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      where, E is the energy of incident X‐ray beam (in keV) undergoing total reflection, ρ is the density in g/cm³ of the second medium, and Z and A are the atomic number and atomic mass of the second medium, respectively. The above formula holds true only for the X‐ray energies above the absorption edges of the medium [8–10].

      It is clear from the above elaboration that for a glancing angle lower than the critical angle, there will be no refraction and the X‐ray beam falling on the sample is totally reflected from the second medium back to first medium. This means that the penetration of X‐rays in the medium, when the glancing angle is below the critical angle, is negligible and this is very advantageous for TXRF analysis. Thus, three physical quantities: critical angle, reflectivity, and penetration depth, are important parameters for TXRF analysis. The critical angle is already discussed in detail above.

The reflectivity of a medium is defined as the ratio of the intensity of the reflected beam and that of the incident beam. In TXRF analysis, a reflectivity of about 100% is desirable. For X‐rays falling on a medium at an angle greater than that of critical angle, reflectivity is low but it increases drastically at the critical angle and reaches almost 100% when the angle of incident is slightly lower than the critical angle. Penetration depth, which is defined as that depth of a homogeneous medium that a beam can penetrate, and its intensity is reduced to 1/e, or 37% of its initial value. As mentioned above the penetration depth in TXRF geometry is very small (only a few nanometers). Both these factors, high reflectivity and low penetration depth avoid the interaction of X‐rays with the medium and hence, the scattering. Therefore, spectral background is very low in TXRF.

Table 4.1 Some sample support materials suitable for use in TXRF analysis and their characteristics for TXRF analysis. Data from Klockenkämper [10].

Support material	Critical angle for Mo Kα (17.44 keV) (Degrees)	Reflectivity for Mo Kα (17.44 keV) at critical angle	Critical angle for WLα (8.39 keV) (Degrees)	Reflectivity for W Lα (8.39 keV) at critical angle
Plexiglas	0.08	0.932	0.16	0.879
Glassy Carbon	0.08	0.939	0.17	0.884
Boron Carbide	0.10	0.93	0.21	0.876
Quartz	0.10	0.855	0.21	0.734
Platinum	0.28	0.394	0.58	0.453
Gold	0.26	0.387	0.55	0.448

The critical angle given in the above equation depends on the energy of the X‐ray beam and on the material being used as a reflector. It is a very small value in the range of 0–1°. The critical angles and reflectivities at critical angles for some materials with respect to different X‐ray sources, e.g. Mo Kα and W Kα X‐rays have been given in Table 4.1 [10]. Since the critical angle is so small, it is imperative that the X‐rays used for sample excitation travel parallel to each other in the X‐ray beam and do not have any diversion. Special type of line focus X‐ray tubes are used for this purpose. Moreover, sufficient smoothness and flatness of the supports are essential to maintain such a small angle. Different types of sample supports are used in TXRF analysis. The sample supports to be used for TXRF analysis should be flat, highly reflecting, and made from pure and chemically inert material. In addition, they should be either of a low Z material, e.g. quartz, Plexiglass or a pure single element metal, (e.g. gold) to ensure that there are no more than a few characteristic X‐ray lines from the support material and hence, interference with the analyte lines from the sample are kept to a minimum. Further, the supports should be reusable and easy to clean so that TXRF analysis is simple and economical.

Now, if a thin film of sample having a thickness of a few nanometers is placed on the sample support, at the intersection of incident and totally reflected beams, the path of incident beam shall not change much, as the thickness of the sample on the support is insignificant. However, the incident beam shall excite the sample. After the beam is totally reflected from the support, the totally reflected beam shall also pass through the sample and its path shall not change much, once again due to the negligible thickness of the sample. This beam shall also excite the sample. Thus, the sample is excited doubly, one by the incident beam and again by the totally reflected beam, whereas in XRF the sample excitation is done only once by the incident beam which is absorbed in the sample as well as the sample support. This is a simple explanation of TXRF and has been shown in Figure 4.3. Due to this phenomenon (TXRF), the intensity of the characteristic X‐rays coming out of the sample shall be approximately double that of when the X‐rays fall at an angle greater than the critical angle (off TXRF condition). Since the critical angle is dependent on the falling X‐ray beam energy on the support, the beam falling on the sample should be preferably monochromatic. If the beam is not monochromatic, the X‐rays of energy lower than the cutoff energy shall be totally reflected but those above the cutoff energy shall not be totally reflected, and will penetrate deep into the sample support, thus increasing the spectral background. These high energy X‐rays shall not excite the sample efficiently as they shall be far away from the absorption edges of the analytes and thus, are merely a nuisance. Due to this reason, a monochromatic beam of X‐ray is required for the ideal sample excitation. For producing a monochromatic beam, a monochromator ‐ either a synthetic multilayer or a crystal cut from a particular plane ‐ is chosen [10]. Multilayer monochromators have high reflectivity, are rugged, and thus are preferred compared to crystals which have low reflectivity and do not possess much strength. However, some researchers have found that during monochromatization, a large part of the X‐ray beam is cut off and remains unutilized, concluding that it is better to use polychromatic excitation where the increase in spectral background is compensated for by the high flux excitation by the polychromatic beam accompanied by a simple instrumentation. Nonetheless, monochromatic excitation is ideal for TXRF analysis [11].

Figure 4.3 A representation of the advantages of TXRF analysis involving efficient sample excitation, X‐ray detection and reduction in spectra background compared to that of EDXRF background.

      4.4.3 TXRF Instrumentation for Trace Element Determination

For trace element determination in TXRF, monochromatic X‐rays coming from an X‐ray source, as mentioned above, are passed through a slit so that a parallel beam of X‐rays

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