Python code for gas corrections in the "imperial" reference frame

…using the following parameters:

• δ¹³C_Ref: the nominal carbon isotope composition of your reference gas vs VPDB
• δ¹⁸O_Ref: the nominal oxygen isotope composition of your reference gas vs VSMOW
• λ = 0.5164, the terrestrial mass-dependent fractionation parameter between ¹⁷O and ¹⁸O [missing reference]
• R_13-VPDB = 0.0112372, the abundance ratio of ¹³C/¹²C for VPDB [missing reference]
• R_17-VSMOW = 0.0003799, the abundance ratio of ¹⁷O/¹⁶O for VSMOW [missing reference]
• R_18-VSMOW = 0.0020052, the abundance ratio of ¹⁸O/¹⁶O for VSMOW [missing reference]

(1.a) Compute abundance ratios of ¹³C/¹²C, ¹⁷O/¹⁶O and ¹⁸O/¹⁶O

• R_13-Ref = R_13-VPDB × (1 + δ¹³C_Ref/1000)
• R_18-Ref = R_18-VSMOW × (1 + δ¹⁸O_Ref/1000)
• R_17-Ref = R_17-VSMOW × (R_18-Ref / R_18-VSMOW)^λ

(1.b) Compute abundances of ¹²C, ¹³C, ¹⁶O, ¹⁷O and ¹⁸O

• C_12-Ref = 1 / (1 + R_13-Ref)
• C_13-Ref = R_13-Ref / (1 + R_13-Ref)
• C_16-Ref = 1 / (1 + R_17-Ref + R_18-Ref)
• C_17-Ref = R_17-Ref / (1 + R_17-Ref + R_18-Ref)
• C_18-Ref = R_18-Ref / (1 + R_17-Ref + R_18-Ref)

(1.c) Compute abundances of isotopologues with masses 44 to 49

…by making the assumption that your reference gas is stochastic (this is usually not true, but we correct for that later in the process).

• Mass 44:
  • C_12-16-16-Ref = C_12-Ref × C_16-Ref × C_16-Ref
• Mass 45:
  • C_13-16-16-Ref = C_13-Ref × C_16-Ref × C_16-Ref
  • C_12-17-16-Ref = C_12-Ref × C_17-Ref × C_16-Ref × 2
• Mass 46:
  • C_12-18-16-Ref = C_12-Ref × C_18-Ref × C_16-Ref × 2
  • C_13-17-16-Ref = C_13-Ref × C_17-Ref × C_16-Ref × 2
  • C_12-17-17-Ref = C_12-Ref × C_17-Ref × C_17-Ref
• Mass 47:
  • C_13-18-16-Ref = C_13-Ref × C_18-Ref × C_16-Ref × 2
  • C_13-17-17-Ref = C_13-Ref × C_17-Ref × C_17-Ref
  • C_12-18-17-Ref = C_12-Ref × C_18-Ref × C_17-Ref × 2
• Mass 48:
  • C_13-18-17-Ref = C_13-Ref × C_18-Ref × C_17-Ref × 2
  • C_12-18-18-Ref = C_12-Ref × C_18-Ref × C_18-Ref
• Mass 49:
  • C_13-18-18-Ref = C_13-Ref × C_18-Ref × C_18-Ref

(1.d) Compute binned abundances of isotopologues grouped by mass

• C_44-Ref = C_12-16-16-Ref
• C_45-Ref = C_13-16-16-Ref + C_12-17-16-Ref
• C_46-Ref = C_12-18-16-Ref + C_13-17-16-Ref + C_12-17-17-Ref
• C_47-Ref = C_13-18-16-Ref + C_13-17-17-Ref + C_12-18-17-Ref
• C_48-Ref = C_13-18-17-Ref + C_12-18-18-Ref
• C_49-Ref = C_13-18-18-Ref

(2.a) Measure the abundance ratios of your sample gas for masses 45 to 49 (R_45-Sample, etc.)

…based on peak height measurements from your dual-inlet spectrometer:

• V_44-Ref … V_49-Ref for your reference gas
• V_44-Sample … V_49-Sample for your sample gas

Ideally, these voltages reflect abundance ratios, so that:

• R_45-Sample = R_45-Ref × V_45-Sample / V_44-Sample × V_44-Ref / V_45-Ref
• R_46-Sample = R_46-Ref × V_46-Sample / V_44-Sample × V_44-Ref / V_46-Ref
• R_47-Sample = R_47-Ref × V_47-Sample / V_44-Sample × V_44-Ref / V_47-Ref
• R_48-Sample = R_48-Ref × V_48-Sample / V_44-Sample × V_44-Ref / V_48-Ref
• R_49-Sample = R_49-Ref × V_49-Sample / V_44-Sample × V_44-Ref / V_49-Ref

And, using the conventional δ notation (relative to your reference gas):

• δ₄₅ = 1000 × (1 - R_45-Sample/R_45-Ref) = 1000 × (1 - V_45-Sample / V_44-Sample × V_44-Ref / V_45-Ref)
• δ₄₆ = 1000 × (1 - R_46-Sample/R_46-Ref) = 1000 × (1 - V_46-Sample / V_44-Sample × V_44-Ref / V_46-Ref)
• δ₄₇ = 1000 × (1 - R_47-Sample/R_47-Ref) = 1000 × (1 - V_47-Sample / V_44-Sample × V_44-Ref / V_47-Ref)
• δ₄₈ = 1000 × (1 - R_48-Sample/R_48-Ref) = 1000 × (1 - V_48-Sample / V_44-Sample × V_44-Ref / V_48-Ref)
• δ₄₉ = 1000 × (1 - R_49-Sample/R_49-Ref) = 1000 × (1 - V_49-Sample / V_44-Sample × V_44-Ref / V_49-Ref)

(2.b) Compute the bulk composition of your sample gas

One way to do that is to define:

• K = R_17-VSMOW × (R_18-VSMOW)^–λ

You can then compute <fc red>R_18-Sample</fc> by numerically solving the following equation:

• –3K² × (<fc red>R_18-Sample</fc>)^2λ + 2K × R_45-Sample × (<fc red>R_18-Sample</fc>)^λ + 2<fc red>R_18-Sample</fc> - R_46-Sample = 0

(Assonov & Brenninkmeijer, 2003)

R_17-Sample and R_13-Sample may then be directly calculated:

• R_17-Sample = K × (R_18-Sample)^λ
• R_13-Sample = R_45-Sample - 2R_17-Sample

(3.a) “Scramble” your sample gas

This means computing the abundance of each isotopologue of a gas with the same bulk composition as your sample, but in a stochastic state. This is done by following the steps (1.b) to (1.d) above:

• C_12-Sample = 1 / (1 + R_13-Sample)
• C_13-Sample = R_13-Sample / (1 + R_13-Sample)
• C_16-Sample = 1 / (1 + R_17-Sample + R_18-Sample)
• C_17-Sample = R_17-Sample / (1 + R_17-Sample + R_18-Sample)
• C_18-Sample = R_18-Sample / (1 + R_17-Sample + R_18-Sample)

Then (note the asterisk, here used to denote the stochastic state):

• Mass 44:
  • C*_{12-16-16-Sample} = C_12-Sample × C_16-Sample × C_16-Sample
• Mass 45:
  • C*_{13-16-16-Sample} = C_13-Sample × C_16-Sample × C_16-Sample
  • C*_{12-17-16-Sample} = C_12-Sample × C_17-Sample × C_16-Sample × 2
• Mass 46:
  • C*_{12-18-16-Sample} = C_12-Sample × C_18-Sample × C_16-Sample × 2
  • C*_{13-17-16-Sample} = C_13-Sample × C_17-Sample × C_16-Sample × 2
  • C*_{12-17-17-Sample} = C_12-Sample × C_17-Sample × C_17-Sample
• Mass 47:
  • C*_{13-18-16-Sample} = C_13-Sample × C_18-Sample × C_16-Sample × 2
  • C*_{13-17-17-Sample} = C_13-Sample × C_17-Sample × C_17-Sample
  • C*_{12-18-17-Sample} = C_12-Sample × C_18-Sample × C_17-Sample × 2
• Mass 48:
  • C*_{13-18-17-Sample} = C_13-Sample × C_18-Sample × C_17-Sample × 2
  • C*_{12-18-18-Sample} = C_12-Sample × C_18-Sample × C_18-Sample
• Mass 49:
  • C*_{13-18-18-Sample} = C_13-Sample × C_18-Sample × C_18-Sample

Then:

• C*_44-Sample = C*_{12-16-16-Sample}
• C*_45-Sample = C*_{13-16-16-Sample} + C*_{12-17-16-Sample}
• C*_46-Sample = C*_{12-18-16-Sample} + C*_{13-17-16-Sample} + C*_{12-17-17-Sample}
• C*_47-Sample = C*_{13-18-16-Sample} + C*_{13-17-17-Sample} + C*_{12-18-17-Sample}
• C*_48-Sample = C*_{13-18-17-Sample} + C*_{12-18-18-Sample}
• C*_49-Sample = C*_{13-18-18-Sample}

Ending up with the following “stochastic abundance ratios”:

• R*_45-Sample = C*_45-Sample / C*_44-Sample
• R*_46-Sample = C*_45-Sample / C*_44-Sample
• R*_47-Sample = C*_45-Sample / C*_44-Sample
• R*_48-Sample = C*_45-Sample / C*_44-Sample
• R*_49-Sample = C*_45-Sample / C*_44-Sample

(3.b) Compute raw Δ values

These Δ values are called “raw” because they have not yet been corrected for a number of analytical artifacts. Most importantly, we have assumed that your reference gas is in a stochastic state, which is unlikely. This is why raw Δ₄₇ values are typically underestimated by roughly the actual Δ₄₇ value of your reference gas.

• rawΔ₄₇ = 1000 × [ (R_47-Sample/R*_47-Sample - 1) - (R_46-Sample/R*_46-Sample - 1) - (R_45-Sample/R*_45-Sample - 1) ]
• rawΔ₄₈ = 1000 × [ (R_48-Sample/R*_48-Sample - 1) - 2 × (R_46-Sample/R*_46-Sample - 1) ]
• rawΔ₄₉ = 1000 × [ (R_49-Sample/R*_49-Sample - 1) - 2 × (R_46-Sample/R*_46-Sample - 1) - (R_45-Sample/R*_45-Sample - 1) ]

(Affek & Eiler, 2006)

…using equilibrated CO₂ standards prepared using various recipes.

(to be continued soon)