So, in octavio part two the conclusion was that the pinch coil configuration (current in the outer coils opposite to the current in the core) cannot create a virtual cathode.
So the only configuration left now is the core with focus coils:
To understand the idea, take a look at the following 2D analogy (not a cross section, just an analogy, you can click the image to see it enlarged):
As explained in octavio part two the electrons will slow down because they loose energy when they are accelerated/de-accelerated by the electric field between the focus coils and the core, and when spiraling in the magnetic field, bremsstrahlung and synchrotron radiation losses.
As explained in part two, to compensate for the losses an AC voltage can be superimposed on the DC potential of the focus coils, this will also introduce extra losses (bunch induction, explained in part two), which can be compensated by the same AC power.
This will create an oscillating virtual cathode, the idea is to create a situation where the DC current on the high voltage positive core is almost zero.
The voltage between the wehnelt cylinders and the focus coils should be negative most of the time, only when the AC power is at it´s positive peak the electric field between the wehnelt cylinders and the focus coils should allow some electrons to be injected into the system, the AC power maximum positive peak is the moment when most of the electrons are near the focus coils, the negative space charge in the center of the focus coils at that moment can prevent too much electron injection (in other words, the input current is self regulating when the focus coils have the right negative DC bias).
Some preliminary calculations show that the resonant frequency can easily get higher than two gigahertz, but it will be lower when the magnetic field is stronger. Feeding AC power to such a construction at such high frequencies is another challenge.
I've been thinking about a self-oscillating octavio, but naturally every oscillation will get it's AC power by slowing down electrons, keeping the electrons at a high speed will always dissipate AC power. Any self oscillation (some kind of buncher/catcher or reflex-klystron-like construction) should be between the the electron guns and the focus coils, not between the focus coils and the core.
Still lots of calculations and simulation to do, and I have to find a way to make it self-oscillating.