Nuclear Instruments and Methods 203(1982) 503-508

503North-Holland Publishing Company

DETERMINATION OF THREEPARAMETERS OF ADEPTH PROFILE OF F(`I`FIGN ATOMSIN BULK MATERIAL USING PIXE ANALYSISPart II: Influenceof the errors of theoretical quantities which are used in the evaluati .

mocedure

M.GERETSCHLÄGERJohannes-Kepler-Unirersiliit Linz, lnstitm Jür E.vpcrinuvunllrl(rrik. A-41110(.Htz, ausrria

Received 18 March 1982

The mean depth(a). the mean width (b) and the relative concentration (r) of an unknown distribution of foreign atoms in bulkmaterial can be determined by relative X-ray yield measurements. It is shown by me-ins of a set of case studies for rectangulardistributions of foreign Cu-atoms in Ag-bulk to whichextent errors in the theoretical description of theX-ray production processandthestopping of projectiles contribute to errors in the result . Errors of Ih" . 6`P and 7% for a, G, and c, respv-.lively, arise from theuncertainty of the theoretical description of the ionization processprovided that one uses corrections of the theoretical ionization crosssection and corrects forenergy loss straggling andsmallanglemultiple scattering effects. However, ;in additional large uncertainty isbrought into evaluation by the uncertainty of the stopping power. Measurements on Ag--Cu-Agsandwich targets are presented.

1. Introduction

During the last decade particle induced X-ray emis-sion (PIXE) has become a routine analytical method for-asuring trace element concentration in complextargets [1,2] . There have also been some attempts toobtain information about thedepth distribution of traceelements (3-11] in complex targets from PIXE measure-ments. In an earlier paper [11], we developed a methodto determine three parameters of a distribution fromPIXE measurements: the mean depth (a), the meanwidth (b) and the relative concentration (c) of theforeign atoms in a bulk. In this earlier paper (11] wealso compared our PIXE measurements with Ruther-ford back-scattering (RBS) measurements . Unfor-tunately, we found a systematic disagreement forparameter b, which we could not explain satisfactorily.Since the evaluation procedurewhich is necessary to getthe three parameters of a depthdistribution from X-rayyield measurements is rather complicated [11,12] andsince it is therefore difficult to see how measurementerrors and errors of theoretical quantities contribute toerrors in the result, we discussed in Part 1 of this paper[13] the effect of experimentai errors. Now we considerthe effect of errors of theoretical quantities which areused in the evaluatioA procedure.

2. Accuracy of the method

The theory of our method and the evaluation proce-dure have been described in detail before [11,13) . The

0167-5087/82/0000-0000/$02.75 O 1982 North-Holland

central point in this procedure is the evaluation of eq.(1), which represents the X-ray yield Y(E,) measured bya detector, if a target with a trace element concentrationn(x), where n(x) is the number of foreign atoms perunit volume, is bombarded with protons of energy E, :

1~(E, rlfxn(x)u(E(x,E,))t

X exp( -apx )dx.

In eq. (1), tl represents the absolute X-ray detectionefficiency including the fluorescence yield, E(x, E,) theenergy of protons at a depth x, o(E(x, E,)) the corre-sponding ionization cross section, a a factor whichdepends on proton beam-target-detector geometry, andp the X-ray absorption coefficient.

If we now consider the error contribution of thetheoretical quantities in eq . (1), we find that only thequantity a(E(x, E,)) contributes to the error ofsince our evaluation method [11,13] is based on X-rayyield ratios (Y(E, )I Y(E, )) and therefore n cancels outand errors in the absorption factor exp(-apx) can beneglected because of the same reason . The energy E(x,E,) at a depth x is related to the proton impact energyE, by

E(x,E,)-E,-f.'S(E(x'))dx',(2)o

where S is the stopping power ")f protons in the targetmaterial . We therefore have to consider error contribu-tions of the ionization cross section a(E) and of S(E).Our aim is to show the error contributions of the

5,04

theoretical description of the ionization process and ofthe stopping of ihcprojectiles to the result by means ofa set of case studies for rectangular distributions offoreignCu-atoms in Ag-bulk for some "optimum" pro-ton impact energy combination (see ref. 13), since itseems impossible to show it in a general form, i.e. . forall possible parametercombinations of all possible pro-ton impact energy combinations for all possible elementcombination,

..1 . Ivnc,rrr,m pr,H'rss arul cnergr lorr mraggliug

The computer prograr, (121 which calculated ourtheoretical \-raN yields by numerical integration of eq.(1), and which used in the original version PWBA crosssections of Basbas et al . [141, was changed and it nowused ionization cross sections given by the ECPSSRtheory of Brandt and Lapicki I IS]. The ECPSSR theoryin general predicts ionization cross sections very wellbut at very low ionenergies the ECPSSR cross sectionsare tow high 116-18] . From a comparison of all experi-mental ionization cross sections available in the litera-ture 1191 to the ECPSSR cross sections [151, it can beseen that at fixed-.scaled icon velocity ~=2r tiBx r_ s1ueref. 16) the deviation of the ECPSSR cross sectionsfrom averageexperimental crosssections is universal forall target elements . Fig. 1 shows 119] the ratio of experi-mental and ECPSSR cross sections as a function oft. Itis evident from fig. 1 that the ECPSSR cross section hasto he corrected especially, if (me uses low proton impactenergies, which is necessary to obtain minimum experi-mental errors 1131, We corrected o(E) by multiplying itwith the function A(f(E(x. E,))) = ni ft, using a=50 "ft == 2.5 for ~ ,0.2 and a =- 0.98, (3 = 0.053 for t>0.2 .Theuncertaintyof this correction is smaller than 5% forj >0.3 but rises to about 30%,, at Q= 0.1 .

If a proton Cream with initial energy distributionJ,rl E,, F.) (fig . 2) penetrates a target it will suffer a mean

.M. Geretsrhlüger ; D-rnriparlon af threeparanrer, "rs qf udep,hpr,file

Fig. l . Ratios of mean values of all experimental ionizationcrosssections available in the literature (17] to ECPSSR crossscctions 114) as a function of scaled ion velocity t:.

p,~É=E.-dE

AE=f,'S( E(x'))dx',

Fig. 2 . Broadening of the energy distribution of a proton beamdueto energy loss straggling.

energy loss which is given by

E

and a broadening of its energy distribution because ofthe energy loss straggling [20]. At a depth x we willtherefore have an energy distribution ft((E,-AE). E)and we should use in eq. (1) a mean ionization crosssection

4'.(E)fl((E,-AE), E)dE

Oil-, -JE)=°

-fxf((E,-AE), E)dEo

instead of the ECPSSR cross "ection a(E(x, E,)) . Toaccount for the energy loss straggling effect we in-o-duce a correction function

B(E(x,E,))=s,"suo(E)

FIw.6',1 -SSE

exp[- (E'- E( x,E;)) /S2Z]dE'X

S2 2xra[E(x,E,)]

where S2 is the standard deviation of the stragglingdistribution . Evidently we used in eq . (3) a Gaussiandistribution which is for our purposes close enough tothe true straggling distribution [21,22]. For S2 we usedBohr s formula [231 although Bohrs value is slightly toohigh. In order to avoid excessive computer time wecalculated in eq . (3) o(E) according to the SCA theoryof Laegsgaard et a(. [241 . In the low energy range theSCA predictions are too small [16-181 but in eq . (3)only the energy dependence of a(E) within theintegra-tion limits is important and therefore we feel that theuncertainty of our correction function B(E) is smallerthan 20% Instead of eq. (1), we therefore now have

Y(E,)=r1fown(x)o(E(x .E;))A(Z[E(x,E,)1)

XB(E(x,E;))exp(-agx)dx . (la)

?.2. Stappfg pau,er and .rnulll angle nutltiple sca!lerinq

For the numeric integration of eq. (1) one has todivide the target into very thin layers of thickness :5xand calculate the mean proton energy E. in every layer.We used the stopping power values of Andersen andZiegler [251 to calculate the energy EI of target layer j,using

F,=Et , -S(EI . t),1x ;

(4)

starting of course with E,)= E,. Error contributions ofS( E) will be discussed at the end of this section .

The stopping power accounts for the mean energyloss of ions within the target,butat least for thick targetone also has to consider small angle multiple scatteringeffects: If a proton beam penetrates matter its angulardistribution is broadened by scattering processes atlarge impact parameters (fig. 3) . The true path length l .of an individual ion is longer than the correspondingtarget thickness X, and therefore the energy loss of thision will be enhanced by the factor L/x= I/cos a-- C.It has been shown [26,27] that the reduced half widthd, ; . : of multiple scattering distributions depend on re-duced target thickness r by the simple formula &, _ =

41/2 "' if the interaction potential is of type V(r) ar_"'. Sigmund and Winterbon 127] calculated smallangle scattering distributions using more realisticThomas-Fermi- and Lenz-Yensen-type potentials andpresented their results in tabulated form . For simplicity,we approximated their results by 61 ,,_ = tc�,4 +- x �, using=0.25, xt,=0 for 4<5 and x", - 0.115, h�=0.6 for

T>5. The reduced half width &,,., of the scatteringdistribution was converted to the standard deviation ofthe angular distribution as assuming Gaussian distribu-

M. Gerelrchliiger / DetennBtallon n,f tlare parnntetrrs of a depllt pn,file

Fig . 3. Broadening of theangular distribution of aproton beamdue small angle multiple scattering.

tion, and from that we calculated the multiple scatteringcorrection factor for layerj, which is

t

The proton energy in layerj is nowE __ E,

I -S

é',

t ).Ix(, .

(4)

To demonstrate the influence of all corrections on X-rayyields mentioned above, we plotted in fig. 4 theratios ofcorrected Cu K X-ray yield to uncorrected one for aproton impact energy E, =250 keV a. t function ofa -- b/2 which is the depth of the first atomic layer ofthe foreign material (Cu) in the bulk (Ag) . Because ofthe very strong proton energy dependence of the ioniza-tion cross section at those low energi,,, for fixed valuesa - h/2 these ratios arc nearly inde,rondent of paratue-ter h. As can be seen from fig. :, the energy lossstraggling correction (!3) dominates at large values ofa -- /)/2 and !hecorrection of the ECPSSR cross section(a) is important in the whole regi :m. The small anglescattering correction (C) does not have tux) much in-flucncc on the X-ray yield .

In order to see to what extent these corrections in eq.(1) change the mean depth (a) and the mean width (1))we calculated theoretical yield ratios for sonic protonimpact energy combinations for c" = 1 and various com-binationsof a and h using thevarious corrections in eq.(I). Using these yield ratios and the uncorrected database we determined the corrected values a and /) asdescribed in Part 1 of this paper113] .

From theseresultswe obtained thecorrection factorst n, :, Dt,) for a and h. Figs. 5, 6 and 7 show as anexample the results for an "optimum" [131 proton int-pact energy combination of 750, 350 and 250 keV. Fig . 5

l8

â 0.8

0.6-

0'l

0.7"

C

!2 3

5 6 7 8 o b/2 ,kA'

505

Fig. 4. Ratios of corrected to uncorrected X-ray yields as afunction of the depthof the first foreignatomic layer (a - h ./2)

fora proton impact energy of E=250 keV . A : correction of theECPSSR cross sections, B: energy loss straggling correction. C:multiple scattering correction.

506

a , F aBC

co5f

Fig. 5 . Correction factor for the mean depth ( n, l as a functionof a for the proton impact energy triple 750, 350 and250 keV.A: correction of the ECPSSR cross sections ; B: energy lossstraggling correction; C: multiple scattering correction . Corre-lation to parameterbis indicated by L forb =11000A+"r2a,and by S for h =1000A.

r.1

aeGcs[

a`EFig.6. Correction factor for themean width D,, as afunction ofhfor theproton energy triple 750, 350, 250keV. B: energy !n«straggling axnYtion ;C: multiple scattering wrrection .

ta,

Reri

Q6~11 2 3 <

5 6 7 8

b &A]

dl. G-hliiger , : Determinatiorr rJ th-" par<anegrr.e u/ a depth hr,,/il,

2 3 <

5 6 -, 8

b AA]

-IBC

__ 3. .

e-

a/r,al

Fig. 7, Correction factor for themean width D,, as afunction ofbfor theproton energy triple 750, 350, 250 keV.A: correctionof the ECPSSR cross sections; B: energy loss straggling corme-tion;C: multiple scattering correction.

shows the r.sulting correction for the mean depth ( D� )as a function of a. Since there is a correlation of D� toparameter Ir also, we draw curves which are labelled byletters L and S, respectively. S means that b= 1000 Aand L means that b= 11000 A or b=2a, whichever issmaller. It can be seen that the energy loss stragglingcorrection is significant only for a > 8000 A and thatsmall angle scattering effects can be neglected . Th":correction of the ECPSSR cross sections, however, isimportant in the whole rage of a. Fig. b shows theinfluence of the energy loss straggling correction and ofthe small angle scattering correction, respectively, uponh. The small angle scattering correction is significantonly for very thin and deep layers. Energy loss stragg-ling, however, changes b very strongly. Fig. 7 shows theeffect of all corrections on b . For a--8000 A thestraggling correction and the ECPSSR correction nearlycancel each other.

As far as the theoretical description of the ionizationprocess is considered we feel that the parameters a, band c can be determined with errors smaller than 1%,6% and 7%, respectively, provided a > 2000 A and b >2000 A . if one uses corrected ECPSSR cross sectionsand takes energy loss straggling into account . Theseerrors in a, h, and c which arise from the final uncer-tainty in the theoretical description of the ionizationprocess are comparable to errors arising from experi-mental yield measurement errors [13] using "optimum"proton impact energy combinations. However, we havefinally to consider the mean energy loss within thetarget, which enters eq. (1) by means of eq . (4) . Chestopping power values of Andersen and Ziegler areconsidered [25] to be accurate to about 5% at 500 keV.At lower energies the accuracy deteriorates to about10% to 20%. In order to see to which extent changes inthe stopping power enter into values of a and b weagain calculated theoretical yield ratios forsome protonimpact energy combinations forc= 1 and variouscom-binations of a and b using changed stopping powers ineq . (4) : S.(E)=kS(E) . One set of kS(E) calculationswas carried out using k= 1 .1, for an otherset we usedk= 1/(0 .679+0.111 log E), where E is given in keV,which results in an energy-dependent increase of thestopping powerof 10% between 750 and 100 keV start-ing with k (750 keV)= 1 .00 . Using this yield ratio andthe uncorrected data base we determined a and b asdescribed in Part I of this paper [13] and obtainedfinally the changing factors (F., F) for a and b. Figs . 8and 9 show as an example the results for the protonimpact energy combination of 750, 350 and 250 keV. Infig. 8, which shows the influence of changes in stoppingpower on the mean depth a, one can see that for anenhancement of the stopping power by 10% (k= 1 .1)themean depthis also enhanced almost exactly by 10%.Thesituation for the energy-dependent enhancement ofthe stopping power is less trivial ; the mean depth a is

1

7

3

1

5

6

7

8

a [kA!

Fig. 8 . Changing factor for the mean depth F,, as a function of afor the proton energy triple 750, 350, 250 keV, using changedstopping power S,(E)= S( Elk. Correlation to parameter h isindicated by L forh=1 1000 A or 2a and by Sforh - 1000 A.

enhanced by 1% to 7`"e, depending on a and, as indi-cated by labels L and S, on h (L means h== 11000 A orb = 2a, whichever is smaller. S means h= 1000 A) .Fig. 9 shows the influence of changes in stopping poweron the width b. For an enhancement of the stoppingpower by 10% the width b is also enhanced by 10`iexcept for very thin and deep layers. An energy-depen-dent enhancement of S, however, reduces b strongly forsmall values of b which is in contrast to the enhancement of a (see fig. 8). For 6000 A<h< 11000 A and3500 A <a < 8500 A there are only small changes in lz .

In conclusion, we find that the limited accuracy inthe stopping powervalues [23] can introduce an error of4% to 8% for themean depth and errors of about 20 to60% for the width and hence for th " relative concentra-tion respectively if one uses a proton impact energyE< 300 keV in his proton impact energy combination.If, however, one uses proton impact energies E>400keV, these errors are estimated to be 2% to 4"n for a and

tz

ae ~o-so .o

h1. (ieret.celdüxer

Deterzninathur of threeparameim q( a deptir pr,)file

1

2

3

4

5

6

7

8

b IkA)

Fig. 9 . Changing factor forthemean width F,, as a function of hfor the proton energy triple 750, 350, 250 keV, using char3edstopping power S,(E)= S(E)k.

10

to 30% for h and c, provided ii>2000 A antih>2000 A.

3. Measurements and results

>07

To demonstrate our method we determined theparameters ez and b of Ag-Cu-Ag sandwich targets .where a and lr were depth and width of the foreign (('ittatoms . We used energy analyzed protons of 270 to 7511keV and measured the Cu K X-ray yields. The detailedexperimental setup has hcen described before [28[ . Thetargets were prepared by vacuum evaporation of pureAg and Cu layers onto AI backings. The lateral honto-g-ncity was checked by means of Rutherford hack-scattering (RBS) and was found to deviate in the worstcase by *- 25 from the average layer thickness over anarea of 2 cm= . From all evaporated layers we also pre-pared separate reference targets, so we were able tocheck their thickness by RBS, The experimental errorswere according to table I in ref. 13.

Theexperimental resultsof our X-raymethod (PIV)are compared to Rutherford hack-scattering (RBS) mea-surements in table 1, For PIXE we used five differentproton impact energies and for evaluation six differentenergy triples. The entries labelled as Sa and Sb give theexperimental errors, and in parenthesis the errors date toinaccurate theoretical quantities are given, which aremainly determined by stopping power errors if oneassumes the error estimates of ref . 25 . For proton im-pact energy combinations which arenear an "optimum"combination (see Part I of this paper [13]) the experi-mental errors are frequently smaller than the errors due

theoretical quantities. For the RBS measurements weused 750 and 500 keV protons, The RBS results front750 and 500 keV protons, respectively, differ systemati-cally fiorn each other which indicates errors in the usedstopping power values: the mean depth obtained fromthe RBS measurements using500 keV protons are about3% to 6% larger than those using750 keV protons. Thewidth slows an opposite energy dependence : the 5(K)keV values are about 3% smaller than those using 750keV protons. The errors of the RBS measurementsinclude evaluation errors and errors of target inhomo-gencity. The agreement between the PIXE and RBSresults is satisfactory especially if one looks at the 500keV RBS results.

We also re-evaluated the measurements which wepresented in an earlier paper [Ill using all correctionsmentioned above and we now found satisfactory agree-ment between the PIXE and RBS results for both themean depth and the width of the foreign layer .

508

Reference+

M. Geretschläger ; Determirtatimt of three parameters of a depth p-file

Table 1Comparison of results front X-ray method with Rutherford back-scattering method. Proton impact energies E, are given in keV, valuesfor a, Sit, hand Sh are given in Â. Estimated errors are ako given (see text)

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Tauget E, : PIRE RBS

750 750 750 750 750 750 750 500500 450 300 500 450 500270 270 270 300 300 450

1 a 8430 8550 8600 8420 8500 8400 7910 8425Sa I50 (250) 100 (250) 860 (250) 150 (170) 100 (170) 400 (120) 180 190h 3910 4300 4460 3300 3600 1200 4210 4100SA 650 (400) 450 (400) 1200 (400) 650 (320) 450 (320) 4500 (200) 100 100

2 u 6100 6130 6300 6070 6,00 6050 5660 6090Sa 170 (200) 160 (200) 760 (200) 170 (130) 165 (130) 700 (100) 130 140b 2960 3400 3800 2300 2700 1200 4450 4330ab 1300 (200) 1200 (200) 2300 (200) 1400 (160) 1300 (160) 6000 (130) 100 100

3 a 10580 10700 11000 10480 10550 10350 10180 10520Sa 490 (210) 230 (2W) 600 (210) 280 (150) 230 (150) 500 (120) 230 240b 8280 9620 9500 7550 7700 6700 8490 82508h 460 (800) 450 (800) 1400 (800) 660 (600) 450 (600) 4200 (400) 195 190

4 a 8150 8170 8370 8200 8121 8600 7840 8080Sa 200 (250) 190 (250) 860 (250) 200 (170) 190 (170) 680 (120) 180 185

h 7810 7860 7920 7650 7450 10300 8620 83506b 650 (160) 600 (160) 1800 (160) 650 (130) 600 (130) 4-00 (100) 200 190