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Stopping power and energy straggling for small and large energy losses of MeV protons transmitted through polyester films
Nuclear Instruments and Methods North-Holland, Amsterdam
in Physics
Research
84 (1984) 1-12
STOPPING POWER AND ENERGY STRAGGLING FOR SMALL AND LARGE ENERGY LOSSES OF MeV PROTONS TRANSMI’I-IED THROUGH POLYESTER FILMS * A. L’HOIR
and D. SCHMAUS
Groupe de Physique des Solides de I’Eeole Normale Received
4 October
SupC;ritwre, Uniuersiti
Pans 7, i’ pluce Jussieu,
75251 Paris Cedex 05, Frunce
1983
The stopping power S(E) and the energy straggling Q’(E,, AE) of 0.5 MeV to 2 MeV proton beams in polyester (C,,HsO,) have been measured in transmission geometry. S(E) is found to be 2% to 4% lower than the Andersen and Ziegler tabulations, assuming the Bragg’s additivity rule. The energy straggling a’( E,, TQ was measured at various beam energies E,, mean energy losses m and target thicknesses (from 5.6 pm to 47 pm). The ratio fJ/s2,, (where 0, refers to the free electron model in the small energy loss approximation) of the order of 1.07 for small energy losses (d/E, = 0.05), increases to about 1.5 to 2 for large energy losses (AE/E, = 0.75). For not too small beam energies (E, 2 700 keV) and for our targets. a nearly universal curve O(Ei, AT)/O, is obtained as a function of the mean relative energy loss d-/E,. For d-/E, 6 0.7, the beam energy spread curves which were nearly energy loss spectra. Our experimental results 0 Gaussian, were compared to theoretical predictions 9,, valid for nearly symmetrical are from 0 to 7% larger than the theoretical values B, up to A E/E, = 0.8. The surface roughness of the targets was checked and its influence on the experimental results was generally between 1% and 5%, depending on the targets and on E,.
1. introduction This paper is the first part of a work devoted to the experimental study of the interaction of proton beams with polyester films. In the present paper, experimental results on the- energy loss spectra (in particular the mean energy loss AE and the energy straggling 0’) of the beams transmitted through targets perpendicular (B = 0 ’ ) to the beam direction are presented and compared to the theory. In ref. [I] (the second part of the work), the lateral spread of the beam in the solid targets is determined from a careful analysis of transmission energy loss spectra through highly tilted targets (8 f 0”). For both studies, it was important to minimize unwanted effects such as surface roughness, porosity, thickness inhomogeneity or texture of the targets. From this point of view, rather thick organic targets represent a favourable case. The polyester films (C,,H,O,) studied in the present work are used in our group as absorbers in nuclear microanalysis experiments [2] to stop unwanted backscattered light particles such as protons, deuterons or alphas to avoid pile-up problems. As will be shown in the present paper, these amorphous films are very uniform, of high stability, and the influence of the surface roughness on the energy spread of ion beams * Work supported
by the Centre National de la Recherche Scientifique (R.C.P. no. 157) and presented at the 10th International Conference on Atomic Collisions in Solids, Bad Iburg (Germany), July 1983.
0168-583X/84/$03.00 Q Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V
transmitted through these films (generally in the 6 to 24 pm thickness range) is small and can be evaluated with reasonable precision. In our energy domain (0.5 to 2 MeV) and with the target thicknesses considered, the mean relative energy losses a/Ei were between 0.05 and 0.8. Our study extends therefore from small energy losses, which is the domain generally used for energy straggling measurements, to relatively high energy losses. In the latter case, where interaction cross sections vary along the path of the particles, overall Poisson statistics no longer apply and the usual theories have to be revised. This can be simply performed for not too high energy losses [see eq. (34)], provided that the stopping power S(E) in the medium is known. A precise determination of the stopping power of protons in polyester was therefore necessary. The results on S(E) are given in section 4. Section 5 is devoted to the energy straggling measurements; in this section, a derivation of the energy straggling calcu__ lation when A.5 -c E, no longer applies is also presented, and the influence of the surface roughness of the targets is discussed. It must be pointed out that the present paper and the lateral spread study [I] are strongly linked. When a target is perpendicular to the beam (B = 0°) the influence of the lateral spread of the ions on the energy loss spectra is negligibly small (the case of the present paper); when the target is tilted (8 f 0 ’ ), the lateral spread increases the energy spread, especially at large 8. The lateral spread was therefore indirectly determined
2
A. L’Hoir, D. Schmaus / Proton energy losses in polyester films
by comparing spectra recorded at 0 = 0’ (presented work) with spectra recorded with tilted targets (for more details of the procedure, see ref. [l]).
2. Experimental set up Monoenergetic proton beams were produced by the 2 MV Van de Graaff accelerator of the Ecole Normale Superieure. The accelerator was calibrated using the numerous narrow resonances of the Al(p, y) nuclear reaction [3]. Protons transmitted through the targets were analysed at an angle of 0’ using a 25 mm2 area silicon surface barrier detector from Ortec, located at 111 cm from the target and provided with a 0.6 mm diameter diaphragm. Two 0.2 mm diameter nickel diaphragms located in the accelerator pipe at 74 cm from each other ensured a sufficiently low beam current in the vacuum chamber with no significant angular or energy spread. For the type of experiments considered here, counting rates of the order of lo3 per second could easily be achieved. Targets and detector could be translated perpendicularly to the beam directions using stepping motors. More details about the experimental set up are given in ref. [4]. In these experiments a model 142A charge sensitive preamplifier, a model 572 linear amplifier and a model 444 gated biased amplifier from Ortec were used. The amplitude of the pulses were analyzed by means of a 8192 channels ADC from Laben and energy spectra were recorded using an Informatek online computer system. Calibration of the electronics was performed by detecting directly the beam particles, the energy of which was well known. A very small variation of the keV/channel value with energy was found and taken into account in the data reduction. The uncertainty in the keV per channel values was estimated to be _tOS%, considering the very high stability of the electronic set-up.
3. The targets The polyester foils used in our experiments had the stoichiometry C,,H,O,. They were obtained from Rhone-Poulenc (trade mark Terphane), without aluminum coverage (Terphane is very similar to Mylar from DuPont). Six different rectangular foils of about 6, 12 and 24 pm thicknesses were used, with areas of about 500 cm2. The mean thicknesses xi (j = 1 to 6) of by weighing, these foils (in g cmm2) was determined using a Mettler balance, with a lo-’ g uncertainty (the mass of the foils being of the order of 1 g) and by measuring their areas. In most cases, the precision on the xi was of the order of +0.5%, i.e. much better than for example using the RBS technique in the case of thin films. Pieces of these sheets were mounted on eight
27 X 27 mm steel frames; some of the eight targets so obtained (indexed by n) were constituted of two or four superimposed foils. For precise, absolute stopping power measurements, a determination of the target thickness homogeneity on a macroscopic scale is necessary while for energy straggling measurements, a precise knowledge of the uniformity of the targets on a microscopic scale (surface roughness) is required. The thickness uniformity of the targets and of the initial sheets was therefore hence evaluated on four different scales: i) on the scale of the beam diameter (0.2 mm), ii) on the scale of the size of a target (- 1 cm), iii) on the scale of the dimensions of the initial sheets (- 10 cm), iv) on a microscopic scale (surface roughness); this point is discussed in section 5. For the first two cases, the very sensitive method described in ref. [5], using as a tool angular multiple scattering of the beams transmitted through the targets was chosen. An overall thickness variation below + 0.5% was observed for one of the targets (target no. 3, see table l), when translating the latter by & 5 mm, using a stepping motor with 0.125 mm steps, the beam diameter being 0.2 mm. A comparison of the energy loss measurements on our various targets indicates that this conclusion is valid for all the targets (see section 4.3). The third point happened to be more involved; the non-uniformity on the large scale (- 10 cm) of the weighed Terphane sheets was evaluated from energy loss measurements and taken into account in the following way. Assuming the results of the weighing and area measurements to be valid, we took the thicknesses x: of the targets (n = 1 to 8) to be XL or in some cases two or four times these values. Taking one of the targets as a reference (n = 1), we determined a stopping power curve S,(E) from energy loss measurements on that target at various beam energies E,:S,(E) does in fact depend on x:, but only the shape of the stopping power curve S,(E) is of interest at this point. Energy loss measurements were also performed on the seven other targets at various beam energies. Letting x” be the actual thicknesses of the targets, these energy loss measurements give the ratios x”/x’ with a F 1% precision including the uncertainties on the mean energy loss determinations (in such relative measurements, errors on the energy calibration nearly cancel) and thickness variations for a given target (see above). Calculations of these ratios were performed by numerical integration of E,
JE,-AE_dE/‘S,(E), -
where E, is the beam energy and AE the mean energy loss in a given target n: eq. (1) gives a thickness proportional to x”. When comparing the ratios x:/.x: and x”/x’ a maximum of 5.2% discrepancy appears, which gives an order of magnitude ( f 2.5%) of the thickness
A. L’Hotr,
3
D. Schmaus / Proton energy losses in polyester films
Table 1 2
3
4
n
1
x: (pm) Precision (5%)
5.8 0.6
2 x 6.03 1.5
4 x 5.45 0.6
X”/XI
1
1.023
1.026
C/d xn (w)
5.625
11.96
21.66
5
11.94 0.4
2x11.94 0.4
1.032
I
23.1 0.8
1.024
11.95
variations of the sheets on the large scale (- 10 cm). The absolute values of the xR were finally determined from the known ratios x”/x’ by setting (x”) = (x:), where the average is taken over the eight targets, with weighting factors taking into account the precision on the x: (hence on the XL) and the fact that for targets with two or four superimposed foils long range thickness variations are minimised. This procedure lowers the influence on the final result of the thickness variations at long range from i 2.5% to f 1%. Values of the x: together with the precision with which they were measured, values of the ratios x”/x, divided by x:/xi and best estimated thicknesses x” are given in table 1. The thicknesses are expressed in pm assuming a 1.395 g.cm m3 density. From the above considerations, the precision on the x” for a given target and beam impact point was estimated to be better than *2.5% for all the targets. These experimental results are in fact independent of the density of Terphane. However, the thicknesses of the various initial foils were also mechanically measured. The thickness (in pm) so obtained agree within f 3% with the final x” values.
6
1.052
23.12
23.51
8 2x23.1 0.8 1.048
46.96
23.24 1.1 1.043 23.51
as: Sag [ln( G&G)
(3)
-Ll>
where S(E) is expressed in eV for 1OL5atoms per square centimeter, E is the proton energy in keV, I is the excitation energy in eV and where L stands for all the above mentioned correction terms. For our polyatomic solid targets (C,,H,O,), an expression equivalent to eq. (3) may be used if Bragg’s additivity rule is assumed. Setting N, as the number of atoms, Z’ as the atomic numbers, and Z, as the excitation energies of the various elements in our polyatomic targets, Z, may be replaced in eq. (3) by a mean atomic number Z, given by: cN,Z’,
g,cN,= I
I
which gives in our case Z, = 4.545, and replaced by a mean value 7 averaged as:
I may
be
(5) I
I
Similarly, one has for the mean value z of the correction term: 4. Stopping
power measurements
_Z,L~N,=~Z’N,L,, I
4.1. Theoretical survey
For non-relativistic protons (Z, = 1, velocity o), the stopping power S(E) in a monoatomic medium with atomic number Z, and N atoms per unit volume can be expressed as (Bethe formula, in ref. [6]): 4rrZ2e4
S(E)=----’
d
NZ2[lo+=)
-t].
(2)
where -e and m are respectively the charge and rest mass of the electron, I the target atoms mean excitation energy and C/Z, the so called inner shell correction term. The Bethe formula, based on the first order Born approximation, was improved by introducing correction terms proportional to Zf by Bloch, [7] and more recently, correction terms proportional to Z: (Ashley et al., [S], Lindhard, [9]). For our work, in which non-relativistic protons are considered, eq. (2) may be expressed
I
where L, is the correction term for the element indexed by i. Theoretical values of the Z, are only available for very low Z atoms and generally the values proposed in the literature are obtained by fitting the theory to experimental results, and hence generally including chemical binding effects. From ref. [6], one finds values of i in the 71.3 eV to 76.5 eV range, depending on the compounds containing H, C or 0 atoms considered. 4.2. Experiments The stopping power measurements and the precise determination of the target thicknesses X” were, as described in section 3, intimately connected. The experimental stopping power curve S,,,(E) for 0.3 to 2 MeV protons in Terphane was determined using the ap-
proximation: S(E)
= S( E, -a/2)
=d,‘x”,
(6) where Ei is the beam energy, A E the mean energy loss in the target of thickness xn and Ei - A E/2 the arithmetic mean energy of the beam in the target. Eq. (6) is an excellent approximation in the Bethe region, as can be seen in what follows. For a stopping power variation law S(E) = k/E (where k has a constant value) eq. (6) is rigorous as one has:
-
Mylar. Their experiments and calculations indicate a slight increase of A E( 8) with the angle of detection 8. For Mylar, very similar to our Terphane foils, this increase was below 1% even at relatively large 0 values. When averaging dE(B) over 0, one may conclude from their and our experiments that measuring AE( B = 0) will lead to an underestimation of A E and hence S(E) of less than 0.5%. This correction which is by all means small and was hence known with poor precision, has been neglected here.
-
and hence S(E, -AE/2)=AE/x”. For a stopping power S(E) following a k/E log (E/X) law (where k and X have constant values), which is nearly the case for MeV protons in low Z materials, deviations from eq. (6) were numerically calculated using the tabulated exponential integral function. For AE/Ei below 0.6 and for E, - AE 21 X, eq. (6) was found to overestimate E by less than 0.05% while in the extreme case A E/E, = 0.9, the overestimation rises to 2%. These results, vafid for decreasing stopping powers in the Bethe region, no longer apply in the Iower velocity region, where S(E) increases with E: when A E/E, is not small, corrections to eq. (6) have to be added. In these experiments, the energy losses A E were in the 100 keV to about 1 MeV range with electronics set for to 1.5 keV per channel. In order to take into account the slight asymmetry of the energy loss spectra (see section 5), the mean energy losses A2fi;were determined by considering the middle of the fwhm of the peaks. Typically, the fwhm of the peaks were of the order of 20 channels and the precision on the TE including the precision of the energy calibration was better than iOS%. Most of the experimental spectra (for this section) were recorded using target no. 1 (xn = 5.625 pm). At high energies however (E > 1 MeV), for better precision on the A E, thicker targets were also used. The mean energy losses dE measured in our experiments correspond to protons emerging from the targets at angles 0 near zero (the detection solid angle corresponds to 0 = 0 + 0.3 mrd). The stopping power S(E) [eq. (3)] is, however, defined with no conditions on the angular deflexions undergone by the particles. In order to evaluate to what extent our experimental procedure (very small solid angle) influences the measured A E, -the analysing detector was translated and variations - of AE with B were studied: no significant increase of A E with 6 was observed within our experimental precision ( & 0.5%) for various protons energics and various polyester targets. The maximum @values in this study corresponded to counting rates 10 to lo2 time smaller than at 0 = 0. A systematic study on this subject has been undertaken by Sakamoto et al. [lO,ll] for 7 MeV protons in various solid materials, in particular copper and
Our experimental results Se_,{ E) obtained from different targets (which are, by definition of the x”, selfconsistent on an average) were compared to S(E) given by eq. (3) with Z, = 2, = 4.545 and i = j = 73 eV. Setting S,,,(E) = S(E), values of the correction term L in eq. (3) were obtained as a function of E: they are plotted in fig. 1. L is found constant between - 0.8 MeV and 2 MeV (L = 0.294 f 0.015, where u, = 0.015 is one standard deviation). Below E - 0.7 MeV, L increases with E. Bars, corresponding to a change of t 1% of S(E) are also indicated in fig. 1: fluctuations arising from thickness inhomogeneities for a given target (various impact points), and uncertainties in the dE (determination of the position of the peaks on the energy spectra and energy calibration) appear to be always below 1.1% with a 0.5% mean square deviation. Finally, the stopping power S(E) of 0.3 to 2 MeV protons in C,,H,O, is given by (S(E) in eV for 10” atoms per cm2 and E in keV):
s,,,(E)=
(8)
~z[In(&j-‘.],
o.4t
0
.5
1.
1.5
2.
E CMeV 1 Fig. 1. Dimensionless correction term L [see eq. (8)] in the approximate Hethe stopping power formula S(E) for 0.3 to 2 MeV protons in polyester (C,,H,O,) obtained from energy toss measurements on polyester targets of various thicknesses: (+) 5,625 ym. (0) thicker targets (see table 1). Bars indicate the incidence on L of 5 1% deviations in the energy loss measurements. The solid line is an estimated fit to the experimental points; above 0.8 MeV a least square fit was used.
5
A. L’Hoir, D. Schmaus / Proton energy losses in polyester films
E -401.
> ; -
1
0
8
I
1
L
0.5
1
1.5
2
20*
I
E (Mel’)
Fig. 2. Experimental stopping power curve S(E) for 0.3 to 2 MeV protons in polyester (C,,H,O,): (0) experimental results; the solid line is obtained from eq. (S), with the variation of L given in fig. 1. Only the experimental results obtained with a 5.625 pm thick target are represented. Results of Andersen and Ziegler tabulations assuming Bragg’s additivity rule are also plotted (+).
where L can be directly obtained from the solid line in fig. 1. S(E) in keV/pm is obtained from eq. (8) by multiplying by 9.622. The precision on .S( E) depends directly on the precision on the thicknesses X” (*2.5%). Moreover. the measurements of the X” and of S(E) being completely interconnected, our conclusion is that the precision on .&,(E) [eq. (8)] is also f 2.5%. the shape of &r(E) curve being known with a better precision, of the order or less than 5 1%. (i.e. all the ratios S(E,)/S(E,‘), where E, and E: are arbitrary chosen between 0.3 and 2 MeV are known within + 1%). It must be pointed out that the particular values of 2, and j which enter into eq. (8) are not experimentally determined; consequently, the curve L(E) (fig. 1) does not represent a precise measure of the inner shell and Zfand Zf correction terms; however, if the averaging method leading to the 2, value is assumed to be valid, changing significantly the value of I in the logarithnli~ term may be exactly compensated by adding a small constant value to L(E) and in this assumption, the values of L(E) proposed may appear significant, re-
membering however that a & 1% error on SeXp(E) enters very enhanced on the precision of L(E) (see fig. 1). The experimental stopping power curve S,,,(E) for 0.3 MeV to 2 MeV protons in C,,H,O, is represented in fig. 2, where only experimental values obtained from target no. 1 are represented (the experimental results obtained from the other targets are completely in agreement with the results on target n = 1). Our results are compared in fig. 2 with tabulations of Andersen and Ziegler [12] assuming Bragg’s additivity rule and no departure from the C,0H804 stoichiometry: our results are systematically lower than theirs. A more precise comparison is given in table 2: our results are from - 2% at low energies, to 4% above 700 keV, lower than those given by their tabulation. An agreement in shape (which is independent of the x”) is then no completely achieved. Moreover, the - 4% discrepancy in the 0.7 to 2 MeV energy range is larger than our claimed experimental precision ( + 2.5%) and may be significant. In conclusion, when high precision is not necessary, the stopping power of organic compounds such as C,,H,O, (Terphane or Mylar) for 0.3 to 2 MeV protons may be calculated from Bragg’s additivity rule and from tabulations such as ref. [12]. However, as was already observed for MeV 4He+ ions in low Z compounds and in particular in oxides [13], a better agreement with experimental results is obtained by slightly lowering the tabulated stopping power of protons in gaseous oxygen. Moreover, the stopping power of carbon for protons may also depends on the compounds studied as was also observed in the case of helium ions 1141 and of protons at low energies [15] in hydrocarbon gases. In our case (0.3 to 2 MeV protons) deviations from Bragg’s additivity rule are, if any, very small; they may be however larger at lower energies. The stopping power curve S(E) [eq. (S)] is extensively used in the next section devoted to energy straggling measurements for small or large mean energy loss A E.
5. Energy
straggling
5. I. Theory
For an ideal target consisting of MAR randomly distributed scattering centers per unit area and at low
Table 2 E (MeV)
0.3
0.4
0.6
0.8
1.0
1.5
2.0
7.291
6.152
4.738
3.912
3.372
2.542
2.063
7.464 - 2.4
6.291 - 2.2
4.909 - 3.6
3.514 -4.2
2.635 - 3.7
2.150 -4.2
sc*p (F)“’ our results S’(E)“’ Andersen and Ziegler (.s/S’1)X 100 ai In eV for 10” atoms/cm2.
4.09 - 4.5
6
A. L’Hoir, D. Schmaus / Proton energy losses in polyester jllms
density .N”, the successive collisions suffered by an incident ion of energy E losing on average an energy A E small compared to E (AE = h’ARjTda(E, T) = hfARK,(E), where do(E, T) is the differential cross section for an energy transfer T) are independent and follow Poisson statistics. The variance of the energy loss spectra is then given by [16]: 02=JIrdRjT2da(E,
T)=XARK,(E).
(9)
When the number of significant collisions is large, the energy loss spectrum is Gaussian, with a variance 02. For light energetic particles such as protons with a velocity u large compared to the orbital velocities of the electrons of the target atoms, a2 is dominated by the collisions with the electrons of the target atoms (elastic collisions may be neglected) and moreover, do(E, T) can, to a good approximation, be calculated from the Coulomb interaction potential when we consider the target electrons as free and at rest. This leads to the famous simple expression [ 161;
/
Q2 = a; = NA RZ24nZ;e4, i.e. K,(E)
(IO) = K, = 4rZfe4
independent of E and where NAR is the number of target atoms per unit area. This simple relation was shown experimentally to be valid [17] for protons in gaseous neon (Z, = 10) provided that E 2 500 keV, while for E ,< 500 keV, one has Q2 < Qi. However, D2 is not necessarily lower than a;: more refined treatments (Fano [6], Chu [lg]) but which, however, still neglect charge state fluctuations and the bunching of electrons, i.e. multielectron excitations in the same atom predict at not too low velocities a slightly larger straggling than Dg for MeV protons in very light materials. In our case (polyatomic solid targets, constituted of light elements), if independence of successive collisions is assumed, the Bohr result [eq. (lo)] can easily be modified. a; = AR&rZfe4~
N, Z’ = 4nZ;e4NA
hypothesis of completely independent successive collisions is no longer valid: the energy transfer T for a given collision with energy E depends on the previous collisions via E. Calculation of the energy spectrum g( E, x)(where x is the depth) in such a situation, very similar in its numerical treatment aspect to the calculations of the range of an initially monoenergetic beam in matter [19], is generally performed using transport equations [20,21]. This technique leads to the determination of the moments of the energy spectrum and the latter is calculated from its moments via some hypothesis about its shape. Though already derived in the literature [20,21], it seems to us interesting to reformulate with a different approach the derivation of the basic equations leading to the calculation of the first moments of g( E, x). Let us consider the energy spectrum g(E, x) of a particle beam at depth x in a given medium. With the condition:
R&,
dEg(E,x)=l,
(12)
g(E, x) appears as a probability density. The problem is to determine how g( E, x) transforms when the depth increases from x to x + Sx. For this, let us consider (see fig. 3) the particles at depth x with a well defined
g(E,x)
-\ ___~________ ____-__ ‘/
dE
~
--__-----_z IE
T
,gfE,x)dE
I I ,
I
a(E;E,&x)j
(11)
where N, and the mean value Z2 have already been defined in section 4.1. In what follows, eq. (ll), based on the free electron model with Coulomb interaction potential between the protons and the target electrons, neglecting charge state fluctuations of the protons and energy straggling from elastic collisions, and assuming randomly distributed electrons (no bunching) and Poisson type statistics was chosen for comparison with our experimental results as a basis of comparison. When the mean energy loss AE ceases to be small compared to the incident energy Ei (i.e. for thick targets) the differential cross section do( E, T) where E = E, AE varies along the path of a given particle and the
E
E”
Fig. 3. Energy spectrum g(E, x) of a beam at depth x in a given medium (upper curve); the particles with a well defined energy between E and E +dE are represented by a delta function S(E) of mass g(E, x)dE. When travelling from x to x + 6x the delta function transforms in the lower solid curve; this transformation is described by a( E’, E, 6x). Both hatched areas are equal to g( E, .x)d E. The lower dashed line represents the energy spectrum of the beam g( E’, x f Sx) at depth x + 6.x. This spectrum is the sum of the contribution of all the partial energy spectra (lower solid line).
A. L’Horr,
D. Schmaus /
Proton energy losses
energy, between E and E + dE. The number of such particles is proportional to g( E, x)d E. Very similarly to the diffusion processes under a field, when they travel from x to x + Sx, the energy of these particles is spread out with a mean value shifted (see fig. 3) towards low energies. This process can be summarized as: g( E, x)dE6(
E) -g(
E, x)dEu(
E’, E, 6x),
(13)
where S(E) is the delta function located at E and where u( E’, E, Sx), the propagator, embodies all the information (shift and spread) for particles of an initially well defined energy E travelling a path length 8x in the medium considered *. a( E’, E, 6x) gives the shape and position of the broadened curve of fig. 3: the number of particles considered (proportjonal to g( E, x)d E) being constant between x and x + 6x. one has: I
a(E’,
E,Sx)dE’=l.
(14)
Integrating over all the energies spectrum at depth x + 6x
E, we obtain
the energy
g(E’,x+Sx)=f~(E,x)a(E’,E,Sx)dE.
05)
Eq. (15) can be considered as a transport equation. It must be pointed out that, at this step, no hypothesis have been made on the slowing down processes; for example, the number of collisions between x and x + 6x may be infinite (infinite cross sections) and a(E’, E, 8x) may not be Gaussian. We now assume 6x to be small i.e. a small mean energy loss between x and x + 8x. In this approximation the mean value of a(E’, E, Sx) can be calculated by using the stopping power value S(E): j
Eb(E’,E,Sx)dE’=E-S(E)Sx.
(16)
Here S( E)Sx represents the mean energy loss of particles of energy near E, between x and x + 8x. Calculation of the mean energy E(x) at depth x can now be performed. By use of the definition of E and of eqs. (15) and (16) one has: E(x
+6x)
= /g(
=
f
E, x)dEjE’a(
g( E, x)[E
in polyester
I
films
which for 8x = 0 gives: E(x)
=jEg(E,
x)dE.
(18)
Subtracting (18) from (17), dividing 6x tend towards zero, were find dE(x),‘dx=
by 6x and letting
+(E)g(E,x)dE.
(19)
We now make the assumption of fully independent collisions between x and x+&x (this is completely justified as 6x is assumed to be small), and we assume a finite variance for a( E’, E, Sx). Necessarily, the variance of a( E’, E, x) has the following form: u,‘(E,
6x) = u’f E)Sx,
(20)
i.e. u, of the energy beam E(x).
is proportional to Sx. Here, a’(E) is the variance energy loss per unit path length for particles of E. Calculation of the variance fZ2(x) of the at depth x is very similar to the calculation of One has:
E2(x
+ Sx)=jdEg(E.x)
jdErE%(Er,
with E2(x+Sx)=f22(x+Sx)+(~(x+Sx))2
(22)
and ~=~~2~(E,x)d~=~2(x)+(~(~))~.
By definition /
(23)
and using eqs. (20) and (16) one can write
a(E’,E,Sx)E’2dE’=o’(E)Sx+[E-S(E)Sx]2. (24)
Subtracting (23) from (21), dividing by Sx, letting 8.x tend towards zero and using eq. (24) we obtain the implicit equation dti’(.x)/dx
=/cr2(
E)g(
E, x)dE
(25)
E’, E, 8x)dE’
- S( E)Sx]dE,
E, 6x),(21)
(17)
Successive collisions are assumed fully independent in a given slice of matter 6.x (this hypothesis, which lead to eq. (9), is only strictly valid in a dilute gas). With this assumption. successive collisions are independent from slice to slice, and the departure from eq. (9) at large x arises only through the energy E of the particles in do(E, T): energy loss is a Markovian process.
It must be pointed out that eqs. (19) and (25) from which B(x) and Q’(x) can be calculated were derived withour using Taylor expansion series and with minimum assumptions. An easy calculation of E(x) and Q2(x) can be carried out from eqs. (19) and (25) by approximate methods. This consists of expanding the stopping power S(E) and the energy straggling per unit path length u*(E) in Taylor series around the mean energy value E(x): S(E)=S(E(x))+
E ‘E-~~x)~~~(~(x)), ,I-=1
(26)
8
A. L’Hoir, D. Schmaus / Proton ener~):losses in po!wster films
cJ’(E)=o*(E(x))+
a:(
yxqg(qx))
f
x = 0) =
0)
a$(6(x))=s’(6(x))j-F-Ip’(E)dE/s’(r)
n = I
(34)
(27) Setting p,(x) for the nth central moment of g( E, x) (p, = 0, p2 = a*(x)), one finds from eq. (19) for the mean value E(x):
(28) Similarly,
from eq. (25) one finds
One can see from eqs. (28) and (29) that an exact calculation of E(x) and of U2(x) implies in fact the calculation of all the moments of g( E, x) and a perfect knowledge of the variations of S and a2 with E. However it has been shown [21] that approximating eq. (28) by -dE(x)/dx=S(E(x))
(30) in the case of not too large energy losses (A E/E, 5 0.7), i.e. not too large moments p(x) (n 2 2) and in the Bethe region, overestimates E(x) by less than 0.1%. The zero order approximation in eq. (29) yields dS2*(x),‘dx
= u2(E(x)),
(31)
which essentially predicts a thickness proportional variance. Contrary to eq. (30) eq. (31) is only valid for small energy losses. This explains the fact that expanding eq. (29) to the first order yields a good but not an excellent approximation. Let us first notice that in eq. (29) o*(E) varies very slowly with E for MeV protons in light materials (in the free electron model approximation, u2 is constant): neglecting (d”o*/dE”)~,, (n 2 2) in eq. (29) therefore appears as a fairly good approximation. In fact, deviations from the thickness proportional variances essentially arise from the variations of S(E) with E. The first order approximation in eq. (29) (i.e. -p3(d2S/dE2)(E(x)) and higher order neglecting terms) gives the following differential equation (32) All the factors in eq. (32) can be expressed as a function of _??by inserting the approximation (30); leading to (33) which
has the solution
(with the condition
s2$( E,) =
a result already derived in ref. [22] with the notation So+)
=K:(E(x))~~:)K*(E)dE/Kf(E)
(35)
completely identical to eq. (34): eq. (34) transforms into eq. (35) when choosing for the unit length, one electron per unit area (with our definition of K,). Expanding S(E) and u2( E) to higher order moments would relate a’(x) to higher order central moments of g( E, x) in particular to p3 which describes its asymmetry: eq. (34) is hence valid only if the skewness y3 = p3/ti3 is small, i.e. if the energy spectrum is nearly Gaussian. From ref. [21] and for mean relative energy losses AE/E, in the 0.1 to 0.7 range, which is the case in our experiments. the skewness y3 is negative and below 0.2 in our energy domain. According to ref. [21] for such low skewnesses the approximate solution 9r given by eq. (34) gives the variance of the energy spectra with good precision. In the next section, our experimental results on’ the energy straggling of 0.5 to 2 MeV protons for 0.05 5 A E/E, 5 0.8 are compared to eq. (34) where S(E) is our experimental stopping power SeXP(E) (section 4.3) and where for a*(E) the Bohr result [equ. (ll)] was used. In our calculation of 52,, the unit length chosen was lOI atoms per cm2; this choice gives (eq. (11)): o*(E) = CT*= 1.184 eV* for lOI atoms per cm*. In the present experiments (MeV protons, i.e. in the Bethe velocity domain), s2$ increases faster than the target thickness x, contrary to the usual result (9) (3* proportional to x), valid for small x. This behaviour may easily be understood. Qualitatively, particles at a given depth x, with energies E < E(x) (low energy tail of g( E, x)), i.e. with S(E) > S( E( x)), lose on average, more energy than particles with E > E(x). g( E, x) broadens therefore faster than predicted by eq. (9). On the other hand, for smaller velocities, when S(E) is an increasing function of E (typically, helium ions below 0.5 MeV), g( E. x) broadens more slowly than predicted by (9) for large x. 5.2. Experimentul
results and discussion
5.2.1. Energy spectra Two typical energy spectra recorded as described in sect. 2, are represented in fig. 4. They were obtained respectively with a 1.7 MeV proton beam transmitted through a 23.5 pm thick target (target - no. 8 of table 1) with a relative mean energy loss AE/E, = 0.351 and with a 1.21 MeV proton beam transmitted through a 21.66 pm thick target (target no. 3 of table 1) with A E/Ei = 0.681. These two spectra are not fully symmetrical, with a more pronounced asymmetry (tail towards
A. L’Horr,
610
570
D. Schmaus / Profon ener~)? losses in
650
CHANNELS I
I
I
I
I
I
b
0 100
180
1LO
CHANNELS Fig. 4. Energy spectra of MeV proton beams transmitted through polyester (C,,H,O,) targets. The solid line is the best estimate of an overall fit to these spectra by the convolution product of a Gaussian with a one sided-exponential factor [eq. (36)]; the asymmetry of these convolution products is described by the dimensionless ratio u,/~~. a) 1.72 MeV protons transmitted through a 23.5 pm thick target, with a mean relative energy loss AE/E, = 0.35 and L,,, = 47.2 keV (fwhm). The solid line corresponds to uOo/10 = 2.0 (1.35 keV per channel). b) 1.21 MeV protons transmitted through a 21.66 pm thick target (four superimposed foils), with AE/E, = 0.68 and L,,, = 65.9 keV (fwhm); o,,o/~,, = 1.4 (2.10 keV per channel).
polyesterfilms
9
when u~O/TJ~= 2 and 6.8% smaller when u,,/~~ = 1.4 i.e. when the asymmetry is more pronounced. For the two curves of fig. 4, a satisfactory fit was obtained (see solid lines in fig. 4) with uO/q,, = 2 (upper curve) and u,,O/~~= 1.4 (lower curve), yielding respectively y3 = -0.18 and yj = -0.39. These yX estimated values are less than two times larger than the theoretical one [21]; on this basis, and considering that for all our experimental spectra, the ratio u,,O/~~was always larger than 1.4 (i.e asymmetries were relatively small), eq. (34) was considered to give a good theoretical estimate of the variance of the energy loss spectra. The influence of the surface roughness of our targets on the widths of the energy spectra was estimated as follows. The widths L,,, of spectra recorded with a - 24 pm thick target (target no. 8 of table 1) and with a relatively small mean energy loss A E/E, (this is the case for the first spectrum of fig. 4) were compared to the widths L’,,2 of energy spectra recorded with the same A E/E, and with a - 4 X 6 pm thick target (target no. 3 of table 1 made up of four superimposed foils). Assuming a Gaussian shape, the variances Q* and Q” of these spectra were determined (D = L,,,/2.355) and compared to the Bohr results [eq. (ll)] s2, and &. The ratio U/D’, was systematically larger than Q/Q, by 2 to 3%, for E, of the order of 1.5 MeV (Q’/Q’.. = (1 + e)D/s2,): these discrepancies may totally be attributed to the fact that for the 4 x 6 pm target, the protons energy spread is influenced by eight crossings through foil surfaces as compared to two for a target constituted of a single foil. Let us assume the independence of successive collisions (AE/E, small), and that the roughness is the same for all the targets and that the variance S2f of the energy spread arising from the surface roughness for a proton passing through two surfaces has a constant value (Qf depends mainly on the stopping power S(E): for small A E/E,, the energy is assumed to be nearly constant and hence also S(E)). Calling s2:, the true variance of the energy spread (i.e. if there is no surface roughness) one may write
low energies) for the second one. Estimation of the skewness of these experimental curves have been obtained by fitting them to the convolution product h(E) of a Gaussian by a one sided exponential factor: h(E)=A
exp(-E2/20,Z)
(37)
One necessarily has Q:Js2: = Q:f/@ see sect. 5.2.2.). One may also write:
* [exp(-E/qo)Y(-E)],
(same A EE,,
(36) where A is a constant and Y the Heaviside unit step function. The variance of h(E) is tii = 0,’ + 11: and its third order central moment is pX = - 2173,;the skewness of h(E) is hence yj = -2/(1 + u,‘/$,)‘/‘. This type of curves for which the dimensionless quantity u,,/Q describes the asymmetry has already been used in ref. [23]. The full width at half maximum Ll,2 of h(E) is smaller than 2.35552, (valid for Gaussian curves): 2.4% smaller
a’2/s2’,2 = (1 + +*S22/S2;
= (1+ 2c)P/O2,
(38)
with l from our experiments around 1.5 MeV of the order of 2.5 X 10e2 (c depends on the energy E;). From the above two equations, one finds (s2: -=z Q2): s2; = -f&2,
(39)
where Q2 is the variance of the energy spread corresponding to a single = 24 pm thick target. In conclu-
10
A. L’Hoir,
D. Schmaus / Proton energy losses in
sion, for a single = 24 pm thick target, and for Ei = 1.5 MeV, the widths L,,2 of the recorded energy spectra are broadened by about 0.8% (with c = 2.5 X lo-*) by the surface roughness; for a 4 x 6 pm thick target (four superimposed 6 pm thick foils) or for a single 6 pm thick target the broadening is about 3.3% in the same energy range. These broadening effects vary with the energy range considered via the stopping power S(E); they are roughly proportional to S(E), i.e. they are larger at low energies than at high energies. The experimental energy resolution also contributes to the broadening of the recorded energy spectra, especially at low target thicknesses and small mean relative energy losses AE/E. In our experiments the energy resolution for protons which is dominated by the electronic noise was nearly Gaussian and its full width at half maximum (fwhm) was about 7 keV, measured when analysing directly the energy of the beam. For our thinnest targets (5.62 ,um) and at high energy (A E/E, -C 0.1) i.e. for the smaller energy straggling, the measured fwhm of the energy spectra were of the order of 21 to 22 keV, i.e. markedly larger than the energy resolution. The latter was however always taken into account in the data reduction. (From the above considerations, cooling the detector was not of prime importance.) 5.2.2. Energy straggling Energy spread curves for eight different targets (see table 1) and for various proton beam energies were recorded. Their fwhm were measured with precision (typical counts per channel at maximum were of the order of lo4 and fwhm of the order of 20 to 35 channels). The variances of the energy loss spectra were calculated assuming a Gaussian shape and were corrected for the energy resolution according to section 5.2.1. this may underestimate D by about 2% to 7% according to the amount of asymmetry of the spectrum considered. However this procedure appeared to us more consistent for a comparison with eq. (34), which is valid for nearly Gaussian shapes. The experimental values a divided by Da [eq. [ll]) are plotted in fig. 5 as a function of the mean relative energy loss AE/E,, for eight different targets with thicknesses varying from 5.6 pm to 47 pm and for A E/E; in the range 0.05 to 0.8. For each experimental Q/Q,, a theoretical value 0, was calculated from eq. (34) with o*(E) given by eq. (ll), i.e. independent of the energy (Bohr model) and S(E) obtained from our experimental results (eq. 8). It happens that for E, > 700 keV and for all the A E/E, considered, the theoretical ratios D,/O, follow a nearly single curve as a function of A E/Ei within f 1% (solid line in fig. 5). The dashed line in fig. 5 corresponds to Ei < 700 keV and to experiments performed with the 5.63 pm thick target. This result (i.e. only two theoretical curves 0,/O, as a function of A E/E, for comparison with our experimental results) may appear as for-
polyester films
.
3.63
q
23.6
.
23.7
0
47.0
rm
Fig. 5. Experimental standard deviation, Q, corrected for the energy resolution of the energy spectra of 0.5 to 2 MeV proton beams transmitted through eight different polyester (C,OH,O,) targets (see table l), normalized to the Bohr result Q, ((eq. (11)). D was obtained from ti = L,,,/2.355, i.e. by assuming Gaussian spectra. The ratios Q/Q, are plotted as a function of the mean relative energy loss AE/E, of the beams, and are compared to the theory Q,/O, [eq. (34)], valid for not too large AE/E,: the dashed line corresponds to experiments pertarget (o), with beam energies E, from 2 E, corresponding large a/E,; the solid line (superimposed on the calculated dashed line at small A E/E,) corresponds to experiments performed on the seven other targets with Ei between 2 MeV and 1.2 MeV. formed
on the thinnest
MeV to 0.5 MeV, to small
tuitous: in fact, for stopping power S(E) varying as (k/E) In E/h where X has a constant value, it may be shown that s2,/0, is a function of AE/E, and also of El/X. However, this latter dependence is weak when E, is very large as compared to h. Moreover, the (k/E) In E/h law is not strictly valid: experimentally the term L is found to decrease below - 800 keV. This implies that A is not strictly constant but decreases at low energies. Above - 800 KeV, eq. (8), where L is found constant (L = 0.294), may be reduced to: &,,(E)=pln$, keV
(40)
where h = 44.96 keV = 33.51 keV x exp (0.294). For E, = 2 MeV and A E/Ei = 0.5 one finds Q,/D, = 1.36, while for Ei = 1.5 MeV and AE/E, = 0.5 one finds D,/D, = 1.34, which indicates a slight dependence of 52,/Q, on E/h when Ei > A. This explains the nearly single law (the solid line in fig. 5) s2,/s2, = f(dE/E,) at high energies. The fact that h decreases
A. L’Hoir, D. Schmaus / Proton energy losses in polyester films
11
6. Conclusion
ing in the range 0.2 to 0.4. The influence on g( E, x) of the surface roughness of the targets is shown to be small in our measurements. The widths of the spectra increase with the thickness x of the targets more rapidly than x’j2, especially at large x (i.e. large A E/E,), indicating that successive collisions suffered by the protons are not independent. This deviation from the x1j2 law can be theoretically predicted, using a transport equation. For the theoretical calculation, a free electron model was assumed for large energy transfers T to target electrons and classical mechanics was used (/T2do(E, 7’) = 4nZfe4, independent of E). For small energy transfers T, where both free electron model and classical mechanics fail, a detailed analysis of do( E, T) can be avoided in the large collision number approximation: da( E, T) at low T enters in the calculations only through /Tda( E, T), i.e. through the stopping power S(E), which was measured experimentlly in the 0.3 to 2 MeV range. Our experimental S(E) for protons in C,,H,O, is smaller than but very near (2 to 4%) to S(E) obtained from tabulations of Andersen and Ziegler [12] assuming Bragg’s additivity rule. When comparing to the Bethe theory in this approximation, a correction term L (including shell corrections and Z:, Zf terms) can be extracted which is constant (L = 0.29) between 0.8 MeV and 2 MeV and which increases with E below 0.7 MeV. These results indicate a very slight influence of chemical bindings on S(E) for MeV protons in the organic compound considered (E > 0.3 MeV). The theoretical calculation of the widths of the energy spectra g( E, x) at a given depth x is particularly simple if Gaussian shapes are assumed and if higher order moments are neglected in the calculation of the variance O2 of g( E, x). Our experimental results were compared to theory in this approximation. An excellent agreement is found if contribution to the widths of g( E, x) of the surface roughness of the targets (here of the order of 1 to 5%) is removed from the experimental results: our experimental D are only 2.5% larger than the calculated 9, for mean relative energy losses A E/E, in the 0.05 to 0.8 range, indicating that for a precise prediction of the fwhm of energy loss spectra at large A E/Ei very simple theories apply. In addition to these results on the comparison between theories and experimental results, our measurements of the energy loss spectra of MeV protons in polyester may be of use in Nuclear Microanalysis where in a great number of cases, absorbers are used to avoid pile-up problems [2]: Terphane (or Mylar) is particularly well adapted for this purpose.
The energy spectra g( E, x) of 0.3 to 2 MeV protons transmitted through 5.6 to 47 pm thick polyester films (C,,H,O,) with mean relative energy loss A E/E, in the 0.05 to 0.8 range are slightly asymmetrical, with a tail towards low energies, the skewness of the spectra vary-
The authors would like thank J. Chainey from Rhome Poulenc for providing the Terphane polyester films and for mechanical thickness measurements. Fruitful discussions with G. Amsel and C. Cohen were also greatly appreciated.
when E, decreases (below - 800 keV) obviously attenuates the influence of the dependence of L?,/s2, on E/h. This may explain the quasi-scaling (dashed line) found at lower energies. It may be pointed out that when calculating the ratios tit/L”,, the actual thicknesses of the targets cancel. In effect, the experimental S(E) is inversely proportional to the measured thicknesses (table 1) of the eight targets used: multiplying all the targets thicknesses by a constant factor (Yone finds a new value of S(E), S’(E) = S( E)/a, leading from eq. (34) to a new value of tit, L$ = aJ2$. Similarly, tii transforms as f&i = ,yL?i and hence a$,&“, is independent of a: only the shape of the stopping power S(E) is influencing the ratio Q,/O,; in particular for a constant S(E), one has 9, = 9,. It follows that the precision of the calculation of Or/O, is excellent, depending only on the shape of S(E) determined experimentally with a *l% precision. Our experimental results s2/s2, are from 0 to 7% larger than values calculated from eq. (34) with u2( E) independent of E( free electron model). The experimental results obtained with the 5.63 pm thick target and the 4 X 5.415 = 21.66 pm thick target, for which the surface roughness effect is maximum give as expected, larger straggling values than for the other targets. After correction for the estimated surface roughness effect, the experimental results are - 2.5% above the theoretical results as averaged over all the experimental results, with a + 2.5% spread. Though very small, this discrepancy may be significant (large number of experimental results, with eight different targets), in particular at small A E/E, where Or/O, is very near to 1 and the approximation leading to eq. (34) cannot be doubted. The larger experimental straggling may partially be attributed to an underestimation of the surface roughness of the targets and partly to effects unaccounted for, as more refined theories predict slightly lower or higher energy loss straggling than Qi according to the protons energies and the target Z value; these corrections are however small especially for E 2 1 MeV and low Z materials. For a better comparison with the theory, chemical bindings and density effects might also be taken into account. From these considerations it did not appear necessary to us to further refine the calculation of Or/Q,: eq. (34) which neglects the asymmetry of the energy spectra g( E, x) appears to describe with precision the width of g( E, x) in the Gaussian approximation up to A E/E, 5 0.8.
12
A. L’Hoir, D. Schmaus / Proton energy losses m polyester films
References [l] D. Schmaus and A. L’Hoir, Nucl. Instr. and Meth. B (1984) in press. See also: D. Schmaus and A. L’Hoir, Nucl. Instr. and Meth. B2 (1984) 156 (Proc. 10th Int. Conf. Atomic collisions in solids). [2] G. Amsel, J.P. Nadai, E. d’Artemare, D. David, E. Girard an J. Moulin, Nucl. Instr. and Meth. 92 (1971) 481. [3] P.M. Endt and C. Van der Leun, Nucl. Phys. A 214 (1973) 1. [4] D. Schmaus and A. L’Hoir, Nucl. Instr. and Meth. 194 (1982) 75. [5] D. Schmaus, A. L’Hoir and C. Cohen, Nucl. Instr. and Meth. 194 (1982) 81. [6] U. Fano, Ann. Rev. Nucl. Sci. 13 (1963) 1. [7] F. Bloch, Ann. Phys. (Leipz.) 16 (1933) 285. (81 J.C. Ashley, R.H. Ritchie and W. Brandt, Phys. Rev. B 5 (1972) 2393. [9] J. Lindhard, Nucl. Instr. and Meth. 132 (1976) 1. [lo] N. Sakamoto, N. Shiomi and R. Ishiwari, Phys. Rev. A 27 (1983) 810. [ll] R. Ishiwari, N. Shiomi and N. Sakamoto, Phys. Rev. A 25 (1982) 2524. 1121 H.H. Andersen and J.F. Ziegler, Hydrogen stopping
[13]
[14] [15] [16] [17] [18] (191 [20] [21] [22] [23]
powers and ranges in all elements (Pergamon, New York, 1977). A. L’Hoir, C. Cohen, and G. Amsel, in Ion Beam Surface Layer Analysis, eds., 0. Meyer, G. Linker and F. Kappeler (Plenum, New York, 1976) p. 965. S. Matteson, E.K.L. Chau and D. Powers, Phys. Rev. A 14 (1976) 169. J.T. Park and E.J. Zimmerman, Phys. Rev. 131 (1963) 1611. N. Bohr, Mat. Fys. Medd. Dan. Vid. Selsk (1948) no. 8. F. Besenbacher, H.H. Andersen, P. Hvelplund and H. Knudsen, Mat. Fys. Medd. Dan. Vid. Selsk. (1981) no .9. W.K. Chu, Phys. Rev. A 13 (1976) 2057. J. Lindhard, M. Sharff and H.E. Schiott, Mat. Fys. Medd. Dan. Vid. Selsk. 33 (1963) no. 14. K.R. Symon, Thesis (Harvard University, Cambridge, Mass., 1948). C. Tschalti, Nucl. Instr. and Meth. 61 (1968) 141. C. Tschalar and H.D. Maccabee, Phys. Rev. B 1 (1970) 2863. G. Amsel, C. Cohen and A. L’Hoir, in Ion Beam Surface Layer Analysis, eds., 0. Meyer, G. Linker and F. Kappeler (Plenum, New York, 1976) p. 953.
in Physics
Research
84 (1984) 1-12
STOPPING POWER AND ENERGY STRAGGLING FOR SMALL AND LARGE ENERGY LOSSES OF MeV PROTONS TRANSMI’I-IED THROUGH POLYESTER FILMS * A. L’HOIR
and D. SCHMAUS
Groupe de Physique des Solides de I’Eeole Normale Received
4 October
SupC;ritwre, Uniuersiti
Pans 7, i’ pluce Jussieu,
75251 Paris Cedex 05, Frunce
1983
The stopping power S(E) and the energy straggling Q’(E,, AE) of 0.5 MeV to 2 MeV proton beams in polyester (C,,HsO,) have been measured in transmission geometry. S(E) is found to be 2% to 4% lower than the Andersen and Ziegler tabulations, assuming the Bragg’s additivity rule. The energy straggling a’( E,, TQ was measured at various beam energies E,, mean energy losses m and target thicknesses (from 5.6 pm to 47 pm). The ratio fJ/s2,, (where 0, refers to the free electron model in the small energy loss approximation) of the order of 1.07 for small energy losses (d/E, = 0.05), increases to about 1.5 to 2 for large energy losses (AE/E, = 0.75). For not too small beam energies (E, 2 700 keV) and for our targets. a nearly universal curve O(Ei, AT)/O, is obtained as a function of the mean relative energy loss d-/E,. For d-/E, 6 0.7, the beam energy spread curves which were nearly energy loss spectra. Our experimental results 0 Gaussian, were compared to theoretical predictions 9,, valid for nearly symmetrical are from 0 to 7% larger than the theoretical values B, up to A E/E, = 0.8. The surface roughness of the targets was checked and its influence on the experimental results was generally between 1% and 5%, depending on the targets and on E,.
1. introduction This paper is the first part of a work devoted to the experimental study of the interaction of proton beams with polyester films. In the present paper, experimental results on the- energy loss spectra (in particular the mean energy loss AE and the energy straggling 0’) of the beams transmitted through targets perpendicular (B = 0 ’ ) to the beam direction are presented and compared to the theory. In ref. [I] (the second part of the work), the lateral spread of the beam in the solid targets is determined from a careful analysis of transmission energy loss spectra through highly tilted targets (8 f 0”). For both studies, it was important to minimize unwanted effects such as surface roughness, porosity, thickness inhomogeneity or texture of the targets. From this point of view, rather thick organic targets represent a favourable case. The polyester films (C,,H,O,) studied in the present work are used in our group as absorbers in nuclear microanalysis experiments [2] to stop unwanted backscattered light particles such as protons, deuterons or alphas to avoid pile-up problems. As will be shown in the present paper, these amorphous films are very uniform, of high stability, and the influence of the surface roughness on the energy spread of ion beams * Work supported
by the Centre National de la Recherche Scientifique (R.C.P. no. 157) and presented at the 10th International Conference on Atomic Collisions in Solids, Bad Iburg (Germany), July 1983.
0168-583X/84/$03.00 Q Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V
transmitted through these films (generally in the 6 to 24 pm thickness range) is small and can be evaluated with reasonable precision. In our energy domain (0.5 to 2 MeV) and with the target thicknesses considered, the mean relative energy losses a/Ei were between 0.05 and 0.8. Our study extends therefore from small energy losses, which is the domain generally used for energy straggling measurements, to relatively high energy losses. In the latter case, where interaction cross sections vary along the path of the particles, overall Poisson statistics no longer apply and the usual theories have to be revised. This can be simply performed for not too high energy losses [see eq. (34)], provided that the stopping power S(E) in the medium is known. A precise determination of the stopping power of protons in polyester was therefore necessary. The results on S(E) are given in section 4. Section 5 is devoted to the energy straggling measurements; in this section, a derivation of the energy straggling calcu__ lation when A.5 -c E, no longer applies is also presented, and the influence of the surface roughness of the targets is discussed. It must be pointed out that the present paper and the lateral spread study [I] are strongly linked. When a target is perpendicular to the beam (B = 0°) the influence of the lateral spread of the ions on the energy loss spectra is negligibly small (the case of the present paper); when the target is tilted (8 f 0 ’ ), the lateral spread increases the energy spread, especially at large 8. The lateral spread was therefore indirectly determined
2
A. L’Hoir, D. Schmaus / Proton energy losses in polyester films
by comparing spectra recorded at 0 = 0’ (presented work) with spectra recorded with tilted targets (for more details of the procedure, see ref. [l]).
2. Experimental set up Monoenergetic proton beams were produced by the 2 MV Van de Graaff accelerator of the Ecole Normale Superieure. The accelerator was calibrated using the numerous narrow resonances of the Al(p, y) nuclear reaction [3]. Protons transmitted through the targets were analysed at an angle of 0’ using a 25 mm2 area silicon surface barrier detector from Ortec, located at 111 cm from the target and provided with a 0.6 mm diameter diaphragm. Two 0.2 mm diameter nickel diaphragms located in the accelerator pipe at 74 cm from each other ensured a sufficiently low beam current in the vacuum chamber with no significant angular or energy spread. For the type of experiments considered here, counting rates of the order of lo3 per second could easily be achieved. Targets and detector could be translated perpendicularly to the beam directions using stepping motors. More details about the experimental set up are given in ref. [4]. In these experiments a model 142A charge sensitive preamplifier, a model 572 linear amplifier and a model 444 gated biased amplifier from Ortec were used. The amplitude of the pulses were analyzed by means of a 8192 channels ADC from Laben and energy spectra were recorded using an Informatek online computer system. Calibration of the electronics was performed by detecting directly the beam particles, the energy of which was well known. A very small variation of the keV/channel value with energy was found and taken into account in the data reduction. The uncertainty in the keV per channel values was estimated to be _tOS%, considering the very high stability of the electronic set-up.
3. The targets The polyester foils used in our experiments had the stoichiometry C,,H,O,. They were obtained from Rhone-Poulenc (trade mark Terphane), without aluminum coverage (Terphane is very similar to Mylar from DuPont). Six different rectangular foils of about 6, 12 and 24 pm thicknesses were used, with areas of about 500 cm2. The mean thicknesses xi (j = 1 to 6) of by weighing, these foils (in g cmm2) was determined using a Mettler balance, with a lo-’ g uncertainty (the mass of the foils being of the order of 1 g) and by measuring their areas. In most cases, the precision on the xi was of the order of +0.5%, i.e. much better than for example using the RBS technique in the case of thin films. Pieces of these sheets were mounted on eight
27 X 27 mm steel frames; some of the eight targets so obtained (indexed by n) were constituted of two or four superimposed foils. For precise, absolute stopping power measurements, a determination of the target thickness homogeneity on a macroscopic scale is necessary while for energy straggling measurements, a precise knowledge of the uniformity of the targets on a microscopic scale (surface roughness) is required. The thickness uniformity of the targets and of the initial sheets was therefore hence evaluated on four different scales: i) on the scale of the beam diameter (0.2 mm), ii) on the scale of the size of a target (- 1 cm), iii) on the scale of the dimensions of the initial sheets (- 10 cm), iv) on a microscopic scale (surface roughness); this point is discussed in section 5. For the first two cases, the very sensitive method described in ref. [5], using as a tool angular multiple scattering of the beams transmitted through the targets was chosen. An overall thickness variation below + 0.5% was observed for one of the targets (target no. 3, see table l), when translating the latter by & 5 mm, using a stepping motor with 0.125 mm steps, the beam diameter being 0.2 mm. A comparison of the energy loss measurements on our various targets indicates that this conclusion is valid for all the targets (see section 4.3). The third point happened to be more involved; the non-uniformity on the large scale (- 10 cm) of the weighed Terphane sheets was evaluated from energy loss measurements and taken into account in the following way. Assuming the results of the weighing and area measurements to be valid, we took the thicknesses x: of the targets (n = 1 to 8) to be XL or in some cases two or four times these values. Taking one of the targets as a reference (n = 1), we determined a stopping power curve S,(E) from energy loss measurements on that target at various beam energies E,:S,(E) does in fact depend on x:, but only the shape of the stopping power curve S,(E) is of interest at this point. Energy loss measurements were also performed on the seven other targets at various beam energies. Letting x” be the actual thicknesses of the targets, these energy loss measurements give the ratios x”/x’ with a F 1% precision including the uncertainties on the mean energy loss determinations (in such relative measurements, errors on the energy calibration nearly cancel) and thickness variations for a given target (see above). Calculations of these ratios were performed by numerical integration of E,
JE,-AE_dE/‘S,(E), -
where E, is the beam energy and AE the mean energy loss in a given target n: eq. (1) gives a thickness proportional to x”. When comparing the ratios x:/.x: and x”/x’ a maximum of 5.2% discrepancy appears, which gives an order of magnitude ( f 2.5%) of the thickness
A. L’Hotr,
3
D. Schmaus / Proton energy losses in polyester films
Table 1 2
3
4
n
1
x: (pm) Precision (5%)
5.8 0.6
2 x 6.03 1.5
4 x 5.45 0.6
X”/XI
1
1.023
1.026
C/d xn (w)
5.625
11.96
21.66
5
11.94 0.4
2x11.94 0.4
1.032
I
23.1 0.8
1.024
11.95
variations of the sheets on the large scale (- 10 cm). The absolute values of the xR were finally determined from the known ratios x”/x’ by setting (x”) = (x:), where the average is taken over the eight targets, with weighting factors taking into account the precision on the x: (hence on the XL) and the fact that for targets with two or four superimposed foils long range thickness variations are minimised. This procedure lowers the influence on the final result of the thickness variations at long range from i 2.5% to f 1%. Values of the x: together with the precision with which they were measured, values of the ratios x”/x, divided by x:/xi and best estimated thicknesses x” are given in table 1. The thicknesses are expressed in pm assuming a 1.395 g.cm m3 density. From the above considerations, the precision on the x” for a given target and beam impact point was estimated to be better than *2.5% for all the targets. These experimental results are in fact independent of the density of Terphane. However, the thicknesses of the various initial foils were also mechanically measured. The thickness (in pm) so obtained agree within f 3% with the final x” values.
6
1.052
23.12
23.51
8 2x23.1 0.8 1.048
46.96
23.24 1.1 1.043 23.51
as: Sag [ln( G&G)
(3)
-Ll>
where S(E) is expressed in eV for 1OL5atoms per square centimeter, E is the proton energy in keV, I is the excitation energy in eV and where L stands for all the above mentioned correction terms. For our polyatomic solid targets (C,,H,O,), an expression equivalent to eq. (3) may be used if Bragg’s additivity rule is assumed. Setting N, as the number of atoms, Z’ as the atomic numbers, and Z, as the excitation energies of the various elements in our polyatomic targets, Z, may be replaced in eq. (3) by a mean atomic number Z, given by: cN,Z’,
g,cN,= I
I
which gives in our case Z, = 4.545, and replaced by a mean value 7 averaged as:
I may
be
(5) I
I
Similarly, one has for the mean value z of the correction term: 4. Stopping
power measurements
_Z,L~N,=~Z’N,L,, I
4.1. Theoretical survey
For non-relativistic protons (Z, = 1, velocity o), the stopping power S(E) in a monoatomic medium with atomic number Z, and N atoms per unit volume can be expressed as (Bethe formula, in ref. [6]): 4rrZ2e4
S(E)=----’
d
NZ2[lo+=)
-t].
(2)
where -e and m are respectively the charge and rest mass of the electron, I the target atoms mean excitation energy and C/Z, the so called inner shell correction term. The Bethe formula, based on the first order Born approximation, was improved by introducing correction terms proportional to Zf by Bloch, [7] and more recently, correction terms proportional to Z: (Ashley et al., [S], Lindhard, [9]). For our work, in which non-relativistic protons are considered, eq. (2) may be expressed
I
where L, is the correction term for the element indexed by i. Theoretical values of the Z, are only available for very low Z atoms and generally the values proposed in the literature are obtained by fitting the theory to experimental results, and hence generally including chemical binding effects. From ref. [6], one finds values of i in the 71.3 eV to 76.5 eV range, depending on the compounds containing H, C or 0 atoms considered. 4.2. Experiments The stopping power measurements and the precise determination of the target thicknesses X” were, as described in section 3, intimately connected. The experimental stopping power curve S,,,(E) for 0.3 to 2 MeV protons in Terphane was determined using the ap-
proximation: S(E)
= S( E, -a/2)
=d,‘x”,
(6) where Ei is the beam energy, A E the mean energy loss in the target of thickness xn and Ei - A E/2 the arithmetic mean energy of the beam in the target. Eq. (6) is an excellent approximation in the Bethe region, as can be seen in what follows. For a stopping power variation law S(E) = k/E (where k has a constant value) eq. (6) is rigorous as one has:
-
Mylar. Their experiments and calculations indicate a slight increase of A E( 8) with the angle of detection 8. For Mylar, very similar to our Terphane foils, this increase was below 1% even at relatively large 0 values. When averaging dE(B) over 0, one may conclude from their and our experiments that measuring AE( B = 0) will lead to an underestimation of A E and hence S(E) of less than 0.5%. This correction which is by all means small and was hence known with poor precision, has been neglected here.
-
and hence S(E, -AE/2)=AE/x”. For a stopping power S(E) following a k/E log (E/X) law (where k and X have constant values), which is nearly the case for MeV protons in low Z materials, deviations from eq. (6) were numerically calculated using the tabulated exponential integral function. For AE/Ei below 0.6 and for E, - AE 21 X, eq. (6) was found to overestimate E by less than 0.05% while in the extreme case A E/E, = 0.9, the overestimation rises to 2%. These results, vafid for decreasing stopping powers in the Bethe region, no longer apply in the Iower velocity region, where S(E) increases with E: when A E/E, is not small, corrections to eq. (6) have to be added. In these experiments, the energy losses A E were in the 100 keV to about 1 MeV range with electronics set for to 1.5 keV per channel. In order to take into account the slight asymmetry of the energy loss spectra (see section 5), the mean energy losses A2fi;were determined by considering the middle of the fwhm of the peaks. Typically, the fwhm of the peaks were of the order of 20 channels and the precision on the TE including the precision of the energy calibration was better than iOS%. Most of the experimental spectra (for this section) were recorded using target no. 1 (xn = 5.625 pm). At high energies however (E > 1 MeV), for better precision on the A E, thicker targets were also used. The mean energy losses dE measured in our experiments correspond to protons emerging from the targets at angles 0 near zero (the detection solid angle corresponds to 0 = 0 + 0.3 mrd). The stopping power S(E) [eq. (3)] is, however, defined with no conditions on the angular deflexions undergone by the particles. In order to evaluate to what extent our experimental procedure (very small solid angle) influences the measured A E, -the analysing detector was translated and variations - of AE with B were studied: no significant increase of A E with 6 was observed within our experimental precision ( & 0.5%) for various protons energics and various polyester targets. The maximum @values in this study corresponded to counting rates 10 to lo2 time smaller than at 0 = 0. A systematic study on this subject has been undertaken by Sakamoto et al. [lO,ll] for 7 MeV protons in various solid materials, in particular copper and
Our experimental results Se_,{ E) obtained from different targets (which are, by definition of the x”, selfconsistent on an average) were compared to S(E) given by eq. (3) with Z, = 2, = 4.545 and i = j = 73 eV. Setting S,,,(E) = S(E), values of the correction term L in eq. (3) were obtained as a function of E: they are plotted in fig. 1. L is found constant between - 0.8 MeV and 2 MeV (L = 0.294 f 0.015, where u, = 0.015 is one standard deviation). Below E - 0.7 MeV, L increases with E. Bars, corresponding to a change of t 1% of S(E) are also indicated in fig. 1: fluctuations arising from thickness inhomogeneities for a given target (various impact points), and uncertainties in the dE (determination of the position of the peaks on the energy spectra and energy calibration) appear to be always below 1.1% with a 0.5% mean square deviation. Finally, the stopping power S(E) of 0.3 to 2 MeV protons in C,,H,O, is given by (S(E) in eV for 10” atoms per cm2 and E in keV):
s,,,(E)=
(8)
~z[In(&j-‘.],
o.4t
0
.5
1.
1.5
2.
E CMeV 1 Fig. 1. Dimensionless correction term L [see eq. (8)] in the approximate Hethe stopping power formula S(E) for 0.3 to 2 MeV protons in polyester (C,,H,O,) obtained from energy toss measurements on polyester targets of various thicknesses: (+) 5,625 ym. (0) thicker targets (see table 1). Bars indicate the incidence on L of 5 1% deviations in the energy loss measurements. The solid line is an estimated fit to the experimental points; above 0.8 MeV a least square fit was used.
5
A. L’Hoir, D. Schmaus / Proton energy losses in polyester films
E -401.
> ; -
1
0
8
I
1
L
0.5
1
1.5
2
20*
I
E (Mel’)
Fig. 2. Experimental stopping power curve S(E) for 0.3 to 2 MeV protons in polyester (C,,H,O,): (0) experimental results; the solid line is obtained from eq. (S), with the variation of L given in fig. 1. Only the experimental results obtained with a 5.625 pm thick target are represented. Results of Andersen and Ziegler tabulations assuming Bragg’s additivity rule are also plotted (+).
where L can be directly obtained from the solid line in fig. 1. S(E) in keV/pm is obtained from eq. (8) by multiplying by 9.622. The precision on .S( E) depends directly on the precision on the thicknesses X” (*2.5%). Moreover. the measurements of the X” and of S(E) being completely interconnected, our conclusion is that the precision on .&,(E) [eq. (8)] is also f 2.5%. the shape of &r(E) curve being known with a better precision, of the order or less than 5 1%. (i.e. all the ratios S(E,)/S(E,‘), where E, and E: are arbitrary chosen between 0.3 and 2 MeV are known within + 1%). It must be pointed out that the particular values of 2, and j which enter into eq. (8) are not experimentally determined; consequently, the curve L(E) (fig. 1) does not represent a precise measure of the inner shell and Zfand Zf correction terms; however, if the averaging method leading to the 2, value is assumed to be valid, changing significantly the value of I in the logarithnli~ term may be exactly compensated by adding a small constant value to L(E) and in this assumption, the values of L(E) proposed may appear significant, re-
membering however that a & 1% error on SeXp(E) enters very enhanced on the precision of L(E) (see fig. 1). The experimental stopping power curve S,,,(E) for 0.3 MeV to 2 MeV protons in C,,H,O, is represented in fig. 2, where only experimental values obtained from target no. 1 are represented (the experimental results obtained from the other targets are completely in agreement with the results on target n = 1). Our results are compared in fig. 2 with tabulations of Andersen and Ziegler [12] assuming Bragg’s additivity rule and no departure from the C,0H804 stoichiometry: our results are systematically lower than theirs. A more precise comparison is given in table 2: our results are from - 2% at low energies, to 4% above 700 keV, lower than those given by their tabulation. An agreement in shape (which is independent of the x”) is then no completely achieved. Moreover, the - 4% discrepancy in the 0.7 to 2 MeV energy range is larger than our claimed experimental precision ( + 2.5%) and may be significant. In conclusion, when high precision is not necessary, the stopping power of organic compounds such as C,,H,O, (Terphane or Mylar) for 0.3 to 2 MeV protons may be calculated from Bragg’s additivity rule and from tabulations such as ref. [12]. However, as was already observed for MeV 4He+ ions in low Z compounds and in particular in oxides [13], a better agreement with experimental results is obtained by slightly lowering the tabulated stopping power of protons in gaseous oxygen. Moreover, the stopping power of carbon for protons may also depends on the compounds studied as was also observed in the case of helium ions 1141 and of protons at low energies [15] in hydrocarbon gases. In our case (0.3 to 2 MeV protons) deviations from Bragg’s additivity rule are, if any, very small; they may be however larger at lower energies. The stopping power curve S(E) [eq. (S)] is extensively used in the next section devoted to energy straggling measurements for small or large mean energy loss A E.
5. Energy
straggling
5. I. Theory
For an ideal target consisting of MAR randomly distributed scattering centers per unit area and at low
Table 2 E (MeV)
0.3
0.4
0.6
0.8
1.0
1.5
2.0
7.291
6.152
4.738
3.912
3.372
2.542
2.063
7.464 - 2.4
6.291 - 2.2
4.909 - 3.6
3.514 -4.2
2.635 - 3.7
2.150 -4.2
sc*p (F)“’ our results S’(E)“’ Andersen and Ziegler (.s/S’1)X 100 ai In eV for 10” atoms/cm2.
4.09 - 4.5
6
A. L’Hoir, D. Schmaus / Proton energy losses in polyester jllms
density .N”, the successive collisions suffered by an incident ion of energy E losing on average an energy A E small compared to E (AE = h’ARjTda(E, T) = hfARK,(E), where do(E, T) is the differential cross section for an energy transfer T) are independent and follow Poisson statistics. The variance of the energy loss spectra is then given by [16]: 02=JIrdRjT2da(E,
T)=XARK,(E).
(9)
When the number of significant collisions is large, the energy loss spectrum is Gaussian, with a variance 02. For light energetic particles such as protons with a velocity u large compared to the orbital velocities of the electrons of the target atoms, a2 is dominated by the collisions with the electrons of the target atoms (elastic collisions may be neglected) and moreover, do(E, T) can, to a good approximation, be calculated from the Coulomb interaction potential when we consider the target electrons as free and at rest. This leads to the famous simple expression [ 161;
/
Q2 = a; = NA RZ24nZ;e4, i.e. K,(E)
(IO) = K, = 4rZfe4
independent of E and where NAR is the number of target atoms per unit area. This simple relation was shown experimentally to be valid [17] for protons in gaseous neon (Z, = 10) provided that E 2 500 keV, while for E ,< 500 keV, one has Q2 < Qi. However, D2 is not necessarily lower than a;: more refined treatments (Fano [6], Chu [lg]) but which, however, still neglect charge state fluctuations and the bunching of electrons, i.e. multielectron excitations in the same atom predict at not too low velocities a slightly larger straggling than Dg for MeV protons in very light materials. In our case (polyatomic solid targets, constituted of light elements), if independence of successive collisions is assumed, the Bohr result [eq. (lo)] can easily be modified. a; = AR&rZfe4~
N, Z’ = 4nZ;e4NA
hypothesis of completely independent successive collisions is no longer valid: the energy transfer T for a given collision with energy E depends on the previous collisions via E. Calculation of the energy spectrum g( E, x)(where x is the depth) in such a situation, very similar in its numerical treatment aspect to the calculations of the range of an initially monoenergetic beam in matter [19], is generally performed using transport equations [20,21]. This technique leads to the determination of the moments of the energy spectrum and the latter is calculated from its moments via some hypothesis about its shape. Though already derived in the literature [20,21], it seems to us interesting to reformulate with a different approach the derivation of the basic equations leading to the calculation of the first moments of g( E, x). Let us consider the energy spectrum g(E, x) of a particle beam at depth x in a given medium. With the condition:
R&,
dEg(E,x)=l,
(12)
g(E, x) appears as a probability density. The problem is to determine how g( E, x) transforms when the depth increases from x to x + Sx. For this, let us consider (see fig. 3) the particles at depth x with a well defined
g(E,x)
-\ ___~________ ____-__ ‘/
dE
~
--__-----_z IE
T
,gfE,x)dE
I I ,
I
a(E;E,&x)j
(11)
where N, and the mean value Z2 have already been defined in section 4.1. In what follows, eq. (ll), based on the free electron model with Coulomb interaction potential between the protons and the target electrons, neglecting charge state fluctuations of the protons and energy straggling from elastic collisions, and assuming randomly distributed electrons (no bunching) and Poisson type statistics was chosen for comparison with our experimental results as a basis of comparison. When the mean energy loss AE ceases to be small compared to the incident energy Ei (i.e. for thick targets) the differential cross section do( E, T) where E = E, AE varies along the path of a given particle and the
E
E”
Fig. 3. Energy spectrum g(E, x) of a beam at depth x in a given medium (upper curve); the particles with a well defined energy between E and E +dE are represented by a delta function S(E) of mass g(E, x)dE. When travelling from x to x + 6x the delta function transforms in the lower solid curve; this transformation is described by a( E’, E, 6x). Both hatched areas are equal to g( E, .x)d E. The lower dashed line represents the energy spectrum of the beam g( E’, x f Sx) at depth x + 6.x. This spectrum is the sum of the contribution of all the partial energy spectra (lower solid line).
A. L’Horr,
D. Schmaus /
Proton energy losses
energy, between E and E + dE. The number of such particles is proportional to g( E, x)d E. Very similarly to the diffusion processes under a field, when they travel from x to x + Sx, the energy of these particles is spread out with a mean value shifted (see fig. 3) towards low energies. This process can be summarized as: g( E, x)dE6(
E) -g(
E, x)dEu(
E’, E, 6x),
(13)
where S(E) is the delta function located at E and where u( E’, E, Sx), the propagator, embodies all the information (shift and spread) for particles of an initially well defined energy E travelling a path length 8x in the medium considered *. a( E’, E, 6x) gives the shape and position of the broadened curve of fig. 3: the number of particles considered (proportjonal to g( E, x)d E) being constant between x and x + 6x. one has: I
a(E’,
E,Sx)dE’=l.
(14)
Integrating over all the energies spectrum at depth x + 6x
E, we obtain
the energy
g(E’,x+Sx)=f~(E,x)a(E’,E,Sx)dE.
05)
Eq. (15) can be considered as a transport equation. It must be pointed out that, at this step, no hypothesis have been made on the slowing down processes; for example, the number of collisions between x and x + 6x may be infinite (infinite cross sections) and a(E’, E, 8x) may not be Gaussian. We now assume 6x to be small i.e. a small mean energy loss between x and x + 8x. In this approximation the mean value of a(E’, E, Sx) can be calculated by using the stopping power value S(E): j
Eb(E’,E,Sx)dE’=E-S(E)Sx.
(16)
Here S( E)Sx represents the mean energy loss of particles of energy near E, between x and x + 8x. Calculation of the mean energy E(x) at depth x can now be performed. By use of the definition of E and of eqs. (15) and (16) one has: E(x
+6x)
= /g(
=
f
E, x)dEjE’a(
g( E, x)[E
in polyester
I
films
which for 8x = 0 gives: E(x)
=jEg(E,
x)dE.
(18)
Subtracting (18) from (17), dividing 6x tend towards zero, were find dE(x),‘dx=
by 6x and letting
+(E)g(E,x)dE.
(19)
We now make the assumption of fully independent collisions between x and x+&x (this is completely justified as 6x is assumed to be small), and we assume a finite variance for a( E’, E, Sx). Necessarily, the variance of a( E’, E, x) has the following form: u,‘(E,
6x) = u’f E)Sx,
(20)
i.e. u, of the energy beam E(x).
is proportional to Sx. Here, a’(E) is the variance energy loss per unit path length for particles of E. Calculation of the variance fZ2(x) of the at depth x is very similar to the calculation of One has:
E2(x
+ Sx)=jdEg(E.x)
jdErE%(Er,
with E2(x+Sx)=f22(x+Sx)+(~(x+Sx))2
(22)
and ~=~~2~(E,x)d~=~2(x)+(~(~))~.
By definition /
(23)
and using eqs. (20) and (16) one can write
a(E’,E,Sx)E’2dE’=o’(E)Sx+[E-S(E)Sx]2. (24)
Subtracting (23) from (21), dividing by Sx, letting 8.x tend towards zero and using eq. (24) we obtain the implicit equation dti’(.x)/dx
=/cr2(
E)g(
E, x)dE
(25)
E’, E, 8x)dE’
- S( E)Sx]dE,
E, 6x),(21)
(17)
Successive collisions are assumed fully independent in a given slice of matter 6.x (this hypothesis, which lead to eq. (9), is only strictly valid in a dilute gas). With this assumption. successive collisions are independent from slice to slice, and the departure from eq. (9) at large x arises only through the energy E of the particles in do(E, T): energy loss is a Markovian process.
It must be pointed out that eqs. (19) and (25) from which B(x) and Q’(x) can be calculated were derived withour using Taylor expansion series and with minimum assumptions. An easy calculation of E(x) and Q2(x) can be carried out from eqs. (19) and (25) by approximate methods. This consists of expanding the stopping power S(E) and the energy straggling per unit path length u*(E) in Taylor series around the mean energy value E(x): S(E)=S(E(x))+
E ‘E-~~x)~~~(~(x)), ,I-=1
(26)
8
A. L’Hoir, D. Schmaus / Proton ener~):losses in po!wster films
cJ’(E)=o*(E(x))+
a:(
yxqg(qx))
f
x = 0) =
0)
a$(6(x))=s’(6(x))j-F-Ip’(E)dE/s’(r)
n = I
(34)
(27) Setting p,(x) for the nth central moment of g( E, x) (p, = 0, p2 = a*(x)), one finds from eq. (19) for the mean value E(x):
(28) Similarly,
from eq. (25) one finds
One can see from eqs. (28) and (29) that an exact calculation of E(x) and of U2(x) implies in fact the calculation of all the moments of g( E, x) and a perfect knowledge of the variations of S and a2 with E. However it has been shown [21] that approximating eq. (28) by -dE(x)/dx=S(E(x))
(30) in the case of not too large energy losses (A E/E, 5 0.7), i.e. not too large moments p(x) (n 2 2) and in the Bethe region, overestimates E(x) by less than 0.1%. The zero order approximation in eq. (29) yields dS2*(x),‘dx
= u2(E(x)),
(31)
which essentially predicts a thickness proportional variance. Contrary to eq. (30) eq. (31) is only valid for small energy losses. This explains the fact that expanding eq. (29) to the first order yields a good but not an excellent approximation. Let us first notice that in eq. (29) o*(E) varies very slowly with E for MeV protons in light materials (in the free electron model approximation, u2 is constant): neglecting (d”o*/dE”)~,, (n 2 2) in eq. (29) therefore appears as a fairly good approximation. In fact, deviations from the thickness proportional variances essentially arise from the variations of S(E) with E. The first order approximation in eq. (29) (i.e. -p3(d2S/dE2)(E(x)) and higher order neglecting terms) gives the following differential equation (32) All the factors in eq. (32) can be expressed as a function of _??by inserting the approximation (30); leading to (33) which
has the solution
(with the condition
s2$( E,) =
a result already derived in ref. [22] with the notation So+)
=K:(E(x))~~:)K*(E)dE/Kf(E)
(35)
completely identical to eq. (34): eq. (34) transforms into eq. (35) when choosing for the unit length, one electron per unit area (with our definition of K,). Expanding S(E) and u2( E) to higher order moments would relate a’(x) to higher order central moments of g( E, x) in particular to p3 which describes its asymmetry: eq. (34) is hence valid only if the skewness y3 = p3/ti3 is small, i.e. if the energy spectrum is nearly Gaussian. From ref. [21] and for mean relative energy losses AE/E, in the 0.1 to 0.7 range, which is the case in our experiments. the skewness y3 is negative and below 0.2 in our energy domain. According to ref. [21] for such low skewnesses the approximate solution 9r given by eq. (34) gives the variance of the energy spectra with good precision. In the next section, our experimental results on’ the energy straggling of 0.5 to 2 MeV protons for 0.05 5 A E/E, 5 0.8 are compared to eq. (34) where S(E) is our experimental stopping power SeXP(E) (section 4.3) and where for a*(E) the Bohr result [equ. (ll)] was used. In our calculation of 52,, the unit length chosen was lOI atoms per cm2; this choice gives (eq. (11)): o*(E) = CT*= 1.184 eV* for lOI atoms per cm*. In the present experiments (MeV protons, i.e. in the Bethe velocity domain), s2$ increases faster than the target thickness x, contrary to the usual result (9) (3* proportional to x), valid for small x. This behaviour may easily be understood. Qualitatively, particles at a given depth x, with energies E < E(x) (low energy tail of g( E, x)), i.e. with S(E) > S( E( x)), lose on average, more energy than particles with E > E(x). g( E, x) broadens therefore faster than predicted by eq. (9). On the other hand, for smaller velocities, when S(E) is an increasing function of E (typically, helium ions below 0.5 MeV), g( E. x) broadens more slowly than predicted by (9) for large x. 5.2. Experimentul
results and discussion
5.2.1. Energy spectra Two typical energy spectra recorded as described in sect. 2, are represented in fig. 4. They were obtained respectively with a 1.7 MeV proton beam transmitted through a 23.5 pm thick target (target - no. 8 of table 1) with a relative mean energy loss AE/E, = 0.351 and with a 1.21 MeV proton beam transmitted through a 21.66 pm thick target (target no. 3 of table 1) with A E/Ei = 0.681. These two spectra are not fully symmetrical, with a more pronounced asymmetry (tail towards
A. L’Horr,
610
570
D. Schmaus / Profon ener~)? losses in
650
CHANNELS I
I
I
I
I
I
b
0 100
180
1LO
CHANNELS Fig. 4. Energy spectra of MeV proton beams transmitted through polyester (C,,H,O,) targets. The solid line is the best estimate of an overall fit to these spectra by the convolution product of a Gaussian with a one sided-exponential factor [eq. (36)]; the asymmetry of these convolution products is described by the dimensionless ratio u,/~~. a) 1.72 MeV protons transmitted through a 23.5 pm thick target, with a mean relative energy loss AE/E, = 0.35 and L,,, = 47.2 keV (fwhm). The solid line corresponds to uOo/10 = 2.0 (1.35 keV per channel). b) 1.21 MeV protons transmitted through a 21.66 pm thick target (four superimposed foils), with AE/E, = 0.68 and L,,, = 65.9 keV (fwhm); o,,o/~,, = 1.4 (2.10 keV per channel).
polyesterfilms
9
when u~O/TJ~= 2 and 6.8% smaller when u,,/~~ = 1.4 i.e. when the asymmetry is more pronounced. For the two curves of fig. 4, a satisfactory fit was obtained (see solid lines in fig. 4) with uO/q,, = 2 (upper curve) and u,,O/~~= 1.4 (lower curve), yielding respectively y3 = -0.18 and yj = -0.39. These yX estimated values are less than two times larger than the theoretical one [21]; on this basis, and considering that for all our experimental spectra, the ratio u,,O/~~was always larger than 1.4 (i.e asymmetries were relatively small), eq. (34) was considered to give a good theoretical estimate of the variance of the energy loss spectra. The influence of the surface roughness of our targets on the widths of the energy spectra was estimated as follows. The widths L,,, of spectra recorded with a - 24 pm thick target (target no. 8 of table 1) and with a relatively small mean energy loss A E/E, (this is the case for the first spectrum of fig. 4) were compared to the widths L’,,2 of energy spectra recorded with the same A E/E, and with a - 4 X 6 pm thick target (target no. 3 of table 1 made up of four superimposed foils). Assuming a Gaussian shape, the variances Q* and Q” of these spectra were determined (D = L,,,/2.355) and compared to the Bohr results [eq. (ll)] s2, and &. The ratio U/D’, was systematically larger than Q/Q, by 2 to 3%, for E, of the order of 1.5 MeV (Q’/Q’.. = (1 + e)D/s2,): these discrepancies may totally be attributed to the fact that for the 4 x 6 pm target, the protons energy spread is influenced by eight crossings through foil surfaces as compared to two for a target constituted of a single foil. Let us assume the independence of successive collisions (AE/E, small), and that the roughness is the same for all the targets and that the variance S2f of the energy spread arising from the surface roughness for a proton passing through two surfaces has a constant value (Qf depends mainly on the stopping power S(E): for small A E/E,, the energy is assumed to be nearly constant and hence also S(E)). Calling s2:, the true variance of the energy spread (i.e. if there is no surface roughness) one may write
low energies) for the second one. Estimation of the skewness of these experimental curves have been obtained by fitting them to the convolution product h(E) of a Gaussian by a one sided exponential factor: h(E)=A
exp(-E2/20,Z)
(37)
One necessarily has Q:Js2: = Q:f/@ see sect. 5.2.2.). One may also write:
* [exp(-E/qo)Y(-E)],
(same A EE,,
(36) where A is a constant and Y the Heaviside unit step function. The variance of h(E) is tii = 0,’ + 11: and its third order central moment is pX = - 2173,;the skewness of h(E) is hence yj = -2/(1 + u,‘/$,)‘/‘. This type of curves for which the dimensionless quantity u,,/Q describes the asymmetry has already been used in ref. [23]. The full width at half maximum Ll,2 of h(E) is smaller than 2.35552, (valid for Gaussian curves): 2.4% smaller
a’2/s2’,2 = (1 + +*S22/S2;
= (1+ 2c)P/O2,
(38)
with l from our experiments around 1.5 MeV of the order of 2.5 X 10e2 (c depends on the energy E;). From the above two equations, one finds (s2: -=z Q2): s2; = -f&2,
(39)
where Q2 is the variance of the energy spread corresponding to a single = 24 pm thick target. In conclu-
10
A. L’Hoir,
D. Schmaus / Proton energy losses in
sion, for a single = 24 pm thick target, and for Ei = 1.5 MeV, the widths L,,2 of the recorded energy spectra are broadened by about 0.8% (with c = 2.5 X lo-*) by the surface roughness; for a 4 x 6 pm thick target (four superimposed 6 pm thick foils) or for a single 6 pm thick target the broadening is about 3.3% in the same energy range. These broadening effects vary with the energy range considered via the stopping power S(E); they are roughly proportional to S(E), i.e. they are larger at low energies than at high energies. The experimental energy resolution also contributes to the broadening of the recorded energy spectra, especially at low target thicknesses and small mean relative energy losses AE/E. In our experiments the energy resolution for protons which is dominated by the electronic noise was nearly Gaussian and its full width at half maximum (fwhm) was about 7 keV, measured when analysing directly the energy of the beam. For our thinnest targets (5.62 ,um) and at high energy (A E/E, -C 0.1) i.e. for the smaller energy straggling, the measured fwhm of the energy spectra were of the order of 21 to 22 keV, i.e. markedly larger than the energy resolution. The latter was however always taken into account in the data reduction. (From the above considerations, cooling the detector was not of prime importance.) 5.2.2. Energy straggling Energy spread curves for eight different targets (see table 1) and for various proton beam energies were recorded. Their fwhm were measured with precision (typical counts per channel at maximum were of the order of lo4 and fwhm of the order of 20 to 35 channels). The variances of the energy loss spectra were calculated assuming a Gaussian shape and were corrected for the energy resolution according to section 5.2.1. this may underestimate D by about 2% to 7% according to the amount of asymmetry of the spectrum considered. However this procedure appeared to us more consistent for a comparison with eq. (34), which is valid for nearly Gaussian shapes. The experimental values a divided by Da [eq. [ll]) are plotted in fig. 5 as a function of the mean relative energy loss AE/E,, for eight different targets with thicknesses varying from 5.6 pm to 47 pm and for A E/E; in the range 0.05 to 0.8. For each experimental Q/Q,, a theoretical value 0, was calculated from eq. (34) with o*(E) given by eq. (ll), i.e. independent of the energy (Bohr model) and S(E) obtained from our experimental results (eq. 8). It happens that for E, > 700 keV and for all the A E/E, considered, the theoretical ratios D,/O, follow a nearly single curve as a function of A E/Ei within f 1% (solid line in fig. 5). The dashed line in fig. 5 corresponds to Ei < 700 keV and to experiments performed with the 5.63 pm thick target. This result (i.e. only two theoretical curves 0,/O, as a function of A E/E, for comparison with our experimental results) may appear as for-
polyester films
.
3.63
q
23.6
.
23.7
0
47.0
rm
Fig. 5. Experimental standard deviation, Q, corrected for the energy resolution of the energy spectra of 0.5 to 2 MeV proton beams transmitted through eight different polyester (C,OH,O,) targets (see table l), normalized to the Bohr result Q, ((eq. (11)). D was obtained from ti = L,,,/2.355, i.e. by assuming Gaussian spectra. The ratios Q/Q, are plotted as a function of the mean relative energy loss AE/E, of the beams, and are compared to the theory Q,/O, [eq. (34)], valid for not too large AE/E,: the dashed line corresponds to experiments pertarget (o), with beam energies E, from 2 E, corresponding large a/E,; the solid line (superimposed on the calculated dashed line at small A E/E,) corresponds to experiments performed on the seven other targets with Ei between 2 MeV and 1.2 MeV. formed
on the thinnest
MeV to 0.5 MeV, to small
tuitous: in fact, for stopping power S(E) varying as (k/E) In E/h where X has a constant value, it may be shown that s2,/0, is a function of AE/E, and also of El/X. However, this latter dependence is weak when E, is very large as compared to h. Moreover, the (k/E) In E/h law is not strictly valid: experimentally the term L is found to decrease below - 800 keV. This implies that A is not strictly constant but decreases at low energies. Above - 800 KeV, eq. (8), where L is found constant (L = 0.294), may be reduced to: &,,(E)=pln$, keV
(40)
where h = 44.96 keV = 33.51 keV x exp (0.294). For E, = 2 MeV and A E/Ei = 0.5 one finds Q,/D, = 1.36, while for Ei = 1.5 MeV and AE/E, = 0.5 one finds D,/D, = 1.34, which indicates a slight dependence of 52,/Q, on E/h when Ei > A. This explains the nearly single law (the solid line in fig. 5) s2,/s2, = f(dE/E,) at high energies. The fact that h decreases
A. L’Hoir, D. Schmaus / Proton energy losses in polyester films
11
6. Conclusion
ing in the range 0.2 to 0.4. The influence on g( E, x) of the surface roughness of the targets is shown to be small in our measurements. The widths of the spectra increase with the thickness x of the targets more rapidly than x’j2, especially at large x (i.e. large A E/E,), indicating that successive collisions suffered by the protons are not independent. This deviation from the x1j2 law can be theoretically predicted, using a transport equation. For the theoretical calculation, a free electron model was assumed for large energy transfers T to target electrons and classical mechanics was used (/T2do(E, 7’) = 4nZfe4, independent of E). For small energy transfers T, where both free electron model and classical mechanics fail, a detailed analysis of do( E, T) can be avoided in the large collision number approximation: da( E, T) at low T enters in the calculations only through /Tda( E, T), i.e. through the stopping power S(E), which was measured experimentlly in the 0.3 to 2 MeV range. Our experimental S(E) for protons in C,,H,O, is smaller than but very near (2 to 4%) to S(E) obtained from tabulations of Andersen and Ziegler [12] assuming Bragg’s additivity rule. When comparing to the Bethe theory in this approximation, a correction term L (including shell corrections and Z:, Zf terms) can be extracted which is constant (L = 0.29) between 0.8 MeV and 2 MeV and which increases with E below 0.7 MeV. These results indicate a very slight influence of chemical bindings on S(E) for MeV protons in the organic compound considered (E > 0.3 MeV). The theoretical calculation of the widths of the energy spectra g( E, x) at a given depth x is particularly simple if Gaussian shapes are assumed and if higher order moments are neglected in the calculation of the variance O2 of g( E, x). Our experimental results were compared to theory in this approximation. An excellent agreement is found if contribution to the widths of g( E, x) of the surface roughness of the targets (here of the order of 1 to 5%) is removed from the experimental results: our experimental D are only 2.5% larger than the calculated 9, for mean relative energy losses A E/E, in the 0.05 to 0.8 range, indicating that for a precise prediction of the fwhm of energy loss spectra at large A E/Ei very simple theories apply. In addition to these results on the comparison between theories and experimental results, our measurements of the energy loss spectra of MeV protons in polyester may be of use in Nuclear Microanalysis where in a great number of cases, absorbers are used to avoid pile-up problems [2]: Terphane (or Mylar) is particularly well adapted for this purpose.
The energy spectra g( E, x) of 0.3 to 2 MeV protons transmitted through 5.6 to 47 pm thick polyester films (C,,H,O,) with mean relative energy loss A E/E, in the 0.05 to 0.8 range are slightly asymmetrical, with a tail towards low energies, the skewness of the spectra vary-
The authors would like thank J. Chainey from Rhome Poulenc for providing the Terphane polyester films and for mechanical thickness measurements. Fruitful discussions with G. Amsel and C. Cohen were also greatly appreciated.
when E, decreases (below - 800 keV) obviously attenuates the influence of the dependence of L?,/s2, on E/h. This may explain the quasi-scaling (dashed line) found at lower energies. It may be pointed out that when calculating the ratios tit/L”,, the actual thicknesses of the targets cancel. In effect, the experimental S(E) is inversely proportional to the measured thicknesses (table 1) of the eight targets used: multiplying all the targets thicknesses by a constant factor (Yone finds a new value of S(E), S’(E) = S( E)/a, leading from eq. (34) to a new value of tit, L$ = aJ2$. Similarly, tii transforms as f&i = ,yL?i and hence a$,&“, is independent of a: only the shape of the stopping power S(E) is influencing the ratio Q,/O,; in particular for a constant S(E), one has 9, = 9,. It follows that the precision of the calculation of Or/O, is excellent, depending only on the shape of S(E) determined experimentally with a *l% precision. Our experimental results s2/s2, are from 0 to 7% larger than values calculated from eq. (34) with u2( E) independent of E( free electron model). The experimental results obtained with the 5.63 pm thick target and the 4 X 5.415 = 21.66 pm thick target, for which the surface roughness effect is maximum give as expected, larger straggling values than for the other targets. After correction for the estimated surface roughness effect, the experimental results are - 2.5% above the theoretical results as averaged over all the experimental results, with a + 2.5% spread. Though very small, this discrepancy may be significant (large number of experimental results, with eight different targets), in particular at small A E/E, where Or/O, is very near to 1 and the approximation leading to eq. (34) cannot be doubted. The larger experimental straggling may partially be attributed to an underestimation of the surface roughness of the targets and partly to effects unaccounted for, as more refined theories predict slightly lower or higher energy loss straggling than Qi according to the protons energies and the target Z value; these corrections are however small especially for E 2 1 MeV and low Z materials. For a better comparison with the theory, chemical bindings and density effects might also be taken into account. From these considerations it did not appear necessary to us to further refine the calculation of Or/Q,: eq. (34) which neglects the asymmetry of the energy spectra g( E, x) appears to describe with precision the width of g( E, x) in the Gaussian approximation up to A E/E, 5 0.8.
12
A. L’Hoir, D. Schmaus / Proton energy losses m polyester films
References [l] D. Schmaus and A. L’Hoir, Nucl. Instr. and Meth. B (1984) in press. See also: D. Schmaus and A. L’Hoir, Nucl. Instr. and Meth. B2 (1984) 156 (Proc. 10th Int. Conf. Atomic collisions in solids). [2] G. Amsel, J.P. Nadai, E. d’Artemare, D. David, E. Girard an J. Moulin, Nucl. Instr. and Meth. 92 (1971) 481. [3] P.M. Endt and C. Van der Leun, Nucl. Phys. A 214 (1973) 1. [4] D. Schmaus and A. L’Hoir, Nucl. Instr. and Meth. 194 (1982) 75. [5] D. Schmaus, A. L’Hoir and C. Cohen, Nucl. Instr. and Meth. 194 (1982) 81. [6] U. Fano, Ann. Rev. Nucl. Sci. 13 (1963) 1. [7] F. Bloch, Ann. Phys. (Leipz.) 16 (1933) 285. (81 J.C. Ashley, R.H. Ritchie and W. Brandt, Phys. Rev. B 5 (1972) 2393. [9] J. Lindhard, Nucl. Instr. and Meth. 132 (1976) 1. [lo] N. Sakamoto, N. Shiomi and R. Ishiwari, Phys. Rev. A 27 (1983) 810. [ll] R. Ishiwari, N. Shiomi and N. Sakamoto, Phys. Rev. A 25 (1982) 2524. 1121 H.H. Andersen and J.F. Ziegler, Hydrogen stopping
[13]
[14] [15] [16] [17] [18] (191 [20] [21] [22] [23]
powers and ranges in all elements (Pergamon, New York, 1977). A. L’Hoir, C. Cohen, and G. Amsel, in Ion Beam Surface Layer Analysis, eds., 0. Meyer, G. Linker and F. Kappeler (Plenum, New York, 1976) p. 965. S. Matteson, E.K.L. Chau and D. Powers, Phys. Rev. A 14 (1976) 169. J.T. Park and E.J. Zimmerman, Phys. Rev. 131 (1963) 1611. N. Bohr, Mat. Fys. Medd. Dan. Vid. Selsk (1948) no. 8. F. Besenbacher, H.H. Andersen, P. Hvelplund and H. Knudsen, Mat. Fys. Medd. Dan. Vid. Selsk. (1981) no .9. W.K. Chu, Phys. Rev. A 13 (1976) 2057. J. Lindhard, M. Sharff and H.E. Schiott, Mat. Fys. Medd. Dan. Vid. Selsk. 33 (1963) no. 14. K.R. Symon, Thesis (Harvard University, Cambridge, Mass., 1948). C. Tschalti, Nucl. Instr. and Meth. 61 (1968) 141. C. Tschalar and H.D. Maccabee, Phys. Rev. B 1 (1970) 2863. G. Amsel, C. Cohen and A. L’Hoir, in Ion Beam Surface Layer Analysis, eds., 0. Meyer, G. Linker and F. Kappeler (Plenum, New York, 1976) p. 953.