Asymmetry effects in atom-molecule collisions I

Physica 30 1459-1464

Reuss, J. 1964

ASYMMETRY

EFFECTS IN ATOM-MOLECULE COLLISIONS I by J. REUSS

Fysisch Laboratorium

v.d. Katholieke

Universiteit

Nijmegen,

Nederland

synopsis For rotating molecules, scattered by particles of radial symmetry, the total elastic collision cross section is investigated. It is assumed, that the scattering takes place in a region, where a weak magnetic field is applied. Asymmetric terms in the intermolecular potential are taken into account as a small perturbation. The final result can be used to analyze experimental data obtained in scattering experiments with polarized molecular beams.

1. Introduction. The interaction between two collision partners can be described by an intermolecular potential V(r). Generally V(r) contains asymmetric terms, if collisions between molecules and atoms are considered. Through those terms the interaction depends on the orientation of the molecular axis with respect to the direction of the relative velocity of the collision partners. As a consequence, the total collision cross section is influenced by the rotational state of the molecule. The rotational state can be charactterized by two quantum numbers j and mj, where tt[i(i + l)]* is the rotational angular momentum of the molecule and Fzmj its projection on a fixed direction. We define mj as measured in the direction of the incident molecule, in the center-of-mass system. Thus, the total cross section can be written as c++z~; i’mi), where imj characterize the rotational state before the collision and j’mi after the collision. Arthurs and Dalgarno 1) have analyzed the scattering of a rigid rotator. However, right in the beginning of their work the cross section c is averaged over all possible mf-numbers and summed over all possible mj-numbers. This procedure gives the total cross section if no magnetic field is applied in the scattering region. We shall re-analyze the problem under the assumption that there is a magnetic field in the scattering region. Its direction shall coincide with the direction of the incident molecule, in the center- of mass system. Therefore, the component AM of the total angular momentum I%[J(J + l)]* is conserved during the collision. In a partial wave analysis .I is composed of j and I, where I is the quantum number of the angular momentum of the -

1459 -

1460

J. REUSS

considered partial wave. J can assume the values between ]Z-jl and 1 + j. There is a restriction for the magnetic field in the scattering region in order to keep our analysis applicable. We shall assume that J remains a good quantum number during the collision. This is true if the splitting of the energy levels by the magnetic field is small in comparison with the level distance in a zero field. Our results will be specialized for the case j = j’ and mj = mi, i.e. we shall only consider the total elastic collision cross section. Berkling e.a.2) have shown that under normal experimental conditions the transition probability for rnj --f mi # m&s extremely small. Collisions with j’ # j can be suppressed by use of sufficient slow molecular beams. Thus, the conditions under which our results will be derived, can easily be met experimentally. The starting point of our analysis is an expression for o(jmj; jmr) which is obtained by combination of equations (12) and (15) of ref. 1 followed by the integration u(jmj; jmj) = /do(kmf; jm#). This expression is: Jl+i

*[(2h+

Ja+i

Jlii

I; iz”-za(2J1+ 1)(2J2+ 1)* =$* JI=O 5 Ja=O 5 h=lJ~--51 C I2 Zz=IJz-ii Z'=lJl-jl

&q;jmj)

1)(x12+

*T*J’mf(jll;

1)3’*

j12).TJam’(jlz;

jl’).

(1)

Here T is related to the S-matrix by TJ*(jl; j’l’) = d,&lf-SJM(jZ; j’Z’). The Clebsch-Gordan coefficients of ref. 1 are replaced by the more symmetric 3-j symbols. A& is the momentum of the incident molecule. k = jkj is conserved during the collision, according to our assumptions j = j’ and rni = mj. We shall closely follow the work of Art hurs e.a.1). At first we shall derive the S-matrix in the distorted-wave approximation for a potential with a small deviation from radial symmetry. Then, using some Racah algebra we shall evaluate all summations in equations (1) except one and discuss the result in general terms. 2. The S-matrix. Using the distorted-wave approximation Art hurs derive the expression for the S-matrix (equ. (36) and (37) of ref. 1) : SJ(jZ; jZ’) = exp [;(v&! + $)I

*{&~~(1+2i&3-22i/?~}.

e.a.

(2)

As a general result S is found independent of M and diagonal in J . r&! is a real phase shift, which appears in the solution Z@(Y) of the equation w2

H 2P

ds --+ dr 2

/‘(/I + 12

1)

-

k2

+


J>

1

W?(Y) = 0

ASYMMETRY

W$“(Y)

must

EFFECTS

satisfy

the

IN ATOIvI-MOLECULE

boundary

COLLISIONS

1461

I

condition

w$ --f sin(kr+‘n + 7:). Y-+CO expression of Arthurs e.a. and is

Jiz is defined (-) times the equivalent t% corrected for a dimensional factor to m

,q

=

_

5-.;/4~(7)


~lvliz’;

J>

W;“(7)

V(7)

=,go

V,(7)

d7.

(3)

0

The

potential

V(r)

can be expanded

into

P#*i’),

where

the Pt are Legendre’s polynomials and P *P’ is the cosine of the angle between the direction of the relative velocity of the collision partners and the direction of the molecular axis. With Arthurs e.a.1) we define

= f~(jZ’;jZ; J). It follows ft(jZ’; jZ;J)

= (-

l)l-l’-J*

(:

:

6)(1,

fi(jZ’; jZ; J).

=igoV$(7)

“d 6)

(2j+

One finds:

l)l*-

1) [(2Z+l)(2Z’+

*W(jZjZ; Ji)

(4)

W is a Racah

coefficient as defined in ref. 3, for example. Equation (2.35) in j j i ref. 3 gives 0 0 0 = 0 for i = 1, 3, 5, . . . . Therefore, neither r,$ nor @ ( > depend on VI(Y), V47), . . . . SJ(jZ; jZ’; J) and u(jm,; jmj) are not influenced by a term with an odd index in the expansion of V(r). This general result could be experimentally tested by a measurement of the total elastic cross section when polarized Hz- and HD-molecules are scattered by noble gas atoms. Hereafter we put V(7) = Vo(7) + V2(7) Ps(?*P’), what is equivalent with using the four first terms of the expansion of V(7). Then is for I’ # I:

,8$’

=

-

$+/2(jl’;

ji; 1).

/W3”(7)

V2(7)

WF(7)

dr

(5)

0

SJ(jZ; jZ’)does not depend on /?$ as can be seen from equation (2). Therefore, from now on equation (5) shall be used to determine /?$?. F’s(r) is assumed to be small in comparison with Vo(7) for all relevant values of 1. We look for the influence of Vs(r) on SJ and a(jm;r; jmj) in a first order approximation. &Jiz itself is a small quantity, which vanishes for vs(r) 3 0; in first order we can replace it by a simpler expression. wl(7) shall be a solution of A2

H 2P

ds --+ dr 2

Z(Z+ 72

1)

-k2)+Tlgo]w&)

=0

I. REUSS

1462 with UI&) --f sin(kr-@n T+ca

+ 71). Thus we have in first order t!l$? = /$? with

In the same approximation one finds with Takayanagi (ref. 4) r$ = ?ZZ+ + #l;‘“. As gs;‘”is regarded to be small in comparison with qr we may use the approximation exp[i$] = (1 + i/?[‘) exp [iqr]. Thus, if Vs(r) is taken into account in first order as a small perturbation one finds (6) 3. The total collision we derive:

4im3;imd = a--

cross section.

an 00 k2

With the help of equations *

(1) and (6)

Jii

c J_i, (2J-tl) p+w'+w Jzo z,,z=,

.@$‘[6zzflsin 27j9,--sin ~2’ COS(~~-~~(Z’-Z))], where 476 O” &=--x(21+ k2 1-0

1)sin2yz

(7)

&!’ contains f&jZ' ; jZ; J) as a factor. From equation (4) follows that fz(jZ’;

jZ; J)

= 0 unless I’ = 1, Z f

= 0 unless I’ = I, Z -j= 2. This is because

(see ref. 3 table III). If we use this result and sum over I’ in equation it results :

.[wJ+l)(t ; -;J .@*3sin .

2 (7)

2qr + [(2Z+ 1) (2Z+5)]**

I.

i z-+2 J *&!~s+sin(~r + lj)r+s) --ml > ( mJ 0

(8)

ASYMMETRY

With

EFFECTS

IN ATOM-MOLECULE

COLLISIONS

rl,l’ =TZPQ(Y) wrs(r) Vs(r) dr this last expression

I

1463

can be written:

0

.fs(jZ;

jZ;J)

- I?j.sin 2r~z+ [(2Z+ 1)(21+5)-j*

As a general x (2J +

result

one finds

x u(jmj; jm;r)/(2j +

1) fi(jZ; jZ; J) = 0 unless “,! = 0

(see ref.

1) = 00 because

1) and

J

of

I: (21 + I) m,

(I ; -!J (L,; -“m,> =6w*

Thus, without

a magnetic

field

in the

scattering region the total elastic collision cross section is not influenced by the term Vs(r)Ps(?*3’), in our first order approximation. This result was already given by Art hurs e.a. (ref. 1). Equation (8) can still further be simplified. We use the contraction rule (see ref. 3 appendix II, .for example) : F (2J+

l)(-

*

l)J-+

i _-y ZJ (_a _J (; “g Qw

il; J4 =

(a$ TE)G; a>.

Then follows as our final result : Cr(jm,; jq)

= uo-

-

87c,~

%%3

’

3m;-j(j+l) (2j+3)(2j-1)

O” Z+l -I 2Z+3

;FO

sin 273 + 3(1+2) i?j+’ ..sin(qr + 7r+s)

(9)

This expression can be analysed by numerically computing the phase shifts and the integrals F’“,“‘.Vs(r) ca n be approximated by a Lennard- Jones 12-6 potential. The potential parameters can be fitted to experimental results. v”,“’ can be split into two terms and calculated before the potential parameters are determined. In a forthcoming paper we shall evaluate equation (9) with the help of the Massey and Mohr approximation. Bennewitz e.a. (ref 5) have measured the total elastic collision cross section of a polarized TlF beam, scattered by noble gas atoms. Our results will be compared with the discussion of Bennewitz e.a. on their experimental data. Received 18-2-64

1464

ASYMMETRY

EFFECTS

IN ATOM-MOLECULE

COLLISIONS

I

REFERENCES 1) Arthurs, 2)

A. M. and Dalgarno,

Berkling,

K., Schlier,

3)

Brink,

4)

Takayanagi,

D. M. and Satchler,

5)

Bennewitz,

K., Progr.

A., Proc.

Roy. Sot. A256

Ch. and Toschek, G. R., Angular

theor.

H. G., Kramer,

Phys.

(1960) 540.

P., Z. Phys. 168 (1962) 81. Momentum,

Oxford

University

Press 1962.

11 (1954) 557.

K. H., Paul,

W. andToennies,

J. P., Z. Phys. 177 (1964) 84.