Nuclear Instruments and Methods 198 (1982) 527-533 527 North-Holland Publishing Company

D O U B L E S C A T T E R I N G IN R U T H E R F O R D B A C K S C A T r E R I N G S P E C T R A

A. W E B E R , H. M O M M S E N , W. S A R T E R a n d A. W E L L E R * Institut fiir Strahlen- und Kernphysik der Universiti~t Bonn, Nussallee 14-16, D-5300 Bonn, Fed. Rep. Germany

Received 7 December 1981

The origin of the low energy tails of Rutherford backscattering peaks was investigated. Systematic measurements of this background point to double or multiple scattering occurring in the target. Therefore a model of double scattering was developed, which predicts the height and form of the background in dependence on the projectile energy, the target material and target thickness in good agreement with experimental data. This confirms that mainly double scattering is the reason for the low energy background in RBS spectra.

1. I n t r o d u c t i o n

The Rutherford backscattering method (RBS) for elemental composition analysis and depth profiling is generally well known [t]. Projectiles accelerated to en- ergies around 1 MeV/nuc leon are directed onto a sam- ple and the energy spectrum of the backscattered par- ticles is measured. This spectrum is influenced by three basic factors: the energy loss of the projectiles in the sample material, the Rutherford cross section describing the probability of scattering and the kinematics of the scattering. Since these factors are known for most ele- ments with sufficiently high accuracy, quantitative mass analyses in dependence on the depth below surface of a sample are possible with RBS. This is done best and fastest by simulating the spectrum with a computer programme [2,3]. In fig. 1 a typical RBS-spectrum from a pure self-supporting Au-foil is shown together with the calculated simulation called RBS.

The agreement between experimental and theoretical backscattering peak can easily be improved by taking into account the finite detector resolution (6 keV in this case) and the influence of microscopic inhomogeneities of the target on the low energy side of the peak [4]. The reason or reasons for the appearance of the low energy tail of the backscattering peak and for the rising back- ground at the lowest energies are not fully understood. In this region no backscattering events at all should be seen according to the RBS-simulation. For the analysis of a small amount of a light element in the presence of a heavy one this low energy tail has to be kept as low as possible, since it limits the sensitivity of the RBS-method for the light element. Furthermore the heavy element's tail introduces an uncertainty in the determination of

* New address: Kernforschungsanlage, IGV, 5170 Jiilich I, Germany

the total backscattered intensity in the peak. For opti- mum analytical results with RBS the origin of the low energy tail should be known to treat the data correctly.

Only a few authors mention the problems connected with the low energy tail. In ref. i low energy projectiles in the beam due to scattering on the collimating slits or deficiencies of the detection system are considered as possible reasons for the low energy particles measured in RBS spectra. Blewer [5] for his experimental set-up traced the background back to a halo of the beam, which was backscattered from the target holder. Scherzer et al. [6] as well as Biewer think that besides these experimental effects double or multiple scattering in the target may play a role. The experimental findings are summarized by Chu et al. [1]: The height of the tail is

Z "n 8 l03

10 2

10

*,, iii .... 1.111 1 .... i . . . .

Au 610 pg/cm 2

RBS ~ JuE

, il ' " i 6 6 " ' 3 6 6 ' ' '

E [keV]

Fig. l. Typical Rutherford backscattering spectrum from a Au foil 610 /tg/cm 2 =0.316 p,m thick and the simulation of the backscattering peak (RBS) (proton incoming energy 400 keV, scattering angle v a= 156 °, beam charge normalized to I /~C, energy scale I keV/channel).

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

528 A. Weber et al. / Double scattering in Rutherford backseattering

approximately proportional to the total number of counts in the signal of the thin film (targeO. A reduction of the tail heights is... observed with increasing incident beam energy. This reduction can also be obtained by use of protons rather than 4He ions.

Energy straggling of the projectiles penetrating the target material cannot account for this tail according to the known theories [7]. The straggling causes only a small broadening of the low energy side of the back- scattering peak, in most cases smaller than the effect of the microscopic target inhomogeneities mentioned above.

In this paper we present our experimental investiga- tion of the low energy background in RBS-spectra. The aim was to prove that plural and multiple scattering is the reason for its occurrence. We describe a model of double scattering, which is able to predict in a quantita- tive way the height and form of the tail.

2. Systematic measurement of low energy background

RBS-analyses in our laboratory are done with a LC-400 (High Voltage Engineering) Van de Graaff accelerator with protons in the energy range 150-400 keV. This low energy is especially well suited to in- vestigate the low energy tail, since its height increases with decreasing energy.

A first objective was to test our apparatus for clean background-free working. The set-up is a slightly mod- ified version of that described in ref. 3. The proton beam is well collimated to a diameter of about 1 mm with 2 holes 2 m apart defining the beam divergence (0.100 °) and additionally an antiscattering diaphragm to suppress slit scattering. The selfsupporting target foils are mounted on large Al-target holders with a 14 mm central hole and stand perpendicular to the beam direction. The Faraday cup made from carbon is not seen by the Si-surface barrier particle detector placed at an scattering angle of 156 ° . The quality of the beam and the set-up was checked by the following experiments:

a) measurement of an empty target frame, b) measurements of the background in dependence

on the target thickness, c) measurements of the background in dependence

on the beam intensity. The spectrum with the empty target frame taken

with protons of energy 400 keV showed only a few background counts at energies below 50 keV probably due to detector or power line noise. To check the dependence of the background on the target thickness d, a set of Au-foils was used. In these preliminary measurements the targets have been inclined by 30 ° and the backscattering angle was i 50 °. The spectra look like the one shown in fig. 1, which was taken with the standard set-up described above with v ~= 156 °. The

background does not change its general U-form, but rises strongly with target thickness d about proportional to d 2~5 as shown in fig. 2 for the two backscattering energies 50 keV and 100 keV. To check the proper working of the detection system several RBS-spectra on different targets with vailing beam intensity have been taken and no change in the observed background up to 15% dead time of the electronic system could be found.

Experiment (a) proves a "clean" experimental ap- paratus. The background is produced only in connec- tion with the actual measurement. From (c) additionally follows, that the background is not produced in the detection system by incomplete charge collection or other deficiencies in the electronics. The strong rise of the background with target thickness [experiments (b)] shows, that the background is not due to low energy particles in the beam caused e.g. by slit scattering. These would result in a linear increase of the background with thickness. The same argument holds for scattering of the backscattered particles on the edges of the di- aphragm of the detector.

In further measurements the behaviour of the back- ground in dependence on the beam energy and the target material was investigated. A few spectra from these runs will be shown later. The background increases strongly with decreasing energy and with increasing atomic number of the target material. The strong energy dependence points to an explanation of this background

Background per 0.SkeV and laCb

200 0 0 ~ H*---,, Au, E =/,

10o

5O

20

10

5

2

1 I | i i i i I 100 200 300 /,00 500 600

d pg/cm 2

Fig. 2. Dependence of the low energy background on the target thickness d for 400 keV protons on AU. The statistical error of the measured points is below 4%. The curves for 150 keV and 200 keV not shown are well described by about d 24 too. These experiments were done with a backscattering angle d, = 150" and a target inclination angle of 30 °.

A. Weber et al. / Double scattering m Rutherford backscattering 529

by plural Rutherford scattering. The dependence on the target thickness and the atomic number proves that this plural or multiple scattering occurs inside the target.

3. Double scattering model

At first in a very simple approach the assumption of double scattering of the projectiles in the target and its effect on the energy spectra will be discussed. The low energy background is supposed to originate by a two- fold deflection of the particles by about 90 ° each in the target foil. After the first scattering the particle moves parallel to the foil surfaces and can loose a large amount of its energy, before it is deflected by the second scatter- ing into the detector.

With this simple model several predictions already can be made:

a) The low energy background increases quadrati- cally with target thickness: The number of particles being deflected parallel to the target boundaries in- creases linearely with thickness as does the solid angle in which the second scattering may occur (fig. 3).

b) Since both scatterings are governed by the Rutherford law, the low energy background should scale

4 4 4 with Ztarget, Zparti~lc and l/Epartid e. C) The U-form of the background can be under-

stood. Starting at the low energy side of the back- scattering peak, the count rate in the tail declines, since with increasing path length of the particles in the target foil the solid angle open for the second scattering de- creases (fig. 3). The minimum in the background is reached, if this decrease is made up by the Rutherford cross section for the second scattering growing as I / E 2

with decreasing energy of the particles along their path. The strongly rising background at low energies reflects this cross section behaviour.

This simple model, especially the scaling (b) does not take into account the exact stopping power values,

to detector

Fig. 3. Schematic diagram of simple double scattering model to illustrate the dependence of the background intensity on the target thickness and the path length of the projectiles.

especially their dependence on target material and re- maining energy. Furthermore for an exact simulation of the background the much more complex geometry for double scattering, the exact solid angles and particle tracks occuring inside the target should be considered. For each specific target this task can only be solved with the help of a computer programme for practical reasons.

A more accurate analysis of the double scattering reveals a specific problem. It concerns the exact defini- tion of what is to be taken as single and double scatter- ing. The passage of a projectile near or through a single atom will already yield an - although in most cases infinitesimal - deflection. The concept of "single scattering" is not an exact description of reality, but comprises already the assumption that small angle scattering events only have a small influence on the number and energy of the backscattered particles. This is proven to be well fulfilled by several RBS simulation programmes [2,8] and their high performance in describ- ing the backscattering peaks.

In a similar way the notion of "double scattering" must be defined such that essentially only two scattering events influence the flight path of the projectile. The angles of these two scatterings add up to a total deflec- tion ~ in the direction to the detector. If e.g. a deflection of greater 10 ° is considered to be essential and all scatterings with smaller angles are ignored, events with the first or second deflection angle below this limiting angle still will be counted as single scattered. This means, that in any calculation of the effects of double scattering this limiting angle must be fixed beforehand. Therefore in this model the calculations will be divided artificially into two parts to generate two complemen- tary energy spectra:

I) the spectrum, that contains all double scattered particles defined in the above way (RBSDO),

2) the complementary spectrum of particles thought of being only single scattered including all these events with one scattering angle below the limiting angle (RBSSI).

The two spectra add up to a theoretical back- scattering spectrum including double scattering (DRBS) which can be compared to experimental and conven- tionally calculated RBS spectra. The effect of the limit- ing angle on the spectra will be discussed after a more detailed description of the DRBS computer programme.

4. Double scattering computer programme DRBS

To simplify the scattering geometry and with it to shorten computing time the target foil is assumed to stand perpendicular to the beam direction. The energy loss of the projectiles as a function of path length inside the target is calculated [9] and stored as table in the

530 A. Weber et aL / Double scattering in Rutherford baekscattering

computer memory. The target foil is thought to consist of layers ID of equal thickness parallel to the surface. In each layer the numbers of particles scattered into the different cones with opening angles a > amln, the limit- ing angle, are calculated and the beam intensity for the next layer is reduced by these primarily scattered par- ticles. The situation is illustrated in fig. 4. The angle fl of the second scattering, which leads the particles in the direction to the detector, can be determined from the angle a and the azimuth ~0 of the first scattering:

fl = arccos ( - sin(180 ° - ~ ) cos cp sin a

- c o s ~ c o s ( 1 8 0 ° - ~ ) } ,

with 0 = total backscattering angle as for single scatter- ing. While the particles move along each a-cone, the intensities scattered into the detector are calculated, until the target boundaries are reached. The sequence of integrations carried out in the programme is thus:

1) target layers ID, 2) angles a,,in < ~ < 180 °, 3) particle path IS along the a-cone. For the first and second scattering the kinematical

factors are considered and the energy loss of the par-

/ D ,I-- - -

/

/ / /

/ i / / /

/

~ 'BEAM

/ /

L/ / T

I D

Fig. 4. Geometry of double scattering inside a target: A the beam enters the target; B particles in the layer ID are scattered primarily by an angle a with the azimuth ¢p; C after travelling a distance IS on the a-cone the particles are scattered a second time by an angle fl into direction of the ditector; D the particles leave the target and move on to the detector: T target cut into layers ID perpendicular to the beam direction; O total scattering angle.

ticles along their total path length in the target is taken into account. The programme adds up all the intensities in direction to the detector to the double scattering spectrum RBSDO.

The symmetry of the geometrical structure and the advantageous properties of the Rutherford scattering cross section permit a tabulation of the angle dependent parts of the cross section - already averaged over the azimuth dependence fl(9~) - before starting the various integrations. Thus is could be achieved that only basic operations appear within the integration loops. With that the computing time could be kept within acceptable limits without loosing precision by an integration grid too coarse. For actual spectrum simulations a layer thickness ID of 5 p ,g /cm 2, an a-angle interval of 0.72 ° and a path step width IS of 10 # g / c m 2 was found to give results sufficiently close to calculations with finer grids. On a small XVM computer (Digital Equipment Corporation) the simulation of a spectrum from a Au- target 610 # g / c m 2 = 0.316/~m thick takes now only 50 min with such a grid choice.

Alter calculation of the double scattering spectrum RBSDO the complementary single scattering spectrum RBSSI is calculated and added to the spectrum yielding the total backscattering calculation DRBS. In a third step the conventional backscattering calculation RBS as already shown in fig. I is done for comparison. All spectra may be printed, stored on tape or depicted one by one or simultaneously on display.

In fig. 5 these simulations are shown for the experi- ment of fig. 1. The low energy background in the spec- trum having the expected U-form is described entirely by the double scattering spectrum RBSDO. The main part of the backscattering peak is described by the

Z

0 (.>

lO 3

10 2

10

Au 610 pg/cm 2

DRBS

, , , i , , , , i . . . . i . . . . i . . . .

100 200 3oo ,~00 E [keV]

Fig. 5. Spectrum simulation of the experiment shown in fig. 1. RBS: conventional RBS simulation as shown in fig. 1, RBSDO: double scattered part of the total DRBS-simulation. It de- scribes the low energy background completely and the back- scattering peak partly.

A. Weber et al. / Double scattering in Rutherford baekscattering 531

RBSSI spectrum. The remaining intensity in the peak results from double scattering. Its contribution to the peak increases strongly with decreasing scattered en- ergy, i.e. with depth below surface of the target. This is the reason for an enhancement on the low energy side of the peak in the sum spectrum DRBS in comparision to the RBS spectrum which already described the ex- perimental backscattering peak well (see fig. 1). This enhancement must be ignored since in this region the programme is proven to be invalid (see below). The small peaks in the curves are a consequence of the geometrical grids of integration.

5. Discussion and validity of the double scattering model

The spectrum simulations according to this double scattering model are done in an absolute way. No fitting parameters are used. The only parameter in the pro- gramme is the limiting scattering angle, which was in- troduced in the model to distinguish clearly between the single and double scattering cases. This angle is mathematically necessary too, since the Rutherford cross section diverges at small scattering angles. At projectile energies of 400 keV the deviation of the scattering cross section from the Rutherford law is measurable already at scattering angles below about 15 ° [10]. The limiting angle has to be choosen for the first and the second scattering and needs not necessarely to be equal. For the first scattering the limiting angle a.,~n gives the distribution between the scattered intensity into the cones and the beam. With 15°~<a,,~,<30 ° the true distribution in general is well reproduced. The choice of this angle influences the intensity in the spectrum only little, since all particles not scattered will reach the next targef layer. For the second scattering the limiting angle flmi, serves for the division of the particles into the single or double scattered parts. The programme part for double scattering RBSDO handles all the particles with angles of the second scattering fl > fl,,i,. Particles, which start in such a direction a, cp that they need only a second scattering with an angle f l < f l , , ~ are not calculated by that part. They are considered in the complementary part RBSSI as well as the particles, which are in the first scattering assumed to remain in the beam and then scattered directly into the detector by the total angle 0. The choice of the limiting angles therefore influences mainly the partition into the two complementary parts RBSSI and RBSDO in the region of the low energy side of the backscattering peak, but has only little effect on the intensity of the low energy background. This is demonstrated in fig. 6, lower part, where the relative height of the background is shown in dependence on the limiting angle flmi, for two target foils with different thickness. The calculated back- ground at 150 keV backscattered energy does not change

2.5

2.0

h rnax.DRBS h max. RBS

1.5 ~ Au .610 pg/cm 2 ~ . ~ 80pglcrn2

1.0 '~- . . . . . . . . . . .

" X ~ , , hB I select e

0.~ limiting angle ~ X

I I/e I I o .2 . .6 .8 1:o ;.,

limiting angle ~rnin (rodian]

Fig. 6. Upper part: enhancement of the low energy side of the calculated backscattering peak in dependence on the limiting angle flmi," Plotted is the ratio of maximum peak heights h n,~.rmBs to h m,,,.RBS for two target foils with different thick- ness for a proton incoming energy of 400 keV and a scattering angle of 156 ° . Lower part." change of the height of the back- ground at 150 keV backscattered energy in the same set of calculations.

up to about flmin = 400 for the thicker Au-target. The range of this plateau turns out to be determined mainly by the target thickness. With decreasing target thickness the background intensity is due more and more to large secondary scattering angles fl as seen from the curve for the Ag-target with the extended range of constant back- ground up to about flmin = 45°' In the upper part of the figure the effect of the choice of a small limiting angle tirol, on the backscattering peak is shown. The strong increase of the Rutherford cross section at small scatter- ing angle produces an artificial enhancement of the low energy side of the peak, which is not found experimen- tally. So a smaller limiting angle flmi, results in an increased peak intensity, but does not change the height of the low energy background. This increase is stronger for higher Zt~rget material. Thus for the simulation of the background in RBS spectra a wide range of the angle fl,nin for the second scattering is permitted in the model. The angle flmi, can be chosen about equal to Otmi n and both should be large enough to guarantee the correct- ness of the Rutherford cross section formula.

From the description of the effects of the limiting angles it becomes clear, that not all energy regions of the DRBS backscattering spectrum are simulated with the same quality by our double scattering model. By several test calculations the following conditions for the validity of the model were ascertained:

l) The particle flux inside the target has to be de- termined mainly by the first scattering. This is the case

532 A. Weber et al. / Double scattering in Rutherford backscattering

if the total scattering cross section for the second scattering is small.

2) The probability for triple and multiple scattering must be neglegible.

In the following the validity of the double scattering model is discussed in the different energy regions of the backscattering spectrum:

a) Low energy part of the backscattering peak: Par- ticles in this region cover only a slightly increased distance in the target compared to single scattered par- ticles. After the first scattering they have already about the direction to the detector and need only to be scattered by a small angle fl with corresponding high cross section. So they violate both conditions. The spec- trum simulation is strongly dependent on the choice of the limiting angle for the scattering in this energy region (upper part fig. 6). The low energy side of the peak can not be simulated absolutely by the model, no unambigu- ous prediction is possible. Nevertheless the experimen- tally found enhancement of the low energy side of the peak compared to the simple RBS simulation (see fig. 7, 14 in ref. 1 or fig. 15 in ref. 3) could be fitted by the double scattering model and thus understood.

b) High energy part of the background: In this region which is often called the low energy tail of the back- scattering peak, both conditions are fulfilled by con- struction of the model. This is confirmed by additional calculations described below (part c).

c) Low energy part of the background." The very low energy of the scattered particles in this part must be due to long travelling distances inside the target relative to the target thickness. During their way parallel to the target surfaces the particles will suffer with high proba- bility one or several small angle scatterings, so that a high number will be lost and not reach the detector. This decrease of particle flux along the path IS con- tradicts conditions 1 and 2. To take it into consideration in the programme is difficult for the following reasons: (1) Especially for thin target foils the angles of these scatterings are so small, that the Rutherford cross sec- tion formula overrates the true scattering probability. (2) the "diffusion" of particles along this path will not be described well by only one additional (triple) scatter- ing. (3) The complex geometry of this plural scattering will take a too long computing time for practical calcu- lations.

An attempt was made to take into account the particle loss along the path IS by an additional scatter- ing and thus to determine the intensity only due to pure double scattering in the sense of the model. These calculations show clearly that indeed the high energy part of the background (b above) represents particles scattered twice whereas the low energy part (c above) is strongly reduced. Its presence in the DRBS simulation and the agreement of this calculation with the experi- ments show, that the neglect of a flux reduction in the

model along the path IS is a good approximation for the missing multiple scattering parts. The flux and energy distribution of the particles in a real target must be close to the representation in the DRBS simulation. The number of particles scattered out of the path seems to be approximately equal to the number scattered in by multiple scattering, so that this model is successful in simulating the complete U-form of the background.

6. Spectrum simulations and conclusion With the restrictions mentioned above the DRBS

programme up to now can be used to calculate back- scattering spectra of one elemental target foils. To show the extent of agreement, simulations of several spectra from foils made of different material together with the experimental data are depicted in figs. 7-9.

All data were taken with a proton beam of 400 keV at a scattering angle of 156 ° with a Si(Li) surface barrier detector with solid angle of 0.20 msr. About 100 counts/keV in the low energy background were col- lected. All spectra are normalized to a beam charge of 1 /~C and transformed to a length of 500 channels with 1 keV/channel. The peak at the right is a pulser signal.

The simulations shown are calculated absolutely. The exact choice of the limiting angles used in the calcula- tions is of minor influence. The theoretical spectra are given without smoothing and for ideal detector resolu- tion to show clearly the results of the double scattering model. The agreement between theoretical and experi- mental spectra can be improved easily by applying a smoothing procedure and a convolution with the finite detector resolution.

In fig. 7 the data of fig. 1 from a Au foil 610/~g/cm 2 thick are shown again with the DRBS simulation (limit-

Z "a

O ¢.~

10 3

Au 6101xj/cm 2

lO 2 ~

1 100 200 300 400

E [keV]

Fig. 7. RBS spectrum from a Au foil 610 p.g/cm 2 -0.316 ttm thick and DRBS simulation (experimental conditions as given in fig. I).

A. Weber et al. / Double scattering in Rutherford backscattering 533

8 - I03=

10 2 ,

I0

Ag 380pglcm 2

' ' 1 ' ' ' , 1 . . . . i . . . . " , ' '

100 200 300 400 E 1keV]

Fig. 8. RBS spectrum from a Ag foil 380 #g/cm 2 =0.362 /~m thick and DRBS simulation (experimental conditions as given in fig. 1). Below 100 keV the simulation underestimates the experimental data.

ing angle for the first scattering amin= 15 °, for the second scattering flm~n = 230) • Fig. 8 shows the spectrum from an Ag target with thickness 380 # g / c m 2 (Otmi n = 15°, flmin = 12°) • The increase of the experimental back- ground at the lowest energies in the spectrum can partly be explained by noise in the particle detection system. In fig. 9 the spectrum of a Cu target 250 /~g /cm 2 thick is

, , , , i . . . . I , , l l l . . . . I . . . .

1 0 3 Cu 250 ~ug/cm 2

i01 ~ ORES.

, , , , , , , f , , , , ,

1~ 2oo'3~ ~o E [l(eV]

Fig. 9. RBS spectrum from a Cu foil 250 /.tg/cm 2:0.279 ~m thick and DRBS simulation (experimental conditions as given in fig. 1). At the energies around 220 keV backscattering peaks due to possible oxide layers on the surface of the Cu target are expected.

depicted (Otmi n = 15 °,/~min = 12°) . The additional counts around the energy 220 keV are most probably due to a surface oxidation of this target.

Our DRBS-model describes the experimental low energy background well for all ,the cases investigated. The absolute height and the form of this background is well reproduced for different target materials and thick- nesses. We conclude that the low energy tail of the backscattering peak is due to double scattering in the target. The increasing background at the lowest energies arises physically from multiple scattering, but still can be described by the double scattering model. The back- ground depends in a complex way on the proton incom- ing energy and energy loss in the target, on the target material and the actual geometrical particle tracks oc- curring inside the target foil determined mainly by its thickness. For each specific case the tail and the back- ground can be predicted with our model. In further work we will investigate if simple analytical expressions for the dependence on the separate quantities can be found. The possibility to predict the background per- mits to estimate the sensitivity of the RBS method for light elements in heavy target foils.

References

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

[2] A. Weller, Diplomarbeit, Bonn (1979) (in German). [3] H. Mommsen, T. Mayer-Kuckuk, W. Sarter, P. Schtirkes

and A. Weller, Rutherford-Rt~ckstreu-Analysen mit Proto- nen, Forschungsberichte des Landes Nordrhein-Westfalen Nr. 2999 (Westdeutscher Verlag, Opladen, 1980) (in Ger- man).

[4] A. Weller, H. Mommsen and W. Sarter, Proc. 7th Div. Conf. on Nuclear physics methods in material research, eds., K. Bethge et al. (F. Vieweg, Braunschweig, 1980) p. 352.

[5] R.S. Blewer, in Ion beam surface layer analysis, vol. 1, eds., D. Meyer et al. (Plenum Press, New York, 1976) p. 185.

[6] B.M.U. Scherzer, P. Beffgesen, M.-A. Nicolet and J.W. Mayer, in ref. 5, p. 33.

[7] W.K. Chu, in Ion beam handbook for material analysis, eds., J.W. Mayer and E. Rimini (Academic Press, New York, 1977).

[8] B.M.U. Scherzer, private communication. [9] H.H. Andersen and J.F. Ziegler, The stopping and ranges

of ions in matter, vol. 3 (Pergamon Press, New York, 1977).

[10] H.H. Andersen, J. B~'ttiger and H. Knudsen, Phys. Rev. A 7 (1973) 154.