Well, assume the electron beam begins roughly on axis of the solenoid and
well outside the solenoid so the magnetic field is low compared to the field
at the center of the coil. Picture the lines of magnetic field close to the
axis - at the center of the coil they are lines parallel to the axis but as
they leave the coil they slowly diverge. If the transverse energy of the
electron beam is "low" so that the cyclotron orbit diameter of an electron
about a particular line of magnetic field is small compared to the diameter
of the solenoid, the electron will spiral about that line of force and
follow it into the solenoid. Since the lines of force converge as they
enter the solenoid the electron beam is condensed, by the ratio of the
magnetic field at the electron beam source to the full field at the center
of the solenoid. The cyclotron orbit diameter is also reduced by the field
ratio. The system is symmetrical so the beam expands as it leaves the
solenoid, regaining its initial diameter as the field falls to the initial
value. This is a simple qualitative picture, of course. Because the
magnetic field lines are not exactly parallel a small retarding force is
generated. The greater the magnetic field gradient and the greater the
transverse energy of the beam the greater this force, so if you have a low
enough energy along the solenoid axis and a large enough transverse energy
and field gradient the electron will be reflected back out of the solenoid -
this is a "magnetic mirror". Conversely, a beam with high energy along the
axis, centered on the solenoid, with very low transverse energy, will
penetrate the solenoid with virtually no loss (so every magnetic mirror
leaks a little right on axis).
On one model of Nicolet (then Waters, Extrel, Finnigan) Fourier transform
mass spectrometer the filament was located outside the main field and on
axis. The filament was a strip of rhenium ribbon 0.03" wide by about 0.3"
long, in a field of about 3/7 tesla. There was a small, 2 mm orifice at the
center of the field at 3 tesla, and it was quite easy to align the filament
so the entire beam passed through this hole with no loss, with beams of 5-50
uA at 70 eV. Actually, so long as the filament was within a disc of about
0.5" diameter the beam would clear the hole. If the beam were compressed by
the field ratio it would be about 0.042" x 0.004" (about 1 mm x 0.1 mm),
half the size of the hole. In one apparatus I built the filament was moved
another two or three feet further away, and since the total range of
electron filament adjustment I had was about 1" I was actually unable to
misalign it enough to make the beam hit the edge of the hole. At 70 eV and
these currents the space charge of the electron beam is basically negligible
so each electron follows its own path with no interaction with other
electrons. At lower energies and higher currents the space charge acts to
spread the beam laterally, especially in the low magnetic field portion of
the electron trajectory, and to increase the distribution of the energy
along the axis. This smearing all acts to make the magnetic mirror more
efficient, and at low enough energy the beam doesn't penetrate at all.
Experimentally I once moved a filament from full field out to about 1/8 of
full field and measured the maximum current collected on a solid plate
behind that orifice hole, and on the orifice plate, at constant filament
power as the extraction voltage was varied. The higher the magnetic field
at the filament the higher the space charge limited current that can be
extracted from the gun, while the lower the field the more the beam will be
compressed as it traverses the solenoid so the greater the current that will
pass through the 2 mm orifice. At 70 eV it really didn't matter so far as
current goes but the alignment was much less critical at the lower field, so
the Nicolet position worked very well. Down at about 7 eV the best position
was about 10" from the orifice plate, where the field was estimated (from
memory here) at about 90% of full field. Space charge in the gun was
clearly dominating but a little magnetic compression could still be
achieved. Going all the way down to 1 or 2 eV the closer to full field the
better, in terms of current through the orifice. At this low energy space
charge between the filament and the extraction grid was all-important.
Anyway, just thought you might find some old experiments interesting
.
-----
Regards,
Carl Ijames
"Jamie M" wrote in message
Hi,
How does a solenoid act to focus a beam of electrons that are passed
axially through the coil?
cheers,
Jamie
