Neutron Scattering

The probability that a freely joint chain with z monomers talçes a conformation with the endto-end
vector R, is determined by the Gaussian distribution according to
3/2 2
W R,z -
3 ex - 3R (14 .5)
(2Tczb2) pC 2zb 2 )
On the statistical average the mean end-to-end distance is zero and its
is linear proportional to the number of monomers z .
=0 =R~=b 2 z
mean square deviation
The radius of gyration is given as R 2 = R2 /6 . A linear "real" chain is different from a freely
joint chain in so far as neighboring monomers are correlated. This effect is considered by the
parameter c z in Rg = c z b 2z/6 or expressed by the statistical segment length according to
a =bF and the radius of gyration Rg =a2 z / 6 or respectively the mean square end-toend
distance R,2, =62 z determined from scattering experiments . So, a statistical segment
length of polystyrene is determined as 6Ps = 6.8A. As in Figure 14 .2 a realistic chains can
therefore be represented by a polygon with vectors of segment length a .
The form factor of a linear chain measured in a SANS experiment is determined from the
sum of the phase factors from the monomers and an averaging over all possible chain
configurations according to
D
z
p(Q ) =
z2 Y,< e iQR i _ R i) >
(14 .6)
The meaning of the vectors R becomes clear from Figure 14 .2 . In a macroscopically large
sample the number of polymers is sufficiently large, in order to describe the polymer
conformation with the probability distribution of Eq .(14 .5) . The average value of the phase
factors between the positions i and j within the chain is calculated according to
< expiQR;j >= fd 3 R ;jw(R ij Ji - jl)exp(l2llij )= WJ(Q )
For the form factor one gets
Wij(Q) = exp(-li- jl 62
Q2) = Wa ll-jl .
14-4