Radionuclides equation

These are the most basic equations needed in order to utilize U-Th series radionuclides as time-dependent tracers. In turn, they can provide an apparent age of a process or a geochronological framework. 

This equation has been used for estimating migration velocities of radionuclides (e.g. 66). Here Pr is the density of the rock (kg/m3), p the density of water, e the fissure porosity, af the specific surface of fissures in the bedrock (m2/m3) and ap the specific surface of particles used in the Kd determinations (m2/m3). The distribution coefficient Kd represents ar. equilibrium value for the particular rock under the pertinent conditions. 

An important consideration is the relative importance of the two processes that supply radionuclides to the dissolved and adsorbed inventories from within the host rock minerals. The recoil term in Equation (1), bsiA,pPR, can be compared to the weathering... 

As a noble gas, Rn in groundwater does not react with host aquifer surfaces and is present as uncharged single atoms. The radionuclide Rn typically has the highest activities in groundwater (Fig. 1). Krishnaswami et al. (1982) argued that Rn and all of the other isotopes produced by a decay are supplied at similar rates by recoil, so that the differences in concentrations are related to the more reactive nature of the other nuclides. Therefore, the concentration of Rn could be used to calculate the recoil rate for all U-series nuclides produced by a recoil. The only output of Rn is by decay, and with a 3.8 day half-life it is expected to readily reach steady state concentrations at each location. Each measured activity (i.e., the decay or removal rate) can therefore be equated with the input rate. In this case, the fraction released, or emanation efficiency, can be calculated from the bulk rock Ra activity per unit mass ... 

The ICRP (1994b, 1995) developed a Human Respiratory Tract Model for Radiological Protection, which contains respiratory tract deposition and clearance compartmental models for inhalation exposure that may be applied to particulate aerosols of americium compounds. The ICRP (1986, 1989) has a biokinetic model for human oral exposure that applies to americium. The National Council on Radiation Protection and Measurement (NCRP) has also developed a respiratory tract model for inhaled radionuclides (NCRP 1997). At this time, the NCRP recommends the use of the ICRP model for calculating exposures for radiation workers and the general public. Readers interested in this topic are referred to NCRP Report No. 125 Deposition, Retention and Dosimetry of Inhaled Radioactive Substances (NCRP 1997). In the appendix to the report, NCRP provides the animal testing clearance data and equations fitting the data that supported the development of the human mode for americium. 

The vertical scavenging model also allows one to predict the distribution of particulate radionuclide profiles. Following Craig et al. [53] the particulate phase activity would be given by the solution of the equation ... 

Equation (9) is quite generalized and allows for variations in the mixing rate, K, and in situ density, p, with depth in determining the activity-time relationship of radionuclide profiles. However, in all commonly used models, equation (9) is further simplified using assumptions such as K, p and S to be... 

Solution of equation (10) which involves sedimentation in the presence of mixing and that of equation (11) which contains the sedimentation term only, are exponential in nature. The major conclusion which arises from this is that the logarithmic nature of the activity-depth profiles by itself is not a guarantee for undisturbed particle by particle sediment accumulation, as has often been assumed. The effects of mixing and sedimentation on the radionuclide distribution in the sediment column have to be resolved to obtain pertinent information on the sediment accumulation rates. (It is pertinent to mention here that recently Guinasso and Schink [65] have developed a detailed mathematical model to calculate the depth profiles of a non-radioactive transient tracer pulse deposited on the sediment surface. Their model is yet to be applied in detail for radionuclides. )... 

The retardation equation can also be applied to inorganic soluble substances (ions, radionuclides, metals). But here we have to consider, in addition to the sorption or ion exchange process, that the speciation of metal ions or ligands in a multi-... 

Hydrolysis. NMR results show that TBT carboxylates undergo fast chemical exchange. Even the interfacial reaction between TBT carboxylates and chloride is shown to be extremely fast. The hydrolysis is thus not likely to be a rate determining step. Since the diffusivity of water in the matrix is expected to be much greater than that of TBTO, a hydrolytic equilibrium between the tributyltin carboxylate polymer and TBTO will always exist. As the mobile species produced diffuses out, the hydrolysis proceeds at a concentration-dependent rate. Godbee and Joy have developed a model to describe a similar situation in predicting the leacha-bility of radionuclides from cementitious grouts (15). Based on their equation, the rate of release of tin from the surface is ... 

If the decay constant of the parent radionuclide is much lower than that of the daughter radionuclide (i.e., Ai << A2), equation 11.32 can be reasonably reduced to... 

In equation 11.77, 4 is a numerical constant depending on the geometry and decay constant of the parent radionuclide. If the half-life of the parent is long in comparison with the cooling period, A takes a value of 55, 27, or 8.7 for volume diffusion from a sphere, cylinder, or plane sheet, respectively. If decay rates are faster, A progressively diminishes (see table 1 in Dodson, 1973, for numerical values). 

This relationship is transformed into an equality by introduction of a proportionality constant, 1, which represents the probability that an atom will decay within a stated period of time. The numerical value of A is unique for each radionuclide and is expressed in units of reciprocal time. Thus, the equation describing the rate of decay of a radionuclide is... 

The rate of decay of a radionuclide is often discussed in terms of its half-life. The half-life (T1/2) is tiie time required for one-half of a given number of atoms of a radionuclide to decay. Mathematically, when t = Tm, N= 0.5 x N . Substituting these values into Equation (8.3), we find... 

Because the short-lived nuclides are extinct, a different approach must be taken to use them as chronometers. Equation (8.9) cannot be used to calculate a date because the number of parent nuclides, N, is zero and the equation is undefined. However, if a short-lived nuclide was homogeneously distributed throughout a system, then one can determine the order in which objects formed within that system based on the amount of radionuclide that was present when each object formed. The oldest object would form with the highest amount of the radionuclide relative to a stable isotope of the same element, and the youngest will have the lowest amount. Obviously, no chronological information can be obtained about objects that formed after the radionuclide has reached a level too small to detect the radiogenic daughter isotope. 

Table V presents the leaching data for Melt Glass 3 only. Glasses 1 and 2 were omitted from this interim report since they were both older than 3 and provided lower sensitivity for radionuclide detection. The data in Table V were calculated using the following equation (12)...

Any radionuclide is characterised by its half-life r whose value is independent of the type of decay products that are created. Half-life is defined as the time required (from initial time t = 0) for the decomposition of half the atoms in the sample. The law of radioactive decay allows calculation of the number of atoms N left at time t in a population with N0 atoms initially. The integrated form of this law is given by the following equation ... 

Land Subsurface Burst. Everything which was said above about land surface burst applies exactly to the aerial cloud particle population produced by a land subsurface burst in which an aboveground fireball appears. However, a third component of the particle population is found. This component appears to result from soil material which interacted with the fireball at high temperature but which was separated from the fireball early, before the temperature had fallen below the melting point of the soil materials. The particles in this component have diameters ranging from tens of microns to several centimeters and have densities which are apt to be quite low compared with those of the original soil components. The relative abundance of radionuclides in this component is quite constant from sample to sample and is independent of particle size. If we indicate by subscript 1 this third component and by 2,3 the aerial cloud components, radionuclide partitioning can be described by a series of equations of the forms... 

Air Burst. Particles produced by condensation are spherical, and range in size from about 0.01/x to perhaps 20/x in diameter. The particles are metal oxides and exhibit densities in the range of the metal oxides of the condensable materials vaporized by the detonation. Radionuclide partitioning can be described (2) by a series of equations of the form... 

The equation in this form states that in the samples analyzed the distribution of any radionuclide, A, can be expressed as a linear combination of the distribution of two species, a refractory and 137Cs. Therefore, we need to determine only refractory distribution, 137Cs distribution, and mass distribution with particle size and the distribution of all other isotopic species for which aA values are known and can be calculated. The refractory species used is 155Eu. It has a half-life of 1.811 years and two easily resolved gamma photopeaks so that its abundance as well as that of 137Cs can be determined readily by gamma spectrometry. 

Measurements have been made (see Figures 1—4) on sets of samples obtained with aircraft from debris clouds resulting from atmospheric explosions. In many cases it has been observed that the behavior of the active products was determined by the behavior of two and only two different radionuclides and that for an entire set of samples from a single event, the amount of nuclide i in sample j was given, usually to within 10%, by an equation such as... 

The determination of the rank of a matrix is fairly simple and straight-forward. Unfortunately, the orthodox methods applied to a matrix such as A in Equation 2 give an answer which is exact mathematically but useless physically, namely that the rank of A is the number of radionuclides measured or the number of samples analyzed, whichever is less. This unfortunate result arises from presence of experimental imprecision in the elements of A. One must therefore rewrite Equation 2 in the form... 

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