Multiple collisions in molecular beam scattering experiments

Chemical Physics55 (1981) 169-176 North-Holland Publishing Company

MULTIPLE COLLISIONS IN MOLECULAR BEAM SCATTERING EXPERIMENTS N.F. VERSTER, H.C.W. BEIJERINCK, JM. HENRICHS and PMA. VAN DER RAM PhysicsDeportment. Eindhoven Universi@ ofTechnology, Eindhoven, The NetherIan& Received 11 February 1980; in final form 14 November 1980

Due to the forward peaked differential cross section for elastic atom-atom scattering the effect of multipIe c~Eisians has to bc considered in *he analysis of crossed beam measurements of the total cross section and especially of the small angle differential cross section at large values of the beam attenuation. At angles B = 80. with 00 the quantum mechanical scaling angle of the elastic differential cross section, ihe cotrection for the latter case amounts to 20% at beam attenuations I/Io = exp(-1). Firstly, a careful analysis of the probabiities for single and multiple scattering is given, resulting in an expression for the measured beam signals which is correct for all values of the beam attenuation. The probability for mul.tiplescatteringis then cab&ted for an inversepowerpotentialV(r) = -C,r-‘, with s = 4 through s = 7, which include both the case of ion-atom scattering (s = 4) and atom-atom scattering (s = 6). The results are given as effective differential cross sections an(e) for n-fold scattering. They are described by a singie, simpie analytical function with four free pararneters that have been determined for R = 2,3 and 4 by a least squares method The o,(e) are normal&d to th.? total cross

1. Iutioduction

In the last decade the development of high intensity supersonic beams as targets in molecular beam scattering experiments has evolved rapidly. The increased imight into the process of super~~n.k beam formation has resulted in well detiued scattering targets with a Hughvalue of the den&-length product- For many systems of scattering partners a large attenuation ‘of the primary beam can easily be achieved experirnentaUy. This development can be f&vourable for the quality of scattering experiments. The -use of setondary beams which give a large attenuation requires, however,.a clear insight into the proeessof multiple scattering.-This holds especially for a measurement.of

r&o of relative values of Q end u(6). We will show that at angles 0 z=0 o the contribution of multiple scattering amounts to 20% at a beam attemuation of exp(- 1). The influence of multiple scattering on the beam attenuation in a measurement of Q can in gen-

eral be neglected [6,?]. In this case elastic small angle scattering gives a correction AQ due to tlmtiuite angaku resolution, multiple scattering increases AQ by.’ LO-1576 at an attenuation exp(-1). In this paper we treat the subject of multiple scatteriug in detail, including a discussion of the method of calculation. Numerical results are given for an inverse Power potential V(r) = - C,r*, with s ranging Prom 4 to 7, b.ased on the semiclassical value of the cross sections Q and u(8) [ 1,s ,9]- The final expression for the detectcr signal includes the angular resolution Yur~ction-r&Y).Our definition depends on the cm. scattering angle and differs from the usual deftition [IO,II];

Thi! same method has also been-applied in the analysk of meesuuementi of the cross section for &All augle r+atioGU~

heia&c

scattering of CsF

l

2. Elastic small angle scattering

The strongly forward peaked character of ~(0) is clearly demonstrated by the fractional cross section Q&e), defmed as

2.i. The scattekg cross section The differential cross section o(B) for elastic scattering of molecules is strongly peaked in the forward direction. At small angles 8 < Be diffraction phenomena dominate the differential cross section, analogous to the diffraction pattern of a diffuse disc with an area equal to the total cross section Q_ The characteristic angle 13~for this diffraction contribution is @en by 0,, = (47r/.rc*Q)‘o _

(1)

For the value of Q the semiclassical Schiff-LandauLifchitz theory [89] can be used as a reference, resulting in Q, = 3.70 X IO-r8 m’ and Be = 1.17 X lo-* for Ar-Ar at a relative velocity of 500 m/s [I]_ For slightly larger angles the scattering phenomena are fully descriied by classicaI mechanics, i.e. the equivalent of refraction, and we fmd o(e)

a

e-*-*f=.

Fig. 1 shows u(B) for an inverse power potential V(r)aI-‘withs=6,~~veninref. [I].

1.The reduced diffazntial aoss section for.ao iover& :

Q&7) = j! o(B’) 2+ sin 0’ de’

_

0

In fig. 2 we gve Qrr for s = 6. One half of the total cross section

falls within the cone 0 < 2eo, i.e. a

solid angle of 4.30 X lo4 for the numerical example given above. Roughly speaking this central half of the total cross section corresponds to particles which have a chance to be scattered back into the beam by subsequent collisions while the other half corresponds to particles which are irrevocably lost. An accurate description of this latter half is not necessary for our purpose, and thus we may use small angle formulae throughout. This means that the complicated behaviour of u(6) at large angles can be ignored here. The freedom to use small angle approximations is essential for the treatment of multiple collisions as given in this paper. There is an analogy with the scattering of moonlight by a haze. The intensity of the direct light is reduced by scattering but a large fraction of this light is found back in a narrow halo. The remainder

N.F. Versteret d / Multiple collisionsin mo!ecu!arhem scatteringexperiments .isscattered over larger angles and is considered as lost.

Large angle phenomena, though beautiful, are ignored2.2. Attenuation due to single scattering Consider a measurement of the total cross section Q [6,7]. The primary beam moves along the x-axis with a velocity ur and is crossed by a target beam with a velocity vz and a density in the scattering region nr(x) The relative velocity isg. The target will be described by ,$,

4 = @/ui>Jn,cx,

69

dx.

The factorg!ur results from the fact that the target molecules are not stationary_ For the most simple experiment we assume both an infiniteiy high angular resolution and a weak target (a -=Z1). The detector signal 1 is attenuated by a fraction $Q to &YlrfO) =

1-

5Q.

For an arbitrary target density this generalises to 1($)/1(O) = exp(-a)

= 1 - .~YQ + )E’Q’ - . .. .

A real experiment has a ftite angular resolution. For simplicity we assume that particles still reach the detector when they are deflected over a centre of mass angle 0 4 Bexp. For a weak target the attenuation becomes GW@) Qexp

= I - @an

=j

=

1-



a(@ 2 A sin 6 de = Q - Qfr(8exp).

(3)

QP

In a good experiment err/Q
fl + -

171

2.3. The deflection o

In this paper deflections are always given in the c-m. system. The symbol w describes the deflection, i.e. a displacement on the unit sphere, and is shorthand for (0, $). The solid angle e!ement is d’o = sin 0 de do. In the present case of small angle scattering 0 Q 1 we remain in the north-pole area which can be replaced by the tangent plane v&h polar coordinates 13(radius) and 0- The vector w now represents a displacement vector in this plane and the resulting deflection w from two successive deflections o, and wa is found by vector addition (fig. 4). Integration over all directions becomes integration over ‘he whole tangent plane. 2.4.17re transmission function q(w) This function is defined for the experiment shown in fig. 3 as “the probability that a particle, coming from the source passes the central diaphragm and reaches the detector While undergoing a cm. deflection-w” divided by “the probability that a particle coming from the source reaches the detector without deflection and in the absence of the central diaphragm?_ In these probabilities the emittance distribution of the source (homogeneous or otherwise) and the sensitivity distribution of the detector are implied. The detector signal in the situation of the denominator will be denoted by I,,_ This definition differs from some other definitions [IO] by the use of cm. deflections As a resnlt q depends on the velocity ratio v&. Elsewhere we describe a Monte-Carlo method to calculate TJfor ail ratio’s v& in one run [I I]. The fractional cross section Qf, must be replaced by the more general qu%n-

(4

The termd&& gi&s the contribution of molecules

&lch haveundergone a single collision over less tbari I3exp. For an arbitrary target density we must also look at molecules which come within BeXpafter multiple : colli&ns. The restof this paper dealswith j.be correct -.ex&msionof eq. (4) for an arbitrary taigit’dknsity. T $Ioreover, we *..aIlo*.for a generaltmnsni&on ftin@ion,?.. FirsS’yF. discuss .tJiode$lec_tioniw and the_:

SOUfce

yw&

detector

Fiw 3. Geometryfor a meayxementof the total co& section p and-thesmallangledifferentkla&s s&ion o(S). The detect& signalis measureda.Sa functionof the transversepesitionof tie cent& diaphtigm.

N.F. Serster et aL j Multiple

I72

cdlisi~ns in molecular beam scattering experiments

tity AQ. defmed as (5) br the case of elastic scattering by an isotropic potential u is independent of @so that AQ = s u(0) T(0) 2nB rk’ , 0 ri(e) = (2rr)-’ 7

r#‘, 4) d$

-

As long as all de5ection.s are srnail the velocity ur remains unaltered. As a result we need not correct _for _ an increased trajectory length while the lab-cm. transformation is the same-for all collisions. This means that, although in general deflections must be added in the lab system, we now may add cm. deflections to a total cm_ de5ection. The above, simplifications are no longer valid after a large angle collision. In that case, however, the particle is lost anyhow so that a correct treatment is not essential-. From the given basic probabilities we calculate the probability

0

For a general transmission function eq. (4) is written as r(o)/r,

= n(O) 2

4wo

= exp(-tQl

h(O) + E AQ I-

(6)

The first term gives the signal from particles which have suffered no collisions, the second term gives the contribution from particles which reach the detector although they have suffered a small angle collision. We have performed measurements of the total cross section Q and of the small angle differential cross section o(0) by u&g two centraI diaphragms, the 5rst one well centred with q(O) = 1 and AQ Q Q, the second slightly off axis or annular with ~(0) = 0 [4,551_ The fmt measurement gives Q with a correction due to u(6), .the second gives ~(0) with a correction due to Q (see fig_ 3).

3. The probability for mu!tiple scattering For the calculation of the probabilities we ditide the scattering region in N intervals of equal weight extending fromxi to xicl as given by

R,( or, ___,co,) d2 01 --- d* o,, that a molecule makes exactly n collisions, the fist with a deflection or, the next with a deflection oe etc. This requires N - n intervals without.scattering and n intervals with scattering over oi in d* oi. There are N!/(N - n)! choices of these n intervals out of N. These choices contain, however, all n! permutations of the scattering sequence while R, corresponds to the gwen sequence or, 02, .__. The total number of correct realisations is thus ($). Each realisation has a scattering probability

[’ - CW'lQIN-"

v (EDI 4-d

d*-i-

Multiplying with (3 and taking the limit N * 00 results in R,(oi,

__.,on) =.e-@(l/n!)

cn II doi) i

_

(7)

Integration of &over all oi gives P,, the probability for exactly n collisions (for simplicity we_assurne that all deflections are small angle collisions) with the result .-

173

N-F. Vets& et al_/MultiplecoEsiottsin molecularbarn seatieringexperimenfs

section. Substitution of eq. (11) in eq. (10) gives the final expression for the detector signal

R, over all variables Wi with the constraint

This constraint is dealt with in the usual way. New variables e$ are introduced according to 0: = Oi for i < n and e$ = C$, c+_ The constraint becomes e$, = o = constant. Tbe integration over R, can now be restricted to the (n - I) variables o;, .._, ok-r, resulting in

or, for a better comparison with eq. (), I/IO =

exp(-EQI hK0 + WQmd t

AQmult=jvW

~mult(4 d*o 9

%d~)

rcm”-%!l

= $

(14b)

%(~)

By deftition the zero order function is equal to Se(o) = e+Q s(o),

(I4a)

(W (9)

with 6(a) the two-dimensional normal&d deltafirnc-

with the cross section CJ,~~(O) describing all single and multiple collisions.

tion. Tbe detector signal I follows as J We -= n$o In(a)

S,(o)

d2W .

(10)

This expression describes the whole process of ,multiple scattering and replaces eq. (6). To achieve more insight, however, into the relation between eq. (10) and eq. (6) we rewrite the scattering probability S,(o) in teams of a new quantity oh(w) which is a generalized cross section for n-fold scattering_ The required relation is [cf. eq. (7)] S,(o)

= emEQ{@Q)“-r/n!}

5 UJW) _

(11)

From this definition of u,Jo) and from the equations foi S,, and R, fohow the properties of a, ob(e$=

Q8(-),

(124

&i=

o(o),

Wb)

k.i?l ..

d*01;

:-.

4. Numerical evaluation of the cross section for multiple scattering The functions u,(0) can be calculated by the recursion relation in eq. (12~) once the differentialcross section u(0) is known. In ref. [l] we have given an analytical expression which approximates u(0)/@!) for an inverse power potentiaI V(r) a f5 in the sm$ angle region 0 < 46e. This expression depends oniy on

s and S/f& In this paper we use Q/0$ as a scaling factor for CT&). By using u(O) = If(O) I* and the optical theorem Q = (k/k) hnf(0) from quantum meclutnical scattering theory and at&(O)] = 42 - zr/(s - 1) from

the Schiff-Landau-Lifkhitz G(O)= {41r ;os2 [rr/(s - l)]

model 189 J we derive

3-l

Q/6’, _

(1%

The differential cross section is scaled as

WC). (12dj Fo~~@IBe) -weuse the approximation of ref. 113,

-resultingin-_.-

..

:.

;

174

N.F. Verster et al. / Mulriple cd&ions

in molecular beam scattering experimetits

0

1. I

2

’

I

tile,

3

I

-

Fig. 4. The deflections WI and 02 add up to the resulting deflection w. The length B2 of wz foliows from 8 1,01, and 0 -

with ce given by eq. (15). The coefficients Ci from eq. (16) for s = 4 through s = 7 are given in table 1 and are indicated by R = 1, i.e. the differential cross section for single collisions_ This function has bee.lused as input for the calculation of o&7) with n = 2,3 and 4. The general character of (I has already been discussed, one half of the vohrme of the kernel u(w)@ is concentrated on a disk with radius 2Be. The remainder decreases rather slowly so that some care in the numerical evrduation is necessary. This is emphasized by the fact that the small angle treatment is not possible for the hard sphere limit s 4 = as the integral of o(0) over the tangent plane diverges. For s = 6 the convergence is slow but sufficient. The convolution can be written as

x

uP2@,

OI,@I)I@I

*I

d&

,

where ive have chosen G = (0, O), o1 = (SI, I#& a2 = (ff2, @a) as shown in fig. 4. The value of q!!ais unhnportant. From cer + o1 = e9 follows

e,=(e*+e:

-2288,

tisQR.

As a fmt step we have ‘tabulated the integral

Fig_ 5. The differential cross sections u,(B) for multiple scattering, as defined in eq. (12) and calculated with the differential cross section from fa :. as input The dotted lines give the simple analytical approximation in eq. (18).

integrals need to be evaluated as K(0, B 1) =K(6 I,@. The equation for cm becomes

For 6, %.i!Io and e1 9 f3 the integrand decreases as V3-*/s. For an integrand f(0) with the asymptotic character 8- we use the scheme

iV.F. Verster et al. /Multiple

175

co!lisions inmolecular beam scatterihg experiments

Table1 . for n = 4. At 0 = 3Be the effect ranges from 2.2% for Numericalvaluesofthecoefficientsi;ltheanalyticalapproxin=2 to ll%forn=4. ma!ionsofeqs.(l~ and(l8)foro,(e).Thevaluzsforn= 1 The analytical approximation of eq. (16) for c = aretheinputvalues.Fors=.!throughs=7theyate from u1 can also be used to represent cr2, u3 and u4. The ref. [l],fors=4.0and4.5 theyarecalculatedbythesame results aregivenin tablel-Basedon allcheckswe method are confident that these results are correct within 3% &Tn/Q~Co[l-C* siIl(c~x)iC~X]-~-l~ for 0/d, < 3. The results for s = 6 are given graphiwitix = (6/eo)* [eq. (16)] cally in fig. 5. The dotted straight lines represent a cc0 erp(-hx) [es. (18)1. simpler approximation for small B/B0 given by n

s

1 1 1 1 1 1 1

4.0 4.5 5.0 55 6.0

2 2 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4

=0

=I

=2

c3

0.318 0.205 0.159 0.136

3.568 3.247 3.394 3.544

0.494 0.545 0.547 o-554

4.592 3.496 3.216 2975

0.122 0.112

3.750 3.999

0.556 0.554

2.940 2.964

7.0 0.106 4.264 4.0 0.0724 O-468 4.5 0.0479 0.472 5.0 0.0380 0.510 5.5 0.0331 0.551 6.0 0.0300 0598 65 0.0285 0.652 7.0 0.0265 0.110

0.551 0.480 0.499 0.484 0.474 0.464 0.454 0.446

3.025 0.878 0.694 0.633 0.614 0.615 0.625 0.642

0.642 0.466 0.393 0.356 0.335 0.327 0.320

4.0 45 5.0 5.5 6.0 65 7.0

0.0297 0.0189 0.0145 0.0123 0.0108 0.0099 0.0092

0.130 0.131 0.140 0.149 0.161 0.175 0.191

O-453 O-446 0.429 0.420 0.412 0.406 0.400

0.356 0.288 0.265 0.257 0.257 0.261 0.267

0.324 0.245 0.215 0.203 0.195 0.194 0.191

4.0 '45 5.0 5.5 6.0 65 7.0

0.0153 0.039 0.0091 0.031 0.006i 0.039 0.0055 p.041 0.0047_ 0.044 O-0042 0.049 0.0038 0.054

0.445 0.431 0.414 0.407 0.401 0.395 0.390

0.189 0.192 0.154 0.152 0.141 0.136 0.137 _ 0.130 0.137 0.125 0.138 0.125 0.141 0.125

6.5

dl

W9 The

coefficient dr is aiso given in table 1_

5. Concluding remarks The importance of multiple scattering can be seen fromtbenumerical results.Asaf~texamplewetake a measurement of Q with an angular resohrtion eeXp= 0.5 eo, an intense cross beam giving EQ = 1 and s = 6. The correction AQJQ from eq. (4) or eq. (6) due to single elastic scattering follows from fig. 2 as 8%. Double scattering increases A& with a term [see eq. (1411. AQ2 = f(W

[es(OYor(O)]

AQI = 0-1XQl

AQI -

For .$Q= 1 the correction of 8% increases to 9%, multiple scattering is thus barely of influence for this type of experiments. The situation is different for a measurement of the small angle differential cross section a@). Double scattering gives a relative correction $(.$Q) 0&3)/u,(d) which is 12%l’or r;Q = 1,O = 0 and 20% for $Q = i ,8 = Oo.For these experiments multiple scattering is thus important even for ,medium cross beam densities EQ = 0.2. Multiple scattering should also be considered in other high precision experiments which show a strong attenuation of the primary beam. The main reason is that the factor exp(-a) overstimates the attenuation_exce~t in those cases where the-angular resohrtion is riiuch better than Bo_A moditication of exp(-a) to ex+@&,) with QuP = Q - A@ is in general not correct in higher order because the correction terms_& AQ)‘-and higher are not linear +I the transmSon.ft&tion 7$ A secoudary reasoRt0 con: : sider_multi~le&&tering is the hare effect. The nar- / -.. row haio due &l&st@ s&ah angle ~&tterin,g &fuses :. f+e~&t+J+i&lprofiI~ [4]. :. .-.

176

N-F. Venter et al. /Multiple collisions in molecular beam scattering experiments

References [l] H.C.W. Beijetick, P.M.A. yan dzr Kam, W.J.G. Thij=n and N.F. Verster, Chem. Phys. 4.5 (1980) 225. 121 R. Heibing and H. Pauly, 2. Phys. 179 (1964) 22. [3} S. Stolte, Ph.D. Thesis, University of Nijmegen, The Netherlands (1972). [4] (a) J.M. Henrichs, Ph.D. Thesis, Emdhoven University of Technology, The Netlicriancis (1579); @) J.J. Everdij, Ph9. Thesis, Eiidhoven University of Technol&, The Netherlands (1976). [Sj H.C.W. Beijerinck, P&A. van der Kam, D. Bol, W. Thijssen and N.F. Verster, Book of Abstracts, Xth International Conference on the Physics of Electronic and Atomic Collisions, Vol. 1 (CEN, Paris, 1977) p- 84.

[6] H. Pauly, in: Physi& [7j [8] [9j 1101

[ 111

chemistry, Vol. VIB, e&W. Jost (Academic Press, New York, 1975) p.553. J-P- Tommies, in: Physical chemistry;VoL VIA, ed. W. Jest (Academic Press; New York, 1975) p; 227. L-1. S&it-f, Phys. Rev. 103 (1956) 443. LD- Landau and E&f_ Lifchitz, Quantum mechfia (Pergamon Press, New York, 1959) p. i46. (a) P. Kusch, J. Chem. Phys. 40 (1964) 1;. (b) F. van Busch, Z. phys 193 (1966) 412;. (c) F. van Busch, J. Phys. B.: At. Mot Phyr 8 (1975) 1440. P.M.A. van dei Kam, H.C.W. Beijerinck, W.J.G. Thijssen, J.J. Everdij and N-F. Verster, Chem. Phys.. to be pubIished.