section summarizes what $g^{(1s)}(q,t)$ and $g^{(1)}(q,t)$ actually reveal about particle displacements. Extended derivations have appeared previously in two earlier papers, refs and , and are not repeated here. For probe diffusion, the intermediate structure factor is always determined by the displacement distribution function, namely the average in eq \[eq:g1sandr\] can be written as $$g^{(1s)}(q,t) = \int_{-\infty}^{\infty}  d(\Delta x)  \exp(i q \Delta x) P(\Delta x, t).
        \label{eq:g1sPDelta}$$ $P(\Delta x, t)$ is taken to be properly normalized.

On taking a Taylor series expansion of the exponential in powers of $q$, reflection symmetry, namely $P(\Delta x, t) = P(-\Delta x, t)$, eliminates all terms odd in $q$. As a result, $g^{(1s)}(q,t)$ and its logarithm are necessarily power series in $q^{2}$. The coefficients of the $q^{2n}$ are generated by the even moments $\langle(\delta x)^{2n}\rangle$ of $P(\Delta x, t)$. As shown previously[@phillies2005a], the lead terms of an expansion for $g^{(1s)}(q,t)$ are $$g^{(1s)}(q,t) = N \exp\left( - \frac{1}{2} q^2 \langle \Delta x(t)^{2} \rangle + \frac{1}{24} q^{4}( \langle \Delta x(t)^{4} \rangle - 3\langle \Delta x(t)^{2} \rangle^{2}) - {\cal O}(q^{6})\right).
      \label{eq:g1sanddisplacements}$$ All even moments $\langle(\delta x)^{2n}\rangle$ are required for the complete expansion.

It was early shown that quasielastic scattering from a binary solvent: macromolecule system determines the mutual 