energy-loss straggling and higher-order moments

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Volume 96A. number 7 PHYSICS LETTERS 18 July 1983

ENE RGY-LOSS STRAGGLING AND HIGHER-ORDER MOMEN TS

OF ENERGY-LOSS DISTRIBUTIONS FOR PROTONS

E. KGHRT i and R. WEDELLSektio n Physik , Bereich 06, Humbol dt- Univ ersit iit Berlin, 1040 Berlin, GDR

Received 17 December 1982

Using the classical collision theory for two moving charged particles the energy-loss straggling and the skewness of an

energy-loss distribution for protons passing a medium are calculated. It is shown that the motion of the target electronsis responsible for significant deviations from Bohr’s results in the low-energy region.

The slowing down of ions passing a medium is the result of individual collision events, the number of which isgoverned by statistical laws. Therefore a primary sharp energy distribution is broadened. For constructing such adistribution, which plays an important role e.g. for the range of ions in ion implantation, one must know their in-finite set of moments. Of practical interest are often only the mean energy loss (first moment), the energy strag-gling (second moment) and the skewness (third moment).

Whereas the energy loss of charged particles is investigated by many authors the straggling !Y12 as not receivedsuch a broad attention (for a review see ref. [ 11). Bohr [2] has given the high-energy limiting case

a; = 4nZ;Z2e4NAR, (1)

where Zl and Z2 are the atomic numbers of the projectile and target atoms, respectively, e is the charge of anelectron, N the number of target atoms per unit volume and AR the target thickness. One of the assumptionsmade by Bohr is a high projectile velocity u1 compared to the electron velocity in the target atom u,, whichbreaks down for low and medium projectile energies. For considering this breakdown Lindhard and Scharff [3],Bonderup and Hvelplund [4] and Chu [5] extended Bohr’s theory by applying a correction factor. It takes intoaccount that for low and medium energies not all target electrons effectively take part in the stopping process.

For the skewness z3 one gets according to Bohr [2]

$t = 4nZ:Z2e4meu;NAR,

where me is the electron mass.(2)

In our papers [6,7] we used the differential scattering cross section do/dT of two moving particles by Gerjuoy[8] to calculate the stopping power, which was derived for an elastic collision of two moving particles. The inelas-tic character of the collision will be approximately considered by the cut-off of the transferred energy and thePauli principle, but not directly in the collision process. Therefore, the obtained formulae can only be used fornot too small projectile velocities ul. In this letter we calculate the second and third moments of an energy distri-bution on the basis of Gerjuoy’s cross section. We concentrate on such light ions that the charge-exchange effects[9] and nuclear collision effects [2] can be neglected and on such energies that the electron motion becomesmore pronounced. For an ion mass ml much greater than the electron mass me one can derive from Gerjuoy’s

’ Present address: Institut fur Kosmosforschung der AdW, Rudower Chaussee 5,1199 Berlin, CDR.

0 031-9163/83/0000-OOOOo/$ 03.00 0 1983 North-Holland 347



Volume 96A, number I PHYSICS LETTERS 18 July 1983

analysis the following cross section assuming an isotropic velocity distribution of the target electrons in the labo-ratory system:

du/dT = $r(Zfe4/T3v;) X [3(r1;)~ + $1, O<TGb,

X [(IJ; +IJ~)~+(u,-LJ~)~]/~u,, b<T<a,

X 0, otherwise, (3)

where a = 2meul (ul + ue) is the maximum transferred energy, u; = (UT - 2T/rr~~)~~ and uk = (u,’ + ~T/vz,)~~are the velocities after the collision with an energy transfer T and b = 2meul (u 1 - ue). The energy-loss stragglingand the skewness are then

a

a2 =NZ,AR s T2do, X3 =NZ,AR j T3da,u u

where U is the minimum excitation energy. Using eq. (4) and performing the integrations one obtains

a2 =!!,ZfZ2c4NAR X[$ln(~)+$ln[4/3($-- l)] -$(-$F)+ 1)) O<UGb,

(4,5)

1 u:+ su$q63 - 3h2 + 86 b<U<a,

x 0, otherwise,

where /I = I/U, I = 4 meui is the kinetic energy of a target electron and 6 = (1 - l/@1/2 and

(6)

E/d- E/&U-

Fig. 1. Energy-loss straggling (in reduced units) versus proton Fig. 2. Skewness of the energy-loss distribution (in reducedenergy. units) versus proton energy.

348



Volume 96A, number 7 PHYSICS LETTERS 18 July 1983

I I 1 1

0 102 ml ml 4w m

E/keV-

Fig. 3. Energy-loss straggling of protons in Ar (solid curve,this theory; dotted curve, theory by Chu [S]; dashed curve,theory by Lindhard and Scharff [31; l experimental results[lOI>.

p-Kr

V / I I I

0 103 ml 300 400 x0

E/keV-

Fig. 4. Energy-loss straggling of protons in Kr (e experlmen-tal results from ref. [lo]; 0 experimental results from ref.[ill).

z3 = 4nZ;Z2e4m,v;NAR X [#&I~)~($, - $S2 - ;s”) t 11, Q<UGb,

X [$Q/u,-(1/6/3)u$ul t&(1 -Q4(1 +46)(ue/tQ4 +;I, bGU<a,

X 0, otherwise. (7)

Both results give in the limit case ul > u, Bohr’s formulae (1) and (2).Figs. 1 and 2 illustrate the dependence of straggling in units of ZiZ,NAR X 10-12eV2 cm2 and of the skew-

ness in units of NARZ: Z2 UX lo-l2 eV2 cm2 on the ion energy E in units of eU, where E = ml/mp and mp isthe proton mass. It is seen that the straggling and the skewness are significantly influenced by the electron motionin the lowenergy region, but this influence decreases for increasing order of the moments.

In figs. 3 and 4 theoretical and experimental [4,10] straggling values of protons in argon and krypton arecompared. Using for U experimental binding energies [ 11,121, and setting 0 = 1 (kinetic energy equal to potentialenergy) and summing over all shells one obtains satisfactory agreement with experiment. The calculated momentscan be used to construct the energy-loss distributions of ions [ 131.

References

[ 1] M.A. Kumakhov and F. Komarov, Energy losses and ion ranges in solids (Harwood, 1981).[2] N. Bohr, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 18 (1948) No. 8.[3] J. Lindhard and M. Scharff, K. Dan.Vidensk. Selsk. Mat. Fys. Medd. 27 (1953) No. 15.[4] E. Bonderup and P. Hvelplund, Phys. Rev. A4 (1971) 562.[S] W.K. Chu,Phys. Rev. Al3 (1976) 2057.[6] E. Ktihrt and R. Wedell, Phys. Lett. 86A (1981) 54.[71 E. Ktihrt and R. Wedell, Phys. Stat. Sol., to be published.[8] E. Gejuoy, Phys. Rev. 148 (1966) 54.[9] 0. Vollmer, Nucl. Instrum. Meth. 121 (1974) 373.

[lo] F. Besenbacher, J. Heinemeier, P. Hvelplund and H. Knudsen, Phys. Lett. 61A (1977) 75.[111 E. Storm and H.J. Israel, Nucl. Data Tables A7 (1970) 565.[121 Landoldt-BornsteIn, Zahlenwerte und Funktionen aus Physik, Astrophysik, Geophysik und Tech&, Vol. 1 (Springer,

Berlin, 1950).[ 131 K.B. Winterbon, P. Sigmund and J.B. Sanders, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 37 (1970) No. 14.

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energy-loss straggling and higher-order moments

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