Anode erosion and electron beam energy estimates

LPP has just started sending out technical reports to supplement the monthly reports, as many would like to know the technical details. Unfortunately, this may be a bit confusing. Those not interested in the details need not read further. In general, these reports will be elaborations on points already mentioned in the monthly report.

The significance of this anode report is as follows; it is basically good news in that it shows we are converting a lot of the energy into the beam. The erosion will decrease, we expect, when the electron energy gets higher and the beam penetrates more into the copper, dispersing the energy better.

Since December, the depth of the hole that was machined into the anode, the central electrode, has been measured with a depth gauge four times.  The results are shown at the end of this note.  Since we have not accurately measured the length of the entire electrode—we will do this next time we open the chamber—we cannot directly measure the erosion off the lip of the electrode.  But since the electrode can’t get any longer, we can estimate the minimum erosion from the times when the distance from the lip to the edge decreased.  By adding this to the measured increase in the depth of the hole, we can estimate a minimum rate of erosion.  This turns out to be about 18 microns per pinch.  If we assume the beam is only exactly as wide as the hole—a ride of 1.4 cm (it may be somewhat larger), the minimum amount of erosion per pinch is 0.011 cc or 0.1 gm of copper.  To raise this much copper to vaporization temperature and then boil it takes 570 J of heat.  Some copper could be removed by melting, forming droplets and the droplets falling off the surface, which is pointed downwards.  However, other copper was surely heated far above its boiling point by high energy particles, so this energy estimate is still conservative.  Since the ion beam must contain the same energy, the average pinch contains 1.1 kJ.  For comparison, the energy in the magnetic field, assuming an inductance of about 5 nH, is 1.2 kJ for a peak current of 700kA.  The kinetic energy of the plasma should be similar for a total available energy of 2.4 kJ.

Is it reasonable that the e-beam is depositing so much energy in the anode surface?  The depth of penetration of an electron is 0.1 E1.5/p microns, where E is electron energy in keV and p is density.  So a penetration of 18 microns would mean average electron energy of 140 keV.  Of course one could have higher energies, spread out over a greater depth, but with more energy.

From the total energy estimate and the electron energy, we can get an estimate of 4 mC for the total charge in the beam.  If we then assume that the beam is spreading out linearly and covers a 1.4 cm radius circle at a distance of 4 cm, we can expect that at the upper Rogowski coil, we will get a total charge passing of 0.28 mC.  Interestingly, while we are still working on getting the noise down in the beam Rogowski coil, one seemingly clear signal from shot 040209 did contain 0.13 mC, not far from our estimate.  While we can’t be sure this is not noise until we get both beam Rogowskis working and their noise down, this particular pulse would have originated at about the same time as an X-ray pulse observed by the Far Time of Flight (or FTF) instrument.

Observations (distances in inches):

Hole radius: 0.55, half-radius: 0.27

Original depth of anode hole: 1.575

Dec 12 – edge 1.568, center 1.587, center erosion 0.012, total >0.019, 18 pinches

Feb 12 – edge 1.565, half 1.580, center 1.591, center erosion 0.04 total >0.07, 10 pinches

Mar 22 – edge 1.593 half 1.595 center 1.609 center erosion 0.018 center-half 0.003, 25 pinches

Apr 1 – edge 1.581 half 1.581 center 1.597 center erosion 0 center-half 0.002 total >0.014, 18 pinches

Average erosion per pinch – 0.0007 inch/pinch

Lip erosion > 0.022”

Edge erosion > 0.050”

Half erosion > 0.050”

Center erosion >0.066”

Assume beam radius is 0.55”, 20 degrees

18 microns penetration of electrons into copper

0.011 cc of copper vaporized

0.1 gm of copper vaporized

569 J heat of vaporization plus heating to boil for copper

E-beam width about 20 degrees

Plasmoid minimum energy 1.1 kJ

Total magnetic energy at 0.7 MA: 5 nH, 1.2 kJ, total energy 2.4 kJ