Round Robin test for Total Reﬂection X-Ray Fluorescence Analysis using preselected and well characterized samples

A TXRF round robin test using preselected micro- and nanoscaled samples has been performed and shows excellent results, demonstrating the strength and reliability of the TXRF method when decoupled from unfavorable sample preparations.


A GIXRF measurement
Grazing incidence X-ray fluorescence (GIXRF) uses a small glancing angle for exciting samples to X-ray fluorescence. Typically, angles of approximately 0.1°a re used to achieve insight into some special properties of materials and the distribution of elements near the surface. In most cases GIXRF is offered at Synchrotron facilities [1,2] due to the excellent control of the beam geometry used for excitation.
Total reflection X-ray fluorescence can be described as measuring an EDXRF spectrum at a fixed glancing angle Θ T XRF . The angle for the primary beam used for excitation to X-ray fluorescence should be below the critical angle of external total reflection of the X-rays used. Best condition can be achieved at about 70 % of the critical angle. The spectrum displays the intensity versus the energy of photons. GIXRF uses an angular scan of the primary beam at very small glancing angles. The signals of preselected energies are recorded along the angular scan. In a certain way TXRF is a single static view of one position of GIXRF containing all the energies detected.
On flat surfaces and for glancing angles below the critical angle of external total reflection, the interference of the incoming and reflected beam result in an X-ray standing wave (XSW) field [3]. The intensity distribution of the XSW field is very sensitive on a variation of the incident angle of the excitation radiation. It directly influences the emitted fluorescence line intensity of an element in a sample located at the surface, revealing information about the vertical distribution [4,5] with respect to the sample surface. In Figure 1, simulations of Ni layers with several thicknesses on quartz glass substrates show the development of the XSW field intensity inside of the respective layers as a function of incident angle.
For thin films, the enhancement below the critical angle of total external reflection of the substrate can be up to a factor of about 4 (in this simulation at 0.1075°for an excitation energy of 17.44 keV). For particle-like distributions over a height region of 100 nm, a factor of about 2 over a larger range of incident angles below the critical angle of total external reflection arises. Above the critical angle, the XSW field collapses and the relative intensity becomes one. Surface roughness and the divergence of the incident beam averages out the sinuous behavior and can reduce the XSW field intensity [6,7]. Figure 2a shows the experimentally determined XSW field intensity from a 50 ng µL droplet sample for an excitation energy of 10.5 keV, measured with monochromatized synchrotron radiation at the FCM beamline [8] in the PTB laboratory at BESSY II. There is no difference in the characteristic of the intensity as a function of the incident angle except for Ni, allowing for the assumption of one film, where all elements are mixed. The enhancement factor I XSW below the critical angle of total external reflection is about 1.4 to 1.8 for the elements Ga, Sc and Mn, being in line with an assumed surface roughness of several 10 nm as a model parameter for the simulation of the respective XSW field intensity. Ni shows an even lower enhancement factor. This can be explained by an excitation radiation induced fluorescence radiation of instrumentation materials,  Figure 1: Simulation of the intensity distribution of the XSW field for Ni layers with several thicknesses at 17.44 keV incident photon energy. The intensity distribution of Si from the glass substrate was simulated for the same XSW field and shows the evanescent part above the critical angle of external total reflection. It has been upscaled for a better comparability.
reaching the X-ray detector by a second scattering process e.g. at the substrate, or alternatively by a direct substrate contamination. The Ni intensity as a function of the incident angle would then be a combination of the signal of the Ni from the droplet on the substrate surface and the excitation radiation induced fluorescence radiation of instrumentation materials, which is independent on the incident angle. For µL droplet samples, a GIXRF measurement at a position of the substrate without sample has been performed and shows a constant Ni background for all incident angles. This background measurement was used for a correction of the GIXRF measurement for Ni originated from the sample. With current generation table-top TXRF instrumentation it is also possible to perform a GIXRF measurement for a qualitative validation of the curve shape and to ensure the optimal adjustment of the incident angle. In Figure  2b a table-top GIXRF measurement for a similar sample is shown. Here, the excitation energy is 17.44 keV (Mo-Kα) and the respective critical angle of total external reflection is about 0.1°. Comparison to the synchrotron-radiation-based GIXRF measurement shows similar development of the curves and illustrates the performance of the table-top instrumentation. Above the critical angle of total external reflection, the signal of the background is increasing rapidly in comparison to the fluorescence lines and can negatively affect the deconvolution. This can be seen most notably in the Sc count rate, where the run of the curve follows the substrate signal. The total measurement time of the table-top GIXRF measurement shown here was rather short with about 75 min, already suitable for a qualitative analysis of the sample.

B Lateral measurement
Based on the GIXRF measurement with monochromatized synchrotron radiation, the incident angle for the lateral measurement was chosen to be 0.1175°, which is well below the critical angle of total external reflection for SiO 2 at 10.5 keV (about 0.168°). For this incident angle, the experimentally determined enhancement factor I XSW is 1.4 to 1.8, described in detail in section A. In order to use I XSW for the quantification at every vertical lateral position of the sample, the assumption was made that the formation of the XSW field is similar at every vertical lateral position. This assumption has been experimentally verified for one of the samples.
The vertical size, particularly the full width at half maximum (FWHM) value of the incident beam with a Gaussian distribution, was about 100 µm and was experimental determined with a knife-edge scan [9,10]. The size of the beam in the propagation direction is drastically increased from a horizontal beam size of about 270 µm to a footprint of about 130 mm because of the TXRF-conditions with an incident angle of 0.1175°. It is important to have a footprint of the incident beam larger than the sample area in this direction because it is essential for the absolute quantification to excite the whole material in propagation direction while doing a lateral scan of the sample in the vertical direction. Figure  3a shows the normalized fluorescence photon flux of the elements Ga, Ni, Mn and Sc from a 50 ng µL droplet sample as a function of the vertical position.   [11,12] from Sherman [13] (for the determination of the mass deposition mi/FI with the unit area FI of the element i) were used for a selectable sample size in propagation direction in order to obtain a value of the mass deposition for each element of interest, multiplied by the same preselected sample size and the step size of the lateral scan in vertical direction.    The step size of the vertical lateral scan has to be smaller than the vertical beam size in order to ensure that the whole material of the µL droplet is excited. Here, the step size was chosen to be 20 µm. The resulting overlap of the excitation is considered by a numerical integration in the vertical direction for the reference-free quantification.

C Footprint impact on the reference-free quantification
The footprint of the incident beam is larger than the size of the sample in the direction of the propagation of the beam, which ensures an excitation of the whole material in this direction for every vertical position, shown in Figure 4a with an ideal Gaussian distribution as an example. In this case, reference-free TXRF has the advantage to be nearly independent from the information of the precise knowledge of the sample size in the direction of the beam propagation, explained in the following: Due to the very small incident angle, the quotient Q int of part of the total excitation intensity over a preselectable projected length l (equation 1) normalized to the same length l (equation 2) is almost constant over a large range. Descriptively, Q int is the quotient of the illuminated area and the solid angle of detection defined by this area. This quantity varies by less than 2 % below a length value of 40 mm, which is more than the size of the quartz glass substrate of all samples used in the Round Robin activity. In practice, this theoretical maximum value for preselected length is further reduced (here to about 8 mm) in view of the effective solid angle of detection constraints defined by the detector diameter and angular acceptance [11].
The limit of Q int towards zero is a constant. Due to the normalization of the Gaussian distribution to the maximum value, this constant is equal one. For the uncertainty budget, one may use the maximum Q int deviation of 2 % as a conservative estimate, assigned to the fraction of the incident beam intensity used for the determination of the mass deposition.
With further increase of the length, the integral intensity of the beam footprint is reaching the total excitation intensity and the quotient is decreasing. In Figure 4b, this quotient Q int is shown as a function of the length l for a horizontal beam size of 270 µm FWHM, an incident angle of 0.1175°and a respective footprint of about 130 mm FWHM in propagation direction. For a correct quantification, it is crucial that this chosen length is within the almost constant range of the quotient, but the precise length of the sample does not have to be known.
Here, a length of 3 mm has been chosen for the quantification, being close to the actual size of the µL droplet and sufficiently apart from the two limitations described above. Note that the chosen length (equivalent to preselected sample size) directly has an effect on the fraction of the incident photon flux used for the determination of the mass deposition, but the mass in Figure 3b is almost independent on the chosen length. It gets canceled out when multiplying the mass deposition (determined with the chosen length) and the chosen length itself.

D Barplots
Shown in barplots are the results of the different droplets and thin film samples from each laboratory for the respective elements. The uncertainty budget has three contributions, the statistical uncertainty of independent measurements, the uncertainty of the internal standard and an estimation of the uncertainty of the instrumentation. Variations in the total uncertainty are due to different statistical uncertainties of the measurements.

E Boxplots
Before the box plots of all measurements are shown, a principal comparison between a boxplot and a probability density function of a Normal distribution is shown in Figure 15 [14].50 % of all results are located in the range of Q l and Q u . Figure 15: Boxplot and a probability density function of a Normal distribution [14]. Q 1 and Q 3 are similar to Q l and Q u , respectively.

F Uncertainty budgets
In the following tables, the uncertainty budgets for the reference-free TXRF quantification of the Sc, Mn, Ni and Ga are listed, calculated with the GUM Workbench [15]. The element-specific mass within the dried-in droplet was determined at one vertical lateral position. For the fraction of the incident beam in propagation direction (defined by the chosen integration length of 3 mm, i.e. the preselected sample size) P 0,f rac , an uncertainty of 2 % was estimated. The uncertainty contribution of the numerical integration in both directions is negligible for the uncertainty of the total mass. According to GUM, the uncertainty index h(x i ) for non-correlated quantities is defined as follows: Here, x i is the input quantity, u is the total uncertainty and u i is the uncertainty contribution of the input quantity. The uncertainty contribution u i is the product of the standard uncertainty u(x i ) and the sensitivity coefficient c i of the respective input quantity. The sensitivity coefficient c i can become negative. From this it follows that the uncertainty contribution u i can become negative as well.  Figure 3b) and for the total mass of Sc. The standard uncertainties with the symbol ≤ are estimated. d x is the step width of the vertical scan and length y is the integration length (preselected sample size) in propagation direction of the beam.  Table 2: Uncertainty budget for the reference-free TXRF quantification of Mn at one exemplary vertical lateral position x (here: −44.0 mm, see Figure 3b) and for the total mass of Mn. The standard uncertainties with the symbol ≤ are estimated. d x is the step width of the vertical scan and length y is the integration length (preselected sample size) in propagation direction of the beam.  Table 3: Uncertainty budget for the reference-free TXRF quantification of Ni at one exemplary vertical lateral position x (here: −44.0 mm, see Figure 3b) and for the total mass of Ni. The standard uncertainties with the symbol ≤ are estimated. d x is the step width of the vertical scan and length y is the integration length (preselected sample size) in propagation direction of the beam.  Table 4: Uncertainty budget for the reference-free TXRF quantification of Ga at one exemplary vertical lateral position x (here: −44.0 mm, see Figure 3b) and for the total mass of Ga. The standard uncertainties with the symbol ≤ are estimated. d x is the step width of the vertical scan and length y is the integration length (preselected sample size) in propagation direction of the beam.