential preventing their separation. The alpha particle is trapped in a potential well by the nucleus. Classically, it is forbidden to escape, but according to the (then) newly discovered principles of quantum mechanics, it has a tiny (but non-zero) probability of “tunneling” through the barrier and appearing on the other side to escape the nucleus. Using the tunneling mechanism, Gamow, Condon and Gurney calculated the penetrability of the tunneling α particle through the Coulomb barrier, finding the lifetimes of some α emitting nuclei. The main success of this model was the reproduction of the semi-empirical Geiger-Nuttall law that expresses the lifetimes of the α emitters in terms of the energies of the released α particles. It must be noted, that other common forms of decay (e.g. beta decay) are governed by the interplay between both the nuclear force and the electromagnetic force.
Special Reference: W.S.C. Williams. Nuclear and Particle Physics. Clarendon Press; 1 edition, 1991, ISBN: 978-0198520467.
Geiger-Nuttall Law
Geiger-Nuttall law is a semi-empirical law that expresses the lifetime (half-life) of the alpha emitter in terms of the energy of the released alpha particle. In other words, it states that short-lived isotopes emit more energetic alpha particles than long-lived ones. This rule was formulated by by Hans Geiger and John Mitchell Nuttall in 1911 prior to the development of the theoretical formulation. Geiger-Nuttall law can be mathematically expressed as:
where a and b are empirical constants that are found from logarithmic plots of experimental data. Rα represents the linear range of alpha particle, thus it is a direct measure of kinetic energy of the alpha particle. The width of the resonance (Γ) is generally related to the mean lifetime (τ) of the excited nucleus by the relation: Γ = ℏ / τ
Conservation Laws in Alpha Decay
In analyzing nuclear reactions, we apply the many conservation laws. Nuclear reactions are subject to classical conservation laws for charge, momentum, angular momentum, and energy(including rest energies). Additional conservation laws, not anticipated by classical physics, are:
Certain of these laws are obeyed under all circumstances, others are not. We have accepted conservation of energy and momentum. In all the examples given we assume that the number of protons and the number of neutrons is separately conserved. We shall find circumstances and conditions in which this rule is not true. Where we are considering non-relativistic nuclear reactions, it is essentially true. However, where we are considering relativistic nuclear energies or those involving the weak interactions, we shall find that these principles must be extended.
Some conservation principles have arisen from theoretical considerations, others are just empirical relationships. Notwithstanding, any reaction not expressly forbidden by the conservation laws will generally occur, if perhaps at a slow rate. This expectation is based on quantum mechanics. Unless the barrier between the initial and final states is infinitely high, there is always a non-zero probability that a system will make the transition between them.
For purposes of analyzing non-relativistic reactions, it is sufficient to note four of the fundamental laws governing these reactions.
Conservation of nucleons. The total number of nucleons before and after a reaction are the same.
Conservation of charge. The sum of the charges on all the particles before and after a reaction are the same
Conservation of momentum. The total momentum of the interacting particles before and after a reaction are the same.
Conservation of energy. Energy, including rest mass energy, is conserved in nuclear reactions.
Reference: Lamarsh, John R. Introduction to Nuclear engineering 2nd Edition
Alpha Decay – Q-value
In nuclear and particle physics the energetics of nuclear reactions is determined by the Q-value of that reaction. The Q-value of the reaction is defined as the difference between the sum of the restmasses of the initial reactants and the sum of the masses of the final products, in energy units (usually in MeV).
Consider a typical reaction, in which the projectile a and the target A gives place to two products, B and b. This can also be expressed in the notation that we used so far, a + A → B + b, or even in a more compact notation, A(a,b)B.
When describing alpha decay (a reaction without projectile), the disintegrating nucleus is usually referred to as the parent nucleus and the nucleus remaining after the event as the daughter nucleus. The total rest mass of the daughter nucleus and of the nuclear radiation released in an alpha disintegration, mDaughter + mRadiation, is always less than that of the parent nucleus, mparent. The mass-energy difference,
Q = [mparent – (mDaughter + mRadiation)]c2
appears as the disintegration energy, liberated in the process. For example, the Q-value of typical alpha decay is:
The disintegration energy of about 5 MeV is typical kinetic energy of alpha particle. In order to fulfill the law of conservation of momentum, most of the disintegration energy must appear as the kinetic energy of alpha particle. After an alpha or beta decay, the daughter nucleus is often left in an excited energy state. In order to stabilize itself, it subsequently emits high-energy photons, γ-rays.
Robert Reed Burn, Introduction to Nuclear Reactor Operation, 1988.
U.S. Department of Energy, Nuclear Physics and Reactor Theory. DOE Fundamentals Handbook, Volume 1 and 2. January 1993.
Paul Reuss, Neutron Physics. EDP Sciences, 2008. ISBN: 978-2759800414.
See also:
Radioactive Decay
We hope, this article, Alpha Decay – Alpha Radioactivity, helps you. If so, give us a like in the sidebar. Main purpose of this website is to help the public to learn some interesting and important information about radiation and dosimeters.
Alpha decay is a quantum tunneling process. In order to be emitted, the alpha particle must penetrate a potential barrier. This is similar to cluster decay, in which an atomic nucleus emits a small “cluster” of neutrons and protons (e.g. 12C).
Theory of Alpha Decay – Quantum Tunneling
Alpha decay (or α-decay and also alpha radioactivity) represents the disintegration of a parent nucleus to a daughter through the emission of the nucleus of a helium atom. This transition can be characterized as:
As can be seen from the figure, alpha particle is emitted in alpha decay.
Among the variety of channels in which a nucleus decays, alpha decay has been one of the most studied. The alpha decay channel in heavy and super heavy nuclei has provided information on the fundamental properties of nuclei far from stability, such as their ground state energies and the structure of their nuclear levels.
Alpha decay is a quantum tunneling process. In order to be emitted, the alpha particle must penetrate a potential barrier. This is similar to cluster decay, in which an atomic nucleus emits a small “cluster” of neutrons and protons (e.g. 12C).
The height of the Coulomb barrier for nuclei of A « 200 is about 20-25 MeV. The alpha particles emitted in nuclear decay have typical energies of about 5 MeV. On the one hand an incoming 5 MeV alpha particle is scattered from a heavy nucleus and it cannot penetrate the Coulomb barrier and get sufficiently close to the nucleus to interact via the strong force. On the other hand, a 5 MeV alpha particle bound in a nuclear potential well is able to tunnel that same Coulomb barrier.
By 1928, George Gamow (and independently by Ronald Gurney and Edward Condon) had solved the theory of alpha decay via quantum tunneling. They assumed that the alpha particle and the daughter nucleus exist within the parent nucleus prior to its dissociation, namely the decay of quasistationary states (QS). A quasistationary state is defined as a long-lived state that eventually decays. Initially, the alpha cluster oscillates in the potential of the daughter nucleus, with the Coulomb potential preventing their separation. The alpha particle is trapped in a potential well by the nucleus. Classically, it is forbidden to escape, but according to the (then) newly discovered principles of quantum mechanics, it has a tiny (but non-zero) probability of “tunneling” through the barrier and appearing on the other side to escape the nucleus. Using the tunneling mechanism, Gamow, Condon and Gurney calculated the penetrability of the tunneling α particle through the Coulomb barrier, finding the lifetimes of some α emitting nuclei. The main success of this model was the reproduction of the semi-empirical Geiger-Nuttall law that expresses the lifetimes of the α emitters in terms of the energies of the released α particles. It must be noted, that other common forms of decay (e.g. beta decay) are governed by the interplay between both the nuclear force and the electromagnetic force.
Special Reference: W.S.C. Williams. Nuclear and Particle Physics. Clarendon Press; 1 edition, 1991, ISBN: 978-0198520467.
Geiger-Nuttall Law
Geiger-Nuttall law is a semi-empirical law that expresses the lifetime (half-life) of the alpha emitter in terms of the energy of the released alpha particle. In other words, it states that short-lived isotopes emit more energetic alpha particles than long-lived ones. This rule was formulated by by Hans Geiger and John Mitchell Nuttall in 1911 prior to the development of the theoretical formulation. Geiger-Nuttall law can be mathematically expressed as:
where a and b are empirical constants that are found from logarithmic plots of experimental data. Rα represents the linear range of alpha particle, thus it is a direct measure of kinetic energy of the alpha particle. The width of the resonance (Γ) is generally related to the mean lifetime (τ) of the excited nucleus by the relation: Γ = ℏ / τ
Robert Reed Burn, Introduction to Nuclear Reactor Operation, 1988.
U.S. Department of Energy, Nuclear Physics and Reactor Theory. DOE Fundamentals Handbook, Volume 1 and 2. January 1993.
Paul Reuss, Neutron Physics. EDP Sciences, 2008. ISBN: 978-2759800414.
See also:
Alpha Decay
We hope, this article, Theory of Alpha Decay – Quantum Tunneling, helps you. If so, give us a like in the sidebar. Main purpose of this website is to help the public to learn some interesting and important information about radiation and dosimeters.
Geiger-Nuttall law is a semi-empirical law that expresses the lifetime (half-life) of the alpha emitter in terms of the energy of the released alpha particle. In other words, it states that short-lived isotopes emit more energetic alpha particles than long-lived ones. Radiation Dosimetry
Geiger-Nuttall Law
Geiger-Nuttall law is a semi-empirical law that expresses the lifetime (half-life) of the alpha emitter in terms of the energy of the released alpha particle. In other words, it states that short-lived isotopes emit more energetic alpha particles than long-lived ones. This rule was formulated by by Hans Geiger and John Mitchell Nuttall in 1911 prior to the development of the theoretical formulation. Geiger-Nuttall law can be mathematically expressed as:
where a and b are empirical constants that are found from logarithmic plots of experimental data. Rα represents the linear range of alpha particle, thus it is a direct measure of kinetic energy of the alpha particle. The width of the resonance (Γ) is generally related to the mean lifetime (τ) of the excited nucleus by the relation: Γ = ℏ / τ
By 1928, George Gamow (and independently by Ronald Gurney and Edward Condon) had solved the theory of alpha decay via quantum tunneling. Using the tunneling mechanism, Gamow, Condon and Gurney calculated the penetrability of the tunneling α particle through the Coulomb barrier, finding the lifetimes of some α emitting nuclei. The main success of this model was the reproduction of the semi-empirical Geiger-Nuttall law that expresses the lifetimes of the α emitters in terms of the energies of the released α particles.
Robert Reed Burn, Introduction to Nuclear Reactor Operation, 1988.
U.S. Department of Energy, Nuclear Physics and Reactor Theory. DOE Fundamentals Handbook, Volume 1 and 2. January 1993.
Paul Reuss, Neutron Physics. EDP Sciences, 2008. ISBN: 978-2759800414.
See also:
Alpha Decay
We hope, this article, Geiger-Nuttall Law, helps you. If so, give us a like in the sidebar. Main purpose of this website is to help the public to learn some interesting and important information about radiation and dosimeters.
Alpha Decay – Q-value. The disintegration energy of about 5 MeV is typical kinetic energy of alpha particle. In order to fulfill the law of conservation of momentum, most of the disintegration energy must appear as the kinetic energy of alpha particle. Radiation Dosimetry
Alpha decay (or α-decay and also alpha radioactivity) represents the disintegration of a parent nucleus to a daughter through the emission of the nucleus of a helium atom. This transition can be characterized as:
Alpha decay is a quantum tunneling process. In order to be emitted, the alpha particle must penetrate a potential barrier. This is similar to cluster decay, in which an atomic nucleus emits a small “cluster” of neutrons and protons (e.g. 12C).
The height of the Coulomb barrier for nuclei of A « 200 is about 20-25 MeV. The alpha particles emitted in nuclear decay have typical energies of about 5 MeV. On the one hand an incoming 5 MeV alpha particle is scattered from a heavy nucleus and it cannot penetrate the Coulomb barrier and get sufficiently close to the nucleus to interact via the strong force. On the other hand, a 5 MeV alpha particle bound in a nuclear potential well is able to tunnel that same Coulomb barrier.
Alpha Decay – Q-value
In nuclear and particle physics the energetics of nuclear reactions is determined by the Q-value of that reaction. The Q-value of the reaction is defined as the difference between the sum of the restmasses of the initial reactants and the sum of the masses of the final products, in energy units (usually in MeV).
Consider a typical reaction, in which the projectile a and the target A gives place to two products, B and b. This can also be expressed in the notation that we used so far, a + A → B + b, or even in a more compact notation, A(a,b)B.
When describing alpha decay (a reaction without projectile), the disintegrating nucleus is usually referred to as the parent nucleus and the nucleus remaining after the event as the daughter nucleus. The total rest mass of the daughter nucleus and of the nuclear radiation released in an alpha disintegration, mDaughter + mRadiation, is always less than that of the parent nucleus, mparent. The mass-energy difference,
Q = [mparent – (mDaughter + mRadiation)]c2
appears as the disintegration energy, liberated in the process. For example, the Q-value of typical alpha decay is:
The disintegration energy of about 5 MeV is typical kinetic energy of alpha particle. In order to fulfill the law of conservation of momentum, most of the disintegration energy must appear as the kinetic energy of alpha particle. After an alpha or beta decay, the daughter nucleus is often left in an excited energy state. In order to stabilize itself, it subsequently emits high-energy photons, γ-rays.
Robert Reed Burn, Introduction to Nuclear Reactor Operation, 1988.
U.S. Department of Energy, Nuclear Physics and Reactor Theory. DOE Fundamentals Handbook, Volume 1 and 2. January 1993.
Paul Reuss, Neutron Physics. EDP Sciences, 2008. ISBN: 978-2759800414.
See also:
Alpha Decay
We hope, this article, Alpha Decay – Q-value, helps you. If so, give us a like in the sidebar. Main purpose of this website is to help the public to learn some interesting and important information about radiation and dosimeters.
Parent Nucleus – Daughter Nucleus. In nuclear physics and physics of nuclear decays, the disintegrating nucleus is usually referred to as the parent nucleus and the nucleus remaining after the event as the daughter nucleus. Radiation Dosimetry
Parent Nucleus – Daughter Nucleus
In nuclear physics and physics of nuclear decays, the disintegrating nucleus is usually referred to as the parent nucleus and the nucleus remaining after the event as the daughter nucleus.
Parent Nucleus – Daughter Nucleus in Alpha Decay
For example, the alpha decay represents the disintegration of a parent nucleus to a daughter through the emission of the nucleus of a helium atom. This transition can be characterized as:
As can be seen, the atomic number (as well the mass number) of the daughter nucleus is lower than the atomic number of the parent nucleus.
By 1928, George Gamow (and independently by Ronald Gurney and Edward Condon) had solved the theory of alpha decay via quantum tunneling. They assumed that the alpha particle and the daughter nucleus exist within the parent nucleus prior to its dissociation, namely the decay of quasistationary states (QS). A quasistationary state is defined as a long-lived state that eventually decays. Initially, the alpha cluster oscillates in the potential of the daughter nucleus, with the Coulomb potential preventing their separation. The alpha particle is trapped in a potential well by the nucleus. Classically, it is forbidden to escape, but according to the (then) newly discovered principles of quantum mechanics, it has a tiny (but non-zero) probability of “tunneling” through the barrier and appearing on the other side to escape the nucleus. Using the tunneling mechanism, Gamow, Condon and Gurney calculated the penetrability of the tunneling α particle through the Coulomb barrier, finding the lifetimes of some α emitting nuclei.
Parent Nucleus – Daughter Nucleus in Beta Decay
If a nucleus emits a beta particle, it loses an electron (or positron). In this case, the mass number of daughter nucleus remains the same, but daughter nucleus will form different element. This transition can be characterized as:
Parent Nucleus – Daughter Nucleus in Gamma Decay
In nuclear physics and nuclear chemistry, the various species of atoms whose nuclei contain particular numbers of protons and neutrons are called nuclides. Nuclides are also characterized by its nuclear energy states (e.g. metastable nuclide 242mAm). Nuclides with equal proton number and equal mass number (thus making them by definition the same isotope) but different energy states are known as isomers. We usually indicate isomers with a superscript m. As a result, in the gamma decay parent and daughter nuclei have equal proton number and equal mass number, but differ in their energy states. This transition can be characterized as:
Robert Reed Burn, Introduction to Nuclear Reactor Operation, 1988.
U.S. Department of Energy, Nuclear Physics and Reactor Theory. DOE Fundamentals Handbook, Volume 1 and 2. January 1993.
Paul Reuss, Neutron Physics. EDP Sciences, 2008. ISBN: 978-2759800414.
See also:
Atomic Nucleus
We hope, this article, Parent Nucleus – Daughter Nucleus, helps you. If so, give us a like in the sidebar. Main purpose of this website is to help the public to learn some interesting and important information about radiation and dosimeters.
Beta decay or β decay represents the disintegration of a parent nucleus to a daughter through the emission of the beta particle. Beta decay is governed by the weak interaction. Radiation Dosimetry
Beta decay or β decay represents the disintegration of a parent nucleus to a daughter through the emission of the beta particle. This transition (β– decay) can be characterized as:
If a nucleus emits a beta particle, it loses an electron (or positron). In this case, the mass number of daughter nucleus remains the same, but daughter nucleus will form different element.
Beta particles are high-energy, high-speed electrons or positrons emitted by certain types of radioactive nuclei such as potassium-40. The beta particles have greater range of penetration than alpha particles, but still much less than gamma rays. The beta particles emitted are a form of ionizing radiation also known as beta rays. There are the following forms of beta decay:
Negative Beta Decay – Electron Decay. In electron decay, a neutron-rich nucleus emits a high-energy electron (β– particle). The electrons are negatively charged almost massless particles Due to the law of conservation of electric charge, the nuclear charge must increase by one unit. In this case, the process can be represented by:
Positive Beta Decay – Positron Decay. In positron decay, a proton-rich nucleus emits a positron (positrons are antiparticles of electrons, and have the same mass as electrons but positive electric charge), and thereby reduces the nuclear charge by one unit. In this case, the process can be represented by: An annihilation occurs, when a low-energy positron collides with a low-energy electron.
Inverse Beta Decay – Electron Capture. Electron capture, known also as inverse beta decay is sometimes included as a type of beta decay, because the basic nuclear process, mediated by the weak interaction, is the same. In this process, a proton-rich nucleus can also reduce its nuclear charge by one unit by absorbing an atomic electron.
The emission of electrons was among the earliest observed decay phenomena. The inverse process, electron capture, was first observed by Luis Alvarez, in vanadium 48. He reported it in a 1937 paper in Physical Review.
Uranium 238 decay chain comprises alpha and beta decays.
In a nuclear reactor occurs especially the β− decay, because the common feature of the fission products is an excess of neutrons (see Nuclear Stability). An unstable fission fragment with the excess of neutrons undergoes β− decay, where the neutron is converted into a proton, an electron, and an electron antineutrino. A free neutron also undergo this type of decay. A free neutron will decay with a half-life of about 611 seconds (10.3 minutes) into a proton, an electron, and an antineutrino (the antimatter counterpart of the neutrino, a particle with no charge and little or no mass).
Theory of Beta Decay – Weak Interaction
Beta decay is governed by the weak interaction. During beta decay one of two down quarks changes into an up quark by emitting a W– boson (carries away a negative charge). The W– boson then decays into a beta particle and an antineutrino. This process is equivalent to the process, in which a neutrino interacts with a neutron.
As can be seen from the figure, the weak interaction changes one flavor of quark into another. Note that, the Standard Model counts six flavours of quarks and six flavours of leptons. The weak interaction is the only process in which a quark can change to another quark, or a lepton to another lepton (flavor change). Neither the strong interaction nor electromagnetic permit flavour changing. This fact is crucial in many decays of nuclear particles. In the fusion process, which, for example, powers the Sun, two protons interact via the weak force to form a deuterium nucleus, which reacts further to generate helium. Without the weak interaction, the diproton would decay back into two hydrogen-1 unbound protons through proton emission. As a result, the sun would not burn without it since the weak interaction causes the transmutation p -> n.
In contrast to alpha decay, neither the beta particle nor its associated neutrino exist within the nucleus prior to beta decay, but are created in the decay process. By this process, unstable atoms obtain a more stable ratio of protons to neutrons. The probability of a nuclide decaying due to beta and other forms of decay is determined by its nuclear binding energy. For either electron or positron emission to be energetically possible, the energy release (see below) or Q value must be positive.
Example of Beta Decay
Free Neutron
A free neutron is a neutron that is not bounded in a nucleus. The free neutron is, unlike a bounded neutron, subject to radioactive beta decay.
The free neutron decays into a proton, an electron, and an antineutrino with a half-life of about 611 seconds (10.3 minutes). Source: scienceblogs.com
It decays into a proton, an electron, and an antineutrino (the antimatter counterpart of the neutrino, a particle with no charge and little or no mass). A free neutron will decay with a half-life of about 611 seconds (10.3 minutes). This decay involves the weak interaction and is associated with a quark transformation (a down quark is converted to an up quark). The decay of the neutron is a good example of the observations which led to the discovery of the neutrino. Because it decays in this manner, the neutron does not exist in nature in its free state, except among other highly energetic particles in cosmic rays. Since free neutrons are electrically neutral, they pass through the electrical fields within atoms without any interaction and they are interacting with matter almost exclusively through relatively rare collisions with atomic nuclei.
It must be noted, a proton cannot decay in free space. That is, a proton can undergo decay, but only inside a nucleus. It is due to the conservation of baryon number that has been established as a basic principle of physics. This principle provides basis for the stability of the proton. Since the proton is the lightest particle among all baryons, the hypothetical products of its decay would have to be non-baryons.
The study of beta decay provided the first physical evidence for the existence of the neutrino. The discovery of the neutrino is based on the law of conservation of energy during the process of beta decay.
Tritium is a radioactive isotope, bur it emits a very weak form of radiation, a low-energy beta particle that is similar to an electron. It is a pure beta emitter (i.e. beta emitter without an accompanying gamma radiation). The electron’s kinetic energy varies, with an average of 5.7 keV, while the remaining energy is carried off by the nearly undetectable electron antineutrino. Such a very low energy of electron causes, that the electron cannot penetrate the skin or even does not travel very far in air. Beta particles from tritium can penetrate only about 6.0 mm of air.
Tritium decays via negative beta decay into helium-3 with half-life of 12.3 years.
Energy Spectrum of Beta Decay
In both alpha and gamma decay, the resulting particle (alpha particle or photon) has a narrow energy distribution, since the particle carries the energy from the difference between the initial and final nuclear states. For example, in case of alpha decay, when a parent nucleus breaks down spontaneously to yield a daughter nucleus and an alpha particle, the sum of the mass of the two products does not quite equal the mass of the original nucleus (see Mass Defect). As a result of the law of conservation of energy, this difference appears in the form of the kinetic energy of the alpha particle. Since the same particles appear as products at every breakdown of a particular parent nucleus, the mass-difference should always be the same, and thekinetic energy of the alpha particles should also always be the same. In other words, the beam of alpha particles should be monoenergetic.
It was expected that the same considerations would hold for a parent nucleus breaking down to a daughter nucleus and a beta particle. Because only the electron and the recoiling daughter nucleus were observed beta decay, the process was initially assumed to be a two body process, very much like alpha decay. It would seem reasonable to suppose that the beta particles would form also a monoenergetic beam.
To demonstrate energetics of two-body beta decay, consider the beta decay in which an electron is emitted and the parent nucleus is at rest, conservation of energy requires:
Since the electron is much lighter particle it was expected that it will carry away most of the released energy, which would have a unique value Te-.
The shape of this energy curve depends on what fraction of the reaction energy (Q value-the amount of energy released by the reaction) is carried by the electron or neutrino.
But the reality was different. The spectrum of beta particles measured by Lise Meitner and Otto Hahn in 1911 and by Jean Danysz in 1913 showed multiple lines on a diffuse background, however. Moreover virtually all of the emitted beta particles have energies below that predicted by energy conservation in two-body decays. The electrons emitted in beta decay have a continuous rather than a discrete spectrumappeared to contradict conservation of energy, under the then-current assumption that beta decay is the simple emission of an electron from a nucleus. When this was first observed, it appeared to threaten the survival of one of the most important conservation laws in physics!
To account for this energy release, Pauli proposed (in 1931) that there was emitted in the decay process another particle, later named by Fermi the neutrino. It was clear, this particle must be highly penetrating and that the conservation of electric charge requires the neutrino to be electrically neutral. This would explain why it was so hard to detect this particle. The term neutrino comes from Italian meaning “little neutral one” and neutrinos are denoted by the Greek letter ν (nu). In the process of beta decay the neutrino carries the missing energy and also in this process the law of conservation of energy remains valid.
Conservation Laws in Beta Decay
In analyzing nuclear reactions, we apply the many conservation laws. Nuclear reactions are subject to classical conservation laws for charge, momentum, angular momentum, and energy (including rest energies). Additional conservation laws, not anticipated by classical physics, are:
Certain of these laws are obeyed under all circumstances, others are not. We have accepted conservation of energy and momentum. In all the examples given we assume that the number of protons and the number of neutrons is separately conserved. We shall find circumstances and conditions in which this rule is not true. Where we are considering non-relativistic nuclear reactions, it is essentially true. However, where we are considering relativistic nuclear energies or those involving the weak interactions, we shall find that these principles must be extended.
Some conservation principles have arisen from theoretical considerations, others are just empirical relationships. Notwithstanding, any reaction not expressly forbidden by the conservation laws will generally occur, if perhaps at a slow rate. This expectation is based on quantum mechanics. Unless the barrier between the initial and final states is infinitely high, there is always a non-zero probability that a system will make the transition between them.
For purposes of analyzing non-relativistic reactions, it is sufficient to note four of the fundamental laws governing these reactions.
Conservation of nucleons. The total number of nucleons before and after a reaction are the same.
Conservation of charge. The sum of the charges on all the particles before and after a reaction are the same
Conservation of momentum. The total momentum of the interacting particles before and after a reaction are the same.
Conservation of energy. Energy, including rest mass energy, is conserved in nuclear reactions.
Reference: Lamarsh, John R. Introduction to Nuclear engineering 2nd Edition
Beta Decay – Q-value
In nuclear and particle physics the energetics of nuclear reactions is determined by the Q-value of that reaction. The Q-value of the reaction is defined as the difference between the sum of the restmasses of the initial reactants and the sum of the masses of the final products, in energy units (usually in MeV).
Consider a typical reaction, in which the projectile a and the target A gives place to two products, B and b. This can also be expressed in the notation that we used so far, a + A → B + b, or even in a more compact notation, A(a,b)B.
When describing beta decay (reaction without projectile), the disintegrating nucleus is usually referred to as the parent nucleus and the nucleus remaining after the event as the daughter nucleus. The emission of a beta particle, either an electron, β–, or a positron, β+, changes the atomic number of the nucleus without affecting its mass number. The total rest mass of the daughter nucleus and of the nuclear radiation released in a beta disintegration, mDaughter + mRadiation, is always less than that of the parent nucleus, mparent.
The mass-energy difference,
Q = [mparent – (mDaughter + mRadiation)]c2
appears as the disintegration energy, liberated in the process. For example, the Q-value of typical beta decay is:
In the process of beta decay, either an electron or a positron is emitted. This emission is accompanied by the emission of antineutrino (β- decay) or neutrino (β+ decay), which shares energy and momentum of the decay. The beta emission has a characteristic spectrum. This characteristic spectrum is caused by the fact that either a neutrino or an antineutrino is emitted with emission of beta particle. The shape of this energy curve depends on what fraction of the reaction energy (Q value-the amount of energy released by the reaction) is carried by the massive particle. Beta particles can therefore be emitted with any kinetic energy ranging from 0 to Q. After an alpha or beta decay, the daughter nucleus is often left in an excited energy state. In order to stabilize itself, it subsequently emits high-energy photons, γ-rays.
Robert Reed Burn, Introduction to Nuclear Reactor Operation, 1988.
U.S. Department of Energy, Nuclear Physics and Reactor Theory. DOE Fundamentals Handbook, Volume 1 and 2. January 1993.
Paul Reuss, Neutron Physics. EDP Sciences, 2008. ISBN: 978-2759800414.
See also:
Radioactive Decay
We hope, this article, Beta Decay – Beta Radioactivity, helps you. If so, give us a like in the sidebar. Main purpose of this website is to help the public to learn some interesting and important information about radiation and dosimeters.
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.
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.
