**Isotope Abundances in Godiva Criticality
Calculations**

The PFC Godiva criticality calculation shown in Example 8.4
of Volume 1, pages 259-267, used the metal density and isotopic fractions given
in Los Alamos report LA-4208 (reference 8 of chapter 8) to compute the system
multiplication. The isotopic abundances
presented in the

For an element containing a mixture of isotopes the number density of isotope i is given by

N_{i}
= ρ_{i}N_{0}/A_{i} (1)

where r_{i} is the partial density of isotope i in the
mixture, N_{o} is Avagadro’s number, and A_{i} is the atomic
mass of isotope i. The atom fraction of
isotope i is thus

f_{i}^{a} = N_{i}/∑N_{i}
= ρ_{i}/A_{i}/∑(ρ_{i}/A_{i})

The mass fraction, on the other hand, is given by

f_{i}^{m} = ρ_{i}/∑ρ_{i}

where Sr_{i} = r is the total density of
the mixture and the sum is taken over all of the isotopes of the element in the
mixture. We then have

N_{i}
= f_{i}^{m}ρN_{0}/A_{i} (2)

Comparing eqns 1 and 2 we have

f_{i}^{m} = f_{i}^{a}A_{i}/∑
f_{i}^{a}A_{i}

For polyisotopic elements it is clear that the mass
fractions are equal to the atom fractions only if A_{i}/Sf_{i}^{a}A_{i}
= 1. In practice the latter is
approximately true for high atomic mass elements. However, even for uranium, with A_{i}
values very close to 235 and 238 for the dominant isotopes, the mass fractions
differ slightly from the atom fractions.
The mass and atom fractions for Godiva uranium, obtained using the above
equations and the data given in Example 8.4, are shown in Table 1.

Table 1. Atom and Mass Fractions for Godiva Uranium

i |
f |
f |

235 |
0.9386 |
0.9393 |

238 |
0.0614 |
0.0607 |

The number density of uranium atoms in Godiva metal is given
on p. 261 of Volume 1 as 0.04815 atoms/barn-cm.
This result was obtained assuming the mass fractions were equal to the
atom fractions, atomic masses of exactly 235 and 238, and using 0.6023 x 10^{24}
for Avagadro’s number:

N = N_{0}ρ/A ≈
0.6023 x 18.80/(0.9386 x 235 + 0.0614 x 238) ≈ 0.04815

For this number density and the mass fractions of Table 1 the calculation of the number densities shown in Table 8.33 gives

For U-235:
0.9386*0.04815E24 = 0.04519359E24 atoms/cm^{3}

For U-238:
0.0614*0.04815E24 = 0.00295641E24 atoms/cm^{3}

The mass percent enrichment used is thus:

0.04519359*235.0439/(0.04519359*235.0439 + 0.00295641*238.0508) = 93.79%

The calculation of a critical mass of 52.58 ± 0.44 kg in Example 8.4 is therefore the result for a mass percent enrichment of 93.79% rather than the actual 93.86% enrichment for Godiva.

The “correction” for the use of 93.79% mass enrichment rather than 93.86% can be estimated by using an empirically derived relationship[1] which is that the critical mass of a bare sphere of enriched uranium is inversely proportional to the enrichment in atom percent to the 1.7th power. Thus, since the critical mass calculated for 93.79% was 52.58 ± 0.44 kg, the correct value for 93.86% would be:

CritMass = 52.58*0.9386^{1.7} / 0.9393^{1.7 }=
52.51 kg,

since 93.86 mass percent enrichment corresponds to 93.93 atom percent enrichment. The statistical uncertainty remains ± 0.44 kg but this does not reflect any uncertainty resulting from the approximation in reference 1. Thus the “correction” for increasing the enrichment from 93.79% to 93.86% is to reduce the critical mass by 0.07 kg.

Another correction can be calculated based on a more
accurate calculation of the atomic number density for the uranium. The figure used in the text is 0.04815E24
atoms/cm^{3}. At the time the
Hanson-Roach cross sections were calculated, the atomic masses and Avogadro’s
number were based on the “Physical Scale” whereby the mass of the oxygen-16
isotope was assumed to be exactly 16 AMU.
The data of the time (for example from Semat[2])
gave the atomic mass of U-235 as 235.12517 and the atomic mass of U-238 as
238.13232. Avogadro’s number on the
physical scale was calculated as 6.02472E23.

Using the above, the atomic mass of the 93.86% enriched uranium (again assuming incorrectly that the number is atom percent) would then be:

0.9386*235.12517 + 0.0614*238.13232 = 235.30981

The number density would then be:

18.80*6.02472E23/235.30981 = 0.048134E24

Thus the number density for the calculation in Example 8.4 should have been 0.048134E24 rather than 0.04815E24. The effect of this can also be estimated. As is intuitively obvious, especially from Monte Carlo calculations of criticality, the size of a critical sphere of uniform material is dependent on the (number density) times (radius), or as is often written, on ρR. The distances can be expressed in terms of mean free paths. The actual dimensions do not affect the criticality calculation. For example, if the density of the material in a critical sphere were doubled, then the radius would need to be divided by two to maintain a critical configuration. The amount of material in the critical sphere is a variable, however, since it depends on the density times the volume, which is proportional to the radius to the third power.

The critical radius obtained in the calculation was 8.74 cm. With the above correction it would have been 8.7429 cm and this would have added about 0.05 kg to the critical mass.

Note: The use of more current data has little effect on the calculations. Using current data from NIST:

The atomic mass of U-235 is 235.0439.

The atomic mass of U-238 is 238.0508.

Avogadro’s number is 6.0221415E23.

The above are all based on the atomic mass of C-12 as 12.

Assuming 93.86% enrichment based on atom percent would give the following:

The effective atomic mass would be

0.9386*235.0439 + 0.0614*238.0508 = 235.2285

The number density would then be:

18.80*6.0221415E23/235.2285 = 0.04813E24