Interpretation of the SANS Data

5.7.4. Interpretation of the SANS Data

In a so-called homogeneous glass, the concentration fluctuations, which influence the spontane- ous formation of ‘clusters’, are uncorrelated. Therefore, they should lead to spatially and temporally random appearance of the nucleation centres throughout the glass matrix. The growth of the new crys- talline phase, however, modifies the solute ‘super-saturation’ ratio in the neighbourhood of growing particles by forming a ‘depleted zone’ extending from the particle surfaces [43]. The concentration at the surface maintains a dynamic equilibrium with the particle radius, but the gradient and the extent of the ‘depleted zone’ depend upon the diffusion rate of solute material and the age of the particle. It has been suggested that the presence of the ‘depleted zones’ can deform both the ‘spatial and temporal randomness’ of the nucleation process by lowering the probability of nucleation near existing nuclea- tion sites, thereby creating short range order between the particles and eventually a saturation of the particle number density occurs [44].

When the diffusion rates are high, the ‘depletion zones’ can overlap to fill up the entire space available. All the particles then become influenced by the combined concentration profiles set up by the neighbouring particles. When the local concentration falls below a critical level, some particles will redissolve. The most affected will be the smaller particles close to or surrounded by the larger neigh- bours [5, 44]. This simply implies that, since the concentration of solute is greater around smaller parti- cles than around larger ones, there is a diffusion of solute to the larger particles from the smaller ones, and these will eventually redissolve [5].

Let us mention very briefly about the theory of small angle neutron scattering (SANS), which is relevant for a study of nano particles, as described below :

By considering the fluctuations of electronic density, it is known that the local density [ ρ(r)] of a material is not uniform in the microscopic scale, but it shows a small deviation with respect to the average density ( ρ 0 ) as :

ρ(r) = ρ 0 + Δρ

It is convenient to decompose the fluctuations in their spatial Fourier components Δρ x as :

⎛⎞ 1 Δρ x = ⎜⎟ Δρ ( ) exp ( r − ix r dr .)

⎝⎠ V ∫ v

where V is the volume of the system. For a dilute system of N identical particles of volume V, having a uniform electronic density ρ′

contained in a matrix of density ‘ ρ’, the intensity of the scattered radiation at small angles as a function of the scattering vector Q = 4 π sin θ/λ (where θ = scattering angle and λ = wavelength), is written as :

(5.19) where, I e is the scattered radiation by an electron and Δρ = (ρ – ρ′) is the difference of the electronic

I(Q) = I ( Δρ) 2 e NV 2 (1 – 1/3 R 2 G .Q 2 + . . . .)

densities of the particles and the glass matrix. The form of the scattering curve , i.e. I(Q) depends on the electronic gyration radius R G , defined

by the density ρ(r) as follows :

r 2 ρ () r dr

ρ () ∫ r dr

MAGNETIC PROPERTIES

215 For a homogeneous particle of density ρ(r) = ρ of radius R S , it can be written as :

R G =R S (5.21)

The intensity, I(Q), is represented by Guinier approximation as :

(5.22) However, as shown later in this case, the main feature of the SANS spectrum is a well-defined

I(Q) = I e ( Δρ) 2 NV 2 exp(– 1/3 R 2 G .Q 2 )

peak, which is characteristic of the ‘inter-particle interference’ effects [2, 4, 45]. In the absence of the long-range order, the curve of I(Q) at Q > Q max is not strongly modified by the inter-particle interfer- ence, but it is characteristic of ‘scattering’ from the isolated particles for which the mean particle size

can be determined within the limits of “Guinier” approximation, i.e. Q R G < 1.2, from the slope of the

linear plot of ln I(Q) vs. Q 2 .

From the maximum of the ‘scattering curve’, a characteristic wavelength ( Φ = 2π/Q max ), which is interpreted as the ‘mean distance’ between the small precipitates, and hence a relative ‘number den- sity’ of particles can be obtained (N = Φ –3 ) [2, 4, 45]. Therefore, the above three parameters (R G , Φ, N) can be obtained from the simple evaluation of the scattering curves of the SANS spectrum. So, now we know : what we can get from SANS study on nano materials ?