Neutron Scattering - JUWEL - Forschungszentrum Jülich

KWS-1 & KWS-2 5
mass of the protein is 1.43 × 10 4 g/mol. The number density of the protein is therefore n/V =
0.02g/cm 3 /(1.43 × 10 4 g/mol) =1.40 × 10 −6 mol/cm 3 =8.42 × 10 −7 ˚A −3 . For a simple cubic
packing the typical distance is given by ℓ = 3� V/n. For a hexagonal packing the typical
distance is ℓ = 6√ 2 3� V/n. Both calculated distances are to be compared with the measured
one.
7 Theoretical Background for Cylindrical Micelles
Now a formula for the macroscopic scattering cross section of an infinitely long cylinder with
a finite radius will be discussed. This cylinder is assumed to be infinitely long, because the
experimental Q-window is not sufficient to observe the entire particle as a whole.
So our scattering formula does not have a Guinier region for the entire cylinder. Moreover, the
question about the orientation of the cylinder arises, because this particle is no longer isotropic.
Simple sample preparations (like in the case of the lab course) do not impose a preferred orientation
of the particles and they are oriented arbitrarily and therefore scatter isotropically. The
presented macroscopic scattering cross section was calculated with an orientational average,
which however will not be discussed in detail. One finds:
dΣ
dΩ (Q) =(Δρ)2 · φcyl · Acyl · π
Q ·
�
2 J1(QR)
�2 QR
You get individual factors with the following meanings: (a) contrast, (b) the concentration of
the cylinders, (c) the cross-sectional area of the cylinder, (d) a form factor part for infinitely
long cylinders and (e) a form factor part of the shape of the cross-sectional area, which is
simply a circle. Since the Guinier region of the entire cylinder slides out of the Q-window, the
observable particle volume is reduced to a cross section Acyl of the cylinder. In addition, the
total form factor can be expressed as the product of two terms. This is possible if the entire
structure can be expressed as a convolution of two simpler structures. In this case, this is an
infinitely thin infinitely long cylinder and a flat disk perpendicular thereto. The convolution is
a cylinder with a finite cross section. The infinitely long cylinder is described by the term π/Q.
While having strong oscillations for finite lengths, in the limiting case this factor yields just a
power law – similar to the Porod-law. The orientation-averaged disc has a rather complicated
expression with a first-order Bessel function. The first zero is at Q =3.832/R. This factor can
also be regarded as a separate form factor with its own Guinier region. In this case, eq. 1 would
be written as follows:
dΣ
dΩ (Q) =(Δρ)2 · φcyl · Acyl · π
�
· exp −
Q 1
4 Q2R 2
�
These two principal areas of Guinier regions derive from a scale separation. On the one hand,
the cylinder is very long, and the first Guinier region disappears at very low Q. On the other
hand, one finds a Guinier region of a circular disk of radius R in our Q-window. Both Guinier
regions are clearly separated in reciprocal space. It is just that the observed Guinier region is
covered by an additional form factor for an infinitely long, infinitely thin cylinder. The radius of
(1)
(2)