Stopping power and straggling of 0.2-2.0 MeV protons and 0.3-3.1 MeV 4He ions in erbium

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Stopping power and straggling of 0.2-2.0 MeV protons and 0.3-3.1 MeV 4He ions in erbium

J.C. Oberlin, A. Amokrane, H. Beaumevieille, J.P. Stoquert, R. Perrier de la Bathie

To cite this version:

J.C. Oberlin, A. Amokrane, H. Beaumevieille, J.P. Stoquert, R. Perrier de la Bathie. Stopping power and straggling of 0.2-2.0 MeV protons and 0.3-3.1 MeV 4He ions in erbium. Journal de Physique, 1982, 43 (3), pp.485-491. �10.1051/jphys:01982004303048500�. �jpa-00209417�

Stopping ^{power and} straggling ^{of 0.2-2.0 MeV}

protons ^{and 0.3-3.1 MeV} 4He ions in erbium

J. C. Oberlin, ^A. Amokrane, ^H. Beaumevieille, ^{J. P.} Stoquert

Centre des Sciences et de la Technologie Nucléaires, ^BP 1017, Alger-Gare, Algeria

and R. Perrier de la Bathie

Centre National de la Recherche Scientifique, Laboratoire Louis Néel,

Avenue des Martyrs, 38042 Grenoble Cedex, ^France

(Reçu ^le 23 juillet ^1981, accepté ^{le 24 novembre} 1981)

Résumé. ²⁰¹⁴ La section efficace de perte d’énergie, ^{et la} dispersion ^en énergie ^des protons ^{et des ions} 4He dans des couches minces d’erbium, évaporées ^sous vide, ^{ont été} déterminées en utilisant la technique ^de rétrodiffusion des

particules incidentes, ^{dans les domaines} d’énergie ^de 0,2 à 2 MeV pour les protons ^{et de} 0,3 ^à 3,1 MeV pour les ions 4He. Les résultats expérimentaux concernant les sections efficaces de perte d’énergie ^{ont été} ajustés par la formule de Brice, ^{et l’erreur commise sur ces mesures est} de l’ordre de ± 3 %. Les valeurs trouvées sont en accord

avec celles prévues par les calculs semi-empiriques ^de Ziegler. ^Les résultats obtenus pour la dispersion ^en énergie

des protons ^sont proches ^des prévisions de la théorie de Bethe et Livingston. ^{Le disaccord entre les} points expéri-

mentaux et théoriques pour la dispersion ^en énergie ^{des ions 4He est} expliqué par l’influence de la ^non uniformité de l’épaisseur ^{de cible.}

Abstract. ²⁰¹⁴ The stopping ^{power cross} section and energy straggling ^of protons ^and 4He ions in erbium films

evaporated ^{in vacuum, were} determined using ^a backscattering technique in the energy range from 0.2 ^to 2.0 MeV for protons, ^{and 0.3 to} 3.1 MeV for 4He ions. The experimental stopping power data ^were fitted by ^{means of}

Brice’s formula, and agree with Ziegler’s semiempirical calculations. The accuracy of the measurements was esti- mated to be ± ³ %. The energy straggling results for protons ^{in erbium are in} agreement ^{with the} predictions ^of

Bethe and Livingston. Disagreement between the experimental and theoretical points for energy straggling ^of 4He ions is explained by the influence of the thickness non uniformity ^{in the} target.

Classification Physics ^Abstracts

61.80M

1. Introduction. ^- There are

only

^{a few}

experi-

mental data

concerning

^the

stopping

power of protons and 4He ions in erbium, ^{in the} energy range from 0.2 to 3.0 MeV

[1,

2], ^{and no} results for energy

straggling

of these

particles

^{are available.}

As for other rare-earth metals, it is rather difficult to obtain thin films of erbium

by

^vacuum

evaporation,

to

weigh

them in order to determine the areal

density,

and to insert the

samples

^{into the}

scattering

^chamber.

Langley

and Blewer

[I I

measurements have been made

by

^ion

backscattering technique,

and the oxygen contamination of the erbium thin films was mini- mized

by in

^situ

deposition

of the reactive metal, ^the

areal

density

^{of the}

layer

^was determined

by

quartz

crystal

resonators.

Only

^seven

stopping

power ^cross section data were obtained for each

particle.

^The

stopping

powers of Knudsen et al. [2] ^{were measured}

relative to the

stopping

power of silver,

using

^{the ion}

backscattering technique

^{on a}

layer

^{of erbium of}

thickness

larger

^{than the} range of the ions,

deposited

on a

glass plate

and covered with a

protective gold layer

of thickness 200-500 A. The

stopping

power ^cross section values of

Langley

and Blewer differ from these achieved

by

Knudsen et al. Therefore, it appears ^to be necessary ^to have more new measurements.

In this work, ^we report ^{results of}

stopping

power obtained

by

^a

backscattering

^{method with a} proton

or 4He ion beam on thin erbium films

deposited

^onto

thick aluminium

backings.

^To avoid oxidization of the

chemically

reactive element

during

^the

weighing

process, the thickness of the

evaporated

^{film was}

deduced from the

yield

^{of the} backscattered

particles.

We also determine _energy

straggling

^of protons, ^while the too

high

values for _energy

straggling

^{of incident}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01982004303048500

486

4He ions can be

explained by

^{thickness non uni-}

formity

^{in the}

evaporated

film. These measurements

were

performed

^{in the} energy range from 0.2 to 2.0 MeV for protons, ^{and from 0.3 to 3.1 MeV for} 4He ions.

2. Experimental. ^{- The ion}

backscattering

^measu-

rements described here were carried out

using

^the

3.7 MV Van de Graaff accelerator of the Nuclear Sciences and

Technology

^{Centre of}

Algiers

^{for 4He}

ions of

energies higher

^{than 2.0 MeV,} and the 2 MV Van de Graaff accelerator of the ^« Centre d’Etudes Nucleaires ^» of Grenoble for protons ^and ’He ions of

energies

below 2.0 MeV.

The

magnetically analysed

^{beam was} reduced with

a system of collimators to a section of _{1 square} mm at the entrance of the

scattering

chamber. The target ^was surrounded

by

^a

liquid nitrogen

^cold copper trap ^which

provided cryogenic pumping

and avoided

gradual building

up of carbon contamination ^on the

samples.

A biased

chopper

(+ ¹⁸⁰

V) intercepted

^{one third}

of the incident beam

just

before it reached the target.

So, ^{the most}

important inaccuracy

ⁱⁿ

measuring

^the

ion beam current is due to the

integrator

calibration, and can be evaluated to be less than 2

%. During

the whole

experiment,

^the

intensity

of the ion beam

was maintained between 1 and 10 nA. The

particles

scattered on the target ^were detected with a silicon surface barrier detector

(ORTEC,

FWHM rr 14 keV for 4He

ions)

^{located at 150°}

(laboratory)

^{from the}

incident beam, ^{at a} distance of 14 cm from the

sample.

The influence of the energy response of the detector has been taken into account in the determination of the

particle

energy per channel as a function of the proton ^or 4He ion _energy. The

resulting

^correction

in the lower _energy

region

^{was about 2}

%

for 300 keV 4He ions

compared

^{to 2.0 MeV 4 He ions.}

It was necessary ^to maintain a vacuum of the order of 10-6 torr in the

scattering

^chamber

during

^{the run}

to reduce the oxidization of erbium. The

angle

^bet-

ween the incident beam and the normal of the target

was 50 to rule out a

possible

^{error due to the micro-}

crystalline

^{structure of the}

evaporated

^erbium

layer.

The targets ^were

prepared

^{in the «} Laboratoire Louis-Neel » of the C.N.R.S. in Grenoble. The thin film

samples

^{of erbium were}

evaporated

^{onto thick}

buffed aluminium substrates. The

evaporation

^vacuum

was between 1 and 2 x 10-’ torr in order to avoid

pollution

^{of the}

evaporated

material. The erbium was

heated

by

^a

high frequency

^{source and}

kept

^{up in}

levitation. A titanium trap ^was

employed

^{to absorb}

oxygen ^traces in the

evaporation

^chamber.

To determine the areal

density

of the erbium

layer

(687 ^{x 1015}

at/cm2 )

^{used to measure the 4He ions}

stopping

power, the elastic

scattering

^{cross section of}

a

particles

^{on Er,} which follows the Rutherford law,

was

applied

^{However, the}

screening

effect of the electrons was taken into account

by

^{means of the}

correction

given by 1’Ecuyer et

^al.

[3].

^{As a check on}

this method, ^{a thin}

gold

target ^{was also}

evaporated

onto a thick aluminium substrate. The areal

density

and the

stopping

^{cross section, s, of 4He ions in}

gold

were deduced at two different

energies

(1.3 ^and

1.8 MeV). ^{Our 8 values were found to be 1.5}

%

^and

1.9

%

lower,

respectively,

^than

Ziegler’s [4].

The thickness of the erbium

layer

^{used to deter-}

mine the

stopping

^cross section of protons

(1797

^{x 1015}

at/cm’)

^{was obtained}

by measuring

the energy width of a +He

backscattering

spectrum ^{on this} film, ^and

by employing

the values of 4He ions

stopping

^cross

section found in the first part of this work.

3. Data analysis. ^- The energy width, AE, ^{of a .}

backscattering

spectrum ^is

given by :

where N Ax is the areal

density

of the erbium

layer,

kE the fractional energy obtained after ^a

backscattering event at energy

^{E, 4E) the}

stopping

^cross section, E; ^and Eo average

energies

of ions before and after

scattering, 0;

^the

angle

between the incident beam and the normal to the target,

00

^the

laboratory

^scat-

tering angle.

A

typical

^{4He ion}

backscattering

spectrum ^from

an erbium thin film on an aluminium substrate is shown in

figure

^{1. The two inflexion}

points,

^at

energies E

1 and

E2, correspond

^{to 4He ions}

scattering

^from

the back and the front surfaces of the target, respec-

tively.

Then, ^AE

= E2 -

El.

If we consider that the resolution of the detector does not

depend

^{on the} energy in our energy range, the

depth

resolution deterioration between the two

Fig. ^{1. -} Energy spectrum of 2.0 MeV incident 4He ions backscattered on an erbium film evaporated ^{on an alumi-}

nium substrate. No noticeable impurities ^are detected. The

backscattering geometry indicates the incident energy Ei,

the detected energies E2 ^and Ei correspond ^{to a} scattering

event on the front and the rear surfaces of the erbium layer respectively. Oi ^{is the} angle between the normal of the surface of the film and the incident beam, 0, ^the backscattering angle.

inflexion

points

^is

mainly

^{due to} the energy

straggling, particularly

^{in the case of 4He ions} spectra.

The method

employed

^{in the}

peak analysis

^of

backscattering

spectra is described in reference [5].

It consists in a fit of

experimental points

^{to a theore-}

tical spectrum

depending

^{on six} parameters, ^{which is} the convolution

product

^{of a}

straight

line between the two energy

points E1

^and E2 ^{and a Gaussian}

function of variable mean standard deviation

a(E).

Figure

^{1 shows an}

example

^{of the}

resulting

^{fit to the}

data.

In order to determine the

experimental stopping

power ^cross sections, ^we

proceed

^{as follows.}

For each spectrum J, ^with incident energy

E;(J),

we can define the total energy loss

DE(J),

^the

energies

lost

by

^{the beam}

particles along

^the inward and outward tracks :

AE;(J)

^and

AEo(J), respectively,

^the

average

energies

before and after

scattering E;(J)

and

Eo(J).

^The

stopping

^cross section values

for

^the

incoming

^and

outgoing paths E;(E

^and

eo(Eo)

^can

be obtained with an iterative method,

by

^the

following equations :

With the conditions :

and

the _average

energies

^are

given by :

and

For the first iteration, ^we

take-qi

^{= qo = 1. So,}

we have

^a

^plot

of n values of

i;i(E.)

and n values of

eo(Eo)

^where

n is

the number of spectra. ^{We fit these}

ei(E.)

^and

eo(£o)

data with Brice’s formula

[6], by

^a

least _square method. Also, ^{we define two functions}

S;(E)

^and

So(E). During

^{the 1 th iteration (10 1),}

the values _qi and _qo become

and

where

S;(E)

^and

So(E)

^are calculated in the (I ^- 1) ^th

iteration. After a few number of iterations, ^{we have}

S;(E)

⁼

So(E).

With this method, ^{it is}

possible

^to

deduce two

experimental stopping

cross ^section values _per spectrum, ^at

energies E;(J)

^and

Eo(J)

respec-

tively.

^The program RALENT for the calculation of

S (E)

used in the present ^{work can} be obtained from the authors.

For the

straggling

measurements, ^{an iterative} method is not used since the variation of

straggling

with _energy is _very slow. Then, the energy

straggling

parameter, 6E, ^{at the mean} energy

is obtained

by :

where

6Ei

^and

bE2

^are the full widths at half maximum

(FWHM)

associated with the standard deviations at

energies

E, ^and E2,

respectively (Fig. 1).

As

6E2(j) is

the detector resolution at energy

E2(j)

for each spectrum J, the resolution at energy E is deduced from the curve

bE2 = f (E2).

^{For 4He ions,}

the resolution is ^a constant within the

experimental

error, and for protons, ^{it is an}

increasing

function of energy.

As a further correction, ^{it is} necessary ^to consider the

dependence

^of energy

straggling

^on energy loss

[7].

We define the « total »

straggling,

QT, which includes this

dependence,

^{and the} true >>

straggling,

^Q,

obtained when the _energy loss effect is removed. Then

According

^{to reference} [7]

where E3 is the incident _energy on the back surface of the erbium film.

4. Results and discussion. ^- 4.1 4He IONS STOP- PING POWER. ^- The

experimental stopping

^cross

section values for helium ions are

plotted

^{as a function} of 4 He ions _energy in

figure

2, ^{and tabulated in table I.}

Our results differ from

Langley’s

[1]

by

^{less than 1}

%

in the energy range 0.9 to 2.3 MeV, ^and

by

^3.5

%

^{in the}

lowest _energy

region.

^Knudsen’s

experimental points [2]

^{do not} agree with our values,

particularly

^below

1 MeV.

According

^to Knudsen, ^the

protective

gold

layer evaporated

^{on the rare} earth material may introduce some uncertainties in the.lowest

region.

The solid curve in

figure

² represents ^{a least} square fit of our data to Brice’s three parameter formula with the resultant coefficients : a ⁼ 0.342 5 ; ^{z =} 3.019 9 and

n ⁼ 2.782 0.

488

Table 1. ^- Experimental stopping ^cross sections,

oJ’helium

^{ions in erbium}

Fig. ^{2. -} Stopping ^{cross sections of 4He ions in erbium,}

versus energy. Our experimental points ^are obtained with a

387 x 1015

at/cm2

target deposited ^{on an aluminium}

substrate. The full line represents ^{our data} fitting using

Brice’s formula. We have also mentioned Ziegler’s ^calcu-

lations and the experimental results of Knudsen et al. and these of Langley and Blewer.

The

comparison

^{of our} results with the

semiempi-

rical calculations of

Ziegler [4]

shows that there is a

shift in _energy of about 70 keV for the maximum value of the

stopping

^cross section, ^{and a}

good

agreement (less ^{than 1.5} %) ^{for 4He ion}

energies higher

^than

1.3 MeV. In the lower

region (below

⁴⁰⁰ keV), ^the

discrepancy

^{is as}

high

^{as 4}

%.

The _accuracy of the

experimental

measurements

depends mainly

^{on the} uncertainties in the atomic areal

density

^{of the}

evaporated

^thin film, and in the determination of the measured energy width AE.

These errors are estimated to be 2.5

%

^{and 1}

%, respectively. Stopping

^cross section values are obtained with an accuracy better than 3

%.

4.2 PROTONS STOPPING POWER. ^- Our

stopping

cross section data are

plotted

^{as a} function of proton energy in

figure

3, and tabulated in table II. The fit obtained

by using

Brice’s formula

[6],

with resultant coefficients : a ⁼ 0.417; ^{z =} 2.691 and n ⁼ 5.441, is in

good

agreement with Andersen and

Ziegler’s semiempirical

calculations

[8],

^while

discrepancies

with

Langley’s

measurements

[1]

^{become more}

impor-

tant above 1 MeV.

There is a

large disagreement (up

^{to 20}

%)

^between

our

experimental

data and Knudsen’s values

[2]

^below

300 keV proton energy which can be

explained by

^the

same consideration as for

discrepancies

^{in the case of}

the helium ion

stopping

power. However, ^{as seen in}

figure

3, ^the proton

stopping

^{cross sections} agree very well with those of Knudsen et al. above 500 keV. At these

higher energies,

the data of Knudsen et al. did not suffer from the uncertainties caused

by

^the

protective gold

^{film on} top of the erbium. On the contrary, ^the substrate was here carbon, ^{and the} oxygen ^content

was determined

directly

^{from the}

high

energy spectra and corrected for. Hence, ^the

good

agreement ^{with the}

Table II. ^-

Experimental

stopping ^cross sections, oj ’pro tons ^{in erbium.}

Fig. ^{3. -} Stopping ^cross sections of protons ⁱⁿ erbium,

versus energy. Our experimental points ^are obtained with

a 1 797 ^x 1015 at/cm’ target deposited ^{on an aluminium}

substrate. The full line represents ^{our data} fitting using

Brice’s formula and the dotted line represents Andersen and

Ziegler calculations. The experimental values of Langley

and Blewer and these of Knudsen et al. are also mentioned.

data of Knudsen et al. at

higher

proton

energies

^is

taken as direct evidence for the films to be

sufficiently

oxygen free.

The _accuracy of our measured cross sections is estimated to be 4 % ^{in this case,} since the areal

density

is determined

by

using ^{4He ions}

stopping

^{cross section}

values.

4.3 STRAGGLING. ^- The

straggling

results for protons ^{are seen in}

figure

^4.

They

^{can be}

compared

with the theoretical

predictions

of the Bohr

[9],

^Bethe-

Livingston [10]

and Lindhard-Sharff [11] ] ^theories.

Let us

designate Qlh

the theoretical

straggling

^after

traversing

^a target ^{of areal}

density

^{N Ax.}

Fig. ^{4. -} Straggling ^of protons ⁱⁿ erbium, ^versus energy

(relative units). Comparison ^{of our results to the} Bohr,

the Bethe and Livingston and the Lindhard and Scharff

predictions.

The Bohr’s

theory,

valid for low

charge projectiles

of

high

energy

(higher

than 100 keV for protons)

gives :

where

Z,

^and Z2 ^{are the} atomic numbers of the incident

particle

^{and the} target atoms,

recpectively.

Following

^the

theory

^{of Bethe and}

Livingston [10],

the _energy

straggling

^is

given by :

where Z’ is the total number of effective electrons, Ii

the _average excitation _energy of the

Zi

electrons in the ith atomic orbit,

and ki

^{are constants all}

equal

^to

1.

^The

summation extends over all shells for which 2 mv2 > ij (m is the electron mass and v the incident

particle

velocity). According

^to Comfort et al. [12], ^the

energies

490

li ^are

replaced by

ph v; ^where hVi ^{are the} X-ray ^critical

absorption energies

for the ith shell

[13],

^and p is evaluated from the sum rule

given by Livingston

^and

Bethe [10] and rewritten

by

Sternheimer

[14].

Lindhard and Scharff [11]

give

^the

expression

where _x is a reduced _energy variable :

v is the

projectile velocity

^and vo the electron

velocity

in the first Bohr orbital of a

hydrogen

atom, ^and

U j

is the

stopping

number which can be

approximated by :

The comparison ^{of our} proton data with the calculated values obtained

by

these theories shows

(Fig.

4) ^{that the}

Bethe and

Livingston

formalism is in reasonable agreement ^with

experiment.

^{However, a noticeable}

discrepancy

^- up ^to 15 % - appears in the _energy

region

^{of 0.5 to 0.8} MeV, ^{which is}

larger

^{than the}

accuracy of the measurements

(less

^{than 10}

%).

We have not

reported

^the

experimental points

^{of the}

energy

straggling

^{of 4He} ions in erbium.

They

^are

located from

20 %

^{to 30}

%

^{and from 6}

%

^to

12 % higher

than the Bohr’s

prediction

^{for the} targets of thicknesses 687 ^x 1015

at/cm2

^{and 1 797 x 1015}

at/cm2,

respec-

tively.

As is discussed in reference

[15],

^{if we} suppose that the

discrepancy

^{is due to the non}

uniformity

^{of the}

thickness of the

evaporated

film, the measured _energy

straggling,

Slm, ^{can be related to the true}

straggling by :

where

u2(N

Ax) ^{is the corrected term due to non}

uniformity

Here, 62 ^is

independent

of the incident

particle,

^and

represents the thickness fluctuation of the target, ^and

Assuming

^{that the} energy

straggling

^{for 4He ions is}

given by

the Bethe and

Livingston

formalism, ^the

value of -a is about 20 ^x 1015

at/cm2

for both the

targets of 687

^x

1015 at/cm2

^{and 1797 x 1015}

at/CM2.

Then, ^{if the« true »} _energy

straggling

^for protons ⁱⁿ the erbium film of thickness 1 797 x 1015

at/cm2

^is

calculated

including

this value of 6, the results differ from the measured _energy

straggling by

²

%.

This correction can be considered as

negligible

^for

protons, but it is _very

important

^{for 4He ions or}

heavier ions (about ²⁰

%

^{for 4He ions in our} case).

5. Conclusion. ^- The

stopping

power ^{cross sec-} tions in erbium for protons ^{in the} energy range 0.2 to 2.0 MeV and for 4He ions in the energy range 0.3 ^to 3.1 MeV are

reported.

Our results are in agreement with

Ziegler’s semiempirical

calculations within 3

%,

except in the lowest _energy

region

^{for 4He ions where}

the

discrepancies

^{are about 4}

%.

^This agreement ^{is of} the same order ^as with

Langley

^{and Blewer’s}

experi-

mental data, while the

discrepancies

in the lowest

region (less ^{than 1 MeV for 4 He ions and 300 keV}

for

protons)

^{with the} results of Knudsen et al. can be

explained by

the influence of a thin

protective gold

film on the

evaporated

erbium thick

layer.

The _energy

straggling

^of protons in erbium films is in agreement ^{with the}

predictions

of the Bethe and

Livingston theory.

^{Due to non}

uniformity

effects of the target, ^{it was not}

possible

^to deduce values of energy

straggling

^{of 4He ions in erbium from} our measure- ments. However, it has been shown that these effects do not affect the results for protons.

Acknowledgments.

^- The authors would like to thank Dr. E.

Ligeon

^{from the}

Department

^{of Solid}

State

Physics

of the «Centre d’Etudes Nucleaires » in Grenoble for his research facilities and valuable advice.

References

[1] LANGLEY, ^{R. A. and} BLEWER, ^R. S., Nucl. Instrum.

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