Section 2.3.5.1.5. Intensity variation with take-off angle

The intensity of the characteristic line spectrum emerging from the tube depends on the anode element, voltage, and take-off angle ψ. The depth of penetration of the electrons in the anode is approximately proportional to kV²/ρ, where ρ is the density of the anode metal. The path length L of the X-rays to reach the surface depends on the depth D at which they are generated and the take-off angle, . Self-absorption in the anode causes a loss of intensity that increases with D and decreasing ψ. The intensity of Cu Kα radiation at 50 kV as a function of take-off angle is shown in Fig. 2.3.5.2(d). This effect has been described in a number of publications: Green (1964), Brown & Ogilvie (1964), Birks, Seebold, Grant & Grosso (1965), Parrish (1968), and Phillips (1985). Because of the self-absorption, the wavelength distribution varies slightly with take-off angle (Wilson, 1963, pp. 61–63).

The optimum kV and mA operating conditions are not sharply defined and the range can be determined with a powder reflection or by using small apertures in the direct beam with balanced filters and pulse-amplitude discrimination. The intensity is measured at various voltages, keeping the current constant and converting the data to constant power. Typical experimental curves relating Cu Kα intensity to kV for various ψ's are given in Fig. 2.3.5.2(c). At 50 kV, the intensity doubles by increasing ψ from 3 to 12° (although the projected width of the focal spot also increases). The effect is much larger for Cr Kα and W Lα because of their higher absorptions. The linear region of I versus V is relatively short and increases with ψ. At small ψ's, I is virtually independent of V and could decrease with increasing voltages; increasing the current would give a greater increase using the same power. For a tube with maximum power values of 60 kV, 55 mA and 2200 W, the relative intensities of Cu Kα are about 100 for 40 kV/55 mA, 88 for 50 kV/44 mA and 74 for 60 kV/37 mA. However, the filament life decreases with increasing current and most manufacturers specify a maximum allowable current.

The intensity distribution reaching the specimen is not uniform over the entire illuminated area. In the direction normal to the specimen axis of rotation, one end of the specimen views the X-ray tube focus at an angle ψ − (α/2) and the other at ψ + (α/2), where α is the angular aperture of the entrance slit [Fig. 2.3.5.2(d)]. The intensity differences are determined by ψ and α_ES so that the centre of gravity does not coincide with the geometrical centre. The dependence of the diffracted-beam intensity on the aperture of the entrance slit α_ES, therefore, may also be nonlinear. For example, at ψ = 6°, the intensity difference at the ends of the specimen is 9% for α_ES = 1°, and 44% for α_ES = 4°; the corresponding numbers for ψ = 12° are 2 and 10% respectively.

Although increasing ψ increases the intensity, it also increases the projected width and may increase the widths of the reflections (§2.3.1.1.5). The brightness expressed as also decreases rapidly. When one is working with small apertures, as in grazing incidence and the analysis of small samples, the brightness becomes a very important factor in obtaining the maximum number of counts. For example, the intensity at ψ = 12° is twice that at 3° but the brightness is one half [Fig. 2.3.5.2(d)]. However, it should be noted that the smaller the take-off angle the greater the possibility of intensity losses due to target roughening.