From: les.holladay@ALZA.COM
  To  : rasmb@bbri.harvard.edu
  Date: Fri, 5 Sep 1997 10:22:26 -0700

Re: question about Wiener skewing...

Les -

The matter is not simple, but I will try and give a few ideas. 

Yours question concerns the the steepness of fringe gradient one can get away
with before Wiener skewing effects become detectable. The simple answer,
computed by Lloyd (*Optical Methods in Ultracentrifugation, Electrophoresis and
Diffusion* p.88 Clarendon, 1974) is 6.7 fringes per mm in a 10 mm cell, if a
precision of 0.02 fringes is desired. 

However, life is more complicated than that. There are TWO *anomaly* terms in
the Svensson equation which relates fringe shift to dn/dr, the first being an
aperture effect term, the second the so-called Wiener skewing term (a
terminology derived from Schlieren peaks). Only with Schlieren optics is it
possible to set both of these anomaly terms to (effectively) zero, by focussing
on the 2/3rds plane - if you can find it with sufficient accuracy (see below).

Rayleigh interference optics use, of necessity, a light source expanded in
precisely the plane which makes the aperture anomaly worse. To make it zero, you
need to focus on the mid-plane of the cell - which is also the plane at which
the fringes are in optimal focus. If you then stick to the Lloyd condition and
are not looking for precision below 0.02 fringes, you are OK.

But what happens if you increase the gradient, and reduce the optical pathlength
? From the look of the equation (which in both terms involves the pathlength to
one order greater than the gradient, this should be beneficial. HOWEVER - this
is assuming that one can hit the desired focal position with increased precision
- the condition here is in relative pathlength space, NOT absolute space. Or to
put it more simply, if you go from a 12 mm to a 3 mm cell, it gets 4 times
harder to find the plane you want. If you cannot hit the 2/3rds plane spot on,
then Wiener skewing will be serious at (say) 20 fringes/mm - and even if you do
hit it on the nail, then aperture effects cause fringe blurring, and with steep
gradients thats bad news for resolving them, especially with pixel-based
systems.

Even with a precision focussing mod - which I feel the XL-I should have - you do
not have it easy doing high conc/high gradient stuff using interference optics.

Of course this is a major reason why we have locally developed on-line Schlieren
optics, which have a precision similar to interference yet can easily cope with
any gradient short of one which sends the light outside the optics !

{we expect to have first results from a Schlieren XL-I within around 2 months
from now}

All best

Arthur Rowe

***************************************************
Dr Arthur J Rowe
Director
UK National Centre for Macromolecular Hydrodynamics
Leicester Laboratory
Adrian Building
University of Leicester
Leicester LE1 7RH    UK

Tel: +44 (0)116 252 3448
Fax: +44 (0)116 252 5602
ajr@leicester.ac.uk
***************************************************