The presence of higher harmonics in the incident X-ray beam distorts XAFS spectra. The amount of harmonics sometimes changes during the measurement of spectra. When the amount of harmonics changes smoothly during a scan, distortions may not be apparent in the spectrum. Even if distortions do not appear in the spectra, spectra measured using X-rays containing higher harmonics are incorrect. Thus, we need to be sure to eliminate harmonics.

How much the harmonics need to be reduced is described. The signal intensities I₀ and I measured by ionization chambers areHere, c_m(I₀, m) is the ratio of the mth harmonic to the fundamental (m = 1) for I₀ and c_m(I, m) is that for I. α(I₀, m) denotes the detection efficiency for I₀ and α(I, m) is that for I. The term exp(−μ_mt) is the transmission of the mth harmonic through the sample. The mth harmonic has an energy which is m times higher than the fundamental, and yields a current that is m times larger in the ionization chamber. This phenomenon is expressed as m in the second terms of equations (1) and (2). In principle, the second harmonic is not generated by Si(111) and Si(311) double-crystal monochromators, which are widely used, because of the extinction rule. Only the third harmonic (m = 3) should be considered, as harmonics with m > 3 are negligible.

In the case in which monochromator crystals are strongly distorted by clamping or heat load, a weak second harmonic may be generated. If there is second-harmonic contamination, the beamline optics should be set to reject the second harmonic.

A measurement of Ti K-edge XAFS is taken as an example to discuss the influence of the third harmonic. Let us assume that ionization chambers for I₀ with a 140 mm electrode and I with a 280 mm electrode are filled with N₂ and with 85% N₂ and 15% argon, respectively.

When a titanium foil of 6.3 µm (μt = 2) thickness is measured with an incident X-ray containing 0.1% third harmonic, the apparent absorption becomes smaller by ∼0.007 (∼0.36%) compared with the true value, as listed in Table 1 and shown by solid lines in Fig. 1. It becomes worse when measuring a thicker, 12.5 µm (μt = 4), foil, when the apparent absorption is smaller by ∼0.045 (∼1.2%) compared with the true value (Table 1 and dotted lines in Fig. 1). These effects lead to serious errors.

Figure 1 (a) The influence of the third harmonic on the difference in apparent and true absorption coefficients. (b) Enlargement of (a) for the range 0–1%. Solid, dashed and dotted lines denote μt = 2, 3 and 4, respectively.

Extended X-ray absorption fine-structure (EXAFS) oscillations are also affected by the presence of third harmonics. Let us assume that we have a sample with μt = 4 and Δμt = 1. The incident X-ray contains 0.1% third harmonic, which fluctuates by 10% and the ratio of the harmonic decreases to 0.09%. The absorption (χ) then changes by 0.004. One might think that this is small, but actually it is large. This change affects EXAFS oscillations of k²χ at k = 12 Å⁻¹ by ∼0.58. When EXAFS oscillations of k³χ are used, the change is ∼6.9. The magnitudes in this k range for bulk solid samples are usually ∼0.5 for k²χ and ∼5 for k³χ; thus, the influence considered here is not negligible. The amount of third-harmonic contamination must be reduced to less than 10⁻⁵ in order to suppress the change in k²χ to ≲0.01 (or that of k³χ to ≲0.1). This level is equivalent to about 2% of the true oscillatory magnitudes of bulk solid standard samples such as copper and CuO₂, for example. Because real samples would show weaker EXAFS oscillations, it is suggested that the third-harmonic contamination level should be reduced to 10⁻⁶.

Relatively low-energy X-rays, for example 4–7 keV, from hard X-ray synchrotron sources may contain a considerable fraction of higher harmonics. The former BL-20B at Photon Factory, a beamline that belonged to the Australian National Beamline Facility, emitted 1% of the third harmonic at 5 keV compared with the fundamental (Tran, Barnea et al., 2003; de Jonge et al., 2006). This was reported to be `much to their surprise' (Tran, Chantler et al., 2003).

Handling higher harmonics is one of the key factors in order to obtain correct XAFS spectra. Much attention has been paid to harmonics, and many solutions and diagnostics have been suggested (Barnea & Mohyla, 1974; Jach et al., 1987; Sainctavit et al., 1988, 1989; Oversluizen et al., 1992). A paper summarizing and discussing methods of handling higher harmonics (Chantler et al., 2012) was presented at the Q2XAFS2011 conference held in KEK, Japan, which was aimed at the standardization of XAFS.

There are two major methods to reject higher harmonics: two-mirror systems and detuning. Two-mirror systems were developed and have been widely used for harmonics rejection (Sainctavit et al., 1988, 1989; Lamble, 1995). Tabletop two-mirror systems have also been developed for use in experimental hutches (Latimer et al., 1995).

The specular reflectivity of materials has a sharp cutoff beyond a certain X-ray photon energy. In other words, mirrors can work as low-pass filters in this energy region. The second of the two mirrors is set in parallel to the first mirror, and the beam direction is maintained. Based on this idea, placing mirrors in X-ray paths under a certain condition results in a situation in which the fundamental passes through and the higher harmonics are blocked. The critical angle for total reflection θ_t is of the order of a few milliradians for the X-ray range in which we usually measure XAFS spectra (Sainctavit et al., 1989), although θ_t depends somewhat on the energy of the X-rays and the elements in the materials. When we choose nickel as a mirror material and fix the mirror at 10 mrad (0.5729°) to the incident X-ray, for example, the X-ray reflectivities at 5 and 15 keV are 0.863 and 1.86 × 10⁻³, respectively. Using a two-mirror system, the resulting intensities of the incident X-rays at 5 and 15 keV are 0.74 and 3.5 × 10⁻⁶, respectively. Thus, the third harmonic is cut off sufficiently and its influence becomes negligible.

Another way to reject higher harmonics is simply to detune double-crystal monochromators (Jach et al., 1987; Hart & Rodrigues, 1978; Mills & Pollock, 1980; Hashizume, 1983; Bonse et al., 1983; Schulte-Schrepping & Drube, 2001). Higher harmonics are drastically reduced by slightly detuning the crystals. The amount of detuning is of the order of the Darwin width associated with the harmonic to be rejected and is usually about several arcseconds (Mills & Pollock, 1980). According to a ray-tracing study, the third harmonic is reduced to less than 10⁻² of its fully tuned condition by detuning by 5 arcsec under a certain condition at a storage-ring energy of 3.5 GeV (Hou, 2005). Detuning can be adopted at beamlines where a two-mirror system is not available. In general, the use of two-mirror systems is recommended rather than detuning because two-mirror systems are fixed and stable once they have been set to a certain experimental condition.

Ti K-edge raw absorption and processed spectra of titanium foil, measured on BL-9C, Photon Factory, KEK, are shown in Fig. 2. The raw absorption spectra are shown in Fig. 2(a), their X-ray absorption near-edge structure (XANES) in Fig. 2(b), their EXAFS oscillations k²χ(k) in Fig. 2(c) and their Fourier transforms (FTs) in Fig. 2(d). These spectra were measured under different conditions as follows. The I₀ ionization chamber was filled with a mixture of 70% helium and 30% nitrogen. The I ionization chamber was filled with 100% nitrogen for the blue and orange spectra. The blue spectra were measured with a 50% detuned monochromator in order to reduce the harmonics, while the orange spectra were recorded with the fully tuned monochromator, under which condition the harmonics were contained in the incident X-rays. The red spectra were also recorded with the fully tuned monochromator, but the gas in the I ionization chamber was changed to a heavier mixture of 85% nitrogen and 15% argon. Under the heavier gas condition, higher harmonics affect XAFS spectra more noticeably. These conditions are summarized in Table 2.

Figure 2 Ti K-edge (a) raw absorption spectra, (b) normalized XANES, (c) EXAFS oscillations k2χ(k) and (d) their Fourier transforms measured under different conditions. The measured conditions are as follows. Blue lines, 50% detuned monochromator and nitrogen ionization chamber for I; orange lines, fully tuned monochromator and N2 ionization chamber for I; red lines, fully tuned monochromator and 85% nitrogen + 15% argon ionization chamber for I.

In Fig. 2, the blue spectra are reliable and were measured with a 50% detuned monochromator to eliminate harmonics. The raw absorption spectrum (orange line) in Fig. 2(a) might appear to be fine, but it is not. The background shows an irregular shape because of harmonics contamination. The XANES has a small but obvious spike at ∼5060 eV. There are more serious effects on the EXAFS analysis. The EXAFS oscillation shows some spikes and entirely different oscillatory structures, in particular in the high-k region. The strange oscillatory structures yield ghost peaks in the FT spectrum, which are clearly recognized at ∼1.2 and 1.7 Å. Much worse than the ghost peaks, the main peak at ∼2.5 Å shows a strange shape with a shoulder structure. When this peak is analyzed, we may obtain a result with two titanium sites in the first shell, which is definitely wrong. We should carefully reject the higher harmonics in order to obtain correct results.

When a heavier gas is flowed into the I ionization chamber, the effects of harmonics become stronger and the spectra become worse. At a glance, the red spectrum in Fig. 2(a) is strange and there is a distinct spike at ∼5060 eV in the XANES. The EXAFS oscillation is terrible. There are many spikes and apparently strange structures. This will be transformed to the FT spectrum, in which considerable ghost peaks appear. Incorrect conclusions would be made in the presence of the contaminating harmonics.

Thus, harmonics seriously contaminate XAFS spectra. It is not possible to correctly analyze contaminated XANES spectra. EXAFS oscillations and their FTs show ghost structures and peaks. Their fitting yields incorrect results. We must carefully reject the harmonics and avoid their effects.

A direct way to detect harmonics is to measure the elastic scattering of the incident X-ray using a solid-state fluorescence detector. The third harmonic peak will be observed if it has sufficient intensity, and we can compare its intensity with that of the fundamental. The scattering cross sections of the fundamental and harmonic should be taken into account in order to make a quantitative comparison, as they differ from each other.

In practice, however, we are not really interested in the exact intensity of the harmonic, but we would like to avoid its influences. It is important to check whether the harmonic appears and affects the XAFS spectra. In order to check for this, there are some approaches to detect and reduce the presence of harmonics.

We can check whether there is a serious amount of harmonics by placing an aluminium foil before the I₀ ionization chamber. A detailed procedure for this method has been reported (Tran, Barnea et al., 2003; de Jonge et al., 2006; Chantler et al., 2012) and is called an `inline' measurement (Tran, Barnea et al., 2003). The `inline' method with a `daisy wheel' enables us to estimate harmonics easily and effectively (Tran, Barnea et al., 2003; de Jonge et al., 2006).

For Ti K-edge measurements, the fundamental is ∼5 keV and the third harmonic is ∼15 keV. A 50 µm aluminium foil reduces the intensity of 5 keV X-rays to 8% of the original intensity, while 15 keV X-rays are only reduced to 90%. That is, the intensity ratio c₃(I₀, 3) (see equation 1) of the harmonic (15 keV) to the fundamental (5 keV) increases ten times.

By comparing XAFS spectra before and after placing aluminium foils, it can be checked whether or not harmonics effects are present in the spectra. If it is necessary to change the intensity ratio c₃(I₀, 3), further aluminium foil of 100 µm or greater in thickness can be used. Aluminium foil of about 100 µm in thickness attenuates 5 keV X-rays to ∼0.7% and 15 keV X-rays to 80%, which means that the ratio of the harmonic increases a hundred times.

Another way to vary the effect of harmonics is to change the gases in the ionization chambers. The detection efficiencies of the harmonics increase when the ionization chambers are filled with heavier gases. Comparing XAFS spectra measured using ionization chambers filled with different gases can inform us of contamination by harmonics. If the incident X-ray contains non-negligible harmonics, different structures appear in the XAFS spectra. If the XAFS spectra are completely the same, there is no contamination by harmonics. It would be safer to choose a lighter gas in order to avoid contamination by harmonics.

It is of critical importance that ionization chambers are entirely stable. The high voltage, gas pressures, gas-flow rates, gas compositions and temperature must be sufficiently constant. When the gas pressures fluctuate, the gas densities in the ionization chambers also fluctuate. These fluctuations cause fluctuations in the XAFS spectra, which make them noisy and unreliable. In addition, instability in temperature leads to instability or drift of experimental setups such as slits, sample stages and measurement instruments. It is vital that the mechanical setups are stable and traceable, particularly when small-sized beams are used.

Slits and apertures must be set at their correct positions. Otherwise, the energy calibration and resolution may be affected. A possible explanation is as follows. A misalignment of the I₀ slit affects the radius of the Ewald sphere and the energy shifts. Also, the effective `thickness' of the Ewald sphere increases on moving the slit position and the energy resolution is more strongly affected. A further cause of rsolution broadening can be a larger than expected beam divergence. Ti K-edge XANES spectra of titanium foil are shown in Fig. 3(a) on changing the position of the I₀ slit. The solid line shows a spectrum measured with the correct I₀ slit position. The dashed, dotted and dash–dotted line spectra were recorded with the I₀ slit moved by 500, 750 and 1000 µm in the horizontal direction, respectively. The pre-edge peak (∼4964 eV) region is expanded in Fig. 1(b) to show the influence of the misalignment on the quality and the reliability of the spectra.

Figure 3 (a) Ti K-edge XANES spectra of titanium foil measured on changing the position of the I0 slit. The solid line spectrum was measured with the correct I0 slit position. The dashed, dotted and dash–dotted line spectra were recorded with the I0 slit moved by 500, 750 and 1000 µm in the horizontal direction, respectively. (b) The energy range 4960–4970 eV was expanded to observe the pre-edge peak closely. Influences of the mis­alignment on the spectra were recognized. Shifting and broadening of the pre-edge peak were observed on changing the I0 slit position.

It is evident that the spectra shift to high energy on moving the I₀ slit position. When the I₀ slit was moved by 1 mm (dash–dotted line) the shift was about 1 eV, which is a very significant value. The reliability of the energy axis is crucial to perform XANES analyses. We may obtain incorrect results in, for example, valence (charge) estimation. Worse than this, we might not become aware of mistakes. The misalignment heavily influences the pre-edge peaks. Not only shifting but also broadening of the peak was observed. The height of the pre-edge peak (4964 eV) was greater than that of the shoulder peak (4968 eV) in the spectrum with the correct I₀ slit position. The pre-edge peak, however, apparently shrank and became lower than the shoulder peak when the I₀ slit was moved by 1 mm.

Thus, it is essential to set slits and apertures at their correct positions. The I₀ photon energy changes on their misalignment, which leads to energy shifts and peak broadening.

Inhomogeneity of samples cause considerable distortions in EXAFS, in particular in high k ranges, and can limit the usable ranges for EXAFS oscillations. Samples should be prepared uniformly and homogeneously. When the samples consist of particles, the particles should be sufficiently small compared with the absorption length of the X-rays of interest. The size effect on EXAFS amplitudes has been reported (Lu & Stern, 1983) and particles finer than 400 mesh can depress the EXAFS amplitude to ∼90% of its true value. Self-absorption effects cause similar depressions in XAFS spectra measured in fluorescence yield (FY) mode (Pfalzer et al., 1999). A useful sample holder has been presented to obtain depression-free XAFS spectra from concentrated samples using the FY mode (Abe et al., 2016).