FDC cathode tension measurement - a few numbers

[Eugene Chudakov <gen@jlab.org>]

The cathode film has to be stretched to a certain tension. The optimum
tension is yet to be specified. However, it is unlikely that this tension would be
much lower than the tension of the existing prototype, perhaps a factor of 2 lower.

The prototype tension is about 660N/m. In fact, it can be different in
two projections: along the strips and perpendicular to them. The strains
(elongations) seem to be different: 0.25% perpendicular to the strips and
<0.2% along them.
The 5um copper layer provides about 80% of the tension, while kapton
provides 20%. A 2um copper layer will still dominate, but the Young's
modulus will become more isotropic than it is with the 5-um prototype.

In order to measure the tension in situ, while stretching, one can consider
3 methods:
1) Vibration. The frequencies of the radialy-symmetric modes are:
  f=B0/R*SQRT(T/sigma)/(2pi), where
    R=0.5m - radius of the foil,
    T - tension = 660 N/m
    sigma - surface density = 25e-6*1.3e3+5e-6*9e3*4/5=0.069kg/m**2
    B0 - zeros of the J0 Bessel function: 2.40, 5.52 ...
      we need the fundamental mode B0=2.40

  f=75 Hz for 5um copper
  f=91 Hz for 2um copper

  For T=200 N/m, 2um copper: f=50 Hz

2) Put a ring on the foil and measure the sagitta with the existing
   laser device. The shape is Z=W/(2pi*T)*log(R/r), where W is the weight.
   For Z=1mm, T=660 N/m and r=150mm: W=3.4N, m=350 g

2) Use an overpressure under the foil and measure the sagitta with the existing
   laser device. The shape is Z=P/(4T)*(R**2-r**2), where P is the pressure.
   For Z=1mm, T=660 N/m and r=0mm: P=10.5N/m**2=1e-4 bar

Eugene