Ian W. Hamley - Small-Angle Scattering

(1.42)

Here n is the number density of cylinders, ν is usually treated as a fit parameter and Pcyl ( q ) is the form factor of a cylinder of radius R and length L (see also Table 1.2):

(1.43)

Here J 1( x ) denotes a first order Bessel function.

Table 1.1 Sequences of Bragg reflections for common structures.

Structure	Reflections	Positional ratio
Lamellar	(001),(002),(003),(004),(005),(006)	1 : 2 : 3 : 4 : 5 : 6
Hexagonal	(1,0),(1,1),(2,0),(2,1),(3,0),(2,2)	1 : : :
Body‐centred cubic	(110),(200),(211),(220),(310),(222)	: : :
Face‐centred cubic	(111),(200),(220),(311),(222),(400)	: : :
Bicontinuous cubic Primitive cubic ‘plumber's nightmare’	As above	As above
Bicontinuous cubic ‘double diamond’	(110),(111),(200),(211),(220),(300)	: : :
Bicontinuous cubic ‘gyroid’	(211),(220),(321),(400),(420),(332),(422)	: : :

It is possible to compute ν from an equation for osmotic compressibility [17, 18]:

(1.44)

Here B = πR 2 Ln , C = 4 πr 3 n /3, and D = πRL 2/2, where n is the number density [17].

For lamellar or smectic structures a number of structure factors have been proposed, based on the fluctuations of the layers. Figure 1.6illustrates the thermal fluctuations that arise from the flexibility of the layers in a lamellar system. These fluctuations destroy true long‐range order in all one‐dimensional systems according to the Landau‐Peierls instability [20].

Figure 1.6 Fluctuations in the layer positions in a lamellar structure that are characterized by the membrane stiffness.

The structure factor for a stack of N fluctuating layers can be written as [21]

( 1.45)

The term σ is the width parameter in a Gaussian function

(1.46)

This is used to account for polydispersity in the number of layers and a suitable choice is for a sufficiently large N . In Eq. (1.45), N diffis a diffuse scattering term arising from uncorrelated fluctuations of layers.

Different models have been proposed to describe the thermal fluctuations and hence the structure factor terms Sk . In the thermal disorder model, each layer fluctuates with an amplitude Δ = 〈( dk − d ) 2〉, where d is the average lamellar spacing. The structure factor is that of an ideal one‐dimensional crystal multiplied by a Debye‐Waller factor: