Brindley and Brown, pages 274 - 300

**XRD identification of mixed-layer clay minerals**

Recall that mixed-layer clays are identified by the presence of
an non-rational series of reflections. This concept of understanding non-rational series was pioneered by J. Mering (1949)
X-ray diffraction in disordered layer structures Acta crystallographica
vol:2 pg:371 and has become to be known as Mering's Principles. To learn more, go to Drits and McCarty (1996).

The two variables that determine the
nature of the diffraction are:

1. The proportions of layer types

2. The ordering of the sequence.

The position of the irrational reflections occur between the
nominal positions of the (00*l*) peaks of each member of
the mixture.

The position of a reflection is fixed by the proportions of
end-members.

The designation for a reflection is given by the contributing
(00*l*)'s.

**Example:** For a randomly mixed-layer I/S (*e.g.*
IS50R0), the composite peak that results from the (002) of the
illite and the (003) of the smectite is designated the (002)_{I}/(003)_{S} or (002)_{10}/(003)_{17}. Remember, the layer with the smaller d-value
is listed first. Sometimes, for clarity, the approximate
d-spacing value for the respective layer type is appended as a
subscript (see above and example below).

In the example below, illite has a repeat
of ~10Å and the smectite (in the ethylene glycol saturated
state) has a repeat of ~17Å. The position of the discrete
reflections of illite and smectite are marked on the
diffractogram below with the red and blue lines, respectively.

**Peak Widths**

Note that the closer the end-members are to a composite peak,
the sharper the peak shape becomes. This is exemplified above in
the (003)_{10}/(005)_{17} reflection for the I/S.

The farther the end-members are from each other; the broader the
composite peak shape becomes.

Therefore, in addition to the occurrence of non-rational series,
mixed-layer clays can be further identified by the occurrence of
peaks with variable peak widths (FWHM).

**Example:** Regularly ordered I/S 70 R1 (IS70R1). In the
example below, note that the composite peaks that result from
two closely-spaced higher-order reflections are narrow. This is
seen in the (001)_{10}/(003)_{27} reflection. Note the (002)_{10}/(006)_{27}
reflection is broad.

Patterns can be thought of as a random
mixture of rectorite and illite.

Note positions of where the Rectorite super-structure peaks
would be (i.e., 001*, 002*...).

Some smectite layers are followed by more than one illite layer.
Because this is R1 (*i.e*., the reach back or probability
of an S following an I is determined by only one layer) there is
a random probability for the occurrence of layers beyond one
layer.

In the case where the "reach back" involves three nearing-neighbors, a large superstructure is created.

Superstructure of ISII = 10Å + 10Å + 10Å + 17Å = 47Å.

For the special case where ratio of the
abundance of layer type **A** to layer type **B** is exactly 3:1 (*i.e.,* **P** _{A} =
0.75) and the ordering scheme is R = 3 (*i.e.*, **P** _{A·B·A·A
} = 1) then this becomes a discrete
mineral phase. For ISII this mineral is named tarasovite
and it has a basal d-spacing with a 47Å repeat.

**Example:**
Regularly ordered I/S 90 R3 (IS90R3). The example below shows
the effect of non-nearing neighbor ordering.

Peak at 11.3Å is the composite (001)_{10}/(004)_{47} reflection

- d-spacing of the (001) = 10 +/- 0.05 Å.
- d-spacing of (060) = 1.50 +/- 0.01 Å.
- (002) intensity greater than (001)/4 (suggests aluminum-rich octahedral sheet, Fe causes decrease).
- Calculate the "illite ratio" (I
_{r}) where:

- .

- The I
_{r}index uses the intensities of the first and third order basal reflections. If expandable smectitic layers are present, then the relative intensities will be influenced by mixed-layering. An I_{r}greater than 1 indicates smectitic layers.

Caution: Line broadening also comes from effects of crystallite size, strain, and XRD optics (slit size, misalignment...).

Step 1. Suppose that there is a mixture of two layer types, (

Step 2. Take the ratio of the small d(001) to the large d(001),

Step 3. Multiply this ratio by

*e.g.,*

1 x 0.704 = 0.074

2 x 0.704 =1.408

Step 4. Determine the deviation of each number from the nearest
integer.

*e.g.,*

1 - 0.704 = 0.294

1.408 - 1 = 0.408

This is easy to set up in a spreadsheet. In Excel the syntax is as follows:

- Enter respective layer type
**A**and**B**(001) d-values into cells B1 and B2 - Let cell C2 = B1/B2
- Let cell B4 = A4 * $C$2
- Let cell C4 = ABS(ROUND(B4,0)-B4)
- Copy formulas from row 4 to the rows below.

D001A |
10 | Ratio |

D001B |
14.2 | 0.704 |

Order | Order x ratio | Q |

1 | 0.704 | 0.295 |

2 | 1.408 | 0.408 |

3 | 2.112 | 0.112 |

4 | 2.816 | 0.183 |

5 | 3.521 | 0.478 |

6 | 4.225 | 0.225 |

7 | 4.929 | 0.070 |

8 | 5.633 | 0.366 |

9 | 6.338 | 0.338 |

10 | 7.042 | 0.042 |

Step 5. The Q values predict the relative widths of the
mixed-layer type where:

Q = 0.000 - no line broadening at all (only that due to
crystallite and instrument effects).

Q = 0.500 - maximum breadth possible.

Step 6. Determine line widths for near-discrete phase and
correct for instrumental line broadening (this is done
mathematically by analyzing observed peak and a defect free
sample with large coherent domains such as NIST SRM-660 XRD reference material LaB_{6}).

Step 7. Plot Q versus peak width, corrected for by cos(θ) to eliminate angle dependent particle size
broadening. Recall the Scherrer equation;

Where:

L = Mean crytallite dimension (Å)

λ = Wavelength of radiation

K = constant (near unity)

β = FWHM (in radians i.e., measure °2θ then multiply by π/180)

θ = angle of incidence

Therefore

β (radians) = λK/cosθ

β (2θ) = β (radians)* 180 /π

In the example below from Moore and Reynolds used calculated data, whre they assume a mean defect distance of 10 unit cells with the largest being 50 unit cells.

Step 8. The slope of Q versus corrected
width is related to the percentage of layer types in the
mixed-layer clay.

If all the peaks have identical widths, then the line will plot
vertically and there is only one layer type (*i.e*., Q is
meaningless).

Note: The above figure is modified from Moore and Reynolds.
It is included for use only by the students in this class. Do not
reproduce without permission from the authors.

Here is an example for illite/smectite with 90% smectite layers
and R0 ordering.