Characteristic and Continuous Components of the Emission Spectrum From an X ray Tube

Geology and Mineralogy Applications of Atomic Spectroscopy

Michael E. Ketterer , in Encyclopedia of Spectroscopy and Spectrometry (Third Edition), 2017

XRF

X-ray absorption and emission spectra originate from electronic transitions of inner-shell electrons; these transition energies are only slightly affected by the physical or chemical form of the element of interest. The transition energies of K (n=1) and L (n=2) inner-shell electrons are closely related to the atomic number, as was discovered in the early twentieth century by Moseley. XRF spectra can be generated by bombarding a sample with a focused monochromatic X-ray beam; monochromatic X-rays are produced by ∼10–100   keV electron bombardment of a metal target in an X-ray tube. The incident X-rays interact with the sample atoms, exciting inner-shell electrons into higher energy states. Subsequently, X-rays of other energies, characteristic of the elements present in the sample, are emitted. Similar spectra can be generated using in vacuo electron bombardment of solid samples, as is done in electron microscopy. X-ray spectra generated by either method afford both qualitative and quantitative information; qualitative indication of the presence of an element stems from the appearance of expected X-ray lines of specific energies. Quantitative analysis is possible because the emission intensity is proportional to atomic concentration, although matrix effects exist due to differences in sample transparency to the incident electrons/photons or emitted X-rays. The matrix effects are strongly associated with the sample's bulk density, geometry, and average atomic number; several successful computational methods such as 'fundamental parameters' allow for routine, robust quantitative analysis. Two types of spectrometers are in common use: 'dispersive' spectrometers based on diffracting crystals, that is, an X-ray monochromator, and 'nondispersive' spectrometers based on semiconductor detectors with energy-discrimination capabilities. The latter are more common and less expensive; these are usually referred to as 'energy-dispersive' spectrometers. Both XRF and its closely related technique, scanning electron microscopy with energy-dispersive X-ray analysis (SEM-EDX), are well suited for analysis of major and minor elements, with some capabilities for trace (<100   ppm) measurements. XRF and SEM-EDX have two strong points: first, SEM-EDX has excellent capabilities for microanalysis with spatial resolution <1   μm limited by the focusing of the electron beam and the sample interaction volume. Second, XRF and, for conducting samples such as metals, SEM-EDX are nondestructive. These techniques can be used as an initial screening analysis without preparation, or when nondestructive analysis is strictly necessary (i.e., in archaeometry). Both XRF and SEM-EDX have broad elemental coverage (elements with Z=11 and higher are routinely analyzed), although some spectral overlaps exist between the K lines of light elements and the L lines of heavier elements. An overview of the applications of XRF to the analysis of environmental samples (soils, water, sediments) is given in the study by Melquiades and Appoloni.

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DV-Xα Calculation of X-Ray Emission Spectra

Jun Kawai , in Advances in Quantum Chemistry, 1998

1 Introduction

X-ray emission spectra of solids and molecules are methods of measuring electronic structure of matter [1–5]. The x-ray emission spectra reflect the occupied electronic structure as shown in Fig. 1 , while the x-ray absorption spectra reflect the unoccupied molecular orbitals (MO). These x-ray spectra represent local (L) and partial (P) electron density of states (DOS) because of the electric dipole selection rule, and thus the x-ray spectroscopy is a powerful tool to study the electronic structure of matter. The development of synchrotron radiation facilities makes it possible to measure the x-ray emission spectra more easily with using a resonance condition[6,7], and we can get information on the detailed electronic structure of matter using the characteristics of synchrotron radiation such as resonance and polarization.

Fig. 1. Measured sulfur Kβ (K-M _2,3) x-ray emission spectrum of Na₂SO₄ and the assignment of x-ray transitions for the SO₄ ^{2   −} cluster. The x-ray intensity reflects the S 3p atomic orbital component in the molecular orbitals.

(Taken and modified from Kawai et al. [1].)

The calculation of x-ray emission spectra of molecules or solids are one of the most successful applications of the discrete variational (DV) Hartree-Fock-Slater (Xα) MO method using cluster approximation [8–10], which was originally coded by Ellis and his coworkers [11–14] based on Slater's Xα exchange potential [15]. The DV-Xα method has several advantages for the calculation of x-ray transition process as follows.

Firstly, it can calculate the inner-shell hole state. The atomic orbitals as well as the molecular orbitals are relaxed by the inner-shell or valence shell hole in the self-consistent field (SCF) calculation. This is because the DV-Xα method uses SCF numerical atomic orbitals as the basis set in an atomic potential in a cluster and thus the orbitals contract or expand according to the hole potential. They also self-consistently contract or expand due to the formation of chemical bonds. Figure 2 shows the Fe 3d atomic orbital used as one of the basis functions in the DV-Xα method for the ground state or the 1 s ^{−   1} hole state. It is found that the 3d wavefunction contracts due to the core hole potential. The Gaussian type orbital (GTO) basis sets are usually fixed and thus the relaxation effect due to the core hole is included either by the change of MO coefficients or by the configuration interaction. Thus the expression of core hole state by the fixed basis function requires a large basis set. On the other hand, the numerical atomic basis functions are self-consistent themselves, and thus are good eigenfunctions of the Hamiltonian. Therefore the DV-Xα method usually needs only the minimal basis set for the MO calculations.

Fig. 2. Iron 3d atomic orbital used in the DV-Xα method as a basis function. Solid line: ground state, and dotted line: 1 s ^{−   1} hole state.

Figure 3 shows the electron density of states of [MnO₆]^{10   −} for the ground state and the 1 s ^{−   1} core hole state [16]. This cluster is a model cluster of MnO. In the ground state, the Mn 3d and O 2p are separated in energy and the hybridization is week. Thus MnO is an ionic solid. In the 1 s ^{−   1} core hole state, however, the Mn 3d orbital energy becomes as deep as that of the O 2p and they hybridize strongly each other. Thus we can know, using the DV-Xα method, that the initial state of x-ray emission (or final state of photoionization) of MnO is no longer ionic but covalent.

Fig. 3. Electron densities of states of [MnO₆]^{10   −} cluster for the ground state (left) and the 1 s ↑^{−   1} hole state (right) calculated by the spin-included DV-Xα method. The shadowed area indicates oxygen 2p component.

(Taken from Suzuki [16].)

The second advantage of the DV-Xα method is that the basis functions of the DV-Xα method are atomic orbitals. Thus the number of nodes is exact as shown in Fig. 4 , where Si 2 s GTO in GAUSSIAN method is compared with the numerical basis function used in the DV-Xα method [17]. The use of the atomic orbital wavefunction makes it possible to perform the direct calculation of the electric dipole matrix elements, e.g. <   1s| r| 2p>, using the DV-Xα MO, yielding better result than when using a GTO basis MO.

Fig. 4. Comparison of atomic basis functions of Si 2 s used in the DV-Xα (solid line) and GAUSSIAN 6-31G* (broken line).

(Taken and modified from Nakamatsu [17].)

The third advantage of the DV-Xα method is in the cluster approximation. The core hole state as the initial state of x-ray emission or final state of x-ray absorption is treated as an impurity atom in solid when using the band theory. However, since the core hole potential usually affects the next nearest neighbor atoms at most, it is enough for the calculation of x-ray emission spectra to use the clusters which include the second nearest neighbor atoms to the x-ray emitting atom as is described below.

The fourth advantage is that the precision of the DV calculation is comparable to the experimental precision. The energy resolution of x-ray optics is at most 0.01   eV, which is the comparable value to the DV-Xα precision. Thus the infrared spectra cannot be reproduced by the DV-Xα method.

We have calculated for these several years the x-ray emission spectra of solids and molecules and we have found that the calculation of x-ray emission spectra is one of the most successful applications of the DV-Xα method. In the present paper, results of our ongoing research, as well as published results are described.

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X-Ray Emission Spectroscopy, Methods

George N. Dolenko , ... Jolanta N. Latosińska , in Encyclopedia of Spectroscopy and Spectrometry (Third Edition), 2017

Characteristic X-Ray Emission Spectra

Characteristic X-ray emission spectra consist of spectral series (K, L, M, N…), whose lines have a common initial state with the vacancy in the inner level. Labels of basic X-ray transitions are shown in Figure 2. All electron levels with the principal quantum number n equal to 1, 2, 3, 4, etc. are named as K, L, M, N etc. levels and denoted with corresponding Greek letters and digit indexes. The electron transitions which satisfy the dipole selection rules

Figure 2. Scheme of the most important X-ray emission transitions; n, I and j are correspondingly the principal, orbital and total quantum numbers of K, L₁, L₂, L₃ levels, etc.

[1] $\text{Δ}l=1;\text{Δ}j=0,1\left(j=\pm \frac{1}{2}\right);\text{Δ}n\ne 0$

are most intense. The dependence of X-ray emission line energy on atomic number Z is defined by Moseley's law:

[2] $h\nu \sim {\left(Z-\sigma \right)}^2$

where Z is the atomic number and σ is the shielding constant, which varies from series to series. Therefore, any X-ray emission spectral line is the finger-print of an element.

With X-ray emission excitation by electron bombardment (primary emission) all emission lines of the ith series appear when the X-ray tube voltage U exceeds the ionization potential of the ith level (V _i ). At higher U the intensity of all lines of the ith series, I _i , increases because the electrons penetrate deeper into the target substance and, therefore, the number of excited atoms in the target increases. In the V _i <U<3V _i region, the intensity obeys the rule I _i ∼(U−V _i ) ². With a further increase in U X-ray emission begins to be absorbed by the target atoms; therefore the increase in I _i is reduced. At U≥11 V _i , I _i decreases because now most of the excited atoms are so deep in the target that their emitted radiation is absorbed by the target substance.

X-ray emission spectra are usually excited by X-ray photons because most chemical compounds are decomposed by electron bombardment. With X-ray emission spectra excitation by photons [secondary emission or fluorescence (XFS)] the fluorescent line intensity depends on the exciting photon energy hν and I _i =0 if hν<V _i . All lines of the ith series appear if hν=V _i ; however, I _i decreases little with further increase in hν. Therefore, to excite X-ray fluoresence one must use a target that contains a substance with intense characteristic X-ray lines whose energy just exceeds eV _i . Using the continous radiation of an X-ray tube with a target consisting mostly of heavy elements it is possible to excite X-ray fluorescence.

The intensity of a characteristic X-ray spectrum (both primary and fluorescent) depends on the probability p _r of a radiation transition in the atom having the vacancy in the ith level. The value of p _r is determined by the total probability of photon emission when this vacancy is filled by outer electrons. However, with a probability p _A the vacancy may be filled by outer electrons without radiation as the result of the Auger-effect (see Figure 1). For the K series of medium and heavy elements p _r>p _A, for the light elements p _r<p _A. For all other series of any elements p _r<<p _A. The ratio f=p _r/(p _r+p _A) is called the yield of characteristic radiation.

However, X-ray characteristic lines appear because of single atom ionization; in X-ray emission spectra weaker lines are found to occur as a result of binary (or multiple) atom ionization when two (or more) vacancies are formed simultaneously in different electron shells. If, for example, one vacancy is formed in the K shell of atoms and filled by electrons belonging to the L_2,3 shell, atoms emit an Êα_1,2 doublet. If another vacancy is formed simultaneously which too is filled by electrons from the L_2,3 shell, then the final state will have a binary ionization L_2,3L_2,3, and would correspond to the emission of radiation with energy exceeding that of the Êα_1,2 doublet. As a result, in an X-ray emission, spectrum a short wavelength Êα_3,4 doublet, called a satellite of the main Êα_1,2 doublet, would appear. Because of such processes of multiple ionization X-ray emission, spectra may have a large number of satellites of the main lines. Usually, the satellite intensity is some orders of magnitude less than that of the main lines. However, if target atoms are bombarded by heavy ions with great energy, the probability of multiple atom ionization becomes higher than that of single ionization. Therefore, in this case the intensity of the main emission lines is essentially less than that of the satellites.

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X-ray Absorption Spectroscopy in Biology (BioXAS)

Martin C. Feiters , Wolfram Meyer-Klaucke , in Practical Approaches to Biological Inorganic Chemistry, 2013

In the X-ray emission spectrum of a first row transition metal (such as the Mn ²⁺ represented in Figure 6C, left), the Kα₁ and Kα₂ lines are well resolved and more intense than the Kβ₁ and Kβ₃ lines, which are not resolved, by an order of magnitude; these are in turn more intense than the Kβ satellite lines Kβ_2,5 and Kβ″. Not indicated in Figure 6C, but usually present for transition metals which have a total electron spin S   ≠   0 (such as Mn²⁺), is the Kβ′ line at slightly lower energy than the Kβ_1,3 line. This results from emission from the metal 3p level combined with a spin flip of a 3d electron and is therefore sensitive to the spin state of the metal ion. Of the Kβ satellite lines, the cross-over emission line Kβ″ is extremely sensitive to the nature of the coordinating ligands, because it involves emission from the ligand's 2s level to the metal's 1s core hole, and allows one to distinguish O from N and C ligands. This ligand identification is of interest as it gives information complimentary to that obtained from EXAFS, which can typically not discriminate between coordination by ligands from the same row of the periodic table, see text. Examples are the variation in the number of O ligands to Mn in the so-called Kok cycle in Photosystem II (Yano and Yachandra, 2008), and the identification of central atom bound to Fe in the Fe, Mo cofactor of nitrogenase as C (Lancaster et al., 2011). A typical theoretical approach calculation of the multiplet 'ligand field multiplets' to interpret Kα and Kβ main lines, and molecular orbital theory for the Kβ satellites (Glatzel and Bergmann, 2005); both are outside the scope of this chapter.

FIGURE 6C. Overview of ligand edge, L edge, and X-ray emission transitions; the y-axis only gives relative energies and is not to scale. Left panel, effect of 'low-Z' ligands (C, N, O, F) on the X-ray emission of a Mn²⁺ complex. K edge excitation (blue) leads to a 1s core hole intermediate state (central green oval), which can emit X-ray fluorescence at various wavelengths. The final state obtained with Kα₁ fluorescence (red box) is identical to that obtained by direct L₃ edge excitation (red). Right panel, illustration of probing mixed 3p orbitals of Cl or S ligands with a transition metal's 3d orbital, in this case the singly occupied 3d _{x² -y²} orbital of Cu²⁺ (other Cu 3d orbitals grouped together for clarity), by either K edge or L₃ edge XAS.

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High Energy Ion Beam Analysis

Geoff W. Grime , in Encyclopedia of Spectroscopy and Spectrometry, 1999

Quantification of PIXE spectra

Figure 2 shows that a PIXE spectrum consists of peaks which are approximately Gaussian sitting on top of a continuous background. The background-subtracted area of each peak is proportional to the concentration of the associated element, and a number of software packages are now available to perform the processing required to extract the areas and convert these to true concentrations.

For a sample consisting of a thin film (so that proton energy loss and X-ray absorption can be neglected), the yield of characteristic X-ray photons of energy E _x from element Z induced by particles of energy E is given by

[2] $N=C_{\text{z}}Q\text{Ω}\epsilon (E_{\text{x}})Y(Z,E)$

where N is the total number of photons detected using a detector of solid angle Ω, Q is the total beam charge, C _z is the concentration of the element and ε(E _x) is the dependence of detector efficiency on X-ray energy. Y(Z, E) is the yield of the interaction expressed as counts per unit of concentration per unit of charge per unit of solid angle. This, in turn, is derived from the ionization cross-section (the probability of creating a vacancy) and the fluorescence yield (the number of photons emitted in each X-ray line for each vacancy).

In more realistic thick samples, the calculation must also take account of the energy loss of the particle as it penetrates the sample and also the absorption of the X-rays as they emerge from the sample. To do this, a knowledge of the bulk composition of the sample is required (together with any variation with depth) and both the stopping power of the ion in the matrix and the X-ray attenuation coefficient must be known. The calculation then involves a numerical integration of the total X-ray yield from each sublayer of the sample.

All of the physical parameters involved in this calculation (ionization cross-section, fluorescence yield, stopping power, X-ray attenuation and detector efficiency) have been measured and parameterized with sufficient accuracy that quantitative PIXE analysis may be carried out without routine reference to standards once the fixed system parameters have been determined. A small number of computer programs have been developed to carry out PIXE spectrum processing, and special mention must be made of GUPIX, developed at the University of Guelph, Ontario, which embodies the most detailed physical model of the PIXE process and also offers the capability to iterate the sample matrix composition (including user-defined 'invisible' elements such as oxygen) until the matrix composition matches the values obtained from the peak areas.

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Ion beam analysis for cultural heritage

Chris Jeynes , in Spectroscopy, Diffraction and Tomography in Art and Heritage Science, 2021

5.3 Elastic backscattering spectrometry

In the discussion of PIXE we have ignored them so far, but of course atoms do have nuclei, and the incident ion beam can scatter from these nuclei. This scattering is elastic and (like billiard balls) can be directly backwards (180 degrees scattering angle) for a head-on collision. Of course, detecting such events is impossible since then the detector would have to be in the incident beam! But you can get quite close to 180 degrees, and it turns out that there are good analytical reasons to use large scattering angles. Fig. 5 shows a scattering angle of 150 degrees: this sample (from the recent restoration of the Rosslyn Chapel near Edinburgh [35]) is a 19th century soda-lime glass whose composition was not known beforehand.

Fig. 5. Elastic backscattering spectrometry of Rosslyn glass. Top (linear scale): EBS total spectrum from a 19th century soda-glass using 3080 keV protons and a backscattering angle of 150 degrees. The O non-Rutherford signal dominates the spectrum. The feature labelled "R" shows the effect in the spectrum of the 2663 keV resonance in the O signal. Partial spectra are shown for O, Na, Si, Ca (respectively purple, blue, orange, green). Top (inset): detail of the EBS signal above the O edge. Centre &amp; bottom (log scale): EBS of minor elements Al, K, S (purple, green, dark green); RBS of minor elements Pb, Ba, As, Mn (red, blue, orange, magenta).

From Fig. 5A of C. Jeynes, V.V. Palitsin, M. Kokkoris, A. Hamilton, G.W. Grime, On the accuracy of Total-IBA, Nucl. Instrum. Methods Phys. Res. B 465 (2020) 85–100 (© Elsevier B.V., 2020, by permission).

Even though the glass is uniform in composition, Fig. 5 is clearly a complicated spectrum: there are no less than 16 measurable elements present in the glass! How are such spectra to be interpreted? And when would we need such data?

At the beginning of the 20th century, the big questions in physics included trying to work out what the mysterious α, β, and γ rays were. Ernest Rutherford's team in Manchester were watching how α rays passed through gold foil, and they noticed α particles scattering at all angles including more or less straight backwards! They were greatly surprised since by that time they already knew that an α particle was He⁺⁺ (Rutherford got the Chemistry Nobel Prize for this in 1908), and they knew that there were protons and electrons in atoms (they did not know about neutrons then), but they thought these positive and negative charges were all mixed up in the atom, since unlike charges attract (J.J. Thompson's "plum pudding" model was proposed in 1904). But Rutherford recognised that this newly observed scattering behaviour meant that the atomic nucleus (including all the protons) had to be both very small and also the same polarity as the α particle: by simply pretending that the electrostatic force (like charges repel) was responsible he was able to account entirely for the observations. The Rutherford model of the atom was published in 1911 [36], and Rutherford backscattering spectrometry (RBS) refers to this analytical model. To sum up: RBS assumes point charges in a Coulomb field.

But this must be an approximation! Neither the α particle nor the atomic nuclei of the sample are "point charges": the α particle has a radius 1.68   fm and the O nucleus has a radius 2.71   fm [37]. These values may be small, but they are not nothing! And the ion beam does have a high energy! It turns out that when the incident particle is going fast enough, it comes close enough to the scattering nucleus for the nonzero sizes to become important. In this case you cannot pretend that Rutherford's approximation is good enough anymore, and you have to do a proper quantum mechanics calculation to work out what happens. This gets complicated: an overview with references to the technical literature can be found in Ref. [38].

Fig. 5 shows this complicated non-Rutherford behaviour: there is a pronounced "dip" in the O signal, and the Al signal has got lots of peculiar wiggles in it. Actually in this case, only the signals from the heaviest elements, including lead (Pb), are Rutherford. Looking at the literature, elastic scattering spectra are quite often (but incorrectly) called "RBS" when actually EBS (non-Rutherford) spectra are in view. In fact, especially for cultural heritage applications where proton beams are usually used, the particle spectra cannot be correctly interpreted unless they are treated as EBS (not RBS) spectra. Fig. 9 of Jeynes and Colaux [38] shows that for 2 MeV protons, elastic scattering is not Rutherford for all elements lighter than Fe.

To interpret EBS spectra we need to know what the probability is of a certain energy particle scattering from a given nucleus into a given direction. This probability is known as the "differential scattering cross-section" σ′, which for RBS and the centre of mass frame of reference is a very simple formula (Eq. 1 in [39]):

$\sigma \prime ={\left\{Z_1{Z}_2{e}^2{cosec}^2\left(\theta /2\right)/\left(4E\right)\right\}}^2$

where Z ₁ and Z ₂ are the atomic numbers of the incident and target nuclei, e is the charge on the electron, θ is the scattering angle and E is the beam energy at the scattering event. But for EBS σ′ must be measured : it is not possible to calculate it ab initio sufficiently accurately for analytical purposes (and in any case, the calculations are very difficult).

The good news is that in the last couple of decades an enormous amount of work has been done both to catalogue all the scattering cross-section measurements done over the last 50 years or so (and make new measurements) [40, 41], and to fit these data to nuclear scattering models [42]. This means that although Fig. 5 shows that EBS for glass is complicated by the need for extensive and detailed scattering cross-section data, these data now exist in a form convenient for analysts. Therefore, EBS is now feasible in a way that simply was not available last century.

However, given the difficulties, why should curators want EBS spectra from their precious samples when they already have detailed information from PIXE? The first and trivial answer is that they are there for free! When the proton beam strikes, the sample responds in multiple ways: IBIL, PIXE, EBS (and the other responses including PIGE we will discuss below). You only have to put in the detector to collect the data! And given that the sample is precious, so are the data.

But in fact, we want the EBS data specifically when we suspect interesting things going on at the sample surface (such as corrosion effects, or surface coatings—gilding, etc.: we will discuss various such examples in Section 6). To understand this, consider the differences between PIXE and EBS spectra (compare Figs. 3 and 5). The abscissa on the PIXE spectra is in units of "keV," where for the EBS spectra it is in units of "channel number". This is because PIXE intrinsically gives an energy spectrum, but EBS intrinsically gives an energy loss spectrum. For PIXE the characteristic lines have their own characteristic energies which automatically calibrate their own energy scale. But for EBS the energy scale has to be calibrated externally. The signal for PIXE is the energy (colour) of X-ray photons, and photons are either absorbed or not as they pass through the material. If they are not absorbed their energy does not change. But for EBS the signal is the energy of the scattered particle, which steadily loses its energy as it passes through matter. That means that EBS is sensitive to the elemental depth profile in a way the PIXE spectrum is not.

This means that the EBS spectrum is more complicated than the PIXE one. If we ask the spectrum, what sample generated you? there is an analytical answer for the PIXE spectrum, but the EBS spectrum will answer: this question is mathematically ill-posed. Programs to automatically handle PIXE data (and the very similar XRF, SEM-EDX, SEM-WDX, EPMA ^b spectra) have been standard for over 20 years. But fitting EBS spectra in general (including fitting the elemental depth profile) must use advanced numerical methods (see the "Topical Review" of Jeynes et al. [43] describing the DataFurnace code: to date there is no other code capable of fitting PIXE   +   EBS datasets self-consistently; the use of DataFurnace for such Total-IBA was reviewed in 2008 [44], in 2012 [58], and in 2020 [12]).

Note that fitting a spectrum (i.e., extracting from the spectrum the composition of the sample that generated it) is the inverse operation to simulating the spectrum one would get from that sample. Of course, simulating the EBS spectrum is "easy" (provided one knows the scattering cross-sections) and suitable programs have been available since the 1970s, but the inverse operation—fitting it—is actually ill-posed mathematically.

We now need to look in more detail at the EBS spectrum in Fig. 5. Remember that this is the spectrum of a simple piece of uniform glass, for which the only interesting question from the ion beam analysis point-of-view is, what is my (elemental) composition? Therefore, the complexities of the spectrum are entirely due to the technique not the sample, which is pedagogically very helpful. There are four crucial aspects to understand:

1.

Elemental edges. Looking at the logarithmic plot of the whole spectrum it is clear that there are four main steps, due to (i) the heavy group of elements (mostly Pb), (ii) the intermediate group of elements (mainly Ca), (iii) the matrix elements (mainly Si), and (iv) oxygen. In an elastic collision the recoiled particle takes more energy from the incident particle the more closely their masses match. Think of a ping pong ball bouncing off a billiard ball. The little ball will bounce very fast and the big ball will hardly move. But if the little ball is a bit heavier—say a tennis ball—it will not come back so fast and the big ball will roll away more. So here, the incident proton bounces back from the lead atom with almost all its initial speed, but it bounces off the oxygen atom much slower.

2.

Mass-depth ambiguity. The spectrum is continuous! Compare channels 341 and 421 (the O and Pb edges). If you detect a backscattered proton with an energy that puts it in channel 421, it must have scattered from a Pb atom at the surface of the glass. But if you detect a backscattered proton with an energy that puts it in channel 341, it could have scattered from a O atom at the surface of the glass, or it could have scattered from a Pb (or other) atom buried in the sample. The incident proton would have had to go through some glass to get to it, losing energy on the way, and then having scattered it would have to come back along a (slightly different) path, losing more energy. This is why we say that the EBS spectrum is an energy loss spectrum. It is this mass-depth ambiguity that makes inverting the spectrum an ill-posed problem. And what makes interpreting EBS spectra notoriously difficult.

3.

Mass closure properties. Glass is always essentially a mixture of oxides. In this case there is no boron or lithium so that the EBS spectrum sees all the elements in the sample—especially including the oxygen! Note that this is not the case for RBS spectra, for which the scattering cross-section goes with Z ²—the square of the atomic number. So for RBS the step height ratio of the O:Si signals for this sample (with an O:Si ratio of 2.4) would be (2.4) * (8/14)²  =   0.8. But for this EBS spectrum the O:Si step height ratio is about 3. This is because the scattering cross-section for 3 MeV protons on O at this angle is about seven times what it would be if it were Rutherford scattering (protons on Si is not Rutherford either). The sensitivity to the light elements may be greatly enhanced in EBS.

But now, with all the signals visible, the spectrum has a mass closure property that PIXE spectra do not have (since PIXE does not usually detect elements lighter than about Na). That is, the composition of the sample is determined unambiguously by the spectrum. It turns out (see [14] for details) that this mass closure property allows the analyst to correct the spectrum for over- or underestimated scattering cross-section functions. So the inset of Fig. 5 showing the details of the Si signal also shows that the fit is systematically below the data. That is, the spectrum is illegal, strictly speaking: it cannot be fitted properly by any composition under the given assumptions—which of course include the scattering cross-section functions used. Putting this another way, the spectrum allows one to extract correct cross-sections (this has been known and used since the 1980s: see [12] for details). In this case, even though it is clear that the scattering cross-sections for Si are underestimated by about 5%, the sample composition is obtained with a much lower uncertainty of about 2%.

4.

Synergy with PIXE. In this case it is clear that the trace elements are undetectable by EBS, although they are easily measured by PIXE. Actually, in this case EPMA (i.e., SEM-WDX) was used to obtain the minor elements Al and Na (as well as the matrix elements—but not O). In principle both are obtainable with PIXE but were not in this case. Na can be measured successfully by EBS, but not so accurately as EPMA. It is important to stress the synergy between PIXE and EBS: each is strong where the other is weak.

The value of the backscattering spectrum is obvious where one wishes to measure (for example) gilding layer thicknesses, as Ortega-Feliu et al. [45] have done recently. But EBS is invaluable wherever there is compositional variation near the surface (corrosion layers for example), as we will discuss in the following text. Such cases have been regularly ignored up to now because the spectra are difficult, requiring the analyst to make use of the EBS-PIXE synergy. This has become possible only quite recently (see the review of IBA depth profiling by Jeynes and Colaux [38]).

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Experimental and Computational Methods in Supramolecular Chemistry

C. Tedesco , M. Brunelli , in Comprehensive Supramolecular Chemistry II, 2017

2.04.3.4 Line Fitting and Whole Powder Pattern Fitting

XRPD patterns arise by the convolution of instrumental and sample factors. Instrumental contributions include both effects related to the geometry of the diffractometer (axial divergence, sample displacement, etc.) and to the shape of the X-ray emission spectrum (the X-ray-source size, the angle of divergence of the incident beam, etc.). Sample contributions include particle size distribution, nonhomogeneous strain, and textures.

The idea of pattern fitting is at the basis of the Rietveld method: in its beginning the method was applied to low resolution, constant wavelength, neutron powder diffraction data, whose line shapes are well described as simple Gaussian functions considering the instrumental and sample contributions to the pattern as a whole.

With the application of the Rietveld method to XRPD data, it became evident that simple Gaussian functions would be inadequate and other functions were devised to achieve an empirical description of the line shape, as pseudo-Voigt (a linear combination of Gaussian and Lorentzian functions) or Pearson VII functions.

A different approach to the problem of pattern fitting is proposed and applied by taking into account every contribution (i.e., instrument geometry, radiation source, sample size, etc.) to the pattern by considering instead fundamental parameters, that is, sample and slit lengths, receiving slit and source widths, horizontal divergence, primary and secondary Soller slits angles, etc.

Indeed the use of a phenomenological description of the diffraction pattern is standard practice in Rietveld refinement, but a fundamental parameters approach would probably become more and more used as the research in the field of line profile analysis advances. ^{40 , 41}

Fitting of single Bragg peaks is useful for several different purposes, that is, determination of peak positions for indexing purposes, stress characterization, and determination of microstructural parameters.

Whole powder pattern fitting can be applied with or without a structural model, in the former case this leads to the Rietveld method for structure refinement, ^{15 , 16} in the latter case this leads to whole pattern decomposition methods, as Pawley ¹⁷ and Le Bail ¹⁸ methods.

Whole pattern decomposition methods maybe considered as variants of the Rietveld method in the absence of a structural model.

Pawley, in 1981, showed that it was possible to determine the integrated intensities by applying a least-squares analysis of the XRPD pattern, as in the case of Rietveld refinement, considering as variables the peak areas themselves. ¹⁷

His approach involves the least-squares fitting of the diffraction pattern by adjusting the background parameters, cell parameters, peak shape parameters, and integrated intensities. The method assumes that all the reflections have independent variable intensities. Therefore, during a least-squares refinement, although the total area of a group of overlapping peaks can be well determined, the individual intensities can assume any value (including negative ones), which could lead to completely unrealistic results. This problem was mitigated by constraining the individual intensities to be close to the "mean value" calculated for overlapping peaks.

Subsequently Le Bail, in 1988, ¹⁸ proposed to assign arbitrarily equal values to the integrated intensities of overlapping reflections, which are no longer treated as least-squares variables and are never refined. In this way each least-squares cycle is very fast since the matrix remains small.

Initially the intensity values are treated as calculated values as if they had been derived from a structural model, then it is applied the Rietveld procedure for partitioning the observed peak intensity among the hkl reflection that contribute to it.

The least-squares procedure will minimize the difference between the observed and calculated y(i) intensities and will provide better "calculated" integrated intensities that will be used in the subsequent cycles until convergence is reached.

It must be noted that in case of severe overlapping (as it occurs systematically in high symmetry systems or as it may occur at high angular values), both methods will tend to give erroneous values and care must be taken in using such data for structure solution purposes.

Noteworthy, the final values of the pattern R factors in a Pawley ¹⁷ or Le Bail ¹⁸ refinement, R _p or R _wp, will represent a lower limit for the pattern R factors in subsequent refinement procedures.

Whole pattern decomposition methods maybe used to refine the cell parameters and to help in space group determination, but mainly is used to extract the integrated intensities for structure solution purposes.

Also in cases where the extraction of the integrated intensities is not needed for structure solution, whole pattern decomposition methods are useful to establish the initial values of the profile parameters to be used.

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X-ray absorption and emission spectroscopy in biology

Martin C. Feiters , Wolfram Meyer-Klaucke , in Practical Approaches to Biological Inorganic Chemistry (Second Edition), 2020

As an example of a photon-in–photon experiment with incident energy ω _in and emitted energy ω _out, the analysis of the 2p to 1s emission stimulated by 1s-3d absorption of a 3d transition metal is given in Fig. 7.7, left. The accurate measurement of ω _out in X-ray emission spectra requires element-specific analyzer crystals that have a high-energy resolution, close to the natural linewidth. In the Johann approach, this is achieved by point-to-point scanning using a crystal bent with radius 2R, which is placed with the sample and detector on a Rowland circle with radius R to ensure that radiation from the sample that is selected by the crystal hits the detector (Fig. 7.7, right). In a wavelength dispersive arrangement, X-rays emitted by the sample are diffracted and the photons of different energies detected separately with a position-sensitive detector.

Figure 7.7. (Left) Principle of a hard X-ray photon-in/photon-out experiment. (Right) Experimental setup at a storage ring beamline, with the sample, analyzer crystals, and detector in Rowland circle geometry. The arrows indicate the motion of the components when a spectrum of the emitted X-rays is taken.

Source: Adapted from Glatzel, P., Sikora, M., Fernández-García, M., 2009. Eur. Phys. J. Special Top. 169, 207–214.

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