J . Phys. Chem. 1984,88,4479-4484
4479
Alignment of I, Rotation in a Seeded Molecular Beam William R. Sanders and James B. Anderson* Department of Chemistry, The Pennsylvania State University, University Park, Pennsylvania 16802 (Received: October 4 , 1983)
A significant alignment of the axes of rotation of I2 molecules is observed in the free jet flow from a slit fed with I2 seeded into CO, at low pressures. In experimental measurements of the degree of polarization of laser-induced fluorescence the maximum alignment observed, expressed as the ratio of the I, molecules with axes perpendicular to the flow to those with axes parallel to the flow, is estimated to be 3:2. At higher pressures, such as those normally used in producing seeded molecular beams, the alignment effect is small, 1:l within a few percent.
Introduction The possibility of alignment of molecular rotation in molecular beams from nozzle sources was first suggested by Steinfeld, Korving, and co-workers.' The question of alignment is important in the interpretation of the results of many experiments with molecular beams since alignment may affect, for example, reaction cross section. If the degree of alignment can be made large with nozzle sources, the effect of alignment on reaction cross sections might be conveniently studied. Sinha, Caldwell, and Zare2 found a significant alignment of rotation for Na2 for the Na/Na2 system. Korving et al.3a and Visser et al.3bfound a much smaller effect for the pure 1, system. Both groups29 suggested a greater degree of alignment might be realized with seeded beam systems. We previously reported4 trajectory calculations fro the Ar/12 system indicating significant alignment of I, with seeding. We report here experimental measurements for the C02/12 system using laser-induced fluorescence to determine the degree of alignment of I, rotation. That molecules become aligned in gaseous transport has been known for many years. Many indirect measurements of alignment have been made by examining changes in gaseous transport coefficients during application of external fields-the Senftleben-Beenaker Recent summaries may be found in ref 2 and 3a. The conditions for significant alignment of molecular rotation are a nonisotropic distribution of relative velocities for the molecules with respect to their collision partners and an interaction potential dependent on molecular orientation. In the expansion of gases in free jets nonisotropic velocity distributions are produced which favor relative velocities parallel to the jet For gas mixtures such as those used in producing seeded molecular beams the effect is enhanced for collisions between light and heavy species.13 For low values of P,@, the nozzle pressure-diameter product, the phenomenon of velocity slip occurs and light and heavy velocities differ significantly. This suggests that under optimum conditions the nozzle expansion of a gas mixture may produce a high degree of molecular alignment. In the experiments by Sinha et al., the degree of polarization of laser-induced Na, fluorescence indicated the ratio of the number (1) J. I. Steinfeld, Department of Chemistry, Massachusetts Institute of Technology; J. Korving, Kamerlingh Onnes Laboratorium, Leyden. (2) M. P. Sinha, C. D. Caldwell, and R. N. Zare, J. Chem. Phys., 61,491 (1974). (3) (a) J. Korving, A. G. Visser, B.
S.Douma, G. W. 't Hooft, and J. J. M. Beenakker, in Proceedings of the Ninth International Symposium on Rarefied Gas Dynamics, DFVLR Press, Gottingen, 1974, paper C 3. (b) A. G. Visser, J. P. Bekooy, L. K. van der Meij, C. de Vreugd, and J. Korving, Chem. Phys., 20, 391 (1977). (4) W. R. Sanders and J. B. Anderson, Chem. Phys. Lett., 47,283 (1977). (5) H. Senftleben, Phys. Z . , 31, 822, 961 (1930). (6) H. Engelhardt and H. Sack, Phys. Z . , 33, 724 (1932). (7) M. Trautz and E. Foschel, Ann. Phys., 22, 223 (1935). (8) H.Senftleben and H. Gladisch, Ann. Phys., 30, 713 (1937). (9) J. J. M. Beenakker, G. Scoles, H. F. P. Knaap, and R. M. Jonkman, Phys. Lett., 2, 5 (1962). (10) J. J. M. Beenakker, Lect. Notes Phys., 31, 414 (1974). (11) J. B. Anderson and J. B. Fenn, Phys. Fluids, 8, 780 (1965). (12) J. B. Anderson, Gasdynamics, 1 (1974). (13) J. B. Anderson, Entropie, 18, 33 (1967).
0022-3654/84/2088-4479$01 S O / O
of molecules with axes of rotation (angular momentum vectors) parallel to the flow to those with axes perpendicular the flow. The observed maximum (Ito 11) was about 3:2. In the experiments by Korving et al. and Visser et aL3the total fluorescence intensity of 1, was measured with and without an applied magnetic field to alter the axes of rotation. The maximum ratio (Ito 11) observed was about 1.05:l. Our trajectory study of I, in Ar gave a ratio (Ito 11) of about 3:l. The C02/12system considered here is certainly not the optimum system for obtaining a high ratio of I to /I axes. The angular dependence of the intermolecular potential is far greater for many other systems. It is, however, representative of a number of systems of interest and it has major advantages of experimental convenience for measurements of alignment as well as for cross-beam and other studies. The results obtained are from measurements of the degree of polarization of the fluorescence of I, excited to the B3n, state with the 5145-A line of a C W argon-ion laser.
Polarization of Fluorescence and the Spatial Distribution of Molecules The polarization of fluorescent light is a sensitive function of the excitation-observation geometry and of the spatial properties of the fluorescent substance. We use the generalized geometry for fluorescence experiments described by Sinha et al., (see Figure 1). Here the fluorescence observation direction 0,axis) is perpendicular to the flow direction ( z axis). The exciting laser beam propagates along the x axis with the plane of polarization forming an angle a with respect to the z axis. Measurements of the fluorescence intensity are resolved into two components, I,, plane-polarized along the molecular beam axis, and Zx, planepolarized along the incident laser beam. A generalized degree of polarization which is a function of the angle a is defined as The dependence of this function on the spatial orientation of the molecules can be derived by considering the probability of exciting a given molecule, its rotational motion, and its probability of emitting light of a given polarization in a particular direction. This can be averaged over a distribution of molecules to result in an equation relating the measured degree of polarization to some distribution function describing the spatial orientation of the molecules. For diatomic molecules with large rotational angular momenta (J 2 8 for 12),14the transition moment of the molecule can be replaced by a classical dipole oscillator, 1.1. Its relationship to the molecular frameI5 depends on the type of transition involved.16-'* For I, excited a t 5145 A, resonance fluorescence involves two (14) P. Pringsheim, "Fluorescence and Phosphorescence", Interscience, New - York. - ----,1965. -- --(15) R. N. Zare, J . Chem. Phys., 45, 4510 (1966). (16) P. P. Feofilov, "The Physical Basis of Polarized Emission", Consultants Bureau, New York, 1961.(17) R. N. Zare, Mol. Photochem., 4, 1 (1972). (18) D. A. Case, G. M. McClelland, and D. R. Herschbach, Mol. Phys., 35, 541 (1978).
0 1984 American Chemical Society
4480 The Journal of Physical Chemistry, Vol. 88, No. 20, 1984
Sanders and Anderson
-
[
Moleculor Direction
Flow
Direction
I
1
-sin 0 cos @
cos 0
(3)
11
Figure 1. Generalized geometrical arrangement with a varying polarization of the incoming light beam for studying fluorescence polarization of molecular beams.
+
sin e sin @
The expression for the intensity of fluorescence polarized parallel to the incident light beam, I,, is
Fluorescence Observotion Direction
Molecular
@ sin $ sin @ cos $ cos e -sin @ sin $ cos @ cos $ cos 0 sin e cos $
-cos
cos @ cos $ sin @ sin $ cos 0 A = sin@cos$ + cos @ sin $ cos 0 sin e sin $
=
Spexpernit.x
~ ( J do I
(4)
where P,, is the probability of exciting a molecule, Pernit,, is the probability of a molecule emitting fluorescence in the y direction plane-polarized parallel to the x direction, and n(e) is a distribution function for the J vectors of the molecule, integrated over all space. In performing the above integration, J and p are taken as unit vectors initially pointing along the z and x axes, respectively, and x’, y’, z’ are calculated from [;;]=Ak]
1‘
These become x’ = cos 4 cos 3, - sin 4 sin 3, cos 0
(6) (7) (8) The probability of excitation, P,, is (e.p)2 where e is the unit vector of the electric vector of the exciting laser beam. Since it can have an angle a to the z axis we have ( e , ~= ) ~(y’cos a z’sin a ) 2 (9)
+
y’ = sin 4 cos 3, cos 4 sin $ cos 0 z’ = sin 0 sin rl,
v
+
Observotion Direction
a
Substituting for y ’ and z’ from eq 7 and 8 and integrating rl, for the molecular rotation yields Pex= 7r cos2 a sin2 4 + 7r cos2 a cos2 4 cos2 0 + 27r sin a cos a cos 4 sin e cos e 7r sin2 a sin2 0 (10)
V
Exciting Light a rn
+
Figure 2. Relationship of lab-centered coordinates ( x , y, z) to molecular coordinates (x’, y’, z ? .
The probability that emission will occur in t h e y direction polarized parallel to the x axis is
perpendicular transitions between Z and II states (Z -,II -,2). There are also P, R transitions (AJ = fl).19 The first classification places the dipole oscillator a t right angles to the internuclear axis of the molecule while the second requires that it lie at a right angle to the angular momentum vector, JeZo Since the dipole oscillator lies in the plane of rotation, calculations of absorption and emission probabilities must include an averaging over the rotational motion of the molecules. These must be done independently because the lifetime of the excited state is longer than a rotational cycle indicating that there is no relationship between the position of the dipole at absorption and emission. Using the dipole oscillator, J vector, and molecular axis, we defined a new coordinate system for the molecule (see Figure 2). With laboratory coordinates x , y, z as before, the molecular coordinates are x’, y’, z’with the J vector of the molecule along the z‘axis, p along the x‘axis, and the molecular bond along the y’ axis. Using Euler’s angles, one finds the coordinates x’, y’, z’ are related to x , y, z by eq 2 where A is the rotation matrix (eq 3).
A distribution n(0) of the form n(0) = di d2 COS 0 (12) is chosen because this can be related to the distribution function, f(O), used in our trajectory calculations4 f(0) = 1 + s(c0s 0 - 1/2) (13) which can be rewritten f(0) = [ l - (1/2)s] s cos 0 (14) or f ( 0 ) = d l d2 cos 0 (15) Equation 4 can now be written as I, = I j j P , , x‘[dl d2 cos 01 sin 0 d0 d$ drl, (16)
F]=AF]
(2)
+
+
+
+
Substituting for P,, from eq 10 and for x’ from eq 6 and performing the integration gives I , = 7r3 sin2 a[(12/15)dl + (13/24)d2] + r3COS2a[(12/15)d1 + (1/3)d2] (17)
In a similar way, I, becomes Here 0 is the angle between the J vector and the molecular flow direction and rl, is the angle through which the dipole oscillator rotates. (19) L. A. Hackel, K . H. Casleton, S. G. Kukolich, and S. Ezekiel, Phys. Rev. Lett., 35, 568 (1975). (20) M. McClintock, W. Demtrder, and R. N. Zare, J. Chem. Phys., 51, 5509 (1969).
Z, = 7r3 sin2 a[(12/15)dl
+ (1/3)d2] +
r3COS’ a[(16/15)di
+ (1/3)d2]
(18)
Substituting eq 17 and 18 into eq 1 results in P ( a ) = (-sin2 a[(5/24)d2] + cos2 a[(4/15)di])/ (sin2a[(24/15)d1 (21/24)d2] + cos2 a[(28/15)dl (2/3)d21) (19)
+
The Journal of Physical Chemistry, Vol. 88, No. 20, 1984 4481
I2 Rotation in a Molecular Beam Expansion Chamber
r(
Pump
l2 or l 2
/COP
"
