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Resolution function for small angle neutron scattering

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## Resolution function for small angle neutron scattering

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Resolution function for small angle neutron scattering

Khaled Gad Mostafa ashshoushsupervisor Dr.Alexander Kuklin

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 1015 n/cm2 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

- 2.5 neutrons per event
- 1 neutron consumed in sustaining reaction
- 0.5 absorbed
- high power load per neutron (~ 180 MeV)

- 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)

Universitatea din Bucuresti, Facultatea de Fizica, Septembrie 2008

REGATA

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.

3-Neutron energies are comparable to normal mode

energies in materials (for example phonons, diffusive

modes). Neutrons are good probes to investigate the

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.
- 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

EQUIPMENT

- 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 mm2
- Detectors of 3He (home made)
- Detector for direct beam of 6Li (home made)

YuMO Spectrometer: 1 – reflectors;

2 – chopper; 4,6 – collimator;

8 – sample table; 11,12 – detectors;

14 – direct beam detector

IBR – 2 Reactor

Sizes of interest = “large scale structures” = 1 – 300 nm or more

kS

QS

SANS Approach

ki

2 θ

S1≈ 2 S2

DETECTOR

S1

Δθ

3m – 16m

1m – 15m

SSD

≈

SDD

Optimized for ~ ½ - ¾ inch diameter sample

SANS PRINCIPLE

- 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

J. Texeira, Introduction to Small Angle Neutron Scattering Applied to Colloidal Science, Kluwer Academic Publishers, Netherlands, 1992

FORM AND STRUCTURE FACTORS

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

CONTRAST VARIATION METHODS

- 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

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

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:

σQis the Q standard deviation.

2. VARIANCE OF THE Q RESOLUTION

In order to express σQ, differentiate Q on both sides:

Take the square:

SANS resolution has three contributions

Geometry part

Wave part

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 .

AKNOWLEDGEMENTS

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.

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