Opened 4 years ago
Last modified 3 years ago
#973 new defect
sticky hard sphere fails for q<epsilon
Reported by: | pkienzle | Owned by: | |
---|---|---|---|
Priority: | minor | Milestone: | SasView 4.3.0 |
Component: | SasView | Keywords: | |
Cc: | Work Package: | SasModels Model Issues |
Description
Looking at the sticky hard sphere code it should converge in the limit of q → 0
Here is the calculation:
S(q) = 1 / (A^2 + B^2) => A^-2
for
A = 1 + 12 eta [A1 + A2 - A3] => 1 + 12 eta [alpha/3 + beta/2 - lambda/12] B = 12 eta [B1 + B2 - B3] => 0 A1 = alpha [(sin(qr) - qr*cos)/(qr)^3] = alpha/3 3j1(qr)/qr => alpha/3 A2 = beta [(1 - cos(qr))/(qr)^2] => beta/2 A3 = lambda/12 [sin(qr)/qr] => lambda/12 B1 = alpha [ 1/(2qr) - sin(qr)/(qr)^2 + (1-cos(qr))/(qr)^3] => 0 B2 = beta [ 1 / qr - sin(qr)/(qr)^2 ] => 0 B3 = lambda/12 [ (1 - cos(qr))/qr ] => 0
where
alpha = (1 + 2 eta - mu)/(1 - eta)^2 beta = (mu - 3 eta)/(2 (1-eta)) mu = lambda eta (1-eta) lambda = (-b - sqrt(b^2 - 4 a c))/2a a = eta/6 b = tau + eta/(1-eta) c = (1 + eta/2)/(1 - eta)^2 eta = Vf/(1-epsilon)^3
and
r = 2 Re / (1 - epsilon)
using the input parameters:
Re is radius_effective, Vf is volume fraction, epsilon is perturbation and tau is stickiness.
We already have an accurate calculator for 3 j1(x)/x and sin(x)/x
Need to check accuracy of the following for low q, possibly replacing them with the Taylor expansion
[1-cos(x)]/x^2 = 1/2 - x^2/24 + x^4/720 - x^6/40320 + ... 1/(2x) - sin(x)/x^2 + (1 - cos(x))/x^3 = x/8 - x^3/144 + x^5/5760 - x^7/403200 + ... 1/x - sin(x)/x^2 = x/6 - x^3/120 + x^5/5040 - x^7/362880 + ...
Change History (3)
comment:1 Changed 4 years ago by butler
- Milestone changed from SasView 4.2.0 to SasView 4.3.0
comment:2 Changed 3 years ago by butler
- Work Package changed from SasModels Redesign to SasModels New Model
comment:3 Changed 3 years ago by butler
- Work Package changed from SasModels New Model to SasModels Model Issues
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