Jim Hylen

February 7, 2003

DRAFT - Leakage Current between Horn Stripline Conductors

The purpose of this note, which is currently a work in progress, is to
estimate or at least set a bound on the expected leakage current between
horn stripline conductors, to determine if a design modification is
necessary. The cause of the leakage current is the beam ionization of
the air between the striplines.

Two concerns are possible:

(i) breakdown (sparking)

(ii) diversion of enough current from horn to affect beam focusing

The NuMI horn current peaks at 200 kA, and if the leakage current was
greater than 0.5% one would need to do a more detailed circuit model to
see if the magnetic focusing field would be affected.

**NuMI Environment.** There are four 8 inch wide striplines feeding the
horn, each with about 1 m length in the high radiation area, as
illustrated in Figure 1. Figures 2 and 3 show the current pulse and
voltage between striplines for the high energy and low energy neutrino
beam configurations of the horns. The beam spill lasts for a maximum of
10 micro-seconds, and occurs at the peak of the current pulse. At that
time there is about 50 Volts across the 3/8 inch air gap between
stripline conductors. A MARS model of the stripline beam energy
deposition has not been done, however the radiation environment near the
end of the horn ranges from order of 1 J/g/spill near the beamline
(Byron Lundberg MARS model of cross hair system) to 0.09 J/g/spill at
the horn outer conductor and 0.02 J/g/spill at the horn/module support
hanger (C. James MARS model of horn clamp) to 0.007 J/g per beam spill
at the bottom of support module (Igor S.Tropin MARS model of module).
The stripline starts basically at the radius of the horn outer
conductor, and 0.07 J/g is taken as a rather conservative estimate of
the average beam energy deposition.

**Breakdown.** With reference to concern (i), the general lore is that
ionizing radiation causes faster initiation of breakdown, but does not
significantly reduce the voltage at which breakdown (caused by
avalanching of charges) occurs. Our 50 Volt/cm field is far too low to
cause breakdown. Even the maximum potential in the system, which is less
than 1000 Volts at the power supply, is an order of magnitude below what
would be required to cause arcing between stripline plates. Further
confidence in this statement is provided by a test done by F.M.
Bieniosek where he set up parallel plates 3 cm apart in the high
radiation area after the AP0 target (Pbar Note #556 Summary of Results
from the Beam Sweep Test Module, 1-June-95). With 16 kV potential
difference between the plates, and nearly an order of magnitude higher
particle flux than expected for the NuMI stripline region, there was no
sign of avalanching. Data from the Bieniosek Note are reproduced in
Figure 4.

**Leakage Current.** In addressing concern (ii), we can first repeat the
calculation Bieniosek did to roughly estimate the expected current for
his configuration (F.M. Bieniosek, A Beam Sweeping System for the
Fermilab Antiproton Production Target, FERMILAB-TM-1857, August 1993).
The calculation is based on a simple balance between ionization rate and
recombination rate giving the charge density (see formula (i)), combined
with a parameterization of the electron drift velocity (see formula
(ii)). Table 1 lists the parameters as given in the TM, as modified for
the conditions of the highest and lowest voltage data points of the
test, and as modified for the conditions of the NuMI stripline. For the
NuMI calculation, the drift velocity is taken from Figure 5 rather than
formula (ii), as NuMI is below the E/p region where that formula is a
good description; additionally the reduction in drift velocity due to
the magnetic field is estimated for NuMI conditions.

As shown in the table, this calculation overestimated the current in the
test by factors of 2 to 7. A further problem is that the data in Figure
4 show a higher than linear behavior on E/p, whereas the calculated
value is proportional to the electron drift velocity which is less than
linear in that region. As noted in the TM, the current also showed a
somewhat stronger dependence on intensity than the square root
dependence in the plasma model (see formula (viii)).

One possible effect is due to space charge. The electron drift velocity
is of order 1000 times larger than the positive ion velocity, and unless
electrons can be ejected from the negatively charged plate at the rate
that they captured on the positively charged plate, the process becomes
space charge limited on a very short (nanosecond) time scale. During the
beam spill, this may be provided by delta rays knocked out of the
negatively charged plate by the beam spray, but I have not yet attempted
to calculate this rate.

Because the calculation does not track the data very well, the
extrapolation to the NuMI conditions by the above calculation may not
give one confidence. A reasonably straight-forward bound is set by the
following argument. The NuMI conditions have a smaller V, smaller E/p,
and smaller beam radiation level, but a factor of 250 larger surface
area. If we take the lowest test point of 10 amps leakage, scale up by
the area, and take credit for a no worse than square-root dependence on
the beam intensity, we get a bound of 375 Amps. Given that the NuMI E/p
is an order of magnitude smaller, the current is likely to be
substantially less than this.

**Available mitigation.** Two available mitigation methods, if the above
currents are deemed to be too large, are:

(i) insulate by aluminum striplines by anodizing them. One might still
    leave the high stress bend regions bare, as the anodized surface is
    more prone to cracking.

(ii) tune the beam spill to occur slightly after the peak horn current,
     so that the voltage between the sripline plates is zero. The
     resulting 2 to 3% drop in horn focusing power has only a minor
     effect on the beam neutrino intensity, but does make the beam
     spectrum somewhat more sensitive to precise timing between beam
     spill and horn pulse.

**Conclusion.** At this point we are not motivated to take mitigating
action, as neither breakdown or leakage current appear to be a
significant problem.

# Appendix: Formulae

Balance between ionization and recombination of electrons. (Assumes no
avalanche).

(dU/dt)/W~i~ = n~e~^2^ (i)

U = beam energy deposition per unit volume per spill

t = time (length of beam spill)

W~i~ = mean energy to produce an electron-ion pair

= recombination coefficient

n~e~ = number of electrons per unit volume

Electron drift velocity as parameterized in TM-1857

V~d~ = 7x10^5^ + 5x10^5^ (E/p) cm/sec (ii)

(E/p) = (Electric field / pressure) in volts/cm/Torr

Momentum-transfer collision frequency for electron-neutral collisions

~m~ = e E / (m~e~ V~d~ ) (iii)

E = electric field

Plasma conductance

= n~e~ e^2^ / m~e~ ~m~ (iv)

e = electron charge

m~e~ = electron mass

"Resistance" between plates from TM-1857

R = 4 a / ( L d ) (v)

a = width of gap between plates

4 = relation between applied voltage and field for sweeping magnet?

Leakage current between plates

I = E L d (vi)

L = length of stripline

d = width of stripline

I = n~e~ V~d~ A (vii)

A = L d = area of stripline

I = ( (dU/dt) / ( W~i~) )^1/2^ V~d~ A (viii)

Magnetic field between striplines

B = ~0~ I~S~ / d (ix)

I~S~ = current in stripline

Drift angle in magnetic field, for orthogonal B and E

Tan(~B~) = 2 B V~d~ / E (x)

V~d~' = V~d~ cos(~B~) (xi)

~B~ = drift angle w.r.t. E field direction

V~d~, V~d~' = drift velocity without and with effect of field

  ------------------------- ----------- ---------- ----------- ----------- -----------
                            units       TM-1857    Bieniosek   Bieniosek   NuMI
                                                   test Hi     test Lo     stripline

  W~i~, energy to produce   eV          33         33          33          33
  ion pair in air                                                          

  Beam protons on target /  pot/pulse   5.00E+12   2.60E+12    1.80E+12    4.00E+13
  spill                                                                    

  U, deposited energy       J/g         1.5        0.78        0.54        0.07
  density / spill                                                          

  air denisity              g/cm3       1.20E-03   1.20E-03    1.20E-03    1.20E-03

  Conversion factor         J/eV        1.60E-19   1.60E-19    1.60E-19    1.60E-19
  Joules/eV                                                                

  beam spill length         sec         1.60E-06   1.60E-06    1.60E-06    8.00E-06

  dU/dt                     eV/cm3sec   7.03E+21   3.66E+21    2.53E+21    6.38E+19

                                                                            

  Total N~e~ electron       e/cm3       3.41E+14   1.77E+14    1.23E+14    1.55E+13
  produced                                                                 

  production dN~e~/dt       e/cm3/sec   2.13E+20   1.11E+20    7.67E+19    1.93E+18

  Recombination coefficient cm3/sec     2.20E-06   2.20E-06    2.20E-06    2.20E-06

  equilibrium N~e~          e/cm3       9.84E+12   7.10E+12    5.90E+12    9.37E+11

                                                                            

  applied Voltage           V           2500       16000       2000        50

  a (gap)                   cm          3          3           3           0.9525

  E=V/a                     V/cm        833        5333        667         52

  E/p                       V/cm/Torr   1.10       7.02        0.88        0.07

  Vd                        cm/sec      1248246    4208772     1138596     438141

  Drift distance during     cm          2.0        6.7         1.8         3.5
  spill length                                                             

                                                                            

  e                         coulomb     1.60E-19   1.60E-19    1.60E-19    1.60E-19

  m~e~                      kg          9.10E-31   9.10E-31    9.10E-31    9.10E-31

  collision freq. ~m~       1/sec       1.17E+12   2.23E+12    1.03E+12    2.11E+11

  conductance               mho/m       2.36E-01   8.96E-02    1.61E-01    1.25E-01

                                                                            

  L, plate length           cm          30         21.59       21.59       800

  d, plate width            cm          3          3           3           20.32

  R=4a / sigma L d          ohm         **57**                              

  R Without inductive       ohm                    51.69       28.71       0.05
  factor of 4                                                              

  leakage current           amps                   **310**     **70**      1068

  *measured current*        amps                   *160*       *10*         

  * *                                              * *         * *          

  Stripline current per     amps                                           50000
  quadrant                                                                 

  d, stripline width        m                                              0.2032

  ~0~                       Tesla m /   1.26E-06                           1.26E-06
                            amp                                            

  B between stripline       Tesla                                          3.10E-01
  plates                                                                   

  Tan (~B~)                                                                0.518

  ~B~ (the drift angle)                                                    0.478

  cos (~B~) = V~d~' / V~d~                                                 0.888

  Leakage current including Amp                                            **842**
  B field                                                                  
  ------------------------- ----------- ---------- ----------- ----------- -----------

Table 1. Calculation of leakage currents due to ionization by beam
showering.

![](media/image1.jpeg){width="5.997916666666667in"
height="6.190972222222222in"}

Figure 1. Stripline through radiation shielding block to focusing horn.

![](media/image2.png){width="5.997222222222222in"
height="4.543055555555555in"}

Thslcm_2-1-2001.cir Run date 2-3-03, krb

Figure 2. Plot of Voltage on the terminals of Horn-1 (Green -- 235 V.)
and Voltage across stripline conductors at remote release clamp (Red --
237 V.) vs: Horn-1 current (Blue) for the Target Hall installation with
the full length stripline (LE, ME, and HE taps). (Ken Bourkland, 2/3/03)

![](media/image3.png){width="5.977083333333334in"
height="4.499305555555556in"}

Thslcm_2-1-2001.cir Run date 2-3-03, krb

Plot of Voltage on the terminals of Horn-1 (Green -- 253 V.) and Voltage
across stripline conductors at remote release clamp (Red - 255 V.) vs:
Horn-1 current (Blue) for the Target Hall installation for truncated
length stripline (completed only to Horn-2). (Ken Bourkland, 2/3/03).

![](media/image4.jpeg){width="5.997916666666667in" height="4.475in"}

Figure 4. Leakage current between parallel plates as a function of
applied voltage, for two proton beam intensities. (from F.M. Bieniosek,
Pbar Note #556).

![](media/image5.jpeg){width="4.8in" height="3.7069444444444444in"}

Figure 5. Electron Drift Velocity as function of E/p.

![](media/image6.jpeg){width="5.996527777777778in"
height="4.365277777777778in"}

Figure 6. Magnetic field around stripline at peak current. (Chris
Jensen)