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In a conversation with Peter Fritschel yesterday afternoon we discussed
the possibility that the improvement that has resulted from backing off
the earthquake stops on the LLO ITMY might be due to reduction in the
noise from fluctuating charge layers on the viton and fused quartz. The
charge layers are caused by the now well known problems of the viton 
touching the SiO2 and transferring charge because of the difference in
Fermi energy of the electrons in the two materials. (A possible fix
was suggested several years ago by putting SiO2 tips on the Viton in
the earthquake stops, but this was never implemented).

The mechanism being considered is the idea that charge transferred 
does not remain totally immobile but rather wanders by hopping around
on the suface to reduce the local surface charge density. The process
is discontinuous but can be characterized by a relaxation time tau.
The time is determined by the ratio of the dielectric constant to the
electrical conductivity of the material. On a fused silica surface,
it can be weeks to months. The spectral density of the fluctuating force
on the fused silic surface is

                          2    1/2     <F>
               F(f) = (-------)   *  -------
                        pi*tau          w

where:
 
F(F) =  the spectral density of the force dynes/sqrt(Hz)

tau  =   charge relaxtion time seconds

<F>  =  the average DC electrostatic force on the mirror dynes

w    =  2*pi*frequency

The formula above assumes that w >> 1/tau and is discussed in a note I wrote several years
ago on noise from fluctuating electrostatic charges

In the case of the LLO ITMY assume that the average force is given by
        
                       a
             <F> = ---------
                   (d0 + x)^2

which is appropriate if the charge transferred from the Viton tip is initially well
localized on fused silica, effectively a point charge when the surfaces separate.
d0 is the distance of closest contact and x is the distance between the surfaces
under non contacting conditions. d0 will be a few microns while x is a few millimeters.

a and d0 are estimated by noting that the mirror has been pulled over to the viton 
stop and with some judicious guess about the difference in Fermi energy, say 20 volts.

The displacement noise in this formulation becomes

                       2   1/2         a             3 x 10^-11
            x(f) = (-------)   *  -------------  =  ------------   cm/sqrt(Hz)
                     pi*tau       m * x^2 * w^3         f^3

where

tau = 1 week = 6 x 10^5 sec
a   = 1 x 10^-2 dyne*cm^2
d0  = 3 microns = 3 x 10^-4 cm
m   = 1 x 10^4 grams
x   = 4 mm = 4 x 10^-1 cm


Estimated values of the displacement noise

         f frequency Hz   x(f) cm/sqrt(Hz)

          40                3 x 10^-16

          50                1.5 x 10^-16

         100                1.9 x 10^-17


Not so crazy, many of us have noted the 1/f^3 dependence in this region of the
spectrum. If this holds up, it may be useful to back off all the
earthquake stops (or better still do the fix recommended several years ago,
to make the contacting surfaces of the same material.)
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