2167
J . Phys. Chem. 1993,97, 2167-2171
A Pocket Model of Seeded Supersonic Beamst Susan DePaul,*David Pullman,'J and Bretislav Friedrich' Department of Chemistry, Hamard University, Cambridge, Massachusetts 021 38 Received: August 4, 1992; In Final Form: December 3, 1992
We present a simple model of molecular beams produced by a supersonic expansion of binary gas mixtures with one component diluted. The model leans on the nonequilibrium sudden-freeze dynamics developed for neat gas expansions. It provides handy expressionsfor quantities such as terminal temperatures, velocity slip, enrichment factor, and beam intensity. A comparison with measured properties of I 2 seeded in rare gas beams illustrates the reach of the model.
Introduction Theoretical understanding of single-component supersonic beams had reached a satisfactory level already in the 1970s. Simple theories such as the sudden-freezemodel' proved useful and handy in predicting terminal properties of molecular beams on a semiquantitative level. In spite of their wide use and application, molecular beams produced by expanding gas mixtures are much less understood. Although numerical solutions to the Boltzmann equation for binary gas mixtures are a ~ a i l a b l e , simple ~ - ~ models which would allow quickestimates of the properties of two (or more)-component beams are still lacking. The need for a better understanding of two-component beams has arisen recently in connection with studies of alignment of rotational angular momentum due to a supersonic expansion. It was demonstrated that substantial alignment can be achieved depending on the parameters of the expansion.6 The alignment turned out to be driven by the velocity difference of the expanding species (velocity slip) and to depend on the rotational cooling, also related to the velocity slipU6 A special class of two-component beams is constituted by the so-called seeded molecular beams: one component (the seed gas, S) has a much smaller number density, ns, than the other (the carrier gas, C), nc. The disparity in the number densities, ns > OSS, so that most collisions suffered by the seed gas are collisions between the seed and carrier gas molecules. Therefore, the seed gas contributes only negligibly to the relaxation of the mixture, and the expansion of the carrier gas can be treated as a nearlypure gas expansion; the seed gas is embedded in the flow field of the carrier gas and essentially adjusts to the state of its environment. Hence, the seed gas parameters are almost entirely determined by the flow field of the carrier gas. Let us assume that a temperature can be ascribed to any degree of freedom of either the seed gas or carrier gas molecules at any point of the flow field, Le., that we have translational, rotational,
' Dedicated to Dudley R. Herschbach on the occasion of his 60th birthday. 1 Senior
Student, Harvard College. Present address: Department of Chemistry, Massachusetts Institute of Technology, Cambridge, MA 021 39. To whom correspondence should be addressed.
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and vibrational temperatures of both components for all values of z = Z / d , the distance from the nozzle (2)expressed in units of the nozzle diameter (d).
Velocity Distributions The fastest relaxation of the carrier gas is generally that of its translational degrees of freedom.' Therefore, the (parallel) translational temperature, T,,, of the carrier gas plays the role of the local-equilibrium temperature to which the temperature of all other degrees of freedom of either component relaxes. Translation is also the last degree of freedom whose relaxation is capable of supporting the expansion; having "stepped out" the gas ceases to expand. At an equilibrium state with temperature T,the translational temperatures of the carrier and seed components, Ttrr and Ttr,s, are the same, equal to the temperature T. When there is a flow in the system, the individual temperatures Tir,c and Ttr,sare generally different. In what follows we write 2'ir.C TCand Ttr,s TSfor brevity. In analogy to the relation for the flow velocity of a neat-gas isentropic expan~ion,'~~ we assume that the flow velocities, uc and U S , are given by uc = u, [ 1 - ( Tc/ To)]''2
us = u,[l
(1)
- (Ts/To)]'/2
where the flow velocity u, for a perfect expansion from a stagnation temperature TOis the same for both components, evaluated with the average mass of the mixture, m:
m = ms[no,s/(no.s+ no,c)l + mc[no,c/(no,s+ no,c)l (3) Here n0.s and n0.c are the stagnation number densities of the seed and carrier gas components, and mS and mc are their respective masses. Then U m = [27ckTo/
