Nuclear Instruments and Methods 204 (1983) 559-563 559 North-Holland Publishing Company

B A C K G R O U N D IN R U T H E R F O R D B A C K S C A T I ' E R I N G S P E C T R A : A S I M P L E F O R M U L A

A. W E B E R a n d H. M O M M S E N

Institut fi~r Strahlen- und Kernphysik der Universiti~t Bonn, Nussallee 14-16, D-5300 Bonn, Germany

Received 11 June 1982

Proton Rutherford backscattering spectra show at low energies a background often called low energy tail that consists of double-scattered particles. A complete prediction of this background is possible with our computer programm DRBS. From extensive calculations we derive a simPle background formula. It predicts the background intensity in the most important part of the spectrum for nearly all targets absolutely within 20% for proton energies up to 1 MeV. We further present experimental evidence at proton energies of 400 keV and below. The validity ranges of the formula and the double-scattering model are discussed.

1. Introduction

Protons of about 0.3 1 MeV kinetic energy back- scat tered from a thin target give a large peak in the energy spectrum due to single Ruther ford scattering. This peak is well unders tood in terms of the Ruther ford cross section, the energy loss of the protons and the kinematics of the scat ter ing [1]. Ruther ford backscat ter- ing (RBS) measurements can therefore be used for e lemental analysis and dep th profil ing of targets.

The analysis of multi- layered or mixed element targets is impaired - especially at low b o m b a r d i n g energies - by a low energy tail following the highest energy peak and extending to the lowest energies. Fig. 1 shows a typical observed RBS peak with its low energy background, taken with a "c lean" experimental set-up, i.e. a monoenerget ic beam and negligible slit scattering.

In a previous paper [2] we have pointed out that this background can be fully explained by a double-scatter- ing process in the target. After a first scattering, par- ticles may travel long distances inside the target and may be deflected by a second scattering in the direct ion of the detector. Our computer program DRBS, taking into account the exact geometry of all possible double- scattering cases, was shown to give excellent agreement between simulated and exper imental spectra. A full descr ipt ion of the double-scat ter ing model and the DRBS program, its parameters and its validity can be found in ref. 2. The program requires nearly one hour on a Digital Equ ipment Corpora t ion X V M computer to simulate the spect rum of a typical one-elemental target and is therefore ra ther unsui table for everyday use.

In different appl icat ions of the RBS analysis tech- nique the exact knowledge of the double-scat ter ing background is important . Examples are: (a) A h igh-Z target foil is analysed for l ight-element

contamina t ion , present ei ther only on the surfaces or th roughout the target. The peak of the light element will appear at energies slightly below the heavy element 's

backscat ter ing peak. (b) The (e.g. radia t ion induced) diffusion of a heavy

COUNTS 100000

10000

1000

100

10

d =140pg/cm2

F , Eout

I~ ,, , 100 200 300 400 500

keY

Fig. 1. Typical RBS spectrum from a thin gold target. Experi- mental parameters: collected charge normalized to 10 ~C, detector solid angle 0.195 msr, scattering angle ~ = 156 °. The channel width of the spectrum is 1 keV. Note the high, narrow peak consisting of single-scattered particles and the broad background caused by double-scattered particles. Here, this background shows a broad "flat bottom" in contrast to the strong U-shape evident in a thick target background (see fig. 6). The bar indicates the range where the prediction formula is usable (0.74 e~<0.9).

0 1 6 7 - 5 0 8 7 / 8 3 / 0 0 0 0 - 0 0 0 0 / $ 0 3 . 0 0 © 1983 Nor th -Hol l and

560 A. Weber, H. Mommsen / RBS background formula

element layer on a light backing is studied. Knowledge of the heavy element's double scattering background in the gap between the heavy and the light element's RBS peaks is essential.

To allow a quick estimate of the height of the double-scattering background in a RBS spectrum, a simple formula is presented in this paper. It is derived from extensive DRBS calculations and a theoretical study of the scaling properties of the double-scattering model and predicts the background height absolutely.

2. T h e b a c k g r o u n d f o r m u l a

At first a data base of about 70 complete RBS spectra was accumulated using the DRBS program and varying target material, thickness, proton incoming en- ergy and scattering angle. Most of the calculations are done for the elements Cu, Ag and Au for different thickness/ incoming energy combinations as shown in figs. 2, 3 and 4. To get a better idea of the Z-depen- dence, some spectrum simulations for Sr, Ce and U targets are included in the data base. In all these spectra minimum background occurs at approximately 0.7 Eo, ,, where Eo,, is the low energy edge of the single scattering peak taken at half-maximum and therefore the high energy limit of the background in the spectrum.

Furthermore, the shape of the background is similar in all cases, as the "f lat-bot tomed" U-shape is just stretched or compressed according to Eo. ' (fig. 6). We denote the backscattered energy of the region of interest in the background in units relative to Eou t by a parame- ter e (fig. 1).

According to the examples given in the introduction the most interesting region of the background in spectra of targets with Z > 25 starts at e = 0.7 - which corre- sponds approximately to the position of a carbon peak - and extends to higher e. As upper limit e = 0 . 9 is

taken, which corresponds to the position of a chlorine peak. At higher e most experimental background is due to target inhomogeneities, energy straggling and the finite detector resolution (fig. 1 and refs. 1, 3).

From considerations explained in sect. 3 we chose the following statement to describe the height of the background for this e-range:

Bkgnd = f ( e ) ( s i n 0 / 2 ) A E B d c Z D,

with f ( e ) a linear function, 0 the scattering angle, E the proton incoming energy, d the target foil thickness and Z the atomic number of the target element. The un- known exponents A to D are determined from the calculated data base by a fitting procedure.

We first evaluated our data at e = 0.73 and 0 -- 156 ° and optimized the parameters B, C and D to get mini- mum chi-square. We found a broad optimum for the parameters, covering about ÷ 5% of the respective best values. Within this optimum, best values for B, C and D are strongly interdependent.

The found best values reproduce the E, d and Z dependence of our data base with a standard deviation of 20%. Of 57 predictions, 43 are within + lo of the DRBS calculation, 56 within -+20.

13 DRBS calculations with 130 ° ~<0~ < 170 ° for the three major target elements were then used to find the exponent A. The function (sin 0 ' / 2 ) A w a s normalized to agree with all our ~ = 156 ° calculations. The back- ground formula predicts these 13 DRBS spectra with a standard deviation of only 7%. We did not include extremely thick targets in this ~ variation as one would choose a scattering angle 0/> 150 ° experimentally under these conditions.

Finally the background of our data points at e = 0.9 was evaluated and an average increase of 16% over the e = 0.73 background was found. This determines the linear function for the weak e-dependence.

The final optimized background formula for back-

pg/cm2 [.~ Cu I000~ "

,oo • •

,oo ,

200 zOO 600 800 Energy[keV ]

Tar et ,~ ~ \

moo ~ \\

8 0 0 ~ < • • •

200~

T a r g e t m ~ , ~ A u

• . :

. . :

•

200 400 600 800 200 /.00 600 800 Energy[keV] Energy[keY]

Figs. 2, 3, 4. The dots represent target thickness/energy combinations of the performed DRBS calculations for copper, silver and gold targets. The white areas indicate the estimated validity range of our background formula. In the shaded areas, validity is impaired by multiple scattering (m.s.) resp. particle loss (p.l.) as explained in sect. 4.

A. Weber, H. Mommsen / RBS background formula 561

ground from a target placed perpendicular to the beam is:

Bkgnd = (1.68 E 7 + 5.12 E 7. e) (sin 0 / 2 ) -157

)< E -5.762 . d2 .552 . Z 1.378,

with Bkgnd: number of protons double-scattered into a solid angle of 1 msr, into a 1 keV interval in the spectrum, per /~C collected charge; 0.7 < E ~< 0.9 energy of the above 1 keV interval in units of the low energy edge Eou t of the single scattering peak. As "thin" targets at high energy show a pronounced "flat bot tom" in the background, e = 0.7 values could then be used at lower e as well; 130 ° ~0~< 170 ° scattering angle for single scattering; the fit is optimized for 0 = 156°; E energy of incoming protons in keV; d target thickness in /~g/cm2; 25 < Z ~< 92 atomic number of target material.

The permitted E / d combinations are shown in figs. 2, 3, 4. In sect. 4 a discussion of the complete validity range is given. The estimated accuracy of the back- ground prediction is +20%.

3. Discussion of the found exponents

At first sight the dependence of the background height especially on the scattering angle, the energy E and the atomic number of the target material is unex- pected. Without repeating all the features of the double-scattering model presented in ref. 2, in the fol- lowing we will give some plausible arguments for these dependences.

Let us assume a very thin target and consider the first scattering of an angle of about 90 ° . The thin target will produce no significant energy loss and therefore all first scatterings will occur at the beam energy E. The number of particles subjected to a first Rutherford scattering consequently will vary with the beam energy as E -2. These particles travel parallel to the target surfaces to suffer an energy loss of ( 1 - e ) E = ( 1 - e)Eout, since for a thin, heavy target Eou t ~ E. Assuming constant specific energy loss over the range of E, we find that the path length the particles have to travel in the target to reach energy eE is proportional to E. As the solid angle subtended by those atoms of the target that can cause a second scattering at eE varies as (path length) l, this indicates a factor of E - l in the number of background counts. A thick target with a significant energy loss over its thickness as compared to E will yield a stronger path length variation and a stronger decrease of the background with E. Finally, the second scattering cross section follows an E -2 dependence which gives us a total variation of the background with the proton energy that is somewhat stronger than E -5.

The basic Z-dependence of the background would be Z 4 according to the Rutherford law, simply taking two subsequent Rutherford scatterings at constant number of scattering centers. With constant thickness meaning in our case constant area weight (~g /cm2) , a higher-Z target with atomic mass A ~ 2 Z contains fewer scatter- ing centers p e r / ~ g / c m 2. The number of particles from the first scattering therefore will be approximately pro- portional to Z. The number of atoms available over a given solid angle for a second scattering will be subject to the same reduction; the probability for a second scattering also varies as Z. Therefore a Z-dependence of Z 2 is expected. Since the position in the spectrum eEou t for the background height prediction and the channel width are fixed, the effects of different stopping power, i.e. the variation of the path length and simultaneously the layer thickness for the second scattering with chang- ing Z cancel in the chosen e-range. Since the atomic weight A increases stronger than linearly with Z, the total Z-exponent will be somewhat below 2.

The dependence of the background on target thick- ness d was already mentioned in ref. 2. The number of particles scattered parallel to the target surfaces will increase proportionally to d. Similarly the solid angle in which these particles can find a target atom for a second scattering will increase - -d . This d2-dependence holds true for cases where the target thickness is smaller than the path length the particles must travel inside the target to lose (1 - e)E energy, because then all first scattering angles will be around 90 ° and the cross section will be approximately constant. A very thick target would allow particles scattered in the forward direction in the first scattering to travel such a path length, that they lose sufficient energy to reach the detector with energy eEou t after a second scattering. Due to the forward-peaked Rutherford cross section there will be an explosive increase in these double-scattered protons from a cer- tain thickness onward. The program DRBS, which takes the complete geometry into account, produces such an increase. This increase is also observed experimentally at large target thicknesses. The range of the thickness fit has been selected to give reliable predictions with a single exponent for d. Any excess of the thickness exponent over the expected value of C = : 2 can be considered an approximation of the geometrical effect over the selected range.

Due to the complicated way the two scattering an- gles add up to a total of 0 no simple explanation of the found 0-dependence is possible. We note that the ex- tremely high exponent is mainly determined by the few additional calculations at 0 = 130 ° and 0 - - 1 4 0 °. As these include more forward angles for the second scattering, the background increase seems reasonable; but for the same reason, selection of the cutoff angle flmin in the model [2] might influence the DRBS results. We did not check for such an influence.

562 A. Weber, 14. Mommsen / RBS background formula

As pointed out in the discussion of E -57, increasing e will increase the solid angle for a second scattering, resulting in a background proportional to e.

4 . V a l i d i t y r a n g e o f t h e f o r m u l a

There are two basic factors limiting the validity of the formula:

(1) The ability of the fit formula to reproduce the DRBS results: The estimated validity ranges for target thickness/energy combinations are indicated in figs. 2, 3, 4 as white areas. The target thickness range includes all cases in which the high energy single-scattering peak of the main element leaves enough space in the spec- trum for a second peak at low energy to be analysed (upper boundary) and in which the background is high enough to be considered (lower boundary). This thick- ness range of the formula as well as the chosen e-range are smaller than the DRBS ranges as explained above.

(2) The validity of the double-scattering model itself and therefore of the DRBS program, which was used to generate the data base: The shaded areas in figs. 2, 3 and 4 are not well understood. In the upper region at not too large target thicknesses a DRBS simulation will still be good, but may easily exceed the formula predict- ion by 200%. The width of the DRBS validity range is unknown, as ultimately at very great target thicknesses multiple scattering becomes dominant. We therefore refer to this domain in figs. 2, 3, 4 as the multiple scattering area. The lower shaded region, named particle loss area, is governed by a possible decline in particle flux as explained in ref. 2, This effect also limits the e range to approximately the range considered here. In the double-scattering model the particle flux distribu- tion in the target is determined by the first large angle scattering and not disturbed thereafter. This may not hold true when the path length of the particle in the target becomes long as compared to the target thickness. In this case, a large part of the particle flux may be deflected out of the target prematurely by small angle collisions and lost. This particle loss may occur at low e or at high energy with thin targets. In ref. 2, sect. 5, we have found that multiple scattering compensates for this effect with thick targets at low e. This compensation would not be possible with thin targets at high energy. Under particle loss conditions the fit formula as well as DRBS will overstate the background. At present there is no experimental or mathematical proof for this effect. The particle loss areas in figs. 2, 3, 4 may or may not extend to higher thicknesses at higher energies. Experi- mental work in this area, although essential, will be difficult because of the low background rate. It should be remembered that the validity of the DRBS program is not assured experimentally for scattering angles 0 < 150 ° and for proton energies > 400 keV.

5 . E x p e r i m e n t a l c o n f i r m a t i o n

The background formula was verified using a LC-400 (High Voltage Engineering) Van de Graaff accelerator with a maximum proton energy of 400 keV. A descrip- tion of the experimental set-up and the way measure- ments are done can be found in ref. 2. We only add here a discussion of the target properties that must be con- trolled in a valid check on the background:

(1) Target thickness should be known with high precision. The best results are obtained by analysing the single-scattering peak. The minimum error by this method can be held below 5%. Although the analysis can be more exact in theory, a real target foil will always show some surface roughness. As the thickness dependence of the background is d 255, a 5% error in the target thickness will cause a 13% error in the back- ground prediction. As our understanding of the back- ground increases, "double-scattering" analysis might be- come a superior tool in thickness determination.

(2) At around e = 0.79 an oxide layer on the targers rear will produce single-scattered counts. Therefore no- ble metals should be used. We chose Cu, Ag and Au, but only the gold targets showed no impurities.

(3) In the same way, carbon contamination on the surfaces can cause single-scattered counts from e = 0.73 upward. The low background rate requires a high de- posited charge. Working at a vacuum of about 3 × 10 _6

mbar with a LN 2 trap in the scattering chamber, we sometimes found the usefulness of our targets reduced after only three spectra were taken.

The experiments were limited to a proton energy range of 200 keV < E ~< 400 keV and to a fixed scattering angle of ~ = 156 ° . Fig. 5 shows the 14 target thickness/energy combinations of the measured gold targets. They make a check of thickness and energy

Target thickness lag/cm 2

500

/.00

300

200

tO0

0

H*-.-p Au O _

• • • O _

I n n a a I n I i I I 200 300 400

Energy [keV]

Fig. 5. The dots represent the target thickness/energy combina- tions of our measurements on gold. The targets thicker than 600 /~g/cm 2 already show a strong deviation from the back- ground prediction at E = 340 keV.

A. Weber, H. Mommsen / RBS background formula 563

dependence of the background possible. The Z-scaling was checked by measuring one silver and one copper target. Unsuitable target thickness (Ag) and strong oxidation (Cu) prevented an energy variation. Spectra from these targets are shown in ref. 2. Count rates were averaged over a range of about e = 0.70-0.80 to reduce the statistical error in each case to below 3%.

From these measurements the following results are obtained:

The Au background follows the background formula with a standard deviation of 18%. Averaged experimen- tal background values are 95% of the formula predict- ion. Omitting one extreme value (140 / . tg/cm 2 at 400 keV, which is 40% too high) the spread is reduced to 9%, the remaining values average to 91% of the formula prediction.

COUNTS H¼Au, E:/,OOkeV tO0000

I0000

1000

I00

lO

tO0 200 300 400 keV

Fig. 6. Comparison of gold measurements with background formula prediction for various target thicknesses. Experimental parameters are as in fig. 1. The spectra have been averaged twice over three channels. Indicated target thickness has been determined from area and width of the single scattering peak. The formula prediction is absolute. A linear dependence of the background on e is clearly confirmed.

The background of the silver target (d = 380 # g / c m 2) is 91% of the prediction at E = 400 keV. At the same energy, the background from the copper target (d = 250 /~g/cm 2) exceeds the prediction by 30%, but this must be attributed to a strong oxide layer as visible in fig. 9 of ref. 2.

Finally fig. 6 shows a comparison between four gold spectra ( d = 215, 290, 610, 670 ~ g / c m 2) and the abso- lute formula prediction at 400 keV. The linear e-scaling is in agreement with the experimental evidence.

Because DRBS is unable to calculate the background of targets that are not placed perpendicular to the beam direction, we have not commented on the effect of target rotation. We believe that with # = 180 ° any target rotation would increase the background. At v ~ = 156 ° we found experimentally that inclining the targets by 30 ° against the perpendicular position increased the back- ground by 25%.

6. Conclusion

We have demonstrated that the double-scattering background in RBS spectra can be easily predicted for most applications. The background formula, for exam- ple, justifies background subtraction routinely done in quantitative RBS analysis.

Furthermore, the double scattering model can now be subjected to rigorous checks without the need for extensive calculation. Especially at high energies such work would clarify the transition between double scattering and processes of higher order in the back- ground of RBS spectra.

References

[1] W.K. Chu, J.W. Mayer and M.-A. Nicolet, Backscattering spectrometry (Academic Press, New York, 1978).

[2] A. Weber, H. Mommsen, W. Sarter and A. Weller, Nucl. Instr. and Meth. 198 (1982) 527.

[3] A. Weller, H. Mommsen and W. Sarter, Proc. 7th Div. Conf. on Nuclear physics methods in materials research, eds., K. Bethge, H. Baumann, H. Jex and F. Rauch (Vieweg, Braunschweig, 1980).