Derivation of expansion factor in Svergun's Crysol Paper

[phoenix]

Dear Crysol Users,

We have a question regarding the derivation of overall expansion factor in your Crysol paper (Svergun et al, 1995, J. Appl. Cryst.). While we tried to derive the overall expansion factor [Eqn 13] from the dummy atom form factor equation [Eqn 12] in Crysol paper, we got a slightly different constant factor rather than the one reported in Crysol paper. The constant factor (4pi/3) in [Eqn 13] is raised to the power of (3/2) in Crysol paper, but when we tried to derive the same expression we got (4pi/3) raised to the power of (2/3). Please refer the attached file for details. We tried to imagine the possibilities, either this must have been a typo error or the power of the constant factor is empirically adjusted (parametrized) to get a good fit with the experimental data. We would be very thankful if someone could help. Thank you.

Sincerely
prabha

[HLee]

I heard that the CRYSOL paper has typo(s) on the equation(s).

[SaxsMax]

Indeed, it should be "2/3" rather than "3/2".
Please note however, that in the present Crysol application,
the linear term Vj is decoupled from the exponent.
In other words, Vj values of individual atomic groups are used for the linear term
and the exponent depend only on the effective atomic radius,
so that the exponential part of the overall expansion factor G(s) is abandoned.

gj(s) = Vj*exp(-1/(4PI)*s^2*(4/3PI)^(2/3)*r0^2)
where s = 4PI*sin(theta)/lambda.
Vj is defined by the atomic group type
and r0 in the exponential part varies from 1.4A to 1.8A during fitting.
The total volume is adjusted within the limits of
0.925*VDWVOL to 1.075*VDWVOL, where VDWVOL is a sum of individual Vj