Small-Angle Scattering. Ian W. Hamley

      Here J₁(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 = πR²Ln, C = 4πr³n/3, and D = πRL²/2, 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 1.6 illustrates 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_diff