I think you have just explained how thermionic emission
works...
Explaining the emission was not my purpose. But Joe's comment
helped me to understand the cloud of "space charge" near a hot
cathode. I always wandered about the reason for the lingering of
electrons near the cathode. How is the cloud formed? I would
expect electrons to travel away, according to initial velocities,
even if there were no positive anode (in a simple diode). They
repel each other and should not stay near the cathode after being
"liberated" from it. Here is how can explain the cloud now.
The main point is that electrons in the cloud are not at all
"liberated". Only those electrons whose velocities are "in the
Boltzmann tail" are liberated. Those whose mean velocities
are as "small" as 10^5 or 10^6 m/s (great majority) are just
dancing back an forth in the electric field of "exposed ions".
Thermal equilibrium implies that they receive kinetic energies
from vibrating ions (and from collisions with other electrons),
move away from the surface and fall back toward it. A "cloud"
is the region where this happens. Something like rocks above
an erupting volcano; not an electrostatic situation
The dominant motion in the cloud is along the radii because,
unlike electrons inside the metal, outer elections have ions on
one side only. In that sense there is more randomness inside
than outside. Accelerating (dancing) electrons radiate waves
and that is how an object is cooled by radiation. But that is
another topic. The main point is that clouds are dancing
electrons, randomly driven oscillators.
Does this make sense to you? I do not remember seeing this
explanation in books describing vacuum tubes. What is a
better way of explaining electron lingering?