5 - Lecture notes for Clay Mineralogy


Required reading:

Moore and Reynolds, Pages 77-103
Brindley and Brown, pages 128-135, 225-261


Theoretical treatment of X-ray Diffraction

It is possible to calculate the diffraction pattern (i.e., coherent interference pattern) for any given crystal structure given:


Miller Indices - a symbolic vector representation for the orientation of an atomic plane in a crystal lattice and are defined as the reciprocals of the fractional intercepts which the plane makes with the crystallographic axes. For more on Miller indices click here.


Scattering effects

Atoms scatter radiation (with a wavelength equal to that of incident radiation) in all directions (like a beacon). The efficiency (f) is the result of scattering from individual electrons. The nucleus of the atom, although charged has an extremely large mass and can not be made to oscillate due to incoming radiation. Intensity of coherent scattering is inversely related to the square of the mass of the scattering particle. Coherent scattering is primarily attributed to the electrons.  An atom having Z electrons will have a scattering proportional to Z times the amplitude of a single electron.

f = Amplitude of a wave scattered by atom / Amplitude of wave scattered by an electron

Scattering efficiency is also controlled by direction of scattering.  The figure below shows the phase shift that results from scattering from two different regions of the electron cloud. In the forward direction there is no phase shift. As the angle increases, so does the phase shift.



As a consequence,decreases with increasing angle of reflection. The figure below shows the change in f for commonly encountered ions in clay minerals. The atomic scattering factor values are plotted as a function of 2θ. The plot below is for Cu Kα radiation. Click here for an Excel spreadsheet that allows you to change the wavelength of radiation (data from Cullity, 1978). In fact, the patterns below look similar to the pattern you would get for a monatomic gas of that particular element. Ask yourself, what would the Earth's atmosphere look like for a particular wavelength of radiation?



Scattering from a unit cell

Recall that rows of atoms cause scatter in specific directions resulting in constructive interference (i.e., coherent scatter).

For the case of clay minerals, the approach is greatly simplified. The morphological nature of clay minerals is such that they can easily be prepared to orient their crystallographic axes (the ab plane) relative to the X-ray beam. This is called "preferred orientation" (as opposed to random orientation).

We now want to describe this diffraction effect from a unit cell in a crystal.  If the clays are oriented, then we can consider this to be a one-dimensional diffraction problem.

The scattering from a unit cell (F) is always less than the total sum of atoms in the unit cell because the rays that the atoms scatter are out of phase with each other.   F is call the structure factor and is therefore, a measure of the intensity of the diffracted X-ray beam.

To find F, the sum of the amplitudes of each atom in the unit cell must be determined.

The sum of amplitudes must be adjusted by the amount of phase difference due to the location of the atoms in the unit cell. Recall that the phase difference is related to (1) the wavelength, (2) the angle of incidence, (3) the position of the atom planes and (4) the number and type of atoms in each plane. An example is given the figure below.


Under the Bragg condition, the phase shift (φ) resulting from the path length differences between rays 1' and 2' (Δ1'2') can describe as:
  Δ1'2' =  λ  = 2d001 sinθ          ( e.g.,   λ = 1.54049  Copper K-alpha radiation :  kaolinite  d001  = 7.167    12.40 =  2θ)
The path length difference between rays 1' and 3' is less than the path length difference between rays 1' and 2'.  By proportion it we see the path length difference between rays 1' and 3' is scaled by fractional distance of the plane relative to the ray 2' plane. This path length difference is expressed as:

Δ1'3' = λ *  z /d001       (e.g., given  z  = 2.35, then Δ1'3' = 1.54059 * 2.35 / 7.167 = 0.505 )

where z is the distance from the top plane of atoms (the origin) reflecting ray 1 to the plane reflecting ray 3.

Recall that (u, v, w) are the fractional coordinates for any position within a 3-D unit cell.  d001 is related to the c lattice parameter by the angle of the c-axis (β), where d001 = c sin(β).       (e.g.,  c = 7.37 and  β  = 104.5 and d001 = 7.167)

z /d001 = w     (e.g.,   z  = 2.35 / 7.167  = 0.3279)

where:  w is the fractional coordinate.

The path length difference then becomes:

Δ1'3' = λ * w    (e.g.,  1.54059 * 0.3275 = 0.505 )

The phase shift (in radians) is then given by:

φ = 2 π Δ / λ   

The phase shift for rays 1 and 2, where  Δ1'2'  =  λ  creates the condition for maximum constructive interference
φ1'2'   = 2 π Δ1'2' / λ = 2 π    (e.g.,  2 π * 1.54059  / 1.54059 = 2 π )

The phase shift for rays 1 and 3, where  Δ1'3'  =  λ * z  creates the condition for partial destructive interference.
φ1'3'   = 2 π Δ1'3' / λ   =   2 π z l  / c   (e.g.,  2 π * 0.505  / 1.54059 = 0.655 π )

For any order of indice,

d00l = c/l   (c = unit length and l is miller indice for plane)   (e.g., d002 = 7.37 / 2 = 3.584)

The fractional coordinates w = z/c

φ = 2 π l w

Recall that (u, v, w) are the fractional coordinates for any position within a 3-D unit cell.

For the 3-D case of (hkl)

φ = 2 π (h u + k v + l w)

Phase differences between the scattered waves (all with the same wavelength) can be determined mathematically by a structure factor function where:

F (hkl) = Σn f n    e(i φn)   =  f1  e(i φ1) +  f2  e(i φ2) + ... fn   e(i φn)

where:

We use the identity:

e(i φn) = cos φn + i sinφn

to yield

F = Σn fn cosφn + i Σn fn sinφn

If there is a center of symmetry in the unit cell and the origin for the calculation can be placed at that point then the sine series goes to zero and the complex number is eliminated. This elimination is not essential to the theoretical development of the structure factor. It's just being eliminated here to help streamline the example and simplify the calculations.

Therefore the above equation becomes,

F = Σn fn cosφn

We can expand the phase of the wave (φn) by letting

φn = 2πl (zn/c)

where:

let:

F = ΣnPn fn cos[2πl (zn/c)]

The fact that F can be negative or positive is not detectable in the X-ray experiment. The only thing that we measure with a detector is the intensity or magnitude. Therefore, squaring F eliminates its sign. What the detectors sees then is |F|2 .

F is a discontinuous function (i.e., it is defined by the integer l).

In order to consider the structure factor over a range of angular space (i.e., make it a continuous function) we return to Bragg's Law.

nλ = 2d sin θ

let:

solve in terms of l

                                          l = 2 c sin θ/ λ

by substitution:

G = Σn Pn fn cos[4π zn(sinθ/λ)]

Where:

 

The interference function (Φ)

Under ideal conditions all the diffraction takes place at the Bragg angles of reflection.

Diffraction effects (in one dimension) due to scattering from a grating can be described by an interference function:

where:

If N = 1 then Φ = 1 at all angles. Bragg reflection cannot occur from a single scattering center.

If N = 100, then Φ (at the ideal Bragg angle) is large. At the same time Φ is small away from the Bragg angle. In other words the peaks are very intense and narrow.

The example below are graphical solutions to the interference function using values of 8 2θ, λ = 1.54049 and values of N from 2 to 5.



Click here for an Excel spreadsheet that will let you play with the variables N, D, and λ

Lorentz-Polarization factors (Lp)

Polarization factor (p) - The X-ray beam that exits the tube is unpolarized (analogous to light coming from the sun).  Low angle scattering causes polarization of the beam (analogous to light reflecting off a lake). The polarization factor accounts for increase scattering at low angles. Various workers have conducted experiments and fit their results to theory and found the scattering intensity (Ip) due to polarization is proportional to (1 + cos2  θ)/2. This is taken from theoretical study and is known as the Thomson equation for scattering of an X-ray beam from a single electron. 

Lorentz factor (L) - The X-ray beam that exits the tube is also not strictly monochromatic nor parallel (some divergence occurs). These factors in combination with motion of the crystal (as noted by Klug and Alexander) contribute to a planes "opportunity" to reflect (i.e, planes that make an angle with the rotation axis are in a reflecting position longer than those parallel to the axis, hence disproportionate intensities will be observed).  The Lorentz factor is related to the volume of sample irradiated as a function of angle.

The number of crystals exposed to the beam is also a factor. Therefore we need to consider scattered beams from random powders differently from single crystals.  The single crystal form of the Lorentz factor is sin 2θ. The random powder form of the Lorentz factor is (sin θ sin 2θ). We will see later, that from a practical standpoint it is almost impossible to achieve a perfectly oriented sample or a complete randomly oriented sample. The approach will be to use some mixture or blending of the single crystal form and the powder ring distribution factor (ψ).  Lp = (1 + cos2  θ) ψ / (sin θ). For random powders ψ is proportional to 1/(sin θ). For a single crystal ψ is constant. Here is a spreadsheet to graph the functions. The development of this theory is given by Reynolds (1986).


If you really want to know a lot more about Lp factors, then seek the reference below.

Reynolds R. C. (1986) The lorentz-polarization factor and preferred oreinetation in oriented clay aggregates: Clays and Clay Minerals 34(4), 359-367.


The complete kinematic XRD equation

I(2θ) = Lp |G|2 Φ



Effect of scattering domain size and defects

A characteristic of clay minerals is such that the size of their coherent scattering domains is typically small and there are imperfections or stacking defects between adjacent domains (i.e. N is small).

The result of numerous stacking faults or small domain sizes is known as particle size broadening and can be inferred from the peak widths (as seen in the interference function above).

X-ray diffraction peak widths are commonly measured by their Full Widths at Maximum Half-height (FWMH).

Under certain conditions the FWMH can be used to estimate the distribution of "particle sizes" (this really means their domain sizes, not physical particle diameters).

This is expressed by the popular Scherrer Equation which assumes all reflections along a line normal to the reflecting plane. The equation also ignores strain broadening, which is due to variation in lattice parameters.

L =λ K / β cosθ

where:

Example: Case of (101) reflection for the quartz structure.

let:

then:

L = (1.54059 x 1 ) / [(0.1 x 3.14 / 180) x cos (13.3)] = 1300 (130 nm)

With a d-spacing of 3.3446 for the (101) this translates to 388 coherent scattering domains.


Click here for a set of notes that discuss sources of XRD error

Click below for a very detailed set of notes produced by Robert H. Blessing.
Part I
Part II