Small$-$angle neutron scattering $($SANS$)$ was used to find the form factor $P(q)$ for a narrow

distribution poly$($ethylene$-$alt$-$propylene$)($PEP$)$ sample with $M=700000gmo{l}^{-1}$ in the

melt. The plot below is in a Kratky format where the experimental values of $P(q)$ are

multiplied by $q^2$ and $N.$

a$.$ First, make a reasonable estimate of $R_g$ for this polymer without using the

scattering data, to get a sense of the $q{R}_g$ range where most of the data lie. For

PEP, $M_0=70gmo{l}^{-1}$ and $b=8.$

b$.$ Take the appropriate limit of the Debye function to find the very simple asymptotic

result in this regime.

c$.$ Use this result and the data to determine the statistical segment length for PEP.

d$.$ Calculate the characteristic ratio for PEP.