Stopping ratios of 50–300 keV light ions in metals

NUCLEAR

INSTRUMENTS

AND

METHODS

168

(1980)

33-39;

O

NORTH-HOLLAND

PUBLISHING

CO.

STOPPING RATIOS OF 50-300 keV LIGHT IONS IN METALS P. M E R T E N S and Th. KRIST

Hahn-Meitner-lnstitut .Eir Kerrc[brschung Berlin GmbH, Glienicker Sir. 100, D-IO00 Berlin 39, Germany

With 50-300 keY H +, He +, 7Li +, and Be + stopping cross sections and stopping ratios are measured for carbon, a l u m i n i u m , copper, silver, and gold. The thickness of the target foils is calibrated with 100-200 keV He +, using the stopping cross sections fitted in ref. 1. Based on this thickness calibration the stopping cross sections for H +, 7Li+, and Be + can be measured with an error smaller than 7%. Foil experiments prove to be a very appropriate method for determining ratios of stopping cross sections. These ratios can be evaluated to a precision of 2--4%. The stopping ratios were found to be independent of the target material or the ion within 7% for most ion-target combinations, i.e. the simplified stopping function can be verified within this limit.

1. Introduction The still growing application of ion implantation and radiation damage research has generated new interest in the stopping process of energetic particles in matter. Range and radiation damage produced by the particles are directly related to this process. Radiation damage, i.e. the production of collision cascades finally resulting in point defects, is attributed to elastic or nuclear collisions, which transfer kinetic energy to the target atoms. The range of a particle in matter is determined by nuclear collisions as well as inelastic or electronic collisions. The inelastic fraction of the energy transfer in a collision is due to ionization or internal excitation of the colliding particles. In the energy range investigated here these inelastic processes are by far dominating, so that a nuclear contribution will be neglected. The electronic stopping cross section Se has been shown to oscillate as a function of the ion and target nuclear charge Z1 and Z2 at fixed ion velocity2'3). In the inelastic stopping theories of Lindh a r e ) and FirsovS), Se is a monotonous function of Z~ and Z2, i.e. Se is not oscillating. Modifications of these theories incorporating more realistic electronic distributions result in an oscillatory behaviour of S~ 6-8). To obtain absolute values for Se, however, certain parameters have to be fitted to experimental data. So the need for experimental data is not reduced by this kind of approach. No self-consistent and easily applicable theory being available, efforts have been made in relating experimental data to each other by scaling l a w s l ' 9-11). Making use of scaling laws will enable the interpolation to stopping cross sections of experimentally not available ions or stopping media, thus reducing the need for

experimental data. The most comprehensive and accurate effort relating experimental data by scaling laws has been made by Zieglerl). In this work he successfully scaled experimental stopping cross sections for hydrogen to those of helium by applying the simplified stopping function. The applicability of this simplified stopping function is the crucial point in the scaling procedure. As this function cannot be theoretically derived and directly verified, only the resulting consequences can be experimentally checked.

2. The simplified stopping function The simplified stopping function is discussed in detail in ref. 1. Here only two consequences will be pointed out that can be experimentally examined. In the simplified stopping function the ZI- and Z2dependence of Se are separated:

Se(Zl, Z2, v) = P(Z*, v)" T(Z2, v).

(1)

Z~ denotes the effective projectile charge. On the basis of eq. (1), the stopping ratio is the same for any value of Z2: SeA ( AZl, Z2, v) = P(Az~, V). T(Z2, v) S B~ (BZ ~ , Z 2 , v ) P(BZ~, v) T(Z2, v)

= R(AZ1, BZ 1 , v).

(2)

This relation can be applied to deduce an experimentally not available cross section SeA, ifR (for any material) and S~ are known"

S~A ( AZx , Z2, v) = R(AZ1, 8Zl, 0 . S o~ (Bz ~ , z 2 , v ) . (3) If, on the other hand, the ratio Q(Zf, Z~, v) for one ion in the two media with Z~ and Z b is given, $2 can be derived with the help of S~, if Z~ is I. S T O P P I N G

POWER AND STRAGGLING

34

P. M E R T E N S

AND TH. KRIST

assumed to be independent of the target material: S~(Z, , Z~z, v) _ P ( Z * , v) T(Za2, v) = Q ( z ~ , z ~ , v); s (z, , T(Z , (4)

= Q(Z ,

v).

(5)

Thus, using eq. (3) it is possible to deduce S~ for another ion, with eq. (5) for another material. The validity of a target-independent ratio R and projectile-independent Q will be tested for H +, He +, Li +, and Be + by our experiments.

3. Experimental The magnetic spectrometer is shown in fig. 1. As this instrument has already been described in ref. 14, we will only briefly report on some modifications. Before entering the target chamber the ion beam was scanned now by means of x-y deflection plates to achieve the low ion beam density used for this experiment. The amplitude of the scan was controlled via CAMAC digital-to-analog converters, so that the ion beam on the monitoring cup and the target was kept constant during the measurement. Working with a well defined ion current may be of importance, if residual gas desorption or sputtering become relevant. In the measurements presented here sputtering of the targets could be neglected, as the ion dose on the target amounted only to 30 nC (ion current ~1 nA, duration of a run ~30s), resulting in sputtering of only a fraction of a monolayer for the complete measurement. For all ions investigated here the same set of targets was used in order to enhance the precision of the resulting stopping ratios.

The most delicate point in performing energy loss measurements is the production and treatment of the foils. They were produced by vacuum deposition on glass discs covered with a thin soap film. The evaporation unit employed, built in UHV-technology, is equipped with two electron guns, two quartz crystal oscillators, a shutter, and a watercooled 12-position-target wheel. During evaporation the cooling of the targets is very helpful, as the decomposition of the soap film by heat can be largely suppressed. The evaporation rate on the quartz monitors is electronically controlled and kept constant within ___.05 A/s. The thickness display on the oscillator control unit supplies only an approximate measure for the foil thickness. Deviations up to 30% from the actual thickness are not unusual. By thickness calibration by energy loss measurements with 100-200 keV He + it was found, however, that for each material a correction factor can be used which is the same for all the different target thicknesses within 1%. Typical evaporation rates were 3-5 A/s for all materials. During the foil production the pressure in the vacuum system increased up to 1 × 10 - 6 torr. Target thickness calibration was performed with 100-200 keV helium ions, making use of Ziegler's fit ~) as reference. This calibration could have been accomplished by weight, too, but changes in weight of the soap film tended to falsify the results. All energy loss measurements were carried out at a vacuum of about 4 x 10 _8 torr. To avoid a noticeable contribution of particles to the measured energy loss spectra that might have been channeled in the small crystallites forming the foils, the spectro/ I

w ndo

electrostatic quadr upote ens

defiechon plates ;\ ,_ \

I e

O

scanned ion beam man tar cun , r I

,

I ~ r - ~ l l

~

'

. . . . . . . . . . . . l. . . . . . . . . . . [] |

~

transmitted ions ~- 19 target f o i l ~ 1 "~':\ (500- 5QOA) \

~

[

I

'

I F|

--

analysing -. ~ -rnulhplier magnet X, i '~,~ I/-- ~ II ~ -- - P - a r a ~~Jq y / / __ ___Jexternally cup / \ adjus~ableslts

( ~'l

,~ :;-,-

[~

.,~TJ"

I

~ -,~0"~18

~

~-

.~

I

°

II I

Z-

measuring

J ~

~ J DANFYSIK

L~J

Isweep

[

amplifier,

T t

•

J

| ~

I

[ ~ -

scan

Idc amplifier t-/

amplitude

--

-

control

Fig. 1. The magnetic spectrometer with automatic beam control.

-

voltage

am .f~"~ I"~"~1"~"~ f,i~r

,

L[

Imagnet ~l Jpowersuppyl

L ~ rain

~-[~Z]

i

--

iI

~DAC I

I

uAu-t ^

I

J

MCA

. in

fext.contr t ~, ~ data i

CAMAC

I I

i I

I m~Imult'scai'ng F ii

i i

I

.ss

STOPPING RATIOS

meter was tilted at 5 ° out of the direction o f the incoming beam 15). 4. Data evaluation The energy losses were derived as described in ref. 13 and attributed to an average ion energy E = E 0 - ½AE (E0-- initial energy). Comparing the energy losses obtained for foils of different thickness, no contribution of a surface oxide film could be recognized. Thus, the stopping power was simply taken to be Se= AE/d (d = film thickness) instead of S e = (AE2- AEI)/(d2-di) for two significantly different values of d, and d2, as would have been necessary in the presence of a thick oxide film. No nuclear correction has been applied to Se. For the lowest ion energies and the heaviest targets the contribution of nuclear stopping amounts to about 10% of the electronic stopping cross section, if all collision parameters between 0 and ~ are taken into account. As has been pointed out by Hvelplund and Fastrup'V), the nuclear contribution is strongly attenuated for ions that are transmitted in a forward direction through a thin foil or a gas cell. Particles once deflected significantly out of the forward direction have only a minor chance to be scattered back into the acceptance angle of the spectrometer. As a consequence the nuclear contribution to stopping is reduced to about 1% of Se (see also ref. 14). This error being noticeably smaller than other errors encountered here, no correction for nuclear stopping was performed. The errors comprised in the stopping cross sections and stopping ratios are mainly due to errors in foil thickness calibration and energy loss determination, foil inhomogeneities and a coverage of residual gas on the foil's surfaces. The foil thickness calibration has been carried out with 100, 140 and 200 keV helium ions. Considering the usually slightly differing energy dependence of our data and the stopping cross sections fitted by Ziegler as well as the uncertainty of this fit, an inaccuracy of about 5% has to be conceded for the thickness calibration. Being the quotient of two energy losses, the stopping ratios are not influenced thereby. Shifts in the accelerating voltage and small temperature effects in the magnetic spectrometer add up to an error of typically 300 eV for the AE-values. As generally AE is of the order of 10 to 30 keV the relative error for AE amounts to 1 - 3 % , the resulting error for the stopping ratios to 2 - 4 % . For some targets (C, AI) it cannot be prevented completely that tiny wrinkles happen to be on the target foils.

35

Caused by the limited reproducibility of the target wheel positioning and the small acceptance angle of the spectrometer (i.e. the spectrometer may " s e e " different parts of the foil in different runs) additional fluctuations of AE may therefore be observed. These fluctuations directly influence the stopping ratios. They are usually in the order of 1%. Since every run was repeated at least once, it was discovered that the second and consecutive runs yielded a smaller energy loss than the first run. This fact was assigned to an ion beam induced desorption of residual gas from the target surfaces. At our standard pressure of 4 × 10 -8 torr this effect amounted to an average difference of about 150 eV for 300 keV helium ions. Between the second and a further run there was no difference in energy loss. Using the stopping cross section for 300 keV He + in an (assumed) nitrogen layer from ref. 1 again, this coverage can be calculated to be two atomic layers thick. Foils that are analyzed immediately after being mounted (p < 1 0 6 torr) were found to have significantly higher energy losses during the first run (1-2 keV more). After the first desorption by the ion beam the equilibrium coverage slowly built up in typically an hour. The corresponding desorption rates are approximately 100 for the desorption of two monolayers, 1000 for the target that just had been exposed to atmospheric pressure. All the stopping cross sections reported in the next section were taken from targets, whose residual gas layer was desorbed immediately before, so that a falsification of the data can be ruled out. The overall error for the stopping cross sections amounts to about 6%, the error for the stopping ratios to 2 - 4 % . 5. Results and discussion 5.1. STOPPING CROSS SECTIONS In fig. 2 the results of our measurements with H +, He + , 7Li+, and Be + are compiled for the targets C, AI, Cu, Ag, and Au. Every measurement was carried out using at least three different foils. The special purpose of this work being to evaluate stopping ratios, the hydrogen measurements were restricted to energies below 100 keV. As the foils have been calibrated with He + , the helium stopping cross sections cannot be taken as an absolute measure. Within the error limits, the cross sections for lithium agree with the ones reported by us in an earlier paper'S). It must be noted that these former results were based on a weight calibration of foils which yielded larger errors than the recent meaI. 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

36

P.

40 60

~

100

H, ÷ , . .

200 ' 80'-60--

40 60

100

MERTENS

200

-

-

40--

i

AND

TH.

KRIST

TABLE

1

Stopping cross sections (interpolated) for 10~,0 keV/amu Be ÷, 7 L i + , H e + and H + (Se in 10 15eVcm2/atom).

10

15

Energy (in keV/amu) 20 25 30

Gold S e (Be) S e (Li) S e (He) S e(H)

55 39 33

64 51 39.5

74 59 44

83 65 49

93 72 54 22.7

Silver S e (Be) S e (Li) S e (He) S e(H)

64 50 35

76 58 41

86 66 47

96 74 52

Copper S e (Be) Se (Li) S e (He) Se(H)

38 30 23

51 40 27.5

60 47 31

s u r e m e n t s . O u r d a t a for h y d r o g e n a g r e e w i t h i n a few p e r c e n t w i t h Z i e g l e r ' s fits in t h e c a s e o f f o u r targets. F o r a l u m i n i u m o u r c r o s s s e c t i o n s are a b o u t 15%o h i g h e r . T h e l i n e s d r a w n in fig. 3 r e p r e s e n t a g r a p h i c a l i n t e r p o l a t i o n t h a t is u s e f u l for d e d u c i n g t h e s t o p p i n g ratios. T h e r e s u l t s o f t h i s i n t e r p o l a t i o n are t a b u l a t e d in t a b l e s 1 a n d 2. T h e s e t a b l e s g i v e t h e d a t a t h e s t o p p i n g r a t i o s are d e r i v e d f r o m . A d d i tionally the cross sections have been approximated by a fit S d l 0 -15 e V c m 2 / a t o m ] = ao Ep, (E in keV). T h e c o e f f i c i e n t s a0 a n d p are s u m m a r i z e d in t a b l e 3. C o m p a r e d to t h e e x p o n e n t s p o b t a i n e d in ref, 13 t h e r e is a g o o d a g r e e m e n t a m o n g C , A I , a n d C u for H e + a n d C , C u , a n d A u for 7Li+ F a c i n g t h e d e v i a t i o n s for t h e o t h e r t a r g e t s it m u s t b e s t a t e d t h a t t h e s e fits are n o t p h y s i c a l l y r e l e v a n t a n d c a n constitute only a rough approximation. On the o t h e r h a n d , t h e e n e r g y r a n g e s for t h e fits w e r e n o t identical.

Aluminium S e (Be) S e (Li) S e (He) S e(H)

50 39 28

56 46 32.5

Carbon Se (Be) S e (Li) S e(He) S e (H)

30.5 23 19

35 28 21.5

.5

20-u~ ~......,,_._.,,__o~ E[keV]

>~ 4 u~

7Li +

E [keV]~

-80 ,,--o/ ~.~ / o.--°"

/'*';

Be +

- - - - 40

>~

,j,J"

E [keV] - 20 ElkeV]~ 40 60 100 2'00' 40 60 100 200 electronic stopping cross sec- stopp,ng med,o carbon ~, aluminum

tion Se for H~ He+, 7Li+ond Be + silver

•

I

gold

] t

/.

._

_

o_ master curve -~ (J E Z l e g [ e r )

I

2

-

1

i '

°~

.

;

C

-

-T i

.

.

.

~, c o r b o n I • Qlumlndrn , copper

40

79 58 24

86 62 25

105 81 57 25.3

90 62 26.8

99 66 28

68 52 34

74 58 37 17.4

63 40 18.2

69 44 19

63 50 36.5

67 55 40

72 60 42 23.3

65 45 23.8

70 48 24

40 31 23.3

45 33.5 25

50 37.5 26.5 14

40 28 14.5

44 29.5 14.9

copper

•

Fig. 2. Electronic stopping cross sections S e for 50-300 keV H + , He + , 7Li+. and Be + in C, AI, Cu, Ag and Au.

,

35

"~ I

.go,o

10 100 1000 ion energy [ keY/emu]

Fig. 3. Se(He+)/Se(H+) stopping ratios and Ziegler's master curve l).

TABLE

2

Stopping cross sections (interpolated) for 45-70 keV/amu H + and He + (Se in 10-15 eVcm2/atom). Energy (in keV/amu) 55 60 65

45

50

70

Gold Se (He) Se (H)

66 26

69 27

72 27.6

75 28

78 28.5

80 29

Silver S e (He) S e (H)

71 29

76 29.6

80 30.2

84 31

87 31.5

92 32

Copper S e (He) Se(H)

46.5 19.7

49 20.3

52 20.9

55 21.3

56 21.7

58 22

A luminium Se (He) Se (H)

50 24.1

51.5 24.2

53 24.2

54 24.2

56 24.2

58 24.2

Carbon S e (He) Se(H)

30.5 152

32 15.5

33 15.7

34.5 15.9

36 16

37 16.2

STOPPING RATIOS

37

TASLE 3 Coefficients for the power approximation of the stopping cross sections: Se[10 -15 eV cm2{atom]= aoEP (E in keV). Ion

Energy range (keV)

H+ He + 7Li+ Be +

30-100 40-280 60-280 90-280

Carbon

Stopping medium Copper

Aluminium

Silver

Gold

ao

P

ao

P

ao

P

ao

P

a0

P

9.16 5.42 3.16 4.23

0.13 0.34 0.46 0.44

23.7 6.84 5.81 10.49

0.00 0.38 0.44 0.34

4.68 4.57 2.48 2.37

0.35 0.47 0.49 0.62

11.27 5.71 4.21 8.43

0.25 0.49 0.56 0.45

9.84 6.32 3.77 6.62

0.25 0.45 0.55 0.47

5.2. STOPPING RATIOS

The stopping ratios which are evaluated as a function o f energy in k e V / a m u can be directly derived from tables 1 and 2 by dividing corresponding stopping cross sections. All stopping ratios for H +, He +, 7Li+, and Be + are compiled in tables 4 and 5. Restricted by the limited energy range of our accelerator, in some cases only one or a few

ratios can be given. If the simplified stopping function (1) were strictly fulfilled stopping ratios Se(A)/ Se(B) should be the same for all targets. For Be + and 7Li+ this consequence can be verified within a deviation of 7%. For Be + and He + the deviations are about ± 1 0 % , and ± 7 % too if copper is discarded (see below). At 30 keV H + is already near the energy region of its maximum stopping

TABLE 4 Stopping ratios for 10-40 keV/amu H +, He +, 7Li+, and Be +.

S e (14)/Se (B)

Stopping medium

10

15

20

Energy (in keV/amu) 25 30

35

40

1.33 1.42 1.57 1.43 1.41

1.36 1.45 1.58 1.44 1.43

1.39 1.50 1.57 1.46 1.49

3.17 3.20 3.33 2.58 2.68

3.29 3.35 3.46 2.73 2.76

3.44 3.53 3.63 2.92 2.95

2.38

2.42 2.31 2.20 1.89 1.93

2.48 2.36 2.32 2.00 1.98

Se(Be) Se(Li)

Au Ag Cu AI C

1.41 1.28 1.27 1.28 1.32

1.25 1.31 1.28 1.22 1.25

1.25 1.30 1.28 1.26 1.29

1.28 1.30 1.31 1.22 1.34

1.29 1.30 1.28 1.20 1.33

Se(Be) S e (He)

Au Ag Cu A1 C

1.67 1.83 1.65 1.78 1.60

1.62 1.85 1.85 1.72 1.63

1.68 1.83 1.94 1.73 1.72

1.69 1.85 2.00 1.68 1.80

1.72 1.84 2.00 1.71 1.89

Se(Be) Se(H )

Au Ag Cu AI C

Se(Li) Se(He)

AU Ag Cu AI C

Se(Li) Se(H )

Au Ag Cu AI C

Se(He) Se(H )

Au Ag Cu AI C

4.10 4.15 4.25 3.09 3.57 1.18 1.43 1.30 1.39 1.21

1.29 1.41 1.45 1.41 1.30

1.34 1.40 1.52 1.37 1.33

1.33 1.42 1.53 1.38 1.34

2.25

2.12 1.80 1.89

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

38

P. MERTENS AND TH. KRIST

TABLE 5

Se(He+)/Se(H +)

Stopping ratios Stopping medium Carbon Aluminium Copper Silver Gold

for 45-70 keV/amu.

Energy (in KeV/amu) 55 60 65

45

5,0

2.00 2.07 2.36 2.45 2.54

2.06 2.13 2.41 2.57 2.55

2.10 2.19 2.49 2.65 2.61

2.17 2.23 2.58 2.71 2.68

2.25 2.31 2.58 2.76 2.74

70 2.28 2.40 2.63 2.87 2.76

cross section for carbon and aluminium. As this maximum cross section is reached at different energies (in keV/amu) for hydrogen and helium, the ratios S~(A)/Se(H+) cannot be expected to be the same for all materials [validity of eq. (1) restricted to +_ 15% in these cases]. Se(Be+)/Se(H*), Se(Li+)/ /S,(H +) and Se(He+)/Se(H+) are equal for Cu, Ag and Au within 4%. The same holds for Se(TLi+)/ /Se(He +) within 7%. Thus it can be stated that for the energy range (10-30keV/amu) investigated here the target independent stopping ratios can be experimentally verified within a deviation of 7% for most of the ion-target combinations, i.e. the simplified stopping function is applicable within this error. Considering the importance of this energy range for practical purpose and the errors usually encountered in energy loss measurements, the simplified stopping function and its application as demonstrated by Ziegler prove to be a valuable tool in estimating stopping cross sections. A comparison between the above stopping ratios and the ones in ref. 13 shows good agreement for C, AI, Ag and Au. At 30keV/amu both results differ from each other only by a few percents. As the energy resolution of the spectrometer has been improved since the first experiment was carried out, the data given above are the more accurate ones. A

special problem may be connected to the copper measurements. The stopping ratios measured for this material differed up to 10% from those for the other materials and differed at higher energies from Se(TLi+)/S~(He+) in ref. 13. A systematic error in the measurements can be excluded because all the different targets were analyzed in succession so that the error should be apparent in all materials. In figs. 3 and 4 our stopping ratios for all ions and all materials (including copper) and Ziegler's master curves 1'13) for Se(He+)/Se(H+) and Se(Li+)/Se(H *) are plotted. For Se(He+)/S~(H*) our stopping ratios are symmetrically distributed around Ziegler's master curve which is based on ratios which deviate in parts strongly from this fit. As the error implied in our data is significantly smaller (2-4%) than their deviation from the master curve, these deviations are a measure for the limited validity of the simplified stopping function. For the ratio Se(TLi')/ S~(H +) Ziegler's master curve almost exactly agrees with our data for carbon and aluminium, while the ratios for copper, silver and gold are about 20% above the curve. It must be stated, however, that for 7Li* the fit is based on a smaller quantity of data. In these plots no deviations of our measurements for copper become obvious. In table 6 stopping ratios Se(a)/S~(b) are presented for one ion in distinct target materials. Not taking into account the ratios for hydrogen if carbon or aluminium are concerned (see above), these ratios turn out to be independent of the ion within 7%. Summing up the results for S~(A)/Se(B)(different ions) and S~(a)/Se(b) (different targets) it can be stated that the simplified stopping function can be verified in most cases within 7% for 10-40 keV/ amu H +, H e - , 7Li+, and Be + in carbon, aluminium, copper, silver, and gold. If one of the stopping TABLE 6 Some stopping ratios for different target material at 30 keY/ ainu.

6

2 0

Se(a)/Se(b)

(J ~ ZIegler ]

_

_

~

copper

• s,[v~r

/

]

,gold

10

100 1000 ion energy [keVlomu]

Fig. 4. Se(7Li+)/Se(H +) stopping ratios and Ziegler's master curvel3).

Se(Au)/Se(Ag ) Se(Ag)/Se(Cu) Se(Cu)/Se(AI) Se(A1)/Se(C) Se(Au)/Se(Cu) Se(Au)/Se(AI) Se(Au)/Se(C)

Ion Be +

7Li+

He +

H+

0.86 1.42 1.03 1.44 1.26 1.29 1.86

0.89 1.40 0.97 1.60 1.24 1.20 1.92

0.95 1.54 0.88 1.58 1.46 1.29 2.04

0.91 1.45 0.75 1.66 1.30 0.97 1.62

STOPPING RATIOS

cross sections is near its m a x i m u m as a function of energy (e.g. 100 keV H + in C, A1), the validity of eq. (1) is reduced to about _+ 15%. The authors are indebted to Prof. H. Wollenberger for stimulating discussions and continuous support. We are grateful to P.-R. V01z for supplying the thin metal foils.

References l) j.F. Ziegler, Helium stopping powers and ranges in all elements (Pergamon Press, New York, 1977). 2) See refs. 1-21 in Nucl. Instr. and Meth. 149 (1978) 149. 3) D. Ward, H. R. Andrews, I. V. Mitchell, W. N. Lennard and R. B. Walker, Can. J. Phys., 57 (1979) 645. 4) j. Lindhard and M. Scharff, Phys. Rev. 124 (1961) 128.

39

5) O. B. Firsov, Sov. Phys. JETP 9 (1959) 1076. 6) W. Neuwirth, W. Pietsch and R. Kreutz, Nucl. Instr. and Meth. 149 (1978) 105. 7) W. Pietsch, U. Hauser and W. Neuwirth, Nucl. Instr. and Meth. 132 (1976) 79. 8) D.J. Land, J. G. Brennan, D.G. Simons and M. D. Brown, Phys. Rev. A16, No. 2 (1977) 492. 9) W. Whaling, Handbuch Physik 34 (1957) 192. 10) L.C. Northcliffe, Ann. Rev. Nucl. Sci. 13 (1963) 67. 11) U C. Northcliffe and R.F. Schilling, Nucl. Data Sect. A7 (1970) 233. 12) H. H. Andersen and J. F. Ziegler, Hydrogen stopping powers and ranges in all elements (Pergamon Press, New York, 1977). 13) p. Mertens, Phys. Rev. AI9 (1979) 1442. 14) p. Mertens, Nucl. Instr. and Meth. 149 (1978) 149. 15) p. Mertens, Thin Solid Films, 60 (1979) 313. 16) j.p. Biersack, E. Ernst, A. Monge and S. Roth, Rep. Hahn-Meitner-lnstitut, Berlin, No. HMI-B 175. 17) p. Hvelplund and B. Fastrup, Phys. Rev. 165 (1968) 408.

I. STOPPING POWER AND S T R A G G L I N G