determination of the best experimental conditions for using the α parameters method to calculate...

Nuclear Instruments and Methods in Physics Research B73 (1993) 71-84 North-Holland

RlOMl B Beam Interactions

with Materials&Atoms

Determination of the best experimental conditions for using the a! parameters method to calculate matrix corrections for infinitely thick targets (TTPIXE)

J.M. Delbrouck-Habaru a, G. Weber a,*, P. Aloupogiannis b, G. Robaye a and I. Roelandts ’ a Institut de Physique Nu&aire Expt%nentale, Uniuersite’ de Li;ge, BlS, Sart-Tilman, B-4000 Lit?ge, Belgium ’ Institute of Materials Science, National Research Center, “Detnocritos”, GR-153, 10 Ag. Paraskeui, Athens, Greece

’ Institut de Giologie, Uniuersite’ de Lisge, Belgium

Received 26 June 1992 and in revised form 28 August 1992

The correction method for PIXE data described previously is based on the experimental determination of an (Y parameter linking two independent phenomena: X-ray absorption and proton energy loss. Using the a parameters, corrections can be calculated without making any hypothesis about the matrix composition. In this paper the best experimental conditions for obtaining the a parameters in the case of infinitely thick samples (TTPIXE) are studied. General tables and curves allowing an easy use of the method are given. Reference materials are used to test the precision of the method.

1. Introduction

In previous papers [l-4], we have established a method for correcting PIXE data for matrix effects: the (Y parameters method. This method is based on the definition of a so-called (Y parameter characterizing a given X-ray energy and a particular matrix at a given proton energy. (Y is defined as follows:

a = ~/Q-G), (1)

where p is the X-ray absorption coefficient and S(E,,) is the stopping power of the incident proton beam at the energy E, in the considered matrix. The main advantage of the LY parameter method is that, in con- trast with other methods, it allows corrections to be calculated without any hypothesis about matrix composition. In fact, RBS allows one to get additional information about the major element concentrations but, contrary to PIXE, it needs (Y particles and two separate measurements are thus necessary. This method was first used to calculate matrix corrections for intermediate thickness targets. In a previous paper [5], we have demonstrated the possibility of extrapolating the (Y parameters method to the infinitely thick targets

Correspondence to: J.M. Delbrouck-Habaru, Institut de Physique Nucliaire Exp&imentale, Universitt? de Li&ge, B15, Sart-Tilman, Litge B-4000, Belgium. * Research Associate of the National Fund for Scientific

Research (Belgium).

(TTPIXE): we use two PIXE measurements performed either at different incident beam energies or in different geometrical situations. This allows the calculation of the cx coefficient using the ratio r of the normalized numbers of the X-rays of the same element detected in the two measurements. This study has shown that the choice of the experimental conditions strongly influ- ences the precision of the method. The purpose of the present paper is to summarize a deep discussion leading to the proposition of a “best choice” of experimental conditions, to give the results of experimental valid- ity tests realized on different types of reference materials and to provide some tables and curves facilitating the use of the method.

Some general considerations are to be made at the beginning; PIXE analysis is an experimental method and therefore two physical facts must be kept in mind. Firstly, it is not practical to change the angular position of the Si(Li) detector after each measurement because such detectors (with their liquid nitrogen containers) are large, fragile and heavy and moreover they need some thin window in the target chamber for allowing X-rays to hit them. Secondly, if one chooses to use couples of PIXE measurements performed at different proton energies, it is necessary to change the accelera- tor settings very frequently which is time-consuming and also introduces new experimental problems, problems namely related to changes in secondary electrons emission rates with energy, and other total charge measurement difficulties. The obviously simplest mea-

0168-.583X/93/$06.00 0 1993 - Elsevier Science Publishers B.V. All rights reserved

72 J.M. Delbrouck-Habaru et al. / Matrix correction for TTPIXE

surement process consists of a rotation of the target, keeping incident energy and detector position constant. In this paper we shall study the best conditions for this last procedure.

2. Theory

Using the Folkmann [6] formula for the stopping power S(E) and the (Y formalism [l-3], the intensity Zi detected for a characteristic X-ray emitted by an element i in a given matrix is equal to:

z,= AQCi /

E”u(Eo-u)‘(~) du

’ S(E”) 0 [(+u)/E,]P ’ with

(2)

sin(p +ej) 1 I PU

sin 8, ( )I 2E, ’ (3)

where A is a factor taking into account the detector solid angle and its efficiency; Q is the incident proton charge and ci is the concentration of the element i in the matrix. The variable u, always positive, is equal to u = E, -E where E, is the incident proton beam energy, which is also used as the reference energy in the Folkmann formula for the stopping power. The angles p and ej are defined in fig. 1 and p is a constant equal to - 0.65 [l]. The X-ray production cross section vi(E, - u) is calculated using the adjust- ment proposed by Paul [7].

The integral in the relation (2) is a function of the (Y parameter, of the incident energy E, and of the geometrical parameters p and 0, and it is called [l] the matrix factor fi,(~, E,, /3, 0,).

To simplify the discussion, we set:

sin( p + 0,) c,=c(aT P> ej) =a sin 8, 9

I

fij( a, Eo, P, ‘j) =.fi( Eo, Cj)

= / 060Pi(Ea-u)T(~~ Cj) du, (7)

AQci ” = S( E,)

p.fij( a, Eo, PT ej).

It is clear that the matrix factor fij relative to an X-ray emitted by an element i is depending on the corresponding cr parameter and on the geometrical conditions j through the C parameter only. This fact is interesting for the commodity of the formalism.

Let us call “thin target” a target where the incident beam energy loss and the X-ray absorption are negligi- ble. Taking into account the definition of ci = mi/m and the fact that the intensity of the same superficial mass (m,) thin target is equal to

zi,corr =AQmi’+i( Eo) 9 (9)

it is possible to define a total correction factor CF as done by Aloupogiannis [l] for intermediate thickness samples:

‘i,corr = Z,(CF) -’

- 450 mm .

Havar

homogeneizer

Fig. 1. Definition of the geometrical parameters and schematic representation of the experimental setup (not to scale).

J.M. Delbrouck-Habaru et al. / Matrix correction for TTPIXE

F(C)

1

0.6

0

0 5 10 15 20 25

Fig. 2. Variation of F as a function of C.

with

CF= fij(a, ECl, Pf ej>

m~i(ECl)S(EO) ' (10)

where mi and m correspond now to the part of the target really reached by the incident proton beam. Thus

E, dE m= 0 S(E)’ /-- (11)

3. Analytical procedure

We call I, and Z2 the measured intensities for one element i corresponding to the two geometries of fig. 1

P p=rc/a I

Si(Li) Ii7 ...

I ... ......................... X-rays .:::::::::: _ ........................................ .............................................

.;.;.I. ... ............... ............

.......................................... ......................................

: il .......................................... ......................................... .... ..................................... ........................................

......................................... ... .................................................. ................... ‘(. .....

81 -n/2 sample

I

73

with the same experimental conditions, namely the same E,, and p, and Q, = Q2. The experimentally determined ratio r of the two intensities is thus equal to

4 fi(a, 6, Pt ‘4) r = 5 = f*(a, E,,, p, e,) = r(a)7 (12)

withO<r<l. As the experimental parameters E,,, the incident

energy, ~3, the angle between the proton beam and the detector, and 0,, the emergence angles, are well known, the relation (12) gives r as a function of LY for a given element i. It is not possible to solve eq. (12) analytically because the functions fj(~, E,,, /3, 0,) are not simply analytical expressions of (Y but the solution can be found numerically using the Newton algorithm [5].

P p=n/2 1

Si(Li)

7-l

Fig. 3. Illustration of the two extreme geometrical situations.

74

30

25

20

15

10

5

0

J.M. Delbrouck-Habaru et al. / Matrix correction for TTPIXE

GFrl

- I3 = 75”

- I3 = 60” or 90”

- 6 = 450 or 105”

- 6 = 300 or 1200

- 6 = 150 or 135”

Eo = 2.5 MeV Element q sulphur

Fig. 4. Example of the effect of the choice of the angle /3 between the detector and the beam on the factor GFr,.

3.1. Influence of the dr error on the matrix factor

The experimental ratio r of the two intensities is always obtained with a given precision which introduces a certain error in the calculation of the (Y coefficient and of the matrix factor f,. In a previous paper [5], we have shown the importance of the relative error on r in the calculated concentration. Beside the statis-

tical fluctuations on the counting rates, a large contri- bution to the error on r could be due to errors on the measurement of the incident beam intensity. It is thus important to calculate the effect of an error dr/r on the precision of the concentration of an element which is directly depending on df/f by relation (8). Our discussion begins with a search for the geometrical conditions giving the most accurate determination of

10

8

6

Fig. 5. Illustration of the theoretical advantage of using grazing incidence and emission angles.

J.M. Delbrouck-Habaru et al. / Matrix correction for TTPIXE 75

the matrix factor f supposing that the only cause of error is the limited precision obtained on the r factor

determination. If we remember that:

dfj dcu

(j= 1 or 2),

1 dfj -- dfj dr fj da

,=I 1 df2 1 df, -- --- fz da fl da

Setting

we obtain

dfj x=

dr F, --= r F2--F,

%CZrj r

with

GFrj = & (j=lor2). (17) 2 1

(j=lor2),

(13)

(14)

(15)

(16)

GFrj (GF for geometrical factor) is thus the coefficient which shows how the geometrical conditions change the initial error on the experimental ratio r into the final error on the f matrix factor.

The relation (16) shows clearly that:

GFr, = GFr, - 1. (18)

3.1.1. Choice of the Oj and of the constant /3 angles Fig. 2 shows the variation of F as a function of C

for different elements: the variation is very similar for all the elements usually detected, ranging from S to

Cu. It is evident from eq. (17) that the best conditions to minimize the error factor GFr, consist of having the smallest F, and the largest F, - F2. Fig. 2 shows that this is obtained by working in the two experimental situations corresponding for the first (j = 1) to C nearly .equal to 0 and for the second (j = 2) to C very large. If we recall the definition (6) of the parameter C, the first geometry corresponds to a very small angle of incidence and an emergence angle 0, close to r/2, (p + Oi = r; /3 = a/2), in the second geometry, the

emergence angle O2 is nearly equal to 0. The two cases represent two opposite absorption conditions: in the first one, the X-rays are created near the target surface

and emitted perpendicularly to the surface, and the X-rays absorption is minimum; in the second one, the X-rays generated in the depth of the sample are detected in a direction nearly parallel to the surface, and the X-absorption is thus maximum (fig. 3).

At this point we must remember that, due to the anisotropy of the bremsstrahlung, the peak-to-background ratio is better when the X-rays are detected backwards [8]. On the other hand, the use of very small incidence and emergence angles supposes a very care- ful evaluation of surface irregularities and grain effects [9]. The choice of the best experimental conditions will be a compromise between different conflicting require- ments. We have chosen to work with incidence and emergence angles never smaller than 15”.

With these restrictions on the experimental angles and following the C parameter definition (6), the best experimental conditions correspond to:

sin 165” Cl =(Y

sin( 165” - /3)

and

c,=(Y sin(/? + 15’)

sin 15” ’ (19)

If we study the influence of the angle /3 on the factor GFr,, we can see that the optimum value is 75”. The values of p, which are symmetrical around 75”, give the same value for GFr,. The effect of the choice of p on the factor GFr,, when the limits of the incidence and emergence angles are chosen to be equal to 15”, is illustrated in fig. 4, in the case of S, for (Y ranging from 0 to 15 which covers the majority of matrix compositions and for E, equal to 2.5 MeV. We can see that if we choose to work at /3 = 45”, where the bremsstrahlung is less important than at p = 75”, the

GFr, factor increases by 30% only. It is very accept- able and allows to obtain a good peak-to-background ratio in the experimental spectra.

The chosen limit of incidence and emergence angles is of very large influence on the GFr, factor as shown in fig. 5 for S and p = 45”: for (Y ranging from 5 to 15, GFr, is multiplied respectively by = 2 and by = 4 when this limit varies from 5” to 10” and to 15” for E, = 2.5 MeV. For homogeneous targets of very low granulometry and presenting a perfectly flat surface, it is thus very interesting to lower the incidence and emergence angles limits. However, several experimental (Y determinations on real targets of known composition, made at incidence and emergence angles lower than 15”, have given wrong results corresponding to additional random absorption phenomena, probably due to target surface irregularities.

76 J.M. Delbrouck-Habar~ et al. / Matrix correction for TTPIXE

GFrl 10 -

Element : sulphur

B = 450

-+- Eozl.5 MeV - Eo~2.0 MeV

- Eo~2.5 MeV

Fig. 6. Variation of GFr, as a function of (Y for different values of the incident protons energy E, in the “best geometrical conditions”.

If we remember the definition (1’7) of GFr, and the F variation as a function of C shown in the fig. 2, we see that it is more interesting to lower the incidence than the emergence angle. This effect suggests to use an incidence angle as small as possible, 15” for our experimental samples to avoid surface irregularities. On the contrary, the influence of e2 on GFr, is less important. We therefore choose to work at & equal to 40”, a value which does not damage the GFr, too

much and gives better statistics in the second measurement.

3.1.2. CLroice of the occident beam emgy E,

Another important factor is the proton beam energy. Fig. 6 shows the GFr, variation as a function of the (Y parameter for different proton beam energies. In the cx range from 8 to 15, the GFr, value is multiplied by 2 and 4 when the proton energy varies from 1.5 to

1.2

0.6

Fig

Normalized X-Ray YIELD

t I I t

-+-Iron : calcul.

Matrix : IRSID

0 0.5 1 1.5 2 2.5

7. Normalized thick target X-ray yields when the incident proton energy increases.

a

6


GFrl

-sulfur Eo = 2.5 MeV -potassium

-calcium P = 45”

-C-titanium

-+iron

+-copper

Fig. 8. Evolution of GFr, as a function of a for different elements in the “best geometrical conditions”.

2.0 and to 2.5 MeV in the finally chosen geometrical conditions.

The main reason to work at E, = 2.5 MeV is that the X-ray production cross sections increase with the proton beam energy allowing much shorter measurements to obtain a given statistical accuracy. Fig. 7 shows the variation of the X-ray yields, normalized at E, = 2.35 MeV, for S and Fe in IRSID matrix at

p = 45” and 0 = 95”. The curves represent calculated yields and the points correspond to the experimental values. We must choose the experimental value of E,, to avoid on the one hand too high a value of the GFr, factor and on the other hand too bad statistics in the case of e2 = 40” corresponding to a high X-ray absorption.

The GFr, factor examples are given as a function

-GFrl -GFP -GFo -GF6 2 10

8

6

Eo q 2.5 MeV

Element q sulphur

p q 45”

e1 q 120”

el= 40”

1

0.8

0.6

0 5 10 15

Fig. 9. A comparison of the magnitudes of the different geometrical factors in the “best experimental conditions”. In this case, GFr, is more than 10 times greater than the others. Note that the left-hand vertical scale is for GFr, and that the right-hand

vertical scale is for GF/3, GF0, and GFB,.


of the (Y parameter in fig. 8, for different elements, in the case where E,, = 2.5 MeV, /I = 45”, 0, = 120” and 0z = 40”.

3.2. Influence of the d/3, de, and de, errors on the matrix factor

The definition (12) of the ratio r indicates that r is not only a function of (Y and E,,, but also of the angular variables p, 0, and 0z:

r = r(o, E,,, P, or, 0,). (20)

An error dy on the experimental setting of one of the angular variables induces an error da on the a-parameter:

;da + &dy = 0. (21)

The subsequent error on the matrix factor f, is easy to calculate by:

df, af, da afl -_=--+- dy aa dy ay ’ (22)

Adopting the same type of notations as previously, GF for geometrical factor, followed by /?, 8i or 0, for characterizing the geometrical variables, the results for elementary errors d/3, de,, de,, given in degrees, are respectively:

df, - = d@“‘GFP fl

(23)

with

FIFZ GFp=ii- sin( 0r - 0,)

180 F2 - F, sin( p + 13,) + sin( /3 + 0,) ’ (24)

df, fl = d@“‘GFB,, (25)

with

= FIFZ GF0, = -~

sin p

180 F, -F, sin(p + 0,) + sin(0,) ’ (26)

df, - = dB:“‘GF6’ fl

29 (27)

with

F,F2 GFe,= -L- sin p

180 F2 - F, sin(P + 0,) + sin(0,) ’ (28)

In our working conditions (/3 + 0, = 165”, 0, = 40”), the value of GFr, is one order of magnitude greater than the different angular correcting factors: GF/3, GF0, and GF0, (see fig. 9). Let us recall that GF@, GF0, and GFB, are the relative errors on the fl factor for a 1” error on the corresponding angular variable. The incidence and the emergence angles must

be lowered as far as to a few degrees, so the different factors may become of the same magnitude as GFr,; their importance will thus be always very limited with respect to GFr,.

In order to test the method, (Y parameters have been determined experimentally for six different matrices (biological and geological) in the form of thick pellets of reference materials. As their composition is known, it is also possible to calculate the (Y using fundamental parameters (p and S(Q). The two sets of values can thus be compared.

4. Experimental

4.1. Setup

In order to position the samples very accurately, a special revolving target holder has been realized. The smooth rotation was facilitated by the use of ball-

bearings which eliminate the lash associated with O- ring seals. The angular position was easily set with a precision of 0.1”. Given that the target was rotated about a vertical axis situated in the vertical target surface plane and that the incident proton beam and the detected X-rays were in a horizontal plane, we choose to use a vertical rectangular entrance collimator of 1.5 x 6 mm2 allowing a much better geometrical definition of the target beam spot than if we made use of a round collimator of the same aperture. The im- provement is especially important in situations where the proton beam incidence is grazing. It is also of importance that the proton beam axis exactly crosses

the sample rotation axis situated in the sample surface plane in order to be sure that the distance between the beam spot and the Si(Li) detector remains constant. The important geometrical characteristics of the setup are given in fig. 1.

The Si(Li) detector of 12 mm2 area is placed in a chamber wall cavity at an angle of 135” (lab), corresponding to j? = 45”, at a distance of 70 mm from the centre of the target spot. As there is no contact between the detector and the chamber in order to avoid mechanical vibrations transmission, a small uncertainty on /3 is possible. The maximum error on the mean /3 value is 0.8”.

4.2. Beam charge measurement

As the quality of the method depends directly on the precision of the calculation of the ratio r, a great accuracy is needed in the measurement of the number of incident protons. The different solutions have re- cently been reviewed by Johansson and Campbell [lo]. Given that we had to perform couples of measurements with different targets orientations, the evident


choice to avoid secondary electrons problems was to use the method consisting of the detection of the protons backscattered on a rotating vane upstream from the specimen. The reproducibility of the charge measurement is more important here than the absolute value since we are dealing with a ratio r. A series of

measurements performed using a Faraday cup with suppressor as a reference and without any target in the chamber showed a reproducibility of 1%. We also verified that the backscattering detector was ade- quately shielded against other protons than those directly scattered by the chopper.

Table 1 a (MeV-‘1, r and CF-’ correlated values for usually detected elements (P to Cal in PIXE analysis in the following experimental conditions: E, = 2.35 MeV, /3 = 45”, 0, = 120” and e2 = 40”

a P s Cl K Ca

r = f2 /fl CF-’ r = f2 /fl CF-’ r = f2/fl CF-’ r = f2/fl CF-’ r = f2 /f, CF-’

0 1 4.226 1 4.448 1 4.663 1 5.066 1 5.253 0.2 0.8793 4.363 0.8827 4.588 0.8858 4.805 0.891 5.213 0.8933 5.402 0.4 0.7817 4.502 0.7874 4.729 0.7926 4.949 0.8016 5.362 0.8054 5.553 0.6 0.7021 4.643 0.7094 4.874 0.7161 5.096 0.7276 5.513 0.7325 5.706 0.8 0.6367 4.787 0.645 5.02 0.6527 5.245 0.6659 5.666 0.6716 5.861 1 0.5824 4.934 0.5915 5.169 0.5997 5.396 0.6141 5.821 0.6203 6.018 1.2 0.5372 5.082 0.5466 5.32 0.5552 5.549 0.5702 5.978 0.5768 6.177 1.4 0.4991 5.233 0.5087 5.474 0.5175 5.705 0.5329 6.138 0.5396 6.338 1.6 0.4668 5.387 0.4764 5.629 0.4853 5.862 0.5009 6.299 0.5077 6.501 1.8 0.4393 5.542 0.4488 5.787 0.4577 6.022 0.4732 6.462 0.48 6.666 2 0.4156 5.7 0.4251 5.946 0.4338 6.184 0.4492 6.627 0.456 6.832 2.2 0.395 1 5.86 0.4044 6.108 0.413 6.347 0.4282 6.794 0.4349 7.001 2.4 0.3774 6.021 0.3864 6.272 0.3949 6.513 0.4098 6.963 0.4164 7.171 2.6 0.3618 6.185 0.3707 6.437 0.3789 6.68 0.3935 7.133 0.4 7.342 2.8 0.3481 6.351 0.3567 6.605 0.3648 6.849 0.3791 7.305 0.3854 7.516 3 0.336 6.519 0.3444 6.775 0.3522 7.02 0.3662 7.479 0.3724 7.691 3.2 0.3252 6.689 0.3334 6.946 0.3411 7.193 0.3547 7.655 0.3608 7.868 3.4 0.3157 6.861 0.3236 7.119 0.331 7.368 0.3443 7.832 0.3503 8.046 3.6 0.3071 7.034 0.3148 7.294 0.322 7.544 0.335 8.011 0.3408 8.226 3.8 0.2993 7.21 0.3069 7.471 0.3139 7.722 0.3265 8.191 0.3322 8.407 4 0.2924 7.387 0.2997 7.649 0.3065 7.902 0.3188 8.373 0.3243 8.59 4.2 0.2861 7.566 0.2932 7.829 0.2998 8.083 0.3118 8.556 0.3172 8.774 4.4 0.2803 7.746 0.2872 8.011 0.2937 8.266 0.3054 8.741 0.3107 8.96 4.6 0.2751 7.928 0.2818 8.194 0.2881 8.45 0.2995 8.927 0.3046 9.147 4.8 0.2703 8.112 0.2769 8.379 0.283 8.636 0.2941 9.114 0.2991 9.335 5 0.266 8.297 0.2723 8.565 0.2783 8.823 0.2892 9.303 0.294 9.525 5.5 0.2565 8.767 0.2625 9.037 0.2681 9.297 0.2783 9.781 0.2829 10.004 6 0.2488 9.246 0.2544 9.517 0.2597 9.778 0.2693 10.266 0.2736 10.491 6.5 0.2424 9.732 0.2477 10.005 0.2526 10.268 0.2617 10.758 0.2658 10.984 7 0.237 10.226 0.242 10.5 0.2467 10.764 0.2553 11.256 0.2592 11.483 7.5 0.2325 10.727 0.2372 11.002 0.2417 11.267 0.2498 11.76 0.2535 11.989 8 0.2287 11.235 0.2331 11.51 0.2373 11.775 0.2451 12.271 0.2486 12.5 8.5 0.2254 11.748 0.2296 12.024 0.2336 12.289 0.2409 12.786 0.2443 13.015 9 0.2225 12.267 0.2265 12.543 0.2303 12.808 0.2373 13.306 0.2405 13.536 9.5 0.22 12.791 0.2239 13.066 0.2275 13.332 0.2342 13.83 0.2372 14.06

10 0.2179 13.319 0.2215 13.594 0.225 13.86 0.2314 14.358 0.2343 14.589 10.5 0.216 13.852 0.2195 14.127 0.2228 14.392 0.2289 14.89 0.2317 15.121 11 0.2143 14.388 0.2176 14.662 0.2208 14.928 0.2267 15.425 0.2293 15.656 11.5 0.2128 14.928 0.216 15.202 0.2191 15.467 0.2247 15.964 0.2272 16.195 12 0.2115 15.471 0.2146 15.744 0.2175 16.008 0.2229 16.505 0.2254 16.736 12.5 0.2104 16.016 0.2133 16.289 0.2161 16.553 0.2213 17.049 0.2236 17.28 13 0.2093 16.565 0.2121 16.837 0.2148 17.1 0.2198 17.596 0.2221 17.826 13.5 0.2084 17.116 0.2111 17.387 0.2136 17.65 0.2185 18.145 0.2207 18.375 14 0.2075 17.669 0.2101 17.939 0.2126 18.201 0.2172 18.695 0.2194 18.925 14.5 0.2067 18.224 0.2092 18.493 0.2116 18.755 0.2161 19.248 0.2182 19.478 15 0.206 18.78 0.2084 19.049 0.2108 19.31 0.2151 19.803 0.2171 20.032


4.3. Electronic dead iime compensation 5. Results

In order to take dead times into account, we fed a 50 Hz pulser through the detection system and into a separate scaler. Each PIXE and each backscattering spectrum contained thus a pulser peak. The chopper vane intercepted the beam during 10% of the irradia- tion time. The dead time correction was calculated keeping in mind that there was no loss of pulser impulsions collection in the PIXE spectrum when the chopper intercepted the proton beam. The opposite consideration is true for the backscattering spectrum.

4.4. Sample preparation

The powdered samples were carefully reground in an agate mortar, transferred to an evacuable die and pressed at 4 x lo8 Pa in a hydraulic press. A polypropylene film was placed between the pellet and the pressing surface to prevent any contamination and removed afterwards. The pellets were 1-2 mm thick with a 20 mm diameter.

In accordance with the results of the above mentioned studies, we have performed series of ten couples of PIXE measurements on six reference materials pellets: a metallurgical sample: IRSID [ll], two biological samples: NBS 1571 (orchard leaves) and NBS 1577 (bovine liver) [12] and three geological samples: GXR4 [13], BR [14,15] and GSN [16]. The experimental parameters were E, = 2.35 MeV, after homogenization, /I = 45”, 8i = 120” and e2 = 40”. For this set of parameters, we have established tables 1 and 2, which give, for different elements from S to Zn, the related values of (Y, r and CF-‘. The main interest of these tables is that they are not dependent on the matrix composition and are thus of general use. Introducing an experimentally determined r value in these tables, it is possible to obtain the corresponding (Y and the CF-’ factor by interpolation. It is very rapid and adequate for a large range of r.

In a previous paper [5], we presented another way for treating the problem. Instead of calculating a table

Table 2 a (MeV-‘1, r and CF-’ correlated values for usually detected elements (Ti to Zn) in PIXE analysis in the following experimental conditions: E, = 2.35 MeV, /3 = 45”, 0, = 120” and e2 = 40

a Ti Mn Fe CU Zn

r=f2/fl CF-’ r=f2/fl CF-’ r = fz /fl CF-’ r = f2 IfI CF-’ r = f2/fl CF-’

0.0 1 5.6 1 6.05 1 6.184 1 6.538 1 6.64 0.2 0.8972 5.752 0.9018 6.207 0.903 6.342 0.9062 6.7 0.9071 6.802 0.4 0.8121 5.906 0.82 6.366 0.8222 6.502 0.8276 6.862 0.8292 6.966

0.6 0.7412 6.063 0.7514 6.526 0.7543 6.664 0.7614 7.027 0.7634 7.131 0.8 0.6816 6.221 0.6935 6.688 0.6968 6.827 0.7051 7.194 0.7074 7.298

1.0 0.6312 6.381 0.6442 6.852 0.6479 6.992 0.657 7.362 0.6596 7.467 1.2 0.5882 6.543 0.602 7.018 0.6059 7.159 0.6156 7.531 0.6183 7.638 1.4 0.5514 6.707 0.5657 7.186 0.5697 7.328 0.5797 7.703 0.5825 7.81 1.6 0.5196 6.873 0.5341 7.355 0.5382 7.498 0.5485 7.875 0.5514 7.983 1.8 0.492 7.04 0.5066 7.526 0.5107 7.67 0.5211 8.05 0.524 8.158 2.0 0.4679 7.21 0.4825 7.699 0.4866 7.843 0.497 8.226 0.4999 8.335 2.2 0.4468 7.381 0.4612 7.873 0.4653 8.018 0.4757 8.403 0.4786 8.513 2.4 0.4281 7.553 0.4424 8.048 0.4464 8.195 0.4567 8.582 0.4595 8.692 2.6 0.4115 7.727 0.4256 8.225 0.4296 8.373 0.4397 8.762 0.4425 8.873 2.8 0.3967 7.903 0.4105 8.404 0.4145 8.552 0.4245 8.944 0.4273 9.055 3.0 0.3834 8.081 0.397 8.584 0.4009 8.733 0.4107 9.127 0.4135 9.239 3.2 0.3715 8.259 0.3849 8.766 0.3887 8.915 0.3983 9.311 0.401 9.424 3.4 0.3608 8.44 0.3738 8.949 0.3776 9.099 0.387 9.497 0.3897 9.61

3.6 0.3511 8.622 0.3638 9.133 0.3675 9.284 0.3768 9.683 0.3794 9.797 3.8 0.3422 8.805 0.3547 9.318 0.3583 9.47 0.3674 9.871 0.3699 9.985 4.0 0.3342 8.989 0.3463 9.505 0.3499 9.658 0.3588 10.061 0.3613 10.175 4.2 0.3268 9.175 0.3387 9.693 0.3421 9.846 0.3509 10.251 0.3533 10.366 4.4 0.32 9.363 0.3317 9.882 0.335 10.036 0.3436 10.442 0.346 10.558 4.6 0.3138 9.551 0.3252 10.073 0.3285 10.227 0.3369 10.635 0.3392 10.751 4.8 0.308 9.741 0.3192 10.264 0.3224 10.419 0.3306 10.829 0.3329 10.945 5.0 0.3027 9.932 0.3136 10.457 0.3168 10.612 0.3248 11.023 0.3271 11.14

Table 3


Comparison of “theoretical” and experimental (Y values (MeV-‘1 for six reference materials

P S Cl K Ca Ti Cr Mn Fe Cu Zn

Biological reference materials NBS 1571 theor 2.27 1.59 0.81 0.7 0.5

*0.14 kO.1 If: 0.05 f 0.04

ew 2.06 1.44 0.72 0.66 - 0.07 &O&6 f 0.02 f 0.015

NBS 1577 theor [2.98] 2.07 1.55 0.81 0.66 0.37 +0.18 *0.12 *0.10 f 0.05

exp 2.77 2.08 1.37 0.78 f 0.08 + 0.04 + 0.03 + 0.015 L L L

Geological reference materials BR theor 9.44 6.6 3.43 2.65 2.18

+0.2 +0.16 +0.13

ew 3.37 2.58 2.03 - +0.1

GSN theor 10.11 7.07 3.66 0.065 f + 0.06 3.06 1.86

f 0.22 +0.18 +0.11

em 3.58 2.92 1.82 - * * 0.05 + 0.02

GXR-4 theor 10.26 7.49 3.88 3.25 1.93 +0.6 +0.24 *0.19 +0.12

ew 10.1 3.75 3.09 1.78 - + 0.5 + 0.06 + 0.06 f 0.035

IRSID theor 11.95 9.93 6.09 4.75 3.24 (876-l) +0.72 +0.6 f 0.36 *0.3

exp 11.75 10.75 6.6 4.92 1.52 --- * 0.05 + 0.02 + 0.015 + 0.09 f u.uz - - L L ~

- - - - - f 0.72 f 0.38 *0.2 *0.1

0.3 0.23 0.18 0.1 0.08

0.22 0.17 0.13 0.07 0.06

1.4

1.14

1.17

2.09 *0.12

1.78

1.1 + 0.065

1.1 + 0.08 0.9 + 0.05

0.96 0.095 + 0.92

1.66 *0.1

1.67

0.88 + 0.055

0.84 + 0.03 0.72 f 0.04

0.7 +0.01 0.74 f 0.045

0.72 0.02 1.33

f 0.08 1.3

0.72 0.6

0.45 0.36

0.47 + 0.03

0.46 0.03 1.6

+0.1

0.38

1.32 + 0.08

1.35 ^ ^_

of r values corresponding to a series of given (Y, which is straightforward, we obtained the (Y corresponding to an experimentally determined r by an iterative calculation (Newton algorithm). In that case, we sometimes encountered convergence problems if the chosen iteration step was too large or if the initial (Y value needed to start the iteration process was not near enough to the real one.

As Zi,,,rr =Z,(CF)-‘, tables 1 and 2 allow to calculate the Zi,corr which represent the corrected intensities. These values, normalized to a given number of incident protons and compared to a thin calibration curve, give the corrected absolute mass concentrations if the stopping power S(E,) of the matrix is known and the relative mass concentrations in the other case. It is worth mentioning here that in many problems studied using PIXE, the relative concentrations which allow

the calculation of the EF (enrichment factors) are what matters. Examples can be found in pollution studies, in archeometry and also in geology.

Table 3 allows the comparison of the experimentally obtained (Y and the‘corresponding “theoretical” values. These “theoretical” (Y’S have been calculated us-

ing the certified matrix composition for each reference material, the S(E) from Anderson and Ziegler [17] and the absorption parameters p from Leroux and Thinh [18]. The experimental LY in this table correspond to chemical elements whose concentration in the different samples was sufficient to obtain rapidly a statistically good value for the r ratio.

The errors of the order of 6% accompanying the “theoretical” (Y values reflect the accuracy of the S(E) tables and the Z.L tables as discussed by Johansson and Campbell [lo]. These errors would be somewhat larger

Table 4 Effect of 5% residual moisture on the S(E) and LY values in IRSID reference material at E, = 2.35 MeV

S(E,I S Cl K Ca Ti Cr Mn Fe

IRSID 876-l 85.7 11.95 9.93 6.09 4.75 3.24 2.09 1.66 1.33 IRSID + 5% H *O 88.3 11.23 9.33 5.71 4.43 3.02 1.95 1.54 1.24

Cu Zn

1.6 1.32 1.48 1.23

82 J.M. Delbrouck-Habaru et al. Matrix correction for TTPIXE

if the uncertainties in the data concerning the concentrations in the reference materials and the hetero- geneities of the real samples were taken into account.

The errors coupled to the experimental values are deduced from the study of the standard deviations of the series of ten r values previously evoked. The experimental errors in table 3 are the standard error on the mean of the ten values. The good agreement is obvious.

It is important to lay stress on the following point: considering the results of table 3, one can remark that the experimental (Y values are almost always slightly below the theoretical ones. That could be attributed to a slight moisture of the pellets. The stopping power of hydrogen is indeed about twice the value for other elements. As an illustration, table 4 shows for the IRSID matrix, the S(E) and (Y values calculated from fundamental parameters and the corresponding values with the assumption of 5% Hz0 in the matrix.

6. Stopping power

The problem of normalization leading to absolute concentrations is a general one in PIXE [lo]. It is often resolved by making hypotheses about the composition of the matrix and by calculating S(E) using fundamental parameters.

The (Y parameters method offers a more attractive and satisfying possibility to get information about S(E). Actually it has been shown in a previous paper [5] that (Y and S(E) are generally very much correlated. The physical explanation can be found in the LY definition

of relation (1) which shows that the (Y is inversely proportional to S(E). The stopping power of a matrix increases and the absorption coefficient p decreases when its light elements concentration grows, and the lighter the matrix, the lower the (Y. This general trend is, however, not respected when the (Y considered corresponds to an element whose radiation is near an absorption discontinuity for the matrix studied. It is thus convenient to use the (Y corresponding to several elements to estimate the matrix stopping power: for a given matrix and for each detectable element i, one can obtain an (Y~ and, using the correlation, a corresponding value for S,(E). The arithmetic mean value S(E), of all the S,(E), obtained after eliminating the values corresponding to important discontinuities effects, is generally concordant with the real value within less than 8% as it will be shown hereafter.

We have searched for an analytical expression for the correlation between S(E) and (Y values. For this purpose we have chosen 76 known matrices represent- ing a very large range of compositions. The matrices used are oxides and other chemical compounds: Na,O, MgO, Al,O,, SiO,, P,O,, K,O, CaO, TiO,, MnO, Fe,O,, H,O, CO*, KCl, K,O, CaCO,, Fe, CH,, U,O,, a series of NBS reference materials 1121: NBS 1571 and 1577 (biological), NBS 1648, 1633, 1633A, 1633a/ MOD, 1648/MOD, (environmental), NBS 436, 58a (steels and alloys), BAS reference materials [19]: 464, 243/4, 577-1, 587-1, 180/2, 344, 578-1, 178/2 (a large range of alloys), a series of geological reference materials: GH, BR, DR-N, UB-N, BX-N, BE-N, GS-N, IF-G, GXR-4, AL-l, FK-N, SGR-1, SU-1, a series of geo- chemical materials from Mason [20]: earth bulk, crust,

S(E) (MeV*cm’/ g )

I I 0 10 a(MeV’) I 5

Fig. 10. Relation S(E) = a - b In (Y for nine elements behveen S and Zn.


igneous rocks, sandstone, limestone, sediments, river and sea waters, man, marine invertebrates, wood, peat, lignite, bituminous coal, anthracite, zircon, some color pigments: Ti, Ca white, red lead, yellow Cr, green Cr, blue Fe,(Fe(CN),),, and some compounds illustrating matrices often studied by PIXE as Havar, Mylar, IRSID, Hoepfner, potteries and volcanic lava.

Using the known compositions we have calculated S(E) and (Y values from the fundamental parameters [17,18]. The correlation function between S(E) and the (Y is of course slightly different from one element to the other but its general form may be written as follows:

S(E)=a-b In a

with

(29)

a = 345.6 - 17.13 Z + 0.287 Z2 (r = 0.998),

b = 109.9 - 4.57 Z + 0.0661 2’ (r = 0.994),

where Z is the atomic number of the corresponding element.

The fig. 10 gives the curves corresponding to relation (29) for nine elements from S to Zn. These curves

are directly useable for determining S(E) if a series of (Y parameters are experimentally measured.

In order to test the quality of the method for the determination of S(E), use has been made of the above mentioned curves to obtain for each of the 76 matrices the arithmetic mean value S(E), of ail the individual S(E), obtained for each element i. We call here S(E), the stopping power calculated using the matrix composition and the database.

For our 76 matrices the distribution of the differ- ences [S(E) - S(E),]/S(E) shows that the mean error in the determination of the stopping power by this method is about 8%.

7. Discussion and conclusions

The (Y parameters method allows the calculation of the matrix effects correction factors; it can be applied to infinite thickness samples. In this case, the (Y’S are obtained using two PIXE measurements performed in two geometrical configurations. We have defined the best practical experimental conditions. For a given

Table 5 CF-‘, dr/r (o/o), da/a (%) and dCF-‘/CF-’ (%I correlated values for the detected elements in the six reference materials

P s Cl K Ca Ti Cr Mn Fe Cu Zn

NBS 1571 CF-’

dr/r [%I da/a [%I dCF-‘/CF-‘[o/o]

5.99

5.3 10 2.9

NBS 1577 CF-’ 6.33 6.1

dr/r [%I 4.1 2.8

da /a [%I 7.7 5.2 dCF-‘/CF-‘[%I 2.8 1.5

BR CF-’

dr/r (o/o) da/cu (%I dCF-‘/CF-‘[%I

GSN CF-’

dr/r (%) da /a (%o) dCF-‘/CF-’ [%I

GXR4 CF-’ 13.7

dr/r (%) 3.0 da /a (%o) 15.8 dCF-‘/CF-’ [%I 12.5

IRSID CF-’ 15.47

(876-l) dr/r (%) 2.8 da/a (o/o) 18.0 dCF-‘/CF-‘[%I 14.9

5.74 5.6 5.75

5.7 2.4 2.5 11.5 7.1 7.5 2.5 0.8 0.8

5.68 5.65 - - 3.6 2.4 7.6 7.1 1.45 0.8.

7.80 7.32 7.24 6.93 6.86 -- 4.7 3.9 3.9 7.4 3.0 9.5 7.6 7.9 3.6 2.3 2.00

7.99 7.63 7.06 --- 1.7 2.3 1.7 3.5 4.5 3.5 1.4 1.60 0.80

8.15 7.77 7.02 --- 2.4 3.0 2.9 5.0 6.0 6.1 2.1 2.2 1.3

14.66 10.86 9.45 - - - 2.2 3.2 2.6

18.8 8.5 2.70 1.00

6.82 6.75 -- 9.1 1.3

27.3 4.3 3.3 0.4

6.76 6.91 - - 2.2 3.3 7.2 13.7 0.7 1.0

7.5 7.41 7.24 7.8 7.77 =

-- 2.0 2.2 s.s 2.2

11.1 9.3 6.1 8.0 4.3 5.3 20.7 5.4 8.8 5.8 3.2 1.6 0.9 0.85 3.6 0.80

84 J.M. Delbrouck-Habaru et al. ’ Matrix correction for TTPIXE

element, the (Y and the correction factor can be imme- diately extracted from the general purpose tables 1 and 2 by simply introducing the experimentally obtained value of the ratio r of two peaks areas. Fig. 10 allows a determination of the matrix stopping power necessary to calculate absolute concentrations. Our method, based on direct experimental measurements, is less dependent on the database uncertainties than the other correction methods.

References

ill Dl

P. Aloupogiannis, Ph.D. thesis, Paris (1988). P. Aloupogiannis, G. Robaye, I. Roelandts, G. Weber, J.M. Delbrouck-Habaru and J.P. Quisefit, Nucl. Instr. and Meth. B14 (1986) 297.

131

[41

The accuracy of our method, that is the degree of agreement between the experimental (Y and the theoretical ones, is summarized in table 3. The precision in the determination of one (Y value using only one cou- ple of measurements is presented in table 5. In this table we give for each of the six reference materials studied and for each element the total matrix correction factor CF- ‘, the experimental standard error dr/r

in % for one measurement of r, the corresponding calculated standard errors da/a in % and dCF-‘/CF-’ in %. One can see that even for large values of the total correction factor CF-‘, its standard error remains of the order of a few percent even if the corresponding error dcu/cx in % is sometimes much larger.

P. Aloupogiannis, G. Robaye, I. Roelandts and G. We- ber, Nucl. Instr. and Meth. B22 (1987) 72. P. Aloupogiannis, G. Robaye, G. Weber, J.M. Del- brouck-Habaru and I. Roelandts, X-ray Spectrom. 19 (1990) 133.

El

It is important to remind that if one wants to calculate absolute concentrations, other factors of the same order of magnitude appear. They refer to the precision of the thin target calibration curve and of the matrix stopping power. The homogeneity and the rep- resentativity of the sample also play an important role. These error factors are common to all TTPIXE analy- ses. The main advantage of our method is that no hypothesis about unknown sample composition is necessary to get the matrix correction factors with a very good precision.

161

171

[81 191

no1

[ill

ml

1131

[141

1151

[161 [171

P. Aloupogiannis, G. Weber, J.P. Quisefit, J.M. Del- brouck-Habaru, I. Roelandts, M.C. Rouelle and G. Robaye, Nucl. Instr. and Meth. B42 (1989) 359. F. Folkman, Ion Beam Surface Analysis, vol. 2 (Plenum, New York, London, 1976) p. 747. H. Paul, Nucl. Instr. and Meth. B3 (1984) 5 and Erratum Nucl. Instr. and Meth. B5 (1984) 554. K. Ishii and S. Morita, Int. J. PIXE 1 (1990) 1. J.L. Campbell, R.D. Lamb, R.G. Leigh, B.G. Nickel and J.A. Cookson, Nucl. Instr. and Meth. B12 (1985) 402. S.A.E. Johansson and J.L. Campbell, PIXE: A novel technique for elemental analysis (Wiley, New York, 1988). Euro-echantillon Type 876-1, Certificat d’analyse Irsid, Maizieres-Its-Metz Irsid (octobre 1977). E.S. Gladney, B.T. O’Malley, I. Roelandts and T.E. Gills, NBS Special Publication 260-I 11 (1987). ES. Gladney, C.E. Burns and I. Roelandts, Geostan- dards Newsletter 8 (1984) 119. M. Roubault, H. de La Roche and K. Govindaraju, Sciences de la Terre 15 (1970) 351. K. Govindaraju and H. de La Roche, Geostandards Newsletter 1 (1977) 67.

iI81

Acknowledgement

We are indebted to the Institut Interuniversitaire des Sciences Nucleaires (Belgium) for financial sup- port.

I191

BOI

K. Govindaraju, Geostandards Newsletter 8 (1984) 173. H.H. Andersen and J.F. Ziegler, in: The Stopping Power and Range of Ions in Matter, vol. 3 (Pergamon, New York, 1977). J. Leroux and T.P. Thinh, Revised Tables of X-Ray Mass Attenuation Coefficients (Corp. Scientifique Claisse, Quebec, 1977). BAS (Bureau of Analysed Samples, Ltd., NewhamHall, Newby, Middelsbrough, Cleveland, England TSS-9EA) SRM cat 550 (1987). B. Mason, Principles of Geochemistry (Wiley, New York, 1952).

determination of the best experimental conditions for using the α parameters method to calculate...

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