Small Angle Neutron Scattering

5.7.3. Small Angle Neutron Scattering

The ‘Small Angle Neutron Scattering’ (SANS) technique shows some advantages in following the ‘growth process’ of the precipitates in the bulk materials. Three particular advantages are generally exploited :

1. The range in scattering vector (Q), which is possible to be studied, include the region around Q = 5 × 10 –4 /nm, which usually falls between the scope of Small Angle X-ray Scattering (SAXS) and Light Scattering (LS) measurements.

2. The low absorption of neutrons even at long wavelengths (~ 1 nm) allows an easy investiga- tion in the transmission of a bulk glass, which is heated directly in situ in the neutron beam,

i.e. a dynamic on-line study.

3. The scattering contrast between the precipitates and the host glass matrix, particularly glasses containing transition metal TM oxides, like a basalt glass.

The striking effects have been revealed by SANS study in different glass-ceramics like basalt, cordierite etc. [2] during the early stages of the growth of the precipitates, i.e. the formation of just ‘nano’ particles. As a result of the continuous viscosity-temperature relation, the rate of crystal forma- tion in glasses can be put on a ‘favourable’ time-scale to follow the ‘kinetics of growth’ simply by working at appropriate temperature.

The ‘Small Angle Neutron Scattering’ (SANS) technique was first used by Roth and Zarzycki [42] on samples heat-treated at different temperatures, and by Bandyopadhyay and Zarzycki [3] for the dynamic study of nucleation and crystallization behaviour at the high-flux reactor facilities created at Institut Laue Lngevin (Grenoble, France) for such studies to be able to compare with those of SAXS data on various systems, and also for the early nucleation and crystallization studies, when the size of

213 the nuclei or crystallites are in the very small nano range [2-4]. The readers with a knowledge of Fourier

MAGNETIC PROPERTIES

transform should appreciate the following part on the theoretical considerations for SANS study, where we take a real example of a material.

How Fourier transform comes into this picture ?

The intensity of the scattered radiation in both cases of X-rays and Neutrons at small angles as a function of the scattering vector k is proportional to the square of the modulus of the Fourier transform of the ‘fluctuations’ of the local ‘scattering power density’, P(r). As a real example, for a two compo- nent system like soda-silica glass, it is expressed as :

P(r) = N(r){C(r) b 1 + (1 – C(r)b 2 } – <N(r){C(r) b 1 + (1 – C(r)b 2 }> (5.12) where, b 1 and b 2 are scattering coefficients for stoichiometric silica and soda units respectively, C(r) is

the local mole concentration in silica, and N(r) is the local value of the number of the stoichiometric units per unit volume. Since silica and soda units have the same molar mass (M ≈ 60), N(r) can be simply taken as proportional to the local density ρ(r) as :

(5.13) where, N = Avogadro’s number. The term in the bracket < > represents the ‘average’ value of the first term over the volume of

N(r) = (N/M) ρ(r)

the sample. In the case of X-rays, it can be readily shown that b 1 =b 2 , since both units of silica and soda have the same number of electrons (i.e. 30). Hence, P(r) is reduced and can be written as :

P(r) = (N/60) [ ρ(r) – <ρ(r)>] b 2 (5.14) Thus, in the soda-silica glasses, the SAXS only measures the ‘density fluctuations’, and it is

insensitive to the ‘concentration fluctuations’. In the case of neutrons, the values of b 1 and b 2 are given as :

b 1 = 1.576 × 10 –12 cm

and,

b 2 = 1.280 × 10 –12 cm

Hence, the SANS depends on both density and molar concentration fluctuations, which are ex- pressed as :

P(r) = (N/60)[ ρ(r) – <ρ(r)>]b 2 + (N/60)[ ρ(r)c(r) – <ρ(r)c(r)>](b 1 –b 2 ) (5.15) Thus, a comparison of SAXS and SANS data could separate the two effects. The solid states

actually represent ‘bounded atomic association’, and the diffracted intensity is a consequence of the coherently scattered wave interference. The neutron scattering can be coherent (both elastic and inelas- tic) and incoherent. By neglecting the incoherent term, the differential cross section for neutron coher- ent scattering is given for a general case by the Fourier sine transform as :

(d ∞ σ/dΩ)

coh = N<b> 2 {1 +

4 π r 2 [ ρ(r) – <ρ(r)>] sin (Kr)/Kr dr}

The term in the large bracket involves the structure factor incorporating the concentration and density terms of the individual scattering atoms. It is seen from this equation that the RDF of atomic density [4 πr 2 ρ(r)] is the Fourier transform of the diffraction intensity. Then, from the Fourier inverse transform, the density and concentration fluctuations (local) can be estimated in an amorphous material with an interface of phase separation. Hence, the importance of Fourier transform is again seen for SAXS and SANS problems.

NANO MATERIALS