Required reading: Moore and Reynolds,
174175, 261297, 359371
Brindley and Brown, pages 249267
XRD identification of mixedlayer clay minerals
The terms interlayering, mixedlayer and interstratification all
describe phyllosilicate structures in which two or more layer
types are vertically stacked in the direction parallel to c*.
Nomenclature  It's important to note that (with
exception to regularly ordered 5050 interstratified clays)
there is no "accepted" nomenclature for mixedlayer clays.
However, for simple binary
mixed systems, an easy shorthand notation can be established
that is also relatively unambiguous. The hierarchy is as
follows:
A shorthand notation to denote the binary mixedlayer system system above would be:
ABXXRY
where:
A = Capital initial of the smaller dspacing mineral/group name
RY = Reichweite or ordering scheme.
Examples of R commonly used in the calculation of mixedlayer systems include:
Example of shorthand notation for mixedlayer
type.
IS20R0 is an illitesmectite with 20% illite type layers and 80% smectite type layers, that are randomly interstratified.
In the special case where the proportions of each layer type are equal and they are ordered in alternating sequence (ABABABABAB..., i.e., XX = 50 and Y=1) then specific new mineral names are given.
Example:
IS50R1 = rectorite.
How do you recognize the presence of interstratification?
If you examine a crystal structure that repeats its basal
reflections at periodic spacing it "obeys" Bragg's Law
n λ = 2dsinθ
where, the d's occur as integral series.
This series is referred to as a rational series of
reflections.
One method to assess the rationality of a series is look at the
standard deviation of the reflections "normalized" by their
order.
An example of chlorite is shown below.
The values of d in the table below have been taken from the CuKα diffractogram above (higherorder reflections are not plotted above).
d (00l) Å 
l 
l * d 
13.939 
1 
13.939 
7.0197 
2 
14.039 
4.6916 
3 
14.074 
3.5243 
4 
14.097 
2.8204 
5 
14.102 
2.3876 
6 
14.325 
2.0020 
7 
14.014 
1.5678 
9 
14.110 
1.4124 
10 
14.124 

mean 
14.091 

std dev. 
0.105 

CV 
0.75% 
Note the small standard deviation. Bailey (1980, Am. Min. v67 p394) has suggested that any series with a coefficient of variation (CV) of less than 0.75% constitutes a discrete phase.
CV is defined as (100 x stdev) / mean
In the case of the regularly ordered 50/50
mixedlayer clays, the two layer types will combine to form a super
structure (equal to the sum of the two layer dimensions).
These result in very low angle reflections (i.e., 2 
3.5° 2θ for Cu Kα radiation).
Statistical treatment of sequences with two layer types
One must consider (1) the composition of the layer types and (2)
the probability of a given junction of layer types (i.e.,
interface).
In a two component system with layer types A
and B let,
P _{A} = fraction of A
P _{B} = fraction of B
then,
P_{A} + P_{B} = 1
There are therefore four possible junction probabilities:
P_{A}._{B }, P_{B}._{A}, P_{A}._{A}, P_{B}._{B}
P_{A.B}is therefore the junction probability of layer type
B following layer type A.
It does not specify the probability of finding an AB pair.
The probability of finding an AB pair is product of the fraction of A
and the junction probability of layer type B
following layer type
A. This is designated,
P_{AB} = P_{A}P_{A}._{B}
Either an A or a B must follow an A, therefore,
P_{A}._{A} + P_{A}._{B} = 1
and either an A or a B must follow a B, therefore,
P_{B}._{A} + P_{B}._{B} = 1
and the probability of finding a AB pair is the same
as finding a BA
pair,
P_{AB} = P_{BA} = P_{A}P_{A}._{B} = P_{B}P_{B}._{A}
or
P_{A}._{B} = P_{B}._{A} P_{B} / P_{A}
Here we have six variables with four independent equations.
Therefore, by giving any two variables the complete system is
described.
Usually provided are:
1. The compositional parameter (P_{A} or P_{B} )
2. One junction probability (e.g., P_{A}._{A} )
Example 1:
Example 2: Using illitesmectite. IS60
Note: This treatment only applies to sequences
that are affected by its nearest neighbor. Layer sequences are
defined by three particular types, including:
The random case is specified by equal junction probabilities of any layer being followed by an A, which in turn is equal to the amount of layer type A.
P_{A}._{A} = P_{B}._{A} = P_{A}
and likewise, there is are equal junction probabilities of any
layer being followed by a B, which in turn is equal to the amount of layer type
B.
P_{B}._{B} =P_{A}._{B} = P_{B}
P_{A}._{A} = 0, if P_{A} < 0.5
or
P_{B}._{B} =, 0 if P_{A} > 0.5
The range of P_{A}._{A} from P_{A}._{A}= 0 > P_{A}._{A} = P_{A} describes
conditions from perfect to random interstratification.
If P_{A}._{A} > P_{A}, then A and B are separated into completely discrete domains (i.e., a physical mixture).
Ordering type 
Conditions 
Random 
P_{A}._{A} = P_{A} 
Ordered 
P_{A}._{A} = 0, if P_{A} < 0.5 _{} 
Segregated 
P_{A}._{A} = 1, if P_{A}._{A} > P_{A} 
The frequency of occurrence for any arrangement of layers into a crystallite is found by using the junction probabilities and compositions. For example the 6crystallite illitesmectite sequence ISSISI is given by;
P_{I} P_{I}._{S} P_{S}._{S}P_{S}._{I} P_{I}._{S} P_{S}._{I}
When the layer sequence is random, then the frequency of occurrence simplifies* to
(P_{I})^{n}I . (P_{S})^{n}^{S}
where n_{I}_{ }=
3 is the number of illite layers and n_{S}
= 3 is the number of smectite layers in the ISSISI example above.
*see Bethke,
C.M. and Reynolds, R.C. (1986) Clays and Clay Minerals v.34,
224226 for more detail about the mathematical basis for
this simplification.
As noted by Reynolds (1980) the nature of nonnearest neighbors is more complicated but follows the same logic.
Here's an example of ordering that considers three nextnearest neighbors in the same 6crystallite illitesmectite sequence ISSISI as above.
P_{I} P_{I}._{S} P_{IS}._{S} P_{ISS}._{I} P_{SSI}._{ S} P_{SIS}._{I}
In addtion to the single junction probabilities and compositions, ternary junction probabilities are also needed. In our example, these are given as,
Now there are 8 variables and 6 equations, therefore only two junction probabilities will be needed to satisfy the system. One must come from a set containing I as the nearestneighbor and one comes from a set containing S as the nearestneighbor
Here are the equations that describe the higher order parameters for a four layer model.
There are 13 equations above and 16 varibles. Consequently 3 values must be given that occur in the last 5 equations. The last 5 equations must consider II and IS as nearestneighbors. The one additional value must contain SI as nearestneighbor pairs. Assuming values for P_{SII}._{I} , P_{ISI}._{I }, and P_{SSS}._{I} will allow calculation of all probabilities.
Reynolds (1980) notes that calculation of
thriceremoved neighbors requires 7 variables to be defined.
Remember! Random ordering and/or nonequal layer proportions
produce irrational reflection series.