Ar Ar Calculations

ArAr Age Calculations

F-valve Calculation

# user supplied values
# ic/decay corrected intensities
a40=  #Ar40
a39=  #Ar39
a38=  #Ar38
a37=  #Ar37
a36=  #Ar36
k4039= #(40/39)K production ratio
ca3937= #(39/37)Ca
k3739= #(37/39)K
k3839= #(38/39)K
ca3637= #(36/37)Ca
ca3837= #(38/37)Ca
cl3638= #(36/38)Cl
lambda_cl36= # Cl36 decay constant
decay_time= # time since irradiation
atm3836 = # trapped Ar38/Ar36
trapped_4036= # trapped Ar40/Ar36

# ca-correction
# ca37 = a37 - k37
# ca39 = ca3937 * ca37
# k39 = a39 - ca39
# k37 = k3739 * k39

# k39 = a39-ca3937*(a37-(k3739*k39))
#    = a39-ca3937*a37+k3739*k39*ca3937
# k39*(1-k3739*ca3937) = a39-ca3937*a37

k39 = (a39-ca3937*a37)/(1-k3739*ca3937)
k37 = k3739 * k39
k38 = k3839 * k39

ca37 = a37 - k37
ca39 = ca3937 * ca37

ca36 = ca3637 * ca37
ca38 = ca3837 * ca37

# cl-correction
# calculate atm36, cl36, cl38

# starting with the following equations
# atm36 = a36 - ca36 - cl36

# m = cl3638*lambda_cl36*decay_time
# cl36 = cl38 * m

# cl38 = a38 - k38 - ca38 - ar38atm
# ar38atm = atm3836 * atm36

# rearranging to solve for atm36
# cl38 = a38  - k38 - c38 - atm3836 * atm36

# cl36 = m * (a38  - k38 - ca38 - atm3836 * atm36)
#     = m (a38  - k38 - ca38) - m * atm3836 * atm36
# atm36 = a36 - ca36 - m (a38  - k38 - ca38) + m * atm3836 * atm36
# atm36 - m * atm3836 * atm36 =  a36 - ca36 - m (a38  - k38 - ca38)
# atm36 * (1 - m*atm3836) = a36 - ca36 - m (a38  - k38 - ca38)
# atm36 = (a36 - ca36 - m (a38  - k38 - c38))/(1 - m*atm3836)

m = cl3638 * lambda_Cl36 * decay_time
atm36 = (a36 - ca36 - m*(a38 - k38 - ca38)) / (1 - m * atm3836)

atm40 = atm36 * trapped_4036

k40 = k39 * k4039

rad40 = a40 - atm40 - k40
f = rad40 / k39

Age Calculation

  lambda_k =  # total 40K decay constant
  f = # F-value e.g Ar40*/Ar39K
  j = # J-value e.g. neutron flux parameter
  age = lambda_k ** -1 * ln(1 + j * f))

Apply Fixed (37/39)K

"""
    x=ca37/k39
    y=ca37/ca39
    T=s39dec_cor

    T=ca39+k39
    T=ca37/y+ca37/x

    ca37=(T*x*y)/(x+y)
"""

k3739 = # (37/39)K
ca39 =  # (39/37)Ca

x = k3739
y = 1 / ca3937

ca37 = (a39 * x * y) / (x + y)

ca39 = ca3937 * ca37
k39 = a39 - ca39
k37 = x * k39

Decay Factors

"""
    McDougall and Harrison
    p.75 equation 3.22

    the book suggests using ti==analysis_time-end of irradiation segment_i

    mass spec uses ti==analysis_time-start of irradiation segment_i

    using start seems more appropriate
"""

    dc37 = # Ar37 decay constant
    dc39 = # Ar39 decay constant

    a = sum([pi * ti for pi, ti, _, _, _ in segments])

    b = sum([pi * ((1 - math.exp(-dc37 * ti)) / (dc37 * math.exp(dc37 * dti)))
         for pi, ti, dti, _, _ in segments])

    c = sum([pi * ((1 - math.exp(-dc39 * ti)) / (dc39 * math.exp(dc39 * dti)))
         for pi, ti, dti, _, _ in segments])


    df37 = a / b
    df39 = a / c

Abundance Sensitivity

s40 = # m/e=40 intensity

# correct for abundance sensitivity
# assumes symmetric and equal abundant sens for all peaks
n40 = s40 - abundance_sensitivity * (s39 + s39)
n39 = s39 - abundance_sensitivity * (s40 + s38)
n38 = s38 - abundance_sensitivity * (s39 + s37)
n37 = s37 - abundance_sensitivity * (s38 + s36)
n36 = s36 - abundance_sensitivity * (s37 + s37)

Integrated Age Weighting Factors

Variance

    W_i = 1 / Ar39K_i_sigma**2

Volume

    W_i = (Ar39K_i / Total_Ar39K * Ar39K_i_sigma)**2