Energy distribution functions of low energy ions in silicon absorbers measured for large relative energy losses

N U C L E A R I N S T R U M E N T S AND METHODS

I38 ( I 9 7 6 )

331-343;

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N O R T H - H O L L A N D P U B L I S H I N G CO.

ENERGY DISTRIBUTION F U N C T I O N S OF L O W ENERGY IONS IN S I L I C O N ABSORBERS MEASURED FOR LARGE RELATIVE ENERGY L O S S E S BEREND WILKEN

Institut fiir Stratosphiiren-Physik, Max Planck Institut fiir Aeronomie, 3411 Lindau/Harz, IV. Germany and T H E O D O R E A. FRITZ

National Oceanic and Atmospheric Administration, Environmental Research Laboratories, Space Environment Laboratory, Boulder, Colorado 80302, U.S.A. Received 29 July 1976 The energy straggling of low energy hydrogen and aHe ions in silicon has been measured in extremely thin surface barrier detectors for energy losses of 5-99% of the initial incident energy of the ion. A distinct maximum in the variation of the width of the energy distribution o f the ions has been found as a function o f the residual particle energy, ER, with which the ions emerge from the silicon absorbers. The value of ER when these maximum distribution widths are observed has been compared with the kinetic energy Tm for which the specific energy loss has its maximum value in silicon. Reasonable agreement has been found.

1. Introduction Charged heavy ions passing through absorbing material lose their energy primarily by electrostatic interaction with electrons bound to the stopping atom and by direct collisions with the atomic nucleus. Due to the statistical nature of the energy loss process a monoenergetic energy distribution in an incoming particle beam does not remain a delta function after the particles have passed through a given absorber thickness. Broadening of the initial energy distributions function results from the fact that the total energy loss of a charged particle crossing an absorbing layer consist of a large number of independent interactions between the incident ion and electrons~). The resulting fluctuations in energy loss are generally known as energy straggling. As shown by Lewis 2) this process is intimately related to range straggling. The energy transferred in a single interaction is a statistical process and is described by an energy dependent collision probability function2). For low energy particles the energy distribution function should rapidly approach a symmetric Gaussian shape as a function of increasing energy loss. This is due to the fact that the maximum energy that a non-relativistic heavy ion of mass M and kinetic energy T can transfer to an electron of mass m in a single collision is rather small and is of the order of 4(m/M)T,,-2x 10-3% with • denoting the particle's energy per nucleon. Therefore, even a small total energy loss of a slow heavy ion is the result of a large number of collisions which, on the average, transfer a very small amount of the initial ion energy. In general,

the energy distribution function of a given species of incident particle passing through a given thickness of absorbing material shows not only increasedfluctuations but also a significant skewness as the relative energy loss in the absorber becomes large 3' 4). The response function of a charged particle spectrometer is in general composed of contributions of various noise sources within the detector and the subsequent electronics. If an ionization process in the detector is used to convert the primary particle energy into a pulse height, the statistical energy straggling process represents an additional source for signal fluctuations which limits the ultimate resolution of the particle detector. In particular, the detection and analysis of heavy ions can be effected significantly by these intrinsic fluctuations e.g. Fritz and CessnaS). The present paper is concerned with energy loss fluctuations for low energy ions of hydrogen and helium in various thicknesses of silicon absorbers in the range from 3 to 15/~m with emphasis on large relative energy loss of the penetrating ions. The fluctuation process in silicon is of particular interest since silicon solid state detector assemblies are often used for multi-parameter analysis of heavy ion spectra in laboratory and space flight application. The overall characteristics of the energy loss of heavy ions in these thin silicon absorbers have been discussed previously by Wilken and Fritzr).

2. Experimental To investigate fluctuations in the energy loss of low

332

BEREND WILKEN AND THEODORE A. FRITZ

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a Manufacturer supplied data. b Total system noise.

energy ions o f h y d r o g e n a n d h e l i u m in thin silicon layers a d v a n t a g e has been t a k e n o f recent i m p r o v e m e n t in s e m i c o n d u c t o r technology. C o m m e r c i a l l y available extremely thin silicon surface b a r r i e r detectors were used to m e a s u r e the pulse distributions o f ions t r a n s m i t t e d t h r o u g h the detectors. These thin detectors were m o u n t e d in front o f a second silicon surface

b a r r i e r d e t e c t o r with the geometric s e p a r a t i o n d ( d ~ 4 m m ) o f the two detectors being d e t e r m i n e d only b y the i n d i v i d u a l transmission m o u n t s o f the detectors. T h e b a c k d e t e c t o r o f the telescope configuration h a d a thickness o f 50 p m a n d an a r e a o f 50 m m 2. Details o f the d e t e c t o r telescope are shown in fig. 1. T h e total silicon thickness o f the two solid state detectors was

ENERGY DISTRIBUTION

larger than the range in silicon of all ions investigated. The data presented below are collected with different thin front detector elements. Characteristic parameters of all detectors used for this investigation are summarized in table 1. The response of the two detectors in the telescope configuration was investigated for protons and heliumions for particle energies up to 2 MeV/charge. The detector telescope was mounted in a vacuum chamber and maintained at a sufficiently constant temperature of +18°C within _+2°C. The detector signals were routed into separate preamplifier and amplifier chains outside the chamber walls. A block diagram of the follow-on electronics is shown in fig. 1. Energy calibration of all detector-amplifier systems were performed routinely with a calibrated precision-Hg pulser. The pulse outputs of each of the detector systems were interchangeably connected to a pulse-height analyzer to expand the pulse distribution into 4096 channels. The ions used for this investigation were accelerated with a 2 MeV Van de Graaff machine. The particle beam diameter was 1 mm. Due to the placement of beam defining apertures, the ion beam illuminated a fixed area on the detector surface as the ion species and energy was varied. Therefore, thickness variations of the detectors over scale sizes larger than 1 mm were not of significance to the present study. [Measurements relating to the thickness variation over the entire active area of these detectors have been discussed by Wilken and Fritz6)]. The uncertainties in the measurements introduced by thicknesses variations over a distance of ~ 1 mm on the detector surfaces are small compared to other uncertainties associated with the measurements. The active areas of the detectors are given in table 1. It was possible to extract from the ion source of the accelerator both singly charged hydrogen and helium ions and doubly charged helium ions. Helium was used in its isotopic form of 3He. During the runs, the ion beam was monitored with a thick silicon surface barrier detector with very good low noise performance (system f w h m = 8 . 9 k e V ) . With this monitor the energy spread of the ion beam was found to be approximately constant and equal to 4.3 keV for all particle energies above 100 keV. It was possible to rotate the monitor detector into the particle beam between the runs with the detector telescope. All detectors which were used to investigate the fluctuation process were operated in a fully depleted mode. Thin surface barrier detectors are depleted by only a few volts. Due to the very large field strength within their sensitive volume these detectors have to be carefully protected against even small overbias

333

FUNCTIONS

because of plasma breakthrough. The bias voltage of the thin detectors has therefore been monitored with an electrometer to maintain a constant voltage cross the detector even if the ambient temperature changed by a few degrees. With the system described above it was possible to investigate the fractional energy loss of hydrogen and helium ions in each of the two detectors, as well as the spread of the energy loss spectra in each detector and the transmission rate through the entrance detector. The channeling of ions between crystal planes can produce significant differences in the rate of energy loss (and total particle range) between channeled and unchanneled ions. In addition, for very heavy ions this same effect can produce pronounced differences in the pulse-height linearity and energy resolution. As noted previously commercially procured detectors were used in the present study. The manufacturerers have stated that the silicon wafers were cut from the crystal at an angle which minimized channeling effects. (The stated angle was within 6 ° of the crystal 110 axis but this was not checked independently in the present study.)

3. Results and interpretation Fig. 2 shows the energy loss curve for hydrogen ions in a 4.6/~m thick silicon detector (D1) and the response of the second detector to penetrating ions. The maximum energy loss in the thin detector occurs at an incident particle energy E m ( p ) = 4 2 5 k e V . Particles start to penetrate the first detector for incident energies slightly smaller than Em (p) as is indicated by a !

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Eo(keV)-Fig. 2. Energy loss curves for protons in silicon measured with detector DI (4.6/tm) (indicated as dots) as the front element in a detector telescope. Maximum energy loss is found for an incident energy Em(p)=425keV. Energy loss in the second detector is indicated by crosses.

334

BEREND W I L K E N AND T H E O D O R E A. F R I T Z

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Fig. 3. Energy distributions in silicon (detector D1) for six energies below and above the energy E,.(p) II 425 keY with Era(P) defined as the particle energy associated with maximum energy loss in detector D1.

measurement of the percentage of particles transmitted through the thin detector6). The response of detector D1 for six incident proton energies slightly below and above the energy Era(p)= = 425 keV is shown in fig. 3. The width and shape of the pulse distributions vary in a distinct manner as the particle energy is increased and becomes larger than Era(P). Gaussian curves have been fitted to the energy loss spectra to show deviations from a symmetric pulse distribution. As is shown in fig. 3, the pulse spectra are Gaussians for particle energies low enough to be completely absorbed in the thin detector D1. With increasing beam energy the distribution first starts to develop a low energy tail, and then shows a strong broadening. The width of the distributions approaches a constant value at sufficiently high beam energies after passing through a clear maximum. The broadest pulse distribution occurs approximately at a particle energy of 450 keV which is close to the energy associated with a transmission rate of 50% through the thin detector6).

Fig. 4 shows the energy loss versus incident energy curve for 3He-ions measured with a 3.8/am thick front detector (D2) in the telescope. Maximum energy is deposited in detector D2 at an energy Em(He)= = 1230 keV. Six typical pulse spectra measured with detector D2 are shown in fig. 5. As in the case for hydrogen ions, a significant increase in signal dispersion is observed when the aHe-ions start to penetrate detector D2. Again, asymmetric distributions appear for beam energies slightly above Era(He)= 1230 keV. However, in contrast to the proton measurements, no high energy tail is observed in the spectra.

4. Data

analysis

The position of the centroid and the width of the distribution of the energy spectra measured with the two detectors were computed using the numerical values of the measured data points. Conversion into the proper energy scale was obtained by calibrating all detector-amplifier systems with a Hg-pulser and charge

ENERGY DISTRIBUTION FUNCTIONS terminator. The position of the centroid o f the energy spectra expressed in energy units were used to produce energy loss versus incident ion energy curves as shown in fig. 2 and 4. No corrections have been made for energy lost in the various aluminum and gold layers of the surface barrier detectors (the assumed thickness of these layers is given in table 1). The measured full width at half-maximum r/r of the energy distribution spectra is comprised of contributions from a variety o f noise sources within the detector-amplifier system, as well as the spread in energy of the incident particle beam and the fluctuations in the energy loss process described by the straggling parameter r/. The total distribution width is given by ~2 =r/D2..{_ ~/E 2+ r/2-t-r/2,

(1)

where t/a , r/E and r/b denote the noise contributions of the particle detector, amplifier chain and accelerator beam respectively. According to Tschal/ir 3,4) the evolution in the energy distribution along the path of an ion travelling in an absorbing material can be described by a function f ( E , x). A schematic representation of the different phases of this function as the particles progress through the absorbing material is presented in fig. 6. The straggling parameter r/is then related to the standard deviation ~ of this distribution function by I

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Eo (keV) Fig. 4. Energy loss curves for aHe-ions in silicon measured with detector D2 (indicated by dots) as the front detector in the telescope configuration. Maximum energy loss occurs for an incident particle energy Era(He) = 1230 keV. Energy loss in the second detector is indicated by crosses.

(2)

with E the mean kinetic energy of the incident ion after traversing the path length x. The value of r/ is strongly dependent on the ratio of the energy loss A E in the absorbing material to the incident particle energy E o. It is therefore informative to consider the value o f t / f o r two situations, A E / E o = 1 and A E / E o < 1. For the first case the incoming beam of ions is completely stopped within the front detector and essentially all of the energy is collected regardless of fluctuations in the effective specific energy loss process along the path of the particle. In this case there is energy straggling due to the statistics of the electronic ionization process in the detector and the effects of nuclear collisions on the energy loss of the incident ion7-9). rl2 ( A E / E = 1) = ~/2 + r/2.

(3)

The electronic contribution r/~ is dependent on the magnitude of energy deposited in the detector. Goulding et al. 1o) give an expression for the statistics of charge production in silicon r/e = 1.75E ~ (keV),

(4)

with incident energy E given in MeV. For E = 500 keV this source contributes 1.3 keV in energy spread. The parameter r/, on the other hand, is approximately independent of the incident ion energy. As has been shown by Lindhard and Nielsen 9) rl, = 0.7Z~ A ~ (keV),

l

(E-E) 2 dE,

(5)

where Z and A are the nuclear charge and mass number of the incident ion. For protons r/n is approximately 0.7 KeV and for 3He it is approximately 4.3keV. For A E / E o < I , a portion of the incoming beam penetrates the front detector and the statistics associated with fluctuations in the specific energy loss begin to dominate ~/. This effect is the subject of the remainder of the paper. Due to the relatively large noise, r/o, o f extremely thin detectors and their large detector capacitances, the total system noise, r/s =(r/~+~/~)~, was typically well above 20 keV (table 1). Measurements were made of the spectral line width r]T = [~]2 ..l_r]2 ..1_/~2 (he/Eo

= ])]~,

of the incident protons which did not penetrate the thin front detector, and of the line width rh of the precision test pulser. No statistically significant difference was found in the values of ~/T and r/S in this

336

BEREND W I L K E N A N D T H E O D O R E A. F R I T Z

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case. It is therefore assumed that the quantity (r/~+~/2=+r/~)½ is negligible compared to r/s~>20 keV. From the previous discussion this is expected since for protons r/b~4.3 keV, r/,~0.7 keV, r/¢ = 1.3 keV, and 2 2 so (~/b+r/,+P/=Z) ½ 4.6 keV which is much less than % ;~ 20 keV. Essentially the same situation was found for 3He-ions.

Therefore the quantity r/2 = r/2r-r/s2

defines the energy straggling parameter rl in which the contribution of rI(AE[Eo< 1) must dominate.

5.

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(6)

The

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parameter

q

The computed straggling parameter, r/, is plotted in figs. 7 and 8 as a function of the incident particle energy for hydrogen and helium ions respectivgly. The fraction of energy deposited within the silicon layer of the thin front detector obtained from energy loss measurements (figs. 2 and 4) has been introduced into figs. 7 and 8 to demonstrate the evolution of the energy loss dispersion with the fractional energy loss of the incident ion. For sufficiently high beam energies ions emerging from the thin front detector have also been detected with the back detector. As has been mentioned above this back detector was in all cases thick enough to absorb completely the residual ion energy. The straggling parameter r/ was derived from the width of the

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Fig. 7. Straggling parameter ~ for protons passing through a silicon absorber (DI) as a function of the incident ion energy. Also shown is the fractional energy loss in the active volume of D1, dE/Eo, obtained from the energy loss curves in fig. 2. Maximum energy loss in D1 occurs for the beam energy Era(P) = 425 keV. E~ denotes the incident particle energy with maximum dispersion.

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Fig. 8. Straggling parameter r/for 3He-ions passing through a silicon absorber (D2) as a function of the incident beam energy as well as the fractional energy loss in the active volume of D2, ztE/Eo, obtained from the energy loss curves in fig. 4. Maximum energy loss in D2 occurs for the beam energy Era(He) = 1230 keV. Maximum dispersion is observed at an energy E 0. sk is the skewness of the energy distributions.

338 --

BEREND W I L K E N AND THEODORE A. F R I T Z 160

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Fig. 9. Straggling parameter t / f o r hydrogen and helium (3He)-ions in silicon absorbers as a function of the fractional energy loss in the various thin surface barrier detectors.

energy spectra ~/r, of the back detector by quadratic subtraction, i.e. eq. (6), of r/s = 12 keV (table 1). As is demonstrated in figs. 7 and 8 the results from the back detector do agree reasonably well with the values obtained from the respective front detector despite the rather different system noise levels. As is shown in figs. 7 and 8 the straggling r / o f the energy loss suddenly increases when the incident ion starts to penetrate the silicon layer of the front detector. After passing through a maximum with increasing particle energy the signal spread falls off to a constant value for higher incident beam energies. The particle energies Era(p) and Era(He) defined by the experimental energy loss characteristics (figs. 2 and 4) for hydrogen

and helium ions are introduced in figs. 7 and 8 to indicate that the maximum energy straggling does occur at slightly larger incident energies than the energies associated with the maximum energy loss of the respective ion. In terms of fractional energy loss (ArJEo) within the thin silicon detector the maximum in the value of r/ occurs at smaller values of AE/Eo (i.e., larger incident energies) than the maximum value of the total energy loss, AE. The results for helium ions show a more significant increase in straggling than is measured for protons. The maximum dispersion observed for helium ions is almost twice as large as that for hydrogen ions. In fig. 9 the straggling parameter r/ obtained from

ENERGY 160

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Fig. 10. Straggling parameter ~/for hydrogen and aHe-ions in silicon absorbers normalized to unit thickness and unit charge. The theoretical predictions of Bohrl), Cranshaw2=), and Lewis2) are shown. Experimental values of Tschalar and Maccabc¢14) and Sykes and Harris 19) are shown for comparison (the values of ref. 15 are divided by a factor of two). For the present measurements charge exchange may become effective for fractional losses exceeding 50%. measurements with helium and hydrogen ions is shown as a function of the fractional energy loss AE/Eo for a variety of different thin detectors. To remove the dependence on the absorber thickness and particle species, the quantity

6. The distribution skewness

(pd) ½z(Z/A) ½

sk = ~ - E_ 1/z

is plotted in fig. 10 as a function of AE/Eo, where z is the charge number of the incident ion, and (pd), Z and A denote thickness, charge, and atomic number of the absorbing material.

Here ~ is the median energy o f the pulse distributions and E-1/z and E+I/2 are the energies for which the distributions have fallen off to one half of its maximum

The shape of the pulse distributions (fig. 5) are approximately Gaussians with the exception of a small energy range close to the penetrating energy. The skewness has been defined by Tschal~ir 4) as (7)

E+1/2-~"

340

BEREND WILKEN

A N D T H E O D O R E A. F R I T Z

value. The skewness sk of the distributions for 3He-ions is shown in fig. 8. As can be seen the largest deviation from symmetry occurs between the energy Era(He) and the energy associated with the maximum of the line width. 7. Discussion

The energy loss process for a charged particle is a stochastic process and the actual energy distribution of a monoenergetic particle beam entering an absorber and travelling a distance x is described by the distribution function f(E, x) which was discussed in connection with eq. (2) and presented schematically in fig. 6. The shape and width of the function f(E, x) is determined by the so-called collision probability P(E, Q)dQ, which describes the probability that a particle of energy E will suffer an energy loss between Q and Q + d Q in a single collision. In general the collision probability is assumed to be of the form2):

P(E, Q) = const.- 1/E. 1/Q2" dQ.

(8)

At the beginning of the path traversed by a monoenergetic beam when the energy loss is still small, the particle energy E can be considered a constant over the width off(E, x). As a result, the shape and width off(E, x) is dominated by the Q-dependence of eq. (8) yielding an asymmetric energy distribution function (fig. 6). This region of the energy straggling process is characterized by the importance of single collisions with larger than mean energy-losses. The statistics of ionization loss with the limitations described has been treated by Bohrl), Landau H) and Vavilov 12). As the particles lose more energy and the width of f(E, x) grows, the particle energy E can no longer be considered a constant over the width of the distribution f(E, x) and the E-dependence of the probability function P(E, Q) starts to influence the shape of f(E, x). The evolution of the distribution in the region of large total energy loss for heavy charged particles has been considered most recently by Tschal~ir3,a) in detail. He was able to show that for fractional energy loss exceeding 80%, the loss of even a small fraction of the particles can seriously effect the distribution function, particularly its skewness. Tschal~ir's theoretical predictions for the shape and width of the energy distribution function f(E, x) of charged particles is based on the assumption that charge exchange and binding effects can be neglected in the ionization protess. However for low energy ions charge exchange will modify significantly the E-dependence of the specific energy loss of heavy ions. As shown by

Northcliffe13) the effect of electron capture and loss by the incident ion on the ionization process can adequately be described by the concept of the effective charge ze~ = ~z, where z is the nuclear charge number of the ion and ~ is a velocity dependent charge parameter. The specific energy loss formula (dE/dx) is then proportional to the quantity (~z)2. According to Northcliffe13) ~ is essentially equal to 1 for proton energies above 0.5 MeV and for helium ions with energies e,,'~eeding 2 MeV. The binding energy of the electrons in the stopping material will effect the energy loss process if the energy transferred from the ion to the stopping electron is not much larger than this binding energy. It can be estimated that a minimum ion energy of z ~ 1 MeV/nucleon is required for the K-electrons in aluminum to participate in the energy loss process. Tschal/ir and Maccabee 14) measured the energy distribution function f(E, x) of high energy protons and helium (4He) ions after passing them through aluminum absorbers of various thicknesses. Beam energies E = 19.68 MeV, 49.10MeV for protons and 78.8 MeV for 4He-ions were used. The observed energy distributions show very good agreement with the predictions of Tschal~ir 3, 4)The AE/Eo dependence of the straggling parameter r/ derived from the measurements of Tschal~ir and Maccabee is presented in fig. 10 in comparison with the results obtained from the present measurements at lower ion energies. Good agreement is found for fractional energy losses AE/Eo <~70%. For AE/Eo> 70% the distributions measured by Tschal~ir and Maccabee show considerably larger straggling than the distributions obtained from low energy protons and 3He ions. As is indicated in fig. 10 by the cross-hatched area, the residual energy ER of the hydrogen and 3He-ions used in the present investigation falls below the energy limit for which charge exchange becomes an important effect. In the energy range used by Tschal/ir and Maccabee the ions emerge from the absorber with energies ER> 3 MeV for protons and F~> 10 MeV for ~ particles indicating that charge exchange is, in fact, negligible in these measurements. The pronounced maximum in the straggling parameter ~/observed at high fractional energy losses AE[Eo and the subsequent narrowing of the distribution function has also been found by Sykes and Harris t 5). These authors observed a comparable energy bunching effect when passing monoenergetic ~-particles ( E = 5.4 MeV) from an 241Am source through aluminum absorbers with different thicknesses. The resulting energy distribution of the emerging at-par-

ENERGY

DISTRIBUTION

ticles has been measured with a semi-conductor detector. This same technique has been used by Sykes and Harris 16) and A1-Bedri et al. tT) to measure the straggling in a variety of solids and gaseous absorbers up to fractional energy loss large enough to show the energy bunching effect. Comfort et al. t a) investigated the straggling in metal foils using a-particles from a 21p0 source (E=8.78 MeV) and the results are in fair agreement with Sykes and Harris 16) although the range of absorber thicknesses did not extend far enough to demonstrate the bunching. Several other authors 19-2t) studied the straggling of protons and a-particles in various gaseous absorbers but most of these studies do not extend to sufficiently large relative energy losses to show the maximum in stragging. The results of Sykes and Harris t 5) for the straggling of a-particles in aluminum are compared in fig. 10 with the present results for silicon showing a similar functional variation with the fractional energy loss for AE[Eo <~80%. However, the values of Sykes and Harris appear to be consistently larger than the present results and the values derived from the measurements of Tschal~ir and Maccabee as well. This could reflect the anomalously high straggling found in aluminum by Sykes and Harris 16) and Comfort ~8), but it should be noted that the results of Tschal/ir and Maccabee 14) do not show the discrepancy between silicon and aluminum. Detailed theoretical predictions for the straggling parameter at low particle energies and for large fractional energy losses are not available at present. Several estimates for the straggling have been published in the past. The earliest expression for the straggling parameter r/ has been derived by Bohr ~) using the principle of the Central Limit Theorem of Statistics: r/2 = 4(2 In 2) 47re4z2ZNAx,

(9)

where e is the electronic charge, z is the charge number of the ion, and N, Z, and Ax are the density (atoms/cm3), the charge number and the thickness of the absorbing material. Cranshaw 22) obtained an equivalent expression for the fluctuation in energy loss by combining the collision spectrum [eq. (8)] with a normal distribution for the number of collisions in a given path length dx:

7c

2.35 klnEM/Em]

where AE is the mean energy loss in dx, Era and Em are the maximum and minimum energy transferred to

FUNCTIONS

341

an electron in a collision with the travelling ion. Lewis 2) proposed an expression for the distribution width r/derived from the observation that the number of collisions in the path length Ax is described by Poisson's law (11) where M, E and AE are the mass, energy and energy loss of the ion, respectively, m is the electron mass and r/2 represents expression .... (9). In fig. 10 the straggling predicted by Bohrl), Cranshaw 22) and Lewis 2) for protons in silicon is compared with the experimental results. For larger energy loss eqs. (10) and (11) have been numerically integrated using the tabulations of Northcliffe and Schilling23). It is apparent that the measured dependence of q on the fractional energy loss AE/E o is not predicted by the theoretical expressions. However, the expressions proposed by Bohr I) and Cranshaw 22) appear to apply in the limit of small fractional energy loss. Sykes and Harris 15) proposed that the steep decrease in straggling at extremely large fractional energy losses can be explained using the energy dependence of the specific energy loss function dE/dx at low particle energies. As is shown in fig. 11 the E-dependence of dE/dx changes slope with increasing energy E and reaches a maximum at an energy Tm- Near the end of the path of the incident ion in the absorber, the instantaneous energy E will be less than Tm and therefore be in an energy range which has a positive slope in the dE/dx function. In this energy range particles in the energy distribution with higher than average energy E will lose more energy than particles with energies lower than E in a given path length increment. As a result the distribution width will tend to decrease while the particles proceed through the absorbing material. This suggestion would imply that the maximum straggling should be observed for particles emerging from the absorber with a residual energy ER equal to TinThe energies ER for protons and 3He-ions have been evaluated from the present measurements by determining for a given thin detector the incident energy E~ for which maximum straggling is observed (figs. 7 and 8). The residual energy ER associated with the incident energy E~ was then obtained from energy loss measurements as shown in figs. 2 and 4. The stopping power and range tabulations of Northcliffe and Schilling 23) have been used for corrections due to

342

BEREND WILKEN AND THEODORE A. FRITZ

Tm

Helium (He4) Ions in Nitrogen

1.5

Helium (lies) Ions in AI

1.0 Tm LUX

"ol'=

Helium (Hee) Ions in Cu

05

Hydrogen lent in AI

100

500

1000

1500 T(keV) - - , , -

Fig. 11. Specific energy loss of hydrogen and helium ions in various absorbers as a function of ion energy2a). TABLE2 Comparison of the residual particle energy ER with the instantaneous kinetic energy Tm associated with the maximum in the specific energy loss function. Particle

Absorber

E~ ( k e V )

Reference

Tm ( k e V )

Reference

80+= 67.5+ = 63 + a 396 + b 600 1100 750 1400 1000 1250 550 850 500

23 24 25 23 23 25 23 25 25 23 25 23 25

1H

Si

604-13

present work

aHe 4He

Si AI

360 570

present work 15

4He

Cu

580

16

4He

Air

635

21

4He

Kr

595

17

• Corrected for silicon absorber. b Corrected for aHe-ions in silicon absorber. energy loss in contact layers. The m e a n values o f the energies ER f o u n d for p r o t o n s a n d 3He-ions in silicon are c o m p a r e d in table 2 with values for the energy Tm determined from theoretical z 3) a n d experimental 24, 2 s)

stopping power curves. Also included in table 2 are those values for ER which could be o b t a i n e d from published m e a s u r e m e n t s 1s-t 7, 2t) of ~-particle straggling in solid a n d gaseous absorbers.

ENERGY DISTRIBUTION FUNCTIONS

8. Conclusion I n view o f the little i n f o r m a t i o n available o n the stopping power for h e l i u m ions below 3 M e V the agreement between ER a n d Tm exhibited in table 2 is strong evidence that the increase in line b r o a d e n i n g a n d its s u b s e q u e n t decrease with increasing fractional loss ztE/Eo can be explained by the energy dependence o f the specific energy loss dE/dx at low energies. A m e a s u r e m e n t o f the straggling p a r a m e t e r r/as a function of the incident particle energy can possibly provide i n f o r m a t i o n o n the kinetic energy T m associated with the m a x i m u m value o f dE/dx for a given a b s o r b e r a n d ion which has n o t been studied in detail previously. The results presented in this p a p e r are a n o u t c o m e o f work initiated in s u p p o r t of the N A S A Applications T e c h n o l o g y Satellite-6 program. W e are pleased to acknowledge the significant c o n t r i b u t i o n s o f Mr. S. Brown o f N A S A / G S F C d u r i n g the data a c c u m u lation phase o f this study.

References 1) N. Bohr, Phil. Mag. 30 (1915) 581. 2) H. W. Lewis, Phys. Rev. 85 (1952) 20. 3) C. Tschal~ir, Nucl. Instr. and Meth. 61 (1968) 141. ,L) C. Tschal/ir, Nucl. Instr. and Meth. 64 (1968) 237. 5) T. A. Fritz and J. R. Cessna, IEEE Trans. Aerospace Electrons Systems (Nov. 1975).

343

6) B. Wilken and T. A. Fritz, Nucl. Instr. and Meth. 121 (1974) 365. 7) E. L. Haines and A. B. Whitehead, Rev. Sci. Instr. 37 (1966) 190. s) j. Lindhard, M. Scharff and H. E. Schiott, Kgl. Dan. Vid. Selsk. 33, no. 14 (1963). 9) j. Lindhard and V. Nielsen, Phys. Lett. 2, no. 5 (1962) 209 lo) F. S. Goulding, D. A. Landis and R. H. Pehl, Semiconductor nuclear particle detectors and circuits (National Acad. of Sci., Washington, D.C., 1969). 11) L. Landau, J. Exp. Phys. (USSR) 8 (1944) 201. lz) p. V. Vavilov, Zh. Exper. Teor. Fiz. 32 (1957) 320 [Transl. JETP 5 (1957) 749. 13) L. C. Northcliffe, Phys. Rev. 120 0960) 1744. ~4) C. Tschal~irand H. D. Maccabee, Phys. Rev. B1 (1970) 2868. 15) D. A. Sykes and S. J. Harris, Nucl. Instr. and Meth. 94 (1971) 39. 16) D. A. Syke ,~.nd S. T. Harris, Nucl Instr. and Meth. 101 (1972) 423. ~7) M. B. AI-Bedri, S. J. Harris and D. A. Sykes, Nucl. Instr. and Meth. 106 (1973) 241. is) j. R. Comfort, J. F. Decker, E. T. Lynk, M. O. Swint, A. R. Quinton and R. A. Blue, Phys. Rev. 150 (1966) 249. ~9) D. L. Mason, R. M. Prior and A. R. Quinton, Nucl. Instr. and Meth. 45 (1966) 41. zo) j. j. Ramirez, R. M. Prior, J. B. Swint, A. R. Quinton and R. A. Blue, Phys. Rev. 179 (1969) 310. 21) E. Rotundi and K. W. Geiger, Nucl. Instr. and Meth. 40 (1966) 192. 22) T. W. Cranshaw, Progr. Nucl. Phys. 2 (1952) 271. 23) L. C. Northcliffe and R. F. Schilling, Nuel. Data A7 (1967). 24) S. D. Warshaw, Phys. Rev. 76 (1949) 1759. zs) W. Whaling, Itandbuch der Physik, vol. 34 (1958) p. 204.