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Calculations of Ion-Atom Differential Scattering Cross-Sections Using Inverse Power Potential


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Indian J, Phy». 48, 968-972 (1974)

Calculations of ion-atom differential scattering cross-sections using inverse power potential

S. N. Ghosai. akd (Miaa) M. Das

Mass Spectrometer Research Laboratory, Department of Physics, Presidency College, Calcutta










Differential elastio scattering cross-sections in ion-atom or atom-atom collisions can in general be calculated, using classical mechanics, except at very small scatterii^ angles, provided the form of the interaction potential is knoum. In case where the interaction potential can be expressed by an inverse power law, calculations of the cross-sections are possible under certain circumstances using the Mott-Smith expansion for the scattering angle in terms of the impact parameter. Usually the first term in the expansion is used for calculating the cross-sections at small scattering angles. In the present work, the method of calculation has been extended, considering upto second order terms in the Mott-Smith expansion. Comparison with experimental data, when the form of the potential is accurately known, shows bettor agreement than cross sections calculated from first order Mott-Smith formula.

1* Introduction

Experimental studies on the differential elastic scattering cross-section of mono- energetic ions and atoms from atomic systems are capable of yielding quantitative information about the interatomic force-fields. It has been found that different models of the potential can reproduce a given set of experimental data (Mason

& Vanderalioe 1962). It is therefore necessary to have some a priori knowledge of the potential


to get its exact form. On the other hand if a physically realistic model, such as a screened Coulomb potential is chosen then the ex­

perimental data can be used to determine the parameters of the potential.

Sometimes suitable mathematical models for the potentials can be used in interpreting the experimental data. The merit of such semi-empirical potentials are judged by their mathematical simplicity, the accuracy with which they re­

present the actual interaction and their theoretical basis.

One such semi-empirical potential which has bem widely used is the inverse power law potential, both attractive and repulsive. Mott-Smith (1960) has ob­

tained an expression for the angle of scattering due to collision between two atomic or mdeoular systems as a function of the impact perameter in the fmm of an infinite power series, assuming an inverse power intermolecnlar potential of the




Io n -a to m differen tial Gross-sections


for n > 2, both for C positive (attrewjtive) and C negative (repulsive). This ox- preasion can be used to calculate the differential elastic scattering cross-section (dixjdCi) with the help o f the usual classical formula. Calculations of the croas- seotions at small angles can be carried o6t (Hasted 1972) by using only the first term of the expansion in Mott-Smith formula. Tn the. prostmt work we have extended these calculations upto the seco||d term of the Mott-Smith formula using repulsive potential and have comparcMl wfth experimental data the (dajdQ,) values calculated for those atomic systems for which the known interaction potentials can be well roi^resented by an inverse power law formula satisfying the Mott- Smith criterion o f n > 2. }


2. Thborbticai. Oonsidebation

The Mott-Smith expression for the deflection function 0 in C.M. system for a relative velocity g o f the colliding partners having a reduced mass for a repulsive potential is

d _ y I ^

t f r a n f - H D \ . « ) ’ (1)

where b is the impact parameter. The above series converges for b > whore bo is a critical impact parameter as defined below.

The classical expression for differential elastic scattering cross-section can be expressed as (Hasted 1972)

dcr ^ I

dtl sin 0 ^ dO

where /? is the reduced impact paranietei’ given by the relation t Uffi \\ttl

I E )


... (3)

78 T l/ »

and = bj/i.

fieru as (Kihara et al 1960)


n J,

[ V

it can be shown that for email angles o f scattering, i.e., for large impact parameters ( 6 » 6 „ ) .

1 - I t <-<■ 1.


970 S. N . G hoshal an d M. D as

Hence for small angles of scattering, only the first few terms will be of importance.

Previous workers have taken only the first term (< = 1) in the expansion for


and have obtained the expression for the differential scattering cross-section in the C.Sf. system as follows (Hasted 1972)


d a - L \

~ « L 2e J ... (4)

where and

e =


c „ = v»^r ( )/r(»/2).

Considering upto the second term (i.e., t = 2) of the expansion,


can be written as e _ E ! n 2 ± i ) r ( } )

r(»/2) ' 2r ( » - i ) '

Differentiating eq. (5) with respect to /?, we got after some simplification


^dfi r ( « + i ) r ( i ) 1 i-> (fi)

^ ~ d e ~ M^l 2n*^ r f n - i ) • (? J From oqs (2) and (6), we get for small


/^.\ /?* f , r { n + m i ) 1 1 - '

c.u ~ r

2n*/?»"»f(w-l) • "S J ‘

Now putting the value of


from eq. (3) and using the first order approximation

^ / » r (n /2 )]* 'V * ^ » we get,

l ^ Z ] _ » [ ( » - i ) c . c f v „ . , M , [ i r ( ./2 ) ■[-.


» I 2« j I 0 .( n - l) r ( » - l) 1 ' '

3. Disousszok

A'*^-\-A acaUering

We have compared the differential scattering cross sections calculated from eq. (7) with those calculated on the basis of the first order approximation of eq.

(1) in the case of argon ion-argon atom collision at 25Kev. The experiments were peorformed by Fuls

a al

(1957) at the energies 26, 50 and 100 Key. Lane


Everhart (I960) have used these, experimental data to deduce the interaction between




and have obtained repulsive potentials in idl the three oases


at; sinftll nuclciar soporations and h,8»V6 givon th,ose in grap)i,ioal forms. After suitably magnifying these graphs, we have tried to fit them by inverse power law expressions o f the form V = The best fit for the potential deduced from the 26 K ev data is found to be given by

F(r) = I70r‘'2.2^v,

where r is measured in angstroms. This fatisfios the Mott-Sinith criterion for the applicability o f eq (1) since the exponei^ n > 2. In the other two cases, the potentials deduced by Lane & Everhart (|960), though expressible by an inverse power law formula, do not satisfy this cr|toj*ion.

Io n -a to m differen tial Gross-sections

97 J

6 (d tg re e i)

Fig. 1. Differeutial elastic scattering cross sections of A - + A collisions at 26 Kev ion energy calculated from Mott-Smith formula, using both first order and second order approximations.

The oroas-seotions calculated with the help of eq. (7) for

for 26 Kev are plotted in figure 1. along with those calculated on the ^ first order approximation of eq. (1). In this figure the

by Ererhwrt

et al

(1966) on the basis of an exponentially

tial as also the experimental cross-sections of Fuls

«t d

(196 ) are s own.



S. N. Ghoshal and M. Das

It is fotmd that the second-order calculations reported here are in better agreement with the experimental values than the flrat-order calculations at all angles. In fact the agreement is excellent at small angles upto 9° and again between 19° and 24°. In the intermediate region the experimental points arc scattered and seem to indicate a hump. The correction term in the second order formula eq (7) shifts the calculated values in the right direction.


Everhart E., Stone 0. & Carbone R. J. 1965 Phyi. Rev. 99, 1287

Fub E. N., Jones P. R., Ziemba F. P. StEverhart E. I9S7 Phye. Rev. 107, 704.

Hasted J. B. 1972 Phyeict of Atomic CoUiaions, Chap II., p 88-89, (Butterworths, London).

Kihara I., Taylor M. H. & Hirsohfolder J. O. 1960 Phye. Fluida 3, 716.

Lane Q. H. & Everhart E. 1960 Phya Rev. 120, 2064.

Mason E. A. A Vanderslice J. T. 1962 .4tof»tc and Molecular Proceaaca, Chap XVII, p 66.*{, Edited by D. R. Bates (Academic Press, New York).

Mott-Smith H. M. 1960 Phya. Fluida 3, 721.


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