Punto de Equilibrio

In a crystalline material, if the Bragg condition is satisfied for equivalent atoms in adjacent unit cells, then it must also be satisfied by the equivalent atom in every other unit cell, and constructive interference results. If it is not satisfied, even by a very small fi^action, then completely destructive interference will result: assuming that there are a large number o f unit cells then for every atom there will always be an atom positioned to scatter completely out o f phase with the first. Coherent scattering therefore only occurs in a series o f sharp peaks (the Bragg reflections). These will be seen at regular intervals in Q space corresponding (equation 4.11) in real space to the separation o f planes o f atoms within the unit cell. Peaks are denoted by {hkl) where h, k and / are the Miller indices which give the orders with respect to each axis o f the unit cell.

The intensity o f each Bragg reflection is dependant on the phase relationship o f all the atoms in the unit cell, which in turn depends on their position along the axis o f the cell parallel to Q. The summation o f the wavefimctions for scattering fi’om all atoms in the unit cell is given by (fi’om equation 4. lb)

S(.0) = 'LKe‘^"

(4 1 2)

n

■^(0 =

COS(0„) +/

S“ ( 0 n )

(4 13)

«

We can also express this as an integral over the length o f the unit cell

d

^ iS ) =

J

0

where z is the position in a unit cell o f length d and p(z) is the scattering density profile. If the unit cell has a centre o f symmetry, which is generally the case for clay minerals, then the sine term must equal zero. This is because the sine term is anti-symmetric, so the integral o f this term over the first and second halves o f the unit cell will have the same magnitude but opposite signs. In this special case equation 4.14 simplifies to

d

S{Q ) = J p{z)qos{Qz) dz (4.15)

0

This relationship between the scattering density profile and the structure factor is illustrated in figure 4.4 and the accono^anying text. The coherent scattering, which is also called the Bragg intensity, is the amphtude o f this wave

I{Q ) = S \ Q ) S{Q ) (4,16)

From equation 4.16 it follows that the observed coherent scattering must always be positive. As a result o f this we can never observe the sign o f the structure factor. Furthermore, because constructive interference only occurs at the Bragg reflections we can only observe the magnitude o f the structure factor at values o f Q for which the Bragg condition is satisfied. Because o f this loss o f information it is not possible to calculate a scattering density profile directly firom the coherent scattering and so a fitting procedure is required. The fitting procedure used for the experiments described in this thesis is described in chapter 5.

0.8

0.6

0.2

0.6

0

20

Figure 4.4. Plot o f the scattering density profile (solid line; arbitrary units on the y axis) along the c* axis for a hypothetical clay gel with a layer spacing of 40Â. Peaks near r = 0 and 2 = 40Â represent atoms in the clay layers; peaks at 10, 20 and 30Â represent three possible positions o f an atom in the interlayer. The fimction cos(Qz) is shown for

Q values corresponding to the (001) (dashed line) and (002) Bragg reflections (sohd line with crosses). S(Q) at a Bragg reflection is given by a summation over the density profile weighted by the appropriate fimction. Atoms in the centre of the cell will therefore make a negative contribution to the (001) and a positive one to the (002). Atoms at one quarter and three quarters of the unit cell length however will make no contribution to (001) and a negative contribution to the (002). It is therefore possible to locate species in the interlayer fi'om the changes in Bragg intensities accompanying a change o f the scattering profile. In neutron diffraction this is achieved by isotope substitution.

We have so far treated the atomic separation in only one dimension, that parallel to the scattering vector (figure 4.3). It is therefore hr^ortant to consider what this dimension represents for different materials. In the case o f a single crystal the distribution o f atoms on the z axis is dependant on the orientation o f the sample with respect to Q. DifiFerent orientations o f the crystal can therefore lead to diffraction from different sets o f planes o f atoms in the crystal. In a crystal where one axis o f the unit cell is orthogonal to the others it is possible to ahgn the sanq)le with this axis parallel to Q and so investigate the structure solely along that axis. In the case o f clays this means that they can be ahgned to study the structure perpendicular to the clay surface (along the c* axis). This is an inq)ortant advantage as it greatly simplifies the analysis o f a complex, multicomponent system It does involve the loss o f information on the structure parallel to the surface, but as the structure developed in the electrical double layer is expected to be manifested principally along the c* axis this is not so important.

In a powder or a hquid or amorphous satrple there is no long range orientational order and whatever the relative orientation o f beam, sample and detector we investigate the structure in ah three (equivalent) dimensions simultaneously.