Hydrogen and helium stopping powers of rare-earth metals

NUCLEAR

INSTRUMENTS

AND

METHODS

168

(1980)

41-50;

(~

NORTH-HOLLAND

PUBLISHING

CO.

HYDROGEN AND HELIUM STOPPING POWERS OF RARE-EARTH METALS H. KNUDSEN, H. H. ANDERSEN and V. MARTINI*

Institute of Physics, University of Aarhus, DK-8000 Aarhus C, Denmark

An improved method for measuring relative stopping powers for light energetic ions in highly reactive materials is used to obtain data for 200-1000 keV hydrogen and 600-2000 keV helium ions in the rare-earth metals La, Ce, Pr, Gd, Dy, Ho, Er and Yb as well as in Sn and Bi. The stopping powers were measured relative to the stopping power of silver. Using as a normalization the semiempirical silver stopping powers of the data compilations by Andersen and Ziegler and by Ziegler, we compare our results with their semiempirical fits for the other target materials investigated here and find satisfactory agreement.

1. Introduction During recent years, there have been increasing efforts to obtain detailed and accurate knowledge of the stopping power of light energetic ions in matter. This is due partly to the importance of such information for testing basic models of ion-penetration phenomena and partly because a knowledge of ion stopping powers is essential for several surfacelayer-analysis techniques such as those using Rutherford backscattering and nuclear reactions. The accuracy of these methods is often limited by the accuracy of the stopping powers applied. To fulfil this need, Andersen and Ziegler ~) and Ziegler 2) published comprehensive tabulations of experimental data as well as semiempirical fits for hydrogen and helium stopping powers in all elements. These tabulations greatly improved the situation and have found widespread use. However, their data base show remarkable vacancies, especially for the rare-earth materials. This is unfortunate, partly because it makes the semiempirical fits more uncertain for these targets, partly because the atomic-number dependence of the stopping power for these elements constitutes a good test of the theoretical models. The reason for the lack of accurate experimental data is to a large extent that the rare earths oxidize readily, hence making it difficult to obtain sufficiently pure targets. Recently3), we reported on an improved method for accurate measurements of relative stopping powers of light ions in solids. The technique is based on the fact that the height of the energy spectrum belonging to particles Rutherford-back-

* Permanent address: University of Bonn, West Germany.

scattered from the target surface is inversely proportional to their stopping power. The main advantages of the method are that it eliminates problems with beam- and solid-angle normalization, it makes possible a collection of accurate data via computer online control and analysis, it produces low background spectra, and, finally, target composition and possible effects stemming from target texture may be investigated in situ. In the previous paper3), we discussed in detail the experimental technique and the method of data evaluation. Furthermore, we presented measurements of the ratio between the stopping powers of silver and copper for hydrogen and helium ions of 200-2000 keV energy. As the results were in agreement with the semiempirical values of refs. 1 and 2, and hence with a rather large amount of experimental data, we thereby checked the applicability of the method. In ref. 3 we also discussed the application of our technique to the measurement of the stopping power of light ions in highly reactive materials such as the ra|e earths. We showed that targets, which, after evaporation of the reactive material onto a backing, in the same process had been covered with a thin protective layer of gold, could be used in the measurements, and we presented data for the stopping power of lanthanum for 200-1000 keV protons. Here we present stopping powers for 200-1000keV hydrogen ions and 600-2000keV helium ions of the rare-earth metals La, Ce, Pr, Gd, Dy, Ho, Er, and Yb measured relative to the stopping power of silver. Applying the semiempirical values of refs. 1 and 2 as normalization, we have converted the raw data into absolute stopping powers, and these are, in turn, compared with both I. S T O P P I N G POWER AND S T R A G G L I N G

42

H. K N U D S E N et al.

the few other existing data and with the semiempirical results of refs. 1 and 2. As the experimental arrangement and technique, as well as the data evaluation, have already been discussed in detail, in this paper we shall put the main emphasis on those details that are specific to the use of rare-earth targets as well as on a presentation and discussion of the data.

2. Experimental The experimental arrangement is shown schematically in fig. 1 of ref. 3. The ion beam from the Aarhus 2 MV Van de Graaff accelerator impinges on the target after having passed through the central opening of an annular solid-state detector. By means of an elaborate slit system, this detector is protected against stray particles from both sides. Only a small annular region of optimum energy resolution is left unshielded. The detector arrangement is shown in detail in fig. 2 of ref. 3. Ten targets were mounted at fixed positions around the circumference of a wheel, which was rotatable by a step-motor drive. Four targets were irradiated in sequence, each for a period of 2 s, while the backscattered-particle spectrum was collected in a quadrant of a multichannel analyzer allotted to the target in question. The cycle was repeated more than one hundred times, thereby averaging over beam fluctuations. With this method, solid-angle normalization as well as beam-current measurements were unnecessary. Measurements were usually performed at perpendicular incidence, but using the same cycling procedure as described above, we could irradiate one target with the beam incident at - 3 . 1 °, 0 °, and +3.1 ° and thereby investigate possible effects stemming from target texture4). In no case did we observe any significant influence from such effects. The targets used for the stopping-power measurements consisted of a layer of rare-earth material, of thickness larger than the range of the light ions, deposited onto a glass plate and covered with a thin (200-500 ,~) protective gold layer. We also used thick targets of silver, tin, and bismuth for normalization purposes as well as for the purpose of investigating the influence of straggling and multiple scattering. Finally, in our investigation of target composition, we used targets consisting of a very thin layer of rare-earth material, deposited onto a carbon backing and covered with a gold layer of the same thickness as that on the other rare-earth

targets. All the targets were prepared by standard vacuum evaporation, and the protective layers were deposited without breaking the vacuum. When using coated targets, it is necessary to measure the gold-layer thickness as well as to investigate whether the protection against oxydation was efficient. This could be accomplished with the same experimental setup as that used for the stopping-power measurements. Irradiating the targets with 2 MeV helium ions, we obtained backscattering spectra, in which the gold peak was resolved in front of the rare-earth edge. Combining the information of such spectra with a rather accurate beamcurrent measurement, we could find the thickness of the gold layer. The results of these measurements are shown in table 1. The content and distribution of oxygen in our targets was found via backscattering spectra, using 2 MeV hydrogen ions impinging on thin, goldcovered, carbon-backed rare-earth targets. For protons of this energy, the backscattering cross section for oxygen atoms is a factor of 3.8 greater than the Rutherford one, which enhances the sensitivity. An example of such a spectrum is shown in fig. 6 of ref. 3. From this spectrum, the mean oxygen content was found to be 5.5 at. %. However, most of the oxygen is seen to be positioned in a thin (probably fully oxidized) layer at the rare-earth surface, and in ref. 3, it was shown that this layer does not influence our results. The content of oxygen in the bulk of our thick targets is smaller than or equal to that given by the minimum value of the " o x y g e n feature" in spectra like that discussed here. Oxygen contents found in this way are also given in table 1. TABLE 1 Targets. Projectile

H, H, H He H, H, H, H, H, H, H, H,

He He

He He He He He He He He

Target Z Ag Sn La La Ce Pr Gd Dy Ho Er Yb Bi

47 50 57 57 58 59 64 66 67 68 70 83

Au layer (A)

N(O)/N(Z)

190 180 500 590 450 500 460 500 490

1.6 1.9 8.1 5.0 5.6 4.8 15.3 9.7 5.3

(at. %)

HYDROGEN

AND HELIUM

i

3. Data analysis When comparing backscattering spectra obtained from two target materials (1 and 2), using the same number of beam particles and identical experimental geometries, we can find the ratio between the respective stopping powers S~ and $2 from 3) (S E1 o )

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600 keV Hydrogen ions~

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Ag +720~, Au (,)

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(1)

S2(Eo) (?k~,'+k,)a2Y,(k, Eo)

z

Here, E0 denotes the initial energy of the projectiles, 7 is the ratio between the path length of the outgoing and incoming particles, k is the kinematic factor3), cr is the elastic-backscattering cross section, and Y ( E ) is the number of backscattered particles detected with energy between E and E + d E . Eq. (1) is valid provided the k values are close to one, in which case the relative difference in energy between incoming and backscattered particles is small and the energy dependence of the stopping power can be accurately approximated by the power law, SCE) oc E ~ .

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In our experiment, where only very light ions and scattering angles close to 180 ° are used, the values of k are very close to unity3). From eq. (1) it can be seen that the ratio between the stopping power of the target material and that belonging to the normalization material is given by the ratio between the yields per unit energy of particles backscattered from the target surfaces. These yields were found by fitting straight lines to the upper energy part of the spectra and extrapolating these lines to the position of the high-energy edges. In the case of the silver targets, this procedure was uncomplicated, but for the coated, rareearth targets, the upper part of the spectra was changed due to the presence of the protective layer. The effect can be seen in fig. 1, where we have shown spectra for 600 keV hydrogen ions backscattered from a coated and a pure silver target receiving identical doses. The high-energy edge for the coated target is shifted towards lower energies due to the energy loss in the gold layer, which appears as a large peak in front of the edge. However, at slightly lower energies, the spectrum is closely identical to that belonging to the uncoated target; hence we analyzed the rare-earth spectra in the same way as we did the uncoated target spectra with the only exception that the straight line was extrapolated to the position where the step was calculated to be positioned if there had been no gold layer. Comparing the extrapolated yields obtained for

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Fig. 1. Backscattering spectra for 600 keV H ions impinging on a coated and an uncoated silver target.

coated and pure silver targets in the way described above, we found small differences, the yield for coated targets being slightly larger. For hydrogen projectiles, these differences were small except for the very lowest ion energy used here, where the protective layer introduces some uncertainty into the data extraction. For helium projectiles, however, the differences were larger and had to be corrected for. The correction as obtained via the silver-target investigation is proportional to the thickness of the protective layer and increases with decreasing projectile energy. The size of the correction is given in table 3. The effect is thought to be caused by multiple scattering of the projectiles in the heavy protective layer. In eq. (1) we used the Rutherford-backscattering cross section as there are no anomalies due to nuclear interaction for the heavy target atoms and projectile energies used here. Electron screening effects are negligible for the measurements presented in ref. 3 but have to be corrected for for some of the data presented here. Such screening effects have been studied by L'Ecuyer et al. s) and by Andersen et al. 6) and give rise to the correction, 48.97 z Z ; 1 - 1 O'Ruth

-

1-~

EcM(eV)]

(3) '

where z and Z are the projectile and target atomic I.

STOPPING

POWER

AND STRAGGLING

44

H. KNUDSEN et al.

TABLE 2 Hydrogen-ion stopping powers. Target

E (keY)

Oxygen correction

(%~ Ag

La

Ce

Pr

Gd

Dy

Ho

Er

Yb

200 300 400 600 800 1000 200 400 600 800 1000 400 600 800 1000 200 400 600 800 1000 200 300 400 600 800 1000 200 400 600 800 1000 200 400 600 800 1000 200 300 400 600 800 1000 200 300 400 600 800 1000

0.5 0.5 0.5 0.5 0.4 2.6 2.5 2.4 2.3 1.6 1.6 1.5 1.5 1.4 1.8 1.8 1.8 1.6 1.6 1.4 1.8 1.6 1.5 1.4 1.4 5.8 5.1 4.8 4.6 4.4 3.7 3.2 3.1 3.1 2.9 2.8 1.9 1.8 1.8 1.7 1.6 1.5

a/aRuth

correction

S(Z)/S(Ag)

S(Z)

_+1 a

(eV cm2/ 1015 atoms)

(%)

1.2 0.6 0.4 0.3 0.2 0.7 0.4 0.3 0.3 1.4 0.7 0.5 0.4 0.3 2.0 1.4 1.0 0.7 0.5 0.4 2.2 1.2 0.7 0.6 0.5 2.4 1.2 0.8 0.6 0.5 2.5 1.7 1.3 0.9 0.7 0.5 2.8 1.9 1.4 1.0 0.7 0.6

1.31 _+0.05 1.20 _+0.02 1.25 _+0.02 1.20 --0.02 1.19 _+0.04 1.17 _+0.02 1.14 +0.03 1.23 _+0.06 1.14 _+0.07 1.13 _+0.06 1.18 _+0.03 1.18 _+0.03 1.206_+0.016 1.27 _+0.03 1.07 _+0.07 1.14 _+0.03 1.16 _+0.02 1.20 _+0.02 1.25 _+0.02 1.22 _+0.03 1.21 _+0.06 1.12 _+0.03 1.19 +0.02 1.226_+0.015 1.28 -+0.02 1.14 +_0.04 1.07 _+0.02 1.09 _+0.02 1.17 +_0.03 1.25 _+0.03 0.95 _+0.06 1.07 _+0.04 1.20 _+0.04 1.18 _+0.02 1.22 +_0.02 1.23 +0.03 0.86 -+0.04 1.04 +0.03 1.07 _+0.02 1.13 _+0.02 1.19 +0.02 1.23 _+0.02

32.329 28.707 25.979 21.459 18.538 16.548 42.3 31.1 26.9 22.4 19.6 30.5 24.5 22.8 18.9 36.5 30.7 25.3 22.4 21..1 34.5 32.7 30.2 25.7 23.2 20.2 39.0 29.0 25.6 22.7 21.1 36.7 27.7 23.3 21.7 20.6 30.7 30.7 31.3 25.3 22.7 20.3 27.8 29.9 27.8 24.2 22.1 20.3

a a a a a a

a Normalization values taken from ref. 1. For E~<500 keV, the values were extracted from tables 1 and 2 of ref. 1 (' medium energy fit '). For E > 500 keV, the ' Bethe fit ', eq. (13) of ref. 1, was used.

number, respectively. The size of this correction as applied to eq. (1) is given in tables 2 and 3. To use eq. (1), it is necessary to know the proper values of v. However, as the k values in this experiment are all close to unity, there is only a weak dependence on these parameters, and a rough estimate is sufficient. We used v values extracted from semiempirical tabulations 1'2) or found experimentally from the slopes of the backscattering spectra3). The small amount of oxygen present in our rareearth targets was discussed in the previous section. It gives rise to a correction to the measured rareearth stopping powers SRE,me,s, which can be shown to be given by (

SRE .....

SRE

--

NO sO "~_-1

1 -'[- NRESRE]

,

(4)

where No and NRE are the atomic densities of oxygen and rare earth, respectively, while So and SRE are the corresponding stopping powers. The magnitude of this correction, which in all cases studied here, is smaller than 6%, is given in tables 2 and 3. Most of the experimental uncertainties that normally influence stopping-power measurements are negligible in this investigation due to the special technique applied and because we measure relative values. Hence, uncertainties stemming from beamenergy calibration, dose measurements, differences in solid angle, etc., are negligible. Sources of small uncertainties are the corrections applied. We estimate the correction for oxygen contamination to be accurate to within 20%, which gives a negligible uncertainty in the stopping-power data, except for the holmium targets that contain 15.3% of oxygen, which introduces up to 1% uncertainty into the stopping-power data for this target material. The gold-layer correction and the correction on the backscattering cross section are so small that even appreciable errors do not give noteworthy uncertainties for the stopping-power results. The main source of error stems from the fit and extrapolation procedure and is mostly of statistical nature. In the worst case, the combined experimental uncertainty on the relative stopping-power values is - 1 0 % , but mostly it is - 3 % . In tables 2 and 3 the uncertainty is given as ± one standard deviation. 4. Results and discussion The stopping powers of 200-1000 keV hydrogen ions and 600-2000 keV helium ions for the eight

45

H Y D R O G E N AND H E L I U M S T O P P I N G P O W E R S TABLE 3 Helium-ion stopping powers. Target

Ag

La

Ce

Pr

Gd

Dy

Ho

Er

Yb

E (keV)

600 800 1000 1300 1600 2000 600 800 1000 1300 1600 2000 1000 1300 1600 600 800 1000 1300 1600 2000 600 800 1000 1300 1600 2000 600 800 1000 1300 1600 2000 600 800 1000 1300 1600 2000 600 800 1000 1300 1600 2000 600 800 1000 1300 1600 2000

Oxygen Correction (%)

0.6 0.6 0.6 0.6 0.6 0.6 2.7 2.6 2.6 1.7 1.7 1.7 1.6 1.6 1.5 1.9 1.9 1.9 1.8 1.8 1.7 1.8 1.8 1.8 1.7 1.6 1.6 6.0 6,0 5.8 5.6 5.2 5.1 4.0 3.9 3.8 3.6 3.5 3.3 1.1 1.0 1.0 1.0 1.0 1.0

G/O'Ruth correction (%)

0.8 0.6 0.5 0.4 0.3 0.2 0.5 0.4 0.3 1.0 0.7 0.6 0.4 0.4 0.3 1.4 1.0 0.8 0.6 0.5 0.4 1.5 1.2 0.9 0.7 0.6 0.5 1.6 1.2 1.0 0.8 0.6 0.5 1.7 1.3 1.0 0.8 0.6 0.5 1.9 1.4 1.1 0.9 0.7 0.6

Au-layer correction (%)

1,1 0.9 0.7 0.5 0.3 0.1 1.8 1.1 0.6 2.8 2.3 1.8 1.1 0.6 0.2 2.8 2.3 1.8 1.1 0.6 0.2 2.8 2.3 1.8 1.1 0.6 0.2 2.8 2.3 1.8 1.1 0.6 0.2 2.8 2.3 1.8 1.1 0.6 0.2 2.8 2.3 1.8 1.1 0.6 0.2

S(Z)/S(Ag)

S(Z)

_+ 1 o-

(eV cm2/lO 15 atoms)

1.47_+0.05 1.39_+0.03 1,27-+0.03 1,28-+0.03 1,20+0.03 1,21 _+0.03 1.34+_0.04 1.18--0.03 1.21 _+0.02 1.28_+0.14 1.22_+0.05 1.23_+0.07 1.20_+0.10 1.15+0.04 1.18-+0.02 1.10-+0.05 1.35+0.08 1.21 -+0.03 1.18_+0.03 1.17-+0.03 1.16+0.03 1.02 ± 0.06 1.15+_0.05 1.07-+0.05 1.12-+0.05 1.11-+0.03 1.12-+0.02 0.97-+0.04 1.01 _+0.06 1.03 ± 0.03 1.03 -+ 0.03 1.06_+0.03 1.07_+0.03 1.24_+0.11 1.27_+0.07 1.14_+0.03 1.13_+0.04 1.17_+0.04 1.12_+0.03 1.22_+0.09 1.19-+0.06 1.25-+0.05 1.03 -+0.04 1.10_+0.04 1.04-+0.03

107.19 111.35 111.38 107.60 102.12 94.64 158 155 141 138 123 115 149 127 124 137 136 137 130 117 112 118 150 135 127 120 110 109 128 120 120 113 106 104 112 114 111 109 101 133 141 127 I22 120 106 131 133 139 111 113 98

a a a a a a

a Normalization values, taken from ref. 2.

I. S T O P P I N G P O W E R A N D S T R A G G L I N G

46

H. K N U D S E N

rare-earth metals measured in this work relative to the corresponding silver stopping powers are given in tables 2 and 3. The relative values have been converted into absolute stopping powers, using the absolute semiempirical silver stopping powers of refs. 1 and 2. These normalization values as well as the resulting rare-earth stopping powers are also given in the tables. In ref. 1 it is stated that the accuracy of these semiempirical values for hydrogen ions in silver is - 1 % at 1000 keV, decreasing to N3% at 600 keV, while at 500 keV, the accuracy is - 5 % , becoming larger, the smaller the energy. For the helium semiempirical stopping powers of silver, ref. 2 states no accuracy, but the fitted curve is the mean curve of a rather large amount of experimental data, which scatter +_5% at 600keV and +_2% at 2000 keV around the curve. The accuracy of the

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helium normalization values used here can also be judged via a comparison with the recently published results by Fontell and Loum@irvi7). They agree with the values of ref. 2 to within 2% in the energy interval used here. The absolute stopping powers obtained via this normalization are shown in figs. 2 and 3, where they are compared to the semiempirical results of refs. 1 and 2 and to the few existing measured data 8-12) for these materials. From fig. 2 we observe that our hydrogen-ion data in general agree with the semiempirical curves. This lends strong credit to the interpolation procedure, from which these curves were obtained. On the other hand, for all the target materials, there is a tendency that the data of the present work are slightly lower than the semiempirical curves at the lowest energies used here. However, this tendency

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might be enhanced due to the influence from the protective layer which, as mentioned previously, may introduce some uncertainty into the 200 keV results. Comparing our data with the other experimental data, we find good agreement. A notable exception are the results for cerium and ytterbium of Sirotinin et al.8), which are obviously incorrect. These authors do not report any use of protection against oxidation for their targets, and their stopping powers are closer to the values for the rateearth oxides.

Fig. 3 shows that for helium ions, there is good agreement between our data and the semiempirical results, especially for ion energies higher than 1000 keV. It can further be seen that our own data in general agree with the sparce experimental data already obtained for these target materials. However, there seems to be too large a scatter of our experimental data for helium ions with energies below 1000 keV, in particular for the heavier target materials. There is also some indication of the same problem for the hydrogen data of fig. 2. To investi1. S T O P P I N G

P O W E R AND S T R A G G L I N G

48

n. KNUDSEN et al.

gate whether effects s t e m m i n g from multiple scattering or energy straggling might be the source of this excessive scatter, we measured the stopping powers of tin and bismuth relative to silver because these two target materials have lower and higher atomic mass, respectively, than the rare-earth elements. The experimental method was the same as that used for the rare-earth m e a s u r e m e n t except that no protective layer was used. The results of these m e a s u r e m e n t s are shown in figs. 4 - 7 . As can be seen, our tin and bismuth stopping powers agree very well indeed with both semiempirical results and other experimental data, thus excluding multiple scattering and energy stragggling as the sources of error. Hence, at this stage we believe that the reason [

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why the data for helium ions with energies below 1000 keV tend to be somewhat unreliable is to be found in the influence of the protective layer or perhaps in effects s t e m m i n g from an excessive rareearth-surface oxygen concentration. However, the details of such effects are not yet understood. One of the more d e m a n d i n g checks of theoretical calculations of stopping powers as well as of semi-

HYDROGEN i

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25 1000keY

AND HELIUM STOPPING POWERS

r

i

H-IONS

A e~

O

S

~o o

20

>~ to - .... [

I

50

60

Andersen, Ziegter Lindhard, Winther I

(19771 (196/.I I

70

80

Fig. 8. The experimental data of this work for 1000 keV H ions compared to the semiempirical result of ref. 1 and the theoretical calculation of ref. 22. The relative stopping powers have been normalized to the Ag result of ref. 1 (arrow).

empirical fits is that they should show the right dependence on the target atomic number Z. In figs. 8 and 9, we have plotted the stopping powers presented in this work for 1000 keV hydrogen ions and 2000 keV helium ions, respectively, as a function of target atomic number. They were all normalized to the silver stopping power, which is indicated by an arrow. In the figures, the experimental data are compared to the semiempirical results 1'2) as well as to the theoretical calculation of Lindhard and Winther 22) as tabulated by Ziegler and Chu23). For the hydrogen projectiles, the semiempirical results is clearly in better agreement with the data than is the theoretical result. This is also the case for the helium ions, but here a simple renormalization of the data will remove the discrepancy. However, such a renormalization is not justified by the large amount of experimental data l

I

I

I

I

2000 keY He-IONS

~_

,2°

S o 100

to

/"

90 80

50

t

60

- -

Ziegler

....

Lindhard, Winther

~

70

H977) (196/.)

i

80

Fig. 9. The experimental data of this work for 2000 keV He ions compared to the semiempirical result of ref. 2 and the theoretical calculation of ref. 22. The relative stopping powers have been normalized to the Ag result of ref. 2 (arrow).

49

on which the present normalization is based. We are therefore left with the conclusion that the theoretical results are too low for the experimental parameters we are concerned with here. As a more important conclusion, however, we can state that the interpolation schemes used in refs. 1 and 2 apparently work well for the rare earths. A direct interpolation between the experimentally well-documented materials silver and gold would have yielded values in the tables significantly lower than the present experimental data, while the interpolation based on the trend of theoretical mean-ionization potentials fits well. An independent check such as that presented here enhances the credibility of the tables for other materials where experimental data are uncertain or entirely missing. The authors are grateful to J. Chevallier for preparing the targets used in this work. One of us (VM) gratefully acknowledges a grant from the N A T O Research Organization.

References () H. H. Andersen and J. F. Ziegler, Hydrogen stopping powers and ranges for all elements (Pergamon Press, New York, 1977). 2) j . F . Ziegler, Helium stopping powers and ranges in all elements (Pergamon Press, New York, 1977). 3) H . H . Andersen, H. Knudsen and V. Martini, Nucl. Instr. and Meth. 149 (1978) 137. 4) H . H . Andersen, K. N. Tu and J. F. Ziegler, Nucl. Instr. and Meth. 149 (1978) 247. 5) j. L'Ecuyer, N. Matsunami and J . A . Davies, Nucl. Instr. and Meth. 160 (1979) 337. 6) H. H. Andersen, F. Besenbacher, P. Loftager and W. M611er, to be published. 7) A. Fontell and M. Lourmaj~irvi, Phys. Rev. B, to be published. 8) E.I. Sirotinin, A . F . Tulinov, A. Fiderkevich and K.S. Shyshkin, Radiat. Eft. 15 (1972) 149. 9) H . H . Andersen, H. Simonsen, H. Sorensen and P. Vajda, Phys. Rev. 186 (1969) 372. 10) R.A. Langley and R. S. Blewer, Nucl. Instr. and Meth. 132 (1976) 109. l() K. W. Lin, H. G. Olson and D. Powers, Phys. Rev. B8 (1973) 1881. (2) W. K. Chu, J. F. Ziegler, I. V. Mitchell and W. D. Mackintosh, Appl. Phys. Lett. 22 (1973) 437. 13) D. W. Green, J. N. Cooper and J.C. Harris, Phys. Rev. 98 (1955) 466. 14) A. Valenzuela, W. Meckbach, A . J . Kestelman and J.C. Eckardt, Phys. Rev. B6 (1972) 95. ]5) H. B~itzner, Ann. Phys. 25 (1936) 233. 16) R. Ishiwari, N. Shiomi, S. Shirai and Y. Uemara, Bull. Inst. Chem. Res. Kyoto Univ. 52 (1974) 19. I. S T O P P I N G

POWER AND STRAGGLING

50

H. K N U D S E N et al.

17) S. Matteson, E. K. L. Chan and D. Powers, Phys. Rev. AI4 (1976) 169. 18) S. Rosenblum, Ann. Phys. 10 (1928) 408. 19) W. K. Chu and D. Powers, Phys. Rev. 187 (1969) 478. 2o) j. A. Borders, Radiat. Eft. 21 (1974) 165.

21) j . C . Eckardt, Phys. Rev. AI8 (1978) 426. 22) j. Lindhard and A. Winther, Mat. Fys. Medd. Dan. Vid. Selsk. 34 No. 4 (1964). 23) j. F. Ziegler and W. K. Chu, At. Data Nucl. Data Tables 13 (1974) 463.