Imaging in Chemical Dynamics - ACS Publications - American

Chapter 15

Downloaded by UNIV MASSACHUSETTS AMHERST on September 20, 2012 | http://pubs.acs.org Publication Date: October 18, 2000 | doi: 10.1021/bk-2001-0770.ch015

Imaging the Atomic Orientation and Alignment in Photodissociation 1,3

1

1

Eloy R. Wouters , Musahid Ahmed , Darcy S. Peterka , Allan S. Bracker , Arthur G. Suits , and Oleg S. Vasyutinskii 1,4

1

1,5

2,5

Chemical Sciences Division, Ernest Orlando Lawrence Berkeley National Laboratory, Berkeley, CA 94720 Ioffe Physico-Technical Institute, Russian Academy of Sciences, 194021 St. Petersburg, Russia

2

A rigorous theoretical connection is established between experi mental measurements of the photofragment orientation and align ment and the underlying photodissociation dynamics. Labora tory and molecular-frame angular momentum state multipoles are derived as a function of photofragment recoil angles. These state multipoles are expressed i n terms of orientation and align ment anisotropy parameters, which contain information on ex cited state symmetries, coherence effects, and nonadiabatic i n teractions. To demonstrate the power of our theoretical method, it is applied to experimental data obtained with velocity map ion imaging and Doppler techniques in both diatomic (RbI and Cl ) and polyatomic systems (NO and N O ) . Strong recoil-frame alignment and orientation has been observed, as well as coherence effects and long-range nonadiabatic interactions. 2

2

2

Introduction Vector correlations i n the photodissociation of molecules have attracted the in terest of experimentalists for a long time (Jf). Initially (2,3), the correlation of 3

Current address: School of Chemistry, Cantock's Close, University of Bristol, Bristol BS8 ITS, United Kingdom Current address: Naval Research Laboratory, Code 6876,4555 Overlook Avenue SW, Washington D C 20375 Corresponding authors (e-mail: [email protected] and [email protected])

4

238

© 2001 American Chemical Society

In Imaging in Chemical Dynamics; Suits, A., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 2000.


239

the fragments' recoil velocity vector ν with the laser polarization direction e was probed, i.e., the photofragment angular distribution or velocity anisotropy, characterized by the familiar anisotropy parameter β J More recently, another important vector correlation has been measured, that between the e vector and the projection of the angular momentum J of the photofragments on the spacefixed Ζ axis (^-7). These vector correlations are characterized by the moments of the magnetic sublevel distribution: the population, which is independent of the magnetic sublevel rrij distribution, the orientation, which is proportional to the dipole moment of the ensemble and implies a nonstatistical rrij distribu tion, or the alignment, which is proportional to the quadrupole moment of the ensemble and implies a nonstatistical \rrij\ distribution (#). Important infor mation about the dissociation dynamics, the shape of the potential curves, the symmetries of excited states and the role of nonadiabatic interactions can be obtained from a detailed analysis of these vector correlations. Another notable aspect is the correlation between the photofragment recoil direction v , and the photofragment angular momentum J , or the angular distribution of the angular momentum polarization (9-12). This has been studied in considerable detail for photofragment rotational angular momentum, where the experiments can sometimes provide insight into the broad features of the dissociation dynamics and the nature of the transition state. Investigations of this v - J correlation for photofragment atomic orbital po larization, have begun recently in several groups, using either the ion imag ing technique (13-15), in which these effects can be dramatically evident, or Doppler (16-18) or ion time-of-flight profiles (19). These studies have the po tential to provide insight into the underlying photophysics i n the frame of the molecule. In much of the recent literature, coherences, i.e., the off-diagonal elements of the density matrix, are assumed to vanish and only the diagonal elements of the density matrix, the magnetic sublevel populations, are inferred. Lately however, studies in our laboratory (20-23), and i n the Zare labora tory (24~26) have shown that these coherence effects are by no means negligible, and in fact may be used to provide new insights into the photodissociation dy namics. These studies build on the theoretical foundation provided by Siebbeles, Vasyutinskii and coworkers (27,28), however, each experimental group has pro vided a similar description of the photodissociation mechanism in terms of a different set of alignment parameters. In this chapter we will provide the reader with a comparison of the two sets, and equations to convert the one set into the other. We feel that this will solve the confusion that the two comparable sets might have created. As an overview of our recent work, in this chapter we will first describe the theoretical apparatus developed for photofragment orientation and alignment. In this comprehensive frame work, photodissociation processes will be expressed in terms of orientation and alignment anisotropy parameters, each related to a specific dissociation mechanism. A demonstration of its power will be shown in f This parameter will be called βο in this chapter for reasons explained below. In Imaging in Chemical Dynamics; Suits, A., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 2000.

240


a few experimental applications: orientation has been measured in the photodis sociation of R b l ; photofragment alignment is analyzed in the case of diatomic photodissociation (CI2), as well as i n the photodissociation of polyatomics ( N O 2 and N 2 O ) . For a detailed discussion of those examples we refer the reader to our recent publications (20-23,29).

Theory: Two-photon ion imaging spectroscopy of polarized atomic photofragments Photofragment orientation and alignment angular distributions Laboratory frame In this chapter, we consider a generic molecular photodissociation event in which a molecule AB produces fragments A and Β with angular momenta JA and J'B, respectively. Each fragment can either be an atom or a molecular radical. The differential excitation cross section matrix elements ^ ^ ( 0 , 0 ) give the proba bility of photofragment A flying in a direction specified by the polar angles 0, φ with components m , m of JA along the space-fixed Ζ axis (see Figure 1). The diagonal elements of the matrix (m — m') give the probability of producing the fragment with a specific angular momentum JA and component m , while the off-diagonal elements (τηφτη') describe the coherence between states with dif ferent m quantum numbers (8). The initial and the final total angular momenta of the molecule are and J , respectively. 1

Figure 1. Space-fixed reference frame for a diatomic molecule

AB.

It is convenient to express the excitation matrix elements σ^^θ,φ) in terms of the angular momentum polarization irreducible cross sections σ^£%\θ, φ) = ο- (θ,φ) which are spherical tensors of rank Κ and component Q where κο

}


241

Q = -K...K

(8,30,31)


The photofragment differential cross section (eq 1) for one-photon fragmenta tion, obtained with first-order perturbation theory for electric dipole transitions in the axial recoil approximation, is (27)^ 1 ) 1 / 2

a (M) =

Σ

Kq

E t - ^ ' W e )

kd,£

a

/o(0,0) + 2 / o ( l , l ) ι / (l,l)-/ (0,0) 2

a

2

=

V(JAY

2

/o(0,0) + 2 / o ( l , l ) '


247

Re[/ (1,0)] 2

1

72 = 2V3VUA)-

(18)

/ο(0,0) + 2 / ( 1 , 1 ) ' 0

Im[/ (1,0)3 2

1

= 2V3VUA)-

7 2

/o(0,0) + 2 / ( l , l ) ' 0

2


/ (i,-i)

V6V(j ) - ι

η =

2

A

/o(0,0) + 2 / o ( l , l ) ' 1

2

with V(j)=5{j(j + l)/[(2j + 3)(2j - l)]} /' . The parameters and their physical range are listed in Table II. The ranges were calculated assuming maximum possible alignment of the atomic fragment in the molecular frame (28). They do not depend on the specifics of the molecule, which can either be diatomic or polyatomic, and have the advantage that for any type of reaction the parameters cannot be outside the ranges; these most likely will be smaller in less general cases, as in the experimental examples further on. Note, that the ranges presented i n our publications (21-23) for the 772 and 72 anisotropy parameters differ from those given in Table II. The reason is that Table II. Range of the rank Κ — 2 anisotropy

parameters.

Range 1 2j-l . . . . 5 * ' * 5(j-H) ^

—

52

, f

1 Sa n

_ (2j-l)(2j+3) (2j~l) 20j(j+l) ' * ' 5(j+l) -

2

J

lf 3

i n t e

, Sa

g

e r

half-integer " g

h a ! t

, n t e

W ^ - ^

e r

^ an integer

«2

_ m r l M ^ ^ m

. .. W-ajgff+'>

72

2j-l

Γ(1+/?ο)(2-/?ο)Ί ]

1 / 2

0

2

-( ~ft>)

m

2

( -g°)

1/2

if

1/2

1/2

2J-1 r(l+j3o)(2-g )] 50+Ϊ7 [ 2 J

r

2 j - l \(1+0ο)('2-βο)] * ' ' 5(j+l)

_1 Γ(1+/?ο)(2-/?ο)1 1 Γ(1+/3ο)(2-/3ο)1 5 I 2 J '"5 1 2 72

teg

1

0

7 / J / ) ·

(

3 0

)

The quantum numbers ji, j and jf designate total angular momentum of ini tial, excited (intermediate or "virtual") and final states of the photofragment, respectively. T h e ji, %, and 7/ are sets of all other fragment quantum num bers excluding projections. The presence of both j and j' as well as j and 7g is a result of coherent sums over different intermediate excited states. T h e factor S( yiji j je, y j ,jfjf) contains reduced matrix elements and the energy denominator of second order time-dependent perturbation theory: e

e

/

,

1 e

,

,

e

e

e

e

S(jiji, 7e je, le3e > Ifjf ) = {lf3f\\d\\leje){lfjf\\d\\iJ^ (Eei-hy

+ iTWiEji-hv-iTfi)

K

'

}

Eqs 28-31 can be used for any polarization of the probe light and any experimen tal geometry. They are equivalent to those given by Kummel and coworkers (48) and by Docker (49), except that all projection information and laser polarization dependence have been factored out. The practical convenience of this modifi cation lies i n the complete separation between the scalar linestrength factor S* and the tensor quantities in the photon-atom dot product. In this form, the qualitative dependence of the signal on laser polarization can be studied without reference to linestrengths. We now write expressions for the 2+1 Resonance-Enhanced Multiphoton Ionization (2+1 R E M P I ) signal for four experimental geometries, correspond ing to the linear probe laser polarization along the axes Χ , Y , Z , and to the right/left hand circularly polarized dissociation light propagating along the Ζ axis. Later on, we will define these signals as Ιχ, Ιγ, Izi and respec tively. Using eqs 28 and 29, with polarization tensor components 1 2 £00 = Ei = 0; E = -^=; E i = 0; E22 = 0 for e = e 2

ki

q

1

20

1

Eoo =

E

Eoo =

EIQ = -~=; E

tq

= 0; E

=

20

20

2

z

1 E i = 0; E ± 2

2

= ±TJ

2

= - - ^ = ; E21 = 0; E ±2 2

=0

FOR

E

e

e

= x> Y

for e = e ,


R

255 we obtain I

(32)

= C [PoPoo + P-2P20 + PiPio]

z

Pi

Ιχ,γ

= C^PoPoo - γ

{/320 T \/6Re[p 2]} 2

(33)

+ ~ {ipio Ψ 2\/ÏÔRe[p42] + \/7ÔRe[p 4]} Downloaded by UNIV MASSACHUSETTS AMHERST on September 20, 2012 | http://pubs.acs.org Publication Date: October 18, 2000 | doi: 10.1021/bk-2001-0770.ch015

4

IR,L = C [ΡξPoo ± PÏPio

+ P2P20 ± P3P30 + PAPAO]

(34)

where

p

=

2

P4 =

°20 '

7=&ο.·> "1

Λ/5

22 I '

2

^—S 3vl4 ' 4 2

(35)

C

Pn =

Pi

Ρξ ~

6 ^ 0 + ^ 2 2

^ 2 1

^ 2 2 j ,

Imaging in Chemical Dynamics - ACS Publications - American

Recommend Documents