Slope Year for the UPb Dating Method and Its Applications

The slope year tslope for the U-Pb dating method is given as ( ) 235 238 235 238 1 ln slope t k = − λ λ λ λ , where λ238 and λ235 are the decay constants for 238U and 235U, respectively, and k is the slope of the tangent line at a point on either the Concordia or Discordia line. These two lines are determined by the initial 206(7)Pbi concentrations in minerals. If 206 207 Pb Pb 0 i i = = , the line is the Concordia. However, if 206 207 Pb 0 Pb 0 i i ≠ ∧ = ( ∧ is the logical operator “and”, also known as the logical conjunction), 206 207 Pb 0 Pb 0 i i = ∧ ≠ or 206 207 Pb 0 Pb 0 i i ≠ ∧ ≠ , the line is Discordia. The Concordia line is of the form ( ) 238 235 206 238 207 235 Pb U Pb U 1 1 p p p p + = − λ λ (where p stands for the present), while the Discordia line has the form ( ) 206 238 207 235 Pb U Pb U p p p p k b × = + (where k and b are the slope and intercept of the straight line, respectively).

These nuclear reactions occur in host minerals, such as zircon (ZrSiO 4 ), and are the basis of the U-Pb dating method in geology [5]- [8].In a mineral, Pb and U isotopes obey the exponential decay law: where the subscripts i and p represent the initial measurement time and the present, respectively, and t is the age of the mineral [1] [6].
The coordinates n( 206 Pb p )/n( 238 U p ) (n, the number of isotopes in the bracket) as the ordinate and n( 207 Pb p )/ n( 235 U p ) ratios as the abscissa form the Pb/U ratio diagram (Figure 1).Samples formed t years ago plot on either the Concordia or Discordia lines [9]- [12].For instance, the classical Discordia line was discovered by Ahrens (1955) from Equation (2).
To interpret the Discordia line, conventional theories have proposed: 1) this line was caused by Pb loss or U gain after formation of the host mineral [9] [11]- [17], 2) the upper intersection of the Discordia and Concordia lines represents the crystallization age of the mineral [12] and 3) the lower intersection of the Discordia and Concordia lines represents the metamorphic age of the mineral [14].
However, previous theories are not tenable when used in the following cases: 1) the lower intercept point is negative or 2) no upper intercept point exists.For instance, in Zheng et al. (2012) (Figure 1), all zircons in YX1 from Yingxian lamproites were found to be discordant and yielded a lower intercept age of −370 ± 690 Ma.According to conventional theories, this age indicates that the samples will experience a metamorphic process in a distant age.In addition, in Zheng et al. (2012), all zircons in HBxa from Hebi basalt are also discordant, but yield no upper intercept age.According to conventional theories, these data indicate that the samples did not crystallize until the present.Apparently, the explanations do not conform to the objective facts: the samples are in front of scientists now.New studies should thus focus on resolving these discrepancies.
Herein, the slope years t slope s for the U-Pb dating method for the Concordia and Discordia lines are presented, and a method for estimating values for t slope from the experimental data is proposed.In addition, four examples are presented to illustrate the application of the proposed method.

Slope k and Slope Year Tslope
In mathematics, the variance on the ordinate is a function of the variance on the abscissa [50] from Equation (4).Equation ( 6) divided by Equation ( 7) gives This equation indicates that if t is determined, the value of k is a constant (Table 1) since t ≥ 0, 0 < k ≤ 0.1575.In addition, the slope monotonically decreases with increasing time t (Figure 3).
If k is determined (see Section 2.6), the slope year is given by rewriting Equation ( 9): ( )

Initial 206(7) Pbi Concentrations in Minerals
If the values for t slope , 206 (7) Pb p and 235 (8) U p are known, the initial 206 (7) Pb i concentrations in minerals can be determined using the following: ( ) which are derived from Equations ( 1) and ( 2).Clearly, the concentrations are greater than or equal to zero: ( )

Mathematical Expressions for the Concordia and Discordia Lines
The initial 206 (7) Pb i isotope concentrations determine the mathematical expressions for the general graph in Figure 2.This relationship can be demonstrated using assumed samples formed at the same time t with specific initial conditions.Assume there are three samples The mathematical expressions are given by solving the first-order differential Equation (9) using Equations ( 3) and ( 4 The general solution of Equation ( 16) is Since the concentrations of 206   (Equation ( 14)), Equation ( 15) is not an elementary function and the solution to it cannot be obtained using elementary integral calculus.This difficulty can be overcome in the following manner.
Since k is a constant when t is given (Table 1), the solution to this equation is where k and b are the slope and intercept of the line, respectively.This equation shows that the general curve in Figure 2 is a straight line, i.e. the Discordia line.Equation ( 21) is consistent with the initial condition (Equation ( 14)).If k = 0.15751 (at t = 0) is applied: This equation indicates that 1) in the geological system, 206 Pb i / 238 U i monotonically increases with increasing 207 Pb i / 235 U i from samples 4 to 5 to 6 (Figure 4(b)) and 2) these two ratios for the three samples cannot simultaneously be zero.points plot on the origin (0, 0) where the Concordia line begins (Equation ( 19)).As time increases, the slope of the curve decreases from 0.15751 (0 Ma) to 0.06870 (1000 Ma) to 0.02997 (2000 Ma) and finally to 0.01307 (3000 Ma) (Table 1) and b) samples 4, 5 and 6 (Figure 4(b)), for which when t = 0 the ( ) points plot on discordant lines with different slopes, and the slope of each line decreases from 0.15751 (0 Ma) to 0.06870 (1000 Ma) to 0.02997 (2000 Ma) and finally to 0.01307 (3000 Ma) (Table 1).

Methods for Determining k from Experimental Data
For n ( ) where 238 235 0.15751 λ λ = . The mean slope for all the n points is then b) If the n data points plot on the Discordia line (Figure 4(b)), the slope can be determined using the least squares method [51].This method gives a linear function for the points:

Error Propagation
For a function

( )
, , , f f x y z =  , where x, y and z are independent variables, the error (1σ) is given by where x σ , y σ and z σ are the standard errors for x, y and z, respectively [51].
According to Equation ( 27), the standard error for t slope (Equation ( 10)) is  Then the values for k σ are given as follows.
a) For concordant data, the standard error of the ith slope (Equation ( 23)) is and the standard error of the mean slope (Equation ( 24)) is b) For discordant data, the standard error of k in Equation ( 26) is See proofs of this equation in Appendix A.

Applications
To demonstrate the validity of our work, four examples are illustrated (Table 2 and Figure 5).Table 2 includes original Pb/U isotope ratios from the published literature along with the slope years (i.e.U-Pb ages) when the samples were formed.
The first example comes from Qinghu granite in the Nanling Range, South China [44].The Pb/U ratios in this granite are the concordant type (Figure 5(a)) [44].The slope and slope year were calculated using Equations ( 24) and (10), respectively, and found to be k Concordia = 0.13792 ± 0.00025 and t slope = 160 ± 2 Ma (  The k and t slope values for the three discordant examples described in the introduction were also calculated using Equations ( 26) and (10), respectively.For the Zimbabwe uranium deposit (Figure 5(b)), the slope was k Discordia = 0.03950 ± 0.00178 and slope year was t slope = 1668 ± 55 Ma.For amphibolites in the Yingxian lamproite (YX1, Figure 5(c)), the slope was k Discordia = 0.06779 ± 0.00564 and slope year was t slope = 1016 ± 100 Ma.For Hebi amphibolites (HBxa, Figure 5(d)), the slope was k Discordia = 0.010734 ± 0.00196 and slope year was t slope = 3237 ± 220 Ma.

Conclusion
A method for determining the slope year for the U-Pb dating method and initial 206 (7) Pb concentrations in samples was described.It was also found that if no 206 (7) Pb isotopes are initially present in minerals, the Pb/U ratios plot on the Concordia line.On the other hand, if 206 (7) Pb isotopes are initially present in minerals, the Pb/U ratios plot on the Discordia line.Therefore, the Discordia line is not the result of Pb loss or U gain.Furthermore, methods for determining the slope year using experimental data were also proposed and applied to data on four samples previously described in the literature.These results demonstrate that our approach is useful for geological research.
where σ is the standard error of i Y or ( ) ( ) The square root of this equation is the 1σ error of k.
are the decay constants for 238 U and 235 U, respectively, and k is the slope of the tangent line at a point on either the Concordia or Discordia line.These two lines are determined by the initial 206(7) Pbi concentrations in minerals.is the logical operator "and", also known as the logical conjunc- Q is the heat, β denotes the beta decay and He stands for the element Helium.The decay constants λ for 238 U and 235 U are

Figure 1 .
Figure 1.Pb/U ratio diagram.This diagram shows the predicament for conventional theories.The Concordia (blue, colour for online version) and classical Discordia (black) for Zimbabwe samples (black diamond points) (Ahrens, 1955) are illustrated.This Discordia and Concordia intercept at A and B, for which the meanings in conventional theories are shown in the lower-right corner.Two counter-examples to traditional theories are also shown: HBxa (hexagon points and red Discordia, Zheng et al. (2012)) and YX1 (right triangle points and green Discordia, Zheng et al. (2012)).See discussions in text.

Figure 3 .Figure 4 .
Figure 3. Plot of the slope k versus time t.
which is the expression for the Concordia line.b)For samples 4, 5 and 6, because of the existence of the variances in Consider a geological body (containing samples 4, 5 and 6) with continuous 206 Pb i , 207 Pb i , 238 U i and 235 U i distributions.are continuous variables[50].Looking back to the original differential Equation (

( ) 206 7
Pb i also determines the histories of the ( ) data points on the Concordia and Discordia lines.In Figure4, the histories are shown for a) samples 1, 2 and 3 (Figure4(a)), for which when t = 0, the ( ) proofs for k Discordia in Appendix A.

σ
is the standard error of the slope.

Figure 5 .
Figure 5. Present slope years (with 1σ error) for (a) Qinghu granite, (b) a Zimbabwe uranium deposit, (c) Yingxian amphibolites and (d) Hebi amphibolites.All data points except Zimbabwe are plotted with 1σ error bars.The norms of the residuals (R 2 ) for the least squares fits are illustrated, and the slopes (with 1σ errors) are given.In (a), the red diamond indicates the mean value for all the measured data and the tangent line at this point coincides with the Concordia line.

Table 1 .
Values of the slope for specific years.
Pb i and 207 Pb i are both zero at t = 0, the result is 0 = 0 + C; thus, C = 0. )