Anisotropic well depth parameters for for N2Ar, O2Ar, N2Kr, and O2Kr from total differential cross sections

Volume 88. number 2

CHCMICAL

PHYSICS

ANISOTROPIC WELL DEPTH PARAMETERS FOR N+, FROM TOTAL DIFFERENTIAL

30 APrd 1982

LETTERS

02-Ar,

N+,

AND O+

CROSS SECTIONS

G. ROTZOLL

Rcccivcd 26 January 1982,in

fmd form 5 March 1982

Laboratory angular distnbutlons of N2 -Ar,

02 -Ar,

N2 -Kr,

and 02 -Kr xc analyzed in terms of anlsolroplc polcn!d

models and compared to prcv~ously pubhshcd potcnt~als. All systems show some dxnpmg of the ranbow structure which is cvplaincd by the xusotropy

of the potential well depth.

I. Introduction Recently it has become possible to analyze total differential scattering data of atom-molecule systems [ 1-41 m terms of anisotropic potentials by using the infmlte order sudden (IOSA) or other equivalent approximations [S-8]. The simplest way to do this IS to calculate total differential cross sectrons in the center-of-mass (CJII.) system, transform these into the laboratory (lab) system as III the elastic case and average over experimental condltlons. This procedure wll be adequate when the melastic components contained in the total differential cross section produce only small velocity changes. When larger velocity changes occur, it 1s necessary to calculate individual 0’ +-l) c.m. cross sections and transform them separately mto the lab system [9]. For the systems in this work the simple method of transformatron is used. The justification for this lies in the restriction to low scattering angles. It has been shown in a coupled-states calculation for N2-Ar by McGuire [lo], that the low-angle region for the model potential surface employed is dominated by elastic scattering. Additional test calculations for the other systems and other potential surfaces have been performed by the author using the IOSA. These calculations all show that, for the angular range measured, it is possible to use the total differential cross section in the c_m.-lab transformation without introducing noticeable errors. 0 009-2614/82/OOOCLOOOO/.$O2.75 0 1982 North-Holland

All systems mvestlgated in tlus work have heen measured before in crossed molecular beams [I I 1. Analysrs, however, was restrlcted to spherically syrnmetric potential scattenng at that time. In the meantime, for some of the systems, anisotropic potentials have been proposed in the literature. These potent& and simple model potent& will bc used m comparing measurements and fully averaged calculatrons rn this work.

2. Experimental and dara analysis Angular distnbution measurements were perfonned in the usual way by crossing two nozzle beams and measuring the in-plane scattered density of the hghter partrcle as a function of the lab an& with a mass filter detector. The measured rdinbow structure allows the determination of parameters characterizing the potential well depth and Its anisotropy. There is not much sensillvity to parameters characterizing the well posltion (I.e. T,,,) and Its amsotropy and therefore these parameters have always been futed at reasonable values. For comparison wtth a computed result based on an assumed potentral it IS necessary to average over expenmental conditions. Since information about the potential well depth anisotropy is to be extracted from darnpmg of the rainbow structure, broadening effects

due to expertmental resolution must bc correctly taken into account.

Volume 88. numbcr 2

CHELIICAL

30 Aprd 1982

PHYSICS LETTERS

terns, we define an average well depth by

The most strmghtforward way to do this is by a Monte Carlo method. Because the averaging procedure using the cm. total diffcrenttal cross section is the same as in the spherically symmetric case, the method dcscnbed before [ 121 for elastic scattering was used. The quantities needed rn the averaging program were obtained from the geometry of the apparatus and from tune-of-flight measurements of the beam veloctty datrrbutions.

I

SE; E(Y)dcos7, _I being the body-fiied angle of orientation and well depth anisotropy parameter a = 2(e1 - et)/(2~t + et,), the subscripts Iiand 1 labeling the drrectrons along and perpendicular to the molecular axis. The definition of the well position parameters fm and b is analogous. The parameters of all potentials used in this work are listed in table 1. Fig. I shows a comparison between experiment and calculattons based on the mdrcated potentials for N2-Ar. No calculations were performed for the electron gas potential since the radial functions of ttus potential are not given rn analytical form. It can be seen that the PLBC and KDV potential do not fit the data very well, the first one having a value of Z which is too small and the second one a value which is too large. In order to obtain a better fit the simple anisotropic modification of the LennardJones (LJ)-12,6 potentral proposed by Pack [S] was used: 7

3. Results and discussion 3 1. N2-Ar Two empirical anisotroprc potentials for this system have been used In the literature. The first is the potential of Patter@ et aJ. (PLBC) [ 131, the second the one of Kistemaker and de Vries (KDV) [ 141. More recently, an electron gas model potential has been cdculated [ 15,161. Because all these potentials have different analyttc forms tt is necessary to compare them through some charactenstic parameters. Following the definition of Pack [S] for atom-dtatomic sysTable 1 Potcntralpannictcrs far varrouspotcntrah a)

b

Ref.

3.92 2.85

0 06 0.09

1131 I141

-0 25 -0.26

3.70 3.70 3.90 3.71

0.06 0.06

tlris work tlris work 1111

-0.20

3.72

0.06

this work

3.72

-

this work

SyStelll

PoIcntlal

N2-AI

PLBC RDV LJ -12,6

1.69 2.03

-0.20 -0 33

(a) U-12.6 (I) U-12.6 (I) PV

1.85 1.85 1 88 1.90

pv (3)

1.99

Oz-Ar

Nz-Rr

02-Kr

Fill

I171

PV (1)

1.99

LJ-I?,6 (I)

2.03

LJ-12.6

2.75

-0.25

4.00

009

th

U-12,6(i)

2.15

-

4.00

-

this

U-12,6 (I)

2.19

-

4.05

-

work work 1111

LJ-12,6 (a) U-12.6 (a) U-12,6 (I) U-12,6(i)

2.33 2.35 2.35 2.33 --

-0.19 -0.25

3.60 4.00 4.00 4.05

0 07 0.09 -

(181 ttus work this work IJJI

3) P 1sgiven rn 1O-‘4 erg and Ft,, in A.

(a)

-

3.90

-

ill1

The pmmetcaa and b are dcfiied in the te\t. Potential funcuons used bolh in lsotrop~c form are denoted by (I) and (a), respcctlvely. Only Z and II were varied in obtaining tils to the expenmental data. Estimated uncertainties arc for 95% confidence lmuts. Uncertaintxs in T are *-0.04 except for 01 -Ax where it is f 0.02; uncertainty in 11is +005.

180

CHEMICAL PHYSICS LCITCRS

Volume 88. number 2

30 April 1982

lation but it is seen that the measured distrrbutlon is flatter in the rainbow region than !he isotropic calculatron. This devration allows the estimation of the well depth anisotropy patameter u. It should be noted that the parameter E of the effective Isotropic potcntial must not be confused with the spherically symmetric component (the Vo term of a Legendre polynomial expansion) of the anisotropic potential. The E value of tlus tcml is always lower than the effcctivc E and cannot be used to give an approximate dcscnption of differential cross sections m terms of an ISOtropic potential.

3.2. 02-Ar For this system, a sophisticated anisotropic potenttal

of the

ESMSV-type

with 12 parameters for the

VO term and 5 parameters for the I’, temr, obtained ,FY

LEFF

LR5

9

6

3

II

SCRTTERIIJG

PWLE

I2

IDE.31

by a multiproperty analysis, has recently been published by Pirani and Vecchiocattivi [ I7j. The calculation based on this potential (PV) is compared with expenment in fig. 2. An obvious deficiency of this potential is an average well depth which is too small. This discrepancy with dlfferentlai cross sections was already noted by Prrani and Vecchiocattivi when they compared the experimental data of Tully and Lee [ 111 imth calculations based on their potential. Since the same deviation is found in the present work, it is ncc-

Tg.

1. Comparisonof c~culntions (solid bncs) bzwd on the indintcd potentials wth mczwemcnt (squares)for Nz-Ar. VAN is artisotroptcand VJSF isotpptc U-12,6 potcnktl calculntion. Mean colhs~oncncrgy E = 1.24 X 1O-‘3 erg.

essary to modify the average C value. To retain the radial form of the PV potential and simplify the 3Nsotropic part the following potential was used in the data fitting:

with

f&(x) being the isotropic potential of Pirani and Vccchiocattivl in reduced form and ~(7) and r,(r) defied as abovc. im was fiicd at 3.72 A, the value of the Vo term of the PV potential and b set equal to 0 06 obtained from the complete PV potcntlal. The best fit is obtained with Z = 1.99 X IO-14 erg and a = -0.20 and shown as VAN in fig. 2. As anticipated, a much better agreement between experunent and calculation results by increasing the average well depth. The corresponding isotropic potential fit with E and r,,, independent of 7 is shown as VEFF below the anisotropic Iit. As in the case of N2-Ar, the effective E is the same as Z of the anisotropic potential and the

E(y) = P[ 1 + aP*(cos r)] , ‘m(~)

= irn 11 + bP2(cosT)1.

The well posltion parameters Trn and b were fued at 3.70 W and 0.06 respectively, because the data are insensitive to the precise values, as mentioned before, and E’and II varied to produce the best match between experiment and calculation. The best fit so obtained is labeled VAN in fig. 1. The curve labeled VEFF is the best fit with an isotropic U-12,6 potential. The E value is identical with 5 from the anisotropic calcu-

181

Volume 88, number 2

CHChllCAL

PHYSICS LETTERS

30 April 1982

C7-FP

“I

0

NZ-l’,R

4 2

1

LfiB

6 SCPTrERlNC

8

10 RNCLE

I?

14

P

0

IOECI

Fg. 2 Compartson between calculations (sobd hncs) and e\pcrrmcnt (squares) for 02-Ar PV is the oqind potential of ref. [ 171. VAN is an anisotrop~cmoddication of the Vo potcntid of ref. [ 171 (see text) and VEFF is l’o only wth the same c as VAN. Mean collision energy E= 1.23 X LO-‘” erg.

anisotropy shows up only in some damping of the rainbow structure. 3.3. Iv,-Kr

For thissystem apparently no anisotropic potential has been suggestedsince the work of Tully and Lee [l l] . Calculationswere performed with the anisotropic and isotropic U-12,6 potential. The results are shown in fig. 3 with the potential parameters given in table 1. im = 4.0 A was taken from ref. [l l] and b = 0.09 estunated from the N2 bond length. Again both fits yield identical values for Z and E, respectnely, and the data show more damping 111the rainbow region than the isotropic calculation. 182

0

16

3

6 LRB

9

scnrrmIw

12

I5 ANGLE

I6

ZI

24

[oEcI

3. Comparison of arusotropic (VAN) and isotropic (JEFF) U-12,6 potenti calculation (solid lines) with measurement (squares) for Nz -Kr. hican collision energy E= I.3 1 X 1O-‘3 erg. TIN.

3.4. 02-Kr For 02-Kr an anisotropic potential model has recently been suggested on the basis of total cross-section measurements [IS]. The potential form used is of the W-12,6 type and therefore similar to the anisotropic W-12,6 potential employed in this work. Calculations were performed only with this latter potential but the result is readily compared to that of ref. [18]. Fig. 4 shows the best fits in the anisotropic and isotropic case, respectively. The general comments apply here as well as for N,-Kr. im = 4.0 A was again taken from ref. [ 1 l] and b set equal to 0.09. The well depth parameters are compared to those derived from the potential of ref. [18] in table 1. The agreement, especially in 0, is very good.

Volume 88, number 2

CHEMICAL PHYSICS LCITCRS

30 Aprd 1982

ate damping of the rainbow structure. The well depth anisotropy is in fact not large enough to cause a shift of the rainbow structure to larger scattering angles (see ref. [S] for this effect). This is the reason why identical values are obtained for Z and E in an anisotropic and Isotropic potential analysis, respectively.

02-KR

4. Conclusions

ViFF

cl

6

3

LRB

Kg. 4. Comparison U-12.6

poknthl

(squares) for Oz-Kr. erg.

9 SCRTTEPINC

ofamsotropic

I?

I5 RNGLE

irl

21

2-l

iOEG1

(VAN) and

isotropic (VCrF)

cdculatlon

(sohd hncs) with mcasurcmcnt hlcan colhslon energy t’= 1.28 A lo-I3

As menhoned

in section 2, the well position parameters Frn and b cannot be estimated very accurately from total differenttal cross sections and therefore have been fmed at reasonable values deduced from other sources of information. This is especially true of b, for which the measurement of well-resolved rapid osdations

The investigated atom-diatomic systems all cxhibit a well-resolved rainbow structure. These rambows allow the determination of the average potential weU depth, both with anisotropic calculations using the IOSA and with isotropic calculations ncglecting the anisotropy. The average or effective potential wcU depth is the same for both types of calculation. The E’or E values obtained in this work all fall withm the error limits of Tully and Lee [I 11. All data, however, show more damping of the rainbow structure than can be explained by experimental rcsolution assuming spherical potential scattering. This deviation between isotropic calculations and experimental data was used to obtain estlmatcs of the well depth amsotropy. All systems have moderate well depth anisotropies with negative u values glvmg the most stable configuration with the atoms perpcndlculnr to the bond a_Gsof the molecules.

or inelastic transitions

Acknowledgement The author wishes to thank Dr. F. Vcccluocattwi the work about the O,-Ar and 02-Kr potentials prior to publication. The help of M. Pauluth with some of the measurements is gratcfully acknowledged. The calculations were performed at the Regionales Rechenzentrum fir Nledersachsen. for communicating

would be neces-

=Y-

The data, however, are most sensitive to the average well deprh I?,which can be seen from the estimated uncertainties in table 1. Less sensitivity exists to the well depth anisotropy parameter o, which results in larger error estimates for this quantity (see table 1). This is because the rather weak anisotropies for the present atom-diatom systems produce only a moder-

References [ 1I U. Buck, V. Kharc and M. IMc, Mol. Phys. 35 (197’G) [ 21 ~l?Kd G A. Parker and A. Kuppermann. Chcm. Phys. Letters;9 (1978) 443.

[ 3 1 hl. Kell,

J.T. Shnka

and A. Kuppcrmtnn,

J. Chem.

Phys. 70 (1979) 541.

183

Volume 88, number 2 14 1 U. Buck, F. Ccstcrmann

[S] [6] [7] [S] [9] (IO] (1 I]

184

CHEMICAL PHYSICS LLTTERS

and H. Pauly. Chem. Phys. 50 (1980) ‘17. U. Buck and V Kharc, Chcm. Phys 26 (1977) 215. G A. Fxkcr and R T Pack, J. Chem. Phys. 68 (1978) 158.5. R. Goldflam, S. Green, D.J. Kouri and L. hlonchtk, I. Chem Phys 69 (1978) 598. R.T Pack, Chcm. Phys. Letters 55 (1978) 197 C. Rotaall and A. Lu’bburt, J. Chcm Phys. 71 (1979) 2275. P. hfcCuue, J.Chem. Phys 66 (1977) 1761. F P TuUy and Y.T. Lee. J Chcm Phys. 57 (1972) 866, and rcfcrenccs thcrcm.

30 Apnll982

[ 121 A. Liibbert nnd C. Rotzoll. J. Phys. El1 (1978) 63. 1131 M.D. Pattcngill, R.A. LaBudde, R.B. Bernstein and C.F. Cur&s, J. Chem. Phys. 55 (1971) 5517. [ 141 P.G. Kistemakcr and A.E. de Vries, Chcm. Phys. 7 (1975)371. [lS] Y.S. Kun. J. Chem. Phys. 68 (1978) 5001. [ 161 S. Lee and Y.S. Kim, J. Chem. Phys 70 (1979) 4856. [ 171 F. Pirani and F. Vecchiocattivl. Chem. Phys. 59 (1981) 387. [ 181 F. Rram, F. Vecchtouttd. J.J H. vanden Btcsenand C.J.N. van den hfclJdenbcrg, J. Chem. Phys. 75 (1981) 1042.