Re: C14 in diamonds

Friar Broccoli wrote:
> Hi David Ewan Kahana:
>
> I have a question for you which is completely off topic for this
> thread and which I forgot to ask you before.
>
> Is the orbital velocity of electrons around the atomic nucleus
> directly tied to the speed of light?
>

As other people noted, the idea that electrons are orbiting
around the nucleus something like planets in the solar
system is an old picture of atomic structure which isn't
valid for most of the typical states you find atoms in.

But basically the answer I am going to give is no, the
average (rms) speeds of electrons in atoms are not directly
tied to the speed of light. Theoretically their are other
factors that contribute as well, and you have to know how
those change to say anything useful in answer to your
question, which also needs to be rephrased somewhat
before it makes sense.

To expand on the issue of why it isn't very sensible to talk
about orbital velocities:

Bohr and Sommerfeld did make a first step towards atomic
theory, which involved quantizing the angular momentum they
imagined was present in what they still thought of as
something like classical orbital motion in Rutherfords
picture of the atom. But a better, more comprehensive theory
was pretty quickly constructed, and it doesn't generally
allow for such an interpretation of the electron's states.

The states are called atomic orbitals in fact, to
distinguish them from classical orbits, and they are
solutions of the Schroedinger equation corresponding to
probability waves. Squaring the density of probability
waves (wave function) gives the probability density for an
electron. In general, you can at best talk about the
expectation value of the velocity of the electron in some
atomic state. According to the theory, you aren't guaranteed
to find the electron moving with that velocity, even if you
could measure it directly.

Now there do exist some weird states for the hydrogen atom
involving superpositions of a large number of nearly unbound
states, for which the motion of the centroid of the quantum
mechanical distribution of positions can be shown to follow
the classical equations of motion for the position of an
electron moving in the coulomb potential of the hydrogen
atom, but these states are extremely special; one has to go
to very great lengths to make them experimentally.

Quantum mechanical systems, in other words, even ones as
small as atoms, generally may have a classical limit, in
which they for some purposes behave approximately
classically.

But for practical purposes atoms almost always require a
fully q.mechanical description. So for most purposes you
don't want to think in terms of orbital velocities.

Average squares of the velocities, or equivalently, average
kinetic or potential energies, or rms radii of the electron
orbitals, or energy levels of the bound states in atoms, are
all sensible quantities to talk about, though.

If we see those levels, or those features of atoms changing
with time, it would indicate changes in what we call some of
the fundamental constants of nature.

> In other words, if the speed of light had been much higher in
> the past, would electron velocities likewise have been higher?
>

First you have to consider what you mean by saying that the
speed of light was higher in the past. When people say
they've measured the speed of light they are really talking
about assigning, by some specific procedure, dimensionful
units to this quantity.

So for example, the speed of light we always say has units
of distance divided by time. Now if you look at actual
experiments that measure the speed of light, you'll see that
one needs generally to have a standard of distance and a
standard of time to do the measurement. Using some kind of
timing device, one measures the travel time of light over a
known distance. Before doing this people must have already
defined standard units for time, such as seconds, and
standard units for distance such as meters. Nothing in
nature says what should be the length of a meter, nothing
says how long a second should be. These are simply arbitrary
numbers that people make up, and that we tie to specific
events and specific objects in nature.

So we first have to say exactly what we mean by the
numerical value of these units. For example, the accepted
definition, by an international convention, used to be that
the standard meter was 1 ten millionth of the distance from
the North Pole to the equator, and a platinum-iridium bar of
exactly one meter length was made up and carefully stored in
a vault in Paris, to be measured with the temperature at 0 C
and thus to define the standard meter. The standard second
was once defined as 1/86400 of the mean duration of a solar
day.

In other words, people used to tie the size of the basic
units of time to the motion of the earth around the sun, and
the basic units of distance to the size of the earth.

We have other definitions now. The second is instead defined
as follows:

`The second is the duration of 9,192,631,770 periods of the
radiation corresponding to the transition between the two
hyperfine levels of the ground state of the cesium 133
atom.'

But never mind those details for now. The principle is
exactly the same as it was before.

So in other words, people used to tie the size of the basic
units of time to the motion of the earth around the sun and
on its own axis, and the basic units of distance to the size
of the earth. These celestial properties were though to be
very basic when the definitions were made up,

Now suppose, using the old conventions, that I did a
measurement of the speed of light by some method and came up
with a number for the speed, and you, a couple of decades
earlier, had done the exact same measurement with the exact
same distance and time standards and had come up with a
different, larger number for the speed.

What does this result mean, assuming both of our experiments
were checked and replicated and there were no errors, and
that the difference in the measurements was outside the
experimental errors? Does it mean that the speed of light
has decreased? Or maybe it means instead that the length of
a meter has increased? Or possibly, does it mean that the
second has become a shorter time period? Maybe the earth
became larger in the meantime, maybe it is now rotating more
slowly on its axis?

So you can see that there's a problem in interpretation. In
fact, we could always reabsorb a change in the speed of
light into a change of the basic units of time or of
distance, or more generally into a change in the
relationship between basic time scales and basic distance
scales.

In this sense, the speed of light can be seen to serve
simply as a conversion factor between time scales and
distance scales.

In fact, in all theories that have local Lorentz invariance
as a symmetry, this basic role is played by the speed of
light. If local Lorentz symmetry happens to be broken in
some theory, then it becomes a whole different question,
since in that case a change in the speed of light may well
become physically meaningful independent of changes in
distance and time scales.

The modern definition of the meter takes advantage of the
local Lorentz invariance that most physicists believe to be
a very general property of nature. It might be broken,
though, and please note: there's nothing preventing someone
from saying that it is broken or making some specific theory
of how it is broken. But there is no evidence whatsoever
that local Lorentz invariance is ever actually broken at
all, none so far. So we now define the standard meter as
follows:

`The meter is the length of the path travelled by light in
vacuum during a time interval of 1/299,792,458 of a
second.'

The effect of that definition, of course, is just to fix the
speed of light in vacuum as exactly 299,792,458 m/s. In this
case, the speed of light just converts between the distance
units and the time units.

A way to try to get around the problem of interpretation
I've stated is to define dimensionless quantites that are
given as ratios of the dimensionful ones. So for example, to
see whether the radius of the earth had changed we might
compare its radius to the radius of the sun, or the moon.
If both sizes changed at once and by the same amount, of
course we couldn't see any difference in the ratio. But if
the earth became larger while the moon stayed the same size,
we might be able to detect that.

Let me return to the question about the energy levels of
atoms. A very accurate theory of the energy levels and the
wavefunctions of single electron atoms exists. It's based on
the theory due to Dirac. It is not a complete treatment of
the problem, even for such simple atoms, because there are
further very subtle effects that need to be included, having
to do with the fact that the electron, and the force between
the electron and a nucleus really need to be treated as
quantum mechanical fields. In addition there are effects due
to the finite size of the nucleus, to its spin and mass, and
others which are neglected in first approximation. We can
put all of them in however.

Nevertheless Dirac's equation gives an extremely good first
approximation to the observed energy levels of single
electron atoms, and their properties. I'm going to write out
the formula for the energy levels in Dirac's theory, just to
illustrate a point. The formula is:

E(n,j) = m c^2 [ 1 + { (Z * alpha) / (n - (j+1/2) + ( (j+1/2)^2 - (Z
* alpha)^2)^(1/2) ) }^(1/2) ]

c here is the speed of light.

But notice how the constant c^2 appears in the equation. It
appears in combination with another dimensionful constant
m. m here is the mass of the electron. More generally, if we
had some other negatively charged particle than an electron
orbiting around the nucleus, then its mass would appear
instead. But in a normal single particle atom, m is the mass
of the electron.

The constant Z is dimensionless: it's a pure number, an
integer in fact, which tells you how many positive charges
are contained in the nucleus of the atom. So we would have
Z=1 for a hydrogen atom, Z=2 for a singly ionized helium
atom, and so on.

The numbers n and j are also pure numbers, which describe
the quantum mechanical bound states of the atom. n is a
positive integer, while j is a positive half-odd integer.

The constant alpha is also a pure number, whose definition
I'll discuss in a moment. But simply notice for now that
it's the product of two pure numbers, Z and alpha, that
determines the spacing of the energy levels in Dirac's
theory (the difference in energy between levels having
different values of n and j).

This level splitting is what is actually observable when we
do spectroscopy of atoms, and observe the light that they
emit. The energy of the photons emitted, hence the
frequency of the light, is determined by the energy level
spacing of the atom.

Now the pure number alpha is a specific constant, called the
fine structure constant that is a combination of the charge
of the electron, the speed of light, and another
dimensionful constant h, called Planck's constant. Planck's
constant in fact relates frequencies to energies, while the
charge of the electron basically determines the strength
(interaction energy) of the force between the electron and
the nucleus due to one being positively charged and the
other negatively charged. alpha is defined to be:

alpha = e^2 / (h_bar * c); and h_bar = h / (2 * pi).

Numerically the value of alpha works out to be close to
1/137. There is actually a further subtlety that I'm
sweeping under the rug by calling alpha a constant, but lets
forget it for now. The correct value to be used in the case
of an ordinary atom is certainly very near the measured
value of 1/137.

So c appears in the Dirac formula in two places: out in
front with m as an overall multiplicative factor, and also
under square root symbols through alpha, changing the
dependence of the levels on n and j.

If you expand this formula in powers of (Z * alpha),
assuming that this is a pretty small number, you'll see that
the first term is just the energy due to the rest mass of
the electron. The terms in (Z*alpha)^2 then give a splitting
depending on n and j, and then there are terms of order
(Z*alpha)^4 and so on. The lowest order terms depending on n
and j actually prove to be independent of c.

Now imagine changing c by a small amount over time. It
means by the way, that whatever theory describes this
change, is one that violates local Lorentz invariance. So
this might well also completely invalidate Dirac's theory
....

In other words ... someone who wants to propose such a
theory in which c changes had better tell us *exactly* what
that theory says about how atomic energy levels change, and
they had better do so in at least as precise a way as Dirac
did.

But, suppose that Dirac's theory remained perfectly valid as
c changed, and m, e and h all remained constant too: then
you would expect to see no change at all to the lowest order
in (Z*alpha).

You would have to measure accurately enough to see
differences in the fourth order terms to detect any
variation. But even if you did see some change there, the
interpretation of that change would remain ambiguous,
because you couldn't tell for sure with just that one
measurement, whether m, e, or h might have changed or
whether c had changed.

On the other hand, if the combination alpha itself changed
and the Dirac theory remained valid, then you could see
quite unambiguous behaviour by measuring very accurately the
overall spectrum of light emitted and absorbed by the atoms
.... because basically the spectrum depends on the energy
splittings and the splittings depend DIRECTLY on alpha.

> And a bonus question: Would that extra electron velocity
> translate to greatly increased chemical reaction rates?
>
> Since, I am sure you can guess the intent of this question,
> any other comments you see fit to make will be greatly
> appreciated.
>
>

There are different classes of c-variation theories around.

Some theories are proposed by young earth creationists to
try to deny the great age of the cosmos: these all involve
very large changes in c in a very short time period. All of
the theories of this kind that I've heard of fail, and fail
spectacularly.

Then there are some people working on cosmology who have
proposed that c may have been much faster in the very, very
early universe.

These are theories that violate local Lorentz invariance,
and are based on something other than General
Relativity. They are proposed mostly to solve well known
theoretical problems with the initial conditions in the big
bang ... the so-called horizon problem in particular. The
problem is: Why is the temperature of the CMBR precisely the
same on opposite sides of the sky, since in the early the
regions that emitted the radiation would have been causally
disconnected from eachother? There was no natural
explanation for this fact in the standard hot big bang
model. It had to be built in as part of the initial
condition. The widely accepted theory now is that there was
an inflationary period very early in the universe, so that
regions on the opposite side of the sky would have been in
causal contact in the early universe. That neatly solves the
theoretical problem. But not everyone is happy with the
solution and everything else that flows from it. So some
have proposed the idea that c varies. Personally I find
these proposals a bit ad hoc, but one can't really rule them
out. In any case, what these workers have to say is of no
help whatever for young earth creationists. None of them are
proposing any detectible *current* variations in c.

Finally, there is some experimental evidence, controversial,
but apparently real enough, that the fine structure constant
alpha, which I was just discussing, may have varied in the
very distant past, and by a very small amount. Personally my
opinion on this is that there are problems with the
interpretation of the data, since the theory required is for
the levels of atoms containing many electrons and it is
nothing like as simple as even the overly simplified version
of the theory of one electron atoms that I presented.

It's necessary, you see, to calculate in theory just how the
energy levels of atoms with many electrons are expected to
change as alpha changes. This is highly non-trivial, and
then you must make the best fit of the theory to spectra
observed in far distant clouds of gas, with all of the
attendant problems that exist in observational astronomy.

The last two are real scientific work and will give
no help whatever to young earth creationism, even
if the ideas turn out to be true.

David

.

Relevant Pages

  • Re: Electron-orbits, light-wavelength and amplitude
    ... Atoms contain electrons. ... an electron move from a higher energy orbit of the atom to a lower ... distance the electron falls. ... Does this mean that a fall of greater distance produces a shorter ...
    (sci.physics)
  • Interpreting LEED images
    ... I do not know the height of the electron gun so I can ... Where n is the diffraction order, d is the distance between the atoms, ... initial beam diameter is 1 micrometer. ...
    (sci.physics)
  • Re: Electron-orbits, light-wavelength and amplitude
    ... Atoms contain electrons. ... an electron move from a higher energy orbit of the atom to a lower ... distance the electron falls. ... Does this mean that a fall of greater distance produces a shorter ...
    (sci.physics)
  • Re: Time. /My opinion./
    ... and the planetary time is absolute for them? ... One law of SRT says that speed of light quanta ...  isn't the Maxwell's theory / SRT without electron. ... It means that different masses and speeds can create ...
    (sci.logic)
  • Re: My New Website
    ... Describing heavy atoms requires more computational time than ... > in pioneering scince at the beginning we are not 'nit pickers' ... Did I ever mention "lots of free parameters"? ... But what I have seen are e.g. observations of the electron density ...
    (sci.physics)