J. Phys. Chem. 1991,95, 8129-8136
JJKM)states for which the Stark effect is suitable for both focusing and orienting the state.48 The energy of states that can be focused increases with the electric field strength; molecules in such statea move to the axis of a hexapolar field, where the field is weakest. More abundant are states whose energy decreases with the field strength: these are defocused and so discarded. The method advocated here does not involve focusing. By cooling much of the beam to l o w 4 states and using a strong field, we strive to put a large fraction of the molecules into pendulum states. For such bound states, the energy decreases with field strength, so these states could not be focused in a multipolar field. Indeed, we use only a uniform field to avoid defocusing these states. We thus discard the focusing field instead of the defocusing molecules. Experiments using molecules in pendulum states must be conducted in a strong electric field, sufficient to maintain the hybridization of rotor states that produces the angular localization. This requirement will for some purposes be awkward or prohibitive, but many experiments can accommodate or even exploit the presence of a strong field. For instance, collisions of oriented or aligned molecules with a neutral beam of atoms or molecules can be studied by sending the beam through suitable grids in the ele~trodes.4~Likewise, spectroscopy and electron diffraction can be carried out with modest elaborations. For molecules containing quadrupolar nuclei, a strong field suppresses unwelcome complications, by uncoupling the nuclear spin from molecular rotat i ~ n . ~ OEspecially attractive is the opportunity to induce spec(48) Ramsey, N. F. Moleculur Beams; Oxford University Press: London, 1956. (49) Herman, 2.; Birkinshaw, K. Err. Bunsen-Ges. Phys. Chem. 1973,77, 566. (50) Xu, Q.-X.;Quesada, M. A.; Jung, K.-H.; Mackay, R. S.;Bernstein, R. B. J. Chem. Phys. 1989,91,3477. Gandhi, S.R.; Eernstein. R. B. J. Chem. Phys. 1990, 93, 4024.
8129
troscopic transitions between the pendulum states. Note Added in Proof. Recently we demonstrated pendular states of molecules in a diagnostic spectroscopic experiment?' A molecular beam of IC1 seeded in H2 was passed between a pair of parallel plate electrodes and illuminated within the gap with orange-red light from a pulsed dye laser (Lambda Physik, EMG 202/FL 3002E). Excitation spectra with resolved rotational structure were obtained for the X lZ(u"=O) --* A 311(v'=19) vibronic band of P5Cl. We found that all aspects of the fieldinduced spectra conform to the theoretically predicted behavior of the pendulum/pinwheel states. For instance, the extent of orientation for the pendular ground state of 13sCl(X 'Z(u"=O)) as derived from the measured Stark shifts corresponds to (cosd ) = 0.55 a t an electric field strength of 20 kV/cm. Likewise, comparison of the observed and predicted intensity variations of individual transitions provides striking evidence for pendular states; this is particularly clear-cut because for the X and A states of IC1 the dipole moment has opposite sign.
Acknowledgment. This work is dedicated to the memory of Richard B. Bernstein. H e pursued his beloved field with verve and zest and left us an inspiring legacy of orienting ideas. We thank Jill Cheney for checking some of the calculations, William Klemperer for discussions of quadrupolar interactions, and Jim Duff for help with the trajectory program. For support of our efforts to align and orient molecules, we are grateful to the Corporate Science Laboratory, Exxon Research and Engineering Co., and the National Science Foundation. Registry No. I,, 7553-56-2; H,, 1333-74-0; D,, 7782-39-0; He, 7440-59-7; Ne, 7440-01-9; Ar, 7440-37-1. (51) Friedrich, B.; Herschbach, D. Nurure, in press.
Tight Focusing of Beams of Polar Polyatomlc Molecules via the Electrostatic Hexapole Lens Victoria A. Cho* and Richard B. Bemsteint Department of Chemistry and Biochemistry, University of California, Los Angeles, California 90024 (Received: December 21, 1990; In Final Form: April 24, 1991)
Interest in orientational control of molecule-surface scattering has stimulated consideration of tight focusing of oriented molecule beams. Various hexapole field configurations are compared for their ability to provide the highest practicable flux densities of focused (and oriented) molecules onto the smallest possible element of target surface. The two important criteria are ( I ) the demagnification factor (Le., the ratio of the size of the focused image relative to that of the beam source) and (2) the focused fraction (Le., the fraction of the beam flux entering the hexapole that is focused on the target element). Using these considerations one can design configurations to maximize beam flux densities at microscopic targets. Focusing experiments with beams of CH3Cl and CHF3 using various combinations of hexapole lenses are reported, confirming several of the key results of the present analysis. Using two hexapole fields of lengths I , and 1, in series with the voltage on each lens controlled + 12) for achieving tight focusing. independently is found to be superior to the traditional single hexapole field of length lo (4,
I. Introduction There is a growing interest in experiments on molecular beam scattering from well-characterized surfaces, especially insofar as they yield information on the dynamics of gassurface interactions.' Recently there have been reports of the experiments employing oriented molecule beams, which have demonstrated appreciable steric effects in molecule-surface scattering and s t i ~ k i n g . ~Such . ~ experiments would require both tight focusing of the beam and high flux densities a t the surface; thus it would be desirable to work with small target areas, ultimately achieving 'Deceased.
0022-3654/91/2095-8 129$02.50/0
submicrometer resolution. High flux density can compensate for a smaller target area and provide adequate scattered flux, e.g., (1) See,for example: (a) Somorjai, G. A. Chemistry in ?'bo Dimensions; Cornell University Press: New York, 1981. Gerber, R. B. Chem. Rm. 1987, 87.29. (c) Gadzuk, J. W. Annu. Reu. Phys. Chem. 1988,39,395. (d) Tully, J. C.; Cardillo, M. J. Science 1984, 223, 445. (2) (a) Kuipers, E. W.; Tenner, M. G.; Kleyn, A. W.; Stolte, S.Nature 1988,331,420. (b) Tenner, M. G.; Kuipers, E. W.; Kleyn, A. W.; Stolte, S. J . Chem. Phys. 1988,89,6552. (c) Kuipers, E. W.; Tenner, M. G.; Kleyn, A. W.; Stolte, S.Phys. Rev. Lett. 1989,62, 2152. (d) Kleyn, A. W.; Kuipers, E. W.; Tenner, M. G.; Stolte, S. J . Chem. Soc., Faraduy Trans. 2 1989,85, 1337. (e) Fecher, G.; Volkmer. M.; Bilwering, N.; Pawlitzky, B.; Heinzmann, U. J . Chem. SOC.,Furaduy Trans. 2 1989,85, 1364.
0 1991 American Chemical Society
Cho and Bernstein
8130 The Journal of Physical Chemistry, Vol. 95, No. 21, 1991
for time-of-flight analysis. It is noteworthy that the very technique which makes possible molecular orientation, namely, the electrostatic hexapole lens! is also a beam focusing device. Letokhov et al. and Cohen-Tannoudji et aL5 have already proposed methods of tight focusing of atomic beams and have successfully deflected and collimated such beams with appropriate configurations of laser radiation fields. However, it is a nontrivial problem to extend the laser method to focusing of beams of molecules, and so it is worthwhile to consider the applicability of the electrostatic hexapole lens focusing technique to the tight focusing of molecular beams. Section I1 is an application of hexapole lens focusing theory to multiple thick lens systems and also presents equations and graphs relevant to the performance of a wide range of hexapole lens configurations for tight focusing. Section 111 summarizes experiments with the U.C.L.A. three-lens system, testing some of the predicted focusing characteristics. Section IV presents the experimental results, comparing with the calculations. Section V is a discussion of the implications of the present work with respect to the practical realization of tight focusing of molecular beams. Appendix A provides details of the theory for the most general two-lens system; Appendix B derives alternative expressions for the magnification factor for the one-lens system; Appendix C derives a simple expression for the focal length of a two-lens system.
symbols are defined in text.)
11. Computational Study
Thus, the radial displacement r within the lens with initial values of radial displacement ri and radial velocity ti can be expressed as
A. Theoretical Background for Hexapole Lens Focusing. The elementary theory of focusing, state selection, and orientation of polar molecules by the electrostatic hexapole lens has been amply discussed.6 Here we review the essentials bearing on the tight focusing issue. One well-known feature of the theory6 is that symmetric-top molecules with first-order Stark effects take either sinusoidal or exponentially diverging trajectories within the hexapole field. (Rotational states with the appropriate sign of the KM product follow the desired sinusoidal path.) A related but more esoteric feature is that, in this case, the hexapole field behaves like a thick optical lens. The beam source (nozzle orifice) and the final collimator, both outside the field, are analogous to the object and image, respectively, in a light-optical system. For a symmetric-top molecule in a hexapole field the first-order Stark energy is
where J, K , and M are the usual (symmetric-top molecular) rotational quantum numbers, p the molecular electric dipole moment, r the radial displacement from the field axis, ro the inside (3) (a) Curtiss, T.J.; Bernstein, R. B. Chem. Phys. Let?. 1989,161, 212. (b) Mackay, R.S.;Curtiss, T. J.; Bernstein, R. B. Chem. Phys. Le??.1989, 164,341. (c) Mackay, R.S.; Curtiss, T. J.; Bemstein, R. B. J. Chcm. Phys. 1990,92,801. (d) Curtiss, T. J.; Mackay, R. S.; Bernstein, R. B. J . Chcm. Phys. 1990.93,7387. (e) Tenner, M.E.; Gcuzemk, F. H.; Kuipers, E. W.; Wiskerke, A. E.; Klyen, A. W.; Stolte. S.;Namiki, A. Chcm. Phys. Lett. 1990, 168, 45. (0 Ionov, S.I.; LaVilla, M. E.; Mackay, R. S.;Bernstein, R. B. J . Chcm. Phys. 1990, 93,7406. (g) Ionov, S.1.; LaVilla, M. E.; Bemstein, R. B. J . Chem. Phys. 1990, 93, 7416. (4) Kramer, K. H.;Bernstein, R. B. J . Chem. Phys. 1965, 42, 767. (5) (a) Lctokhov, V. S.;Minogin, V. G.; Pavlik, B. D. Opt. Commun. 1976, 19,72. (b) Balykin, V. I.; Letokhov. V. S.;Orchinnikov, Yu. B.; Sidorov, A. I.; Shul’ga, S. V. Opt. Lett. 1988, 13,958. (c) Balykin, V. I.; Letokhov, V. S. Op?. Commun. 1987,64, 151. (d) Salomon, C.; Dalibard, J.; Aspect, A.; Metcalf, H.; Cohen-Tannoudji, C. Phys. Reo. Lett. 1987. 59, 1659. (6) Sec. for example, ref 3 and the following: (a) Brooks, P. R.; Jones, E. M.;Smith, K. J. Chcm. Phys. 1969,51,3073. (b) Jones, E.M.;Brooks,P. R. J . Chcm. Phys. 1970,53,55. (c) Brooks, P. R. Science 1976,193,ll. (d) Stolte, S.Ber. Bunsen-Ccs. Phys. Chcm. 1982,86,413. (e) Bemstein, R. B. Chemical Dynamics oia Molecular Beam ond Luscr Techniques; Oxford University: New York, 1982; Section 3.7. (f) Stolte, S.In Aromic and Moleculor Beom Methods; Scoles, G., Ed.;Oxford University: New York, 1988; Chapter 25. (g) Choi, S.E. Ph.D. Thesis, University of California, Los Angeles, 1987. (h) Gandhi, S.R. Ph.D. Thesis, University of California, Los Angeles, 1988.
Figure 1. Geometry of a hexapole lens represented as a thick lens. Ho and Hi denote the entrance and exit principal planes, respectively. (Other
radius of the hexapole, and Vothe so-called rod voltage.69’ Thus, the radial force on the molecule is F,
(=?)
= -mw2r
where m is the molecular mass and w the angular frequency of the radial motion defined as (3)
r ( z ) = ri cos ( p z )
+
(2)
sin @z)
(4)
where Y ( = Y , ) is the component of velocity along the field axis (z axis) and p ( E O / V ) is a so-called lens c o n ~ t a n t . ~ . ~ Since the molecules follow a straight-line trajectory [Le., r(z) = ri + (ki/v)z] in field-free space, the complete trajectory of the one-lens system consisting of the field-free space from the beam source to the hexapole entrance (.object distance” a), the region of the field within the hexapole lens (length lo),and the field-free space from the hexapole to the detector (“image distance” b) can be compactly written9*I0as
Equation 5 can also be expressed in terms of optical parameters” 0 l a = 1)(0 l‘)(&u)
(h)
(A
!‘)&
where f is the focal length of the lens and equal to [p sin @lo)]-’, b ’ r b di, and a’= a do with do = di = [l -cos @ l O ) ] / [ psin @lo)]. Here do (di) is the distance between the “physical” entrance (exit) and the entrance (exit) principal plane of the lens (seeFigure
+
+
1).
The focusing conditionsg is that the upper right element of the product of the 2 X 2 matrices in eq 5 should be zero: i.e. a cos @Io) - abp sin @lo) + p-l sin @lo) + b cos @lo) = O (6) or
(7) For recent reviews, see: Parker, D. H.; Bernstein, R. B. Annu. Rcu. Phys. Chem. 1989,40, 561. ( 8 ) Bromberg, E. E.; Roctor, A. E.; Bernstein, R. B. J . Chem. Phys. 1975, 63. 3287. (9) For prior work involving the electrostatic quadrupole for focusing of polar diatomics, see: (a) Bennewitz, H. G.; Paul, W.; Schlier, Ch. Z . Phys. 1955, 141, 6. (b) Berg. R. A.; Wharton, L.; Klemperer, W.; Bachler, A.; Stauffer, J. L. J . Chcm. Phys. 1965,13. 2416. (IO) See, for example, Klein, M.V.; Furtak, T. E. Optics, 2nd ed.;Wiley: New York, 1986; Section 3.3. ( 1 1) Penner, S.Reo. Sci. Insrrum. 1961, 32, 150.
The Journal of Physical Chemistry, Vol. 95, No. 21, 1991 8131
Tight Focusing of Oriented Molecule Beams Note that, in the limiting case of a = 6 = 0, eq 6 becomes sin @lo) = 0 and the ‘effective length” l2 1of the entire system, which is given by sin ( P I ) = 0, is simply equal to the hexapole length 1,. In general, the effective length 1 is defined by 1= */p (7) where p is the lens constant of the single-loop (“half-wave”) trajectory which satisfies eq 6 for given values of a, b, and 1,. The magnification factor A of the focused beam is then given by the upper left element of the product of the 2 X 2 matrices in eq 5 A (=r/rJ = cos @lo)- bp sin @Io) (8)
or & . = 1 - (6’/fl
(8’)
Now, replacing w2 with ( ~ pin) eq~ 3 results in an expression for the rod voltage V, as an explicit function of the lens constant p . For a one-lens system, the rod voltage V, needed for focusing a symmetric-top molecule in a certain JKM state is V, = ap2
(9)
where
and the value of p is determined from eq 6. B. Multiple-LensSystems. The preceding results for a single hexapole (thick lens) configuration are k n ~ w n . *However, ~~ what follows is a generalization to deal with multiple-lens systems for focusing neutral polyatomic molecules which has not been previously studied. Consider a multiple-lens system, in which hexapole field number 1 (of length I,) is followed by a field-free13region (spacing s2), then a second hexapole (of length 1,) followed by a space s3 to the third hexapole (of length 13), and so on to the nth field. We use a and b, as before, to denote the first and last field-free distances. We recall that the matrices in eq 5 are linear transformations of the initial variables ri and (bi/v). Thus, in the combined (multiple lens) system, such linear transformations are carried out cumecutively and thus can be described in terms of the product of the individual matrices (of eq 5 ) in the sequence of the physical arrangement (target to beam source collimator). The generalized form of eq 5 is then of the form
where pn is the lens constant for the nth hexapole. As in the one-lens case, the focusing equation is obtained by setting the upper right element of the product of all the 2 X 2 matrices to be zero. The corresponding magnification factor Jll as a function of the p i s is given by the upper left element of the product of all the 2 X 2 matrices. Practical examples are presented in section IID. (12) (a) Gandhi, S. R.; Curtiss, T. J.; Xu, Q.-X.; Choi, S.E.; Bernstein, R. B. Chem. Phys. Left. 1%, 132,6. (b) Gandhi, S. R.; Xu, Q.-X.; Curtiss, T. J.; Bemstein, R. B. J. Phys. Chem. 1987, 91, 5437. (c) Gandhi, S. R.; Bernatein, R. B. J. Chem. Phys. 1987,87,6457. (d) Xu. Q.-X.; Jung, K.-H.; Bemtein, R. B. J. Chem. Phys. 1988.89,2099. (e) Xu, Q.-X.; Qucsada. M. A.; Jung, K.-H.; Mackay, R. S.;Bernstein. R. B. J. Chem. Phys. 1989, 91, 3477. (0 Gandhi, S.R.; Bernstein, R. B. J. Chem. Phys. 1990,93,4024. (g) Gandhi, S.R.; Bernstein, R. B. Chem. Phys. Left. 1988, 143, 332. (13) The absence of any E field between any two hexapole lenses would caw the randomization of a quantum number M. Since only symmetric-top molecules in a IJKM) state where the sign of the product KM is negative can follow a sinusoidal trajectory and thus get focused by a hexapole field, the randomization of M before entering the second hexapole ICM incurs a loss of some of the molecules selected and focused by the first hexapole lens. Nonetheless. this problem can be circumvented by using a homogeneous E field between the two hexapole lenses (see refs 6c and 6g).
For a multiple-lens system (of n hexapoles with the inside radius ro),a set of rod voltage Vovalues (V,,,, V0,2, ..., Vo,n]is required to achieve focusing. In other words, the overall focusing in the multiple-lens system is the result of concerted participation of n individual lenses and requires a unique set of p values Ip,, p2, ..., p,) to satisfy the focusing equation. The voltage on each lens Voa is related to its own lens constant pn by V0.n
= a~n2
(1 1)
where
as before. C. Flux Density. If the hexapole lens system acts as a perfect focuser, in the limiting (low pressure) case in which there is no loss of molecules throughout the entire focusing trajectory, the “target flux” (number of molecules s-l) received at the target, F,, should be equal to the “source flux”, F,,of the rotational state entering the allowed cone of acceptance of the hexapole, usually governed by a “source collimator” following the skimmer. In this situation,*the flux density (number of molecules s-l per unit area) at the target, J,, is very simply related to the flux density at the source collimator, J,:
(12) Obviously, the tighter the focusing (Le., lAl b. Thus, for tight
Cho and Bernstein
8132 The Journal of Physical Chemistry, Vol. 95, No. 21, 1991
