Application of Macro X-ray Fluorescence Fast Mapping to Thickness Estimation of Layered Pigments

Computing the ratio of Lα line emission over Lβ line emission for lead can provide an estimate of azurite layer thickness. The energy intervals selected for the two lines were as follows:

A visual representation of a spectrum with highlighted ROIs is shown in Figure 7, left, with red bars and green bars delimiting, respectively, Lα and Lβ peaks. The spectral peaks obtained from these regions are also shown superimposed in Figure 7, right.

The novelty of the present work lies in the application of the ratio method [69] within a fast scan, that is, when we are dealing with a low count rate for each single spectrum, as is typical in MA-XRF; in this case, even if we expect quite large errors, we demonstrate that the method is highly useful to check the average thickness of pictorial layers. Indeed, the map obtained upon integrating the whole spectrum and the one that originated from integrating around the Lα lead peak are shown in Figure 8, upper row. In both images, especially in the one referring to the whole spectrum, the small holes (about 2 mm²) left from the sampling are evident. This aspect is quite interesting as it allows us, when applied to real cases, to clearly highlight the presence of an eventual lack of painting materials, and thus, to conduct mapping of the conservation state. This approach also allows us to compare MA-XRF with imaging methods [35].

The map in the left part of the middle row in Figure 8 was obtained considering the sulfur Kα (2.3 keV) and Kβ (2.5 keV) lines from gesso primer on the integral of the spectrum between 2.2 keV and 2.6 keV. In this map, the organic finishing layers are also clearly underlined and give a different result depending on their composition and thickness. This allows us to discriminate different surface layers on the same color/part of a painting, and thus, to speculate about the presence of restorations. In this way, MA-XRF proves once again to be competitive against imaging analyses, and can be considered a useful tool each time a surface light element layer must be evaluated, as well as for detecting the presence of patinae in metallic historical objects [70,71,72]. Indeed, considering emissions at such low energies, and thanks to the high performance of the IRIS X-ray detector, it is even able to highlight dark stains in mockups with oil as a binder due to oil spreading in the primer.

From the ROIs reported in Figure 7, the integrals were computed, and then, from the ratio of the integrals of the two emission lines, the thickness was calculated with the inverted formula obtained from the calibration procedure, shown in Equation (3). The output is represented, as in the right part of the middle row in Figure 8, via heatmap plotting. The plot was created using the Python open-source library Seaborn.

• The obtained image was transposed with respect to the actual object;

• The support is represented as a mid-thickness layer, which is incorrect.

• The information from the image was stored using the Python Pandas open-source library, exploiting the DataFrame built-in object. DataFrames are created as tables that collect keys in the form of [row]

  . On the contrary, the IRIS software creates a matrix of the type
  [row][spectrum], and populates it starting from the bottom left, scanning towards the right and ending at the top right. Therefore, it is necessary to lock the columns and range on the rows.

• Since the wood support presents noise in the lead line regions, due to backscattered radiation from the excitation source, the contrast can be highly increased considering a threshold to be overcome by at least one of the ROIs’ integrals. That threshold was set as the number of bins in the ROI times a constant of 1.1.

The output from the fast acquisition and from the high-spatial-resolution acquisition obtained after these implementations are shown, respectively, in the lower row of Figure 8. The parameters of the acquisitions are described in Table 2.

From both the fast-scanning and high-resolution acquisitions, the thickness map was thus successfully obtained. It is worth recalling that fast and high-resolution scans are performed with different integration times and collimator diameters. Due to inhomogeneity of the substrate, the difference in collimator diameters may affect the output result of the thickness estimation, since the averaging effect of a smaller collimator is reduced compared to a larger one, producing different results, as evident in Figure 8.

The mean estimated thicknesses of azurite of the three samples spread in oils (A1, A2 and A3 in Figure 1) are, respectively, A1 = 93 ± 59 µm, A2 = 63 ± 63 µm and A3 = 96 ± 58 µm. The expected thickness for azurite layers, reported in Section 3.2, are 37 ± 7.4 µm for A1, no azurite layer for A2, and 18 ± 1.5 µm for A3. For the A1 layer, the thickness estimation is coherent, inside the error, with the one measured from the cross section. For A2, a layer made of ultramarine blue alone (no azurite present, so no Cu signal besides the background), the result is negligible, and this indicates the absence of the investigated pigments and the Cu signal to be below the detection limit of the spectrometer. Moreover, in cases whereby a Cu signal is present from impurities of the materials layered, the calibration obtained in this work would obviously not be useful for calculating the lapis lazuli layer thickness. For the A3 layer, the obtained average thickness is highly overestimated and not in good agreement with measured one. In this peculiar case, the azurite layer is not the upper layer, but it is covered by an ultramarine blue layer. Considering the chemical composition of lapis lazuli [73], which can be written as (Na,Ca)₈(SO₄,S,Cl)₂(AlSiO₄)₆, and also considering the dilution in linseed oil, which can be considered to be linoleic acid [45], the variation in the absorption coefficient, respectively, for the Lα and Lβ Pb lines does not significantly affect the ratio. Once again, we cannot use the obtained calibration for calculating the thickness of the lapis lazuli layer.

A fundamental aspect to be taken into account is that, as already indicated in Section 3.1, underlying lead white layers in paintings do not have infinite thickness [74]; in extreme cases, the lead-based preparatory layer can be so thin that no signal from lead would pass the azurite layer, even if it is a rare situation. It is worth noting that the overestimation of the layer thickness is due, in part, to the strong hypothesis made when performing the calibration: the lead support must be sufficiently thick to be approximated as infinite for X-ray penetration compared to the pigment layer thickness. That is, the actual lead white layer could produce fewer L signals. It has been demonstrated [68] that the value for Au (La/Lb), due to self-attenuation, does not change in an appreciable manner with thickness, reaching an almost constant value for a thickness of about 20 μm, implying that the results for the A1 layers are less affected by this aspect. As shown from the invasive analysis results [45], the lead white layer is typically thinner than the blue layer, so the fluorescence from the lead cannot build up completely, resulting in Lβ line underestimation and subsequent thickness overestimation. This effect can be effectively mitigated by calibrating the instrument on lead white pigment layers instead of a thick block of metallic lead. For metals, it has been demonstrated that when the incident radiation is sufficiently higher than the absorption edge, the ratio in the case of infinite thick and thin samples can be calculated as the ratio of the linear attenuation coefficient (in cm⁻¹) at the energy of its Lα with that one at the energy of its Lβ [69]. This could suggest a correction factor to use for refining of our method.

A further aspect to be taken into account to explain the difference from the measured thickness of the A3 sample is the percentage of binder in the pictorial layers, which may vary with the different adsorption properties of the pigments themselves, i.e., azurite and lapis lazuli may require different oil concentrations to be conveniently applied. In this regard, the results obtained in the already quoted work [45] on the same mockups also show an overestimation of the azurite layer when it is supposed to be in a mixture with 90% binder, which is the case for our stand-alone calibration samples.

We must thus keep in mind that this kind of calculation must forcibly be considered for estimations, as it is also an estimation of the measured average value over a hand-layered pigment; in fact, the pigment layers spread out by a brush by the artist’s hand, as in our case, makes the layers not constant in thickness. Indeed, the non-uniform layers are highlighted in the reported thickness maps and are also evident from the large errors reported in the thickness determination obtained by cross-section measurements [45].