The Talk.Origins Archive Post of the Month: March 2001

Subject:    Re: Decay Rates
Newsgroups: talk.origins
Date:       March 28, 2001
Message ID: 3AC26990.8BD15C41@bnl.gov

Robert Carroll wrote:
>
> "Sverker Johansson" <lsj@hlk.no.hj.spam.se> wrote in message news:3ABF147C.A7391DD5@hlk.no.hj.spam.se...
>
> > Robert Carroll wrote:
> >
> > > "James R. Hofmann" <jhofmann@fullerton.edu> wrote in message news:3ABB8556.74FBDB89@fullerton.edu...
> > > > Any comments on this AIG article on altered decay rates? It postulates
> > > > (very) different conditions in order for the rate involved to be
> > > > different, but it probably will get a lot of publicity.
> > > >
> > > > http://www.answersingenesis.org/docs2001/0321acc_beta_decay.asp
> > > >
> > > The argument seems to be that the electron cloud surrounding a nucleus
> > > would provide a strong electromagnetic barrier to a beta particle being
> > > ejected from the nucleus.
> >
> > Not quite. The nucleus itself has an electric field which holds on
> > to electrons around it. With the full complement of normal electrons,
> > there is no room for the new beta-decay electron near the nucleus,
> > so the decay process has to supply enough energy to boost the electron
> > out from the field of the nucleus. If there are no electrons in the
> > K shell, then the decay process only needs enough energy to get an
> > electron from the nucleus into the K shell, which can be significantly
> > less.
>
> Right. I was overlooking nuclear charge. This seems to be the inverse of
> K-electron capture. I'm surprised at the large changes in decay rate,
> though.

You're right, this process is in fact very like K-capture, but time-reversed, and with all of the other atomic electrons removed from the picture.

The argument presented in the article is simply bizarre. Anyone who reads it for very long does so at the risk of suffering severe brain damage. It contains confusion at near toxic levels. In my opinion a quite careful effort has been made here to misdirect the reader, and there is an implicit assumption that the readers will be unsophisticated.

The factor of 10⁹ enhancement of the decay rate in fully stripped ¹⁸⁷Re is indeed extremely surprising on the face of it. I was surprised to hear of that myself. But it's important to remember that the decay of ¹⁸⁷Re is not really a very typical beta decay. More on this later.

The first hint that something quite special is up and that the author is attempting to mislead you about it, even if you didn't know anything at all about the physics of beta decay, is provided by Woodmorappe himself, when he refers to the ¹⁶³Dy system, which is stable as a neutral atom, but has been observed to decay to ¹⁶³Ho quite quickly when it is fully stripped. Why doesn't Woodmorappe point out that this is an incredible enhancement of the beta decay rate by a factor of infinity? Isn't that a much more spectacular effect than a mere nine orders of magnitude?

Woodmorappe doesn't want you to think about this question for too long, so he doesn't make it a central point. He wants you just to believe that all beta decays will be affected in just this way by stripping, and he later suggests that the variation probably might extend to alpha decays as well. This is why he first goes so far as to provide a spurious explanation for the broad phenomenology of beta decay lifetimes.

In the mind of any physicist who has ever calculated a nuclear beta decay, a partial explanation for the effect would already be forming or would be fully formed already, by the time that Woodmorappe mentions ¹⁶³Dy.

K-electrons, beta decay electrons, and any other electrons which find themselves deep inside multi-electron atoms do in fact all have a `barrier' to being excited to higher energy levels. But one doesn't generally talk about a barrier in this case, because the actual potential for electrons doesn't have a barrier. It is a purely attractive potential, with close to a 1/r dependence right up to the edge of the nucleus, changing over to an r² dependence in the region of constant charge density inside.

Remember that this is a beta decay: it is essentially a weak process resulting from a zero range interaction. It is very different from an alpha decay, in which the competition between the long range repulsive Coulomb forces inside the nucleus and the attractive short range strong interactions produce an actual barrier that an alpha particle must penetrate in order to escape from the nucleus.

Repeating it once more, all of the electrons feel the attractive Coulomb force from the nucleus, corrected by screening due to other electrons, and the repulsion of the other electrons. The Pauli principle operates, so that an inner electron cannot be excited to any of the occupied levels above it. All but the very highest levels in a multi-electron atom, in its ground state, are filled with the maximum possible number of electrons: no more electrons can be put into these states. To be excited, any electron must be given energy sufficient to get above the Fermi level in the atom. To within a few eV, the Fermi level will coincide with the continuum. At least enough energy must be given to a decay electron then, that it can reach an unoccupied bound state in the new atom (which has one more unit of positive charge on its nucleus), otherwise the decay will be energetically forbidden. In most beta decays, much more than this amount of energy is available.

Coulomb corrections to the electron wavefunction are always present when calculating beta decays, but though they are certainly substantial in certain regions of phase space, they are not generally responsible for such spectacular effects as are seen here.

But the account Woodmorappe gives of the mechanisms is not to be taken seriously. One can safely ignore what little he writes about the details. Here is a choice expository passage in which he beautifully illustrates his willful ignorance of the subject:

Not to put to fine a point on it, but at this point the discussion is already complete crap. It is true in all the incidental details, but it is all essentially irrelevant. After this point the discussion in the article degenerates even further. Do not even try to learn about beta decay from this man.

I think his rather clear suggestion, here, is that beta decay electrons are somehow held inside the nucleus by the Coulomb force, that otherwise they would easily escape, and that that is the root cause of certain very long predicted and observed, rather than `inferred' beta decay lifetimes.

What he says is completely backwards. He pretends that the special case is the general case, he says nothing useful about the underlying mechanisms, and he is wrong in all of his conclusions as well as his subsequent mis-application of the ideas to radioactive dating of rocks.

In the great majority of neutral atom beta decays one can do reasonably well by ignoring the Coulomb attraction of the nucleus for the decay electron, as well as the repulsion of the atomic electrons for the decay electron. These are usually small corrections to the process, because the energy available from the change of the nuclear state, which always occurs in a weak nuclear decay, is generally much larger than the change in the atomic binding energy. Known beta decays have endpoint energies a wide range: but most typically these fall between a few hundred and several thousands of keV. `Crashing through the walls of a well' is just not an issue for the electrons emitted in beta decays.

The decay electron is almost always simply emitted into the continuum, and the chance of capture into an atomic bound state is very small. The Pauli principle forbids the decay electron from being captured into a deeply bound state of a multi-electron atom, since the inner orbitals generally remain fully occupied in the daughter atom. This statement is almost always true despite corrections for non-orthogonality of the atomic wave-functions in the daughter atom, due to the change of the nuclear charge. Capture into an outer orbital is generally quite strongly suppressed due to the weak binding of outer electrons and the small wavefunction overlap with the decay electron.

It might be worth pointing out a few more simple facts about the phenomenology and the theory of beta decay. Beta decay is in the present context treatable theoretically as if it resulted from a zero-range, current-current interaction, which transforms a proton (neutron) bound within a nucleus into a neutron (proton), with the simultaneous creation or absorption of an electron (positron) and a neutrino (anti-neutrino). The naturally occurring nuclear beta decays were very early on shown experimentally to be directly associated with transitions between discrete stationary states of the parent and the daughter nucleus, most usually a transition from the ground state of the parent to the ground state or a low lying excited state of the daughter.

Depending on the details of the nuclear structure, such a process may or may not require a large rearrangement of the nuclear state, and may or may not release a lot of energy. If the only change required in the nuclear state is a change in the charge state, or equivalently, the z-component of the isospin, and a readjustment of the nuclear well due to the change in nuclear Coulomb energy, the transition is generally called super-allowed. Such transitions are the most favoured possible beta decays, and they typically have small lifetimes, once one corrects for the basic underlying energy dependence of weak decays.

This energy dependence, by the way, is very strong. For large enough total decay energies, the dependence is roughly as (W₀)⁵ where W₀ is the endpoint electron energy.

The premier example of a super-allowed beta decay is of course the decay of the neutron in free space into the proton, with a lifetime of about 1000 seconds. Superallowed decays fall into a group with the lowest possible (ft) values. Actually one really discusses log₁₀ (ft), where t is the half-life and f is a theoretical factor which corrects for the widely differing total energies of nuclear beta decays.

The real explanations for sometimes very long beta decay half-lives which are predicted by theory and observed in nature (not `inferred') in quite a few naturally occurring, neutral, beta unstable atoms is that these atoms can now be seen to fall into two general classes. The classes are not mutually exclusive.

The first class includes those decays where the nuclear matrix element is large or at least not unusually small, but there is simply not very much energy available for the decay.

The second class includes cases in which the nuclear matrix element for the transition is extremely small, though there may or may not be ample energy available.

The first class includes certain allowed (as opposed to super-allowed) transitions, as well as some so-called forbidden transitions of various orders. Allowed transitions are those which can still occur when the spatial dependence of the electron and neutrino wavefunctions across the nucleus is ignored. To within about 1 percent, this is actually a good first approximation in most beta decays. Other transitions for which we must look to higher orders in the expansion of the wavefunctions are suppressed by additional factors on the order of 100, and are these are thus called forbidden transitions. The order of forbiddenness is related to the order in the expansion of the electron wavefunction in powers of the momentum at which the first contributions to the decay are obtained.

Selection rules for the allowed decays are Delta-J = 0 with no change of parity for so-called Fermi or vector transitions, and Delta-J=0,1 with no change of parity for Gamow-Teller transitions. Transitions with higher Delta-J or a change of parity are always first or higher order forbidden.

The total energy available for this beta decay which Woodmorappe concentrates on is tiny. It is the decay of the 5/2+ ground state of neutral ¹⁸⁷Re to the 1/2- ground state of 187 Osmium with an endpoint energy of W₀ = 2.6 keV. This decay is a so-called unique (meaning only one operator in the expansion connects the two nuclear states) first forbidden transition. The Delta-J is 2, and there is a change of parity. These factors together account for the very long lifetime of the neutral atom.

Just above the ground state in 187 Osmium, at only 9.75 keV, lies the 3/2- first excited state. Decay to this state from 187 Rhenium, if possible, would still be a first forbidden transition, but because Delta-J is only 1, it is a non-unique first forbidden transition, which is somewhat more favoured than a unique one. However decay to this state is not even a possibility in the neutral system at normal temperatures: it is energetically forbidden.

Now, the critical point to understand here is that in a very large atom, like Rhenium or Osmium, the total Coulomb binding of the atomic electrons is not at all a small number. It is especially not a small number in comparison to the tiny endpoint energy of this particular beta decay. The total electronic binding is in fact on the order of 400-500 keV. In addition, the binding is about 20 keV larger in Osmium than it is in Rhenium, due to the extra unit of nuclear charge. It's not very hard to make rough estimates of these numbers knowing just a very little about atomic physics.

Furthermore, the binding of a K-electron in these systems is approaching 90 keV in the stripped, hydrogen-like atom, though in the neutral atom it will be somewhat less due to screening from the second K-electron. So if we apply our normal intuition about beta decays here to estimate what might happen to the lifetime of the system when 75 bound atomic electrons have been stripped away and say confidently, nothing much at all, we will be sunk. The nuclear energy levels have been shifted relative to each other by a considerable amount and the energetics of the decay clearly has to be reconsidered.

In the end, the transition to the 3/2- first excited state with a bound K-electron becomes energetically allowed, and that is the dominant decay mode for the stripped system. There is much more energy available for the decay: about 60 keV versus 2 keV in the neutral system, and that, together with the increased overlap due to the smaller change in J is quite sufficient to account for 9 orders of magnitude enhancement of the decay rate. Beta decay into the continuum, interestingly, is not even energetically allowed in the stripped atom.

So we see that the systems for which this sort of thing is a very important effect are quite special. To find them, one must comb through hundreds of known beta decays, and come up with the few that have happen to have small Q values, which are comparable to the changes in atomic Coulomb binding when going from stripped or highly ionized atom to neutral atom.

The general rule, however, is approximately this: for the most part, nothing extremely spectacular will happen to total beta decay rates of most beta unstable atoms (electron emitters), even if the atoms are totally stripped. Moreover, because of the nature of quantum mechanics in a coulomb potential, it will be necessary to nearly completely strip the atoms in most cases, to be able to see any effect at all. Eliminating a valence electron will simply not be enough, and that is all that can be achieved at any reasonable temperature.

It is interesting that Woodmorappe completely omits any discussion in this article of the case of Potassium 40, which is unstable against positron emission, K-capture, to Argon 40 and by electron emission to Calcium 40. This is the relevant system in the well known Potassium-Argon dating technique. The dominant decay mode for the positron emission here is a third forbidden Delta-J=4 transition, with a change of parity. The total decay lifetime of the branch to Argon is about 1.28 billion years. The available energy is however, much larger, the endpoint being W₀ ~= 1320 keV. The effect of completely stripping the atoms on the decay rate in this system, though certainly different from zero, will be far less than it was in Rhenium.

The same applies to yet another case, and I wonder even more why Woodmorappe has ignored this one. Consider the odd-odd nucleus 186 Rhenium, which beta decays by electron emission to the neighboring nucleus 186 Osmium. Considerations of atomic binding energies are very nearly the same as for the case we just went through in detail. This transition is from the 1- ground state of Rhenium 186, and has a branch of about 75% to the 0+ ground state of Osmium 186 (which, being an even-even nucleus, is much more bound than is Osmium 187). There is also a 23% branch to the first (2+) excited state of Osmium, as well as smaller branches to two higher excited states. Both transitions are first forbidden, Delta-J=1, with a parity change. The endpoint energy of the transition to the 2+ state, however, is about 930 keV, and that to the ground state is nearer to 1100 keV. Again, we can expect much more modest effects to occur when the systems are totally ionized.

I will now conclude with a few remarks on the relevance of this silly and massively dishonest article to radioactive dating and geological time scales. I am doing no more than to repeat points that others have made here, but I have added a couple of numbers, just for fun.

Large atoms as we know'em and like'em, namely at all temperatures important for questions of rock formation, can be thought of as being essentially neutral when it comes to calculating their beta decays.

It is these kinds of atoms that make up rocks, whether molten or solid, and that of course includes the rocks in Woodmorappe's head. One does not typically find Rhenium atoms in charge states like 75+. It took quite a few talented people working at a complicated and expensive facility, using an accelerator like the one at GSI, to produce a usable number of these exotic objects for their experiment. To see just how absurd the discussion Woodmorappe gives of the earth's origins actually is, it's worth making a couple of simple order of magnitude estimates.

First, the gravitational binding energy of the earth can be roughly estimated from the formula for a uniform sphere:

Taking approximate values r=6500 km, m=6x10²⁴ kg, and G = 6.67 x 10^-11 m³ / kg / s², this gives:

What sort of conditions are required to make 75+ the expected charge state of Rhenium? Here I am going to play very fast and loose with my estimates. If the separation energy of the first electron in Rhenium is about 9 eV, and that of the last is about 90 keV, that suggests a total binding energy of about 500 KeV for all of the electrons. To separate the last electron we thus need a temperature at least on the order of 10⁹ K, while smaller temperatures would suffice for ionizing the rest of the outer electrons. We shall need to approach charge states of 72+, 73+ or more preferrably 74+, I'd bet, in order to see very strong effects on the beta decay lifetime. If the K-shell is completely empty in Osmium, then capture to the L-shell is energetically allowed, but it is greatly suppressed over K-capture. So perhaps T = 10⁸ K might be sufficient. To approach this kind of temperatures in the current universe, we shall need to make a descent into the core of a supergiant star. Or perhaps we could wait around for the shock wave of a supernova explosion to hit us. So while the result discussed in the article concerning bound state beta decays of fully ionized Rhenium seems possibly to be very interesting for astrophysics, it is certainly quite irrelevant for any estimates of the age of terrestrial rocks.

To make this point a little clearer, if it isn't clear enough already, consider that the binding energy fraction for the electrons in neutral Rhenium is by my above estimate on the order of:

Thus, in the process of raising the entire planet earth to the temperature necessary to make 75+ the expected charge state for Rhenium so that it could then quickly decay into Osmium, before the earth cooled, God therefore also must have made the earth gravitationally unbound. The whole planet would simply have exploded into a cloud of plasma which would even yet be expanding into space. A cloud at this temperature, having the mass of the earth, could never have coalesced to form the earth.

Unless, of course, the hand of God squeezed the plasma back into place, or He also adjusted the gravitational coupling constant ...

Now, if one is the sort who is happy with that kind of explanation, then why should one bother vomiting forth a totally botched article on the fascinating and complex physics of exotic nuclear beta decays, in an effort to make this religious point? Why wouldn't one simply assert that it's clear that God put every single atom right into its present place, and that the angels are still pushing all of the tiny little electrons around in their classical orbits? That at least would be a much more honest statement of one's actual beliefs.

> > > If this were true, K-electrons (and other low
> > > energy electrons in multi-electron atoms) would have much the same barrier
> > > to being excited to higher energy levels,
> >
> > They do.
> >
> > > making it much more difficult to
> > > achieve that plasma that Woodmorappe referred to.
> >
> > That's why we don't have Rhenium plasmas at any Earthly conditions.
> >
> > > The reference to nuclear
> > > particles "crashing through" a potential barrier serves to illuminate the
> > > crudity of his understanding of tunneling. It is a thoroughly dishonest
> > > piece of work.
> >
> > Yes, it's dishonest, but not because the physics in it is wrong.
> > The process is well known to operate inside stars.
> > It's the application to Earthly rocks that's dishonest.

Hi Sverker. I agree with everything you've said about the physics, but I think you're being much too kind here about Woody. (Of course, that's not very kind at all when you're calling him dishonest.) He has quoted some correct results of physics, but there isn't very much in what he himself says about the physics that's correct and neither has he drawn any really correct conclusions as far as I can tell.

I hate to answer my own post, especially when people have looked kindly on it, but what I said in the following paragraph requires a minor erratum:

>It is interesting that Woodmorappe completely omits any discussion in this
>article of the case of Potassium 40, which is unstable against positron
>emission, K-capture, to Argon 40 and by electron emission to Calcium 40. This
>is the relevant system in the well known Potassium-Argon dating
>technique. The dominant decay mode for the positron emission here is a third
>forbidden Delta-J=4 transition, with a change of parity. The total decay
>lifetime of the branch to Argon is about 1.28 billion years. The available
>energy is however, much larger, the endpoint being W₀ ~= 1320 keV. The
>effect of completely stripping the atoms on the decay rate in this system,
>though certainly different from zero, will be far less than it was in
>Rhenium.

I should amend this discussion of the A=40, Argon-Potassium-Calcium system: I went over it a little too quickly. There are some mistakes in what I said in the paragraph above, which don't affect any overall conclusions, but which are actually interesting for evaluating Woodmorappe's article.

The lifetime I quoted here was, of course, the total decay lifetime for the neutral atom, including all of the decay modes. The branch to ⁴⁰Ca actually accounts for about 89% of the decays. The other 11% of the decays almost all occur by K-capture to ⁴⁰Ar.

The total atomic electron binding in these systems can be estimated from the values in potassium (Z=19). I estimate that the total binding of the atomic electrons here is about 15 keV, while the binding of a K-electron in the stripped atoms is about 5 keV.

The first excited state (0+) of ⁴⁰Ca lies well above the ground state at 3352 keV, and it can just be ignored here. The first excited state (3-) of 40 K is quite low lying at about 30 keV, but it too can safely be ignored at normal temperatures. All the decays thus occur from the ground state of Potassium 40.

The endpoint energy I quoted, 1320 keV, is that for the dominant decay mode of the (4-) ground state of neutral ⁴⁰K. This mode is actually electron emission to the (0+) ground state of ⁴⁰Ca, not positron emission to the (0+) ground state of ⁴⁰Ar. The difference in the atomic masses of 40 Potassium and 40 Argon is 1503 keV: so that is the total energy available for the decay which is of most interest in the dating technique.

Positron emission to the ground state is energetically allowed and does occur, but as it turns out, only rarely. K-capture to the ground state dominates positron emission to the ground state, and both of these are dominated by decays to the first excited state (2+) which is at 1460 keV. The endpoint energy for positron emission is W₀=489 keV, quite a bit less than what I said. But it is not enough less that we have to worry about energy shifts due to binding of atomic electrons in going to the stripped system: these are, both relatively and absolutely speaking, far smaller than in the Rhenium-Osmium case.

On the Argon side of the diagram, I've pointed out there are two states to consider. There is the 0+ ground state, to which the Q-value in the neutral system is 1503 keV, and there is also the 2+ first excited state, which lies 1460 keV above the ground state, so with a Q-value of 43 keV. The transition to the 2+ first excited state has a smaller Delta-J and is only first forbidden. Even though the Q value is small, K-capture to this state is the dominant mode for producing Argon-40. Positron emission is energetically forbidden in the transition to the first excited state, and all decays to the ground state are strongly suppressed by the nuclear matrix elements despite the larger available energy. The first excited state then decays to the ground state by emitting a 1460 keV photon (it's an E2 transition.) There can also be various associated X-rays, internal conversions, and Auger electrons. I won't get into discussing all of these subtleties.

Considering these facts, we can see that the fully stripped system is naively expected to have the same decay lifetime, to within about 10%. The 10% change comes about because fully stripped Potassium has no K-electrons. K-capture is therefore not a possible decay mode for an isolated fully stripped Potassium atom. The widths for positron and electron emission into the continuum are not much affected, but K-capture is gone. If the atom still had one bound electron though, then the mode would still be allowed.

The conclusion would appear to be that fully stripped, isolated Potassium 40 hardly ever decays to Argon 40 at all. The decay rate should go essentially to zero, exactly the opposite of the behaviour which Woodmorappe trumpets proudly in the case of Rhenium.

Of course, under realistic and imaginable conditions, where Potassium or Rhenium could actually be fully stripped, namely in very hot neutral plasmas, we should have to also consider other reactions, such as capture of continuum electrons, as well as possibly contributions from additional low lying excited states of the various nuclei. This statement is valid for Rhenium 187 as well.

If these channels are opened up, it will likely make the total changes in production rates, at least for Potassium/Argon, rather smaller than what is naively predicted, or observed for the isolated atoms.

But these are problems of nucleosynthesis, not of radioactive dating, and that is perhaps a good place to end my erratum.

Modifications of Nuclear Beta Decay Rates

Post of the Month: March 2001