Neutron Scattering

where rit denotes the monomer distance and P is the characteristic monomer length . This
is the same expression as for a random walk but with the time t replaced by I i - j I .
Because for not too small distances ri9 is the sum of many random variables the central
limit theorem is applicable and the final distribution of the distance is a Gaussian :
3/2 (_ 3ri j 2 \1I
g(ràj - ( 27r(ri j 2)) exp 2 (rij 2 )/ .
(5 .16)
Application of equation (5 .8) (radially averaged as (5 .10)) yields the so-called form factors
of the monomer
exp P(Q) = .~ 47r ri j 2d3ri j sin Qri~
JV2
9(ri j ) - -Q2 l i - j I~2) (5 .17)
Q i7
where N is now the number of mouomers . Analogous to the preceding derivation of (5 .9)
we take the diagonal part out of the sum and couvert the double sum into a single sum
over all differences k - Ii - j I :
P(Q) _
Cl + 2 ~ (1 -
N/
exp -Q 2 k~2 l
~ .
k--/1
(5 .18)
Here it is taken account that in contrast to (5 .9) not all pairs are equally probable but an
index distance k occurs 2(N-k) times in the chain .
one obtains
Converting this sum into an integral
P(Q) = z
(e-z - 1 + z) -- D(z) with z = Q26
~ 2 . (5 .19)
The expression D(z) is usually called the Debye function .
It describes the scattering of a
single polymer coil in the melt which is labeled e .g . by isotopic contrast . Figure 5 .6 shows
the scattering cross section of protonated polystyrene in a deuterated polystyrene matrix .
The solid curve represents a fit with equation (5 .19) .
At large scattering vectors the leading
asymptotic term of D(z) is 2/z and P(Q) becomes proportional 1/Q 2-characteristic
for a Gaussian random walk . In a so-called Kratky plot (Q2 - dor/dÇZ vs . Q) one expects
a plateau at high scattering vector Q . Figure 5 .7 shows this plateau for polystyrene .
At very large Q values deviations occur again which signalize the breakdown of Gauss
statistics for small distances .
s
Here the monomers are simply considered as "big atoms" neglecting their inner structure
.
One has to keep in miud that the thus obtained results only represent the actual
scattering law for small scattering vector when 27r/Q is larger than the size of a monomer .
5-9