A molecular beam machine for the measurement of the scattering of polar diatomic molecules

A molecular beam machine for the measurement of the

scattering of polar diatomic molecules

Citation for published version (APA):

Everdij, J. J. (1976). A molecular beam machine for the measurement of the scattering of polar diatomic

molecules. Technische Hogeschool Eindhoven. https://doi.org/10.6100/IR101263

DOI:

10.6100/IR101263

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Published: 01/01/1976

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A MOLECULAR BEAM

CHINE FOR THE

MEASUREMENT OF TH . SCATTERING OF

POLAR DIATOMIC

OLECULES

A MOLECULAR BEAM MACHINE FOR THE

MEASUREMENT OF THE SCATTERING OF

POLAR DIATOMIC MOLECULES

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGESCHOOL EINDHOVEN, OP GEZAG VAN DE RECTOR MAGNIFICUS, PROF. DR. IR. G. VOSSERS, VOOR EEN COMMISSIE AANGEWEZEN DOOR HET COLLEGE VAN DEKANEN, IN HET OPENBAAR TE VERDEDIGEN OP

VRIJDAG 5 MAART 1976 TE 16.00 UUR.

DOOR

JACOBUS JOHANNES EVERDIJ

Dit proefschrift is goedgekeurd door de promotoren

prof.dry N.F. Verster en pröf. dr. ir. H.L. Hagedoorn

J.J. Everdij, Thesis, Eindhoven University of Technology, 1976.

ERRATA.

1. In all equations in Chapter 2 and Appendix A Planck's constant h must be replaced by

h.

2. The polarizabilities ~ and A accuring in equation (2.6) and (2.10), res-pectively, which .do not carry a subscript, have to be replaced by ~ .1. and

'"'.1."

3. In equations (2.9) and 2.10) are brackets absent:

=

:~

[1

+ ••.••. _P3 (cos

y)}J

4. In Tab.le 2.2. the numerical values of A(s) and C(s) are wrcmg. The correct values are: A(s) B(s) C(s) s 5 2.261 - 3.393 0.3951 5. Equation (4.18) reads: 5.5 2.013 - 3.544 0.3755 6 1.889 - 3.751 0.3582 6.5 1.834 - 3.999 0.3434

6. Equation (5.44) and the four following lines must be replaced by:

M/F = 7f y1 0 2 v(v-l)(x -.1)/x 2 4

where x

=

g/vl"

Te

correction reaches its maximum value for x

=

12.

The width of the skimmer s l i t (0. 777 mm) . . . (etc.).

7. Equation (5.45) reads:

-5

Aan mijn ouders

Aan Tiny

CONTENTS

1 , INTRODUCTION

1.1. Aim of the experiment 1. 2. 2. 2.1. 2.2. 2. 2.1. 2.2.2. 2. 2. 3. 2.2.4. 2.3 2. 3 .1. 2.3.2. 2.4. 2. 4.1. 2.4.2. 3. 3.1.

Contents of the thesis

SCATTERING THEORY Introduetion

The intermolecular potentlal (IP) Introduetion

Long range dispersion part of the IP Long range induction part of the IP

General representation of the angular dependent IP Theory Df the elastic scattering

Elastic scattering in classical mechanics Quantummechanical treatment of the elastic scattering

Inelastic scattering Formal theory

Evaluation of the transition probability for small scattering angles

STATE SELECTION Introduetion

3.2. IMPROVED SPACE FOCUSING OF POLAR DIATOMIC MOLECULES IN A SYSTEM 'OF QUADRUPOLE AND HEXAPOLE FIELDS 3.2.0. 3. 2.1. 3.2.2. 3.2.3. 3.2.4. 3.2.5. 3.2.6. 3.2.7. Abstract Introduetion

Theory of focusing and its limitations in a quadrupale field

Compensation of the spherical aberration Numerical results for an actual lens Apparatus Experimental results Acknowledgements 3 5 5 5 5 7 11 13 15 15 18 27 27 31 37 37 39 39 39 40 46 47 49 49 52

3.3. 3. 3 .1. 3.3.2. 3.4. 3.4 .1. 3.4.2. 3.4.3. 3.5. 4. 4.1. 4.2. 4.3. 4. 3.1. 4.3.2. 4.4. 4. 4.1. 4.4.2. 4.4.3. 4.5. 4.6. 4. 6. 1. 4.6.2. 4.6.3. 4.6.4. 4. 7. 4. 7 .1. 4.8. 4.8.1. 4.8.2. 4. 8. 2. 0. 4.8.2.1. 4. 8. 2. 2.

Some special topics in the lens design Exact numerical calculation and tabulation of the Stark effect

Practical realization of two dimensional multipole fields

The actual lens systems Dimensions

Construction

Performance of the state selectors

Chromatical aberration and influence of the velocity selector

TECHNICAL DETAILS OF THE APPARATUS The molecular beam quality

Design of the apparatus General provisions The cryopumping facility The computer facility Primary beam souree Construction

Temperature-control system

Effusion behaviour Primary beam monitor Secondary beam

General features of a supersonic beam

Construction

The gashandling system Performance

The velocity selecter Control of the selecter The detection system Introduetion

DESIGN OF A MOLECULAR BEAM SURFACE IONIZATION

DETECTOR IN COMBINATION WITH A QUADRUPOLE MASSFILTER

Abstr>act

Intr>oduction

Description

of the detector system

53 53 56 59 59 63 65 67 71 71 73 74 74 75 75 76 77 81 83 83 83 85 87 89 91 91 93 93 95 95 95 96

4. 8. 2. 3.

Theory

4.8.2.3.1. Behaviour of a eylindrical wire in a

homogeneaus electric field

4.8.2.3.2. The complete ion

optical

system

4.8.2.4.

Experimental behaviour of the detector

4.8.2.4.1. Detection using quadrupale massfilter

4.8.2.4.2. Sensitivity

4.8.2.5.

Conclusion

4.8.2.6.

Acknowledgments

4.9. 4.10. 5. 5.1. 5.2.

Diaphragms in the experiment Automation of the apparatus

EVALUATION OF THE CROSS SECTION MEASUREMENTS Introduetion

Fundamental set-up of a scattering experiment 5.3. The transformation between center of mass and

97 97 101 103 103 107 107 108 109 111 115 115 116

laboratory variables for small scattering angles 118 5.4. 5. 4.1. 5.4.2. 5.5. 5.6. 5. 6. 1. 5.6.2. 5.6...3. 5.6.4. 6. A. Angular resol~tion

Method for the measurement

or

the differentlal con tribution

Calculation of the -angular resolution function Velocity resolution

Preliminary scattering measurements and analysis Beam profile

Expe~imental velocity dependenee of the elastic total cross sectien between ~sF and Ar

Measurement of the differential cross sectien b~tween CsF and Ar

Analysis of the differentlal measurements

CONCLUSIONS Appendix Reierences Summary Samenvatting Nawoord 119 119 122 130 133 133 134 136 139 145 147 151 155 157 159

1. INTRODUCTION

1.1. Aim of the experiment

The interaction between two molecules is ruled by an inter-molecular potential (IP). Such a potential depends in a complicated way on the distance r between the molecules and the orientation of the molecules.

A variety of experimental and theoretical methods can be used to determine the IP. Each method, however, probes only a limited aspect of the IP and a complete survey of the intermolecular behaviour is, if possiblé, only obtained by fitting all the data together.

This thesis describes an experimental method to determine the long range, angular dependent part of the IP between a polar diatomic molecule and a spherical symmetrie partner. The method contains the study of the scattering behaviour of the molecules in a crossed beam experiment. The primary beam consisting of polar diatomic molecules at thermal veloeities {_. 0.1 eV), is selécted in a specified rotational state by means of an electrostàtic, inhomogeneous field befere the scattering center, where it crosses the (supersonic) secondary beam under an angle of '90°. By means of a secend State selecter, followed by a velocity sel~ctor and a partiele detector, we study the consequences of th~ scattering process on the primary beam, i.e. the behaviour of the total and differential elastic cross sections plus the ~ransition probability of a cellision induced transition to another rotational state.

The 1.ntermolecular potential, in particular the angul-ar dependent terms, plays an important rele in such processes as the thermal

relaxation of gases, chemical reaction and the supersonic flow of molecules.

The theory -of the cellision between two spherical symmetrie particles (for instanee rare gas atoms), although difficult, is intensively studied and reasonably known, whereby a wide variety of experiments is. available for comparison.

The exact scattering theory of our cellision partners is yet too complex, although a large number of 'practical' and. more or less difficult theories are developed during the last years (JOH 66, GOR 6~

SAM 69, LEV 71, GOR 73). The quantity of produced theoretica! papers, however, has no correspondence with the quantity of available

experimental results. Only a few experiments are available for inter-pretation (BEN 64, BEN 69, TOE 65, STO 72, MOE 74). Our aim is to add a piece of experimental information for the determination of the intermolecular potential, whereby a relative simple 'practical' theory is used for the interpretation.

To o6tain valuable information from the measurements, the following conditions have to be fulfilled:

- a high angular resolution for the total cross sectien measurements to limit the corrections on the experimental data,

- a knowledge of the differentlal cross sectien in order to calculate the correction mentioned in the _preceding point,

- a sensitive primary beam detector for the detection of the very small beam intensities resulting from the small transition probabilities,

- a stable primary beam to allow long measuring times,

- a very intense and high quality cross beam to get a measurable inelastic scattered signal,

- a thorough knowledge of the focusing properties of the electre-static state selectors, especially concerning the behaviour of the differentlal transmission,

a large aperture of. the state selecters for a maximum transmitted intensity especially for the (1,0) rotational state.

This thesis deals for a large part with the experimental equip-ment which has been developed to fulfill the primary requireequip-ments. The experimental results on scattering are restricted to the elastic scattering of unselected molecules and a qualitative observation of the transitions between the rotational states (2,0) ~ (1,0) of the CsF molecule after scattering with argon.

1.2. Contents of the thesis

In chapter 2 the theory of the elastic and inelastic cellision is discussed. A derivation is given of these terms in the inter-molecular potential which play an essential role in the elastic and inelastic scattering processes relevant to our experiment. Classical as well as quantummechanical expressions of the total and differential cross sections for the elastic scattering are derived. By means of a half classical theory the transition probability between two

rotational states is calculated.

Much attention is paid to the design and construction of a set of state selecters in chapter 3. The improved working of the selecters is shown in various measurements.

In chapter 4 a survey is given of the experimental set-up. All devices incorporated in the apparatus are reviewed. A description of the generalprovisionsbeing present in the Eindhoven molecular beam section, and forming an essential part of the apparatus, has been copied from Beijerinck (BEY 75). The most critical devices, seen in the light of the experimental requirements, have got special

attention such as the primary beam source, the secondary beam and the detection system.

Chapter 5 deals with the first scattering measurements. First of all we derive how the total elastic cross section can be extracted from the scattering experiments. Special attention is paid on the correction due to the angular resolution. A new measuring method is introduced enabling us to study the elastic differential scattering at small deflection angles. The evaluation of the measurements to get the differential cross· section is discussed in detail. The first

scattering measurements are presented together with a preliminary analysis.

2. SCArrERING THEORV

2.1. Introduetion

In this chapter we study the elastic and inelastic scattering between a polar diatomic molecule and an atom. A detailed study of the intermolecular potential, ruling the scattering process, is given in sectien 2.2.

A generalized Van der Waals potential satisfies for the

description of the elastic scattering. In sectien 2.3. we shall derive the expressions for the elastic total and differential cross sections in the classical as well as the quantummechanical approximation.

In sectien 2.4. the inelastic scattering is studied for a transi-tion between two rotatransi-tional states of the molecule (in its vibrational ground state). It will turn out that these transitions are evoked by the angular dependent parts of the potential.

2.2. The intermolecular potential (IP)

2.2,1. Introduetion

Befere the problem of two colliding molecules can be approached from the theoretical side, the IP must be known. First of all we investigate therefore the types of potential terms which play a rele in the scattering process.

Usually the interaction is divided into a long range attractive and a short range repulsive part. Because of the fact that all scattering processes, being studied in our experiment, deal with small deflection angles, only the long range part of the IP is

scanned and the interactions are very weak. For this reasen we do not pay further attention to the short range part of the potential in this thesis.

In the most simple case we have two neutral spherical cellision partners (for instaneetwoof the rare gas atoms). The long range IP

resulting from dispersion farces, is then spherically symmetrie and is in good approximation given by (PAU 65)

V(r)

=

(2 .1)

and represents the dipole-dipole dispersion interaction. The constant

c

₆ is called the Van der Waals constant. An a priori value of C6 can be obtained by a perturbation calculation, a tractable approximation reads (LON 30, MAR 39)

(2 .2)

where

ai

and

Ii

are the polarizability and the ionization energy, respectively of the ith scattering partner.

With decreasing distance the higher order dispersion farces also have to be taken into account (dipole-quadrupole dispersion inter-actions etc.), resulting in a series

V{r)

=

(2. 3)

Replacing one of the two scattering partners by a polar diatomic

molecule with a permanent dipale moment ~ has two consequences. In the first place also dipole-induced multipale moment interactions appear in the IP. Generally one refers to them as the induction terms of the potential, in contrast with the mutual induction terms, contributing to the dispersion farces. These farces depend on the angle between

the separation vector ~ of the two partners and the axis of the polar diatomic molecule. Secondly we see that also the dispersion farces become dependent on this angle since the polarizability of the

diatomic molecule must be described by the tensor (a// , a _

instead of a scalar.

a

The appearance of angular dependent terros in the long range potential, i.e. the combination of the dispersion potential (treated in sectien 2.2.2.) and the induction potential (treated in sectien 2.2.3.), allows callision induced transitions between the rotational

states in the molecule. The aim of the experimental set-up, described in this thesis, is to measure these inelastic processes,

The experiment also requires a detailed knowledge of the elastic scattering process. In good approximation this process is described by the over all directions averaged potential of the form

V

=

(2.4)

Empirically i t is found that

s

has the value

s

=

6 or a slightly deviating value. Only with special provisions i t is possible to measure the very small influence of the anisotropy (P_{2 (cosy)} dependencel of the pótential on the elastic total cross section

(BEN 64, STO 72) of a state selected molecule.

2.2.2. Long range dispersion part of the IP

In the usual approach of the long range part of the !P the charge distribution of the molecule, either permanent or induced, is expanded in multipele moments. The electrastatic interaction between the multi-poles of the two molecules then gives the various terros in the IP. The following property of a multipele expansion is of special importance:

The value of a multipele moment depends in general on the choice of the reference point. The lowest moment, in our case the dipole moment, is however independent of this choice. The next higher moment, in our case the quadrupele moment, can he reduced to zero by the proper choice of a reference point.

Due to the fact that in the molecule

X

Y a dipole has been induced,

there exists a reference point (M), obviously lying on the molecular axis, where the quadrupele moment varrishes (see figure 2.1). M is sametimes called the center of dispersion forces. It may be expected that the IP has its most simple form with respect to the point

M.

On the ether hand we want to describe the relative motion of the partners with respect to their center of masses and the rotatien of the

molecule around its center of mass

C.

The distance between

C and

M is denoted as ó , from which is

supposed that i t is in first instanee independent of the orientation of the molecule.

y

_Yo

A

CM· 6

co-

d

Figure 2.1 Coordinate systems used for the description

of

the

inter-molecular potential of the collision between the molecule XY

and

atom

A. C

is the internal center of maas of the molecule, M the center of

dispersion forces

and

D the dipole center.

\ \

'

isopotential curve a ~...,.-_,1::. a

i

potential deviation V Cr +fl) ·V Cr)

0 2Jt

b

Figure 2.2 Part a. shows th

e

influence of a shift (M

C

) of the

referen

ce

p

o

int on the intermolecular potent

i

al. With r

e

spect to point

M the potential

is isotrope

and

an

isopotenti~l

curve

(

a)

is

spherically symmetrie. With respect to point C (shift MC

<< r~

we

have in first approximation the same radial dependenee

(

dott

e

d

isopotential curve (b)). The correction

is given

by

the angular

dependent distance

8 (

figur

e

part b) •

The dispersion part of the IP taken with respect to the point

M

is in good approximation again

v.'lisp

=

-C. / 6

M 6 rM (2.5)

and spherically symmetrie. We have to add an angular dependent term descrihing the non-isotropie behaviour of the molecular polarizability. The potential reads then (PAU 651 BUC 67), neglecting

P4

and higher order terms where and

q2,6

=

(CJ./1 - CJ.

CJ.M

=

(a//

+ 2CJ.

(2.6)

)/(3 CJ.M)

is the anisotropy factor,

)/3 is the mean molecular polarizability

3

2

CJ.A

CJ.M

IA

IM

(4n~o)2

IA

+

IM

is the Van. der Waals constant in the same approximation as in equation

(2.2).

If this potential is slightly displaced over a distance

o

(

o

<< rM ) which is equivalent to an opposite and equal displacement of the atom,

(isopotential curve a in figure 2.2.a) in first instanee the same radial dependenee of the potentialis observed (isopotential curve b). Obviously also a P₁(cosy) dependent correction term must be intro-duced (figure 2.2.b). The displaced potential V is easily calculated

c

by taking the first terms in a Taylor series expansion

~isp (r, y)

c

avM (r, y)

=

VM(r,y) + (rM-r) 'ilrM + (cosyM - cosy) oVM(r, y) ---".._-'---- + ... <>COSYM

Within good approximation we have

2 ö

cosyM - cosy

=

3

r

(P/cosy) - 1)

(2. 7)

(2.8a)

which results in the potential ~isp (r, y)

c

(2.9)

For most molecules the terms with

~

q_{2, 6 can be neglected} because of the small value of

q

₂J_{6 ("'} 0.25).

The position of the center of dispersion forces

M

is to some extent uncertain. A practicable choice seems to be the center of the charge of the electron cloud (REE 71). Even very asymmetrie molecules like CsF still have nearly spherically symmetrie electron distribu-tions, so the midpoint of the molecule is a very reasonable choice of

M.

For the molecule CsF for instance, the distance ó is then 0.88 ~ calculated with a Cs-F equilibrium distance

re

=2.345 ~. Assuming an interaction distance of about 8 ~ _{, the coefficient of the P1 (cosy)} term inside the square brackets becomes about 0.7.

For comparison we give ···also the dispersion·-potential -as de.rived by Buckingham (BUC 67), using a strictly formal treatment of the multipele interactions. He gives

x

(2.10) where

All

and A~ are the higher order polarizabilities of the molecule, representing a combination of both the value of the dipole moment induced by a field gradient and the value of the quadrupele moment induced by a uniform field, parallel and perpendicular to the molecular axis, respectively.

This potential shows where our description in terros of a fixed displacement ó fails. In our approximation we assume that there exists

a direct relation between the anisotropies of the polarizability a and the higher order polarizability

A

of the molecule. This is in general not the case. The consequence is that there exists no unique center of dispersion farces relative to which the r-7 contribution is

zero for all values of the angle y. Herman (HER 66) introduced the center of dispersion farces by taking the point where the term r - 7 P 1 (cosy) vanishes. Unfortunately, reliable va lues for Al

I

and

A

_~

are not yet available for any molecule and an estimate of the order of magnitude of the potentlal terros is scarcely possible. Therefore we prefer the potentlal of equation (2.9) which gives in our apinion a clear insight in the problem and a correct order of magnitude of the terms.

2.2.3. Long range induction part of the IP

Because the molecule is assumed to have a permanent electrical dipale moment ~ , a reference system can be chosen where the q uadru-pele moment

Q of the molecule vanishes. The origin lies on the

molecular axis and is usually called the dipole center (point D in figure 2.1). The value of the displacement DC

=

d fellows immediately from the definitions of the dipale and quadrupele moment (BUC 67)

lJ

=

I

e . r.

i

-z. ----1-(2 .11)

~

I

2 2 Q

=

_Qzz

=

e. (3z. - r .

i

-z. -z. -z. (2 .12)

ei

is the i-th element of charge at the point

!i

relatively to the chosen origin. The value of d is

d

=

Q I (2)J) (2 .13)

The electric field ~)J due to the dipale at the place of atom A , induces a dipale moment

.l!,q

=

~ ~IJ in the atom. The interaction of the

field of the dipale

.l!,q

with the dipale of the molecule causes an interaction energy

(1 + P₂(cosy)) (2.14)

Table 2.1 Estimated or known values of the quantities appearing in the intermolecular potential between Ar and CsF or TlF. Quantity I lJ Q Unit a) 10- 29 Cm 10-40 _Cm 2 -1 _V (eV) (D) Ar 25.12 (15.68) d) 1.82 (1.64) h) CsF 16.0 (10) c) 2.630 (7.884) f) - 5.72 (-3.57) g) 3.69 (3.32) c) 4.6 TlF 17.9 (11. 2) 1.410 (4. 228) - 3.75 (-2.34) 6.97 (6.27) 8.7 e) f) g) i) 2 -1 10-40 Cm V (4.1) (7 .8) j) q2,.6 1' e ê d

C6

(calc) 2 aA lJ ( 4Tre ) 2 0 10- 10 m 10- 10 m 10- 10 m 10-79 Jm6 3.2 (2.9) 0.122 c) 2.3453 f) 0.88 1.09 80 51.0 222 231 6.1 (5.5) j) 0.122 j) 2.0844 f) 0.86 1. 33 161 14.7 78 204 (g =503 m/s) (g = 305 m/s) a) For the conversion between the esu system and the SI system the

following relations are used:

9 -9 2 -2 -1 2 2 -1

1 C = 3 10 esu, 4Tre = 1/9 10 C m N = esu cm dynes

-18

°

-29

1 D = 10 esu cm = 1/3 10 Cm

b) When changing from the esu to the SI system one should first replace the polarizability a by à= a/(4Tre0 ). The quantity à has

the dimension (length) 3 • c) estimated value. d) (HOD 64) e) (MUR 66). f) (LAN 74). g) (HON 73). h) (ROT 59). i) (SIL 27). j) (BEN 64). k) With

the help of equation (5.1) we find:

c

6

(ex~)

=

hg

(Q(g)/8.083) 2"5 (fors= 6); for the total cross sectien we take experimental values (TOE 74).

Expanding this potential part with respect to the variables r and y , defined relative to the center of mass

C

1 in a similar manner as described in section 2.2.2. yields the result

. d ~n (r, y)

=

12 2 etA \l

6

_{2 (1} + _P2(cosy}} T'

(4TTE )

0

The secend term may be interpreted as the quadrupole-induced dipole interaction and is usually written (PAU 65)

d

~nd

QD

For CsF we find with

Q

1.09

R.

At a distance r

(2. 16)

3.57

eR

2 and \l

=

1.642

e~ the shift

8

R

the value of the secend

term in equation (2.15) exceeds that of the first term. This means that for a correct description of the potential function further terros in the Taylor series expansion (r-8and higher) should be included. However, for a description of the interaction only the first terms are of interest.

2.2.4. General representation of the angular dependent lP

For the IP a notatien in Legendre polynomials has been chosen because of the fact that in future calculations the rotational behaviour of the molecule has to be treated quantummechanically, implying the expansion of the rotator wave functions in spherical harmonies, which are very-closely reiated to the Legendre polynomials.

In the previous sections (2.2.2. and 2.2.3.) we obtained the first angular dependent terros via a displacement of the crigin of the

reference system and taking the first terms of the Taylor series expansion. This method can be generalized in order to obtain the higher order terms of the expansion in the powers of r-l as well as

Figure 2.3 Classicalelastic scattering in a Lennard-Jones potential.

Top: Lennard-Jones potential

(equation (2.21)).

Middle: Classical trajectories of

the incident particles for

different impact parameters b. at

2

an

energy~g

=

2c.

Bottom (right): Classical

scattering angle

9

as a function

of the impact parameter b

(equation (2.20)).

Bottom (left): The three

eontributions to the classical

differential cross section.

The

numbers 1. 2

and 3

refer to the

different contributing impact

parameters in S(b

)

. (BEY 75)

100 10 (T(-1)-) 0.1 arb. units -1t

the Legendre polynomials. General expressions for the terms are evaluated by (REE 71a, MON 72).

The total long range, angular dependent potential may be written in the following form

V(r. y)

=

V (r) +

L

0 L=l

(2. 17)

with

(2 .18)

For the most important induction and dispersion terms, discussed in sections 2.2.2. and 2.2.3. the coefficients a are

n,L

- c6 2

I

2

a6,0

=

- CI!A IJ (4TrE ) (2 .19a) 0

6

_-~

2 2

a?. 1

=

- 6C6 - ::g:6 oq2 6 ₅ Cl! A IJ / (4TrE0 ) (2.19b)

J 2 2 a6. 2

=

c6 q2.6 - CI!A IJ /(4rre ) (2 .19c) 0 - 24 c 24 2 2 a7,3

=

oq2 6

- 5

CI!A IJ d / (4TrE₀) (2.19d) 5 6 J

In table 2.1 we have presented a survey of the presently known or estimated numerical values of the quantities appearing in the IP of the scattering partners CsF-Ar and TlF-Ar.

2.3. Theory of the elastic scattering

2.3.1. Elastic scattering in classical mechanics

·The treatment of the scattering problem is simplified by a trans-formation to a reduced reference frame where an incoming partiele with reduced mass IJ =

m

₁

m

₂

/

(m

_{1 +}

m

₂J moves with a position

~

=

~l - ~_{2 and a velocity} ~

=

~l - ~_{2 in a fixed potential field}

V

(r) ,

where

(m. •

!'. •

v.)

are the mass, position and velocity of the

- t. -z. -z.

partiele i in the laboratory (LAB) system, rèspectively. Scattering results obtained in this reduced system are equivalent to those in the center of mass (CM) system where the center of mass is at rest.

In the case of a spherically symmetrie potential

V(r),

the

scattering angle 6(b) for an incident partiele with impact parameter b is given by 00 e(bJ

=

11 - 2

f

r c

2

2

2 2

~ b

dl"/(r

(1 -

2V(r)/( 11g )- b

r ) ) (2.20)

where re is the classical turning point, i.e. the zero of the

denominator in the integrand. In figure 2.3 the general behaviour of scattering angle 8 as a function of b is plotted for a Lennard-Jones potential (LEN 24)

12 6

V(r)

=

E

((rmfrJ

-

2(rmfrJ

)

(2. 21)

where E and rm are the depth and the radius of the potential well minimum.

The differential cross section o(6) is defined as the intensity per unit of solid angle divided by the incident flux and is given by

0(8)

=I

bi

I

(sin

e

1:!1~

)

i

(2.22)

where the summatien is carried out over the three branches sketched in figure 2.3 (bottom).

In our experiments we only cbserve particles which are scattered over a very small angle

e .

We make the essential assumption that this small angle is always due to a large value of the impact parameter. We ignore the contribution of the small impact parameters (between the regions 2 and 3), the so-called glory region. The small deflections at large impact parameters can be calculated by the 'high energy'

approximation. The trajectory of the incident partiele is assumed to remain unchanged under the influence of the potential. The deflection of the trajectory is solved as a perturbation of the straight line trajectory.

We use for the interaction potential the long range attractive potential with the general form

V(r)

=-

C

/r8 _{(see equation (2.4)).}

8

gives the deflection function

(2. 23)

where f(s)

=

lrr r({s-1)/2) jr(s/2) (2.24)

áome numerical values of f(s) can be found in table 2.2.). We may calculate the differential cross sectien with the help of equation

(2.22), using only

i=

1. The result is 1 2(s-1) f(s) Cs a

re

J

₌

( 2 c~ s IJg The total cross sectien Q is defined

7f

Q =

2rr

f

a{e)sinS dS 0

;2/s

e-(2s+2)/s (2. 25) as (2.26)

and is singular due to the infinite range of the potential. Sametimes an 'incomplete' total cross sectien is introduced, defined as

a(e) 2rr sine dS

=

rr b2 (X )

0

where b(X_{0 )} is the impact parameter corresponding with X₀ We find Q(

x )

0 2(s-1) f{s) Cs) 218

=

7f (---~----~. )J

rl

xo

(2 .27) (2.28)

The expression e(b) may be used until the deflection angle becomes sa small that the uncertainty relation between momenturn and position sets a limit. This is in the region where

8crit ~ h j ()J g b) (2.29)

2.3.2. Quanturn meéhaniéal treatment of the elastic scattering The interaction potential again is assumed to have the general

form

V(r)

= -

C lr

8 _{since the same experimental restrictions hold.}

s

In the standard treatment of the scattering by a central potentia~

the scattering amplitude is introduced. It is defined by the behaviour of the wave function of the relative motion at a large interpartiele

separation r (after scattering)

where z

=

rcose is the coordinate in the axial direction

f(6) k

the scattering amplitude ~ g

I

h is the wave number.

The elastic differential cross sectien is given by

a(e)

=

lf(e)

12

and the elastic total cross section by

11 · --"Q-=

·

j

lf(6) 12 2ir sin6 d6 0 (2.30) (2. 31) (2. 32)

After expanding ~ in Legendre polynomials (method of partial waves)

the scattering amplitude can be expressed in terms of the phase

shifts

nz(k)

f(eJ

=

~k

"'

_{2in 2}

L

(l+t) (e - 1) P~(cose) ~=o (2.33)

The phase shift n~ is equal to the phase shift between the solutions

of the l-th order radial wave equation in the perturbed and unper-turbed case, respectively.

The radial wave equation can be solved with the assumptions that

we have large

l

and/or small

nz

values, i.e. small scattering angles

8 < 0.1 (BER 66).This Jeffreys-Born approximation yields for the phase shifts

=

f(s)

C

I

(h

g

b

8 -

1

_J

where

b

=

rl+~J

I

k (2. 35)

and f(a) was given in equation (2.24).

This approximation is similar to that used in the 'high energy' calculation of

e(b).

The phase shift

n

z

can be interpreted as one half of the phase deficiency between the phase of the unperturbed straight line trajectory and the perturbed one at large values of the impact parameter b (BER 66, BEY 75). This is known as the semi-classica! equivalence relationship.

The derivation of an expression for the total cross section is a straightforward matter, using the optica! theorem. This theerem states that

4n

Q

= k

Im (f(o)) (2. 36)

where f(o) is the value of the scattering amplitude at

a

=

0. We will calculate f(o) by replacing the summatien by an integration over all l-values in equation (2.33).

Prior to the calculation of Q and cr(e) we introduce some reduced parameters. These are

c

ks

ll(s-1) Ijl

=

(-8--) 2 (2. 37) J.Jg 1'

=

u

tjl

I

k (2 .38) and

a

=

Ijl -1

u

(2.39)

The quantity Ijl is a scale factor for the l-values. It is chosen such that l-values in the neighbourhood of l

=

Ijl mainly contribute to the sum in the expression for the scattering amplitude (equation (2.33)). The phase shift expressed in units of Ijl is

) (l+~J 1-s

nl

=

f(s Ijl

and has a value close by one if

l

=

Ijl •

The quantities

ru

and

9u

are scale factors for length and angle,

respectively. In table 2.3 the expression for

9u

is compared with

other angular units which are in vogue.

The imaginary part of the scattering amplitude, required for

calculation of

Q ,

reads

Im(f(o))

=

(2f(s))2l(s-1)

_s-1

f

(

2n)-(s+l)l(

8 -

1

) (

1-cos

(

2n)) d(2n)

0

The integral can be solved analytically and gives the result

Im(f(o))

=

'''r

n(2f(sJJ 2

1(

8 -1 )

I

(

4sin(

n )

rr~

1

JJ

'~' u 8-'1

8-This yields the expression for the total cross sectien

Q

=

F(s) r>2 u

where F(8}

=

n 2 (2f(8}) 2

1(

8 - 1 )

I

(sin( 8:1) r( 8:1))

Some numerical values of F(8) are given in table 2.2.

(2. 41)

(2.42)

(2 .43)

The differential cross sectien at 9; 0 can be calculated in a

similar way. The evaluation of

Re(f

(o

))

is carried out similar to

that of the imaginary part. With this result we find

and

Re ( f( o))

I

Im{f(

o))

=

tg(~1)

s-o(o)

=

(1 + tg2 (--n--)) k 2

(

~)

2

8-1

4n

(2.44)

(2 .45)

For very small angles 9 the differential cross sectien o(9) may be

calculated by replacing the Legendre polynomials

Pz

(

co

s6)

in the

scattering amplitude by the corresponding series expansion. For large

values of l this may be represented by

(l+~)

2

₉₂

₊

_{(l+

_~J

2

_{- l+~}}

₉₄

_{+ •.••}

The final result in which all terms containing

e

4 are omitted, equals (BER 66) where a(S) a(o) f 2(s)

=I

tg(2n1J(r(2/(s-1)))2/r(4/(s-1)) n

s-(see table 2.2 forsome numerical values of

f

₂(s)).

Substitution of the angular unit 8u yields

a( 8) 1- /7(s)(8/8u) 2

OTöJ

=

"

where

(2f(sJ) 2/(s-l) r(2/(s-1)) cos(n/(s-1)) fis) = ar/4/(s-1)) cos(2n/(s-1})

(see table 2.2 forsome numerical values of

J

₃(s)).

Sometimes one expresses equation (2.49) in the form

(2.47)

(2.48)

(2.49)

(2.50)

(2. 51)

for easy calculation,but only the first two termsin the series expansión of the exponential function are reliable! An analytica! function describing cr(S) for somewhat largervalues of 8 cannot be derived with the same method because of the fact that the integration

4 over the range of l-values explodes if terms of the order 8 and higher are included in the series expansion of the Legendre pöly-nomials. Even the term with 8 2 is doubtful as the coefficient

f

₂(s)

becomes infinite fors 5 (equation (2.48)).

Unfortunately, there exists a 8 region where neither equation (2.49) nor the classical expression of the differential cross sectien (equation (2.25)) are valid. The latter reads for smallangles in the classical region in reduced form

=

f4{s) r2 $2 (8/8 )-2(s+l)/s

Table 2.2 Numerical values of a number of coefficients depending on the power of the general interaction potential.

s 5 5.5 6 6.5 f(s) 0.6667 0.6243 0.5890 0.5591 F(s) 9.0930 8.5036 8.0828 7.7695 2 'TT 2.0000 l+tg (-1)

_s-

1.7041 1.5279 1.4130 f/s) 6.6524 4.1402 3.3208 f3(s)

_"'

1.1254 0.6658 0.5133 f/s) 0.3907 0.3406 0.3010 0.2690 A(s) 2.182 2.280 2.413 2.573

B(sJ

0.546 0.555 0.557 0.555 C(s} 2.010 1. 913 1.889 1.908 where (2. 53)

(see table 2.2 forsome numerical values of

f

_{4 {s)).}

In the intermediate region the differential cross sectien has to be

calculated numerically, carrying out the summatien over l-values in

the scattering amplitude directly (VER 75). It turns out that for an

accuracy of cr(S) better then 0.1 %we must include the terros up to

l _{ma x}

=

lOw . The region 0.8

w

< l < 1.2

w

gives the main contribution.

A difficulty arises in this calculation due to the fact that the

contribution of the terms with l < 0.8

w

(ran.dom phase region,(BER 66))

shows a fluctuation in cr(S) in the order of 5% if the value of

w

is slightly changed.

This problem can be avoided by convoluting the differential cross

exp - (2.54)

2

where w~ is the varianee of the functionl w~ << ~₀ • We then have

a (e, ~ J

=

f

a(e, ~) D(~ - ~ J d~

exp o o (2.55)

This is a legitimate eperation since the quantity ~ depends directly on the relative velocity g determined by an experiment and always provided with a certain distribution. After the calculation of

a (81 ~ ) is carried out per 8-value and fora few values of

w

1

exp o ~

we may extrapolate to W~

=

0. This yields the numerical value of a(8) where the fluctuations have been eliminated,

We have calculated a(8) in this way for a number of s-values. In figure 2.4 we have plotted a(8) for s 6 (curve al 1 together with the

differential cross sectien calculated in equation (2.51) (curve cl. There exists a streng discrepancy at somewhat higher values of

e.

In a previous experiment (BUS 67) this discrepancy was found in a

phenome-nological way and could be minimized by a correction factor in the exponent of the expression for the differential cross section

(curve d). They found

a(8)

=

a(o) exp - (8/8

'*'*

) 2

where

a**=

1.20

e*

(correction factor 1.20)and

(2.56)

(2.57)

We refer to table 2.3 for a comparison between the various angular units presented here.

A formula which describes the exact curve of a(8) remarkably well in both the quantummechanical and the classical region is given by

§ 0

:5

r-1

2

~ Ol ~ ~ -2

i

0.1 -3 - . : . . _ 0 5 10 15 20 25°•01

_

_,2,_,a

Figure 2.4

Curve

(

a) gives the

exact,

quantummeehaniea

Z

differentiaZ

cross

seeti

on

for

s =

6. Curve

ibJ

gives the differen

-

ÜaZ

cross

seetion in the eZassieaZ approximation.

The

straight Zines (e)

and

(d)

are the

resuZts

of a

first order

eaZeulation (equation (2.51)) and the

best exponentiaZ fit with experiments

(

(BUS

67),

see

equation

(2.56)),

respeetiveZy.

a(e)

=

a(o) (1 + A(s)(eje J2 + B(s) sin(C(sJ(eje J2 JJ-(s+l)/s

u

u

(2. 58)

For some values of s the coefficients

A(s),

B(s)

and

C(s)

are given

in table 2.2.

So far we have derived analytical (or numerical) expressions for the elastic total and differential cross sections assuming a long range, attractive potential of the form

-C

/rs . It is of interest to

8

know in what way this quantities behave i f anisotropies or higher order multipele interactions in the potential are taken into account.

Two cases are considered.

Firstly a potential with a higher order interaction

V(r) _(2.59)

which gives a correction (BER 66)

6Q/Q

(2.60)

in the total elastic cross section.

With the values

B =

1 ~ and

Q

=

500 ~ we find a correction 0.4%. The secend case, including an anisotropy in the potential, is of interest if we consider the scattering of a molecular beam with state selected molecules, quantumnumbers(j, m). With a potential of the

ferm

(2. 61)

the correction becomes (REU 64, STO 72)

6Q/Q

=

0.1 q2 6

_,

(2 .62)

We assume that the quantization axis is definéd by an externally applied electr.ic field.

For the values q_{2 6}

_,

=

0.12 ,

Q

500 0 A 2 , (J, . m) (1,0) we then have

6Q/Q

= 0.5%.

Both corrections do net contribute significantly to the value of the total elastic cross section. However, there are experiments with state selected beams which are sensitive to the anisotropy in the potential (BEN 64, STO 72). The experiments arebasedon the fact that due to the anisotropy the attenuation of the main beam changes if the orientation of the quantization axis (external field) is changed relatively to the direction of the relative velocity

g.

We ·will net discuss this subject since i t lies out of our experimental

Table 2.3.a. Some used angular units(2.3.a) are given together with the relation between the values (2.3.b)

as a function of s

6

u

, (this work)determined by the intersectien between

the classica! relation 6(b} (equation (2.23)) and the uncertainty relation (equation (2.29))

-1

, (this work) defined as~ (equation (2.37))

. ~ 2

, (HEL 64), chosen such that a(e)=a(o)exp-(6/6)

(equation (2.51))

(BUS 67), phenomenological angle which fits scattering data in the best way with

.

**

2

a(6} = a(o) exp-(6/6 )

9

F(s}~

1 crit _(2(s-1) f(s))l/(s-1) _(k2Q)Ji 6 _u = (F(s))Jz (k2Q)Ji

e*

=

(16

rr/

f

2

(s))~

(k2Qy'i 6** = 1. 20 6*

Table 2.3.b. 8 5 5.5 6 6.5

e

.{_e

0.6580 crJ. u 0.6814 0.7014 0.7187

e*;e

0 u 0.9426 1.2256 1. 3958

e**; e

0 u 1.1312 1.4707 1.6749 2.4. Inelastic scattering 2.4.1. Formal theory

The potential used in the calculation of the total and differen-tial inelastic cross sectien has the general ferm (equation (2.17))

where V(r, y)

=

V (r) + V'(r, y) 0 (2 .63) V'(r, y)

=

L

VL(r) PL(cosy) L=1 (2.64)

z

x

Figure 2.5 Coordinates used in

the

quantummeahaniaal

desarip-tion of

the

aollision between

the

moleaule

XY and atom

A.

In

the referenae system

(x

,y,z)

the veetor

a

denotes the

direetion

of the

mo

leaul

ar

axis

with spheriaal

aoordinates

e

and

~

,

~

is

the separation

veator

between

the aenter of

masses

of

XY and

A

and

has spheriaal

aoordinates

n

and

S

y is

The Hamiltonian descrihing the interaction process is given by

H

=HeM + Hrel + Hrot +

V(r,

y) (2.65) The first term describes the motion of the overall center of mass and is of no interest here. The second term gives the kinetic energy of the relative motion of the two colliding particles. The third term describes therotationalenergyof the molecule. We use spherical coordinates with ~ = (r, a, B) for the relative position and

a=

(8, ~) for the direction of the molecular axis (figure 2.5). The last term,

V(r,

y) couples the relative motion described by

~rel (r, a, B) with the rotation of the diatomic molecule, described by ~rot(e, ~).

For the analysis of this problem we use what we like to call the half-classical model (EVE 73a) . Practically the same description can be found by other authors under the name 'classical path' method

(GOR 73). It is basedon the following assumptions:

a. the influence of the angular dependent part of the IP on the relative motion can be neglected, this motion stays unaware of the rotatióriäl "S"taté of the molecule,

b. the now independent, relative motion can be adequately described by the classical mechanics, respectively by a wave packet with dimensions small compared to the intermolecular distance r, c. the small scattering angle 0 is due to a correspondingly large

impact parameter b (weak collision and no glory contribution). The classical solution of the relative motion gives the trajectory

(r{t), a(t), B(t)). In conneetion with point c. we will take an unperturbed straight line trajectory.

The quantummechanical rotation is now described by ~rot(8,~,t)

and the time dependent Hamiltonian

H = Hrot + V(r(t), y(a(t), B(t), e, ~)) (2.65a)

As a reference we first take the case V

motion of a rigid rotator described by

H

_rot

=

0, i.e. the unperturbed

where I

=

moment of inertia of the rotator and

-

·

n

2 A2

=

operator for the square of the total angular momentum. The solution of the Schrödinger equation is given by

(2.67)

where ~

re.

~) are the spherical harmonies, being the eigen functions J

of the equation, j and

m

are the usual quanturn numbers. Commonly h2

2I

is called the rotational constant

B.

The coupling between the rotator and the angular dependent part of the potential is described by the time dependent wave equation

=

(H

+

V'(

r

(t), y(a(

t

),

B

(t)))

~ rot

=

h

_i

(2.68)

we may treat

V'(r,y)

as a small perturbation. We expand ~ in eigen states ~

r

e.

~) of the· unperturbed rotator

J

~

=

I

j,m

leading to the equation

d C •1 I J

m

dt

i

=-h

i

=-h

(2.69) ~ +i(E.-E.,Jt/h .,

'i

V

'i.

!. c j m e J J <J . m J m>

j,m

I

(2.70)

j,m

The bracket can be split with the help of the spherical harmonie addition theerem as

<j' m'IV'(r, y)jj m>

=

We shall use the abbreviation

., , .

m'

R(L,J ,m ,J,m )

= (-)

121:1

<j'

m'i~re,

<t>Ji

j

m>

= .j ( 2j I+ 1) ( 2j + 1) Ij I L j ) / j I L j \

\o o o1

(

-m'

Mmj

From the properties of the 3j-symbols we derive

(2. 72)

a. j ' + L + j

=

even and j', Land j satisfy the triangle relation

L=

j j ' - j j ,

j j ' - j + 2 j , .•.. , j ' + j

b. -m 1 _{+ M + m}

=

_{0 or M}

=

_{m 1 - m}

Due to the last condition the summatien over

M

may be dropped.

Selection rule (a) says that a transition Aj = + 1 only can be caused by potential terms, containing Legendre polynomials of odd order and ój

=

0, ~ 2 only with even order.

A first order salution for the transition probability reads:

H., I • dt) 2

J m J m (2. 73)

This equation gives a goed approximation of the transition probability

u

Ie.,

.1

«Ie. 1.

J m J m

The complete expression determining the transition probability is

*

f

Hj' m'

j m

dt -co

_ _ 1-h. \{-)M-m' R(L • . , , . )( (L-M)!

)t

=

L .~ J ,m ,J"m (L+M)!

L

x

f

e

i (E

J

.-E .,)t

J

/h VL{r(t)}

~(cosa(t))

e-iMB(t) dt (2.74)

The inelastic total and differentlal cross sections can be obtained from the elastic total and differentlal cross sections via the expresslons

Q(inel)

=

j m j ' m' Q (el) P.

0(inell(e). =

J m j' m' a (el) (e) . J

m

(2.76) (the angle e in equation (2.76) refers to the scattering deflection angle in the CM system, in contrast with the notatien used in the remaining chapters of this thesis, where

e

is used for the CM

deflection angle and e for the LAB deflection angle. This is done in order to avoid confusion with the variable

e

meaning in this chapter the polar angle of the molecular axis.)

2.4.2. Evaluation of the transition probability for small scattering angles

In our experiment we measure particles scattered over a very small angle. The elastic cellision (section 2.3.) has been treated as a perturbation of a straight line trajectory. For scattering angles larger than a certain angle ecrit the semi-classica! theory converges to the classica! theory where a defined relation 0(b) exists.

We now assume that a partiele scatteredover an angle 0 > Gerit , has fellewed a trajectory with an impact parameter b(

e

).

We estimate the matrix-element for an inelastic transition assuming a straight line trajectory.

We distinguish between two practical cases, namely a trajectory either parallel or perpendicular to the z-axis, which is defined by an externally applied electric field (see figure 2.6).

Firstly we introduce two reduced variables, namely E ; 'collision time'/'rotational period' ;

IE.

.

J

reduced time scale

u ;

'time'/'collision time'

a. Parallel trajectory

E. 1

I

b/hg and a J

gt/b.

/ 2

In this case we have cosa ; u / l+u , B independent of u and

1' ; b ll+u2 •

The potential then reads

z

(al 9 A y z y (b)

Figure 2.6 Two

d

i

fferent measuring situations.

In

case

(a) the

trajectory is

parallel

to the z-axis,

i.e.

~1/z.

Then

the

~m

=

0

transition is nearZy forbidden,

whiZe

~m

=

+ 1

is aZlowed. In

case

(b), ~

z, the reverse soft selection rule holds.

The evaluation of the integral gives an answer which is an expression

depending only on the parameter

e

and the value of

s .

It is given in

the notatien

1{

J

Hj'm'jmdti

where

=

12

R(L,j

',m',j,m) V(L,

L

piMir_u_J

(2.78) V(L,

IMI,

s)

=

~

as,L

L 2

= _

ib / ( L- M !

J

e +iSur. ---:::--11+,-=l,;+,:::Ü_ du

hg

L+ M

,

-oo

(l+u2)s/2

(2.79)

and S = sign (E .-E. ,) ,

J

J

Fora transition (1,0) + (2,0) mainly caused by a potential term 7

(a_{7, 1}!r) _P1(cosy) we have

V(l,O,?) =~-a? 1 e;

e

-e; 7T (3 + Je; 2

48 + e; )

hg

b6

(2.80)

and the transition probability is

P.

J,m +

j',m'

(2.81)

For CsF we find with -19 o7

a

_{7 1}= 833 x 1

o

J A -24 0

B

=

3.66 x 10 J , g

=

750 m/s,

b

=

7.5 A and • -5 p1 0+2 à

=

6.9 x 10

the following values e;

=

4bB/hg

=

0.14 and

b. Perpendicular trajectory

I

.

2

I

2

r

= b

1+u ,

cosa = cosa /

l+u

and cos6

0 . 2 2

uilsin

a +

u

0 Defining Ü(L, M,s) (2 .82)

We have for the transition probabiiity:

z

=

II

R(L,

j',m', j,m)

Ü(L,

IMI, sJI

2

L

(2.83)

Figur

e 2

,

7

IZZus

t

r

at

ion

o

f the

6m s

e

le

c

tion ruZe. I

f

6m

=

~

1 we

have for the case aj

/

z : 6Lz

=

h

(con

s

e

rvation of anguZar m

o

m

e

nturn

6g

=

h

I

(~b)

). Th

e c

o

n

servat

io

n

o

f

e

n

e

rgy

g

iv

e

s tha

t

óg

=

e:

6

g a

nd

thus 6L

=

e;

h.