What is it that feeds our battle, yet starves our victory?

January 6 Tapes?

Where are the tapes? Anyone, Anyone? Bueller? Johnson??

Paging Speaker Johnson…this is your conscience calling you out on broken promises.

Evading Reality

Many things the Left believes are simply not true. Right now the focus is on the size and scope of our government, and the many many billions of dollars the government has been spending on no-one-knew-what. None of that money is going to a key role of government. Which, after all, has the sole purpose of protecting rights.

And if you, Leftist Lurker, want to dismiss this as dead white cis-male logic…well, you can call it what you want, but then please just go fuck off. No one here buys that bullshit–logic is logic and facts are facts regardless of skin color–and if you gave it a moment’s rational thought, you wouldn’t either. Of course your worthless education never included being able to actually reason–or detect problems with false reasoning–so I don’t imagine you’ll actually wake up as opposed to being woke.

As Ayn Rand would sometimes point out: Yes, you are free to evade reality. What you cannot do is evade the consequences of evading reality. Or to put it concretely: You can ignore the Mack truck bearing down on you as you play in the middle of the street, you won’t be able to ignore the consequences of ignoring the Mack truck.

And Ayn Rand also pointed out that existence (i.e., the sum total of everything that exists) precedes consciousness–our consciousnesses are a part of existence, not outside of it–therefore reality cannot be a “social construct” as so many of you fucked-up-in-the-head people seem to think.

So much for Leftist douchebag lurkers. For the rest of you, the regular readers and those lurkers who understand such things, well here we go for another week of WINNING against the Deep State.

I confess that the novelty has not worn off.

Justice Must Be Done.

The prior election must be acknowledged as fraudulent, and steps must be taken to prosecute the fraudsters and restore integrity to the system.

Yes, we won this time around. Not only did we win, we got to KEEP that win instead of having it stolen from us.

But no one should imagine that that’s the end of electoral fraud. Much work needs to be done to ensure it doesn’t just happen again next time around. And incidentally to rescue those states currently in the grips of self-perpetuating fraud, where the people who stole the last election, make sure it’s easier to steal the next one.

This issue, though it’s not front-and-center right now, is not going away, and if we ignore it, we’ll pay the price. See the article above about the consequences of evading reality.

Lawyer Appeasement Section

OK now for the fine print.

This is the WQTH Daily Thread. You know the drill. There’s no Poltical correctness, but civility is a requirement. There are Important Guidelines,  here, with an addendum on 20191110.

We have a new board – called The U Tree – where people can take each other to the woodshed without fear of censorship or moderation.

And remember Wheatie’s Rules:

1. No food fights
2. No running with scissors.
3. If you bring snacks, bring enough for everyone.
4. Zeroth rule of gun safety: Don’t let the government get your guns.
5. Rule one of gun safety: The gun is always loaded.
5a. If you actually want the gun to be loaded, like because you’re checking out a bump in the night, then it’s empty.
6. Rule two of gun safety: Never point the gun at anything you’re not willing to destroy.
7. Rule three: Keep your finger off the trigger until ready to fire.
8. Rule the fourth: Be sure of your target and what is behind it.

(Hmm a few extras seem to have crept in.)

Spot Prices.

Kitco Ask. Last week:

Gold $2,985.50
Silver $33.87
Platinum $1004.00
Palladium $988.00
Rhodium $5,700.00
FRNSI* 143.424-
Gold:Silver 88.146-

This week, markets closed as of 3PM MT.

Gold $3,024.40
Silver $33.10
Platinum $988.00
Palladium $984.00
Rhodium $6,100.00
FRNSI* 145.305+
Gold:Silver 91.372-

Gold was actually up in the 3040s Wednesday and Thursday but dropped on Friday, (which has been a common pattern for years). Platinum went nowhere. Silver is actually down. My understanding is that gold’s rise has been driven by central bank purchases. Since they don’t bother with silver, that explains why silver is basically tango uniform. The gold:silver ratio has been above 100 before, and I would be surprised if it doesn’t get there again soon.

*The SteveInCO Federal Reserve Note Suckage Index (FRNSI) is a measure of how much the dollar has inflated. It’s the ratio of the current price of gold, to the number of dollars an ounce of fine gold made up when the dollar was defined as 25.8 grains of 0.900 gold. That worked out to an ounce being $20.67+71/387 of a cent. (Note gold wasn’t worth this much back then, thus much gold was $20.67 71/387ths. It’s a subtle distinction. One ounce of gold wasn’t worth $20.67 back then, it was $20.67.) Once this ratio is computed, 1 is subtracted from it so that the number is zero when the dollar is at its proper value, indicating zero suckage.

The Math Behind Radiometric Dating

This is going to be a bit brutal for those who are math phobic. For the rest it will reward careful attention.

Radiometric dating involves, at the very least, measuring the quantities of parent isotopes and daughter isotopes. In some situations it gets more complicated than that, and to be honest those situations are actually the usual ones.

Introducing Uranium-Lead Dating

So let’s take an actual case as an example, a method called uranium-lead dating, because the parent isotope is uranium, and the daughter isotope is lead. Actually there are two sets of parent-daughter isotopes in this case: uranium-238 with daughter isotope lead-206, and uranium-235 to lead-207. (Check: The differences between the mass numbers must be a multiple of 4, because a change of 4 is the effect alpha decay will have. And yes it looks like I got the right numbers. I saw a video recently that swapped the lead isotope numbers; an easy mistake to make.) This makes it a favorite because you can do two datings with one sample.

For simplicity we will start by considering only the uranium-238 to lead-206 pairing. Most dating methods use an isotope that decays directly into the daughter product. U-238 does not do this. It decays into thorium-234 by alpha decay, then there are a chain of thirteen more decays (there are alternate paths, but all are thirteen steps long, not including that first alpha decay) for total of fourteen steps before it gets to lead-206.

[Check this one too. The difference in mass numbers is 238-206=32; dividing by four that is eight alpha decays. But eight alpha decays reduces the number of protons by sixteen, so if that’s all that happened uranium-238 would become osmium-206. (Uranium is element 92; 92-16=76; 76 is osmium.) In order to actually end up with lead (82), we need to, somewhere along the way, get six additional protons; we do that with negative beta decay (β^–) which turns a neutron into a proton. So six beta decays are needed. Eight alpha decays + six negative beta decays = 14 total decays.]

Is this huge number of decays a problem? In principle it could be, but in this case it’s not. The initial uranium-238 to thorium-234 decay is very slow, with a half life of 4,468,000,000 years. Compared to this the others are practically instantaneous, with uranium-234 to thorium-230 requiring 245,000 years and thorium-230 to radium-226 requiring 75,400 years. Radium-226 to radon-222 has a half life of 1600 years. All of the other decays happen in less than a year and some take less than a second. So, basically, once a uranium-238 atom cuts loose and spits out an alpha particle, it’s going to be a lead-206 atom in less than a couple of million years, tops; which in comparison to the half life of uranium-238 (4,468 million years) is negligible.

Half Lives and the Radioactive Decay Equation

So, let me remind you about half lives. This is the amount of time it takes for half of the atoms in the sample to decay. It’s a little tricky wrapping one’s head around this at first. Surely, if half of the atoms are gone in 4,468 million years, the other half ought to be gone in another 4,468 million years. But it doesn’t work that way.

Each atom is independent of the others, and any given U-238 atom has a totally random 50 percent chance of going “kablooey” sometime within the next 4,468 million years. It could be right now while you’re watching it, or it could be 4,467.999 million years from now. If it doesn’t happen between now and then, guess what? You are still looking at an atom that has a 50 percent chance of going “kablooey” in the next 4,468 million years. The past history doesn’t matter. If the atom is 100,000 million years old already, versus created last year, it still has the same changes of blowing up in the same period of time.

Go to multiple atoms in a sample; billions, trillions or quadrillions of them [a one-gram sample of uranium-238 has 2.53 sextillion (or 2.53 trillion billion) atoms in it]. Since each individual atom has a 50 percent chance of blowing up in the next 4,468 million years, half of them, you don’t know which ones in advance, but half of them will do so. OK, so let’s say your very distant descendant takes your one gram sample and separates out all of the lead and intermediate decay products (the other things on the chain), and he has half a gram of uranium-238. His past does not matter; he has a half-gram sample of uranium-238 and half of it will decay in the next 4,468 million years, leaving his distant descendant with a quarter of a gram of uranium. (And maybe by that time an honest Leftist will have been born.)

The following GIF is a simulation of radioactive decay, with four samples each of four and four hundred atoms. The number at the top is the number of elapsed half lives. (It runs a bit fast unfortunately, so watch closely.)

Another way to talk about it is to say that, for a given isotope, the number of decaying atoms in some time interval is proportional to the current number of atoms.

In fact since the decaying atoms reduce the size of the sample, the number of decays is the rate of change of the size of the sample. Twice as many decays? Twice as fast a reduction.

In order to truly understand radiometric dating we have to understand this, and be able to express it mathematically. And I want you to truly understand it.

Intense Math Alert! Go to the bolded paragraph below to skip this.

Expressing this semi-mathematically, using ∝ for “is proportional to”:

(Number of decays in a specified time interval) ∝ (Current size of sample)

Or:

(rate of change of the sample size) ∝ (Current size of sample)

Or a bit more formally, with N(t) being the number of atoms in the sample at time t, expressed in calculus notation:

−dN/dt ∝ N

(The negative sign is because the change is in the downward direction, yet we will want to use a positive constant when we introduce it shortly.)

Or we can make it an equation by creating a proportionality constant, λ, (Greek lower-case lambda); in this case it’s called the decay constant.

−dN/dt = λN

And rearrange just a bit:

dN/N = −λdt

Welcome to the world of differential equations. This one is easy to solve, since the only way a function can be its own derivative is if the function is e^t, and if you want a function to be its own derivative but multiplied by some number is for the function to be something like e^kt., in which case the derivative will be ke^t. So taking advantage of this fact, we get:

Key Equation: N(t) = N₀e^−λt

Where N₀ is the size of the sample at some particular time, and N(t) is the size of the sample at some earlier or later time, t.

[e is the base of the natural logarithms and I talk about it (and logarithms) here: https://www.theqtree.com/2023/05/20/2023%C2%B705%C2%B720-joe-biden-didnt-win-daily-thread/ and here: https://www.theqtree.com/2023/06/17/2023%c2%b706%c2%b717-joe-biden-didnt-win-daily-thread/.]

This concerns the so-called parent nuclide. The daughter product increases, of course. What’s the formula for that? It’s convenient that the number of atoms does not change; even if fewer and fewer of them are the parent isotope. In other words, the total number of atoms, parent + daughter, is always N₀ no matter how much (or how little) decay has happened. That means that to get the number of daughter atoms, you can simply subtract the number of parent atoms from the original number of parent atoms. I’m using subscripts p and d here to indicate number of parent and daughter atoms

N_d(t) = N₀ − N_p(t)

Substituting in the Key Equation for N_p:

N_d(t) = N₀ − N₀e^−λt

But this just begs to be simplified a bit:

N_d(t) = N₀(1 − e^−λt)

Earlier I was talking in terms of half lives, but we don’t see that here; we see this funky lambda constant instead. Can we get to a formula that uses half lives?

Yes but before we proceed I should say something about λ. We’re eventually going to want to put an actual numerical value on it, but it’s important to note that this is a number that applies to some given time interval. A second, a year, a million years. This number is actually the fraction of the sample that decays in whatever time interval you choose. So if a trillionth of the sample decays in one second, λ is one trillionth (0.000000000001), but to express that in years, you need to multiply it by 60 × 60 × 24 × 365.25, and if you want it to express it for millions of years, you have to multiply it again by another million. In this case we’re dealing with geology and our λ values will be set for million-year units.

The first step is to simply invert λ, defining a new constant τ:

τ = 1/λ

This gives you the average lifetime for an atom of the parent isotope, in whatever unit (seconds, years, millions of years, whatever) that you used for λ.

Note that this is not the same as half life. Half life is the time it takes for half of the atoms to go kablooey, but that’s not the average time it will take for one to do so. Some atoms ( one in 1024) will survive ten half lives, and they pull the average up. But it’s easy to get to the half life, t_1/2 from here–multiply by the natural log of 2 (about 0.693):

t_1/2 = τ ln(2) = ln(2) / λ

The reverse process:

λ = ln(2) / t_1/2

And it turns out that there’s a version of the Key Equation with the half life…actually two versions that are equivalent to each other.

Cumbersome Half Life Key Equation: N(t) = N₀e^{−ln(2)t/t_1/2}

This is ugly so it gets simplified as follows:

Simple Half Life Key Equation: N(t) = N₀ 2^-(t/t_1/2)

This one is intuitive in terms of half lives. Raise 1/2 to the power of the number of half lives that have elapsed to get the fraction of atoms remaining, then multiply by the original number. Or equivalently raise 2 to the negative power of the number of half lives.

So do we prefer working with λ or with t_1/2? Scientific calculators come with an e^x key; they never come with a 2x key–which forces us to work with that cumbersome formula above if we want to use half lives. (Honestly I’d write a program if it were a programmable calculator: store the half life in memory, and input the time, let the program do all the steps in the Cumbersome equation.) On the other hand I am having a very difficult time finding a table of isotope decay rates; half lives are easy to find.

END OF INTENSE MATH but be aware there will be a lot of applying the equations from here on out. That’s basically arithmetic, though, not differential equations.

For those of you rejoining us here, I’m going to repeat the formulas and definitions:

Decay Rate Key Equation: N(t) = N₀e^−λt

Cumbersome Half Life Key Equation: N(t) = N₀e^{−ln(2)t/t_1/2}

Simple Half Life Key Equation: N(t) = N₀ 2^-(t/t_1/2)

N(t) is the number of atoms of the parent isotope (the one that’s decaying away) at any given time t. N₀ is the original number of atoms of the parent isotope.

t_1/2 is the half life. λ is the decay constant, the fraction of the sample that decays in some specified time (seconds, or millions of years, as appropriate). Dividing 1 by λ gives τ, the average lifetime of an atom in the sample.

I rarely see λ used. But for uranium-238 it’s: 4.916 x 10^-18 (when working in seconds) and 1.551 x 10^-10 (when working in years). If you use the latter number you need to supply t in years, not seconds–which, let’s face it, is more likely what you want to do anyway. In fact, this is geology (or have you forgotten?): You probably want millions of years, in which case λ for uranium-238 is 1.551 x 10^-4.

In fact, let me supply the numbers for U-235, U-238, and thorium-232, since we’re going to be mentioning them at some point below.

(I here took the liberty of using “engineering” numbers rather than strict scientific notation (where the power is set so only one number is left of the decimal point) so that you could compare the decay rates more readily. The engineering mode uses powers that are multiples of three so it’s easy to write out a metric prefix, e.g, 1.32 × 10⁴ watts (scientific notation) becomes 13.2 × 10³ watts (engineering notation) or 13.2 kW (very readily read off the engineering notation).)

OK, there is some more math but at least I’m not slinging differential equations any more:

Determining Age (The Simple, Ideal Case)

Since we are dating a sample rather than predicting how much will be left after some time, these formulas are backwards. Instead of telling us how much is left after a known time has elapsed, we expect to know how much is left, and want to know how much time it took to get to that point.

Rearranging the Decay Rate Key Equation we get:

t = ln(N/N₀) / –λ

or, getting rid of the minus sign by taking the natural log of the reciprocal instead:

The Dating Equation: t = ln(N₀/N)/λ.

(“ln” is the natural log; the logarithm to the base e.)

OK so how do we apply this?

In principle, we can analyze some rock to determine how much U-235, Pb-207, U-238, Pb-206, and for bonus, Th-232 and Pb-208 is in it. We will want to know numbers of atoms–or at least the ratio of the number of atoms, not weight, but it’s easy enough to convert. We can then use the last formula above three times, remembering that (for the U-235 case) N₀ is actually the sum of the U-235+Pb-207 numbers, whereas N is just the U-235 count. So you have two (or three, if you are checking thorium as well) pairs of numbers; run the calculation with each one. You now have three answers; if they are all the same, you’re in business.

Here’s an example: A rock that happens to be 704 million years old. You don’t know this (real science doesn’t have the answers in the back of the book), but you want to, so you take a tiny sample. For now we’ll assume no pre-existing daughter nuclides, no losses of any atoms over time from the sample, and no contamination of the sample. You put that sample into a mass spectrometer, which vaporizes the sample, ionizes the atoms, and sends them past a magnet at high speed. The atoms, being charged, will follow curved paths past the magnet. Heavier atoms will be bent less by the magnet. We put a detector downstream and it notes how many atoms hit where an atom of mass 206 should strike. Also for 207, 208, 232, 235, and 238.

What does our scientist see? It so happens I picked that number for a reason; it’s the half life of U-235. So our scientist will see equal numbers of U-235 and Pb-207, say 5 million of each. It doesn’t matter, it’s the ratio between the two that matters, and in this case it’s 1:1. That will probably make him smile because he won’t even have to pull out his calculator for that one–he will already know the answer. But he decides to check that, so what about U-238 and Pb-206? He will see 11.54 atoms of Pb-206 for every hundred atoms of U-238. Say, 1 billlion atoms of U-238 and 115.4 million atoms of Pb-206 But in order to use the formula above, he needs N₀/N, the ratio of the remaining parent atoms (1 billion) divided by the total number of atoms involved in this (parent + daughter), to the number of remaining atoms. So what he wants is (1 billion + 115.4 million)/1 billion = 1,115,400,000/1,000,000,000 = 1.0577. When he plugs that into the dating equation t = ln(N₀/N)/λ, being careful to use the value of λ for U-238, he gets 703.98 million years. Now he’s really happy because his numbers match. If he checks Th-232/Pb-208 he’ll get (regardless of the actual amounts) 3.534 atoms of lead-208 for every hundred atoms of thorium-232. Using this, he gets N₀/N of 1.0353, divides by λ for Th-232 (49.33 × 10⁻⁶), hits the e^x key on his calculator…and the age comes out as 704.1 million years. Now he’s grinning ear to ear, because he took three sets of measurements, independent of each other, and got the same result every time. What could be easier? (Actually a lot of things could be easier; actually analyzing the sample in order to get those six numbers is painstaking work.)

That’s the third grade version of what is called “uranium-lead dating.”

[In the light of a joke Pat Frederick made last time, uranium-lead radiometric dating is when her father shoots you full of lead or depleted uranium because he caught you shooting his daughter full of something else. But that’s the high-school level version.]

How To Deal With Non-Ideal Cases

I said this was the third grade version. That’s because I made a bunch of assumptions for this case. I said so before, and now I am going to repeat them.

1. No initial daughter nuclides (in other words the rock contained no lead when it was formed).
2. Neither a) any lead nor b) any uranium (or thorium) leached out of the rock since it was formed, since that will mess up our ratios when we measure them.
3. No contamination of the sample either from natural processes or as it is collected and analyzed.

The problem is when dealing with rocks it’s never that tidy, though it can get close. We cannot simply assume that the sample started out with no daughter isotope in it. Or make any of those other assumptions, at least not without justification.

So when uranium-lead dating is done (in the lab, not in her house), it’s usually done with the mineral zircon. The chemical formula for this is ZrSiO₄. It’s a silicate of zirconium, element #40. Here is an insanely nice specimen of zircon:

But hold on here. There’s no uranium in this mineral!

There’s no uranium, if it’s pure. However these crystals form in a mass of molten rock (magma), and this is Planet Earth unfiltered and unpurified. There will, therefore, almost always be impurities in it. (This is why pure white diamonds are very valuable; they have little to no impurities in them and that is a rare situation indeed.)

Zircon crystals will form in any igneous rock as it solidifies from magma. In fact, they’re practically the first thing that will form. (This is good because it’s easier for a crystal in still-liquid medium to reject impurities than it is, if almost everything surrounding it is already solidified.) They typically end up being the size of sand grains, and so any sizeable igneous rock will have a number of them in it. Zircons are also harder than quartz, with a Mohs hardness of 7.5 vs. quartz’s 7.0. (They will scratch steel and glass.) Which means they will be hard to damage, and can erode out of an igneous rock and end up incorporated in a sedimentary rock, essentially undamaged.

As it happens, if there is any uranium (either isotope, it doesn’t matter because we’re doing chemistry at the moment) in the magma, it can be incorporated in the zircon quite readily. So can thorium. The uranium atoms end up as part of the crystal, replacing some of the zirconium. But the cool thing about it, and the reason we want zircon crystals, is that lead is rejected as the crystal forms. So the innards of the new crystal will contain some uranium, and no lead whatsoever.

Furthermore, uranium won’t leach out of the zircon crystals over time.

That’s handy! And sedimentary rocks will have the crystals too, once they erode from igneous rocks. The thing to remember about the crystals in sedimentary rock is that dating those crystals will not give you the age of the sedimentary rock, but rather the age of some igneous rock that eroded, the grains from which became part of the sedimentary rock. They will put a maximum on the age of the sedimentary rock, but no minimum (maybe it formed last Tuesday).

This is a bit of a pain, because we find fossils, including index fossils, almost entirely in sedimentary rock. It’s not insurmountable, but that’s a topic for a future post. For now let’s stick to igneous rock.

If we take care to use the innards of the zircon crystal, rather than the surface, and run our lab like a clean room, we can reduce the possibility of contamination. If there is contamination in spite of all of these, we won’t get consistent answers (as we did in our example above) and can just disregard the results from that sample.

We’ve taken care of every assumption, except for 2a: We don’t know whether or not any lead leached out of the crystal after it was formed. And that does happen. Zircon crystals don’t like lead, so they’ll push it out if they have a chance, while hanging on to the uranium and thorium that’s still left.

One thing that can cause this is if the zircon is heated to over 900 C after it is formed, like happens when the rock it is part of is transformed into a metamorphic rock. Also, ironically, the uranium’s radiation can actually damage the crystal as it decays, allowing lead to leak out at lower temperatures. (And remember, every uranium-238 atom that decays emits radiation fourteen times!)

OK so how do we deal with this?

Let’s go back to our example of the 704 million year old unicorn rock that had no issues with it. Make it a rock with some number of zircon crystals, and we’ll use the crystals.

The crystals will all have different rates of lead loss, slightly different but different enough for this to work. So let’s take a specific crystal, and let’s say it has lost fifty percent of its lead.

How does a scientist use this to date the rock?

Let’s first see what he measures.

In our previous example, the scientist saw 5 million atoms each of U-235 and Pb-207. This time, though, unbeknownst to the scientist, half of the lead leaked out before the sample was collected. So what he sees instead is 5 million atoms of U-235 and 2.5 million atoms of Pb-207. It looks like N₀ is 7.5 million (not 10 million) so he does the division N₀/N and comes up with 3/2 (instead of 2).

He plugs that into the Dating Equation:

t = ln(N₀/N)/λ

using λ for U-235 and gets: 411.8 million years (probably rounding it to 412).

We happen to know this is wrong. He doesn’t.

Not yet, anyway.

Next, he works U-238 and Pb-206. This time he sees a billion atoms of U-238, but instead of 115.4 million atoms of Pb-206, he only sees 57.7 million. So he is dividing 1,057,700,000 by 1,000,000,000, and getting 1.0577. Plugging that into the Dating Equation with the decay constant for U-238, he gets 361.597…okay, 362 million years. Doing the same exercise with thorium (I’ll save you the gory details) yields an age of 355.1 million years, or 355 million.

These numbers are all over the freaking map. None of them are even remotely right…and this guy has been doing this sort of work for more than two weeks so he knows that this variation actually means he’s nowhere close, and he’s dealing with crystals that lost some of their lead.

What to do now?

Concordia Diagrams to the Rescue

Analyze another zircon crystal, and another. Let’s say the next one has only 40 percent loss of the lead. In that case the U-235 date is 477 million years, the U-238 date is 432 million years, and the thorium date is 425 million years.

He accumulates at least a few of these sets of data.

OK now the mathematical tool comes in. Bring out a sheet of graph paper and label the horizontal Pb-207/U-235 and the vertical Pb-206/U-238.

Before doing anything with the data collected, we have a bit of prep to do. We have to draw a line that shows the values where the two dates are in “concord” (agreement) with each other, and this will be called a “concordia diagram”

For instance, for an age of 100 million years, a perfect rock (like in my first example) will show 90.6 percent U-235 and 9.377 percent Pb-207. Dividing the two we get a ratio 0.1035. The same ideal rock would be 98.5 percent U-238 and 1.54 percent Pb-206. (We’re going to ignore the thorium for now), the ratio is 0.0156. So put a mark at 0.1035 on the horizontal and 0.0156 on the vertical. Label that mark “100.”

Do this for a bunch of different ages and you get this sort of curved line, with different points on the line labeled with different ages. It goes up to the right, but instead of being a straight line it bulges slightly upward.

Note this is a bit different from the graphs we made in Algebra 1. There, we had the independent variable along the horizontal axis, and the dependent variable on the vertical axis. This time we have two dependent variables, those two ratios, off of one independent variable (the age that would give those ratios under ideal circumstances). We are plotting the two dependent variables against each other.

(You don’t have to do all of this math every time, because the points are always the same. I would bet that it’s in tables. In fact, I wouldn’t be surprised if there was graph paper with this line pre-printed on it, though in the modern digital age that may have fallen by the wayside.)

In fact, I just used a spreadsheet to compute the values for 100, 200, 300…and so on up to 1000 million years, and plotted them. (I wasn’t able to get the spreadsheet to label the points.) In this case, 100 million years is at the lower left, and 1 billion years is at the upper right.

OK, now that you have this blank template, plot all of your measurements on it. For the first sample, there were 2.5 million atoms of Pb-207 and 5 million of U-235, so the ratio is 0.5. That’s your “x” value. And there were 57.7 million Pb-206 atoms versus 1 billion U-238 atoms, so the ratio is 0.0577. That’s your y value. Plot a point at (0.5, 0.0577)

Repeat for the other samples. These data points should lie along a straight line (if it’s not exact there are mathematical methods to find the “best fit” line). Extend it, and it will cross the curved line in two places. The upper right intersection represents the original ratios, you then can backtrack to figure out what age that point on the curved line represents. And you will get 704 million years, which is the actual age of the rock you pulled the zircon crystals from.

I can’t seem to get sample points onto the graph I just uploaded, but what I can do is show you an actual concordia diagram. This one was used to date rocks from the Klamath Mountains in Northern California. In this case as you can see the age is 461.17 +/- 31 Ma. (Spoiler: This turns out to be the middle of the Ordovician period. My example 704 million year old rock would, if real, come from the Cryogenian period.)

The upshot of this is, the concordia diagram lets you use the fact that there are two measurements to account for loss of daughter isotope, provided you can take multiple samples (with different amounts of loss) from the same rock.

What about the thorium-to-lead part? One could use Pb-208/Th-232 in a concordia diagram, instead of one of the two lead/uranium isotopes, but thorium decays more slowly so its ratios are smaller and a tiny variation in the measurement leads to a bigger variation in the date. The two uranium-lead numbers are more sensitive, so they get used instead.

There is a closely related method called Lead-Lead dating. I’ll cover that next time. Meanwhile, you’re probably wondering. What’s the oldest rock we’ve found?

Quit Holding Out On Us

The oldest “hit” found using zircons and uranium-lead dating so far is some zircon crystals taken from a rock in the Jack Hills in Western Australia, north of Perth. The crystals were found in a sedimentary rock, so the rock as a whole is younger than the zircons, which came out of an igneous rock that eroded a long time ago.

And that number came out to be: 4,404 +/- 8 million years.

Remember that this is a minimum age for the Earth. We’ll improve on it.

Here’s a picture of a Jack Hills rock:

It is believed that this rock (as a combined entity) is about 3 billion years old (that’s not a very precise number, but that’s the point; it’s hard to date sedimentary rocks), but obviously it’s made of older stuff, including those ancient zircon crystals.

I want to close by emphasizing that uranium-lead dating has been used countless times, and between that and other methods of dating I’ll be covering soon–also used countless times–we have built up a consistent notion of Earth’s age and ages for events during Earth’s “lifetime.” This isn’t a one-off that was then uncritically accepted.

Other, slightly newer zircons (4.3 billion years old) from the same area have had their oxygen atoms examined and the isotopic mix there (O-16 vs. O-17 vs. O-18, all stable) implies there was already liquid water on Earth’s surface.

And accepting these numbers isn’t a “presupposition” (as some people would claim) because the numbers are the results of a lot of evidence and scientific investigation.

If you want to dispute numbers in the millions and billions of years, you are going up against, quite literally, tons of hard evidence.

What is it that feeds our battle, yet starves our victory?

Speaker Johnson: A Reminder.

And MTG is there to help make it stick.

January 6 tapes. A good start…but then nothing.

Were you just hoping we’d be distracted by the first set and not notice?

Are you THAT kind of “Republican”?

Are you Kevin McCarthy lite?

What are you waiting for?

I have a personal interest in this issue.

And if you aren’t…what the hell is wrong with you?

Fun Quote

(HT Aubergine)

This is amazing. This is glorious. Summon a surgeon – it’s been a little over a week and you’re supposed to call the doctor after just four hours.

From Kurt Schlichter, who can certainly write a good rant (https://townhall.com/columnists/kurtschlichter/2025/01/30/trumps-winning-streak-is-totally-discombobulating-the-democrats-n2651308)

Yep, Kurt has noticed that lots of people are getting twanging schadenböners.

And you do not have to be male to get this kind of böner.

Hat tip to Scott (I think–if it wasn’t Scott it was 4GodAndCountry) for this video, which implies a LOT of schadenböners in our future.

Lawyer Appeasement Section

OK now for the fine print.

This is the WQTH Daily Thread. You know the drill. There’s no Poltical correctness, but civility is a requirement. There are Important Guidelines,  here, with an addendum on 20191110.

We have a new board – called The U Tree – where people can take each other to the woodshed without fear of censorship or moderation.

And remember Wheatie’s Rules:

1. No food fights
2. No running with scissors.
3. If you bring snacks, bring enough for everyone.
4. Zeroth rule of gun safety: Don’t let the government get your guns.
5. Rule one of gun safety: The gun is always loaded.
5a. If you actually want the gun to be loaded, like because you’re checking out a bump in the night, then it’s empty.
6. Rule two of gun safety: Never point the gun at anything you’re not willing to destroy.
7. Rule three: Keep your finger off the trigger until ready to fire.
8. Rule the fourth: Be sure of your target and what is behind it.

(Hmm a few extras seem to have crept in.)

Spot (i.e., paper) Prices

Last week:

Gold $2,858.10
Silver $31.20
Platinum $953.00
Palladium $945.00
Rhodium $5,100.00
FRNSI* 137.261-
Gold:Silver 91.606-

This week, 3PM Mountain Time, Kitco “ask” prices. Markets have closed for the weekend.

Gold $2,911.50
Silver $32.60
Platinum $974.00
Palladium $934.00
Rhodium $6,000.00
FRNSI* 139.844-
Gold:Silver 89.310-

Palladium is below platinum again…but look at rhodium, which has gone up nine hundred bucks!

The people who bloviate on this sort of thing for a living (if this is all I did I’d starve to death) claim the precious metals are “consolidating” with gold in the 2910-2920 range while the stock indices go down. At least silver is up relative to gold!

*The SteveInCO Federal Reserve Note Suckage Index (FRNSI) is a measure of how much the dollar has inflated. It’s the ratio of the current price of gold, to the number of dollars an ounce of fine gold made up when the dollar was defined as 25.8 grains of 0.900 gold. That worked out to an ounce being $20.67+71/387 of a cent. (Note gold wasn’t worth this much back then, thus much gold was $20.67 71/387ths. It’s a subtle distinction. One ounce of gold wasn’t worth $20.67 back then, it was $20.67.) Once this ratio is computed, 1 is subtracted from it so that the number is zero when the dollar is at its proper value, indicating zero suckage.

Latest Flerfer Goofiness

OK unlike last week I’m going to try to supply some actual content. (Last week was nonstop busy.)

Our friend Fkatzoid is back. Watch him duck and weave when he’s asked where the south pole is (at 16:45).

Will Duffy is trying to make the point that whether you head south from Africa, South America, or New Zealand, you end up at the same place when you get to the south pole. According to the Gleasons’s Map, however, the South Pole isn’t a point, it’s a circle approximately 60,000 miles in circumference, so that shouldn’t happen. Either Fkatzoid is an even bigger idiot than he showed himself to be last time, or he’s trying very hard to evade having this pointed out to his audience.

Later on at about 2:31:45…apparently Lisbeth (who went to Antarctica) is on the verge of joining Mark Sargent’s channel; MC Toon begs her not to ruin her life doing so.

Just a few minutes later, you see someone named JK trying to find a video proving that people who try to go to Antarctica will be intercepted by any of a number of different navies and turned back as soon as they sail across 60 S latitude. He claims there are many of these videos; he eventually finds the one Will Duffy expected–taken in the Bass Strait between Tasmania and mainland Australia, nowhere near Antarctica. Listen to McToon’s rant at 2:45:22.

Here’s another debate with Duffy destroying someone I’ve never heard of named Nathan Thompson. (Not to be confused with Nathan “where are the guns” Oakley.)

Two Birds…One Stone

OK this one is going to seem like geology…then physics…then back to geology. It’s a good illustration of how all of human knowledge about the natural world is interconnected. Sometimes great progress is made when people in two different fields get together; sometimes a new discipline even is formed–recent work has done much to highlight the effects of living organisms on the geology of the Earth…yes, our rocks would be different if there were no life on earth (and there’d be no geologists to notice, of course).

An Extremely Inadequate Intro to Mineralogy

Let’s take a very brief and incomplete (and likely incompetent, as I am out over my skis here) look at mineralogy.

I’ve talked about rocks a lot but not so much about what they’re made of. If you look closely–perhaps it will take a microscope–at an igneous rock (one that cooled from the molten state) you’ll see it’s made up of a bunch of different kinds of crystals. Crystals form when a chemical compound comes out of solution and the individual molecules line up in a regular array.

Some rocks are just one big crystal. Others are multiple crystals of the same thing.

The compounds that make the crystals are minerals (and one of their characteristics is how the crystals are shaped).

What are those compounds? Let’s set the stage a bit. If you take the outer layer of the Earth, the Earth’s crust, and analyze a completely average piece of it…it’s 46.1 percent oxygen by weight. Oxygen! There is much, much, much more oxygen beneath your feet than above your head in the air. Oxygen is also the third most abundant element in the universe as a whole–after hydrogen and helium.

Coming in second at 28.2 percent is silicon. Then aluminum at 8.23%, iron at 5.63%, calcium at 4.15%, sodium at 2.36%, magnesium at 2.33% potassium at 2.09%, titanium at 0.565%…and everything else is at 1/7th of a percent or less. At the bottom end you have rhenium at 7/10ths of one part per billion. (However two gases, krypton and xenon, also show up at even lower percentages, and a bunch of transient radioactive elements are lower still than that.)

The ones at the top of the list don’t ever show up in pure elemental or “native” form; they’re pretty reactive. Minerals will be largely (but very luckily for us, not completely) formed of these elements.

The elements in general are divided into groups according to the “Goldschmidt Classification.” The groups are “lithophile” (rock loving), “Siderophile” (iron loving), “chalcophile” (bronze loving), and “atmophile” (atmosphere loving). The group an element is in is a huge determiner of its fate. Lithophile and chalcophile elements both appear predominantly near the Earth’s surface, in the crust; with the chalcophile elements often combining with sulfur. Siderophile elements largely sank, with almost all of the iron, towards the Earth’s core.

(There is a very slick wikipedia graphic for this, a periodic table colored by Goldschmidt classification…but it’s actually a table rather than an image and I was unsuccessful in getting it copied over here. Link: https://en.wikipedia.org/wiki/Goldschmidt_classification )

There are officially 6,118 mineral species known to man today. Minerals must be naturally occurring and forming by natural or geological processes, must be a solid substance (with the exception of native mercury). Water and carbon dioxide are not minerals even when they show up embedded in rocks, but water ice in glaciers is a mineral. A mineral must have a well defined crystal structure. (This ends up excluding things like obsidian which don’t have a crystal structure.) And the chemical composition must be well defined. However, that could include mixtures of similar compounds; sometimes one element will substitute for another of similar size and chemistry to one extent or another.

There are a number of different ways minerals can be classified, based on hardness, color, crystalline structure, cleavage (i.e. which planes it will split on most cleanly), specific gravity (galena, a lead ore, is very dense, for instance–over seven times that of water whereas the typical rock is in the 2.5-3.5 range)…and by chemistry. But this is far from straightforward, since nothing is pure. For instance a mineral whose structure is largely silicon will often have an aluminum atom substituted for the silicon; sometimes this is a regular substitution, making a distinct chemical series.

Minerals fall into a number of different groups; the most common by far is silicates; these are minerals formed by different arrangements of the [SiO₄]^4- tetrahedron, one silicon atom surrounded by four oxygen atoms.

This shouldn’t be too much of a surprise, after all oxygen and silicon make up most of rocks. There are a simply staggering number of ways to combine these tetrahedra at the corners (where an oxygen atom will end being shared by two tetrahedra); chains, rings, lattices…just for instance:

The most basic of these is quartz, consisting of nothing but silicon and oxygen. Since each oxygen atom is shared by two silicon atoms, the formula ends up being SiO₂.

Quartz, when absolutely pure, is clear as glass. Different impurities will give it colors, smoky quartz and amethyst being examples, but there are many more.

And in some cases other elements are interspersed with the tetrahedra, or sometimes the silicon is partially replaced by other elements. This can alter the structure as well as the composition.

The most common of the silicates are a grouping called the feldspars, where Al3+ substitutes for the Si4+, but this creates a charge imbalance that requires other elements added in as cations. You end up with [AlSi₃O₈]^– or [Al₂Si₂O₈]^2-. In other cases silicates can form in sheets, like mica.

If you haven’t realized this by now, it turns out that silicates are bewilderingly complex. For a deep dive: https://en.wikipedia.org/wiki/Silicate_mineral.

In other groups we have native elements. For example gold, silver, and copper appear in native form as nuggets. There are also platinum nuggets. But also there are diamonds and graphite, both forms of native carbon. Sulfur also appears near volcanic vents. And in many cases the nuggets aren’t pure but are alloys, but still a lump of metal, rather than a “rock.”

Next we have sulfides, compounds of metals and sulfur, famously iron pyrite (fools gold), red cinnabar (a mercury ore). Sometimes tellurium, arsenic, or selenium will substitute for some or even all of the sulfur.

Oxides are metals combined with oxygen, such as hematite (iron), bauxites (aluminum), magnetite (iron again).

Halides are those where a halogen (fluorine, chlorine, bromine, iodine) is the main anion; table salt is the most common example, with chlorine combined with sodium.

Carbonate minerals have a carbonate [CO₃]^2- group in them. They will react with acids; so field geologists will often have a small vial of acid to test for them. The most common is calcium carbonate…also known as calcite, the main component of limestone. This is weakly soluble in water, leading to the formation of cave systems.

There are sulfates (distinct from sulfides mentioned above) with the sulfate anion, [SO₄]^2-, combined with something else.

The last common group is the phosphates, with a [PO₄]^3- unit, combined with something else. These minerals are what our bones and teeth are made out of.

It’s a gigantic mess, honestly; and it gets more and more complicated when it turns out that a mineral can be a mixture of, say, two different sulfates mixed together; the formula ends up including a bracket with two or three different atoms specified because they are intermixed in some proportion.

I have not even scratched the surface of this topic (and those familiar with hardness testing will see the pun). I may not have said anything wrong in this section, but even if we’re that fortunate, I’m sure a real mineralogist would find much to complain about, important things left out, inconsistent “depth” of the dives I took, and so on. I know I said next to nothing about crystal structure and I may try to rectify that some day.

Back to the Historical Narrative

OK so back to the story: By the mid 1800s geology had made huge strides to systematize the variety of rocks and landscapes we see here on Earth. Geologists had even developed the ability to describe what had happened in the past in some arbitrarily picked location. Glaciers, lakes, oceans, desert…all had left telltale signs in the rock. They saw a world of mostly slow change…but with the occasional disasters, local in scope not worldwide.

They had even realized the Earth must be far older than previously thought; the events they could read in the rocks simply could not have happened fast enough to fit within a few thousand years of time.

Impressive work. There were obviously a lot of unsolved problems (like how it could be possible that former sea floor bottom ended up high in the Alps), but still a lot learned.

Physics and astronomy (closely associated with each other) were pretty much the most successful and advanced branches of scientific knowledge. Were astronomers and physicists at least somewhat impressed with what geologists had come up with?

Perhaps but in one key respect the answer was probably more like, “You gotta be shitting me.”

You see the physicists and astronomers of the mid 1800s couldn’t possibly see how the Sun could be old; if the Sun weren’t old neither could the Earth be old. There was simply no way to power the sun for those lengths of time. However the geologic evidence was simply overwhelming.

Beyond suspecting that geologists were smoking something that was distinctly not a mineral (and vice versa from the geologists’ point of view), there was little that could be done. Tons of hard evidence (i.e., rocks) vs. quite well established kinetics and thermal physics. Neither of them could be shaken.

So what was the cause of this disconnect?

In 1862, William Thomson (1st Baron Kelvin…after whom the Kelvin scale is named), published calculations that assumed the Earth had started out completely molten, then computed how long it would take to cool to what we see today. His answer was 20 – 400 million years. OK, that seems a bit low to geologists, but not horrifically so. (He did not account for convection inside the Earth, which would increase the number…nor for other factors he simply couldn’t have imagined, which I’ll get to.)

The big problem was that he also computed how long the sun could have been shining at its present brightness, if it derived all of its energy from gravitational contraction. And that answer was 20 million years. It agreed with his Earth calculation at the low end so that made sense–Thomson probably reasoned that the Earth therefore had to be 20 million years old, but geologists (and biologists) simply couldn’t believe the Earth was that young. Other physicists (Hermann von Helmholz and Simon Newcomb) got similar values of 22 and 18 million years, respectively.

Other possible sources of solar energy were combustion and impacting comets and asteroids. The first was ridiculous. If the entire Sun, huge as it is, were a burning pile of coal, it would be gone within a couple of thousand years at the rate it would have to be burning to be as luminous as it is. This is not even long enough to carry us from the Great Pyramid to Julius Caesar, much less to today. Asteroid impacts sounds more promising, until one realizes there’d have to be so many of them that surely Earth would be catching a lot more of them than we actually are getting. And it was only good enough for a few hundred thousand years. The other flaw was that the sun would be increasing with size as more matter accumulated in it, and that imposed a strict time limit too…after a certain amount of time the Sun would simply be bigger than we see it.

Another tack taken by physicists and astronomers was to use the moon. George H. Darwin (son of Charles “the” Darwin) was an astronomer, figured out that if the Earth and Moon had split apart while still molten, tidal forces would have created our current situation with a 24 hour day after 56 million years. This may not look like it to you, but given the sorts of approximations that both Darwin’s and Kelvin’s calculations enailed, that’s actually close enough to Kelvin’s number that it appeared that both of them were likely on the right track. (When two totally different methods of computation give similar answers, that’s a powerful argument that the actual answer is pretty close to the ones we computed.) Yet another tack, computing how long it took for the oceans to accumulate the salt they contain, based on erosion of rocks, gave an answer of 80-100 million years for the age of the oceans.

When you see an apparent contradiction like this, something is missing from your mental picture. Or perhaps you have a wrong premise. Because an actual contradiction cannot exist.

And, as it turned out, one mineral, when it was discovered, turned out to be the beginning of the path not just to resolving this, but fulfilling another thing that was on the geologists’ wish list–one they never thought they’d get. Like the kid who doesn’t bother asking Santa for the really expensive toy for Christmas…but Santa read his mind and he gets it.

The mineral is an oxide, one called pitchblende. This was first described in 1772 by F. E. Brueckmann. In 1789 M. Klaproth worked with this stuff and discovered the element uranium.

[Uranium oxide has been in mosaic glass from Roman times; clearly they’d found some of the ore and experimented to see what it would do to color glass. However, we don’t have written records of the Romans recognizing it as a distinct material.]

Here’s some nice big crystals of pitchblende:

Uranium was nothing special, just another of a bunch of metals being discovered around this time. Along with such other favorites as cobalt, nickel, manganese, tungsten, niobium, tantalum, and chromium. Curiously, pitchblende also includes some lead, without fail. No such thing as “pure” uranium oxide pitchblende. That seems kind of weird because lead and uranium are chemically quite different.

Flawed analyses led to uranium’s atomic weight being calculated at 156 or so; later on the mistake was realized and the atomic weight was corrected to 238, far above anything else known at the time. Kind of interesting to geeks; no one else cared.

In 1895 this changed. And so did the world.

Henri Becquerel was trying to see if uranium salts, known to fluoresce in visible light, also fluoresced in X ray frequences. (X Rays had been discovered the year before by Röntgen.) [As a reminder, fluorescent things will glow in bright colors for a while after being exposed to ultraviolet light. This can actually be used to identify some minerals. Becquerel wanted to see if they also emitted X-rays alongside the visible light.] He’d expose the compounds to sunlight, then set them next to wrapped photographic plates. If the plates fogged, he would conclude the uranium compounds were giving off X-rays after being “charged” by the sun. Then he had days of cloudy weather, so he put the uranium salts and wrapped plates in a drawer while he awaited sunny days. Ultimately he decided, what the Hell, and developed the plates without exposing the uranium salts to sunlight, and found that they had fogged anyway. Well this was new!! Further experimentation established that uranium emitted strong radiation, all the time, no matter what.

Even more experimentation established that the uranium was turning into lead as it did this. Which is why pitchblende always has some lead in it, even though lead is very different from uranium, chemically.

This led to our current picture of the structure of an atom–which up to then had not been proven to exist. (The final piece of proof was supplied by Albert Einstein in 1905, the Annus Mirabilis)

That is a very long story. Detailed here (9 – End of Classical Physics (Rays & Radiation)):

2021·06·26 Joe Biden Didn’t Win Daily Thread

And here (13 – Ernest Rutherford):

2021·08·07 Joe Biden Didn’t Win Daily Thread

And here (17 – Nuclear Physics Finds a Hammer):

2021·09·04 Joe Biden Didn’t Win Daily Thread

And here (19 – Antimatter):

2021·09·18 Joe Biden Didn’t Win Daily Thread

And here (20 – The Little Neutral One (Neutrinos)):

2021·10·02 Joe Biden Didn’t Win Daily Thread

One key thing to note is that this new “radioactivity” was extremely energy intense, far more so than burning coal, and now we had a hints of a power source that would allow the sun to shine for hundreds of millions–even BILLIONS–of years.

And this is indeed the case, as described here (22 – Powering Stars):

2021·10·23 Joe Biden Didn’t Win Daily Thread

And the world was never the same, because this ultimately led to nuclear weapons.

But for our purposes here, the main effect is that now there was no more contradiction about how old the Earth might be. The Sun could indeed be old enough for an old Earth.

And Now We Can Measure It

Surprise! We also now had a way to measure the age of some rocks, to put actual numbers on things.

To explain this adequately (given the fact that there are charlatans out there who try to fling mud on this, and some of you believe them), I’m going to try to do a Science For Senators review of radioactivity and nuclei. It’s a bit densely packed since I’m not telling a story here. (The story was in all those posts above.)

Matter is made up of atoms, very roughly a hundred picometers (a picometer is a trillionth of a meter) across. Most of this volume is taken up by electrons (which have a negative electrical charge) that are bound to a positively charged nucleus (plural, nuclei). The nucleus contains almost all of the mass of the atom yet occupies a space only a few femtometers (a femtometer is a quadrillionth of a meter across); roughly 1/10,000th the diameter of the atom as a whole.

The nucleus, in turn consists of protons–positively charged particles–and neutrons–neutrally charged particles. Other than the charge, these two particles are very similar to each other–the neutron is just a bit more massive–and they’re collectively referred to as nucleons. (Neutrons are blue, protons red in the diagram below…but they don’t actually have color and they’re not actually shaped like little hard spheres, so the diagram is notional.)

As it turns out the number of protons in a nucleus (the “atomic number”) determines what chemical element it is. One proton: hydrogen. Six: carbon. Eight: oxygen. Twenty-six: iron. Forty-seven: silver. Seventy-nine: gold. Eighty-two: lead. Ninety-two: uranium. (Plus all of the other numbers in between of course.) In order to balance out, an atom will have the same number of electrons as protons, at least until it starts sharing or even giving or taking electrons with, to, or from other atoms–which is what chemistry is all about.

The number of neutrons, on the other hand can vary, even within an element. Just for instance, most uranium nuclei have 146 neutrons in them, but some have only 143. This has very little effect on the chemistry, but it is possible to very painstakingly sort these out. The two different types of uranium are described by their mass numbers, the total number of nucleons. 92+146=238, and 92+143=235; uranium-238 and uranium-235, respectively. These different-weight forms of the same element are called isotopes.

As it turns out radioactivity, when it happens, happens to nuclei. There are two main kinds of radioactivity that matter for our purposes here, alpha decay and beta decay.

Alpha decay is when a large nucleus basically pukes up a helium nucleus (containing two protons and two neutrons–mass number of 4). Since the nucleus gives up two protons in doing this, it changes to another element; this should therefore happen five times as uranium turns to lead, changing the atomic number from 92 to 82. Except that that’s not actually right; it turns out to be eight times. That’s because uranium-238 is becoming lead-206; that’s a difference of 32 mass units and eight alpha decays does that.

The reason the atomic number changes by ten rather than 16 (two per alpha decay) is that there is also beta decay. In this kind of radioactive decay, a neutron turns into a proton, ejecting an electron (which flies off into the distance, so you can basically forget about it) and a neutrino (which flies off away forever, so you can really forget about it). The effect is to leave the mass number unchanged…but it increases the atomic number by one (we have one more proton than we used to). To make up the discrepancy noted above, uranium, in turning to lead, must undergo six beta decays.

Technically speaking what I just described is negative beta decay, because it spits out a negatively charged particle. The reason why one might to be anal about this is that there’s actually a different kind of beta decay that may come into play, though…and that’s positive beta decay, where a proton spits out an anti-electron (“positron”–yes, this is antimatter) and turns into a neutron (the exact opposite change from the first kind of beta decay). This causes the nucleus to go down one in atomic number, again without changing the mass number.

Uranium and thorium (atomic number 90) undergo alpha decay, as do a lot of the things they turn into on the way to becoming lead (as do many of the intermediate elements in between and on the way). A lot of the intermediate products undergo beta decay. That’s all stuff at the high end of the periodic table, though.

It turns out that a lot of much lighter elements…ones we thing of as stable…are at least partially made up of isotopes that do one or the other form of beta decay (there are dozens of examples). Even potassium has a long-lived isotope (potassium-40 or ⁴⁰K) that decays, in fact it can decay two different ways: negative beta decay or “electron capture” where a proton absorbs an electron. The first turns it into calcium-40, the second turns it into argon-40.

Our atmosphere is about one percent argon, and that argon is almost all argon-40. The sun’s argon–which presumably came from the nebula that condensed to form the solar system–is almost all argon-36, which leads to the conclusion that none of the Earth’s original argon is still around, and all of the argon in the atmosphere is actually from the decay of potassium-40.

There is just one thing I haven’t mentioned yet. Alpha and beta decay occur at constant rates. The rate is different for each nucleus, but constant for that nucleus. (All sorts of attempts have been made in laboratories to change the rate…with one oddball exception, absolutely nothing happened.) It’s a proportional thing; over some period of time, half of the atoms of some radioactive isotope will decay. You’re then left with a sample half the size of your original sample…and half of that will be gone after you wait the same period of time again. And so on. This period of time is known as the half life, because half of the atoms are gone after that period of time.

Of the things I’ve touched on, here are their half lives: Uranium-235: 703.8 million years. Potassium-40: 1251 million years. Uranium-238: 4458 million years. And Thorium-232: 14,050 million years.

And now maybe you can see how this might be useful to geologists. Find a rock with some uranium, thorium, or potassium in it. (Potassium most likely; it’s common compared to the others.) Then determine how much “daughter” product is in the rock. It helps if the daughter product is such it wouldn’t have been in the rock when it solidified. E.g., a zircon crystal, which might pick up uranium impurities as it crystalizes, but will positively reject lead atoms. Any lead in the zircon crystal can only have come from uranium decay. Count atoms (yes, you might have to literally count atoms) to determine how much daughter product there is, versus parent isotope. Figure out how much decay has taken place and compare to the half life.

You now know the age of the crystal. Not the relative age, the absolute age, of the crystal.

But there are a lot of details with this (including the fact that dating sedimentary rock is dicey), and I will cover some of them next time. These details, when fully considered only serve to make these methods rock solid.

Notice

I have a complex project coming up IRL, and I absolutely have to reallocate my “spare” time. This will mean less laughing at online flerfs, but it also means science posts will be infrequent and/or unpredictable.