On classical small angle scattering from many-particle targets

Volume 96A, number 4

PHYSICS LETTERS

27 June 1983

ON CLASSICAL SMALL ANGLE SCATrERING FROM MANY-PARTICLE TARGETS N. NE~KOVIC, B. PEROVIC and D. ~IRIC Laboratory for Physics, Boris Kidrt'~'Institute, P.O. Box 522, 11001 Belgrade, Yugoslavia Received 5 January 1983 Revised manuscript received 18 April 1983

The problem of classical elastic small angle scattering from many-particle targets is considered. Special attention is devoted to the effect of multiple scattering. The conditions for which this effect is not observable and those for which it is especially pronounced are found. The obtained results are applied to the ease of scattering of a charged particle from an electric dipole.

The problem of scattering from many-particle targets appears in investigations of a great number of processes on atomic and nuclear levels. In many experiments the forward scattering is much more pronounced. On the other hand, in the case of small scattering angles the theory simplifies considerably. In this paper we shall treat the problem of classical elastic small angle scattering from many-particle targets. The problem will be treated as a real many-body problem. Let us mention briefly some of the work on classical elastic small angle scattering relevant to this paper. Cross [1] and Gentry [2] considered the problem of ion-molecule scattering, treating the molecule as a point dipole. Gryzifiski [3 ] investigated the scattering : from a set of charged particles. He showed that in the case of scattering from a dipole the differential scattering cross section had a singularity, and that it stayed singular after averaging over the dipole orientations. The scattering from more than two particles was treated using the multipole expansion of the interaction potential. The problem of electron-molecule scattering was considered by Dickinson [4]. The molecule was treated as a point dipole, and the classical approach was combined with the Born approximation. The scattering of ions (atoms) from molecules was also studied by Dickinson and Richards [5,6] and McFarlane and Richards [7]. The approach was via the asymptotic expansion of the interaction potential. Dickinson and Richards considered the cases of homonuclear and 0 031-9163/83/0000-0000/$ 03.00 © 1983 North-Holland

heteronuclear molecules, and found that the differential scattering cross section, averaged over the molecule orientations, had the singularities; those singularities were connected to the characteristic values of the scaitering angle and interaction potential. In the case of heteronuclear molecules [6] one maximum in the averaged differential scattering cross section was found too. McFarlane and Richards investigated the case of tetrahedral molecules. They showed that the averaged differential scattering cross section had one maximum. Demkov [8] studied the problem of scattering from harmonic anisotropic potentials. Special attention was devoted to the scattering from a set of charged particles. It was shown that in the case of N charged particles the differential scattering cross section had 2N - 2 singularities, and that its singular character did not vanish after averaging over the target orientations. That had been shown before by Komarov and Shcherbakov [9] for the scattering from two charged particles. Dernkov also considered the possibilities of observing the singularities in the differential cross section in ion scattering from molecules, and atoms of a crystal surface. We shall assume that the incident velocity of the projectile is parallel to the y axis. Since the scattering angle is small, we can use the momentum approximation [10]. Then, the x and z components of the seattering angle are

183

Volume 96A, number 4

PHYSICS LETTERS Oi =

+oo

1

0x = - ~

f

[W(p x , y , p z ) / b p x l d y ,

_ ot~ +~,

0z = - ~

1

f

[OU(Px,y, pz)/bpz] dy,

where E is the incident energy, and Px and Pz define the minimal distance of the projectile from the origin; U is the interaction potential of the projectile and the target. For potential U we have N

U(Px,Y, Pz)=i~l.= Ui(Px,Y, Pz) , where the summation i~ performed over the particles of the target;N is their total number. Consequently, the components of the scattering angle are N

N

Ox = i ~ Oxi ,

Oz = i ~ Ozi

o = I--J-I'

bOx bOz where J = bpx bpz

box bOz bpz bpx

( J is the jacobian of the functions Ox(Px, Pz) and Oz(Px, Pz)) [8,9]. The jacobian J can be written as J=Jo

+J

' ,

where N

So --

"=

N

J;;,

s' =

•

The sum of the jacobians Ji! and J/i describes the coupling between the ith and jth particle of the target as seen by the projectile. Thus, from the point of view of differential scattering cross section the collision of N + 1 particles under consideration does not reduce to N independent collisions of two particles, i.e. there exists the effect of multiple scattering. It should be emphasized that the quantity J ' (which describes the effect of multiple scattering) contains only the terms corresponding to double scattering. This can be attributed to the fact that the scattering angle is small. It is clear that the equation

J'=0,

OJ'/Opz = 0 ,

(2)

determine the points at which the effect of multiple scattering is especially pronounced. The rainbow lines, i.e. the lines in the Px Pz plane along which the differential scattering cross section is infinite [8,9], are given by the equation J = 0.

(3)

It should be noted that in the case of a harmonic interaction potential the rainbow lines degenerate into the so called focal points [8,9]. The equations

bJ/apx = O,

bS/bpz = O,

(4)

define the points at which the differential scattering cross section has extreme values. Let us now consider the above described lines and points in the case of scattering of a charged particle from an electric dipole. The positions of the positive and negative charges of the dipole are (0, O, 0) and (0, 0, --d), respectively. In this case the terms J0 and J ' are

N

i=1 ]=i+1

(1)

gives the lines in the PxPz plane along which the effect of multiple scattering is not observable. The equations

OJ'/Opx = O,

(angles Oxi and Ozi correspond to the Rh particle of the target). Thus, from the point of view of components of the scattering angle the collision o f N + 1 particles under consideration reduces to N independent collisions of two particles. Certainly, this is a consequence of the fact that the scattering angle is small. In this case the differential scattering cross section is

1

1/IJiil

27 June 1983

(Jii ÷ sji) ;

Jo = -(qQ/E)2(I[P p4 + 1/pn4)

(5)

and

Ji] is the jacobian of the functions Oxi(Px, Pz) and Oz] (Px, Pz). The jacobian Jii describes the scattering from the ith particle of the target; the differential scattering cross section corresponding to this particle is

184

j ' = _2(qQ/E)2(l[pp4pn4)(2d2p2x _ p p 2 p n 2 ) , (6) where q and Q are the charges of the l~rojectile and the dipole, respectively, pp2 = px2 + Pz and pn2 = p2 + (Pz + d) 2" It is a clear that eqs. (1) and (6) give

27 June 1983

PHYSICS LETTERS

Volume 96A, number 4 I

I

I

I

Pz +~ d = O '

I

0.8

p2 +Pz(Pz + d ) = 0 , 4d2p2 _ pp2pn2 = 0 . O0

Pz,d -0.8

-1.6 I

I

-1.6

I

I

-0.8

I

I

0.0

I

I

08

1.6

Px, d

Fig. 1. The lines along which the term J ' is equal to zero. 2d2p2 _ pp2pn2 = 0 . The lines determined by this expression are given in fig. 1 (these are the lines along which the effect of multiple scattering is not observable). It can be shown that the first o f eqs. (2) yields in this case

The lines defined by these expressions are also given in fig. 2. It is easy to see that these four and the other three lines shown in this figure have five crossing points; their coordinates are Px = 0 and Pz = - t d , and Px = -+(~ + ~X/~)l/2d and Pz = ' ~ d (these are the points at which the effect o f multiple scattering is especially pronounced). It should be emphasized that all these points lie on the line along which the differential scattering cross sections corresponding to the positive and negative charges o f the dipole are equal to each other [eq. (5)]. Fig. 3 gives the absolute value of term J ' as a function of parameter Px for Pz -- - 21_d • For Px = 0 and +1.54d term J ' is maximal while forPx = _+0.36d it is minimal. It is evident that for Px = 0 the function IJ ' l has the absolute maximum, i.e. at point Px = 0 and Pz = -½d the effect o f multiple scattering is pronounced at the most. In the case under consideration the jacobian J reads

Px=O J = -(qQ/E)2(d2/pp4pn4) [4p ] + (2p z + d) 2 ] •

and 2d2pp2pn2 _ (pp2 + pn2)(4d2 p2x _ pp2pn2) = O. The lines determined by these expressions are given in fig. 2. Similarly, the second o f eqs. (2) yields in this case

0.8

oo

o+,o - 0.8

1

I

/

I

l

I

i

i

I

I

x ~ "

i/

%

/

,f ,L--t .........

%\

1//

//ilkX',

........

\

\\

This jacobian is equal to zero [eq. (3)] for Px = 0 and Pz = - 2td. Thus, in the case under consideration there is one focal point [3,8,9]. It is important to note that the obtained focal point coincides with the point at which the effect o f multiple scattering is pronounced at the most. For the x and z components of the scattering angle corresponding to the focal point we have 0x = 0 and 0 z = - (qQ/E) 4/d. The first o f eqs. (4)

103

I

I

I

I

I

I

I

I

\

I---'

, 101

x

J'l, 2(-~) 2 10-~

-1.8 -1.6

-0.8

0.0

0.8

1.6

p~, d

Fig. 2. The lines along which the partial derivatives of the term J ' with respect to the parameters Px (solid line) and Pz (dashed line) are equal to zero.

1.6

08

0.0

0.8

1.6

Px, d Fig. 3. The absolute value of term J ' as a function of parameter Px for Pz = -~d. 185

Volume 96A, number 4 I

I

PHYSICS LETTERS I

I

/

I

I

I

27 June 1983 102

1

0.8

I

I

I

}

I

I

I

I

I

I 1,6

10' 0.0

Pz, d

....

, ...............

O, (~qz)210 °

-0.8

/zL ///

.J

10-~ -1.6 I

I

-1.6

I

-G8

I

L

0,0

I

0.8

i

I

1.6

10

Px, d

Fig. 4. The lines along which the partial derivatives of jacobian J with respect to parameters Px (solid line) and Pz (dashed line) are equal to zero.

I -1.6

l

I -0.8

I

I 09

I

I 0.8

Px,d Fig. 5. Dependences of the differential scattering cross section on parameter Px for Pz = - l d with (solid line) and without (dashed line) the term describing the effect of multiple scattering.

gives in this case

Px=O, 2pp2pn2 _ (pp2 +pn2)[4px2 + (2pz +d)2] = 0 . The lines defined by these expressions are shown in fig. 4. The second of eqs. (4) yields in the case under consideration

Pz + ½d = 0 , pp2pn2 _ [/,2

+Pz(Pz+ a)l[ap 2

+

(2pz + a ) 2] = 0 .

The lines determined by these expressions are also given in fig. 4. It is evident that these and the other two lines shown in this figure have, beside point Px = 0 and Pz = -½d, two crossing points; their coordinates are Px = +d/(2x/3) and Pz = - ~ d (these are the points at which the differential scattering cross section has extreme values). We want to emphasize that these characteristic points also lie on the line along which the differential scattering cross sections corresponding to the positive and negative charges of the dipole are equal to each other. The x and z components of the scattering angle at these two points are 0x = 0 and Oz = - ( q Q / E)3]d, and the differential scattering cross section is o = [E/(qQ)] 2d4/27. Fig. 5 gives the dependence of the differential scattering cross section on parameter Px for Pz = - ~ . It is evident that for Px = +-0.29d this quantity is minimal. In this figure we also give the differential scattering cross section when term J ' is not 186

taken into account, i.e. when the effect of multiple scattering is neglected. It is clear that the fine structure o f the differential scattering cross section under consideration is to be attributed to the effect of multiple scattering. It should be noted that the above discussed focal point could be observed in ion scattering from an alkali covered metal surface. Namely, in this case the surface is covered with dipoles [11,12], and ions penetrating the first atomic layer could thus experience an additional focusing. We conclude with the remark that the approach described in this paper has been found to be very useful in studying the problem of ion channeling in thin crystals [13]. We would like to thank Drs. R.K. Janev and P. Gruji6 for helpful discussions. This work is a part of a research program supported by the Scientific Community of Serbia and the Research Fund of the Serbian Academy of Sciences. References

[1 ] [2] [3] [4] [5]

R.J. Cross, J. Chem. Phys. 46 (1967) 609. W.R. Gentry, J. Chem. Phys. 60 (1974) 2547. M. Gryzifiski, J. Chem. Phys. 62 (1975) 2610. A.S. Dickinson, J. Phys. B10 (1977) 967. A.S. Dickinson and D. Riehards, J. Phys. B12 (1979) 3005.

Volume 96A, number 4

PHYSICS LETTERS

[6] A.S. Dickinson and D. Richards, J. Phys. B14 (1981) 3663. [7] S.C. McFarlane and D. Richards, J. Phys. B14 (1981) 3643. [8] Yu.N. Demkov, Soy. Phys.-JETP 53 (1981)63. [9] I.V. Komarov and A.P. Shcherbakov, Vestn. Leningr. Univ. 16 (1979) 24 (in Russian).

27 June 1983

[10] L.D. Landau and E.M. Lifshits, Mekhanika (Nauka, Moscow, 1965) p. 73 (in Russian). [11 ] I. TerziE, N. N~skoviE and D. C~iriE,Surf. Sei. 88 (1979) L71. [12] I. TerziE, D. 6xic, N. Ne]koviE, V.S. Chernysh and B. Perovi~, Nucl. Instrum. Methods 170 (1980) 509. [13] N. NetkoviE and O.H. Crawford, to be published.

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