A limiting parameter in energy-dispersive X-ray spectrometry (EDXS) in a transmission electron microscope (TEM) is the magnitude of the solid angle of the EDXS detector: Characteristic X-rays are emitted equally distributed into the whole space, but only a small part within the detector solid angle is detected. A larger solid angle results in a larger collection efficiency of X-rays and a higher sensitivity of the detector.
A geometrical calculation of the solid angle is difficult since it is based on the knowledge of the detector geometry [1,2], which is not very well known. An experimental procedure was described by Egerton et. al.  [3], using NiO where density and composition are known, measuring the number of characteristic X-rays of the Ni-K line and applying values for the ionization cross section and fluorescence yield. This measurement was made using the Ni-K line only. In addition reliable values for ionization cross sections and fluorescence yields are hard to find and typically exhibit severe uncertainties. Therefore we did an extensive literature research to obtain the most accurate, state-of-the-art values.
 
We use the following experimental approach to determine the solid angle: The intensity of an X-ray line depends on several parameters which are related to the specimen, the measurement setup and to the detector including the solid angle (Fig.1). The latter can be determined via obtaining values for all these variables either through searching for databases and literature or experimentally. A literature research reveals the databases NIST 164 [4] for ionization cross sections, EADL [5] for fluorescence yields and relative X-rays line intensity ratios from Scofield [6] improved by values from Aßmann et. al. [7] and Wendt [8] as reliable sources. We use X-ray lines of six elements (Al, Si, Ti, Ga, As, Sr) via EDXS for the solid angle determination. In addition the measurement of the specimen thickness is necessary. Therefore we work with a special sample configuration using the focused ion beam instrument to produce a lamella with a rather uniform specimen thickness for the actual measurement and a conical, circular symmetric rod of each specimen. The latter is used for the experimental determination of the inelastic mean free path λ [9]. Due to the circular symmetric shape of the rod the mean free path can be measured by directly linking it to the absolute thickness/diameter of the rod (Fig. 2) and thus the thickness of the lamella can be calculated accurately using the t/λ method.

In our case we work with a FEI Titan³ 60‑300 equipped with the ChemiSTEM technology. Hence we have four windowless SDD detectors symmetrically placed around the optical axis (Fig. 3) and we are using the high visibility holder from FEI.

With the profound knowledge about the reliability of ionization cross sections, fluorescence yields and relative X-ray line intensity ratios were are able to determine the solid angle of our EDXS system properly. We show that the solid angle for each detector is between 0.15 to 0.17 sr, which roughly corresponds to the manufacturer value of 0.175 sr per detector.

[1]          Zaluzec, Microsc. Microanal. 2014, 20, 1318–1326, .

[2]          Conway, Nucl. Instrum. Methods 2010, 614, 17–27.

[3]          Egerton et al., Ultramicroscopy 1994, 55, 43–54.

[4]          Llovet et al., J. Phys. Chem. Ref. Data 2014, 43, 13-112.

[5]          Perkins et al., LLNL Report 1991, UCRL-50400, vol. 30.

[6]          Scofield, “Radiative Transitions” in: Ionization and transition probabilities, 1975, Acad. Press, New York.

[7]          Aßmann et al., Spectrochim Acta B 2003, 58, 711-716.

[8]          Wendt, Microchim Acta 2002, 139, 195-200.

[9]          Kothleitner et al., Microsc. Microanal. 2014, 20, 678-686 .

The authors acknowledge the Austrian Research Promotion Agency FFG (project 850220) for funding.

Figures:

Figure 1: Intensity of an X-ray line: The calculation of the intensity depends on several parameters which originate from the specimen, the measurement setup and the detector system. To determine the solid angle, data for all parameters need to be known.

Figure 2: Determination of the inelastic mean free path λ via the rod: The inelastic mean free path λ (bottom) can be calcutated using a t/λ map (top) and the thickness of the rod from the HAADF image (middle).

Figure 3: Scheme of the used EDXS system: The four windowless SDD detectors are symmetrically placed around the optical axis. The solid angle of one detector is marked in orange.

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