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This page further illustrates the use of the Three Phase ISM tool. It assumes you have considerable familiarity with the tool. If you do not, you should first try out the beginner and the ¹²⁹I/¹²⁷I tutorials.

The particular problem addressed in this tutorial is that of the concentration of ²³⁸U and ²³²Th in the interstellar medium and its three phases. These two isotopes are of interest because they are both long-lived, so they help us infer the duration of Galactic nucleosynthesis. Furthermore, they are both actinides; thus, they are only produced by the r-process of nucleosynthesis.

Solar Abundance Ratio

The solar abundances today of ²³⁸U and ²³²Th are 0.00893 and 0.0335, respectively, on the scale Si=10⁶. This gives an abundance ratio of 0.00893/0.0335=0.267. We may now determine the abundance ratio at the time of the Sun's birth 4.567x10⁹ years ago. ²³²Th has a half life of 1.405x10¹⁰ years. The age of the Solar System is thus 0.3246 ²³²Th half lives, and the abundance of ²³²Th 4.56x10⁹ years ago was 0.0335x2^0.3251=0.0420. Similarly, the ²³⁸U half life is 4.468x10⁹ years, and we may calculate that the ²³⁸U abundance was 0.0181 at the time of the Sun's birth. This means the ²³⁸U/²³²Th abundance ratio at the beginning of the Solar System was 0.0181/0.0420=0.431. We may use the Three Phase ISM tool to try to best fit that number.

Galactic Disk Age

Open the Three Phase ISM Tool and enter the species information for ²³²Th and ²³⁸U (the half lives are in the preceding paragraph). We now seek the age of the Galactic disk. Since the Universe is now known to be 13.7 x 10⁹ years old, and because we seek the abundance ratio at the time of the Sun's birth 4.567x10⁹ years ago, choose a Galactic disk age of 13.7x10⁹ minus 4.567x10⁹ = 9.1x10⁹ years. Galaxies may not have begun to form until 10⁹ years after the Big Bang, so the better Galactic disk age may be 8.1x10⁹ years. Enter that number. For Delta and k, enter the default values 2x10⁸ years and 3, respectively.

Production Ratio

We now seek a production ratio. The actual production ratio must be obtained from a detailed calculation of r-process nucleosynthesis. Such calculations tend to give results ranging from 0.4 to 0.7. We choose here to follow the simple route of assuming equal abundances of short-lived progenitors of the two isotopes and adding the abundance they contribute to ²³²Th and ²³⁸U.

We begin with ²³²Th. The following isotopes alpha decay into ²³²Th in a time short compared to the ²³²Th half life: ²³⁶U (100%), ²⁴⁰Pu (100%), ²⁴⁴Pu (99.88%), ²⁴⁸Cm (91.61%), and ²⁵²Cf (96.91%). Mass numbers greater than A=252 all spontaneously fission and do not contribute to ²³²Th. The numbers in parenthesis give the percentage of the progenitor that actually alpha decays--the remainder is lost to fission. These numbers are obtained from the Nuclear Wallet Cards database. Adding these up, including ²³²Th itself, and accounting for loss due to fission, we find 1 + 1 + 1 + 0.9988 x ( 1 + 0.9161 x ( 1 + 0.9691 ) ) = 5.8 progenitors. Note that we account for the fact that 91.61% of ²⁴⁸Cm decays into ²⁴⁴Pu and then 99.88% of this new ²⁴⁴Pu alpha decays.

For ²³⁸U, the progenitors are itself (100%), ²⁴²Pu (100%), ²⁴⁶Cm (99.97%) and ²⁵⁰Cm (18%). Adding these up we find 1 + 1 + 0.9997 x (1 + 0.18) = 3.2 progenitors. The resulting ²³⁸U/²³²Th production ratio is then 3.2/5.8 = 0.55. Enter this number in the Production Ratio panel form.

Results

For the remainder of the input, the masses in the three phases and the mixing times between the phases, enter the default values Mass_1=0.3, Mass_2=0.3, Mass_3=0.4 and Time_1=Time_2=1.0e7. Here are the results of the calculation. The average ISM abundance ratio is 0.47. This is also true for all three phases since the mixing time is so short compared to the lifetimes of the two species. Interestingly, this number exceeds 0.431, the value we inferred from meteoritic data for the initial solar system ratio. A lower production ratio will solve this discrepancy, which may thus be telling us about the r-process, but perhaps there other solutions. Try the tool to find out.