Method suitable for dating uranium-lead isotopes and deducting common lead

Abstract

The invention belongs to the field of isotope theory and application, and particularly relates to a method suitable for dating uranium-lead isotopes and deducting common lead, which comprises the following steps: deducting a third-stage age t2 calculation formula according to a U-Pb isotope three-stage age formula, and calculating t2, alpha2 and beta2 by adopting uranium-lead isochrone fitting; establishing a regression equation according to a U-Pb isotope three-stage age formula, and calculating a regression equation coefficient; establishing a transcendental equation according to a regression equation coefficient calculation formula, and calculating t1 and t2 by adopting a Newton iteration method; calculating alpha 1, beta 1 and mu 1 according to t1, t2 and alpha 2; and calculating an age error, comparing and verifying t2, and discussing the reliability of a result. The obtained result conforms to the isotope theory, isotope dating composed of lead isotopes can be accurately measured, the method has good reliability and measurement precision, and the problems that common lead is not reliable, dating is difficult to conduct after multiple times of opening of a system, and common lead is deducted can be effectively solved.

Description

Technical Field

[0001] This invention belongs to the field of isotope theory and application, specifically relating to a method applicable to uranium-lead isotope dating and ordinary lead deduction. Background Technology

[0002] Uranium-lead isotope dating is widely used to calculate the ages of rock and mineral formation and hydrothermal activity. It is commonly obtained using whole-rock and single-mineral thermoelectric mass spectrometry (TIMS) and micro-area in-situ scale methods such as laser ablation and ion probe microanalysis (LA-ICP-MS, SHRIMP, SIMS, etc.).

[0003] The TIMS method uses a double dilution method followed by mass spectrometry measurement. 204 Pb, 206 Pb, 207 Pb, 208 Pb abundance values ​​were obtained by measuring U and Pb content data using inductively coupled plasma mass spectrometry (ICP-MS). The corresponding age was calculated using computer software based on U-Pb isotope theory. (This is not considered here.) 232 Th and decay products 208 Pb (hereinafter the same).

[0004] In-situ methods, without damaging the sample, utilize techniques such as lasers, ion excitation, and secondary ionization, interpolating standards to measure lead content based on a common lead content reference standard. 204 The number of Pb atoms was determined by isotopic evolution models and comparison with standard samples to obtain the single-point U and Pb content. 207 Pb / 206 Pb, 207 Pb / 235 U (corrected lead), 206 Pb / 238 U-value (corrected for ordinary lead) is used to fit and calculate isotopic age.

[0005] In U-Pb isotope dating, there are several basic definitions: ① Homogenization. The isotopic composition of lead was consistent throughout the early stages of Earth's formation, and was identical to that of troilite meteorites; this is called primordial lead. Subsequent changes in lead isotopic composition are the result of adding varying amounts of radiogenic lead to primordial lead; ② Ordinary lead (primordial lead). Primordial lead plus radiogenic lead from the separation of the parent and daughter radioactive materials, with no subsequent change in the number of lead atoms.

[0006] To address the dating of multi-stage open systems, a "stage-episode" model for U-Pb systems is defined: when the opening of a U-Pb system occurs only during a brief period of certain major geological events, this "brief" geological event is called an "episode," and the time interval between two episodes is called a "stage." During a "stage," uranium and lead remain in a closed system. Based on this model, a three-stage U-Pb model can be used:

[0007] 206 Pb / 204 Pb = 206 Pb / 204 Pb0+μ0(e λ8t0 -e λ8t1 )+μ1(e λ8t1 -e λ8t2 )+μ2(e λ8t2 -1)…(1)

[0008] 207 Pb / 204 Pb = 207 Pb / 204 Pb0+(μ0 / 137.88)×(e λ5t0 -e λ5t1 )+(μ1 / 137.88)×(e λ5t1 -e λ5t2 )+(μ2 / 137.88)×(e λ5t2 -1)…(2)

[0009] 206 Pb / 204 Pb0, 207 Pb / 204 Pb0 represents the time (t0) when the Earth was formed. 206 Pb / 204 Pb, 207 Pb / 204 Pb atomic ratio; μ0, μ1, and μ2 correspond to the evolution under closed conditions at t0, t1, and t2, respectively, to the present day. 238 U / 204 Pb atomic ratio, μ2 is the currently measured value; λ8 and λ5 are respectively... 238 U、 235 The decay constant of U; e λ8t0 e λ8t1 e λ8t2 These are from t0, t1, t2 to the present. 238 U decay factor; e λ5t0 e λ5t1 e λ5t2 These are from t0, t1, t2 to the present. 235U decay factors; μ0 / 137.88, μ1 / 137.88, and μ2 / 137.88 represent the decay factors from t0, t1, and t2, respectively, under closed conditions to the present. 235 U / 204 Pb value.

[0010] Because the variation in the isotopic composition of ordinary lead is constrained by the radioactive parent nuclide and age, ordinary lead must be subtracted when directly calculating apparent age and conformity line age using isotopic analysis data. Ordinary lead is usually obtained by testing a single mineral from which the radioactive parent and daughter nuclide separated at that moment, such as the lead isotopic composition of galena. However, even if such a single mineral is found, it is questionable whether it represents the isotopic system of the target being tested.

[0011] Ordinary lead can be identified using a single-stage model, but since most geological bodies in nature belong to the U-Pb multi-stage evolution system, this method obviously has significant problems.

[0012] Micro-area in-situ method utilizes standard materials to... 204 The method of obtaining age by fitting nonlinearity in Pb-corrected data does not fully conform to the isotope principle.

[0013] In addition, during in-situ micro-area testing, the scale of the measuring point is limited. Even if the correct mineral is selected, it cannot be guaranteed whether the target point contains lead from mixed minerals or minerals affected by multiple isotopic opening events. The lead isotope composition at this point may be of mixed origin, and it is obviously inappropriate to calculate the age based on this.

[0014] Currently, the main methods for U-Pb dating and the corresponding commonly used methods for subtracting common lead are:

[0015] (1) Isochronous diagrams are used to determine the legal year, avoiding deductions for ordinary lead. 206 Pb / 204 Pb- 238 U / 204 Pb (Y-axis - X-axis, hereinafter the same) uranium-lead isochron method, 207 Pb / 204 Pb- 206 Pb / 204 Pb lead-lead isochron method, these two methods are mainly applicable to TIMS test data; 207 Pb / 206 Pb- 238 U / 204 The Tera-Wasserburg inverse coincidence line for Pb is applicable to both TIMS and in-situ methods. The intersection of the isochronous and inverse coincidence line fitted lines with the Y-axis represents ordinary lead. 207 Pb / 206 Pb value.

[0016] All three methods directly fit and calculate the isotopic age using test values. The uranium-lead isochron method can also obtain the atomic ratios of common lead isotopes at the same time as obtaining the age. 206 Pb / 204 The conditions for using Pb are: ① the samples are formed simultaneously, have the same common lead composition, and achieve lead isotope homogenization within the geological body; ② the age calculated using the sample test results is the age at which the isotopic system last remained closed.

[0017] (2) The age calculation software uses an iterative method to avoid common lead and obtain age; TIMS method measurement data 206 Pb / 238 U— 207 Pb / 235 The U-congruence line method. This method works by using the uranium-lead isotope dating formula to obtain a congruence line (curve), with the Y-axis as... The X-axis is Because the U-Pb system is easily opened, radiogenic lead loss or uranium acquisition can occur, resulting in age inconsistencies. The least squares method is used to fit the inconsistency line, and the formula for the inconsistency line is: 206 Pb / 238 U=k×* 207 Pb / 235 U+b, where 206 Pb and 207 Pb is the radiogenic Pb that has been subtracted from ordinary lead. The intersection of the non-constitutive line and the congruent line represents the age. Using computer software, an iterative age method is employed: inputting a hypothetical age, calculating and returning an age, until the difference between two consecutive ages is less than a certain value, such as 0.5 Ma, which is the intersection age of the fitted line and the congruent line. Therefore, this method directly inputs U, Pb content, ... 204 Pb, 206 Pb, 207 Pb, 208 Pb abundance values ​​do not need to be adjusted for common lead.

[0018] The conditions for using this method are: ① The minerals were formed simultaneously and simultaneously subjected to short-term geological events (episodes), accompanied by the removal or substitution of lead or uranium; ② The minerals had the same common lead composition when they were formed, and common lead correction is required when fitting the calculation; ③ The age calculated from the sample test results is the age at which the isotopic system was last closed.

[0019] (3) Micro-area in-situ method 204 Pb correction; based on standard calibration combined with software calculations, 204 Pb correction directly subtracts ordinary lead, and the test results can be directly applied to fit the nonlinearity to obtain the intersection age.

[0020] (4) Single-stage pattern method and anti-consistency line method to obtain ordinary lead and age, etc.

[0021] However, each of the above methods has its advantages and disadvantages, and each has its own problems. In particular, for multi-stage open systems, they may not fully comply with the basic principles of U-Pb isotopes. Summary of the Invention

[0022] The purpose of this invention is to provide a method applicable to uranium-lead isotope dating and ordinary lead subtraction. This method is based on the U-Pb three-stage theory and uses mathematical methods to obtain the ordinary lead isotope composition and corresponding age at different stages. The results obtained are consistent with the isotope theory and can accurately measure the isotope dating of lead isotope composition. It has good reliability and measurement accuracy, and has the advantages of convenient data processing and universal applicability. It can effectively avoid the problems mentioned above, such as unreliability of ordinary lead, difficulty in dating multiple open systems, and ordinary lead subtraction.

[0023] Technical solution to achieve the purpose of this invention:

[0024] A method applicable to uranium-lead isotope dating and ordinary lead deduction includes:

[0025] Step 1: Based on the three-stage age formula of U-Pb isotopes, derive the formula for calculating the third-stage age t2, and use uranium-lead isochron fitting to calculate the third-stage age t2 and the ordinary lead values ​​α2 and β2;

[0026] Step 2: Based on the three-stage age formula of U-Pb isotopes, establish a regression equation and calculate the coefficients of the regression equation;

[0027] Step 3: Based on the formula for calculating the coefficients of the regression equation, establish the transcendental equation and use Newton's iteration method to calculate the second stage age t1 and the third stage age t2.

[0028] Step 4: Based on the second stage age t1, the third stage age t2, and the common lead value α2, calculate the common lead values ​​α1 and β1. 238 U / 204 Pb atomic ratio μ1;

[0029] Step 5: Calculate the age error, compare and verify the t2 age obtained in Step 1 and Step 3, and discuss the reliability of the results.

[0030] Furthermore, in step 1, the U-Pb three-stage age evolution formula is obtained based on the U-Pb three-stage age formula, and the U-Pb three-stage age evolution formula is as follows:

[0031] α1=α0+μ0(e λ8t0 -e λ8t1 (3)

[0032] α2=α1+μ1(e λ8t1 -e λ8t2 (4)

[0033] α3=α=α2+μ2(e λ8t2 -1)………………………………………(5)

[0034] β1=β0+μ0 / 137.88(e λ5t0 -e λ5t1 )………………………………(6)

[0035] β2=β1+μ1 / 137.88(e λ5t1 -e λ5t2 )………………………………(7)

[0036] β3=β=β2+μ2 / 137.88(e λ5t2 -1)………………………………(8)

[0037] In equations (3) to (5), α = α3, α0, α1, and α2 correspond to the current (measured value), t0, t1, and t2 times, respectively. 206 Pb / 204 Pb value, where t0 is the time of Earth's formation, t1 is the second stage age, and t2 is the third stage age; in equations (6) to (8), β = β3, β0, β1, and β2 correspond to the present, t0, t1, and t2 times respectively. 207 Pb / 204 Pb values; μ0, μ1, and μ2 represent the evolution under closed conditions at t0, t1, and t2, respectively, to the present day. 238 U / 204 Pb atomic ratio, μ2 is the currently measured value; λ8 and λ5 are respectively... 238 U、 235 The decay constant of U; 1 / 137.88 is 235 U has evolved to the present day and 238 The ratio of the number of atoms of U is a constant of 1 / 137.88.

[0038] Further, in step 1, the calculation of the third-stage age t2 and the ordinary lead value α2 using uranium-lead isochronous fitting includes: constructing a scatter plot dataset of uranium-lead isochronous points with α as the Y-axis and μ2 as the X-axis; performing weighted least squares linear regression on the scatter plot dataset to fit a straight line equation and obtain the slope b, intercept a, and linear correlation coefficient R; when the absolute value of the linear correlation coefficient R is greater than or equal to 0.99, the slope b = (e λ8t2-1) Calculate the age t2, and the intersection of the fitted line with the Y-axis is the intercept a, which is the common lead isotope ratio α2 at t2;

[0039] The calculation of the common lead value β2 using uranium-lead isochron fitting includes: constructing a scatter plot dataset of uranium-lead isochrons with β as the Y-axis and μ2 / 137.88 as the X-axis; performing weighted least squares linear regression on the isochron dataset to fit a straight line equation and obtain the intercept a and the linear correlation coefficient R; when the absolute value of the linear correlation coefficient R is greater than or equal to 0.99, the intersection of the fitted line and the Y-axis, i.e., the intercept a, is the common lead isotope ratio β2 at t2.

[0040] Furthermore, in step 2, the established regression equation and the formula for the regression equation coefficients are as follows:

[0041] β=b1×α+b2×μ2+a…………………………………………(I)

[0042]

[0043]

[0044] In the formula, α and β are respectively 206 Pb / 204 Pb, 207 Pb / 204 The test value of Pb, b1 and b2 are the coefficients of the regression equation, and a is the constant term.

[0045] Further, step 2 specifically involves: using SPSS software, inputting the measured U-Pb isotope data, establishing a regression equation with α and μ2 as independent variables and β as the dependent variable, and calculating the coefficients b1 and b2.

[0046] Furthermore, in step 3, the established transcendental equation is:

[0047] 1+137.88×(b2-b1)+137.88×b1×e λ8t2 =e λ5t2 …..……………………………….…(V).

[0048] Further, step 4 specifically involves: calculating the values ​​of α1 and μ1 according to equations (3) and (4) based on the second-stage age t1, the third-stage age t2, and the common lead value α2, where the α1 value should represent the t0~t1 stage and is obtained by superimposing the radiogenic lead value of α0 on this stage; calculating β1 according to equation (6), where the β1 value should represent the t0~t1 stage and is obtained by superimposing the radiogenic lead value of β0 on this stage; according to ( 238 U / 204 Pb) / ( 235 U / 204 Pb) The current ratio is 137.88, and the calculated 235 U / 204 Pb value.

[0049] Furthermore, in step 5, the method for calculating the age error is as follows:

[0050] ① When the original measurement data gives experimental errors, these errors are substituted into the age equation to directly calculate the age error;

[0051] ② The least squares method is used to calculate the standard deviation and mean square error of the fitted straight line and the actual scatter data, and then substituted into the age formula to calculate the age error;

[0052] ③ When fitting the age through a transcendental equation using the Newton iteration method, according to the regression equation regression calculation, the standard deviation and mean square error are obtained, and then substituted into the age iteration software to obtain the age error result.

[0053] Furthermore, in step 5, the t2 ages calculated in steps 1 and 3 are compared and verified. The specific steps include:

[0054] The t2 age obtained by combining the binary linear regression and the iteration method in step 3 is compared and verified with the t2 age calculated by fitting the isochron in step 1; when the ages are consistent within the error range, they are credible values; when the obtained ages are inconsistent, the age results are discussed according to the specific situation; when the ages obtained by these two methods are mathematically consistent, it is also necessary to compare with the geological conditions and the existing ages to ensure the reliability of the ages.

[0055] Furthermore, in step 5, according to the calculated age results, the reliability of the results is discussed as follows:

[0056] ① If t1 > t2, and the results calculated by the isochron and the binary linear regression combined with iteration are consistent within the error range, the age is reliable and belongs to a three-stage or more than three-stage evolution system;

[0057] ② If t1 < t2, the age is meaningless, or it is a U-Pb two-stage evolution system. Taking t1 as the second-stage age, at this time, the t1 calculated by the isochron method should be consistent with the t1 calculated by the iteration method;

[0058] ③ If t1 = t2, it is a U-Pb two-stage evolution system, and the t1 calculated by the isochron method should be consistent with the t1 calculated by the iteration method.

[0059] The beneficial technical effects of the present invention are as follows:

[0060] 1. This invention calculates the age and corresponding common lead isotope composition based on the U-Pb isotope principle. The results are accurate and reliable, and are characterized by standardization, proceduralization, and ease of operation. This method has been well applied in the calculation of U-Pb isotope ages of granites and metamorphic rocks, and satisfactory results have been obtained by comparing with geological facts and other isotope ages.

[0061] 2. This invention can be extended to multi-stage systems with more than three stages when calculating U-Pb isotope ages and ordinary lead using the TIMS method.

[0062] 3. When applying the micro-area in-situ method, this invention requires accurate measurement. 204 Only after determining the abundance of Pb can it be utilized. However, single-point measurements of U and Pb content can still be used. 207 Pb / 206 Pb, 207 Pb / 235 U、 206 Pb / 238 U-values ​​were fitted with Pb-Pb isochronous lines to obtain 207 Pb / 206 Initial Pb value (ordinary lead) and corresponding age. Attached Figure Description

[0063] Figure 1 The diagram shows the Newton-based iterative solution of the transcendental equation in an embodiment of the present invention. Detailed Implementation

[0064] The present invention will now be described in further detail with reference to the accompanying drawings and embodiments.

[0065] This invention mainly relates to the calculation of U and Pb contents and lead isotope composition data by whole-rock, single-mineral dissolution sampling and micro-area in-situ methods. It is applicable to geochronological calculations of U-Pb isotope “episode-stage” open and closed systems, and has the characteristics of conforming to the basic principles of isotopes, reliable results and convenient use.

[0066] This invention addresses the challenges of multi-stage U-Pb evolution systems. Based on a three-stage evolution model of uranium-lead isotopes, it establishes formulas for ordinary lead and age calculations. Employing methods such as binary regression linear solutions and Newton's iterative solution of transcendental equations, it obtains U-Pb isotope ages and corresponding ordinary lead calculations for complex systems. The calculation results are verified using other methods and geological facts. Ultimately, a method for ordinary lead subtraction and age calculation is established for complex U-Pb isotope systems with multiple open periods, providing an accurate, reliable, and effective approach for isotope age calculations. This method is applicable to U-Pb isotope dating and ordinary lead calculations with complete measurement data and can be extended to complex isotope systems with more than three stages. The calculation process and results conform to basic isotope theory, and the invention's results possess universal applicability and comparability.

[0067] This invention provides a method for uranium-lead isotope dating and ordinary lead deduction, specifically including the following steps:

[0068] Step 1: Based on the three-stage age formula of U-Pb isotopes (Formula 1), derive the formula for calculating the third-stage age t2, and use uranium-lead isochron fitting to calculate the third-stage age t2 and the common lead value α2. 206 Pb / 204 Pb), β2( 207 Pb / 204 Pb):

[0069] The three-stage formula for U-Pb isotopes is as follows:

[0070] 206 Pb / 204 Pb = 206 Pb / 204 Pb0+μ0(e λ8t0 -e λ8t1 )+μ1(e λ8t1 -e λ8t2 )+μ2(e λ8t2 -1)…(1)

[0071] 207 Pb / 204 Pb = 207 Pb / 204 Pb0+(μ0 / 137.88)×(e λ5t0 -e λ5t1 )+(μ1 / 137.88)×(e λ5t1 -e λ5t2 )+(μ2 / 137.88)×(e λ5t2 -1)…(2)

[0072] 206 Pb / 204 Pb0, 207 Pb / 204 Pb0 represents the time (t0) when the Earth was formed. 206 Pb / 204 Pb, 207 Pb / 204 Pb atomic ratio; μ0, μ1, and μ2 correspond to the evolution under closed conditions at t0, t1, and t2, respectively, to the present day. 238 U / 204 Pb atomic ratio, μ2 is the currently measured value; λ8 and λ5 are respectively... 238 U、 235 The decay constant of U; 1 / 137.88 is 235 U has evolved to the present day and 238The ratio of the number of atoms of U is a constant of 1 / 137.88.

[0073] Based on the U-Pb three-stage age formula (Formula 1), the U-Pb three-stage age evolution formula is obtained as follows:

[0074] α1=α0+μ0(e λ8t0 -e λ8t1 (3)

[0075] α2=α1+μ1(e λ8t1 -e λ8t2 (4)

[0076] α3=α=α2+μ2(e λ8t2 -1)………………………………………..(5)

[0077] β1=β0+μ0 / 137.88(e λ5t0 -e λ5t1 (6)

[0078] β2=β1+μ1 / 137.88(e λ5t1 -e λ5t2 )………………………….…….(7)

[0079] β3=β=β2+μ2 / 137.88(e λ5t2 -1)……………………………….(8)

[0080] In equations (3) to (5), α = α3, α0, α1, and α2 correspond to the current (measured value), t0, t1, and t2 times, respectively. 206 Pb / 204 Pb value, where t0 is the time of Earth's formation, t1 is the second stage age, and t2 is the third stage age; in equations (6) to (8), β = β3, β0, β1, and β2 correspond to the current (currently measured value), t0, t1, and t2 times, respectively. 207 Pb / 204 Pb value; other symbols have the same meaning as in equations (1) and (2).

[0081] The third-stage age t2 and the common lead value α2 were calculated using uranium-lead isochron fitting, including:

[0082] For equation (5), the straight line is directly fitted using the correlation function method in the Excel table. α and μ2 are both measured values. A scatter plot dataset of uranium-lead isochrones is constructed with α as the Y-axis and μ2 as the X-axis. Weighted least squares linear regression is performed on the isochrone scatter plot dataset to fit the straight line equation and obtain the slope b, intercept a, and linear correlation coefficient R. When the absolute value of the linear correlation coefficient R is greater than or equal to 0.99, the slope b = (e λ8t2 -1) Calculate the age t2, and the intersection of the fitted line with the Y-axis is the intercept a, which is the common lead isotope ratio α2 at t2.

[0083] The calculation of the common lead value β2 using uranium-lead isochronous fitting includes: directly fitting the straight line of equation (8) using the correlation function method in the Excel table, where β and μ2 are both measured values, and constructing a scatter dataset of uranium-lead isochronous points with β as the Y-axis and μ2 / 137.88 as the X-axis; performing weighted least squares linear regression on the scatter dataset of isochronous points to fit the straight line equation, and obtaining the slope b, intercept a, and linear correlation coefficient R; when the absolute value of the linear correlation coefficient R is greater than or equal to 0.99, the slope b = (e λ5t2 -1) Calculate the age t2, and the intersection of the fitted line with the Y-axis is the intercept a, which is the common lead isotope ratio β2 at t2.

[0084] Step 2: Based on the three-stage age formula of U-Pb isotopes (Formula 1 and Formula 2), establish a regression equation and calculate the coefficients of the regression equation.

[0085] The established regression equation and its coefficients are as follows:

[0086] β=b1×α+b2×μ2+a…………………………………………(I)

[0087]

[0088] In equations I-IV, the symbols have the same meaning as in the previous formulas, where μ2 is t2. 238 U / 204 The ratios of Pb evolution to the present day, α and β are respectively 206 Pb / 204 Pb, 207 Pb / 204 The test value of Pb, b1 and b2 are the coefficients of the regression equation, and a is the constant term.

[0089] In one specific implementation, SPSS software is used to input the measured U-Pb isotope data of a sample, with α(X1) and μ2(X2) as independent variables and β(Y) as dependent variable, and a regression equation is established using Equation I to calculate the values ​​of coefficients b1 and b2.

[0090] Step 3: Based on the regression equation coefficient calculation formula (III), establish the transcendental equation and use Newton's iteration method to calculate the second stage age t1 and the third stage age t2.

[0091] The established transcendental equation is:

[0092] 1+137.88×(b2-b1)+137.88×b1×e λ8t2 =e λ5t2 ……..………………………….….(V)

[0093] The symbols in equation V have the same meaning as in the previous formula. Since equation V is a transcendental equation, conventional methods cannot yield an algebraic solution. Therefore, Mathematica 5 software is used, and Newton's iteration method is employed to obtain analytical solutions for t1 and t2. (When performing Newton's iteration analysis on a transcendental equation, three cases can be obtained: two solutions, one solution, and no solution. The case with two solutions is shown in Example 2.) Figure 1 (Obtain analytical solutions for t1 and t2) and discuss the age results. If t1 and t2 are greater than the Earth's age or are negative, it indicates that the samples they represent do not belong to the same U-Pb system, are not on the same isochronous surface during fitting, and the results are incorrect. The original data needs to be re-analyzed and screened.

[0094] Step 4: Based on the second stage age t1, the third stage age t2, and the common lead value α2, calculate the common lead values ​​α1 and β1. 238 U / 204 Pb atomic ratio μ1.

[0095] Compare the t2 values ​​calculated in steps 1 and 3. If they are equal, the ages are consistent. If they are not equal, compare the single-stage ordinary lead values ​​calculated according to t2. The value that matches is used for subsequent calculations.

[0096] Based on the t1, t2, and α2 values ​​obtained in step 1 or step 3, calculate the α1 and μ1 values ​​according to equations (3) and (4), where the α1 value should represent the t0 to t1 stage and is obtained by superimposing the radiogenic lead of this stage on α0; calculate β1 according to equation (6), where the β1 value should represent the t0 to t1 stage and is obtained by superimposing the radiogenic lead of this stage on β0; according to ( 238 U / 204 Pb) / ( 235 U / 204 The current ratio of Pb is 137.88, which can be calculated. 235 U / 204 Pb value.

[0097] In this step, the α0 and β0 values ​​correspond to the period from Earth's formation to the attainment of a homogeneous uranium-lead isotope reservoir. 206 Pb / 204 Pb,207 Pb / 204 The original Pb values ​​are typically taken as 4470 Ma, 4550 Ma, 4570 Ma, etc., and the corresponding α0, β0, and μ0 values ​​can be found in the literature.

[0098] Step 5: Calculate the age error, compare and verify the t2 ages obtained in Step 1 and Step 3, and discuss the reliability of the results.

[0099] The methods used to calculate age error include:

[0100] ① When the original measurement data gives experimental error, these errors are substituted into the age equation to directly calculate the age error;

[0101] ② Using the least squares method, calculate the standard deviation and root mean square of the fitted straight line (isochronous line and non-contrary line) and the actual scattered data, and then use the age formula to calculate the age error.

[0102] ③ When fitting the age using the transcendental equation (Formula V) and the Newton-Raphson iteration method, the standard deviation and root mean square error are obtained by regression calculation based on the regression equation (Formula I). ​​These are then substituted into the age iteration software to obtain the age error result.

[0103] In one specific implementation, the steps for calculating the age error using the least squares method include:

[0104] We performed a weighted least squares linear regression on the isochronous scatter dataset to fit the linear equation and obtain the slope b, slope mean square error Sb, intercept a, and intercept mean square error Sa.

[0105] Based on the slope b and its root mean square error Sb, the upper error limit ΔT1 and lower error limit ΔT2 of age t2 are calculated using the error propagation formula at twice the root mean square error.

[0106] The error propagation formula is:

[0107] △T1=1 / λ8ln(b+2Sb+1)-t2……………………………………(9)

[0108] △T2=1 / λ8ln(b-2Sb+1)-t2…………………………………….(10)

[0109] In the formula, 2Sb represents the uncertainty range of the slope b at the 2σ confidence level. Typically, ΔT1 is positive and ΔT2 is negative.

[0110] Based on the upper error limit ΔT1 and the lower error limit ΔT2, the final age value T is obtained:

[0111] T represents the final age value with a mean squared error of 2.

[0112] Compare and verify the t2 ages obtained from Step 1 and Step 3. The specific steps are as follows:

[0113] Compare and verify the t2 age calculated according to the isochron (straight line) fitting in Step 1 (Equation (5)) with the t2 age obtained by the binary linear regression combined with the iterative method in Step 3 (Formula (V));

[0114] Compare and verify the t2 age obtained by the binary linear regression combined with the iterative method in Step 3 with the t2 age calculated according to the isochron (straight line) fitting in Step 1. When the ages are consistent within the error range, they are credible values; when the obtained ages are inconsistent, discuss the age results according to the specific situation. When the ages obtained by these two methods are mathematically consistent, it is also necessary to compare with the geological conditions and existing ages to ensure the reliability of the ages.

[0115] Discuss the reliability of the results according to the calculated age results, as follows:

[0116] ① If t1 > t2 and the calculation results according to the isochron (Equation (5)) and the binary linear regression combined with iteration (Equation (V)) are consistent within the error range, the age is reliable and belongs to a three-stage or more than three-stage evolution system;

[0117] ② If t1 < t2, the age is meaningless or it is a U-Pb two-stage evolution system. Take t1 as the age of the second stage. At this time, the t1 calculated by the isochron method (Equations (4) and (5)) should be consistent with the t1 calculated by the iterative method;

[0118] ③ If t1 = t2, it is a U-Pb two-stage evolution system. The t1 calculated by the isochron method (Equations (4) and (5)) should be consistent with the t1 calculated by the iterative method.

[0119] Example 1

[0120] Take the uranium pre-enrichment and enrichment in the black rock series in Jishou area, Hunan as an example for age calculation, as follows:

[0121] Using Equation (5), the calculated 206 Pb / 204 Pb- 238 U / 204 Pb isochron age t2 is 98.7 Ma (α2 = 206 Pb / 204 Pb1 = 17.96, β2 = 207 Pb / 204 Pb1 = 15.82), which is consistent with the single-stage initial Pb isotope value of ~100 Ma.

[0122] A three-stage evolution model of U-Pb was set up. The second stage age t1 was 1207.1 Ma and t2 was 99.76 Ma, calculated by binary regression and age iteration. The common lead isotope ratio corresponding to t1 was consistent with that of the single-stage common lead Pb at ~1200 Ma.

[0123] The results show that the common lead isotope ratios calculated based on the fitting results are consistent with the single-stage common lead isotope ratios calculated for the second and third stages (t1, t2) in the U-Pb three-stage system. Furthermore, the age calculated using the method of this invention is consistent with the age of regional tectonic evolution events.

[0124] Example 2

[0125] The SPSS binary linear regression equation and Mathematica 5 age iteration calculation are described in detail below:

[0126] Step 1: Data input, as shown in Table 1.

[0127] Table 1 SPSS Data Input Table

[0128]

[0129] Step 2: SPSS Statistical Analysis

[0130] Statistics calculates regression scores based on the input X1, X2, and Y.

[0131] Step 3: Output the results and interpret them.

[0132] The output results are shown in Table 2 (N=5).

[0133] Table 2. Coefficients

[0134]

[0135] Dependent Variable:Y

[0136] In this embodiment, X1 and X2 are used as independent variables, and Y is the dependent variable. A regression equation is established using the full input selection method. In Table 2, Unstandardized Coefficients represent the unstandardized regression coefficients, and t is the t-value for the hypothesis test when the partial regression coefficient is 0. Sig represents the significance level (probability) of the hypothesis test when the partial regression coefficient is 0, which is the probability that it is greater than the mean squared error F-value (F = regression mean square / residual mean square). In the analysis of variance, when the regression equation contains different independent variables, the hypothesis that the regression coefficient is 0 is rejected if the significance probability values ​​are all less than 0.001. In this example, Beta is the standardized regression coefficient. A regression equation is considered meaningful when at least one of the partial regression coefficients Beta is significant (e.g., the value of 0.998 in this embodiment).

[0137] The regression equation is Y = 0.116X1 - 0.001X2 + 14.275, that is, b1 = 0.116; b2 = -0.001; a = 14.275.

[0138] Step 4: Substitute the values ​​into Mathematica 5 software to iteratively calculate the t1 and t2 ages in the U-Pb isotopic three-stage evolution system according to the exponential equation (V-form). Then, substitute the results back into the three-stage formula to obtain the values ​​of α, β, and μ.

[0139] Graphs of Newton's iterative solutions to transcendental equations, such as Figure 1 As shown. Figure 1 The intersection of the straight line and the curve is the value of X(t). The graph shows two solutions, corresponding to ages t1 and t2 (0.9976 × 10⁻⁶ respectively). 8 and 1.2071×10 9 ).

[0140] The present invention has been described in detail above with reference to the accompanying drawings and embodiments. However, the present invention is not limited to the above embodiments, and various changes can be made within the scope of knowledge possessed by those skilled in the art without departing from the spirit of the present invention. All contents not described in detail in the present invention can be derived from existing technologies.

Claims

1. A method applicable to uranium-lead isotope dating and ordinary lead deduction, characterized in that, include: Step 1: Based on the three-stage age formula of U-Pb isotopes, derive the formula for calculating the third-stage age t2, and use uranium-lead isochron fitting to calculate the third-stage age t2 and the ordinary lead values ​​α2 and β2; Step 2: Based on the three-stage age formula of U-Pb isotopes, establish a regression equation and calculate the coefficients of the regression equation; Step 3: Based on the formula for calculating the coefficients of the regression equation, establish the transcendental equation and use Newton's iteration method to calculate the second stage age t1 and the third stage age t2. Step 4: Based on the second stage age t1, the third stage age t2, and the common lead value α2, calculate the common lead values ​​α1 and β1. 238 U / 204 Pb atomic ratio μ1; Step 5: Calculate the age error, compare and verify the t2 age obtained in Step 1 and Step 3, and discuss the reliability of the results.

2. The method for uranium-lead isotope dating and ordinary lead deduction according to claim 1, characterized in that, In step 1, the U-Pb three-stage age evolution formula is obtained based on the U-Pb three-stage age formula. The U-Pb three-stage age evolution formula is as follows: α1=α0+μ0(e λ8t0 -e λ8t1 )……………………………….…………….(3) α2=α1+μ1(e λ8t1 -e λ8t2 )……………………………….…………….(4) α3=α=α2+μ2(e λ8t2 -1)………………………………………..(5) β1=β0+μ0 / 137.88(e λ5t0 -e λ5t1 )……………………………….…….(6) β2=β1+μ1 / 137.88(e λ5t1 -e λ5t2 )……………………………….…….(7) β3=β=β2+μ2 / 137.88(e λ5t2 -1)……………………………….(8) In equations (3) to (5), α = α3, α0, α1, and α2 correspond to the present, t0, t1, and t2 times, respectively. 206 Pb / 204 Pb value, where t0 is the time of Earth's formation, t1 is the second stage age, and t2 is the third stage age; in equations (6) to (8), β = β3, β0, β1, and β2 correspond to the present, t0, t1, and t2 times respectively. 207 Pb / 204 Pb values; μ0, μ1, and μ2 represent the evolution under closed conditions at t0, t1, and t2, respectively, to the present day. 238 U / 204 Pb atomic ratio, μ2 is the currently measured value; λ8 and λ5 are respectively... 238 U、 235 The decay constant of U; 1 / 137.88 is 235 U has evolved to the present day and 238 The ratio of the number of atoms of U is a constant of 1 / 137.

88.

3. The method for uranium-lead isotope dating and ordinary lead deduction according to claim 2, characterized in that, In step 1, the calculation of the third-stage age t2 and the ordinary lead value α2 using uranium-lead isochronous fitting includes: constructing a scatter plot dataset of uranium-lead isochronous points with α as the Y-axis and μ2 as the X-axis; performing weighted least squares linear regression on the scatter plot dataset to fit the linear equation and obtain the slope b, intercept a, and linear correlation coefficient R; when the absolute value of the linear correlation coefficient R is greater than or equal to 0.99, the slope b = (e λ8t2 -1) Calculate the age t2, and the intersection of the fitted line with the Y-axis is the intercept a, which is the common lead isotope ratio α2 at t2; The calculation of the common lead value β2 using uranium-lead isochron fitting includes: constructing a scatter plot dataset of uranium-lead isochrons with β as the Y-axis and μ2 / 137.88 as the X-axis; performing weighted least squares linear regression on the isochron dataset to fit a straight line equation and obtain the intercept a and the linear correlation coefficient R; when the absolute value of the linear correlation coefficient R is greater than or equal to 0.99, the intersection of the fitted line and the Y-axis, i.e., the intercept a, is the common lead isotope ratio β2 at t2.

4. The method for uranium-lead isotope dating and ordinary lead deduction according to claim 2, characterized in that, In step 2, the established regression equation and the formula for the regression equation coefficients are as follows: β=b1×α+b2×μ2+a…………………………………………(I) In the formula, α and β are respectively 206 Pb / 204 Pb, 207 Pb / 204 The test value of Pb, b1 and b2 are the coefficients of the regression equation, and a is the constant term.

5. A method for uranium-lead isotope dating and ordinary lead deduction according to claim 4, characterized in that, Step 2 specifically involves using SPSS software, inputting the measured U-Pb isotope data, establishing a regression equation with α and μ2 as independent variables and β as the dependent variable, and calculating the coefficients b1 and b2.

6. A method for uranium-lead isotope dating and ordinary lead deduction according to claim 4, characterized in that, In step 3, the transcendental equation established is: 1+137.88×(b2-b1)+137.88×b1×e λ8t2 =e λ5t2 …..………………………….….(V)。 7. A method for uranium-lead isotope dating and ordinary lead deduction according to claim 6, characterized in that, Step 4 specifically involves: calculating the values ​​of α1 and μ1 based on the second stage age t1, the third stage age t2, and the ordinary lead value α2, according to formulas (3) and (4). The value of α1 should represent the t0 to t1 stage and is obtained by superimposing the radiogenic lead of this stage on α0. β1 is calculated according to equation (6), where the value of β1 should represent the t0~t1 stage, obtained by superimposing β0 with the radiogenic lead of this stage; according to ( 238 U / 204 Pb) / ( 235 U / 204 The current ratio of Pb is 137.88, calculated as follows: 235 U / 204 Pb value.

8. A method for uranium-lead isotope dating and ordinary lead deduction according to claim 6, characterized in that, In step 5, the method used to calculate the age error is as follows: ① When the original measurement data gives experimental error, these errors are substituted into the age equation to directly calculate the age error; ② The least squares method is used to calculate the standard deviation and root mean square error between the fitted straight line and the actual scattered data, and then the age error is calculated by substituting them into the age formula; ③ When fitting the age using the transcendental equation and Newton's iteration method, the standard deviation and root mean square error are obtained by regression calculation based on the regression equation. These are then substituted into the age iteration software to obtain the age error result.

9. A method for uranium-lead isotope dating and ordinary lead deduction according to claim 6, characterized in that, In step 5, the t2 ages calculated in steps 1 and 3 are compared and verified. The specific steps include: The t2 age obtained by the binary linear regression combined with the iterative method in step 3 is compared and verified with the t2 age calculated by fitting the isochrones in step 1. When the ages are consistent within the error range, they are considered reliable values. When the obtained ages are inconsistent, the age results are discussed according to the specific circumstances. When the ages obtained by the two methods are mathematically consistent, it is still necessary to compare them with geological conditions and existing ages to ensure the reliability of the ages.

10. A method for uranium-lead isotope dating and ordinary lead deduction according to claim 6, characterized in that, In step 5, according to the calculated age results, the reliability of the results is discussed as follows: ① If t1 > t2 and the results calculated by the combination of isochron and binary linear regression through iteration are consistent within the error range, the age is reliable and belongs to a three-stage or more than three-stage evolution system; ② If t1 < t2, the age is meaningless or it is a U-Pb two-stage evolution system. Taking t1 as the age of the second stage, at this time, the t1 calculated by the isochron method should be consistent with the t1 calculated by the iteration method; ③ If t1 = t2, it is a U-Pb two-stage evolution system, and the t1 calculated by the isochron method should be consistent with the t1 calculated by the iteration method.

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