Rg of Sphere
data
angle
30
35
40
45
50
55
60
65
70
75
90
105
110
115
120
125
130
135
Large
Small
210.2
132.8
147.7
120.1
103
103.5
73.3
90.9
42.6
80.2
25
71.7
13.7
63.8
5.7
58.4
1.9
51.1
0.9
46.8
2.9
32.6
5.3
28.6
5.2
27.3
4.8
27.1
5
25.8
5
25.6
5
25.7
5.2
25.4
The data above represent light scattering intensity measurements for two different
spheres, Large and Small. They are made on a conventional rotating arm instrument, so
the very first thing you have to do is multiply the intensities by sin(). That will give you
the “real” intensities that can be used for analysis. Solvent intensity has been subtracted
out already.
Arrange your answers around these numbered points. Make it easy for us to follow by
numbering sections of your answer!
1. Plot both data sets in the style of Guinier on a single plot (see Guinier HowTo:
http://macro.lsu.edu/HowTo )
2. Try to obtain Rg and R of both spheres. You may not be able to use all data points
at high angles because some may not adhere to the linear Guinier model equation
(the Guinier equation only applies to the initial slope of the plot).
3. For the smaller spheres, what is the uncertainty in slope as a percentage of the
slope itself? (100 slope/slope).
4. For the smaller spheres, what is the uncertainty in Rg as a percentage of Rg?
(Before you answer, please note: this is an unusual case in which propagation
behaves in a strange way; you have to work out the propagation and show us how
you did it!).
5. As noted in part (2) above, not all the data points may fit the linear Guinier
prescription, which is only valid at low angles (i.e., qRg < 1). Unfortunately, data
at very low angles are difficult to obtain, but it’s a shame to throw away the data
at high angles just because it doesn’t fit the simple Guinier theory. We don’t have
to! If we know the shape is spherical, then we can use all the data points instead
of throwing away ones the high angles. Do this by using Solver (or other
nonlinear least squares fitting package) to fit the data (after correcting for
scattering volume) to the particle form factor equation for spheres:
2
3

I (q)  I (0)  3 (sin x  x cos x)
x

where x = qR (not Rg) and I(0) is the intensity at zero scattering angle. You will
set up a solver to fit I(0) and R. You should also add a smooth curve showing the
fitted equation (of course, you need the solver-found values at the discrete points,
too). From the fit, you should be able to obtain better values of R (and then Rg,).
At least they should be better for the large sphere. Compare the Guinier and
solver values and make it easy for us to find that! Here is your chance to practice
what we learned about nonlinear least squares fitting a few weeks ago!
Grading Criteria
Plot
/4
Rg and R of both /6
Slope uncertainty /4
Propagate
/6
Solver
/10