Fission product yield

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Uranium-235 is an isotope of uranium that differs from the element’s other common isotope, uranium-238, by its ability to cause a rapidly expanding fission chain reaction, i.e., it is fissile. It is the only fissile isotope found in any economic quantity in nature. It was discovered in 1935 by Arthur Jeffrey Dempster.

If at least one neutron from U-235 fission strikes another nucleus and causes it to fission, then the chain reaction will continue. If the reaction will sustain itself, it is said to be critical, and the mass of U-235 required to produce the critical condition is said to be a critical mass. A critical chain reaction can be achieved at low concentrations of U-235 if the neutrons from fission are moderated to lower their speed, since the probability for fission with slow neutrons is greater. A fission chain reaction produces intermediate mass fragments which are highly radioactive and produce further energy by their radioactive decay. Some of them produce neutrons, called delayed neutrons, which contribute to the fission chain reaction. In nuclear reactors, the reaction is slowed down by the addition of control rods which are made of elements such as boron, cadmium, and hafnium which can absorb a large number of neutrons. In nuclear bombs, the reaction is uncontrolled and the large amount of energy released creates a nuclear explosion.

The fission of one atom of U-235 generates 200 MeV = 3.2 × 10<sup­>-11 J, i.e. 18 TJ/mol = 77 TJ/kg. However, approximately 5% of this energy is carried away by virtually undetectable neutrinos. [1]

The nuclear cross section for slow thermal neutrons is about 1000 barns. For fast neutrons it is in the order of 1 barn. ^[1]

Only around 0.72% of all natural uranium is uranium-235, the rest being mostly uranium-238. This concentration is insufficient for a self sustaining reaction in a light water reactor; enrichment, which just means separating out the uranium-238, must take place to get a usable concentration of uranium-235. Pressurised Heavy Water Reactors, other heavy water reactors, and some graphite moderated reactors are known for using unenriched uranium. Uranium which has been processed to boost its uranium-235 proportion is known as enriched uranium, different applications require unique levels of enrichment.

The fissile uranium in nuclear weapons usually contains 85% or more of ²³⁵U known as weapon(s)-grade, though for a crude, inefficient weapon 20% is sufficient (called weapon(s)-usable); even less is sufficient, but then the critical mass required rapidly increases. However, judicious use of implosion and neutron reflectors can enable construction of a weapon from a quantity of uranium below the usual critical mass for its level of enrichment, though this would likely only be possible in a country which already had extensive experience in developing nuclear weapons. The Little Boy atomic bomb was fueled by enriched uranium. Most modern nuclear arsenals use plutonium as the fissile component, however U-235 devices remain a nuclear proliferation concern due to the simplicity of this nuclear weapon design.

Uranium-235 has a half-life of 700 million years.

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• Enriched uranium
• Nuclear fuel cycle
• Nuclear power
• Nuclear reprocessing
• United States Enrichment Corporation
• Uranium market

• Table of Nuclides
• US DOE Handbook 1019/1-93

• Uranium | Radiation Protection Program | US EPA
• NLM Hazardous Substances Databank – Uranium, Radioactive

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Editor-In-Chief: C. Michael Gibson, M.S., M.D. [1]

The half-life of a quantity, subject to exponential decay, is the time required for the quantity to decay to half of its initial value. The concept originated in the study of radioactive decay, but applies to many other fields as well, including phenomena which are described by non-exponential decays.

The term half-life was coined in 1907, but it was always referred to as half-life period. It was not until the early 1950s that the word period was dropped from the name. ^[1]

The table at right shows the reduction of the quantity in terms of the number of half-lives elapsed.

It can be shown that, for exponential decay, the half-life <math>t_{1/2}</math> obeys this relation:

<math>

t_{1/2} = \frac{\ln (2)}{\lambda} </math> where

• <math>\ln (2)</math> is the natural logarithm of 2 (approximately 0.693), and
• λ is the decay constant, a positive constant used to describe the rate of exponential decay.

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

<math> t_{1/2} = \ln (2) \cdot \tau </math>

Main article: Exponential decay–Applications and examples

The constant <math>\lambda</math> can represent many different specific physical quantities, depending on what process is being described.

• In an RC circuit or RL circuit, <math>\lambda</math> is the reciprocal of the circuit’s time constant. For simple RC and RL circuits, <math>\lambda</math> equals <math>1/RC</math> or <math>R/L</math>, respectively.
• In first-order chemical reactions, <math>\lambda</math> is the reaction rate constant.
• In radioactive decay, it describes the probability of decay per unit time: <math>dN = \lambda N dt</math>, where dN is the number of nuclei decayed during the time dt, and N is the quantity of radioactive nuclei.
• In biology (specifically pharmacokinetics), from MeSH: Half-Life: The time it takes for a substance (drug, radioactive nuclide, or other) to lose half of its pharmacologic, physiologic, or radiologic activity. Year introduced: 1974 (1971).

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

<math>T_{1/2} = \frac{\ln 2}{\lambda _1 + \lambda _2} \,</math>

or, in terms of the two half-lives <math>t_1</math> and <math>t_2</math>

<math>T_{1/2} = \frac{t _1 t _2}{t _1 + t_2} \,</math>

i.e., half their harmonic mean.

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

<math>N(t) = N_0 e^{-\lambda t} \,</math>

where <math>N_0</math> is the initial value of <math>N</math> (at <math>t = 0</math>)

When <math>t = 0</math>, the exponential is equal to 1, and <math>N(t)</math> is equal to <math>N_0</math>. As <math>t</math> approaches infinity, the exponential approaches zero. In particular, there is a time <math>t_{1/2} \,</math> such that

<math>N(t_{1/2}) = N_0\cdot\frac{1}{2}. </math>

Substituting into the formula above, we have

<math>N_0\cdot\frac{1}{2} = N_0 e^{-\lambda t_{1/2}}, \,</math>

<math>e^{-\lambda t_{1/2}} = \frac{1}{2}, \,</math>

<math>- \lambda t_{1/2} = \ln \frac{1}{2} = – \ln{2}, \,</math>

<math>t_{1/2} = \frac{\ln 2}{\lambda}. \,</math>

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

Most appropriate to validate the concept of half-life for radioactive decay, in particular when dealing with a small number of atoms, is to perform experiments and correct computer simulations. See in [2] how to test the behavior of the last atoms. Validation of physics-math models consists in comparing the model’s behavior with experimental observations of real physical systems or valid simulations (physical and/or computer). The references given here describe how to test the validity of the exponential formula for small number of atoms with simple simulations, experiments, and computer code.

In radioactive decay, the exponential model does not apply for a small number of atoms (or a small number of atoms is not within the domain of validity of the formula or equation or table). The DIY experiments use pennies or M&M’s candies. [3], [4]. A similar experiment is performed with isotopes of a very short half-life, for example, see Fig 5 in [5]. See how to write a computer program that simulates radioactive decay including the required randomness in [6] and experience the behavior of the last atoms. Of particular note, atoms undergo radioactive decay in whole units, and so after enough half-lives the remaining original quantity becomes an actual zero rather than asymptotically approaching zero as with continuous systems.

• Exponential decay
• Mean lifetime
• Elimination half-life
• For non-exponential decays, see half-life in the article Rate equation

• System Dynamics – Time Constants

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