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Famous mathematical constants include the ratio of circular circumference to diameter, π = 3.14 …, and the natural logarithm base, e = 2.718 …. Students and professionals can often name a few others, but there are many more buried in the literature and awaiting discovery. How do such constants arise, and why are they important? Here the author renews the search he began in his book Mathematical Constants, adding another 133 essays that broaden the landscape. Topics include the minimality of soap film surfaces, prime numbers, elliptic curves and modular forms, Poisson–Voronoi tessellations, random triangles, Brownian motion, uncertainty inequalities, Prandtl–Blasius flow (from fluid dynamics), Lyapunov exponents, knots and tangles, continued fractions, Galton–Watson trees, electrical capacitance (from potential theory), Zermelo's navigation problem, and the optimal control of a pendulum. Unsolved problems appear virtually everywhere as well. This volume continues an outstanding scholarly attempt to bring together all significant mathematical constants in one place.
Two alkali-tin-silicate (ATS) glasses have been prepared at Argonne National
Laboratory (ANL) as part of our ongoing research in radioactive waste glass
development. These glasses dissolved 5% and approximately 7% Pu. Early
corrosion test results indicate that Pu-bearing ATS glass is extremely
durable. The initial goal in this project concerned equally both the
solubility of Pu and the durability of the ATS glasses; however, our primary
emphasis has changed recently to maximizing the loading of Pu in the glass.
ATS-based glasses, using Th(VI) and Ce(III) as surrogates for Pu(IV), are
now being investigated to increase the solubility of Pu without
substantially sacrificing the durability of the current ATS formulations.
The solution data from various corrosion tests on the original Pu-containing
ATS glasses are also presented.
Spent nuclear fuel (SNF) is unstable under oxidizing conditions. Although
recent studies have determined the paragenetic sequence for uranium phases
that result from the corrosion of SNF, there are only limited data on the
potential of alteration phases for the incorporation of transuranium
elements. The crystal chemical characteristics of transuranic elements (TUE)
are to a certain extent similar to uranium; thus TUE incorporation into the
sheets of uranyl oxide hydrate structures can be assessed by examination of
the structural details of the β-U3O8 sheet type.
The sheets of uranyl polyhedra observed in the crystal structure of
β-U3O8 also occur in the mineral billietite
where they alternate with α-U3O8 type sheets.
Preliminary crystal structure determinations for the minerals ianthinite,
and “wyartite II” (mineral name not approved by IMA committee on mineral
indicate that these phases also contain β-U3O8 type
sheets. The β-U3O8sheet anion topology contains
triangular, rhombic, and pentagonal sites in the proportions 2: 1:2. In all
structures containing β-U3O8 type sheets, the
triangular sites are vacant. The pentagonal sites are filled with
U6+O2 forming pentagonal bipyramids. The rhombic
dipyramids filling the rhombic sites contain U6+O2 in
billietite, U4+O2 in
ianthinite, and U4+O3 in “wyartite-II” (in which one
apical anion is replaced by two O atoms forming a shared edge with a
carbonate triangle of the interlayer). Interlayer species include:
H2O (billietite, “wyartite II”, and ianthinite),
Ba2+ (billietite) Ca2+ (”wyartite II”), and
CO3−2 (”wyartite II”); there is no interlayer in
β-U3O8. The similarity of known TUE coordination
polyhedra with those of U suggests that the β-U3O8
sheet will accommodate TUE substitution coupled with variations in apical
anion configuration and interlayer population providing the required charge
This paper aims to discuss the applicability of the classical matrix diffusion model against the integrated body of new data obtained by different methodologies on several samples of three granite boulders. The matrix diffusion model was tested against observations from the upper (most weathered in contact with air) and lower (fresh in contact with the ground) part of a boulder block. A U(VI) enrichment up to nearly 300 ppm (compared to about 10 ppm background concentration) mostly as uranophane was observed in the zone between the weathered and fresh rock. U-series disequilibrium studies indicated that most of U has been accumulated recently, about 10 000 years ago .
High interconnected porosity (total porosity of > 1% and up to about 5.5% in altered minerals) characterizes the weathered zone (upper part), whereas the maximum porosity values in the fresh zone (lower part) of the rock are about 0.4 – 0.6%. Stable isotope studies δ18O and δ2H confirm that the mineralogical changes observed in the weathered upper part are due to old hydrothermal events. That is, the alteration is much older than uranium accumulation. Mössbauer spectroscopy showed that the Fe(III) content of the biotites from the upper to the lower part decreases from 30% in the weathered zone to 17% in the fresh rock, thus indicating possible redox control for the observed U precipitation. Fission track studies showed that secondary U(VI) also occurs within minerals grains (especially plagioclase) in the upper part.
Mathematical simulations indicate that matrix diffusion alone is not enough to reconstruct the past U accumulation. The simulated concentrations derived from U concentration in pore water multiplied by Kd are clearly too small, indicating apparent insufficiency of the Kd approach. However, even with only matrix diffusion, the simulations roughly reconstruct the observation that U levels are clearly higher in the upper part of the boulder than in the lower part.