Scanning Electron Microscopy and X-Ray Microanalysis

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needed. Fortunately, the complex three- dimensional dis-

tribution can be reduced to a one- dimensional problem for

the calculation of absorption, since the path out of the speci-

men towards the X-ray detector only depends on depth. The

φ(ρz) curves discussed previously give the generated X-ray

distribution of X-rays in depth (See. Figs. 19.8, 19.9, and

19.10).. Figure 19.12 shows calculated φ(ρz) curves for Cu

K-L 3 X-rays in pure Cu for initial beam energies of 10, 15, and

30 keV. The curves extend deeper (in mass depth or depth) in

the sample with increasing E 0. The φ 0 values also increase with

increasing initial electron beam energies since the energy of

the backscattered electrons increases with higher values of E 0.

The X-rays which escape from any depth can be found

by placing the appropriate path length in the X-ray absorp-

tion equation for the ratio of the measured X-ray intensity,

I, to the generated X-ray intensity at some position in the

sample, I 0 :

II/e 0 =−xp ()μ/ρρ()t

(19.18)

The terms in the absorption equation are (μ/ρ), the mass

absorption coefficient; ρ, the specimen density; and t, the

path length (PL) that the X-ray traverses within the speci-

men before it reaches the surface, z = ρz = 0. For the purpose

of our interests, I represents the X-ray intensity which leaves

the surface of the sample and I 0 represents the X-ray inten-

sity generated at some position within the X-ray generation

volume. Since the X-ray spectrometer is usually placed at an

acute angle from the specimen surface, the so-called take-off

angle, ψ, the path length from a given depth z is given by

PL = z csc ψ, as shown in. Fig. 19.13. When this correction

for absorption is applied to each of the many layers Δ(ρz) in

. Fig. 19.12 Calculated φ(ρz) curves for Cu K-L 3 in Cu at E 0 = 10 keV,
20 keV, and 30 keV; calculated using PROZA
. Fig. 19.13 Schematic diagram of absorption in the measurement
or calculation of the φ(ρz) curve for emitted X-rays. PL = path length,
ψ = X-ray take-off angle (detector elevation angle above surface)

Cu

K-L 3 = generation

φ(ρz) = distribution

0.5 μm 1 μm1 μm

E 0 = 10 keV E 0 = 20 keV E 0 = 30 keV

. Fig. 19.11 Monte Carlo simulations (Joy Monte Carlo) of the X-ray generation volume for Cu K-L 3 at E 0 = 10 keV, 20 keV and 30 keV. The sites of
X-ray generation (red dots) are projected on the x-z plane, and the resulting φ(ρz) distribution is shown

Chapter 19 · Quantitative Analysis: From k-ratio to Composition