D. RUSSELL HUMPHREYS, PH.D.*
STEVEN A. AUSTIN, PH.D.
ANDREW A. SNELLING, PH.D.
INSTITUTE FOR CREATION RESEARCH**     

JOHN R. BAUMGARDNER, PH.D.
LOS ALAMOS NATIONAL LABORATORY^**

KEYWORDS

Helium, Diffusion, Radioisotopes, Age, Dating, Nuclear Decay, Zircon, Biotite, Accelerated Decay

ABSTRACT

Two decades ago, Robert Gentry and his colleagues at Oak Ridge National Laboratory reported surprisingly high amounts of nuclear-decay-generated helium in tiny radioactive zircons recovered from Precambrian crystalline rock, the Jemez Granodiorite on the west flank of the volcanic Valles Caldera near Los Alamos, New Mexico [9].  Up to 58% of the helium (that radioactivity would have generated during the alleged 1.5 billion year age of the granodiorite) was still in the zircons.  Yet the zircons were so small that they should not have retained the helium for even a tiny fraction of that time. The high helium retention levels suggested to us and many other creationists that the helium simply had not had enough time to diffuse out of the zircons, and that recent accelerated nuclear decay had produced over a billion years worth of helium within only the last few thousand years, during Creation and/or the Flood.  Such acceleration would reduce the radioisotopic time scale from megayears down to months.

However, until a few years ago nobody had done the experimental and theoretical studies necessary to confirm this conclusion quantitatively.  There was only one (ambiguously reported) measurement of helium diffusion through zircon [18].  There were no measurements of helium diffusion through biotite, the black mica surrounding the zircons.  In 2000 the RATE project [14] began experiments to measure the diffusion rates of helium in zircon and biotite specifically from the Jemez Granodiorite.  The data, reported here, are consistent with data for a mica related to biotite [17], with recently reported data for zircon [19] and with a reasonable interpretation of the earlier zircon data [18].  We show that these data limit the age of these rocks to between 4,000 and 14,000 years.  These results support our hypothesis of accelerated nuclear decay and represent strong scientific evidence for the young world of Scripture.

Figure 1 - Zircons from the Jemez Granodiorite. Photo by R.V. Gentry

1.  INTRODUCTION

A significant fraction of the earth's radioactive elements, particularly uranium and thorium, appear to be in the granitic rock of the upper continental crust.  Uranium and thorium tend to be localized in the granites inside special minerals such as zircon (zirconium silicate, ZrSiO₄).  Zircon has high hardness, high density, and high melting point, often forming microscopic, stubby, prismatic crystals with dipyramidal terminations (Fig. 1, commonly grayish, yellowish, or reddish brown.  Atoms of uranium and thorium within cooling magma replace up to 4% of the normal zirconium atoms within the lattice structure of zircon as it is crystallizing.  The radioactive zircon crystals often become embedded in larger crystals, such as mica, as magma cools and solidifies.

As the uranium and thorium nuclei in a zircon decay, they produce helium.  For example, uranium-238 (²³⁸U) emits eight alpha particles as it decays through various intermediate elements to lead-206 (²⁰⁶Pb).  Each alpha particle is a helium-4 nucleus (⁴He), consisting of two protons and two neutrons.  Each explosively expelled ⁴He nucleus eventually comes to a stop, either within the zircon or in the surrounding material.  There it quickly gathers two electrons and becomes a neutral helium atom.

Helium is a lightweight, fast-moving atom that does not form chemical bonds with other atoms.  It can diffuse through solids relatively fast, meaning that helium atoms wiggle through the spaces between atoms in a crystal lattice and spread themselves out as far from one another as possible.  For the same reason it can leak rapidly through tiny holes and cracks, making it ideal for leak detection in laboratory vacuum systems.  The diffusion and leakage rates are so great that believers in the billions of years had expected most of the helium produced during the alleged 4.5 billion years of the earth's existence to have worked its way out of the crust and into the earth's atmosphere long ago.

But the helium is not in the earth's atmosphere!  When non-specialists hear that, they usually assume that (A) helium has risen to the top of the atmosphere as it would in a balloon, and (B) most of the helium has then leaked from the top of the atmosphere into space.  However, assumption (A) is wrong, because unconfined helium spreads throughout the atmosphere from top to bottom, like any other gas. 

On assumption (B), the simple kinetic theory of gases says the loss of neutral helium atoms into space would be much too small to account for the missing helium.  In 1957 Melvin Cook, a creationist chemist, pointed out this problem in the prestigious journal Nature [4].   In 1990 Larry Vardiman, a creationist atmospheric scientist, calculated that even after accounting for such slow leakage into space, the earth's atmosphere has only about 0.04% of the helium it should have if the earth were billions of years old [23].

Until the 1970’s, uniformitarians (see next section) had no good answer.   However in recent decades they have been trying to evolve one.  Satellite data [1,13] show that ions (electrically charged atoms) of helium (and other gases) move back and forth along the earth’s magnetic lines of force above much of the atmosphere.  Some space plasma physicists theorize that storms of particles from the sun blow the helium ions loose from the lines of force outward into space frequently enough “to balance the [assumed] outgassing from the earth’s crust” [16].  The theory is very complex, and no creationist expert in the field has yet reviewed it to see whether it is well founded.

Rapid helium leaks into space are essential to uniformitarians, but slow leaks are not essential to creationists.  If the leakage turns out to be slow, it would bolster our case here.  But fast leakage would not damage our case.  The next section offers evidence for a much simpler explanation of the missing atmospheric helium: most of the radiogenic (nuclear decay generated) helium has not entered the earth’s atmosphere.  It is still in the earth’s crust and mantle—much of it still in the zircons.  In this paper we argue that the helium has not had enough time (less than 14,000 years) to escape the zircons, much less the crust.

2.  THE HELIUM IS STILL IN THE ZIRCONS

In the 1970's, geoscientists from Los Alamos National Laboratory began drilling core samples at Fenton Hill, a potential geothermal energy site just west of the volcanic Valles Caldera in the Jemez Mountains near Los Alamos, New Mexico (Fig. 2).  There, in borehole GT-2, they sampled the granitic Precambrian basement rock, which we will refer to as the Jemez Granodiorite.  It has a radioisotopic age of about 1.5 billion years, as measured by various methods using the uranium, thorium, and lead isotopes in the zircons themselves [25].  The depths of the samples varied from near the surface down to 4.3 kilometers, with temperatures from 20°C to 313°C.  The Los Alamos team sent some of these core samples to Oak Ridge National Laboratory for isotopic analysis.

Most of the zircons were in biotite [10], a black mica common in granitic rock.  At Oak Ridge, Robert Gentry, a creationist physicist, crushed the samples (without breaking the much harder zircon grains), extracted a high-density residue (because zircons have a density of 4.7 grams/cm³), and isolated the zircons by microscopic examination, choosing crystals about 50-75 μm long.  The zircon masses were typically on the order of micrograms.  The Oak Ridge team then heated the zircons to 1000°C in a mass spectrometer and measured the amount of helium 4 liberated.  In 1982 they published the data in Geophysical Research Letters [9].  Table 1 details their results.

Table 1.  Helium Retentions in Zircons from the Jemez Granodiorite

The first column itemizes the samples analyzed.  The second and third columns show the depth and temperature of each sample in situ.  The fourth column shows the volume (at standard temperature and pressure) of helium liberated in the lab per microgram of zircon.

The fifth column is the ratio of the observed quantity of helium Q (total number of helium atoms in the crystal) to the calculated quantity Q₀ that the zircons would have accumulated and retained if there had been no diffusion.  The Los Alamos team measured the amount of radiogenic lead in zircons 2.9 km deep in the same borehole and same granodiorite [25], and the Oak Ridge team confirmed those figures with their ion microprobe [10].  Because the various decay chains generate an average of 7.7 helium atoms per lead atom produced, Gentry and his colleagues were able to calculate Q₀ from the amount of lead in the zircons.  In doing so, they compensated for the estimated loss of alpha particles emitted from near the edges of the zircons out into the surrounding material.

The Oak Ridge team estimated that uncertainties in calculating Q₀ might limit the accuracy of the ratio Q/Q₀ to ±30%.  The sixth column of the table shows the resulting estimated errors in the ratios.

Samples 1 through 6 came from the granodiorite, but sample zero came from larger zircons in a surface outcrop of an entirely different rock unit.  For that rock unit U/Th/Pb information was not available, making an estimate of Q₀ not feasible.  Lacking a ratio, we cannot use sample zero in the calculations.

Samples 5 and 6 had the same amount of helium.  Gentry and his colleagues noted that helium emerged from those samples in shorter bursts than the other samples, indicating a different distribution of helium within those zircons.  In section 6, we will show that the amount of helium from sample 5 is just about what would be expected from the trend in the cooler samples.  But we allow for the possibility of its error being considerably larger than the cooler samples.

According to the thermal behavior outlined in the next section, we would ordinarily expect that the hotter sample 6 would have much less helium than sample 5.  The fact that the helium content did not decrease suggests that some additional effect may have occurred which limited the outflow of helium from the zircon.  In section 6 we suggest a likely explanation.

The above considerations suggest that we can use samples 1 through 5 in a theoretical analysis with ordinary diffusion.  We will treat sample 6 as a special case.

Samples 1 through 3 had helium retentions of 58, 27, and 17 percent. The fact that these percentages are high confirms that a large amount of nuclear decay did indeed occur in the zircons.   Other evidence strongly supports much nuclear decay having occurred in the past [14, pp. 335-337].  We emphasize this point because many creationists have assumed that "old" radioisotopic ages are merely an artifact of analysis, not really indicating the occurrence of large amounts of nuclear decay.  But according to the measured amount of lead physically present in the zircons, approximately 1.5 billion years worth — at today’s rates — of nuclear decay occurred.  Supporting that, sample 1 still retains 58% of all the alpha particles (the helium) that would have been emitted during this decay of uranium and thorium to lead.

It is the uniformitarian assumption of invariant decay rates, of course, that leads to the usual conclusion that this much decay required 1.5 billion years.  Uniformitarianism is the prevalent belief of this age that “all things continue as they were from the beginning” [II Peter 3:4], denying the possibility of any physical interventions by God into the natural realm.  Uniformitarians interpret scientific data to support their idea of cosmic and biological evolution during billions of years of imagined time.  We maintain that their interpretations are a distortion of observational data all around us.  As the Bible predicted [II Peter 3:5-6], uniformitarians willingly ignore “elephant in the living room” evidence for a recent creation and a worldwide catastrophic flood.   In this paper we will include their assumption of billions of years of time and solely natural processes in the uniformitarian model we construct for diffusion.

Getting back to the helium data, notice that the retention levels decrease as the temperatures increase.  That is consistent with ordinary diffusion: a high concentration of helium in the zircons diffusing outward into a much lower concentration in the surrounding minerals, and diffusing faster in hotter rock.  As the next section shows, diffusion rates increase strongly with temperature. 

In later sections, we will show that these large retentions are quite consistent with diffusion taking place over thousands of years, not billions of years.

3.  HOW DIFFUSION WORKS

If the reader is not very familiar with diffusion and wants to know more, we recommend a very clear little book, Atomic Migration in Crystals, written for non-experts [11].  Figure 3, adapted from that book [11, p. 39, Fig. 23], illustrates how an atom diffuses through a solid crystal lattice of other atoms.   Figure 3(a) shows a helium atom initially at position A, surrounded by a cell of lattice atoms.  The lattice atoms repel the helium atom, tending to confine it to the center of the cell, where the repulsion balances out in all directions.  Heat keeps the atoms of the lattice vibrating at its various resonant frequencies. The vibrating atoms continually bump into the helium atom, jostling it from all sides.  The higher the temperature, the more vigorous the jostling.

Every now and then, the lattice atoms will bump the helium atom hard enough to push it into the "activated" position B, midway between cells.  The lattice atoms must give the helium enough kinetic energy to overcome the repulsive potential energy barrier between the cells, which we have shown in Figure 3(b).  This required amount of kinetic energy, E, is called the activation energy.  If the lattice atoms have given any more energy than E to the helium atom, it will not stop at position B.  Instead, it will continue on to position C at the center of the adjacent cell.  The helium atom has thus moved from one cell to the next.

If there is an initially high concentration of helium atoms in one part of the crystal, these random motions will eventually spread — i.e., diffuse — the helium more uniformly though the crystal and out of it.  Let us define C(x, y, z, t) as the concentration, the number of helium atoms per unit volume, at position (x, y, z) at time t.  Many textbooks show that when diffusion occurs, the time rate of change of C is proportional to the “sharpness” of the edges of the distribution of helium, or more mathematically, proportional to the Laplacian of C, ѲC :

(1a, b)

Equation (1a), called the "diffusion equation", occurs frequently in many branches of physics, for example to describe heat conduction in solids.  Specialists in the diffusion of atoms through materials call it "Fick's Second Law of Diffusion".  The factor D, the diffusion coefficient (or “diffusivity”), has dimensions of cm² (or m²) per second.  (Most of the diffusion literature still uses centimeters and calories instead of meters and joules).  Very often it turns out that at high temperatures, the diffusion coefficient depends exponentially on the absolute temperature T (degrees kelvin above absolute zero):

(2)

where R is the universal gas constant, 1.986 calories per mole-kelvin (8.314 joules per mole-kelvin).   The constant D₀ is independent of temperature.  The “intrinsic” activation energy E₀ typically is between 10 and 100 kilocalories per mole (about 40 and 400 kilojoules per mole).  Section 10 discusses how these quantities are related to the geoscience concept of closure temperature, and it shows why the concept is irrelevant to our conclusions.

If the crystal has defects such as vacancies in the crystal lattice, impurities, dislocations, or grain boundaries, then the diffusion coefficient equation will have a second term related to the defects:

(3)

The defect parameters (D₁ and E₁) are almost always smaller than the intrinsic parameters (D₀ and E₀):

(4)

The typical Arrhenius plot in Figure 4(a) shows how the diffusion coefficient D of eq. (3) depends on the inverse of the absolute temperature, 1/T.  Because the plot uses a logarithmic scale for D and a linear scale for 1/T, each term of eq. (3) manifests itself as a straight line in the temperature region where it is dominant.  (Plotting with T instead of 1/T would make the lines curved instead of straight.) The slopes are proportional to the activation energies E₀ and E₁.  The intercepts with the vertical axis, where 1/T is zero, are the parameters D₀ and D₁.

The intrinsic line has a steep slope and a high intercept, while the defect line has a shallow slope and a low intercept.  Starting on the right-hand side of the graph, at low temperatures, let us increase the temperature, moving to the left.   When the temperature is high enough, we reach a region, the “knee”, where the two terms of eq. (3) are about equal.  To the left of that region, at high temperatures, the intrinsic properties of the crystal dominate the diffusion.  To the right of the knee, at lower temperatures, the defects dominate.  Because defects are very common in natural crystals, this two-slope character is typical [11, pp. 102, 126].

For a given type of mineral, the location of the knee can vary greatly.  It depends on the value of D₁, which depends on the amount of defects in the particular crystal.  The more defects there are, the higher D₁ is.  If we increase the number of defects, the defect line moves upward (keeping its slope constant) on the graph, as Figure 4(b) illustrates.

In the case of zircons containing radioisotopes, the main cause of defects is radiation damage, so highly radioactive (“metamict”) zircons will have a large value of D₁, causing the defect line to be higher on the graph than for a low-radioactivity zircon.

4.  EARLY ZIRCON DATA WERE AMBIGUOUS

In 1970 Sh. A. Magomedov, a researcher in Dagestan (then part of the Soviet Union) published diffusion data for radiogenic lead and helium in highly metamict (radiation-damaged) zircons from the Ural Mountains [18].  These were the only helium-in-zircon diffusion data we could find during an extensive literature search we did in 1999.

Magomedov was mainly interested in lead diffusion, so he did not list his helium data explicitly in a table.  Instead he showed them in a small graph, along with data for lead diffusion and electrical conductivity, s.  His label for the ordinate was ambiguous: “ln(D, s)”.  In scientific literature “ln” with no further note usually means the natural logarithm (base e).  The common logarithm (base 10) is usually shown as “log”.  If we assume Magomedov was reporting ln_eD, the resulting diffusion coefficients would be very high, as the triangles and dotted line near the top of Figure 5 show.  The RATE book shows that interpretation [14, p. 347, Fig. 6].  Another interpretation is that Magomedov was reporting ln_e(D/a²), where a is the effective radius of his zircons, about 75 μm.  As Figure 5 shows (circles and thin solid line near middle), that still gives rather high diffusion rates in the temperature range of interest to us.

Based on those supposed high rates, we assumed in our first theoretical model [14, pp. 346 –348] that the zircons were a negligible impediment to helium outflow, compared to the minerals around them.  But in 2001 we received a preprint of a paper [19] listing new helium diffusion data in zircons from several sites in Nevada.  Figure 5 shows some of that data (Fish Canyon Tuff sample FCT-1) as a line of solid dots.  These data were many orders of magnitude lower than our interpretation of Magomedov’s graph.  The Russian data would agree with the Nevada data if we re-interpret Magomedov’s label as meaning “log₁₀D”, the common logarithm of D.  Figure 5 shows that interpretation near the bottom (squares and thick solid line).  The small difference between the high-slope “intrinsic” parts of the Russian and Nevada data is easily attributable to site-to-site differences in composition.  The nearly horizontal part of the Russian data is probably a “defect” line due to much radiation damage (see end of previous section).

The new data and our new interpretation of the old data imply that zircon is not a negligible impediment to helium diffusion.  In this paper we have changed our theoretical model to account for that fact.  As we will show in later sections, our new interpretation of the Russian data is still five orders of magnitude too high for uniformitarian models.  But it is quite compatible with creationist models and time scales.

5.  DATA FOR MINERALS FROM THE JEMEZ GRANODIORITE

Measurements of noble gas diffusion in a given type of naturally occurring mineral often show significant differences from site to site, caused by variations in composition.  For that reason it is important to get helium diffusion data on zircon and biotite from the same rock unit (the Jemez Granodiorite) which was the source of Gentry’s samples.  Accordingly, in 2000 the RATE project commissioned such experimental studies.

Los Alamos National Laboratory was kind enough to give us core samples of granodiorite from the same borehole, GT-2, and from a similar depth, about 750 meters.  The geology laboratory at the Institute for Creation Research extracted the biotite using heavy liquids and magnetic separation.   Using similar methods, Activation Laboratories, Ltd., in Ontario, Canada, extracted the zircons and chose three of them for isotopic analysis.  Appendix A gives those results, which agreed fairly well with the lead-lead dates published by Los Alamos National Laboratory for the same site [25].  We reserved the rest of the zircons, about 0.35 milligrams, for diffusion measurements.

Through a small mining company, Zodiac Minerals and Manufacturing, we contracted with Kenneth A. Farley of the California Institute of Technology (Division of Geological and Planetary Sciences) to measure the diffusion coefficients of the zircon and biotite from the Jemez site.  He is a recognized expert on helium diffusion measurements in minerals, having many publications related to that field.  As we wished, Zodiac did not tell Farley they were under contract to us, the goals of the project, or the sites of the samples.  We have encouraged him to publish his measurements and offered to send him the geologic site information if he does so.  Appendices B and C list his data in detail.

Figures 6(a) and 6(b) are Arrhenius plots of the most relevant data for zircon and biotite, respectively.  The zircon data are from the Jemez Granodiorite in New Mexico, the Fish Canyon Tuff in Nevada, and the Ural Mountains in Russia (the re-interpreted Magomedov data).  The first two studies are for essentially the same size crystals (average length ~60 μm, a @ 30 μm, sect. 6).  The Russian study was for crystals ~150 μm long.

The biotite data are from the Jemez Granodiorite.  Those, and similar data we obtained (see Appendix B) for biotite from the Beartooth Gneiss in Wyoming, are the only data for that mineral we know of.  For comparison to the biotite data, we have also included published data for muscovite, a different mica [17].

Notice that all the sets of zircon data agree fairly well with each other at high temperatures.  At 390ºC (abscissa = 1.5), the Russian data have a knee, breaking off to the right into a more horizontal slope for lower temperatures.  That implies a high number of defects (see sect. 4), consistent with the high radiation damage Magomedov reported.  The Nevada and New Mexico data go down to 300ºC  (abscissa = 1.745) with no strong knee, implying that the data are on the intrinsic part of the curve.  A least-squares fit of eq. (2) to the New Mexico (Jemez Granodiorite) zircon data gives the following diffusion parameters:

(5a)

However, there appears to be a slight decrease of slope in the data below 450ºC.  Later on we will need a fit at temperatures below that.  The best-fit parameters from 440ºC down to 300ºC are:

(5b)

Because the New Mexico zircons are radioactive, they must have some defects and should have a knee at some lower temperature than 300ºC.  We have recently requested that Farley get additional data from 100ºC to 300ºC.  But as of February 2003, we do not have reliable data for that range.

The muscovite and biotite data are consistent with each other.  In the low temperature range of interest, the New Mexico biotite has a somewhat higher diffusion coefficient than the zircons.  That means the biotite, while not being negligible, did not impede the helium outflow as much as the zircon did.

6.  A NEW CREATION MODEL

We need a theoretical framework in which we can interpret the diffusion data of the previous section.  As we mentioned at the end of section 4, in our first creation model we wrongly assumed that the zircons were a negligible impediment to the helium diffusion.  In this section we construct a new creation model.

As before, the creation model starts with a brief burst of accelerated nuclear decay generating a high concentration C₀ of helium uniformly throughout the zircon (like the distribution of U and Th atoms), but not in the surrounding biotite.  After that the helium diffuses out of the zircon into the biotite for a time t.  As in our previous model, we chose t = 6000 years.  The time is short enough that the additional amount of helium generated by normal nuclear decay would be small compared to the initial amount.  We assume the temperatures to have been constant at today’s values.  We will show in section 7 that this assumption is generous to uniformitarians.

Because the biotite diffusion coefficients are not too different from the zircon coefficients, we should have a model accounting for two materials.  Diffusion in zircon is isotropic, with helium flowing essentially at the same rate in all three directions.  Diffusion in biotite is not isotropic, because most of the helium flows two-dimensionally along the cleavage planes of the mica.  But accounting for anisotropy in the biotite would be quite difficult, so we leave that refinement to the next generation of analysts. To keep the mathematics tractable, we will assume spherical symmetry, with a sphere of zircon of effective radius a inside a spherical shell of material having an outer radius b, as Figure 7 shows.  Then the concentration C will depend only on time and the distance r from the center

Let us consider the values we should assign to a and b.  Magomedov’s zircons were between 100 and 200 μm long [18, p. 263], for an average length of about 150 μm.  He assigned the crystals an effective radius of half the average length, or 75 μm.  Gentry selected zircons between about 50 μm and 75 μm, for an average that we will round off to 60 μm.  Half of that gives us an effective radius for our analysis of the Jemez zircons:

(6)

Biotite in the Jemez Granodiorite is in the form of flakes averaging about 0.2 mm in thickness and about 2 millimeters in diameter.  Because the cleavage planes are in the long direction, and diffusion is mainly along the planes, the diameter is the relevant dimension for diffusion.  That gives us an outer radius of:

(7)

Because b is more than 32 times larger than a, the disk-like (not spherical) volume of biotite the helium enters is more than 1000 (~32 squared) times the volume of the zircon.  This consideration affects the boundary conditions we choose for r = b, and how we might interpret sample 6 (see sect. 2), as follows.

Suppose that helium could not escape the biotite at all.  Then as diffusion proceeds, C would decrease in the zircon and increase in the biotite, until the concentration was the same throughout the two materials.  After that C would remain essentially constant, at about 0.001 C₀.  The fraction Q/Q₀ remaining in the zircon would be about 0.001, which is just what Gentry observed in sample 6.

So a possible explanation for sample 6 is that diffusion into the surrounding materials (feldspar, quartz), and leakage (along grain boundaries) was slow enough (during the relatively short time t) to make the outflow of helium from the biotite negligible.  For that sample, the temperature and diffusion coefficient were high enough for helium to spread uniformly through both zircon and biotite during that time.

Our measurements (see Appendix B) showed that the helium concentration in the Jemez biotite at a depth of 750 meters was small, only about 0.32 ´ 10^-9 cm³ (at STP) per microgram.  Taking into account the difference in density of biotite and zircon (3.2 g/cm³ and 4.7 g/cm³), that corresponds to almost exactly the same amount of helium per unit volume as sample 6 contained.  That suggests the zircon and biotite were near equilibrium in sample 6, thus supporting our hypothesis.

At lower temperatures, for helium retentions greater than 0.001, C in the biotite would be lower than C in the zircon.  In that case the boundary at r = b would not significantly affect the outflow of helium from the zircon.  We will assume this was approximately true for sample 5 also, but not for sample 6.  To simplify our analysis for samples 1 through 5, we will assume the usual boundary condition, that the concentration C(r) falls to zero at radius r = b:

(8)

For the initial conditions, we assume that the concentration is a constant, C₀, inside the zircon, and zero outside it:

(9a, b)

After time zero, there also must be continuity of both C and helium flow at r = a.  We need a solution to the diffusion equation, (1), in its radial form, for the above boundary conditions.  In 1945, R. P. Bell published such a solution for the corresponding problem in heat flow [2, p. 46, eq. (4B)].  His solution, which is mathematically complex, allows for different diffusion coefficients in the two regions.  We will simplify the solution considerably by making the diffusion coefficients the same in both regions.  Because the diffusion coefficient of biotite is somewhat higher than that of zircon at the temperatures of interest, our solution will have slightly slower (no more than 30% slower) helium outflows and correspondingly longer times than the real situation.  But because