X-RAY SPECTROMETRY, VOL. 18, 101-104 (1989)

Quantitative X-Ray Fluorescence Analysis of Specimens with Rough Surfaces : Monte Carlo Approach

H. Ebel and Ch. Poehn Institut fur Angewandte und Technische Physik, Technische Universitat Wien, Wiedner HaupstraDe 8-10, A-1040 Vienna, Austria

In general, the application of the fundamental parameter method in quantitative x-ray fluorescence analysis is restricted to specimens with plane and smooth surfaces. For such surfaces, the correlation of path lengths x of incident tube photons and y of escaping fluorescent photons is well defined. For rough surfaces this correlation is described by distribution functions. Thus, for x1 < x < x p a probability density &v),,,,, allows the probability of y values to be described within the range y A < y < y e . From the results it is now possible to describe probability distributions of rough surfaces by means of Monte Carlo calculations.

DERIVATION OF THE EQUATION FOR PRIMARY AND SECONDARY EXCITATION

A beamlet of cross-section d2q = dadb impinges on the surface and penetrates to a depth which is defined by the path length x within the specimen (see Fig. 1). In order to quantify the absorption, the energy E of photons and the number n(E) dE of pseudo- monochromatic photons per second and unit area within the energy range from E to E + dE are intro- duced. Thus, the rate of photons within the beamlet out of the mentioned energy range is given by dZqn(E) dE. On its path x, some of the photons are either photoab- sorbed or scattered. Since the probability of scattering is smaller relative to that of photoabsorption and more- over scattered photons still contribute to the production of fluorescent x-rays, we use instead of the mass absorp- tion coefficient p the photoabsorption coefficient T,, , where E and c characterize the photon energy E and the ‘composition’ c of the multi-element specimen. Thus, after a path x a number of photons exp(-z,,,p,x), where pc is the density, still exist. As these photons pen- etrate an additional distance dx some of them are pho- toabsorbed in the volume defined by d2q and dx. The number of photons absorbed in d3v is given by

d4z, = d 2 d E ) dE ~xP(-T,, PC x)zE, c PC dx (1) A multi-element specimen consists of n > 1 different

chemical elements j (1 < j < n). As the characteristic radiation of element i is of interest, Eqn (1) has to be verified in order to describe photoabsorptions in element i of the specimen:

d4z,, = d2qn(E) dE exp( - zE, pr x)ci z,, pc dx (2)

with d4Z,, j = d4Z, and ZE, = $ CjZE, j Ci iS the j = 1 j = 1

amount of element i given as a weight fraction (or lOOc, in weight percent). Element i may be represented in the analytical procedure by a characteristic line of photon

energy Ei = hvi = hc,/li. This means that there is no need for an additional radiation index in the following consideration. Radiation i is due to an electron tran- sition from an occupied level i, to a level i, with a vacancy. Photoabsorption in i is obtained by multipli- cation of Eqn (2) by (Si - l)/Si. Besides the jump ratio Si , the quantity wi describes the efficiency of relaxation by the emission of fluorescent radiation and the tran- sition probability p i of electrons for the specific line ( i 2 to il). Consequently, the rate d4zi of production of i radiation in d3u follows:

d4zi = d2qN(E) dE exp( -TE, pc X) si - 1

x W i P i (3) x ci z,, pc dx x ~

Si Some of these photons are detected. Since no angular correlation for incident and fluorescent photons exists, the solid angle R of detection defines by R/4n the number of detectable fluorescent photons out of d3u. After a loss of exp( - zi, pc y) along the escape path y, d4zi(R/4n) exp( -z i , pc y) fluorescent photons from element i leave the surface. A percentage E~ of them is really detected; E~ is the efficiency of detection for photons of energy E,. These detected i photons are given by

d4ni = d2qn(E) dE exp( - T,, pc x)

Now it becomes possible to deal with the procedure which is necessary to obtain d3ni, d2ni, dni and finally the measured characteristic i count rate, n,. As one assumes no variation of the roughness profile normal to the plane of Fig. 1 from zero to depth b, it is possible to replace db by b and d2q = da db by dq = da b. The extension from da to a is depicted in Fig. 2. It causes a change from a single escape length y to a variety of y values within ymin < y < y,,, . Therefore, a probability

0049-8246/89/030 10 1-04 $05.00 0 1989 by John Wiley & Sons, Ltd.

Received 15 October 1988 Accepted 7 January 1989

102 H. EBEL AND Ch. POEHN

Figure 1. Rough surface with an impinging beamlet of cross- section da db (da parallel and db normal to the plane of the page). Path length, x; interaction in the volume da db dx = d3v; observa- tion of the fluorescent beamlet with escape path y.

density 4(y) has to be introduced, which describes by @(y) dy a quantity that is directly proportional to the probability for an escape length within y and y + dy. Moreover,

{ymax4(y) dy = 1 (5 ) Ymin

must be valid. Now, Eqn (4) has changed to d2ni = abn(E) dE exp( - tE, pc x)ci tE,

si - 1

Si x pc dx x - x mi Pi(s2/4Z)&i

x Iymax+(Y) exp( - t i , c Pc Y) dy (6) Ymin

Further information can be gained from Fig. 2. 4(y), ymin and y,,, must depend on x. For decreasing x, ymin and y,,, also decrease and further the position of &(y) shifts to lower y values. As a consequence, (pb) should be replaced by 4 ( ~ ) , and ymin and Ymax by yrnin,, and y,,, x , respectively. Now, it is possible to introduce the variation of x from zero to infinity:

fl(E).dE -a -b

Y" \

dx\

Figure 2. As Fig. 1, but with da replaced by a finite value a.

(7)

A final step is performed when integrating over all ener- gies E of primary photons within Eedge, i and the maximum photon energy E, = eV,:

si - 1 Si x wi pAW4n)~i n, = abcipc x -

Ymax, I

x 1 4(Y)x exp(-ti, c Pc Y) dy (10) Ymin, x

describes the measured i fluorescence count rate for a specimen with a rough surface (note: ni has been replaced by ni,P in order to define the count rate of i fluorescent radiation due to 'primary' excitation).

First this result is verified by replacing the rough surface by a plane and smooth surface. As the angle of incident photons with respect to the surface normal is tl and p for the escape of detected x-rays, the depth t is given either by t = x cos tl or by t = y cos p. Thus y can be expressed by x:

and the probability density becomes a delta function. This means that Eqn (9) changes for plane and smooth surfaces to

n(E)t,, i dE IE0 Eedgs, i ni, p = Kici P c

exp(-zE,cpcx)

x exp[ - ti, pc x cos cc/cos p)] dx

= Ki ci pc

Equation (12) is the well known expression for primary excitation as published by Sherman' and Shiraiwa and Fujino.' Equation (10) has been derived on the assump- tion of a roughness profile that does not change with variation of the position normal to the plane given in Figs 1 or 2. However, it is simply a statistical problem. As a sufficient number of roughness profiles have been

XRFA OF ROUGH SPECIMENS 103

Computer Scale

Figure 3. Flow chart for the experimental determination of @ ( y ) , .

evaluated, the resulting distributions +(y), will not vary significantly when a further roughness profile of the same specimen surface is added for evaluation. There- fore, this assumption should be acceptable.

EXPERIMENTAL

The flow chart is depicted in Fig. 3. Roughness profiles can be measured by surface analysers. A digital output can be used directly as input for the computer. Other- wise, the chart strip with the trace is placed in a scanner and the signal is used for evaluation.

In general, the scales in the direction of registration and depth of the roughness profile are not identical.

This problem requires either a transformalam to identi- cal scales or conversion of a and /? into a’ and /?.

A further step in the evaluation of the roughness pro- files must be an estimation of the maximum depth of x-ray emission from photoabsorption of incident and fluorescent x-rays. This restricts the area of evaluation.

As by means of a random number generator a first pair of 5 and 7 values has been produced, it is possible to define point PI(<,, q l ) and to draw from there lines to the surface in the x and y directions. When contin- uing the generation of points, a number of x-y pairs are obtained and can be stored. They are sorted for x values from x1 to x2 = x1 + Ax and the corresponding y values are arranged in steps of ymin, to ymin, , + By, ymin, , + Ay to ymin, , + 2Ay, etc. This results in a histo- gram. The width of the distribution depends, of course,

XI = 0 pin x2 = 30 pin

I I 75 150

75 150

75 150 Y C p ’

Figure 4. Measured distributions of a smooth surface.

104 H. EBEL AND Ch. POEHN

x i = 0 ylm x2 = 30 prn

x l = 30 pm x2 = 60 pm I

1

75 150

xl = 60 plm x2 = 90 prn

75 150

x l = 90 plm x2 = 120 pm I ' 75 1 so

Y C p I Figure 5. Measured distributions of a rough surface.

on the thickness Ax of the layer. For a practical applica- tion the integrals are replaced by sums and only the normalization X ~ ( Y ) ~ ~ , x2.Ay = 1 has to be performed.

Secondary excitation is a correction term which con- tributes about 10% to the measured signals. Hence the roughness correction is a second-order correction and it is recommended that for secondary excitation the equa- tions for plane and smooth surfaces as described by Sherman' and Shiraiwa and Fujino' are used.

The distributions depicted in Figs 4 and 5 were obtained from a smooth and a rough surface, respec- tively. The angles of incident x-rays and detected fluo- rescent x-rays towards the surface were 45". In Fig. 5 the average depth of the roughness profile was k 10 pm. For the specific examples a maximum x-ray emission of 100 pm led to a restriction of x values to 120 pm. For a practical application the measured distributions have to be normalized to an area of unity.

REFERENCES

1. J. Sherman, Specfrochim.Acfa 7, 283 (1955); 11, 466 (1959). 2. T. Shiraiwa and N. Fujino, Jpn. J . Appl. Phys. 5, 886 (1966); Boli. Chem. SOC. Jpn. 40,2289 (1967).