Perfect Crystal Diffractometer for Ultra Small-Angle Neutron Scattering
Slit Smearing:
The narrow q-resolution produced by the double crystal diffractometer exists only in the horizontal direction. The analyzer collects neutrons over several degrees in the vertical direction. The vertical Q resolution is +/- ΔqV = 0.117 Å-1. The slit smeared cross-section can be calculated by
After making standard background corrections and absolute scaling the data, the above slit smeared cross-section dΣ S/dΩ(q) is obtained. The desmeared cross-section can than be obtained by several different inversion methods. All these methods exhibit instability in the solution exhibited in all inversion problems.
Guinier-Law Scattering:
The Guinier approximation to scattering in the central region predicts scattering following
where RG is the Guinier radius. If the above holds for all q, the slit smeared cross-section is
In the limit of ΔqVRG >> 1 , the above relation simplifies to
Slit smearing does not change the Guinier radius, but does reduce the cross-section by the factor of
.In practice, the Guinier approximation never extends to large enough q for this approximation to be truly valid. This approximation is still useful for a quick "first order" determination of RG and dΣS/dΩ(0).
Calculating SAS Intensity:
In planning your experiment, it is useful to calculate the expected detector count rate based on your estimation of the scattering cross-section dΣ/dΩ(q). See the previous section for correcting for slit smearing. The detector count rate as measured by the main detector is
where εD is the detector efficiency, Ibeam is the count rate of the neutron beam on the sample, T is the sample transmission, dS is the sample thickness, and ΩAnalyzer = 2.7x10-7 Sr is the solid angle collected by the analyzer.
The detector efficiency corrected beam intensity εDIbeam is the count rate measured from the empty beam with analyzer set at q=0.
Example: Latex spheres in heavy water
A 0.1 vol. % solution of 1.0 μm diameter polystyrene latex spheres is suspended in D2O. The scattering length density contrast is Δρ = 4.95x1010 cm-2. The sample is placed in a dS = 1.0 cm thick cell, having a transmission T = 0.58 x 0.89 = 0.52. To maximize the intensity, the larger USANS liquid cells are used with 44 mm diameter sample aperture yielding a beam intensity εDIbeam = 57,000 s -1.
Guinier Approximation:
The forward cross-section at q=0 is calculated to be dΣ/dΩ(0) = 1.28x106 cm-1Sr-1. Slit smearing reduces the cross-section by a factor of 93 to dΣ/dΩ(0) = 1.37x10 4 cm-1Sr-1 (see slit smearing section for determination of factor). The scattered intensity is than calculated as I(0) = 109 s-1, a suitably strong signal.
Power-law Approximation:
The slit smearing of the Porod region can be calculated using the power-law approximation described in the slit smearing section. The interfacial surface area of the particles per unit sample volume SV = 6Φ/D = 60 cm-1, where is the volume fraction and D is the particle diameter. The Porod region scattering is than dΣ/dΩ(q) = [2π(Δρ)2SV]q-4. The slit smeared cross-section is dΣS/dΩ(q) = (1.96x10-7Å-3cm-1)q -3. The scattered intensity is IS(q) = (1.57x10-9Å-3s-1)q -3. At q = 0.001 Å-1, the scattered intensity IS(q=0.001 Å-1) = 1.57 s-1, a factor of 20 above the instrument background.
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Last modified 21-March-2007 by website owner: NCNR (attn: )
