Experimental and theoretical asymmetry parameters for photoionization of H2 showing interference from the Q1 and Q2 doubly excited states

T. J. Reddish1, A. Padmanabhan1, M. A. MacDonald2, L. Zuin2, J. Fernández3 and F. Martín4,5

 1 Department of Physics, University of Windsor, 401 Sunset Avenue, Ontario, Canada, N9B 3P4

2 Canadian Light Source, 101 Perimeter Road, Saskatoon, SK, Canada, S7N 0X43 Dep. Química Física I, Facultad de Ciencias Químicas, Universidad Complutense de Madrid, 28040 Madrid, Spain

4 Departamento de Química, Modulo 13, Universidad Autónoma de Madrid, 28049 Madrid, Spain5 Instituto Madrileño de Estudios Avanzados en Nanociencia (IMDEA-Nanociencia), Cantoblanco, 28049 Madrid, Spain

Dissociative Photoionization Process (DPI) in HDissociative Photoionization Process (DPI) in H22

esHHesHXHh ggg 1122

12

esHHepHXHh uug 1222

12

esHHesHQHXHh gg**

g 112212

12

esHHesHQHXHh gg**

g 112222

12

esHHepHQHXHh uu**

g 122222

12

(2)

(5)

The Q1( ) & Q2 ( ) Rydberg series converge to the two lowest repulsive ionic

states of H2+ and , respectively.

These Rydberg states can autoionize to the bound ground state ion, or dissociatively ionize, or produce neutral fragments [2].

uu p22 ug p22

Fig 1: The potential energy curves of the H2 and H2

+ systems with the shaded area representing the ionisation continuum [3] . In the energy region h = 31-35 eV, are the Q1 (red curves) and Q2 (blue curves) states of & symmetry designated by full and dashed curves, respectively.

u1

u1

• In this study, we focus on the region between h = 31-35 eV where DPI can proceed directly or via the doubly excited neutral states that promptly autoionize.

• These processes provide routes to DPI and result in ion and electron angular distributions that display the hallmark characteristics of quantum mechanical interference [1].

• What is particularly dramatic is that such interference effects are prominently evident even in the case of randomly orientated H2 molecules.

(a) (b)

Listed above are competing processes relevant for DPI. The energy difference between the H2

+ & states is ~ 17eV in the Franck

Condon (FC) region. If H2 is non-resonantly ionized into these two final states, the emitted photoelectrons will have different energies and process (1) and (2) would be readily distinguishable.

uu p22 gg s 12

nlp u ,2 nlp u ,2

Fig 2: Semi-classical pathways for DPI by a 33 eV photon for processes (3-5).(a)Process (3): Resonant DPI through the lowest Q1 state (b) Processes (4,5): Resonant DPI through the lowest Q2 state leading to either H2

+

or states. uu p22 gg s 12

(3)

(1)

(4)

[1] Martín, F et al Science 2007 315 629[2] Fernández, J and Martín, F New J. Phys 2009 11 043020[3] Fernández, J and Martín, F Int. J. quant. Chem 2002 86 145

Toroidal SpectrometerToroidal SpectrometerThe parameters were measured using a toroidal photoelectron spectrometer [4]. Electrons emitted in the plane orthogonal to the photon beam are focused on to the entrance slit of the toroidal analyzer. Energy analyzed electrons emerge from the toroidal exit slit to be focused on to a 2-dimensional position-sensitive detector, so preserving the initial angle of emission. The spectrometer was oriented so that electrons emitted at 0 and 90 to were both included in the final image. The photon energy resolution was ~10 meV at ~33 eV and the (angle-averaged) electron energy resolution was measured as 100 meV (FWHM) using He+ (n =

2) photoelectrons.

Fig 3 : A schematic diagram showing the configuration of the two (partial) toroidal analyzers. Only the 180º analyzer was used for this experiment.

Perpendicular Plane Geometryk , k1 & k2

[4] Reddish et al 1997 Rev. Sci. Instrum. 68 2685

The emission of photoelectrons from a random distribution of atoms or molecules has a characteristic differential cross section that is expressed in terms of a asymmetry () parameter when using 100% linearly polarised

light [5]: (6)

Here is the photoionization cross section for a particular ionic state and is the angle between the polarisation axis, , and the direction of the ejected electron. Our spectrometer has its symmetry axis about the photon beam direction, ,not , and using standard equations given in [6], Eqn (6) is modified in the frame where z is along to be:

(7)

In this work, we take the ratio of two angular distributions of separate processes obtained under the same spectrometer tuning conditions and polarization state, Stokes parameter S1 (= 1 for 100% linearly polarized light) [8]. Followed by rigorous mathematical calculations and appropriate approximations, the ratio becomes:

(8)

is the azimuthal angle, whose origin lies on the major axis of the polarization ellipse. k is a constant defined by the following integral (9),

(9); is the mean efficiency over the

range and by inspection; .The efficiency function, , is obtained using a photoionization process with a known parameter and S1 for a given photoelectron energy. k is obtained from (9) and we take S1 = 0.98.

1cos3

21

4cos1

42

2

Pd

d

k

2121

2

2111

1

11

12

2

1

2

331

4

2

331

4

314

1

314

1

cosSS1

cosSS1

kS

kS

I

I

dkkd

2

1

2

1

2cos

Photoelectron Angular DistributionsPhotoelectron Angular Distributions

10 k

[5] Dehmer J L and Dill D Phys Rev A 1978 18 1 164[6]Cooper J and Zare R N Lectures in Theoretical Physics vol ll c (New York: Gordon and Breach) 1969 p317-37 [7] Schmidt V Electron Spectrometry of Atoms using Synchrotron Radiation, (Cambridge University Press) 1997 pp 41-45, 364-366.

2

122

11 cos14

3cossin

2

331

41

4

,SSS

d

d

Ratio

145 180 215 250 2850.4

0.7

1

1.3

1.6

Emission Angle

145 180 215 250 2850.4

0.7

1

1.3

1.6

Emission Angle(a) (b) (c)

145 180 215 250 2850.4

0.7

1

1.3

1.6

Emission Angle

Fig 4: Ratio of angular distributions of the experimental data fitted with weighted least squares fit using (8). ratio fitted for h = 31 eV at photoelectron energies, a) 6.84 eV and 6.64 eV b) 5.44 eV and 5.24 eV c) 9.04 eV and 8.84 eV

145 180 215 250 2850.7

1

1.3

1.6

1.9

145 180 215 250 2850.4

0.7

1

1.3

1.6

Emission Angle

Ratio

145 180 215 250 2850.4

0.7

1

1.3

1.6

Emission Angle

2H

Data Acquisition and AnalysisData Acquisition and AnalysisAt a given h , the angle-dispersed photoelectron yield is recorded at each photoelectron energy for a fixed number of counts . The raw images are processed and the angular distributions are histogrammed in 5º intervals.

Beginning with the calibration point(s), the variation of with Ek is found by sequentially performing a weighted LSF of the

yield, where Ek = 0.2 eV. For a

given , the uncertainty in

is between ± (0.02 – 0.06), corresponding to the relative uncertainty of the ‘channel-to-

channel’ variations.

The theoretical curve is not convoluted with the experimental photoelectron energy resolution For the values below Ek ~ 10 eV there is also contribution due to low energy ‘background’ electrons, which increases as Ek → 0 eV and suppresses the amplitudes of the oscillations.

,,,,22 kkk EkHEEkkH EIEEI

kE

kk EE

2H

2H

Fig 5: Variation of βH2 with Ek for h =

31, 33 & 35 eV; close coupling calculations (black), measured data (red). Blue error bars on the at 9.9 and 13.9 eV indicate the uncertainty in the overall scale; Red error bars show the relative statistical uncertainty.

0 2 4 6 8 10 12 140.5

0

0.5

1

1.5

2

e

(a) h= 31 eV

0 2 4 6 8 10 12 14 160.5

0

0.5

1

1.5

2

e

h= 33 eV(b)

0 2 4 6 8 10 12 14 16 180.5

0

0.5

1

1.5

2

e

h= 35 eV(c)

Electron Energy (eV)

The comparison with theory reveals, for the first time, the presence of the predicted oscillations in βH2

as a function

of Ek. There is a remarkable agreement in the phase and frequency of the oscillations at all three photon energies; the only minor exception being at ~13 eV in the h = 35 eV data.

Theoretical AnalysisTheoretical Analysis

Fig 6: The dominant contribution to the total (blue) is due to H2

+ channels; the H2

+ contributes only at low electron energy, as expected.

uu p22

gg s 12

gg s 12 Fig 7: Variation of the electron asymmetry parameter, , associated with the H2

+ ionization channel with electron energy for h = 33 eV. The black dashed curve is the result of our full ab initio calculations. (a) Top panel shows the dominant ℓ = 1 partial wave contribution. (b) Bottom panel shows the individual contributions of the 1Q1

1u+ and

1Q21u amplitudes together with their

coherent superposition, which gives rise to oscillations in .

Funding Agencies:

Email contact : reddish@uwindsor.ca

The experiments were performed at the Canadian Light Source (CLS) , VLS PGM beamlineThe experiments were performed at the Canadian Light Source (CLS) , VLS PGM beamline