M2L11d.txt

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That a given electric field states so to speak,
just disappear, they are no longer stable,
and this is a process, which is called field ionization.
The phenomenon is that sufficiently strong
electric fields ionize the atom.

And whenever there is a single model
and I can give you an analytic answer, I try to do that,
because I feel a lot of our intuition
is shaped by understanding simple models.
And the simplest model for field ionization
is just a classical model by calculating
what is the settle point in the combined
potential of the nucleus, which is a Coulomb potential,
and the external magnetic field.
So many features of the experiment
can be understood by this simple free-line derivation.
So we have a potential.
One part of it is the Coulomb potential and focusing
on one special direction here.
And then, in addition, we apply an electric field.
And the electric field creates a linear potential.
And if I take the sum of the two, at large distances
it's the electric field which dominates,
then the Coulomb potential takes over.

So that's how it looks like.
So now we have the situation if you would put in atoms
and we would look at the energy eigenvalues, at this point,
this is the maximum excited state in the atom, which
is still stable.
So what I want to derive for you is
what determines sustainability is simply the center point.
When the binding energy of the excited state for which we
use the [INAUDIBLE] formula is not stronger
than the position of the center point,
the atom becomes unstable and it becomes field ionized.
And we'll discuss a little bit later
if this really applies to real atoms.
The quick answer is for lithium and all of the other atoms,
it applies for hydrogen. It doesn't, because hydrogen
is too many [? symmetries ?], too many exact degeneracies.

OK.
So the total potential is the Coulomb potential
plus the electric potential.

What we need is the position of the settle point where
we have a maximum in this one.
This is a one dimensional cut, and this one dimensional cut
has a maximum at this position.
And by taking the derivative of the total potential,
you immediately find this to be of that value.
And now what we are calculating next
is, what is the potential energy at this point.
And this is E to the three halves.
And now what you want to do is, we
want to postulate that for field ionization,
this should be able to the binding
energy of the electron, which is nothing else
than the [INAUDIBLE] constant divided by N squared.

OK.
Now here we have the square root of the electric field
from this calculation.
So that means the critical electric field will scale
as one over N to the fourth.
And this is famous scaling, which
can be found in many textbooks that the critical electric
field for ionization equals-- and now
that's the beauty of atomic units--
it is one over 16 n to the fourth.
Beautiful formula derived from the settle point criteria.
Of course, what I mean here is-- and if you do the derivation,
it's in atomic units-- which means
in units of the atomic unit of the electric field, which
is E over the [INAUDIBLE] radius squared.
So it's a simple model.
It's an analytic result. The question is, is it valid?
Does it make any sense?
And the answer is yes, but in a quantum mechanical problem,
you would actually use Schrodinger's equation
as your potential.
But then the onset of field ionization
comes when tunneling becomes possible for this area.
But it is the nature of tunneling
that if you're a little bit too low, tunneling is negligible.
You may have ionization rates of one per millisecond or so.
And if you just go a little bit closer,
it becomes exponentially larger.
So therefore, this scaling is very, very accurate,
because the transition where you go
from weak tunneling to strong tunneling
to spilling over the barrier, it's
a very narrow range of electric fields.
But yes, people have looked at it in great details
and to calculated corrections due to tunneling.
So these are quantum corrections to the classical threshold
which I just calculated.

But now in hydrogen, a lot of in error mixing matrix elements,
matrix elements due to the electric field
between nL states vanish.
Hydrogen is just too pure, too precise.
There are [INAUDIBLE] quantum numbers
where you can exactly [? dianalyze ?] hydrogen
electric fields, and you find some stable states, which
do not decay, and they above the classical threshold
we have just calculated.
So as Dan Kleppner would have said,
"The simplest of all atoms is the most complicated
when it comes to field ionization,
because it has a lot of stable states
above the classical barrier."
So you can sort of envision that their
will be orbits, which are just confined to this region
and the electron never samples the settle point.
And if you look at it this diagram on the Wiki,
these are actually calculations for hydrogen,
which include ionization rates.
You will find that the states, which
are the ones which go down, which are on the down hill
side of the electric field.
You see all of this here marked and on set for ionization.
And then you see a rapid increase
in the ionization rate.
But you also see those hydrogenic state,
which go upward in energy.
And they refused to ionize because of the symmetry
of [INAUDIBLE] coordinates and the things I've mentioned.
Anyway it's too special to spend more time here in class on it,
but I just think you should at least know qualitatively
what is different for hydrogen.