Quantum Interference as the Source of

Stereo-Dynamic Effects in NO-Rare Gas Scattering

A. Gijsbertsen, C.A. Taatjes*, D.W. Chandler*, H.V. Linnartz and S. Stolte

Department of Physical Chemistry,

De Boelelaan 1083, 1081 HV Amsterdam

vrije Universiteit amsterdam

*Combustion Research Facility, Sandia National Laboratories, Livermore, California 94550

Outline

1. Introduction

2. Quasi-Quantum Treatment

3. Ion Imaging Experiments

4. Differential Cross Sections (DCSs)

5. Parity Effects

6. Conclusions and Outlook

Introduction

La se r b e a m

Prim a ry b e a m va lve (16 % N O /Ar)

Se c o nd a ry b e a m Va lve (Ar)He xa p o le sta te se le c to r

Le ns syste m a ndp ho to m ultip lie r

O rie nta tio n fie ldLig ht b a ffle s

oriented 21/2 NO ( j = ½, = -1) + R

21/2 NO ( j’, ’ ) + RWith R = Ar, He, D2,...

The NO molecules are rotationally excited due to collisions with rare gas atoms.

Laser Induced Fluorescence (LIF) is used to measure the amount of molecules present in a particular rotational state after collision: it provides the total collision cross section .

The steric asymmetry S is given by:

Introduction

NO-Ar, Etr 500 cm-1

NO-He, Etr 500 cm-1

Sif

Sif

Introduction

N-end j = odd dominatesO-end j = even dominates

Introduction

Introduction

A close coupling treatment reproduces experimental Sif:

- it showed that the oscillatory Sif is due to the anisotropy in the hard shell of R-NO potential*.

- it offers no explanation for undulative dependence upon j’ !

Our goal is to construct a quasi-quantum mechanical model to: obtain more information about the physical background of the steric asymmetry

*(Alexander, Stolte, J. Chem. Phys. 112 (2000) 437)

Introduction

Rainbow undulation for atom-atom scattering are caused by the pathway interference of 3 “rays” with different impact parameters:

H. Pauly et al. 1966

Quasi-Quantum treatment

The state selected wave function contains all NO orientations.

Assuming a hard shell the scattering angle is determined only by the angle between the surface normal and the incoming momentum ħk.

At fixed an infinite number of “rays” with different impact parameters b interfere, due to different path lengths.

Equipotential shell surface at Etr.

Quasi-Quantum treatment

Quasi-Quantum treatment

The kinematic apse n points perpendicularly to the hard shell. The projection of the rotational angular momentum (mj) is conserved along n.

The difference between the “hard shell” trajectory and the virtual pathway through the center of mass yields the phase shift .

^

^^

k||

k

k

k’ (j = 0)

k’ (j = 1)

k’ (j = 2)

k’ (j = 3)

k’ (j = 4)

k’|| (= k||)

k’

Assuming a hard shell, only the momentum component perpendicular to the shell (k) can be transformed into rotation.The scattering angle depends on:1. Spacing between rotational states2. Angle between incoming momentum and apse.

Quasi-Quantum treatment

Quasi-Quantum treatment

Quasi-Quantum treatment

The phase shift for several rotational states, as function of n.

In this case cos()=-1.

O-end N-end

The asymptotic solution of the Schrödinger equation at large distance, can be expressed as:

The differential cross section relates to the dimensionless scattering amplitude:

Remind, in the hard shell model, mj is conserved along the kinematic apse: mj’ =mj in the apse frame.

Vdiff is ignored, so ’=.

Quasi-Quantum treatment

The scattering amplitude resulting from the hard shell model can be expressed as:

Where w takes care of the non-isotropic shape of the shell:

with:

The conservation of flux is taken care of by introducing C()

Note the elimination of the quantum numbers l and l’ !

Quasi Quantum treatment

Quasi Quantum treatment

Flux conservation correction C()2

Note that:

After some algebra one finds:

The distinguishes between the orientations.

If positive Head collisionsIf negative Tail collisions

Quasi Quantum treatment

Quasi Quantum treatmentThe orientation dependent DCS can be written as:

Note that:

in which “+” denotes N-end and “–” O-end collision;

From which follows: !

Increasing j’ j’+1, switches the orientation preference!

Quasi Quantum treatment

Quasi Quantum treatment

max

cos(w)cos(w)

O-end preference

Quasi Quantum treatment

-

Quasi Quantum treatment

A non-oriented NO wave function has parity

the DCS follows as:

Note parity-pairs of similar DCSs, that can be observed:

etc.

:

NO source chamber

He source

XeCl excimer laser

dye laser

308 nm, 5 mJ

226 nm, 1 mJ He

Hexapole

NO

collision chamber

Experiments

Hexapole state selected NO collides with He at Ecoll 500 cm-1:

Crossed 1+1’ REMPI detection

excitation 226 nmionization 308 nm

NO (j=½, =½, =-1) NO ( j’, ’, ’ )

50

100

150

200

250

300

150 200 250 300 350

200

250

300

350

400

450

Experiments

To test our setup, some 2% NO was seeded in the He beam.

The NO beam consists of 16 % NO in Ar.

This image reflects the velocity distributions for both our pulsed beams.

vNO

vHe

*

j’ = 1.5 j’ = 2.5 j’ = 3.5 j’ = 4.5 j’ = 5.5 j’ = 6.5

j’ = 7.5 j’ = 8.5 j’ = 9.5 j’ = 10.5 j’ = 11.5 j’ = 12.5

*

Parity conserving: p’ = p = - 1

Experiments

Marked images are from Q-branch transitions that are more sensitive to rotational alignment and show more asymmetry.These images were omitted for the extraction of the DCS.

*****

j’ = 1.5 j’ = 2.5 j’ = 3.5 j’ = 4.5 j’ = 5.5 j’ = 6.5

j’ = 7.5 j’ = 8.5 j’ = 9.5 j’ = 10.5 j’ = 11.5 j’ = 12.5

Parity breaking: p’ = - p = 1

Experiments

Parity conserving: p’ = p = - 1, DCSs

j’=1.5 j’=2.5 j’=3.5

j’=4.5 j’=5.5 j’=6.5

[o] [o] [o]

[o] [o] [o]

[Å2]

[Å2][Å2] [Å2]

[Å2][Å2]

Parity conserving: p’ = p = - 1, DCSs

j’=7.5 j’=8.5 j’=9.5

j’=10.5 j’=11.5 j’=12.5

[o] [o] [o]

[o] [o] [o]

[Å2]

[Å2][Å2] [Å2]

[Å2][Å2]

j’=1.5 j’=2.5 j’=3.5

j’=4.5 j’=5.5 j’=6.5

[o] [o] [o]

[o] [o] [o]

[Å2]

[Å2][Å2] [Å2]

[Å2][Å2]

Parity breaking: p’ = - p = 1, DCSs

j’=7.5 j’=8.5 j’=9.5

j’=10.5 j’=11.5 j’=12.5

[o] [o] [o]

[o] [o] [o]

Parity breaking: p’ = - p = 1, DCSs

[Å2]

[Å2][Å2] [Å2]

[Å2][Å2]

Recall that the Quasi Quantum Treatment yields the following propensity rule depending on the parity

These parity-pairs of similar DCSs are seen in experimental results, the ratios within the pairs can be verified using HIBRIDON results.

Parity Effects

p’ = p = - 1

p’ = p = - 1

p’ = - p = 1

The ratios between differential cross sections within parity pairs, is close to what the Quasi- Quantmum Treatment (QQT) predics.

For large j the agreement becomes worse.

Parity Effects

Conclusions and Outlook

1. Quasi quantum mechanical treatment that eliniminates l and l’ appears to be feasible for inelastic scattering.

2. The oscillatory dependence of S upon j’ can be explained as a quantum interference that invokes the repulsive part of the anisotropic potential.

3. An interference induced propensity rule of the DCS follows from our treatment and is seen experimentally. A physical interpretation of the DCSs emerges.

4. Measurements of orientation dependence of the DCSs will be attempted.

5. Is it possible to invert oriented DCSs to PESs?

Questions?

j’ = 4.5, R21

velocity mapping

ions

Extractor MCP's

+

Molecules in a certain rotational state (after collision) are ionized using 1+1’ REMPI and the ions are projected onto the detector, providing a 2D velocity distribution.

+

+

+

+repellor

velocity mapping

The velocity distribution is recorded with a CCD camera. Ion images show the angular dependence of the inelastic collision cross sections of scattered NO (j’, ’, ’) molecules.

Voltages: Vrepellor = 730 VVextractor = 500 V

Sensitivity: S = 7.7 m/s / pixel

NO beam velocity: vNO = 590 +/- 25 m/s He beam velocity: vHe = 1760 +/- 50 m/s

Images are: - 80 x 80 pixels- averaged over 2000 laser shots (@ 10 Hz)

Some parameters

Forward scattering ( = 0): Backward scattering ( = ):

vNO

vHe

DCS extraction

Extraction of differential cross sections (dcs’s) fromimages:

1. Calculate the center(pixel) of the scattering circle2. use intensity on an outer ring of the image as trial dcs

3. Use the trial dcs to simulate an image4. Improve the dcs, minimizing the difference between

simulated and measured image

Step 3 and 4 are repeated until the simulated an measured images correspond well enough.

NO-He, P11 (’=1/2, ’=1)

12-03-2004

j’ = 1.5 j’ = 2.5 j’ = 3.5 j’ = 4.5 j’ = 5.5 j’ = 6.5

j’ = 7.5 j’ = 8.5 j’ = 9.5 j’ = 10.5 j’ = 11.5 j’ = 12.5

j’ = 1.5 j’ = 2.5 j’ = 3.5 j’ = 4.5 j’ = 5.5 j’ = 6.5

j’ = 7.5 j’ = 8.5 j’ = 9.5 j’ = 10.5 j’ = 11.5 j’ = 12.5

12-03-2004

NO-He R21 (’=1/2, ’=-1)

NO-He R11 Q21 (’=1/2, ’=1)

15-03-2004

j’ = 1.5 j’ = 2.5 j’ = 3.5 j’ = 4.5 j’ = 5.5 j’ = 6.5

j’ = 7.5 j’ = 8.5 j’ = 9.5 j’ = 10.5 j’ = 11.5 j’ = 12.5

NO-He Q11 P21 (’=1/2, ’=-1)

17-03-2004

j’ = 1.5 j’ = 2..5 j’ = 3.5 j’ = 4.5 j’ = 5.5 j’ = 6.5

j’ = 7.5 j’ = 8.5 j’ = 9.5 j’ = 10.5 j’ = 11.5 j’ = 12.5

NO-He P12 (’=3/2, ’=1)

15-03-2004

j’ = 1.5 j’ = 2..5 j’ = 3.5 j’ = 4.5 j’ = 5.5 j’ = 6.5

j’ = 7.5 j’ = 8.5 j’ = 9.5 j’ = 10.5 j’ = 11.5 j’ = 12.5

NO-He R22 (’=3/2, ’=-1)

15-03-2004

j’ = 1.5 j’ = 2..5 j’ = 3.5 j’ = 4.5 j’ = 5.5 j’ = 6.5

j’ = 7.5 j’ = 8.5 j’ = 9.5 j’ = 10.5 j’ = 11.5 j’ = 12.5

NO-He P22 Q12 (’=3/2, ’=-1)

15-03-2004

j’ = 1.5 j’ = 2..5 j’ = 3.5 j’ = 4.5 j’ = 5.5 j’ = 6.5

j’ = 7.5 j’ = 8.5 j’ = 9.5 j’ = 10.5 j’ = 11.5 j’ = 12.5

NO-He Q22 R12 (’=3/2, ’=1)

j’ = 1.5 j’ = 2..5 j’ = 3.5 j’ = 4.5 j’ = 5.5 j’ = 6.5

j’ = 7.5 j’ = 8.5 j’ = 9.5 j’ = 10.5 j’ = 11.5 j’ = 12.5

15-03-2004

R21

R21

P11

P11