l Baseline left-parenthesis q right-parenthesis equals integral Subscript 0 Superscript pi slash 2 Baseline left-bracket StartFraction sine left-parenthesis one half italic q upper L cosine alpha right-parenthesis Over one half italic q upper L cosine alpha EndFraction period StartFraction 2 upper J 1 left-parenthesis italic q upper R sine alpha right-parenthesis Over italic q upper R sine alpha EndFraction right-bracket squared sine alpha normal d alpha"/>
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 |
As |
|
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