The Marines, Sailors, and Soldiers Who Died in Afghanistan

Unless otherwise mentioned, these people are Marines.

All died in Afghanistan, serving us, their lives expended stupidly by His Fraudulency.

That does not take away from the debt we owe them, in fact it increases it. We owe it to them to remove His Fraudulency and his cohorts from power.

Justin Allen, 23

Brett Linley, 29

Matthew Weikert, 29

Justus Bartett, 27

Dave Santos, 21

Jesse Reed, 26

Matthew Johnson, 21

Zachary Fisher, 24

Brandon King, 23

Christopher Goeke, 23

Sheldon Tate, 27

Max Soviak (USN)

Rylee McCollum, Wyoming

David Lee Espinoza, 20,  Texas

Sgt. Nicole Gee

SSGT Ryan Knauss (USA)

Normal Introduction

Another week, another deluge of BS from the White House and from the Controlled Opposition.

The Audit continues.

The collapse of the Covidschina continues.

No doubt much will be said about those today. (And I have missed a lot this past week.)

To my mind the audits are the last hope for a within-the-system fix to what happened last November. “Within the system” meaning the audits find fraud, the various states decertify the results, and some dang judge rules that Biden must step down and Trump must be installed.

That last step is crucial. The way our system works, “fraud” isn’t a fact until some “competent authority” (i.e., meaning “one that has jurisdiction,” not “one that won’t end up with an ice cream cone on its forehead”) rules it is so. That must happen before the system will accept that the election is vitiated by fraud. No finding of fraud means, as far as they are concerned no fraud, no fraud means nothing vitiated. We sit and fume, because the system has failed.

I’ll leave it to you to decide how likely you think it is that a judge will rule against the Left given the riots that would likely endanger his/her family.

As for the military stepping forward and doing the job instead? Well, that’s technically “outside of the system” and besides…this military, that’s being made woke as we speak?

What do we do in the likely event that fraud is found, but no judge will find it to be “fact” as far as the Federal Government is concerned? I keep hoping someone will come up with a suggestion, and so far “general strike” (H/T Scott) is the only one I’ve seen.

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. (This doesn’t necessarily include deposing Joe and Hoe and putting Trump where he belongs, but it would certainly be a lot easier to fix our broken electoral system with the right people in charge.)

Nothing else matters at this point. Talking about trying again in 2022 or 2024 is pointless otherwise. Which is not to say one must never talk about this, but rather that one must account for this in ones planning; if fixing the fraud in the system is not part of the plan, you have no plan.

This will necessarily be piecemeal, state by state, which is why I am encouraged by those states working to change their laws to alleviate the fraud both via computer and via bogus voters. If enough states do that we might end up with a working majority in Congress and that would be something Trump never really had.

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

Last week:

Gold 1781.50
Silver 23.13
Platinum 1000
Palladium 2354
Rhodium 17,100

This week, 3 PM MT on Friday, markets closed for the weekend

Gold $1817.80
Silver $24.08
Platinum $1016.00
Palladium $2498.00
Rhodium $18,400.00

Everything was much lower this week (except for rhodium and palladium which had been creeping upward), but suddenly on Friday, gold jumped 24 bucks, silver 44 cents, platinum $33, palladium $22, and rhodium $100 (that’s a very small move for rhodium, by the way; it usually moves a lot more in whatever direction it is going). So now it’s a net improvement for the week.

Part XVI – De Broglie, Schrödinger, and Heisenberg

Very flaky connection. Who knows whether I’ll be able to finish this.

I can’t figure out a coherent history of the 1920s; apparently a lot of stuff was happening simultaneously. So I will treat three different threads as though they were independent. They weren’t.

De Broglie

(Which is pronounced “Deh Broil-ee” or at least was by my physics professor–the four alternatives given for pronunciation in Wikipoo are different from this), put 2 and 2 together.

We had already established that light, even though a wave of energy, has particle behavior (that was largely Einstein in 1905). And we also knew, thanks to Max Planck (1905), that the particles (“photons”) had energy proportional to their frequency.

But we also knew that matter is basically equivalent to energy, that thanks to Einstein as well in 1905.

Since light was energy and could behave as both a particle and a wave, could matter, which was equivalent to energy, also behave as both a particle (which you’d expect from matter) and a wave (which you would not expect)?

De Broglie said, in essence, “yes,” in his doctoral thesis in 1924.

Here is the Planck-Einstein relation:

E = h ν

…which relates the energy of the photon to its frequency ν (Greek letter nu) and Planck’s constant, h.

And light turned out to have momentum (p), too, based on the energy E or wavelength λ:

p = E / c = h / λ

You can simply rearrange this last to get:

λ =h / p

And of course p is mass times velocity, mv.

The implication is that any chunk of matter is a wave, at least while it is in motion. However, if you consider the very tiny size of Planck’s constant, 6.6 x 10^-34, and realize it is being divided by, say 100 (a 100 kg object moving at one meter per second, for instance) for any sort of object you will deal with in your daily life, the wavelength (6.6 x 10^-36 meters) is very, very small, much, much smaller than an atomic nucleus (roughly 10^-15 m). Immeasurably small. The wavelength is about the same size, in relation to that nucleus, as the nucleus is in relation to the Earth.

But for something very, very light, like, say, an electron…you might get a reasonable wavelength. If de Broglie isn’t just talking out his ass.

An electron’s mass is about 9.1 x 10^-31 kg. So assume (for the sake of example) one is travelling roughly at one hundredth the speed of light ( 3 x 10⁶ m/s, and plug that into λ =h / p and you get:

λ = 6.6×10^-34 / ( 3 x 10⁶ * 9.1 x 10^-31 ) = 242 x 10^-12 meters

Now the diameter of a hydrogen atom is about 62 x 10^-12 meters. Its circumference is therefore roughly 195 x 10^-12 meters…which is pretty doggone close to this wavelength actually, considering I just made a guess as to how fast an electron might be moving.

Standing Waves (Not Really a Digression)

If you have ever plucked a guitar (or violin) string, you’ll have noticed it moves in a certain fashion, the top left in the diagram below:

It turns out that the other modes shown in the diagram also exist to a certain extent. You can also have standing waves (of sound) in an organ pipe or any other wind instrument.

Now back to the electron. If it is a wave, and it’s orbiting around a nucleus, the wave has to mesh cleanly with itself after one orbit around the nucleus. Look at the top right string, which is a full wavelength. So if that string was actually arranged around a circle, instead of a straight line, at any given point it would look like a smooth wave; because the end of the wave would be consistent with the beginning of the wave where it joins, it’d be a smooth wave travelling in a circle. But if the electron were to have a different wavelength, it couldn’t be in that orbit, because the wave wouldn’t be smooth–there’d be a “break” in it somewhere along the circumference.

That’s only the beginning of the argument, but it will eventually turn out that de Broglie’s hypothesis, that an electron is a wave as well as a particle, ends up explaining why electrons can only assume certain orbits in an atom. We knew, thanks to Bohr and the spectroscopists, that it did do so, but didn’t know why it followed the rule. Remember that the quantum theory was just an arbitrary-seeming restriction on various processes that seemed to work out. Now we had some hint as to why the restriction exists. It was the only way for electrons to form standing waves.

Of course it’s all very well for him to propose a hypothesis that seems whacky but seems to match the facts. But it’d be nice to do an experiment that proves that electrons can behave as waves, and Clinton Davisson and Lester Germer did so at Western Electric (later Bell Labs) in 1927, sending a beam of electrons through a sample of nickel and seeing a diffraction pattern form.

A diffraction pattern is a property of waves.

De Broglie won the 1929 Nobel Prize for Physics.

Erwin Schrödinger

But this tidy little solution was way oversimplified.

An electron is not a one-dimensional string, tied down at two points, but left free to vibrate. It instead is a wave free to move in three dimensions, but caught in a potential well, attracted to a nucleus, the more strongly the closer it got.

It’s possible to write the equations of those strings’ waves, and doing so gets you a nasty mess full of trigonometry. (Nasty, that is, to the mathematically dis-inclined.)

It’s also possible to write a three dimensional equation, using complex variables, to describe the waves an electron can make when bound to a nucleus by the electrical force.

It’s even possible to write yet a different equation, which the first equation must satisfy in order to work. That may have made your head spin just a bit, so let me back off and explain something about higher mathematics.

Arithmetic and algebra deal with functions, you plug a number in and out comes another number, that is termed “a function of” the first number. f(x) = x² is one example; plug 3 in and out comes 9.

Calculus, and even higher forms of math, actually works on functions, not on numbers; giving you another function (which you can then plug a number into).

For example, if you want to know how fast f(x) = x² changes as you change x, you can do something called “taking the derivative” to the first function, and you get another function, f'(x) = 2x. You can then plug your number into that equation, and know that not only is x² = 9 when x is 3, but you know how fast x² is increasing at that point: it’s increasing at a rate of 6 for every 1 increase in x.

In 1925 Schrödinger postulated, then in 1926 published an equation, which involves a lot of calculus, into which you plug your proposed equation for an electron wave. If it balances out, you have the equation for a wave an electron might assume. There are many possibilities for what the electron is actually doing (depending on its energy, for instance), and yours might just be the one it’s following.

[Linguistic aside: Schrödinger is sometimes spelled Schroedinger, where oe is an acceptable substitute for ö. Sometimes it’s rendered “Schrodinger” which is technically incorrect. To form the sound the Germans mean when they write ö, prepare to say “eh” as in bet but round the lips (like you do when saying “oooh”) when you say it. To some people it sounds a bit like “er” and I’ve even see “teach yourself” books that said that was how to do it. Cuppa Covfefe will no doubt amplify or correct me. Meanwhile, the same physics prof who taught me how to butcher de Broglie pronounced it “shraydinger” which would make sense…if his name was Schrädinger/Schraedinger.]

So here it is. I had to work with a simplified version of this back in college, and I do not remember how to do it; in fact certain aspects of the notation don’t make sense to me at all, so I’ve forgotten much.

Note that i, the square root of minus one, (the unit “imaginary” number) appears at the left hand side of the thing. Ψ is the actual electron wave equation, if it meets the condition shown, it’s a possible equation for an electron.

Remember that this is being done in three dimensions, and that the value of the wave function Ψ is itself a complex number, i.e., the sum of a real number and an imaginary number, a+bi. (Yes, we did a lot with complex numbers…in fact electrical engineering would be damned near impossible without them.)

In college, I had to work with a problem called “particle in a box” where the particle had free reign of a small region of space (in one dimension); at the edges of the region the potential went up to infinity, meaning the particle couldn’t go past those points. The answer was a standing wave, just like the ones in the diagram above.

When solved in three dimensions, for electrons orbiting a nucleus, you get these possibilities:

Schrödinger is most famous for his cat. Well, actually, it wasn’t his actual cat (I don’t know if he even owned any, or served as staff for any), but it was a facetious thought experiment he proposed to ridicule a certain interpretation of quantum mechanics. But that was in 1935, far in the future, but it does touch on quantum uncertainty, which brings us to…

Werner Heisenberg

In 1927, Werner Heisenberg put forward the uncertainty principle. It states that even in principle it is impossible to know a particle’s position and momentum perfectly. You could know one of them very accurately, but then you’d know the other one quite imperfectly. You can multiply the two uncertainties together, and the product will be greater than h/4π, or ℏ/2. Even if your measuring apparatus is very accurate the uncertainty cannot be less than ℏ/2.

This introduced some spookiness into quantum mechanics.

Up until now, everything physics had produced was fundamentally deterministic. If you knew the state of a system at a certain time, you could, in principle (though it would take a YUGE computer a YUGE amount of time) figure out what state it would be in at some time in the future…or what state it had been at some time in the past. it would be like playing a movie forward and backward.

Along comes Heisenberg, and says you cannot know the state of a system at a certain time. If you know where the constituents are, you don’t know how fast they are moving. If you know how fast they are moving, you don’t know where they are.

And it turns out, even the particles themselves don’t “know.”

The electron wave, it turns out, actually describes the probability of the electron particle being at various places. According to one interpretation (the one that is dominant today) called the “Copenhagen Interpretation”, the electron can be anywhere that wave function says, but at some point it will interact with something, and then it will assume a definite position. (Please note, not the same as “assum[ing] the position.”)

This is seriously weird stuff. And I’m going to leave it to a competent physicist to talk about some of the ramifications. Richard Feynman was once voted (by currently practicing physicists) as the 7th greatest physicist of all time. And his lectures are famous…they used to be for sale at dead tree bookstores (e.g., Borders), and I’m sure they’re available in printed form on the web. (I’d look but it’s already nine minutes before 10 PM my time.) This video is almost an hour long, but good.

It was this sort of thing that brought about Einstein’s quote that God does not play dice with the universe, and also led Schrödinger to propose the cat experiment, which purported to show that the Copenhagen interpretation of quantum mechanics led to a situation where a cat was neither dead nor alive but both but neither until you opened the box and looked.

But we do not need to open the box to know that Saturday is bacon day around here.

And that Joe Biden didn’t win.

[I must apologize for this article; I doubt it made anything clear at all. However, there’s an old saw about how anyone who thinks they understand quantum mechanics just shows his ignorance. I don’t know that there is actually any way at all to make this stuff clear; it’s utterly counter to anyone’s intuition to the point where intelligent/geeky people just flatly refuse to accept it, despite the fact that it has been experimentally verified again, and again, and again….]

Obligatory PSAs and Reminders

China is Lower than Whale Shit

To conclude: My standard Public Service Announcement. We don’t want to forget this!!!

Remember Hong Kong!!!

中国是个混蛋 !!!
Zhōngguò shì gè hùndàn !!!
China is asshoe !!!

China is in the White House

Since Wednesday, January 20 at Noon EST, the bought-and-paid for Joseph Biden has been in the White House. It’s as good as having China in the Oval Office.

Joe Biden is Asshoe

China is in the White House, because Joe Biden is in the White House, and Joe Biden is identically equal to China. China is Asshoe. Therefore, Joe Biden is Asshoe.

But of course the much more important thing to realize:

Joe Biden Didn’t Win

乔*拜登没赢 !!!
Qiáo Bài dēng méi yíng !!!
Joe Biden didn’t win !!!

His Fraudulency

Joe Biteme, properly styled His Fraudulency, continues to infest the White House, we haven’t heard much from the person who should have been declared the victor, and hopium is still being dispensed even as our military appears to have joined the political establishment in knuckling under to the fraud.

One can hope that all is not as it seems.

I’d love to feast on that crow.

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.

Nothing else matters at this point. Talking about trying again in 2022 or 2024 is hopeless otherwise. Which is not to say one must never talk about this, but rather that one must account for this in ones planning; if fixing the fraud is not part of the plan, you have no plan.

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 $1780.60
Silver $23.83
Platinum $1034
Palladium $2736
Rhodium $20,200

This week, markets closed as of 3PM MT.

Gold 1781.50
Silver 23.13
Platinum 1000
Palladium 2354
Rhodium 17,100

Gold has actually moved around a bit, but the end result was a tiny gain over the week. Silver has dropped significantly (about 3 percent by eyeball). Platinum was well under a thousand yesterday, and has recovered…some. Palladium is down over ten percent. And Rhodium is getting its ass kicked; it dropped 1900 dollars on Friday alone.

From Special to General

Introduction

Let us start off by recapping our list of “as of 1894” mysteries and conservation laws, and bring things up to date including the Bohr atom and the work done on justifying the periodic table (much of which happened well beyond 1913). Otherwise, we’re at about 1913 now.

Let’s recap/update those lists.

1. Conservation of mass
2. Conservation of momentum
3. Conservation of energy
4. Conservation of electric charge
5. Conservation of angular momentum
6. (ADD:) Conservation of mass-energy

The following mysteries were unanswered at the end of 1894.

1. Why was the long axis of Mercury’s orbit precessing more than expected, by 43 arcseconds every century? Was it, indeed, a planet even closer to the sun? If so, it’d have been nice to actually see it.
2. Why was Michelson unable to measure any difference in speed of light despite the fact we, being on planet Earth that is orbiting the sun, had to be moving through the medium in which it propagates?
3. What makes the sun (and other stars) shine (beyond the obvious “they shine because they’re hot” answer). What keeps the sun hot, what energy is it harnessing?
4. How did the solar system form? Any answer to this must account for how the planets, only a tiny fraction of the mass of the solar system, ended up with the vast majority of the angular momentum in the system.
5. What is the electrical “fluid” that moves around when there is an electric current, and that somehow seems imbalanced when we perceive that an object has a charge? Were there both negative and positive fluids, or just one fluid that had a natural neutral level; below it was negative (deficit), above it was positive (excess)?
6. Why are there so many different kinds of atoms? How did electrical charges relate to chemistry? How is it that 94 thousand coulombs of charge are needed to bust apart certain molecules (though it often had to be delivered at different voltages depending on the molecule)?
7. Why were the atomic weights almost always a multiple of hydrogen’s? Why was it never quite a perfect multiple? Why was it sometimes nowhere near to being a multiple?
8. Why does the photoelectric effect work the way it does, where it depends on the frequency of the light hitting the object, not the intensity?
9. Why does black body radiation have a “hump” in its frequency graph?

In just 20 years we had come a long way. Out of nine mysteries, only three were completely left open, and another was mostly solved. And even mystery number 3 had tantalizing hints.

More Developments in Special Relativity

A few weeks ago–actually the last time I used this particular eagle–I described the four Big Papers Einstein published in 1905. Two of them had to do with what today we call “Special Relativity.”

What made it “special”? Did it ride the short bus to school?

What made it special was that it only applied to a very specific case, the case where the frames of reference are not accelerating. Constant speed, even high speed, isn’t an issue, but if there’s any sort of acceleration, it’s a different ball game.

General relativity doesn’t have this restriction. Special relativity turns out to be a special case of general relativity.

1915 was the year Einstein first put forward general relativity, which means that historically speaking, with the last article taking us up to that about then dealing with subatomic physics, this is the right time to take up general relativity.

But there had been some developments in special relativity in the meantime. Einstein hadn’t really thought about relativity from a geometric point of view, but many others, including his former math professor Hermann Minkowski, did. They pointed out that if you simply consider time as being a fourth dimension, a lot of things fell into place.

This does make some sense. After all, if you and I agree to meet at the corner of Pikes Peak and Cascade on the 14th floor of the Holly Sugar Building (which isn’t called that any more), we’ve specified a meeting place in three dimensions…latitude, longitude (the streets run north/south east/west in that part of town), and elevation (14th floor). Or coordinates…a triplet of them…can be used to define any location in space once you’ve defined the coordinate system (and it doesn’t even have to be a cubical grid either; cylindrical or spherical coordinates can work). You need three coordinates, though, because space is three dimensional. You can get by with two if you implicitly specify the third (in this case, surface level could be assumed; that’s probably a good idea when dealing with ships).

But if you and I arrange a meeting place in this manner, we’re committing a Bidenesque screwup: Because we also need to specify a time. So really, you need four coordinates, three space coordinates x, y, and z, and a time coordinate, t.

When you specify all four, you’ve defined what physicists call an event. And you’re doing it in terms of spacetime.

And so, it turns out that special relativity fits well with the concept of spacetime and works in four dimensions. This was pointed out by Minkowski.

But there was a difference! And it becomes most manifest when considering interval. The interval is the distance between two events.

If you are using a “Cartesian” (cubic grid) coordinate system, the difference between two points in space is an extension of Pythagoras. In two dimensions, on a Cartesian grid, the distance between two points is simply the difference between their x-coordinates, squared, plus the difference between their y coordinates, squared, then take the square root of all that.

distance (2 dimensions) = sqrt( ( x₁-x₂ )² + (y₁-y₂)²)

It’s precisely equivalent to a²+b²=c². (And note, it doesn’t make any difference whether you subtract point 1 from point 2, or vice versa. Sure, you will get opposite signs depending on the order, but those get wiped out when you square the differences.)

To move up to three dimensions, you can square the two dimensional distance again, then square the difference in the third coordinate. But when you do that, it’s algebraically equivalent to just squaring all three differences, adding them together, then taking the square root:

distance (3 dimensions) = sqrt( ( x₁-x₂ )² + (y₁-y₂)² + (z₁-z₂)²)

So it stands to reason that for four dimensions you’d square the time difference as well, like this, right?

distance (spacetime) = sqrt( ( x₁-x₂ )² + (y₁-y₂)² + (z₁-z₂)² + (t₁-t₂)²) (wrong, don’t do this)

Well, it might stand to reason, but it’s wrong.

First there’s one issue to clear out of the way: time is measured in seconds and distance is measured in meters; by simply taking a difference in time and jumbling it in with three differences in meters, you are mixing apples and roadcones.

It turns out that with spacetime, a distance of d = ct is equivalent to a duration of t. In other words a one second time difference is equivalent to a distance of 299,792,458 meters. So when doing this computation, if you divide your space distances by the speed of light, you get units of seconds, and now the four “pieces” of the equation all match units. You’ll have to multiply the result by c again to get back to meters.

So let’s imagine two events at the same x and y, but with z differing by 299,792.458 meters, and t differing by one second. Dividing all of the space coordinates by c, you get the x and y differences = 0, the z difference being 1 second, and of course the t difference is 1 second.

Incidentally a difference is often denoted by Δ, the Greek letter delta, so we can say Δx=0, Δy=0, Δz=1, and Δt=1. It’s a lot more convenient, and amongst techie types “delta” is often slang for “change” or “difference.” (“What’s the delta in the cost of gas switching from the orange guy with the mean tweets to His Fraudulency?” for instance.)

So square everything and get 0, 0, 1, 1, add them together to get 2, take the square root, and the interval is 1.414 seconds, or about 424 million meters, right?

Well, no. The BIG difference is that with space time you subtract the space components from the time component!

Here is the correct formula:

distance (spacetime) = sqrt( (t₁-t₂)² – ( x₁-x₂ )² – (y₁-y₂)² – (z₁-z₂)² )

Note that the time difference is first and all the space differences are subtracted from it.

So in this case the interval is zero seconds; the two ones cancel.

(Equivalently, you could multiply the time by c and work entirely in meters, rather than seconds…but that would have made the arithmetic ugly.)

Now there’s only one thing that can get from that first event, to that second event. The one thing that can move 299,792,458 meters in one second, and that, of course, is light in a vacuum.

But the light, in doing so, covers no interval. Which means that the light beam perceives no distance traveled and no time elapsed! But if you remember the time and distance dilation formulas from the last time we talked about special relativity, that’s what we would expect. At light speed, both effects cause the elapsed time and traveled distance (from the point of view of the light beam) to reach zero.

So what we have here is a geometric model of special relativity.

OK, let’s play another game here. Let’s make the space distance twice as much as it was before, while leaving the time distance 1. You end up with Δx=0, Δy=0, Δz=2, Δt=1.

Plugging that in, we get sqrt( 1² – 0² – 0² – 2² ) = sqrt( 1 – 4 ) = sqrt( -3 ).

Now you can’t take the square root of -3 and get a meaningful distance (or time) out of it. What the spacetime model is telling you is you cannot get from one event to the other. If you could, it would be by traveling faster than the speed of light. So the spacetime model has built into it a rationale for not being able to exceed the speed of light in a vacuum.

Einstein didn’t use this in 1905, but he adopted it shortly thereafter. (I wonder if Minkowski ever told his former student how proud he was of him.)

Minkowski invented the spacetime diagram, where the vertical axis is time, and the horizontal axis is space. Objects traveling on this diagram cannot do so at a slant of less than 45 degrees (that implies traveling faster than c), light itself moves at a 45 degree slant on the diagrams.

An interesting consequence of spacetime is that everything moves at exactly the same speed through it. You, sitting in your chair reading this are traveling through time purely, at one second every second. Move fast enough, and your motion becomes predominantly through space and you are moving slower through time. The second motion is called spacelike because most of the motion is through space, and time slows down signficantly, the first motion is called timelike not because it’d be snarky to refer to it as “sitting on your ass” but because most of the motion is along the time axis.

More Einsteinian Thought Experiments

Spacetime, it turns out, is the easiest way of dealing with general relativity. Not that it’s easy.

I actually wasn’t that far off when I talked about special relativity riding the short bus. The math involved with it is an absolute breeze compared with the math in general relativity. It’s a major event when someone is able to solve the general relativity equation for a certain specific scenario. In fact, I will be honest with you: I don’t understand the math. I never got exposed to tensors; I just have a vague idea that they’re sort of like matrices (which are a power tool in mathematics I do know something about), but not quite.

So with that, I can’t comprehend the real situation then try to explain it to you. I have to rely on the same science-for-senators handwaving that you’ve probably already seen. As such, I’ve been half-dreading writing this post.

But, it does start with Einstein’s doing thought experiments, so at least that part should be comprehensible if I am doing my job right. [Only later will you see the wild leap I can’t justify.]

The supposition this time is that if you were in a locked chamber, no way to see in or out, and were feeling earth-normal gravity, you’d be unable to distinguish it from being in a room that is being accelerated ‘upward’ at g, the acceleration due to gravity. The rules of physics would be the same; any experiment you could carry out would have the same result.

That doesn’t seem too unreasonable. If you drop your four hundred ounce gold brick on your foot in either scenario, it will hurt just as much, just as quickly.

But this does lead to differences with the conventional understanding when you deal with light.

The conventional understanding is that light has no mass, so gravity should not act on it. A beam off your laser pointer should travel in a straight line no matter how strong the gravity is.

On the other hand, if you’re in a room that’s under acceleration, it feels like gravity, there’s an obvious up and a down. But you should be able to tell the difference between an accelerating room and one experiencing gravity, because if you fire your laser pointer horizontally, and the room is accelerating, you should see the beam bend. That is because the beam of light is moving vertically at the same speed you are, but once it has left the laser pointer, it doesn’t speed up in the vertical direction, but you and the room do, so you see the beam drop.

So if the room is feeling gravity, the beam shouldn’t bend because the force of gravity on a massless object should be zero, but if the room is being accelerated, the beam should bend, because the room is moving faster than it was before, by the time the beam hits the wall.

But if Einstein is right and there really is no way to tell the difference, then either both beams need to move in a straight line, or both beams need to bend. In the second case, light is affected by gravity even though it has no mass.

You need really strong gravity to see this, though. Or a long distance. Because light crosses any normal everyday distance in microseconds or even nanoseconds, and if it’s going to “drop” due to gravity, well, gravity only gets to act on it for a few billionths of a second. Plug that in to d=1/2at² and it’s almost nothing.

OK, but there is a concrete prediction. A light beam going by a massive object, should bend a bit. This is testable with great difficulty.

Here’s another: If light is affected by gravity, light traveling upward has to lose energy, just like a thrown baseball loses kinetic energy (trading it for potential energy) and slowing down. But light cannot lose energy by slowing down, its speed in any particular medium is a constant.

It can lose energy another way, however. Remember E = hv? (Where ν is the frequency?)

So the light, climbing in a gravity field, should decrease in frequency. That’s the only way it can lose energy. Similarly, light going “downhill” should increase in frequency to gain energy.

There’s an alternative way of looking at this though. Imagine that light beam in the accelerating room, firing upward from the floor. By the time the beam reaches the ceiling, the ceiling has sped up, so there’s a doppler shift in the wavelength, towards the red. Since you can’t tell this case from a room feeling “real” gravity, in that room the light has to redshift too.

This is gravitational red shift. Visible light becomes redder as it moves uphill. Again, this effect is tiny on Earth, but it’s measurable today (I don’t think it was measurable using 1915 equipment).

Hiding inside that effect is another.

Imagine someone on the surface of earth, shining a light straight up. He blinks, and then a second later he blinks again. In the meantime, about 600 trillion wavelengths of the light are emitted.

Someone, up in space, will see the same sequence of events. Blink, 600 trillion wavelengths, then a blink. But the light is red shifted when he sees it. 600 trillion wavelengths takes more than a second to pass by him, because the frequency has dropped.

Therefore he sees it take more than a second between the two blinks. From his point of view, time is running slower down on earth than it is for him in space.

This is gravitational time dilation.

So these are concrete, comprehensible predictions to see whether an accelerating reference frame, where effects happen due to inertia, is truly the same as one with gravity (effects due to mass).

But when Einstein followed the math…it got interesting. And I’m going to have to state it without trying to justify it. Sorry. Complicated business!

Gravity, it turns out, isn’t a “force” like electromagnetism is. It turns out that any object not being accelerated by a real force (like a rocket motor), travels a straight line in space time, the shortest distance between two events. If you think it’s curving because, for instance it’s a space probe doing a “flyby” of Jupiter, it’s because spacetime is curved.

OK, now this takes time to wrap one’s brain around, and if you fail at it you’re in very good company. How does space itself actually bend? Objects bend in space, space itself, can’t bend, there’s nothing to bend.

Nevertheless it does. Not just in Einstein’s thought experiments, but in reality.

Einstein used his new concepts to compute the orbit of Mercury.

Remember there had been a long-standing mystery about Mercury. It orbits the sun in a markedly elliptical orbit, and under Newtonian two-body gravity, the long (or “major”) axis of the ellipse should never change direction. But in fact it does change direction. Some of this can be shown to be due to the other planets’ pulling on Mercury constantly. But not all. After subtracting all of that out, the major axis still shifts by 43 arc seconds every century. That’s an angle about three quarters the width of a quarter set out at a hundred yards, and it takes a century (about 400 revolutions of Mercury about the sun) for it to make that shift.

People had theorized that an undiscovered planet closer to the Sun than Mercury could be perturbing Mercury’s orbit, but it would be frustratingly difficult to see such a planet so close to the Sun.

But when Einstein did the computation with his modified law of gravity, he found that an object orbiting that close to a very massive object like the sun…would see a shift of exactly this amount!

The net effect of Einstein’s new law of gravity is that near very massive objects, gravity’s effect is slightly greater than an inverse square law. Which means that at perihelion (closest approach) gravity is a bit stronger than Newton would expect. However, Kepler’s second law still applies (a line from the sun to the planet sweeping out equal areas in equal times) because it depends on the conservation of angular momentum. So this manifests itself as the elliptical orbit behaving like something out of a Spirograph set.

OK, so Einstein had made one prediction he could test himself. But to be really solid science, predicting new phenomena (rather than just being a possible explanation of a known phenomenon) would be good.

Testing General Relativity

The light bending, doppler effect, and time dilation effects were something that had not been seen before, had not been predicted by any other theory, and if seen would be otherwise unexpected; i.e., a successful prediction by this theory…three successful predictions, actually.

As it turned out, the light bending was the easiest. For this you can use a large massive body that’s between you and stuff of known position, if the position of those background objects appears shifted near the body, you have gravity (from the massive body) bending the light coming to you from the background objects.

This is a job for the Sun. As seen from earth, it moves against the background (it’s really the Earth moving), which is a known pattern of stars. We’ve got plenty of star maps taken when the sun is nowhere along the line of sight (in fact when the sun is behind the mapper, because he’s doing this at night and the sun is below his feet somewhere). So we just need to see if the stars seem shifted (away from the sun as it turns out) when the sun is on the line of sight to the stars.

Did I mention earlier the sun is bright? This makes it impossible to see stars that are almost behind it.

Except during a solar eclipse, when the moon neatly covers the sun!

There was a solar eclipse in 1919. Astronomer Arthur Eddington took photographs, not of the corona (as people usually do during eclipses) but of the stars near the Sun. The elegant mathematical reasoning of Albert Einstein was put to the test. (If you don’t find it elegant, it’s because you haven’t seen and understood the math; I haven’t understood it myself, so I’m taking other peoples’ word for it that its elegant.)

It was hard to measure accurately enough to truly nail it down, but the stars’ apparent position had indeed shifted and the measured effect was consistent with General Relativity.

This was big news. I mean, really big news. It made the newspapers read by regular people. This was when Einstein became famous outside of scientific circles.

Today, we can see entire galaxies bending the light of galaxies behind them. In fact, there’s a spectacular instance of two almost-perfectly-lined up galaxies causing the background galaxy to look like a ring, known as an Einstein Ring:

The gravitational redshift took longer. For this, the ideal situation is a bright, massive, small object (small is better because the gravity is more intense, and a white dwarf, which is a sun-sized star that has run out of nuclear fuel and collapsed down to the size of the earth, is ideal. It still shines brightly because it will take millions or even billions of years to cool off, but it has a very strong gravitational field. As early as 1925, someone attempted to measure the gravitational redshift off of the star Sirius B (see my article on Sirius A and B: https://www.theqtree.com/2020/01/01/another-sirius-tale-of-two-stars/), but other scientists pointed out there was too much glare from Sirius A (which is, after all the brightest star in the nighttime sky). Finally in 1954 Popper got a good measurement off of 40 Eridani B and confirmed this prediction. It’s also possible now to measure the shift in frequency of gamma rays going up several stories here on Earth.

The gravitational time dilation can be measured by two different atomic clocks at different elevations. Eventually, the lower one will fall behind the upper one.

Most famously, the GPS constellation of satellites demonstrates both special relativity time dilation, and general relativity time dilation.

The GPS system works by having each satellite sending out time signals. Their position at any time can be computed by your GPS receiver, so it’s just a matter of comparing the signals from at least four (but even more is better), noticing the differences of the times in the signals, turning that into different distances from the satellites, then doing a lot of geometry to triangulate, and figure out where the receiver must be.

Extremely accurate time sources on the satellites are an absolute necessity. If one is off by ten nanoseconds, your position will be off by ten feet (light travels roughly a foot per nanosecond).

The satellites are moving quickly, which means a clock on that satellite will seem, from down here on earth, to be ticking more slowly due to special relativity time dilation. (Not much more slowly, but enough to be measurable with modern atomic clocks.) They are also higher so due to gravitational time dilation, our clocks should run more slowly than the GPS satellite ones. The two effects are in opposite directions, so they will tend to cancel each other out. The gravitational effect is the larger of the two, so from our standpoint the GPS clocks look like they’re running faster than they would to someone actually on one of the satellites. In fact, it will run 38 microseconds per day faster than you’d expect without either time dilation effect. That would be enough to throw position calculations off by several miles…after one day.

This effect is real, it does happen. What the GPS engineers do is slow the satellites’ clocks down to compensate. That way in orbit when they speed up (as seen by us), we see the clocks ticking off normal seconds, and so if you drive your car into the Mississippi river when trying to get to Pikes Peak, it’s not the fault of GPS.

GPS wasn’t designed for the purpose of testing general relativity, but there are a couple of rather more detailed predictions involving a phenomenon called “frame dragging” (which I am not even going to try to explain, because I want to publish this this week, not sometime in October) that have been confirmed by satellites deliberately launched to test general relativity.

General relativity has met every test thrown at it. It’s real. Spacetime bends. And objects move along the shortest possible path through spacetime.

As famously put by John Archibald Wheeler (1911-2008, a veteran of the Manhattan Project) in 2000, “Spacetime tells matter how to move; matter tells spacetime how to curve.”

I debated whether to put a “rubber sheet” diagram in this post. They’re very problematic. Yes, you can see how an object might follow a curved path on the rubber sheet, which is supposed to be how gravity works, but the rubber sheet is itself bent by gravity pulling on an object. If you can’t ignore that, you’re going to be hung up on the fact that (demoed) “gravity” is caused by (real) gravity. I decided, ultimately, not to do it even though I could write disclaimer after disclaimer that it’s a visualization tool only, not an explanatory one. (And I believe I hear Wolf breathing a sigh of relief.)

But one doesn’t need a rubber sheet diagram to know that Joe Biden didn’t win.

And, in case you didn’t notice…we can cross mystery number one off the list. Thanks, Herr Doktor Einstein!

Obligatory PSAs and Reminders

China is Lower than Whale Shit

Remember Hong Kong!!!

中国是个混蛋 !!!
Zhōngguò shì gè hùndàn !!!
China is asshoe !!!

China is in the White House

Since Wednesday, January 20 at Noon EST, the bought-and-paid for His Fraudulency Joseph Biden has been in the White House. It’s as good as having China in the Oval Office.

Joe Biden is Asshoe

China is in the White House, because Joe Biden is in the White House, and Joe Biden is identically equal to China. China is Asshoe. Therefore, Joe Biden is Asshoe.

But of course the much more important thing to realize:

Joe Biden Didn’t Win

乔*拜登没赢 !!!
Qiáo Bài dēng méi yíng !!!
Joe Biden didn’t win !!!

His Fraudulency

Joe Biteme, properly styled His Fraudulency, continues to infest the White House, and hopium is still being dispensed even as our military appears to have joined the political establishment in knuckling under to the fraud.

All realistic hope lies in the audits, and perhaps the Lindell lawsuit (that will depend on how honestly the system responds to the suit).

One can hope that all is not as it seems.

I’d love to feast on that crow.

The Lindell Reports

It sounds worse that most of us imagined. And we have good evidence (if placed before a judge who understands probability, combinatorics, and statistics (three closely-connected branches of mathematics).

The question is, now that we have this, what’s next?

Can we get more states to do forensic audits? It will be tougher in states where the auditors themselves ended up in their positions of authority through cheating!

Even if not, it’s good to go into whatever comes next with the certitude that we were and are right about…

Joe Biden Didn’t Win. And neither did Hoe, and neither did half the craptastic Dems out there. RINOs might have won the general because at that point voters had a choice between a definite Dem and a maybe-not-as-bad “Republican.” But how many got in due to a corrupted primary?

We have to do our best to force this to stick and force “them” to pay attention to it!

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.

Nothing else matters at this point. Talking about trying again in 2022 or 2024 is hopeless otherwise. Which is not to say one must never talk about this, but rather that one must account for this in ones planning; if fixing the fraud is not part of the plan, you have no plan.

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.)

(Paper) Spot Prices

Last week:

Gold $1763.90
Silver $24.48
Platinum $985.00
Palladium $2712.00
Rhodium $21,150.00

This week, 3PM Mountain Time, markets have closed for the weekend.

Gold $1780.60
Silver $23.83
Platinum $1034
Palladium $2736
Rhodium $20,200

This might be a good time to buy silver. On the other hand it could drop even m0re.

Electrons Get Quanta

If you’ll recall, last time I mentioned that in 1911 van den Broek suggested that an atom’s place in the periodic table depended on the positive charge of the nucleus; when that charge was expressed as a positive-signed multiple of e, you had a simple integer number which is that atom’s atomic number. I then said it was merely an idea for about two years, and then I left you hanging.

I’m going to pick up that thread, but I’m going to do it my way: I’m going to back up a bit and follow another thread to that same place.

As of 1900, chemists were pretty sure they were missing eight elements on the periodic table. Because they didn’t know how many lanthanides (“rare earths”) actually existed (some guesses ran as high as 25) and simply had no idea what was going on there, they didn’t know how many they were missing. (We now know that lanthanum through lutetium is fifteen elements inclusive; chemists back then knew twelve in that range, and suspected there were more.)

Remember in 1900 they didn’t know about atomic number. They did have the periodic table, and it had holes in it that were clearly missing elements, but the lanthanides didn’t seem to fit into that scheme at all so they were a big question mark.

In 1901, europium–a lanthanide whose atomic weight was between samarium and gadolinium–was discovered, and then in 1902-03 actinium was discovered during investigations of the radioactive decay chains. (From the radioactive decay series, astatine, francium and protactinium were not known yet as of 1911, but the first two were “known” holes in the table, below iodine and cesium, and protactinium was probably suspected–it’s hard to tell because back then chemists didn’t realize the actinides were like the lanthanides. My extensive discussion last week was based largely on current knowledge.)

1906 saw the discovery of lutetium, at the time the heaviest of the rare earths.

So in 1911, van den Broek came up with the concept of the atomic number. And the periodic table was pretty “tidy” right up through barium, but after barium were the lanthanides. So I believe they were able to assign every element up to barium atomic numbers, with barium at Z=56. There was a gap at Z=43. Then with an unknown number of lanthanides, it would be impossible to assign an actual number to the first known element after the lanthanides, tantalum, but we knew what group tantalum was in, so we could basically restart counting from there, identifying more holes. Two spaces to the right, under that hole for Z=43, was another hole. Then a hole under iodine and a hole under cesium, as previously mentioned.

Protactinium was discovered in 1913, so we may not have realized it at the time but everything from radium (directly below barium) on up was known.

In 1913 the picture became a lot clearer. Henry Moseley (a student of Rutherford’s), in 1913 was doing x-ray spectroscopy on a variety of elements and measuring the wavelengths. He noticed a fairly simple mathematical relationship between the atomic number (where known) and at least one of the x-ray wavelengths. From this he formulated Moseley’s law. (I’d quote the law here, but although the formula is simple, explaining what the symbols meant would be a royal pain.)

So now the guesswork was gone. Moseley could zap even a rare earth metal with his x ray device, and calculate its atomic number. Lanthanum was 57. Lutetium was Z=71. We had, without realizing it, already nearly completed the list in between: Cerium (58), praseodymium (59), neodymium (60), samarium (62), europium (63), gadolinium (64), terbium (65), dysprosium (66), holmium (67), erbium (68), thulium (69), ytterbium (70), and lutetium (71). Only #61 was missing. {Yes I am enough of a geek to known those by heart.)

So now that numbers could be assigned to every element and not just the first 56, we knew we were missing #43 (right below manganese), #61 (a rare earth), #72, #75 (below #43), #85 and #87. Uranium came in at #92 and was the last element.

Moseley’s law was consistent with the Bohr model of the atom, which was put forward that year (just two years after the Rutherford model).

And the Bohr model is our main topic today, but I will finish Moseley’s story first. Sadly, it won’t take long.

It sure looked like Moseley was destined for bigger and better things, and he had certainly earned himself a Nobel Prize for putting the atomic number on a solid footing. But World War I broke out the next year and Moseley volunteered. He was sent to Gallipoli in modern day Turkey and was killed on August 10, 1915. The Nobel Prize committee gave no award for physics in 1916. We can only speculate, but it seems as if they intended to give that award to Moseley but as they do not give posthumous awards, had to change their plans.

The Bohr model of the atom is actually considered a modification of the prior Rutherford model, which was unsatisfactory for a number of reasons. So it’s technically the “Bohr-Rutherford” model, but most just call it the Bohr model, after the Danish physicist Neils Bohr (1885-1962).

Why was the Rutherford model unsatisfactory? Chief among the issues was that if it were accurate, no atom would last more than about ten billionths of a second. Since I am writing this, and you will soon be reading this, and you and I are both made up of atoms that haven’t collapsed yet, there’s clearly a disconnect.

The Rutherford model supposed that the negatively charged, light electrons orbited the much more massive and very tiny positively charged nucleus. It didn’t discuss orbital periods of the electrons, or anything like that, so it wasn’t very specific. But that wasn’t the big issue.

The problem is that any electric charge that is being accelerated will emit electromagnetic energy. And electrons in orbit about a nucleus are constantly being accelerated. Remember that an object in motion will continue moving at that speed and direction unless acted on by an outside force (this goes back to part 1). An outside force, of course, will cause an acceleration. Since the electrons are following a curved path, they are being accelerated.

Calculations at the time based on Maxwell’s equations showed that it would take about ten billionths of a second for an orbiting electron to radiate away all of its kinetic energy, causing it to spiral in and plow into the nucleus.

How to solve this problem?

Well, there was a sketchy tool in the physicist’s tool kit that essentially functioned by forbidding certain values of energy, or momentum. If this tool could be applied here, then an electron in an orbit would be unable to drop downward, unless it took a big step downward all at once. And there’d be a minimum orbital energy it could not drop below.

That tool was quantum theory. It’s not the same quantum theory that we have today. As I hinted, it basically functioned as an overlay on classical physics, forbidding certain values of some parameters. It had been used by Max Planck to explain the black body spectrum in 1900, and it had been invoked by Albert Einstein to explain the photoelectric effect in 1905 (for which he eventually won the Nobel prize–for this, not for relativity!).

Energy came in fixed quanta, and these quanta’s sizes were always related somehow to Planck’s constant, which is:

h = 6.62607015×10⁻³⁴ J⋅Hz⁻¹

Or equivalently (since a hertz is a “per second”):

h = 6.62607015×10⁻³⁴ J⋅s

This turns out to have the same dimensions as angular momentum. A joule is a kg⋅m²/s², or as a dimension rather than units, m⋅d²/t². Multiply that by time to match Planck’s constant and it’s m⋅d²/t. Angular momentum is speed, times mass, times the distance from the central point around which angular momentum is being measured, or (d/t⋅m⋅d) which is also m⋅d²/t.

However h is defined in terms of full revolutions, and angular momentum operates in radians, so we really need h/2π, a number that turns up so often, it has it’s own symbol, ħ, pronounced “H-bar” and often known as the “reduced Planck constant.” It’s equal to 1.054571817…×10⁻³⁴ J⋅s. Or, since we are talking about atoms here, the preferred units are in terms of electron volts, so the reduced Planck constant is 6.582119569…×10⁻¹⁶ eV⋅s

So if the angular momentum of electrons in an atom were restricted to multiples of ħ, it could keep the main descriptive feature of the Rutherford model (electrons orbiting about the nucleus) while solving the problem of having them spiral into the nucleus, radiating energy all the while. The lowest possible orbit would be the one where the angular momentum was equal to ħ, the next one up (higher energy), 2ħ, and so on.

Well, it’s a fine idea, but does it actually make things look the way they really are?

Let’s work with hydrogen. One electron, one proton. No other electrons to cause complications because they repel the first electron.

Assuming a circular orbit (so that the requisite cross product becomes equal to multiplying distance by velocity), the angular momentum of the electron is going to equal its mass, times its velocity in orbit, times its distance from the nucleus:

m_evr = nħ

The n is the integer multiplier and is now known as the principal quantum number.

Well, we know one of these, the mass. But we can actually express the velocity needed to maintain a circular orbit, in terms of distance and the attractive force between the proton and the electron (which we know), so that gets us down to one unknown. And we can eventually work our way down to figuring that when n is 1, the orbital radius is 0.0529 nanometers (billionths of a meter) for a hydrogen atom (one electron orbiting one proton).

OK, so by analogy with orbital mechanics, the lowest energy orbit is indeed this n = 1 orbit. What could make the electron move out of that orbit?

The hydrogen atom could actually hit another hydrogen atom, transferring kinetic energy to the electron, enough that it could jump to n=2. Thus a hot hydrogen gas, where the kinetic energy of the atoms is higher, could result in electrons being “jumped up” to higher orbits. So, basically, heat can do it.

Or the electron could absorb a photon with enough energy to make the jump.

And if in a higher orbit, how could an electron drop? It could do so by emitting a photon. But it would be a photon that contains precisely the energy difference between the two orbits! .

Remember that E = h ν for light (that last letter being Greek “nu” not a “vee”). So if we know the energy difference, we should be able to figure out the frequency, ν of the photon, then get to its wavelength in nanometers. For wavelengths between 400 and 770 nanometers, the photon will be visible to our eyes and will have a certain exact color.

The lowest orbit has the minimum energy. Just like with astrodynamic orbits, the energy is set to zero at a distance of infinity, and becomes more and more negative the closer the orbit gets to the nucleus, so the energy of the minimum orbit (n=1) is -13.6 eV. The second orbit (n=2) is at -3.4 eV, the third (n=3) is -1.51 eV, and so on, approaching but never equaling zero. So an electron in the third orbit can shed a photon and drop all the way down from -1.51 eV to -13.6 eV, a difference of 12.1 eV. This corresponds to a wavelength of 102.57 nm. That’s an ultraviolet wavelength.

But how about dropping from n=3 to n=2? That difference is about 1.9 eV. And that corresponds to a wavelength of 656.3 nm, which is visible light.

That number no doubt leaped out at someone. And when they computed the numbers for jumping from n=4 to n=2, then n=5 to n=2, and so on, those numbers looked familiar, too.

They were the wavelengths of light in the hydrogen emission spectrum. This is known as the Balmer series, all the lines you get from dropping from some higher n down to 2.

The series of lines corresponding to dropping down to n=1 is called the Lyman series, and as previously indicated, they’re all ultraviolet.

So now we have an explanation of the hydrogen emission spectrum.

Maybe there was something “real” behind this quantum buggery!

The Bohr atom model stopped here. It explained hydrogen very well, but it couldn’t, by itself, cope with more than one electron. However its underlying principles do hold for other cases.

What Moseley had done was identify, via his X ray work, the transition down to n=1, which in heavier atoms is in the x-ray band. This gets progressively more energetic as the charge in the nucleus increases, such that one can actually tell what the nuclear charge is from the x ray wavelength. So this, too, validated the Bohr model in principle, at least insofar as the Bohr model assumes quantum effects are in play.

I’m going to carry this story through (in a grossly oversimplified way) to the present day, except I won’t delve too deeply into the quantum mechanical aspects of it–quantum theory turns out to be seriously weird but this wouldn’t begin to become apparent until about 1925. So far (as of the 1910s), this bowdlerized version where it just arbitrarily restricts what can happen in an otherwise classical physics realm was working pretty well (this is now called “old quantum theory”).

The n=1, n=2, n=3 and so on principal quantum numbers were named electron shells. It became apparent as time went on, though, that each of these shells contained subshells, according to a simple rule: The 1st shell consisted of one subshell, the second shell had two subshells, and so on. The subshells got labeled s, p, d, and f. This arose from quantum mechanical considerations.

Each subshell can only hold a certain number of electrons. An s subshell could hold 2 electrons, a p subshell 6 electrons, a d subshell 10 electrons, and an f subshell 14 electrons. We’ve never dealt with a fifth subshell, but it would probably be labeled g, with 18 electrons. Each goes up four electrons. This, too, arose from quantum mechanical considerations.

The subshells are in turn divided into orbitals holding 2 electrons each, but I won’t tread there. (And again, quantum mechanical considerations).

So, the following subshells exist: 1s, 2s, 2p, 3s, 3p, 3d, 4s, 4p, 4d, 4f, and so on.

Electrons are added to the lowest energy shell that isn’t already full. That’s whether you’re creating an ion by adding extra electrons, or just trying to get a large atom up to its normal complement.

Let’s take oxygen as an example. It has eight protons in its nucleus, it will want eight electrons in its shells.

So the first two electrons go into the 1s subshell. Then the 2s subshell gets the next two electrons. Finally, the four remaining electrons go into the 2p subshell, which could accept another two electrons if they were available.

Now let us consider iron, Z=26. The first eight electrons go like oxygen’s. The next two fill up the remainder of the 2p subshell, after which we move on to the 3s subshell, which takes two more electrons (12 so far). 3p takes up another six electrons (18 so far). You might expect that now we will move to the 3d subshell…but that turns out to be wrong. The 3d subshell’s energy is actually slightly higher than the 4s subshell, so we will fill the 4s before the 3d. Electrons 19 and 20 go into the 4s subshell, then the last six electrons do go into the 3d subshell. If we were to continue, the next subshell to fill would be the 4p subshell.

So we’re seeing at the end a sequence where we fill a 2 electron s subshell, a 10 electron d one, then a six electron p one. If we were to carry on to lead (Z=82), we’d encounter our first f subshell, 4f, right after the 6s subshell but before the 5d subshell; lead takes us into the 6p subshell.

The numbers 2, 6, 10, and 14 might be tickling your brain trying to be noticed. If not, perhaps their successive sums will: 2, 8 (2+6), 18 (2+6+10) and 32 (2+6+10+14).

These are the lengths of the rows on the periodic table. In fact, if you look at the table, the left hand side is a “tower” two elements wide–corresponding to the s subshell. The left side is a block six elements wide–corresponding to the p subshell. The central skinny part is ten elements wide, and corresponds to the d subshell. Looking at the two rows that are “footnoted” below the main body of the table, those are usually depicted as 15 units wide, but they are supposed to tuck into a square in the third column, so one of those 15 squares really belongs to the d block. The other 14 are the f subshell. (By the way, chemists argue over whether the first or last of the fifteen is the one in the d-block; they seem to have recently decided to go with the last one of the fifteen.)

So the very shape of the periodic table reflects the shells and subshells, which in turn derive from quantum principles.

The periodic table is on a firm footing now. Atomic number is on a secure footing, We now even understand those elements whose atomic weights aren’t close to integers. We just don’t know why they aren’t exact integers yet.

Obligatory PSAs and Reminders

China is Lower than Whale Shit

Remember Hong Kong!!!

中国是个混蛋 !!!
Zhōngguò shì gè hùndàn !!!
China is asshoe !!!

China is in the White House

Since Wednesday, January 20 at Noon EST, the bought-and-paid for His Fraudulency Joseph Biden has been in the White House. It’s as good as having China in the Oval Office.

Joe Biden is Asshoe

China is in the White House, because Joe Biden is in the White House, and Joe Biden is identically equal to China. China is Asshoe. Therefore, Joe Biden is Asshoe.

But of course the much more important thing to realize:

Joe Biden Didn’t Win

乔*拜登没赢 !!!
Qiáo Bài dēng méi yíng !!!
Joe Biden didn’t win !!!

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.

Nothing else matters at this point. Talking about trying again in 2022 or 2024 is hopeless otherwise. Which is not to say one must never talk about this, but rather that one must account for this in ones planning; if fixing the fraud is not part of the plan, you have no plan.

The Audit

The Audit is definitely heating up. Let’s see if the Opposition manages to squelch it and its consequences. I’ll be honest; I expect it to be ignored by anyone capable of ordering Biden/Harris to step down.

Nevertheless, anything that can be done to make Biden look less legitimate is a worthy thing!

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 $1815.20
Silver $25.56
Platinum $1053.00
Palladium $2747.00
Rhodium $19,500.00

This week, 3PM Mountain Time, markets have closed for the weekend.

Gold $1763.90
Silver $24.48
Platinum $985.00
Palladium $2712.00
Rhodium $21,150.00

Gold was up in the 1810s all week up to Friday morning, but tanked HARD on that day, down $41.20. Everything took a beating, honestly, except rhodium which went up.

Part XIII – Rutherford On A Roll

We left off, circa 1903, having discovered radioactivity and the electron, and making quite a bit of progress with them.

To try to recap (and there are a few things in this so-called “recap” that I should have mentioned earlier, but didn’t), an electron is a negatively charged particle about 1/1830th the mass of a hydrogen atom, which up to then had been the lightest thing known to exist. They could be knocked off of atoms in a Crookes tube and they would then form what was called a cathode ray (yes, the same “cathode ray” in those big tubes in those old boxy TVs). It is possible to strip one electron off a hydrogen atom, at which point the remaining piece of the hydrogen atom (called an ion) had a positive charge that balanced the electron’s negative charge. The atom as a whole was neutral, charge 0; the individual pieces also added up to 0. Even though there was plenty of mass left in the ion, easily enough for hundreds more electrons, no one could get a second electron to come out of a hydrogen atom.

Thomson, the discoverer of the electron, suggested that atoms were fairly solid spheres of positive electrical charge with little electron inclusions that could be knocked out to ionize the atom; this was called the plum pudding model of the atom.

Radioactivity had been discovered in 1896. Uranium and thorium, it turns out, are radioactive. Radioactivity turned out to consist of three types of rays, alpha, beta, and gamma.

Alpha rays turned out to be identical to doubly-ionized helium, i.e., helium from which two electrons had been stripped (and there was no sign of being able to strip away a third electron from helium). Helium itself had been discovered on Earth back in 1895, trapped in a uranium ore; its atomic mass was four times that of hydrogen. Clearly the helium had begun as alpha particles, then combined with electrons in the ore to become helium gas. The charge of an alpha particle is 2e.

Beta rays turned out to be high-speed electrons. Their charge, of course, is –e.

Gamma rays turned out to be electromagnetic radiation, extremely strong electromagnetic radiation, like X-rays on steroids. Gamma rays, like all photons, have no electrical charge at all.

Alpha rays could be stopped by a sheet of paper. Beta rays could penetrate many sheets of paper, but would be stopped by a thin sheet of metal. Gamma rays required a lot of shielding to stop.

Uranium (atomic weight ~238) and thorium (atomic weight ~232), which had just been discovered to be radioactive, were the heaviest known elements, roughly 238 and 232 times as massive, atom for atom, as hydrogen. The Curies discovered that uranium ore was four times as radioactive as the ores it contained; they were able to isolate two new elements, radium (atomic weight 226) and polonium (atomic weight 210), by processing tons of the ore pitchblende.

It was also clear that a pure block of refined uranium would grow more radioactive over time, eventually reaching a level significantly higher than before, but not nearly as high as the ores.

In radioactive decay, the total amount of energy released, relative to the mass, turned out to be staggeringly huge, thousands if not millions of times more than what was released by burning chemicals. In 1904 Ernest Rutherford (who had named the three types of radiation, and who is the star of today’s story) suggested that radioactivity could provide enough energy to power the sun for the many millions of years necessary for Darwinian evolution to take place. (Previously known sources of energy were woefully inadequate; it was one of the 1895 mysteries I listed.)

At the time atomic weight was considered to be a defining characteristic of an element. This would cause some confusion for a few years.

Some stuff I should have covered previously, but didn’t:

The electric charge of an electron is about -1.602 x 10^-19 coulombs. This is a negative number (because Benjamin Franklin arbitrarily picked one kind of charge to be positive and the other negative, and when the electron was discovered, it happened to be the one he tagged as negative), so, perhaps a bit counterintuitively, physicists define the minimum charge e to be +1.602 x 10^-19 coulombs, i.e., -1 times the charge of an electron. Physicists, in fact, find it far more convenient to use e as the unit of electric charge when talking about atoms, that way they don’t have to sling 10^-19s everywhere.

And they do something similar for energy. Just like a falling weight generates kinetic energy (a mass being attracted to another mass by gravity, speeds up that mass), an electron responding to one volt of electrical potential generates a certain amount of energy, which is defined to be an “electron volt.” This is abbreviated eV (which spell checkers will try to “fix” the capitalization of). This ends up being 1.602 x 10^-19 joules. (Notice it’s the same factor, 1.602 x 10^-19. This is a consequence of the way the joule, coulomb, and volt are defined.) Energy at the atomic level, particularly when dealing with chemical energy, tends to be a convenient, human-relatable number of electron volts.

And a reminder: An atomic mass unit was defined, in 1898, as 1/16th the mass of an oxygen molecule. This was very close to the mass of a hydrogen atom, but because oxygen reacted with more things, it was easier to use it as a yardstick. [This definition has since been modified, for reasons I’ll explain below.] It was equal to 1.6604675209 x 10^-27 kilograms. (This is slightly different from today’s value.) It was abbreviated “amu.” Atomic weights were expressed in amu’s, so oxygen’s atomic weight was 16.0000, and hydrogen’s was almost exactly 1.0: In 1949, under this definition, it was measured at 1.008 amu. (At least, according to a 1951-52 CRC handbook–well, it’s a book that fits King Kong’s hand–that I happen to own.)

OK, so that, I believe, catches us up.

A Plethora of Radioactive Elements?

Scientists continued to investigate radioactivity. They would find more and more elements, distinguished by their atomic masses, in both uranium and thorium ores.

Even as early as 1900-1903 Rutherford was involved in this effort. Looking at thorium “emanations” with his student Frederick Soddy, they discovered thorium x and a gas, thoron. At first they thought these were special forms of thorium, but then they realized these were not thorium. By 1903 they had concluded that these emanations were the result of thorium changing into another element. This was a very bold conclusion, since chemists up to now had believed elements were immutable, that such things were alchemist balogna. (And under normal circumstances this was true…but radioactivity was something fundamentally new, and certainly nothing like what the alchemists had thought of.)

So perhaps these new elements could fill in the large gap between bismuth and thorium in the periodic table? Well, they could, but it turned out that in fact, there were way too many of them. Realistically between lead and uranium there was room for nine elements, and we already had five of them: bismuth, polonium, radium, radon (which was basically the thoron gas) and thorium. But just in uranium ore there seemed to be about thirty of them (based on my count looking at a chart in Wikipoo–perhaps they had found fewer than that before they figured out what was actually going on). Thorium ores brought in another ten or so.

But it was very, very difficult to separate out these putative elements. For instance Soddy in 1910 showed that mesothorium, atomic weight 228, radium, atomic weight 226, and thorium X, atomic weight 224, were impossible to separate chemically, as if they were the same element. But how could that be so when the atomic weights were different? Trying to place these elements in the table led Soddy and Kazimierz Fajans to independently come up with the notion of radioactive displacement in 1913. Basically, this stated that an alpha decay reduced an atom’s mass by about four amu (the mass of the alpha particle), and also moved it two places to the left on the periodic table. (If such a thing were to happen to (say) nickel, it would become iron, which is two spots to the left of nickel. But it won’t.) A beta decay left the mass almost unchanged (the mass of the electron that gets kicked out is relatively insignificant), but moved the element one place to the right. (If an atom of palladium were to undergo a beta decay, it would become silver. This has happened under very special circumstances, ones that won’t affect the palladium bullion I hope you own.) Gamma decay had no such effect; apparently it was just a way to get rid of energy.

For this work Rutherford won the 1908 Nobel Prize for Physics.

But he hadn’t even got started yet.

The Isotope

Now if one used the radioactive displacement principle, it appeared that two or more different “elements” could occupy the same place on the periodic table. The three I named above all fit in the same square, directly under barium. Because they occupied the same place, they were termed isotopes, from Greek for “the same place.”

So you had “elements” of different mass that otherwise behaved identically. At this point chemists decided that the mass wasn’t as important as the behavior, and swallowed the concept of two different atomic weights representing the same element, rather than insisting they must be different elements solely because of different atomic weights. Atomic weight wasn’t necessarily a crucial characteristic of an element, particularly when it came to ones extracted from radioactive ones.

In 1912, meanwhile, J. J. Thomson, who had discovered the electron in 1897 (with some help from Rutherford, it turns out) wasn’t done yet, had ionized neon (which was the tenth element listed on the periodic table at the time) in a Crookes tube and magnetically and electrically deflected its ions, the same way that he had deflected electrons in 1897, to determine the ions’ charge to mass ratio. He was quite surprised to see these ions, which should have weighed in at about 21.18 amus, went to two different locations! Some were deflecting more than others, because they were lighter than those others.

Assuming that they were singly ionized, with one electron removed (it takes a lot more energy to take the second electron off than it did the first), one group of ions had an atomic weight of almost exactly 20, the other had an atomic weight of almost exactly 22. The atomic weight of neon had been measured as 20.179, which made it one of those cases where the atomic weight was not almost a whole number, but now it looked like that was actually an average value. Most neon had atomic weight of almost exactly 20, but some came in at about 22, and the weighted (ahem) average was 20.179.

So now, even perfectly ordinary stable elements had isotopes, and this time no one thought these must be two different elements because the weights are different. In modern terms neon consists of a mix of neon-20 and neon-22.

I have mentioned in the past that many elements had “atomic weights” or “atomic masses” that were almost a perfect multiple of hydrogen’s. These mostly turn out to be elements with exactly one isotope in nature, or perhaps more than one isotope but one of them is much, much more common than the other(s).

Hydrogen, it turns out, has two isotopes found in nature, hydrogen-1 and hydrogen-2. Hydrogen-1 is overwhelmingly common, hydrogen-2 is rare, a bit more than one atom in ten thousand hydrogen atoms is hydrogen-2.

For various reasons, the isotopes of hydrogen actually ended up with “real” names–not true for any other element! Hydrogen-1 is called protium and hydrogen-2 is called deuterium.

The actual atomic mass of hydrogen is a bit higher than the atomic weight of pure protium expressed in kilograms, because the tiny amount of deuterium pulls the average up.

If, in an alternate universe, the atomic mass unit had been defined differently so that hydrogen–mixed hydrogen–got an atomic mass of 1 unit, this would actually have been slightly higher than the atomic mass of pure protium, because the occasional deuterium atom pulls the average up.

But in the real world, the atomic mass unit was defined to be 1/16th the atomic weight of oxygen. So oxygen was 16.0000 by definition. Hydrogen ended up being a hair more than 1.008. Could the excess be due to the deuterium? Not so fast. Oxygen, it turned out in 1919, consists of three isotopes. Oxygen-16 is overwhelmingly more common than oxygen-17 and oxygen-18. But even if you set pure oxygen-16’s atomic weight to 16.00 by definition, and then look at the atomic weight of pure protium, pure protium doesn’t come in at precisely 1.000. There’s still this slight tendency to be off just a bit from integers. At the time no one knew why, but they knew about it well enough to talk about a mass defect. But at least now, we understood the elements that were way off from being whole integer atomic weights–they were mixtures of isotopes. So this is a partial answer to one of our mysteries.

Physicists often discussed different isotopes of the same element. Chemists rarely did back then. Physicists used the whole number to label them, rather than the exact number. This whole number was termed the “mass number” and had the symbol A (from German Atomgewicht). I’ve been using these mass numbers a lot so far, and will continue to do so.

So we have three things with similar-sounding names. There’s the atomic mass unit (amu), almost (but not quite) equal to the mass of a hydrogen atom. There’s an atomic weight, measured in atomic mass units, which represents the mass of the atom. But there is also a mass number, which is a rounded version of the atomic weight, for a specific isotope. Hydrogen’s atomic weight is 1.0008, but the mass number of its most common isotope was just simply 1. When doing ordinary chemistry weighing out reactants the atomic weight is used to compute the number of moles of each reactant. When talking about isotopes, the mass number is used, without fail.

(Looking ahead a little: In the 1920s physicists began using a physical atomic mass unit, that really was based on oxygen-16 rather than mixed oxygen. To distinguish it from the other one, the prior one was called the chemical atomic mass unit–which the chemists kept on using. And then it turned out that oxygen obtained from water had a slightly different isotope mixture and hence real atomic weight, than oxygen extracted from the air. So the chemists’ unit was based on a foundation of quicksand. But even using the physical amu, the atomic weight of a pure isotope was still never a clean, perfect integer, except for oxygen-16.

(But now we had two slightly different units with very similar names. In 1961 they compromised, and created the “unified atomic mass unit” (symbol u, also called the dalton, symbol Da) that was 1/12th of the mass of a carbon-12 atom. This was closer to the chemists’ standard than to the physicists’.

(No matter what standard was chosen, however, the only isotope that had a perfect integer mass was the reference isotope. All others were off, just a bit.

(But that was all in the future. Let’s return to our story, back to 1912.)

The Nucleus

Backing up just a couple of years from there, there had been another very important discovery in 1909 by Ernest Rutherford. He was collaborating with Hans Geiger (who is definitely a counter) and Ernest Marsden.

They used a beam of alpha rays (which, as a reminder, are heavy and positively charged) to bombard a very thin layer of gold foil. They were pretty much expecting those alpha particles to plow through the “plum pudding” atoms. Instead, though most indeed cruised right through the gold atoms as if nothing were there, a very few of them bounced away at sharp angles, repelled by an intense and concentrated positive charge. Some even bounced back towards the beam source! Rutherford said, in a very famous quote: “It was quite the most incredible event that has ever happened to me in my life. It was almost as incredible as if you fired a 15-inch shell at a piece of tissue paper and it came back and hit you.”

In 1911 Rutherford argued that those alpha particles were bouncing off an atomic nucleus. This meant that an atom consisted almost entirely of empty space. All of that positive charge (and almost all of the mass of the atom) was in a tiny, tiny, very dense body about 1/10,000th the width of the atom; the rest of the space was the domain of the electrons, which orbited the nucleus much like planets orbit the sun, except in this case the attractive force wasn’t gravity, but the attraction between the positively charged nucleus and the negatively charged electrons. This was a new model of the atom, called the “Rutherford Model.” Rutherford is credited with discovering the atomic nucleus.

And in fact that number understates things; according to modern measurements entire atoms can be anywhere from 26,000 to 60,000 times as wide as their nuclei. Which works to to be anywhere from 17.6 – 216 trillion times the volume.

Atomic Number

Later that year, Antonius van den Broek proposed that the sequential location of each element in the periodic table was equal to its nuclear charge, this charge (in units of e) was the atom’s atomic number. This fit well for hydrogen, which could only have one electron stripped off, leaving a +1e charged nucleus behind. And for helium, which could be ionized twice leaving a +2e charged nucleus behind. They were the first and second elements listed in the table. However, we couldn’t strip every atom down to a bare nucleus to see its charge; the heavier the atom the harder it was to do that.

This was a new concept. Chemists had talked about the atomic weight of an atom, never its number. You could list the elements in the order they appeared in the periodic table, of course (accounting for the very few unfilled “holes” in the grid), but the place on the list wasn’t considered terribly significant. But now it appeared as if charges came in discrete quantities, and given that one could only remove one electron from a hydrogen atom, and two from the atom with the next higher weight, the implication was that this nucleus had a specific charge, an integer multiple of the charge of an electron (but with the opposite sign). So hydrogen’s atomic number was 1, helium’s was 2. Lithium’s was 3. And so on, through carbon (6), oxygen (8), aluminum (13), iron (26), zinc (30), rhodium (45), silver (47), tin (50), platinum (78), gold (79), lead (82), thorium (90), and uranium (92), to give some examples. (However the exact numbers for anything above the upper fifties really weren’t certain at this point.)

This was only a suggestion…until about two years later. I will pick that story up next time, because it actually ties in more with electrons, and this week I want to concentrate on the nucleus. Suffice it for now to say that van den Broek was absolutely right. I’m going to reference the concept of atomic number, abbreviated Z (from German Zahl, ‘number’), from here forward.

The Proton

So, let’s continue Rutherford’s story. In 1917 he ran some more experiments. He fired alpha beams into air (which is mostly nitrogen), and detected hydrogen ions. After refining his experiment, he realized that the alpha particles were reacting with the nitrogen. When he reported his results in 1919, he claimed that the alpha particle had simply knocked a hydrogen nucleus out of a nitrogen nucleus, reducing the nitrogen nucleus’ charge (and atomic number) and weight by one and thereby turning it into carbon. Nitrogen-14 was seemingly becoming carbon-13, a rare (but stable) isotope of carbon, which is mostly carbon-12.

But by then we had cloud chambers and could see some forms of radioactivity and ions leaving trails through the chamber. In 1925, Rutherford examined some cloud chamber tracks of this reaction, and he realized he was totally wrong about what was happening. The alpha particle wasn’t bouncing off the nitrogen nucleus after knocking one proton out of it. No, it was disappearing. What was in fact happening was the nitrogen nucleus, 7 positive charges, total mass 14, was absorbing the alpha particle.

I mentioned, up above, the principle of radioactive displacement. An atom, spitting out an alpha particle moves two places to the left on the periodic table. That means its atomic number decreases by two. The atomic mass drops by four.

Absorbing an alpha particle has exactly the opposite effect. The atomic number increases by two, and the atomic mass increases by four. So the nitrogen-14 was becoming fluorine-18.

Immediately upon becoming fluorine-18, the nucleus then shed a proton, which was the hydrogen ion that Rutherford saw. This turned it into oxygen-17, stable but uncommon (most oxygen being oxygen-16).

But in the meantime, people had decided that that hydrogen nucleus was a basic particle, and it was named the proton. It’s regarded as having been discovered in 1919, since that was the first time it was seen to exist having come from some source other than hydrogen gas. or in 1920 when someone suggested it might be an elementary particle. Rutherford, as the discoverer, got to name it.

William Prout, clear back in 1815, had suggested that the other elements might be built up, somehow, from hydrogen, and now it looked like he was at least partly right. Hydrogen indeed consisted of a single proton, mass 1, and an electron, and other elements apparently had 2, 3, 4 or more protons, all the way up to uranium with 92 of them–each with a matching electron. You couldn’t just bundle hydrogen atoms together to get other kinds of atoms, but conceivably, if you separated the electrons and protons, then combined the protons, and put the electrons back in place, you could get larger atoms.

In fact, Rutherford had suggested both the name “proton” and the name “prouton” for this particle, the latter to honor Prout. (The English would have pronounced “prouton” as if it rhymed with “grout on”, and the French would have made it rhyme with “crouton” so we dodged a bullet of linguistic confusion there.)

The proton’s mass is 1.007 amus (using the modern AMU scale). Again, maddeningly close to a whole number. But because of this, the proton looked like the underpinning for atomic number but it couldn’t be the underpinning of atomic mass. That’s because, to take an example, oxygen’s nucleus has eight protons in it, but a mass of sixteen, twice as much as the protons. Uranium is even more out of whack. It has 92 protons, but its most common isotope has a mass of 238, leaving 146 mass units unaccounted for! Why? We didn’t know, yet.

In 1920, Rutherford voiced a suggestion. He thought that the excess mass consisted of a number of protons and electron pairs, bound to each other to make a net neutral bundle. So an oxygen-16 nucleus actually contained sixteen protons, but eight of them were bundled with, and masked by, electrons. The net positive charge is eight, and that’s critical because it requires eight orbiting electrons to balance out, and those eight orbiting electrons are responsible for oxygen’s chemical properties. So the chemical nature of an atom ultimately depended on the number of protons not in these bundles.

This actually made quite a bit of sense. Remember beta decay? This is where a nucleus can spit out an electron. The electron has a single negative charge. In order to make up for that loss, the nucleus has to gain a positive charge; it’s as if a new proton were appearing. But if Rutherford’s idea were correct, rather than a proton and an electron being magically created, one of these bound pairs was breaking apart, freeing the electron and unmasking the hidden proton.

Another thing arguing in Rutherford’s favor was the fact that whatever-it-is that was left over in the nucleus had a mass that was nearly that of a whole number of protons; it would make sense for the missing ingredient to be that number of “masked” protons.

Physicists would spend the 1920s thinking that the nucleus consisted of a number of protons equal to the mass number A, plus a bunch of nuclear electrons, which left a net number of “unmasked” protons equal to Z. With some mysterious “mass defect” making the total mass slightly off.

But there were some theoretical difficulties with this…which I will take up in a future installment.

Who Cares About Isotopes?

Until late in the last century, chemists almost never concerned themselves with differing isotopes. That’s because oxygen-16’s chemical behavior is nearly indistinguishable from oxygen-17’s. Because the oxygen-17 is a bit heavier, it’s perhaps a tiny bit slower to react than oxygen-16, but not much. If you were to liquefy oxygen-16 and oxygen-17, then measure their boiling points, the oxygen-17 would require a slightly higher temperature to boil, because it would take just a little bit more energy to kick those heavier oxygen-17 atoms into vapor. Melting and boiling points are in fact the biggest difference a chemist might see…if he had separated samples to work with in the first place. And chemical means of separation were simply untenable; they were too much alike.

Water made with oxygen-17 and oxygen-18 evaporates a bit less readily than water with oxygen-16, so rainwater tends to be slightly richer in oxygen-16 than seawater (and this is part of the reason we had to stop defining the atomic mass unit as 1/16th of mixed oxygen–the mix could differ depending on where you got the oxygen from).

The chemical differences between protium (hydrogen-1) and deuterium (hydrogen-2) are actually significant, due to the fact that proportionally, the difference is greater than for any other pair of isotopes. Water made out of deuterium (“heavy water”) instead of protium actually melts at 4C, rather than 0C. I’ve seen a video of a heavy water ice cube sunk to the bottom of a glass of cold (regular) water. It’s not going to melt as long as that water is properly chilled. Note that I said the bottom of a glass of cold water. It doesn’t float because it’s heavier than regular ice and heavier even than regular water. (Now, if it were in a glass of heavy water, it would float.)

And of course, heavy water, because of its significantly different chemical behavior, is toxic when pure.

Other than that, for “traditional” chemistry, isotopes just didn’t matter.

Today things are a bit different. Mass spectrometers–which are the descendant of Crookes tubes, designed to ionize, accelerate, and deflect atoms and molecules to see how much they deflect and thus figure out the masses–are relatively cheap, and they can read out absolute numbers of “hits” at each possible mass. So one can run a sample of water through one of these and get a very precise notion of the isotopic composition. Now, you can tell whether a sample of water was rain water or ground water. Or you can analyze a sample of metal and be able to tell where it was mined, because it turns out each mine has a slightly different isotopic mix of the metal. Or one can prove that CO₂ was added to champagne artificially, because the CO₂ used has no carbon-14 in it (whereas the carbon dioxide in fermentation does).

Incidentally, if you’ve ever had TSA swab your luggage then stuff the swab into a machine which tells them you aren’t carrying explosives–that device is a mass spectrometer.

That’s today. But back in 1910, chemists didn’t give a rip about isotopes. Physicists studying radioactivity, on the other hand, knew that “which isotope is this?” could make all the difference in the world. And that’s even more true today too, now that we can artificially make all sorts of radioactive isotopes that don’t exist in nature. We now have to concern ourselves with radioactive hydrogen-3 (“tritium”), cesium-137, iodine-131 and strontium-90…and these were elements that were never radioactive in the days of the Model T and the Wright Flyer.

In 1910 we were just starting down this road. Remember, Rutherford had made fluorine-18 and oxygen-17 artificially.

Decay Chains

Keep this in mind as we go back now to uranium (atomic number Z=92) and thorium (Z=90). Remember that whole process of figuring out the pieces of an atom started in part because of the discovery of radioactivity, a property of these two elements in particular.

At the time of today’s story, had become quite clear that when there was radioactivity, one kind of atom was changing into another, this is called “decay.”

Uranium and thorium decay very slowly, or I should say, uranium-235, uranium-238, and thorium-232 decay very slowly (as I said, the isotope matters). It’s a statistical process. When you are looking at one uranium-235 atom, it could decay a second from now…or it could wait a billion years. There’s no way to know when it will happen, but it’s almost a stable nucleus; it’s very, very unlikely to blow in the next second. And if that atom is still around in a billion years, someone watching it then is just as unlikely to see it go kablooey in the next second as you are today.

I’m going to get on a soap box here, for just a minute. Let’s say you watch someone flip a coin 20 times and it comes up tails each time. Do you think, “wow, it’s overdue to come up heads, I’ll bet it comes up heads next time?” If so, you have a “naive” view of probability. The more sophisticated view is that, since the tosses are independent events they aren’t affected by each other. The chance is 50/50 of heads next time, no matter how many times in a row it has come up tails just now. But then, there is the cynic’s view. He doesn’t believe the odds are fifty/fifty either. But he doesn’t figure it’s overdue to come up heads; he figures the coin probably is crooked; perhaps tails on both sides! And he might have a point there. The smart bet, if you’re not allowed to examine the coin, is probably to bet on “tails.” But, if the coin really is fair, the 50/50 view is correct.

Similarly, for the chances of an unstable nucleus going kablooey in the next second, or minute. A billion years from now, provided your unstable nucleus hasn’t gone kablooey in the meantime and it’s still around, it’s just as likely to not go kablooey in the next second, as it is to not go kablooey in the next second today.

At an individual atom level, radioactivity isn’t predictable. But, if you take a large number of atoms of one of these three isotopes (or of any unstable isotope for that matter), you can make some predictions.

You can say, for instance, that any large sample of uranium-235 will be half gone in about 700 million years. Half of the atoms (no way to predict beforehand which specific ones) will have decayed to something else. Does that mean that the other half will decay in another 700 million years? Absolutely not. If you start with a pound sample of uranium-235, after 700 million years, you now have a half-pound sample of uranium-235, now mixed in with a bunch of impurities to be sure, but a half pound sample nonetheless, and half of that sample will decay in the next 700 million years.

700 million years is the half life of uranium-235. Similarly, uranium-238 has a 4.5 billion year half life, and thorium-232 comes in at 14 billion years.

You get one guess as to who discovered the concept of a half life in 1907. I’ll give you a tiny hint: He did it using one of the short-lived isotopes in the thorium decay chain, one that was deposited by decaying radon gas.

Thorium-232’s half life is about three times that of uranium-238. As you can imagine, given a godzillion uranium-238 atoms, and a godzillion thorium-232 atoms, you’ll see three times as many decays in a day from the uranium as from the thorium. But it also scales by quantity; two godzillion thorium-232 atoms will produce twice as many decays in a day as one godzillion will. And three godzillion thorium-232 atoms will produce as many decays in a day as one godzillion uranium-238 atoms. Keep this in mind–the ratio of the half lives is same as the ratio of quantity, for the same number of decays to occur from samples of two different isotopes.

[A “godzillion” is a highly technical word someone made up once for a really large number. He used it to describe the national debt when it was a lot smaller than it is now. However, even today’s national debt pales next to the number of atoms in a mole (which would be 600 sextillion or so). I decided to adapt the term rather than just say “zillions” or “jillions.”]

When an atom of (say) thorium spits out an alpha particle, it actually changes to another element and another isotope; it is decaying. If the new isotope is also unstable, it too will decay, again and again until the result is a stable nucleus. Eventually the starting thorium-232 nucleus will have become a lead-208 nucleus.

OK, with thorium being Z=90 and lead being Z=82, we can do a little bit of accounting-style sleuthing. The difference between these two masses–the change in A–is 24. That’s the equivalent of six alpha particles. In fact, since the only mode of decay that changes an atomic weight is alpha decay, we expect exactly six alpha decays to occur during this process.

But going from thorium to lead would involve changing Z by eight, which is something you’d get from four alpha decays at two apiece. Six alpha decays, absolutely required by the mass change, give you a reduction of Z by 12, and so it looks like you’d end not with lead-208 but rather platinum-208 (which if it even exists, surely isn’t stable).

Beta decays come to the rescue. They move you one element to the right, without changing the mass. So if you figure that the total number of alpha decays is six, reducing Z by 12, but then throw four beta decays into the mix, increasing Z by four, it balances; the net reduction of Z is 8. The total set of reactions boils down to:

Thorium-232 (Z=90, A=232) – 6 alphas (Z=12, A=24) – 4 betas (Z=-4, A=0) = Lead-208 (Z=82, A=208).

(Remember when subtracting the four betas, you are subtracting a negative number, which means to add the opposite positive number.)

If you look at the detailed sequence of events, this is exactly what happens. Thorium-232 decays by alpha particle to radium 228 (Z=88, A=228 one alpha decay so far). Radium-228 then undergoes a beta decay to get actinium-228 (Z=89, A=228, alpha, one beta so far). Actinium-228 undergoes another beta decay to get thorium-228 (Z=90, A=228; one alpha, two betas so far).

Let’s pause here to look at the half lives. The original thorium-232 has a fourteen billion year half life. That means that (on a percentage basis) very, very little of it decays in (say) one day. The radium-228 has a 5.7 year half life. The actinium-228 has a 6.1 hour half life. The thorium-228 has a … wait for it! … 1.9 year half life. (It’s thorium, but it’s not thorium-232 and that makes all the difference in the world when it comes to half lives.)

If you started with a pure thorium-232 sample and waited about ten years, a certain amount of radium-228 has accumulated. As it accumulates, you can detect more and more decays of it (because there is more and more of it over time. But it won’t accumulate forever: It turns out that after a few years of building up, there’s now enough of it that it’s decaying about as fast as it’s being created. So you should be able to see based on our discussion above that, given thorium-232’s half life is three billion times as long as radium-228’s, when there is one radium-228 atom for every three billion thorium-232 atoms, then they’ll both produce the same number of decays. But the radium-228 doesn’t go away, because it’s being replenished by the thorium-232 decays. Since the amount isn’t changing over time the radium-228 is in equilibrium with the thorium-232. (The thorium-232 is slowly going away, of course, as it does so it will produce slightly less radium-228 during a given time, so the radium-228 will decline at the same percentage rate. But people don’t live long enough to see this happen, not with a 14 billion year half life!) Equilibrium is reached in something like 1 1/2 or two half lives of the daughter isotope.

Similarly for the actinium-228–because it has a much shorter half life than radium-228, it reaches equilibrium with the radium-228 almost instantly. And so on down the chain. Once everything is at equilibrium, there is one decay of each daughter isotope, for each decay of a thorium-232 atom. This is why a “pure” sample of thorium actually grows more radioactive right after it’s made.

So back to that chain. It continues. Thorium-228 alpha decays to radium-224 (Z=88, A=224, two alphas, two betas so far). Radium-224 alpha decays to radon-220 (Z=86, A=220, three alphas, two betas so far). Radon-220 alpha decays to polonium-216 (Z=84, A=216, four alphas, two betas so far). Polonium-216 alpha decays to lead-212 (Z=82, A=212, now five alphas and two betas so far).

Lead-212 is lead, and lead dug out of the ground is stable, but lead-212 is not stable. It’s an unstable isotope, a very unstable one in fact. Its half life is 10.6 minutes.

The next step is a beta decay, lead-212 becomes bismuth-212 (Z=93, A=212, five alphas, three betas). We now have just one alpha and one beta decay left to get to lead-208. But now, the path splits. We can either do the alpha decay first then the beta decay (thallium-208 (Z=81), then lead-208) or the other way round (polonium-212 (Z=84), then lead-208).

All of these decays from thorium-228 onwards have half lives of days or less, one even has a half life of less than a millionth of a second. So once the thorium-228 reaches equilibrium with its great-grandparent thorium-232, the rest of the chain ends up in equilibrium in just a few days.

The diagram below summarizes this whole process. And it uses a notation I haven’t used yet. So far when I’ve named an isotope, I’ve done it as [element name]-[mass number]. But you can also use a superscript before the element symbol like this: ²³²Th. Superscripting is a bit of a pain in the ass in the WordPus editor (and besides you might not know all the symbols), so I didn’t do it this way. It can even be taken a step further (and is, in the diagram below). You can put the atomic number Z as a subscript before the symbol, like this: ₉₀Th. (Or you can do both. And I do mean you can do both. I can’t. If I try, I get something like this: ²³²₉₀Th. I can’t get the super and subscripts one over the other.)

Technically the atomic number is superfluous, thorium is by definition atomic number Z=90. But it’s helpful for all the non-geeks out there who don’t have the numbers memorized.

(Even chemists don’t usually know all of the atomic numbers, nor do they know all of the symbols; I watched one give a lecture on this very sort of thing, and when he showed the symbol Pa, he called it “palladium” (it’s actually protactinium, atomic number Z=91; palladium’s symbol is Pd and its atomic number is Z=46 and its price is almost three thousand dollars an ounce. The symbol was right, his verbal reading was wrong). Chemists will know the common elements like sulfur (16, S), plus ones they themselves are personally working with…unless they’re complete geeks, in which case they’ve memorized them all. By the way, if you ever run into someone claiming to be an organic chemist and they don’t know that carbon’s atomic number is Z=6, he’s a faker. Actually, he’s a lying sack of bearded dragon shit. Run, do not walk, away, from this person, and do not believe him if he tells you that the sky is blue; don’t even believe him if he says that Joe Biden lost.)

One last thing to note about the thorium decay series. Every single isotope on it has a mass number A that divides by four. The starting number divides by four, and any time the mass number changes, it changes by four, so it will always be divisible by four.

The other two decay series have uranium in them. Uranium has two long-lived isotopes, and they are each at the beginning of their own decay chains. You can walk through them if you so desire, but I’m just going to put up the diagrams. The first is the “Uranium decay series” starting with uranium-238:

Every one of these isotopes’ mass numbers, when divided by four, leaves a remainder of 2. Therefore, none of these isotopes appears in the thorium decay series, and none of these appear there either. Never the twain shall meet.

Note that one of the intermediates is uranium-234 with 245,000 year half life. If you (personally) start out with pure uranium-238, you won’t live long enough to see it come into equilibrium with its daughter isotopes, because uranium-234 decays too slowly. Over about the next half million years, ²³⁴U will build up in the sample and then be in equilibrium. Everything downstream from it is much faster. You will see, rather quickly, the intermediate thorium and protactinium 234 isotopes reach equilibrium, though.

The uranium-235 series is actually called the “actinium decay series” to avoid confusion with the other uranium decay series. It includes the longest-lived actinium isotope, actinium-227.

All of these isotope mass numbers, when divided by four, leave a remainder of 3. They therefore won’t appear in either of the first two series, or vice versa.

There ought to be a fourth series, one where all the mass numbers leave a remainder of one when divided by four. Right?

Well, there was. A long time ago. The problem is no isotope in that series (which we can reconstruct today since we can make artificial isotopes) has more than a 2,140,000 year half life. That’s much shorter than the uranium and thorium isotopes in the other series. That isotope is neptunium-237 (Z=93). One of its daughters is uranium-233, with a half life of 159,200 years. Everything else in that series is shorter, much shorter.

If there was any neptunium-237 on earth when it first formed, ten half lives (21.4 million years) would have reduced it to 1/1024th of its original amount. Another ten half lives would have reduced it to less than a thousandth of a thousandth, or less than a millionth of the original amount. A total of eighty half lives would be enough to reduce an entire mole of neptunium to less than one atom on average, an undetectably small concentration, especially since the neptunium probably started out as a minor constituent of whatever rock it was in, to begin with. (Realistically, fifty half lives is probably enough to escape detection by modern equipment.) Seventy half lives is about 170 million years.

There was either never any neptunium-237 when the earth formed, or the earth is at least 170 million years old. In fact, there are a lot of isotopes with even longer half lives (like plutonium-244, half life roughly 80 million years) that do not exist in nature, and the same logic applies: either that isotope was never around, or the earth is hundreds of millions of years old, or even older–plutonium-244’s absence implies billions of years.

Returning to the “neptunium decay series,” because it has no sufficiently long lived isotope, it is extinct. When we started making isotopes artificially, we eventually found neptunium-237, and uranium-233, and all the others, and could then figure out what the neptunium decay series looked like. But back in the 1910s, this was all well in the future.

[Actually, oddball nuclear reactions sometimes create a trace of these isotopes in uranium ore, but that’s an almost immeasurable trace, and clearly not remnants of an original stock.]

The second to last product of the neptunium decay series is bismuth-209. It was long thought to be a stable isotope, but fairly recently it was discovered to have a half life of 19 quintillion years-almost a million years for every dollar of our national debt. It is so weakly radioactive that it might as well be stable, and its radioactivity is consequently almost impossible to measure. When it bestirs itself to do so, it decays to thallium-205, which is unfortunately quite stable. I say unfortunately, because thallium is extremely toxic. There is actually plenty of thallium-205 out there already, but it has to almost all be original or primordial stock, because hardly any bismuth-209 has decayed in a mere few billions of years.

Summing it up

Radioactivity was discovered in 1896. At that time, the words electron and proton didn’t exist. Atoms were indivisible things. Twenty years later, we knew that last bit was wrong, and we were well on our way to knowing the real nature of matter. In large part thanks to Ernest Rutherford.

OK. Next time, we take one step out, back into the realm of the electrons.

Obligatory PSAs and Reminders

China is Lower than Whale Shit

Remember Hong Kong!!!

中国是个混蛋 !!!
Zhōngguò shì gè hùndàn !!!
China is asshoe !!!

China is in the White House

Since Wednesday, January 20 at Noon EST, the bought-and-paid for His Fraudulency Joseph Biden has been in the White House. It’s as good as having China in the Oval Office.

Joe Biden is Asshoe

China is in the White House, because Joe Biden is in the White House, and Joe Biden is identically equal to China. China is Asshoe. Therefore, Joe Biden is Asshoe.

But of course the much more important thing to realize:

Joe Biden Didn’t Win

乔*拜登没赢 !!!
Qiáo Bài dēng méi yíng !!!
Joe Biden didn’t win !!!