Resolution function for small angle neutron scattering Khaled Gad Mostafa ashshoush supervisor Dr.Alexander Kuklin

Neutron sources

Since the early days of neutron scattering, there has been an insatiable demand for higher neutron fluxes. Neutron sources are based on various processes that liberate excess neutrons in neutron rich nuclei such as Be, W, U, Ta or Pb. Presently, the highest fluxes available are around a few n/cm 2 sec. Even though various neutron sources exist, only a few are actually useful for scattering purposes. These are: 1- continuous reactors 2- spallation sources 3- some other neutron sources.

Most of nuclear reactors is continuous fission mode reactor which neutrons are one of the fission products, the intensities of the neutron at the sample and the detector are as in the following fig 1-Continuous reactors

Universitatea din Bucuresti, Facultatea de Fizica, Septembrie 2008 High energy incoming particle (typically protons) Heavy metal target (Ta, W, U) Neutron cascade >10 neutrons per incident proton low power load per outgoing neutron (~ 55 MeV) 2.5 neutrons per event 1 neutron consumed in sustaining reaction 0.5 absorbed high power load per neutron (~ 180 MeV) Spallation Fragmentation

IBR-2 Pulst Fast ReactorREGATA

Advantage of neutron scattering 1- Neutrons interact through short-range nuclear interactions. They are very penetrating and do not heat up (i.e., destroy) samples. 2-Neutron wavelengths are comparable to atomic sizes and inter-distance spacing. 2- Neutron wavelengths are comparable to atomic sizes and inter-distance spacing. 3-Neutron energies are comparable to normal mode 3- Neutron energies are comparable to normal mode energies in materials (for example phonons, diffusive energies in materials (for example phonons, diffusive modes). Neutrons are good probes to investigate the modes). Neutrons are good probes to investigate the dynamics of solid state and liquid materials. dynamics of solid state and liquid materials.

Disadvantage of neutron scattering 1- Neutron sources are very expensive to build and to maintain. 2-Neutronsources are characterized by relatively low fluxes compared to x-ray sources (synchrotrons) and have limited use in investigations of rapid time dependent processes. 2- Neutron sources are characterized by relatively low fluxes compared to x-ray sources (synchrotrons) and have limited use in investigations of rapid time dependent processes. 3-Relatively large amounts of samples are needed: typically 1 mm-thickness and 1 cm diameter samples are needed for SANS measurements. This is a difficulty when using expensive deuterated samples or precious (hard to make) biology specimens. 3- Relatively large amounts of samples are needed: typically 1 mm-thickness and 1 cm diameter samples are needed for SANS measurements. This is a difficulty when using expensive deuterated samples or precious (hard to make) biology specimens.

Sizes of interest = “large scale structures” = 1 – 300 nm or more Mesoporous structures Biological structures (membranes, vesicles, proteins in solution) Polymers Colloids and surfactants Magnetic films and nanoparticles Voids and Precipitates

IBR – 2 Reactor YuMO Spectrometer: 1 – reflectors; 2 – chopper; 4,6 – collimator; 8 – sample table; 11,12 – detectors; 14 – direct beam detector Reactor parameters: Mean power 2 MW, in pulse 1500 MW Pulse frequency of 5 Hz Spectrometer parameters: Wavelength 0.5 A to 8 A Size range of object 500 A – 10 A Size of beam on sample 8 – 22 mm 2 Detectors of 3 He (home made) Detector for direct beam of 6 Li (home made) EQUIPMENT

Δθ Sizes of interest = “large scale structures” = 1 – 300 nm or more SANS Approach QSQS kiki kSkS SSD SDD ≈ S 1 ≈ 2 S 2 Optimized for ~ ½ - ¾ inch diameter sample 2 θ S1 3m – 16m 1m – 15m

A typical sans result is a graphic of the Scattering Intensity function of a wavevector Q Q is defined as where Q = wavevector Theta = scattering angle Lambda = Wavelength of incident beam The scattering intensity is defined as: Where I(Q) = scattering intensity Phi = density of particles in volume P(Q) = form factor S(Q) = structure factor SANS PRINCIPLE J. Texeira, Introduction to Small Angle Neutron Scattering Applied to Colloidal Science, Kluwer Academic Publishers, Netherlands, 1992

FORM FACTOR Concerns each particle and is related to its nuclear density Usually defined as: Where F(Q) is defined as: Rho is the density of scattering length of the sample It can be calculated using a simple formula (given here for heavy water): Where b is the scattering length of deuterium respectively oxygen STRUCTURE FACTOR Is related to the spatial distribution of the centres of mass Is usually defined as Where R is the position vector of a particle inside the compound FORM AND STRUCTURE FACTORS

Contrast variation is used when the sample being studied is made up of a series of compounds with close scattering lengths One of the most important advantages of SANS spectroscopy is the ability to change contrast by isotope substitution The most common form of substitution is changing hydrogen compounds with deuterium ones Another interesting situation appears when a mixture of normal and deuterated solvents are obtained in colloidal suspensions in such a way that the background scattering length of the solvent is “erased”. This is called contrast matching We can take as an example a sample containing three compounds and, using contrast matching, we can erase the contrast between two parts allowing us to analyze the third compound CONTRAST VARIATION METHODS

THE SANS INSTRUMENTAL RESOLUTION

Instrumental smearing affects SANS data. In order to analyze smeared SANS data, either desmearing of the data or smearing of the fitting model function is required

THE RESOLUTION FUNCTION

1. THE RESOLUTION FUNCTION Instrumental smearing is represented by the following 1D convolution smearing integral (suitable for radially averaged data): the 1D resolution function is defined as a Gaussian function: σ Q is the Q standard deviation.

2. VARIANCE OF THE Q RESOLUTION In order to express σ Q, differentiate Q on both sides: Take the square:

VARIANCE OF THE Q RESOLUTION Geometry part Wave part SANS resolution has three contributions Gravity part

SANS is a powerful method for condensed matter investigation for objects of sizes between 1 nm to 100 nm – therefore it can be considered a nanoscale procedure The IBR – 2 reactor at the JINR is adequate for SANS machine Several applications for SANS exist in the fields of Biology, Chemistry, Polymers, Ferrofluids, etc. The resolution of the device depends and the quality of the information depend on considering the errors due to wavelength,finite width of the detector cell,finite time of the detector and collimation system. Conclusion

The authors would like to acknowledge the following: Kuklin Aleksandr Ahmed Islamov Balasoiu Maria Raul Erhan All of the above from the YuMO Group, Condensed Matter Department We would also like to extend our regards to the organizer of this Practice and all members of the JINR involved with this project. AKNOWLEDGEMENTS

Thank you for attention!