# Re-Os Decay#

High-Temperature Geochemistry

# import relevant modules

import numpy as np
from math import log10, floor

# create our own functions

# function to round a value to a certain number of significant figures
def round_to_n_sf(value, no_of_significant_figures):
value_rounded = round(value, no_of_significant_figures-1-int(floor(log10(abs(value)))))
if value_rounded == int(value_rounded):
value_rounded = int(value_rounded)
return value_rounded



## Re-Os Decay System#

${^{187}Re \longrightarrow {^{187}Os} + \beta^-} \qquad t_{\frac{1}{2}} = 42.3\,Gyr$

The $$Re$$-$$Os$$ decay system is unusual in many respects and it has evolved into an important tool for some particular geochemical applications.

Most parent and daughter elements of long-lived radioactive decay systems are incompatible lithophile elements ($$Rb$$, $$Sr$$, $$Sm$$, $$Nd$$, …).

$$Re$$ and $$Os$$ are siderophile ($$Fe$$-loving) elements, such that they are concentrated in the Earth’s core. As a result, most terrestrial rocks have low $$Re$$ and $$Os$$ abundances of $$<10\,ppb$$. Both elements also have chalcophile tendencies and in the Earth’s mantle $$Re$$ and particularly $$Os$$ are concentrated in magmatic sulfides.

Overall, $$Re$$ is a mildly incompatible element (similar to the HREE, $$D \approx 0.1$$), whereas $$Os$$ is generally strongly compatible ($$D \approx 10$$) during partial melting of the mantle. As a result, partial melting can generate very strong fractionations of the $$Re/Os$$ ratio.

As $$Re$$ and $$Os$$ differ in their geochemical properties from the parent and daughter elements of other common decay systems, the $$Re$$-$$Os$$ system is very useful for chronological applications that are difficult to accomplish by other means, e.g., dating of iron meteorites, precious metal ores, and some ultramafic rocks (e.g., komatiites). Komatiites are very high degree mantle melts ($$F > 20\%$$) with relatively high $$Re$$ and $$Os$$ abundances of ~$$1$$ to $$5$$ $$ppb$$. Hence, they are comparatively easy to analyze for $$Re$$-$$Os$$ isochron studies. The $$Re$$-$$Os$$ decay system also provides one of the few means to date iron meteorites, which are essentially devoid of lithophile trace elements, such as $$Rb$$, $$Sr$$, $$Hf$$, and the REE.

# Re-Os decay equation
# each return depends on what we want to find from the equation
def Re_Os_decay_equation(Os187_Os188_ratio, initial_Os187_Os188_ratio, Re187_Os187_ratio, t):
decay_const_Re = 1.64 * 10**-11  # yr^-1  # decay constant of Re-187
if Os187_Os188_ratio == '?':
return initial_Os187_Os188_ratio + Re187_Os187_ratio*(np.exp(decay_const_Re*t)-1)
elif initial_Os187_Os188_ratio == '?':
return Os187_Os188_ratio - Re187_Os187_ratio*(np.exp(decay_const_Re*t)-1)
elif Re187_Os187_ratio == '?':
return (Os187_Os188_ratio - initial_Os187_Os188_ratio)/(np.exp(decay_const_Re*t)-1)


## Problem Set 6 - Question 2#

What is the $$Os$$ isotopic composition of a pure osmiridium grain that was formed in and removed from a chondritic mantle $$1\,Gyr$$ ago? The $$Os$$ isotope evolution of the chondritic mantle is described by the following present-day parameters: $${^{187}Os}/{^{188}Os} = 0.125$$ and $${^{187}Re}/{^{188}Os} = 0.401$$.

# Question 2

# Given values
Os187_Os188_ratio = 0.125  # present-day
Re187_Os187_ratio = 0.401  # present-day
t = 1 * 10**9  # yr
# calculate Lu176/Hf177 ratio
initial_Os187_Os188_ratio = Re_Os_decay_equation(Os187_Os188_ratio, '?', Re187_Os187_ratio, t)  # at t = 1 Gyr

The intial Os-187/Os-188 ratio (1 Gya) of the Os grain is 0.118369.