Lecture 11-13: Radioactive Decay

1. Mechanisms
2. The decay equation
   1. Isochrons
   2. Sm-Nd system
   3. Pb-Pb dating (age of the Earth)

We acknowledge and respect the lək̓ʷəŋən peoples on whose traditional territory the university stands and the Songhees, Esquimalt and W̱SÁNEĆ peoples whose historical relationships with the land continue to this day.

Mechanisms of radioactive decay.

The decay equation.

The decay equation.

Half life

U-238 decay chain

U-Series dating of corals

Isochrons

Recall: $$ N_0 e^{-\lambda t} = N$$

N_0 = 1 # parent isotope initial
LAMBDA = 1e-4 # decay constant
t = np.linspace(0, 1e5, 1000) #time (from now to 100 thousand years ago)
N = N_0 * np.exp(-LAMBDA * t) # parent isotope
D = N_0 - N #descendant isotope

plt.figure(figsize=(5, 5))
plt.plot(t, N, label="Parent", alpha=1, lw=3)
plt.plot(t, D, label="Descendant", alpha=1, lw=3)
plt.legend(loc="best")
_ = plt.gca().set_xlabel("Time")
_ = plt.gca().set_ylabel("Amount")

Batch melting: $$\dfrac{C_S}{C_0} = \dfrac{D}{F+D(1-F)}$$

def batch_S(F, D, Co): #batch melting equation for solid
    return (D*Co) / (F + D * (1 - F))

D_n = np.array([1,2,3,4]) #a list of partition coefficients
F = 0.8 #melt fraction
N_0 = batch_S(F, D_n, 1) #C_l for each F

plt.figure(figsize=(6, 5))
plt.plot(D_n, N_0, "o")
_ = plt.gca().set_xlabel("Bulk partition coefficient D$_n$")
_ = plt.gca().set_ylabel("N$_0$")

N = [n * np.exp(-LAMBDA * t[:100]) for n in N_0] # the decay equation for each starting N
C = [(n - NS) for n, NS in zip(N_0, N)] # the descendant isotope (initial minus current)

plt.figure(figsize=(6, 5))
plt.plot(N_0, [0, 0, 0, 0], "o", alpha=1)
_ = [plt.plot(a, b, label="F=" + str(c), alpha=1) for a, b, c in zip(N, C, D_n)]
plt.legend(loc="best")
_ = plt.gca().set_xlabel("Parent") 
_ = plt.gca().set_ylabel("Descendant")
_ = plt.gca().set_xlim([0, 3])
_ = plt.gca().set_ylim([-0.1, 2])

np.random.seed(12)
N = [n * np.exp(-LAMBDA * t[:50]) for n in N_0] # the decay equation for each starting N
Ic = 0.5*np.ones(4) # some fixed amount of initial child isotope
C = [i + (n - NS) for i, n, NS in zip(Ic, N_0, N)] # the descendant isotope (initial parent minus current)

plt.figure(figsize=(6, 5))
plt.plot(N_0, Ic, "o")
_ = [plt.plot(a, b, label="F=" + str(c)) for a, b, c in zip(N, C, D_n)]
plt.legend(loc="best")
_ = plt.gca().set_xlabel("Parent")
_ = plt.gca().set_ylabel("Descendant")
_ = plt.gca().set_xlim([0, 2])
_ = plt.gca().set_ylim([-0.1, 2])

Ic = 0.5 + np.array(np.random.normal(0.0, 0.2, 4)) # some random amount of initial child isotope
C = [i + (n - NS) for i, n, NS in zip(Ic, N_0, N)] # the descendant isotope (initial parent minus current)

plt.figure(figsize=(6, 5))
plt.plot(N_0, Ic, "o")
_ = [plt.plot(a, b, label="F=" + str(c)) for a, b, c in zip(N, C, D_n)]
plt.legend(loc="best")
_ = plt.gca().set_xlabel("Parent")
_ = plt.gca().set_ylabel("Descendant")
_ = plt.gca().set_xlim([0, 4])
_ = plt.gca().set_ylim([-0.1, 2])

Utility

Sm-Nd decay and model ages (the chondritic uniform reservoir or CHUR)

$^{147}$Sm decays to $^{143}$Nd through alpha decay with a decay constant of $\lambda^{147}$=$6.54\times10^{-12}$

Age of the Earth

Age of the Earth

Age of the Earth

Age of the Earth

Age of meteorites and the earth. Claire Patterson 1956