From: Arthur Rowe <arthur.rowe@nottingham.ac.uk>
  To  : rasmb@alpha.bbri.org
  Date: Mon, 23 Oct 2000 16:54:11 +0000

squashes, pathlengths & aberrations

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Greetings all !

A few points which I think need making on the topic of large gradients and
optical aberrations:

The problem arises because a dear, familiar assumption, which is incredibly
convenient to make, happens to be an approximation only. The assumption is
that the change in optical pathlength (delta-S) between the 2 cells
(solution and solvent) in a Rayleigh interferometer is related in a linear
fashion to the refraction difference (dn = n - n0) between the contents of
the two cells (at given r, of course). 

i.e.  delta-S  =  a * (n - n0)   . . . . .  where a is the optical
pathlength in the cell(s).

Well, that isn't the case.  As Svensson showed a long time ago, there are
2nd and 3rd order terms in a to be considered, which involve the refraction
gradient  (dn/dr) to the 1st and 2nd power respectively. Both terms involve
a variable r, which is the fractional distance of the plane of focus from
the entrance window of the cell, divided by the optical pathlength of the
cell. So for r = 0.66667 - the so-called "two-thirds" plane, the physical
plane of focus will be located at 8 mm into a 12 mm pathlength cell.  The
second order term goes to zero at r = 0.5:  the third order term ditto at r
= 0.666667. 

Does this matter ?  Well, the amount of serious consideration which has been
given to the issue is minimal.  Peter Lloyd, years ago, reckoned that if you
were working with a resolution of 0.02 fringe, then the 3rd order term
becomes significant at 7.6 fringes/mm gradient.  Many people today use
better resolution and steeper gradients than that. However - he was assuming
the optics were focussed on the mid-plane of the cell, whereas most everyone
now would try for the 2/3rds plane. 

So - what happens when you check out these higher order terms,  and can you
get around the problem by using shorter optical pathlengths (as Walter and
Allen have just asserted) ?  I have dusted off some old calculations and
done a bit of tidying up and checking out - here are the results:

(1) Peter Lloyd's conditions are not totally clear, but basically I get the
same result as he does under his conditions. So my coding up seems to check
out.
...
(2) If it assumed we can hit the 2/3rds plane of focus within 0.5 mm, then
for a low-speed equilibrium run, SIGMA = 2, the errors caused by the 3rd
order term in optical pathlength do not exceed 1.5%, near the base of the
cell (10 fringes/mm in a 2 mm column)
...
(3) . . . but at intermediate/high speed equilibrium, with SIGMA = 6, we are
talking up to 9% error at 25 fringes/mm
...
(4) it is true that if  one considers the case of given, constant dn/dr then
as the pathlength goes down, the above errors go down rapidly. However, this
is not the way one actually works. The whole purpose of using shorter
optical pathlengths is to be able to use more concentrated solutions !  When
you code up on this basis, assuming that dn/dr values go up linearly with
concentration used, which itself changes with (1/pathlength), the you find
there is no actual advantage at all, so far as the aberrations are
concerned.  Obviously - as Allen Minton points out - you do of course then
at least get some light through, and  get a pattern, which you can interpret
it as you will.
...
(5) if anyone has persevered this far, they will have noticed my silence on
the second order term. This partly because I do not have a decent value to
hand for the objective aperture (in the plane of extension of the
interference slit) of the XL-I, but chiefly because I am still trying to
understand how its possible for two terms which are algebraically additive
can give rise to different physical effects !!  I will just say that using a
plausible value for beta (objective aperture) the 2nd order term is not
negligible, and can vary in a rather weird way.

So - how does all this affect the results you get from analyses of AUC runs
?  I guess I really ought to do some simulations here, and get the whole lot
published, if anyone out there in the journal world will take it.  The only
point I want to make at the moment is that you CANNOT simple say these are
negligible effects. 

Oh, and regards absorption optics, I am grateful to Tom Laue for mailing to
me the comment that dn (and thus dn.dr) is wavelength dependent, so life is
even that little bit more complicated !

Enough for now - I think I feel the need for a large beer coming on.

All best to everyone out there

Arthur

*****************************************************
Arthur J Rowe
Professor of Biomolecular Technology
University of Nottingham
School of Biological Sciences
Sutton Bonington
Leicestershire LE12 5RD   UK

Phone/voicemail       +44 (0)115 951 6156
Phone/fax             +44 (0)115 951 6156/7
email                 arthur.rowe@nottingham.ac.uk
                      arthur.rowe@connectfree.co.uk
Web                   http://www.nottingham.ac.uk/ncmh/business
*****************************************************

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<HTML>
<HEAD>
<TITLE>squashes, pathlengths & aberrations</TITLE>
</HEAD>
<BODY BGCOLOR=3D"#FFFFFF">
<FONT SIZE=3D"2">Greetings all !<BR>
<BR>
A few points which I think need making on the topic of large gradients and =
optical aberrations:<BR>
<BR>
The problem arises because a dear, familiar assumption, which is incredibly=
 convenient to make, happens to be an approximation only. The assumption is =
that the change in optical pathlength (delta-S) between the 2 cells (solutio=
n and solvent) in a Rayleigh interferometer is related in a linear fashion t=
o the refraction difference (dn =3D n - n0) between the contents of the two ce=
lls (at given r, of course). <BR>
<BR>
i.e.  delta-S  =3D  a * (n - n0)   . . . . .  where a is the optical pathleng=
th in the cell(s).<BR>
<BR>
Well, that isn't the case.  As Svensson showed a long time ago, there are 2=
nd and 3rd order terms in a to be considered, which involve the refraction <=
/FONT><I>gradient</I><FONT SIZE=3D"2">  (dn/dr) to the 1st and 2nd power respe=
ctively. Both terms involve a variable r, which is the fractional distance o=
f the plane of focus from the entrance window of the cell, </FONT><I>divided=
</I><FONT SIZE=3D"2"> by the optical pathlength of the cell. So for r =3D 0.6666=
7 - the so-called "two-thirds" plane, the physical plane of focus =
will be located at 8 mm into a 12 mm pathlength cell.  The second order term=
 goes to zero at r =3D 0.5:  the third order term ditto at r =3D 0.666667. <BR>
<BR>
Does this matter ?  Well, the amount of serious consideration which has bee=
n given to the issue is minimal.  Peter Lloyd, years ago, reckoned that if y=
ou were working with a resolution of 0.02 fringe, then the 3rd order term be=
comes significant at 7.6 fringes/mm gradient.  Many people today use better =
resolution and steeper gradients than that. However - he was assuming the op=
tics were focussed on the </FONT><I>mid</I><FONT SIZE=3D"2">-plane of the cell=
, whereas most everyone now would try for the 2/3rds plane. <BR>
<BR>
So - what happens when you check out these higher order terms,  and can you=
 get around the problem by using shorter optical pathlengths (as Walter and =
Allen have just asserted) ?  I have dusted off some old calculations and don=
e a bit of tidying up and checking out - here are the results:<BR>
<BR>
(1) Peter Lloyd's conditions are not totally clear, but basically I get the=
 same result as he does under his conditions. So my coding up seems to check=
 out.<BR>
...<BR>
(2) If it assumed we can hit the 2/3rds plane of focus within 0.5 mm, then =
for a low-speed equilibrium run, SIGMA =3D 2, the errors caused by the 3rd ord=
er term in optical pathlength do not exceed 1.5%, near the base of the cell =
(10 fringes/mm in a 2 mm column)<BR>
...<BR>
(3) . . . but at intermediate/high speed equilibrium, with SIGMA =3D 6, we ar=
e talking up to 9% error at 25 fringes/mm<BR>
...<BR>
(4) it is true that if  one considers the case of given, </FONT><U>constant=
 dn/dr</U><FONT SIZE=3D"2"> then as the pathlength goes down, the above errors=
 go down rapidly. However, this is not the way one actually works. The whole=
 purpose of using shorter optical pathlengths is to be able to use more conc=
entrated solutions !  When you code up on this basis, assuming that dn/dr va=
lues go up linearly with concentration used, which itself changes with (1/pa=
thlength), the you find there is no actual advantage at all, so far as the a=
berrations are concerned.  Obviously - as Allen Minton points out - you do o=
f course then at least get some light through, and  get a pattern, which you=
 can interpret it as you will.<BR>
...<BR>
(5) if anyone has persevered this far, they will have noticed my silence on=
 the second order term. This partly because I do not have a decent value to =
hand for the objective aperture (in the plane of extension of the interferen=
ce slit) of the XL-I, but chiefly because I am still trying to understand ho=
w its possible for two terms which are algebraically additive can give rise =
to different physical effects !!  I will just say that using a plausible val=
ue for beta (objective aperture) the 2nd order term is not negligible, and c=
an vary in a rather weird way.<BR>
<BR>
So - how does all this affect the results you get from analyses of AUC runs=
 ?  I guess I really ought to do some simulations here, and get the whole lo=
t published, if anyone out there in the journal world will take it.  The onl=
y point I want to make at the moment is that you CANNOT simple say these are=
 negligible effects. <BR>
<BR>
Oh, and regards absorption optics, I am grateful to Tom Laue for mailing to=
 me the comment that dn (and thus dn.dr) is wavelength dependent, so life is=
 even that little bit more complicated !<BR>
<BR>
Enough for now - I think I feel the need for a large beer coming on.<BR>
<BR>
All best to everyone out there<BR>
<BR>
Arthur<BR>
<BR>
*****************************************************<BR>
Arthur J Rowe<BR>
Professor of Biomolecular Technology<BR>
University of Nottingham<BR>
School of Biological Sciences<BR>
Sutton Bonington<BR>
Leicestershire LE12 5RD   UK<BR>
<BR>
Phone/voicemail       +44 (0)115 951 6156<BR>
Phone/fax             +44 (0)115 951 6156/7<BR>
email                 arthur.rowe@nottingham.ac.uk<BR>
                      arthur.rowe@connectfree.co.uk<BR>
Web                   http://www.nottingham.ac.uk/ncmh/business<BR>
*****************************************************<BR>
</FONT>
</BODY>
</HTML>

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