An Overview of XRF Basics
1. Fundamental Principles
1.8 Diffraction in crystals
1.8.1 Interference
Electromagnetic radiation displays interference and diffraction effects due to the nature of its waves. "Interference" is the property of waves to overlap each other and, under certain circumstances, to cancel out or amplify each other.
Amplification takes place when waves of identical wavelength have zero phase difference (coherence), i.e. when "wave maxima" and "wave minima" overlap in such a way that minima meet minima and maxima meet maxima. This is precisely the case when the phase difference Δλ is zero or a multiple of the wavelength λ, i.e.:
Δλ = nλ |
n = 0, 1, 2, ... |
"n" is referred to as the "order" (Fig. 12):
Fig. 12: Amplification resulting from the effects of interference
Where the phase difference is one half of the wavelength, that is where n = 1/2, 3/2, 5/2, ... , wave maxima coincide with wave minima resulting in total cancellation (Fig. 13). When a number of waves of the same wavelength propagating in the same direction interfere with each other under continuous phase shift, only the coherent among them will be amplified. In total, the rest will almost completely cancel each other out.
Fig. 13: Cancellation resulting from the effects of interference
1.8.2 Diffraction
From what we experience every day we know that light generally travels in straight lines. This corresponds with the notion of light as a beam of particles (photons, quanta). We know from ocean waves that when a wave series travels through a hole smaller than the wavelength, the waves exiting the hole spread out to the sides. Light displays the same wave characteristic. The deviation of light from its travel in a straight line is called diffraction.
There are numerous applications for the effects of diffraction. In wavelength dispersive XRF we are mainly interested in diffraction in reflection grids. Often used in the optical range (λ = 380 - 750 nm) are mirror lattices produced by spacing grooves at equal distances in reflecting metal surfaces. This is not possible in the X-ray field because the wavelengths involved are around 2 to 5 orders of magnitude smaller (λ = 0.02 - 11 nm). Very much smaller lattice distances, such as those found in natural crystals, are required for X-ray diffraction in the reflection grid.
Diffraction is a prerequisite for wavelength dispersive XRF. After excitation of the elements in the sample (by X-rays), a blend of element-characteristic wavelengths (fluorescence radiation) leaves the sample. There are now two methods in XRF of identifying these various wavelengths. Energy dispersive XRF calls on the assistance of an energy dispersive detector that is able to resolve the different energies. Wavelength dispersive XRF utilizes diffraction effects of crystal to separate the various wavelengths. The detector subsequently determines the intensity of a particular wavelength. The procedure will be covered in detail in the following sections.
1.8.3 X-ray Diffraction From a Crystal Lattice, Bragg's Equation
Crystals consist of a periodic arrangement of atoms or molecules that form a crystal lattice. In such an arrangement of atoms you generally find numerous planes running in different directions through the lattice points (atoms, molecules), and not only horizontally and vertically but also diagonally. These are called lattice planes. All of the planes parallel to a lattice plane are also lattice planes and are a set distance apart from each other. This distance is called the lattice plane distance "d."
When parallel X-rays strike a pair of parallel lattice planes, every atom within the planes acts as a scattering centre and emits a secondary wave. All of the secondary waves combine to form a reflected wave. The same occurs on the parallel lattice planes for only very little of the X-ray wave is absorbed within the lattice plane distance, d. All these reflected waves interfere with each other. If the amplification condition "phase difference = a whole multiple of wavelengths" (Δλ = nλ) is not precisely met, the reflected wave will interfere such that cancellation occurs. All that remains is the wavelength for which the amplification condition is precisely met. For a defined wavelength and a defined lattice plane distance, this is only given with a specific angle, the Bragg angle (Fig. 14).
Fig. 14 Bragg's Law
Under amplification conditions, parallel, coherent X-ray light (rays 1, 2) falls on a crystal with a lattice plane distanced d and is scattered below the angle θ (rays 1', 2'). The proportion of the beam that is scattered on the second plane has a phase difference of 'ACB' to the proportion of the beam that was scattered at the first plane. Following the definition of sine:
| 'AC' | = sinθ | or | 'AC' = d sinθ |
| d |
The phase difference 'ACB' is twice that, so:
'ACB' = 2d sinθ
The amplification condition is fulfilled when the phase difference is a whole multiple of the wavelength λ, so:
'ACB' = nλ
This results in Bragg's Law:
| nλ = 2d sinθ | Bragg's equation |
n = 1, 2, 3, ... |
reflection order |
Fig. 15a: 1st order reflection: λ = 2d sin θ1
Fig. 15b: 2nd order reflection: 2λ = 2d sin θ2
Fig. 15c: 3rd order reflection: 3λ = 2d sin θ3
Figures 15a, b and c illustrate Bragg's Law for the reflection orders n = 1, 2, 3.
On the basis of Bragg's Law, by measuring the angle θ, you can determine either the wavelength λ, and thus chemical elements, if the lattice plane distance d is known or, if the wavelength λ is known, the lattice plane distance d and thus the crystalline structure.
This provides the basis for two measuring techniques for the quantitative and qualitative determination of chemical elements and crystalline structures, depending on whether the wavelength λ or the 2d-value is identified by measuring the angle θ (Table 3):
Table 3: Wavelength dispersive X-ray techniques
| Known | Sought | Measured | Method | Instrument type |
|---|---|---|---|---|
| λ | d | θ | X-ray flourescence | Spectrometer |
| d | λ | θ | X-ray diffraction | Diffractometer |
In X-ray diffraction (XRD) the sample is excited with monochromatic radiation of a known wavelength (λ) in order to evaluate the lattice plane distances as per Bragg's equation.
In XRF, the d-value of the analyzer crystal is known and we can solve Bragg's equation for the element-characteristic wavelength (λ).
1.8.4 Reflections of Higher Orders
Figures 15a-c illustrate the reflections of the 1st, 2nd, and 3rd order of one wavelength through the different angles θ1, θ2, θ3. Here, the total reflection is made up of the various reflection orders (1, 2, ... n). The higher the reflection order, the lower the intensity of the reflected proportion of radiation. How great the maximum detectable order is depends on the wavelength, the type of crystal used and the angular range of the spectrometer.
It can be seen from Bragg's equation that the product of reflection order 'n = 1, 2, ...' and wavelength 'λ' for greater orders, and shorter wavelengths 'λ* < λ' that satisfy the condition 'λ* = λ/n', give the same result.
Accordingly, radiation with one half, one third, one quarter etc. of the appropriate wavelength (using the same type of crystal) is reflected through an identical angle 'θ':
1λ = 2(λ/2) = 3(λ/3) = 4(λ/4) = ...
As the radiation with one half of the wavelength has twice the energy, the radiation with one third of the wavelength three times the energy etc., peaks of twice, three times the energy etc. can occur in the pulse height spectrum (energy spectrum) as long as appropriate radiation sources (elements) exist (Fig. 16).
Fig. 16 shows the pulse height distribution of the flow counter using the example of the element hafnium (Hf) in a sample with a high proportion of zircon. The Zr Kα1 peak has twice the energy of the Hf Lα1 peak and appears, when the Hf Lα1 peak is set, at the same angle in the pulse height spectrum.
Fig. 16: 2nd order reflection (n=2)
1.8.5 Crystal Types
The wavelength dispersive X-ray fluorescence technique can detect every element above the atomic number 4 (Be). The wavelengths cover the range of values of four magnitudes: 0.01 - 11.3 nm. As the angle θ can theoretically only be between 0° and 90° (in practice 2° to 75°), 'sin θ' only accepts values between 0 and +1. When Bragg's equation is applied
| 0 ‹ | nλ | = sinθ ‹ +1 |
| 2d |
This means that the detectable element range is limited for a crystal with a lattice plane distance d. Therefore a variety of crystal types with different '2d' values is necessary to detect the whole element range. Table 4 shows a list of the common crystal types.
Lithium fluoride crystals that also state the lattice planes (200, 220, 420) are identical to the following names:
| LiF(420) | = | LiF(210) |
| LiF(220) | = | LiF(110) |
| LiF(200) | = | LiF(100) |
Besides the 2d-values, the following selection criteria must be considered when a particular type of crystal is to be used for a specific application:
- resolution
- reflectivity (→ intensity)
Further criteria can be:
- temperature stability
- suppression of higher orders
- crystal fluorescence
| Crystal | Name | Element Range | 2d-value (nm) |
|---|---|---|---|
| LiF(420) LiF(220) LiF(200) Ge InSb PET ADP TlAP XS-CEM XS-55 XS-N XS-C XS-B |
Lithium fluoride Lithium fluoride Lithium fluoride Germanium Indium antimonide Pentaerythrite Ammonium dihydrogen phosphate Thallium biphthalate Specific Structure Multilayer [W/Si] Multilayer [Ni/BN] TiO2/C LaB4C |
≥ Ni Kα1 ≥ V Kα1 ≥ K Kα1 P, S, Cl, Ar Si Al - Ti, Kr - Xe, Hf - Bi Mg F - Na Al - S N - Al, Ca - Br N C B (Be) |
0.1801 0.2848 0.4028 0.653 0.7481 0.874 1.064 2.576 2.75 5.5 11.0 12.0 19.0 |
1.8.6 Dispersion, Line Separation
The extent of the change in angle Δθ upon changing the wavelength by the amount Δλ (thus: Δθ/Δλ) is called "dispersion." The greater the dispersion, the better the separation of two adjacent or overlapping peaks. Resolution is determined by the dispersion as well as by surface quality and purity of the crystal.
Mathematically, dispersion can be obtained from the differentiation of the Bragg equation:
| dθ | = | n |
| dλ | 2d cosθ |
It can be seen from this equation that dispersion (or peak separation) increases as the lattice plane distance d decreases.
Examples:
The 2θ-values of the Kα1 peaks of vanadium (V) and chromium (Cr) are farther apart when measuring with LiF(220) than when measuring with LiF(200).
The 2θ-values of the Kα1 peaks of sulphur (S) and phosphorus (P) are farther apart when measuring with the Ge crystal than when measuring with the PET crystal.
| Crystal type | 2d-value (nm) | 2θ (El1) (degrees) | 2θ (El2) (degrees) | Difference (degrees) |
|---|---|---|---|---|
| LiF(220) | 0.2848 | 107.11 (Cr) | 123.17 (V) | 16.06 |
| LiF(200) | 0.4028 | 69.34 (Cr) | 76.92 (V) | 7.58 |
| Ge | 0.653 | 110.69 (S) | 141.03 (P) | 30.34 |
| PET | 0.874 | 75.85 (S) | 89.56 (P) | 13.17 |
The following describes the characteristics of the individual crystal types divided into "standard crystals," "multilayer crystals" and "special crystals."
1.8.7 Standard Types, Multilayers
LiF(200), LiF(220), LiF(420)
LiF crystal types exist in a variety of lattice planes (200, 220, 420, etc.). In the sequence (200) → (220) → (420), resolution increases and reflectivity decreases (Fig. 17).
Fig. 17: Intensities of the crystals LiF(220) and LiF(420) in relation to LiF(200). (Intensity LiF(200) = 1)
LiF(200):
A universally usable crystal for the element range atomic number 19 (K) onwards; high reflectivity, high sensitivity.
LiF(220):
Lower reflectivity than LiF(200) but higher resolution; can be used for the element range atomic number 23 (V) onwards; particularly suitable for better peak separation where peaks overlap.
Examples of the application of the LiF(220) for reducing peak overlaps:
Cr Mn U |
Kα1,2 Kα1,2 Lα1 |
– – – |
V Cr Rb |
Kβ1 Kβ1 Kα1,2 |
LiF(420):
One of the special crystals; can be used for the element range atomic number 28 (Ni or Co Kβ1) onwards; best resolution but low reflectivity.
Figure 17 shows a reflectivity of only 10% of that of LiF(200) for LiF(420) in the energy range around 10 keV.
PET:
A universal crystal for the elements Al to Ti (K-peaks), Kr to Xe (L-peaks) and Hf to Bi (M-Peaks).
The PET is the crystal with the greatest heat-expansion coefficients, i.e. temperature fluctuations are most noticeable here.
Multilayers XS-55, XS-N, XS-C, XS-B
Multilayers are not natural crystals but artificially produced "layer analyzers." The lattice plane distances d are produced by applying thin layers of two materials in alternation onto a substrate (Fig. 18). Multilayers are characterized by high reflectivity and a somewhat reduced resolution. For the analysis of light elements the multilayer technique presents an almost revolutionary improvement for numerous applications in comparison to natural crystals with large lattice plane distances.
Fig. 18: Diffraction in the layers of a multilayer crystal
XS-55:
The most commonly used multilayer with a 2d-value of 5.5 nm for analyzing the elements N to Al and Ca to Br; standard application for measuring the elements F, Na and Mg.
1.7.8 Special Crystals
The term "special crystals" refers to crystal types and multilayers that are not used universally but are employed in special applications.
LiF(420):
See, for example, "standard types" description of the LiF crystals (200, 220 and 420).
Ge:
A very good crystal for measuring the elements S, P and Cl. In comparison to PET, Ge is characterised by a higher dispersion and a more stable temperature behavior. Ge suppresses the peaks of the 2nd and 4th order, in particular.
Ge is especially suitable for differentiating between sulphide and sulphate, such as in samples of cement.
ADP:
In practice, ADP is only used for the analysis of Mg and has a higher resolution with a significantly lower reflectivity compared to the multilayer XS-55. ADP is therefore required where interference peaks can occur such as in the case of low Mg concentrations in an Al matrix.
TlAP:
TlAP has high resolution but low reflectivity and is recommended for analyzing F and Na if the resolution of XS-55 is insufficient (e.g. where Na is overlapped by the Zn-L peaks in Zn-rich samples).
Disadvantages are the limited time, toxicity, and high price.
InSb:
InSb is a very good crystal for analyzing Si in traces as well as in higher concentrations (e.g. glass). It replaces PET and is used wherever high precision and great stability is required. The disadvantages are the limited use (only Si) and the high price.
XS-N:
XS-N is a multilayer with a 2d-value of 11.0 nm, specially optimized for nitrogen.
XS-C:
XS-C is a multilayer with a 2d value of 12.0 nm, specially optimized for carbon.
XS-B:
XS-B is a multilayer with a 2d-value of 19.0 nm, specially optimized for boron and is equally suitable for the analysis of Be.
Which multilayer crystal is the most suitable for analyzing the very light elements?
Fig. 19a shows that XS-B is the best crystal for analyzing boron (B), naturally with a corresponding coarse collimator (at least 2° opening). A compromise for analyzing boron is the XS-160 crystal when carbon (C) also needs to be measured with the same crystal.
For the analysis of carbon (C) the XS-C crystal provides a sharper peak and a better ratio of peak/background intensities, which means that better sensitivities can be achieved (Fig. 19b). To apply the XS-55 for analyzing carbon should be exceptional in case of having no XS-C or XS-160. Only very high concentrations (several tens of percent) of carbon can be determined with XS-C. In case of determining carbon with XS-55 using a "standardless" precalibrated XRF routine, remember to select a very slow scanning speed (long measuring time) for carbon or to select the peak/background measurement mode.
Nitrogen (N) is best analysed using XS-N. If needed, XS-55 can also be applied (Fig. 19c). XS-B, XS-C and XS -160 are not suitable for analyzing nitrogen.
Oxygen (O) and all "heavier" light elements have to be analysed with XS-55, which gives the best resolution and the best peak/background ratio (Fig. 19d).
Fig. 19a: XS-B is the best multilayer crystal for analysing boron (B).
Fig. 19b: The XS-C multilayer crystal is suitable for the determination of carbon.
Fig. 19c: Nitrogen (N) is best analysed using XS-N.
Fig. 19d: Oxygen (O) and all "heavier" light elements have to be analyzed with XS-55.
1.8.9 Curved Crystals
Whereas flat crystals are used in sequence spectrometers, multichannel spectrometers principally employ curved crystals.
The curvature of the crystals is selected in such a way that by applying slit optics the X-ray entrance slit is focussed by the curved crystal onto the exit slit. This allows higher intensities in a space-saving geometric arrangement.
Different types of crystal curvature are used for focussing. The most commonly used are the curvatures that follow a logarithmic spiral (Fig. 20a) and the Johansson curvature (including grinding) (Fig. 20b).
|
|
