The Resolution of Small Angle
Neutron Scattering (SANS):
Theory and the Experimental
Authors:
•E. L. Maweza (University of Fort Hare in SA)
•A. KUKLIN (Supervisor: JINR in Dubna)
Table of Contents
1.
2.
3.
4.
5.
6.
7.
8.
9.
Introduction
Theory and Literature Review
Experimental Setup
Description of the Equipment
Sample Characterization
Experimental Procedure
Results and discussion
Conclusion
Acknowledgements
Introduction
• The choice of Small Angle Neutron Scattering (SANS) as a
technique to investigate the structure of materials was
based at its efficiency in determining their structural
properties at length range 10 to 1000 Å.
• The SANS experiments require a wide range of
momentum transfer (Q range) to determine reliable
structural properties of materials.
• Frank’s Laboratory for Neutron Physics currently uses a
modernized two-detector (“old” and “new”) system in order
to increase the Q-range of the instrument.
Theory and Literature Review
• The intensity of the scattered neutron beam is
given by I Q  PQS Q
– P(Q)
:
Periodicity function – Form factor.
2
 dV
• Definition: PQ  F Q , where F Q  





e

– S() :
Inter-particle function - Structure Factor.
i Q. r
0
ParticleVolume
• Definition:
S Q  
exp iQ  R  R  

 
and Q( ,  ) 
,
k1
k2
4

k2
Q
k1
Figure: 1: Schematic representation of a scattering experiment.
sin

2
Bragg’s equation for crystallite
periodicity and size
• Bragg’s Equation is given by

n  2d sin
2
• Combining the Bragg’s equation with momentum
transfer we obtain the periodicity of the crystal.
2
d
n
Q
• The size of the crystallite is given by the DeBye’s
equation. D  k
w cos( / 2)
• The wavelength of the neutrons is given by
t
  3.85 
L
Experimental Setup
1.
The two-reflector
system.
2.
The reactor with the
moderator.
3.
The chopper.
4,5. The first collimator.
6,7. Vacuum cube.
8.
The second collimator.
9,11. Table for the sample
holder , sample holder
10. The water bath
thermostat
12,14.Vanadium Standards
13. First detector
15,16. Second detector
17. The direct neutron beam
detector
Figure 2: The two detector YuMO spectrometer
http://flnp.jinr.ru/135/
Description of the Equipment
• The YuMO two-detector system uses 8 homemade
ring wire detectors with central holes:
– Old detector : 200 mm central hole.
– New detector : 80 mm central hole.
• SANS experiments are carried out in two stages.
– The study of the sample in the beam without vanadium
standards.
– The sample with both vanadium standards in the beam.
Sample Characterization
• The sample for this project is Silver Behenate
powder (“AgBE”).
– Chemical Formula: CH 3 CH 2 20 COOAg 
– Made up of small plate-like crystals
– Surface dimensions: (0.2-2.0 µm) and thickness ≤ 1000 Å
• The long-period spacing obtained from literature is
given by d001  58.378 Å.
Experimental Procedure
• The primary aim is to obtain periodicity and the
size of the AgBE crystallite.
• Origin data analysis program was used to treat the
results obtained from the SANS program.
• The data obtained and analyzed covers the neutron
scattering observed by detectors from the 2nd to the
7th ring.
• The peaks occur where the diffraction of the AgBE
crystals take place.
Results and Discussion
3.0
2.5
Gauss fit of BelOr4sm_B
Gauss fit of BelOr4sm_B
Intensity, cm
-1
2.0
1.5
1.0
0.5
0.0
-0.5
0.0
0.1
0.2
0.3
0.4
0.5
-1
Q, (Å )
Figure 3: Illustration of the periodicity of AgBE by Lorentz
Approximation.
0.6
Periodicity by Gaussian
Approximation
Literature Values
Gaussian Approximation: Peak 1
Gaussian Approximation: Peak 2
60
59
58
Periodicity, (Å)
57
56
55
54
53
52
51
50
2
3
4
5
6
Ring No.
Figure 4: Illustration of the periodicity of AgBE by Gaussian
Approximation.
7
Periodicity by Lorentz
Approximation
Theoritical Values
Lorentz Approximation: Peak 1
Lorentz Approximation: Peak 2
60
Periodicity, (Å)
58
56
54
52
50
2
3
4
5
6
Ring No.
Figure 5: Illustration of the periodicity of AgBE by Lorentz
Approximation.
7
The size of the AgBE crystal
8000
Average Size of Crystal, Å
7000
DGi1A
DLi1B
6000
5000
4000
3000
2000
1000
0
2
3
4
5
Ring No.
Figure 6: Illustration of the size of AgBE crystallite.
6
7
Analysis
• Periodicity values that are in agreement with values
obtained by other authors were expected for AgBE
because it has been adopted as calibration standard.
• The considerable deviation that was observed is
attributed to systematic errors like:
– Time of delay must be calculated more precisely
• (not by “vision” as we did.)
– Asymmetry of the peaks (as shown in figure 3).
• The size of the crystallite clearly becomes constant
for bigger rings showing better resolution.
Conclusion
• The characteristic parameters of AgBE were
obtained.
– The periodicity ≤ 58 Å
– The size of crystallite was about 7300 Å.
.
• It was shown, that time of delay obtained from
raw spectra must be corrected.
• In this case we have good agreement with another
authors.
• The AgBE is suitable as calibration sample.
Conclusion
• Standard procedure of SAS program gives us the
Gaussian resolution value.
• Both the Gaussian and Lorentz distribution is
suitable for low resolution of SANS method.
• For averaging data using Gaussian distribution one
should be careful.
References
• Teixeira, J. (1992) “ Introduction to Small Angle Neutron Scattering Applied to
Colloidal Science”. Structure and Dynamics of Strongly Interacting Colloids and
Supramolecular Aggregates in Solution. Kluwer Academic Publishers.
• Cser, L.(1976)”Investigation of Biological Macromolecular Systems With Pulsed
Neutron Source- A Review”. Brookhaven. Symp. Biol. (27) VII3 – VII29.
• Keiderling, U., Gilles, R., Wiedernmann, A., (1999) “Application of Silver Behenate
Powder for the Wavelength Calibration of a SANS instrument- a comprehensive
study of experimental setup variations and data processing techniques”. J. Appl.
Cryst., 32., 456 – 463.
• A. J.Kuklin, A. KH. Islamov, V. I., Gordelly (2005), Two-Detector System for SmallAngle Neutron Scattering Instrument. Neutron News. V. 16, 16 -18pp
Acknowledgements
1. JINR SA Representation (Dr. Jacobs and Prof.
Lekala)
2. YuMO Team
i.
ii.
iii.
iv.
v.
Raul Erhan
Oleksandr Ivankov
Dmitry Soloviov
Andrey Rogachev
Yury Kovalev
Helpful definitions
w  wresolution  wideal
wresolution
D
Cons tan t

k
w cos( / 2)