State Multipoles Research Articles

Velocity Dependence of Rotationally Inelastic Cross-Sections

Techniques are described for obtaining the energy-dependence of state to state rotational energy transfer (RET) cross-sections in thermal cell experiments through velocity selection by Doppler shift. The case of polarised excitation and detection is discussed in detail and expressions presented which enable the state multipoles of population, orientation and alignment to be obtained from appropriate RET experiments. It is found that the translational order present in a non-isotropic collision environment, such as that resulting from excitation with a narrow linewidth laser, causes state multipoles of different rank to become coupled. Experimental results are presented for the energy dependence of cross-sections for state to state transfer of population and of orientation in Li2-Xe collisions. The way the relative velocity vector changes as a result of a collision may also in principle be studied using a high resolution probe (e.g. a second laser) of the velocity spread following RET.

A theoretical study of alignment effects in collisions of N2 with a Ag surface

We report rotationally inelastic transition probabilities for N2 scattered from a rigid, flat as well as a weakly corrugated Ag surface. For this system Sitz, Kummel, and Zare [J. Vac. Sci. Technol. A 5, 513 (1987)] have recently measured the population state distribution and the quadrupole and hexadecapole alignment of the rotational angular momentum vector with respect to the surface normal. These alignment ratios are directly proportional to the coefficients in a Legendre expansion of the classical angular momentum spatial distribution function. Quantum mechanical expressions for this distribution function in terms of state multipoles of the density operator are presented, and then the classical limit is obtained by introducing asymptotic expressions for the vector coupling coefficients which appear in them. The effect of lattice symmetry on the collisional reorientation of the incident rotational angular momentum vector is investigated through close-coupled scattering calculations based on a model interaction potential. Specular scattering is dominated by ΔM selection rules. The alignment of the rotational angular momentum vectors is analyzed in terms of these selection rules and the azimuthal dependence of the interaction potential.

A scattered-electron-cascaded-photon coincidence experiment on the electron impact excitation of the 32Djstates of atomic hydrogen

Out of the scattering plane angular correlation measurements have been made of the Lalpha radiation arising from the cascade of the 32Dj states of atomic hydrogen to 22Pj detected in coincidence with electrons with n=3 energy loss. The experimental results, together with previous measurements in the authors' laboratory, constitute a complete determination of all the second-order state multipoles characterising the excitation. Data are presented at incident energies of 54.4 and 100 eV and at electron scattering angles theta e of 20 and 25 degrees.

Preparation of polarized excited rare gas atom statesR(np 5(n+1)s) andR(np 5(n+1)p) by laser optical pumping

On the basis of rate equations we have theoretically analyzed all optical pumping schemes, which arise from the rare gas transitionsR(np 5(n+1)s,J i =0, 2) toR(np 5(n+1)p,J f =1, 2, 3) for linearly, circularly and unpolarized light propagating in one direction. Analytical formulae are given for the time-averaged or steady state populations of the magnetic substates and their state multipoles, as a function of the partial decay rateA fi , the natural lifetime τ of the upper level, and the laser induced pumping rateb. Numerical values of these quantities are presented for the case of actual interestR=Ne.

A polarisation correlation study of the excitation of the H2d3Πu(ν1=0, N1=1) level by electron impact

Full details are given of an experiment in which the three normalized Stokes' parameters characterizing the decay of the electron impact excited d3Piu (v1 = 0, N1 = 1) rotational level of H2 were determined in an electron-photon coincidence experiment at 25 eV impact energy and over the angular range of 20-60 deg. An analysis shows how these parameters are related to the state multipoles characterizing the excited state. The rather low degree of coherence which was observed is consistent with various depolarizing effects, namely the averaging over magnetic sublevels which occurred in both the excitation and decay channels and also the perturbations due to fine and hyperfine structure relaxations.

State dependence of rayleigh scattering from an=2 state of hydrogenlike atoms

We consider Rayleigh scattering from a hydrogenlike atom in an arbitrary excitedn=2 state, and we investigate theoretically the dependence of the scattered radiation intensity and Stokes parameters on the state multipoles for the case of unpolarized incident radiation. Because in then=2 case Rayleigh scattering can not be accompanied by the change of the electron angular momentum, only 10 out of the 16 state multipoles influence the scattered radiation attributes. Our study reveals the existence of a measurable quantity which is determined only by Rayleigh scattering from 2p states. For the particular case of excitation by electron impact, some quantitative predictions are made, at the photon scattering angle ϑ=π/2, based on values for the state multipoles extracted from the literature (Blum and Kleinpoppen, Band). The vicinity of Balmer α and Lyman α resonances are studied in detail.

State multipoles for the n=1 to n=2 excitation of atomic hydrogen by electron impact

Multi-pseudostate close-coupling has been used to calculate parameters which describe coherence between the 1s-2s and 1s-2p excitation at 54.4, 100, 200, 300 and 350 eV. Phase conventions are carefully specified. The connection between state multipoles and the elementary parameters used is explicitly detailed. A complete tabulation of all the n=2 excitation parameters at energies from 54.5 to 680 eV is available from the authors on request.

Complete determination of the state multipoles of rotationally resolved polarized fluorescence using a single experimental geometry

When laser radiation is used to prepare single rovibronic levels in molecules, the excited state Mj distribution is invariably polarized. In many such experiments the polarization of the excited state is ignored, which is an inadequate basis for accurate work as much valuable detail is lost. A better approach is a completely polarization resolved experiment in which the preparation, dynamics, and detection of the excited state polarization components (the state multipoles JJρKQ) are fully described. A treatment of polarized excitation in terms of the state multipoles JJρKQ is presented and consideration of excited state symmetry indicates that a common experimental geometry for linearly and circularly polarized excitation is feasible. A complete determination of the state multipoles (K=0,1,2) is shown to be possible within a single experimental geometry. It is shown that neglect of polarization phenomena can lead to ambiguities in the interpretation of some experiments.

Scaling of state multipoles in rotationally inelastic transfer

Laser excitation of molecules in thermal cells gives rise to an anisotropic distribution of m states and thus fluorescence from these states, and from levels populated by inelastic collisions, is polarized. Neglect of fluorescence polarization can give rise to misleading conclusions concerning population transfer in the excited state. Precise measurements of the transfer of population, orientation, and alignment, through polarization ratios, can provide an excellent means for testing the various fitting laws for rotationally inelastic transfer through modified ‘‘sudden’’ scaling laws. The justification for this is fourfold: rotational line strength factors and laser power fluctuations do not affect the degree of polarization; thirdly, the effect of multiple collisions is less significant on the inelastic polarization transfer than it is on the population transfer as measured in conventional total fluorescence intensity measurements; and fourthly, the leading parameters present in fitting laws cancel out, reducing the number of fitting parameters by one. Scaling laws appropriate to polarization measurements are derived and various fitting relations are tested against Li*2/He and Li*2/Ar rotationally inelastic polarization data.

Total characterisation of the excited-state density matrix in laser fluorescence experiments

The axial symmetry of the tensor density matrix elements created by the absorption of linearly polarised radiation is demonstrated. It is shown that a full determination of the state multipoles of a polarised array is possible using a single experimental geometry.

Scaling of state multipoles; a critical test of the scaling relations

Scaling laws for rotational energy transfer may be written for transfer of polarisation as well as for transfer of population. Here we derive an expression, based on the sudden scaling relations, for the transfer of stare multipoles following energy transfer. Some current fitting laws are tested using Li 2 */He, Li 2 */Ar circular polarisation data. We find that polarisation ratios provide a more stringent test of scaling and fitting relations than do unpolarised intensities and that the power gap law yields the best fit for both sets of data examined here.

Transfer of state multipoles in open shell molecule–open shell molecule scattering

General expressions are derived for the transfer of state multipoles in open shell polyatomic–open shell polyatomic rotationally inelastic collisions. These are then used to write expressions for the circular polarizations of spin and rotationally resolved spectral emission lines. Using the distorted wave Born approximation (DWBA), circular polarization ratios are calculated for the specific case of inelastic H+NH2 collisions. The potential is decomposed into a spin-dependent term and an exchange term using tensor methods to facilitate the calculation. Alternatively, the scattering may be considered to take place on singlet or triplet surfaces. An interesting feature of the results is that reorientation appears to be faster on the triplet surface than on the singlet.

Integrated state multipoles and plane asymmetry

The relationship between the integral alignment and orientation coefficients and the polarizations of the decay photons measured in an experiment where the scattered electrons are not observed is expounded. The determination of some integrated state multipoles is shown to provide also a simple checking of the plane symmetry invariance of an atomic collision.

Observation of integrated state multipoles in collisional excitation of Hg atoms by polarized electrons

Impact excitation of Hg atoms (61 S 0→63 P 1) by polarized electrons from a GaAs source has been studied by measuring the polarization of light emitted by the excited 63 P 1 state. The polarization of light which is related to recently introduced integrated state multipoles for 61 S 0→63 P 1 excitation depends strongly on the transverse spin polarization in the vicinity of resonances near excitation threshold. A description of our GaAs source of polarized electrons is given.

Theory and physical importance of integrated state multipoles

We consider the polarisation properties of light emitted by heavy atoms after excitation by polarised electrons. It is discussed what information on spin-dependent forces can be obtained from the measurements.

Relation between atomic coherent-state representation, state multipoles, and generalized phase-space distributions

The relationship between the atomic coherent-state representation of Arecchi et al. [Phys. Rev. A 6, 2211 (1972)] and the state multipoles is established. The state multipoles are used to develop a theory of generalized phase-space distributions for angular momentum (collective atomic) systems. The general theory for angular momentum systems is shown to have many features in common with the general theory for boson systems [Phys. Rev. D 2, 2161 (1970)]. These generalized phase-space distributions contain as a special case the coherent-state representation of Arecchi et al. The applications of the generalized phase-space distributions and state multipoles to the dynamical problems and to the calculation of multitime correlations are given. State-multipole techniques are used to give a brief discussion of the master equation describing cooperative resonance fluorescence.

Generalized reorientation cross section for cylindrically symmetric velocity distributions

Generalized reorientation cross sections are derived for the case of atom–molecule collisions where the molecules initially have a velocity distribution cylindrically symmetric about an axis in the laboratory reference frame. This spatial ordering of the velocity can come about, for instance, by exciting molecular electronic states with a light source whose linewidth is much narrower than the Doppler-broadened absorption line. A simple kinetic theory can be set up in terms of state multipoles that are not completely irreducible; the resulting reorientation cross sections are only slightly more complex than the cross sections occurring in a spherically symmetric velocity field. Two approximations are investigated: a McGuire–Kouri mj-conserving model and a semiclassical model where the orientation of the rotation plane is conserved. The import of the generalized cross sections for several types of experiment and the applicability of the approximate models are discussed.

The influence of hyperfine coherence, elastic and inelastic collisions on state multipoles in laser-excited diatomic molecules

The observed polarization of single-rotational-level fluorescence in diatomic molecules results from the time evolution of the state multipoles set up in excited molecular arrays. Hyperfine coupling affects the observed polarization and we present a theory which treats this phenomenon for two differing cases, those of Li2* and I2*. In the former case hyperfine coherences are important and the theory encompasses this effect. The effects of elastic and rotationally inclastic collisions are considered and cross-sections for destruction of orientation are evaluated. Results on rotationally inelastic transfer are used to evaluate current atom-diatomic molecule scattering theories.

Transfer of state multipoles in excited A 1Σ +u7Li 2 following rotationally inelastic collisions with He: Experiment and theory

Circularly polarized dye laser radiation is used to prepare rotational levels j = 1 to j = 20 of the A 1Σ + u excited state of 7Li 2 with well defined values of the state multipoles K = 0, 1 and 2. Inelastic collisions with helium atoms populate other j levels and we have measured the circular polarisation ratio of emission, C, from these levels. C is plotted versus final j′ for each value of Δ j from +2 to +18 and a family of curves is obtained which may be used as a critical test of current theories. The results are interpreted in terms of cross sections σ K for transfer of the state multipoles under isotropic collision conditions. The observation of substantial polarisation following inelastic collision indicates that the σ K are dominated by certain restricted scattering channels. Relative magnitudes of the multipole cross sections are calculated using the “ l-dominant”, “restricted Δ m j channels” nd the Born approximation. These cross sections are then used to calculate C.

Electron - polarised-photon coincidence study of H2(1Σg+-1Πu) excitation

The first electron-photon coincidence data for molecules are presented. A considerable degree of coherence in the emitted radiation is observed even though the rotational structure is not resolved. The Stokes parameters obtained are related to the state multipoles defining the excited state.