n-Vector correlations in collision dynamics with atomic orbital

Boulder, Colorado 80309-0440 (Received: January 2, 1992; In Final Form: April 15, 1992) ... an extensive expos6 of the necessary mathematical formalis...
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J. Phys. Chem. 1992, 96,6136-6146

6136

FEATURE ARTICLE n-Vector Correlations in Collision Dynamics with Atomic Orbital Alignment: The Importance of Coherence Denoting Azimuthal Structure for n 1 3 Jan P.J. Driessent and Stephen R.Leone*.* Joint Institute for Laboratory Astrophysics, National IItstitute of Standards and Technology, and University of Colorado, Department of Chemistry and Biochemistry, and Department of Physics, University of Colorado, Boulder, Colorado 80309-0440 (Received: January 2, 1992; In Final Form: April 15, 1992)

An introduction is presented of vector correlations in collision experiments involving atomic orbital alignment or orientation. At present, aligned or oriented species can be prepared (or probed) with “relative”ease using polarized laser beams. First, an extensive expos6 of the necessary mathematical formalisms for two- and three-vector correlations is given, and then experimental examples for atomic Ca are discussed to elucidate the theory. It is demonstrated both theoretically and experimentally that azimuthal structure about the initial relative velocity vector (called coherence) becomes important when three or more vector quantities are controlled in the collision process.

I. Ioboduction: Vector Correlations Collision processes are very sensitive to vector quantities. The most obvious vector quantity in a collision is the initial relative velocity, vi. With the availability of crossed-beam apparatuses control over the direction of the relative it is possible to have pr& velocity vector. If we consider elastic conditions of structureless species, there is only one additional vector quantity that may be resolved experimentally, the final relative velocity v f . The correlation between the initial and final relative velocity vectors can be determined in a differential scattering experiment, where both velocities are resolved. This correlation can select collision processes within certain well-defined impact parameter ranges. By analyzing the scattering as a function of impact parameter range, one can obtain accurate potential information.I4 Atomic collision processes that attract recent interest involve structured states, characterized by the quantum state b,m) for the total electronic angular momentum, j , and magnetic quantum number, m. The potential curves governing the colliiion dynamics are now characterized by a magnetic substate dependence. In most early experimental situations this m dependence is not resolved, because one encounters a statistical average of the varidus substates. The availability of polarized laser beams has made it possible to manipulate the alignment (symmetric m distribution) or orientation (asymmetric m distribution) of these structured species and to study their role in collision dynamics.s-21 For example, a p orbital can be prepared asymptotically parallel (I; state) or perpendicular (n state) in a collision. Interesting magnetic substate dependences can thus be obtained for the collision process, which may reveal important features of the potential cur~es.2~-~ Further possibilities of polarized laser beams may be utilized by preparing two collision partners in an aligned (oriented) state prior to the collision event. Furthermore, it is possible to probe the alignment (orientation) distribution after the collision for either one or both of the structured species. Therefore, it is evident that the number n of relevant vector quantities can easily range from n = 3 to n = 6, or higher if we consider molecular targets. ‘Present address: Physics Department, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands. *Staffmember, Quantum Physics Division, National Institute of Standards and Technology, Boulder, CO 80309.

0022-3654/92/2096-6 136$03.00/0

In a Ycompletenscattering experiment all the relevant vector quantities would be controlled and manipulated with respect to each other.28 In general, however, not all vector quantities are controlled in the experiment. To control each individual vector quantity, additional experimental effort is required, in terms of additional lasers and/or specific apparatus design. The experimental results become more numerous and more detailed, but in general this also results in a significant reduction of experimental signal. The latter effect is the main reason most vector correlation experiments with atomic orbital alignment have thus far resolved only two vectors, and in only a few cases three vectors. In a typical two-vector correlation experiment, an aligned (oriented) atomic state is collided with a structureless (isotropic) partner, e.g., a ground-state rare-gas atom. The angle between the relative velocity and the laser polarization is varied, resulting in magnetic substate dependent results. In threevector correlation experiments various combinations of three controlled vectors have been reported. Some of these three-vector experiments employ an aligned (oriented) state in a differential scattering apparat ~ s . ~ In~this ~ case ~ ~theJ three ~ - controlled ~ ~ vectors are the two velocity vectors, vi and vf, and one laser polarization vector [linear, E, (or circular, k,)]. In another type of three-vector correlation experiment, two independently aligned (oriented) atoms are scattered (e.g., the associative ionization studies of Na(3p)13J7). Here the three controlled vectors are vi and two laser polarizations E, (or k,) and E2 (or k2). Recently, we have embarked on a three-vector experiment as One polarized laser El(or k,) is used to prepare an aligned (oriented) Ca(4s5p,lPI) state which scatters with He with a well-defined relative velocity vi. With a second polarized laser beam E2(or k2) we probe the alignment (orientation) distribution of the final Ca(4~5p,~P,)state. Many different alignment effects have now been reported, and interest in this area is increasing. The theoretical formalisms for many of these experiments are not readily available in the literature and are rather scattered in many places. The present paper attempts first to give a cohesive expos6 of this subject by pointing out the diverse possibilities of n-vector correlation experiments with atomic orbital alignment. It is not our intention to be complete in reviewing the broad field of available alignment data. Rather we review the various mathematical formalisms that have been employed in this field and then discuss just a few examples from the work in our lab. In presenting the theoretical expressions, 0 1992 American Chemical Society

The Journal of Physical Chemistry, Vol. 96, No. 15, 1992 6137

Feature Article it is also our aim throughout this article to visualize the underlying mathematical concepts of alignment studies as much as possible with simple vector (or orbital) plots, which can be understood without the equations.

obtained by taking the "outer product" of the state vectors p = I x ) ( X I . In the v , m ) representation this results in the following matrix form:Z9

II. Matbematical F o ~ s m We first introduce the general mathematical concepts and ideas behind atomic orbital alignment in scattering theory. The following sections summarize briefly the necessary mathematical tools, and references to more detailed literature are given where appropriate. The reader can skip sections which are already familiar. In section 1I.F we draw together the previous sections to demonstrate the importance of coherence for n L 3. A. "Pure States" v e " "Mixed States". To perform scattering calculations, a quantum mechanical representation of the initial state is required. Normally, this is done by a state vector Ix). Problems arise, however, if we deal with a degenerate state, since the magnetic sublevels are not necessarily populated independently of each other. To take this effect into account, we must introduce the concept of "pure states" and "mixed states". A more detailed review of this concept is given by B l ~ m Let . ~ ~the state vector I x ) denote a "pure state". A pure state is described as a linear superpositionof magnetic basis states p,m) and represents a vector in (2j 1)-dimensionalspace:

+

where the coefficients a,,,have both a well-defined absolute value and a well-defined phase relation with respect to each other. Of course,the coefficients a,,,depend on the choice of the coordinate frame. The term pure state is often interpreted as a single magnetic suktate (in a specific choice of the coordinate frame). This is not necessarily true, as will be described in section I I C , when superposition states of several magnetic sublevels are discussed. The opposite, however, is true: a single magnetic substate is always a pure state. The linear superposition of eq 1 is referred to as a fully coherent superposition. Full coherence implies that a well-defined phase relation exists between the individual basis states in any coordinate frame. In statistical equilibrium, structured species have a natural abundance where all magnetic substates p,m) are equally populated, independent of the choice of coordinate frame. In this situation we can give only a statistical weight distribution W,,, for the basis states Ix,,,): +i

w,,, = (2j + 1)-1; 2

W, = 1

m--j

In general the basis states Ixm)are individual magnetic sublevels p,m). Experimentally this results in a statistical averaging over the possible substates p,m), and alignment resolved scattering information cannot be obtained. Such a state cannot be represented as a linear superposition of magnetic substates with a definite phase relationship, like a pure state (eq 1). States which are not pure are called "mixed states". A mixed state with a statistical population distribution [ W,,, = (2j l)-l] is the extreme opposite of a pure state and is often referred to as a fully incoherent state. The population distribution of a mixed state need not be statistical [W,,, # (2j 1)-*],in which case we have a partially coherent state. B. Deasity Matrix Represent.tioa. An alternative and highly useful method of characterizingpure and mixed states is through the density matrix representation. It describes the preparation steps which have been performed experimentally and therefore contains all the information about the quantum mechanical state of the scattering partners. In the special case of a pure state we can e x w the state vector Ix) as a h e a r combination of magnetic substates p,m) (eq 1). The density matrix operator p is then

+

+

L+l..

.

=

The diagonal terms pmmof the density matrix give the probability of finding a particle in the corresponding p,m) level (population). The off-diagonal elements p,,,,,, ( m # m? contain the information about the phase-relation (coherence) between the various basis states. Both the well-defined absolute value (diagonal elements) and the well-defined coherence of the a,,,-coefficients (off-diagonal elements) are now contained in this matrix representation. The spatial structure of the excited state is fully characterized by the density matrix. The diagonal elements p,,,,,, represent the structure along the z axis (but with cylindrical symmetry about z) and the off-diagonal elements pd (m # m? represent structure in directions other than z. Clearly, the density matrix has different values in different coordinate frames. For instance, the density matrix will be diagonal only in a coordinate frame where the excited state is cylindrically symmetric about the quantization axis (z axis). In the case of a mixed state the density matrix is obtained by taking a weighted sum of the pure state density matrices for the different basis states Ix,,,) (see eqs 2 and 3): P

= CwmIxm)( X m I m

(4)

In the special case of a fully incoherent state [ W,,, = (2j + l)-l], the phase relation between any two basis states is arbitrary in any coordinate frame. Therefore, the off-diagonal elements of the density matrix must be zero, independent of coordinate representation, and the density matrix is proportional to the identity matrix [pmd = 6,,(2j + l)-l]. The density matrix representation provides a powerful mathematical tool to describe the quantum mechanical state for both pure states and mixed states. It contains all the information about the scattering species. To appreciate the strength of the density matrix formalism, the reader is referred to the work of B l ~ m . ~ ~ C. Optical Pumpirag. The use of polarized laser beams in a collision experiment does not necessarily guarantee the preparation (or probing) of pure states. When both the upper and lower level are degenerate (j > 0), there are several independent excitation processes which couple different sets of magnetic substates. The excited state is then a mixture of independently populated states, a so-called mixed state. On the other hand, if one of the states is nondegenerate (j = 0), only one set of magnetic substates is coupled by the laser radiation and we deal with a pure state. There are other possibilities to excite a pure state, even if both states are structured levels (j> 0). One possibility occurs when pulsed lasers are employed in the excitation step and the lower level is already in a pure state. This is because the time duration of the laser pulse can be typically much shorter than the lifetime of the states involved. Therefore, there is no time for populating neighboring magnetic substates in the lower level incoherently through spontaneous emission. A second possibility to prepare pure states is to remove the degeneracy of the structured species by applying an external magnetic field and making use of the Zeeman effect. One-Photon Excitation Processes. In many experimental studies a single laser beam is used to prepare an aligned/oriented

6138 The Journal of Physical Chemistry, Vol. 96, No. 15, 1992 m= - 1

-

0 +1 l-r-

I

a)j=O+j=l

; B=O

w,=o

m = -3

-2

-1

0 +I

+2 +3

m= -3

-2

-1

0 +I

+2 +3

w,=o

0

0

1

0

0 0

1

0

Figure 1. Examples of pure and mixed states prepared with a linearly polarized laser beam. The thick arrows indicate laser absorption and the thin arrows represent spontaneous emission. With no magnetic field, the excited state is (a) a pure state in case of a nondegenerate lower levello and (b) a mixed state for a degenerate lower In these cases the laser polarization El serves as the quantization axis. (c) An external magnetic field B may remove the degeneracy of the structured lower level and a pure state can be ~repared.~’Because the B vector serves as the quantization axis, the quality of the laser polarization is not important.

atomic state. The experimental data will depend on the angle 0 between the laser polarization vector and the relative velocity vector. In Figure 1 we show some experimental examples of excited atomic states, which can be prepared with a single linearly polarized laser beam. Figure l a shows excitation from a nondegenerate lower level (j = 0 - j = l), which is encountered in state changing collisions of a Ca p state.” Since the lower level (lS0) is a pure state, the upper level (‘P1) is prepared in a pure state as well. With an appropriate mathematical formalism it can be shown that the experimental cross section will vary smoothly with angle 0 as a cos (20) function [in tensor notation, rank-2 alignment28-30]. Figure 1b shows an excitation process which couples two degenerate levels = 2) = 3). Experimentally, this excitation process is encountered in alignment studies with Na atoms8J3J7 and with Ne at0ms.~9’~ In this situation mixed states are prepared. When continuous-wave (cw)lasers are used, a nonstatistical weight distribution W, is obtained after a few cycles of photon absorption and spontaneous emission, as indicated in Figure lb. To avoid the nonstatistical averaging of the experimentalresults by the factors W,,one may use the Zeeman effect to remove the degeneracy of the upper and lower level. This has been done in the case of Ne by applying an external magnetic field of =222 Ge31This excitation scheme is depicted in Figure IC. Individual Zeeman levels (pure states) can be excited. The alignment (or orientation) of the structured species is determined by the laser frequency and the orientation of the B field, which serves as the quantization axis for the magnetic sublevels. The quality and the arrangement of the laser polarizations are not important. Multiple-Photon Excitation Processes. In more advanced alignment studies two or more pulsed laser beams are used to prepare an aligned (oriented) state. Excited states with more complex symmetries can now be prepared. The laser polarizations can have different mutual orientations, allowing for preparation of a greater variety of wave functions which describe the excited state. Two-photon sequential excitation has been used in near-resonant charge transfer of Na(4d,2D5/2)16 and in state changing collisions of Ca(4p2,’D2).19Experimental results are obtained in both cases for a parallel and perpendicular arrangement of the two linear polarization vectors. The shape of the electron charge cloud allows for angular behavior up to cos (40) in the experimental data [d states: rank-(2j = 4) alignment]. Three-photon sequential excitation has been applied in state-changing collisions of Ca(4s4f,’F3)?O In this case the linear laser polarizations are arranged either all three parallel or with one polarization perpendicular to the others. Angular behavior up to cos (60) has been demonstrated

v

Driessen and Leone

-v

d) ( Y T . - ~ - YI)/D T

c> y7.0

Figure 2. Spherical parts of the wave functions for d and f orbitals prepared by two and three linearly polarized laser beams, respectively. The thick arrows denote the arrangement of the laser polarizations. The spherical harmonics, yj,oand (q,-l- l 3 / d 2 ,represent excited states in the photon frame, which is chosen such that the x-z plane corresponds with the plane of the figure (containing all laser polarizations).

in this case [f states: rank-(2j = 6) alignment]. Substantial experimental complication is encountered in the two-photon experiment with sodium,16 because of the fine and hyperfine structure. The excited Na(4d,2D5/2)is described by a mixed state. In the multiple photon excitation of calcium, pure states can be prepared since no fine or hyperfine structure (zero nuclear spin) is present in the ground state of Ca(4s2,’So). Figure 2 shows two different angular parts of the wave functions for both the d state (j = 2) and f state (j = 3) of Ca as they are experimentallyprepared by a multiple photon e x c i t a t i ~ n . ’ The ~*~~ prepared wave functions are pure states which can be written as linear combinations of basis states v , m ) . The angular parts of the wave functions are mathematically described by spherical harmonics correspondingto Y2,0and the superposition state (Y2,-’ - Y2,’)/1/2 for the d states, and Y3,0and the superposition state (Y3,-1 - Y3,1)/1/2 for the f states. Note that these superposition states are pure states, but not individual magnetic substates in any frame (see section 1I.A). The superposition state (Y2,-’ - Y2,’)/1/2 is symmetric for rotations over ~ / 2 and , the corresponding experimental curves show a pure rank-4 alignment [ao+ u4cos (4@)].19 This suggests that it is possible in general to prepare an excited state with an N-photon excitation process = 0) = N ) that demonstrates pure rank-2N alignment [ao a2Ncos (2Np)l. This may be achieved by orienting the N linear laser polarizations in a single plane at mutual angles T/N. In the Appendix it is shown that the resulting excited state is symmetric for rotations T/N in that plane, resulting in pure rank-2N alignment. D. “Collision Frame” versus “Photon Frame”. In section 1I.C we have shown that the laser excited state is most easily described in a coordinate frame where the laser polarization vector E (or k) serves as the quantization axis. We will refer to this coordinate frame as the “photon frame”. Experimentally, the photon frame(s) can be arbitrarily oriented with respect to the initial relative velocity vector vi. Scattering calculations usually refer to a “collision frame”, where the velocity vector vi serves as the quantization axis. The nuclear motion of the two colliding atoms is quantum mechanically represented by a plane wave dkeR.The direction of the incident wave vector k (Ilq) is an axisof rotational symmetry of the scattering problem if total cross sections are measured. The collision frame is usually preferred in scattering calculations

+

-v

The Journal of Physical Chemistry, Vol. 96, No. 15, 1992 6139

Feature Article because the plane wave dkSR obtains its most simple representation in this frame as an expansion in a series of Legendre polynomials p I (cos e):30932

h

where 1 represents the orbital angular momentum quantum number for the nuclear motion, and yl(R) is an R-dependent expansion coefficient. Note that this expression does not depend on the azimuthal orientation angle 6 in the collision frame. Only spherical harmonics Yl,mwith magnetic quantum number m = 0 contribute, which are 6 independent. Classically, this corresponds with the fact that the orbital angular momentum vector wi X b is perpendicular to the relative velocity q, with b the impact parameter and p the reduced mass. The preference for the collision frame in scattering calculations is also apparent from a semiclassical point of view. In the collision process a large range of impact parameters b can be recognized. Semiclassical trajectories have to be calculated for all these impact parameters. By choosing the collision frame, these trajectories have to be calculated only as a function of lbl and not as a function of azimuthal angle 6 of the impact parameter b. In principle all coordinate frames are equivalent; each representation can be transformed into any other. Although the collision frame is usually preferred in scattering calculations, other coordinate frames may be more appropriate for an understanding of the collision physics. For differential scattering experiments with collisional excitation, Hermann and Herte133suggested two coordinate frames with the quantization axis perpendicular to the collision plane (which contains vi and vf), a “natural frame” and an “atomic frame”. When the initial states have no preferred direction, as in collision-induced alignment studies, the reflection symmetry in the collision plane is a conserved quantity. Thus basis states can be chosen which already have this reflection symmetry and fewer basis states have to be taken into account. In differential scattering experiments with initial structured species, the natural frame (or atomic frame) is still useful if the laser excited atomic states are prepared with a well-defined reflection symmetry.14 This implies that the laser polarization vector should not be arbitrarily oriented with respect to the collision plane. E. Rotation Operator: “Photon Frame” “Collision Frame”. All structured species have to be represented in an unique coordinate frame, usually the collision frame (see section 1I.D). To relate quantities in the collision frame (zwl),when the structured species are given in the photon frames (zph), an appropriate rotation operator R is used. We will use the “passive” transf~rmation,~~ Le., the laser excited state in the photon frame is not actively rotated to the collision frame but rather written as a superposition state in the collision frame. An extensive study of rotation operators is given by Messiah32and by We use the notation of Brink and S a t ~ h l e r A . ~transformation ~ from one coordinate frame to another can be characterized by three Euler angles a, 8, and y:

-

R(a,B,r)= RZ(4 Ry(@>Rz(r) R(a,B,’Y)P,m)ph= CDm’m(a,@,T)P,m’)wI m‘ = Ee-iatn‘&’mtm(B)e-’YmP,m ’) col m’

XP

Figure 3. The Euler angles (/3,?) serve as polar angles for the relative velocity yi in the photon frame. The azimuthal angle y is important only for excited states with three characteristic axes, as is indicated here for the d state of Figure 2b (Y2,-,- Y 2 , 1 ) / d 2 .

direction. This can be recognized in eq 6 through the real Wigner-d functions ddm(@). An individual magnetic sublevel p,m) is an eigenstate for rotations y about the quantization axis with an eigenvalue d i y m . If only a single j,m level is populated, the first rotation y only results in an overall phase-factor e-’Ymand an arbitrary value of y = 0 can be chosen (e.g., the spherical parts Yzoand Y3pin Figure 2, parts a and c, respectively). Therefore, two Euler angles a and 0, are sufficient to describe the coordinate frame transformation. This reflects the fact that we are rotating an azimuthally symmetric state which has one characteristic axis. If the initial state is a superposition of magnetic sublevels, the third euler angle y cannot be neglected. The superpositionstates (Y2,-1 - Y 2 , 1 ) / dand 2 (Y3,-1- Y3,l)/d2of Figure 2, parts b and d, demonstrate this very clearly. The angle y is now related to the azimuthal orientation of the perpendicular laser polarization with respect to the plane formed by the two quantization axes (zph and z,J. This is visualized in Figure 3, where the quantization axis zwlis shown for the superposition state (Y2,-1- Y 2 , J / d 2of Figure 2b. The angle y is required for excited states with three characteristic axes. It is important to know whether the aligned/oriented state has one or three characteristic axes: three characteristic axes only appear in N 2 2 photon excitation processes with at least two nonparallel laser polarization vectors. The angle a determines how the xWland ywl axes are oriented in the collision frame. The phase-relation (coherence)between the DAlm(a,&y) coefficients for the superposition state of eq 6 depends on this angle a. This angle a can be set at arbitrary values unless we are dealing with three or more controlled vector quantities in the experiment (see section 1I.F). Since the density matrix is a convenient way to represent both pure states as well as mixed states (section II.A), it is important to know how the density matrix transforms under the rotation of the coordinate frame. The expansion coefficients Dklm(a,@,y) in eq 6 are the elements of the rotation matrix which can be multiplied by the state vector Ix) of eq 1:

[’

Q-j,-,(agPTy)

Wa,P,r)=

o’+j,-Ja,P,r>

(6)

The expansion coefficients Dim(a,/3,y) of the rotated superposition state are the elements of a (2j + 1) X (2j + 1) unitary matrix, called the rotation matrix. The rotation operator R is equivalent to first a rotation R,(y) about the z p h axis, then a rotation ai,(B) about they axis, and finally a rotation Rz(a)about the rotated zwIaxis. The rotations a and y are azimuthal rotations, where the quantization axis does not change. Therefore, they affect only the phase relation of the LYAtm(a,&y)coefficients, as can be recognized in eq 6. The absolute value of Dim(a,/3,y) is affected only by the rotation B, in which the quantization axis changes

i

IX’)col

]

Q-j,+ja,P,Y) ’

i

(7)

Q+j,+ba,P*d

= DIx>,h

Mathematically eqs 6 and 7 are equivalent. From eqs 3,4, and 7, it follows that the density matrix p transforms under the coordinate transformation by p’col

= D(a,@,r)pphDt(a,@,r)

(8)

We now have the mathematical equipment to obtain the quantum mechanical representation of laser excited states in the collision frame. Thus we can construct a representation for the total

-uriessen . .ana Leone

6140 The Journal of Physical Chemistry, Vol. 96, No. 15, 1992

t ,

/

f

/

a

....

.

-I

Ycol

XCOI

t

,

Xed

a) two vector correlation : a@,) = a@,)

a) two vector correlation

b) three vector corelation Figure 4. Relevant angles in n-vector correlation experiments. (a) n = 2. Only the angle 6 between the two vectors is important. The azimuthal angle a is arbitrary. (b) n = 3. Three angles are necessary to characterize the mutual orientations of the three vectors: two polar angles 8, and &, and one azimuthal difference angle A a = a2- a,.

scattering process in the collision frame and scattering calculations can be performed and/or experimental parameters can be derived. F. Importance of Azimuthal Structure (or Coberence) for n 2 3. Consider an alignment experiment where only two vectors are controlled. For crossed-beam experiments, these two vectors would most likely be vi = zWl,and one laser polarization El = zph(or kl). The angle a used in the coordination transformation represents the azimuthal orientation of zph in the collision frame. Thii is visualized in Figure 4a. Since there are only two vectors, the angle a can be set to arbitrary values and the results (e.g., cross section values) remain unchanged. In the density matrix representation only the off-diagonal elements pmd ( m # m') vary with angle a according to (eq 8). Therefore, the final cross section expression cannot contain terms proportional to pmm,( m # m'). Only population information given by the diagonal elements pmm is important. If we consider three (or more) controlled vectors, the angle a is no longer arbitrary and azimuthal structure (or coherence) becomes observable. In this case, parameters are necessary that describe how the phase relationship in the initial state transfers into phase relationships in the final state, which is distinctly separate from population information. The relative velocity vector vi is directly related to zWI.The directions of the two other vectors can be specified in this collision frame. For this, three angles are necessary: &, p2, and A a = a2- aI, as is indicated in Figure 4b. The expression for the cross section will depend on all three angles pl, 02,and Pa. One angle is an azimuthal difference angle Aa, which is related to the azimuthal structure (or coherence) of the process. The A a dependence in the cross section expression comes about only through the terms which are proportional to the off-diagonal elements of the density matrix pmmt(m' # m ) . Of course, if one (or both) vectors are parallel to vi (zWl),the azimuthal difference angle A a can take on arbitrary values again and coherence is no longer important, i.e., azimuthal structure is no longer observable in the experiment. To better appreciate the effect of the angle Aa, visualize the collision with one or two p orbitals (see Figure 5 ) , which are aligned perpendicular to the relative velocity vi (=zEol).In the case where only one p orbital is involved (see Figure sa), we recognize two controlled vector quantities (vi and the symmetry axis of the p orbital). The angle a can be chosen such that we have a px orbital [(YI,-, - Y 1 , , ) / d 2or] a py orbital [i(Yl,-l+

b) W vector correlation : a(px9px> f a@,,py) Figure 5. Vector correlations visualized with p orbitals aligned perpendicular to the relative velocity vector. (a) In a twcwector correlation only one p orbital is involved. The azimuthal orientation of the p orbital is arbitrary: u(p,) = u(py). (b) In a three-vector correlation two p orbitals are involved. Azimuthal orientation of the two p orbitals is no longer arbitrary: u(p,,p,) f u(p,,p,). Y 1 , , ) / d 2in] the collision frame. The corresponding density matrices are given by

Both collision procases are identical and must have the same cross section expressions. Therefore, the off-diagonal elements p-l,l and pl,-l do not contribute to this expression. In the case where two p orbitals are involved (see Figure 5b), there are three controlled vector quantities (vi and the two symmetry axes of the p orbitals). In this case an azimuthal difference angle Aa between the two p orbitals is required for a complete description. In Figure 5b two cases are shown. In the case where A a = 0, two px orbitals are colliding with each other (or equivalently two p,, orbitals). When ha = a/2, a px orbital collides with a py orbital. These two cases, Pa = 0 and A a = a/2 are clearly not identical. The P a dependence of the cross section expression can come about only through terms proportional to the off-diagonal elements p-l,l and pl,+ of the density matrices for the two colliding p orbitals, since these are the only elements which depend on Act (eq 9). Thus we conclude that the coherence terms, which describe azimuthal structure, become important in n 1 3 vector correlation experiments.

111. Collision Theory The collision dynamics of scattering processes are governed by interaction potential curves (e.g., L: and II curve crossingsll). These potential curves are usually described in a coordinate frame with the internuclear axis R as the quantization axis. Because the internuclear axis changes its direction continuously along a typical trajectory R = R ( t ) , this so-called "molecular frame" is body-fixed. In the molecular frame, 1;1 is the magnetic quantum number. Therefore, the potential curves have Q labels: V,(R). Experimentally, we do not determine these potential curves directly in collision studies. Rather, crossed-beam experiments give scattering information in terms of cross section values. To link the experimental data with the desired potential curves, collision theory must be used. There are two distinct steps in this link, shown in Figure 6. First, a quantum (or semiclassical) formulation for the scattering process is used to obtain scattering which

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Feature Article

termine the transition probability for a transition If) li). This wave function \k is a solution to the Schrijdinger equation (Ho V)Q = E*, where the total Hamiltonian contains the interaction terms (V # 0). Various techniques are used to calculate the scattering amplitudes, ranging from full quantum mechanical treatment'2*2s-27to semiclassical model^.^^^^^ In a full quantum treatment the Schradinger equation is solved in a certain well chosen set of basis states Id),characterized by a quantum number P for the total angular momentum, which is a conserved quantity in the scattering process. This results in a scattering matrix ((p;plSld),from which the desired scattering amplitudes can be obtained. Because the SchrMinger equation is time independent,it is hard to obtain dynamical information about how and where the If) li) transition takes place. Because of this nature these calculations constitute essentially a "black box". Recently, a quantum flux method has been proposed for alignment studies in which the evolution of a wave packet is studied.35 The redistribution of flux is followed. This type of calculation may remove the "black box" character and give more detailed insight into the dynamical nature of the collision. Because of the "black box" character of the quantum mechanical treatment, semiclassical models are also very popular. The semiclassical approach comes down to separating the coupled electronic-nuclear motion into an electronic motion, which is generally treated quantum mechanically, and a nuclear motion, which is treated classically. Solving the latter problem results in a classical trajectory R = R(t). The physical processes govemed by the interaction potentials V,(R) are calculated in the (bodyfmed) molecular frame. Coordinate transformations are necessary to obtain scattering amplitudes in the (spacefured) collision frame. Because of their nature, semiclassical models do not necessarily give the correct phase of the scattering amplitudes. Thus, semiclassical calculations may give incorrect results for collision processes where coherence (phase information) plays a role, which occurs if n 2 3 vectors are controlled in the experiment (see section 1I.F). This is demonstrated in the calculations of coherence terms for the Na-He where the quantum mechanical and semiclassical results are compared. An important feature of semiclassical models is the spatial evolution of the structured species, since the Q-dependentinelastic processes depend critically on this effect. During the collision the two scattering partners pass through regions where different Hund's case coupling schemes The Q-dependent interaction potentials V,(R) result in a torque acting on the aligned state causing it to try to preserve its alignment with respect to the internuclear axis. This effect is called "locking" to the molecular frame. The finite velocity of the collision partners limits the effect of this torque. Long interaction times (low velocities) tend to preserve the initial alignment with respect to the internuclear axis (locking), while for short interaction times (high velocities) the aligned state tends to remain unaffected by the torque (remains space fixed). The interpretation of this phenomenon remains an important topic in the literature.'4Js,2324*27,36 Recently, a precession picture has been suggested to describe l o ~ k i n g . ' ~ ,Semiclassical ~~.~~ calculationsfor the Na-He system can readiiy reproduce quantum oscillations in the total angular momentum dependence of the state transitions, confirming the validity of the precession model.24936 In the case of a p = 1) state the energy splitting between the Z and II states causes a torque, resulting in the precession of the p = 1) vector about the internuclear axis with a frequency

+

-

h' quantum mechanical calculations h' semiclassical trajectory calc.

- ,

I

flotli)

I

#2) tensor omrator5

p expanded in

#1) density matrir Piit P f r

X 0Rlf)tli)li')

'

P k q 'kq

pk'q' 'k'q' Qk'q'ckq

-

In the absence of other effects, this precession would result in a conserved quantum number Q in the body-fmed molecular frame. However, the internuclear axis R = R(t) rotates during the collision with an angular velocity B = dB/dt: e N (I + y 2 ) h / ~ R 2= vib/R2 (13)

e

When wp > e (low velocities), the coupling is strong and the p = 1 ) vector becomes body-fixed. Depending on the type of semiclassical model, one may choose a sharp boundary between two regions where a body-fixed or a space-fixed description of the structured species is applicable,15 or one may actually calculate the precession of the state vectors involved according to eq 12.24.36 Basic Cross Sections. Using either quantum mechanical or semiclassical calculations, one obtains scattering amplitudes. Basic cross sections can be constructed from these amplitudes according to

where li) = pl,ml)p2,m2) and If) = p3,m3)p4,m4) represent the magnetic substates of the structured species in the collision frame for the initial and final state, respectively. From this equation we can recognize two types of basic cross sections: ulf)tli)=

(a) conventional: (b) coherence:

ulf)lp)+li)li’) &li)

IV;,-,i,12 do

=

~&lifllf~)-li~) f

(15)

do

&-lit)

The conventional cross sections are real positive quantities, which represent how the state-to-state scattering processes affect the population of the basis states in the transition If) li). The coherence cross sections are complex parameters which describe how the coherence between the initial basis states li) and li’) affects the coherence between the final basis states If) and If‘). In other words, they describe how preferred directions other than the z axis are affected. Depending on the number of resolved vector quantities, the number of basic cross sections can become quite large.41 The scattering amplitudes are given byf :?)(e,@), where (e,@) represents the scattering direction in t t e colksion frame. One can show that the azimuthal dependence of this scattering amplitude goes as e-i(md)@.If the final relative velocity vector uf is resolved, the integral in eq 14 is taken only over the resolved scattering directions. Otherwise, one has to integrate over 47r solid angle, resulting in an azimuthal @ integration of

-

21

uIf)tp)-li)lil)

a

S,

e-iI(m~-m~’)+(m~-m~)-(m,-m,3-(mr-m4’)l~

&#,

0 (16)

if ( m l - ml’) + (m2- m i ) # (m3- m3’)

+ (m4- mq))

thus reducing the number of nonzero basic cross sections. B. Step 2 Slate Preparation Denrity Matrices. In this section we will describe how the basic cross sections ulf)lp).,i)li,) are related to the experimentally determined cross section u,,,. We need to know only how the structured species are prepared or probed in the experiment. This is easily done in the density matrix representation (section 1I.B). The experimental cross section is now given by (see Figure 6) uexpt(Bi;W

=

C

all1...)

ti) = Lilml)li2m2)

If)

= Li3m3)li4m4)

PiiJPfp X blr)lp)-li)li’)

-

Piit

= PmlmllPm2m2,

PIP

= Pm3m31Pm4mi

(17)

Consider again a two-vector correlation experiment where only one structured species b l , m , )is initially resolved. In that case three of the four density matrices in eq 17 are represented by identity matrices (section 1I.B). Because the off-diagonal elements are zero, we only obtain nonzero contributions in eq 17 if m2 = mi..m3 = m3‘, and m4 = m i . In addition, eq 16 results in the reqwement mI = ml’ in order to have nonzero basic cross sections.

Driessen and Leone Therefore, off-diagonal elements of the density matrix describing p l , m l )are multiplied with zero-valued basic cross sections and azimuthal structure (coherence)cannot be observed in a two vector correlation experiment. In the case of a three vector correlation experiment, we have either two density matrices with nonzero off-diagonal elements and one velocity vector ui or one density matrix with nonzero off-diagonal elements and two velocity vectors yi and vf, i.e., with basic cross sections obtained by integration over the scattering direction (not over the full solid angle). In both cases the final cross-section expression will contain nonzero terms which are proportional to the off-diagonal elements of the density matrix. A three vector correlation experiment with two density matrices will be discussed in section 1V.B; an example of a three-vector correlation study with one density matrix can be found in eq 14 in a paper by Reid et where the off-diagonal elements of one density matrix are multiplied by two non-identical transition matrix elements. In principle, one can treat an (n - 1)-vector correlation as a special case of an n-vector correlation experiment. If we replace the density matrix of one structured species by the identity matrix (mixed state with statistical weight distribution), it means that we no longer control that correspondingvector quantity. Because the off-diagonal elements of the identity matrix are zero, this results in a sisnificant reduction of relevant parameters (basic cross sections) which are necessary to describe the collision process. Spherical Tensor Operators. The cross section a,, is written in terms of basic cross sections ulr)lp)cli)li,) in eq 17. {he angular structure of the cross section comes about through the density matrices piiTand p p , as they are transformed from the photon frame(s) into the collision frame, which may be arbitrarily oriented with respect to each other. Another way of showing the angular structure of the cross section is by decomposing the density matrices p in a basis set given by the spherical tensor operators T ,which behave under rotations as the spherical harmonics Yk~6,@):14,16,22,28-30 2J

P

=

+k

k=O p - k

PkqTkq

(18)

The Tkcare called state multipoles and the expansion coefficients Pkq, which now represent the excited state, are called multipole moments. The state multipole Tooremains unaffected by rotations and can therefore only represent the identity matrix and the corresponding multipole moment poo is proportional to the total state population. The multipoles of rank k = 1 and k = 2 represent the orientation and the alignment of the atomic state, respectively. In the transformation of the density matrix of eq 8, there appears the product of two rotation matrix elements Udm(a,&~), which can be rewritten with the Clebsch-Gordan series [(3.105) in ref 301. This results in a series of rotation matrix elements ok,,,(a,B,r) with k ranging from 0 up to 2j, detailing the k summation in eq 18. The number of multipole moments P k q is equal to (2j + 1)2,which is equal to the number of parameters necessary to characterize the full density matrix. Therefore, the descriptions of a system in terms of either density matrix elements or multipole moments are e q u i ~ a l e n t .To ~ ~determine ~~~ one is to determine the other. Since the angular behavior of the state multipoles Tkqis given by the spherical harmonics Ykg(O,+), one can expand the cross section expression uecxp, in terms of angular structures, given by Y&q(e,@) Yp,JO,@)multiplied by the multipole moments P&qpp< and a tensorial cross section Qw-kq (see Figure 6), which is a linear combination of basic cross sections ulf)lp)yi) Thus, the cross section ueXp, can be represented in two equivalent ways: in terms of tensorial cross sections Qvd-kq or in terms of basic cross sections ulf),y)-,i~liT). Because the basic cross sections are directly related to the scattering amplitudes, they may be more useful than the tensorial cross sections QYttkq. However, the number of relevant basic cross sections can become quite large and it is not always possible to determine all these cross sections individually from the experimental data. In that case the spherical tensor representation may be considered more useful to analyze the data.

The Journal of Physical Chemistry, Vol. 96, No. 15, 1992 6143

Feature Article IV. Examples of Experiments In the following sections, we give two examples from experiments in our laboratory: a two-vector correlation for an inelastic energy transfer from an f state and a three-vector correlation for energy transfer from one p state to another, where both initial and final alignment are defined. A. Two-Vector Correlation Experiments. Two-vector correlation experiments have been carried out in our laboratory to investigate changes of atomic Ca states induced by collisions with nonreactive rare-gas atoms.' 1~19*20 One, two, or three linearly polarized laser beams of different colors have been used to prepare the p, d, or f atomic orbitals, respectively, with different initial geometries (see Figure 2). Even though multiple lasers may be employed, the excited state represents one controlled vector quantity. The other controlled vector quantity is the relative velocity vi. Thus, the experimental cross section aexpt(fl) only contains terms proportional to the diagonal elements pmm(0) of the density matrix in the collision frame (population): aexpt(6)

= Cpmm(P)Ulml; C p m m ( P ) = 1 m

m

(19)

where almlrepresents the basic relative cross section for an individual magnetic sublevel p,m) in the collision frame, and 6 is the one relevant angle between the two controlled vectors (see Figure 4b). The excited Ca states are pure states (see section 1I.C). In the case of a mixed state the cross section ue?&3) would also contain a summation over the weight distribution W,,according to eq 4. Using eqs 6-8 we can calculate the diagonal elements pm,(/3): for

Pmm(6)

for

= p,o> = Y2l4n,-l(6) - djm,+l(6)12

la>ph

la)ph

= {py-l ) - p,+l

)]/fi

(20)

Because the superposition states (yi,-, - 5 , 1 ) / d 2 have three characteristicaxes for j 2 2, the Euler angle y cannot be neglected (seesection 1I.E). In our particular experimental setup the relative velocity vector vi lies in the plane containing the perpendicular laser polarization^^^*^^ (Figure 2b,d). From Figure 3 it is evident that we have to use y = 0 in the derivation of pmm(@). The angular structure of a,, (6) is determined through the real Wigner d functions dd,(/3) o?eq 6, resulting in the general form

There are (j + 1) possible expansion coefficients aZn,making it possible to extract all (j + 1) basic cross sections alml for the collision-induced state transfer. As an example we discuss the experiments with Ca(4s4f,lF3), which is prepared in a three-photon sequential excitation.20 In this experiment we study the transition Ca(4s4f,lF3) He Ca(4p2,lSo) + He + AE = 557 cm-' (22) The short radiative lifetimes of the initial and final Ca states enable us to study this process by collecting the fluortscence radiation. The angular structure of the cross section is determined by four nonzero basic cross sections dml,with m = 0, f l , f2, and f3, and includes terms up to cos (66). These four cross sections serve as weight factors in eq 19 and can be determined by a least-squares fit of the data. A useful way to visualize the data and the fit is by making use of a polar in which the angular structure is decomposed into p,,(@) according to eq 19. The cross section is plotted radially outward from the center, and the alignment angle 0 serves as the polar angle. The angular structure of the diagonal elements pmm@) is represented in Figure 7a,b for the Y3,o and (Y3,-1- Y3,J/d2 excited Ca states, respectively. The total population given by the sum Cp,,,,(@) = 1 is independent of angle 8, resulting in the circular shape. In Figure 7c,d we have plotted the cross section data, together with a weighted sum of the diagonal elements p,,(P), as deter-

+

-

0 m = 0

n m = + l

m=+2

a m = k 3

Figure 7. (a) and (b): Deconvolution of the total population into the m-sublevel populations, given by p,,(B), for the Y3,0and (Y3,-]Y 3 , 1 ) / dexcited 2 Ca states, respectively. (c) and (d): Deconvolution of the cross section data into a weighted sum of pm,(/3)dml, for the Ca states of (a) and (b), respectively. The angle 0is the one relevant angle between the photon frame (ph) and the collision frame (col).

mined in a least-squares fit. For the Y3,oexcited Ca state, it is obvious that the data points suggest a larger weighting factor for m = 0 and m = f l , resulting in the vertically elongated shape of Figure 7c. The last-squares fit gives the relative numbers 1.34, 1.25, 0.75, and 0.83 for the basic cross sections dml,with m = 0, f1, f2, and h3, respectively.20 A remarkable propensity for the m = 0 cross section ao is observed. For the initial state 'F3 a set of four potential curves V,(R) is necessary, labeled 2, II, A, and a, for Q = 0, 1,2, and 3, respectively. The final state ISo, which is exclusively Q = 0, is characterized by one potential V,. Curve crossings between the initial and final state potential curves can only occur for Q = 0, which may explain the larger cross section ao for m = 0. B. Threevector Correlation Experiments. Recently we started a three-vector correlation experiment,21studying the process Ca(4s5p,'P1) + He C a ( 4 ~ 5 p , ~ P+~He ) + AE = 156 cm-' (23) One polarized laser beam El (or k,) is used to prepare the initial state Ca(4s5p,IP1), which scatters with He at a well-defined relative velocity yi. So far, this setup mimics the two-vector correlation experiment as used for p orbitals." Then a third controlled vector quantity is added: a second polarized laser beam E2(or k2) is used to probe the alignment (orientation) distribution of the final Ca(4~5p,~P,) state, by excitation to the Ca(3d2,3P1,2) levels (laser-induced fluorescence probing). As pointed out in the theory section, the number of cross section parameters increases dramatically by adding the third controlled vector quantity. In the two-vector situation" only two basic cross with m = 0, and m = f1. The ratio sections contribute to of these cross sections is experimentally determined to be al/ao 1.6. In the three-vector setup we need 14 basic cross sections alf)l~)+li)lit); 8 conventional type (rea1,positive) and 6 coherence type (complex), leading to a total of 20 cross section parameters. Because of reflection symmetry only 15 parameters are necessary to characterize these 14 basic cross sections.41 The angular setup of the three controlled vectors is now characterized by three relevant angles &, 02,Aa (see section II.F, Figure 4b). The possibilities to vary these angles in the experiment are more numerous. However, in the present experimental setup,

-

6144 The Journal of Physical Chemistry, Vol. 96, No. 15, 1992

Driessen and Leone

col

A

I

,

I

,

0

45

90

1

0 -135

-90

-45

P,

[degrees] Figure 9. Alignment data obtained for the angular arrangement of Figure 8. Two probe transitions are used, resulting in essentially different alignment curves (0) 4s5p,3P2 3d2,3P2;( 0 )4s5p,3P2 3d2,3P1.

-

Figure 8. Orbital plot visualizing the three-vector correlation setup for the angular configuration PI = 7r/2, and A a = 0. Both linear laser polarizations lie in the x-z plane, guaranteeing A a = 0. In this setup the initial Ca(’P1) state corresponds to a px orbital. Because azimuthal symmetry is broken in the probe step, a p,, orbital (Aa = 7r/2) will lead to a different cross section. TABLE I: Cross-section Parameters Obtained in the Three-Vector Correlation ExperimentO PI = 0; ha = unspecified; varying P2 EO 9)to )-IO) IO) all)ll)-lo)lo) 1.18 f 0.09 ~l2)l2)+tO)lO) 1.32 f 0.09

PI = ~ / 2 A; a

= 0; varying

P2 2.32 f 0.27

ato, lO)-t I ) I1 )

b ~ - l ~ ~ - l ~+ + ~ l ~~ l~ ~l ~~ l ~ + ~ l ~ 2.60 ~ l f ~ 0.25 l / ~ bl-2)l-2)-~l)ll) Re 611)I-1 )-I1

+ p12)12)+11)11)1/2

)I41 Re al2)lo)-ll )I-I)

1.71 f 0.14 -2.52 f 0.31 -0.31 f 0.31

Error bars denote two standard deviations.20 Re u stands for the real part of the complex-valued coherence cross section.

the two laser beams are both collinear and perpendicular to the relative velocity.21 Therefore, angular variation of A a is limited to a fixed value A a = 0 (or A a = ~ / 2 if, one laser beam is circularly polarized). Additionally, we have the possibility to switch between two probing schemes 3P2 3P1and ’P2 3P2, which essentially investigate different subsets of magnehc sublevels in the photon frame of the probe laser. This allows us to determine 13 of the 15 cross section parameters. An extensive discussion of this three vector correlation experiment is given elsewhere, both experimentally2I as well as theoretically.41*44 We will only elaborate on one particular setup to exhibit the importance of coherence: /!I1 = ~ / 2 A, a = 0, and varying f12 using linear laser polarizations El and &. This angular setup is depicted in the orbital plot of Figure 8. By choosing & = ~ / 2 the , initial state can be visualized as a px orbital. The other fixed angle A a = 0 restricts the variation of E2 to the plane containing vi and El (x-z plane in Figure 8). From Figure 8 it is obvious that the azimuthal symmetry about yi is broken. This indicates that the cross section (PI = ~ / 2B2, , A a = 0) contains coherence cross sections. Five conventional cross sections u1m2)lmz)cll )11), and two coherence cross sections ull),-l)cll),-l) and a12) o)cll)I-l) appear in this expression. +he experimental results for ucxpt(0, = ~ / 2 B2, , A a = 0 ) are given in Figure 9 for the two probing schemes, 3P2 3P1and 3P2 3P2 In a least-squares fit to these data we are able to determine five cross-section parameters, which are given in Table I. In this table we also present the two parameters obtained for the angular configuration with = 0. This situation differs from the one in Figure 8 because now a pz orbital is initially prepared. Because the pz orbital is parallel to vi, the collision process demonstrates

-

-

-

-

-+

,

azimuthal symmetry about yi. The azimuthal difference angle A a can take on arbitrary values again ( A a = unspecified) and coherence is no longer important. This can also be recognized from the density matrix for the pz orbital, which does not have any nonzero off-diagonal elements. In this situation only conventional cross sections contribute and no coherence cross sections appear in Table I for b1= 0. For a more detailed discussion of this experiment the reader is referred elsewhere.21 Coherence cross sections can be interpreted as difference cross sections. This was first recognized and pointed out by Duren et al.’ In the cross-section expression of eq 17 the coherence cross sections are multiplied by off-diagonal elements of the density matrix pmd ( m # m?. If we change the coherence of the initial setup (Le., changing Aa), only the off-diagonal elements pmd are affected. The populations (given by pmm) remain unaffected. Therefore, the difference cross section u ( A a l )- u(Acu2)for two coherence cases, Aal and Aa2, will only contain coherence cross sections. For example, changing the px orbital in Figure 8 to a pv orbital results in a different cross section. This corresponds to a change of A a = 0 to A a = a/2. The off-diagonal elements pl,-l and p-l,l change sign for a px and a py orbital (see eq 9). Thus the difference cross section Abexptwill only contain terms proportional to (see eq 17) b ~ , - l ( ~ x-)P ~ , - I ( P= ~ )- l ~ p m ~ m j ~ l m 2 ) ~ m ~ ’ ) ~ ~ l(24) )~-l) This interpretation may be helpful in understanding the nature of coherence cross sections. A difference cross section is easier to interpret than some arbitrary complex-valued parameter, given by eq 15. Further theoretical work is underway to explore the interpretation of coherence cross sections.

V. Conclusion Alignment (orientation) studies in crossed-beam apparatuses have become an important tool in the investigation of collision dynamics. Structured species can be controlled in the experiments with “relative” ease by using polarized laser excitation. The measurement of vector correlations in collision processes can provide important information about the magnetic substate dependence of the potential curves governing the collision dynamics. Many different alignment studies have been reported ranging from two- to three-vector correlations in a crossed-beam setup or a differential scattering experiment. There are even reports of four-vector correlation experiment^.^^$^^ In this paper we have attempted to present an overview of the various mathematical formalisms that are employed in this field and several relevant examples. The complexity of the theoretical expressions necessitates a clear distinction between the various aspects of the theory. First, a unique space-fixed coordinate frame has to be chosen. Second, quantum mechanical or semiclassical scattering calculations are

The Journal of Physical Chemistry, Vol. 96, No. 15, 1992 6145

Feature Article

-v =

Appendix. Pure Rank-2N Alignment in N-Photon Excitation p = 0) N)

a) N=2

b) N=3

N=4

d) N=5

C)

Figure 10. Spherical parts of the wave functions for the symmetric states = N), for N = 2-5. The dotted lines indicate the arrangement of the

-

It is possible to excite a pure state by multiply absorbing N photons: p = 0) = N ) . Pulsed lasers must be used, so that the effect of spontaneous emission is negligible during the laser pulse (section 1I.C). We arrange the N linear laser polarizations in a single plane at mutual angles ?r/N. Each photon is a single magnetic sublevel 11 ,O), in its own photon frame where E, serves as the quantization axis. Without loss of generality we choose they axis in these photon coordinate frames to be perpendicular to the plane containing all laser polarizations. The excited state = N ) is calculated by performing vector addition of the N absorbed photons. Between consecutive excitation states a rotation about they axis is necessary to transform to the next photon frame. Because the vector addition results in angular momenta ranging from (j = 0) to (j = N), we need a projection operator PN to select the correct angular momentum for the excited state p = N ) . Thus, in the first excitation step we excite

Absorbing the second photon Il,O),, results in

N linear laser polarizations. The excited states are superposition states of spherical harmoncis. Using eq 1, we can write these superposition states in the following vector notation: (a) N = 2, 1 / 4 2 [0,1,0, -1,Ol; (b) N = 3, 1/4[0,4 3 , 0 , - 4 1 0 , 0, 4 3 , 0 ] ;(c) N = 4, 1/4[0,1,0, - 4 7 , 0 , 4 7 , 0 , -1,OI; (d) N = 5 , ‘/jz[O, 4 2 0 , 0 , -4240,O, 4504,O, -4240, 0, 4 2 0 , 01.

necessary to relate the potential curves (input) with basic cross sections, which are directly related with the scattering amplitudes (output). A third and final aspect is the analysis of the experimental data in terms of cross-section parameters. Only the first and final aspect are necessary to obtain cross section parameters from a least-squares fit to the experimental data points. The second step actually provides a test for the potential curves and dynamical models used in scattering calculations. The calculated basic cross sections can be compared with the fitted cross section parameters. The number of cross-section parameters necessary to describe the experimental data increases very rapidly as the number n of controlled vector quantities is increased in the experiment. This increased number of parameters can provide a much more rigorous check for potential curves and the dynamical models. However, experimentally it may become impractical to determine all these cross section parameters. In that case a spherical tensor representation may be more useful to analyze the data. We have demonstrated in section 1I.F that azimuthal structure or coherence (off-diagonal elements in the density matrix representation) becomes important if three or more vector quantities are controlled in the experiment. In a 2-vector correlation experiment only one polar angle 0 between the two vectors is relevant. The azimuthal orientation of either vector about the other vector is not important. If we now add more controlled vector quantities (n 1 3), each vector has to be described by two extra angles; one polar angle and an azimuthal orientation with respect to the other (n - 1 ) vectors. This azimuthal angle characterizes the coherence of the structured species, and cannot be neglected for n 1 3. Acknowledgment. We gratefully thank Visiting Fellow Larry Eno for the many valuable discussions about the interpretation of coherence effects. We highly appreciate his careful reading of an early draft of the manuscript and his remarks about the theoretical formulations. In addition, we thank Robert Parson and Alan Gallagher for careful reading of the manuscript and providing many valuable suggestions for improvement. We also would like to mention the computer package MATHEMATICA, which enabled us to visualize (and understand) the coherence effects and many symmetry aspccts of the collisions. Furthermore, we would like to acknowledge the National Science Foundation for support of this work.

Here Ryis defined by eq 6. The final state p = N)ENcan thus be written as

In eqs A1-3 the excited states are described in the uncoupled representati~n.~~ To apply the projection operator PN,the excited state needs to be written in the coupled representation. For this transformation we need Clebsch-Gordan coefficient^.^^ This means that none of the excitation steps can be saturated, since saturation implies that all transitions p,m) p+l,m) are equally strong and not weighted by Clebsch-Gordan coefficients. This excited state is now symmetric for rotations over ?r/N about the y axis:

-

p = N)EI = Ry(a/N)V = N

E N

= PNfi{Ry((N n= 1 + 1 - n)r/N)I1,O>EJ

(n

-

n’ + 1 ) and (n’ = 0 )

= -PN$l{Ry((N = -p = N)EN

-

(n’ = N)

- n??r/N)I1, O ) E n p + j

-

(A4)

The minus sign arises from the substitution (n’= 0) (n’= N ) , which corresponds with Ry(?r)ll,O)E -Il,O), and it reflects the fact that all the neighboring lobes of the symmetric state are opposite in sign. Thus the quantum mechanical state which describes the collision process is symmetric for rotations of the excited state p = N ) over an angle ?r/N in the plane containing the linear laser polarizations. Therefore, the experimental data will reproduce over angles R / N and a pure rank-2N alignment [ao+ a2N cos (2N0)] will be observed. In Figure 10, we have depicted the spherical part of these symmetric wave functions for the cases of N = 2-5. The symmetric arrangement of the N linear laser polarizations is indicated by the dotted lines. Note that the lobes of the wave functions are not aligned along the polarization vectors for even N values. The nodal planes, however, are always perpendicular to the polarization En. This reflects the fact that each individual absorbed photon 11 ,O)En has a nodal plane perpendicular to its quantization axis. It is not necessary for the N polarizations to be arranged in a chronological order. Any permutation of the N polarization vectors

6146 The Journal of Physical Chemistry, Vol. 96, No. 15, 1992

is permitted, and the same symmetric excited state prepared. References and Notes

= N ) E ,is

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