P a g e | 1 U‐Pb Data Reduction Equations version 27 A. Inputs Measured Parameters r206_204m r207_204m r208_204m r207_206m r206_208m r204_205m r206_205m r207_205m r208_205m r202_205m r238_235m r233_235m r238_233m r233_236m Tracer Solution r206_204t r207_204t r208_204t r207_206t r206_208t r204_205t r206_205t r207_205t r208_205t r202_205t r238_235t r233_235t r238_233t r233_236t r235_205t concPb205t concU235t Physical Constants lamda230 lamda231 lamda232 lamda235 lamda238 gmol204 gmol205 gmol206 gmol207 gmol208 gmol235 gmol238 Lab Parameters Other Fraction‐ specific Input r206_204b r238_235s r207_204b tracerMass r208_204b zirconCase rho206204b_207204b r206_204c rho206204b_208204b r207_204c rho207204b_208204b r208_204c r238_235b r206_207c alphaPb rTh_Umagma alphaU commonPbAge labPbBlankMass ar231_235sample labUBlankMass P a g e | 2 B. Intermediate Parameters 1. Pb 1a. tracer alphaPb molPb205t 1b. Pb blank r204_205fc molPb204b molPb206b molPb207b molPb208b molPb204c molPb206c molPb207c molPb208c blankPbGramsMol blankPbMass 1c. radiogenic Pb, sample Pb molPb206r molPb207r molPb208r molPb206s molPb207s molPb208s 2. U 2a. tracer molU235b molU238b molU235t molU238t molU233t alphaU 2b. sample U molU235s molU238s 3. Ages 3a. radiogenic isotope ratios r206_238r r207_235r r207_206r 3b. radiogenic isotope ages age206_238r age207_235r age207_206r 3c. Th and Pa correction molsU age206_238_Th molTh232s delta206Pb_Th r207_206r_Th age207_206_Th age207_235_Pa molPa231s delta207Pb_Pa r207_206r_ThPa age207_206_ThPa 4. Isochron ratios r207_206s r204_206s r204_207s r238_206s r238_207s r235_207s r238_204s r235_204s 5. Outputs 5a. Pb calculations blankPbMass InitCommonPbMass totCommonPbMass radToCommonTotal radToCommon206 radToCommon207 radToCommon208 percentDiscordance concU concPb r208_206r_Th rTh_Usample alphaU P a g e | 3 1. Pb 1a. tracer If tracerType is ‘mixed 202‐205‐233‐235’, or ‘mixed 202‐205‐233‐236’ If fcPb is false 1
3
1
202_205
202_205
If fcPb is true 0 End If for mixed 202Pb‐205Pb tracer, calculate linear coefficient of fractionation correction based on the mean 202
Pb/205Pb ratio; if the fractionation correction occurred externally (e.g. in Tripoli), don’t perform it again in Redux; if a double‐spike is not used, alphaPb for a mono‐isotopic tracer is taken from LabData. 205
205 ·
moles of 205Pb in the tracer 1b. Pb blank 204_205
204_205
· 1
measured, fractionation‐corrected 204Pb/205Pb If user selected zircon case: 204
206
207
208
1 ·
206_204 ·
207_204 ·
208_204 ·
205
205
205
205
·
·
·
·
204_205
204_205
204_205
204_205
204_205
204_205
204_205
204_205
moles of 204Pb, 206Pb, 207Pb, and 208Pb in laboratory blank total lab Pb blank mass, grams; total common Pb mass, grams; initial common Pb mass, grams P a g e | 4 204
206
207
208
0 0 0 0 moles of initial common 204Pb, 206Pb, 207Pb, and 208Pb Else (If user did not select zircon case): If user selected Stacey‐Kramers common Pb model: if commonPbAge > 3700 206_204
207_204
208_204
207_206
9.307 7.19 e
7.19
10.294
e
137.88
29.487 33.21 e
207_204 / 206_204 .
e
P A
.
e
P A
.
e
P A
first stage of Stacey‐Kramers model, for samples older than 3700 Ma else 206_204
207_204
208_204
207_206
11.152 9.74 e
9.74
12.998 e
137.88
31.23 36.84 e
207_204 / 206_204 .
e
P A
.
e
P A
.
e
P A
second stage of Stacey‐Kramers model, for samples younger than 3700 Ma Else (if user did not select Stacey‐Kramers common Pb model) use user‐input common Pb ratios End If P a g e | 5 1·
208_204 ·
204
206_204 ·
206
207_204 ·
207 208 grams of laboratory blank Pb per 1 mol 204Pb 204
205 ·
204_205
204_205 total moles of 204Pb in lab blank and initial common Pb if molPb204tc > labPbBlankMass/blankPbGramsMol if total common Pb mass > laboratory Pb blank mass, partition common Pb between initial common Pb and laboratory blank 204
206
207
208
1 ·
206_204 ·
207_204 ·
208_204 ·
/ / / / moles 204Pb, 206Pb, 207Pb, and 208Pb in (average) laboratory blank 204
206
207
208
204
206_204 ·
207_204 ·
208_204 ·
204 204 204 204 moles of initial common 204Pb, 206Pb, 207Pb, and 208Pb total laboratory blank mass is maximum allowable laboratory blank (set by user) P a g e | 6 204 · 1 ·
204
208_204 ·
208 206_204 ·
206
207_204 ·
initial common Pb mass, grams; total common Pb mass, grams else Use the math (data reduction and error propagation) for zircon case = true. end if (inequality) End If (zircon case) 207
P a g e | 7 1c. radiogenic Pb, sample Pb 206
205 ·
206_205
· 1
206_205
206
206 radiogenic moles 206Pb in sample = 206Pb(total – tracer) – 206Pb(blank) – 206Pb (initial common) 207
205 ·
207_205
· 1
2·
207_205
207
207 radiogenic moles 207Pb in sample = 207Pb(total – tracer) – 207Pb(blank) – 207Pb (initial common) 208
205 ·
208_205
· 1
3
208_205
208
radiogenic moles 208Pb in sample = 206Pb(total – tracer) – 206Pb(blank) – 206Pb (initial common) 206
205 ·
206_205
· 1
206_205
206 moles 206Pb in sample = 206Pb(total – tracer) – 206Pb(blank) 207
205 ·
207_205
· 1
2·
207_205
207 208_205
208 moles 207Pb in sample = 207Pb(total – tracer) – 207Pb(blank) 208
205 ·
208_205
· 1
3
moles 208Pb in sample = 206Pb(total – tracer) – 206Pb(blank) 208 P a g e | 8 2. U 2a. U blank, tracer 235
·
235
1
238_235 ·
238
moles of 235U in laboratory blank 238
238_235
235 moles of 258U in laboratory blank 235
238
235 ·
238_235
235 moles of 235U, 238U in the tracer 2b. Sample U If (tracerType = ‘mixed 205‐233‐235’ or ‘mixed 202‐205‐233‐235’ or ‘mixed 205‐233‐235‐230Th’) and ~fcU 233
233_235
moles of 233U in the tracer 235 P a g e | 9 270_267
1
265_267
1
2
238_235
18 _16
233_235
2
233_235
18 _16
233_235
2
270_267
18 _16
265_267
2
265_267
18 _16
265_267
238_235
1
233_235
1
equations for oxide correcting and de‐oxide correcting for use in propagating error in r18O_16O. 3
235
238_235
/ 233_235
233
5
5
238_235
238_235
moles 235U in sample = complicated (see explanatory text) 238
235
238_235 moles 235U in sample 1
2
233
2
233_235
235
alphaU for double‐spikes with 233U and 235U. End If Else 235
235
2
235
238_235
235
2
238
238
P a g e | 10 If tracerType = ‘mixed 205‐235’ or ‘mixed 208‐235’ and ~fcU use alphaU for mono‐isotopic tracer, which is input in LabData End If If tracerType = ‘mixed 205‐233‐236’ or ‘mixed 202‐205‐233‐236’ and ~fcU 1
3
1
233_236
233_236
alphaU for mixed 233U‐236U tracer run as metal End If If fcU 0 U ratios are already fractionation‐corrected End If 235
238
238
1 3
238_235
1
3
moles 235U in sample = complicated (see explanatory text) 238
235
moles 235U in sample End If
238_235 238_235
235
238_235
235
P a g e | 11 3. Ages 3a. radiogenic isotope ratios 207_206
206_238
207_235
208_206
207
206
207
208
/
/
/
/
206 238 235 206 207
Pb/206Pb, 206Pb/238U, and 207Pb/235U ratios of sample after blank, tracer, and initial common Pb subtraction 3b. radiogenic isotope ages 1
206_238
238
· log 206_238
1 206
Pb/238U age, in years. log() is the natural logarithm. 1
207_235
235
· log 207_235
1 207
Pb/238U age, in years; log is the natural logarithm Newton‐Raphson Method for iterative solution of 207Pb/206Pb age: 206_238 for i = 1 to 10 ·
10
207_206
238_235 ·
1
·
1
11 238_235 ·
·
10/
next i 207_206
11 1
·
·
235 ·
238_235 ·
·
·
1
1 ·
238_235 ·
238 ·
·
P a g e | 12 3c. Th and Pa correction 238
235 total moles of U in sample Newton‐Raphson Method for iterative solution of Th‐corrected 206Pb/238U age: 206_238 for i = 1 to 100 10
11
206_238
238
230
1
232
1
10/
208
208
_
1
238
11 next i 206_238 _
232
208 /
_
_
1 moles of 232Th in sample, back‐calculated using 208Pb measurement and Th‐corrected 206Pb/238U age 206 _
238
_
moles of 206Pb corrected for initial 232Th disequilibrium _
1 1 _
1
238
230
1
P a g e | 13 206_238 _
206 _
/
238 206
Pb/238U ratio corrected for initial 232Th disequilibrium 207_206 _
207 /
206 _
ratio of 207Pb/206Pb, with 206Pb corrected for initial 238U ‐ 230Th disequilibrium Newton‐Raphson Method for iterative solution of Th‐corrected 207Pb/206Pb age: U
207_206 for i = 1 to 100 ·
10
207_206 _
238_235 ·
1
·
1
11 238_235 ·
·
10/
11 next i 207_206 _
1
·
·
235 ·
238_235 ·
·
·
1
1 ·
238_235 ·
238 ·
·
P a g e | 14 207_235 _
1
235
log 207_235
1
235
231
log 1
231_235
1
207
Pb/235U age corrected for initial 231Pa disequilibrium 207_
_
235
235
231
_
moles of excess or deficit 207Pb from initial 235U ‐ 231Pa disequilibrium 207_206 _
207
207_
/
206 ratio of 207Pb/206Pb, with 207Pb corrected for initial 235U‐231Pa disequilibrium 207_235 _
207
207_
/
235 ratio of 207Pb/235U, with 207Pb corrected for initial 235U‐231Pa disequilibrium 231_235
1 P a g e | 15 Newton‐Raphson Method for iterative solution of 231Pa‐corrected 207Pb/206Pb age: 207_206 for i = 1 to 100 ·
10
207_206 _
1
·
238_235 ·
1
11 238_235 ·
·
1
·
·
235 ·
238_235 ·
·
·
10/
11 next i 207_206 _
207_206 _
207
207_
/
206 _
radiogenic 207Pb/206Pb ratio, corrected for both 230Th and 231Pa initial disequilibrium 1
1 ·
238_235 ·
238 ·
·
P a g e | 16 Newton‐Raphson Method for iterative solution of 231Pa‐ and 230Th‐corrected 207Pb/206Pb age: 207_206 for i = 1 to 100 ·
10
207_206 _
238_235 ·
1
·
1
11 238_235 ·
·
1
·
·
235 ·
238_235 ·
·
·
1 ·
238_235 ·
1
10/
11 next i 207_206 _
4. Isochron ratios 207_206
204_206
204_207
238_206
238_207
235_207
238_204
235_204
207 /
204 /
204 /
238 /
238 /
235 /
238 /
235 /
206 206 207 206 207 207 204 204 isotope ratios for use in Pb/Pb and U/Pb isochrons; ratios are of blank‐ and tracer‐corrected quantities.
238 ·
·
P a g e | 17 5. Outputs 5a. Pb Calculations 204 ·
208 ·
204
206 ·
206
207 ·
207
208
total lab Pb blank mass, in grams 204 · 1 ·
208 ·
204
206_204 ·
206
207_204 ·
207
208_204
total common Pb mass, grams; initial common Pb mass, grams 206 206 207 207 208 208 total grams of radiogenic Pb in fraction (including 208Pb); total Pb mass (blank, common, and initial) /
ratio of radiogenic Pb (including 208Pb) to common Pb fraction output 206
207
208
206
206
206
207
208
207
207
208
208
ratios of radiogenic to common Pb for individual isotopes: 206Pb, 207Pb, 208Pb P a g e | 18 100
100
206_238
207_206
fraction output data table output percent discordance 235
10
235
238
238
concentration of U, ppm _
10
concentration of Pb, ppm, radiogenic + initial common + laboratory blank _
10
concentration of Pb, ppm: radiogenic Pb _
10
concentration of Pb, ppm: laboratory blank + initial common Pb 206_
10
206
206
206
206
concentration of 206Pb, ppm: radiogenic + initial common + laboratory blank 206_
10
206
206
concentration of 206Pb, ppm: radiogenic Pb 206_
10
206
206
206
concentration of 206Pb, ppm: laboratory blank + initial common Pb 206_204
205 ·
206
206_205
· 1
206_205 /
Pb/204Pb ratio, tracer‐ and fractionation‐corrected only 204
204 P a g e | 19 5b. U calculations 232
_
238
235
ratio of Th/U in the sample, calculated from 208Pb 236
233
233 / 233_236 233
235
235
235
236
235
moles of 236U in tracer; approximate U mass from tracer 236
238
238
238 238