3 - Lecture notes for Clay Mineralogy


Suggested reading:

X-ray Diffraction Principles

The nature of X-rays. Two types of radiation: white radiation and characteristic radiation.

The electromagnetic (EM) spectrum.

The production of X-rays - In practice, X-rays are produced by streaming electrons across an extremely high voltage potential (15-45 Kv with the common stationary anode). The voltage is applied to a filament (typically a tungsten cathode) in a vacuum. The electrons are then accelerated into a metal target (typically a copper anode). The energy released results in two types of X-radiation.

The first type is known as white radiation and consists of a broad, continuous spectrum containing many wavelengths of radiation. It is a result of the very rapid deceleration of electrons as they encounter the strong electric fields of target metal. As the electrons collide they lose energy (often designated ΔE) and that energy goes into making X-ray photons. That energy, ΔE is related to the frequency of the X-ray radiation by Planck's Constant,

White radiation intensity as a function of energy potential (voltage) as shown in figure below from Klug and Alexander (1994). Original paper by Ulrey (1918)

Recall electrons orbiting an nucleus are tightly bound. When source electrons strike these outer shell/orbital electrons, the electrons get bounced out of position (in other words the electrons undergo an energy transition). This event is immediately followed by another electron dropping back toward the nucleus (much like dropping a book on the desk). The loss in energy appears as an emitted photon with a characteristic frequency.

transitions

The energy difference between electron levels are quantum (i.e., discreet) and the energy released will depend upon the number of protons and neutrons in the nucleus and the shell from which the electron was displaced.

In geology labs, copper and cobalt are commonly used targets. Copper radiation is "brighter", meaning it will produce higher intensity than cobalt radiation. However, if copper radiation encounters material with a high iron content, then a high background signal can be accompany the diffracted signal. Cobalt radiation is preferred for high iron-bearing samples. Since there are only two possible sites in the L shell of copper, there are only two (slightly different) energy transitions: Kα1, from the outer most L shell and Kα2 from the next lower shell.

The result is the production of very intense monochromatic radiation. This is the radiation that we take advantage of when doing X-ray crystallography. The figure below is modified from Klug and Alexander (1974) and shows the characteristic peaks for the target molybdenum. The green and red curves simply show the white radiation expected for lower potentials.


Exercise:

Given:
h = 6.63 x 10-34 J s
c = 3.0 x 108m s-1
λCuKα1= 1.54059



Find: ΔE for the CuKα1 transition (in units of KeV).

Hint: 1 eV = 1.6 x 10-19J

Here's a link to help with the conversion of energy units.

Solution:


X-ray absorption or "Things that can happen to characteristic radiation when it encounters matter".

1. Part of the
characteristic radiation is transmitted through at the same incident wavelength.
2. Scattering, which can occur two ways:


    a. Compton (incoherent) - caused by elastic collision of photon and electron.
    b. Bragg (coherent) caused by re-radiation at the same wavelength as the incident beam.

3. Heat
4. Fluorescent characteristic X-rays
5. Tertiary X-rays (more white radiation)


In differential form the decrease in intensity of the X-ray beam can be expressed as:
dI/dx = - Io
or
dI/Io= - dx

Where: I is intensity
is the linear absorption coefficient
x is the distance traversed

Upon integration:

I = Ioe-x

The linear absorption coefficient is proportional to the density (ρ ) which means that the quantity μ/ρ or (*) is constant for a particular set of elements and is constant regardless of physical or chemical state (i.e., phase state).

This relationships allows us to modify the "Lambert Law" to handle the case for mixtures of chemical compounds.

I = Ioe(-/ρ) ρ x
or
I = Ioe-(*) ρ x

Where * is the mass absorption coefficient.

The * for a particular element will vary depending upon the wavelength of radiation absorbed. The "absorption edges" shown in the figure below mark the point in the frequency (i.e., energy or wavelength) scale where the X-rays can eject an electron from one of its orbitals. An edge is formed in the curve, where absorption drastically drops. The example below from Klug and Alexander (1994) is for platinum.

Pt Edges



Matching the energy edge for some materials (i.e., transmitting the X-ray through thin foils of a metal) with the characteristic radiation is a common way to increase signal to noise (i.e., a form of filtering). The figure below is modified from Klug and Alexander (1994) is an example of a single filter where the edge falls just short of Kα line of the target generating the characteristic radiation. The result is a filtering of the Kβ radiation. The numbers written 194, 180, and 173 are the equivalent energies in eV for the Mo Kβ, Zr-edge, and Mo Kα transitions, respectively.


Beta filter


Mass absorption (cm2/g) and densities (g/cm3) for commonly encountered elements for Cu Kα radiation (λ = 1.542 ) and  Co Kα radiation (λ = 1.790 ) at room temperature. From International Tables for X-ray Crystallography

Absorber Cu
* cm2/g
ρ (g/cm3)
Co
* cm2/g
H 0.3912 0.08375 x 10-3 0.3966
Li 0.477 0.533 0.659
C
4.219
2.27 (graphite)
6.683
N 7.142 1.165 x 10-3 11.33
O 11.03 1.332 x 10-3 17.44
F 15.95 1.696 x -3 25.12
Na 30.3 0.966 47.34
Mg 40.88 1.74 63.54
Al 50.28 2.7 77.54
Si 65.32 2.33 100.4
K 148.4 0.862 222
Ca 171.4 1.53 257.2
Ti 202.4 4.51 300.5
Mn 272.5 7.47 405.1
Fe 304.4 7.87 56.25
Rb 106.3 1.53 159.6
Sr 115.3 2.58 173.5
Cs 325.4 1.91 (-10C) 483.8
Ba 336.1 3.59 499
U 305.7 19.05 446.3




Absorption characteristics of a sample have practical importance for the following reasons...

Calculation of * for the clay mineral kaolinite - Al2Si2O5(OH)4, using CuKα radiation


μ* = Σ μ*i W i

Constituent Elements
Atomic Weight
Weight % in Kaolinite
* cm 2/g
* (wt%)
H x 4
4
1.6
0.3912
0.01
Al x 2
54
20.9
50.28
10.5
Si x 2
56
21.7
65.32
14.17
O x 9
144
55.8
11.03
6.15
Totals
258
100
127
30.8

Click here for an example of how to use *.