ХИМИЧЕСКАЯ ФИЗИКА, 2004, том 23, № 2, с. 36-40
ЭЛЕМЕНТАРНЫЕ ФИЗИКО-ХИМИЧЕСКИЕ ПРОЦЕССЫ
УДК 539.186
CALCULATION OF DIFFERENTIAL OPTICAL COLLISION SPECTRA
© 2004 r. F. Rebentrost
Max-Planck-Institut für Quantenoptik, 85748 Garching, Germany Received 16.11.2002
With the aim to obtain information on a sub-collisional level optical collision have been investigated experimentally in gases as well as by the more recent molecular beam approach with differential detection. In both cases the coupled-channels theory of optical collisions provides an accurate framework for the interpretation of spectra and differential cross sections. We outline the basis of the method and present calculations for the NaAr and KAr systems. Further applications are the electronic energy transfer in the LiNe(3ö) system and the characterization of the Feshbach resonances in KAr.
1. INTRODUCTION
We consider an optical collision
A(nS) + X + hv —► A(nP) + X
where A is an optically active atom (alkali) and X a (structureless) perturbed (rare gas). Using the view of lineshape theory one can give a different physical interpretation depending on the relation between detuning À = (v - v0)/c and the duration of a collision Tc (~1 ps). In the impact region àtc <§ 1 the absorption is due to collisional broadening of the atomic line. In the far wing àtc > 1 the absorption is related to the existence of Condon points Rc
Uf ( Rc ) - Ui ( Rc ) = hv
(1)
i.e. the resonance condition in fulfilled during a collision involving the potential Ui and Uf. This fact makes optical collisions of interest for the study of the collision process itself. In early applications Eq. (1) was used to obtain potential differences and eventually the potentials themselves from the recorded quasistatic profiles. The real challenge following from Ea. (1) is however the unique possibility to excite a collision pair under controlled conditions (electronic potential term, internuclear distance). Of interest are in particular the properties of the collision complex prepared by the pump laser and of the excited A* state at the end of the optical collision (population of the fine-structure levels, orientation and alignment of the magnetic sublev-els). Even more complex investigations involving fsec pump-probe techniques of a transient collisional system seem presently within reach.
An optical collision involves a three-body interaction of the two colliders and one photon. Thus one may view this process either as a collisional-induced absorption or a collision in the presence of an optical laser field. We define
dN
a*
dt
= k3 NaNX N ф
(2)
where k3 is a three-body rate constant (cm6/s). Then q = = k3/c (cm5) is the absorption cross section per collision pair. On the other hand k3N0 (cm3/s) is the collision rate at a photon density N$ corresponding to a laser field
amplitude E0 = Jl n h v N $. Further one defines a collision cross section by a = k3N0/v for a monoenergetic
distribution (v = -TliTg). Similar for a thermal distribution aT = {av)/vT with vT = J&kT / . The calculation of optical collision cross section is here of interest only for the weak-field limit. In principle using perturbation theory the required scattering S matrices follow from the evaluation of a matrix element
Sfi(e, A) = -inEo(e + hA)№(e)) (3)
of the dipole operator However the wave functions corresponding to the lower and upper manifolds coupled by the optical field will in general be multichannel wave functions. The calculation of wave functions can however be avoided by including also the photons into the Hamiltonian
H = H a + H x + у ax + H ф + V
ax, ф-
(4)
2. COUPLED-CHANNELS THEORY OF DIFFERENTIAL OPTICAL COLLISIONS
The coupled-channels approach based on Eq. (4) provides an accurate and complete treatment of optical collision spectra. The optical collision is hereby treated via the dynamics on the dressed lower and upper molecular terms and the Condon condition is equivalent to a crossing, Fig. 1. The theory fully accounts for nonadi-abatic interactions during the collision, e.g. due to rotational (decoupling) and spin-orbit interactions. The recent development in the investigation of optical collisions using the molecular beam method and a differential detection scheme allows a test between theory and experiment at the level of the differential cross sections
U, cm-1 1000
500
-500
-1000
4 8 12
Fig. 1. Potential scheme for an optical collision.
[1,2]. These differential cross sections contain new and unique information on intermolecular potentials, geometry of collision complexes [3] and nonadiabatic interactions.
We expand the wave function in the form
¥ - R-1 £ |TMlJ)\n)u&JR) (5)
T MlJn
where |TMlJc represents the electronic (J) and the nuclear rotational motion (l), |n) the state of the photon field, and the functions u(R) describe the radial nuclear motion. Here the internal wave function is a Hund's case e basis with T and M being the total molecular angular momentum and its space-fixed projection. After substitution into the Schrodinger equation with the
Hamiltonian of Eq. (4) we obtain a set of coupled-channels equations for the radial functions
" d2 2
—2 + k2 (R) -
dR2
l ( l + 1 )' -------R---2-------
U T MlJn( R )
_ 2m ^ VTMlJn (p)
- £ v t m r J- n uT m r J-n(R) •
T M l J' n'
(6)
The scattering matrices will be defined for a basis of atomic dressed states. In the perturbative case the asymptotic behavior of the wave function can be chosen to be independent of the projection M quantum number
T lJ
U t
TJJ ( R ) ~ 8 TT 8 ll8 JJ W(kJR) -
16
R, a. u.
rJTij j n dx T T i'J'hr(kJR )
(7)
where ht and j are Bessel functions. From the expression for the scattering amplitude
fJ,
(Rk») -r £ £
,mn ^ Jm
.ln- l
T n T ln lm0 ml
x Jnlnmnml |TnMn)(Jlmml\TM) x
(8)
x Y^ (kn) Ylm (R) T
T MlJn
Tn M n ln J n nn
one obtains the differential optical collision cross sections
dGJ0m0 ^ Jm^= |//0m0 ^ Jm(R k0)| •
The full T-matrix required in Eq. (7) is obtained from the reduced T-matrix of Eq. (8) which is independent of M0 and M by standard transformation.
sin 6 do/dO 15000 r
10000
5000
1500
1000
500
180° 0
60°
120°
180°
6CM
Fig. 2. Differential optical cross sections for KAr. Excitation is to the BX (left figure) and to the An (right figure) states, respectively.
0
x
XHMHóECKAü OH3HKA tom 23 № 2 2004
sin 6 da/de 30000
20000 -
10000 -
2000
1500
1000
500
180° 0
60°
120°
180°
6CM
Fig. 3. Differential optical cross sections for KAr.
\ г
\¿
0
о
10 20 30 10 20 30 10 20 30
Scattering angle
Fig. 4. Dependence of the differential optical cross section on the laser polarization. The field vector rotates in the collision plane. Suppression of oscillations is observed when the field vector is perpendicular to one of the Condon vectors.
3. DIFFERENTIAL OPTICAL COLLISION CROSS SECTIONS IN NaAr AND KAr
The most remarkable feature of the differential optical cross sections is the oscillatory structure which has also been verified experimentally. The origin of these Stueckelberg-like oscillations is well understood in terms of the potential scheme of Fig. 1. As an optical transition can occur at a Condon point on either the in-
going or outgoing direction two types of pathes with different phases arise. For a given scattering angle these pathes belong usually to different impact parameters and the dynamics inside the Condon region is either on the ground state or on the excited state. As Figs. 2 and 3 show the oscillatory structure is very regular in cases where only a single Condon point exists and the transition is to a repulsive BE state. An almost quantitative
ХИМИЧЕСКАЯ ФИЗИКА том 23 № 2 2004
a
10°r
10
1-2
10
1-4
10
i-6
2Pn-3Dn
3P
500
1000
1500
2000
8, cm-1
Fig. 5. Energy dependence of 3D and 3P fractional optical excitation cross section. The strong variation seen at low collision energies is due to quasibound levels.
semiclassical theory is based on calculating classical trajectories and the corresponding phases [1]. Furthermore the oscillatory structure depends very sensitively on details of the intermolecular potentials and allows testing or even improving of potentials with an accuracy of a few wave numbers. Excitation of attractive potentials like the An state can result in more complicated structures since interference from more than two pathes for a given scattering angle is possible or more than one Condon point exists. In this case it will depend also on the experimental resolution if structure in the differential cross sections is observable. Figs. 2 and 3 show also examples for excitation in the red wing where in favor-
able cases a pronounced but much less regular structure is seen.
The oscillatory structure of the differential cross section provides a unique way to visualize the geometry of a collision complex. For the simplest case of a single Condon point the two contributing trajectories will have different directions of the internuclear axis at the instant of the optical excitation (Condon vectors). The transition dipole may be either along the internuclear axis (Z-Z) or perpendicular (Z-n) to it. Therefore the contribution of each path will also depend on the direction cosine between Condon vector and the optical field vector if linearly polarized light is used. In Fig. 4 an example for the considerable change of the structure is given. It shows further that the oscillatory structure may completely be suppressed by choosing the direction of the polarization perpendicular to one of the Condon vectors. A detailed analysis of this behaviour can be used to determine the Condon vectors and the corresponding geometry of the collision complex experimentally.
4. ELECTRONIC ENERGY TRANSFER
IN LiNe(2P-3D) OPTICAL COLLISIONS
Electronic excitation transfer 3D-3P in optical collisions in the LiNe(2P-3D) system was investigated using new data on the potentials and nonadiabatic coupling terms [4]. This sy
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