Equilibrium secular

The system that satisfies Eq. (5.25) is in secular equilibrium. Since the activity denoted by parenthesis is defined as the product of decay constant and the amount of radioactive nuclides, 

Only in this chapter, parenthesis represents activity. 

Thus the sufficient condition to reach secular equilibrium in an undisturbed geological system is enough isolated time for the system. The time to reach secular equilibrium is normally five half-lives of the daughter nuclides. For h, the half-life for is 75,000 years, and after 375,000 years, or five half lives of °Th, the system reaches equilibrium. Similarly, the half-life for Ra is 1,600 years, and Th-Ra secular equilibrium can be reached after five half-lives of a, or 8,000 years. 

If there is no initial in the system, that is, = 0, using the activity notation, C Th) = and =, then from 

If there are initial nuclides in the system, using the activity 

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An understanding of the concept of secular equilibrium is also important for such age determinations using U-Th series radionuclides. Owing to the longevity of Th, and the number of parent atoms remains essentially... 

Owing to the stability of the uranyl carbonate complex, uranium is universally present in seawater at an average concentration of ca. 3.2/rgL with a daughter/parent activity ratio U) of 1.14. " In particulate matter and bottom sediments that are roughly 1 x 10 " years old, the ratio should approach unity (secular equilibrium). The principal source of dissolved uranium to the ocean is from physicochemical weathering on the continents and subsequent transport by rivers. Potentially significant oceanic U sinks include anoxic basins, organic rich sediments, phosphorites and oceanic basalts, metalliferous sediments, carbonate sediments, and saltwater marshes. " ... 

This situation, when the activity of the higher atomic number nuclide, the parent, is equal to the activity in the next step in the chain, the daughter, is known as radioactive equilibrium (also referred to as secular equilibrium). Thus, secular equilibrium between a parent and a daughter implies an activity ratio of 1. 

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... 

Based on Equation (3), in the case of a system where there is an initial disequilibrium in the chain (namely A,jNj A,2N2), it is generally stated that the system returns to secular equilibrium after -six half-lives of the daughter. The wide variety of parent-daughter pairs allows disequilibria to provide temporal constraints over a wide range in time scales (Fig. 3). 

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. 

Figure 4. Return to secular equilibrium of activity ratio with no initial Ra. The return to...

Adaughterx/weff-daughter and the daughter will be effectively supported by a greater amount of parent than that in secular equilibrium. These dynamical effects will result in greater U-series fractionation than expected in static systems. 

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. 

Immediately that the parent is incorporated into the crystal it will start to decay to the daughter. At secular equilibrium the (radio)activity of parent and daughter, will be equal, such that ... 

U-series disequilibria are powerful for studying mantle melting because secular equilibrium between all members of the chain should exist for any parcel of solid mantle upwelling from depth. Prior to formation of any fluid phase (early formed melt or volatile rich fluid), upwelling material will have been solid for times that are much longer than... 

Measurement of U-series disequilibria in MORB presents a considerable analytical challenge. Typical concentrations of normal MORB (NMORB) are variable but are generally in the 50-150 ppb U range and 100-400 ppb Th range. Some depleted MORB have concentrations as low as 8-20 ppb U and Th. The concentrations of °Th, Pa, and Ra in secular equilibrium with these U contents are exceedingly low. For instance, the atomic ratio of U to Ra in secular equilibrium is 2.5 x 10 with a quick rule of thumb being that 50 ng of U corresponds to 20 fg of Ra and 15 fg of Pa. Thus, dissolution of a gram of MORB still requires measurement of fg quantities of these nuclides by any mass spectrometric techniques. 

If we assume that the material coming in the box is in secular equilibrium, this equation can be further simplified and multiplied by yield activity ratios ... 

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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