Decay chain

The analysis of steady-state and transient reactor behavior requires the calculation of reaction rates of neutrons with various materials. If the number density of neutrons at a point is n and their characteristic speed is v, a flux effective area of a nucleus as a cross section O, and a target atom number density N, a macroscopic cross section E = Na can be defined, and the reaction rate per unit volume is R = 0S. This relation may be appHed to the processes of neutron scattering, absorption, and fission in balance equations lea ding to predictions of or to the determination of flux distribution. The consumption of nuclear fuels is governed by time-dependent differential equations analogous to those of Bateman for radioactive decay chains. The rate of change in number of atoms N owing to absorption is as follows ... 

In these cases what is usually measured is not the time of the original formation of the rocks, but the time at which the parent and daughter elements were last separated. That is, if the rocks were remelted at some point in their history in a manner that removed the daughter elements, this would be the age measured. For the measurements this is especially important. The daughter is a gas and thus could escape at any time when it was not sealed in. The decay sequences of Th, and all have gaseous members in their decay chains (see Tables 1 and 2), but the final members are solids. 

ThSiO "Th and Th are present in naturally occurring uranium Th and Th occur in uranium minerals as members of the decay chain. The remaining isotopes are formed upon neutron bombardment of those isotopes discussed, or by charged particle bombardment of various targets. 

Figure 1 Chart showing the decay chain of the U-Th decay series isotopes. Vertical arrows define alpha (a) decays while beta (/ ) decays are illustrated by diagonal arrows...

Th decay chains. The grayscale reflects half-life, with darker grays for longer half-lives. 

A useful analogy for understanding secular equilibrium is visualizing a decay chain as a series of pools of water (Fig. 2). These pools eventually lead to a continuously filling pool representing a stable isotope of lead (either ° Pb, ° Pb or ° Pb). Over the timescale... 

Figure 2. Analogy of U-series decay chain with a series of tanks feeding into each other. See text for description.

The previous section showed that if the decay chain remains undisturbed for a period of approximately 6 times the longest half-lived intermediate nuclide then the chain will be in a state of secular equilibrium (i.e., equal activities for all the nuclides). The key to the utility of the U-series is that several natural processes are capable of disrupting this state of equilibrium. 

The diffusion of U and Th within a solid is, in general, very slow due to their large size and charge (Van Orman et al. 1998). Even at mantle temperatures, it is expected that a solid will not fully equilibrate with the surrounding phases (fluid, melt or other solid phases) if solid diffusion controls the equilibration. As yet, there have been no direct determinations of diffusion coefficients for any other decay chain element. 

Measurement of specific activity. The half-life of a nuclide can be readily calculated if both the number of atoms and their rate of decay can be measured, i.e., if the activity A and the number of atoms of P can be measured, then X is known from A = XP. As instrumentation for both atom counting and decay counting has improved in recent decades, this approach has become the dominant method of assessing half-lives. Potential problems with this technique include the accurate and precise calibration of decay-counter efficiency and ensuring sufficient purity of the nuclide of interest. This technique provides the presently used half-lives for many nuclides, including those for the parents of the three decay chains, U, U (Jaffey et al. 1971), and Th. 

Secular equilibrium materials. For materials that have remained a closed system for sufficient time that secular equilibrium has been achieved, the half-lives of nuclides within the decay chain can be calculated from the relationship A,pP = A,dD. If the atom ratio P/D is measured, and one of the decay constants is well known, then the other can be readily calculated. Limitations on this approach are the ability to measure the atom ratios to sufficient precision, and finding samples that have remained closed systems for a sufficient length of time. This approach has been used to derive the present recommended half lives for °Th and (Cheng et al. 2000 Ludwig et al. 1992). 

The solution to the general decay equations is often given in textbooks (e.g., Faure 1986). However, this solution is given for initial abundances of the daughter nuclides that are equal to zero. In the most general cases, the initial abundances of the daughter nuclides are not equal to zero. For example, in many geological examples, we make the assumptions that the decay chain is in secular equilibrium. The solutions of these equations can also be used to solve simple box models of U-series nuclides where first order kinetics are assumed. 

It may be worth mentioning the analogy between the laws of radioactive equilibrium in decay chains (Ni/Ti = N2/T2 =... Ni/Ti) and those of successive magma reservoirs at steady state (i.e., with the same input and output rates O, where Mi/xi = M2/X2 =... Mi/xi ), as illustrated in Figure 5. 

Krishnaswami S, Turekian KK, Beimett JT (1984) The behavior of h and the decay chain nnclides... 

Here, I review a one-dimensional model for melt moving relative to solid following the work of Spiegelman and Elliott (1993). The physical model described provides the parameters used in the equations tracking residence times differences for decay chain nuclides and thus generating disequilibria. Assuming steady state, the transfer of mass between the solid and melt is described by ... 

An important property of the U-series decay chains is that the daughter and parent... 

For the decay chain the change in a, with position in a ID melt... 

Figure Al(a) shows the constant value of porosity used in the analytic model (dashed curve), compared to the porosity distribution for a ID melt column in which the upward flux of melt is required to remain constant (see Spiegelman and Elliott 1993). The solid curves in Figure Al(b) show values of ct, calculated from equations (A12-A14) along the (dimensionless) length of the melting column for the decay chain with a constant porosity of 0.1% and solid upwelling velocity of 1 cm/yr.

This section describes the continuous flux melting model used in Bourdon et al. (2003) and has many similarities with the model of Thomas et al. (2002). A significant difference is that the model described here keeps track of the composition of the slab as it dehydrates. This model is based on mass balance equations for both the mantle wedge and the slab. We assume secular equilibrium in the U-series decay chain initially ... 

Since Ra and " Ra are both produced by recoil from the host mineral, it might be assumed that the production rates are equal. However, the relative recoil rates can be adjusted by considering that the parent nuclides near the mineral surface may not be in secular equilibrium due to ejection losses i.e., the activity of Th may be lower than that of Th due to recoil into groundwater of the intermediate nuclide Ra. Krisnaswami et al. (1982) calculated that the recoil rate of " Ra is 70% that of Ra if radionuclides are depleted along the decay chain in this way. 

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